STUDIES IN INTERFACE SCIENCE

Drops and Bubbles in Interfacial Research

S T U D I E S IN I N T E R F A C E S C I E N C E

SERIES E D I T O R S D. M 6 b i u s and R. M i l l e r

Vol. I

Dynamics of Adsorption at Liquid Interfaces Theory, Experiment, Application

by S.S. Dukhin, G. Kretzschmar and R. Miller

Vol. 2

An Introduction to Dynamics of Colloids

by J.K.G. Dhont

Vol. 3 Interracial Tensiometry

by A.I. Rusanov and V.A. Prokhorov

Vol. 4 New Developments in Construction

and Functions of Organic Thin Films edited by T. Kajiyama and M. Aizawa

Vol. 5 Foam and Foam Films

by D. Exerowa and P.M. Kruglyakov

Vol. 6 Drops and Bubbles in Interfacial Research

edited by D. M6bius and R. Miller

Drops and Bubbles in lnterfacial Research

Edited by

D. MOBIUS Max-Planck-lnstitut fur Biophysikalische Chemie

P.O. Box 2841 G6ttingen Germany

R. MILLER Max-Planck-lnstitut fur Kolloid- und Grenzfl~ichenforschung

Rudower Chaussee 5 Berlin-Adlershof

Germany

I 9 9 8

ELSEVIER

A m s t e r d a m - L a u s a n n e - N e w Y o r k - O x f o r d - S h a n n o n - S i n g a p o r e - Tokyo

ELSEVIER SCIENCE B.V. Sara Burgerhartstraat 25

P.O. Box 2ii, IOOO AE Amsterdam, The Netherlands

ISBN: o 444 82894 X

�9 I998 Elsevier Science B.V. All rights reserved.

No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by

any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publisher, Elsevier Science B.V., Copyright & Permissions Department, P.O. Box 52i,

I OOO AM Amsterdam, The Netherlands.

Special regulations for readers in the U.S.A.- This publication has been registered with the Copyright

Clearance Center Inc. (CCC), 222 Rosewood Drive, Danvers, MA, 01923. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the U.S.A.

All other copyright questions, including photocopying outside of the U.S.A., should be referred to the

copyright owner, Elsevier Science BV, unless otherwise specified.

No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a

matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein.

This book is printed on acid-free paper.

Foreword

The shape of drops and bubbles and their specific properties make them a centre of interest of

many interfacial scientists. There are classical methods for measuring the surface and interfacial

tension, such as the maximum bubble pressure technique or the drop volume method.

Milestones in the development of the theoretical basis were set by Thomas Young in 1805 and

P.S. de Laplace in1806. On the basis of an analysis of the forces acting at an interface, they

derived a description of both the capillary elevation and the contact angle. While Young only

described the phenomena qualitatively, Laplace developed a very clear and well developed

physical and mathematical theory without any analytical solution, given in 1830 by Gauss. This

theory, named Gauss-Laplace-Equation is the basis for all coming methods on surface and

interfacial tension.

In the early years of this century many methods had been established as standard methods.

However, a number of physically principle questions were under discussion at that time.

Remember the discussion of Lohnstein and Tate in 1905 through 1913 about the correction

factors for the drop volume methods. In recent time this method was further developed to make

it applicable at drop times less than one second, which brings along additional problems

connected with the hydrodynamics of the drop formation and detachment. Also, in presence of

surfactants, surface rheological parameters come into play and have to be considered.

Another method, the maximum bubble pressure technique, used in literature for many decades,

has had a renaissance also very recently due to the availability of high precision pressure sensors

which made this method faster and more accurate and allow nowadays measurements of

dynamic surface tensions down to adsorption times of some hundred microseconds.

The shape of a pendent drop as another example gives access to the interfacial tension and

various developed sol, ware packages such as ADSA or ASTRA supply simultaneously the area

vi

and volume of the studied drops. This makes the pendent drop method very variable and allows

even interfacial relaxation experiments with a single liquid drop.

The spinning drop has been especially developed for ultra-low interfacial tensions. Experiments

have shown that such ultra-low tensions are important for the stability of emulsions and lead

under certain conditions even to a spontaneous emulsification. However this technique has a

number of limitations and needs attention when used at higher interfacial or surface tensions.

A completely new methodology for studying the dilational rheology of adsorption layers has

been developed only during the last 25 years, the oscillating bubble technique. This method

allows to determine the dilational rheology of adsorbed layers and simultaneously the exchange

of matter of these adsorbed molecules or other relaxation mechanisms. This technique has been

applied for so interesting scientific problems like the compression/expansion of layers formed

by lung surfactants. It is also a method of choice for fast matter exchange or interfacial

relaxation processes for example of mixed protein/surfactant systems the intrinsic mechanisms

of which is not understood so far.

The properties of rising bubbles in surfactants solutions is another highly interesting system in

which bubbles play a key role. The rising characteristics of a bubble in a liquid are significantly

influences by adsorbed layers of surfactants or polymers. This makes the phenomenon

interesting for example for the purity test of drinking water or for the progress of purification

procedures in water treatment plants. The same scenario is the basis for a technology of large

economic and ecological importance - the flotation. The efficiency of this technology is

controlled by the interfacial properties of the floating bubbles and their interaction with the

particles to be flotated.

Several of the described techniques are available as commercial instruments. The book will

serve as an up to date guide to understand the methods, to show their advantages and

disadvantages, to specify the range of application and to explain their experimental limits. For

vii

other methods just laboratory set-ups exist and the book will give insight into how to design

them and what are the key questions to be solved in order to get accurate results. Especially

these prototype instruments give access to completely new phenomena or allow studies of

particular properties, such as it is true for the oscillation mode analysis of drops, the force

measurements between two liquid phases or the shape analysis of growing drops using modern

light fibre techniques.

The are more fields of application of bubbles and drops, for example in medicine and biology.

The lungs are constructed by bubbles, by so-called alveols. The functionality of alveols and their

mechanical properties are still not fully understood, although live is based on the perfect

functioning of the lungs since millions of years.

Another very practical problem closely linked to drops is the metallurgy. Metals or alloys during

their processing are usually liquids. This fact is used to study their properties - to determine the

surface tension of metals from the shape of a molten metal drop, to understand reactions on the

drop surface etc.

The book contains 15 chapters dedicated to either of the above mentioned topics. In Volume 1

of this series (Dynamics of Adsorption at Liquid Interfaces) few of the classical methods have

been described more from the point of view of interfacial dynamics so than only a small overlap

with this books exists. Moreover, there is a basic introduction of the surface and interfacial

tension methods in Volume 3 (Surface Tensiometry). However, chapters on these methods

given here do not introduce into the physical basics but describe in detail the most recent

developments and how to use them correctly. Also experimental examples are given to

demonstrate the data interpretation properly.

After a general introduction into the topic the chapter about the mechanics of axisymmetric

liquid menisci gives an overview of the theoretical background for all drop and bubble

experiments. Subsequent chapters are dedicated to drop and bubble methods. The chapters

viii

about the three main drop experiments provide the theoretical basis, a description of

experimental set-ups, specific advantages and disadvantages, correction and calibration

problems, experimental examples, and their interpretation: pendent and sessile drop, drop

volume, and spinning drop technique.

The chapter about capillary pressure methods summarises different techniques and gives

examples of applications, for instance measurements under microgravity conditions in space

experiments. This type of methods comprises bubbles and drops.

The maximum bubble pressure technique as a particular capillary pressure method will be

described with emphasis on the decisive developments in the last five years, during which this

method was developed as the most reliable one for extremely short adsorption times, down to

the range of milliseconds and less. Problems connected with aerodynamics and hydrodynamics

are discussed and used to show the limits of this widely used standard method.

The oscillating bubble technique has been first developed about 25 years ago and does exist still

in form of individual set-ups only. However it provides a number of information not available by

other techniques, for example about the dilational rheology of adsorption layers and relaxation

processes at the interface. Especially the frequency interval spans a range which is of large

importance for many surfactants. The theory as well experimental details and results will be

presented and compared with other techniques.

The description of rising bubbles in surfactant solutions will contain the hydrodynamic basis as

well as the theoretical description of the effect of interfacial layers on the movement of bubbles.

Besides the theoretical basis also experimental data and the relevance for practical applications

will be presented, such as water purification, flotation processes etc.

The chapter about lung alveols demonstrates how important bubbles built by biological

membranes are in everyday life. The relevance for medicine and biology as well as model studies

will be discussed.

ix

An important example for the application of drops is the metallurgy, where the surface tension

of metals and alloys is an important parameter for many applications. Also the chapters on drop

shape analysis by using fibre technique and on force measurements between emulsion droplets

are of much practical relevance. These chapters however will close the present book although a

number of further interesting fields could be added, such as on formation and stabilisation of

emulsions, or drops in shear fields, thin films between drops or bubbles, are extensive enough to

fill separate books and thus will be excluded here. A book especially dedicated to "Foams and

Foam Films" has been published within this series.

This book will be interesting for all users of commercial instruments based on principles such as

drop volume, spinning drop and maximum bubble pressure techniques. Also for those scientists

and engineers using unique or home-made set-ups for example of the pendent or sessile drop

technique will find the state of the art of the respective methods and comparison with other

techniques. The different examples of application are interesting for a number of groups, in

material sciences as well as life sciences. For those scientists starting up research in the field of

characterisation of liquid interfaces this book will serve as guide to find the right methods and to

realise the certain peculiarities inherent in specific liquid systems.

For the very valuable support during the whole preparation of the book manuscript we want to

express our gratefulness to Dr. Martina Bree and Sabine Siegmund.

September 1997 D. M6bius R. Miller

This Page Intentionally Left Blank

Contents xi

J. Gaydos The Gauss Laplace Equation for liquid axisymmetric meniscii

P. Chen, D.Y. Kwok, R.M. Prokop, O.I. del Rio, S.S. Susnar and A.W. Neumann Axisymmetric Drop Shape Analysis (ADSA) and its Applications 61

R. Miller and V.B.Fainerman The drop volume technique 139

A.M. Seifert The spinning drop tensiometry 187

L. Liggieri and F.Ravera Capillary pressure tensiometry with applications in microgravity 239

V.B. Fainerman and R.Miller The maximum bubble pressure tensiometry 279

K.D.Wantke and H.Fruhner The oscillating bubbles method 327

S.S. Dukhin, R. Miller and G. Loglio Physico-chemical Hydrodynamics of Rising Bubble 367

R. Herold, R. Dewitz, S. Schfirch and U. Pison Pulmonary Surfactant and Biophysical Function 433

A.Passerone and R.Ricci High temperature tensiometry 475

A.V. Nguyen and H. Stechemesser Dynamics of the Impact Interaction between a Fine Solid Sphere and a Plane Gas-Liquid Interface 525

P.D.I. Fletcher Interactions of Emulsion Drops 563

N.D.McMillan, V.Lawlor, M.Baker, and S.Smith From stalagmometry to tensiography; The definition of the instrumental, software and analytical requirements for a new departure in drop analysis. 593

Subject Index 707

This Page Intentionally Left Blank

Drops and Bubbles in Interfacial Research D. M6bius and R. Miller (Editors) �9 1998 Elsevier Science B.V. All rights reserved.

THE LAPLACE EQUATION OF CAPILLARITY

J. Gaydos

Department of Mechanical and Aerospace Engineering,

Carleton University, 1125 Colonel By Drive, Ottawa, CANADA

Contents

Introduction

The Historical Laplace Equation of Capillarity

The Excess Concept and the Planar Surface Fundamental Equation

Equilibrium Conditions

The Free Energy Representation

Free Energy and Alternative Curvature Measures

The Non-Moderately Curved Surface of Gibbs

Developments After Gibbs

A General, Second-Order Laplace Equation of Capillarity

Numerical Integration of the Generalized Laplace Equation of Capillarity

Acknowledgements

References

Appendix

Derivation of Principal, Mean and Gaussian Curvatures

The Classical Laplace Equation of Capillarity

Numerical Integration of the Classical Laplace Equation of Capillarity

The Non-Classical Laplace Equation of Capillarity

Introduction

Interfacial physics is a rich area of study with many practical manifestations and significant inherent complexity in the underlying two-dimensional behaviour. Structures formed from aggregates of self-assembled amphiphiles may show a variety of forms and properties ranging from ordered arrays of micelles to disordered, bicontinuous microemulsions. As developments in the characterization of microemulsions, micellar solutions and mesomorphic phases advanced there was a corresponding need to explain the variations in properties and phase behaviour that accompanied these systems. A theoretical study of this behaviour often begins with a characterization of both the shape and energetic state of the interface. An integral part of this approach is a formulation of both the fundamental equation for the surface free energy and the form of the Laplace equation of capillarity. Often one has terms in the energy that depend on both the area [e.g. surface tension] and the curvature. Even in situations were one is concerned with sub-structure models of multi-layer surfactant or biological films an appropriate choice of 'curvature measure' is important if the transition zone between two adjacent bulk phases is to be modelled properly as either one or many two-dimensional surfaces.

In this introductory chapter we i) survey the historical developments that led to the original explanation of the connection between interfacial bending and the pressure jump across a surface, ii) provide a detailed account of the surface excess concept and its role in the proper definition of a dividing surface fundamental equation, iii) formulate the basic variational problem that characterizes all capillary systems, iv) discuss the important role of the free energy formulation in capillary variational problems, v) survey alternative curvature measures, vi) contrast the form of alternative curvature measures with the attempts by Gibbs and other workers to develop a non- moderately curved capillary theory with a corresponding generalization of the classical Laplace equation of capillarity [cf. Table 1], vii) provide a generalization of the classical Young-Laplace equation that is completely second-order in the principal curvatures [cf. Eq. (90)], and viii) provide numerical examples of the influence of first- order bending energy effects upon the shape of axisymmetric pendant or sessile drops [cf. Figs. 4-9].

The Historical Laplace Equation of Capillarity

The study of phenomena that are, in some manner, influenced by the presence of a liquid-fluid interface is as old as the first recorded observations of water rising in a small capillary tube by Leonardo da Vinci (1490). Much later, Honoratus Fabry (1676) found that the elevation rise of water in a glass capillary tube was inversely proportional to the tube's radius. The first measurements of capillary rise were by Francis Hawksbee (1709) and a physician by the name of James Jurin (1719). They attributed the rise of water elevation to an attraction between the glass and the water. The important concept of interfacial or surface tension was introduced by J. A. von Segnar (1751) who ascribed the surface tension to attractive forces of extremely short range between different, but adjacent, portions of the liquid. John Leslie (1802)

demonstrated that the attractive force between the glass wall of the capillary tube and the thin layer of liquid in contact with it could be both normal to the tube wall and responsible for the capillary rise. However, it was not until the significant investigations of T. Young and P.S. Laplace that a proper formulation of the phenomena was established. Young (1805) proposed a theory whereby two forces, one attractive and the other repulsive, acting between fluid bodies in a surface were responsible for the existence of interfacial tension. Subsequently, he concluded that on a curved surface a net force, proportional to the surface's mean curvature, must act on a superficial body to force it towards the center of curvature of the surface. Laplace (1806) obtained essentially the same result as Young, but with the important difference that he expressed the result [for a spherical surface] via the mathematical relation

27 P =Pm + (1)

R

where the pressure P at a point in the interior of a liquid is given by the sum of a constant 'molecular pressure,' denoted by Pro, and a term that includes both the surface

1 tension 7 and the the radius of curvature, -~-, of the spherical surface. Even though

Laplace's initial assumption about the density being uniform within the transition zone was wrong, Poisson (1831) was able to demonstrate that the form of Eq. (1) remains unaltered. 1'2 The significant work of J.W. Gibbs (1876-8) created a "pure statics of the

effects of temperature and heat." 3 His approach placed the static equilibrium behaviour of the transition zone on a sound conceptual basis while demonstrating that Eq. (1) applies for a "moderately curved dividing surface ''4 representation of the interfacial zone. Many subsequent descriptions of capillary phenomena have relied upon the form of the Laplace equation of capillarity; i.e., A P =3'J where A P is the pressure difference across an interface that separates adjacent bulk phases and J is the mean curvature, 5 to properly characterize both the static balance of forces across an interface and the interfacial linear momentum.

In this chapter, we present various 'curvature measures' that have been employed to evaluate the degree of surface bending and their relationship to both the generalized theory of capillarity and the form of the corresponding equilibrium Laplace equation of capillarity. We shall not examine the appropriateness of employing Gibbs' 'surface- excess' approach to the transition zone from a molecular point of view,6-19? nor explain in any detail sub-structure models of multi-layer surfactant or biological films, 2~ nor be concerned with the dynamic behaviour of the transition zone. 47-82 However, for most of these situations, an appropriate choice of 'curvature measure' is still important if the transition zone is to be modelled as either one or many two- dimensional surfaces.

t The references listed in this paragraph have been included to provide a sense of the scope of activity rather than a complete, historical- ly accurate listing of relevant publications.

The Excess Concept and the Planar Surface Fundamental Equation

To develop a generalized thermodynamic formalism which takes into account the interfacial regions, [i.e., the confluent zones between two fluid phases], it is convenient to start by considering the smoothed volume densities of internal energy, entropy and mass of the i th component, i.e. u (v), s (v) and p(V) throughout the whole fluid system.

u lV)' (v) !v) will be functions of In any equilibrium state, the actual densities , s and p position. The superscript (v) on these quantities implies that it is reasonable to discuss these quantities as continuum densities defined at a point in space. An analogous designation, employing the superscript (a), will be used for dividing surface densities. These volume densities will vary slowly through each bulk phase because of the influence of body forces like gravity but may vary quite rapidly across each interface. Using the methodology developed by Gibbs we may represent each interface by a single mathematical surface or div id ing surface. To avoid the 'empty' spaces between the bulk fluid phases which results from this reduction of the interface to a strictly two-dimensional boundary it is necessary to extrapolate the bulk properties from the interior of each fluid phase right up to the dividing surface. The extrapolations are performed such that the densities u (v) , s (v) and p~V) on either side of the dividing surface conform with the bulk fundamental equations and with the influence of gravity, however, they are uninfluenced by the proximity of the other bulk phase. As a consequence, one may define the excess quantities u~ v) , s~ v) and p~V~ by the relations

u~ v) ( r ) : u (v) (r)-u(oo v) ( r ) (2)

s~ v) ( r ) = s (v) (r)-s(oo v) ( r ) (3) and

p~V! ( r ) = p ~ V ) ( r ) - p ~ ( r ) (4)

where r represents the position vector and the subscript infinity symbol indicates an idealized bulk phase density based on an infinite bulk phase without surfaces. In other words, the actual densities u (v) s (v) and p!V) will, in general, be different from the ideal densities u(oo v) , s~ ) and 9~! that one would have for an exclusively bulk system that is uninfluenced by external boundaries. These excess volume densities [i.e., the actual volume density at location r in excess of the extrapolated one] are zero outside or sufficiently f away from the interface. Integration of these excess quantities u~ v) , s ~ v) and p~! along a path which is directly across the interface yields the

complete or total excess amount for the interface at that location. These total excess quantities, which are attributed to the dividing surface, are denoted by u (a) , s (a) and p~a) and are commonly called the surface excess densities of the internal energy, entropy and mass of the i th component. It should, however, be realized that to a certain extent, the surface densities u (a) , s (a) and 9!a) depend on the integration path across the interfacial zone and they must be evaluated at an interface location which is sufficiently far away from any contact line. The dividing surface, which is initially constructed as a geometrical surface of bulk separation, may be transformed into a thermodynamic, autonomous system governed by a suitable fundamental equation for the interface which is dependent only on excess or surface quantities. To quote Defay

and Prigogine 83

This fundamental difference between bulk phases and surface phases is taken account of by expressing the properties of bulk phases in terms of variables relating solely to these phases, while the properties of surface phases are presumed to depend not only on the variables describing the surface, but also on the variables which define the state of neighbouring bulk phases. Bulk phases are said to be au tonomous , while surface phases are non-au tonomous . This distinction loses its importance for equilibrium states, since then the intensive variables characterizing one phase fix, through the equilibrium conditions, the intensive variables characterizing all the coexisting phases.

Ultimately, this means that we assume that it is physically meaningful to be able to discuss surface densities defined at a point on the dividing surface and that it is reasonable to treat the surface phase as a mathematical surface of zero thickness amenable to differential geometry. Thus, surface densities defined at a point in the dividing surface will be considered in exactly the same manner as volume densities defined at a point in the bulk fluid. The suitability of this approach for irreversible, unsteady situations has motivated many authors to seek alternative descriptions of the interfacial region based upon either a singular surface 47~57'62'69'71'74 or a thin, three-

dimensional zone with a designated constitutive relation, e.g., van der Waals fluid. 84-86 However, for equilibrium or near equilibrium situations the Gibbs' dividing surface approach is both simple and geometrically intuitive. As far as the geometric variables are concerned, the fundamental equation for bulk phases is complete since a volume region has no extensive geometric variables [besides its volume], and hence no geometric point-variables upon which the volume densities, such as u (v), could be assumed to depend. Likewise, we require that a fundamental equation for surfaces be complete as far as the geometric variables are concerned. We do not seek any additional variables [besides u (a) , s (a) and p!a) where i = 1 , 2 , . . . , r ] other than geometric ones since the corresponding properties would have to be considered also in the fundamental equation for bulk phases and the resulting theory would be more general [e.g. electrocapillarity] than presently desired. A specific density form of a fundamental equation for surfaces can now be set up by analogy with the corresponding bulk phase expression.

For a planar dividing surface we can see that a surface domain in two-dimensional space [analogous to a volume region in three-dimensional space] has no extensive geometric properties other than its surface area. Therefore, the complete fundamental equation for planar surfaces is identical to the one suggested by Gibbs more than a century ago, namely

u(a) : u ( a ) [ s ( a ) p t a ) p ~ a ) . . . ,P~a) ]. (5)

where u (a), s (a), pt a), �9 �9 �9 are the total excess internal energy, entropy and component densities assigned to the dividing surface; as such, they are defined as quantities per unit area. The corresponding intensive parameters, are given by

for the temperature,

T=I~u(a) 1 ~S (a) (a) {P i }

(6)

[~u (a) 1 gi = ~9}a) s(a) , {Pj~i(a) } (7)

for the i th component chemical potential and, using the surface version of the Euler relation, by the specific surface free energy expression

r

T = u(a) - T s (a) - ~ gi 9}a). (8) i=1

where g represents the surface tension for a planar interface. In definition (7), the subscript { [3?) i } indicates that all surface densities, except the density 9! a), are held constant during the differentiation. The corresponding extensive or total quantities are defined in analogy with the bulk phase definitions; however, all integrations are carried out over the surface instead of the volume.

Fundamental relations may also be developed for both linear phases and point phases. In the remainder of this chapter we shall be concerned with alternative forms for the specific surface fundamental equation and corresponding mechanical equilibrium conditions, e.g., forms of the Laplace equation of capillarity.

Equilibrium Conditions Any particular configuration of the total system in which the thermodynamic

parameters are distributed in compliance with the fundamental equations and also in compliance with the constraints on and within the system is called a possible state of the system. In our case, this means that we maintain the total entropy and the total mass of each component in the system as a constant. The fundamental equations, such as Eq. (5) for each surface region in the system, determine and describe the thermodynamic states in all parts of the fluid system, while the minimum principle is a necessary condition which allows determination of the equilibrium states from the multitude of thermodynamic states allowed by the governing fundamental equations. Mathematically, the thermal, chemical and mechanical equilibrium conditions are obtained through application of the calculus of variations.

Gibbs applied the criterion necessary for equilibrium of a volume region to the internal portion of a fluid system with the condition of isolation imposed by enclosing the internal portion of the composite system within an imaginary envelope or bounding wall. 87 Following his approach, we may write the necessary condition for equilibrium of a composite system with volume, area, line and point phases as

I I 0 ~) Et _(v~ v (Mlv~ ) (9) ~'t '

where E t = Ut + ~ , represents the total internal energy and external field energies of the composite system. The expression for the total internal energy, Ut, is given by Eq.

(12) below while the corresponding expression for the mass potential, ~ , , is given by replacing u (v) with 9 (v) ~, u (a) with 9 (a) ~, u (l) with 9 (l) ~p and U (~ withM (~ ~p where ~)(r) represents the potential energy associated with the external field. The three subsidiary conditions, denoted by the subscripted quantities above, are necessary if one requires that the variational problem remain equivalent to the problem stated by Gibbs for an isolated composite system. In other words, an isolated system does not permit the transfer of either heat, mass or work across its outer boundary. If these restrictions are imposed on our system and on the formulation of the variational problem which accompanies the system, then we must force all dissipation processes to vanish, restrict the total mass of each species in the system to remain fixed and require that all outer boundary variations that would perform work be zero. We impose the first condition that all dissipation processes vanish in the composite system by requiring that the total entropy remain fixed. Imposition of the second condition simply requires that the mass of each species remain constant. The final boundary condition, which requires that no virtual work be possible on the outer wall, requires that

8 r I~aw, j~ = 0 (10)

and

5fi [(Lw.k}-- 0 (11)

where r is a position vector denoting the point of interest, fi is an outward-directed unit normal on an arbitrary intemal surface, {Aw, j } denotes the union of all intemal surfaces that would intersect the bounding wall during a variation and { Lw, k } denotes the union of all intemal contact lines that would intersect the bounding wall during a variation. Condition (10) fixes the 'imaginary' bounding wall by imposing the condition that all intemal surfaces remain unvaried along the bounding wall while the second condition fixes the unit normals to the dividing surfaces along all contact lines which contact the bounding wall.

The outer wall may have arbitrary shape, however, to insure that the total intemal energy Ut is unambiguously determined it is necessary to place certain geometric constraints on the manner by which internal surfaces, lines and points contact the outer wall. Specifically, it shall be required that no portion of a dividing surface, with the exception of its boundary lines or points, [i.e. no amount of its area] lie on the outer wall. In addition, it shall also be required that no segment of a dividing line, with the exception of its end points, [i.e. no amount of its length] may lie on the outer wall. Finally, it is necessary to require that a dividing point not be a outer wall point. If any of these conditions are violated, then one would obtain a constrained variation or the mechanical equilibrium conditions for the dividing surfaces, lines or points would be connected to the geometric shape of the imaginary bounding surface of the composite fluid system.

The total energy is divided into parts which belong to the bulk, surface, line and point regions of the composite system. If the total number of bulk phases, dividing surfaces, dividing lines and dividing points inside the composite system are denoted by the symbols Vi ,Aj ,L k and Pl, respectively, then it is possible to write the total intemal energy of the system as

V i Aj L k U t = ~_~ f f f u ( V ) d V + ~_~ ~f u(a) d A + ~., f

m = l Vm m = l Am m = l L m u (l) dL + ~ U (~

m = l Pm (12)

where Vm denotes a particular volume region with a particular specific internal energy out of a total of Vi volume regions which contribute to the composite system. Likewise, A j , Lk and Pl denote the total number of dividing surfaces, lines and points. The subscripts on the symbols Vi, A j , Lk and Pz acquire values, in general, such that i ;e j ~: k ~: l. However, these seemingly unrelated quantities are in fact connected by a topological or combinatorial quantity )~ which is called the Euler characteristic.88-92 t

Upon solution of the variational problem (9) one finds that the condition of thermal equilibrium in isolation is

T = T . (13)

where the equilibrium temperature T is the same in all bulk phases, dividing surfaces, and linear regions. Similarly, considering the chemical components to be independent, with no chemical reactions permitted, one finds that the conditions of chemical equilibrium for each component are

ILLi + ~ ) -- gi for i = 1 ,2 , . . . , r (14)

throughout the system, where ~i are the equilibrium chemical potentials of the chemical constituents of the system at the reference surface, ~ ( r ) = 0.

In addition to the thermal and chemical equilibrium conditions that are given by Eqs. (13) and (14) there are mechanical equilibrium conditions for each liquid-fluid interface [i.e. dividing surface] and for each dividing line. When the surface fundamental equation has the functional dependence indicated in Eq. (5), then the condition of mechanical equilibrium for each dividing surface is given by the classical form of the Laplace equation of capillarity. 4 Alternative expressions or models for the transition zone yield different forms of this relation.

The Free Energy Representation

The fundamental equations for bulk, surface, etc. regions describe the thermodynamic states in all parts of the fluid system, while the minimum principle determines only the equilibrium states possible from the multitude of thermodynamic states permitted by the fundamental equations. Various forms or representations of the minimum principle and the fundamental equations are possible; cf. Eq. (9) for one example. The connection between the various expressions of the fundamental equation [i.e. the thermodynamic potentials] is performed by means of a mathematical technique known as a Legendre transformation. 9397 Using this technique, parameters defining the fundamental equation may be replaced by their corresponding intensive quantities. Therefore, in essence, it becomes possible to design the thermodynamic formalism so

t For any compact surface in three-dimensional space the Euler characteristic Z is related to the geometric genus of the surface gs by the relation X = 2 ( 1 - gs ). Furthermore, if the surface can be segmented and represented by a large number of regions or patches, then the number of vertices Pt, edges Lk and patches Aj are related to the Euler characteristic by the expression ~ = Aj - Lk + Pt. A surface which is representable in this fashion is known as a differential geometric surface.

that parameters like the entropy, volume, or interfacial area, which are not easily manipulated experimentally, may be replaced by quantities like the temperature, pressure, and surface tension which are easier to control. We shall consider some of the alternative Legendre transformed versions of the fundamental equations for capillary systems.

As noted by Callen, 93 the energy formulation [internal energy plus gravitational as given in Eq. (9)] is not really suited for capillary systems because the representation does not take advantage of the thermal equilibrium present in the system [i.e. temperature is constant throughout and known]. The next thermodynamic potential to consider is the Helmholtz function. In this representation the entropy as an independent variable is replaced by the temperature, which is kept constant throughout the system. The Helmholtz function is "admirably ''98 suited to assure thermal equilibrium since the search for configurations that are at complete equilibrium is reduced to the identification of configurations that already are at thermal equilibrium. However, the equilibrium principle for the Helmholtz function still requires fixed component masses inside a fixed system volume which eliminates the possibility of considering open systems. If the Helmholtz function is used, the desired constant pressure within each phase and the composition of the phase can only be obtained indirectly. The next thermodynamic potential, the Gibbs function, is rejected immediately for capillary systems because it requires that each pressure be controlled by a pressure reservoir. 99 This is impossible for a small bubble or drop phase surrounded by another larger fluid phase since it is obvious that the smaller phase does not have a pressure reservoir. At this point, the well-known thermodynamic potentials have been exhausted. Thus, to no surprise, it is the Helmholtz function that is usually selected when treating capillary systems.

Conceptually, the relevant Legendre transformations have not really been exhausted because neither the Helmholtz nor the Gibbs potentials considers the possibility of changes in mass or mole numbers, and hence the possibility of chemical equilibrium with one or more components, expressed by the equality of chemical potentials. Thus, the thermodynamic potential in which the independent variables "entropy" and "masses" of the individual chemical constituents are replaced respectively by the temperature and the chemical potentials is a suitable fundamental equation for investigating capillary systems. This thermodynamic potential, often called the grand canonical potential and denoted by ~2, does not seem to have been used that often in the field of thermodynamics [i.e., Gibbs refers to it once without a name], although it is well-known in statistical mechanics. 1~176 When it comes to capillary systems, there are many instances of either the Helmholtz or the Gibbs potentials being used in applications where the free energy or grand canonical potential would have been more suitable and appropriate.t

Consequently, the conditions of thermal and chemical equilibrium which are the same throughout many capillary systems may be used beforehand to reduce the minimum energy [internal energy plus gravitational] problem described above in Eq.

t However, for closed, isothermal surface systems such as red blood cells the Helmholtz function is employed with the side constraint or condition that the surface mass remain fixed, usually stated as the requirement that the surface area remain constrained. 36.39

10

(9). Evidently, in the reduced minimum problem, the state of complete equilibrium is sought only among those thermodynamic states that already are in thermal and chemical equilibrium. Thus, using the equilibrium conditions which exist between the temperature and the chemical potentials throughout the system [i.e. Eqs. (13) and (14)] we may write the grand canonical potential density for the bulk phase as

co(v) _ u(V) _ T s ( V ) _ ~ _ . g i p ! v ) i ( 1 5 )

where all quantities are to be evaluated at the equilibrium temperature T = T and chemical potentials ~1, i = g i - ~) ( i = 1 , 2 , �9 �9 �9 , r ). In essence, Eq (15) defines a Legendre transformation from the specific volume internal energy u (v) to the specific grand canonical potential

co(v) = co(v)[ T, g l , lbl,2, " ' " , ~l,r ] (16)

which is the specific free energy representation of the fundamental equation for bulk phases which are known to be in thermal and chemical equilibrium. Expression (16) simultaneously replaces the entropy density by the temperature and the mass densities by the chemical potentials as the independent parameters in the fundamental equation. The differential form of the fundamental equation is obtained by taking a total differential of Eq. (15) and using the expression for d u (v) to obtain

do) (v) = - s (v) d T - ~p!V) d g i . (17) i

A comparison with the Euler relation

P = T s (v) +EiLtip! v) - u (v)

i

and Eq. (15) yields

(18)

co(v) _ _ p (19)

which shows that the negative of the pressure in a bulk phase is the expression for the specific free energy. Alternatively, the quantity 03 (v)dV = - P d V may be interpreted as representing the work done on the bulk system when there is an associated volume change dV. The contribution of the bulk phases to the total free energy k'2 t i s then written as

~ m D

~'2(Vm) _ CO(v) [--~-, ~1,1_ t~, ~ 2 - d~, " ' " , ~l ,r--f ~ ] d E . ( 2 0 ) vm

where Vm denotes a particular volume region; contrast this term with the first term on the right-hand side of Eq. (12). Expression (20) is considerably reduced in the sense that the independent functions of o3 ~v), which remain in the integrand, are known so that to (v) becomes a known function of position through the given external potential

( r ). However, to evaluate f2 (v") one still needs to know the exact functional relation for the fundamental equation, 03 (v) ( r ).

The reduction of the dividing surface portion of f2t may be carried out in complete analogy with that of the bulk phase. The conditions of thermal and chemical equilibrium permit one to use Eq. (5) to write that

O.)(a) = u ( a ) -- T s ( a ) - ~_~gi 13} a) ( 2 1 ) i

which introduces the specific free energy representation of the fundamental equation for surfaces as

(D (a) -- 0.) (a) [ T, ~1.1 , g 2 , " ' " , g r ] . (22)

The differential form of Eq. (22) is given by

do) (a) = - s (a) d T - ~ 9 } a) dgi (23) i

and, from Eqs. (8) and (23), the surface version of the Euler relation is given by

CO (a) - ]t (24)

which defines the specific free energy of a dividing surface. Thus, only in the restrictive case of a flat interface, will the surface free energy co (a) be equal to the surface tension 7. The contribution of the dividing surfaces to the total free energy function f2t becomes

~"2(am) -- o)(a) [--T- , ~tl - dO, ~t2 - * , " ' " , ~ r - (~ ] d a ( 2 5 ) Am

where A m denotes a particular dividing surface and the integrand becomes a known function of (~ ( r ) on each dividing surface in the system. Once again, the functional expression for 0,) (a) ( r ) remains unknown.

Reduction of the total free energy f2t into its separate geometric contributions when the system also contains linear and point phases follows directly and in an analogous manner to that of the bulk and surface phases discussed above. After a suitable reduction, the total free energy ~"2 t remains a thermodynamic potential with the same extremum properties [yielding the same solution] as any other suitable thermodynamic potential. Mathematically, the difference between the total energy and the total free energy extremum formulations is that the constraints in the first definition [namely that the total entr__opy and masses remain constant] are replaced by the subsidiary conditions T = T [a constant] and gi 4 - ~ ) - [.t i [a constant] in the second definition such that both problems yield the same solution. The transformations between such conjugate extremum problems are also known as involutory

transformations. 1~ Finally, the advantage of employing the free energy f2t is that there is a direct connection between the variation 8 f2t and the virtual work.

The modified, free energy integral is given by the expression F

~2t = U t + ~'2d? - ~" S t - ~ ~ i M t i (26) i=1

where k and ki are the Lagrange multipliers for the total entropy constraint and the i th component total mass constraint. Any variation of the total free energy ~ t , together with the boundary conditions (10) and (11), is handled as an unconstrained problem. The Lagrange multipliers may be evaluated from the boundary conditions. The final equilibrium conditions are obtained by eliminating the Lagrange multipliers using the

12

constraint conditions that the total entropy and the total mass of each component must remain fixed.

Accordingly, the variation of the total free energy may be written as V i Aj Lk Pz

a t = 2 ~ ~2~ ) -t- E ~ ~"2(m a) -t- E ~ ~2(m/) q- E ~ ~'2(m ~ -- 0 (27) m =1 m =1 m =1 m =1

where ~ r

ff2~ I I I ~0(v) dV where co(v/ (v) co,v) ~. (vl !v) = = U + -- S -- ~ ~i I 9 , (28) Vm i=1

/ -

II ~'2(m a) ~(a) dA w h e r e co(a) (a) c o ~ a ) ~ (a) !a) = -- U + -- S -- ~ ~i 0 ' (29) A m i =1

and

~ r

a(lm) I ~(l) dL where co(l) (l) g) (l) (l) = = u +co -)~s - ~)~iP~ (30) L m i =1

r

~'2(~ ) = U (~ + ~"2~ ~ - ~" S(~ - ~_. ~i M} ~ (31) i=1

The solution of the variational problem posed by Eq. (27) will depend critically upon the choice of parameterization and upon the generality of the functional expressions which are adopted for the free energies co (v) , co(a), 03(0 and f2 (~

Free Energy and Alternative Curvature Measures

The earliest attempts at solving the problem posed by Eq. (27) [i.e., determining the mechanical equilibrium conditions that would render the integrals stationary] usually considered a capillary system as a composite system of at most three bulk phases with three surface phases and one contact line of mutual intersection. Any mobile interface that existed between adjacent deformable bulk phases was considered to possess an energy that was proportional to the surface area of the interface. In virtually all cases, this proportionality factor was treated as a constant or uniform tension on the surface. The only real exception to this state of affairs, until the studies

102 of Buff and Saltsburg -107 and Hill, 1~ was the impressive fundamental capillarity

work of Gibbs. 1~ The mechanical equilibrium condition for a single thin film surface system [the simpliest surface system to consider] arises from the solution to the variational problem stated in Eq. (27) when it is approximated by

~a(ma) = ~) { f l ~(a) dA } = 0 (32) Am

~(a) where 03 - 3 ' ; a constant. Equation (32) simplifies approximately to a problem which renders the area of the interface a minimum, or

13

8 II dA = 0. (33)

When solved, this problem yields a minimal surface of negligible thickness and mass whose mean curvature, J = c 1 + c 2, vanishes. If the surface bounds a phase of fixed volume, a constraint must be added to the variational problem; that is,

8{IITdA - I I IAPdV} = O (34)

where A P represents the Lagrange multiplier for the fixed volume constraint. A formulation based on Eq. (34) leads to a surface of constant, but not vanishing, mean curvature and a Laplace equation of capillarity in the form A P = y J . The unique properties of these surfaces, with either fixed or zero mean curvature, soon captivated the imagination and interest of many mathematicians. For both the fixed and zero mean curvature situations, the variational problems were restricted by fixing the position of the boundary so that no boundary conditions occur and by excluding the constraint of fixed volume. In addition, alternative surface integral expressions such as

8 II j2 dA = 0, (35)

designed by Poisson (1812) in the nineteenth century to characterize the potential

energy of a membrane, started to appear. 11~ Another example, was provided by Casorati (1889) 111

8 II (j2 -2K)dA - 0 (36)

where K = c lC2 is the Gaussian curvature. It might be argued, as was done by

Nitsche,112 that a more appropriate surface integral to investigate would be

5 I I O')(a) dA = 0 (37)

where o~ (a) denotes a positive, symmetric but not necessarily homogeneous function of the curvatures J and K; that is, 03 (a) =03 (a) ( J , K ) . Polynomial examples are 03 ( a ) = a + b J 2 - c K , 1 1 2 with both b and c much less than a or

CO (a) -- b (J - Jo )2 + c K .113 If 0,) (a) = I t / ( J ) - c K , then the Euler-Lagrange equation, which is a necessary condition for the variation of the surface integral to vanish, is given by 112

+ ( j = o (38)

where Abz denotes the Beltrami-Laplace operator. 114 For the special case 03 (a) = j2 the differential equation (38) reduces to

, 0 39,

and was derived by Schadow (1922). 115 Regardless what particular expression is

14

adopted for the surface energy, the Euler-Lagrange equation for the variational problem

~) I I O')(a) ( J ' K ) dA = 0 (40)

is lengthy and involves fourth-order derivatives of the position vector for the surface. Recent mathematical investigations have centered on the expression (35) and its higher dimensional extensions. The case of surfaces with non-fixed or free boundaries "requires the discussion of appropriate boundary conditions and has not attracted much attention so far . ' '112 '116 Recent extensions and elucidations of Gibbs' and Buff's efforts, which consider non-fixed boundary conditions with volume constraints, by Murphy, 117 Melrose, 118-12~ Cahn and Hoffman, 121'122 Helfrich, 113 Boruvka and

Neumann, 123 Scriven et a1.,124'125 Rowlinson and Widom, 9'126 Alexander and Johnson, 127'128 Shanahan, 129-132 Neogi et al., 133'134 Markin et al., 135'136 Povstenko, 137

Kralchevsky e t al . 138-143 and Eriksson and Ljunggren 144-146 have been primarily directed at the determination of the appropriate mechanical equilibrium conditions across a surface [i.e. the Laplace equation of capillarity] andat a contact line boundary [i.e. either Young's equation or Neumann's triangle relation] for quite general differential geometric surfaces. However, a certain amount of contention among these investigators has occurred over the particular functional expression that one might expect for the free energies.

The Non-Moderately Curved Surface of Gibbs

Part of the difficulty with the selection of a suitable curvature measure to describe surface bending stems from the original suggestion of Gibbs 1~ that one consider an area A a to be considered as sufficiently small so that it may be considered uniform throughout in its curvature and in respect to the state of surrounding matter so that the expression for the variation of the surface energy will be determined not only by the variables in Eq. (22), but also by the variations of its principal [orthogonal] curvatures 8 c I and 8 c 2, such that

8~2 (A) "- TSA + C 1 ~C 1 + C2 ~c2 (41)

or

1 1 8~2 (A) -- TSA + 7 ( C 1 + C 2 ) 8 ( c 1 + c 2 ) + - f ( C 1 - C 2 ) 8 ( c 1 - c 2 ) (42)

where the principal curvatures are related to the principal radii of curvature by the I 1

relations R 1 = and R 2 = respectively. The superscript (A) symbol on the free Cl c2

energy f2 (A) indicates that the variation of the surface tension contribution arises from the expression 76(A ) rather than from the more general expression 6 ~7dA. In the latter case, the superscript (a) symbol is attached to the free energy f2 (a) For a "moderately curved" dividing surface, as considered by Gibbs, this distinction is not necessary. The curvatures, r 1 and c 2, are assumed to be uniform on the surface piece A a . The variables C 1 and C2 represent the energy inherent in the bent, non-planar

surface. Gibbs never provided an expression for his form of the dividing surface fundamental equation, but proceeded immediately to show that it is possible to select a position for the dividing surface where higher-order bending effects are insignificant. However, if we proceed towards a fundamental equation and write down a relation based on Gibbs' expression we would obtain, in the free energy notation, the relation

~-2(A)-- ~"2(A) I T , A , ~ i , C l , c21 �9 (43)

The other equilibrium expressions implied by Eq. (43) are

d ~ (a) = ydA + C ldc 1 + C2dc2 (44)

and

f2 (A) = TA (45)

where Eq. (44) is integrated at constant principal curvatures to yield Eq. (45) since both principal curvatures are intensive. Gibbs did not expand upon the physical meaning of the quantities C1 and C2 but proceeded immediately to eliminate any consideration of these terms by shifting the dividing surface to the surface of tension position defined by

the condition C 1 + C 2 = 0.147 It is not surprising that he eliminated these dependencies on the curvature almost immediately since he was primarily interested in investigating the effects of capillarity for systems which are "composed of parts which are

approximately plane" 148 or for those common situations in which "our measurements are practically confined to cases in which the difference of the pressures in the

homogeneous masses is small." 149 Furthermore, from the partial derivative

I ~)~'2(A) ] [ ~)(oj(a) A) ] = o)(a) 0,4 = 0,4 (46) T, l.t/, cl ,C2 T, gi, c1 , r

it may be seen that this quantity represents the average specific free energy density of the interface [sometimes referred to as the average specific grand canonical potential] or o3 (a) and not the surface tension T. In addition, it was made quite clear by Gibbs that 150

The value of T is therefore independent of the position of the dividing surface, when this surface is plane. But when we call this quantity the superficial tension, we must remember that it will not have its characteristic properties as a tension with reference to any arbitrary surface. Considered as a tension, its position is in the surface which we have called the surface of tension, and, strictly speaking, nowhere else.

In the current vemacular this means that 0.) (a) = ]t only at the surface of tension position where T is a pure tension. At any other position, the equality between eo (a) and T will not hold since the specific free energy of the surface will also contain energetic curvature contributions. Throughout his analysis Gibbs was very much aware of the constraints which he imposed on his formalism. Thus, when he considered, for instance, the surface tension % he was very careful to distinguish between its value at the surface of tension and its value at any other dividing surface location. It would seem quite apparent that Gibbs had no intention of generalizing his analysis beyond

capillary systems with moderate curvatures. 151 The specific point is that one should

16

employ a definition of the "superficial tension" or specific surface free energy that yields a quantity which is a pure tension at the "surface of tension" dividing surface location. At any other position of the dividing surface, the quantity 7 remains as a pure tension so that there is a distinction between the surface tension 7 and the specific surface free energy o3 (a). In the discussion which follows on non-uniformly curved interfaces, we shall express the specific surface free energy as a simple summation of Gibbs' planar surface tension 7 and a symmetric function of the curvatures J and K. This approach will simplify the formal thermodynamic description enormously. One primary advantage of writing o3 (a) as

03 (a) = ~ 4- tD~ a) ( J , K ) (47)

is that the surface tension portion, i.e. 7, of the expression remains unambiguously in its definition while remaining identical to the commonly measured experimental quantity [for planar or nearly planar surfaces].

Developments After Gibbs

The next significant developments after Gibbs were those of Buff, 102"107 Hill 1~ and somewhat later Murphy 117 and Melrose. 118-12~ Collectively, they provided a fundamental extension to Gibbs original work and a potential re-interpretation of his surface free energy expression to

d ~ (A) = y d A + A (--C-ldCl + C--2dc2 ) ~ (48)

where the bending moments C 1 and C 2 were assumed constant. In the general case, one could imagine situations where the bending moments C1

and C 2 could vary from point to point on the surface. Perhaps, a variation could arise from inherent inhomogeneities in the properties of the system or it could result from modifications induced by the effects of the contact lines on the surface. Irregardless of the particular cause, the adjustment to Gibbs original work was to assume that

m ~ m C 1 dA = C1 A and C2 dA = C2 A . (49)

In this form, three distinct consequences arise: i) it becomes impossible a priori to consider systems which may possess gradients in the bending moments on the surface, i.e. surface gradients V2 C__21 and V2 C2 are both zero_by definition along the surface, ii) for a given temperature T and chemical potential ~i state the bending moments are related by

~)C2 = ~)C1 (50) A,Cl A,c2

which implies bending moment symmetry in the principal directions, and iii) the Laplace equation of capillarity acquires the form1~

AP + p(a)fi �9 V~ = 7(c 1 + C2 ) -- C---1 c 2 - C 2 c2" (51)

where p(a) is the total surface excess density, defined by

17

F

P (a) = ]~ P} a), (52) i=1

fi is a unit vector [outward] which orients the dividing surface and V~ is the gradient of the external field potential. We have introduced the symbol ~, to indicate that this quantity is not equivalent to the planar or nearly planar surface tension that is commonly measured in laboratory experiments [cf. Table 1 for a comparison of definitions].

For a spherical surface [i.e., c 1 = c 2 everywhere on the surface],a rather unique situation arises because the bending moments are equal; that is, C1 = C2 = C by Eq. (50), and the Laplace equation of capillarity simplifies to

z x e + p ( a ~ f i ' 7 ~ - 7c(Cl +c2)-C (c~ +c~ ) (53) or

A P + p(a)fi. Vd~ = ' y c J - C ( J 2 - 2 K ) . (54)

where ~/c denotes a quantity defined for a surface of fixed curvature. In this situation, some authors have suggested that "this is the special case of a plane interface ''152 suggested by Gibbs 153 because the "unrigid dividing surface (C-- = 0) coincides with

the surface of tension. ''152 Subsequently, it was suggested that Eq. (54) was "not

accurate for highly curved regions" 154 and that for a transversely uniform interface the appropriate surface excess energy should be given by 15s

d ~2 ( a ) = T dA + A ( -Cj d J + --CK d K ) . (55)

where the two intrinsic and invariant [unlike the difference Cl - c 2 expression in Eq. (42)] surface curvature measures are the mean curvature J = c 1 + c_ff_.,2 and the Gaussian curvature K = c 1 c2. Using our notation, the bending modulus Cj and the torsional modulus CK are defined by the integral expressions

~f Cj da = Cj A . (56)

and

~f CK dA = C--K A . (57)

which yields the corresponding generalization of the Laplace equation of capillarity to the form 117

A P + p(a)a- V~) = ~ J - C j ( J 2 _ 2 g ) - C----K J g . (58)

In this form, two distinct consequences arise: i) as in the case above, it remains impossible to consider systems which may possess gradients in the bending moments, and ii) it was discovered many years ago by Poisson 156'157 that the coefficient C--/< should be absent from the final form of the Euler-Lagrange equation that leads to the Laplace equation of capillarity and that this conclusion applies whether or not the surface system has a boundary or it is closed. Furthermore, by the Gauss-Bonnet

89 91 theorem - it is possible to demonstrate that if the surface is an orientable, compact

18

surface of sufficient 'smoothness' or continuity that the surface integral is a topological invariant; that is,

~fK dA 2 = constant (59) /1; a

where ~ is the Euler characteristic [cf. earlier footnote]. The important point is that the value of the integral will remain fixed provided the surface system does not change its genus. Genus is the topological property of 'holeiness' such that the genus of a sphere is zero, of a torus one, et cetera. Thus, for surfaces which are continuous, orientable, of positive genus and without self-intersection [e.g. Klein bottles are excluded] the statement in Eq. (59) applies. Unusual but acceptable surfaces include both Schwarz's and Neovius' periodic minimal surfaces which partition space into two equal, infinitely connected, interpenetrating sub-volumes. 6'124 Mathematically, the condition stated in Eq. (59) permits one to conclude that the three variational expressions

( C l +c2 dn = 0 (60)

and

f f ( C l + : o (61)

~) ( C 1 -- C 2 dA = 0 (62)

are equivalent [when the boundary is fixed] because they all provide an integrand which depends upon the square of the mean curvature and a linear term for the Gaussian curvature which does not change provided the Euler characteristic does not change.

In their sequence of papers, 123'158-161 Boruvka et al. supposed that both Buff and Melrose had mixed-up extensive and intensive thermodynamic quantities. To avoid any ambiguity, they explicitly defined the total mean and Gaussian curvatures as extensive curvature terms using the local mean J and Gaussian K curvatures to obtain

[ [ J dA (63) Y 3.1

and

~ K dA . (64) K

If one employs these definitions, it is no longer necessary to restrict one's consideration to surface systems which have zero bending moment gradients as was necessary with both expressions used in Eqs. (48) and (55). However, in order to be able to compare the Boruvka et al. formulation to previous efforts we shall restrict our interest to the case of a homogeneous dividing surface, i.e. a dividing surface where both surface gradients V2 C1 and V2 C2 are zero along the surface. With this restriction, the corresponding differential and integrated forms are

19

ds "2(A) - dU (A) - TdS(A) - ~ _ ~ i dM! A) = y d A + Cj d fl + CK dg( i

and

~"2 (A) = U (A) - T S (A) - ~_~i M! A) = y A + Cj .~ + CK X. i

and the Laplace equation of capillarity is given by

A P + o(a)I I �9 Vd~ - ~ J + 2 K Cj .

(65)

(66)

is straightforward and given by 158-160

03 (a) -- u (a) - T S (a) - ~_~gi D! a) = ' y+ C j J + C K g i

(70)

and, in differential form, by

do3 (a) - - s (a) d T - ~p!a) dgi + Cj dJ + C K dK i

(71)

where all quantities are defined locally on the dividing surface. When the surface is planar or nearly planar, the expression for o3 (a) simplifies to that given in Eq. (24). For a non-uniformly bent surface, Eq. (70) represents the energy required to bend a planar surface by an approximation that includes the first two differential invariants of the surface. The form of the expression in Eq. (70) is not unlike the energy density

expressions obtained per unit area of the middle surface of a plate or shell. 162 In both cases, an integral across the middle surface yields two terms, one proportional to the mean curvature J and another proportional to the Gaussian curvature K. One may also a)" consider the bending energy 03) = Cj J + CK K to represent the energy required per unit area to bend a surface away from a planar reference surface/configuration. Thus, the surface tension 7 is a measure of the change in free energy with change in area at constant mean and Gaussian curvature. The bending moment Cj is a measure of the change in free energy with change in mean curvature at constant area and Gaussian curvature and will be of importance when the excess pressure distribution is an odd

function about the reference dividing surface. 16~ This might be imagined to occur for surface active long-chained molecules which are non-symmetric about their mid-point when present at an interface. The second bending moment CK is, from the mechanical point of view, the second moment of the excess pressure distribution about the

reference dividing surface 16~ and represents a change in free energy with change in

If one had not assumed that the surface mechanical potentials y, Cj and CK were constant along the dividing surface, then the condition of mechanical equilibrium across each dividing surface would be 123

AP + 9(a)fi �9 V~ - ~J + 2 K Cj - V~ Cj - K V~'( V2 CK ) (68)

where V~ and V2 are surface differential operators. 5'123

The corresponding definition for the specific or density form of the surface free energy, based on the fundamental equation

03 (a) = 0.)(a) [ T, g l , g 2 , " ' " , g r , J , K ], (69)

(67)

20

Gaussian curvature at constant area and mean curvature. Under these conditions, a straightforward calculation shows that changes in Gaussian curvature 5 K are equal to

1 2 e i t h e r - - - j - S D where D is the deviatoric curvature [defined below] or " - I F

_1,t 8 ( H 2 - K ) where H - J /2 . Alternatively, if one had selected the expression

o~h')_ j 2 _ 4K or 03~ a) = H 2 - K initially one would have found that this quantity is zero for all spheres and spherical caps. As a consequence, the free energy o3~ a) would represent the energy required per unit area to bend a surface away from a spherical reference configuration [e.g. spherical microemulsions or symmetric bilayer membranes] and the systems would be characterized by either their surface area or enclosed volume. 163

As with all the previous relations, i.e. Eqs. (51), (54) and (58), relation (68) expresses the balance which exists in equilibrium between the internal surface forces and the forces external to the dividing surface [namely the pressure difference] when gravitational effects are present. However, unlike previous relations, Eq. (68) is not restricted to uniformly curved systems because the curvature potentials Cj and CK are local variables rather than global averages, cf. the average definitions in Eqs. (49), (56) and (57).

In 1990, the first of a long series of papers 138-146 based on the work of

Kralchevsky 139 commenced. Their work is a blend of the original curvature expression of Gibbs and the explicit extensive total curvature definitions of Boruvka et al. In particular, they opted to define the surface mechanical work per unit area, our specific free energy 03 tal, as

do3 (a) = "f d c~ + B d H + 19 d D . (72)

where d a = d ( A a ) is the relative dilation of the area element A a of the dividing Aa

1 1 surface, H = ~- ( c I + c 2 ), B is the associated bending moment, D = ~- ( c 1 - c 2 ) is

the deviatoric curvature and 19 is the associated bending moment. Their expressions are defined locally so that their bending moments, i.e., B and 19, are not necessarily uniform across the surface. Under similar 'homogeneous dividing surface' assumptions as those used to simplify Eq. (68) to Eq. (67) their form of the Laplace equation of capillarity is given by 139'145

A P +0(a ) f i .V~) = 2 g l H - B ( H 2 + D 2 ) - 2 O H D . (73)

or, after using the expression K = H2 _ D 2, the slightly modified expression

A P +9(a)fi. Vd? = 2 ~ n - B ( 2 H 2 - K ) - 2 o n ~ / ( n 2 - K ) . (74)

which would seem to imply the presence of both the Gaussian curvature and a bending moment related to K [cf. Eq. (76) below] in the final expression, contrary to Poisson's earlier discoveries. 156

Despite the apparent similarity in form between Eqs. (71) and (72), especially when comparing the third term of (71) and second term of (72) on the right-hand side, it has been claimed that the bending moments are not related by simple expressions

21

Table 1

Synopsis of the Various Superficial Tension Definitions

Author Equation Expression

Buff (51) ~t -- y - k - C 1c1 + C 2c2

m

Buff (54) Yc - Y + C ( c 1 + c 2 )

Murphy (58) y = y + C j J + C K K

A

Boruvka etal. (67) 7 - Y

Kralchevsky etal. (73) ~/ - 7 + B H + O D

such as 2 B - Cj. Instead, the more complicated relationships 145

B = Cj + 2CK H (75)

and

0 - - 2 C K D (76)

have been derived which seem to combine the definitions of the bending moment or curvature potential with the local curvatures. This coupling of curvature and bending moment definitions is not necessary if the obvious definition 2B = Cj is used in combination with the definition164

O

Y - 7 + B H + O D . (77)

Substitution of this expression for ~ into Eq. (73) causes the term with O to drop-out directly so that when 2 B is set equal to C j , one recovers the earlier expression (67) for the Laplace equation of capillarity. Thus, rather thoan the cumbersome approach using definitions (75) and (76) it is possible to represent 7 as a Legendre transformation of 7. In Table 1 we present a brief synopsis of the relations that exist between the various definitions, i.e. qt in Eq. (51), 7c in Eq. (54), ~, in Eq. (58), y in Eq. (67) and ~ in Eq. (73), and the surface tension Y that Gibbs defined for planar or nearly planar surfaces. Substitution of either Murphy's ~/ expression from Table 1 into Eq. (58) or Kralchevsky's ~/expression from Table 1 into Eq. (73) simplifies both Laplace relations to the form given in Eq. (67). Eq. (67) has two primary advantages: i) it is the most compact representation, and ii) the surface tension ~, definition of Gibbs for planar or

22

nearly planar surfaces occurs explicitly. However, it is not the most general expression that may be derived for a homogeneous dividing surface.

A General, Second-Order Laplace Equation of Capillarity

A significant body of research on the behaviour of vesicles, membranes and microemulsions has been performed with consideration of an energetic contribution to the free energy that is proportional to the squared term ( J - Jo )2.20-46 AS this term is second-order in the principal curvatures it would seem arbitrary to exclude this term from consideration while including the Gaussian curvature. Thus, we write the expression for co~a), see Eq. (47), in the form

CO (a) = '~ + C j J + C H ( J - Jo )2 + CK K (78)

where the bending moments [i.e. curvature potentials] Cj and CK are defined in an analogous manner to the definitions in Eq. (70). We shall denote the factor CH as the Helfrich curvature potential [Helfrich used the symbols kc for 2 CH and kc for CK ].113 The final quantity, Jo, represents the spontaneous curvature. Several comments need to be made about this choice of free energy: i) the contribution to the free energy has been limited to energetic terms upto the second-order in the curvature with the understanding that inclusion of higher order terms would involve quantities such as J 3, H 3, j K , H K et cetera, 124 ii) from stability considerations one has y _ > 0 , iii) based on the order of the curvature, the magnitude of the various bending moments should be related by [ Cj I > I Cn l >- I Cgl >-O, iv) inclusion of the third term, involving the coefficient CH, provides an energetic mechanism for describing and obtaining the shape of surface systems such as vesicles, bilayer membranes, microemulsions, et cetera, and v) for a symmetric, bilayer membrane with two identical sides Jo = O.

Intrinsic anisotropy occurs in a membrane because of the reluctance of amphiphilic molecules to dissolve in the surrounding aqueous phase. 165-167 Using realistic approximations for material behaviour one may express the principal membrane tensions by the following elastic constitutive relations: 1 I Is]

E1 -- ~ 0)~lJ + e (79) T

and

1[ 1 + (80

T

where el and e2 are the principal tensions, ~ is the isotropic tension in the membrane, ~1 and ~,2 are the principal extension ratios, fs is the membrane elastic (strain) free energy density due to shear force resultants and T is the temperature. These forms were derived by Skalak et. al. 168 using the non-linear, large elastic deformation approach of Green et. al. 169'170 Many additional elucidations of membrane elasticity

and stress have followed. 171-174 Equations (79) and (80) describe the constitutive

23

behavior of the membrane, [i.e., they represent isothermal equations of state], and are valid for reversible thermodynamic states where viscous dissipation is insignificant. The membrane is considered to be isotropic in the plane of the surface and the free energy density depends on just two independent variables which characterize the deformation. Instead of the principal extensions, one could define alternative variables to characterize the deformation. Recognizing that the membrane tensions are approximately four orders of magnitude less than the area compressibility modulus in

elastic extensional deformation produced by membrane shear 175 it would seem reasonable to treat the surface area of the membrane as constant and to define suitable deformation variables accordingly. One particularly useful alternative involves the variables

Aa = ~1 ~,2- 1 and Ab - - 1 (81) 2k l k2

where Aa symbolizes the fractional change in area of a surface element [equals zero when the membrane area is constant] while Ab represents the extensional deformation or distortion of an initial circular surface element into an ellipse at constant area. The choice of variables in (81) is preferable to one based on the product of the extension ratios, ~1 ~2, and the sum of the squares of the extension ratios, ( k2 + k~ ), because the variables Aa and Ab are independent, [e.g., consider the case k l = k2 ]. When these variables are selected the principal tensions are given as:

E1 = ~ Aa + T, Ab 2 ) 2 k2 ~) Ab T, Aa (82)

and

~2 = 0 Aa + T,Ab 2~12~2 c)Ab T,Aa (83)

These two expressions can be used to define an isotropic tension e as the mean of the principal tensions which arises because of surface area change and a maximum shear resultant ~s associated with surface extension at constant area that is one-half of the magnitude of the difference between the principal tensions. The alternative forms are 176

= ~ e l + e 2 = OAa (84) T, Ab

and

s = ~ E1 -- s -- 2~2 ~2 ()Ab (85) T, Aa

The static resistance to extensional deformations of the membrane surface may be represented by the shear modulus coefficient, gs, defined as

24

Its = 3 Ab (86) T, Aa

which is, in general, a function of the invariants Aa and Ab and the temperature. In a m

similar manner, it is possible to define the initial isotropic tension or surface tension eo of the membrane in the reference state as

~o = 0 Aa (87) T, Aa = 0, Ab

and the isothermal area compressibility modulus Kc as

Kc = 0 Aa (88) r,Aa =0,13

where the modulus Kc relates changes in isotropic tension to small fractional changes in area relative to the initial state. For membrane systems which are closed to material transport into and out-off the surface 177'178 it is possible to define the initial, undeformed state as the state were eo = 0. The combination of experimental values for the two moduli Its and K together with the functional expressions (84) and (85) provide the first-order, isothermal constitutive relations for the membrane.

Consequently, if one slightly modifies the variational problem in Eq. (34) to include the possibility that the surface area A could remain constant, one obtains the variational problem

where ~ denotes the surface's or membrane's lateral tension and A P represents the pressure difference across the interface. Employing the expression for co ca) from Eq. (88) permits one to determine the appropriate Laplace equation of capillarity for a bilayer vesicle with surface area A and enclosed volume V. A tedious manipulation yields the Laplace equation of capillarity, at the same level of generality as Eqs. (58),

179 (67) or (73), as

AP +p(a)fi. Vd? = ( , y + E ) J _ C H ( J _ j o )(j2 - 4 K +JJo) -CHAbl J +2KCj (90)

where Abl denotes the Beltrami-Laplace operator. 114 It should be anticipated that 7 or eo will be approximately zero for these cases where e is non-zero.

At present, most numerical schemes for solving Eq. (90) have been limited to axisymmetric geometries under the side conditions [appropriate for vesicles] that 7 = 0 and Cj = 0.180-187 Analyical investigations are likewise limited to the shapes of

spheres, cylinders, a Clifford toms and its conformal transformations. 188'189 In the next section we propose a scheme for evaluating the shape of a pendant or sessile drop when one assumes that the dominant correction to the surface free energy is, to first-order, just Cj J . This represents an approach that is consistent with the assumption adopted by Buff et al. 1~176

25

Numerical Integration of the Generalized Laplace Equation of Capillarity

When the capillary system is axisymmetric and the specific surface free energy o (a) is a constant, i.e., 3', the Laplace equation of capillarity is given by the relatively simple expression 190

3'J = 2 3 ' + A p g l ~ ( r C : 0 ) _ ~ ( r = 0 ) ] . (91) Ro t. J

which may be numerically integrated when the natural boundary condition Ro is given as input. 191 In Eq. (91), A 9 represents the density difference across the interface, z = ~ ( r ) is an axisymmetric function which defines the position of the surface, r is an independent coordinate that measures the radial distance from the axis of symmetry to a point on the surface such that ~ ( r e 0 ) - ~ ( r = 0 ) represents an elevation

1 difference, and is the radius of curvature of the surface at the axis of symmetry

Ro location ( r , z ) = ( 0 , 0 ). However, if o (a) is generalized to the particular choice

03 (a) = "~ + C j J , (92)

then the Laplace equation of capillarity, given by Eq. (67), becomes

V J + 2 C j K = Ro V - Ro j + A p g ~ ( r ~ O ) - ~ ( r = O . (93)

In this particular form, the Laplace equation of capillarity may be numerically integrated once suitable values for the physical parameters have been selected to show the effect of the non-classical bending moment Cj upon the surface shape.

To avoid difficulties with vertical gradients and infinite derivatives which may arise in functions which are given by explicit equations, [e.g., in the form z = ~ ( r ) ] , the Laplace equation of capillarity was not directly integrated as given in Eq. (93) but was converted to a dimensionless, parameter-dependent expression. The relation is derived in Appendix A and is given by

sin0 a 0 ~ - 1 - + _ ~ x

= (94) dY 1 + 2~j sin0

X

where all lengths, including the arc-length s , are made dimensionless using the capillary constant c, defined by

A9 g c - . (95)

3' The dimensionless lengths are given by

Y = s c 1/2 �9 X = r c 1/2 ; Bo = Ro c 1/2 ; E = Z C 1/2 (96)

26

Figure 1 Profile curve shapes, from the solution to Eq. (94), for a sessile drop em- ploying dimensionless (X, E ) and actual ( r , z ) coordinates. The origin corresponds to the apex or top of the drop. The complete drop surface for any curve is obtained by revolving the curve in question about the vertical axis. The volume equals the region enclosed by the drop surface and a horizontal plane which intersects the contact line [i.e., end-point of the profile curve] of the drop. All curves are obtaining by starting the integration at the origin and progressing until the contact angle, 0l, equals r~. Dif- ferent curves have different initial radii of curvature, B o , at the origin. Dimensionless data, used to generate these curves, is tabulated in Table 2 along with corresponding data for a water-air sessile drop whose physical parameters were: m p(lv) _ _ 103 kg m- 3, @lv) = 0.072 J m- 2 and c = 136250 m- 2.

5 . 0 i i 1 i l I Bo = 0.316

II 4.0 [ / ~ Bo = 0.562 [ I ] t !

~J i I11 ~ LIQUID B o = 0.794 ~3.o ,, /

l',2',

0 0 ' i ' ' . 0 . 0 1 . 0 2 0 3 . 0 4 0

Dimensionless Radial Coordinate X = ~c r 5 . 0

27

I , i i

] R o = 0.86 mm Ro = 1.52 mm

O. 010 // i I I I LIQUID Ro = 2 . 1 5 mm

l

/ ~ = 3.41 mm

Q ~ / ~ 4 = .82 mm

_.,..i)/// 0 . 0 0 0 , , , L . . . . , ,

0 . 0 0 0 0 . 0 0 5 0 . 0 1 0

v

t,4

r,.) 0 . 0 0 5

4.-.' r >

Radial Coordinate r (m)

Figure 2 Characteristic profile shapes, from the solutions to Eq. (94), for a pendant drop using both dimensionless ( X , E ) and actual ( r , z ) coordinates. The ( X , E) origin corresponds to the bottom of the drop. The complete drop surface for any profile curve is obtained by revolving the curve in question about the vertical axis. The dimensionless volume, y(v), equals the region enclosed by the drop surface and a horizontal plane which intersects the contact line [i.e., end-point of the profile curve] of the drop. All curves are obtaining by commencing the integration at the origin. Different curves have different initial radii of curvature, B o , at the origin, but the same Bo values used to generate the sessile drop curves in Fig. 1 were used here. A small arrow on the vertical axis indicates the inflection point beyond which the solu- tion [indicated by a dashed curve] is not physically realizable.

28

while the dimensionless bending moment constant bj is defined by

VT-cq ^

C j ~ ~ ,

7 (97)

Finally, the corresponding dimensionless quantities for the drop's surface area and volume are

y(a) = A c ; y~v~ = V c 3/2 (98)

Typical axisymmetric sessile and pendant drop shapes, for the classical situation where Cj = 0, are shown in Figs. 1 and 2 using both dimensionless (X, ~ ) and actual ( r , z ) coordinates. In both of these figures, the parameters, corresponding to a water- air interface, were selected as: A g(lv) = 103 kg m -3 and @lv) = 0.072 J m -2. Each curve in Fig 1 represents a distinct sessile drop profile. The complete drop surface for any profile curve is obtained by revolving the curve in question about the vertical axis [i.e. the axis of symmetry]; cf. shaded region of the smallest sessile drop in Fig. 1 for the zone that is revolved about the vertical axis. The dimensionless volume y(v) [which equals V c 3/2 where V is the actual volume] equals the region enclosed by the drop surface and a horizontal plane which intersects the contact line [i.e., end-point of the profile curve] of the drop. Each curve represents a surface with a different radii of curvature at the origin; either Bo [dimensionless] or Ro [actual]. The sessile drop integration, in all cases, begins at the origin and is stopped when the contact angle, 0l, reaches 180. ~ Alternatively, the integration is stopped when the turning angle, 0, equals n. Associated data for all curves in Fig. 1 is given in Table 2. Area and volume information is for a sessile drop whose contact angle equals n. The same radii of curvature values used in Fig. 1 to generate the sessile drop curves were also used in Fig. 2 to generate the pendant drop shapes. Once again, one must revolve the curve of interest in Fig. 2 around the vertical axis to create the full axisymmetric surface. A small arrow on the vertical axis indicates the inflection point beyond which the solution [indicated by a dashed curve in Fig. 2] is not physically realizable. The pendant and sessile drop profile curves in Fig. 1 and 2 are generic of form and illustrative of the dimensions commonly observed in practice.

Numerical solutions, using the same protocol, to more complicated arrangements are also possible. Figure 3 shows numerical solutions for the axisymmetric profiles of a double pendant drop arrangement, consisting of two immiscible liquids, suspended from the tip of a vertical syringe [with a radius of = 0.001 m ]. The liquids considered were water [inner licjuid] and n-heptane [outer liquid] with dissolved octadecanol [concentration = 10 -~ g / m l or less] in air at 20 ~ C. The numerical solution was designed to simulate the behaviour of a double pendant drop, created from a co-axial syringe arrangement, 192 as the outer drop slowly evaporates to leave an insoluble monolayer, i.e. octadecanol, on the surface of the inner pendant drop. Via expanding and contracting the inner drop's size, measuring the effective surface tension and the drop's surface area, it is possible to measure the surface pressure isotherms of the film. 192'193 In Fig. 3, the syringe's wall diameter is explicitly shown and the water- heptane interface is plotted as coming right-out from the tip of the syringe. Usually a hanging pendant drop will not have such a convenient experimental attachment to the

29

Table 2

Sessile Drop Data for Profile Curves in Fig. 1

Bo Ro ( m ) y(a) A ( m 2) y(v) V ( m 3)

0.316 8.56x 10 -4 1.165 8.55• 10 -6 0.121 2.40x 10 -9

0.562 1.52• -3 3.240 2.38x 10 -5 0.576 1.15x10 -8

0.794 2.15 x 10 -3 5.609 4.12 x 10 -5 1.344 2.67 x 10 -8

1.259 3.41 • 10 -3 10.584 7.77x 10 -5 3.6il 7.18x 10 -8

1.778 4.82x 10 -3 15.819 1.16x 10 -4 6.752 1.34x 10 -7

supporting syringe tip unless one; i) carefully cleans the syringe, ii) rasps the inside of the syringe's wall to facilitate liquid channeling down the inner wall and iii) gently nudges the syringe to vibrate the drop into position.

Extensive numerical calculations show that a non-zero value for the bending moment Cj causes the surface profile to flatten and the contact angle to decrease for the case of a sessile drop on a fiat surface. Figure 4 provides a typical illustration of the influence of the bending moment Cj on the profile of a [dimensionless] sessile drop [with CK set to zero]. The complete drop surface for any profile curve is obtained by revolving the curve in question about the vertical E axis [the axis of symmetry]. Each profile curve terminates at a different end-point since they each enclose the same volume, i.e., y(v) = 0.0041473434. It is also apparent from this figure that a non-zero, positive value for Cj causes the surface to resist bending and to flatten itself out, subject to the constraint of fixed volume, in resistance to the tendency of the surface tension ~, to pull the system into a spherical shape. Furthermore, the flattening tendency of a surface with Cj > 0 also manifests itself by reducing the contact angle, 0l, which the sessile drop would form on its solid support.

For the purposes of illustration, a liquid-vapour sessile drop system with the following physical parameters: A 9 = 103 kgm -~, 3' = 0.072 J m -2 and B o - 0 . 1 was selected. Other choices are possible. These parameters designate a water drop system, near room temperature, and of a size that approximates the characteristic size of sessile drops [i.e., Ro is approximately 0.27 mm ] encountered in many laboratory situations. Numerical integration was performed in each case until the contact angle, 0z, equaled 180 degrees. Comparison of the same system with different values of Cj

30

0 . 0 0 2 0

0 . 0 0 1 0

(I.)

0 0

> 0 . 0 0 0 0

LIQUID

, ~ ~ - - - - Syringe Side Wall

I I

~ VAPOUR

I I I !

'~ i ~ Water/Heptane Interface

// ! !

I I

i I i I I I

/ / ~ Heptane/Air Interface j l i i J I

i I i I s I i __..;;'.-"

-O.Or}lO_____ - , J , , , , i I

0.0000 0.0010 0.0020

Radial Coordinate r (m)

, I , , ,

0 . 0 0 3 0 0 . 0 ( ) 4 0

Figure 3 Numerical solution for an axisymmetric double pendant drop consisting of water [inner drop] and immiscible n-heptane [outer drop] suspended from the same syringe tip; nominal radius 1 m m . Characteristic physical parameters, at 20 ~ C, for a water-heptane-air system without any octadecanol are: AOw a = l O00kgm -3 Aph a = 684 k g m - 3, 7wh = 0.0502 J m - 2 and '~ha = 0.0201 J m - 2. For the profiles il- lustrated, the inner water drop has a volume of 2.27 mm 3, the next outer heptane profile encloses 4.38 mm 3 (excludes water volume) while the outermost profile en- closes a heptane volume of 5.91 mm 3.

31

Table 3

Influence of Cj upon the Contact Angle of a Sessile Drop with fixed Dimensionless Volume, y(v) = 0.0041473434

Cj (J/m ) Ol ( o )

10 -12 175.00 10 -7 142.59 10 -5 46.62

1.715 • 10 -5 16.81

requires that one select an arbitrary dimensionless volume, y(v), and then find the same volume, possibly by interpolation, for other systems with different values of Cj. In this way, it is possible to compare the effect of non-zero Cj values on the shape or profile of a sessile drop whose volume is the same in all cases. Figure 4 plots four constant volume curves of differing Cj determined in this fashion. In each case, the volume y(v) enclosed by the curve was 0.0041473434 and the other curves were plotted from the position ( 0 , 0 ) to that value of (X, E ) which enclosed the designated volume y(v). For example, the C j - 10-12 j m-1 curve travels from the origin to a point where the contact angle 0l = 175. ~ The classical case of Cj -O, corresponds to a sessile drop with contact angle of 180 ~ and dimensionless volume y(v) of 0.0041474.

The influence of a non-zero bending moment on the magnitude of the contact angle may be tabulated from the sessile drop profiles curves. For example, if the normalized volume y(v) is selected as 0.0041473434, then the contact angle 0l corresponding to this volume would be 175 ~ when Cj = 1 0 - 1 2 J m - 1 . At other positive values for Cj the sessile drop profile flattens and the contact angle decreases according to the results presented in Table 3. It should be realized that one is not restricted to a volume of y(v) _ 0.0041473434 but could just as easily have considered a value less than this for comparison. For example, if one had selected the volume y(v) = 0.0012919995 instead of y(v) = 0.0041473434, then the results would be given as in Table 4. Other choices of y(v) are possible as are other sets of curves from other choices for B o.

As a consequence of these results, it is possible to appreciate situations in which both the line tension and the bending moment Cj may influence the measured contact angle. If there are geometric arrangements in which the radius of curvature or the system's properties are such as to permit Cj to have a perceptible influence, then an attempt to measure the line tension from a measurement of the contact angle would be

ambiguous. 194'195 However, in virtually all cases in which the interface is open with respect to mass transport from the adjacent bulk phases the magnitude of Cj is expected to be quite small so that any effect would be imperceptibly small. The

32

0.00

II

[i] -0.05

�9 �9

ro -o .1o

o

r ~

~ -o.15 �9 �9 I,,,,,I

r ~

?5 -0 .20

0.(

1 ' I '

5 - 1 = . - Jm

1 0 l _ ~ C J = l • lO-5 jm-1

LIQUID

Cj = 1 • 10 -7 Jm-1

10-12 jm -1 I l i ! i

~0 0.10 0.20 0.30

Dimensionless Radial Coordinate X - ~ c r

Figure 4 Influence of the bending moment, Cj > 0, on the profile of a sessile drop [dimensionless]. The (X, E) origin corresponds to the apex or top of the drop. The complete drop surface for any profile curve is obtained by revolving the curve in ques- tion about the vertical E axis. The dimensionless volume, y(v), equals the region en- closed by the drop surface and a horizontal plane which intersects the contact line of the drop, i.e., end-point of the profile curve. All curves are obtaining by commencing the integration at the origin and progressing until they enclose the same volume; in this case, y(v) = 0.0041473434. Consequently, each profile curve terminates at a dif- ferent end-point and at a different contact angle, 0l.

33

Table 4

Influence of Cj upon the Contact Angle of a Sessile Drop with fixed Dimensionless Volume, y(v) _ 0.0012919995

Cj (J/m ) Ol ( o )

10 -12 75.00 10 -7 74.20 10 -5 33.89

1.715 • 10 -5 12.47

situation is somewhat different for interacting condensed membranes. These structures, usually formed from mixtures of diacyl-chain lipids and other amphiphilic constituents, exhibit surface cohesion with restricted surface compressibility. As a result, when the specific free energy of a membrane surface is changed [e.g., by adding electric charges or by screening charges by electrolytes] what occurs is a slight contraction or expansion in surface density until a new equilibrium configuration is achieved. The relatively closed, with respect to mass transfer, nature of these membrane structures means that if the membrane is constrained it will exhibit both resistance to area dilation and to deformations which cause bending. According to Evans 8 "this bending rigidity is dominated by elastic expansion of one layer of the bilayer relative to compression of the adjacent layer when the membrane is curved". Furthermore, "the differential tension between layers produces a membrane torque or stress couple about contour lines in the surface". 196 But, even for these kinds of systems E v a n s 196 estimates that the resistance to bending is extremely small [i.e., his estimate yields a value of Cj = 10-11 ~tN ] and that it "offers little visible resistance to deformation for vesicles with diameters greater than 10 -6 meters. ''197 However, it should be realized that there is still a great deal of uncertainty about the range of magnitudes that are possible for the bending moment. Recent dynamic measurements 198 at a frequency of 5 GHz indicate that this bending [or rigidity modulus] might be 103 times larger than previously believed for surfactant layers in swollen lyotropic lamellar liquid-crystal phases. Even if this value for Cj had been used to calculate a profile curve for Figure 4 the difference between the Cj = 0 and Cj = 10 -8 ~N profiles would have been much less than the thickness of the lines plotted in Figure 4. Therefore, it seems reasonable to assume that for relatively large, pure liquid sessile drops that the bending moment Cj does not yield a perceptible effect on either the surface profile or the contact angle. However, for surfactant systems one may need to be more cautious about dismissing the importance of the bending moment.

Using Bo =0.1 at the origin, it is possible to solve Eq. (94) and to generate corresponding pendant drop shapes; cf. Fig. 5, for the situation Cj > 0. The Fig. 5

34

profile shapes are not restricted to a particular volume enclosure or to the requirement that they begin at a particular syringe wall radius. The counterpart of Fig. 5 is the Cj = 0 situations illustrated in Fig. 2. If a volume constraint were imposed, then the profile curves would appear as shown in Fig. 6. Obviously, the comparison which is of most importance is the one where profiles of differing Cj are suspended from the same syringe tip; cf. Fig. 7. Figure 7 illustrates this situation. All profiles commence from a fixed [constrained] syringe tip location, i.e. Rc- ~f--CCrc = 0.05 (dimensionless), and terminate when the profile curve intersects the vertical axis. An increase in Cj permits the pendant drop to enclose greater volume and to decrease its radius of curvature at the origin. A careful observation shows that with increasing Cj a fixed volume pendant drop will pull itself upward, against gravity, and push itself outward at points above the center of mass elevation.

35

t',4

II [ I ]

-O.20 .=~

o O

. ~ = , 1 4 - . a

-0 .40 r.,,r r.,/'3

o ,~,-~ r..t3

~ ~-..4

C~

Pendent Drop under the I n f l u e n c e of Bending Moment Constant Rc=O.05 (dimensionless), surface t e n s i o n - O . 0 7 2 d / m * * 2

.... ' ! ' I 1 1 ' I '

" " , . . .

\ 'l

|

i i

t

i '

~ t

\

\

I

I t

R c - 0.05 (dimensi0nless)

7 - 0 . 0 7 2 Jm- 2

LIQUID I i

t

t

I

,,' Cg - o.o . . . . . . . . . . . . . C j - 1 x 10 - 7 J m -

i i j

..'" Cj - 1 x 1 0 - 6 j m - 1

................. .-'""" Cj -- 1 X 1 0 - 5 Jm-

V A P O U R

i ! A - 0 . 6 0 , l , l , l .

0.00 O. 10 0.20 0.50 0.40 0.50

Dimensionless Radial Coordinate X - q~c r

Figure 5 Illustration of the influence of a non-zero bending moment, Cj > 0, on the shape of an axisymmetric pendant drop. The complete drop surface for any profile curve is obtained by revolving the curve about the vertical axis. However, profile curve points beyond an inflection point are not physically possible. Initial radius of curvature, in all cases, was Bo = 0.1.

36

t~ I> II

[I]

0 0

Q) ~

>

(D

0 ~

0.20

0.10

Pendent Drop under the Influence of Bending Moment constant V=0.004251322556734, Bo=0.1, surface tension=O.072

I ' I ' 1 ' i ' i '

B o - 0.1

Y - 0 . 0 7 2 J m - 2 C j - 0 .0

y(v ) _ 4 .231 x 10 - 3 Cj - 1 x 10- 7 Jm - 1

C j - 1 X 1 0 - 6 j m - 1

0.00 0.00 0.05 0.10 0.15 0.20 0.25

Dimensionless Radial Coordinate

-' ~, . Cj - 1 • 10 -5 J m - 1

~, ..

"" ..... .i............."'"", V A P O U R

X - ~ c r

Figure 6 Illustration of the influence of a non-zero bending moment, Cj > 0, on the shape of an axisymmetric pendant drop subject to a volume constraint. Each profile curve terminates at a different end-point because they are required to enclose the same [dimensionless] volume, i.e. y(v) = 0.004231322556734. Initial radius of curvature, in all cases, was Bo - 0.1.

37

Figure 8 illustrates the manner by which the radius of curvature, Ro, is influenced by the bending moment, Cj , when the syringe tip constraint is fixed at rc = 0.1 ram. Three distinct plots are shown; corresponding to situations where the surface tensions are: 30 mJ m -~, 50 mJ m -2 and 72 mJ m -2. The curves on each plot represent fixed drop volume situations. For large drops, the influence of Cj on Ro is small. As the drop becomes smaller the fixed volume curves begin to deviate from near linearity [i.e. they acquire a concave down curvature] at lower, positive values of Cj. Figure 9 illustrates another aspect of this interesting effect with two plots of Ro versus Cj. In both cases, the syringe tip radius is fixed at rc = 0.1 mm and the two plots show curves of surface tension with different drop volume. Once again, if the drop is larger, then the influence of Cj is smaller. However, it is also true to state that if the magnitude of the surface tension is larger, then the influence of Cj is correspondingly smaller. Thus, it would appear reasonable to consider dimensionless ratios, such as:

or y(v)

to characterize drops that might be influenced by bending moment affects.

38

Pendent Drop under the Influence of Bending Moment

0.8 t , 4

II

[ I 1

0.6

.~= n73 o o

0.4 -

>

"-'~ 0 2 - ' �9

-~"I

?5 0.0

- - 0 .

constant Bo=0.1, surface tension=O.072 ' ' I ' I

Bo - 0.1

Y - 0 . 0 7 2 J m - 2

/ s-"'"'"'" /

Cj - 0 .0

"")3 ,,,,,,I- s"

1-- 1 L 1

0.00 0.10 0.20

Cj - 1 x 10 - 7 Jm-1 . . . . . . . . . . . . . C j - 1 x l O - 6 j m - 1

Cj - 1 x 10 - 5 Jm -1

\

i,

! ! i

/ / /I

./ i I /

V A P O U R

0.30

D i m e n s i o n l e s s R a d i a l C o o r d i n a t e X - ~ c r

Figure 7 Illustration of the influence of a non-zero bending moment, Cj > 0, on the shape of an axisymmetric pendant drop subject to a syringe tip constraint. Each profile curve begins at the same syringe tip location, i.e. R c = 0.05 (dimensionless) , and terminates when the profile curve intersects the vertical axis. The pendant drop with the largest volume, y(v), is the drop with the largest, positive bending moment.

CJ VS Ro fo r D i f f e r e n t D rop V o l u m e s Tip Radius Rc=I OE-4 (m), Oarnrna=O 03 (d/rn"2)

, , ,

P~

- P~ ~'o8. o ~ . / / ~

r c = 1 x 1 0 - 4 m

31 = 0 . 0 3 J m - 2

0 00000 0 0000000 0 0000005 0 0000010 0 0000015 0 0000020

C j ( J m - 1 )

0 00020

0 00010

CJ VS Ro f o r D i f f e r e n t D rop V o l u m e s

Tip Radius Rc=I OE-4(m), Oemma=O072 (J/m**2)

r c = 1 x 1 0 - 4 m

= 0 . 0 7 2 J m - 2

0 00000 0 0000000 0 0000010 0 0000020 0 0000030 0 0000040 0 0000050

CJ ( J m - 1 )

CJ VS Ro f o r D i f f e r e n t D r o p V o l u m e s Tip Radius Rc-10E-4 (m), Camrna=O 05 (J/rn**2)

o ooo3o , , '

p.

p ~176 1 o 0002o - * #os.l" "~*

b-

/ /toj o 00010 ~ . . ~ ' " / ~

' ' @ % " r c = 1 x 1 0 - 4 m

= 0 . 0 5 J m - 2

o ooooo i i o ooooooo o 000001 o o 0000020 o ooooo3o

C j ( J m - 1 )

Figure 8 Plots of R o versus Cj for pendant drops of differing volume when the sur- face tension changes. The region of the plots where the fixed volume curves deviate from near linearity is the region where Cj effects are most pronounced.

39

40

CJ VS R o f o r D i f f e r e n t V a l u e s o f G a m m a

Drop Volume= 1.0E- 11 (m*"3), Tip Radius Re= 1.0E-4 (m) 0.00014 1 , '

D r o p Volume = l x l O-1 l m 3

r c = l x l O - 4 m

0.00012

0.00010

0.00008

0.00006 . . . . I , J ,

0.000000 0.000010 0.000020 0.000030

Cj (Jm -1)

0.00025

0.00020

0.00015

0.00010

CJ VS R o f o r D i f f e r e n t V a l u e s o f G a m m a

Drop Volume=4.0E-11 (m**3), T ip Rad ius Rc=I .0E-4 (m) i i ,

D r o p V o l u m e = 4 x 1 O- 11 m 3

r c = 1 x lO-4m

0.00005 , J , l , 0.00000 0.00005 0.00010 0.00015

C j ( Jm -1 )

Figure 9 Plots of R o versus Cj for pendant drops of differing surface tension when the syringe tip radius is fixed. The two plots illustrate the situation when the drop volume is changed. The region of the plots where the surface tension curves deviate from near linearity is the region where Cj effects are most pronounced.

41

Acknowledgements

This work was supported by the Natural Science and Engineering Research Council of Canada (NSERC) through grant OGP 0155053 and by Carleton University through a GR-5 Grant. The author would also like to thank S.S. Chetty, who compiled Table 1, for many informative discussions. In addition, the author has benefited from discussions with P. Chen, L.A. English, E.A. Evans, D. Kwok, A.W. Neumann, S. Treppo and the late L. Boruvka. Finally, the assistance of N. Lui, in the preparation of Fig. 3, and D. Kwok, in the preparation of Figs. 4-9, is gratefully acknowledged.

References

(1) J.W. Strutt (Lord Rayleigh), Philos. Mag. (5 th" series) 16 (1883) 309.

(2) J.W. Strutt (Lord Rayleigh), Philos. Mag. (5 th" series) 30 (1890) 285. (3) C. Truesdell (Ed.), Rational Thermodynamics, 2 nd edn, Springer-Verlag, New

York, 1984, p. 20. (4) J.W. Gibbs, The Scientific Papers of J. Willard Gibbs, Vol. 1, Dover, New York,

1961, pp. 55-371. (5) We shall define the mean curvature as the sum of principal curvatures; that is,

J = Cl + c2. This definition is found in many of the older texts, such as: C.E. Weatherbum, Differential Geometry of Three Dimensions, 2 Vols., Cambridge University Press, London, 1930.

(6) L.E. Scriven, Nature 263 (1976) 123. (7) H. Brenner, J. Colloid Interface Sci. 68 (1979) 422. (8) E.A. Evans and R. Skalak, Mechanics and Thermodynamics of Biomembranes,

CRC Press, Boca Raton, Florida, 1980. (9) J.S. Rowlinson and B. Widom, Molecular Theory of Capillarity, Clarendon

Press, Oxford, 1982. (10) H.T. Davis and L.E. Scriven, Adv. Chem. Phys. 49 (1982) 357. (11) J.S. Rowlinson, Chem. Soc. Rev. 12 (1983) 251. (12) A.I. Murdoch, Int. J. Eng. Sci. 25 (1987) 883. (13) S. Andersson, S.T. Hyde, K. Larsson and S. Lidin, Chemical Reviews 88 (1988)

221. (14) D.M. Anderson, H.T. Davis, L.E. Scriven and J.C.C. Nitsche, Adv. Chem. Phys.

77 (1990) 337. (15) E.M. Blokhuis and D. Bedeaux, J. Chem. Phys. 97 (1992) 3576. (16) B.V. Derjaguin, Adv. Colloid Interface Sci. 40 (1992) 191. (17) J. Israelachvili, Intermolecular & Surface Forces, 2 nd edn, Academic Press,

1992. (18) M. Napiorkowski and S. Dietrich, Phys. Rev. E 47 (1993) 1836. (19) A.I. Murdoch, On the Physical Interpretation of Fluid Interfacial Concepts and

Fields, in: Modelling Macroscopic Phenomena at Liquid Boundaries, W.

42

Kosinski and A.I. Murdoch (Eds.), Springer-Verlag, Wien, 1991, pp. 5-63. (20) Y.C.B. Fung and P. Tong, Biophys. J. 8 (1968) 175. (21) J.N. Israelachvili, J. Chem. Soc. (London), Faraday Trans. II, 72 (1976) 1525. (22) P.R. Zarda, S. Chien and R. Skalak, J. Biomech. 10 (1977) 211. (23) H. Wennerstrgm and B. Lindman, Phys. Rep. 51 (1979) 1. (24) J.N. Israelachvili, S. Marcelja, and R.G. Horn, Q. Rev. Biophys. 13 (1980) 2. (25) C.A. Miller and P. Neogi, AIChE. J. 26 (1980) 212. (26) D.S. Dimitrov, Prog. Surf. Sci. 14 (1983) 295. (27) V.A. Parsegian and R.P. Rand, Ann. New YorkAcad. Sci. 416 (1983) 1. (28) A.G. Petrov and I. Bivas, Prog. in Surface Sci. 16 (1984) 389. (29) J.C. Eriksson, S. Ljunggren and U. Henriksson, J. Chem. Soc. Faraday Trans. H

81 (1985) 833. (30) R. Nagarajan, Langmuir 1 (1985) 331. (31) Ch. Taupin, New Ideas for Microemulsions Structure: the Talmon-Prager and de

Gennes Models, in: Physics of Amphiphiles: Micelles, Vesicles, and Microemulsions, Proc. of the Int. Sch. of Phys. <Enrico Fermi>, V. Degiorgio and M. Corti (Eds.), North-Holland, New York, 1985, pp. 757-767.

(32) S.J. Chen, D.F. Evans, B.W. Ninham, D.J. Mitchell, F.D. Blum and S. Pickup, J. Phys. Chem. 90 (1986) 842.

(33) D. Exerowa and Z. Lalchev, Langmuir 2 (1986) 668. (34) E. Ruckenstein, J. Colloid Interface Sci. 114 (1986) 173. (35) L.A. Turkevich, S.A. Safran and P.A. Pincus, Theory of Shape Transitions in

Microemulsions, in: Surfactants in Solution, Vol. 6, K.L. Mittal and P. Bothorel (Eds.), Plenum Press, New York, 1986, pp. 1177-1191.

(36) E. Evans and D. Needham, J. Phys. Chem. 91 (1987) 4219. (37) S. Ljunggren and J.C. Eriksson, J. Chem. Soc., Faraday Trans. II 84 (1988) 329. (38) M.L. Robbins, J. Bock and J.S. Huang, J. Colloid Interface Sci. 126 (1988) 114. (39) S. Leibler, Equilibrium Statistical Mechanics of Fluctuation Films and

Membranes, in: Statistical Mechanics of Membranes and Surfaces, D. Nelson, T. Piran and S. Weinberg (Eds.), Jerusalem Winter School for Theoretical Physics, Vol. 5, World Scientific, 1989, pp. 45-79.

(40) D.J. Mitchell and B.W. Ninham, Langmuir 5 (1989) 1121. (41) L.T. Lee, D. Langevin and R. Strey, Physica 168A (1990) 210. (42) E. Evans, A. Yeung, R. Waugh and J. Song, Dynamic Coupling and Nonlocal

Curvature Elasticity in Bilayer Membranes, in: The Structure and Conformation of Amphiphilic Membranes, R. Lipowsky, D. Richter and K. Kremer (Eds.), Springer-Verlag, Berlin, 1992, pp. 148-153.

(43) M.M. Kozlov and W. Helfrich, Langmuir 8 (1992) 2792. (44) M.M. Kozlov, M. Winterhalter and D. Lerche, J. Phys. H (France) 2 (1992) 175. (45) U. Seifert, L. Miao, H.-G. D~Jbereiner and M. Wortis, Budding Transition for

Bilayer Fluid Vesicles with Area-Difference Elasticity, in: The Structure and Conformation of Amphiphilic Membranes, R. Lipowsky, D. Richter and K. Kremer (Eds.), Springer-Verlag, Berlin, 1992, pp. 93-96.

43

(46) A.I. Rusanov, Colloids and Surfaces A76 (1993) 7. (47) L.E. Scriven, Chem. Engin. Sci. 12 (1960) 98. (48) J.C. Slattery, Chem. Engin. Sci. 19 (1964) 379. (49) J.C. Slattery, Industrial and Engineering Chemistry Fundamentals 6(1) (1967)

108. (50) L. Waldmann, Z. Naturforschg. 22 (1967) 1269. (51) R. Ghez, Surf Sci. 4 (1966) 125. (52) G.M.C. Fisher and M.J. Leitman, Arch. Ration. Mech. Anal. 30 (1968) 225. (53) W.O. Williams, Arch. Ration. Mech. Anal. 49 (1972) 225. (54) M.E. Gurtin and A.I. Murdoch, Arch. Ration. Mech. Anal. 57 (1975) 291. (55) K. Wilmanski, Arch. Mech. 29(3) (1977) 459. (56) T.S. S~5rensen, Instabilities Induced by Mass Transfer, Low Surface Tension and

Gravity at Isothermal and Deformable Fluid Interfaces, in: Dynamics and Instabilities of Fluid Interfaces, T.S. S~5rensen (Ed.), Lecture Notes in Physics, Vol. 105, Springer-Verlag, Berlin, 1979, pp. 1-74.

(57) G.P. Moeckel, Arch. Ration. Mech. Anal. 57 (1975) 255. (58) A.I. Murdoch, Q. J. Mech. Appl. Math. 29(3) (1976) 245. (59) J. Kovac, Physica 86A (1977) 1. (60) A.R. Deemer and J.C. Slattery, Int. J. of Multiphase Flow 4 (1978) 171. (61) A.M. Albano, D. Bedeaux and J. Vlieger, Physica 99A (1979) 293. (62) L.G. Napolitano, Acta Astronautica 6 (1979) 1093. (63) J.S. Dahler, Microscopic Theory oflnterfacial Dynamics, in: Continuum Models

of Discrete Systems, University of Waterloo Press, Waterloo, 1980, pp. 345-361. (64) J.-F. Dumais, Physica 104A (1980) 143. (65) A. Grauel, Physica 103A (1980) 468. (66) A.M. Waxman, J. Non-Newtonian Fluid Mech. 9 (1981) 235. (67) P. Fergola and A. Romano, Ric. Mat. 32 (1983) 221. (68) D. Ronis, Physica 121A (1983) 1. (69) D. Iannece, Int. J. Eng. Sci. 23 (1985) 449. (70) W. Kosinski, Thermodynamics of Singular Surfaces and Phase Transitions, in:

Free Boundary Problems: Applications and Theory, A. Bossavit, A Damlanian, A.M. Fremond, (Eds.), Vol. 3, Pitman, Boston, 1985, pp. 140-151.

(71) W. Kosinski, Field Singularities and Wave Analysis in Continuum Mechanics, Ellis Horwood with PWN Polish Sci. Publ., Chichester, 1986.

(72) D. Bedeaux, Adv. Chem. Phys. 64 (1986) 47. (73) R.C. Desai and M. Grant, Dynamics at a Liquid-Vapour Interface, in: Fluid

Interfacial Phenomena, C.A. Croxton, (Ed.), John Wiley & Sons, Toronto, 1986, pp. 135-164.

(74) F. Dell'isola and A. Romano, Int. J. Eng. Sci. 25 (1987) 1459. (75) T. Alts and K. Hutter, J. Non-Equilib. Thermodyn. 13 (1988) 221. (76) M.E. Gurtin, Arch. Ration. Mech. Anal. 100 (1988) 275.

44

(77) J.C. Slattery, Interfacial Transport Phenomena, Springer-Verlag, Berlin, 1990. (78) D.A. Edwards, H. Brenner and D.T. Wasan, Interfacial Transport Processes and

Rheology, Butterworth-Heinemann, Boston, 1991. (79) W. Kosinski and A.I. Murdoch (Eds.), Modelling Macroscopic Phenomena at

Liquid Boundaries, Springer-Verlag, Wien, 1991. (80) M.E. Gurtin, Thermomechanics of Evolving Phase Boundaries in the Plane,

Clarendon Press, Oxford, 1993. (81) G.M. Mavrovouniotis and H. Brenner, Phil. Trans. Roy. Soc. (London) A345

(1993) 165. (82) Y.D. Shikhmurzaev, Physica 192A (1993) 47. (83) R. Defay and I. Prigogine, Surface Tension and Adsorption, A. Bellemans,

collab, with D.H. Everett, trans., Longmans, Green & Co., London, 1966, p. xvii. (84) M. Slemrod, Arch. Ration. Mech. Anal. 81 (1983) 301. (85) J. Serrin, Q. Appl. Math. 41 (1983) 357. (86) J.E. Dunn and J. Serrin, Arch. Ration. Mech. Anal. 88 (1985) 95. (87) J.W. Gibbs, op. cit., Ref. 4, p. 62, 278. (88) R. Courant, What is Mathematics?, Oxford University, Toronto, 1947, pp. 258-

264. (89) Abraham Goetz, Introduction to Differential Geometry, Addison-Wesley,

Toronto, 1970, pp. 256-258. (90) M.M. Lipschutz, Schaum's Outline Series: Theory and Problems of Differential

Geometry, McGraw Hill, Toronto, 1969, pp. 242-246. (91) R.S. Millman and G.D. Parker, Elements of Differential Geometry, Prentice-

Hall, Englewood Cliffs, New Jersey, 1977, pp. 188-191. (92) R. Osserman, A Survey of Minimal Surfaces, Dover, New York, 1986, pp. 85-

86. (93) H.B. Callen, Thermodynamics, 2 nd. edn., John Wiley, New York, 1985. (94) J.G. Kirkwood and I. Oppenheim, Chemical Thermodynamics, McGraw-Hill,

New York, 1961. (95) A. Miinster, Classical Thermodynamics, John Wiley, New York, 1970. (96) H. Reiss, Methods of Thermodynamics, Blaisdell, Toronto, 1965. (97) L. Tisza, Generalized Thermodynamics, M.I.T. Press, Cambridge, MA, 1966. (98) H.B. Callen, Thermodynamics, 1 st. edn., John Wiley, New York, 1960, p. 107. (99) H.B. Callen, op. cit., Ref. 98, p. 115. (100) R.C. Tolman, The Principles of Statistical Mechanics, Dover, New York, 1938,

p. 511. (101) R. Courant and D. Hilbert, Methods of Mathematical Physics, 2 Vols.,

Interscience Publ., New York, 1937. (102) F.P. Buff, J. Chem. Phys. 19 (1951) 1591. (103) F.P. Buff, J. Chem. Phys. 23 (1955) 419. (104) F.P. Buff, J. Chem. Phys. 25 (1956) 146.

45

(105) F.P. Buff and H. Saltsburg, J. Chem. Phys. 26 (1957) 23. (106) F.P. Buff and H. Saltsburg, J. Chem. Phys. 26 (1957) 1526. (107) F.P. Buff, The Theory of Capillarity, in: Encyclopedia of Physics, S. FliJgge

(Ed.), Vol. 10, Structure of Liquids, Springer-Verlag, Berlin, 1960, pp. 281-304. (108) T.L. Hill, J. Chem. Phys. 19 (1951) 1203. (109) J.W. Gibbs, op. cir., Ref. 4, pp. 219-331. (110) S.D. Poisson, Memoire sur les Surfaces Elastiques, Mem. Cl. Sci. Mathem. Phys.

(Inst. de France), Part 2, 1812, 167-225. (111) F. Casorati, Acta. Math. 14(1890-1) 95. (112) J.C.C. Nitsche, Lectures on Minimal Surfaces, Vol. 1, Cambridge University

Press, New York, 1989, pp. 23-24. (113) W. Helfrich, Z. Naturforsch. 28C (1973) 693. (114) C.E. Weatherbum, Differential Geometry of Three Dimensions, 2 Vols.,

Cambridge University Press, London, 1930, p. 243. (115) G. Thomsen, Uber konforme Geometrie. Abh. Math. Sem. Univ. Hamburg, 3

(1924) 31, see p. 56. (116) U. Dierkes, S. Hildebrandt and A. KiJster, Minimal Surfaces, 2 Vols., Springer-

Verlag, New York, 1992. (117) C.L. Murphy, Thermodynamics of Low Tension and Highly Curved Interfaces

(PhD. Thesis), University of Minnesota, Department of Chemical Engineering, University Microfilms, Ann Arbor, MI, 1966.

(118) J.C. Melrose, Industrial Engin. Chem. 60(3) (1968) 53. (119) J.C. Melrose, Thermodynamic Aspects of Capillarity, in: Applied

Thermodynamics, K.-C. Chao (Ed.), American Chemical Society, Washington, DC, 1968, pp. 249-266.

(120) J.C. Melrose, Thermodynamics of Surface Phenomena, in: Proc Int. Conf. on Thermodynamics, Cardiff, UK, 1970," P.T. Landsberg (Ed.), Butterworths, London, 1970, pp. 273-286.

(121) D.W. Hoffman and J.W. Cahn, Surf. Sci. 31 (1972) 368. (122) J.W. Cahn and D.W. Hoffman, Acta. Metall. 22 (1974) 1205. (123) L. Boruvka and A.W. Neumann, J. Chem. Phys. 66 (1977) 5464. (124) L.E. Scriven, Equilibrium Bicontinuous Structures, in: Surfactants in Solution,

Vol. 2, K.L. Mittal (Ed.), Plenum Press, New York, 1977, pp. 877-893. (125) R.E. Benner Jr., L.E. Scriven and H.T. Davis, Structure and Stress in the Gas-

Liquid-Solid Contact Region, in: Structure of the Interfacial Region, D.H. Whiffen (Ed.), Vol. 16, Faraday Symposia of the Chemical Society, London, 1981, pp. 169-190.

(126) J.S. Rowlinson, J. Chem. Soc. Faraday Trans. II 79 (1983) 77. (127) J.I.D. Alexander and W.C. Johnson, J. Appl. Phys. 58 (1985) 816. (128) W.C. Johnson and J.I.D. Alexander, J. Appl. Phys. 59 (1986) 2735. (129) M.E.R. Shanahan and P.G. de Gennes, Equilibrium of the Triple Line

Solid~Liquid~Fluid of a Sessile Drop, in: Adhesion 11, K.W. Allen (Ed.), Elsevier Applied Science, New York, 1987, pp. 71-81.

46

(130) M.E.R. Shanahan, J. Adhesion 20 (1987) 261. (131) M.E.R. Shanahan, Revue Phys. Appl. 23 (1988) 1031. (132) M.E.R. Shanahan, J. Phys. D 23 (1990) 321. (133) P. Neogi, M. Kim and S.E. Friberg, J. Phys. Chem. 91 (1987) 605. (134) P. Neogi and S.E. Friberg, J. Colloid Interface Sci. 127 (1989) 492. (135) V.S. Markin, M.M. Kozlov and S.L. Leikin, J. Chem. Soc. Faraday Trans. H 84

(1988) 1149. (136) M.M. Kozlov and V.S. Markin, J. Colloidlnterface Sci. 138 (1990) 332. (137) Y.Z. Povstenko, J. Colloid Interface Sci. 144 (1991) 497. (138) I.B. Ivanov and P.A. Kralchevsky, Mechanics and Thermodynamics of Curved

Thin Films, in: Thin Liquid Films, I.B. Ivanov (Ed.), Marcel Dekker, New York, 1988, pp. 49-129.

(139) P. Kralchevsky, J. Colloid Interface Sci. 137 (1990) 217. (140) P.A. Kralchevsky and I.B. Ivanov, J. Colloid Interface Sci. 137 (1990) 234. (141) T.D. Gurkov and P.A. Kralchevsky, Colloids and Surfaces 47 (1990) 45. (142) P.A. Kralchevsky, T.D. Gurkov and I.B. Ivanov, Colloids and Surfaces 56

(1991) 149. (143) P.A. Kralchevsky, J.C. Eriksson and S. Ljunggren, Adv. Colloid Interface Sci.

48 (1994) 19. (144) J.C. Eriksson and S. Ljunggren, J. Colloid Interface Sci. 152 (1992) 575. (145) S. Ljunggren, J.C. Eriksson and P.A. Kralchevshy, J. Colloid Interface Sci. 161

(1993) 133. (146) J.C. Eriksson and S. Ljunggren, J. Colloid Interface Sci. 167 (1994) 227. (147) J.W. Gibbs, op. cit., Ref. 4, p. 225. (148) J.W. Gibbs, op. cir., Ref. 4, p. 228. (149) J.W. Gibbs, op. cit., Ref. 4, p. 232. (150) J.W. Gibbs, op. cit., Ref. 4, p. 234. (151) It would seem reasonable to conclude that Gibbs was aware of invariant

curvature measures; namely, the sum of curvatures C l + C2 and the product of curvatures C l c2 based on his text: J.W. Gibbs, Vector Analysis, Charles Schribner' s Sons, New York, 1901.

(152) A. Sanfeld, Thermodynamics of Surfaces, in: Physical Chemistry: An Advanced Treatise, W. Jost (Ed.), Vol. 1, Academic Pr., New York, 1971, pp. 245-291, see p. 260.

(153) J.W. Gibbs, op. cit., Ref. 4, p. 227. (154) C.L. Murphy, op. cit., Ref. 117, p. A2-4. (155) C.L. Murphy, op. cit., Ref. 117, p. B4-11. (156) S.D. Poisson, op. cit., Ref. 110, pp. 224-226. (157) J.C.C. Nitsche, op. cit., Ref. 112, p. 25. (158) L. Boruvka, Y. Rotenberg and A.W. Neumann, J. Phys. Chem. 89 (1985) 2714. (159) L. Boruvka, Y. Rotenberg and A.W. Neumann, Langmuir 1 (1985) 40.

47

(160) L. Boruvka, Y. Rotenberg and A.W. Neumann, J. Phys. Chem. 90 (1986) 125. (161) Y. Rotenberg, L. Boruvka and A.W. Neumann, Langmuir 2 (1986) 533. (162) P.M. Naghdi, The Theory of Shells and Plates, in: Handbuch der Physik, C.

Truesdell (Ed.), Vol. 6a/2, Mechanics of Solids II, pp. 425-640, Springer-Verlag, Berlin, 1972, p. 471.

(163) B.Y. Chen, Proc. Amer. Math. Soc. 40 (1973) 563. (164) Eq. (13) of Ref 145 should read

(H 2 + D 2 ) B + 2 H D O = C j ( 2 H 2 - K ) + 2 C K H K

with a plus sign in front of the final term. (165) J.N. Israelachvili, S. Marcelja and R.N. Horn, Q. Rev. Biophys., 13 (1980) 121. (166) V. Degiorgio and M. Corti (Eds.), Physics of Amphiphiles: Micelles, Vesicles,

and Microemulsions, Proc. of the Int. Sch. of Phys. <Enrico Fermi>, North- Holland, New York, 1985.

(167) J. Meunier, D. Langevin and N. Boccara (Eds.), Physics of Amphiphilic Layers, Springer-Verlag, Berlin, 1987.

(168) R. Skalak, A. Tozeren, P.A. Zarda and S. Chien, Biophys. J., 13 (1973) 245. (169) A.E. Green and W. Zerna, Theoretical Elasticity, 2 nd. edn., Clarendon Press,

Oxford, 1968. (170) A.E. Green and J.E. Adkins, Large Elastic Deformations, 2 nd. edn., Clarendon

Press, Oxford, 1970. (171) M.A. Peterson, J. Math. Phys., 26 (1985) 711. (172) C. Maudarelli and D. Zeev, D., 1986, AIChE Symposium Series No. 252, 82

(1986) 179. (173) S. Ljunggren and J.C. Eriksson, J. Chem. Soc., Faraday Trans. 2, 84 (1988) 329. (174) M.M. Kozlov and V.S. Markin, J. Colloid Interface Sci., 138 (1990) 332. (175) E.A. Evans and R. Skalak, op. cit., Ref. 8, p. 186. (176) E.A. Evans and R. Skalak, op. cit., Ref. 8, p. 76. (177) E.A. Evans and R. Waugh, J. Colloid Interface Sci., 60 (1977) 286. (178) J.C. Eriksson, Thermodynamics of Bilayer Lipid Membranes, in: The Modem

Theory of Capillarity, F.C. Goodrich and A.I. Rusanov (Eds.), Akademie-Verlag, East Berlin, 1981, pp. 193-208.

(179) J. Gaydos, Colloids and Surfaces 114 (1996) 1. (180) H. Dueling and W. Helfrich, J. Phys. (Paris) 37 (1976) 1335. (181) J. Jenkins, Math. Biophys. 4 (1977) 149. (182) M. Peterson, J. Appl. Phys. 57 (1985) 1739. (183) S. Svetina and B. Zeks, Euro. Biophys. J. 17 (1989) 101. (184) J. Bemdl, J. K~is, R. Lipowsky, E. Sachmann and U. Seifert, Europhys. Lett. 13

(1990) 659. (185) L. Miao, B. Fourcade, M. Rao, M. Wortis and R. Zia, Phys. Rev. A 43 (1991)

6843. (186) U. Seifert, Phys. Rev. Lett. 66 (1991) 2404.

48

(187) U. Seifert, K. Bemdl and R. Lipowsky, Phys. Rev. A 44 (1991) 1182. (188) Ou-Yang Zhong-can, Phys. Rev. A 41 (1990) 4517. (189) Ou-Yang Zhong-can, Phys. Rev. E 47 (1993) 747. (190) J. Gaydos and A.W. Neumann, Thermodynamics of Axisyrnmetric Capillary

Systems, in: Applied Surface Thermodynamics, A.W. Neumann and J.K. Spelt (Eds.), Marcel Dekker, New York, 1996, pp. 53-108.

(191) S. Hartland and R.W. Hartley, Axisymmetric Fluid-Liquid Interfaces, Elsevier, New York, 1976.

(192) L.A. English A Pendant Drop Technique for Measuring n - A Isotherms, M.Sc. Thesis, Carleton Univ., 1996.

(193) D.Y. Kwok, D. Vollhardt, R. Miller, D. Li and A.W. Neumann, Colloids and Surfaces A88 (1994) 51.

(194) J. Gaydos and A.W. Neumann, J. Colloid Interface Sci. 120 (1987) 76. (195) J. Gaydos and A.W. Neumann, Line Tension in Multiphase Equilibrium Systems,

in: Applied Surface Thermodynamics, A.W. Neumann and J.K. Spelt (Eds.), Marcel Dekker, New York, 1996, pp. 169-238.

(196) E.A. Evans, Colloids and Surfaces 43 (1990) 327, see p. 330. (197) E.A. Evans, op. cir., Ref. 194, p. 332. (198) S. Mangalampalli, N.A. Clark and J.F. Scott, Phys. Rev. Lett. 67 (1991) 2303. (199) W.H. Press, S.A. Teukolsky, W.T. Vetterling and B.P. Flannery, Numerical

Recipes in Fortran, 2 nd. edn., Cambridge University Press, New York, 1992, p. 703.

(200) S. Hartland and R.W. Hartley, op. cit., Ref. 191, p. 696. (201) J. Gaydos, Implications of the Generalized Theory of Capillarity, Ph.D. Thesis,

Univ. of Toronto, 1992, Appendix 3L.

Appendix A

The expressions for the principal radii of curvature, the mean curvature and the Gaussian curvature for a surface of revolution described by the function z = ~ ( r ) are derived. Once derived, these expressions are substituted into the Laplace Eq. (67) and a dimensionless expression derived that may be integrated using a fourth-order Runge- Kutta method. The classical Laplace Eq. (A-22) and associated numerical approach is discussed to facilitate direct comparison with the approach adopted for the generalized Laplace Eq. (A-56).

Derivation of Principal, Mean and Gaussian Curvatures 5

A surface of revolution may be generated by the rotation of a plane curve [i.e., z = ~ ( r ) ] about an axis in its plane. If this axis, which is known as the generator of the surface, is taken as the z-axis and we let r denote the perpendicular distance from

49

the axis to the curve, then the coordinate of any point on the surface of revolution may be expressed by the parametric relations

x - r c o s ~ , y = r s i n ~ , z = ~ ( r ) (A-l)

where the longitudinal angle ~ is the inclination of the axial plane through the given point to the ( r , z )-plane. The parametric curves ~ = constant are the meridian lines or intersections of the surface by the axial planes while the curves r = constant are the parallels or intersections of the surface by planes perpendicular to the z-axis.

First-order derivatives of the position vector r of any point on the surface of revolution when the surface is described in terms of the parameters r and ~ are

rr = r ' = (cos~, sin~, ~') (A-2)

and

r~ = ( - r sin~, r cos~, 0 ) . (A-3)

These derivatives permit one to determine the first-order fundamental forms [cf. any monograph on differential geometry for more details] as

f "x

E - 1 + [ ~ ' ] 2 , F = 0 , G - r 2 ; H 2 = r 2 L l + [ ~ ' ] 2 ~ (A-4)

Furthermore, the second-order derivatives

rrr = r " = ( 0 , 0 , ~ " ) , rrr = ( - s i n ~ , cos~, 0 ) , rr162 = ( - r c o s ~ , - r sin~, 0 ~A-5)

enable one to derive the second-order fundamental forms as

L r ~" = T2 = r = M = 0 N r2~ ' 3~,~,, H ' ' H ' H 2 (A-6)

The unit normal to the surface is given by

~l = ( - ~ ' r cos~),- ~ ' r sin~), r ) (A-7) H

whereupon the derivatives of the unit normal with respect to the parameters become

fir = a , = _ ~ " ^ ~ ' ( 1 + [ ~ ' ] 2 ) 3 / 2 r ' and nr = - r ( 1 + [ ~ , ] 2 ) 1 / 2 re (A-8)

The F = 0 expression alone yields the fact that the parallels cross the meridians orthogonally. Since both F and M vanish identically the parametric curves are the lines of curvature. Consequently, the equation for the principal curvatures reduces to

r ( l+[~ '12 )2c2 - ( l+[~ '12 ) l / 2 I r~"+~ ' ( l+[~ '12 ) l c+~ '~" = 0 (A-9)

where upon the roots of this equation become principal radii of curvature given by ~,,

e l = ( 1 +[~ ' ]2 )3/2 (A-10)

and

50

C 2 -- (A-11) r ( 1 + [~,]2 )1/2

The first expression is the curvature of the generating curve while the second expression is the reciprocal of the length of the normal intercepted between the curve and the axis of rotation. Since Eq. (A-8) also shows that the parametric lines (the parallels and meridians of the surface) are the lines of curvature the expressions for c 1 and c 2 could have been determined from Eq. (A-8) directly.

The expressions for the principal radii of curvature permit one to evaluate the mean and Gaussian curvature as

1 + 1 _ r ~ " + ~ ' ( l + [ ~ ' ] 2 ) J = C l + c 2 - R 1 R2 - r ( 1 +[~,]2)3/2 (A-12)

and

1 _ ~'~" B

K = ClC2 = R 1 R 2 r ( 1 + [ ~ ' ] 2 ) 2 (A-13)

Alternatively, one may choose to express J and K in terms of the angle between the tangent to the surface and the horizontal. From the relation

~ ' - tan0 (A-14)

between the slope of the ~ surface function and the tangent angle 0 we may derive the relation

~" = sec 20 d 0 dr (A-15)

Substituting relations from (A-14) and (A-15) into Eqs. (A-12) and (A-13) yields

dO sin0 J = cos0 ~rr + r (A-16)

and

s in0 cos0 dO K = (A-17)

r dr

It is possible to express these quantities in terms of the arc-length s , which shall be used subsequently as an incrementing parameter in the numerical procedure when the Laplace equation of capillarity is integrated, by relating the horizontal and vertical coordinates ( r , ~) to the arc-length using the relations

dr - cos 0 (A-18)

ds

and

dz = sin 0 (A-19)

ds

Employing these derivatives permits us to write the mean and Gaussian curvatures in terms of the arc-length as

51

and

dO sinO J - +

ds r (A-20)

sinO dO K - (A-21)

r ds

The Classical Laplace Equation of Capillarity

The number of parameters present in the Laplace equation of capillarity may be reduced by using the capillary constant, defined by Eq. (95), to make all lengths dimensionless. This approach also has the physical advantage that one usually knows the physical properties of the system under investigation. Alternatively, one could try

1 to dimensionalize with respect to the factor -R-o--o ' however, this involves the, usually

unknown, value for the radius of curvature at the apex of the drop. Thus, by multiplying all length scales by the factor ~ we may write all lengths, the surface area and volume as dimensionless quantities; cf. Eqs. (96) and (98). Substituting these dimensionless variables in to the classical form of the Laplace equation of capillarity, Eq. (91), yields the dimensionless version

dO 2 sin 0 - + E - ~ . (A-22)

dY B X

The variables X, E, Y and 0 are related geometrically by

dX dE = cos0 ; - sin0

dY dY

and the variation in area y(a) and volume y(v) are given by

dy(a) dy(v) = 2 n X ; - rtX 2sin0

dY dY

(A-23)

(A-24)

The natural boundary conditions [defined at the apex of the drop] are

dO sin 0 1 - - (A-25)

dY X B

where X, E, Y, 01, y(a) and y(v) are all zero at this location. Expressions (A-22), (A-23) and (A-25) in combination with the natural boundary condition at the apex of the drop constitute a complete specification of the problem. Numerical integration along the drop profile proceeds using the dimensionless arc-length as the adjustable parameter. Typical sessile and pendant drop profile shapes are shown in Ref. 191.

Numerical Integration of the Classical Laplace Equation of Capillarity

As similar considerations and expressions also apply when integrating the non- classical Laplace equation of capillarity, we shall briefly detail the key expressions that are involved. A procedure to perform this numerical integration using the dimensionless arc-length Y was established by Hartland and Hartley. 191 They carried-

52

out the integration using a fourth-order Runge-Kutta method which incrementally increased Y in controlled [i.e., adjustable] steps AY. Runge-Kutta methods propagate a solution over an interval by combining the information from several Euler-style steps and then using the information obtained to match a Taylor series expansion up to some higher order. While not computationally efficient or sophisticated Runge-Kutta methods are almost always successful. 199 For a step increment of AY the corresponding increments in the variables 01, X, E, y(a) and y(v) are given in a fourth-order Runge- Kutta method as

A0 = ~ A01 [1] + 2 A01 [2] + 2 A01 [3] + A01 [4] (A-26)

and

AX - -~- AX[1 ] + 2AX[2 ] + 2AX[3 ] + AX[41

11AE +2AE +2AE +AE 1 A ~ = Z [1] [2] [3] [4]

1 I AY~]t + 2 AYt~] + 2 AYt~] + AYt~] 1 Ay(a) = --g-

(A-27)

(A-28)

(A-29)

Ay(v ) = 16 [ AY]]] + 2AYk~] + 2AYk~ t + AYk~]I (A-30)

where the subscript square brackets [] are used to denote the points at which evaluation occurs during each step. For the classical case, the specific terms in these expressions are given by

sin~ 1 A01 [1] - -~- + E - X AY (A-31)

AX[1 ] = cos 0 AY (A-32)

AE [11 = sin 0 AY

= 2 xAv

AYt~ t = ~X2sin0AY

I 2 AE[I] s in(0+ 1/2A0[l] ) A01 [2] - -~- + E -~ 2 X + 1/2 AX [1]

I 1 ] AX[2I = cos 0+~A0[ l l AY

l AY

(A-33)

(A-34)

(A-35)

(A-36)

(A-37)

53

and

[ ' 1 AE[2] = sin 0 + ~ A 0 [ I ] AY

[ 1 1 AY~3] = 2~ x + - f A x t i ~ /,Y

1 A v ~ t = ~ x + ~xt i l

2 AE [21 A013] - -~- + E + 2 X + 1/2 AX [21

E l ; AN[3 ] = cos 0+-~-A012] AY

E l l A~ [3] - sin 0 + ~ A0[21 AY

AY]~] - 2~ x + Ax ~ v

1 1 AY)~ t = rc X + AX sin 0 + A0121 AY

I 2 AE sin ( 0 + A013] ) 1 AY A01 [41 = g + E + [31 - X + z~X[3 ]

~L3([4] = c o s I 0 + A 0 [ 3 ] I AY

AE[41 = sinI0+A0[311 AY

~ ~ = ~ [ x + ~ ~ 1 ~

E ' 1 sin 0 + ~ A0[1 ] AY

sin ( 0 + 1/2 A012] ) ]AY

(A-38)

(A-39)

(A-40)

(A-41)

(A-42)

(A-43)

(A-44)

(A-45)

(A-46)

(A-47)

(A-48)

(A-49)

I ;2 I 1 AYt~ ] = rt X + AX[3 ] sin 0 + A013 ] AY. (A-50)

A slightly modified, double precession version of Hartland and Hartley's fortran program 2~176 was created based on these relations. 2~ A comparison, in double precision, of our results with their calculations of the situation they describe in Table 10.2a on pg. 670 of their monograph 191 permits us to categorically state that there is no difference, to the level of significance calculated by Hartland and Hartley, between the results which they obtained and published in Table 10.2a and the results which we obtained.

54

We shall not dwell on the details and implementation of the classical solution but shall only note that additional details, including a flowchart, may be found in their monograph.

The Non-Classical Laplace Equation o f Capillarity

The dimensionless quantities defined above for the classical Laplace equation of capillarity may be used to express the non-classical Laplace equation of capillarity, Eq. (93), in an analogous, dimensionless form.

Replacing the mean and Gaussian curvatures by their equivalent expressions in terms of arc-length s [cf. Eqs. (A-20) and (A-21)] and dividing through by y yields

dO sin0 Cjo sin0 dO 2 Ap g Cjo 1 I + 2 - + z - 2 ~ - - - T �9 ( A - 5 1 )

ds r "~ r ds Ro T ~I Ro

Defining the dimensionless physical constant as

4TGo bj = (A-52)

Y and substituting this constant into Eq. (A-51) gives

dO + sin0 +2 bj sin0 dO 2 bj/4~-c ds r -~c r ds - Ro + c z - 2 ~ Ro ]2 (A-53)

Dividing through by ~ c and rearranging yields

dO sin0 sin0 dO 2 cj (A-54) + ~ C r + 2 c j 'S'FT"~- -- v c r v c ua = ~ + ~ c Z -- 2 c77 s 4Tc R-------r. o

or

dO sin0 sin0 dO 2 cj + ~ + 2 ~j = - - + ~" - 2 (A-55)

dY X X dY B - B 2

after the definitions in Eq. (A-22) are used. Factoring and rearranged Eq. (A-55) gives the final desired result; the non-classical Laplace equation of capillarity expressed in terms of dimensionless coordinates as

_E1 sin0 dO B " X

= (A-56) dY 1 + 2~j sin0

X

The variables X, E, Y and 0 are related geometrically by the expressions in Eq. (A-23) and the variation in area y(a) and volume y(v) are given in Eq. (A-24). The natural boundary condition [defined at the apex of the drop] requires that the principal radii of curvature be egual at the apex so that Eq. (A-25) still applies. Finally, the quantities X, E, Y, 0, yta) and y(v) are all zero, once again, at this location. This now yields a completely specified problem in which numerical integration along the drop profile

55

using the dimensionless arc-length as the adjustable parameter may be performed in a similar manner to that of the classical Laplace equation of capillarity.

Numerical Integration of the Non-Classical Laplace Equation of Capillarity

A fourth-order Runge-Kutta method as proposed by Hartland and Hartley 191 has been used to integrate this differential equation, however, it is necessary to change each of the particular expressions in Eqs. (A-31), (A-36), (A-41) and (A-46) to reflect the change in the form of the governing differential equation. Once these expressions are modified they may be used in the expressions given by Eqs. (A-26) through (A-30) to calculate the corresponding increments in the variables 0, X, E, y(a) and y(v). The particular expressions, which correspond to Eqs. (A-31), (A-36), (A-41) and (A-46) are

sin0 B X

AY (A-57) A0[I] = sin ( 0 )

l + 2 ~ j X

[ @] AE[1] sin(0 + 1/2A011] ) 2 1 - + '~q B " 2 X + 1/2 AX[1 ]

- AY (A-58) A012] - sin ( 0 + 1/2 A0[1 ] )

1 + 2 b j X + 1/2 AX[1]

I _~1 AE[2] sin(0 + 1/2A012] ) ! 1 - +_~+ B 2 X + 1/2 53( [2]

A0131 = sin ( 0 + 1/2 A012] ) AY (A-59) l + 2 b j

X + 1/2 AX[2]

and

sin o+ o 3, -~- 1 - + E + A E [3] - X + A X [3]

A0141 = sin ( 0 + A013] ) AY (A-60) 1 + 2~-j

X + AX[3]

A program based on the method and approach of Hartland and Hartley 191 was implemented using the modified steps listed above for A0[I] through A014]. The program was run several times with various values for Cj. A straightforward generalization of this procedure may be applied to the situation where either CK or CH are non-zero [cf. Eqs. (67) and (90)]. 2~ A Fortran routine is available from the author. The reason for this choice is as follows: It is generally believed that the specific free energy 03 (a) of a liquid-vapour interface is very nearly constant, i.e., co (a) "" it, until one reaches surfaces with sufficiently high curvature, say Ro < 10 --6 meters. Given these

56

conclusions, one may estimate the relative magnitude of Cj to CK in the limit as Ro goes to zero to show that CK is significantly smaller than Cj. Consequently, the dominant correction to the classical Laplace equation of capillarity is furnished via a non-zero mean curvature bending moment Cj.

57

Glossary

Aa Fractional change in area of a surface element. Ab Extensional deformation or distortion of an initially circular surface element into

an ellipse at constant area, i.e. Aa = 0. c Capillary constant; cf. Eq. (95). c I c 2 Principal curvatures at a point on a dividing surface. ~j Dimensionless bending moment constant; cf. Eq. (97). fs Membrane elastic (strain) free energy density. fi Outward directed unit normal. r Position vector of a point within the capillary system. s (a) Specific entropy density on a dividing surface. s q) Specific entropy density on a dividing line. s (v) Specific entropy density within a volume phase. s(oo v) Specific entropy density within an idealized volume phase without an

interface(s). s~ v) Excess specific entropy assigned to a dividing surface; cf. Eq. (3). u ta) Specific internal energy density on a dividing surface. u (l) Specific internal energy density on a dividing line. u (v) Specific internal energy density within a volume phase. u(oo v) Specific internal energy density within an idealized bulk phase without an

interface(s). u~ v) Excess specific internal energy assigned to a dividing surface; cf. Eq. (2). {Aw,j }Union of all internal surfaces that would intersect a system's bounding wall

during a variation in position vector; cf. Eq. (10). Bo = ~ R o Dimensionless radius of curvature at the origin. C1 C2 Principal curvature potentials; cf. Eq. (41). C 1 + C2 = 0 Gibbs' criteria for the placement of a dividing surface at the surface of

tension location. C1 C2 Average bending or curvature potentials based on the principal

curvatures; cf. Eq. (49). CH Helfrich curvature potential; cf. Eq. (78). Cj CK Bending or curvature potentials based on the mean, J , and Gaussian

curvatures, K ; cf. Eq. (65. Cj CK Average bending or curvature potentials based on the mean and

Gaussian curvatures; cf. Eqs. (56)/(57). V2 C 1 V2 C 2 Surface gradients of the curvature potential. D = 1/2( c 1 - c 2 ) Deviatoric curvature. E t - U t + ~d~ H -J /2 J - C l + C 2

J K =Cl c2 K~

Total composite energy of the capillary system. Alternative designation for the mean curvature. Mean curvature of a point on the dividing surface; cf. Ref. # (5). Total (extensive) mean curvature; cf. Eq. (63). Gaussian curvature of a point on the dividing surface. Compressibility modulus; cf. Eq. (88).

58

K Total (extensive) Gaussian curvature; cf. Eq. (64). {Lw, k }Union of all internal contact lines that would intersect a system's bounding wall

during a variation in position vector; cf. Eq. (11). M~ v) Total mass of the i th component within a composite capillary system. P Pressure at a point within the interior of a liquid. Pm Molecular pressure. A P Pressure jump across an interface. R 1 R 2 Principal radii of curvature of a point on the dividing surface. S~) Radius of surface curvature at the origin of an axisymmetric capillary system.

Total entropy of a composite capillary system. T Temperature; cf. Eq. (6). T Equilibrium temperature; cf. Eq. (13). Ut Total internal energy of a composite capillary system. V Total volume of a composite capillary system. X = 4-C-C r Dimensionless radial distance from the axis of symmetry to a point on

the profile of a axisymmetric surface. Y = 4~c s Dimensionless arc-length measured from the origin of an axisymmetric

surface along the surface profile. y(a) = A c Dimensionless surface area of an axisymmetric capillary system. y(v) _ V c 3/2 Dimensionless volume of an axisymmetric capillary system. __el e2 Principal membrane tensions; cf. Eqs. (79)/(80). e Isotropic membrane tension; cf. Eq. (84). eo Isotropic or surface tension; cf. Eq. (87). es Shear tension; cf. Eq. (85). ~, Surface tension in the sense of Gibbs' original definition and the specific surface _ free energ__y density of a planar dividing surface; cf. Eq. (8). qt=~,+ Cl__C 1 + C2c2 Buff's surface free energy definition; cf. Eq. (51). )/c = ~' + C (c 1 + c 2) Buff' s spherical surface free energy definition; cf. Eq. (53). ~, = ~, + Cj J + CK K Murphy's surface free energy definition; cf. Eq. (58).

= ~, Boruvka etal. surface free energy definition; cf. Eq. (67). ~, = ~, + B H + O D Kralchevsky etal. surface free energy definition; cf. Eq. (74). ~-1 ~,2 Principal extension ratios of the membrane; cf. Eqs. (79)/(80). ~i Lagrange multiplier; cf. Eq. (26). ~1, i Chemical potential of the i th component; cf. Eq. (7). ILI, i = gi + (~ Equilibrium chemical potential; cf. Eq. (14). kts Shear modulus coefficient; cf. Eq. (86).

( r ) Explicit surface coordinate for an axisymmetric capillary surface defined by the function z = ~ ( r ).

~ (a) Total surface density on a dividing surface; cf. Eq. (52). (a) Surface density of the i th component on a dividing surface. ~,

9~, ~] Excess volume density of the i th component assigned to a dividing surface; cf. Eq. (4).

9(oov! Volume density of the i th component within an idealized bulk phase without an interface(s).

9~ v) Volume density of the i th component.

59

A 9 Density difference across an interface. q~ ( r ) Potential energy of an external field.

Euler characteristic; cf. Eq. (59). ~(a) = u(a) _ Ts(a) _ ~_~1, i ~)!a)

i Specific grand canonical potential of a surface phase; cf. Eq. (21).

o3(V) = u(V) - T s ( V ) - ~ i O! v) = - P

i Specific grand canonical potential of a bulk phase; cf. Eq. (15).

Abl Beltrami-Laplace operator; cf. Eq. (38). = ~ z Dimensionless vertical coordinate in an axisymmetric capillary system.

f2(m a) Grand canonical potential of the m th dividing surface; cf. Eq. (29) and text after

~(l)

~'~t f2,

Eq. (42). Grand canonical potential of the m th dividing or contact line; cf. Eq. (30). Grand canonical potential of the m th dividing point; cf. Eq. (31). Grand canonical potential of the m th volume phase; cf. Eq. (28). Total grand canonical potential of a composite capillary system. Total external field energy

This Page Intentionally Left Blank

Drops and Bubbles in Interfacial Research D. M6bius and R. Miller (Editors) �9 1998 Elsevier Science B.V. All rights reserved. 61

AXISYMMETRIC DROP SHAPE ANALYSIS (ADSA) AND ITS APPLICATIONS

P. Chen, D.Y. Kwok, R.M. Prokop, O.I. del Rio, S.S. Susnar and A.W. Neumann

Department of Mechanical and Industrial Engineering, University of Toronto, Toronto, Ontario,

Canada M5 S 3 G8

CONTENTS

1. Introduction

2. Axisymmetric Liquid-Fluid Interfaces (ALFI)

2.1. Axisymmetric Drop Shape Analysis - Profile (ADSA-P)

2.2. Axisymmetric Drop Shape Analysis - Diameter (ADSA-D)

2.2.1 Contact angle greater than or equal to 90 ~

2.2.2 Contact angle less than 90 ~ .

3. Surface and Interfacial Tension Measurements

3.1. Measurements on Static Drops

3.1.1. Liquid-Vapor Surface Tension

3.1.2. Liquid-Liquid Interfacial Tension

3.1.3. Ultralow Interfacial Tension

3.1.4. Pressure Dependence of Interfacial Tension

3.1.6. Surface Tension of Polymer Melts

3.2. Measurements on Dynamic Drops

3.2.1. ADSA as a Film Balance

3.2.2. Dynamic Surface Tension of Surfactant Solutions

3.2.3. Dynamic Surface Tension of a Mixed Solution of Protein and Small Molecules

3.2.4. ADSA-CB: Captive Bubble Method in Lung Surfactant Studies

4. Contact Angle Measurements

4.1 Contact Angle Measurements on Smooth Solid Surfaces by ADSA-P

4.1.1 Static Contact Angles

4.1.2 Comparison of ADSA-P and an Automated Capillary Rise at a Vertical Plate Technique

4.1.3 Low-Rate Dynamic Contact Angles

4.1.4 Determination of Line Tension from the Drop Size Dependence of Contact Angles

4.2 Contact Angle Measurements on Smooth Solid Surfaces by ADSA-D

4.3 Contact Angle Measurements on Rough and Heterogeneous Solid Surfaces by ADSA-D

5. References

6. List of Symbols

7. List of Abbreviations

62

1. INTRODUCTION

Numerous methodologies have been developed for the measurement of contact angles and

surface tensions as outlined in Refs. [1-4]. Liquid surface tension measurements commonly

involve the determination of the height of a meniscus in a capillary, or on a fibre or a plate.

Contact angles are most commonly measured by aligning a tangent with the profile of a sessile

drop at the point of contact with the solid surface. Other notable methods are the Wilhelmy

slide and the capillary rise technique. An overview of such techniques reveals that in most

instances a balance must be struck between the simplicity, the accuracy, and the flexibility of the

methodology.

An alternative approach to obtaining the liquid-vapor or liquid-liquid interfacial tension and/or

the contact angle is based on the shape of a sessile or pendant drop. In essence, the shape of a

drop is determined by a combination of surface tension and gravity effects. Surface forces tend

to make drops spherical whereas gravity tends to elongate a pendant drop or flatten a sessile

drop. When gravitational and surface tension effects are comparable then, in principle, one can

determine the surface tension from an analysis of the shape of the drop. Figure 1 shows two

pendant drop images of a 0.02 mg/ml bovine serum albumin aqueous solution at 37~ image

(a) was acquired at time zero, with a corresponding surface tension of 70.24 mJ/m 2, and image

(b) was acquired at time 400 s, with a corresponding surface tension of 54.22 mJ/m 2.

The advantages of pendant and sessile drop methods are numerous. In comparison with a

method such as the Wilhelmy plate technique, only small amounts of the liquid are required.

Drop shape methods easily facilitate the study of both liquid-vapor and liquid-liquid interfacial

tensions. Also, the methods have been applied to materials ranging from organic liquids to

molten metals and from pure solvents to concentrated solutions. There is no limitation to the

magnitude of surface or interracial tension that can be measured: The methodology to be

presented here works as well at 103 mJ/m 2 as at 10 .3 mJ/m 2. Measurements have been

satisfactorily made over a range of temperatures and pressures. In addition, since the profile of

the drop may be recorded by photographs or digital image representation, it is possible to study

interfacial tensions in dynamic systems, where the properties are time-dependent.

63

Fig. 1. Pendant drop images of a 0.02 mg/ml bovine serum albumin aqueous solution at 37 ~ image (a) was

acquired at time zero, with a corresponding surface tension of 70.24 mJ/m 2, and image (b) was acquired

at time 400 s, with a corresponding surface tension of 54.22 mJ/m 2.

The Laplace equation is the mechanical equilibrium condition for two homogeneous fluids

separated by an interface. It relates the pressure difference across a curved interface to the

surface tension and the curvature of the interface:

7 + - A P

where 7 is the interfacial tension, R1 and R2 are the two principal radii of curvature, and AP is

the pressure difference across the interface. In the absence of any external forces other than

gravity, AP may be expressed as a linear function of the elevation:

= + ( a o ) g z

where AP0 is the pressure difference at a reference plane, A 9 is the density difference between

the two bulk phases, g is the gravitational acceleration, and z is the vertical height of the drop

measured from the reference plane. Thus, for a given 7, the shape of a drop may be determined

(via R1 and Rz). The inverse, i.e., determination of the interfacial tension 7 from the shape, is

also possible in principle, although this is a much more difficult task.

64

Mathematically, the integration of the Laplace equation is straightforward only for cylindrical

menisci; i.e., menisci for which one of the principal curvatures, 1/R, is zero. For a general

irregular meniscus, mathematical analysis would be very difficult. For the special case of

axisymmetric drops, numerical procedures have been devised. Fortunately, axial symmetry is not

a very significant restriction for most sessile drop and pendant drop systems.

The earliest efforts in the analysis of axisymmetric drops were those of Bashforth and Adams

[5]. They generated sessile drop profiles for different values of surface tension and radius of

curvature at the apex of the drop. The determination of the interfacial tension and contact angle

of an actual drop was accomplished by interpolation of tabulated profiles. Hartland and Hartley

[6], also collected numerous solutions for determining the interfacial tensions of axisymmetric

fluid-liquid interfaces of different shapes. A computer program was used to integrate the

appropriate form of the Laplace equation and the results were presented in tables. The major

shortcoming of these methods is in data acquisition. The description of the surface of the drop

is accomplished by the measurement of a few preselected points. These points are critical since

they correspond to special features, such as inflection points on the interface, and must be

measured with a high degree of accuracy. Also, for the determination of the contact angle, the

point of contact with the solid surface, where the three phases meet, must be established.

However, these measurements are not easily obtained. In addition, the use of these tables is

limited to drops of a certain size and shape range.

Maze and Burnet [7, 8] developed a more satisfactory scheme for the determination of

interracial tension from the shape of sessile drops. They utilized a numerical nonlinear

regression procedure in which a calculated drop shape is made to fit a number of arbitrarily

selected and measured points on the drop profile. In other words, the measured drop shape

(one half of the meridian section) is described by a set of coordinate points and no particular

significance is assigned to any one of the points. In order to start the calculation, reasonable

estimates of the drop shape and size are required, otherwise the calculated curve will not

converge to the measured one. The initial estimates are obtained, indirectly, using values from

the tables of Bashforth and Adams. Despite the progress in strategy, there are several

deficiencies in this algorithm. The error function is computed by summing the squares of the

horizontal distances between the measured points and the calculated curve. This measure may

65

not be adequate, particularly for sessile drops whose shapes are strongly influenced by gravity.

For example, large drops of low surface tension tend to flatten near the apex. Therefore, any

data point which is near the apex may cause a large error even if it lies very close to the best

fitting curve, and lead to considerable bias of the solution. In addition, the identification of the

apex of the drop is of paramount importance since it acts as the origin of the calculated curves.

Rotenberg et al. [9] developed a technique, called Axisymmetric Drop Shape Analysis-Profile

(ADSA-P), which is superior to the above mentioned methods and does not suffer from their

deficiencies. ADSA-P fits the measured profile of a drop to a Laplacian curve. An objective

function is formed which describes the deviation of the experimental profile from the theoretical

profile as the sum of the squares of the normal distances between the experimental points and

the calculated curve. This function is minimized by a nonlinear regression procedure yielding

the interfacial tension and the contact angle in the case of a sessile drop. The location of the

apex of the drop is assumed to be unknown and the coordinates of the origin are regarded as

independent variables of the objective function. Thus, the drop shape can be measured from any

convenient reference frame and any measured point on the surface is equally important. A

specific value is not required for the surface tension, the radius of curvature at the apex, or the

coordinates of the origin. The program requires as input several coordinate points along the

drop profile, the value of the density difference across the interface, the magnitude of the local

gravitational constant, and the distance between the base of the drop and the horizontal

coordinate axis. Initial guesses of the location of the apex and the radius of curvature at the

apex are not required. The solution of the ADSA-P program yields not only the interfacial

tension and contact angle, but also the volume, surface ,area, radius of curvature, and contact

radius of the drop. Essentially, ADSA-P employs a numerical procedure which unifies both the

method of the sessile and pendant drop. There is no need for any table nor is there any drop

size restriction on the applicability of the method.

Cheng et al. [ 10] automated the methodology by means of digital image acquisition and image

analysis. Pictures of sessile or pendant drops are acquired using a video camera attached to a

computer, where image analysis software automatically extracts several hundred coordinates of

the drop profile, which in turn are analyzed by ADSA-P to compute surface tension.

66

Recently, ADSA-P has been rewritten [11], implementing more efficient, accurate and stable

numerical methods in order to overcome convergence problems of the original program for very

low interfacial tensions with well-deformed drop shapes [ 11 ]. Also, two additional optimization

parameters were introduced: the angle of vertical misaligment of the camera and the aspect ratio

of the video image. With these revisions, the accuracy and the range of applicability of ADSA-P

have been further improved.

For contact angle determinations, with most techniques it becomes increasingly difficult to make

measurements for flat sessile drops with very low contact angles, say below 20 ~ . The accuracy

of ADSA-P also decreases under these circumstances since it becomes more difficult to acquire

accurate coordinate points along the edge of the drop profile. For these situations, it is more

useful to view a drop from above and determine the contact angle from the contact diameter of

the drop. Initially, Bikerman [12] proposed to calculate the contact angle from the contact

diameter and volume of a sessile drop by neglecting the effects of gravity and assuming that the

drops are sections of a sphere. Obviously, this simple approach is only applicable to small drops

and/or to very large liquid surface tensions. A modified version of ADSA, called Axisymmetric

Drop Shape Analysis-Contact Diameter (ADSA-CD), was developed by Rotenberg and later

implemented by Skinner which does not ignore the effects of gravity [13]. ADSA-CD requires

the contact diameter, the volume and the liquid surface tension of the drop, the density

difference across the liquid-fluid interface, and the gravitational constant as input to calculate

the contact angle by means of a numerical integration of the Laplace equation of capillarity,

Eq. (1).

It has been found that drop shape analysis utilizing a top view is quite useful for the somewhat

irregular drops which often occur on rough and heterogeneous surfaces. In these cases, an

average contact diameter leads to an average contact angle. The usefulness of ADSA-CD for

averaging over irregularities in the three phase contact line proved to be such an asset that it

became desirable to use it instead of ADSA-P for large contact angles as well. Unfortunately,

for contact angles above 90 ~ the three phase line is not visible from above. For such cases, yet

another version of ADSA has been developed by Moy et al. [14] called Axisymmetric Drop

Shape Analysis-Maximum Diameter (ADSA-MD). ADSA-MD is similar to ADSA-CD;

however, it relies on the maximum equatorial diameter of a drop to calculate the contact angle.

67

ADSA-CD and ADSA-MD have been unified into a single program called Axisymmetric Drop

Shape Analysis-Diameter (ADSA-D).

This chapter provides an account of these ADSA methodologies. It contains a description of

the numerical algorithms and their implementation. The applicability of ADSA is illustrated

extensively for the investigation of surface tension measurements with pendant and sessile drops

and contact angle experiments with sessile drops using both ADSA-P and ADSA-D.

2. AXISYMMETRIC LIQUID-FLUID INTERFACES ( A L F I )

The classical Laplace equation of capillarity describes the mechanical equilibrium conditions for

two homogeneous fluids separated by an interface. For axisymmetric interfaces it can be written

as the following system of ordinary differential equations (ODE) as a function of the arc-length

s, as shown in Fig. 2 [9]:

dr - cos (la)

ds

dz - sin 0 (lb)

ds

dO sinO - 2 b + c z - - ( l c )

ds x

dV - 7rx 2 sin 0 (1 d)

ds

dA --2~x (le)

ds

~(0)- z(0)- 0(0)- v(0)- A(0)- 0 (lf)

where b is the curvature at the origin of coordinates and c=(Ap)g/7 is the capillary constant of

the system. 0 is the tangential angle, which, for sessile drops, becomes the contact angle at the

three-phase contact line. Although the surface area A and the volume V are not required to

define the Laplacian profile, they are included here because of their importance and the fact that

they can be integrated simultaneously without a significant increase of computational time.

68

Z

0 __X ~ ~ J f

s I

I

. . . . . . . . . . " , 2 i )

/

Fig. 2. Coordinate system used in the numerical solution of the Laplace equation for axisymmetric liquid-fluid

interfaces (ALFI).

For given values of b and c, a unique shape of a Laplacian axisymmetric fluid-liquid interface

can be obtained by simultaneous integration of the above initial value problem (IVP). However,

there is no known analytical solution for this IVP except for very limited cases, and a numerical

integration scheme must be used. There exist several numerical methods to solve systems of

ODEs for IVPs and considerable research is still devoted to this subject [ 15]. One of the most

efficient and flexible methods is the fifth and sixth order Runge-Kutta-Verner pair, DVERK,

written by Hull, Enright and Jackson [ 16, 17].

A computer program called ALFI was written [11], implementing the DVERK numerical

integration scheme to generate Laplacian profiles of pendant and sessile drops of any size

(controlled by the apex curvature b) and surface tension (specified by the capillary constant c)

by integrating the IVP (1). Some of the features of ALFI are:

�9 The volume V and surface area A are computed simultaneously with the drop profile.

The integration can be stopped when any given values of s, 0, x, z, V or A are reached,

allowing the computation of drop profiles of any specified contact angle, volume, surface

area or size. The integration also terminates if 0 reaches 180 ~ (sessile drops) or becomes

negative (pendant drops).

The inflection point of pendant drops is accurately computed, which is useful for testing and

evaluating drop profile methods.

69

* The origin of the coordinate system can be translated and rotated arbitrarily, and the

coordinates can be scaled in the horizontal and vertical directions. This feature permits the

comparison between theoretical and experimental drop profiles which have generally an

arbitrary origin of coordinates, can be vertically misaligned due to a vertical misalignment of

the video camera and have an arbitrary magnification.

. The profile coordinates can be randomly perturbed in the normal direction allowing the

simulation of experimental errors, which can be used to evaluate ADSA methods.

As mentioned before, ALFI generates complete Laplacian profiles from values of b and c by

integrating the IVP (1). The inverse process of determining b and c (from which ~/and contact

angle 0 can be easily computed) based on drop profile characteristics is a more difficult task and

forms the basis of the ADSA methods described in the following sections.

2.1. AXISYMMETRIC DROP SHAPE ANALYSIS- PROFILE (ADSA-P)

The ADSA-P methodology to determine interfacial properties by means of a numerical fit of

several arbitrary drop profile coordinates to the Laplace equation was originally developed by

Rotenberg et al. [9]. The current version of ADSA-P [ 11 ] uses the same strategy as the original,

i.e. a non-linear least-squares optimization, but with a slightly different definition of the

objective function (see below) and implementing more advanced numerical methods, as

described below. This method is applicable to sessile and pendant drops.

The strategy utilized is to construct and minimize an objective function E, defined as the sum of

the weighted squared normal distances between any N profile coordinates and the Laplacian

profile (IVP 1), as seen in Fig. 3:

N

E - Z w i e i (2a) t=l

e~=-~ ~ - -~ - ] (2b)

where w, is a weighting factor, (X,, Z, ) are the measured drop coordinates, and (x~, z, ) are the

Laplacian coordinates closest to (X~, Z~). Currently, w, is set equal to 1.0 until more studies are

available on the effect of weighting factors. By introducing the generally unknown origin (x0, z0)

70

and angle of rotation of the system of coordinates or, and scaling factors on both coordinates

(Xs, Z, ), the individual error can be written (dropping the subscript i) as:

1 e - - ( e 2 + e 2) (2c)

2

e x = x - x o - X ~ . X cosot + Z ~ . Z sin c~ (2d)

e z = z - z o - X s Y sin ot - Z , Z cosc~ (2e)

The objective is therefore to compute the set of M optimization parameters a that minimizes (2),

where a = [b c x0 z0 ot X, Zs ]r or any subset of it. It should be noted though that only one

of the scaling factors, X~ or Z,, can be optimized simultaneously with b and c for the solution to

be unique. Generally, one of the scaling factors is known from the experimental setup and can

be held constant while optimizing the other to correct for the aspect ratio to calibrate the optical

system. The rotational angle et can also be optimized to correct for the rotational misalignment

of the camera for calibration purposes.

x o x

,,

r~ ,zi)

Z L d i

Fig. 3. Definition of error function parameters for the ADSA-P optimization problem.

The optimization problem can be written as

N

min E(a) - ~7~ w~e (a) a i=1

71

which is a multi-dimensional non-linear least-squares problem that requires an iterative

optimization procedure. When the minimum has been found, the optimization parameters

determine the Laplacian profile that best fit the given profile, from which y and other properties

can be readily computed.

Evaluating E for a trial set of a, i.e. for each optimization iteration, involves determining the

minimum (normal) distance from the Laplacian curve to each experimental point. This is done

using a one-dimensional Newton-Raphson iteration to solve, for each i-th point,

de f ( s) - d s - e cosO+e sinO-O

There exist several numerical methods to solve optimization problems. Among them, Newton's

method is well known for its second-order convergence if the initial values are very close to the

solution, but it is unpredictable otherwise, particularly for multi-dimensional problems. To

overcome this problem, several Newton-like algorithms have been developed with more

advanced convergence strategies. The original ADSA-P used Newton's method with

incremental loading to approach the solution, but this approach is computationally expensive

and its convergence is not guaranteed. A more efficient and globally convergent method for

non-linear least squares optimization is the Levenberg-Marquardt method, as implemented in the

MINPACK library by Mor6 and Wright [19]. The current version of ADSA-P employs a

combination of Newton's and Levenberg-Marquardt methods. Very often, as in the case of time-

dependent studies, the results from a previous run can be used as initial values and Newton's

method can be used to take advantage of its fast convergence, but it is aborted as soon as

divergence is detected. If good initial values are not available or if Newton's method fails, the

Levenberg-Marquardt method is then used.

As with any non-linear numerical method, the optimization parameters must be initialized with

approximate values of the solution. Good initial values for the curvature at the apex b, and the

origin of the system of coordinates x0 and z0 can be found by a least-squares elliptical fit of

several points near the drop apex, and the rotational angle et and the scaling factors X~ and Zs

are generally known from the experimental setup. The capillary parameter c is initialized using

an estimated surface tension value, but the method will converge even with a bad initial guess.

72

In practice, the drop profile coordinates (X~, Zi ) are extracted from digital images of pendant or

sessile drops using edge detection techniques as implemented by Cheng et al. [ 10]: By applying

the well-known Sobel operator on the digital image, the pixel coordinates of the drop edge can

be obtained as those pixels with a maximum Sobel value, following the contour of the drop from

one end of the drop to the other. This procedure yields profile coordinates with pixel resolution,

which are limited by the resolution of the digital image (usually 640 by 480 pixels). A more

accurate subpixel resolution can be obtained by means of a cubic-spline fit to the pixel values

across the interface to find the position of the interface as implemented by Cheng, or by a

quadratic polynomial fit of the Sobel values across the interface to find the position with

maximum Sobel value that represents the drop edge.

2.2. AXISYMMETRIC DROP SHAPE ANALYSIS- DIAMETER ( A D S A - D )

The ADSA-D methodology to compute contact angles 0 from the contact or maximum

diameter D (usually measured from a picture of the drop looking from above) and volume V of

sessile drops with known surface tension y, was originally developed by Skinner et al. [ 13] and

Moy et al. [ 14]. The current implementation by del Rio [ 11] uses the numerical solution of the

Laplace equation as a boundary value problem (BVP), as described below.

There are two cases to consider, depending on the contact angle: 1) contact angles greater

than or equal to 90 ~ and 2) contact angles less than 90 ~ which represent two separate BVPs. In

the first case the maximum diameter corresponds to the equatorial diameter of the drop (at 0 =

90 ~ and in the second case the maximum diameter corresponds to the three-phase contact line

(see Fig. 4).

2.2.1 Contact angle greater than or equal to 90 ~

Rewriting Eqs. (la)-(1 c) as functions of x, considering the curvature b as a new variable and

with the BCs as seen in Fig. 4a, the Laplace equation can be written as the following BVP for

contact angles greater than or equal to 90~

sin 0 dO _ 1 2b + cz - (3a) dx cos0 x

dz - t a n 0

dr

db - 0

dx

z(O)-- o(o)- o;

where R=D/2 is the maximum (equatorial) radius.

7C o(R) -

(a)

0 X

x = R

. . 0 = x / 2

V=Vc

0 = 0 c

73

(3b)

(3c)

(3d)

(b)

0

R i

X

x=R

V = V c

0 = 0 c

Fig. 4. Boundary conditions for ADSA-D boundary value problems. (a) Contact angle greater than or equal to

90~ (b) contact angle less than 90 ~

74

BVP (3) completely defines the Laplacian shape; its solution gives directly the profile shape for

0 < x < R, and the constant value of the apex curvature b. The contact angle can then be

computed by integrating the IVP (1), for the known values of b and 7, past the maximum

diameter, stopping when the computed volume reaches the drop volume, V~.

2.2.2 Contact angle less than 90 ~ .

Similarly, as seen in Fig. 4b, the Laplace equation for contact angles less than 90 ~ can be written

as the following BVP:

sin t3 d e _ 1 2 b + c z - ~ (4a) dx cost3 x

dz - tan 0 (4b)

dr

d V - 7rx 2 tan 0 (4c)

dr

db - 0 ( 4 d )

dr

z ( 0 ) = 0 ( 0 ) = v ( 0 ) = 0; v (R) = (4e)

where V~ is the total volume of the drop. BVP (4) completely defines the Laplacian shape. There

is no need for an additional numerical integration since the contact angle can be obtained simply

from the value of 0 at x = R.

To initialize ADSA-D it is necessary to determine, for given values of Vc and R, whether the

contact angle is greater than or equal to 90 ~ (BVP 3) or less than 90 ~ (BVP 4). On occasions,

the user can give this information as input, but in many cases, especially for contact angles near

90 ~ it is not known. The approach implemented in the program is: 1) If the user knows whether

the drop is wetting or non-wetting, solve the respective problem and exit; otherwise 2) assume

that the contact angle is greater than or equal to 90 ~ and solve BVP (3) for the given R,

compute volume V90 at 0 = 90 ~ by integrating IVP (1). If V90 <- Vc the initial assumption was

correct, compute the contact angle and exit; otherwise, 4) solve BVP (4) for contact angle less

than 90 ~ .

75

The non-linear BVPs (3) and (4) must be solved numerically. The current version of ADSA-D

uses a finite difference method with collocation formulas, as implemented in the COLSYS

library by Ascher et al. [ 18]. The program requires to be initialized with approximate values of

the solution. For the case of contact angles greater than or equal to 90 ~ , the profile and the

curvature are initialized with an elliptical approximation, and for contact angles less than 90 ~ the

solution is initialized with zero values and the curvature with 1/R. It was found that these rough

approximations are sufficient for the method to converge in most cases. However, a

continuation algorithm [ 18] was implemented to guarantee convergence to a solution in case of

an initial failure of COLSYS, using the capillary constant c as the continuation parameter. Care

is taken in the numerical implementation to avoid the discontinuity of the BVP (3) at 0 = 90 ~

and the algorithm succeeds for any contact angle, including 0 = 90 ~

In practice, the drop diameter can be obtained from digital images of the drop, acquired with the

camera positioned vertically, looking at the sessile drop from above. The drop volume can be

measured with a micrometer syringe.

3. SURFACE AND INTERFACIAL TENSION MEASUREMENTS

As mentioned above, ADSA is a powerful experimental tool to measure surface properties.

Surface or interfacial tension is a necessary parameter in surface physical chemistry and

biomedical engineering. In this part of the chapter, the various applications of ADSA-P to

surface tension measurements are illustrated. These applications are divided into two

categories: 1. experiments on static drops, where the pendant or sessile drops maintain a

constant volume and the surface or interfacial tensions are measured; 2. experiments on dynamic

drops, where the volume of the pendant or sessile drops is varied and the interfacial tension

response to the surface area change is measured. To some extent, the main objective of the

static drop experiments is the determination of the equilibrium surface tension; however, from

the perspective of this chapter, this is a dynamic (time-dependent) measurement, and it includes

monitoring the surface tension variation over the equilibration of the surface. In the dynamic

drop experiments, the surface area of the drop is varied by changing the drop volume using a

motorized syringe; the pattern of the interfacial tension response is then analyzed, which reflects

the dynamics of molecular movements and interactions at the interface.

76

In static drop measurements, we will illustrate: (1) the high accuracy of ADSA surface tension

measurements; (2) interfacial tensions of two mutually saturated liquids; (3) ultralow interfacial

tensions between oil-rich phases and aqueous solutions; (4) the pressure dependence of

interfacial tensions; (5) the temperature dependence of interfacial tensions; and (6) the surface

tensions of polymer melts. In the dynamic drop measurements, we will present: (1) ADSA as a

film balance; (2) dynamic surface tensions of surfactant solutions; (3) surface tension responses

to area changes of mixed solutions; and (4) ADSA-CB, where a captive bubble method is

employed to study lung surfactant systems.

3.1. MEASUREMENTS ON STATIC DROPS

3.1.1. Liquid-Vapor Surface Tension

In this section, the high accuracy of ADSA-P surface tension measurements is demonstrated

through the surface tension determination for several liquid-vapor interfaces. The experimental

set-up for ADSA-P is shown schematically in Fig. 5 [9-11,20]: The sample is a pendant drop

formed at the tip of a capillary or a sessile drop generated from the bottom on a flat solid

surface. Light from a Newport light source passes through a frosted glass diffuser and onto the

sample. A CCD video camera (Cohu 4800 CCD monochrome) attached to a microscope (Leitz

Apozoom) is connected to a digitizer. The digitizer (Parallax Graphics XVideo) grabs the

frames and digitizes the acquired image into 640 x 480 pixels with 256 gray levels. The 0 gray

level represents black, while 255 represents white. A Sun SPARCstation 10 computer receives

the digitized image and performs the necessary calculations.

The surface tension of 16 liquids measured by the pendant-drop method are summarized in

Table 1 [21 ]. All experiments were performed at room temperature, 21 ~ It should be noted

that the temperature coefficients for alkanes is about 0.1 mJ/m2/~ the errors may well be due

to temperature fluctuation of the sample cell. Hence, the ultimate accuracy of ADSA-P should

be better than that listed in Table 1.

77

Fig. 5. Schematic of an experimental set-up for pendant drop and sessile drop experiments.

Table 1. Surface tension 7 of 16 liquids (measured by the pendant drop method).

Liquid

Decane

Dodecane

Tetradecane

Hexadecane

trans-Decalin

cis-Decalin

Ethyl cinnamate

Dibenzylamine

Dimethylsulfoxide

1-Bromonaphthalane

Diethylene glycol

Ethylene glycol

Thiodiglycol

Formamide

Glycecol

Water

(mJ/m 2)

23.43

25.44

26.55

27.76

29.50

31.65

38.37

40.63

43 58

44.01

45.04

47.99

54 13

57 49

63 11

72.75

+ 95% confidence limits (mJ/m 2)

0.02

0.02

0.05

0.04

0.06

0.05

0.03

0.09

0.08

0.06

0.07

0.02

0.11

0.08

0.06

0.06

78

3.1.2. Liquid-Liquid Interfacial Tension

ADSA's ability to measure surface tension is not restricted to liquid-vapor systems; it can also

be used to measure liquid-liquid interfacial tension. The interfacial tension between two liquids

is important in many industrial and biomedical systems. Examples can be seen in emulsions or

dispersions of a large number of chemical engineering products such as paints, detergents and

lubricants since these dispersions have specific requirements as to solubility, rheology and

colloid stability [22-24]. In biological systems and emulsions, the relevant interfacial tension is

that of the hydrocarbon/water interface [3,25,26].

When two liquids come in contact, mutual dissolution starts. Equilibrium interfacial tension

measurements need to be done while the two liquids are mutually saturated. To guarantee this

mutual saturation, a special set-up [27], schematically shown in Fig. 6, is used in the pendant

drop method. To measure the interfacial tension between the two liquids, a pendant drop of

liquid 1 is formed in liquid 2, where liquid 1 is assumed to be heavier than liquid 2.

Fig. 6. Experimental set-up for (a) interfacial tension measurements of liquids 1 and 2, (b) surface tension

measurements of liquid 2, and (c) surface tension measurements of liquid 1, where liquid 1 is heavier

than liquid 2 [27]. a - quartz cuvette, b - metal tubing, and c - teflon capillary.

79

To measure their respective surface tensions, air bubbles are formed in the two liquid phases.

Nineteen liquids saturated with doubly distilled water were used in the ADSA-P experiment

[27]. Before the measurement, each liquid was allowed to saturate with water inside a test tube,

at an approximate volume ratio 1:1, for at least 12 hours. Since the density difference across

the liquid-fluid interface is required as input information for ADSA and since the density will be

affected by the mutual solubility, an accurate measurement of the density of both the liquid and

the water saturated with each other is required. The density measurement was performed with a

digital density meter (Anton Paar DMA 45) with temperature control at 25.0 + 0.1 ~

The results for the liquid-water interfacial tension, 7l~ , of all liquids saturated with doubly

distilled water are shown in Table 2. Also included in Table 2 are the liquid surface tension, 7l,

(saturated with water) and the water surface tension, 7~, (saturated with the respective liquid).

It is seen from this table that the 95% confidence limits are of the order of 0.01 mJ/m 2, typically

an order of magnitude better than many other surface tension measuring techniques [3 ].

Table 2. Surface and interfacial tensions of liquids saturated with water, respectively.

Liquid 7w (mJ/m 2)

7l 95% (mJ/m 2) confidence

limits Diethyl ether 16.98 0.12 Isopropyl ether 17.28 0.03 Heptane 19.78 0.07 Octane 21.07 0.02 Decane 22.95 0.01 Ethyl formate 23.33 0.19 1-pentanol 24.80 0.05 1-hexanol 26.05 0.12 Hexadecane 26.23 0.13 1-octanol 26.88 0.06 1-decanol 27.50 0.12 trans-decalin 29.59 0.04 Ethyl acetoacetate 3 0.45 0.12 cis-decalin 31.28 0.09 Diethyl oxalate 31.66 0.14 Ethyl cinnamate 37.96 0.13 Methyl salicylate 38.91 0.16 Dibenzylamine 40.36 0.11

30 71 36 97 70 49 71 00 71 46 27 96 24 33 28 59 72 11 28 44 28 18 66.74 35.57 62.92 35 57 56.59 65.05 54.45

95% 7lw 95% confidence (mJ/m 2) confidence limits limits 0.09 7.36 0.02 0.20 17.88 0.02 0.06 50.66 0.04 0.11 50.83 0.04 0.17 51.07 0.11 0.20 3.15 0.13 0.11 4.45 0.01 0.24 7.13 0.002 0.14 52.24 0.11 0.04 8.44 0.01 0.09 8.61 0.05 0.20 37.62 0.04 0.08 2.72 0.01 0.06 31.48 0.08 0.08 9,96 0.12 0.16 19.75 0.04 0.26 23.84 0.24 0.25 16.08 0.23

....... 4 3 . . 6 7 ................ 0 . . .1_2 ........................ 12_....1_2 ............. 0:_.1___1. ...................... 2 3 . 2 . 5 . . ........... ..0.-.!..6. ..............................

80

3.1.3. Ultralow Interfacial Tension

In many emulsion or microemulsion systems, the interfacial tension between the oil-rich phase

and the aqueous solution is very low (or ultralow), which presents considerable difficulties for

many experimental methodologies. The most commonly employed approach for measuring

ultralow interfacial tension is the spinning drop technique [28]. However, ADSA has also been

used to study these systems and possesses a number of advantages over the spinning drop

technique: higher accuracy, more versatile environmental control (high pressure and

temperature) and ability to study time-dependent effects [29].

The use of a drop shape technique such as ADSA to measure ultra low interfacial tension is

complicated by a number of factors: When the interfacial tension is low, a pendant drop may fall

off the supporting capillary since gravity overpowers the adhering force due to interfacial

tension; or, film leakage may start if a monolayer forms at the interface. Film leakage occurs

when the surface active molecules spread from the liquid-fluid interface onto the surrounding

solid. To overcome these difficulties, an inverted sessile drop has to be used. The experimental

set-up is shown in Fig. 7 schematically.

Fig. 7. Schematic of an apparatus to form inverted sessile drops for ultralow interfacial tension measurements [29]. 1. Inverted pendant drop, 2. Inverted sessile drop, 3. Steel capillary, 4. Glass surface, 5. Teflon screw, 6. Teflon support, and 7. Quartz cuvette. System 1: Liquid 1 - AOT in solution of NaCl/water; Liquid 2 - n-Heptane. System 2: Liquid 1 - Oleic acid in olive oil; Liquid 2 - Solution of NaC1 and NaOH.

81

In the figure, an inverted sessile drop of liquid 1 (with lower mass density) is formed from a U-

shaped steel capillary onto a glass surface, while the second liquid (with higher mass density)

surrounds the sessile drop. The entire system is enclosed in a quartz cuvette [29].

Two ultralow interfacial tension systems are illustrated: (1) oleic acid in olive oil and aqueous

solution of NaC1 and NaOH; (2) Dioctyl Sulfosuccinate (AOT) in aqueous solution of NaC1 and

n-heptane at three different concentrations of AOT [29].

3.1.3.10leic Acid in Olive Oil and Aqueous Solution of NaC1 and NaOH

The interfacial tension results for lmM oleic acid in olive oil with 0.15 M of NaC1 and lmM of

NaOH aqueous solution are shown in Fig. 8.

1.0

0.8

r

0.6 O ~

cD [..,

"U 0.4

0.2

0.0

Y

I ~ I ~ I

200 400 600 Time [Sec]

800

Fig. 8. Interfacial tension vs. time for 1 mM of oleic acid in olive oil in the aqueous solution of 0.15 M of NaC1

and 1 mM of NaOH [29].

82

The interfacial tension decreases approximately from 0.8 to 0.2 mJ/m2 in 10 minutes. This

decrease in the interfacial tension reflects the surface equilibration [29].

3.1.3.2 Dioctyl Sulfosuccinate (AOT) in Aqueous Solution of NaC1 and n-Heptane

Figure 9 shows the interfacial tension results for three different concentrations of AOT in

aqueous solution of 0.0513 M NaC1 and n-heptane. The interfacial tension decreases from

about 0.25 to 0.06 mJ/m 2 in three minutes at the concentration of 0.410 mM for AOT in

aqueous solution of 0.0513 M NaCI and n-heptane (Fig. 9a). At this concentration, the

equilibrium of interfacial tension is not reached within the experimental period. For the 0.415

mM AOT in aqueous solution of NaC1 and n-heptane, the interfacial tension declines from about

0.05 mJ/m 2 to an equilibrium value of 0.01 mJ/m 2 in 12 minutes (Fig. 9b). When increasing the

AOT concentration to 0.420 mM in aqueous solution of NaC1, the interfacial tension decreases

from about 0.026 mJ/m 2 to an equilibrium value of 0.006 mJ/m 2 within two minutes (Fig. 9c).

0 .25 ( ~ . , . , . , , ,

0 .20

r

"~ 0 .15 O

"~ 0 .10

0 .05

0 .00 ' ~ ' ~ ' ' ' ' '

50 100 150 2 0 0 21

Time [Sec]

Fig. 9a. Interfacial tension vs. time for 0.410 mM of AOT in aqueous solution of 0.0513 M of NaC1/water and n-

heptane [29].

0.05

0.04

0.03 . ~

't~ 0.02 o

0.01

0

0 0

(s

".',J"-" "4_~.~,E,_5'~ " " - ' ~ ' ~

0 . 0 0 . . . . . ' . . . . . . . . J ' l

200 400 600 Time [See]

800

83

Fig. 9b. Interfacial tension vs. time for 0.415 mM of AOT in aqueous solution of 0.0513 M of NaC1/water and n-

heptane [29].

0.030

0.025

0.020

.o = 0.015 o

0.010

0.005

0.000

0 0

0 0

�9 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

9~n .... a~n , . ~ , J . . . . I . . . . . . . . I 60 80 1 O0 i 20 140 �9 - , i r %#

Time [See]

Fig. 9c. Interfacial tension vs. time for 0.420 mM of AOT in aqueous solution of 0.0513 M of NaCl/water and n-

heptane [29].

84

From Fig. 9, we can see a concentration dependence of the interfacial tension: increasing the

AOT concentration decreases both the final interfacial tension value and the time required to

reach equilibrium. Since the system at the AOT concentration of 0.410 mM (Fig. 9a) did not

reach equilibrium, only the results at the other concentrations of AOT (Figs. 9b and 9c) could be

compared to those published by Aveyard et aL [28] who used the spinning drop technique: The

interfacial tension values reported by Aveyard et al. were estimated from their graph; this

interpolation produced: 0.01 and 0.003 mJ/m 2 for 0.415 and 0.420 mM of AOT in the aqueous

solution of 0.0513 M NaC1 and n-heptane, respectively. This is in good agreement with the

results shown in Figs. 9b and 9c.

3.1.4. Pressure Dependence of Interfacial Tension

Due to the fact that the pressure dependence of interfacial tension, Oy/OP, is quite small

(typically of the order of 10 l l m), a very accurate method of measurement is necessary. There

are a number of techniques which can be used to measure interfacial tension, but drop shape

methods are best suited [30,31]. For pressure dependent measurements with ADSA-P, a

specially designed pressure/temperature (P/T) cell was used (Fig. 10) [20]. The cell consisted of

a 316 stainless steel cylinder with 25 mm thick optical glass windows fitted at each end. All

lines and connections were 316 stainless steel. The cell was rated at 350 bars (5000 psi) and

200~ An Eldex HPLC pump was used to pressurize the system by pumping water into the

P/T cell and to form a pendant drop of n-decane at the tip of a stainless steel needle immersed in

water.

The pressure and time dependence of the interfacial tension measured by ADSA-P at 21.5~

are presented in Fig. 11 [31]. Initially, the system was pressurized, then a new drop was

formed, and time-dependent readings were made using ADSA to acquire and analyze the drop

images. The procedure was repeated at a higher pressure. A time dependence for the interfacial

tension 7 was observed at each pressure P. This is probably due to the presence of surface

active impurities which migrate to the liquid-liquid interface causing a decrease in the interfacial

tension. The isochronic 7-P plot revealed a linear relationship between 7 and P in the pressure

range studied. The slope of 7 against P was found to be 23.29 + 0.48 • 10 12 m at the 95%

confidence level [31]. The observed pressure dependence was found to be significant at the

85

99% confidence level. This attests to the suitability of the apparatus and the ADSA

methodology for the measurement of interfacial tension as a function of pressure. Moreover,

the P/T cell readily facilitates the study of the temperature dependence of interfacial tension (see

below).

1 7 fl

9

1

I

, ' a 16

2

16

Fig. 10. Schematic of the pressure/temperature cell. 1. Light Source, 2. Diffuser, 3. Pressure/Temperature Cell, 4. Optically Flat Glass Windows, 5. Pendant Drop, 6. Microscope and CCD Camera, 7. To Digitizer, 8. Discharge and Relief Valves, 9. Pressure Transducer, 10. Pressure Transducer Indicator, 11. Isolation Valves, 12. Tee, 13. Bulk Fluid Line, 14. Drop Fluid Line, 15. Intermediate Cell Filled with Alkane, 16. From the HPLC Pump, 17. To the Discharge Container [31].

52.4 !

52.2 -

E

4" 52.0 -

P = 3.8 bara * ~t~ a P = 132 bara o

P = 241 baxa [] ~ . a P = 336 bara �9

a P = 359 bara A �9 tx

~I n �9 A

% o o

51 .8 - o

o o

o

51"6 i ~ ~** �9 ,

51.4 i 0 2~o

[] O [] []

[]

�9 �9 �9 �9 ,

~ 86o l ~ ~ioo t (s)

Fig. 11. Measurement of the pressure dependence of the interracial tension, ?, of a drop of n-decane immersed in

water at constant temperature (21.5~ The time dependence of 3' may be attributed to system

impuri t i es [31].

86

3.1.5. Temperature Dependence of Interfacia,1 Tension

The temperature dependence of interfacial tension at ambient pressure is, comparatively,

straightforward experimentally. As a demonstration, the study on the interfacial system of an

aqueous protein solution and an alkane is presented.

In biological and biotechnological research, the surface activity of protein is often measured by

its interfacial tension [32-35]. Temperature dependent studies allow the detection of

conformational changes of protein adsorbed. Since the temperature coefficient of interfacial

tension represents surface entropy, such measurements contain information about protein

surface structure [32]. Moreover, knowledge of the interfacial tension of the aqueous protein

solution is relevant to biological systems at temperatures near body temperature. Compared to

many techniques used for measuring interfacial tensions [35-37], ADSA requires a much smaller

amount of the sample liquid, which is clearly advantageous for biological systems in which the

materials may be in limited supply. Also, since the experimental system is smaller, it is easier to

control environmental conditions such as impurities, pressure and temperature [20].

55

50

45

40

"~ 35

._~ o 30

1::

.c_ 25 @ ~ ~

20

i

0

0 30 ~ run a �9 30 ~ run b 0 37 ~ run a �9 37 ~ run b <3 50 ~ run a �9 50 ~ run b

@ @ @ | @ | | | | | | | | 0 r r r �9 ~ r r r r r r r e r <<<<<<<<<<<<~e,~<<<<<<<<

I i I i

2000 4000 time t (s)

6000

Fig. 12. Interfacial tension of 0.02 mg/ml aqueous solution of HA and decane at 30, 37 and 50~ [32].

87

The protein used in this illustration is human albumin (Sigma, USA) and the oil phase is decane

(Caledon Laboratories LTD. Canada). The experimental set-up for the temperature cell is

shown in Fig. 10. The interfacial tension of a 0.02 mg/ml aqueous human albumin solution and

decane was measured in a temperature range from 20 to 60~ The error limits for the

interfacial tension were + 0.02 mJ/m 2 at the 95% confidence level. Figure 12 shows the typical

interfacial tension measurements at three temperatures 30, 37 and 50~ as a function of time.

Two runs were plotted for each temperature to illustrate the reproducibility of the measurement.

2 5 , , , , , , , . , . . . . .

23

E ~- 21 0 . _ c ....,

. _ 0

1 9

17

0",,.

" ' " '0

",0 "',,.

Q ""-. ,

"C) , ,

"C)... " - - .

""G,., " ' " ' -Q ....

.......... 0

�9

I i 1 i I i 1 i 1

20 30 40 50 60 temperature T (~

Fig. 13. "Equilibrium" interfacial tension vs. temperature [32].

Two domains are clearly observed in these isotherms: (1) at the beginning (first few minutes),

the interfacial tension declines sharply; (2) after a few minutes, the interfacial tension changes

rather slowly and approaches a steady value. In order to compare the data at different

temperatures, we need to establish a criterion for equilibrium interfacial tension; hence, the

emphasis of the data analysis is on the second domain of the isotherm where the slope of the

interfacial tension 3' against time t is declining constantly. Based on the fact that when dT/dt = O,

the system has reached equilibrium, we identified the smallest value of dT/dt reached at all

88

temperatures and considered the y corresponding to this smallest value to be an approximation

of the equilibrium value [32].

With dy/dt set at 0.0001 mJ/m2/s for all temperatures, the corresponding 7 values are plotted

against temperature (Fig. 13). The interfacial tension decreases from about 24 to 17 mJ/m 2 as

the temperature increases from 20 to 60~ [32].

3.1.6. Surface Tension of Polymer Melts

The surface tension of polymer melts is important in many technological processes, such as

wetting, adhesion, polymer blending and the reinforcement of polymers with fibers [38-44].

ADSA-P can be used for measuring the surface tensions of polymer melts at elevated

temperatures, e.g., above 170~ [45].

The polymer used in the demonstration is polypropylene (Phillips: HNZ-020, Mw: 318,000).

Before the experiment, a filament of a given polymer was extruded to be 1 mm in diameter and

cooled at room temperature. To avoid contamination, the extruded polymer and the glass

capillary, which was used to hold the pendant drop of the polymer melt, were cleaned with

ethanol. Then, approximately five mm of the polymer filament was inserted into the end of the

glass tube. To prevent degradation of the polymer drop, argon gas was introduced into the

temperature (T) cell during the entire experiment. After placing the polymer in the T cell, the

exposed portion of the polymer filament started to melt, forming a pendant drop at the tip of the

glass capillary. To ensure the complete melting of the polymer, image acquisition of the drop

profile was started approximately 30 min after the sample insertion. A sequence of pictures

were then acquired at about one picture every two min. When the surface tension obtained

from ADSA-P appeared to be constant, the temperature was changed and the surface tension as

a function of time was again studied [45].

Figure 14 shows the surface tension results of polypropylene at four different temperatures:

210, 200, 190, and 180~ It is apparent that the surface tension of polypropylene melt drifted

up from about 10 to 19 mJ/m 2 in about an hour and then remained essentially constant. The

constancy in the surface tension at each temperature indicates that no thermal degradation took

place within that time interval. The error bars shown in Fig. 14 are the 95% confidence limits

calculated from ADSA-P [45].

89

It is surprising that the surface tension increased initially toward equilibrium, since equilibrium

of most liquid-fluid systems is reached by a decrease of the surface tension. Such an initial

increase in surface tension suggests a mechanism different from surface relaxation. From a

thermodynamic perspective, this indicates that the polymer surface does not act as an

independent phase: Presumably, bulk relaxation occurs, which decreases the free energy of the

drop so far that the system can afford the concomitant increase in surface tension [45].

i , i i | i I '

22.0 [

180~ _ )'= 21.13 + 0.02

/ ~ 270~ . . . . 7= 20.63 + 0.02 ] g 210~ ~'= 20"15 + 0"03

18.0 . . . .

14.0

I I I 1 i I J

10.0 1 2 3 4 5

Time (hr.)

Fig. 14. Surface tension vs. time for polypropylene melts at different temperatures. Time t = 0 in the x-axis

represents t = 1/2 hr. after the experiment was started. The initial increase in the surface tension

suggests a different mechanism from surface relaxation (see text). The error bars are the 95%

confidence limits calculated from ADSA-P [45].

3.2. MEASUREMENTS ON DYNAMIC DROPS

The ADSA applications illustrated above are based on static drop experiments, where the drop

volume is fixed. The following examples describe surface tension measurements on dynamic

drops where the drop surface area is varied by adjustment of the drop volume using a motorized

syringe. The fact that ADSA computes surface tension, drop volume and surface area

simultaneously makes the technique suitable for dynamic studies. In the dynamic drop

90

experiment, the surface tension response can be used to study the surface adsorption and

desorption, and molecular interactions during surface equilibration.

3.2.1. ADSA as a Film Balance

The change of the drop volume in a controlled manner combined with the monitoring of the

interfacial tension and surface area changes can be utilized to make evaluations corresponding to

film-balance measurements [46-54]. The possibility of using surface tension measurements to

obtain the surface pressure depends on the well-known relation 7t = y0 - y, where rt is the surface

pressure, 70 is the surface tension of the pure liquid, and y is the surface tension of the liquid

covered with the monolayer. In a film balance, the monolayer film is expanded and compressed

by a floating barrier separating the pure liquid from the liquid covered with the monolayer. The

corresponding compression and expansion of the film can also be performed similarly by

decreasing and increasing the volume of a pendant drop. The experimental scheme is as

follows: Initially, a few pictures of a pendant drop of the pure liquid are taken to determine 70.

The desired amount of the insoluble surfactant is weighed and dissolved in a solvent (such as

heptane), and a known amount of the surfactant solution is deposited onto the surface of the

drop. Upon evaporation of the solvent, the drop carries an insoluble monolayer. A sequence of

images of the drop profile are acquired while the drop volume is decreased continuously until

the drop becomes quite small. Subsequently, the drop volume is increased to the original value.

To ensure reproducibility of the results, the same cycle of compression and expansion is

repeated. The measured surface tensions and surface areas can be transformed into the

corresponding surface pressure as a function of the area per molecule by using the above

relation and the known amount of insoluble surfactant on the drop surface. A typical result for

a film of purified octadecanol on water with alteration of the surface area at the rate of 7.2

A2/molecule-minute is illustrated in Fig. 15.

It is apparent that the two runs are quite similar, illustrating the reproducibility of the results.

Moreover, these measurements are in close agreement with film-balance results of the same

sample of octadecanol [46]. However, ADSA offers several distinct advantages over the

conventional film-balance methodology for determination of surface pressures. First, only small

quantities of liquid and spreadable material are required. Second, both liquid-vapor and liquid-

91

liquid interracial tensions can be studied [48]. Third, environment control (contamination,

temperature, and pressure) is a relatively straightforward matter. Fourth, a much larger range of

rates of change in surface area can be obtained [46,47].

50.0 -

40.0

30.0

0

,.~ 20.0 r~

10.0

Compression

Orun 1 •run2

Expansion

0.0 22.0 23.0 24.0 25.0 26.0 27.0 28.0 29.0

Area per molecule (Angstrom square per molecule)

Fig. 15. Expansion and compression of a purified monolayer of octadecanol on a pendant drop of water.

Measurement of the liquid surface tension and drop surface area result in surface pressure measurements which closely resemble film balance measurements of the same system [46].

Recently, ADSA-P as a film balance has been applied to study the rate dependence of the

collapse pressure for an octadecanol monolayer on a water surface [47]. With the motorized

syringe to change the drop volume automatically [55], one can easily change the rate of

molecular area compression and expansion:

area 1 a rea ) ( m o l e c u l e ) ( m i n ) = t m o l e c u l e )~ min

92

by either changing the surfactant amount spread on the surface or changing the speed of surface

area variation. Varying the motor speed setting while keeping fixed the amount of surfactant

spreading on the water surface, three pressure-area (rt-A) isotherms were obtained at three

different compression rates (Fig. 16).

60.0

50.0

40.0 r

E

30.0 t . .)

= 20.0 r.~

10.0

' I I ' i ' I I

o o 5.71 A2/molecule-min. l Ill = = 11.6 A2/molecule-min.

~ A A 22.8 A2/molecule-min.

I

I -% I l l

I l l

I l l l l l l l l

i i I I I I 1 8 ~ 20.0~~,~~,22.0 ' 24.0

\ Limiting Molecular Area

Molecular Area at Collapse [A2/molecule]

[AZ/molecule]

Fig. 16. Surface pressure (mJ/m 2) vs. area per molecule (A2/molecule) of pendant drops with an octadecanol film

compressed at 5.71, 11.6 and 22.8 A2/(molecule min). These isotherms were established with the same

amount of surfactant (1.3 • 1014 octadecanol molecules) but were compressed at different motor speed settings:

4, 5 and 6. Collapse pressures of 53.0 + 0.5 mJ/m 2 were obtained, independent of the motor speed.

Extrapolations for the lilm'ting molecular areas and the molecular area at collapse are shown [47].

93

The actual rate of change in drop surface area for a given motor speed setting was calculated by

using linear regression in the linear region of the drop surface area-time curve, which is one of

the outputs of ADSA [47]. From the above figure, it is seen that no dependence of the collapse

pressure on the compression rate was found; however, the shape of rt-A isotherm varies

significantly with the compression rate: increasing the compression rate shifts the lower part of

the isotherm to the right [47].

3.2.2. Dynamic Surface Tension of Surfactant Solutions

As mentioned before, ADSA can be readily applied to the study of the time evolution of profiles

of pendant drops. With a motorized syringe to change the drop volume automatically, periodic

area changes can be obtained with arbitrary amplitudes. Various dynamic aspects of interfacial

tension phenomena can be investigated. In this section, we demonstrate the dynamic surface

tension response of sodium dodecyl sulfate (SDS) solutions to periodic surface area changes.

Many industrial processes such as mineral floatation, enhanced oil recovery, and the production

of foam or emulsions are examples of non-equilibrium processes that can be greatly affected by

the addition of surfactant [56-59]. Several techniques exist for measuring dynamic surface

tension of surfactant solutions. The conventional film balance [60] can be used only to measure

the film pressure of insoluble surfactants as a function of surface area, but it has several

additional drawbacks: due to its size, this film balance requires large amounts of surfactant, and

good environmental control can be difficult; the rate of area compression and expansion is also

quite limited. Franses and co-workers [61-63] have made use of a pulsating bubble

surfactometer (PBS) [64-67] to measure the dynamic surface tension of a variety of surfactant

systems. However, the pulsating bubble surfactometer is not capable of varying the peak to

peak change in surface area AA. (Although the conventional film balance can vary AA, it cannot

be used to study soluble surfactants such as SDS.) With ADSA-P and a motorized syringe, all

of these difficulties can be overcome.

In Fig. 17, the first 200 s of data of a typical oscillating drop experiment with a 8 mM SDS

solution are shown. It should be noted that both ADSA and the pulsating bubble technique use

an indirect approach to varying the surface area by changing the drop or bubble volume. This is

in contrast to the direct method of area change used in the conventional film balance. From

94

Fig. 17, we can see that there is a certain amount of scatter in the surface tension plot while the

surface area and volume plots exhibit quite regular saw-tooth patterns. The surface tension plot

against time in Fig. 17 appears somewhat less symmetric, having broad peaks and narrow

troughs. This was also observed by Chang and Franses [61 ].

37 .0 . . . . . . . . . , - - ~ . . , . . .

"4 35.0 te

A A A A

A 33.0 . . . . ' . . . . I . . . . I . . . .

~o 0.20

0.10 0.020

i . . . . I . . . . ! . . . . I . . . .

on

v

(D

�9 ~ 0.010 >

____o.oo'i . . . . i . . . . i . . . . i . . . . 0.0 50.0 100.0 150.0 200.0

Time (s)

Fig. 17. Surface tension 7 (mJ/m2), surface area A (cm 2) and drop volume V (cm 3) versus time t (s) for All =

50%, 8 mM SDS solution [68].

Upon a change AA of surface area, the peak-to peak value (Tm~ - 7m,, = AT) of the surface

tension response can be determined as a function of AA. It has been predicted [62] that A7

should increase as AA increases, but no prediction was made about the functional relationship

between A7 and AA. The surface tension response to surface area change is due to the

compression or expansion of the interface, which alters the surface concentration of surfactant

95

to values above or below equilibrium. It is reasonable to anticipate that there should be a larger

tension response to a larger area excitation. As a first observation of this type of relationship,

Fig. 18 shows A7 versus A,4 for three different concentrations of SDS. A linear relationship

between A7 and d,4 is observed. Also from Fig. 18, both d~, and the slope of the AT-AA curve

decrease as the concentration increases.

2.00

o 1.50

O 0,-i

1.00

0P,,t

ell

az 0.50 r,.)

i ' ' ' i ' ; a ' i ' ' ' i '

a.~mM i

i .,/./'' / . . A 16 mM

/ ~

i ~//71 I "

0 12mM

0 . 0 0 : , , I , , , I : , , I , i : I , , ,

0.0 20.0 40.0 60.0 80.0 100.0 Percent Change in Area AA (%)

Fig. 18. Surface tension response A3, (mJ/m 2) versus percent change in surface area AA (%) for 8, 12 and 16 mM

SDS solutions [68].

It is reasonable to assume that there are two competing effects occurring as the interface is

compressed or expanded. One effect is surface dilation as the surface area increases, and

similarly surface concentration increases as the surface contracts. The other is adsorption and

desorption of the surfactant at the surface, i.e., processes which will oppose the surface tension

96

change induced by area changes. The linearity of the plots in Fig. 18 suggests that under the

present experimental conditions, adsorption/desorption processes are overpowered by the

dilation type of effects due to surface area change.

3.2.3. Dynamic Surface Tension of a Mixed Solution of Protein and Small Molecules

As demonstrated above, we can study surface tension response to a saw-tooth type of area

change by periodically changing the drop volume with a motorized syringe. The pattern of the

surface tension response to the same saw-tooth area variation depends on the substance(s)

forming the interface; hence a mixed solution will behave differently from either of the pure

liquids; we shall show, below, that this idea can be used to investigate the surface interaction

between macromolecules and small molecules, such as protein and a smaller organic molecule.

The example given here is a study of a bovine serum albumin (BSA) (Sigma Chemical Co., St.

Louis, MO, USA) aqueous solution mixed with dimethyl sulfoxide (DMSO) [69]. Two types of

samples were prepared: (a) BSA aqueous solution at a concentration of 0.02 mg/ml; (b) 1.0 ~tl

DMSO added to 1.0 ml BSA solution.

Figure 19 shows the pattern of the surface tension ~, response to the saw-tooth type of area

change for BSA aqueous solution. We note that there is a transition in the pattern of the ),

response in early stages, from an initial rather symmetric peak shape to a skewed one, in

response to the symmetric saw-tooth pattern in the area variation. The skewed, asymmetric

pattern becomes steady atter 60 s. Figure 20 shows the asymmetric shape more clearly at later

times (note arrows). In general, within each cycle, the dynamic surface tension 7 increases as

the surface area A expands (due to the reduction in the surface concentration), and 7 decreases

when A shrinks. The ~ response shows two kinks (see arrows), one each in the branches of the

surface expansion and compression.

Figure 21 illustrates the ~/response to the surface area variation of a system in which DMSO

was added to the BSA aqueous solution at a concentration of 1.0 ktl DMSO to 0.02 mg/ml BSA

water solution.

c~

r o

[/3

O

C ~ o o

o 0

~-b

=~

0

r.r

0 0

g ~

g ~

_-!.

~ g

N

r~

0 ~

g

~ g

~1~

~,,.~

,

~ o

' '~

cm2)

Surfa

ce T

ensi

on 7

(mJ/

m2)

~

&re

a A

o o

o .o

~ .~

ol

o~

"1>..,

"t>.

.

JJ

Jz

~~

,l

,,

,.

, ,

I ..

..

a,,,

,d

~,,,

.I .

...

I,,

,I.

~ = o (3

)

-,..,

l:::r'

o o I:="

.-.

"----

o i,- h

r../3

> r.~

r.~

0

Are

a A

(cm

2)

Sur

face

Ten

sion

7 (

m J

/m2)

o

o o

o l ~

ol

o~

...,

I io

~

:~

01o

o o

o ...

. I ..

.. I .

... I

.... I

....

I'

98

75

65 t- .O_

~, 55 I - o 3::: ~t~ ; , I , , I ~ , I , , l , , I , ,

0.4

0.3 <

0.2 <

0.1 0 30 60 90 120 150 ' 180

~ 7 5

v ~" 65

a~ 55 I-- o

�9 t:: 45

.-.. 0.4

~ 0.3

~0 .2

0.1 180 210 2 4 0 2 7 0 300 330 360

T i m e (s)

Fig. 21. Surface tension 7 response to the area change of the DMSO and BSA solution at a concentration of 1.0

~tl DMSO/0.02 mg BSA in 1.0 ml water. The area change is the same as in Fig. 19 (sawtooth shaped),

but for space consideration it is omitted. A transition is shown in the 7 response, from the initial

symmetric pattern to the later asymmetric one [69].

A significant pattern change is observed in the 7 response. The surface tension initially does not

respond at all to the area variation. Then, beginning after 30 or 40 s, the surface tension shows

responses which gradually increase in amplitude and have a rather narrow but symmetric trough.

After approximately 180 s, the peaks start becoming asymmetric, and towards the end of the

experiment, 360 s, the shape of the 7 curve has become very similar if not identical to the one

that was observed in pure BSA aqueous solution (Fig. 20). Fig. 21 provides a detailed picture

of the asymmetry in the 7 pattern at late stages, comparing well to Fig. 20.

99

' I ' I ' I ' I ' I ' I ' : !

7o J -2

6o

5o o~

= 40 co

0.4

E o 0.3 <

.~ 0.2

0.1 290 300 310 320 330 340 350 360

Time (s)

Fig. 22. Asymmetric Y oscillation of Fig. 21 on an expanded time scale in late stages of the experiment for the

DMSO/BSA solution. The pattern of the 7 cycling is similar to that of the pure BSA solution (Fig. 20).

The arrows point to kinks [69].

The curves in Figs. 18-22 can be explained as follows. At early stages, both protein and the

small molecules diffuse to the water-air interface. Because of their small size, the DMSO

molecules have a much higher diffusivity than the large protein molecules. Consequently, they

adsorb at the surface at a faster rate in this competitive adsorption. Therefore, in the early

stages of the experiment, the surface properties are mainly governed by the small organic

molecules. In general, a change in surface tension is associated with the change in surface

concentration. An increase in the surface area will likely induce a decrease in the surface

molecular concentration, and hence an increase in the surface tension ,{. Conversely, a decrease

in the surface area will be followed by a decrease in Y. However, the amplitude of the Y

oscillation is also related to the desorption of the adsorbed molecules, in addition to the area

variation. If the surface molecules can desorb from the surface sufficiently fast so that the

molecules can quickly adjust their surface concentration to maintain a constant value while the

surface is compressed, then, the surface tension, which is determined mainly by the molecules

100

left at the surface, will show little variation in response to the area change. In Fig. 21, the small

DMSO molecules dominate the molecular population at the surface in early stages, and

consequently small amplitudes in the 3' oscillations are observed.

With the passage of time, more and more protein molecules diffuse to and adsorb at the surface.

From Figs. 21 and 22, the pattern of the 3' response to the area change becomes similar to that

of the pure BSA (Figs. 19 and 20). This indicates that the 3' behavior is governed mainly by the

protein molecules at the surface at late stages. It stands to reason that the small molecules,

during surface compression, will continue to be desorbed; but during surface expansion, they

will now be replaced by protein molecules, which are not as readily desorbed during

compression. Hence, we may infer that the small molecules are squeezed out of the surface,

with mainly protein molecules occupying the surface in late stages.

3.2.4. ADSA-CB: Captive Bubble Method in Lung Surfactant Studies

As illustrated in the previous section, in the presence of a surface film, surface tension will

change as the interracial area is decreased and increased. Without proper design, at low surface

tension, this type of system could suffer from film leakage [70-74]. Film leakage is due to a

fundamental surface thermodynamic principle: at low surface tension the surface active

molecules can spread from the liquid-air interface onto the surrounding solid, thereby decreasing

the free energy of the system. Film leakage has been demonstrated to occur in the Langmuir-

Wilhelmy film balance [75], and has been observed in the Pulsating Bubble Surfactometer [76].

Film leakage can lead to surface behavior which has been erroneously thought to be intrinsic to

the system under study.

The only way to eliminate film leakage is by removing the potential pathway through which the

surface active molecules can leave the air-liquid interface. The captive bubble geometry

accomplishes this by holding a bubble of air captive at the top of a chamber filled with sample

liquid. In this system, there is no need to pierce the bubble with any capillaries. A hydrophilic

ceiling ensures an aqueous layer between the solid ceiling and the bubble, leaving the air-liquid

interface completely intact.

When measuring the surface tension of lung surfactant using the captive bubble method, two

new problems occur: One, the surface tension can reach as low as 0.5 mJ/m 2, which makes the

101

captive bubble extremely flat. This sometimes makes Rotenberg's ADSA algorithm [9]

malfunction as the radius of curvature at the bubble apex becomes very large. Two, the

contrast between the drop and background is not adequately sharp, which causes Cheng's edge

detection algorithm [ 10] to fail. To solve the first problem, the new second generation ADSA

[11 ], as described earlier in this chapter, has been employed, which is capable of analyzing flat

drops with near zero apex curvature. No limitations have been found in this improved version

of ADSA, and it is considerably more efficient than the original ADSA algorithm [9]. As a

solution to the second problem, a different image analysis scheme using image thresholding with

polynomial smoothing has been implemented as a temporary approach until more advanced edge

detection algorithms are developed. Despite the known limitations of the thresholding method,

good results have been obtained [74].

3.2.4.1 The ADSA Captive Bubble Chamber

The ADSA captive bubble chamber [74] comprises two quartz viewing windows which are

secured on both sides of a metal plate. A section of this plate has been removed, forming the

side walls of the chamber (the end walls are the viewing windows). The windows are placed in

between the metal plate and two metal end plates. Seals are ensured by O-rings on both sides of

the windows. Four sets of bolts are used to fasten the whole assembly together. Two lateral

holes were drilled through the end plates to allow water circulation for temperature control of

the chamber. As shown in Fig. 23, the section hole of the middle metal plate has straight edges

on the sides and bottom. The top was designed such that a glass piece with a concave surface

could be held in place, thereby providing a glass "ceiling" for the chamber. Glass was chosen to

ensure an aqueous layer between the captive bubble and the ceiling, leaving the air-liquid

interface completely intact. The glass piece was obtained by cutting an optical lens.

Three ports were made to provide access to the chamber (Fig. 23). One port was designed for

the temperature probe which remains in place during the experiment. Fittings are used to

connect a Teflon capillary to the second port of the chamber. The capillary, in turn, is attached

to a motorized syringe. The chamber internal pressure is changed by pumping liquid in or out.

The last port is used to form an air bubble in the sample chamber using a microsyringe.

102

3 I I i

2 6

Fig. 23. Schematic of the section hole of the metal plate forming the side walls of the ADSA-CB test chamber.

(1) Captive bubble, (2) pulmonary surfactant solution, (3) glass ceiling, (4) microsyringe port, (5)

syringe port, (6) temperature probe port [74].

3.2.4.2 Surface Tension Measurements for Lung Surfactant

The surfactant used for this demonstration was bovine lipid extract surfactant (BLES|

Biochemicals Inc., London, Ont. Canada). BLES was supplied as a suspension, containing the

phospholipids of natural surfactant (27 mg/ml) and surfactant associated proteins: SP-B and SP-

C. The suspension was gently stirred and 1.5 ml was diluted in a 10 ml flask with 0.9% NaCl

solution, resulting in a phospholipid concentration of 400 ~tg/ml in the diluted surfactant

solution.

The cycling of the interface between the air bubble and the lung surfactant solution was

performed at a rate of 25 seconds per cycle. The amount of compression was approximately

80% by volume, corresponding to about 65% change in surface area. The experiment was

conducted at a temperature of 37.0 + 0.1 ~

103

Figure 24 shows three dynamic cycles of a typical experiment where the interface was

compressed sufficiently to achieve collapse of the surface film. Although not entirely obvious in

Fig. 24, close examination of a region of minimum surface tension (Fig. 25) showed an intricate

pattern.

3 0 I " . . . . . . . . I ' ~

25

E 20 " 3 E

c- O -~ 15 r C1) b-

O "12 -~ 10 CO

, , , , , , . . . . . . , .... ~ " _ ! ' , . . . . . . . ,

2 !~ :

, . I , . 1

10 20 I , _ _ _ ! , I . . . . . . . . L . . . . . 1 , I

0 30 40 50 60 70 Time (s)

Fig. 24. Surface tension as a function of time for the first three cycles of a captive bubble experiment. Note that

these compressions were sufficient to achieve collapse. The error limits shown are the 95% confidence

levels [74].

During compression, at 523.5 s, the surface tension y was approximately 0.5 mJ/m 2, and half a

second later increased to ~ 1.5 mJ/m 2. This indicates that expulsion of DPPC molecules from

the film (i.e., collapse) occurred, resulting in a surface tension increase of about 1 mJ/m 2. At

524 s, the interface was still being compressed and between this time and one second later the

surface tension decreased to, and remained at, ~ 1.0 mJ/m 2. Then between 525 and 526.5 s,

when the bubble was between compression and expansion, ADSA failed to provide data, despite

the images being acquired. This might be due to the bubble being non-Laplacian in shape [74]:

104

At these very low surface tensions, sudden oscillations of the drop where observed presumably

are due to collapse of the film.

6.0

5.0

e l

E "~ 4.0 E

v

c -

O

r - 3.0 F--

0

1E "- !

u) 2.0

1.0

- - - ' i . . . . . D - - I r" . . . . . I" T 1- ' I ' ...... I

C O r i - l p r e s s i o r l T e x p a n s ~ n

'SS'II

0 . 0 ~ ~ - - - - 1 L ~ I ~ I a I i I i

519 520 521 522 523 524 525 526 527 528 529 530 Time (s)

Fig. 25. Detailed surface tension as a function of time of a collapsing film. The missing points are probably due

to the non-Laplacian shape of the bubble (see text). The error limits shown are the 95% confidence

levels [74].

The rather large error limits at 7 = 0.5 mJ/m z in Fig. 25 may indicate that the bubble was in the

midst of one of these sudden movements, and not quite in its equilibrium shape. In addition, in

the interlude between the compression and expansion, the images could not be processed by

ADSA, again indicating the deviation of the bubble shape from mechanical equilibrium [74].

4. CONTACT ANGLE MEASUREMENTS

In the vast majority of contact angle studies in the literature, the method used is direct

measurement of sessile drops. To an extent, the quality of these measurements relies very much

on the skill of the experimentalist. The measurements also pose considerably strain on the eyes.

105

Recent developments in image analysis and processing employed by Axisymmetric Drop Shape

Analysis-Profile (ADSA-P) have increased the accuracy and reduced the subjectivity

considerably. The contact angle determination from ADSA-P has many advantages over the

conventional way of putting a tangent to the sessile drop at its base. ADSA-P determines not

only the contact angle 0 but also the liquid-vapor surface tension 7t~, volume V, surface area A,

and three-phase contact radius R of the drop.

4.1 CONTACT ANGLE MEASUREMENTS ON SMOOTH SOLID SURFACES B YADSA-P

4.1.1 Static Contact Angles

The application of ADSA-P to sessile drop contact angles requires the solid surface to be

smooth and homogeneous so as to ensure that the sessile drop is axisymmetric. On carefully

prepared solid surfaces, Li et al. [21,77] have performed static contact angle experiments and

found that a contact angle accuracy of better than + 0.3 ~ can be obtained. The static advancing

contact angle experiments were performed by supplying test liquids from below the surface into

the sessile drop, using a motor-driven syringe device. A hole of about 2 mm in the center of

each solid surface was required to facilitate such procedures. To begin the experiment, an initial

liquid drop was carefully deposited on the surface, covering the hole on the surface. This is to

ensure that the drop will increase axisymmetrically in the center of the image field when liquid is

supplied from the bottom of the surface and will not hinge on the lip of the hole. Liquid was

then pumped slowly into the drop from below until the three-phase contact radius was about 0.4

cm. Aiter the motor was stopped, the sessile drop was allowed to relax for approximately 30 s.

to reach equilibrium. Then 3 pictures of this sessile drop were taken successively at intervals of

30 s. More liquid was then pumped into the drop until it reached another desired size, and the

above procedure was repeated [21 ]. These procedures ensure that the measured static contact

angles are indeed the advancing contact angles.

Figure 26 shows these contact angle results, by plotting the values of 7~v cos0 vs. 7~v for a large

number of pure liquids with different molecular properties on three carefully prepared solid

surfaces: FC-721-coated mica, Teflon (FEP) heat pressed against quartz glass slides and

polyethylene terephthalate (PET). The FC-721 surface was prepared by a dip-coating technique.

FC-721 is a 3M company "Fluorad" brand antimigration coating designed to prevent the creep

106

of lubricating oils out of bearings. When coated onto a suitable substrate, it dries to a film of

low surface energy on which even silicon oils spread. Teflon FEP (fluroinated ethylene

propylene) surfaces were prepared by a heat-pressing method. The material was cut to 2 • 4 cm,

placed in between two glass slides, and heated pressed by a jig in an oven. PET (polyethylene

terephthalate) is the condensation product of ethylene glycol and terephthalic acid. The surfaces

of PET films were exceedingly smooth as received and only cleaned before using. Details of the

preparation of the solid surfaces can be found elsewhere [21 ].

o eo

4 0 . 0 . , , , . ,

2 0 . 0

0 . 0

- 2 0 . 0

- 4 0 . 0

FEP DMSO \ \ 1-Bromonaphthalene - - - - . . \ /

Diethylene Glycol

Hexane / / ~ ~ , . . ~

Metha

l-Bromonaphthalene / ~ , ~ ~

oFC~77~11 [[~] Diethylene Glycol

IIFEP [21] APET [21]

a I J I , I

2 0 . 0 4 0 . 0 6 0 . 0

7t, (mJ/mz) 80.0

Fig. 26. A plot of 71vCOS0 versus 3'1v on a FC-721 surface; data are from Refs.[21] and [77]. The smoothness of

the curves indicates that the values of 7tvCOS0 depend only on ~[lv and 7s~.

In Fig. 26, because these curves are so smooth, one has to conclude that the values of 3,tvcos0

depend only on 71v and 7.~ [21,77,78]; and because of Young's equation, the value of 7st can be

expressed as a function of only 71v and 7 .... This is in good agreement with theoretical results on

the thermodynamic phase rule [79-82] and the equation-of-state approach for solid-liquid

interfacial tensions [83-86]. Focusing on the experimental contact angle data of hexane,

107

methanol and decane on the FC-721 surface shows that the value of 7lv cos0 or the contact angle

responds only to surface tension of the liquid, and not directly to the intermolecular forces

which give rise to that surface tension [78]. These findings contradict the basic postulates of the

surface tension component approaches: Fowkes approach [87,88] and the Lifshitz-van der

Waals/acid-base approach [89,90]. Similar conclusions can be drawn for DMSO, 1-

bromonaphthalene and diethylene glycol on the FC-721 surface and DMSO, 1-

bromonaphthalene and diethylene glycol on the FEP surface [78]. It should be noted that the

accuracy of the contact angle is important here; the patterns obtained in Fig. 26 might be blurred

if a conventional goniometer type of contact angle technique with + 2 ~ in accuracy were used.

In addition, the quality of the solid surface in these experiments plays an important role: the

solid surfaces have to be very smooth, homogeneous, and inert so as to ensure that the

measured advancing contact angle is indeed a Young contact angle (i.e., a contact angle which

can be used in conjunction with Young's equation), and that 7.s.v is constant.

Since 7~l is shown to be a function of only 7tv and 7.s.v and that the geometric mean relation,

7~, = Y,~+ 7.,~ - 2 (7,~ 7sv )~/2, (5)

is well-accepted in the literature, Li et al. [21 ] showed that

1/2

cos 0 =-1 + 2 7.~ exp {- [3 (7lv- 7~.~)2 } (6) "~ Iv ,,

where 13 reflects the fact that the square root term in Eq.(5) is too large [21,91,92]. The values

of 7s~ and 13 can be determined from experimental contact angles and liquid vapor surface

tensions using a multi-parameter least-square analysis. It was found that the values of 13 were

virtually independent of the three solid surfaces used. Thus, a mean 13 value of 0.0001247

(m2/mJ) 2 was obtained. The resulting solid surface tensions for the FC-721, FEP, and PET were

found to be 11.78 mJ/m 2, 17.85 mJ/m 2, and 35.22 mJ/m 2, respectively [21].

4.1.2 Comparison of ADSA-P and an Automated Capillary Rise at a Vertical Plate Technique

It is apparent that ADSA-P is capable of producing very accurate contact angle data, of a

quality comparable to that of an automated capillary rise at a vertical plate technique [93-95]. In

several instances, ADSA contact angle measurements [21,77] are available for the same systems

108

for which capillary rise measurements were performed [93]. The contact angle determination

from the capillary rise at a vertical plate relies on the following equation

Apgh 2 sin 0 = 1 - (7)

271v

Equation (7) was obtained by the integration of the Laplace equation of capillarity when the

vertical plate is infinitely wide [94]. Knowing the density difference, Ap, between liquid and

vapor, the acceleration due to gravity, g, and the liquid-vapor surface tension, 7iv, the contact

angle, 0, can be obtained from a measurement of the capillary rise, h. The task of measuring a

contact angle has thus been reduced to a measurement of a length, which can be performed

optically with a very high degree of accuracy by means of a cathetometer These procedures

have been automated recently [93-95]. On a smooth and homogeneous solid surface, the three-

phase line in the central part of the plate will be straight and hence independent of edge effects.

This finding implies that the assumption of an infinitely wide vertical plate is fulfilled for

ordinary liquids of a surface tension of, say, 71v < 100 mJ/m z for plates only 2 cm wide.

Results from both techniques for FC-721 and FEP are summarized in Table 3.

Table 3. Comparison of measured contact angles 0 (degrees) using the automated capillary rise technique and ADSA-P for the two solid surfaces, FC-721 and FEP.

FC-721 FEP

Liquids ADSA-P" Capillary Rise c ADSA-P' Capillary Rise c

Tetradecane 73.31 -1- 0.14 73.5 + 0.1 72.96 + 0.21 b

Hexadecane 75.32 + 0.27 75.6 + 0.1

Dodecane 69.82 + 0.25 70.4 + 0.1

Dimethyl formamide

52.51 + 0.23 52.5 + 0.1

53.75 + 0.22 53.9 + 0.1

47.96 + 0.21 47.8:1:0.1

68.52:1:0.21 68.6 + 0.2

The error limits are 95% confidence limits " Source: Ref. 21. b Source: Ref. 77. r Source: Ref. 93.

109

The capillary rise results presented are the average of 10 contact angles measured at 10 different

velocities of the three-phase contact line ranging from 0.08 to 0.49 mm/min. It should be noted

that the contact angle results from ADSA were measured at zero velocity of the three-phase

contact line. There is excellent agreement between the contact angles from the two techniques.

The choice of method depends on the specific application and is largely a matter of convenience

and equipment availability.

4.1.3 Low-Rate Dynamic Contact Angles

Recently, Kwok et al. [96] performed dynamic contact angle experiments using ADSA-P on a

well-prepared FC-722-coated mica surface. FC-722 is also a 3M company "Fluorad" brand

"fluorochemical" coating and is chemically very similar to the FC-721 used by Li et al. [21,77].

A dip-coating technique was used to coat the FC-722 on freshly cleaved mica surfaces. Before

dip coating, a hole of about 2 mm was drilled in the center of each 25 by 50 mm mica surfaces.

This allows formation of the sessile drop by pumping the liquid from below the solid surface,

using a motorized syringe mechanism. To perform static/dynamic contact angle measurements

for sessile drops, a motor driven syringe, similar to that used by Li et al. [21,77], was employed

in the experimental set-up shown in Fig. 5. The schematic of this mechanism is shown in Fig. 27.

The procedure was similar to that used for static drops: an initial liquid drop of about 0.3 cm

radius was carefully deposited, covering the hole, and liquid was supplied from the bottom of

the surface by a motor syringe. The syringe mechanism pushes the syringe plunger, leading to an

increase in drop volume and hence the three-phase contact radius. The velocity of the three

phase line could be set by adjusting the speed of the motor. A sequence of pictures of the

growing drop was then recorded by the computer typically at a rate of 1 picture every 2-10

seconds, until the three-phase contact radius was about 0.5 cm or larger. The procedure used

here is different from that by Li et al. [21,77] in that the contact angles measured by Li et al.

[21,77] were static angles, i.e. contact angles at zero velocity of the three-phase contact line.

Since ADSA-P determines the contact angle and the three-phase contact radius simultaneously

for each picture, the advancing dynamic contact angles as a function of the three-phase contact

radius (i.e. location on the surface) can be obtained. In addition, the change in the contact angle,

volume, surface area, and the three-phase contact radius can also be studied as a function of

110

time. The actual rate of advancing can be determined by linear regression, by plotting the three-

phase contact radius over time.

Fig. 27. Schematic of a motorized syringe mechanism for dynamic contact angle experiments.

It should be noted that this procedure of measuring contact angles as a function of the three-

phase contact radius has an additional advantage: the quality of the surface is observed indirectly

in the measured contact angles. If a solid surface is not very smooth, irregular and inconsistent

contact angle values will be seen as the three-phase contact line advances. When the measured

contact angles as a function of surface location are essentially constant, the mean contact angle

for a specific rate of advancing can be obtained by averaging the contact angles.

Figure 28 shows an experimental result of this dynamic/static contact angle experiment for

cis decalin. As the drop volume increases from 0.56 cm 3 to 0.72 cm 3, the three-phase contact

111

line advances from about 0.36 cm to 0.41 cm at a rate of 0.412 mm/min. A sequence of drop

images was acquired after the motor was stopped. As can be seen in Fig. 28, the contact angle

is independent of rate, at low rates of advancing. This result suggests that the low-rate dynamic

contact angle Ody, is identical to the static contact angle O.,.tat �9 It was found that Ody,, = O,,.tat for a

velocity of the three-phase contact line of the order of 0.1 mm/min [96].

80.00

79.50

79.00 < > < >

78.50

78.00

Dynamic Static

0.41

0.39

0.37

0.35

' I ' 1 '

I ~ I L

0.72

0.68 E ,~, 0.64

0.60

0.56 0.0 150.0

Stop Motor

. . . . z I , I ,

50.0 100.0 Time [sec.]

Fig. 28. Results of the dynamic/static contact angle experiments of cis-decalin on a FC-722-coated mica surface [96]. This result suggests that the dynamic contact angle Ody,, is identical to the static contact angle Ostat. It also reconfirms the experimental protocol used by Li et al. [21,77] to measure static contact angles.

112

This result validates the experimental protocol used by Li et al. [21,77] to measure static

contact angles. This is also in good agreement with recent work [93] to determine low-rate

dynamic contact angles by the automated capillary rise technique [94]: using an automated

capillary rise technique, it was shown that low-rate dynamic contact angles at velocities of the

three-phase contact line ranging from 0.08 to 0.49 mm/min are identical to the static contact

angles.

Low-rate dynamic contact angles of 17 pure liquids with different molecular properties were

performed on this FC-722-coated mica surface [96]. Figure 29 shows a typical dynamic contact

angle result for water. It can be seen that after an initial increase of drop volume at constant

three-phase contact radius R, the contact angle 0 reaches its appropriate advancing mode, i.e. as

the drop volume increases from 1.3 cm 3 to 1.4 cm 3, the water contact angle increases from

about 108 ~ to 120 ~ This is due to the fact that even carefully putting an initial drop from above

on a solid surface described above can result in a contact angle somewhere between advancing

and receding. Therefore, it takes time for an initial drop front to advance. At 120 ~ , the three

phase line sticks momentarily before the three-phase contact line starts to move as V increases.

Further increase in V causes the three-phase contact line to advance at essentially constant 0 of

about 118 ~ Increasing the drop volume in this manner ensures the measured 0 to be an

advancing contact angle. From linear regression, it was found that the drop front was advancing

at 0.143 mm/min and a mean water contact angle of 118.4 ~ was obtained.

It should be noted that although the accuracy of the automated capillary rise technique is

comparable with that of ADSA-P, the amount of information obtained by the former technique

is limited. The capillary rise technique requires, a priori , the liquid-vapor surface tension 7lv as

input information and computes only the contact angle 0; the latter technique, however,

computes not only the contact angle 0, but also 3'tv �9

Figure 30 shows a summary of these low-rate dynamic contact angle results for 17 pure liquids

on a FC-722-coated mica surface, in a plot of 7l~ cos0 versus 7~. The values of 7~ cos0 again

change smoothly as 7t~ increases, regardless of intermolecular forces. Thus, intermolecular

forces do not have any independent effect on the contact angles. This finding is in excellent

agreement with those from Li et al. [21,77] who worked with static angles.

' - ' 2 e~O

1 2 5 . 0 . , ' , ' , ' ' '

1 2 0 . 0

1 1 5 . 0

1 1 0 . 0

1 0 5 . 0 a I , I , I ~ I i

0 . 5 6

0 . 5 2

0 . 4 8

0 . 4 4

I ' I ' I ' I '

I J I ~ I A I ,

113

1 . 8 0

1.60

1.40

1 . 2 0 , I ~ 1 ~ I L I ,

0.0 100.0 200.0 300.0 400.0 500.0

Time [sec.]

Fig. 29. Results of the low-rate dynamic contact angle experiment of 1-pentanol on FC-722-coated mica surface

[96]. The rate of advancing for the three-phase contact line is 0.143 mm/minute. The mean contact

angle is found to be 118.41 o.

4.1.3.1 Comparison between ADSA-P and a Goniometer Technique

In the literature, however, the contact angle patterns shown in Figs. 26 and 30 are not always

reported; considerable scatter in the contact angles may be obtained and might suggest

additional degrees of freedom. Such measurements are normally performed using a goniometer-

114

sessile drop technique of which the contact angle is obtained manually by putting a tangent to

the sessile drop at its base. To clarify this, ADSA-P and a goniometer-sessile drop technique

were employed recently by Kwok et al. [97] to measure, respectively, low-rate dynamic and

static contact angles on two copolymers in order to explain the discrepancies between the

results in Figs. 26 and 30 on the one hand, and those in the literature on the other.

C'4

r ~ 0 r

30.0

20.0

10.0

0.0

-10.0

-20.0

-30.0

20.0

' I ' I ' i ' i ' I

Decane / E t/h YDlibenn;lrnaatrneine

/1-Pentan~ / / DMSO

/ trans-Decalin / / / / / // //~ 1-Bromonapthalene

~ r / / / / , ~ Diethylene Glycol

/ / / thy,en lyco,

/ / 1 D e ! a n C i ; -Decalin ~ ~ ' ~ ~ ~ i ~ d h 2 1 1 1 2 s 1

/ Hexadecane _ j ~ . ~ i ~ ~2-.~

Formamide / - - - ~ _____________'~

Glycerol �9 FC-722 [96]

Water

j I , I , I , I , I

30.0 40.0 50.0 60.0 70.0 71~ (mJ/m2)

80.0

Fig. 30. A plot of3qv cosg versus 7iv on a nearly perfect FC-722-coated mica surface. Data are from Ref. [96]. This

experimental result suggests that the values of 71v depend on only 7iv and 7.,.~, but not directly on

intermolecular forces.

Two typical contact angle experiments of formamide on the poly(propene-alt-N-(n-alt-

propyl)maleimide) copolymer are shown in Fig. 31 at different rates of motion of the three-

phase contact line. It can be seen in Fig. 3 l a that as the drop volume increases initially, the

115

contact angle increases from 60 ~ to 63 ~ at essentially constant three-phase contact radius, R. As

the drop volume continues to increase, t3 suddenly decreases to 60 ~ and the three-phase contact

line starts to move. As R increases further, the contact angle decreases slowly from 60 ~ to 54 ~

Focusing on the surface tension-time plot, it can be seen that the surface tension of formamide

decreases with time. One possible explanation is that dissolution of the copolymer by formamide

occurs, causing the liquid-vapor surface tension to change from that of the pure liquid. Similar

behavior for a different experiment (at nearly the same rate of advancing) is shown in Fig. 3 lb.

It is an important question to ask which contact angles one should use for the interpretation in

terms of surface energetics. Since chemical or physical reactions such as polymer dissolution

change the liquid-vapor, solid vapor and solid-liquid interface (interfacial tensions) in an

unknown manner, such a question is very difficult to answer. Because we are unsure whether or

not the solid-vapor surface tension, 7.~.v, will remain constant and whether Young's equation is

applicable, these contact angle data should be disregarded for the interpretation in terms of

surface energetics, since all contact angle approaches [84-90] assume the constancy of 7~v and

the validity of Young's equation.

Other experimental results of diiodomethane are shown in Fig. 32. It can be seen in Fig. 32a that

initially the apparent drop volume, as perceived by ADSA-P, increases linearly, and the contact

angle increases from 45 ~ to 65 ~ at essentially constant three-phase contact radius. Suddenly, the

drop front jumps to a new location as more liquid is supplied into the sessile drop. The resulting

contact angle decreases sharply from 65 ~ to 40 ~ . As more liquid is supplied into the sessile drop,

the contact angle increases again.

Such slip/stick behaviour could be due to non-inertness of the surface. Phenomenologically, an

energy barrier for the drop front exists, resulting in sticking, which causes 9 to increase at

constant R. However, as more liquid is supplied into the sessile drop, the drop front possesses

enough energy to overcome the energy barrier, resulting in slipping, which causes t3 to decrease

suddenly. It should be noted that as the drop front jumps from one location to the next, the

drop will not remain axisymmetric. Such a non-axisymmetric drop cannot meet the basic

assumptions underlying ADSA-P, causing errors, e.g., in the apparent surface tension and drop

volume. This can be seen from the discontinuity of the apparent drop volume (the physical

volume was steadily increasing throughout the experiment as the liquid was pumped into the

116

drop) and apparent surface tension with time as the drop front sticks and slips. A similar

experiment for a new surface, with similar results, is shown in Fig. 32b.

57.0

52.0

E 47.0

42.0

64.0 ,,-.2. e~o

60.0 "t3

56.0

0.54

'~" 0.50

~: 0.46

0.42

1.05

0.95 r

0.85

0.75

(a) solid surface #1 ' I ' i ' i

rate = 0.332 mm/min. R = 0.998

50 100 150 Time (sec.)

2()0

(b) solid surface #2 , , , , .

' I ' t I I ' _

(

R = 0.997 1 ,, 1 ~ J - ~

C

50 100 150 200 Time (sec.)

Fig. 31. Low-rate dynamic contact angles of formamide on poly(propene-alt-N-(n-propyl)maleimide) copolymer

surface measured by ADSA-P [97]. The decrease in the '~lv suggests dissolution of the copolymer by the

liquid.

Obviously, the observed contact angles in Fig. 32 cannot all be Young contact angles, since 7,v,

3'lv (and 7,z ) are constants, so that because of Young's equation, 0 ought to be a constant. In

117

addition, it is difficult to decide unambiguously at this moment whether or not Young's equation

is applicable at all because of lack of understanding of the slip/stick mechanism. Therefore,

these contact angles should not be used for the interpretation in terms of surface energetics.

Similar results were also found for the poly(propene-alt-N-(n-hexyl)maleimide) copolymer [97].

52.0 50.0

E 48.0 E '-" 46.0

44.0

80.0 "~ 70.0 ",= 60.0

50.0 40.0

0.50

0.46

0.42

0.38

0.90 ~ 0.80 ~" 0.70

0.60

(a) solid surface #1 (b) solid surface #2 I ' I '

t i I ,

, I t I ,

, I ~ 1 A

0 200 400 Time (sec.)

Z Z Z Z

Z Z Z Z Z

6()0

, t t t , t ,

t i I , I 0

50 100 150 200 Time (sec.)

Fig. 32. Low-rate dynamic contact angles of diiodomethane on poly(propene-alt-N-(n-propyl)maleimide)

copolymer measured by ADSA-P [97]. It should be noted that not all contact angles from this slip/stick

behaviour can be used in conjunction with Young's equation (see text).

It was found that the contact angles and/or the liquid-vapor surface tensions for some solid-

liquid systems do not remain constant [97]. Upon elimination of these contact angles shown to

118

be meaningless in the ADSA-P study, the contact angle patterns of those in Figs. 26 and 30

emerge (see Fig. 33). The curves in Fig. 33 all follow the same trend as seen with the results

obtained for more inert polar and non-polar surfaces [21,77,93,96] (in Figs. 26 and 30).

6 0 . 0 , , , , . , . , ' ,

t"q

r/3 o

5 0 . 0

40.0

3 0 . 0

2 0 . 0

1 0 . 0

0 . 0

- 1 0 . 0

1-Bromonaphthalene

Thiodiethanol

lycerol

cis_/Decalin ~ ~ ; t e r

Dlethylene Glycol Glycerol / ~ ~

�9 Poly(propene-alt-N-(n-hexyl)maleimide) [97] U] Poly(propene-alt-N-(n-propyl)maleimide) [97] Water

- 2 0 . 0 , j , I , i , l , I , 3 0 . 0 4 0 . 0 5 0 . 0 6 0 . 0 7 0 . 0 8 0 . 0 "[Lv (mJ/m2)

Fig. 33. Excluding the inconclusive contact angle data, the values of 7lvcos0 from ADSA-P change smoothly with 3qv for the poly(propene-alt-N-(n-propyl)maleimide) and poly(propene-alt-N-(n-hexyl)maleimide) copolymers, regardless of intermolecular forces which give rise to the surface tensions. Data are from Ref.[97].

For comparison purposes, Kwok et al. [97] also measured advancing contact angles using a

conventional goniometer technique. The procedure is as follows: A sessile drop of about 0.4 -

0.5 cm radius was formed from above. The three-phase contact line of the drop was then slowly

advanced by supplying more liquid from above through the capillary which was always kept in

contact with the drop. The maximum (advancing) contact angles were measured carefully from

the leR and right side of the drop and subsequently averaged. It was found that the contact

angles observed by the goniometer technique and ADSA-P are virtually identical for solid-liquid

systems which have essentially constant contact angles. However, for solid-liquid systems which

are complex, only the maximum (advancing) contact angle is normally recorded by the

119

goniometer technique. While pronounced cases of slip/stick behaviour can indeed be observed

by the goniometer, it is virtually impossible to record the entire slip/stick behaviour manually.

While these maximum contact angles are the advancing contact angles, they cannot be used for

the interpretation in terms of surface energetics due to obvious reasons discussed earlier. The

distinctions and differentiations made in the ADSA-P study are not possible in the goniometer

study. Of course, a contact angle thus recorded by the goniometer should agree with maximum

angles obtained by ADSA-P. For example, for diiodomethane on the poly(propene-alt-N-(n-

hexyl)maleimide) copolymer surface, the goniometer value is 98 ~ in agreement with the maxima

in the entire slip/stick pattern ADSA-P results (0 ~ 96 ~ in Fig. 32.

6 0 . 0

5 0 . 0

4 0 . 0

3 0 . 0 cq

~ ' 2 0 . 0

�9 o 1 0 . 0

0 . 0

- 1 0 . 0

�9 Poly(propene-alt-N-(n-hexyl)maleimide) [97] [] Poly(propene-alt-N-(n-propyl)maleimide) [97]

[]

O 0 []

- 2 0 . 0 ' ~ ' ' ' ' ' ~ ' ' 3 0 . 0 4 0 . 0 5 0 . 0 6 0 . 0 7 0 . 0 8 0 . 0

7tv (mJ/m2)

Fig. 34. The values of 71v cos0 vs. 71v for the poly(propene-alt-N-(n-propyl)maleimide) and poly(propene-alt-N-(n- hexyl)maleimide) copolymers using contact angle data measured by a conventional goniometer technique. Data are from Ref.[97]. Due to the scatter, one might want to conclude (erroneously) that the values of 71v cos0 may depend on various parameters such as intermolecular forces, in addition to 71v and 7~.

It is obvious that a simple contact angle technique, e.g. a goniometer, cannot reflect the

complexities of solid-liquid interactions such as slip/stick mechanisms and physico-chemical

reactions. In the cases where the liquid-vapor surface tension of the sessile drop decreases due

to dissolution of the surface, it is impossible for a goniometer type of technique to detect

120

changes in the liquid-vapor surface tension. Thus, conventional goniometer contact angle

measurements are liable to produce a mixture of meaningful and meaningless contact angle data,

with no criteria to distinguish between the two.

If the contact angle data from the goniometer study are plotted as Ylv cos0 vs. Ylv (see Fig. 34),

no obvious functional dependence can be observed. One might wish to conclude from Fig. 34

that the values of 7lv cos0 could depend on various parameters, in additional to 7l~, such as

specific intermolecular forces. The picture changes drastically (see Fig. 35) upon elimination of

the contact angles shown to be meaningless in the ADSA-P study.

t"q

�9 r

6 0 . 0 . , . ,

5 0 . 0

4 0 . 0

3 0 . 0

2 0 . 0

1 0 . 0

0 . 0

- 1 0 . 0

' i ' i ' 1 '

�9 Poly(propene-alt-N-(n-hexyl)maleimide) [97] 5 Poly(propene-alt-N-(n-propyl)maleimide) [97]

- 2 0 . 0 , , , , , l , , , ' , 30.0 40.0 50.0 60.0 70.0 80.0

7tv (mJ/m2)

Fig. 35. The values of '~lv cos0 vs. 'Ylv for the poly(propene-alt-N-(n-propyl)maleimide) and poly(propene-alt-N-(n-

hexyl)maleimide) copolymers using contact angle data by a conventional goniometer technique, after

the elimination of the contact angle data shown to be inconclusive in the ADSA-P study. Data are from

Ref.[971.

121

4.1.3.2 Comparison between Low-Rate Dynamic Contact Angles Interpreted by ADSA-P and

an Automated Polynomial Fit Program (APF)

With respect to the above comparison, one might argue that the ADSA-P results and tangents

to sessile drops might be misleading, in that the real difference between the two types of

experiments was that of a fairly sophisticated and automated low-rate dynamic contact angle

measurement and a very simple if not crude static measurement. To explore this thought, del

Mo et al. [98] reported an automated method to put tangents to the same drop images on which

ADSA-P operates. Thus, low-rate dynamic contact angle experiments were performed and

interpreted separately by ADSA-P and an automated polynomial fit program (APF). The APF

program was developed to estimate the contact angle by fitting a tenth order polynomial to a

drop profile which is extracted from a digitial image, using the scheme of Cheng et al. [ 10].

Two types of solid surfaces were used: FC-722-coated silicon wafer surface and a non-inert

(hydrophilic) poly(propene-alt-N-(n-methyl)maleimide) copolymer surface.

An example of such experiments is shown in Fig. 36 for hexane on a FC-722-coated silicon

wafer surface. As can be seen in Fig. 36a, increasing the drop volume, V, linearly from 0.40 cm 3

to abou t 0 .45 cm 3 increases the apparent contact angle, 0, from about 44 ~ to 54 ~ at essentially

constant three-phase contact radius, R. This increase in the contact angle has been explained

above and is due to the fact that even carefully putting an initial hexane drop from above on a

solid surface can result in a contact angle somewhere between advancing and receding. Further

increase in the drop volume causes the three-phase contact line to advance, with 0 essentially

constant as R increases. A mean ADSA-P contact angle of 50.89 + 0.14 ~ was obtained. The

APF contact angles for this experiment are also shown in the same figure. It can be seen that the

initial increase in the contact angles was also observed by the APF scheme. The mean APF

contact angle is found to be 51.40 + 0.39 ~ in good agreement with that from ADSA-P. A

similar experiment is also shown in Fig. 36b, with similar results.

Two different experimental results for formamide are shown in Fig. 37, at different rates of

motion of the three-phase contact line. It can be seen from the ADSA-P results in Fig. 37a that,

as the drop volume increases initially, the contact angle increases from about 40 ~ to 42 ~ and the

surface tension decreases from about 58 mJ/m 2 to 54 mJ/m 1. As the drop volume continues to

122

increase, the contact angle decreases from about 42 ~ to 39 ~ Similar contact angle behaviour at

a different rate of advancing is also shown in Fig. 37b. These contact angle patterns have been

described above. One possible explanation is that dissolution of the copolymer occurs, causing

the liquid-vapor surface tension to change from that of the pure liquid. It should be noted that a

general trend of such contact angle patterns was also observed by the APF program, but with

larger scatter. These contact angle data should be disregarded for the interpretation in terms of

surface energetics. It should be noted that either ADSA-P and APF enables one to distinguish

meaningful contact angles from meaningless ones, to some extent. While the three-phase contact

radius, drop volume and drop surface area can in principle be estimated by the former technique,

ADSA-P has the additional advantage that 7tv is also computed.

(b) solid surface #2 ' I ' 1 ' I ' 1 1 T

20.5

~ ~9.5

E 18.5

17.5

54.0 52.0 50.0

,=:, 48.0 46.0

0.50 "~" 0.45

0.40 0.35 (

0.85 0.75

E 0.65

0.55 0.45 (

0.0

rate = 0.431 mm/min. R = 0.992

_ _

(a) solid surface #1

, I , I , I ,

~- o o ADSA: 0 = 50.77 + 0.15 ~ - ~ APF: 0 = 50.79 + 0.25 ~

rate = 0.458 mm/min. _ ~

" = ~ -_ (

50.0 100.0 150.0 200.0 200.0

( , I , I ~ I ,

50.0 100.0 150.0 Time (sec.) Time (sec.)

Fig. 36. Low-rate dynamic contact angles of hexane on a FC-722-coated wafer surface measured by ADSA-P and

APF [98]. Good agreement was found between the two techniques.

123

It was found [98] that only formamide and thiodiethanol on the poly(propene-alt-N-(n-

methyl)maleimide) copolymer do not result in constant contact angles and/or constant liquid-

vapor surface tensions. If these inconclusive contact angles are omitted, smooth curves emerge,

by plotting the values of 7lv cos0 vs. 7~.

(a) solid surface #1

62.0

e q 58.0

54.0

50.0 [ ~ , I , I ,

42.0 --:.. 41.0

40.0

39.0 SA 38.0 ' I ~ t , -

0 . 6 6

0.56 E

0.46

0.36

1.4

1.0 m

0.6

(b) solid surface #2 I ' i ' i T

0.2

e, = APF

100.0

I ,

O

t

0.0 200.0 300.0 400.0 50.0 100.0 Time (sec.) Time (sec.)

62.0

58.0

54.0

50.0

44.0 43.0 42.0 41.0 40.0

0.(36

0.56

0.46

0.36

1.4

1.0

0.6

0.2 I , . .

150.0 200.0

Fig. 37. Low-rate dynamic contact angles of formamide on a poly(propene-alt-N-(n-methyl)maleimide)

copolymer measured by ADSA-P and APF [98]. The decrease in both the liquid-vapour surface tension

and the contact angle suggest dissolution of the copolymer by formamide. A decreasing trend of the

contact angles was also observed by the APF scheme, with larger scatter. These contact angles are

disregarded for the interpretation in terms of surface energetics.

Figure 38 shows this plot using the mean contact angles from both ADSA-P and the APF

scheme as well as contact angles from other studies [21,77,93,96,97]. It can be seen in this

figure that, using the mean contact angles from either technique, the values of 7lv cos0 all change

124

smoothly with 7Zv for the (non-polar) FC-722-coated surface and for the (polar) poly(propene-

alt-N-(n-methyl)maleimide) copolymer, regardless of intermolecular forces. When comparing

with the other maleimide copolymer surfaces, a consistent change in the wettability is observed

as the side chain changes from hexyl to methyl groups. These patterns are in excellent

agreement with previous results for other polymer surfaces. Clearly, contact angle patterns

different from those in Figs. 26, 30, 33, 35 and 38 are due to bad experimentation, poor surface

quality, and/or lack of inertness of the solid surface.

' I ' I ' 1 '

50.0

30.0

-~ 10.0

o o -10.0

-30.0

poly(propene-alt-N-(n-methyl)maleimide)

poly(propene-alt-N-(n-propyl)maleimide)

poly(propene-alt-N-(n-hexyl)maleimide)

FC-721, and FC-722 ~

C)FC-721-coated mica, ADSA-P [21] ~ <~FC-721-coated mica, ADSA-P [77] -~FC-721-coated mica, automated capillary rise [93] VFC-722-coated mica, ADSA-P [96] A FC-722-coated silicon wafer, ADSA-P [98] ~FC-722-coated silicon wafer, APF [98] <~poly(propene-alt-N-(n-hexyl)maleimide), ADSA-P [97] [>poly(propene-alt-N-(n-propyl)maleimide), ADSA-P [97] Opoly(propene-alt-N-(n-methyl)maleimide), ADSA-P [98] Vpoly(propene-alt-N-(n-methyl)maleimide), APF [98]

- 5 0 . 0 , ~ , I , J , 0 . 0 2 0 . 0 4 0 . 0 6 0 . 0 8 0 . 0

7tv (mJ/m2) Fig. 38. The values of ~lv cos0 vs. ~lv for an inert FC-722, and a non-inert poly(propene-alt-N-(n-

methyl)maleimide) copolymer [98], together with other contact angle results from other studies

[21,77,93,97]. Excluding the inconclusive contact angle data, the values of ~'Iv cos0 all change smoothly

with 3'iv, independent of which of the experimental methods used. A coherent change in the wettability

can be observed as the side chain of the maleimide copolymer changes from hexyl to methyl groups.

125

4.1.4 Determination of Line Tension from the Drop Size Dependence of Contact Angles

ADSA-P can also be used to study drop size dependence of contact angles in terms of line

tension on carefully prepared solid surfaces through the modified Young equation [99,100]

(y V~v cos0 = ~/tv- ~ - --a (8)

/x

where cy is the line tension and R is the radius of the three-phase contact circle. As can be seen

in this equation, as R --~ 0% Eq.(8) reduces to the classical Young equation with the equilibrium

contact angle 0| Combining Eq.(8) with the classical Young equation yields

cos 0 . . . . + cOS0oo (9) ?iv R

This equation predicts that, for a given solid-liquid system, the cosine of the contact angle will

be a linear function of 1/R. Thus, the line tension ~ can be determined from the slope of cos0

versus 1/R, using experimental contact angle data as a function of the three-phase contact

radius.

71.5 -

71.0-

d 70.5

<

70.0 - O

O

69.5 0.1

t t . . . . !

0'.2 0.3 0.4 0'.5 0'.6

R (cm) Fig. 39. Dependence of contact angle on the drop contact radius for dodecane on the FC-721 surface [103].

126

Recently, ADSA-P has been applied in such a study and it has been found that line tension

values are of the order of 1 ~tJ/m [101-104]. Figure 39 shows a typical drop size dependence of

contact angles for dodecane on a FC-721-coated mica surface. The plot of cos0 versus 1/R for

glycerol on FC-721 surface is shown in Fig. 40. As can be seen, the contact angles change by

about 1 ~ to 2 ~ over a few millimeters of the contact radius. Thus, conventional contact angle

measurement techniques with an accuracy of one or two degrees would fail to detect such a

drop size dependence.

-0.34 -0.34

-0.36 O U

Run 1, r = 0.95 a = 5.24 (lzl/m)

I |

-0.38 -0.38

| | t t !

1 2 3 4 5 6

~ D

~" -0.36 O

~ :I: ~ Run 2, r = 0.84 2T z a - 5.62 (M/m)

inQ

! | | | |

1 2 3 4 5 6

1/R (l/cm) 1/R (l/cm)

Fig. 40. Cosine of the contact angle vs. the reciprocal of the drop contact radius for glycerol on the FC-721

surface; r is the l inear correlat ion coefficient and cr is the l ine tension [ 103].

4. 2 CONTACT ANGLE MEASUREMENTS ON SMOOTH SOLID SURFACES B Y ADSA-D

For the case of very low contact angles (e.g. below 20~ it becomes increasingly difficult to

measure contact angles with most techniques; Fig. 41 shows a typical example of a low contact

angle system. For such a case, the precision of ADSA-P is decreased since it becomes more

difficult to acquire coordinate points along the edge of the drop profile.

This deficiency can be overcome by using ADSA-CD (contact diameter) [13] and ADSA-MD

(maximum diameter) [105], where the drop is viewed from above. The choice between these

two techniques depends on the values of the contact angle: ADSA-CD is particularly suited for

the case of contact angles less than 90 ~ and ADSA-MD for contact angles greater than 90 ~

127

Both programs have been unified into a single program called Axisymmetric Drop Shape

Analysis- Diameter (ADSA-D). Figure 42 shows a schematic of the experimental set-up for

ADSA-CD and ADSA-MD. A typical example of a water drop on the FC-721-coated mica

surface, having a contact angle greater than 90 ~ is shown in Fig. 43.

Fig. 41. (a) Side and (b) top view of a sessile drop with a contact angle larger than 90~ (c) side and (d) top view

of a sessile drop with a small contact angle.

128

v

digitizer monitor

microscope and video camera

/IX /IX

computer terminal

i A D s I D I Fig. 42. A schematic of the experimental set-up for ADSA-CD and -MD study.

Fig. 43. (a) Profile and (b) top views of a water sessile drop on an FC-721-coated mica surface. Computed

contact angles are 117.08 + 0.13 ~ from ADSA-P and 117.34 ~ from ADSA-MD [14].

As can be seen, the observed diameter from above does not represent the contact diameter, but

rather the maximum diameter. Thus, ADSA-CD cannot be used in this situation and A D S A - M D

(or the unified ADSA-D) has to be employed. It was found that there is virtually no difference in

129

the measured contact angle on carefully prepared solid surfaces between ADSA-P and ADSA-

CD, and between ADSA-P and ADSA-MD (see Tables 4 and 5). It can be seen that both

ADSA-CD and ADSA-MD are able to yield contact angle accuracy of + 0.3 ~ on well prepared

solid surfaces [ 13,105].

Table 4: Comparison of contact angles (in degrees) obtained with ADSA-CD and ADSA-P" [131.

S ub strate/liquid ADS A-CD ADS A-P

1. VM/EG 84.1 + 0.4 83.2 + 0.6

2. VM/EG 84.9 + 0.2 84.7 + 0.5

3. VM/EG 85.2 + 0.3 85.2 + 0.5

1. SM/EG 84.3 + 0.3 83.6 + 0.4

2. SM/EG 84.3 :t: 0.3 84.3 + 0.6

1. SM/UN 22.7 + 0.2 21.8 + 0.7

2. SM/UN 21.2 + 0.2 22.2 + 0.6

3. SM/UN 21.5 + 0.2 22.2 + 0.5

" Substrate: Siliconized glass (VM and SM are specific type of substrate); liquid: ethylene glycol (EG) or undecane (UN); the error limits are 95% confidence limits for ADSA-CD and ADSA-P;

Table 5" Results from ADSA-MD and ADSA-P for the same water sessile drops on FC-721 coated mica surface at 23~ [ 105]. The error limits are 95% confidence limits.

Drop Volume (ml) Maximum Contact angle (deg.) diameter (cm)

ADSA-MD ADSA-P

1 0.0892 0.6728 117.34 117.08 + 0.13

2 0.0894 0.6735 117.63 117.20 + 0.13

130

4. 3 CONTACT ANGLE MEASUREMENTS ON ROUGH AND HETEROGENEOUS SOLID SURFACES BY

ADSA-D

ADSA-D can also be used for contact angle measurements on rough and heterogeneous solid

surfaces. It should be noted that measuring contact angles on rough surfaces such as biological

materials is generally difficult. This is due to the fact that such surfaces usually present not only

small contact angles, but also morphological and energetic imperfections, leading to

irregularities of the three phase contact line, as seen in Fig. 44. It will be difficult and dubious to

measure contact angles on such drops by finding a tangent of the drop profile at a three-phase

contact point. However, ADSA-D avoids this problem and provides averaging by analyzing

drop contact area.

Fig. 44. Sessile drops of water on a layer of T. ferrooxidans. The contact angles calculated for two different drops

using ADSA-CD were (a) 12.7 ~ and (b) 11.3 ~

Fig. 45. Schematic of the determination of the perimeter of a fictitious sessile drop on a video screen using a

cursor controlled by a mouse.

131

As an illustration, contact angle measurements on layers of several bacterial strains were

measured using ADSA-D. Using ADSA-D, a video image of the drop contact area can be

digitized semi-automatically, using a "mouse" on the video screen. The pattern of selection of

points on the three-phase contact line is characterized schematically in Fig. 45.

Table 6: Water contact angles (degs.) on a coated steel measured by ADSA-D.

Drop # Contact angle

1 100.7

2 101.4

3 98.8

4 92.5

5 99.7

Table 7: Water contact angles on heat treated and untreated wood surfaces (elm) [108].

Heat treated Untreated

Drop # Contact Contact angle Drop # Contact Contact angle Diameter (cm) (degrees) Diameter (cm) (degrees)

1 0.371 66.8 1 0.467 41.9

2 0.372 66.5 2 0.477 39.6

3 0.372 66.3 3 0.475 40.0

4 0.375 65.5 4 0.473 40.4

5 0.375 65.5 4 0.471 40.9

6 0.373 65.9

7 0.376 65.0

95% Conf. 0.373 + 65.9:1:0.6 0.473 + 0.005 40.6 + 1.1 Levels 0.0002

An average drop diameter was then calculated from these values. ADSA-D then calculates the

contact angle based on such a drop diameter and the input values for the drop volume, the

density and surface tension of the liquid. It was found that contact angles of water on piliated

132

E. coli RDEC-1 and non-piliated RDEC-1 strains are 20.8 + 0.9 ~ and 10.6 + 0.5 ~ respectively

[106].

Another illustration for the use of both ADSA-D to measure contact angles on imperfect solid

surfaces is shown as follows: It was proposed to investigate the hydrophobicity of the intestinal

tract at different sections for different ages of rabbits [ 107]. Since the surfaces of the intestine

are, of course, imperfect and rough, conventional goniometer technique cannot be used. ADSA-

D was employed to measure water contact angles on these surfaces. It was found that the

intestinal hydrophobicity is altered by maturational changes, regional differences and mucosal

inflammation. For example, a water contact angle of 53.2 +_ 8.4 ~ was obtained on the proximal

colon of suckling rabbits and of 93.2 + 6.7 ~ on proximal colon of adult rabbits [ 107].

It should be noted that the application of ADSA-D on rough and heterogeneous solid surfaces is

not limited only to biological solid surfaces. Recently, ADSA-D has been applied to measure

contact angles on other imperfect solid surfaces: coated steel, heat treated and untreated wood

(ash and elm). Table 6 shows the measured contact angles of different water drops on different

parts of a coated steel using ADSA-D. Table 7 shows water contact angles on heat treated and

untreated wood surfaces (elm) [108]. It can be seen in Table 7 that the difference in wettability

between pretreated and untreated woods (elm) are manifested in the values of the contact

angles, as calculated from ADSA-D. The accuracy and consistency of these water contact

angles illustrates the capability of ADSA-D to measure contact angles on rough and

heterogeneous solid surfaces.

5. REFERENCES

1. J.F. Padday, in"Surface and Colloid Science," Vol. 1, Matijevic, ed. W'dey, New York, p. 101, 1968.

2. D.S. Ambwani and T. Fort Jr. in "Surface and Colloid Science," Vol. 11, R.J. Good and

R.R. Stromberg, eds. Plenum Press, New York, p.93, 1979.

3. A.W. Adamson, "Physical Chemistry of Surfaces," 5th edition, Wiley, New York, 1990.

4. A.W. Neumann and R.J. Good, in "Surface and Colloid Science," Vol. 11, R.J. Good and

R.R. Stromberg, eds. Wiley, New York, p. 31, 1979.

5. F. Bashforth and J.C. Adams, "An Attempt to Test the Theory of Capillary Action,"

Cambridge University Press and Deighton Bell & Co., Cambridge, 1883.

6. S. Hartland and P~W. Hartley, "Axisymmetric Fluid-Liquid Interfaces," Elsevier Amsterdam, 1976.

133

7. C. Maze and G. Bumet, Surface Science, 13, 451, 1969.

8. C. Maze and G. Burnet, Surface Science, 24, 335, 1971.

9. Y. Rotenberg, L. Boruvka and A.W. Neumann, J. Colloid Interface Sci. 93, 169, 1983.

10. P. Cheng, D. Li, L. Boruvka, Y. Rotenberg and A.W. Neumann, Colloids Surfaces, 43, 151, 1990.

11. O.I. del Rio and A.W. Neumann, J. Colloid Interface Sci. submitted, 1996.

12. J.J. Bikerman, Ind. Eng. Chem. Anal. Ed. 13, 443, 1941.

13. F.K. Skinner, Y. Rotenberg and A.W. Neumann, J. ColloM Interface Sci. 130, 25, 1989.

14. E. Moy, P. Cheng, Z. Policova, S. Treppo, D. Kwok, D.R. Mack, P.M. Sherman and

A.W.Neumann, Colloids Surfaces, 58, 215, 1991.

15. T.E. Hull, W.H. Enright and K.R, Jackson, "Runge-Kutta Research at Toronto," Applied

Numerical Mathematics, accepted, 1996.

16. T.E. Hull, W.H. Enright and K.R. Jackson, "User's Guide for DVERK- a Subroutine for

Solving Non-Stiff ODEs," Technical Report 100, Department of Computer Science,

University of Toronto, 1976.

17. W.H. Enright, D.J. Higham, B. Ox~en and P.W. Sharp, "A Survey of the Explicit Runge-Kutta

Method," Technical Report 291, Department of Computer Science, University of Toronto, 1995.

18. U.M. Ascher, R.M. Mattheij and ILD. Russell, '~Numerical Solution of Boundary Value

Problems for Ordinary Differential Equations," Prentice Hall, Englewood Cliffs, NJ, 1988.

19. J.J. Mot6 and S.J. Wright, "Optimization Software Guide," SIAM, Philadelphia, PA, 1993.

20. S. Lahooti, O.I. del Rio, P. Cheng and A.W. Neumann, in "Applied Surface

Thermodynamics," A.W. Neumann and J.K. Spelt, eds. Marcel Dekker, New York,

chapter 10, pp 441-507, 1996.

21. D. Li and A.W. Neumann, J. Colloid Interface Sci. 148, 190 (1992).

22. P.G. de Gennes, J. Phys. Chem. 94, 8407, 1990.

23. B. Cabane and R. Duplessix, J. Phys. 43, 1529, 1982; 48, 651, 1987.

24. M.J. Rosen, "Surfactants and Interfacial Phenomena," 2nd ed. Wiley, New York, 1989.

25. F. MacRitchie, "Chemistry at Interfaces," Academic Press, San Diego, New York, 1990.

26. J.D. Andrade, "Surface and Interfacial Aspects of Biomaterial Polymers," Vol. 2, Plenum,

New York, 1985.

27. D.Y. Kwok, W. Hui, R. Lin and A.W. Neumann, Langmuir, 11, 2669, 1995.

28. R. Aveyard, B.P. Binks, T.A. Lawless and J. Mead, Can. J. Chem. 66, 3031, 1988.

29. D.Y. Kwok, P. Chiefalo, B. Khorshiddoust, S. Lahooti, M.~ Cabrerizo-Vilchez, O.I. del Rio and

A.W. Neumann, in"Surfactant Adsorption and Surface Solubilization," ACS Symposium Series

615, P,. Sharma, ed. American Chemical Society, Washington, D.C., Chapter 24, 1995.

30. S.S. Susnar, C.J. Budziak, H.A. Hamza and A.W. Neumann, International Journal of

Thermophysics, 13, 443, 1992.

134

31. S.S. Susnar, H.A. Hamza and A.W. Neumann, Colloids Surfaces A: Physicochem. Eng. Aspects, 89, 169, 1994.

32. M.A. Cabrerizo-Vilchez, Z. Policova, D.Y. Kwok, P. Chen and A.W. Neumann, Colloids Surfaces B: Biointerfaces, 5, 1, 1995.

33. M. Paulsson and P. Dejmek, J. Colloidlnterface Sci. 150, 394, 1992.

34. P. Suttiprasit, V. Krisdhasima and J. MacGuire, J. Colloidlnterface Sci. 154, 327, 1992.

35. D.E. Graham and M.C. Phillips, J. Colloidlnterface Sci. 70, 415, 1979.

3 6. P. Suttiprasit, V. Krisdhasima and J. MacGuire, J. Colloidlnterface Sci. 154, 316, 1992.

37. A.J.I. Ward and L.H. Regan, J. Colloidlnterface Sci. 78, 389, 1980.

38. R.J. Roe, V.L. Bacchetta and P.M.G. Wong, J. Phys. Chem. 71, 4190, 1967.

39. R.J. Roe, J. Phys. Chem. 72, 2013, 1968.

40. S. Wu, J. Colloidlnterface Sci. 31, 153, 1969.

41. S. Wu, J. Phys. Chem. 74, 632, 1970.

42. W.W.Y. Lau and C.M. Bums, J. Colloid Interface Sci. 45, 295, 1973.

43. S. Wu, in "Polymer Interface and Adhesion," Marcel Dekker, New York, 1982.

44. K. Grundke, P. Ulhmann, T. Gietzelt, B. Redlich, H.-J. Jacobasch, Colloids Surfaces A: Physicochem. Eng. Aspects, 116, 93, 1996.

45. D.Y. Kwok, L.K. Cheung C.B. Park and A.W. Neumann, Po~ner Sci. andEng, subrrfitted, 1996.

46. D.Y. Kwok, D. Vollhardt, R. Miller, D. Li and A.W. Neumann, Colloids Surfaces A: Physicochem. Eng. Aspects, 88, 51, 1994.

47. D.Y. Kwok, B. Tadros, H. Deol, D. Vollhardt, R. Miller, M.A. Cabrerizo-Vilchez and

A.W. Neumann, Langmuir, 12, 1851, 1996.

48. J. Li, R. Miller, R. WOstneck, H. M6hwald and A.W. Neumann, Colloids Surfaces A: Physicochem. Eng. Aspects, 96, 295, 1996.

49. R. WOstneck, P. Enders, Th. Ebisch and R. Miller, Thin Solid Films, 298(1997)39.

50. J. Li, R. Miller and H. M6hwald, Thin Solid Films, 284-285(1996)357.

51. J. Li, R. Miller, R. WOstneck and H. M6hwald, Colloids Surfaces A: Physicochem. Eng. Aspects, 114, 113, 1996.

52. J. Li, R. Miller, R. WOstneck and H. M6hwald, Colloids Surfaces A: Physicochem. Eng. Aspects, 114, 123, 1996.

53. J. Li, tL 1Wlller, D. Vollhardt, G. Weidemann and H. M6hwald, Colloid Polymer Sci. 274, 995, 1996.

54. J. Li, V.B. Fainerman and R. Miller, Langmuir, 12, 5138, 1996.

55. S.S. Susnar, Ph.D. Thesis, University of Toronto, Toronto, Canada, in preparation.

56. R. Miller, G. Loglio, U. Tesei and K.-H. Schano, Adv. Colloid Interface Sci. 37, 73, 1991.

57. R. Miller, R. Sedev, K.-H. Schano, C. Ng and A.W. Neumann, Colloids Surfaces, 69, 209,1993.

58. R.L. Kao, D.A. Edwards, D.T. Wasan and E. Chen, J. Colloidlnterface Sci. 148, 247, 1992.

135

59. C.H. Chang, N.-H.L. Wang and E.I. Franses, Colloids Surfaces, 62, 321, 1992.

60. G. Loglio, U. Tesei and R. Cini, Ber. Bunsen Ges. Phys. Chem. 81, 1154, 1977.

61. C.H. Chang and E.I. Franses, J. Colloid Interface Sci. 164, 107, 1994.

62. C.H. Chang and E.I. Franses, Chem. Eng. Sci. 49, 313, 1994.

63. A. Pinazo, M.R. Infante, C.H. Chang and E.I. Franses, Colloids Surfaces A: Physicochem.

Eng. Aspects, 87, 117, 1994.

64. G. Enhorning, J. Appl. Physiol.: Respir. Environ. Exercise Physiol. 43, 198, 1977.

65. G. Kretzschmar and K. Lunkenheimer, Ber. Bunsen Ges. Phys. Chem. 74, 1064, 1970.

66. K. Lunkenheimer and G. Kretzschmar, Z. Phys. Chem. 256, 593, 1975.

67. K. Lunkenheimer, C. Hartenstein, R. Miller and K.-D. Wantke, Colloids Su,,faces, 8, 271, 1984.

68. S.S. Susnar, P. Chen, O.I. del Rio and A.W. Neumann, Colloids Surfaces A: Physicochem.

Eng. Aspects, 116, 181, 1996.

69. P. Chen, Z. Policova, S.S. Susnar, C.R. Pace-Asciak, P.M. Demin and A.W. Neumann,

Colloids Surfaces A: Physicochem. Eng. Aspects, 114, 99, 1996.

70. S. Schtirch, H. Bachofen, J. Goerke and F. Possmayer, J. Appl. Physiol. 67, 2389, 1989.

71. S. SchOrch, F. Possmayer, S. Cheng and A.M. Cockshutt, Am. J. Physiol. 263, L210, 1992.

72. S. Schfirch, H. Bachofen, J. Goerke and F. Green, Biochem. Biophys. Acta, 1103, 127, 1992.

73. S. SchOrch, D. Schtirch, T. Curstedt and B. Robertson, J. Appl. Physiol. 77, 974, 1994.

74. R.M. Prokop, A. Jyoti, M. Eslamian, A. Garg, M. Mihaila, O.I. del Rio, S.S. Susnar, Z. Policova

and A.W. Neumann, Colloids Surfaces A: Physicochem. Eng. Aspects, accepted, 1996.

75. J. Goerke and J. Gonzales, J. Appl. Physiol. 51, 1108, 1981.

76. G. Putz, J. Goerke, H.W. Taeusch and J.A. Clements, J. Appl. Physiol. 76, 1425, 1994.

77. D. Li, M. Xie and A.W. Neumann, Colloid Polymer Sci. 271,573, 1993.

78. D.Y. Kwok, D. Li and A.W. Neumann, Colloids Surfaces A'Physicochem. Eng. Aspects,

89, 181, 1994.

79. R. Defay, "Etude Thermodynamique de la Tension SuperficieUe", Gauthier-Villars, Paris, 1934.

80. R. Defay and I. Prigogine, "Surface Tension and Adsorption," (Bellemans, A. in Collab.

with Everett, D. H. (Transl.)), Longmans-Green: London, 1966.

81. D. Li, J. Gaydos and A.W. Neumann, Langmuir, 5, 1133, 1989.

82. D. Li and A.W. Neumann, Adv. Colloid Interface Sci. 49, 147, 1994.

83. C.A. Ward and A.W. Neumann, J. Colloid Interface Sci., 49, 286, 1974.

84. O. Driedger, A.W. Neumann and P.J. Sell, Kolloid Z. Z. Polym., 201, 52, 1965.

85. A.W. Neumann, R.J. Good, C.J. Hope and M. Sejpal, J. Colloid Interface Sci. 49, 291, 1974.

86. J.K. Spelt and D. Li, in "Applied Surface Thermodynamics," A.W. Neumann and J.K.

Spelt, eds. Marcel Dekker, New York, chapter 5, pp 239-292, 1996.

87. F.M. Fowkes, J. Phys. Chem. 66, 382, 1962.

136

88. F.M. Fowkes, Ind. Eng. Chem. 12, 40, 1964.

89. C.J. van Oss, M.K. Chaudhury and R. Good, J. Chem. Revs. 88, 927, 1988.

90. R.J. Good and C.J. van Oss, "The Modem Theory of Contact Angles and the Hydrogen Bond

Components of Surface Energies," in "Modem Approaches to Wettability: Theory and

Applications," M. Schrader and G. Loeb, eds. Plenum Press, New York, pp 1-27, 1992.

91. G.C. Maitland, M. Rigby, E.B. Smith and W.A. Wakeham, "Intermolecular Forces: Their

Origin and Determination," Clarendon Press, Oxford, 1981.

92. D. Li and A.W. Neumann, J. ColloM Interface Sci. 137, 304, 1990.

93. D.Y. Kwok, C.J. Budziak and A.W. Neumann, J. ColloMInterface Sci. 173, 143, 1995.

94. D.Y. Kwok, D. Li and A.W. Neumann, in "Applied Surface Thermodynamics," A.W.

Neumann and J.K. Spelt, eds. Marcel Dekker, New York, chapter 9, pp 413-440, 1996.

95. C.J. Budziak and A.W. Neumann, Colloids Surfaces, 43, 279, 1990.

96. D.Y. Kwok, R. Lin, M. Mui and A.W. Neumann, Colloids Surfaces A'Physicochem. Eng.

Aspects, 116, 63, 1996.

97. D.Y. Kwolg T. Gietzelt, K. Gnmdke, R-J. Jacobasch and/kW. Neamaann, Langmuir, 13,2880,1997.

98. O.I. del Rio, D.Y. Kwok, R. Wu, J.M. Alvarez and A.W. Neumann, Langmuir, submitted, 1996.

99. L. Boruvka and A.W. Neumann, J. Chem. Phys. 66, 5464, 1977.

100. J. Gaydos, Y. Rotenberg, L. Boruvka, P. Chen and A.W. Neumann, in "Applied Surface

Thermodynamics," A.W. Neumann and J.K. Spelt, eds. Marcel Dekker, New York,

chapter 1, pp 1-51, 1996.

101. J. Gaydos and A.W. Neumann, J. ColloM Interface Sci. 120, 76, 1987.

102. D. Li and A.W. Neumann, Adv. Colloid Interface Sci. 39, 347, 1992.

103. D. Duncan, D. Li, J. Gaydos and A.W. Neumann, J. Colloid Interface Sci. 169, 256, 1995.

104. J. Gaydos and A.W. Neumann, in "Applied Surface Thermodynamics," A.W. Neumann

and J.K.Spelt, eds. Marcel Dekker, New York, chapter 4, pp 169-238, 1996.

105. E. Moy, P. Cheng, Z. Policova, S. Treppo, D.Y. Kwok, D.R. Mack, P.M. Sherman and

A.W. Neumann, Colloids Surfaces, 58, 215, 1991.

106. B. Drumm, A.W. Neumann, Z. Policova and P.M. Sherman, J. Clin. Invest. 84, 1588, 1989.

107. D.R. Mack, A.W. Neumann, Z. Policova and P.M. Sherman, Amer. J. of Physiology, 262,

171, 1992.

108. M. Kazayawoko, A.W. Neumann and J.J. Balatinecz, Wood Sci. Technol. accepted, 1996.

6. LIST OF SYMBOLS

137

A

AA

E

Mw

P

AP

APo

R

R

R

R1

R2

T

V

g~

X,

Z, a

b

C

ei

h

S

Wi

Xi

Zi

7

Yl

%,

%,,

surface area

surface area change

objective function

molecular weight

pressure

pressure difference across the interface

pressure difference at a reference plane

maximum (equatorial) radius

radius of curvature

three-phase contact radius

principal radius of curvature

principal radius of curvature

temperature

volume

drop volume

scaling factor on the x-coordinate

scaling factor on the z-coordinate

a set of M optimization parameters

curvature at the origin of coordinates

capillary constant (=Apg/y)

half a squared normal distance between the drop profile and the Laplacian curve

capillary rise

arc-length

weighting factor

Laplacian coordinate

Laplacian coordinate

angle of rotation of the system of coordinates

surface or interracial tension

liquid surface tension

liquid-vapor surface tension

solid-vapor surface tension

solid-liquid surface tension

water surface tension

liquid-water interface tension

138

O

O

Ode.

O stat

O~

Ap

surface pressure

contact angle

tangential angle

dynamic contact angle

static contact angle

equilibrium contact angle for the classical Young equation

density difference between the two bulk phases

line tension

7. LIST OF ABBREVIATIONS

ADSA

ADSA-P

ADSA-CB

ADSA-CD

ADSA-MD

ADSA-D

ALFI

APF

BC

BVP

BSA

COLSYS

DMSO

DVERK

FEP

VP

MINPACK

ODE

PET

SDS

Axisymmetric Drop Shape Analysis

Axisymmetric Drop Shape Analysis-Profile

Axisymmetric Drop Shape Analysis-Captive Bubble

Axisymmetric Drop Shape Analysis-Contact Diameter

Axisymmetric Drop Shape Analysis-Maximum Diameter

Axisymmetric Drop Shape Analysis-Diameter

Axisymmetric Liquid-Fluid Interfaces

automated polynomial fit program

boundary conditions

boundary value problem

bovine serum albumin

a numerical library by Ascher et al. (see Ref. 18) dimethyl sulfoxide

a numerical library by Hull et al. (see Ref. 16)

fluroinated ethylene propylene

initial value problem

a numerical library by Mor6 and Wright (see Ref. 19)

ordinary differential equations

polyethylene terephthalate

sodium dodecyl sulfate

Drops and Bubbles in Interfacial Research D. M6bius and R. Miller (Editors) �9 1998 Elsevier Science B.V. All rights reserved.

THE DROP VOLUME TECHNIQUE

139

Reinhard Miller* and Valentin Fainermanw

* Max-Planck-Institut ffir Kolloid- und Grenzfl~ichenforschung, Rudower Chaussee 5,

D- 12489 Berlin-Adlershof, Germany

w Institute of Technical Ecology, blv. Shevchenko 25, Donetsk, 340017, Ukraine

C o n t e n t s

1. Historical development - from a stalagmometer to an automatic device 1.1. The principle of a stalagmometer 1.2. The general principle of a drop volume method 1.3. Lohnstein's theoretical basis for drop volume experiments 2. The drop volume experiment 2.1. Experimental set-up 2.2. Measuring procedures 2.3. The design of a capillary 2.4. Correction factors 2.5. Optimal experimental conditions 2.6. Advantages and disadvantages of instrument designs 3. Hydrodynamics of drop formation 3.1. Experimental evidence 3.2. Experimental corrections 3.3. A simple correction model based on a drop detachment time concept 3.4. Irregularities in drop formation 3.5. Experiments with higher viscous liquids 4. Theory on adsorption at the surface of growing drops 4.1. Approximate solution 4.2. Diffusion theory 5. Experimental results 5.1. Measurements of pure liquids 5.2. Dynamic surface tension of surfactant solutions 5.3. Dynamic interfacial tensions of various systems 5.4. Comparison with other techniques 6. Summary and Conclusions 7. References 8. List of symbols

140

Amongst the numerous methods for measurement of the surface tension of a liquid or the

interfacial tension between two liquids the drop volume method has gained a reputation as a

standard technique [ 1, 2]. Its major advantage is that it can be applied to both liquid/gas and

liquid/liquid interfaces. Although its experimental conditions and theoretical desription are well

established for a standard range of drop formations time it has only been very recently that a

number of peculiarities have been observed and discussed.

The precursor of this method is the so-called stalagmometer method. Basically it consists of

counting the number of drops formed from a definite amount of liquid detaching from a

capillary. This drop number is then compared with values obtained for liquids of known

interfacial tension [3, 4]. The stalagmometer method is still used in many laboratories for initial

estimation of the interfacial tension of liquids.

The theoretical basis of the drop volume method was founded at the beginning of this century

by Theodor Lohnstein [5 - 9]. This theory is the basis for all further refinements which has made

the method into one of the most frequently used techniques over the years. Therefore there are

different automated devices on the market for the study of interfacial tension of various liquid

systems [10-25]. The modern devices have many advantages compared with other commercial

methods: easy handling, easy temperature control in a wide range, applicable at all liquid/fluid

interfaces without any modifications, no disturbing wetting effects as observed in the ring or

plate tensiometry [26].

1. H I S T O R I C A L D E V E L O P M E N T - F R O M A S T A L A G M O M E T E R TO AN A U T O M A T I C D E V I C E

The origin of the drop volume or drop weight method is possibly the experiment of the

pharmacist Tate in 1864 [27]. He tried to use drops as a measure of a liquid volume in order to

dose liquid medicine. As the result of his studies, Tate formulated several laws. Among these

laws the proportionality between the volume of a drop and the capillary radius and the capillary

constant are the most important once. Thus, Tate postulated that the weight W of a drop detaching from a capillary is proportional to the product of capillary radius ro,p and surface

tension 7.

W = 27t r.p7 (1)

This is known as the law of Tate and has been used for a long time to determine the surface

tension of liquids although experiments demonstrated that it was only a rough estimation.

Lohnstein [5 - 9] especially criticised this law and made a series of calculations to establish a

basis for an accurate theory.

141

Lord Rayleigh [28] has already suggested that Eq. (1) has to be modified by introducing a

correction factor F in order to obtain an accurate relationship

W = 2~ ro~?F

where F is a function of r,p and the capillary constant a

(2)

a = X/2~//Apg . (3)

Based on this proposal Lohnstein developed a theory which induced a systematic improvement

of the experimental technique and which is accepted now as the theoretical background.

1.1. THE PRINCIPLE OF A STALAGMOMETER

The stalagmometer can be seen as the most primitive version of the drop volume method. It

allows only a very rough estimate of the surface tension of a liquid. The principle is depicted

schematically in Fig. 1.

2

Fig. 1 Schematic of a stalagmometer; M1 and M2 are marks, L container with liquid, C capillary

The container of the stalagmometer made from glass is filled with the liquid under study. The

lower part consists of a narrow capillary in order to control the liquid flow velocity. When the

liquid flows out of the instrument, the number of drops formed at the capillary end are counted,

142

during this time the liquid level in the container is noted between the two marks M1 and M2.

The surface tension y is then obtained from a simple relationship

y n~y~p~

np (4)

where n and p are the number of drops and the density of the liquid under study, and 3's, ns, and

p~ are the respective parameters of a known standard liquid. It is easy to check that this

principle allows only an estimate of the surface tension. When using different standard liquids,

the obtained surface tension will change significantly.

1.2. THE GENERAL PMNCIPLE OF A DROP VOLUTt4E METHOD

With the drop volume technique an accurate determination of the volume of a drop formed at

the tip of a given capillary is obtained. The measuring procedure is realised by means of a

precise dosing system which forms drops continuously at the capillary. In Fig. 2 the subsequent

stages of this process is shown schematically.

From stage 1 to 3 the drop grows until it becomes unstable (stage 4) and detaches. The

detachment process is completed at stage 5 and the entire process of drop formation starts

again. Due to the force balance between the acceleration due to gravity and interfacial tension,

the critical drop volume correlates directly with the interfacial tension and the density difference

Ap of the two adjacent phases and is given by

27trcap~, N VApg" (5)

The factor 27trcap is the circumfei'ence of the drop where the interfacial tension y acts and

counterbalances the force VApg. As the drop does not detach directly at the tip of the capillary

but at its neck, Eq. (5) needs to be corrected. An illustration of about how this correction has to

be made is given in Fig. 3 [29].

Along the line BB ~ the drop detaches and the diameter at this location has to be used for the

force balance. Finally the following relationship is obtained with the help of which the interfacial

tension can be calculated from the measured drop volume:

Aogv ~/= 27t r~p f (6)

9

where f is a correction factor specified later (cf. paragraph 2.5.).

Fig. 2

C)

J

Schematic of subsequent stages of a drop formation and detachment

1

143

Fig. 3 Schematic of a drop in the moment of detachment

Over a long time the drop volume principle was practised via the stalagmometer method [4],

while first attempts of automation were undertaken by use of a light barrier for counting the

drops. Until recently the majority of apparatuses in many scientific labs were self-made set-ups,

with some even semi- automated versions designed by several authors [14, 18, 19]. Now

besides various individual laboratory designs, two main automated commercial apparatus are

144

available based on the advantageous drop volume principle, the TVT1 from LAUDA/Germany

and the DVT10 from KrOss/Germany. The advantages and disadvantages of the two instruments

will be discussed in paragraph 2.6.

1.3. LOHNSTEIN'S THEORETICAL BASIS FOR DROP VOLUME EXPERIMENTS

An important step in the historical development from a simple stalagmometer to the drop

volume method as a modern automated instrument of today was made at the turn of this

century, mainly by the German physicist Theodor Lohnstein.

In his paper of 1906 Lohnstein [5] a basic understanding of the drop volume method and a

criticism of the so-called law of Tate are given. The law of Tare is the basis of the calculations

of the surface tension from stalagmometer experiments and given by Eq. (4). He made detailed

calculations of the volume of detaching and residual drops as a function of the capillary radius

and the capillary constant a and found systematic deviations from the law of Tate.

Lohnstein also criticised the out of date explanations given at that time in the most famous

Textbook on Practical Physics of Kohlrausch [30]. In the same year Kohlrausch [31] gave a

response to the criticism of Lohnstein which contained several contradictions. This however

made Lohnstein to publish a second supplement (still in the same year 1906) to explain his

theory again in detail [7]. He first apologises in his paper for any comments Kohlrausch could

have understood as criticism: "Es hat mir nat0dich durchaus fern gelegen, an der Darstellung,

die Hr. Kohlrausch in seinem Lehrbuch der Praktischen Physik dem Thema widmet, eine

irgendwie pers6nlich gemeinte Kritik Oben zu wollen;...". However, Lohnstein again presents

new accurate calculations to support his theory of the drop volume method and compares these

results with experimental data from literature.

Nowadays such a lively and almost real time discussion in a scientific journal is hard to imagine

due to the more and more increasing time taken for publication of an article. Moreover the

subjective argumentation and the conversation like style of papers makes reading these historical

sources worthwhile. It is unfortunate for the scientific community that many papers at that time

had been published in German, the main scientific language at that time. Thus it is the privilege

of German speaking colleagues to enjoy the art of scientific discussion common at the turn of

the century.

2. THE DROP VOLUME EXPERIMENT

The design of an instrument based on the drop volume principle is comparatively simple and can

be organised in a modem laboratory in less than one day. For a manual seVup the only thing one

145

needs is an accurate syringe equipped with a needle of a certain diameter and design. This

syringe then needs to be mounted somehow vertically, and the experiment can immediately

begins.

Fig. 4 Manual design of a drop volume method

From knowledge of the volume of a drop detaching from the tip of the needle, i.e. from reading

the syringe the surface tension of the liquid under study can be calculated. The commercial

instruments are based on this principle, with the reading of the drop volume being made

automatically.

2.1. EXPERIMENTAL SEP-UP

As mentioned before there are a wide variety of designs of instruments based on the drop

volume principle. The important part of any instrument is an accurate volume determination,

typically arranged by a dosing system of constant liquid flow and an accurate timer. If one

knows the dosing rate and the time for the formation of a drop, the volume can be calculated

accurately.

In Fig. 5 the principle of a drop volume apparatus is shown as an example. The motor

controller-encoder system linked with the syringe provides a constant and accurate dosing rate

while the light barrier is used to detect each detaching drop. Thus the time in between two drop

signals multiplied by the dosing rate gives the drop volumes. The dosing system is linked via an

146

interface to the serial port of a PC. The PC software controls complete measurement programs,

i.e. drop volume measurements for a liquid can be performed at different dosing rates. After

each measurement the surface tension as a function of time is calculated and plotted as a graph.

3

4

pulges

pulses

Fig. 5 Principle of an automated drop volume instrument, according to the TVT1 of LAUDA, Germany,

1 - capillary, 2 - syringe, 3 and 4 motor controller-encoder system, 5 - drop, 6 - temperature control

jacket, 7 - light barrier, 8 - electronic interface, 9 - IBM PC

The other main type of automated instruments, the DVT10 of Kr0ss, Germany, is designed in a

similar way but it is controlled by a special microprocessor system independent of any PC.

Further data interpretation, however requires typing surface tension and time values into a

computer, which can be a very tedious.

2.2. MEASURING PROCEDURES (DYNAMIC, Q UASISTA TIC)

Three different measurement modes can be employed with the drop volume method which can

yield different data. However, when all peculiarities of each measuring procedure are

considered, the results obtained by the different procedures should be the same. This will be

demonstrated in paragraph 5.

2.2.1. Dynamic measuring mode

The dynamic version of the drop volume method is the classical procedure for the measurement

of surface and interfacial tensions of pure liquids. This mode consists of creating a continous

formation of drops at the tip of a capillary by means of an accurate dosing system. The

interfacial tension is calculated from the average volume measured for several subsequent drops.

147

From experimental experience it has been noted that the first two drops should not be used for

the calculation of the average volume as their volumes could be affected by uncontrolled

conditions prior to the start of the measurement. Beginning with the third drop the drop

volumes are registered and finally the average volume, the interfacial tension and the standard

deviation can be calculated.

By tuning the dosing rate it is possible to study the system at different ages of the interface. This

is of importance for systems containing surface active substances such as surfactants and

proteins. As such compounds start to adsorb at a freshly formed interface the interfacial tension

is a function of the drop formation time. The theoretical basis of this adsorption process at the

surface of a growing drop will be discussed in paragraph 4.

The interfacial tension of pure liquids is independent of the drop formation time. Nevertheless,

at small drop times a systematic increase in the obtained interfacial tensions can be observed.

This is caused by the dynamic conditions of the measuring procedure and must be corrected. In

paragraph 3 these necessary corrections will be discussed.

2.2.2. Quasi-static measuring mode

The quasi-static mode is suitable for the study of solutions of surface active agents, such as

surfactants, biosurfactants, artificial and natural polymers. The principle is based on the

adsorption process of surface active compounds at the interface leading to a continuous

decrease in interfacial tension. If a drop is formed in such a way that its drop volume is small

enough for the moment the drop will remain at the tip of the capillary. While keeping now this

drop volume constant the interfacial tension decreases due to increasing adsorption and after a

certain period of time it will reach a value which corresponds to the critical volume. Now the

drop detaches. Of course, if the initial drop volume was set too small it never corresponds to a

critical value within the available surface tension interval and the drop will never detach. A

certain strategy and experience is needed for manual performance of such experiments, however

some software have implemented this procedure and hence provide an automatic strategy.

When the drop falls off the time interval between drop formation and detachment can be

registered and the next drop can be formed in the same way. If the time for drop formation is

small compared to the total lifetime of the drop the whole procedure can be called quasi-static.

This means during the adsorption process the area of the interface was constant over the almost

entire time interval. This is shown schematically in Fig. 6.

V

148

time

Fig. 6 Schematic of the drop formation in a quasi-static measuring mode

In case a the time for the initial formation of the drop is negligible compared with the total life

time of the drop, while in case b it is of the same order and the experiment cannot be called

quasi-static. The lifetime in a quasi-static experiment approximately corresponds to the

effective age of the interface.

Due to our knowledge this measuring procedure was first applied by Addison and co-workers

[32] and later used by other authors [11, 15, 33-36] to determine the adsorption kinetics of

surfactants without the need of considering an area change.

This measuring mode is only applicable to systems containing surface active substances,

otherwise a decrease in interfacial tension with increasing drop age cannot be expected. A

successful application of this procedure to "pure" solvents can be explained only by an

unsufficient grade of purity of the solvent. Surface active impurities present in the solvent act

in the system like surfactants and therefore the quasi-static measurements succeed. This case is

in general not desirable.

2.2.3. Static measuring mode

In addition to the two already mentioned measuring modes there is a so-called "static" mode. In

this version the drop formation consists of different stages. At the beginning the drop is formed

with a rather high speed. Then the dosing rate is decreased with increasing drop volume. By

this measuring mode interfacial tensions can be obtained in a large time interval without

hydrodynamic influences because the final dosing rate just before the drop detachment is always

very low. An interpretation of the resulting data however is very difficult because of lack of any

149

appropriate theory. Nevertheless a good approximation can be made: the static mode produces

interfacial tensions on a time scale of the effective surface age which is in between those of the

dynamic and the quasi-static version. Thus taking 50% of the drop formation time as the

effective age is a good approximation.

The three measurements modes of the drop volume technique are implemented in the so,ware

of the TVT1 from LAUDA. In the classical dynamic version the drop time can be chosen by

selecting the appropriate dosing rate. In the quasi-static mode a large variety of selections are

available. In its present version the sottware of the TVT1 provides the surface tension as a

function of drop formation time and does not recalculate the effective surface age nor corrects

the hydrodynamic effects. The DVT-10 from KRrOSS allows for the classical dynamic version in

a large interval of drop formation time.

2.3. THE DESIGN OFA CAPILLARY

The capillaries play a major roll on the performance of measurements. The cylindrical shape for

the capillary has proved to be the most suitable. The material has to be chosen so that it is

wetted by the dosed liquid within the syringe. In that case one can be sure the drop is formed at the outer circumference of the capillary and the outer diameter can be taken as 2 r , p (Fig. 7).

Fig. 7 Wetting conditions at the tip of capillaries with different shape

It has to be ensured that the drop is not formed at the inner circumference or somewhere in

between which can happen under certain wetting conditions. Taking capillaries with conical

shape an uncontrolled formation of the drop can happen under certain conditions again and the effective radius roap where the drop detaches is unknown. Therefore, capillaries with a conical

shape should be avoided.

150

2.4. CORRECTION FACTORS

Calculation of the interfacial tension from a measured drop volume can be performed via Eq. (8)

if the correction factor f is known. The shape and size of a drop is controlled by the capillary radius r,p, the drop volume V, the interfadal tension % the difference Ap of the two fluids, and

the acceleration due to gravity g. Therefore, the correction factor will depend on these

parameters. From the Gauss-Laplace-Equation it becomes evident that it is the capillary

constant a which effects the correction factor more.

Lohnstein [5-9] calculated the correction factors f as a function of a and r,p first. His physical

model considered the residual drop after detachment as having the same contact angle as the

drop before it detaches. In later work Freud and Harkins [37] and Hartland and Srinivasan [38]

demonstrated the accuracy of Lohnstein's calculations and improved them.

Scientists nowadays may wonder why comparatively large differences between the calculated

data appeared. To understand this, one has to take into consideration that Lohnstein had no

computer at his disposal but made all the calculations, the numerical solution of a set of non-

linear differential equations, by hand supported only by some co-workers. Then the obtained

accuracy of Lohnstein's results become impressive. Hartland for his recalculations used a

computer, most modern at that time. Today, such calculations can be performed on a highly

efficient PC, having a performance orders of magnitude higher than the "supercomputers" 20

years ago.

Hartland specified the interval where Lohnstein's calculations were accurate. He also analysed

how deviations of Lohnstein's calculations from experimental data of Harkins and Brown [39]

and from Hartland's recalculations can be explained. Hartland showed that for values of r,p / a > 0.7 the assumption of equal contact angles of the drop before and aRer the detachment

is not sufficient to describe the situation. In addition to the surface tension force, an excess

pressure force has to be considered in order to calculate accurate correction factors.

Wilkinson [40] discussed several approaches for obtaining accurate correction factors and showed that the correction factors can be presented in the form r,p / a = f(r,p / V~/3). Based on

experimental data [37, 39] Wilkinson expressed r,p / a by the following polynomial :

lea p //a - z(A + z(B + z(C + zD))) + E

with

(7)

z = r.p / V 1/3, (8)

and the values for the coeffidents A to E:

151

A = 0.50832, B = 1.5257, C = -1.2462, D = 0.60642, E = -0.0115. (9)

A table which summarises the values of reap / a = f(r~p / V 1/3) calculated from Eq. (7) in the

interval 0.0267 < r , p / a < 1.465 are given in different papers and books [40 - 42]. Its

calculation is easy on a PC using the polynom of Eq. (7). Combining Eqs (7) and (8) the

interfacial tension can be calculated:

V = a2Apg/2. (10)

The sotlware of commercial drop volume instruments calculates the interfacial tension in the

following way. From the averaged drop volume V (or the volume of a single drop) first the value z = r,p / V 1/3 is calculated and then via Eq. (7) the value of r~ap / a. As rca p is known the

value for a and finally via Eq. (10) ~/is obtained.

A more detailed discussion of the corrections factors and the analysis of the force balances

acting on a drop in the moment of detachment has been given by Rusanov and Prokhorov [41 ].

2.5. OPTIMAL EXPERIMENTAL CONDITIONS

In the past measurements of high accuracy required capillaries with large tip radii rr [39]. This

of course allows for large drop volumes which can be measured with greater accuracy. However

this is not the most advantageous range for drop volume measurements as was demonstrated by

Earnshaw et al. [43 ] very recently in a first detailed error analysis.

The relative uncertainty of the interfacial tension c~ /3' is given by (when the covalence terms

are neglected)

- -X-0-0J + I+V + l+3v , , (11)

where t~ i are the uncertainties on the parameters reap, A o and V, respectively. These

experimental uncertainties are known and can be given easily. However, the second and third

terms on the fight hand side of the relationship which contain the correction factor f and its

derivative f ' are a function of reap/V 1/3 and exist only as experimental data or as numerical

values from theoretical calculations as discussed in the previous paragraph. Thus the overall accuracy mainly depends on the fight choice of the radius r,p such that the total relative

uncertainty is minimum.

The error analysis of the whole problem has been made by Earnshaw et al. [43] in great detail

and was based on the experimental data of Harkins and Brown [39] and Wilkinson [40]. Of

course tha analysis could have been based on the theoretical calculations made by Lohnstein

152

[5, 6] and later improved by Hartland and Srinivasan [38] but this would lead to the same final

conclusions. Due to the structure of the terms in Eq. (11) the highest accuracy can be reached

reap f ' , ~ m <0 This will reduce the second and third term to a when they are negative, i.e. VV 3 f .

minimum. Thus the most appropriate interval for drop volume experiments would be

re,p VV 3 <0.85 This is different from the recommendation made by Harkins and Brown, who

reap suggested 0.6 < ~ < 1.2, due to the small variation in the correction factor f in this interval.

To translate the results obtained by Earnshaw et al. into experimental dimensions, optimum

conditions are fulfilled for surface tension measurements of aqueous surfactant solutions using a capillary with a tip radius roap <0.2 cm. For reap =0.3 cmhowever, at surface tensions

~/< 40mN / m the value reap /V1/3 > 0.85, i.e. beyond the optimal region. For interfacial tension

measurements for example at the water/alkane interface (A9 ~ 0.3g/cm 3 ) optimal conditions

are reached even for larger tip radii. Only for extremely low interfacial tensions of few mN/m

the optimal range is left with the larger capillaries. The less the density difference is, the larger is

the volume of a drop for a certain capillary, and hence the smaller is the value of reap / VV3.

Although the optimal range of the method can be established by very small tip radii easily there

is of course a limit given by the accuracy of volume measurements. Thus the use of capillaries of

too small radii is not recommended either [44].

2.6. ADVANTAGES AND DISADVANTAGES

The commercial versions of the drop volume methodadvantages of have a lot of advantages in

comparison with other surface tension methods. First of all this method is characterised by easy

handling and easy temperature control in a wide range. Its accuracy is comparatively high,

especially when applied to liquid/liquid interfaces, which is possible without any modifications.

There are also no disturbing wetting effects, as it is observed in the ring or plate tensiometry

[26].

Another important advantage of the drop volume method is only a small amount of solvent and

solute are necessary for the performance of measurement. Often special surfactants are available

only in "homeopathic" amounts, especially in case of biosurfactants or those surfactants

synthesised in a lab on a small scale. This method is suitable for dynamic adsorption studies

where interfacial tension as a function of interfacial age from parts of a second up to about half

an hour are available.

153

However, equilibrium interfacial tension values are not available if the adsorption of a surfactant

is very slow and takes several hours [45]. Also, at small drop times the so-called hydrodynamic

effect has to be taken into consideration, which requires additional experiments. This procedure

is explained in detail in the subsequent paragraph.

Another disadvantage was observed in the studies of irregularities of drop formation at very

short drop formation times - only the average of the volume of several drops is available. To

detect the peculiarities discussed below (cf. paragraph 3.4) the volumes of single drops had to

be determined and only by designing a special experimental set-up could these observations be

made. Using a standard commercial instrument the experiments produced highly scattering data,

for which the reason was completely unclear.

3. HYDRODYNAMICS OF DROP FORMATION

The principle of the drop volume method is dynamic in character and therefore used frequently

for rather small adsorption times. However, if used at drop formation times of less than say 10

seconds so-called hydrodynamic effects come into play and have to be taken into consideration

in order to obtain real dynamic surface tensions. These hydrodynamic effects have been

observed by different authors [4, 46- 50] and lead to significant changes in the measured drop

volumes and hence in the surface tension values.

Davies and Rideal [49] discussed two main effects influencing the formation of a drop at and its

detachment from the tip of a capillary, the so-called "blow-up" and "circular current" effects

blow-up circular current. The first effect increases the detaching drop volume and simulates a

higher surface tension while the second effect leads to an earlier break-off of the drop and

results in an opposite effect. However, the two effects act in different time windows, as it was

discussed in [42]. While the "circular current" leads to an early drop break-off at drop times less

than 1 second, the blow-up effect is effective up to 10 seconds of drop formation time.

3.1. EXPERIMENTAL EVIDENCE

The surface or interfacial tension of pure liquids should be independent of time, except in a time

range where molecular rearrangements are possible. For water this should be well below

10 .6 seconds. Thus, drop volume experiments have to yield drop volumes for water constant

over the entire drop formation time interval available. This however is not the case and

significant deviations occur as demonstrated in Fig. 8.

154

76

75

~ '74

73

72

71

% 0

~ 4~IG o 0 t~$~**e-~ ~ 6 [] ,

I I I I I

0 10 20 30 40 5O

t[s]

Fig. 8 Measured apparent surface tensions of pure water using capillaries with different tip radii; rm,= 0.254 mm (~), 0.504 mm (Fl), 0.633 nun (O), 1.051 mm ('~'), 1.322 nun (O), 1.701 mm (O)

The measured drop volumes have been used to calculate the surface tension via Eq. (6). There is

an increase in surface tension at shorter drop times which is much larger than the experimental

accuracy. The larger the tip radius the larger is the deviation from the expected constant value

(dotted line). In order to estimate the true surface tension values the drop volumes have to be

corrected. Several attempts have been made to perform this task and are published in the

literature.

3.2. EXPERIMENTAL CORRECTIONS

To quantify the hydrodynamic effect Kloubek et al. [4] measured the drop volume of pure

liquids at small drop formation times and presented an empirical relationship for V(t)

V(t) = V e + k / t (12)

The uneffected drop volume Vo can be obtained from fitting experimental data with this

equation while K is an empirical constant. As an example data for pure water are given in Fig. 9.

The intersection of the line with the ordinate yields the drop volume Vo which corresponds to a

volume not effected by the hydrodynamics and k is the slope of the line.

Another result was obtained by Jho and Burke [46]. Based on a large number of experiments

with pure liquids they found a dependence of the drop volume on the drop formation time in the

following form:

155

V( 0 -- We _[_ kt -3/4 (13)

where V e is again the uneffected drop volume and k is a regression coefficient obtained by a

linear regression of the experimental data. They found that k depends on surface tension, density

difference, and tip radius, but an effect of the viscosity of the liquid was not obtained. The

results were confirmed by similar experiments performed by van Hunsel [47]. In all the works

no physical model has been presented to explain the statistically proved relationship.

36-

35,5

35

34,5

f

I I I I I

0,1 0,2 0,3 0,4 0,5

1/t [l/s]

Fig. 9 Plot V as a function of 1/t according to Kloubek et al. [4] to determine the hydrodynamic correction

for the drop volume of pure water; rc~ = 1.051 mm

3. 3. A SIMPLE CORRECTION MODEL BASED ON A DROP DETACHMENT TIME CONCEPT

The analysis of the complete process of drop formation and detachment shows that a drop after

is has reached its critical volume and starts to detach is for a certain time still connected with the

liquid flow through a liquid bridge. During this time, necessary for the act of detachment itself, a

certain amount of liquid flows additionally into the detaching drop. A schematic of the flow

pattern of a detaching drop from a capillary is shown in Fig. 10.

From this it becomes evident that the detaching drop volume is always larger than the critical

volume. Under the conditions that the act of detachment happens under laminar conditions and

neglecting of "circular current" effects can be made the additional volume should be

proportional to the dosing rate. This is also reflected by the empirical formula given in [46].

156

Fig. 10 Schematic of the flow of liquid into a detaching drop

From the given physical model the drop volume Va pumped additionally into the detaching drop during the time t o (detachment time)can be calculated from

Va = to Q, (14)

where Q is the liquid flow rate. According to the experimental conditions the flow rate is given

by

q = v e/(t-to) = v(t)/t, (15)

where V(t) is the drop volume measured at a drop formation time t. From Eqs (14) and (15) the

following expression is finally obtained [48]

V(t) - V~ (1 + to/(t-to) ) = V~ t / (t-to). (16)

V~ is the undisturbed or corrected volume which has to be used for the calculation of interfacial

tensions. As the presented physical picture also holds for surfactant solutions, the hydrodynamic

effect has to be corrected in the same way.

The effect of the capillary radius r~p on measurements of surface tensions of water was shown

in Fig. 8. The bigger the capillary radius the more pronounced is the apparent increase in surface

tension. The analysis shows that the drop detachment time t o is a linear function of the capillary radius r,p. By fitting the data of Fig. 8 to the relation for V(t) given by Eq. (16), the drop

detachment time as a function of re, p can be calculated. The results of the fitting process are

presented in Fig. 11.

157

74,5 O Q

O

O

o o o

El

~ 7 3

O

72,5 o o

�9 B �9 �9 O

71,5 * I I -

5 10 15

13 I"1 ffl

O

�9 �9 u �9

- - - - - + J t I I

20 25 30 35 40 t Is]

Fig. 11 Fitting of Eq. (16) to measured apparent surface tensions of pure water at small drop formation time

using capillaries with different tip radii:

reap = 0.254 mm (11), 0.504 mm ([5), 0.633 mm (~), 1.051 mm (~), 1.322 mm (0) , 1.701 mm ( 0 )

A linear regression of these data leads to the following equation, from which the drop

detachment time to can be calculated [48]

t o =cz + [3reap. (17)

Further studies have shown that the absolute value of surface tension influences the

hydrodynamic effect and consequently t o. This effect also described in [48] is however not so

large and can be neglected in most cases. The data given in Fig. 12 in the form of V(t) / Vo as a

function of 1/t demonstrates this statement.

Although the surface tension of the studied systems varies from 72.6 mN/m for pure water to

22 mN/m for pure ethanol the slope of the curves, representing approximately the value for to

is almost constant. Consequently, as a good approximation for liquids having a viscosity

p < 5 mm2/s the parameters c~ and 13 have the following values:

oc = 0.008 s and 13 = 0.041 s/mm. (18)

Actually, these kind of experiments have to be made prior to adsorption kinetics studies with

systems different from a water/air interface. If one intends for example to study dynamics of

adsorption at a water/liquid interface the kind of dependencies as shown in Fig. 12 have to be

158

measured (cf. for example [50]). It is very important that one does not simply apply a

correlation like Eq. (12) or (13) to experiments in a completely different interracial tension

range. From the data in Fig. 12 it becomes clear that only in a dimensionless plot the slope is

constant while the parameters in Eqs. (12) and (13) do depend on the surface tension.

Compared with Eq. (12) the slope of the data in Fig. 12 corresponds to k = Vot o.

1,1

1,08

1,06

~ 1,04

1 , 0 2

1

0,98 0

[]

I I I I

5O 100 150 2OO

t (s)

Fig. 12 Relative drop volume V(t) / V c as a function of 1/t for different ethanol/water mixtures measured at

20~ rca p = 0. ram; ethanol content = 5% (m), 10% (El), 15% (0), 20% (+)

The correction for the hydrodynamic effect in surface tension measurements has not been

implemented in the software of either of the commercial instruments yet. The user thus has to

correct the data afterwards. Although the software provide surface or interracial tension values

for such a correction the drop volume measured has to be corrected by using the above

proposed procedure, and then, using the right correction factor f, the tension y can be

calculated.

This whole procedure is quite time consuming and requires a small programme on a PC.

However, as an acceptable approximate one can also start with the tension value calculated

uncorrected for the hydrodynamic effect. This value can be multiplied then by the factor

Vo _ 1 _ t o 1 a+13rcaP V(t) t t (19)

to obtain the corrected surface tension. This procedure assumes that the correction factor f does

not change significantly in the interval [Vo, V(t)].

159

The act of detachment until the total separation of the drop from the residual liquid is controlled

by a liquid flow process and should be influenced also by the liquid's viscosity. This effect will

be discussed separately below.

3.4. IRREGUZAMTIES IN DROP FORMATION

At even smaller drop times, shorter than the range of the "blow-up" effect which in general

increases the volume of detaching drops, other phenomena can be observed. In this range

irregularities can be observed which appe~x to be of chaotic character. More careful studies

showed that these chaotic irregularities are rather regular and highly reproducible. To study

these effects neither of the available commercial instruments are applicable as the volume of

single drops cannot be measured with a sufficiently high accuracy. For this reason Fainerman

and Miller [51] designed a special apparatus with which single drops at very small drop

formation time could be studied (Fig. 13).

Fig. 13 Schematic of a drop volume method designed for fast drop formation experiments; 1 - liquid

container,

2 - liquid flow capillary, 3 - capillary, 4 - light barrier, 5 - electronic interface, 6 - computer

Using a container for the liquid under study (1) and a flow capillary of a certain length (2) the

drop formation time can be controlled over a wide range and allows also very fast drop

formations. The accurate time measurements for the drop formation was arranged with a light

barrier (4) and a specially designed electronic interface (5) as the computer clock is not accurate

enough. Keeping the container at a constant height the dosing rate of the liquid to the capillary

(3) changes very slowly as the liquid level decreases. The larger the container cross section the

slower is the change in the dosing rate.

160

0,78

0,76

0,74 rao

~. 0,72

0,7

0,68

0,66

Fig. 14

0 50 100 150 200 250 300 350 400 450 500

Measured drop times as a function of the drop number obtained with a continuously changing flow rate;

5-10 -6 mol/cm 3 SDS, container diameter 60 mm, tip diameter 7.8 mm; according to [51]

r

0,34

0,33 t I I I I I

210 215 220 225 230 235 240

Fig. 15 Periodic pattern of measured drop times as a function of drop number at a constant flow rate; 0.5 g/1

Triton X-165, container diameter 60 mm, tip diameter 7.8 mm; according to [51]

The results for a solution of SDS (sodium dodecyl sulphate - the possibly most frequently

studied surfactant world-wide) are shown in Fig. 14. A linear dependence of drop t ime against

drop number was expected (number of subsequent drops).

As a first impression the results look chaotic, as emphasised in the beginning. However, there

are ranges of comparatively regular drop time changes between ranges of high "scatter".

161

Analysis of these ranges of high scatter shows that there is a kind of bifurcation - altemating

larger and smaller drops. Reconstructing exactly the same conditions the whole pattern can be

repeated, as it was demonstrated in [51 ]. In some cases complex pattern can even be obtained.

Such case is shown in Fig. 15, obtained for a Triton X-165 solution.

As one can see, the sequence of smaller and larger drop from drop number 212 trough 217 is

exactly repeated by drop number 218 through 223 and again 224 through 229. After this

another pattern starts which has not such a sequence of 6 drops with changing volumes

following a definite but unknown law.

Fig. 16 Schematic of the situation of drop detachment from a capillary; AA' and BB' are locations of possible

drop separation

The physical reason for the drop time or drop volume bifurcations could be the following.

There are capillary waves at the residual drop which are not damped out completely before

the next drop is ready to detach. In the moment of drop detachment, oscillations of the liquid

bridge between the residing and the detaching drop exist, as shown schematically in Fig. 16.

Thus the break-off process of the liquid bridge does not take place at the line A-A', as

expected under slow flow conditions due to the balance of capillary and gravity forces, but

rather at the place B-B' where the fastest development of the modes of the capillary waves

are located.

As a results of the presented findings we can conclude that measurements in a drop time region

of such irregularities are not useful. Such regions are however difficult to detect with the

existing commercial instruments. Only a standard deviation of the measured data higher than

162

usual are a hint of such a range. Measurements in this range should be avoided, except one

wants to study effects like those reported here

3.5. EXPERIMENTS WITH HIGHER VISCOUS LIQUIDS

Literature data on the influence of liquid viscosity on the hydrodynamics of drop detachment

allows only qualitative speculation. The first results of a systematic study of liquids over a large

viscosity interval (up to several hundred mmVs) display remarkable features [52]. These effects

show clearly that at high viscosities the uncertainty of measurements rises up and data for short

adsorption times are less accurate. However, the general physical picture drawn in paragraph

3.3 remains valid. As an example the drop volumes measured for water-glycerol mixtures as a

function of 1/t are depicted in Fig. 17.

0,028

0,026 ~ I ~ t I I i I

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7

1/t [l/s]

Fig. 17 Drop volumes as a function of 1/t measured for 75% glycerol in water; TVT1 (11), drop volume

instrument described in Fig. 13 (Tq)

In [52] it is shown that at sufficiently low viscosities the dependence of drop volume on drop

time is quite regular and yields drop detachment times comparable to those for water. With

increasing viscosity the hydrodynamic effect increases and levels off. This tendency was

observed with capillaries of larger and smaller tip radii. Therefore it can be concluded as being

a genuine one [52] and a formalism is given how to consider tip radius reap and viscosity of the

liquid in the correction of measured drop volumes.

For liquid/liquid interfaces no systematic results exist so far on the effect of liquids' viscosities

which can be generalised. It is obvious that t o strongly depends on the viscosities and at the same

time on other conditions of a liquid/liquid interface (density difference, interfacial tension).

For the time being it is recommended to perform measurements of such systems only for drop

163

times t > 30 s as at these drop times hydrodynamic effects can be avoided under most practical

measuring conditions. If one needs data on small adsorption times the hydrodynamic effects

have to be studied in detail, following the instructions given above.

4. THEORY OF SURFACTANT ADSORPTION AT THE SURFACE OF GROWING DROPS

Drop volume experiments are extremely suitable for adsorption kinetics studies of solutions

containing surface active agents. Solutions of amphiphilic agents, classical surfactants as well as

biosurfactants, soluble in water or other solvents, and macromolecules of natural or artificial

origin can be investigated in order to determine their adsorption mechanisms.

A quantitative data analysis of drop volume experiments requires quantitative theories of the

adsorption process at a surface at rest (quasi-static version) or at the surface of a growing drop.

A similar problem has to be solved also in other experiments based for example on growing

drops or bubbles, such as bubble and drop pressure measurements with continuous, harmonic or

transient area changes [42].

A theoretical model of the adsorption kinetics of amphiphilic molecules at fluid interfaces, for

example surfactants at the aqueous solution/air or solution/organic solvent interface, can be

easily formulated. The first physically founded model for interfaces with constant interfacial area

was derived by Ward and Tordai [53]. It is based on the assumption that the time dependence of

interfacial tension, which can be directly correlated to the interfacial concentration F of the

adsorbing molecules, is caused by the transport of molecules from the bulk to the interface. In

absence of any convection or external disturbances this transport is controlled by diffusion. The

result of the so-called diffusion controlled adsorption kinetics model has the following form

[53]: ]1/2 - c(O, t-x) dffx ], (20) r(t) - 2 [c~ X/i 0

where D is the diffusion coefficient and c o is the surfactant bulk concentration. This integral equation describes the change of F(t) as a function of time t, and bulk concentration c o. c(0, t)

is the bulk concentration close to the surface, the so-called subsurface concentration. The

application of Eq. (20) to dynamic surface tensions 7(0 requires numerical calculations [54, 55].

A simple equation derived by Sutherland [56] can be used as a first approximation but its range

of application is very limited [57].

The models of adsorption kinetics mentioned are only valid for interfaces with constant

interfacial area. This is fulfilled only in the quasi-static mode of the drop volume method after

164

the stage of fast drop formation. In the classical, dynamic mode in which the drops are

continuously growing until detachment, this condition is not fulfilled and the change of the drop

surface area as well as the flow inside the drop has to be taken into consideration. The first

complete physical model considering a radial flow inside the drop was derived by Pearson and

Whittaker [58]. In analogy to Eq. (21) given by Ward and Tordai [53] the following integral

equation can be derived [59]:

0 ;/7t7/3 (of 3

(~ t 7/3 -- ~,) 1/2

d;~. (21)

A numerical analysis of this rather complex integral equation showed that the rate of adsorption

at the surface of a growing drop with a linear volume increase, as it is the case in drop volume

experiments, is about 1/3 of that at a surface with constant area [59]. This can be seen from the

dependencies given in Fig. 18 which represent the results calculated from Eqs (20) and (21)

using the same adsorption parameters.

1.0

0.5

i / " i /

t /

6 4

Fig. 18 F(t)/F0 -dependencies calculated from Eq. (20) (dotted line) and Eq. (21) (solid line); according to

[59]

From experience of adsorption kinetics studies, the approximation for the effective age of 1/3 of

the drop formation time is sufficiently accurate to interpret dynamic interfacial tensions [60-64].

This means also that the same approximation 1/3 can be acceptable for other kinetic models. In

the subsequent paragraph some experimental examples will be given to demonstrate the validity

of this approximation.

165

4.1. APPROXIMATE SOLUTIONS

The use of an equation as complex as Eq. (21) requires a lot of numeric approximate solutions

are very favourable. The first model to describe the adsorption at the surface of a growing drop

was derived by Ilkovic in 1938 [65]. The boundary conditions were chosen such that the model

corresponded to a mercury drop in a polarography experiment. These condition however are not

suitable for describing the adsorption of surfactants at a liquid drop surface.

In the late forties Addison and co-worker [34] made the attempt to take into consideration the

area change caused by the drop growth. This procedure was a semi-empirical process, and

consisted in a step-wise calculation of F(t) from the equation of Ward and Tordai (20) in a

certain time interval, and a stepwise correction of the surface coverage inversely proportional to

the area increase in the same time interval. This way of area change consideration however does

not take into account any flow in the bulk phase and hence is only an estimate. A similar

procedure was later elaborated by Kloubek [63]. In both cases, the trend to a thinner diffusion

layer due to the flow inside the drop was neglected, and the data obtained overestimated the

effect of the growing drop area [42].

Delahay et al. [61, 62] used the theory ofllkovic and derived an approximation suitable for the

description of adsorption kinetics at a growing drop. The relationship was derived only for the

initial period of the adsorption process

F(t) = 2% 4 3Dt 77t (22)

This relationship already indicates a correlation between the rate of adsorption at a growing

drop surface and a stationary interface: the adsorption at a growing drop surface is 3/7 times

slower. This result is close to 1/3 discussed before.

4. 2. DIFFUSION THEORY

A first attempt to present a complete description of the adsorption process at a growing drop

surface was made by Pierson and Whittaker [58] who presented the initial and boundary

condition problem based on a diffusion as well as mixed diffusion-kinetic-controlled adsorption

mechanism. The numerical evaluation of the problem however was not satisfactory and in

contrast to experimental findings. Even the simple approximation of Eq. (22) was not

reproduced.

The diffusion equation for describing the transport inside or outside a spherical drop or bubble

has the following form:

166

0c 0c _.0'c _2ac) - - q - V r -" 1.) ( ~ ' - q- & &- r & "

(23)

For a surfactant solution drop, this diffusion equation applies in the interval 0 < r < R, where R

is the radius of the drop. For a solvent drop or a bubble in a surfactant solution, the diffusion

process happens outside the drop or bubble respectively, and Eq. (23) holds in the interval

R < r < oo. A schematic of the situation is shown in Fig. 19.

diffusion layer compression due to drop growth

" I

diffu~on layer extension due to diffusion

I 8D Fig. 19 Schematic representation of diffusion layer changes inside a drop during adsorption

The drop growth is connected with a flow inside the drop and an area expansion. The flow

inside the drop can be assumed to be radial. Although the area stretching is not totally isotropic

no significant deviations are expected and, therefore, no flow tangential to the interface is

initiated. Nevertheless, the stretching of the interface simultaneously leads to a stretching of the

adjacent liquid layers. As a result, the concentration profile caused by diffusion is compressed

(cf. [42] chapter 4). Two processes then overlap and are directed opposite to each other:

diffusion layer compression due to the enlargement of the drop, and diffusion layer expansion

due to the drop growth accompanied by a dilation of the adsorption layer coverage. All these

processes have to be taken into consideration in current theories.

The radial flow, which is a good approximation of the real flow pattern, is given by [66]

R 2 dR (24) V r r 2 dt

167

In analogy to the Ward & Tordai equation (20) the non-linear integral equation (21) was

derived [59]. Instead of a radial flow a turbulent flow also was considered in reference [67] for

the discussion of kinetic data obtained from dynamic drop volume experiments.

A more advanced theoretical analysis of the adsorption process at growing drop surfaces was

performed by MacLeod & Radke [68]. In contrast to the theory discussed above they do not

assume a point source at the beginning of the process but instead a finite drop size. On the basis

of an arbitrary dependence R on t a theory of diffusion- as well as kinetically-controlled

adsorption was then derived. In addition to the diffusion equation (23) the following boundary

condition is proposed:

OF + D 0 C - F d l n A . . . . r - R(t). (25) 0t Or dt '

The sign of the first term on the right hand side of Eq. (25) depends on whether the diffusion

takes place inside or outside the drop. The final result is given by the equation

i (dc(O' t)) to d l n A dr __(R(t))~~ " dt dt o - F ~ (26) d t - 0 j*t (R(~))4 d~) 1 / 2 dt

t 0

This very complex equation takes into consideration any function of R(t), and hence any A(t),

resulting from experiments with growing drops or bubbles. For a spherical geometry one obtains

h R(t) = ~v + _ (27)

2h 2

A(t) = 27trcap h (28)

where the drop height h as the specific parameter in the two relationships is given by

Z ~ 9 )2 ) 2 ( 3 6 + (Qt+Vo h = x (Qt + v o ) + ( - 1 ) i r ~ i=l

1/3 (29)

Here Q is the liquid volume flow rate, and Vo is the initial volume of the residual drop.

Eq. (26) can be used only via numerical calculations. Some model calculations have been

performed by MacLeod and Radke [68] which demonstrate the suitability of the theory for drop

volume as well as growing drop experiments.

For long drop formation times the initial state of the drop surface does not significantly affect

the dynamic surface or interfacial tension and the assumption of an initially freshly formed

168

surface for the analysis of experimental dynamic drop volume data. In contrast, for fast drop

formations, i.e., far from the equilibrium state, the effect of initial adsorption on dynamic surface

tensions is not only significant, but may even lead to a non-monotonous change of the surface

tension due to the expansion of the drop surface [68].

The fact that at short adsorption time the dynamic interfacial tension for Triton solutions

decreases faster than predicted from the diffusion model for freshly formed surfaces, [24, 69]

can be ascribed to an initial adsorption at the drop surface of the residual drop after drop

detachment. A theoretic model for diffusion controlled adsorption at a fiat surface with initial

load was proposed first by Cini et al. [70] and later generalised to the case of a growing bubble

surface for any values of initial adsorption by Joos and Van Uffelen [71, 72]. The result has the

general form of the equation of Ward and Tordai Eq. (20)

2./-~ f4 7 r(-t)f(t)=r~+ .~/Tjo [Co-C,(X-gQ]d(-~ ) (30)

F d is the initial adsorption values at the drop surface, f(t)= (1 + m) n , o~ and n are constants,

So x - [f(t)] dt, )~ is the integration variable. For only small deviations from equilibrium an

analytical solution of Eq. (30) was derived in [72] which can be applied to growing bubbles and

drops as well [25].

The form of the function fit) in Eq. (30) was suggested by Joos and Van Uffelen [72] to

describe the variation of the area of a growing drop

A ( t ) - A0(1 + c t t ) n (31)

For a spherical drop growing at constant liquid flow n - 2/3, while n = 1 corresponds to a linear

rule for the area growth of cylindrical drops forming at the tip of large capillaries. Images of

growing drops represented for example by Pierson and Whitaker [58] give evidence of the

approximate character of Eq. (31). It appears that Eq. (31) for n - 2/3 is satisfactory only for

rather thin capillaries (less than 1 mm in diameter) and large interfacial tensions. This case

corresponds to the technique of growing drops [68, 73, 74]. In the drop volume method wide

capillaries with diameters of few millimetres are normally used [ 16, 24, 58, 75, 76].

MacLeod and Radke [68] used the equilibrium adsorption value at a given concentration F~q as

the initial adsorption value F 0 while in the model given in [72] it was assumed that F 0 has a

value between 0 and Foq. It is rather difficult to precisely specify this value for the drop volume

method. After a drop has reached a critical volume the drop surface area increases rapidly. It

follows from the data presented in [58] that after the break-off of the liquid bridge the area of

169

the residual drop at the capillary, plus the area of the detaching drop, is less than the surface of

the stretched drop prior to the break-off. The capillary radius, the density difference of the liquid

phases and the interfacial tension determine the relative area before and after break-off of the

liquid bridge. This effect, however, has not yet been studied. The expansion of the surface and

its subsequent compression can produce an adsorption at the residual drop and a corresponding

interfacial tension that differs significantly from that obtained at the critical moment before the

drop starts to detach. Neglecting these possible expansion/compression effects one can assume that F 0 - F(t) , where F(t) is the dynamic adsorption corresponding to the drop lifetime t.

The interfacial tension change with time at a growing drop as given by Joos and van Uffelen

[72] has the form

(1 +czt) ~ - ( 1 - ~ ) Ay(t) -~'(t) - Yeq - {Bx/(2n + 1)a

4(1 +(zt)2n+' - 1 (32)

where y eq is the equilibrium surface tension, ~ = (F(t)/U~) 2, Q = (ro-r(t))/r(0,

B-RTF:2q 4 ~ - ( d y ( t ) ' ~ l

Co ~d(te]/2))t-~~176 (33)

R is the gas constant, T is the temperature, and t of = t / (2n + 1) is the effective adsorption time.

Eq. (33) follows immediately from Eq. (32) for at >> 1. The initial drop has a size less than a hemisphere with the radius equal to the capillary radius reap [68] so that the drop area can be

given by A o -1.5rtr~ap. The area of a drop after the break-off of the liquid bridge can be

described by

A(t) = (471;) 1/3 (3V) 2/3 + A o ~ 4 . 8 3 V 2/3 + A o (34)

where V is the drop volume. The value of at in Eq. (32) can be obtained from Eq. (31)

(A( t ) l 1/n

czt- \ - ~ 0 J -1 (35)

Thus for any time t the value of c~ can be calculated from Eqs. (34) and (35). If we assume a

Langmuir-Szyszkowski adsorption isotherm and inteffacial tension equations, the parameters

and ~ can be expressed via the values of the dynamic and equilibrium interfacial pressures, H(t)

and 1-I~q as

[- exp(-II(t)/RTF~) 7~ ~-I 1- ] (36)

-L J

170

exp(-II(t)/RTF.~) - exp(-H~q/RTF~o) ~=

1- exp(-H~/RTF~) (37)

where Hoq = Y o-Yoq, 1-I(t)= Yo-y( t ) , Yo is the inteffacial tension of pure liquids, F~ the

limiting adsorption value.

Working with liquid/gas interfaces one can assume that the surfactant is present only in the

liquid phase, except when it is highly volatile. For liquid/liquid systems however, the typical

situation is that a surfactant is soluble in both adjacent liquid phases. In this case the distribution

of the surfactant between the two phases has to be considered. If the surfactants is in a

distribution equilibrium between both adjacent liquid phases, the diffusion coefficient in Eq. (33)

represents the effective value and is defined by the equation [24]:

Def- ( ~ 1 -[- K ~ 2 ) 2 (38)

Here K = Co~ / Co2 is the equilibrium distribution coefficient of the surfactant between the phases

1 and 2 (subscripts 1 and 2 refer to water and oil phase, respectively).

60 50

r ' - - - i

4o ,...., 30 < 20

10 0

0 200 400 600

t [ms]

Fig. 20 Determination of the dependence of drop volume as a function of time using a fast video technique,

symbols - data obtained from video images, solid line - Eq. (35), according to [79]

If the distribution equilibrium is not established prior to an adsorption kinetics experiment, the

transfer across the interface between the two liquids has to be considered. A theoretical analysis

of such systems using a Langmuir isotherm

C o r ( t ) = r~

aL + Co (39)

as well as experimental results obtained from pendent drop experiments have been published

very recently [77, 78].

171

Very recently fast video studies of the drop formation process have been performed in order to

verify Eq. (35) [79]. The results are shown in Fig. 20 in terms of drop volume as a function of

time. As can be seen, the experimental data are well described by Eq. (35) until the drop starts

to detach. After the start of the detachment process a remarkable deviation between the data

points and the theoretical curve can be observed. The time interval referring to the detachment

process of the drop is of the order of 65 ms, in excellent agreement with the data obtained in

paragraph 3.3.

5. EXPERIMENTAL RESULTS

As previously stated the drop volume technique can be used for measurements of surface and

interfacial tensions of pure liquids and solutions of surface active substances, such as surfactants

and polymers. No modifications are necessary when changing from surface tension to interfacial

tension measurements between two immiscible liquids. In the following some examples of the

use of the method are described to demonstrate its accuracy and the large variety of problems to

which it can be applied.

38

37

36 �9

0 10 20 30 40 n

I

50

Fig. 21 Statistics of individual drops measured for water, three runs at different drop formation times

The software of commercial instruments, for example the TVT1, usually provides surface

tension data calculated from averaged volumes of two and more drops. The surface tension data

for water given in Fig. 21 are obtained for individual drops under constant conditions of drop

formation. The small deviation of the individual values from each other underline the high

precision of drop volume measurements [29].

The deviation of individual values from the mean value is less than 0.1 mN/m, an accuracy in the

same order as that of other methods such as the ring or plate tensiometry.

172

5.1. MEASUREMENTS OF PURE LIQUIDS

When summarising the advantages of the drop volume method the easy temperature control was

one of the advantages emphasised. The temperature dependence of surface tension of water in

the interval 15 - 50~ was performed in [29] using an automated drop volume tensiometer. The

obtained data together with results from other authors [80- 82] using different techniques are

presented in Fig. 22.

74

73

72

& 70 69

68

67 I I ,, I I I

10 20 30 40 50 60

T[~

Fig. 22 Temperature dependence of the surface tension of water; data from TVT1 (R), Cini et al. [80] ([]),

Gittens [81] (e), Kayser [82] (~,)

The large differences between the methods, much larger than the accuracies of the individual

methods, show that the results depend strongly on the procedure of temperature control. As the

chamber around the capillary is small and definitely sealed the results obtained with the TVT1

should be more reliable.

Another example to demonstrate the capacity of the drop volume technique is the following

experiment at a liquid/liquid interface. The measured interfacial tension between commercially

available decane and water is represented in Fig. 23.

Within the measured time interval of 2 minutes a distinct decrease of 7(0 is observed. The

plateau value for t ~ oo is much below the value expected from literature (51.8 mN/m [74]).

This effect can be explained by two simultaneously acting processes: the decrease of 7(0 due to

hydrodynamic effects similar to those discussed at the liquid/air interface and the adsorption of

impurities at the interface which also leads to a decrease in interfacial tension. However, the

mechanism of hydrodynamic effects can be assumed to be caused by the same physical reason

173

and the presence of impurities is mainly responsible for the decrease of y(t) over the whole

period of time given in Fig. 23.

52 -A

51,5

50,5

~ 50

49,5

49

mAk AAk �9 �9 k

�9 �9

0 0 0

I I I I I

0 50 100 150 200 250

t [s]

Fig. 23 Interfacial tension between water and decane at 20~ commercial decane (41,), decane after

purification by distillation (l),decane after purification by rinsing through a column (A), data from

[291

These particular measurements are also a simple check procedure of the purity of solvents. The

sample used in Fig. 23 is not of sufficient purity for interfacial studies. Distillation does not

remove the disturbing surface active impurities. Only a repeated rinsing of the alkane through a

column filled with activated Alumina oxide yields a highly purified solvent suitable for

adsorption studies.

5. 2. DYNAMIC SURFACE TENSION OF SURFACTANT SOLUTIONS

In this section a small selection of experiments will be proposed and discussed to demonstrate

the wide variety of systems possible to study. The surface active agents used in the studies are

of different nature, for example water soluble surfactants, chloroform soluble lipids, water

soluble proteins. The motivation of such measurements range from obtaining general

understanding of surfactant adsorption, to biomedical and biophysical model investigations of

lipid membranes to questions arising from technologies in food or pharmaceutical production

based on proteins.

In Fig. 24 the interfacial tension data measured for a lecithin dissolved in chloroform at the

interface to water using the drop volume (filled circular) and pendent drop techniques (open

circular). The large divergence between the two curves (the larger time values for the same y

174

correspond to the drop volume method) is caused by the different effective surface age of the

drops in the two methods. In contrast to where in the pendent drop the interfacial area does not

change the drop in drop volume measurements is formed under the condition of a constant

liquid flow and hence the drop surface is expanding. This area change and the flow inside the

drop have to be considered, as discussed in the next paragraph. The results obtained from the

drop volume experiments, corrected for the hydrodynamic effect according to Eq. (19), and 3

using the effective time t ef --''7 t as a good approximation for growing drop experiments, are

also shown in Fig. 24. With this the two methods are in excellent agreement. Such good

agreement between different methods is a crucial point in interfacial studies where omen

complementary methods have to be used to span over a large range of surfactant concentration

and hence a large time interval. Comparison with other techniques will be further discussed

below.

The results given in Fig. 24 are interesting for understanding the formation of lipid monolayers

at a water surface by spreading from a chloroform solution. It was shown in [83] that most of

the spreading experiments are done with lipid solutions in chloroform above a critical

aggregation concentration. The present aggregates may possibly have an effect on the

morphology of the formed monolayer.

~]5"

30

Z5

'~ ZO-

O" 15-

5 0

q l �9

I D Q Q Q O e O

O O O

, o . |

20 110 GO IBO 100 lZO Ii, 0

tCs3

Fig. 24 Dynamic interfacial tension ),(t) of 210 5 mol/l DPPC at the chloroform/water interface measured with

different techniques: (O) ADSA; (e) TVT, (11) TVT corrected due to hydrodynamic and growing

drop effects; according to [83]

To demonstrate the interpretation of adsorption kinetics data the results given in the next two

figures will be used. For decyl diethyl phosphine oxide solutions of two different concentrations

the results obtained from dynamic and quasi-static drop volume experiments are presented in

Fig. 25.

175

The data are plotted as y(1/4~) which due to the diffusion theory yields a linear course for

large t [84]. This is true for both the dynamic and quasi-static mode. Only the final slope of the

curves should differ by a factor of ~f7/3 ~ 1.5. This qualitative picture is obviously fulfilled by

the data in Fig. 25. Moreover, both sets of experimental data extrapolate to the same

intersection point with the ordinate which is the approximate value of the equilibrium surface

tension. This is another important qualitative argument for a successful experiment. A

quantitative interpretation of dynamic surface tension data however requires additional

information: accurate knowledge of the adsorption equilibrium, given by an adsorption

isotherm.

6 5 - - �9

63 - � 9 n - �9 o "

55

61 r - - - I

sg i _ _ _ s

57

53 I I I I I 0,0S 0,1 0,15 0,2 0,25

1/'x/t [ 1/~/s]

!

0,3

Fig. 25 Dynamic surface tension of a decyl diethyl phosphine oxide solution of concentration Co= 10 "8 mol/cm 3

(liD), Co=10 -7 mol/cm 3 (,'~'); dynamic mode (lie,), quasi-static mode ([3'~'), data from [16]

For the surfactant homologues of alkyl dimethyl and alkyl diethyl phosphine oxides the

Langmuir isotherm, Eq. (39), describes the adsorption equilibrium very well. For the studied

decyl diethyl phosphine oxide the following isotherm parameters result [85]:

Food = 3.3-10 -10 mol/cm 2, a L = 1.0-10 -8 mol/cm 3.

From rearrangements of Eqs. (39) and using Gibbs' fundamental equation the following

relationship between 7(0 and F(t) results [42]"

r(t) = ro (1 + aL/Co)(1 - exp[(v(t)-%)~Troo]). (40)

176

Thus via Eq. (40) for each measured 7(t)-value (quasi-static mode) a corresponding F(t)/F o -

value can be calculated. The equilibrium surface concentration F o results directly from the

isotherm Eq. (39). By variation of D the value of F(t) can be fitted to the experimental ~,(t)-

value via Eq. (20). In this way for each measured value 7(t) a diffusion coefficient D results,

from which the mean value D can be calculated.

The interpretation of data obtained from measurements in the dynamic mode can be performed

in two different ways. Either a fitting of F(t) has to be performed via Eq. (21). This however

requires large numerical calculations. Another way is to calculate the age of the drop surface

from the drop formation time t. r. The measured 7(t,r)-values are therefore related to a surface

age t = 3/7 t. r. The resulting dependence 7(3/7.t.r)---7(t ) can then be interpreted as if it was

obtained via the quasi-static measuring mode.

70' i j �9 u �9 �9

�9 m.l ~

65 ~ [ ] - �9

55

50

45 I I I I I 0 0,1 0,2 0,3 0,4

1 N [ s ]

Fig. 26 Dynamic surface tension of three dodecyl dimethyl phosphine oxide solutions: concentrations are

c o = 2-10 -8 mol/cm 3 01H),310-8 mol/cm 3 (~,r mol/cm 3 (~kA);

(II~I,A) - dynamic mode, ([3,r - quasi-static mode, data from [16]

For three concentrations of dodecyl diethyl phosphine oxide the experimental data 7(tr) as well

as the result of fitting are given in Fig. 26. The solid lines are theoretical curves calculated with

a mean diffusion coefficient of D = 3.5 -10 -6 cmVs. Good agreement between theory and

experiment results. It can be deduced that the studied dodecyl dimethyl phosphine oxide adsorbs

diffusion-controlled.

177

5. 3. DYNAMIC INTERFACIAL TENSIONS OF VAMOUS SYSTEMS

The drop volume technique is one of the few suitable methods for measurement of dynamic

interfacial tensions of liquid/liquid systems. As underlined before no special modifications are

needed for these experiments. However, a number of peculiarities have to be considered for

such systems, the most important of them being the solubility of surfactants in both adjacent

liquid phases. There is a striking difference in studies at a liquid surface where only very few

surfactants show a comparable phenomenon, the evaporation from the adsorption layer. If the

surfactant is soluble in both phases but adsorbs only from one (typically from the aqueous

phase) the surfactant is transferred across the interface and desorbs into the oil phase.

In general there are three cases for the adsorption process at a liquid/liquid interface:

a)

b)

c)

the surfactant is present in the water phase only;

the surfactant is present in the oil phase only;

the surfactant is present in both phases with an equilibrium surfactant concentration

distribution.

The theoretical model for the cases a) and b) is a generalisation of the theory of Ward and

Tordai [53] and was first proposed by Hansen [54]:

c,o + (41)

Case c) is described by the Ward and Tordai equation (20) using instead of D the effective

diffusion coefficient Def as defined by Eq. (38).

As an example to demonstrate the solubility of a surfactant in water and also in the adjacent

nonane phase measurement with Triton X-45 solutions were performed as follows. At the

beginning the container in which the drops of the Triton solution are formed contains only pure

nonane. The amount of aqueous solutions were about 300 ml while the amount of nonane was

10 ml to 20 ml. The experimental results are shown in Fig. 27. Each of the five subsequent runs

comprises 500 drops. The drops are formed such that for each flow rate 10 drops were formed

and averaged starting with the largest flow rate. During the first four runs the obtained 7(0-

curves change while the fifth and further runs do not. This state refers to the case of adsorption

from both adjacent phases, i.e., during the measuring procedure the equilibrium distribution of

the Triton was reached between the nonane and water phases. Only experiments for this case

allow a quantitative interpretation as the experimental conditions can be simulated by the

theoretical model.

178

a l low a quanti tat ive interpretation as the exper imental condit ions can be s imulated by the

theoret ical model .

Exper imenta l results for solutions o f other Tritons have been reported in [24]. F r o m these

studies it was concluded that the distribution coefficient for Tri ton X-45 is significantly higher

than for Triton X-405.

50

45

40 z

35

30

25 0,4

Fig. 27

dpr#q qn' o

nnqh

O~d[l~do~ ,~ q:~:l c~l~ m n n q n �9 ~j~.,,. "- q.%.__ ~ man n m

"'q#'~,,:~,,x, * ~ . * . . [] cp _

�9 . . , . o o~..<,.<,,, o

[ ]

I I I I I I I ,I

0,6 0,8 1 1,2 1,4 1,6 1,8 2

t [ s ] Dynamic interfacial tension of Triton X-45 solutions as a function of xf{; c o = 1.2. 0 -8 mol / cm 3,

5 subsequent runs; according to [24]

50

45

40

i ._.a

30

25

Fig. 28

20

iPil~neimminmm ~ ~ Damn �9

-,-u,-, ~j.n[~]::):art:~ 4p.,.,~41~,b~g:~ 4i~ 4 1 , ' H I �9 �9 �9 �9

I I t - t --, t

0,3 0,5 0,7 0,9 l,l 1,3 1,5 1,7

t [ s ]

Dynamic interfacial tension of a Triton X-45 solution as a function of x/t; c o = 2.4 0 -8 mol / cm 3 for

the three different cases a) (m), b) ( , ) , and c) (El); and in absence of Triton X-45 (0), according to [24]

179

To visualise the significant differences of dynamic surface tensions measured for the three cases

discussed above, the results of experiments with Triton X-45 are reported in Fig. 28. It is

obvious that for case c) the adsorption process is the fastest as adsorption takes place from both

adjacent liquid phases. In the other two cases a) and b) the adsorption is much slower due to

only adsorption from one phase and moreover due to the loss of adsorbed molecules by

desorption into the second liquid phase. It was emphasised in [24, 77, 78] that dealing with

liquid/liquid interfaces one always faces the problem that surfactant molecules are soluble in

both adjacent liquids and hence adsorption from one phase generally leads to a transfer across

the interface.

5. 4. COMPARISON WITH OTHER TECHNIQUES

In order to demonstrate the reliability and accuracy of the drop volume method it is suitable to

compare it with other techniques operating over similar experimental windows. There are

several methods which have a partial or even complete overlap in the time interval. For example,

the pendent drop method almost covers the same time range. The maximum bubble pressure

technique however has a time window from less than 1 ms up to some ten seconds.

75-

70-

65

6 0 -

55-

50

4 5 -

40-

35-

.

[ ]

3 0 I I I I I

0,001 0,01 0,1 1 10 100

t [s]

F i g . 29 Dynamic surface tensions of two undecyl dimethyl phosphine oxide solutions measured by using the

drop volume method (TVT1, [] '~') and the maximum bubble pressure technique (MPT2, II , ) ;

concentrations are 210 -7 mol/cm 3 (11 [] ) and 710 .7 mol/cm 3 (4),~,)

There are quite a number of other methods available for such comparison. Detailed description

of the pendent drop and maximum bubble pressure methods is given in others chapters of this

180

book. Other methods not based on liquid drops or bubbles have been discussed in particular in

[41, 42].

As an example data are presented here measured with the drop volume and the maximum bubble

pressure method. These two methods have an overlap in the time range from part of a second

up to about 100 s. Fig. 29 shows the results for solutions of a surfactant model system, an alkyl

dimethyl phosphine oxide at the water/air interface. There is excellent agreement between the

data which are presented in co-ordinates of surface tension versus effective adsorption time.

Moreover the hydrodynamic correction for the drop volume method given by Eq. (19) has been

applied to correct surface tensions.

Additional results of dynamic surface tension studies for dimethyl dodecyl phosphine oxide

solutions at a considerably high concentration is presented in Fig. 30. In contrast to Fig. 29

some differences arise in the data in Fig. 30. The dynamic surface tensions measured with the

drop volume method are significantly lower than the data obtained by the maximum bubble

pressure method.

The main reason for the discrepancy is due to the initial load of the residual drop after the

preceding drop is detached. This effect is smaller in bubble pressure experiments as the final

bubble surface area expansion is considerably larger. The results of the drop volume

experiments agree with Eq. (32), and this agreement improves with decreasing y-?~ or

increasing time, which is obviously the consequence of the approximate nature of Eq. (32).

Accurate agreement can be expected if Eq. (30) is used with the correct value of the initial

adsorption value F a [87].

75 T 70 -; 65-- 6o -F 55 -

5o

~" 45- 40- 35 30

+ -I~- +

Fig. 30. Dynamic surface tension of DMDPO solution for c o = 1.10 "7 mol/cm 3, measured by the MPT2 (x), the TVT1 (A) and the ring tensiometer TEl (+), respectively; the lines are calculated according to Eq. (21) (dotted) and to Eq. (32) (solid), respectively

-2 -1 0 1 2 3 4

Ig(t), [s]

181

As a comparison data measured at a liquid/liquid interface with the pendent drop method can be

used. Results obtained with both methods for DPPC at the chloroform/water interface were

given in Fig. 24. A good agreement between the two methods was established alter all necessary

corrections of drop volume data had been performed.

However, in general, difficulties and peculiarities show up when investigating the adsorption

kinetics at a liquid/liquid interface due to the solubility of surfactant in both adjacent phases. A

number of results on such systems have been reported very recently elsewhere [24, 25, 77, 78].

6. SUMMARY AND CONCLUSIONS

The drop volume method as described here in detail is a very powerful method for studies of

adsorption kinetics. Most significant is that it can be easily used also for liquid/liquid interfaces,

which is not trivial for a number of other methods. However, data obtained with this technique

have to be interpreted correctly in order to obtain reliable results. The method has restrictions

for example with respect to the drop formation time. If drops are formed too fast the measured

drop volumes are no longer a measure of the surface tension alone but are in addition governed

by chaotic effects leading to so-called drop volume bifurcations.

Data corrected with respect to drop growth and hydrodynamic effects provide excellent

agreement between this method and others. Differences between this and other methods can be

obtained when the initial load of the drop surface of the studied system is significant. Then a

correct data interpretation is possible only if this initial adsorption is properly considered in the

theory.

An interpretation of experimental data with respect to the adsorption mechanism was

demonstrated and it was shown that the studied systems follow a diffusion controlled adsorption

model. This conclusion is correct also for many surfactant systems. If one analyses the state of

the art of adsorption kinetics studies one can conclude that this mechanism is true for almost all

surfactants and any other model should be seen as exception and proved sufficiently.

In addition it was shown how fast video technique can be of help to give an insight into the

elementary processes of drop formation and detachment. In particular the evolution of the

growing drop can be determined accurately and also the drop detachment time can be obtained

by this technique, which is in excellent agreement with values determined from the time

dependence of the drop volume of pure liquids. In [79] it was even possible to visualise the

surface waves postulated to be responsible for the drop volume bifurcations in the process of

fast drop formation. One can be sure that this technique will give further direct information

182

necessary for a better understanding of phenomena in the drop volume technique still unclear so far.

7. REFERENCES

1. Padday, J.F. in E.Matijevic (Ed.): Surface and Colloid Science, Vol. 1, Wiley-Interscience,

New York-London-Sydney-Toronto 1969, pp 39

2. Goodrich, F.C., in E.Matijevic (Ed.): Surface and Colloid Science, Vol. 1, Wiley-Interscience,

New York-London-Sydney-Toronto 1969, pp 1

3. Rowe, E.I., J. Pharm. Sci. 61 (1972)781

4. Kloubek, J., Friml, K. and Krejci, F., Czech. Chem. Commun., 41(1976)1845

5. Lohnstein, T., Ann. Physik, 20(1906)237

6. Lohnstein, T., Ann. Physik, 20(1906)606

7. Lohnstein, T., Ann. Physik, 21 (1907) 1030

8. Lohnstein, T., Z. phys. Chem., 64(1908)686

9. Lohnstein, T., Z. phys. Chem., 84(1913)410

10. Tornberg, E., J. Colloid Interface Sci. 60(1977)50

11. Tornberg, E., J. Colloid Interface Sci. 64(1978)391

12. Joos, P. and Rillaerts, E., J. Colloid Interface Sci. 79(1981)96

13. Tornberg, E. and Lundh, G., J. Colloid Interface Sci. 79(1981)76

14. Carroll, B.J. and Doyle, P.J., J. Chem. Soc., Faraday Trans. 1, 81(1985)2975

15. Nunez-Tolin, V., Hoebregs, H., Leonis, J. and Paredes, S., J. Colloid Interface Sci.

85(1982)597

16. Miller, R. and K.-H.Schano, Colloid Polymer Sci. 264(1986)277

17. Babu, S.R., J. Colloid Interface Sci. 115(1987)551

18. Doyle, P.J. and Carroll, B.J., J. Phys. E: Sci. Instrum., 22(1989)431

19. Miller, R. and Schano, K.-H., Tenside Detergents 27(1990)238

20. Paulsson, M. and Dejmek, P., Journal Colloid Interface Sci. 150(1992) 394

21. Miller, R., Hoffmann, A., Hartmann, R., Schano and K.-H., Halbig, A., Advanced Materials,

4(1992)370

22. Ewart, H.A. and Hyde, K.E., Journal Chem. Education, 69(1992)814

23. Wawrzynczak, W.S., Paleska, I. and Figaszewski, Z., Journal Electroanalytical Chemistry

and Interfacial Electrochemistry, 319(1991) 291

24. V.B. Fainerman, S.A. Zholob and R. Miller, Langmuir, 13(1997)283

25. S.A. Zholob, V.B. Fainerman and R. Miller, J. Colloid Interface Sci., 186(1997)149

26. Lunkenheimer, K. and Wantke, K.-D., Colloid Polymer Sci. 259(1981)354

183

27. T. Tate, Phil. Mag., 27(1864) 176

28. Lord Rayleigh, Phil. Mag., 48(1899)321

29. Miller, R., Hoffmann, A., Schano, K.-H., Halbig, A. and Hartmann, R., Seifen Ole Fette

Wachse, 118(1992)435

30. Kohlrausch, F.W., Lehrbuch der praktischen Physik, 1905

31 Kohlrausch, F.W., Ann. Physik 20(1906)798

32. Addison, C.C., J. Chem. Soc. (1946)579

33 Addison, C.C., Bagot, J., McCauley, H.S., J. Chem. Soc. (1948)936

34. Addison, C.C. and Hutchinson, S.K., J. Chem. Soc. (1948)943

35. Addison, C.C. and Hutchinson, S.K., J. Chem. Soc. (1949)3387

36. Addison, C.C. and Hutchinson, S.K., J. Chem. Soc. (1949)3406

37. Freud, B.B. and Harkins, W.D., J. Phys. Chem. 33(1929)8

38. Hartland, S. and Srinivasan, P.S., J. Colloid Interface Sci., 49(1974)318

39. Harkins, W.D. and Brown, F.E., J. Amer. Chem. Soc., 41(1919)499

40. Wilkinson, M.C., J. Colloid Interface Sci., 40(1972) 14

41. Rusanov, A.I. and Prokhorov, V.A., Interfacial Tensiometry, in "Studies of Interface

Science", Vol. 3, D. MObius and R. Miller (Editors), Elsevier, Amsterdam, 1996

42. Dukhin, S.S., Kretzschmar, G., and Miller, R., Dynamics of Adsorption at Liquid Interfaces,

in "Studies in Interface Science", D. MObius and R. Miller (Eds.), Vol. 1, Elsevier,

Amsterdam, 1995

43. Earnshaw, J.C., Johnson, E.G., Carroll, B.J. and Doyle, P.J., J. Colloid Interface Sci.,

177(1996)150

44. Kaufman, S., J. Colloid Interface Sci., 57(1976)399

45. Kretzschmar, G. and Miller, R., Adv. Colloid Interface Sci., 36(1991)65

46. Jho; C. and Burke; R., J. Colloid Interface Sci., 95(1983)61

47. Van Hunsel, J., "Dynamic Interfacial Tension at Oil Water Interfaces", Thesis, 1987,

University of Antwerp

48 Miller, R., Schano, K.-H. and Hofmann, A., Colloids Surfaces A, 92(1994)189

49. Davies, J.T. and Rideal, E.K., Interfacial Phenomena, Academic Press, New York, 1969

50 van Hunsel, J., Bleys, G. and Joos, P., J. Colloid Interface Sci., 114(1986)432

51 Fainerman, V.B. and Miller, R., Colloids & Surfaces A, 97(1995)255

52. Miller, R., Bree, M. and Fainerman, V.B., submitted to Colloids & Surfaces A

53 Ward, A.F.H. and L.Tordai, J. Phys. Chem. 14(1946)453

54. Hansen, R.S., J. Phys. Chem., 64(1960)637

55 Miller, R. and Lunkenheimer, K., Z. phys. Chem., 259(1978)863

56. Sutherland, K.L., Austr. J. Sci. Res., AS(1952)683

184

57. Miller, R., Colloids Surfaces, 46(1990)75

58. Pierson, F.W. and Whittaker, S., J. Colloid Interface Sci., 52(1976)203

59. Miller, R, Colloid Polymer Sci. 258 (1980) 179

60. Davies, J.T., Smith, J.A.C. and Humphreys, D.G., Proc. Int. Conf. Surf. Act. Subst.,

2(1957)281

61 Delahay, P. and Trachtenberg, I., J. Amer. Chem. Soc., 79(1957)2355

62. Delahay, P. and Fike, C.T., J. Amer. Chem. Soc., 80(1958)2628

63 Kloubek, J., J. Colloid Interface Sci., 41 (1972) 1

64. Fainerman, V.B., Koll. Zh., 41 (1979) 111

65 Ilkovic, D., J. Chim. Phys. Physicochem. Biol., 35(1938)129

66. Levich, V.G., Physicochemical Hydrodynamics, Prentice-Hall, Englewood Cliffs, New

York, 1962

67. Fainerman, V.B., Zh. Fiz. Khim., 42(1983)457

68 MacLeod, C.A. and Radke, C.J., J. Colloid Interface Sci., 166(1994)73

69. Liggieri, L., Ravera, F. and Passerone, A., J.Colloid Interface Sci., 169(1995)226.

70. Cini, R., Loglio, G. and Ficalbi, A., Ann. Chim. (Rome) 62(1972)789

71 Joos, P. and Van Uffelen, M., J. Colloid Interface Sci., 155(1993)271.

72. Joos, P. and Van Uffelen, M.,, J.Colloid Interface Sci., 171 (1995)297.

73 Nagarajan, R. and Wasan, D.T., J.Colloid Interface Sci., 159(1993)164.

74. Passerone, A., Liggieri, L., Rando, N., Ravera, F. and Ricci, E., J.Colloid Interface Sci.,

146(1991)152. 75. Van Hunsel, J. and Joos, P., Colloids Surfaces, 24(1987)139.

76. Van Hunsel, J. and Joos, P.,Colloid Polymer Sci., 267(1989)1026.

77. Ferrari, M., Liggieri, L., Ravera, F., Amodio, C. and Miller, R., J.Colloid Interface Sci.,

186(1997)40 78. Liggieri, L., Ravera, F., Ferrari, M., Passerone, A. and Miller, R., J.Colloid Interface Sci.,

186(1997)46 79. Makievski, A.V., Miller, R. and Fainerman, V.B., Fast video experiments on drop and

bubble formation processes, in preparation

80. Cini, R., Loglio, G. and Ficalbi, A., J. Colloid Interface Sci., 41 (1972)287

81. Gittens, G.J., J. Colloid Interface Sci., 30(1969)406

82. Kayser, W.V., J. Colloid Interface Sci., 56(1976)622

83. Li, J.B., Fainerman, V.B. and Miller, R., Langmuir, 12(1996)5138

84. Makievski, A.V., Fainerman, V.B., Miller, R., Bree, M., Liggieri, L. and Ravera, F.,

Colloids & Surfaces A, 122(1997)269

85. Lunkenheimer, K., Haage, K. and Miller, R., Colloids & Surfaces, 22(1987)215

185

86. MacLeod, C.A. and Radke, C.J., J. Colloid Interface Sci., 160(1993)435

87. Miller, R., Zholob, S.A., Makievski, A.V., Joos, P. and Fainerman, V.B., Remarks on the

interpretation of data from the dynamic drop volume method, Colloids & Surfaces A, in press

8. LIST OF SYMBOLS

A

Ao a

c

Co

D

Deft

Eo f

g

h

K

k

kad

kdes k~ kd

N

n

P

P~ Q R

R(t)

r o rcap

T

t

tdrop

to tof

area of the interface [cm 2]

interfacial area at time t=0 [cm 2]

capillary constant as defined in Eq. (3)

bulk concentration [mol/cm 3]

equilibrium bulk concentration [mol/cm 3]

diffusion coefficient [cmVs]

effective diffusion coefficient [cmVs]

thermodynamic surface dilational modulus [mN/m]

correction factor

acceleration constant [cm/s 2]

drop height [cm]

equilibrium distribution coefficient

slope of the dependence V(1/t n)

rate constant of adsorption [cm/s]

rate constant of desorption [ I/s]

rate constant of micelle formation [I/s]

rate constant of micelle dissolution [ I/s]

Avogadro number

number of drops

pressure [N/m 2]

capillary pressure [mN/m 2]

liquid volume flow rate [cm3/s]

gas law constant [ 8.314 g cm2/(s 2 mol K)]

drop radius [cm]

radius of curvature of a bubble or a drop at its apex [cm] radius of capillary tip

absolute temperature [K]

time [s]

drop detachment time

drop detachment time effective adsorption time [s]

186

U

V

V~

V r

X

X

Y

Y Z

flow velocity [cm/s]

volume [cm 3]

corrected drop volume [cm 3]

radial flow velocity [cm/s]

dimensionless coordinate in x direction

direction normal to the interface [cm]

dimensionless coordinate in y direction

direction tangential to the interface [cm]

radial coordinate in a spherical system

F eq

Fo

3I

~o

'~ eq

F I - ~ o - ~

ot coefficient for hydrodynamic correction

J3 coefficient for hydrodynamic correction

A0 density difference between two adjacent phases

F = F (t) / F o dimensionless adsorption

adsorption at time t--0 [mol/cm z]

equilibrium of adsorption [mol/cm z]

adsorption of component i [mol/cm 2]

equilibrium surface concentration [mol/cm 2]

maximum surface concentration [mol/cm 2]

interfacial tension [mN/m]

interfacial tension of the pure solvents [mN/m] is the equilibrium surface tension [mN/m]

surface pressure [mN/m] I 'Ieq

|

"1~ D

"Cad

f~

equilibrium surface pressure [mN/m]

dimensionless time uncertainties on the parameter i

relaxation time of the diffuse part of the electric double layer [s -1]

characteristic time of the adsorption process

area disturbance (d In A/dt)

Drops and Bubbles in Interfacial Research D. M6bius and R. Miller (Editors) �9 1998 Elsevier Science B.V. All rights reserved. 187

SPINNING DR01' TENSIOME TR Y

ANDREAS M. SEIFERT

Philipps-Universit~it, Fachbereich Physikalische Chemie und Wissenschaffliches Zentrum

for Materialwissenschaften, Hans-Meerwein Stral3e, 35032 Marburg, FRG

CONTENTS

1 Introduction

1.1 The principle of spinning drop tensiometry

1.2 A brief survey Properties and applications of the spinning drop technique

Theoretical background: Equilibrium shape and stability of rotating drops

2.1 Equilibrium shape in gyrostatic equilibrium

2.1.1 The balance between interfacial and centrifugal forces

2.1.2 General case: Ellipsoid-like droplets

2.1.3 Limiting case: Cylindrical droplets and Vonnegut's equation

2.1.4 Determination of interfacial tension from SDT data

2.2 Stability of rotating liquid threads

2.3 Approach to gyrostatic equilibrium

Practical aspects of spinning drop tensiometry

3.1 Spinning drop tensiometers

3.1.1 The spinning drop interfacial tensiometer SITE 04

188

3.1.2 Sample preparation and spinning drop tensiometers for viscous liquids,

polymer melts and solutions

3.2 Determination of the true droplet radius

Non-equilibrium spinning drop techniques

4.1 Relaxation of stable droplets

4.1.1 Phenomenological approaches

4.1.1.1 Upper and lower bounds for interracial tension

4.1.1.2 Extrapolation techniques

4.1.1.3 'SDT relaxation spectroscopy'

4.1.2 Theoretical approaches

4.2 Spontaneous break-up of liquid threads

42.1 Non-rotating threads: The breaking thread method

42.2 Break-up of rotating liquid threads

43 Studies on the effect of mass transfer across interfaces

43.1 Mass transfer across the interface of a 'stable' droplet

43.2 The effect of mass transfer on break-up and coalescence

Concluding remarks

List of symbols

7 References

189

1. INTRODUCTION

The spinning drop technique (SDT) has been developed to measure extremely low interfacial

tensions (from 10 -6 mNm -1 to 10 mNm -1 ). It uses profile analysis of deformed droplets similar

to the pendent drop method. Unlike in pendent drop experiments, where the droplets are de-

formed by the gravitational force, SDT is based on the balance of centrifugal and interracial

forces in rapidly rotating systems.

Although the study of rotating droplets goes back to the middle of the last century [ 1, 2], SDT

was first proposed by Vonnegut [3] in 1942. Since Vonnegut's original work, approximately

thirty papers have been published describing spinning drop tensiometers and discussing the

theoretical background, the range and limitations of SDT applications. In addition, numerous

publications show that today SDT is a well established standard tool for the investigation of

systems exhibiting low interfacial tension and relatively low viscosities. SDT is applied in

emulsifier development, cleanser and surfactant formulation, insoluble phase reactivity optimiza-

tion, qualitative analysis of solubility limits and in the study of adsorptive behaviour at phase

boundaries. Today, spinning drop tensiometers are available as commercial instruments at least

for the above mentioned applications. Although basically simple and versatile, SDT has a

number of limitations and needs attention, even when it is applied to 'simple' systems.

The first SDT investigations on the interfacial tension of viscous polymer melts were described

by Patterson et. al. [4] thirty years after the publication of Vonnegut's original paper. Since this

time, various attempts have been made to adopt SDT for the investigation of polymer melts.

The results were very promising - at least in principle. Nevertheless, SDT is far from being a

standard method in polymer science due to problems with the sample handling and the long

relaxation times required to achieve equilibrium.

Apart from purely tensiometric applications SDT has been found to be a versatile tool for

surface and interface science. It allows the study of adsorption phenomena and even permits the

'simulation' of spontaneous structure formation processes, e. g. the break-up of liquid threads

and the coalescence of droplets.

The present contribution reviews both standard and non-standard SDT applications. After a

brief description of basic principles and properties, the equilibrium properties of a rotating drop,

190

i. e., its shape and its stability, are considered in detail (Sect. 2). Sect. 3 deals with experimental

aspects of SDT: Both commercial and laboratory SDT set-ups are introduced. Problems arising

from sample preparation (particularly in the case of highly viscous polymers) and the

determination of the droplet dimensions are discussed. Finally, Sect. 4 deals with promising non-

equilibrium spinning drop techniques.

1.1. THE PRINCIPLE OF SPINNING DROP TENSIOMETRY

Consider a droplet of phase 1 (density Pl ) immersed in a surrounding matrix phase 2 (density

P2 > P~). It is furthermore assumed that this droplet-matrix system is enclosed in a cylindrical

glass tube rotating about its horizontally oriented long axis (Fig. 1).

Fig. 1 Schematic representation of equilibrium shapes at different angular velocities

At low angular velocity co the droplet is centered about the axis of rotation, its shape still being

nearly spherical. At higher but still moderate velocities the drop stretches along the axis until

interfacial and centrifugal forces balance each other and an ellipsoid-like equilibrium shape is

observed. At sufficiently high velocities the shape of the droplet can be approximated by a cylin-

der with rounded ends. For both ellipsoid-like and cylindrical drops the ratio ~//Ap (interfacial

tension ~/, density difference Ap = 92-Pa ) is simply obtained via the profile analysis of the

equilibrium shape. In the case of a cylindrical droplet, the interfacial tension can be calculated

from Vonnegut's equation [3 ]

Apo 2ro 3 (1) ~ - - - ~

4

191

where r 0 is the radius of the cylindrical part of the droplet. The determination of interfacial

tension from ellipsoid-like drop shapes is more elaborate and requires numerical procedures.

Fig. 2 Block diagram of a contemporary spinning drop tensiometer: (1) sample cell, (2) furnace, (3) window,

(4) motor drive, (5) velocity controller, (6) clock generator for strobe lighting, (7) illumination,

(8) optical microscope, camera, (9) image processing system, (10) temperature controller

Vonnegut's original experimental work was performed on a small metal working lathe with a

maximum speed of 3000 min -1 . His investigations on the surface tension of liquid Wood's metal

(t) were largely qualitative due to the lack of equipment for producing constant and accurately

known speeds of rotation. Today, modern spinning drop tensiometers incorporate the following

components (Fig. 2):

�9 sample cell

�9 driving unit and controller for angular velocity

�9 heating unit and temperature controller

�9 illumination, stroboscopic lighting

�9 devices for profile analysis, optical components, camera, image processing system

192

l. 2. A BRIEF SURVEY: PROPERTIES AND APPLICATIONS OF THE SPINNING DROP TECHNIQUE

In the following a brief outline of SDT properties, its range of applications and its limitations is

given.

�9 SDT permits the measurement of interfacial tensions in the range from 102 - 10 -6 mNm -1

(depending on the density difference and droplet volume).

�9 SDT is suitable for the measurement of both interfacial and surface tension. The measurement

of surface tension requires the profile analysis of a gas bubble.

�9 SDT yields the ratio 7 / Ap. A density difference Ap = P2- 91 > 0 between the droplet (91)

and the surrounding matrix (p2) is required.

�9 For the detection of the droplet dimensions or profile, respectively, at least a slight difference

between the optical properties of the droplet and the matrix is required. The matrix phase has to

be sufficiently transparent.

�9 One of the major advantages of SDT is that the surface of the bubble is closed and no

question of contact angle arises.

Prior to a summary of more SDT advantages some characteristic problems and difficulties,

which may seriously limit the range of this method, should be mentioned:

�9 Compared to pendent drop devices, the design and manufacturing of a spinning drop tensio-

meter is more complex. (Sect. 3.1).

�9 At low angular velocities the effect of gravity causes the droplet to depart from the cylinder

axis. This buoyancy effect is counteracted by centrifugal forces and the viscosity of the matrix

phase. Thus, at sufficiently high velocities or high viscosities, respectively, the disturbing influ-

ence of gravity becomes negligible, and the system approaches gyrostatic equilibrium, i. e., it

performs a rigid body rotation, and the shape of the drop can be calculated exactly (Sect. 2.3).

�9 The sample cell acts as an cylindrical lens. Since in Vonnegut's equation the droplet radius is

raised to the third power, careful radius calibration is a prerequisite for reliable experimental

results (Sect. 3.2).

�9 The tube filling procedure often proves to be a frustrating task, since apart from surface

tension studies the inclusion of air bubbles must be avoided. The capillary must be carefully

sealed to prevent the entrance of air or the escape of volatile components during operation.

193

Particularly when dealing with polymers serious problems arise from sample preparation due to

the high viscosities and outgassing of the samples (Sect. 3.1.2).

The major advantages of SDT are less obvious and merit particular emphasis:

�9 The analysis of the droplet profile is simple. Only a few parameters (diameter, length) have to

be determined (Sect. 2.1).

�9 Apart from problems with the preparation of the samples, the method should be well adapted

for the investigation of rather viscous liquids, e. g. polymer solutions and melts at elevated

temperatures. Equilibrium drop shapes are achieved even in viscoelastic systems. Extrapolation

techniques may be applied reducing the time of measurement in systems exhibiting long re-

laxation times (Sect. 4.1.1.2).

Fig. 3 a) Cylindrical droplets observed in polystyrene-methylcyclohexane, b) ~//Ap versus co for both an ellipsoid-

like (1"1) and a cylindrical droplet (O), Seifert [6], experiments performed with the SITE 04 (Sect.

3.1.1). The interfacial tension value obtained at 1200 min -1 is too low due to departure from gyrostatic

equilibrium (see also Sect. 2.3).

�9 Under 'normal' experimental conditions the shape of the droplet is stable, i. e., small varia-

tions of the shape determining parameters will cause small changes of the droplet dimensions

(Sect. 2.2). Even very slender droplets as they are observed in ultralow tension systems, are

stable against small disturbances of their shape. This proves to be an enormous advantage since

disturbances due to thermal fluctuations or vibrations introduced by the motor drive are inevi-

tably present in SDT experiments. In addition, the stability of the droplets permits various re-

194

laxation studies, e. g. the investigation of dynamic interfacial tensions (Sect. 4.1) and (surface)

rheological properties.

�9 In contrast to the pendent drop technique, the 'external' force deforming the droplet may be

varied continuously by changing the angular velocity. Thus, droplet profiles may be analyzed at

different speeds of rotation in order to reduce experimental error and to make sure that the

system is sufficiently close to gyrostatic equilibrium (Fig. 3). By changing the speed of rotation

the area of the bubble may be controlled for the investigation of surface films. Compared to a

Langmuir film balance, however, the variation in surface area is too small to be used for

evaluating full surface pressure-area isotherms [5].

0 �9 | �9 | �9 | �9 i �9 | �9

a)

35

~30

25 20 4O 60 80 100 120

T[~

3.2

3.1

3.0

2.8

,

140 60

�9 ! �9 |

I

80 100 120 T [~

Fig. 4 Another example: Temperature dependence of a) surface tension of mixtures of polyethylene glycole,

M=400, and polypropylene glycole, M=425, (O) 5 %, (e) 10 %, (r-I) 20 %, (11) 60 % weight fraction

PPG 425 and b) interfacial tension of a PEG 10000-PPG 4000 blend. The tensiometer used is presented

in Sect. 3.1.2, Seifert [6].

�9 SDT is suitable for a number of interesting studies far beyond 'simple tensiometry'. Although

rotating droplets are stable under normal experimental conditions, it is possible to induce the

instability of extremely extended cylindrical threads in order to investigate the break-up of liquid

filaments, as it is observed during the decomposition and processing of incompatible polymer

blends (Sect. 4.2). Furthermore a special arrangement of the two-phase system even allows the

study of droplet coalescence. In addition, SDT should be suitable for investigations on

deformation and migration of droplets in the presence of an electric field [7], droplet migration

due to Marangoni flow in concentration and temperature gradients.

195

2. THEORETICAL BACKGROUND: EQUILIBRIUM SHAPE AND STABILITY OF ROTATING DROPS

The calculation of both ellipsoid-like and cylindrical equilibrium drop shapes and its application

to SDT measurements is presented in Sect. 2.1.

Particularly when systems exhibiting extremely low interfacial tensions are investigated, the

rotating droplets achieve thread-like shapes with very large length-to-diameter ratio. From non-

rotating systems it is known that cylindrical liquid threads spontaneously break up into smaller

droplets. Such disintegration processes would seriously disturb SDT investigations, but fortuna-

tely are not observed during standard SDT measurements. In Sect. 2.2 the stability of rotating

liquid threads is considered.

The theoretical analysis of the droplet shape and stability is strictly valid only if the rotating

system is in gyrostatic equilibrium, i. e., if the system is at rest with respect to a co-rotating

frame. Lack of gyrostatic equilibrium results from a number of disturbing influences, particularly

from buoyancy effects due to the density difference between droplet and surrounding liquid, and

can seriously limit the range of SDT applications (Sect. 2.3).

2.1. EQUILIBRIUM SHAPE IN GYROSTATIC EQUILIBRIUM

The general mathematical solution for the shape of rotating droplets is complex. If, however,

the effect of gravity is neglected the treatment is considerably simplified. In addition, it is

commonly assumed that both the droplet and the matrix phase are incompressible. Princen &

Zia & Mason [5] (in the following referred to as PZM) were the first to present both a

theoretical and experimental analysis of arbitrary drop shapes. An alternative approach was

proposed by Slattery & Chen [8]. Earlier theories on the shape of rotating drops are - though

correct in principle- less suitable for SDT applications, e. g. Ref. [9]. Apart from Vonnegut's

original paper a simplified treatment, which most directly yields Vonnegut's equation, was

presented by Couper et al. [ 10].

The equilibrium shape of rotating droplets is affected by

�9 interfacial tension y

�9 density difference A9 = 92 -9~ between the droplet (1) and the surrounding matrix (2)

�9 angular velocity co

�9 droplet volume V

196

It is possible to calculate the droplet shape for arbitrary values of the above parameters.

Conversely, for a given value of the angular velocity the radius and the length of the droplet are

sufficient to evaluate the ratio 7 /A9 by the PZM method described below. In general, the

droplet volume must not be known. To obtain the value of interfacial tension, the density

difference has to be determined by suitable techniques.

The equilibrium shape is not affected by the viscosities and the dimensions of the sample cell.

The relaxation times, however, strongly depend on the rheological properties of the liquids and

even on the dimensions of the capillary (Sect. 4.1).

While most SDT investigations deal with two-phase systems, Seeto et al. [ 11] applied SDT to

systems of three liquid phases. They describe procedures for the formation or the removal of a

third phase during spinning, classify the various possible distributions of the phases and thin-film

states and discuss their stability.

2.1.1. The balance between interfacial and centrifugal forces

According to Fig. 5a cylindrical coordinates z, r (x, y in PZM's original notation) are introduced,

assuming that the left-hand end of the droplet is placed at the origin of the coordinate system.

Now, the shape of the drop can be represented as a function r=r(z). The length of the semiaxes

is denoted by z 0, r 0 . Let 0 be the angle between the normal of the interface at z, r and the

negative z-direction. For reasons of symmetry it is sufficient to consider r(z) in the range

0 < Z < Z 0 .

The pressure P2 in the matrix phase increases with the distance r from the axis of rotation:

p2m 2r2 P2 = P0 + ~ (2)

2

At r = 0 inside the drop Laplace's equation yields

2~ pl(r = 0) = P0 + ~ , (3)

rorig

where rong is the radius of curvature of the interface at the origin (a in the original notation of

PZM).

197

3.0 , , , , ,

rO "', z

i

5 z0 > ,'

a)

25

20

1.5

1.0

0.5

0.0 , o.o 05 1'.5 ' . a ' '

Z

c~=0.59

~ c~=0.45 b)

~.0

Fig. 5 a) Coordinale system used by PZM, b) Solulions of Eq. (10b)

Thus, at r > 0 inside the drop

2 . 2 27 9~0 t

Pl - P0 + + r 2 ong

and the pressure difference at the interface

27 Apco 2r2 A p - P l - P: - -

r 2 ong

is balanced by the capillary pressure

27 Apco 2 r 2

rorlg 2 = ' ) / (K 1 + K 2 ) -

Since the principal curvatures can be expressed by

d 2 z / d r 2 d sin0

K1 [ 1 + (dz / dr) 2 ],/2 dr 2

dz / dr sin 0

r[1 + (dz / dr) 2 ]~ ~ r

we obtain

d s i n 0 sin0 + - - - 2 - otR 2

dR R

with the dimensionless radius R - r / ror,g and the shape determining parameter

(4)

(5)

(6)

(7)

(8)

198

2 3 Ape0 rong - ~ (9)

2y

may assume values in the range 0 ___ a ___ 16/27 (see below), where the limiting cases a = 0

and a - 16/27 correspond to a spherical and an infinitely extended cylindrical shape,

respectively. Integration of Eq. (8) leads to

(a) s i n 0 - RI1 - -~-- 1 (b) tan0 - d Z

The solution of this differential equation yields the shape of the rotating droplet in the

dimensionless form Z = Z(R) where Z = z/rorig (Fig. 5b). The 'true' shape z = z(r) can be

obtained by expressing the radius rong as a function of y / Ap, co and V.

2.1.2. General case: Ellipsoid-like droplets

For a detailed discussion on the case of ellipsoid-like drops the reader is referred to the original

paper of PZM. After replacing the unknown radius ror,g by the radius rsp h a couple of equations

is obtained which allows the drop shape to be computed for arbitrary values of ~ using tables

of the elliptic integrals, rsp h is the radius of a spherical droplet having the same volume as the

deformed droplet (r in the original notation of PZM).

In practice, the use of ct as a shape determining parameter combination proves to be

disadvantageous since it is subject to a 'saturation effect': Cylindrical drops of arbitrary length-

to-diameter ratio are associated with values ct ~ 16 / 27. Hence, it is convenient to introduce the

dimensionless number

Ap~ 2 - r 3 ( 1 1 ) sph Cr3ph 4y

including only parameters which are experimentally relevant. Unlike or, Cr3ph may assume

arbitrary positive values. Fig. 6a shows the dependence between tx and Cr3ph .

199

0.7

0.6 . . . . . . . . . . . . . . . . . . . . .

0.5

o . 4 t / - . . - _- ....

spl~cal: a= 0

o.lV a) O.O,t~ . . . . . . .

0 1 2 3 4 5 3

Crsph

4.0

3.5

3.0

2.5 d

2.0

~'~' 1.5

1.0

0.5

, ! | , i | ,

i b) / "

; 2

, ~

r 0 / rsp h . 0 ' i i i i

o 3

Crsph

Fig. 6 Variation of a) the parameter o~ and of b) the droplet dimensions with crs3ph, dotted lines: Eqs. (14a, b, c)

For given 7, AP and co the degree of droplet deformation can be affected by the volume V.

Even at small values of c-Apco 2/43' the drop approaches a cylindrical shape if only its

volume and hence crs3ph are chosen to be sufficiently large. Conversely, nearly spherical droplets

are obtained even in ultralow tension systems if their volume is very small.

Tab. 1 reports the value of the most important droplet parameters for various values of ot or

Cr3ph, a graphical representation is given in Fig. 6b.

2.1.3. Limiting case Cylindrical droplets and Vonnegut's equation

Although Vonnegut's equation can be obtained more directly by the variational procedure of

Couper et al. [ 10], it is instructive to derive this fundamental expression from the PZM theory.

If R - R0, it is obvious that sin e - 1, and Eq. (10a) simplifies to

otR3o -4Ro +4 - 0. (12)

At high angular velocities the cylindrical part of the droplet is characterized by d sin e / dR = 0,

e - rc /2 and R - R o and Eq. (8) becomes

otR30 -2R0 +1 = 0. (13)

The combination of Eq. (12) and Eq. (13) yields for long cylindrical droplets R 0 - 3 /2 and the

highest possible value O ~ m a x - - 1 6 / 27 of the shape determining parameter.

200

__ 1 m p ( . 0 2 3 Combining the above expressions immediately leads to Vonnegut's equation 7 z r 0 . For

cylindrical droplets Tab. 1 or Fig. 6b, respectively, can be continued by

(a) z~ 2 crs3ph + 1 r0 , 3 ,-1/3 z 0 2 -- _ tC r sph ) rsp h 3"l, Crsph)3 -1/3 (b) rsp h - (c) --r o ----(crs3ph3 + 1) (14)

Using a procedure similar to that of Couper et al., Seifert [6] has shown that Vonnegut's equa-

tion is valid even in the case of rotating gas bubbles (not depending on the specific gas proper-

ties).

2.1.4. Determination of interfacial tension from SDT data

The PZM theory was originally developed for the determination of interfacial tension from the

length and the volume of the drop avoiding the measurement of droplet diameter. At least in

principle this is a good idea. In practice, however, it is found to be very difficult to prepare

droplets of a given and accurately known volume.

For a given equilibrium shape - ellipsoid-like or cylindrical - the interfacial tension can easily be

determined from the droplet length 2z 0 and/or its diameter 2r 0 . According to Silberberg [12],

Vonnegut's equation is applicable if the length-to-diameter ratio z o / r o is larger than 3.5, which

is confirmed by the PZM theory. This 'critical' value, however, is chosen arbitrarily and depends

on the desired accuracy. If the length-to-diameter ratio is large enough the droplet diameter or

its radius, respectively, is the only profile parameter required.

If the droplet shape is ellipsoid-like the determination of both its length and its diameter is ne-

cessary, and the value of the interfacial tension can readily be obtained from Tab. 1:

�9 Calculate z 0 / r 0 and determine the corresponding values of (z 0 / rsph) and (cr3h).

�9 Since 2z 0 is known, the radius rsp h (and hence the droplet volume) is obtained from

(z 0 / rsph). In principle, r~p h can also be determined from (r 0 / r~ph). This, however, is not recom-

mended, since the experimental error of r 0 is usually larger than the error of z 0 due to the prob-

lems arising from the radius correction (Sect. 3.2).

201

�9 The desired ratio y / Ap results from y co 2 r3ph

AO 4 (crs3ph)

I f the density difference is determined by suitable techniques the value o f interfacial tension is

obtained.

Tab. 1 Table of characteristic droplet dimensions rewritten from PZM [5]. The second column of the original

table is omitted.

ot cr~3ph Z~ r~ z~

rspla rsph ro

0 0 0.05 0.0263 0.10 0.0557 0.15 0.0888 0.20 0.1265 0.225 0.1476 0.250 0.1703 0.275 0.1951 0.300 0.2222 0.325 0.2521 0.350 0.2854 0.375 0.3227 0.400 0.3653 0.425 0.4146 0.450 0.4727 0.475 0.5435 0.500 0.6330 0.525 0.7536 0.550 0.9354

1.000 1.009 1.018 1.029 1.042 1.048 1.056 1.063 1.072 1.082 1.092 1.104 1 117 1 132 1 150 1 171 1 198 1,234 1,287

1.000 1.000 0.996 1.013 0.990 1.028 0.985 1.044 0.980 1.063 0.976 1.074 0.973 1.085 0.969 1.098 0.965 1.1 ! 1 0.960 1.126 0.955 1.143 0.950 1.162 0.944 1.184 0.937 1.209 0.928 1.238 0.919 1.275 0.907 1.321 0.892 1.384 0.869 1.481

Zo r o z o O~ crs3ph m

rsph rsph ro

0.555 0.9854 1.301 0.863 0.560 1.043 1.318 0.857 0.565 1.111 1.338 0.849 0.570 1.192 1.361 0.840 0.575 1.296 1.390 0.828 0.580 1.435 1.429 0.814 0.5825 1.528 1.455 0.804 0.5850 1.648 1.488 0.792 0.5875 1.817 1.534 0.776 0.5900 2.105 1.613 0.751 0.5910 2.314 1.669 0.734 0.5920 2.739 1.781 0.702 0.5922 2.944 1.834 0.688 0.5924 3.227 1.907 0.670 0.5925 3.555 1.990 0.651 0.59255 3.869 2.068 0.634 0.59257 4.161 2.140 0.620 0.59258 4.453 2.209 0.606

1 508 1 539 1 576 1.621 1 678 1.756 1 809 1.878 1.977 2.148 2.275 2.538 2.667 2.846 3.059 3.261 3.452 3.645

Drop length measurements are sometimes disturbed by a slow movement o f the drop towards

the end of the tube, owing to instabilities in the tube and improper horizontal alignment.

Compensat ion o f this effect can be achieved by measuring the drop length twice, f rom the left to

right and vice versa.

The table o f P Z M can simply be extended [5, 13]. For the accurate and reliable evaluation o f

SDT data it is sufficient to use adequate interpolation techniques.

202

2. 2. STABILITY OF ROTATING LIQUID THREADS

This section deals with the stability of infinitely long rotating liquid threads embedded in a coro-

tating matrix. Non-rotating threads as well as cylindrical droplets of finite length are included as

limiting cases. The treatment presented below (Seifert & Wendorff [14]) is purely static, i. e.,

the resulting stability criterion only yields information whether the thread is stable or not. In the

case of instability the thread spontaneously decays into a series of isolated droplets. The break-

up dynamics are discussed in Sect. 4.2.

co 0 r0~_ ~ _ a)

I I

b)

Fig. 7 Part of the a) undisturbed, b) disturbed cylindrical liquid thread

As in the preceding section both the matrix and the thread phase are regarded as incompressible

and the system is assumed to be in gyrostatic equilibrium. In addition, the following assumptions

are made:

�9 We consider the stability of a rotating cylindrical liquid thread with circular cross section and

an undisturbed radius r 0 . The thread is assumed to be infinitely long.

�9 The shape of the thread is disturbed by a sinusoidal modulation of its diameter, the amplitude

13 of the disturbance being small and its wavelength ~, arbitrary (Fig. 7). Due to thermal

fluctuations and inevitable vibrations introduced by the motor drive such disturbances are

always present.

The disturbed cylinder radius r(z) may be written as

r(z) - r0 + 13sinkz, (15)

203

with the 'wave number' k = 2rt/9~ and an average radius ?o:

I lY 7~ r~ 2 (16)

The difference between 7 o and the undisturbed radius r o arises from the requirement of volume

conservation:

rcr02~, - 7rf[~0 + psinkz]2dz. (17) 0

The rotating system is associated with an effective potential, which includes contributions from

the interfacial and rotational energies:

1 2 U - YA-TIco (18)

For small amplitudes [3 the interfacial area A per wavelength is given by

A - A~ +/r[32[ 21r2r~ ~'1 ~, 2ro , (19)

while the moment of inertia I is found to be

I= I o -~r132Apr~)~ ; (20)

A o and I 0 denote the undisturbed quantities (13 = 0 ).

The system is stable against disturbances of a given wavelength ~,, if the potential U = U(13)

passes through a minimum at 13 = O. From this condition we obtain the desired stability

criterion:

Ape02___~3 ro2k 2 + r~ > 1. (21)

This result is identical to Rosenthal's stability criterion [9]. Rosenthal's approach is based on a

hydrodynamic stability analysis for the case of liquids of negligibly small viscosity. Even if

allowing for effects of viscosity Eq. (21) remains valid. The stability of the thread is independent

of the viscosities, since a rigid body rotation is considered. In case of instability, disturbances of

the cylindrical shape are accompanied by a decrease of potential energy. Hence, a surplus energy

204

arises enabling the disturbance to grow. Then the viscosities determine the dynamic properties

of the break-up, i. e., the growth law 13(0 of the amplitude and the wavelength )~d of the

disturbance actually observed.

Eq. (21) can be represented graphically by a stability diagram (Fig. 8). In this diagram the arc of

a quarter circle represents the borderline between stability and instability. The stability of the

rotating thread depends on the dimensionless parameter

laoro ~ g - m (22) 3'

A straight vertical line at g < 1 intersects the borderline at (r0k)c = 2rtr 0 / )~c, and the thread is

unstable with respect to small disturbances of wavelengths larger than 9~. Non-rotating threads

( e - - 0 , dotted line in Fig. 8a) are unstable against disturbances of wavelengths larger than

)~ - 2rtr 0 . This result was first derived by Plateau [2] and immediately follows from Eq. (19).

Instability arises for purely 'geometrical reasons', since for disturbances with )~ > )% the interfa-

cial area becomes smaller. The critical wavelength is not affected by the value of the interfacial

tension.

stable

1

rok a) rok

stable

0 1 0 2 2

b)

rok c) L, -,i I

stable I I I I I

0 1

~,. 0

stable

rok L I I

I

1 2 0 2

d)

Fig. 8 Stability diagram (see text)

205

Rotation has a stabilizing influence, and the critical wavelength becomes infinitely large as

approaches unity. For ~ > 1 the thread is stable against disturbances of arbitrary wavelength.

Now we consider droplets of finite length, their length-to-diameter ratio being sufficiently large,

so that end effects may be neglected. It is assumed that the droplet has achieved its equilibrium

shape. Hence, Vonnegut's equation is applicable, and it is found that ~ - 2. Obviously, the

droplet is unconditionally stable (broken line in Fig. 8a). Next we examine disturbances of this

equilibrium due to variations of A9, ~{ or m. Changes of interfacial tension and density difference

may arise, for instance, from the variation of' temperature, mass transfer across the interface and

adsorption phenomena. While these changes are relatively slow, angular velocity can be altered

very rapidly. The parameter ~ indicates departures from equilibrium: Immediately aider changing

Ag, ~, or m it is found that ~ ~ 2. Under 'normal' experimental conditions (see below) the

rotating system responds by readjusting the droplet shape until finally a new equilibrium shape

(a new equilibrium radius) is achieved. If 2 < ~ ( 1 < ~ < 2 ), e. g. due to an increase (decrease)

of angular velocity, the actual diameter of the droplet is larger (smaller) than its final equilibrium

diameter. The droplet stretches (contracts) along the axis of rotation while its diameter becomes

smaller (larger) until finally a new equilibrium shape is attained, where again ~ - 2. During the

relaxation process the shape of the droplet remains stable since 1 < ~ (Fig. 8b, c), and reliable

measurements of interfacial tension are guaranteed by the robust mechanical equilibrium of

drops in standard SDT applications.

It is possible, however, to induce instability and to observe the break-up of droplets after a rapid

change of the rotational speed, if the final angular velocity m r < m, / 2 is less than half the initial

velocity m~ (Fig. 8d). Then, ~ < 1, and the rotating thread is unstable against disturbances with

wavelengths larger than the critical value obtained from Eq. (21). The dynamics of such

spontaneous break-up phenomena are further discussed in Sect. 4.2.

2.3. APPROACH TO GYROSTATIC EQUILIBRIUM

If at lower speeds of rotation the length-to-diameter ratio of the droplet is smaller than about

3.5, Vonnegut's equation yields incorrect results and the PZM method described above should

be used. But, unfortunately, not even the PZM analysis can hold strictly at small rotational

speeds. The main reason is the presence of gravity, which is neglected in the formulas. Gravity

206

gives rise to buoyancy effects since the density of the droplet is inevitably smaller than the

matrix density. In this situation a strict rigid body rotation cannot be established. Gyrostatic

equilibrium, however, can be approached asymptotically as co--4 ~ where gravity becomes

negligible. In addition, buoyancy effects play a minor role or are even negligible if the viscosity

of the matrix phase is sufficiently high.

Departures from gyrostatic equilibrium were shown experimentally (and partly by simple

theoretical treatments) to be associated with [ 15, 16]

�9 complex flow fields (free shear layers, Ekman type boundary layers, Taylor columns) both

outside and inside the droplet owing to the interaction of buoyancy and Coriolis forces,

�9 the displacement of the droplet axis from the axis of the sample cell in both upward and

horizontal (!) direction,

�9 a lag of the droplet surface behind the rotational speed of the capillary,

�9 bizarre unexplained drop shapes and

�9 a fall-off in the apparent values of the interfacial tension at lower rotational speeds.

Beside these phenomena buoyancy effects may cause droplet migration towards either end of

the sample cell, therefore the axis of rotation should be placed accurately in horizontal direction.

Other probable causes for droplet migration are density gradients due to temperature variations

which, of course, can drive convective motions inside the rotating tube or droplet migration due

to Marangoni flow. Lack of diffusion and chemical equilibrium in multicomponent systems is

another likely cause of inconsistent SDT results.

' Vonnegut PZM, ] t a p p

I I

1

CO

Fig. 9 Dependence of apparent interfacial tension on angular velocity. The volume is decreased from (1) to (3)

207

Buoyancy effects limit the range of SDT applications at low rotational speeds, and the basic

question may be stated as follows: When is gyrostatic equilibrium approached closely enough so

that Vonnegut's equation or the PZM analysis are actually applicable?

For both ellipsoid-like and cylindrical droplet shapes the simplest operational criterion is the

constancy of the ratio 7 /A9 when 03 is varied. (Since a change of 03 is accompanied by a

change in interfacial area, there may be a time interval before the equilibrium interfacial tension

is reestablished.) This test has been described by Manning & Scriven [15] and Currie & Van

Nieuwkoop [ 1 6]. Fig. 9 shows a schematic plot of apparent interfacial tension data versus 03

(see also Fig. 3b). At low angular velocities the apparent interfacial tension falls off due to lack

of gyrostatic equilibrium. Hence, the measured values obtained from this region should be dis-

carded. Droplets of larger volume were found to provide reliable data even at lower rotational

speeds. Unfortunately, the evaluation of ~/app(03)-CUrves is time consuming when the system

equilibrates slowly after an area change.

One might suppose that gyrostatic equilibrium is established if the entire drop surface is an

equilibrium surface. To the author's knowledge this test has not been described in the literature.

As pointed out by Manning & Scriven, however, the overall resemblance to an equilibrium

shape, though necessary, is not a sufficient condition. At least in exceptional cases interfacial

tensions obtained by SDT could also be validated by showing agreement with interfacial tension

values obtained by other tensiometric methods.

The theory of spinning drops apart from gyrostatic equilibrium is not highly developed.

According to Manning & Scriven the degree to which gyrostatic equilibrium is approached is

controlled by ratios of forces and accelerations and by several other dimensionless groupings of

variables. For instance, the ratio 03 2r 0 / g , the rotational Froude number, should be very large

compared to unity as already stated by Torza [ 17]. Dimensionless analysis and the fluid mecha-

nical background yield a suitable set of independent dimensionless numbers for describing

spinning drop behaviour. Such considerations, however, may support but cannot take the place

of detailed experimental and theoretical analysis.

Up to now, a reliable procedure permitting the correction of apparent interfacial tension values

is not available. At low rotational velocities the interfacial tension values predicted by PZM are

2O8

very sensitive to small changes in the length-to-diameter ratio of the drop, and small changes in

this ratio produced by buoyancy effects will produce large changes in the apparent value of the

interfacial tension. Hence, a theory is required which takes into account the interaction between

viscous forces both inside and outside the drop, together with the interfacial force balance

equations governing the resulting deviations of the droplet shape. The correction factor would

inevitably incorporate the rheological properties of the liquids, thus leading to a rather

impractical correction procedure.

Than et al. [18] described a spinning rod tensiometer, basically a spinning drop tensiometer

provided with a solid rod at the axis of rotation. The rod pierces the drop and can help to

overcome problems arising from buoyancy effects, it reduces spin-up time and droplet migra-

tion. The effects of contact lines are shown to be negligible under suitable conditions. The idea

of spinning rod tensiometry is promising, but still needs further refinement.

3. PRACTICAL ASPECTS OF SPINNING DROP TENSIOMETRY

Beside the theoretical background outlined in the preceding section spinning drop tensiometry

includes a series of practical considerations, e. g.

�9 the design and operation of spinning drop instruments and auxiliary devices (optical

instrumentation, image processing, velocity and temperature controllers etc.),

�9 sample handling, particularly filling procedures,

�9 correction of observed droplet diameters,

�9 evaluation of interfacial tension values from SDT data (see Sect. 2),

�9 error analysis and comparison with literature values obtained by other tensiometric methods.

A detailed discussion of the above subjects, however, is far beyond the scope of this contri-

bution. In Sect. 3.1 some general aspects of modern spinning drop tensiometers are described. A

commercial standard instrument and a laboratory set-up for applications in polymer science are

presented. In addition, Sect. 3.1.2 briefly sketches problems associated with the preparation of

polymer samples. For a further description of sample preparation techniques and operating

procedures the reader is referred to [4, 13, 15 - 17, 19 - 21 ].

209

The apparent droplet diameter is measured optically, and the cylindrical lens effect of the sample

cell must be compensated by a correction factor to arrive at the true diameter. Various

alternatives have been suggested to avoid the problems associated with the detection and

correction of the diameter, for instance, the evaluation of interracial tension from the length and

the volume of the droplet [5, 17]. In general, however, the preparation of drops of accurately

known volume is more difficult than the determination of the droplet diameter. Hence, SDT

investigations cannot be made without the radius correction procedures discussed in Sect. 3.2.

The evaluation of interfacial tensions from SDT data has already been treated in Sect. 2.1.4.

Interracial tensions obtained from spinning drop tensiometry are compared with literature values

in [4, 5, 17, 19, 20, 22].

An excellent investigation on the overall precision of SDT experiments is presented by Seeto &

Scriven [22]. For drops of small diameter the accuracy of the measured ratio 7 / Ap mainly

depends on the error introduced by the determination of the droplet radius. The error in angular

velocity (= 0.1% ) dominates the total error if droplet diameters are large. In [22] an 'error map'

is given allowing for errors in droplet diameter, angular velocity and density difference. The

latter may become dominating if the densities of the droplet and the surrounding phase are very

close.

3.1. SPINNING DROP TENSIOMETERS

Since Vonnegut's original paper a number of spinning drop tensiometers has been described in

the literature [6, 13, 17, 20 - 23]. They mainly differ in

�9 the range of accessible rotational speeds (up to 25000 min -~ )

�9 use of ball or air bearings. Air bearings are particularly suited for high operating temperatures

and practically remove rotational vibrations. In addition, the heating of ball bearings and hence

the disturbing effect of axial temperature variations is avoided.

�9 the coupling between the motor and the tube holder. Problems arising from vibrations

introduced by belt couplings are overcome by use of direct drives and magnetic couplings.

�9 the range of accessible operating temperatures (from room temperature - no thermostatting

facilities- up to 350 ~

210

�9 the thermostatting facility: not available, thermostatted air or oil bath, furnaces equipped with

electrical heating elements

�9 sample cells, various lengths and diameters, fixed or removable, sealed with stoppers or

screws etc.

In several papers the design of spinning drop instruments is reported in great detail [ 17, 20, 23 ],

among them even a portrayal of a 'multicell' spinning drop tensiometer developed by Fink &

Hearn [23], a curious SDT device permitting the simultaneous investigation of six pairs of

liquids.

3.1.1. The spinning drop interfacial tensiometer SITE 04

The commercially available spinning drop tensiometer SITE 04 [24] is designed particularly for

the investigation of systems of low or moderate viscosity (e. g. polymer solutions) at 'moderate'

temperatures. Although its range of applications is limited, this instrument is state-of-the-art as

it is obvious from numerous papers concerned with surfactant and emulsifier development etc.

[25 - 37]. A schematic sketch of the SITE 04 is presented in Fig. 10. The rotating assembly is

belt driven (not shown). The instrument operates at continuously variable speed up to 10000

min -1 (optional 20000 min -1 ). In contrast to other spinning drop tensiometers the capillary of

the SITE 04 is not closed. The open ends of the tube are connected to non-rotating reservoirs

containing the matrix liquid. This permits a simple filling procedure if sufficient amounts of the

heavier phase are available. The sample cell is thermostatted by an oil bath allowing

measurements at temperatures up to to 100 ~

Prior to analysis, the denser liquid is poured into an elevated glass reservoir. Valves are opened

and the dense liquid flushes through the capillary. The valves are then closed and the capillary is

set in rotation. A droplet of the lighter phase is injected into the rotating capillary through a

septum using a syringe. After a few seconds, the droplet appears in the field of vision. Its

diameter is then measured using a built-in microscope. If the drop cannot be stopped exactly in

the field of vision, the inclination of the groundplate has to be altered by handwheel.

To repeat the measurement, the valve controlling flow of the dense phase is opened and the

droplet is removed by the heavy phase flowing through the capillary. Afterwards, a new droplet

211

can be injected. It is often necessary to allow a few minutes for the two phases to equilibrate,

especially when working at elevated temperatures. The capillary remains fixed in the instrument.

Fig. 10 Components of the SITE 04 (schematic): (1) measuring microscope, (2) oil bath for temperature control,

(3) inlet, (4) syringe, (5) septum, (6) seal, (7) bearing mount, (8) housing, (9) illumination, (10) droplet,

(11) matrix phase, (12) capillary, (13) outlet, (14) window (redrawn from [24])

The SITE 04 is factory equipped for two axes of observation. The measuring microscope is

built on the horizontal axis. The vertical axis allows mounting of a second set of optics for video

or photography of the drops. The standard instrument is fitted with two fluorescent tubes, one

on the horizontal and one on the vertical viewing axis. This permits flexible illumination without

eye strain, even over long time periods. Optional stroboscopic illumination provides a more

accurate diameter determination and observation of details on the interface.

If only small amounts of matrix liquid are available the use of an optional 'small quantity set'

('SQ') reduces the amount of heavy phase required to about 1.0 ml, the volume of the capillary.

The SQ set is particularly recommended for measurements at temperatures above 60 ~ and/or

speeds above 7000 min -1 .

The preparation of polymer-solvent systems is described below.

212

3.1.2. Sample preparation and spinning drop tensiometers for viscous liquids, polymer melts

and solutions

The SITE 04 described above does not permit studies on polymer blends. To date, a standard

instrument for SDT applications in polymer science is not available. In principle, SDT devices

for polymer melts are designed very similar to those for samples of low or moderate viscosities.

Serious problems, however, arise from high rotational velocities, operating temperatures up to

300 - 350 ~ and thermal insulation between the hot sample cell and the 'cold assemblies'

(bearings, driving unit and optical devices). But it is particularly the lack of both adequate

sample cells and preparation techniques which often makes SDT studies on molten polymers a

frustrating task.

SDT experiments on polymers were first reported in 1971 by Patterson et al. [4] who studied in-

terfacial tensions of several viscous systems at room temperature and surface tensions at

elevated temperatures. Not until 1986, a 'subsequent' study on interfacial tensions in polymer

blends was published: Elmendorp & De Vos [20] investigated the surface tensions of PMMA

and HDPE as well as interfacial tensions of PE-PS, PE-Ny 6, PE-PMMA and PS-PMMA pairs.

The samples were introduced under vacuum into the glass tube as a tightly fitting rod consisting

of a slice of the one polymer between two cylindrical pieces of the denser polymer and brought

to the required temperature in the apparatus itself.

Some years later, a very detailed description of an adequate filling procedure was given by

Joseph et al. [21 ]: To make the matrix phase, two short rods from the heavier polymer were cut

from the stock, their diameters chosen so that they fit snugly inside the glass tube. The ends of

the cylinders were accurately squared. Into one of them, a small cylindrical hole with a square

bottom was drilled. The drop phase consisted of a short cylinder of the lighter polymer with a

diameter that was slightly smaller than the hole drilled in the matrix rod. The two matrix rods

and the drop cylinder were cleaned, the latter was inserted into the matrix hole and any excess

was trimmed with a razor blade. Then, the two matrix pieces were inserted into the glass tube.

The capillary was mounted on the tensiometer and spun up. Upon melting, the two matrix

pieces fused together and the drop formed a cylindrical shape with rounded end caps. Originally,

a vacuum was applied to the polymer glass tube assembly during heating to remove from the

213

sample any entrapped air or volatile components. Further experiments showed that this

degassing was unnecessary with the improved machining of the polymers to perfect fit. The

newly developed spring loading provided an adequate pressure in the molten polymers, and no

air bubbles were observed during the experiments.

Of course, such a procedure does not at all fit the requirements of routine measurements and

proves to be completely impracticable when only small amounts of material are available. In

general, the preparation of polymer blends is yet an unsolved problem.

Fortunately, the preparation of polymer-solvent systems is much less problematic, but also

requires special care as decribed by Heinrich & Wolf [19] in a paper presenting studies on the

interfacial tensions in polystyrene-cyclohexane and polystyrene-methylcyclohexane systems.

They used the SITE 04 described in Sect. 3.1.1. The standard filling procedure (filling of the

tube with the matrix phase first and then injecting a droplet of the less dense phase) does not

work if the viscosity of the matrix phase is very high, and, hence, the authors propose two

alternative procedures: In one type of experiment they introduced the homogeneous solution at

higher temperature (UCST-behaviour!) and subsequently cooled down to the equilibrium

temperature. In the other case two coexisting phase were jointly put into the tube at the

temperature of interest. It is essential to choose the total composition such that the volume of

the droplet phase remains sufficiently small and isolated droplets can be observed. Of course,

these procedures require the detailed knowledge of the cloud point curves. The formation of

macroscopic drops from a mist of disperse droplets is achieved by rotating the tube and tilting it

alternately from one side to the other (see also Ref. [37]).

SDT instruments for polymer science are described in some detail by Elmendorp & De Vos

[20], Joseph et. al. [21], Seifert [6] and Seifert & Wendorff [14]. Although the tensiometer of

the last mentioned authors is not state-of-the-art due to the use of ball beatings and a belt drive

(currently being replaced by air bearings and a magnetically coupled drive), the design of the

rotating assemblies has proven to be most advantageous. It permits easy mounting of the

capillary and guarantees smooth running and safe operation. Presently, rotational speeds up to

15000 min -1 and operating temperatures up to 300 ~ are accessible. The instrument is

equipped with two separate shafts, one of them driven by an electric motor, the other simply

supporting the sample cell and co-rotating via adhesion. The rotational speed of the shafts is

214

detected by optical sensors providing 180 pulses per revolution, and their speeds are compared

to signal slipping of the sample cell. The axial position of the driven shaft (5a in Fig. 11) is fixed

while the co-rotating shaft (5b) can be shifted in axial direction to permit quick and easy

insertion of the sample cell. The co-rotating shaft is spring-loaded providing pressure at the

contact areas between sample cell and the conical holes at the end of the shafts. Gentle pressure

is sufficient to avoid slipping of the capillary. The capillary (80 mm long, 6 mm i. d.) is closed by

seals partly equipped with plungers similar to those described by Joseph et al. [21 ].

The furnace was designed in such a way that the front part is removable and, hence, permits an

easy access to the rotating assemblies. Excellent thermal insulation was achieved by a ceramic

housing surrounding the furnace. The metal core and the ceramic material are separated by an

air gap. At elevated operating temperatures, the gap may be flooded with cooled air. In

addition, the hot tips of both shafts are manufactured from a high performance ceramic material.

Fig. 11 Spinning drop tensiometer for viscous liquids and melts [6, 14], a) photograph of the interior with front

part of the furnace removed, b) details of the sample cell (schematic, not to scale): (1) groundplate, (2)

furnace, (3) window, (4) beating mounts, (5a) guiding shaft, (5b) driven shaft, (6) ceramic housing, (7)

glass tube, (8) stopper (details omitted), (9) seals, (10) conical hole

215

Various SDT studies using the tensiometer described above are reported in [6], some examples

are presented in Fig. 4.

3.2. DETERMINATION OF THE TRUE DROPLET RADIUS

The glass capillary acts as an cylindrical lens, and the true droplet radius r 0 has to be calculated

from the apparent radius rap p using a magnification factor

S ~ rap p / r 0 . (23)

The determination of the actual droplet radius requires special care, since in Vonnegut's

equation the radius is raised to the third power. Hence, a small fractional error in r 0 produces

about triple that fractional error in interfacial tension.

The factor M can be found by measuring the apparent diameters of calibration wires or of low-

density, cylindrical bodies of known diameters that float in the sample liquid and, therefore,

center themselves on the axis of the rotating tube [ 15, 22]. M, however, may vary considerably

with the refractive index of the matrix phase and hence with temperature and composition, so

that the experimental evaluation of M can become a time consuming task. Nevertheless, this

calibration procedure is prefered by many SDT users.

Fig. 12 Schematic sketch of a calibration body used by Seifert & Wendorff [6]

The correction factor M can be obtained alternatively by measuring the refractive indices of the

liquids. Silberberg [ 12] was the first to show that in the ideal case of a perfectly cylindrical tube,

M is given by

M d = n--2-2, (24) n o

provided that the droplet and the sample cell are concentric, n 2 is the refractive index of the

matrix liquid, n o is the refractive index of the medium outside the tube, e. g. air. Eq. (24) is re-

stricted to the case of diffuse light illumination (hence M d) which is often realized in practice

216

since most spinning drop devices incorporate light sources which are large compared to the

diameter of the capillary.

Puig et al. [38] confirmed Silberbergs's equation by an analysis based on the ray-tracing tech-

niques of geometric optics. The magnification factor is found to be independent of both the re-

fractive index and the wall thickness of the glass tube. This proves to be very convenient, since

the refractive index of liquids can be measured more quickly and accurately than calibration

measurements are performed. Eq. (24) was confirmed experimentally by measuring the apparent

diameter of cylindrical floats of known diameter in liquids of refractive indices ranging from

1.34 to 1.66. M d agreed with the value of n 2 / n o with maximum deviations of 1.4 % arising

from the eccentricity of the inner and outer cylinder surfaces of the tube.

In reality no tube is perfectly cylindrical. Most significant are departures of the inner and outer

surfaces from concentricity, azimuthal variations in the radii are generally smaller. Precision-

bore sample tubes were found to have diametrical eccentricity causing up to 0.5 % error in

apparent drop diameter. The detailed theoretical analysis of this non-ideal case is reported in

[39]. Due to non-uniformities in the glass wall, the instantaneous image of the drop is shifted

and distorted, particularly for drop diameters less than 100 ~m the image can be shitted

vertically up to 30 % away from the rotational axis, once per revolution. Since the eye is unable

to resolve the instantaneous image of the drop a synchronized strobe light is required to avoid

positional blurring. Moreover, biprecison tubing is needed to guarantee reliable measurements

[22].

Coucoulas et al. [40] discussed in detail the effect of diffuse and parallel light illumination of the

spinning tube on the correction factor. They have found that the optical arrangement of a

spinning drop tensiometer can be improved by using parallel light illumination, which in practice

is simply realized by a slit across the diffuse source parallel to the rotating tube. The wide bright

bands defining the droplet shape in diffuse light illumination now become sufficiently narrow to

give a sharp and well defined picture of the droplet surface. The correction factor Mp for

parallel light illumination, however, differs from the factor given in Eq. (24), it depends also on

the tube and droplet dimensions. For ro~ t / r~ > 3 or r 0 < ri, / 2 (rout, rin : outer, inner diameter

of the capillary) the correction factors Mp and M d are approximately equal. Large differences

217

between Mp and M d arise in the case of large drops and thin-walled capillaries.

In three-phase systems one may find the situation of a droplet inside another drop [ 11 ]. The ex-

perimental determination of the magnification factor M for the inner drop is difficult, if not im-

possible. Extending their single drop analysis Puig et al. [38] have shown that

n2 (25) M d =

n 0

where the medium (2) is now the outer droplet (the 'matrix' in which the inner droplet is

suspended), i. e., the magnification factor is independent of intermediate media which are not in

direct contact with the drop.

4. N O N - E Q U I L I B R I U M SPINNING DROP T E C H N I Q U E S

The theory of non-equilibrium spinning drop phenomena is not highly developed, although

recently several promising theoretical studies on transient phenomena have been published. The

models, however, still suffer from a very limited range of applicability. Up to now, only a few

papers report on experimental investigations of non-equilibrium phenomena. Nevertheless, there

are good reasons to examine the dynamics of rotating droplets as shown by the following

examples.

�9 Particularly if interracial tensions of polymer systems at elevated temperatures are to be

investigated, it is desirable to shorten the time of measurement by an extrapolation of the

transient state avoiding the degradation of the samples. An extrapolation method was first

proposed by Patterson et al. [4].

�9 In more complex systems the interracial tension may be time-dependent due to the adsorption

dynamics of interfacially active agents [41 ]. Hence, transient droplet behaviour is controlled by

both viscous retardation effects and changes of interracial tension due to the diffusion of

adsorbant to the interface. For instance, such phenomena play an important role in the

development of emulsifiers, cleanser and surfactant formulation. The optimization of immiscible

polymer blends and the modification of polymer surfaces often involve the synthesis of

compatibilizers (e. g. block- or statistical copolymers) and evaluation of their effect on the final

blend morphology or the surface properties [42]. These optimization processes could be

218

simplified if suitable experimental and theoretical methods were available; they should permit the

'desmearing' of spinning drop data to achieve the true adsorption dynamics.

* The blend morphology of incompatible polymers in the molten state is strongly influenced by

the interfacial tension between the liquid phases. Interfacial phenomena are of importance both

in blends of totally incompatible components and in partially miscible blends subject to phase

separation [43 - 45]. In addition, the morphology of the blends is crucially affected by the

rheological properties of the blended components. Since the evolution of real multiphase

structures often is too complex to be understood in full detail, it is desirable to study structure

formation at geometrically simple model systems. SDT not only allows a reliable determination

of interfacial tensions but, far beyond, permits the investigation of typical structure formation

processes, e. g. the spontaneous break-up of liquid threads or the coalescence of droplets [6, 14,

46].

�9 Apart from studies on the rheological behaviour of bulk liquids [21, 47], SDT can even be

used to measure the two interfacial viscosities if small oscillations are imposed on the angular

velocity. Slattery et al. [48] presented a theoretical paper on 'spinning drop interfacial viscome-

try'. Their numerical results show that this technique is suitable for the investigation of surface

viscosities at liquid-gas but not at liquid-liquid interfaces. Compared to other methods used to

determine surface viscosities, SDT offers advantages if interfaces are to be investigated at eleva-

ted temperatures.

The organization of this section is rather arbitrary. After examining the dynamics of stable

droplets (Sect. 4.1), we consider the break-up of highly elongated drops (Sect. 4.2), and finally

deal with the effect of mass transfer across interfaces and its impact on structure formation

(Sect. 4.3).

4.1. RELAXATION OF STABLE DROPLETS

In Sect. 2.2 a criterion for the stability of a rotating droplet was discussed in detail. It was found

that under 'normal' experimental conditions the shape of the droplet remains stable even if the

equilibrium situation is disturbed, i. e., a new equilibrium shape is approached by simply read-

justing the droplet dimensions. In contrast to non-rotating systems both the stretching and the

contraction of rotating drops are reversible (Fig. 13).

219

4.1.1. Phenomenological approaches

Because of the high viscosity of most commercial polymers, equilibrium profiles are obtained

only after long times with the risk of thermal degradation of the polymers [49]. A 'theory' of

upper and lower bounds for interfacial tension [50] and a theoretically based method of

exponential fitting [21 ] have been developed by Joseph et al. to overcome the problems of slow

approach to equilibrium. In addition, a model is presented allowing for the effect of time-

dependent interfacial tension in the presence of soluble or insoluble adsorbants [6, 51 ].

4.1.1.1. Upper and lower bounds for interfacial tension

Joseph et al. [50] proposed the investigation of upper and lower bounds for interfacial tension

because the equilibrium interfacial tension is not a robust function and depends on the presence

of impurities. In very viscous systems it is inconvenient to wait long enough to attain

equilibrium values. Nevertheless, the bounding method may be applicable.

Joseph et al. defined a relaxation function which can easily be measured in experiments: The

maximum diameter (D(t,c0) in the notation of the original paper) or the radius r(t, c0) of the

evolving bubble at a fixed value of the angular velocity co should approach asymptotically the

equilibrium value r 0 . Applying Vonnegut's equation an interfacial tension relaxation function,

which should approach asymptotically the equilibrium interfacial tension ~0 (independent of co

and r(t = 0, co ) ) is obtained.

Two types of relaxation, I and II, are considered:

I. Overly large initial diameter: r(0, c0) > r 0 (co)

II. Overly small initial diameter: r(0,c0) < r 0 (co)

The relaxation functions of group I lie above 70, all those in group II lie below 7o. The relaxa-

tion curves depend on r(0, c0) and co. They have a common asymptote 70 (Fig. 13). Of course,

the idea of a tension relaxation function needs further study from a more theoretical point of

view. A subsequent paper ofHu & Joseph [52] deals with this subject (see Sect. 4.1.2).

A 'method of upper and lower bounds' has been already proposed by Princen et al. [5]: They

found that an air bubble in a 0.1% solution of Carbopol 940 (Goodrich Chemical) reached a

steady but not an equilibrium shape. The final droplet dimensions depended on whether the final

220

speed was reached by increasing or decreasing 03. In such systems the appearance of upper and

lower steady state dimensions indicates the presence of yield stresses which prevent the

establishment of an equilibrium shape.

( ) ( ) group I

) group II

~

+ . a

t ~

Fig. 13 Relaxation functions of group I and group II, after [50]

4.1.1.2. Extrapolation techniques

�9 ! �9 !

upper bounds

bounds

group I, , I , , ,

time t

Observing transient droplet shapes and using correct extrapolation techniques may considerably

shorten the time of measurements. The approach to the equilibrium shape yields both a cha-

racteristic time "t of relaxation and interfacial tension 7. The first experimental SDT in-

vestigation on polymeric systems and relaxation phenomena was performed by Patterson et al.

[4]. They observed that after a short induction period there was a linear relationship between the

logarithm of the drop length l(t) and the reciprocal of time (Fig. 14a). Hence, by extrapolating

1/t--~ 0, logl(t)--~ logl 0 the (logarithm of the) equilibrium drop length should be obtained

within a short time (e. g. minutes instead of hours), if the extrapolation procedure were correct.

Similar to Patterson et al., Verdier [53] plotted the logarithm of the drop diameter against the

reciprocal of time. He also found a straight line which could be extrapolated to infinite time.

Joseph et al. [21], however, have shown that the straight line plots of Patterson et al. and

Verdier correspond to a curious relaxation function, which is inconsistent with theory as shown

by the following argument: If l(t) and 10 denote the actual and the equilibrium length of the

drop, respectively, the linear relation between logl(t) and 1 / t can be expressed as

221

"c logl 0 - logl(t) - - (26)

t

with some constant z. Then,

l(t) - 10 exp(-'c / t) . (27)

The relaxation functions, however, are necessarily exponential with decay like e x p ( - t / x ) and

not like e x p ( - ~ / t ) , if the departure from equilibrium is sufficiently small and the equilibrium

does not depend on time. Hence, it is supposed that in the final stage

r(t) - r0(t ) + (r, - r0 )exp{- ( t - t~) / x) (28)

with an initial radius r - r ( t - ti), and thus, fitting an exponential to the data should yield r 0

and x.

120 t , " , , ,

1.18 ~ %KI a)

1"16 f %,,

g ,-,4 t ~

1.101 D \ ? O

1.%! \ , \ []

1. ' 1'0 z 0 3.0

1000/t [s -l] 4.0

error

b)

Ngion of str~gl~t line pl6t

1/t

Fig. 14 a) Variation of the logarithm of drop length with reciprocal of time for a drop of polyisobutylene in

polydimethylsiloxane (redrawn from [4]), b) schematic: l(t) - exp(-t/z) represented in a log l(t)-l/t plot

Joseph et al. converted Patterson et al.'s original data to obtain the time dependence of the

droplet radius and then fitted this data to Eq. (28). They found that Patterson's straight line in

the plot is nearly identical to their fitted exponential provided that 1 / t is not too small. For

small values of 1 / t , small deviations arise. Hence, for the two methods the extrapolation to the

equilibrium radius leads to different, but not greatly different results (Fig. 14b). The method of

exponential fitting, however, seems to be more adequate as it is confirmed by simulation studies,

and works well in all cases.

222

4.1.1.3. 'SDT Relaxation spectroscopy'

Next, the frequency domain analogue to the exponential relaxation function is introduced on the

basis of a simple 'control circuit model' (Seifert [6]). This model can readily be extended to

allow for the effect of variable interfacial tension due to the presence of soluble surfactants.

Consider a rotating system, which initially is in equilibrium. The parameters r0, 03 0 and 70 refer

to the undisturbed equilibrium state. The system may be disturbed simultaneously by small

changes of angular velocity A03 (t) = 03 (t) - 03 o and interfacial tension AT( t )=y ( t ) -7o

resulting in a small change of the droplet radius Ar(t)= r ( t ) - r 0 . For simplicity, the density

difference between droplet and matrix phase is taken to be constant. From Vonnegut's equation

we obtain

lr0 Ar ( t ) - 2 r o k03(t)+ AT(t) (29) 3o,0

First we consider the case of constant interfacial tension, and the second term on the right hand

side of Eq. (29) is dropped. In the following, A03(t) and At(t) will be interpreted as 'input sig-

nal' and 'output signal' of the overall spinning drop system (Fig. 15).

Am "-[ Fv(S) -I aco + Fv(s)

Ar

a) b) FA(S)

Ar

Fig. 1 5 Block diagrams of spinning drop systems: a) 'simple' system, b) system allowing for adsorption

dynamics

As expected, a small increase of angular velocity is accompanied by a small decrease of the

droplet radius. Eq. (29) does not account for the delayed response of the output Ar(t). The

most simple model consistent with the exponential relaxation function suggested above is de-

scribed by an ordinary linear differential equation for Ar(t) :

d 2 r 0 (30) X v - Ar + A r - Ao. dt 3 o 0

223

The Laplace transform of Eq. (30) yields the transfer function of the rotating system:

Ar(s) _ 2 r o 1 (31) F v ( s ) - Ao3 (s-----) - 3 o30 1 + l:vS

Seifert [6] investigated the frequency response of an air bubble in a viscous silicone oil by a

sinusoidal modulation of the angular velocity:

Am(t) - Am m a x cos{fit}. (32)

The resulting change of the bubble radius was also found to be sinusoidal as it is expected from

the theory of linear response:

Ar(t) - Armax(~) cos{~t + ~)(~)}, (33)

with both the amplitude Arma• and the phase lag (b(fl) depending on the modulation fre-

quency fl. The Bode diagrams in Fig. 16 show a fairly good agreement with the theoretical

prediction and confirm the assumption of an exponential relaxation function. Although very pro-

mising in principle, 'SDT relaxation spectroscopy' proves to be a time consuming method which

presumably does not lend itself for routine measurements.

1

0" 101.1

. . . . . . . . ! . . . . . . . .

. . . . . . . . 1 . . . . . . . . 10 f~/f~

20

0 b)

-80 ~ ~

' ~ 1 7 6 . . . . . . . . i . . . . . . . . 7o

Fig. 16 Bode diagrams: Frequency response of a rolating droplet-matrix system: a) amplitude, b) phase lag, ~c

is the cut-off frequency of the 'low-pass filter'

In the presence of soluble adsorbants the behaviour of the rotating system is generally more

complex due to the fact that now the interfacial tension may vary with time. A relatively simple

224

situation, however, is given when in low viscosity systems the adsorption dynamics are very

slow compared to viscous retardation effects: Then, at any instant, the rotating system is in

mechanical equilibrium, and the slowly varying droplet dimensions immediately indicate the

variation of interfacial tension with time (Fig. 17). In viscous systems the actual droplet dimen-

sions do no longer represent the actual value of interfacial tensions due to a superposition of

both viscous and adsorption dynamics. Hence, the droplet dimensions l(t) or r(t) have to be

'desmeared' to arrive at the true interfacial tension relaxation function y(t).

The relaxation functions of group I lie above 3'0, all those in group II lie below 3'0. The relaxa-

tion curves depend on r(0,03) and 03. They have a common asymptote 70 (Fig. 13). Of course,

the idea of a tension relaxation function needs further study from a more theoretical point of

view. A subsequent paper ofHu & Joseph [52] deals with this subject (see Sect. 4.1.2).

A 'method of upper and lower bounds' has been already proposed by Princen et al. [5]: They

found that an air bubble in a 0.1% solution of Carbopol 940 (Goodrich Chemical) reached a

steady but not an equilibrium shape. The final droplet dimensions depended on whether the final

speed was reached by increasing or decreasing 03. In such systems the appearance of upper and

lower steady state dimensions indicates the presence of yield stresses which prevent the

establishment of an equilibrium shape.

55

45 0

~'50

qzL

' ' ' ' 8 0

t [min]

17 Time-dependent surface tension of an aqueous solution of polyethylene glycole (5 wt % PEG 200) Fig.

0 , , ,

Seifert [6] has proposed a phenomenological model allowing for the adsorption of both soluble

and insoluble components. The model is based on the theory of linear response, and the rotating

225

droplet-matrix system is considered as a feedback control system (Fig. 16b). The transfer

function Fv(s ) describes the purely 'viscous response', while F A (s) in the feedback loop re-

presents the adsorption dynamics, i. e., it describes the changes of interfacial tension due to a

variation of the droplet dimensions and, hence, of its interfacial area. Then, the transfer function

of the overall system can be expressed by

Ar(s) Fv(s ) = (34)

Ao3 (s) 1 + F v (s) F A (s)

If Fv(s ) is known it is possible to distinguish between changes of the droplet shape due to

viscous retardation and time-dependent interfacial tension due to diffusion of compatibilizers to

the interface. In a forthcoming paper these ideas are discussed in more detail [51 ].

4.1.2. Theoretical approaches

An adequate hydrodynamic analysis of spinning drop behaviour is difficult, even if 'simple'

systems (Newtonian liquids, constant interfacial tension) are considered, and even if gravity is

neglected and assumed that gyrostatic equilibrium is maintained during the relaxation process.

Hsu & Flumerfelt [47] were the first to propose the spinning drop technique for the investiga-

tion of rheological properties. They modelled the flow problem as the deformation of the drop

under a squeeze of the surface and derived an evolution equation for the radius of the drop. The

flow inside and outside the drop was assumed to be purely extensional, shear stresses at the

surface of the drop were neglected. Comparison with experimental data shows that these

simplifications are generally inadequate, hence the applicability of the Hsu-Flumerfelt theory is

restricted to a very limited range of experimental situations.

More recently, Joseph et al. [21 ] have developed a theory of relaxation for both Newtonian and

non-Newtonian fluids. They modified the derivation of Hsu and Flumerfelt by taking into

account the inertia terms and the shape of the end tips of the droplet and derived a relaxation

function for systems of Newtonian liquids sufficiently close to their equilibrium shape. Their

result, however, lacks agreement with their experimental data and yields a faster relaxation than

actually observed.

226

To date, the direct numerical simulation of the full Navier-Stokes equations for the rotating

droplet-matrix system by Hu & Joseph [52] is the most exciting study since it excellently fits the

situation of real spinning drop investigations of highly viscous Newtonian systems. Their

simulation results yield in-depth information on the flow both inside and outside the droplet. The

flow inside the drop is found to be essentially extensional. The shear stress on the drop surface,

which was neglected in the papers mentioned above, proves to be very important. The

relaxation of the droplet radius is basically exponential. Based on the numerical results the

authors developed a simplified theory of relaxation, which takes into account the shear stresses

on the cylindrical drop surface. The exponent of relaxation depends on the interfacial tension,

the equilibrium radius of the drop, the viscosities of the droplet and the surrounding liquid, the

length-to-diameter ratio of the drop and on the ratio of the radius of the sample cell to the

radius of the drop. This theory agrees better with experimental observation than the theory of

Joseph et al. [21]. Transient measurements of the drop shape can be used to extract the

rheological properties as well as the interfacial tension of the system.

In a very recent paper Lister & Stone [54] examined the various contributions to the viscous

stresses resisting droplet deformation using scaling arguments. They identified a number of

asymptotic flow regimes ('bubble, pipe, sliding-rod, toffee-strand limits'). For the limit of negli-

gible interfacial tension effects, similarity solutions are developed. Their analytical results are in

good agreement with numerical simulations based upon a boundary-element solution to the full

viscous flow equations.

4. 2. SPONTANEOUS BREAK-UP OF LIQUID THREADS

In contrast to the preceding section, we will now deal with the dynamic behaviour of unstable

liquid threads or thread-like extended drops, respectively. In Sect. 2.2 it was shown that non-

rotating threads are unstable against disturbances of wavelengths larger than 2nr0, where r 0

denotes the radius of the undisturbed cylindrical thread. In addition, rotating cylindrical drops

were found to be unconditionally stable if either equilibrium is established or the shape

determining parameters are changed sufficiently slowly. It will be shown, however, that

instability may be induced. As a consequence, the rotating thread decays into a series of isolated

227

drops with nearly spherical shape. This phenomenon is closely related to the decay of non-

rotating threads.

4.2.1. Non-rotating threads: The breaking thread method

The determination of polymer interfacial tension by the spontaneous break-up of liquid threads

was first proposed by Chappelear [55] and subsequently used by Rumscheidt & Mason [56],

Pakula et al. [57] and Elmendorp [44, 58]. In the last years, this method achieved renewed

interest [45, 59]. In contrast to the pendent or the spinning drop technique, the breaking thread

method does not require a density difference between the two phases, which is ot~en difficult to

measure with sufficient accuracy. Apart from the embedded fiber retraction method (short

fibers) used by Carriere et al. [60, 61], the 'classical' breaking thread method is based on the

investigation of very long polymer threads embedded in a polymer matrix, where the disperse

and the matrix phase are in the molten state. Both methods are dynamic since there is no

equilibrium near to a thread. The breaking thread method does not use thread break-up - as it

might be expected - but concentrates on the initial stages of interfacial tension driven defor-

mation. It is commonly assumed that the purely Newtonian theory of Tomotika [62] applies,

and effects of elasticity and gravity are neglected. In addition, the simultaneous contraction of

long fibers as well as end-pinching phenomena observed in the experiments are not considered

in the theory.

6

r

4 2

8

2

, i i ,

:2 ; 5 4 log (lh / ~)

Fig. 18 Dominating wavelength obtained from Tomotika's theory plotted versus the logaritm of the viscosity ratio

228

The complete set of equations obtained from Tomotika's theory is too lengthy to be reported

here in detail. It is based on a linear hydrodynamic stability analysis. For the early stage of the

break-up process an exponential growth ]3(t)- exp(t/x) of the disturbance amplitude (comp.

Sect. 2.2) is found:

z- ~ = y (1 - x 2 ) ~(x, ~t~ / g2). (35) 2r0~t2

la 1 and g2 are the viscosities of the thread and the matrix, respectively, x = r0k is the dimen-

sionless wavenumber. For the equations defining the function ~(x,~ h /g2) the reader is re-

ferred to the original paper [62].

If the thread is stable (unstable), negative (positive) values of x -1 are obtained. Thus, the theory

predicts the existence of the critical wavelength ~ = 2zcr 0 in agreement with the static analysis

presented in Sect. 2.2. In addition, a dominant wavelength )~a > )~ is found. )~d is the

wavelength for which the growth factor x --~ becomes maximum, depending on the viscosity

ratio g~ / g2, but not affected by interfacial tension. It merits particular emphasis that 3~ d passes

through a minimum at g~ / g2 ~ 1. As g~ / g2 --~ 0 or g~ / g2 --~ oo the wavelength )~a tends to

infinity. If the viscosities are known, interfacial tension can be determined from the experimental

observation of the growing amplitude 13(0 and the dominant wavelength )~d.

4.2.2. Break-up of rotating liquid threads

In Sect. 2.2 it was shown that a rapid change of angular velocity may induce instability if the

final rotational velocity is less than half the initial velocity. Seifert & Wendorff [ 14] were the

first to study the decay of rotating liquid threads in a SDT instrument. The spontaneous break-

up was investigated at highly extended cylindrical droplets (10 mm long, 0.1 mm radius) obser-

ved in a polymer-solvent system close to the critical point of mixing. The rotating system was

allowed to equilibrate at an initial velocity o i = 3500 min -~ , and instability was induced by a

rapid change of velocity down to a final velocity of = 1000 min -1. The growth of the

disturbance and the final break-up into a series of isolated droplets are shown in Fig. 19. For this

experiment a critical wavelength )~r = 0.76 mm was calculated from Eq. (21). The droplets

229

were found to be at equal distances indicating a dominant'wavelength X a = 1.56 mm. In

addition, smaller droplets - usually referred to as 'satellites'- are observed. These satellite drops

are well known from experiments on non-rotating liquid threads implying that both situations

are closely related [63 ].

Tomotika's theory can easily be extended to the case of a rotating liquid thread, if it is assumed

that the system remains in gyrostatic equilibrium even during break-up. If the process is

sufficiently slow this assumption should be justified. The effect of centrifugal forces only enters

the boundary condition for the normal stress difference at the interface, and the modified linear

stability analysis yields

12 1;-1 . . . . 2robt2 3/ (I)(x, ILl, 1 / ~1, 2 ). (36)

rhis result is in agreement with the stability condition Eq. (21).

Fig. 19 Break-up of a rotating liquid thread after a rapid decrease of angular velocity (time interval 30 s)

More recently, similar experiments have been described by Quirion & Pageau [64] who have

investigated the static and dynamic properties of liquid threads in blends of polyethylene glycole

and polypropylene glycole. These authors discuss in detail the competition between capillary

waves and end-pinching phenomena.

230

4. 3. STUDIES ON THE EFFECT OF MASS TRANSFER ACROSS INTERFACES

SDT particularly lends itself for the study of mass transfer phenomena across droplet interfaces.

In Sect. 4.3.1 the investigations of Heinrich & Wolf [65] are described providing valuable

insight in the dynamics of extraction processes. In polymer science such processes have only

recently been developed.

Structure formation in polymer blends is usually considered assuming a spatially uniform

interfacial tension. Interfacial tension gradients, however, may arise due to a non-uniform

temperature distribution or a spatial variation of the interfacial composition. They are of major

importance in non-equilibrium systems where heat or mass is transferred across the interfaces,

and they considerably affect structure formation processes, e. g. via Marangoni flow [66, 67].

4.3.1 Mass transfer across the interface of a 'stable' droplet

Heinrich & Wolf [65] investigated the reaction of a phase separated polymer solution on a rapid

rise in temperature. Their measurements yield an apparent interfacial tension as a function of

time and demonstrate how new phase equilibria are achieved. They started from an equilibrium

situation at an initial temperature with two phases coexisting inside the sample cell. The overall

composition of the polymer-solvent system was chosen such that a small increase in temperature

was sufficient to leave the miscibility gap. Hence, the mixture became homogeneous after

sufficiently long times, i. e., the drop completely dissolved but remained 'stable', i. e., it did not

decay into smaller droplets. The homogenization process involves the diffusion of polymer and

solvent molecules across the interface. In the first stage the transport of matter is driven by the

gradient of the chemical potential within the interfacial layer leading to the establishment of a

local equilibrium. In a subsequent step diffusion is driven by the gradient of chemical potential

outside the interfacial area. In the final stage the system has reestablished equilibrium, it is now

homogeneous and the droplet has completely dissolved.

4.3.2 The effect of mass transfer on break-up and coalescence

By means of SDT experiments Seifert et al. [46] investigated the mass transfer in ternary oligo-

meric systems and its impact on the break-up of liquid threads and the coalescence of droplets.

They started from incompatible components A and B, mixing them thus yielded a two-phase

structure. A third component C, compatible with A and B, was introduced (A, B, C: several

231

polyethylene and polypropylene glycoles). The compatibilizing effect of C was confirmed by

interfacial tension measurements. Adding C to the binary blend A/B leads to a considerable

decrease of the equilibrium interfacial tension.

Ternary blends, in which interfacial tension gradients may arise, can be obtained by the

following procedure: blending A and C yields a homogeneous mixture, afterwards injected to

form a droplet in a surrounding matrix of pure component B. Since C is also compatible with B,

diffusion of C across the interface is observed, and the transfer may produce interfacial tension

gradients due to non-uniform concentration of C along the interface. An analogous procedure

leads to transfer from the matrix into the disperse phase, if pure component A is injected into a

homogeneous blend B/C.

Fig. 20 Preparation of the 'droplets'-matrix system for the study of coalescence

The theoretical and experimental background of the mass transfer experiments are reported in

[6, 46]. It is obvious from the experimental results that structure formation in a viscous ternary

model system is considerably affected by a combination of interfacial and transport phenomena.

In particular, specific transport processes arising due to the departure from phase equilibrium

permit the control of morphology via the Marangoni effect. The mass transfer was found to

stabilize or destabilize the shape of cylindrical liquid threads depending on the direction of

transport (from the thread into the matrix or vice versa).

232

A special arrangement of the two-phase system even allows the simulation of coalescence. Both

the measurement of interracial tension and the simulation of thread break-up are based on the

investigation of single drops completely surrounded by the matrix liquid. In contrast, the two-

phase system may be prepared according to Fig. 20 in order to study droplet coalescence: Two

threads of disperse phase liquid extend from the stoppers at the end of the capillary to the

interior of the sample cell. The distance between their semispherical ends may be controlled by

the variation of angular velocity. Thus, it is possible to approach the cylinders in a well defined

manner. In the region of contact the matrix phase is squeezed out until finally the remaining

liquid layer becomes unstable. It is obvious that this situation is very similar to the coalescence

of two isolated drops [68]. In the presence of mass transfer the time required for coalescence

was found to be remarkably shorter or longer (depending on the direction of transport)

compared to systems in phase equilibrium.

5. CONCLUDING REMARKS

It has been shown that SDT is a well established method for the investigation of systems exhibi-

ting even extremely low interracial tension and relatively low viscosities. Such systems can be

studied using commercially availabe spinning drop tensiometers. The application of SDT to

molten polymers is less advanced, nevertheless very promising, and requires further refinement.

The evaluation of interracial tension values from the droplet profile is simple, if the system is

sufficiently close to gyrostatic equilibrium, and the stability of the droplet shape guarantees

reliable experimental results.

Non-equilibrium spinning drop tensiometry has been shown to be of major importance if highly

viscous liquids are to be investigated or if interracial tension varies with time, e. g. due to ad-

sorption phenomena. Recently, several theoretical and experimental studies on time-dependent

droplet behaviour have been published. The results are encouraging, but still suffer from a very

limited range of applicability.

Furthermore, SDT has been found to permit investigations far beyond 'simple tensiometry'. It

allows the investigation of structure formation processes (break-up of liquid threads, coales-

cence of droplets) even in complex systems far from phase equilibrium. The methods, however,

are relatively new and require further development.

233

6. LIST OF SYMBOLS

LA TIN SYMBOLS

A, A 0 surface area of disturbed, (0) undisturbed liquid thread

C~h dimensionless parameter determining the droplet shape, c - Apco 2

4T - ~ , see also %h

transfer functions of spinning drop 'subsystems': (v)iscous response, (a)dsorption

I, I o moment of inertia of disturbed, (0) undisturbed liquid thread

k = 2rt / )~ wavenumber

1 = l(t), 10 length of droplet, (0) equilibrium length

M, Ma, Mp radius correction factor: M = rap p / r , (d)iffuse, (p)arallel lighting

n0, n2 refractive index of the (0) medium outside the capillary, (2) matrix phase

P0, Pl, P2 pressure in the (0) matrix at r = 0, (1) droplet, (2) matrix at r > 0

radial cylindrical coordinate

r = r(z) shape function (cylindrical coordinates)

r ( t ) , r(t, co) time-dependent droplet radius

R = R(Z) with R = r / rong and Z = z/rong: shape function in dimensionless form

equilibrium radius of ellipsoid-like and cylindrical droplets, radius of

an undisturbed cylindrical liquid thread

R 0 = r 0 / ror~g equilibrium radius in dimensionless form

average radius of a disturbed liquid thread

r ~ (app)arent droplet radius

r i initial droplet radius

rin , rou t (in)ner, (out)er radius of glass capillary

234

ro~ radius of the droplet at the (orig)in

rsph radius of a sphere with volume equal to the volume of the rotating droplet

U mechanical potential

V droplet volume

X=rok normalized wavenumber

axial cylindrical coordinate

Z 0 half of the equilibrium drop length

GREEK SYMBOLS

dimensionless parameter determining the droplet shape

amplitude of sinusoidal disturbance (Sect. 2.2)

interfacial or surface tension

70 equilibrium interfacial tension

Y app apparent interfacial tension (Sect. 2.3)

Ap = 192 --191 density difference

Ar(t), Am (t) small deviations from equilibrium or average values

Ay (t), A~(t)

Armax, Aft)max amplitudes of sinusoidally modulated radius and angular velocity

dimensionless parameter characterizing stability of rotating liquid threads

angle between normal to the droplet interface and z-axis

K 1 ~ K2 principal curvatures of droplet interface

wavelength of disturbance, (c)ritical, (d)ominating

235

~'~1, ~J'2 viscosity of (1) droplet or thread, (2) matrix

Pl , 102 density of (1) droplet or thread, (2) matrix

"~, "I~ V characteristic time, due to (v)iscous effects

= ~)(f2) phase lag

function defined by Tomotika, see Ref. [62]

�9 (x, ~, / ~t~)

03, 031, 03f angular velocity of the rotating system, (i)nitial, (f)inal

modulation frequency, (c)ut-off

7. REFERENCES

1. Beer, A., Poggendorffs Annalen der Physik und Chemie, 96 (1855) 210

2. Plateau, J. A. F., "Statique Experimentale et Theorique des Liquides ...",

Gauthiers-Villars, Paris, 1873

3. Vonnegut, B., Rev. Sci. Instrum., 13 (1942) 6

4. Patterson, H. T., Hu, K. H., Grindstaff, T. H., J. Polym. Sci. C, 34 (1971) 31

5. Princen, H. M., Zia, I. Y. Z., Mason, S. G., J. Colloid Interf. Sci. 23 (1967) 99

6. Seifert, A. M., Ph.D. thesis, Technical University of Darmstadt, FRG, 1993

7. Garton, C. G., Krasucki, Z., Proc. Roy. Soc. A, 280 (1964) 211

8. Slattery, J. C., Chen, J. D., J. Colloid Interf. Sci., 64 (1978) 371

9. Rosenthal, D. K., J. Fluid Mech. 12 (1962) 358

10. Couper, A., Newton, R., Nunn, C., Colloid Polym. Sci. 261 (1983) 371

11. Seeto, Y., Puig, J. E., Scriven, L. E., Davis, H. T., J. Colloid Interf. Sci., 96 (1983) 360

12. Silberberg, A., Ph.D. thesis, University of Basel, Switzerland, 1952

236

13. Cayias, J. L., Schechter, R. S., Wade, W. H., in "Adsorption at Interfaces",

ACS Symposium Series, 8 (1975) 234

14. Seifert, A. M., Wendorff, J. H., Colloid Polym. Sci. 270 (1992) 962

15 Manning, C. D., Scriven, L. E., Rev. Sci. Instrum., 48 (1977) 1699

16 Currie, P. K., Van Nieuwkoop, J., J. Colloid Interf. Sci., 87 (1982) 301

17 Torza, S., Rev. Sci. Instrum., 46 (1975) 778

18. Than, P., Preziosi, L., Joseph, D. D., Arney, M., J. Colloid Interf. Sci., 124 (1988) 552

19. Heinrich, M., Wolf, B. A., Polymer, 33 (1992) 1926

20. Elmendorp, J. J., De Vos, G., Polym. Eng. Sci., 26 (1986) 415

21 Joseph, D. D., Arney, M. S., Gillberg, G., Hu, H., Hultman, D., Verdier, C.,

Vinagre, T. M., J. Rheol., 36 (1992) 621

22. Seeto, Y., Scriven, L. E., Rev. Sci. Instrum., 53 (1982) 1757

23 Fink, T. R., Hearn, D. P., Rev. Sci. Instrum., 49 (1978) 188

24. KrOss GmbH, Spinning Drop Tensiometer SITE 04, 22453 Hamburg, FRG

25. Zhou, J. S., Dupeyrat, M., J. Colloid Interf. Sci., 134 (1990) 320

26. Plucinski, P., Nitsch, W., J. Colloid Interf. Sci., 154 (1992) 104

27. Vermeulen, M., Joos, P., Ghosh, L., J. Colloid Interf. Sci., 140 (1990) 41

28. Koczo, K., Lobo, L. A., Wasan, D. T., J. Colloid Interf. Sci., 150 (1992) 492

29. Aveyard, R., Binks, B. P., Fletcher, P. D. I., Lu, J. R.,

J. Colloid Interf. Sci., 13 9 (1990) 128

30. Aveyard, R., Binks, B. P., Lawless, T. A., Mead, J.,

Can. J. Chem., 66 (1988) 3031

31. Aveyard, R., Binks, B. P., Lawless, T. A., Mead, J.,

J. Chem. Soc. Faraday Trans., 1, 81 (1985) 2155

237

32. Aveyard, R., Binks, B. P., Mead, J.,

J. Chem. Soc. Faraday Trans., 1, 83 (1987) 2347

33. Aveyard, R., Binks, B. P., Mead, J.,

J. Chem. Soc., Faraday Trans., 1, 82 (1986) 1755

34. Hofman, J. A. M. H., Stein, H. N., J. Colloid Interf. Sci., 147 (1991) 508

35. Nickel, D., Nitsch, C., Kurzend6rfer, P., von Rybinski, W.,

Progr. Colloid Polym. Sci., 89 (1992) 249

36. Kutschman, E. M., Findenegg, G. H., Nickel, D., von Rybinski, W.,

Colloid Polym. Sci., 273 (1995) 565

37. Heinrich, M., Wolf, B. A., Macromolecules, 25 (1992) 3817

38. Puig, J. E., Seeto, Y., Pesheck, C. V., Scriven, L. E.,

J. Colloid Interf. Sci., 148 (1992) 459

39. Seeto, Y., M. S. thesis, University of Minnesota, Minneapolis, Minnesota, 1976

40. Coucoulas, L. M., Dawe, R. A., Mahers, E. G., J. Colloid Interf. Sci., 93 (1983) 281

41. Dukhin, S. S., Kretzschmar, G., Miller, R., "Dynamics of Adsorption at Liquid

Interfaces", Studies in Interface Science, l, Elsevier, Amsterdam, 1995

42. Noolandi, J., Hong, K. M., Macromolecules, 15 (1982) 482

43. Utracki, L. A., "Polymer Alloys and Blends", Hanser Publishers, Munich, 1989

44. Elmendorp, J. J., Ph.D. thesis, Technical University of Delt~, Netherlands, 1986

45. Janssen, J. M. H., Ph.D. thesis, Technical University of Eindhoven, Netherlands, 1993

46. Dappa, A. M., Seifert, A. M., Wendorff, J. H.,

Z. f. Physikalische Chemie, 184 (1994) 219

47. Hsu, J. C., Flumerfelt, R. W., Trans. Soc. Rheol., 19 (1975) 523

238

48. Slattery, J. C., Chen, J.-D., Thomas, C. P., Fleming III, P. D.,

J. Colloid Interf. Sci., 73 (1980)483

49. Wu, S., "Polymer Interface and Adhesion", Marcel Dekker, New York, 1982

50. Joseph, D. D., Arney, M., Ma, D., J. Colloid Interf. Sci., 148 (1992) 291

51 Seifert, A. M., in preparation

52. Hu, H. H., Joseph, D. D., J. Colloid Interf. Sci., 162 (1994) 331

53 Verdier, C., Ph.D. thesis, University of Minnesota, Minneapolis, Minnesota, 1990

54. Lister, J. R., Stone, H. A., J. Fluid Mech., 317 (1996) 275

55. Chappelear, D. C., ACS Div. Polym. Chem. 5/2 (1964) 363

56. Rumscheidt, F. D., Mason, S. G., J. Colloid Sci. 17 (1962) 260

57. Pakula, T., Grebowicz, J., Kryszewski, M., Polymer Bulletin 2 (1980) 799

58. Elmendorp, J. J., Polymer Eng. Sci., 26 (1986) 418

59. Elemans, P. H. M., Janssen, J. M. H., Meijer, H. E. H., J. Rheol., 34 (1990) 1311

60. Carriere, C. J., Cohen, A., Arends, C. B., J. Rheol., 33 (1989) 681

see also: Elemans, P. H. M., Janssen, J. M. H., J. Rheol., 34 (1990) 781

61 Carriere, C. J., Cohen, A., J. Rheol., 35 (1991) 205

62. Tomotika, S., Proc. Roy. Soc. (London), A150 (1935) 322

63 Tjahjadi, M., Stone, H. A., Ottino, J. M., J. Fluid Mech., 243 (1992) 297

64. Quirion, F., Pageau, J., J. Polym. Sci. B, 33 (1995) 1867

65. Heinrich, M., Wolf, B. A., Macromolecules, 26 (1993) 6106

66. Berg, J. C., Canadian Metallurgical Quarterly, 21 (1982) 121

67 Miller, C. A., Neogi, P., "Interfacial phenomena", Surfactant Science Series, 17,

Marcel Dekker, New York, 1985

68. Elmendorp, J. J., Polym. Eng. Sci., 26 (1986) 1332

Drops and Bubbles in Interfacial Research D. M6bius and R. Miller (Editors) �9 1998 Elsevier Science B.V. All rights reserved. 239

CAPILLARY PRESSURE TENSIOMETRY AND APPLICATIONS IN MICROGRAVITY

Libero Liggieri and Francesca Ravera

Consiglio Nazionale delle Ricerche, Istituto di Chimica Fisica Applicata dei Materiali,

Via De Marini 6, 16149 Genova - Italy

Contents

1 Introduction

2. Working Principle of the Capillary Pressure Tensiometry

3 The Pressure Derivative Method

3 1 Pressure Derivative Results

4. The Expanded Drop Method

4.1 Interface Dilation

4.2 Expanded Drop Results

5. The Growing Drop Method

5.1 Growing Drop Results

6. Other Capillary Pressure Methods

7. Critical Experimental Aspects

7.1 Pressure Measurement

7.2 Determination of the Pressure Offset

7.3 Capillary Wettability

7.4 Drop Deformation

8. Interpretation of the Capillary Pressure Tensiometry Data

8.1 Expanded Drop Data Interpretation

8.2 Growing Drop Data Interpretation

9. Experiments in Microgravity: the FAST Facility

10. References

11. List of Symbols

240

1. INTRODUCTION

Surface/interracial tension, y, has its origin in the anisotropy of the molecular interactions in the

interracial layer caused by the enormous density gradients in a very thin layer (~10 .7 cm) [ 1 ].

According to that, the pressure tensor is homogeneous in the core of the bulk phase, while its

normal (PN) and tangential components (PT) become different in the interfacial layer. These

concepts are formalized in the Bakker equation which expresses the surface tension of a plane

layer as

+ o o

--O0

where z is the coordinate normal to the interface. This equation has been generalized for

interfaces with arbitrary geometry [2,3 ].

This close link with the microscopic features of the interracial layer makes surface/interracial

tension one of the most important parameters in the study of the physical-chemistry of interfaces

both from the equilibrium and dynamic point of view.

In multi-component systems, at equilibrium, the surface tension is linked to the adsorption at the

interface of each component which is classically described in the Gibbs model [4] as a surface

excess concentration.

In the particular case of dilute solution of a single surfactant the adsorption F is linked to the

surface tension and to the bulk concentration, c, by the Gibbs adsorption isotherm

1 r = - ~ (2)

Rg r c3(ln c)

where Rg is the gas constant and T the absolute temperature. This equation gives direct access

to the value of F by equilibrium measurements.

An important field of investigation concerns the dynamic aspects of the adsorption in surfactant

solutions [5]. These studies can give important information about the surfactant, the surface,

their mutual interaction and the interaction with the bulk. They also have an impact on an

increasing number of technologic applications. The main experimental tool for investigating

these phenomena is the measurement of dynamic surface/interracial tension.

From a mechanical point of view, surface tension sets a pressure difference, called capillary

pressure (APcap), across curved interfaces

241

APcap - Y + (3)

where R I and R 2 are the principal radii of curvature of the interface.

Several techniques for indirectly measuring S.T. through the effects of the capillary pressure, for

example the drop shape [6-9] and the capillary rise [ I0] methods, have been in use for many

years. Other methods are based on the balance of capillary forces with the liquid weight as the

Du NoiSy Ring, Wilhelmy Plate (pulling methods) and Drop Weight/Volume [ 11,12].

Although all these methods were initially conceived for measuring static surface tension, they

have been adapted with variable success to measure dynamic surface tension [ 13 ].

Over the last few years, the development of pressure transducers for liquids, which are able to

measure very low pressures with great accuracy, has given the possibility to evaluate surface

tensions by the direct measurement of the capillary pressure. Therefore a new class of methods

has been developed using this approach called Capillary Pressure (CP) tensiometry. These

methods are nowadays one of the most promising and flexible tensiometric tool for dynamic

adsorption studies.

CP tensiometry is especially helpful for studying liquid/liquid interfaces. Indeed, only a few of

the traditional surface tension measurement techniques can be suitably used for such interfaces,

this is one of the reasons for the small number studies published on this important subject.

Basically the same experimental apparatus - the Capillary Pressure Tensiometer (CPT) - can be

used according to different experimental methodologies to access different physicochemical

aspects characterizing the interface of a single sample.

The Pressure Derivative method [14] has thus been conceived for measuring the interfacial

tension of pure liquids. To study the dynamic aspects of the adsorption the Growing

Drop/Bubble [15-19] and the Expanded Drop [20-23] methods have been developed. The

rheological properties of the interface can be studied by the Oscillating Bubble [24-26] and the

low amplitude Stress/Relaxation.

A schematic classification of these methods is given in Table 1 on the basis of their properties.

The Static Maximum Bubble Pressure method has been the first CP method developed.

However its dynamic version [27,28] is essentially an integral method in which the surface

tension evolution is deduced by the bubbling times.

242

Table 1" Principal Capillary Pressure methods.

Method

!

Maximum Bubble Pressure

Pressure Derivative

Growing Drop

Expanded Drop

Low Amplitude Stress-Relaxation

Oscillating Drop

Main accessible quantities

Static and dynamic interfacial tension

Static interfacial tension

Static and dynamic interfacial tension

Dynamic interfacial tension

Dynamic interfacial tension and surface dilational modulus

Surface dilational modulus

Mechanical state of the interface

Dynamic

Quasi-Static

Quasi-Static

Static

Static / Quasi- Static

Dynamic

Ref.

26,27

13

14-18

19-22

23-25

2. WORKING PRINCIPLES OF THE CAPILLARY PRESSURE TENSIOMETRY

The pressure difference AP across a stretching spherical interface, can be derived by the normal

stress balance at the interface as

2y 4K dR 4(gi -~te) dR AP - --~ + R2 dt R dr (4)

where R is the drop radius, K the surface dilational viscosity and gi and ge are the internal and

external viscosity, respectively. For an interface at mechanical equilibrium (dR/dt=0) the

equation reduces to the Young-Laplace equation and AP corresponds to the capillary pressure.

The second and third terms arise from the dynamic viscosity of the surface and the bulk, and are

negligible for slow variation of the drop radius. The contribution due to the bulk viscosity is

negligible for dR/dt<<~/(gi-ge ) which for surfactant aqueous solutions means at worst

dR/dt<<103cm/s. The second term of equation (4) can also be neglected as far as

(dR/dt)/R<<~//~:, which for most common systems is (dR/dt)/R<<l 0 4 S "1.

243

Thus, under these mechanical quasi-equilibrium hypotheses, the Young-Laplace equation holds

at any time

2y(t) APcap(I)- R(t) (5)

From this equation, y can be evaluated as a function of time by the simultaneous measurement

of the capillary pressure and of the drop radius. This is the principle at the basis of the CP

tensiometry. In practice, a small drop/bubble of a fluid is formed at the tip of a capillary, by

means of a liquid supplier system inside another fluid. The pressure difference between the two

fluids is monitored while stimuli (for example, variation of the drop area) are applied. At the

same time the drop radius is measured by direct imaging or calculated from the injected liquid

volume. Finally, the surface tension evolution is reconstructed by applying equation (5).

However, in gravitational environment drops and bubbles are generally not spherical, their exact

shape being determined by the competition between the liquid weight and its surface tension.

The relative influence of these two effects on an interface with a characteristic length scale d,

between two fluids with density difference A 9 can be characterized by the Bond number

Bo = ~ (6) Y

where g is the gravity acceleration.

For Bo=0, liquid interfaces have a shape with a uniform mean curvature, and, in particular,

drops are spherical. This can be obtained either for isodense fluids (A9=0) or for g=0. The latter

condition can be realized in a reference system in free falling, where the weight is compensated

by the inertial force. This particular environment is today accessible aboard space vehicles and

platforms, where levels of residual acceleration close to zero (10 -2 + 10 -5 m/sec 2) are achieved.

This condition is called microgravity. Thus, microgravity emphasizes the capillary effects with

respect to the buoyancy ones, and this is one of the reasons for which today an increasing

number of experimental studies on liquid surfaces utilize this environment [29 -31 ].

Moreover, due to the strong attenuation of the gravity driven convection, the study of all those

phenomena involving mass transport can largely benefit from this purely diffusive experimental

environment where transport is driven only by gradients of the chemical potential.

244

For these reasons microgravity also represents an ideal tool for studying the dynamic aspects of

adsorption of soluble surfactants and the CP tensiometry is the most suitable technique for these

kind of studies in this environment, both for liquid-liquid and liquid-vapor interfaces.

However, provided that the Bond number is sufficiently small, CP tensiometry can also be used

in normal laboratory conditions.

The CPT has consequently been employed with several configurations and with different

methodologies to measure the interfacial tension of pure liquids and for studying the dynamics

of adsorption on different time scales both on earth and in microgravity. Hereafter some of

these methodologies are described in detail, discussing the critical aspects and the main

experimental results.

3. THE PRESSURE DERIVATIVE METHOD [ 14]

This method has been developed for measuring the surface/interracial tension of pure liquids and

it is based on the simple evidence that Eq. (5) states a linear relationship between the capillary

pressure and the drop curvature (l/R) whit a slope of 2y. Thus, by collecting the (AP,1/R) data

during the growth of a drop of a pure liquid, y can be calculated by fitting a linear relationship.

The CPT used by Passerone et. al. is shown in Fig. 1, and it has been developed principally for

interracial tension measurements. The cell is made up of two principal bodies connected by the

capillary and containing the two liquids. The cell has been fully built in PTFE, PCTFE and glass

to improve the cleaning and filling procedures. The capillary is hand made from a Pyrex glass

pipe drawn on a flame, cut perpendicularly to its axis and then carefully fine grounded. The drop

is formed on the inner radius cz which is typically in the range 0.25 + 0.35 mm.

The pressure signal is measured with a single transducer placed in contact with the liquid

forming the drop. Indeed, the variation in the pressure difference between the two phases is

essentially due to variations in the capillary pressure since all other hydrostatic contributions

remain constant. The signal is sampled with the typical frequency of 25 Hz by a PC board.

A precision syringe with a motorized piston driven automatically by a PC, is inserted in the

lower part of the cell and allows the drop to grow with a controlled flow rate in the range 10 -2

- 10 -1 mm3/s.

245

The whole apparatus is housed inside a thermostatic chamber allowing the temperature to be

controlled with the precision of 0.1 ~

Fig. 1: Sketch of Capillary Pressure tensiometer for Pressure Derivative and Expanded Drop experiments used in Ref.[21]. a) pressure transducer; b) injection system; c) liquid 1; d) gas reservoir; e) gas injection valve; f) liquid 2; g) capillary; h) optical window. During the Pressure Derivative runs, the gas reservoir is also filled by the liquid 1.

In each experimental run, a drop of the liquid 1 grows in the liquid 2 at constant flow rate at the

tip of the capillary.

The typical signal for the pressure difference between the two liquids, APm, recorded in this kind

of experiment is shown in Fig. 2. According to the variation of the drop curvature, the capillary

pressure increases until the maximum value is attained when the drop is hemispherical. Then the

capillary pressure decreases under the effect of the decreasing curvature.

APm, is linked to the capillary pressure through a constant term Poff, containing all hydrostatic

contributions and any other constant pressure term internal to the measurement device.

ap - + eo:: (7)

1900

1800

<! 1700

1600

Drops are formed at constant flow rate w, thus, having passed the hemisphere, the radius can be

calculated at any time t by equating the supplied volume with the volume of a spherical cap

having radius R and constant base radius et

27t(R3 ot3'~ 3rt4R2 _ a 2 ( 2 R 2 +Or2 w ( , - , . ) - - j+ , (8)

where t* is the time at which the drop is hemispherical.

246

0 20 40 60 80 100

t (s)

Fig. 2 Pressure signal recorded using the Pressure Derivative method, obtained with o~=0.247 mm and w=8.710 -2 mm3/s, for water-hexane. The maximum in the signal corresponds to the hemispherical drop.

By using Eqs. (7) and (8), it is possible to obtain a set of ( APca p, l/R) data from the decreasing

part of the pressure signal. Finally, the interfacial tension is evaluated by means of Eq. (5) as half

the slope of the best fitting straight line to this set of data.

It is worth noting that the pressure term Poff is also evaluated by the best fit procedure together

with ~, thus the method does not require any calibration with respect to the pressure offset.

Moreover, since the curvature data are derived from the supplied liquid volume, the drop image

is not required and the procedure can be used, in principle, also for opaque liquids.

247

In spite of the apparent simplicity of the method, some difficulties arises from the evaluation of

the experimental parameters needed for calculating the drop radius by Eq. (8). Indeed, only the

flow rate w can be easily and directly measured while the other parameters require an indirect

evaluation.

To minimize the error on the determination of the maximum (t*,AP*) in the pressure signal

caused by the noise, the data around the peak are smoothed by the fitting of a cubic polynomial.

This leads to a fair evaluation of such point.

Moreover, the base radius of the drop can differ from the inner radius of the capillary, due to a

possible small spreading of the drop contact line on the tip of the capillary. The evaluation of

that parameter is a critical item. In fact, when R is calculated from equation (8), even small

errors on the evaluation of ot can lead to remarkable deviations from the linearity of the AP vs.

1/R relationship (see Fig. 3) and then to errors in ~,.

Another problem in the evaluation of the drop radius from the supplied volume is due to the

small amount of gas that could remain trapped inside the liquid forming the drop when filling the

measurement cell. While the drop is growing, the capillary pressure varies and, as a

consequence, this gas changes its volume accordingly. Thus, an additional flow term must be

taken into account by adding an ideal gas expansion term to the right-hand side of Eq (8), which

becomes

w t - 3 3 2 y / ~ +

where Pext is the total external pressure and V* the volume of the trapped gas at the maximum

capillary pressure 2y/c~.

The evaluation of both the parameters ot and V* cannot be made by direct measurements, thus a

particular strategy has been set-up. As the correct values of cz and V* are the only ones which

warrant a linear relationship between capillary pressure and curvature, the adopted values are

those maximizing the correlation coefficient of the linear fitting of the (APcap, I/R) data.

The contour plot of the correlation coefficient vs. ot and V*, obtained by the linear fitting of a

theoretical signal, really presents a maximum in correspondence of the correct values of the c~

and V* . The presence of noise in the signal smoothes the maximum and slightly shifts its

248

position. However, as this shift is random, statistics can be applied to different experimental

runs to gain some accuracy in the determination of c~ and V*.

300

13. 200 Q.

0 13_

100

_

_ a b c d -

_

_ e _

_

_

_

_

0 5 10 15 20 25 30 35 40

1/R (cm -1)

Fig. 3: Prediction of the effect of the error on a on the evaluation of the APeap,1/R curve using Eq (8): a) 8~ =- 0.1 �9 b) 8~=-0.05 �9 c) 8~=0" d) 8~=0.05 �9 e) 8o~=0.1mm. The theoretical signal was generated with ~=0.3mm and y=50dyne/cm.

In reality, the measurement cell can be optimized in order to make V* negligible. In this case,

the speed and accuracy of the maximization procedure is largely improved since only the

parameter ot needs to be evaluated.

An example of the final (APcap,1/R) data calculated with the parameters obtained by the above

procedures is given in Fig. 4.

3.1 PRESSURE DERIVATIVE RESULTS.

Some values of interfacial tension measured by the Pressure Derivative method are reported in

table 2. These values are the average of 10-15 experimental runs.

The absolute values of y are in agreement with the literature and the values of the standard

deviations show a reproducibility of the measurements ranging between 0.5 and 1.5%.

249

One of the table columns shows the ratio between the interfacial tension by the PD method and

that obtained by the Maximum Bubble Pressure (MBP) method on the same signal.

Indeed, the maximum capillary pressure, 2"~/o~, is evaluated by the maximum pressure in the

measured signal, AP*, and by the value of Poff produced by the fitting procedure. This provides

an internal reference test for the measurement of ~.

400

300

lU O

a.. 200

100

5 10 15 20 25 30 35 40

I /R (cm -1) Fig. 4: APcap vs. I/R data obtained from the signal of Fig 2.7 is a half of the slope of the best fit straight line.

Table 2: Measured values of the interfacial tension of some organic liquids vs. water at 20 ~ by the Pressure Derivative method. The reference values are taken from literature.

Organic liquid phase

hexane

paraffin oil

dioctyl-phtalate

benzene

1-bromonaphtalene *

"~PD (dyne/cm)

50.8

51.0

29.5

34.1

38.5

Stand. Deviation (dyne/cm)

0.3

0.9

0.4

0.3

7MBP (dyne/cm)

49.9

29.4

33.4

38.5

Ref.values (dyne/cm)

49.4 + 51.1

49.0 (at 25 ~

33.3 + 34.9

42.0

*) 2 runs in the MITE microgravity experiment.

250

To test the effectiveness of the method in weightless conditions, the experiment MITE

(Measurement of Interfacial Tension Experiment) was performed in 1990 in a sounding rocket

of the European Space Agency. These rockets put a payload in a parabolic trajectory which

provides about 7 minutes of microgravity condition.

During this test the interfacial tension of a system, namely ct-bromonaphtalene/water, with large

difference of density was measured. Although the system was not pure enough, and showed a 7

value about 10% lower than the literature data [32], the comparison of the results with the

internal MBP reference (see table 1) have shown that the method is sufficiently reliable.

In 1994, a new experiment was performed, again in a sounding rocket, with a new module and

experiment design (MITE-2). In this experiment, the PD was applied to measure the pure

water/hexane interfacial tension to be used as reference value for a parallel experiment

(described hereafter) dealing with the adsorption kinetics of Triton X-100. In this case the

results were satisfying and in agreement with the literature.

4. THE EXPANDED DROP METHOD [20-23].

The ED method is a development of the classical stress/relaxation methods which allows the

study of the time dependence of the dynamic interfacial tension for surfactant at liquid/liquid or

liquid/air interfaces, starting from a "fresh" surface (F=0) until the achievement of the

adsorption equilibrium, while the surface area is kept constant.

The fresh interface is obtained with good approximation by a fast expansion of a drop (or

bubble) inside a surfactant solution ; in this way the surfactant layer adsorbed at the interface is

strongly diluted. After the expansion, the dynamic interfacial tension is obtained by recording

the capillary pressure, and by using Eq. (5).

This method allows adsorption processes with characteristic times from a few seconds to several

minutes to be followed.

Fig. 5 schematically describes the working principle of the ED methods, by showing the

variation of the principal quantities involved. During the fast expansion, 7 reaches its maximum

value - almost that of the pure liquid - while the capillary pressure has a sharp decrease due to

the large increase of the drop radius. From that point (t=0) the drop area is kept constant and,

251

due to the progression of the adsorption process, the interface relaxes towards its equilibrium

state, causing a continuous decrease of the capillary pressure.

Yo

7(t*) 7eq

interfacial .................. i, i i ) ~ . . . , tension

.

/ ~ p ~ . . . . . . . . . . . , i

1 , APm(O)

capillary pressure

APeq ...........................................................................................

a(o) ...................... ,

:; drop �9 ,

: : radius

( Z . . . . . . . . . . . . 'r.a-.--_ ,:' '

t* 0 time

v

Fig. 5 �9 Scheme of the variation of the main quantities involved in the Expanded Drop method.

In the set-up by the authors [21 ], the CPT sketched in Fig. 1 is again used where the drop is

slowly increased to reach the hemisphere, then the interface is quickly expanded in about 0.2 s

with a relative variation of its area of the order of 50.

A CCD camera, connected to a video recorder, monitors the experiment. The drop radius R(0)

and the other geometrical parameters at the end of the expansion are measured from the TV

images.

252

Fig. 6 shows a typical pressure signal recorded during a kinetic adsorption experiment.

The dynamic interfacial tensions are obtained from these measured pressure/time data. Indeed,

by Eq. (5) and (7) it is

'y(t) - (l~~ + Poff ) R(O) 2 (10)

13_ v

E 13_

4000

3900

3800

3700

3600

, , , , l , , , , I , , , , I , , , , I , , , , I , , , , I , , , , I , l , , I , , , ,

- / /

, , , , I , , , , I , , , , I , , , , I , , , , I , , , , I , , , , I , , , , I , , , , 0 20 40 60 80 100 120 140 160

t(s)

Fig. 6 :Typical pressure signal recorded during an Expanded Drop experiment [21]. The sharp decrease corresponds to the expansion of the interfacial area.

Other ED studies can be found in literature. Breen [23] used the same method to study the

adsorption of oil soluble polyethers at water/oil systems. In his paper, few details are given

about the experimental set-up and the methodological aspects, concentrating more on the

experimental results.

An apparatus based on the same principle is presented in detail by Soos et al.. In this article a

versatile CPT is presented where the fine control of the drop formation allows many CP

methodologies to be implemented, among them the ED.

253

4.1 INTERFACE DILATION

To obtain an initial condition for the adsorption process with F(0)=0, the interface has to be

expanded as widely and as quickly as possible. There are however, a number of constraints to

cope with. Firstly, turbulence in the fluids must be avoided to keep a diffusive regime for the

surfactant transport. Moreover, when drops of liquid are expanded, the large over-pressure

which has to be imposed to obtain the needed flow rate trough the capillary may become

incompatible with the accuracy needed for the pressure measurements.

In the CPT described by Soos at al. [18] drops are formed and expanded in a finely controlled

way by using a syringe driven through an effective system of linear actuators and levers.

However, in the reported ED experiments the interfacial area in expanded within some seconds

to obtain a relative variation of the order of 10. In that case at the end of the expansion F is

significantly different from zero. Indeed, in their plots can be observed that adsorption starts

with 3'(0) of about 10% lower than the value of the pure liquids.

In the paper by Breen et al. [23] there are no details about the drop expansion, however,

according to the sketched apparatus, a direct injection seems to be used.

An alternative way for expanding the drop is described in Ref. [21 ]. When the liquid in which

the drop is formed is open to atmospheric pressure, the fast expansion can be obtained by

introducing a given volume of gas into a specific reservoir in contact with the liquid forming the

drop (see Fig. 1). In fact, under particular conditions this system exhibits unstable

behavior [33]: after the drop has reached the hemispherical shape, a sudden increase in its

volume is observed. This expansion, which is revealed by a sharp decrease in the capillary

pressure (see Fig. 6), is explained by the fact that, after reaching the hemisphere, any increase in

the drop volume (i.e. in the drop radius) causes a decrease in the pressure acting on this gaseous

volume, which causes a further increase in the drop volume. Thus an explosive growth of the

drop volume can be triggered which stops only when a new equilibrium state is established. This

phenomenon is similar to the "explosive stage" of the bubbling from orifices except for the

bubble detachment which is avoided by the experimental conditions.

The mathematical modeling of a growing drop/gas-bubble system [33] has shown that its main

properties can be described by the Bubble Stability Number (BSN)

254

277-dgextOt 4 B S N - 8V *Y(t *) (11)

Where V* is the gas volume at the maximum pressure (hemispherical drop). The system is

unstable when BSN<I which, due to the strong dependence on the capillary radius, can be easily

obtained with V* of the order of 1 cm 3 in the normal experimental conditions (ot~0.2 mm ).

Moreover, ifBSN < 10 -1 , the relative variation of the surface area obtained depends only on the

BSN since it is well approximated by

A o - A * 2 1 ~ - - - - 1 (12) A* q2 q

where A 0 is the final area, A* is the area of the hemisphere and q is given by

81q 3 - 16~q 2 - 16~q- 16~ - 0 (13)

where ~ is the BSN value.

Thus, if ~/(t*) and Pext are known, the expansion of the drop interface can be fixed by choosing

the values of V* and or, according to Eqs.(12) and (13).

The dynamics of the drop expansion is controlled by the Poiseuille flow through the capillary,

thus it is limited by the viscosity of the liquid forming the drop. For liquids with cinematic

viscosity of the order of 1 cSt (i.e., water, hexane, decane) an expansion time of less than 0.3 s

can be easily achieved with AA/A* of the order of 50.

An analysis of the expansion dynamics has been presented in Ref. [21] and showed that the

expansion occurs in conditions which are safe with respect to the onset of turbulence inside and

outside the drop.

There are, however, some drawbacks in the utilization of this procedure due to the variation of

the gas volume with pressure and temperature. The temperature can be, in principle, controlled

with the desired precision. On the contrary, the system is intrinsically subjected to a variable

capillary pressure during the adsorption kinetics. This causes a continuous variation of the gas

volume. For this reason, slight variation of the curvature radius has to be considered in Eq.(10)

which now reads

7(l) -- (I~P m (I) + Poff ) R(I) ( | 4) 2

255

R(t) can be calculated by assuming the pertbct behavior of the gas and equating the drop and

gas volume variations from the time t=O.

1 l t 4"tt,R3(t)- R3(O), - V ] + Pext (15) 3

where, the drop being much larger than the hemisphere, its volume has been approximated with

that of a sphere. The comparison of the calculated R(t) with the radius data measured by the

video image at different times, has proven the validity of this correction.

The gas volume, and, as a consequence, the drop volume also vary due to temperature

fluctuation. This induces a noise in the capillary pressure, due to the fluctuations in the drop

curvature and in the surface area, which can become the main source of error. To minimize this

effect, a trade off is needed between the gas volume used to trigger the drop expansion and the

acceptable temperature fluctuation. But, as the gas volume enters the definition of the stability

conditions and of the extent of the drop expansion, the temperature fluctuations may impose an

effective operational limitation.

However, for thermal fluctuation in a range of + 0.1 ~ by choosing ot and V*, it is possible to

suitably fix the BSN value to obtain a satisfactory surface dilation of the drop and to keep the

measurement precision into an acceptable range.

As far as gas/liquid interfaces are concerned, the utilization of this method for expanding the

interface is not suitable. In that case, in fact, we are in a situation similar to the MBP

experiments where is BSN<<I because of the value of V* and a very large expansion occurs

responsible for the bubble detachment.

An effective way to avoid all the problems arising from the compressibility of gas inside the

experimental cell is to form the drop/bubble by "sucking" it into an incompressible closed

environment. This can be easily achieved by designing a CPT where the capillary is embedded in

a closed cell completely filled with liquid and to which the injection system is connected. In that

case the drop/bubbles are formed by sucking instead of injecting liquid. An alternative method

for driving the drop formation/expansion is to exploit the changes of volume of a piezoelectric

rod as sketched in Fig. 7, while a traditional syringe system provide the coarse adjustment of the

drop.

256

The piezoelectric rods today commercially available allow volume variations of the order of 10

mm 3 and they have an almost instantaneous response which permits to expand the interface as

wide and as fast as needed.

A similar configuration has been already used for Oscillating Bubbles experiments in Ref. [34]

and also allows the liquid/air systems to be studied.

These concepts are now at the basis of new CPT's ; this gives a wide flexibility in the use of this

technique and allows the drop area to be finely controlled, which is today one of the major

requirements for the utilization and development of new CPT methods.

This design is particularly useful for microgravity experiments. Here, in fact, due to evident

containment reasons the system has to be closed. In this case, the containment causes a strong

attenuation of the described gas instability, therefore expanding the interface in this way is not

efficient.

Fig. 7 : Sketch of a Capillary Pressure tensiometer using a piezoelectric rod for controlling the surface area. The syringe pump provides for a coarse adjustment.

4.2 Expanded Drop Results

In Ref. [18], Soos et al. studied using the ED method the adsorption process of BRIJ 58

(C16E20 polyoxyethylene) at water/dodecane interface. As the aim the paper was essentially to

describe the performance of their CPT, there is no interpretation of these data.

257

A more detailed study is given by Breen [23] who adopted the ED method to investigate the

dynamics of adsorption of homologous series of polypropylene glycols and triols at

water/heptane and water/toluene interface. These surfactants being soluble in the oil phase,

heptane and toluene have been used to investigate the effect of aliphatic and aromatic solvents.

According to this study, all these surfactants have a dynamics of adsorption which is not

controlled by diffusion.

5 0 ' ' ' " ' 1 ' ~ ' ' l " J l ' ' ' ~ " ~ ' 1 ' ' ' ' " l ' l ' ~ ' ' " l ' l �9 �9 O ~ O0

40

-

~ :30 -

20 -

I I I I I I I l l l l l l l , I I I , , , l l ' I ~ l l l l l l I , I l l l l , l -

0.1 1.0 10.0 100.0 1000.0

t ( s )

Fig. 8 : Example of Expanded Drop experiment results [22]: dynamic interfacial tension by the of Triton X-100 adsorbing at a fresh water/hexane interface. The dashed lines represents the equilibrium values. o) c~176 l 0 -9 m o l / c m 3 , 1'eq=22.38 dyne/cm; e) c~176 10 .8 mol/cm 3 ' 7eq= 18.34 dyne/cm.

Liggieri et al. [21,22] have studied the adsorption process of Triton X-100 at water/hexane

interface as a function of the surfactant concentration. A plot of the measured dynamic

interfacial tension is shown in Fig. 8. The Freundlich surface isotherm well describes the

equilibrium behavior of this system [35]. The interpretation of the dynamic interfacial tension

data has shown that for this system adsorption is controlled by diffusion. This result has been

confirmed [36] by a microgravity experiment performed aboard of the Texus-33 sounding

rocket by the European Space Agency. This experiment had been planned by the authors with

the principal aim of testing the performance of the MITE-2 module. This is composed by two

CPT suitable for use in microgravity, where the experiments can be carried out from earth by

258

remote control. Although it has suffered some technical problems, this microgravity ED

experiment has shown the validity of using CP tensiometry in weightless conditions.

5. THE GROWING DROP METHOD [15-19]

In the applications of the CP tensiometry described up to now, Eq. (5) is used in the two

particular cases in which either the surface tension (PD method) or the drop curvature (ED

method) are constant. In other applications, like the expanding or growing drop (GD) methods

developed respectively by Nagarajan et al.[15] and McLeod et al.[16], and later used and

improved by Zhang et. al. [ 19] and by Soos et al.[ 18], the capillary pressure is monitored while

the surface area is increasing. In these methods APcap changes due to the variation of the drop

radius and of the interracial tension caused by the dilation of the surface which put the system in

a state out of the adsorption equilibrium. These techniques are suitable for measuring dynamic

interracial tension in liquid-liquid or liquid-air systems.

The experimental methodology is similar to that of the PD method: the capillary pressure is

monitored while a drop is enlarged at constant flow rate at the tip of the capillary. The choice of

the flow rate depends on the characteristic time for adsorption and spans from 10 -4 to 1 mm3/s.

The resulting area dilation is small enough to make rheological effects vanishing.

In the setups here described (see Fig. 9), the capillary is immersed inside an other liquid (or air)

from the upper side of the cell.

In the version by Nagarajan et al.[ 15], oil drops and a glass capillary are used. The initial state

consists of an oil drop at equilibrium of adsorption, with a radius much larger than the capillary

tube, in order to ensure nearly radial expansion. Then the drop is enlarged with a slow volume

rate (of the order of 10 -2 mm3/s) and the whole variation of the drop radius during the

experiment does not exceed 20 % of the initial radius. Only in this way, in fact, can a first order

approximated theory be used to interpret the dynamic interfacial tension data obtained during

the dilation of the surface. At any time the drop radius is evaluated from the injected volume by

using the expression of the volume of the spherical cap similar to Eq. (8), referred to the initial

drop volume, v 0, instead of the hemisphere volume.

259

In the experimental methodology presented by McLeod et al. [16] the capillary pressure is

monitored from the time corresponding to the hemispherical drop, and also in this case the

curvature radius of the drop is calculated from the injected volume.

Fig. 9 : Schematic set-up of the Capillary Pressure tensiometer used in refs. [15-19, 23] for Growing Drop and Expanded Drop experiments.

The analytical solution of Eq. (8) with wt*=v0 has been used for the curvature radius

R - ~ - q+ +q_)+(q+ +q_ (16)

with

The typical behavior of the dynamic interfacial tension sketched in Fig. 10 is found, with a

maximum which is linked to the coexistence of the adsorption process and the depletion of the

surface due to the drop expansion. The same authors [ 17] have developed a model to interpret

the experimental data which quantitatively explains this behavior. This theory is based on the

diffusion controlled adsorption applied to radially growing drops.

260

The main innovative point of the work of Zhan et al. [ 19] is the measurement of the curvature

radius together with the capillary pressure. To determine this they use a high speed recording

system which allows the acquisition of the drop image and the associated value of the pressure

with a sampling rate of the order of 10 3 Hz. With this technique, the growing drop tensiometer

can be used for systems with characteristic time of adsorption of the order of 10 -2 s.

In practice, in a typical experimental run dynamic surface tension measurements are made in a

periodic flow situation in which drops are formed, grow and detach from the capillary.

7max

7eq

Ix

interfacial

tens ion

. . . . . . . .

i

0

drop

radius

t ime

Fig. 10: Sketch of the typical behavior of the dynamic interfacial tension for Growing Drop experiments according to the methodology described in Ref. [16]. The characteristic maximum in 7 is made possible by the coexistence of the adsorption process and the depletion of the surface due its expansion.

5.1 GROmNG DROP RESULTS.

Many experimental data obtained with the growing drop tensiometer about dynamic interfacial

tension of liquid-liquid and liquid-air systems are presented in refs. [ 16] and [ 17]. In particular

Triton X-100 at water-air and at water-dodecane is studied and from the interfacial tension

evolution data considerations about the time of adsorption of these surfactants are made. Other

surfactants used are SDS and dodecanol at liquid-air interface.

261

In Ref. [ 17], a theoretical model has been developed and used to simulate the dynamic behavior

of the interfacial tension on a growing drop with different boundary and initial conditions. The

comparison between theory and experiments allows the obtaining of information about the

mechanism governing the adsorption of these surfactants and about their diffusion coefficients.

In particular relatively close agreement is found for dodecanol aqueous solution in the presence

of evaporation.

Data about adsorption properties of BRIJ 58 at water-dodecane are given in Ref. [15],

moreover from these data a determination of the Gibbs elasticity [37] is also made.

The experimental results obtained in Ref. [ 19] are relevant again to the Triton X-100 and SDS

at water-air. However because this set up allows lower times of adsorption to be analyzed,

these surfactants are studied at higher concentration.

In Ref. [ 18], the growing drop technique is used both for the dynamic interfacial tension of a

soybean oil-water system and for the dynamic tension of the same oil film between two water

phases.

6. OTHER CAPILLARY PRESSURE METHODS

The experimental possibilities offered by CP tensiometry are so wide that besides those

described above many other methods have been developed or are still under design.

The Oscillating Drop is now a well assessed experimental method for studying the interface

rheology. It allows the possibility of characterizing the surface dilational modulus [26] as a

function of the frequency in a wide range (about 0.01 + 100 Hz.). This is done by measuring the

phase shift between area harmonic disturbances and the capillary pressure signal.

Another possibility for accessing the rheological aspects of the interface by CP tensiometry is to

study the response of surface tension to non harmonic interracial area. These low amplitude

stress/relaxation experiments have already been performed with the pendant drop technique [38,

39] by Loglio et al. and now a CP tensiometry version of these experiments to be performed in

microgravity [40] has been proposed. In this experiment, expansion-relaxation-compression-

relaxation cycles with relative amplitude of the order of 5% are imposed on the surface area.

During these cycles the dynamic surface tension is measured from the capillary pressure. As an

262

example, in fig. 11 is schematically shown the response of 3' to a trapezoidal variation of the

surface area.

An interesting application of CPT in the study of liquid films has been given by Soos et. al. [ 18].

By a simple procedure, they formed at the tip of the capillary a soyabean oil film with water at

the two sides. They then measured the dynamic film tension during its contraction.

So far CP tensiometry has been quite exclusively used in the field of soluble surfactants.

However, this technique is also suitable for studying the dynamics of insoluble monolayers.

In fact, the drop can be used in principle as a Langmuir trough apparatus, provided an effective

procedure for the deposition of the monolayer is set.

AA f ~

J tl tl+t 2 2tl+t 2 time

Fig. 11: Sketch of the response of surface tension to a trapezoidal area change according to Low Amplitude Stress/Relaxation experiments.

7. CRITICAL EXPERIMENTAL ASPECTS.

Some critical items of CP tensiometry like the evaluation of the drop radius for the PD and GD

methods or the expansion of drop interface in the ED methods, have been treated as an inherent

aspect of the methodology. There are, moreover, other critical items which have a general

character. In particular the pressure measurement and the determination of its capillary

contribution, the wetting properties of the capillary by the liquids and the effects of the

263

deformation due to gravity needs a specific discussion since they are important in all the CP

methodologies.

7.1 Pm~ssu~ MEASUP~4ENr

The magnitude of capillary pressures measured by CP tensiometry are typically below few

hundreds Pa. However, the measured pressure also contains hydrostatic contribution (see next

chapter) which considerably enlarge the range in which the measurement has to be made.

From the strict point of view of the measurement the ideal characteristics for pressure

transducer are obviously the lowest allowed full range, high accuracy and repeatability in the

measurement. The dynamic aspects of the measurement are also important for high frequency

applications (e.g. Oscillating Bubbles). There are however, other needs linked to

methodological, technical and experimental aspects which may become even more important.

Firstly, the change of dead volume due to the pressure variation must be negligible compared

with the typical bubble/drop volume. This constraint is generally in contrast with the necessity of

low full range and high accuracy.

Secondly, the chemical inertia and compatibility of the transducer, its easy assembling and

cleaning are mandatory needs in particular for liquid/liquid tensiometry.

Thus a trade-off has to be found between all these constraints in the choice of a suitable

transducer.

For example, the capacitive transducers warrant today the higher available performance in the

pressure measurement for liquids. However, they are not generally suitable for liquid/liquid

applications due to their complex piping. Moreover, their cantilever mechanism is not suitable

for high frequency applications.

The most appealing compromise is today the pressure transducer based on the implanted ions

strain-gauges and piezoelectric technology. They are easy to mount, to clean and to flush, and

they are suitable for high frequency applications. Their sensitive elements are relatively stiff. On

one hand this warrants a very small change of the dead volume, while on the other, it decreases

their measurement performance. They give an overall accuracy of the order of 1 Pa. However,

by a careful calibration of these transducers in the working range, accuracy can also be

enhanced.

264

As the error in pressure measurements is usually absolute, the evaluation of ~/ results less

accurate when low capillary pressures are involved. An analysis of the error on the 7

measurement by CP tensiometry is given in Ref [18]. However, most of the CP methods are

devoted to the study of dynamic aspects and data are interpreted by fitting procedures. In this

case, the significant information is contained in the whole y(t) signal while the precision on the

single surface tension measurement is less important.

7. 2 DETERMINATION OF THE PRESSURE OFFSET.

As stated by Eq.(7), the pressure measured in a CPT is made up of the capillary pressure and a

constant pressure term essentially arising from hydrostatic heads and, in some cases, by viscous

losses. These contributions to the pressure signal have to be carefully evaluated to obtain

reliable surface tension measurements. Although particular arrangements of the pressure line

may be able to compensate the hydrostatic contribution, they can not compensate for viscous

losses. Therefore, internal calibration procedures are more effectively used.

In Ref. [ 18] Poff is taken as the pressure measured for the flat interface when the drop comes out

of the capillary. However, this straightforward method for evaluating the pressure offset cannot

be generally applied since in some cases the relative wettability of the capillary by the two

liquids does not allow a flat interface to be formed.

Another way to measure the offset is described by MacLeod et al. [ 16]. In their CPT, the liquid

1 forming the drop is always the heavier one. The capillary enters from above inside a cell filled

with the fluid 2 (gas or liquid depending on the experiment). Liquid 1 also partially fills the

bottom of the cell for saturation purposes.

The measured pressure can be expressed as

27 o1+ where 91 and 92 are the densities of the liquids. APm~ is the pressure measured when, at the end

of the experiments, the level of the heavier liquid is raised to touch the capillary. This

configuration is used as a reference since there is a "zero" capillary contribution. The differences

(hl_hlO) and (h2-h2 ~ are between the levels of the two liquids during the experiments and in the

reference configuration. These heights are measured by a cathetometer.

265

The same method for the determination of the pressure offset has been used by Zhang et al.

[19].

Two indirect ways to evaluate Poff from the recorded pressure signal during ED experiments

are reported in Ref. [21 ]. The first method requires to record the pressure signal until the system

has reached the adsorption equilibrium, the corresponding measured pressure being APeq. Then,

if the interface is at equilibrium just before the expansion, when the drop is hemispherical,

equating the two interfacial tensions one obtains

~ *a - APeqR - ( 1 9 )

Po f f - R - o r

where AP* is the measured pressure at the hemisphere.

The evident advantage of this method is that only the geometrical parameters have to be known.

However, it requires severe hypotheses about the equilibrium of the system that often cannot be

met. In particular, some care has to be spent in obtaining the hemispherical drop at equilibrium.

Moreover, this method requires the adsorption process to have a relatively short characteristic

time (of the order 100 seconds): in this case APeq can be accurately evaluated.

An alternative way for calculating Poff does not require any equilibrium hypothesis, thus it can

be applied to kinetics with longer characteristic times and without waiting for the equilibrium

before the expansion. However, in this case, the knowledge of the surface equation of state is

needed.

Indeed, the offset can be calculated by solving the following set of equations, formed by the

Laplace equation and the equation of state, respectively just before (t=t *) and after (t=0) the

interface expansion, and by a relationship for the surface dilution

~(t *)- ~,(r(t *)) r(0)-

(X

( ~ *-Poff ) y - r(t*) (20) R(o)

Poff )--- i-- -

A* r(o)- ~-o r(t *)

The last equation implies that the number of adsorbing molecules is assumed negligible during

the expansion, which is justified for processes with large characteristic time.

266

The unknowns F(t*), F(O), 3'(t*), 3'(0) can be eliminated to obtain an expression for Poff. As an

example, when linear or Freundlich isotherms are considered, solving the set (20) leads to

R(o)21 R(o) 3 e~,o 1 - e -A,r'*+A,'-,(0) 3 ~2 Poff = R(0)3 (21)

1 - 2 ~ ~3

where 3'o is the interfacial tension of the pure system.

For Langmuir isotherm an explicit form of Pof f cannot be obtained and the numerical solution of

the set (20) has to be implemented.

7.3 Capillary Wettability

When a drop is growing on a capillary tip or the surface tension changes in a drop at constant

volume, the wettability properties of the fluids on the tip can be a very critical point.

If the liquid forming the drop wets the capillary better than the outer phase, an advancement of

the three phases contact line from the internal edge of the tip to the external one can occur, and

in some cases, when the wettability of the inner phase is strong the contact line can moves along

the external wall of the capillary.

These problems are especially important in the GD and in the PD methods, where the drop

radius is calculated from the injected volume. In these cases, even a small advancing of the

contact line can affect the evaluation of the radius. However, also in the case where the drop

radius is determined by direct measurement the climbing of the drop along the tip must be

avoided.

Moreover, because the wetting properties are strongly dependent on the interfacial tension

between the two fluids, during an adsorption process the contact line can move even if the

volume of the drop is kept constant, like for example in the ED methods, especially if high

concentration surfactant solutions are used.

Particular care in the choice of the materials with respects to the wettability properties is

therefore needed in all the applications of the CP tensiometry, and sometimes particular

mechanical or chemical treatment is performed on the capillary tube used.

267

If a drop of oil is formed inside an aqueous phase with a glass capillary, the wettability

properties favour the pinning of the contact line at the internal edge of the tip [ 14,15,21 ]. In

Ref. [18], to further improve the hydrophilicity of the glass, a chemical etching with

hydrofluoric acid is used on the external part of the capillary tube.

In the case of a drop of aqueous solution inside an oil, a silane coating process can be used to

make the glass capillary hydrophobic [23]. However, in this latter case capillary tubes built in

stainless steel are more often used [ 16] because these materials are better wetted by the most

used organic oils (alkanes, aromatics, alcohols) than water.

The control of the contact line is more difficult during the experiments concerning liquid drop in

air, because usually the liquid forming the drop wets the capillary. In refs. [ 16,17] the tip face of

a stainless still capillary has been roughened with a 600 grain abrasive. The growing drop is so

obtained with the contact line pinned at the external edge of the tip for aqueous solutions below

the cmc. For solutions above the cmc or for organic oils an ulterior sheathing with Teflon is

used to avoid the drop climbing.

7. 4 DROP DEFORMATION

Eq. (5) is exact only for spherical interfaces, i.e. for Bo=0. In all other cases, due to the drop

deformation, the mean curvature of the interface is not uniform and the capillary pressure in

each point of the interface varies according to Eq. (3).

Drops and bubbles used in CPT have Bond numbers which typically span in the range from 0 to

some 10 -1. To analyze in detail the performance of the CP tensiometry in this range of Bo an

accurate study of the deformation induced by gravity is needed.

In reference [ 16] MacLeod et al. calculated the ratio between the mean curvature and the radius

of the equivalent sphere for a pendant drop as a function of the Bond number, respectively at

the apex, at the equator and at the capillary tip. The drop deformation results practically

negligible as far as Bo<l 0 -2. However, when this value is surpassed, the ratios at the apex and at

the tip rapidly deviate from 1. At Bo~l 0 -1, this deviation is already of the order of 10 %. The

deviation is less important for the equatorial ratio but still exists and it is of the order of few

percents.

268

Although these results have been obtained for a pendant drop-like shaped interface, they can

also be applied, at least as order of magnitude, to sessile drop-like interfaces.

To describe the deformation of a bubble/drop it is more useful to introduce, instead of the Bond

number, the parameter

Apgb 2 [3 - (22)

7

where b is the radius of curvature at the drop/bubble apex, and Ap is the difference between the

internal and external densities, g has to be taken with the positive sign when directed toward the

inside of the drop and with the negative one in the opposite case.

As a consequence, 13 is equal to zero for spherical drops, it is negative for pendant drops and

emerging bubbles and it is positive for sessile drops and captive bubbles.

This parameter is the "shape factor" of the Bashforth-Adams [6] equation which describes in

differential form the shape of axis-symmetric menisci.

This latter equation is obtained by stating the equilibrium of the pressures at the genetic

interface point

2y ka~ ---if- + Apgz (23)

where z is vertical coordinate referred to the apex and directed toward the inner of the drop.

As far as CP tensiometry is concerned, the value of [3 corresponds more or less to the value of

the Bond Number.

The value of [3 together with the drop volume and the base radius, univocally determines the

shape of a drop/bubble. Therefore, during the adsorption this shape changes according to the

variation of ~/and b. However, in the range of [3 concerned by the CP tensiometry, the shape

dependence on y is very weak so that at fixed volume the shape does not change appreciably

during adsorption.

If imaging techniques are available, according to Eq. (23) ~, could be evaluated by using the

curvature radius at the apex in Eq. (5), and by adding to Po~ a variable hydrostatic term

Apgh, h being the drop height. This method has been applied by the authors to the ED

experiments described in [21]. The evaluation of b has been made by considering that for Ij31

269

lower then about 0.1 the drop meridian section is fairly approximated by an ellipse and b is given

by

ac~ 2

b-h(a_4a2_ot2) (24)

where a is half the equatorial diameter of the drop.

The drop volume being kept constant, the drop shape can be assumed not to change. Therefore

b has been calculated by the values of h and a measured by the video image just after the drop

expansion.

8. INTERPRETATION OF THE CP TENSIOMETRY DATA

The equilibrium properties of surfactants at liquid-liquid or liquid-air surfaces can be studied by

using the Gibbs model [5]. As surfactant solutions are in general very dilute, F can be identified

with the amount of molecules per unit area at the surface and Eq. (2) can be used.

If an ulterior model for the adsorbed layer is assumed (for example the Langmuir or Frumkin

model), a surface equation of state can be obtained that provides a relation between y and F. By

using this latter together with Eq. (2) a F-c surface isotherm can be obtained.

When the system is put out of the adsorption equilibrium, for example by a surface stretching, a

dynamic adsorption process starts. During this process the relationship between F and y

provided by the equation of state is still assumed to be valid but this is not true in principle for

the surface isotherm [4].

Two time dependent processes contribute to the dynamics of adsorption. One is the transfer of

the surfactant from the sub-layer to the interface, i.e. the adsorption kinetics. The other is the

diffusion in the bulk layer caused by the surface that behaves like a sink for the surfactant. The

first process is usually much faster than diffusion; thus for most surfactant species the diffusion

is the controlling mechanism of the dynamics of adsorption. In this case the isotherm is also

valid during the dynamic process and states a relationship between F and the concentration of

the portion of solution just close the surface usually called sub-layer concentration, Co. In other

words the surface is locally at equilibrium with the sublayer.

27O

However, especially when surfactant molecules with a complex structure are under

consideration, the characteristic time of the adsorption kinetics may become comparable with

that of diffusion. In this case, the whole process is called "mixed kinetics".

In particular, suitable modeling of the adsorption kinetics problem [41-44] allows one to

calculate the surface activation barriers for adsorption and desorption from the experiments.

Whatever the mechanism underlying the surfactant transport, the theoretical description of

adsorption dynamics always starts from the expression of the mass balance at the interface

dn dt - AOn (25)

which expresses the fact that the number of molecules at the interface of area A varies due to

the net flux of surfactant molecules at the interface, �9 n. This flux depends, in general, on the

adsorption F itself and on the concentration c s in the bulk and can be made explicit through the

assumption of the surface model.

To get an expression for F(t), equation (25) has to be coupled with the Fick equation [45]

describing the surfactant diffusion in the bulk and the resulting set of equations has to be solved

with suitable initial and boundary conditions.

The natural symmetry for the interpretation of the CPT data is the spherical one. A great

simplification can be introduced however by observing [46] that the one-dimensional geometry

can be suitably used for describing the diffusion problem when the drop radius is much larger

than the thickness of the diffusion boundary layer 6=(D'Cc) ~. Where D is the surfactant diffusion

coefficient and ~c is the characteristic time for the adsorption process. This condition is verified

for most surfactant solutions.

Being n=FA, Eq. (25) can be rearranged to obtain

dr" dln A (26)

dt - O n - F dt

8.1 EXPANDED DROP DATA INTERPRETATION

The expanded drop method concerns the problem of the adsorption at a fresh interface (F(0)=0)

at rest. In this case the second term in the right hand side of Eq. (26) is zero.

271

In this case, if the adsorption kinetics is only controlled by diffusion, (I) n is the diffusive flux at

the interface and for mono-dimensional diffusion, the problem is described by the classical

Ward-Tordai equation:

t

r(t) - 2c = ~ - ~ co (x) dz

0

(27)

where c ~ is the surfactant concentration at infinite (far from the interface) and the sub-layer

concentration, Co, is linked to F through the surface isotherm, because of the local equilibrium

hypothesis.

For mixed kinetics adsorption according to the approach proposed in Ref. [44] one obtains

similar equation

1 f c o (1:) ~ g ( q ) d q - - dx (28)

0 0

where g is a function of F related to the surface isotherm and, thus, specific to the surface model

describing the system. For the Langmuir model, for example is g(F)=(1-F/F ~) where F ~ is the

saturation adsorption.. D a is an apparent diffusion coefficient which takes into consideration the

existence, if any, of an energetic barrier to adsorption ea- This coefficient is defined as

D a - D e x p ( - 2~T ) (29)

where D is the ordinary diffusion coefficient of the surfactant into the bulk phase, k is the

Boltzman constant and T the absolute temperature. In this case c0(t ) has not the physical

meaning of sub-layer concentration but, due to the formal treatment it is again linked to F

according to the isotherm.

8. 2 GROWING DROP DATA INTERPRETATION

In Ref. [ 15] an analysis for a radially expanding drop inside a surfactant solution is presented.

The classical diffusion controlled adsorption is assumed in a first order analysis valid for small

dilation. This theory describes quite well the experimental data obtained with dilations of the

order of 10%.

272

A more general theory in which there are no restrictions about the area dilation has been given

in Ref. [17] where a model by Ilkovic [47] has been extended. Assuming diffusion controlled

adsorption, for a spherical drop with variable radius, R(t), the mass balance (5) becomes [48]

d r _ R2 - r ~ (30) dt \ dt J~ I ! dt 0 [R(~)]4 d~

These theoretical treatments have been developed under the hypothesis of 6<<R which greatly

simplify the diffusion problem. However, especially for very dilute solutions or when systems

with transfer of surfactant are considered [49], this condition can not be met and the problem

has to be stated in a more realistic geometry. In most of these cases, an explicit evolution

equation for F can not be found and the complete diffusion problem has to solved, often by

using numerical techniques [50].

9. EXPERIMENTS IN MICROGRAVITY : THE FAST FACILITY.

Although adsorption itself is not perceptibly affected by the gravity field, there are several

reasons to study this phenomena in microgravity, which concern its dynamic aspects in

particular.

Some of them have been already described, in particular the enhancement of the diffusive effects

against the convective ones.

Moreover, beside the discussed problems related to the calculation of the drop curvature in

deformed drops, this deformation has an important influence on the interpretative aspects.

Indeed, in dynamic adsorption experiments, the drop shape is in principle intrinsically changing.

This has two negative effects. Firstly the drop area also changes at constant volume, so the

geometry of the diffusive path is changed. Although, the drop area can be adjusted by acting on

its volume, however, this causes a further variation of the geometry.

On the other hand, using the correct geometry is an important need for data interpretation.

These are typical problems of the Dynamic Drop Shape methods where drops are significantly

non spherical according to their large values of Bo. However, they can become important also

for CP tensiometry when Bo>0.1.

273

Fig. 12 : Schematic drawing of the Capillary Pressure tensiometer cells of the FAST facility.

Also all these effects can be avoided by using CP tensiometry in microgravity conditions.

Therefore this environment will allow the possibility of studying the dynamics of adsorption in

very "clean" condition closer to the interpretative schemes now available. This should ease the

comprehension of the phenomena and the development of the theories and their testing.

The FAST project (acronyms for: Facility for Adsorption and Surface Tension Studies) was

born as a proposal for exploiting in this field the huge opportunities of microgravity

experimentation that will be available aboard the International Space Station. The FAST facility

[40] represents the realization of this project. This is a largely improved evolution of the

sounding rocket modules MITE and MITE-2, which have been flown respectively aboard

MASER 4 and TEXUS 33. It gives the possibility of an automatic experimentation about

dynamics of adsorption for medium time (days) microgravity missions.

274

The core of the FAST facility are two CPT where independent experiments can be performed. A

functional sketch of these tensiometer is given in Fig. 12.

During the experiments a fine control of the drop/bubble interfacial area is obtained by varying

the volume of a piezoelectric rod immersed in the upper liquid while a syringe provides for

coarse movements.

The system being closed, the measured pressure signal is the differential pressure between the

two fluids.

A special device allows the increase of the surfactant concentration over about two decades by

the controlled injection of small quantities of a stock solution.

The drop radius and height are monitored by an on-line TV imaging elaboration software.

An active thermostatic system allows the experiment temperature to be changed in the range 15

+ 50 ~ with an accuracy of 0.1 ~

Due to the high sensitivity of surface tension even to small traces of impurities, special care has

been paid in the mechanical design, in the selection of the cell materials of the cells and of all the

parts in contact with the fluids in order to improve the cleaning procedures and to minimize any

possible chemical contamination. The main body of the cell is built in vitreous silica and a

special Teflon (PCTFE), stainless steel and highly inert fluoro-elastomers is used for the

accessory parts. To avoid residual gas bubbles the liquids are out-gassed and an under-vacuum

procedure is used to fill the cells.

The data acquisition and control soitware allows the experiments to be performed essentially by

automatic procedures, the intervention of an earth operator by remote control being planned

only for emergency procedures and for the start-up operations.

The majority of the data will be stored on-board while a little control data will be sent to the

ground. The TV video link will also be partially available on earth for control purposes.

In these CPT it is possible to implement all the described CP methodologies essentially by

changing the software.

The described setup has been conceived as a module to be flown aboard the NASA Space

Shuttle. In the first mission hundreds of Expanded Drop, Oscillating Bubble and low amplitude

Stress/Relaxation experiments will be extensively performed during some 100 hours of available

microgravity time.

275

10. REFERENCES

1. Rowlinson J.S. and Widom B., "Molecular Theory of Capillarity", Clarendon Press, Oxford 1982.

2. Goodrich F.C., in "Surface and Colloid Science" ser., Matijvic E. (ed.) vol.1, p. 1., Wiley- Intersci.,NewYork 1969

3. Liggieri L., Sanfeld A., Steinchen A., Physica A, 206 (1994) 299.

4. Defay R., Prigogine I., Bellemans A. and Everett D.L., "Surface Tension and Adsorption", Longmans & Green, London 1966

5. Dukhin S.S., Kretzschmar G., Miller R., Dynamics of Adsorption at Liquid Interfaces, in "Studies in Interface Science", Mobius D., Miller R. (Eds.), Vol. 1, Elsevier, Amsterdam, 1995

6. Bashforth F., Adams J.C., An Attempt to Test the Theory of Capillary Action, Cambridge Univ. Press, Bell & Co., 1892.

7. Rotemberg Y., Boruvka L., Neumann A.W., J. Colloids Interface Sci., 93 (1983) 169.

8. Liggieri L., Passerone A., High Temperature Tech., 7 (1989) 82.

9. Pallas N.R., Harrison Y., Colloids and Surfaces, 43 (1990) 169.

10. Padday. J.F., in "Surface and Colloid Science" ser. (Matijevic E. ed.) vol 1, p.101, Wiley- Intersci., New York 1969.

11. Tornberg E., J. Colloid Interface Sci., 60 (1977) 50.

12 Van Hunsel J., Bleys G. and Joos P., J. Colloid Interface Sci., 114 (1986) 432.

13 Franses E.I., Basaran O.A., Chang C.-H., Curr. Opinion in Colloid & Interface Sci., 1 (1996)296.

14 Passerone A.,Liggieri L.,Rando N.,Ravera F. and Ricci E., J. Colloid Interface Sci.,146(1991)152.

15 Nagarajan R. and Wasan D.T., J. Colloid Interface Sci., 159 (1993) 164.

16 MacLeod C. A. and Radke C. J., J. Colloid Interface Sci., 160 (1993) 435.

17. MacLeod C. A. and Radke C. J., J. Colloid Interface Sci., 166 (1994) 73.

18 Soos J. M., Kokzo K., Erdos E. and Wasan D.T., Rev. Sci. Instrum, 65 (1994) 3555.

19 Zhang X., Harris T. and Basaran O. A., J. Colloid Interface Sci., 168 (1994) 47.

20. Ravera F., Liggieri L. ,Passerone A., Steinchen A., in "Proceedings of the first European Symposium on Fluids in Space", ESA SP-353 (1991) 213.

21. Liggieri. L., Ravera F., Passerone A., Journal of Colloid and Interface Sci., 169, (1995) 226.

22. Liggieri L., Ravera F., Passerone A., Sanfeld A., Steinchen A., Lecture Notes in Physics, 467 (1996) 175.

23. Breen P.J., Langmuir, 11 (1995) 885.

24. Lunkenheimer K.; Kretzschmar G., Z. Phys. Chem. (Leipzig), 256 (1975) 593.

276

25 Wantke K.-D., Miller R., Lunkhenheimer K., Z. Phys. Chem. (Leipzig), 261 (1980) 1177

26 Fruhner H., Wantke K.-D, Colloids and Surfaces A, 114 (1996) 53.

27 Hua, X. Y. and Rosen, M.J., J. Colloid Interface Sci., 124 (1988) 652.

28 Mysels, K.J., Colloids and Surfaces, 43 (1990) 241.

29 Walter H.U. (ed.), Fluid Sciences and Material Science in Space, Spfinger-Veflag, Berlin 1987.

30 Myshkis A.D., Babskii V.G., Kopachevskii N.D., Slobozhanin L.A., Tyuptsov A.D., Low- Gravity Fluid Mechanics, Spfinger-Verlag, Berlin 1988.

31. Ratke L., Walter H., Feuerbacher B. (eds.), Materials and Fluids under Low Gravity, Lecture Notes in Physics vol.464, Springer, Berlin 1996.

32. W.D. Harkins, G. L. Feldman, J. Am. Chem. Soc., 44 (1922) 2665.

33. Liggieri L., Ravera F. and Passerone A., J. Colloid and Interf. Sci., 140 (1990) 436.

34. Fruhner H., Wantke K.D., Colloids and Surfaces A, 114 (1996) 53.

35. Liggieri L., Passerone A. and Ravera F., J. Colloid Interface Sci., 169 (1995) 238.

36. Liggieri L., Ravera F., Passerone A., in "Proceedings of The Second European Symposium on Fluids in Space - Naples - April 1996" (Viviani A. ed.), p. 135.

37. Lucassen-Reynders E.H., "Physical Chemistry of Surfactant Action", Surfactant Science set. vol 11, Marcel Dekker, NewYork 1981.

38. Loglio G., Tesei U., Pandolfini P., Cini R., Colloids and Surfaces A, 114 (1996) 23.

39. Chen P., Policova Z., Susnar S.S., Pace-Asciak C.R., Demin P.M., Neumann A.W., Colloids and Surfaces A, 114 (1996) 99.

40. Liggieri L., Ravera F. , Passerone A., in "ESA Symposium Proceedings on Space Station Utilization- Darmstad, Germany, Sept. 1996", ESA SP-385 (1996), p. 265.

41. Baret, J.F., J. Colloid Interface Sci., 30 (1969) 1.

42. Ravera F., Liggieri L., Steinchen A., J. Colloid and Interface Sci., 156 (1993) 109.

43. Ravera F., Liggieri L. Passerone A., Steinchen A., J. Colloid Interface Sci., 163 (1994) 309.

44. Liggieri. L., Ravera F., Passerone A., Colloids and Surfaces A, 114 (1996) 351.

45. Jost W., "Diffusion in Solid, Liquid, Gases", Academic Press Inc.- New York 1952.

46. Levich V.G., "Physicochemical Hydrodynamics", Prentice-Hall London 1962.

47. Ilkovic D., J. Chim. Phys. Phys. Chim. Biol., 35 (1938) 129.

48. Newman J., in "Electroanalytical Chemistry" (Bard A.J. ed.), vol 95, p. 173, Dekker New York 1973.

49. Ferrari M., Liggiefi L., Ravera F., Amodio C., Miller R., J. Colloid Interface Sci., 186(1997) 40.

50. Liggiefi L., Ravera F., Ferrari M., Passerone A., Miller R., J. Colloid Interface Sci., 186 (1997) 46.

277

11. LIST OF SYMBOLS

a equatorial half diameter of a drop

A surface area of the drop

A0 surface area at t=0

A* surface area of the hemispherical drop

b curvature at the drop apex

Bo Bond number

BSN Bubble Stability Number

c surfactant concentration

c ~ bulk concentration

Co sub-layer concentration

d characteristic length scale of the interface

D diffusion coefficient

D a apparent diffusion coefficient

g gravity acceleration

g(F) characteristic function of the surface model

h drop/bubble height

hi-hi ~ level difference during the experiment and at the reference of liquid i (GD tensiometer)

n number of adsorbed molecules

PN, PT normal and tangential components of the pressure tensor

AP differential pressure

APc.p capillary pressure

APm

AN*

APeq

Poff

Next

q, q+

R

measured differential pressure

measured differential pressure maximum (at the hemisphere)

measured differential pressure at equilibrium (ED method)

pressure offset

total external pressure

dummy variables

drop radius

Rg gas constant

278

R1,R2

t

t*

V0

V*

W

Y0

7eq

7max

F

gt

Ap

~D

principal curvature radii

time

time of the maximum in the PD and ED pressure signal

absolute temperature

initial drop volume in the GD tensiometer

volume of the trapped gas

volumetric flow rate

vertical drop/bubble coordinate

tip radius

drop/bubble shape factor

thickness of the diffusion boundary layer

net flux at the interface

surface/interfacial tension

interfacial tension of pure system

equilibrium interfacial tension

maximum interfacial tension (GD method)

Gibbs adsorption

surface viscosity

bulk viscosity

density difference

characteristic time for diffusion

dummy variable

BSN value

Drops and Bubbles in Interfacial Research D. M6bius and R. Miller (Editors) �9 1998 Elsevier Science B.V. All rights reserved. 279

T H E M A X I M U M B U B B L E P R E S S U R E T E N S I O M E T R Y

V.B. Fa inerman "~ and R. Miller*

w Institute of Technical Ecology, Blv. Shevchenko 25, Donetsk, 340017, Ukraine

Max-Planck-Institut far Kolloid- und Grenzfl~chenforschung, Rudower Chaussee 5,

D- 12489 Berlin-Adlershof, Germany

Contents

1. Introduction 2. Principal types of devices 3. The design of maximum bubble pressure tensiometers 4. The dynamics of growing bubbles 5. The theory of the maximum bubble pressure method 5.1. Effects related to the deviation of the bubble from a spherical shape 5.2. Effects related to the viscosity and inertia of the liquid 5.3. Gas flow regime and the resistance of the capillary 5.4. Deadtime 5.5. Meniscus hydrodynamic relaxation time 5.6. Bubble lifetime 5.7. Effective adsorption time 5.8. Stopped flow procedure 6. Problems related to the experimental technique 6.1. Influence of the measuring system and bubble volume 6.2. Hydrophilic and hydrophobic capillaries 6.3. Effect of capillary length 7. Comparison of MBPM with other methods 8. Examples of experimental results 8.1. Solutions of surfactants 8.2. Natural water 8.3. Biological liquids 9. References 10. List of symbols

280

1. INTRODUCTION

The maximum bubble pressure method (MBPM) for measurements of surface tensions was

proposed by Simon [1] about 150 years ago. For a long time this method was believed to be

rather complicated and unreliable applied to any pure liquids, and in particular to solutions [2].

During the last 20 years, however, the method was considerably developed and improved. More

than 200 publications have been published on theoretical and experimental problems connected

with the MBPM and its use in studies of various systems. The history of the method was

reviewed in detail by Mysels [3], while shorter descriptions can be found elsewhere [4, 5].

This chapter deals with the physico-chemical and hydrodynamic processes taking place at

various stages of the growth of a bubble and its separation from a capillary. Particular emphasis

is made on theoretical problems

- surface tension calculation from the measured excess pressure

- splitting of time interval between consecutive bubbles into lifetime and deadtime,

- calculation of these characteristic times involving inertial and viscous properties of liquid

and gas, non-stationarity of flows, etc.,

as well as experimental details

- measurements of pressure and bubble formation frequency,

- optimisation of the geometry of capillary and measuring system,

related to the application of the MBPM. Both theoretical considerations and experimental

implementation become much more complicated when the method is used for studies in the

short time range, i.e. in the millisecond and submillisecond time range. However, this is exactly

the time range which attracted recently the great attention to the MBPM as it promises new

important physico-chemical results. Some particular results for various systems obtained by

using the MBPM are presented in the last sections.

The successful development of both theory and experimental technique of the MBPM,

especially the automation of measurements and calculations, and the increasing demand for

studies of dynamic surface phenomena in a time range shorter than one second (including a

number of industrial and biological applications) had resulted in at least three types of

281

commercial tensiometers: Sensadyne PC500L (Chem Dyne Research Corp., USA), BP1 and

B P2 (Kr0ss, Germany), and MPT 1 and MPT2 (Lauda, Germany).

We believe that the results presented here will contribute both to an improvement of the

commercially available devices, and to a better understanding of the method by the users,

helping them in the application of the MBPM and in a correct interpretation of the results.

2. PRINCIPAL TYPES OF THE DEVICES

Simon [1] used a single capillary in his experiment; therefore the immersion depth of the

capillary into the liquid was measured to account for the hydrostatic pressure. This depth would

have not to be measured if the capillary tip is at the liquid level [6] or if two capillaries (narrow

and wide) are used, as proposed by Sugden [7]. However, to ensure fast separation of the

bubble from the capillary tip the immersion depth has to exceed the diameter of the separating

bubble. Therefore the method is unsuitable for measurements in a short time range [6]. Sugden's

idea was further developed by others leading to modifications of the method based on the

measurement of the immersion depth difference for capillaries with different diameter in the

same liquid, for two identical capillaries in different liquids, etc. [8- 12]. Due to our opinion,

set-ups with two capillaries can be successfully used in studies of pure liquids (cf. for example

[ 13, 14]). However for surfactant solutions, due to the dynamic character of the surface tension,

significant errors can arise and an explicit measurement of the capillary immersion depth into the

liquid or its adjustment to a given depth is preferable.

One of the MBPM advantages is the possibility to study the surface tension in micro-volumes of

a liquid, which is particularly important in studies of biological liquids (cf. Section 8.3). As

example a measuring cell developed for measurements with very small liquid volumes was

described recently [ 15, 16].

The main parameter measured in the MBPM is the pressure. Instead of water manometers used

earlier [17- 21], all hand made and commercially available tensiometers employ electrical

pressure transducers [22- 35]. In addition to the options related to data processing by a

computer (using in addition AD-transducers), the advantages of this technology are high

sensitivity and low inertia of these pressure transducers. This makes it possible even to use them

282

for the measurement of the bubble formation frequency [22- 25, 27- 31]. This method of

frequency registration is most suitable when the volume of the measuring system V s is small,

only few orders of magnitude larger than the volume of a separating bubble V b. In this case

however, at high bubble frequencies, the error in the measured maximum pressure increases

[28, 33]. Therefore additional and independent methods were proposed to measure the time

interval between successive bubbles, namely stroboscopic [19, 20, 36], photoelectric [26, 35],

conductometric [21, 32] and acoustic [33, 34] sensors. Some devices employ also the

measurement of gas (air) flow rate and bubble volume [19- 21, 26, 31- 34, 36]. It will be

shown later that the gas flow rate and bubble volume is important when the results of the

measurements have to be presented in terms of effective surface lifetime. While all other

commercially available devices determine the bubble frequency directly from the pressure signal,

the MPT1/MPT2 tensiometer from Lauda is equipped with an additional gas flow rate

transducers and an independent sensor system (acoustic, conductometric, or photoelectric) to

measure the bubble formation frequency.

3. THE DESIGN OF MAXIMUM BUBBLE PRESSURE TENSIOMETERS

As an example the tensiometer MPT2 is shown schematically in Fig. 1.

4

I . ,

11

16

5

. _ u [ I

12 / 15

I

m B

Fig. 1. Schematic design of the tensiometer MPT2 from LAUDA

283

The air coming from the micro-compressor 1 is cleaned in a fine purification filter 2. Also other

gases can be employed using a gas bottle connected to the compressor via a reducer. The gas

flows through the two electromagnetic valves 3 into the gas flow capillary 4 in which the air

flow rate is measured by the differential pressure transducer 8. The excess pressure within the

system is measured using the pressure sensor 9.

The measuring cell is shown in details in Fig. 2. It comprises the thermostat jacket and the

vessel for the liquid under study (not shown in Fig. 2), the capillary holder 5, the measuring

capillary 1 and the ring 3 used to fix the distance doo between the capillary and the electrode 2.

The electrode 2 and another electrode immersed into the liquid comprise the electrode system

used for the measurement of bubble frequency in electro-conductive liquids, typically aqueous

solutions. The second independent system which monitors the bubble formation frequency in

any type of liquid is the acoustic system employing a high sensitive microphone 5 (cf. Fig. 1).

Fig. 2. Schematic of a measuring cell for the MBPM

When the bubble touches the electrode 2 (detail see Fig. 2, [66]) the resistance between the

electrodes increases, while the collapse of the gas cavity at the moment of bubble separation

creates a sound wave registered by the microphone. The electric signals from the pressure

transducer 9, air flow sensor 8, the electrodes 16 and the microphone 5 are transferred to the

measuring unit 10 connected to a computer 11 via an AD-converter 7. Instead of the electrode

284

system, a photoelectric registration system can be used. The electromagnetic valves 3 and the

compressor 1 are operated by the computer via the interface 6.

The total gas volume in the present instrument (the volume between the gas flow capillary 4 and

the measuring capillary 15, cf. Fig. 1) compiises about 35 cm 3, of which ca. 30 cm 3 is provided

by an additional vessel with the built-in pressure sensor 9. For the volume of a separating bubble

(3+4).10 .3 cm 3 and a capillary radius of about 0.0075 cm the maximum pressure drop in the

system during a bubble separation does not exceed 0.5 %.

The above described design is most complex among the commercial instruments, but has some

advantages as demonstrated below. The simple set-up for a MBPM comprises a gas delivery

system 1, a pressure sensor 9, the capillary 15, and the connection of the instrument via an

electronic unit 10 to a computer 11.

The immersion depth needs to be determined accurately only if the capillary diameter is large.

Thus, the LAUDA instrument does not require large accuracy in the immersion depth as the

capillary diameter is much smaller than that in the other instruments. In the Sensadyne

instrument two capillaries instead of one are used to avoid the determination of the immersion

depth. In the Krass instrument the immersion depth is determined automatically with high

precision so that no calibration of this parameter is necessary.

The calibration and testing of the devices, the measurement and calculation procedures are

typically automated in the commercial devices. Their computer programs provide a number of

measurement regimes, for example with continuous, constant (for the technologic processes

control) or stopped (for the measurements in long time range, see Section 5.8) air flow. Also the

relative change of the air flow rate (dlnL/dt) and probing frequency (excess pressure, gas flow

rate, time interval between the separating bubbles) can be tuned in some softwares.

4. THE DYNAMICS OF GROWING BUBBLES

Fig. 3 [41 ] illustrates the dynamics of a bubble at the tip of a capillary immersed in water. The

internal capillary surface is hydrophobised by n-octadecyl trichlorsilane CH3(CH2)17SiC13.

Therefore penetration of liquid into the capillary after separation of a bubble can be excluded.

285

Fig. 3. States of meniscus hydrodynamics for a hydrophobic capillary; a) bubble starts separation, b) intermediate state of separation, c) bubble separated and subsequent bubble formed

The bubble growth at the capillary tip can be separated into main stages as follows. At the

moment of the air bridge collapse, that is, when the bubble separates, the meniscus curvature

radius is approximately equal to the radius of the separating bubble r b (r b ~ reap). At this time

moment the next bubble begins to grow. During the time interval t~l the radius of curvature of

the meniscus approaches the capillary radius, and the meniscus itself moves into the capillary by

some depth h (forward meniscus motion), depending on the properties of internal capillary

surface. At the next stage during the time tl2 the meniscus driven by excess pressure in the

system moves to the end of the capillary (reverse meniscus motion). The following stage of the

growing bubble evolution during which the bubble radius decreases and becomes equal to the

capillary radius (for narrow capillaries) is determined by the air flow: the greater the air flow L

is, the shorter is this time interval tl3. The time interval between the collapse of the air bridge and

the moment when r - reap is called bubble lifetime t I = til + tl2 + t13 [42]. The last stage of the

growing bubble evolution is the so-called deadtime interval t a between the moments of

maximum pressure (r = reap) and the bubble separation. During this time the pressure in the

growing bubble decreases, enabling fast bubble growth due to excess pressure in the system.

For hydrophilic capillaries the flow pattern is more complicated as shown in Fig. 4 (cf. [41 ]).

The position of the meniscus in the capillary, and the variation of the meniscus curvature and

pressure in the growing bubble and measuring system are shown schematically in Fig. 5 for

instruments with large system gas volume, i.e. at largeratio Vs / Vb.

286

Fig. 4. Meniscus hydrodynamics for a hydrophilic capillary.

Fig. 5. Bubble formation, variation of bubble radius r and pressure within the bubble Pb and measuring system Ps during the lifetime and deadtime (schematically); according to [42]

It was mentioned above that any oscillation amplitudes of the pressure in such a system Ps do

not exceed 0.5 %. The maximum pressure is achieved at the end of the lifetime period; at this

moment the pressure in the bubble Pb differs only slightly from Ps. Throughout the deadtime and

287

at the initial stage of the lifetime the pressure in the bubble decreases sharply. The scheme

shown in Fig. 5 does not reflect a number of fine effects which will be discussed below, e.g. the

gas pressure oscillations after the collapse of the air bridge for short and wide capillaries, or the

inertial effect of precritical maximum pressure overcome. Also for the case of long and narrow

hydrophilic capillaries the symmetric meniscus shown in the scheme does not exist.

To summarise, the time interval between successive bubbles t b consists of the lifetime h and the

deadtime t d. The lifetime t 1 can in general be subdivided into three further components. The

lifetime theory, as applied to the MBP method, has to answer the following principal questions:

(1) how can the time interval t~ be calculated from t b measured in the experiment, and (2) what

are the conditions at which the inequality tl3 >> tll + t~2 is valid. Another point of interest in the

short lifetime MBPM theory is the calculation of the excess pressure Ps- Pb during the lifetime

interval.

5. THE THEORY OF THE MAXIMUM BUBBLE PRESSURE METHOD

This section presents the state of the art of the theoretical basis for the MBPM. Parts of the

theory were developed only very recently. There are still some significant points open so that a

complete theoretical description of this methodology is not reached yet. However, powerful

groups are involved in a further refinement of existing models as well as solving problems

unclear so far. Some of the results presented here refer to instruments of special design. For set-

ups of different design modified models will have to be formulated and solved. This will be the

topic of future work.

5.1. EFFECTS RELATED TO THE DEVIATION OF THE BUBBLE FROMA SPHERICAL SHAPE

The surface tension y can be calculated from the values of maximum capillary pressure P and

capillary radius ro using the Laplace formula:

rr 7 = ~ f (1)

2

f is a correction factor which accounts for the deviation of the bubble from a spherical shape.

The capillary pressure P can in turn be expressed via the excess maximum pressure in the

288

measuring system Ps, the hydrostatic liquid pressure PH = ApgH and the excess pressure P d that

arise between the measuring system and the bubble due to dynamic effects (aerodynamic

resistance of the capillary, viscous and inertial effects in the liquid etc.). Here A9 is the

difference between the densities of liquid and gas, g the gravity and H the immersion depth of

the capillary into the liquid. Therefore:

P = P~ - P . - Pd ( 2 )

The correction factor f was calculated by Sugden [43] and tabulated as a function of the ratio

roap/a, where a is the capillary constant:

a - (2~//Apg) '/2 (3)

Sugden's tables were later transformed into a polynomial form by a number of authors

[22, 44 - 46]; for example by Bendure [22]:

f - a , + + + . . .

The values of the polynomial coefficients are a0=0.99951 , a~ =0.01359, aa=-0.69498 ,

a 3 =-0.11133, a 4 =0.56447 and a5=-0.20156. Another expression for f was proposed by

SchrOdinger [47]. Various corrections to the Laplace equation were analysed by Mysels [3]. It

follows from Eq. (4) that for capillaries with a radius reap < 0.02 cm the value of f differs from

unity only insignificantly. If narrow capillaries are used in the MBPM studies (reap < 0.01 cm is

fulfilled for the standard capillaries of the MPT1 and MPT2 from LAUDA as well as BP 1 and

BP2 from Kr0ss), this correction can be neglected; hence the formula for the calculation of

surface tension can be expressed as:

leap

However, if larger capillaries are utilised, as it is provided for the BP1 and BP2 (data for the

Sensadyne instrument have not been available by the authors), a correction according to Eq. (4)

is needed.

289

5. 2. EFFECTS RELATED TO THE VISCOSITY AND INERTIA OF THE LIQUID

The expansion of the bubble surface and the displacement of the meniscus into the liquid bulk

during the lifetime can contribute to the excess pressure P d. Based on the model developed in

[48, 49], Keen and Blake [50] have considered the effect of bulk and surface dilational viscosity

on the growth dynamics of the bubble at the capillary tip immersed into a liquid. It was shown,

in particular, that the effect of dilational viscosity k s is significant for very large values of

ks/r~ap B > 1000 (~t is the bulk viscosity of the liquid). According to the calculations performed

by Kao et al. [35], the excess pressure in the growing bubble arising from the bulk viscosity of

the liquid can be expressed as:

BL Pd - ~:r 3 (6)

where r is the current bubble radius, L the gas flow rate. At the final lifetime stage it can be

assumed that L-~rc2ap(rcap/tl). Except of a numerical factor Eq. (6)transforms into an

approximate expression obtained in [51 ] from the Stokes' law

3B Pd - (7)

t~

This relationship agrees qualitatively with the experiments performed with highly viscous

liquids; however, a logarithmic dependence of Pd on ~t better matches the experimental data

[51 ]. The contribution of the dilational surface viscosity to Pd was estimated in [35] as:

ksL Pd - zcr 4 (8)

or, for the final lifetime stage in the MBPM

ks Pd - (9)

rc,ptl

The values of surface dilational viscosity for sodium dodecyl sulphate and octanoic acid

presented in [35] (ks=105-10 4 mN s m 1) show that its contribution to Pd in the MBPM is

negligible for the millisecond and submillisecond time range.

290

The liquid inertia contribution to Ps was estimated by Dukhin et al. [52]. It was shown that the

inertial effects are most significant at the final lifetime stage and does not amount more than one

percent with respect to P. The liquid inertia contribution can be neglected if

r 3 PL7 cap 8

(321.11)2 <<I (10)

where PE is the liquid density, rl the dynamic viscosity of the gas, 1 the capillary length, 8 a small

factor (8 << 1). The inequality (10) is only violated for very wide and short capillaries.

5. 3. GAS FLOW REGIME AND THE RESISTANCE OF THE CAPILLARY

The hydrodynamic theory for MBPM was developed by Dukhin et al. [42, 52, 53] and

Koval'chuk et al. [54, 55]. After the gas bridge collapses (see Figs. 3 and 4) the gas excess

pressure in the capillary at the orifice Pb is significantly lower than the excess pressure at the

opposite side Ps (cf. Fig. 5). However, at this moment the gas velocity in the capillary reaches its

maximum value. The gas from the measuring system flows into the capillary driven by the

pressure difference between the capillary ends and by the gas inertia. This leads to a smoothing

of the pressure profile along the capillary.

The excess pressure in the capillary P(x,t) was expressed analytically as a function of the time t

elapsed from the gas bridge collapse and the coordinate x measured along the capillary axis. For

x = 0, i.e. at the capillary orifice, the following expression is obtained in [53 ]

2 exp(_ t /xo)~ k + -~ cos t~3 k + sin t13 k (11) P (x , t ) - Ps 1- 7 k=O-

with

- - , O k , 0 ( 1 2 )

where v is the sound velocity, % is a characteristic frequency, x0 is the characteristic time of the

oscillations, v the gas kinematic viscosity.

The analysis of Eq. (11) (and a more general expression for the pressure dependence on the

coordinate obtained in [53]) shows that the regime of the process within the capillary is

determined by the dimensionless parameter K = m0x 0 defined in Eq. (12). For K > 1 the pressure

292

Fig. 6 shows the dependence of the excess pressure near the meniscus P(x,t) calculated from

Eq. (11) for capillaries with various geometric characteristics. It is seen that for K < 1 the

pressure increases monotonously, for K - 1.4 a single, just noticeable oscillation takes place,

while for K - 5.65 high frequency pressure oscillations arise.

The important consequence of the oscillation process, as follows from Fig. 6, is a faster

smoothing of the pressure throughout the capillary; therefore the value of P d decreases when

K 2 ~> 1. This conclusion is supported by the data presented in Fig. 7 (cf. [57]). For capillary 1

(K 2 = 2), where the oscillations do not arise (see curve 3 in Fig. 6), the value of Pd = 2Ay/rcap

for the short time range amounts to 10 % or more of the capillary pressure P = 27/rca p (A7 is

the difference between the apparent and real 7 value). At the same time for the capillaries 2 to 4

(K 2 > 10) the Pa value does not exceed 1 to 2 % of P. Note that the Poiseuille resistance for the

capillaries 1 and 4 are equal as they have the same values of l/rc4p. Therefore one can reduce

significantly the aerodynamic resistance contribution to the P a value by a proper choice of the

capillary geometry.

Recent theoretical results obtained by Koval'chuk [56] enable one to further reduce this

aerodynamic resistance contribution in the value of P a. In this study the interrelation between

lifetime and deadtime was analysed for the case that the gas velocity in the capillary at the end

of the deadtime period acts as the initial condition for the lifetime period, and vice versa. It was

shown that for short and wide capillaries (r~2,p/1 > 10 "4 cm) the excess pressure is reduced

significantly due to the high initial velocity of the gas. Moreover, the theory developed in [56]

predicts the possibility of maximum pressure inertial overcome. Under certain conditions the

formation of a bubble becomes possible even when the pressure within the system P~ is lower

than the capillary pressure. The apparent surface tension for water, measured by the MPT2 for

z /1 = 1 4.10 -4 cm) is shown z /1 - 4.8.10 .5 cm) and B (K 2 - 81 rc,p two capillaries A (K 2 - 20, r,p , .

in Fig. 8. It is seen that for capillary B (the condition r~ /1 > 10 .4 cm is fulfilled) the value of P d

does not exceed 0.5 % P, and is 3 to 4 times lower than that for capillary A, in spite of the fact

that fast pressure oscillations (K 2 )) 1) take place for either capillary. Thus the hydrodynamic

MBPM theory enables one to determine the conditions which minimise the aerodynamic

component of the excess pressure Pa. Then, as proposed in [33, 51], using an empirical

291

P(x,t) shows oscillations. The amplitude of the first oscillation at x = 0 and K>> 1 is

approximately equal to Ps. For K < 1 the pressure in the capillary smoothes without any

oscillations (aperiodic regime). It follows from Eq. (12) that the process regime is determined

mainly by the geometric characteristics of the capillary - the shorter and wider the capillary is,

the higher is the value of K. Therefore the pressure oscillation regime can exist in short

capillaries, while for long capillaries the aperiodic pressure smoothing regime takes place

because oscillations are fading away rapidly.

Fig. 6. Dependence of the gas excess pressure P(0,t) at the capillary end immersed into water on the dimensionless time value ~ = t(av/r~) for different K2= 0.5 (curve 1),1 (curve 2),2 (curve 3),

8 (curve 4) and 32 (curve 5); according to [53]

83 82 81 8O

~, 79 ~ 78 ~.. 77 ~ 76

75 74 73 72

[]

[]

I

$ .

-4 -3 -2 - 1 0 1

lg time [s]

Fig. 7. Apparent dynamic surface tension of water at 18 ~ (To = 73 mN/m) measured by MPT1 for various capillaries: K 2 -- 2 (11), 11.9 ([~), 88.2 (§ and 18.0 (<k); according to [53]

293

correction of the measured Ps value obtained for pure liquids, one can entirely exclude Pd and

comes up with the relation P - Ps - PH.

74

72

71

0,0001

! []

0,001 0,01 0,1 1 10

tef [sl

Fig. 8. Apparent dynamic surface tension of water for two different capillaries: A (FIlI) and B (Alk) (see text)

5. 4. DEADTIME

First theoretical calculations of the deadtime t d w e r e performed in the Poiseuille approximation

for a gas flow through a capillary of length 1. The differential equation for the rate of bubble

growth due to the pressure difference between the two capillary ends can be expressed then by

(cf. [57])

dr rc4p(p~ - PH - 27/r)

dt 321rl r2 (13)

Integrating this equation one can calculate the deadtime, i.e. the time during which the bubble is

growing rapidly so that its radius increases from reap to the radius r b of the separating bubble

_ 3211"1 [ l~ rb q s 7* ~rb ~2q (14)

t d - r~ap(P _pH)L3~r~p) +r~ap(P~-PHlk, rr J

The first term in the right hand side of Eq. (14) describes the gas expansion into the infinite

space, while the second term corresponds to the capillary pressure in the growing bubble. The

value of surface tension for a growing bubble 7" during the deadtime, which enters the second

term, is in fact unknown for surfactant solutions. The analysis performed in [58] had shown that

for solutions the value of Y* in Eq. (14) lies between the equilibrium value Too and the dynamic

294

value of y at t = t 1. Another important conclusion of this analysis is that the variation of y* in the

range y > y* >_ % does not affect the t d value. Thus y* can be substituted by y in Eq. (14) and for

the Poiseuille approximation one obtains (cf. [33, 58])

L( 3reap I 15, td--tb'k----~ 1+2 r b

where kp is the Poiseuille equation constant for a capillary not immersed into the liquid

(L = kpP), L the gas flow rate, P = Ps- PH, tb the time interval between successive bubbles. A

more rigorous deadtime theory developed in [52, 54-56] had shown that the corrections related

to the non-stationarity of the gas flow through the capillary and to the effect arising at the initial

section of the capillary, do not exceed a few per cent of the t d value calculated from Eq. (15).

These corrections lead to an increase in the actual t d value as compared to that calculated from

Eq. (15). This conclusion agrees with the experimental data [58] which show that the t d value

calculated from this expression is systematically lower by ca. 5 % than that obtained

experimentally.

5.5. MENISCUS HYDRODYNAMIC t~LAXA TION TIME

The hydrodynamic relaxation time t h represents the sum of the first two components of the

lifetime: t h = tll + tl2 (cf . Section 4), that is the times of the forward and reverse meniscus motion

(Fig. 5). For short capillaries of hydrophobic internal surface the liquid penetration depth h into

the capillary is small (Fig. 3), while for a hydrophilic internal surface the value h is of the order

of the capillary radius (Fig. 4). Therefore for hydrophilic capillaries the time interval t h can

contribute significantly to t I value. The values of t n and h for hydrophilic capillaries were first

estimated in [42]. It was shown that for the aperiodic regime (K < 1) h amounts to 2 - 3 % of

the capillary length, and for long and narrow capillaries tll c a n achieve 10 .3 s, while for short and

wide capillaries the forward meniscus motion time is very small, tll < 10 -5 S.

The dependence of the reverse meniscus motion time tl2 and maximum penetration depth h on

the capillary length I are shown in Fig. 9. These dependencies were calculated numerically from

the equations describing the liquid and gas motion within a capillary in the aperiodic regime

[54]. It is seen that the values h and t12 increase with capillary length. The time t12 c a n achieve

295

even some milliseconds. Thus, the employment of long and hydrophilic capillaries are not

suitable for measurements in the millisecond time range. For the hydrophilic capillary an

approximate analytic expressions for h and t h was given in [54]

h2 = 128W 12 (16) 71~ 4 ~Kpa reap

= l P 8PvI2 th - - tll + t12 ~ 2 V 2 r c a p 2 + ~ 3 p d 2 2 (17) rcapV

where Pa is the atmospheric pressure, and K - 1.4 is the adiabatic constant.

tta [10Ss ] 600

,/ 400

/ I i/'.. 2oo

2 .ff i"

I I I I 0

O i 2 3 4 i [crn]

h==E~m]

I I I I

O ) 2 3 4 l[cm]

, / /

/

(a) Co)

Fig. 9. Dependence of tl2 (a) and maximum meniscus penetration depth into the capillary (b) as a function of

the capillary length 1 at Pd/P -- 0.05 for rca p -- 0.005 cm (1) and rca p - 0.01 cm (2); according to [54]

Note that the value of h does not depend on the dynamic excess pressure in the system,

P o = Ps- Pn" P, while the forward and reverse meniscus motion times depend strongly on the

ratio P/Po- The larger the excess pressure is, the lower is the meniscus hydrodynamic relaxation

time. P O in turn depends on the capillary geometry and the gas flow regime, as explained in

Section 5.3. Comparing the experimental dependence of Po on t I with the theoretical results

296

P d(th) one can see that for the aperiodic regime in long narrow capillaries t h is close to the

lifetime, while for short capillaries the inequality t h << t I holds (cf. [53 ]).

0.06

0.04

0.02

0.00

I i

\ !

\ \

"~-------_,._,_.__ ._....,

I I I ~ I I

0 10 20 30 40 50 60

t t [105s]

Fig. 10 Dependence of relative excess pressure Pd/P on meniscus hydrodynamic relaxation time for a capillary of 1 cm length

A relationship for the meniscus hydrodynamic relaxation time in very short and wide capillaries

(K>> 10) was derived in [55]. This is the case when the liquid does not penetrate into the

capillary alter bubble separation, i.e. the case of short hydrophilic or hydrophobic capillaries. It

was shown that

4vp1 ~ 4m 2 + 1 lm + 19 2 2m th -- T L 6(m+ 1) 3 + (m+ 1) 4 l n ~ + m + l

1 - m 2 - 2 m ~ [ m 2 (m + 1)4~m(m+ 2) L arctan ~ m + 2 + arctan 11t 4 m ( m + 2 )

(18)

It is seen that even for very low values of excess pressure t h does not exceed 1 ms [55]. For

capillaries short enough so that the ratio Pd/P - (1 +2) %, the value of t h < 0.1 ms. It means that

where p is the gas density, m - Pd/P. Eq. (18) leads to very small values of t h. The results of

calculations from Eq. (18) for capillaries of 1 cm length are shown in Fig. 10.

297

short and wide capillaries can be used for the MBPM in the milli- and submillisecond range of

bubble lifetime.

5. 6. BUBBLE LIFETIME

One can be sure that a comprehensive lifetime theory will be derived in the near future, based on

the recent developments in the hydrodynamics of the MBPM. This theory would enable one to

calculate t~ from the known geometry of the capillary, the volume of the measuring system Vs,

the gas flow rate L and the total bubble formation time t b. At present, however, the procedure

generally used to determine the lifetime is to calculate the difference between the deadtime t a

and t b. The bubble time can be found from the experiment - - either explicitly (by fast video

technique [19, 29, 59]), or using the method first proposed by Kloubek [36]. The method of

Kloubek assumes that for high frequency bubbles formation the deadtime interval becomes equal

to the time interval between successive bubbles. In the dependence of Ps versus tb, the transition

from the regime t a < t b to the regime t a - t b is marked by a sharp pressure increase in the

measuring system (cf. [36, 57]). Therefore in the moment when the pressure jump takes place

t b = t a. This method, however, does not account for the dependence of t a on tb, first discovered

by Austin et al. [ 19].

3000-

2800 -

2600-

2400 -

"~' 2200-

2000-

1800 -

1600

1400

[]

121

n/3•l Es149 D o O D D D

I I I

0,02 0,04 0,06

O O

1200 I I I I I 0 0,08 0,1 0,12 0,14 0,16

L [cm3/s]

Fig. 11 Dependence of pressure in the measuring system on the air flow rate for a 0.2 % Triton X-100 solution, roap = 0.0084 cm, 1 = 6 cm (m), 3 cm ([3) and 1.5 cm (~)

298

More precise is a combined experimental/theoretic procedure which is based on Eq. (15)

(cf. [32, 34]). This procedure assumes that t d depends on t b and P. The dependence of excess

pressure in the measuring system on the gas flow rate L for capillaries of various length are

shown in Fig. 11.

All these curves possess the transition point in the co-ordinates Pc and L c. The linear sections of

the curves in the region L > L c are described by the Poiseuille equation and correspond to the jet

regime of gas expansion from the capillary. For L < L c the flow results in a formation and

separation of individual bubbles with t I > 0. However, in the transition point (L = Lc) the lifetime

vanishes, therefore the time interval between successive bubbles becomes equal to the deadtime.

For constant r b one can combine Eq. (15) for the two cases L = L c and L < L c and obtains

LP~ t d = t b ~ (19)

LcP

It follows from gq. (19) that ([32-34])

t~ - t b - t d - t b 1- Lop) (20)

Thus the procedure developed in [32-34] provides a calculation of the lifetime using the

parameters tu, P = Ps- PH and L, all available from the experiment. The co-ordinates of the

transition point in the dependence of P as a function of L can be easily calculated by the

algorithm based on the Poiseuille equation. Eq. (20) accounts for the dependence of t I o n t u (or,

more precisely, on P) which is predicted by Eq. (15). The value of t a usually increases with tu,

because in this case P decreases (for solutions). Eq. (19) was derived assuming constant r b or

V b. This can be achieved in the experiment if the distance between the capillary and the

electrode (or another body) dc~ is fixed (cf. Fig. 2). When this distance dc~ is unrestricted, which

is the case for all bubble tensiometers except MPT1/MPT2 (cf. [20, 24, 26-28, 59]), the

Eqs. (19) and (20) will not be applicable.

It was mentioned in Section 5.4 that Eq. (15) disregards the non-stationarity and the effects

introduced by the initial section of the capillary. However these non-linear effects can be almost

entirely avoided if the co-ordinates of the transition point are used in the derivation of Eqs. (19)

and (20) from gq. (15).

299

5. 7. EFFECTIVE ADSORPTION TIME

If surfactant solutions are studied, the surface of the bubble expands during the last (third) stage

of the lifetime period. Various types of surface deformation are characteristic also to other

methods used to measure the dynamic surface tension of solutions, such as the drop volume,

oscillating jet, or dynamic capillary methods. To compare various methods, and also to make

use of a unique standard independent of specific features of the method, the results of dynamic

surface tension measurements are usually represented as function of so-called effective

adsorption time [4, 60]. The diffusion-controlled adsorption kinetics for the present process,

expressed via the effective time, transforms into the equation for diffusion-controlled adsorption

kinetics at a non-deformed plane surface, given by the classical Ward and Tordai equation [61 ]:

(Dt'] v2 (D) 1/zt F-2c0\ rt) - 2 Ic(0, t-~)d(~, 1/2) (21)

0

where F is the dynamic adsorption, D is the diffusion coefficient, c o and c(0, t) are the bulk and

subsurface concentrations, respectively, and ~ is a dummy integration variable. If convection

and surface deformation are taken into consideration for a radially symmetric drop or bubble

growth under constant flow rate conditions, then the dynamic adsorption obeys the equation

[61]:

(3Dt'~l/2 F - 2Co~-~-) - (D)~/2 t-2/3 i c(0' (7/3)" 9~3/7) o ~.~-~-)i/3 d)~ (22)

where z - (3 /7 ) t 7/3. One can easily see that Eq. (22) can be transformed into Eq. (21)

substituting

3 t e f - -~ t (23)

The numerical coefficient in Eq. (23) is defined by the surface area deformation rule. For drops

or bubbles growing under a constant flow rate it follows that (cf. [61,62]):

dlnA 2 e - - (24)

dt 3t

300

where A is the surface area. If the relative dilation rate 0 is described by more general

expression

0 - c t t -1

with c~- const, then according to [62] the effective time can be represented as

(25)

t (26) tef - 2~ + 1

Eq. (23) is a special case of the more general expression (26) for o r -2 /3 . The case of relative

dilation corresponding to the expression

no~ 0 - (27)

l + m

is considered in [63] where n and ot are constants.

The calculations of effective lifetime of the bubble surface during the stage of bubble lifetime

were performed in [36, 64, 65]. For the case of a hydrophilic capillaries with a geometry

corresponding to the condition K 2 > 10 or hydrophobic capillaries (in both cases the liquid does

not penetrate into the capillary after bubble separation) the pressure within the bubble

throughout the whole lifetime stage remains constant [36, 64]:

Pb -- (2y,/rr cos% - const (28)

where y, and % are the current instantaneous values of surface tension and the contact angle

during the lifetime stage (0 < t < tl). Equation (28) is the basic expression for the calculation of

the relative bubble surface dilation rate. The following expression was derived in [64]"

I< t9 - . - (29) y~(1 + sin%) sin% dx J

where y is the dynamic surface tension at x - h. The dependence of 0 on y and x is rather

complicated and do not obey the simple relationships (25) and (27). Assuming that the bubble

surface area increases during the lifetime stage is small (in fact, the largest possible area

variation is f r o m 7~r~a p for 1: - 0 t o 7~1"~Zap for "C - t l ) , the finite variation in bubble surface area was

analysed in [64, 65] to estimate 0. It was shown that within a sufficient accuracy the relative

301

dilation rate and effective time value can be expressed by Eqs. (25) and (26). In particular, the

following expression for a was derived in [64]:

2sinq~~ (30) 2 + sin q~0

where q~0 -arccos(7/70), ~/0 is the surface tension of the solvent. For 7 -70 the value of ct is

zero, i.e. the bubble surface is virtually not deformed. For surfactant solutions with 7/y0 < 0.8 the

value of ~ ~ 2/3, a result similar to that obtained for a growing drop, tef-(3/7)t . Note that

Eq. (30) is valid only for short and wide capillaries, when t h ~< t I.

In order to estimate the initial load of the bubble surface one has to analyse the expansion

process during the deadtime stage. In this stage the air flow rate can be assumed to be nearly

constant, because the pressure within the bubble drops rapidly at the initial time moment. For

this period the effective time can be expressed by Eq. (23). However, due to the additional

deformation of the bubble during the final stage of the deadtime (cf. Fig. 5), the numerical factor

in Eq. (23) becomes smaller than 3/7 [66]

td,ef _< to/3 (32)

TO summarise the discussion on the effective time, it is necessary to emphasise the approximate

character of this parameter. The model of the gas and liquid flow normal to the surface used to

derive Eq. (26) is not completely correct in all cases. However, this approximation simplifies the

solution of the whole problem because the convective term in the diffusion equation is

proportional to 0 [62, 67]. In addition, the pure diffusion-controlled adsorption model does not

describe all experimental results obtained for concentrated solutions [68-71]. Only for a barrier

controlled adsorption mechanism the surface dilation effect is accounted for in the boundary

condition at the surface (cf. for example [4, 60]), and for the case of growing drop tel = (3/7) 1/2 t

was obtained, [69]. All experimental results concerning dynamic surface tension measurements

by the MBPM are represented in this chapter in functions of t~f (Eq. (23)). In the subsequent

paragraphs this subscript 'ef' will be omitted, so that t always refers to the adsorption time or

effective surface age.

302

5. 8. STOPPED FLOW PROCEDURE

Usual surface tension measurements with the MBPM are performed under a stationary gas flow

through the capillary and the bubbles are generated with constant frequency. In a longer time

range, tens and hundreds of seconds, technical problems arise connected with the generation of

bubbles at low frequency. If the volume of the measuring system V s is large enough compared

to the volume of a separating bubble Vb, then the measurements in the long time range can be

performed in a so-called stopped flow regime [72]. For a surfactant solution, the bubbles do not

cease to generate immediately after the air supply to the measuring system is stopped. The

pressure drop in the system due to the separation of a single bubble is

AP-h v~ Vs

which corresponds to a surface tension decrease of

(32)

rePaY b A~ - (33)

2v~

Further bubble separation will be possible only if the bubble lifetime and the solution

concentration are sufficient to reduce the surface tension by the value l-I = ? 0 - ?i-] - A?, where

7,-1 is the surface tension for the preceding bubble. As the surface pressure increases with each

successive bubble, the time interval between two bubbles increases as well. For the MPT1 the

value of A7 is a few tenths of mN/m. The efficiency of the method can be increased by using a

larger volume of the measuring system. The formation of bubbles terminates when II ~ 70 - 7oo.

In the stopped flow regime the MBPM is capable to measure the surface tension of solutions for

a lifetime up to 100 s and more.

6. PROBLEMS RELATED TO THE EXPERIMENTAL TECHNIQUE

The results of surface tension measurements using the MBPM for pure liquids and, especially,

for surfactant solutions, depend on the experimental conditions. Some of these problems of the

experimental technique were summarised by Mysels [3, 28]. The measured surface tension is

affected by the capillary inclination angle and the intensity of the liquid agitation in the

measuring cell [3, 73], the capillary immersion depth [ 16], the diameter and wettability of the

303

vessel containing the liquid [74- 77]. For foaming solutions a difference in the results is

obtained depending on whether the surface tension measurements are started from minimal or

maximal gas flow rates, see [78, 79]. From experience with the MBPM is becomes clear that the

reproducibility of the results depends significantly on the stability of the process of bubble

formation and on the bubble volume. A stable bubble formation, without termination and bubble

series formation, is promoted by a relatively large ratio Vb/V~, long capillary, high quality of the

capillary tip prepared by cutting or splitting, and hydrophilicity of the tip. In addition, to

enhance the separation and rising of bubbles, the external diameter of the capillary tip should be

somewhat smaller than the diameter of the separating bubble, while the immersion depth should

exceed this diameter. Small inclination of the capillary (up to 10 ~ also improves the

performance. And finally, the condition V b = const which means a volume of separating bubbles

independent of the air flow rate and surface tension, is extremely important. This independence

can be achieved by installing an electrode or any other sharp body opposite to the capillary tip,

as shown in Fig. 2. The distance between the capillary and electrode doe controls the critical

diameter of the separating bubble dbo , which can be defined from the balance of the capillary and

buoyancy forces

�9 \ 1/3

db~ - (12r~y/Apg} (34)

To obtain correct values of surface tension of a solution, especially in the short time range, the

values of the characteristic parameters of the tensiometer, in particular VJV b and bubble volume

V b are to be chosen in an optimum way. The measured results are also strongly affected by the

properties of the internal surface of the capillary and the capillary length 1. The results of the

studies concerning the influence of the above parameters on the measured surface tension [41 ]

and the recommendations on their optimum choice will be presented now.

6. l. INFLUENCE OF THE MEASURING SYSTEM AND BUBBLE VOLUME

During the fast bubble growth in the stage of the deadtime the pressure within the measuring

system decreases. At the end of this time period the pressure drop is

304

P. r~p Vb - - - ( 3 5 )

P~ 2? V~

that is, the value of the pressure drop is determined by the ratio VJV~. This expression is valid

for large lifetimes, when t b )> t d. For short lifetimes the actual value of ~ is lower than that

calculated from Eq. (35). Expression (35) gives the theoretical value of the pressure drop within

the system; the value measured by a pressure sensor agrees with the calculations only if e < 1%.

This condition is achieved, in particular, in the MPT2 tensiometer (V~ = 35 cm 3) for V b < 4.

10 -3 cm 3. If the bubble volume is increased, or V~ is decreased (MPT2 tensiometer allows to

remove a built-in vessel; this results in a reduction of the system volume down to 5 cm3; for the

other commercial instruments no information about the system volume is given). Then the

theoretical ~ value increases to a few per cent, while the values obtained from the experiment

are several times lower, due to the inertia of the pressure sensor which becomes significant for

fast pressure oscillations (bubbles formation) in the system. Therefore the measured P value in

small volume system for high frequency of bubbles formation can be lower than the capillary

pressure. It follows then that the calculated ? for this system will be lower than that for the

system possessing larger volume.

The results of dynamic surface tension measurements for 0.2 % Triton X-100 solution for

various V b values using the standard MPT2 tensiometer (V~ = 35 cm 3) and without the

additional vessel (V s = 5 cm 3) are shown in Figs. 12 and 13 [41 ]. In all experiments the same

hydrophobic capillary was used (reap = 0.0084 cm, 1 = 1.5 cm). One can see that for V~ = 35 c m 3

the increase of V b from 2.1.10 .3 to 6.10 .3 cm 3 does not affect the results, while for V~ = 5 cm 3

the 3' is by few mN/m lower, and a dependence of 3' on V b in the range from 1.5-10 -3 t o 6.

10 .3 cm 3 is evident.

For very large bubble volumes the measured surface tensions are strongly underestimated, as

one can see from curves 3 and 4 in Fig. 12. The mechanism of this phenomenon was discussed

in [66]. It follows from Eq. (19) that the deadtime increases almost proportionally with the

increase in bubble volume. For example, when a capillary of radius r~ v = 0.0084 cm and

length 1 = 1.5 cm was employed, then for V b = 2.1.10 .3 cm 3 a deadtime of t d = 20 ms is found,

while for V b = 1.8.10 .2 cm 3 the deadtime is 200 ms. According to Eq. (31), the respective

305

effective deadtimes for the bubble surfaces are 7 and 70 ms. It is easy to see that, while an

effective lifetime of 7 ms corresponds to an almost clean surface ( 7 - %, curves 1 and 2 in

Fig. 12), a time of 70 ms corresponds to Y ~ 60 mN/m. This value is close to that characteristic

for t I --~ 0 obtained for large bubbles (curves 3 and 4 in Fig. 12).

75

70

65

60

~ 55

50

45

40 -

35 -

30 0,001

�9 �9 [] ,.D~ mmi~mm:d~~:~_

%~A'**, %

~>~>~ ~ []

-v v~ %% o

~c~o ,c

I I I

0,01 0,1 1

t[s]

Fig. 12. Dynamic surface tension of a 0.2 % TritonX-100 solution at Vs-35 cm 3, V b =2.1.10"3 cm 3 (m), 6.10 -3 cm 3 ([3), 1.2.10 -2 cm 3 (+)and 1.8.10 -2 cm 3 (<7)

Thus the increase in Vb/V S results in errors in the measured surface tension values due to the

increase of pressure oscillations amplitude in the system. The increase of the separating bubble

volume itself leads to an increase in the deadtime. These effects distort the results of surface _

tension measurements for solutions and restrict the applicability of the MBPM under these

conditions to dilute solutions and large t~ values. On the contrary, to study the surface tension of

concentrated solutions in the milli- and submillisecond time range, one has to decrease t a. To

achieve this, one can either decrease V b, or increase Lo by employing shorter or wider capillaries

[66].

306

75

70

65

g: 60 z E

~- ' 55

5 0 -

4 5 -

�9 �9 � 9 i l i ~

40 I I I

0,00l 0,01 0,l 1

t Is]

Fig. 13. The same as in Fig. 12 for V s - 5 cm3; V b - 1.6-10 -3 c m 3 (11), 3.5-10 -3 cm 3 (0), 6.10 .3 crn 3 (§

8.10 -3 c m 3 (~'); the l ine corresponds to curve (B) f rom Fig. 12

6. 2. HYDROPHILIC AND HYDROPHOBIC CAPILLAMES

The properties of the internal surface and the tip of a capillary employed in the MBPM vary in a

wide range. Completely hydrophilic glass capillaries [19, 22, 24, 26,32 - 34,59], capillaries

possessing a hydrophobic internal surface [28, 29, 41, 60] and completely hydrophobic Teflon

capillaries [23, 27, 30] were used for measurements. The properties of the internal surface of a

capillary affect the maximum rise height h of a liquid in the capillary after bubble separation (cf.

Figs. 3 and 4). However, it was shown in Section 5.5 that for short and wide capillaries K >> 10,

and h is negligibly small. For the capillaries studied in [41] (rc,p- 0.0084 cm, 1 = 1.5 cm) the

value of h determined from high-speed video images was (l+2).rcap. The results of dynamic

surface tension measurements of 0.2 % Triton X-100 solutions using capillaries with

hydrophobic and hydrophilic internal surfaces are compared in Fig. 14.

One can see that for equal V b the measured 3' are higher when using a hydrophilic capillary than

for a hydrophobic capillary. The same kind of results are obtained for small and large bubble

volumes (curves 4, Fig. 12 and 14).

307

Similar dependencies are obtained also for a small system volume (V~ = 5 cm 3, [41]). The

increase of 7 can be related to the expansion of the liquid surface during the reverse meniscus

motion during the stage q2. When the meniscus moves backwards with the velocity dh/dt, a

liquid film may remain on the internal capillary surface. In that case the relative surface dilation

velocity is equal to [33]

d lnA 2 dh

dt rca p dt (36)

The expansion of the surface decreases the Triton X-1 O0 adsorption, leading to an increase in 7

for given lifetime as compared to a non-deformed surface.

75

70

�9 �9 II II Ii []

65 2; & ~- 60

55

O O O O O O O O ~

50 I I I

0,001 0,01 0,1

tef IS]

O

Fig. 14. Same as in Fig. 12 for a hydrophilic capillary; V b - 1.8.10 -3 cm 3 (11), 3.5.10 -3 cm 3 (U), 5.10 -3 cm 3 (§ and 1.8.10 -2 cm 3 (~'); the line corresponds to curve (D) from Fig. 12

6. 3. EFFECT OF CAPILLARY LENGTH

In the MBPM typically narrow capillaries (reap=0.005 -- 0 .015cm) are employed

[19, 22 - 30, 36, 41, 60], which makes it possible to neglect gravitational corrections discussed

in Section 5.1. However the length of capillaries is varied within a rather wide range, from a few

tenths of centimetre up to some centimetres. The effect of the capillary length and the

wettability of its internal surface on the measured surface tension was studied in [41]. Both

308

hydrophilic and hydrophobic capillaries with a radius of reap = 0.0084 cm were used and the

length of the capillaries 1 was varied from 1 cm to 12 cm. The dynamic behaviour of the liquid

meniscus within a long hydrophobic capillary is essentially the same as in a short capillary, see

Fig. 3. For intermediate and high bubble frequencies the penetration depth of water into a

capillary of 6 cm length does not exceed 0.2 reap. However it is seen from Fig. 15 that the

pattern of the liquid flow within a hydrophilic long capillary is rather different [41 ].

On the final stage of bubble separation the liquid penetrates into the capillary from the rear side

of the separating bubble and rises along the internal capillary surface in form of an expanding

drop. During this process both the volume and the surface of the drop are increasing rapidly

until the whole cross-section of the capillary is filled by the liquid. The symmetric meniscus

formed in this way subsequently rises to a significant height. The maximum height h depends on

the capillary length and the bubble frequency. With the decrease of t b the value of h somewhat

decreases. No substantial relative pressure decrease ~ was found in the systems employing long

capillaries, which is possibly related to the high aerodynamic resistance of these capillaries. In

long narrow capillaries an aperiodic gas flow regime is established, as shown in Section 5.3.

This results in a prolongation of the meniscus hydrodynamic relaxation time, mainly during the

reverse motion phase described by Eq. (17). The dependence of 3/on t I for long hydrophilic and

hydrophobic capillaries (V~ = 35 cm 3) is shown in Fig. 16 [41 ].

Fig. 15. Dynamics of the liquid flow into a long hydrophilic capillary (1 = 6 cm, t b = 0.5 s); a) bubble reaches ist critical state, b) bubble starts separation and air flows into the capillary from one side, c) air flows in further, d) bubble keeps separating while a meniscus within the capillary is formed

309

The results obtained for the hydrophobic capillaries agree satisfactorily with the standard curve

presented in Fig. 12 - for relatively large bubbles the value of 7 slightly decreases. For the

hydrophilic capillaries however, the discrepancy between the results is rather dramatic. There is

a sharp decrease in surface tension for t I ~ 0, while the total slope of the curve is lower than

that of the standard curve. Thus for large t I the values of ~ exceed those characteristic for the

standard capillary.

The results in Fig. 16, and also Fig. 11 showing P as a function of L, for capillaries of various

lengths, allow to explain this difference between the curves obtained for hydrophilic and

hydrophobic long capillaries. It is seen from Fig. 13 that the value of Lo decreases with

increasing 1. Therefore the longer the capillary is, the lower are the L values in the transition

point. The lower the Lo is, the longer is the deadtime at a given V b and the lower is the surface

tension value at the beginning of the lifetime (at the end of the deadtime period). The arguments

according to which the values y decrease due to the increase of ta agree well with the results

obtained for long hydrophobised capillary. An additional sharp decrease of y at tl ~ 0 for long

hydrophilic capillaries can be related to the process of dilation and subsequent fast compression

of the liquid surface within the capillary, see Fig. 15.

75L 70

65

6o

'~' 55

5O

45

4O

0,001

�9 [] [] �9 t []

II, I I I

0,01 0,1 1

tef [S]

Fig. 16. Same as in Fig. 12 for a hydrophobic ( N o + ) and hydrophilic ( ~ ) capillary of 6 c m length;

V b = 2.5-10 -3 cm 3 (m), 4.5-10 -3 cm 3 ([3), 8.10 -3 cm 3 (+), 3.10 -3 cm 3 (~'), and 7-10 -3 cm 3 (~);

the solid line corresponds to curve (O) from Fig. 12

If during the time of about 4 ms the surface expands 10 times, and compresses thereupon

rapidly (for 1 ms) back to its initial size, then an additional effective lifetime of 30 to 40 ms can

310

be ascribed to the new bubble surface. The fact that the slope of the Y dependence on t I for

hydrophilic capillaries is less pronounced, can be explained by the reverse meniscus motion. The

duration of this reverse motion increases with the increase of l and q, because the liquid

penetrates deeper into the capillary, see Eqs. (16) and (17). Therefore the longer the hydrophilic

capillary is, the greater is the lifetime part spent on the expansion of the liquid during the reverse

meniscus motion. This expansion results in a slower decrease of y as compared to a hydrophobic

capillary.

Therefore the results of dynamic surface tension measurements using MBPM essentially depend

on the characteristics of the capillary employed. Reliable results can be expected for relatively

short and wide capillaries possessing a hydrophobic internal surface. The application of

hydrophilic capillaries with the same geometry (re 2/1 > 5.10 .5 cm) does not lead to significant

errors in the determined y. The error resulting from hydrophobic long capillaries (re 2/1 < 2.

10 .5 cm) is also moderate. These capillaries however are characterised by increased t d values and

therefore cannot be used for the study of concentrated surfactant solutions. The use of

hydrophilic long capillaries leads to significant errors in the measured dynamic surface tensions.

The results obtained with these capillaries do not reflect the actual system behaviour but depend

entirely on complicated hydrodynamic and physico-chemical processes which take place inside

the capillary. The relation between the volume of a separating bubble and the measuring system

volume affects the accuracy of dynamic surface pressure measurements - a ratio of

VJV b > 5000 is recommend. In addition, the volume of a separating bubble has not to be too

large, otherwise increased t d values would restrict the MBPM applicability to weak concentrated

solutions and long times.

7. COMPARISON OF M B P M WITH OTHER METHODS

The results of dynamic surface tension measurements for surfactant solutions obtained from the

MBPM are compared now with the data from other methods, such as Wilhelmy plate, dynamic

capillary, inclined plate, strip, drop volume and oscillating jet [4, 21, 33, 64, 65, 80 - 83]. Good

agreement is found when the data are represented as the function of the effective time (see

Section 5.7).

311

In a number of studies an implicit comparison of data can be performed by comparing the

diffusion coefficients or adsorption-desorption rate constants calculated from the experimental

data. Also in these cases a satisfactory agreement was attained [62, 68-72, 84 - 90].The dynamic

surface tensions for Triton X-100 solutions of various concentrations measured by MBP,

oscillating jet and inclined plate methods are summarised in Fig. 17 and Fig. 18 [33]. The last

two methods have been described in detail elsewhere [91, 92] and have partly a common time

window with the MBPM.

75 -[

6 5 -

,__, 60 - :c: "--- 5 5 - Z .E. 5 0 - 7"--

4 5 -

4 0 -

3 5 -

30

_•++ + - r+,_n

I +,-.. ! +

A []

�9 i +

i I I a

I I I I I I

0,00 0,10 0,20 0,30 0,40 0,80 0,90

t[sl

[] + U [ ] +

D + []

�9 A �9 �9

t I I

0,50 0,60 0,70

Fig. 17 Dynamic surface tension of Triton X-100 solutions, measured by the methods of oscillating jet ( i + ) , inclined plate (DA) and maximum bubble pressure ( � 9 at concentrations of 0.2 ( i D + ) and 0.5 g/1 (+Ai,,)

70 -+

6O

5:5

7" 50 0_!

45 ~11~ '

4 0 ~<)O ~+ B i ' ~ l i l

35 we,%,

30 I t

0 0,01 0,02

+ + + +

I I I I I I I

0,03 0,04 0,05 0,06 0,07 0,08 0,09

t [ s ]

Fig. 18 Dynamic surface tension of Triton X-100 solutions, measured by the methods of oscillating jet ( i + ) , and maximum bubble pressure (+A) at concentrations of 2 ( i + ) and 5 g/l (+A)

312

The capillary used in the studies with the MBPM was wide and short enough (reap = 0.0088 cm,

1 = 1.5 cm) to prevent liquid inflow after bubble separation (re 2/1 > 5-10 5 cm) and fast pressure

oscillation in the capillary (K 2 >) 1). It is seen from the curves presented in Figs. 17 and 18 that

good agreement exists between the experimental methods.

8. EXAMPLES OF EXPERIMENTAL RESULTS

In order to explain the capacity of the MBPM and the way of data interpretation, a number of

experimental results are presented in this section. First results for typical surfactant solutions are

presented to demonstrate the main application field of this methodology. The maximum bubble

pressure method provides dynamic surface tension data in the shortest adsorption time available

now. This became clear already from the comparison of experimental results in the preceding

section.

As a second important field of application the test of liquids, mostly water, concerning its

content in surfactant is shown. This group of experiments is important when natural water, for

example from rivers or lakes, is tested against pollution by surfactants. On the other hand,

industrial water is treated to replace for example surface active compounds and the technology

needs sensitive measures to control the purification procedure.

Finally a group of experiments dynamic surface tensions of the biological liquids, such as blood

or urine, are given and possible future applications discussed in terms of screening test to

analyse diseases and their therapy.

8.1. SURFACTANT SOLUTIONS

Rehbinder [93, 94] and subsequently Adam and Shute [95] were the first to apply the MBP

method to measure the dynamic surface tension of surfactant solutions, followed by many others

[69]. As very recent results, the micelle dissociation kinetics studies based on dynamic surface

tensions of micellar solutions [32, 65, 82, 96, 97], investigations of general rules of adsorption

kinetics for ionic and non-ionic surfactants [85, 86, 98, 99] and their mixtures [100, 101],

studies of the adsorption mechanism for lower normal alcohols [88] and oxyethylated anionic

surfactants [ 102], unusual adsorption behaviour of Tritons with different numbers of EO groups

313

[83, 103] and alkyl dimethyl phosphine oxides possessing long hydrocarbon chains [ 104] should

be mentioned here.

Fig. 19 shows the dependence of dynamic surface tension on t -1/2 for Triton X-100 solution at

various temperatures [ 103 ]. The results for t > 2 s are obtained by the stopped flow procedure

and fit quite well with the data obtained for a constant gas flow. For large time Eq. (21)

transforms into the well-known approximation [60, 62, 64, 98, 105]

dY I R T F 2 ~ (37) dt-l# t-,~o- 2c0

From the slope of the linear part of dy/dt -v2 at t --> oo, as shown in Fig. 18, it can be concluded

that the adsorption mechanism of Triton X-100 in the long time range is diffusion controlled.

The values of F calculated from Eq. (37) coincide with the equilibrium values obtained from the

corresponding adsorption isotherm. The intersection points of these straight lines with the

ordinate axis yield the equilibrium surface tension values % which agree well with the data

published elsewhere [ 103 ].

The capabilities of the MBPM for studies of processes in the millisecond and submillisecond

range of surface lifetimes are illustrated in Fig. 20. The measuring cell and the capillary used for

these studies have been designed such that bubbles could be generated with a frequency of

almost 100 Hz, so that a deadtime of 10 ms was reached [66].

75 70

65 60

E ---- 55 7" 5o

45 4O

35 3O

I l l i l I �9 i j l i �9 I I I �9 �9 ,..jil aa iliWi i D [3 D D 4 , n r i

. . . D I D ` D D . �9 ~O $ �9 [] ,. * <><><> <

~A A A A

I

AA

A

, t A <> A A A

I I I I

0 1 2 3 4 5

1/qt [1/qs]

Fig. 19 Dynamic surface tension as 3r (1 / 4]-) plots of Triton X- 100 solution c o = 0.155.10 -6 mol / cm 3 at different temperatures: 30~ (m), 40oC (n), 50oC (,), 60oc (0), 70~ (Ik)

314

75- 73 71 69

r-----I E 67 --....

7" 65 E

63 61 59 57 55

D ~ O ~

- O 0 O

- A

_

A A A Z~

0

�9 A

A

�9 m e o []

<> O <> <}

A �9 �9 A

A Z~ Z~

[] [] [] [] D O �9 �9 ~ r �9

O O O r

�9 �9 �9 A A �9

A A A A A ,,X A

I I i i I

0,001 0,002 0,003 0,004 0,005

t[s]

Fig. 20 Dynamic surface tension ofpropanol solutions in the time range t < 5 ms; Co = 2 (to), 4 (41,), 8 ('~'), 16 (A), 32 (A) 10-Smol/cm 3

The experimental dependencies of ~ on t for aqueous solutions of low molecular alcohols

(propanol to pentanol), in particular in presence of substances which form (fructose) or break

(urea) the water structure were found to be in a good correspondence with results obtained

earlier [88]. The adsorption mechanism of these surfactants is non-diffusive. To achieve

agreement between the experimental data and theoretical results, a non-equilibrium adsorption

layer model was developed, where the concentration gradient of the diffusion layer causes the

increased 1' values observed experimentally [88].

The dynamic surface tension for the solutions of tridecyl dimethyl phosphine oxide C13D/V[PO ,

pentanol, and mixture of these two surfactants are illustrated in Fig. 21. One can see that for

mixture at times of about 0.1 s, small increases in 7 take place (by 0.5 mN/m or less), followed

by a sharp decrease.

It was shown in [ 101 ] that the anomalous dependence of t' on t is characteristic to surfactant

mixtures where the partial molar area of the main component (in the present case propanol) is

lower than the partial molar area of the second component (C13DMPO) the concentration of

which is significantly smaller while the surface activity is significantly higher. This results in a

depletion of the first component by the second one from the adsorption layer causing the

extremum in the ~(t) dependence. This phenomenon, which seems to conflict with

315

D

on the basis of the well-known principle of

75

70

65

6O

5O

45

m mm mm,mmlm-mlmm mm mu ~Imn

~~IsJ~e~s o ~ ooooo []

I I

thermodynamics, can be explained

Braun-Le Chatelier for adsorption [ 106].

40 I

0,01 0,1 1 10

t [s]

Fig. 21 Dynamic surface tension for aqueous solutions of tridecyl dimethyl phosphine oxide c o = 2.10 -7 mol/cm 3 (m), pentanol c o = 4.10 -3 mol/cm3(rq), and their mixture (119

This principle also govern the state of surfactant molecules at an interface. If a molecule can

vary its partial molar area at the interface, then at small surface pressures H the state with the

maximum partial molar area is more probable [83, 103, 106]. This conclusion follows from the

relation between the surfactant adsorption values in state 1 (with maximum partial molar area

031) and state 2 (with minimum partial molar area c02) [107]:

/ t (H 2 ~ F1 031 -032 exp R~ F2 - exp 03 z

Here mr. is the average partial molar area. As an example the adsorption behaviour of such a

system is shown in Fig. 22 in form of dynamic surface tensions of tetradecyl dimethyl phosphine

oxide solutions C14DMPO [104]. The employment of three complementary methods (MBPM,

drop volume and ring method) made it possible to extend the experimental time range over

seven orders of magnitude.

316

Fig. 22 Dynamic surface tension for aqueous solutions of tetradecyl dimethyl phosphine oxide, c o = 1.25.10 -8 mol/cm 3 from MBPM ([3), drop volume method (A) and ring method ('~'). Calculated values from the diffusion model assuming no reorientation at the surface (dashed line), or possible reorientation(dotted line and solid line); approximate model (dotted), rigorous model (solid)

The C14DMPO molecules are rather asymmetric and the ratio of molar areas between the two

s t a t e s (.D1/(l) 2 - 3. The theoretical curves calculated from two diffusion models (one allowing for

and other neglecting the reorientation) show that the C14DM~PO molecule can indeed vary its

molecular area: for low 1-I values the hydrocarbon chain can orient along the surface.

8. 2. NATURAL WATER

An important field of application of the dynamic tensiometry is the monitoring of natural and

industrial waste waters with respect to trace quantities of surfactants. For such studies the

MBPM can be used successfully. As an example the standard version of the MPT1 tensiometer

operated in the stopped flow regime can be used when equipped by an external gas volume of

300 cm 3 connected to the measuring system via a capillary (1 = 5 cm, reap =0.02 cm). The

inclusion of this additional gas volume results in an increased sensitivity in the stopped flow

regime (cf. Section 5.8). However, the data obtained for distilled water are somewhat distorted

yielding to an apparent decrease of 3' by 0.1 mN/m in the time range from 0.5 to 15 s. This is

caused by the resistance of this additional capillary (11 in Fig. 23). The results of the dynamic

317

surface tension studies for water of the river Kalmius flowing through the Ukrainian industrial

city Donetsk are illustrated in Fig. 23.

70,9

70,8 r----i

2 70,7 --._.

z U. 70,6

70,5

70,4

70,3

mml~)~)~O)~~m m �9 �9 ~

[] [] []

[]

�9 �9 mmm

[ ] D

[33 D D D

[]

[ ]

I I I I I I I

1 2 3 4 5 6 7

~/t [~/sl

Fig. 23 Dynamic surface tension for distilled (m) and river (t3) water measured by a special equipped MBPM using the stopped flow regime (see text)

For the short time Eq. (21) can be transformed into the following relationship [4, 69, 98]

d7 I - - 2 R T c 0 ~ (39)

From this approximation one can see that the slope of the line 7--y(t 1/2) is proportional to the

surfactant concentration. The plot presented in Fig. 23 was used to estimate the surfactant

concentration in the river water. The result obtained is c o = 1.5-10 -9 mol/cm 3, which for mean

molecular mass 300 g/mol corresponds to 0.45 mg/1. This value agrees well with the chemical

analysis data for the ionic surfactants contents of 0.1 to 0.8 mg/1 in the river water.

8. 3. BIOLOGICAL LIQUIDS

A completely new and very encouraging area of application of the MBPM is the measurement

of dynamic surface tensions of biological liquids, i.e. the blood, urine, and cerebrospinal,

synovial, amniotic and other human body liquids. Systematic investigations in this area were first

performed by Kazakov et al. [ 108-110] and Sinjachenko et al. [ 111 ]. The dynamic tensiograms

of blood from healthy males of various age are presented in Fig. 24.

318

75

~-, 70 E

Z .E.

65

60

O m iRma �9 nmmmmiD D D ud~3 �9 �9

URn DIDO [] DD~

% <> 0<> <>o~z,~,~* G O < > ,

D

I I I I I I

-8 -6 -4 -2 0 2 4

In t[s]

Fig. 24. Dynamic surface tensions of blood from healthy male patients of different age: 54 (i), 52 ([3), 27 (§ and 20 ('~) years

One can see that the surface tension of blood of young men is relatively low and increases with

the age. In these experiments a special measuring cell was employed containing ca. 1 ml of the

liquid. This cell allows to study any kind of human or other biological liquids. Some

characteristics of the dynamic tensiogram were found to be informative enough, in particular,

the dynamic surface tension for t = 0.01 s and 1 s, the equilibrium surface tension %, the slope

of the straight line 3,(t 1/2) at t--~ 0 and the slope of the straight 1 ne 7(t 1/2) for t ~ oo. The

physical meaning of the last two characteristics becomes clear from Eqs. (37) and (39). The age

dependence of the tensiometric data is consistent: for example, for males below 20 the value of

at t - 0.01 s is 67+1 mN/m, while for men above 50 this value is 72+1 mN/m. The value

TE=d~//dt 1/2 for the same ages decreases from 15+1 mN.ml.s 1/2 to 10+1 mN.ml.s "I/2,

respectively.

It is known that various pathologies result in a variation of the concentration and composition of

proteins, lipids, peptides, glycerides, nucleic and amino acids, etc. This leads to changes in the

dynamic tensiograms for biological liquids. The tensiometric data for ill persons compared with

corresponding averaged data for health men differ significantly. The results obtained are rather

promising, enabling one to foresee a future application of the dynamic tensiometry among other

modern differential diagnosis methods. This is illustrated by the data obtained for various forms

of glomerulonephritis. For example, for the acute glomerulonephritis the TE value for blood

319

does not vary with respect to the standard data, while for urine this value decreases by 60 % to

the average. On the contrary, for the Genoch glomerulonephritis (gemorragical vasculit) no

variations in the urine are found, while the TE value for blood increased by 20 %. For the

chronic glomerulonephritis and lupus glomerulonephritis (system red lupus) the TE value was

subject to variations in both liquids: 20- 30 % increase in the blood was accompanied by

20 - 40 % decrease in the urine. Variations of other dynamic tensiogram characteristics also had

shown opposite trends for various forms of the glomerulonephritis. These effects have been

statistically tested, and representative sampling have shown high correlation.

For all the pathologies mentioned above the tensiometric characteristics correlate with the

biochemical composition of the blood. It was shown, however, that in most cases the

information derived from the dynamic tensiograms does not only duplicate that obtained from

usual biochemical analyses of the same biological liquids. In this connection it is interesting to

examine the plots represented in Fig. 25. One can see that immediately alter the kidney

transplantation the form of the blood dynamic tensiogram undergoes significant changes. After

one month, however, the tensiogram characteristics return to values usual for health persons of

that age. Various pathologies affect significantly the form of the dynamic tensiograms for other

biological liquids as illustrated in Fig. 26. The tensiograms of cerebrospinal fluid of a patient

suffering from a thorax trauma are presented.

80-

75

70 -ff 65

~- 60

55

_ �9

% 50 I I I I I

0,001 0,01 0,1 1 10 100

tof, [s]

Fig. 25 Dynamic tensiograms of blood before (11) and after a kidney transplantation: immediately ([3), one week (A) and one month (A) after transplantation.

320

80

70

60

so

40

3O I I

0,1 1 10

tef [S]

I

100

Fig. 26 Dynamic tensiograms of cerebrospinal fluid of a patient suffering from a thorax trauma before (I), immediately (U), and one week (A) after operation.

The y values, which are very low before the operation, increase somewhat after the operation,

and return to values characteristic for a health person already after one week. While the studies

of the mechanism of such significant changes in the dynamic surface tension behaviour of

biological liquids in pathologic states are yet on their initial stage, one can hope that together

with the commercialisation of dynamic tensiometers the efforts of scientists in medicine and

biology will increase significantly to come closer to a solution of this problem.

9. REFERENCES

1. M.Simon, Ann. Chim. Phys. 32(1851)5.

2. J.F.Padday, in Surface and Colloid Science, Matijevic (Ed.), Vol. 1, Wiley - Interscience,

New York, 1969, p. 101

3. K.J.Mysels, Colloid Surfaces, 43(1990)241.

4. R.Miller, P.Joos and V.B.Fainerman, Adv. Colloid and Interface Sci., 49(1994)249

5. A.I.Rusanov and V.A.Prokhorov, Interfacial Tensiometry, in Studies in Interface Science,

Vol.3, D.M6bius and R.Miller (Eds.), Elsevier, Amsterdam, 1996

6. R.C.Brown, Philos. Mag., 13(1932)578

7. S.Sugden, J. Chem. Soc., 121(1922)858

8. F.M.Jaeger, K.Ned. Akad. Wet.Versl. Gewone Vergad. Atd. Natuurkd., 23(1914)330;

Z. Anorg.Chem., 101(1917)1.

321

9. EL.Warren, Philos. Mag., 4(1927)358

10. P.P.Pugachevich, Zh. Fiz. Khim., 38(1964)758

11 P.T.Belov, Zh. Fiz. Khim., 55(1981)302

12. K.Lunkenheimer, R.Miller and J.Becht, Colloid Polymer Sci, 260(1982)1145

13 R.Razouk and D.Walmsley, J. Colloid Interface Sci., 47(1974)515

14 J.L.Ross, W.D.Bruce and W.S.Janna, Langmuir, 8(1992)2644

15 I.N.Feldman, I.V.Malkova, V.I.Sokolovskij and R.A.Zaturenskij, Zh.Prikl. Khim.,

53(1980)1594

16 K.H.Cuny and K.L.Wolf, Ann. Phys. Leipzig, 17(1956)57

17 R.Kuffner, M.T.Bush and L.J.Bircher, J. Am. Chem. Soc., 79(1957)

18. A.M.Kragh, Trans. Faraday Soc., 60(1964)225

19. M.Austin, B.B.Bright and E.A.Simpson, J. Colloid Interface Sci., 23(1967)108

20. J.Kloubek, Tenside, 5(1968)317

21 V.B.Fainerman and S.V.Lylyk, Kolloidn. Zh., 44(1982)598

22. R.L.Bendure, J. Colloid Interface Sci., 35(1971)23 8

23 T.E.Miller and W.C.Meyer, American Laboratory, (1984)91

24. X.Y.Hua and M.J.Rosen, J. Colloid Interface Sci., 124(1988)652

25 S.G.Woolfrey, G.M.Banzon and M.J.Groves, J. Colloid Interface Sci., 112(1986)583

26. P.R.Garrett and D.R.Ward, J. Colloid Interface Sci., 132(1989)475

27. D.E.Hirt, R.K.Prud'homme, B.Miller and L.Rebenfeld, Colloids Surfaces, 44(1990) 101

28. K.J.Mysels, Langmuir, 2(1986)428; 5(1989)442

29. C.D.Dushkin, I.B.Ivanov and P.A. Kralchevsky, Colloid Surfaces, 60(1991)235

30. C.P.Hallowell and D.E.Hirt, J. Colloid Interfaces Sci., 168(1994)281

31 Tz.H.Iliev and C.D.Dushkin, Colloid Polymer Sci., 270(1992)370

32. V.B.Fainerman, Colloids Surfaces, 62(1992)333

33 V.B.Fainerman, R.Miller and P.Joos, Colloid Polymer Sci., 272(1994)731

34. R.Miller, V.B.Fainerman, K.-H.Schano, W.Heyer, A.Hofmann and R.Hartmann, Labor

Praxis, NS(1994)

35. R.L.Kao, D.A.Edwards, D.T.Wasan and E.Chen, J. Colloid InterfaceSci., 148(1992) 247

36. J.Kloubek, J. Colloid Interface Sci., 41 (! 972)7

322

37. G.Papeschi, S.Bordi and M.Costa, Ann. Chim., 71(1981)407

38. Z.N.Markina, N.M.Zadymova and O.P.Bovkun, Colloids Surfaces, 22(1987)9

39. L.L.Schramm and W.H.F.Green, Colloid Polymer Sci., 270(1992)694

40. C.D.Holcomb and J.A.Zollweg, J. Colloid Interface Sci., 154(1992)51

41. S.V. Lylyk, A.V. Makievski, V.I. Koval'chuk, K.-H. Schano, V.B. Fainerman, and R.Miller,

Colloids & Surfaces A, in press

42 S.S.Dukhin, N.A.Mishchuk, V.B.Fainerman and R.Miller, submitted to Colloids Surfaces A

43 S.Sugden, J. Chem. Soc., 125(1924)27

44 B.N.Volkov and L.D.Volyak, Zh. Fiz. Khim., 46(1972)598

45 C.H.J.Johnson and J.E.Lane, J. Colloid Interface Sci., 47(1974)117

46 I.S.Kisil', A.G.Mal'ko and M.M.Dranchuk, Zh. Fiz. Khim., 55(1981)177

47. E.Schr6dinger, Ann. Phys., 46(1915)413

48. N.N.Oguz and A.Prosperetti, J. Fluid Mech., 257(1993)111

49 K.S.Avramidis and T.S.Jiang, J. Colloid Interface Sci., 147(1991)262

50. G.S.Keen and J.R.Blake, J. Colloid Interface Sci., 180(1996)625

51 V.B.Fainerman, A.V.Makievski and R.Miller, Colloids Surfaces A, 75(1993)229

52. S.S.Dukhin, V.B.Fainerman and R.Miller, Colloids Surfaces A, 114(1996)61

53 S.S.Dukhin, A.I.Koval'chuk, V.B.Fainerman and R.Miller, submitted to Colloids Surfaces A

54. V.I.Koval'chuk, S.S.Dukhin, V.B.Fainerman and R.Miller, submitted to Colloids Surfaces A

55. V.I.Koval'chuk, S.S.Dukhin, V.B.Fainerman and R.Miller, submitted to J. Colloids

Interfaces Sci.

56. V.I.Koval'chuk, S.S.Dukhin, A.V. Makievski, V.B.Fainerman and R.Miller, submitted to

J. Colloids Interfaces Sci.

57. V.B.Fainerman, Kolloidn. Zh., 41(1979)111

58. V.B.Fainerman, Kolloidn. Zh., 52(1990)921

59. T.S.Horosov, C.D.Dushkin, K.D.Danov, L.N.Arnaudov, O.D.Velev. A.Mehreteab and

G.Broze, Colloids Surfaces A., 113(1996)117

60. S.S.Dukhin, G.Kretzschmar, R.Miller. Dynamics of Adsorption at Liquid Interfaces.

Theory, Experiment, Application, in "Studies in Interface Science". D.M/Sbius and

R.Miller (Eds.),Vol. 1, Elsevier, Amsterdam, 1995

61 R.Miller, Colloid Polymer Sci., 258(1980) 179

62. P.Joos and E.Rillaerts, J. Colloid Interface Sci., 79(1981) 96

63 P.Joos and M.Van Uffelen, J. Colloid Interface Sci., 171 (1995)297

64. V.B.Fainerman A.V.Makievski and P.Joos, Zh. Fiz. Khim. 67(1993)456

65. A.V.Makievski, V.B.Fainerman and P.Joos, J. Colloid Interface Sci., 166(1994)6

66. V.B.Fainerman and R.Miller, J. Colloid Interface Sci., 175(1995) 118

67. F.Van Voorst Vader, Th.Erkelens and M.Van den Tempel, Trans Faraday Soc.,

60(1964) 1170

68. G.Bleys and P.Joos, J.Phys.Chem., 89(1985)1027

69. V.B.Fainerman, Usp. Khim., 54(1985) 1613; Russ. Chem. Rev., 54(1985)948

70. C.-H.Chang and E.I.Franses, Colloids Surfaces A, 100(1995)1

71 B.Li, G.Geeraerts and P.Joos, Colloids Surfaces A, 88(1994a251

72. V.B.Fainerman and R.Miller, Colloids Surfaces A, 97(1995)65

73 K.J.Mysels and R.E.Stafford, Colloids Surfaces, 36(1986)289; 41(1989)385

74. C.Huh and E.L.Scriven, J. Colloid Interface Sci., 30(1969)325

75 J.E.Lane, J. Colloid Interface Sci., 42(1973)145

76. G.A.Bottomley, Aust. J.Chem., 27(1974)2297

77 R.Campanini, A.Swanson and S.K.Nicol, J.Chem. Soc., Faraday Trans.1, 72(1976)2638

78 J.Kloubek, Colloid Polymer Sci., 253(1975)754

79. V.B.Fainerman, S.V.Lylyk and V.D.Jamilova, Kolloidn. Zh.,49(1987)509

80. J.Van Hunsen and P.Joos, Colloids Surfaces, 24(1987)139

81 J.Kloubek, J. Colloid Interface Sci., 41 (1972) 17

82 G.Geeraerts and P.Joos, Colloids Surfaces A., 90(1994)149

83 V.B.Fainerman, A.V.Makievski and P.Joos, Colloids Surfaces A., 90(1994)213

84. R.P.Borwankar and D.T.Wasan, Chem. Eng. Sci., 41 (1986) 199

85 X.Y.Xua and M.J. Rosen, J. Colloid Interface Sci., 141(1991)180

86. P.Joos, J.P.Fang and G.Serrien, J. Colloid Interface Sci., 151 (1992) 144

87. V.B.Fainerman and S.V.Lylyk, Kolloidn. Zh., 44(1982)1141

88. V.B.Fainerman and R.Miller, J. Colloid Interface Sci., 178(1996) 168

89. P.Joos and G.Serrien, J. Colloid Interface Sci., 127(1989) 97

323

324

90. G.Geeraerts, P.Joos and F.ViI16, Colloid Surfaces A, 75(1993)243

91. P.Van den Bogaert and P.Joos, J.Phys.Chem., 83(1979)2244

92. R.Defay and G.Petrr, In Surface and Colloid Science, E.Matijevic (Ed.), Vol. 3, Wiley,

New York, 1971

93. P.A.Rehbinder, Z.Phys.Chem., 111(1924)447

94. P.A.Rehbinder, Biochem. Z., 187(1927) 19

95. N.K.Adam and H.L.Shute, Trans. Faraday Soc., 31(1935)204; 34(1938)758

96. S.D.Dushkin, Tz.H.Iliev and Y.S.Radkov, Colloid Polymer Sci., 273(1995)370

97. K.Danov, P.Vlahovska, T.Horozov, C.Dushkin, P.A.Kralchevsky, A.Mehreteab and

G.Broze, J. Colloid Interface Sci., 183(1996)223

98. V.B.Fainerman, A.V.Makievski and R.Miller, Colloids Surfaces A, 87(1994)61

99. J.Eastoe, J.Dalton, P.Roqueda, D.Sharpe, J.Dong and J.R.P.Webster, Langmuir

12(1996)2706

100. V.B.Fainerman and R.Miller, Colloids Surfaces A, 97(1995)65

101. V.B.Fainerman and R.Miller, Langmuir, 13(1997)409

102. A.V.Makievski, R.Miller, G. Czichocki and V.B.Fainerman, Colloids Surfaces A, in press

103. V.B.Fainerman, R.Miller and A.V.Makievski, Langmuir, 11 (1995)3054

104. A.V. Aksenenko, A.V.Makievski, R.Miller and V.B.Fainerman, submitted to Colloids

Surfaces A

105. A.V. Makievski, V.B. Fainerman, R. Miller, M. Bree, L. Liggieri and F. Ravera, Colloids

& Surfaces A, 122(1997)269

106. P.Joos and G.Serrien, J. Colloid Interface Sci., 127(1989)97

107. V.B.Fainerman, R.Miller, R.WOstneck and A.V.Makievski, J.Phys.Chem., 100(1996)7669

108. V.N.Kazakov, V.B.Fainerman, O.VSinjachenko, R.Miller, P.Joos, S.V.Lylyk, A.E.Ajko,

D.V.Trukhin and M.V.Ermolaeva, Archive for Clinical and Experimental Medicine,

4(1995)3

109. V.N.Kazakov, O.V.Sinjachenko, V.B.Fainerman, E.E.Barinov, R.Miller, M.V.Ermolaeva

and I.A.Sidorenko, Archive for Clinical and Experimental Medicine, 5(1996)3

110. V.N.Kazakov, Ju.A.Talalaenko, O.V.Sinjachenko, V.B.Fainerman and R.Miller, Medical

and Social Problems of Family, 1 (1996)47

111. O.V.Sinjachenko, V.N.Kazakov, V.B.Fainerman, M.V.Ermolaeva, Ju.A.Talalaenko,

I.A.Sidorenko and D.V.Trukhin, School Fundament Med. J., 2(1996)87

10. LIST OF SYMBOLS

325

A

c O

D

g

H

K = o0"1; 0

kp

L L

P a

Pb

Pd

PH

P~

R

rb

rc~p

T

- surface area

- capillary constant

- bulk concentration

- diffusion coefficient

- bubble diameter, also distance

- correction factor in the Laplace equation

- gravity constant

- capillary immersion depth

- penetration depth of liquid into the capillary

- dimensionless parameter

- Poiseuille equation constant

- dilatational surface viscosity

- gas flow rate

- capillary length

- capillary pressure

- atmospheric pressure

- excess pressure in the bubble

- excess dynamic pressure

- hydrostatic pressure

- excess pressure in the measuring system

- gas constant

- current value of bubble radius

- radius of a separating bubble

- capillary radius

- temperature

- time

- time interval between successive bubbles

326

t d

TE = dy/dt 1/2

tef

th

t 1

%

Vs

- deadtime

- dynamic tensiogram characteristic

- effective adsorption time

- hydrodynamic relaxation time

- lifetime

- bubble volume

- volume of measuring system

- sound velocity

- coordinate along the capillary axis

s = A P / P S

F

Y0

Y~

11

0 = d In A/dt

H

PL

~ - t(4v/r~)

~0

(%

- relative pressure drop

- adsorption

- surface tension

- surface tension of the solvent

- equilibrium surface tension of a solution

- gas dynamic viscosity

- relative dilation rate

- dynamic viscosity of liquid

- kinematic viscosity of gas

- surface pressure

- gas density

- liquid density

- current time

- dimensionless time

- characteristic time of pressure oscillations

- contact angle

- partial molar surface area

- characteristic frequency of pressure oscillations

Drops and Bubbles in Interfacial Research D. M6bius and R. Miller (Editors) �9 1998 Elsevier Science B.V. All rights reserved.

The Oscillating Bubble Method

327

K.-D. Wantke and H. Fruhner

Max-Planck-Institut ftir Kolloid- und Grenzfl~ichenforschung, Rudower Chaussee 5,

13-12489 Berlin-Adlershof, Germany

Contents

1. Introduction

2. Some remarks on surface rheology

3. Oscillating bubble devices

3.1. Version 1: Excitation of the system via the gas chamber

3.2. Version 2: Excitation of the system via the solution chamber

4. Theory of spherical bubble oscillations in surfactant solutions

4.1. Force balance relations

4.2. Dynamic surface tension

4.3. Comparison of measurements with isotherm calculations

5 Modifications of the standard model

5 1. Intrinsic viscosity

5 2. Modification of the model for ionic surfactant solutions

5.3. Influence of the deviation of ideal sphere

6 Experimental examples

7 List of symbols

8 References

328

1. INTRODUCTION

Investigation of the mechanical properties of soluble adsorption layers of surfactants,

polymers, and their mixtures, provides information about the flow properties of liquid

interfaces. The knowledge of these surface rheological properties is important for the

understanding and controlling of processes in which surface deformations take place such as

the coalescence of foams and drops in emulsions, thin films flow, flotation, and high-speed

coating processes in the manufacture of photographic materials. Surface rheological properties

effect all these technological processes but the exact mechanism is not yet fully understood.

For example, coating experiments have demonstrated that the dilational properties of

gelatin/anionic surfactant adsorption layers were of crucial importance in maintaining the

stability of the liquid bridge formed between the lower edge of the slide hopper and the moving

web [ 1 ]. Other authors [2, 3 ] demonstrated that the surface rheological properties, especially

the surface elasticity, played a major role in stabilizing foams. The importance of the

rheological properties of adsorption layers at thin liquid films between emulsion droplets is not

doubted, but experimental measurements of these properties are very scarce.

In particular the dilational properties of adsorption layers have not been sufficiently

investigated. The main reason for this is that there are not many convenient experimental me-

thods to observe all possible relaxation processes over a wide frequency range. Surface shear

properties have been more extensively investigated despite the dominance of the surface

dilational properties in processes involving expansion or compression of surfaces.

The dilational properties of the surface include elastic, viscous and transfer effects. Separation

of surface effects from the influence of the bulk is complicated. This is due to the simultaneous

action of the diffusion process between the surface and the bulk and the intrinsic surface

rheological effects. The diffusion exchange of matter gives rise to an apparent dilational

viscosity [4]. When a soluble adsorption layer is compressed and expanded the change in

surface tension is determined by a change in concentration, in other words the adsorption

kinetics and the kinetics of molecular rearrangement at the surface. Therefore, size and

chemical structure of the surfactant molecules are decisive factors. If the surface deformation is

faster than the diffusion exchange of surfactant molecules between the surface and bulk, the

surface layer behaves like an insoluble monolayer.

In general there are two different procedures for measurement of dilational properties. The first

is measurement of the dynamic surface tension 7(0 aRer a change in surface area (relaxation

effects). The second procedure is that based on a harmonic deformation of the surface.

Investigations of relaxation processes in soluble adsorption layers using a technique based on

transient changes in surface area have been described by Loglio et al. [ 5-7 ], Joos et al. [ 8 ] or

329

Fang et al. [9, 10]. Another well known method is the pendent drop technique. This method

determines interfacial tensions from analysis of the drop profile to within an accuracy

+0.1 mN/m [11]. A modification of the pendent drop technique is the drop pressure

experiment which allows measurement of the pressure difference Ap across the curved surface

[12-14, chapter 4]. The pressure transducer which is mounted in the measuring chamber, has

a high time resolution and allows measurement of dynamic surface tensions in time scales of

less than 1 ms.

There are different techniques available for measurements of dilational properties based on the

principle of harmonic changes of area. These operate over different frequency intervals and use

different excitation principles. For example, the capillary wave technique generates

mechanically or electrically harmonic waves and measures the damping factor and the

wavelength of the propagated wave. It operates in a frequency range of 30 Hz to 4 kHz

[15-18]. Attachment of high frequencies is advantageous for investigation of some surface

rheological problems. A critical point is that the wave parameters must be determined very

precisely. The wavelength for example must be known to within an accuracy of better than 1%.

This is very difficult to achieve experimentally. A related technique is the light scattering from

thermally excited capillary waves [ 19-21 ] which also needs a complicated model.

Another established technique is the oscillating barrier method. This is a modified Langmuir

trough with two symmetrically oscillating barriers and a Wilhelmy plate used for measuring

the dynamic surface tension [22 - 24, 10]. A disadvantage of this method is that it works only

over a small frequency range of 0.005 Hz to 0.5 Hz, therefore the method is limited to

insoluble monolayers or soluble surfactants at low concentrations.

With the oscillating bubble method which will be described in more detail later, the dilational

elasticity, dilational viscosity and the diffusional exchange of matter between the adsorption

layer and bulk can be determined in a frequency range of 0.001 Hz to 400 Hz. Various

modified experimental setups have been used by several authors [25-29], all of them were

developed for investigation of the rheological properties of adsorption layers at the air/water

interface. However, using a modified measuring head, the dilational properties of an oil/water

interface can also be determined by this method. Such measurements are useful to investigate

correlation's between the surface properties and the emulsion stability.

Another application of the oscillating bubble technique is the investigation of the dynamic

surface tension of lung surfactants [28, 29, chapter 10] which will not be discussed in this

chapter. Such systems are much more complicated and the rheological models described in this

chapter need to be modified. However the dynamic surface tension of such systems,

determined by a specific form of oscillating bubble technique, has been applied to characterize

lung surfactants. One of the modifications of this technique, the captive bubble method, has

330

been also developed for this purpose [30]. The problem of the lung surfactant is reviewed in

[31].

2. SOME REMARKS ON SURFACE RHEOLOGY

The theoretical model necessary for the evaluation of all oscillating bubble measurements is

based on the well known theory of the dynamic behavior of surfactant solutions [ 14 ]. Here we

consider the problems in the framework of a general surface rheology. This framework allows

a uniform treatment of surface and bulk properties because it describes the flux processes and

the force balance conditions from a general point of view.

Special problems arise in the theoretical description of such systems due to the fact that a

surface or an interface can be considered as a two-dimensional, highly compressible phase of

variable composition, while the adjacent bulk phase is approximated as a three-dimensional

incompressible homogeneous fluid. Therefore the dilational properties have a greater influence

at the surface than within the bulk. Under static conditions, the composition, tension state, and

the shape of the interface are determined by the laws of equilibrium thermodynamics

(cf. Chapter 1). This means that an external disturbance of such a system, e.g. by stretching or

heating, leads not only to a new tension state, but also to relaxation processes as a consequence

of the disturbed thermodynamic equilibrium. The tension state is a function of its history which

means that the system exhibits typical viscoelastic behavior. Therefore, in most cases an

independent two-dimensional model of the interface is an unrealistic idealization. Nevertheless,

the tension state and the transport equations of the components are useful basic notations for a

model describing the mechanical behavior of an inhomogeneous fluid system. The relation

between these two notations is the main topic of investigation in rheology.

For a simple homogeneous phase, the relation between the force balance and motion equation

is well known as in Hooke's equations for an elastic body or pressure tensor of a frictional

fluid, whose divergence leads to the Navier-Stokes equations [32]. As long as the volume

elements are so big that the assumption of a homogeneous material is realistic, additional

effects, as a consequence of an inhomogeneous substructure, can be taken into account by

additional terms in the force balance equations. An example of this is the use of a time-

dependent functional in the Navier-Stoke equations describing a Maxwell model of a

viscoelastic fluid [33]. Detailed investigation of such effects is difficult because the

determination of the local tension state, e.g. by exploration of the substructure, is uncertain.

However, the dynamic surface tension provides direct information about a state of tension in a

molecular dimension of one space direction, and such measurements allow experimental

investigations of particular questions concerning properties of complex fluids. For example, it

331

is easy to study the influence of the specific substances, like surfactants, on this state of

tension.

An advantage of harmonic deformation measurements is that there are different techniques

which altogether cover a wide frequency range. For low frequencies, the barrier motion within

the Langmuir trough is used as a standard method and for high frequencies, the light scattering

of thermic waves is studied. However, there is a lack of reliable methods to cover intermediate

frequencies. Therefore over the last few years, some groups have tried to improve the

oscillating bubble method for investigation of the dilational properties of fluid interfaces. The

main problem of all dynamic methods is the elimination of the bulk influence. In this respect

with oscillating measurements, particularly with the oscillating bubble method, the influence of

the bulk is easily eliminated by a calibration procedure.

As mentioned above, we consider an air/fluid or a fluid/fluid interface as a two-dimensional,

compressible phase of variable composition and the adjacent bulk phase as a three-

dimensional, incompressible homogeneous fluid. The tension state within the bulk phase is

then given by the general pressure tensor of a frictional flow [ 32 ] :

----PS,k -1- -[- - [ -~ik K - - (1)

All symbols are explained in section 7. For an incompressible homogeneous fluid (0vl/0x 1 = 0)

the divergence of the stress tensor completed by the inertial terms leads to the Navier-Stokes

equations in the form

9 + (vVv - -grad p + TIALv (2)

if there is no external force acting. This homogeneous force balance equation must be replaced

within or near the interface by equations which take into account the inhomogeneous structure

and the transfer effects.

To describe fully dynamic processes exact knowledge of local details near the interface is

important. The simplest way to do this is the formal introduction of a two-dimensional pressure

tensor. If the interfacial tension is only caused by the molecules of a monolayer a tensor can be

defined in this layer [27, 34, 35]. However, this procedure raises some problems. A two-

dimensional state of tension requires normally an expanded third dimension with equal tension

in all cut planes, e. g., the elastic tension within a rod. This special condition is not fulfilled for

an interface. Besides, with a two-dimensional pressure tensor we cannot describe a shear

332

tension between the bulk and the interface. Therefore, we must keep in mind that the force

balance within the surface is not only given by the divergence of this tensor as in the case of a

homogeneous bulk phase, but it also needs additional terms which can be considered as

external forces. They must describe such shear effects and the influence of the boundary

condition resulting from the bulk stress tensor [32, 36, 37].

By taking into account this fact it is possible to introduce a two-dimensional pressure tensor

according to Eq. (1)

T~ LT;I T~

where the components are functions o f the coordinates x 1 - x1 (~,~), x2 - x2(~,~), x3 - x3(~,~) at

the surface. With the assumption of a homogeneous fluid interface and an appropriate

definition of the constants K' and q' the formal transformation of the tensor Eq. (1) leads to

3 ( ) T:~, - Eo t~ io t~ ,k aik(~/ nu(K'--T1 I )(V s "vS))+ n ' (Vs Vs nU(VsVS)T)i k .

i,k=l (4)

The transformation matrix ot v i results from the normal and tangential unit vectors at the surface

and Eq.(4) describes the projection of the general stress tensor on the surface in Cartesian

coordinates indicated by i, k. Another coordinate system can be introduced if appropriate. For a

plane surface Eq.(4) leads to the standard Eq.(1) in two-dimensional Cartesian coordinates. An

alternative description of the state of surface tension has been given in [38] on the basis of the

standard equation of the elasticity theory, however, for soluble monolayers the fluid model is

more realistic. The term Y in Eq.(4) is the surface tension. According to the standard

assumption this tension depends on the local surface concentration. Therefore an

inhomogeneous deformation of the surface causes a tension gradient (Gibbs-Marangoni effect).

The simple geometry of an ideal spherical bubble results in a homogeneous deformation which

is ideal for the investigation of dilational properties. This is because there are no lateral effects

present and the separation of the influencing parameters is easy. Therefore, we discuss details

only for the important case of an ideal sphere which is the basic model of the oscillating bubble

method (section 4.).

333

3. OSCILLATING BUBBLE DEVICES

3.1 VERSION 1: EXCITATION OF THE SYSTEM VIA THE GAS CHAMBER

All oscillating bubble devices are characterized by an excitation system and measurement of

the resulting change in pressure. The source of the excitation and the pressure transducer can

be located within the air or fluid phase. Here we discuss two examples of both versions. Fig. 1

shows a scheme of the experimental apparatus developed by Lunkenheimer [25, 39]

according to the theoretical outlines, of [40 - 43 ].

3

L

,< W:--- I - s o l u t i o n - -~ . . . .

Fig. 1. Schematic cross-section of the oscillating bubble apparatus of Lunkenheimer [39]: (1) rod of the electro

dynamic excitation system; (2) silicon rubber membrane; (3) temperature controlled chamber; (4)

capillary; ro : radius of of the bubble at the pole; AH0 amplitude of the bubble oscillation at the pole; R

capillary radius; V0 volume of the air chamber.

A small bubble is formed at the tip of a capillary which is immersed in the liquid system and

connected with a gas chamber. The gas chamber is subjected to harmonic oscillations, via a rod

and a membrane, by an electrodynamic excitation system. With this setup the excitation

voltage of the system, the cross sectional area, and the amplitude AH at the top of the bubble

were measured as a function of the excitation frequency at constant bubble parameters (size,

radius, and amplitude AH). The frequency range is 1 Hz - 150 Hz. From these measurements

information about the dynamic properties of the surface can be obtained by use of an

appropriate model described in the next chapter. Calibration measurements_are required with

this system.

334

The contour of a static bubble can be determined by the Young-Laplace equation. Numerical

integration reveals that for small variations in the bubble size, there is a linear relationship

between changes in the following interrelated parameters: volume V and surface area of the

bubble A, radius of curvature at the top of the bubble r0, and distance H. That means we can

replace AA by AA = const.*AH. These calculations have been verified with an optical setup

[39,41].

If the largest dimension of the gas volume is small compared to the length of the corresponding

acoustic waves, the pressure p is given by the adiabatic equations of state. The magnitudes AH

of a surface point at the top of the bubble, the pressure oscillation Ap and the phase shill

between these two oscillations are measurable quantities. With these three parameters we

obtain the "effective" surface elasticity which includes elastic, viscous and transfer effects

according to Eq.(33), (53), or (65). In the first version of Lunkenheimer, measurement of

pressure within the gas chamber was not possible, and had to be determined with the aid of the

gas equation and the amplitude of the excitation volume AV1 (cf.. Eq.(20)). However, the

results were not very accurate. Introduction of a pressure transducer can improve the situation.

In addition the system has a characteristic frequency which is caused by the elasticity of the gas

chamber and approximately determined by apparatus constants V0 and B (cf. Eq.(23) and

Eq.(26)). For measurement of surface rheological properties the characteristic frequency must

be considerably higher than the excitation frequency. Therefore, the gas volume V0 should be

small and the volume AV1, which is proportional to the moving mass, should be large. Besides,

the excitation within the gas chamber requires a higher magnitude of the pumped volume and

therefore an appropriate excitation system must be used. For low frequencies this is

unproblematic, however in the high frequency range, a system with excitation within the

solution is more stable.

A main problem for reproducible measurements is the requirement of a defined position of the

three phase contact line at the tip of the capillary. For the stability of the three phase contact

line, a sharp edge at the tip of the capillary and special wetting properties are very important.

A new version of an oscillating bubble apparatus (Fig. 2. and 3.) with the excitation within the

gas chamber was proposed and designed by K. Stebe and co-workers [27, 44, 45]. They also

excite the system within the gas chamber and measure the gas pressure by using a pressure

transducer. A quartz cell is filled with a surfactant solution in which a bubble is formed at the

tip of an inverted needle. A gas cell contains a piezoelectric piston and a piezoresistive pressure

transducer.

L./

|

syringe

335

Fig. 2. Schematic of the bubble formation apparatus of Johnson and Stebe [45]: it consists of two valves: one to

form the bubble (A) and one (B) to expose the pressure transducer (C) and the piezoelectric piston (D);

the bubble is formed via a syringe pump.

lamp

A/I) board digitizer boa rd

lamp CCD camera

, i i I I

e l

-.I 1,1 | _!._= vressure L ~ , I txansducer ~ ' ~

-0: ..... , , bubble o !

~ I o I

I: i o I t

i photodiode

Fig. 3. Schematic of the oscillating bubble apparatus of Johnson and Stebe [45]

The piston is driven sinusoidally by a function generator and causes the bubble radius to

oscillate. An optical system records the shape of the bubble and the magnitude of the bubble

oscillation. The frequency range of this apparatus is 0.001 Hz to 5 Hz. The sphere is

336

approximately completely formed and therefore the assumptions of the theory are better

fulfilled. However, the gas chamber leads to an additional elasticity and a viscous effect if the

radius of the capillary connecting the bubble with the gas cell is too small. To evaluate the

mass transfer kinetics of a surfactant a phase angle between the oscillation of AA and Ap is

measured. The authors did not calibrate and, instead, the final equation for the evaluation of the

measurements is based on a few theoretical calculations with the aid of the isotherms. Besides,

they do not use their final equations in the form of the Lucassen/v. d. T. model according to

Eq.(53).

3.2 VERSION 2: EXCITATION OF THE SYSTEM VIA THE SOLUTION CHAMBER

Recently, other devices have been developed which use pressure transducers for monitoring

pressure changes inside the solutions [26, 46-52]. From the oscillating bubble versions which

were proposed over the last few years we will mention two: The pulsating bubble

surfactometer used by Chang and Franses [ 26, 46 ] (commercial instrument from Electronetics

Co., Amherst, New York, according to the bubble method of Enhorning [28]) and the

oscillating bubble method of Fruhner [50, 51 ]. Fig. 4. shows a sketch of the pulsating bubble

surfactometer. I

Pulsating Rod

Fig. 4. Schematic of the pulsating bubble method of Chang and Franses [26].

A chamber is filled with the surfactant solution. A capillary is placed in the top of the chamber

and is open to the atmosphere. The bubble is formed at the tip of the capillary in the solution

and the pressure is measured by a pressure transducer inside the chamber. The oscillations of

337

the bubble surface are caused by a pulsating rod located at the bottom of the chamber. The

bubble radius oscillates from a minimum value of 0.40 mm to a maximum value of 0.55 mm at

frequencies between 0.01 Hz and 2 Hz. The pressure responses of this system exhibit the

influence of higher harmonic waves which couple to the linear term. It is possible due to the

large change in radius which does not allow the non-linear terms in Eq.(10), (38), or (46) to be

neglected. The authors have also not introduced the complex elasticity modulus as a relation

between the change in surface tension and the relative change in surface area. Therefore, the

comparison with other experimental results is difficult.

A similar principle suitable for higher frequencies, was proposed by Fruhner [50, 51]. The

principle of this oscillating bubble method is shown in Fig. 5. and 6. A closed measuring

chamber which is temperature controlled, is filled with the surfactant solution. A small

hemispherical bubble is produced at the tip of a capillary with an inner diameter of about 0.05

cm. For stability of the three-phase contact line, a sharp edge at the tip of the capillary and

special wetting properties are also very important in this case. Via a piezoelectric driver

connected to the measuring chamber, the bubble volume and consequently the bubble surface

and radius are subjected to sinusoidal oscillations. Changes in the bubble radius and surface

area produce sinusoidal changes of the pressure in the measuring chamber. These pressure

changes are monitored by a sensitive pressure transducer which is mounted at the bottom of the

chamber. Its electrical signals are measured so as to obtain the pressure amplitudes and the

phase difference q~' between the motion of the driver and the sinusoidal changes of the

pressure in the chamber (Fig. 5).

Fig. 5. Oscillating bubble setup of Fruhner: (1) cap with capillary; (2) walls of the measuring chamber; (3)

solution, (4) piezoelectric driver; (5) low-pressure quartz transducer; (6) amplifier and measuring

instruments for pressure amplitudes and phase angles; (7) frequency generator.

338

Calculation of the pumped volume requires only a constant of the driver to be determined.

Calibration measurements show that a driver monitor is not necessary. The piston directly

follows the excitation voltage with reference to amplitude and phase [ 50 ]. Over the frequency

range of 1 Hz - 400 Hz the pressure transducer has a linear response to the driver. The

piezoelectric driver can also produce other area changes such as step-type or square-pulse,

which are of interest in transient relaxation experiments.

With this technique two possible shapes for the meniscus can be formed as shown in Fig. 6. On

the left side of the figure the meniscus is concave. This shape corresponds to an oscillating

bubble. On the right side the meniscus is convex which corresponds to an oscillating drop. The

disadvantage of this form is a considerably higher effect of inertia and viscosity of the liquid in

the short capillary on the pressure amplitudes. On the other hand, this shape enables optical

measurements at the oscillating surface, and investigations of dilational properties of

monolayers after spreading of an insoluble substance on the surface of the drop.

curved surfa capillary tube

_ . _ c h a m b e r ring

pressure t r a n s d u c e r I I

Fig. 6. Schematic diagram of the oscillating bubble measuring chamber of Fruhner

A cross section of the radial oscillating bubble is shown in Fig. 7. The hemispherical shape of

the bubble is ideal for the determination of dynamic surface tension. The bubble geometry is

monitored by means of an optical system. The amplitude, AH, at the top of the bubble is about

25 ~tm. This is related to a relatively small change in surface area of about 10%. In the

presence of adsorption layers, changes in surface area produce changes in surface tension A~/.

The pressure transducer allows measurement of the resulting changes in pressure. The surface

elasticity values can then be calculated from the pressure amplitudes, the pumped volume and

the area changes of the bubble of known geometry.

339

Fig. 7. Cross-section of the radially oscillating bubble

At higher frequencies inertia effects must be considered (Eq.(10)). As a first approximation all

dynamic effects of the bulk phase are neglected. Then, in the case of small sinusoidal changes

in area of the bubble surface, the measured pressure amplitude can be written as the sum of a

radius component and a contribution caused by changes in the surface tension A~. The dynamic

pressure can be described by

A p - 2),Ar 2A~ (5)

(r0) ~ r0

with r0 ___- 0.5(rl + r2). The first term is the radius component. The second term is used for

calculation of the dilational properties. Here A% the change in surface tension, also

incorporates the elastic and the viscous contributions.

To maintain accuracy of performance it is important to retain the hemispherical form. By using

this shape the values of the change in radius, Ar, go through a minimum, and the pressure

amplitudes reach values close to zero for Ay = 0 (pure water). In the presence of adsorption

layers one obtains pressure amplitudes which are directly proportional to A~/ for low

frequencies. The dilational properties of adsorption layers at the oil/water interface can also be

investigated by use of a modified measuring head.

4. THEORY OF SPHERICAL BUBBLE OSCILLATIONS IN SURFACTANT SOLUTIONS

4.1 FORCE BALANCE RELATIONS

As previously mentioned the main problem of surface rheology studies is the separation of the

influence of the adjacent bulk phases. Therefore, a complete description of the dynamics of the

system is desirable.

340

Here we start with the assumption that the bubble is formed as a complete sphere. In this case

the theoretical description of the problem is simple. Fig. 8. illustrates such an arrangement. The

measurement chamber, the bubble and the air pump have the shape of spheres with identically

fixed centers. This assumption is difficult to be realized experimentally, however, the

theoretical results can be easily transformed into equations for a real system. If the excitation is

located in the center of the gas phase (Fig. 8, left), the fluid chamber must be open for the

input and output of the solution. In the case where the chamber wall excites the system (Fig. 8,

right), the air center should have an input and an output to keep the air pressure constant. Such

conditions are necessary for application of the formulas to a real experimental system.

Arl Fig. 8. Schematic diagrams of an ideal spherical oscillating bubble with excitation within the gas volume (left)

or excitation within the solution volume (right).

For the radial geometry, the flux velocity has only a radial component ~ Ad << r0, [ 40-46, 50 ] )

r~ v0(t) v0(t) -ic01Arlexp(io~t) V r - - v(r, t) - 7 ' (6)

where r is the radial coordinate, r0 the bubble radius, v0 the flux velocity of at the surface, and

Ar = IArl exp(ico t). (7)

the change of the bubble radius with the amplitude I Ad. According to Eq.(1) the stress tensor

of the fluid bulk phase (r > r0) in a spherical coordinate system has the components

0V r - - p + 2 r l ~ , (8)

341

~ -~q~ - -p +2rl v . (9) r

Neglecting the influence of gravity and using the Eq.(6), the divergence of this tensor

completed by the inertial terms leads to the special form of the Navier-Stokes equation_.

0(c3v 2v) 0p (c3v __~_) +-2rl-~ & + - r - & - p ~ - + v . (10)

Then the pressure p at a measurement point r~ on the chamber wall is given by integration of

Eq.(1 O) over r in the form

AA p = p~ +m 2 B ~ (11) A

where the constant B is described by

B - pA ~ls(r)ldr_ pAro(1 - roll Ar [ - k A r , "7[[Jl~'l

(12)

because the magnitude of the displacement Is(r)[, represented by

ro Is(r)l- 7- Y [Ar[, (13)

is independent of the frequency due to the incompressibility of the bulk phase and for small

[Arl the second inertial term v c3v/Or ~ 0 in Eq.(10) is negligible, ps describes the pressure near

the bubble surface. Within the gas phase the pressure pg is uniform as long as the size of the

bubble is small compared to the wavelength of the acoustic wave of the chosen frequency.

There is no contribution due to shear as a result of the geometry, and the tensor Eq.(4) is

reduced to the components

T:I = T22 = 7' (14)

342

The surface velocity vector v s in Eq.(4) is determined by the variation of the relative distance

between two molecules at the surface during the deformation, and therefore with the aid of the

continuity equation we obtaine the relation

~ '= r ( r ) + K' ( v , . v ~) = v ( r ) + K ' - ~ 1 d(AA)

A dt (15)

and for the normal stress balance on the surface [27]

g f 2y' arr --arr =-- (16)

r~

In Eq.(16) the right side can be interpreted as the limit value of an integral over a divergence of

a pressure tensor in a curved volume with vanishing thickness and croo = cr~ = y ' . With

Eq.(8) this condition leads to the relation

0v 2 ! m

p~ - p g - 2~! Or - rb 7 (17)

which can be split into a static equation of the form

2 ps - p g + 9 g h - ro,(h ) 7 with r b - r o ( h ) + A r - r o +Ar , (18)

and a dynamic one

0v 2 2 Aps - Apg - 21"1 Or - r o AT'+ - ( )'r0" 2 ~Ar. ( 1 9 )

Eq.(18) is the Young-Laplace relationship. Solution of this is not necessary because the

influence of gravity on r0' is negligible in the dynamic Eq.(19) and we can use the mean value

r0. However, this calculation should be performed for the determination of the relation between

Ar and AA or AH which for small amplitudes is linear.

Experiments with an oscillating bubble setup should provide information on the relative change

in the surface area AA/A, and the change in pressure Apg or Ap measured at a point within the

system. Additional information provided is the pumped volume which can be used for the

343

verification of the pressure measurement (excitation within the gas volume) or for an

independent measurement of the surface area change (excitation within the fluid). These

measurements are the basis for the determination of the dynamic surface properties with the aid

of an appropriate model.

We assume that the pumped volume in the model of version 1 (excitation in the gas volume) is

created by a pump at the center C of the bubble (Fig. 8, left). The dimension of the gas volume

V0 must be small compared with length of the corresponding acoustic waves. Then the pressure

pg is given by the adiabatic gas equations

_(K, K, ) pg - pg 1 - ~ A V I - ~ A V 2 .

Vo Vo

Here

(20)

AV~ = -cI)(f) exp(i(m t + q~' (f))) (21)

denotes the change of the gas volume caused by a pump in the center C and

AV: = (d-d-~l AA (22)

is the resulting change in the bubble volume, q~(f) should be adjusted so that approximate equal

values for AV1 are obtained in all cases.

In the device of Lunkenheimer (Fig. 1), no pressure transducer was installed within the closed

air chamber. Therefore, the relation between Apg and AV2 could not be tested experimentally.

The uncertain determination of the apparatus constants V0 and AV1 possibly explains large

values of the surface elasticity obtained by this device. The installation of a pressure transducer

within the air chamber will allow replacement of the apparatus constant AV1/V0 by direct

pressure measurements. However a viscous effect caused by the gas capillary of a real system

is not detected, and leads to an additional term in Eq.(20). This can be eliminated by calibration

measurements.

In version 2 the gas volume is infinite and the gas pressure is kept constant by an input and

output in C (Fig. 8, right). The common formula describing the dynamic term of the gas

pressure of version 1 (ver. 1) and version 2 (ver. 2) is given by

344

.V~~(f)exp(iot+qa,(f)) K*Av0 AAA-~V]-~] exp(i~ ver. 1

Apg - . (23)

0 ver.2

For determination of the pressure within the solution by the ideal spherical model, we assume

that the chamber is either open (ver. 1) or closed (ver. 2). In the first case the pressure

measured at the output remains approximately constant (p = const.) and, therefore, the dynamic

pressure of the fluid near the surface has the form (cf. Eq.(11))

A p ~ - - o 2BI--~I exp(iot). (24)

For the closed chamber, Eq. (11) includes the inertial term and an additional dynamic term Ap.

Therefore, the dynamic contribution of the bulk pressure at the surface is given by

2BAA ~ 0 ver.1 Ap~ - - o + ( 2 5 )

A [ Ap ver. 2J

and the force balance Eq.(19) reads

f -Apg ver. 1~ 2B AA 2 2~ 4i~!o - o - - - A y ' + A r - A r ( 2 6 )

[ Ap ver.2 J A r 0 ~ r 0

o r

~-Apg ver. 1 _ 2 e(f, c) + F(f) + g(f, c (27) [ Ap ver. 2 r o A

with

e(f, c) - E(f, c)exp(iq~(f, c)) - A A~' (28) A A '

[(r-U >) Ar g(f, c ) - 2 (r /- To)- 4io (11-1"10 ~ a , (29) r0

and the calibration function

2T0 Ar F(f) - 0 2B- 4iorl0 __Ar A + ~ ~ A. (30) r 0 AA (r0) 2 AA

345

Eq. (27) describes the relation between the oscillation of the pressure at the pressure transducer

within the gas volume or within the fluid and the oscillation of the surface area. The magnitude

of both oscillations and the phase shift between them can be measured. The relationship

between the change in surface area and the radius is normally controlled by an optical system.

If the magnitude and the phase angle of the excitation function, ~(f) and q)'(f), of the gas

chamber or the pumped volume of the fluid are recorded, an additional verification of the

results is possible.

For pure water, the elasticity s(f,c), and the function g(f,c) are zero because the dynamic

surface tension is equal to the static surface tension. Then, with the aid of Eq.(27) a calibration

function F(f) can be determined from experiment. Such calibration measurements are necessary

particularly if the shape of the real bubble deviates from the ideal sphere (section 5.3.).

In the case of a half sphere, the radius of the bubble is equal to the capillary radius, the value of

Ar/AA is negligible and the influence of the correction function g(f,c) is very small. Here the

bubble oscillation is only stable in version 2. In addition, the center of the bubble is not fixed

and one must use a solution with a small elasticity for a practicable calibration measurement.

However, this case of the half sphere radius is the most convenient operating point for accurate

measurements.

With the knowledge of the functions F(0 and g(f,c), it is possible to separate quite easily the

contribution of the dynamic surface tension in the force balance equation (27) by a complex

elasticity modulus eft, c). It involves elastic, transport and viscous properties of the surface

which must be separated by a fit procedure.

4. 2 DYNAMIC SURFACE TENSION

For the investigation of s(f, c) we need a model describing the dynamic processes at the surface

which have an influence on the surface tension state. The standard assumption is that the

dynamic surface tension ~{ is only a function of the surface concentration F(cs), resulting from

the equilibrium isotherm and the viscous effect being neglected. That means ~/' = ~{(F) and for

small deformations AA, this equation allows the following Taylor expansion for a solution with

one surfactant component [43 ]

- d ~ - r ( r ) + ~ A r (31)

dF

and therefore Eq.(27) reads

346

e ( f , c ) - d7 d l n r d In F d In A (32)

An extension of these formulas can easily be carried out to include several surfactants by

introduction of partial differential quotients and sums [53]. For the application of the last

differential quotient a clear relation between F and A is required (F = n(A)/A). This is

determined by the diffusion properties of the bulk phase. For oscillating processes such a

relation is always established. In Eq.(32) the term

dy ~o(C)- d lnF (33)

is the classical Gibbs elasticity and therefore is

dr/C_ dA) ~(f, c) - -~o(C) d t / \ A dt " (34)

The complex modulus eft, c) depends on the diffusion exchange and the deformation rate. The

change in number of molecules, n, at the surface is given by mass conservation which is

represented at the surface (y = r0 - r _< 0) by

1 dn= 1 d ( A ( t ) r ) = _ D c3_.c_c] (35) A dt A dt 0yl y=0

This change must be equal to the difference in the adsorption and desorption rate. To estimate

these rates we use an established assumption about the adsorption/desorption process. The

adsorption rate P should be proportional to the amount of unoccupied surface. It is also of first

order in the bulk concentration immediately adjacent to the surface, cs. The desorption rate Q is

of first order in surface concentration. The standard expressions for these functions are [27,

44]

P - ct exp(-E. / RT)cs (Fo~ - F) , (36)

Q - ~' exp(-E b / RT)F,

where Ea and Eb are energies of activation of adsorption and desorption respectively, and the

current is

1 dn

A dt - - ~ = Jr - P(cs, F) - Q(cs, r ) . (37)

347

This difference must be equal to the bulk diffusion at the surface which follows from the

sinusoidal solution of the diffusion equation. The diffusion equation in spherical coordinates

reads

Oc(r,t) Oc(r, t ) 1 0 ( ' r 2 0 )) "a t - V r ~ - - D c(r, t (38)

0t Or r-r 0r \ ~rr "

By using the coordinate y - r0 - r _< 0, the convection term becomes small near the surface for

small oscillating amplitudes ] Al~, and for radii which fulfill the condition x/c0 / (2D)r 0 >> 1, we

can replace Eq.(38) by

_ 0 2 c ( y , t ) 0c(Y,0t t_____~) _ D 0Y z (39)

We need the sinusoidal solution of this equation according to the boundary condition Eq.(35)

with a sinusoidal change in AA. If we take into account only linear terms of Eq.(10), (37) or

(38) the dynamic term of the solution of the diffusion problem must have the form

m e - U 1 exp[(1 + i)k'y + ic0t] (40)

and also the change in adsorption

AF - u: exp(k0t + qo*). (41)

For small magnitudes these formulas are correct because all higher terms are negligible. The

experimental results show that for I AA]/A <0.05, there are no higher harmonic contributions [50]. Then we can develop Eq.(37) for close to the equilibrium state F = F(E) which is

determined by the condition

P(g, F) = Q(e, F). (42)

Considering the changes of the concentrations Ac and AF as independent variables, the

development of Eq.(37) has the form [27, 44]

Jr -- dcAc + drAF (43)

3 4 8

where

dc 0P 0Q , dr c=~ c=~ F = ~ F=~

(44)

However, for a sinusoidal excitation a rela6on exists between the surface concentration F and

the subsurface concentration cs which can be described by

, dF dF dc* I AF - AF(c (c~)) - dc-----r (c* - U) - dc* dc~ Ic.,-_~ (c~ - U). (45)

Here the introduction of the concentration c*, which also fulfills the equilibrium isotherm

equation F = F(c*), is only a formal interim step. However, using this relation the current

Eq.(43) reads

c" Jr - d~(c~-c*)+dc(C* - U ) + d r ~ - c i~.__ ~ - - ) (46)

where dF/dc* results from the equilibrium isotherm equation. Using this formula all further

calculations become much simpler because all deviations from the standard case, which is the

diffusion controlled process, are described by the factor dc~/dc*. We obtain with Eq.(3 5), (40),

and (46)

d~Ac* = D(1 + i)k' Ac s + d~Acs (47)

or (~ = c, Acs = Ac(y = 0))

h ( f , c ) - dc------z-~ - d / (d +(1 +i)403D / 2 ) - h( f ,c ) - Ih( f , c)lexp(il3' (f,c)). dc* c

(48)

From Eq.(3 5) follows

s dA = -D ac~[ dr (49) A dt 0y ]y=o dt

and with the relation

dF dF dc* c3c~

dt dc* dc~ 0t

349

(50)

the Eq.(34) leads to { }1 1

s(f, c) - - d ln----F d r dc ---7 \ c~ J y=o " (51)

Therefore the complex elasticity module has the form

{ l + i dc* 2~D} -1 g0 (c)(1 + 42glh(f, c)1 e x p ( i ( r t / 4 - ~ (f, c)))) eft, c) - go (c) 1 + ~ h(f, c) =

i - -~ 1 + 2~]h(f, c)[(cos(~ (f, c) + sin(13' (f, c)) + 2~2 [h(f, c)l 2

(52) or for h(f,c) - 1

1+ r + ir m~__~ _ D(dC~ 2 s(f, c) - s o (c) 1 + 2( + 2Q 2 ' ~ -- ' m~ \ d F J ' (53)

with the magnitude

E(f, c) - e~ (54) x/1 + 2( + 2( 2

The function h(f,c) = dcs/dc* describes the influence of the kinetic effects, in other words the

molecular exchange between surface and subsurface. For d c >> 4 ~ D the function h(f, c) is

approximately 1 and Cs--- c*. Thus we have bulk diffusion controlled behavior (see Eq.(48)). If

the kinetic effects are fast enough (h(f,c) - 1) then the isotherm equation F = F(cs) is valid for

all times as assumed in the Lucassen/van den Tempel (v. d. T.) model or Ward and Tordai

model [14, 22, 23]. Then Eq.(52) gives the Lucassen/v. d. T. modulus Eq.(53). For small

frequencies, the values of the function h(f, c) are approximately 1 in all cases, and the low

frequency limit of the complex elasticity is (Eq.(53), (58), (59))

i)dF 4- ~ F~~/~ e ( f , c ) : e 0 (c) (1 + - ~ c - - - (1 + i ) R T - - c --" (ss)

350

From this it can be inferred that the phase angle is 45 ~ According to Eq.(48) the magnitude of

the complex function h(f,c) is less than 1 if the kinetic effect must be taken into account. The

influence of the kinetic effects is important when the constants have the following magnitudes

d c ~ D >_0.1. (56) 0.1*d~ __4~-D, ~ 203

The high frequency limits of e(f,c) and E(f,c) are the Gibbs modulus in all cases. They are

independent of h(f,c) because the ratio of the change in number of molecules and the total

variation of the surface concentration reduces at higher frequencies according to

1 dn/ /d r = (-1 + i)h(f, c)dc~a l/ D (57) A d t / - - ~ -d--ff v2o3

In this case there is only a very small diffusion exchange and an approximately insoluble

behavior of the monolayer. Nevertheless, the magnitude of the change in bulk concentration

Acs = (dcJdF)AF is constant as long as the condition h(f, c) _-- 1 remains valid for higher

frequencies too. However, you must keep in mind that the high frequency limit of Acs is also

zero and the exchange between surface and subsurface ceases completely if the frequency is

high enough. The formulas, given in [27, 44, 45], can be also transformed in our notations with the aid of a

relation between F, c*, and c~ which is justified for an oscillating deformation of the surface.

The advantage of this formal step is obvious: dc*/dF is a property of the equilibrium isotherm.

It describes the disturbance of the thermodynamic equilibrium by a change of the surface

concentration. With knowledge of the function h(f, c), we can easily recognize the disturbance

within the bulk and between the bulk and the subsurface. Besides, the formal step allows the

use of the well established Lucassen/v. d. T. modulus in a small modified form.

4. 3 COMPARISON OF MEASUREMENTS WITH ISOTHERM CALCULATIONS

Neglecting the kinetic effects the complex elasticity can be determined in principle with the aid

of the isotherm. However in a higher concentration range near the CMC, the changes in F are

very small. In this range the value of the theoretical Gibbs elasticity becomes uncertain because

Eq.(33) can be written as

351

_r .c ...r .c - -c + c

due to the adsorption equation

F - - -S - - c dy (59) RT dc

Therefore e0 is determined by the differential quotients of the isotherm resulting from a fit

procedure. In general, the difference between the mathematical idealization and the real

physical process increases with further differentiation of such a function, and so after the

second derivation the error in Eq.(58) is large. Besides, in the linear range of the T-log(c)

isotherm near the CMC, the surface concentration is approximately constant due to Eq.(59).

Therefore, the value of dc/dF calculated with the aid of the isotherm increases unrealistically

(Table 1, 2, 3). That is a critical point for the proof of models which describe the dynamic

surface tension behavior because the central parameter of all these models is m0 (Eq.(52)). Here

the situation is improved if the measurement of the surface elasticity reaches a constant level at

higher frequencies. Then we can use an experimental value ern (C) instead of the Gibbs

elasticity e0 resulting from the isotherm. The oscillating bubble method allows these

considerations because the frequency range (to 400 Hz) is large enough to reach the constant

level for most surfactant solutions. However, you must bear in mind that in this case the

differential quotient is replaced by the ratio of differences. If we assume that the adsorption

F(c) calculated with the T-log(c) isotherm and the insoluble limit era(c) from the oscillating

bubble experiments are more exact than the isotherm values dc/dF, it is better to transform

Eq.(58) with the aid of Eq.(59) into

A___~c = c~ m (c) (60 ) AF (F(c))2 RT

for investigations of dynamic effects. If all properties on the right side of the equation are

available the use of this formula improves the situation in the critical range.

Another important point for explanation of the difference between experimental and theoretical

Gibbs elasticity is the following: All previous considerations are based on the assumption that

the surface tension depends only on surface concentration T = T(F) and this surface

concentration is determined by the Gibbs adsorption equation. The Gibbs model introduces a

fictive plane which divides the two adjacent bulk phases. The surface concentration results

352

from the difference between the complete Immber of molecules and the sum of the molecules

within the adjacent homogeneous bulk phases. Besides, the Gibbs-Duhem relation between the

chemical potentials must be fulfilled. With the additional condition that the number of

molecules of the solvent is vanishing at the interface (Fs = 0) the position of the interface and

the interfacial concentration F of surfactant is defined in a theoretical manner by Eq.(59). For

equilibrium considerations the assumptions of the model are not critical in most cases,

however, for the solution of the diffusion problem this means that the surface concentration is

located in a monolayer which has only an influence on the surface tension. With this

geometrical assumption, the boundary condition Eq.(35) is formulated. If this precondition is

not fulfilled, and the area which has an influence on the surface tension is not so small, we

must introduce a volume model like the Guggenheim model [54] to solve the diffusion

problem. Then the surface concentration used in the boundary Eq.(35) is given by Fv = N/A.

Here N is the number of surfactant molecules in a thin volume. This Fv is different from the

concentration F defined by the adsorption equation (59). However, for an oscillating motion of

the system we can introduce again a functional dependence F = F (F~ (0) which we need for the

relation between the diffusion process and the isotherm equation. This formalism includes as

well the equilibrium state as the non-equilibrium state of the surface volume. With such a

relation it is possible to transform Eq.(32) and (33) into

e ( f , c ) - d3, d l n r d lnF v (61) d lnF d lnF v dlnA

o r

dr e 0 (c)= -q( f --+ 0% c ) ~ (62)

d lnF

Here is

d lnF q(f,c) = ~ (63)

d lnF v

By this step the theoretical considerations remain valid if we also substitute (Eq.(50))

h(f, c) - d____F_F dc__._~ d____cc = d___~c dF~ (64) dF v dc*' dF dF~ dF

All final equations must be invariant with respect to thickness changes in the surface phase.

However, this thickness must be small compared to the wavelength of the diffusion wave

(Eq.(40)). Both functions, h(c, f) and q(c, f) can be determined by fit-procedures with the aid of

353

the bubble measurements. Unfortunately, there is a lack of relevant experiments at the moment

for a detailed discussion of such effects. It is not yet clear whether the difference between the

theoretical and experimental values of the Gibbs elasticity near the CMC are an effect of the

mathematical procedure as discussed above, or an effect of the physical model which neglects

the influence of the thickness of the interface. The consequence of both interpretations is the

same. Only the correction function q(f, c) and h(f, c) must be introduced and then the formulas

remain valid.

5. MODIFICATIONS OF THE STANDARD MODEL

5.1. INTRINSIC VISCOSITY

Not all experimental results of the oscillating bubble method can be described by the standard

formula Eq.(53). Some highly concentrated surfactant solutions and solutions of ionic

surfactants exhibit a typical viscous behavior. The explanation of this effect is difficult. The fit-

procedure shows that the surface elasticity of such solutions is given by the equation

e(f,c) e 0(c) I+Q'+iQ' - + ioK' ( 1 - u(f) ) . (65) 1 + 2Q' +2Q '2

The formal application of Eq.(4) or (15) leads to u(f) - 0. However, the experimental results

are better described by the function u(f) = exp(-Kf) because the systems exhibit the viscous

behavior only at higher frequencies in some cases (section 6). A possible explanation is the

following. For higher concentrations we apply the volume model to describe the surface

behavior. This means that rearrangement processes are necessary because the equilibrium

distribution of the water molecules and surfactant molecules within the surface are different for

different expansions of the surface area. If these kinetic processes are fast enough in

comparison with the time resolution of the frequency range, no additional pressure is necessary

for the realization of the deformation of the surface. The momentum balance is zero in this case

like in a normal diffusion process. If for higher frequencies the equilibrium status within the

surface volume is not immediately reached, maybe an additional pressure is necessary for the

realization of the deformation which causes the change of the surfactant density in the surface

area. Then such density changes lead to a contribution of the dilational viscosity in the pressure

tensor. Only within the surface area, and not in an incompressible homogeneous bulk phase,

are density changes of the components possible and the intrinsic surface viscosity can be an

effect of the inhomogeneous structure. This consideration is only an hypothesis.

354

5. 2. MODIFICATION OF THE MODEL FOR IONIC SURFACTANT SOLUTIONS

For description of the dynamics within the ionic double layer during an oscillating

deformation, we must combine the model of Gouy Chapman, the diffusion equation, and the

conservation law of mass at the interface. Here we consider the case of positively charged

surfactant molecules and a negative counterion. According to the Gouy Chapman theory the

equilibrium concentration c § and c within the sublayer is given by [55 - 57]

~+ - c o exp(-eW(y) / kT), U- - c o exp(eW(y) / kT) (66)

where ~F(y) is the electrostatic potential. This potential can be calculated with the aid of the

Poisson-Boltzmann equation

ed2_____~_ ~ : %2 sinh(e~) (67) kTdy 2 k kTY

by well known steps [55, 56]. % = ex/8nc0/SdkT is the reciprocal Debye length and Sd the

dielectric constant. After two integrations and rearrangement we obtain the equation (y < 0)

exp(egt / 2kT)= 1 + tanh(e~ o / 4kT) exp(~,y) 1 - tanh(e~F0 / 4kT) exp(Xy)

(68)

for the calculation of the potential, xg0 is the potential at the surface (y = 0). The diffusion

equation in the presence of an electric field has the form

Oc -+_D • 0 Oc-+_+c • (69) ot Oy oy oy

with w = e~ / kT and the mass conservation equation at the interface reads

An additional condition results from the Poisson equation

(70)

a2w- e2 (c+-c-) 0y 2 SdkT

3 5 5

For sinusoidal excitation the solution function must have in the first order the form

c + - g+ + Ac + exp(io3t)

c- - g- + Ac- exp(icot) (72)

and the electrostatic potential function

w(y) - W(y) + Aw(y) exp(i03t). (73)

Because Aw can be replaced by an integral

(0Aw e 2 ~ T io (Ac§ - A c )dy' )ds + Aw(0) (74)

the diffusion equation in first order is of the form

02Ac -+ c3Ac • c ~ 025 ic~ - D • @2 + ~ - - + Ac• Oy Oy Oy ~

(_~_(Ac + _ = d_ I y )11 (75) e 2 + 0-C -+ (e kT 0Aw - j (Ac + - Ac- )dy' + Ac-)+ Oy e 2 0y y=0 0 ekT

Solution functions Ac+(y) and Ac(y) of this system can be obtained by a numerical procedure

using the boundary condition Eq.(70). This was calculated in [ 56 ] with a different approach to

the solution function. If we assume that only the monolayer contributes to the surface tension

the resuking equations have also the form of the Lucassen/v. d. T. modulus with a transfer

function h(f,c).

5. 3. INFLUENCE OF THE DEVIATION FROM IDEAL SPHERE

All previous theoretical derivations are based on the assumption that the bubble is an ideal

sphere with a fixed center. No experimental system exists which fulfills this demand. However,

a complete theoretical description of a real bubble system is very complicated. For an

incomplete sphere the flux in the bulk phase does not have the form of Eq.(6) and the shape of

the bubble deviates slightly from an oscillating sphere shape as a result of the bulk pressure.

However, using a calibration procedure a detailed investigation of most of the deviation from

the ideal case can be avoided. This is because the force balance is fulfilled at all surface points

356

and therefore at the top or the bottom of the bubble too. For these points the surface model is

the same as in the case of an ideal sphere with a mobile center and a change of area. The

influence of the bulk flux can be determined by a calibration measurement. At a different

surface point the contributions of the bulk and of the surface to the force balance are changed,

however these two changes compensate each other and we must only consider the point at the

bottom. Therefore, the optical control of the shape at the bottom of the bubble and calibration

measurements are very important for exact experimental results. If this is carried out

modification of the model for an incomplete sphere is not necessary and , therefore it is

possible to use the more stable incomplete sphere without problems.

6. EXPERIMENTAL EXAMPLES

Figs. 9 - 11 demonstrate the frequency dependence of the "effective" surface elasticity of

different surfactant solutions measured with the apparatus of Fruhner. Examples of typical

cases to date are included. Most surfactant solutions exhibit the characteristic behavior of the

Lucassen/v d. T. modulus according to Eq.(53). For example, the surface elasticity of the low

concentrated solutions of decyl dimethyl phosphine oxide (Fig. 9) and dodecyl dimethyl

phosphine oxide (Fig. 10) show that characteristic behavior.

50

45

40 3- Z �9

~ , 7 - . ~ ~o:~. ~ = 0 ~ 3 0 . �9 . . ,

2o //~--~" ~ ~ = ~ "

15 , S ~ j -~ 'p=s~

,o- I

5 t S q~=24" 01 , ~ ' ~ ' ~ '

0 100 200 300 frequency (Hz)

400

Fig. 9. Magnitude and phase angle of the complex elasticity of decyl dimethyl phosphine oxide: m calculations of E(c,f) = Is(f,c)l according to Eq.(66) with the parameters e.m, oam, K' of Tabl. 1; �9 magnitude Eft, c) and phase angle q~ = q~(f,c) of measured ela..Cticity of a 2.5"10 -3 M solution; A, q0 of a 1"10 -3 M solution; Y, q0 of a 5"10 .4 M solution; O, qo of a 3 '10 -5 M solution.

357

At high frequencies , a constant level is reached wi th vanishing phase angle. The points o f

in tersect ion also agree with the theory. The parameters o f these slopes can be de termined f rom

a f i t -procedure (described by am and 03m~ or by derivat ion o f the equi l ibr ium i so therm

(described by eo and O3o~. Agreement , however , is not so good in some cases, par t icular ly near

the CMC, where the values calculated f rom the i so therm are m u c h higher than the observed

data (Tabl. 1, 2, 3). This quest ion has been discussed in 3.4.

Table 1" Rheological parameters of decyl dimethyl phosphine oxide (K = 0.01, CMC = 2.3 10 .3 M):

parameters \ concentr. 2.5"103 M 1"10 -3 M 5"10 -4 M 1"10 -4 M 3"10 -5 M

eo 'mN/m 410 185 92.6 18.5 5.6

~m: mN/m 12.6 31 36 34 28.1

COo: s'l 112000 17000 1270 7.0 0.56

COm] S-1 90 42 24 5 2.4

K" mN*s/m 0.011 0.015 0.0126 0.0003 0

Table 2: Rheological parameters of dodecyl dimethyl phosphine oxide (K = 0.025, CMC = 2.7 10 -4 M):

parameters \ concentr. 2.5"104 M 1.5" 10 -4 M 1" 10 -4 M 5"10 -5 M 2"10 -5 M

~o " mN/m 2400 1100 735 319 116

am: mN/m 69 65.5 65.5 57 44.7

COo:S 4 202500 12500 2550 169 4.8

COm: s'l 950 37 12 1.5 0.2

~c' :mN*s/m 0.005 0.00027 0.0001 0.00005 0

Table 3" Rheological parameters of hexadecyl trimethyl ammonium bromide (CTAB, K = 0.025, CMC = 1 10 -3 M)

parameters \ concentr. 1" 10 -3 M 6" 10 .4 M 1' 10 -4 M 3" 10 -5 M

~o �9 mN/m 1338 569 24.6 5.51

am: mN/m 36 51 35.5 29.1

COo: s -1 25000 3650 12.3 1.2

COm: S-1 700 140 1.1 0.22

K" mN*s/m 0.0142 0.0104 0.0011 0

3 5 8

A few surfactant solutions with a higher CMC value, e. g. decyl dimethyl phosphine oxide,

show an unexpected behavior. At low concentrations the values of em normally increase, then

Sm goes through a maximum and decreases at concentrations near the CMC. Besides, these

measurements exhibit an intrinsic viscous effect near the CMC, which can be described by

Eq.(65). It is reasonable to assume that for such high concentrations it is not only the

monolayer which contributes to the surface tension. Maybe the influence of the sublayer is

observed in an indirect manner by a higher osmotic pressure caused by the bulk concentration.

If so then we must use the Guggenheim convention and a relation q(c,f) (Eq.(63)) for the

solution of the diffusion problem. In this model one can also explain the decrease of era.

Although F remains approximately constant the value of Fv is growing with increasing bulk

concentration and, therefore in Eq.(63) and (64) dF/dFv becomes small.

70

60 t (p =9~ �9 %=8" �9

Z �9

~E 50 (i)=14 ~ ~ = 2 3 ~

-~ ~ -_ �9 i i / - ~aOO ~ 40 ~ ~=a" = - ~_ .

t (# 39 A "~'~=35~ 30

2O

10

' I ' I ' I

0 100 200 300 f r equency (Hz)

400

Fig. 10. Magnitude and phase angle of the complex elasticity of dodecyl dimethyl phosphine oxide: -- calculations

of E(c,f) = I~(f,c)l according to Eq. (66) with the parameters em, 0am, K' of Tabl. 2; �9 magnitude Eft, c)

and phase angle 9o = q~(f,c) of measured elasticity of a 2.5"10 -4 M solution; Y, 9o of a 1.5"10 -4 M solution;

0 , q~ of a 5'10 -5 M solution; +, q~ of a 2"10 -5 M solution.

The highest concentration of dodecyl dimethyl phosphine oxide (Fig. 10) and in particular

decyl dimethyl phosphine oxide (Fig. 9) exhibits a viscous effect as indicated by a phase angle

at frequencies fv >100 Hz. Another indication of this effect is the rapid growth in stability of

foams with increasing surface dilational viscosity. If the above discussed hypothesis over the

viscous effect is realistic then we obtain from Eq.(48) a value for the kinetic constant do of

359

d c <x/2rtfvD ~ 0.18cm/s. The frequency fv corresponds to the values of or' discussed in

[27].

In Fig. 11 an example of an ionic surfactant is demonstrated. The surface elasticity is similar to

the case of a non-ionic surfactant. At low concentrations we obtain the slope of the Lucassen/

v. d. T. modulus but near the CMC the surface exhibits an intrinsic viscosity as indicated by a

phase angle. Maybe this effect can be explained by the dynamics of the electric double layer

and their influence on the surface tension.

60

uJ 40 1 IP=6~ '

3O

2O

' " " o i i I I ! 0 100 200 300 400

frequency (Hz)

Fig. 11. Magnitude and phase angle of the complex elasticity of CTAB: m calculations of E(c,f) = I~(f,c)l

according to Eq. (66) with the parameters em, C0m, K' of Tabl. 3; O , qo magnitude E(f,c) and phase angle

(p(f,c) of measured elasticity of a 1"10 .3 M solution; A, q~ of a 6"10 .4 M solution; V, q~ of a 1"10 -4 M

solution.

In the last few years it has been possible to improve the oscillating bubble method by new

technical components such as pressure transducers and CCD-cameras. These devices allow an

accurate measurement of the relationship between the change in surface tension and the

deformation of a liquid/air interface. With the aid of a model the dilational surface properties

can be determined from these measurements as similar to the Langmuir trough. However, the

frequency range is much broader and it is possible to verify the assumptions of the theory of

dynamic surface tension for faster processes. The experiments exhibit some new effects.

Therefore an improved theoretical model is necessary which takes into account a detailed

360

description of the kinetics and the dynamic surface tension. In this respect some modified

theoretical models have been also discussed in the last few years, however, these models have

not been tested systematically by experiments.

For experimental investigation of such specific effects like the intrinsic surface viscosity and

the transfer kinetics at a fluid surface, the oscillating bubble method is a powerful tool. By

means of this method a systematic investigation of surface dilational properties of surfactant

solutions can be made for characterizing the properties of more complex systems like foams

and emulsions.

7. LIST OF SYMBOLS

A

AA = lad exp(icot)

B

c = U + A c

c

Ac

Cs - E~ + Acs

c* = E * + Ac*

surface area,

change in surface area,

integration constant,

bulk concentration (static and dynamic term),

equilibrium bulk concentration,

dynamic term of bulk concentration,

subsurface concentration (static and dynamic term),

fictive subsurface concentration (static and dynamic term),

c § = ~+ + Ac + exp(iot) bulk concentration of positive ions (static and dynamic term),

c- - E- + Ac- exp(iot) bulk concentration of negative ions (static and dynamic term),

D, D § D- bulk diffusion coefficients,

do, dr sorption constants,

e elementary charge,

E(f,c) magnitude of the complex elasticity modulus,

Ea, Eb activation energies,

f frequency,

F(f) calibration function,

g(f) correction function,

g gravitation constant,

h(f,c) transfer function,

h dip distance of the bubble,

361

H

AH

j= j r

k

k ' - x/D0) / 2

K

M

n

N

p - ~ + A p

Ps =Ps +Ap~ m - - 0 p~ =p~ +pgh

pg = ~g + Apg

P = P(c~, r)

q(f,c)

Q = Q(cs, F)

r, 8 , ,

rb = r0 '(h)+ Ar

r0

r l , r2

hr

R

s(r)

t

T

T~

T~ s v

u(0

Ul

u2

V

height of the bubble cap,

change of height of the bubble cap,

diffusion current,

Boltzmann constant,

wave number of the diffusion expansion,

fit parameter,

mol*dm "3,

number of surfactant molecules within the surface monolayer,

number of surfactant molecules within a surface volume,

pressure within the bulk (~ static term, Ap dynamic term),

bulk pressure within the subsurface (static and dynamic term),

static term of ps,

gas pressure (static and dynamic term),

adsorption rate,

relation between different defined surface concentrations,

desorption rate,

spherical coordinate system,

radius of curvature of a local bubble area,

mean value of bubble radius,

radial coordinates of different positions (Fig. 6, 7),

change in radius of the bubble,

gas constant,

radial displacement of the fluid,

time,

temperature,

pressure tensor of the surface,

components of the pressure tensor of the surface,

evaluation function,

magnitude of the change in concentration,

magnitude of the change in surface concentration,

volume,

362

V 0

A V 1

AV2

v = (Vl, v~, v~)

v = v r = v(r,t)

v o (t) s

V

w = W + Awexp(kot)

Xl -- X, X2 = y, x3 = z

volume of the gas room,

change in gas volume by excitation,

change in gas volume by bubble motion,

vector of the flux velocity,

radial velocity,

radial velocity of the bubble surface,

vector of the relative velocity at the surface,

potential function (static and dynamic term),

Cartesian coordinates,

Xl(13,z), x:(13,~), x~(13,z)

y = r o - r

C~, C~ v

C~v i

J3,%

13'

Yo

surface coordinates,

coordinate perpendicular to the surface,

parameters of adsorption and desorption,

components of the tangential and normal unit vectors at the surface,

parameters of the surface coordinates,

phase angle of the transfer function,

surface tension (static term and dynamic term),

surface tension of the calibration solution,

dynamic surface tension with viscous contribution,

A3t = A T + •' (1/A)*d(AA)/dt m

F = F + A F

F +, F-

Fs

Fv

F~

~ik

A L

~(f,c)

go

8m

~d

~,~'

dynamic term of surface tension with viscous contribution,

surface concentrations(static and dynamic term),

surface concentration of ionic surfactants,

surface concentration of the solvent,

surface concentration of the Guggenheim model,

saturation adsorption concentration,

Kronneker symbol,

Laplace operator,

complex surface elasticity modulus,

Gibbs elasticity resulting from the isotherm,

experimental values of elasticity for high frequencies,

dielectric constant,

molecular exchange functions,

363

n

rio

rl'

K

K w

K*

p

~ik

(Jrr, I~88, ~q~q~

~rr f, ~rr g

o = 2 f r t

03m

O0

q~(f,c)

q~*

~ ' ( t )

V

(v) ~

V~

bulk shear viscosity,

bulk shear viscosity of the calibration solution,

surface shear viscosity,

bulk dilational viscosity,

surface dilational viscosity,

constant of the adiabatic gas equations,

reciprocal Debye length,

density of the bulk solution,

components of the pressure tensor within the bulk solution,

components of the pressure tensor in spherical coordinates,

radial components of the pressure tensor within the fluid or the gas chamber,

angular frequency,

experimental value of molecular exchange parameter,

molecular exchange parameter,

phase angle of the elasticity modulus,

phase angle of the change in adsorption,

phase angle of the excitation function,

magnitude of the excitation function,

electrostatic potential.

nabla operator,

transposed nabla operator,

gradient operator on the surface.

8. REFERENCES

1. Fruhner, H., Kr~igel, J., Kretzschmar, G., J. Inf. Rec. Mats., 19(1991) 45.

2. Djabbarah, N. F., Wasan, D.T., AIChE J., 31(1985)1041.

3. Wantke, K.-D., Malysa, K., Lunkenheimer, K., Colloids Surfaces. A, 82 (1994)183.

4. Gottier, G. N., Amundson, N. R., Flumerfelt, R.W., J. Colloid Interface Sci., 114(1986)106.

5. Loglio, G., Tesei, U., Cini, R., Rev. Sci. Instrum., 59(1988)2045.

6. Loglio, G., Tesei, U., Miller, R., and Cini, R., Colloids Surfaces, 61(1991)219.

364

7. Loglio, G., Miller, R., Stortini, A., Tesei, U., Degli-Innocenti, N., Cini, R., Colloids

Surfaces A, 90(1994)251.

8. Joos, P., and Van Uffelen, M. J. Colloid Interface Sci., 155(1993)271.

9. Fang, J., Joos, P., Lunkenheimer, K., Van Uffelen, M., J. Colloid Interface Sci.,

155(1993)271.

10. Fang, J., Wantke, K.-D., Lunkenheimer, K., J. Colloid Interface Sci., 182(1996)31.

11 Miller, R., Policova, Z. Sedev, R., Neumann, A. W., Colloids Surfaces, 76(1993)179.

12. MacLeod, C. A., Radke, C. J., J. Colloid Interface Sci., 160(1993)435.

13. Liggieri, L., Ravera, F., Passerone, A., J. Colloid Interface Sci., 140(1990)436.

14. Dukhin, S. S., Kretzschmar, G., Miller, R., "Dynamics of adsorption at liquid interfaces", in

D. M6bius and R. Miller(Eds.)Studies in Interface Science, Vol 1, Elsevier,

Amsterdam, 1995.

15. van den Tempel, M., van de Riet, J. Chem. Phys., 42(1964)2769.

16. Lucassen-Reynders, E. H., Lucassen, J., Adv. Colloid Interface Sci., 2(1969)347.

17. Lucassen-Reynders, E. H., J. Colloid Interface Sci., 42(1973)573.

18. van den Tempel, M., Lucassen-Reynders, E. H., Adv. Colloid Interface Sci., 18(1983)281.

19. Langevin, D., J. Colloid Interface Sci., 80(1981)412.

20. Earnshaw, J. C., McGivern, R. C., McLaughlin, A. C., Winch, P. J.,Langmuir, 6(1990)649.

21. Earnshaw, J.C., Hughes, C. J., Langmuir, 7(1991)2419.

22. Lucassen, J., and van den Tempel, M., Chem. Eng. Sci., 27(1972)1283.

23. Lucassen, J., van den Tempel, M., J. Colloid Interface Sci., 41(1972)491.

24. Lucassen-Reynders, E.H. "Physical Chemistry of Surfactant Action", Surfactant Science

Series, Vol. 11, Marcel Dekker, New York, 1981.

25. Kretzschmar, G., Lunkenheimer, K., Ber. Bunsenges. Phys. Chem., 74(1970)1064.

26. Chang, C.-H., Franses, E. I., J. Colloid Interface Sci., 164(1994) 107.

27. Johnson, D. O., Stebe, K. J., J. Colloid Interface Sci., 168(1994)21.

28. Enhorning, G., J. Appl. Physiol., 43(1977) 198.

29. Hall, S., Bermel, M., Ko, Y., Palmer, H., Enhorning, G., Notter, R., J. Appl. Physiol.,

75(1993)467.

30. Schfirch, S, Bachhofen, H., Goerke, J., Possmayer, F., J. Appl. Physiol., 67(1989)2389.

31. Prokop, R. M., Neumann, A. W., Current Opinion in Colloid Interface Sci., 1(1996)677.

365

32. Landau, L., Lifschitz, E. M., Vol. 6 "Fluid Mechanics", Pergamon Press, New York, NY, 1959.

33. Laso, M. L., 0ttinger, H. C., Phys. BI., 49(1993)121.

34. Scriven, L. E., Chem. Eng. Sci., 12(1996)98.

35. Kao, R. L., Edwards, D. A., Wasan, D. T., Chen, E., J. Colloid Interface Sci.,

148(1992)247.

36. Lucassen-Reynders, E.-H., Lucassen, J., Colloids and Surface A, 85(1994)211.

37. Nadim, A., Borham, A., and Haj-Hariri, H., J. Colloid Interface Sci., 181(1996)159.

38. Miller, R., Wtistneck, R., Kr~tgel, J., Kretzschmar, Colloids Surfaces A, 111(1996)75.

39. Lunkenheimer, K., Hartenstein, C., Miller, R., Wantke, K.-D., Colloids Surfaces,

8(1984)271.

40. Wantke. K.-D., Miller, R., Lunkenheimer, Abh. Akad. Wiss. DDR Nr. IN, Akademie-

Verlag Berlin, Teil 1(1974)439.

41. Wantke, K.-D., Miller, R. and Lunkenheimer, K., Z.Phys. Chem.(Leipzig), 261(1980)1177.

42. Wantke, K.-D., J. Colloid Interface Sci., 41(1991)293.

43 Wantke, K.-D., Lunkenheimer, K., and Hempt, C., J. Colloid Interface Sci., 159(1993)28.

44. Johnson, D. O.,and Stebe, K. J., J. Colloid Interface Sci., 182(1996)21.

45 Johnson, D. O., and Stebe, K. J., Colloids Surfaces A, 114(1996)41.

46 Chang, C. H., and Franses, E. I., Chem. Eng. Sci., 49(1994)313.

47 MacLeod, C. A. and Radke, C. J., J. Colloid Interface Sci., 160(1993)435.

48 Nagarajan, R., and Wasan, D. T., J. Colloid Interface Sci., 159(1993)164.

49 Kao, R. L., Wasan, D. T., J. Colloid Interface Sci., 155(1993)518.

50 Fruhner, H., Wantke, K.-D., Colloids Surfaces A, 114(1996)53.

51 Fruhner, H., Lunkenheimer, K., Miller, R., in L. Ratke and B. Feuerbach(Eds.), Lecture

Notes in Physics, Vol. 464, Springer-Verlag, Berlin 1996, 41.

52. Kitching, S., Johnson, G. D. W., Midmore, B. R., and Herrington, T. M., J. Colloid

Interface Sci., 177(1996)58.

53. Jiang, Q., Valentini, J. E., Chiew, Y. C., J. Colloid Interface Sci., 174(1995)268.

54. Guggenheim, E. A., "Thermodynamics", North-Holland, Personal Library, 1967.

55. Bonfillon, A., Sicoli, F., Langevin, D., J. Colloid Interface Sci., 168(1994)497.

56. Bonfillon, A., Langevin, D., Langmuir, 10(1994)2965.

57. Dukhin, S.S., Miller, R., Kretzschmar, G., Colloid Polymer Sci., 261(1983)335.

This Page Intentionally Left Blank

Drops and Bubbles in Interfacial Research D. M6bius and R. Miller (Editors) �9 1998 Elsevier Science B.V. All rights reserved.

PHYSICO-CHEMICAL HYDRODYNAMICS OF RISING BUBBLE

367

S.S. Dukhin 1, R. Miller 2 and G. Loglio 3

1 Institute of Colloid Chemistry and Chemistry of Water, 42 Vernadsky Avenue, 252680 Kiev, Ukraine

2 Max-Planck-Institut fiir Kolloid- und Grenzfl~ichenforschung, Rudower Chaussee 5, D-12489 Berlin-Adlershof, Germany

3 University of Florence, Institute of Organic Chemistry, Via G. Capponi 9, Florence, Italy

CONTENTS

2.1.

2.2.

2.3.

2.4.

2.5.

2.6.

.

3.1

3.2.

3.3.

Introduction

Qualitative approach

Qualitative description of dynamic adsorption layers and surface retardation

Bubble hydrodynamics and theology of the water-air interface

Classification of regimes of bubble buoyancy based on magnitude of

Reynolds number

Role of dynamic adsorption layer in foam and emulsion technologies

Main stages in the development of physico-chemical hydrodynamics of

bubble

Quantitative consideration- foundation of the theory of diffusion boundary

layer and dynamic adsorption layer of moving bubbles

Steady dynamic adsorption layer at small Re

Dynamic adsorption layer under condition of uniform surface

retardation. The case Pe<<l

Dynamic adsorption layer under condition of uniform surface retardation.

The case P e ~ 1, Re << 1

Theory of dynamic adsorption and diffusion boundary layers of a bubble

with Pe>)1, Re<< 1 and Weak Surface Retardation

368

3.4.

3.5.

3.6.

3.7.

~

5.1.

5 . 2 .

5.3.

6.1.

6.2

6.3.

6.4.

6.5.

6 . 6 .

6.7.

6.8.

6.9.

6.10.

Theory of dynamic adsorption layer of bubble (drop) at Re((1 and strong

surface retardation

The rear stagnant region of a buoyant bubble

Dynamic adsorption layer at slow desorption kinetics

Newest development in the theory of Marangoni retardation of bubble

terminal velocity

Dynamics of rear stagnant cap growth and rising bubble relaxation

Hydrodynamics 'of spherical particles at higher Reynolds numbers

Development of flow field with Reynolds number

Theory of bubble hydrodynamics at negligible surface retardation.

Experimental investigation of bubble rising and deformation in "hyper

clean" water

Dynamic adsorption layer and suffactant influence on bubble buoyancy at

higher Reynolds numbers

DAL at a weakly retarded bubble surface

Conditions of realisation of different states of dynamic adsorption layer

formation for a buoyant bubble

Stagnant ,,ring" model

Finite difference solution of full Navier-Stokes equations for mixed

boundary conditions of stagnant cap

Coupling transfer momentum and transfer of surfactant at large Re

DAL structure and surface retardation at large Re

The effect of surfactant on the rise of" a spherical bubble at high Re

Investigation of micro-flotation kinetics as a method of DAL studies

Boundary layer solution for DAL and flow over upper surface of rising

bubble

Experimental investigation of rising bubble relaxation

Summary

References

List of Symbols

369

1. INTRODUCTION

In Levich's classical monograph "Physico-Chemical Hydrodynamics" (1962) one of the twelve chapters is devoted to drop and bubble movement in liquid media. During the last three decades essential progress has been achieved in this area which is important for both fundamental and applied colloid science.

The movement of bubbles and drops in a liquid is influenced by the kinetics of adsorption and desorption of surfactant molecules at the liquid surface. After a certain time the motion levels off and a steady state is reached in which the hydrodynamic field around the bubble, and in particular the motion of its surface, appears as the driving force of the steady state process. The transient process is more difficult to consider since it entails the necessity to solve non- stationary convective diffusion equations. Investigators have paid most attention to the steady- state process of convective diffusion and adsorption/desorption. In contrast to the adsorption and desorption caused by the deviation from equilibrium between the adsorption layer and the subphase in the initial moment, the total amount of adsorbed substance does not change atter the stationary state of buoyant bubbles has been reached. Thus, rather than a time dependence of adsorption, a qualitatively different characteristic parameter determines the specific nature of adsorption kinetics caused by the hydrodynamic field of a buoyant bubble. Steady-state motion of bubbles induces adsorption-desorption exchange with the subsurface, with the amount of substance adsorbed on one part of the bubble surface being equal to the amount desorbed from another part. Thus, surface concentration varies along the surface of a buoyant bubble taking a maximum value at the rear stagnation point and a minimum at the leading pole (Frumkin & Levich, 1947). This qualitative suggestion relates to a not very large Reynolds numbers. At rather large Reynolds numbers the regularities of the surface concentration distribution along a rising bubble surface changes drastically.

The surface concentration difference between the poles of a bubble is due to its movement. The difference increases with faster movements and disappears in the case of a resting bubble. Therefore, the state of the adsorption layer on a moving bubble surface is qualitatively different from that on a resting one. Such adsorption layers are called dynamic adsorption layers (Dukhin, 1965). Thus, a dynamic adsorption layer (DAL) is an analogy of the time dependence (kinetics) of adsorption initiated by a deviation from equilibrium at a freshly formed surface.

The problem of adsorption kinetics in the case of a buoyant bubble can be transformed into the problem of a dynamic adsorption layer. As a result of such a transformation, the adsorption kinetics is coupled with other processes caused by the angular dependence of adsorption. Surface tension gradients generating the Marangoni effect are connected with a non-uniform adsorption and a feedback arises in this case. The hydrodynamic field around a bubble initiates the adsorption-desorption exchange and leads to a dynamic adsorption layer which retards the motion of the surface and so acts on the hydrodynamic field and the adsorption-desorption exchange.

The coupling of the surfactant mass transfer and the transport of the momentum causes large difficulties in quantifying the DAL theory. At small Re this difficulty diminishes because of the linearisation of the mathematical formulation possible. As a result the DAL theory was

370

developing over almost half a century for the case of the creeping flow. The main results are obtained for small Reynolds numbers and correspondingly they occupy a large place in this Chapter as well (cf. section 3). There is a large unbalance between the theory and the experiment because even a negligible surfactant concentration causes almost complete surface retardation. No experimental data were obtained for the velocity of a rising bubble with a not completely retarded surfaces at small Re. More favourable conditions for the experiment arises now due to three reasons. There is a large improvement in deep water purification that may allow experiments with a mobile bubble surface under creeping flow conditions.

Stebe et al. (1994) showed experimentally that the generation of a surfactant gradient and correspondingly the surface retardation can be prevented for surfactants with sufficiently fast sorption kinetics at concentration near the CMC (critical micelle concentration). Thus investigations in thin water layers can enable one to reveal and measure a bubble velocity increase due to surface mobility. This theory of an unsteady DAL must be used. Meanwhile such a theory has been developed for the steady DAL only. A first quite simple version of an unsteady DAL theory is outlined in section 4.

The DAL study is more realistic for larger bubbles and large Re because trace concentrations of surface active impurities cannot retard the bubble surface movement completely. During many decades the theory for the extreme cases of almost free bubble surfaces and for strong retardation of the bubble surface as a whole was developed (Dukhin 1965, 1966, 1981). The situation has changed drastically during the last two years when much experimental and theoretical data have been published concerning bubble rising at large Re. As a result Chapter 8 in Dukhin et al. (1995) published only 3 years ago must be supplemented by these new valuable results (sections 5 and 6).

2. QUALITATIVE APPROACH

2.1. QUALITATIVE DESCMPTION OF DYNAMIC ADSORPTION LAYERS AND SURFACE RETARDATION

The state of the surface of a floating bubble depends on its size. Surfaces of reasonably large bubbles are mobile. As a result adsorbed surfactants are pulled down to the rear of the bubble, i.e. even under steady-state conditions the value of adsorption on a mobile bubble surface is different from that on an immobile one, F o (at the same surfactant concentration in the bulk).

The leading part of the mobile surface of a floating bubble is stretched, the lowest part is compressed (Levich, 1962). The newly created sections of the surface are being filled with adsorbed substance, in the compressed part of the surface the substance desorbs. The surface concentration on the leading surface of the buoyant bubble is lower than F o which provides a

continuous supply of surfactant (or adsorbing inorganic ions) from the bulk to the stretched surface. The surface concentration on the rear part of the buoyant bubble is higher than F o

which initiates desorption of surfactant. Thus, F(0) increases in a direction opposite to the bubble motion, i.e. from the leading pole (0 = 0) to the rear one (0 = zt), where the angle 0 is counted from the leading pole as shown in Fig. 1.

The supply of surfactant to the leading surface of the bubble and withdrawal of desorbing surfactant into the bulk from the lower half is governed by diffusion and leads to the formation

371

of a so-called diffusion boundary layer adjacent to the surface. Its thickness 8 D is much smaller than the bubble radius ab,

6 D = a b / ~j~e, (2.1)

where Pe is the Peclet number defined by

Pe = a bv / D = Re v / D. (2.2)

Re is the Reynolds number defined by

Re = 2a by/ , (2.3)

where v is the bubble velocity of buoyancy; v = q / 9 ; 9 and 1"1 are density and viscosity of the

liquid, respectively, D is the diffusion coefficient of surfactant molecules.

Fig. 1 Schematic of processes at a rising bubble surface; a) Surface concentration gradient created by interfacial convection, b) Transport process at the bubble surface

The concentration in the diffusion layer on the leading surface of the bubble is less than in the bulk and it increases from the leading to the rear pole like in a dynamic adsorption layer. At a given 0 value, local equilibrium is preserved between the surface concentration F(0) and the concentration adjacent to the surface c(a b ,0),

r ( 0 ) / c(au,0) = r o / Co = K . , (2.4)

where F o is the equilibrium concentration on an immobile bubble surface at a surfactant bulk

concentration of c o, K H is the so-called Henry constant and a measure of the surface activity of

the surfactant.

The situation is rather fine balanced since the motion of the surface has an effect on the

formation of the dynamic adsorption layer, and vice versa.

Adsorption increases in the direction of the liquid motion while the surface tension decreases.

This results in the appearance of forces directed against the flow and retards the surface motion.

372

Thus, the dynamic layer theory should be based on the common solution of the diffusion equation, which takes into account the effect of surface motion on adsorption-desorption processes and of hydrodynamic equations combined with the effect of adsorption layers on the liquid interfacial motion (Levich 1962).

2.2. BUBBLE HYDRODYNAMICS AND RHEOLOGY OF THE WATER-AIR INTERFACE

The theoretical description of a diffusion process of a surfactant towards or away from the surface of a floating bubble, is impossible without information on the floating velocity and the hydrodynamic field around the bubble. The first of these quantities can be found comparatively easily from experiment, whereas the Navier-Stokes equation is used to define the hydrodynamic field around the floating bubble. A solution of the equation must satisfy all boundary conditions at the bubble surface. It should be stated that a general analytical solution of this problem is impossible. This difficulty can be overcome in two extreme cases: when the bubbles rise at small Reynolds numbers (Re < 1, Stokes conditions) or at very large Reynolds numbers (Re ~ 1, potential conditions). In the first case, the inertial term can be eliminated from the Navier-Stokes equation, in the second the viscous term.

If the surface of a drop or bubble is immobile for any reason and the coordinate system is defined as moving together with the bubble, the floating velocity is the same as that of a solid sphere. In particular, at small Reynolds numbers, the bubble movement can be described by Stokes' equation,

2a 2 b

Vst - - ~ g, (2.5)

where g is the acceleration due to gravity. If the drop surface is mobile, a velocity distribution also arises inside the drop. The velocity distribution over the drop surface can be found by a collective solution of the respective Navier-Stokes equation both inside the drop and in the surrounding liquid. The continuity conditions must be fulfilled both for the velocity and the tensor of viscous stresses when passing through the phase boundary. Under these boundary conditions, the Stokes equation (linearised Navier-Stokes equation at very small Reynolds numbers Re ((1) for drops floating up in a liquid was solved by Rybczynski (1911) and independently by Hadamard (1911). Of course, this solution applies also to the case of a floating bubble when its viscosity is set equal to zero (neglecting the effects of the order of the ratio of the gas and liquid viscosities). According to the Hadamard-Rybczynski approximation the velocity of floating bubbles is expressed by the equation

2 13, l a b ( P - ) (2.6) V - - g 3 1"1

9 and 9' are densities inside and outside a droplet. Any mobility of the surface decreases the

velocity difference and the viscous stresses. The result is that the hydrodynamic resistance becomes smaller and the floating velocity of a bubble according to (2.6) increases by a factor of 3/2 as compared to Stokes' Eq. (2.5). In early experiments, under the condition of Re<l, it was found (Lebedev 1916) that small bubbles of diameters less than 0.01 cm behave like rigid spheres since their velocity is described by Stokes' formula (2.5). At the same time, Bond (1927)

373

has found that drops of a sufficiently large size fall at velocities described by Eq. (2.6). To overcome contradictions with the Hadamard-Rybczynski theory, Boussinesq (1913) considered the hypothetical influence of the surface viscosity and derived the following relation,

2 p - p ' rl + rl'+2rls / a b v - - - g a Z b (2.7)

3 1"1 21"l+3q'+2ris /a b

where rl and rl' are viscosities of the liquid inside and outside the droplet, and rlsis the surface

viscosity.

Reopening the discussion around the Hadamard-Rybczynski theory, Levich (1962) claimed that the introduction of a surface viscosity is not important, as suggested by Boussinesq, rather it is the Marangoni effect. Frumkin & Levich (1947) postulated that the anomalous experimental results of the movement of fluid droplets as solid spheres can be explained by the presence of surface-active agents which were swept to the rear of the droplet. The surface concentration gradients, and concomitant interfacial-tension gradients, created thereby were assumed to be responsible for the retardation of the drop velocity. Experimental findings reported later (Gorodetskaya 1949) qualitatively justified Levich's theory.

As both effects arise from the presence of surfactant, the controversy of "surface viscosity effect" against "surface tension gradient effect" was regarded as being nontrivial (Edwards et al. 1991). This controversy was further developed by creating the theory which united both mechanism of surface retardation and bubble rising retardation (Agrawal and Wasan 1979). Recently Stebe and Maldarelli (1994) discussed the effect of surface viscosity and showed that it is otten negligible. Following Stebe and Maldarelli surface viscosity will not be taken into account in this Chapter.

2.3. CLASSIFICATION OF REGIMES OF BUBBLE BUOYANCY BASED ON MAGNITUDE OF REYNOLDS

NUMBER

Data on the velocity of rising bubbles in aqueous surfactant solutions (sodium dodecyl sulphate SDS) published by Okazaki (1964) are presented in Fig. 2.

10-

I

005

3O

V[cm]sec]

?o-

0.1 ablcm]

Fig. 2 Velocity of bubbles floating up in aqueous (distilled water) solution of SDS of different concentrations: 0 (1), 106(2); 1.2 10"5(3); dotted curve - solid bubble in distilled water

374

The dotted curve corresponds to rigid bubbles. It was obtained by recalculations from

experiments on spherical glass balls with a density of 2.37 g/cm 3. Significant surfactant effects

appear for large bubbles. Small bubbles rise as solid spheres, even in thoroughly cleaned liquids:

from a b = 0.03 cm (Re = 36) and at c > 10 -3 M up to a b = 0.065 cm (Re = 182). It is typical

that even at such low concentrations as 10 -6 M the velocity of bubbles is essentially decelerated.

Usually three ranges of Reynolds numbers are introduced, namely small Reynolds numbers

(Re<l), intermediate Re (l<Re<200) and large Re (Re>200). At large Re bubble buoyancy is

complicated by two phenomena,, the bubble shape deformation (Moore 1963, 1965; El Sawi

1974; Benjamin 1987; Miksis et al 1981; Ryskin and Leal 1984; Duineveld 1995) and bubble

path instability (Saffmem 1956; Hartunian and Sears 1957; Batcheler 1967; Tsuge and Hibene

1977; Fan and Tsutchiya 1990; Duineveld 1995).

Both by theory and experiments performed recently by Duineveld (1995) it was established that

these phenomena are important at Re >150-200. Due to bubble surface deformation the

hydrodynamic resistance increases and at rather high Reynolds numbers the terminal velocity

decreases with the bubble dimension increase. (cf. Fig. 2). Naturally the bubble shape

deformation arises earlier than the terminal velocity decrease and manifests itself in the

retardation of the terminal velocity increase and its extremal dependence on bubble dimension.

In this study attention will be paid to the regimes of small and intermediate Reynolds numbers.

The Navier-Stokes equation can be linearised for the creeping flow regime (Re<l) leading to an

essential simplification in the theory. The use of the non-linear Navier-Stokes equation makes

the case of intermediate Reynol6s numbers very difficult for an analytical solution. As to the

experiment at small Reynolds numbers the main difficulty is caused by the high sensitivity of the

bubble surface movement to any surfactant impurities which cannot be controlled. The main

question remains to be answered: is it possible to attain a surfactant concentration low enough

not to affect the surface motion? Meanwhile a positive answer was given recently in respect to

intermediate Reynolds numbers (Fdhila and Duineveld 1996; Park et al. 1995). The viscous

stresses increase with the transition from small to intermediate Reynolds numbers which is

favourable for the bubble surface movement.

2.4. ROLE OF DYNAMIC ADSORPTION LAYER IN FOAM AND EMULSION TECHNOLOGIES

Dynamic adsorption layers differ from equilibrium layers not only by the existence of an angular

dependence but also by the difference in the adsorbed amount averaged over the bubble surface

(Sadhal & Johnson, 1983). Usually, in foam flotation, the surfactant yield is calculated under the

assumption of equilibrium adsorption at the surface of buoyant bubble. The theory of dynamic

adsorption layers leads to substantial changes in the notion of surfactant flotation. Thus, the

375

mechanism of transport at the bubble-solution interface has a substantial effect on the transport

process at the surfactant solution-foam boundary.

A non-equilibrium state of the adsorption layer of bubbles or drops initiates adsorption

processes in foams and emulsions. Transport of surface active substances in foams and

emulsions can additionally be complicated by this dynamic adsorption layer. Specifically, water

purified of surfactants or their fractionation are strongly influenced by a dynamic state of the

adsorption layer (cf. Dukhin et al. 1995, Section 8.8.3).

The problem of dynamic adsorption layers does not only arise in connection with adsorption-

desorption processes, it can have a substantial effect on processes of interaction between

bubbles or drops and thus on coagulation and coalescence processes in foams and emulsions (cf.

Dukhin et al. 1995, Chapter 12).

Consider flotation (cf. Dukhin et al. 1995, Chapter 10), dynamic adsorption layers affect all

stages of the elementary flotation processes, since they affect the hydrodynamic field of a bubble

and thus the trajectory of particles in the neighbourhood as well as the process of its deposition.

The next stage is the thinning of the liquid interlayer between the surfaces of the bubble and the

particle approaching it. Mechanisms due to the influence of the surfactant on film thinning was

also considered recently (cf. Dukhin et al. 1995, Chapter 11). Finally, in the case of the

overlapping fields of the equilibrium surface forces of a bubble and a particle it is again

important to take into account the dynamic state of adsorption layers of bubbles since it controls

the local values of equilibrium surface forces.

The description of dynamic adsorption layers under the condition of bubble/bubble or

bubble/particle interaction is much more complex than the consideration of dynamic adsorption

layers of individual bubbles discussed in the present chapter. It is still more difficult to control

dynamic adsorption layers experimentally under conditions of the above-mentioned interactions.

Because of these experimental difficulties, the role of mathematical modelling is extremely

important in studying coagulation and hetero-coagulation processes in foams and emulsions.

The problem of dynamic adsorption layers becomes very complicated in concentrated foams and

emulsions and is probably of interest for the investigation of the rheology of foams and

emulsions. Thus, a systematic study of the dynamic adsorption layer of an individual bubble is

important not only in itself, but to a larger degree as an investigation of fundamental processes

in a variety of more complex situations spanning a range of fluid technologies.

376

2. 5. MAIN STAGES IN THE DEVELOPMENT OF PHYSICO-CHEMICAL HYDRODYNAMICS OF BUBBLE

Physico-chemical hydrodynamics of bubble and drops attract the attention of many investigators

in different countries over the last fi~y years. Despite the obvious difficulties in contact between

groups in East and West, the main results of investigations published in Russian and English

agree well and complement each other. A prerequisite for this agreement is the fact that all the

theories were developed on the same basis given by the works of Frumldn and Levich.

The founders of the physico-chemical hydrodynamics of bubbles (Frumldn & Levich, 1947)

have derived only the simplest solution of the problem which was formulated at a time where

the conditions of its applicability were not specified. The theory was simplified by three

assumptions: i) the relative surface concentration difference along the mobile bubble surface is

small, ii) the dynamic adsorption layer retards the surface uniformly (i.e., the velocity of any

section of the surface is decreased under its effect by the same factor which makes it possible to

introduce the retardation coefficient ~b), and iii) the variation of surface concentration is strictly

antisymmetric with respect to the equatorial plane which is expressed by a very simple angular

dependence of adsorption. Clearly the dynamic adsorption layer in Frumkin-Levich's description

is only a limiting case which has impelled Derjaguin and Dukhin to carry out systematic studies

of that problem (Derjaguin et al., 1959, 1960; Dukhin & Derjaguin, 1958, 1961 a, b; Derjaguin

& Dukhin, 1960, 1971; Dukhin & Buikov, 1965; Dukhin 1964, 1965, 1966, 1981, 1982, 1983).

A substantial difference to the work of Frumldn & Levich (1947) and Levich (1962) is the

elimination of the three simplifying conditions mentioned above. Thus the problem was

formulated by revealing the diversity of the states of dynamic adsorption layer differing in the

complex angular dependence of adsorption and, respectively, in non-uniform surface retardation

and the possibility of a substantial relative variation of adsorption along the bubble surface.

Considering the conditions Re << 1, Pe )) 1, the main difficulty in coupling the transport of

momentum and the transport of matter was avoided in the limiting cases of weak (cf.

Section 3.3) and very strong retardation (cf. Section 3.4) of the bubble surface. In the former

case the hydrodynamic field of velocities is given and the convective diffusion equation is

defined specifically. The solution is rather difficult, owing to the complexity of the boundary

conditions (cf. Section 3.3). A further simplification is attained by restricting to a low or a high

surface activity of a surfactant. In the first case the relative adsorption difference is small but its

angular dependence is very complex. The second case turns out to be especially interesting since

almost the whole surface is free from surfactant which is pulled down to the rear and forms a

stagnant cap within which adsorption far exceeds the equilibrium value. In parallel with these

limiting cases Dukhin (1965, 1981) derived an integro-differential equation which describes the

structure of dynamic adsorption layers for any surface activity.

377

In the other limiting case of strong retardation, an efficient approximate analytical method was

proposed to enable the coupling of the transport of momentum and matter (cf. Section 3.4).

Here it is also possible to discriminate between the conditions of a slight deviation of adsorption

from equilibrium, and the appearance of two zones on the bubble surface showing strong and

weak retardation. At strong retardation of the entire bubble surface, the relative adsorption

differential is small and the angular dependence is close to that postulated by Levich (1962).

Thus the simple solution by Levich can be realised also at Pe ~ 1, but it is valid only at strong

retardation and not too high surface activity.

The intermediate concentration region is characterised by heterogeneities of the dynamic

adsorption layer structure (Derjaguin and Dukhin, 1959 - 1961). When increasing the surfactant

concentration a stagnant cap is formed and expands, while at decreasing concentration the

weakly retarded zone expands. Thus, a double-zone dynamic adsorption layer arises in the

intermediate concentration region. This problem was first formulated in a work by Savich

(1953) and later much effort has been made for its solution (Harper 1972, 1973, 1974, 1982,

1988, see also section 3.5).

In the dynamic adsorption layer theory of Derjaguin-Dukhin, the hydrodynamic field of a bubble

can be assumed to be known as first approximation, while the more difficult stagnant cap

problem was solved by Sadhal and Johnson (1983).

The rate of the exchange process of surfactant molecules between the surface of a bubble (drop)

and the bulk solution is determined not only by convective diffusion but in the general case also

by the kinetics of the adsorption step itself. A method which takes into account the effect of

adsorption kinetics on the formation of the dynamic layer was developed by Levich (1962).

Using this method, attempts were made to generalise the theory of the dynamic adsorption layer

of buoyant bubbles at weak surface retardation (cf. Dukhin 1965 and section 9.1 in Dukhin et al.

1995). A more general theory was developed recently by He and Maldarelli (1991, cf. section

3.6.)

The new level in the DAL theory for the intermediate concentration region is achieved by

numerical procedures (Chen and Stebe 1996). This more general and exact solution enables

discrimination between the regimes well described by the r.s.c, model and those for which it

does not work (cf. section 3.7.).

The new problem of the DAL growth during the transient stage of a bubble buoyancy is

formulated in section 4. If the transient process is sufficiently slow the unsteady DAL can be

described by the r.s.c, model, and also at any moment when its dimension increases slowly.

Three stages or types of a dynamic adsorption layer effects can be distinguished:

378

i) the dynamic adsorption layer slightly retards the bubble motion;

ii) the dynamic adsorption layer strongly retards bubble motion;

iii) a stagnant cap exists.

This classification discussed in investigations under the restriction Re (< 1 is probably correct as

well as for Re )) 1. At the transition to Re )) 1 where a stagnant cap exists the hydrodynamic

problem has to be solved.

In respect to large Reynolds numbers 30 years ago (Dukhin 1965) until recently (cf. section 8.6.

in Dukhin et al. 1995) the knowledge was restricted to evaluations of conditions for different

states of adsorption layers formation and retardation and to the DAL theory at weak and strong

retardation conditions. An unexpectedly rapid progress has been made during the last 2 years

both in theory and experiment (cf. section 5).

2.6. QUANTITATIVE CONSIDERATION- FOUNDATION OF THE THEORY OF DIFFUSION BOUNDARY

LAYER AND DYNAMIC ADSORPTION LAYER OF MO VING BUBBLES

In connection with the development of the theory of convective diffusion in liquids the

foundation of the theory of diffusion boundary layers and dynamic adsorption layers are given

by Levich (1962) in his works on physico-chemical hydrodynamics.

The mathematical description of the convective diffusion process is based on the solution of the

convective diffusion equation

0c(0,z,t) + v(0, z) grad c(0, z,t) = A Dc(0,z,t), (2.8)

0t

where c(0,z, t) and v(0,z) are the distributions of concentration and velocity of liquid,

respectively. If we reduce this equation to a dimensionless form and assume stationary

conditions the process can be characterised by a dimensionless Peclet number. It is convenient

to express Pe (cf. Eq. 2.2) in terms of the Reynolds number Re and the Prandtl number Pr which

depend only on the properties of the medium,

Pe = Re Pr, (2.9)

where Pr = v/D. Like in the case of a viscous liquid flow at large Reynolds numbers, the role of

molecular viscosity arises in a thin boundary layer, where the liquid is retarded, and the velocity

difference is localised. At large Peclet numbers molecular diffusion manifests itself in a thin

boundary layer which is called the diffusion layer.

379

While the thickness of the hydrodynamic boundary layer 8 o can be estimated by L / R e n', a

similar estimate L / Pe n2 is obtained for the thickness of the diffusion boundary layer 8D, with n 1

and n 2 having values less than unity. Therefore, a rough estimate of the ratio 8D/8 G is Pr -n' at

n 1 = n 2 . The kinematic viscosity of mobile water-like liquids is of the order of v - 10 -2 cm2/s,

the diffusion coefficients of molecules and ions in aqueous solutions are of the order of

D ~ 10 -5 cm2/s, those of macromolecules D ~ 10 -6 cm2/s. Thus, in water and in similar liquids

Pr ~ 10 3 and 8D is much smaller than 8a. This means that conditions for the existence of a

diffusion boundary layer Pe>> 1 can be fulfilled not only at Re>> but also at Re<< .

It should be pointed out that in the formulation of the problem of convective diffusion one

should know the velocity distribution v(z) since it appears on the left-hand side of Eq. (2.8). A

solution of the hydrodynamic problem is possible in the limiting cases of small and large

Reynolds numbers. Therefore, the theory of the boundary diffusion layer for the two limiting

cases Re<(1, Pc<(1 and Re>> 1, Pc>> 1 has to be developed.

In the following, interest is focused on the dynamic adsorption layer and the boundary diffusion

layer respectively, which show a strictly stationary character due to the interplay of surfactant

adsorption at one part of the mobile surface of the bubble (drop) and its desorption from the

other part. Boundary conditions must take into account the convective transfer of surfactant

along the surface and exchange between the surface and the bulk. The corresponding boundary

condition can be regarded as

div~ (F(0)v0 (0) - D~gradr(0)) - -jn (0), (2.10)

where D S is the surface diffusion coefficient, J n is the normal component of the flow density

between the bulk and the surface, v 0 is the velocity distribution over the bubble surface. Thus,

this boundary condition takes into account the process of mass transfer along the surface both

due to convection and surface diffusion. At small adsorption times (high rate of adsorption) it

can be expected that local equilibrium between adsorption F(0) and bulk concentration at the

bubble surface C(ab,0 ) exists. This means that the same functional relation between F(0) and

C(ab,0) as between Fo and Co exists. Far from saturation, this functional relation can be

considered as linear, Eq. (2.4). The adsorption rate Jn is determined by the diffusion rate,

Jn - -D0C(Z'0) I (2.11) (~Z z= a b

Thus, at high adsorption rates the quantities appearing on the right-hand and left-hand sides of

Eq. (2.10) are expressed in terms of the concentration distribution so that Eq. (2.10) becomes a

boundary condition for Eq. (2.8).

380

3. STEADY DYNAMIC ADSORPTION LAYER AT SMALL Re

3.1 DYNAMIC ADSORPTION LAYER UNDER CONDITION OF UNIFORM SURFACE RETARDATION.

THE CASE P e u 1

If the Peclet number is small, Pe<<l, two simplifying approximations are possible (Dukhin, 1965,

1981). As it will be shown in the following, if Pe<<l the surface concentrations differ negligibly

from their equilibrium values,

(F(| - Fo) / Fo - [C(ab,| Co] / Co<<l, (3.1)

where a b is radius of droplet or bubble, and the convection-diffusion equation (2.8) reduces to

the Laplace equation. Thus, the hydrodynamic and diffusion fields may by truncated to the first

two spherical harmonics and their derivatives (terms in sin0 and cos0 only),

2 ab

c(z,| - Co + A c ~ - cosO, r (o ) - ro + ArcosO (3.2)

with

AF = KHAC. (3.3)

With these simplifications, it has been shown that the effect of the adsorption layer on the

droplet hydrodynamics is quantitatively described by the introduction of a retardation

coefficient, and the translational velocity of the droplet, respectively,

Ac 2 Apga2b rl+rl'+Xb and v - (3.4)

Xb - aC 3Vo 3 ~! 21"1 + 3rl'+3Xb

The velocity distribution along the surface is given by

v(0) = v o sin0 (3.5)

where

1 Apga2b (3 .6 ) v~ - 3 2~+ 31]'+3Xb

Explicit expressions for Ac, AF, X b and V o can be derived from the boundary condition (2.10)

2 FoRTK n (3.7) )(~b - - " "3 D + 2D~K. / a b

Ac 2F o v a b 1 v o D = o < ~ ~ P e < < I (3 .8 ) c o Coa b D I + 2 D s / D . F o /coa b v Ds

381

and

2Apga2bK. r(o) - ro - ro [ ~ ] I 2RTFoK. ]cos| (3.9)

3 D + 2Ds 21"1 + 31"1'+ D + 2D~K H / a b

For low surface activity

KH((a b (3.10)

the surface diffusion is negligible with respect to convective surface transport, provided that D

and D s have the same order of magnitude. For high surface activity,

KH))ab, (3.11)

surface diffusion is the dominant factor and determines the adsorption distribution and surface

retardation while the convective surface transport is negligible.

3.2. DYNAMIC ADSORPTION LAYER UNDER CONDITION OF UNIFORM SURFACE RETARDATION.

THE CASE P e ~ 1, Re u 1

Condition (3.1) may not necessarily be fulfilled at Pe)) 1, and the angular distribution dependence

of the adsorption and surfactant concentration is more complex than the functions given in Eq.

(3.2). The concentration distribution must obey the convective diffusion equation. In spite of

this, Frumkin & Levich (1947) have proposed an approximate theory of the diffusion boundary

layer and dynamic adsorption layer at Pe))1, and Re((1 based on condition (3.1) as well as the

simple relation (3.3) characterising the dynamic adsorption layer, and with a uniform surface

retardation. This means that it is permitted to introduce the retardation coefficient.

At large Peclet numbers a thin diffusion boundary layer is formed, so that

Ac cos0 Jn = D ~ (3.12) 5~(0) '

where the postulated angular dependence of the concentration distribution Eq. (3.2) is taken

into account.

Expressing the right-hand side of the boundary condition (2.10) by means of (3.12), it is easy to

conclude that the assumption of simple angular dependencies of concentration and adsorption

(3.2) are not compatible if we also take into account the angular dependence of the diffusion

layer thickness 5 D (| In other words, the Frumldn-Levich approximation corresponds to

simplifications of the boundary condition (2.10) by substituting the averaged value of the

382

diffusion layer thickness into Eq. (3.12). With the mentioned assumptions and simplifications it

is possible to derive the retardation coefficient for Pe>> 1

Zb = 2RTI'o28D / 3DabCo. (3.13)

If in addition to Pe))1 the condition Xb <<3/2rl is valid, the condition for low surface retardation

can be stated in the form

(F ~ / Co )2 Co , (27 / 4)rIDa b / RT5 D. (3.14)

The Frumkin-Levich theory was of great methodological importance and was the basis for more

rigorous considerations at Pe))l by Derjaguin & Dukhin (1959 - 1961). These results are

discussed in the next sections. It turns out that uniform retardation and relatively small variation

of adsorption exist along with other conditions which radically differ from predictions of the

Frumkin-Levich theory.

3.3. THEORY OF DYNAMIC ADSORPTION AND DIFFUSION BOUNDARY LAYERS OF A BUBBLE WITH

Pe,1 , Re((1 AND WEAK SURFACE RETARDATION

When the surface retardation is weak and Eq. (3.14) is fulfilled, the convective diffusion

equation can be transformed into the heat transport equation with constant coefficient (Levich

1962). The field equation is thus a simple one, but the complexity of Eq. (2.10) requires the

investigation of extreme cases which simplifies the boundary condition. The ratio of the second

to the first terms of the right-hand side of Eq. (2.10) may be written as

(Dsv /Dv o sin0 Pe) (c~ lnF /03), so that at Pe))l (except near the poles 0=0, 0 = rt) it can be

simplified by dropping the surface diffusion term

0c 0) v o 0 D-~z (a b - [sin2 0. F(0)]

' a b sin0 03 (3.15)

For lower values of KH, surface concentration changes become relatively small, that this almost

enables us to simplify the fight hand side of boundary condition (3.15) to solve the convective

diffusion equation and to obtain

F o-F(0) C o--C(ab,0) 2 F o Pe'/2 m , . - nV 2 �9 I(0) (3 16 t

ro Co

where the function 1(t3) is given by

I(0) - I cos0' sin0' dO' 0 ~(~ 11/2 (0 , 0 ' ) (3.17)

383

For low surface activity

KH.5 D, (3.18)

Eqs (3.16) and (3.17) lead to the relation (3.1), which specifies that a relatively insignificant

surface concentration change occurs.

For the opposite case,

KH>>fD, (3.19)

the surface concentration over the major part of the bubble surface is considerably smaller than

F o. Later both conditions (3.18) and (3.19) have been used by many authors through the

dimensionless criterion

F F R e 1-n - - ~ (3.20)

C8 C a b

where n is given by the hydrodynamic boundary layer theory as n=l/2.

If (3.19)is valid, i.e. c(ab,0)<<Co, to a first approximation the simplified boundary condition

reads

C(ab,0 ) --0. (3.21)

A solution of the heat transport equation with the boundary conditions (3.2 I) yields (Derjaguin

et al 1959)

28D nl/2 1-- COS_______~0 X/2 + COSe. (3.22) F (0) - ~ c o sin2 0

The integration constant has been adjusted to make F(0) finite at 0-+0. The singularity at

0 ~ 7t is eliminated when surface diffusion is taken into account.

The results for these extreme cases have a simple physical interpretation. The tangential flux of

adsorbed material along a drop or bubble surface is proportional to Fo, and the rate of bulk to

surface exchange is proportional to c o . Furthermore it is clear that at sufficiently small ratios

Fo /Co, the tangential transport is so small and the exchange so large that the adsorption is

practically always in equilibrium. At sufficiently large KH, the exchange of matter is small in

comparison with a strong tangential transport and the steady state surface concentration

distribution differs significantly from the equilibrium.

Later (Dukhin 1965,1981) the integro-differential equation was derived which describes the

DAL at any values of dimensionless criterion.

384

I 1 tc(t"0) 1 KH-1F(0)sin 2 0 - c(ab0)sin 2 0 - m c o x/t --~ ; ~ dt' 0

with m - 25 o ~ / K n related to the conditions (3.18) and (3.19).

(3.23)

This general equation in the limiting cases (3.18) and (3.19) leads to the solutions (3.16) and

(3.22) which have been obtained by independent methods.

The theory of the dynamic adsorption layer on slightly retarded surfaces presented by Derjaguin

et al. (1959; 1960) and Dukhin (1964; 1966) was basically confirmed by Harper (1973), Saville

(1973) has found an additional derivation. The cases of high and low surface activity have been

individually considered in these works. In the formulation of conditions (3.18) and (3.19) a

dimensionless surface activity parameter K H was introduced as the ratio of the amount of

surfactant adsorbed on the surface to the amount dissolved in the diffusion boundary layer.

Harper (1973) and Saville (1973) confirm agreement of their results with those obtained by

Derjaguin and Dukhin (1959, 1960, 1961, 1966) for the limiting cases KH ~ 0 and K n -+ oo

The results in the present paragraph have limits of applicability, in particular those connected

with the application of the boundary layer method. It is well known that results obtained by this

method are unsuitable in the neighbourhood of the rear stagnant point.

Saville (1973) solved the convective diffusion equation numerically and gave the same value of

retardation coefficient as obtained by Dukhin (1965, 1981). Listovnichii (1985) has succeeded

in obtaining simple approximation formulas for the concentration distribution not only along a

bubble surface but also across the diffusion layer, based on numerical solution of Eq. (3.23). He

has also shown that the analytical solutions Eqs. (3.16) and (3.22) deviate from the exact

solution by less than 1%, at m > 10 and m < 0.1.

3.4. THEORY OF DYNAMIC ADSORPTION LAYER OF BUBBLE (DROP) AT Reu 1 AND STRONG SURFACE

RETARDATION

In the DAL theory for weak retardation it was possible to use the equation for conservation of

the surfactant mass only (cf. section 3.3). In a similar way it is possible in a DAL theory for

strong retardation to use the conservation equation for the momentum transfer, i.e. the stress

balance.

The non-uniform surface concentration F ~ establishes a non-uniform surface tension ?' which

resists tangential shearing at the interface,

- _ _ OF) 1 0y 1 0y (3 25) "Cr01r=ab-- a ~ = a o F ~

385

where z~ is the shear stress.

The form of o~/OF in Eq. (3.25) governs the coupling between the surfactant mass transfer and

the stress balance. The dependence of Y(F0) is determined by the adsorption isotherm which

relates F0(C0), and by the Gibbs adsorption equation for the interface. On the droplet interface

the surface tension is assumed to be in local equilibrium with the surface concentration.

The condition v 0 (ab,0)((v enables the application of a successive approximation method to

calculate the velocity, concentration and adsorption fields (Dukhin & Derjaguin, 1961). Since

the velocity of the flow relative to the bubble is much higher than the velocity of its surface, the

following boundary conditions can be taken as the first approximate,

v0(ab,0) = 0. (3.26)

This relation coincides with the boundary condition for a viscous flow around solid spheres. In

this approximation the velocity distribution at Re(<l is expressed by Stokes' formula. From

Stokes' velocity distribution v(z,0), it is easy to calculate the viscous stresses acting on the

surface of the sphere and the equilibrating surface tension gradient

1 dy ~aC~Vz Or0 ~_1 3 ab dO- - ~ + O---z-- z--~ - 2ab qVo sin O. (3.27)

Under the assumption of adsorption equilibrium, the establishment of the viscous stress is

promoted by surface retardation, and the equation follows

N(ab,0) v0 sin0. (3.28)

Here c(a b ,0) characterises the concentration distribution along the surface. Integrating both

sides of the equation and assuming a linear relation between local adsorption F(0) and c(a b ,0)

we obtain the distribution, C(ab,0), and adsorption, F(0), on a strongly retarded bubble

surface,

F(ab,0 ) - F o + S F - -~rlV o cos0. (3.29)

5c and 5F are unknown quantities. Keeping in mind that a diffusion boundary layer is formed at

Pe~l and that Stokes' velocity field can be used at high retardation, the convective diffusion

equation at Re<( 1 and Pe~ 1 can be transformed into the following form (Levich 1962)

386

Solving equation (3.29) under the condition of constant concentration beyond the diffusion

boundary layer, we can determine c(~,0) and an expression for the complete diffusion flow on

the surface of the sphere can be derived. This flow is zero under stationary conditions and leads

to a condition for determining dF from Eq. (3.27)

x---~0

After c(x,0) and F(x,0) have been determined we can calculate the velocity distribution on a

strongly retarded surface using the boundary condition

div~[r(0)Vo(au,0)] = D ) (3.32) x--,0

According to Eq. (3.32) and the expression for c(x,0), we obtain the following result for the

velocity distribution on a retarded surface,

Vo(a ,O)- (3.33) 1+5r'/r'o +(A / Co) cos0'

mADPe'/3 and A = 3 riVo(~c~ -1 with N = 31/3Fo ~-

If the denominator in Eq. (3.33) is set to 1, F(0) is given by et. sin0, with ot = F(0) / s in (0)

almost invariant according to numerical calculations (Dukhin and Derjaguin 1961) and an

equation for X b is obtained which only differs from Eq. (3.13) by a numerical factor. Hence, if

we introduce an additional condition for a slight variation of the adsorption along the surface,

both the sinusoidal velocity distribution and the relation for the retardation coefficient proposed

by Levich (1962) can be verified.

The described theory of strong retardation of a bubble surface by DAL at small Reynolds

numbers was developed by Dukhin & Derjaguin (1961) and Dukhin & Buikov (1965) and

confirmed by Saville (1973). The balance of Marangoni and viscous stresses, given by

Eq. (3.25) as the basis for the determination of the surface concentration distribution, was used

later in the theory of a stagnant cap (cf. section 3.5). This stress balance is otten characterised

qualitatively by a dimensionless number, the Marangoni number (cf. He et al. 1991),

Ma = RTF o / rlV, (3.34)

387

which characterises the ratio of the surface pressure that surfactant molecules exert under

compression to the viscous forces tending to compress the surfactant layer.

h above which the bubble surface as a whole is strongly A formula for the concentration c ,

retarded

h 3 ~lv C c r = - - ~ (3.35)

2 RTK h

follows from Eq. (3.29).

3.5. THE REAR STAGNANT REGION OF A BUOYANT BUBBLE

At low surfactant concentrations the surface of a bubble is not uniformly stagnant. Usually the

main portion of the surface can be weakly stagnant, and only a narrow truly stagnant region can

exist in the vicinity of the rear bubble pole (Savic 1953, Harper 1982). The existence of the

stagnant cap was confirmed experimentally by Savic (1953), Garner & Scelland (1956), Elsinga

& Banchero (1961), Griffith (1962), Horton et al. (1965), Huang & Kintner (1969) and Beitel

& Hedeger(1971).

It is likely that the unrestricted growth of adsorption at 0 ~ n given by Eqs. (3.16) and (3.22)

in both limiting cases was considered by Derjaguin et al. (1959). The necessity of the growth of

F(0) at 0 ~ n is physically obvious since the adsorption layer is compressed. The increase of

F(0) is naturally restricted by parameters which were not taken into account in deriving

Eq. (3.23).

At 0 --~ n the process of dynamic adsorption layer formation is complicated by three factors: the

expansion of the diffusion boundary layer, the necessity to consider surface diffusion, and the

retardation of the surface taking place at 0--~ n even when the complete remaining surface is

essentially not stagnant. The appearance of all the three factors is qualitatively different at high

and low surface activity, respectively. The special features of the process of adsorption layer

formation at 0 ~ n are considered by Dukhin (1965) for each case separately.

The structure of the strongly stagnant region in the vicinity of the RSP was studied numerically

by Davis & Acrivos (1966). An analytical description of dynamic adsorption layers at bubble

surfaces is proposed by Harper (1973, 1982) which is asymptotically suitable for small

dimensions of the stagnant cap. The mathematical apparatus of the theory is simplified in the

case of small dimensions of the stagnant cap region (Harper 1973, 1982) and the results can be

presented more clearly.

388

It is assumed that the surface of the bubble is completely free at 0 < r t - ~ and completely

stagnant at 0 > 7t - ~ . In addition, it is assumed that in the former region the adsorption is much

lower than the equilibrium value and in the latter much higher than F o. This allows the

assumption to be made that the adsorption of surfactants at the free surface takes place in the

surfactant free region, and desorption of surfactants from the surface to the bulk of the solution

with c o ~ 0 in the stagnant region. The size of the strongly stagnant region is determined by the

balance of the total adsorption flow to the bubble surface and the total desorption flow as

shown in Fig. 3.

TAGNANT CUP

Fig. 3 Stagnant cap model

The total adsorption flow to a weakly stagnant bubble surface was calculated by Levich (1962),

J ad -- 5.79 ~-~ab DCo. (3.36)

The function F(0) and total desorption flow can be calculated by using the distribution of

viscous stresses in the strongly stagnant region near the lower pole (Harper 1973),

RT cqF 0v0] v 4m (3.37) = = T I

ab 80 r i t z z=~[ abz tx /1-m 2'

with m = (r t-0) / 0", s = ( z - a b ) / (ab0*) being inner co-ordinates within the stagnant cap. This

method of estimation of F(0) in the case of a strong stagnation was proposed for the first time

389

by Dukhin & Derjaguin (1961) (cf. section 3.4). On the basis of variable transformations the

equation of convective diffusion in the neighbourhood of the RSP can be reduced to the form,

which can be solved analytically. Thus, the total desorption flow of surfactant from the surface

of the stagnant cap can be calculated (Harper 1973),

Equating the total desorption and adsorption flows for Re((1 results in

F -II/16F RTFo -]3,8 v-,.76 L zj j , <3.3=)

. . . . F a b v 11/16.F RTFo ]5/8

The stagnant cap regime has been investigated for arbitrary cap angles by Holbrook and Levan

(1983). In the arbitrary cap angle studies, the surface tension difference between the rear pole

and the clean interface is related to the cap angle and terminal velocity of the droplet.

The creeping flow past a bubble with a stagnant cap has been investigated by Savic (1953),

Davis & Acrivos (1966) and Harper (1973, 1982), Holbrook & Levan (1983). In each case the

difficulty came about in dealing with the mixed boundary conditions of the stagnant cap. The

formulation led to an infinite set of algebraic equations for the coefficients of a series solution.

Savic (1953) truncated the series after the sixth term, while Davis & Acrivos (1966) used 150

terms. Harper (1973, 1982) studied the case of small cap angles and carried out an asymptotic

analysis using oblate spherical co-ordinates. Sadhal & Johnson (1983) generalized the problem

to include both drops and bubbles by allowing internal circulation within the drop.

As a purely hydrodynamic problem, the velocity field due to a stagnant cap at the rear of a

moving drop was solved exactly in terms of an infinite series of Gegenbauer polynomials with

constants depending on the cap angle ~. From this series, an analytical solution F(~) for the

drag exerted on the drop can be obtained, from which the terminal velocity was computed once

the external force on the drop is resolved,

{ E J ; 2n - - - sln3~ + . F ( ~ ) - 4rmVRau 4~(?+ n') 2~ + sin~ sin2~ 1 . + 311' (3.40)

3 2rl +2r i ' J

For the limiting case of ~=0 (no surfactant), equation (3.40) is transformed into the Hadamard-

Rybczynski equation. For ~=~ (completely stagnant interface), a solid sphere behaviour results.

If the drop viscosity becomes infinitely large (1"!'-->oo), a solid sphere drag is also obtained. The

drag force for the special case of a bubble is easily obtained from (3.40) by letting TI' --~ 0. This

yields

390

F(~q]) bubble - 4rtrlURf~-~ I2W + sin �9 - sin 2W - - l sin 3~1+ 1}. (3.41) 3

The cap angle is determined by the adsorbed amount of surfactant necessary to cause a surface

pressure that balances the compression due to the viscous shear on a cap of angle ~.

Sadhal & Johnson (1983) give a dimensionless equation for the difference in shear stresses

exerted by the surrounding phases on the interface in the cap region

1: 2) - k't 0) ) - h(0, ~ ) /~ (3.42) z0(s) z0(s)

where the tangential stress is scaled by rlV/a b , k is the ratio of the droplet to the continuous

phase viscosity, and h(0, ~) is a very complicated function.

The balance of Marangoni and viscous stresses (3.42), reformulated in terms of F, is integrated

to obtain the surfactant distribution and yields F as a function of ~ and the dimensionless

Marangoni number Ma. The surfactant distribution can be integrated over the cap region to

obtain the total amount on the surface, M. The variable M is also computed independently from

the surfactant conservation equations and equating the two expressions ~. Once ~ is specified,

the drag coefficient and terminal velocity can be calculated.

The above procedure was first introduced by Griffith (1962) whose study is incomplete since he

did not use the proper hydrodynamic solution, and later by Sadhal & Johnson (1983) in their

exact solution of the problem. Each of these authors assumed that the surface pressure exerted

by the compression of the surfactant in the stagnant cap may be represented by a linear

isotherm.

3.6. DYNAMIC ADSORPTION LAYER AT SLOW DESORPTION KINETICS

At slow desorption kinetics the surfactant exchange between the surface and the bulk is

controlled by the surfactant transport from the sublayer to the surface and in opposite direction.

It is not necessary to solve the convective diffusion equation in this case because the transport

across the diffusion layer is more rapid and thus the concentration drop is small across the

diffusion layer. This allows to estimate the surfactant concentration within the sublayer by the

bulk concentration (Dukhin 1965, Dukhin et al. 1995, Chapter 9).

The total adsorption rate at a unit surface is

Jr, -- -P(F) + Q[c(ab,O)]

= -ka~F(| + kadC(a b , 19)(1- F(| / Foo)

= -kae~r(| + k,ac o (1- F(| / Foo)

(3.43)

391

where Fo~ is maximum surface concentration. The theory of dynamic adsorption layer of rising

bubble at weak surface retardation and at surfactant transport controlled by sorption kinetics is

rather simple [Dukhin (1965)]. The more general approach is developed by He et al. (1991).

The relation for the total amount adsorbed is obtained by multiplying Eq. (3.43) by 0 and

integrating from 0 to n. The net diffusive flux is equal to zero,

, - % . 1

f 0 lr 0 ., 00, 0

Multiplying both sides of Eq. (3.44) by /F~o and expressing the flux density by Eq. (3.43), one

obtains

i F(O) s inOdO+ic l l F(O)JsinOdO 0 . . . . . . . .

o Foo o Foo

where

(3.45)

c - k~a Co (3.46) kd~ Fo~

is the so-called non-dimensional bulk concentration,

M 2C -M+2C-CM-0 or ~ = ~ (3.47)

2n I + C

and M = 2n~ ~ F(| sin |174 is the dimensionless total adsorbed amount. J0 Fo~

When the surfactant density becomes large, the finite size of the adsorbed molecules gives rise

to strong repulsions between surfactant molecules and generates surface pressures that vary

non-linearly with the surface concentration. The concentration of surfactant in the stagnant cap

will depend principally on the amount adsorbed and the degree of compression exerted by the

viscous forces. If these forces are large enough, they can compress the surfactant to a density

high enough so that a gaseous expression does not apply.

Since the use of the gaseous constitution equation underestimates the surface pressure, Sadhal

and Johnson's result underestimates the cap angle and consequently the drag coefficient. He et

al. (1991) obtained a more realistic value for the cap angle by allowing for non-linear

interactions. In their study, Frumkin's equation of state (Frumkin and Levich 1947, Chapter 2)

Y o - Y - -RTroo ln(1- (3.48)

392

was used to formulate an expression for a non-linear surface pressure. We shall also adopt this

approach.

From the Frumkin equation (3.48) and the Marangoni stress balance of Eq. (3.42), a differential

equation for the surfactant distribution may be obtained

Ma OF = h(O, ~F)/~,. (3.49)

0- r) 00

where Ma is the Marangoni number defined as Ma = RTF~/(rlV); h is the tangential

component of the gradient of liquid velocity distribution (Sadhal & Johnson, 1983); 9~ is the

drag coefficient. The surfactant distribution is obtained by integrating Eq. (3.49) from an

angular position in the cap to W. From the second integration the total (non-dimensional) amount on the surface, M(q j) is obtained. Thus,

L MaZ, 0

M(~)/2rt - ! sin0{1 - exp[ l a~ ' i h(0" ~)d0 ' l ~ 0 0 (3.51)

In obtaining Eq. (3.50), the condition F(W)= 0 has been used.

Combining equations (3.47) and (3.51) results in the following implicit equation for the cap

angle W"

, ,! { (1 + k) - -2 sin0 1- exp Ma~0 (3.52)

As the inner integral in equation (3.52) can be evaluated analytically, solutions for W as a function of k and Ma may be obtained by fixing Ma and qJ and numerically evaluating the outer integral in order to solve for k.

When the Marangoni number is small, the viscous compression forces far exceed the linear surface pressure, and the surfactant is compressed considerably so that non-linear repulsion provides the added surface pressure necessary to balance the viscous action. As a result of the significant compression, relatively high bulk concentrations of surfactant (compared to the large Marangoni regime) are required to achieve large cap angles. These results are shown in Fig. 4.

393

10 z

10 1

10 0

10 -I

lO-Z

lO-a

lO-,J

lO-S

/

J

, f d

. . , , ,

. . . , - 5

m m -

lO

- I I I I I 0 30 60 90 120 150 180

cap angle

Fig. 4. Cap angle W as function of non dimensional bulk concentration C, calculated for a bubble by Eq. (3.52) for small Marangoni numbers Ma: 0; 0.1; 0.5; 1; see Eq. (3.46); according to He et al. (1991)

For small Ma, we do not expect the linear gaseous equation to give accurate results because of

the strong viscous compression. For this case, k(q j ,Ma) is given by Sadhal and Johnson (1983)

in the form which coincides with the leading term in the expansion of the non-linear result for

large Ma. This equation is plotted as the dashed line in Fig. 4 and it shows that the use of the

linear equation of state allows the monolayer to be more compressible.

10 0 -

10-1

C 10-~

10-3

10" '

lO-S

M a = l O . . , , - . . . e - m

. , . - . , - . . , , . , -

. . , . , - /

. , , . , -

100

/ / 1000 / f - " "

.,,,,._ . . . - - . , , - - - ~ ~ - ! / ~---

I / f / / /

/ , / , , , , ,

0 30 60 90 120 150

cap angle

Fig. 5. Cap angle q~ as a function of non-dimensional bulk concentration C, calculated for a bubble by Eq. (3.52) for large Marangoni numbers Ma: 10; 100; 1000; see Eq. (3.46); according to He et al. (1991)

394

A qualitative distinction results in the case in which Ma is very large (Ma># 10), and therefore

the characteristic linear surface pressure forces are much larger than the compressive viscous

shear forces. As a result of this disparity, the adsorbed monolayer cannot be compressed very

much by the viscous forces, and the cap angle increases dramatically with a small increase in the

bulk concentration. This trend is observed as expected in Fig. 5, which represents the numerical

solution of equation (3.52) for k(~F) with Ma equal to 10, 100, and 1000.

If the adsorbed monolayer cannot be compressed very much use of the Henry type adsorption

isotherm is possible. One concludes that this application is justified for large Marangoni

numbers. At a specified Marangoni number the bubble radius should be chosen such that it 2UHRab 2a~g

satisfies the conditions of low Reynolds number R e - ~ - < 1 V 3V 2 "

For the aqueous medium it means a b < (3v 2 /2g)1/3 _ 510 -5 m, which corresponds to a bubble

rising velocity of approximately v = 810 -3 m / s.

Some data concerning F~o are collected in Dukhin et al. (1995), Fdhila and Duinveld (1996). Its

range is .410 -l~ to 1 1 0 -9 mol/cm 2, corresponding to Marangoni numbers from 420 to 3000.

Thus the analysis by He et al. (1991) is very valuable and in particular it enables us to justify the

applicability of the simple Henry isotherm in the theory of bubble rising in water at low

Reynolds numbers and low surfactant concentration. The new regime of r.s.c, formation

proposed by He et al. (1991) is important for liquids having a viscosity much larger than water.

For water and other low viscosity liquids the definition of Ma given above is not reasonable

because Foo is not relevant for the DAL structure and the bubble surface retardation. However, a

modified Marangoni number Ma" can be introduced

Ma'= RTF o / n v ~ (3.53)

and used for the classification of different DAL states (VHR - Hadamard Rybczynski velocity).

At Ma'> 2 / 3 a bubble surface is retarded as a whole, i.e. (~=~, which follows from the theory

of strong retardation (section 3.4). The smaller Ma' the smaller is ~. Very small Ma"

corresponds to a DAL structure described by the theory given in section 3.3.

3. 7. NEWEST DEVELOPMENT IN THE THEORY OF MARANGONI ~TARDATION OF BUBBLE TERMINAL

VELOCITY

In a recent study by Chen and Stebe (1996), the theory of Marangoni retardation and bubble

terminal velocity was improved by a more exact consideration of the physico-chemistry of the

surfactant in parallel with a more exact solution of the hydrodynamic problem. The terminal

velocity of a bubble rising in a surfactant solution has been studied for two nonlinear adsorption

395

models, the Langmuir isotherm, which is based on a maximum surface concentration a

monolayer can attain, and the Frumkin isotherm, which assumes both monolayer saturation and

additional non-ideal interactions between adsorbed surfactant molecules. There is a number of

other kinetic equations developed in literature which can also be used for this purpose (Baret

1969, Miller and Kretzschmar 1980, Chang and Franses 1992, MacLeod and Radke 1994,

Ravera et al. 1994).

Chen and Stebe (1996) quantified the coupling between the surfactant mass transfer and the

stress balance for the first time without application of the r.s.c, model. The basis is Eq. (3.25),

however, the hydrodynamic field is obtained in a more general form and substituted into

Eq. (3.25). According to Happel and Brenner (1973) the Stokes" equation for steady,

axisymmetric, creeping flow in terms of the stream function can be written

E 2 (E2W ~ - 0 (3.54)

Here the E 2 operator is the axisymmetric stream function operator in spherical coordinates,

0 2 sin0 O( 1 O]

Egor -

The velocity components can be expressed in terms of the stream function:

(3.55)

For the derivation boundary conditions were needed. First the uniform velocity field far from the

droplet requires that the stream function obey

limqX2 ) _ 1 r2 sin2 0 r--+m 2 (3.57)

Moreover, at the droplet surface, the tangential velocity components must be continuous, the

normal velocities at the interface are zero, and the normal stress balance on the interface is

replaced by an integral force balance which requires the integral drag force on the droplet to be

balanced by the gravitational force.

The general form of the solution of Eq. (3.54) in spherical coordinates are found by separation

of variables

oo

Z ( n .n+l ,'~ n+2 r-~ -n+3)Cnl /2 (COS0) W(r,O) - Anrn +l~n r +tcnr +Unr n=O (3.58)

1 OVI/(i) 1 C~VI/(i) V (i) = - V~ 0 =

r r2sinO 03 , rsinO & (3.56)

396

where Cn 1/2 (cos0) is the Gegenbauer polynomial of order n and degree -1/2 (Abramowitz and

Stegun 1970).

Using the above given boundary conditions, the stream function can be rewritten in terms of the

unknown coefficients B n:

oo v (r 2 _ r)sin 2 0 W(r ,0 ) - ~ B n (r "-1 - r -n+3 )C: '/2 (cos0)+~ ,--2 (3.59)

The solution (3.59) leads to the expressions (3.60) and (3.61) for angular dependencies for the

surface velocity and surface concentration.

C~ '~ (cos0) v - v 0 (1, 0) = _1 sin0 - 2~--' B. (3.60)

" 2 n=2 sin0

The surfactant concentration F(0) can be expanded in Legendre polynomials,

oo

F(0) - ' ~ amP m (cos0), (3.61) m=0

where Pm (COS0) is the Legendre polynomial of order m.

A procedure for numerical calculation of the coefficients in Eqs. (3.55) and (3.56) exists and

results are discussed in terms of angular dependencies of surface velocity, surface concentration

and Marangoni stress influenced by different parameters. Among many new results two should

be mentioned here. The proposed mechanisms of the surface remobilization are confirmed. In

parallel with regimes described by the r.s.c, model the obtained regime yields a uniform surface

retardation (cf. section 3.3. and 3.4.)

Attention is paid mainly to the cases of Ma=0.5 and Ma=l. This magnitude of Marangoni

number corresponds to very viscous liquids in comparison to water. Interesting new information

concerning DAL on bubble surfaces rising in water can be obtained by a specification of the

theory of Chen and Stebe to very large Ma and very low surfactant concentration. At lower

surfactant concentration a slower adsorption kinetics results. Both the small bubble dimension

and its surface mobility cause a rather rapid adsorption kinetics even at small surfactant

concentration.

4. DYNAMICS OF REAR STAGNANT CAP GROWTH AND RISING BUBBLE RELAXATION

A rising bubble velocity at the moment of its separation from a capillary tip is zero or at least

very small. The transition to the terminal velocity takes some time which can be called the

relaxation time of bubble rising x. This time is a measurable value and its measurement can yield

397

new information about adsorption dynamics. For this propose a theory of rising bubble

relaxation is necessary. This task is even more complicated than the theory of a steady DAL, the

development of which has taken some decades. The problem is purely hydrodynamic in the

absence of surfactant, and not easy at intermediate and high Reynolds numbers (Chen (1974),

May and Klausner (1992), May et al. (1991,1994), May (1994)).

The problem becomes very simple at small Re when only the drag force and the inertial force

have to be taken into account. In addition the assumption of a quasi steady drag force is

necessary. The Hadamard-Rybczynski theory was proposed for the steady movement of drop

(bubble) with free surface which can be used to describe the drag force at any moment. The

equation expressing the balance of gravity force, the inertial force and the drag force is linear

and of first order if the bubble velocity is taken as an unknown function. Its integration yields a

simple exponential time dependence of the bubble velocity. The characteristic time of this

dependence is called the hydrodynamic relaxation time Zh which equals approximately 1 ms at

Re=l.

In surfactant solutions the non steady surfactant flux to the rising bubble surface overlaps with

its non-steady movement caused by pure hydrodynamic reasons. The coupling of physico-

chemical and hydrodynamic non-steady processes is very difficult to quantify. Fortunately for

the most interesting extreme case of bubble rising a decoupling is possible. This relates to the

case of very clean water with very small addition of surfactant. The lesser the surfactant

concentration the smaller is its flux to the bubble surface. As a result the surfactant

accumulation during the accelerated stage of bubble rising is negligible. Thus the bubble velocity

grows up from the initial very small value to VHR velocity during the accelerated stage of bubble

rising if the surfactant concentration is sufficiently small. The further surfactant accumulation on

the bubble surface will be accompanied by a drag force increase. It causes the decelerated stage

of bubble rising. During the decelerated stage the bubble velocity decreases from VHR to the

values according to Stokes.

The consideration of bubble rising relaxation as a two stage process is justified if the time of the

second stage is very long in comparison with Xh. In addition the inertial forces can be neglected

at very weak decelerations. The deceleration stage is very short and therefore difficult to

measure and to quantify except when the surfactant concentration is small. The transition to the

Stokes regime occurs very rapidly due to fast surfactant adsorption.

During the short deceleration stage the separated bubble is quite close to the capillary tip which

causes additional difficulties for a quantification of the deceleration stage. Thus the theory of a

non-steady DAL is the most important and easier for low surfactant concentrations. The lower

398

the surfactant concentration the smaller is the diffusion flux. As a result the surfactant transport

can be considered as diffusion controlled in quiescent media. The situation can change

qualitatively in the present case of a rising bubble with a mobile surface. The diffusion is a rapid

process due to the small thickness of the diffusion layer which can be expressed by Levich's

equation

ab _ ab ab ~ 2~tm (4.1) m

30

This estimation is made for Re=l and correspondingly Pe = Pr =v/D~103, ab~60gm , and the

viscosity and diffusitivity for water. Using the Einstein equation it is easy to estimate the time 2 / 4D which is comparable to Xh necessary for the steady diffusion layer formation 8 D ,

With decreasing concentration the diffusion flux diminishes. However both the diffusion flux

and the flux from the sublayer to the surface are proportional to the concentration and decrease

in a similar way. Thus the transport can be controlled by adsorption kinetics if it is retarded even

at a low surfactant concentration. For example the adsorption of ionic surfactants can be

strongly retarded due to electrostatic repulsion. Especially in the considered case of very low

concentration the surface potential is high and causes a strong suppression of the adsorption

flux. This suppression is enhanced for two-valent ions (Dukhin et al. 1995, Chapter 7).

The r.s.c. (rear stagnant cap) model elaborated for steady DAL overcomes the serious difficulty

in the theory of non-steady DAL. The mechanism of the formation of a r.s.c, under non-steady

conditions is identical: adsorbed surfactant molecules are fully washed out by the liquid flow

from the top of the rising bubble. Therefore the corresponding shear stresses in the adjacent

fluid becomes zero. At the same time the surfactant is collected in the rear zone of the surface

(r.s.c.) to provide its stagnant state. The r.s.c, model assumes a sharp transition between the two

surface regions discussed above. The sharp transition corresponds to large Pe, because the

surface diffusion cannot compete with the strong convection.

The hydrodynamic field as a function of time can be described by the theory of Sadhal and

Johnson using the Marangoni stress balance and the theory of the surface concentration

distribution by He et al. (1991) if the deceleration time is very long in comparison to the

hydrodynamic relaxation time.

By using such a quasi-stationary approach the angle of the r.s.c, location can be obtained as a function of time and bubble velocity ~g(t, v). Thus there is a complicated drag force dependency

on time and velocity

Far, g : 4Zcrlavf[gt(t, v)] (4.2)

399

4 3 In the creeping flow regime the drag force fully balances the Archimed force F A --~gabog.

Equality of both forces yields the following equation

Vf[~t(t, V)]- 1 (4.3)

A dimensionless velocity can be expressed as follows

v v 3Vrl V . . . . . (4.4)

VHR F A / 4 ~ l a a29g

where VHR is the bubble velocity corresponding to the Hadamard-Rybczynski drag force (gt - 0). Note that during the process under consideration the following inequality holds

2 - - < V < 1 ( 4 . 5 ) 3

The bubble velocity as a function of time V (t) is obtained as the solution of Eq (4.3).

The main aim for predicting ~(t, V ) from Eq. (4.3) is to evaluate the amount of the adsorbed

surfactant in two ways. The first deals with the known shear stresses that produces local

gradients of surface tension. Then one can evaluate the local adsorption and the total amount of adsorbed surfactant as a function of ~ and v(M(~t, V)).

The second way is to predict the total amount of the adsorbed surfactant as a function of �9 and time M(~g, t) by consideration of the adsorption kinetics. The equality of the two amounts yields

~t(t, V) for further substitution into Eq. (4.3). For the determination of the velocity time

dependence it is convenient to rewrite Eq. (3.42)

,-..,

Ma dF ----= . . . . hi0, wjV ' ' (4.6) 1 - F dO k - - ]

where the Marangoni number is attributed to the velocity VHR

M a - RTFoo _ ~3RTFoo (4.7) rlv HR a 2 Pg

For the present consideration the bubble is rising in a liquid of a viscosity similar to water. For

this case and at low bulk surface concentration the Marangoni number is very high (cf. section

3.6.) and F~ 1. After linearisation Eq. (4.6) can be easily solved with the help of the condition

F(tF) - 0 which is inherent in the r.s.c, model.

400

q l

j'h(O',v)do' r ( O ' v ) = ~ Ma V (4.8)

The second integration of this value over the r.s.c, region yields the total amount of the

adsorbed substance

M = 47ta~-----~~ ~-aa 0

The definite double integral on the right hand side of Eq. (4.9) yields

, - . o

1VI(v,V)- v ( 2 V - 4 V c o s v - s i n 2 v + 4 s i n v ) 4nMa

(4.10)

By using (3.41) we can rewrite (4.10) as follows

_ 1 . 2 ~ - 4 ~ c o s ~ - s i n 2 ~ + 4 s i n ~ 1 l~i(V) Ma 2 v + s i n v - s i n 2 v - ~ s i n 3 v + 4 x

(4.11)

The evaluation of a surfactant amount can be obtained from an adsorption kinetics model

OF - - + div~(vor)= - k ~ r + k~ (r~ - r)Co (4.12) &

kad Fo kd~, - (Foo - Fo)Co = Keq (4.13)

where F o is the equilibrium adsorption corresponding to the surfactant bulk concentration c o.

Since c o is constant the integration of both sides of Eq (4.12) over the bubble surface yields a

simple result

dM = 1 , (M ~ _ M) (4.14) dt

with M 0 = 4ha2 F0 ' x = Foo -F0 k~dFo

m

kd~F ~ F~oc

The integration of the surface divergence (second term on the left hand side of Eq. (4.12)) over

the bubble surface is zero. With the help of the initial condition M(0)=0 from Eq. (4.14) we

obtain

401

When t ~ oo the amounts of the surfactant approaches the value M 0 in equilibrium. However

the situation happens far from equilibrium. The local adsorption is either more (r.s.c.) or less

("clean" part of the surface) than the equilibrium value. The equation for this time dependence

follows from the comparison of Eqs (4.11) and (4.14) yielding

[ ( _ t ] ] 2~t- 4gtcosgt- sin2~t+ 4 singt N]_I- exp = 1 (4.16)

4n + 2gt + sin ~ - sin 2~t- ~ sin 3~

where

N - ~ F~ Ma - RTF~ (4.17) Foo T~VHR

The physical reason of cancelling Foo is clear. The adsorbed layer cannot be strongly compressed

at small Re and at low bulk concentration the surface concentration has to be small in comparison to Foo (section 3.6).

3,5 Q

3

2,5

.~. i 2 1,5

1

0,5

i I I i i I

0 1 2 3 4 5 6

t/t

Fig. 6 W as characteristic position of the upper bound of the r.s.c, as a function of dimensionless time t / x for different modified Marangoni numbers Ma': (,)- 7; (I)- 2, (A)- 0.7, (,)- 0.2

The time dependencies of the angle can be used to obtain final velocity (Zholkovski et al. 1997)

1

v - [ ( 4 x 2~1/ ~1/ s in2~ q/) 1 + (418) l ~ + s i n - 1 sin3 1

3

The results of that are presented in Figs. 6 and 7 for various Marangoni numbers.

402

1,1] 1

0,9 ~ . . . ~

0,8-1~ ~ " * , ,--~__, ,

0,7 I ~ ~ ~ _ _ . .

0,6 , , ,

0 2 4 6

Fig. 7 Dimensionless bubble velocity V as a function of dimensionless time t/T for different modified Marangoni numbers Ma': 1- 7; 2- 2, 3- 0.7, 4- 0.2

5. HYDRODYNAMICS OF SPHERICAL PARTICLES AT HIGHER REYNOLDS NUMBERS

There is a large similarity in the surfactant transfer at Re<l and Re>>l if Pe>>l while on the

contrary the bubble hydrodynamics differs significantly. At Re<<l the theory of Sadhal and

Johnson yields a general (with the restriction of r.s.c, model) analytical description of the flow

field. At Re>> 1 an analytical theory exists only in the extreme case of a completely free bubble

surface (Levich 1947). This theory allowed Derjaguin and Dukhin to present an analytical

theory for DAL in the case Re>> 1 and for weak surface retardation. During 5 decades there was

no progress in bubble hydrodynamics. The situation changed recently when Fdhila and

Duineveld (1996) made the first attempt to quantify bubble hydrodynamics. These ideas and

methods elaborated for solid sphere hydrodynamics are the basis for further developments in

bubble hydrodynamics. Thus at first the main results of the solid sphere hydrodynamics are

presented (section 5.1). Afterwards attention is paid to the opposite case of a free bubble

surface (section 5.2).

5.1. DEVELOPMENT OF FLOWFIELD WITH REYNOLDS NUMBER

We follow the classic rigid spheres review concerning this topic (Cliff et al. 1978). In the

absence of analytic results, sources of information include experimental observations, numerical

solutions, and boundary-layer approximations. For numerical solution of the Navier-Stokes and

continuity equations in axisymmetric flow, it is useful to introduce the dimensionless stream

function, ~, and vorticity, 03. The vorticity has one component 03 along the azimutal direction,

which is given in terms of V by

sin0 &~ + T N ~ -r03

The rate of change of 03 in the chosen co-ordinates reads

&0 c3 c3 2(c3[-0r03] 0[- 1 r -~- + -~ (ru103) + ~ (u203) - ~ee ~ -~ ] -~ - J + ~ ] r sin0 - -

(5.1)

&0 sin0 ]~ (5.2) J) O3

403

In these non-dimensional equations velocities are made dimensionless with the terminal rise

velocity U~o, and the radial coordinate r with ab, the radius of the bubbles. This introduces the

scaling factors Uo~ / a b for the vorticity and U~a b for the stream function. This mathematical

formulation is more general than that considered in section 3.7, however the set of boundary

conditions is identical. The results of numerical calculations of the hydrodynamic field is

presented in Fig. 8 (Masliyah 1 970). The steady flow to the rear of a sphere is symmetric in the

limit of zero Reynolds number only. Asymmetry becomes progressively more marked as Re

increases. Streamlines and vorticity contours calculated numerically are shown in Figs. 8 and 9.

Fig. 8 Streamlines for flow past a sphere. Numerical results of Masliyah. Flow from right to left values of ~t' indicated, (a) Re=l.0; (b) Re=10; (c) Re=50; (d) Re=100

Fig. 9 Vorticity contours for flow past a sphere. Numerical results of Masliyah. Flow from right to left. Values of Z indicated. (a) Re=l.0; (b) Re=10; (c) Re=50; (d) Re=100

For Re=l, axisymmetry is most apparent in the vorticity distribution. The surface vorticity has a

maximum forward of the equator, while contours remote from the body (9) show convection of

404

vorticity downstream. For Re=l 0, axisymmetry is also apparent in the streamlines (Fig. 8) while

the position of the maximum surface vorticity has moved further forward.

Flow separation is indicated by a change in the sign of the vorticity and first occurs at the rear

stagnant point. Re=20 is the best estimate for the onset of a recirculation flow regime.

In the region 20<Re<130 steady wakes are obtained. As Re increases beyond 20, the separation

ring moves forward so that the attached recirculating wake widens and lengthens. The outer

streamlines also curve less and the vorticity is convected further downstream. Development of

the wake is evident in photographs of flow past a rigid sphere reproduced by Cliff et al. (1978).

The wake changes from a convex to a concave shape at Re=35. The separation angle, measured

in degrees from the front stagnation point is well approximated by

0 s = 180 - 42.5[In(Re/20)] 0.483 (20<Re<400) (5.3)

Predicted and observed wake lengths and wake volumes agree closely for Re=100. For Re> 100,

the excess pressure over the leading surface of the sphere approaches that for an ideal fluid, but

there is little recovery in the wake. As Re increases, the importance of skin friction decreases

relative to form drag.

In the range 130<Re<400 wake instabilities appear. As Re increases beyond about 130,

diffusion and convection of vorticity no longer keep pace with vorticity generation. Instead,

discrete pockets of vorticity begin to be shed from the wake. The Re value at which vortex

shedding begins is often called the "lower critical Reynolds number", although the transition is

much more gradual than this label would imply. At Re=130, a weak long-period oscillation

appears in the tip of the wake. Its amplitude increases with Re, but the flow behind the attached

wake remains laminar to Re > 200. At about Re=270 a large vorticity, associated with

pulsation's of the fluid circulating in the wake, periodically forms and moves downstream

(Seeley 1972). Later some attempts to quantify the flow pattern around a special particle at

intermediate Re were made (Luttrel et al. 1988;Yoon and Luttrel 1991; Yoon 1991; Nquen and

Kmet 1992). In the latest numerical calculations of steady viscous flow past a sphere at high

Reynolds numbers (Fornsberg 1988) there is no revision of the results concerning the region Re

< 100.

Hamielec et al. (1967) computed drag coefficients and wall effects due to a spherical container

for spherical drops (no slip at the interface) and for spherical bubbles (free slip) respectively.

The accuracy of their predictions of the drag coefficients of sphere in an infinite fluid was

limited due to limitations in the computer capacity. For near creeping flow (Re-0.1), the

405

computed wall effects of a spherical container agreed to within 3% with the analytical creeping

flow solution.

A boundary layer theory has been applied to predict fluid velocities with some success for

Re>3000, but with less success at lower Re. The main difficulties are that the pressure

distribution only follows potential flow up to about 30 ~ from the front stagnation point, the

boundary layer thickness is only small relative to the sphere radius at very high Re, and the

tangential velocity in the boundary layer shows a maximum which is greater than the free stream

value. Although an exact solution is available (Schlichting 1968) using the potential flow

solution as the outer boundary condition, it gives velocity and vorticity distributions which are

only realistic within 20 ~ of the front stagnation point. Separation is predicted at 109.6 ~ which

corresponds to observed separation at Re=400, whereas at very high Re where the boundary

layer theory should be more reliable, separation occurs at -~ 81 o. A more reliable treatment was

given by Tomotika (1935) using Pohlhausen's method (Schlichting 1968) and an experimental

pressure distribution. This approach predicts separation correctly at 81 ~ , but the prediction

velocity distribution is again only accurate over the leading part of the sphere (Seeley et al.

1975).

5. 2. THEORY OF BUBBLE HYDROD YNAMICS AT NEGLIGIBLE SURFACE RETARDATION

The specificity of bubble hydrodynamics at Re))l becomes clearer by a comparison of a

circulating spherical bubble with a rigid sphere. For the latter case, the boundary layer is

perceived as a thin layer at the particle surface where viscous forces play a dominant role and

across which the velocity variation is of the order of v outside this layer. The flow departs little

from the irrotational pattern. For the bubble on the other hand, it is not necessary for the outer

fluid to come to rest at the sphere surface. Here flow diviates significantly less from a rotational

motion. As a first guess it might appear that the potential flow could be a valid solution for the

entire external flow field about a circulating bubble. However, the velocity derivatives in that

case would not satisfy the tangential stress boundary condition. Thus a boundary layer must still

exist on the surface, but it is of a rather different kind from that on a rigid body. In particular,

the velocity variation across the boundary layer is only of the order v / 4~--e. In addition, the

boundary layer is much thinner, and remains attached to the surface longer than on a comparable

rigid body. These features are discussed in detail by Levich (1949), Batchelor (1967) and

Harper (1972).

Another important peculiarity is that no boundary layer separation is predicted when there are

no surface tension gradients along the surface. It means that a skin friction drag is negligible.

406

The form drag is determined by calculating the viscous energy dissipation for the potential flow

past a sphere (Levich 1949). This gives

C D = 48 / Re (5.4)

and a simple expression for the rising velocity is obtained

1 go 2 (5 5) v - - - ~ a b P n

The liquid velocity distribution around a rising bubble can be written in perturbation form

~5(z,0) = ~(0~ (z,0) + ~(1~ (z,0) (5.6)

in which f~176 everywhere within the hydrodynamic boundary layer

6a ~ a b / 4 ~ . Here ~(0~ (z,0) is the velocity distribution for the potential flow. For the first

time v~ 1~ was considered by Levich (1962). The derivation contains some errors which were

later corrected independently by Moore (1965) and Dukhin (1965, cf. Dukhin et al. 1995,

section 8.6.2). On the bubble surface we have

( ~ e ] v sin0~]2 + c~ (5 7) V~I) ~ __ 8 2 1 + cos0

Levich's (1962) analytical theory was confirmed by further numerical calculations.

Ihme et al. (1972) determined the flow pattern and drag coefficients of no-slip spheres in an

unbounded fluid for bubble Reynolds numbers in the range 0.1 < Re < 80. Past the sphere a

stationary annular vortex was observed outside the Stokes range. The computed drag

coefficients agreed with experimental values, and an equation was derived which was

experimentally verified up to Re = l04. Haas et al. (1972) made similar calculations but for

spherical bubbles with free slip at their surface for 1 < Re < 200. In contrast to the no-slip

condition, no recirculation behind the bubble was observed. The computed drag coefficients

agreed with theoretical equations for low and high Reynolds numbers.

Numerical simulations of the flow past spherical bubbles situated on the axis of a cylinder

performed by use of the finite element technique with the penalty function method are

accomplished in recent investigations of Hartholt et al. (1994). No-slip or flee-slip boundary

conditions are prescribed on the sphere and no recirculation behind the bubble is observed at Re

up to 530. It is therefore likely that recirculation behind this free-slip bubbles only develops as

the bubble deforms. In Fig. 2 of the paper by Hartholt et al. (1994) it was shown that the

pressure profile on the bubble surface approaches the symmetrical potential-flow profile which

407

means that the flow approximates potential flow as Re becomes large. The same conclusion can

be drawn from the fact that the computed drag force for the 1.2 mm diameter bubble rising at its

terminal velocity (Re=530) is 88.7% of the force which a Newtonian fluid flowing in a potential

flow pattern around a sphere would exert on the sphere (12toga bV, Levich 1962).

In Levich's theory and in the numerical simulation it was assumed, that the bubble preserves its

sphericity up to Reynolds numbers of 500-700. The deformation of a bubble is investigated

usually in dependence of the dimensionless Weber number

We = 2pV2ab / 7 (5.8)

The higher the capillary pressure 27 / a b the smaller is the deformation; the larger the pressure

of the flow the larger is the deformation. Both these regularities are reflected in the Weber

number.

Moore (1963, 1965) extended Levich's results to higher Reynolds numbers and included the

deformation of the bubble. He calculated the deformation of the oblate spheroidal bubble in an

approximate way by equating the normal force balance only at a few points on the bubble

surface. E1 Sawi (1974) and Benjamin (1987) satisfied this boundary condition on the complete

bubble surface. Miksis, Vanden-Broeck & Keller (1981) calculated the bubble shape by a

numerical method and allowed for a deviation from the oblate spheroidal shape. In these three

articles the potential theory was used, resulting in bubbles with fore-aft symmetry, and no-

steady axisymmetric bubble shape was found for We > 3.2.

Numerical calculations of the drag coefficient and bubble shape as a function of the Weber

number for various Reynolds numbers were published by Ryskin & Leal (1984). For large

Reynolds numbers and even at small Weber numbers fore-aft symmetric no longer holds,

resulting in a somewhat different drag coefficient than given by Moore (1965). Owing to

numerical problems the authors were limited to Re < 200, corresponding to a bubble of

0.45 mm in water.

The influence of bubble deformation on bubble velocity is characterised by the expression for

the drag coefficient as given by Moore (1965),

CD- 48G(z)I1 + H(Z)+ ) - Re R e 1/2 " '"

(5.9)

and valid for high Reynolds numbers. Here G(%) and H(X) are functions of the deformation X

with G(%) given by

4O8

1 X4/3 1) 3/2 (Z2 _ 1),/2 - ( 2 - ~2) secx-' G(X) = -~ (~2 - ()C: s e c x - ' - ( ~ 2 - 1)'/2): (5.10)

and H(X) as follows from Table 1 in the paper by Moore (1965).

The direct measurements of deformation can be compared with the deformation found from the

relation between the Weber number and W(~), where the approximate relation given by Moore

(1965), is used,

W(~)-- 4~-4/3(~ 3 +~--2)[~ 2 sec~-(X 2- 1)1/212(~ 2-1) -3 (5.11)

The deformation of the bubble Z is the ratio between the longer and smaller area. It is

established that at ab = 0.91mm and a Weber number of We=3.3 a path instability occurs, i.e.

the bubble starts to zigzag. This phenomena which is important for larger bubbles is beyond the

scope of this Chapter. The production of hyper-clean water is the prerequisite for investigations

of the surfactant influence on bubble rising (cf. section 6.)

For the following discussion a definition of "large" Reynolds number is needed. Re is called

large, if the hydrodynamic boundary layer concept can be applied, and is called intermediate

when Re> 1 holds. Thus the range of large numbers is extended in comparison to the definition

given in section 2.3.

For intermediate Reynolds numbers the error distribution method is quite efficient. Error

distribution (or Galerkin) methods are based on choosing a polynomial for the stream function

in order to satisfy all the boundary conditions together with an integral form of the Navier-

Stokes equation. Hamielec et al. (1961, 1962, 1963) applied Galerkin's method to fluid spheres

up to Re=500. Since inertia terms for the internal fluid were neglected their solution are

restricted to small Re. For 4<Re<100, the following correlation was suggested for the total

drag:

CD - - 3"05(783k2 + 2142k + 1080) Re_0.74 (5.12) (60 + 29k)(4 + 3k)

where k = rl'/ , i.e. k is very small for bubbles. An equation which gives a good fit to the

numerical predictions of drag on spherical bubbles (cf. Haas et al. 1972) is:

C D = 14.9 Re -~ for k ---> 0 and Re > 2 (5.13)

There is a remarkably good agreement between Eqs. (5.12) and (5.13). When a theoretical

dependence is obtained by qualitatively different mathematical procedures the solution must be

reliable. C D represented in Fig. 1 0 have been calculated by using different theories.

4 0 9

100 -

1 0 -

CD

_

0~1 -

~ s' Law

Eq. 5 . 1 ~ ~

- Eq. 5 . r ' 3 ~ �9 -~

0 ,01 I I I

1 10 Re 100 1 0 0 0

Fig. 10 Drag coefficients for bubbles in pure systems; predictions of numerical, Galerkin, and boundary layer theories compared with selected experimental results; O Hamielec et al. 1967, A Brabston and Keller 1975, i Bryn 1949, �9 Haberman and Morton 1953, �9 Redfield and Houghton 1965; according to Clift et al. (1978)

The good agreement between the different equations allows the characterisation of C D by a

single curve. The larger the value of Re the better is the agreement between experimental points

and this theoretical curve, and only for Re<20 do the experimental points deviate from the

theoretical curve. This deviation is most likely caused by water impurities. Note that there is a

big similarity with Okazaki's results (section 2.3).

5.3. EXPERIMENTAL INVESTIGATION OF BUBBLE RISING AND DEFORMATION IN "HYPER-CLEAN"

WATER

The influence of water purity on bubble buoyancy can be seen from the family of curves in

Fig. 71 of Levich's monograph (1962). In the experiment of Goretskaya (1949) bidestilled

water was used and the bubble velocity noticeably exceeded that in higher contaminated water.

Nevertheless the predicted velocity exceeded the measured values by 30%. Levich explains that

the measured bubble velocity is decreased due to trace impurities and due to bubble

deformation.

Kok (1993) elaborated a special system for_ bubble formation and reports a rising velocity of

0.236ms -1 for a bubble of radius ab=0.5mm. Duineveld (1995) used Kok's method of bubble

410

generation. Additionally, he produced "hyper-clean" water with a "Millipore" purification

system. He claims that the velocity of small bubbles is significantly larger then in previous

experiments. Another peculiarity of his results is that no deceleration stage in bubble buoyancy

was observed. The latter is caused by the gradual accumulation of impurities. The centre and the

shape of the bubble projection were determined with image sot~ware routines. The measurement

of bubble velocity and its shape enables the comparison of the experimental data with theories of

Moore and Ryskin & Leal.

40 T 35 -I-

3O r ~

25

20

15

10

0,3 0,5 0,7 0,9 1,1

ab (mm)

Fig. 11 Vertical rise velocity of a bubble in pure water; (11) Moore (1965), (0) Duineveld (1995)

As it seen from Fig. 11 for small bubbles there is a perfect agreement with theory. For the larger

bubbles there is an increasing deviation from theory, which is due to an overestimation of the

deformation in the theories mentioned. Measurements of the shapes of the larger bubbles show

that the bubble no longer exhibits fore-aft symmetry, contrary to what is assumed in the

theories. Duineveld stresses that the neglecting of bubble shape deviation from fore-aft

symmetry leads to an overestimation of the deformation influence on bubble velocity at its larger

dimension. The calculations by Ryskin & Leal (1984) include drag coefficients for bubbles with

Re=100 and 200, as a function of the Weber number. From their results for a bubble with

ab=0.36 mm the rise velocity is 5.7 +0.2cms -1 which is in good agreement with the

experimental velocity of 5.3 + 0.2cms -1 .

411

6. DYNAMIC ADSORPTION LAYER AND SURFACTANT INFLUENCE ON BUBBLE BUOYANCY AT

HIGHER REYNOLDS NUMBERS

6.1. D A L AT A WEAKLY RETARDED BUBBLE SURFACE

To derive a convective diffusion equation, it is important to simplify the expression for vo(0,x )

inside the boundary layer

~o(O,O) vo(O,x ) - v o sinO + ~ x (6.1)

Taking into account Eq. (5.6) and Re>> 1 yields

8v~ _ v~ _ v~176 v~176 = v--~~ sin 0 (6.2)

8 X a b a b a b

According to (6.1) and (6.2), v0 (0, x) changes only slightly within the boundary layer and has

the same angular dependence as for Re<l; Pe>> 1, so that the convective diffusion equation and

the boundary condition (3.15) at Re<<l and Pe>> 1 as well as at Re>> 1 are identical. This implies

that the adsorption fields given by Eqs (3.16) and (3.22) are also valid for Re>> 1.

6.2. CONDITIONS OF REALISATION OF DIFFERENT STATES OF DYNAMIC ADSORPTION LAYER

FORMATION FOR A BUOYANT BUBBLE

The three stages or types of a dynamic adsorption layer have been discussed in section 2.5.

Evaluations are possible by using the concept of hydrodynamic and diffusion boundary layers,

having a thickness 8 s and 8 D respectively, and are independent of angle 0. The evaluations are

simplified under the assumption of either a strong retardation of the surface (section 3.4),

8v

6z z=a b

v ~ (6.3) 8~

or a weak retardation

div~[V(O)vo (0)] ~ F(O)vo (6.4) ab

At first conditions are considered under which the surfactant adsorption at the bubble surface is

strongly retarded,

IF(0)- Fo [((Fo, v o/v<< (6.5)

An estimation of the right-hand side of Eq. (3.27) using Eq. (6.3) results in

412

lqv a b Ir(o)- ol/ o <<1 (6.6) RTF o 6~

which allows an approximation of the bubble surface velocity by conditions (2.10) and (2.11).

[C(ab)-- C01 {F-F0I c o F0v D .~ D ~ - - ~ (6.7)

6 D 6D F0 ab

5 g = a b / P e v3 if the bubble surface is almost completely retarded and 5 D - r o / P e 1/3 in the

opposite case of an almost free bubble surface (Levich 1962). From (6.6) and (6.7) the second

necessary condition of the considered state can be obtained,

v---P-~ ~ ~lDc~ ab ab <<1 (6.8) V RTFo 2 6 D 5~

and the following estimate for X b at Re))1 can be obtained (Dukhin & Derjaguin 1961),

Xb ~ RTF~ 6D 6~ (6.9)

Dco a b ab

New conditions for the formation of the second state of the dynamic adsorption layer formation

of nonionic surfactants is formulated under conditions where the surface concentration slightly

deviates from the equilibrium state, Fo, and the bubble surface is weakly retarded,

[r(0)- r0]/r0.1 at v 0 / v ~ 1 (6.10)

The conditions of a slight deviation of an adsorption layer from equilibrium at Re>> 1 are derived

in the same way as at Re<<l which yields Eq. (3.18). The second necessary condition is that the

viscous stresses on the surface of a rising bubble must be much smaller than the characteristic

value of a strong surface retardation.

v~ v ~ 1 R T f F v (6.11) TI 6z a b - a b -~- <<q 6 G

z = a b

After rearrangements we get

riDc0 a b ab << (6.12) RTF0 2 5 D 5~

Note that the left-hand sides of the inequalities (6.8) and (6.12) are identical which indicates that

the regions where the first and second conditions of DAL formation are valid do not overlap.

413

Now conditions of formation of the third state of dynamic adsorption layers is considered. In

this case the adsorbed surfactant is almost completely dislocated to the lower pole and the main

part of the bubble surface (except for a narrow rear zone) is weakly retarded,

I t (0)- r01/r0 1 at v 0/v<< (6.13)

The derivation of the condition of strong variation of surface concentration along the bubble

surface at Re>> 1 is carried out in the same way as at Re<<l and leads to condition (3.19). The

second necessary condition is (6.11) which can be rewritten in the form,

TIV a b - - > > (6.14)

RT[F(~)- r(o)] 5~

Comparing condition (3.18) with Eq. (3.19) and Eq. (6.13) with (6.6) it becomes clear that

regions of realisation of the different states do not overlap. These conditions are presented in

Fig. 12 in a graphic form.

I 0 - I

K. H IO .2

[era]

10-1

i0-~

i0-~

c "~\~ \

3

I l u l l I I I I I I I I l | I ! I I I I I 1 1 ! I I ! 1 1 1 1 1 I I I I I l l l [ I

I O -a I O-e I O "~' I O '5 I 0 -s

Co[molll]

Fig. 12 Conditions of realisation of different states of dynamic adsorption layer formation of nonionic surfactant. Estimates are given for bubbles with a radius of 0.05 cm. The regions of parameters c o and Fo/C o are as follows: A - slight deviation of surface concentration from equilibrium and strong surface retardation B - slight deviation of surface concentration from equilibrium and weak surface retardation C - almost a complete displacement of adsorbed surfactant to the rear stagnation pole and a weak retardation of the main part of the surface

The regions of the three states of dynamic adsorption layer are separated by curves 1, 2 and 3

which are given by the following respective equations:

414

TIV a b

RTF o 6 =1 (6.15)

1"1 =1 (6.16) Xb

F0

C05D = 1 (6.17)

Regions A, B and C are separated from each other not by lines but rather by wide bands as there

is a change from %" into the condition ".", and vice versa.

6. 3. STAGNANT "RING" MODEL

Andrews and Wong (1995) showed that the "stagnant-cap" model is a consequence of creeping

flow hydrodynamics and that at higher Reynolds numbers, it must be replaced by a "stagnant-

ring" model in which the adsorbed surfactant accumulates around the flow separation point and

not around the rear stagnant. In the upper boundary-layer region the situation is similar to that

at small Re. The derivative of the velocity v' is positive and the resulting shear stress pushes the

interface down around the bubble. F is increasing according to Eq. (3.25) and the interfacial

tension y decreases around the interface. However, in the wake region V" is negative, which

tends to push the interface and its adsorbed surfactant, back up around the bubble. Eq. (3.25)

shows that F is decreasing and ~, increasing around the bubble in this region. The adsorbed

surfactant does not accumulated around the rear stagnation point, where by definition V'=0, and

F reaches a maximum.

The symmetry requires dF/dO = 0 at the rear stagnation point. This does not correspond to a

maximum adsorbed concentration, as in the creeping flow case, but a minimum. Shear from the

recirculation wake is continuously creating a fresh interface at the rear stagnation point. The

adsorbed surfactant concentration at this interface may be large, particularly when the surfactant

concentration in the wake exceeds its CMC (Stebe and Maldarelli, 1994). However, it must be

less than the adsorbed concentration at the flow separation point. At the flow separation point

dF / dO = 0 and V'=0 from Eq. (3.25). This means that there is no net transfer of surfactant to

the interface in the boundary-layer region and the adsorption of surfactant around the front

stagnation point is exactly balanced by desorption from the region further down the interface but

above the flow separation point. The same balance between surfactant adsorption and

desorption must also hold for the wake region.

415

6. 4. FINITE DIFFERENCE SOLUTION OF FULL NAVIER=STOKES EQUATIONS FOR MIXED BOUNDARY

CONDITIONS OF STAGNANT CAP

The problem is an analogue of the investigation by Sadhal and Johnson (section 3.6) for the case

of large Re. For a long period of time it was the main unsolved problem of bubble

hydrodynamics at large Re.

Fdhila and Duineveld (1996) elaborated a numerical algorithm for the problem for

50 < Re < 200. A spherical bubble shape is assumed which causes the restriction Re <200. In

contrast to Andrews and Wong (1995) Fdhila and Duineveld claim that the boundary layer

approximation cannot lead to an analytical solution which is valid for the entire flow field and

particularly the wake. Thus they use the vorticity-stream function formulation in spherical polar

co-ordinates (r, e) where r is the non-dimensional radial distance and e is the polar angle

starting from the front stagnation point. They also use a finite difference technique for solving

two coupled non-linear difference equations, avoiding linearising and decoupling assumptions.

The technique is specified by use of an alternating directions implicit method.

Similar to the case of the flow past solid sphere (section 5.1), the potential flow around a sphere

provides the external flow boundary conditions for the stream function, the vorticity and the

velocities.

On the contrary, mixed boundary conditions are formulated for the bubble surface. The shear

stress is zero on the clean upper part and no-slip condition is fulfilled on the contaminated lower

part

82~/1 --" 2 8r I,_-~ for 0 < 0 < u - O 81.2 r= 1 8 ~ t l

(6.18)

8T5~ ] - 0 f o r ~ : - ~ < 0 _ < ~ : (6.19) o i l r=l

The corresponding boundary values for the vorticity are obtained by using a polynomial fitting

of the stream function in the vicinity of the interface. In the contaminated area the fitting

satisfies the no-slip conditions

2 8~ co ]r--~ = sin----O 8--~ ]r--~ for 0 _< 0 < n - ~ (6.20)

_ 1 82~ for zc-~ < e < ~ (6 21) c o [ : : , - s inO ~Sr -~ I:=a - -

416

The elaborated numerical algorithm is verified by comparison with known theoretical results.

The analytical solutions for the Stokes and potential flow are obtained for very low Re and

Re=200, respectively. Also, comparison w;,th some models for the drag coefficient and the

separation angle gave satisfactory results. Calculations for the drag coefficient are in accordance

with Ryskin & Leal and Moore for spherical bubbles. It is unclear why a comparison with the

equation of Sadhal and Johnson is not made because their theory also corresponds to the mixed

boundary conditions.

6.5. COUPLING TRANSFER MOMENTUM AND TRANSFER OF SURFAC TANT AT LARGE Re

Elaboration of a numerical algorithm for the transfer of momentum (cf. section 6.4) creates the

prerequisite for the extension of the stagnant zone model to large Re. Fdhila and Duineveld

(1996) united their numerical algorithm with the model of He and Maldarelli for surfactant

kinetic controlled adsorption (section 3.6). The Eqs. (3.49) to (3.52) of the hydrodynamic flow specificity manifest itself in the function h(0, ~) for the case of creeping flow using the theory

of Sadhal and Johnson. Based on the numerical algorithm developed by Fdhila and Duineveld,

Eq. (3.52) can be extended to Re in the range 50 < Re < 200.

(1~ F(0) - 1 - exp - Ma ~'~-* co d (6.22)

C 1

- -.F 0F(0) sin0d0 (6.23) C+I 2 -

Through (6.22) this equation depends on the interfacial vorticity and the outer flow. After

simplifications are introduced by the assumption of a cap-angle, the flow problem depends only

on three non-dimensional parameters: the Reynolds number Re, the Marangoni number Ma and

the dimensionless bulk concentration C.

6. 6. DAL STRUCTURE AND SURFACE RETARDATIONAT LARGE Re

Numerical results obtained by Fdhila and Duineveld allow the establishment of the main features

of the DAL structure and surface retardation at 50 < Re < 200. When surfactant molecules

occupy the rear region of the surface (up to ~=45~ a sudden increase of the vorticity results at

the leading edge of the layer at which the no-slip condition is satisfied. Until this stage the drag

is not modified considerably by the increase of shear stress, and a clean behaviour results (of.

Fig. 13).

6 0 -

5 0 -

4 0 -

Oint 3 0 -

2 0 -

1 0 -

I 2

0 , I

0 50 100 150 200

0(~

417

Fig. 13 The interfacial vorticity distribution ~mt (0) at Re=200; 1 - q~=0 ~ 2 - q~=45 ~ 3 - q~=90 ~ 4 - qJ=135 ~ 5 - q~= 180~ according to Fdhila and Duineveld (1995)

Spreading of surfactant over the bubble cap from W=45 ~ to W=90 ~ is accompanied by a huge

increase in the vorticity at the leading edge. This reaches its maximum value at the equator and

contributes significantly to a very high drag. In this angle range, a reduction in the rising velocity

occurs and eventually reaches the behaviour of solid spheres. For cap-angles varying from 90 ~

to 180 ~ the drag coefficient is not significantly affected because the peak of vorticity gets

steeper, its distribution approaches the behaviour of a full rigid surface, and the existence of a

stagnant cap does not influence the drag coefficient considerably.

,4 --

1,2 -

1

0,8

0,6

0,4

0,2

0

2

50 0(~ 100 150

Fig. 14 The inteffacial velocity distribution V mt ( 0 ) at Re=200; 1 - q~=0 ~ 2 - q~=45 ~ 3 - q~=90 ~

4 - q~=135~ according to Fdhila and Duineveld (1995)

418

The tangential velocity of clean bubbles is positive and reaches a maximum near the equator.

When part of the bubble interface is contaminated by surface active molecules, the velocity at

the clean part is nearly unchanged, while it slows down very sharply at the leading edge of the

stagnant cap (Fig. 14).

0,25

0,2

0,15

> 0,1

0,05

0 o

Fig. 15

| i |

50 ~ [~ lOO 150

The bubble rise velocity versus the cap-angle; ab = 0.3 mm (1), ab = 0.4 nun (2); according to Fdhila and Duineveld (1995)

200 180 160 140 120

~.: 100 8O 60 40 20

0 -6 -4 -2 0

log C

Fig. 16 The cap-angle versus the non-dimensional concentration in the bulk; ab = 0.3 mm (1), ab = 0.4 mm (2); according to Fdhila and Duineveld (1995)

This picture is consistent with the vorticity profiles in both the rigid and the free surface. The

numerical results presented in Fig. 15 show the change in the bubble rise velocity versus the

cap-angle for two bubbles sizes, 0.3 and 0.4 mm equivalent radius, in a quiescent surfactant

solution. Note that 70% of the total decrease of the velocity is performed for angles between

419

45 ~ and 90 ~ In Fig.16 the cap-angle is given as a function of the dimensionless bulk

concentration C.

At low concentrations, low stagnant caps are formed up to an angle of 45 ~ Above this value a

small increase in surfactant bulk concentration results in a high increase in the cap-angle until

the entire bubble surface is covered. The results show that the critical concentration for these

bubbles is reached at C~1. For similar Marangoni numbers the Stokes flow gives critical

concentrations C which are two orders of magnitude smaller.

6. 7. THE EFFECT OF SURFACTANT ON THE RISE OFA SPHERICAL BUBBLE AT HIGH R e

In addition to numerical simulation (cf. sections 6.5 and 6.6) the rise velocities as a function of

the concentration in the bulk are measured by Fdhila and Duineveld (1996) using three

surfactants, Triton X-100, Brij 30, and SDS for different bubble sizes, between 0.4mm and lmm

equivalent radius. For the sake of simplicity only two curves are presented in Fig. 17.

0.4

0,3

m "~ 0,2 b.. . . >

0,1

0,35

0,3

0,25

0,2 E

0,15

0,1

0,05

0 1,00E-04

0 , , ! !

1,00E-07 1,00E-05 1,00E-03 1,00E-02 1,00E+O0 C [mol/rrff]

C [mol/m 3]

(a) (b)

Fig. 17 The bubble rise velocity versus the surfactant concentration in the bulk. (a) Triton X-100, (b) Brij 30 and SDS; 1 - ab=l.0 mm, 2 - ab=0.7 mm, 3 - ab=0.5 mm, 4 - ab=0.4 mm; according to Fdhila and Duineveld (1995)

The use of hyper-clean water is an advantage in these measurements. Initial uncontrolled

amounts of impurities can affect the concentration dependence of the bubble rise and can

decrease the reliability of interpretation. For all bubbles a critical concentration range appears.

At this concentration range the bubble rise velocity deviates from that of pure water (W-0) and

approaches the value of a solid sphere (~-z 0. Any further increase in the surfactant

420

concentration does not affect the rise velocity. The steepness of the velocity decrease increases

with increasing bubble size. The critical concentration range and the critical concentration at

which the rise velocity reaches the solid sphere velocity depends on the respective bubble radius.

This is evident because the larger the viscous stress (larger bubble has a larger velocity) the

larger the Marangoni stress, i.e. the higher is the surface concentration.

Beyond the critical concentration the bubble reaches a constant terminal velocity independent of

the surfactant concentration. The critical concentration is a specific property of the surfactant.

Above this concentration the hydrodynamic regime of strong retardation is realised which is not

sensitive to the surfactant nature (cf. section 6.2), and also the drag coefficient must coincide

with that of a solid sphere. This has been observed in the experiments discussed above.

Specific surfactant properties, F~o and kd~ / kad, need to be known from experiment in order to

verify model results. These constants are obtained from the equilibrium surface tension isotherm

and are used in the calculation of the concentration dependence of the rise velocity (Fig. 18).

0,25

�9 �9 41. �9 �9 0,2 �9

0,1 , A

0,05 0 0 0 0 n 1,00E-08 1,00E-07 1,00E-06 1,00E-05 1,00E-04 1,00E-03

c[mol/m']

0,15 , . . . . ,

Fig. 18 The rise velocity of a bubble with an equivalent radius ab=0.4 mm, versus the Triton X-100 surfactant concentration in the bulk; solid line - calculated, symbols - measured; according to Fdhila and Duineveld (1995)

Excellent agreement between the experimental and numerical critical concentrations is achieved

for Triton X-100, while for SDS a small difference is observed which according to the authors (Fdhila and Duineveld 1996) is probably caused by an uncertainty in the determination of Foo

and a L. The distribution of the non-dimensional surface concentration C along the bubble for

different cap-angles is plotted in Fig. 19.

0,5

0,45

0,4

0,35

0,3

F 0,25

0,2

0,15

0,1

0,05

0 I

0 50 200

4

100 150

~[o1

421

Fig. 19 SDS concentration distribution on the bubble surface for a bubble with an equivalent radius of a b =0.4 mm; according to Fdhila and Duineveld (1995)

The surfactant concentration gradient reaches its highest value at the leading edge of the

stagnant cap when it is about 90 ~ , and decreases rapidly for higher bulk concentration. These

effects are caused by the vorticity distribution. In accordance with the qualitative analysis of

Andrews and Wong (1995, cf. section 6.3), the maximum surface concentration is not at the

rear stagnation point but occurs at the flow separation point. However the term stagnant ring

proposed by these authors does not agree with the surface concentration distribution in a

stagnant zone. There is no ring but rather an almost homogenous distribution. Fdhila and

Duineveld assumed that the surfactant exchange between the bulk and the interface is mainly

controlled by the sorption kinetics process. This asymptotic behaviour corresponds to Bi

tending rapidly to zero while ~) tends to infinity more slowly. Bi is the Biot number expressing

the ratio of surfactant desorption to the rate of surface convection given by:

B i - kd~ (6.24) Vooab

Therefore a value of the desorption constant kd= is necessary. This constant can be determined

from dynamic surface tension data, for example from Li et al. (1990). For a bubble of

a b = 500~tm gives r and a lower bound for Bi=6.6104 and Bir ~ 0.05 (~) is defined by

Eq. (3.20)). The uncertainty in k d= contributes mostly to the limitation of the assumptions.

The desorption rate constant k 0= can be probably an order of magnitude larger, hence the

stagnant cap assumption, Bi<< 1, may not be satisfied. The concept proposed recently by Lin et al.

(1994) assumes that surfactants with strong cohesion can adsorb diffusion controlled while a

mixed mechanism controls the desorption. This concept is not easy to verify, especially not with

422

rising bubbles. However, a surfactant transport with diffusion controlled adsorption and kinetic

controlled desorption is simpler and sounds realistic.

Additional information useful for understanding the reliability of the used model of surfactant

transport can be obtained from a primitive evaluation of bubble retardation if the surfactant

transport process as a whole is diffusion controlled. For this purpose the Henry constants must

be evaluated using reasonable data for Foo and a L. It yields values of the Henry constant

K H = 5 * 10 -5 m for Triton X-100 and K H = 2 * 10 -6 m for SDS. The critical concentration for

retardation for Triton X-100 can be evaluated from Fig. 17 as c = 10 -8 mol / cm 3 compared with

c = 5 10 -l~ mol / cm 3 obtained from Fig. 12.

The retarded desorption enhances the stagnant cap extension and bubble retardation. It may

explain why the critical concentration for kinetic controlled surfactant transport is so small in

comparison with that for diffusion controlled transport. On the another hand the critical

concentration may be underestimated as characterised by Fig. 12.

The use of a gas-like equation of state underestimates the surface pressure, the cap angle and

the drag coefficient. The form drag coefficient is also not taken into account. In distinction from

the case of small Re (section 3.6) the non-linear decrease in surface tension can be important at

large Re. The dimensionless F cannot be small according to Fig. 19.

6. 8. INVESTIGATION OF MICROFLOTATION KINETICS AS A METHOD OF DAL STUDIES

In physico-chemical hydrodynamics of rising bubbles, much attention is paid to the influence of

the surfactant concentration on the bubble velocity. Meanwhile the theory of DAL has an

independent significance, as it was underlined in section 2.4. Particles mainly collide with the

upper surface of a rising bubble. Thus the DAL structure in a broad vicinity of the front stagnant

point is important. As it is seen from Figs. 15 and 16, information about this section of the DAL

cannot be obtained from bubble velocity measurements, as it is not sensitive to the stagnant cap

extension of the cap-angle into the range 120<~<180.

It was shown that the DAL structure determining the degree of retardation of surface motion

has a strong effect on the deposition of small particles on a bubble surface. In this case it is

important that Reay & Ratcliff (1973, 1975) have demonstrated the possibility of the

experimental determination of the collision efficiency with rather high accuracy. Thus,

investigations of deposition of small spherical particles on buoyant bubbles can be an efficient

method for the experimental verification of' theories of the DAL, its retardation effect of the

bubble surface, and the hydrodynamic field a bubble changing under the effect of DAL.

Emphasised by Dukhin et al. (1995) it was supported by the results of Figs. 15 and 16.

423

As a first approximation the investigation of processes in the vicinity of the front stagnant point

is possible without exact information about processes in the wake region. This was postulated in

a recent paper by Andrews and Wong (1995) who underlined the importance of this problem for

flotation.

6. 9. BOUNDARY LAYER SOLUTION FOR DAL AND FLOW OVER UPPER SURFACE OF RISING BUBBLE

As mentioned in Section 5.1 the boundary layer theory as a whole is not sufficient to describe

the flow around a bubble. Meanwhile some results concerning the DAL and the flow in the

vicinity of the front stagnant point have been obtained which can be of importance for a

quantification of the particle transport to bubble surfaces. The theory of Andrews and Wong

(1995) is based on Eqs. (2.10) and (3.25) and an integral boundary theory for surfactant and

momentum transport. Eq. (2.10) can be simplified after its transformation according to the

integral boundary theory for the concentration boundary layer. The concentration profile in the

boundary layer can be approximated by a fourth-order polynomial which satisfies the known

limits in the sublayer and at the boundary with bulk. Two unknown parameters exist, the

diffusion layer thickness and the concentration profile shape factor. The velocity profile in the

boundary layer is given by the conventional Pohlhausen fourth-order polynomial, modified to

allow for motion of the interface. Again two unknown parameters are included, the

hydrodynamic boundary layer thickness and the velocity profile shape factor.

The complete model consists of four non-linear differential equations for the adsorbed surfactant

concentration. These equations are solved simultaneously by Andrews and Wong (1995) via five

highly non-linear algebraic equations. The limiting cases correspond to a fluid sphere and a solid

sphere, respectively. As expected the boundary layers are thinner on a bubble with a mobile

interface and the momentum thickness tends to zero for a completely fluid sphere. Also, on the

fluid sphere, a fresh interface is continuously created at the front stagnation point, hence less

surfactant can be accumulated there. If the surfactant concentration decreases the model should

approach the fluid-sphere velocity profile. A feature of this profile is that the interface velocity is

larger than the velocity at the outer edge of the boundary layer. Further development seems

possible because the results discussed are encouraging and relevant for flotation.

6.10. EXPERIMENTAL INVESTIGATION OF RISING BUBBLE RELAXATION

The terminal velocity of a single bubble rising in a column of 4 m length of pure water and

surfactant solutions was the subject of a recent study by Sam et al. (1996). They showed that

the axial rising velocity varies with height after release. Typical velocity profiles in water (tap or

distilled) and a surfactant solution are shown in Fig. 20.

424

35 ~ o F o o o

OOo o

2s

T 2o

15

10 0 100 200 300 400

h [cm]

Fig. 20 Velocity profile in water alone (O) and frother (A) (DF 250) solution showing maximum, deceleration stage and, for the frother solution, a constant velocity stage

Initially the bubble accelerates and then reaches a maximum velocity a~er less than 0.5s. After

this, the bubble starts to decelerate because of adsorption of surface active molecules (impurities

in the case of water or surfactants in case of the solution). The maximum velocities for bubbles

with 0.15cm diameter are compared with Duineveld's experimental data. In the experiments by

Zhang et al. (1996) and Sam et al. (1996) the velocity is 1-2crn/s higher which is difficult to

understand. Duineveld claims the absence of deceleration due to impurities in his measurements

because of the high purity of his water. As it is seen from Fig. 20, the velocity deceleration is

very slow even in tap water, and almost invisible at a distance of 50cm, which is the height of

Duineveld's set-up. However, for smaller bubbles deceleration should be more rapid and 50cm

would be sufficiently long enough to observe the deceleration in Duineveld's experiments.

According to both experiments by Sam et al. and Zhang et al. with increasing surfactant

concentration the time to reach constant velocity (adsorption time) decreases but the steady

velocity is not significantly affected. As it was emphasised above the terminal velocity at

considerable surfactant concentrations corresponds to maximal bubble retardation, i.e., to the

velocity of a solid sphere identical for solutions of different surfactants. The higher the

surfactant concentration the more rapid is the adsorption and hence the transition to a terminal

velocity.

The theory of rising bubble relaxation for the case of small Re was described in Section 4 but it

does not exist for large Re. Unfortunately the opposite situation exists with respect to

experiments. It does not exist for small Re but some groups started experimental investigations

with larger bubbles. Aybers & Tapuccu (1969) measured rise velocities for different bubble sizes

as a function of the distance travelled. Their results show a maximum rise velocity and aider that

a decrease. This decrease was relatively large for small bubbles and small for large bubbles,

425

clearly indicating the existence of surface-active impurities. Suzin and Ross (1984) have also

observed the height dependence of bubble rising velocity. In the experiment by Loglio et al.

(1989) the dependence of the rising velocity of bubbles of 2-3 mm in diameter in aqueous

surfactant solutions was studied. For this purpose the rising velocity was measured at different

heights inside a Pyrex glass column 140 cm high. As one would expect, the velocity decreases

since the amount of adsorbed surfactant is greater the longer the path length. For the same

reason the velocity decreases with increasing surfactant concentration. These results are given in

Fig. 21 as a function of the ratio between rise time in solution of a given concentration and rise

time in pure water.

1,2

1,15

O ~, 1,1

1,05

1-~ 0

| i |

50 100 150 h [cm]

Fig. 21 Variation of the rising time ratio as a function of the rising distance for aqueous solutions of decyl dimethyl phosphine oxide at two concentrations: 2 mg/1 (~) and 4 mg/1 (11), bubble radius ab=l.8 mm

One can conclude that a path of the order of several meters is required for the stationary state of

the adsorption layer on bubbles of that size. This was confirmed by the recent experiment of

Sam et al. (1996) and Zhang et al. (1996). During the adsorption process the bubble surface is

partially free because a maximum retardation corresponds to the terminal velocity. This

conclusion is important for flotation kinetics because the surface retardation suppresses the

particle flux to the bubble surface during flotation.

Measurements of clean spherical bubble trajectories rising in a stagnant liquid have been

accomplished by Park et al. (1995). The terminal Reynolds number and Weber number range

from 13 to 21 and 0.03 to 0.69, respectively. The authors emphasise that good agreement

between measured and calculated bubble velocities have been obtained, hence the bubble surface

was mobile. However, they do not comment on a possible deceleration stage but discuss only

426

the acceleration stage. The results are valuable to any case because the bubble surface is cleaner

during the acceleration stage.

The investigation of both deceleration and acceleration stages is complicated by the

manifestation of the history force. In Mei et al. (1994), an expression for the total force on a

clean bubble executing a rectilinear motion was derived and is symbolically given as

dv 4 7Za3b9 , - FB~ + FQs(t ) + FH(t ) + FAM(t ) + FFs(t ) (6.25)

The left-hand side is the inertial force of the bubble which is negligibly small owing to a small

bubble density 9"- The terms on the right-hand-side are the buoyancy force FB~, quasi-steady force FQs(t ), history force FH(t ), added-mass force F ~ ( t ) , and the force due to free-stream

acceleration FFs(t ) .

Difficulties in the theory are caused by the history force which represents the fading memory

effect of the relative acceleration between the bubble and the fluid. The time dependence of

bubble velocity can be calculated numerically according Eq. (6.25) for suitable experimental

conditions. Consideration of the history force can improve the agreement between calculated

and measured time dependencies. An agreement could be interpreted as a proof for a weak

retardation of the bubble surface.

7. SUMMARY

At small Reynolds numbers effective approximate analytical methods allow to characterise

different states of dynamic adsorption layers quantitatively: weak retardation of the motion of

bubble surfaces, almost complete retardation of bubble surface motion, transient state at a

bubble surface between an almost completely retarded and an almost completely free bubble

surface. Unfortunately, the measurement of bubble terminal velocity in water cannot be used for

the experimental verification of these theories because uncontrolled impurities in water

immobilise a small bubble surface almost completely without any addition of surfactant. On the

contrary, bubble surfaces can be rather clean in the moment of its formation and separation from

a capillary tip. Thus the rising bubble velocity relaxation caused by the DAL formation can be

measured. So far any experiment on the deceleration stage of bubble rising is absent for small

Re. There is a large success in deep water purification that probably will allow to perform rising

bubble relaxation experiments with a rather mobile bubble surface at small and intermediate Re.

Correspondingly, main theoretical tasks concerning the bubble rising relaxation at small Re.

The DAL study is more realistic for large bubbles and large Re because trace concentrations of

surface active impurities cannot retard the bubble surface movement completely. During many

427

decades the theory for the extreme cases of almost free bubble surfaces and for strong

retardation of the bubble surface as a whole existed only. The first attempt to extend the r.s.c.

model to large Reynolds numbers has been made only very recently and valuable information

about the DAL properties are received. Simultaneously it must be emphasised that due to many

reasons these results are the first step only in this most important direction. Many restrictions

are introduced to overcome severe mathematical difficulties. It is obvious that the r.s.c.

modelling for large Re under the condition of diffusion controlled and mixed adsorption kinetics

is even more difficult and remains to be done. Numerical results have been presented only for a

rather narrow range of values of the parameters significant for the r.s.c, structure. Finally it

turned out that the exact experimental verification of the proposed theory could not been done

because of the uncertainties in the value of the kinetic rate constant kdos.

The new important trend is the combination of DAL studies with the investigation of its role in

flotation. When this important role is understood and confirmed experimentally the attention to

physico-chemical hydrodynamics of rising bubbles will increase and more efforts will be

undertaken to solve the remaining mathematical tasks. On the other hand flotation kinetics is

more sensitive to the DAL structure than the bubble rising velocity. The exactness of the

experimental test of the DAL theory can be essentially improved at the transition to

investigations of flotation and microflotation kinetics.

8. REFERENCES

Abramowitz, M. and Stegun, I.A., "Handbook of Mathematical Functions" Dover, New York, 1970

Agrawal, S.K. and Wasan, D.T., Chem. Eng. J., 18(1979)215

Andrews G.F. and Wong, S.L., Ind. Eng. Chem., 34(1995)1371

Aybers, N.M. and Tapuccu, A., "W~irme- und Stofi~bertragung", 2(1969)118

Baret, J.F., J. Colloid Interface Sci., 30(1969)1

Batchelor, G.K., Fluid Dynamics, Cambridge Univ. Press, 1967

Benjamin, T.B., J. Fluid Mech., 181 (1987)349

Beitel, A. and Heideger, W.J., Chem. Eng. Sci., 26(1971)711

Bond, W., Phil. Mag., 4(1927)889

Boussinesq, M.J., Comp. Rend., 56(1913)983, 1035, 1124

Brabston, D.C. and Keller, H.B., J. Fluid Mech., 69(1975)179

Bryn, T., David Taylor Model Basin Transl. 132(1949)???

Chan, C.H. and Franses, E.I., Colloids & Surfaces, 69(1992)189

Chen, J. and Stebe, K.J., J. Colloid Interface Sci., 178(1996)144

428

Chen, J.L.S., J. Appl. Mech., 41 (1974)873

Clarke, A.N. and Wilson, P.J., Foam Flotation, New York and Basel, Marcel Decker, 1983

Cliff, R., Grace, J.R. and Weber, M.E., Bubbles, Drops and Particles, New York, Academic

Press, 1978

Davis, R.E., Acrivos, A., Chem. Engn. Sci., 21 (1966)681

Derjaguin, B.V. and Dukhin, S.S., in ,,Issledovania v Oblasti Poverkhnostnykh sil",

B.V. Derjaguin (Ed.), Nauka 1967, in ,,Research in Surface Forces", Vol. 3, Consultant Bureau,

New York and London 1971

Derjaguin, B.V. and Dukhin, S.S, Trans. Inst. Mine. Metal., 70(1960)221

Derjaguin, B.V., Dukhin, S.S. and Lisichenko, V.A., Zh. Phys. Chim., 33(1959)2280

Derjaguin, B.V., Dukhin, S.S. and Lisichenko, V.A., Zh. Phys. Chim., 34(1960)524

Duineveld, P.C., J. Fluid Mech., 292(1995)325

Dukhin, S.S., Thesis, Moscow, Institute of Physical Chemistry, 1965

Dukhin, S.S., in "Modern Theory of CapiUarity", Berlin, Akademic Verlag, 1981

Dukhin, S.S., Kolloidn. Zh., 26(1964)36

Dukhin, S.S., in "Research in Surface Forces", Vol.2, New York and London, Consultant Bureau, 1966

Dukhin, S.S., Kolloid. Zh., 44(1982)896

Dukhin, S.S., Kolloid. Zh., 45(1983)22

Dukhin, S.S., Kretzschmar, G. and Miller, R., "Dynamics of Adsorption Liquid Interfaces", in

,,Studies in Interface Science", Vol. 1, D.MObius and R.MiUer (Eds.), Elsevier, Amsterdam, 1995

Dukhin, S.S. and Buykov, M.V., Zh. Phys. Chim., 39(1965)913

Dukhin, S.S. and Derjaguin, B.V., Kolloidn. Zh., 20(1958)705

Dukhin, S.S. and Derjaguin, B.V., Zh. Phys. Chim., 35(1961)1246, 1453

Edwards, D.A., Brenner, T. and Wasan, D.T., "Interfacial Transport Processes and Rheology",

Butterworth-Heinemann Publishers, 1991

Elsinga, E.R. and Banchero, J.T., AIChE J., 7(1961)394

E1 Sawi, M., J. Fluid Mech., 62(1974) 163

Fan, L.S. and Tsutchiya, K., ,,Bubble Wake Dynamics" in ,,Liquids and Liquid-Solid

Suspensions", Butterworth-Heinemann, Stoneha, 1990

Fdhila, R.B. and Duineveld, P.C., Phys. Fluids, 8(1995)310

Fornberg, B., J. Fluid Mech., 190(1988)471

Frumkin, R.B. and Levich, V.G., Zh. Phys. Chim., 21(1947)1183

Garner, F.H. and Skelland, A.H.P., Eng. Des. Equip., 48(1956)51

Gorodetskaya, A.B., Zh. Phys. Chim., 23(1949)7

429

Griffith, R.M., Chem. Eng. Sci., 17(1962)1057

Haas, U., Schmidt-Traub, H. and Brauer, H., Chem. Ing. Tech., 44(1972)1060

Haberman, W.L. and Morton, R.K., David Taylor Model Basin Rep. 802(1953)

Hadamard, Comp. Rend., 152(1911 ) 1735

Hamielec, A.E., Ph.D. Thesis, Univ. of Toronto, 1961

Hamielec, A.E. and Johnson, A.I., Canad. J. Chem. Eng., 40(1962)40

Hamielec, A.E. and Johnson, A.I., Canad. J. Chem. Eng., 40(1962)41

Hamielec, A.E. and Johnson, A.I. and Houghton, AIChE J., 13(1967)220

Hamielec, A.E., Storey, S.H. and Whitehead, J.H., Can. J. Chem. Eng., 41 (1963)246

Happel, H. and Brenner, H., "Low Reynolds Number Hydrodynamics", Noordhoof, Leyden, 1973

Harper, J.F., Adv. Appl. Mech., 12(1972)59

Harper, J.F., J. Fluid Mech., 58(1973)58, 539

Harper, J.F., J. Mech. Appl. Math., 27(1974)87

Harper, J.F., Appl. Sci. Res., 38(1982)343

Harper, J.F., J. Mech. Appl. Math., 41 (1988)203

Hartholt, B.G.P., Hoffmann, A.C. and Janssen, L.P.B.M., Z angew Math Phys, 45(1994)

Hartunian, R.A. and Sears, W.E., J. Fluid Mech., 3(1957)27

He, Z., Maldarelli, C. and Dagan, Z., J. Colloid Interface Sci., 146(1991)442

Holbrook, J.A. and Levan, M.D., Chem. Eng. Commun., 20(1983)171,273

Horton, Y.J., Frish, T.R. and Kintner, R.G., Canad. J. Chem. Eng., 43(1965)143

Huang, W.S. and Kintner, R.G., AIChE J., 15(1969)735

Ihme, F., Schmidt-Traub, H. and Brauer, H., Chem. Ing. Tech., 44(1972)306

Kok, J.B.W., J. Mech. Fluids B, 4(1993)541

Lebedev, A.A., ZhRPhHO, Fiz Otd., 48(1916)3

Levich, V.G., Zhur. Exper. i Teoret. Fiz. 19(1949)18

Levich, V.G., "Physico-Chemical Hydrodynamics", Prentice-HaU, Englewood Cliffs, New York, 1962

Lin, S.Y., McKeigue, K. and Maldarelli, C., Langmuir, 10(1994)3442

Listovnichy, A.V., Kolloidn. Zh., 47(1985)512

Loglio,G., Degli-Innocenti,N., Tesei,U., Cini,R. and Qi-Shan, W., Nuovo Cimento, 12(1989)289

Luttrel, G.H., Adel, G.T. and Yoon, R.H., Proe. 16 Int. Miner. Process Congress, Foresberg,

K.B.E. (Ed.), Stockholm, Elsevier, Amsterdam, Vol 2, 1988

MacLeod, C.A. and Radke, C.J., J. Colloid Interface Sci., 166(1994)73

Masliyah, J.H., Ph.D. Thesis, Univ. of British Columbia, Vancouver 1970

Mei, R., Lawrence, C.J. and Adrian, R.J., J. Fluid Mech., 233(1991)613

430

Mei, R., J. Fluid Mech., 270(1994)133

Mei, R. and Klausner, J.F., Phys. Fluids A, 4(1992)63

Mei, R., Klausner, J.F., Lawrence, C.J., Phys. Fluids, 6(1994)418

Mikis, J.M., Van den Broeck, J. and Keller, J.B., J. Fluid Mech., 108(1081)89

Miller, R. and Kretzschmar, G., Colloid Polymer Sci., 258(1980)85

Moore, D.W., J. Fluid Mech., 16(1963)161; 23(1965)749

Nquen Van and Kmet, S., Int. J. Miner. Process, 35(1992)205

Okazaki, S., Bull. Chem. Soc. Japan, 37(1964)144

Park, W., Klausner, J. andMei, R., Experiments in Fluids, 19(1995) 167

Ravera, F., Liggieri, L., Passerone, A. and Steinchen, A., J. Colloid Interface Sci., 163(1994)309

Reay, D. and Ratcliff, G.A., Can. J. Chem. Eng., 51 (1973) 178

Redfield, J.A. and Houghton, G., Chem. Eng. Sci., 20(1965)131

Rybczynski, Bull. de Cracovie A, (1911)40

Ryskin, G. and Leal, L.G.,J. Flui Mech., 148(1984) 19

Sadhal, S.S. and Johnson, R.E., J. Fluid Mech., 126(1983)126, 237

Sam, A., Gomez, C.O. and Finch, J.A., Int. J. Mineral Processing, 47(1996)177

Savic, P., Nat. Res. Counc. Can., Div. Mech. Engng. Rep. MT-22, 1953

Savill, D.A., Chem. Eng. Sci., 5(1973)251

Schlichting, H., "Boundary Layer Theory", 6th ed. McGraw-Hill, New York, 1968

Seeley, L.E., Ph.D. Thesis, Univ. of Toronto, 1972

Seeley, L.E., Hummel, R.L. and Smith, J.W., J. Fluid Mech., 68(1975)591 Sotskova, T.Z., Bazhenov, Yu.F. and Kulski, L.A., Kolloidn. Zh., 44(1982)989; 45(1983)108

Suzin, Y. and Ross, S.J., J. Colloid Interface Sci., 103(1984)578

Stebe, K.J., Lin, S.Y., and Maldarelli, C., Phys. Fluids, 3(1991)3

Stebe, K.J. and Maldarelli, C., J. Colloid Interface Sci., 163(1994) 177

Tomotika, S., Aeronaut. Res. Rep. Memo, 1678(1935)

Tsuge, H. and Hirino, S.L., J. Chem. Engng. Japan, 10(1977)66

Yoon, R.M. and Luttrel, G.H., Miner. Process. Extr. Metal. Rev. 5(1991) 101

Zhang, Y., Gomez, C.O. and Finch, J.A., submitted to Int. J. Mineral Processing, (1996)

Zholkovski, E., Kovalchuk, V., Miller, R. and Dukhin, S.S., submitted to J.Colloid Interface Sci.

9. LIST OF SYMBOLS

A area of the interface [cm 2] a b bubble radius [cm] a L constant of the Langmuir isotherm [mol/cm 2] a F constant of the Frumkin isotherm [mol/cm 2]

C

C

Co

Cel

D

Deft

v.o Fo (t) Fn(t)

F M(t)

g H = h / a p

h~r

KH k

karl

kdes L

Pe

P~ R

Re

r o

St

T

t

V

x

Y z

9~b

5D 6a

r = r ( t ) / r o

ro

dimensionless surfactant concentration [-] bulk concentration [mol/cm 3]

equilibrium bulk concentration [mol/cm 3] electrolyte concentration [mol/cm 3]

diffusion coefficient [cm2/s]

effective diffusion coefficient [cmVs] buoyancy force less gravitational force quasi-steady force

history force

added-mass force

force due to free-stream acceleration

acceleration constant [cm/s 2] dimensionless thickness

critical thickness

Henry constant = Fo/c o [cm] Boltzmann's constant [ 1.3 8 1 0 -23 J/mol/K]

rate constant of adsorption [cm/s]

rate constant of desorption [ l/s]

characteristic length

Pecklet number

capillary pressure [mN/m 2]

gas law constant [ 8.3 14 g cm2/(s 2 mol K)]

Reynolds number

radius of curvature of a bubble or a drop at its apex [cm]

Stokes number

absolute temperature [K]

time [s]

buoyancy bubble velocity

direction normal to the interface [cm]

direction tangential to the interface [cm]

radial coordinate in a spherical system

coefficient of retardation

coefficient of retardation for a bubble surface

thickness of the diffusion layer [cm]

hydrodynamic thickness [cm] dimensionless adsorption

equilibrium surface concentration [mol/cm 2]

431

432

r~

Y

Yo

rl" rls

k v= r l /p

I I = y o - 7 0

P p"

"Cad

Indices

ad

b

c r

des

diff

P

P S

S

st

St x

Y Abbreviations

CMC

DAL

r . s . c .

RSP

SDS

maximum surface concentration [mol/cm 2] interfacial tension [mN/m]

interfacial tension of the pure solvents [mN/m]

bulk viscosity of liquid medium

bulk viscosity of bubble or drop surface viscosity

drag coefficient dynamic viscosity

surface pressure

angle coordinate in a spherical system

density of liquid medium[g/cm 3]

density of the bubble or drop

thickness of the interfacial region [cm]

characteristic time of the adsorption process

angle characterising the rear stagnant cap

adsorption

bubble

critical

desorption

diffuse part of the electric double layer

potential flow particle

Stokes flow

surface

static

Stern

normal to the interface

tangential to the interface

critical concentration of micellisation

dynamic adsorption layer

rear stagnant cap

rear stagnant point

sodium dodecyl sulphate

Drops and Bubbles in Interfacial Research D. M6bius and R. Miller (Editors) �9 1998 Elsevier Science B.V. All rights reserved.

PULMONARY SURFACTANT AND BIOPHYSICAL FUNCTION

433

Ralf Herold, 1 Regine Dewitz, 2 Samuel Schiirch 3 and Ulrich Pison 2

l'2Charit6-Virchow-Klinikum, Departments of Paediatrics 1 and Anaesthesiology, 2

Medical Faculty of Humboldt-University, Augustenburger Platz 1, D- 13344 Berlin,

Germany.

3Respiratory Research Group, Faculty of Medicine, University of Calgary, Calgary,

Alberta, Canada.

Contents

.

2.

2.1

2.2

3.

3.1

3.2

3.3

3.4

4.

5.

Surface-related effects in pulmonary physiology

Pulmonary surfactant as material responsible for biophysical lung function

Pulmonary surfactant components

Pulmonary surfactant regulation

Biophysical functions of pulmonary surfactant

Background

Spreading, adsorption, quasi-static and dynamic cycling

Methods

Perspective

Acknowledgements

References

434

1. SURFACE-RELATED EFFECTS IN PULMONARY PHYSIOLOGY

In human and mammalian lungs, physiological functions rely extensively on specific properties

of surfaces and surface actions. The extraordinary plasticity and stability of the lungs reside in

its construction of a multitude of similar anatomical units, the alveoli. The alveoli are also the

smallest functional units, supplying maximum free surface area in combination with minimum

volume occupied. Via the large alveolar surface area, gas exchange occurs during respiration,

the vital characteristic of warm-blooded species. The non-collapsing alveolar volume, a steady

gas reservoir, turns the heavy respiration gas flow towards an alveolar diffusion-controlled

process.

Several biochemical-biophysical properties contribute to the superior construction of the

alveolus. A thin, liquid lining layer, specially composed of an aqueous subphase and a lipid

film-like surface, covers the inner surface in the alveoli. The surface material is known as

pulmonary surfactant which is still under investigation since detection of its influence on

respiration [219] and its primal discovery [32, 158], because its working mechanisms have not

yet been resolved satisfactorily. In this regard, scientific interest is stimulated by the

complexity of the pulmonary surfactant composition, alveolar geometry, lung forces

interactions, and special surface-active properties of the alveolar lining layer. The latter are

characterised in general by almost stable, nearly nullified surface tensions observed under only

moderate surface area compression.

This chapter mainly outlines current knowledge on pulmonary surface and surfactant

investigations and targets at modelling of the surface-related interactions between pulmonary

surfactant and alveoli. The first part includes a discussion of biological components and

functions of pulmonary surfactant and of methods available and suitable for its investigation.

The second part focuses on the difficult translation of concepts and findings into a

prospectively useful model of alveolar function. Major areas of disagreement and uncertainty

will be indicated.

Surface-related effects, as discussed in several other chapters of this book, apply to many

aspects of biological functions as far as in the lung, diverse and highly concentrated lipids and

proteins constitute a system with complex physicochemical properties. The lung's alveoli are

the prime site of surface-related effects in the human body because there is no other interface

as large and no other substances (air and pulmonary surfactant) as different in density, although

435

several other fluids are also surface-interacting at their respective anatomical interfaces

(cerebrospinal fluid, urine, eye lens, joints). The pulmonary surfactant system is outstanding

because there is reasonable understanding of the surface-related, i.e. biophysical function. In

addition, substitution of exogenous surfactant is a first choice therapy in lung disease states

with surfactant deficiency. Therefore, pulmonary surfactant research is not only basic science,

but may lead to surfactant production for exogenous substitution in man that surmounts

limitations of animal material resources, danger of allergy, variant pharmacological

preparation, inactivation, and cost-restricted availability.

To perform its function as the organ for gas exchange, the lung of adult male interfaces directly

with its environment through a surface of approximately 150 m 2 in an adult human (Table 1).

This extensive surface area is necessary for gas exchanges but encompasses two basic

functional demands. The gas exchange surface has to be kept available during breathing and

defended against infectious agents and irritants that may be present in air or blood. Pulmonary

surfactant, a highly specialised system found in all mammalian lungs, may do both. It helps to

keep the gas exchange surface available during breathing by its surface tension reducing

properties, and may represent early lung defence strategies of the host avoiding air or blood-

borne pulmonary infection and lung tissue injury.

The investigation of pulmonary surfaces started with experiments on excised lungs. It has

repeatedly been shown that in excised lungs, inflation-deflation curves are different between

air filled and liquid filled lungs. This finding was soon attributed to the presence or absence of

the liquid-air interface. Many experiments were then carried out with this interface either

abolished by various detergents or substituted by various fluids of varying surface tension.

Surface tension was principally found to be surface area dependent in the lung, and additionally

showed distinctive inspiration and expiration responses.

After deducing patterns of volume and surface tension responses to pressure imposed on whole

lungs, investigators turned to separate alveoli, which could be investigated with even a greater

number of methods.

The alveolar surface itself came under scrutiny by measuring its surface tension and by directly

visualising its regional environs. An important finding is that alveolar geometry is very similar

even in very different regions of the lung, where blood and gas flow are indeed different. Still,

most of the biophysical research on surfactant has been carried out by using classical

monolayer methods and theories, because it was not possible to study a working pulmonary

436

alveolar surfactant film in vivo. The principle properties of such a pulmonary monolayer

surfactant in vitro have been formulated and are reiterated in more detail later. In short, the

surface film is enriched during its formation and cyclic compression to a mixture of abundant

highly compactible lipid molecules and some intertwining specific proteins, which can store

that much energy as compared to its subphase that surface tension almost nullifies. These

properties of a monolayer evolve also under cyclic surface area changes in vitro.

Table 1: Physiological characteristics of lungs. Parameters and values according to findings in adults and new-bores. Conventional units given in last column; to convert from CmH20 to kPa, multiply with 0.098, from mmHQ to kPa, multiply with 0.133.

Parameter Value in an adult

Diameter of alveolus 120 i i i

Total number of alveoli 300.106 t i

Total alveolar surface i

70-150

Value in a term Unit new-bom

i i

50 gm i

24.10 s !

2.8 m 2 l !

Lung (elastic) compliance 100 5 1.4 1.3

ml CmH20I

ml CmH2o-1 kg "1 i i i I I i

Lung (flow) resistance 5.5 68 CmH20 S 11-1 i i ! i i

Alveolar ventilation 4200 400 ml min -1 i i i i i

Tidal volume 450 20 ml I i i

Total lung volume 5000-7000 (total lung capacity, TLC) i

i i

Functional residual volume 1200 (functional residual capacity, FRC)

i i

200 ml i

I i i 60 ml

Oxygen diffusion constant 16 5 mlo 2 min -1 mm~ I I I I I

Carbondioxide diffusion constant 20 1.5 mlco 2 min -1 mmve I I I I I

Work of breathing 25 1.5 kg cm min -1 I I i I I

Respiration frequency 20 ' 40 min -1 I I I I

Lung perfusion 5 4.5 I min -1 m -2

Between the findings on whole lung pressure-volume relationships and on in vitro monolayer

surface tension-surface area relationships, a cleft of desired knowledge of the actual

distribution, composition, and function at the molecular level exists. To some extent, there is

now information available to bring together the aforementioned discrete findings. Importantly,

the alveolar lining layer (and its subphase) has been found continuous throughout the whole

lung [ 14].

437

A basic concept of lung 'construction' is the extension of its inner surface by multiple

subdivision of the contained air [35]. Lung function then includes exchanging oxygen and

carbon dioxide across these membrane-like surface subdivisions, according to the Fick laws.

Exchange takes place by gas diffusion from red cells within capillaries (smallest blood vessels,

radius about 2 lam), via the capillary membrane, interstitial fluid (about 3 gm), the alveolar

cell's basal membrane, the alveolar cell itself, the cell's apical (top) membrane, the hydrophilic

subphase of the alveolar lining layer (area-weighted average in rats at 80% TLC, 0.2 gm [ 14]),

and the surface film with its thickness in the nanometer range. These distances as well as the

available area are rate limiting for gas diffusion whereas the red cells' capillary transit time is

limiting the exchange by convection. It is important that the abundant capillaries form a very

dense meshwork around each alveolus, which is created in between the fixed subdivisions of

an elastic collageneous scaffold. The alveolar scaffold can be imagined as a large sponge with

evenly distributed wholes. Such a picture of parts of the lung can even imaginarily be extended

to reflect the elastic behaviour of the lung, the self-retraction forces, or alveolar fluid oedema.

The basic surface-related properties of each single lung's alveolus relates to this scaffold

framework; in specific, there are long fibers of stable collagen which suspend sheet of elastic

collageneous tissue on which flat alveolar cells rest [236].

A fundamental working hypothesis of the current research on surface-related effects in lungs is

that in essential no other interface than the alveolar lining layer-air interface is contributing to

the reduction of forces in low-volume states of the lung. This concept is supported by model

studies on excised lungs in which the alveolar surface was fixed at known values by either

rinsing or filling the lungs with hydrophobic agents (e.g., perfluorocarbon) [7_].

In the next section, the composition and regulation of pulmonary surfactant as material

responsible for biophysical lung function, which encompasses the surface-related effects in

lungs, are discussed.

2. PULMONARY SURFACTANT AS MATERIAL RESPONSIBLE FOR BIOPHYSICAL LUNG

FUNCTION

The alveolar epithelium in the lung is covered with a liquid, the alveolar lining layer. This

alveolar lining layer consists of an aqueous subphase covered by a film of pulmonary

surfactant. In principle, the surfactant film reduces the air-liquid interracial tension, thereby

preventing alveolar collapse and preserving the gas exchange surface. The aqueous subphase

438

provides a storage medium for surfactant components. Into this medium surfactant components

are secreted from type II cells, adsorb to and desorb from the air-liquid interface, change their

morphological structures, and interact with cells of the host defence system (for example,

alveolar macrophages).

The importance of the lung air-liquid interfacial film for lung mechanics has been recognised,

since Kurt von Neergard in 1929 attributed the differences in recoil forces between fluid- and

air-filled lungs to the action of surface tension [219]. Pattle [158] and Clements [32] first

demonstrated that a substance providing very low surface tension is present in pulmonary

oedema fluid and lung extracts. Pattle considered the very low surface tension he discovered in

lung oedema fluid a significant anti-pulmonary oedema factor [159]. Clements measured

surface tension of lung extracts at different surface areas in vitro [35]. He found that surface

tension decreased during film compression and increased during its expansion. His

experimental results supported the view of "anti-atelectasis" activity of pulmonary surfactant,

and enabled a new concept of alveolar stability that depend on variable surface tensions. Direct

measurements of surface tension in intact lungs [193] and energy analysis based on well

defined structure-function relationships of isolated, air-filled and perfused lungs [236]

confirmed that the surface tension within alveoli at lung volume levels of normal breathing is

very low indeed and changes with alveolar size.

There is also experimental and theoretical evidence that surfactant modulates alveolar defence

reactions of the host against invading pathogens [144, 165], and increases the mucociliary

clearance of foreign particles [42]. Surfactant stabilises small airways against airway closure

[110], balances lung fluids [25, 33], and provides a certain protection against physico-chemical

assaults. These functions of pulmonary surfactant are related to the classic, biophysical

function of the surfactant system, which is its surface tension reducing property. This surface

tension reducing property will be discussed in the following paragraphs.

Although pulmonary surfactant in general terminology and also in the scope of this review is

equivalent to the alveolar surfactant described above, surface-active material is present in other

anatomical structures of the lung. In airways, surface-active material derives from local

secretion, but its composition and relation to alveolar surfactant have not been established.

Airway surfactant reduces the airway surthce tension [192], enhances clearing of foreign

particles [62], provides airway patency in vivo [52], in lung grafts [27], and in vitro [127], and

reduces airway reopening pressure [163].

439

Much of the surfactant research has been stimulated by the finding that surfactant is altered in

new-borns and adults with the respiratory distress syndrome. The respiratory distress syndrome

of (premature) new-borns [5] is mainly characterised by both a deficiency of quality and

quantity of surfactant and an anatomical immaturity of the alveoli, which are dependant on

sequential gene activation and gene expression [19, 50, 94, 137]. The successful treatment of

this condition (which can hardly be called a disease because it merely reflects the degree of a

new-born's maturity for life) has fortunately become possible by substituting surfactant with

exogenous surfactants and by inducing gene activation with cortisone-like drugs and maybe

other hormones. The other form, the acute respiratory distress syndrome [4], is characterised by

a consequential functional impairment of the pulmonary surfactant that can be the endpoint of

different pathogenic pathways [164]. The starting point is not surfactant deficiency, but lung

damage from either material transported by the airways (such as aspirated meconium or

aggressive liquids and gases) or from blood-borne substances (such as fibrinogen, reactive

oxygen species or degrading enzymes released by inflammatory cells). Blood-borne substances

are pathogenic for example during septic infections with blood leakage from capillaries into

alveolar spaces or during trauma-induced lung contusion. The treatment of this disease is

directed to curing the pathogenic condition whereas the consecutive surfactant damage may be

alleviated by surfactant replacement. For reviews, cf. [ 124, 204].

Research of other disease states that are associated with lung surfactant but are not associated

with the respiratory distress syndromes has become increasingly productive. These may include

diverse conditions such as pulmonary oedema, interstitial pneumonia, hypersensitivity

pneumonitis, asthma, allergies, inborn genetic disorders of surfactant-specific proteins, and

alterations by dust or smoking. Related studies mostly are mostly conducted in vitro by

investigating material gathered from individuals by aspirating liquid from bronchoalveolar

lavage or from tracheal contents. In specific, pulmonary oedema and clinically related diseases

can be differentiated by determining the proportion of alveolar involvement which is positively

correlated with the SP-A levels in aspirates [166, 202]. Low SP-A levels occurring solely in

idiopathic interstitial pneumonia may account for the patients' susceptibility to infections by a

diminished host defence [ 103]. SP-A levels may be elevated independently from cell numbers

in hypersensitivity (allergic) pneumonitis [38], and both SP-A and SP-D bind to the main dust

mite allergens [223] and thus could mitigate allergic lung reactions. This notable modulation of

host defence function by SP-A had already been demonstrated in vitro to result from

440

interactions with carbohydrate molecules or with pulmonary macrophages [165]. So far, the

search for a specific disease originating from a non-functioning surfactant specific protein

identified a mutation of the SP-B-gene to be responsible for a rare, but lethal form of neonatal

respiratory distress syndrome [31, 147], whereas SP-A-mutant animals showed increased

susceptibility to bacterial infections [ 119]. With regard to smoking, there were decreased levels

of SP-A, SP-D [104], and SP-B [208] found in bronchial lavages that may be associated with

development of chronic disease based on an altered host immune system.

These and other related studies have also promoted the much-wanted disclosures of functional

connections and dependencies between different surfactant components (as discussed later).

Apart from the more readily analysable immune modulation by surfactant proteins, there is also

evidence of host defence suppression by surfactant lipids [3]. Primary surfactant lipid changes

occur mainly through oxidation by endogenous-derived or aerosol-borne reactive molecules

(for example, [134]), leading to non-specific respiratory distress, or through deposition of

material that can not be cleared from alveoli and thus chronically stimulates lipid secretion, as

in silicosis. However, secondary surfactant lipid composition changes are entailed in the acute

respiratory distress syndrome and may govern the clinical picture [77]. As an indication of

integral surfactant malfunctioning without structural or compositional changes, a reversal of

the characteristic lung hysteresis in well-known pressure-volume-tracings may be linked to the

sudden infant death syndrome [99]. Apart from the studies on natural diseases related to

surfactant summarised here and reviewed with a focus on the respiratory distress syndrome

elsewhere [164], there exists a body of findings related to non-natural conditions. These

conditions are caused by more or less specifically surfactant inactivating agents or by using

reconstituted surfactants (that is, self designed with regard to composition and structure).

Several reviews on the natural surfactant system [40, 68, 160], on non-natural surfactants [24,

102, 188], and on the history of surfactant [37, 150, 211 ] are recommended to the reader. We

will now continue with detailed information conceming the composition, regulation, and

functions of the natural surfactant system.

2.1. PULMONARY SURFACTANT COMPONENTS

Surfactant forms an insoluble film at the surface of the alveolar lining fluid modifying the

surface tension in a manner that depends on alveolar surface area. Besides the surface film

itself, surfactant has many other structures in the alveolus, including the contents of the

441

lamellar inclusions in pulmonary type II epithelial cells, an unusually ordered vesicular

structure called tubular myelin, and other vesicular and lamellar structures with quite varied

dimensions and structures (Table 2).

Table 2: Types of lipid aggregates discernible in pulmonary surfactant. The four main types are natural components of surfactant and can to a certain extent be separated by centrifugation. Some distinguishing characteristics are listed in rows. Legend: ++ important, + present, (+) presumably present, - not present.

Surfactant Proteins SP-A SP-B SP-C

II Univesicular body

(+) (+) (+)

Multilamellar Lamellar Tubular body body Myelin

+ (+) ++ + + +

§ +

Phospholipids PC (+) Fatty acids +

Divalent ions Calcium Morphology Shape globular ellipsoid

Size Significance intracellular transport pre-degradation

+

4(+) [233]

Metabolic state Number per cell Extracellularly Surface activity Reference

§

. l-t-

lattice lamellar =1 I~m

exocytosis 100

+

pre-adsorption

..H-

Homogenous preparations of each form of surfactant have been difficult to isolate and to

prepare in vitro. The numerous biochemical studies on surfactant fractions that are enriched to

various degrees in the different surfactant structures suggest that their lipid composition is

relatively similar, the protein composition is quite variable [131,240]. One exception of this

general statement exists: the actual surface film in the lung changes its lipid composition and

structure during the dynamic compression to low non-equilibrium but mechanical stable

tensions on expiration and expansion of the film on inspiration [67, 88].

The heterogeneity of surfactant structure is clearly relevant to the interpretation of the results of

surfactant compositional analysis. Despite these concerns, surfactant prepared from whole lung

or lavage fluid in many different species is surprisingly similar, varying only during

development [120] or under pathological conditions, usually lung parenchymal injuries the

peak of which is ARDS [72, 85, 168]. In adults, it contains approximately 90% lipid and 10%

protein by weight.

Pulmonary surfactant lipids More than 80% of the pulmonary surfactant lipid is phospholipid [ 15, 115]. The most abundant

phospholipid is phosphatidylcholine (60-80%, dependent on species and methods of isolation

and detection employed). Approximately 60% of the phosphatidylcholine are disaturated, with

more than 90% of the acyl groups being palmitic acids [26]. L-ot-dipalmitoyl-

phosphatidylcholine (DPPC) is one of the most important surfactant components for lowering

surface tension [67, 149]. Still, this one component alone was not sufficiently beneficial as a

surfactant substitute in new-borns with respiratory distress [29]. DPPC makes up only 40% of

all phosphatidylcholine [101, 109]. Other required lipid components that were found in

- - +

[63, 233] [156] [93, 209]

442

significant amounts in surfactant are the negatively charged phospholipids (at pH 7, the

phosphate group is PO4-) phosphatidylglycerol and phosphatidylinositol which have several

hydroxyl groups in the head group. These phospholipids make up about 11% of the

phospholipid fraction and may substitute for each other [21], as supposedly in the neonatal

respiratory distress syndrome, where phosphatidylglycerol is diminished [83]. The function of

these hydrophilic phospholipids is to facilitate the adsorption of DPPC (a zwitterionic at pH 7),

which would otherwise be too slow to correspond to physiological needs. In fact, the

interactions between all phospholipids result from their respective equilibrium surface tension

values that determine if a molecule enters or leaves the surface at a given surface pressure.

Fig. 1: Captive-bubble-surfactometer. This instrument can be used to measure dynamic surface tension at the air-water interface. Surface area and bubble volume can be varied and surface tension is determined from bubble geometry analysis [190]; 1 - piston, 2 - 1% agarosis gel, 3 - captive bubble with surfactant film, 4 - stir bar, 5- surfactant suspension, 6 - stainless steel base, 7 - plug

443

For instance, the latter phospholipids are molecules with unsaturated fatty acids having a lower

melting point than DPPC, which renders their air-water films less compactible. This

mechanism of film formation is also working during tidal breathing at rest, without maximum

inspiratory surface expansion [194]. The equilibrium surface density (F ~ ) of DPPC is about

2.31018 mol m -2 [190].

It has repeatedly been asserted that the alveolar surface consisting of a film of such

phospholipid molecules must be regarded as a solid [12], because the body temperature is

below the film's melting temperature. However, the properties of insoluble molecules at a

surface are more complex and specifically require the differentiation of varying states of

molecular orientation and ordering relative to the surface, inhomogeneous molecular surface

coverage and formation of disperse system domains. In an air-water surface, the phospholipids

are oriented with a variable tilt to the surface [95, 139]; their hydrophilic headgroups are in the

sub-surface and their fatty acid chains protrude into the air space, because of the insolubility of

phospholipids in an aqueous subphase. Concerning film formation and structure of pulmonary

surfactant lipids, several studies so far have investigated the effects of hypophase lipid

concentration, divalent cations, temperature, diffusion hindrance, surface area, subtype

composition, and pre-formed lipid aggregates. Some of the basic findings are summarised here.

Biophysical properties of various (pulmonary) lipids were first collected with a Wilhelmy

balance [35, 98], then using a pulsating bubble [51, 247], and recently by a captive bubble

system [190] (Fig. 1). With increasing subphase lipid concentration, the adsorption time of

both natural and extracted phospholipids from pulmonary surfactant decreases about

exponentially (and asymptotic to critical micellar concentration) [118, 174]. In addition to

determining the concentration value, which is problematic with less soluble molecules, it might

be necessary to determine the aggregation status of the lipids, e.g. by turbidity determination

[80]. Divalent cations have long been recognised to stimulate adsorption [89], but it has been

difficult to separate their effect on proteins from that on lipids. In addition to chain length and

saturation, a phospholipid's isotherm depends on the headgroup charge, which in turn depends

on the general ionic strength of the subphase; sodium ions are usually 150 mM. Calcium ions

are required for protein lipid interaction and are usually 2-5 mM. The diffusion resistance is

usually minimised in experiments by providing subphase convection, mostly through stirring,

but experimental data are not available. Surface area changes induce surface tension changes,

which depend on the rate of surfactant adsorption from the bulk phase. In an interpretation of

pulmonary surfactant isotherms of surface tension vs. area, the area may correspond to the

alveolar area, which is stabilised against further reduction in area at a given volume by the

surface film having a low compressibility. The different subtypes in pulmonary surfactant

444

become evident in electron microscopy of differentially centrifuged samples [74, 174].

Centrifugation at 1,000-60,000 g, 20-60 rain, sediments the most surface active material

[ 154]. This consists of large aggregates, which adsorb quickly and form stable surfaces films.

Other sedimentation accelerations result in different aggregates that are assumed intermediate

metabolic forms [ 117, 240].

With regard to lipid aggregation, the phospholipids are not only known to undergo several

phase transitions (between liquid and solid) in a two-dimensional arrangement, but also adopt

different three-dimensional arrangements, like liposomes (micelles), tubular arrays [160], and

membrane-like layers [171, 196, 249]. It is intriguing to speculate that these arrangements also

exist in vivo and correspond to multilamellar vesicles, tubular myelin, and a multilamellar

surface structure (a surface-associated surfactant reservoir). The function of cholesterol (about

8% of pulmonary surfactant lipids) in pulmonary surfactant is not yet known, but cholesterol

may preserve a two-dimensional mixture of different surface phospholipids in a given surface

which otherwise are expected to separate laterally into different phases [249].

Lung lipid concentrations and surfactant pool size are generally lower in premature new-borns

and are quite variable according to the method of measurement employed. For example, the

alveolar saturated phosphatidylcholine content may be as low as 1.4 mg kg 1 bodyweight [ 178],

and the phospholipid concentration in the alveolar subphase as high as 120 mg ml 1 [174]. As

an estimate, exogenous phospholipid amounts of 3 mg kg l bodyweight probably suffice for

complete alveolar surfactant replenishment [84]. Thus, the surfactant metabolism (that is, both

production and removal) must be tightly regulated as the turnover of surfactant may reach 40%

of total surfactant per hour [88].

The surface area cycling typical of respiratory cycles can be regarded as a model procedure for

the extracellular metabolism of pulmonary surfactant [74], and it leads to loss of large and

generation of small lipid aggregates. This conversion may also be provoked in vivo [107],

depending on the amount and velocity of surface area changes during artificial ventilation.

Small aggregates (e.g. vesicles) cannot promote surface tension reduction as much and as

quickly as large aggregates like tubular myelin. A recently defined enzyme and inhibitors may

participate in the conversion process [73] that may also involve surfactant protein (SP-B)

degradation [215].

Several physico-chemical alterations of pulmonary surfactant may lead to global functional

impairment; for instance, lipid peroxidation from free radical production reduces surface

445

activity [63], pH values above 7.4 spoil artificial, but not natural surfactant [79], temperatures

above 40~ entail protein degradation [93], and endogenous or exogenous nitric oxide disturb

alveolar surfactant synthesis and function [80, 81 ]. These limiting aspects for normal surfactant

function have not yet been reviewed comprehensively but should be accounted for in

experiments. In the following, properties of l,he pulmonary surfactant proteins are reviewed.

Pulmonary surfactant proteins

Beside the phospholipids (see above), which contribute fundamentally to the surface lowering

activity of pulmonary surfactant, specific proteins are also associated with surfactant: SP-A,

SP-B, SP-C, and SP-D [108, 162, 170]. The complete amino acid sequences of these proteins

have been determined. In addition, partial amino acid sequences have been used to demonstrate

in a given protein some of the functional domains responsible for molecular interactions

important for surface tension changes. Several surface properties have thus been studied in

one-component solutions of protein derivatives in surfactant for basic comparison with the

effect of other proteins (e.g., albumin). Yet, it appears more important to conduct studies on the

quasi catalysing effect of surfactant proteins in mixtures with phospholipids on surfactant

activity. The effects of these proteins on surface film formation are exemplified in Table 3 and

their properties are discussed in the subsequent paragraphs.

SP-A is the most abundant surfactant-associated protein. It has apparent molecular masses of

26-36 kD under reduced conditions. The heterogeneous molecular mass results from

differences in posttranslational modification, especially variable degrees of glycosylation [91,

231]. Under unreduced conditions, the molecular mass of SP-A is approximately 700 kD

[ 116], suggesting that in its native state SP-A is highly oligomerised, consisting of probably 18

similar or identical subunits. The complete amino acid sequences of dog [20], human [58,

229], rat [182] and rabbit SP-A [22] have been deduced from their respective eDNA

sequences. The amino-terminus region of the mature protein contains 7-10 amino acids

depending on the species, followed by a collagen like sequence characterised by a series of 24

repeats of the triple sequence Gly-Xaa-Yaa where Yaa is frequently 4-hydroxyproline [93].

The collagen like series of reiterated sequences is disrupted in triplet 13 by the insertion of an

additional residue, which is also a feature of the complement protein C lq [220] and the

mannose-binding proteins [48]. Human SP-A is encoded by presumably two different genes

and one pseudogene, which are located on chromosome 10 [56].

446

In the lung, SP-A is expressed in alveolar type II epithelial cells, and in some species, it may

also be synthesised in non-ciliated cells of respiratory bronchioles (Clara cells). SP-A also

regulates the amount of surfactant in alveoli through a receptor-mediated process on type II

pneumocytes [47, 241,243] and it augments macrophage dependent defence reactions through

a receptor mediated process [213] or through direct antigen binding [212]. The biophysical

function of SP-A is mainly to enhance the surface activity of surfactant under limiting

conditions such as low concentration of phospholipids [195], and to catalyse the sub-surface

formation of lipid structures such as tubular myelin [82, 119, 217]. SP-A promotes the

adsorption of surfactant to the interface [90, 169], especially from a surface associated

reservoir [196]. The interaction of SP-A with lipids can be inhibited by nitration of amino

acids [82] and it may interact not only with phospholipids, but also with neutral lipids that are

present in significant amounts in alveolar surfactant [248, 249]. In addition, SP-A decreases the

inhibitory effect of blood proteins on surface activity [36]. The formation of the largest

identifiable lipid aggregate in the alveolar subphase, namely tubular myelin, requires both

SP-A and SP-B [235] and may occur in vitro in aqueous solutions with calcium. SP-A mutant

and completely SP-A deficient mice lacked formation of tubular myelin, but showed an

otherwise normal postnatal survival and pulmonary function [119].

SP-B and SP-C are the two hydrophobic surfactant proteins. They have distinctive masses

under non-reduced conditions of 18-20 and 5-6 kD respectively. The nominal molecular

masses for the proteins under reduced conditions however are very similar (5-8 kD). The

amino acid sequences for both proteins have been derived from cDNA molecules [64, 90, 228],

predicting a precursor of approximately 42 kD for SP-B and one of approximately 21 kD for

SP-C. The human SP-B gene is localised on chromosome 2 and expressed in the lung by both

type II cells and Clara cells. The human SP-C gene is localised on chromosome 8. The

developmental and corticoid-induced metabolism of SP-B is notable for the complexity of the

genetic elements involved [18, 96, 130, 132], whereas the metabolism of SP-C has not been

established, but it seems to start earlier during gestation [9, 50, 112, 238].

The principle function of the hydrophobic surfactant proteins SP-B and SP-C is the promotion

of a low and stable surface tension in alveoli. SP-B and SP-C achieve this function through

interactions with the surfactant lipids and SP-A. However, SP-B's predominant surface-related

effects are to accelerate the adsorption of phospholipids [226] by facilitating the insertion of

lipids into the surface [ 152]. This may involve changes of the free energy balance between the

447

lipid aggregates in the subphase and the their transformed state in the surface film at the air-

water interface [200]. SP-B probably also equilibrates irregular lipid distribution during surface

expansion by means of fusion of lipid domains [172]. The pronounced ability of SP-B to

interact with lipids is assumed to be due to repeated long alternating hydrophobic and short

positively charged amino acid sequences [36, 131,232]. These may associate with negatively

charged phospholipids such as phosphatidylglycerol [39]. A synthetic SP-B fragment analogue

has been efficacious in an animal model [179], although similar fragments induced different

lipid-protein conformations [78]. However, the molecular mechanisms of the SP-B-lipid

interactions are not yet sufficiently known [71, 126, 157]. As already pointed out above, a

complete deficiency in SP-B due to genomic mutations is usually fatal within hours after birth

[ 148]. Although it is not yet known if SP-B's complete or partial [ 10] deficiency or its genetic

differences in adult population subgroups [59] are really major causes of respiratory distress

syndromes, these conditions underscore the functional significance of SP-B [230]. Without

SP-B present, there is a marked disturbance of extracellular surfactant with regard to protein

secretion, altered phospholipid composition, altered macromolecular aggregates, and the

absence of tubular myelin [43, 86, 147]. SP-B deficiency in mice leads to death by respiratory

distress syndrome if the deficiency is complete [31]. If the deficiency is partial, postnatal

survival is normal but pulmonary functioa is impaired because the number of alveoli is

reduced, lung compliance decreased, and air trapping increased [30].

In comparison with SP-B, SP-C is more effective in promoting adsorption and film stability at

low minimal surface tensions in studies of reconstituted pulmonary surfactants [ 177]. This and

another study [225] also showed that posttranslational modifications of SP-C itself (such as

acylation/palmitoylation) are essential to its interaction with phospholipids and must be

considered by in vitro studies. SP-C plus SP-A were the most potent promoters of

incorporation of sub-surface surfactant material into the surface [ 191 ]. SP-C seems to remain

in the surface, along with lipids under higher lateral surface pressure than SP-B. The presence

of SP-C leads to a decrease of surface liquid-condensed lipid domains via predominantly

hydrophobic interactions [ 140], which could translate in vivo into increased surface elasticity.

This study showed that calcium ions induce liquid-condensed surface film packing of

negatively charged (such as phosphatidylglycerol), but not zwitterionic (such as DPPC)

phospholipids. Furthermore, SP-C is less effective in preventing surfactant inhibition by blood

proteins than SP-B [199]. Preliminary data on therapeutic use of pulmonary surfactants with

448

synthetic SP-C like proteins, produced by recombinant bacteria [92] or synthesised in varying

lengths [210], demonstrated similar efficacy, irrespective of the dissimilar protein origin.

The principle changes in the alveolar surfactant's lipid aggregates under the influence of SP-A,

SP-B, and SP-C as described above were also demonstrated by electron microscopy [235].

Some data on the proteins' effects on adsorption of phospholipids have been compiled in

Table 3.

Table 3: Different suffactant proteins contribute differently to the rate of phospholipid adsorption to an air-liquid interface. PA, palmitic acid; PG, phosphatidylglycerol; Ca 2§ calcium present (1.5-5 mM) or absent; "y~n, minimum surface tension at end of adsorption; t'/~n, half-time of adsorption as deduced from published data. *The reversal of surfactant inhibition as a function of SP-A concentration has been studied even earlier [36]. **The co-operative functions of SP-A and SP-B [90] and of SPoA and SP-C [114] on interfacial film formation have been reported before.

Surfactant Protein

SP-A

SP-B

SP-C

SP-B+C

"SP"-A+B+C "SP"-A "SP"-B+C "SP"-A+B+C

Protein Origin

bovine

bovine

bovine de-palmitoylated palmitoylated

bovine

synthetic peptides

Concentra- tion [% of

lipid weight]

0 I 4

0 0.13 0.33 0.67 1.0 1.3

0 5

1.5+1.5 5+1.5+1.5

Lipids added Lipid con- Ca 2+ t,/,mi n Reference [weigth ratio] centration is]

bovine lipid extract 0.05 mg m1-1 + 486 [195]* surfactant + 27

+ 2

DPPC:PG (7:3) 0.1 mg m1-1 - 546 [246]** DPPC:PG (7:3) + 123 PG - 857 PG + 272

DPPC:PG (7:3) 10 mg m1-1 + 131 [177] + 65 + 32

calf lung pulmonary 0.25 mM + 493 [227] phospholipids + 366

+ 261 + 153 + 60 + 27

DPPC:PG:PA (7:3:1) 2.5 mg m1-1 + 85 [222] + 60 + 4 + 5

Most but not all consider SP-D (formerly CP-4) as the fourth protein associated with

pulmonary surfactant [121, 162]. This protein is synthesised in alveolar type II cells and the

Clara cells of respiratory bronchioles [41 ]. SP-D is the only surfactant protein that is not lung

specific, since gastric mucosa expresses SP-D mRNA and immunostaining and protein blot

analysis has demonstrated the presence of this protein in mucus-secreting cells of the gastric

mucosa [55]. SP-D is a 43 kD protein and forms a structural subunit as a trimer but may exist

as a dodecamer (516 kD) in the alveolar lining fluid. The primary structure of both rat [201]

and human SP-D [128] have been deduced from their respective cDNA clones. Both rat and

449

human SP-D are 355 amino acids long with 72% homology. Human SP-D has a short amino

terminal of 25 amino acids containing two vysteins followed by a collagen-like domain

comprising 177 amino acids with 59 uninterrupted repeats of Gly-Xaa-Yaa and a carboxyl

terminal of 153 amino acids that represents a carbohydrate recognition domain [161, 181 ].

Interestingly, both rat and human SP-D show greater homology to conglutinin (66% homology)

than to SP-A (36% homology). However, SP-D's metabolism is different from that of SP-A

[49]. In principle, SP-D levels do not increase in all conditions leading to SP-A level increase

(such as chronic lung diseases). Only a fraction of 10% of SP-D is associated with pulmonary

surfactant phospholipids [162]. Apparently, SP-D functions are confined to normal host

defence [87] and abnormal immune responses [223]. SP-D properties were reviewed recently

[40].

2.2. PULMONARY SURFACTANT REGULATION

The overall synthesis of surfactant phospholipids and surfactant proteins is under multifactorial

control, and it is regulated (i.e., increased or decreased) by substances including

glucocorticoids, thyroid hormone, retinoic acid, insulin, growth factors, sex hormones,

catecholamines, and cAMP. In cultured lung cells, glucocorticoids stimulate fatty acid

synthesis, phosphatidylcholine synthesis and both SP-B and SP-C synthesis [137]. After

several clinical trials, it is a standard procedure to prevent a respiratory distress syndrome

(expected surfactant deficiency in prematurity) with glucocorticoids antenatally (from 23 rd

week of gestation on [111]), and possibly also postnatally [224]. In addition to corticoid

induced gene activation, thyroid hormone treatment may in future be used to amplify this effect

[50, 73, 115]. Retinoic acid, which generally stimulates cell maturation, also enhances

surfactant protein gene expression in vitro [23] (and alveoli formation in rats [133]). All

surfactant components are synthesised in lung type II cells [15, 41,221, 234]. Some of the

surfactant specific proteins are also produced in Clara cells. This surface-active material is

involved in stabilising small airways, in facilitating mucociliary movement, and in modulating

host defence mechanisms, but its functional importance for the pulmonary surfactant system is

not known.

It is remarkable that the working lung regulates surfactant release, in analogy to the developing

lung regulating surfactant synthesis. In addition, the surfactant proteins and among these

especially SP-A and SP-C, also take part in (feedback) control regulation of surfactant uptake

and degradation [ 106]. In alveolar type II cells, surfactant is stored in 100-150 lamellar bodies

per cell. Multiple signals result into secretion of surfactant into the alveolus. One secretion

stimulus is stretching of type II cells, a stimulus that typically occur during breathing and

pronounced with sigh [237]. Such stimulus-secretion coupling probably helps to match

ventilation-perfusion distribution in mammalian lungs [ 145]. Surfactant is secreted as lamellar

body into the alveolar lining fluid but converted into various macromolecular structures

450

including tubular myelin, the structure which may be one of the immediate sources of

phospholipids for the surface film [232]. The structural changes of surfactant are probably

triggered by calcium [89].

The kinetics of surfactant phospholipids moving from the various structures in the alveolar

subphase to and into the surface film is not yet known, but the surface-related DPPC flux during

breathing has been estimated. The amount of DPPC adsorbing in the time of a single inspiration

is about the content of 500,000 lamellar bodies [88] (Table 2), as estimated from the

distribution and flux in the adult rat lung [242] assuming that all of the DPPC secreted into the

alveolus becomes part of the surface film. This also implies the requirement for a tight

regulation of the extracellular surfactant pool, i.e. production and elimination [240]. Type II

cells under resting conditions secrete about 10-40% of their total intracellular surfactant content

per hour [241, 245]. An equally active clearance pathway must exist to prevent excessive

accumulation of surfactant in alveoli. Three principal routes of surfactant clearance have been

identified [44, 207]. The major route of surfactant clearance is re-uptake into type II cells. This

process is probably controlled through SP-A-type II cells interactions via a specific cell-surface

receptor and might lead to reutilization of at least some of the material. The second pathway is

through alveolar macrophages that remove surfactant via phagocytosis (about 10%) [ 16]. The

smallest amount of surfactant is cleared up the airways (about 1%). Another metabolic

pathway, which is less well known quantitatively, is the extracellular conversion between

different lipid aggregate structures in alveolar lining layer through working alveoli and

converting enzymes [216]. In terms of material turnover, the regulation of surfactant

metabolism enables that the biological half-life of phosphatidylcholine is over 40 h long in

surfactant. For more detailed information on surfactant metabolism and turnover the reader is

referred to recently published reviews [ 15, 17, 18, 40, 57, 88, 241 ].

3. BIOPHYSICAL FUNCTIONS OF PULMONARY SURFACTANT

3.1. BA CKGR O UND

A quite distinctive feature of the pulmonary (and maybe also of the extrapulmonary) surfactant

as a biological system is that it is functionally active and can be studied in a cell-free

environment, and that its components do not require a cell machinery for providing energy,

enzyme-coupled metabolising, ion gradient conserving, or receptor-mediated signalling. The

surfactant is formed and transformed in various macromolecular aggregates, it buffers

unexpected contaminants, and responds to physical signals such as tidal volume change. Of

course, cells are needed to synthesise and secrete the surfactant components and to form an

anatomical space for the distribution of the surfactant molecules.

451

In this section, we will describe the surface-related properties of surfactant as deduced from

studies on lungs and from in vitro analyses. Background information on this topic shall help to

understand some of the problems that hindered a complete understanding of the mechanical

effects of pulmonary surfactant. A significant interaction between alveoli and surface-active

material probably begins already in lung morphogenesis [129], which is also under complex

genetic control [ 19]. From a central core portion, the alveoli are created by dichotomic space

divisions of which about 2 TM occur before and 2 6 o c c u r atter birth. These alveoli share essentially

the same structural properties but may vary slightly in volume and surface area [183]. In a

volume of only 1 ml of lung tissue, the surface area available for gas exchange is about 300 cm 2.

As indicated above, the alveolar lining layer and its surface are presently at the centre of interest

with regard to the surface tension properties of the lung. There are a few well-known models

that try to explain the functioning of this surface in situ. Pattle suggested from his observations

with very stable bubbles in lung foam that only insoluble proteins would contribute to the

surface film [158]. He suggested that surface tension was almost nullified in the bubbles.

Clements extended these findings to dynamic situations like breathing. He established a lung

model in which surface tension varies with surface area [32]: A phospholipid film would

constitute a complete surface coverage with varying molecular packing density. In another

model, the phospholipid film not only covers the alveolar surface, but is also suspended the

form of foam bubbles in the openings to the airways [ 183]. Gas exchange between alveoli and

airways would occur by diffusion through this suspended film, for which there is no generally

accepted evidence. If the alveolar surface is regarded as a solid-rigid membrane that resists

alveolar collapse [ 11 ], the difficult notion of near zero surface tension of an aqueous alveolar

lining layer can be avoided. A recent, but not yet settled model turns away from the

aforementioned monolayer theories and proposes that the surface film may be composed of

phospholipid multilayers [188, 249]. The surface layer may become homogeneously thick to

form multilayers upon compression and on expansion, may convert to monolayer form [ 171].

By surface area compression, phospholipid and water molecules compete more for surface

occupation, and water molecules are expelled under high lateral pressures. Although in some

experimental settings super-physiologic area compressions may have been used, phospholipids

in this way definitively accumulate in the surface and lower surface tension.

Is it not clear, if surface tension in the normal lung can be higher than about 30 mN m l at all.

The thickness of the aqueous subphase may vary between alveoli [14] and the uniformity of

surface tension throughout lung has not been established. On the other hand, surface tension

values measured directly in various lung alveoli were comparable [187]. In an inspiratory

position at 80% TLC, surface tension may rise to about 18 mN m l [ 187] (Fig. 2) and at end

452

expiratory position, surface tension was found to be less than 2 inN m ~, and the capillary

meshwork bulged into the alveolus, as characteristic for low surface tension (Fig. 3). . . .

Fig. 2: Morphology of several alveoli (A) and abundant capillaries (C) in a rabbit lung rinsed with FC-77 and fixed at 40% TLC during deflation. Air spaces are wide, surfaces are smooth, and fibres in the alveolar opening are stretched. Total surface area compares to the low values of detergent-rinsed lungs. Surface tension is

16 mN m -1. Surfactant appears to be arranged in multiple pleats, as indicated by the arrangement of the capillaries. From reference [7] with permission.

Then, there are corners and recesses which exhibit high capillary forces (due to the small

curvature radius), but which are effectively compensated for by surfactant fluid accumulating in

the same place [8]. This corner surfactant accumulation also protects from blood capillary

rupture [141 ]. Conversely, the large planes between the corners have a high curvature radius

and thus exert much reduced capillary forces. This results in an interaction of the alveolar

surfactant film with the alveolar geometry, as further explained in the next paragraph by model

investigations.

Interactions between the surfactant film and the geometry of lung structures during

development have been studied in rats [142]. They revealed age-dependent changes in the

respiratory system in which these interactions became apparent. Results of animal models are

customarily thought suitable to generalise on humans, which might not always be justified.

Some known properties of human lungs, airways, and alveoli are given in Table 1.

453

Fig. 3: Morphology of several alveoli (A) and abundant capillaries (C) in a rabbit lung filled with hexadecane and fixed at 40% TLC during deflation. Alveolar surfaces are corrugated by the bulging capillaries and total surface area is large, which are situations comparable to that of saline-filled lungs. Surface tension is <1 mN m 1 as measured in experiments with hexadecane and lipid extract surfactant in a captive-bubble-surfactometer. From reference [7] with permission.

In excised lungs, volume changes during breathing were defined as oscillatory changes in order

to investigate elasticity and resistance [205], which are mechanical properties of the lung.

Elasticity was found decreased in the presence of a surface film. Tissue resistance, but not

airway resistance, increased with oscillating frequency in the absence of a surface film, pointing

to the importance of surfactant-mediated structural properties of alveoli.

In airways as modelled in different experimental settings, the presence of surfactant reduces

compliant and meniscus collapse. Compliant collapse occurs in airways colliding under reduced

pull by surrounding structures, e.g. at low lung volumes. However, the time to compliant

airway collapse is several times larger then the time of expiration because the surface (Gibbs)

elasticity is amplified by surfactant [ 153 ]. The expiration time constant in a given individual is

the product of lung compliance and resistance, and at least three times this constant is necessary

to empty the lung without air trapping. In respiratory distress syndrome, this constant is

reduced (as surfactant is deficient and compliance reduced), favouring shorter times of

454

expiration, whereas in meconium aspiration, it is increased (as resistance increases by

obstruction), requiring longer times of expiration that endanger airway collapse as pointed out

above. Meniscus collapse is airway obstruction by liquid bridging, which is also affected by

surface elasticity. In such a case, the measured pressure for re-opening the airway was found to

correspond to a surface tension of about 35 mN m 1 in the airway [ 143 ], which is similar to the

surface tension measured in large airways [62]. In another study, surfactant in a narrow tube

provided a continuous fluid flow, whilst other substances produced a blocking, staggered fluid

movement [127]. Airways' and parenchymal properties were simultaneously modelled using

different foam preparations under shear stress [163]. It was studied how tethering forces

interact with airways and correctly estimated that elevated transpulmonary pressures are

required during breathing in respiratory distress syndrome and emphysema (i.e., loss of alveoli).

In alveoli, interactions between geometry and surfactant function have been shown

experimentally as well. Basic concepts started from a Laplacian alveolar shape, that is a more or

less spherical configuration. This lead to the provisional, but important notion that surfactant

induces reduction of surface tension and that thus the pressure required to keep an alveolus

open decreased to values feasible in man [69]. Then, the importance of the interdependence of

alveoli was recognised [ 135] and indeed, alveoli do not empty into one another due to different

radii because they are fixed side to side to anatomical structures. Accordingly, alveolar shape

becomes irregular, probably polyhedral. A noted model of surfactant action by Wilson [236]

was derived from electron micrographs: Interstitial sheets between alveoli are suspended from

alveolar openings and are stretched to planes by surface tension. The overall mechanical lung

properties derive from the sum of surface tension effects, and not significantly from the planes'

tissue properties. Experimental data are limited, but some support comes from a theoretical

study, which calculated that hysteresis in lung volume change is the average of hysteresis

behaviour of all microstructural components [ 113]. During lung volume expansion, lung recoil

pressure increases to enable expiration without additional force, and this pressure is

proportional to the value of (actual surface tension)/(total lung volume) 3. Assuming small non-

homogenous volume changes, low surface tension values (<10 mN m -1) are required for

positive compliance values (which are a prerequisite to lung stability) and to overcome recoil

pressures (the sum of tissue and surface forces) [206].

It is also possible to analyse lung pressure-volume curves without assuming a certain alveolar

geometry [203]. So derived alveolar surface tension-lung volume curves showed that most of

the behaviour of lungs is attributable to that of the surface layer. This is supported by a study on

dog lungs in vivo that employed a measurement technique for the ratio of surface area to lung

volume [1]. It was found that the surface layer displays a larger stress hysteresis under area

changes than the lung tissue under volume changes.

455

3.2. SPREADING, ADSORPTION, Q UASI-STA TIC AND DYNAMIC CYCLING

The biophysical behaviour of pulmonary surfactants includes spreading, adsorption, and non-

equilibrium states that could be measured in vitro during quasi-static and dynamic cycling.

Spreading and adsorption lead to (local) equilibrium states of molecule number by surface area.

In the lung, the spreading process occurs probably only when exogenous surfactant is

administered into larger airways or when endogenous liquid or solid barriers in airways are

removed [53, 62]. The way by which exogenous surfactant spreads could be described as

follows: A bolus droplet volume of surfactant extends into the airways' spaces, thereby

increasing its surface area and decreasing its surface curvature radius from the droplet radius to

the surrounding airways' radius, which entails a reduction of surface tension. Eventually,

exogenous surfactant is intended to reach the alveoli (in new-borns after estimated 12 s [53])

and to spread onto the alveolar surface, in order to substitute for endogenous surfactant which

normally would form the alveolar film by adsorption from the subphase. Due to the endogenous

pulmonary surfactant present in airways (although in lesser amounts than in alveoli), equilibrium

surface tension is normally about 32 mN m ~ in airways [62, 123]. This airway surfactant can

spread onto solid particles and other airway contaminants to facilitate their clearance [62].

Substances are pulled into the subphase by surface forces for uptake by macrophages. The

thickness of the airway surfactant subphase is non-homogenous and it is varying with airway

diameter and lung volume [244], as is also the case with alveoli. The interaction between

endogenous and exogenous has not yet been quantified or resolved [75]. The properties of

pulmonary surfactant may also be exploited to deliver pharmaceutics by a spreading surfactant

carrier [214, 23 9].

While the spreading process occurs in vivo presumably during therapeutic interventions, the

adsorption process occurs during every breathing cycle. The term adsorption rate does not refer

to the actual number of molecules inserted into the interfacial film, but is used widely for the

change of surface tension by time, which can readily be measured in vitro. It is assumed that this

surface tension change follows (at least empirically) an exponential decay function of the time,

corresponding to a (slow) one-phasic behaviour of lipid dispersion adsorption and a biphasic

behaviour of the adsorption of lipids with proteins or with organic solvents [222]. In vitro, the

adsorption process is expected to be predominantly diffusion-controlled, because the subphase

is stirred in most experimental settings in order to increase convection and to decrease

diffusion-resistance. Still, as comparative data are not available, the dynamic surface tension

during adsorption is often described only by empirical functions (Table 4). In vivo, adsorption

also depends on the material flow in the subphase. This material convection has been found to

be sufficiently large in airways [53, 244] and alveoli [14] to respond to pressure and volume

changes that induce the adsorption process.

456

Table 4: Parameters employed for quantifying surface properties. Calculations adapted from the respective reference. QSI, quasi- static isothermal cycling; DYN, dynamic non-equilibrium cycling; ADS, adsorption isotherms; PVl, lung pressure-volume isotherms. ~,, surface tension; r, film density; A, surface area (Ao, before adsorption); t, time; eq, equilibrium; k~s, adsorption rate constant; kd,,s, dedsorption rate constant; ~,~, half-time of adsorption process i.

Parameter Application Calculation

Extremes of surface tension QSI, DYN, ADS 7"min , }"max

Reference

Surface tension span QSI, DYN, ADS AX = )/max - Ymm

Monolayer collapse rate QSI l dA [98] at constant surface tension kA = D . A ~ Monolayer collapse rate QSI l f , ' / ~ [98] at constant surface area ky - �9

Yeq - Y o o

Monolayer compressibility QSI, DYN 1 dA [98] Cy - ~ . Aay

Surface elasticity (e) ADS Ao ~' = A y - ~

d A

Stability index (is) 1 QSI ' ] i s = 2 " ) 'max - Ymin

i }"max + }/mm

[35]

Rate constant ADS of adsorption At' . . . . Ir I t - rm'n " ~ t - kads m , n (Feq - Ft )dt

[174]

Empirical kinetic ADS of adsorption (I) Y, = Yeq + Z a'e-t/k' " t,,/2 = 0.69/k,

I

[97, 222]

Empirical kinetic ADS of adsorption (11)

AR, adsorption rate, is the inclinination of the tangent to the sigmoid curve phase of the surface tension vs. time curve ([AR]=mN m-Is-l). x, lag time is intercept of AR tangent with time axis at 7=0 mN m -1 ([T]=s).

[173]

Tissue elastic force PVI

Empirical kinetic of adsorption (111) and desorption

PT = ~ Zmax

E, coefficient of tissue elasticity (500 mN m-l); Vpt, alveolar volume at a given transpulmonary pressure; Vmax, maximum alveolar volume

PVI d n kad s = kae s =

d t A . ( y t - y , q )

k a d s = 5 - s . c m 2 . m N

kay,,. = 1 . 1 0 . 6 m

s . c m 2 �9 m N

[35]

[66]

Lift-Off QSI Surface area reduction during first compression at which isotherm deflects from baseline [%]

[222]

Metastable Region QSI Surface area proportion where surface tension remains below 10 mN m -1 [%]

[222]

457

An effective surfactant must adsorb rapidly, within a few seconds to around 25 mN m "l, the

plateau or equilibrium surface tension [67]. Rapid adsorption toward equilibrium means that

adsorption will keep the surface tension in the lung near the equilibrium value during

inspiration to total lung capacity (TLC). This will keep the surface component of lung recoil

pressure as low as possible, which helps to minimise the work of breathing [ 176]. Goerke [65]

pointed out that the capacity for surfactant to adsorb rapidly is indirectly observed on ventilated

patients in the operating room, where the anaesthesiologist sometimes is forced to ventilate a

patient at low tidal volumes for prolonged periods.

This mode of ventilation is often accompanied by an increase in inflation pressure that is likely

due to microcollapse related to film overcompression and loss of surface activity. In this

situation, a single breath to TLC restores normal inflation pressure by allowing new surfactant

to adsorb within seconds to the depleted film. The restored film is then compressed again to a

low tension as the original low pressure at functional residual capacity (FRC) is approached

during deflation. In a closely related situation in ventilated dogs, Mead and Collier [136]

restored volume at FRC and compliance by a single breath to TLC.

The prerequisite of fast adsorption into the air-liquid interface for an effective surfactant is

further supported by the recent animal experiments reported [151] indicating that lung

surfactants which show faster adsorption retes in vitro, produce lower mean airway pressure

and larger lung compliance. Adsorption studies conducted on purified natural surfactant and on

bovine and porcine lipid extract surfactant have demonstrated that, at subphase concentrations

of 1 mg m1-1 or higher of phospholipids, all of these preparations achieve the equilibrium

surface tension of 22-25 mN m -1 within a few seconds [ 189, 195, 197].

Spreading and adsorption refer to surfactant material movement into the surface, a process that

takes place also using quasi-static and dynamic cycling techniques. With cycling techniques,

also material movement within the surface can be studied, at surface tensions below those

achieved by spreading or adsorption. Measurements of surface tension has been made by two

techniques: (1) by placing microdroplets of a variety of test fluids onto alveolar surfaces [ 187,

193, 194] and (2) by washing procedures involving pressure-volume studies after filling the

lungs or covering the alveoli with test liquid films [6, 61, 206]. The microdroplet method

showed that surface tension varied with lung volume. For excised lungs held at 37~ and

deflated to FRC (Table 1), the minimum surface tension is below 1 mN m -1 at FRC.

Remarkably, this minimum surface tension can be obtained under quasi-static conditions; that

458

is, a minimum compression rate is not required to achieve near zero tension. In addition, this

low surface tension is extremely stable [ 187], similar to values inferred from pressure- volume

curves on air-filled lungs where the surface tension rose only 1-2 mN m -1 during 20 min at

FRC [ 111 ]. The maximum quasi-static surface tension during a pressure-volume loop between

FRC and TLC is approximately 30 mN m -1 [8, 194].

The direct measurements of alveolar surface tension agree well with results obtained by the

model of Wilson and Bachofen [236], which is based on an energy analysis of well-defined

structure-function relations of isolated, air-filled, and perfused lungs. These studies have been

possible because of the excellent preservation for microscopy of the lung microstructure by the

perfusion fixation technique in combination with morphometric analysis perfected by Weibel

and his associates during the past 20 years. The combination of morphometric analysis with

determination of alveolar surface tension has allowed Bachofen and his co-workers [8] not

only to determine the surface tension at FRC (minimum) and TLC (maximum) but also at

intermediate lung volumes on the deflation as well as the inflation limb of the pressure-volume

curve. For lungs inflated from zero pressure, the surface tensions on the deflation limb were

considerably lower than on the inflation limb: at FRC about 1 mN m -I on deflation, whereas on

inflation the tension was approximately 6 mN m l . The corresponding alveolar surface areas

were 63 and 47%, respectively, of the area at TLC.

These values demonstrate that at low volumes, at the surface tension of 6 mN m -1, the alveolar

surface area is approximately 16% smaller than at the surface tension of 1 mN m -~, suggesting

that surfactants" having minimum surface tensions of 5 or 6 mN m -I on deflation will not

sufficiently stabilise alveolar surface area. This may have important implications for the quality

control for a surfactant to be used for replacement therapy. Only if minimum surface tension at

low volumes can reach a value near zero will there be the optimal alveolar area available for

the gas exchange. The correlation studies between lung microstructure and alveolar surface

tension further demonstrated that the area compression is about 30% from TLC to FRC,

corresponding to surface tensions of about 30 and 1 mN m -l, respectively [8]. In addition, the

studies performed on excised rabbit lungs demonstrated that pressure-volume and surface

tension-area hysteresis is substantial only in excised lungs inflated from volumes lower than

FRC or from about zero transpulmonary pressure. In healthy, normally breathing lungs, the

pressure-volume hysteresis is relatively small [6]. This corresponds to the very small hysteresis

is the lining film is re-expanded from FRC [8]. Thus, a good quality surfactant investigated in

459

vitro after a few cycles that feature compressions only to minimum surface tension, that is,

without overcompression and collapse should not show appreciable hysteresis during quasi-

static or dynamic cycling [ 189, 197]. However, a reciprocal correlation of hysteresis area with

surfactant function has been found in surfactant inhibition, e.g. in respiratory distress syndrome

[167]. The term quasi-static is used here in accordance with physiological manipulations

employed for pressure-volume relations for excised lungs. Quasi-static cycling involves a

series of small discrete alterations in bubble area where the surface film is allowed to partially

"relax" during the compression-expansion process. These processes are not quasi-static

processes as defined in thermodynamics where a quasi-static process is an idealised concept

that is defined as a succession of equilibrium states [28].

In addition, minimum surface tension of approximately 1 mN m "1 can be readily obtained by

reducing the surface area of the film by 20-25% during initial quasi-static or dynamic film

compression. The compressibility of these films at 15 mN m -1 is less than 0.010 mmN -1, which

is close to the value of pure DPPC film [67, 190]. Since lipid extracts and natural surfactants

contain only about 50% DPPC, the low compressibility observed with adsorbed surfactant

films during the initial compression suggests that surfactant films are highly enriched in DPPC

during film formation [197]. It appears that lipid extract surfactants, which do not contain

SP-A but do contain SP-B and SP-C, have surface activities essentially indistinguishable from

that of natural surfactants containing SP-A when tested at high concentrations. However, at

lower lipid concentrations, SP-A has the capacity when combined with SP-B and SP-C of

accelerating the adsorption process and enriching the film in DPPC, thereby reducing

compression requirements and stabilising the film [197]. Thus, SP-A, in addition to the many

functions described at the beginning of this chapter, plays also an important role in maintaining

mechanical stability of the terminal air spaces of the lung when the surfactant concentrations

are limiting.

The structure of surfactant films and the manner in which they can become enriched in DPPC

is not known. However, recent studies by Bachofen and co-workers [188] have indicated that

the surfactant extracellular film present in rabbit lungs in vivo is too thick to be a monolayer.

These observations have led to the concept that the surfactant surface phase is composed of

more than a single monolayer and that the surplus material might act as a surface-associated

reservoir for the formation of a surface-active film [188]. In depletion experiments in which

the original lipid extract surfactant suspension was removed by washing with salt solution, we

460

showed that material in excess of a single monolayer remains in a "surface-associated

reservoir". Whether all of this excess material can be moved from the reservoir into the surface

action film is not known. The results from the depletion experiments using lipid extract

surfactant show that there is material for at least three monolayers in the interface that can

contribute to the surface activity. Quite similar to previous results from experiments with lipid

extract surfactant and SP-A [ 195], the addition of SP-A to lipid extract surfactant appears to

assist in maintaining the quality of the surface-active film. The film compressibility was lower

and there was a total amount of surfactant material equivalent of about five monolayers at the

bubble interface compared to about three monolayers for lipid extract surfactant without SP-A.

The addition of SP-A could lead to the formation of tubular myelin structures [209, 235].

Tubular myelin remains closely associated with the interface, as shown in electron microscopic

studies by Bastacky et al. [14]. Still, the importance of SP-A has become less clear after

finding that SP-A deficient mice have a functionally normal respiratory system, even though

tubular myelin is deficient [119].

Alveolar surfactant also reduces the transpulmonary pressure necessary for maintaining a

certain functional residual capacity in the lung to about 5 kPa. From a study on artificially

ventilated animals, surfactant was found to increase the lung's passive elasticity (i.e., static

compliance, Table 1), and the maximum dynamic compliance due to steeper pressure-volume-

curves at low lung volume [70]. This is an analogue to the surface tension-surface area

tracings in which with adequate methods, there is a large surface tension change induced by a

rather small surface area change (up to 1.4 mN m 1%-1 [ 188]). During tidal breathing at rest,

the surface area change induced by the volume ventilated is only about 4-12% [183].

The ability of serum proteins to adversely affect pulmonary surfactant and the physiological

consequences of such inactivation are well documented [ 100, 180]. Holm [ 100] suggested that,

by adsorbing rapidly, serum proteins could monopolise the surface, thereby interfering with

phospholipid adsorption. It appears that some serum proteins interact with surfactant

components, although the inhibitory mechanism is not clear. Addition of SP-A to lipid extract

surfactants counteracts the inhibitory effects of serum proteins, presumably due in part to the

ability of SP-A to enhance adsorption [36, 218]. Washout experiments with the captive-

bubble-surfactometer revealed that in the presence of serum proteins, the area of the

compressed bubble at minimum surface tension remains essentially unaltered with repeated

cycling. This demonstrates that under these conditions additional surface-active material

461

cannot be recruited into the surface-active film. Consequently, there is either no formation of a

surface-associated reservoir in the presence of calf serum protein or the material associated

with the interface cannot be transformed from an inactive state into a surface active state. It

remains to be demonstrated by morphological analysis whether or not this inactivation has

structural consequences, i.e. whether the film shows increased polymorphism compared to

normal surfactant layers. The functional inactivation of most surfactants by fibrinogen (in

different dose-dependent manners) [198] occurs by plasma leakage into the alveolus. This

entails clotting of surfactant phospholipids together with fibrinogen into large insoluble

structures, which may at least in vitro be reversed by clot-lysis so that surfactant function is

restored [76].

3.3. METHODS

As indicated before, advances in assessing surface properties of pulmonary surfactant films

have become feasible by the recent development of methods that more reliably can quantify the

working principles of pulmonary surfactant. New methods exist not only for classical,

biophysical function surface studies in vitro (captive bubble surfactometer, Brewster angle

microscopy [45], laser ellipsometry [171]), but also for surface studies in vivo (laser light

scattering stereology [ 138], alveolar bubble microscopy and cinemicroscopy, advanced in vivo

lung function testing). The identification of molecular species behaviour has become possible

with methods such as infrared spectroscopy [46], UV/IR-excitation of native phospholipid

layers or preparations [39, 117], and fluorophore release from lipid aggregates [ 172].

With regard to the methods employing bubbles or suspended flat films, it is to be recognised

that both types of surfaces are entirely flat from the point of molecular size in proportion to

surface size. As the pulmonary surfactant is composed of water insoluble molecules, its

mechanical properties such viscosity and elasticity are probably best studied using oscillating

area changes [125, 155], which can also be applied to lung ventilation to investigate tissue

resistance and lung elasticity [205]. Still, these properties are not easily correlated to a certain

lung sub-function. At least, it can be stated that there are increasingly fewer applications of the

Wilhelmy balance that has been criticised for a number of years [13].

Besides studying supported mono- or multilayer films (as described above), pulmonary

surfactant suspensions may also give rise to foam, which consists of stable self-organising

films. These foam films may in fact correspond to in vivo alveolar bubbles and films [184],

462

and in vitro film experiments are predictive of in vivo physiological function [54]. Recently,

molecular processes in foam films similar to that in supported monolayer films were observed

[122, 146].

Still, a complete understanding of the mechanical effects of surfactant in the lung has been

hindered by several methodological problems: The behaviour of surfactant films in vitro, e.g.,

in the Wilhelmy balance or in the pulsating bubble surfactometer, is different from that of the

alveolar lining layer [67]. Although minimum surface tensions below 5 mN m 1 in the

Wilhelmy balance or below 1 mN m 1 in the pulsating bubble surfactometer can be obtained at

a relatively high speed of film compression, these low surface tensions have much less stability

(surface tension versus time at minimum tension) than the alveolar film. The alveolar film

itself has a far greater stability, as calculated from pressure-volume studies on lungs of intact

animals or on excised lungs [ 105, 136]. These results are in line with the surface tension by the

microdroplet spreading technique [187, 194]. The interesting collapse dynamics at minimum

surface tension have not yet been resolved. It was suggested that during the surface expansion,

higher surface tension values correlate with the amount of surface (over-) compression, that is

the material excluded [79, 221 ].

An examination of the results on surfactant activity obtained with various methods

demonstrates clearly that the criteria for "good" surface activity depends on the instrument

employed and for a particular instrument on the way a certain test is performed. Although the

therapeutic efficacy of a particular surfactant preparation has to be tested in animal models of

surfactant deficiency [60], in vitro studies provide valuable guidelines for choosing or

preparing a physiologically active surfactant. A standardised approach with a particular

instrument may give results with regard to surface activity that correlate well with

physiological parameters measured in animals even if the approach does not mimic the

situation in the lung.

Unfortunately, the standardisation of a particular approach to measure surface activity for a

given instrument is quite difficult. This can be demonstrated for the Wilhelmy balance and the

pulsating bubble surfactometer. Goerke and Gonzales [68] carefully primed the Teflon walls

and barrier of their balance with long-chain saturated phosphatidylcholine and a solution of

lanthanum chloride to reduce film leakage. In addition, continuous Teflon ribbons standing on

edge were used in the 1960s by Clements in rectangular troughs [34] and with rhombic frames

[185]. In our experience, even these ribbon balances do not contain films from natural

463

surfactant or lipid extract surfactant unless they are primed by over-compressing (collapsing)

several films of disaturated phosphatidylcholine at near zero minimum surface tension prior to

the experiments. Thus, minimum surface tension and film stability depend on the careful

preparation of the surface balance because films that produce low and stable minimum surface

tensions in a leak proof system show instability in a leaky balance.

Similarly, special precautions have to be taken when the pulsating bubble surfactometer is

used. Putz and co-workers [176] recently improved the performance of the pulsating bubble

surfactometer by keeping the capillary of the sample chamber dry. They demonstrated by video

observations that this modified instrument performs better than the unmodified version,

because keeping the capillary dry prevents surface films from occupying part of the capillary

surface during cycling. Additionally, the preparation of the surface-active material to be tested

has to attain a certain degree of purity and contamination by foreign substances, e.g. from

experimental containers, have to be avoided. This wanted preparation may be called a surface-

chemically pure but there is no known standardised procedure with reference to pulmonary

surfactant.

It appears that the film area changes, especially in the unmodified pulsating bubble

surfactometer, during pulsations are not well defined, which may lead to erroneous

interpretations of the results obtained with this instrument. For example, some surfactant films

have to be pulsated in the pulsating bubble surfactometer for 10-20 minutes at 20 cycles min l

in order to obtain near zero surface tensions (e.g., see ref. [247]), whereas the same surfactant

films produce near zero surface tensions on a first quasi-static or dynamic compression in the

captive bubble surfactometer [195]. The necessity for prolonged pulsations to achieve near

zero surface tension in the pulsating bubble surfactometer for certain films compared with

those that need only a few pulsations to obtain low minimum surface tensions is usually

interpreted as the necessity to produce refinement of the dipalmitoyl-phosphatidylcholine in the

former films, whereas the latter films show already a behaviour close to that of DPPC. Since

the amount of film area compression might be uncertain, especially in the unmodified pulsating

bubble surfactometer, the notion of film refinement or squeeze-out or film purification has to

be carefully scrutinised.

The recent use a leak proof captive bubble system [189] has facilitated the collection of more

reliable data describing surface tension-area relations than by using more conventional

methods. By comparing the results obtained in the pulsating bubble surfactometer with those in

464

the captive bubble surfactometer, Putz et al. [176] concluded that the captive bubble

surfactometer should be used where accurate film area compressions are required to explain

mechanisms that govern surfactant surface activity near zero surface tension, film stability,

film compressibility, and hysteresis. Theoretic considerations of bubble deformations have

been extended to captive bubbles [2]. Separate versions of a captive bubble system have been

described recently [97, 175]. Both instruments have been further developed for convenience

and reliability [ 176, 186, 197] and produce equivalent results.

3.4. PERSPECTIVE

The outlook on forthcoming studies on pulmonary surfactant is promising, as even more

researchers are using more methods to disentangle the complexity of this system. In addition,

theoretic grounds are now paved by considering aspects of mass and energy transfer, defining

interfacial energy models, computing interfacial molecular arrays, and integrating anatomical

and functional lung components. These may eventually clarify the actual compositions of the

alveolar surface, its sub-surface, and the subphase at a given functional situation of the lung.

4. ACKNOWLEDGEMENTS

This work was supported by the Deutsche Forschungsgemeinschaft (Pi 165/5-2).

5. REFERENCES

1. Akahori T, S Suzuki, M Suzuki et al.: Dynamic behavior of alveolar surface-to-volume ratio in live dogs by light-scattering stereology. J Appl Physiol 1993, 75:1624-1629

2. Andrade JD, RN King, DE Gregonis et al.: Surface characterization of (poly)hydroxyethyl methacrylate and related polymers. I. Contact angle methods in water. J Polym Sci: Polym Symp 1979, 66:313-336

3. Ansfield MJ and BJ Benson: Identification of the immunosuppressive components of canine pulmonary surface active material. J Immunol 1980, 125:1093-1098

4. Ashbaugh DG, DB Bigelow, TL Petty et al.: Acute respiratory distress in adults. Lancet 1967:319-323

5. Avery ME and J Mead: Surface properties in relation to atelectasis and hyaline membrane disease. Am J Dis Child 1959, 97:517-523

6. Bachofen H, J Hildebrandt and M Bachofen: Pressure-volume curves of air- and liquid-filled excised lungs - surface tension in situ. J Appl Physiol 1970, 29:422-431

7. Bachofen H, S Schtirch and F Possmayer: Disturbance of alveolar lining layer: effects on alveolar microstructure. J Appl Physiol 1994, 76:1983-1992

8. Bachofen H, S Schtirch, M Urbinelli et al.: Relations among alveolar surface tension, surface area, volume, and recoil pressure. J Appl Physiol 1987, 62:1878-1887

9. Ballard PL, R Ertsey, LW Gonzales et al.: Transcriptional regulation of human pulmonary surfactant proteins SP-B and SP-C by glucocorticoids. Am J Respir Cell Mol Biol 1996, 14:599-607

10. Ballard PL, LM Nogee, MF Beers et al.: Partial deficiency of surfactant protein B in an infant with chronic lung disease. Pediatrics 1995, 96:1046-1052

465

11. Bangham AD: Lung surfactant: how it does and does not work. Lung 1987, 165:17-25

12. Bangham AD, CJ Morley and MC Phillips: The physical properties of an effective lung surfactant. Biochim Biophys Acta 1979, 573:552-556

13. Barrow RE and BA Hills: A critical assessm,~'nt of the Wilhelmy method in studying lung surfactants. J Physiol Lond 1979, 295:217-227

14. Bastacky J, CYC Lee, J Goerke et al.: Alveolar lining layer is thin and continuous: low-temperature scanning electron microscopy of rat lung. J Appl Physiol 1995, 79:1615-1628

15. Batenburg JJ: Surfactant phospholipids: synthesis and storage. Am J Physiol 1992, 262:L367-L385

16. Bates SR and AB Fisher: Surfactant protein A is degraded by alveolar macrophages. Am J Physiol 1996, 271: L258-L266

17. Beers MF and AB Fisher: Surfactant protein C: a review of its unique properties and metabolism. Am J Physiol 1992, 263:L151-L160

18. Beers MF, H Shuman, HG Liley et al.: Surfactant protein B in human fetal lung: developmental and glucocorticoid regulation. Pediatr Res 1995, 38:668-675

19. Bellusci S, Y Furuta, MG Rush et al.: Involvement of sonic hedgehog (shh) in mouse embryonic lung growth and morphogenesis. Development 1997, 124:53-63

20. Benson BJ, S Hawgood, J Schilling et al.: Structure of canine pulmonary surfactant apoprotein: cDNA and complete amino acid sequence. Proc Natl Acad Sci USA 1985, 82:6379-6383

21. Beppu OS, JA Clements and J Goerke: Phosphatidylglycerol-deficient lung surfactant has normal properties. J Appl Physiol 1983, 55:496-502

22. Boggaram V, K Qing and CR Mendelson: The major apoprotein of rabbit pulmonary surfactant. Elucidation of primary sequence and cyclic AMP and developmental regulation. J Biol Chem 1988, 263:2939-2947

23. Bogue CW, HC Jacobs, DW Dynia et al.: Retinoic acid increases surfactant protein mRNA in fetal rat lung in culture. Am J Physiol 1996, 271:L862-L868

24. Boncuk-Dayanikli P and HW Taeusch: Essential and nonessential constituents of exogenous surfactants. In: Surfactant therapy for lung disease. Robertson B and HW Taeusch (eds.). Marcel Dekker, New York 1995, 217-238

25. Bredenberg CE, GF Nieman, AM Paskanik et al.: Microvascular membrane permeability in high surface tension pulmonary edema. J Appl Physiol 1986, 60:253-259

26. Brown ES: Isolation and assay ofdipalmityl lecithin in lung extracts. Am J Physiol 1963, 207:402-406

27. Buchanan SA, MC Mauney, VI Parekh et al.: Intratracheal surfactant administration preserves airway compliance during lung reperfusion. Ann Thorac Surg 1996, 62:1617-1621

28. Callen HB: Thermodynamics. John Wiley, New York 1960

29. Chu J, JA Clements, EK Cotton et al.: Nenoatal pulmonary ischemia. Part I: clinical and physiologic studies. Pediatrics 1967, 40:709-782

30. Clark JC, TE Weaver, HS Iwamoto et al.: Decreased lung compliance and air trapping in heterozygous SP- B-deficient mice. Am J Respir Cell Mol Biol 1997, 16:46-52

31. Clark JC, SE Wert, CJ Bachurski et al.: Targeted disruption of the surfactant protein B gene disrupts surfactant homeostasis, causing respiratory failure in newborn mice. Proc Natl Acad Sci USA 1995, 92: 7794-7798

32. Clements JA: Surface tension of lung extracts. Proc Soc Exp Biol Med 1957, 95:170-172

33. Clements JA: Pulmonary edema and permeability of alveolar membranes. Arch Environ Health 1961, 2: 280-283

34. Clements JA: Sixth Bowditch lecture: Surface phenomena in relation to pulmonary function. Physiologist 1962, 5:11-28

35. Clements JA, RF Hustead, RP Johnson et al.: Pulmonary surface tension and alveolar stability. J Appl Physiol 1961, 16:444-450

466

36. Cockshutt AM, J Weitz and F Possmayer: Pulmonary surfactant-associated protein A enhances the surface activity of lipid extract surfactant and reverses inhibition by blood proteins in vitro. Biochemistry 1990, 29: 8424-8429

37. Comroe JH: Premature science and immature lungs. Am Rev Respir Dis 1977, 116:127-135

38. Cormier Y, E Israel Assayag, M Desmeules et al.: Effect of contact avoidance or treatment with oral prednisolone on bronchoalveolar lavage surfactant protein A levels in subjects with farmer's lung. Thorax 1996, 51:1210-1215

39. Creuwels LA, LM van Golde and HP Haagsman: Surfactant protein B: effects on lipid domain formation and intermembrane lipid flow. Biochim Biophys Acta 1996, 1285:1-8

40. Crouch E, P Brown-Augsburger, K Hartshorn et al.: Molecular assembly and multimerization of surfactant protein D: cellular mechanisms and functional consequences. Appl Cardiopul Pathophysiol 1995, 5:16-18

41. Crouch E, D Parghi, SF Kuan et al.: Surfactant protein D: subcellular localization in nonciliated bronchiolar epithelial cells. Am J Physiol 1992, 263:L60-L66

42. De Sanctis GT, RP Tomkiewicz, BK Rubin et al.: Exogenous surfactant enhances mucociliary clearance in the anaesthetized dog. Eur Respir J 1994, 7:1616-1621

43. deMello DE, S Heyman, DS Phelps et al.: Ultrastructure of lung in surfactant protein B deficiency. Am J Respir Cell Mol Biol 1994, 11: 230-239

44. Desai R, TD Tetley, CG Curtis et al.: Studies on the fate of pulmonary surfactant in the lung. Biochem J 1978, 176:455-462

45. Discher BM, KM Maloney, WR Schief, Jr. et al.: Lateral phase separation in interfacial films of pulmonary surfactant. Biophys J 1996, 71:2583-2590

46. Dluhy RA, KE Reilly, RD Hunt et al.: Infrared spectroscopic investigations of pulmonary surfactant. Surface film transitions at the air-water interface and bulk phase thermotropism. Biophys J 1989, 56:1173-1181

47. Dobbs LG, JR Wright, S Hawgood et al.: Pulmonary surfactant and its components inhibit secretion of phosphatidylcholine from cultured rat alveolar type II cells. Proc Natl Acad Sci USA 1987, 84:1010-1014

48. Drickamer K, MS Dordal and L Reynolds: Mannose-binding proteins isolated from rat liver contain carbohydrate-recognition domains linked to collagenous tails. Complete primary structures and homology with pulmonary surfactant apoprotein. J Biol Chem 1986, 261:6878-6887

49. Dulkerian SJ, LW Gonzales, Y Ning et al.: Regulation of surfactant protein D in human fetal lung. Am J Respir Cell Mol Biol 1996, 15:781-786

50. Durham PL, EJ Nanthakumar and JM Snyder: Developmental regulation of surfactant-associated proteins in rabbit fetal lung in vivo. Exp Lung Res 1992, 18:775-793

51. Enhoming G: Pulsating bubble technique for evaluating pulmonary surfactant. J Appl Physiol 1977, 43: 198- 203

52. Enhoming G, A Yarussi, P Rao et al.: Increased airway resistance due to surfactant dysfunction can be alleviated with aerosol surfactant. Can J Physiol Pharmacol 1996, 74:687-691

53. Espinosa FF, AH Shapiro, JJ Fredberg et al.: Spreading of exogenous surfactant in an airway. J Appl Physiol 1993, 75:2028-2039

54. Exerowa D, Z Lalchev, B Marinov et al.: Method for assessment of fetal lung maturity. Langmuir 1986, 2: 664-668

55. Fisher JH and R Mason: Expression of pulmonary surfactant protein D in rat gastric mucosa. Am J Respir Cell Mol Biol 1995, 12:13-18

56. Floros J, S DiAngelo, M Koptides et al.: Human SP-A locus: allele frequencies and linkage disequilibrium between the two surfactant protein A genes. Am J Respir Cell Mol Biol 1996, 15:489-498

57. Floros J and AM Karinch: Human SP-A: then and now. Am J Physiol 1995, 268:162-165

58. Floros J, R Steinbrink, K Jacobs et al.: Isolation and characterization of cDNA clones for the 35-kDa pulmonary surfactant-associated protein. J Biol Chem 1986, 261:9029-9033

59. Floros J, SV Veletza, P Kotikalapudi et al.: Dinucleotide repeats in the human surfactant protein-B gene and respiratory-distress syndrome. Biochem J 1995

467

60. Fujiwara T and B Robertson: Pharmacology of exogenous surfactant. In: Pulmonary Surfactant: from molecular biology to clinical practice. Robertson B, LMG Van Golde and JJ Batenburg (eds.). Elsevier, Amsterdam 1992, 561-592

61. Ganesan S, SJ Lai-Fook and S Schtirch: Alveolar liquid pressures in nonedematous and kerosene-washed rabbit lung by micropuncture. Respir Physiol 1989, 78:281-295

62. Gehr P, M Geiser, V Im Hof et al.: Surfactant and inhaled particles in the conducting airways: structural, stereological, and biophysical aspects. Microsc Res Tech 1993, 26:423-436

63. Gilliard N, GP Heldt, J Loredo et al.: Exposure of the hydrophobic components of porcine lung surfactant to oxidant stress alters surface tension properties. J Clin Invest 1994, 93:2608-2615

64. Glasser SW, TR Korthagen, T Weaver et al.: cDNA and deduced amino acid sequence of human pulmonary surfactant-associated proteolipid SPL(Phe). Proc Natl Acad Sci USA 1987, 84:4007-4011

65. Goerke J: Surfactant and lung mechanics. In: Pulmonary Surfactant: From Molecular Biology to Clinical Practice. Robertson B, LMG Van Golde and JJ Batenburg (eds.). Elsevier, Amsterdam 1992, 165-192

66. Goerke J: Prediction of mechanical responses to surfactant therapy. In: Surfactant therapy for lung disease. Robertson B and HW Taeusch (eds.). Marcel Dekker, New York 1995,349-369

67. Goerke J and JA Clements: Alveolar surface tension and lung surfactant. In: Handbook of Physiology: The Respiratory System. Macklem PT and J Mead (eds.). American Physiological Society, Bethesda, MD 1986, 247-261

68. Goerke J and J Gonzales: Temperature dependence of dipalmitoyl phosphatidylcholine monolayer stability. J Appl Physiol 1981, 51: 1108-1114

69. Goerke J and S Schtirch: Mechanical properties of the alveolar surface. In: The Lung, Scientific Foundation. Crystal RG and JB West (eds.). Raven Press, New York 1991, 821-827

70. Gommers D, C Vilstrup, JAH Bos et al.: Exogenous surfactant therapy increases static lung compliance, and cannot be assessed by measurement of dynamic compliance alone. Critical Care Medicine 1993, 21:567-574

71. Gordon LM, S Horvath, ML Longo et al.: Conformation and molecular topography of the N-terminal segment of surfactant protein B in structure-promoting environments. Protein Sci 1996, 5:1662-1675

72. Gregory TJ, WJ Longmore, MA Moxley et al.: Surfactant chemical composition and biophysical activity in acute respiratory distress syndrome. J Clin Invest 1991, 88:1976-1981

73. Gross NJ: Inhibition of surfactant subtype convertase in radiation model of adult respiratory distress syndrome. Am J Physiol 1991,260:L31 l-L317

74. Gross NJ and KR Narine: Surfactant subtypes of mice: metabolic relationships and conversion in vitro. J Appl Physiol 1989, 67:414-421

75. Grotberg JB, D Halpern and OE Jensen: Interaction of exogenous and endogenous surfactant: spreading-rate effects. J Appl Physiol 1995, 78:750-756

76. GOnther A, M Kalinowski, A Elssner et al.: Clot-embedded natural surfactant: kinetics of fibrinolysis and surface activity. Am J Physiol 1994, 267:L618-L624

77. Gtinther A, C Siebert, R Schmidt et al.: Surfactant alterations in severe pneumonia, acute respiratory distress syndrome, and cardiogenic lung edema. Am J Respir Critical Care Med 1996, 153

78. Gustafsson M, G Vandenbussche, T Curstedt et al.: The 21-residue surfactant peptide (LysLeu4)4Lys(KL4)

is a transmembrane alpha-helix with a mixed nonpolar/polar surface. FEBS Lett 1996, 384:185-188

79. Haddad IY, BA Holm, L Hlavaty et al.: Dependence ofsurfactant function on extracellular pH: mechanisms and modifications. J Appl Physiol 1994, 76:657-662

80. Haddad IY, H Ischiropoulos, BA Holm et al.: Mechanisms of peroxynitrite-induced injury to pulmonary surfactants. Am J Physiol 1993, 265:L555-L564

81. Haddad IY, S Zhu, J Crow et al.: Inhibition of alveolar type II cell ATP and surfactant synthesis by nitric oxide. Am J Physiol 1996, 270:L898-L906

82. Haddad IY, S Zhu, H Ischiropoulos et al.: Nitration of surfactant protein A results in decreased ability to aggregate lipids. Am J Physiol 1996, 270:L281-L288

468

83. Hallman M, BH Feldmann, E Kirkpatrick et al.: Absence of phosphatidylglycerol (PG) in respiratory distress syndrome in newborn. Pediatr Res 1977, 11: 714-720

84. Hallman M, TA Merritt and K Bry: The fate of exogenous surfactant in neonates with respiratory distress syndrome. Clin Pharmacokinet 1994, 26:215-232

85. Hallman M, R Spragg, JH Harrell et al.: Evidence of lung surfactant abnormality in respiratory failure. Study of bronchoalveolar lavage phospholipids, surface activity, phospholipase activity, and plasma myoinositol. J Clin Invest 1982, 70:673-683

86. Hamvas A, FS Cole, DE deMello et al.: Surfactant protein B deficiency: antenatal diagnosis and prospective treatment with surfactant replacement. J Pediatr 1994, 125:356-361

87. Hartshorn KL, KB Reid, MR White et al.: Neutrophil deactivation by influenza A viruses: mechanisms of protection after viral opsonization with collectins and hemagglutination-inhibiting antibodies. Blood 1996, 87:3450-3461

88. Hawgood S: Surfactant: composition, structatre, and metabolism. In: The Lung, Scientific Foundation. Crystal RG and JB West (eds.). Raven Press, New York 1991,247-261

89. Hawgood S, BJ Benson and R Hamilton, Jr.: Effects of a surfactant-associated protein and calcium ions on the structure and surface activity of lung surfactant lipids. Biochemistry 1985, 24:184-190

90. Hawgood S, BJ Benson, J Schilling et al.: Nucleotide and amino acid sequences of pulmonary surfactant protein SP 18 and evidence for cooperation between SP 18 and SP 28-36 in surfactant lipid adsorption. Proc Natl Acad Sci USA 1987, 84:66-70

91. Hawgood S, H Efrati, J Schilling et al.: Chemical characterization of lung surfactant apoproteins: amino acid composition, N-terminal sequence and enzymic digestion. Biochem Soc Trans 1985, 13:1092-1096

92. Hawgood S, A Ogawa, K Yukitake et al.: Lung function in premature rabbits treated with recombinant human surfactant protein-C. Am J Respir Crit Care Med 1996, 154:484-490

93. Hawgood S and K Shifter: Structures and properties of the surfactant-associated proteins. Annu Rev Physiol 1991, 53:375-394

94. Hellqvist M, M Mahlapuu, L Samuelsson et al.: Differential activation of lung-specific genes by two forkhead proteins, FREAC-1 and FREAC-2. J Biol Chemistry 1996, 271:4482-4490

95. Helm CA, P Tippmann Krayer, H M6hwald et al.: Phases of phosphatidyl ethanolamine monolayers studied by synchrotron x-ray scattering. Biophys J 1991, 60:1457-1476

96. Henry M, M Ikegami, T Ueda et al.: Surfactant protein B metabolism in newborn rabbits. Biochim Biophys Acta 1996, 1300: 97-102

97. Herold R, H Banger and U Pison: Assessment of pulmonary surfactant function using a captive bubble surfactometer. Colloids Surfaces A 1996, 114:211-219

98. Hildebran JN, J Goerke and JA Clements: Pulmonary surface film stability and composition. J Appl Physiol 1979, 47:604-611

99. Hills BA, IB Masters, JC Vance et al.: Abnormalities in surfactant in sudden infant death syndrome as a postmortem marker and possible test of risk. J Paediatr Child Health 1997, 33:61-66

100. Holm BA: Surfactant inactivation in adult respiratory distress syndrome. In: Pulmonary Surfactant: From Molecular Biology to Clinical Practice. Robertson B, LMG Van Golde and JJ Batenburg (eds.). Elsevier, Amsterdam 1992, 665-684

101. Holm BA, Z Wang, EA Egan et al.: Content of dipalmitoyl phosphatidylcholine in lung surfactant: ramifications for surface activity. Pediatr Res 1996, 39:805-811

102. Holm BA and AJ Waring: Designer surfactants. The next generation in surfactant replacement. Clin Perinatol 1993, 20:813-829

103. Honda Y, T Fujishima, Y Imai et al.: Surfactant protein A in bronchoalveolar lavage fluid from patients with idiopathic interstitial pneumonia. Nippon Kyobu Shikkan Gakkai Zasshi 1996, 34:1326-1330

104. Honda Y, H Takahashi, Y Kuroki et al.: Decreased contents of surfactant proteins A and D in BAL fluids of healthy smokers. Chest 1996, 109:1006-1009

469

105. Horie T and J Hildebrandt: Dynamic compliance, limit cycles, and static equilibria of excised cat lung. J Appl Physiol 1971, 31:423-430

106. Horowitz AD, B Moussavian and JA Whitsett: Roles of SP-A, SP-B, and SP-C in modulation of lipid uptake by pulmonary cells in vitro. Am J Physiol 1996, 270:L69-L79

107. Ito Y, RA Veldhuizen, LJ Yao et al.: Ventilation strategies affect surfactant aggregate conversion in acute lung injury. Am J Respir Crit Care Med 1997, 155:493-499

108. Johansson J, T Curstedt and B Robertson: The proteins of the surfactant system. Eur Respir J 1994, 7: 372- 391

109. Kahn MC, GJ Anderson, WR Anyan et al.: Phosphatidylcholine molecular species of calf lung surfactant. Am J Physiol 1995,269:L567-573

110. Kamm RD and RC Schroter: Is airway closure caused by a liquid film instability? Respir Physiol 1989, 75: 141-156

111. Kattner E, B Metze, E Waiss et al.: Accelerated lung maturation following maternal steroid treatment in infants born before 30 weeks gestation. J Perinat Med 1992, 20:449-457

112. Kelly SE, CJ Bachurski, MS Burhans et al.: Transcription of the lung-specific surfactant protein C gene is mediated by thyroid transcription factor 1. J Biol Chem 1996, 271: 6881-6888

113. Kimmel E, M Seri and JJ Fredberg: Lung tissue resistance and hysteretic moduli of lung parenchyma. J Appl Physiol 1995, 79:461-466

114. King RJ: Comparative properties of reassembly lipoproteins formed with surfactant proteins A and C. In: Basic Research on lung surfactant. Wichert Pv and B Mtiller (eds.). Karger, Basel 1990, 36-47

115. King RJ and JA Clements: Lipid synthesis and ~urfactant turnover in the lungs. In: Handbook of Physiology: Circulation and Nonrespiratory Functions. Fishman AP and AB Fisher (eds.). American Physiological Society, Bethesda, MD 1985, 309-336

116. King RJ, D Simon and PM Horowitz: Aspects of secondary and quaternary structure of surfactant protein A from canine lung. Biochim Biophys Acta 1989, 1001:294-301

117. Knells G, MK Ahmed, RM Das et al.: Fourier-transform infrared spectroscopic analysis of rabbit lung surfactant: subfraction-associated phospholipid and protein profiles. Chem Phys Lipids 1995, 77:193-201

118. Kobayashi T and B Robertson: Surface adsorption of pulmonary surfactant in relation to bulk-phase concentration and presence of CaCI 2. Respiration 1983, 44:63-70

119. Korfhagen TR, MD Bruno, GF Ross et al.: Altered surfactant function and structure in SP-A gene targeted mice. Proc Natl Acad Sci U S A 1996, 93:9594-9599

120. Kulovich MV, M Hallman and L Gluck: The lung profile: normal pregnancy. Am J Obstet Gynecol 1979, 135:57-63

121. Kuroki Y, M Shiratori, Y Ogasawara et al.: Characterization of pulmonary surfactant protein D: its copurification with lipids. Biochim Biophys Acta 1991, 1086:185-190

122. Lalchev ZI, RK Todorov, YT Christova et al.: Molecular mobility in the monolayers of foam films stabilized by porcine lung surfactant. Biophysical J 1996, 71: 2591-2601

123. Lee MM, S Schtirch, SH Roth et al.: Effects of acid aerosol exposure on the surface properties of airway mucus. Experimental Lung Research 1995, 21:835-851.

124. Lewis JF and AH Jobe: Surfactant and the adult respiratory distress syndrome. Am Rev Respir Dis 1993, 147:218-233

125. Li J, R Miller and H M6hwald: Characterisation ofphospholipid layers at liquid interfaces. 2. Comparison of isotherms of insoluble and soluble films of phospholipids at different fluid/water interfaces. Colloids Surfaces A 1996, 114:123-130

126. Lipp MM, KYC Lee, JA Zasadzinski et al.: Phase and morphology changes in lipid monolayers induced by SP-B protein and its amino-terminal peptide. Science 1996, 273:1196-1199

127. Liu MY, LM Wang, ELi et al.: Pulmonary surfactant will secure free airflow through a narrow tube. J Appl Physiol 1991, 71: 742-748

470

128. Lu J, AC Willis and KB Reid: Purification, characterization and cDNA cloning of human lung surfactant protein D. Biochem J 1992, 284:795-802

129. Lubkin SR and JD Murray: A mechanism for early branching in lung morphogenesis. J Math Biol 1995, 34: 77-94

130. Luzi P, M Anceschi and DS Strayer: Glucocorticoid responsiveness conferred by a cloned DNA binding protein. Receptor 1995, 5:93-103

131. Magoon MW, JR Wright, A Baritussio et al.: Subfractionation of lung surfactant. Implications for metabolism and surface activity. Biochim Biophys Acta 1983, 750:18-31

132. Margana RK and V Boggaram: Functional analysis of surfactant protein B (SP-B) promoter. Sp 1, Sp3, TTF- 1, and HNF-3alpha transcription factors are necessary for lung cell-specific activation of SP-B gene transcription. J Biol Chem 1997, 272:3083-3090

133. Massaro GD and D Massaro: Postnatal treatment with retinoic acid increases the number of pulmonary alveoli in rats. Am J Physiol 1996, 270:L305-310

134. Matalon S, V DeMarco, IY Haddad et al.: Inhaled nitric oxide injures the pulmonary surfactant system of lambs in vivo. Am J Physiol 1996, 270:L273-280

135. Mead J: Mechanical properties of lungs. Physiol Rev 1961, 41:281-330

136. Mead J and C Collier: Relation of volume history of lungs to respiratory mechanics in anesthetized dogs. J Appl Physiol 1959, 14:669-678

137. Mendelson CR and V Boggaram: Hormonal control of the surfactant system in fetal lung. Annu Rev Physiol 1991, 53:415-440

138. Miyazawa N, S Suzuki, T Akahori et al.: Effects of surfactant on the in vivo alveolar surface-to-volume ratio. J Appl Physiol 1996, 80:86-90

139. Mtihwald H, A Dietrich, C Bohm et al.: Domain formation in monolayers. Mol Membr Biol 1995, 12:29-38

140. Nag K, J Perez-Gil, A Cruz et al.: Spontaneous formation of interfacial lipid-protein monolayers during adsorption from vesicles. Biophys J 1996, 71:1356-1363

141. Namba Y, SS Kurdak, ZX Fu et al.: Effect of reducing alveolar surface tension on stress failure in pulmonary capillaries. J Appl Physiol 1995, 79" 2114-2121.

142. Nardell EA and JS Brody: Determinants of mechanical properties of rat lung during postnatal development. J Appl Physiol 1982, 53:140-148

143. Naureckas ET, CA Dawson, BS Gerber et al.: Airway reopening pressure in isolated rat lungs. J Appl Physiol 1994, 76:1372-1377

144. Neumeister B, S Woerndle and P Bartmann: Effects of different surfactant preparations on bacterial growth in vitro. Biol Neonate 1996, 70:128-134

145. Nieman GF, CE Bredenberg, WR Clark et al.: Alveolar function following surfactant deactivation. J Appl Physiol 1981, 51: 895-904

146. Nikolova A, D Exerowa, Z Lalchev et al.: Thermal transitions in dimyristoylphosphatidylcholine foam bilayers. Eur Biophys J 1994, 23:145-152

147. Nogee LM, DE deMello, LP Dehner et al.: Brief report: deficiency of pulmonary surfactant protein B in congenital alveolar proteinosis. N Engl J Med 1993,328:406-410

148. Nogee LM, G Gamier, HC Dietz et al.: A mutation in the surfactant protein B gene responsible fo~ffatal neonatal respiratory disease in multiple kindreds. J Clin Invest 1994, 93:1860-1863

149. Notter RH and JN Finkelstein: Pulmonary surfactant: an interdisciplinary approach. J Appl Physiol 1984, 57: 1613-1624

150. Obladen M: History of surfactant research. In: Pulmonary Surfactant: from molecular biology to clinical practice. Robertson B, L van Golde and J Batenburg (eds.). Elsevier, Amsterdam 1993, 1-8

151. Ogawa A, CL Brown, MA Schlueter et al.: Lung function, surfactant apoprotein content, and level of PEEP in prematurely delivered rabbits. J Appl Physiol 1994, 77:1840-1849

471

152. Oosterlaken-Dijksterhuis MA, HP Haagsman, L van Golde et al.: Characterization of lipid insertion into monomolecular layers mediated by lung surfactant proteins SP-B and SP-C. Biochemistry 1991, 30:10965- 10971

153. Otis DR, Jr., M Johnson, TJ Pedley et al.: Role of pulmonary surfactant in airway closure: a computational study. J Appl Physiol 1993, 75:1323-1333

154. Oulton M, J MacDonald, DT Janigan et al.: Mouse alveolar surfactant: characterization of subtypes prepared by differential centrifugation. Lipids 1993, 28:715-720

155. Park SY, C-H Chang, DJ Ahn et al.: Dynamic surface tension bahavior of hexadecanol spread and adsorbed monolayers. Langmuir 1993, 9:3640-3648

156. Pastrana-Rios B, CR Flach, JW Brauner et al.: A direct test of the "squeeze-out" hypothesis of lung surfactant function. External reflection FT-IR at the air/water interface. Biochemistry 1994, 33:5121-5127

157. Pastrana-Rios B, S Taneva, KM Keough et al.: External reflection absorption infrared spectroscopy study of lung surfactant proteins SP-B and SP-C in phospholipid monolayers at the air/water interface. Biophys J 1995, 69:2531-2540

158. Pattle RE: Properties, function and origin of the alveolar lining layer. Nature 1955, 175:1125-1126

159. Pattle RE: Surface lining of lung alveoli. Physiel Rev 1965, 45:48-79

160. Perkins WR, RB Dause, RA Parente et al.: Role of lipid polymorphism in pulmonary surfactant. Science 1996, 273:330-332

161. Persson A, D Chang and E Crouch: Surfactant protein D (SP-D) is a divalent cation-dependent carbohydrate binding protein. J Biol Chem 1990, 265:5755-5760

162. Persson A, D Chang, K Rust et al.: Purification and biochemical characterization of CP4 (SP-D), a collagenous surfactant-associated protein. Biochemistry 1989, 28:6361-6367

163. Perun ML and DP-r Gaver: An experimental model investigation of the opening of a collapsed untethered pulmonary airway. J Biomech Eng 1995, 117:245-253

164. Pison U, J B6ck, S Pietschmann et al.: The adult respiratory distress syndrome: pathophysiological concepts related to the pulmonary surfactant system. In: Surfactant therapy for lung disease. Robertson B and HW Taeusch (eds.). Marcel Dekker, New York 1995, 167-197

165. Pison U, M Max, A Neuendank et al.: Host defense capacities of pulmonary surfactant - evidence for "non- surfactant" functions of the surfactant system. Eur J Clin Invest 1994, 24:586-599

166. Pison U, U Obertacke, W Seeger et al.: Surfactant protein A (SP-A) is decreased in acute parenchymal lung injury associated with polytrauma. Eur J Clin Invest 1992, 22:712-718

167. Pison U, W Seeger, R Buchhorn et al.: Surfactant abnormalities and adult respiratory failure. Am Rev Respir Dis 1989, 140:1033-1039

168. Pison U, W Seeger, R Buchhorn et al.: Surfactant abnormalities in patients with respiratory failure atter multiple trauma. Am Rev Respir Dis 1989, 140:1033-1039

169. Pison U, K Shifter, S Hawgood et al.: Effects of the surfactant-associated proteins, SP-A, SP-B and SP-C, on phospholipid surface film formation. Prog Respir Res 1990, 25:271-273

170. Possmayer F: A proposed nomenclature for pulmonary surfactant-associated proteins. Am Rev Respir Dis 1988, 138:990-998

171. Post A, AV Nahmen, M Schmitt et al.: Pulmonary surfactant protein C containing lipid films at the air-water interface as a model for the surface of lung alveoli. Mol Membr Biol 1995, 12:93-99

172. Poulain FR, S Nir and S Hawgood: Kinetics of phospholipid membrane fusion induced by surfactant apoproteins A and B. Biochim Biophys Acta 1996, 1278:169-175

173. Putman E, LA Creuwels, LM van Golde et al.: Surface properties, morphology and protein composition of pulmonary surfactant subtypes. Biochem J 1996, 320:599-605

174. Putz G, J Goerke and JA Clements: Surface activity of rabbit pulmonary surfactant subfractions at different concentrations in a captive bubble. J Appl Physiol 1994, 77:597-605

175. Putz G, J Goerke, S Schtirch et al.: Evaluation of a pressure-driven captive bubble surfactometer. J Appl Physiol 1994, 76:1417-1424

472

176. Putz G, J Goerke, HW Taeusch et al.: Comparison of captive- and pulsating-bubble surfactometers using lung surfactant. J Appl Physiol 1994, 76:1425-1431

177. Qanbar R, S Cheng, F Possmayer et al.: Role of the palmitoylation of surfactant-associated protein C in surfactant film formation and stability. Am J Physiol 1996, 271:L572-580

178. Rebello CM, AH Jobe, JW Eisele et al.: Alveolar and tissue surfactant pool sizes in humans. Am J Respir Crit Care Med 1996, 154:625-628

179. Revak SD, TA Merritt, CG Cochrane et al.: Efficacy of synthetic peptide-containing surfactant in the treatment of respiratory distress syndrome in preterm infant rhesus monkeys. Pediatr Res 1996, 39:715-724

180. Robertson B: Animal Models of neonatal surfactant dysfunction. In: From Molecular Biology to Clinical Practice. Robertson B, LMG Van Golde and JJ Batenburg (eds.). Elsevier, Amsterdam 1992, 459-484

181. Rust K, L Grosso, V Zhang et al.: Human surfactant protein D: SP-D contains a C-type lectin carbohydrate recognition domain. Arch Biochem Biophys 1991, 290:116-126

182. Sano K, J Fisher, RJ Mason et al.: Isolation and sequence of a eDNA clone for the rat pulmonary surfactant- associated protein (PSP-A). Biocl~m Biophys Res Commun 1987, 144:367-374

183. Scarpelli EM and AJ Mautone: Surface biophysics of the surface monolayer theory is incompatible with regional lung function. Biophys J 1994, 67:1080-1089

184. Scarpelli EM, AJ Mautone and MR Chinoy: Intraalveolar bubbles and bubble films: I. Formation and development during the first 48 hours of extrauterine life in rabbits. Anat Rec 1996, 244:344-357

185. Schoedel W, H Slama and E Hansen: Zeitabhfingiges Verhalten von Filmen von oberflfichenaktivem Material aus Lungenalveolen. PflUgers Arch 1971, 322:336-346

186. Schoel WM, S Schtirch and J Goerke: The captive bubble method for the evaluation of pulmonary surfactant: Surface tension, area and volume calculations. Biochim Biophys Acta 1994, 1200:286-290

187. SchUrch S: Surface tension at low lung volumes: dependence on time and alveolar size. Respir Physiol 1982, 48:339-355

188. SchUrch S and H Bachofen: Biophysical aspects in the design of a therapeutic surfactant. In: Surfactant Therapy for Lung Disease. Robertson B and HW Taeusch (eds.). Marcel Dekker, New York, 1995, 3-32

189. Schtirch S, H Bachofen, J Goerke et al.: Surface properties of rat pulmonary surfactant studied with the captive bubble method: adsorption, hysteresis, stability. Biochim Biophys Acta 1992, 1103:127-136

190. Schtirch S, H Bachofen, J Goerke et al.: A captive bubble method reproduces the in situ behavior of lung surfactant monolayers. J Appl Physiol 1989, 67:2389-2396

191. Schtirch S, H Bachofen, R Qanbar et al.: Incorporation of surface-associated surfactant material into the surface active film. Appl Cardiopul Pathophysiol 1995, 5:106-107

192. SchUrch S, P Gehr, V I m Hof et al.: Surfactant displaces particles toward the epithelium in airways and alveoli. Respir Physiol 1990, 80:17-32

193. Schtirch S, J Goerke and JA Clements: Direct determination of surface tension in the lung. Proc Natl Acad Sci USA 1976, 73:4698-4702

194. Schtirch S, J Goerke and JA Clements: Direct determination of volume- and time-dependence of alveolar surface tension in excised lungs. Proc Natl Acad Sci U S A 1978, 75:3417-3421

195. Schtirch S, F Possmayer, S Cheng et al.: Pulmonary SP-A enhances adsorption and appears to induce surface sorting of lipid extract surfactant. Am J Physiol 1992, 263:L210-218

196. Schtirch S, R Qanbar, H Bachofen et al.: The surface-associated surfactant reservoir in the alveolar lining. Biol Neonate 1995, 67 Suppl 1:61-76

197. Schtirch S, D Schtirch, T Curstedt et al.: Surface activity of lipid extract surfactant in relation to film area compression and collapse. J Appl Physiol 1994, 77:974-986

198. Seeger W, C Grube, A GUnther et al.: Surfactant inhibition by plasma proteins: differential sensitivity of various surfactant preparations. Eur Respir J 1993, 6:971-977

199. Seeger W, A Gtinther and C Thede: Differential sensitivity to fibrinogen inhibition of SP-C- vs. SP-B-based surfactants. Am J Physiol 1992, 262:L286-L291

200. Shifter K, S Hawgood, HP Haagsman et al.: Lung surfactant proteins, SP-B and SP-C, alter the thermodynamic properties of phospholipid membranes: a differential calorimetry study. Biochemistry 1993, 32:590-597

473

201. Shimizu H, JH Fisher, P Papst et al.: Primary structure of rat pulmonary surfactant protein D. cDNA and deduced amino acid sequence. J Biol Chem 1992, 267:1853-1857

202. Shimura S, T Masuda, T Takishima et al.: Surfactant apoprotein-A concentration in airway secretions for the detection of pulmonary oedema. Eur Respir J 1996, 9:2525-2530

203. Smith JC and D Stamenovic: Surface forces in lungs. I. Alveolar surface tension-lung volume relationships. J Appl Physiol 1986, 60:1341-1350

204. Spragg RG and RM Smith: Pathology of the surfactant system of the mature lung. Am J Respir Critical Care Med 1997, 155:756-760

205. Stamenovic D and GM Bamas: Effect of surface forces on oscillatory behavior of lungs. J Appl Physiol 1995, 79:1578-1585

206. Stamenovic D and JC Smith: Surface forces in lungs. II. Microstructural mechanics and lung stability. J Appl Physiol 1986, 60:1351-1357

207. Stem N, S Riklis, M Kalina et al.: The catabolism of lung surfactant by alveolar macrophages. Biochim Biophys Acta 1986, 877:323-333

208. Subramaniam S, JA Whitsett, W Hull et al.: Alteration of pulmonary surfactant proteins in rats chronically exposed to cigarette smoke. Toxicol Appl Pharmacol 1996, 140:274-280

209. Suzuki Y, Y Fujita and K Kogishi: Reconstitution of tubular myelin from synthetic lipids and proteins associated with pig pulmonary surfactant. Am Rev Respir Dis 1989, 140:75-81

210. Takei T, Y Hashimoto, T Aiba et al.: The surface properties of chemically synthesized peptides analogous to human pulmonary surfactant protein SP-C. Biol Pharm Bull 1996, 19:1247-1253

211. Tiemey DF: Lung surfactant: some historical perspectives leading to its cellular and molecular biology. Am J Physiol 1989, 257:L1-L12

212. Tino MJ and JR Wright: Surfactant protein A stimulates phagocytosis of specific pulmonary pathogens by alveolar macrophages. Am J Physiol 1996, 270:L677-688

213. van Iwaarden F, B Welmers, J Verhoef et al.: Pulmonary surfactant protein A enhances the host-defense mechanism of rat alveolar macrophages. Am J Respir Cell Mol Biol 1990, 2:91-98

214. van 't Veen A, D Gommers, JW Mouton et al.: Exogenous pulmonary surfactant as a drug delivering agent: influence of antibiotics on surfactant activity. Br J Pharmacol 1996, 118:593-598

215. Veldhuizen RA, K Inchley, SA Heam et al.: Degradation of surfactant-associated protein B (SP-B) during in vitro conversion of large to small surfactant aggregates. Biochem J 1993,295:141-147

216. Veldhuizen RA, J Marcou, LJ Yao et al.: Alveolar surfactant aggregate conversion in ventilated normal and injured rabbits. Am J Physiol 1996, 270:L152-158

217. Veldhuizen RA, LJ Yao, SA Heam et al.: Surfactant-associated protein A is important for maintaining surfactant large-aggregate forms during surface-area cycling. Biochem J 1996, 313:835-840

218. Venkitaraman AR, SB Hall, JA Whitsett et al.: Enhancement of biophysical activity of lung surfactant extracts and phospholipid-apoprotein mixtures by surfactant protein A. Chem Phys Lipids 1990, 56:185-194

219. von Neergard K: Neue Auffassung tiber einen Grundbegriff der Atemmechanik. Die Retraktionskratt der Lunge, abhfingig von der Oberflfichenspannung in den Alveolen. Z Ges Exp Med 1929, 66:373-394

220. Voss T, H Eistetter, KP Sch/ffer et al.: Macromolecular organization of natural and recombinant lung surfactant protein SP 28-36. Structural homology with the complement factor Clq. J Mol Biol 1988, 201: 219-227

221. Walker SR, MC Williams and B Benson: lmmunocytochemical localization of the major surfactant apoproteins in type II cells, Clara cells, and alveolar macrophages of rat lungs. J Histochem Cytochem 1986, 34:1137-1148

222. Walther FJ, R David Cu, C Leung et al.: A synthetic segment of surfactant protein A: structure, in vitro surface activity, and in vivo efficacy. Pediatr Res 1996, 39:938-946

223. Wang JY, U Kishore, BL Lim et al.: Interaction of human lung surfactant proteins A and D with mite (Dermatophagoides pteronyssinus) allergens. Clin Exp Immunol 1996, 106:367-373

224. Wang JY, TF Yeh, YC Lin et al.: Measurement of pulmonary status and surfactant protein levels during dexamethasone treatment of neonatal respiratory distress syndrome. Thorax 1996, 51: 907-913

474

225. Wang z, O Gurel, JE Baatz et al.: Acylation of pulmonary surfactant protein-C is required for its optimal surface active interactions with phospholipids. J Biol Chem 1996, 271:19104-19109

226. Wang Z, O Gurel, JE Baatz et al.: Differential activity and lack of synergy of lung surfactant proteins SP-B and SP-C in interactions with phospholipids. J Lipid Res 1996, 37:1749-1760

227. Wang Z, SB Hall and RH Notter: Roles of different hydrophobic constituents in the adsorption of pulmonary surfactant. J Lipid Res 1996, 37:790-798

228. Warr RG, S Hawgood, DI Buckley et al.: Low molecular weight human pulmonary surfactant protein (SP5): isolation, characterization, and cDNA and amino acid sequences. Proc Natl Acad Sci U S A 1987, 84:7915- 7919

229. White RT, D Damm, J Miller et al.: Isolation and characterization of the human pulmonary surfactant apoprotein gene. Nature 1985, 317:361-363

230. Whitsett JA, LM Nogee, TE Weaver et al.: Human surfactant protein B: structure, function, regulation, and genetic disease. Physiol Rev 1995, 75:749-757

231. Whitsett JA, G Ross, T Weaver et al.: Glycosylation and secretion of surfactant-associated glycoprotein A. J Biol Chem 1985, 260:15273-15279

232. Williams MC: Conversion of lamellar body membranes into tubular myelin in alveoli of fetal rat lungs. J Cell Biol 1977, 72:260-277

233. Williams MC: Vesicles within vesicles: what role do multivesicular bodies play in alveolar type II cells? Am Rev Respir Dis 1987, 135:744-746

234. Williams MC and B Benson: Immunocytochemical localization and identification of the major surfactant protein in adult rat lung. J Histochem Cytochem 1981, 29:291-305

235. Williams MC, S Hawgood and RL Hamilton: Changes in lipid structure produced by surfactant proteins SP- A, SP-B, and SP-C. Am J Respir Cell Mol Biol 1991, 5:41-50

236. Wilson TA and H Bachofen: A model for the mechanical structure of the alveolar duct. J Appl Physiol 1982, 52:1064-1070

237. Wirtz H and M Schmidt: Ventilation and secret:~on of pulmonary surfactant. Clin Investig 1992, 70:3-13 238. Wohlford Lenane CL, PL Durham and JM Snyder: Localization of surfactant-associated protein C (SP-C)

mRNA in fetal rabbit lung tissue by in situ hybridization. Am J Respir Cell Mol Biol 1992, 6:225-234 239. Wolfson MR, JS Greenspan and TH Shaffer: Pulmonary administration of vasoactive substances by

perfluorochemical ventilation. Pediatrics 1996, 97:449-455 240. Wright JR and JA Clements: Metabolism and tumover of lung surfactant. Am Rev Respir Dis 1987, 135:

426-444 241. Wright JR and LG Dobbs: Regulation of pulmonary surfactant secretion and clearance. Annu Rev Physiol

1991, 53:395-414 242. Wright JR, RE Wager, RL Hamilton et al.: Uptake of lung surfactant subfractions into lamellar bodies of

adult rabbit lungs. J Appl Physiol 1986, 60:817-825 243. Wright JR, RE Wager, S Hawgood et al.: Surfactant apoprotein M r = 26,000-36,000 enhances uptake of

liposomes by type II cells. J Biol Chem 1987, 262:2888-2894 244. Yager D, T Cloutier, H Feldman et al.: Airway surface liquid thickness as a function of lung volume in small

airways of the guinea pig. J Appl Physiol 1994, 77:2333-2340 245. Young SL, SA Kremers, JS Apple et al.: Rat lung surfactant kinetics: biochemical and morphometric

correlation. J Appl Physiol 1981, 51:248-253 246. Yu SH and F Possmayer: Effect of pulmonary surfactant protein B (SP-B) and calcium on phospholipid

adsorption and squeeze-out of phosphatidylglycerol from binary phospholipid monolayers containing dipalmitoylphosphatidylcholine. Biochim Biophys Acta 1992, 1126:26-34

247. Yu SH and F Possmayer: Adsorption, compression and stability of surface films from natural, lipid extract and reconstituted pulmonary surfactants. Biochim Biophys Acta 1993, 1167:264-271

248. Yu SH and F Possmayer: Effect of pulmonary surfactant protein A (SP-A) and calcium on the adsorption of cholesterol and film stability. Biochim Biophys Acta 1994, 1211:350-358

249. Yu SH and F Possmayer: Effect of pulmonary surfactant protein A and neutral lipid on accretion and organization of dipalmitoylphosphatidylcholine in surface films. J Lipid Res 1996, 37:1278-1288

Drops and Bubbles in Interfacial Research D. M6bius and R. Miller (Editors) �9 1998 Elsevier Science B.V. All rights reserved.

HIGH TEMPERATURE TENSIOMETRY

475

A.Passerone and E. Ricci

ICFAM-CNR, Via de Marini 6, 16149 Genova, Italy

Contents

.

1.1 1.2 1.3 1.4 1.5 2. 2.1 2.2 2.2.1 2.2.2 2.2.3 2.2.4 2.2.5 2.2.6 2.3 3. 3.1 3.1.1. 3.1.2. 3.2 3.2.1 3.2.2 4. 4.1 4.2 5. 6.

Theoretical background Thermodynamic definitions Surface excess properties Effect of temperature Effect of composition The Marangoni effects Surface tension and adsorption measurements Experimental determination of liquid-vapour surface tension Brief survey of experimental methods to measure surface tension at high temperatures Surface shape methods: sessile drop and pendent drop Capillary rise Maximum bubble pressure Wilhelmy plate; Du NoOy ring Levitated drop Drop weight Surface tension values of liquid metals Gas-liquid exchange Oxygen transfer at liquid metal-vapour interfaces Experiments under protective atmospheres (low Po2 values) Experiments under a vacuum Evaluation of the characteristic times to reach steady state conditions "Small" drops "Large" drops Sessile drop experiments: an example of standard procedure The apparatus ASTRA (Automatic Surface Tension Real Time Acquisition system) Acknowledgements References

476

In the second part of the 18 th century and the very first years of the 19 th scientists have been

especially interested in the explanation of the basic principles of "capillarity".

The years 1805-06 should be considered as a milestone: two important contributions appeared,

namely by Thomas Young [ 1 ] and by P.S. de Laplace [2], who, on the basis of a deep analysis

of the influence of forces acting at the interfaces, drew a correct description of both the

capillary elevation and the contact angle. In the paper by T. Young not one equation was put

explicitly, but the phenomena were described in quite a thorough, sometimes obscure, way.

Laplace's work, on the other hand, is very clear and well developed both from a physical and

mathematical point of view. In this work, the analytical form of a capillary surface, still the

basis of all measurement techniques of surface tension, was first derived. The Laplace

equation, which relates the pressure difference AP across a point of a curved interface to the

mean curvature at that point through the surface tension ~/:

AP=?' + (1)

does not admit, in general, any analytical solution, as confirmed in a rigorous way by Gauss [3].

This means that only particular solutions can be sought, or numerical techniques have to be

applied to get numerical results. A substantial contribution to the utilisation of the Laplace

equation to measure surface and interfacial tension was made, in the year 1883, by the classical

work of F. Bashforth and C. Adams [4] , who, by applying a finite differences method,

succeeded in preparing extended tables which report, in a non-dimensional form, the shape,

volume and other parameters of sessile-pendent drops. The accuracy of these tables (to the

sixth decimal place!) made them an invaluable means for experimenters, until about fifteen

years ago, when computers and video imaging techniques became readily available, relegating

this monumental piece of work to the Scientific Archaeology.

A classic contribution to understanding both the historical development of the theories of capillary

action and summarising the theoretical developments already obtained can be found in the

celebrated paper by J.C. Maxwell which appeared in the Enciclopedia Britannica in 1878 [5].

If the attention of the former experimenters was attracted by "ordinary" liquids, such as water,

organics, mercury, in the course of this century more and more applications demanded data on

477

high temperature systems, thus encouraging the development of new experimental techniques

and of sophisticated procedures to measure interfacial tensions of molten systems and their

wettability characteristics when in contact with solid bodies.

One of the first and more thorough attempts to utilise the Bashforth and Adams work is that

published by B.F. Ellefson and N.W.Taylor in 1938 [6], devoted to the measurement of the

surface tension and density of molten glasses. Their approach, also from the experimental point

of view, can be considered a pioneering one, because the experimental design and the treatment

of the experimental data is, basically, the one used nowadays in sessile drop experiments

(technological improvements excepted). Further important work on high temperature materials

has been done in eastern Countries: a thorough review up to 1965 can be found in

Semenchenko's book [7] and in many reports by Eremenko's school [8]. In the same years, in

France, P. Kozakevitch developed an original approach to the measurement of surface tension

both from the point of view of the experimental set up and from that of data treatment [9]. In

this case, a systematic approach to the effect of oxygen and of other trace impurities on the

surface tension of pure metals as well as an efficient way to use the Bashforth and Adams

tables were proposed.

A special challenge, in high temperature surface tension measurements, is represented by the

intrinsic reactivity of all materials against all "containers" (remember that also the

"atmosphere" around the specimen is a "container"). This means, as can be shown on the basis

of thermodynamic arguments, that the surface tension values obtained in many experiments do

not really refer to "pure" substances, but rather to alloys or solutions, especially taking into

account that certain surface-active elements, have a dramatic effect on surface tension even if

present in vanishing traces.

As a consequence, very sophisticated techniques had to be developed, like the "levitating drop",

to avoid both the aforementioned problems and to utilise energy transfer methods to heat up the

specimen that were clean enough and not invasive (electron beam, induction, optical heating etc.)

The present paper is intended firstly to present, after a short reminder of thermodynamic

definitions, the most commonly used techniques for surface tension measurements with some

details of the most interesting of them for high temperature applications, secondly some recent

478

results on the evaluation of the influence of extemal factors, like the surrounding atmosphere,

on the determination of the surface tension of molten systems.

1. THEORETICAL BACKGROUND.

Surface and interfacial tensions are physical quantities that retain a sound meaning if measured

both at equilibrium or under dynamic conditions. In this paper, reference will only be made to

liquid systems, reasonably far from the critical point.

In view of the fact that it is not the aim of this paper to give a specialised thermodynamic

definition of the processes involved or to approach the dynamics of adsorption, only few basic

concepts will be treated, referring the interested reader to other specific treatises [ 10,11 ].

1.1 THERMODYNAMIC DEFINITIONS.

A simple relationship between surface tension and surface free energy can be deduced using the

classical Gibbs model describing a two-phase capillary system in equilibrium with a planar

interface, as shown in Fig. 1.

The presence of the interface introduces an additional work term ~/df2 associated with an

increase in surface area dr2 so that a change in the internal energy is given by:

dU = - P d V + YdS + ~ pidni + y df2 (2)

the variation of the Helmholtz free energy F=U-TS is then:

dF = - P d V - SdT + ~/ l~dn i + y df2 (3) i

where S is the entropy, T the absolute temperature, P the pressure (for a planar interface

between the two phases o~ and [3, P,=P~=P), V the volume, ~ti the chemical potential of the

component i (at equilibrium ].l, ia-'~i It ) and ni the number of moles of the ith component.

From eq. 3 the surface tension ~/ may be defined as"

7' - ( 4 )

T , V , n t

In the Gibbs treatment [ 12] of capillary systems, bulk phases are maintained homogeneous up

479

to a dividing surface assumed to be of zero thickness. Local concentrations of each extensive

property, such as free energy or number of atoms, are defined constant up to the geometrical

surface in each bulk phase. Accordingly, the balance of any extensive property is assigned to

the dividing surface (Fig. 1)

13 phase

interface

phase

O .

1

c . 1

Rea l S y s t e m

Fig. 1 Gibbs model of a capillary system.

( Volume V ~)

13 phase

t~ phase

( Volume V a)

c . !

n ~ r !

G i b b s M o d e l

Ot

C. !

C. !

1.2 SURFACE EXCESS PROPERTIES.

We denote by Ci a the concentration of the component i in the bulk c~ phase, far from the

interface. In the Gibbs model this concentration is considered to be constant up to the dividing

surface (corresponding volume V ~) and defines a number of moles ni a= ci a V a. The excess

number of moles which has to be assigned to the dividing surface to maintain the total mass

balance may therefore be written:

ni ~ = n i - ni ~ - ni 13 (5 )

where ni is the total number of moles of the component i in the system.

Dividing by s the area of the surface, we obtain the Gibbs adsorption of component i, F i ,

which may be either positive or negative:

Fi = ni~/s (6)

480

In a similar way, from the classical expression of the free energy, the surface excess free energy

F ~ can be defined as:

F ~ U ~ TS ~ (7)

where U ~ and S ~ are the surface energy and surface entropy of the system, respectively.

The variation of free energy corresponding to the bulk ~ phase in the Gibbs model is:

dF a = - P dV a - S a dT + Ei ~l,i dni (8)

and similarly for the [~ phase.

By subtracting the contributions dF ~ and dF ~ from eq. 3 we obtain the excess free energy

associated with the dividing surface:

dF ~ = - S ~ dT + 3' dr2 + Ei P,i dn~ ~ (9)

At constant temperature, the excess surface free energy F ~ is a first order homogeneous

function of f2 and ni ~ Thus, from eq. 9 and using the Euler theorem, the surface tension may

be written

3, = F~ - E i ~l,i 1-'i (10)

This relationship shows that, in general, the surface tension cannot be identified with the

surface free energy per unit area; however, this is true in the case of a pure liquid:

~, = F~ = f ~ (11)

The validity of this relationship is limited to a pure system (liquid or solid) in which the surface

and bulk chemical potentials remain equal after an equilibrium displacement. In particular, for

solids [ 13] this condition restricts the validity of eq. 11 to high temperatures.

Differentiating eq. 10 and substituting dF ~

adsorption equation:

from eq. 9 we obtain the well known Gibbs

dy = - S~ d T- Zi Fi d~i (12)

At constant temperature and using the Gibbs-Duhem relationship for the c~ and [3 bulk phases,

eq. 12 can be rewritten as:

dy = - ~ F , l d / 1 i = - Fi _ F1 i ,=2 i=2 C 7 - e l fl

where Fil is the relative adsorption of constituent i with respect to the constituent 1.

(13)

481

1.3 EFFECT OF TEMPERA TURE.

A general expression for the temperature coefficient d~,/dT may be deduced from eq. 12

d y _ S ~ ,~ d/2~ dT - - -~')-- - Fi dT (14)

In a one-component system and choosing a dividing surface corresponding to zero adsorption,

we have

dy S ~ - - ( 1 5 )

dT

In this case, the temperature coefficient of the surface tension is equivalent in magnitude, but

opposite in sign, to the excess surface entropy per unit area S ~, a quantity which is an

indication of the surface disorder. At the critical temperature, the surface tension must be zero;

consequently, the surface tension of a system composed of a pure liquid should decrease with

increasing temperature. This point, which is valid only if dealing with equilibrium surface

tension, has sometimes been questioned by experiments which seemed to show an "increase" of

? with temperature. However, this fact can only be attributed to the presence of pollutants

(oxygen in particular) in the system under study. Empirical relationships have been proposed to

correlate the surface tension variations with temperature [14,15] as, for example, the E6tv6s

equation [ 15]"

/~p~_/1Vl 2/3 7LV = K(T~ - T) (16)

where M is the molar weight, PL the liquid density and Tc the critical temperature. The constant K

is about 2.1 erg K l for many organic liquids but approaches 0.64 erg K l for liquid metals [ 16].

A different behaviour can be expected when treating multi-component systems. The Gibbs

adsorption equation may be written

482

so.

d T - - f2 - . Fi �9 . c ~ j dT (17)

o r

d T - - Z r , s, -Zr , +, dN, i , . c ~ j d T

(18)

B

where S, is the partial entropy of the ith constituent in the bulk. The last term in eq. 18 is

proportional to N'/N where N' is the number of surface atoms and N the total number of

atoms in the bulk phase.

Then, eq. 18 can be reduced to:

dT - - - )--'~ F, S, (19) i

Thus, the surface tension coefficient may be negative or positive, according to the

thermodynamic behaviour of the solution.

Positive values have been observed for several binary or ternary liquid alloys (Fig.2) and

interpreted on the basis of a strong surface segregation [ 17,18].

- i i i i I - -

r Atomic */. Sn

e--IE 650 t ~ ~ i ~ . O 3 .74 .66 ~ S ~ < ~ ~ ---~ -~.os

.~ ~ ~ , o . o ~ c -

~ Goo[ ~ ~ ! 6 7 _ ~ .93 - �9 ,,-- t~ ..,...--.--r,--.~ 36

, , / I - -

t~ 550

500 I ! I I ! _ O 100 200 300 400

Temperat u re / ~

Fig.2 Effect of temperature on the surface tension of liquid Sn-Ga alloys (after [ 19]).

483

1.4 EFFECT OF COMPOSITION.

For binary alloys, at constant temperature, the Gibbs adsorption equation reduces to:

d y = - F A d/2 A - FBd/2 B (20)

In the case of a liquid-vapour interface at equilibrium (or = liquid phase, [3 = vapour phase) eq.

13 can be simplified neglecting the concentration in the gas phase

( X , ) dy =-F~,Ad~ = - F~ - ~ F ~ d~B (21)

where XA, XB are the bulk molar fractions of the liquid phase.

In the case of a dilute solution and assuming that the Henry law is valid in the solute B we

have:

1 d7 RT dlnX B --FB'A -FB (22)

In dilute solutions XB~0, and the relative adsorption is essentially the solute adsorption.

Consequently, the more the solute lowers the surface tension the more it segregates to the

surface. This equation is frequently used to interpret the effect of surface active additives. Some

applications are discussed in [20] with reference to the high temperature range and in [21]

with respect to the behaviour of surfactants in low (ambient) temperature applications.

1.5 THE MARANGONI EFFECTS.

When an interface between two immiscible fluids is subjected to a temperature (or

concentration ) gradient, its interfacial tension varies from point to point: these gradients along

the surface induce shear stresses on it that result in fluid motion. For a pure liquid in a

temperature field parallel to the surface there is flow from the hot end toward the cold end.

Since the bulk fluids are viscous, they are dragged along: thus, bulk fluid motion results from

interfacial temperature gradients (thermocapillary effect). The velocity profile is a function of

the vertical distance of the free surface (or the interface) from the base plane. The maximum

velocity occurs at the surface with a magnitude of the order of

484

OT'AT V - - - 23)

OT p v

where v represents the kinematic viscosity, 9 the density. The surface can flow at velocities

ranging from some mm/s to several cm/s, depending on the surface tension temperature

coefficient and on the liquid viscosity (Table 1). The velocity so calculated represents a

maximum value, which may drop to lower values as a function of thickness of the liquid layer

and of other experimental parameters. Indeed, experiments on liquid tin have shown a velocity

of the order of 0.2 cm/s at around 560 K [22].

Table 1. Marangoni velocities (cm s ~ ) in liquid metals for AT=I K, at melting point (after eq. 23).

Ag

Au

Cu

4.08

3.23

4.69

Cs

Ga

Ge

8.66

4.72

34.7

Hg

Li

Mg

11.9

22.0

26.9

Na

Sb

Si

12.1

3.33

10.5

The phenomena arising from interfacial tension gradients are collectively termed "Marangoni

effects", after the Italian physicist Carlo Marangoni (1840-1925) who published a series of

papers on this subject between 1871 and 1878 [23-25]. In the same period other scientists were

attracted by these kinds of movements, giving qualitative interpretations of such phenomena.

For example, the first correct explanation of the spreading of a drop of alcohol on the surface of

water, experimentally observed along the walls of a glass containing strong wine ("the tears of

wine") was given in 1855 by J. Thomson [26], who attributed these movements to gradients in

surface tension originated from local evaporation of alcohol.

These phenomena can be found in systems where liquids are subjected to temperature gradients

or in the presence of mass transfer as in liquid-liquid extraction, in bubbles and drops

migration, in spreading of lubricants and so on. Marangoni motion is responsible for wall

erosion in glass-melting furnaces, affects crystal growth processes and foam stability in many

ways, and, in general, the stability of liquid layers.

It is clear that Marangoni effects become predominant under weightlessness (microgravity)

conditions, where the ratio between body and surface forces becomes very small. Indeed, in

485

recent years, a new impetus for the study of thermal and solutal Marangoni flows has come

from the utilisation of the space environment [27]. This because these flows are extremely

relevant in containerless processing of liquids, bubble and drop management, liquid transfer

operations and so on.

The technological implications of the Marangoni effects are vast and very important. As

already mentioned, they span from crystal growth processes to biology, from chemical

engineering to metallurgy. In this particular field, it should be mentioned that welding

procedures are highly affected by surface tension driven motions: the penetration of the liquid

phase depends, to a great extent, on the surface and bulk movements of the liquid pool. The

presence of surface active elements, like oxygen and sulphur in liquid iron (Fig. 3) change in a

dramatic way the liquid surface tension and its temperature dependence (see refs. reported in

[28] ), so that the liquid can flow not only with very different velocities but also the direction of

flow can be reversed, due to changes of the surface tension coefficient from negative to positive

[291.

1900

1800

1700 E Z 1600 E c o 1500

i ~. 1400

8 ~: 1300

or) 1200

1100

1000

i , 1

-1" N ._

0

= L 0.02 0.04 0.06 0.08

Atomic %

o

i q L

0.1 0.12 0.14

Fig. 3 Effect of non-metallic additions on the surface tension of liquid iron at 1820 K [30].

As a consequence, different weld pools morphologies are obtained, where the depth of the

fusion zone is determined by the movements of the molten alloy [31,32]. Typical, computed

shapes of molten pools of steels, with different sulphur contents heated by a high-intensity laser

486

beam are reported in Fig. 4 [33].

Similarly, during cooling and solidification, the droplets which form when a metallic alloy

undergoes a monotectic separation, are propelled by the Marangoni flow which develops at the

liquid-liquid drop/matrix interface, towards the higher temperature regions of the container.

This happens because the interface flows towards the cold region and the whole drop, due to

the resulting viscous drag, moves in the opposite direction. This effect has been used to

counteract the sedimentation of the denser droplets due to gravity by suitably shaping the steep

temperature gradients during continuous casting of metallic alloys. In this way, a homogeneous

dispersion of the two phases can be readily obtained [34,35].

Fig. 4 Schematic computed shape of a molten pool of a steel sample heated from above (top center). a) High bulk sulphur content, pool depth=l mm, b) Low bulk sulphur content, pool depth=0.09 mm.

487

2. SURFACE TENSION AND ADSORPTION MEASUREMENTS

2.1 EXPERIMENTAL DETERMINATION OF LIQUID-VAPOUR SURFACE TENSION

Liquid systems take a shape which minimises their total energy. At large Bond numbers

(Bo P g L2 - where 9 is the density, g the acceleration due to gravity, L a characteristic length

Y

of the system and y its surface tension), the contribution of the potential gravitational energy

completely overshadows the capillary contribution. Only at small Bond numbers, for example

when the system is very small (capillary systems) or when the residual acceleration becomes

zero, as in microgravity conditions does the action of the surface tension forces impose

particular shapes to the entire free surfaces and interfaces. This effect is largely used as a basis

for many measurement techniques. However, surface tension effects can always be observed

locally, like menisci at a three phases line junction, or like Marangoni motions, when thermal

or compositional gradients exist on the surface itself.

The surface tension value retains its full thermodynamic significance, as a general property of

the system under study, only if measured under true equilibrium conditions. Measurements out

of equilibrium, giving information about the kinetics and the mechanisms of mass transfer to

and though the surface, need the exact conditions under which the measurements have been

made to be specified, their significance being restricted to these transient conditions.

For these reasons, measurements of the surface tension of pure substances must be made under

saturated vapour conditions, taking particular care that even the smallest quantity of foreign

components is avoided. In multi-component systems mechanical and chemical equilibrium

must be established before measurements start; this can take some minutes with liquid metallic

alloys and even several hours when dealing with highly viscous liquids, like glasses, polymers,

or solids [ 13,36].

Thus, the measurement of surface tension can be made under "static" or "dynamic" conditions.

By definition, the "pure dynamic surface tension" is that measured at the instant of the

formation of a new surface. Physically, this means that liquid with the bulk composition is

drawn to the surface and no time is allowed for adsorption processes to take place. Following

the Gibbs model [ 10] this can be stated by saying that the 'relative adsorption' is zero. After the

488

first non-diffusive process of forming the fresh surface, diffusion takes place, allowing the

system to reach its equilibrium condition by modifying the surface composition. During this

process, the surface tension varies according to the generalised Gibbs adsorption equation:

~y - - s ~ 6 T - ~ Fi~il i a q- C' Z oc'irt~x~ -I- C" Z ~i"t~X~' (24) i i i

where s ~ is the excess surface entropy per unit area, c' and c" the total molar concentration in

the two adjacent phases, e i' and ei" the cross chemical potentials, defined by Defay [37] as the

derivative of the surface free energy with respect to the concentration of component i in the

bulk phase. Under equilibrium conditions the cross chemical potentials become zero.

Thus, when an interface between a fluid phase and a surfactant solution is created (fresh

interface), the value of the interfacial tension decreases from its initial value, (Pure Dynamic

Interfacial Tension, 70), to its value at equilibrium, (Static Interfacial Tension, )'eq).

At time to, when the "fresh" interface is formed, the concentration of the solution is uniform in

the whole bulk phase, up to the interface. The fresh surface is a sink for the surface active

molecules and, for t>t 0, the adsorption process starts up, inducing a diffusion profile in the

neighbouring phases. The surface concentration increases until it reaches the equilibrium value

with the bulk concentration, corresponding to the vanishing of the diffusion profile (equal bulk

and surface chemical potentials).

Eq. 4 clearly shows that the surface tension value depends on the instantaneous composition of

the surface layer: this composition can be evaluated directly by means of infrared spectroscopy,

electron spectroscopies, low angle X-ray diffraction and so on, or, indirectly, by surface tension

measurements: in these ways the tracing of all the mass transfer processes between the surface

and the adjacent phases is also possible.

2.2. BRIEF SURVEY OF EXPERIMENTAL METHODS TO MEASURE SURFACE AND INTERFACIAL TENSIONS

A T HIGH TEMPERA TURE

2.2.1. Surface shape methods: sessile drop and pendant drop

The two methods are treated here together because, in principle, they utilise the same physical

phenomenon and are described by the same equations.

489

A drop, either sessile or hanging , takes on a shape which is the result of the equilibrium

between the action of the gravitational field, which tends to flatten or elongate it, and the action

of surface tension, which requires the smallest surface area.

The drop shape is described by the Laplace equation which can be rewritten in the more

explicit form proposed by Bashforth and Adams [4]

- ~ [ 1 s i n a i 2 + = R/b + x / b J (25)

pgb 2 where f l - b is the radius of curvature at the drop apex and R, d~ the radius of

curvature and the inclination of the tangent at the point P (x,z) (Fig. 5).

This equation has no analytical solutions, so the parameters that appear in it, the surface tension

in particular, must be determined by fitting this equation to the experimental drop shape.

" ' " ' " ' " ' ~ .~o N

O X__

/ z O

•

Fig. 5 Meridian section of a sessile and a pendent drop.

The sessile/pendant drop method has attracted, in the last 100 years, a large number of

scientists have been concentrating their work in two main directions. The first was an attempt

490

to solve eq.25 numerically and to present the results in the form of handy reference tables

[4,38,39], or to try to find approximate solutions of eq.25 by using these tables in a suitable manner

[9-40-42]. The second direction concerned an attempt to improve the methods for measuring the

shape of the drop with the highest possible precision and to design experimental set-ups which could

guarantee extremely clean and controlled physico-chemical conditions [43-45].

Today, the development of computerised numerical calculation and computer aided imaging

techniques [46-53] is making the sessile/pendant drop method more and more reliable, accurate

and, as a consequence, more attractive for surface tension measurements. This method is

particularly suitable for high temperature systems and for systems where equilibrium is

established only after a long time. Some details of the experimental procedures and of

automatic measurement techniques are given in Section 4.

2.2.2. Capillary rise

In this method, the surface tension is determined by measuring the level to which the liquid

under study rises when allowed to enter a capillary tube of specified radius (Fig. 6).

Fig. 6 The capillary rise technique.

The exact formula for the capillary rise was given by Sudgen [54]:

bH = 2 ),/(pg) (26)

where H is the height of the bottom of the meniscus in the capillary above the level of the free

surface of the liquid and b is the radius of curvature of the meniscus itself. Tables have been

491

developed with which eq. (26) can be solved by successive approximations [55].

This method is most suitable for low temperature systems, and when the contact angle on the

capillary wall can be assumed to be zero.

2.2.3. Maximum Bubble Pressure

This method consists in measuring the pressure necessary to force a bubble of gas or a drop of

a second immiscible liquid through a capillary into the liquid under investigation.

This method was proposed by Simon [56], and then thoroughly investigated by Sudgen [57],

and, more recently, by Mysels [58].

When a bubble, or a drop, is forced into the matrix liquid, the capillary pressure which

develops inside it due to the curvature of the capillary interface, passes through a maximum.

In a gravitational field, the drop shape is distorted with respect to the sphere, and the surface

tension is usually calculated by the relationship :

1 * I 2 ( g p a ) l(g2p2 a ) l 7 =~-aP 1-~ - ~ - ~ p*2 (27)

where g is the acceleration due to gravity, a is the radius of the capillary, P the density

difference between the two phases and P* the maximum pressure value. This method has been

successfully used for pure liquids at low and at high temperatures (liquid metals). However,

when other components are present, extreme care must be used in order to be sure that

conditions as close as possible to equilibrium are established.

2.2.4. Wilhelmy plate - Du Noiay ring

Wilhelmy [59] first proposed to measure the surface tension of a liquid by measuring the

maximum force needed to pull a vertical plate from the surface of a liquid. Du Notiy [60]

proposed the use of a ring of wire mounted horizontally and other investigators have used

solids of different shapes and dimensions [61-63].

All these methods aim at measuring the maximum weight of the meniscus formed by the plate (

or ring, rod etc.) when it is raised above the level of the free liquid surface. This weight is

exactly counterbalanced by the surface tension force only if the plate, or the ring wire have a

492

zero thickness and the liquid completely wets it.

The surface tension is calculated by the following equations:

?= mg/2(L 1 +L0) Wilhelmy (28)

? = (mg/4rtR) F Du Notiy (29)

where mg is the maximum force, L1 the plate perimeter, L0 an end correction depending on the

thickness of the plate, R the radius of the ring and F a correction factor which can be found in

special Tables as a function of the system geometry [64].

These methods are particularly suitable for pure liquids and when the interfacial equilibrium is

attained in a fast way. An interesting modification is to determine the surface tension by

making only slight deformations of the surface [65] so that a minimum stress is imposed to it.

These methods can seldom be used in high temperature systems unless a special set-up is used

with particular attention to the choosing of the materials in contact with the liquid phase [66].

2.2.5. Levitated drop

The levitated drop technique [67-69] is based on the fact that a freely floating drop (for

example, suspended in an electromagnetic field) will develop natural oscillations about its

equilibrium spherical configuration [70], (Fig. 7).

/ 1 Prism

I i

-~ Image of drop profile

I G O S - - - ~ M I

Reverse-wound~. . ~ ) / M o l t e n drop induct ion coil ~

e.__....~- S d i ca tube

I 1 Fig. 7 Schematic layout of a levitated drop apparatus.

493

The frequency x of oscillation of the drop of mass M is related to its surface tension by the

Rayleigh equation [71]:

3~r Mx 2

~ " - 1(1-1)(1+2) (30)

where 1 represents the normal modes of the oscillations. The fundamental mode 1=2 yields the

Rayleigh frequency XR.

The levitation results are sensitive to many factors, such as gravitational distortions,

oscillations amplitude, rotation of the drop etc., which must be taken into account in order to

obtain reliable surface tension values [72,73].

For aspherical, non-rotating drops, the 1=2 mode is shifted and split into three peaks, whereas

for rotating drops five peaks arise. Recently, accurate mathematical treatments have been

developed [74,75] allowing reliable surface tension measurements to be made by this

technique, taking into account the aforementioned shifts and splitting, and the translational

frequency of the drop in the sustaining magnetic field.

This method is ideal for high melting point metals and alloys, because it allows the drop to be

examined without any contact with supporting solids. By this way, also very reactive materials

can be studied or high temperatures can be attained without polluting the liquid under study by

contact with a container material.

Moreover, the absence of solid walls reduces by a great extent the possibility of heterogeneous

nucleation. Thus, if the liquid surface can be maintained free of solid oxides, this technique has

also been used for measuring surface tension and viscosity in liquid metal systems in the

undercooled region, and, in particular, in microgravity conditions [76-79].

The levitation of the drop can be obtained also by means of acoustic waves [80] or by a laminar

gas flow [81 ].

2.2.6. Drop weight method

This method represents an alternative to the levitation technique when dealing with transition

metals at their melting point, for which hardly any containers can be used due to their high

reactivity. The method consists in evaluating the surface tension from the weight of the liquid

494

drop issuing from an orifice at the end of a vertical tube or, in the case of metals, at the end of a

rod of the metal itself, suitably heated up to the melting point. Tate, in 1864, proposed [82] the

formula mg=2rtR) ,, where m is the mass of the drop and R the radius of the metallic rod.

However, as shown by Guye and Perrot [83] and by Harkins and Brown [84] the drop breaks

away following a complex procedure: this means that the real volume of the falling drop must

be derived by solving the Laplace equation. In this way, Harkins and Brown were able to

introduce into the Tate law a correction factor which is a function of the ratio V/R 3, where V is

the detached drop volume and R the rod radius. Thus the correct formula reads:

mg = 2rcR)qv F (31)

The factors F were tabulated for a large range of drop volumes [85]. Recently, Vinet et al. [86]

have presented an extension of the Harkins tables, on the basis of a careful analysis of the

detachment conditions, when applying the drop-weight method to the measurement of the

surface tension of high melting metals.

Further details on this method and a thorough discussion on its possible applications can be

found in ref. [87].

2.3. SURFACE TENSION VALUES OF LIQUID METALS

The surface tension of pure metals has been measured over the last 100 years using all the

methods mentioned in the previous paragraph and, occasionally, using other techniques

dedicated to specific applications. Reviews were compiled reporting these data (88-91) and,

sometimes, critically evaluating the results as a function of experimental accuracy, materials

purity etc. It is often assumed that the highest surface tension values should be retained.

This implicitly assumes that all impurities which segregate to the surface always decrease the

surface tension: this is thermodynamically true, as demonstrated in the previous chapters.

However, high values can result from other sources of errors: among them, errors in the

determination of the metal density, which are reflected directly in the surface tension value,

systematic errors due to image analysis and to computing procedures, and so on.

Until the availability of data in which the surface tension value is associated to the

measurement of the surface composition (experiments are already under way), the above

495

criteria can be used, coupled with a careful evaluation of" i) the experimental conditions under

which the measurements have been performed; ii) the purity of the metals; iii) the real

composit ion of the gaseous and solid (support) environment .

Table 2 . Surface tension values of pure metals, qt = A- B (T-Tmp).

I ~ M e t a l A B

(mN m -l) (mN m "1 K -1)

Aluminium 890 0.182 Mercury 498 0.215

Bismuth 389 0.097 Nickel 1834 0.376

Cadmium 637 0.15 Palladium 1482 0.279

Chromium 1642 0.20 Platinum 1746 0.307

Cobalt 1928 0.44 Potassium 118 0.065

Copper 1374 0.26 Rhodium 1925 0.664

Gadolinium 664 0.058 Silicon 775 0.145

Gallium 724 0.072 Silver 955 0.31

Germanium 1162 0.18 Sodium 203 0.08

Gold 1162 0.18 Tin 586 0.124

Indium 561

Iridium 2264

0.095 i

!

0.247 i I

Tantalum 2010 at T=3273 K **

Titanium 1525 at T=1953 K **

Iron 1909 0.52 Tungsten 2310 at T = 3683 K **

Lanthanum 728 0.10 Uranium 1552 0.27

Lead 471 0.156 Vanadium 1855 at T=2183 K **

Lithium 399 0.15 Zinc 817 0.227

Magnesium 577 0.26 Zirconium 1430 at T=2123 K**

Data taken from ref. [92], (**) taken from ref.[86].

The data reported in table 2 have been taken from the recent review by Keene [92]. Where

possible, the average of the highest values published for each element has been taken, fitting

them to a linear equation. Values for high melting metals have been taken from ref.[86].

496

As mentioned previously, the evaluation of surface tension at high temperatures of the metallic

systems, comes from complex experimental situations which require:

�9 careful preparation and control of each experiment;

�9 full knowledge of all physico-chemical parameters (pressure, composition, vapour

pressure etc.) of the system under study;

�9 careful interpretation of the phenomena occurring between the liquid metal surface and

the gaseous environment, which have to be interpreted from a dynamic point of view and

not only, as often occurs, in terms of thermodynamic equilibrium.

As the main gaseous contaminant of liquid metals is oxygen, the studies of the interplay of all

possible mechanisms of oxygen mass transport at the interface of liquid metals-gas systems are

fundamental. The definition of the factors controlling the oxygen mass transport through the

liquid metal interface as a function of the boundary conditions also assume an important role

also in the definition of the efficiency of technological processes. As a consequence, several

effects have to be considered with respect to their capability of influencing the surface tension

values: metal evaporation, bulk diffusion, surface adsorption, chemical reactions in the gas

phase, mass transfer, competition between diffusion and kinetics of the gaseous species under

the different experimental environments (e.g. inert gas or vacuum), and chemical reactivity

with reference to competitive bulk and surface reactions.

Theoretical models have been developed taking into account all these effects in terms of the

interactions between the liquid metal surfaces and the gaseous phases as a function of the

different experimental conditions. These models allow the best working conditions to be

singled out and the oxygen behaviour and its actual concentration near the surface to be

predicted.

The principal arguments of some of these models will be presented in the following sections,

pointing out the role of oxygen with respect to surface segregation of metallic alloys and to gas-

liquid exchange.

3. GAS-LIQUID EXCHANGE

3.1. O X Y G E N TRANSFER AT LIQUID METAL - VAPOUR INTERFACES

When measuring the surface tension of liquid metals as a function of oxygen concentration one

of the critical points of the experiments is to know exactly the oxygen equilibrium

497

concentration in the bulk phase. This can be done either by a direct measurement inside the

bulk liquid, or by deriving this quantity from the knowledge of the actual oxygen concentration

near the surface of the liquid metal. The former is difficult to implement in surface tension

measurements, usually made with small quantities of the liquid metal. The latter requires a full

kinetic-fluid dynamic description of the system to be made.

Theoretical descriptions on the basis of kinetic and transport theories, relating the mass

exchange between liquid metals and the surrounding atmosphere at different oxygen partial

pressures have been made [93,94] both under inert atmosphere (He fluxing with an inlet

pressure of ~ 0.1MPa) and under a vacuum (total pressure lower than 1 Pa in Knudsen-regime).

When oxygen is transported by a gas-carrier flow, the boundary conditions are determined by

the presence of a counter-flow of metal vapours at high temperature and by chemical reactions

in the gas phase. Studies on the effect of oxygen partial pressure on the vaporisation rate of

metals [95] have shown that an enhancement of the rate of vaporisation of metals is verified

which depends linearly on the oxygen content of the surrounding gaseous atmosphere. There

are two mechanisms regulating this phenomenon: the former is a chemical process which

involves the formation of volatile metallic compounds (the oxide), the latter is a transport

process. When the metal vapours come into contact with the oxygen present in the gas-carrier,

the two gaseous phases interact forming the stoichiometric oxide: this process enhances the

fluxes of oxygen and metal vapours. In steady conditions, at a fixed temperature, the formation

of an oxide "fog" in the vapour phase around the surface can be observed. On the other hand,

far from the surface, the oxygen partial pressure is considered as constant. Thus, when the

oxygen partial pressure increases, the distance through which the metal vapours are transported

should decrease, i.e. the vaporisation rate increases.

3.1.1. Experiments under protective atmospheres (low Po2 values).

Recent studies [94,96,97] point out the relative importance of the different parameters that can

influence the behaviour of metal-oxygen systems and suggest how to control them in order to

explain and to drive the liquid-vapour interacting process. A dynamic description [94] allows

the oxygen fluxes near the liquid-vapour interface to be quantitatively evaluated, taking into

account the influence of oxidation reactions in the gas phase on the diffusion of the two vapour

498

species, the oxygen and the metallic vapours.

When a liquid metal comes into contact with a gaseous flux, four situations (A,B,C,I)) (Fig.8)

corresponding to four characteristic reaction regimes [98] are possible:

A) A counterdiffusion of metal vapours and oxygen is set up, with an oxygen adsorption rate at

the liquid metal surface depending on the partial pressure of oxygen in the gas-carrier. This is

the regime of slow reactions where, practically, diffusion is not disturbed by oxidation

reactions. This regime applies also to the cases where oxidation reactions are

thermodynamically impossible.

B) The molecules of oxygen which approach the liquid surface react with the metal following

the reaction

(gas) _. ]k/l(~(cond) (32) M (c~ + V 0 2 . . . . 2v

This is the regime of fast reactions with excess oxygen, where the diffusion time of metal

vapours is greater than the reaction time, so that the oxidation reaction consumes all metal

vapours and a layer of oxide is formed at the liquid surface.

C) Regime of fast reactions with excess metal vapours. At variance with case B, the metal

vapours are in large excess and oxygen cannot reach the liquid surface because nearly all

molecules are suddenly consumed to form the oxide "fog" very far from the liquid surface at

the limit of the flow pattern i.e. the reaction acts as a barrier for oxygen. The oxygen diffusion

times are greater than the reaction time.

D) Regime of instantaneous reactions. The reaction takes place at a defined distance from the

liquid surface. At this "reaction surface ~:" the equilibrium partial pressure of oxygen is

reached. The reaction times here are smaller than the diffusion times for both gaseous species.

From a quantitative point of view it is possible to define these different conditions by using

two characteristic parameters:

Thiele Modulus �9 = 8{[vKPM/(PtotDo2)]} 1/2 (33)

and e = (vPMDo2)/(Po2Do2) (34)

499

where PM is the vapour pressure of the liquid metal, Po2 is the oxygen partial pressure in the

gas-carrier, Do2 and DM are the oxygen and the metal vapour diffusivities, v and K are

respectively the stoichiometric coefficient and the kinetic constant of the oxidation reaction in

the vapour phase, and 6 is the distance between the surface and the gas fluxing phase which is

unaffected by the presence of the liquid surface.

Fig.8 Schematic layout of the four reaction regimes.

is a dimensionless quantity referred to a process in which the reactants reach via diffusion

the site where the reaction itself takes place. The square of this quantity defines the ratio

between the reaction and the diffusion rates, while ~ compares the magnitudes of the fluxes of

matter in the two opposite directions.

The four reaction regimes are defined by the following quantitative conditions:

Regime A) (I)2(( 1 (I)2 (( g

500

Regime B) e <( 0 2 <( 1/e

Regime C) 1 << �9 2 <( ~;2

Regime D) (I) 2 >> 1/~ 0 2 ))~3 2

Consequently, for any given set of temperature, pressure and flow rate values, a certain metal

system can be represented by a defined point in a diagram where log e is plotted against log (I) 2

as shown in Fig.9. It must be remembered that (I) 2 and e are calculated with reference to a well

defined reaction, with its own stoichiometry.

The only values of these two parameters that must be taken into consideration are those

corresponding to couples of values of Po2 and Temperature for which the reaction products

are thermodynamically stable.

2O

Regime A The reaction is slow with respect to diffusion

10

I

I ,

-10

-20 -10

Regime C / ~

~fas:reactions with excess . / /

/ / Regim,

~"sta"~~ I

0 10 20

lg~ 2

Fig.9 Stability diagram (Thiele diagram)

All the information contained in a stability diagram T/POE [99,100] can be suitably translated

on the plane lg t92 / lg e (Fig.9)" the position and the width of this selected zone are defined by

the thermodynamic properties of the system, and in particular, by the chemical stability of the

species taking part in the process considered.; all the points which lie outside this zone no

longer have any physical significance.

501

The results of such a procedure are presented in Fig 10 and Fig. 11 (assessed data about

stability of both pure Sn and Ni metals and their different oxides can be found in literature

[101]).

Fig 10 Kinetic diagram for Sn. The representative points are bound by the isotherms at the metal melting point, by the 105 Pa (1 bar) line and by the oxide thermodynamic stability limits. Isobars of oxygen partial pressures and (vertical) isotherms are also shown.

Fig. 11 Same as Fig 10, for Ni.

502

Moreover, it can be easily demonstrated that varying the value of oxygen partial pressure a

family of straight lines is obtained whose separation is linear with AP. These isobaric lines

allow the interpolation and extrapolation of data to be easily made with reference to the oxygen

imposed partial pressure.

On the basis of the different domains so defined, the most important quantity, the oxygen flux

No2, can be evaluated as a function of experimental quantities. Moreover, when No2 is known,

it is also possible to calculate a minimum characteristic time Om to form an oxygen monolayer.

As an example, No2 and Q~m values are reported in Tab. 1 for different metals exposed to a slow

flux (1 l/h) of a carrier gas containing 1 ppm of 02. These data show that, even if the oxygen

partial pressure is thermodynamically sufficient to oxidise the metal, in many cases (at high

temperature) the values of fluxes No2 are so small and the corresponding values of the reaction

times Orn so long that the metal surface can be considered "clean".

This is in agreement with Nogi et al. [102] who pointed out that the adsorption of surface

active impurities, oxygen in particular, on liquid metals is influenced by the degree of their

vaporisation, i.e. by metal volatility. They have shown experimentally that no variation of the

surface tension occurs at high temperature by changing the oxygen content in the gas

atmosphere. The diagram in tig. 12, shows the temperature dependence of the surface tension

of molten Zinc.

The Zinc surface tension in the purified hydrogen atmosphere (full line) decreases linearly with

increasing temperature. On the contrary, the surface tension in hydrogen saturated with the

water vapour at 273 K increases with increasing temperature near the melting point. This effect

is strongly dependent on the gas flow rate (curve a=70 cm3/min; curve b=120 cm3/min).

This behaviour depends on the relative quantities of metal vapours and oxygen in the

atmosphere: at high temperatures the metal vapours are able to remove oxygen even for high

gas flow rates [ 103]. The consequence of this effect is a cleaner surface of the liquid metal.

Further confirmation of these effect can be found in the work performed by S.Krishnan [ 104]

on the spectral emissivity of liquid metal surfaces.

503

850

Z E E 800-

c .g o~ c a)

I-- cD o m 750- I::

700 650

Fig. 12

i a..-'""

! / /

:m.p. 7~o 8~o 9~o ~o~

Temperature (K)

Surface tension of molten zinc as a function of temperature in different flow regimes (after [ 102]).

3.1.2. Experiments under a vacuum.

When the problem of oxygen mass transfer under very low total pressures is approached, it has

to be taken into account that the mean free path of the molecules in the gas phase is large with

respect to the dimensions of the reactor (Hertz - Knudsen diffusion regime)

As a consequence, when metals able to form volatile oxides are considered [105-108], the

oxide vapours can be removed from above the metal surface by pumping or by condensation on

the reactor walls: these processes determine significant displacements of the oxidation

equilibrium at the interface gas-liquid, by enhancing the rate of the condensed oxide

vaporisation.

This effect definitely needs to be taken into account when a theoretical model of the system is

made. On the other hand, it can be usefully exploited to purify the metal melt through "vacuum

refining" procedures [ 109].

Let's refer to a liquid metal sample, placed in a closed chamber at a temperature T, under a total

pressure ['tot lower than at least 1 Pa. The oxygen residual pressure Po2.g, is considered

504

constant. The limit of solubility [110]of oxygen in the liquid metal (= Xo,s) given above, the

formation of the first stable oxide in the condensed phase is favoured thermodynamically

(eq.32).

When Xo, ~ = Xo, s, i.e. when the metal and the oxide co-exist, the atmosphere composition is

fixed at constant T for a system at equilibrium, and can be calculated from thermochemical data

[101] if the equilibria in the metal - oxygen system are known [105,106]. In particular, the

vapour pressures Pj, s of the species containing oxygen (suboxides) resulting from thermal

decomposition of the oxide depend exclusively on T.

If the saturation degree "s" of the bulk liquid with respect to dissolved oxygen is defined as the

ratio between the oxygen activities �9 s = - ao, l

ao , s , the oxygen partial pressure in equilibrium with

a molar fraction of dissolved oxygen Xo, 1 can be expressed as the product between s and the

saturation p r e s s u r e Po2,s"

=s2P (35) Po2,1 o2 ,s

An analogous relationship for the jth oxide partial pressure is obtained:

Pj,l = s2v' Pj,s (36)

The oxygen mass transfer between gas and condensed phase can then be formalised through the

evaluation of the global flux to and from the surface, which can be expressed by means of the

overall thermodynamic driving force, that is the difference between the equilibrium partial

pressure and the actual ones, indicated by the subscript "g".

,ot (s Po ) 02 = ,s -- Po2,g - PJ,g J

(37)

where the quantities Ko2, and K j, are global kinetic coefficients for the mass transfer and vj

the stoichiometry coefficient of the jth oxide formation reaction.

There are two different kinds of contributions to the oxygen total flux, one given by "free"

505

molecular 0 2 and the other by the j species, where oxygen is "linked" as oxides. When

equilibrium is established between the gas and the condensed phase, all pressure gradients

disappear, and the net flux is equal to zero; but steady state conditions in which no variations of

composition with time take place can be attained also out of equilibrium_( equal incoming and

outgoing oxygen fluxes) [94].

The oxygen transfer process can be formally split into individual "elemental" processes,

namely:

a) mass transfer inside the liquid phase, from the bulk liquid with an activity ao, ~ = Sao, s to the

liquid close to the interface ( ao, ~ = S~ao, s ).

b) Liquid - interface mass transfer, from the layer at saturation S I to the interface.

c) Interface - gas mass transfer, from the layer at saturation S I to the gas just overhanging the

liquid surface, at a partial pressure Po2,~ �9

d) Mass transfer inside the gas phase, from the layer at Po~,~ to that at partial pressure Po,,g �9

At very low total pressure the mean free path of the molecules can be assumed to be much

greater than the characteristic dimensions of the gas phase container. As a consequence, no

significant pressure gradients can be established in the gaseous environment, and it can be

assumed that: Po2.~ ~ Po2.g. Thus, under the conditions considered, the transport phenomena in

the gas phase are not a controlling stage of the oxygen exchange.

Similar considerations apply to volatile oxides, whereas the total oxygen flux at the interface

must take into account the contribution of the evaporation of oxides.

When, in addition to the vacuum, the presence of a wall cold and/or infinitely extended with

respect to the liquid metal sample is imposed at a certain distance from the interface, a

quantitative condensation of the metal and oxide vapours can be invoked, so that further

considerable simplification can be introduced : Pj,g ~ PM,g "~ 0

This is likely to happen mainly when the temperature of the chamber wall is considerably

lower than that of the sample; as a consequence, the vapour pressure of the elements and the

506

compounds condensed on the cold wall can be neglected.

When at least one of these conditions is verified, a steady state saturation degree Sst can be

defined"

Po2 ,g

Po2s+ v j - - vj Pjs ' " a 0 2 ' s~

where M M

m - Mo,

This equation relates the bulk oxygen activity in steady state condition to Po2,g at a definite T,

upgrading the commonly used Sievert's model [11] for gas dissolution with specific

arguments on kinetics and oxides vaporisation. According to eq.38, "vacuum" operating

conditions and vapour condensation can determine a lowering of the oxygen content of liquid

metals with respect to normal gas - liquid equilibrium conditions. The importance of this

lowering, due to the presence of the factor ,~] .... P , . . . . , depends on the oxides volatility. J

By introducing a slightly different formalism, the conditions of residual Po2 and T can be

determined and these are required to obtain a specific value of Sst, in order to carry out

experiments on or measurements on a system whose real physico-chemical conditions

(oxidising or reducing) are known. It can be shown [94] that a steady - state condition with

respect to the oxygen exchange is reached when the 0 2 residual pressure is equal to the

quantity:

~ (m+) p (s)=s Po + v j - - v, 02 ,s . g~02

Pj,s s2"j (39)

p E o2 can be regarded as the "effective oxidation pressure": it is the 0 2 pressure that has to be

imposed on a liquid sample (e.g. liquid drop) to maintain, in a steady state, an oxygen activity

ao, l in the liquid phase

507

According to this equation, starting from a system with an initial saturation degree So,, vacuum

refining with respect to oxygen can be obtained in two ways:

�9 by lowering the 0 2 residual pressure in the system at constant T, to an oxygen pressure

Po2.g --PEo2 (Sfin)< P E02 (S,n)" this condition is often not attainable experimentally, due to the

extremely low values Po2 would need to reach for certain metal oxides.

�9 by modifying the temperature [ 111-113] making the oxide volatility increase. This requires

a previous estimation of the vapour pressures derivatives with T for both oxides and metal,

in order to be sure that their ratio does not become unfavourable with increasing

temperature.

It is worth pointing out that the effective feasibility of such a process depends not only on

oxygen partial pressure and volatility of oxides, but also on that of the metal.

If the vapour pressure of the metal is much greater than that of the oxides, the oxygen loss

from the liquid may be lower than that of metal; as a consequence, oxygen impurities

concentrate in the liquid.

/ s, 1/ In the particular case of a metal completely saturated with oxygen s = - - = , eq. 39 takes s

the form:

pE =Po2 + ~ J - - m + P. (40) o~,s v, v~ ~,s ,s ~O2

This equation defines the oxygen effective saturation pressure PoE s , and it is the actual value

of the residual oxygen pressure that must be imposed over the condensed phase in order to

saturate it.

This quantity applies both to oxidation of liquid and solid metals under a vacuum (with

quantitative vapours removal), and it is frequently much higher than the equilibrium saturation

value, as shown in Fig. 13 for the AI - O system.

When Po2.g > pE is imposed, a thermodynamically stable oxide layer, thickening with time, 0 2 ,S

508

is expected to appear on the surface. This can be defined as "coating regime of oxidation".

When, on the contrary, Po2,g < PoEs, the following phenomena are expected:

a liquid metal dissolves oxygen proportionally to the pressure imposed, according to eq.39;

a solid metal able to form volatile suboxides enters a regime of "active corrosion"; the

surface reacts with oxygen, forming compounds that evaporate rapidly, determining a

consumption of the sample.

This can be regarded as a "non - coating regime of oxidation", so that the PoE: VS. T curve in

Fig. 13 represents the "transition curve" between the two regimes. This figure shows also the

results obtained by Goumiri and Joud [ 114], who investigated on the oxidation kinetics of both

liquid and solid aluminium surfaces previously cleaned by ion - sputtering under the following

conditions"

For solid AI: Po2 ~ 10 -~8 MPa ; T = 293 K

For liquid AI" Po2 ~ 10 -17 M P a ; T = 973 K

-10 A t. =E -20

,,...,. -30

lie

g. -40 g ~ -50

0

-60

-70

Surface o x ~ al

,,

l ....

m Eft. Press.

.... Eq. press

0 500 1000 1500 2000 TemDerature (10

Fig. 13. Transition curve between coating (surface oxidation) and non-coating (oxide removal) regimes. The upper line represents the effective oxidation pressure, the lower one the thermodynamic oxygen saturation pressure. The two full triangles represent the experiments reported in [ 114].

The representative point of the solid sample lies in the coating regime zone of the diagram; on

the contrary, liquid A1 is not expected to be coated in the given experimental conditions.

509

These theoretical predictions agree closely with the experimental observations: a rapid

oxidation of the solid surface previously sputtered was in fact reported by Goumiri and Joud.

On the contrary, no sign of contamination was observed in the time of the experiment on liquid

Aluminium.

3.2. EVALUATION OF THE CHARACTERISTIC TIMES TO REACH STEADY STATE CONDITIONS

To reach steady-state conditions requires a time interval which is a function of systems

parameters. The evaluation of this time is of great relevance to surface tension measurements.

Interfacial characteristic times (for diffusion, and to reach stationary conditions) and the trend

of the oxygen interfacial composition vs. time can be calculated, starting from a local oxygen

mass balance in the liquid metal drop under Knudsen conditions, and taking into account the

total mass balance [ 115 ].

The results of this study show that, if the diffusion time within a drop of radius R is defined as

[98]:

R 2 t D _ , (41)

9D o

st the time tst necessary for a homogeneous drop to reach the steady state composition x o from

0 the initial concentration x o is:

1 Rc~ t s t - 3 - 6 - ( x ~

No2,I (42)

3.2.1. "Small" drops

We define the drop as "small", when the diffusion time within the drop is very small (related

to the system parameters), so that the drop composition can be taken as uniform and the

condition t D ~( tst is verified. From a geometrical point of view, this condition is equivalent to :

By comparing the characteristic times to. . , a s defined in eq. 41, and tst, two subclasses of

systems can be distinguished: "small" and "large" drops, depending on the different

mechanism of the oxygen transport in the liquid phase.

510

R << 3DoCl (xO _x~) (43) 0

N o2,I

Thus, for "small" drops, the characteristic time is calculated according to eq.42 and the steady-

st is achieved at the same time both at the interface state condition: x o = Xo, ~ = x o = const ,

and in the bulk.

3.2.2. "Large" drops

The drop is defined "large" when tD>> tst and eq.43 becomes:

ooc, (xO s,) R >> -xr0 - Xo

IN 0 2 ,I

(44)

In this case the diffusion within the liquid metal drop is the controlling process.

The time tst c a n be evaluated, in this case, by using the relationship, whose derivation can be

found in [ 115] :

Rc~xo, s 1 = . (45)

Pj , s tst 3k~176 I+X-~j~

Po ,s

where the different parameters have the same meaning as in the definition of the "effective

oxidation pressure PoE s (eq. 39 ).

The characteristic time tst for some metallic systems of technological interest has been

calculated by using eq. 20, considering, as a first approximation, the ratio of the condensation

coefficients ~ and ao2 to be equal to one if the efficiency of the evaporating process is

assumed to have about the same value for both species.

As an example, the values referred to the liquid tin are reported in Table 3.

The values of tst as a function of temperature at a total pressure Ptot = 10"1 Pa for a set of

selected metals are reported in Tab. 4. By comparing the reported values of tst calculated under

fixed conditions of temperature ( Tm<T<Tb ) and pressure for different liquid metal drops

511

having constant dimensions, with the estimated [116] corresponding values of oxygen

diffusion times in the liquid metal (tD --101 - 103 S) , the two subclasses can be recognised:

"small drops" when tst >>tD and "large drops" when tst <<to (shadowed cells), as defined by

the theoretical model. A tin drop, for example, can be considered "small" up to a temperature

of about 700 K , but for temperatures above 1000 K it can be considered "large". Drops of In

or Pb can be considered "small" in the overall range of temperature here considered. On the

other hand, a drop of Ge can always be considered "large".

Table 3 Characteristic times for liquid tin.

Table 4 tst as a function of temperature at a total pressure P t o t - l0 ~ Pa for a set of selected metals.

For clarification, Fig. 14 shows that the classification in "small" or "large" drops, not only

depends on the drop's radius but mostly on the liquid metal temperature. The trend of tst as a

function of the radius of a liquid tin drop is shown at three fixed temperatures. The trend of to

512

is also reported: the values of the diffusion time have been calculated at the boiling temperature

on the basis of available literature data [8] and extrapolated for T < 1023 K. It can be noticed

that despite its small radius, at a temperature near the boiling point, a liquid tin drop is

classified as "large", whereas near the melting point the drop is "small". For the intermediate

temperatures (e.g.: 870 K in Fig. 14) some tst curve may cross the corresponding tD curve

and a fixed radius can be singled out which determines its the belonging to the "small" or

"large" drop class.

1E+12

1E+10

1E~OG

IE+Oe

X 1 E ~

i 1E+02

IE~O

1E02

. . . - - - - - - -" " " " " " w l z l ( ~ K ) w ie l (a'to ~ mN(11MK) ID IS] (5t0 K- 11MK)

1 E - g e . , . , . , . , . o - , - - - " ' ~ 1 7 6

1 1 ~ 1 ~ . . . . . . . . . . 1E,.04 1Lr'-0~ 1E..l~

i l m l

Fig. 14 Diagram showing the influence of temperature on the characteristic times for a tin drop as a function of drop radius. Above the tD grey line the transport process is diffusion controlled (="large" drop).

4. SESSILE DROP EXPERIMENTS: AN EXAMPLE OF STANDARD PROCEDURE,

The Surface Tension measurements of liquid metals and alloys, made by the Sessile Drop

technique, follow procedures which may vary from one laboratory to another. This fact, with

some other reasons which will be discussed in the following, may be considered one of the

main reasons for the large scatter in the literature data which is usually found even for the

"simplest" pure liquid metal.

513

Hereafter, an attempt to summarise the most important steps to be followed in this kind of

measurements is outlined and briefly discussed.

4.1. THE APPARATUS.

The ideal apparatus for surface tension measurements at high temperatures should be made up

of the following parts, and should perform in some special ways:

a) A high-vacuum chamber, both for vacuum and controlled atmosphere experiments.

b) A heating element with the lowest possible polluting potency: for T < 1700 K Pt

resistors can be used, also under slightly oxidising conditions. BN coated graphite

elements can also be used. For vacuum or reducing conditions tungsten can be used, or

vitreous carbon. A cleaner way of transferring the heating energy from outside the

chamber is induction heating (here again, with a vitreous carbon susceptor if working

under reducing conditions) or image fumaces, which are to-day able to reach

temperatures of 2000 K.

c) A homogeneous temperature distribution around the molten sample.

d) A perfectly aligned optical line, providing background illumination (homogeneous and

stable in time), optical filters, to obtain monocromatic light and to cut infrared emissions,

a high quality objective lens for the photographic camera or the TV camera, providing a

magnification factor larger than 1, and a distortion-free image.

e) Diagnostic elements: vacuum gauges, quadrupole residual gas analyser, an ion-gun to

clean the liquid metal surface, temperature sensors etc.

f) The possibility to make simultaneous measurements of the liquid metal surface by Auger

spectroscopy is of the highest relevance. However, this requirement suffers from severe

limitations due to metal evaporation, which make it feasible only in a few selected cases.

g) Electrochemical oxygen sensors should also be present in the chamber, to record the

actual residual oxygen partial pressure, whose influence has been clearly shown in the

previous paragraphs.

h) The liquid drop must be perfectly levelled and axisymmetric. To reach this latter

condition a special cup can be used as shown in

514

i) Fig. 15. This would also help in the obtaining of high apparent contact angles, due to the

pinning effect of the sharp edges.

j) A shape factor 13 >>2 should be sought in order to minimise the measuring incertitude

[42]. To this end, diagrams like the one shown in Fig. 17 can be used, where, for a given

base radius, the volume necessary to obtain a certain beta value and a desired apparent

contact angle can be estimated.

j) The metal composition must be perfectly known. This point represents the real obstacle

to the reproducibility of measurements, in view of the extremely high surface-activity

many elements have for liquid metals and alloys.

k) Absence of any reaction between the solid support and the liquid phase. To this end solid

supports like vitreous carbon, monocrystalline sapphire, boron nitride alumina and other

high stability ceramic compounds have to be chosen. If reactions cannot be avoided on

the basis of rigorous thermodynamic arguments, other methods should be used, like the

levitating drop, the pendent drop or the drop-weight techniques.

1) A suitable and highly performing computational technique. As already mentioned in the

previous sections, several procedures are available to-day. As a further example, the one

currently in use in the authors' laboratory is here presented briefly.

............. I ..........................

Fig. 15 Cross section of the sapphire cup used in surface tension measurements. R = 6 mm, d=4 mm.

515

Fig. 16 Sessile drop volume as a function of [3 for different apparent contact angles. Base radius R=4mm.

4.2. ASTRA (AUTOMATIC SURFACE TENSION REAL TIME ACQUISITION SYSTEM) [49].

In the introduction it was noted that the Laplace equation cannot be solved analytically, but

requires numerical integration methods, which are now easier and faster thanks to the use of

computers. Moreover, the availability of computer-aided image processing systems makes it

possible to get the drop's profile automatically so that the measurement of 3' has become very

fast, allowing the monitoring of ~/even under dynamic conditions.

ASTRA is an experimental methodology and an integrated software to get and process data of

drop shape profiles to determine surface and interfacial tension and contact-angle values. Due

to its high performances in terms of time of acquisition and reliability, it is particularly suitable

for both static and dynamic measurements. Indeed, by using ASTRA it is possible to reach up

to two interfacial tension measurements per second, having access to dynamic measurements

over very large time scale. ASTRA is currently used both for liquid metals and for liquid

systems at room temperature.

For the acquisition and processing of the drop's image, the IMAGING System is used (Imaging

Technology Inc., Woburn, Massachusetts, USA). It consists of an AD/DA interface processor

516

connected to a Personal Computer, which digitises each TV frame coming from a B/W TV

camera into 512 X 512 image points (pixels), assigning to each pixel a grey level value ranging

from 0 (black) to 255 (white). In this way, the image frames are digitized into a 512 X 512

matrix whose ~j element, on choosing a Cartesian reference compatible with the standard

matrix notation, is the average grey level of the pixel with its bottom-right comer at the

coordinate point (i, j). More than 200-300 points are generally acquired, instead of the usual

10-20 points provided by measurements made with a microscope.

The images coming from the TV camera can be processed in real time, or stored in order to be

used later. A routine library allows the A/D interface to be linked to FORTRAN programs to

achieve the image processing.

The most important sources of error related to this method are: intrinsic TV-camera distortion;

discretisation process; evaluation of the system magnification factor. TV cameras using traditional

tubes produce distorted images unsuitable for our purposes; CCD cameras give rise to image

distortions less significant than the errors due to the discretisation process.

Owing to the discretisation of the CCD camera and to the digitisation into pixels, the edge of

an acquired object has an indeterminacy of one pixel. Consequently, the maximum resolution

allowed by the present system is of the order of D/512, where D is the typical dimension of the

object image. For drops of about 1 cm, this error is equal to about 20~tm. To improve this limit

a procedure to obtain a Sub-Pixel Accuracy (SPA) is used.

This procedure is based on the calculation of the profile point as the "center of mass" of the

distribution of grey variation around the profile, by the formula :

E Xi~ii i where x are the coordinates of the points and 8 the grey variations.

A knowledge of the magnification factor of the system used is essential for deriving the real

drop profile coordinates from the acquired ones.

The image acquired has two different magnification factors, ay and ax, in the vertical and

horizontal directions, respectively. This anisotropy is due to the discretisation of the standard,

517

rectangular TV frame, whose dimensions are in the typical ratio of 4.5/6, into 512 X 512 pixels.

Thus, the digitisation process introduces an anisotropic metric transformation, M, of the real

images.

Denoting b y f the ay to ax ratio, M can be rewritten as:

:axE'o ;] ,46 Eq. 46 clearly shows that the 'physical' magnification factor can be split into two contributions:

ax, which represents the magnification factor due to optical devices (TV camera, lenses, etc.)

and f , which is the anisotropic magnification factor due to digitisation.

The ax value can easily be obtained by acquiring a horizontal dimension of a reference object

and by calculating the ratio of the acquired dimension to the real one.

It is very useful to measure a• after putting the drops to be measured on a support of known

dimensions, using this support as a reference object (Fig. 17).

Fig. 17 Snapshot of a molten gold drop sitting in a sapphire cup.

The f value is an intrinsic constant of the TV camera and of the A/D interface system, and, in

principle, it could be calculated from technical data. The value o f f is, however, extremely

important; in fact an error of 0.01 on f may cause an error of about 2% on the calculated ST

values. For this reason, suitable experimental methods should be adopted to define it with the

greatest accuracy.

On the other hand, all methods based on the acquisition of the profile of an object of known

horizontal and vertical dimensions are not precise enough, leading to errors onfof the order of 0.01.

518

The evaluation o f f is obtained by imaging the straight profile of a razor blade inclined to an

exactly known angle W and by calculating the slope of the acquired straight line by a linear fit

to the coordinates of the line points. The f value matching the angle q~ is calculated as the

mean of 20 measurements, with a standard deviation of the order of 0.0004, which introduces

a very small error on the calculated values of the surface tension.

The ASTRA method can be used to process real drop images, or photographic high-resolution

negative images of molten, metallic drops used in order to keep some documentation and to

make comparisons with the traditional method for profile acquisition with a microscope. In this

case high-resolution film (Kodak Technical Pan Film) is used to obtain dimensional stability

and high image contrast.

The CCD TV camera is equipped with a standard high quality photographic lens. To allow for

measurement repeatability, the focusing, focal length and diaphragm of the optical system were

kept fixed. All the optical devices are mounted on an optical bench.

A typical system magnification factor is equal to about 400 pixels/cm. As mentioned earlier,

this factor is calculated for each photo.

For the sake of clarity the software process is presented here subdivided into three parts

(modules): acquisition module; data pre-processing module; ST calculation module. The three

modules are written in standard FORTRAN 77 and communicate through data files, but in the

case of dedicated applications, they can communicate via the common memory areas, in order

to accelerate the processing step.

4.2.1. Acquisition module

After a preliminary phase, which is necessary to focus and align the drop, etc., and in which the

images are continuously monitored, a frame is frozen and processed to get the drop-profile

coordinates.

Each line of an image is scanned and the edge position is roughly determined as the point

where the absolute variation in the grey level is a maximum. The exact position of the

maximum in the grey level distribution around this point is then determined by the SPA

procedure. This method is usually slow, but it has been accelerated by limiting the scanning

area next to the drop profile.

This module also acquires the dimensions of the support to calculate the magnification factor

ax. The profile coordinates so obtained are stored in the bulk memory for further processing.

519

4.2.2. Data pre-processing module

This module allows systems formed by different metals and at different temperatures to be

examined. Variable data (density, temperature, parameters estimation, etc.) are entered at this stage.

The profile coordinates, previously acquired, are converted by the M matrix to the real ones.

Some drop parameters (e.g. maximum diameter, apex coordinates, apex curvature) are also

evaluated.

4.2.3. Surface tension calculation module

The Surface Tension is estimated, basically, by using the non-linear regression method

proposed by Maze and Burnet.[46]. This method uses eq.25, expressed in terms of the arc

length s of the drop profile, with

dx = cosq) ds

dy = sinq) ds

dcI) = 1/R ds

The drop profile can therefore be described in terms of the two independent variables r and s.

To take into account the effect of errors on the drop's coordinates, the following substitution is

also made

z = z+~, where e is the error in the Z - positioning of the drop apex.

The program makes a non-linear estimation of the parameters: J3 = Pg/7, b,

As input data, it uses a series of drop diameters and an initial estimation of the parameters.

To get S. T. values from automatic data, the original programme has been modified in order to

deal with a larger number of diameters, to follow more closely the logical trends of the new

techniques, and to obtain outputs of particular parameters and quantities (drop volume, shape

factor, contact angle, etc.).The processing time required is variable, depending on the speed of

convergence and on the type of computer. A processing time of less than one second can easily

be achieved on PC Pentium 200MHz.

520

5. ACKNOWLEDGEMENTS

The Authors wish to thank Profs. P. Costa and E. Arato (ISTIC Univ. Genova) and Drs. P.

Castello and L. Nanni for their contributions to the various part of the work reviewed in this

paper. Special thanks also to Dr. L. Liggieri (ICFAM-CNR) who designed and wrote the

ASTRA procedure reported in Section 4.2.

6. REFERENCES

1. T. Young, Phil. Trans. Royal Soc. 95, 65 (1805).

2. P.S. de Laplace, Suppl. au X livre de M6canique C61este, Courier, Paris, 1806.

3. K. Gauss, Princ. Gener. Theoriae Fig. Fluid., Gottingen (1830), Dove's rep. Bd.V, p.49.

4. F. Bashforth and C. Adams, An attempt to test the Theories of Capillary Action, Cambridge

Univ. Press, 1883.

5. J.C. Maxwell, "Capillary Action", Enciclopedia Britannica, 9th ed., 1878

6. B.S.Ellefson and N.W. Taylor, J. Am. Ceramic Soc. 21, 193 (1938).

7. V.K. Semenchenko" Surface phenomena in metals and alloys" Pergamon Press, Oxford, 1965.

8. See for ex. : "The role of surface phenomena in metallurgy" (V.N. Eremenko ed.),

Consultant Bureau, New York, 1963.

"Physical chemistry of surface phenomena and wettability" (V.N.Eremenko ed.) Naumka

Dumka, Kiev, 1971

"Physical chemistry of surface phenomena at high temp." (V.N.Eremenko ed.) Naumka

Dumka, Kiev, 1971

9. P. Kozakevitch and G. Urbain, M6m.Sci.Rev.Metall. 58,p.401,p.517,p.931 (1961).

10. R. Defay ,I. Prigogine, A. Bellemans and D.H.Everett, "Surface tension and adsorption",

Longmans, London, 1966

11. C.H.P. Lupis, "Chemical thermodynamics of Materials" Elsevier, Amsterdam, 1983

12. J.W. Gibbs, "The collected works" vol. 1, Thermodynamics .Yale Univ. Press, New Haven,

Connecticut 1928

13. L.E. Murr "Interfacial phenomena in metals and alloys", Addison-Wesley, 1975

14. E.A. Guggenheim J.Chem. Phys. 13,253 (1945)

15. R.V. E6tv6s, Wied Ann. 27, 456 (1886)

16. A.V. Grosse J. Inorg. Chem. 24,147 (1962)

521

17. J.C. Joud, N. Eustathopoulos, A.Bricard and P. Desr6, J.Chim. Phys., 9, 1290 (1973); and

7-8,113 (1974)

18. J.C. Joud and P. Desr6, Acta Astronautica 8-5,407 (1981).

19. Kh.I. Ibragimov and A.G. Nal'grev, Fiz.Lhim.Gran.Kaz.Koutac. Faz., Naukova Dumka,

Kiev, p. 85, 1976.

20. N. Eustathopoulos and J.C. Joud, Current Topics in Materials Science, (E. Kaldis ed.)

North Holland, Amsterdam, vol. 4, 1980, p. 281.

21. S.S. Dukhin, G. Kretzschmar and R. Miller, "Dynamics of adsorption at Liquid Interfaces",

Studies in Interface Science (D. Moebius and R. Miller Eds.),vol. 1,Elsevier, Amsterdam, 1995.

22. P. Tison, D. Camel, I Tosello and J.J. Favier, in "Hydromechanics and heat/mass transfer

in microgravity", (V.S. Avduevski ed.), Gordon and Breach, Amsterdam, p. 121, 1991

23. C. Marangoni, Ann. Phys. und Chemie (Poggendorff), 143,337 (1871).

24. C. Marangoni, Nuovo Cimento ser.2, 5/6, 239 (1872).

25. C. Marangoni, Nuovo Cimento ser.3, 3, 50 -97-193 (1878).

26. J. Thomson, Phil. Mag ser.4, 10, 330 (1855).

27. L.G. Napolitano and A. Viviani, First Intern. Microgravity Congress, Bremen, 1990.

28. B.J. Keene, Int. Materials Rev. 33,1 (1988).

29. P.J. Desr6 and J.C. Joud, Acta Astronautica, 8, 407 (1981).

30. P. Kozakevitch, Liquids: Structure, Properties and Solid Interactions (T.J. Hughel ed.),

Elsevier, Amsterdam, 1965

31. C.R. Heiple, J.R.Roper, R.T. Stager and R.J.Aden, Welding J., 62, 72 (1983)

32. H.A.Papazian, High Temp. Sci. 21, 41-48 (1986)

33. G. Tsotridis, H. Rother and E.D. Hondros, Naturwissensch. 76, 216-218 (1989).

34. S. Diefenbach, L. Ratke, B. Prinz and H. Ahlbom, in "Immiscible Liquid metals and

organics" (L.Ratke ed.), Inf~176 Germany, p.291, 1993

35. H.Walter, Microgravity News from ESA 7, 2 (1994)

36. S. Wu, "Polymer interface and adhesion", Marcel Dekker, New York,1982, p. 257

37. R. Defay, "Thermodynamique de la Tension Superficielle", Gauthier-Villars, Paris, 1971

38. S. Hartland and R. Hartley R., "Axysimmetric fluid-liquid interfaces", Elsevier,

Amsterdam, 1976

39. S. Fordham, Proc.Royal Soc.(London) 194A,1 (1948).

522

40. N.E. Dorsey, J. Wash. Acad. Sci. 18,505 (1928).

41. D.W.G. White, "A supplement to the tables of Bashforth and Adams", Mines Branch, Phys.

Metall. Div., Rep.PM-I-67-4, May 1967

42. R. Sangiorgi, G. Caracciolo and A. Passerone, J.Mater.Sci. 17,2895 (1982)

43. D.W.G.White, Trans.Amer.Soc.Metals 55,7571 (962)

44. R. Sangiorgi, C. Senillou and J.C. Joud, Surf.Sci. 202,509 (1988)

45. S.C. Hardy, Surf.Sci.71,6021(985)

46. C. Maze and G. Burnet, Surf.Sci. 13,451 (1969); and Surf.Sci. 24,335 (1971)

47. H.H.Girault, D.J. Schiffrin and B.D.V. Smith, J.Electroan.Chem. 137,207 (1982); and

J.Colloid Interface Sci. 101,257 (1984)

48. Y. Rotenberg, L. Boruvka and A.W. Neumann, J.Colloid Interface Sci. 93,169 (1983)

49. L. Liggieri and A. Passerone, High Temp.Techn.7,80 (1989)

50. P. Cheng, D. Li, L. Boruvka, Y. Rotenberg and A.W. Neumann, Colloids and Surf. 43,151 (1990)

51. N.R. Pallas and Y. Harrison, Colloids and Surf. 43,169 (1990)

52. Yu.V. Naidich and N.F. Grigorenko, J. Mater. Sci. 27, 3092 (1992)

53. F.K. Hansen, J.Colloid Interface Sci. 160, 208 (1993)

54. S. Sudgen, J.Chem. Soc. 120, 1483 (1921)

55. J.F. Padday, in "Surface and Colloid Science" (E.Matijevic ed.) Wiley-Intersci., New York,

1969,vo1.1, p.101-149

56. M. Simon, Ann.Chim.Phys. 32, 5 (1851)

57. S. Sudgen, J.Chem.Soc. 121,858, (1922). ibidem 123,271(1924)

58. K.J. Mysels, Colloids and Surf. 43,241 (1990)

59. L. Wilhelmy, Ann. Physik 119,177 (1863)

60. L. Du Nofiy, J. Gen. Physiol. 1,521 (1919)

61. R.L.Tiede, Ceramic Bulletin 51,539 (1972)

62. Z. Ugarcic, D.K. Vohra, E. Atteya and S. Hartland, J. Chem. Sot., Faraday Trans. 1, 77, 4 (1981)

63. J.F. Padday, J.Chem.Soc.,Faraday Trans.1 75,2827 (1979)

64. C. Huh and S.G. Mason ,Colloid and Polym.Sci. 253,566 (1975) (Suppl. tables)

65. B. Maijgreen and L. Odberg, J Colloid Interface Sci 88, 197 (1982)

66. D. Chatain, C. Vahlas and N. Eustathopoulos, Acta Met. 32, 227 (1984)

67. M.E. Frazer, W.K.Lu, A.E. Hamielec and R. Murarka ,Metall. Trans.2,817 (1971)

523

68. R. Murarka, W.K.Lu and A.E. Hamielec, Can.Met.Quart.14,111 (1975)

69. B.J.Keene, K.C. Mills, A. Kasama, A. McLean and W.A.Miller, Metall. Trans., 17B,159 (1986)

70. H. Lamb, "Hydrodynamics", Cambridge Univ.Press, London, p.428,1906

71. Lord Rayleigh, Proc. Royal Soc. 29,71 (1879)

72. H. Soda, A. McLean and W.A. Miller, Metall.Trans. 9B,145 (1978)

73. S. Krishnan, G.P.Hansen, R.H. Hauge and J.L.Margrave, Metall. Trans. 19A,1939 (1988)

74. D.L. Cummings and D.A. Blackburn, J. Fluid Mech. 224, 395 (1991)

75. I. Egry, J. Mater. Sci. 26, 2997 (1991)

76. I. Egry, B. Feuerbacher, G. Loh6fer and P. Neuhaus, Proc.VII Eur.Symp.on Mater.and

Fluid Sci.in microgravity, Oxford,Sept.1989 ESA SP-295,257,1990

77. I. Egry and J. Szekely, XXVIII COSPAR Plen.Meet., The Hague, The Netherlands, June

25-July 6 1990, Paper S. 11.3.8

78. S. Sauerland, G. Loh~fer and I.Egry, Thermochimica Acta 218, 445 (1993)

79. I.Egry, G.Jacobs, E. Schwartz and J. Szekely, Int. J. of Thermophys. 17,1181 (1996)

80. C.J.Hsu and R.E.Apfel, J.Colloid Interface Sci. 107,467 (1985)

81. L. Elasmi and F. Feuillebois, XXVIII COSPAR Plen. Meet., The Hague, The Netherlands,

June 25-July 6,1990, Paper S. 11.1 .P. 17

82. T.Tate, Phil. Mag. 27, 176 (1864)

83. P. Guye and F i . Perrot, J. Chim. Phys.Arch. Sci.Phys. Nat. 15, 178 (1903)

84. W.D. Harkins and F.E. Brown, J.Am. Chem.Soc. 41,503 (1919)

85. W.D. Harkins and F.E. Brown, Int. Critical Tables, vol.IV, McGraw-Hill,New York,1928

86. B. Vinet, J.P. Garandet and L. Cortella, J.Appl. Phys. 73, 3830 (1993.

87. R. Miller and V. Fainerman, this Volume, pp 139

88. O. Flint, J. Nuclear Materials 16,233(1965)

89. G. Lang, Z. Metallk. 67, 549(1976)

90. M.J. Murtha and G. Burnet, Ames Lab. ERDA, Iowa State Univ. Rep. IS-3829, 4/1976

91. L.D. Lucas, Techniques de l'Ingenieur 7, M67,1(1984)

92. B.J. Keene, Intern. Mater. Rev. 38, 157(1993)

93. E. Ricci, A. Passerone, P. Castello and P. Costa, J..Mater. Sci. 29, 1833 (1994)

94. P. Castello, E. Ricci, A. Passerone, and P. Costa, J..Mater. Sci. 29, 6104 (1994).

524

95. E.T.Turkdogan,P.Grieveson,L.S.Darken, J. Phys.Chem. 67,1647 (1963)

96. E. Ricci ,A. Passerone and J.C. Joud, Surface Science 206,533 (1988)

97. A. Sanfeld ,A. Passerone,E. Ricci and J.C. Joud, I1Nuovo Cimento 12D,353 (1990)

98. G.Astarita,"Mass Transfer with Chemical reactions", Elsevier, Amsterdam, 1967

99. Y. Austin Chang and Ker- Chang Hsieh "Phase Diagrams of Ternary Copper- Oxygen-

Metal Systems" ASM Int., Metals Park, Ohio, 1989

100. T.B. Massalski, J.L. Murray, L.H. Bennet, H. Baker, "Binary Alloy Phase Diagrams", ASM

International, Metals Park, Ohio, 1986

101. O. Knacke, O. Kubaschewski and K. Hesselmann, "Thermo-chemical Properties of Inorganic

Substances - Second Edition" Springer Verlag, Verlag Stahleisen m.b.H. D~seldorf, 1991

102. K.Nogi,K.Ogino,A.McLean and W.A.Miller, Metallurgical Trans.B 17B,163 (1986)

103. P.A. Distin and S.G. Whiteway ,Can.Met.Quart. 9,19 (1970)

104. S. Krishnan and P.C. Nordine, Phys. Review B 47,11780 (1993)

105. H.H. Kellogg, Trans. of the Met. Soc. of AIME 263,602 (1966)

106. D. Beruto, L. Barco and G. Belleri, Ceramurgia inter. 1, 87 (1975)

107. C. Wagner, J. of Applied Phys. 29, 1295 (1958)

108. E.A. Gulbransen, K.F. Andrew and F.A. Brassart, J. of Electrochem. Soc. 113,834 (1966)

109. L.Brewer and G. M. Rosemblatt, Trans. ofthe Met. Soc. of AIME 224,1268 (1962)

110. S. Otsuka and Z. Kozuka, Trans. Jpn. Inst. Met. 22,558 (1981)

111. J.J. Brennan and J.A. Pask, JAm. Cer. Soc. 51,569 (1968)

112. L. Coudurier, J. Adorian, D. Pique and N. Eustathopoulos, Rev. Int. des Hautes Temp.

et des Refractaires 21, 81 (1984)

113. V. Laurent, D. Chatain, C. Chatillon and N. Eustathopoulos, Acta Met. 36, 1797 (1988)

114. L.Goumiri and J.C. Joud, Acta Met. 30, 1397 (1982)

115. P. Costa, E. Ricci, E. Arato and L. Nanni, submitted to J. Mater. Sci.

116. for In: P. Rajendera, P.Chhabra and K.Roy, Z. Metallkde. 79, 64 (1988)

for Sn: T.A. Ramanarayanan and R.A. Rapp, Met. Trans 3, 3239 (1972)

for Pb: R. Szwarc, K.E. Oberg and A. Rapp, High Temperature Science 4, 347 (1972)

Drops and Bubbles in Interfacial Research D. M6bius and R. Miller (Editors) �9 1998 Elsevier Science B.V. All rights reserved. 525

DYNAMICS OF THE IMPACT INTERACTION BETWEEN A FINE SOLID SPHERE AND A

PLANE GAS-LIQUID INTERFACE

Anh V. Nguyenl~'2 and Hansjoachim Stechemesser I

1Max Planck Institute for Colloids and Interfaces, Research Group at the Freiberg University of

Mining and Technology, Chemnitzer Strage 40, D-09599 Freiberg, Germany

2Centre for Advanced Numerical Computation in Engineering and Science (CANCES), The

University of New South Wales, Sydney 2052, Australia

Contents

1. Introduction

2. Experimental Methods of Studying Impact Interaction

2.1. The Captive Bubble Methods

2.2. The Pendant Drop Method

2.3. New Method of Studying Impact Interaction between an Plane Gas-liquid Interface and a

Solid Sphere

3. Theoretical Models

3.1. Philippoff' s Model

3.2. Evans' Model

3.3. Scheludko's Model

3.4. Schulze's Model

4. Comparison of the Theoretical Models With the Results of the Experiments with Plane Gas-Liquid Interfaces

5. New Model Development: the Interface Extend Effect on Impact Dynamics

5.1. Force Balance on a Sphere Oscillating with a Gas-Liquid Interface

5.2. The Approximation Problem: an Extension of Derjaguin's Formula

5.3. Transition angle or, maximum depth H and restoring force F

5.4. Collision Time 6. Summary

7. Acknowledgement

8. List of Symbols

9. References

526

1. INTRODUCTION

Bubbles and gas-liquid interfaces have been the focus of both scientific and technological interests.

Physical chemists and interracial scientists have been interested in measuring the interfacial tension

or studying many interracial transport phenomena since the last century. These topics are well

described in many chapters of this book. In this chapter, a different topic is taken up. We are

concerned with the interaction between a fine solid particle and a gas-liquid interface before an

equilibrium establishes between them. This interaction is of the dynamic nature and occurs only

within a very short time period, just a few seconds, but is significant to many areas. One of the

classical examples is the flotation process. Another example is the measurement of the colloidal

forces between a fine solid sphere and a gas-liquid interface which is the very recent focus of

scientific research using the experimental novel devices based on the principle of the atomic force

microscopy. This interaction forms the subject matter of this chapter.

In flotation, fine particles are selectively attached on the surfaces of air bubbles rising through a

flotation slurry (a fine particle suspension), and are thereby concentrated or separated. The process

of froth flotation has been extensively used in mineral industry for separation and selective

concentration of valuable minerals from ores. The recently expanded application of flotation finds in

the de-inking process of the recycling of old and used papers. Interaction between a solid particle

and a gas-liquid interface controls the efficiency of the flotation process and has been considered in

detail (Schulze, 1984; 1989; 1993). The bubble-particle interaction involves many stages which are

commonly referred to as the before-contact and alter-contact interactions (Scheludko et al., 1963;

1968/1969; 1971). The before-contact bubble-particle interaction is strongly affected by the long-

range hydrodynamics when the bubble and the particle are far from each other whereas is controlled

by interfacial physics, e.g., interfacial dynamics, capillarity and colloidal interaction if they are close

enough. The first step of the before-contact interaction has been studied most extensively (e.g.

Derjaguin and Dukhin, 1959; Schulze, 1989; Nguyen, 1994). The next elementary step of the

before-contact interaction is the drainage of the intervening liquid film which is formed between a

particle and a bubble. This drainage behaviour has been studied in great detail, on experimental

level (e.g. Blake and Kitchener, 1972; Schulze and Birzer, 1987; Hewitt et al., 1993) and

527

theoretical level as well (for a review see Ivanov, 1988). The drainage of the intervening liquid film

on a hydrophobic solid surface is unstable and results in a rapture of the film in a time, called the

induction time. The difference between the drainage in the experiments and in flotation is that the

occurrence of the film rapture in flotation depends, in addition to solid surface hydrophobicity, on a

time within which a bubble and a particle can stay in a close position. This time is called contact

time. If the contact time is shorter than the induction time, the bubble and the particle will be

separated before the intervening liquid film ruptures. In this case, the particle cannot be attached

onto the bubble and the bubble-particle interaction is ineffective.

Generally, a solid particle, under the influence of different forces, may approach a bubble under

different conditions. It may meet the bubble surface under different angles of their movement

directions. If a bubble and a particle approach each other on a straight line, the momentum of their

encounter is the biggest. In this case, the gas-liquid interface is usually strongly deformed. The

corresponding bubble-particle interaction is called the impact interaction and the contact time the

collision time. Another extreme case is the case when a bubble and a particle meet each other

without any bubble surface deformation. The corresponding contact time is referred to as the

sliding time.

The measurement of the forces of the interaction between a gas-liquid interface and a solid particle

at small separations is of central importance to gain a fundamental understanding of many aspects

of the complex behaviour of the colloidal system. Surface forces of the colloidal interactions have

been studied extensively in recent years, with the aid of the atomic force microscope (Butt, 1994;

Ducker et al., 1994 and Fielden et al., 1996). Clearly, the deformability of gas-liquid interfaces is an

important factor in their colloidal interactions with a solid particle. In addition, the deformation is

also very important in the determining the behaviour of numerous systems such as emulsions and

foams.

In this chapter, the deformation of a gas-liquid interface due to a fine solid sphere (of the order of

100 gm in diameter) approaching will be investigated in the context of bubble-particle impact

interaction in flotation. However, the results of the deformation studies will have a strong

implication in quantifying colloidal forces acting between a fine solid sphere and a gas-liquid

528

interface. The bubble size is usually much larger than the particle size in flotation, hence the

physical picture of the impact interaction has commonly been approximated to a solid particle

approaching a gas-liquid interface that is almost plane. We will follow this approximation and

restrict ourselves to the case of the approaching of a solid particle in the liquid phase under the

influence of the particle gravity.

2. EXPERIMENTAL METHODS OF STUDYING IMPACT INTERACTION

2.1. THE CAPTIVE BUBBLE METHODS

The first experimental work aimed at investigating bubble-particle impact interaction was done by

Philippoff (1952), using a captive bubble blown in a cavity bored in a paraffin block. The solid

particles were of regular cylindrical shape and filed from short lengths of steel wire. This material

allowed the particles to be suspended by an electromagnet directly over the captive bubble. When

the current was switched off, the particle dropped onto the bubble surface. The bubble-particle

impact interaction was photographed together with the area of the top of the bubble, using a high-

frequency camera operating at about 1000 flames per second. The film velocity was timed by using

a spark discharge of 120 sparks per second simultaneously photographed on the bubble image. The

displacement of the particle and the bubble surfaces at impact location was evaluated in dependence

of time (Fig. 1). From these time-displacement curves the collision (contact) time was determined

as the time between the first moment of the bubble surface deformation and the next moment when

the curves of the bubble and the particle were the nearest. The special particle shape of a regular

cylinder was chosen to fulfil the assumption made in the impact interaction modelling by the author

(Philippoff, 1952). This particle shape is obviously not typical in flotation. The measured collision

times are about 15% longer than the calculated times.

Another method for measuring contact time, using a captive bubble was reported by Schulze

(Schulze and Gottschalk, 1979, 1981; Schulze, 1984). A captive air bubble was formed on the top

of a capillary in a glass tube (1) of an inner diameter greater than 40 mm (Fig. 2). A swarm of

particles was fed above the captive bubble through a small tube (4). The particle trajectories around

the captive bubble and the bubble-particle contact interaction were recorded by means of a

529

stroboscopic camera (2, 8, 9) and an electronic timer device (3). Liquid flow velocity in the tube (1)

was regulated using a pump (5). Hence, for a captive bubble, different relative velocities between

the bubble and the liquid phase and different bubble Reynolds numbers were achieved.

r

I

4 . . . . . % . B U B B L E

<( l-- u) 2 o

,D----- 12 MS.- - - , "

r ' " E OF ~ ' CO"TACT L l l~ ~

. . . . . . . . . . . . .

i

_ �9 ~ ........a

0 I 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0

N U H B E R OF F R A H E S ( . 9 8 M S . )

Fig. 1. Displacements of a particle and a bubble surface vs. time during their impact interaction (Philippoff, 1952).

The size of the cylindrical particle (a piece of steel wire) is 740 ~tm in diameter and 770 ~tm in height.

From the pictures recorded by means of the stroboscopic photography, bubble-particle contact time

(sliding time or collision time) can be determined. Generally, contact time experiments carried out

with this technique do not yield very accurate information about collision time, which is relatively

short compared with sliding time. Since the gas-liquid interface deformation is difficult to

investigate in detail, this method is not convenient if one wants to measure collision time as the time

between the first and the last moments of the first local bubble surface deformation.

530

bottotini

# 9

' P M

solution

2 --7

I V/7//~d k~M

i 6

5

v

Fig. 2. Schematic diagram of an apparatus for studying the interaction between a captive bubble-and a particle of a

particle swarm by Schulze (1984).

The collision contact time can be, in principle, evaluated using a high speed camera which

photographs the interaction between a rising bubble and the particles under the flotation-related

531

conditions (Spedden and Hannan, 1948; Whelan and Brown, 1955/1956; Slavnin, 1960). However,

collision time cannot be very accurately measured by this method because, as mentioned above, the

deformation of the bubble surface is impossible to accurately detect.

2.2. THE PENDANT DROP METHOD

Collision contact time is defined as half the period of the local bubble surface oscillation during

particle impact. To measure this time one needs to detect the local bubble surface deformation.

Stechemesser et al. (Stechemesser, 1989; Bergelt et al., 1992) measured the collision time on the

basis of this principle. A gas-liquid interface (7) of a pendant drop was formed at the bottom of a

glass capillary (10) as shown in Fig. 3. Deformation of the gas-liquid interface was caused by

impact of a single particle inside the pendant drop, observed by a microscope (11) and laterally

recorded by a high speed camera (13) and stored in a computer (14). Collision time was determined

by means of Fourier analysis of the deformation course (Fig. 4).

Fig. 3. Schematic set-up for studying impact interaction using a pendant drop interface (Bergelt et al., 1992).

Since the impact interaction is laterally observed, experiments with an initially plane gas-liquid

interface are difficult to carry out (the lateral view of the interface is identical with that of the

532

contour of the end of the capillary tube). Resolution of lateral imaging of a shallow interface is not

very good. The deeper the gas-liquid profile, the better analysis of the deformation course. For a

good deformation course analysis, the pendant drop method of measuring collision time needs

either big and/or heavy particles or big pendant drops in order to produce sufficient lateral imaging

resolution.

Fig. 4. Local deformation of a pendant drop interface of distilled water by impact of a lead sphere of 500 ~tm radius

(Bergelt et al., 1992).

2.3. NEW METHOD OF STUDYING IMPACT INTERACTION BETWEEN AN PLANE GAS-LIQUID INTERFACE

AND A SOLID SPHERE

The gas-liquid interface of bubbles in flotation is almost convex, i.e., curved inwards in the liquid

phase whereas the interface of pendant drops is curved inwards in the gas phase. The impact

interactions in the big pendant drops are therefore the reverse of those in flotation. In bubble-

particle impact interaction in flotation, the convex liquid surface of air bubbles can be considered as

plane compared with the particle size. It would be significant to experimentally investigate this

approximation of the flotation impact interaction. To achieve this aim, a new experimental method

has been developed.

D

I li: I-I =1~ I ICM 405

/

533

Fig. 5. Schematic diagram of the experimental set-up for investigating impact interaction between a particle and an

initially plane air-liquid interface (Nguyen et al., 1997a). This equipment consists of an ICM 405 inverse microscope

(Carl Zeiss, Oberkochen, Germany) (1); a capillary tube (3); a through-illumination lamp (4); and a CCD high-speed

video system (SPEEDCAM, Weinberger AG, Dietikon, Switzerland). The high-speed video system consists of a

CCD high-speed camera (2); a video monitor (5); a control system for the camera (6); and a computer (7).

A plane gas-liquid interface is produced at the bottom of a capillary tube (3) as shown in Figs. 5

and 6. The impact interaction between a particle and a plane gas-liquid interface is observed

underneath the interface and recorded by means of a high-speed video system. The new principle is

based on the video imaging of the grey levels of the gas-liquid interface. The grey levels depend on

the intensities of the light beam reflection from the interface (Fig. 7). They are therefore measures

of interface deformation. If the interface is plane, the video image of the interface is white, and its

grey level is maximum. When the interface is curved by an impact of a particle, the reflection

becomes weaker. The grey levels of the video images therefore decrease proportionally to the

interfacial deformation. The video images are recorded with a high-speed camera and digitised.

534

Fig. 6. Schematic diagram of the capillary tube used in the experimental studies of the impact interaction between a

particle and an initially plane air-liquid interface. The inner diameter of the capillary tube is approximately 8 mm.

With the ferromagnetic steel sphere we can move one particle from the horizontal to the vertical part of the capillary

tube (by means of a permanent magnet). After opening the metal-ball valves by means of the permanent magnet, the

particle settles under gravity. Before dropping onto the gas-liquid interface, the particle passes through a narrowed

part of the capillary tube (centring tube) and impacts the plane air-liquid interface at its centre.

535

Fig. 7. Reflection of incident light beams from a plane air-liquid interface (a) and from a curved interface (b) which

occurs as a result of its deformation due to the impact of a particle. The solid lines describe the incident light beams

and the dotted lines the reflective light beams.

140

�9 ~: 120

,- 100 1,=

J3l ~- 80 < C -~ 60 > (D "~ 40 (1) k . .

(.9 2O

.*%

t v

AJ

0 10 20 30 40 50 60

Time in Miliseconds

Fig. 8. An example of grey levels (arbitrary unit) measured as a function of time (experimental series No. Z3N413).

536

With two high-speed cameras, the image frequency can be changed from 265 to about 6000

pictures per second. Depending on the measuring frequency, the video images can be digitised at a

resolution from 128 x 16 to 256 x 256 pixels. The time resolution can be changed from 0.179 ms to

3.77 ms. Grey levels are assigned to points or areas on the video images. For a point or a small area

on the video image the grey levels are distributed as a function of time (Fig. 8). Collision time,

defined as the time between the first and the last moments of the first local bubble surface

depression, is determined by means of Fourier analysis.

3. T H E O R E T I C A L MODELS

3.1. PHILIPPOFF'SMODEL

--L J- - __-- -1--, Liquid

F Fig. 9. Depression (solid line) of an initially plane (dotted line) gas-liquid interface by impact of a cylindrical particle

of diameter D (Philippoff, 1952).

The first model for dynamics of bubble-particle impact interaction and collision time was given by

Philippoff (1952) using the approximation of particle impact against a plane gas-liquid interface as

illustrated in Fig. 9. Philippoff's model was developed for a cylindrical particle, probably for the

sake of simplicity. The concept that the gas-liquid interface acts as an elastic body under impact of

the particle is employed. The elasticity, defined as the restoring force on a mechanical deformation,

is caused by the interfacial tension and is the result of the principle of the minimum of free surface

537

energy. The elasticity together with a mass determines the frequency of oscillation which gives the

following expression for collision time:

to - re , (1)

where m is the mass and E is the elasticity. Eq. (1) follows from the frequency of a system of one

degree of freedom. The key to predicting collision time by Eq. (1) is the elasticity E which can be

calculated, by the above Philippoff definition, by

F E - (2)

H '

where F is the restoring force and H is the depression of the cylinder under the gas-liquid interface

(Fig. 9). Phillippoff calculated the force F using the well-known Poisson equation (Poisson, 1831)

F - rc DZ Afi .g. H + reD. o-. sin a (3) 4

and applied Poisson's solution to Young-Laplace's equation to quantify the deformation of the gas-

liquid interface (Poisson, 1831)

H - D s i n a { l n 4L } - ~ - - - y . (4)

Here, A6 is the difference of the liquid and gas densities, y = 0.5772 is the Euler constant. L is the

capillary length defined by:

For an air-water interface at 20~ (Aft = 5 - Pvapour ~ 1000 kg/m 3, o- = 72.4 mN/m), the capillary

length is equal to 2.72 mm. Inserting Eqs. (2) - (5) into Eq. (1) yields

tc - , (6) 2o" ~PhilippoS

where the factor is given by

~eh~1~ppo~ - ~ + ln(4L / D ) - y

538

Eq. (6) is valid for any cylindrical particle. However, in his example calculation, Philippoff (1952)

considered a cylindrical particle of a height equal to its diameter, somewhat like a cube. In this case,

Eq. (6) is simplified by

= ~ D3 p~ ~/z

where p is the particle density.

3.2. EVANS' MODEL

In 1954 Evans published another model for impact dynamics of a particle against a gas-liquid

interface. The local bubble surface was treated as that from a plane air-water interface deformed to

the shape of a spherical particle as in Fig. 10 because, again as with Philippoff's assumption, in

flotation the bubbles are much larger than the particles. The gas-liquid interface outside the local

interaction was considered as plane and not deformed. A "disjoining" liquid film was considered to

be formed between the particle and the air-water interface during the impact of the particle. Evans

considered the dynamics of the impact interaction to be simple-harmonic vibration and calculated

the collision contact time using methods similar to those in Philippoff's model. The elasticity E was

calculated by

1 dW E -

H dH'

where W is the work done due to increase in surface area

Increase in area = rcDH-(rcDH-zcH 2) ;r/-H 2

W = or.increase in area = zccrH 2 .

Finally Evans (1954) obtained the following model for collision time

(9)

(10)

(11)

[D3p t c - - zc~ ]-~. (12)

Evans correctly stated in writing that the increase in surface area approximately equals the area of

the spherical segment of the distorted interface minus the area of a circle of radius a (Fig. 10).

However, the approximated calculation given by Eq. (10) is incorrect. Instead, the increase in

539

surface area would be 2rcaH-rca 2 - 2rcaH-rc(DH-H2). Since D is much larger than a and H, the approximation of the surface area increase to rcH 2 is not true. Moreover, Evans' calculation of

the energy due to the interface distortion is not complete. The energy consists of many components

including, at least, pressure to the distorted volume and interfacial tension to the distorted interface

area which leads to (Fig. 10), using Eq. (9)

1 [ | {D}22-3cosct+cos3ct~o.D 24 (13) E - ~.Lsin2 a -

The second term in the product on the right-hand side in this equation is identical with the restoring

force which can be also derived on the basis of the force balance approach in the next subsection.

The simple geometry in Fig. 10 gives H =/3(1 - cosa) / 2 or cosa = 1- 2H / D. Inserting this into

Eq. (13) yields

14 { H } I ~ / / ] ~ D ~ 2 l{H}2{L}23~cr_4rccr+O{H/D } (14) E - - 4 - 2 ( D J L L J +3

The corrected Evans' collision time

fz 3P t~- - rc~ 2 - ~ (15)

is shorter than Evans' collision time Eq. (12) by the factor 1/~/2 as earlier reported by Schulze et al.

(1989).

Evans' model was modified by taking into account the effect of the bubble and the particle motion

on collision time by Ye and Miller (1988; 1989)

[D3p t~- 7c~ 1 - ~ �9 ~bre (16) { }1,: 2 12 V2cr, o ~r. - 1 + -- arcsin 1 + (17) z)ag2(p-ay

These two equations are adapted to the context of this present chapter. Unfortunately, the above

incorrectness which occurred in Evans' model was not improved by Ye and Miller (1988; 1989). As

Liquid

can be seen in the next section, this modification does not change collision time very much

compared with Evans' original model.

a

f I

i

Particle i

I |

I

Gas

540

D

i

,d I (x

, i H

Fig. 10. Depression (solid line) of an initially plane (dotted line) gas-liquid interface by impact of a spherical particle

according to Evans (1954).

3.3. SCHELUDKO 'S MODEL

Another modification of Evans' model was proposed by Scheludko et al. (1976) who theoretically

argued that the gas-liquid interface should deform smoothly, as shown in Fig. 11. The authors used

the following approximation for predicting the maximum depth of the interface depression through

Derjaguin' s formula (Derjaguin, 1946):

{~ 4L/R I } H-R 1 ( l + c o s a ) s i n a - y sin2 a . (18)

The dynamics of the impact was considered to be the harmonic oscillation under the action of the

restoring force

F = 2zccrR sin 2 a . (19)

541

Finally the following collision time model was presented (Scheludko et al., 1976)

t~ - -~--~ . qk S~h~l~ako 7V V 12O" " qk S~hel.ako

I 1 S~he1~ako -- In 8L / D sin a(1 + cosa) - 7"

(20)

(20a)

Liquid - Particle

D

- o o S /"<4 I +oo

H :~r I / ~ ~ ~'~ Non-s'pherica Gas / meniscus

Transition Spl~erical point meniscus

Fig. 11. Depression (solid line) of an initially plane (dotted line) gas-liquid interface by impact of a spherical particle

(Scheludko et al., 1976; Schulze et al., 1989).

The factor given by Eq. (20a) does depend on the polar angle ot as illustrated in Fig. 11. This means

the improvement by Scheludko et al. (1976) remains semi-analytical, needing the experimentally

determined ot to calculate collision time. However, the authors quantitatively showed that the

collision time predicted by Eq. (20) would be approximately four times greater than that by Evans'

model.

3 . 4 . S C H U L Z E ' S M O D E L

The idea raised by Scheludko et al. (1976) was further developed experimentally and theoretically

by Schulze et al. (1989). Deformation of the gas-liquid interface due to impact of a spherical

particle is treated as smooth profile as shown by Scheludko et al. (1976). The deformed interface is,

542

however, divided into two parts as shown in Fig. 11. The first part (the inside meniscus), where the

intervening liquid film is formed, is spherical. The outside meniscus is non-spherical. These two

parts of the meniscus profile are separated by the transition angle a. This deformation treatment is

reasonable with the experimental observations by Schulze et al. (1989). Note that the spherical

meniscus deformation corresponds to the deformation suggested by Evans (1954) and not treated in

Philippoff's model. The additional mass of liquid adjacent to the oscillating particle is taken into

consideration in modelling contact time by Schulze et al. (1989).

The force due to interfacial deformation, given by Eq. (19), is evidently difficult to express as a

function of the maximum depth of the deformed interface H (Schulze et al., 1989). Due to this fact,

the particle oscillation equation cannot be analytically solved to predict collision time. A semi-

analytical model for collision time was obtained (Schulze et al., 1989) on the basis of a simple

harmonic oscillation

~e~r ~,~., (21) t~ = V 4o" " ~s~1~ - ~ 24o-

where me H is the effective mass of an impacting particle. The parameter j in Eq. (21) accounts for

the effect of the co-oscillating liquid adjacent to the oscillating particle, and depends on the

frequency of particle oscillation(Schulze et al., 1989), i.e., on collision time. However, no analysis

has been performed, and the valuej = 1.5 was chosen in all calculations in Schulze et al. (1989).

- ~ + ~ . ( 2 2 ) m ~ f f - m ( l + j 6 / p ) m l + 2 p 2p

where/3 is a dimensionless number which accounts for the portion of effective mass due to particle

oscillation (Stokes, 1851; Lamb, 1924 and Landau and Lifshitz, 1975). Stokes' formula for 13

during the first oscillation can be expressed by

rcD26 r 2 _ (23)

18ptc

This parameter can be employed as the criterion for the retention or neglect of the oscillating virtual

mass effect on the dynamics of the oscillating sphere. For oscillations of low frequency, fl << 1 and

of high frequency, fl >> 1. The collision time is of the order of milliseconds. Thus, the frequency of

543

oscillation of the impacting particle in the first depression might be relatively high (order of kHz).

The oscillating effective mass may have little effect on collision time.

Despite of the approximations performed by Schulze et al. (1989) the dynamics of impact

interaction remains non-linear. The factor in Eq. (21) still depends on the transition angle, like the

factor in Scheludko's model

~b Sch,1~" = D a m

{ 1 . y +1 n DocmeLtl/2' (24,

where am is the maximum transition angle measured in radians and was experimentally determined

by the pendant drop method described in subsection II. 3. Some experimental results are given in

Table 1.

Table 1. Mean values of the experimental results for maximum transition angle OCm (Schulze et al., 1989) and

calculated ~/ISchulz e by Eq. (24)

Systems

Methylated glass spheres; air-water

interface

Methylated glass spheres; air-aq.

solution of 10 ~tmol/1Na dodecyl

sulphate interface

cr [mN/m] D [gm] a m [degr.] ~k3chulz e

448 + 36 13.2 + 2.0 3.11 72

887 + 80 18.1 + 1.5 2.77

65 887 + 80 18.2 + 1.8 2.75

Glass spheres; air-aq, solution of 10 448 + 36 12.6 _+ 2.1 3.10

gmol/1 dodecylamine hydrochlorid 60

& 10 mmol/1KC1 interface 887 _+ 80 18.6 _+ 2.7 2.73

With experimental data of the pendant drop technique for glass particles considered, the factor

given by Eq. (24) was approximated by (Schulze, 1993)

~Schulze --1.71- 0.461nD, (25)

where D is given in cm.

544

4. COMPARISON OF THE THEORETICAL MODELS WITH THE RESULTS OF THE

EXPERIMENTS WITH PLANE GAS-LIQUID INTERFACES

All of the above-mentioned theoretical models were developed for the impact dynamics of a solid

sphere against a plane gas-liquid interface - an approximation of impact dynamics in flotation. In

this section, these models will be compared with the results of the compatible experimental systems,

i.e., of the impact of a small solid sphere against a plane gas-liquid interface as described in

subsection II. 3.

The experiments were carried out with two types of spherical particles: the soda-lime-glass spheres

(also called ballotini) and the spheres artificially prepared from an alloy of 60% Sn (by weight) and

40% Pb. The density of these particles was determined by the pycnometric method. Density of the

glass particles was 2500 kg/m 3 and of the Sn-Pb particles 8400 kg/m 3. The cleaned glass spheres

were hydrophilic. Some experiments were carried out with hydrophobic glass spheres which were

made hydrophobic by silanation. Different hydrophobicities, as measured by the contact angle

against the water, were achieved by changing the time of silanation. The advancing contact angles

were changed from 20 ~ to 70 ~ Different hydrophobicities did not experimentally indicate any

significant difference of collision time. They are, for this reason, not specifically given here.

The experimental data for collision time of the glass and Sn-Pb particles versus their diameters are

given as points in Figs. 12 and 13, respectively. Theoretical collision time models are indicated by

different curves. The sizes of the particles were determined by the inverse microscope and, in some

cases, by a micrometer, immediately after each experiment. These experiments were carried out

with an air-water interface. The surface tension of the water, measured by the Wilhelmy plate

technique, was about 72.4 mN/m.

As can be seen from Figs. 12 and 13, the experimental data are higher than the collision time

theoretically predicted by the available models. The modification of Evans' model by Ye and Miller

(1989) does not improve Evans' collision time model significantly. Evidently, the correct restoring

force should be F = ~ro-Dsin 2 a . The angle a measures the dimension of spherical deformation.

For the same depth of interface depression, the transition angle a Evans' model is the biggest. The

collision time predicted by the corrected Evans' model is therefore the smallest, compared with all

the available models.

545

Fig. 12. Comparison between available theoretical collision time models and experimental data for hydrophilic (filled symbols) and hydrophobic glass spheres (unfilled symbols).

Fig. 13. Comparison between available theoretical collision time models and experimental data (squares) for Sn-Pb particles (hydrophilic).

546

The semi-analytical model of Schulze et al. (1989) still predicts a collision time smaller than the

experimental data, as shown in Figs. 12 and 13. One might expect that the restoring force employed

in all the available models of Philippoff (1952), Schulze et al. (1989) is bigger than the force which

acts on the particles in the experiments. This implication is very probable because these models are

derived on the basis of the assumption of unlimited interface deformations whereas the plane

interfaces in our experiments are bound by the wall of the capillary tube. The interface deformations

in our experiments probably could not extend to infinity as theoretically expected. In the following

section a model will be developed to take this effect into consideration.

5. NEW MODEL DEVELOPMENT: THE INTERFACE EXTENT EFFECT ON IMPACT

DYNAMICS

An external meniscus is formed when an infinite horizontal plane interface is deformed by a

vertically axisymmetric body. To support the infinite horizontal plane interface, the downer phase is

always denser than the upper phase. The deformation of a gas-liquid interface due to a solid particle

approaching in the liquid phase in the context considered here cannot be an external meniscus as

has been assumed; because the phases are in the reverse situation: the upper phase (the liquid

phase) is denser than the downer phase (the gas phase). Gravitational forces tend to pull the liquid

phase downwards from the interface equilibrium configuration. Obviously, such an equilibrium

configuration of infinite extent cannot be experimentally created. In our experiments this

configuration is supported by the walls of the capillary. Our model system is illustrated in Fig. 14.

The deformed interface is treated as smooth profile as in Scheludko's and Schulze's models, and

consists of two parts. The spherical meniscus is formed with the intervening liquid film whereas the

non-spherical meniscus is between the two bulb phases. These two parts of meniscus profile are

separated by the transition angle a.

5.1. FORCE BALANCE ON A SPHERE OSCILLATING WITH A GAS-LIQUID INTERFACE

The general equation of oscillation of impacting particles can be written as follows:

m - - ~ - + F a + F - m 1- g - O , (26)

547

where m is the mass of particle; t is time; g is the acceleration of gravity. Fd is the damping force

depending in general on the oscillation velocity, dH/dt. Stokes (1851) derived the following

expression for Fd for a small sphere:

m{ ~ ~ ~} dH [1 3-~fl2 } Fa - ~ + ~2/9 + 3rcctD - - ~ + . (27)

~ 2b

iiiiiii

/ Panicle Liquid

_ _ _

\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \

\ \ \ \ \ \ \ ~ \ \ \ \ \ " \ \ \ \ \ \ \ \ \ \ \ \ \ \

Fig. 14. Depression (solid line) of an initially plane (dotted line) gas-liquid interface bounded by a capillary by

impact of a spherical particle. The axis of symmetry of the interaction in the cylindrical co-ordinate system is

identical with the direction of gravity, b denotes the radius of the capillary, o~ is the transition angle which separates

the deformed interface into spherical and non-spherical menisci, a is the radius of the transition points.

The parameter fl depends on oscillation frequency and can be expressed by Eq. (23) for the first

impact oscillation. Substituting of Eq. (27) into Eq. (26) we obtain the motion equation given by

Schulze et al. (1989) where meff is the particle effective mass generally given by Eq. (22). The

remaining problem is how one can evaluate the restoring force F. Scheludko et al. (1976) and

548

Schulze (1989) gave the expression for this force by Eq. (19). Using the force balance approach we

obtain a more general expression which shows that Eq. (19) is a good approximation for a small

sphere, namely when D/L << 1.

The sphere in Fig. 14 is under the influence of different forces. One of the obvious forces is that due

to the vertical components of the interfacial tension, -2rccrasin a . Recall that the vertical axis of

the reduced cylindrical co-ordinate system depicted in Fig. 14 is identical with the direction of

gravity and downward forces are therefore taken as positive. The next forces are due to the action

of gravity, olumer~ r, . (p-6)g+VolumerA r, "(P--6v~pour)g. Since the sum of the two volumes of

the particle above and below the level TT' gives the particle volume, one can rewrite these

gravitational forces by means of the particle mass, m [ 1 - ( 6 / p ) ] g + VolumerAr,gA6. In addition to

the mentioned forces, the particle is subjected to an extra force due to the excess Laplace pressure

acting on the top and bottom levels of the volume of revolution of the free (non-spherical)

meniscus, rz~ 2. APr - rcbZ. Ap E where AP denotes the Laplace pressure at the corresponding levels.

It is noteworthy that for an external meniscus, AP E = 0, AP r = ASgh r where hv is the depth of the

transition point T, the last force reduces to the well-known upward hydrostatic force. The net of

these forces gives the particle motion equation since the particle is not at equilibrium

m - - ~ + F e = -2~-crasin a + m 1 - g + VolumerAr, gA6 + Jra2APr - ~bzAPR. (28)

The same particle motion equation can be deduced using an integral approach for calculating the

forces acting on the particle through Gauss' theorem. Comparison of Eqs. (26) and (28) yields

F = 2zccrasin a - VolumerAr, g A f i - 7rct2 APT + ~b2 Apg . (29)

The non-spherical meniscus is governed by Young-Laplace's equation of a pendant interface

d(rsin f~) cr = & o + h g A S , (30)

rdr

where .(2 is the acute angle of meniscus inclination to the horizontal as given in Fig. 14. A simple

calculation based on Eq. (30) indicates that Eq. (29) expresses a simple law: the restoring force F is

equal to the buoyancy of the liquid volume enclosed by the deformed meniscus revolution. If the

549

meniscus is considered to be an external interface, APe = 0 and Eq. (29) reduces to Eq. (19) - an

approximation of Eq. (29) of the order of (D/L) 2 However, as discussed above, the deformed

meniscus cannot be an external meniscus, but a pendant drop interface. The Laplace APe differs

from zero and even substantially contributes to the restoring force F. In the next subsection we will

solve Eq. (30) to quantify the restoring force.

5. 2. THE APPROXIMATION PROBLEM: AN EXTENSION OF DEtUAGUIN 'S FORMULA

Evidently that Derjagiun's formula (Derjaguin, 1946) for the depth of a distorted non-spherical

meniscus given by Eq. (18), which was developed for an external meniscus, cannot be employed

here. To solve Eq. (30) we employ the matched asymptotic approximation method which was

employed by Derjaguin (1946). This approximated solution was described in Nguyen et al. (1997c).

Some features and results will be summarised below.

Transforming Eq. (30) into the terms of the depth of the meniscus, h, at the radial distance r yields

a differential equation of the second order. The meniscus profile is then mathematically separated

into two characteristic regions, called the outer and the inner regions. The outer region is close to

the capillary wall and characterised by small meniscus angular inclinations whereas the inner region

is close to the transition point where the condition of small angular inclinations is not fulfilled,

however the condition of a small radius of the transition point compared with the capillary constant

can be applied. The differential equation of capillarity can be approximately solved for these two

characteristic regions by the method of asymptotic matching. The following useful results are

obtained:

h r = a In - y + Y0 (Z) -t

A P E - ~ j~(z)JO(Z)-Yo(Z) ~-5 -s ina

J: (Z) g: (Z) 4 ( Z ) sin a (31)

(32)

AP r = hr gA(~ + AP E (33)

550

b Z - L ' (34)

where Z is the dimensionless radius of the plane bounded interface, i.e., the dimensionless inner

radius of the capillary (Fig. 14). Yn and J, are the Bessel functions of the first kind and second kind

of order n, respectively and, for the sake of simplicity, will be used hereafter in the notation without

the variable. Ignoring the effect associated with the interface radius in Eq. (31) yields Derjaguin's

formula (Derjaguin, 1946).

5. 3. TRANSITION ANGLE a, MAXIMUM DEPTH H AND t~STORING FORCE F

The balance of the depths of menisci at the transition point reads (see Fig. 14)

h r - H - D(1- cosa) (35) 2

0.02

0.015

0.01

0.005 f

..,.-

J

o 0. oo 0.02 o. 04 o. 06 o. 08 0.10

H/L

Fig. 15. Numerical results for variation of the restoring force F, made dimensionless with 2zcrL, against the maximum depth of meniscus H/L: D/L = 0.036, 0.12, 0.22, 0.4 (upwards) and interface radius Z = 1.47 (= 4mm/2.72mm). This corresponds to the dimensional range 98 ~tm _< D < 1080 ~tm and H < 270 ~tm. Diamonds

describe the variation given by Eq. (37).

Inserting Eqs. (31-(33) and (3 5) into Eq. (29) yields the following prediction for restoring force:

I ( t } I L } z -1 +3 c~ c t - 2c~ ct (36) F = 1 - Z2 D 2sin2a ~ J o - g o J 1 DH s i n 2 a - z o D 4 16-s ~ J1 - 2---~- 24

551

Eq. (36) shows that restoring force strongly depends on, in addition to particle diameter, maximum

depression depth, transition angle and interface radius. Solving differential Eq. (26) in terms of H is

probably the best way of predicting impact dynamics. This requires an explicit expression of the

force F as a function of H. Direct evaluation of this functional dependence of F(H) is impossible

because of the complex, non-linear dependence of a on tl, as given by Eq. (36). This dependence

can be obtained using a numerical method. If the maximum depth H and the particle diameter D are

known, we can predict the transition angle a for a given interface radius Z by numerically solving

Eq. (35) and the corresponding restoring force by Eq. (36). Some results are illustrated for the F-H

dependence in Fig. 15. It is found that the F-H dependence, as illustrated in Fig. 15, can be best

fitted by using a two-parameter model of the logarithmic function of the independent variable H

(Ratkowsky, 1990). For convenience, we rewrite the dependence by

F - 2 JzcrH[co 2 + 2cos. In(H/L)] , (37)

where co and ~ depend on the dimensionless parameters D/L and Z (Fig. 16). For the case Z = 1.47,

the dependence of the two numerically fitted parameters can be expressed by Eqs. (38) and (39)

co - 2~ In co + 1.5~ - 0.555 + 0.0481 In --D (38) L

c - 0.0210 + 0.00346 In D . L (39)

These approximations are valid for the conditions: 0.036 <_ D/L <_ 0.4. We are interested in these

expressions because they occur in the collision time model presented in the following subsection.

The approximation given by Eq. (37) is within a 5%-error compared with the original expression

for the restoring force given by Eq. (36).

5. 4. COLLISION TIME

We introduce, analogously to the fluid dynamics of deformable particles like bubbles or drops, three

characteristic dimensionless parameters. First, the Bond number of collision, Bo, which represents

the ratio of the apparent gravity of an impacting particle to the interfacial tension force

552

27rcrL (40)

Second, the Weber number of collision, We, which represents the square root of the ratio of the

kinetic energy of the impacting particle to the surface tension energy

We = ~ 2~crL 2 ' (41)

where V is the particle velocity relative to the gas-liquid interface before impact interaction. Third, the

capillary number of collision, Ca, which relates the viscous force to the interfacial tension force

3 ~ ~ 1 +-~ V 3r + 3fl)V Ca - = (42)

2.2rccrL. We 8crL We

0.53 0.02

0.50

0.47 "1"

= 0.44 i

r i 0.41

0.38

J

y 0.018

0.016

0.014

0.012

0.01

0.35 0.008 0.00 0.10 0.20 0.30 0.40

D/L

Fig. 16. Dependence of the fitting parameters in Eq. (37) on the dimensionless diameter D/L for the case Z = 1.47.

Squares describe the dependence for cand diamonds for (co- 28 lnco+ 1.5e).

We make the particle motion equation (26) dimensionless as follows

553

d2y dY co2 + 2Ca ~ + �9 Y - Bo - -2coc. Y In Y (43) d r 2 d r

H Y = - - (44)

L

t V r - ~ - - (45)

We L

Differential Eq. (43) is subjected to the following initial conditions:

Y(0) = 0 (46)

~=0

The differential Eq. (43) is of high non-linearity and analytically solved in only a few cases. Since

the term c is small (see Fig. 16), we can solve this differential equation approximately. This

approximate method is described in Nguyen et al. (1997c). The approximate solution to the

differential Eq. (43) governing the particle oscillation during the first impact can be expressed by

y ( v ) _ A . e x p ( _ C a v ) . c o s ~ ( c o 2 - C a 2 +coef ~ } Bo ~ , ~/~oS-Ca 5 - )v+O +-~--+O(c). (48)

Here, A and 0 are the amplitude and the phase of particle oscillation during the first interface

depression. The functionf depends on the oscillation amplitude, A, and the phase, 0

f l+ ln (A A cos 0) 1 In 1 - s inO {2 Ca } { Caa } (49) - - - + + i _-Ca 2 �9 cos20 + O 2 �9 2 cos 0 2 - 2 cos 0 ~/co co - Ca 2

With numerical values considered (Table 2), the capillary number and the oscillating effective mass

have little effect on collision time for particles of diameter D < 1000 gm. Neglecting all the small

terms, dimensionless collision time can be approximately predicted by

" o(c.) c.) -t~ = ~ + - + �9 - ~ (50) 03 + s. f co - 2s In m + 1.5s + s In Bo do Nguyen

and dimensional collision time by

t C - t l + 2 f l . ( 2 p + 6 ; ' (51)

554

where t2 is the dimensional collision time which is independent of oscillating effective mass, i.e.,

when the effect of oscillating effective mass on collision time is completely ignored

t*~ Jr ID3(2p+6) (52)

~Nguyen 24O"

The dimenionless parameter 13" is given analogously to Eq. (23)

I ~D~# (53) f l * - 18M~

The factor in Eq. (52) depends on the dimensionless radius of the plane gas-liquid interface, 2", and

the dimensionless particle diameter, D/L. For the capillary used in our experiments described in

subsection II. 3 (2 "= 1.47) this factor is simply given by (see Eqs. (38), (39) and (50))

~Nguyen 0-555+0-04811nD { D } - - - + 0.0210 4- 0.00346 In In Bo. (54) L

Table 2. Calculated collision time between a spherical particle and an initially plane air-water interface

bounded by a capillary tube of inner radius b -- 4 mm (L = 2.72 mm).

........ D-i~m) ........................... t;-*?ms) .................................................................................................................................... tc (ms) Eq. (52) Eq. (51) Eq. (48) t

Density P = 2500 k ~ m 3 50 0.28 0.33 100 0.70 0.81 200 1.75 1.97 400 4.31 4.73 600 7.28 7.90 800 10.55 11.37 1000 14.08 15.08

0.34 0.81 1.93 4.70 7.98 11 69 15.77

Density p = 7000 kg/m 3 50 0.42 0.46 0.45 100 1.05 1.13 1.10 200 2.61 2.77 2.70 400 6.42 6.72 6.72 600 10.85 11.30 11.56 800 15.74 16.33 17.08 1000 21.00 21.73 23.19

~Results of the numerical solution to Y(v) - 0.

555

Comparison of the collision time theoretically predicted by Eq. (54) with the experimental data

obtained in the subsection II. 3. is illustrated in Figs. 17 and 18 and shows a good agreement

between the experimental data and the new model for collision time. This indicates that the

deformation of a gas-liquid interface due to a solid particle approaching in the liquid phase in the

context of impact interaction cannot be an external meniscus as has been assumed. The free (non-

spherical) meniscus is a pendant drop interface. As can be seen in Table 2 and Figs. 17 and 18 the

density of the solid phase strongly influences collision time. The bigger and the heavier the particle

the longer the collision time is. The velocity V, which is needed to calculate the data in Table 2 and

Figs. 17 to 19 is l~he terminal settling velocity (Nguyen et al., 1997d)

v= 1+ )0 (55 18p (1 + 0.079Ar ~ 749 755

Ar is Archimedes number defined in the list of symbols. This prediction was experimentally verified

with the impact experiments described in the subsection II. 3 (Nguyen et al., 1997d). As shown in

Table 2, the collision time calculated by simplified Eq. (51) agrees well with results of the numerical

solution to Eq. (48).

The last term in the braces on the right-hand side in Eq. (51) presents the effect of oscillating

effective mass on collision time. This term contributes a very small portion (no more than 10%) to

collision time. Within this error one may ignore the oscillating effective mass effect and obtain a

simpler approximation given by Eq. (52).

To see what extent the initially plane gas-liquid interfaces influence the impact dynamics, different radii of

the interface (i.e. the inner radius of the capillary) are employed to calculate the restoring force given by

Eq.(36) and the collision time in a similar way to that described above for the experimental case of the

interface radius ~ = 1.47. The following results are obtained for 1.1 _< Z -< 2.1:

co - 2sln co + 1.5s -(0.869 - 0.214Z)+ (0.0954 - 0.0317Z) In D (56) L

s -(0.0399 - 0.0126Z) + (0.00786 - 0.00290Z)ln---D. (57) L

556

16

r~ 12

op~ l

8 =

o j , I

o j , I I=,,,,,I

4

,v'

J . /

0 200 400 600 800 1000

Particle Diameter (~tm)

Fig. 17. Comparison between the collision time model given by Eq. (51) and experimental data obtained in the

subsection II. 3 for hydrophilic (unfilled squares) and hydrophobic (filled circles) glass spheres.

20

16 r ~

12 op~ l

o 8 o ~

o ~

~ 4

f /

- - t

f

0 200 400 600 800

Particle Diameter (~tm)

Fig. 18. Comparison between the collision time model given by Eq. (51) and experimental data (squares) obtained in the subsection II. 3 for Sn-Pb particles (hydrophilic).

557

15

,~, 12 r ~

9 om

1= o 6 o j . I r ,~

o~ pH i

3

! |

---E}--- 800 x 600

400 - -0- - - 200

- - I ! - - 1 0 0

,____,.----4

t"---

S

, , . , - - - - - - )le, - ~ ' - ' - "

= i " i " - -

1 1.2 1.4 1.6 1.8 2 2.2

Dimensionless Interface Radius, b/L

Fig. 19. Dependence of collision times on the radius of initially plane gas-liquid interfaces, Z = b/L for the particles of density of 2500 kg/m 3. The legend shows the particle diameters in p.m. The collision times are calculated using Eqs. (56) and (57).

The effect of the radius of the plane interfaces on the dynamics of impact interaction is illustrated by

collision time in Fig. 19. The bigger the particle the bigger, the effect of the interface extent on

collision time. Obviously, the gas-liquid pendant interface cannot be of infinite extent.

Experimentally, we cannot create a pendant interface at the bottom of the capillary tube whose

inner radius is three times bigger than the capillary length L. The range 1.1 _< Z -< 2.1 is of

experimental interest. It should be noted at this point that the pendant interface can, in principle,

generate a radiated wave propagating along its surface by deformation due to the particle impact

since both the Bessel functions Jn and Yn in Eq. (31) are wave-like. However, the wave length is

larger than the extent of the interface which could be experimentally created. As a result, we cannot

experimentally observe any wave cycle of the interface during the particle impact, but only a part of

it. The energy loss connected with this wave is therefore taken into full consideration in terms of

the restoring force in the present theory. Moreover, the theoretical consideration by Schulze et al.

(1989) is absolutely correct since the energy loss is introduced into the theory in the energy balance.

558

The energy loss is connected with the energy dissipation in the intervening liquid film. The film

drainage is very strongly affected by hydrodynamic interaction as we cannot see any difference in

collision time between hydrophobic and hydrophilic particles (Figs. 12 and 17). Regardless of

hydrophobic or hydrophilic particle, we observe no rupture of the intervening liquid film at the first

interface depression. Two sub-processes, namely, the particle oscillation and the thinning of the

intervening liquid film are dependent. We hope to present some results of the interaction between

the two sub-processes in the future.

6. S U M M A R Y

The dynamics of the impact interaction between an initially plane gas-liquid interface and an

approaching solid sphere in the liquid phase is a non-linear oscillation problem. The deformed gas-

liquid interface cannot be an external meniscus as has been assumed, but is a pendant interface. The

deformation has to consist of two regions: (1) a region of strong hydrodynamic interaction with the

particle where the intervening liquid film is formed with a spherical meniscus and (2) a region of

free interface where no intervening liquid film is formed. The energy of the deformation of the last

region significantly contributes to the total deformation (restoring) energy. All of the collision time

models in flotation theory which ignore this kind of deformation predict therefore collision time

smaller than the experimental data.

7. ACKNOWLEDGEMENT

This work was supported by the German Research Council (DFG) and the Alexander von

Humboldt Foundation.

8. LIST OF SYMBOLS

A

Ar

Bo

Ca

Amplitude of particle oscillation, Eq. (48)

Archimedes number (=J6g(p-8)/~)

Bond number defined by Eq. (40)

Capillary number defined by Eq. (42)

559

D

f E

F

Fa

g

h

hr

H

J

J.

L

m

meff

r

t

tc

tc*

V

We

Y

Y.

W

Particle diameter

Function given by Eq. (49)

Elasticity of a bubble surface

Restoring force of a gas-liquid interface

Damping force given by Eq. (27)

Acceleration of gravity

Depth of a deformed gas-liquid interface

h at the transition point T (Figs. 11 and 14)

Maximum h

Parameter which accounts for the mass of liquid adjacent to an oscillating particle, Eq. 22

Bessel function of the second kind and of the order n

Capillary length defined by Eq. (5)

Particle mass

Effective mass of an impacting particle

Radial distance

Time

Contact time during impact (collision time)

Collision time calculated by completely neglecting the oscillating effective mass, Eq. (52)

Particle and gas-liquid interface relative velocity before impact interaction

Weber number of collision defined by Eq. (41)

Dimensionless H defined by Eq. (44)

Bessel function of the first kind and of the order n

Work done due to increase in surface area in Evans' collision time model

Greek letters

Z

am

Transition angle (Figs. 11 and 14)

Dimensionless interface radius defined by Eq. (34)

Maximum a

Dimensionless parameter defined by Eq. (23)

Dimensionless parameter defined by Eq. (53)

560

6

A6

oe

0

Y

lu

0

P

o

(o

Liquid density

Density difference between liquid and vapour

Numerically fitted constant in Eq. (37)

Dimensionless parameter used with different subscripts in many collision time models.

Euler number (= 0.5772)

Liquid viscosity

Phase of particle oscillation, Eq. (48)

Particle density

Surface tension of the free gas-liquid interface (non-spherical meniscus)

Dimensionless time defined by Eq. (45)

Dimensionless collision time

Numerically fitted constant in Eq. (37)

Meniscus angular inclination to the horizontal.

9. REFERENCES Blake, T.D. and Kitchener, J.A., J. Chem. Soc. Faraday Trans. 1, 68 (1972) 1435.

Butt, H.-J., J. Colloid Interface Sci.. 166 (1994) 109.

Derjaguin, B.V., Dokl. Akad. Nauk SSSR, 51 (1946) 517, (in Russian).

Derjaguin, B.V. and Dukhin, S.S., Izv. Akad. Nauk SSSR, OTN: Metalurgiya i Toplivo (Div. Tech.

Sci." Metall. and Fuel), 1(1959) 82 (in Russian).

Ducker, W.A., Xu, Z. and Israelachvili, J.N., Langmuir, 10: 3279.

Evans, L.F., 1954. Industr. Engng. Chem., 46 (1994) 2420.

Fielden, M.L., Hayes, R. and Ralston, J., Langmuir, 12 (1996) 3721.

Hewitt, D., Fornasiero, D., Ralston, J. and Fisher, L., J. Chem. Soc., Faraday Trans., 89(1993)817.

Ivanov, I.B. (Editor), Thin Liquid Film. Fundamentals and Applications. Marcel Dekker, Surfactant

Science Series, Vol. 29, New York, NY, 1988, 1125 pp.

Lamb, H., Hydrodynamics. Camb. Univ. Press, London, 5th Edition, 1924, 687 pp.

561

Landau, L.D. and Lifshitz, E.M., Course of Theoretical Physics (Volume 6): Fluid Mechanics.

Pergamon Press, Oxford, 1975, 618 pp. (translated from the Russian by J.B. Sykes and

W.H. Reid).

Nguyen, A.V., J. Colloid Interface Sci., 162 (1994) 123.

Nguyen, A.V., Schulze, H.J., Stechemesser, H., and Zobel, G., Int. J. Miner. Process. 50 (1997a)

97.

Nguyen, A.V., Schulze, H.J., Stechemesser, H., and Zobel, G., Int. J. Miner. Process. 50

(1997b) 113.

Nguyen, A.V., Schulze, H.J., Stechemesser, H., and Zobel, G., Int. J. Miner. Process. (1997c)

(accepted for publication).

Nguyen, A.V., Stechemesser, H., Zobel, G., and Schulze, H.J., Int. J. Miner. Process. 50 (1997d)

53.

Philippoff, W., Trans. Amer. Inst. Min. Engrs., Min. Engng., (1952)386-390.

Poisson, S.D., Nouvelle Theorie de l'Action Capilaire. Bachelier et Fils, Paris, 1831, pp. 236-240.

Ratkowsky, D.A., Handbook of Nonlinear Regression Models. D.B. Oven, Coordinating Ed.,

Statistics: Textbooks and Monographs, Vol. 107., Marcel Dekker, New York, 1990, 241

Schulze, H.J., and Gottschalk, G., Investigation of the hydrodynamic interaction between a gas

bubble and mineral particle in flotation. In: Develop. Miner. Process., D.W.Fuerstenau,

Advisory Editor, Vol. 2 "Proc. 13th Int. Miner. Process. Congr., Warsaw" (J. Laskowski,

Editor), Elsevier, Amsterdam, 1979, pp. 63-85.

Schulze, H.J., and Gottschalk, G., Aufbereitungstechnik, 22 (1981) 254.

Schulze, H.J., Physico-chemical Elementary Processes in Flotation - An Analysis from the Point of

View of Colloid Science Including Process Engineering Considerations. D.W Fuerstenau,

Advisory Ed., Develop. Miner. Process., Vol. 4, Elsevier, Amsterdam, 1984, 384 pp.

Schulze, H.J. and Birzer, J.O., Colloids and Surfaces, 24 (1987) 209.

562

Schulze, H.J., Hydrodynamics of bubble-mineral particle collisions. In: Miner. Process. Extract.

Metall. Rev., Vol. 5 "Frothing in Flotation: The Jan Leja Volume" (J.S.Laskowski, Editor),

Gordon & Breach, New York, 1989, pp. 43-76.

Schulze, H.J., Radoev, B., Geidel, Th., Stechemesser, H., and T6pfer, E., Int. J. Miner. Process.,

27 (1989) 263.

Schulze, H.J., Flotation as a heterocoagulation process: possibilities of calculating the probability of

flotation. In: Surfactant Science Series, M.J.Schick, Consulting Editor, Vol. 47

"Coagulation and Flocculation: Theory and Application" (B.Dobi~ts, Editor), Marcel

Dekker, New York, 1993, pp. 321.

Scheludko, A., Kolloid-Zeitschrifl und Zeitschrift fiir Polymere, 191 (1963) 52.

Scheludko, A., Radoev, B. and Fabrikant, A., Annuaire de l'universit6 de Sofia "Kliment

Ochridski", Facult6 de Chemie, 63 (1968/1969) 43.

Scheludko, A., Tschaljowska, S., Fabrikant, A., Radoev, B. and Schulze, H.J., Freiberger

Forschungshefle, A484 (1971) 85.

Scheludko, A., Toshev, B. and Bojadiev, B., 1976. Attachment of particles to a liquid surface. J.

Chem. Soc. Faraday Trans. 1, 72: 2815-2828.

Slavnin, G.P., Investigation of Mineral Particle Flotation by High Speed Cameras. Gosgortekhizdat

Publisher, Moscow, 1960, 39 pp. (in Russian).

Spedden, H.R. and Hannan, Engng. Min. J., 149 (1948) 95.

Stechemesser, H., Freiberger Forschungshefte, A790 (1989) 81.

Stokes, G.G., Trans. Cambridge Philos. Soc., 9 (part 2)(1851) 8.

Whelan, P.F. and Brown, D.J., Trans. Instn. Min. Metall. 65 (1955-1956) 181.

Ye, Y and Miller, J.D., Coal Preparation, 5 (1988) 147.

Ye, Y and Miller, J.D., Int. J. Miner. Process., 25 (1989) 199.

Drops and Bubbles in Interfacial Research D. M6bius and R. Miller (Editors) �9 1998 Elsevier Science B.V. All rights reserved. 563

INTERACTIONS OF EMULSION DROPS

P.D.I. Fletcher

Surfactant Science Group, Department of Chemistry, University of Hull, Hull HU6 7RX, U.K.

Contents

1. Introduction

2. Emulsions

3. Colloidal forces between liquid surfaces

4. Interaction of an oil drop with an oil-water interface

5. Attractive interactions between two emulsion drops

6. Interactions between drops in concentrated, bulk emulsions

7. Summary

8. References

9. List of symbols

564

1. INTRODUCTION

Emulsions occur very widely in a large range of chemical products including, inter alia, food,

agrochemicals, inks and paints and pharmaceutical and cosmetic products. Because of their

technological importance, the formation, breakdown and other properties of emulsions have

been studied extensively [see, for example, refs. 1 - 7]. Although a number of aspects of

emulsion systems are qualitatively understood, there is still insufficient detailed knowledge to

allow quantitative and predictive control of many emulsion properties.

The main focus of this chapter concerns the colloidal interactions between emulsion drops which

are one of the major factor influencing emulsion stability. Following the theme of this volume,

we describe a number of (mainly) recent experimental approaches using small liquid drops for

the determination of the interactions relevant to emulsion systems. The chapter is organised as

follows. Following a brief description of the main breakdown processes of emulsions, the

different types of colloidal forces between liquid surfaces are described. Experimental methods

for the determination of these interactions are then described for emulsion drops in three

different configurations, (i) the interaction of a single drop with an oil-water interface, (ii) the

interaction of a drop with a second drop and (iii) drop interactions in a highly concentrated

emulsion. The necessary background together with limitations and possibilities of each approach

are discussed.

2. EMULSIONS

Emulsions consist of a dispersion of two immiscible liquids, commonly water with an apolar oil.

Although multiple emulsions are possible, the main types contain either water drops in oil (w/o)

or oil drops in water (o/w) with drop sizes generally in the range 0.1 to 10 ~tm. The formation

of an emulsion from two, bulk liquid phases involves the creation of a large oil-water interfacial

area with an unfavourable free energy cost equal to 7dA where 7 is the oil-water interfacial

tension and dA is the increase in interfacial area. Hence, emulsions require energy input for their

formation and are thermodynamically unstable. In the absence of adsorbed films coating and

stabilising the drops, emulsions rapidly separate into bulk oil and water phases. Thus, kinetically

stable emulsions require the presence of surface active species (surfactants, polymers or

565

particles) which adsorb at the oil-water surfaces of the drops and provide resistance to the

processes of drop growth and phase separation.

For an emulsion system containing equal volumes of oil and water plus a stabilising surfactant,

the type of emulsion formed (water-in-oil or oil-in-water, w/o or o/w) depends primarily on the

nature of the surfactant. The equilibrium phase behaviour (i.e. before emulsification) of such

two-phase mixtures containing equal volumes of oil and water together with surfactant is as

follows [4,8,9]. Hydrophilic surfactants may form either micelles or o/w microemulsion

aggregates in the water phase, which coexists with excess oil. Hydrophobic surfactants form

reversed micelles or w/o microemulsion aggregates in the oil phase, which co-exists with an

excess water phase. Emulsification of these two-phase systems yields an o/w emulsion when the

surfactant is hydrophilic with micellar or o/w microemulsion aggregates in the continuous

aqueous phase. A hydrophobic surfactant gives a w/o emulsion with reversed micellar of w/o

microemulsion aggregates in the continuous oil phase. Thus, the guiding principle to predict

emulsion type is that the emulsion type is the same as the type of surfactant aggregate produced

at equilibrium. The reader is referred to references [4,6,10] for a fuller discussion of these

points. It is relevant to note here that emulsions with surfactant concentrations in excess of that

required to fill the drop surfaces generally consist of ~tm sized emulsion drops dispersed in a

continuous phase which itself contains a concentration of nm sized surfactant aggregates.

The main mechanisms leading to emulsion instability are illustrated schematically in Figure 1.

Creaming or sedimentation of the drops is driven by gravity and causes the emulsion to separate

into regions of high and low droplet volume fraction. It can be easily reversed by simple shaking

of the emulsion. For an isolated, rigid, uncharged drop of size large enough to be unaffected by

Brownian motion (radius > 1 ~tm) the velocity of sedimentation/creaming v is given by Stokes'

law.

v = 2a2Apg/9rl (1)

where a is the drop radius, A 9 is the density difference between the drop and continuous phase,

g is acceleration due to gravity and ~1 is the viscosity of the continuous phase. In real emulsions,

neighbouring drops hinder sedimentation/creaming and the velocity decreases with increasing

566

drop volume fraction [5]. Equation 1, applicable to low drop volume fraction emulsions, shows

that creaming/sedimentation rate increases strongly with increasing drop radius.

Figure 1. Breakdown processes of emulsions.

Ostwald ripening is the process whereby large drops grow at the expense of smaller ones due to

mass transport of the (sparingly) soluble dispersed phase through the continuous phase. The

thermodynamic driving force for thb, process arises from the greater tendency of the dispersed

material present in small drops to dissolve in the continuous phase because of the higher Laplace

pressure in the small drops. A form of the Kelvin equation gives the radius-dependent solubility.

RTln(S/S| = 2TVm/a (2)

where S and Soo are the values of the solubility of the dispersed component from a drop of radius

a and when the interface is planar (i.e. when a = oo) and Vm is the molar volume of the dispersed

component. The rate of Ostwald ripening is given by [ 11,12]

d<a>3/dt = 8VTSooD/9RT (3)

where <a> is the mean drop radius and D is the diffusion coefficient of the dispersed component

when dissolved in the continuous phase. Since the values of Vm, T and D for most emulsion

systems do not vary by more than one order of magnitude, the prime, system-dependent

567

parameter determining the rate is the solubility of the dispersed component in the continuous

phase [ 13, 14].

Emulsion drops adhere together or flocculate if the forces of attraction between the drops are

sufficient to overcome the random thermal motions of the drops. If the attractive forces are

weak, the process may be reversible. The colloidal forces responsible for flocculation and their

experimental determination form the main focus of this chapter and are discussed in detail in the

following sections. The occurrence of flocculation strongly affects the structure and properties

of emulsions. In a dilute emulsion, flocculation causes the average particle size to increase and,

from equation 1, this leads to an increased rate of sedimentation or creaming. In concentrated

emulsions, strong adhesion of the drops may lead to the formation of a gel-like network of

flocculated drops giving big changes in the rheological properties of the emulsion [5].

Coalescence is the irreversible fusion of two or more drops together to form a single larger drop

and is thought to occur in the following stages. Firstly, the droplets diffuse together as

undeformed spheres. Closer contact results in droplet deformation and the formation of a thin,

approximately planar thin film, which may drain and thin. The final stage of film rupture and

consequent fusion of the droplets may occur if the film thickness reaches a sufficiently low,

critical value. At the critical rupture thickness, the film is thin enough to be ruptured by random

surface fluctuations, the amplitude of which is thought to be controlled by the rheology of the

adsorbed layers and the colloidal forces between the surfaces [ 15, 16].

In real emulsions, the more than one of the processes described above commonly occur

simultaneously and each process generally affects the others. Hence, overall emulsion stability is

a highly complex affair and the formulation of emulsions of the desired stability must normally

be determined using experiments guided by colloidal principles. A major consideration of

relevance, particularly to flocculation and coalescence, is the nature and strength of the colloidal

interactions between emulsion drops as discussed in the next section.

3. COLLOIDAL FORCES BETWEEN LIQUID SURFACES

The range and magnitude of the interaction between approaching emulsion drops can be

discussed either in terms of the energy (V) required to bring the drops from infinity to a finite

separation distance or the force (F) between the drops at that separation. Additionally, emulsion

568

drops are liquid and can deform to produce a thin emulsion film between the drops. Hence, one

also has to distinguish between the force or energy as a function of separation of the

undeformed, spherical drops and the force or energy as a function of the thickness of the

emulsion film. For the initial approach of the drops, one must consider the spherical drops. In

the limit of large, highly deformed drops and thin films, the interactions across the film dominate

the total interaction. For small drops with low deformation and thick films, contributions from

both the film and the drop must both be considered [ 17].

For most of the experimental techniques to be discussed here, we consider large, highly

deformed drops for which the interactions are dominated by the thin emulsion films. It is

therefore most convenient to express the interactions in terms of the film disjoining pressure H

which is defined as the negative differential of the film free energy (per unit area) E with film

thickness, i.e. H = -(dE/dh)T. I-I can also be considered as the repulsive force per unit area

perpendicular to the film surfaces. The disjoining pressure is positive for repulsive forces and

negative for attractive forces. The total force (F) is equal to the product of the disjoining

pressure and the film area and the energy of interaction per unit area of the film (E) may be

obtained by integration of II as a function of film thickness h.

h

E - I FIdh (4) oo

The total disjoining pressure contain contributions from van der Waals forces (1-I~aw),

electrostatic (H~l), a variety of short range forces (H~) and forces arising from the presence of

micellar, reversed micellar or microemulsion aggregates within the continuous phase of the

emulsion. The total disjoining pressure is generally assumed to be the sum of all

contributions [ 18].

Van der Waals forces are always attractive for symmetrical oil-water-oil or water-oil-water

emulsion films. The simplest treatment gives the following expression for the non-retarded

disjoining pressure as a function of film thickness [ 18].

1-Iva w = -A / 6zch 3 (5)

569

where A is the Hamaker constant for phases of water (oil) interacting across a film of oil

(water). For typical alkane-water emulsion films, the Hamaker constant is around 5 x 10 21 J,

approximately one order of magnitude weaker than for foam (air-water-air) films [19]. For

systems where the oil and water components have similar dielectric dispersion properties

(mainly reflected in the values of refractive index), the Hamaker constant, and thus the van der

Waals attractive forces, can be vanishingly small. Equation 5 is a simplification and more exact

treatments need to consider the following points.

(i) The Hamaker constant is determined by the dielectric dispersion properties of the

components of the system summed over a wide frequency range and can be separated into zero

frequency and finite frequency contributions, Av=0 and Av>0 respectively. For emulsion films,

both contributions are typically of similar magnitude. The finite frequency component Av>0 is

subject to retardation for film thicknesses greater than approximately 5 nm and is thus weaker

than predicted by equation 5. The zero frequency term is not subject to retardation but, for the

case of water films, is subject to screening by electrolyte present in the aqueous phase. A fuller

discussion of these points can be found in refs. [ 18, 19].

( i i ) Emulsion films contain adsorbed monolayers of surfactant at the oil-water interfaces

bounding the film. For film thicknesses smaller than or equal to the thickness of the adsorbed

monolayer, Ilvdw is mainly determined by the properties of the monolayers and the continuous

film component. For larger film thicknesses, 1-Ivdw is mainly determined by the bulk oil and water

phases. Approximate equations for Ilvdw as a function of film thickness in films containing

adsorbed layers are given in ref. [ 18].

Oil-water-oil emulsion films experience electrostatic repulsion owing to the surface charges of

the oil-water interfaces bounding the water film. This is tree not only for systems containing

ionic surfactants but also for films stabilised by non-ionic surfactants. As for oil-water interfaces

in the absence of adsorbed films of surfactant [20], non-ionic surfactant monolayers are thought

to acquire a surface charge by the differential negative adsorption of positively and negatively

charged ions [21]. For surfaces of equal surface charge density in the presence of a 1:1

electrolyte, I-Iol is given by

570

Hel-64ckT t a n h 2 ( ~ ] exp(-tch) (6)

where c is the bulk concentration of electrolyte, e is the electronic charge, ~0 is the surface

potential and K is the reciprocal Debye length, which (for a 1:1 electrolyte) is

[ecokT

where e is the relative permittivity of the film liquid and ~0 is the permittivity of free space [ 18].

Equation 6 is valid for separations larger than the thickness of the diffuse double layer, i.e.

h > rc 1. For smaller separations, approximate equations valid for low surface potentials (less

than approx. 25 mV) for the cases of either constant surface potential or constant surface charge

density can be found in refs. [18, 22]. For higher surface potentials and short separations a

numerical calculation must be used [23]. The simple treatments described here neglect short-

range effects associated with, inter alia, the "discreteness" of the surface charges and the finite

sizes of the electrolyte ions and discussion of these refinements to the simple theory can be

found in refs. [ 18, 22, 24].

At small separations, of the order of a few molecular diameters, surfaces can experience a

variety of short-range forces. Emulsion film surfaces with adsorbed monolayers of surfactant are

not perfectly smooth but show thermal fluctuations of two main types. Within the monolayers,

individual surfactant molecules are expected to show fluctuations in the extent to which they

protrude from the average monolayer position. Using energy parameters derived from

monomer-micelle exchange kinetics, the mean protrusion lengths are estimated to be in the

range 0.08 - 0.3 nm [25]. In addition to the "protrusional" fluctuations, the monolayers are also

expected to show wave-like undulations on larger length scales. The mean amplitudes of such

waves are expected to be inversely related to the magnitude of the interracial tension

(proportional to 7 "~/2 [26]). In the presence of adsorbed surfactant, oil-water interracial tensions

are typically 10 - 1 mN m ~ for common surfactants but may drop to ultralow values

(< 10 .3 mNm ~) for microemulsion forming systems. Hence, emulsion drop surfaces are expected

to be relatively rough with amplitudes of the thermal fluctuations of the order of nm. (We note

here that foam films with much higher tensions (typically 30 mN m -~) should have significantly

571

lower roughness.) For film thicknesses less than the amplitudes of the fluctuations, decreasing

thickness leads to suppression of the different fluctuations (entropically unfavourable) giving

rise to repulsive protrusion, undulation and peristaltic forces [ 18]. The disjoining pressure due

to protrusion forces is predicted to decay approximately exponentially with film thickness with a

decay length of the order of a fraction of a nm. The undulation and peristaltic disjoining pressure

contributions are predicted to decay as 1/h 3 and 1/h 5 respectively [ 18].

Overlap of the chains of adsorbed monolayers gives rise to short range forces which may be

either attractive or repulsive depending on whether the solvent is a good or bad one for the

adsorbed species. In a good solvent the force is repulsive, ("steric" or "overlap" force). In a bad

solvent, "bridging" of the two surfaces by adsorbed species gives an attractive force at relatively

large separations, which may switch to repulsion at shorter separations when overlap is high. In

the case of adsorbed polymer monolayers, the range of these interactions may be many tens of

nm whereas for surfactant monolayers the range is much smaller. A detailed discussion of such

forces can be found in ref. [27].

For atomically smooth, hard surfaces short range oscillatory forces associated with solvent

ordering at such surfaces have been observed [ 18]. Since emulsion drop surfaces are relatively

rough and deformable, such molecular-scale oscillatory forces may not be important in emulsion

films. Solvation effects are only likely to be important for surfactant monolayers in that they

affect the energy required to cause overlap of surfactant monolayers. Overall, short range

contributions to the total disjoining pressure of emulsion films stabilised by surfactant

monolayers remain poorly understood and controversial but are likely to be repulsive and

steeply decaying functions of the film thickness with ranges of the order O. 1 - 1 nm, (see, for

example, ref. [28]).

We now consider forces in emulsion films arising from the presence of surfactant aggregates in

the continuous phase. For film thicknesses less than the diameter of the surfactant aggregates,

the aggregates are excluded from the film. This leads to a lower osmotic pressure within the film

relative to the bulk continuous phase outside the film. This causes an attractive depletion

interaction [29] and the resultant contribution to the disjoining pressure (l'-Idep) is given by

I-Ia~p = 0 for h > d

572

w h e r e Posm is the osmotic pressure arising from the surfactant aggregates of diameter d in the

continuous phase. For the case where the surfactant aggregates behave as hard spheres, Posm is

6~bkT(1 + ~b + ~b2 - ~b3 t Po~m-- mt3 (l_q~) 3 (9)

where ~ is the volume fraction of aggregates in the continuous phase. Depletion effects caused

by the addition of surfactant aggregates in the continuous phase have been observed for both

o/w and w/o emulsions. [31,32].

In addition to the attractive depletion interaction seen for h < d, the surfactant aggregates can

also give rise to an oscillatory contribution to the disjoining pressure (Hose) for h > d caused by

the formation of "stratified" layers of the particles owing to the presence of the film surfaces.

The disjoining pressure isotherms show a series of maxima and minima which decay in

amplitude with increasing film thickness. The period of the oscillations is approximately equal to

the particle diameter. The number and amplitude of the oscillations increase with particle

volume fraction. Kralchevsky and Denkov [33] have derived a zeroth-order, semi-empirical

analytical expression for 1-Iosc confined between smooth hard walls.

( 2 ~ ( d ~ h l-Io,~ - Posm c o s ~ / e x p / ~ - ) (10)

' - ~ d , ) ~ d i d 2 d 2

where dl and d2 are functions of d and ~ defined according to

= , 2 + 0.23728A~b + 0.63300(A~b al d ~ 3

d 2 0.48663 Jr d - A----~ - 0.42032 and A~b - ~ - - ~b (11)

Such oscillatory forces induced by the presence of surfactant aggregates have been observed for

both foam and emulsion films [34-38]. Although the treatment provides a reasonable description

of the number and film thicknesses for the maxima in FIo~ observed in foam film thinning

experiments [33], it remains to be seen whether the magnitude of Ho~ is given correctly for

l'-[dep = -Posm for h < d (8)

given by [30]

573

emulsion films. Equations 9 and 10 apply in the case of hard spheres confined between smooth

hard walls whereas emulsion drop surfaces are deformable and rough.

This brief discussion of colloidal forces in emulsion films provides a basis for qualitative

understanding many effects seen in emulsions, e.g. stability changes of non-ionic emulsion drops

induced by the addition ionic surfactant and salt [39, 40] and depletion flocculation induced by

the addition of high concentrations of surfactant aggregates in the continuous phase [31, 32].

However, there have been relatively few quantitative tests of these theories for emulsion

systems. The lack of detailed knowledge of (particularly) the short range forces remains an

obstacle to quantitative understanding of droplet coalescence. In the following sections of this

chapter we describe some (mainly) recent experimental approaches using liquid emulsion drops

for the quantitative determination of the interaction between drop surfaces.

4. INTERACTION OF AN OIL DROP WITH AN OIL-WATER INTERFACE

Interactions between liquid surfaces have traditionally been studied using the experimental

techniques pioneered by Sheludko [41 ] and by Mysels and Jones [42]. In these methods a thin

film is formed in a capillary cell in the contact region between two (nearly) hemispherical liquid

surfaces. The thin film radius is typically of the order of 100 ~tm, far larger than likely to be

formed by the contact between two gm emulsion drops. The vast majority of studies of this type

have been made for foam (i.e. air-water-air) films (reviewed in refs. [ 15, 41, 43-46]) together

with a few reports of "pseudo-emulsion" (i.e. water-oil-air or oil-water-air) films [47-49].

Experimental studies of various aspects of emulsion films are described in refs. [50-65].

Previous measurements of disjoining pressure for emulsion films have been made in two ways

using Sheludko cells. Firstly, film thickness has been determined for a single value of the

disjoining pressure held constant at a value of the order of a few tens of Pa. Secondly, the

variation of disjoining pressure with film thickness (the disjoining pressure isotherm) has been

deduced from measurements of the film drainage rate where the calculation relies on the

assumption of particular hydrodynamic conditions within the film. As discussed for example in

ref. [55] this latter, indirect method commonly yields results that do not agree with equilibrium

methods. Thus, there is a need for a technique that allows the direct measurement of the full

disjoining pressure curve as a function of the film thickness. This can be achieved using the

H

liquid surface forces apparatus (LSFA) (described below) which contains a number of features

in common with an apparatus developed to examine the adhesion forces between biological cells

and solid surfaces [66].

The LSFA in the configuration used to study oil-water-oil emulsion films is shown schematically

in Figure 2 [67].

micropipette

piezo 1 I I

I ~ o i l

L water

reflectance microscope objective

- q laser 1 ) piezo2 ) [

Figure 2. Schematic diagram of the LSFA.

574

1

oil

oil ----7

water

Figure 3. Detail of the micropipette assembly with attached mirror arm and view of the oil drop at the pipette

tip close to the oil-water interface.

A small oil drop is supported on the tip of a fine glass micropipette (Figure 3) which is held

initially a few ~tm below the oil-water interface. The other end of the micropipette is

575

connected to a manometer filled with the same oil. The radius of curvature of the oil drop ro

(typically of the order of 10 ~tm) is controlled by the balance of the Laplace pressure inside the

drop and the applied pressure head controlled by the manometer height H.

Hog = 2yro (12)

where 9 is the oil density. Using this arrangement, the minimum oil drop radius that can be

achieved is equal to the internal radius of the capillary tip re, reached at a maximum pressure

corresponding to Hmax. Varying H between 0 and Hmax causes ro to vary from infinity to re.

Increasing H to a value greater than Hmax causes oil to flow from the capillary. The highest

accessible pressure is dependent on 7 and re and is typically a few thousand Pa.

Using the left piezo translator (piezo 1), the micropipette and oil drop can be moved up to the

oil-water interface causing the drop and interface to deform and leading to the formation of a

thin water film between the apex of the drop and the interface. The oil drop experiences a force

and this causes the horizontal shaft of the micropipette to deflect vertically downwards. A

mirror, mounted on a horizontal extension arm attached to the horizontal shaft of the

micropipette, is positioned at the focus of a laser reflecto-optical device. The laser and detector

of the reflected intensity are mounted on a second piezo translator (piezo 2). The laser is

maintained at a vertical height such that the laser beam impinges on the bottom edge of the

mirror attached to the pipette. In this position, the reflected intensity is approximately half the

maximum reflected intensity obtained when the laser spot is positioned centrally on the mirror.

The measured intensity is kept constant by continual adjustment of the vertical height of the

laser using a feedback circuit control of piezo 2. In this way, the laser height accurately (within

approx. 200 nm) follows the vertical movement of the mirror arm. When the oil drop is far from

the interface and experiences no force, the movement of the laser (piezo 2) matches the

movement of the oil drop (controlled by piezo 1). When the drop is close to the oil-water

interface, the force experienced causes the horizontal arm of the micropipette to deflect

vertically. The difference in movement of the pipette and mirror arm (measured using accurate

position transducers attached to both the pipette and laser) yields the deflection. The force is

obtained from the measured deflection and the separately measured force constant of the

576

micropipette. The accuracy of the force measurement is mainly determined by stray vibrations

and is approx. +5 nN.

The thickness and radius of the emulsion film formed when the apex of the drop contacts the

oil-water interface are determined by reflectance microscopy using computer analysis of the

captured video images. Details of the analysis used to obtain the thickness are given in ref. [67].

The variation of reflected light intensity with film thickness is such that the accuracy of the

thickness determination is approx. +2 nm for films thicker than 10 nm and +5 nm for thinner

films. The radius of the film can be measured to approx. +0.2 pm.

As described above, the LSFA can be used for the direct measurement of the force as a function

of distance as a ~tm sized oil drop is moved against an oil-water interface. The thickness of the

oil-water-oil emulsion film formed by contact of the drop with the interface is expected to be

determined by the disjoining pressure within the film (FI) which is related to the applied

hydrostatic pressure applied to the pipette as follows. The geometry of the deformed oil drop in

contact with the oil-water interface is shown in Figure 4. In general, the shape of the oil drop is

determined by a combination of gravity and capillary forces. However, for the systems

considered here, the capillary length (= (y/Apg) 1/2, where Ap is the density difference of the two

liquid phases) associated with the oil drop is of the order of millimetres whereas the oil drop

radius is of the order of micrometers. Under these conditions gravity forces can be neglected

and the radius of curvature of the oil drop not in contact with the interface remains constant at

ro. For the film region at the apex of the drop, the total tension of the film is simply 27 since the

film contains two oil-water interfaces. Since the hydrostatic pressure P must be balanced by the

Laplace pressure, the radius of curvature of the film rf is simply twice ro. Therefore, the

disjoining pressure H in the film is given by 1-I = P/2. In principle, there is a small excess tension

in the emulsion film due to the disjoining pressure. However, this is generally less than

0.1 mN m 1 and can usually be neglected.

From the considerations given above, it is evident that the disjoining pressure H and hence the

film thickness should remain constant when an oil drop is pressed up to an oil-water interface at

constant hydrostatic pressure P. This has been confirmed experimentally as illustrated in

Figure 5 which shows the radial interference intensity profiles measured through the centres of

the films for two different pipette positions. The pipette vertical position px (measured at

577

piezo 1) is taken to be zero when the apex of the undeformed oil drop first contacts the oil-

water interface. In each intensity profile, the central flat region corresponds to the film and the

outer interference fringes correspond to the meniscus region. Comparison of the two intensity

patterns shows that upward movement of the pipette at constant pressure P causes the film

radius to increase with no change in reflected intensity, i.e. no change in film thickness.

:z x ' 0~'

I l rc / ~ 0 / /

,~rf /

/

/

I

I

I

I

I

I

I

I

I

I

I

I

(

Figure 4.

3.00

IL

2110

1.00

0131

The geometry of the deformed oil drop in contact with the oil-water interface.

4110

.00

3~0

R

2.00

1110

' ' 0"08 ' 8'110 ' 8~ 16.00 ~0

Figure 5.

16.00

Radial interference intensity profiles (expressed as the normalised ratio R of the interference intensity divided by the intensity of reflection from a single oil-water interface) for dodecane-water- dodecane films stabilised by AOT. The pressure is 6 17 Pa and the pipette radius is 12.8 mm. The left profile corresponds to Px = 0.5 ~tm (close to first contact with the interface) and the right corresponds to px = 4.2 Ixm.

578

We now consider how the force F and film radius r vary with pipette position px at constant

pressure P. For an undeformed drop far from the interface, the drop height b (the distance from

the end of the pipette to the drop apex is given by

2r o - x /4r 2 -4r~ b - 2 (13)

Simple geometry gives the radius r of the circular emulsion film as

r - ~ / - b 2 - x z + 2 r b - 2roX + 2 b x (14)

The angle 0 made by the oil-water monolayer to the vertical at the contact line with the

emulsion film is

O = c o s - ' ( r / r o ) (15)

For films in which there is a net attractive free energy of interaction between the two oil-water

interface, a nonzero contact angle will be present between the film and adjoining meniscus. As

discussed in ref. [68], for such films the value of this contact angle must be added to e. We

consider here only repulsive films for which e is given by equation 15. The product of the

emulsion film perimeter and the vertical component on the tension gives the downward force F

exerted on the pipette by the interfacial tension.

F = 27vry cos 0 (16)

The vertical distance z between the film perimeter and the flat level of the oil-water interface

(i.e. far from the drop) is

2r / cos 0 - 4(2r / cos 0) 2 - 4 r 2 z = (17)

2

Experimentally, F and r are measured as a function of pipette position p~ relative to a zero

position taken to be when the apex of the drop first contacts the oil-water interface. The overall

movement of the pipette takes accounts of the displacement of the oil-water interface (z), the

deflection of the pipette due to the force (= F/f~ where f~ is the pipette force constant) and the

change in length x.

p = ( b - x ) + z + F / f ~ (18)

100

The series of equations 13 - 18 together with knowledge of P, 7 and rc allow the calculation of

the variation of F and r with px and the complete profile of the drop in contact with the oil-water

interface using no adjustable parameters. An example of data is shown in Figure 6 where it can

be seen that good agreement with theory is obtained. A drop profile calculated for the same

conditions as Figure 6 is shown in Figure 7.

80

12

Z 60

~- 40 o

20

f i

w . x _ w

Figure 6.

3

1

I I I

0 1 2 3 4 0 1 2 3 4

rg m

10

Variation of force (left plot) and film radius (fight plot) with pipette position 1~ for dodecane-water-

dodecane films stabilised by the anionic surfactant AOT. The conditions were as follows: ~, = 18 mN m 1, P

= 2696 Pa and rc = 12.8 ~tm. The solid lines are calculated as described in the text.

-2

-20

1 1 I I I

Figure 7.

579

-15 -10 -5 0 5 10 15 20

Calculated profile of an oil drop in contact with the oil-water interface. Experimental conditions are

as for Figure 6 with p• = 3.22 ~tm. The axes show vertical and horizontal dimensions in ~tm.

580

Experiments in which pipette position px is varied at constant P allow the measurement of the

"capillary" force associated with the deformation of the drop and oil-water interface but do not

reveal the direct interaction forces between the liquid surfaces. This can be done by measuring

the film thickness (independent of px) as a function of P (and hence H). As an example of data

obtained in this way, disjoining pressure isotherms (H versus film thickness h) for dodecane-

water-dodecane emulsion films stabilised by the non-ionic surfactant n-dodecylpentaoxyethylene

glycol ether (C12E5) are shown in Figure 8.

1500

1000

r ~ r ~

e~ 500

0 "N'

Pod

-500 I 1 I I

0 20 40 60 80 100

film thickness/nm

Figure 8. Disjoining pressure curves for dodecane-water-dodecane emulsion films stabilised by 0.02 mM C12E5

with [NaC1] equal to 0.1 (open circles), 0.7 (filled circles), 8.55 (open triangles), 51.3 (filled

triangles) and 136.7 mM (open squares). The solid lines are calculated as described in the text.

The solid curves are fitted lines calculated on the basis that H = l'-Ivdw -I- I ' I e l (equations 5 and 6)

using only the surface potential as an adjustable parameter. The variation of the surface potential

with NaC1 concentration, shown in Figure 9, is similar to that found for emulsion films stabilised

by non-ionic surfactants [21 ].

>.

ra~

0

O. 1

0.08

0.06

0.04

0.02

581

I I I

0.0001 0.001 0.01 0.1 1

[NaCl]/M

Figure 9. Variation of surface potential for emulsion films stabilised by 0.02 mM C12E5 with NaC1 concentration.

Further examples of disjoining pressure isotherms for a range of stabilising surfactants can be

found in ref. [69]. We have also used the LSFA to examine attractive (adhesive) forces between

liquid surfaces [68].

In conclusion, the LSFA allows the direct measurement of force and film thickness and radius as

functions of both the drop position and disjoining pressure. A special feature of the LSFA is that

the drop size is comparable to that found in bulk emulsions. For these smaller drop sizes,

"dimple" formation within films as seen for films of larger dimensions [16] are absent. The

LSFA is useful for the study of disjoining pressures of range greater than approx. 5 nm since the

optical method for film thickness measurement used is not accurate for thinner films. We have

described here measurements made under static (equilibrium) conditions. In principle, the LSFA

could be used to examine hydrodynamic forces present under conditions when the drop is

moved rapidly toward the oil-water interface.

5. ATTRACTIVE INTERACTIONS BETWEEN TWO EMULSION DROPS

When two emulsion drops have a net attractive energy of interaction but do not coalesce then

they spontaneously adhere together and show a finite contact angle co between the emulsion film

and the adjoining meniscus (Figure 10).

582

Figure 10. Spontaneous adhesion of two attractive (but non-coalescing) drops showing the formation of a finite

contact angle co between the emulsion film and adjoining meniscus.

The energy per unit area E required to bring the surfaces together from infinity to their

equilibrium separation hoq is obtained by integration of the disjoining pressure curve according

to equation 4. A (metastable) equilibrium separation hoq is obtained when the disjoining pressure

in the film is equal to the Laplace pressure in the drops (= 2y/R) and the slope dH/dh at h = hoq

is negative. If these conditions are not met either the drops do not adhere (when the interactions

are repulsive) or else coalescence occurs (when the net interactions are attractive and the height

of the disjoining pressure barrier is low). Schematic disjoining pressure curves illustrating some

of the different possible situations are shown in Figure 11.

The interaction energy E has units of energy per unit area and can be considered to be the

excess film tension arising as a result of interactions (E = ]tfilm - 2~). E is related to the film

contact angle 03 by

E - 2y (cos co - 1)

1 0 0 0

500 - _ _

() ~ 20 3 -500 -

-1000

h/rim

1 0 0 0

500 - _

0 )

-500 -

- 1 0 0 0

(19)

10 20 3

h/nm

Figure 11. Schematic disjoining pressure curves showing the equilibrium disjoining pressure equal to the capillary pressure of the drops (horizontal dashed line). In the left hand Figure the total disjoining pressure contains contributions from a repulsive short range interaction combined with an attractive van der Waals interaction which gives heq = 2 nm, corresponding to the intersection of FI~r with the capillary pressure. In this case, the net energy of interaction is negative and the contact angle is finite. The right hand Figure shows the effect of introducing an electrostatic repulsion barrier giving heq = 10 nm, an overall positive (repulsive) energy of interaction and hence a contact angle of zero.

583

The magnitude of the contact angle is related to both E and 7. The values of E found for liquid

surfaces are generally in the range 0 - 1 mJ m 2 (equivalent to mN rel). For foam films where 7

is typically of the order of 30 mN m 1 contact angles are small (less than a few degrees). The

same magnitude of E in emulsion films where the tension is typically of the order of 1 mN m 1

produces large contact angles of tens of degrees, which can be easily measured.

Figure 12. Schematic diagram (side view) showing microscopic measurement of contact angles between adhesive

emulsion drops.

As described in a series of papers by Princen and co-workers [70-73], the technique for

measuring contact angles of emulsion films is experimentally very simple (Figure 12). A coarse

emulsion with drop radii in the range 5 - 100 ~m is prepared (usually simple hand shaking is

sufficient), transferred to a microscope slide with a shallow well and covered with a cover slip.

It is important to ensure that the emulsion is sufficiently diluted with continuous phase such that

single adhering drop doublets can be isolated within the field of view of the microscope. For oil

drops in water the oil drops cream to reach the underside of the cover slip. Drop doublets with

equal sized drops are selected for measurement as this ensures the two drop centres lie in the

same horizontal plane. The microscope focus is set to lie in the plane of the drop centres and the

image is captured using a video camera with digital frame grabber. The most accurate method to

obtain the contact angle from the image is to computer fit the entire profile of the drop doublet;

584

however, reasonable accuracy is obtained by simple manual measurements directly from the

image.

Using the method outlined above, Aronson and Princen [72] were able to measure the attractive

energies of interaction for a range of anionic surfactants in the presence of high concentrations

of various electrolytes. A similar adhesive interaction with a similar magnitude of E has been

seen for foam films stabilised by the same surfactants [74, 75]. The molecular origin of this

interaction (which is highly dependent on the nature of the salt and surfactant headgroup) in not

completely clear but is thought to be associated with the formation of an ordered two

dimensional structure within the low water content thin film [72, 75].

In addition to probing the strength of this (relatively strong) salt induced interaction, the

emulsion film contact angle method should be useful to determine other types of attractive

interactions and their modification by repulsive interactions. Following Aronson and Princen

[72], we illustrate this point with some theoretical calculations as follows. We consider an

uncharged droplet system for which electrostatic interactions are absent. Initially, we suppose

the interactions to contain contributions only from van der Waals forces and short range forces

represented as an effective "hard wall" interaction of range equal to heq, i.e. FI~ = 0o for h < heq

and 0 for h > heq. Using the simple equation 5 for Hvdw, we obtain the following equation for co

in terms of the Hamaker constant A, the tension 7 and heq.

(" A co- cos 1- 2

24~,hs (20)

Figure 13 shows the calculated variation of co with heq for A = 5 x 10 "21 J (typical for alkane-

water systems) and y = 1 mN m 1.

It can be seen that measurement of the contact angle allows an estimate of heq and thereby the

effective range of the repulsive short range forces responsible for the prevention of coalescence.

In principle, measurements with different drop sizes (for which the Laplace pressure is different)

allow the determination of the variation of heq with the disjoining pressure in the film of

equilibrium thickness. The approach outlined here can be easily extended to include other types

of forces including electrostatic, depletion and oscillatory interactions.

100

80

60 O B

"~ 40

20 I I

0 1 2 3

585

heq/nm

Figure 13. Variation of emulsion film contact angle with equilibrium film thickness, calculated according to

equation 20.

6. INTERACTIONS BETWEEN DROPS IN CONCENTRATED BULK

EMULSIONS

The LSFA allows measurement of the disjoining pressure isotherm for emulsion films of

thickness greater than approx. 5 nm and for disjoining pressures up to a few thousand Pa. The

contact angle method gives a quantitative measure of the total energy of interaction per unit film

area for adhesive but non-coalescing drops. Neither technique gives information about the

magnitude of the disjoining pressure barrier that must be overcome before film rupture and drop

coalescence can occur. Bibette and co-workers [76, 77] have developed an interesting and

promising approach to this problem using osmotic compression of mono-disperse emulsions.

Figure 14. Schematic diagram of osmotic compression of mono-disperse emulsions.

As shown schematically in Figure 14, a dispersion of mono-disperse o/w emulsion drops is

contained within a semi-permeable membrane and equilibrated with an external solution

containing concentrated polymer. Attainment of osmotic equilibrium causes the aqueous phase

to be withdrawn from the emulsion into the external polymer solution leading to closer packing

586

and deformation of the emulsion drops. In the experiment, the concentration of free surfactant is

maintained equal on both sides of the semi-permeable membrane to avoid leaching of the

surfactant out of the emulsion.

Osmotic compression of the emulsion to a volume fraction higher than that corresponding to

random close packing of spherical drops (63%) causes the drops to deform with the formation

of fiat films between the drops. The Laplace pressure in the deformed drops is equal to 2T/rb

where rb is the radius of curvature of the regions of the drop outside the fiat film regions.

Because the drops are mono-disperse the film regions between the deformed drops are planar

and the disjoining pressure within the films is equal to the Laplace pressure. Increasing the

applied osmotic pressure increases the drop deformation and decreases rb until a critical value is

reached when the droplet Laplace pressure is sufficient to overcome the disjoining pressure

barrier to film rupture II*. Thus, drop coalescence (observed microscopically) is found only

above a critical osmotic pressure Poem*. In the limit of large drop sizes (typically greater than a

few gm), the disjoining pressure in the films is equal to the applied osmotic pressure and hence

measurement of Poem* yields l-I*. There also exists a critical initial drop radius for small drops for

which the Laplace pressure of the undeformed drop (i.e. 27/r~ where r~ is the initial, undeformed

drop radius) is equal to l-I*. Drops of radius equal to or smaller than this critical radius coalesce

as soon as they contact, i.e. under very low osmotic compression. Hence, the "stability map" for

the emulsions with respect to applied osmotic pressure and drop radius has the form shown

schematically in Figure 15 [77].

P ~ UNSTABLE

STABLE

Drop radius

Figure 15. Schematic "stability map" for an emulsion under osmotic compression showing the stability limits at

high and low drop radius.

587

As shown by Bibette and co-workers [76, 77], the boundary between stable and unstable

behaviour may be determined by microscopic examination of the emulsions for various drop

sizes and values of P .... Measurement of either Posm* for large drops or the critical drop radius at

zero Posm provides independent determinations of the value of I-I* for the particular emulsion

system. In principle, theoretical approaches for the calculation of drop shape in concentrated

emulsions under osmotic compression (see, for example, [78]) could be used to extract the

additional information of the critical rupture thickness of the emulsion film at the critical

osmotic compression, particularly if the emulsion volume fraction is measured independently.

For an o/w emulsion system of silicone oil drops stabilised by the anionic surfactant sodium

dodecyl sulphate with an ionic strength of 0.01 M, Bibette at al. have determined that l-I* is

approx. 1.2 atm. This value is obtained both by measurement of large drops or by determination

of the critical (lower) drop size. The magnitude of l-I* shows reasonable agreement with the

height of the disjoining pressure barrier for a 1-I versus h isotherm calculated using the known

values for the Hamaker constant (4 x 10 zl J) and surface potential (-45.5 mV) for the emulsion

system [79]. Given this encouraging agreement for this single system, it would be useful to

extend this type of measurement to a wider range of systems in order to quantitatively test

theoretical predictions of coalescence stability.

7. SUMMARY

We have described three experimental methods using drops for the measurement of different

aspects of the interactions between emulsion drops. The LSFA allows measurements of the

capillary force and drop deformation of a gm sized emulsion drop as it is pressed up to an oil-

water interface. The forces and drop deformations are well described by theory using no

adjustable parameters. The LSFA can also be used to determine disjoining pressure isotherms

for film thicknesses 5 - 100 nm and YI values in the range 100 - 2000 Pa. The emulsion film

contact angle method developed by Princen at al. can be used to determine the interaction

energy per unit area for adhesive, non-coalescing drops. It has been successfully used to

characterise a specific, salt induced interaction but could be extended to quantify other forces.

The osmotic compression method of Bibette et al. can be used to measure the magnitude of the

disjoining pressure barrier to emulsion film rupture and drop coalescence. The different

techniques yield complementary information and offer future promise to gain an increased

quantitative and detailed understanding of the interactions of emulsion drops and their role in

determining the emulsion stability.

588

0

.

5.

,

7.

,

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

REFERENCES

Theory and Practice of Emulsion Technology, Ed. A.L. Smith, Academic Press,

London, 1976.

Encyclopedia of Emulsion Technology, Ed. P. Becher, Marcel Dekker Inc., New York.

Vol. 1, Basic Theory (1983), Vol. 2, Applications (1985), Vol. 3, Theory, Measurement

and Applications (1988) and Vol. 4, Emulsification, Stability and Rheology (1996).

Macro- and Microemulsions. Theory and Applications, Ed. D.O. Shah, Amer. Chem.

Soc., Washington DC, 1985.

Emulsions and Solubilisation, K. Shinoda and S. Friberg, Wiley, New York, 1986.

An Introduction to Food Colloids, E. Dickinson, Oxford University Press, Oxford,

1992.

B.P. Binks, Royal Society of Chemistry Annual Reports C, 1996, 92, 97.

Emulsions and Emulsion Stability, Ed. J. Sjoblom, Surfactant Sci. Series Vol. 61,

Marcel Dekker Inc., New York, 1996.

R. Aveyard, Chemistry and Industry, 20 July 1987, 474.

M. Kahlweit and R. Strey, Angew. Chem. Int. Ed. Engl., 1985, 24, 654.

B.P. Binks, Langmuir, 1993, 9, 25.

I.M. Lifshitz and V.V. Slezov, J. Phys. Chem. Solids, 1961, 19, 35.

C. Wagner, Zeit. Electrochem., 1961, 35, 581.

A.S. Kabalnov, A.V. Pertzov and E.D. Shchukin, J. Colloid Interface Sci., 1987, 118,

590.

A.S. Kabalnov, K.N. Makarov, A.V. Pertzov and E.D. Shchukin, J. Colloid Interface

Sci., 1990, 138, 98.

Thin Liquid Films, Fundamentals and Applications, Ed. I.B. Ivanov, Surfactant Science

Series, Vol. 29, Marcel Dekker Inc., New York, 1988.

K.P. Velikov, O.D. Velev, K.G. Marinova and G.N. Constantinides, J. Chem. Soc.

Faraday Trans., 1997, 93, 2069.

K.D. Danov, D.N. Petsev, N.D. Denkov and R. Borwankar, J. Chem. Phys., 1993, 99,

7179.

Intermolecular and Surface Forces, J.N. Israelachvili, 2 "d Ed., Academic Press, London,

1992.

D.B. Hough and L.R. White, Adv. Colloid Interface Sci., 1980, 14, 3.

K.G. Marinova, R.G. Alargova, N.D. Denkov, O.D. Velev, D.N. Petsev, I.B. Ivanov

and R.P. Borwankar, Langmuir, 1996, 12, 2045.

21.

22.

23.

24.

25.

26.

27.

28.

29.

30.

31

32.

33

34

35

36

37

38

39

40.

41

42.

43

44.

45.

46.

589

E.D. Manev and R.J. Pugh, Langmuir, 1991, 7, 2253.

The Colloidal Domain where Chemistry, Physics and Technology Meet., D.F. Evans and H. Wennerstrom, VCH, New York, 1994.

D.Y.C. Chan, R.M. Pashley and L.R. White, J. Colloid Interface Sci., 1980, 77, 283.

V.N. Paunov, R.I. Dimova, P.A. Kralchevsky, G. Broze and A. Mehreteab, J. Colloid

Interface Sci., 1996, 182, 239.

E.A.G. Aniansson, J. Phys. Chem., 1978, 82, 2805.

Physicochemical Hydrodynamics, V. Levich, Prentice Hall, Eaglewood Cliffs, N.J.,

1962.

Th.F. Tadros, Chap. 6, Steric Interactions in Thin Liquid Films, in ref. 15, pp. 331.

I.G. Lyle and G.J.T. Tiddy, Chem. Phys. Lett., 1986, 124, 432.

S. Asakura and F. Oosawa, J. Chem. Phys., 1954, 22, 1255.

N.F. Carnahan and K.E. Starling, J. Chem. Phys., 1969, 51, 635.

M.P. Aronson, Langmuir, 1989, 5,494.

B.P. Binks, P.D.I. Fletcher and D.I. Horsup, Colloids and Surfaces, 1991, 61, 291.

P.A. Kralchevsky and N.D. Denkov, Chem. Phys. Lett., 1995, 240, 385.

E.D. Manev, S.V. Sazdanova and D.T. Wasan, J. Dispersion Sci. and Technol., 1984, 5, 111.

A.D. Nikolov and D.T. Wasan, J. Colloid Interface Sci., 1989, 133, 1.

V. Bergeron and C.J. Radke, Langmuir, 1992, 8, 3020.

A.D. Nikolov and D.T. Wasan, Langmuir, 1992, 8, 2985.

O. Krichevsky and J. Stavans, Phys. Rev. E, 1997, 55, 1.

G. Stalidis, A. Avranis and D. Jannakoudakis, J. Colloid Interface Sci., 1990, 135, 313.

A. Avranis and G. Stalidis, J. Colloid Interface Sci., 1991, 143, 180.

A. Sheludko, Adv. Colloid Interface Sci., 1967, 1,391.

K.J. Mysels and M.N. Jones, Disc. of the Faraday Soc., 1966, No. 42, 42.

J.S. Clunie, J.F. Goodman, and B.T. Ingham, in Surface & Colloid Science, Ed.

Matijevic, E., Vol. 3, Wiley, New York, 1971.

Coagulation and Stability of Disperse Systems., H. Sonntag, and K. Strenge, (translated

by R. Kondor), Halsted Press, New York, 1972.

R. Aveyard and B. Vincent, Prog. in Surface Sci., 1977, 8, 59.

D. Exerowa, D. Kashchiev, and D. Platikanov, Adv. Colloid Interface Sci., 1992, 40,

201.

590

47.

48

49

50

51

52

53.

54.

55.

56.

57.

58.

59.

60.

61.

62.

63.

64.

65.

66.

67.

H. Sonntag, N. Buske and H. Fruhner, Kolloid-Z. Z. Polym. 1972, 250, 330.

L. Lobo and D.T. Wasan, Langmuir, 1993, 9, 1668.

V. Bergeron and C. Radke, J. Colloid Polym. Sci., 1995, 273, 165.

M. van den Tempel, J. Colloid Sci., 1958, 13, 125.

H. Sonntag, J. Netzel and H. Klare, Kolloid Zeit. and Zeit. Polymere, 1966, 211, 121.

J. Netzel, and H. Sonntag, Abh. Dtsch. Akad. Wiss. KI. Chem., Geol. Biol., 1966, 6A, 589.

D. Platikanov and E. Manev, Proceedings of the IVth International Congress on Surface

Active Substances, Volume II, Physics and Physical Chemistry of Surface Active

Substances, Ed. Overbeek, J.Th.G., Gordon and Breach, London, 1967, p. 1189.

H. Sonntag J. Netzel and B. Unterberger, Spec. Discuss. Faraday Soc., 1970, 1, 57.

T.M. Herrington, B.R. Midmore and S.S. Sahi, J. Chem. Soc. Faraday Trans. 1, 1982,

78, 2711.

L.R. Fisher and N.S. Parker, Faraday Discuss. Chem. Soc., 1986, 81,249.

I.B. Ivanov, S.V.K. Chakarova and B.I. Dimitrova, Colloids and Surfaces, 1987, 22,

311.

H.J. Muller, B.B. Balinov and D.R. Exerowa, Colloid Polym. Sci., 1988, 266, 921.

L.R. Fisher, E.E. Mitchell and N.S. Parker, J. Colloid Interface Sci., 1989, 128, 35.

O.D. Velev, T.D. Gurkov and R.P. Borwankar, J. Colloid Interface Sci., 1993, 159,

497.

O.D. Velev, A.D. Nikolov, N.D. Denkov, G. Doxastakis, V. Kiosseoglu and G. Stalidis,

Food Hydrocolloids, 1993, 7, 55.

O.D. Velev, T.D. Gurkov, Sv.K. Chakarova, B.I. Dimitrova, I.B. Ivanov and R.P.

Borwankar, Colloids and Surfaces A : Physicochemical and Engin. Aspects, 1994, 83,

43.

O.D. Velev, T.D. Gurkov, I.B. Ivanov and R.P. Borwankar, Phys. Rev. Letts., 1995, 75,

264.

O.D. Velev, G.N. Constantinides, D.G. Avraam, A.C. Payatakes and R.P. Borwankar, J.

Colloid Interface Sci., 1995, 175, 68.

K. Koczo, A.D. Nikolov, D.T. Wasan, R.P. Borwankar and A. Gonsalves, J. Colloid

Interface Sci., 1996, 178, 694.

G.W. Francis, L.R. Fisher, R.A. Gamble and D. Gingell, J. Cell Sci., 1987, 87, 519.

R. Aveyard, B.P. Binks, W-G. Cho, L.R. Fisher, P.D.I. Fletcher and F. Klinkhammer,

Langmuir, 1996, 12, 6561.

68.

69.

70.

71

72.

73

74.

75

76.

77.

78

79

o

<a>

a

A

b

C

d

D

e

E

F

f~

g

h

H

k

P08II1

px

r

591

W-G. Cho and P.D.I. Fletcher, J. Chem. Soc. Faraday Trans., 1997, 93, 1389.

B.P. Binks, W-G. Cho and P.D.I. Fletcher, Langmuir, submitted for publication.

H.M. Princen, M.P. Aronson and J.C. Moser, J. Colloid Interface Sci., 1980, 75, 246.

M.P. Aronson and H.M. Princen, Nature, 1980, 286, 370.

M.P. Aronson and H.M. Princen, Colloids and Surfaces, 1982, 4, 173.

H.M. Princen, Colloids and Surfaces, 1984, 9, 47.

J.A. de Feijter, J.B. Rijnbout and A. Vrij, J. Colloid Interface Sci., 1978, 64, 258.

J.A. de Feijter and A. Vrij, J. Colloid Interface Sci., 1978, 64, 269.

J. Bibette, D.C. Morse, T.A. Witten and D.A. Weitz, Phys. Rev. Lett., 1992, 69, 2439.

J. Bibette, Langmuir, 1992, 8, 3178.

H.M. Princen, Langmuir, 1986, 2, 519.

D.N. Petsev and J. Bibette, Langmuir, 1995, 11, 1075.

LIST OF SYMBOLS

mean emulsion drop radius

emulsion drop radius

Hamaker constant

undeformed oil drop height

salt concentration

diameter of surfactant aggregate

diffusion coefficient

electronic charge

energy of interaction per unit area of film

force

micropipette force constant

acceleration due to gravity

film thickness

manometer pressure head

Boltzmann constant

osmotic pressure

micropipette vertical position

radius of emulsion film perimeter

592

R

go

S

Soo

T

t

V

V

v~

Z

II

to

7

n

K

0

Gas constant

drop radius of curvature outside the film region (osmotic compression method)

internal radius of micropipette tip

radius of curvature of emulsion film

initial, undeformed emulsion drop radius (osmotic compression method)

radius of curvature of oil drop in LSFA

solubility (drop radius dependent)

solubility

absolute temperature

time

energy of interaction of approaching drops

velocity of sedimentation or creaming

molar volume

vertical distance between micropipette tip and emulsion film perimeter

vertical distance between emulsion film perimeter and planar oil-water interface

film disjoining pressure

relative permittivity ( dielectric constant)

permittivity of free space

volume fraction

oil-water interfacial tension

viscosity

reciprocal of the Debye screening length

angle of meniscus to vertical

density

film contact angle

surface potential

Drops and Bubbles in Interfacial Research D. M6bius and R. Miller (Editors) �9 1998 Elsevier Science B.V. All rights reserved.

FROM STALAGMOMETRY TO MULTIANALYSER TENSIOGRAPHY: THE

DEFINITION OF THE INSTRUMENTAL~ SOFTWARE AND ANALYTICAL

REQUIREMENTS FOR A NEW DEPARTURE IN DROP ANALYSIS.

593

N.D. McMi l l an 1, V. Lawlor 1, M. Baker 1 and S. Smith 2

School of Science, Regional Technical College, Carlow, Ireland

Carl Stuart Ltd., Tallaght Industrial Park, Tallaght, Dublin 24, Ireland

Contents

.

1.1

1.1.1

1.1.2

1.2

1.2.1

1.3

2.

2.1

2.1.1

2.1.2

2.1.3

2.1.4

2.1.5

2.2.

2.2.1

2.2.2

2.2.3

Introduction

Background

Form of the Tensiograph Data

The Science of Tensiography

Instrumental Requirements

Instrumental Engineering

Test Results on Instrumental Performance and Drop Head Design

Instrument for Physical Measurements

Theoretical Background

Surface Tension

Viscosity and Molecular Weight

Absorbance and Turbidity

Refractive Index

Tensiograph Electrochemical Measurements - A Beginning

Testing of the Instrumental Capability

Surface Tension, Dynamic Surface Tension, Gas Sensing and Monitoring of

Surface-Liquid Interactions by Drop Methods

Viscosity and Molecular Weight

Turbidity and Particle Size

594

2.2.4

2.3

3.

3.1

3.1.1

3.1.2

3.1.3

3.2

3.2 1

3.2 2

3.2 3

3.2 4

3.2 5

Refractive Index

Instrumental Review or Work To date

Instrument for Fingerprint Analysis

Theoretical Background

Introductory Comments on M and I-Functions

Definition of M-functions

Weighting of M-Functions

The I-functions- Associated Statistics for tensiograph Functions

Basic Quantities and Concepts of I-functions

The I-functions for trace Times

Conceptual Basis of the Height I-functions

Relationship Between the Tensiograph and Rainbow Peak Variations

Height I-function Resolution

3.2 6 Null Conditions for the I-function Algorithms

3.2.7 Tensiograph Height I-functions

3.2.8 Rainbow Peak Height I-functions

3.2.9 Percentile Point and Area I-functions

3.2.10 t-test Analysis

3.2.11 Experimental Testing of I-function Analysis

3.3 Fingerprint and D Functions

3.3.1 Definition of the D Functions

3.3.2 Analysis using the D Functions

3.4 Testing of the Fingerprint Capability

3.4.1 Experimental Results on M-Functions

3.4.2 Experimental Results on I-functions

3.4.3

4

41

4.2

43

5

6.

7.

Applications of Fingerprinting

Summary

Physical Measurements and M-functions

I-functions for Sub-Sensitivity Analysis

Conclusions

References

List of Symbols

List of Figures

595

1. INTRODUCTION

The tensiograph is a new instrument that has been developed on the principles of the old

established stalagmometric instruments recently advanced most notably by Miller et al. [ 1 ] and

his group. McMillan et al [2] have discussed work on both an LED single bandwidth and a

multi-wavelength tensiograph based on the University College Dublin eleven fibre portable

multi-channel spectrometer (PMS). This study was concerned solely with the use of the

instrument for the analysis of disease in synovial fluid.

The first part of this chapter deals with the interpretation of signals obtained from the

tensiograph (called elsewhere the fibre drop analyser) with liquid delivered to the drop head by a

stepper pump to produce what is called here the tensiotrace. It describes fully the apparatus and

measurements made with the instrument using both stepper pump and constant head liquid

delivery system to obtain a number of physical and chemical measurements with a basic drop

head. It also details the promising investigations into modification of the apparatus to provide an

electric field around the head to carry out some preliminary work on electrochemical

measurements of liquids.

In this part of the chapter the studies by McMillan et al. [3] into the use of a second method of

tensiograph analysis based on the interpretation of the trace obtained from a vibration of the

drop are described for relatively low viscosity samples. The vibration drop trace (vdt) is

produced by a mechanical shock delivered to the drop head. This trace should also be able to

fingerprint a sample as the vdt appears to provide a unique signature of the liquid under test. In

low viscosity liquids a vibration trace occurs at the start of an tensiotrace resulting from the

oscillation of the drop after it separates from the drop head. In their most recent communication

McMillan et al [4] have described a fully engineered system capable of both tensiotrace and vdt

analysis.

In the second part of this chapter a detailed description is provided of the theory fingerprinting

tensiotraces together with the software algorithms developed to execute this analysis. The

fingerprinting is based on M(matching), D(discrimination) and I(indicator) functions that are

defined and an example given of the application of each function to a real industrial problem.

596

The instrument has been used now in a mJmber of applications areas other than body fluid

analysis, namely, sugar analysis [5], brewing [6], distilling [7] and pollution monitoring [8]. It is

clear from current work that there are a wide range of other potential application areas for the

tensiograph such as in polymer science and particularly for adhesive manufacture; in food

analysis used to analyse oils and other liquid products; in pharmaceutical fingerprinting for the

forensic identification of drugs; and finally ink monitoring and manufacture probably for quality

control applications.

A practical software engineering solution for the instrument has slowly emerged through an

evolutionary process of instrument development and software implementation. It is believed that

there now exists an acceptable user interface for the tensiograph which has been briefly

described in a recent paper by McMillan et al. [9]. Given that the mechanical engineering and

software engineering objectives have now been met, it is hoped that the tensiograph will shortly

begin to find widespread applications in industry. The present template of software will greatly

facilitate the rapid tailoring of software foi any later targeted industrial, medical or scientific

application. This software development for the existing tensiograph will shortly be described by

McMillan et al. [ 10] providing the various flow diagrams of the program development and the

working algorithms for the data capture, data analysis and user interface which will include a

discussion of the use of neural networks in tensiographic applications.

1.1 BA CKGR 0 UND

1.1.1 Form of the Tensiograph Data

The tensiograph is an instrument based on the fibre drop head for the simplest case of a two

fibre system. Light from a tungsten or LED source is injected into the drop through the source

fibre and the signal picked up by the collector fibre is delivered to a phototransistor/photodiode

or CCD detector at the end of the second collector fibre.

The tensiograph operates by recording just one single tensiotrace which is scissored from the

incoming MD detector signal produced from the light collected from the collector fibre. This

signature trace is obtained by recording the opto-electronic signal between the falling of two

drops from the head. This is achieved using software control to extract this information from the

temporal signal. The data acquisition is triggered by the control signal of the opto-eyes. To

597

achieve this scissoring a "trigger drop" is formed and falls from the drop head and the opto-eyes

which are situated below the drop head are initially triggered to begin the data acquisition. The

recording of the signal received at the detector on the end of the source fibre then proceeds until

the second drop, the "measurement drop" falls. The data for this measurement drop is then

stored in the archival system of the computer after conversion to a digital form by the AID card.

The trace recorded for just one single drop is known as the tensiotrace and this is a unique

fingerprint of the liquid.

\ Signal (volts)

Tensiograph Peak

Rainbow Peak (colour) Shoulder (refractive index) Peak

Separation Vibration

~ (viscosity)

'%

.............................................. Drop period (surface tension & density)

Time (seconds)

Fig. 1. Typical tensiotrace showing the characteristic features.

The principal features of the tensiotrace are illustrated by Figure 1 which shows the data

obtained for a water sample. Water is taken as the reference liquid for most applications as with

most other established analytical techniques. This figure shows a typical steady drop growth

tensiotrace displaying all the principal features of a tensiotraces. The principal features of the

trace seen here are the separation vibration, rainbow peak, the tensiograph peak and drop

period. The drop head can be designed to remove any coupling of the signal from the initial

period of drop growth. Such a head then produces a signal lit~ off from the base line at the

598

rainbow peak which may be advantageous as this produces a well defined analysis point. This

figure indicates what information can be obtained from the various features of the tensiotrace.

The principal drop head design used in the present study is referred to as the concave polymer

drop head. Detailed design drawings of the head are shown in Figure 2(b). The design employ

standard 1 mm polymer fibre which are polished using 0.3 micron diamond paper before

mounting in the drop head.

I I I

{a}

C

b a

] 1

b |

El '

(c)

! I

(b}

Fig. 2 a) Drop head design for polymer fibre cylindrical head. b) Drop head design for polymer fibre concave

drop head. c) Drop head design for silica fibre drop head.

The size of the drop head is very critical for the present work of multi-measurand

characterisation. The diameter of the head is 9mm with the fibres separated at 6mm. The fibres

are positioned with a jig to give a standard tensiotrace and just protrude a small distance from

the concave base. A HPLC capillary is used to deliver the liquid to the head which is glued into

the head at the centre. Ideally the head should be designed such that it wets (i.e. the suspended

liquid covers the entire lower surface of the drop head) when liquid is delivered to this head

599

without any mechanical stimulation or human intervention. The second desirable characteristic

of a drop head for multi-measurand analysis is that the tensiotrace should exhibit both rainbow

and tensiograph peaks for all liquids, and not just tensiograph peaks, as is the usual situation for

any badly designed head. For alcohols and similar liquids which produce drops shaped like an

inverted bell curve then the trace exhibits only a large rainbow peak and a small portion of the

tensiograph peak. These traces are very different to those of water which have the long water

pendant drop shape that produce the standard type of trace described above. However, the

alcohol trace can be made to look more like that of water trace simply by the expedient of

moving the fibres closer to the centre of the drop head.

1.1.2 The Science of Tensiography

The science of tensiography can perhaps be simply stated as the extraction of physical, chemical

and product information from the instrumental tensiotraces. This analysis can be done largely by

the software described below. The tensiotrace is a temporal opto-electronic trace that arises

from the modulation of the light as it is coupled between source and detector fibre in the drop.

It is produced by the changes in the reflected light paths as the drop grows from the liquid

added to the drop head by the stepper pump or other supply. The liquid drop on the fibre head

develops gradually in size until it falls off to be replaced by a second growing drop and so forth

repeatedly. It is found from experiment that the tensiotraces are reproducible within usual

instrumentation and signal-to-noise limits. For measurement reasons it has been demonstrated

that the drop growth during the recording of an tensiotrace should be slow enough that the

force conditions on the drop remains in a quasi-equilibrium state throughout almost the entire

drop cycle. This condition only fails for a very small time interval involved in the drop

separation process at the end of the tensiotrace. If the quasi-equilibrium condition is maintained,

then the tensiotraces will be indistinguishable with regards to the peak heights and general

tensiotrace shape, whether a drop is delivered by the stepper pump, or alternatively by the

constant head delivery. Obviously the drop period will be different for both methods of delivery

because the flow rate to the drop head will differ for both methods. In practice this quasi-

equilibrium condition requires that the drop periods should exceed 40 seconds.

The tensiograph can provide quantitative information on the shape of the drop, its refractive

properties, its colour, and with modified instrument heads other information such as its

6OO

turbidity, electrochemical properties (including pH) and perhaps other user specific information

such as the stringability of the liquid in polymer solutions can be obtained from the standard

information or a slightly modified device. The viscosity of the liquid can be obtained in a number

of ways with gravity feed delivery systems that also allow the measurement of the molecular

weight of the liquid and also for example from studying the opto-electronic signal from liquid

drops forced into vibration by mechanical shocks or using a shaker. The existing instrument has

sottware that allows measurements to be made on drop evaporation and an environmental

control chamber allows this measurement to be made under various conditions of saturated

vapour. The environmental chamber also allows the drop to be used to measure the properties

of the surrounding vapour or gas [ 11]. The complex form of the tensiotrace is used as a

fingerprint of a liquid and recent work by McMillan et al [12] have developed substantially the

theory of fingerprint analysis which is also detailed extensively here in this chapter.

1.2 INSTRUMENTAL REO UIRF_A4ENTS , . . , ,

1.2.1 Instrument Engineering

Figure 3 shows the schematic diagram of the present design of a fully functional tensiograph

with both multi-wavelength and LED operation [ 13]. The drop head is fitted onto the end of a

steel tube filled with thermal compound shown in this figure. The drop head is pushed from

above into a temperature block which has been designed to provide also an environmental

chamber for the drops as they are delivered at the drop head. The drop head design shown here

in this figure pushes from above into the glass environmental tube that is fitted into the heater

block from below. The glass fitting is vented to the atmosphere via a tight, but not vacuum

fitting, at the top of the glass which fits against the aluminium block. This venting stops the

build up of any excess pressure as liquid is delivered to the drop head as the development of

such a pressure change would effect the drop growth shape and hence change the tensiotrace.

The tube can be charged with the specific liquid under test if the bleed valve on the glass unit is

shut to thereby produce a saturated vapour around the drop to reduce or stop the evaporation

of volatile drops. A saturated atmosphere for the measurement drop may thus be obtained by

simply squirting some liquid into the glass unit before the measurement with the tap of the glass

unit closed and then leaving for an appropriate time for the volume to become saturated. It is

601

clearly important that the fitting between the glass block and the aluminium is tight enough to

maintain the saturated vapour.

The liquid can be delivered to the drop head via a Paar Scientific DPRT density meter. The

density meter may be housed in an aluminium block that is bolted onto the main temperature

block and thereby maintained at the same temperature as the drop head. Alternatively the

density measurement can be made at ambient and a correction applied to this to obtain the value

at the measurement temperature. For some applications the DPRT is by-passed and the liquid

sample is delivered directly to the drop head from the stepper motor pump. Directional valves

V1 and V2 shown in Figure 3 enables a manual selection of the sample delivery path to be made

so the sample can be delivered either to the drop head directly or via the DPRT. The liquid can

alternatively be delivered from the constant head apparatus by selecting the appropriate position

of valve V3. A short piece of Teflon HPLC 2.8 mm Teflon tubing is fitted above the drop head

which ensures that the liquid flow from the constant head is such that even for the maximum

head the liquid drops are slowly delivered and drop times are in excess of 40 seconds as

required for the quasi-equilibrium conditions of drop growth. The constant head can feed liquid

to the drop head under gravity feed. This involves a specially designed glass unit with an upper

chamber to which the liquid sample is added, connected to the lower constant head chamber,

which is fed from this upper reservoir and maintained at a constant head by the overflow.

The light coupled back from the detector fibre in the drop head is connected to the detector and

then amplified before being passed to a DASH 16 Junior A/D card which digitises the signal for

analysis by the computer. The signal acquisition is triggered when a drop falls by the infra-red

beam of the optical eyes being intercepted.

The optical eyes are fitted to the glass unit in the temperature block which also records the drop

period by timing drop fall events.

With the CCD system a tungsten or halogen light source is used and SMA connectors deliver

the light to the drop head with 200 micron multimode fibre. An Ocean Optics SD 1000 fibre

spectrometer is used to detector the coupled radiation picked up from reflections and other

coupling mechanisms with the liquid drop via a second collector fibre mounted in the drop head.

602

Figure 3. Diagram of the tensiograph setup based on a block heater and spring vibration damping.

The various components are all mounted in a housing as seen in Plate 1 which shows at the

front left the Hamilton Microlab M OEM stepper motor pump used to deliver the liquid sample

The sample to be measured is placed in a buzz bath situated in front of the OEM pump for

degassing before being injected into the system. The sample is situated just below the pump.

The instrument panel has controls to allows the selection of the heater block temperature and

secondly to operate of the buzz bath via a push button.

The main component pieces are mounted with Farrat RM rubber metal isolators that are loaded

in shear to damp vibrations of the drops. The heating block/environmental chamber stands on

Farrat CR11 spring mounted anti-vibration feet that are each loaded with more than 10.0 kg and

the pump is fitted to the front panel with isolation mounts. These mountings damp all vibrations

above 3.4 Hz. Since drops typically resonate at about 10Hz, the mountings afford some

vibration protection but occasionally small vibrations may be observed in the tensiotrace. These

603

vibrations result in small variations in the drop periods and some additional noise in the trace.

For sensitive measurements the pump may be removed, using the quick release screws on the

front panel, and placed on an isolation table.

Experiments were conducted with the pump isolated from the drop head which stood on a

vibration free table and with the apparatus ctescribed above using the drop period which is the

most sensitive tensiograph measurand for detecting vibrations. It was found that the standard

deviation of drops measured with both set-ups were approximately the same and it was

therefore concluded that the vibration design was effective and that no further instrumental

arrangements were necessary when using the stepper pump. This conclusion does not hold good

for gravity fed tensiography.

1.3 TEST RESULTS ON 1NSTR UMENTAL PERFORMANCE AND DROP HEAD DESIGN

Two principal drop head designs have been used by the authors which are referred to here as the

cylindrical polymer and concave polymer drop head, the former for a 7-9 mm diameter. Detailed

design drawings of the heads are shown in Figure 2(a) and (b) respectively. Both designs

employ standard 1 mm polymer fibre. It should be noted that the experimental study described

below into optimised drop head design however is also applicable to silica fibres. It is found that

the size of the drop head is critical effects the form of the tensiotrace. The best type of head

from the point of view of manufacture is the cylindrical drop head.

Ideally the head should be designed such that it wets (i.e. the suspended liquid covers the entire

lower surface of the drop head) when liquid is delivered to this head without any mechanical

stimulation or human intervention. The second required characteristic of a drop head for multi-

measurand analysis is that the tensiotrace should exhibit both rainbow and tensiograph peaks for

all liquids, and not just the single tensiograph peaks, as is the usual situation for any badly

designed head.

The ideal drop head for the instrument would be a flat cylindrical design. Such a head enables

surface tension measurements to be made that are based on the established stalagmometric

methods. The polymethylmethacralate (PMMA) fibres used for the LED-wavelength

measurement tensiograph are 1 mm in diameter and an investigation was made into the form of

the tensiotrace obtained for drop heads of various diameters with these fibres. The drop heads

were all polished flat with a specially designed polishing puck using a series of diamond papers

604

and finishing with a diamond paper of 0.1 micron size. The fibres in all cases were positioned

diametrically opposed to each other across the centre of the drop head. The drop heads were

made from nylon 66 which was found in practice to be both very resistant to contamination and

damage, but could also be easily machined. It has been found that nylon is a very good drop

head that resists most chemicals, is easily wetted and is simple to machine.

Plate 1. Tensiograph with control panel, stepper pump and buzz bath on the exterior of the housing. All the other

components are housed inside the cabinet with the exception of the constant head accessory which clips

onto the back panel of the housing when required.

Figure 4 shows a series of tensiotraces obtained at 20C for water with flat drop heads of 5.5, 6

and 7mm diameter. Larger flat drop head diameters are unable for all liquids to support a drop

that hangs fully around the circumference of the drop head. As can be seen by reference to the

tensiotrace shown in Figure 1 for a 9mm concave drop head, these tensiotraces from the

cylindrical drop head contain only the tensiograph peaks with no rainbow peak. In fact from the

series of tensiotrace shows that the coupling between the fibres for these drop heads

progressively develops the so called tensiograph peak, from a rather structureless peak in the

5.5 mm head, to one with a structural shoulder in the 7mm head. These peaks have in earlier

papers been attributed by McMillan, Fortune et al [ 14] to different total internal reflection paths

of coupled rays between the source and collector fibres.

605

5.Smm lo!

0 1000 2000 3000 time

6mm

1!f

7mm 10

8

0 1000 2000 3000 0 1000 2000 3000 time time

Fig. 4. Water tensiotraces at 20C recorded on cylindrical drop heads with 5.5, 6 and 7mm diameters.

6mm drop heads 40% ethanol

0 0 500 1000

time 60% ethanol

40% ethanol

_, Qt 0 500 1000

time 60% ethanol

0 0 500 1000

time 100% ethanol

okA'---J 0 500 1000

time 100% ethanol

D ~5 5

0 0 0 500 1000 0 500 1000

time time

Drop head A Drop head B

Figure 5. Series of tensiotraces for 40, 60 and 100% v/v solutions of ethanol and water for two of the drop head

different drop heads made to the same specifications showing variation in form of trace.

A rainbow peak was however obtained for a 6 and 7 mm flat drop head for aqueous solutions of

ethanol. The tensiotraces recorded for 40, 60 and 100% v/v aqueous ethanol samples are shown

in Figure 5 recorded not with Siemens detectors but here with a 850nm Honeywell sweetspot

LED-detector pair taken on the 6 mm drop head. As can be seen in these tensiotraces, a

pronounced rainbow peak has developed and this peak demonstrates the characteristic

behaviour reported in earlier work, namely that its size is determined by refractive index. Later

606

in this chapter a detailed discussion is given on the dependence of peak sizes with refractive

index and absorbance.

In this investigation of drop head manufacture the limiting size of the concave drop heads was

found to be 9 mm diameter because the drops for any larger drop head diameters start to touch

the side of the environmental tube in the heater block and therefore will not fall properly to

intercept the opto-switch eye used to measure accurately drop periods. All concave drop heads

wet as all liquids are forced to the edge of the drop head by the hollow under edge of the drop

head. The concave drop heads however unfortunately have a greater problem with trapped

bubbles with regards to the cylindrical drop head. It is therefore absolutely necessary to degas

samples when working with concave drop heads.

It was discovered from experiments with the concave drop heads, that for the large majority

liquids a rainbow peak is present in the tensiotrace. For this reason the concave drop head has

been selected in the initial design of the tensiograph as the drop head for general use. This

decision was arrived at despite the fact only the cylindrical drop head can be employed for

stalagmometric measurements of surface tension.

An experimental investigation of the present method of drop head manufacture for cylindrical

drop heads was made using as a test of reproducibility for manufacture, the ratio of the rainbow

and tensiograph peaks for a series of v/v % concentrations for ethanol solutions on two

identically manufactured 6 mm cylindrical drop heads. The actual procedure of manufacture was

as follows, the turning was done on a precision collet lathe and the holes were subsequently

drilled with a digital readout Bridgeport milling machine with a special fitting with a special 180 ~

movement made to produce diametrically opposed fibre holes in the drop head. The accuracy of

the machine itself is +0.005 ram. It is believed that with the movement of the very fine drill that

the holes are positioned to an accuracy better than + 0.02 mm.

Another major advantage of the flat drop head was hoped to be that they would be readily

manufactured to produce identical tensiotraces for a specific test liquid. Drop heads could then

be made to be disposable. In this practical investigation into the engineering of reproducible

drop heads it was found that the relative sizes of the peaks in the resulting tensiotraces were

markedly different. The differences were a large factor of more than 10% in peak heights

607

between the tensiotraces for the two drop heads. It therefore follows that improved methods

such as injection moulding are necessary for the manufacture of disposable drop heads. At this

point of instrumental development of the tensiograph if the drop head is changed then the

measurement conditions are disturbed and comparisons can not then be made with any

quantitative reproducibility. This situation is analogous to that in chromatography with regard to

columns.

In summary therefore, it has been demonstrated that it is not possible to produce wetting flat

(both PMMA polymer and silica) fibre drop heads that universally exhibit a rainbow peak as the

diameters of such heads would be too large to support drops. The only practical design of drop

head for refractive measurement capability therefore has been found to be the concave drop

head described above.

2. INSTRUMENT FOR PHYSICAL MEASUREMENTS

2.1 THEORETICAL BA CKGRO UND

2.1.1 Surface Tension

The usual stalagmometric method [15] of measuring surface tension can be used with the

tensiographic method but these methods require the use of a cylindrical drop head.

Investigations into the form of the tensiotrace produced by a range of drop head diameters by

McMillan et al [16] for multimode fibre showed for all liquids that no cylindrical drop head

existed which produced traces containing both rainbow and tensiograph peaks. For all diameters

however tensiograph peaks were obtained.

These stalagmometric methods are subject to considerable error which can be as much as

0.25%. In more recent times [ 17] there has been a renewed interest in the technique with efforts

to improve the range of applications and also the correction factors used in these techniques.

The corrections that apply to cylindrical drop heads do not apply to concave drop heads and the

tensiography requires the use of calibrations for standard solutions to produce curves of !

measured drop times versus surface tension. The advantage of such a method however is that

the accuracy of electronic/computer timing can then be exploited to the full and it is found that

608

accuracies of better than 0.01% have been obtained from calibrations based on measurement

reproducibilities of 0.005%.

All the standard methods of measuring dynamic surface tension are available to the tensiograph

as the standard types of cylindrical drop heads can be used with this instrument. The recent

major advances in dynamic surface tension instrumentation are due to such people as Miller et

al. [ 18]. Obviously these instrumental advances are highly relevant to the potential development

of the tensiograph for such dynamic studies. The fact that the tensiograph gives a graph (even

when this is only the single tensiograph peak) rather than simply a measurement of drop period

means that it is capable of monitoring temporal changes of the drop shape as the surfactant

molecules arrive at the surface of the drop. The analysis of the tensiotrace therefore has the

potential to provide subtle dynamic surface tension information that may supplement the limited

drop period information of the existing instruments.

2.1.2 Viscosity and Molecular Weight

2.1.2.1 Gravity Feed Studies - The Theory of a New Viscometric Effect in Drop Science

The viscosity of the sample can be obtained from the drop period of the liquid delivered from a

constant head delivery system following the measurement of drop volume from the stepper

pump controlled liquid delivery. The drop volume of both the pump delivered and gravity

delivered drops are the same for slowly delivered drops. From the drop period of the gravity fed

system the flow rate can then be obtained once the drop volume is known. The flow to the drop

head in the gravity feed is principally determined by the impedance of the capillary which

delivers the liquid to the drop head and provides a laminar flow which is principally determined

by the viscosity of the test liquid. McMillan et al. [19] found a new flow dependence in their

system produced by the drop curvature and the resulting Laplace-Young force which slightly

modifies this uniform gravity flow.

The delivery from a constant head is in essence determined by the impedance offered to the flow

by the PEEK capillary connecting the drop head to the liquid supply and this forces the sample

delivered to the head into the laminar flow regime. Using some simple assumptions of

Poiseuille's type flow and assuming basic stalagmometric equations for slow drop delivery with

609

drop periods (T1) in excess of 40s. McMi!lan et al. [20] and in an unpublished report [21]

obtained the general form of flow to be of the form;

-1/3 2/3 1/3 -1

1/~t = (q /k l ) ( l+k3(3 /4rc ) [ 0.148 VD + 0.279 r VD - 0.166 r ] ) (1)

where 1/g the kinematic fluidity measured in reciprocal viscosity units m2s "l, q is the flow rate

ml s 1, VD is the drop volume in ~_tl, r h is the drop head radius measured in mm, and the kl, kz,

and k3 are all constants.

This equation shows that the delivery is determined not only by the gravity head but is modified

by the Laplace-Young pressure which produces here the drop volume dependence. The

constants in this equation include the gravity head which is constant throughout. The flow from

the constant head is modulated by the Laplace-Young pressure of the drop and specifically with

drop radius as this varies throughout the drop cycle. Consequently, the flow varies throughout

the drop cycle because of this variable Laplace pressure. This pressure variation can however be

approximated to a constant flow integrated over the drop head period corresponding to an

average drop radius for the drop period. The drop radius depends physically (with slow drop

delivery) on only the physical ratio ~,/19, where ~ is the surface tension measured in mN/m and P 3

is the density measured in kg/m

The complex nature of the flow dependence here does not commend this approach for general

drop viscometry, but this method may find application in specific applications such as synovial

fluid study were it might be employed in the determination of molecular weight of biological

macromolecules(also this may find application in polymer science) by the Mark Houwink

method [22] since this method requires exceedingly dilute concentrations of the sample in a

solvent. The Mark Houwink equation is;

[ 1"1 ] = k MI3 (2)

where [ rl ] is the limiting viscosity number which is defined by [r/] - Limit~ -~ 0[(r//r/0-1)]/c,

and c is the concentration kg/m 3, 1"1 is a the absolute viscosity in Pa s and the subscript indicates

the measurement on the solvent, M is the molecular weight, k is an empirically derived constant

with units m3kg 1. 13 is the Mark Houwink constant which is dimensionless for such molecular

610

weight measurements the surface properties and the density of the liquid would in effect be

constant since the sample is in a highly diluted state. Consequently, it appears that the drop

period would then only be a function of the viscosity. This technique may perhaps be of some

relevance to the study of synovial fluid since this contains large biological molecules and can

only be obtained in small volumes. The sample must be diluted for this measurement.

The reproducibility for such a measurement of molecular weight using this technique would be

better than 0.1% with a best value of 0.025 % given that calibration standards of the requisite

quality were available. The accuracy would be approximately the same as for the viscosity

measurements.

Figure 6. Tensiotrace of water recorded with a concave drop head. The liquid delivery has been stopped at a

position half way up lhc leading edge of the tensiograph peak and the drop hammer used six times to

excite vibration tensiolraces.

2.1.2.2 Possibilities for Vibrating Drop Fibre Rheometry

A large body of work exists on this facet of drop science of which the work of Lu and Apfel

[23] is perhaps a good example. However the theoretical problem of modelling this complex

behaviour of drops is very daunting. The present work suggests that there are ways of

611

circumnavigating around the formidable theoretical problems by analysing with FFT methods

the vibration traces in order to isolate specific frequencies. From studying these individual signal

components approaches have been suggested that simplify the problem.

Figure 6 shows the tensiotrace of water at 20C with the two important characteristics of the

rainbow and tensiograph peaks. The liquid delivery here has been stopped atter a single trace at

a point half way up the leading edge of the tensiograph peak. A series of six vdts can be seen

that are produced by repeated mechanical shocks.

The vibration that follow the drop separation peak is referred to as the separation vibration and

results from the modulation of opto-electronic coupling between the source and collector fibre

resulting from the damped vibrations of the remnant drop. The vdts produced by mechanical

shocks with the liquid delivery stopped at a position corresponding to the leading edge of the

tensiograph peak seen in Figure 6 are referred to as tensiotrace vibrations. Similar traces can be

produced with the drop volume set such that the signal is positioned half way up the rainbow

peak and these vibrations are referred to as rainbow vibrations. Vibrations are only observed in

liquids with relatively low viscosity approximately below 50 cP, but the single separation peak is

seen in most tensiotraces with these concave drop heads as this peak arises from the drop

separation process rather than any drop vibration. This peak is observed even in quite viscous

liquids such as glycerine provided that the drop head is level.

The separation vibration has the form approximately of a half wave rectified tensiograph

vibration as may be seen from this figure. Closer examination shows there are significant

differences however between these traces and the vdt. This vibration has an asymmetric form

because it appears on the base line of the tensiotrace and produces vibrations that are rather

more complex structure than either the tensiotrace or rainbow vibrations.

From the experimental evidence collected in this study and by mathematical fitting of the data

for the tensiograph and rainbow vibrations it would appear that the experimental data in all

cases can be fitted by the Loran function which has a slightly modified form to the standard

form that is given by

S = S O t 2 exp(- ~t) [A sin( ~t)] [A' sin( o~'t)] [A 1 sin( COlt)] [A n sin( COnt)] [A m sin( COmt)l (3)

612

where S is the measured signal and S O is the reference signal with both measured in volts; A is

the amplitude of the first dominant harmonic component and, A' the amplitude of the second

dominant harmonic that are seen to beat in some signals; A1 and 2% are the amplitude of the 1st

and nth components with frequencies above the fundamental that have angular frequencies of c0~

and COn; Am is the amplitude of the low frequency modulation component that have angular

frequencies of C0m; t 2 is the function which gives the rising feature on the front edge of the

envelope; and ot is the decay constant which determines the rate of decay of the envelope on its

trailing edge.

In many cases the fit to these vdts are obtained with only the first three terms and one

modulation frequency or as with the case of the envelopes that exhibit beat patterns with only

the first four terms.

2.1.3 Absorbance and Turbidity

2.1.3.1 LED Measurement

The shape of a slowly developing pendant drop is determined only by the surface tension and

gravitational forces acting on the drop [24]. Light injected into a drop from a LED source fibre

this emerges from the flat end of the fibre almost as a uniform cone of light. The angle of this

cone is determined the numerical aperture of the fibre and this in turn is determined here by the

refractive index of the liquid. Light paths around inside the drop have been studied with camera

systems by McMillan et al. [25] and tensiotrace is the result primarily of a series of total internal

reflection (TIR) reflections inside the drop which were qualitatively described in this early study.

Both the drop shapes and final drop voh,~me vary from sample to sample, along with the

numerical aperture which is determined by the refractive index of the sample liquids. It follows

that there can therefore be no one fixed and equivalent measurement position in the drop cyle

for all drops. What is required for a perfect absorbance reading position in the drop cycle would

be for the path lengths of light coupling between the source and detector fibre to be identical for

both reference and test liquids. Theoretically we know that such an equivalence is impossible. It

has been found never-the-less that it is possible to perform quantitative spectroscopic

measurements of high accuracy and repeatability with nearly all the liquid systems tested to date

613

despite this variability in drop shape and size. The measurement position adopted has been the

tensiograph peak maxima which works very well as a practical solution.

The basic procedure therefore for measurement of tensiograph absorbance is to assume(an

approximation) that the tensiograph peak maxima corresponds to an equivalent position from

drop to drop. The peak height is shown in Figure 7 and a tensiograph absorbance is defined

thus;

A = log(HR/HM) (4)

where HR is the size of the tensiograph peak for the reference liquid and HM is the size of the

tensiograph peak for the measurement trace. Both quantities are measured in volts.

It is convenient to work with Tensiograph Units (TUs) in these measurements which are defined

as follows;

TU = HM / HR (5)

where the subscripts M and R corresponds respectively to the measurement liquid and reference

liquids respectively.

Figure 8a shows a graph of rainbow peak height of sucrose solutions measured at 20C in TUs

plotted against standard refractive index values at 589nm. Figure 8b of rainbow peak TUs

versus refractive index for these seven hydrocarbons shows that this refractive index variation of

the rainbow peak is a general dependence for non-absorbing liquids. These result has been

confirmed by other workers and many subsequent measurements. Absorbance complicates this

simple refractive index relationship with TUs and this is dealt with below. It is obvious that both

absorbance and turbidity can be defined in terms of reciprocal TUs.

It is found that despite the very small variability in drop shapes explained above that the size of

the tensiograph absorbance obeys very closely the Beer Lambert Law where;

- d l = ( 6 ) A( t ) e. c IA,,p~hs log(TU)

here ~ is the tensiograph molar absorptivity and the integral represents the sum of the path

length connecting the source and collector fibres at the tensiograph peak maximum. Here c is

the concentration measured in moles.

614

Fig. 7 Convention for heights and times of peaks used in the tensiograph analysis.

2 4

2 . 1 5

I - ~ " 1 .9 e. .$ z

1 6 5 D.

.a 1 . 4

1 . 1 5

0 . 9 I ~ ~ I ~ I

1 3 3 1 3 3 5 1 3 4 1 3 4 5 1 3 5 1 3 5 5 1 . 3 6

R e f r a c t i v e I n d e x

1 . 3 6 5

Fig. 8a Measured values of rainbow peak height of sucrose solutions at 20C plotted against refractive index for

a 950nm tensiograph.

615

7 -

6 -

5 - t-,

" ~ 4 - 8. ~: 3- o r "~ 2- iv,

1

0

i;

I t t I I I

1.3 1.32 1.34 1.36 1.38 1.4 1.42 1.44

Refractive Index

Fig. 8b Measured values of rainbow peak height for various liquids at 20C plotted against refractive index for

950nm LED tensiograph.

The situation is very similar if their are scattering centres in the solution and a tensiograph

turbidity can be defined thus;

A(t)- ('C.C)IAUpa,h.,dl --1og(TU) (7)

where "t is the tensiograph turbidity coefficient.

2.1.3.2 CCD Measurement

The PMS-tensiograph can be used as a comparative spectrophotometer. The transmitting fibre

can illuminate the drop with white light and the collecting fibre, connected to the PMS observes

any absorbing or fluorescence effects. As in the case of a spectrophotometer, comparative

measurements against suitable references give the best sensitivity and stability. To illustrate the

capability for comparative spectro-photometry in the drop, measurements have been made on a

series of solutions of rhodamine-b dye in water.

The PMS has an effective bandwidth of 5 nm; it measures at all wavelengths simultaneously, and

its multi-fibre inputs can be utilised to simultaneously monitor the illuminating light source. The

PMS measurements are compared with absorbance change measured on a Pye Unicam SPG

616

Spectrophotometer. The measurement sensitivity of the PMS- tensiograph can thereby be

referenced to absorbance units. The features in the rhodamine-b spectra used in this calibration

testing of the PMS-tensiograph are the fluorescence at 600 nm and the water absorbance band

at 550 nm which when taken in a ratio gives a direct means of monitoring the concentration of

the dye by spectral radiometric methods. Zalloum et al. [26] have demonstrated the use of

similar techniques for determining the level of rhodamine-b in remote sensing applications in

esturine water.

2.1.4 Refractive Index

In the LED-tensiograph described by McMillan, Fortune et al [27] with the 1 mm PMMA fibre

pairs set close to the edge of the drop head, there existed two distinct types of peaks in the

tensiotrace, the rainbow peak and the tensiograph peaks. A typical tensiotrace for the concave

drop head is shown above in Figure 1 with all the various important features of this trace

labelled. The rainbow peak appears early in the trace and usually follow closely the separation

peak which marks the beginning of the trace and which is associated with the drop separation.

In low viscosity samples vibrations occur in tensiotrace following the drop separation and these

are referred to as separation vibrations.

The rainbow and tensiograph peaks can be seen also in the tensiotrace for all the six other

liquids shown in Figure 9. The liquids are (a) 40% v/v ammonium chloride (b) ethylene glycol

(c) 20% v/v acetic acid (d) 20% v/v methanol and water (e) 60% v/v ethanol and water (f)

acetic acid. With the LED-tensiograph the two distinct features of the rainbow and tensiograph

peaks seen in all these six tensiotrace have been shown to be generally associated with the

remnant drop phase and the pendant drop phase. The first tensiograph studies reported that

there exists a general dependence of refractive index with the ratio of the rainbow peak height

measured as a ratio of peak heights in the reference and sample liquids.

The explanation offered by these authors for the form of these tensiotraces was based on an

analysis of simultaneously recording images of drop shapes and a tensiotrace. This imaging

study provided a qualitative understanding of the tensiotrace in terms of the types of couplings

which were based on descriptions of ray couplings between the source and collector fibres. The

most significant peaks in the tensiotrace opto-electronic signal arise from T1R couplings. The

617

main portion of the rainbow peak was shown to be produced in the remnant drop phase by a

symmetric three TIR coupling.

Fig. 9. Tensiotraces for a series of six liquids recorded at 20C on the 950nm LED-tensiograph (a) 40% v/v

ammonium chloride (b) ethylene glycol (c) 20% v/v acetic acid (d) 20% v/v/methanol-water (e) 60%

v/v ethanol-water (f) acetic acid.

618

An interesting and important result was obtained from an imaging shape analysis of a number of

remnant drops from different liquids. This study revealed that all remnant drops had shapes with

no measurable differences when studied using a standard PAL CCD system. This observation

can be understood from simple physical force considerations, since in the remnant drop the

surface forces dominate the drop gravitational forces. The practical implications of this

observation are significant remembering that the rainbow peak occurs as a rule in the remnant

drop. If this drop phase produces an indistinguishable drop shape it can be thought of as a

perfect natural cuvette. Differences in sizes of rainbow peaks therefore would in such a situation

in the first instance be controlled only by the NA of the liquid fibre system and thus the

refractive properties of the liquid. Unfortunately the situation for the standard concave drop

head is not universal one for all liquids as the rainbow peak develops only in the remnant drop

phase with drop heads that have fibres placed closer to the centre than in the standard drop head

described above. The rainbow peak was shown to be associated on its rising edge with a

symmetric triple 3R TIR reflection process within the drop, one TIR reflection of either side of

the drop and one TIR reflection off the base of the drop near its centre. A 4R reflection

develops on the falling edge as the signal begins to uncouple. The rainbow reflections are

associated with reflections from either side of the drop separated by one close to the apex of the

drop base.

The tensiograph peaks in this imaging study were shown always with water type drops(and

many other liquids) to be associated with the pendant drop phase and were also to be produced

by triple TIR reflections. The tensiograph peak coupling reflections develop after the rainbow

reflections had largely uncoupled due to drop growth. The other structure in the tensiograph

peaks(as many as five peaks have been observed) arise from higher order TIR reflections. These

tensiograph peaks typically contain three distinct peaks and are produced by TIR couplings that

arise in the pendant drop. The initial 3R TIR reflections that produced the rainbow peak are

completely decoupled and the rainbow 4R TIR are uncoupling before the tensiograph couplings

begin to develop. Generally there is an overlap of rainbow and tensiograph peak couplings as

can be seen in most of the tensiotraces shown in Figure 9. The rising edge couplings for this

peak are of the 4R TIR type. The tensiograph reflections are associated with reflections

generally from the base of the drop rather than the restricted drop centre as with the rainbow

619

reflections. The tensiograph peaks contain information on drop shape because in the pendant

drop phase the shape is determined by a balance of surface and gravitational forces rather than

just the former.

2.1.5 Tensiographic electrochemical measurements - a beginning

McMillan et al. [28] reported a very substantial drop period variation with electric field. This

original work was done without any strict control on fringing of fields with a fixed drop head

placed in a parallel plate capacitor. This work was quite satisfactory for the utilitarian purpose

of obtaining a drop period measurement that correlated with pH in a series of industrial sucrose

solutions but it was experimentally limited. This study provided a physical explanation of drop

charging by induction for the observed effect.

The presence of ions in solutions can effect a wide variety of characteristics of the solvent. This

can manifest itself as a change in viscosity, local density and surface tension [29]. The change in

viscosity can arise as a result of the solvent molecules experiencing a viscous drag as they pass

an ion in solution concentrated locally around an ion. The change in density is as a result of the

extra mass per unit volume of solution. The surface tension may change as a result of the

presence of ions at the surface of the drop. The surface charge will give rise to electrostatic

interactions at the drop surface which can effect the surface tension of the solution. All of these

effects potentially give rise to measurable changes in the tensiotrace.

The earlier reported drop period changes in an electric field have been observed in the present

work for a range of other solutions and these effects would appear to be largely produced by the

changes in surface tension of the solutions and could perhaps in a better controlled experiment

be described by a modified electrocapillarity equation [30, 31, 32] of the type well known in the

Dropping Mercury Electrode (DME). It has been observed once again that in this series of

measurements on ionic solutions and buffers, that deionised water gave the greatest change of

drop period being approximately halved (0.55) with field in the range 0 to 5 V/mm while this

effect shows a general decreasing tendency with increased ionic strength of the solutions.

The presence of either an electric field or potential around the drop should give rise to ion

movement towards or away from the surface of the drop. Such a change in surface charge will

620

give rise to associated changes in surface tension which will effect both the drop period of the

tensiotrace and the form of the peaks in the tensiotrace.

73

72.9

72.8 A

E 72.7 z

72.6 _~ 72.5

8 72.4

72.3

72.2

72.1

�9 Reference Surface Tension (mN/m)

[] Measured Surface Tension (mN/m)

" " ' -L inear (Reference Surface Tension (mN/m))

. . ,_.,__.-------- r ' -

72 I I I 0 5 10 15

Concentration w/w sucrose(%)

20

Fig. 10. Graph of standard and measured surface tension for aqueous sucrose solutions measured at 20~

2.2 TESTING OF THE INS TR UMENTAL CAPABILITY

2.2.1 Surface tension, dynamic surface tension, gas sensing and monitoring of interactions of

solid-liquid interfaces by drop methods.

The measurement capability of the tensiograph for surface tension measurements has not been

properly investigated. Principally this is because of the existence of very large body of work on

stalagmometry which is now conveniently summarised in this volume. However a reasonable

number of measurements have been taken simply to demonstrates the instruments capability

with regard to surface tension measurement. Figure 10 shows a graph of surface tension of

aqueous sucrose solutions plotted together with the values obtained from standard tables. These

measurements were taken with a standard cylindrical drop head using the opto-eyes to record

the drop periods. This specific measurement problem is included here in preference to other

measurements because this is felt to be a good test of the capability of the tensiograph for

surface tension measurements. As can be readily seen concentration over the range 0-20% w/w

concentration but these test solutions have densities that are quite different. Drop periods

621

consequently vary considerably but the measured surface tension stays fairly constant. The

biggest deviation in the set of reading is at 20% concentration and this is 0.18% which is inside

the error in the stalagmometric correction factor used here which is 0.25%.

I I I I

-5 -4 -3 -2 -1 0

Log(concetration) [g/ml]

Fig. 1 la Graph of measured surface tension of sodium lauryl sulphate at 20C against logarithm of concentration.

47

45

�9 ~ 43 o .o_. E '- ' 41

~ 39

D 37

35

} } } } }

I I I I I I I

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

T1 ^(-3/4)

Fig. 1 lb A standard McGee plot of/mole sodium lauryl sulphate at 20C obtained from measurements with the

Microlab M stepper liquid delivery system.

622

The instrument can of course be used to carry out studies of dynamic surface tension and the

software has been fully developed to allow such studies. The measurement procedure is to

determine a series of drop periods with the stepper pump delivering drops at a succession of

pump speeds incrementally increased from the slowest delivery rate. It is observed that the

pump speed varies the drop volume in solutions such as sodium lauryl sulphate (C12H25NaO4S)

as seen in Figure 11 a which shows the measurement of surface tension for this solution versus

concentration for slowly delivered drops using a standard cylindrical drop head. Figure 1 l b

shows a standard McGee plot for this sample. There is an interesting and unexplained effect

associated with the pronounced dip that occurs in this plot. This sample shows some unusual

behaviour with respect to drop repeatability for the three positions marked as A, B and C.

Repeat readings were then taken with pump speeds fixed at respectively 2.4, 6.5 and 7.5~tl/s.

For these speeds the drop effects using tetramethyl benzidine solutions as a chromogenic

collection liquid to detect pC12 -~ 900 ppbv in a time of 66 seconds.

Stalagmometry has been principally concerned with studying dynamic surface tension effects

and with the associated dynamics of molecular species at the drop surface but then only with the

resulting effects of these surface interactions at an air or vapour liquid periods stayed

approximately constant in value, in the second the drop period drifted down over six drops by

some 14% and in the last speed it drifted up by 5%. interface. Dasgupta and co-workers have

recently reported some very exciting results with regards to gas.

Recent studies conducted by McMillan et al. [35] have shown some interesting effects between

an acetyl nylon drop head surface and a commercially obtained stout. Figure 12 shows the

surface tension variability observed on a series of drops following injection of the sample onto a

newly cleaned drop head. The drop times slowly settle showing what is believed to be a process

of drop head coating with some active molecular component of the stout, most probably an

enzyme. This effect was not however observed when using acetyl nylon drop heads and the

effect can be easily reproduced as washing of the drop head produces a repeat settling of the

drop times. Once the drop times settled the readings were reasonably steady (on a sample of

Guinness a standard deviation of 0.01876 on a 65.536s drop period a ratio of 349) but these

readings were not as good as those observed without this surface coating of the head (a sample

of commercially obtained Irish Distillers whisky had a standard deviation of 0.0589 and a drop

623

period of 42.602 giving a ratio of the two of 723) using nylon 66. This effect perhaps offers a

new method of studying the active components of a beverage and may be investigated further in

the future.

67

66

65 A

"o o 64 . ~ , L

=. 63 o L

62

61

60 I I I I I I

1 3 5 7 9 11 13 15

Drop number

Fig. 12. Variation observed in successive drop periods for commercially obtained stout on a newly cleaned drop

head of acetyl nylon showing drop periods settling with incremental coating of head from some active

species in the beer.

2.2.2 Viscosity and molecular weight

2.2.2.1 Gravity Feed Studies - Experimental Observation of a New Viscometric Effect in Drop

Science

To test the physical assumptions used to derive the form of the flow equation from a gravity

feed through a capillary, and in particular the surface tension dependence of this flow as

described by Equation (1), measurements were taken on acetone-water samples because these

provide a series of samples with a very diverse range of surface tension.

Samples were delivered from the gravity feed operated from a constant head throughout and the

flow rate was determined for each using the CCD-tensiograph set up. The results in Figure 13

624

show by the curve in the dynamic or kinematic fluidity versus flow rate (1/Vt vs q) plot revealing

a very marked surface tension dependence as revealed by the curve for the series of labelled

acetone-water results as the values move between the two end-points surface tension(20-30

mN/m) sit close to the full line. Intermediate solutions sit between these two extremes.

2.5

A 2

O 4 i

<

E 1.5

. i

" O ~

. i to=.

o 1 ~

E c -

a 0.5

r

J "'I

./,." ...--." . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0 2 4 6 8 10

Flow rate (micro-litres per sec) Fig. 13. Plot of dynamic fluidity versus flow rate for a series of liquids measured at 20~ representing

respectively water (0%) and acetone (100%). Liquids of relatively high surface tension (70-50mN/m)

such as water sit close to the dotted line shown here and the low surface tension(20-30mN/m) sit close

to the full line. Intermediate surface tension solutions sit between the two extremes.

It has been found experimentally that the samples being delivered from the gravity feed produce

repeatable drop period (To) measurements to within the resolution of the computer timer of a

286 PC. This very high time precision shows that the tensiograph is potentially a very powerful

analytical technique.

A range of 21 samples were measured, namely water, glycerol, acetone solutions, sucrose

solutions and a set of viscosity standard liquids, and it has been found that the results can be

empirically fitted with a correlation coefficient of 0.9885 to a relationship of the form given in

Equation (1) while a purely empirically established formula of the form;

625

1/~t = a 0 + a I q + a 2 VD- a 3 q3 (8)

The data had been obtained without any temperature control or undue analytical experimental

care and the excellent nature of the data fit suggest strongly that this method may be a useful

experimental technique for investigating liquid properties.

The instrumental reproducibility for this measurement can be easily estimated from Equation (8)

and it is recalled that;

V D = q T D (9)

Fig. 14a. Series of tensiograph vibrations for (a) water (b) ethylene glycol (c) toluene (d) 40% w/w sucrose

solutions.

The reproducibility is approximately given by ('t/TD) from Equation (9), where x is the error in

the drop time measurement, if only the most significant term is here considered. For a minimum

626

drop period of 40s and with a PMS-tensiograph A/D conversion time of 40ms gives a

reproducibility of approximately 0 .1%, as all drop period measurement are limited by 25 Hz

acquisition rate of the CCD. For the single wavelength tensiograph employing a

Flytech(Taiwan) MD with a 42 ~s conversion time, the measurement error obtained where a 0 =

0.182588 a 1 = 0.179961 a2= 0.005840 a 3 = 0.000869 fits the data with a correlation

coefficient of 0.9942.

2.2.2.2. Experimental Observations on Vibration Tensiograph Rheometry

The detailed qualitative description of the vdts seen in Figure 14(a) and (b) which shows

respectively a series of rainbow and tensiograph vibrations has been given by McMillan et al

[40] The principal features of these vdts are the characteristic Loran shape seen for (a) the

627

rainbow vibration of 10% w/w sucrose solution with the characteristic rise and decay of the

envelope of vibrations. The modulation of the envelope by a low frequency harmonic is seen in

(b) for the rainbow vibration of acetic acid and in (c) for the tensiograph vibration of ethylene

glycol. A more complex pattern is observed for water (d) which shows a grouped pattern of

vibrations.

The vibrations are extremely sensitive to the tilt of the drop head and specifically this is

illustrated rather dramatically by the changes in the complex pattern of water from simply tilting

the drop head by 3 degrees. Four tilts directions are used in this study which are called Tilt 1, 2,

3 and 4. Tilt 1 has the collector fibre higher while Tilt 2 has the source fibre higher. Tilts 3 and 4

are in the orthogonal directions and should produce identical traces if the fibres are mounted

precisely in the head. Figure 15 shows the rainbow vibrations for the four directions with 3

degree tilts.

The fast Fourier analysis (FFT) of the complex trace of water recorded with a level drop head

reveals a dominant frequency of 110 Hz, but this analysis shows that this trace is the produced

from several modal admixtures, numbering more than six.

Tilting the drop head has the practical effect of reducing a considerable amount of the modal

components and Figure 15 shows the very clear beat pattern for Tilt 1 which may be fitted with

just two harmonic components of c0/2rc of 10 Hz and c0'/2rc of 10.35 Hz with a ratio A/A' of

unity. Experiments with CCD cameras using a shaker has shown that the water drop resonance

frequency for the concave drop head was close to 10Hz. The fit for the obverse tilt of the drop

head for the rainbow peak can be made with the same two frequencies but this time with a ratio

A/A' of 1.05. It should be noted that this vdts is asymmetric and has its envelope maxima

displaced. The displacement is a clear feature of this trace despite the difficulty of seeing this

here because the top of the vibration in the base line is clipped. Also the average signal position

is approximately zero on the ordinate near the start of the vibration. This settles to close to 1.0

near the end of the vibration. These features have not at this point been accurately modelled but

are obviously are significant features that may hold several clues to the mechanisms involved

here.

628

The tilt experiment on the drop head appear for water to indicate the presence of two dominant

modal frequencies which are being sampled by the reflected light, probably near the base of the

drop, in a ratio that depends on the position of these reflected rays on the inside of the drop

surface. This modulation of the light changes from position to position. It does appear from this

study that there are two dominant modal frequencies which are encoding the light coupling with

amplitude modulation through the mechanism of variation of the light being reflected near the

base of the drop in a ratio which depends on the exact position of these reflected rays. A very

recent discovery was made with a drop head which had been damaged by a solvent. The fibres

were accidentally foreshortened by an amount that fortuitously positioned them such that they

became incredibly sensitive to sensing the vibrations on the surface of the drop. It had been

observed for many years that the stepper pump produced micro-vibrations along the length of

the tensiograph with a frequency of precisely 1.666Hz.

Fig. 15. Water rainbow vibrations for drop head tilts of 3 degrees for four orthogonal tilt directions. Tilts 3 and 4

are equivalent. Tilt 1 is for collector fibre high and Tilt 2 has collection fibre low.

629

Water ...... /

/

!

I:thanol

I th> I c n c g l .~co l !\

/ \ \

a

( ;tu nncg,~

/ ~-- ..C~I~ .

l l i~ll Ihstillers ~ htske.~ / /\,.~

660nm 770nm 950nm

Fig. 16. Tensiotraces for water, ethanol, ethylene glycol, Guinness and Irish Distillers whiskey recorded with an

LED tensiograph at 30C at respectively 660, 770 and 950nm.

630

These vibrations are associated with a mechanical resetting of the pump gearing and they look

exactly like those shown in Figure 6 but are of course very much smaller. These vibrations in

water droplets produce a vibration resonance on the tensiograph peak near its maxima of

approximately 1/4 of the full signal maxima. The duration of the vibrations on the trace

observed in these vdts was approximately 3 seconds The plot of the vibration amplitude with

drop volume is useful in itself as it is defines the sensitivity of this drop to vibrations with drop

volume. It must be emphasised that the drop vibrations on this head were visually the same as

on any other drop, it was just that the fibres were positioned exactly to monitor the surface

vibrations on the drop by the light modulation. As an aside, it might be worth pointing out that

perhaps such a drop head could even be employed for earthquake monitoring in that the drop

has the advantage of both low inertial mass and would also being capable to capture all the

frequencies of the tremor.

2.2.3 Absorbance and Turbidity

2.2.3.1 LED Absorbance

LEDs have been used extensively throughout the development work for this new instrument to

provide the source radiation. It is well ~ nown that LEDs have a bandwidth of typically 50nm

and the three Siemens sources selected because they offered the best practical solutions for the

polymer system, have half-power bandwidths of 55nm(950nm), 38nm(770nm) and

30nm(660nm). Figure 16 shows the tensiotraces for a number of samples at 30C for these three

wavebands and five liquids. It should be noted that the outputs of the three LEDs are not

matched and neither is the responsivity of the detector the same for each LED.

Water has been adopted as the standard trace for this technique. It can be seen that despite the

variabilities in this system from LED to LED the water tensiotraces are very similar in every

feature. The 770nm trace differs from that of the 950nm trace by a mere couple of percent,

while the 660nm trace is only reduced by something approaching 10%. The other sets of traces

for ethanol, ethylene glycol, Guinness stout and Irish Distillers whiskey all show radical

reductions in the size of the traces from 950nm to 660nm. Interestingly, the "black stuff"

Guinness is almost transparent at 950nm, but is obviously a very dark liquid in the visible at

660nm. Ethylene glycol at 950nm has a substantially broader peak than at either 770nm or

631

660nm. In summary therefore we can say that these results show that the tensiotrace can be very

sensitive to wavelength changes in two regards, firstly, the magnitude of the tensiotrace signal,

and secondly, in some cases the actual form of the tensiotrace.

2.2.3.2. Spectral Absorbance

The measurements on the rhodamine-b solutions of increasing dilution were taken on the PMS-

tensiograph and the SPG 500 laboratory spectrophotometer. Figure 17 shows a series of

corrected spectra recorded with the PMS-tensiograph for a 10 second integration on a

stationary drop. The ratio of the PMS-tensiograph intensities measured at both 600nm and

550nm were recorded giving I(600nm)/I(550nm) and compared to direct absorbance

measurements on the SPG 500 spectrophotometer. The comparison of the band ratio and the

spectrophotometer absorbance as a function of corresponding dye concentration are shown in

Figure 18. It might be noted here that the value obtained for the SPG 500 at 101 ppb was

obtained from the signal descending into the noise at this point and this marked the very limit of

sensitivity of this commercial instrument. The PMS-tensiograph was capable of making very

good measurements at this rhodamine-b dilution. Pure water had for the PMS-tensiograph an

intensity ratio I(600nm)/I(550nm) of 2.65.

Fig. 17. A series of corrected spectra of aqueous rhodamine-B solutions taken for three concentrations of the dye.

Series 1, 2 and 3 are respectively for lppm, 100ppb and 10ppb.

632

The PMS instrument had a recorded noise level of 0.0048 bits, which has been taken to mark

the limit of sensitivity of the instrument. From a simple calculation, an instrument sensitivity of

0.04 ppb of rhodamine-b dye is indicated. This is a factor of 250 times better than the practically

determined limit of sensitivity of the SPG 500 as is more in line with the sensitivity levels

achievable with luminescence spectrometers.

The PMS-tensiograph clearly does not possess the ideal geometry for such spectral absorbance

measurements and it is obvious that a standard optrode arrangement could be fitted above the

drop head to carry out this measurement in a standard way. It is never-the-less of some real

significance that drop absorbance measurements can be done to such a impressive standard of

sensitivity and this experimental fact has some important consequences for other tensiotrace

measurements that are discussed below.

Fig. 18. Graph of absorbance taken with a standard laboratory UV-visible spectrophotometer and an intensity

ratio[I(600)/I(550)] taken on the PMS-tensiograph.

The accuracy that can be obtained with the PMS-tensiograph for the spectral absorbance

measurement is obviously dependent upon the calibration of the radiometer, but if it is assumed

that this can be done without penalty to accuracy from the calibration itself then the results

indicate that an accuracy better than 110 -3 A units is possible even at the lower range of the

PMS-tensiograph sensitivity range with a sensitivity calculated from the noise limit of probably

633

better than 2510 -6 A units if a full 10 second integration is used. These results were achieved

with a PMS working with a 8-bit CCD and the instrument has since been upgraded to a 12-bit

CCD which would yield a much improved performance with regards to the spectral absorbance

measurement.

2.2.3 Turbidity and Particle Size

The turbidity of milk solutions have been extensively studied and some of the most relevant

work by Elicabe and Frontini authors [36], Podkamen and Cuminetsky [37] and McCrae and

Lepoetre [38]with the tensiograph and corresponding reference measurements taken on a UV-

visible spectrophotometer. The results shown in Figure 19 are for tensiograph absorbance

measured at 950 and 660nm. The range of measured concentrations was here limited to 2% v/v

because of the limit of about 0.4 A-units placed on the absorbance range for the instrument

working without signal averaging. The 2% milk solution corresponded to a turbidity of 1.75 T

units.

Figure 19 simply demonstrates that the tensiograph is capable of measuring turbid solutions

using an absorbance measurement of the signal. The result is interesting because it shows that

there is a much greater absorbance for the measurement with the 660nm LED than that with the

950nm LED. This result can be easily understood from the Tyndall-Rayleigh scattering law of

1 / ~ 4 . The effect of this wavelength dependence of the scattering can also be clearly seen in the

fact that the Beer's law relationship does not hold for the measurement with the 660nm LED

beyond a concentration of 0.4%. Obviously the conclusion drawn from this is very preliminary

result is that the tensiograph can be used to investigate particle sizes with the CCD-tensiograph

and this work is presently being pursued.

2.2.4 Refractive Index

2.2.4.1. Basic Measurements

The rainbow peak was so named by McMillan et al. [35] because it height and other features are

determined primarily by the refractive index of the drop. This discovery of this peak was made

after consciously seeking the drop head analogue of the concentration of rays in a water droplet

that produces the rainbow. In the latter a full range of impact parameters (positions on the back

of a drop) in a water droplet produces a range of ray paths inside the spherical droplet and an

634

observed concentration of light close to the minimum deviation or Descartes ray is produced.

These ray concentrations produced from the subtraction of the rays from the initial uniform

beam a resulting Alexander Dark Space seen below the bright bow produced from the

concentrated rays.

1 . 2

=, O8

u r,-

0 6

"~ 0 4 e-

0.2

0 02 04 06 08 1 12 14 16 18 2

Concentration (%volume)

___. 050am I 660nm I Linear (950nm) I

Fig. 19. Graph of tensiograph absorbance for milk solutions 0-2% v/v.

The original flat drop heads did not show this feature which was eventually obtained by

producing a concave drop head and moving the source and collector fibre spacings until this

"sweet spot" position was found for this focused ray concentration. The analogy between the

process in the rainbow droplet and the fibre head is visually very clear if the drop volume is such

that the suspended drop on the head is almost spherical.

The numerical aperture (NA) of the fibre is determined by the refractive index of the fibre which for

PMMA is 1.49 at sodium D wavelength. Since the wavelength dependence of polymer is small the

value for 950nm will not be less than this by more than 5%. The emission cone from the fibre of less

than a metre will be very uniform as modal patterns only develop over extended lengths of fibre.

635

1.25

O

~" 1.2 .>

, . , , ~ 1.15 - E

z ~ 1.1

12. ~- ~= 1.05 5 " = O e -

~= 1 n,,

0.95

k X Rainbow Peak Height ]

I + Refractive Index I "" "

. . . . . . Linear (Refractive index) i . o - - I - ~

. . . . . Linear (Rainbow Peak I + . - " " ~ / - Height ) . r / x

= o.~... ~ "

t I t I

1.341

1.34 o r

1.339 a

1.338 = , m ,,m ..=_~ 0 1.337 ~ o i f =

- ~ .336 ~ =

tD -1.335 :~ Cl 0 , , u

- 1.334 N,,,

1.333 r

1.332 0 1 2 3 4 5

Fruc tose concent ra t ion , % (w/w)

Fig. 20. Graph of rainbow peak height measured and refractive index plotted against concentration of fructose % w/w.

As has been explained in the corresponding section above in this chapter the shape of the

remnant drop is the same for all liquids. Given that the rainbow peak is actually formed in the

remnant drop, it follows that the integrated path length in Equation (5) will be almost the same

for each liquid. This situation of almost a fixed shaped drop is analogous to having a cuvette of

a fixed dimension to hold the liquid sample between the source and collector fibres set at a fixed

distance. Calculations on the variation in solid angle for the emission cone of a fibre with

refractive index of the liquid shows a slowly curving and almost linear plot decreasing from

about 2.7 to 1.9 for changes in refractive index values of 1 to 1.5. Such a situation will give rise

to an increasing coupling between source and collector fibres with increasing refractive index.

As one can see from Figure 20 showing a plot of TUs against refractive index for a series of

fructose solutions in the range 0-5%, this theoretically almost linear predicted behaviour with

refractive index is in fact observed. This restricted range of concentrations for fructose was in

fact selected because it is close to the limits of sensitivity of the instrument for this measurand.

An extremely linear plot of TUs for the rainbow peak versus refractive index for sucrose(and

other sugars) was obtained in the range 0-50% w/w (1.333-1.37). Given that the sample does

636

not possess a strong chromophore then a general relationship appears to hold for this linear

dependence as can be seen from Figure 8a. The sugar plots fit onto this line as do a very large

number of other samples tested over several years of measurement with the instrument.

2.2.4.2 CCD Measurements

Measurements using the PMS-tensiograph have recorded the first multi-wavelength silica fibre

tensiotraces and revealed a number of interesting observations which will improve the limited

understanding of the nature of the tensiotraces. Figure 21 shows a series of water tensiotraces

recorded in the range 450 to 800 nm for two symmetrically positioned fibres close to the edge

of the drop head such as indicated in Figure 2(c) by the positions cc'. It was observed that these

tensiograph peaks arose in the pendant drop phase of the drop cycle and they are clearly typical

in form to the tensiograph peaks reported by the earlier work with PMMA fibres. When the

family of tensiotraces where scaled it was found that they all matched very closely to better than

5%. This results demonstrates that water is a very good standard for tensiograph analysis.

Using computer scaling it was observed that within the measurement accuracies these peaks

were all identical in shape. This close similarity of the tensiograph peaks shape arises from the

fact that these are all produced by TIR reflections from the base of the drop and since the drop

base shape are the same, the tensiograph peaks are as a consequence indistinguishable. This

observation shows that the variation in refractive index with wavelength is so small here that the

strong wavelength effect of the NA results in traces without any measurable changes between

the wavelengths.

Water has a strong absorbance band in the infra-red and it would be as a consequence this

would produce some shape variations for different wavelength tensiotraces. Coupling path

lengths in the drop vary throughout the drop cycle and should result in differences in the

attenuation of the signal. Since no measurable differences in any of the tensiograph peak shapes

has been observed across the spectrum from 400-850nm it must be assumed that the

tensiograph peak signal must be approximately two orders of magnitude bigger than the

absorbance signal than are reported above.

The variation in the NA of the source fibre has been shown to be important in determining the

coupling for the tensiograph peaks, since the coupled energy between the source and collector fibres

637

is determined by the emission cone and the collection area of the end of the collector fibre. McMillan

et al [41] had sought to explain the linear relationship between the TUs and the refractive index

observed in sucrose solutions and other liquids, in terms of this coupling mechanism. While this

mechanism is part of the explanation, it is not possible using this exclusively to explain the observed

differences in behaviour in the rainbow and tensiograph peaks.

Fig. 21 A series of water tensiotraces recorded in the range 450 to 800nm for two symmetrically positioned

fibres close to the edge of the drop head.

The PMS-tensiograph work reported in this study involved two drop heads, the first which shall be

designated by # 1 and the second #2, both of which are illustrated in Figure 2(c). The heads have the

same diameter with only the single fibre pair at a position that corresponds closely as possible to

position cc' for this first drop head. The tensiotrace in Figure 21 from the drop head #2 contains no

recognisable rainbow peak which is an observation that has been explained above.

Figure 22 shows the simultaneous tensiotraces recorded for water with drop head #1 for fibre

positions aa' and cc' revealing some interesting features. The tensiograph peaks seen the cc'

tensiotrace is markedly different from that seen in Figure 21 and this demonstrates the sensitivity of

the positioning of the fibres in the drop head. It appears that a small rainbow peak there is being

recorded but with sugar solutions this drop head did not produce the characteristic growth of this

rainbow peak with increasing refractive index as seen with the concave drop head. The sharp peak

638

seen in the aa' tensiotrace is discussed below but it should be noted that this is associated with the

drop separation as this corresponds with the very end of the tensiograph peak.

Fig. 22. Two simultaneously recorded tensiotraces for water at 20C and recorded at 650nm on the drop head #1

for the fibre pairs aa' and cc'.

This work suggests a new distinction between the processes involved in the rainbow and

tensiograph peaks as revealed by the CCD camera study of drops with constructions of ray

639

couplings. The rainbow couplings appear to involve a focusing of the rays while the tensiograph

couplings are produced with defocused rays.

2.2.4.3 Tensiotrace Studies with the Concave Drop Head

The actual form of the tensiotrace is can be seen visually to be a characteristic of a liquid and

these are a unique fingerprint of the liquid. As has been explained elsewhere, within the limits of

signal-to-noise of the present apparatus, the tensiotrace is a totally reproducible analytical trace

determined by the various properties of the liquid drop. The very diverse forms of the

tensiotraces are illustrated in Figure 9 which gives the tensiotrace of some six different liquids

taken with the LED tensiograph that have been chosen to illustrate the variability of tensiotrace

form. The following comments on these six traces may be also useful in highlighting some of the

features of the tensiotrace.

(a) 40% ammonium chloride solution is very similar to water, but the rainbow peak has been

greatly increased relative to the tensiograph peak. This tensiotrace exhibits a separation

vibration and a single rainbow and well formed double tensiograph peak.

(b) Ethylene glycol 40% v/v aqueous solution has a very similar trace to the above, but with

only the recovery peak and no vibration because it has a higher viscosity than the 40% v/v

ammonium chloride. The trace has been selected to illustrate the similarity with (a)

(c) Petroleum spirit has a trace displaced fight to the very end of the drop cycle with what

appears to be a large rainbow peak and a small single tensiograph peak.

(d) Methanol 20% v/v aqueous solution has a very broad rainbow peak to have a double

structure with a narrow double tensiograph peak.

(e) Acetone 40% v/v aqueous solution has a very rounded rainbow peak connected to a

very narrow tensiograph peak which just shows signs of being a double. The peaks are both at

the end of the tensiotrace.

(f) Acetic acid has very flattened rainbow peak and a large double tensiograph peak. It

might be noted that the actual size of the rainbow peak signal is comparable to that of petroleum

spirit.

640

As has been indicated here the actual form of the tensiotrace depends in the first instance for a

specific fibre pair on the shape of the drop, and this drop shape is determined in a quasi-

equilibrium slowly moving drop, by the surface tension density ratio. The drop shape determines

a whole range of features of tensiotrace seen in Figure 9, including the peak shapes and also

peak positions within the drop period. The viscosity of the liquid controls the form of the

separation vibration and it is important to note that if the liquid has a viscosity below about

50cP then this vibration is not present. The relative size of the peaks are both determined by the

refractiee index of the liquids because this progressively narrows the source fibre cone being

injected into the drop. The relationship between the various dependencies of the two sets of

peaks in the tensiotrace is discussed more fully below with respect to the I-functions. It is only

necessary here to make clear that the size of both sets of peaks are also determined by the

presence of absorbing species in the drop and also by any scattering or turbid material which

also attenuates the signal. The drop period is determined by the surface tension density ratio.

All the tensiotraces shown in Figure 9 are for liquids with relatively low absorbance that do not

effected the size or shape of the tensiotrace, but as explained above the attenuation of the

optoelectronic signal has been demonstrated to have a measurable effect on the tensiotrace. The

presence of either absorbing or scattering losses will reduce the coupling, but since the path

length varies continually throughout the drop period this absorption effect will also be observed

in subtle changes in the form of the tensiotrace.

The PMS-tensiograph has been used to study of variation in form of the tensiotrace with

position of the fibre pairs. Figure 2c shows the drop head used in this study with fibre pairs aa',

bb' and cc', which were positioned in the drop head specifically with the intention of avoiding

any coupling except between the corresponding fibre each rotated with respect to the next fibre.

Measurements conducted on a series of aqueous sucrose and acetone solutions were made and

the tensiotraces are shown in Figure 23 and 24 respectively. An obvious general comment on

these results is that the form of the tensiotrace is dependent upon the position of the fibre pairs

in the drop head to a very considerable extent.

It has been found that the positioning of the silica fibre in the drop head is much more sensitive

with regards to the form of the tensiotrace than the polymer fibre. It is found that this positional

sensitivity arises principally because of the smaller fibre diameter rather than their slightly lower

N.A. The top set of traces shown in these figures are for fibre positions cc' and this geometry

641

was chosen specifically with the intentions of obtaining an tensiotrace similar to that obtained

with the polymer fibres and the drop head shown in Figure 2b. The resulting tensiotraces for the

sucrose solutions have however only tensiograph peaks, but the acetone solutions appear to

show for the 10% v/v solution a very small rainbow peak and this rises very considerably for

the 30% and 40% v/v solutions. It is clear that the positioning of the silica fibre to pick up the

rainbow coupling in this drop head design requires the fibres to be placed rather closer to the

centre than for the corresponding polymer drop head which has a 50% larger diameter.

1 2 e e l e ~ 3 u c ~ o . e L 2 0 0 4 e ~

e 2 e o ~ e e s o I

x e o ~

~ s o e

B

2 . , . .....

i s

s a e T l~ ,m <m�9

~ ~r ''~ :*~ l ~ e

~ . ' J ~ , ~ . ~ , ~ , , ~ , ~ ''r~'r176162 ~ --_" : ~ e , ,

e . t e e s 2 o e 3 e e o

T , . - - < . )

_.~ " t ee

m

6 8

T i n , am (am)

s e e

T I , .~ , . . . < . >

6 e :.,,~ 8 ~ c : !,-. o . . , , .

.... 1 Ge l , l ooen

it 3 ~ 0

4 m w

~ . 3 e r i e

2 m e

l o e

__ I

e

o

6 Q e

Fig. 23. Sets of tensiotraces for fibre positions cc', bb' and aa' taken on 10%, 40% and 60% w/w aqueous

solutions of sucrose at 20C recorded on the PMS-tensiograph.

642

4 ~ m

m

z o o

T l m l (m )

s o m . ~ o ~ ~qw~o~9 4 o 0 F

""~ /1 ~ : " r : " r t ~ :"r : " r

ZO; ,4 p~ ee ' t o new ZGg l J . - .

i = " 1 - A

*"!- t \ IS I I * I l l : 211 * I

T I e , ~ l C o )

, t r ee ~ IE~Q 3 o l

T l i i i o ( enD)

= 'S lD ZO :4 ~ o m t o hen 2 .40 Z, l l ~ p r i e s t o h m r " "

---, ,.,~,,,.. <.~- - . ,.-'.,.. ,..-~- -

6QO 3Q: ,4 ~o ,B 4~ QwQ 4 0 0 F 4 o ~ ~wt~n~

- - - r x ""F \ z o o p , ~ " " : !: J L " ' " F _J k..

T tL ~4D q : m ) T l ~ e Cm)

/ i ""F l ] ~ " ' F ""r i ] " " ' r

~ , . e e = . o 3 * * ~ ,.&e , ,& . T tl ~4D (m) T tL ~GD l lm )

m

__L 3w l

1 ~ g m

4 s 4 t

7 - 3 m �9 �9 n L o I 2 8 a 3 ~ 1

T IL~o (~ ) 7 1 ~ e Cm)

Fig. 24. Set of tensiotraces for 10%, 20%, 30% and 40% v/v aqueous acetone solutions at 20C for fibre pairs

respectively cc', bb' and aa' recorded on the PMS-tensiograph.

The sucrose tensiograph peaks seen in all the tensiotraces for position cc' all have four

resolvable peaks that are in fact more clearly defined than for the corresponding polymer fibre

drop head.

643

For the fibre pair bb' the sort of triple coupling could perhaps be responsible for the large peak

which becomes resolved as a double for 30% acetone-water. Increasing refractive index with

increasing sucrose concentration produces an increasingly narrow emission cone. This appears

to produce the wide peak that sits up against a very narrow peak which can be seen just before

the drop separation. This result is interesting as it suggests that the fibre positions bb' can be

used to monitor the refractive index. These traces are all so very different that it is obvious that

these bb' tensiotraces can be used visually to identify the solution.

The coupling produced by the of fibres at position aa' are perhaps the most interesting of the

sets of tensiotraces produced by the three fibre positions as they possibly show the first evidence

of surface guided waves in the drops. Ray tracing studies for fibres in position cc' suggested the

existence of surface guiding. The schematic mechanism of light injection that could perhaps

produce surface guided waves and this would coincided with the drop separation. This guiding

occurs in water type drops just at the point when the drop necks. No camera studies have yet

been done on low surface tension drops and this effect has not yet been studies on these. As the

drop falls off the drop head a flash of light can be seen visually illuminating the surface of the

drop. If these surface guided waves are in fact being picked up by the detector fibres then these

would account for the peak seen in the tensiotraces for sucrose for both positions bb' and aa',

and also for the acetone solutions for position aa'.

If the spike seen in these tensiotrace indeed result from the coupling of surface guided light

then this guided light should be brought successively into the collector fibres a', b' and c' as the

drop progressively necks. A sequential coupling should be produced with a short but measurable

time delay between the position of the signal for each fibre beginning with c', and ending with a'.

The series of spikes at the end of the trace actually do appear in all the sucrose solutions to be

sequential in time, although these peaks are only just discernible for the fibre pair cc'. These

results perhaps are the first observation of surface guided light capture in pendant drops. The

applications of such a technological capability is enormous. This observation would, if

confirmed by subsequent work, be by far the most significant potential application of the

tensiograph. The acetone tensiotraces shown in Figure 24 also show this feature for the fibre

pair aa'. Spectral surface spectroscopy is very important, and if confirmed by later work, would

allow the PMS-tensiograph to be used for surface chemistry applications.

644

2.2 INSTRUMENTAL REVIEW OF THE WORK TO DATE

Table 1 reviews the results obtained in this work which have in most part been discussed and

explained above. The results shown are from measurement except where shown as estimated

values which these are clearly indicated.

The actual S/N recorded with the LED-tensiograph is typically 300:1 if care is taken to avoid

vibrations, but a very much better S/N is obtained with the PMS-tensiograph which has been

isolated from vibrations and employing its various integration capabilities. The measurements on

refractive index reported by McMillan et al. [42] have been largely confirmed here on a range of

other liquids using the LED-tensiograph, although only limited details of this work have yet

been reported. With the this instrument it has been shown that accuracies of 0.002 or better are

possible in this measurand, given the observed change in the rainbow peak height ratio of

samples with respect to a water reference.

Preliminary work done here with the PMS-tensiograph is based firmly on the instrumental

developments of Walsh [42] While the CCD tensiograph study has revealed rainbow features in

the tensiotraces this only represents an indication of the refractive index measuring capability of

the PMS instrument. Insufficient work has been done at this point to categorically confirm this

measurement capability, or indeed properly estimate the quality of this instrumental

determination of this measurand. The practical problem that has stopped a full investigation at

this stage of the phenomenon of rainbow peak dependent with refractive index for the PMS-

tensiotrace is the considerable difficulty in engineering a drop head that will show in a

reproducible way these features. New jigs are being devised to help position the silica fibres in

the concave drop heads.

The projected value shown in the Table 1 for the refractive index accuracy and reproducibility is

therefore based on the S/N obtained by the PMS-tensiograph and the observed changes seen in

the size of the rainbow peaks with this LED-tensiograph. These considerations have been

developed on the basis of Zalloum and O'Mongain's work [43].

There are other potential ways of determining the refractive index from the tensiotrace, and in

particular, the observed rainbow external to the drop offers a very direct possibility for this

measurement. A visible rainbow has been observed on a screen placed beneath and to one side

645

of the drop with white light coming out of a source fibre. A PMS collector fibre have been

placed at this position in an attempt to record the external tensiotrace. This measurement should

give a refractive index method based simply on Snell's Law and knowledge of the physically well

defined shape of drops. At this stage the only work done on these external tensiotraces has been

for water and some indication of the potential quality of this measurement is given in this Table.

The PMS-tensiotraces reported here have a spectral measurement capability and have shown

variations in peak positions with refractive index for external tensiotraces. This preliminary

investigation suggests that such a method can be devised for this measurand.

Table 1. Table showing the measurement capabilities of the tensiograph based on either the results or projections

of the present work (Estimated values are indicated).

Measurand

Absorbance

Refractive Index

Viscosity(Head)

Viscosity(Vib. drop)

Surface Tension

Dynamic ST

Evaporation(drop

volume)

Evaporation(tensiotra

ce monitoring)

Refractive

Index(Inner fibres)

Accuracy Repeatability

10 "4 Aunits 25x10 "4 Aunits

>10 -5 >10 -5

>0.1% >0.1%

>0.2%(12 bit) >0.1%

0.25% i 2.5x10-4%

0.25% 2.5x10-4%

0.1 p l 0.1 Ixl

0.01 IXl 0.01 IXl

>10-5 >10-5

Sensitivity Range Comment

25x10 "6 Aunits

>6x10 -7

>0.01%

0.05%

2.5x10"6%

2.5x10-6%

0.1 ~tl

0.001 IX 1

>6xlO "7

0-2 (PMS)

0-0.4(LED)

A units

> 1.3 and above

1-1000cp

<50cp

5-75mNm

Est. 5-75mNm

Values

Est. 0.01 - 1 Ix l/s

Est. 0.01 - 1 Ix l/s

> 1.3

Better than standard I

laboratory UV-vis on

sensitivity

Comparable to Abbe

Refractometer

Values obtained from

tests

Drop vibrations occur

only in range up to 50

cp

Values from tests and

cf Lauda Analyser

From tests and

reported accuracies of

Lauda Analyser

Values from three

tests*

Values from three

tests*

Only demonstrated

for the LED-

tensiograph

*Details of this measurement are to be found in McMillan et al. [44].

646

However good reproducibility of measured drop volumes with the Hamilton pump has been well

demonstrated with the tensiograph. When care is taken, drops show no measurable difference in

volume. This perfect reproducibility is due to the fact that the volume resolution of the pump is

greater than the statistical variation in the drop volumes being dispensed. With syringes of 100

~tl and using a 1000 stepper liquid dispenser the volume resolution is 0.01 ~tl and it is this

objective instrumental limit determines the accuracy of the technique. With new 2000 step

pumps and other developing pump innovations this accuracy is set to be improved. Analogue

gravity techniques could clearly be devised that would enable a much greater accuracy for

surface tension measurements to be obtained than using the present set-up.

The work on viscosity has shown that the gravity feed technique can be employed for a very

wide range of viscosity measurement. It is significant that this is a method which is very well

suited to measurements on low viscosity samples which can not be easily measured by other

techniques. In particular the technique is ideally suited for the viscosity measurements of highly

dilute solutions of macromolecules in solvents to determine molecular weights of these

macromolecules.

The drop vibration viscosity measurement has not been investigated or demonstrated yet, but

this work has shown that this is possible. Qualitative understanding of the complex patterns

observed with mechanically excited tensiograph vibrations have been offered that could be

developed to provide a method of measuring viscosity. Given that the decay can be fitted to

better than 1%, which has been demonstrated, then the projected value of the viscosity will be

of about this accuracy. The obvious advantage of this technique is that the sample size required

is perhaps only 200 ~tl.

3. INSTRUMENT FOR FINGERPRINT ANALYSIS

3.1 THEORETICAL BA CKGR O UND

3.1.1 Introductory Comments on M- and I-Functions

The fingerprinting approach conceptually separates each feature of a trace(tensiotrace) and

quantifies the match between a reference and a test traces in terms of a normalised tensiotrace

M-function and an associated statistical "indicator" I-function. Only those specific features of

the trace that are associated with a physical property of the test liquid are used in this computer

647

matching because this enables the resulting I-values to be referenced against standard methods

using traditional instruments.

A new method has also been developed for data search routines to obtain matches of

tensiotraces. The search algorithms are based on the series of normalised M-functions that are

defined below. Brief details of an application of M-functions are given for identification of beer

and the use of the I-functions are offered as a solution for a very difficult problem in the brewing

industry of admixtures of foreign product.

3.1.2 Definitions of M-Functions

Figure 1 shows the tensiotrace for a typical water based trace that have been used in the

software to produce normalised search functions for fingerprinting. Less complex traces such as

those exhibited by the alcohols are then handled effectively by the software. The features of the

tensiotrace so used are respectively:

�9 Drop period

�9 Rainbow peak height and time

�9 Tensiograph peak heights and times(typically there are three peaks but frequently more are

recorded)

�9 Data points

�9 Areas

In addition the density(obtained from the Paar DPRT and not from the tensiograph) has been

used as a physical quantity from which statistical information may be inferred and this quantity

will be included in the discussion as if it is really an integral part of the tensiograph.

Five functions give information on various aspects of the tensiotrace all of which are normalised

to give a better conceptual indication of their relative importance. It should be noted that the

figures in the software are printed in black when the reference is greater than the test trace and

red when the test is the larger. When the data is printed for a paper report the values are printed

as absolute ratios with the values greater than 1 when the test trace is the greater. It has been

found that there is a practical advantage to this normalised approach when using certain

648

graphical representations. More specifically when fingerprinting this is an advantage as this

always shows how close the two values are on a normalised scale regardless of which trace is

actually is the greater.

Fig. 25. Reference and test tensiotraces P and Q respectively showing height measurements of the mth data

point.

Figure 7 gives details of the labelling of a tensiotrace for two traces P, the reference trace, and

Q, the test trace. The peak heights and peak times that are associated with the test trace are

indicated by the dashed terms. A normalised function that can be defined for drop period is

simply;

TD or TD' (1 > M 1 > 0) (10) M 1 - f ( T D ) - TD, T D

where TD = drop period of sample P and TD' is the drop period of sample Q.

A peak time function can be similarly defined;

T~ or Tin'(1 M 2 > 0 ) (11) M -ftL)-L, L >

649

where m = 1, 2 or 3 to label individual peaks for the situation shown in Figure 25, but might be

greater or lesser for various drop head geometries where Tm is peak time of the m th peak for

sample Q. Usually there are at most three peaks, namely the rainbow, and two tensiograph

peaks. In some traces there are several other peaks.

A peak height function can be similarly defined;

/ 4 / 4 ' M 3 = f ( H m ) = or (1 > M 3 > 0) (12) /-/~' /-/m

where as before m = 1, 2, 3 etc. for individual peaks. Here Hm is the peak height the m th peak

for sample P and Hm' is the peak height for sample Q.

Figure 25 shows a schematic situation of an overlap of two traces indicating the situation of the

n th series of points which have a signal value respectively hn and h'n for the traces P and Q. A

normalised "analogue point" function can be defined for the traces from the sum of the ratio of

these signal sizes summed over the entire series of points for what will hereafter be referred to

as the long period of the traces, N ifN > N' and N' i f N ' > N. This point function can be defined

as followed for the long period;

- - o r , 7 "=N~ lh , , , ) (13a)

M4~ - f ( h , ) - ~" N m=l

where O> ~ or - - <1 k. hm J h' m

An alternative "binary point" function can be defined using a tolerance set on (1) of off (0)

summation thus over the long period;

m=NorN' A M4b = f (h~) = 2 (13b)

m=-I N

(h'm" ~ (hm" ~ ~ o r , 7 1 w h e r e A - l i f L > ~ h m ) ~,hm) <

L might be selected for instance set to 0.95 to check for ratios in the range 0.95 to 1.

650

Finally an area overlap integral can be defined for the tensiograph which is illustrated in

Figure 26. The shaded area O represents the overlap of the two traces and the dotted line

indicates the total area T of the two traces for the long period N or N' depending on which is the

greater normalised area (A) function can be defined thus;

M 5 - f(A)-(A--~r ) (14)

Fig. 26. Reference and test tensiotraces P and Q respectively showing overlap area of traces.

The tensiograph does incorporates a four figure Paar DRPT density meter and this information

is also available to the user in most experimental situations. An M-function can be defined for

this property of the liquid in a similar way to any of the tensiograph functions thus;

-(P--~t)or(~r~ with 1 > M 6 > 0 (15a) M6 - f (p)

19 = relative density of the liquids with the subscripts r and t applying to the reference or test

trace respectively. This relationship reduces if water is used as the reference liquid to;

M6 = Or or 1/Or (15b)

651

3.1.3 The Weightings of the M-functions

The six tensiotrace "M" functions, namely M1, M2, M3, M4a, M4b and M5 actually give the user

the ability to conceptually analyse the data in the sense that each function has its own clearly

defined conceptual basis being associated with just one specific characteristic of the tensiotrace

which is itself associated primarily with one physical property of the liquid. The rainbow peak is

closely associated with the refractive index, the drop period with the surface tension, and the

tensiograph peak with the colour measured as usual from the absorbance value at the LED

wavelength. There are a number of other very clearly differentiated points on the tensiotrace

that could equally be used for a trace analysis. These points include (i) the point at which the

trace rises from the base line (ii) the minimum between the rainbow and tensiograph peaks (iii)

the second tensiograph peak. None of these additional points have been shown to be associated

with a physical property of the liquid and therefore have been excluded from the "M-function

analysis" for the tensiograph.

The differentiation of each M characteristic of the trace thus has the virtue of facilitating the

investigation of samples by pinpointing the principal dependence in the traces. The M-functions

thereby enable the user to attribute a specific change in the trace to a specific property of the

liquid that is changing in the test solution. For studies of surfactant solutions for example, it is

found that the traces are very similar, differing only in regard to the drop period and this

foreshortening of the trace effects in a proportionate way the rainbow and tensiograph peak

times. The functional analysis of the surfactant drop tensiotraces consequently show changes in

only M1 and M2. There are very, very small changes in the M3 functions. Intelligent use of these

functions can therefore be valuable to the analyst, but the full potential of "M-function" analysis

will have to be discussed elsewhere and here this will be restricted to one example.

In the present study a series of tensiotraces were recorded for 1 to 5% w/w solutions of fructose

and Table 2 shows the resulting M-function analysis for these traces measured against the

reference water. The values in this table for water are typical values obtained for measurement

against other tensiotrace for water. Obviously the values should be unity but because of

instrumental noise these are slightly reduced for each M-function. The figures for water in fact

give a base line against which the other values can be assessed. The instrument can of course

652

use any liquid as a reference liquid and it is sensible for example when analysing spirits to use

ethanol as the standard trace because the spirits are more closely related to ethanol than water.

Table 2 - Functional analysis of series of fructose solutions measured at 30~

Sample/

M function

Water

1%

2%

3%

4%

5%

M 1 M2(RP) M2(TP) M3(RP) M3(TP) M4a

0.9993 0.9917 0.9987 0.9809 0.9962 0.9222

0.9973 0.9864 0.9977 0.9488 0.9943 0.9981

0.9956 0.9929 0.9964 0.9034 0.9699 0.9964

0.9895 0.9882 0.9918 0.8522 0.9595 0.9901

0.987 0.9991 0.9900 0.8128 0.949 0.9877

0.9815 0.9954 0.9955 0.7414 0.9122 0.9822

M5

0.9753

0.9658

0.9201

0.8797

0.855

0.7917

M6

0.9998

.9962

.9922

.9885

.9863

.9108

Fig. 26a Graph of M1 (drop period) function for fructose showing that the function is determined by the ratio of

the surface tension to density.

653

Fig. 26b Graph of UV-vis and tensiograph absorbance at 950nm.

This specific sugar was taken as a measurement problem because this is at the limit of the

existing instruments instrumental resolution. The range in variations of the properties of these

liquids vary for surface tension by 0.05%, for specific gravity by 1.81% and for refractive index

by 0.76%. Any variation in the absorbance of the solutions are below the instrumental resolution

of a standard UV-visible spectrophotometer and will not be considered here. The value shown

in italic in this table indicate that values of the reference are less than those of the test trace. In

the software package this situation is indicated by a red figure while others are printed in black.

The graph of the M-functions can be used to provide a qualitative plot of various physical

properties of the test liquids. Figure 26a shows the M1 function for the fructose plotted against

the ratio of surface tension to specific gravity. This plot reveals the general form of the surface

tension / density relationship in the range 0-5% w/w as can be seen from the plot of this

dependence on this graph. The M3a function applied to the rainbow peak if plotted against

concentration for fructose gives the general form of the refractive index

variation for these samples as can be seen by the plot on this graph. The results of the M3b

function applied to the tensiograph peak does not give the absorbance variation because these

654

are non-absorbing solutions at 950nm and the 8% variability in this peak revealed by the M

analysis is due to the interfering refractive index variability of the tensiograph peak. The errors

shown in this figure are the range taken as three twice the standard deviations.

The measurement capability of the tensiograph has been investigated with this data and an

analysis made using statistical tests in Excel for all the various capabilities of the tensiograph to

produce the various estimates.

The absorbance measurement capability of the tensiograph has been investigated using a

separate study with copper sulphate solutions was conducted and the instrumental capability of

the tensiograph for this measurement determined. Tensiograph absorbance is defined above in

Equation (4). Figure 26b shows the tensiograph absorbance of the copper sulphate

measurements plotted together with the UV-vis absorbance taken at 950nm. The drop period

results plotted here of course correspond to the M3(T1) function which are given in Table 3. The

logarithm of the reciprocal of this M function gives the tensiograph absorbance. In summary

then it has been show that these M-functions contain information about the physical properties

of the sample.

Table 3. Summary of the results obtained from the tensiograph trace analysis for copper sulphate analysi~

Sample/ M-

function

o.oo5 M

0.01 M

0.03 M

0.04 M

M1 M2(R) M2(T1) M3(R) M3(T1) M4b M5

0.8507 0.8535 0.8550 0.9180 0.9252 0.2531 0.4663

0 8773 0.8778 0.8655 0.9235 0.7481 0.2778 0.5021

0.9184 0.9319 0.9022 0.2623 0.1705 0.1976 0.1939

0.9563 0.9338 0.8662 0.1858 0.0840 0.1958 0.1714

The analysis of the data from the copper sulphate solutions give a good estimate as to the

instrumental performance of the tensiograph for various trace features and these results are

summarised in Table 3. These results were used to provide the numerical basis for the I-function

algorithms discussed below. We can see that there is little difference in the M3(R)value for the

two low concentrations but the tensiograph value M3(T1) has begun to show a large decrease.

655

The M1 and M2 values show small variations compared to those of the M3 variability obviously

because of the dominant effect from absorbance in these traces. The points ratios M4 are lower

than the area ratios M5 for the two high values and interestingly are higher for the high

concentration measurements. The points are individually assessed to give a value of 1 or 0, that

is they are a digital measure. The area value is an analogue value. This example is included here

to illustrate some subtle variabilities in a set of traces with the objective of suggesting that this

type of analysis might find applications in many fields.

3.2 THE I FUNCTIONS - THE FORENSIC TEST OF TENSIO TRACE M A T C H I N G

3.2.1 Basic Quantities and Concepts for 1-Functions

The tensiograph is perhaps unique (single bandwidth) in that it has an unequalled amount of

differentiable information contained within its opto-electronic signal. Two approaches have been

integrated into the analysis of the tensiotrace, firstly, trace feature ratioing using M-functions

which has been described above, and secondly, 1-function analysis. The 1-function is a pseudo-

statistical function produced by the tensiograph software from a test and reference trace. The I-

value generated from the algorithm expresses the certitude for the fingerprint identification.

Alternatively, the I-value can be looked on as the estimate of the statistical chance of a false-

positive for a match of the test and reference trace.

An I-value is only returned by the tensiograph analysis package if a definite match has been

obtained for the test trace and a reference trace. The software informs the user that "This is a

fingerprint match". A non-zero value "indicates" that the instrument is unable to distinguish any

differences between the two traces. The magnitude of the value quantifies how good the match

is and depends on the quality of both the test and reference traces. Large I-values shows that the

odds against a chance of a false-positive is very high. Consequently, large I-values are produced

by matches of complex multi-peaked tensiotraces displaying good signal-to-noise. Conversely,

small 1-values result from matches of traces with only one peak and with a poor signal-to-noise.

This function, will find an application in identifying and quantifying changes in similar liquids,

such as for example beer as it ages, or in differentiating one batch of this product from another.

As soon as there are any real measurable differences between the test sample and the aged

656

product, then the I-value would become zero as the liquids would then be forensically

differentiated.

There are two types of I-functions, the "universal" and the "measurement", known as the

statistical and measurement I-values. Both the "statistical I-value" and the "measurement I-

value" are generated by algorithms. The former employs a universal range of properties of

liquids known as a default range. The latter is derived from the experimental data from the

specific measurement in hand. The statistical I-value is the one which predicts what the

statistics would be if a set of experimental trials were conducted on the test liquids.

Corrected values can be obtained for either set of the I-values. The actual relevant ranges of

each property of the test liquid can be inputted into the software. Such ranges of properties

would for example be known from quality control measurements. These values tune the

resulting I-value value for the measurement problem of the analyst. The corrected I-value thus

obtained therefore applies uniquely to the specific measurement problem.

Small I-value are consequently expected for the analysis of both sugars and Guinness samples

because as can be seen from Table 4 these both have very restricted ranges. Small corrected I-

value values will arise directly from the narrow ranges of the properties of absorbance, surface

tension, refractive index and density for these sets of test conditions. Large corrected I-values

are anticipated for the identification of an unknown alcohol testing for example against the

entire homologous series of unbranched "mono-ols" because of the very wide range of

properties for the family of alcohols. The statistical I-values for every analysis will always be

significantly larger than the measurement I-value. The corrected value of either will always be

smaller than the uncorrected value.

There are a large number of applications for which statistical information relating to the analysis

could be of fundamental importance. The present work has been directed towards developing

computer algorithms that give statistical information for tensiograph fingerprint analysis based

on just a reference and test measurement. It is obvious that statistical information could not be

furnished from what is the ultimate limitation on sample size viz. one. The statistical 1-value is a

pseudo-statistics generated by a computer algorithm. This I-value has been designed to try and

generate a number that would match the statistic obtained from a proper sampling of the liquid

657

if the sample size was statistically significant, i.e. above thirty samples. At this early stage of the

instrumental development for the tensiograph, no comprehensive studies have been possible and

therefore computer algorithms have been developed which produce what are highly

conservative estimates of statistical probability. It is planned that better "l-functions" will

subsequently be developed based on extended experimental studies of the use of tensiograph

fingerprinting.

There are a number of experimentally determined values that will be necessary to establish from

the data set before the I-value estimates can be arrived at. These experimental data parameters

are:

1. signal-to-noise of both traces (8S)

2. data acquisition rate (t)a)

3. data averaging (ND)

4. MD resolution (b)

3.2.2 1-functions for Trace Times

The conceptual basis for the "Time" I-functions are all based on the drop period I-function. For

slowly delivered drops the drop period is dependent on the ratio of the surface tension and the

density. In the present apparatus the drop period is measured by the optical eyes and this

measurement is reproducible in its operation. The standard deviation of any drop period

measurements is approximately a fixed value. An experimentally determined value of this

standard deviation is used in the software. Usually, a typical value for the standard deviation of

this drop period is assumed for measurements. The user, can however, put into the software a

real value from their own measurements and thus over-ride this approximate value when this

proves necessary. The drop period, for any specific measurement problem, such as say in quality

control, will vary over some range. Physically, this range is determined by the variability in the

surface tension and density values of the test samples. This limiting range of drop periods is

known as the 'gamut' of values. Table 4 gives the ranges of the typical ranges of the relevant

physical properties as found in laboratories. These values have been used to define the gamut.

We have discounted here in defining the gamut untypical liquids such as mercury.

658

Now in any practical situation, such as for example the analysis of beer, the range of the values

of surface tension and density that will be encountered in practice from sample to sample will be

very narrow when compared to the gamut. Further restrictions on the time 1-value range are

then necessary to produce the 'working range' for drop period. The working range is estimated

as being a fractional part of the full range.

The sottware allows in fact any range to be inputted. A full product range to be used in the

analysis of beer might in fact include all the properties of the constituent parts of the product

such as alcohols, enzyme solutions, sugars etc. On the other hand, a very much narrow range of

properties might be used for an actual analysis situation that pertains only to the production

records of stout. A default range however is offered based on the gamut given by the values in

Table 4.

For a given drop period, TD, and given a measured standard deviation, ~, we can conservatively

take the tensiograph time resolution (RD) of this measurement to be given by;

Ro = TD/(6or) (16a)

The tensiograph resolution is the number produced by dividing the drop period test range by the

+3~ (giving 6~ ranges) error ranges for the drop period measurement. The range of drop

periods are divided into increments each of which represent a statistically differentiated drop

period measurement that can not overlap its neighbour. The '6c criteria' used here, has however

been taken to make this values as conservative as possible.

Considerable experimental efforts have gone into reducing the value of ~. Studies have been

made to determine a good typical value for ~. The experimental reproducibility of drop times

for a drop head manufactured without any polish (finished with a drill to produce the concave

drop head) has been determined by extensive experimental measurement using the system

described here and found to be in the range 20 to 70 ms over ten drop measurements for a wide

range of liquids. Comparative measurements have also been made on drop heads manufactured

with the help of Engis Ltd. Gillingham, Kent, UK with a 0.2 micron polished drop heads with no

measurable improvement in the reproducibility of the drops. It has thus been found that the basic

manufacturing method is adequate for drop time measurements which are the touchstone for

659

this technique. The value used in the software is 70ms to ensure truly conservative values from

the algorithms.

To illustrate the discussion below, the numerical values of three specific problems will be used

here. Firstly, the very simple analysis problem of the identification of a branched mono-ol from

the homologous mono-ols. This study of alcohol is 'a discrete tensiograph analysis problem',

because these pure alcohols have discrete properties. Secondly, the problem of measuring a

fructose solution in the range 0 to 5 % w/w, which 'a continuous tensiograph analysis problem'

but for a very simple unitary solution. Finally, 'a real industrial tensiograph problem' of

measuring Guinness stout will be discussed. These examples are employed here to illustrate the

function of I-values and to provide concrete examples for the full range of problem types.

Table 4. Table of ranges for three measurement problems. The universal range was taken from the CRC

Handbook and general scans of values checking liquids that might be found commonly in chemistry

laboratories. ST:- Isopentane to sodium hydroxide(36%); SG :- Hexane to Carbon tetrachloride; RI:-

Trifluoroacetic acid to Diiodomethane.

Sample/Property

0-5% Fructose

Straight Chain

Alcohols

Guinness

Full

ranges(default)

Surface Tension

(mN/m) (~/1-]t2)

72.75-

72.7875(0.05%)

22.07-28.30(22%)

40.5-45(10%)

13.72-101(86%)

Specific Gravity

(D1 - D2)

1-1.0181(1.81%)

0.791-0.827(4.35%)

1.005-1.016(1.08%)

0.6548-

1.584(58.7%)

Refractive Index

(q l - qz)

1.333-1.3402(0.76%)

1.329- 1.4290(7%)

1.006-1.0102(0.42%)

1.283-1.749(26.6%)

The ranges of the relevant physical properties for these three sets of test liquids are given In

Table 4 together with an approximation to a full range for properties of liquids found in a typical

laboratory situation.

For water based solutions such as sugars the tensiograph time resolution has been measured at

551, 1050 for alcohols, and 77 for Guinness with the pump housed in the instrument cabinet and

660

vibrations being transmitted in an attenuated way to the drop head. Average values of resolution

for sugar solutions were measured on earlier systems with the instrument standing on a vibration

isolation table giving an average value of ! 14(ranges of values were in fact obtained from 1000

down to below 50). Significantly this value is lower than with the present system. The only

vibrations in this experimental set-up would have come from the Hamilton stepper pump

through the liquid delivery tubing. It has been concluded that the vibrations in the present

apparatus are not effecting the drop period in any tangible way and that the improved

preparation of drop heads has produced the better tensiograph drop period resolution.

We know that the full or universal range of surface tensions and densities for laboratory liquids

are bounded. Table 4 shows that the relevant values taken from the CRC Handbook range from

13.72 to 101 mN/m and the density ranges are found to be 0.5548 to 1.584. It is in fact the ratio

of these two quantities that determine the drop period. The longest drop period observed in the

present studies was for a sugar solution with a drop period of 103.5 seconds produced by ratio

of surface tension to density of 73.72. The smallest drop period observed was for ethanol of 40

seconds and a tension/density ratio of 27.9.

However, to make the argument easier to follow let us adopt a rough approximation that drop

volumes on the apparatus are to a very good approximation related to surface tension by a

factor of 1.4 and the density does not effect the final answer. Using the gamut of values given in

Table 4, the largest drop size that would be encountered would be 141.4 gl and the smallest

19.2 ~tl. It is the range 19 to 141 gl therefore that must be divided up into 6or sections( not the

range 0 to 141 gl) to obtain the possible resolution range of the drop period. The statistical

indicator is calculated from the drop period range (ATD) for the measurement. The resulting

resolution range(ARD) is called the tensiograph discrimination and is defined as follows;

ARD = ATD/(6~) (16b)

The discrimination is defined so that it will correlate to the experimentally determined statistical

value for a 'false-positive' match of a reference and test liquid. It should be remembered that a

statistic for such false-positive identifications can only be generated from properly conducted

statistical study using a set of experimental measurements. The statistical indicator is just that,

an indication of what such a statistic would be. These numbers can be very useful to an analyst.

661

The resolution, RD, can however be used to generate another useful number for the analyst in

some experimental studies and is called the measurement indicator. The software gives both the

statistical and measurement I-values. The user has to be knowledgeable about the differences

between these values if they are to interpret properly the numerical information that the

instrument is providing.

A theoretical requirement for statistical I-value analysis is that the algorithm should lead to a

probability of unity if the range becomes narrowed to such an extent that the tensiograph would

be totally unable to discriminate between two test liquids. There is one other requirement for the

corrected value of the I-value in that probabilities are usually expressed in integer fashion. It has

been thought desirable for the computed value to always be presented as an integer. Obviously,

immediately the I-value falls below unity, then the probability must become zero. In such

circumstances the result informs the user that the liquid are 'not a fingerprint match'.

The criteria for non-matching and forcing the respective I-values to a zero is;

Abso lu t e {T D - T' D } >= 6o- (16c)

where the symbols have the meanings as defined above for Equation (6a) and the dash indicates

the reference trace.

The band factor (K1) is obtained from the very good assumption that the drop period is

represented closely by the ratio of surface tension to density[44] The band factor for the drop

period I-value is given by;

K, = { (~/1/D~ - ~/2/D:)/(~/M,x/DM:x- 7Mi,/DMin) } (16d)

where here 71 and 7z are the bounding range of surface tension maxima and minima values

associated with the specific test being conducted.

This is a suitable formulation for the "band factor" and has been devised to satisfy that the

practical requirements discussed above are met. Only in this discussion of the drop period will

the actual form of this factor be explained. Elsewhere in the discussion, it will be assumed that

the form of the band factor will be understood from extending the argument based on the drop

period.

662

These user can enter values if a corrected I-value is to be obtained. Corrected I-values can be

obtained for either the statistical or the measured I-value. In this discussion we will use the

example of the analysis of Guinness to illustrate the point. Assuming that the production records

show that range of the surface tension values obtained for Guinness was 40.5 to 45 mN/m the

range of the specific gravity is 1.005 to 1.016 then the value of K1 can be computed from the

values in Table 4 and gives a value of 0.0932. This value K1 therefore tells us what we already

intuitively know, that is, Guinness has a greatly reduced drop period variability when compared

to the full range of liquids.

The statistical I-value here is now easily evaluated. The statistical I-value of the drop times TD

and TD' of the reference and test liquids will be produced if the criteria of an overlap of RD is

met. We can conservatively calculate the statistical I-value (P1) from the following;

P1 = integer part of { 1 + ( A R D - 1) * KI } (16e)

The discrimination ARD is obtained from Equation (16b) with a value for ATD of 63.5s. This value is

obtained simply from the fact that the largest measured drop time working with a 250 ~tl syringe and

the standard 9ram diameter drop head has been 103.5s and the shortest has been 40s.

This value P1 has a limit of unity when the user enters a Test Specific(TS) range that is narrow

enough such that ~:l becomes small enough to drive the second term in Equation (16d) to less

than unity. Physically, such a narrowing of the TS range would be saying for example that if the

Guinness batches are all identical. The odds on a false-positive match between batches is then

unity i.e. all batches for a perfect quality control situation are identical in every detail and the

tensiograph is unable to distinguish one batch from the other. Experiments conducted on

Guinness samples obtained from commercial outlets have shown that the statistical I-value is

about 120. The odds of a false-positive match between batches is a little over 100:1.

Consequently, once in approximately 100 fingerprints on batches the test sample will be falsely

identified. The tensiograph has at present three LEDs producing different tensiotraces for which

for Guinness are markedly different. Clearly the fingerprint match for Guinness would be much

more secure if three such matches were obtained. The statistical I-value here for drop time and

this Guinness study was 5.6 obtained from the following calculation {1 + (77-1) * 0.06065}.

663

The drop period therefore contributes about 5:1 on the odds of identifying a batch of Guinness.

Overall here that other trace features contribute another 20:1 to the odds.

For the three examples in Table 4 above, working with the regular instrumental set-up, and

using the appropriate K1 values respectively for sugars, and alcohols are

0.0199{(.00043+0.01948)/2} and 0.055{(0.0714+0.0387)/2}. For purpose of illustration here,

we shall use the ARD value for 5% fructose and the alcohols of respectively 551 and 1050 since

these are values obtained experimentally. These give approximately values of P1 of

10.5{ 1+550"0.0199 } and 57.8{ 1+1049"0.055}. These numerical examples are included to aid

understanding of the operation of the I-values.

Experiments to fingerprint sugars are described below. These show in practice that, as one

would expect from common sense and the small calculated value of K1 here, it is a very difficult

experimental problem to separate sugars at low concentrations using the tensiotrace at a single

waveband. On the other hand, it is obvious that the relatively large value of P1 for the alcohol

problem would suggest that an tensiograph identification would be very simple. This is indeed

the case.

The derivative of the tensiotrace signal (S) has been used to identify all the positions of the

maxima in the trace. Repeat measurements with the existing tensiograph have shown that there

are radical differences in the reproducibility for the measurement of the position of different

peaks in the tensiotrace. This arises because of the different shapes of the various peaks. The

rainbow peak is usually very flat and is quite different in form from the tensiograph peak. The

resolution of the system for finding the two peak times and the peak heights are given for the

four liquids in Table 5.

The rainbow and tensiograph peak time measurement I-values are determined from the

tensiotrace. The differential of the trace is used to determine a slope at the peak position and the

reciprocal is taken to be a measure of the peak width. This determination is done empirically

from data, but has then been encoded into an algorithm. The whole tensiotrace is

concerteenered and foreshortens by the drop period variation of a liquid. This can be understood

clearly by giving consideration to a trace produced of a surfactant solution added to water. The

trace is very similar in form to that of water but the drop period can be easily halved by the

664

addition of surfactants. This effect foreshortens both the rainbow and tensiograph peak times.

The measurement 1-value is therefore proportional to the respective peak time for the specific

peak in question divided by the width of the peak, times the two correction factors applied to

the drop period.

Table 5. Table of typical values of M values for samples with standard deviation shown in brackets for 5%

fructose, alcohols and Guinness measured in all cases against water whose measured values are given.

The number of samples taken for water was 35 and for all the others was 10.

Sample/ Trace feature

Water (Reference)

Fructose

(5% w/w)

Alcohols

(Amyl)

Guinness

Drop period

101.98(s)

(0.046)

0.986

(0.0073)

0.46973

(0.0062)

0.654

(0.00125)

Rainbow Peak Time

43.97S

(0.67)

0.997

(0.011)

0.8361

(0.00212)

0.77

(0.00104)

Rainbow Peak Height

1169 (Arbitrary units) (15)

1.2104

(0.004)

13.07

(0.15)

1.49

(0.0088)

Tensiograph Peak Time

88.18(s)

(0.046)

0.99

(O.OOO6)

No peak in the tensiotrace

0.723

(0.0012)

Tensiograph Peak Height

2871

(Arbitrary units) (12)

1.069

(0.003)

No peak in the tensiotrace

0.6412

(0.0065)

The peak position is taken to be determined to a precision given by "a peak position resolution"

obtained from the tensiotrace by the computer algorithm. This algorithm determines the position

from the slope of the derivative trace at the peak maxima. A smaller error in this time is

produced by a sharper peak. The time resolution is determined by the reciprocal of the

derivative slope at the peak maxima multiplied by a constant introduced to experimentally match

the computer value to some trial experiments made on a range of traces.

Atl-k2/(-~) ( t = T , ) (17a)

for peak times Tn where typically n = 1, 2 and 3 and where k2 is an experimentally determined

constant (units Vls2). Since the whole trace length is dependent upon the drop time, any

variability in the drop time has repercussions on the other trace time measurands. It has been

665

found to be convenient to factor into this peak width calculation the standard deviation of the

drop period. In short therefore, a poor reproducibility of the drop period will result in a

deterioration in the rainbow and tensiograph peak times.

At,-(k2*a)/(-~) ( t = T , ) (17b)

Experiments on the typical rainbow peaks give conservative probabilities of chance matches

between the traces of 10:1. For the usual much finer tensiograph peaks values as large as

280: l(for details here see the discussion below on the I-functions) are obtained. The standard

deviation default value for the present system has been taken to be 0.07 for the drop period.

With a value adopted in the software of 10 .2 for o * k2 in Equation (7a) widths for the rainbow

and tensiograph peaks of water 5.9s and 0.35s respectively were determined by the software

algorithm. These values corresponded to very conservatively determined experimental values

taken from the traces by eye. This algorithm results in a I-values of 3:1 and 40:1 respectively

which are both very much lower than the hand and eye values obtained by graphical methods. It

is concluded that the algorithm produces safe values for peak width and corresponding I-values.

This value corresponds to an I-value for the drop periods of 138 determined by the algorithms

which when compared to the value for the tensiograph peak is smaller by a factor of 3 which

agrees with the experimental evidence. The drop periods are determined from experimental

optical eye measurements which have a small standard deviation compared to that of the

software differentiation of the A/D data from the fibres which determine the trace peak position.

The precision with which peaks can be determined is obviously rather arbitrary and depends on

a the exact form of the computer algorithm employed, the S/N and the actual peak shape.

Table 4 gives some indications as to the accuracy of this determination. Also a large value of k2

has been taken which ensures that in the present work the algorithm generates a truly

conservative value.

The drop periods are determined from experimental optical eye measurements which have a

small standard deviation compared to that of the software differentiation of the A/D data from

the fibres which determine the trace peak position.

666

The precision with which peaks are determined is obviously rather arbitrary and depends on the

exact form of the computer algorithms employed, the S/N and the actual peak shape. Table 4

gives some indications as to the accuracy of this determination. Also a large value of k2 has been

taken which ensures that in the present work the algorithm generates a truly conservative value.

For practical purposes, to the estimate of probability of matching the peaks of two curves at

times Tn and Tn' with peak position resolutions of ATn and AT'n respectively, the larger of the

two values has been taken so as to ensure that the most conservative estimate is obtained. The

value of this constant has been arrived at by giving careful consideration to experimental curves

and it has been assumed that the measurement I-value probability P'2 of the matching of peak

times T1 to Tn is given by;

P*2 -- { [ k n ( R 1 - 1 ) ] + [ k 2 2 ( R z - 1)]+ ..... +[k2,(R~- 1)]} (17c)

where;

R1 = T1/ ATI; Rz = Tz/ ATz etc. (17d)

dashed terms are employed here if the values of the drop peak resolutions for these are greater

than for the undashed term. Here the constant terms are respective band constants for the drop

times and are given by;

k2l = rainbow peak factor = Cl * K1 (17e)

where Cl is an empirically entered constant. This allows for the fact that the drop eye

measurements (that generate the first term in this equation) might require different constants and

hence;

k~ = first tensiotrace peak factor = c2 * K1 (17f)

where c2 is an empirically entered constant to allow for the fact that the A/D measurements

taken from the tensiotrace are of a lower accuracy than the drop eye measurements.

k~ = second tensiograph peak factor etc.

The software averages the data to produce readings in time that are smoothed. The data

acquisition period, At,, is determined by the software and is:

667

Ata - ND (17g) ~a

To ensure that the software probability calculation for P2 is not contravened by the data

acquisition condition adopted for smoothing

At >> Ata (17h)

The value of the empirical constants have been carefully chosen to give software values that

while being conservative represent in some realistic and proportional way the actual probability

of a match of peaks of two traces.

The statistical indicators for the peak periods are derived in a different way to the measurement

indicator. The statistical indicator is obtained directly from the drop period statistical indicator

given in Equation (16e). Experimentally it has been found that the rainbow peak can occur in

liquids over a time-range of 70% of the drop period. The peak sometimes appears near the start

of the tensiotrace and sometimes near the end. Its position in the tensiotrace is dependent on

form of the drop for the test liquid. The tensiograph peaks are however always located near the

end of the tensiotrace and over a more restricted time-range within the drop cycle. The range of

position for the tensiotraces within the drop period is 20%. The other factor in the calculation of

the statistical indicator for peak periods is the peak widths which were calculated with

algorithms described above relating to Eqs. 17.

The statistical indicator for the peak positions is calculated from the drop period indicator as

follows;

P2 = P1 * [ 0.7 U21 + 0.2 K22 + 0.2 K23 ..... + 0.2 Kzn] (17i)

where; K21 = constant for the first peak (usually the rainbow peak) = K (cy/AT1) and

K22 = constant for the second peak (usually the tensiograph peak) = K (cy / AT2) and so on. The

constant K is an empirically derived constant that relates the standard deviation of drop periods

and peak width to the experimentally derived value for P2. This constant has been estimated

from experimental trials over a long period on a wide range of measurements. The above

considerations apply only to the standard concave drop head.

668

3.2.3 Conceptual Basis of the Height I-functions

The basic conceptual basis for the Height I-functions comes from the fact that the various

tensiograph peak heights can vary only within certain physical limits. Physically we know that

the height of the rainbow peak is largely determined by the refractive index of the sample, but

this height is to some small extent attenuated by any absorbance of the solution. The tensiograph

peaks conversely, are more sensitive to absorbance effects than to refractive index effects. In the

experimental section of this chapter these effects are quantified for water based solutions.

The tensiograph peak has a size that can vary from any maximum value to zero as the

absorbance effect can completely attenuate the signal to pull the tensiograph peak into the

noise. Given the S/N of a test trace has been experimentally determined, then we might imagine

that this entire peak height may be divided into 6.8S 'height boxes' to give what is referred to

here as the Full Range Resolution. This resolution determined for the test trace can be thought

of as the number of the boxes that might be possible candidates to locate the position of the

comparison trace. The question to be answered is what is the likelihood of a match in height

between the test and reference traces. The traditional approach in statistics of considering boxes

into which balls can be placed here is the perfect analogy to employ. The number of boxes are

given by calculating the resolution of the reference liquid. The balls correspond to the position

of the test trace in that its position with respect to the boxes defined by the resolution determine

into which the peak goes.

Certain complications arise to make this a difficult question to answer. The Full Range

Resolution is modified firstly by the fact that there are refractive index variations from sample to

sample. These variations change the form of the tensiotrace to some extent. In a simple minded

way, these variations can be accounted for by increasing the size of the 'height boxes'. There is a

corresponding reducing in the resolution. This refractive index effect should as a rule for most

practical situations be only a slight perturbation on the I-value calculation. The fact is however,

that the user in most typical measurement problems, will only expect a restricted range of

absorbances from sample to sample. This means that there will only be a restricted range of peak

height variation. Obviously then it follows that the number of possible boxes will be restricted

and the described above for the tensiograph peak. The question then once again arises as to

669

what is the likelihood of a match in height between the test and reference traces. The actual

number of statistically discrete boxes obtained for any given rainbow peak will in fact be once

again in a practical quality control situation be restricted. The restrictions here however are

more than will be observed for the tensiograph peak. The refractive index variability of this

rainbow peak is the first concern. It is found that there is a non-zero lower limit of these

statistically discrete boxes for this peak as the lowest practical refractive index values still

produce a peak height that is not zero. A universal range factor r'u is introduced to limit the

number of these imaginary statistical boxes to those from the upper limit corresponding to the

highest possible refractive index value of 1.749, down to the lowest level that can be occupied

which is determined by the lowest peak height that is found to be 1.283(as can be seen from

Table 4). This 26.6% refractive index range translates in practice in the tensiograph to peak

height variations of approximately 200 times in the rainbow peak height variation. The value of

r'u is taken to be an experimentally determined constant.

Fig. 27a Tensiotraces recorded at 950nm for a series of copper sulphate solutions 0.0025, 0.005, 0.01, 0.02 and 0.03 M.

The user may know that they are dealing with a set of samples with a restricted range of

refractive index values. Using once again the example of beer analysis to illustrate the point, a

670

narrow range of such values is expected for this measurement. The universal range factor must

consequently be restricted by the User Rainbow Peak Range Factor 8N. The resulting double

restriction produces what is called here the Practical Rainbow Peak Range. The fact that the

colour of the liquid can vary is also something however that concerns this calculation, because if

the liquid can have a range of colour variability and these variations clearly can not be separated

from the refractive index effects. However, because the colour variations are smaller than the

refractive index effects, these must be viewed here as an interference of the refractive index

value.

Fig. 27b Tensiotraces recorded at 660nm for a series of copper sulphate solutions 0.0025, 0.01, 0.03, 0.1 M.

If there is in fact an absorbing species in the sample, then the rainbow peak will be reduced in

size, but this will not of course in any way effect the refractive index variability of this peak

height. Never-the-less, a correction factor must be applied to the 1-value to take effect of this

variability in absorbance values. If the user knows the possible range of colours of the sample

then the so ,ware calculates a correction to reduce the 1-value probability as the colour here is

an interference on the actual measurement being considered. The sizes of the resolution boxes

are increased to account for the effect.

671

Fig. 28a Graphs of rainbow and tensiograph absorbance measured at 950nm

Fig. 28b Graphs of rainbow and tensiograph absorbance for LED-tensiograph measured at 660nm.

672

3.2.4 Relationship Between the Tensiograph and Rainbow Peak Variations

The behaviour of the rainbow and tensiograph peaks are different and attempts have been made

to quantify these differences experimentally and to theoretically describe these differences by

giving consideration to the two extreme cases. It is sensible to begin with the simplest of these

cases, namely that in which solutions show no refractive index variations but possess an

absorbing species. To illustrate this case a solution of copper sulphate was used as this has a

strong absorbance at the present LED-tensiograph operating wavelengths of 660nm and 950nm.

This effect was fully explored with solutions in the range up to 0.05M. A set of tensiotraces are

shown for these solutions in Figure 27a and 27b at the two wavelengths of operation. Figure

28a and 28b shows the rainbow absorbance A1 and tensiograph absorbance A2 values for

respectively 950nm and 660nm of all copper sulphate solutions of various concentrations up to

0.1M. The results have been fitted with a second order polynomial because of the saturation in

the plots from the limit of the Beer's law region.

The tensiograph absorbance AN was defined in Equation 4 and 5 above. It will be seen that for

the absorbance measurement that;

(dA2 / de ) > (dA, / dc ) (18)

because the path length of the light in the pendant drop is greater that that of the light in the

remnant drop. The rainbow peak signal develops for all water type drops and in most other

cases in the remnant drop phase and the tensiograph peak to the pendant drop phase. The linear

range for these effects is limited as with the situation that pertains to Beer's law in the usual

UV-visible analysis. The current study shows that this corresponds roughly to 0 to 0.4 A-units

for the present 950, 770 and 660 nm operation.

An increase in refractive index of a sample will increase the coupling between the source and

collector fibre. This effect can be represented by a reciprocal coupling coefficient (~ N) and this

can be defined in a very similar way to the tensiograph absorbance but which is in fact the

reciprocal relationship to this. This coefficient is defined formally as;

~ = log( HN~R n) (19)

673

where HRN is the size of the rainbow peak reference signal which is usually water, HN is the size

of the tensiograph peak test trace signal, N is the number of peaks and here 1 will be the most

usual suffix as this corresponds to the rainbow peak.

Figures 29a and b show a series of tensiotraces for sugar at 950 and 660nm operation.

Figures 30a and b show s the graph of the results for both the rainbow peak ~,1 and for the

tensiograph peak ~,2 both plotted against refractive index(q). As can be seen the results show a

marked variability in behaviour of the peaks in that;

(d ~l/drl) > (d ~2/drl) (20)

because it would appear that the rainbow peak produces a coupling mechanism which is focused

and therefore strongly influenced by the refractive properties of the liquids. Alternatively, the

tensiograph coupling is defocused and therefore only to a much lesser extent dependent on the

refractive properties.

The refractive index coupling relationship is very linear in the range 1.3 to 1.42 but this

increases quite sharply above this value. This effect has been described elsewhere by McMillan

et al. [44]. The relative effects of the absorbance dependence and the refractive index

dependence is defined by this present work in a general way for the first time.

The two graphs shown in Figures 28 and 30 allow us to see the respective variability of the two

effects on the traces and we can see that;

d A660/dc = ~660 * d ~,660/dn (21)

where experimentally we have determined that ~660 = 5.8 for the 660nm operation. This is

determined from the ratio of the slopes of the absorbance and refractive index graphs which are

respectively 8 and 1.38. The units of ~660 are A-units per mole. This factor gives the numerical

constant to relate the concentration measurement to the refractive index variation. In effect the

constant for the refractive plot equalises the slopes of this graph so that this matches the slope

produced by the absorbance graph. This knowledge allows us to see how the two effects relate

one to the other, the absorbance effect diminishes the size of the trace and the refractive index

increases the size of the trace, these constants express the numerical relationship between the

two effects.

d A950/dc = ~950 * d ~,950/dn (22)

674

Fig. 29a. Tensiotraces recorded at 950nm for a series of sucrose solutions 0-40% w/w.

Fig. 29b Tensiotraces recorded at 660nm for a series of sucrose solutions 0-40% w/w.

The experimental value of ~950 found for the tensiograph peak is 1.16 taken from slopes of the

absorbance and reciprocal absorbance graphs of 2.8 and 2.413 for the 950 operation It will be

noted that the 660nm operation is markedly different to that at 950nm because in the first

instance the radical difference in absorbance of the blank being 8 at the lower wavelength and

only 2.8 for the 950nm operation.

675

Fig. 30a Graphs of reciprocal absorbency at 950nm for a series of sucrose solutions 0-40% w/w.

Fig. 30b Graphs of reciprocal absorbance at 660nm for a series of sucrose solutions 0-40% w/w.

3.2.5 Height 1-function Resolution

The algorithm for determining the probability of a peak-height match can be expressed in a very

similar function to the drop-time formulation. The difference between the two situations is that

the statistics of drop-time depend on the A/D acquisition rate while the peak-height in the first

instance depends on the noise (SVn) of the system. The software gives two types of I-functions.

676

The first, known as the universal resolution is related to a typical optimised value. The second

value, known as the measurement resolution relates to actual height recorded for the specific

measurement in hand. The measured height resolution an d the universal height resolution give

the user information on how well optimised the measurements being taken are.

The measurement height resolution (R) is derived from the experimental data set. The digitised

data points that comprise the trace are here represented by the subscript n. The practical

incremental division of a peak is the resolution (RN) is given by;

RN HN H' - o r N (23) 6v. By'~

The dash represents here the test trace while the undashed represents the reference trace. For the

system with signal smoothing from averaging the improved peak-height division is given by

RN = ~ or ~iV n 8V' n (24)

where ND is the number of data points averaged. In the present sottware this is set either to 10,

25, 50 or 100 to give signal enhancements of 3, 5, 7 or 10.

The resolution is the fundamental quantity of the Height indicator values. This value can be

modified to take account of various physical effects on the trace to generate an I-value to

correlate strongly with statistical measurement trials on liquid samples.

Using the Siemens SFH 458 plastic fibre optic divide and detector with lmm PMMA fibres and

a 9mm diameter concave drop head and no signal averaging a typical tensiograph peak S/N ratio

was found to be about 15:1. For signal averaging with 100 points the incremental division is

approximately 300:1 for the tensiograph peak and for water with a rainbow peak approximately

1/3 of the size of the tensiograph peak the incremental division is about 50:1.

The universal height I-value (RN) has been taken to be 15:1 as this is a very conservative figure

which should be met in a practical measurement situation. The user can compare values derived

from the measured height and S/N by clicking between the universal and measured values.

677

3.2.6. Null Conditions for the I-function Algorithm

The algorithms developed for the matching of traces, include conditions for setting any

'Individual I-value' for any chosen trace features to zero, if a matching criteria is not met. The

'Cumulative I-value' will go to zero if any of the individual test criteria for the drop-time, peak-

positions or peak heights is outside of a specified range of acceptability. If any of these

conditions produce a zero Individual I-value, because of one of the matching criteria being

contravened, then the two liquids being compared are not the same with regard to some

measurable physical or chemical property. Consequently, the Cumulative I-value is set to zero.

It should be understood that the instrumental criteria here being defined for these null conditions

are with respect to both the instrumental capabilities of the drop eye system and the MD for the

opto-electronic signal.

The criteria for non-matching and forcing the respective individual I-values to zero are respectively;

Drop times;

Absolute{T N - T ' N } >= 6AtN

Absolute{T N - T ' N } >= 6AtN / ~/ND

{No -averaging} (25a)

{Averaging} (25a)

where the symbols have the meanings as defined for Equations (7a) and (7c).

Drop Heights;

Absolute{H N - H ' N } >= 6b'I-I {No-averaging} (26a)

Absolute{H N - H ' N } >= 66H / ~/ND {Averaging} (26b)

The matching criteria for the range is three times the standard deviation for both traces. A factor

of three standard deviations implies that the range will enclose 99.7% of all values. The user will

be visually alerted by the so,ware if the matches are outside acceptable limits. A screen flags

warns the user to the failure of the matching to reside within these experimentally established

conditions.

3.2.7. Tensiograph Height I-functions

Given that the conditions are realised for Equations (13a) and (13b) then an I-value will be

generated. The height resolution given in Equations (11 a) and (1 l b) must be modified by three

factors to give the height I-value. In the case of the tensiograph peak I-value the 'tensiograph

678

peak universal range factor' (gt) goes to unity as experimentally we know that the peak height is

reduced to zero by a large absorbance of the solution. This factor is given by;

gt = (H2 . . . . - H2,,,. ) /H2 .... = 1 (27)

where Hzm~x = maximum size of the tensiograph peak, Hzm,,, = minimum size of the

tensiograph peak = 0

The second factor that restricts the I-value is the absorbance range that applies to the

measurement problem. The tensiograph has at present a very useful, but restricted upper range

of absorbance capability. The maximum useful absorbance measurement limit (Alimit) is 0.41 A

units. Obviously, the user must restrict their absorbance range to within this measurement range.

The software checks to ensure that the inputted user's range complies with this limit and

A l i m i t (0.41) > An (2S)

The absorbance user range factor Aa is given by;

Aa = ( AH- A, ) / Alimit (29)

The final correction factor for this I-value is the refractive index correction factor (Fn). This

factor is obtained from the variation in tensiograph peak height with refractive index. The

preliminary investigation into this correction factor used the measured dependence of the

tensiograph peak with refractive index for 0, 10, 20, 30 and 40% w/w sucrose solutions. The

slope of the resulting peak height-refractive index graph was found to be p, = 0.452. This

consequently gives a refractive index correction factor of;

�9 a = 1 / (1+ pa [ 1 - A N ] ) (30a)

where AN = (rlr~- Yll)/rllimit. The value of l]limit is determined from the limiting range of refractive

index values that can be measured with the tensiograph and is taken from Table 3 and has a

value 0.466. Combining the factors given in Eqs. 14, 16 and 17 we obtain the final tensiograph

height I-value functions for the universal and measurement values are respectively P2 and P2 ;

P2 = R2Aa~nandP2 = R2Aa~n (31)

679

3.2.8 Rainbow Peak Height I-functions

As with the above case of the tensiograph peak I-values, the universal and measurement

rainbow peak I-values (Pl and P1) can be similarly calculated using corrections to the rainbow

peak resolution. As with the tensiograph peak, there are three correction factors, but in this case

all three are active with none being unity. The rainbow peak universal range factor gr is given

by;

g r - - (Hlmax- H1 ram)/ -- approximately 0.75 (32) /Hlmax

The refractive index user range function is given by;

This value then used to get a refractive index correction factor of;

O~ = 1 / (1 + Pr [ 1-AN]) (3 3)

and empirically it has been found that a value of 10 for pr gives reasonable changes in the I-

values with practical ranges for refractive index variations.

The final correction is to account for the absorbance effect which will reduce the height

tensiograph peak. Any absorbance variability acts as a perturbation on this measurement. The

user inputs an absorbance range AA which is given by (AH - Al) and in quality control situations

this range of variability will be well known to the QC laboratory. The use of Beer' s law can then

be used to determine the absorbance correction factor which is;

FA = 1 /log IAA (34)

This gives the universal and measurement I-value function for the rainbow peak height as

respectively;

P1 = R1 gr (I)~FA and P1 = R1 gr ~ F A (35)

where as with the tensiograph peak the universal resolution for the rainbow peak (R1) is taken

as a standard value to be 15:1 obtained from typical optimised values.

3.2.9 Percentile Point and Area I-value

The analysis of the point data is done by dividing all the trace data up into a 100 average point

array. Consequently, this analysis produces a quantity known as the percentile point M function

680

giving a percentile point I-value. The percentile point resolution for the measurement could then

be defined simply as;

100

R' , - Z H'f133 677 (36) i=1

Here the height of each of the 100 individual sets of averaged data points are divided by the

experimentally determined noise value which is multiplied by a factor and this ensure a

conservative estimate of the resulting I-value.

A fundamental problem here presents itself. Since the data points are not independent of each

other, the value generated by this algorithm is incredibly large and typically produces figures in

excess of 105~ Obviously this is an unrealistic statistical figure. The tensiotrace signal is

produced by the interaction of light with the test drop and its form is fundamentally derived

from the various combinations of physical properties of the liquid drop. These properties are

refractive index, absorbance, density and surface tension (viscosity for constant head delivered

measurements). Logically, it is an inescapable conclusion, that an I-value extracted from the

trace comparison using the software, should not exceed the statistical values that would be

obtained from the analysis of each of the constituent properties of the test liquid employing the

various separate analytical instrumental methods to measure these physical properties. If we

consider the resolution and reproducibility of the instrument as presently constructed for these

various measurements, then a rough estimate of a I-value would be arrived at. This information

is presented in Table 6. The final column in this Table indicates which liquid was used in the

specific study of the measurand. The refractive index values in this table were derived from a

study made of fructose at the limit of resolution of the technique for refractive index

measurement. The figure quoted for the Specific Gravity are for the Paar Density Meter and

relate to experimental observations from a wide angle of samples.

Consequently, the maximum possible cumulative total for the I-values for the tensiograph(not

using the information from the Paar Density Meter) would be 2.52"10 ~2 and this value would be

higher by a factor if the instrument was used in conjunction with the Paar density meter. It is

obvious that this individual point I-function algorithm does not generate logically sensible values

because the result 10 51 is unrealistic. A different approach is necessary.

Table 6. Values of Estimates of the Resolution for Measurands of The Tensiograph and ~ e d Estimates at I-values.

681

Measurand/

Factor

Refractive Index

Specific Gravity

Specific Gravity

Absorbance

Surface Tension

Resolution

0.0005

0.0003

0.0002

0.002

O. lmN/m

Reprodu- cibility

0.0001

0.0001

O. 0001

0.001

0.01 mN/m

U-Range

0.466

0.9292

0.9292

2 A-units

87.28mN/m

U-I-value

933

3097

4646

1000

873

Measurement used to determine data

Fructose

Beer

Paar Specification

Copper Sulphate

Sugars

The Percentile Point I-value should not therefore be (given the present experimental resolution)

greater than this limiting value of 2.52* 1012. For consistency, this value should also taken to be

the value for the maximum value of the Percentile Area I-value since both points and areas

conceptually measure the same thing. As a consequence only one value in the software is

generated for both the points and area measures. This value is taken as the Resolution for both

the points and area measures hence;

(Rp)Max = (RA)Max = 2.52"1012 (37)

As with the earlier discussion on these matters, the points (and Area Function) are also subject

to a correction factor. For this specific case, the correction factor can be obtained from the

extension of the results given in Equations 7, 16 and 20. The Percentile Points I-value (Pp)is

obtained as follows;

Pp=integer part of{ 1 + (Rp-1) * ~:1" {(An- AI)/(Am,x - &~)} * {(tin- rll) / (riM, x- rl~r~) (38)

where the symbols have their usual meaning. Obviously, the Percentile Points I-value is set to

zero if any of the conditions given in Equation 6, 13 or 14 are obtained.

Despite the 'point matching' and 'area matching' being conceptually the same, the points and area

match algorithms are different. Consequently, the computer generates different quantitative

information from each algorithm. These differences, though small, show that the I-values are

looking at the data set in slightly different ways. It may therefore be useful for the user to have

682

available all this computer information. This quantitative information gives conceptual the same

information about the two traces. It has been shown in some practical situations that both sets

of information are very useful. The user can therefore seek to use the algorithmic differences for

their own specific ends, for example, if it is discovered that the digital point algorithm yields a

higher correlation coefficient with some measured product parameter, then obviously this

discovery would encourage this user to click off both the analogue point and area matching and

rely only on this Digital Point I-function.

As in the case of the Point I-function considered above, the Area I-function must also be defined

as a percentile quantity to allow it to maintain a close correlation with the Point I-values. There

is a condition at which the Area I-value is set to zero. This limit for the null condition indicates

that when the two traces are compared there is a limiting stage of divergence. A move one way

will produce a non-zero positive I-value and the move the other way will show a message box

on the screen to indicate that no match was found. This trace is called "the null condition trace".

The conditions that must be met for this null.

~ S t ~ l t ~ h ' ~

/

Atoo ~ R~to t

a) b)

The final stage in obtaining an Percentile Area I-function requires some conceptual explanation.

Suppose that the I-value for each of the hundred averaged data points used for this percentile

analysis, can be represented the resolution, RA {this would be (RA)Max for an ideal trace}. All

Fig. 31 a) Tensiotrace with 3" 8V noise range limits of matching criteria; b) Graph of area indicator factor

plotted against difference in area between the test and reference condition are defined, as with the Point

I-value, by Equations 5, 6, 12 and 13. If any of these criteria apply then the Area 1-value is set to zero.

683

the various points at these 100 positions for both the reference and test trace are identical. The

calculation of the resolution assumes that each of the 100 position yields a value that is

independent of its neighbour. This assumption at first sight appears to be a good one as typically

a tensiotrace comprises more than 20,000 data points and thus the 100 positions on the trace

selected for this calculation of resolution are each separated by approximately 200 data points.

As explained above this is not the case.

If the test points of the trace vary in by more than 66V both, above or below the actual data

points in the reference trace, then this would be deemed to have strayed absolutely outside the

criteria adopted here for a match of the two points. Suppose that an area is calculated for a trace

which is produced by adding this incremental quantity to the reference trace, then this trace

would be "the null condition trace" and has an area that will just give a Percentile Area I-value

of zero. Figure 3 l a shows the situation for the two limiting traces shown drawn above and

below the reference trace. We actually use the incrementally increased trace, because as can be

seen, there is a possibility that the incrementally reduced trace will have negative contributions.

If the incrementally increased trace has an area represented by AMax then the area difference that

just corresponds to this null condition trace will be;

AANull = AMax - AM ...... d (39)

Logically it is obvious that the value of the Percentile Area I-value varies from a maximum,

when the two traces are identical, to just zero at the allowable limit to the trace differences,

namely AANull. This situation is shown in Figure 3 l(b). This condition is produced simply as

follows;

P5 = RA(1- AAMeas/AANun ) (40)

where AAMeas is the difference measured in the area between the reference and the test trace.

3.2.10. t-test Analysis

A second and more fundamental way of looking at this problem matching the data sets of the

reference and test traces is to use the t-test. The tensiograph software executes a t-test on the

data and gives a result to the user for each analysis. The 100 sets of averaged data points from

the reference and test traces are treated as being paired data sets in the standard way. Tests on a

684

sample of 35 traces, taken under the same conditions, were used to determine typical standard

deviation values for the data points. It was found that these varied little across the entire data

field and a standard value has consequently been adopted for this measurement error. A

standard difference between the data sets are calculated in the usual way using the difference

square, but once again a normalised value is used. An M-value is defined using the paired data

sets by~

M 4 c : f(hm) : ~'~m=l~176 ~.,~1 (41)

where hm is the size of the reference trace of sample P and h'm is the size of the test trace Q for

the mth point of the data.

The sol, ware establishes from this 'point' M 4c function the goodness of fit of the data sets.

Typical fit of 99.65% are obtained for repeat measurements. This M-value has been designed so

that it correlates strongly with the o~ value obtained from the t-test. The analysis is then

extended to obtain the cz value using the following steps. Firstly, the averages of the difference

is obtained and from this the value D of is computed.

__ ,~--~ 1 00 dn 1//~00 0 m ,,-..-,100 Dn 1//~00 0 d - ~.,n=l and D - ~--~-1 (42)

where Dn - ( d i - d ) . The standard deviation of the differences is then obtained from summing

the differences between the averages and all the individual differences thus;

It is assumed that the paired differences constitute a random sample from a normal distribution

N(6 , CD), an cz % confidence limit for a mean difference 6 given by;

15 = d + t~ * SD/ X/100 (44)

where t ~ is based on the degrees of freedom which is here 99.

An algorithm is then used to determine the value of ct. This algorithm is based on the t-test

tables for a sample with 99 degrees of freedom which have been analysed to give the following

relationship between t ~ and c~ was obtained;

t ~ - 3.9 - 0.74 [logo~ + 4] - 0.00059 [ logcz + 4] 6 (45)

685

A computer loop is then used starting with a loop value for or, of 0.01 and moving in

incremental steps of 0.1 up to 0.1. The value of t ~ is computed from the value of ot and then 8

is determined from Equation (44). The data set of differences are then checked to see how many

data points lie outside the range value so calculated. The process is repeated until a satisfactory

result is returned or the loop exhausted. If there are no data points outside the range for the first

calculation then the following message is printed, "Data fits are too good for standard t-test as

a (0.01" and if the final loop value of 0.1 is reached then a message is printed "Insufficient data

fit for standard t-test as a)0.1" If the loop finds a situation between these limits in the loop

where the number of data points outside the calculated range is less than that predicted from the

calculation of/5. The loop is ended and the value of ct is printed.

The software gives the user the option to export the data sets for the 100 matched pairs from

the reference and test traces into EXCEL, or other statistics software packages such as SPSS,

to allow the detailed analysis of the data with more advanced statistical methods. The above

computer algorithm determines the values of the standard deviations of the differences for a

reference and test trace for any measurement. A 0.1% ot criteria is adopted to give a 3.16

standard deviation for a 100 data set. If the values for any actual reference and test trace

measurement is then within the t-test(approximating here to the normal) values, determined

from the experimental measurements described above, then a message is indicated by the

software that "The test and reference data sets match within the criteria for a t-test".

3.2.11 Experimental Testing of 1-function Analysis

It is beyond the scope of this discussion, to fully explain the applications of I-functions to

analysis. It must suffice to say here that some pollution studies on water, test measurements on

beverages, preliminary studies on some enzymes and a general range of measurements on

various laboratory liquids have been made. Table 7 shows some values obtained from the I-

value analysis for a selection of samples. It perhaps goes without saying that here the values

have been obtained from two different samples that are a "fingerprint match" and therefore give

non-zero I-values. The individual trace feature I-values thus obtained give an indication as to the

statistical chance of a false-positive for the match of respective trace features. A cumulative I-

value gives a product sum of the five individual I-values. The tensiograph incorporates in it a

686

Paar density meter and this can also be used to produce a further individual I-value value.

However, since this instrumental feature is optional it is not included in this discussion. It will be

noted that the I-values for the beers and cider are all very close in form, and as would be

expected since the traces are themselves very similar. The whiskey and the ethanol are also have

values that are close as again would be expected because the traces are visually quite similar. In

some traces there are further peaks in the tensiotrace that can be analysed with I-value analysis

but since these peaks are not related to other physical properties of the liquids they are not

discussed here.

Table 7 - Probability values for tensiotraces for a selection of liquids obtained from the software.

Sample/I-value P1 P2a P2b P3a P3b ZP

Water 177 2 74 4 74 7.7"106

Ethanol 80 29 21 27 5 6.58"106

Bulmer Cider 114 36 81 11 11 4.02"106

Murphy Stout 116 28 84 7 4 7.6"106

Guinness Stout 114 50 86 9 3 13.4"106

Irish Distiller's

Whiskey

84 23 28 13 3 2.11"106

3. 3 FINGERPRINT D-FUNC TIONS

3.3.1 Definition of the D-functions

The tensiograph can be used either to fingerprint a liquid and to identify this with a trace held in

a reference library. Alternatively, it may be used to show the differences between this trace and

the reference trace. In most quality control applications these trace differences would usually be

visually small. Differences in a tensiotraces could be used for product control which is either in-

line or based on laboratory measurements. Such an application would seek to look for

deviations from the reference defined by a tensiotrace from some standard or reference product.

The objective here would be to ensure that all trace differences were kept within a range.

687

The differentiation of one liquid from another can be done by defining discrimination functions

which yield D-values. These values give a normalised standard measure of the differences in

measurement feature of the trace. The use of milli-differences has been found to be practical for

the tensiograph discrimination functions. The following measures can therefore be defined;

AT D _ ( T D _ TD ,). 1000 (46a) G

AT. - (T n - T. ')* 1000 (47a) T.

A H - (H. - H . ' ) * 1000 (48a) H .

1000 AA - (,4 - A ')* (49a)

A

As before, the assumption here is that the undashed terms are larger than the dashed, but if it is

not the case then the terms are reversed.

The tensiograph software also gives the differences in all the discrimination terms as relative

differences. Each term is plotted against a selected group of terms that scan be clicked on or off.

If all the four terms are selected by clicking them on, then the relative differences are defined as

follows:

ATD (46b) g' (ar +AT.

AT, (47b) (Ar +aT~

_ A H (48b) g3-(Ar +aT~ +M)

AA g4 (AT D +AT. +AH D +AA) (49b)

Any of the four options may be clicking off. Figure 32 shows the histogram for the differences

between the two whiskey products. This histogram shows all the differences in the various trace

features rather than just the four discussed above. This histogram, for all the tensiotrace

components, shows clearly which of the terms is the most important. This information could be

688

very useful. The difference between various whiskey samples studied have shown marked

differences in products in colour (tensiograph peak height), alcohol content (rainbow peak

height) and surface tension/density ratio (drop period). This figure shows the utility of the

relative difference histogram.

3.3.2 Analysis using the Discrimination Functions

Possible applications of the "Discrimination Functions" are here illustrated using a reference

Irish Distiller's whiskey and making measurements against six other whiskey brands, two other

spirits and also ethanol and ethylene glycol. Table 7 is really self-explanatory but here three

points will be briefly noted. Firstly, in the whiskey analysis the "tensiograph peak height"

appears to be the most sensitive and useful function for measuring differences in the products.

Secondly, the drop periods of all the spirits, are nearly all the same and this certainly is a result

that would be anticipated as they have the same proof. Finally, these results suggest that point

and area analysis is the best way of checking the reproducibility of the product.

Fig. 32. Discrimination function differences between an Irish Distiller's whiskey and Bell's whiskey.

689

In an analytical problem, it would probably be sensible to assess the various utilities of the D-

functions before carrying out a lot of analysis. It would certainly be advisable to do this if

routine product measurements were being set up. In the example here discussed, obviously the

area difference values generated are the largest of all the various available parameter values and

consequently these appear to be the most sensitive I-value for this specific product. It would be

hoped that this analysis would allow the user to maintain their production standards to a degree

perhaps not previously considered possible. The single waveband system described here has

been shown to be quite sensitive to product differences. It might be mentioned that a multi-

wavelength system has been developed which will give even greater discrimination. Work is

presently underway to see if product formulas can be obtained using the tensiograph by

experimental admixtures of known components of a product until the D-values have all been

reduced to zero. At this point a recipe formula would have been discovered.

Table 8. Values of milli-part differences with respect to Irish Distillers whiskey taken as the reference liquid.

Sample/ Difference

Bells Blackbush Glenlivet Glenfiddich Lagavulan Paddy Huzzar Hennessy Ethanol Ethylene Glycol

Drop Period

1

1

6 7 10 1 4 1 55 20

Rainbow Peak Period

Tens. Peak Period(l) 0

Rainbow Peak Height 65 10

10 2 4 120 8 4 45 205 10 3 5 115 13 8 3 185 4 1 110 202 12 11 78 75 5 2 85 196 110 0 157 257

25 85 313

Tens. Peak Height(l) 180

31

3.4 TESTING OF THE FINGERPRINT CAPABILITY

3.4.1 Experimental Results on the M-Functions

Points Area

410 170 300 150 406 195 412 180 353 181 502 198 370 205 403 188 606 720 421 511

approaches that have been developed to date.

It is obviously not possible here to detail all the work on tensiograph fingerprinting. In the

following three sections the example of the analysis of beer will be used to illustrate the types of

690

A study was conducted to investigate the ability of the instrument to differentiate beers. Fifteen

brands of beer (a good range of stouts, ales and lagers) and two ciders were used. The samples

were all degassed ultrasonically (for 15 minutes). The degassed samples were sequentially run,

cleaning the system thoroughly each time a new sample was introduced. Three repeat traces

were run for each brand of beer.

The resulting traces were analysed by computer, comparing all files obtained. The software

compares two selected tensiotraces based on six M-functions. For each of these quantities, the

software compares the two traces by dividing the lesser by the greater. An overall %-match

figure was obtained which was simply the average of all six %-match figures obtained.

The %-match figures were obtained for three repeats of all the products, producing a

51 x51-matrix. The figures for identical traces were ignored (because they were simply 100%).

== = 40

u . I

89 90 91 92 93

i __

94 95 96 97 98

Upper Cell Boundary (%-match)

99 I O0

Fig. 33a Histogram of non-matching samples. The results from a full set of commercial products tested against

all others.

The satisfactorily fingerprint of these beers requires that there should be high %-matches

between repeated tensiotraces of the same beer and comparatively low %-matches for those of

different beers brands. Problems arose where either of these criteria were not met. In order to

691

identify where these problems arose, histograms of the two types of data were prepared

separately, one for the repeat or matching samples (Figure 33a), the other for the non-matching

samples (Figure 33b).

As can be seen from the two histograms, there is overlap of the two distributions. This overlap

arose in the preliminary data because the measurements were performed without vibration

isolation which resulted in noisy traces. This study shows that even with this overall crude

approach the technique can fingerprint the majority of product. The overlapping samples were

identified for further analysis which is described below.

Problem samples

89 90 91 92 93 94 95 96 97

Upper Cell Boundary (%-match)

-I 7Iq 9 8 9 9

Fig. 33b Histogram of matching samples of beer. Note the small set of problem samples separated from the main

distribution of samples.

An important feature of the histogram of non-matching samples is that three separate peaks can be

seen. A detailed analysis of the various component parts of these histogram peaks showed that in a

broad generalisation, that the first small peak in the histogram resulted from ciders matching with

stouts, the second peak from beers matching with lagers, and the third and largest peak from stouts,

beers, ciders and lagers matching with other similar products. The actual details of this analysis show

finer detail than this generalisation permits and a bar chart has been produced to show the various /

ranges for all the various %matches. The resulting plot is similar to that used to identify functional

groups in infra-red analysis. In an analogous way to the infra-red charts the tensiograph charts might

692

be used to identify a product, or sample (the method applies not only to beverages), by matching a

traces of a sample against a standard trace. The percentage match of the product might then be

identified from its position on this tensiograph bar chart.

The preliminary study on beers showed the necessity of improved vibration isolation and the

equipment was modified consequently by the addition of the vibration mounts described above in the

description of the apparatus. This modification radically improved the S/N and tensiotraces were

obtained for all the problem beverages. These problem beverages were Holstein, Bulmer's, Guinness,

Carlsberg, Labatt's, Newcastle, Beck, Foster's, LA beer and Harp.

F i g u r e 43 - Measurements Showing Differences Between Traces of Problem Beers

Fig. 34. Plot of tensiograph features for a series of measurements on 'problem' beers. Each column represents

one measurement.

Five repeat readings were obtained for each brand of the problem beverages. This time,

however, with the experience of analysing the results of the first study, a more efficient

analysis method was used. Rather than compare all the traces to each other, they were instead

all compared to a water reference. Furthermore, rather than simply taking the average result of

the various %matches, a multi-dimensional approach was used.

693

For each beverage tensiotrace, the five %matches of the trace compared to the water reference

were obtained. For each beverages used, the %matches were plotted and the resulting graph is

shown in 43. As can be seen from this figure, the beverages which were poorly separated in

earlier preliminary analysis have now all been satisfactorily separated although three, namely

Carling, Labatt's and Carlsberg are still very similar. The columns of points all obviously

correspond to an individual bottle of a beverage. The scatter in values in the same beverage

shows some information on the quality control of the products. Visually, it is possible to see

from this plot that some products have better QC than others.

3.4.2 Experimental Results on I-Functions

The three beers - Carling, Labatt's and Carlsberg all gave poor separation even using this

improved M-function graphical analysis. The actual separation is resolved using the I-value.

Each trace was tested with the corresponding traces of the other beers and with its own repeat

readings. In all cases the software indicated that the different beers were deemed to be "Not a

fingerprint match" while the same beers were identified as "A fingerprint match". The I-value

analysis is really an objective way of testing the information presented graphically in Figure 34,

but here it gave a definitive decision on the identification. This result is very promising because

the differences between these products are really very small indeed and it is believed that the

brewers themselves would be unable to identify definitively these products one from the other in

a blind test.

The I-value analysis obviously depends on the specific tensiotraces of the sample and reference

liquids, but here some example of the I-values for a range of typical samples obtained with the

single waveband polymer tensiograph system are given in Table 9.

3.4.3 Applications of D-Functions

The application of discrimination functions are wide and the technique is powerful. Figure 35

shows the results the rainbow peak height D-values plotted against concentration of milk. The

two graphs shows that the scattering losses are greater for 660nm than for 950nm as would be

expected from the Tyndall scattering in milk. These plots could be used for quantitative

purposes, but the D-functions would find more application in quality control. The D-values

selected used here is that for the rainbow peak, but a very large and systematic variation is seen

694

also for the tensiograph peak (being respectively 38% and 76% reduction for the 660nm and

950nm tensiotraces). For milk, a small drop period variation is observed for this range of

concentrations giving a 30% variation. The rainbow peak times and the tensiograph peak times

are of no use in the analysis of milk. These typically have only a 20% variability and are rather

inconsistent. Both the area and point D-values are very disappointing in this analysis problem

and show little variability at the concentrations above 1.6 %. In summary, it is necessary to

select the best D-function for a specific application problem. While for milk, the peak heights

have been shown to be the best, for surfactants measurements the most obvious choice would

be the points D-value as these are the most sensitive for low concentrations problems. Every D-

function analysis has to be done on its merits and there is no universal answer to an analysis

problems.

4. SUMMARY OF TENSIOGRAPH ANALYSIS

4.1. PHYSICAL MEASUREMENTS

The tensiograph has the ability to deliver simultaneously several measurements from one

tensiotrace from one drop of a test solution. The instrument is a multi-analyser. Perhaps the best

way here to illustrate this capability without undue fuss is to use the example of some stout

measurements. Figure 36 shows measurements taken on six stout samples, a concentrate

labelled "Concentrate 5" by the brewer which had to be diluted by a factor of 20 times in water

to bring this within the measurment range of absorbance, and the results from the reference

liquid water. The histogram show the values of the surface tension, density, refractive index and

colour for each sample. The measurement on each sample took only a few minutes and the

histogram was subsequently obtained directly from the software in a moment. The results on the

specific gravity included here were obtained using the Paar density meter which is incorporated

in the tensiograph described above. This meter is only an optional part of the multi-analyser

tensiograph. It should be remembered however that the density of the sample has to be

determined in order to obtain a measurement of the surface tension. It should also be noted that

this histogram with normalised values of the physical properties, was obtained directly from the

corresponding M-values.

695

Table 9 Table of ranges found for I-values of certitude for typical samples from LED fiber tensiograph with values included for tensiotrace statistical functions and projected values for measurand and statistical functions.

Symbol Tensiotrace feature I-value range Comment

/Detail

P1 Drop Period 10 - 650

P2a Rainbow 1 - 120

Drop period measured with

pump delivery

Drop times

P2b Drop times

P2c Drop times

Tensiograph main peak ,20 - 650

Tensiograph secondary peak ,20 - 650

Rainbow ! 2 - 100 P3a . Peak heights

P3b Tensiograph main peak ,2 - 300 Peak heights

P3c Tensiograph secondary peak ,2 - 300 Peak heights

P4* Points . 105 - 1012 Peak heights

Ps* Area . 105 - 1012 Peak heights

P6 Surface tension 20 - 100

P7 Refractive index 20 - 120

P8 Viscosity 180 - 650

5nm Range

Measured from rainbow

peak

Drop period measured with

constant head

P9 Co lou r 75- 150 Minimum peak size in range

of Tensiotraces from 200 -

1100nm with 6nm

: increments

These results were deemed by a commercial brewer to be satisfactory. From the histogram it is

possible to see immediately the highly reproducible value for refractive index, and the then

increasingly wide variability in respectively the absorbance, gravity and surface tension

measurements. It is not necessary here to discuss the quality control implications of these

results, but it should be obvious to the reader that this is a good representative analysis of useful

measurands and all were obtained on small volumes. Samples 1 to 5 are various types of a stout

and these results show a set of comparisons on the product. The concentrate was too dark for

696

tensiograph analysis at 950nm with the tensiograph and was consequently diluted and its results

with water form a second distinct set of results differing only in colour and gravity. The

absorbance of the reference water is obviously zero.

800

700

~" 600 u (::

500 ._

"r= "2" 400 .9

300 a

200

100

/ "

0 0 2 0 4 0 6 0 8 1 1.2 14 1.6 1.8 2

Concentration v/v %

, 6 6 0 n m

�9 950nm

Fig. 35. Graph of cumulative D-function for tensiograph measurements at 660nm and 950nm for milk in the

range 0-2% v/v.

4.2 D-VALUE AND .[-VALUES PLOTS FOR SUB-SENSITIVITY ANALYSIS

The tensiograph has been shown to be capable of providing useful statistical information on

several samples studied to date. This technique has more potential than has been discussed here

and this fact can be demonstrated by showing D-value analysis of a heated and admixed stout.

If the multi-analyser gives a zero D-value then instrument finds that the samples tested are

indistinguishable. The instrument reports "A fingerprint match". If the test liquid is slowly

heated and the physical properties of the sample change then logically there must be a point at

which the D-value grow to the point at which the instrument deems the trace to have deviated

to such an extent that it will report "Not be a fingerprint match". The I-value will work in the

opposite direction to the D-value and will switch from giving a non-zero result to a zero value.

Figure 37(a) shows the plot of D-values for stout in the range 30~ to 31.4~ fingerprinted

against the reference of itself measured at 30C. This plot shows the error ranges for the

697

D-values. The very large change of 150 D-units(15% as the instrument uses milli-parts for the

analysis at present) in 0.4~ shows the ultra sensitivity of the technique. The multi-wavelength

tensiograph will employ micro-part discrimination functions and greatly extend the capability of

the instrument.

Fig. 36. Histogram of normalised results of four physical measurements on six Guinness samples measured

against the water standard.

The second example here of the use of these D-functions is to tackle the industrial problem of

admixtures of beers in returned kegs. This problem is both an expensive and big technical

problem for brewers. The study described above on fingerprinting beers did not include a

discussion of Beamish in Guinness because stouts were the subject of a separate study. These

two stouts are exceedingly similar and consequently the problem of admixtures of these

products was selected specifically because it is such a difficult one to solve and would show the

technique to its fullest. Mixtures of 20, 40, 60 and 80% v/v were made of the products and five

tensiotraces recorded for each mixture and the products themselves. Taking a single Beamish

tensiotrace as the reference then cumulative D-values were taken for each trace. The six sets of

readings were then individually averaged and the standard deviations of each obtained.

698

Figure 37b shows the results of average D-values plotted against %v/v concentration of

Guinness for this set of admixtures. These results suggest that the multi-analyser could be a very

useful tool in identifying fraud in the brewing industry. Obviously much more comprehensive

studies of this problem are required to prove that the multi-analyser indeed can solve this

commercial problem, but this work certainly demonstrates that the instrument has some proven

capability in this regard.

250

200

150

D-value 100 }

}

} k

k } } }

} } } }

0 I I I I I I

30 30.2 30 4 30 6 30.8 31 31 2

T em peratu re(C

I ~ D-value I

Fig. 37a D-values for Guinness stout as it is heated from 30~ to 31.4~ measured using the tensiotrace of the

stout at 30C as the reference.

In terms of tensiograph terminology when the I-values go to zero then they are said to be

"uncoupled". It should be clear that because various trace features depend on different physical

and chemical properties of the liquid some will decouple with temperature, others with

wavelength variation and others with vapour pressure etc. It should be reported here that a

0.1 ~ change in temperature set the cumulative I-value to zero in all cases, but as can be seen

from the plots shown in Figures 37 the D-values changed more slowly. Sensitivity is defined as

the smallest detectable difference in a signal and here the D-values when they change from zero

in fact are the measure of sensitivity for the multi-analyser. The I-values for the ground zero

situation for D-values will be typically 106 and will decrease from this maximum. It has been

699

shown that these I-values are in fact statistically very reproducible values. The changes in the I-

value therefore gives sub-sensitivity analysis of changes in a liquids properties and this aspect of

the technology may prove to be very useful tool for the analysts.

450

350

3 0 0

D-v~Q"

150"

50"

0 I I I I I I I I I

0 10 20 30 40 50 60 70 80 90 100

%~v

�9 D-value / ~ P t 3 1 h l [l')_v~l I1~] �9 ,

Fig. 37b. Plot of D-values for admixtures of Beamish in Guinness.

4. 3 CONCLUSIONS

The tensiograph has been shown to have considerable fingerprinting capability and the task of

quantitating this instrumental performance has begun. The first phase of the necessary

experimental studies needed to produce algorithmic formations for the various statistical

functions has been completed. At this time only very conservative estimates of the actual

statistical quantities associated with the various trace features are being employed in the

algorithms to generate the I-values.

It is important to stress that there is a very considerable amount of work necessary to establish

properly the statistics associated with tensiograph analysis and at this point a short cut to these

statistics is suggested with the introducing the concept of 1-functions. The authors are not

however suggesting that these are a satisfactory substitute for properly evaluated statistics and

at this point a series of experiments are being conducted with the objective of developing

statistical measures associated with the instrument. Given this reservation, it is clear that the I-

700

functions give very useful quantitative analytical information and their utility has been

demonstrated in studies on water pollution, beer analysis, spirit analysis and this work has

shown the practical utility of this approach.

The actual capability of the tensiograph so far demonstrated suggests that this instrument may

have considerable advantages over other existing methods. In particular in this respect, the short

analysis time for the instrument may be of some considerable practical importance. Only four

application studies have so far been developed. These are the measurement of the properties of

labelled synovial fluid for the diagnosis of disease in this body fluid, and the analysis of beer,

spirits and pollution. The published results relate to two different instruments, the first a LED-

polymer fiber tensiograph, and a second, more advanced multi-wavelength silica fiber

instrument. The present instrument combines both instruments into the single unit.

It is perhaps important to point out that the work has been backed up with some very solid

software development. Windows standard code has been developed to facilitate all the

theoretical concepts discussed above. This software is being developed in line with the on-

going experimental studies, and the work has been directed specifically in recent timers to

quantitating tensiograph statistics. The software should make a wide range of further

applications possible in the next period and because of the extensive software development this

work should be relatively straightforward.

5. REFERENCES

1. R.Miller, A.Hofmann, R. Hartmann, K. Schano, and A. Halbig, Adv. Mater. 4 (1992) 370

2. N.D.McMillan, E.O'Mongain, J.Walsh, L.Breen, D.G.E.McMillan, M.J.Power, J.P.O'Dea,

S.M.Kinsella, M.P.Kelly, C.Hammel, D.Orr, Optical Engineering 33 (1994) 3871

3. N.D.McMillan, F.Feeney, M.J.P.Power, S.M.Kinsella, M.P.Kelly, K.W.Thompson and

J.P.O'Dea, Instrumentation Science & Technology 22(1994) 375

4. N.D.McMillan, M.Baker, S.Smith, D.Lane, R.Corden, J.Hanrahan, K.Thompson, K.Boylan

and M.Bree, "The instrumental engineering of a fibre drop analyser for both quantitative

and qualitative analysis with special reference to fingerprinting products for industruial

alcohol and sugar manufacture", in Sensors and their Applications VII, E. A.Augousti,

Institute of Physics Publishing, (1995) 260

701

5.N.D.McMillan. O.Finlayson, F.Fortune, M.Fingleton, D.Daly, D.Townsend, D.D.G.McMillan

and M.J.Dalton, Meas. Sci. Technol. 3 (1992) 746.

6. N.McMillan, Studies of stouts and beers, Private Consultancy Report, Carlow (July 1997)

7. N.McMillan, Studies of whiskey, Private Consultancy Report, Carlow (August 1997)

8. N.McMillan, Tensiograph studies of pollution in rivers and water courses, Private

Consultancy Report, Carlow (September 1997).

9. N.McMillan, M.Baker, V.Lawlor and S.Smith, The multi-analyser software platform, The

1996 Proceedings of the Optical Enginering Society of Ireland, Carlow(1997).

10. N.McMillan and E.Cass, "Neural nets processing of fingerprint data from the multi-analyser

tensiograph", Private Report, Carlow (August 1997).

11. H.Lui and P.K.Dasgupta, A renewable liquid droplet as a sampler and windowless optical

cell. An automated sensor for gaseous chlorine, Analytical Chemistry, 67 (1995) 4221.

12. N.D.McMillan, Part 1: M and D-function analysis for the multi-analyser tensiograph. Part 2:

I-Function forensic fingerprinting with the multi-analyser tensiograph. In preparation.

13. A.Augousti et al, Fiber drop analyser study of glycerine and sugar solutions, unpublished,

Internal Physics Department Report, Kingston University, Surrey, England.

14. op. cit. Note 2.

15. J.L.Lando and H.T.Oakley, Colloid Interface Sci., 25(1967) 526

16. N.D.McMillan, P.Davern, V.Lawlor, M.Baker, K.Thompson, J.Hanrahan, M.Davis,

J.Harkin, M.Bree, P.Goossens, S.Smith, D.Barratt, R.Corden, D.G.E.McMillan and

D.Lane, Colloids and Surfaces A 114(1996)75-97

17. S.S.Dukhin, G.Kretzschmar and R.Miller, Dynamics of adsorption at liquid interfaces:

theory, experiment, application, Vol. 1, Studies in Interface Science, D. MObius and R.

Miller (Eds.), Amsterdam and Oxford, Elsevier 1995

18. R.Miller, A.Hofmann, R.Hartmann, K.H.Schano and A.Halbig, Adv. Mater. 4 (1992) 370

19. N.McMillan, E. O'Mongain, J.E.Walsh, D.Orr and V.Lawlor, Proc. S.P.I.E.

20005 (1993) 216

20. ibid

21. Private Report by J.Walsh, N.D.McMillan and O'Mongain, unpublished UCD, Dublin 1992.

702

22. M.Kurata, M.Iwama and K.Kamada, "Viscosity-molecular weight relationships, IV-1 to IV-

3 and also IV-7 to IV-45" in Polymer Handbook, J.Brandrup and E.H. Immergut, Eds.,

Wiley, New York (1948)

23. H.Lu and R.E.Apfel, AIP Conference Proceedings, 197 (1988) 81

24. F.Bashforth and J.Adams, An attempt to test the theory of capilliary action, Cambridge

University Press, 1882.

25. N.D.McMillan, O.Finlayson et al, Meas. Sci. Technol. 3 (1992) 746

26. Zalloum, E.O'Mongain, J.E.Walsh, S.Danaher and L.Stapelton, Int. J Remote Sensing,

14(1993) 2285.

27. N.D.McMillan, O.Finlayson et al, Rev. Sci. Instrum., 63 (1992) 216

28. N.D.McMillan, Opto Laser Europe (1993), 42.

30. M.R.Wright, "The nature of electrolyte solutions" Ed. J.Thompson, MacMillan, London

1988

31. D.C. Graham, Chem Rev. 41 (1947) 441

32. R.Parsons, Mod. Asp. Electrochem. 1 (1954) 103

33. D.M.Mohilner, Electroanal. Chem. 1 (1966)241

34. op.cit. Note 14.

35. N.D.McMillan, M.Reddin, V.Lawlor and R.Jordan, A new approach to the information

processing of multivariate data from the multi-analyser tensiograph obatined on

commercial beers, In preparation.

36. op.cit. Note 3.

37. G.Elicabe abd G,.Frontini, J.Colloid and Interface Science 181 (1996) 669.

38. L.I.Podkamen and S.G.Cuminetsky, Izvestiya Akademii Nauk SSSR Fizika Atmosfery I

Oceana 17 10 (1981) 1116-1120.

39. C.H.McCrae and A.Lepoetre, International Dairy Journal 6 (1996) 247.

40. op.cit Note 14.

41. ibid.

42. J.E.Walsh, Spectral reflectance and crop efficiency. PhD Thesis, U.C.D., Dublin (1992).

43 op.cit Note 26.

44. W.D.Harkin and F.E.Browne, J.Am. Chem. Soc. 41 (1919) 499.

703

6. LIST OF SYMBOLS

A

A(t)

amplitude of the vibration of the first fundamental frequency

Tensiograph absorbance measured in A-units

A t

A.

amplitude of the vibration of the first fundamental frequency

amplitude of the n ~ overtone of the drop

c concentration measured in mol

c concentration measured in mol

Oil

dn

specific gravity and is dimensionless

difference of differences (d, -- d )

difference value between the two n th data points in the reference and measurement data

sets

average of the differences

FA absorbance correction factor and is dimensionless

gn dimensionless rainbow peak universal range

gl discrimination function for the drop period

g2 discrimination function for peak times

g3

g4a

discrimination function for peak heights

analogue discrimination function for points

g4b digital discrimination function for points

g5

g6

discrimination function for area

discrimination function for density

H tensiograph signal measured in V

HR, HM tensiograph signal for the Reference and Measurement trace measured in V

L binary setting factor

1 path length measured in cm

M molecular weight

704

M1 M-function for the drop period

M2 M-function for peak times

M3 M-function for peak heights

M4a analogue M-function for points

M4b digital M-function for points

M5 M-function for area

M6 M-function for density

N number of points in the data series

ND data averaging constant, dimensionless

P~ I-value for the drop period

P2 I-value for peak times

P3 I-value for peak heights

P4

P6

I-value for points and area measurements(combined)

I-value for density

RD drop period resolution and is dimensionless

drop head radius measured in mm

SD standard deviation

t time measured in s

TD drop period measured in s

TU dimensionless Tensiograph Unit

VD drop volume in gl

VD drop volume measured in ml

Aa absorbance user range, dimensionless

AA absorbance user range function, dimensionless

AANuI! null absorbance range and is measured in A-units

AN refractive index user range function, dimensionless

705

Atn peak width of the n th peak measured in s

(I) a refractive index correction factor for the tensiograph peak, dimensionless

*r refractive index correction factor for the rainbow peak, dimensionless

A binary point function, dimensionless

ot t-test confidence limit

13 Mark Houwink constant, dimensionless

8H signal-to-noise of tensiotraces measured in V

7 surface tension measured in m N/m

11 absolute viscosity in Pas, the subscript indicates the measurement on the solvent

�9 l refractive index, dimensionless

[1"1] limiting viscosity number

wavelength measured in nm

kinematic fluidity measured in reciprocal viscosity units m2s "l

O'

On

angular frequency of the second fundamental frequency of the drop

angular frequency of the n th overtone of the drop

co angular frequency measured in rads/s

~, reciprocal tensiograph absorbance and is measured in A-units

tensiograph linking constant between the tensiograph absorbance and the reciprocal

absorbance

density measured in kg m -3

tensiograph turbidity coefficient measured in mol "l m "1

tensiograph molar absorptivity measured in mol ~ m "1

A/D resolution measured in s 1

Oa

flow rate ml s -~

data acquisition rate measured in s 1

H o tensiograph signal at time t=0 measured in V

This Page Intentionally Left Blank

SUBJECT INDEX

absorbance 612

acetone solutions 642

added-mass force 426

ADSA as a Film Balance 92

ADSA Captive Bubble Chamber 104

ADSA-CB: Captive Bubble Method in

Lung Surfactant Studies 103

adsorption and desorption flow 388

adsorption at the surface of a growing drop

163

adsorption dynamics 217

adsorption kinetics mechanisms 422

adsorption kinetics model 400

adsorption mechanisms 163

adsorption processes in emulsions 375

adsorption processes in foams 3 75

adsorption rate 346

advantages of drop volume method 152

alcohols 316

ALFI 68

alkyl diethyl phosphin oxide 174

alkyl dimethyl phosphine oxide 179

alkyl dimethyl phosphine oxides 315

alveolar shape 454

alveoli 434

apparent dynamic surface tension of water

295

apparent interfacial tension 230

approximate solutions 165

Archimed force 399

707

ASTRA- Automatic Surface Tension Real

Time Acquisition system 515

attractive interaction 581

Automated Polynomial Fit Program (APF)

126

Axisymmetric Drop Shape Analysis

Diameter (ADSA-D) 67, 72

Profile (ADSA-P) 65, 69

Contact Diameter (ADSA-CD) 66

Maximum Diameter (ADSA-MD) 66

axisymmetric drops 64

Axisymmetric Liquid-Fluid Interfaces

(ALFI) 67

Bashforth and Adams tables 477

Beer's law 672

bifurcations in drop volume 161

biological liquids 320

biophysical function 438

blow-up effect 153

boundary value problem BVP 73

bovine lipid extract surfactant 105

bovine serum albumin (BSA) 99

break-up dynamics 202

break-up of liquid filaments 194

bubble deadtime 288

bubble lifetime 286, 299

bubble rising relaxation 396

bubble rising retardation 373

bubble surface velocity 412

bubble time 288

708

bubble velocity 399

bubble volumes 308

buoyancy effect in SDT 192

buoyancy force 426

buoyant bubble 368

BVP 75

calibration measurements 338

capillary constant 141

capillary design 149, 437

capillary length 537, 557

capillary number 553

capillary number of collision 552

capillary rise 111

capillary rise technique 490

capillary wave technique 329

capillary waves 229

captive bubble method 329, 528

captive bubble surfactometer 461

characteristic times for liquid tin 511

circular current effect 153

circulation in a drop 389

coagulation processes 375

coalescence of droplets 218, 567

coating processes 328

collision time 527, 537, 541

commercial drop volume tensiometers 143

comparison between ADSA-P and a

goniometer technique 117

comparison of MBPM with other methods

312

complete sphere 340

complex elasticity module 349

concave drop head 639

concentration dependence of the interfacial

tension 86

contact angle measurements 108

on rough and heterogeneous solid

surfaces by ADSA-D 136

on smooth solid surfaces by ADSA-D

133

drop size dependence 131

contact time 527, 538

contamination by surfactants 418

continuity equation 342

convective diffusion equation 378, 384

copper sulphate analysis 654

correction factor in SDT 208

creaming 565

critical aggregation concentration 174

critical wavelength 228

cycling techniques 457

cylindrical droplets 199

DAL at weekly retarded bubble 411

damped vibrations of the remnant drop

611

Debye length 354

decyl dimethyl phosphine oxide 356

deformation of a gas-liquid interface 527,

546, 555

density meter 601,650

depletion interaction 571

depth of a deformed gas-liquid interface

559

depths of menisci 550

Derjaguin' s formula 540, 550

design of a drop volume instrument 144

desorption rate 346

detachment time 156

determination of the true droplet radius

215

diffusion boundary layer 378, 411

diffusion controlled adsorption kinetics

163,348

diffusion equation 165, 347

diffusion layer 371,378

diffusion layer compression 166

diffusion of adsorbent 217

diffusional exchange 329

diffusive flux 391

dilation of the adsorption layer 166

dilational elasticity 329

dilational properties 328

dilational viscosity 328, 329

dimethyl sulfoxide (DMSO) 99

discrimination functions 688

disjoining pressure 568

dissolution of the copolymer 119

dodecyl dimethyl phosphine oxide 356

dosing system 145

DPPC 442

drag coefficient 390

drag force 398

drop detachment 143, 155

drop fibre rheometry 610

drop formation 143

drop head design 598

drop head tilt 628

drop period 623, 647, 652

drop volume method 139, 153

correction factor 142

709

comparison with maximum bubble

pressure 180

drop weight method 493

droplet coalescence 194

droplet migration due to Marangoni flow

206

dynamic adsorption layer 368

dynamic measuring mode 146

dynamic surface tension

measurements using MBPM 312

of a mixed solution of protein and small

molecules 99

of surfactant solutions 96

dynamic tensiogram 320

dynamics of growing bubbles 285

effect of capillary length 309

effect of gravity in SDT 192

effective adsorption time 301

effective age 148, 164, 174

effective diffusion coefficient 177

effective mass 542, 547, 553

effects of viscosity 203

effects of viscosity in SDT 203

Einstein equation 398

elasticity 536

electrochemical measurements 619

ellipsoid-like droplets 198

emulsion film 581

emulsions 328

end-pinching p~,enomena 227

E/3tv/Ss equation 481

equilibrium interfacial tension 90

error analysis 151

710

ethanol solutions 606

ethanol-water mixtures 158

Evans' model 539, 544

experiments on dynamic drops 76

experiments on static drops 76

fibre drop analyser 595

filling procedures 208

film leakage 103

film thickness 569

film thinning 375

fingerprint analysis 646

fingerprinting 595, 686

flocculation 567

flotation 375,423,526, 538

flow inside a drop 166

foams 328

force balance 339, 539, 546

at a pendent drop 142

between interfacial and centrifugal forces

196

acting on a drop 151

Fourier analysis 627

frequency domain in SDT 222

front stagnant point 423

Frumkin's equation of state 391

furnace in SDT 214

gas exchange 434

gas kinematic viscosity 292

Gauss Laplace equation 150

gelatin/anionic 328

general principle of the drop volume

method 142

geometric mean relation 110

Gibbs adsorption equation 483,488

Gibbs elasticity 346

Gibbs' fundamental equation 175

Gibbs model of a capillary system 479

Gibbs-Duhem relation 352, 480

Gibbs-Marangoni effect 332

goniometer technique 122

Gouy Chapman theory 354

gravitational force 426

grey level 533

Guggenheim model 352

Guinness samples 697

gyrostatic equilibrium in SDT 192

Hadamard Rybczynski velocity 394

Hadamard-Rybczinski approximation 372

Hamaker constant 569

Helmholtz free energy 478

hemispherical bubble 337

Henry constant 371

history force 426

human albumin 89

hydrodynamic analysis of spinning drop

225

hydrodynamic boundary layer 379, 411

hydrodynamic effects in drop formation

153

hydrodynamic field 403

hydrophilic capillaries 302, 308

hydrophobic capillaries 308

hydrophobised capillary 285

impact interaction 527, 538, 552

inclined plate 313

incomplete sphere 355

individual bubble regime 300

inertia contribution to the bubble

detachment 291

inertial terms 341

initial value problem (IVP) 68

interaction between a solid sphere and an

interface 525

interface extend effect on impact dynamics

546

interfacial tension 192

from SDT 200

gradients 230

pressure dependence 86

temperature dependence 88

water- decane 173

inverted sessile drop 81

ionic double layer 354

ionic surfactant 354

isotherm equation 349

jet regime 300

kinetically stable 564

kinetics of adsorption and desorption 368

Langmuir isotherm 170

Langmuir trough 329

Laplace equation 63, 73, 289, 489

Laplace pressure 586

Laplace's equation 196

law of Tate 140

levitated drop 492

limitations 192

limitations of SDT 192

711

line tension 131

linear hydrodynamic stability analysis 228

lining layer 434

liquid/liquid interface 162, 181

liquid-liquid interfacial tension 79

liquid-vapor surface tension 77

low-rate dynamic contact angles 112

Lucassen/van den Tempel model 349

lung air-liquid interfacial film 438

lung mechanics 438

lung pressure-volume relationships 436

lung surfactants 329

lungs 434

Marangoni effects 483

Marangoni flow in SDT 194, 230

Marangoni number 386

Marangoni retardation 394

Marangoni stress 386

Marangoni velocities 484

Mark Houwink equation 609

mass transfer across interfaces 230

mass transfer kinetics 336

mass transfer, effect on break-up and

coalescence 230

matching samples of beer 691

maximum (advancing) contact angles 122

maximum bubble pressure method 491

maximum bubble pressure tensiometers

283

Maxwell model 330

melts 193

meniscus collapse 453

meniscus hydrodynamic relaxation 296

712

methanol-water mixtures 617

microemulsion 565

mobile bubble surface 369

modified Young equation 131

molecular reorientation 318

molten gold drop sitting in a sapphire cup

517

monolayer collapse 106

MPT2 tensiometer 283

multi-dimensional non-linear least-squares

problem 71

multi-wavelength 600

mutual saturation 79

Navier-Stokes equation 226, 330, 341,

371,415

non-equilibrium spinning drop phenomena

217

non-equilibrium state of the adsorption

layer 375

non-linear regression 65

non-Newtonian fluids 225

non-rotating threads 204, 227

non-steady DAL 397

non-uniform surface tension 384

numerical aperture 634

objective function 65, 69

one-dimensional Newton-Raphson

iteration 71

optimization iteration 71

ordinary differential equations (ODE) 67

oscillating barrier method 329

oscillating drop 338

oscillating jet 313

Ostwald ripening 566

particle oscillation 542, 553, 558

Peclet number 371,378

pendant drop method 531

peristaltic forces 571

permittivity of a film 570

PET (polyethylene terephthalate) 109

Philippoff's model 538, 542

phosphatidylglycerol 442

phosphatidylinositol 442

phospholipid DPPC 174

physico-chemical hydrodynamics 366,

376

piezoelectric piston 334

piezoresistive pressure transducer 334

PMS-tensiograph 632

Poiseuille approximation 296

Poisson-Boltzmann equation 354

polymer melts 212

polymer solutions 193

polypropylene 91

Prandtl number 378

precision of SDT experiments 209

presence of an electric field 194

pressure difference at the interface 197

pressure oscillation regime 292

pressure oscillations amplitude 307

pressure tensor 331, 332

pressure transducers 282

pressure-area (rt-A) isotherms 95

principles of capillarity 476

protrusion 571

pulmonary surfactant 434

pulmonary surfactant lipids 441

pulmonary surfactant proteins 445

pulsating bubble method 336

purity of solvent 173

PZM analysis 195, 200

quasi-static measuring mode 147

radial flow inside the drop 164

rainbow peak 672, 679

rainbow peak height 647

range of applications of SDT 192

rate dependence of collapse pressure 94

ray-tracing techniques 216

recirculating wake 404

recirculation flow 404

refractive Index 616

regime of fast reactions with excess metal

vapours 498

regime of fast reactions with excess

oxygen 498

regime of instantaneous reactions 498

regime of slow reactions 498

relaxation of stable droplets 218

resistance of the capillary 292

respiration 434

respiratory distress syndrome 439

restoring force 536, 544, 550

retardation coefficient 376, 380

Reynolds number 371

rheological properties in SDT 196

rheology of emulsions 375

rheology of foams 375

713

rhodamine-b 615

rising bubble 366

rising bubble relaxation 423

rotating droplets

equilibrium shape of 195

mathematical solution 195

rotating liquid threads 202, 228

rotating liquid threads, infinitely long 202

rotational speeds 209

scaling factors 70

Scheludko's model 540

Schulze's model 541

SDS 160, 42O

SDT properties 192

SDT, stability criterion 203

sedimentation 565

sessile drop experiments 512

shape determining parameter 197

shear stress 3 85, 3 90

Sievert's model 506

sinusoidal modulation 202

sinusoidal modulation of the angular

velocity 223

skin friction 404

slip/stick behaviour 119

Snell's Law 645

sodium dodecyl sulfate 96, 373, 621

spectral absorbance 631

spectro-photometry 615

spinning drop technique 189

spinning rod tensiometer 208, 210

spontaneous adhesion 582

714

spontaneous break-up of liquid threads

218, 226

spontaneously decay of liquid threads 202

spreading of surfactant 417

stability criterion for liquid threads 202

stability diagram 204

stable bubbles 451

stagnant cap 389

stagnant ring model 414

stalagmometer 140, 603

static contact angles 108

static measuring mode 148

steady dynamic adsorption layer 380

Stokes' damping force 547

stopped flow procedure 304

stopped flow regime 319

streamlines 403

stress balance 342

strongly retarded bubble surface 411

submillisecond time range 307

Sugden's tables 290

superposition of both viscous and

adsorption dynamics 224

surface activity 384

surface and interfacial tensions 488

surface concentration 401

surface concentration gradient 370

surface convection 421

surface diffusion 387

surface excess free energy 480

surface films 194

surface pressure 93

surface relaxation 91

surface retardation 373

surface tension 192

gradient 3 73, 385

measurements for lung surfactant 105

of liquid iron 485

of liquid Sn-Ga alloys 482

of molten systems 478

of molten zinc 503

of polymer melts 90

temperature dependence 172

response functions 98

surface viscosity 372

surface viscosity effect 373

surfactant distribution 390

surfactant metabolism 444

surfactant release 449

synovial fluid 610

systems of three liquid phases 196

table of PZM 201

Teflon FEP (fluroinated ethylene

propylene) 109

tensiograph 595

tensiograph peak heights 647

tensiograph peaks 618

tensiograph rheometry 626

tensiograph units 613

tensiotrace 682

terminal bubble velocity 374

terminal velocity 390

tetramethyl benzidine 622

thermal fluctuations 193

thermal fluctuations in SDT 202

thermally excited capillary waves 329

thermocapillary effect 483

thermodynamically unstable 564

Thiele diagram 500

three phase contact line 334

Tomotika's theory 229

total adsorbed amount 400

total drag 408

trace impurities 409

transfer across the interface 179

transient droplet shapes 220

transport at the bubble surface 370

Triton X- 100 178, 313, 315,420

Triton X- 165 160

Triton X-405 178

Triton X-45 177

t-test Analysis 683

tube filling procedure in SDT 192

turbidity 612

Tyndall scattering 693

Tyndall-Rayleigh scattering 633

ultralow interracial tension 81

ultralow tension systems 193

undulation 571

upper and lower bounds for interracial

tension 219

van der Waals forces 568

vertical plate technique 111

vibration drop trace 595

video image method 533

viscoelastic fluid 330

715

viscometric effect 608, 623

viscosity effect on drop detachment 157,

162

viscosity effects on bubble formation 290

viscous behavior 353

viscous energy dissipation 406

viscous liquids 193,212

viscous stress 385

Vonnegut' s equation 190, 199, 215

vorticity contours 403

vorticity-stream function 415

Ward and Tordai equation 177, 349

water purity 409

wave-length 204

weakly retarded bubble surface 412

Weber number 407, 552

weighting factor 70

wetting conditions 149

wetting properties 334

whiskey samples 689

Wilhelmy plate 491

Ye and Miller's model 544

Young-Laplace equation 334, 342

This Page Intentionally Left Blank