From Clays to Living Cells and Tissues
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Series H: Cell Biology, Vol. 64
Mechanics of Swelling From Clays to Living Cells and Tissues
Edited by
Theodoros K. Karalis Democritos University of Thrace Department of Civil Engineering 67100 Xanthi Greece
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest Published in cooperation with NATO Scientific Affairs Division
Proceedings of the NATO Advanced Research Workshop on Mechanics of Swelling: From Clays to Living Cells and Tissues held at Corfu (Greece) from July 1-6, 1991
ISBN-13: 978-3-642-84621-2 e-ISBN-13:978-3-642-84619-9 001: 10.1007/978-3-642-84619-9
Library of Congress Cataloging-in-Publication Data Mechanics of Swelling: from clays to living cells and tissues / edited by Theodoros K. Karalis.
(NATO ASI series. Series H, Cell biology; vol. 64) "Proceedings of the NATO Advanced Research Workshop on Mechanics of Swelling: from Clays to Living Cells and Tissues held at Corfu (Greece) from July 1-6,1991." Includes bibliographical references and index.
Additional material to this book can be downloaded from http://extra.springer.com.
ISBN-13 978-3-642-84621-2 1. Edema--Congesses. 2. Tissues--Mechanical properties--Congresses. 3. Swelling soils--Congresses. I. Karalis, Theodoros K., 1940-. II. NATO Advanced Research Workshop on Swelling Mechanics: From Clays to Living Cells and Tissues (1991 : Kerkyra, Greece) III. Series. RB 1A4.M43 1992 574.19'1--dc20
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© Springer-Verlag Berlin Heidelberg 1992 Softcover reprint of the hardcover 1st edition 1992
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It is the work of an educated man to look for precision in
each class of things insofar as the nature of the subject admits
Aristotle
Souvenir photograph of the NATO ARW participants held in Corfu 1-6 July, 1990. By numbers: (1) Wendy Silk, (2) Pierre-Gilles de Gennes, (3) John R. Philip, (4) Giovanni Pallotti, (5) Alex Silberberg, (6) Evan A. Evans, (7) Theodoros K Karalis, (S) Alice Maroudas, (9) Francoise Brochard- Wyart, (10) de Barios, (11) Pedro Verdugo, (12) Peter J . Basser, (13) Dennis Pufahl, (14) Jacob Israelachvili, (15) Stefan Marcelja, (16) Sidney A. Simon, (17) J.P.G Urban, (1S) Peter R. Rand, (19) S.D. Tyerman, (20) Paul Janmey, (21) Joe Wolfe, (22) Paolo Bernardi, (23) James S. Clegg, (24) Claude Lechene, (25) John Passioura, (26) Larry L. Boersma, (27) Thanassis Sambanis, (2S) Kenneth R. Spring, (29) Philippe Baveye, (30) Avinoam Nir, (31) Pierre Cruiziat, (32) Kostas Gavrias, (33) Panayiotis Kotzias, (34) Frank A. Meyer, (35) Wayne Comper, (36) J . M. A. Snijders, (37) Yoram Lanir, (3S) Rolf K Reed, (39) Adrian Parsegian, (40) Aris Stamatopoulos, (41) George F. Oster, (42) Leonid B. Margolis, (43) Makoto Suzuki, (44) A. P. Halestrap, (45) Charles A. Pasternak, (46) Jersy Nakielski, (47) Louis Hue
Preface
Mechanics of Swelling is crucial in any decision process in a diversity of problems in
engineering and biological practice. II is part of the control steps following identification of
certain materials preceding the' organization that should be made before an industrial action
is undertaken. It includes research on all aspects of osmotic phenomena in clays, plants, cells
and tissues of living systems, gels and colloidal systems, vesicle polymeric systems, forces
between surfactants, etc. It embraces also research on oedema, obesity, tumours, cancer and
other related diseases which are connected with oncotic variations in the parts of a living
system.
The ancient Greeks investigating our physical univcrse, by wrestling with logical
reflections, continuously asking and asking again themselves, they found a certain exodos and
in this particular problem too. Tumor, dolor, calor, rubor, was a maxim already known by
Hippocrates and later-on by the Romans. Presently, major scientific achievement has been
attained in the last decades by biologists, medics doctors and physicists, but it st ill remains
difficult to explain the cause, the cvolution and the various details concerning oncotic
variations in our animate world. Presumably, different generic errors and environmental , factors operate at different stages in the development of these events through multiplicati'on,
differentiation, aggregation, localized proliferation and cell death; each of which is itself a
complex system of biomedical activities. Oedema, obesity, swell ing of cells and tissues,
teratogenic mechanisms, etc. remain of intense clinical importance and the contributions to
explain these diseases have been the subjects of widespread research and observations.
From the inanimate world, the fact that certain materials swell was already known in
antiquity. Ceramic manufacture could not be achieved without knowledge of the used clays'
swelling potential. The Egyptians isolated stones from the rocks by carefully filling holes in
the rock with dried wood and then pouring water over it. The force developed by the swelling
wood made the stone burst. Among other examples the swelling of certain resins in aqueous
solutions and of soils at different moisture content was known earlier.
Silicates and certain crystalline substances swell and/or lose water without ceasing
to behave like crystals. Haemoglobin crystals arc capable of swelling taking up and losing
water without any change in their apparent (microscopic) homogeneity. Hair and textile fibres
swell when placed in water. The way they swell depends on thcir chemical constitution which
affects their behaviour during dyeing and finishing. Knowledge of fibre swelling behaviour
VIII
would be valuable, but it has proved hard to ascertain because of their small dimensions. Few
textile fibres are shorter than about a centimetre in length whilst many are much longer and
few fibres are more than one thousandth of a centimetre in breadth and many are smaller.
These dimensions make it comparatively easy to measure longitudinal swelling and difficult
to measure transverse swelling. Fibres are allotropic and transverse swelling cannot be directly
inferred from the longitudinal value. If swelling values could be determined in both directions
their ratio would be a valuable index in measuring their allotropy.
Keratin or high molecular weight materials when placed in a suitable solvent swell to
an equilibrium value determined by the solvent, the temperature and the nature of the
polymer. Furthermore, for a long time now certain clay minerals (montmorillonite) have
attracted interest and found large uses because of their ability to adsorb large amounts of
water. The role of Polysaccharides in Corneal swelling is also substantial. The problem of
hydration of the cornea has received considerable attention in the past because of its
correlation with corneal transparency. Cornea stroma isolated and immersed in an aqueous
solution swells excessively and consequently loses its transparency.
The systematic and scientific study of the dimension changes of solids was first
discussed by Cauchy in 1828. His outstanding masterpiece was entitled "Sur l' equilibre et Ie
mouvement d' un system des points materiels sollicites par les forces d'attraction ou de
repulsion mutuelle". However, Cauchy was concerned only with explaining dilation and/or
condensation of a solid when it was solicited by external forces. His objective was to study
elastic tension and/or condensation of a solid subjected to bending and torsion which were
pointed out earlier by Leonard Euler (1755) and much earlier by Galilei (1564-1642), Mariote
(1620-1684), Leibnitz (1646-1716), Robert Hooke (1635 - 1703), Jacob Bernoulli
(1654-1705), Thomas Young (1773-1829) and many others.
All these studies concern changes in the dimensions of a solid and not with the
interaction of the solid with fluids, specifically when an amount of liquid is taken up by
certain materials. The systematic work from this point of view was done by the earlier
botanists during the period 1860-1880. Deluc (1791) made some valuable contributions on
this subject but it is only through the work of Vide Nageli (1862), Reinke, Pfeffer, Hugo de
Vries and others that swelling really started to be modelled. According to them, a solid is said
to swell when it takes up a liquid whilst at the same time: (i) it does not lose its apparent
homogeneity, (ii) its dimensions are enlarged and (iii} its cohesion is diminished; the latter
statement outlining the fact that a material instead of being hard and brittle becomes soft and
flexible. Therefore, the flow through a material imbibing water and producing swelling is
clearly distinct from capillary imbibition, such as is shown by a solid having many fine
capillary canals e.g., a piece of birch etc. Such a solid taking up liquids, remains clearly
microscopically inhomogeneous but its dimensions do not change and their cohesion is not
drastically diminished through imbibition of the liquid.
IX
However, despite the remarkable research by the earlier botanists and other
contemporary workers, it still remains difficult to have a complete picture of swelling via
phase interaction, phase transition, mechanical and thermal considerations. It is difficult to
understand why certain materials swell taking up liquids and still not lose its cohesion entirely
but only partially. This and other facts are the reason why swelling was discussed in all the
fields of our natural world in this workshop.
Throughout the meeting our aim was to elucidate the physical meaning and
implications of the concepts which already apply to swelling behaviour in most of the
materials and systems of our natural world. We tried to promote the mechanics of swelling
entering upon a new stage, which is less empirical and where the experimental study of better
defined objects was guided rather by more quantitative theories than by qualitative "rules" or
working hypotheses. The mechanics of swelling as was discussed in this workshop, may serve
as an example of this development reviewing most of the crucial aspects of swelling in nature
and to develop as far as possible a quantitative theory giving as a result a clear, concise and
relatively complete treatment. The material in this volume is arranged in six parts which cover
swelling in soils, plants, cells, tissues and gels. Developments in various techniques are drawn
in part six. The booklet ends with a subject index.
It was the general opinion of all who attended the Conference that this Advanced
Research Workshop was important and valuable to their on-going research projects and
everyone saw the need of having further exchanges on the same subject. The lectures and the
short papers presented here are of exceptional quality and I would like to thank all the
contributors for their efforts. As an editor I had the pleasurable opportunity of becoming
familiar with all the contributions and to interact with their authors. It is with great pleasure
that I extend my sincerest congratulations along with those of the participants of this ARW
to Pierre-Gilles de Gennes who has become a Nobel prize laureate. Furthermore, let me
acknowledge the gratitude of all the participants to the NATO Scientific Committee for its
generous support and worthwhile goal of bringing together scientists from many countries.
Also I which to extend a word of appreciation to the Greek Institutions for their financial
support and to my wife Photini for her efficient secretarial work.
I also feel very proud to have had the confidence of all the participants who supported
the organization of this Advanced Research Workshop held in Corfu from 1-6 July 1991 and
I hope that the papers presented in this Springer Verlag's edition will lead to further advances
in the Mechanics of Swelling.
Theodoros K. Karalis
Flow and volume change in soils and other porous media,
and in tissues John R. Philip
Water movement and volume change in swelling systems
David E. Smiles and J. M Kirby
Thermodynamics of soils swelling non-hydrostatically
Theodoros K KaraUs
media: theory and application to expansive soils
Philippe Baveye
The osmotic role in the behaviour of swelling clay soils
S. L. Barbour, D. G. Fredlund and Dennis Pufahl
PART 2. PLANT GROWTH
formulated and tested?
On the kinematics and dynamics of plant growth
Wendy Kuhn Silk
Regeneration in the root apex: Modelling study by means of
the growth tensor
from a developed example
i-iii
1
3
33
49
79
97
141
143
165
179
193
XII
Physical principles of membrane damage due to dehydration and freezing
Joe Wolfe and Gary Bryant
Ion channels in the plasma membrane of plant cells B.R. Terry, Stephen D. Tyerman and G.P. Findlay
The effect of low oxygen concentration and azide on the water
relations of wheat and maize roots
Stephen D. Tyerman, W. H Zhang and B.R. Terry
The expansion of plant tissues
205
225
237
MDCK cells under severely hyposmotic conditions
James S. Clegg 267
Evan Evans
Athanassios Sam ban is
structure and motility
Paul A. Janmey, C. Casey Cunningham, George F.Oster, and Thomas P. Stossel
Control of water permeability by divalent cations
Charles A. Pasternak
Mechanisms involved in the control of mitochondrial volume and
their role in the regulation of mitochondrial function
Andrew P Halestrap
293
315
333
347
357
379
XIII
The deformation of a spherical cell sheet: A mechanical model of
sea-urchin gastrulation
Epithelial cell volume regulation
Dieter Haussinger and Florian Lang
From glycogen metabolism to cell swelling Louis Hue, Arnaud Baquet, Alain Lavoinne and Alfred J. Meijer
Cell spreading and intracellular pH in mammalian cells Leonid B. Margolis
PART 4. FUNCTION AND DEFORMATION OF TISSUES
The role of tissue swelling in modelling of microvascular exchange
D.G.Taylor
Interstitial fluid pressure in control of interstitial fluid volume during normal conditions, injury and inflammation
391
405
413
433
443
455
457
Rolf K Reed, H. Wiig, T. Lund, S. A. Rodt, M E. Koller and G. Ostgaard 475
Swelling pressure of cartilage: roles played by proteoglycans and Collagen
Alice Maroudas, J. Mizrahi, E. Benaim, R. Schneiderman, G. Grushko 487
Changes in cartilage osmotic pressure in response to loads and their
effects on chondrocyte metabolism
Interstitial macromolecules and the swelling pressure of loose connective tissue
Frank A. Meyer 527
A mixture approach to the mechanics of the human intervertebral disc
H. Snijders, J. Huyghe, P. Willems, M Drost, J. Janseen, A Huson
Blood-tissue fluid exchange-transport through deformable, swellable porous
systems
Yoram Lanir
The influence of boundary layer effects on atherogenesis in dialysis
treatment patients
577
PART 5. BLISTERS, FORCES BETWEEN PARTICLES, PHASE
TRANSITIONS OF GELS AND FLOW IN DEFORMABLE MEDIA 593
Blisters
Pierre-Gilles de Gennes
Origin of short-range forces in water between clay surfaces and lipid
bilayers
functioning biomolecules
membranes
Sidney A. Simon, Thomas J. McIntosh and Alan D. Magid
Polymer gel phase transition: the molecular mechanism of product
release in mucin secretion?
Daniel L. Luchtel
Etsuo Kokufuta, Atsushi Suzuki, and Masayuki Tokita
Power Generation by Macromolecular Porous Gels
Makoto Suzuki, M Matsuzawa, M Saito and T. Tateishi
595
603
623
649
659
671
683
705
xv
Peter Basser
George Oster and Charles S. Peskin
Mechanism of osmotic flow
PART 6. DEVELOPMENTS IN VARIOUS TECHNIQUES
Cell protrusion fonnation by external force
Sergey V. Popov
John Agortsas, MD., Venizelou 100, 67100 Xanthi, Greece.
Peter J. Basser, National Institute of Health, NIH Building 13, Rm 3W13, Bethesda, MD
20892, USA
Philippe Baveye, Cornell University, College of Agriculture and Life Sciences, Bradfield and
Emerson Halls, Ithaca, New York 14853, USA
Paolo Bernardi, MD, Universita degli Studi di Padova, Instituto Di Patologia Generale, Via
Trieste, 75 35121 Padova, Italy
Larry L. Boersma, Soil Science Department, Oregon State University, Corvallis, OR
97331-2213, USA
Francoise Brochard-Wyart, Universite Pierre et Marie Curie (Paris VI), Structure et
Reactivite aux interfaces, Batiment Chimie-Physique 11, rue Pierre et Marie Curie 75231,
Paris Cedex 05, France
James S. Clegg, University of California, Bodega Marine Laboratory, PO Box 247, Bodega
Bay, California 94923, USA
Victoria, 3168, Australia
Pierre Cruiziat, Centre de Recherches Agronomiques du Massif Central, Lab. de
Bioclimatologie, Domaine de Crouelle, 63039 Clermont-Ferrand Cedex, France
Evan A. Evans, University of British Columbia, Department of Pathology, Faculty of
Medicine, 2211 Wesbrook Mall, Vancouver, B.G. V6T 1W5, Canada
XVIII
Kostas Gavrias, Civil Engineer, Amphitrionos 4, 41336 Larisa, Greece
Pierre-Gilles de Gennes, College de France, Physique de la Matiere Condensee, 11, Place
Marcelin-Berthelot, 75231 Paris CEDEX 05, France
A. P. Halestrap, Univ~rsity of Bristol, Department of Biochemistry, School of Medical
Sciences, University Walk, Bristol BS8 lID, United Kingdom
Dieter Haussinger, Medizin Klinik, Universitat Freiburg, Hugstetterstrasse 55, D-7800
Freiburg im BR, Germany
Louis Hue, Unite Horemones et Metabolism Research, UCL 7529, Avenue Hippocrate 75,
B-1200 Bruxelles, Belgium
Nuclear Engineering, Santa Barbara, California 93106, USA
Theodoros K. Karalis, School of Civil Engineering, Democritos Univercity of Thrace, 67100,
Greece
Panayiotis Kotzias, President of the Hellenic Society of Soil Mechanics, and Engineering
Foundation, Isavron 5, 11471 Athens, Greece
Paul Janmey, Massachusetts General Hospital Division of Hematology & Oncology, Bulding
149, 13tb Street, MGH-West, 8tb Floor, Cbarlestown, MA 02129, USA
Yoram Lanir, Technion-Israel Institute of Technology, Department of Bio-Medical
Engineering, The Julius Silver Institute of Bio-Medical Engineering Sciences, Technion City,
Haifa 32000, Israel
Claude Lechene, Harvard Medical School, Brigham and Women's Hospital, Laboratory of
Cellular Physiology, 221 Longwood Avenue Boston, MA 02115, USA
Stefan Marce/ja, The Australian National University, Research School of Physical Sciences,
Department of Applied Mathematics, GPO Box 4, Canberra ACT 2601, Australia
XIX
Alice Maroudas, Technion-Israel Institute of Technology, Department of Biomedical
Engineering, Technion City, Haifa 32000, Israel
Frank A. Meyer, Tel Aviv-Elias Sourasky Medical Center, Department of Rheumatology,
Ichilov Hospital, 6 Weizmann St., Tel-Aviv 64239, Israel.
Jersy Nakielski, Department of Biophysics and Cell Biology, Silesian University, ul.
Jagiellonska 28, 40-032 Katowice Poland
Avinoam Nir, Technion-Israel Institute of Technology, Technion City-Haifa, Israel
George F. Oster, University of California, 201 Wellman Hall, Berkley, CA 94720, USA
Giovanni Pallotti, Professor of Medical Physics, Faculty of Medicine and Surgery, Department
of Physics, University of Bologna, Via Imerio 46, 40126 Bologna, Italy.
Adrian Parsegian, Physical Sciences Laboratory, Division of Computer Research and
Technology, National Institutes of Health, Building 12A, Room 2007, Bethesda, Maryland
20205, USA
John Passioura, CSIRO, Division of Plant Industry, GPO Box 1600, Canberra, ACT 2601,
Australia
Charles A. Pasternak, St. George's Hospital Medical School, University. of London,
Department of Cellular & Molecular Sciences, Division of Biochemistry, Granmer Terrace,
London SW17 ORE, United Kingdom
John R. Philip, Division of Enviromental Mechanics, Centre for Environmental Mechanics,
Black Maountain, Canberra, GPO Box 821, Canberra, ACT 2601, Australia
Sergey Popov, Department of Biological Sciences, Columbia University, Fairchild Building,
Room 915, New York, NY 10027, USA
xx
Canada S7N OWO
Peter R. Rand, Brock University, Department of Biological Sciences, St. Catharines, Ontario,
Canada L2S 3A1
Rolf K. Reed, School of Medicine, University of Bergen, Department of Physiology,
Arstadveien 19 N-5009 Bergen, Norway.
Thanassis Sambanis, Chemical Engineering Department, Georgia Institute of Technology,
Atlanta Georgia 30332-0100, USA
Sidney A. Simon, Duke University Medical Center, Department of Neurobiology, Durham,
Box 3209, North Carolina 27710, USA
Alex Silberberg, The Weizmann Institute of Science, Department of Polymer Research,
Rehovot 76100, Israel
Wendy Silk, Department of Land, Air and Water Resources, Hoagland Hall, University of
California, Davis CA 95616, USA
J. M. A. Snijders, Rijksuniversiteit Limburg, Department of Movement Sciences, P.O.BOX
616, Holland
Kenneth R. Spring, Section on Transport Physiology, Laboratory of Kidney and Electrolyte
Metabolism, National Institutes of Health, National Heart, Lung, and Blood Institute, Bethesda,
Maryland 20892, USA
David E. Smiles, CSIRO Division of Soils, GPO Box 639, Canberra ACT 2601, Australia
Aris Stamatopoulos, Isavron 5, 11471 Athens, Greece
Makoto Suzuki, Mechanical Engineering Lab., Agency of Industrial Science and Thechnology,
Ministry of International Trade and Industry, Namiki 1-2, Tsukuba, Ibazak 305, Japan
XXI
Massachusetts Avenue, Cambridge, MA 02139, Room 13-2153, USA
Perikles S. Theocaris, Member of the Academy of Athens, Professor of N.T.D., PO. Box
77230, 17510, Athens, Greece
S.D. Tyerman, The Flinders University of South Australia, School of Biological Sciences,
Bedford Park, South Australia 5042.
J.P.G Urban, Department of Physiology, University of Oxford, Park Road, Oxford OX1 3 PT,
United Kingdom
Pedro Verdugo, Bioengineering, University of Washington, Seattle WA, USA
Joe Wolfe, School of Physics, University of New South Wales, PO Box 1, Kensington, 2033
Australia
Swelling in Soils
FLOW AND VOLUME CHANGE IN SOILS AND OTHER POROUS MEDIA, AND IN TISSUES
J .R. Philip
GPO Box 821
Canberra ACT 2601
Australia
Some 140 years ago Edward Lear, the English artist and humorist,
visited Corfu and wrote the following:
There was an old man in Corfu
Who never knew what he should do,
So he rushed up and down
Till the sun made him brown,
That bewildered old man of Corfu.
I do not doubt that many of you, too, will let the sun make you brown;
but, unlike that old man, you will all know perfectly well what to do:
namely, to make the most of this unique research workshop. Gathered
together here are scientists from 16 countries and from a great diversity
of disciplines from agronomy to molecular physics, from civil engineering
to physiology and biochemistry, all concerned in one way or another with
the common theme of the mechanics of swelling.
We owe this unusual and valuable event to the imagination, tenacity,
and energy of one man, Professor Theodoros Karalis. He has created this
Workshop virtually single-handed (though he tells me his charming wife
Photini helped with some practical details).
In November 1989 Professor Karalis wrote to tell me of his hopes of
bringing his Workshop into being. I Sai., immediately the compelling logic
behind his vision of a meeting bringing together research scientists from
diverse disciplines, but united in their concern with problems of volume
change. Much illumination may follow from examining both what swelling
phenomena in these various fields have in common, and the ways in which
they differ.
Over my research life I have been somewhat involved in volume-change
problems, both in porous medium physics and in physiology. Most of those
40 years have been spent on simpler systems innocent of the complications
NATO AS! Series, Vol. H 64 Mechanics of Swelling Edited by T. K. Karalis © Springer-Verlag Berlin Heidelberg 1992
4
of volume change, but I have always found my excursions into swelling and
associated problems of equilibrium and flow fascinating and challenging.
My own experience and, I suppose disposition, leads me to concentrate here
on what swelling processes across the board might have in common, and on
their phenomenological characterization and analysis at an appropriate
macroscopic scale. This review of swelling mechanics thus makes no claim
to be encyclopaedic.
1. The Scales of Discourse
In treating the mechanics of porous media (and of cell aggregations), we
must recognise clearly three distinct and separate scales of discourse
(Raats 1965, Philip 1972a, Philip and Smiles 1982): 1) the molecular scale;
2) the microscopic (for fluids, the Navier-Stokes) scale; 3) the
macroscopic (for porous media, the Darcy) scale. Our molecular scale is
precisely that called 'mass-point' or 'microscopic' in continuum mechanics
(e.g. Truesdell and Noll, 1965; Sedov, 1971); and in that sub-discipline
our microscopic scale is confusingly designated 'macroscopic' or
'phenomenological'. Discourse on our macroscopic or phenomenological scale
deals with physical quantities related to averages of analogous quantities
on our microscopic scale, the averages being taken over a volume, or
cross-section, large compared with that of the individual pores of porous
media, or of individual cells of tissues, or large enough to contain many
particles in colloid pastes and suspensions. A formal description of a
suitable averaging process has been given, for example, by Zaslavsky
(1968).
The primary goal of porous medium (and many physiological) studies is
theory on this macroscopic scale: physical observations are most readily
made on this scale, and the practical concern is with processes on this
scale. Usually a full microscopic theory would be needlessly elaborate,
even if it happened to be feasible. Of course microscopic studies are
important in their own right: well-founded theory on this scale may guide
us as to the form macroscopic theory may take, and as to the limits of its
applicability. Macroscopic phenomenological theory must be at least
consistent with what we know on the microscopic scale. Most of this review
concerns the macroscopic scale, but Section 4 on colloid pastes illustrates
the interplay between inquiry on the microscopic scale and theory
formulation on the macroscopic scale.
5
In my view, failure to distinguish clearly between the scales of
discourse has led to confusion and misplaced effort in many studies. It
would be invidious to elaborate. Some examples are enumerated in Philip
(1972a) .
2. Flow in Unsaturated Nonswelling Porous Media
A prime example of phenomenological theory on the macroscopic scale is
the analysis of water movement in unsaturated nonswelling soils and porous
media. The underlying physical concepts were understood by Buckingham
(1907) and the formulation sharpened by Richards (1931). Much basic work
was published in the 1950's (e.g. Childs and Collis-George, 1950; Klute,
1951; Philip, 1954, 1957a, 1957b), and the theory has been in general use
in soil physics and hydrology since about 1960. We describe it in some
detail here, because we shall go on to show how it is generalized to deal
with (at least one-dimensional) problems of equilibrium, flow, and volume
change in swelling media.
We begin, then, with an unsaturated nonswelling porous medium made up
of three components: the rigid solid matrix, a liquid, and a gas. We
outline the theory in the simplification (often justified in the
applications) that gas pressure differences may be neglected. It then
suffices to examine the flow of the liquid component: details of the gas
flow follow from continuity. We deal specifically with the case where the
liquid is water and the gas air (including water vapour): the extension to
other incompressible Newtonian liquids and other gases will be obvious.
2.1 Darcy's law for unsaturated nonswelling media
We may write Darcy's law for saturated media, specialized to water
flow, as
-K \7<IJ • (1)
X is the vector flow velocity, <IJ is the total potential, and K is the
hydraulic conductivity. Expressing potentials per unit weight simplifies
our equations and units: <IJ then has the dimension [length] and K the
dimensions [length] [timej-l. We note that
6
p/(pg) +
where p is the pressure, p the water density, g the gravitational
acceleration, and 2 the potential of the external forces.
Buckingham (1907) suggested that Darcy's law should hold for
unsacuraced media in a modified form with K a function of e, the volumetric
moisture content. Richards (1931), Moore (1939), Childs and Collis-George
(1950), and others confirmed this experimentally and established the
general character of K(e). For obvious physical reasons (Philip 1954,
1957a) K decreases through as many as 6 decades (Gardner 1960) as e decreases from its saturation value through the range of interest.
2.2 Total potential and moisture potential of water in nonswelling media
In (water~wet) unsaturated media the water is not free in the
thermodynamic sense because of capillarity, adsorption, and electrical
double layers (Edlefsen and Anderson 1943, Schofield 1935a). Capillarity
is dominant in wet, coarse-textured media, and adsorption assumes its
greatest importance in dry media. Double-layer effects may be significant
in fine-textured media exhibiting colloidal properties. Buckingham (1907),
a keen disciple of Willard Gibbs, was the first to appreciate that the
conservative forces governing the equilibrium and flow of soil-water are
amenable to treatment through their associated scalar potentials.
We define such potentials relative to the reference state of water (of
composition identical to the soil solution) at atmospheric pressure and
datum elevation z = O. Here z is the vertical ordinate, conveniently taken
positive downward. We then have 2 = -z and
I{I - z .
local interactions between
is the potential of the forces arising from
solid and water, (Philip 1970b). It is not
essential either to know or to specify these forces in detail: it suffices
that I{I can be measured by well-established techniques (Croney, Coleman and
Bridge 1952; Richards 1965; Holmes, Taylor and Richards 1967). In
water-wet nonswelling media I{I 0 at saturation and decreases with e to
very large negative values (typically -10 4m) at the dry end of the moisture
range of interest.
The partial volumetric Gibbs free energy associated with the local
solid-water interaction is pg1Jt and it follows that (in the absence of
solutes) the liquid and vapour systems are connected at equilibrium by the
relation
H exp g1Jt/RT , (3)
where H is relative humidity, R the gas constant for water vapour, and T
absolute temperature. We see that the 1Jt(0) relation presents in different
guise exactly the information conveyed by the adsorption isotherm for water
in the medium.
nonswel1ing media.
(4)
When the relations between K, 1Jt, and 0 are
single-valued, (4) may be rewritten in terms of a single dependent
variable. In terms of 0, the equation is
~t = ~.(D~O) dK aO ot.. - dO dz (5)
Both the moisture diffusivity D, defined by
D K d1Jt/dO ,
and the coefficient dK/dO are, in general, strongly-varying functions of O.
Richards (1931) developed (4). Childs and Geqrge (1948) recognized the
diffusion character of (5) for a horizontal one-dimensional system. Klute
(1951) explicitly derived (5). Philip (1954, 1955, 1957b) extended the
approach to include water transfer in vapour and adsorbed phases in the
same formalism. The strong nonlinearity of Fokker-P1anck equation (5)
cannot be ignored, and progress in unsaturated flow studies has depended
centrally on solution of it and related equations (e.g. Philip 1969a,
1988).
8
We emphasize the macroscopic character and the great generality of this
approach. The macroscopic functions K(O) and W(O) represent a sufficient
characterization of the medium for the purposes of analysis and prediction
of unsaturated flow phenomena. These two functions may be established
directly from routine macroscopic measurements and may be of quite
arbitrary functional form. We are not limited to simplifying assumptions
about the internal geometry of the medium nor to any molecular or
microscopic model of the solid-water interaction. We simply feed the
appropriate K(O) and W(O) functions, whatever their form, into (4) and use
its solutions for analysis and prediction as required.
3. Equilibrium, Flow, and Volume Change in Swelling Media
The foregoing developments have provided a fruitful theoretical
framework for study of the hydrology of nonswelling soils, but the need for
generalization to swelling soils has long been recognized (Philip 1958a).
We now consider some modest steps toward the required extension.
I must warn at once that these extensions are limited in character:
they are, for the most part, restricted to one-dimensional systems; and
they do not purport to treat irreversible structural changes which may
occur in the presence of large enough stresses. Nevertheless, they have
important consequences: they reveal, for example, that many classical
concepts of soil- and ground-water hydrology, based on the behaviour of
nonswelling media, fail completely for swelling soils.
3.1 The extension to swelling media.
At the outset we note that, since K, W, and the moisture ratio,J are
all free to vary in both saturated and unsaturated swelling media, both are
amenable to the same general formulation. ,J is the ratio of the volume
occupied by water to that occupied by particles, and is equal to 0 (1 + e),
where e is the void ratio, defined as the r~tio of void volume to particle
volume. For swelling media it is usually more convenient to work
with,J rather than o. The analysis is simpler and more straightforward for saturated or
two-component (solid, water) systems, since they necessarily exhibit
"normal" volume change (Keen 1931, Marshall 1959) and e '" ,J. Unsaturated
9
or three-component (solid, water, air) systems exhibit "residual" volume
change, with e dependent both on t} and on the normal stress P. Note that
both the particles and the water are taken to be incompressible.
Three new basic elements enter the extension of the flow theory to
swelling media:
A. In unsteady swe+ling systems, the soil particles are, in general,
in motion, so that it must be recognized (Gersevanov 1937) that Darcy's law
applies to flow relative to the soil particles. We therefore replace (1)
with K = K(O) by
Xr= -K(t})'VeIl , (6)
where Xr is the vector flow velocity in the local rest frame of the
particles.
B. For one-dimensional systems involving self-weight and/or surface
loading, (2) must be generalized to include the overburden potentia.l n
(Philip 1969b), so that it becomes
eIl='It+n-z. (7)
It is convenient to take 'It as the "unloaded" moisture potential, and n is
then the contribution to ell due to the normal stress, P. The measured water
pressure in such systems is 'It + n.
greater detail in section 3.2 below.
We discuss overburden potential in
C. Whereas K(O) and 'It(O) provide a sufficient hydrodynamic
characterization of non-swelling media (for nonhysteretic processes), we
now require for unsaturated systems K(t}) , 'It(t}) , and e(t},P), as well as the
particle specific gravity ~s. Figures 1 and 2 show typical 'It(t}) and e(t},P)
relations for a swelling soil. For mineral soils ~s "" 2.7. Saturated
systems need much less elaborate characterization. They require merely
K(t}) , i1(t}) , and ~s.
Unsteady swelling systems may be subjected e~ther to Eulerian analysis
(Prager 1953, Philip 1968) in the physical space coordinate, or to
Lagrangian analysis (Hartley and Crank 1949, McNabb 1960, Smiles and
Rosenthal 1968) in material coordinate m such that
dm = (1 + e)-I dw ' (8)
10
when the datum of m is taken in a plane where particles are stationary.
Here w is the one-dimensional space coordinate. Analogous material
coordinates may be used for two- and three-dimensional axisymmetric
systems.
0.2 0.4 0.6 0.8 1.0 {J-
Fig. 1. Typical relation between moisture potential, W, and moisture ratio, ~, for a swelling soil (Philip 1969b), adopted for illustrative
purposes.
1.0
0.9
O.S
0.7
0.6
0.5
0.2 0.4 0.6 O.S 10 {}-
Fig. 2. The functions e(~,P) and ~(~,P) for the illustrative swelling soil (Philip 1971), e is the void ratio, ~ the apparent wet specific gravity, and P the normal stress. Numerals on the curves denote values
of P(m).
The Lagrangian analysis in material coordinates is much better adapted
to swelling systems and turns out, quite generally, to be simpler and more
manageable than the Eulerian analysis. Note that recognition that the
particles move in unsteady systems is no mere exercise in pedantry: it is
demonstrable that the mass flow with the particles can be of the same order
of magnitude as the Darcy flow relative to the particles (Philip 1968).
3.2 Hydrostatics in swelling media.
The overburden potential n enters the analysis of vertical and loaded
systems. It should be understood that the separation of the
non-gravitational contributions to the total potential (i.e. ¢ + z) into W
and n is, in a sense, arbitrary; but it has the practical advantage that
it enables us to identify, study, and calculate the separate contributions
to water pressure arising from local interaction with the particles (w) and
11
from the normal stress produced by the weight of overlying strata and/or
surface loading (0). It has been shown by Bolt (Philip 1970c, Groenevelt
and Bolt 1972) that
o aP, with a(~,P)
o
(9)
For a vertical column at equilibrium, then, the use of (9) in (7) yields
z
o constant = -2 . (10)
Here P(O) is the vertical stress due to loading at the upper surface z 0
and ~ is the apparene wee specific graviey, defined by
~ (~ + ~s)/(l + e) .
Figure 2
table depth". With ~s and characterizing
functions w(~) and e(~,P) known, (10) may be solved to yield the
equilibrium moisture profile corresponding to a given 2.
Experimental data (Talsma 1977 a, 1977b) and exploratory calculations
indicate that the P-dependence of a is weak and also that ~ is a
slowly-varying function of P; and it appears that the variation of these
functions with P may be neglected without serious error in a P-range of
perhaps 5m (corresponding to the top 2.5 m of the soil) (Philip 1971).
In this approximation Eq. (10) reduces to a linear first-order
differential equation, for which the solution z(~) is available in closed
form. There are three classes of solution giving three distinct types of
equilibrium profile: wet profiles with ~ greatest at the surface and
decreasing as depth increases, separated br a singular profile
with ~ constant and wet apparent specific gravity ~ at its maximum value,
from dry profiles with ~ least at the surface and increasing with depth
(Philip 1969b). These are the results for the practical case with ~s > 1.
Overburden effects in hydrology had previously been recognized by some
authors. Schofield (1935b) saw that ~ should include an overburden
contribution. Coleman and Croney (1952) introduced a "compressibility
factor" a, though their definition, evaluation, and use of a differ from
12
those reviewed here. See also Collis-George (1961) and Rose, Stern and
Drummond (1965).
We note that in saturated, two-component, systems e = {} and O! = 1, so
that (10) reduces to the simpler
z
constant -z .
In this case, with ~ > 1 all solutions are of the "wet" class.
3.3 Steady vertical flows
(11)
Combining (6) and (10) gives the equation for vertical flow. With the
P-dependence of O! and ~ neglected, this is
z
v r = -K [{~~ + ~ [p (0) + J ~dZ]}~ + ~ - 1] (12)
o For steady flows this reduces to a linear differential equation of the
first order in d{}/dz and {}, which may be integrated to yield z({}) in
closed form. The solution is singular under certain conditions: steady
vertical flows are possible only for certain combinations of values of {}o
and {}oo (values of {} at z = 0 and at large z). The details are complicated
(Philip 1969d). For two-component systems (12) assumes the simpler form
For steady flows a quadrature gives z({}) for given vr and {}o.
3.4 Unsteady flows
Applying continuity to (12) and using (B), we obtain the equation for
unsteady flow and volume-change in vertical swelling systems,
a Of
m
a {} [K {dW dO! [J ] } a{}] d [ om 'l'+e d{} + d{} P(O) + ({} + ~s)dm om + d{} K(~O!
m(O)
1)] ~.
(13)
13
m(O) is the m-value corresponding to the soil surface z = O.
The theory of various unsteady processes in vertical systems may be
developed through solution of (13) subject to appropriate conditions: we
have in mind infiltration, capillary rise, drainage, and evaporation; and
also the soil-mechanical processes of consolidation and swelling of layers
so thick that self-weight cannot be neglected. Equation (13) is a
generalization of the nonlinear Fokker-Planck or convection-diffusion
equation: in certain important special cases it reduces to a Fokker-Planck
equation. For example, in two-component systems with e =~, (13) becomes
(14)
of the same general form as the nonlinear Fokker-Planck equation for
infiltration in nonswelling media. Methods developed for studying the
nonlinear Fokker-Planck equation promise to be useful in connexion with
(13) (Philip 1969d, 1970b). For example, certain terms on the right of
(13) are negligibly small at small t for many unsteady phenemona of
interest, so that solutions of the nonlinear diffusion equation
(15)
with
D**(,'} ) (1 + e)-1 K{dllt/d~ + P(O) d(Y/d~}
form the basis of perturbation methods of solving the full equation.
Equation (15) is the counterpart of (13) for horizontal systems. For
two-component and unloaded three-component horizontal systems D** in (15)
is replaced by D*, with
(Smiles and Rosenthal 1968, Philip 1968, Philip and Smiles 1969). The
classical theory of consolidation (Terzaghi 1923)
dllt/d,'} constant and involves a linear diffusion equation.
in Eulerian coordinates, so that mass flmq is ignored.
takes K and
It is expressed
3.5 Applications
Although much further work is required, it is clear that hydrologic
theory based on Eqs. (10), (12), (13) differs profoundly from the classical
theory which takes no account of swelling and neglects the contribution to
~ of the overburden potential O. Perhaps the simplest general statement
one can offer on the influence of swelling is this: the net effect of
gravity on the equilibrium and flow of water in swelling media is
approximately (1 - ~a) times that in nonswelling ones. For a mineral soil
this factor is about -1 in the normal range, increasing to 0 when ~ is
maximum, and approaching +1 as a ~ 0 at small values of {}.
"intuitions" of the hydrologist are therefore invalidated.
Various
Equilibrium moisture distributions for a swelling soil are thus totally
different in character from those for a nonswelling soil (with d{}/dz
necessarily positive and the moisture distribution invariant with respect
to the water table). Attempts to interpret equilibria in a swelling soil
through classical concepts are therefore doomed to failure (Philip 1969c).
This bouleversement carries over to vertical flow processes. For
example, the course of infiltration in a swelling soil evidently has
analogies with that of capillary rise in a nonswelling one, and vice versa:
and evaporation from an initially wet swelling soil does not necessarily
exhibit the sharp transition between constant-rate and falling-rate phases
characteristic of nonswelling media (Philip 1954, 1957c). These
developments have evident practical consequences for groundwater hydrology
(Philip 1971) and irrigation technology (Philip 1972b).
There are, of course, other important applications of this approach to
swelling media. My long-time colleague Dr David Smiles (see for example,
Philip and Smiles 1982, Smiles and Kirby 1991) has been active in applying
the foregoing phenomenological theory to chemical engineering processes
such as filtration and sedimentation.
3.6 Limitations
Exploration of the full implications of the foregoing analysis to
three-component systems has been relatively slow. One impediment has been
the experimental difficulty of making detailed measurements of e ( {} ,P)
(e.g., Stroosnijder, 1976), though the work of Talsma both in the field
(Talsma 1977a) and in the laboratory (Talsma 1977b) is notable.
15
three-dimensional systems requires that we integrate into the analysis an
appropriate macroscopic representation of stress-strain relations in such
systems. In the following section 4 we summarize an attempt to develop
such a representation.
4. Mechanics of Colloid Pastes
In the late 1960 I s it was perceived that the preceding concepts and
analysis of two-component swelling media carryover to the unsteady
behavior of suspensions in such processes as sedimentation, filtration, and
centrifigation (Philip, 1970a). Pastes consisting of dense suspensions of
colloidal particles are typical systems to which the analysis applies.
The dominant particle-water (strictly particle-electrolyte) interaction
in such pastes is that arising from electrical double layers.
therefore apply the Poisson-Boltzmann equation to such systems,
relevant microscopic physical quantities, and integrate
phenomenological results such as the tensorial relations
macroscopic strain and macroscopic stress.
We may
map all
up to
between
A programe of research along these lines was initiated by Philip,
Knight, and Mahony (1985). Much remains to be done, but we indicate
briefly here the character of the work and some of its results.
The connexions between anisotropic strain and anisotropic stress arise
inter alia in the context of swelling soils in the field. An uncracked
swelling soil in the field is constrained to change volume in one dimension
only, the vertical. The horizontal dimensions of a field of swelling soil
do not change with moisture content, but the elevation of its surface does.
As the soil dries and (more or less) vertical cracks open, however, the
individual monoliths between the cracks are free to change volume
three-dimensionally. Change in the constraints on swelling and shrinking
evidently produces a change in the energetics, ahd this is related to the
energetics of cracking.
4.1 Questions from soil mechanics and soil physics
A related motivation was the hope that the work might shed light on
some questions arising in soil mechanics (and indeed soil physics).
16
Although the more thoughtful writers on soil mechanics draw distinctions
between soils of high colloid content and soils such as sands, there
appears to remain a need to test various concepts against what happens on
the microscopic scale in a colloidal soil. We list some questions here.
4.1.1 State of stress of soil-water. Classical soil mechanics considers
the water in soil-water systems to be in a state of isotropic stress; and
soil physicists would generally agree. At one time there were arguments in
the literature about whether the soil water was in compression or in
tension. But double-layer analysis reveals that the water is, in general,
in an anisotropic (tensorial) state of stress: in some circumstances and
at some points it will be in tension in some directions, and in compression
in others.
4.1.2 Transmission of load. Particle to particle contact. With honorable
exceptions, soil mechanics literature envisages the transmission of load
between soil particles as through grain-to-grain contact. As we shall see,
this is by no means essential. Load can be transmitted from charged
particle to charged particle by means of tensoria1 electrostatic Maxwell
stresses in the water.
soil-mechanics concepts of "effective stress" and "pore pressure" tend to
be obscured and confused by the mental picture of both these stresses as
isotropic, and by lack of recognition that particle load transmission
doesn't need particle to particle contact.
4.2 The Poisson-Boltzmann equation in homog~neous particle arrays
Our point of departure is the Poisson-Boltzmann equation applied to the
diffuse double layer:
~2 is the Laplacian in physical space coordinates; ~ is the electrostatic
17
potential; D is the dielectric constant of the solution; ni (0) is the
number per unit volume of ions of species i in regions of the solution at
potential 0; zi is the signed valency; E is the protonic charge; k is
the Boltzmann constant; and T is the absolute temperature. (16) combines
Poisson's equation for the distribution of potential with Boltzmann's
equation for the ionic concentrations. Boltzmann's equation implies that
the total number of ions per unit volume at potential ~,
We apply equation (16) to arrays of identical charged particles.
Minimum energy considerations require that the equilibrium configuration be
a regular particle array. Each particle occupies its own basic cell,
bounded by a surface on which the normal component of the electrostatic
field strength, a~/ap, vanishes. All cells are identical, so the problem
reduces to solving (16) subject to the conditions
o on Al . (17)
Here Ao is the particle surface and Al the surface of its basic cell.
These regular arrays of identical colloidal particles are not just a
convenient fantasy. They actually happen, though of course not in any
exact sense in soils. Colloid crystals occur in nature: for example
ordered arrays of identical spheres of amorphous silica are precursors to
opals (Sanders, 1964). In recent years they have become commonplace in the
laboratory where colloid scientists produce ordered arrays of identical
particles of latex and other polymers (Efremov 1976 , Bartlett, Ottewill
and Pusey 1990).
macroscopic stress tensors
With ~ known from solution of (16) subject to (17), it is a relatively
straightforward matter to analyze the energetics, and the stress and load
distributions in a particle array.
18
4.3.1 Gibbs free energy, variational principle. The Gibbs free energy of
interaction per particle is
v
where the integral is taken over the volume of the cell external to the
particle.
The colloid literature gives elaborate, lengthy, and opaque derivations
of (18). In fact (18) follows immediately from recognition that (16)
yields a variational principle governing ~, namely that the integral of
(18) must be minimized. Formally, (16) is the Euler-Lagrange equation for
- G treated as a functional of ~ (i.e. the differential equation for ~
giving the distribution of ~ minimizing - G defined by (18».
The first term on the right of (18) is associated with an anisotropic
microscopic (Maxwell) stress tensor due to the electric field; and the
second is connected with the osmotic pressure due to the ionic excess in
the double layer.
The local microscopic stress tensor g
(T, ~~ [~~]2 + kT [N(~) - N(O)] ;
D 8~ [~~]2 + kT [N(~) - N(O)] . (19)
(T3 is the component of g in the direction of the electric field with (T" (T2
in directions normal to it. Tensile stress is positive. The Maxwell
stress (the first term on the right) is positive for (T, and (T2 and negative
for (T3. The osmotic term (second on the right) is positive for all three
components.
4.3.3 Macroscopic stress tensor. For exploratory calculations we adopt a
cuboidal array of particles, with a cuboidal basic cell with sides 2a, 2b,
2c in the principal directions of the particle array. The principal
components of the macroscopic stress tensor ~ (defined in terms of averages
over the relevant cross-sections) are then:
19
4.3.4 Debye-HUckel approximation. In the Debye-HUckel (linear)
approximatiori, and with suitable normalization, (18) reduces to the simple
G = - J [(~~)2 + ~2] dv . (21)
v
Applying the divergence theorem and using (17), we get the beautiful result
G (22)
with the normal derivative taken outward. A nice economy is that if we use
orthogonal expansions to solve Debye-HUckel for cylindrical and spherical
particles, G turns out to be simply proportional to the coefficient of the
leading term.
Results are most economically expressed in dimensionless form. When
stresses are normalized so that (20) holds also in the new symbols, (19) is
replaced by
(W)2 + ~2 ;
_ (W)2 + ~2
(23)
Various relevant solutions have been found, and an ongoing program is
planned. To date the maj or body of results is for the very simplest
system: a two-dimensional array of cylindrical particles (dimensionless
radius 1) satisfying the Debye-HUckel equation. Each particle sits in its
own rectangular cell of dimensions 2a x 2c. Among numerous ways of
presenting the results, the following are perhaps the most significant.
20
3
c
a
Fig. 3. Dependence of normalized Gibbs free energy per particle, G, on normalized cell dimensions a and c. Values of G are shown on the bold curves. Lighter curves are hyperbolae ac = constant and are thus curves of
constant cell volume. Note that G(a,c) = G(c,a).
4.4.1 Total Gibbs free energy per particle. Figure 3 shows the variation
of total Gibbs free energy per particle as a function of cell dimensions a
and c. The plot is necessarily symmetrical about the diagonal from lower
left to upper right, since G(a,c) = G(c,a).
Values of G are shown on the bold curves. The lighter curves are
various rectangular hyperbolae ac = constant, representing constant cell
volume. Note that the hyperbolae are always less curved than the constant
G curves. This means that for fixed cell volume G is always more negative
on the diagonal (which represents isotropic volume change) than anywhere
else on a given hyperbola. For fixed cell-volume the excess of G over its
value on the diagonal can be interpreted as strain energy, potentially
available to initiate cracking.
4.4.2 Macroscopic stress tensor. In this two-dimensional system, 8 1 is
the principal macroscopic stress component in the direction of variation of
a, and 8 3 that in the direction of variation: of c. Because of the symmetry
of G(a,c), 8 3 (a,c) = 8 1 (c,a). Figure 4 thus provides the required
information on the nonlinear relations between the strain tensor defined by
a and c and the macroscopic stress tensor ~ .
4.4.3 Details of the macroscopic stress distribution. The maps of G and
8 1 show in global form macroscopic implications of our solutions. There is
21
3
3 a
Fig . 4. Dependence of normalized macroscopic principal stress component SI (in direction of variation of a) on normalized cell dimensions a and c. Nwnerals on the curves are values of SI. Note that SI (c,a) S3(a,c),
where S3 is principal component in direction of variation of c.
also an enormous amount of detail on the microscopic stress distribution
around the particle and in the water. We limit ourselves here to just two
aspects, the principal component of the macroscopic stress in the direction
of the electric field , u 3 , and load sharing between particle and water.
12 20
02 0.4 os 0.8 1.0 12 05 10 1,5 2.0
12 r-~~~rT~ ____________________ -'
10 Z 1
1 "0.9
Fig. 5 . Normalized microscopic principal stress component of the electric field) for v arious cell configurations . curves are values of u 3 . Toned regi ons are regions
(compression).
0"3
u 3 (in direction Nwnerals on the of negative u 3
22
The principal component of microscropic stress normal to the electric
field (Le. roughly parallel to the particle surface), CT 1 is everywhere
positive (tensile). The component in the direction of the field (Le.
roughly normal to the particle surface) may be either positive or negative
(compression). Figure 5 maps CT a for some configurations of the basic cell.
The upper row are isotropic cells: for the quarter-cell 1.2 x 1.2 there is
no compression, for '2 x 2 a small compressive zone, and for 4 x 4 a larg~
one. The two lower maps, plus that for 4 x 4, form a series with the cell
swelling one-dimensionally. Compression occurs in regions of large
gradient of electrical potential and charge density (regions close to one
particle but far from the next particle in that direction).
z
o
2 o
Fraction of total load
lines illustrating how a vertical load is carried by particle. Right: partition of the load between
water and particle.
Figure 6 is a plot of load lines (analogous to streamlines in fluid
mechanics) representing how a given vertical load is shared on various
,horizontal cross-sections between particle and water. Above the particle
the load is, of course, taken 100% by the water; but by the equatorial
plane of the particle, some 85% of the load is carried by the particle.
4.4.4 Two remarks. First we recall that the results presented are based
on the linear Debye-HUckel form of the Poisson-Boltzmann equation. We have
here a striking example of a linear process on the microscopic scale
leading to a strongly nonlinear process on the macroscopic scale.
Second, we observe that, in the unloaded state the water is generally
in tension and the particle in compression. Our particle-water system thus
behaves essentially as a prestressed composite material. A fanciful
analogy is with prestressed concrete, the water behaving like the steel,
the particles like the concrete.
23
5. Equilibrium, Flow, and Volume Change in Cells and Tissues
My first excursion into the mechanics of swelling concerned cells and
tissues (Philip 1958b, 1958c, 1958d). Here I relate that and some
connected work to concepts treated earlier in this presentation.
5.1 Energetics of osmotic cells and tissues.
5.1.1 The classical osmotic cell. Our point of departure is the energetics
of the classical osmotic cell (Starling 1896, Htlfler 1920). We express
this in symbols connected in part to those used here earlier:
iJt(V) T(V) - w(V) (24)
The cell water potential iJt can be regarded as a unique function of cell
volume V when the turgor pressure T and the osmotic pressure of the cell
contents are unique functions of V. These conditions will be met when both
cell osmotic behaviour and cell deformation are reversible.
5.1.2 Total potential and gravitational potential in cells and tissue. In
tissues made up of aggregations of cells, equilibrium, flow, and volume
change are determined by the distribution of total potential ~ throughout
the system.
For systems of this type with the vertical dimension significant,
various physiologists (see, for example, Passioura 1982, Kirkham 1990) have
recognized that ~ includes a gravitational component, so that
iJt(V) + z (25)
where z is the vertical ordinate. It is natural tp take z positive upward,
unlike in the soils context earlier where it was; positive downward. We
use potentials per unit weight of water here also, so the gravitational
component of ~ is just z. (25) is strictly comparable to (2) for
nonswelling porous media.
5.1.3 Overburden potential in cells and tissues. So far as I am aware,
physiologists have not taken the further step of recognizing that, for
24
tissues free to change volume in the vertical, <P should include also the
overburden component n. For such tissues (25) should be replaced by
I{I(V) + z + n (26)
just as, for swelling porous media, (7) replaced (2).
Unfortunately a convincing evaluation of n in tissues seems very
difficult: both estimation of the directional constraints on volume change,
and of the vertical stress against which changes occur, present problems.
A first guess is that n might effectively reduce by 1/3 the variation of <P
due to gravity.
The gravitational effect on the water relations of tall trees may be
rather less than supposed; and the same may hold for the expected
gravitational effect on blood pressure changes due to changes of (animal
and human) posture.
In 1958 I proposed a macroscopic phenomenological analysis of the
propagation of changes of turgor and related quantities such as water
potential, osmotic pressure, and water content, through tissue made up of
aggregations of osmotic cells (Philip 1958c, 1958d).
The initial formulation was rather simplified, but its basic elements
persist in more sophisticated analyses. With the individual cell taken to
be small compared with tissue dimensions, I proceeded to the limit with
cell volume treated as infinitessimal and derived an unsteady diffusion
equation describing the propagation of disturbances through the tissue.
The formulation thus proceeded from the microscopic (cell) scale to
macroscopic phenomenological theory de"aling with quantities averaged over
volumes large compared with the individual cell.
5.2. I Cell volume V, water content (), and water ratio {J. Volume V was
classically used as the independent variable specifying the state of the
osmotic cell. We adopt the convention that each cell is bounded by the
mid-surface of each wall or structure it has in common with adj oining
cells, and that V is the volume contained within that bounding surface.
Within that volume is included, as well as water, the non-aqueous
constituents of the cell, of volume Vn , taken to be constant. Then the
25
volumetric moisture content 0 is 1 - Vn/V, and the moisture ratio {} is
V/Vn - 1. These interpretations of 0 and {} are consistent with their
definitions for porous media.
5.2.2 Dependence of iII, 7, and Ul on 0 or fl. A source of economy in the
analysis is the functional interdependence of various properties. As we
have seen, iII, 7, and Ul are all functions of V and, equally, they are
functions of either 0 or {}. The chain rule for differentiation thus
transforms the equation for propagation of anyone property into that for
any other.
5.2.3 Essential ingredients of the analysis. One essential ingredient of
the analysis is that, at
equilibrium holds on the
scale of the individual cell. With this
requirement met, we may characterize the tissue for our purposes solely by
the following (or equivalent) functionals:
(1) The function iII(O) specifying the energetics of the tissue.
(2) The function K(O) specifying the water-conducting properties of
the tissue. [K(O) enters a macroscopic equation for water
transfer, analogous to (1).]
These two macroscopic functions (or their equivalents) are all we need
to develop the analysis. We may remain ignorant (agnostic) about the
geometry and disposition of cell structures determining K, and also about
the components of iII.
5.2.4 The Eulerian equations. For processes with tissue volume change
relatively small, we obtain a general nonlinear diffusion equation
aO Ot
KdiIl/dO
wholly analogous to (5) for nonswelling unsaturated porous media. [The
gravitational first-order term is missing: it could be included, but we
omit if for simplicity.] Equations for propagation of the other properties
follow simply. They are of the heat-conduction, not the diffusion, form.
26
5.2.5 The Lagrangian equations. When tissue volume change cannot be
ignored, we proceed to a Lagrangian formulation (Philip, 196ge), just as
for swelling porous media. We use ~ rather than 8, and for one-dimensional
systems we employ material coordinate m, defined by
and comparable with (8).
may be treated similarly.
In one dimension we obtain the equation (Philip, 196ge)
(29)
with
5.3 Discussion
The maj or burden of this section 5 has been that, with appropriate
reinterpretation of various quantities, the macroscopic analysis of flow
and volume change in porous media carries over to the same processes in
tissues. One facet is the comparable use of Lagrangian coordinates in the
two systems, an aspect recognized by Raats (1987).
Strictly, equations such as (27) and (29) are nonlinear, but for
tissues the nonlinearities tend to be weaker, and relative volume changes
less, than in some swelling media. The consequence is that the
simplifications of linear equations and Eulerian coordinates may be
justified in some tissue contexts.
Molz (Molz and Klepper 1972, Molz, Klepper, and Browning 1973)
successfully applied a linear and Eulerian' form of the analysis to the
radial system of cotton stem phloem. Molz a~d Klepper (1972) took account
of tissue swelling by allowing the stem radius to be time-dependent. Molz,
Klepper, and Browning (1972) used a constant 'average' radius, but
recognized that a Lagrangian analysis was desirable.
Later Molz (Molz and Ikenberry 1974, Molz 1976) noted that the original
formulation (Philip 1958c) took no explicit account of water transport
within the cell-wall structures, and that its success might therefore
27
occasion some surprise. With a cell model including both vacuole and wall
structures, he found local thermodynamic equilibrium on the scale of the
individual cell and arrived back at the diffusion description. This is of
course consistent with the general analysis of propagation of disturbances
in tissues described in section 5.2 (and especially 5.2.3) above and first
outlined in Philip (196ge).
To this point our treatment has been for tissues in which the solutes
are non-diffusible through the cell membranes. Philip (1958b) and Dainty
(1963) showed that cell dynamics is much complicated by the presence of
diffusible solutes; and the complications carryover to tissue dynamics.
Molz (Molz and Hornberger 1973, Molz 1976) has taken the first steps
towards generalizing the foregoing analysis to embrace solute diffusion.
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WATER MOVEMENT AND VOLUME CHANGE IN SWELLING SYSTEMS
D.E. Smiles and J.M. Kirby CSIRO Division of Soils GPO Box 639 Canberra ACT 2601 Australia
INTRODUCTION
Volume change associated with water movement occurs in many areas of science and technology. In civil engineering for example consolidation of soil beneath man-made structures is a well known phenomenon as is consolidation associated with drainage and exploitation of aquifers. Analogous problems arise also in the chemical engineering processes of sedimentation, centrifugation and filtration where liquid is separated from a consolidating solid in two-phase systems.
These processes are still described using different formalisms and terminology and to different degrees of scientific rigour in the identifiably different disciplines of, for example, soil mechanics, chemical engineering and soil science. In fact these differences also exist within disciplines. Reference to reviews by Leu (1986), Shirato et al. (1986) and Wakeman (1986) for example reveal that the processes of filtration, thickening and expression in chemical engineering are still approached differently despite the fact that the basic physical proce,sses of movement of liquid relative to solid are identical.
Furthermore, acceptance of new approaches to an old problem is often prejudiced by the need to displace physically inaccurate theory which has wide acceptance by virtue of usage and often of regulatory blessing. An example is the linearised, small strain, approximate. approach to consolidation of Terzaghi (1956). The utility of this approach obscured its inherent approximations (which Terzaghi (1923) recognised), despite apparent physical failure to match the prediction based in this incomplete theory (Smiles and Poulos 1969; Smiles 1973). Only in the past decade has there been significant reappraisal of the assumptions of strain and linearity by the industry in question (Gibson et al., 1981). It is distressing to note that this incomplete theory has recently provided basis for "development" in chemical engineering (Shirato et al. 1986).
This paper reviews a general and physically correct approach to one­ dimensional liquid flow and volume chang~ in a two component (solid/liquid) system which is macroscopic (Phi~ip, 1972, 1991, Raats and Klute, 1968a) in the sense that it is based on well defined and measurable properties which represent averages over a material volume which is great relative to the size of the particle or pore. In particular, it requires that the permeability (or hydraulic conductivity) of the system, and the potential of the liquid, be well-defined and measurable functions of the liquid content. The specific gravity of the solid is also required.
NATO AS! Series, VoL H 64 Mechanics of Swelling Edited by T. K. Karalis © Springer-Verlag Berlin Heidelberg 1992
34
The paper also cites experiments which provide support for the approach and identifies practically important situations for which solutions to the flow equation have been explored. Finally, the paper identifies areas where the approach presents challenges for theoretical development and practical application.
PHYSICAL CONSIDERATIONS AND THEORY
The theory depends for much of its insight on seminal papers by Raats and Klute (1968a,b) and Philip (1968), as well as extensive discussion with Drs Raats and Philip who provided impetus to many of the experiments described.
We deal exclusively with one-dimensional, two-phase flow. This provides the simplest system fully revealing the basic phenomena involved. It also encompasses a great many practically important problems and is the one most comprehensively studied. In this system, and without loss of generality we deal with water as the liquid component while most of the experiments deal with Wyoming bentonite as the solid component.
In order to describe unsteady water flow in such a system, where both phases are moving relative to an external observer and to each other, one must formulate continuity statements for each component, and formally describe the physical law that describes their relative motion in response to space gradients of force. Bird et al. (1960) explore these issues in a general sense while Raats and Klute (1968a,b) developed these ideas for water flow in soils.
Material Balance Equations
During one dimensional non- steady flow, equations of continuity may be written for the water (1) and for the solid (2), viz.
(::w] z
(2)
in which Ow and Os are the volume fractions of the water and solid respectively, Fw and F s are the respective volume fluxes relative to an external observer, and z and t are distance and time, respectively.
In a two-phase system it is evident that
(3)
During non-steady flow the flux of water relative to an observer has a component, u, relative to the solid particles, and a "convective"
35
Fw may therefore be
(4)
In (4), Fsl8s is the, average velocity of the solid relative to an observer, and 8w/8 s=f} is the moisture ratio (volume of water per unit volume of solid). In a saturated system, -6 is identical to the void ratio, e: the distinction is maintained to anticipate situations where air may enter the system and -6<e (Philip and Smiles 1969).
Substitution for Fw from (4) in (1), and also -68 s for 8w yields
(5)
Differentiation by parts followed by the elimination of two terms using (2), and division by 8s then yield the equation
(a-6) = _ (1/8 ) (au) _ (F 18 ) (a-6) at s az s s az
z t t (6)
which satisfies both continuity equations (1) and (2), but which focusses on the water component of the system.
Material Coordinates
Philip (1968) and Wakeman (1986) implicitly develop an Eulerian analysis of unsteady flow problems expressed in terms of the space coordinate z consistent with (6). Alternatively, a Lagrangian analysis may be developed recognising that left-hand side of (6), together with the second term on the right, represent the differential of -6 following the motion of the solid (e.g., Bird et al. 1960), that is
with m(z,t) a material coordinate defined by the equations
am az
so that
(7)
(8)
(9)
36
Substitution of (7) in (6) and the use of the first of (8), yields the continuity equation for the water in material space which satisfies material balance for the solid, viz.
Further development of the theory concentrates on (10), and in particular on" the laws of flow necessary to define u.
Darcy's Law for Colloidal Systems
In the system we describe, Darcy's Law describes the volume flux of the water relative to the particles in response to a space gradient of the total potential, ~, of the water (Zas1avsky 1964). Thus
u = -k(6) v- 1 a~/az (11)
In (11) u has units (ms-1), v is the kinematic viscosity of water (m2s-1), and k(6) is the water content dependent permeability. If ~ is expressed as energy per unit mass of water, with SI units Jkg- 1 , then the permeability takes units of (m2). The form of k(6) in systems that change their volume with 6 is well-known.
Total Potential of the Water
In a one dimensional (vertical) swelling material, the total potential of the water is given by
~ - ~(6) + 0 + gz (12)
In (12), g is the acceleration due to gravity, so gz is the gravitational potential of water at z, defined positive upwards relative to a convenient datum; 0 is the overburden potential, the work associated with vertical displacement of the system following addition of unit mass of water at position z; and ~(6) is the water content dependent potential that arises as a result of interaction of the water with the solid surfaces and their geometry. ~(6) is readily measured (Black 1965).
The overburden potential in a two-phase system is defined (Philip 1969a, Philip 1991) by
O(z) = g [(1 'Yw dz + P(O) 1 = gP (13)
where 'Yw is the wet specific gravity of the system, z=zl is its upper surface, P(O) (units L) is any normal surface load.
37
~(z) = .,p(-6) + g [(1 -Yw dz + P(O) 1 + gz
= pw(z) + gz (14)
In (14), pw(z) is the water pressure measured with a manometer fitted with a membrane that permits passage of water but not solid. According to (14), .,p(-6) is the negative of the "effective stress" of civil engineering theory (Aitchison 1961) or the interparticle (solid compressive) pressure of filtration theory (Shirato et al. 1986).
Equation of Unsteady Vertical Flow in a Two-Phase System
Substitution of (12) in (13) and the inclusio