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Gaussian random fields with two level-cuts—Model for asymmetric microemulsionswith nonzero spontaneous curvature?Lise Arleth, Stjepan Marcelja, and Thomas Zemb Citation: The Journal of Chemical Physics 115, 3923 (2001); doi: 10.1063/1.1388558 View online: http://dx.doi.org/10.1063/1.1388558 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/115/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Morphological transition and emulsification failure in globular microemulsions J. Chem. Phys. 131, 094508 (2009); 10.1063/1.3212002 Molecular modeling of surfactant covered oil-water interfaces: Dynamics, microstructure, and barrier for masstransport J. Chem. Phys. 128, 234709 (2008); 10.1063/1.2939123 Derivation of expressions for the spontaneous curvature, mean and Gaussian bending constants ofthermodynamically open surfactant monolayers and bilayers J. Chem. Phys. 118, 1440 (2003); 10.1063/1.1528910 A bending elasticity approach to the three-phase coexistence of microemulsions J. Chem. Phys. 115, 10986 (2001); 10.1063/1.1418730 Structure of droplet microemulsions in the semi-dilute regime J. Chem. Phys. 111, 7646 (1999); 10.1063/1.480090

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Gaussian random fields with two level-cuts—Model for asymmetricmicroemulsions with nonzero spontaneous curvature?

Lise Arletha)

Department of Condensed Matter Physics and Chemistry, Risø National Laboratory, DK-4000 Roskilde,Denmark

Stjepan MarceljaDepartment of Applied Mathematics, Australian National University, Canberra 0200, Australia

Thomas ZembService de Chimie Mole´culaire, CE Saclay, 91191 Gif-sur-Yvette, France

~Received 12 October 2000; accepted 7 June 2001!

The microstructure of a microemulsion is dominated by the thermodynamics of the surfactantinterface between the oil and water domains. As the spontaneous curvature of this surfactantinterface is strongly temperature dependent the microstructure of microemulsions also becomestemperature dependent. In the present work we have assumed that the thermodynamics of theinterface is determined by the Helfrich Hamiltonian and that the interface can be described by twoappropriately chosen level-cuts of a Gaussian random field. It is then possible to express the freeenergy density of the interface as a functional of the spectral distribution of the Gaussian randomfield so that the microstructure which minimizes the free energy can be determined by performinga functional minimization of the free energy with respect to the spectral distribution of the Gaussianrandom field. The two level-cuts are an important feature of the model since they allow us to modelmicroemulsions with nonzero spontaneous curvature and with unequal volume fractions of waterand oil. This again makes it possible to simulate the temperature driven phase inversion of themicroemulsions described above. The model furthermore allows us to predict the microstructure ofthe microemulsion for a given composition of water, oil and surfactant and input parametersH0 , kand k as well as to predict direct space structures and scattering structure factors. Microemulsionswith bicontinuous structures, droplet structures or swollen sponge-like structures are predicteddependent on the input parameters and represented in direct and inverse space. Dilution plots forscattering peak positions are in good agreement with experimental results. ©2001 AmericanInstitute of Physics.@DOI: 10.1063/1.1388558#

I. INTRODUCTION

Experimental work on ternary microemulsion systemsvia measurements of scattering, conductivity and diffusioncoefficients indicates the existence of a microstructure ofwell-defined oil and water domains with typical length scalesof 100–500 Å. The domains are subject to fluctuations on thenanosecond scale, but their average shape cannot yet be pre-dicted for any composition1 and the development of predic-tive models is still a major objective of the physics of com-plex fluids. It is generally accepted that for a system of softinterfaces—i.e., a system where the bending energy is lowcompared to the thermal energy and where there are no im-portant long range electrostatic or steric interactions, thephysics is dominated by the Hamiltonian of the surfactantinterface between the oil and water domains. This can to afirst order be approximated by an effective Hamiltonian of acurvature–elastic interface:2

H5ESdA @k~H2H0!21kK#, ~1!

whereS denotes the interface manifold.H is the mean cur-

vature which is given by12(cx1cy), wherecx andcy are thelocal principal curvatures.H0 is the spontaneous curvature,i.e., the preferred curvature of the unconstrained surfactantfilm. K is the Gaussian curvature, which is given by theproductcxcy . The bending elastic constantk and the Gauss-ian bending elastic constantk have the units ofkT and relatethe curvatures to energy. A self consistent form for a HelfrichHamiltonian which also includes a spontaneous Gaussiancurvature has been suggested by Fogdenet al.3 Even thoughit cannot generally be assumed that the spontaneous Gauss-ian curvature is negligible, we have, for the present work,decided to omit the contributions to the Helfrich Hamiltoniandue to this term.

About ten years ago, Strey and coworkers showed thatthe phase behavior of microemulsions made out of singlechain linear nonionic surfactants, water and oil can be repre-sented by a generic phase prism.4 The cut of the phase prismat equal volume fractions of water and oil has become apopular way of representing the phase behavior of the ter-nary systems. The cut is for obvious reasons often referred toas ‘‘the fish’’ ~see Fig. 1!. Varying the temperature and sur-factant concentration at a constant water to oil ratio allowsthe two basic variables:H0 , the spontaneous curvature andfs , the surface area per unit volume to be varied. The loca-a!Electronic mail: [email protected]

JOURNAL OF CHEMICAL PHYSICS VOLUME 115, NUMBER 8 22 AUGUST 2001

39230021-9606/2001/115(8)/3923/14/$18.00 © 2001 American Institute of Physics


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tion of the relevant expected microstructures are sketched inFig. 1.

According to this description, the system will at lowtemperatures, either phase separate into an oil phase and amicroemulsion phase or form a stable microemulsion depen-dent on the surfactant concentration. In both cases the sur-factant film is curved towards the oil domains which yields acontinuous water domain. The spontaneous curvature of thesurfactant film approaches zero at higher temperatures whichleads to a set of different possible structures of the system. Athigh surfactant concentration a lamellar structure (La) issometimes formed, whereas at lower surfactant concentrationa so-called bicontinuous structure is preferred. The bicon-tinuous structure is a structure where both water and oil do-mains are continuous. If the surfactant concentration dropsbelow a certain concentration, the microemulsion can sepa-rate into a three-phase system consisting of an oil phase, amicroemulsion phase and a water phase. For even highertemperatures the surfactant film will start to curve towardsthe water regions, which, dependent of the surfactant concen-tration, leaves a possibility of either a one-phase water-in-oilmicroemulsion or a two-phase system of water and a water-in-oil microemulsion. The critical pointg0 where the one-phase, two-phase and three-phase regions meet is sometimescalled the point of maximal solubilization. At the corre-sponding temperature,T, the spontaneous curvature of thesurfactant film is zero. For systems of equal volume fractionsof water and oil the mean curvature of the surfactant filmwill also be zero.

In the present paper a structure we have chosen to denotethe diluted bicontinuous structure will be discussed. Thisstructure can be obtained by taking a bicontinuous micro-emulsion with equal volume fractions of water and oil and

zero mean curvature and diluting it with either water or oil.As the composition of the diluted bicontinuous structure isasymmetric in water and oil the mean curvature of the sur-factant film is nonzero even though the spontaneous curva-ture of the film is zero. In Sec. V it will be discussed in moredetail how the diluted bicontinuous structure and the regulardroplet structure differ.

The phase diagram in Fig. 1 represents a microemulsionformed with a general nonionic surfactant. Microemulsionswith ionic surfactants generally show an inverse dependenceon temperature with water-in-oil microemulsions at low tem-peratures and oil-in-water microemulsions at high tem-peratures.4

A number of different models for microemulsions havebeen suggested over the years. When comparing the differentmodels it is important to keep in mind that they have beenmade with very different foci. Some models have been de-veloped for describing the droplet structures while othermodels describe the bicontinuous structures. Furthermoresome models have been developed in order to get tools foranalyzing, e.g., small-angle scattering data while others havebeen developed for explaining the phase behavior of micro-emulsions.

A. Models based on theory from liquid state physics

This group contains several different models which arereviewed in more detail in Ref. 5. The models all have incommon that they describe the microstructure of discreteparticles ~mono- or polydisperse spheres, ellipsoidal par-ticles! interacting with different kinds of potentials~hardspheres, attractive or repulsive!. The liquid state physicsmodels describe the small-angle scattering from dropletmicroemulsions as for example the ionic AOT@bis~2-ethylhexyl!sulfosuccinate sodium salt# microemulsions verysuccessfully6 and allow information about the size, polydis-persity and sometimes the shape of microemulsions dropletsand the type of interaction between them to be deduced fromscattering data. As the models are based on liquid statetheory and have no direct connection to the Helfrich formal-ism, there are only indirect ways of getting informationabout the bending elastic constants7–11 whereas no generalinformation can be obtained about the spontaneous curva-ture.

B. Models of the Talmon–Prager type

This category includes a number of different models allhaving in common that they fill space with different shapedand sized polyhedra, normally cubes or Voronoı¨-cells.12–15

The polyhedra are randomly filled with water or oil accord-ing to the composition of the modeled sample. The surfactantinterface is assumed to be modeled by the interface betweencells with oil and water. In this way the Talmon–Prager typeof models gives a primitive description of the microstructureof bicontinuous or diluted bicontinuous microemulsionswhich can be used for explaining some of the very basicfeatures of both the behavior and the small-angle scatteringpatterns of microemulsions.

FIG. 1. The fish plot. A schematic representation of the phase behaviordependence of temperature and surfactant concentration for a ternary systemof water, oil and nonionic surfactant. 1 denotes the one-phase region wherethe microemulsion is formed, w/o denotes the water-in-oil structure and o/wdenotes the oil-in-water structure. In between the bicontinuous microemul-sion is formed. 2I denotes the part of the two-phase region where the systemseparates into an w/o microemulsion and an excess water phase. 2¯denotesthe part of the two-phase region where the system separates into an o/wmicroemulsion and an excess oil phase. 3 denotes the three-phase regionwhere the system separates into a microemulsion and excess water and oilphases.

3924 J. Chem. Phys., Vol. 115, No. 8, 22 August 2001 Arleth, Marcelja, and Zemb


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C. Disordered open connected „DOC… models

This category of models contains both the original DOCmodel16,17 suggested by Zembet al. and a model of similartype suggested recently by Tlustyet al.18

The DOC ~Disordered Open and Connected! modeltakes as the Talmon–Prager model its starting point in aVoronoı tessellation of space. The Helfrich Hamiltonian isminimized under the constraints of constant area per unitvolume and constant sample composition and assuming thatk andk@kT. In order to avoid unphysically high curvatures,the distance between the points used for generating theVoronoı construction is equal to at least twice the surfactantlength. The model constraints are satisfied in two limitingcases where the interior phase of the microemulsion is placedalong either the edges~DOC-cylinders model! or the facets~DOC-lamellar model! of the Voronoı¨-cells depending onwhether the microemulsion has very strong or very weakcurvature towards water. In the DOC-cylinders model~strong spontaneous curvature! the structure is described bydroplets coalescing via short cylinders19 and a connected net-work structure can be formed. In the DOC-lamellar model~vanishingly low spontaneous curvature! the structure is de-scribed by connected random bilayers and different bilayerstructures including symmetric and asymmetric sponges canbe modeled.20 An analytic expression for the maximum inthe scattering structure factor as well as numerical calcula-tions of the full scattering curve are available.21

Within the frame of the DOC model, the microstructureas well as the connectivity can be predicted, as long as thebending elastic constants remain much larger than kT. How-ever, the ‘‘stiff interface’’ condition is not generally fulfilledand the model breaks down in the case of short single chainsurfactants, where the bending elastic constants are close to 1kT.

Tlusty et al.18 have suggested a model, which as theDOC-cylinders model is based on a picture of the micro-emulsion as having a branched tubular structure. The modelexpression for the free energy takes into account the contri-butions to the curvature energy of the tubes and the junctionsas well as entropy contributions from the junctions and theirdistributions. By changing the microemulsion compositionthe predicted network becomes denser and some of the scal-ing phenomena observed in the three-phase region of themicroemulsion phase diagram can be explained. No directspace structures nor scattering structure factors are predictedby the model. This has the consequence that the model is yetto be testedquantitatively against, e.g., scattering data ormicroscopy pictures. However,qualitativecomparisons withpictures of microemulsions obtained by cryogenic tempera-ture transmission electron microscopy have been made.22

These pictures of nonionic microemulsion systems ofC12E5/water/n-octane show that at certain compositions andtemperatures the microemulsion indeed has a microstructureof interconnected oil-swollen cylinders.

A disadvantage of both the original DOC model and themodel by Tlustyet al. is that only asymmetric bicontinuousmicroemulsions are predicted, i.e., bicontinuous microemul-sions with nonzero mean curvature. This implies that thebicontinuous microemulsion with zero mean curvature ob-

served aroundT cannot be explained by the models.

D. Random surface models

This category of models is based on the ideas of Cahn23

and Berk.24,25 Cahn suggested that the structure of a liquidthat undergoes spinodal decomposition could be described interms of a superposition of standing sinusoidal waves offixed wavelength but random amplitude, direction and phase.By attributing a level-cut value to the hilly landscape thusobtained and assuming that one of the two phases was givenby all the domains above the level-cut value and the othergiven by all domains below the level-cut value he achieved amodel for the structure that bore great resemblance with pic-tures of phase-separating glasses.23

Berk24 adopted this idea and made plausible that it couldalso be used it as a model for bicontinuous microemulsionsystems. He showed how to Fourier transform structuresgenerated according to Cahn’s scheme in order to determinethe scattering patterns and discussed the influence of gener-ating the field by means of a Gaussian distribution of wave-lengths instead of a single wavelength. The same yearTeubner and Strey26 derived the so-called Teubner–Streymodel from Landau theory. In both of these papers the scat-tering peak observed for microemulsions was explained andin the following years the Teubner–Strey model has beenwidely applied when analyzing scattering data on micro-emulsions~see, e.g., Ref. 27!.

Teubner28 proved that the structures generated accordingto the procedures suggested by Cahn and Berk were actuallyrealizations of Gaussian random fields and this allowed himto derive analytical expressions of the mean curvature^H&,the mean squared curvature^H2& and the Gaussian curvature^K& in terms of the level-cut value of the Gaussian randomfield.

Shortly after Pieruschka and Marc˘elja29,30 showed howthe expression for the free energy corresponding to theHelfrich Hamiltonian could be expressed in terms of level-cut Gaussian random fields generated from the spectrumn(k). From a variational minimization of the obtained ex-pression for the Helfrich free energy, they showed that if thephysics of the microemulsion system is governed by theHelfrich Hamiltonian, then the Gaussian random field thatmodels the microemulsion system should be generated by aspectrumn(k) that has the same functional form as theTeubner–Strey model.

Compared to the previously mentioned models theGaussian random field model has the advantage that it isdirectly based on the physics contained in the HelfrichHamiltonian. The structures predicted by the model are notconstrained to be either tubular or lamellar as in the modelsof the DOC type, but can be varied to some extent by chang-ing the spectral density of the generating random field. How-ever, in its present form the Gaussian random field modelstill shares the great disadvantage with the models of theTalmon–Prager and DOC type that the curvature can only bechanged continuously by changing the composition of thesystem.

3925J. Chem. Phys., Vol. 115, No. 8, 22 August 2001 Model for asymmetric microemulsions


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The intention of the present work is to use the formalismdeveloped by Teubner, Marc˘elja, and Pieruschka to set up amodel for microemulsions where both the spontaneous cur-vatures and the volume fractions of water, oil and surfactantcan be varied. We will show how it becomes possible tomodel a phase inversion of the microemulsion microstructurefrom oil droplets in water through bicontinuous to waterdroplets in oil simply by changing the spontaneous curva-ture. Direct space structures and scattering structure factorsare calculated for different sets of values for the bendingelastic constant,k, the Gaussian bending elastic constant,k,and the spontaneous curvature,H0 . Furthermore, predictionsof the microemulsion microstructure as a function of theseparameters are made.

II. BASIC IDEAS

We assume that the physics of the microemulsion is de-scribed by the Helfrich Hamiltonian and that the surfactantinterface can be described by appropriate level-cutsa andbof a Gaussian random field:

s~r!5E dks~k!eikr, ~2!

where the amplitudess(k) are taken from a Gaussian distri-bution centered at zero with standard deviation given by thefunction n~k!:

p„s~k!…51

A2pn~k!e2us(k)u/„2n(k)). ~3!

The level-cutsa and b determine the representation ofthe microemulsion from the underlying Gaussian randomfield. We have assumed that the water and oil domains arebounded as follows:

r is a water domain ifH s~r!>a or s~r!<b, for a.b;

s~r!>a ands~r!<b, for a,b;

r is an oil domain elsewhere ~4!

The definitions are illustrated for a two-dimensional fieldin Fig. 2. The bicontinuous microemulsion with zero meancurvature modeled in the one-level cut case29,30 is obtainedwith equal volume fractions of water and oil and in the limita50 and b56inf. By allowing both the possibilitya.banda,b it becomes furthermore possible to simulate a con-tinuous transition from positive mean curvature to negativemean curvature of the surfactant film. This corresponds to atransition from water droplets in oil to oil droplets in water.It should be noted that for a given composition of the micro-emulsion, the constraint of the fixed volume fraction of thewater has the consequence thatb is a function ofa ~or theopposite!. It should also be noted that even though the levelcut valuesa and b provide a bridge between the Gaussianrandom field and the picture of the microemulsion they donot play a role on their own and are not experimentally ob-servable parameters in real systems.

FIG. 2. An illustration of the two-level-cut modeling principle.~a! Plot of atwo-dimensional Gaussian random field without level-cuts. The differentlevels are indicated by their contour lines.~b! The level-cut values are set toa50 andb5` corresponding to a bicontinuous structure with zero meancurvature and equal surface fractions of water and oil. For the illustrationwater is white and oil is black.~c! A structure with water droplets in oil issimulated by settinga51 andb521. The surface fraction of water in thisstructure is 0.32.



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III. DETERMINATION OF THE GAUSSIAN RANDOMFIELD AND THE LEVEL-CUT VALUES

The free energy density of the Gaussian random fieldwith level-cuts that approximates the bending Hamiltoniangiven by Eq.~1! can be approximated by31,32

f free'S

V@2k^~H2H0!2&1k^K&#2

1

2p2Ekdk k2 ln n~k!

5S

V@2k^H2&24kH0^H&12k^H0

2&1k^K&#

21

2p2Ekdk k2 ln n~k!, ~5!

whereS/V is the surface to volume ratio of the system. Thebrackets^¯& denote the average taken over the Gaussianrandom field. The first four terms of Eq.~5! thus contain thecontributions to the surface energy whereas the last term isour ansatz for the entropy term.

For a two-level-cut model with level-cutsa and b thesurface to volume ratio is given by

S

V5

2

pA^k2&

3@e2a2

1e2b2#[

2

pA^k2&

3@A#[fs , ~6!

where@A# is an abbreviation of@e2a21e2b2

#. This expres-sion is just a generalization of the expression given byTeubner in Ref. 28 for the one-level-cut model.fs is thevolume fraction of the surfactant. Strictly speaking the rela-tion betweenfs and the surface to volume ratio is given by

S

V5

fs

ds, ~7!

whereds is the thickness of the surfactant layer. This relationwill later allow us to attribute an absolute length scale to thesystem.

The moment kn& is in 3-dimensional space calculated asfollows:

^kn&51

2p2E0

kcdk k21n n~k!. ~8!

As the distribution functionn(k) should be normalizedto unity, we have the following constraint:

^k0&5^1&51

2p2E0

kcdk k2 n~k!51. ~9!

The surface to volume ratio is fixed in a thermodynamicallystable microemulsion so we get an additional constraint onn(k) from ~6!:

^k2&53S fsp

@A#2D 2

51

2p2E0

kcdk k4 n~k!. ~10!

The averagesH&, ^H2& and ^K& for the two-level-cutmodel are generalized from the expressions for the one-level-cut model28 as follows:

^x&a1b5

^x&a

Sa

V1^x&b

Sb

V

Sa

V1

Sb

V

5^x&ae2a2/21^x&be2b2/2

e2a2/21e2b2/2, ~11!

where^x&a denotes the average ofx of the surface given bythe a-level-cut.Sa /V is the surface to volume ratio of thesurface given by thea-level-cut, it is proportional to thesurface area of thea surface and is therefore used as aweight factor in the averaging procedure. This means that^H&, ^H2& and ^K& are given by

^H&51

2Ap

6^k2& Fae2a2/22be2b2/2

e2a2/21e2b2/2 G[1

2Ap

6^k2&@B#,

~12!

^H2&51

6^k2&Fa2e2a2/21b2e2b2/2

e2a2/21e2b2/21

6

5

^k4&

^k2&221G

[1

6^k2&†@B2#21‡1

1

5

^k4&

^k2&, ~13!

and

^K&51

6^k2&Fa2e2a2/21b2e2b2/2

e2a2/21e2b2/221G[

1

6^k2&†@B2#21‡,

~14!

where @B# is an abbreviation of @(ae2a2/2

2be2b2/2)/(e2a2/21e2b2/2)# and@B2# is an abbreviation of

@(a2e2a2/21b2e2b2/2)/(e2a2/21e2b2/2)#. By inserting~12!, ~13! and~14! in ~5! and using~8! the free energy can beexpressed as a functional ofn(k). In order to determine thespectrumn(k) which minimizes the free energy given by Eq.~5!, a variational minimization off free with respect ton(k) isperformed. The two constraints~9! and ~10! are taken intoaccount in the minimization via Lagrange multipliers. Theresult of the variational minimization is thatn(k) has thesame functional form as in the case of one-level-cut Gauss-ian random fields,30 namely

n~k!5a

k42b k21c, ~15!

where the constantsa, b andc are modified according to thetwo-level-cut model as follows:

a5p215fs

16k@A#2, ~16!

b52~@B2#21!fs

2p215

16@A#21

2@A#2^k4&

3p2fs2

15@B#p3/2fsH0

2A2@A#2

5

2H0

2

215fs

2p2~@B2#21!k

32k@A#22

5l1

4k, ~17!

and



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c515l2p2fs

16k@A#2. ~18!

The constanta depends only on the level-cut parametersaandb and known parameters. The constantsb andc dependon the undetermined Lagrange multipliersl1 andl2 . b andc can be determined by solving the equation system given by~9! and~10! using the expression forn(k) given by Eq.~15!.Integration of~9! leads to the equation

05

Ab2Ab224ctanh21S A2kc

Ab2Ab224cD

A2Ab224c

2

Ab1Ab224ctanh21S A2kc

Ab1Ab224cD

A2Ab224c2

2p2

a.

~19!

Similarly, integration of~10! leads to the equation

0528kfsp

2

51kc

1

A2~b2Ab224c!3/2tanh21S A2kc

Ab2Ab224cD

4Ab224c

2

A2~b1Ab224c!3/2tanh21S A2kc

Ab1Ab224cD

4Ab224c.

~20!

Here ~19! and ~20! are transcendental equations inb and cand have to be solved numerically for each choice of thelevel-cutsa andb. When this is done we have an expressionfor the spectrumn(k) that minimizes the free energy for agiven set of level-cutsa and b. From n(k) we are able tocalculate the free energy, which is then easy to minimizenumerically with respect to the values of level-cuts. It isnoted that the integral in~10! is not convergent for the spec-tral distribution given by~15!. As in earlier work, a cut-offvalue kc is therefore needed.kc equals 2p/lc , wherelc ,physically, is the shortest wavelength where the first orderexpansion for the elastic theory can be applied. We believethatkc51 nm21 corresponding to alc of about ten molecu-lar widths is a reasonable cut-off value.

IV. CALCULATION OF STRUCTURE FACTORS

Reports of experimental studies of microemulsion sys-tems refer to the concentrations of surfactants~and some-times co-surfactants!, water ~including soluble species suchas salt! and oil. Then a procedure for splitting the spaceoccupied by the microemulsion into polar and apolar vol-

umes relies on identifying a ‘‘polar head’’ and a ‘‘hydropho-bic chain’’ of the surfactant molecule. For the model calcu-lations in the present study we have assumed that half thesurfactant molecule is polar and the other half is apolar. Thevolume fraction of oil referred to in the text,fo , refers to thetotal apolar volume, that is the volume fraction of the pureoil and half the volume fraction of the surfactant. Similarly,the volume fraction of water,fw , refers to the total polarvolume. This means thatfo1fw adds up to unity.

Once the spectrumn(k) with level-cutsa and b thatminimizes the Helfrich free energy is found, we obtain arepresentation of the structure of the microemulsion both indirect and in inverse space.

A. Representation in direct space

In direct space the representation is obtained from theequation~2! where then(k) enters through the distributions(k) and from the level-cut valuesa and b. The softwarethat has been used for generating the direct space represen-tations of the microemulsions in Fig. 6 and 7 was written byRoberts.33

B. Representation in reciprocal space

The reciprocal space representations of the microemul-sion are the structure factors that can be obtained in a scat-tering experiment. The structure factor is calculated fromn(k) according to the following procedure:29 n(k) is Fouriertransformed and the correlation function,g(r ), is obtained.The correlation function of the random Gaussian field withlevel-cutsa and b, Gab, is calculated fromg(r ). Gab isFourier transformed and the structure factor,s(q), is ob-tained. The integrations necessary to determineGab fromg(r ) and to Fourier transformGab are carried out numeri-cally using the very robust Fortran subroutines from theQuadpack library.34

C. Conformations with similar free energy

The variational minimization of the free energy only de-termines the minimizing spectrumn(k). We obtain no infor-mation about how sharp the minimum is and therefore noinformation about what the energy cost of perturbing thisminimal conformation would be. As the minimization of thefree energy with respect to the level-cut valuesa and bis carried out separately after the variational minimizationhas been performed, it is possible to get an estimate of theBoltzmann distribution of the conformations given by thedifferent level-cuts a and b. We have calculated theBoltzmann factors of the different conformations using thefollowing expression:

PB~a!5e[(F free(a)2F free(amin))/pj3/6]

*e[(F free(a)2F free(amin))/pj3/6]da, ~21!

i.e., as the difference between the free energy per unit vol-ume of the conformation given by the level-cut valuea andthe free energy per unit volume of the minimal energy con-formation. The unit volume is taken to be the volume of a



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sphere with diameter equal to the correlation lengthj of thestructure, wherej is calculated from the factorsb and c inthe n(k) using the following expression:26

j51

A1

2Ac1

b

4

. ~22!

In Ref. 35 this approach has been called conformational dif-fusion. However, it does not give the full Boltzmann distri-bution over all conformations but only the Boltzmann distri-bution over the conformations given by the different level-cut values. In spite of this, the averaging provides an ideaabout the effect of conformations with similar free energiesand we have included it in the calculations of the reciprocalspace structure factors presented in the following section.

V. RESULTS

Minimization of the free energy, Eq.~5!, tells us whichmicrostructure is the preferred for one given sample compo-sition and set of parametersk, k andH0 . The sample com-position is fixed once the volume fraction of the water phasefw ~or alternatively the oil phasefo) and the surface tovolume ratiofs are fixed. This gives us five possible param-eters to vary when we want to study the behavior of the freeenergy. With this large parameter space it is an impossibletask to give a complete account of all model predictions, butin the following we will show some examples. In the modelpredictions we have chosen to focus onk51 kT andk52 kTsince most experimental studies of microemulsion systemsindicate that these values are typical for the bending elasticconstants. We will show examples of calculations onmicroemulsions with both symmetric and asymmetric com-positions.

A. Examples of model predictions

Figure 3 contains diagrams of the microemulsion struc-ture as function ofH0 for a microemulsion with equal vol-ume fractions of water and oil but varyingfs , wherefs isdirectly proportional to the surfactant volume fraction.36 Inthe microemulsion part of the three-phase region~see Fig. 1!the mean curvatureH& has been shown to be proportional to(T2T) around T.4 In this part of the phase diagram themean curvature attempts to follow the spontaneous curvatureH0. This means that it is reasonable to assume thatH0 is alsoproportional to (T2T) aroundT. As the temperature depen-dence ofH0 is intrinsic for the surfactant37—in contrast tothe temperature dependence of^H& which depends on boththe surfactant and the composition—it is furthermore reason-able to assume that the proportionality betweenH0 and (T2T) extends into the one-phase region of the phase diagram.This implies that theH0-fs cut through phase space shownin Fig. 3 is comparable to the traditional ‘‘fish’’-cut shownin Fig. 3. However, as the model cannot~yet! be used forpredicting the phase transition lines between one-phase andtwo- or three-phase regions, Fig. 3 is only a diagram ofwhat the microstructure will beif the system forms a micro-emulsion.

In Fig. 3~a! the bending elastic constantk is fixed at 1 kTand in 3~b! k52 kT, the Gaussian bending elastic constantk has been set to a number of different values. The diagramshould be read as follows: Sayk50, find the upper andlower transition line corresponding tok50. Everywhere be-tween the two lines the microemulsion has a bicontinuousstructure with zero mean curvature. In this region the micro-structure of the microemulsion will not change as tempera-ture changes. Above the upper line the microemulsion formswater droplets in oil and below the lower line the microemul-sion forms oil droplets in water. In the model the distinctionbetween bicontinuous and droplet microemulsions are givenby the set of level-cut values~a,b! that minimizes the free

FIG. 3. A model prediction of the phase behavior dependence offs andH0

for microemulsions with equal volume fractions of water and oil. For non-

ionic microemulsionsH0}(T2T) for which reason theH0-scale has beenreversed.~a! The phase behavior of a system withk51 kT and a number ofdifferent values fork has been plotted.~b! A similar plot but fork52 kT.The label ‘‘bicontinuous’’ refers to a bicontinuous structure with zero meancurvature, while w/o~or o/w! refers to a structure of water droplets in oil~respectively, oil droplets in water!.



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energy. If ~a,b!5~0, ! the microemulsion is bicontinuouswith zero mean curvature. Ifa andb have other values say~a,b!5~0.67,20.67! the microemulsion forms droplets. Fig-ures 6~a! and 6~b! are two-dimensional cuts through theGaussian random field models of bicontinuous and dropletstructures generated fork51 kT. Figures 7~a! and 7~b! areprojections of the corresponding 3D structures. Note that thebicontinuous structure given here is in principle similar tothe bicontinuous structure previously described byPieruschka and Marc˘elja.29 Comparing Figs. 3~a! and 3~b!we observe that a largerk gives rise to a more narrow bicon-tinuous microemulsion region.

If k is negative the bicontinuous structure is only fa-vored for low fs and close toH050. The introduction ofspontaneous curvature quickly leads to the formation ofdroplets. The local stability limit evaluated in the absence ofspontaneous curvature,2k,2k, is never reached.38

A positive k would favor a bicontinuous structure in alarger part of the phase space if the structure remained stable.This is not surprising when inspecting the expression for theHelfrich free energy, Eq.~5!. The mean Gaussian curvature^K& will be negative for bicontinuous structures and positivefor droplet structures and the sign ofk therefore determineswhich of these structures has the lowest free energy. How-ever, whenk.0 the free energy is lower for structures ofhigh absolute value ofK&, and the system is likely to phaseseparate.

The model predicts a first order phase transition from thebicontinuous domain to the droplet domain. The transitiontakes place in a transition region where both phases coexistand which lies outside the transition line marked on the fig-ure. For low values offs the transition region is quite ex-tended. With increasinguH0u the droplet domains are gradu-ally closed and the droplets become more and moremonodisperse. The free energy is only weakly dependent onthe level-cut values~a,b! in a rather wide region around thelevel-cut values that minimize the free energy. This gives riseto the easy coexistence of conformations ranging from thebicontinuous structure with zero mean curvature to the mostmonodisperse and curved structure the model can produce.As fs increases the transition region become gradually moreand more narrow, the dependence between the free energyand the level-cut values become more pronounced and fewerconformations are available to the system. For highfs thetransition region is very narrow and a large morphologicalchange is induced by a small increase inuH0u. In the transi-tion region the volume fraction of microemulsion with a zeromean curvature bicontinuous structure quickly drops whilethe volume fraction of the most monodisperse droplets themodel can produce„~a,b!5~0.67,20.67!… increases corre-spondingly. The bending elastic constantk also plays an im-portant role for how the bicontinuous to droplet transitionoccurs. A higherk generally gives rise to a more narrowtransition region.

Figure 4 contains a similar diagram but for a dilutedversion of the microemulsion from Fig. 3, where the volumefraction of the apolar phase~oil plus the apolar part of thesurfactant! is set tofo50.25 while the remaining part,fw ,is constituted by water and the polar part of the surfactant.

This asymmetric microemulsion can take three differenttypes of structures.

For large positive spontaneous curvatures the micro-emulsion will form oil droplets in water which are as mono-disperse and regular shaped as the model allows them to be.This region is labeled o/w in the figure. In Figs. 6~c! and 7~c!a Gaussian random field model of the droplet structure isshown in two dimensions and three dimensions, respectively.On the surface of the droplets small local inclusions/protrusions of a much smaller size than an average dropletcan be observed. These are allowed as long as the bendingconstants are not too high and they consume a significantpart of the available interface.

In a region aroundH050 the microemulsion forms astructure we have chosen to call ‘‘diluted bicontinuous’’ asmentioned in the introduction. Figures 6~d! and 7~d! show

FIG. 4. A similar plot as in Fig. 3 but for a system with asymmetric com-position,fo50.25 andfw50.75. ~a! k51 kT and~b! k52 kT. The struc-tures labeled ‘‘diluted bicontinuous,’’ ‘‘symmetric sponge,’’ and ‘‘asymmet-ric sponge’’ are explained in the text. ‘‘o/w’’ refers to the structure of oildroplets in water also explained in the text.



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the Gaussian random field model of the droplet structure intwo and three dimensions, respectively. Due to the conflictbetween a low spontaneous curvature and the asymmetriccomposition of the sample, an o/w structure with larger andmore irregular shaped droplets than in the droplet micro-emulsion is formed. Everywhere inside the diluted bicontinu-ous region (a,b)5(0.67, ), so one of the level-cuts is in-finite as in the bicontinuous microemulsion with zero meancurvature. However, as the sample composition is asymmet-ric, bicontinuity is not generally guaranteed. The mean cur-vature of this diluted bicontinuous structure is positive, butsignificantly lower than the mean curvature of the dropletstructure described above.

If the spontaneous curvature is large and negative therewill be a competition between the o/w structure preferred bythe asymmetric sample composition and the w/o structurepreferred by the spontaneous curvature of the surfactant film.This leads to a peculiar structure: water-droplets with nega-tive curvature start to grow inside the oil-droplets in order tominimize the curvature contribution to the free energy and amicrostructure that resembles vesicles with an oil-swollendouble layer is obtained@Figs. 6~e! and 7~e!#. The structurecan also be described as an asymmetric sponge structure andis comparable to the multiple emulsion structures~with drop-lets inside droplets! observed on a larger scale. With increas-ingly negativeH0 the size of these objects grow and in thelimiting case a symmetric swollen sponge structure witha52b is formed@Figs. 6~f! and 7~f!#. It should be noted thatthis structure is similar to the swollen L3 structure discussedin Ref. 39. Comparing Figs. 4~a! and 4~b! we note again thata higherk leads to a more narrow region of the diluted bi-continuous structure.

Figure 5 contains diagrams of how the microstructurechanges with temperature for a system with fixed oil to sur-factant ratiofo /fs52.5 and varying oil-droplet concentra-tion fo . Again there are three characteristic structures: Atlow negativeH0 the water in oil structure is preferred, athigh positiveH0 the oil in water structure is preferred, and inbetween there is a bicontinuous or diluted bicontinuous do-main @i.e., ~a,b!5~c, !, where the constantc depends onfo]. We observe that depending on the value ofH0 and kone can obtain an increase or a decrease in the connectivityof the water domains with increasing oil contents contents. Asimilar behavior has been observed experimentally byRushforth et al. in a series conductivity measurements onionic microemulsions.40

Figure 6, which we have already referred to above, con-tains examples of some of the different structures the micro-emulsion system can take. The plots are 2D cross sections ofthe three-dimensional Gaussian random fields generatedfrom the spectrumn(k) with the level-cutsa and b whichminimize the expression for the free energy of the micro-emulsion system@Eq. ~5!#. Figure 7 contains projections ofthe full 3D field of the same structures as Fig. 6.

Figure 8 contains examples of the structure factors thedifferent structures would give rise to in a scattering experi-ment with bulk contrast. Figure 8~a! shows the scatteringfrom a bicontinuous structure formed in a system with equalvolume fractions of water and oil. The scattering curves are

calculated fork51 kT andk52 kT. It is seen that the lowerk value gives rise to a more smooth scattering peak.

Figure 8~b! shows the scattering from the droplet struc-tures formed in the same systems for higheruH0u. For k52kT and with increasinguH0u the transition from the bicon-tinuous to the most monodisperse o/w structure takes placein a very narrow transition region. The scattering patternfrom the droplet structure is shown. Fork51 kT the transi-tion region is more extended. Gradually the large irregularshaped regions formed in the bicontinuous region becomemore and more monodisperse. The scattering patterns result-ing from the droplet structures formed for two different val-ues of H0 are shown. An important feature of the dropletstructure factors is that the scattering peak observed for thebicontinuous system disappears as the structure becomemore and more droplet-like. This was at first quite surprising

FIG. 5. A plot of the microstructure dependence on the oil contentsfo andspontaneous curvatureH0 for a system with fixedfo /fs52.5. The micro-structure is indicated for a number of different values ofk. ~a! k51 kT.~b! k52 kT.



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to us because real droplet microemulsions at these composi-tions normally exhibit a strong scattering peak~see, e.g., Ref.6!. However, the scattering peak observed in real dropletmicroemulsions with bulk-contrast arises from droplet–droplet correlations which are usually due to steric or elec-trostatic interactions between the droplets.5 Such interactionsare not contained in the Gaussian random field model. Fur-thermore, since the picture of the droplets is based on aGaussian random field, once a droplet is observed, the pos-sibility of observing another droplet is larger close to thedroplet than far away from it. This implies that the spatialdistribution of droplets in the model becomes different fromthat of real droplet microemulsions in such a way that noscattering peak is produced. We refer here to the peak occur-ring from the repulsive interaction between droplets. The po-

sition of this maximum is nearly identical for x-ray and neu-tron scattering. This has to be distinguished from thesecondary maximum, mainly visible in systems of sphericalobjects with shell contrast. The position of this secondarymaximum is related to the form factor and is independent ofthe concentration of surfactant.41 Since the droplets modeledin the present case have bulk contrast and are both non-spherical and relatively polydisperse this secondary maxi-mum is not visible.

Figure 8~c! shows how the structure factor evolves withH0 for a microemulsion with a volume fraction of oil,fo

50.25 andk51 kT. Again a clear scattering peak is ob-served for the bicontinuous structure. As for thefo5fw

case this scattering peak is gradually reduced asH0 increasesand smaller and more monodisperse droplets are formed. Asimilar trend is seen whenH0 decreases. The scattering peakis still present for the swollen vesicles/asymmetric spongebut vanishes gradually as the structure with decreasingH0

approaches the symmetric sponge structure. Experimentsshow that real sponge structures also exhibit a scatteringpeak20 which is due to film–film correlations arising from arelatively well-defined thickness of the surfactant bilayer.The thickness of the bilayer simulated by the Gaussian ran-dom field model apparently fluctuates more than that of realbilayers and the scattering peak disappears.

The interaction between adjacent interfaces in the twolevel-cut model is a contact repulsion, instead of havingsome spatial range. Broad peaks are that are observed in realsystems where repulsive interaction is felt at distances typi-cally 1–2 nm from the oil/water interface are not reproducedbecause of this deficiency in the free energy expression ofthe model.

Finally Fig. 8~d! shows how the structure factor of thebicontinuous/diluted bicontinuous structure evolves with thevarying volume fraction of the oil for a system with fixedfs50.1. We observe that the scattering peak graduallymoves to higher and higher scattering angles as the volumefraction of the oil decreases. This is in good agreement withthe smaller and smaller domain size.

B. Validity of the model

Israelachvili, Mitchell and Ninham suggested in the1970s that the so-called surfactant packing parameterp5v/al—wherev is the volume,a is the area per head groupand l is the length of the surfactant—is crucial for the shapeof surfactant aggregates.42,43 Whenp decreases from one to-wards zero the surfactant aggregates will pass through a se-quence of shapes starting with being locally flat as in bilay-ers or vesicles, to form cylindrical micelles for intermediaryvalues ofp and finally to form spherical micelles for lowvalues ofp.

For microemulsion systems the situation is more compli-cated as the structure also depends on the composition andthe bending elastic constantsk and k and in parallel to theHelfrich formalism, both the mean packing parameterp anda spontaneous packing parameterp0 has to be taken intoaccount. In the following discussion we will restrain our-selves to microemulsions with equal volume fractions of wa-ter and oil. For microemulsionsp and p0 are generally re-

FIG. 6. The 2D cross sections of the direct space representation of micro-emulsions with different compositions and spontaneous curvaturesH0 aspredicted by the Gaussian random field model. For the illustrations water isblack and oil is white. The side lengths of the figures are 320pÅ'1000 Å.~a! Bicontinuous microemulsion,fw5fo , fs50.2, k51 kT, k520.5 kT,H050 1/Å. ~b! Droplet microemulsion,fw5fo , fs50.2, k51 kT, k520.5 kT, H050.075 1/Å. ~c! Droplet microemulsion,fo50.25, fs50.1,k51 kT, k520.5 kT, large positiveH0 . ~d! Diluted bicontinuous micro-emulsion,fo50.25, fs50.1, k51 kT, k520.5 kT, H050 1/Å. ~e! Swol-len vesicles/asymmetric sponge,fo50.25, fs50.1, k51 kT, k520.5 kT,H0520.05 1/Å. ~f! Sponge structure,fo50.25, fs50.1, k51 kT, k520.5 kT, large negativeH0 .

FIG. 7. The same structures as in Fig. 6 but projections of the full 3D fields.Note that the side lengths of the cubes in~a! and~b! are only 160pÅ, i.e.,half the side lengths of the squares in Fig. 6~a! and 6~b!. The side lengths of~c!–~f! are 320pÅ as in Figs. 6~c!–6~f!.



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placed by the mean and spontaneous curvaturesH and H0

using the equalitiesp511Hl 1Kl 2/3 and p0511H0l ,where l is the surfactant length, but as a rule of thumb thesequence of preferred conformations still goes from locallyflat structures such as bicontinuous microemulsions withzero mean curvature, through locally cylindrical or tubularstructures, to structures with almost spherical droplets asuH0u increases from 0 to larger values. Structures which arelocally cylindrical cannot be modeled within the Gaussianrandom field formalism. Instead, an increasingH0 , drivesthe microemulsion from a bicontinuous structure with zeromean curvature, though a transition region with large irregu-lar shaped and sized droplets to a structure with as sphericaland monodisperse droplets as the model can produce.

For low values ofH0 where structures which minimizethe mean curvature are favored we expect our model to givea more realistic description of the microemulsion structurethan the models of the DOC-type17,18 which describe thestructures as locally cylindrical or lamellar. But asH0 in-creases to larger values and more curved structures areformed, the description of surface conformations by level-cuts of Gaussian random fields becomes increasingly inaccu-rate and we expect the DOC-type of models to give a betterdescription of the microstructure of the microemulsions. Foreven larger values ofH0 where almost spherical, monodis-perse droplet structures are formed,6,7,11 other factors thanthe physics of the surfactant interface start to play an impor-

tant role for the microemulsion microstructure and theHelfrich formalism is no longer the best choice for describ-ing the microemulsion.

The structures of the microemulsions depend on severalparameters and the value ofH0 for which we expect themodel to be unrealistic depends totally on these parameters.For this reason it is impossible to give a general quantitativemeasure for the value ofH0 where the model breaks down.However, in a system with equal volume fractions of waterand oil andk50 a ‘‘low value of H0’’ would refer to situa-tions whereH0

2,^H2&. In practice,H0 is imposed by salin-ity, temperature etc., whileH is imposed by the volume frac-tion of the components.

C. Comparison to experimental data

In Fig. 9 a part of the temperature versusfs phase dia-gram for fw5fo ~the fish-cut! of the H2O–octane–C12E5

system~reproduced from Ref. 4! have been plotted. The tem-perature scale normally used in this kind of phase diagramsis converted to aH0& scale using the expression4

^H0&51.2231023~ T2T! Å21, ~23!

where it is assumed that^H0& has the same temperature de-pendence asH& around^H&50. The bending moduli of thesurfactant film in this particular system has been determined

FIG. 8. Predictions of the scatteringfunctions for different microemulsionstructures.~a! A bicontinuous structurewith fw5fo , fs50.2, plotted fork52 kT and k51 kT. ~b! Dropletstructure withfw5fo , fs50.2, plot-ted for k52 kT and H050.025 1/Å,for k51 kT and H050.025 1/Å andfor k51 kT and H050.075 1/Å. ~c!The different structures of a micro-emulsion with asymmetric composi-tion fo50.25 andfs50.1 plotted fora symmetric sponge structure, large~negative! H0, a structure with swollenvesicles,H0520.05 1/AA, a dilutedbicontinuous,H050.0 1/AA and for adroplet structure, large~positive! H0 .~d! Dilution series, bicontinuous anddiluted bicontinuous structures, forfixed fs50.1 andH050.0 1/AA. fo

varies from 0.5 to 0.1.



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to k51.0 kT and k520.36 kT.44 The phase diagram hasbeen plotted together with the diagram of the microstructurefor a system with similar parameters.

It is seen that, according to the model, a bicontinuousstructure is favored in the majority of the one-phase region.Only for very high fs a droplet-structure will be favored.This has the consequence that a phase inversion in the one-phase region, from o/w through bicontinuous to w/0, is onlypredicted for highfs in the water/octane/C12E5 microemul-sion. This is in very good agreement with scattering mea-surements of the temperature dependence in the one-phaseregion of the very similar D2O–decane–C12E5 system.45 Themeasurements demonstrate thatI (q) is basically unchangedover a range of temperatures covering the one-phase region.

Most of the scattering data from nonionic microemul-sions published in the literature are measured on systemswith bending moduli and sample compositions not too dif-ferent from the water/octane/C12E5 microemulsion system,where the bicontinuous structure is stable over a large rangeof temperatures in the one-phase region.44 This probably ex-plains why, to our knowledge, there are no reports in theliterature showing experimental evidence for a temperaturedriven structural inversion from o/w through bicontinuous tow/o within the one-phase region, with an associated changein the structure factors measured by small-angle scattering.In order to measure such a transition it will, according to ourpredictions, be necessary to find a microemulsion systemwhere it is possible to go to highfs in the one-phase regionwithout getting in conflict with the lamellar region. Or to finda system with a sufficiently large~negative! k.

Extensive work have been done by Strey and co-workersto study the behavior of the microemulsion phase in thethree-phase region~the body of the fish! and in the two-phase regions below and above the three-phase region.4,27,44

Upon changing the temperature the composition of the mi-

croemulsion phase changes from being more water-rich atlow temperatures to be symmetric in water and oil atT andto be more oil-rich at high temperatures. The so-called char-acteristic length scale,jTS, of the microemulsion domainsize has been determined as a function of temperature byfitting the small-angle scattering data with the Teubner–Streyexpression.4,44 Below and aboveT, jTS is relatively small,thenjTS increases and reaches a maximum atT.4 In terms ofthe position of the scattering peak this has the effect that thepeak moves smoothly from high scattering vectors at lowtemperatures to a low scattering vector atT and back to highscattering vectors at high temperatures. These measurementsare in good agreement with the model. As seen from Figs. 3,4 and 9 a bicontinuous or diluted bicontinuous structure isfavored for sample parameters similar to the ones in thethree-phase region and as seen from Fig. 8~d! the change incomposition of the diluted bicontinuous sample will give riseto a scattering peak which moves to low scattering vectors asthe sample composition approachesfo5fw . Freeze-fracture electron microscopy~FFEM! pictures of the micro-emulsion phase in the three-phase regions4 confirm that theoil domains at low temperatures and the water domains athigh temperatures are relatively large and irregular shapedwhereas a bicontinuous structure is observed for intermediatetemperatures. This is in very good agreement with thebicontinuous/diluted bicontinuous structure predicted by ourmodel. FFEM pictures of microemulsions in the one-phaseregion at temperatures close toT and with fixed surfactantvolume fraction and varying oil–water ratio confirm a simi-lar picture with a microstructural change from bicontinuousto droplets as the sample composition changes from beingsymmetrical to be asymmetrical in water and oil.46

An effective test of a structural model for microemul-sions consists in comparing the scaled peak position,Ps

52pfs /qmaxl, from experiments and model in a dilutionplot,17,47 where l is the surfactant length. Figure 10 showssuch a dilution plot. The scaled peak positions of the Gauss-ian random field model for bicontinuous and diluted bicon-tinuous microemulsions are plotted fork51 kT andk52 kT,respectively. For the calculationsfs is kept fixed at 0.1 whilefo andfw vary.

In models with random placement of the oil and waterdomains, the characteristic length scalej has the followingdependence on the composition:48

j5afo~12fo!

fs / l, ~24!

where a is a numerical constant. The constanta dependsweakly on the particular model of the microemulsion struc-ture. Notice that this formula can be obtained from geometri-cal arguments and need not include any statistical mechanics.

The scattering peak and the corresponding characteristiclength scale,j, have been analyzed for a number of micro-emulsions in the bicontinuous and diluted bicontinuous do-main ~see, e.g., Refs. 27,49!. Fitting data with the Teubner–Strey expression for scattering intensity provides the value ofthe correlation lengthjTS for a particular microemulsion.27 Ifthis length is accepted as the characteristic length scalej, the

FIG. 9. The temperature-fs phase diagram of the H2O–octane– C12E5 sys-tem plotted together with the model predictions of the microstructure. Thetemperature scale of the original fish plot is converted to aH0-scale.mE

denotes the microemulsion region.La denotes the lamellar region. 3 denotesthe three-phase region.



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experiments then suggestj'p/qmax as in Ref. 49. Thescaled peak position is then given asPs'2afo(12fo).The empirical lawj'p/qmax appears natural for symmetri-cal microemulsion compositions where periodicity is of theorder of twice the domain size. However, it does not appearnatural for an imperfect distribution of droplets on a latticeof sizej, where one normally would expect a scattering peakat qmax'2p/j. In Fig. 10 we find that the present modelpredicts a curve for the scaled peak positions which in itsfunctional form is very similar to the random placement ex-pression ~24!. This similarity was already noted byPieruschka32 who compared an approximate calculation validin the scaling regime to the model of Ref. 14 which predictsa behavior ofj similar to ~24!. The empirical identificationj5p/qmax is no longer necessary in the present model, sinceboth j ~or jTS) andqmax are evaluated from the elastic con-stants.

Abillon et al.49 has found a good qualitative agreementbetween Eq.~24! with a56 and the characteristic lengthscales determined from peak positions for a number of mi-croemulsions with different water to oil ratios and differentionic surfactants@respectively, dodecyl trimethyl ammoniumbromide ~DTAB!, sodium dodecyl sulphate~SDS! and so-dium hexadecyl benzene sulphonate~SHBS!#.50 The mea-surements of the peak positions are performed on micro-emulsions in the three-phase region and in the 2-phase regionbelow and above the three-phase region. Instead of passingthrough these phases by varying the temperature, the authorshave varied the salinity of the samples. The experimentallydetermined characteristic sizes are digitized from Fig. 3 in

Ref. 49, transformed into scaled peak positions and plotted inFig. 10 together with Eq.~24! with a56.

In a study of microemulsions with equal volume frac-tions of water and oil and different types of the nonionicCiEj surfactants~alkyl-ethylene-oxides! Sottmanet al.27 hasdetermined the constanta to be 7.16 on average. The peakposition corresponding to this value is also plotted in Fig. 10.

The experimental point corresponding to the peak posi-tions from microemulsions formed in systems with a differ-ent nonionic surfactant falls well on the top of the GaussianRandom field prediction fork51 kT. An average value ofk51 kT for the nonionic microemulsions is in good agree-ment with the experimental findings of Sottmannet al.44

The experimental points for the ionic microemulsion fallin a quite wide band around Eq.~24!. The experimentalpoints coming from microemulsions formed with DTAB arein better agreement with the peak positions of the Gaussianrandom field model fork52 kT than with Eq.~24!. Both theseries of experimental points coming from microemulsionsformed with SDS and with SHBS are asymmetrically distrib-uted aroundfo50.5. This could be explained by a variationin the value fork due to the variation in the salinity of thesamples.51 The plot indicates that the bending rigidityk isgenerally larger in the ionic microemulsions than in the non-ionic microemulsions. This is in good agreement with thetheoretical predictions of Daicicet al.51 A bending rigidityclose tok52 kT disagrees with the experimental findings ofRef. 49. These authors have, using ellipsometry, determinedthe value ofk to be from 0.55 to 1 kT, dependent on thesurfactant type. However, as Gradzielski has pointed outrecently,52 experimentally determined values fork andk stillvary largely from one study to another.

The scaled peak positions of the structure factor of oildroplets in water formed at compositions similar to the onesused above and interacting via a hard-spheres potential cal-culated in the Percus–Yevick approximation5 are also plot-ted. It is interesting to note how well these peak positionscompare to the experimental observations by Abillonet al.49

for fo,0.5. As both the water to oil ratios and the amountof surfactant per unit volume are similar in the two types ofcalculations the models will also be similar in the Porodlimits. This has the consequence that it, in this concentrationrange, is very difficult to distinguish a droplet structure froma bicontinuous/diluted bicontinuous structure if the scatteringdata are only analyzed in terms of the peak position andPorod limit.

VI. CONCLUSION

We have demonstrated that the Gaussian random fieldformalism together with the Helfrich Hamiltonian can beused to formulate a model for microemulsion systems withnonzero spontaneous curvature and asymmetric composition.In the model it is possible to drive a microemulsion withfixed composition from a w/o structure through a bicontinu-ous structure to an o/w structure by changing the spontane-ous curvatureH0 .

By comparing the phase diagrams predicted by theGaussian random field model with experimentally deter-mined phase diagrams it is seen that for typical nonionic

FIG. 10. Scaled peak positions from microemulsions plotted as a function ofthe volume fraction of the apolar phase,fo . Full line: Gaussian randomfield model of diluted bicontinuous microemulsions fork51 kT. Dashedline: the same model fork52 kT. Dotted line: spheres interacting via a hardspheres potential calculated in the Percus–Yevick approximation. Dash–dotted line: curve fitted to experimental results in Ref. 49.* : average scaledpeak position for a number of nonionic microemulsions according to Ref.27. h: microemulsions with SHBS~Ref. 49!. L: microemulsions withDTAB ~Ref. 49!. n: microemulsions with SDS~Ref. 49!.



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microemulsions the bicontinuous or diluted bicontinuousstructure is favored in a very large part of the one-phaseregion. This suggests why a full phase inversion of a one-phase system with fixed composition has, to our knowledge,never been measured for nonionic microemulsion systems.

The direct space microemulsion structures predicted bythe model agree well with freeze fracture microscopy pic-tures of the microemulsion phase in the three-phase region.This suggests that the microemulsion in the three-phase re-gion is of the bicontinuous or diluted bicontinuous kind.Plots of the position of the modeled scattering peak as afunction of concentration are also in good agreement withcorresponding curves determined for experimental systems.

The model is based on the Helfrich Hamiltonian and theGaussian random field formalism. This means that we expectthe model to give a good structural description for structureswith low mean curvature. Film–film and droplet–droplet in-teractions which become significant in, respectively, spongestructures and droplet structures are not included in theHelfrich formalism. Furthermore, locally cylindrical struc-tures cannot be described within the Gaussian random fieldformalism. For these structures we expect other types ofmodels to give more realistic pictures.

ACKNOWLEDGMENTS

The authors wish to thank Jan Skov Pedersen andCarsten Svaneborg for reading and commenting on themanuscript.

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