Bifurcation analysis and liquid–crystal phases in Landau–Ginzburg model ofmicroemulsionA. Ciach Citation: The Journal of Chemical Physics 104, 2376 (1996); doi: 10.1063/1.470933 View online: http://dx.doi.org/10.1063/1.470933 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/104/6?ver=pdfcov Published by the AIP Publishing Advertisement:

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Bifurcation analysis and liquid–crystal phases in Landau–Ginzburgmodel of microemulsion

A. CiachInstitute of Physical Chemistry, Polish Academy of Sciences, Department III, Kasprzaka 44/52,01-224 Warsaw, Poland

~Received 24 July 1995; accepted 27 October 1995!

Landau–Ginzburg model for oil–water–surfactant mixtures with three order parameters is derivedfrom the lattice vector model@Ciach, Høye, and Stell, J. Phys. A21, L777 ~1988!#. In case ofoil–water symmetry all the coupling constants are explicitly expressed in terms of surfactant volumefractionrs , temperatureT, and a parameterg describing the strength of surfactant. The bifurcationline in the parameter space (rs ,T) and the positions of the Lifshitz and the tricritical points arefound for different values ofg. The structure of the ordered phases below the bifurcation isdetermined. For growing surfactant concentration lamellar, double-diamond, simple cubic andface-centered phases are stable; the last one is followed by the tricritical point. For strong surfactantsthe ordered phases appear forrs;10%. We claim that the present model is suitable for weak as wellas for strong surfactants. ©1996 American Institute of Physics.@S0021-9606~96!51205-4#

I. INTRODUCTION

Mixtures containing surfactant have been extensivelystudied theoretically in the recent years.1 Apart from mem-brane and related approaches,2 several lattice,3–8 andLandau–Ginzburg~LG!9–12 models have been introduced.Results, summarized in the review article1 are indeed impres-sive. The major difference between different models is due tothe number of variables~microscopic densities in the latticecase or order parameters in the continuous case!. There aremodels with one,4,9 two5,10,11,12or three3,7,8,13,14order param-eters. In the lattice case the liquid crystal phases other thanthe lamellar phase are stable only in the three-parametermodels.13 For strong surfactants in the CHS model13 the bi-furcation to ordered phases in the mean-field approximationoccurs for low surfactant concentration~;10%!. Therefore,while the mixtures with weak surfactants are well describedby the simpler models, the mixtures with strong surfactantsshould be probably described by the more complex, three-parameter models. In this paper we derive a three-parameterLG model, which we believe is quite general and describesmixtures with weak as well as with strong surfactants.

The powerful Landau–Ginzburg approach in case ofconsidered mixtures has a serious disadvantage. The cou-pling constants~in case of three order parameters there are asmany as 221! are not directly related to physically measuredquantities. Moreover, one can expect that they are not inde-pendent from each other. Therefore, it is difficult to chooseproper variables to study the phase diagrams. Quantitativecomparison between the results and experiment is also diffi-cult; several parameters have to be fitted. In the model de-rived in this paper all the coupling constants are explicitlyexpressed in terms of the temperature, surfactant volumefraction and a single parameter describing the strength ofsurfactant. Thus the results obtained in this model can bedirectly compared with experiment.

In the three order-parameter models the state of the sys-tem is described by the local concentration difference be-

tween oil and water, local concentration of surfactant and bya vector field describing local orientational ordering of sur-factant particles. As already mentioned, in the LG models ofthis kind there are many coupling constants, which are notdirectly related to physically measured quantities, or rathersuch relation is not known. On the other hand, in the latticemodels only few parameters, with direct physical interpreta-tion are present. In the CHS model with oil–water symmetryone parameter characterizes the system and it may be relatedto the length of the surfactant particles in the case of non-ionic surfactantsCiEj . The remaining parameters are thetemperatureT and the chemical potential of surfactantm, orthe concentration of surfactantrs , and a parameter fixing theenergy scale. Because the LG approach has many advan-tages, it is worthwhile to introduce a LG model with thecoupling constants explicitly related to measured quantities,particularly in the systems, where the structure is formed onthe length scale large compared to molecular distances. Onecan derive such a model performing a coarse graining pro-cedure for a ‘‘microscopic’’ model, the CHS model in ourcase. The resulting model will be of course restricted to theparticular model of interactions, thus less general. The num-ber of parameters, however, will be reduced and the phasediagrams will be presented in variables directly allowing forcomparison with experiment.

In this paper we derive a Landau–Ginzburg approxima-tion to the CHS model and perform a bifurcation analysis.We find a bifurcation line, a tricritical point, and a Lifshitztricritical point. For weak surfactants the tricritical point islocated at the line of separation into oil- and water-richphases. Thus the transition to the microemulsion and orderedphases is first order. For strong surfactants the tricriticalpoint is located at the transition to the ordered phases. Wedetermine a structure of the ordered phases stable below thebifurcation. It turns out that the sequence of the liquid-crystalphases is, for growing surfactant concentration, the follow-ing: lamellar phase, double-diamond~dds! phase, double

2376 J. Chem. Phys. 104 (6), 8 February 1996 0021-9606/96/104(6)/2376/9/$10.00 © 1996 American Institute of Physics This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

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simple cubic~dsc! phase, and the double face-centered phase~dfcc!. The regions of stability of the following phases de-crease. The second-order transition to the dfcc phase is fol-lowed by a tricritical point and for higher surfactant concen-tration the transition to ordered phases becomes first order.We should stress that only several~5! structures are consid-ered here, and that it is possible that some different phasesmay be stable in this model. Still, we show that not only thelamellar phase is present, also more complicated structuresappear. Particularly for lower temperatures one may expectstability of different phases, but in the present paper we donot consider this question. To summarize, we derive a LGmodel with coupling constants explicitly related to experi-mentally measured parameters. We show, performing the bi-furcation analysis that this model should be suitable for weakas well as for strong surfactants.

II. CONSTRUCTION OF THE MODEL

We construct the model on a basis of the vector modelintroduced by Ciach, Høye, and Stell3 and generalized byMatsen and Sullivan7 and by Dawsonet al.8 In the first stepwe generalize the lattice CHS model and instead of 2d ori-entations of surfactant particles, compatible with thed-dimensional lattice structure, we considerL orientationsuniformly distributed over the sphere withL→`. The den-sity of surfactant in pointr and orientationvl is denoted by

r l~r !5rs~r ,v l !. ~1!

The sum and the difference of oil and water densities aredenoted bys1 and s2 , respectively. The microscopic densi-ties assume the discrete valuess150, 1, r l50, 1, s250,61. In the close-packing case the total density of surfactantin r is related tos1 by

rs~r !5(l

L

r l~r !512s1~r ! ~2!

and the conditions225s1 must also be satisfied.

In the generalized CHS model the interaction between asurfactant particle in positionr and orientationvl and an oil~water! particle in positionr 8 is proportional to a scalar prod-uct between the vectorsvl and r2r 8. In real systems theinteractions~or rather effective interactions, governing theprobability of configurations of particles! are of course not assimple as that. We may assume, however, that the interac-tions are some functions of the scalar product between theorientation of the amphiphile and the distance between theparticles, and that in the case of oil–water symmetry~bal-anced system! the first approximation to the interactions isjust proportional to the scalar product. More complicatedmodels of interactions could in principle also be considered.The hamiltonian of the generalized model has a form13

H52(r

F(i

d Fb2 ~s1~r !s1~r1ei !1s2~r !s2~r1ei !!

1c(l

L

r lv li•~s2~r1ei !2s2~r2ei !!G1ms1~r !

1mo/ws2~r !G . ~3!

We choose the length unit such that the lattice constant isa51. The unit lattice vectors are denoted byei , fori51,...,d. In this paper we concentrate on the casemo/w50.Generalization to arbitrarymo/w is straightforward.

The Landau–Ginzburg model is constructed in a waysimilar to the way thef4 model for critical phenomena wasconstructed from the Ising model. Coarse-graining procedureleads to construction of an effective thermodynamical poten-tial Veff depending on the order-parameter fields. In this pa-per we simply consider the thermodynamical potential in themean-field approximation13

VMF@s1~r !,s2~r !,r l~r !#

5H@s1~r !,s2~r !,r l~r !#1kT(r

Fs1~r !1s2~r !

2

3 logS s1~r !1s2~r !

2 D 1s1~r !2s2~r !

2

3 logS s1~r !2s2~r !

2 D 1(l

L

r l~r !log r l~r !G . ~4!

Here, s1 , s2 , and r l denote continuous fields, not discretemicroscopic densities, whose equilibrium values minimizeVMF. Equilibrium values of these fields in the uniform phaseare denoted in the following way:

^s1&5 s1 , ~5a!

^r l&512 s1L

, ~5b!

^s2&5 s2 ~5c!

in the case of oil–water symmetry on which we concentratehere, s250. Fluctuations of the fields are denoted byDs1 ,Dr l , ands2 , i.e.,

s15 s11Ds1 , ~6a!

r l512s1L

1Dr l512 s1L

2Ds1L

1Dr l . ~6b!

We assume that the magnitudes of the fluctuations ofthese fields are small, and expandVMF in power series ofs2 ,Ds1 , Dr l about the equilibrium valueV0

MF and obtain

VMF5V0MF1V2

MF1V3MF1V4

MF1••• , ~7a!

with

2377A. Ciach: Landau–Ginzburg model of microemulsion

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V2MF52(

rF(

i

d Fb2 ~Ds1~r !Ds1~r1ei !1s2~r !s2~r1ei !!

1c(l

L

Dr l~r !v li•~s2~r1ei !2s2~r2ei !!G

1kT

2 F Ds12~r !

s1~12 s1!1s22~r !

s11

L

12 s1(l

L

Dr l2~r !G G ,

~7b!

V3MF5

kT

2 (r

F21

3~ s1

221~12 s1!22!Ds1

3~r !

2Ds1~r !s2

2~r !

s12 1

L

~12 s1!2 Ds1~r !(

l

L

Dr l2~r !G , ~7c!

V4MF5

kT

2 (r

F16 ~ s1231~12 s1!

23!Ds14~r !1

s24~r !

6s13

1Ds1

2~r !s22~r !

s13 1

L

~12 s1!3 Ds1

2~r !(l

L

Dr l2~r !

11

6 S L

12 s1D 3(

l

L

Dr l4~r !G . ~7d!

We introduce a vector fieldu~r ! describing the orientationaldegrees of freedom of surfactant particles in the followingway:

Dr l512 s1L

v l–u ~8a!

or

r l~r !5^r l~r !&S 12Ds1~r !

12 s11v l–u~r ! D . ~8b!

The fluctuation of surfactant density in positionr and orien-tation vl consists of a uniform part2@Ds1~r !#/[L], the samein all directions, and a part describing the actual orientationalordering inr , namely,vl–u~r !, which is a ratio between ex-cess surfactant density in directionvl , Dr l , and the averagedensity in arbitrary direction,r l&5(12 s1)/(L). We takenow into account thatL→`, and the sum( l

L may be ap-proximated by

L21(l

L

.L→`~4p!21E dv. ~9!

Using the relations, valid forL→`,

L

12 s1(l

L

Dr l25~12 s1!•L

21(l

L

~v l–u!2

.12 s14p E dv~v–u!25

12 s13

uuu2,

~10a!

S L

12 s1D 3(

l

L

Dr l45~12 s1!•L

21(l

L

~v l–u!4

.12 s14p E dv~v–u!45

12 s15

uuu4,

~10b!

and

(l

L

Dr lv li5~12 s1!•L

21(l

L

~v l–u!v li

.12 s14p E dv~v–u!v i5

12 s13

ui , ~10c!

we can expressVMF in terms ofDs1 , s2 , andu. The fields,however, still depend on the discrete distance. If we restrictour attention to the structure on a length scale large com-pared to molecular distances, as in microemulsion and lyo-tropic liquid-crystal phases, then we may assume that thefieldsDs1 , s2 , andu vary slowly on the length scale of thelattice constanta. In other words, we may assume thata→0.Then, for f being a smooth continuation to real arguments ofa lattice function, we use approximations

(i

d

f ~r1aei !.a→0f •d1a(i

] f

]r i1a2

2 (i

]2f

]r i2 1•••

~11a!

and

f ~r1aei !2 f ~r2aei !52a] f

]r i1••• . ~11b!

The sum over lattice sites can be approximated by an inte-gral. Before we write the final form of the potentialVeff , wehave to pay attention to one more question. In the latticeCHS model no surfactant–surfactant interactions, other thansteric repulsions, provided by the lattice structure are consid-ered. The lattice structure does not allow for changes of theorientation of the surfactant for distances smaller thana ~sizeof molecules!. In the continuous model the pure surfactantsystem should have similar property and the orientation ofamphiphiles should not change for arbitrarily small dis-tances. Roughly speaking for distances smaller thana wedeal with the same molecule, thus the orientation cannotchange. For distances larger thana, on the other hand, onefinds two different amphiphiles, with orientations indepen-dent from each other if the interactions between them areneglected~CHS model!. We thus require that in the puresurfactant system the correlation length for the orientationalorder is just the size of moleculesa. The simplest way tosatisfy this requirement is to add to continuum approxima-tion toVMF a term proportional to~~“–u!21~“3u!2! with anappropriate coupling constant. The Gaussian part of the re-sultingVeff then takes the form

2378 A. Ciach: Landau–Ginzburg model of microemulsion

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V2eff5E dr F12 S kT

12rs2bdD s221 1

2

ba2

2~“s2!

2

11

2 S kT

rs~12rs!2bdDDs1

211

2

ba2

2~“~Ds1!!2

11

2

kTrs3

@ uuu21a2@~“–u!21~“3u!2##

22rsac

3u–“s2G . ~12!

Like in the lattice case we choose the length unit suchthata51. We rescale the fields to obtain dimensionless cou-pling constants. For the rescaled fields we use the standardnotation, namely,

f5s2

A2, ~13a!

r52Ds1

A2, ~13b!

u85uAkTrs3b

, ~13c!

wherer denotes the fluctuation of the local surfactant con-centration@see Eq.~2!#. The resulting effective functionaltakes the form

Veff5~V21V int!•b ~14a!

with

V25E dr F12 a2f21

1

2~“f!21

1

2a2r

211

2~“r!2

11

2@ uuu21~“–u!21~“3u!2#2Ju–“fG ~14b!

and

V int5E dr F 13! ~a3r31b3f

2r1c3uuu2r!11

4!~a4f

4

1a4r41b4f

2r21c4r2uuu21A4uuu4!G . ~14c!

We omit the prime and denoteu8 by u. The meaning of thedifferent terms in Eq.~14! is explained in Ref. 1. The explicitexpressions for the coupling constants are

a252S t

12rs2dD , ~15a!

a252S t

rs~12rs!2dD , ~15b!

J5S 2rsg

3t D 1/2, ~15c!

a352A2tS 1rs2 11

~12rs!2D , ~15d!

b356A2t

~12rs!2 , ~15e!

c3523A2rs

, ~15f!

a458t

~12rs!3 , ~15g!

A4518

5trs, ~15h!

a452tS 1rs3 11

~12rs!3D , ~15i!

b456a4 , ~15j!

c4516t

rs, ~15k!

where

t5kT

band g5S 2cb D 2. ~15l!

g is defined in the same way as in the CHS model and is themeasure of the amphiphilicity of surfactant. We use the pa-rameterb as an energy scale. Only in this way our resultsdepend onb.

Equations~14! and ~15! define the Landau–Ginzburgmodel in terms of thermodynamical parameters and two pa-rameters,g andb characterizing the system. The model dif-fers from the known models in several points. First of all,only gradient terms and no higher order derivatives are in-cluded. Only fourth powers of the order-parameter fields arepresent. The tricritical point can nevertheless be describedwithin the model. Important difference from the other mod-els is that the coupling constants are not independent fromeach other, and their relation to experimentally measuredquantities is explicitly given.

Correlation functions:As in Ref. 12 we split the fielduinto two fields

u5s1t, ~16!

where

“3s50, and“–t50. ~17!

In the Fourier space, fors~k!5* dr s~r !exp~2 ikr ! we havek–t~k!50 ands~k!5i ksi~k!. ThusV2 assumes the form

V25E dkF12 ~a21k2!uf~k!u211

2~11k2!usi~k!u2

2Jksi~k!f~k!11

2~a21k2!ur~k!u21

1

2~ u t~k!u2

1uk3t~k!u2!G . ~18!

Note that the fieldst and r are not coupled to any otherfields. Thus the Gaussian correlation functions between these

2379A. Ciach: Landau–Ginzburg model of microemulsion

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and different fields vanish. The explicit expressions for thecorrelation functions can easily be calculated, with the helpof the square form for the fieldsf andsi . The determinant ofthis form is

D5k41Bk21a2 , ~19a!

where

B511a22J2. ~19b!

The functions interesting for us are

Gf,f~k!511k2

D, ~20a!

Gs,s~k!5a21k2

D, ~20b!

Gf,s~k!52Jk

D, ~20c!

and

Gr,r~k!51

a21k2. ~20d!

Remember that the length unit is chosen so thata51, thusthe cutoff, corresponding to molecular distances iskmax52p/a52p. The largek limit in experiments corre-sponds in the model tok<1, i.e., to small, but not yet mo-lecular distances. Our expressions thus agree with thek24

behavior for the water–water structure factor.

III. BIFURCATION ANALYSIS

Sincea2.a2 for all temperatures, the bifurcation is re-lated to the fluctuations of the fieldsf andsi . The uniformphase becomes unstable for

a250, ~21a!

where the corresponding wave vector of critical fluctuationsis k50. The instability occurs also forD50, and]t/]k505]D/]k. The explicit expressions for the instability and thewave vector in this case are

2Aa21B5052F2S t

12rs2dD G1/21112S t

12rs2dD

22grs3t

~21b!

and

kb25Aa2 . ~21c!

The actual transition corresponds to the higher temperaturefor which the instability occurs. One can easily check thatthe Lifshitz point, separating the transition to two uniformphases and to liquid-crystal phase is

rsL5

3d

2g13dand tL5d~12rs

L!. ~22!

The Lifshitz1 and the disorder1 lines are given byB50 and2Aa2 2 B 5 0, respectively. The bifurcation, the Lifshitz andthe disorder lines are shown in Figs. 1 and 2 forg516 andg550, respectively.

Within the bifurcation analysis we find the tricriticalpoint and determine the structure of the ordered phases stablebelow the bifurcation. It is convenient to rewriteV2 in aform15

V251

2 E dk@A~k!uf~k!u21~11k2!uc~k!u2

1~a21k2!ur~k!u21u t~k!u21uk3t~k!u2#, ~23!

where

FIG. 1. ~a! Bifurcation ~solid!, Lifshitz ~dashed!, and disorder~dotted! linesfor g516. Region of stability of coexisting oil- and water-rich phases isdenoted by o/w. Region of stability of liquid–crystal phases is denoted by lc.The Lifshitz point separating the two regions is denoted by a cross. Betweenthe Lifshitz point and the next cross the lamellar phase is stable. Stabilityregions of the different liquid–crystal phases are separated by the crosses.The last cross is the tricritical point.~b! A portion of the bifurcation line forg516 with indicated regions of stability of the different liquid–crystalphases. The lamellar, double-diamond, double simple-cubic, and doubleface-centered phases are denoted byl , dds, dsc, and dfcc, respectively. Thetricritical point separating the second order~to the left! from the first order~to the right! transitions is denoted by tcp.

2380 A. Ciach: Landau–Ginzburg model of microemulsion

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c~k!5si~k!2Jk

11k2f~k! ~24!

and

A~k!5~11k2!21~k41Bk21a2!. ~25!

Close to the bifurcation, i.e., ford 5 1/4(B 1 2Aa2)→0 andd,0 we have

A~k!.4kb

2

11kb2 d5Abd. ~26!

We introduce the bifurcation parametere 5 Audu. One caneasily verify using Eqs.~23! and ~14c! that close to the bi-furcation f5O(e), c5O(e3), r5O(e2), t5O(e3), andsi5O(e). We thus can approximateVeff by

1

bVeff5( 8

k

1

2@A~k!uf~k!u21~a21k2!ur~k!u2#

11

3! ( 8k1,k2,k3

dKrS (i

3

kiD Fb3f~k1!f~k2!r~k3!

2c3J

2k1–k2~11k1

2!~11k22!

f~k1!f~k2!r~k3!G1

1

4! ( 8k1,k2,k3,k4

dKrS (i

4

kiD Fa4)i

4

f~k i !

1J4A4~k1–k2!~k3–k4!)i

4f~k i !

11ki2G1O~f6!.

~27!

(k8 means summation over vectorsk such thatuku5kb .

A. The tricritical point

Minimalization of Veff @Eq. ~27!# with respect tor~k!gives

r~k!521

3!~a21k2! (k1,k2

8dKr~k11k22k!f~k1!f~k2!

3Fb32c3k1–k2J2

~11k12!~11k2

2!GdKr~ uku2kb!.

~28!

In the above, the relations

J

11k251 and si~k!52kf~k! ~29!

valid at the bifurcation line, have been used. Using the abovewe can write

Veffb215(k

8 1

2Abduf~k!u21V41••• ~30!

and

V451

4! (k1,k2,k3,k4

8dKrS (

i

4

kiD Fa41A4a2~ k1–k2!~ k3–k4!

21

3g~k1,k2!b~k1,k2!b~k3,k4!)

i

4

f~ki!G ~31!

the functionsg andb are defined by

g~ki,kj !5~a21uki1kju2!21, ~32a!

b~ki,kj !5~b32c3ki–kj !. ~32b!

FIG. 2. ~a! Bifurcation ~solid!, Lifshitz ~dashed!, and disorder~dotted! linesfor g550. Region of stability of coexisting oil- and water-rich phases isdenoted by o/w. Region of stability of liquid–crystal phases is denoted by lc.The Lifshitz point separating the two regions is denoted by a cross. Betweenthe Lifshitz point and the next cross the lamellar phase is stable. Stabilityregions of the different liquid–crystal phases are separated by the crosses.The last cross is the tricritical point.~b! A portion of the bifurcation line forg550 with indicated regions of stability of the different liquid–crystalphases. The lamellar, double-diamond, double simple-cubic, and doubleface-centered phases are denoted byl , dds, dsc, and dfcc, respectively. Thetricritical point separating the second order~to the left! from the first order~to the right! transitions is denoted by tcp.

2381A. Ciach: Landau–Ginzburg model of microemulsion

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At the tricritical point bothV2 and V4 vanish. If thebifurcation leads to oil–water separation, then the tricriticalpoint is given bya250 anda45b3/3a2 , or explicitly

rst50.4 and t t51.8 ~33!

just like in the CHS13 and BEG16 models. Forrs.rst the

transition between the disordered phase and the coexistingoil- and water-rich phases is first order. The Lifshitz and thetricritical points coincide,rs

L5rst , giving the Lifshitz tricriti-

cal point for g527/4. This result means that for weak sur-factants, withg,27/4, oil- and water-rich phases coexistwith the uniform phase, i.e., with the microemulsion,whereas for strong surfactants, withg.27/4, three phase co-existence between oil-rich, water-rich and liquid-crystalphases occurs. This result agrees with experiments.1 Forg.27/4 the bifurcation occurs fork.0 if rs.rs

L. The fieldsf~k! corresponding to bifurcation can be written in a form

f~k!5FFw(i

N

dKr~k2kbpi !1w*(i

N

dKr~k1kbpi !G , ~34!

whereuwu51 andpi are unit vectors in different directions.The simplest case corresponds to the lamellar structure forwhich N51 and p describes the direction of modulations.The value ofV45v4F

4 depends on the form off~k!. Thetricritical point corresponds to the lowestrs for which v4becomes negative. Here, we find the approximate position ofthe tricritical point, considering a selection of structures.

In mixtures containing surfactants or lipids rich varietyof liquid-crystal phases has been observed.17 Among themseveral cubic phases, which may coexist with each otherhave been described.17 The most commonly observed cubicphases are characterized by internal surfaces correspondingto D ~diamond!, P ~primitive!, or G ~gyroid! minimalsurfaces.18 In our notationD corresponds to dds andP cor-responds to dsc~double simple-cubic! phases, respectively.One can, however, expect that in addition to already de-scribed structures, different phases can also be formed insuch systems. The experimental verification of complicatedstructures is, however, very difficult. The simplest LG modelwith a single order-parameter leads in addition to the stablelamellar phase to a large number of metastable phases withsurfactant surfaces corresponding to the known minimal sur-faces and also to many new surfaces of high genus.19 Suchcomplicated structures should appear also in the model intro-duced here and some of them may become stable. The de-tailed analysis of the phase diagram and determination of thefirst-order transition lines between different phases is an in-teresting task, which goes, however, beyond the scope of thispaper. Here, we would like to verify, whether the bifurcationanalysis may show the existence of liquid-crystal phasesother than the lamellar phase. Existence of different liquid-crystal phases is characteristic for mixtures with strong sur-factants. In the known LG models, however, the only stableordered phase is the lamellar phase.

In order to find the approximate position of the tricriticalpoint we arbitrarily choose several structures, to which the

bifurcation may lead. The tricritical point can be located atslightly smallerrs . We consider lamellar, dds, hexagonal,dsc, and double face-centered dfcc structures. The last struc-ture is surfactant continuous, whereas oil and water formclosed droplets. The droplets which are nearest-neighbors areof different kind and are separated by a surfactant layer. Forlarge surfactant concentration such a structure may be ex-pected, especially for noninteracting amphiphiles, thereforewe take it into account. For the structures considered heref~k! has a form

~i! Lamellar symmetry

f~k!5F l@dKr~k2kbp!1dKr~k1kbp!#, ~35a!

wherep defines the direction normal to the layers.~ii ! Hexagonal symmetry~water cylinders in oil or vice

versa!

f~k!56Fh(i

3

@dKr~k2kbpi !1dKr~k1kbpi !#, ~35b!

wherepi ( i51,2,3) form equilateral triangle.~iii ! Double-diamond symmetry~intertwined oil and wa-

ter tunnels!

f~k!5FddsF ~11 i !(i

4

dKr~k2kbpi !1~12 i !

3(i

4

dKr~k1kbpi !G , ~35c!

where in this casepi ( i51,2,3,4) form tetrahedron.~iv! Double simple-cubic symmetry~two simple-cubic

lattices shifted with respect to each other by (P/2,P/2,P/2),whereP is a period!

f~k!5Fdsc(i

3

@dKr~k2kbpi !1dKr~k1kbpi !#, ~35d!

wherepi ( i51,2,3) are orthogonal to one another.~v! Double face-centered structure@oil and water drop-

lets, arranged in two fcc-lattices shifted with respect to eachother like in the case~iv!#

f~k!5Fdfcc(i

4

@dKr~k2kbpi !1dKr~k1kbpi !#, ~35e!

wherepi are the same as in the case~iii !.Inserting the above into Eq.~31! gives, after some com-

binatorics, the explicit expressions forv4 for all the phases.The expressions are given in the Appendix. It turns out thatthe dfcc structure leads to negativev4 for the lowest value ofrs , independently ofg. The tricritical points forg516 andg550 are shown in Figs. 1 and 2, respectively. Forrs.rs

t

the transition between the disordered phase and the liquid-crystal phases becomes first order. In our model the micro-emulsion coexists with the dfcc ordered phase. This coexist-ence takes place for high surfactant concentration~forrs;50% and more!. One should remember, however that insuch systems fluctuation-induced first-order phase transitioncan take place and the transition between microemulsion and

2382 A. Ciach: Landau–Ginzburg model of microemulsion

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other ordered phases may turn out to be first order, if thefluctuations are taken into account. We also stress that in ourmodel the interactions between amphiphiles are neglected,thus the surfactant does not form ordered phases.

B. Structure of the liquid–crystal phases

We minimizeVeff just below the bifurcation with respectto r andf. The form ofr is given in Eq.~28!. The condition@dVeff#/@df~k!#50 and relations~28! and ~29! give

~Abd!f~k!51

3! (k1,k2,k3

8dKrS (

i

3

ki2kD)i

3

f~k i !

3dKr~ uku2kb!F13 g~k1 ,k2!b~k1 ,k2!b~k3 ,

2k!1A4a2~ k1–k2!~ k3–k!2a4G . ~36!

For the structures~35! we obtain

Fph2 52~Abd!vph

21bph. ~37a!

The above Eqs.~30! and ~31! give in turn

Vpheff52~Abd!2vph

21bph2 . ~37b!

The subscript ph denotes different phases, ph5l , h, dds, dsc,dfcc. The numerical constantsbph are given in the Appendix.The bifurcation lines with the stable liquid-crystal phases areshown in Figs. 1 and 2 forg516 and 50, respectively. Re-gardless the value ofg; the sequence of the phases is, forgrowing surfactant concentration, the following: lamellar,dds, dsc, and dfcc. The stability regions of the followingphase decrease. The stronger the surfactant, the larger theregion of stability of the structured phases. The orderedphases appear for reasonable surfactant volume fraction.Compare the phase diagrams forC10E5 andC12E6 .

1

IV. SUMMARY

We derived the Landau–Ginzburg model for ternary sur-factant mixtures starting from a particular model of effectiveinterparticle interactions,3 in which the interactions are muchsimplified. In particular, oil-water symmetry is assumed.Therefore the model is not able to accurately describe all theproperties of the real systems, in particular those related tothe different type of interactions between surfactant and oiland surfactant and water. It should be possible to start from amore realistic model of surfactant–water interactions, insteadof the CHS model, and derive a Landau–Ginzburg model ina way described in this paper. The present model should besuitable for a balanced system~vanishing spontaneous cur-vature! and for comparable oil- and water-volume fractions.The advantage of the present model is its relative simplicityand, at the same time, possibility of quantitative comparisonbetween different results and experiment. Only two param-eters can be fitted; one is the amphiphilicity, the other one isthe temperature scale. In this paper we studied the bifurca-tion in some detail. We obtained transitions to various liquid-

crystal phases for reasonable values of surfactant volumefraction ~about 10%–80%!. We note that in the lattice CHSmodel the bifurcation leads only to the dds structure,whereas in the present continuous model several phases arestable below the bifurcation. Different ordered phases appearin mixtures with strong surfactants, and our model is the firstLG model which predicts stability of ordered phases morecomplicated than the lamellar phase. The stability region ofthe lamellar phase is the largest, in agreement with experi-ments. We did not take into account the fluctuations, there-fore we cannot compare quantitatively the portion of thephase diagram we obtained with experiment. In the future weare going to study the full phase diagram and the structure ofmicroemulsion within this model. The water–water andsurfactant–surfactant structure factors can be compared withexperimental results for different (rs ,T) for systems speci-fied by g. One should, however, take into account fluctua-tions to test the quantitative agreement between the resultspredicted by the model and the results obtained in experi-ments. Our preliminary results show that within the modelintroduced here one can describe mixtures with weak as wellas with strong surfactants, varying a single parameterg.More detailed and accurate analysis of the model is neces-sary, however, to verify the quantitative agreement betweendifferent predictions and experiment, and to find the sourceof discrepancies.

ACKNOWLEDGMENTS

I would like to thank Professor Jan Stecki, ProfessorRobert Hołyst, Dr. Anna Maciołek, Dr. Wojciech Go´zdz, andespecially Dr. Andrzej Poniewierski for stimulating discus-sions. This work was partially supported by a KBN Grant~No. 2P303302007!.

APPENDIX

The calculations leading to the explicit expressions forv4 andVeff are tedious, but rather standard. Therefore wepresent here only the results.

~i! The lamellar phase

v l51

36 F9~a41A4a2!2~b32c3!

2

a214Aa222

~b31c3!2

a2G

~A1!

and

bl51

2. ~A2!

~ii ! The hexagonal phase

vh51

12 F45a4127A4a22~b32c3!

2

a214Aa226

~b31c3!2

a2

24~b31

12c3!

2

a21Aa224

~b3212c3!

2

a213Aa2G ~A3!

and

2383A. Ciach: Landau–Ginzburg model of microemulsion

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213.135.41.222 On: Mon, 21 Oct 2013 11:41:10

bh53

2. ~A4!

~iii ! Double-diamond phase

vdds54

9 F45a4129A4a22~b32c3!

2

a214Aa228

~b31c3!2

a2

26~b32

13c3!

2

a2183Aa2

G ~A5!

and

bdds54. ~A6!

~iv! Double simple-cubic phase

vdsc51

12 F45a4121A4a22~b32c3!

2

a214Aa226

~b31c3!2

a2

28b32

a212Aa2G ~A7!

and

bdsc53

2. ~A8!

~v! Double face-centered phase

vdfcc54

9 F81a4133A4a22~b32c3!

2

a214Aa228

~b31c3!2

a2

26~b32

13c3!

2

a2183Aa2

212~b31

13c3!

2

a2143Aa2

G ~A9!

and

bdfcc54. ~A10!

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2384 A. Ciach: Landau–Ginzburg model of microemulsion

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