Hybrid models for the dynamics of an immiscible binary mixture with surfactant molecules

Physica A 167 (1990) 690-735 North-Holland

HYBRID M O D E L S FOR THE D Y N A M I C S OF AN IMMISCIBLE BINARY M I X T U R E WITH S U R F A C T A N T M O L E C U L E S Toshihiro K A W A K A T S U and Kyozi K A W A S A K I Department of Physics, Faculty of Science, Kyushu University 33, Fukuoka 812, Japan Received 3 April 1990 For the purpose of studying pattern formation processes in a binary mixture with surfactant molecules such as the formation of a microemulsion, we propose dynamical models which comprise a continuous field representing an immiscible binary fluid mixture and discrete surfactant molecules. In order to demonstrate the efficiency of such a hybrid approach, computer experiments on pattern formation processes are performed combining the cell dynamics method and the molecular dynamics method.

1. Introduction A surfactant is a molecule which has two chemically distinct bases and, therefore, has an amphiphilic nature [1]. In an immiscible binary fluid mixture, the surfactant molecules align on the interfaces and lower the surface tension. A detergent in an oil-water mixture is a well-known example of such a surfactant. Under certain conditions, the surface tension of the interface becomes almost zero due to the existence of the surfactant molecules; to put it more appropriately, the surface tension does not play a role due to fixed interracial area, and then the interfaces form a random isotropic structure on the mesoscopic scale. In an o i l - w a t e r - s u r f a c t a n t mixture, such a structure has the characteristic length of the order of 100/~ and is called a "microemulsion" [1, 2]. Depending on the volume fraction of water and oil, the microemulsion assumes micellar or bicontinuous structures [2]. Great many papers have been devoted mainly to the study of static aspects of microemulsions. Small angle neutron scattering (SANS) and small angle X-ray scattering (SAXS) techniques are quite suitable for investigating the structure of microemulsion systems [3-5]. The scattering data from the SANS experiments show that the existence of the local order within the range of the order of 100 A. Freeze fracture electron micrographs of microemulsions are also available which clearly demonstrate the micellar and the bicontinuous 0378-4371/90/$03.50 © 1990- Elsevier Science Publishers B.V. (North-Holland)

T. Kawakatsu and K. Kawasaki / Binary mixture with surfactant molecules

691

structures [6, 7]. In order to reproduce these static features of microemulsions, a number of theoretical models have been proposed [1, 2, 8-17]. Aside from such extensive knowledge on the static features of microemulsions, much less is known about the dynamical behavior on the microscopic level which includes, for example, the formation of micelles, fusion and fission of the interfaces, and so on. Intuitively one can expect that since surfactant molecules favor the concentration gradient of the binary fluid, there will be initial acceleration of the phase separation process. However, once stable interfaces are formed, further coarsening practically stops because the total interracial area cannot be further reduced. Recently we have proposed two theoretical approaches, one is a continuum model which is given in our separate paper [18] and the other is a hybrid model of a continuous field and discrete molecules which we have briefly described in our previous note [19]. Each of these two approaches can be a good starting point for investigating dynamics of microemulsions from somewhat different points of view. In this paper, we give the detailed account of the hybrid model and the computational scheme. Some results of the computer experiments are presented to demonstrate the efficiency of our model and our computational scheme where we shall also see the phase separation behavior described above. In the final part of this paper, we give a general formulation of the hybrid model and try to relate the hybrid model presented in this article to the continuum model proposed in our separate paper [18].

2. Model system The dynamics of the phase separation in binary fluid mixtures is usually modeled by the so-called time-dependent Ginzburg-Landau (TDGL) equation [20], in which the binary mixture is expressed by a continuous scalar field defined as the difference between the densities of the two components. Though the model equation of this kind is valid only near the critical point in the strict sense, we often assume that the TDGL-type equations describe well some relevant features of systems quenched deeply into the coexistence region. It is natural to expect that such a TDGL equation will also still be valid to describe binary mixtures in the presence of surfactant molecules and we employ it in our present model. If we denote the difference between the densities of the two components (say A and B) of the binary mixture by a field X ( r ) = - p A ( r ) pB(r), where pA(r) and ps(r) are the densities of the A and B components at position r, respectively, the time evolution of this field X ( r ) in the absence of surfactant molecules can be expressed by the following equation of motion [211:

692

T. Kawakatsu and K. Kawasaki / Binary mixture with surfactant molecules

0_._ X ( r ) = - f Ot

(rl£(X(r)}lr') ~ 8 H

dr'

(2.1)

-LXVZc3(r - r') + VX(r). r(r - r').V'X(r'),

(2.2)

where

(rlf~(X(r)}lr') =

H is the total free energy functional of the system whose explicit form will be given later, and L x is the Onsager kinetic coefficient for mutual diffusion. The tensor 7" is the so-called Oseen tensor describing the hydrodynamic interaction under the Stokes approximation and is defined as ]'(r) ~- ~

1

O~Tflr

(I + H),

(2.3)

where 77 is the shear viscosity, 1 is the unit tensor, r is the magnitude of r and is the unit vector along r. Now we consider the treatment of surfactant molecules. In modeling surfactant molecules, we have to take into account the fact that a surfactant molecule possesses two (in some cases three or more) distinct chemical bases. It is convenient to model such a surfactant molecule as a dumbbell or a rigid rod of length l which has two interaction centers at each of the two ends, one is A-philic and the other is B-philic. Here we assume that the A-philic and the B-philic interaction centers of the surfactant have the same chemical species as A and B components of the binary mixture, respectively, and we assume that the interaction strength of the A-philic (B-philic) center is qA ( q a ) times that of the unit mass of the A (B) component of the binary fluid. Our model surfactant molecule therefore somewhat resembles an electric dipole, where qA and qB correspond to charges. Suppose that there are N s surfactant molecules in the system. We denote the center of gravity of the ith surfactant molecule as r i and the unit vector along the direction (usually called the director) from the B-philic center to the A-philic center of the ith surfactant molecule (ith dumbbell) as si, respectively. In order to express the interaction between surfactant molecules and that between surfactant molecules and the field X ( r ) , we introduce three spherically symmetric interaction potential functions VAA(r), VBB(r) and VAB(r) acting between A - A , B - B and A - B type interaction centers, respectively. With the use of these interaction potentials, we can express the total free energy functional of the system H as H = Hxx + Hxs + Hss.

(2.4)

T. Kawakatsu and K. Kawasaki / Binary mixture with surfactant molecules

693

Here H x x is the free energy of the AB binary mixture in the absence of surfactant molecules and is assumed to have the Ginzburg-Landau form:

H x x = f dr [½DA(Vp,) 2 + ½DB(Vp~)2 + f o ( P , , PB)],

(2.5)

where D A and D B are the positive constants and f0 is the site potential which has two distinct minima, one corresponding to the A-phase and the other to the B-phase, respectively. Hxs in eq. (2.4) describes the interaction energy between the AB-binary fluid and the surfactant molecules. As the position of the A-philic interaction center of the ith surfactant molecule is ri + ½1g~, the interaction energy between this A-philic center and the A-component of the binary mixture is expressed as

f dr qAVAA(r -- (r i + ½lgi) ) pA(r).

(2.6)

The interaction energy between the A-philic center and the B-component, that between the B-philic center and the A-component and that between the B-philic center and the B-component are expressed in similar manner. The second term of eq. (2.4), Hxs, is composed of these four contributions. The last contribution in (2.4), Hss, which describes the interaction between the surfactant molecules, can then be expressed as Hs s = ~ [qAVAA((r, 2 + llgi) -- (rj + ½1gj)) i
+ qaqBVAa((ri + ½1gl) -- (rj -- ½lg~)) + qaqBVaB((ri -- ½lgi) -- (rj + ½lgj)) 2 + qBVBa((r, -- ½lg,) -- (rj -- ½/gj))].

(2.7)

Now we employ the multipole expansion in order to simplify Hxs and Hss. Expanding VAA in eq. (2.6) in a power series in l and retaining terms up to second order in l, we obtain

VAA(r- (r i -F ½1~i)) = VAA(r- ri) -- ½ I V V A A ( r - ri)" si "~- ~I2VVVAA( r -- ri): (sigi) + ~7(13),

(2.8) where •(/3) represents higher-order terms. Applying this procedure on each term in Hxs , we obtain the following expression for Hxs:

T. K a w a k a t s u a n d K . K a w a s a k i / B i n a r y m i x t u r e with surfactant m o l e c u l e s

694

ms f dr pa(r) {[qaVAa(r) + q~VAB(r)]

Hxs=~ i=l

--

l l~ i

"V[ qa Vaa (r ) - qBVAB(r)]

1 2 ^ ^ -}- ~ l ( S i S i ) : V V [ q A V A A ( r )

+ qBVAB(r)]}r=r_ri

1% g + ~ J dr pB(r) {[qAVAB(r) + qBVBB(r)] i=1

½lgi • V[ qAVAB (r) -- quVua (r)]

-

+ ~112,'g^, ~ isg) : VV[qAVAB(r ) + qBVBB(r)]} . . . .

i'

(2.9)

where the subscript r = r - r~ means that r is replaced by r - r~ after the differentiation with respect to r is performed. Similarly the expression for Hss is obtained from (2.7) as

{[qZVAa(r) + qzVBB(r ) + 2qAqaVAB(r)]

Hss = i
+ 1ls6 "V[q2VAA(r)- q2Vuu(r)] + 112(s~Ts~ ): VV[ q2AVAA (r) + qZvBB(r) ] 1 ~2,. + + + ~t t,si/si/ ) :VV[2qAqBVAB(r)]}r=ri_,i ,

(2.1o)

+

where sij -= gi --+gi and the subscript r = r i rj means that the expression in the curly bracket is evaluated at r = r i - r i. Eqs. (2.4)-(2.10) give the complete description of the total free energy of the system. For the sake of simplicity, we here consider only the symmetric case where the system is invariant under the simultaneous exchanges between A and B components and between the two interaction centers of a surfactant molecule. The extension to the asymmetric case is straightforward although we will not go into this problem. Employing this symmetric assumption, we only need the two interaction potential functions t h ( r ) - VAA(r) = VBB(r) and q,(r)~ VAB(r) and we can also choose q -= qA = %" AS is shown in fig. 1, the potentials &(r) and ~b(r) express the interactions between the chemically similar components ( A - A and B - B ) and that between the chemically dissimilar components ( A - B ) , respectively. Then, from (2.4)-(2.10) we obtain the following expressions for the total free energy functional H: -

H ~ Hxx + Hxs + H s s ,

(2.11)

_.L._IIII.

T. Kawakatsu and K. Kawasaki / Binary mixture with surfactant molecules

695

Fig. 1. Schematic representations of 4~(r) and ~(r), the interactions between two surfactant molecules, are presented for the symmetric case. A and B denote the A-philic and the B-phi|ic interaction centers, respectively.

Hxx = f dr [1Dx(VX)2 + f 0 ( X ) ] ,

(2.12)

fo(g)

(2.13)

= - 1 c X 2 -]- 1 L I X 4 '

Hxs=PtsNs + ½ql~,i f d r V - ( r - r i ) g i ' V X ( r ) ,

(2.14)

Hs s = q2 ~ ' [2V+ ( r i j ) + ~1 l 2(sii- sij- ) : VVq~(r/j)+l~ 12(sij+ $i]* ) : VV~b(rij)] /
(2.15)

In deriving (2.14), we have assumed the incompressibility of the fluid where the excluded volume of the surfactant molecule has been neglected, that is, we have assumed that P A ( r ) + pe(r) = constant. The p a r a m e t e r s c and u are positive constants and the following notations have been used:

Dx=-

I(D A +

DB),

Ixs ~ qp ( dr V÷(r)

with p

d

=

PA(r) + pB(r) -- c o n s t a n t ,

V_+(r)--- ~b(r) _ ~b(r) , SiI ~'~ S i q-

Sj and

% =--Ir,

- rj[.

In (2.11)-(2.15) Hxx is the bare free energy of the binary mixture, X(r), in the absence of surfactant molecules, which is taken to have the usual G i n z b u r g L a n d a u form. The second contribution, Hxs, accounts for the interaction between the field and surfactant molecules. T h e first t e r m in eq. (2.14) corresponds to the chemical potential of a surfactant molecule and the second

T. Kawakatsu and K. Kawasaki / Binary mixture with surfactant molecules

696

term expresses the facts that the surfactant molecules prefer to be at the interface where [VX[ is large and that the director of the surfactant molecule tends to be perpendicular to the interface. The last contribution, Hss, expresses the interaction between surfactant molecules. The first term in (2.15) corresponds to the orientation-independent interactions between centers of mass of pairs of nearby surfactant molecules. The second and third terms account for the director-director interactions which prefer the directors to be parallel (see appendix A). One unrealistic assumption entering here is that surfactant molecules and binary fluid can overlap freely. This is somewhat inevitable in our hybrid approach of treating binary fluid and surfactant differently. We could avoid this if we permit p --=PA + P~ to vary so that surfactant molecules and binary fluid avoid each other. We shall not go into this complication in the present paper, expecting that our simplification will not seriously affect the large scale pattern evolution that we are interested in. In order to introduce dynamics of the surfactant molecules, we assume the following phenomenological equations of motion for r i and gi [19]: d OH dt ri = v*(ri) - L p --Or~ '

d--t g ~ = V x v * ( r ) -

L-~s, -

(2.16) ~

" ~ si ,

(2.17)

where L ° and L s are Onsager kinetic coefficients and the local collective velocity field v*(r) is defined as v*(r) =

-f

dr'

r(r -

8H r ' ) . (V'X(r')} a X ( r ' ) '

(2.18)

T(r) being the usual Oseen tensor (2.3). The extra term in (2.17), [(OH/ Ogi).gi]g ~, arises from the constraint Igi] = 1. Eqs. (2.1)-(2.3) and (2.11)(2.18) constitute a complete set of dynamical equations which describes the time evolution of the system. Here we note that, in (2.1), (2.2), (2.16) and (2.17), the cross-coupling effects between the X-field and suffactant molecules, which are expressed by the off-diagonal elements of the Onsager kinetic coefficient matrix, are not taken into account explicitly. Here such a coupling enters into the dynamics only through the free energy H implicitly. A natural simplifying approximation here is to assume that g~ relaxes very fast to its local equilibrium value given by Og i

" Si gi : 0 .

(2.19)

T. Kawakatsu and K. Kawasaki / Binary mixture with surfactant molecules

697

This corresponds to the case of very little friction associated with the rotation of gi, that is, L s tends to infinity, and is a valid approximation in the absence of symmetry-breaking liquid crystalline order. On the other hand, from the computational point of view, solving (2.19) for gi is far from simple, and it is more advantageous to employ (2.17) with a large enough value of L s. In this section, we have presented a hybrid description of the system based on a simplification that the surfactant molecule is a dumbell-like uniaxial molecule. A more general treatment without such a simplification will be given in section 5.

3. Continuum description The total free energy functional of the system given in the previous section is described on a semi-microscopic level in which discrete surfactant molecules and the continuous field of binary fluid coexist. It is useful to see the correspondence between such a semi-microscopic hybrid model and the macroscopic continuum model in which the distribution of surfactant molecules is also described by continuum variables [18].

3.1. Continuum description of surfactant molecules In order to proceed from the discrete description of the surfactant molecules to the continuous description, we introduce the macroscopic variables ps(r), s(r) and O(r) defined as follows: ps(r) --- ~. 8(r - r / i , J

(3.1)

s(r) =- ~'~ ~j~(r - r/)

(3.2)

J

and

O(r) ---- ~ (~jg/J

~ I )8(r - rj) = ~, g/gyS(r - ri) -- ~ Ps(r) I ,

(3.3)

J

where summations are taken over all the surfactant molecules in the system. ps(r) is the number density of the surfactant molecules, s(r) is the local averaged director field and Q(r) is a symmetric and traceless local tensor field which is well known for nematic liquid crystals [22]. Here we note that the vector field s(r) no longer retains its unimodular property which the director of the surfactant molecule gj had. As s(r) is the local average of directors of

T. Kawakatsu and K. Kawasaki / Binary mixture with surfactant molecules

698

surfactant molecules, its magnitude Is(r)l can vanish where the directors are distributed randomly or where there are no surfactant molecules. Using the definitions (3.1)-(3.3), we obtain from eqs. (2.14) and (2.15) the continuum expression for the total free energy as follows:

Hxs=~sNs + ½qlf dr f d,'V_(r-r')VX(r).sO"),

(3.4)

Hs s = q2 f d r f dr' [ V + ( r - r ' ) + ~12VzV+(r - r')] Os(r) Ps(r')

+ kq2,2f dr -

f

{twv+(,.-,-')1: [vvv (,-- ,-')]: s(,.) ,(,-')}.

(3.5)

In (3.5) we have omitted some unphysical terms which arise in the course of transition from discrete to continuum descriptions #1. We further simplify eqs. (3.4) and (3.5) by considering the limit where the range of the interaction 17_+(r) is much shorter than the characteristic length of the spatial variation of ps(r), s(r) and O(r). In this limit, we can effectively replace V+_(r) in (3.4) and (3.5) by c+_6(r), where c_+ are constants. Taking such a limit, we obtain from (3.4) and (3.5) the following results#2:

Hxs =/~sNs + ½qlc_ f dr VX(r) • s(r)

(3.6)

and Hss

= f d r ( q 2c+ {ps(r ) 2 - ~12[Vps(r)] 2} + i q212c+ ps(r ) VV: O(r) - I q 212c- [s(r). V][V- s(r)]).

(3.7)

H e r e we have to mention that the contribution from the entropic effects which originate from various configurations of surfactant molecules is neglected in (3.6) and (3.7). We can, however, expect that such a contribution is rather small because most of the surfactant molecules align on the interfaces and will take a regular configuration. In order to incorporate this entropic contribution into (3.6) and (3.7) phenomenologically, we only have to add a term ,l One encounters some extra terms, such as q2 f dr V÷(0) ps(r), which originate from the (i, i) pair in the double summation Ej Ek (]-k interaction). Such terms are, of course, unphysical because they account for the self-interaction of a single particle. #2 See note added in proof.

T. Kawakatsu and K. Kawasaki / Binary mixture with surfactant molecules

-kBr

f

j dr Jf dg ~s(r, g)In t~s(r, g),

699

(3.8)

where t~s(r, g) is the joint distribution of location r and orientation g of surfactant molecules. This term accounts for the mixing entropy of surfactant molecules. We shall not consider such a term hereafter. With an additional assumption, we can further simplify eqs. (3.6) and (3.7). Namely, we suppose that fluctuations in the directors of the surfactant molecules are very small. Then we can use the following approximations:

s(r) =- ~ gj~(r - r~) = ps(r) g(r)

(3.9)

J

and

O(r) =- ~ gjgiS(r - rj) - l ps(r ) 1 = ps(r) g(r) g(r) - l ps(r ) 1 ,

(3.10)

J

where g(r) is the unit vector along the local average of the directors of the surfactant molecules. This approximation will be valid in the late stage of the phase separation process when the well-defined interfaces, onto which most of the surfactant molecules are adsorbed, are formed. Substituting the approximated expressions (3.9) and (3.10) into (3.6) and (3.7), we obtain

Hxs = ~sNs + ½q l c f dr ps(r) [VX(r). g(r)]

(3.11)

and (

Hss -----J dr q2c+ [Ps(r)] 2 1 _212 ~

+ ~q,

2

2~

^

dr[c+PsO~O~(g~g~)- C_PsS~(O~O~s~)

+ (c+ - c_)Ps(O~a, ps)g~g ~ + (c÷ - ½c)(O~ ps)0~(2 g~s~)] ^ ,

(3.12)

where a~ --- OlOa (a = x, y, z) and g~ is the a-component of the vector g(r) and repeated Greek indices a and/3 imply summation over x, y and z. Integrating by parts and neglecting a surface contribution, which is valid when there is no surfactant molecule on the system boundary, we obtain Hss = f dr q 2c+[ps(r)] 2 + lq212

f dr {lc_p2[(Oj~)(O~g~) + (O~g~)(Oj~)]

+ (c+ - c_)Ps(O~O~ps)gj~ } .

(3.13)

With this expression we can represent the internal degrees of freedom of the

700

T. Kawakatsu and K. Kawasaki / Binary mixture with surfactant molecules

assembly of the surfactant molecules by only one unimodular vector field g. The second term contains (V. g)2, which corresponds to the splay energy of the nematic liquid crystals [22]. The free energy expressions (2.12), (3.11) and (3.13) in some respects resemble the ones which are derived by symmetry arguments in our separate paper [18]. Our semi-microscopic approach described here can, therefore, bridge between the microscopic description and the continuum description. The procedure for obtaining the continuum description of the model system shown here is a very naive one. For a more sophisticated treatment, one may employ the density functional formalism [23, 24]. 3.2. Interface free energy in strong segregation limit With the use of the continuum description of the surfactant interaction energy (3.13), we can derive an interface free energy of the kind proposed by Helfrieh [25]. Now we consider the strong segregation limit where the A and B components of binary mixture separate completely and sharp interfaces are formed. In order to obtain an interfacial description of the free energy, we assume that the interfaces are nearly flat and we denote the unit normal of the interface as fi(a), where a-~ (a 1, a2) is a two-dimensional local coordinate system which parametrizes the interface such that da = da~ da 2 is the area element of the interface, and we take the n-axis along ti(a) at each point on the interface. As surfactant molecules are adsorbed strongly on the interface, the density of surfactant molecules Ps varies rapidly in the n-direction. On the other hand, the surfactant density Ps as well as the director field g(r) varies only slowly along the interface. We can, therefore, employ the following decoupling approximation for Ps: ps(r) = O-s(a) ~'(n),

(3.14)

where ~'(n) is a rapidly varying function which is non-zero only near the interface n --0 and O-s(a) is slowly varying in a and has the meaning of area density of surfactant molecules. Noting that the interface is nearly flat, we assume the equilibrium density profile of surfactant molecules for ~-(n). Substituting (3.14) into (3.13), performing first the integral over n and assuming that ~'(n)---~0 as n--* _+0% we obtain Hss = f da [q2~r~(a) (~7+ - / z ) ] 1 _2t2 ~

+ ~q,

1~

2

da {~c O-s(a) [ ( o j ~ ) ( o j ~ ) + (otjg,)(ot~g~) ]

+ Uaa,(a) gag,,},_0,

(3.15)

701

T. K a w a k a t s u a n d K. K a w a s a k i / Binary mixture with surfactant molecules

where the suffix a = a 1 or a 2 specifies the component along one of the local coordinate axes at a and repeated indices a and a' mean summations over these indices and the subscript n = 0 means that the quantities in the curly brackets are to be evaluated on the interface n = 0. In (3.15) we have further introduced the following quantities: oo

(3.16)

~+_ ~ c+_ f rZ(n) d n , - oo

/z-~¼12(c+-c ) f [r'(n)] z dn ,

(3.17)

l~aa,(a ) ~ (C+ -- C_)O's(a ) [OaOa,Ors(a)] ,

(3.18)

where ~"(n)=-dr(n)/dn and Oa=---O/Oa. If the surface density of surfactant molecules is uniform, we can put O's(a ) ~ or which is independent of a. Then from (3.15) and (3.18) we have

Hss = f da [q2tr2(~+ - / z ) ] (

+ -] q Z t Z J d a { ½ ~

^ ~ ) + (Oeh~)(Ot~h.)]} ¢r2 [(O~n~)(Otjntj

(3.19)

where we identified g(a) with h(a) assuming that the director of the surfactant molecules relaxes rapidly toward h(a) in a time during which the interfacial configuration remains fixed. As is shown in appendix B, we have, in terms of the mean curvature c m and the Gaussian curvature Cg of the interface, the following relations: OaR a = --Cm ,

(3.20)

(O~h~)(O~h~) = c 2m - 2Cg.

(3.21)

Substituting these into (3.19) we finally obtain

Hss f da[q2o2(& =

-

tz)l+ -¼q212 f da[~_

tr 2( c m 2 - Cg)],

which is the well-known expression for the interfacial free energy [25].

(3.22)

702

T. Kawakatsu and K. Kawasaki / Binary mixture with surfactant molecules

3.3. Hydrodynamic effects on pattern evolution Here we estimate the hydrodynamic effects on the pattern evolution in the strong segregation limit. Helfrich's Hamiltonian for a surfactant-adsorbed interface is [25, 18] Her = f [or + ½K(Cm -- CO)2 + YCg] d a ,

(3.23)

which has the general form of (3.22). For domains of size l, we can roughly estimate the parameters as Cm~1-1, Cg~1-2 and also K, Y ~ k B T . The thermodynamic driving force for the interface displacement ~q(a) roughly behaves as

f(a)=

~q(a)

1

1+

,

(3.24)

since the Gaussian curvature term is unchanged for an infinitesimal interface deformation. Also note or ~ k a T / ~ 2 ~ K/~ 2 ,

(3.25)

~: being the interracial thickness (the free energy k ~ T is assigned to each interracial element of the size ~: by the equipartition theorem). Hence K

f(a) ~ ~ l

1+

K

-- -[ (

~:-2 + I - 2 ) .

(3.26)

For the late stage where thermal noise can be neglected, the domain wall velocity v(a) is given by two mechanisms, (1) v = Vhyd due to viscous flow (hydrodynamic effects) and (2) v = vdi. due to diffusion (Lifshitz-Slyozov-like). These are obtained from the balance equation ~R So(a) - f(a) ,

(3.27)

where R is the dissipation rate associated with the domain wall motion v(a). For the cases (1) and (2), we have (1)

Vhy,~(a) = f da' h ( a ) . r ( r a - ra, ) . li(a') f ( a ' ) ,

(3.28)

T. Kawakatsu and K. Kawasaki / Binary mixture with surfactant molecules

(2)

f(AX)

1

2~rL x

Ira - ra,I

vdiff(a')da'

=f(a),

703

(3.29)

where 7" is the Oseen tensor (2.3), and AX is the discontinuity of X across the interface. Order estimates of (3.28) and (3.29) yield for three dimensions l

Lx

Vhyd- "~ f '

1

Vdiff AX 2 l f "

(3.30)

For Vhyd > Vdiff [Vhy~ < Vdiff], the mechanism (1) [(2)] dominates. The crossover length Ic is then (3.31)

lc = ( L X~?) I / e / A X ,

which is generally of microscopic size. Thus, at late stages in three dimensions, the hydrodynamic growth dominates at least in the bicontinuous region as in the case without surfactant molecules [26] (hydrodynamic growth is absent for a collection of spherical droplets). For noise-induced growth due to droplet coalescence, we have [26] V.oise ~ k B ThTl e ,

(3.32)

which is to be compared with kaT

VhYd- _ _

(

~-e

+ /-2).

(3.33)

Thus, we can conclude, as in a simple binary fluid [26], that Vnoise ~

Vhy d

Vnois e < Vhyd

for l

<~ ~: ,

(3.34)

for l > ~.

Thus, so far no new aspect peculiar to the surfactant system is apparent. More complete analyses will require new processes not present in simple binary fluids, such as the diffusion processes of surfactant molecules along interfaces and inside domains. Note that the discussions presented above are valid only in the strong segregation regime where there exist sharp interfaces. Thus, the results obtained here cannot be applied to the early stage dynamics where well-defined interfaces have not yet been formed. In such an early stage, we have to start from the hybrid equations of motion mentioned in section 2 or the continuum equations given in section 3.1. Such a problem is now under investigation.

704

T. Kawakatsu and K. Kawasaki / Binary mixture with surfactant molecules

4. Computer experiments In order to demonstrate usefulness of our model, we performed computer simulation experiments on the pattern formation processes using our hybrid model equations (2.1), (2.2), (2.11)-(2.18), which are a combination of the continuous field X ( r ) and the discrete surfactant molecules. For simplicity we temporarily ignore the hydrodynamic effects, which are described by Oseen tensor 7". Then from eqs. (2.1), (2.2), (2.16) and (2.17), the equations of motion can be written as (4.1)

OX(r, t) = L X v 2 8 H Ot 8X ' d dt

ri =

-

OH L o --Ori ,

(4.2)

and

d _ZslOn _(on,~i)~i dt gi = l_ Ogi Os i

]

'

(4.3)

the two-dimensional versions of which are expected to be valid, for example, for a thin fluid layer between the parallel glass plates in which the convective fluid flow is suppressed. Note that the thermal fluctuating forces are not taken into account in these equations of motion. Thus the randomness comes into our model only through the initial configuration. Hereafter we choose the units of time, length and energy suitably so that all of the three parameters, L x in (4.1), D x in (2.12) and u in (2.13), reduce to unity. In order to integrate eq. (4.1), we use the first order Euler scheme, which corresponds to the usual cell dynamics method [27]. On the other hand, we employ the molecular dynamics method to integrate the equations of motion for the surfactant molecules, (4.2) and (4.3) [28]. Our computer simulation method is, therefore, a hybrid of these two methods. The details of the computational techniques are presented in appendix C. The system is a two-dimensional square box with sides L and a periodic boundary condition is imposed on each side. The system is divided into N × N cells with mesh size a (i.e. a -- L / N ) to represent the field X ( r ) and we put N s surfactant molecules in the system. For the interaction potentials tk and qJ, we temporarily assume the following forms: th(r) = - e x p ( - r )

(4.4)

~b(r) = a e x p ( - r ) .

(4.5)

and

T. Kawakatsu and K. Kawasaki / Binary mixture with surfactant molecules

705

H e r e the parameter a has to be greater than unity in order to compensate the excluded volume of each surfactant molecule and in this work we set a = 5.0. The parameters used for the simulations are chosen as follows: We can set L x = D x = u = 1.0 without loss of generality #3 as mentioned above and we choose L p = 5.0, L s = 1.0, c = 1.0, q = 0.1 and l = 1.0, respectively. In view of the short range nature of the interaction potentials th and qJ, we have chosen the mesh size a to be a = 0.3. The time mesh size At used to integrate the equations of motion was chosen to be At = 0.01 to avoid instability in the scheme. First, in order to confirm that our hybrid scheme correctly takes the interactions between the field and the surfactant molecules into account, we tested the stability of a lamellar structure, which is composed of alternating layers of A-rich phase and B-rich phase separated by the surfactant-adsorbed planar interfaces. In order to prepare lamellar structure, the initial configuration of the field X ( r ) was chosen to be X ( x , y) = sin(kxx ) + ~(x, y ) ,

(4.6)

where k x =- 2 ~ r p / L

(4.7)

is the wave number of the lamellar periodicity perpendicular to its layers and ~(x, y) is a spatially uncorrelated small perturbation which represents thermal agitation. Thus the configuration given by (4.6) and (4.7) expresses the thermally modulated lamellar structure with the 2p layers parallel to the y-direction in the system. For the simulations, two identical configurations given by (4.6) are prepared. For one of these two, to be referred to as case (a), we assign surfactant molecules at uniform spacing on each of the interfaces with their A-philic (B-philic) ends nearer to the A-domain (B-domain) and then small perturbations are imposed on the positions and directors of the surfactant molecules. For the other, called case (b), no surfactants are assigned. Time evolution from these two initial configurations has been computed and the results are presented in fig. 2 for the cases (a) and (b). Parameters selected for these runs are p = 3 and N = 36 for both (a) and (b) and N s = 150 for (a). H e r e we note that the width of a layer is 6a = 1.8 which is of the same order as the interaction range of a surfactant molecule. The left-hand figures of fig. 2 show two-dimensional configurations of the field X ( r ) and the surfactant molecules, where the cells with asterisks and the blank cells denote A-rich cells (positive X ) and B-rich cells (negative X ) , respectively, and a surfactant molecule is indicated by a small square with a short straight line • 3See note added in proof.

706

T. Kawakatsu and K. Kawasaki / Binary mixture with surfactant molecules

a t=O. 0

~:::~

~:::~

~::::~

~::::~

l ]

t=lO.O

]

) 1 l ] 1 ] ] 1 ] l 1 3 l 1

5~i!iB

............. .... ] J

t= 20.0 ~MNNM dN~N~-

MNNNNN ~MNNN~

NNHH ~NM~N~

HHMN~ ~NNNNB"

i ~

~.

- 0 ~

Fig. 2. Time evolution from initial lamellar configurations of the field X(r) is shown for the case (a) with surfactant molecules placed on the interfaces and for the case (b) without surfactant molecules, respectively. The figures on the left-hand side are snapshot pictures of the system. Cells with asterisks and blank cells represent A-rich and B-rich cells, respectively, and the squares with short lines indicate surfactant molecules. The figures on the right-hand side indicate profiles of the field X(r) along the horizontal lines cutting through the centers of the snapshot pictures to the left.

707

T. Kawakatsu and K. Kawasaki / Binary mixture with surfactant molecules

b .mm mm~m~, .mm ~mmmmm

.mmmmm mmmmmm mmmmmmm mmmmmm

mmmmm mmmmmmm mmmmmm mmmmmm

mmmmmm

mmmmmm

m~mmmm

mmmmmm

mmmmmmm

mmNmmm

mmmmmmm

mmmm mmmmmmm ..m. mmmmmmm mmmm mmHmmmm m+m mmmmmm mmm mmmmmm .+m mmmmm m~ mmmmm m+~m mmmmm +~m~m mmmmm ++mmm mmHm +m.mm mmmm ++.m~ mmm =~.mm mmm ~m+~m mmm +mmm~m mmm ,.+m~.m mmm ~mm"mHm mm -+mmm~m mm -+~+mmm mm -~m~mm mm ++~mm~m mm -+++++~mm ,+++.m+ mmmm +-~+~mm mmmm +,~++~m mmmm "+++~-+ mmmmmm +-++~+m ~mmmmm .,~+mmm mmmmmmm +m++mmm .m+mmmmm +mmmmmmm~m+mmmmm -.mmmmmmmmmmmmmmmm ++mwmmmmmmmmmmmmm +mm~mm+mmmmmmmmmm ++mmm~ ~mmmmmmmm -+mmmmm mmmmmmmm

t=O.O

/ t=lO.O

mmmmmm mmmm.m mmmmHm mmmmHm mmmmmm mmmmmm mmmmmm mmm+m ~mmmm mmmmm mmmmm mmm mmm mmm mmm mmm mmm mmHm mmmm mmmmm mmmmmm mHmmmm mmmmmm mmmmmmm mmmmmmmm mmmm~mmm mmmmmmmm m.~mmmmm mmmmmmm mmmmmmm mmmmmmm mmmmmm mmmmmm mmmmmm mmmmmm

t=20.O

mmmmmmmm~mmmmmmmmmm +mmmmmmmmmmmmmmmmm m-.mmmmmmmmwmmm ~mHmmmmmmmmmm mmmmmmm

~NMN

+~X~N~KMNN~

MM~MMH

Fig. 2 (cont.).

708

T. Kawakatsu and K. Kawasaki / Binary mixture with surfactant molecules

which is parallel to the bond connecting the A-philic center and the B-philic center of the surfactant molecule. The right-hand figures of fig. 2 indicate the profiles of the field X(r) along the horizontal line y = L/2. As is shown in fig. 2a, when the surfactant molecules are adsorbed on the interfaces, the initially chosen sinusoidal modulation of X(r) is stable although the modulation amplitude is somewhat reduced. On the other hand, in fig. 2b, the sinusoidal modulation decays rapidly due to the mutual diffusion of the A - B mixture. As thermal noises are neglected in our model, such stability of the layered structure does not come from the so-called Helfrich interaction [29] which originates from the entropic effects. Noting the microscopic scale of the width of a layer mentioned above, we may attribute such stability to the direct interactions between surfactant molecules belonging to the nearby layers, such as the steric hindrance, which are expressed by the second and the third terms of (2.15). In this respect our model is unrealistic since the repulsive Helfrich interaction is well established. To improve on this we will have to include the so far neglected thermal noise although qualitatively the steric hindrance may be regarded as a substitute for the Helfrich force. Next, we perform a series of simulation runs on the pattern formation processes from initial random configurations where the A component, B component and surfactant molecules are completely mixed. For the initial values of the field X(r), a set of spatially uncorrelated normal random numbers with mean value X 0 and the standard deviation o-x are employed. In figs. 3 - 6 we show several examples of the pattern formation processes. For all of these runs, the system is divided into 128 × 128 ceils and the initial standard deviation of the field X(r) is chosen to be o-x = 0.2. In fig. 3, X 0 is taken to be 0.0, which corresponds to equal volume fractions of A and B components and the number of surfactant molecules is chosen to be N s = 512. From an initial random configuration, bicontinuous domains [2, 6, 7] of A-rich phase (dotted cells) and B-rich phase (blank cells) are seen to emerge, which corresponds to the spinodal decomposition of the A - B binary mixture [27]. In order to see the difference between our surfactant system and the usual spinodal decomposition of simple binary mixtures more clearly, we performed another simulation starting from the identical initial configuration of the field X(r) as in fig. 3 without surfactant molecules and the results are shown in fig. 4. Comparing fig. 3 with fig. 4, we note the following three differences between these two cases: 1) The characteristic length of the pattern of fig. 3 (with surfactant molecules) grows at a slower rate than that of fig. 4 (without surfactant molecules). 2) The density fluctuation of the A - B mixture is amplified in the presence of surfactant molecules, which is clearly seen by comparing the profiles of the field X(r) of the two cases.

T. Kawakatsu and K. Kawasaki / Binary mixture with surfactant molecules

709

t=O.O

t =10.0

t= 20.0

Fig. 3. Pattern formation process from an initial state of random mixing of binary fluid mixture and surfactant molecules is shown for the case of equal volume fractions of A- and B-components. In figs. 3, 4, 5 and 6, A-rich cells are indicated by dots rather than asterisks because of the higher resolution of the field X(r) in these figures.

710

T. Kawakatsu and K. Kawasaki / Binary mixture with surfactant molecules

t= 30.0

t=40.O

t=50. O

Fig. 3 (cont.). 3) The interfaces tend to be m o r e flat when surfactant molecules are adsorbed on them and the resulting pattern becomes locally m o r e lamellar-like. These differences can be understood by taking into account the interactions between the field X ( r ) and the surfactant molecules and the interactions between surfactant molecules:

T. Kawakatsu and K. Kawasaki / Binary mixture with surfactant molecules

711

A t= 1 0 0 . 0

(= 2 0 0 . 0

-

iiii!!7 ~ i ! ~

-

m

liiiiiii!iiii'iiiiii!!!i!iii ~

=300.0

,>/, Fig. 3 (cont.). 1) As the surfactant molecules adsorbed on the interface interact repulsively with each other resulting in enormous reduction of the thermodynamic force for coarsening, they prevent further shrinking of the interface, which leads to a slowing down of the growth of the patterns.

712

T. Kawakatsu and K. Kawasaki / Binary mixture with surfactant molecules =:! ;. =i~. : ,:..~ i'b "!, :.:i. = :! ~r<:::

;=~, ="! ==J==".= [ :<';'.:::

t=O.O

.= ' := ,[:'!. : .i.'.~l ..::

i!i~!i!iih~i::~;;;~i~.;;~i;/%.~:!::!~!/~ii;~%i;.;i%~:i:;:;.~Z!;.~:.b~ki!;,~!:~:;~;~!::%:;i;:!~i!i1%.:~i%:i!:;i~::i;~i!!~i]:~;~.~::;~.;
i~;~%i;:;;!:i~i~i:;i:2:!;i`;i~i;~<:ii:;i~!~;=i~!ii~:;~i1~;:!~i!;~;~4;;i~%;~;~;i!?!i

i :~;1:i :}: 5~ii;];2!=:, i:!<' '!ri::=: 5:?i ~;;!! :' i:~
~iiii~Hili~ ~'''''~''~'~,~iiii!iiiiiiiiiiiii!ii!ii!!iiiii!!i!i!!iiii~, ,,,~i;¸¸'=':

t=lO.O

iiiiiiiiiiiiiiiiiilili~ ~ ......,~,~iii{iiiii!iliiiiiiiiiiiiii{iiiiiiiii~'"~iiiiiiiii~!iiiiiiii" iii~'"......... '~!iiiiiiii!= ........."~ ~,~=~iiiiiiiiii~ ~ ~iiiiiiiiiiiiiii~

..........

iiiiiiiiiiii~, ~,~iiiiii{ii~,. ~iii!iiiiiiiiiii!iiiiiiii~ ,iiiiiii!iiii~"'"~ii!iiiiiiil}iiiiiill ~i~i!ii% ~ii~i~i~i~i~i~iii=~]~iii~iii~i~i~i;iiii~iiiiiiiiii[ =

iiiii~iiiiii!i!i!iiiiiii~

:iliiiiiiF

•..............................................

t=20.O

v

'~
.,~,~,

'~,~

A

i!iiiiiiill

Fig. 4. Pattern f o r m a t i o n p r o c e s s f r o m the s a m e initial c o n f i g u r a t i o n o f the field X(r) as in fig. 3, but w i t h o u t surfactant m o l e c u l e s .

2) As the surfactant molecules assist in separating the A component from the B component and stabilizing the interface with greatly reduced effective surface tension, the phase separation of the A - B mixture across the surfactant layer is greatly facilitated.

T. Kawakatsu and K. Kawasaki / Binary mixture with surfactant molecules iii!ii!iiii,

:'~:'=

"~:%~

iii%~:==,,di!i!iiiiiiiiii!i~:

iil

713

t=30.O

:~ii!i~i!!iii~'i~* i~iii~!iilF

ii!ii

i~[riiiiiiij[i,

............."

iigiJliffF[Jiiiii[iMi[ii~ii[eilE[@iii[iii

~iiil;giii[i'

~J[iii[r;e,' ....

:~!i!i!iiiiiiiii! ~'::'~iiii]iJiiiiii~, .......

iilliiillll, :iiiii%iiii!!i~

.~i!ii!iiii:ill

:i~ililli~!~!i;...!iiii!ii!i!iiiili[!.

iiii~i';;'~.

"'::::"*;::~!!

"==: •

~,H~i'~i',',ii~i,,,~, ...... !

!i!~:'

~:

iii

"=:~i i i~ ~ JJi[JJi[J[)iii il j::j iii iiiii jii{ F' ' .... Uil!iil

'Jiiiil!~==,=~iJiiiiiJiJ@i~,

,?,i!iilliiiiiiJliiiiiiiiiiiiiiiiiiiii*,iiii~, ...............

::~;iiii

t=40.O

iii ![!i[JiJ**J i]~=:'"':!iiiJJiiiJ~,iiJiiiii i: ii ii::i i ~="

JiJJj=jjjij~[ijj '~;j;i i;iJ;~i!iiiii~

iiiii!iiiii: iiiiiiiiii" iiii!i!i: .:~iiiiui~ .=~iiii[ii [ii:.~. iiiiii}iii!ii}i~,=~i~iiiE!;i}ii}ii!ii v

............, ............ ~,:~,

i![!iiiii!ii!i~

'~iiii

v

~

~,~, %i'~iiiii ~

:l!i!iiiiBiii!!iiiiiiiiiiiii!iiiiiii!~i=

;~iiii

t=

50.0

Fig. 4 (cont.).

3) The bending elasticity tends to make the interface more flat and the surfactant-adsorbed layers repel each other by the steric hindrance of surfactant molecules. By these interactions the interfaces tend to align in a layered structure like the lamellar phase. Therefore the resulting domains have a locally lamellar-like structure [30].

T. Kawakatsu and K. Kawasaki / Binary mixture with surfactant molecules

714

t= 100.0 !iii~giiiiiii!iiiii!

;iii;;ii;iii!iiiiii~::'"


i:i!i~i~iiiii;iiiiiii~iiiiiiigiiiiiiiiiii

:!!!~iiiili!!iiiiiiii{i;iiiiiii!ii;iiiiiiiiiiiiz~:::"

~ii!ii{iiiiiiii!ii{?ii{[ i!ii" =iiiiiiJiii{iiii!iiiiiiii:

:i

iii

iiiiiiiiiiii}ili~ilr'

iliJiiiiii[iiiiiiii

dil;i~iiii

i: ~

~.

_

~

/~v//-~

ii{ii%ii'i i!i l t t =200.0

Iii il

, ,.,.

~iiFiiiiiiiiiii!iiiiiiiiiiiiiii! iiiii~ iiii;i;ii}iiiiii~' ~,~i}iiiiiiiiiiiiii!iii~H'" ................. "~ii!ii}i}iiil}i~ iiiiiiiiiiiiii[iiii~ ................ =:=:=~=

iiiiiiiiiiii ~

.~il

b~,.

,iiiiim~iiiil}iii

!~:

iiiiiiiiiiiiiiiiiii ~

ii~!!ii#iii!i

:

t=300.O

"

.~!l!l;:~ilib:'':=~!ir]r?i?i}i~??i}i)iiil

....... : ............... ~i=ii]}{iil;ii{ia}{iiiill

:ii!~iii~iiiii!

Fig. 4 (cont.).

Taking these effects into account, we can conclude from our simulation results that the domains created in the phase separation process of the binary mixture tend to be smaller, more sharply defined and more worm-like when surfactant molecules are added to the mixture.

T. Kawakatsu and K. Kawasaki / Binary mixture with surfactant molecules

715

t=O.O

t=lO.O

. .~ ~ .

* "%,~

~,~¢,'~

f ~'t ~ _~

t=20.O

,

~, ~ ~

~>~ "r~ ) , ~ a . ~

-~

Fig. 5. T h e same as fig. 3 for the micellar case w i t h the asymmetric volume fraction ratio A:B=I:3.

We also performed simulation runs similar to those s h o w n in figs. 3 and 4, but with X 0 = - 0 . 5 which corresponds to the asymmetric case of a 1 : 3 ratio of the A to B v o l u m e fractions and the results are s h o w n in figs. 5 and 6, where the number of surfactant molecules are N s = 512 (fig. 5) and 0 (fig. 6),

716

T. Kawakatsu and K. Kawasaki I Binary mixture with surfactant molecules

t=30.O

.,~

, , .

.

%

"~

~

~_

a

.=~. ±= ~

,~'

~-]

t= 40.0

_

.-

~...i:~

t=50.O

~

Fig. 5 (cont.).

respectively. In fig. 5, one can see formation of many small circular micelles (droplets of A-rich phase surrounded by surfactant molecules) [2, 6, 7], while the phase separation does not occur during the time of simulation in the absence of surfactant molecules as shown in fig. 6. Such a difference was

T. Kawakatsu and K. Kawasaki / Binary mixture with surfactant molecules

717

t=lO0.O

f\

t=200.O

Fig. 5 (cont.).

I

already seen in the bicontinuous case shown in figs. 3 and 4. When micelles begin to form, surfactant molecules create large concentration fluctuations of A - B mixture around them, which turn into a seed of the nucleus (micelle). Without surfactant molecules, the nucleation process has to be initiated by the spontaneous thermal concentration fluctuations in the fluid, which must overcome the high free energy barrier, and hence there is much less possibility of occurrence. Note that, in the early stage of the micelle formation process, the micelles have more or less worm-like shapes. Such behavior can never be seen in the nucleation and growth process in simple binary mixtures without surfactant molecules. Needless to say that the formation of such worm-like micelles is due to the high bending elasticity of the surfactant-adsorbed interface and steric hindrance between interfaces. Such worm-like micelles are actually observed experimentally in dilute surfactant solutions and microemulsions [31] and also modeled analytically [32]. We also note that, in the last figure (t = 200.0) of fig. 5, micelles locally form triangular lattice configura-

718

T. Kawakatsu and K. Kawasaki / Binary mixture with surfactant molecules

t=O.O

t =100.0

t= 200.0

Fig. 6. Pattern formation process from the same initial configuration of the field X(r) as in fig. 5, but without surfactant molecules.

tions. Such a configuration can be observed in the oil-water-surfactant system and originates from the repulsive interaction between miceUes produced by adsorbed surfactant molecules.

T. Kawakatsu and K. Kawasaki / Binary mixture with surfactant molecules

719

5. General formulation of the hybrid model In the preceding sections, we described our hybrid model for a simpleminded model system, that is" 1) We assumed that a surfactant molecule has dumbbell-like shape for simplicity. The surfactant molecule was, therefore, assumed to be uniaxial. 2) We limited our considerations to the symmetric system where the system is invariant under the exchanges between the A-philic center and the B-philic center of a surfactant molecule and between the A-component and the B-component, simultaneously. It is certainly useful to generalize our consideration of the hybrid model with a minimum of specific assumptions. In this section, we discuss such a generalization, limiting ourselves, however, to rigid surfactant molecules.

5.1. General expression of the free energy functional As was discussed in section 2, we consider the total free energy functional H consisting of three parts,

H = Hxx + Hxs + Hss,

(5.1)

where Hxx describes a binary fluid with the component densities PA and PB which may be expressed in terms of X---PA- PB and p------PA q- PB" H x x is, therefore, a Ginzburg-Landau type free energy functional of X(r) and p(r),

Hxx = H x x ( ( X } , { p ) ) .

(5.2)

Hss describes surfactant molecules only and Hxs is the interaction between the binary fluid and surfactant molecules. Hxs is assumed to be linear in p/¢ (K = A, B) and additive in surfactant molecules, which amounts to two-body interactions. Thus we can express Hxs as follows: A.B

where r i denotes the center of the ith surfactant molecule, and Fir(r ) is the form factor of a surfactant molecule interacting with the K-component of the binary fluid. Here the subscript i in Fir expresses possible dependence of F on the orientation of the ith surfactant molecule. In order to restrict ourselves to purely dissipative slow time variations, we eliminate p = PA q- PB" Physically, non-dissipative sound propagation occurs

720

T. Kawakatsu and K. Kawasaki / Binary mixture with surfactant molecules

very rapidly and we are permitted to eliminate sound wave modes. This leads to the condition of constant pressure, which differs from the condition of incompressibility. Writing PT --= PA + PB + PS, where Ps is the surfactant density, we have

aPT =

°PT aP + aT - ' ~ V,X,os \ OT / e,X,os \

OX,IP, T,Ps

OPT

ap s ,

(5.4)

where T and P are the temperature and pressure, respectively. If sound waves are eliminated, we have a P = 0 and if a heat bath of constant temperature is attached to the system, we can also take a T = 0 (isothermal processes). If we can further assume that (OPT/OX)p,r,p s = 0 and (OpT/OPs)p,r,x=O, we can impose the incompressibility condition

aPT = 0.

(5.5)

Then (5.3) takes the form

Hxs = H i s + a H s s ,

(5.6)

where

Hxs = ~ f dr S(r) F ; ( r - ri) ,

(5.7)

AHss = ~/ f dr Ps(r) F';(r- ri) ,

(5.8)

F~(r) and F'~(r) being certain linear combinations of FiA and FiB. We associate AHss, contributing to the surfactant-surfactant interactions, with Hss. Putting fl = P T - - P S = P A q- PB into (5.2), we can expand it in powers of Ps,

/-/x.,,- = t-/,,.(x} + Hks + / - G ,

(5.9)

where Hxs is linear in Ps and H i s is quadratic in Ps. We then associate /-/)~s with Hxs and Hss with Hss. Thus finally we find

H = H x x ( X ) + Hxs + H s s , where H x x { X } has the G i n z b u r g - L a n d a u form as before and

(5.10)

T. Kawakatsu and K. Kawasaki / Binary mixture with surfactant molecules

Hxs = ~i f dr X(r) F~(r- ri) = ~ f dr X(r + ri) Fi(r) ,

721

(5.11)

where F~(r) is the form factor describing the interaction of a surfactant molecule with the field X(r), and Hss is the remaining part involving only variables pertaining to surfactant molecules (which does not necessarily mean that Hss is the bare surfactant energy, but may partially contain renormalization effects of fluids which are independent of X(r)). Here we assume that the size of a surfactant molecule is small compared with the distance over which X(r) changes appreciably and thus assume the following expansion:

Hxs = f dr Fi(r) E exp(r'Vi) X(ri) ,

(5.12)

ricO~Or

(5.13)

i

with i .

That is,

Hxs

:

~ (e o + e '1 . V , + a v , "0 '2 "Vi "~-'" ") X(ri) ,

(5.14)

i

where

eio =--f dr Fi(r) ,

, f

el==i

02

dr Fi(r) r ,

(5.15)

(5.16)

/-

=--| dr Fi(r ) rr . J

(5.17)

We now assign three orthonormal unit vectors g~, g~ and g~ to the ith surfactant molecule, which represent orientation of the rigid surfactant molecule. Then, by symmetry, e 0i which is independent of the molecular orientation should not depend on i and hence will be simply denoted as e 0 and we can also write e li = E

P/xS~ ,

(5.18)

0 2;

q t , v S"~'~ i Si •

(5.19)

= ~

T. Kawakatsuand K. Kawasaki / Binary mixture with surfactantmolecules

722

Then

Hxs

= ~i (e°+ ~, p~'g~ "Vi+

½~,~ q~,~(g~"Vi)(g~ " V i ) + ' ' - )

g(ri). (5.20)

Here e0, p~, and q~,~,.., are the constant parameters that characterize a surfactant molecule interacting with the X-field. Next we turn to Hss, which describes the effective pair-interaction between surfactant molecules: Hss = ~]

i
V~j,

(5.21)

where Vq should be invariant under simultaneous translation of r i and r i and ~v also under simultaneous rotations of the vectors u ~ r i - rj and g~ and sj with Ix, g = 1, 2, 3. Therefore, we find the following expansion: 1

2a

^ '

~j = v"(u) + v~(u) r,. (~ + ~';) + v,..,(u) ~7.s~ -]- V2p+b,(u) [ ( l l "

$7)(a

" $~')

-I- ( I I " S ; ) ( l l

" $;')1

2c A ^ ' + v,,,,,Cu) (u. ~'/)(~. s; ) +...,

(5.22)

where the v's are some functions of u --- lu]. We have also written ii ~- u/u. In (5.22), summations over repeated indices IX, I X ' , . . . = 1, 2, 3 are implied.

5.2. General expressions of the equations of motion Next we present a general form of equations of motion for the field X and surfactant molecules. The irreversible entropy production rate per unit volume is written as [33] 1

"a~irr --

1

zJX°~'LX--T~i

OH

Ori

1

•

-t-,.,,'~

Z ~ SO~ ° H ( . 05~-~ - ~v Ai $i~' (5.23)

where the convective velocity (and hence the hydrodynamic effect) was omitted and A~'~ is arbitrary except for the symmetry requirement A~ ~ = A~~ "

(5.24)

Then the A-terms in (5.23) in fact vanish because of the following orthonormality of {g~'}:

T. Kawakatsu and K. Kawasaki I Binary mixture with surfactant molecules

g~.gT=a~.

723

(5.25)

The chemical potential and the continuity equation of the field X are expressed as

8H tXx = ~X

OX Ot - - V . J x .

and

(5.26)

We can then write down the irreversible equations of motion

xs/~ OH - E ~;~: ) J~= - L ~ v . ~ - E L~. oH _ E L,~ i i" i =

~

-t

=-Li

ox

i Vtz x -

~

Ori

L~

°tgri/ ~ - E~

,,s ou ati

Via'x-Li~

i~

i \ Og~

L,~-

-Z

a s #4

(5.27)

(5.28) A,

~,

,

(5.29)

where the numbers )t~ ~ are to be determined by the orthonormality (5.25). Hydrodynamic interactions can be included if fluid anisotropy is neglected in the following way: 1) Make the replacements in (5.27)-(5.29) as

Jx(r) ~ Jx(r) - v*(r) X ( r ) ,

(5.30)

~ , ~ ~, - v * ( r 3 ,

(5.31)

$~'~ S~ + {S~ X [~i X u*(ri)] .

(5.32)

2) The convective velocity v*(r) is determined by

v*(r) = f dr' T(r - r') . [ V ' . / / n d ( r ' ) ] ,

(5.33)

where the non-dissipative part of the stress tensor H nd or, more directly, the force density

fnd(r) : V"//nd(r)

(5.34)

• 4 The Onsager coefficients L can depend in general on {s~'}. Then (5.27)-(5.29) can contain more terms (or dependence on the s's) reflecting anisotropy (e.g. rotation-translation interaction). We neglect such complications here. We also assume that the direct interactions between different surfactant molecules are entirely contained in H.

T. Kawakatsuand K. Kawasaki / Binary mbcture with surfactant molecules

724

will be given by the following consideration. Here we encounter the new situation that we are dealing with the continuous field X(r) and discrete molecules. The part dealing with X(r) is readily found as before [18]. Thus we find //nd

Oh(r)

= OVX(r--~ v g ( r )

-1-//dis,

(5.35)

where h(r) is the free energy density that arises from H x x { X } and Hxs and the subscript "dis" represents the contribution from the discrete molecules. Here we note that Hxs is transformed from (5.20) into

nxs = f

i

I~

+ [VX(r)] • ½ E quv E s~[g; "Vi~(Y /zv

-

-

Yi)I} •

(5.36)

i

Next we obtain the discrete contribution fdi~(r) =- V. Hdis(r )

(5.37)

in the following way. Consider an infinitesimal displacement u(r) of the medium in which discrete surfactant molecules are embedded. Then r i and gff are changed by (5.38) since the local infinitesimal rotation vector is ½V × u(r). The accompanying infinitesimal change in the discrete part of the free energy (~H)dis is (SH)di ' = ~/. ~Tri.Sri+~(OHOH . ~ - ~'.~ A~""gi') • 8g~" ,

(5.39)

where A2 v = h~" is an arbitrary constant arising from the orthonormality (5.25), which implies ^lJ

^v

S ~ ° ~S i =~ S i " 8S~ ~---O .

(5.40)

Thus the A-terms in (5.39) automatically drop out. If we substitute (5.38) into (5.39) and integrate by parts, (5.39) can be rewritten as (gH)dis = -- f fais(r), u(r) dr.

(5.41)

T. Kawakatsu and K. Kawasaki / Binary mixture with surfactant molecules

725

Here fdis(r) can be found after some algebra as

fdis(/')

•

1

r,)

~g--f - ~

X

x

,

(5.42)

where again the A-term is shown to drop out by symmetry Aft = A~". Finally we get

0v'x(r')

Note that the first term on the right-hand side is equivalent to (2.18) provided that h[X(r)] has the usual square gradient form like (2.12):

H[X(r)] =

½DIVXI2 + f(X),

(5.44)

where D is a constant and f(X) is a local potential which does not depend on VX.

6. Conclusion

In this article we proposed a hybrid model, for which it is convenient to investigate dynamics of phase separation processes in binary mixtures in the presence of surfactant molecules. We performed computer simulation runs using this hybrid model, which demonstrate that our model is indeed useful to study dynamical aspects of the surfactant-adsorbed interfaces and their formation processes. Our simulation runs of the binary mixtures with surfactant molecules have shown quite distinct pattern formation processes different from simple binary mixtures without surfactant molecules. In the presence of surfactant molecules, the interfaces, which are easier to form, tend to be sharper and more rigid, which leads to slower growth of the characteristic length of patterns. Here we note that sharpness of the interface is a conseqeunce of the fact that only little or no free energy increase is associated with an increase of the interracial area in the presence of sufficient quantities of the surfactant, which in turn leads to an enormous reduction of the thermodynamic driving force for coarsening as already mentioned. On the other hand, movements of surfactant molecules on the interfaces must be important in such dynamics, Such a process will be taken up in future investigation.

726

T. Kawakatsu and K. Kawasaki / Binary mixture with surfactant molecules

In this work, all the simulations were for the two-dimensional case without hydrodynamic effects. As was discussed in section 3, the hydrodynamic interaction becomes dominant in the late stage of phase separation processes and, therefore, it is necessary to include such effects in our simulation scheme if one were to cover the entire stage of ordering processes. Although hydrodynamic interaction is long-ranged and has a difficulty of divergence in the twodimensional case, it is well-defined in the three-dimensional system. Extension of our simulation scheme and programs to the three-dimensional system is straightforward where inclusion of hydrodynamic effects will be a challenging problem. Many of the simulation runs in this work focused on surfactant-assisted quick phase separation of an immiscible binary fluid. Since the thermodynamic driving force for coarsening diminishes very quickly, what we have seen may well be metastable states with high free energies. Under such circumstances, inclusion of thermal noise should turn out to be quite crucial. Then it would also be possible to study interesting kinetics of morphology changes, which is another future problem. We end this paper by emphasizing utility of the hybrid approach of the kind described here in dealing with complex systems like microemulsions where both microscopic and mesoscopic (or semi-macroscopic) aspects must be taken into account.

Acknowledgements One of the authors (T.K.) would like to express his sincere gratitude to Dr. K. Sekimoto, Mr. A. Ogawa and Mr. T. Koga for fruitful discussions. He also thanks Professor M. Doi, Professor T. Ohta, Dr. R.K. Kalia, Professor C. Knobler, Professor B.J. Alder, Professor X. Sun, Professor S. Komura and Professor I. Hatta for valuable comments and discussions. Computations were carried out at the Computer Center of the National Insitute for Fusion Sciences and the Computer Center of Kyushu University. The work was partially supported by the Scientific Research Fund of the Ministry of Education, Science and Culture of Japan.

Appendix A Here we consider the director-director interaction given by the second and third terms of (2.15).

T. Kawakatsu and K. Kawasaki / Binary mixture with surfactant molecules

727

We consider two surfactant molecules labeled by 1 and 2 where their centers of mass are separated by the distance r and we calculate the director-director interaction between these two surfactant molecules. We take the coordinate origin at the midpoint of the centers of mass of these two surfactant molecules and the x-axis along the vector tiE = r 1 - - r 2. We denote the angles between the x-axis and the directors gi as 0i (i = 1, 2). Now, from (2.15), the interaction between two directors sl and $2 is given, apart from a constant factor q212/4, by h ~ ($12512): VVt~(r12 ) + (s12sl+2) : VV~b(r12) .

(A. 1)

Eq. (A.1) can be rewritten in our coordinate system as h = A , ( c o s 01 - cos 02)2 + B,[1 - cos(01 - 02)] + A , ( c o s 01 + cos 02)2 + B,[1 + cos(01 - 02)],

(A.2)

where we have introduced A , and B,~ as A , =-~b"(r)_ _1 th'(r) ,

(A.3)

B , _= _2 ~b'(r)

(A.4)

r

r

and A , and B , are defined similarly where a prime denotes differentiation. Denoting the extremum of the function h(01, 02) as 0 ( ° ) - (0~°), 0~2°)), the interaction h(O~, 02) can be expanded in Taylor series around 0 (°~ as

h(O,, 02) = h (°) + ½ A 0 - H . A 0 + . - . ,

(A.5)

where A0 ------(01 -- 0~°), 02 -- 0(2°)), /h(o)

h(o)\

H =- ~h(o )

h(o) !

|°'11

\°°12

°'12 |

tr H > 0 ,

(A.7)

°'22 /

and h~°) -- (02h/tgOi O 0 j ) l o = o ( o tremum 0 t°) is given by

(A.6)

)

(i, j = l, 2). The stability condition of the ex-

det H > 0.

The matrix elements of H can be calculated from (A.2) as

(A.8)

T. Kawakatsu and K. Kawasaki / Binary mixture with surfactant molecules

728

hii

=

- 2 ( A 6 + A , ) cos 20 i + 2(A6 - A~,) cos 0i cos 0j +(Be~-B~)cos(0/-0j)

(

i=1,2,

j=

{2 ( / = 1 ) ) 1 (i=2) '

h12 = - 2 ( A ~ - A~0) sin 01 sin 02 - (B6 - B , ) cos(01 - 02).

(A.9) (A.10)

We can readily verify that (01, 02) = (rr/2, ~r/2), which corresponds to parallel orientation of the directors gl and s2, is an extremum of h, (A.2). Substituting 0i = 0j = ~r/2 into (A.9) and (A.10), we obtain the stability condition from (A.8) as tr H = 4(A6 + A~) + 2(Be~ - B~) > 0

(A.11)

det H = 8A~,[2A6 + (B~ - Bo)] > 0 .

(A.12)

and

If we choose the interaction potential introduced in section 4, ~b(r) = - e x p ( - r),

(A. 13)

~0(r) = a e x p ( - r),

(A. 14)

and

with a > 1, (A.11) and (A.12) reduce to trH = 4 ( ( a - 1 ) + -2 ot) e x p ( - r) > 0 F

(A.15)

and det H = 16a(1 + ! ) ( - 1 +

~ ) e x p ( - 2 r ) > 0,

(A.16)

which are fulfilled for any r < a. Thus the directors of two nearby surfactant molecules tend to be oriented parallel to each other when we choose an interaction potential of the type given by (A.13) and (A.14). In figs. 7a-c, we depict three-dimensional representations of the function h(01, 02) with (A.13), (A.14) and a = 5.0 evaluated at r = (a) 0.5, (b) 1.0 and (c) 2.0, respectively. It is clear from these figures that (01, 02)= (~/2, 7r/2) is the unique stable minimum, which corresponds to the parallel orientation of the two directors in

729

T. K a w a k a t s u a n d K . K a w a s a k i / B i n a r y m i x t u r e with surfactant m o l e c u l e s

the direction perpendicular to the separation of the centers of mass of the surfactant molecules, and that this minimum energy configuration becomes more and more stable as the two surfactant molecules come closer.

Appendix B Here we give derivations of the relations (3.20) and (3.21) following ref. [181. Using the Monge representation of an interface as follows: (B.1)

u(r) -- z - f(x, y) = 0 , we readily obtain the following expressions:

h(r) = Vu(r) / IVu(r)l , Cm

=

1

iVul

(B.2)

2

2

[(1 + fy)f~x + (1 + f~)fyy - 2f~fyf~y],

(B.3)

1 2 Cg = ]Vu] 4 (Lxfyy - f xy) , 0~,, -

1

iVu[3 [(1

(0~t/~)(O~t/~) = ] ~

+

(B.4)

2

fy)fxx + (1 + fx2)fyy -- 2fxfyfxy ]

1

q_ ,fy2 ) .f.2. .

[(1

2

+ (1 2

+ (2 + f 2 x + f y ) f x y 1

(o~h~)(O ti~) __ [Vul6 {(1 _~_.f 2y .'~2f2 ...

(B.5)

q_ .f.2.).f 2 yy

- 2 f x f y ( f x x +fyy)fxy], f2)2f2

+ (1+ ....

(B.6)

yy

+ 2(l + f 2 + f2y + 2f2 f2~f2 vx a y taxy -4[(1

+

2

2

2

f y)fxx + (1 + fZx)fyy]fxfyfxy + 2 f x f yf~xfyy } " (B.71

From these expressions, we obtain the following relations: o~a~ = --Cm,

(B.8) 2

(0~fi~)(0~h~) = c m - 2Cg,

(B.9)

730

T. Kawakatsu and K. Kawasaki / Binary mixture with surfactant molecules

20.0

r = 0.5

0.0

- 20.0

02

a

~ ~ - ~ " ~

7t 01

0

20.0

r -- 1 . 0

0.0

- 20.0 2~

b

o

Fig. 7. Three-dimensional representations of the function h(O1, 05) are given for the case of the interaction potentials (A.13) and (A.14) with ot = 5.0 evaluated at r = 0.5 (a), 1.0 (b) and 2.0 (c), respectively.

T. Kawakatsu and K. Kawasaki / Binary mixture with surfactant molecules

731

20.0 r-- 2 . 0

0.0

- 20.0 2~

2~

C

0

Fig. 7 (cont.).

which are satisfied in any representation of the interface. H e r e we have to make some comments on the relation (B.9), which seems to conflict with (3.7) of ref. [18], which states that (O~h~)(O~h~) is equivalent to 2 Cm -- 2Cg. This conflict comes from the selection of the particular representation of the interface which was employed in deriving (3.7) in ref. [18]. H e r e we give an explanation of the reason of this conflict. We define the following three scalar quantities: ll=-(Oah~)(O~h~),

I2-(O~h~)(O~ha)

and

I 3 - - - ( V × t i ) 2,

(B.10)

among which the following identity holds: I, -

I2 = I3 .

(B.11)

As /1, /2 and I 3 are scalar quantities, they are invariant under any rigid (or uniform) rotation of the Cartesian coordinate axes. On the other hand, for a given interface and given h(r) taken at the point r on the interface, the vector field t~(r) generally depends on the choice of the u-field in (B.1) except for r on the interface. For instance, consider the following two choices of the u-field: u,(r) = u(r) ,

(B.12)

732

T. Kawakatsu and K. Kawasaki / Binary mixture with surfactant molecules u2(r ) = B(r) u(r) ,

(B.13)

such that u(r) = 0 but B(r) ~ O, say B > 0, for r on the interface. Then the unit normal of the interface can be expressed in two ways as Vu 1 hi(r)-[Vul[-

Vu [Vu['

(B.14)

BVu + uVB

•U 2

G(r) = -~21 = [BVu + uVB[

(B.15)

"

We see that h I = h 2 only for r on the interface where u = 0. The important point is that 11 , 12 and 13 can in general depend on the choice of the u-field in the above sense since a~h2~ involves, for instance, a quantity like (a~B)(O~u) which does not generally vanish even on the interface. Now, a rotation of the coordinate axes about the origin at a point on the interface can be looked upon as a different choice of the u-function in the above sense. That is, u 1= z-fix,

y),

u 2 = z' - f ' ( x ' ,

y'),

(B.16)

where B =- u2/u 1 # 0 on the interface. Thus 11, 12 and 13 need not be invariant in general under different choices of the Monge representation. However the general result (B.9) suggests that 12 is invariant under different choices of the Monge representation or the u-function although 11 and 13 may not be. We can check this directly using the following formulae:

,L

_

u~

Ivul '

1

o,,L = ~ - ~ (~o. - , L , ~ . ) u , . .

(B.17)

where u,~-~O~u and u~,t3=-O,Ot~u. Now, put u---~Bu in (B.17) and set u = 0 after differentiation. Here and after taking u = 0 is implied in all the final expressions. Then we find B-lO,,O~(Bu)[,=o = b,,ut3 + bt3u,~ + u,,t3 ,

(B.18)

b~=-O~(lnB),

(B.19)

with b----VlnB,

etc.

Therefore we have

( o ~ G ) l ~ o = G[b~ - G ( ~ . o)] + [a~ - G ( ~ "V)]u~/IVul.

(B.20)

T. Kawakatsu and K. Kawasaki / Binary mixture with surfactant molecules

733

Using the identities ho[bo - ha(fl, b)] = 0 and ht3[0t3 - ho(h .V)]u~ = 0, we get (B.21)

=

u--~0

u--~0

This proves the invariance of 12. On the other hand, v

× hl.u

= -h

x h VO (Bu)/BIVu [ = V × hlu - h × b ,

(B.22)

which is not invariant. Thus I, = 12 + (V × h) 2 is not invariant either.

Appendix C Here we present the details of our computational scheme for the hybrid model. In order to express the continuous field X(r), we divide the system (two-dimensional square box) into N x N cells with mesh width a. Next, we transform the partial differential equation (4.1) into a cell dynamics equation of motion as follows [27]:

X(i, j; t+ At) = X(i, j; t) + At 3 L x [ ((SH>)

8H]

(c.1)

where

((A(i, j))) =- ~[A(i + 1, j) + A(i - 1, j) + A(i, j + 1) + A(i, ] - 1)1 + l [ A ( i + 1, j + 1) + A ( i - 1, j + 1) + A(i + 1, j - 1) + A ( i - 1, j - 1)]

(C.2)

for an arbitrary field variable A. On the other hand, we employ the first order Euler scheme for the equations of motion of surfactant molecules. The method for calculating the interactions between surfactant molecules is straightforward. The only difficulty that arises here is in calculating the interaction between the field X(r) and the suffactant molecules. We allow each surfactant molecule to change its position and director orientation continuously, while the value of the field variable X is defined only on discrete lattice points (center of each cell). Without interpolating the field X between lattice points, therefore, we cannot evaluate the interaction between surfactant molecules and the field correctly. Actually, if we do not use any interpolation, the surfactant molecules feel the discreteness of ceils and move as if they were in a periodic potential field which often leads to trapping of suffactant molecules preventing diffusive motion on

734

T. Kawakatsu and K. Kawasaki / Binary mixture with surfactant molecules

the interfaces. In order to avoid such difficulties, we employ the following procedure. Suppose that we are evaluating the force F(r) on a surfactant molecule which is located at r in the cell (i, j). (1) First we evaluate the forces when the surfactant molecule is at the centers of the five cells (i, j ) , (i + 1, j ) , (i - 1, j ) , (i, j + 1) and (i, j - 1). (2) Now we assume the quadratic form for F(r) as F(r) = F(x, y) = C l x 2 + C2Y 2 + C3x + C4Y + C 5 ,

(c.3)

where the C / s are constant coefficients and we have neglected the x y term. Determining the constants in (C.3) with the use of the five values of F obtained in step (1), we can evaluate the force acting on the surfactant molecules at any position r inside the cell (i, j). Although, of course, this procedure will not give us the exact force F(r), however, we can regard (C.3) as the correct interaction potential of our model, rather than the original one described in appendix A.

Note added in proof After submitting this manuscript, we have noticed that the constant c_ in (3.7) is negative according to its definition in (A.13) and (A.14). Thus, integrating the last term of (3.7) by part, we obtain the c _ [ V - s ( r ) ] 2 term, which leads to splay instability on the microscopic level. This short-range instability seems to reflect the stability of the lamellar-like structure of surfactant molecules where directors of the surfactant molecules change on the molecular level. This difficulty m a y be avoided if we first take the strong segregation limit, where surfactant molecules form single layers at interfaces, and then go over to continuum description. We also noticed that actually the value L x = a 2 = 0.09 was used in all of the computations in section 4 due to an error in our p r o g r a m which alters the time scale by this factor.

References [1] For general reviews, see: Physics of Complex and Supermolecular Fluids, S. Safran and N. Clark, eds. (Wiley, New York, 1987); and also: Modern Amphiphilic Physics, A. Ben-Shaul, W. Gelbert and D. Roux, eds. (1990) to be published. [2] P.G. De Gennes and C. Taupin, J. Phys. Chem. 86 (1982) 2294. [3] M. Kahlweit et al., J, Colloid Interface Sci. 118 (1987) 436. [4] L. Auvray, J.P. Cotton, R. Ober and T. Taupin, Physica B 136 (1986) 281. [5] L. Auvray, J.P. Cotton, R. Ober and T. Taupin, J. Phys. (Paris) 45 (1984) 913.

T. Kawakatsu and K. Kawasaki / Binary mixture with surfactant molecules

735

[6] W. Jahn and R. Strey, J. Phys. Chem. 92 (1988) 2294. [7] W. Jahn and R. Strey, Physics of Amphiphilic Layers, J. Meunier, D. Langevin and N. Boccara, eds. (Springer, Heidelberg, 1987). [8] Y. Talmon and S. Prager, Nature 267 (1977) 333. [9] B. Widom, J. Chem. Phys. 81 (1984) 1030. [10] B. Widom, J. Chem. Phys. 84 (1986) 6943. [11] K. Chen, C. Ebner, C. Jayaprakash and R. Pandit, J. Phys. C: Solid State Phys. 20 (1987) 1361. [12] K. Chen, C. Ebner, C. Jayaprakash and R. Pandit, Phys. Rev. A 38 (1988) 6240. [13] S.A. Safran, D. Roux, M.E. Cates and D. Andelman, Phys. Rev. Lett. 57 (1986) 491. [14] D. Andelman, M.E. Cates, D. Roux and S.A. Safran, J. Chem. Phys. 87 (1987) 7229. [15] S.T. Milner, S.A. Safran, D. Andelman, M.E. Cates and D. Roux, J. Phys. (Paris) 49 (1988) 1065. [16] R.G. Larson, L.E. Scriven and H.T. Davis, J. Chem. Phys. 83 (1985) 2411. [17] R.G. Larson, J. Chem. Phys. 89 (1988) 1642. [18] K. Kawasaki and T. Kawakatsu, Physica A 164 (1990) 549. [19] T. Kawakatsu and K. Kawasaki, in: Proc. Yamada Conf. on Strongly Coupled Plasma Physics, S. Ichimaru, ed. (North-Holland, Amsterdam, 1990). [20] P.C. Hohenberg and B.I. Halperin, Rev. Mod. Phys. 49 (1977) 435. [21] K. Kawasaki, Prog. Theor. Phys. 57 (1977) 826. [22] P.G. De Gennes, The Physics of Liquid Crystals (Oxford Univ. Press, Oxford, 1974). [23] R. Evans, Adv. Phys. 28 (1979) 143. [24] M.M. Telo da Gama and J.H. Thurtell, J. Chem. Soc. Faraday Trans. II, 82 (1986) 1721. [25] W. Helfrich, Z. Naturforsch. 28C (1973) 693. [26] E. Siggia, Phys. Rev. A 20 (1979) 595. [27] Y. Oono and S. Puri, Phys. Rev. A 38 (1988) 434. [28] M.P. Allen and D.J. Tildesley, Computer Simulation of Liquids(Oxford Univ. Press, 1987). [29] W. Helfrich, Z. Naturforsch. 33A (1978) 305. [30] C.G. Vonk, J.F. Billman and E.W. Kaler, J. Chem. Phys. 88 (1988) 3970. [31] M.E. Cates, Space-Time Organization in Macromolecular Fluids, F. Tanaka, M. Doi and T. Ohta, eds. (Springer, Berlin, 1989), and references therein. [32] C. Bagdassarian, W.M. Gelbart and A. Ben-Shaul, J. Stat. Phys. 52 (1988) 1307. [33] S.R. de Groot and P. Mazur, Non-Equilibrium Thermodynamics (North-Holland, Amsterdam, 1962).