3292
J. Phys. Chem. 1988, 92, 3292-3301
A Microscopic Lattke Model for Microemulsions Thomas P. Stockfisch and John C. Wheeler* Chemistry Department B-040, University of California, San Diego, La Jolla, California 92093 (Received: July 31, 1987; In Final Form: December 2, 1987)
A microscopic statistical mechanical lattice model for microemulsions is presented whose energy parameters represent interactions among oil, water, and surfactant molecules. Solving the model in the mean-field approximation allows phase diagrams and critical loci to be calculated. Features important in microemulsions are found in the model as well. These include ordinary critical, critical double point, critical end point, and tricritical loci, as well as three-phase equilibrium. Global phase diagrams can be calculated due to the simplicity of the mean-field approach. Lamellar phases are found to be stable at low temperatures.
1. Introduction
Microemulsions are thermodynamically stable mixtures of oil, water, and surfactant, sometimes containing salt and a ‘cosurfactant” as well. They can have very low surface tensions and viscosities and at the same time combine properties of oil and water. This makes them useful in many applications, including tertiary oil recovery,’ modeling of chemical reactions that occur in biological membranes: as a possible low-emission diesel fuel,3 and, using fluorinated hydrocarbons, as a possible blood substitute! The origins of these properties are of considerable scientific interest because they involve complex phase equilibria and critical phenomena. Three-phase liquid equilibrium and tricritical points are examples. In addition, microemulsions can form bicontinuous structures,s in which the oil and water regions interpenetrate. We present in this paper a purely statistical mechanical lattice model for microemulsions, together with its mean-field solution, specific and global phase diagrams, and critical loci. The model exhibits three-phase equilibrium, critical end points, and a tricritical point. In section 2 we define the model and give its Hamiltonian in terms of occupation and spin variables. In section 3 we obtain the thermodynamic potential for the model using a mean-field approximation based on a variational principle. In section 4 we examine the degrees of freedom of the model in light of Gibbs’ phase rule and derive equations for and methods of calculation of phase diagrams. The results of our model including all figures are given in section 5 , where we present, in addition to phase diagrams, results for three-phase equilibrium, tricritical points, and critical end points. Finally, in section 6 we evaluate the strengths and weaknesses of the model and the approximation used to solve it. The reader primarily interested in results may wish to proceed directly from section 2 to section 5 . 2. The Model We propose a lattice model in which cells are assigned either an oil molecule ( 0 ) or a like volume of water molecules (w). Each cell is then of molecular dimensions. In addition to the cells themselves, each face between two neighboring cells can be empty or occupied by one surfactant molecule (s). In effect, there are two interpenetrating lattices, one containing oil and water and the other surfactant. W e call the former the “cell” lattice and the latter the “bond” lattice. E,,, E,,, and E , are nearestneighbor interaction energies between pairs of cells not separated by an intervening surfactant molecule and with occupancy water-water, oil-water, and oil-oil, respectively. E,,,, E,, and E, are the corresponding energies for cell pairs with intervening surfactant. Unlike the earlier microemulsion theories of Talmon-Prager: Jouffroy-Levinson-de Gennes,’ and W i d ~ m ,the ~,~ (1) Bansal, V. K.; Shah, D. 0. In Micellization, Solubilization, and Microemulsions;Mittal, K. L., Ed.; Plenum: New York, 1977; Vol. I, pp 87-113. (2) Letts, K. A.; Mackay, R. A. Inorg. Chem. 1975, 14, 2990, 2993. (3) Friberg, S. E.; Venable, R. L. In Encyclopedia of Emulsion Technology; Becher, P., Ed.; Marcel Dekker: New York, 1983; Vol. I, p 319. (4) Mathis, G.; Leempoel, P.; Ravey, J.; Selve, C . ; Delpuech, J. J . Am. Chem. SOC.1984, 106, 6162. ( 5 ) Scriven, L. E. In Micellizafion, Solubilization, and Microemulsions; Mittal, K. L., Ed.; Plenum: New York, 1977; Vol. 11, pp 877-900.
0022-3654/88/2092-3292$01.50/0
combinations ow, wsw, and os0 are allowed. Of course, we choose comparatively large values for the energies E,,, E,,,, and E,,,. There is an energy associated with bending of the surfactant surface. Because of the lattice, curvature of the surface is concentrated entirely at the edges between adjacent cell faces. Two surfactant-filled faces meeting at an angle have an energy E,,. This is, in effect, a nearest-neighbor repulsion on the bond lattice. To account for the possibility that the surfactant may prefer to bend toward or away from water, there is an asymmetric bending energy, Eb, defined as half the energy difference in bending toward oil instead of water. Real microemulsions often have more than three components. The oil can be polydisperse, the water phase often contains salts, and a cosurfactant is sometimes employed to enhance surfactant action. There are several motivations for keeping the number of model components small. Experimental results are often analyzed in terms of pseudotemary systems. For example, coexistence curve data are often reported in the form of triangular phase diagrams. If we want to study the effects of additional degrees of freedom, we can utilize some of the many model energy parameters. One possible viewpoint is that changing the value of E,, corresponds to varying the activity of cosurfactant. Oil and water are represented by the occupation variables vI. They can take on the values one (representing oil occupancy) and zero (representing water occupancy) and reside on the cell lattice. The variables pl, also have allowed values one and zero, referring to the presence or absence, respectively, of surfactant on the face between cells i and j . The Hamiltonian is then = & W x v l + 4CpIJ + EOWC(vl + VJ - 2vlvJ)(1 - PI/) + I
(11)
(ir)
where Aow is the chemical potential difference between oil and water, As is the chemical potential of surfactant, and C(ij)and C(ijk) are nearest-neighbor sums on the cell and bond lattice, respectively. We work with a linear transformation of the energy parameters: wo
y8(2EOw- E, - E,,)
(6) Talmon, Y.; Prager, S. J . Chem. Phys. 1978, 69, 2984. (7) Jouffroy, J.; Levinson, P.; de Gennes, P. G . J . Phys. (Les Ulis, F r . ) 1982, 43, 1241. (8) Widom, B. J . Chem. Phys. 1984, 81, 1030. (9) Widom, B. J. Chem. Phys. 1986, 84, 6943.
0 1988 American Chemical Society
The Journal of Physical Chemistry, Vol. 92, No. 11, 1988 3293
A Microscopic Lattice Model for Microemulsions The physical meanings and ranges of these parameters are discussed in turn. wo is the cost of mixing oil and water without benefit of surfactant. The adage “oil and water do not mix” implies that wo > 0. w, is the energy involved in moving surfactant from bulk oil or bulk water to an oil-water interface. Since surfactant is amphiphilic, this is an energetically favorable operation and we require that w, < 0. w, (a for asymmetric) is proportional to the energy difference between putting a surfactant molecule in bulk oil as opposed to bulk water. If w, # 0, oil and water do not enter the model on an equal footing and certain symmetries (discussed later) are broken. w1 is (roughly) the energy of exchanging bulk water for bulk oil. w j - w2 is (roughly) the energy of adding surfactant. Finally, since it costs surfaces energy to bend, we require that the quantities (Ess f E b ) be positive. Certain symmetries become transparent if we translate the model into magnetic terminology. To this end we define spin variables s.I I 2 v 1. - 1 Q1 ,1 .
I
&..V - 1
dimensions (apart from logarithmic corrections) but are known to be incorrect for the critical loci. Nevertheless, mean-field approximations are often useful as a guide to the types of phase equilibrium to be expected and serve as an essential first step in the study of new models and phenomena. We have used a standard technique using a variation principle of Gibbs, which has been generalized by Falk.” It involves a density matrix, 5:
kTNf 5 N 4
Tr (5%)
+ kT T r ($ log $)
(6)
Tr represents a trace over all spin configurations, and f is the exact Helmholtz free energy divided by Boltzmann’s constant times the absolute temperature. We write the density matrix as a product of individual spin matrices $ = n$c(si)n$b(‘Jij) i
(7)
(ij)
where the first product and the subscript c refer to the cell lattice and the second product and the subscript b refer to the bond lattice. By minimizing 4 with respect to the functional form of Pc(s)and ;b(b), we obtain a mean-field solution. If sublattice ordering produces a lower free energy, then eq 7 is the wrong choice for 5 and any result based on it is not valid. For example, if a lamellar structure has lower free energy, then we should define $ as
(3)
and also magnetic field variables
(8)
(4) where 4 and q’are the coordination numbers for the cell and bond lattice (6 and 8 for simple cubic). Terms in 7f not containing any spins can be ignored since they cannot influence phase equilibrium. If we replace occupancies by spins, recombine terms, replace interaction energies by the linear transform ( 2 ) , replace chemical potential differences by magnetic fields, and ignore spinless terms, the Hamiltonian becomes 7f=
where now there are separate products over predominantly oil regions (o), predominantly water regions (w), and the interfaces separating these (I). At low enough temperatures or large enough E,, lamellae are bound to form because of the unfavorable surfactant bending energy. Modulations in more than one direction (as occur in columnar or cubic phases), while potentially interesting, are not considered in this paper. One reason is that we would expect columnar phases a t high surfactant concentration where a simple cubic lattice does not provide a very accurate morphology. We are primarily interested in the three-phase liquid equilibrium, which occurs at low surfactant concentrations. We treat the homogeneous-based$ and lamellae-based 5 in turn. 3.1. Homogeneous Solution. In this section we use eq 7 to define the density matrix. Substituting 7f (eq 5 ) into the variation relation (eq 6), we have
N 4 = -Tr $H,Csi - Tr j3Hbcuij+ (ij)
i
1
3. Mean-Field Approximation It would be convenient if we could map the model just defined to a decorated lattice model with oil and water molecules allowed on primary lattice sites, with surfactant molecules allowed on secondary sites, and with only primary cellsecondary cell-primary cell interactions. The advantage of such a decoration would be that it could be further mapped to the three-dimensional king modello for which essentially exact results (including nonclassical critical behavior) have been obtained. Unfortunately, for the second mapping to exist it is essential that there be no secondary cell-secondary cell interactions. The term in the Hamiltonian involving E,, is such an interaction term and thus spoils the mapping. It is known that the essentially exact solution to the decoration interpretation does not have three-phase equilibrium; thus, the condition E,, # 0 is essential if we hope for the model to apply to microemulsions. In this paper we have used a mean-field approximation to solve the model. This means we will get classical critical exponents. These are expected to be correct for the tricritical point in three (10) Wheeler, J. C . Annu. Reu. Phys. Chem. 1977, 28, 411.
(ijk)
4’ + Ed;.) + -Eb 4
(ij)
(ij)
- Tr $
Note that w l , w2, and w3 are subsumed in H, and Hb. This form for 7f corresponds to a (hypothetical) magnetic system with twoand three-spin interaction terms and two magnetic fields.
4
w, Tr
UjjUjk(E,,
Tr
P C r ~ j j Sj (ij)
$Csisj(1 - uij) - w, T r $Csisj(1 + w, Tr
uV)
+
5C(si+ sj)uij + kT T r fi log p (9) (ij)
Because $ is a product of individual spin matrices, the trace and sum can be interchanged. After doing so and dividing by NkT, we have in the mean-field approximation 4 2
f = -h, T r spC(s)- -hb T r
+
&(o)
44’ -(Ess + E b Tr (s$c))(Tr U$b(‘J))’ + 16 94’4 -Eb T r (S$,) Tr ( U $ b ) - p 0 ( T r S$,(S))2(1 - T r 8
4 -Ws(Tr S$,(S))2(1 Tr o$b(a)) 2 q*a(Tr sbc(s))(Tr ufib(u))+ 4 T r $c(s) 1% $c(s) + 2 Tr i’b(0) log
U$b(U))
-
+
Here h,
H , / k T , hb
(11) Falk,
i’b(0)
(lo)
= H b / k T , the bar over energy terms rep-
H.Am. J . Phys. 1970, 38, 991.
3294
Stockfisch and Wheeler
The Journal of Physical Chemistry, Vol. 92, No. 11, 1988
resents division by kT, and, as explained above, we have dropped additional terms not involving spins. To determine the functional form of p, and Cjb, we minimize f with respect to 5, and jjb under the constraints T r p , = T r j j b = I . We have two conditions, the first of which is
f(h,,hb). However, using the easily proved relations hceff tanh hceff= m, tanh-' m,
hbefftanh
hbeff
= mb tanh-' mb
-log 2 cosh tanh-' x
- h s - qW@(Tr sjj,(s))(l - Tr &(a)) qWss(Tr S$,(S))(I T r U f i b ( U ) ) + qwaS T r U&,(U) + 44'-E$ T r (0&,)[2 T r ( U & ) ] log b c ( S ) 1 - h (11) 16
+
+
+
1 1-x2 - log -
2
4
(20)
and defining one more energy transformation
w+ = So + w s
+
w- = \Go - w s
(21)
After determining X and defining the per-site magnetizations
m,
we can at least eliminate the effective fields and solve forf, h,, and hb strictly in terms of magnetizations and energy parameters. We obtain
= Tr s;,(s)
h, =
we have
tanh-I m,
+ qm,(*-mb
- @+)
44'+ qaamb+ -Eb(2mb + mb2) 16
(22) where
%, = s
[
- h,
- qWom,(1
- mb) - qwSmc(1
+ mb) + qqamb+
We use the other minimization condition
1 I -mC2 q 1 -mb2 qWamcmb - log -+ - log (24) 2 4 4 4
+
in a similar manner to determine that
where
Now from the definitions of m, and mb (eq 12) it is straightforward to show
m, = tanh hceff mb = tanh hcff
(18)
where h,"ff (-2/q)%,/s and hCff - % b / b represent respective "effective" magnetic fields on the cell and bond lattices. Substituting the derived expressions for ;, /jb, m,, and mb into the equation for f (eq IO) gives 4
44'-
44'-
f = -hem, - -h m + -Essmb2 + -Ebm,mb(2 2 b b 16 16
+ mb) -
These three equations constitute the mean-field solution to our model in the absence of sublattice ordering. It should be emphasized that although we have f = f(mC,mb;w0,w,,wa,Ess,Eb), nevertheless f is not a Legendre transform in which m, and mb are the natural variables. The form of eq 24 notwithstanding, f is to be thought of as a function of h,, hb, and the energy parameters. 3.2. Lamellar Solution. We now investigate a lamellar solution to the mean-field approximation. We rederive the free energy expression (eq 24) using the alternative density matrix (8). Since the effect on lamellae of lattice type is qualitative and cannot be expressed just in terms of a coordination number, we assume a simple cubic lattice in what follows. We proceed as in section 3 but with the following changes. The sum over cell sites is separated into a sum over those in the predominantly oil layers and a sum over those in the predominantly water layers; the sum over bond sites is separated into a sum over oil layer bond sites, water layer bond sites, and interface bond sites; the sum over nearest-neighbor bond site interactions is separated into a sum of those in the oil region, those in the water region, those between an interface site and an oil region site, and those between an interface site and a water region site. Finally, @ is minimized over pco,,,; &, ,&, and fib,. To perform the sums, we need a number of lattice identities involving the thicknesses of the oil and water layers, represented by do and d,, respectively. The fraction of cell sites in oil layers is do/(do+ dw),the fraction of bond sites in oil layers is (do ' / 3 ) / ( d o d,), and the fraction of bond-bond interactions involving only oil region surfactant is (do- 2 / 3 ) / ( d + o d,). The corresponding fractions for water layers and regions are obtained from the fractions for oil by interchanging doand d,. The fraction of bond sites separating oil and water layers, the fraction of bond-bond interactions involving interface and oil region surfactant, and the fraction of bond-bond interactions involving interface and water region surfactant are all ( 2 / 3 ) / ( d 0+ d w ) . Except for these changes, the calculation is essentially the same as performed in section 3, so we simply state the results.
+
4 4 IiijOmc2(1- mb) - -wSm:(I 2
+ mb) + qwamcmb+ hcefftanh hceff - log 2 cosh hceff+ '(hbeff - log 2 cosh hbeff) 2 (19)
The natural variables off are h, and hb, in the sense that f is minimized at constant h, and hb. It is not possible to eliminate the magnetizations in favor of the magnetic fields to obtainf=
The Journal of Physical Chemistry, Vol. 92, No. 11, 1988 3295
A Microscopic Lattice Model for Microemulsions
TABLE 14
h, = tanh-' (m,) -
E[( d0
do - ; ) m b
+ ;mbi]
(25)
h, = tanh-' (m,,) -
%[ dW ( d , hb = tanh-l (mb)
" +do - x 4
- !j)mbw + i m b i ] (26)
( d o - :)mb W-m,:
+ imbi]+ + 2Wamc0 (27)
W-mcw2+ 2Wam,, (28) 4'-
hb = tanh-' (mbi)+ -Ess(mbo
8
+ mbw)+ W..m,mcw + Wa(mco + mcw) (29)
5 log ( 1
- 7 2 )
!(do-;)
; (*)
+ - log ( 1- 7 2 )
2
log
+
do - - m k + + dwm,,) - i h b[: - -mbi + ( + domc0tanh-I mco + dwmcwtanh-I mcw +
1
0
over all. If we settle on a particular choice of interaction energies, they can be replaced by one temperature variable, leaving three degrees of freedom. Various phase equilibria and critical loci reduce the degrees of freedom further, as summarized in Table I. In the remainder of this section we derive equations for and describe the methods of calculation of phase equilibria for these various equilibria and critical loci. The resulting phase diagrams are presented in section 5, and the reader interested primarily in results may wish to proceed there first. 4.1. Three-phase Equilibrium. Three-phase equilibrium occurs when a microemulsion is in equilibrium with both excess water and excess oil phases. Additional oil or water is rejected by the microemulsion phase. The best candidate for a bicontinuous phase is a middle phase in equilibrium with two other phases. Tertiary oil recovery is based on the ability to achieve very low surface tension between phases.' It has been observed that the conditions for three-phase equilibrium correspond to those for minimum surface tension. In this subsection we consider the symmetric case Eb = Wa = 0 and develop an exact (within the mean-field construct) solution for three-phase equilibrium in our model. We distinguish variables for different phases by adding zero, one, or two primes to them. Note that in eq 24 if the Wa and i?b terms are removed, f is an even function of m,. We therefore anticipate that we can achieve three-phase equilibrium by setting m,' = 0 for one of the phases and m, = -m/ for the other two. m,' = 0 implies that h, must be zero (eq 22). The single prime will be used for the m, = 0 phase variables. This corresponds to equal volumes of water and oil and is thus the microemulsion phase. The expression for hb(m,,mb) (eq 23) together with the phase equilibrium requirement hb = h i implies
+ tanh-'
mb - m i = 0 (31) - mbmb'
1
where we have used the identity tanh-' x - tanh-I y = tanh-'
h,(dom,
3 2 1
7 6 5 5 4 3
"The "global" column refers to the degrees of freedom for the full parameter space with all the interaction energies. For the "particular" column the degrees of freedom are those remaining after w,, w,, E,, Eb, and w, are set t o a particular value.
W-m,Z - qZ,,(mb' - mb)
+
degrees of freedom global particular
feature single phase two-phase equilibrium three-phase equilibrium ordinary critical locus critical end point locus tricritical locus
X - Y -
1 - xy
(32)
From the expressions for h,(m,,mb) and f(mc,mb)(eq 22 and 24) and the requirements h, = h,' and f = f', we get two more equations. (33) qq-m?mb
4 44'+ -q+mCZ - -E,,(mb2 2 16 1
- log 2
4. Phase Equilibrium
We have three degrees of freedom for the three components but lose one due to the requirement that po pw = 1. This requirement can be interpreted as either an incompressible-fluid or an infinite-pressure limit. There are also five degrees of freedom from the energies Wo,W,,E,, Eb. and Waseven degrees of freedom
+
(1 - m,')
- m i 2 )+
4
1 - mb2
+ -4 log =0 1-mmb/2
(34)
Equations 31, 33, and 34 can be solved for the magnetizations in terms of the energy parameters only numerically. Note, however, that all three equations are linear in W+,W-,and E,. We therefore solve for the energies in terms of the magnetizations and obtain after simplification of terms
3296 The Journal of Physical Chemistry, Vol. 92, No. 11, 1988
2m, tanh-' m,
+ q(mb +
mb - m i m ( ) tanh-I 1 - mbmb'
w-m,2
+
+ tanh-' .mb - m i
We next consider three-phase equilibrium when w, # 0 or Eb # 0. Breaking the symmetry requires satisfying six equations, obtained from the requirements h, = h,' = h,", hb = h i = h i ' , andf = f'= f". Although these are again all linear in the energy parameters, there are more equations than parameters and thus there is no exact solution for the energy parameters in terms of the densities. We can eliminate mb and m,,"using the equations for equality of hb in coexisting phases and are left with four equations in the unknowns m,, m:, m,", and m i . The original six equations are
Stockfisch and Wheeler point, but in the remainder of this section we consider only the symmetric case. The technique used to find the tricritical point is power series expansion of the equations for three-phase equilibrium, and we solve for mb,,and W's,t in terms of W'o,l and E,,,,. Since at Tl we have m,' = 0, m, = m,' = md), and mb = m i = mb/' we choose the expansion variables m: and (mb- mi),both of which vanish at TP If we used the implicit equations ( 3 1 ) , (33), and (34)we would have to go to dishearteningly high order before we found a nonvanishing term. However, having solved for E,,, W+,and w- in terms of the magnetizations, we expand eq 35-37. Since the most strongly vanishing denominator in these equations is m:(mb - m i ) , we need keep only the next higher term than this. As a preliminary, we calculate the expansions for tanh-' [ ( x - y ) / ( l - xy)] and In [ ( l - x2)/(1 - y')] in the variable (x - y ) and obtain tanh-'
-X - Y
1 - xy
x-y
1-x2
x2 + y3 (x-yI2 2x + ( x - yl3+ O(x - y ) 4 ( 3 9 ) (1-x) ( 1 - x2)3
1 -x' -2x In -= ( x - y ) 1 -y2 1 -x2
8x3 (1 - x2)2
]+
O(x - y)4
We anticipate that the ratio 44'q(m, - m,')w+ - qm{wa - -Eb(2m(
+ m i 2 ) (38a)
16
4 O = 7w+(mc2 - m12)- *-(m,2mb - mc/2m 0 means that the surfactant is more soluble in water. Bancroft’s rule states that the external subphase is the one in which the surfactant is more soluble. For w, > 0, then, water is external. The model obeys this rule as can be determined by considering the dotted line in Figure 2. On the left axis we start out with surfactant in water. In a physical system, as oil is added we get micelles swollen with oil until we reach the two-phase locus at which p i n t additional oil is rejected to a bulk oil phase. Of course, the mean-field treatment of our model produces no micelles, because any such microscopic object is averaged over. In all of the phase diagrams discussed thus far there is no stable lamellar phase. This has been checked by comparingf(hc,hb) at the same h, and h b and also by comparing the appropriate Legendre transform off a(m,,mb) = f - hcmc - hbmb (73) a t the same m, and mb. If the temperature is lowered, lamellae start to form in the region above the three-phase triangle, as shown in Figure 3. The method for calculating the lamellar phase densities was presented in section 4.5. The triangle in this figure is a region of coexistence of three homogeneous phases with three adjacent two-phase regions. Above the triangle is a region of equilibrium between a homogeneous phase and a lamellar phase. The thin inner band is a one-phase lamellar region. Tie lines are represented by dashed lines. Note that the lamellar phase does Recalling that our not appear for Pb greater than roughly model requires intervening oil or water between surfactant layers (the cell sites cannot be empty), this result is not surprising. Clearly, lamellar phases with high sufactant concentrations ( p b > are not described well by this model. To plot a curve of three-phase equilibrium vs temperature at a particular (and constant) set of energies E,,, w+,w-,w,,and Eb, we must resort to numerical solution. If we choose w, = Eb = 0, we solve eq 37 for mb. That leaves the two equations (36) and (35) and unknowns m, and m i . Curves of the three cell densities p,, p l , and p/ meet at a tricritical point (see Figure 4a,b). To obtain the same plot for w, # 0, we solve eq 38a for mb and m< and satisfy the remaining four equations by numerical adjustment of m,, m l , m?, and m i . Figure 4c,d shows that breaking the symmetry results in the tricritical point being replaced by a critical endpoint. By adjusting E b to balance Wa,we can obtain an asymmetric tricritical point, as shown in Figure 4e,f. Note that these plots are nearly the same as Figure 4a,b, suggesting that Wa and ,!?b have a similar effect on the model. Using eq 52, we can determine the global two-dimensional tricritical locus for the symmetric case (Figure 5). Note that increasing w,/wo at constant E,,/wo moves the tricritical point closer to the oil-water critical point, and increasing E,,/wo at constant w,/wo moves the tricritical point away. Figure 4b shows that is not much different than Pb for the bulk phases, so that if we want the surfactant concentration to be small in these phases,
3300 The Journal of Physical Chemistry, Vol. 92, No. 1 1 , 1988
O8
Stockfisch and Wheeler
t "
r
-2-2
lor-pb
W0IWj
-'l
'
0
'
1
'
";
waJ*c
Figure 6. Global critical end point for the asymmetric case w, # 0, but E b = 0. Curves run at constant ws/wo with values, from upper left to lower left, -0.9, -1, -2, -3, -4,-5, -6, -7, -8, -9, -10. (a) hb vs w,/wo. (b) hc vs w h o .
08
corresponding to the upper part of Figure Sb. Satisfying both these requirements and having E,/w, reasonably small limit the choice of energy parameters to a rather small portion of the global parameter space. The global critical end point locus for the asymmetric case w, # 0, Eb = 0 is shown in Figure 6a,b. These were determined by numerically solving the usual two equations for two-phase equilibrium, plus the one equation from the scheme described at the end of section 4.4. In Figure 6a, the left half of a curve of constant w,/wo meets tangentially with the right half at w,/wo = 0. This is because w, is a symmetry breaking parameter. In Figure 6b, not only does the left half meet the right half tangentially, but the upper half meets the lower half tangentially at h, = 0. This is because h, as well is a symmetry breaking field. These features of both figures are to be expected when it is obEb by -Ebr and si by served that replacing w, by -wa, H,by -Hc, -s, results in no change in the Hamiltonian (eq 5 ) .
---orl
6. Discussion
pc Figure 4. Three-phase equilibrium vs T / T,, = 0.04,Go = 0.111, #s = -0.413. Tow,is the ordinary oil-water critical temperature, discussed in section 4.2. (a) w, = Eb = 0; T / T o w cvs pc. (b) w, = Eb = 0; T/ToWcvs pb. (C) W , = -0.00433,E b = 0; T / T o w cvs pc. (d) W , = -0.00433, Eb = 0;T / Tow, VS Pb. (e) W , = -0.00433, Eb = 0.04;T / TO,, b 0.04;TITowcVS pb. VS pc. (f) W , = -0.00433, E
C16-
'iC
12
i4
!6
2q WoJTi
18
20
"IO
!2
14
16
18
2C
2q woJTt
Figure 5. Global tricritical locus for the symmetric case, w, = Eb = 0. Curves run at constant Ess/wowith values, from top to bottom, 0.0001, 0.03, 0.1,0.3,0.5, 0.7,1.5, 2, and 3. (a) pb tricritical vs 2q(w0/T,). (b) wdwo vs 2q(wo/T,).
then we need Pb,t to be small, corresponding to the lower part of Figure Sa. Phase diagrams with large two-phase lobes, as in Figure lb,c, are characterized by large ws/wo. If we want to avoid such diagrams, we need to choose ws/woto be fairly small in magnitude,
It is encouraging that the homogeneous phases are more stable than lamellar for so much of the phase diagram and such a large temperature range. However, it is possible we have not treated lamellae with complete fairness. In our calculation we characterized the cell magnetization of each lamella by just one variable. This is a reasonable approximation if the thickness is large, but we found that when the homogeneous phase is less stable than a lamellar phase, the layer thickness of lowest free energy is always one. We are currently investigating modulated phases in which each cell layer has its own magnetization. Real systems exhibit three-phase equilibrium with good microemulsion properties over a very narrow temperature range, typically 10 O C . They are bounded below by a lower critical end point in which the microemulsion phase becomes identical with the bulk water phase and bounded above by an upper critical end point in which the microemulsion phase joins the oil phase. This occurs because at low temperature low-entropy hydrogen bonds form between the surfactant and water, whereas a t high temperature the positive entropy of mixing causes the surfactant and oil to be miscible. In our model, which does not take the entropy of bonds into account, a lower critical end point or tricritical point is probably impossible to achieve. This has been proved rigorously for the symmetric case, since there is only one solution for the tricritical point. We are currently devising an extension to remedy this deficiency while preserving the model's microscopic nature. Although the mean-field approximation has been used successfully on many models and it is important to know how well it does even if a more accurate method is worked out, there are reasons to have reservations about its applicability to our model. The first has to do with microscopic structure, especially in the microemulsion phase. Any nonregular (e.g., nonlamellar) structure which separates oil from water within a single phase does not exist in mean field, causing many unfavorable oil-water contacts. For example, in a phase with pc = and pb = 0.1 it is expected that
The Journal of Physical Chemistry, Vol. 92, No. 11, 1988 3301
A Microscopic Lattice Model for Microemulsions 2pc ( 1 - pc)(l - pb), or 4556, of its cell-bond-cell interactions are oil-no surfactant-water. A second reservation concerns the energy parameter E,. We want, this quantity to be interpreted physically as a surfactant bending energy, yet if we rederived the homogeneous solution (section 3 . 1 ) with E,, as a next-nearest-neighbor (or, indeed, any surfactant-surfactant) interaction, the same equations would emerge, apart from an unimportant change in the vaue of 4’. These limitations could be remedied by studying this model in the quasi-chemical approximation, by Monte Carlo simulation, or by real-space renormalization group methods. We hope to pursue these lines of investigation in the future. Acknowledgment. This research was supported in part by the National Science Foundation through NSF Grants CHE 81-19247 and C H E 86-15784 and in part by Proctor and Gamble through a U E R P grant. Appendix I. Proof of the Existence and Uniqueness of the Symmetric Tricritical Point The zeros of polynomial 47 determine the tricritical point. We will show that there is exactly one real root in the region -1 5 mb I1 , given the physical requirements E,,, wo > 0. The polynomial can be written as
P ( x ) E x3
+ + 6 1 +~ 60
(74)
in the coefficients of Q. Therefore, by Descartes’ rule, Q ( x ) has exactly one real root greater than zero and P ( x ) has exactly one real root greater than 1 . Similarly, R(xC0) corresponds to P(x 0 in Q ( x ) corresponds to x > 1 in P ( x ) . Substituting x + 1 for x in P ( x ) and gathering powers of x , we have Q ( x ) = x3
+ 4x2 + ( 5 + 6 1 ) ~+ (2 + bl + 6 0 )
From the definitions of
2
(80)
and 61 the constant term of Q ( x ) is qfEm 4 60 = 2 - 2 49 , s
60
+ 61 +
+
Since E , > 0, the constant term is less than zero. Regardless of the sign of the quantity ( 5 + 6l), there is exactly one sign change
P6
P2
-
P4
2
= 0 (84)
