Langmuir 1995,11,2451-2458
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Phase Equilibria of Balanced Microemulsions John Daicic,” Ulf Olsson, and Hdkan Wennerstrom Division of Physical Chemistry 1, Chemical Centre, University of Lund, P.O. Box 124, S-22100 Lund, Sweden Received January 20, 1995. In Final Form: April 11, 1995@ The flexible surface model is a powerful framework in which to describe the thermodynamics of fluid membrane phases. In the case of balanced microemulsions (and their bilayer analogue, the L3 phase), a general form for the free energy density has been proposed. In this paper, we consider the phase behavior of balanced microemulsions and show that this model is successful in capturing the important features of experimentally determined phase diagrams of systems which exhibit this remarkable phase. 1. Introduction The balanced microemulsion is one of the more exotic phases that appear in surfactant-water-oil systems.l This bicontinuous phase consists of a disordered, connected surfactant monolayer separating the two solvent domains. I t is “ba1anced”in the sense that this monolayer sheet has no preference in curving toward either ofthe two solvents; in other words, it has zero spontaneous curvature. In ternary systems with a nonionic surfactant, for example, temperature may be used to tune the spontaneous curvature to achieve the balanced condition.2 There is a close analogy with the anomalous isotropic L3 (or sponge) phase, the difference being that the sponge consists of a bilayer separating two domains of the same solvent. Over the past decade or so, these two remarkable phases have deservedly received significant attention in the literature. At first glance, a thermodynamic model for them, which in principle requires the evaluation of the partition function of the system, seems a forlorn hope, given their complex topology. Moreover, the observed phase behavior is uncharacteristic of isotropic liquids. Both phases appear in narrow regions of the phase diagram, and have a finite swelling. A powerful tool in reconciling the above geometrical description of these phases with their thermodynamic behavior is the flexible surface model,3 where the monolayer or bilayer membrane is considered as a geometrical surface, the local energy of which is determined by the curvature a t any given point. In principle, this local curvature energy is then integrated over the surface, giving the curvature free energy of the system. In this paper, we will use the flexible surface model as the basis for describing the thermodynamics of the balanced microemulsion. As mooted above, this subject has a rich history in the literature. Talmon and Prager4!5 were the first to provide a thermodynamic model for this bicontinuous phase. Their phenomenological treatment was based upon a subdivision of space into Voronoi polyhedra, each filled randomly with water or oil with a probability proportional to the volume fraction of each component. The surfactant was constrained to lie a t the oil-water interface with constant area per molecule. The Talmon-Prager model was based essentially on entropic effects only and energy considerations enter only through constraints. The entropy was calculated by mixing the oil and water cells randomly while the interfacial conAbstract published in Advance A C S Abstracts, J u n e 15,1995. (1)Surfactants in Solution; Mittal, K., Lindman, B., Eds.; Plenum: New York, 1987. (2)Olsson, U.; Wennerstrbm, H.Adu. Colloid Interface Sci. 1994,49, 113. (3)Helfrich, W. 2.Naturforsch. 1973,28c, 693. (4)Talmon, Y.;Prager, S. J. Chem. Phys. 1978, 69, 2984. (5) Talmon, Y.;Prager, S. J. Chem. Phys. 1982, 76, 1535. @
0743-7463/95/2411-2451$09.00/0
straint was maintained. This led to them predicting a rich phase behavior for the system, including the Winsor I11equilibrium in which distinct phases of microemulsion and both solvents coexist. This is a n essential feature of the characteristic phase diagram which should be captured by a successful model for the phase. However, the way in which the bending energy was evaluated remained, as Talmon and Prager recognized, somewhat artificial. It did not, for example, take into account the local curvature properties of the film, instead localizing curvature a t randomly assigned edges. The next advance was made by de Gennes and Taupin6 and Jouffroy, Levinson, and de Gennes (JLdeG17 They replaced the random polyhedra ofTalmon and Prager with a cubic lattice, again allowing the lattice sites to be filled randomly with water and oil. The lattice size was chosen as thepersistence length, which as originally suggested in ref 6 is a measure of order in membrane systems: large values favor liquid crystal phases, smaller values disordered phases such as bicontinuous microemulsions. The fact that depends exponentially on the bending modulus of the film, K, underscored the importance of characterizing the bending energy properly. JLdeG also removed the interfacial area-per-molecule constraint and showed that phase equilibria should occur a t near-vanishing values of the interfacial tension. The entropy was calculated by random mixing of the cubes. An important difference between the results of JLdeG and of Talmon and Prager was that within the constraints of their model JLdeG were unable to find the Winsor I11 equilibrium. They concluded that the Winsor I11 equilibrium may be found if they incorporated two possible effects: that the solvent domains were to be packed more densely, bringing dispersion forces into play, and that the solubility ofthe surfactant in solvent be considered, this leading to local binding of the film interface near the cloud point. In his contributions, Wid0m*3~departed from a basic assumption of the previous work: rather than assuming the interfacial film to be incompressible (or very nearly so), he instead treated the interface as a n ideal gas of surfactant in two dimensions.8 This is significant in the thermodynamic description of the phase equilibria as the surfactant appears as a proper component rather than through an area constraint. Widom also employed a lattice model, with a variable lattice size, 6, which was minimized for each composition in calculating the free energy. The Widom model successfully captured the Winsor I11 equilibrium. In a further d e ~ e l o p m e n the , ~ also presented a
eK,
eK
(6)de Gennes, P.G.; Taupin, C. J. Phys. Chem. 1982, 86, 2294. (7) Jouffroy, J.; Levinson, P.; de Gennes, P. G. J.Phys. Fr. 1982,43, 1241. (8) Widom, B. J. Chem. Phys. 1984,82, 1030. (9) Widom, B.J. Chem. Phys. 1986, 84, 6943.
0 1995 American Chemical Society
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2452 Langmuir, Vol. 11, No. 7, 1995
model equivalent to the anisotropic next-nearest-neighbor Ising model. Again, the curvature was localized at bends and corners of the lattice units. Ising model treatments of microemulsion systems indeed have an extensive attention in the literature, and a review may be found in the volume by Gompper and Schick.lo Under certain choices of the parameters, such models do recover the Winsor I11 equilibrium. The work of Andelman, Cates, ROUX,and Safran (ACRS)11J2followed the tradition by employing a cubic lattice. Employing the constraint that all ofthe available surfactant lies a t the interface, the lattice size 5 became a variable parameter which was determined by allowing the cells to mix randomly. Curvature was calculated by associating the bending energy of a cube of water or oil with that of a sphere with the same diameter as the lattice size for a given composition. They also incorporated an effect which a t the present time is the subject of intense debate in the literature: that free energy contains a contribution from renormalization of the bending modulus due to thermal undulation effects, so that on a change of length scale E ~ ( 6=) ~ ~ - t ln([/Z)l [ l (1) where K O is the unrenormalized “bare” modulus and 1 is a molecular length. With this renormalization bringing a significant contribution to the curvature free energy, they were able to find the three-phase equilibrium. In fact ACRS claimedll that it was crucial to the apparent success of their model. It determined the minimum amount of surfactant required for a stable microemulsion phase (i.e. the efficiency of the surfactant) and that this quantity scaled a s the inverse of the persistence length, and secondly it provided the stabilizing mechanism for the microemulsion phase relative to lamellae. Later, the work of Golubovic and Lubensky13generalized the ACRS model by including steric entropy so that ordered and disordered phases could be described in a unified way. Another approach, originating with the work of Griffiths and W i d ~ m , ~ and ~ Jlater ~ Kleinert,16 employed a phenomenological Landau free energy expanded to the sixth order in the order parameter. It predicted the Winsor I11 equilibrium, and also tricritical behavior. Gompper and co-workers (ref 17 and references therein) have also couched a description of disordered phases using a Ginzburg-Landau approach. Recently,18-22an interesting approach based upon the theory of random interfaces has proved useful in determining the phase behavior,18 predicting the structure factor and calculating real-space images of disordered bicontinuous phases.21 The entropy is calculated in a manner altogether different than in the work described above: it is the entropy of a band limited random field.21 The curvature contribution is determined from the ensemble average of the local curvatures. This approach
has also provided some new insight into the stability regions of microemulsions as a function of the bending moduli.22 We now turn to the current work. In a recent article,23 two of us argued that within the flexible surface model a general form for the free energy of both balanced microemulsions and the sponge phase can be determined, and here we briefly restate the discussion therein. The first-order behavior of the free energy density, g, can be obtained from a scaling argument first developed by Porte et aLZ4that in short states that dual configurations of the system, that is, configurations differing only by a length scale, enter the partition function with the same statistical weight, so that g = a(Q.$z)3 (2) where QS is the surfactant volume fraction. This is not sufficient to describe the characteristically observed sequence of phase equilibria as a function of composition, as the free energy is monotonic. Wennerstrom and Olsson argued in ref 23 that the right hand side of eq 2 is in fact the leading-order term in a n expansion for the free energy density of the system, as it is obtained from a truncation of the local curvature energy a t harmonic order: g, = 2 K ( H - H o ) ~ EK (3)
(10) Gompper, G.; Schick, M. In Phase Transitions and Critical Phenomena; Domb, C., Lebowitz, J. L., Eds.; Academic Press: London, 1994;Vol. 16. (lUAndelman, D.; Cates, M. E.; Roux, D.; Safran, S. A. J . Chem. Phys. 1987,87, 7229. (12) Cates, M. E.; Andelman, D.; Safran, S. A,; R o n , D. Langmuir 1988,4, 802. (13)Golubovic, L.; Lubensky, T. C. Phys. Rev. A 1990,41, 4343. (14)Griffiths, R.B. J . Chem. Phys. 1974, 60, 195. (15) GriEths, R. B.; Widom, B. Phys. Rev. A 1973, 8, 2173. (16)Kleinert, H. J . Chem. Phys. 1986,84, 964. (17)Gompper, G.; Goos, J. Phys. Rev. E 1994,50, 1325. (18)Huse, D.A.; Leibler, S. J . Phys. France 1988, 49, 605. (19) Teubner, M. Europhys. Lett. 1991, 14, 403. (20) Marcelja, S. J . Phys. Chem. 1990, 94, 7259. (21) Pieruschka, P.;Marcelja, S. J . Phys. II Fr. 1992,2, 235. (22)Pieruschka, P.; Safran, S. A. Europhys. Lett. 1993, 22, 625.
2. Phase Equilibria As mentioned above, the free energy density of the balanced microemulsion generally contains contributions from both the curvature energy and entropy density of the system. To the present, the calculation of these quantities in terms of the physical parameters of the system has proved to be a difficult task. In a recent article,2swe presented a model for the free energy density
+
where H is the mean and K is the Gaussian curvature,Ho is the spontaneous mean curvature, and K and ii. are respectively the bending rigidity and saddle-splay constant of the monolayer. Bending-constant renormalization is not invoked, following the arguments proposed in ref 23. Higher order contributions to this local curvature energy will modify the ideal scaling of eq 2, so that to the next order g = a(@s/z)3 b(Qs/z)5 (4)
+
The observed phase behavior of balanced microemulsions can be understood if the sign of a is negative. The reason for this condition derives from the competing contributions of curvature and entropy to the free energy density: as we will discuss later in the paper, the entropy density may under certain conditions dominate the curvature free energy density, hence driving a to negative values. Clearly, this means that the microemulsion is unstable a t low concentrations, inducing a phase separation with solvent. At higher concentrations, a positive value of b stabilizes the microemulsion. Thus we see a straightforward mechanism to describe the phase behavior of systems which exhibit this phase. The aim of the current paper is to show that the predictions of this very general outline are fulfilled when the phase equilibria of the balanced microemulsion with its commonly observed neighbors (dilute solutions of surfactant in water and oil and, at higher concentrations, the lamellar phase) are calculated for a model ternary system.
(23)Wennerstrom, H.; Olsson, U. Langmuir 1993, 9, 365. (24) Porte, G.; Appell, J.; Bassereau, P.; Marignan, L. J . Phys. Fr. 1989, 50, 1335. (25)Daicic, J.; Olsson, U.; Wennerstrom, H.; Jerke, G.; Schurtenberger, P. J . Phys. II Fr. 1995, 5, 199.
Phase Equilibria of Balanced Microemulsions
Langmuir, Vol. 11, No. 7, 1995 2453
of the sponge phase wherein we considered the dominating role of the mean curvature of the constituent monolayers to the bilayer free energy. In that case, the bilayer symmetry allowed us to obtain the free energy density in terms of the bending modulus and spontaneous curvature of the monolayers. Anegative value for this latter quantity means that the bilayer saddle-splay constant is positive, and this drives the coefficient of the leading-order term in the free energy (identical to that of eq 4) to negative values. In the present microemulsion case, the bilayer symmetry is absent, and it is clear that the monolayer spontaneous mean curvature, being zero, cannot cause the negative value of the coefficient a of the leading term in the free energy density. This is a n important difference between balanced microemulsions and sponge phases. However, some analysis of the specific contributions to the microemulsion free energy density from the mean and Gaussian curvatures and entropy can be made, and we leave these topics for the discussion later in this paper. Again, our aim is to show that the generic scaling form of eq 4 provides a means of describing the behavior of the phase. Firstly, eq 4 must be further generalized to allow departures from equal quantities of water and oil. We introduce @ = QW 112 mS a s a second concentration variable, where QW is the volume fraction of water. The symmetric case for which eq 4 was derived corresponds to @ = 112. The Porte scaling argument, although originally made for @ = 112, is equally valid a t other constant values of @ provided that the restriction of a single continuous monolayer is fulfilled. We can thus write eq 2 as g = u(@)(@s/z)3 (5) Now a t @ = 1/2 we have a symmetric situation so that when expanded to the second order we obtain
+
g=
1
an
+ a2(0 - -
(@s/Z)3
+ bo(Qs/O5
(6)
where we have neglected the @-dependenceof the fifthorder term. How may the values of the expansion coefficients ao, a2, and bo in eq 6 be estimated? One approach is to parameterfit them to experimental values. In a recent article, Kabalnovet described how they were able to measure the chemical potentials of all components along the microemulsion-oil phase boundary in the ternary system ClzE5-water-decane using a n osmotic stress technique, and we shall use their data to estimate the expansion coefficients appropriate to that particular system. The reduced chemical potentials Ai = pilv,, where v1 is the molecular volume of component i, derived from eq 6 are
+
3ao@,2Il3 5bo@,4/15- 4b0(@,/l)5( 7 ~ ) We can obtain estimates for ao, a2, and bo by appropriately (26) Kabalnov, A.; Olsson, U.; Wennerstrom, H.Langmuir 1994,10, 2159.
fitting the experimental data shown in Figures 6-8 of ref 26 to eqs 7a-c. We thus obtainao = -0.68k~T,pz= 2 0 k ~ T , and bd12 = 7 7 k ~ Twhere , we have used 1 = 15A21s20for the ratio a$v, 1 (where a, is the area per molecule in the monolayer) for C12E5. The fact that the coefficient of the leading-order term, ao, is a t least 1 order of magnitude smaller than the higher-order coefficients may be the cause of some concern regarding the validity of eq 6 as an expansion in volume fraction for the free energy density. However, it must be remembered that there are both curvature energy and entropic contributions to this leading-order term, and hence a0 is in reality the difference of two large quantities. The thermodynamic condition for an equilibrium with pure solvent is that the solvent chemical potential in the microemulsion phase vanishes. Using eqs 7a and 7b with this condition gives immediately the parametric equations for the phase boundaries with pure water (+) and oil (-):
In this seemingly innocent equation, there is a surprising amount of information. The two phase boundaries will meet when @ = 112,and this defines the limit ofmaximum swelling of the microemulsion. Further dilution takes us, by the Gibbs phase rule, into the three-phase triangle of the Winsor I11 equilibrium where the microemulsion expels solvent and there is a phase separation. So from the outset it is clear that the model predicts these important features of the phase behavior. All that remains to obtain a full phase diagram is the calculation of the phase equilibrium with lamellae, which should be favored over the microemulsion phase at some point as one concentrates the surfactant. For the model free energy density for the lamellar phase we make the assumption that the undulation energy, as originally modeled by H e l f r i ~ hmakes , ~ ~ the dominating contribution, so
where we have, like others,ll heuristically generalized the original form of Helfrich to account for departures from equal quantities of water and oil. The chemical potentials of each component can then be calculated in a similar fashion as for the microemulsion case, and the phase equilibrium can be determined by equating the chemical potential of each component a t different compositions in each phase. This involves the use of some uncomplicated numerics. We have one parameter that is as yet unspecified, that being the bending modulus K which appears in the lamellar free energy density. In Figure 1, we show the calculated phase diagram when . the lamellar phase is the preferred we set K = ~ B TClearly, phase over a very wide range of concentrations, while the microemulsion phase shows its characteristic features of narrowness both as a function of surfactant concentration and the water-to-oil ratio. As discussed above, the threephase triangle is a natural consequence of the negativity of ao. We show an enlargement of the phase diagram in the vicinity of the single-phase microemulsion region in Figure 2. The points P, Q, R, andxcharacterize the position and width of the single-phase region, and the width of the two-phase region with lamellae along the line of wateroil symmetry. The trajectory P-Xis that where Kabalnov (27) Olsson, U.; Wiirz, U.; Strey, R. J.Phys. Chem. 1993,97,4535. (28)Fukuda, K.;Olsson, U.; Wiirz, U. Langmuir 1994, 10,3222. (29) Helfrich, W. 2.Naturforsch. 1978, 33a, 305.
2454 Langmuir, Vol. 11, No. 7, 1995
Daicic et al.
H,O
n-octane
50
‘0
Figure 3. Experimental phase diagram of the ternary system CloE5-water-octane by Strey,reproducedwith permission from ref 30.
-w
0
Figure 1. Calculated phase diagram for K = kBT, showingthe equilibriawith pure solventand the lamellarphase. The Winsor I11 three-phasetriangle is shaded. Tie lines are also indicated.
Table 1. Values for the Location of the Points P-X, as Designated in Figure 2, Both from the Model for Different Values of K (first four rows) and as Experimentally Determined by Kabalnov et aZ.* for the CI& -Water-Decane System
La ~~~~~~~~
R
La
1 2 5
8~ U E
10
experimental values
?
,‘-
~
_.
. +&h*.-
.
I
Figure2. Enlargement of Figure 1 in the microemulsionregion. PointsP,Q,R, andxcharacterizethe position and width of the
microemulsion“island”. Their values for various choicesof the bending modulus are given in Table 1.
et aZ.26made their osmotic stress measurements. We have calculated the phase diagram for values of K of 1,2,5, and 10 times kBT and given the positions of these points in Table 1,along with the experimentally observed value of point X. Strictly speaking, the expansion coefficients in the microemulsion free energy density are functions of the bending moduli, and in a complete model they should also be changing as we alter the value of the bending rigidity in the lamellar free energy density. However, in a first approximation we keep them fixed, and then compare the position of the pointxwith the experimentally observed value, noting that we must always use the observed value for the limit of maximum swelling, point P, in order to be consistent with our fitting procedure for the microemulsion free energy density expansion coefficients. Referring to Table 1, a value of the bending modulus between 2 k ~ Tand 5 k ~ T would give a match to the experimental value for the position of X. It is now appropriate to comparethe qualitative features of our calculated full phase diagram with an experimentally determined one. Unfortunately, a full diagram is not available for the CuE5-water-decane system. We refer then to a system studied by Strey,3Othat being C1&5water-octane, the phase diagram of which is reproduced (30) Strey, R.Ber. Bunsenges. Phys. Chem.1993,97, 742.
~~
0.066 0.066 0.066 0.066 0.066
0.100 0.090 0.085 0.082
0.154 0.158 0.177 0.202
[0.44,0.0931 [0.45,0.0871 [0.46,0.0821 [0.47,0.0801 [0.46,0.0841
in Figure 3. The general features of the microemulsion “island”,two-phase regions of microemulsion-solvent and microemulsion-lamellae, and the three-phase Winsor I11 triangle of microemulsion-water-oil are nicely featured in both our calculated phase diagram of Figure 1 and the experimental diagram of Figure 3. We do, however, note that some features are not captured by our model diagram. For example, Figure 3 shows three-phase triangles of microemulsion-lamellae-solvent, bounded by two-phase regions of lamellae and solvent. In our model, the lamellar phase swells indefinitely. This clearly unrealistic feature has the consequence that the solvent-lamellae two-phase coexistence is pushed into the corner and the concomitant three-phase triangle degenerates into a line. There are, furthermore, additional phases a t higher surfactant concentrations in the experimental system which are not treated in the model.
3. Discussion The good agreement between the calculated phase diagram in Figure 1 and the experimental diagram of Figure 3 shows that the model free energy of eq 6 is adequate in describing the free energy of the microemulsion. However, we have made use of experimentally determined chemical potentials, so it becomes a valid question as to what new insight is obtained. [Aspointed out in ref 26, measuring chemicalpotentials over a range 0.06
