J . Phys. Chem. 1989, 93, 4243-4253
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Isotropic Bicontinuous Solutions in Surfactant-Solvent Systems: The L, Phase David Anderson,* HBkan Wennerstrom, and Ulf Olsson Department of Physical Chemistry 1. Chemical Center, University of Lund, P.O.B. 124, S-221 00 Lund, Sweden (Received: August 29, 1988; In Final Form: December 20, 1988)
The structure of the isotropic L3phase observed in many surfactant-water or surfactant-water-oil systems is analyzed. It is pointed out that the L3 phase generally appears in equilibrium with a dilute solvent phase on one hand and a lamellar liquid crystalline phase on the other. Irrespective of the detailed chemical nature of the system, the one-phase region is remarkably narrow in one direction, indicating that the thermodynamic degrees of freedom are effectively reduced by one due to an internal constraint in the phase. In accordance with previous work it is argued that the basic structural unit in the L3phase is a surfactant bilayer. Furthermore, we conclude that the L, phase appears when there is a spontaneous mean curvature toward the solvent at the polar/apolar interface. It is shown that, for a system which has such a curvature toward the solvent, the surface formed by the bilayer midplane has a negative average Gaussian curvature ( K ) . By virtue of the Gauss-Bonnet theorem the bilayer under such circumstances has a multiply connected structure. The conclusion is then that, under conditions when there is a spontaneous mean curvature toward the solvent, it is possible to reach a low free energy state by forming multiply connected bilayer structures, as in many cubic phases, rather than planar bilayers. When interbilayer forces are weak, the structure tends to be disordered, leading to an isotropic solution (L,) rather than an ordered cubic structure. To minimize local variations in curvature at the polar/apolar interface, we demonstrate that the midplane surface should be close to a minimal surface. We then show that a certain dimensionless group associated with a given periodic minimal surface has approximately the same value for all of the well-known isotropic minimal surfaces. Assuming a minimal midplane surface, we can then show that, for a given thickness, a bilayer structure with a prescribed area-averaged mean curvature can only exist at a single volume fraction. This explains the internal constraint in the L3 phase, which is manifested in the narrow character of the L3 phase. Applying the equations that express this constraint, and using results from a theory due to Cantor to account for the effect of water/head-group interactions on water penetration, we present fits of these narrow phase-existence regions to the theory and rationalize the temperature dependence of the L, phases in a variety of nonionic surfactant systems. For a microemulsion system the analysis shows that the spontaneous monolayer curvature’increases strongly on the addition of hydrocarbon. The emerging picture of the L, phase is that the solution structure is characterized by a highly connected bilayer, extending in three dimensions, thus appearing bicontinuous in, e.g., NMR self-diffusion experiments, and having an average mean curvature at the polar/apolar interface toward the solvent. The basic driving force forming an L3 rather than a lamellar phase is thus not an entropy increase associated with disorder, as previously suggested, but rather the opportunity to obtain an optimal curvature of the surfactant monolayer.
Introduction Surfactant-water-oil systems show an amazingly rich phase behavior, which is related to the fact that there are a large number of ways to divide space into polar and apolar regions with a given surface-to-volume ratio. Adjacent polar and apolar regions are separated by a film rich in oriented surfactant molecules. The (oil surfactant chains)/(water head groups) volume ratio determines the size of the two subvolumes, while the detailed, chemical nature of the surfactant layer determines the area per polar group and the spontaneous (or “preferred”) curvature of the surfactant monolayer. The optimal aggregate structure in a particular case is determined to a large extent by general physical characteristics, and systems that are chemically very different can show analogous phase behavior. O n e example of a phase that shows the same characteristic behavior irrespective of the chemical details is the so-called L3 phase (sometimes called the “anomalous phase”). This phase has been observed in a number of binary nonionic surfactant-water systems;’-5 a representative phase diagram is shown in Figure 1. The same type of phase is found in some ionic surfactant-water systems in the presence of ~ a l t . ~ An - ~ example from the early
+
+
( I ) Harusawa, F.; Nakamura, S.; Mitsui, T. Colloid Polym. Sci. 1979, 252, 613. (2) Lang, J . C.; Morgan, R. D . J. Chem. Phys. 1980, 73, 5849. (3) Bostock, T . A.; Boyle, M . H.; McDonald, M . P.; Wood, R. M . J. Colloid Interface Sci. 1980, 73, 368. (4) Mitchell, D. J.; Tiddy, G. J . T.; Waring, L.; Bostock, T.; McDonald, M . P. J. Chem. SOC.,Faraday Trans. I 1983, 79, 975. (5) Persson, P. K. T.; Stenius, P. J. Colloid Interface Sci. 1984, 102, 527. (6) Fontell, K. In Colloidal Dispersions and Micellar Behauiour; ACS Symposium Series 9; American Chemical Society: Washington, DC, 1975; p-270. (7) Bellocq, A. M.; Bourbon, D.; Lemanceau, B. J. Colloid Interface Sci. 1981, 79, 419.
0022-3654/89/2093-4243$01.50/0
studies of,Fontel16 on the AOT-NaC1-H20 system is shown in Figure 2. In oil-watersurfactant systems L3 phases can be found that are either rich in water or rich in An example is shown in Figure 3. The same type of phase has been identified also for dipolarI4 and z w i t t e r i o n i ~ lsurfactants ~ and with a triglyceride as The L3 phase is an isotropic solution that is generally in equilibrium with both the dilute solution and a lamellar liquid crystalline phase. It is rather viscous, shows flow birefringence,2 and scatters light strongly,z.6 showing the presence of extended surfactant aggregates. As seen in Figures 1 and 2, the L3phase has a very narrow stability range, with practically only one degree of freedom rather than two as given by Gibbs’ phase rule under the given circumstances. The appearance of an L3 phase is correlated with the presence of a lamellar phase, which strongly indicates that the aggregate structure is locally of a bilayer type. On the basis of detailed diffusion measurements, it was suggested that the phase consists of disordered lamellae.” This qualitative conclusion was given a more quantitative formulation first by Miller and GhoshI8 and (8) Ghosh, 0.;Miller, C . A . J. Colloid Interface Sci. 1984, 100, 444. (9) Ghosh, 0.;Miller, C . A . J. Phys. Chem. 1987, 91, 4528. (10) Kunieda, H.; Shinoda, K. J . Dispersion Sci. Technol. 1982, 3, 233. ( I I ) Kahlweit, M.; Strey, R. Angew. Chem. 1985, 24, 654. (12) Bellocq, A . M.; Roux, D. In Microemulsions: Structure and Dynamics; Friberg, S., Bothorel, P., Eds.; CRC Press: Boca Baton, FL, 1987; p 33. (13) Olsson, U.; Shinoda, K.; Lindman, B. J. Phys. Chem. 1986, 90, 4083. ( 1 4) Laughlin, R . G. Adu. Liq. Cryst. 1978, 3, 99. ( I 5) Marignan, J.; Gauthier-Fournier, F.; Appel, J.; Akoum, F.; Lang, J. J . Phys. Chem. 1988, 92, 440. (16) Kunieda, H.; Asaoka, H.; Shinoda, K. J. Phys. Chem. 1988, 92, 185. (17) Nilsson, P . 4 . ; Lindman, B. J. Phys. Chem. 1984, 88, 4764. (18) Miller, C . A.; Ghosh, 0. Langmuir 1986, 2, 321.
0 1989 American Chemical Society
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60/
I
Anderson et al.
/ W+LAM
LAM
",
60
I\-
/ om+w
I
\
50t LAM 40
w+ ,o
Figure 1. Phase diagram of the binary C12Es-watersystem, adapted from ref 4. For all of the figures in this paper, we use the following notation: LAM (or D), lamellar phase; L1, normal fluid isotropic solution; L2, inverted fluid isotropic solution; L3, the phase that is the subject of this paper; V I (or V), bicontinuous normal cubic phase; V2, bicontinuous inverted cubic phase; H I (or H), normal hexagonal phase; H2, inverted hexagonal phase; W, dilute aqueous phase; S (or XTLS), solid crystalline surfactant.
WATER
20
40
60
80
'
A07
Figure 2. Portion of the phase diagram at 25 'C for Aerosol OT-
NaC1-water, adapted from reference 6. The NaCl scale has been enlarged for clarity. The L3 phase region extends over a wide range of water/AOT ratios at nearly constant salinity and then joins up with the V, phase in a two-phase region. This V, phase is believed to have the Ia3d or "gyrcid" structure. more recently by Cates et aI.l9 The latter paper contains a detailed model for entropy increase on forming disordered, locally bilayer structures from an ordered lamellar structure. Opposing the breakdown of the lamellar structure is the stiffness of the bilayer, which is assumed to have zero spontaneous curvature. However, it has been shown20 that the area-averaged mean curvature in the model of Cates et al. is moderately toward the solvent. I n the present paper we reanalyze the problems of the structure and occurrence of the L3 phase by using differential geometry to describe structures of nonzero curvature. In this way we arrive a t a suggested structure that provides a natural rationalization of the intriguing properties of the L, phase.
Spontaneous Mean Curvature of the Surfactant Layers An ideal surfactant is insoluble in both water and oil, resulting in a self-association of the surfactant molecules, which in the first stage can be considered to lead to the formation of a monolayer film. Depending on the circumstances this film can curve toward the apolar side or toward the polar side, or it can curve on the average toward neither. One of the most useful concepts for the qualitative understanding of phase equilibria in surfactant systems is based on the geometrical characterization of surfactant mole(19) Cates, M. E.; Roux, P.;Andelman, D.; Milner, s. T.; Safran, s. A. Europhys. Left. 1988, 5, 733. (20) Anderson, D.; Davis, T. D.; Scriven, L. E. J . Chem. Phys., in press.
o
0.2
0.4
0.6
0.8
iI
weiqht fraction
Figure 3. Slice, at constant surfactant concentration (16.6%), of the C12ES-tetradecane-water ternary phase diagram as a function of tem-
perature (adapted from ref 13). W, refers to a water-rich microemulsion and 0, to an oil-rich microemulsion. The L3 phase has two branches, one at low oil and high temperature and one at high oil and low temperature. Both of these branches, and the W, and 0, regions, join u p in an apparently continuous fashion to a region of roughly equal volume in the microemulsion, around 45 OC; in this range the microemulsion is probably bicontinuous.1"13 cules suggested by Tartarz1and later developed by Tanfordz2and by Israelachvili and ~ o - w o r k e r s .The ~ ~ ~crucial ~ ~ dimensionless quantity is the v/la ratio formed by the molecular volume u, the molecular length I , and the polar group cross-sectional area a. When u/la equals unity, one has optimal conditions for a lamellar structure, while for L;/la > 1 the surfactant film prefers to curve toward the water and for u/la < 1 the optimal curvature is in the other direction. Although extremely useful for qualitative arguments, it is difficult to use this approach for more quanitative discussions, in particular since the area per polar group a depends on composition and temperature in a complex way. A concept related to the u / l a ratio that has a more general character is the notion of the spontaneous curvature, Ho, of the surfactant monolayer. This was first introduced for amphiphile systems by Helfrichzs when discussing phospholipid bilayer systems. The virture, and the weakness, of this approach is that we can introduce a certain H o while leaving the question of the molecular source of the particular value unanswered. For an ionic double chain surfactant, as, for example, A O T of Figure 2 , u / l a is often close to unity and a lamellar phase is stable over a wide concentration range. A t high water contents, where the electrostatic interactions are s t r ~ n g e s t ,the ~ ~ spon,~~ taneous curvature of the monolayer is toward the apolar region, Le., Ho is positive, while at low water contents with less influence from electrostatic interactions the spontaneous curvature is negative. This leads to the formation of a bicontinuous28 cubic and ultimately to a reversed hexagonal phase on increasing the surfactant content. When salt is added to this system, the electrostatic contribution to the curvature free energy will decrease, driving Ho toward negative values. For a given concentration of salt in the aqueous region, the effect on Ho of the salt will be largest a t (21) Tartar, H. V. J . Phys. Chem. 1955, 59, 1195. (22) Tanford, C. The Hydrophobic Effect; Wiley: New York, 1973. (23) Israelachvili, J. N.; Mitchell, D. J.; Ninham, B. N. J . Chem. Soc., Faraday Trans. 2 1976, 7 2 , 1525. (24) Israelachvili, J. N.; Margelja, S . ; Horn, R. Q. Reo. Biophys. 1980, 13, 121. (25) Helfrich, W. 2.Naturforsch. 1973, 28c, 693. (26) Jonsson, B.; Wennerstrom, H. J . Colloid Interface Sci. 1981, 80,482. (27) Jonsson, B.; Wennerstrom, H. J . Phys. Chem. 1987, 91, 338. (28) Lindblom, G . ; Wennerstrom, H. Biophys. Chem. 1977, 6, 167.
L, Phase in Surfactant-Solvent Systems high water contents, where the electrostatic contributions are largest. With more than 1.5% NaCl in the system the spontaneous curvature is expected to be toward the water over the whole stability range of the L3 phase shown in Figure 2. For nonionic surfactants based on oligomeric ethylene oxide (EO) chains as the polar group, the phase behavior is strongly influenced by temperature, so that the higher the temperature the less hydrophilic the surfactant. As the hydrophilicity, and thus the expected water penetration, decreases with increasing temperature, we expect that the spontaneous mean curvature should decrease. The details of the phase equilibria depend on the number of carbons in the chain and on the number of E O group^,^ but the general pattern is the same in systems showing L3 phases. At lower T values there is a region where the lamellar phase is in equilibrium with a dilute solution. At higher T an L3 phase intervenes in such a way that at the low-temperature end the L3 phase has a higher water content than a t the high-temperature end. For C I Z E Sin Figure 1 one can follow the full behavior from micellar solutions and normal hexagonal liquid crystals at low temperatures to the appearance of a lower critical point (cloud point) and the formation of a lamellar phase on increasing T; then at still higher T , we find an L3 phase and in some systems such as CI6E4,a bicontinuous inverted cubic phase: and finally an L2 phase. Although it is not necessarily true in these systems that the L2 phase consists of inverted micelles, all of these progressions appear to be consistent with a change in mean curvature from toward oil in the normal structures a t low T to toward water in the inverted structures at high T , supporting the hypothesis that the mean curvature is toward water in the binary L, phase. Studies of microemulsion systems, like the one shown in Figure 3, further substantiate this conclusion concerning the change in the sign of Ho. At high water contents (say cy = O . l ) , there are at low temperatures slightly swollen normal micelles in equilibrium with excess oil. On increasing T o n e enters the one-phase microemulsion channel. At the lower end there are highly swollen micelle^.'^^^^ At the high-temperature end, Ho-’ is larger than the radius of the largest monodispersed spheres that can be formed and a dramatic change in aggregate shape and size O C C U ~ S . ~ At still higher T a lamellar phase is formed, and at approximately 45-50 OC in the case of CIZES,we reach the condition Ho = 0, since there the lamellar phase shows maximum stability. By further increasing T, we expect the spontaneous mean curvature H o to become negative, and there one finds the L3 phase. This branch of the L, phase connects to the L, phase in the binary system, making it plausible that H o < 0 also for the binary L3 phase. The conclusions about the Ho values in the ternary system are given further support by the observation that at high a values, where oil is the dominating medium, the behavior is reversed. One finds water droplets in oil at high T, then a lamellar phase, and finally an L3 phase on lowering the temperature. Thus studies at high and low a values give the same conclusions concerning the spontaneous curvature of the surfactant film, namely, that the mean curvature is toward the more abundant solvent. This ”crisscross” pattern, in which the L, phase crosses the main microemulsion channel, showing an opposite temperature dependence, is observed in other similar systems. In the present paper this is interpretted in terms of a fundamental difference in the relationship between the spontaneous mean curvature Ho and the oil/(oil + water) ratio a for the monolayer-microemulsion and bilayer-L3 phase microstructures. In the monolayer case, the mean curvature is toward the less abundant solvent30(whether discrete or continuous), whereas in the case of a bicontinuous bilayer structure the mean curvature is toward the more abundant solvent .*O The phenomenological conditions under which the L3 phase is observed suggests the following conjecture: An L3 phase is formed when the locally preferred structure is a bilayer, but when the surfactant monolayer has a spontaneous curvature toward the (29) Olsson, U.; Nagai, K.; Wennerstrom, H. J Phys. Chem. 1988, 92, 6615.
The Journal of Physical Chemistry, Vol. 93, No. 10, 1989 4245
Figure 4. One mathematical idealization of the surfactant bilayer, in cross section. Given a base surface Sb,one can move a constant distance L away from each point in a direction given by the surface normal at that point, or in the opposite direction, and this defines two displaced “parallel” surfaces. One can imagine the polar/apolar dividing surfaces of the surfactant bilayer a s being well-approximated by these surfaces, with the terminal methyl groups located a t or near the nase surface. In another idealization, the distance L to the displaced surface varies in such a way that the two displaced surfaces are of constant mean curvature. In the cases treated here these two descriptions are very close and lead to the same results.
abundant solvent. To analyze the consequences of this conjecture, we make use of some of the recent advances in the application of differential geometry to the study of surfactant aggregate
structure^.^^^^^^^^^^
~ ,
Curvature Free Energies For a surfactant bilayer one can identify three approximately parallel surfaces, one at the midplane of the bilayer, here denoted the base surface, and two parallel surfaces a distance L on either side of the base surface describing the polar/apolar interface (see Figure 4). We want to assign the curvature energy of the surfactant monolayer in relation to the head-group plane because interactions are strongest in this region. Let HL denote the ~pointwise ~ mean curvature on the parallel surface. The areaweighted average mean curvature ( H , ) over the two displaced surfaces is2034
where ( K ) is the average Gaussian curvature of the base surface. Note that this average ( H L ) is independent of the mean curvature Hbof the base surface, although the mean curvature of each of the two parallel surfaces does depend on Hb. Normally IL*(K)I < 1 (see Appendix), so that if we require ( H L ) to be negative in accordance with the conjecture for the L3 phase presented above, the base surface should have a negative Gaussian curvature. By virtue of the Gauss-Bonnet theorem3s
where xEis the Euler characteristic over a representative patch of surface with area A,. Through (2) a negztive ( K ) necessarily implies that the surface is highly connected. The larger the value of -( HL) is, the larger - ( K ) is and thus -xE and the more connected the surface per unit volume is. Examples of such highly connected surfaces are periodic minimal surfaces.36 The more (30) Anderson, D. Thesis, University of Minnesota, 1986. (31) Scriven, L. E. Nature 1976, 263, 123. (32) Hyde, S. T.; Andersson, S.; Ericsson, B.; Larsson, K. Z . Kristallogr. 1984, 168, 213. (33) Anderson, D.; Gruner, S.; Leibler, S . Proc. N a f l .Acad. U . S . A .1988, 85, 5364. (34) Petrov, A. G.;Mitov, M. D.; Derzhanski, A. Phys. Lett. 1978, 65A, 374. (35) Struik, D. J . Lectures on Classical Dqferential Geometry; Addison & Wesley: Cambridge, MA, 1980. The boundary term has been omitted for simplicity in eq 2. (36) Schwartz, H. Gesammelte marhematische Abhandlungen; VerlagSpringer: Berlin, 1890.
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common minimal surfaces D, P, and the gyroid have, for example, Euler characteristics xEu of -2, -4, and -8 per unit cell, respectively. This shows by a straightforward, but somewhat esoteric, geometrical argument that a bilayer structure with negative average mean curvature toward the solvent can only be formed through building up a highly connected surfactant aggregate. The sole restriction is that branch points with three or four monolayer films meeting are not allowed. For systems with Ho < 0 but lH&l \\y ,
,
,
,
,
,
,
0
-200
XTLS
20
40 60 WT% SURFACTANT
80
50
C14H30
water +C14H30
Figure 8. Resulting fit of the theory to the L, phase region at low oil weight fraction in Figure 3; the surfactant concentration is fixed at y = 0.17. The thick line gives the curve along which (8b) is satisfied. The values of x(7'), To,and c'(=l) are the same as those used in Figure 5a. Estimated values of Hofrom the fit at a = 0.02.0.12, and 0.25 are 0.035, 0.06, and 0.07 A-', respectively.
100
Figure 7. Phase diagrams for two phosphoryl surfactant-water systems,
showing L, phase regions that are substantially linear. For surfactants in which there is low temperature sensitivity of x, the theory predicts a concave upward shape for the L, phase region, as in Figure 6, and in cases of very strong temperature sensitivity of x this can become convex, as in Figure 5. I n these phosphoryl surfactants, the temperature sensitivity of x appears to be intermediate, and this correlates well with the low curvature of these L, phase regions in (a) dodecyl dimethyl phosphate, and (b) dimethyl dodecylphosphonate (both adapted from ref 14).
for large fractions of solvent in the bilayer, the distance between the two opposing polar/apolar interfaces can show large local variations and the picture of a well-defined base surface breaks down. This complication is particularly pertinent for understanding how the L3 phase joins with the main microemulsion channel in Figure 3. However, the behavior a t low and high a values is interesting enough. In Figure 8 we have reproduced the branch of the L3 phase a t low CY values, where water is the abundant solvent. A t T = 73 "C this branch hits the a = 0 axis and joins with the L3 phase of the binary surfactant-water system of Figure 1. Also shown in Figure 8 is a theoretical line giving the fit of the ternary L3 region to the present theory. W e now describe the derivation of this theoretical curve. To begin with, we have used the same expression as in the binary case (( 1 1)) to account for the volume fraction 1 - @, of water in the polar region of the surfactant bilayer. The formulas of Cantor do not, however, apply in the case of the less abundant solvent, oil (tetradecane in Figure 8), because an excess of the solvent was assumed in that theory. In fact, along the curve of interest in Figure 8 the concentration of oil in the apolar region of the bilayer, @, will be taken to be a function only of a. The temperature dependence of @ has been assumed to be negligible, in contrast to the case of water which is present in sufficient quantity to saturate the interface to the concentration given by the Cantor theory, this latter concentration being a strong function of temperature. Specifically, we have taken P to be given by
represents the volume fraction in the sample due to the hydrocarbon portion of the surfactant; Pmaxthus gives the volume fraction of oil in the apolar portion of the bilayer if all of the oil
@HC
40
weight fraction
were located between the surfactant tails. Equation 12 is the simplest possible formula which a t very low oil content puts nearly all of the oil in the interface and a t higher oil contents puts increasing amounts in a separate layer between the ends of the surfactant tails. Given the temperature and concentrations, the values of aJ and are computed from ( 1 1) and (12). Combining (2), ( 5 ) , and (6) for the base surface and inserting the value -2.2 for the dimensionless group yields 2Ao/V = (-2.2(K))1/2. To first order (i.e., ignoring ( K ) L 2 terms), the volumes of the polar and apolar regions are equal to the area of the base surface times the widths of the respective regions. The oil volume is comprised of two contributions, one from the innermost pure oil layer of half-width L - Lo and the rest from the hydrocarbon tail region of the bilayer of which the fraction @ is oil; thus (1 - 7)a' = [ L - Lo + @bHC](-2.2(K))I/'
(13a)
Here a' is the volume fraction of oil over oil plus water. The part of the entire bilayer region that is nonaqueous is everything except the water in the E O region: l-(l-a')(l--y)=
[L-(l-@J)bEO](-2.2(K))1'2
(13b)
For the particular surfactant treated, namely, CI2E5,it is a reasonable approximation to take bHC = 6HC = I8 A. Equations 13a and 13b are then two equations in two unknowns, L and ( K ) . Solving for these and using (1) for ( H L ) yields
( H L ) = -4y2/2.2Lo(1
+
- P)' = -0.0015/(1
@J
+ 95 - 0)'
(A-') (14)
where in the last term we have inserted the value y = 0.17 for the surfactant volume concentration in Figure 8, as well as the estimated value Lo = bHC bHC = 36 A for the bilayer width. All that is necessary now to complete the set of equations is an expression for the spontaneous mean curvature Ho. In the present theory, the changes in Ho are brought about by penetration of water and oil into the head and tail rtgions of the bilayer, thus increasing the effective areas AEO and AHC per ethylene oxide and hydrocarbon chain, respectively; a significantly larger effective area AHC on the hydrocarbon side will lead to a significant mean curvature Ha toward the water. In Figure 9 we show schematically the relation between the areas AEo and A,,, drawn as spherical caps, and the spontaneous radius of curvature Ro = I / H o . The distance X between these hypothetical caps is
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The Jnurnul of'Physrca1 Chemistry, Vol. 93, ,Yo. 10. 1989
R=l/HO
-I
J
Figure 9. Elements involved in deriving the relationship between the effective cross-sectional areas, AEOand AHC. of the ethylene oxide and hydrocarbon portions of the C,,E, molecule, the distance X along the molecule between the points where AEOand AHc are measured, and the spontaneous radius of curvature R = l / H , , .
not entirely unambiguous. but clearly it is between half the total surfactant length and the full length. In the present case where we have taken the value of the surfactant length to be Lo = 36 A, we have taken X = 30 A. Let the superscript (0) refer to the areas AEO and A H c in the absence of solvents. Clearly
=
mJ/(l- d )
(15)
where !I = .4HC(o)/AEo(o).Thus, solving for Ho = R i I gives
-Ho = ( I - [ ( I - / ? ) / Q @ j ] ' / 2 ) / X
(16)
Thc value of' ! I is determined by the condition that, in the binary s>stcm ( c t = $ = 0), Ho = 0 at T = 64.5 O C , where aJ = 0.35 (this value of aJcorresponds to approximately 4.5 water molecules per EO group). This then closes the set of equations. when the condition (HL) = H,, which expresses the working hypothesis of the paper, is enforced. A computer was used to solve iteratively, a t each temperature T of interest, for the value of N at which ( I 4) and (16) lield the same value. As can be seen from Figure 8. the agreement between theory and data is quite good, especially in view of the fact that no attempt was made to improve the quality of the fit by choosing a form of the relation for $ ((12)) which contained adjustable parameters. In fact, since the same formula used in the binary case for aJ (( 1 I ) . with c'= 1 ) was used in the tern'iry case, the only adjustable parameter in Figure 8 is A. and thc results are not sensitive to the value used: since as noted above I8 .I! < X < 36 8( is required, we chose X = 30 .8. Finally v e note that there is an analogous behavior of the L, phase at high ( t values where oil is the abundant solvent. Also, in this case it is necessary to invoke an (Y dependence i n H , to acc'ount for the experimentally observed location of the L, phase tvithin the niodel. At low water contents the EO groups overlap, and this could lead to a n increased tendency to curve toward the oil which i n thic case is the more abundant medium. One can note that the stability range of the lamellar liquid crystalline phase i 4 consistent with this conclusion, i n that a t low ct the lamellar phase extends to high temperatures, while at high N it extends t o IOU temperatures (see Figure 3 ) .
Relative Stability of Lamellar, Cubic. and L3 Phases The 1 phase occurs in a phase diagram as an alternative to 'I Icimellar phase. and i t is important to recognize the factors that inlluence the relativc stability of the two phases. In previous 5tudies"~l 9 it has been emphasized that the I., phase is a disordered Iaiiielkir phaac. uith the implication that the essential factor i'a\oring the formation of d n L, phase is entrop}. Here we have concluded that the niost important factor is the formation of a bil
