1464
J . Phys. Chem. 1989, 93, 1464-1471
is more sensitive to temperature due to the temperature sensitivity of oil uptake between the surfactant chains. This means that the effective surfactant parameter is a sensitive function of temperature.) 6. Conclusions This paper indicates the utility of equations and concepts borrowed from differential geometry for the determination of microstructures of bicontinuous cubic phases. The constraints imposed upon the interfacial structure by the composition and the preferred surfactant molecular geometry set the topology and symmetry of the interface. Unfortunately, accurate measurements of the surfactant chain length in cubic phases are not available. These measurements are needed in order to uniquely prescribe the structure. The microstructures of cubic phases of water mixtures with the surfactants KC14, AOT, GMO, SDS, and C12E06 have been analyzed and compared with earlier proposals (which all invoke genus three surfaces). Except for the SDS/water and AOT/water mixtures, previously suggested structures are consistent with these analyses. The SDS mixture forms a reversed bilayer of genus between three and six. Work is under way to derive new interfaces of symmetry I m j m and genus four to six. The microstructure of the AOT/water mixture appears to be of genus five to eight. Again, we are deriving new periodic minimal surfaces of the measured symmetry and higher topologies than genus three. These structures are thus still open. The formulas and analyses presented here suggest that it is unwise to assume that the microstructure is fixed throughout a single phase. It is certainly not fixed for single phases that occur for large ranges of composition. We have already demonstrated
the large variation in microstructure throughout the microemulsion phase of ternary mixtures of DDAB, water, and 0ils.25*26The calculations on ordered phases suggest that the microstructures of these phases are also sensitive to compositional changes. These changes cannot in general be accommodated by a simple change in lattice parameters (conserving the microstructural topology and symmetry), since such a variation would alter the effective shape of the surfactant molecules (characterized by the effective surfactant parameter). In particular, cubic phases that are formed within a large composition (or temperature) range must exhibit a diversity of microstructures. This imparts a new urgency to the determination of new periodic minimal surfaces and surfaces of constant average curvature. We have not explicitly addressed the fundamental issue of what determines the phase of a mixture. While subtle questions of statistical mechanics must be involved (and these are, in some sense, subsumed within the surfactant parameter), it is remarkable how much local and global geometrical demands alone constrain the structure. The approach used here should be combined with a comparison of theoretical SAXS spectra from suspected structures with actual spectra. Work is under way to determine accurate form factors for periodic bicontinuous structures. The rich variety of periodic bicontinuous structures emerging from mathematical studies must surely be reflected in nature.
Acknowledgment. I thank Hugo Christenson, Krister Fontell, KBre Larsson, and Barry Ninham for ideas and advice concerning this subject. Registry No. SDS, 151-21-3; AOT,577-1 1-7; CIZE06,3055-96-7; KCII, 13429-27-1; GMO, 25496-72-4.
Phase Boundaries for Ternary Microemulsions. Predictions of a Geometric Model S. T. Hyde,* B. W. Ninham, and T. Zembt Department of Applied Mathematics, Research School of Physical Sciences, Australian National University, Canberra, ACT 2601, Australia (Received: March 25, 1988; In Final Form: July 13, 1988)
It is shown that analytical geometric arguments can be used to account for the observed existence region of the L2 phase for a variety of ternary systems. For the systems studied, a surfactant parameter that varies slightly with water content is sufficient to determine phase boundaries and the microstructure throughout the L2 phase.
Introduction The assignment and meaning of microstructure for the class of chaotic fluids called microemulsions has presented a problem. In principle small-angle X-ray (SAXS) or neutron scattering (SANS) ought to have resolved the issue long since. But until recently even the existence of observed peaks in spectra, let along their position, remained unexplained. That is no longer an issue. There have been two lines leading to progress. In one the scattering spectra are derived from a phenomenological effective Hamiltonian description of fluctuations. This approach is apparently quite successful in general.’ It accounts for spectra from a wide variety of systems. But it has some difficulties. Among these the arbitrary truncation of a Landau expansion of the assumed Hamiltonian in an order parameter whose physical meaning is unspecified and the necessity to involve three or more unquantified coefficients are the least. More serious is the fact that such a description says nothing that allows any direct conceptualization of microstructure. ‘Permanent address: Department de Physico-chimie, Centre d’Etudes Nucleaires de Saclay, 91 191 Gif sur Yvette Cedex, France.
0022-3654/89/2093-1464$01 S O / O
A second very different approach based on geometry avoids that difficulty, at least for some systems. These are ternary microemulsions formed from water, various alkanes, and the double-chained cationic surfactant didodecyldimethylammonium bromide (DDAB).24 The systems have the special property that the surfactant is insoluble in both oil and water. Hence the surfactant must reside at the oil-water interface with curvature set by component ratios and the balance of interfacial forces. For these mixtures SAXS and SANS spectra as well as conductivity data have been shown to be quantitatively consistent with the predictions of an explicit random geometric model (DOC model).24 In our earlier work we have assumed that the microstructure throughout a single-phase region is characterized by a single (1) Teubner, M.; Strey, R. J. Chem. Phys. 1987, 87, 3195. (2) Zemb, T. N.; Hyde, S. T.; Derian, P.-J.; Barnes, I. S.;Ninham, B. W. J . Phys. Chem. 1987, 91, 3814.
(3) Ninham, B. W.; Barnes, I. S.; Hyde, S. T.; Derian, P.-J.; Zemb, T.N. Europhys. Lett. 1987, 4, 561. (4) Barnes, I. S.;Hyde, S.T.; Ninham, B. W.; Derian, P.-J.; Drifford, M.; Zemb, T. N. J. Phys. Chem. 1988, 92, 2286.
0 1989 American Chemical Society
Phase Boundaries for Ternary Microemulsions experimental number, the effective surfactant parameter. More exact calculations, detailed in this paper, indicate that the effective surfactant parameter decreases slightly with water content, due to varying hydration effects with water content. The surfactant monolayer forms a disordered surface of fixed average curvature and fixed head-group area, and the connectivity of the structure varies with composition in a predictable way. Once constructed, the spectra can be simulated and are in remarkable agreement with observations throughout the entire L, phase region?+ These two approached seem to be contradictory. For, whatever its faults the first line of attack a t least makes muted appeal to thermodynamics. At first sight the more explicit DOC model does not. In fact however free energetic considerations are implicit in the geometric approach. This is because the surfactant parameter is a measure of the energetics of competition between head-group forces and oil penetration into the surfactant tails. These forces conspire together with mass balance to set the microstructure unambiguously. In this paper a stronger challenge to the random geometric model is put to the test. It will be shown that the experimentally determined existence region of the L2 phase in the ternary phase diagram is consistent with the DOC model, under the proviso that the effective surfactant parameter decreases slightly with increasing water content, corresponding to an increase of the head-group area. The predictions will be illustrated for mixtures of DDAB, water, and the oils cyclohexane, octane, and dodecane that together span a wide range of microstructure. It will he shown further that the same methods can be applied to account for the phase boundaries of the two distinct L2 phases reported for the ternary system hexane, water, and dodecyldecyldimethylammonium hromideS (CI2CI@AB). For this system the DOC model again gives the single-phase boundary of the larger L2 region. For a smaller L2 region that exists a t high surfactant ratios the DOC model is prohibited. The boundaries of this region can he modeled exactly by assuming that the system takes on instead a microstructure constructed from an oil-swollen undulating bilayer of surfactant. Random Geometric Model of Microstructure Before proceeding to the prediction of phase boundaries, we first rehearse the steps necessary to construct the DOC model? The basic notions are these: With water alone the surfactant forms a lamellar phase, Le., the surfactant parameter, ulal, is close to I , where u is the volume of the hydrocarbon tail region, a the head-group area, and I an optimal chain length, usually about 80% of the fully extended chain length! With excess water, so that the head groups are fully hydrated, the area a is set primarily hy steric constraints and remains nearly constant. With added oil, the tails are swollen by oil penetration, giving an effective surfactant parameter (u/al)ew> 1. That parameter is determined by energetics. The surfactant molecules at the oil-water interface take up a surface configuration of constant average curvature, set by the value of the surfactant parameter, independently of composition. The model is an analytic approximation to this surface of constant curvature which, to maximize entropy, exhibits no long-range translational order. The topology of the interface is not fixed. It depends on composition, and the microstructure set by the interface can vary from random monodisperse spheres to multiply interconnected structures. The analytic approximation is a network of spheres connected where necessary by cylinders (Figure I ) . We have called this approximation to the actual surface traced out by the surfactant head groups the disordered open connected (DOC) cylinders model. It leads to analytic expressions for the surface area and volume of internal phase enclosed by the interface. (The calculation of average and Gaussian curvatures to be assigned to our ~
~~
(5) Warr. G. C.;Sen, R.;Evans, D.F.;Trend, 1. E. J. Phys. Chem. 1988, 92, 774. (6) Irraelaehvili, 1. N.; Mitchell, D. I.; Ninham, B. W. J. Chem. Soc., Faraday T~unr.2 1976.72, 1525. Mitchell, D. 1.;Ninham, B. W. J . Ckem. Sa‘..Faraday Tmm. 2 1981, 77.601.
The Journal of Physical Chemistry. Vol. 93. No. 4, 1989 1465
Figure 1. Schematic view of ternary microemulsion micrmtructure according to the DOC cylinders model. The top picture shows the structure with average connectivity 2, at bottom average connectivity 4. approximate representation of the interface is not a simple issue, because of cusps. We deal with this question later.) The model for microstructure can now he put together as follows. The network sphere centers are first located at the centers of hard spheres whose positions are frozen in a random packing. This choice ensures that the interface lacks long-range translational order. The density of the random packing is determined by the composition of the ternary mixture (see later). The hard-sphere radius is related to the chain length of the surfactant molecules and the volume fraction of water, so that neighboring interfacial regions are sufficiently spaced to accommodate the surfactant chains. This steric requirement is critical to the model, provides a characteristic distance for the structure, and gives rise to the broad peak measured in SAXS studies of these systems. Closed polyhedral cells are now formed about each sphere center, where the cell faces are planes that bisect the lines joining centers normally. These cells are of various shapes, and they define a Voronoi polyhedron about each center. The composition of the mixture, together with the preferred local curvature of the surfactant molecules, determines the average coordination number (Z) of the structure traced out by the surfactant interface. The internal tunnel labyrinth of the interface is described by a three-dimensional random net, of average connectivity Z. This net is formed by linking an average of Z neighboring sphere centers. Linked centers must be separated by a face of the Voronoi space partition. Under this construction, a maximum Z of 13.4 can he achieved, corresponding to the average number of faces in a random Voronoi tessellation.’ The DOC structure is formed from the net by placing spheres (radius 1.) centered a t each vertex and surrounding each link in the Z-coordinated net by cylinders (radius rc; Figure 2). Accurate analytic expressions can he derived for the surface area and volume contained within this structure, using statistics collected by Meijering for random Voronoi tessellations.s According to Meijering, the average edge length in the random net is 8.15n2~’,where n denotes the number of net vertices per unit (7) Coretcr, H. S.M. An Inlroduction York. 1969; p 412.
IO
Geomclry, 2nd ed.: Wiley: New
(8) Meijering, I. L. Phirips Res. Rep. 1953, 8, 270.
1466 The Journal of Physical Chemistry, Vol. 93, No. 4, 1989
Hyde et al. fractions. It can be shown9 that this parameter is related to the mean and Gaussian curvatures of the interface ( H and K , respectively) by
( u / a l ) , f f= 1
+ HI + K12/3
(3)
If the interface is curved toward the polar region ( v / a l > l), the average curvature is taken to be positive. It is negative if the preferred curvature is toward the oil (for which u/al < 1). In this paper, we deal exclusively with the former case, so that aint refers to the polar volume fraction (i.e., water plus surfactant head groups and counterions). We can estimate the value of the sphere to cylinder radius ratio ( p ) by constraining the effective surfactant parameter to be the same throughout both structural elements of the interface. Taking the average and Gaussian curvatures for spheres to be 1/ r , and 1/ r : and for cylinders 1 / 2 r , and 0, we get from eq 3 p-l
r = =C rs
3 rs 2(3rs I )
+
(4)
For the typical chain lengths required for this analysis (10-1 5
A), this ratio turns out to be nearly constant and practically equal to a half, up to sphere radii of 150 A.
Figure 2. Construction of the DOC model. A random three-dimensional net is formed by placing vertices at the centers of spheres forming a random hard-sphere packing. The vertices bisected by faces of Voronoi cells constructed about each vertex are linked by edges (top). This net is then decorated by spheres at the net vertices, and cylinders about the net edges (bottom).
volume. The internal volume fraction defined by the DOC structure of connectivity Z is given by
{
8.1:;,24/)2
- (pz -
I
1)'/*Znr, *rcz (1)
where p is the ratio of the sphere to cylinder radii, and 9 is the solid angle subtended by the cylinders where they intersect the spheres, given by
n = 2*( 1 - (1 - 1/ p 2 ) ' / 2 ) The expression eq 1 is derived from the total sphere volume, plus the cylinder volume between spheres. It is exact provided the cylinders do not overlap outside the spheres. Similarly, the surface area per unit volume (specific surface area) is
If the surfactant head groups sit on this surface, the head-group area per surfactant molecule together with the composition sets the specific surface area. The internal volume fraction of the structure-which consists of polar or hydrophobic regions depending on the preferred curvature of the surfactant molecules-is set by the specific volumes of the components and the composition. As already remarked, the preferred curvature of the surfactant in solution is quantified by the "effective surfactant parameter", (o/al)eff,which relates the chain volume per surfactant molecule, ~ the head-group plus any oil adsorbed between the chains ( u ) , to area (a) and the average chain length (l).6 For a given surfactant this parameter can differ with different oils, due to varying degrees of oil uptake between the surfactant chains. The value of this parameter may also depend on water content because the headgroup area is sensitive to hydration effects, especially at low water
The structure of the interface for a given average coordination number ( Z ) is set by the specific surface area and internal volume fraction alone, using eq 1 and 2. Since the coordination number determines the relative contributions of spheres and cylinders to the interfacial structure, the curvature of the interface is a functional of coordination number. (The contribution of the unphysical cusps, which occur where the cylindrical elements join spherical elements, is further discussed below.) So, the average coordination number is determined by the requirement that it is to yield the preferred effective surfactant parameter of the surfactant solution. This construction gives a general trend of decreasing coordination number with increasing water content, consistent with the antipercolation behavior upon water dilution observed in conductivity measurement^.^*^ Further, the DOC model leads to SAXS spectra that are very similar to those m e a s ~ r e d . ~ ? ~ Calculation of L2Phase Boundaries We proceed to find the phase boundaries. Surfactant solutions can form a DOC structure only within a composition region allowed by geometry. Consequently, if microemulsions are exclusively DOC structures, compositional bounds can be calculated for the single-phase L, regions from geometrical constraints. Two criteria govern the formation of these structures. First, the average coordination number is constrained to lie between 13.4 and 0. The upper limit is due to the random hard-spheres construction, while the lower limit corresponds to monodisperse spheres. Second, steric constraints exist that limit the single-phase region further. For the surfactant chains to pack without undue contortions, the cylinders must be longer than about 10 A (which is about the measured chain thickness in a bilayeri0). This minimum length is chosen by assuming a natural chain length of about 13 A and allows for some interdigitation of chains (Figure 3 , top). At low water concentrations, there must be sufficient space between surfactant head groups to prevent excessive repulsive interactions. For the head groups to fit inside the cylinder element, it is necessary that
r, 2
2(head-group volume) a
This gives a minimum cylinder radius of about 3 A for the surfactants used in this study. Allowing for additional electrostatic and hydration effects, a minimum radius of 5 8, is taken as a reasonable lower bound (Figure 3, bottom). Although reasonable, (9) Hyde, S. T., submitted for publication in J . Phys. Chem. (10) Fontell, K.; Ceglie, A,; Lindman, B.; Ninham, B. W. Acra Chem. Scand. Ser. A 1986, 40, 241.
Phase Boundaries for Ternary Microemulsions
The Journal of Physical Chemistry, Val. 93, No. 4, 1989 1467 surfactant
water
FTgure 4. Schematic view of the single-phase microemulsion region for the ternary mixtures investigated. The phase boundaries have been determined by tracking the microstructure along water and oil dilution paths. The low water boundary is set by the requirement that the average connectivity of the structure lie between 0 and 13.4. The high water boundary (AB) is determined by the minimum connectivity of 0 (corresponding to spheres). The minimum oil and maximum surfactant
I I
boundaries are determined by the steric wnstraints.
Figure 3. Top: steric wnstraints utilized to determine the phase boundaries. A minimum cylinder length of 10 A is required to pack the
surfactant chains at either end of the cylinder. Bottom: cylinders must be larger than 5 A radius to accommodate the head groups without excessive repulsive interactions. it must be admitted that this bound is arbitrary. Our choice is justified a posteriori, in that it gives the ohserved phase boundaries for all the oils. Thus, the boundaries of the single L2phase region must lie within the composition region delineated by the following: (i) Z = 0 (maximum water concentration), r > 5 A, cylinder length (I,) > 10 A; (ii) Z 5 13.4 (minimum water concentration), r > 5 A, I, > 10 A; (iii) r = 5 A (maximum surfactant concentration), 1, > I O A; (iv) /, = 10 A (minimum oil concentration), r > 5 A. Experimental determination of the ternary phase diagrams of the DDABlcyclohexanejwater, DDAB/octane/water, and DDAB/dodecane/water systems show that the maximum water concentrations of the single phase L2regions form straight lines of constant water to surfactant concentration ratios4." (Figure 4). According to the DOC model, this region should correspond to a solution of monodisperse water spheres (Z = 0) in oil. This is borne out by SAXS spectra, which are characteristic of monodisperse spheres of the expected radius,)A as well as conductivity data.'.4 The sphere radius is related to the composition by rs = 3*,",/2
(from eq 1 and 2). If & and rpsdenote the water and surfactant volume fractions, us and uh the volumes per surfactant molecule and the head-group volume, and a the head-group area respectively, we have
-3% =
x
3(& + &%/us) +,a/U,
= :-l3: - + -
;I
(5)
It follows from the DOC model that the high water limit of the single-phase microemulsion region must occur at constant water to surfactant volume fraction ratios (which corresponds very nearly to constant water to surfactant weight ratios), since this constraint (I 1) Chen, S.J.: Evans, D.F.;Ninham. 8. W.; Mitchell, D.J.; Blum, F. D.;pickup, S.J. Phyr. Cham. 1986, 90, 842. Evans, D.F.; Mitchell, D.I.: Ninham. B. W. J. Phys. Chem. 1986.90. 2817.
enforces a constant sphere radius and effective surfactant parameter. W e use this radius to determine the effective surfactant parameter a t high water content. Using eq 3, we have for spheres
The effective surfactant parameter for a general DOC structure can be calculated from the variation of the surface area along surfaces parallel to the interface? If the interface (of specific area Eo) is displaced along its normal vectors by distances +d, so that parallel surfaces of specific areas E* are formed, the surface averaged values of the mean and Gaussian curvatures are approximately given by E+ + 2. - ZE0 z+- E. H=-K= 4dZo 264 These formulas effectively smooth the (unphysical) cusps in the DOC structure, which are an unavoidable byproduct of our analytic approximation to the surface of constant average curvature. Exact equations for the area of the parallel surfaces are given in the Appendix. Now assume that this effective surfactant parameter is constant throughout the entire & region for the DDAB solutions. Its value is that inferred when the system forms spheres. This parameter together with the geometric constraints then determines the shape of the L, boundary. Illustrative values of the effective surfactant parameters for several systems considered are given in Table I (higher values). The maximum surfactant and minimum water concentrations have been determined by mlculating allowed DOC structures numerically along many oil and water dilution paths (Figure 4). The internal volume fractions have been calculated for the various dilution paths on the assumption that the head group (including the counterions) occupies 15% of the total surfactant molecular volume (consistent with molecular models). We also assume ideal mixing of the three components. To convert weight and volume fractions, we use standard room-temperature oil densities, a water density of I g cm-l, and a surfactant density of 0.998 g c ~ - ~ . I O The volume per surfactant molecule (Table I) is taken from data collected elsewhere, as is the head-group area (68 Az)." (Note that we assume the same area for both surfactants, and the calculations are not critically dependent on this value.) Chain lengths of 13 and 12 A are used for DDAB and CloC12DAB,respectively, corresponding to about 80% of the maximum lengths expected from the formula of Tanford.6.l2
Hyde et al.
1468 The Journal of Physical Chemistry, Vol. 93, No. 4 , 1989 TABLE I: Dimensions of the Ouaternarv Ammonium Double-Chained Surfactants Analyzed in This PapeP surf. VOI,A’ chain length, A surf./oil/water mix 780 13 DDAB/cyclohexane/water low water content (watermrf. wt ratio -0.2) 780 13 DDAB/octane/water low water (waterwrf. wt ratio -0.7) 780 13 DDAB/dodecane/water low water (watersurf. wt ratio 1 .O) 730 12 CloC,2DAB/hexane/water low water (water:surf. wt ratio -4.0)
eff surf. parameter 1.83-1.53 1.53 1.33-1.18 1.18
-
1.25-1.08 1.08 1.65-1.25 1.25-1.04 1.04
“The effective surfactant parameter relates the surfactant chain volume plus the volume of oil absorbed between the chains (uen) to the chain length ( I ) and head-group area ( a ) via ( u / & ~ The effective surfactant parameter decreases with increasing water content.
Figure 5. Experimentally measured and calculated phase boundaries for the DDAB/cyclohexane/water mixture. The theoretical single-phase boundary is calculated by using the data of Table I, standard oil density, and a head-group area of 68 A2. The effective surfactant parameter varies between 1.53 (high water) and 1.83 (low water content). The dotted curve marks the measured boundary determined by the constraints on the average connectivity of the structure. The dashed curve indicates the portion of the single-phase boundary determined by steric constraints. The numbered arcs through the single-phase region indicate the compositions for which the microstructure is of constant topology (whose connectivity is given by the number on the arc). The structural parameters (according to the DOC model) of the compositions marked A, B, C, and D are given in Table 11. The full curve marks the measured single-phase region of the microemulsion.l0
Results DDAB System. The results of these calculations for the DDAB solutions are shown in Figures 5-7. The predicted boundaries are in qualitative agreement with those measured. The general trends are clearly followed: the L2 region swings toward the water-oil axis as the alkane chain length increases. However, the low water boundaries do not match the measured boundary traces. These can be modeled more exactly by allowing the effective surfactant parameter to increase slightly at low water content. Admitting this possibility, we obtain good quantitative agreement with the low water boundary, again assuming that the effective surfactant parameter is constant along this boundary. Upper limits on the effective surfactant parameter are given in Table I. The boundaries imposed by steric constraints (dashed lines) differ slightly from the expected curves. This is not surprising, given the difficulty in making precise estimates of steric constraints (cutoff radii and cylinder length). It is worth noting that the experimental determination of the boundaries in these regions is itself difficult.” (12) Tanford, C. J . Phys. Chem. 1972, 76, 3020.
Figure 6. Comparison of the observedi0 microemulsion phase boundary and that calculated by using the DOC cylinders model for the DDAB/octane/water system. The boundaries are marked as in Figure 5. The effective surfactant parameter varies between 1.18 (high water content) and 1.33 (low water).
/
\
Figure 7. Observedlo and calculated microemulsion phase boundaries for the DDAB/dodecane/water mixture. The boundaries are marked as in Figure 5 . For this system the effective surfactant parameter varies from 1.08 (high water content) to 1.25 (low water).
The variation of the microemulsion microstructure within a single-phase region of these ternary mixtures is indicated schematically in Figure 4. The expected trend of decreasing average coordination number with increasing water content is also confirmed, particularly at higher water contents. However, at very low water concentrations,
The Journal of Physical Chemistry, Vol. 93, No. 4, 1989 1469
Phase Boundaries for Ternary Microemulsions TABLE 11: Geometries of Some Representative Microemulsion Mixtures in the DDAB/Cyclohexane/Water System' DDAB/ sphere cylinder cylinder cyclohexane/ av radius, radius, water wt fract connectivitv A A (A) 2916813
(B) 26/61/13 (C) 52/22/26 (D) 56 f 3816
3 1
1 3.5
13 22 22 10
11
6
45 30
11 5
16
C 1 0 H ~ 1 C ~ ~ H ~ N (Br CH3)~
11
'The locations of these mixtures within the single-phase region are indicated in Figure 5. the coordination number occasionally increases before decreasing (not indicated on the Figures). This curious behavior had already been inferred from conductivity data.4 The microstructural variation with oil content can be characterized as follows. As the oil content is increased, the topology of the structure remains virtually fixed. However, the cylinder length increases, as do the sphere and cylinder radii. Typical structural dimensions are indicated in Table 11. The DOC solutions are not unique everywhere. This problem is only noticeable at lowest water contents, where a variety of coordination numbers appear to satisfy the geometrical requirements of the structure, as well as the preferred surfactant curvature. In such cases, configurational entropy should favor the structure of lowest connectivity. Consequently, the microstructure is well defined everywhere within the L2 region: both the topology (average coordination number) and the geometry (sphere and cylinder radii, cylinder lengths) can be calculated ab initio from eq 1 and 2. Systems with Two L2 Phases. We turn now to a different system. The C,oC,2DAB/hexane/water mixture exhibits an additional L2 phase, qualitatively different to the DDAB mixtures5 (Figure 8). (This behavior is observed for other systems, e.g., (Clo)2DAB.5)The isolated low water region has no counterpart to the high water L2 phase, which resembles those found in DDAB mixtures. The lower boundary of this higher water CIOCI2DAB L2 phase is also bounded by a line of constant water to surfactant ratio, suggesting that this region again corresponds to a microstructure of polar spheres in oil. The position of this boundary yields an effective surfactant parameter of 1.04 (using eq 5 and 6). Here too, the effective surfactant parameter must increase at low water content, attaining a maximum value of 1.30. The allowed single-phase region calculated for this range of the parameter is shown in Figure 8 (Table I). We deduce that the microstructure within this single-phase region is, just as before, well described by the DOC model, viz., a monolayer of surfactant lying on a surface of constant average curvature and varying topology, curved slightly toward the polar region. It is impossible to generate a further L2 region whose shape resembles that of the smaller additional single-phase region (Figure 8). This region contains very little water. Suppose then that the surfactant parameter could vary markedly a t low water content. Admitting that possibility, we have mapped out phase boundaries for DOC monolayers whose effective surfactant parameters vary between 0.5 and 1.5. Once again, no correspondence with the measured boundaries can be achieved. This circumstance forces us to conclude the microstructure within this region must be very different from that given by the DOC model. Instead the very high surfactant and oil concentrations that occur suggests that the surfactant may here form an oil-swollen bilayer. Such a structure is consistent with the high viscosities of mixtures whose compositions lie in this region. The shape of this region is also reminiscent of cubic phases in ternary phase diagrams of DDAB solutions.13 These consist of bilayers of surfactant lying on translationally ordered minimal surfaces of cubic ~ y m m e t r y . ~ The hypothesis is supported by approximate calculations of the single-phase region of an oil-swollen curved bilayer, provided the (13) Fontell,
K.;Jansson, M., preprint.
Figure 8. The two distinct single-phase microemulsion regions observed in the CloC12DAB/hexane/water ~ystem,~ marked by the full lines. The larger region is modeled by a DOC cylinders model. The dotted (connectivity constraints) and dashed (steric constraints) lines indicate the boundary delineated by this model, using an effective surfactant parameter of 1.04. Substantial agreement between the theory and actuality is achieved if the effective surfactant parameter is increased to 1.25 in the lower water region of this single-phase area. The smaller cigar-shaped region cannot be fitted to a DOC cylinder model. However, this region can be modeled by a curved bilayer, topologically equivalent to a cubic phase. An effective surfactant parameter of 1.65 is used. This increase of the surfactant parameter as with decreasing water content is indicative of a shrunken head-group area at low water concentration.
effective surfactant parameter in this region of the ternary phase diagram is higher than that in the larger L2 region. The effective surfactant parameter of an oil-swollen bilayer is related to the surface-averaged value of the Gaussian curvature of the interface which defines the center of the bilayer (( K ) ) and the thickness of the oil layer ( t ) by9
The total nonpolar volume per surfactant molecule for such a system is given by9
The paraffin volume can be deduced from the composition, and simultaneous solution of eq 7 and 8 for a given effective surfactant parameter determines the oil thickness and the Gaussian curvature. To check that the resulting structure is physically acceptable, we compared the polar and paraffin volumes with those expected from the composition. The volume available for the polar media is calculated from two independent approximations. First, typical surface to volume ratios for minimal surfaces of low topology are used to estimate the total volume available per surfactant molecule. We have chosen normalized surface to volume ratios (S/p/3) between 2.5 and 4.5, which span surfaces of genus 3 to about 9.l49l5 Second, we estimate the polar volume associated with each surfactant molecule (under the assumption that the focal surface of the interface has zero volume) by calculating the volume traced out by a series of parallel surfaces from the interface to the centers (14) Schoen, A. H. NASA Technical Note No. D-5541,1970. (15) Mackay, A. L.; Klinowski, J. Comp. Math. Appl. 1986, 12B, 803.
1470 The Journal of Physical Chemistry, Vol. 93, No. 4, 1989
of curvature of the interface on either side of the surface. It can be established that this approximation is acceptable (within a few percent) for many periodic minimal surfaces of known surface to volume ratio. We get9 Uwlar
=
1
+ (K)(; + l r (9)
Detailed calculations based on eq 7-9 show that a curved bilayer can indeed form with suitable polar and paraffin volumes. The region of the ternary phase diagram in which the structure exists closely follows that of the smaller microemulsion region measured experimentally (Figure 8). The theoretical region fits that determined experimentally most closely if the effective surfactant parameter is about 1.65. This demands a shrunken head-group area of about 55-60 A2, assuming the surfactant chains are about 12 A long. The phase boundaries for this region are determined by the following considerations and assumptions. The minimum oil concentration required for the formation of the single phase (about 28% by weight) is just sufficient to interpenetrate the surfactant chains, so that no bulk oil is present, giving an effective surfactant parameter of 1.65. At an oil weight fraction of about 50%, the single phase demixes. Our calculations indicate that at this concentration, a lower surfactant parameter (1.62) must be adopted to accommodate the structure. This value is unfavorable, given the high oil content. The upper water limit of this region (about 8%) is reached when a sufficient proportion of the hydrated bromide counterions dissolve in the bulk aqueous regions that exist at higher water content. Consequently there will be increased repulsion between the head groups. This interaction forces the head groups apart, and the effective surfactant parameter drops below the value required for formation of the curved bilayer state. It is clear that this model can only be correct if the surfactant has an unusually high surfactant parameter at very low water content. However, the low water boundary of the larger L2 region can be accounted for only by assuming an effective surfactant parameter of 1.30, while the value of this parameter at higher water content is 1.04. Consequently, the effective surfactant parameter is indeed high at low water content, and the value required to form curved bilayers in the smaller L2 region is consistent with this trend. The proposed microstructure in this region is topologically or not equivalent to that which occurs in cubic p h a ~ e s .Whether ~ the microstructure has long-range translational order must be resolved by SAXS studies. Preliminary investigations reveal a very broad peak in the small-angle spectra of these mixtures, indicative of a disordered microstructure of very short correlation length.
Discussion it had already been shown that the In some earlier DOC model gives an explicit quantitative picture of microstructure for the DDAB, alkane, and water L2 phase. SAXS spectra were predicted a b initio and agree with experiment. The surfactant parameter sets curvature at the oil-water interface, and this together with component ratios determines the microstructure. It varies continuously throughout the L2 phase. Conductivity, percolation thresholds, and peak positions in SAXS spectra all emerge from straightforward g e ~ m e t r y . ~ J ’ What has been done here is to carry these arguments further. We have shown that phase boundaries can be accounted for by assuming that the microstructure is set by an effective surfactant parameter, which varies slightly with water content. (Except for very low water content and for surfactants like CIoCl2DABr,whose phase diagrams are radically different.) This is even more bizarre, because what phase exists where in the ternary phase diagram
Hyde et al. ought to be determined by a consideration of free energies of other possible competing phases. The question that arises is not just one of why the DOC model works so well, but rather, why it works at all. The answer can be seen through the following arguments: For surfactant-water systems theories6J8 of self-assembly of micelles depend on mass action, a hydrophobic free energy of transfer of monomer to micelle, the notion of competing surface forces, and geometric packing arguments. Such theories, which provide a global quantitative picture of the self-assembly process, include rod-shaped, globular micelles, vesicles, bilayers and reverse The surfactant parameter (which can be varied for ionic systems with salt, for nonionic with temperature), together with geometric packing are the overriding determinants of microstructure. For oil-water-surfactant systems of the kind considered in this paper, the DOC model represents a direct extension of the same ideas. For the particular surfactant systems chosen, insoluble in oil or water, hydrated head-group areas and counterion concentrations are so large that the area per head group is fixed by steric constraints throughout the entire L2 phase. Hence the only determinant of the curvature at the oil-water interface is the degree of oil uptake into the surfactant tails, and that oil uptake, strongly dependent on alkane chain length, is normally the sole fixed energetic factor of any consequence. So the microstructure really is prescribed once the idea of randomness is built into allowed geometries. Of course the DOC model cannot account for all microemulsions. That has been shown here for the CIOCl2DAB, hexane, and water system. One of the L2 phases that occur is accounted for much as before. But the additional L2 phase is not. The DOC model is here not allowed, and as we have seen, a different structure, analogous to a cubic phase, has to be invoked to predict the phase boundaries. That second L2 phase is amenable to the same kind of analysis9 as that which accounts for cubic phases in these and other systems. But again it is a matter of geometry. The reduction in head-group area required to explain this phase had to involve hydration effects at the low water contents involved. That is not surprising: CloCI2DAB,CloCloDAB,and shorter chained double-chained quaternary ammonium ions are extremely hygroscopic. It is worth noting in passing that attempts to predict phase behavior for other ternary (ionic) systems from the direct calculation of free energies for competing phases via, say, Poisson-Boltzmann theories also fail in precisely the low water content region, where hydration effects are unavoidable. l 6 For the simpler surfactant-water systems such as SDS, CTAB, and other micelle-forming systems, theories of self-assembly can be made semiquantitative in the sense that critical micelle concentration, aggregation numbers, and phase boundaries are determined from free energy calculations, but only for systems where the counterions are strongly “bound”. Specific counterion effects are not accessible by extant primitive model theories. Thus for the DDAB-water alkane system, a change in counterion from Brto CI- or I- induces dramatic changes in phase behavi0r.l’ These changes are obviously related to hydration, the energetics of which process cannot be tackled at this time. In the circumstances one may as well take the surfactant parameter as given, especially when in our case it can be measured, and simply generate the microstructure directly by geometry. That argument takes on especial force if it be borne in mind that no one has yet the faintest idea of how to build topology into statistical mechanics, and the argument works. Our model is not universal. Thus it is apparent that with tetradecane in oil1’ (an alkane that is hardly taken up by the surfactant), a different microstructure must obtain. We shall deal with that elsewhere. Nonetheless it would be surprising to us if recognition (1) that molecules have size and shape and (2) that surfactants do adopt a preferred curvature is not more generally useful in elucidating microstructure in self-assembly processes. (16) Jonsson, B.; Wennerstrom, H. J . Phys. Chem. 1987, 91, 338. (17) Chen, V.; Evans, D. F.; Ninham, B. W. J. Phys. Chem. 1987, 91, 1823. (18) Evans, D. F.; Mitchell, D. J.; Ninham, B. W. J. Phys. Chem. 1984, 88, 6344.
The Journal of Physical Chemisrry. Vol. 93. No. 4, 1989 1471
Phase Boundaries for Ternary Microemulsions
z
Appendix It is known that the area of surface patches on either side and parallel to an interface of mean and Gaussian curvatures H a n d K is given by 6a(fd) = (g(+d))l/’ g(*d) = g(O)(1 2Hd + K&I’ (AI) where dfd)and g(0) are the local surface metria at the parallel surfaces and interface, respectively, and d is the separation between the interface and the parallel surfaces. Consequently
’ td
*
6a(*d) = da(O)(l Hd + K@l (A2) where 6 a ( f d )and 6a(O)are the surface patch areas on the parallel surfaces and the interface? For a surface free of cusps, the normal vector to the surface varies continuously over the surface, so the total area of the parallel surface to the interface is equal to the sum of all the parallel surface patch areas. This means that the parallel surface areas are given per unit volume on either side of the interface,
x(+d),
where u and u define the curvilinear coordinate system on the surfaces, and the integral is evaluated over a unit volume of space. Now, we define the surface-averaged values of the mean and Gaussian curvatures by
Jlufam llu,fae (&(w)Y”
(gO(w))’/’ H(u,u) du du
(H)=
(A4)
du du
(in both expressions the denominator is the specific interfacial area, X(0)). Thus, for a smooth interface, the average values of the mean and Gaussian curvatures are given by 2(+d) X(-d) - 2 2 ( 0 ) X(+d) - E(-d) (K) = (H) = 4d2(O) 2&2(0) (A6) If the interface contains cusps, the area of the parallel surface to the interface differs from the sum of the surface patches, due to the discontinuous variation of the normal vectors at the cusp. The cusps must be smoothed to apply the parallel surface formalism. The contribution of the cusps to the curvature can then be calculated exactly by letting the radius of curvature of the smoothed region approach zero. If the radius of curvature of the smoothed region about the cusps is L, the surface area of this smoothed region is given by the surface area of a section of a torus of major radius (re + e) and minor radius f (Figure 9). Following Willmore,LPwe get
+
2-
area (toroidal patch) =
s
I
((re
”-6
+ + c cos u ) du du =
$ = cos-’
The major radius of the torus remains the same for the smoothed toroidal sections of parallel surfaces on either side of the interface, while the minor radius becomes ( 6 d) and (c - d) for parallel surfaces lying at a distanced from the interface, displaced toward the polar region (inward, -d) and the paraffin region (outward, +d) respectively (Figure 9). In the limit, as the minor radius of curvature of the toroidal region ( e ) approaches zero, this region is equivalent to the cusps a t the junctions of the spheres and cylinders. Thus, the contribution of the cusps to the specific area of the parallel surfaces to the interface is given by
+
2+d(cusps) = 2rZn(-d)(rc@- (-4 sin
@I
2&cusps) = 2rZn(d)(r,@- (d)sin $1
(AS)
+
Similarly, the sphere and cylinder radii become ( r 6)and (r - d)on the outer and inner parallel surfaces, respectively. Further, the cylinder lengths remain constant for all parallel surfaces, as do the excluded areas of the spheres and cylinders. Following the notation of eq 2, we get = ( 4 r ( r , f d)’- OZ(r,
* d)’)n +
6
2rc((rC+ e ) @ - c sin
where
Figure 9. Crdss section through a sphervcylinder join, indicating the cusp geometry. The interface is indicated by the full black lines, while the inner and outer parallel surfaces are indicated by grey lines. The cusp is smoothed by the toroidal section AB, and the minor radius of curvature (e) approaches the cusp as this radius goes to 0. The resulting curvatures are derived in the Appendix.
$1
(A7)
+
r, f r, + f
(19) Willmore. T.J. An Introduction 10 Dif/erentiol Geometry: Oxford University Press: Cambridge. 1959; p 40.
The exact values of the surface-averaged mean and Gaussian curvatures are determined by inserting the specific areas of the parallel surfaces derived from this equation, together with the interfacial specific area (eq 2, main text), into eq A6. Registry No. DDAB, 3282-73-3: CIOCI2DAB. 76476-02-3.
