J . Phys. Chem. 1991, 95,9541-9549 semiquantitative statements concerning the microemulsion under investigation. Thus, low values of K (say below 10-5-10d S cm-I) indicate nonpercolating microemulsions with weakly attractive or repulsive interdroplet interactions and k, below (1-2) X IO9 M-' s-I. On the contrary, high values of K (above IO4 S cm-I) reveal percolating systems with strongly attractive interactions and k , values above (1-2) X IO9 M-' s-l. Likewise, an increase of K with a given parameter and the occurrence of percolation reveals that droplet size, attractive interdroplet interactions, and k, increase with this parameter, the onset of percolation corresponding to k, = (1-2) X IO9 M-' s-I and to an interaction energy of about 2 kT.64 Note that the parameters investigated are as varied as w , i, T , or the oil, surfactant, or alcohol chain length or nature. However, other parameters can be thought of such (64) Hamilton, R.; Billman, J.; Kaler, E. W. Langmuir 1990, 6, 1696. (65) B e l l q , A. M.; Biais, J.; Bothorel, P.; Clin, B.; Fourche, G.; Lalanne, P.; Lemaire, B.; Lemanceau, B.; Roux, D. Adu. Colloid Interface Sci. 1984, 20, 167.
9541
as pressure, ionic strength of the water core, solvents other than water, etc. In the case of water solubility measurements, the information comes from the aspect of the system at a water content exceeding slightly the solubility of water, after equilibrationg." In favorable situations the system will show two well-separated phases. Systems where the volume fractions of the two phases are comparable have their stability controlled by attractive interactions between droplets. Systems where one phase has a very low volume fraction have their stability determined by the spontaneous curvature of the droplet interfacial l a ~ e r . ~Percolation -*~ is observed only in the first type of systems. In view of the above, we suggest that, when starting the study of a new microemulsion system or family of microemulsions, electrical conductivity and water solubility measurements be performed first. This will probably provide a number of qualitative and semiquantitative information that may greatly reduce the number of measurements required on the same system by means of more sophisticated techniques.
Unified Classical and Molecular Thermodynamic Theory of Spherical Water-in-Oil Mlcroemulsions Douglas G. Peck, Robert S. Schechter,* and Keith P. Johnston* Department of Chemical Engineering, The University of Texas, Austin, Texas 7871 2 (Received: April 12, 1991)
A unified classical and molecular thermodynamic model is developed in order to predict the phase behavior and interfacial properties of spherical water-in-oil microemulsions. A modified Flory-Krigbaum theory is used to describe the interactions between the surfactant tails and solvent, while the ionic head-group interactions are treated with the Poisson-Boltzman equation. The interfacial tension and the bending moment of the interface are calculated explicitly. These values are incorporated into a classical thermodynamic framework that is forced to satisfy the Gibbs adsorption equation on the interface, guaranteeing thermodynamic consistency. Given a surfactant molecular architecture, the model predicts the size of microemulsion droplets as a function of the chain length of the alkane solvent. For bis(2-ethylhexyl) sodium sulfosuccinate (AOT) in the solvents propane through decane, the calculated trends agree with experiment and are explained mechanistically at the molecular level. The microemulsion radius increases for the solvents pentane through propane, an unusual behavior that is explained theoretically.
Introduction The surfactant bis(2-ethylhexyl) sodium sulfosuccinate (AOT) has been studied intensively because it has a strong tendency to form reverse micelles and water-in-oil (w/o) microemulsions, without the need for a cosurfactant. Reverse micelles are surfactant aggregates in an oil-continuous phase in which the hydrophilic head groups are oriented toward the micellar core and the hydrophobic tails extend into the oil phase. Aqueous solutions can dissolve into the core and form a water-in-oil microemulsion composed of swollen reverse micelles on the order of 20-1000 A in diameter. These aqueous microenvironments may be used for the extraction of hydrophilic substances selectively from aqueous solutions,l for catalysis in chemical and biochemical reactions,2 and for the production of colloidal materials3 A rich variety of theories has been used to explain microemulsion phase behavior. The influence of variables such as
salinity, temperature, and solvent type can be described qualitatively in terms of binary phase diagrams of the components of the microem~lsion.~In contrast, quantitative theories are required in order to calculate phase diagrams and the size of microemulsion droplets. For example, lattice models have been developed in which each molecule is treated as a difunctional "dumbell" oriented so that hydrophilic and hydrophobic ends form separate regions in the l a t t i ~ e . ~A. ~Flory-Huggins solution model may be used to reproduce a wide variety of types of microemulsion phase diagrams by regressing interaction parameters, which depend upon surfactant concentration? Another quite different approach is based on the tesselation of space by Veronoi cells to describe the onset of percolation.* For systems composed of bicontinuous structures, this model is very useful for treating critical fluctuations. These models do not however account explicitly for the presence of the interface and its consequence according to G i b b ~ . In ~ addition,
( I ) Klein, T.; Prausnitz, J. M. J . Phys. Chem. 1990, 94, 881 I . (2) Shield, J. W.; Ferguson. H. D.; Bommarius, A. S.; Hatton, T. A. Ind. Eng. Chem. Fundam. 1986, 25,603. Gomez-Herrera, C.; Graciani, M. M.; Muiioz, E.; Moya, M. L.; SHnchez, F. J . Colloid Interface Sci. 1991, 141.454. (3) Lianos, P.; Thomas, J. K. J . Colloid Interface Sci. 1987, 117, 505. Robinson, 8. H.; Khan-Lodhi, A. N.; Towey, T. In Structure and Reactiuity in Reoerse Micelles; Pileni, M . P., Ed.; Elsevier: New York, 1989; p 198.
5, 305.
0022-3654/9 1 /2095-9541%02.50/0
(4) Kahlweit, M.; Strey, R.; Schomacker, R.; Haase, D. Langmuir 1989,
(5) Dawson, K. A.; Lipkin, M. D.; Widom, B. J . Chem. Phys. 1988, 88, 5149. (6) Hu, Y.; Prausnitz, J. M. AIChE J . 1988, 34, 814. (7) Rossen, W. R.; Brown, R. G.; Davis, H. T.; Prager, S.; Scriven, L. E. SOC.Pet. Eng. J . 1982, 945. (8) Talmon, Y.; Prager, S.J . Chem. Phys. 1978, 69. 2984.
0 1991 American Chemical Society
9542
The Journal of Physical Chemistry, Vol. 95, No. 23, 1991
the relationships between the parameters in these models and the molecular properties of the components are not well-defined, so that these models are not appropriate for predicting specific phase behavior at present. In this study, we focus on w/o microemulsions composed of spherical drops, which have been described with classical thermodynamics according to Gibbs and O ~ e r b e e k . ~Equilibrium J~ is attained when a balance is stuck between the interfacial energy and the energy due to the dispersion of the m i c e l l e ~ . ~ ~If- Ithe ~ dispersion energy of the micelles is neglected, the interface would assume its "natural curvature", as determined by the interfacial tension and bending moment. If the interfacial tension is small, the free energy change associated with dispersing the drops would favor a large number of small droplets due to the entropy increase. Because energy is required to deform the interface from its natural radius, as represented by the bending moment, there is a limit on how small the droplets may become. A key challenge is to incorporate a molecular model of the interface into the classical thermodynamic framework in a manner that is thermodynamically consistent. Our objective is to develop such a unified model that enables us to predict micelle size as a function of the molecular properties of the surfactant, salt, water, and oil. To accomplish this successfully, it is important that the terms in the modcl should have a clear physical interpretation. Thus, a molccular model of the interface is required to calculate thermodynamic properties such as the surface tension and bending moment from the molecular properties of the pure components. Such a model has been utilized previously,ls but the classical part of the model was not forced to satisfy the Gibbs adsorption equation, the requirement that ensures thermodynamic consistency at the intcrface. Because the new unified model is thermodynamically consistent and its parameters are physically meaningful, it should be appropriate for predicting micelle size and phase behavior. To calculate phase diagrams, two equilibrium criteria will be considered.I0 For a single-phase microemulsion with a given composition, the free energy is minimized to determine the droplet radius. This is commonly called "internal" equilibrium. The internal equilibria of surfactant dispersions in nonaqueous16 and aqueous" media have been explored previously, but our case is complicated by the presence of water in the reverse micelle core. If an oil-continuous microemulsion phase is in equilibrium with an excess aqueous phase, then the chemical potential of water is the same in each phase, a condition referred to as "external" equilibrium. We will examine the effect of the solvent type on the droplet size for both single-phase and two-phase systems. Another major objective of this study is to examine various contributions to the interfacial tension at the molecular level to identify the mechanisms responsible for solvent effects on AOT reverse micelles. We will examine trends in micelle size versus alkane chain length for propane through decane and compare these results with recent experimental data.IE The next paper in the seriesI9 will examine the effect of pressure on the phase behavior of AOT reverse micelles in terms of attractive intermicellar interactions and intramicellar interfacial interactions. In particular, the theory developed in this paper will be utilized to identify the (9) Gibbs, J. W. The Scientific Papers of J . Willard Gibbs; Longmans: Green, New York, 1975; Vol. 1. (IO) Overbeek, J. Th. G.; Verhoeckx, G. J.; Bruyn, P. L. de; Lekkerkerker, H. N . W. J . Colloid Interface Sei. 1987, 119, 422, ( I I ) Rehbinder, P. A. Proc. 2nd Int. Congr. Surj. Acrioiry, London 1957, 1 . 476. (12) Ruckenstein. E.; Chi, J. C. J . Chem. Soc., Faraday Trans. 2 1975, 71, 1690. (13) Miller, C. A.; Neogi, P. AIChE J . 1980, 26, 212. (14) Mukherjee. S.; Miller, C. A.; Fort, T. J . Colloid InrerfaceSci. 1983, 91,223. Jeng, J.-F.;Miller. C. A. In Surfactanrs inSolurion; Mittal, K . L., Lindman, B.;Eds.; Plenum: New York, 1984; Vol. 3, p 1829. (IS) Huh, C. Soc. Pet. Eng. J . 1983, 829. (16) Ruckenstein, E.; Nagarajan, R. J . Phys. Chem. 1980, 84, 1349. (17) Puwada, S.;Blankschtein, D. J . Chem. Phys. 1990, 92, 3710. (18) McFann, G. J.; Johnston, K. P. J . Phys. Chem. 1991, 95, 4889. (19) Peck, D. G.; Johnston. K. P. J . Phys. Chem., following article in this issue.
Peck et al.
INTERFACE
i
i INTERFACE AROUND
1
DROPS INTERACT
I
Figure 1. Conceptual thought process for the formation of a water-in-oil microemulsion.
mechanisms responsible for the pressure effects. Free Energy of a Microemulsion In this section, an expression is derived for the Gibbs free energy of a microemulsion. This expression will be used as a basis for treating a one-phase microemulsion, "internal" equilibrium, and the partitioning of water between phases, "external* equilibrium. Gibbs wrote for two phases, 1 and 2, separated by a spherical interface9 dFsinglc drop
--
-S d T +
xXi dni - P I dV, - P2 dV, + y dA + Ac d ( 2 / r ) (1)
where F is the Helmholtz free energy, S is the entropy, T is the temperature, P is the pressure, Vis the volume, y is the interfacial tension, A is the interfacial area, r is the radius of the spherical interface, and c is the bending moment. As shown in Figure 1, the phases 1 and 2 are oil and water, respectively, and are treated as bulk phases. Two of these independent variables, A and r, appear because of the presence of an interface. It is important to note that Xi is not the chemical potential of component i in a microemulsion phase because p i = (aG/13n,)~,,,#,where Xi = (aG/dni)P.T,nj~i,'.A.
A relationship between PI and P2 can be derived from eq 1 for a sphere:20*2t
Equations 1 and 2 do not depend upon the location of a hypothetical Gibbs dividing surface. It can be positioned so that c is zero, and this radius r, is defined as the surface of tension. Some investigators have chosen to use the surface of tension to avoid considering c.11*12,22723 This approach is appropriate for the case of a single drop, but when a collection of drops is present, the free must be included: energy of dispersing the drops, @A?", d F = -S d T + Chi dni - PI dVl - P2 dV2 + y dA + Ac d(2/r) + d P g p p (3) (20) Bourrel, M.; Schechter, R. S. Microemulsions and Related Systems; Marcel Dekker: New York, 1988. (21) Miller, C . A.; Neogi, P. Interfacial Phenomena. Equilibrium and Dynomic Effects; Marcel Dekker: New York, 1985. (22) Reiss, H. J . Colloid Interface Sci. 1975, 53, 61. (23) Ruckenstein, E.; Krishnan, R. J . Colloid Interface Sei. 1980.76, 188.
The Journal of Physical Chemistry, Vol. 95, No. 23, 1991 9543
Water-in-Oil Microemulsions. 1 We choose to express APi2ta by the Carnahan-Starling equation for hard spheres, along with an attractive intermicellar interaction term:24*2s
where ?td is the number of drops, 4 is the volume fraction of the dispersed phase, ow is the molar volume of water, and w is the mean field attraction parameter. When w > 0, the mean-field interactions are attractive. This expression requires a relation between r a n d 4. If ro is used, this relationship is unknown. It is more convenient to position the dividing surface at the edge of the water core. Now, the relationship between r and 4 is clear and is given by
+
emulsion, shown in Figure 1, analogous to that for normal mi~elles.~'First, the water and oil are formed at P2 and P I , respectively. These states define the X i and account for the PIVI and P2Vz terms in eq 6. The reference states for water and the oil, Xw and A,, are the pure states at P2 and P I ,respectively. The surfactant is in pure form, and this defines 4. Step I1 involves adsorbing the surfactant onto a flat interface between the oil and water. It is convenient to divide y into two contributions: = 7-
+ (7 - 7-1
(8)
where y m is the interfacial tension of the flat interface and (y y-) is the contribution to the interfacial tension due to bending the interface. This separation is convenient because step I1 contributes ymAto F. Then, the interface is bent around the water phase, and this contributes (y - ?")A to F. Next, the drops are allowed to move and interact, as described by Ordinarily, the temperature and the system pressure, P, are specified for phase equilibria calculations. Thus, it is convenient to transform eq 6 to the Gibbs free energy G = F + PV, that is
where n, is the number of water molecules and V = VI V2. This G = PDV - APVI + yA + CXini + AP$Ees (9) definition of the dividing surface does not ensure that c = 0; where the osmotic pressure PD = P PI is due to the interacting consequently, the influence of c must be c o n ~ i d e r e d . ' ~ . ~ ~ ~ ~ drops. This result is similar to another derivationlo except our Previously, empirical expressions have been formulated for c. result accounts for the effect due to the osmotic pressure. Taking One choice is to express c as a function of the natural radius and the differential of eq 9 and then applying the Gibbs-Duhem modulus of rigidity.% However, c should also vary with surfactant relation to the bulk phases and the Gibbs adsorption equation at concentration, as required by thermodynamics.28 In these apthe interfacelo yield proaches, the parameters have not yet been given a molecular interpretation. One attempt in this direction expressed c as a d C , , = PI, d V - AP dV2 y dA Ac d(2/r) dAF$Pm function of electrostatic interactions of the ionic surfactant head EXi dni (IO) groups and solventsurfactant tail interactions.I0 The latter term included an empirical parameter. An advantage of this approach Thus, eqs 2, 4, 5 , and 10 are combined to give is that c was incorporated into a consistent thermodynamic framework. d G , p = PD d V + EXi dni A molecular treatment is needed for the solventsurfactant tail interactions. This has been accomplished in the model of Huh, in which c is expressed as a function of physical and molecular properties.Is However, the classical thermodynamic framework in this model is inconsistent in two ways as discussed in Appendix A. In the present work, we utilize aspects of this molecular model, in conjunction with a thermodynamically consistent classical This relation will provide a basis for deriving the internal and model. external equilibrium criteria. After discussing a means to incorporate the bending moment Equilibrium Drop Size in a One-Phase Microemulsion and APAPta terms, we continue with the derivation of the ex(Internal Equilibrium) pression for the Gibbs free energy. Because F is an Euler function of order 1 in the extensive variables VI, V2,A, and {nil,eq 3 can For a single-phase system composed of surfactant, water, and be integratcd to give oil, six degrees of freedom exist (four from the Gibbs phase rule, and two from the inclusion of r and A). Once T, P, and the F = zX,ni- PIVI - P2V2+ yA + AF$Fa (6) compositions are specified, two degrees of freedom remain. One degree of freedom is eliminated by the spherical geometry we This expression does not explicitly contain the bending moment impose. We require that all surfactant molecules reside on the c, but c does affect the free energy indirectly through its effect interface, so that on y as shown by the Gibbs adsorption equation A = n,A, = 4 d n d (12) dXi - s d T d y = 2cd( 1 /r) (7) where n, and A, are the number of surfactant molecules and area per surfactant molecule, respectively. The geometric constraint where ri is the adsorption of component i, and s is the surface is produced by combining eqs 5 and 12: entropy per unit interfacial area. When calculating the free energy of a microemulsion, one must (13) $ / ( I - 4) = rA,G/3 be careful to avoid double-counting contributions to F through Xi and y.I7 For example, consider the case of surfactantsolvent where is the surfactant concentration in the oil phase on a interactions. If the contribution to F due to these interactions water-free basis. The remaining constraint is eliminated by is included entirely in y, then these interactions cannot be included minimizing G,that is, dGT,p,,n,, = 0. Specification of the comin the A, term for the surfactant. A convenient means to ensure position provides that dni = 0. If the change in volume due to that all contributions are counted only once is to follow the micellizntion is neglected, d V = 0 and d 4 = 0. Then eq 1 1 yields conceptual thought process for forming a water-in-oil micro-
+
+
+
+
+
(24) Overbeek, J. Th. G. Faraday Discuss. Chem. SOC.1978,bS. 7. (25) Huh, C. J . Colloid Interface Sci. 1984, 97, 201. (26) Miller, C. A.; Neogi. P. AIChE J . 1980, 26, 212. (27) Ruckenstein, E. Chem. Phys. Lett. 1983, 98, 573. (28) Lam, A. C.; Falk, N. A.; Schechter, R. S. J . Colloid Inrerface Sci. 1987, 120, 30. (29) Safran, S. A.; Turkevich, L. A. Phys. Reu. Lett. 1983, 50, 1930.
The factor (and/ar), is known from eq 5: (ar/and), = -r/3nd
(15)
9544 The Journal of Physical Chemistry, Vol. 95, No. 23, 1991
Thus, at internal equilibrium
The contribution due to the osmotic pressure is not present in this expression because of the requirement that the volume of the microemulsion phase is constant once composition is specified at constant T and P. This equation may be used to calculate the equilibrium configuration of a microemulsion phase composed of drops once y and c are known as a function of r and As.
Equilibrium between a Microemulsion and an Excess Aqueous Phase (External Eauilibrium) Suppose a three-component microemulsion, phase a, is in equilibrium with an excess aqueous phase /3. Phase a is composed of spherical reverse micelles dispersed in a continuous oil phase. Five degrees of freedom exist in this case (three from the Gibbs phase rule and two for r and A ) . Once f and P are specified, three remain. We will limit the discussion to the case where the surfactant concentration in the aqueous phase is negligible.'8~30 Therefore, once the surfactant concentration in the microemulsion is specified, another degree of freedom is eliminated. The remaining two are removed by the spherical geometric constraint and the requirement of intemal equilibrium (eqs 13 and 16). Now that we are assured that the two-phase system is completely defined thermodynamically, we can solve for the comDositions. The only other composition not known is the water composition in the microemulsion, and it is determined by the condition of phase equilibrium p: = p i . From eq 11
r:: = (E)T.P.nl+, = P p ) TJ'.ni+r + A , +
The derivatives of nd, r. and $ with respect to n, are calculated from eqs 5 and 12. For pure water, its partial molar volume, ij, = u,. Thus, from eq 5, we obtain
The variation in A, with n, can be neglected.I0 Incorporating this assumption, differentiating eqs 5 and 12 with respect to n, and then solving for (an,/an,)T3p,n,zw and (dr/anw)T,P,n,t, yields
Peck et al. Equations 21 and 22 do not explicitly contain any interfacial terms because these terms are eliminated upon substitution of the internal equilibrium criterion. Thus, to calculate p:, internal equilibrium must exist. The reference term A, is the chemical potential of pure water at the internal pressure of the drops, Pz, and thus A, = p;
+ D,(P,
- P)
(23) where 0, is the partial molar volume of water and p: is defined for pure water at the system pressure P. Because D, = u, for pure water, eq 21 can be recast in terms of AP, defined in eq 2, and PD: A, =
+ u w ( 0 - PD)
(24)
7
Equations 23 and 24 can be substituted into 21 along with the condition p; = p i and the knowledge that p, = p: to yield the external equilibrium criterion AP+X=O (25) The systems of interest in this study typically contain salt. To describe such systems, ,we treat the water and salt solution as a pseudo-pure component. This assumption is supported by the observation that the salt concentration in the aqueous core C, is nearly equivalent to that in the aqueous bulk phase For ionic surfactants, the counterions dissociate partially in the micellar core.20,32 Thus, the ion concentration C in the aqueous core contributes to the overall salinity. Given eqs 5 and 12
+
C E elk3x/rA,
(26)
where x is the dissociation constant of the counterions. The approximation that the brine solution is a pseudo-pure component also depends on u, cz 8, in order for the same equilibrium conditions to hold.
Stability of the Microemulsion Phase The attractive micelle-micelle interactions, represented by w in eq 4, can cause the microemulsion to separate into two phases that consist of low, and high, $,,, volume fractions of the dispersed phase. If the bending energy is large relative to kT, each phase will contain micelles of the same size.29 The critical values of w and 4c are determined by satisfying the conditions for spherical microemulsions: (dZAF$Fm/d$2)T,v = 0 and (d3Ap$Fm/d43)T,y= 0. The critical values are wc = 10.6 and & = 0.13. For a given w > wc, 4, and $h are determined through a tangent construction to AF$:- versus 4,25929933 Molecular Model for the Interfacial Tension and Bending Moment To use the equilibrium constraints, eqs 16 and 25, it is necessary to develop molecular models for y and c. It is convenient to define a reference as the interfacial tension of a flat interface and to calculate the change in y with curvature, y - y".Is At constant T and Ai,the change in interfacial tension due to curvature may be obtained by integrating the Gibbs adsorption equation, eq 7:
y - y m = 2 J i i r c d( 1 / r ?
(27)
Equation 3 implies that Combination of eqs 17-20 yields an expression for p: that is simplified by substitution of the criterion for internal equilibrium, eq 16, into eq 17:1° p$ = P D V w
+ hw + uwx
(21)
where
Combination of eqs 27 and 28 yields y-y-=
(F - F ) / A
(29)
where F and F" are the surface free energies of the curved and flat interfaces, respectively. Winsor separated the surfactant
(30) Avcyard, R.;Binks, B. P.; Clark, S.; Mead, J. J . Chem. Soc., Faraduy Trans. I 1986,82, 125.
(31) Fletcher, P. D.I. J . Chem. Soc., Faraday Trans. I 1986.82,2651. (32) Wong, M.; Thomas. J. K.; Nowak, T. J . Am. Chem. SOC.1977,99, 4730. (33) Hou,M.-J.; Shah, D. 0. Longmuir 1987, 3, 1086.
The Journal of Physical Chemistry, Vol. 95, No. 23, 1991 9545
Water-in-Oil Microemulsions. 1 interactions into surfactant-water and surfactant-oil interact i o n ~ . ~ ~As - ~an * extension of this idea, the contributions to y and c are separated into specific contributions:
Y = Tm + (7, - Y",) + (Ye - Yme)
where p,(r) and pm(r) are the segment density distribution functions for curved and flat interfaces, respectively. Combining eqs 29 and 33 and performing a Taylor expansion of the segment density distribution function15 yield
(30) Y.
- Y-. =
ymis the interfacial tension of a flat interface. The change in
interfacial tension due to curvature for the solvent-tail and electrostatic interactions, respectively, is represented by the two terms in parentheses. A model for each term is presented in the following sections. Change in Interfacial Tension with Curvature: Solvent-Surfactant Tail Interactions. To explore the effect of solvents on microemulsion phase behavior, it is necessary to develop a molecular model of the solvent-tail interactions. We choose an approach employed by Huh,I5 although with modifications. The surfactant tail region is represented as a collection of flexible chains, each having one end anchored to the interface. Because we assume that the surfactant tails do not penetrate the aqueous core, the interface is treated as impermeable to the chains. A conventional polymer solution theory such as the Flory-Huggins model assumes a uniform distribution of chain segments throughout the solution. However, for our case, the surfactant tails adsorbed onto the interface are far enough apart (typically, the area per surfactant head group is 50 A235)so that the tail segments are not uniformly distributed throughout the tail region. This condition is treated by the Flory-Krigbaum dilute polymer solution theory,3638 which requires knowledge of the segment probability distribution. The segment probability is determined through the solution of the diffusion equation on a spherical or flat s ~ r f a c e . Then, ~ ~ ~ ~the~ segment probability distribution function is determined by summing the segment probabilities in order to account properly for the presence of the impermeable interface. I On the basis of this theory the free energy change upon mixing solvent and surfactant tails on a micelle is
where A, is the area per surfactant tail group, 1 is the length of a tail segment, and a = 1 . If the surfactant has one hydrocarbon chain per surfactant head group, A, = A,. For the twin-tailed surfactant AOT, A, = 2A,.I5 Change in Interfacial Tension with Curvature: Electrostatic Interactions. A rigorous treatment of the electrostatic contribution has been presented previously.IO We use an approximate solution to the Poisson-Boltzmann equation on the inside of a sphere:Is
2kT AB
ye - yme= -[1.084j3/~r
where K
where x is the Flory interaction parameter, and ut and v, are the volumes of a surfactant tail and oil molecule, respectively. The number of segments per tail molecule is represented by m, and p(r) is the density distribution function for the surfactant tail segments (notice that l p ( r )dr must equal the number of segments on the micelle). The Flory interaction parameter, x , represents the enthalpy of mixing solvent and surfactant tails. If the geometric mean assumption for the unlike-pair parameters is applied
kTx = ~ ~ ( -6 6J2 ,
(32)
where 6, and 6, are the solubility parameters of the oil and surfactant tails, respectively. Thus, x is related conveniently to physical properties by solubility parameters that are known for a variety of solvents.41 The expression for (F,- F",)is the difference between AP,, for flat and curved interfaces. As a result, eq 31 yields
(36) (37)
and e and c are the charge of an electron and the dielectric constant of the aqueous core, respectively. Interfacial Tension of the Flat Interface. The interfacial tension of the flat interface 7'" is a basis for calculating y for a curved interface as shown in eq 30. It is related to yotw,the oil/water interfacial tension:21 7- = Yo/w
-n
(38)
This equation defines II, the spreading pressure, which is a two-dimensional analogue to P. Perturbation theory is convenient for calculating P and, by analogy, The reference state for II is based on an equation of state for hard d i ~ k s ,and ~ ~the , ~total ~ II is given by15
where Ahdis the hard-disk area of the surfactant tailgroup, and a, = d I 2 / 6 . The first term is the reference spreading pressure, and the two terms enclosed in brackets are the electrostatic and solvent-tail interaction terms, respectively. Equation 39 is substituted into eq 38 to obtain the equation of state for 7". Interfacial Tension of the Curved Interface. To describe solvent effects on microemulsions, it is useful to examine the electrostatic, ye, and solvent-tail, y,, contributions to y, the total interfacial tension of the curved interface. Substitution of eqs 34, 35, 38, and 39 into eq 30 yields ye =
(34) Winsor, P. A. Solvent Properties of Amphiphilic Compounds; Butterworths: London, 1954. ( 3 5 ) Eicke, H. F.; Rehak, J. Helv. Chim. Acta 1976, 59, 2883. (36) Flory, P. Principles of Polymer Chemistry; Cornell: Ithaca, NY, 1953. (37) Flory, P. J.; Krigbaum, W. R. J . Chem. Phys. 1950, 18, 1086. (38) Hill, T. L. An Introduction to Statistical Thermodynamics; Addison-Wesley: Reading, MA, 1960. (39) Meicr, D. J. J . Phys. Chem. 1967, 71, 1861. (40) Hesselink, F. Th. J . fhys. Chem. 1969, 73, 3488. (4 1) Barton, A. F. M. CRC H a n d h k of Solubility Parameters and Orher Cohesion Parameters: CRC Press: Boca Raton, FL, 1983.
=(8~e~C/ekT)~/~
j3 = ( ~ e ~ / 2 k T c C A , ~ ) ' / ~
5340
(31)
+ l . 5 8 6 j 3 1 / 2 / ( ~ r ) 2(35) ]
-[1.084/3/~r 2kT As@
(
+ 1 . 5 8 6 p 1 / 2 / ( ~ r -) 2 ] =re2kT )It2/A>
(42) Helfland, E.; Frisch, H. L.; Lebowitz, J. L. J . Chem. Phys. 1961.34, 1037.
9546 The Journal of Physical Chemistry, Vol. 95, No. 23, 1991 TABLE 11: Solvent Properties at 25 "C
TABLE I: Parameters Used To Represent the AOT Molecule
parameter hard-disk area of AOT head group vol per surfactant tail solubility parameter for each tail fraction of dissociated counterions dielectric constant for water
Peck et al.
value
22.38(
-8.97 ( mlral) 2AI2uom1f2al
?)'I [
(
[
+
[ -(
a,kTxul2 -)2]/A: A,
86.0 99.2 116.2 131.6 147.4 163.5 179.9 195.9
12.91 13.37 14.4 14.9 15.3 15.4 15.6 15.8
n-pentane n-hexane n-heptane n-octane n-nonane n-decane
YO/W
dyn/cm 46 48 50 50.7 51.2 51.5 5t.75 52
source
"uO and 6 are from ref 48. yolw are extrapolated from ref 47. and 6 are from ref 41. yolV are from ref 47. "u0 and 6 are from ref 41. yo/s are interpolated from ref 47.
(42)
TABLE III: Calculated Micelle Radius in the Oil Phase for a One-Phase Water-in-Oil AOT Microemulsion at 25 "C with w = 00
- X ~ T U ? -8.97 ( mlral) + A?uom I f 2aI
+)2]
(MPa)'I2
n-butane
+
- 2uom'f2al 3)2]/A: At
y,(enthalpic) = 22.38(
[
TU?
cm'/mol
propane
It is also instructive to express y, in terms of entropic and enthalpic contributions. From eq 41 ?,(entropic) =
6,
00.
solvent
15 A* 100.5 cm3/mol 15.5 MPa'/2 0.23 78.56
micelle radius, A
solvent
micelle radius, A
solvent
(43)
uomlI2al AI Additional Comments Regarding the Bending Moment. The bending moment, c, of the interface can be calculated from y by the r e l a t i ~ n s h i p ~ ~ * ~ ~ 0
(44)
0
with the result c = k T c l ( l / r - c2/cI)
(45)
where CI
=
3.172 -
+
11.19(1 - 2x)u?m1f2al
A2/31f2~2 c2 =
A,ZUO
8.97(1 - 2 x h 2 --1.084 4At2U0 ASK
20
2
Previously, c was expressed as28
where C, is the surfactant concentration, K, is the rigidity modulus, and RN is the natural curvature. Notice the striking similarity between the two functional forms. Even though c in eq 45 does not depend explicitly on C,, it does so implicitly through the influence of AF$P*I as shown in eq 14. The properties y and c may be calculated as a function of r and A with eqs 30,34,35, 38, 39, and 45. These results can then be incorporated into the internal and external equilibrium constraints to determine, at equilibrium, r and A of a water-in-oil microemulsion composed of spherical reverse micelles. An outline of the solution procedure is described in Appendix B. If desired, the value of r may be related to W,,the ratio of water to surfactant molecules, as is evident from combining eqs 5 and 12:
Wo= rA,/3u,
(47)
Predictions of the Molecular Thermodynamic Model
The unified classical and thermodynamic model may be used to predict solvent effects on the size and phase behavior of AOT microemulsions, as a function of the molecular properties of the surfactant and oil. Both one-phase and two-phase microemulsions are discussed. The effect of attractive intermicellar interactions on both micelle size and phase behavior will be investigated. Finally, the mechanisms responsible for the solvent effects on the (43) Murphy, C. L. Ph.D. Dissertation, University of Minnesota, 1966.
4
6
d
10
12
Alkane Carbon Number Figure 2. Calculated Woin the oil phase for a water-in-oil AOT microemulsion in equilibrium with an excess brine phase at 25 "C with w = 0. AOT concentration is 0.034 mol/L in oil. Salinity is 0.4 wt W in water. The ratio of oil to water volumes is I . The alkane carbon number is the number of carbons in a normal aliphatic hydrocarbon: (0)ref 18; (-)
predicted; (- - -) predicted without enthalpic contribution.
interfacial properties will be identified by dissecting y, into its entropic and enthalpic contributions. The representation of the molecular architecture of AOT requires the selection of six parameters, as shown in Table I. The hard-disk area of each surfactant head group was chosen to be 15 A2. For simplicity, we approximated each of the AOT tails as having IO segments. The volume per surfactant tail, ut, was calculated from Bondi volumes to be 100.5 cm3/moLU The two remaining parameters, 6, and x , are not as readily available. The solubility parameter is 15.5 (MPa)lf2for each tail based on a similar molecule, 2-1nethylheptane.~~The results are sensitive to 6,, which means that changes in the surfactant architecture can have a large influence on micelle size. We assigned a universal value of x = 0.23 to achieve good agreement for a wide variety of AOT microemulsion behavior in propane through decane. This value corresponds closely to 0.28, which was estimated from proton and sodium-23 NMR spectro~copy.~~ These pure-component (44) Bondi, A. A. Physical Properties of Molecular Crystals, Liquids, and Glasses; Wiley: New York, 1968. (45) Kaler, E. W.; Billman, J. F.;Fulton, J.; Smith, R. D. J . Phys. Chem. 1991, 95, 458. (46) Wong, M.; Thomas, J. K.; Nowak, T. J . Am. Chem. SOC.1977,99, 4730.
The Journal of Physical Chemistry, Vol. 95, No. 23, 1991 9541
Water-in-Oil Microemulsions. 1
,iL&T--L
O 0.02
-6
Alkane Carbon Number Figure 3. y,(enthalpic) and ?,(entropic) in dyn/cm. Solution conditions are the same as those in Figure 2.
parameters for AOT were fixed and not adjusted from solvent to solvent. Thus, given a surfactant molecular architecture, the model can predict trends in microemulsion phase behavior as a function of the solution conditions. A few other parameters are required to represent the brine and solvent. The value of z is 78.56 for water. The values of yoIware in Table The solubility parameters can be found in tables or calculated from an accurate equation of state (see Table II).4'948 For a given Woin the single-phase region, the radius of AOT reverse micelles is not sensitive to solvent type in the one-phase region, as shown in Table 111. This conclusion is supported e ~ p e r i m e n t a l l y . ~To ~ *change ~ ~ ~ ~ the ~ micelle size with solvent type for a given surfactant concentration, salt concentration, Wo, and A, must change according to eq 13, but the strong electrostatic interactions of the head groups keep A, relatively constant at a given salinity. If the microemulsion is in equilibrium with an excess aqueous phase, the micelle radius is no longer fixed, even for a constant A,. The relationships between the radius and number of drops are given in eqs 12 and 13. Suppose that the surfactant concentration in the oil is fixed. Equation 13 indicates that the radius increases with $I as water partitions from the bulk phase into the micelle core. This causes the number of drops to decrease. Figure 2 compares the predicted and experimental values1* of Wofor reverse micelles of AOT in various solvents in equilibrium with a bulk aqueous phase. The salinity is 0.4 wt ?& NaCI, and at this salinity, previous studies have shown that the AOT concentration in the aqueous phase is negligible.'8.30 Results are not presented for solvents smaller than propane and larger than decane because w/o microemulsions do not form in these solvents at these conditions.I8 There is reasonable agreement between theory and experiment, although the range in Wois smaller for the theory. Notice that Woincreases with increasing alkane carbon number for solvents pentane through decane, a common result described p r e v i ~ u s l y . ' However, ~ ~ ~ ~ ~ ~for~ propane through pentane a new nunusual" behavior is displayed in both theory and experiment, Le., Wodecreases with increasing alkane carbon number. The effects of enthalpic and entropic contributions to the may be examined solvent-tail interactions, which influence Wo, with the model. If the salinity and consequently A, remain constant, then changes in micelle radius with solvent type are due to changes on the oil side of the interface. The usual behavior in pentane through decane has been described in terms of the degree of solvent penetration into the AOT tail region on the micelle.'' Smaller solvents penetrate the tail region more than ~~
~
(47) Good,R. J.; Elberg, E. Ind. Eng. Chem. 1970, 62, 54. (48) Youngblood, E. A.; Ely, J. F. J . Phys. Chem. Re/. Data 1987,16,577. (49) Yazdi, P.; McFann. G. J.; Fox, M. A.; Johnston, K.P. J. Phys. Chem. 1990, 94, 7224. (50) Eastoe, J.; Young, W.K.;Robinson, B. H.; Steytler, D. C. J . Chem. Soc.. Faraday Trans. 1990.86, 2883. (51) Israelachvili. J. N.; Mitchell, D. J.; Ninham, B. W.J . Chem. Soc., Faraday Trans. 2 1976, 72, 1525.
4
'
6
8l
1
0
1 L2
Alkane Carbon Number Figure 4. Variation of the Flory interaction parameter x with solvent. Solubility parameters from Table 11.
4O130 l= 1
A
5
.-3
u)
120
:
110:
:
100
90
' -
8
2
4
10
12
6
Alkane Carbon Number Figure 5. Comparison between the 'natural" and actual micellar radii in n-alkane solvents ((-) actual r, (- --) 'natural" r). Solution conditions are the same as those in Figure 2.
larger solvents. As the degree of solvent penetration into the tail region increases, the volume of the tail region must increase; consequently, the radius increases. This increase in solvent penetration increases the combinatorial entropy of the solvent-tail mixture, as represented by y,(entropic), is shown in Figure 3.14 For comparison, y,(enthalpic) is also included in Figure 3. As expected, the entropic contribution is dominant for pentane through decane, because x is small for these solvents as shown in Figure 4. Equation 42 shows that y,(entropic) is inversely related to uo. This dependence can be discussed in terms of the Flory-Krigbaum theory. As uo increases, the combinatorial entropy of mixing solvent and tails decreases. As a result, there is a significant change in y,(entropic) from propane to decane. As the chain length of the solvent is varied, other properties in addition to the micelle radius change. An important quantity is the natural radius, defined as the equilibrium radius calculated for a single drop when the free energy of mixing of the droplets is neglected. This condition requires c = 0 by inspection of eqs 16 and 25. Consequently, the natural radius corresponds to the surface of tension for this definition. The natural and equilibrium radii are shown in Figure 5. In this section we limit ourselves to the solvents that exhibit "usual" behavior (pentane through decane). According to the model, a natural radius does not exist for decane at these solution conditions, but an equilibrium radius does exist. The difference between the equilibrium and natural radii is 9.4 A in pentane and increases to 26.4 A in nonane. This indicates that AP$Ecsaffects the difference between the e uilibrium and natural radii in these systems. If y is small, AIF$iXRCIcI favors a smaller radius than the natural radius because of the increase in entropy of dispersion as the number of spheres increases. However, the degree that the radius can deviate from the natural radius is limited by the bending moment, c, of the interface. As shown in Figure 6, c is larger for water drops in pentane than in decane. This explains the smaller difference
954% The Journal of Physical Chemistry, Vol. 95, No. 23, 1991
/
9
'
t
5
1
f u
2
4
6
8
10
12
Alkane Carbon Number Figure 6. Bending moment, c, as a function of alkane carbon number. Solution conditions are the same a s those in Figure 2.
between the equilibrium and natural radii in pentane relative to decane. So far, we have not considered the effects of attractive intermicellar interactions, but their existence is well-kn~wn.'~~~~ The potentials between micelles are orders of magnitude larger than the long-range van der Waals potential calculated from the Lifshitz e q u a t i ~ n . ' ~ . ~The ~ . ~predominant ~ contribution to the attractive potential is associated with the overlapping of the tail regions of adjacent reverse micelles, but other contributions such as ion-ion correlations between water pools e x i ~ t . ~ ' *The ~ ~ magnitude of these potentials increases as a phase transition is approached, but these interactions do not affect micellar size for small micelles formed by AOT reverse micelles.56 For the solvents pentane and decane the radius of the micelles is calculated to be 89.2 and 130.6 A, respectively, when w = 0. The minimum value of o required to induce a phase transition due to intermicellar interactions is 10.6.25Therefore, a large value of w is 30, for which the micellar radius is calculated to be 89.2 and 129.3 A for pentane and decane, respectively. Clearly, the micelle size is not affected significantly by the attractive intermicellar interactions, even for the less rigid interfaces formed in decane, although they can affect phase beh a ~ i o r . ~ ~Ina particular, ~' the attractive intermicellar interactions can induce the formation of a middle phase. This behavior is investigated in a companion paper.19 Mechanism of the Unusual Behavior in Propane through Pentane. The unusual decrease in Wofrom propane through pentane may be explained with the model. If only the entropic solvent-tail interactions are considered, the opposite trend is observed as shown in Figure 2. Therefore, the unusual behavior is due to enthalpic contributions to y. Figure 3 compares y,(entropic) to ?,(enthalpic). For the larger solvents, the cohesive energy density of the solvent is sufficiently large that the tails are solvated, and x is small. However, as the solvent size decreases, eventually x increases, as the solvent-tail interactions are not as favorable as solventsolvent and tail-tail interactions. Consequently, solvent will be expelled from the tail region, decreasing the solvent penetration and thus increasing the micelle radius. Conclusions The unified model bridges the gap between the microscopic and macroscopic theories of w/o microemulsions. Most previous theories of w/o microemulsions have focused on either the macroscopic or microscopic level, but not both. Because the unified (52) See for example: Lemaire, B.; Bothorel. P.; Roux, D. J . Phys. Chem. 1983.87. 1023. Billman, J. F.; Kaler, E. W. Longmuir 1990,6, 61 1. Kotlarchyk, M.; Chen, S.-H.; Huang, J. S.; Kim, M.W. Phys. Reo. A 1984,29, 2054. (53) Agterof, W. G. M.; Van Zomeren, J. A. J.; Vrij, A. Chem. Phys. Lett. 1976, 13,363. (54) Bedwell, B.; Gulari, E. J . Colloid Interface Sei. 1984, 102, 88. (55) Luzar, A.; Bratko, D. J . Chem. Phys. 1990, 92, 642. (56) Kotlarchyk, M.: Chen, S.-H.; Huang, J. S.; Kim, M. W. Phys. Reu. A 1984.29.2054.
Peck et al. model considers several important criteria at both levels, it opens up the possibility of predicting the interfacial properties, drop radius, and phase equilibria of microemulsions as a function of surfactant and solvent architecture. At the microscopic level, the location of the dividing surface is defined at the actual drop radius to facilitate calculation of micelle-micelle interactions. The interfacial tension and bending moment are calculated explicitly in terms of readily available surfactant and solvent molecular properties. These interfacial properties are incorporated into a macroscopic thermodynamic framework which accounts for the presence of an interface between the aqueous core and the solvent in a consistent manner. The model predicts the properties of one-, two-, and three-phase microemulsions. It is useful for guiding experimental studies to identify the optimal surfactant structure for various solution conditions, or for a particular surfactant, to determine solvent effects on microemulsion properties. The solvent-tail and electrostatic contributions to the interfacial tension have been examined to identify the mechanism of the observed solvent effects. As the chain length of the alkane solvent decreases from decane to pentane, the micelle size decreases. This is caused by an increase in configurational entropy due to increased solvent penetration of the tails. This is evident in the entropic solvent-tail interaction contribution to y. As the solvent is changed from pentane through propane, the microemulsion radius increases, an *unusual behavior". The mechanism responsible for this behavior has been dimvered with the model. As the cohesive energy density of the solvent decreases to a value well below that of a surfactant tail, the solvent is expelled from the tails and the radius increases. Here, the enthalpic contribution dominates that of the configurational entropy, while the opposite is true for the larger a I kanes . Acknowledgment. This work was supported by the National Science Foundation under Grant No. CTS-89008 19. Any opinions, findings, and conclusions or recommendationsexpressed in this publication do not necessarily reflect the views of the National Science Foundation. Further support was provided by the State of Texas Energy Research in Applications Program, the Camille and Henry Dreyfus Foundation for a Teacher-Scholar Grant (to K.P.J.), and the Separations Research Program at the University of Texas.
Appendix A Here we discuss some inconsistencies in the model of Huh,lS which we have corrected. First, the Helmholtz free energy does not include a AFAFm term, which is necessary to account for the osmotic pressure of the microemulsion. For a collection of spheres dispersed in a continuous medium F = &nsp - PIVI
+ AFS$Ya
('41)
where the subscript sp refers to the spheres. The pressure that the system exerts on the external environment is This proves that the term must be included in the Helmholtz free energy to account for the osmotic pressure due to the dispersion. The other inconsistency is that the differential of the Gibbs free energy does not satisfy the Gibbs adsorption equation. Huh states that dCT,p,kl= Eni dXi + V dF'
(A3)
where F' is a function of 7 , c, and 4. According to Huh, the first term in eq A3 is zero due to the Gibbs-Duhem equation at constant P and T, and the equilibrium constraint is based on this observation. In fact, this is true for components such as oil and water that are not present on the interface. However, for components such as surfactant that reside on the interface, the Gibbs adsorption equation shows that the first term in the equation for surfactant is not equal to zero and is related to differentials of y and c, which reside in the dF' term. Another limitation of Huh's treatment is that the equilibrium criterion is developed for a one-phase microemulsion, but results
J . Phys. Chem. 1991, 95, 9549-9556 based on this criterion are applied to cases for a microemulsion in equilibrium with another continuous phase. The two-phase case requires the minimization of the free energy of the entire system-not the microemulsion phase only. This global minimization can be performed by minimizing the global free energy directly or by equating chemical potentials of components in each phase.
Appendix B The solution procedure is described for a water-in-oil microemulsion in equilibrium with an aqueous phase. The surfactant
9549
concentration and salinity are specified, and the values of r and A, are calculated. (1 ) Guess 4. ( 2 ) Guess A,. (3) Calculate r by eq 13. (4) Calculate y and c by eqs 30, 34, 35, 38, 39, and 45. (5) Test to see if internal equilibrium (eq 16) is satisfied. If not, guess another A, and go to step 3. (6) Now test to see if external equilibrium is satisfied (eq 25). If it is not satisfied, guess another 4 and go to step 2. (7) Test for the stability of the microemulsion.
Theory of the Pressure Effect on the Curvature and Phase Behavior of AOT/Propane/Brine Water-in-Oil Microemulsionst Douglas G. Peck and Keith P. Johnston* Department of Chemical Engineering, The University of Texas, Austin, Texas 7871 2 (Received: April 12, 1991)
Pressure effectson both the curvature and phase behavior of water-in-oil microemulsions (swollen reverse micelles) are predicted with a unified classical and molecular thermodynamic theory developed by Peck et al. (this issue). The theory is used to identify quantitatively the roles of the intramicellar interfacial interactions and micellemicelle interactions. A supplementary molecular model is used to calculate the strength of attractive intermicellar interactions over a wide range of conditions, based on previous small-angle neutron-scattering data. An important distinction is made between systems with a small water-tmil ratio and those where the water-to-oil ratio is much larger, on the order of unity. In the latter the micelle radius is controlled primarily by intramicellar interfacial interactions, specifically the enthalpic propanesurfactant tail interactions. For a small water-to-oil ratio, the micelle radius is limited by attractive micelle-micelle interactions. As pressure increases, the radius increases but eventually reaches a maximum governed by the intramicellar interfacial interactions. There is good agreement between the predictions and experiments over a wide range of water-to-oil ratios.
I. Introduction Typically, temperature and salinity are the variables used to manipulate the phase behavior of a microemulsion composed of a surfactant, oil, and brine. For an ionic surfactant, an oil-in-water (o/w) microemulsion may be converted to a middle-phase microemulsion and finally to a water-in-oil (w/o) microemulsion by increasing salinity or decreasing temperat~re.'-~Phase transitions may also be accomplished by varying the pressure, although extremely large pressure changes are often required in oils such as decane unless the system is already near a phase b o ~ n d a r y .The ~ phase behavior of a microemulsion is a function of two key factors: (1) the intramicellar interfacial interactions that govern the natural curvature of the microemulsion interface, as shown in the classic work of Winsor: (2) micelle-micelle interactions.5 The natural curvature may be defined as the curvature of the interface independent of any repulsive or attractive interactions between the microemulsion droplets. The repulsive micellemicelle interactions are due to the entropy of mixing micelle droplets, which may be treated as hard spheres6 The attractive interactions are due to overlap of surfactant tails.'+ It is likely that pressure may have a much more prevalent influence on the natural curvature and intermicellar interactions in a highly compressible liquid or supercritical fluid such as propane than in decane. If so, pressure may be used to adjust the size and phase behavior of microemulsions. Recently, reverse micelles and w/o microemulsions (swollen reverse micelles) composed of bis(2-ethylhexyl) sodium sulfosuccinate (AOT), water, and in some cases salt (NaCl) have been formed in highly compressible liquids and fluids. In many studies, 'Presented at the symposium entitled "Thermodynamics of Microemulsions in Compressible Fluids", American Institute of Chemical Engineers Fall National Meeting, Chicago, Nov 11-16, 1990.
0022-365419 1/2095-9549$02.50/0
a single-phase w/o microemulsion is prepared with a specified water to surfactant ratio It is well-known that the average (1) Peck, D. G.;Johnston, K. P. J . Phys. Chem., previous article in this issue. (2) Kuneida, H.; Shinoda, K. J . Colloid Interface Sci. 1980, 75, 601. (3) Kahlweit, M.; Strey, R.; Schomlcker, R.; Haase, D. Langmuir 1989, 5, 305. (4) Winsor, P. A. Solvent Properties of Amphiphilic Compounds, Butterworths: London, 1954. ( 5 ) Safran, S.A.; Turkevich, L. A. Phys. Reo. Lett. 1983, 50, 1930. (6) Overbeek, J. Th. G . Faraday Discuss. Chem. SOC.1978, 65, 7. (7) Agterof, W. G . M.; Van Zomeren, J. A. J.; Vrij, A. Chem. Phys. Lett. 1976, 43, 363. Calje, A. A.; Agterof, W. G . M.; Vrij, A. In Micellization, Solubilization and Microemulsions; Mittal, K. L., Ed,; Plenum: New York, 1977; Vol. 2, p 779. (8) Lemaire, B.; Bothorel, P.; Roux, D. J . Phys. Chem. 1983, 87, 1023. (9) Roux, D.; Bellocq, A. M.; Bothorel, P. In Surfactants in Solution; Mittal, K.L., Lindman, B., Eds.; Plenum: New York, 1984; Vol. 3, p 1843. (10) Johnston, K. P.; McFann, G . J.; Lemert, R. M. In Supercritical Science and Technology; Johnston, K. P., Penninger, J. M. L., Eds.; ACS Symposium Series 406; American Chemical Society: Washington, DC, 1989; p 140. (11) Yazdi, P.; McFann, G . J.; Fox, M. A.; Johnston, K. P. J. Phys. Chem. 1990, 94, 1224. (12) Lemert, R. M.; Fuller, R. A.; Johnston, K. P. J . Phys. Chem. 1990, 94, 602 1. (13) Gale, R. W.;Fulton, J. L.;Smith, R. D. J. Am. Chem. Soc. 1987,109, 920. (14) Blitz, J. P.;Fulton, J. L.; Smith, R. D. J . Phys. Chem. 1988, 92, 2707. (15) Fulton, J. L.; Smith, R. D. J . Phys. Chem. 1988, 92, 2903. (16) Fulton, J. L.; Blitz, J. P.;Tingey, J. M.; Smith, R. D. J . Phys. Chem. 1989, 93, 4198. (17) Smith, R. D.; Fulton, J. L.; Blitz, J. P.; Tingey, J. M. J . Phys. Chem. 1990, 94, 781. (18) Tingey, J. M.; Fulton, J. L.; Smith, R. D. J . Phys. Chem. 1990, 94, 1997. (19) Steytler, D. C.; Lovell, D. R.; Moulson, P.S.;Richmond, P.; Eastoe, J.; Robinson, B. H. Int. Symp. Supercritical Fluids, SOC.Fr. Chem. 1988, 67.
0 1991 American Chemical Society
