Thermodynamic description of micellization, phase behavior, and

Thermodynamic Description of Micellization, Phase Behavior, and Phase Separation of ... specific interactions between the various surfactant species p...
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J. Phys. Chem. 1992, 96, 5561-5519

5567

Thermodynamic Description of Micellization, Phase Behavior, and Phase Separatlon of Aqueous Solutions of Surfactant Mixtures Sudhakar Puvvadat and Daniel Blankschtein* Department of Chemical Engineering and Center for Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 021 39 (Received: December 9, 1991; In Final Form: March 24, 1992)

We present a thermodynamic framework to describe micellization, phase behavior, and phase separation of aqueous solutions of surfactant mixtures. The theoretical framework consists of evaluating the Gibbs free energy of the solution which is modeled as the sum of three contributions: the free energy of formation of the various solution species, the free energy of mixing micelles, monomers, and water molecules, and the free energy of interactions between the various solution species. By utilizing the conditions of multiple chemical equilibria and thermodynamic stability, all relevant micellar and thermodynamic properties of the solution can be predicted. The predicted properties include (i) the critical micellar concentration (cmc), (ii) the micellar size and composition distribution including its moments, (iii) thermodynamic properties such as the osmotic pressure and compressibility, (iv) the critical line signaling the onset of phase separation, and (v) the coexisting surface bounding the two-phase region of the phase diagram. We find that the predicted properties i-v can be evaluated from knowledge of two molecular contributions: the free energy of mixed micellization, gd, reflecting intramicellar interactions, and a mean-field intermicellar interaction parameter, C., In this paper, we formulate a simplified phenomenological model for g,, and provide a physical rationalization of the typical values that g d c and Cencan attain for various types of binary surfactant mixtures. Using these we are able to predict qualitative trends of micellar and phase behavior properties as a function typical values of gdc and Ccn, of surfactant type and composition. Specifically, we have derived an expression for the mixture cmc which is identical to the well-known expression derived by Rubingh in the context of the pseudophase separation model using the regular solution theory with an empirical interaction parameter c. Therefore, in our formulation, the molecular basis of the parameter c is clarified. We have also derived expressions for the variations of the optimum micellar composition with solution monomer composition and the weight-average mixed micelle aggregation number with total surfactant composition. In addition, utilizing the conditions of thermodynamic stability, the effects of adding small amounts of a second surfactant on the phase separation characteristics of a single-surfactant solution have been investigated. Specifically, we have derived expressions for the change in the critical temperature with total surfactant composition, as well as for the distribution of the added surfactant between the two coexisting micellar phases. We fmd that all our qualitatiue theoreticalpredictionsreproduce very well the experimentally observed trends in aqueous solutions containing nonionic-nonionic, nonionic-ionic, zwitterionic-ionic, and anionic-cationic surfactant mixtures. In addition, we predict some new interesting trends which have not yet been observed experimentally.

I. Introduction Surface-active compounds used in commercial applications typically consist of a mixture of surfactants because they can be produced at a relatively lower cost than that of isomerically pure surfactants. In addition, in many surfactant applications, mixtures of dissimilar surfactants often exhibit properties superior to those of the constituent single surfactants due to synergistic interactions between the surfactant molecules.' Indeed, in solutions containing mixtures of surfactants, the tendency to form aggregated structures (mixed micelles) can be substantially different than in solutions containing only the constituent single surfactants. For example, the critical micellar concentration (cmc) of a mixture of anionic and cationic surfactants in aqueous solution is considerably lower than the cmc's of each individual surfactant.* On the other hand, antagonistic interactions, in mixtures of hydrocarbon-based and fluorocarbon-based surfactants in aqueous solution, result in mixture cmc's that can be considerably higher than the cmc's of the constituent single surfactants.j In general, specific interactions (synergistic or antagonistic) between surfactants result in solutions of surfactant mixtures having micellar and phase behavior properties which can be significantly different from those of the constituent single surfactants. Consequently, understanding specific interactions between the various surfactant species present in the solution is of central importance to the surfactant technologist. Indeed, in order to tailor surfactant mixtures to a particular application, the surfactant technologist has to be able to predict and manipulate (i) the tendency of surfactant mixtures to form monolayers, micelles, and other self-assembling aggregates in solution, (ii) the properties of the formed aggregates such as their shape and size, (iii) the distribution of the various surfactant 'Present address: Center for Bio/Molecular Science and Engineering, Naval Research Laboratory, Washington, D.C. 20375-5000. *Towhom correspondence should be addressed.

species between monomers and aggregates, and (iv) the phase behavior and phase equilibria of solutions containing surfactant mixtures. In spite of their considerable practical importance, as well as the challenging theoretical issues associated with the description of these complex fluids, solutions of surfactant mixtures have not received the full attention that they deserve. Specifically, previous theoretical studies of mixed micellar solutions have evolved along two very different, seemingly unrelated,fronts. On the one hand, significant efforts have been devoted to understand the mixture cmc,"6 as well as the micellar size and composition distribution.7J On the other hand, very little effort has been devoted to understand the solution behavior at higher surfactant concentrations where intermicellar interactions become increasingly important and control the phase behavior and phase separation phenomena? In view of this, it is quite clear that there is an immediate need to develop a theoretical description of mixed micellar solutions capable of unifying the previously disconnected treatments of micellization and phase behavior, including phase separation, into a single coherent computational framework. The central goal of this paper is to contribute to this much needed theoretical unification. The theoretical approach that we p r o p , inspired by our recent work on single-surfactant solutions,I0 consists of blending a thermodynamic theory of mixed micellar solutions, which captures the salient features of these systems at the macroscopic level, with a molecular model of mixed micellization, which captures the essential physical factors at the micellar level. The resulting molecular-thermodynamic approach provides a valuable tool to predict solution properties of mixed surfactant systems using molecular information that reflects (i) the nature of the surfactant molecules involved in the micellization process and (ii) solution conditions such as temperature and the presence of additives such as salts and urea. As such, the molecular-thermodynamic ap-

0022-3654/92/2096-5561~03.~Q/O0 1992 American Chemical Society

5568 The Journal of Physical Chemistry, Vol. 96, No. 13, 1992 proach may be utilized to design surfactant mixtures for a particular application, as well as to modify and control the resulting micellar solution properties. In this paper, we formulate a thermodynamic free energy model of mixed micellar solutions which captures the essential physical factors governing the behavior of these complex fluids in terms of two molecular contributions: the free energy of mixed micellization, gdc, reflecting intramicellar contributions (see section IID for details), and the mean-field interaction parameter, C,, reflecting the magnitude of intermicellar attractions which are responsible for phase separation (see section IIA and Appendix A for details). It is noteworthy that all the thermodynamic properties of the mixed micellar solution can be computed solely from a knowledge of gmicand C,fp The value of Ceffcan be estimated as explained in section IIIA(b). The value of gmiccan be estimated using a molecular model of mixed micellization which accounts for the detailed molecular structure of the various surfactant species, as well as for the effect of solution conditions. Such a molecular model for gmichas been developed1'J2 recently and has been successfully utilized to predict quantitatively a broad spectrum of micellar and phase behavior properties of aqueous solutions containing binary nonionic surfactant mixtures. Furthermore, a simplified version of this model has also been successfully utilized to predict quantitatively cmc's of aqueous solutions containing anionicnonionic, cationic-nonionic, and anionic-cationic surfactant mixtures.I2J3 The calculation of giC, in the context of the detailed molecular model of micellization, is quite involved and needs to be perjiormed numerically. As a result, it is not possible to obtain analytical expressions for micellar and phase behavior properties of interest. Furthermore, in view of the numerical nature of the resulting quantitative predictions, it is not straightforward to gain a clear appreciation of the relative importance of the various molecular factors contributing to the predicted properties. In view of this, in this paper, we decided to present a simplified model for gmi, (see section IID(c) for details). Although we recognize that with this simplified description of gd our predictions will be qualitative in nature, the proposed approach offers a number of useful features. These include the following: (i) Analytical expressions for many of the micellar solution properties of interest can be derived (see, for example, eqs 41,46, and 48). (ii) Qualitative trends can be predicted without the need of performing rather complex and time-consuming numerical calculations of the type presented in refs 10-12. In other words, the simplified approach proposed here can serve as a useful preliminary screening test in the design and selection of surfactant mixtures of practical importance. (iii) It is possible to rationalize the molecular basis of the predicted trends, without the complicating effects of the numerical calculations. The remainder of the paper is organized as follows. In section I1 we formulate the general thermodynamic framework to describe mixed micellar solutions. Specifically, in section IIA we present the mathematical structure of the various contributions to the Gibbs free energy of the mixed micellar solution and derive expressions for the chemical potentials of the various solution components. In section IIB we derive expressions for the micellar size and composition distribution and its moments, and in section IIC we analyze the phase behavior and phase separation phenomena. An important contribution to the micellar solution Gibbs free energy is the free energy of mixed micellization, gmic, and a simplified model for gdc is developed in section IID. In section I11 we present and discuss qualitative predictions of the thermodynamic framework for aqueous solutions containing binary mixtures of nonionic-nonionic, nonionic-ionic, zwitterionic-ionic, anionic-cationic, and nonionic hydrocarbon-nonionic fluorocarbon surfactants. The predictions include (i) the cmc variation with surfactant monomer composition, (ii) the variation of micellar composition with surfactant monomer composition at the cmc, (iii) the variation of the weight-average mixed micelle aggregation number with total surfactant composition, and (iv) the effects of adding small quantities of a second surfactant on the phase separation characteristics of the micellar solution, including the shift

Puwada and Blankschtein in the critical temperature as well as the distribution of the two surfactant species between the coexisting micellar-rich and micellar-poor phases. Finally, in section IV we present some concluding remarks. 11. Thermodynamic Framework A. Cibbs Free Energy and Chemical Potentials. The thermodynamic formulation used to describe the free energy of a mixed surfactant solution constitutes a generalization of the one developed to describe single-surfactant s01utions.l~For the sake of clarity, in this paper we present a theoretical description of aqueous solutions of a mixture of two surfactants. Extension of the formalism to describe solutions containing additional surfactant species is conceptually similar and therefore will not be discussed. Consider a solution of N , water molecules, NA surfactant A molecules, and NB surfactant B molecules in thermodynamic equilibrium at temperature T and pressure P. If the concentration of the surfactant mixture exceeds its cmc, the surfactant molecules will self-assemble to form a distribution of mixed micelles {N,,,], where N,,, is the number of mixed micelles having aggregation number n and composition a. Note that in such a mixed micelle there are na surfactant A molecules and n( 1 - a) surfactant B molecules. Note also that NA = C,,paN,,, and NB = &n(l - a)~,,,. In the spirit of the multiple-chemical equilibrium descript~on,'~ mixed micelles of different sizes and compositions are treated as distinct species in chemical equilibrium with each other as well as with the free monomers in the solution. The Gibbs free energy of the mixed surfactant solution C is modeled as the sum of three contributions: the free energy of formation Cr, the free energy of mixing G,, and the free energy of interaction Ci. These contributions, as in the single-surfactant case,14are chosen to provide a heuristically appealing identification of the various factors responsible for micelle formation and growth, on the one hand, and for phase behavior and phase equilibria, on the other. The free energy of formation is expressed as

= Nwk'w

+ NAF'A + NBP'B + CnNnagmic(sh,n,a)

(1)

n,a

where pow(T,P),woA(T,P),and poB(T,P)are the standard-state chemical potentials of water, surfactant A monomers, and surfactant B monomers, respectively, at the solution temperature T and pressure P; gmi,(sh,n,a)is the free energy of mixed micellization, which represents the free energy change per monomer associated with transferring na surfactant A monomers and n( 1 - a) surfactant B monomers from water into a mixed micelle of shape sh, aggregation number n, and composition a. The numerical magnitude of g&, which reflects the propensity of a mixed micelle to form and subsequently grow, summarizes the many complex physicochemical factors responsible for mixed micelle formation such as the hydrophobic effect, hydrogen bonding, conformational changes associated with hydrophobic-tail packing in the micellar core, steric and electrostatic interactions between the hydrophilic head groups, and the entropy of mixing the two surfactant species in the mixed micelle.lOJ1 The free energy of mixing the formed mixed micelles, free monomers, and water is modeled by an expression of the form G, = kT[N, In X,

+ EN,,,

In X,,,]

(2)

n,a

where X, = N,/(N, + NA + NB),X, = N,/(N, + N A + NB), k is the Boltzmann constant, and Tis the absolute temperature. -Gm/T is an entropic contribution which reflects the number of ways in which the distribution of mixed micelles, the free monomers, and the water molecules can be positioned in the solution as a function of the solution concentration and composition. The free energy of mixing, as expressed in eq 2, opposes the tendency to form micelles because monomer aggregation reduces the total number of available spatial configurations. The entropy of mixing also opposes the tendency of the micellar solution to phase separate, because of the loss in available spatial configurations associated with this phenomenon. Note that eq 2 is a generalization of the

The Journal of Physical Chemistry, Vol. 96, No. 13, 1992 5569

Aqueous Solutions of Surfactant Mixtures

r expression used with considerable success to d e ~ c r i b e ~ ~ * ' ~ * ~ ~ J ' micellization, phase behavior, and phase separation in singlesurfactant solutions. The free energy of interaction reflects interactions between mixed micelles, water molecules, and free monomers in the solution. At the level of a mean-field type quadratic expansion, in Appendix A we show that this free energy contribution takes the following form The chemical potentials of surfactant A and surfactant B monomers can be obtained from eq 6 by substituting n = 1 and a = 1 (for A) or 0 (for B), respectively, that is where 4 = 'A is the sum of the volume fractions, 'A and &,, of surfactants A and B, respectively, = NA/(NA NE) is the composition of the surfactant mixture, and C,~(CY,~,,) is an effective mean-field interaction parameter for the mixture which is related to the single-surfactant interaction parameters CAW and CBwand a specific interaction parameter CABthrough the folwhere X1A and X I Bare the mole fractions of free surfactant A lowing expression (see Appendix A) and surfactant B monomers, respectively. Using eqs 5-10 for the chemical potentials, below we predict various properties of the Ccfdasoln) = mixed micellar solution. CAWasoln + cBW(1 - asoh) - CABasoln(l - a m l n ) ( G / Y e f f ) B. Micellar Siand Composition Distribution and IQ M o m & When the mixed micellar solution is in thermodynamic equilib(4) rium, the chemical potential p, of a mixed micelle of aggregation number n and composition a is related to the chemical potentials In eq 4, yA = n A / n w and yB= nB/fl,, where n, nA,and nBare of the free monomers through the constraints imposed by the the effective molecular volumes of water, surfactant A, and ~A (1 - asoln)yB. conditions of multiple chemical equilibrium,'* that is surfactant B, respectively, and Teff = a s o l n + Note that, as stressed in the Introduction, the Gibbs free energy pna = ~ W A+ 4 1 - ~ ) P B (11) expressions given in eqs 1-3, as well as all the thermodynamic Equation 11 implies that the chemical potential of a mixed micelle properties derived from it, are uniquely determined by the two having aggregation number n and composition a is equal to the molecular contributions gmic and Ccrp A detailed discussion of sum of the chemical potentials of its constituent na surfactant gmic is presented in section IID, and a physical rationalization of A and n(1 - a) surfactant B molecules. Substituting eqs 6,9, the typical values that gmic and Ccfrcan attain for various types and 10 in eq 11, we obtain the following expression for the of binary surfactant mixtures is presented in sections IIIA(a) and equilibrium micellar size and composition distribution IIIA(b), respectively. The thermodynamic consequences of the proposed free energy formulation are described below. The properties examined include the cmc, the micellar size and composition distribution and its moments, and the phase behavior and phase equilibria. All these where fl = l/kT, flg, = [ f i g ~-c1 - a In a1- (1 - a) In (1 - a l ) ] properties are governed by the proposed Gibbs free energy model is a modified dimensionless free energy of mixed micellization per through the chemical potential of water, p,, and the chemical monomer, X 1 = X1A X I Bis the total mole fraction of free potential of a mixed micelle, of aggregation number n and commonomers in solution, and al = X I A / X is I the composition of free position a,pn,, which are obtained by differentiating the Gibbs monomers in solution. Equation 12 indicates that a delicate free energy, eqs 1-3, with respect to N, and Nn,, respectively. balance between two opposing factors determines the nature of The resulting expressions are given by the micellar size and composition distribution {X,]. The first is the Boltzmann factor, e-'@g*, representing the energetic advantage < 0) of assembling the various surfactant mol(recall that gmic ecules in a mixed micelle, which favors micelle formation. The second factor, XIAnaXIBn(l-,), represents the large entropic disadvantage associated with localizing na surfactant A molecules (each with probability X I A )and n( 1 - a) surfactant B molecules (each with probability X I B in ) a single mixed micelle and opposes micelle formation. It is noteworthy that, with the choice of mean-field interaction potentials adopted in eq 3 (see also Appendix A), the interaction free energy does not affect the micellar kT) + kT(1n X,,, n(X - 1 - EX,,,)) + nap'A + n(l - a)& size and composition distribution. nu The composition a*(n),at which X,, exhibits a maximum for (6) a given micellar aggregation number n, is referred to hereafter as the optimum composition. Note that, in general, a*(n)is a where X = X A + X B ,with X A = NA/(Nw + NA + NE) and X B function of the aggregation number n and can be obtained by = NB/(Nw NA N E ) ,is the total mole fraction of surfactant setting the derivative of X,,, with respect to a equal to zero. in the solution, and the interaction contributions to the monomer Specifically, implementing this procedure with eq 12 leads to the chemical potentials are given by following implicit equation

+

+

+

+

+

p'A

= -3: '[CAW

+

+ $[asolncAW

+ (l

(1 - asodand

- asoln)cBWl(l

- 4) -

6CAB(1 - 4 A ) Ycrr

1

(7)

Using eq 13, the optimum compition for all aggregation numbers can be determined from a knowledge of a, and gmic(n,a).In addition, note that, for large n,X, will exhibit a sharp maximum at a = a* because n appears as a multiplicative factor in the exponential of eq 12, and consequently any small deviations of a from a* will be magnified significantly. This implies that, to

5570 Tke Journal of Physical Chemistry, Vol. 96, No. 13,.1992 leading order in a, the mole fraction of micelles having compositions a # a* will be negligible, and accordingly, E&,, can be approximated by Xna.. Equation 12 for the micellar size and composition distribution is applicable to mixed micelles of all shapes, sizes, and compositions. However, as shown clearly in eq 12, to determine the distribution one needs to know (i) the free energy of micellization g,,&,a) as a function of n and a or, equivalently, g, as a function of n, a, and a I ,(ii) the equilibrium solution monomer mole fraction X I ,and (iii) the equilibrium solution monomer composition a I . Note that conditions ii and iii are equivalent to knowing X I A= a l X l and X I B= (1 - a l ) X I . Note also that X I and a , (or equivalently X I , and XIB)can be found by using eq 12 in the two constraints imposed by the conservation of the total number of surfactant A and surfactant B molecules in solution, that is, NA = CnanaNnaand N B = Enan( 1 - a)Nna,or equivalently XA = asolJ = a1Xl

+ End,,, n.a

(14a)

+ En( 1 - a)Xna (14b)

X , = (1 - aSoln)X = (1 - a l ) X 1

n,a

Given gmiC (or equivalently g,,,), and on inserting eq 12 into eqs 14a and 14b, one obtains two implicit equations for X I and a1 as a function of X and awl,,. Solving these two equations simultaneously one can, in principle, obtain X,(X,a,,,,T,P) and al(X,asoln,T,P), which can then be inserted back into eq 12 to calculate the entire micellar size and composition distribution as a function of X , asoh,T, P, and other solution conditions. Note that, in general, the free energy of micellization gmiC(n,a)will depend on the type and molecular structure of the two surfactant species present in the mixture, as well as on solution conditions such as temperature, pH, and ionic strength. A detailed molecular model to evaluate gmiC has been developed and will be presented elsewhere." However, as explained earlier, to illustrate some general features and prediction capabilities of the thermodynamic framework, in the present paper we develop a simplified analytical phenomenological model for gmiC (see section IID). Having derived an expression for the micellar size and composition distribution {Xnal,we describe next how the various moments of the distribution are related to one another. We define the moment of order k by Mk = EnankXna.Note that the first moment is given by M I= X , and the zeroth moment, Mo = E,,,&,is proportional to the total number of mixed micelles and free monomers in the solution. Note also that the zeroth moment plays an important role in determining the value of colligative properties such as the osmotic pressure (for example, see eq 22). In general, using eq 12, the kth moment can be expressed as

w),

Mk = Ce-lnk(X,)ne-nSg,(a,ai)

(15)

na

-1-

The total derivative of Mk with respect to X I is given by -d M=k - HI

Mk+l

XI

Znk+lXna[f f - - 1 na 1 -a1

1 XI

(16)

Since, as stated above, we assume that, for values of a # a*,X , is negligible, it can be shown (seeAppendix B) that to a very good approximation

By applying the chain rule to eq 17 we obtain dMk Mk+l zdMj Mj+l and by setting k = 1 in eq 17 we obtain

Puwada and Blankschtein pressed solely in terms of the second moment M 2 of the distribution. Note that M 2can be related to the weight-average aggregation number ( t ~of) the ~ mixed micelles through ( n ) w(Xyasoln,T,P) =

M2(X,asoln,T,P)/X

(20) In addition, the relative variance of the distribution is given by the following expression

It is noteworthy that both the weight-average mixed micelle aggregation number and the relative variance can be measured experimentally and through a comparison with eqs 20 and 21 can therefore serve as useful indicators of the applicability of our thermodynamic framework to specific solutions of surfactant mixtures. Note also that eqs 18-21, describing the various moments in the mixed micellar case, are very similar to those developed for single-surfactant solution^.'^ C. Phase Behavior and Phase Separation. Having derived expressions for the micellar size and composition distribution and its moments, the thermodynamic framework is used next to describe the phase behavior and phase equilibria of mixed micellar solutions. In particular, the osmotic pressure T can be related1* to the water chemical potential by T = (pow- pw)/flw,where as stated earlier nWis the effective volume of a water molecule. Using eq 5 we obtain -@Tilw= In (1 - X ) + X - MO(X,asoln,T,P) + Cedasod(@/2~ea) (22) The osmotic compressibility of the solution (a?r/dx);$,,. can be obtained by differentiating eq 22 with respect to X , at constant T, P, and aSoln. This yields

(23) It is interesting to point out that the mathematical structures of eqs 22 and 23 for the osmotic pressure and the osmotic compressibility are identical to those obtained earlier for single-surfactant solutions.14 When solution conditions such as temperature, pressure, or ionic strength are altered in a ternary solution consisting of two surfactant species and water, stability criteria could be violated, and the solution may separate into two or more phases in thermodynamic equilibrium with each other.ls In this paper, we restrict our discussions to two-phase equilibria. The boundary between the stable and unstable regions of a ternary solution is known as the spinodal surface, while that between the two-phase and one-phase regions is known as the coexistence surface. Note that a binary system at fixed pressure will have a single temperature versus concentration spinodal line and coexistence curve. However, a ternary system at fmed pressure will have a family of spinodal lines constituting a spinodal surface and a family of coexistence curves contstituting a coexistence surface in the three-dimensional temperature, total concentration, composition coordinate system. In general, the spinodal surface in a ternary system is describedlg by the following stability condition

(18)

Equations 18 and 19 show that with our choice of Gibbs free energy, eqs 1-3, all the moments of the distribution can be ex-

where G is the Gibbs free energy of the system. On the spinodal surface, the locus of points where the two coexisting phases become indistinguishable is the line of critical points. Note that, at a fixed pressure, there is a single critical point in a binary solution, whereas in a ternary solution there is a line of critical points. Note also that the critical line is the line of tangency between the spinodal surface and the coexistence surface. Thus, at the critical line,

The Journal of Physical Chemistry, Vol. 96, No. 13, 1992 5571

Aqueous Solutions of Surfactant Mixtures in addition to $ = 0, a second stability condition

conditions, is required. Such a model has recently been developed" and has been successfully utilized to make quantitative predictions of a broad spectrum of micellar and phase behavior properties of aqueous solutions containing mixtures of nonionic-nonionic surfactants."J2 needs to be satisfied.Ig A simultaneous solution of the two stability The free energy of micellization is a function of the shape of conditions, = 0 and C#I = 0, yields the entire critical line. the micelle sh, the aggregation number n, and the composition As described earlier,the coexistence surface bounds the unstable a. It is well-known that n can range from very small to very large two-phase region within which a solution spontaneously separates values. Since gmic(sh,n,a)needs to be evaluated for all n and a, the problem becomes computationally intensive. For the purpose into two isotropic phases. The lines connecting the two coexisting phases in the phase diagram are called tie lines, and the critical of illustration, we can simplify the calculations by making the line corresponds to the locus of points at which the length of the following two physically reasonable assumptions. First, as extie lines vanishes. The conditions of phase equilibria'* require plained in section IIB, we approximate E &, by X,*, where a* is the optimum composition of a micelle having aggregation that the temperature, the pressure, and the chemical potentials of each of the components in the solution be the same in the two number n. Second, we evaluate gmiconly for the three regular coexisting phases. Specifically, for the ternary water, surfactant shapes of spheres, infinitesized cylinders, and infinite-sized disks A and surfactant B system, p,( T,P,Y,ay) = pw(T,P,Z,az),pAor bilayers. For the nonregular finite-sized mixed micelles, (T,P,Y,ay) PA(T,P,ZPZ), and P B ( T , P , Y , ~= Y )PB(T,P,Z,~Z), gmic(n,a*)and a*(n) are estimated by linearly interpolating bewhere Y and Z are the total surfactant mole fractions in each tween the g,&,a*) and a*(n)values corresponding to the limiting coexisting phase, respectively, and ayand az are the corresponding regular shapes. surfactant compositions. The three chemical potentials h,p A , (b) Illustrative Example: Finite-Sized Cylindrical Mixed and pe can be expressed as derivatives of the Gibbs free energy Micelles. To describe a finite-sized cylindrical mixed micelle per particle, g = G / ( N , NA NB), and the resulting expressions exhibiting one-dimensional growth, gmic(n,a*)and a*(n) are are given byI9 estimated by interpolating between the optimum values of these quantities for a spherical micelle and an infinite-sized cylindrical ag micelle, that is pw= g - x -

+

+ +

ax

gmic(nra*) = &%(a*cyl) + (nsph/n)

[&%a*sph)

- &y~c(a*cyl)l (28)

and

Using eqs 26a-26c, the three conditions of phase equilibria can be reexpressed in the following mathematically useful forms y,ay) -Y1 ag( T,P, aay

1

ad T , P z , ~ , )

I-

z

aaz

(27a)

a*(n) = a*cyl + (nsph/n)[a*sph - a*cyll (29) where hpb and a*,ph are the aggregation number and composition of the optimum spherical micele, is the composition of the optimum infinite-sized cylindrical micelle, and g"ik andCi;& are the free energies of micellization of the optimum spherical and the optimum infinite-sized cylindrical micelles, respectively. Substituting eqs 28 and 29 in the expression for X , given in eq 12, we obtain

+

Using the Gibbs phase rule,'* a two-phase ternary solution in thermodynamic equilibrium has three independent intensive variables. Therefore, fixing temperature, pressure, and one additional intensive variable the remaining intensive variables are uniquely specified. Accordingly, by fixing one of the four intensive variables Y, 2,ay,and az in eqs 27a-27c, the remaining three can be calculated. In other words, at fixed pressure, the entire family of coexistence curves in the T-X-amIncoordinate system can be generated. D. Modeling the Free Energy of Micellization. ( a ) General Considerations. As described earlier, the free energy of micellization g,,&h,n,a) summarizes the many physicochemical factors responsible for mixed micelle formation and depends on the molecular structure of the surfactants, as well as on solution conditions such as temperature, pH, and the presence of any additives. To illustrate some general features and qualitative prediction capabilities of the thermodynamic framework, below we present a simplified analytical phenomenological model for g&(sh,n,a). Although the predictions based on such a model will be qualitative in nature, major advantages of utilizing a simplified model for g& are that it enables us to obtain analytical expressions for many of the useful micellar solution properties as well as to shed light on the physical basis of some of the observed experimental trends. For more detailed and accurate quantitative predictions of micellar solution properties, a molecular model of mixed micellization, which takes into account the detailed molecular structure of the surfactants as well as the effect of solution

where K = e+, with Ap = nsph[g"$h - g",] kT,and X,, = &Em*. Recall that /3gm= /3gmic- 1 - a In aI- (1 - a) In (1 - a,).Note that the parameter Ap is a growth parameter analogous to the one introducedZ0J4to describe the one-dimensional growth of single-surfactant cylindrical micelles. Clearly, in order to have growth, one expects A p > 0, where an increase in the value of Ap will result in an increase in micellar size. Note also that the concentration Xcll reflects the propensity to form cylindrical micelles and that, 111 the limit of considerable growth, one has Xcyl = cmc. It is possible to show, following an analysis similar to the one utilized in the case of single-surfactant cylindrical m i c e l l e ~ , ~ ~ J ~ that, in the limit ( n ) , >> naph,the following remarkably simple result for X,,,, is obtained

x,,,.= (1 /K)e-n("-"'

(31) Equation 30 shows that, in the limit of significant micellar growth, the micellar size distribution, corresponding to the optimum composition a*,is a monotonically decreasing exponential function of n (for n > h p h ) whose width is directly proportional to ("I2. As stated earlier, if we approximate Ea,,, by X,,,.,the second moment M 2of the distribution (see eq 15 with k = 2) is given by M 2 = nsph 2K'/2312 (32) Similarly, the weight-average mixed micelle aggregation number (see eq 20) is given by

+

( n ) , = M2/X

e

nsph

+ 2(KX)'l2

(33)

Puwada and Blankschtein

5572 The Journal of Physical Chemistry, Vol. 96, No. 13, I992 Note that the analysis of Ben-Shaul et al.' suggests that when the entire size and composition distribution is utilized, instead of only that corresponding to the optimum composition (n), varies as P4 instead of 2,5 as predicted by eq 33. This, of course, does not affect the qualitative conclusions presented in this paper. In addition, the relative variance of the distribution (see eq 21) is given by

w],

u

'/2

wm*},

(34)

(c) Simplified Model for gMc.Below we introduce a simplified model to estimate the free energy of mixed micellization gdc(sh,a) corresponding to the three regular shapes of spheres, infinite-sized cylinders, and infinite-sized disks or bilayers. As illustrated in (b) above, for nonregular finite-sized micelles, gmiC is estimated by linearly interpolating between the optimum gmiC values corresponding to the limiting regular shapes. We model gmiC as the sum of four primary contributions:l03" (i) Hydrophobic free energy gw,fic(sh,a), which represents the free energy gain in transferring the hydrophobic tails from water to the hydrophobic interior of a mixed micelle characterized by a shape sh and composition a. This contribution can be further expressed as gwlfic(sh,a) = gw/hc(a) ghc/fic(sh,a). The quantity gw/hc reflects the free energy gain associated with transferring the hydrophobic tails from water to a bulk phase, having composition a,made from the tails of the two surfactant species (note that, as expected, this contribution is independent of micellar shape), and can be expressed as

+

gw/hc

= a d j h c + (1 - a)g/hc + k T [ a In a + (1 - a ) In (1 - a ) ] + a ( l - a)&!

where &/hc and &/hc are the hydrophobic free energy contributions associated with pure surfactant A tails and B tails, respectively, k T [ a In a (1 - a) In (1 - a)]reflects the free energy of mixing the two tails of the two surfactant species in the mixed micelle, and & ! reflects the strength of the binary interactions between surfactant A and B tails based on the regular solution theory. Note that &/hc and &/hc can be estimated using available solubility data of pure hydrocarbons (or fluorocarbons) in water.10," Note also that d! is typically equal to zero for a mixture of hydrocarbon (or fluorocarbon)-based surfactants1 but is greater than zero3 for mixtures of hydrocarbon- and fluorocarbon-based surfactants which exhibit repulsive antagonistic interactions. The quantity ghc/mic(sh,a), which reflects the free energy loss associated with the reduction in conformational degrees of freedom of the two types of surfactant tails (at composition a)in the constrained environment provided by the micellar core, depends on micellar shape. For computational simplicity, we assume that this contribution is linear in composition, that is, gkImic = a&c/mic + (1 - a)dc/mic, where d c / m i c and d c / m i c are the conformational free energy contributions associated with pure surfactant A tails and B tails, respectively. Note that &c/mic and dCm,c can be computed numerically using a single-chain meanfierb 'approach.IoJ1 (ii) Interfacial free energy g,,(sh,a), which represents the free energy per monomer associated with creating an interface separating the micellar core from the bulk solution. This contribution can be approximated as g, = a d + (1 - a ) z , where d = U A U A and = gag. Note that the interfacial tensions uA and uB associated with the tails of surfactants A and B, respectively, are available experimentally. Furthermore, for a given micellar shape, the areas per monomer uA and aBcan be determinedlo." from the known chemical structures of surfactant A and B tails, respectively. (iii) Steric free energy g,,(sh,a), which represents steric interactions between the surfactant headgroups at the micellar surface. These interactions have been shown1° to play an important role in the micellization of nonionic surfactants. By treating the headgroups at the micellar surface as an adsorbed localized monolayer, it can be shown" that g,, can be approximated by a d + (1 - a)& where = -kT In (1 - q,A/a) and & = -kT 1n (1 - ahB/a) are the steric contributions associated with the headgroups of pure surfactants A and B, respectively. Note that the average headgroup cross-sectional areas ahA and a h B can be

+

estimated from the known chemical structures of pure surfactant A and B headgroups, respectively, and for a given micellar shape, the available area per monomer, a, can be evaluated from the known chemical structures of pure surfactant A and B tails. (iv) Electrostatic free energy gel,(sh,a), required to describe charged surfactants, which represents the free energy associated with creating a charged interface in a sea of counterions. From simple electrostatic considerations2'J3one expects that, to leading order, gel, should be proportional to q2,where q is the net charge per monomer at the micellar surface. Consequently, for a binary surfactant mixture, q = aezA (1 - a)ezB,where zA and zBare the valencies of the two surfactants, and e is the electronic charge. Thus, gel, * a& + (1 + 4 1 - a)dLwhere = K e l d b 2 and d, = K e , d B 2are the electrostatic contributions associated with pure surfactants A and B, respectively, and A,", = -Kels(zA - z ~reflects ) ~ binary electrostatic interactions between surfactants A and B. Note that Kk is a numerical constant which can be evaluated from electrostatic t h e o r i e ~ ' and ~ . ~ depends ~ on micellar shape. In addition, Kdocis expected to be a strong function of salt concentration and should decrease (screening effect) with increasing salt concentration. Note that although in general K,,, may also be a function of surfactant type and composition, for illustrative purposes, in this paper we assume that K,,, is a constant. Combining the four contributions described above, gmiC can be expressed using the following simple relation

+

C,

gmic(sh,a) = agA,ic(sh) + (1 - a)&ic(sh) + a(1 - a)& + k T [ a In a+ (1 - a) In (1 - a ) ] (35) where $,ic(sh) = (d/hc + d c / m i c + d + + d w ) is the free energy of micellization of pure surfactant A, Zic(sh) = (&/hc + &c/mic 8 & #,=) is the free energy of micellization of pure surfactant B, and &,(sh) = & ! + reflects contributions due to specific intramicellar interactions. Note that, as emphasized earlier, $mic,g",,, and $,E can be computed using readily available experimental information about surfactants A and B as well as some numerical calculations.1"'2 A detailed rationalization of typical values that these contributions can attain for various binary surfactant mixtures is presented in section IIIA(a). The modified free energy of micellization g, at the optimum composition a*, namely, g,(sh,a*,al), can be evaluated by inserting eq 35 into eq 13 and using the resulting relation in the definition of g,. Carrying out this procedure, we find that g,(sh,a*,ai) = \

+ + +

( ; ;;) I

&i,(sh)

+ $m:(sh)a*' + kT In 1

kT (36)

where a* can be obtained by solving eq 13 with gmiC given in eq 35. Since the growth parameter Ak = nSph(g",ph - g$) + kT, then upon using eq 36 for an optimum spherical micelle and an optimum infinite-sized cylindrical micelle in this definition we find that Ab' =

nsph

I

(&ic)sph - (&ic)cyl

+ (&&)sph("*sph)' -

111. Results and Discussions

A. Estimation of Molecular Contributions. ( a ) Zntramicellar Conrriburions g A .,giic, and g i t . As explained in section IID(c), dit, g",,, and determine the value of g, (see eq 35). Since the model for gmiC utilized in this paper is an approximate one, we can only predict qualitative trends of micellar solution properties. For this urpose, below we discuss the range of typical values that $,,, and &E can attain for various types of binary surfactant mixtures. In general, micellization requires that the free energy of micellization be negative. In particular, it follows that both $,,,(sh)

2:

AiC,

The Journal of Physical Chemistry, Vol. 96, No. 13, I992 5513

Aqueous Solutions of Surfactant Mixtures TABLE I: Typical Values of the Intramicellar Contribution the Specific Interaction Parameter CAB

surfactant mixture nonionic-nonionic monovalent anionic-monovalent anionic

-k .! ? l k T

-0 -0 nonionic-monovalent ionic -2 to-6 monovalent anionic-monovalent cationic -10 to -25 hydrocarbon-fluorocarbon (nonionics) >O

$2 and

C.n.-.d k T

surfactant type ._

-0 50

nonionic zwitterionic ionic

0. The magnitude of CAB for various surfactant mixtures can be estimated using similar arguments. In anionicanionic or cationic-cationic surfactant mixtures, the molecules interact primarily through electrostatic repulsions, and if the two surfactants have the same valency, then UAB = UAA = UBB > 0. In that case, eq A8 indicates that CAB a -UAA(nA - flB)‘! (QAQB)3/2,which results in negative C A B values. Since C A B is proportional to (Q, - ne)’, ionic surfactants having very dissimilar sizes have larger values of CAB than those having similar sizes.

+

5578 The Journal of Physical Chemistry, Vol. 96, No. 13, I992

In nonionic-ionic mixtures, electrostatic repulsions between the ionic (species A) surfactants leads to very large and positive values of U A A . However, interactions between ionic and nonionic surfactants (present in UAB)or between two nonionic surfactants (present in UBB)are much smaller. Thus, nonionic-ionic mixtures will typically have CAB< 0. In monovalent anionic-monovalent cationic mixtures, both U A A and UBBare large and positive due to electrostatic repulsions. However, attractive interactions between the oppositely charged surfactant A and B molecules leads to a large but negative value of UABwhose absolute magnitude is of the same order as that of U , or UBB.Thus, CAB