Prediction of Critical Micelle Concentrations and Synergism of Binary

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Langmuir 1997, 13, 3968-3981

Prediction of Critical Micelle Concentrations and Synergism of Binary Surfactant Mixtures Containing Zwitterionic Surfactants Anat Shiloach and Daniel Blankschtein* Department of Chemical Engineering and Center for Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 Received February 18, 1997. In Final Form: May 14, 1997X We present a simplified “working model” of a recently developed molecular-thermodynamic theory of mixed surfactant solutions which can be utilized to predict critical micelle concentrations (cmc’s) and synergism of binary surfactant mixtures in which one of the surfactants is zwitterionic. Given (i) the chemical structures of the hydrophilic heads and the hydrophobic tails of the surfactants, (ii) the cmc’s of the pure surfactants, and (iii) the solution conditions such as temperature and the concentration of added salt, the model can predict the βAB interaction parameter characterizing synergism in mixed micelle formation and the mixture cmc as a function of solution composition. The model considers electrostatic interactions as the main contribution to synergism in mixed micelle formation, initially accounting for these interactions at an approximate level. The model applies only to surfactants with linear hydrocarbon tails and neglects synergism due to steric interactions between the surfactant heads or due to the packing of the surfactant tails in the micellar core. The simplified working model is further examined in the context of a more rigorous treatment of the electrostatic interactions, and the electrostatic approximations underlying the simplified model are evaluated and shown to be reasonable. The model predictions are compared to experimental mixture cmc data and to βAB interaction parameters extracted from experimental measurements. Zwitterionic surfactants often exhibit specific interactions when mixed with anionic or cationic surfactants, because zwitterionic surfactants that can donate a proton can acquire a negative charge, while those that can accept a proton can acquire a positive charge. Although the simplified working model does not explicitly account for this specificity, the predictions agree reasonably well with experimental data for mixtures in which the zwitterionic surfactant acquires a partial charge opposite to that of the ionic surfactant with which it is mixed.

I. Introduction Surfactant mixtures are commonly utilized in many surfactant formulations and practical applications, because mixtures often behave synergistically and provide more favorable or desirable properties than the constituent single surfactants.1-6 Zwitterionic surfactants, whose hydrophilic polar heads carry both a positive and a negative charge, are interesting in several respects. They exhibit pH-dependent behavior, display a high foam stability, and are less irritating to the skin than many ionic surfactants.7 Because of these useful characteristics, zwitterionic surfactants are often combined with anionic or cationic surfactants in many consumer products, such as shampoos and detergents.8-10 Mixtures of zwitterionic and ionic surfactants also exhibit interesting rheological * To whom correspondence should be addressed. X Abstract published in Advance ACS Abstracts, July 1, 1997. (1) Holland, P. M.; Rubingh, D. N. Mixed Surfactant Systems; Holland, P. M., Rubingh, D. N., Eds.; ACS Symposium Series 501; American Chemical Society: Washington, DC, 1992; p 1. (2) Holland, P. M. Mixed Surfactant Systems; Holland, P. M., Rubingh, D. N., Eds.; ACS Symposium Series 501; American Chemical Society: Washington, DC, 1992; p 31. (3) Rosen, M. J. Phenomena in Mixed Surfactant Systems; Scamehorn, J. F., Ed.; ACS Symposium Series 311; American Chemical Society: Washington, DC, 1986; p 144. (4) Scamehorn, J. F. Phenomena in Mixed Surfactant Systems; Scamehorn, J. F., Ed.; ACS Symposium Series 311; American Chemical Society: Washington, DC, 1986; p 1. (5) Hill, R. M. Mixed Surfactant Systems; Ogino, K., Abe, M., Eds.; Surfactant Science Series 46; Marcel Dekker: New York, 1993; Chapter 11. (6) Rosen, M. J. J. Am. Oil Chem. Soc. 1989, 66, 1840. (7) Tsubone, K.; Uchida, N.; Mimura, K. J. Am. Oil Chem. Soc. 1990, 67, 455. (8) Tsujii, K.; Okahashi, K.; Takeuchi, T. J. Phys. Chem. 1982, 86, 1437. (9) Lomax, E. G. Amphoteric Surfactants; Lomax, E. G., Ed.; Surfactant Science Series 59; Marcel Dekker: New York, 1996; Chapter 7.

S0743-7463(97)00160-1 CCC: $14.00

behavior. For example, adding an ionic surfactant to a viscous zwitterionic surfactant solution causes an initial increase in the viscosity, followed by a decrease in viscosity at higher ionic surfactant compositions.11 The widespread use of surfactant mixtures which contain zwitterionic surfactants would make a predictive theory for these types of mixtures very valuable. Most existing theories that describe the behavior of surfactant mixtures are either not predictive or do not explicitly treat zwitterionic surfactants. The widely used regular solution theory in the context of the pseudophase separation approximation, for example, uses experimental mixed critical micelle concentration (cmc) measurements as inputs and does not account for the molecular structure of the surfactant.12 Recent molecular-thermodynamic theories for single surfactants have dealt with zwitterionic surfactants,13 but the extensions of these theories to surfactant mixtures have not considered ionic/zwitterionic or nonionic/zwitterionic mixtures.14-16 A notable exception is the mixed micelle theory of Nagarajan,17,18 in which the dipolar electrostatic contribution due to the zwitterionic surfactants is treated explicitly in the context of a capacitor (10) Palicka, J. Amphoteric Surfactants; Lomax, E. G., Ed.; Surfactant Science Series 59; Marcel Dekker: New York, 1996; Chapter 8. (11) Hoffmann, H.; Rauscher, A.; Gradzielski, M.; Schulz, S. F. Langmuir 1992, 8, 2140. (12) Rubingh, D. N. Solution Chemistry of Surfactants, Vol. 1; Mittal, K., Ed.; Plenum Press: New York, 1979; p 337. (13) Nagarajan, R.; Ruckenstein, E. Langmuir 1991, 7, 2934. (14) Nishikido, N. Mixed Surfactant Systems; Ogino, K., Abe, M., Eds.; Surfactant Science Series 46; Marcel Dekker: New York, 1993; Chapter 2. (15) Graciaa, A. P.; Lachaise, J.; Schechter, R. S. Mixed Surfactant Systems; Ogino, K., Abe, M., Eds.; Surfactant Science Series 46; Marcel Dekker: New York, 1993; Chapter 3. (16) Nguyen, C. M.; Rathman, J. F.; Scamehorn, J. F. J. Colloid Interface Sci. 1986, 112, 438. (17) Nagarajan, R. Langmuir 1985, 1, 331.

© 1997 American Chemical Society

Prediction of Critical Micelle Concentrations

model. In our initial, simplified electrostatic description of an ionic/zwitterionic mixed micelle, we also incorporate the capacitor model. In addition, we develop a more rigorous description of the electrostatic free energy of an ionic/zwitterionic mixed micelle, which enables us to clearly identify the conditions under which our simplified electrostatic description and that of Nagarajan are most applicable. Recently, a simplified “working model”19 of a more complete molecular-thermodynamic theory20,21 for the prediction of mixed surfactant solution properties was developed. The complete theory is based on a thermodynamic framework which accounts for mixed micelle formation, the solution entropy of mixing, and interactions between the various species present in solution. This framework is combined with a molecular model of mixed micellization which describes the free-energy contributions driving mixed micelle formation. The resulting theory predicts a broad range of mixed micellar solution behavior, including micellization, micelle growth, and phase behavior. The simplified working model was designed to take advantage of some of the results of the molecularthermodynamic theory, while reducing the number of required inputs and simplifying the complexity of the calculations. The simplified working model predicts19 the mixture cmc as a function of surfactant composition for binary surfactant mixtures in which at least one surfactant is ionic. It also predicts19 the βAB interaction parameter (referred to hereafter as the βAB parameter), which reflects interactions between the two surfactants (A and B) at the micellar level. The simplified working model is much easier to use than the more complete theory. It requires as inputs only the pure surfactant chemical structures and cmc’s, as well as the solution conditions, including the temperature and the concentration of added salt. The pure surfactant cmc’s can be either measured experimentally or predicted from the complete molecularthermodynamic theory for pure surfactants.22-24 As a first step toward predicting properties of binary surfactant mixtures containing zwitterionic surfactants, we have extended this simplified working model19 to predict the mixture cmc’s and the βAB parameters for ionic/ zwitterionic and nonionic/zwitterionic mixed micellar systems. The model could, in principle, also be used to predict the cmc’s and βAB parameters for zwitterionic/ zwitterionic mixed micellar systems, although to our knowledge such systems have not been thoroughly studied experimentally. The remainder of the paper is organized as follows. In section II, we review the essential elements of the simplified working model. In section III, we describe in detail how the simplified model has been modified to include zwitterionic surfactants. In section IV, we present a more rigorous approach for modeling the electrostatic free energy associated with binary mixed micelles which include a zwitterionic surfactant. By comparing this approach to the simplified model, we identify the electrostatic approximations inherent in the simplified model. In section V, we compare our predictions for the mixture (18) Nagarajan, R. Mixed Surfactant Systems; Holland, P. M., Rubingh, D. N., Eds.; ACS Symposium Series 501; American Chemical Society: Washington, DC, 1992; p 54. (19) Sarmoria, C.; Puvvada, S.; Blankschtein, D. Langmuir 1992, 8, 2690. (20) Puvvada, S.; Blankschtein, D. J. Phys. Chem. 1992, 96, 5567. (21) Puvvada, S.; Blankschtein, D. J. Phys. Chem. 1992, 96, 5580. (22) Puvvada, S.; Blankschtein, D. J. Chem. Phys. 1990, 92, 3710. (23) Zoeller, N. J.; Blankschtein, D. Ind. Eng. Chem. Res. 1995, 34, 4150. (24) Zoeller, N. J.; Shiloach, A.; Blankschtein, D. CHEMTECH 1996, 26 (3), 24.

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cmc’s and βAB parameters to available experimental data. In section VI, we present our conclusions. II. The Simplified Working Model Part of the complete molecular-thermodynamic theory focuses on predicting the cmc of a binary mixture of surfactants A and B. In that framework, the following expression was derived to relate the cmc of a binary mixture, CMCmix, to the cmc’s of the pure surfactants, CMCA and CMCB:20,21

R1 (1 - R1) 1 ) + CMCmix fACMCA fBCMCB

(1)

where R1 is the solution monomer composition and fA and fB are the activity coefficients of surfactants A and B, respectively, describing the micellar mixing nonidealities. An identical expression has also been derived in the context of the pseudophase separation approximation.12 The two theories arrive at the same expression for the mixture cmc because in the limit of large micelles having a single composition, the molecular-thermodynamic theory approaches the pseudophase separation approximation.15 Note that at concentrations close to the cmc, the number of micelles is small, and therefore, the monomer composition can be approximated by the solution composition, Rsoln, that is, R1 ≈ Rsoln. Experimentally, Rsoln is a controllable variable, while R1 is not. Equation 1 can be used to calculate CMCmix if the activity coefficients, fA and fB, are known. Many different models can be used to describe the micellar mixing nonidealities, resulting in different forms of the activity coefficients, fA and fB. In the context of the molecular-thermodynamic theory, the following expressions were derived for the activity coefficients:20,21

ln fA ) βAB(1 - R*)2/kT

(2)

ln fB ) βAB(R*)2/kT

(3)

where βAB is the binary interaction parameter described in section I, R* is the optimal micelle composition, at which the free energy of mixed micellization is minimized, k is the Boltzmann constant, and T is the absolute temperature. If βAB ) 0, the mixing is ideal, while a negative βAB parameter indicates synergism and a positive one indicates antagonism in mixed micelle formation. Note that the activity coefficients in eqs 2 and 3 coincide with those used in the regular solution theory and the pseudophase separation approximation, with R* replaced by R, the composition of the micellar pseudophase. The regular solution theory assumes that the excess entropy of mixing is zero. Several authors have noted that this may not be the case for many surfactant mixtures.14,15 Nevertheless, the regular solution theory remains an empirically useful description of mixed micellization. It is important to note that in the pseudophase separation approximation for surfactant mixtures, the βAB parameter is evaluated by fitting experimental mixture cmc data for at least one solution composition,12 using eqs 1, 2, and 3, along with equilibrium pseudophase separation relationships. In contrast, the molecular-thermodynamic theory provides19-21 a method to independently predict βAB, thus eliminating the need to experimentally measure cmc’s of the surfactant mixture at any composition. The predicted βAB can then be used in eqs 2 and 3 to calculate fA and fB. For ionic/nonionic and anionic/cationic mixed micelles, the simplified working model assumes19 that for two surfactants with linear hydrocarbon tails, the main

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Shiloach and Blankschtein

contribution to mixing nonideality arises from the electrostatic interactions between the charged surfactant heads. It is important to note that by making this assumption, the method for predicting βAB described in ref 19 and in this paper is limited to binary mixtures in which both surfactants have linear hydrocarbon tails and heads of similar size. Mixing surfactants that violate these criteria can result in other significant contributions to nonideality which are not accounted for by the model. For example, the model has recently been inappropriately applied to analyze mixtures of nonionic surfactants and bile salts.25 In this case, the bile salts have a nonlinear hydrocarbon tail structure which could introduce nonidealities in the packing of the tails in the micellar core. The same model has also been inappropriately applied to analyze mixtures of Triton X-100 and Tween-80, a surfactant with an unusually large head.26 The head structure of Tween-80 is likely to induce significant steric interactions, thus resulting in nonidealities which the simplified theory neglects. The model developed in ref 19 should only be applied when electrostatic interactions dominate the nonidealities associated with mixed micelle formation. With these considerations in mind, the electrostatic interactions in the simplified model are based on pairwise interactions between charged surfactant heads. According to Coulomb’s law, the electrostatic free energy associated with mixed micelle formation (per surfactant molecule), mix gelec , is proportional to q2, where q is the average charge on a surfactant head (see eq 4 below). If zA and zB denote the valences of surfactants A and B, respectively, and e is the electronic charge, then q ) RezA + (1 - R)ezB, and mix gelec ) Kq2 ) Kelec[RzA + (1 - R)zB]2

(4)

where R is the mole fraction micelle composition which can vary from 0 to 1, and Kelec is a proportionality constant which is assumed to be independent of micelle composition. Note that Kelec has absorbed the term e2 arising from q2, such that Ke2 ) Kelec. Similarly, for pure surfA actant micelles, in which R ) 0 or R ) 1, gelec ) Kelec(zA)2 B and gelec ) Kelec(zB)2. The electrostatic free energy of the mix mixed micelle, gelec , can alternatively be expressed as a combination of the gelec values of the pure surfactant A B micelles, gelec and gelec , and an electrostatic interaction AB term, gelec. Specifically, mix A B AB gelec ) R gelec + (1 - R)gelec + R(1 - R)gelec

(5)

In eq 5, the first two terms represent “energetic” contrimix , while the last term represents an “enbutions to gelec thalpic” contribution. Equation 5 is essentially an expansion in powers of R, truncated at O(R2). Note that if AB A gelec ) 0, the mixing is ideal. By substituting gelec ) B Kelec(zA)2 and gelec ) Kelec(zB)2 in eq 5, equating eqs 4 and AB 5, and then solving for the interaction term, gelec , the following expression is obtained: AB βAB ) gelec ) -Kelec(zA - zB)2

(6)

The interaction parameter, βAB, then corresponds to AB gelec , the excess electrostatic free energy associated with mixed micelle formation. This identity arises explicitly from the complete molecular-thermodynamic theory.20,21 (25) Haque, E. M.; Das, A. R.; Moulik, S. P. J. Phys. Chem. 1995, 99, 14032. (26) Haque, E. M.; Das, A. R.; Rakshit, A. K.; Moulik, S. P. Langmuir 1996, 12, 4084.

Given an appropriate method for calculating Kelec, eq 6 is a central result of the simplified model, allowing the prediction of βAB. By separate examination of the two cases of an ionic/ nonionic surfactant mixture and an anionic/cationic surfactant mixture, the significance of the proportionality constant, Kelec in eq 6, can be better understood. For an ionic/nonionic surfactant mixture, substituting zA * 0 and zB ) 0 (or zB * 0 and zA ) 0) in eq 4, and assuming that Kelec is independent of composition (so that the composition can be set to R ) 0 or R ) 1), we find that Kelec ) A B gelec /(zA)2 or Kelec ) gelec /(zB)2, respectively. For mixtures in which both surfactants are ionic, where zA * 0 and zB A * 0, and zA * zB, eq 6 can be expanded with gelec ) Kelec(zA)2 B and gelec ) Kelec(zB)2 and equated with its original form to obtain

Kelec )

A A B B ( 2(gelec gelec )1/2 + gelec gelec

(zA - zB)2

(7)

where the positive sign corresponds to oppositely-charged surfactants and the negative sign corresponds to surfactants with the same sign valence. The sign convention on the square root term in eq 7 is necessary because gelec, which is always a positive quantity, is proportional to the square of the valence, which may be either positive or negative. Accordingly, taking the square root of the valence should result in the correct sign. Kelec is thus related to the electrostatic free energies of one (for ionic/ nonionic mixtures) or both (for ionic/ionic mixtures) of the pure surfactants. With the formalism expressed in this manner, the electrostatic free energy of the mixed micelle is related to the electrostatic free energies of the pure A B and gelec ) without specifying how surfactant micelles (gelec they are calculated. This approach thus allows flexibility A B and gelec , while retainin the method of calculating gelec ing the same framework for the mixtures. To illustrate this generality, we can alternatively rewrite eq 4 in terms of the electrostatic free energies of the pure surfactants. Specifically, mix A A B B ) R2gelec ( 2R(1 - R)(gelec gelec )1/2 + (1 - R)2gelec gelec (8)

where the positive sign is used if the surfactants have the same sign valence, while the negative sign is used if the surfactants are oppositely charged. This sign convention mix of oppositely-charged surfactants reensures that gelec flects a decrease in the electrostatic interactions due to the partial cancellation of the positive and negative charges. Note that the sign convention in eq 8 is opposite to that in eq 7 because when eq 6 is expanded to give eq 7, the middle term (2zAzB) is negative, while in eq 4, the middle term (2R(1 - R)zAzB) is positive. According to the definition of βAB given by eq 6, if we mix two nonionic surfactants (zA ) zB ) 0), or two ionic surfactants with the same valence (zA ) zB), the interaction parameter will be zero and the surfactants will mix ideally. This is an approximation, which assumes that all contributions to synergism in mixed micelle formation are electrostatic in nature. While it is true that electrostatic interactions contribute most strongly to the mixing nonideality, synergism can also arise from steric interactions between surfactant heads of different sizes, as well as from packing surfactant hydrocarbon tails of different lengths in the micellar core. In certain surfactant mixtures (such as those mentioned above25,26), these additional sources of synergism may be important,


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although their contribution to βAB is expected to be generally smaller than that associated with the electrostatic interactions. III. Electrostatic Free Energy of Mixed Micelles Containing Zwitterionic Surfactants Our main goal in this paper is to extend the predictive capabilities of the simplified working model described in section II to binary surfactant mixtures which include zwitterionic surfactants, while retaining the simplified working model framework. Specifically, we aim to derive an analog of eq 6, which will enable a prediction of the βAB parameter for surfactant mixtures containing zwitterionic surfactants. While for ionic surfactants the electrostatic interactions were described by considering the average charge on a surfactant head, for zwitterionic surfactants we need a different description of the electrostatic free energy. Although the net charge or valence of zwitterionic surfactants is zero, they can interact electrostatically. Indeed, the polar heads of zwitterionic surfactants have both a positive and a negative charge, resulting in a dipole. The electrostatic free energy associated with forming a pure zwitterionic micelle has been modeled in terms of the reversible work of charging a capacitor, whose surfaces are located according to the positions of the positive and negative charges of the zwitterionic dipole.13 As a first approximation, the electrostatic free energy of a mixed ionic/zwitterionic micelle can be decomposed into two parts. The first part corresponds to the Coulombic interactions between the ionic heads, and the second part corresponds to the work of charging the capacitor formed from the zwitterionic heads. A similar analysis applies to nonionic/zwitterionic micelles, in which only the work of charging the capacitor contributes. In other words, for binary surfactant mixtures which include zwitterionic surfactants, eq 4 is modified such that mix ) Kq2 + gcap gelec

(9)

where gcap is the electrostatic free-energy change (per surfactant molecule) associated with charging the zwitterionic capacitor. Expressions for charging a capacitor are available for spherical, cylindrical, and planar geometries.13 For a spherical capacitor, for example

gcap )

2 dsep 2πqcap

a(1 + dsep /rcap)w

(10)

where qcap is the average charge per surfactant molecule on each capacitor surface (see below), dsep is the dipolar distance between the positive and negative charges on the zwitterionic surfactant head (which is also the distance between the two oppositely-charged capacitor surfaces), rcap is the distance from the micelle center to the inner surface of the capacitor, a is the area per surfactant 2 /N, molecule on the inner capacitor surface (a ) 4πrcap where N is the micelle aggregation number), and w ) 4πηw0, with ηw the dielectric constant of water and 0 the permittivity of vacuum. In a mixed ionic/zwitterionic micelle, if we denote the mole fraction composition of the ionic surfactant by R, then the mole fraction composition of the zwitterionic surfactant is (1 - R). Consequently, for a zwitterionic surfactant with monovalent dipole charges, qcap ) e(1 - R). To make the capacitor expression in eq 10 more compatible with the expression for the ionic surfactant in eq 4, the composition dependence of gcap can be isolated from the rest of the expression as follows:

2 gcap ) Kqcap ) Kcap(1 - R)2

(11)

with

Kcap )

2πe2dsep a(1 + dsep /rcap)w

(12)

Note that the proportionality constant Kcap in eq 11 depends only on the capacitor dimensions and the solution conditions. Substituting eq 11 in eq 9, and assuming that the ionic surfactant has a valence zA, we obtain mix ) Kelec(R zA)2 + Kcap(1 - R)2 gelec

(13)

Equation 13 is a central result: it is the extension of eq 4 to ionic/zwitterionic mixed micelles. Here, as discussed in section II, Kelec is essentially gelec of a pure ionic micelle, and Kcap is gcap, or the electrostatic free energy of a pure mix zwitterionic micelle. When we calculate gelec this way, we are superimposing the electrostatic contributions associated with the partial micelles formed separately by the ionic and zwitterionic surfactants. In other words, the work associated with charging each micelle is assumed to be independent. We can then derive an expression which B in eq 5 is analogous to eq 5 by substituting gcap for gelec when the ionic surfactant A is mixed with a zwitterionic surfactant instead of being mixed with a second ionic surfactant B. Specifically mix A AB ) R gelec + (1 - R)gcap + R(1 - R)gelec gelec

(14)

In eq 14, as in eq 5, the first two terms represent “energetic” mix , while the last term represents an contributions to gelec “enthalpic” contribution. The interaction parameter is AB 20,21 . again given by βAB ) gelec In most cases, and in all the surfactant mixtures examined in this paper, the ionic surfactants are monovalent, such that zA ) 1 or zA ) -1 for cationic and anionic surfactants, respectively. Accordingly, we hereafter focus on this case, that is, we take zA ) 1 or zA ) -1 as needed. Following the procedure used to derive eq 6, we let A gelec ) Kelec and gcap ) Kcap in eq 14. Then, equating eqs AB 13 and 14, and solving for gelec yields AB ) -Kelec - Kcap βAB ) gelec

(15)

Equation 15 is also a central result: this expression gives the βAB interaction parameter characterizing synergism in monovalent ionic/zwitterionic and nonionic/zwitterionic mixtures. Note that in a binary mixed micelle composed of zwitterionic and nonionic surfactants, Kelec ) 0 and βAB ) -Kcap is given only by the capacitor term (see eq 15). The relative magnitudes of Kelec and Kcap are determined by the chemical structures of the individual surfactants, but in general, Kelec is larger than Kcap by a factor of 3 to 4, as will be illustrated in section V. The βAB parameter is thus largely influenced by the ionic surfactant with which the zwitterionic surfactant is mixed. Note again that while the simplified “working model” has been extended to incorporate zwitterionic surfactants, its applicability should be restricted to surfactants with linear hydrocarbon tails and to surfactant mixtures in which synergism in mixed micelle formation is dominated by the electrostatic interactions. IV. Rigorous Calculation of the Electrostatic Free Energy of Mixed Micelles Containing Zwitterionic Surfactants A. Derivation of the Electrosatic Free Energy. The simplified electrostatic approach described in the

3972 Langmuir, Vol. 13, No. 15, 1997


previous section is quite straightforward and intuitive, in that the electrostatic contributions of the partial ionic and zwitterionic micelles are simply superimposed. However, the electrostatic calculation for the ionic/zwitterionic mixed micelle can also be carried out more rigorously. For a pure ionic micelle, the electrostatic free energy can be calculated as the reversible work of charging the micelle surface to its final surface charge density. Analogously, mix can in the case of an ionic/zwitterionic mixed micelle, gelec be calculated as the reversible work of charging three surfaces, which we denote as inner, middle, and outer. The positions of the three surfaces are determined by the relative distances of the charged groups from the micellar core/water interface. One surface holds the charged heads of the ionic surfactants. The other two surfaces are defined by the positive and negative charges on the zwitterionic dipoles, with the total charge on each surface equal in magnitude but opposite in sign because each zwitterion has a net zero charge. On each of the three surfaces, the charges are also assumed to be smeared uniformly. In contrast to the simplified electrostatic calculation described in section III, by simultaneously charging the three surfaces, we account for the presence of the partial zwitterionic micelle when we charge the surface of the partial ionic micelle, and vice versa. By carrying out the more rigorous derivation, we can also explicitly identify the assumptions inherent in reducing the more rigorously mix to the sum of a charged surface and a evaluated gelec capacitor contribution, as given in eq 13. Specifically, mix corresponding to our goal is to identify the terms in gelec charging the ionic micelle and those corresponding to the zwitterionic capacitor. Any remaining terms in the mix gelec expression will constitute a quantitative measure of the difference between the simple superposition of the pure ionic and zwitterionic micelles presented in section III, and the more rigorous consideration of their electrostatic influence on each other presented below. The total reversible work of charging the three surfaces to their final surface charge densities is given by27,28 mix Gelec )

∫01[ψi(λ)Qif + ψm(λ)Qmf + ψo(λ)Qof] dλ

(16)

where Qif, Qmf, and Qof are the final total charges on the inner, middle, and outer surfaces, respectively, ψi, ψm, and ψo are the electrostatic potentials at the inner, middle, and outer surfaces, respectively, and λ is a charging parameter which varies from 0 to 1 as the charge simultaneously increases on all three surfaces. Note that the total reversible work of charging the mixed micelle, mix Gelec , is related to the electrostatic contribution to the free mix energy of mixed micellization, gelec , by mix mix Gelec ) Ngelec

Figure 1. Schematic representation of a mixed ionic/zwitterionic micelle, illustrating three charged surfaces at radii Ri, Rm, and Ro . Region 1 is the hydrocarbon core of the micelle, regions 2 and 3 separate the charged surfaces and are assumed to contain no ions, and region 4 is the bulk aqueous solution.

as well as for some of the zwitterionic surfactants considered in this paper. The calculation can also be performed for cylindrical or planar geometries. We divide the mixed micelle into four regions separated by three charged surfaces, as illustrated in Figure 1. Region 1 is the hydrocarbon micellar core, in which we assume no penetration of water or ions because the core is modeled as a liquid hydrocarbon droplet.29 This region is therefore devoid of any electric field and electrostatic potential. Regions 2 and 3 are aqueous and occupy the space between the charged surfaces. In these regions, we assume that there are no ions present, because the dimensions of most ions are larger than the radial widths of these regions. Region 4 is the bulk aqueous solution. To evaluate mix gelec via eqs 16 and 17, we first need to determine the electrostatic potentials at each of the three surfaces. The electrostatic potentials in regions 2 and 3 satisfy the Laplace equation, which is the nonlinear PoissonBoltzmann (PB) equation in the absence of charges. Specifically, using dimensionless variables, in region 2, the Laplace equation is given by30

d2y2 2

dx

where N is the aggregation number of the mixed micelle. The actual charges on the three surfaces are determined by the mixed micelle composition and size. Specifically, the micelle composition determines the relative number of charges on each surface, while the micelle size determines the total number of surfactant molecules in the mixed micelle. The more rigorous electrostatic calculation is illustrated here for a spherical mixed micelle. This geometry is typical for ionic surfactants with a small amount of added salt, (27) Verwey, E. J. W.; Overbeek, J. Th. G. Theory of the Stability of Lyophobic Colloids; Elsevier: New York, 1948. (28) Yuet, P. K.; Blankschtein, D. Langmuir 1995, 11, 1925 and references cited therein.

2 dy2 )0 x dx

(18)

and, in region 3, the Laplace equation is given by

d2y3 2

dx (17)

+

+

2 dy3 )0 x dx

(19)

In eqs 18 and 19, yj ) ezψj /kT is the dimensionless potential in region j (2 or 3), and x ) κR is the dimensionless radial distance, with R the radial distance from the center of the micelle, and κ-1 the Debye screening length, with

κ ) (8πCoe2z2/wkT)1/2

(20)

where Co is the ion concentration. Recall that region 4 is the aqueous solution surrounding the micelle, containing both water and ions. We assume that in this region the ions are distributed according to the Boltzmann distribution. Therefore, the electrostatic potential in this region (29) Tanford, C. The Hydrophobic Effect; Wiley: New York, 1980. (30) Bockris, J. O.; Reddy, A. K. N. Modern Electrochemistry I; Plenum Press: New York, 1977.


Langmuir, Vol. 13, No. 15, 1997 3973

satisfies the PB equation,30 that is

d2y4

2 dy4 + ) sinh y4 2 x dx dx

x2

y2(xm) ) y3(xm)

dy2 )0 dx

( )

d 2dy2 x )0 dx dx

(29)

(30)

Integration of eq 30 yields

x2

dy2 ) C1 dx

(31)

where Cl is an integration constant. A second integration yields

y2 1 ) - + C2 C1 x

(32)

(22) where C2 is another integration constant. Letting y2 ) y2(xi) at x ) xi and y2 ) y2(xm) at x ) xm in eq 32 results in

2. At x ) xm ) κRm,

dy2 dy3 4πσmez - 3 ) dx dx κkT

dx

+ 2x

or

1. At x ) xi ) κRi,

2

2

(21)

Note that, at this point, we have not yet designated which surface corresponds to the ionic surfactant heads and which surfaces correspond to the positive and negative charges of the dipole on the zwitterionic surfactant heads. The derivation so far is thus completely general. In eqs 18, 19, and 21, we have assumed that the surfactant and the added salt are symmetric electrolytes having the same valence, a commonly encountered experimental situation. We next specify the following boundary conditions for eqs 18, 19, and 21

dy2 4πσiez )dx 2κkT

d2y2

(23) (24)

3. At x ) xo ) κRo,

dy3 dy4 4πσoez 3 - 4 ) dx dx κkT

(25)

y3(xo) ) y4(xo)

(26)

4. As x f ∞,

dy4/dx ) 0

(27)

y4 ) 0

(28)

where σi, σm, and σo are the surface charge densities at the inner, middle, and outer surfaces, respectively, and j ) 4πηj0 is the permittivity of region j (2, 3, or 4), with ηj the dielectric constant of region j. Equations 22, 23, and 25 express the variation in electric field across the charged surfaces according to Gauss’s law. Equations 24 and 26 reflect the continuity of the electrostatic potentials across the charged surfaces. The fourth boundary condition indicates that the electrostatic field and potential in region 4 must decay to zero far from the micelle surface. This condition ensures the electroneutrality of the entire system. The rate of this decay depends on the ionic strength of the solution and on the Boltzmann ion distribution. Our solution strategy essentially parallels that of Missel et al. in deriving surface potentials at the micelle surface.31 Since we can find ψo by solving the PB equation in region 4, we would like to relate ψi and ψm to ψo. Expressing these potentials in terms of ψo will greatly simplify the integration of eq 16. To find the required relationships between the electrostatic potentials, we first solve the Laplace equations in regions 2 and 3. The procedure is identical for eqs 18 and 19 and is illustrated here for eq 18. Multiplying both sides of eq 18 by x2 yields (31) Missel, P. J.; Mazer, N. A.; Carey, M. C.; Benedek, G. B. J. Phys. Chem. 1989, 93, 8354.

y2(xi) 1 ) - + C2 C1 xi

(33)

y2(xm) 1 )+ C2 C1 xm

(34)

and

Subtracting eq 34 from eq 33, solving for C1, and substituting the resulting C1 expression in eq 31 yields

dy2 1 xi xm [ y (x ) - y2(xm)] ) d x x2 xi - xm 2 i

(35)

Carrying out the same procedure for eq 19 yields

dy3 1 xmxo [ y (x ) - y3(xo)] ) d x x2 xm - xo 3 m

(36)

Using eq 35 with x ) xi in eq 22 and simplifying yields

y2(xm) - y2(xi) ) -

Si RiκD1 Rm

(37)

where D1 ) Rm - Ri, and Si is the dimensionless instantaneous surface charge density of the inner surface, defined by

Si )

4πσiez 2κkT

(38)

Note that the dimensionless surface charge density, Si, is proportional to the surface charge density, σi. Similar dimensionless surface charge densities are defined for Sm and So, with σm and σo replacing σi, and with 3 and 4 replacing 2, respectively. Equation 37 provides an important relationship which relates the dimensionless potential at the middle surface, y2(xm), to that at the inner surface, y2(xi). Next, we use eq 35, with x ) xm, and eq 36, with x ) xm, in eq 23. Simplifying the resulting expression with the aid of eq 37 yields the following relationship between the dimensionless potential at the middle surface,

3974 Langmuir, Vol. 13, No. 15, 1997


y3(xm), and that at the outer surface, y3(xo)

y3(xo) - y3(xm) ) -

RmκD2 [Si(Ri/Rm)2 + Sm] (39) Ro

Note that in deriving eq 39 we have assumed that 2 ) 3, since regions 2 and 3 are very close to the micellar surface and are likely to have similar dielectric properties. A comparison of eq 39 and eq 37 indicates that their mathematical structures are identical. In eq 37, all the variables refer to the two inner surfaces (Si, Ri, and D1 ) Rm - Ri), whereas in eq 39, all the variables refer to the two outer surfaces (Sm, Rm, and D2 ) Ro - Rm). In particular, note that the quantity in brackets in eq 39 is an effective surface charge density. It reflects the sum of the two dimensionless surface charge densities, Si and Sm, corrected for the fact that the total area of the inner surface is smaller than that of the middle surface by a factor of (Ri /Rm)2. The effective surface charge density arises from applying Gauss’ law to the total charge contained in the sphere of radius Rm, that is,

dy3 dx

|

xm

) -[Si(Ri /Rm)2 + Sm]

(40)

Finally, eq 36 is used in the boundary condition at x ) xo, eq 25, to yield

dy4 dx

|

xo

) -[Si(Ri /Ro)2 + Sm(Rm/Ro)2 + So ]

(41)

Note that in deriving eq 41, we have assumed that 2 ) 3 ) 4, because regions 2, 3, and 4 are all aqueous. Equation 41 provides one of the boundary conditions needed to solve the PB equation, eq 21. The other boundary condition is either eq 27 or eq 28. The expression in brackets in eq 41 is an effective surface charge density at the outer surface, which also accounts for the charges on the two inner surfaces. As explained above, the effective charge density arises from applying Gauss’ law to the total charge contained in the sphere of radius Ro. Given the relationships between y2, y3, and y4, we can mix now calculate Gelec , the total reversible work of charging all three surfaces to their final total charges, increasing the charge on each surface by the same fraction simultaneously. Rewriting eq 16 in its dimensionless form, we obtain mix ) Gelec

∫01[ y2(xi)Qif + y3(xm)Qmf + y4(xo)Qof] dλ

kT ez

λQif ez

λQif 4πez (4πR2i )2κkT

)

R2i 2κkT

mix mix ) Ngelec ) Gelec

D1Qif2

D2(Qif + Qmf)2 +

+ 2Ri(Ri + D1)w 2Rm(Rm + D2)w kT 1 y (x )[Q + Qmf + Qof] dλ (44) ez 0 4 o if

∫

where w ) 2 ) 3 ) 4. Equation 44 is the total work of charging three concentric spherical surfaces to their final charges. The first term in eq 44 corresponds to the work of charging a capacitor on the two inner surfaces, with radii Ri and Rm ) Ri + D1, respectively, whose charge is equal to the final charge on the inner surface, Qif. The second term corresponds to the work of charging a capacitor on the two outer surfaces, with radii Rm and Ro ) Rm + D2, whose charge is equal to the sum of the charges on the two inner surfaces, Qif + Qmf. The last term accounts for the work of charging the outer surface to the final charge given by the sum of the charges on all three surfaces. B. Comparison of the Rigorous and Approximate Electrostatic Free Energies. In deriving the apmix given in eq 13, the charging proximate expression for gelec of each partial micelle (ionic or zwitterionic) was carried mix expression is out independently. The resulting gelec obtained by simply superimposing the electrostatic contributions from the partial ionic and zwitterionic micelles. In the more rigorous derivation presented above, the charging of each partial micelle is not independent. That is, we account for the presence of the zwitterionic surfactants when we charge the surface due to the ionic surfactants, and vice versa. Equation 44 is a general result for the work of charging three charged surfaces to their final surface charge densities. By applying it to an ionic/zwitterionic mixed micelle, we can identify the assumptions and approximamix . First, to tions involved in using eq 13 to calculate gelec facilitate this identification, we convert the total work of charging the mixed micelle given in eq 44 to the work of charging the mixed micelle per surfactant molecule, mix mix gelec ) Gelec /N, that is mix ) gelec

2πqif2 D1

2π(qif + qmf)2D2 +

+ ai(1 + D1/Ri)w am(1 + D2 /Rm)w kT 1 y (x )[q + qmf + qof] dλ (45) ez 0 4 o if

∫

(42)

Note that y2, y3, and y4 in eq 42 are functions of λ. For example, y2 is a function of Si, and because Si is the instantaneous surface charge density, it is a function of λ, that is, σi ) λQif/(4πR2i ), and

Si )

y2(xi) to y3(xm) and then solve the resulting equation simultaneously with eq 39 to express y2(xi) as a function of y3(xo) ) y4(xo). Then, substituting the expressions for y2(xi) and y3(xm) in eq 42, integrating, and simplifying, we obtain

(43)

Analogous relations apply for Sm and So, with Qmf and Rm, and Qof and Ro, respectively, replacing Qif and Ri. We would like to express the potentials, y2(xi) and y3(xm), in eq 42 in terms of y4(xo), because we can calculate this potential by using the PB equation. Therefore, we use eq 39 to relate y3(xm) to y3(xo), noting that y3(xo) ) y4(xo) by virtue of eq 26. Similarly, we use eq 24 in eq 37 to relate

where aj ) 4πR2j /N is the area per surfactant molecule at surface j (i, m, or o) and qjf ) Qjf /N is the final charge per surfactant molecule at surface j (i, m, or o), where N is the total mixed micelle aggregation number. As stated earlier, the charge on each surface arises from smearing the charges on the heads of the ionic or zwitterionic surfactants. Note that due to the presence of the zwitterionic surfactant, the total charge on one of the surfaces will always be equal in magnitude and opposite in sign to the total charge on one of the two other surfaces. The relative locations of the “ionic” surface and the “zwitterionic” surfaces depend on the chemical structures of the surfactants. We can distinguish three cases, depending on the relative locations of the surface associated with the ionic surfactant and the surfaces associated with the positive and negative charges on the zwitterionic dipole. For example, if the charge on the ionic surfactant


Langmuir, Vol. 13, No. 15, 1997 3975

Table 1. Identification of the Three Possible Cases for the Relative Locations of the Surface Arising from the Charge on the Ionic Surfactant (i) and the Surfaces Arising from the Charges on the Dipole of the Zwitterionic Surfactant (z(+) and z(-))a case

Ri

Rm

Ro

I II III

z(+) z(+) i

z(-) i z(+)

i z(-) z(-)

generated values of gelec were then compared with those A calculated by R2gelec . The pure ionic electrostatic free A energy, gelec, can be evaluated in several ways. However, to make the comparison most direct, we made several A was calculated numerically as simplifications. First, gelec the work of charging a pure ionic micelle to its final charge density, that is, R ) 1 and qof ) e, such that A gelec )

a

The radii of the inner, middle, and outer surfaces are denoted by Ri, Rm, and Ro, respectively (see Figure 1).

occupies the outermost surface with radius Ro, then the charges on the zwitterionic surfactant will be located on the two inner surfaces, with radii Ri and Rm. The three cases are listed in Table 1, with z(+) and z(-) indicating a positive or negative charge belonging to the zwitterionic surfactant, and i indicating the charge on the ionic surfactant. Ri, Rm, and Ro denote the radii of the inner, middle, and outer surfaces, respectively. Note that z(+) is always located on a surface inner to that corresponding to z(-), because in most zwitterionic surfactants the positive charge is closer to the hydrocarbon tail than the negative charge. However, the discussion which follows is not affected by the relative locations of z(+) and z(-). The three cases need to be addressed separately, because the differences between the rigorous and the approximate calculations vary from case to case. Case I. Here, the two inner surfaces correspond to the positive and negative charges of the zwitterionic surfactant dipole and the outer surface corresponds to the ionic surfactant. In this case, qif ) -qmf, the second term in eq 45 vanishes, and the last term reduces to the work of charging the outer surface to its final charge, qof. In other words, in case I, using qif ) (1 - R)e,

2πe2D1 kT mix gelec ) (1 - R)2 + ez ai(1 + D1/Ri)w

∫01y4(xo)qof dλ (46)

The charge configuration corresponding to case I is encountered when zwitterionic surfactants possessing a short dipole distance, dsep, are mixed with ionic surfactants for which the distance from the micellar core/water interface to the charge is long. In this case, the result of the rigorous electrostatic calculation, eq 46, maps directly to the result of the simplified calculation, eq 13. With dsep ) D1 and rcap ) Ri in eq 12, eq 13 becomes

2πe2D1

mix gelec ) (1 - R)2 + R2Kelec ai(1 + D1/Ri)w

∫01y4(xo)e dλ

kT ez

A this way, we eliminate any apBy calculating gelec A proximation due to the method of calculating gelec , and the two methods are identical at R ) 1. Second, the ionic strength of the solution, which must be known to evaluate y4(xo) in the integral, is a function of the micelle composition R. In the absence of added salt, it must be determined by an iterative method. To avoid this complication, we performed the calculation in the presence of 0.1 M NaCl, a salt concentration high enough to mask the variation of the ionic strength with R. The difference between the approximate and rigorous values of gelec was found to increase to a maximum of 20% at an ionic composition of R ) 0.1, although at the lower compositions, the value of gelec is very small and the difference between the approximate and more rigorous calculations is insignificant. Case II. This case, in which the ionic surface is between the two surfaces associated with the zwitterionic surfactant (see Table 1), is the most common, in general, and among the mixtures considered in this paper, in particular. Here, qif ) -qof, and in the Kcap expression given in eq 12, we substitute rcap ) Ri, and dsep ) D1 + D2. Due to the zwitterionic surfactant structure, qif is positive and qof is negative, although this information does not affect the discussion which follows. The simplified approach would therefore yield (see eqs 12 and 13)

mix ) (1 - R)2 gelec

2πe2(D1 + D2) ai(1 + (D1 + D2)/Ri)w

+ R2Kelec (49)

To compare eq 49 with the rigorous approach, we can substitute in eq 45 the composition dependence of qi, qm, and qo, the charge per surfactant monomer on each surface. Specifically, qi ) (1 - R)e, qm ) Re, and qo ) -qi ) -(1 - R)e. This substitution yields mix ) (1 - R)2 gelec

2πe2D1

2πe2D2

ai(1 + D1/Ri)w

+

+ am(1 + D2 /Rm)w kT 1 y (x )q dλ (50) ez 0 4 o mf

∫

(47)

The structures of eqs 46 and 47 are identical. In both equations, the second term is gelec, the electrostatic free energy associated with the ionic portion of the mixed micelle. To evaluate the validity of replacing the integral term in eq 46 with the approximate R2Kelec term in eq 47, we need to determine the composition dependence of the integral term in eq 46. That is, if the micelle has an ionic composition given by R, how will gelec depend on R? In eq 47, the dependence on R is explicitly quadratic, given by A . In contrast, in eq 46, it is difficult gelec ) R2Kelec ) R2gelec to extract an analytical dependence of gelec on R, because the potential, y4(xo), must be evaluated by solving the PB equation numerically, and the integral term is also evaluated numerically, with qof ) Re a function of composition. (Note that in the Debye-Hu¨ckel limit, gelec also depends on R2.) Therefore, we have evaluated the integral at different values of R, and the numerically

(48)

Note that the second term in eq 50 does not depend on composition. It can therefore be split into any combination of composition-dependent terms whose sum is composition independent, such as R and (1 - R), or (1 - R)2 and 1 (1 - R)2. In this way, we can factor out a (1 - R)2 dependence from the second term, and then combine it with the first term, such that mix ) (1 - R)2 gelec

2πe2(D1 + D2) ai(1 + (D1 + D2)/Ri)w 2πe2D2

R(2 - R)

am(1 + D2/Rm)w

+

+

∫01y4(xo)qmf dλ

kT ez

(51)

Comparing eqs 51 and 49, we note first that if the micelle is purely zwitterionic, so that R ) 0, in both cases we

3976 Langmuir, Vol. 13, No. 15, 1997


recover the work of charging the zwitterionic capacitor (note that the integral term in eq 51 vanishes because qmf ) Re ) 0). When the micelle is purely ionic, so that R ) 1, the sum of the last two terms in eq 51 is equivalent to charging a capacitor between the middle and outer surfaces, and then charging the outer surface to the final charge density it would have if the ionic surfactants occupied that surface. In general, for R * 0 and R * 1, if we approximate the charging integral in eq 51 by R2Kelec, then at most, the difference between the approximate and mix values is equal to the second term in eq 51 rigorous gelec with R ) 1. For a typical ionic/zwitterionic mixture, this leads to a maximum difference of approximately 15% between the rigorous and approximate electrostatic free energies. Within the context of the simplified model, this difference is acceptable. Case III. Here, the ionic charges occupy the innermost surface, while the zwitterionic charges occupy the middle and outer surfaces. None of the mixed surfactant systems considered in section V fall in this category, and therefore, we do not examine this case in detail. We would like to stress, though, that the differences between the approximate and rigorous electrostatic free energies corresponding to case III are essentially the same as those found for case II described above. The next section describes the prediction of the βAB parameter, where the effect of utilizing the rigorous mix gelec expression on the predicted βAB parameter will be discussed. Specifically, a comparison between the rigorous and approximate electrostatic approaches demonstrates that the approximations linking the two are reasonable. In the context of a predictive theory which is relatively easy to implement, the approximate electrostatic calculation yields acceptable results. V. Predictions of βAB Parameters and Mixture cmc’s A. Molecular Parameters. To predict the βAB parameter for zwitterionic/ionic or zwitterionic/nonionic surfactant mixtures using eq 15, we need to evaluate Kelec for the ionic surfactant and Kcap for the zwitterionic surfactant. A detailed description of the procedure for calculating Kelec is given in ref 19, and will only be briefly described here by summarizing the relevant expressions. As in section II, Kelec corresponds to the electrostatic free energy associated with charging the pure ionic micelle, gelec. This quantity can be evaluated by several different methods;30,32,33 in ref 19, it was calculated using an approximate analytical solution to the PB equation.34,35 For charges which are uniformly distributed on a spherical surface of radius R, gelec is obtained using the following analytical expression:

gelec )

[

]

(1 + (s/2)2)1/2 - 1 s/2 1 + (1 + (s/2)2)1/2 4 (52) 2kT ln sκR 2

2kT ln(s/2 + (1 + (s/2)2)1/2) -

[

(

)]

with κ given in eq 20; and s the dimensionless surface charge density defined by eq 38, and restated here in terms of a, the area per molecule at the surface of charge (σ ) ez/a for a pure ionic micelle): (32) Ohshima, H.; Healy, T. W.; White, L. R. J. Colloid Interface Sci. 1982, 90, 17. (33) Hsu, J.; Kuo, Y. J. Colloid Interface Sci. 1995, 170, 220. (34) Mitchell, D. J.; Ninham, B. W. J. Phys. Chem. 1983, 87, 2996. (35) Evans, D. F.; Mitchell, D. J.; Ninham, B. W. J. Phys. Chem. 1984, 88, 6344.

s)

4π(ze)2 aκkT

(53)

The range of validity and applicability of eq 52 is discussed in detail in ref 19. Due to the approximations made in deriving it, eq 52 is accurate compared to the numerical solution of the PB equation if κR g 0.5. To calculate gelec with eq 52, we need to know R, s, and κ. R is the radius of the surface of charge. We assume that it is equal to the sum of Rcore, the radius of the micellar core, and dch, the distance from the micellar core-water interface to the surface of charge, namely

R ) Rcore + dch

(54)

In eq 54, the value of Rcore is assumed to be equal to the length of the fully-extended hydrocarbon tail in which all the carbon-carbon bonds are in the all-trans configuration. For a tail composed of nc carbon atoms, the first CH2 group is assumed to lie within the hydration region of the hydrophilic head, and therefore, it does not contribute to the hydrophobicity of the hydrocarbon tail. The micellar core radius is then given by29

Rcore ≈ 1.5 + 1.265(nc - 1) (in Å)

(55)

The value of dch can be estimated from the molecular structure of the ionic surfactant head and should include the length of the first CH2 group which is not included as part of the hydrophobic tail. To calculate s, the dimensionless surface charge density, we need to know a, the area per surfactant molecule at the surface of charge. Equation 52 applies for a spherical geometry, which is a reasonable shape to assume for ionic micelles in the absence of a high concentration of salt. For a spherical surface with radius R, a is given by

a ) 4πR2/Nsurf

(56)

where Nsurf is the number of surfactant molecules in the ionic micelle. Assuming that the micellar core is “dry” and has a uniform density equal to that of liquid hydrocarbon,29 it follows that 3 (4π/3)Rcore Nsurf ) Vc

(57)

where Vc is the volume of a hydrocarbon tail in the micellar core, given by Vc ≈ 27.4 + 26.9(nc -1) (in Å3).29 Finally, to calculate κ, we need to know the bulk concentration of ions, C0, consisting of the charged surfactant monomers and the added salt ions. To a very good approximation, the ionic surfactant monomer concentration is equal to the cmc of the ionic surfactant, CMCionic, and therefore

C0 ≈ CMCionic + Csalt

(58)

Using the values of R, a, and C0 obtained from eqs 54-58, we can then find s and κ and use these in eq 52 to calculate Kelec ) gelec (see section II). As mentioned earlier, in the absence of a high salt concentration, ionic surfactants aggregate into spherical micelles, because this shape maximizes the area per head at the surface of charge, and therefore minimizes the electrostatic repulsions between the charged heads. On the other hand, in pure zwitterionic micelles, the dipolar electrostatic repulsions are much weaker, and the surfactants can aggregate into spherical, cylindrical, or


Langmuir, Vol. 13, No. 15, 1997 3977

exp e Table 2. Values of βpred AB and βAB for Nonionic /Zwitterionic and Ionic /Zwitterionic Surfactant Mixtures

ionic or nonionic zwitterionic 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

C10E4 DDAO C12E8 C12BMG DTAB C12BMG C12SO3Na C12BMG C12SO3Na C12AP C12PyrBr C12AP SDS DDAO SDS DDAO SDeS C10SO C10TAB C10SO SDS C12-betaine SDS C14-betaine SDS C14-betaine SDS C16-betaine SDS C14HMP

salt (M)

CMCionic (mM) CMCzwit (mM)

0.0005 Na2CO3

0.1 NaBr 0.1 NaBr 0.0005 Na2CO3

0.01 NaCl 0.01 NaCl 0.01 NaBr 0.01 NaBr

0.8a 1.0a 0.109b 0.552a 16.0b 0.552a 12.4b 0.552a 0.718c 0.53c 2.75c 0.53c 8.0b 2.1b 8.0a 2.0a 23.8c 1.3c 68c 1.3c 4.68a 1.12a 4.68a 0.166a 3.16a 0.126a 3.16a 0.05a 7.2a 0.12a

dch (Å) dsep (Å)

Kelec (kT) Kcap (kT)

1.2 2.5 2.5 2.5 2.6 2.5 2.6 3.8 2.6 3.8 3.8 1.2 3.8 1.2 3.8 1.2 2.5 1.2 3.8 2.5 3.8 2.5 3.8 2.5 3.8 2.5 3.8 8.3

0.46 1.17 3.82 1.17 3.99 1.17 2.52 1.67 2.52 1.67 4.01 0.60 4.60 0.60 3.02 0.56 2.62 0.56 3.62 1.17 3.62 1.24 3.65 1.24 3.65 1.29 4.14 2.65

pred βAB (kT)

exp βAB (kT)

ref

-0.46

-0.8

47

-1.17

-0.9

44

-4.99

-1.3

37

-5.16

-4.5d

44

-4.19

-1.2

37

-4.19

-3.4

37

-4.61

-4.4

37

-4.66

-3.7d

46

-3.58

-4.3

37

-3.18

-0.5

37

-4.79

-5.0d

41

-4.86

-5.4d

41

-4.89

-4.9

45

-4.94

-6.3d

45

-6.79

-2.4d

7

Source of pure surfactant cmc: a experimental data from reference; b tabulated in ref 48; c calculated from single surfactant molecularexp opt thermodynamic theory.22-24 The last column indicates the literature reference for the experimental data. d βAB ) βAB (see text). e The ionic surfactants follow common abbreviations: DTAB is dodecyltrimethylammonimum bromide; C12PyrBr is dodecylpyridinium bromide; SDS is sodium dodecyl sulfate; SDeS is sodium decyl sulfate; and C10TAB is decyltrimethylammonium bromide. The nonionic surfactants denoted by CiEj have i alkyl carbons and j ethylene oxide units. The zwitterionic surfactant abbreviations and chemical structures are given in Table 3.

discoidal micelles. In principle, the shape of the pure zwitterionic micelle should determine the shape of the capacitor used in calculating Kcap. Specifically, for a spherical capacitor, Kcap is given by eq 12, using eq 56 to express it in terms of Nsurf sph K cap )

Nsurfe2dsep 2rcap(rcap + dsep)w

(59)

whereas for a cylindrical capacitor,31,13 the analogous expression is given by

K

cyl cap

2πe2 ) r ln(1 + dsep /rcap) acylw cap

(60)

with acyl ) 2Vc /R, so that Kcap can be calculated independently of the length of the cylinder. The optimal micellar shape can be predicted22-24 using the complete molecular-thermodynamic theory for single surfactants. We thus calculated the optimal shape for each of the zwitterionic surfactants examined in this paper at the specified experimental conditions. According to the predictions, some of the surfactants form spherical micelles, while others form cylindrical micelles. We have found that our predictions of the βAB parameter improve in some cases if we account for the actual shape of the pure zwitterionic micelle. However, in the absence of the complete molecular thermodynamic theory, and in the context of the simplified working model presented here, the shape of the pure zwitterionic micelle is not known a priori. In that case, we have found that assuming that the zwitterionic micelle is spherical gives the best average agreement of the predicted βAB parameters with experi-

ments, and in the absence of detailed knowledge of the shape of the micelle, we recommend assuming that the pure zwitterionic micelle is spherical. The radius of the spherical capacitor, rcap, is then calculated by assuming a fully extended hydrocarbon tail, as for the ionic surfactant. Note that rcap also includes the distance from the hydrocarbon/water interface to the inner charge of the zwitterion, dz, that is

rcap ) Rcore + dz (in Å)

(61)

with Rcore given in eq 55. The distance dz depends on the molecular structure of the zwitterionic surfactant head and includes the length of the first CH2 group not included as part of the micellar core. However, for most of the zwitterionic surfactants considered in this paper, we take dz ) 2.5 Å, because the positive charge of the dipole is on the atom adjacent to the first CH2 group in the hydrocarbon tail. The distance between the shells of the capacitor, dsep, is equal to the distance between the positive and negative charges on the zwitterionic dipole and can be estimated from the chemical structure of the zwitterionic head. Once both Kelec and Kcap are known, the βAB parameter can be predicted using eq 15. A detailed discussion of the procedure for finding the mixture cmc as a function of solution composition is given in ref 19. Briefly, given the predicted βAB parameter, and the optimum micelle composition R*, eqs 2 and 3 can be used to calculate the activity coefficients, fA and fB. These can then be inserted in eq 1, together with the pure surfactant cmc’s, to calculate the cmc of the mixture, CMCmix, as a function of the solution composition, Rsoln. To implement this procedure,

3978 Langmuir, Vol. 13, No. 15, 1997


Table 3. Abbreviations and Chemical Structures of the Zwitterionic Surfactants Examined abbreviation C12BMG C12AP DDAO C10SO C14HMP Ci-betainea a

chemical structure C12N+(C6H4)(CH3)CH2COOC12N+H2CH2CH2COOC12N+(CH3)2OC10S+(CH3)OC12CHOH(CH2)N+(CH3)CH2CH2OP(ONa)OCiN+(CH3)2CH2COO-

Ci denotes a linear hydrocarbon segment with i carbon atoms.

Figure 3. Predicted (line) and experimental (circles) critical micelle concentration as a function of overall surfactant composition, Rsoln for a mixture of SDS (Rsoln ) 1) and C12-betaine (Rsoln ) 0) at 25 °C in 0.01 M NaCl (mixture 11 in Table 2). The experimental cmc values are from ref 41, and the best fit to the pred data gives βopt AB ) -5.0kT, with βAB ) -4.79kT.

Figure 2. Predicted (line) and experimental (circles) critical micelle concentration as a function of overall surfactant composition, Rsoln, for a mixture of SDS (Rsoln ) 1) and C12BMG (Rsoln ) 0) at 25 °C (mixture 4 in Table 2). The experimental cmc values are from ref 44, and the best fit to the data gives βopt AB ) -4.5kT, with βpred ) 5.16kT. AB

R* needs to be determined. The value of R* can be obtained from the molecular-thermodynamic theory by minimizing the free energy of mixed micellization with respect to micelle composition, R. The resulting equation for R* is given by19-21

(

) (

)

βAB R1 CMCB R* ) ln (1 - 2R*) + ln kT 1 - R* 1 - R1 CMCA

(62)

where near the mixture CMC we can safely assume that R1 ≈ Rsoln. B. Comparison with Experimental Data. Table 2 reports βAB parameter values predicted using the procedure described above and compares these to experimental βAB parameter values reported in the literature. All the reported measurements were conducted at 25 °C, except for mixtures 1 and 7, where the temperature was 23 °C, and mixtures 13 and 14, where the temperature was 10 °C. The abbreviations for the zwitterionic surfactants listed in Table 2 are less common, and their structures are therefore given in Table 3. Table 2 also lists the Kcap values of the various zwitterionic surfactants examined, calculated using eq 59. With the exception of C14HMP, which has a very long dsep ()8.3 Å), and C12AP, which also has a longer dsep ()3.8 Å) than most of the zwitterionic surfactants examined, the Kcap values are less than 1.3kT. In contrast, the Kelec values listed in Table 2 are generally approximately 4kT, indicating that the βAB values are dominated by the ionic surfactant interactions. To predict the βAB parameters, only the pure cmc of the ionic surfactant, CMCionic, is needed, because this value is used to estimate the ionic surfactant monomer concentration in determining C0 (see eq 58). However, to

Figure 4. Predicted (line) and experimental (circles) critical micelle concentration as a function of overall surfactant composition, Rsoln, for a mixture of SDS (Rsoln ) 1) and C14betaine (Rsoln ) 0) at 25 °C in 0.01 M NaCl (mixture 12 in Table 2). The experimental cmc values are from ref 41, and the best pred fit to the data gives βopt AB ) -5.4kT, with βAB ) -4.86kT.

predict the mixture cmc as a function of solution composition, the pure cmc’s of both surfactants are needed in finding both R*, using eq 62, and the mixture cmc, using eq 1. The superscript in the cmc column of Table 2 indicates the source of the pure surfactant cmc used in the mixture cmc predictions shown in Figures 2-7, corresponding to mixtures 4, 8, 11, 12, 14, and 15. For these mixtures, experimental data were available for the mixture cmc as a function of solution composition. For completeness, the pure zwitterionic cmc’s of all the surfactants examined are also listed in Table 2. This information would allow generation of the complete curves of mixture cmc as a function of solution composition for all the mixtures in Table 2. In ref 19, predicted βAB parameters were compared to βAB parameters deduced from experimental mixture cmc measurements. The experimental data were gathered from the literature, where, in most cases, experimental mixture cmc values were given as a function of solution composition. This allowed the βAB parameter based on experimental measurements to be calculated in two different ways. First, βAB was calculated separately at


Langmuir, Vol. 13, No. 15, 1997 3979

Figure 5. Predicted (line) and experimental (circles) critical micelle concentration as a function of overall surfactant composition, Rsoln, for a mixture of SDS (Rsoln ) 1) and C16betaine (Rsoln ) 0) at 10 °C in 0.01 M NaBr (mixture 14 in Table 2). The experimental cmc values are from ref 45, and the best pred fit to the data gives βopt AB ) -6.3kT, with βAB ) -4.94kT.

Figure 6. Predicted (line) and experimental (circles) critical micelle concentration as a function of overall surfactant composition, Rsoln, for a mixture of SDS (Rsoln ) 1) and C14HMP (Rsoln ) 0) at 25 °C (mixture 15 in Table 2). The experimental cmc values are from ref 7, and the best fit to the data gives pred βopt AB ) -2.4kT, with βAB ) -6.79kT.

each mixture composition, and then these values were avg arithmetically averaged to obtain a single βAB parameter characterizing the mixture, that is

βavg AB )

1

n

∑(βAB)i n i)1

(63)

where n is the number of compositions examined. Alternatively, the experimentally measured mixture cmc’s were fit as a function of solution composition, to obtain opt parameter, which best fits the data in the leastthe βAB avg opt squares sense. It was noted that βAB and βAB differed by as much as 1kT. For mixed surfactant systems containing zwitterionic surfactants, however, much less experimental cmc data are available in the literature. In most cases, the βAB parameter is reported, but the experimental cmc data from which it was derived is not. It is likely that the experimental βAB parameters reported are calculated by the averaging method, and not by the least-squares method. Since we do not have the necessary experimental cmc data to compare the two methods for all the cases

Figure 7. Predicted (line) and experimental (circles) critical micelle concentration as a function of overall surfactant composition, Rsoln, for a mixture of SDS (Rsoln ) 1) and DDAO (Rsoln ) 0) at 25 °C (mixture 8 in Table 2). The experimental cmc values are from ref 46, and the best fit to the data gives βopt AB ) -3.6kT, with βpred AB ) -4.66kT. opt presented in Table 2, we report βAB for mixtures 4, 8, 11, 12, 14, and 15, corresponding to the mixtures in Figures 2-7. For the other mixtures, we report the βAB parameter avg from the literature, which we assume to be βAB . In opt general, the predicted βAB values agree better with βAB avg than with βAB , although this is not true in all cases. mix If the rigorous derivation of gelec described in the previous section is used to calculate βAB , we find that the βAB parameter calculated this way is dependent on composition. In case I, the composition dependence is embedded in the term associated with charging the ionic portion of the mixed micelle. If that term is approximated by R2Kelec, then the composition dependence vanishes, and the βAB calculated by the rigorous method is identical to that calculated by the approximate method. In case II, AB ) by equating eq 50 with eq 15 and then solving for g elec βAB, we find that

βAB ) -Kelec - Kcap +

D2 Nsurf e2 2-R (64) 1 - R 2Rm(Rm + D2)w

where R is the micelle composition. Here, we have also assumed that the charging term is approximated by R2Kelec. At all compositions, βAB predicted using eq 64 is less pred negative than βAB reported in Table 2. This would exp improve our agreement with βAB for approximately half of the mixtures. Figures 2-7, corresponding to mixtures 4, 8, 11, 12, 14, and 15 in Table 2, compare predicted values of CMCmix with published experimental CMCmix measurements. These cmc predictions were made using the appropriate pred βAB parameters in Table 2, along with eqs 1, 2, 3 and 62. In Figures 2-5, a zwitterionic surfactant from the betaine family is mixed with an anionic surfactant. The predicted CMCmix values follow the experimental trends in all cases, including observed minima in the CMCmix versus composition curves. In Figures 3, 4, and 5, the predicted βAB parameter is less negative than the βAB parameter derived from experimental measurements. The predicted synergism is therefore not strong enough, and the predicted CMCmix values are in general slightly higher than the experimentally measured ones. In Figure 6, the zwitterionic surfactant C14HMP, with a more unusual structure, is mixed with SDS. C14HMP has a very long dsep, resulting

3980 Langmuir, Vol. 13, No. 15, 1997


Table 4. Common Classes of Zwitterionic Surfactants and Their Ability To Accept or Donate a Proton zwitterionic class

chemical structure

proton acceptor?

proton donor?

β-N-alkylaminopropionic acids N-alkylbetaines amine oxides dialkyl sulfoxides

RN+H2CH2CH2COORN+(CH3)2CH2COORN+(CH3)2ORS+(CH3)O-

yes yes yes yes

yes no no no

in a high Kcap value and a βAB parameter that is more negative than the βAB parameter based on experimental CMCmix measurements. The capacitor model used in this paper assumes a rigid dipole perpendicular to the surface of the micelle. If the dipole were allowed to bend, or to assume a finite angle with respect to the micelle surface, Kcap would be smaller and the synergism would be reduced, pred bringing βAB closer to the experimental value. Finally, in Figure 7, DDAO is mixed with SDS. DDAO is highly sensitive to variations in pH, changing from a nonionic surfactant to a cationic surfactant as the pH is decreased. In fact, it has been suggested36 that mixtures of DDAO with anionic surfactants should actually be treated as ternary mixtures of nonionic (protonated DDAO), cationic (unprotonated DDAO), and anionic surfactants. exp paC. Specific Interactions. Examining the βAB rameters listed in Table 2, we can divide the surfactant mixtures into two groups. For mixtures 1, 2, 3, 5, and 10, the magnitude of the experimental βAB is less than 1.5kT, while for all the other mixtures it is greater than 3kT. In contrast, with the exception of mixtures 1 and 2 (which are unusual in that the zwitterionic surfactant is mixed with a nonionic surfactant), the magnitude of all the predicted βAB parameters is greater than 3kT. Therefore, the agreement between the experimental and predicted βAB values for mixtures 3, 5, and 10 is not as good as for the other mixtures. This discrepancy can be understood by examining the structures of the zwitterionic surfactants in mixtures 3, 5, and 10 and comparing these with those in the remaining mixtures. Consider, for example, mixtures 3 and 4. In both cases, the zwitterionic surfactant is C12BMG (see Table 3). The carboxylic acid group on this surfactant can accept a proton, thus neutralizing some of the negative charge and, in effect, acquiring a partial positive charge. The amount of positive charge acquired depends on solution conditions such as pH. The partially positively-charged surfactant then interacts exp ) much more strongly with the anionic C12SO3Na (βAB exp -4.5) than with the cationic DTAB (βAB ) -1.3). In fact, because the low experimental βAB parameter indicates weaker attractive interactions in the DTAB mixture, we can infer an electrostatic repulsion. The difference in the interaction of C12BMG with C12SO3Na and DTAB can also be rationalized by examining the location of the charges.36 On the zwitterionic surfactant, the positive charge is closer to the micellar core. In an anionic/zwitterionic mixed micelle, the anionic charge can easily be close to the positive charge on the dipole, while keeping the hydrocarbon tail entirely inside the core. However, the negative charge on the dipole is farther from the micellar interface. Therefore, to achieve the closest proximity of opposite charges in a cationic/zwitterionic mixed micelle, the tail of the cationic surfactant would have to protrude from the core into the aqueous solution, making mixed micellization less favorable. The simplified theory presented in this paper, however, does not discriminate between anionic and cationic surfactants with the same valence |z| in their interaction with zwitterionic surfactants. Therefore, the predicted βAB parameters for both mixtures are very similar. (36) Weers, J. G.; Rathman, J. F.; Scheuing, D. R. Colloid Polym. Sci. 1990, 268, 832.

Experimental evidence37 indicates that specific interactions may play a significant role in surfactant mixtures containing zwitterionic surfactants. For example, complex formation between zwitterionic and anionic surfactants has been studied by precipitation and surface tension measurements.8,38-40 As described above, zwitterionic surfactants may interact differently with anionic surfactants than with cationic surfactants. A zwitterion that can acquire a proton can eliminate its negative charge and effectively become positively charged. It can then interact more strongly with anionic surfactants. On the other hand, a zwitterion that can donate a proton can become negatively charged and can then interact more strongly with cationic surfactants. The acquisition of a positive or negative charge by a zwitterion depends both on the solution pH and on the surfactant with which it is mixed. However, because only a fraction of the zwitterionic surfactants in a mixed micelle will become positively or negatively charged via this mechanism, the average interaction with an ionic surfactant is never as strong as in the anionic/cationic case. We can therefore think of the zwitterion as having a partial positive or negative charge which is less than the magnitude of the valence of each charge on the dipole. Table 4 lists representative classes of zwitterionic surfactants and their ability to accept or donate a proton. Since the simplified working model presented in this paper cannot capture specific interactions, in view of the results obtained in this paper, we recommend that its use be restricted to those cases in which proton-accepting zwitterions are mixed with anionic surfactants or protondonating zwitterions are mixed with cationic surfactants. Most common applications involving zwitterionic surfactants fall in the former category. VI. Conclusions The theoretical analysis presented in this paper extends the previous simplified “working model” of ref 19 to binary surfactant mixtures which contain zwitterionic surfactants. The only inputs required are the chemical structures of the surfactants, the pure surfactant cmc’s, and the solution conditions. The model is applicable to surfactants with linear hydrocarbon tails at various temperatures and concentrations of added salt. Note that the model neglects synergistic interactions due to steric interactions between the surfactant heads and due to the packing of the surfactant tails in the micellar core. The approximate electrostatic treatment of the simplified “working model” can be mapped to a more rigorous calculation of the electrostatic free energy of mixed micelles containing zwitterionic surfactants. We have shown that the approximations made in relating the more rigorous electrostatic calculation to the approximate one are reasonable. The predictions of the simplified model were found to agree well with experimental βAB parameters and mixture cmc’s in mixtures where the zwitterionic (37) Rosen, M. J. Langmuir 1991, 7, 885. (38) Tajima, K.; Nakamura, A.; Tsutsui, T. Bull. Chem. Soc. Jpn. 1979, 52, 2060. (39) Kolp, D. G.; Laughlin, R. G.; Krause, F. P.; Zimmerer, R. E. J. Phys. Chem. 1963, 67, 51. (40) Rosen, M. J.; Friedman, D.; Gross, M. J. Phys. Chem. 1964, 68, 3219.


surfactant acquires a partial charge opposite to that of the ionic surfactant with which it is mixed. The ability to predict the βAB parameters and the mixture cmc’s for mixtures containing zwitterionic surfactants is important because synergism at this level has been correlated to synergism in several practical applications. For example, in betaine/SDS mixtures, a minimum in the cmc was observed at a solution composition of 0.6 mol fraction zwitterionic surfactant.41 At the same composition, a maximum in the aggregation number and an increase in the viscosity were observed, and a maximum in solubilizing ability was also detected. In DDAO/SDS and C12-betaine/SDS mixtures, a minimum in the diffusion of the surfactant mixture through a collagen film was observed at the same composition corresponding to a minimum in the monomer concentration of the mixture,42 which can be predicted12 given the βAB parameter. Diffusion through collagen is an important indicator of the skin irritancy induced by a surfactant mixture. The ability to predict mixture cmc’s and βAB parameters thus has broader practical implications beyond the utility of the fundamental knowledge. Within the context of the complete molecular-thermodynamic theory, we are currently working on incorporating (41) Iwasaki, T.; Ogawa, M.; Esumi, K.; Meguro, K. Langmuir 1991, 7, 30. (42) Garcia, M. T.; Ribosa, I.; Sanchez Leal, J.; Comelles, F. J. Am. Oil Chem. Soc. 1992, 69, 25.

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predictions for mixtures including zwitterionic surfactants. In addition to the mixture cmc and the βAB parameter, the complete theory predicts micelle shape, size, and composition, as well as micelle and monomer concentrations. Some of these properties have been measured43 for anionic/zwitterionic and cationic/zwitterionic mixed micelles in a self-diffusion study. While the simplified theory presented here gives useful information and is relatively easy to use, the more complete theory may further increase our understanding of the behavior of these complex systems. Acknowledgment. Anat Shiloach is grateful for the award of an NSF Graduate Fellowship. Daniel Blankschtein is grateful to Kodak, Unilever, and Witco for partial support of this work. LA970160X (43) Jansson, M.; Linse, P.; Rymden, R. J. Phys. Chem. 1988, 92, 6689. (44) Rosen, M. J.; Zhu, B. Y. J. Colloid Interface Sci. 1984, 99, 427. (45) Wustneck, R.; Miller, R.; Kriwanek, J.; Holzbauer, H.-R. Langmuir 1994, 10, 3738. (46) Bakshi, M. S.; Crisantino, R.; De Lisi, R.; Milioto, S. J. Phys. Chem. 1993, 97, 6914. (47) Holland, P. M.; Rubingh, D. N. J. Phys. Chem. 1983, 87, 1984. (48) Rosen, M. J. Surfactants and Interfacial Phenomena, 2nd ed.; Wiley: New York, 1978.

Prediction of Critical Micelle Concentrations and Synergism of Binary

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