Effect of Counterion Binding on Micellar Solution Behavior: 1

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Effect of Counterion Binding on Micellar Solution Behavior: 1. Molecular-Thermodynamic Theory of Micellization of Ionic Surfactants Vibha Srinivasan and Daniel Blankschtein* Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 Received February 21, 2003. In Final Form: July 30, 2003 A molecular-thermodynamic theory is developed to model the micellization of ionic surfactants with added electrolytes in aqueous solution, by combining a thermodynamic description of the micellar solution free energy with a molecular model of the micellization process. The molecular model of micellization, which accounts for the various free-energy contributions associated with assembling the charged micelle from its constituent surfactant ions, allows for a fraction of the counterions, released by the surfactant heads and/or any added electrolytes, to bind onto the micelle surface and effectively induce a partial reduction of the micelle surface charge. The bound counterions are modeled as being intercalated among the surfactant heads on the micelle surface of charge. The remaining counterions are distributed according to the Boltzmann equation in the diffuse region, which lies beyond the Stern layer of steric exclusion adjacent to the micelle surface of charge. Expressions for the various free-energy contributions to micelle formation are derived for the general case when multiple counterion species are present in the solution, including inorganic counterions, such as Na+, as well as organic counterions having pendant hydrophobic groups that penetrate into the micelle core, such as the salicylate ion. In our theoretical formulation, the optimal degree of binding of each counterion species onto the charged micelle surface is a predicted quantity, obtained by minimizing the free energy of micelle formation, and depends on the nature of the counterion, including its hydrated size, valence, and lipophilicity. The theory developed is validated by implementing it in the case of the well-studied ionic surfactant system: sodium dodecyl sulfate (SDS) with added NaCl. We find good quantitative agreement between our theoretical predictions of (i) the optimal degree of Na+ binding onto the charged micelle surface, (ii) the electrostatic potential at the micelle surface of charge, (iii) the electrostatic potential at the Stern surface, and (iv) the critical micelle concentration (cmc) and the available experimental results for these properties. In addition, the optimal degree of counterion binding is predicted to vary with the bulk electrolyte concentration and with the micelle shape, consistent with the experimentally observed variations of the counterion concentration in the micelle-water interfacial region. In paper 2 of this series, the molecular-thermodynamic theory developed here is utilized to predict various micellar solution properties, including the cmc, the optimal degree of counterion binding, and the average micelle aggregation numbers, of aqueous solutions of ionic surfactants containing monovalent, multivalent, and/or organic counterions.

1. Introduction Ionic surfactants are the most widely used class of surfactants in both industrial practice and academic research, being utilized in practical applications ranging from household uses, including detergents, pharmaceuticals, and personal care products, to industrial applications involving coatings and lubricants. An increase in the solution ionic strength leads to a lowering of the critical micelle concentration (cmc) of ionic surfactants and to a transition from spherical to cylindrical (rigid rodlike or flexible wormlike) ionic micelles.1,2 This dependence has been attributed to the screening of the electrostatic repulsions between the charged ionic surfactant heads at the micelle surface by counterions released from these surfactant heads and from any electrolytes added to the solution. This strategy to induce micelle growth in ionic surfactant systems, via Coulombic screening by the * To whom correspondence should be addressed. Mail: Department of Chemical Engineering, Room 66-444, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139. Tel: (617) 253-4594. Fax: (617) 252-1651. E-mail: dblank@ mit.edu. (1) Tanford, C. The Hydrophobic Effect: Formation of Micelles and Biological Membranes; Wiley: New York, 1980. (2) Israelachvili, J. N. Intermolecular and Surface Forces; Academic Press: London, 1991.

counterions present in the solution, has been the subject of extensive experimental research3-8 and is also exploited in several commercial applications involving ionic surfactants, including the use of electrolytes as “thickeners” in shampoos, to impart increased viscosity to the product.9 In addition, the micellar properties of ionic surfactants are strongly affected not only by the overall counterion concentration but also by the specific type of counterions released from the surfactant heads and the electrolytes added to the solution. For example, in the case of the anionic surfactant dodecyl sulfate with associated monovalent alkali counterions, the cmc is observed to follow the sequence cmcCs+ < cmcK+ < cmcNa+ < cmcLi+; namely, the cmc is observed to increase with an increase in the size of the hydrated counterion.10 Micelles formed by the cationic surfactant cetyl trimethylammonium bromide (3) Hayashi, S.; Ikeda, S. J. Phys. Chem. 1980, 84, 744-751. (4) Mazer, N. A.; Benedek, G. B.; Carey, M. C. J. Phys. Chem. 1976, 80, 1075-1085. (5) Missel, P. J.; Mazer, N. A.; Benedek, G. B.; Young, C. Y.; Carey, M. C. J. Phys. Chem. 1980, 84, 1044-1057. (6) Mishic, J. R.; Fisch, M. R. J. Chem. Phys. 1990, 92, 3222-3229. (7) Porte, G.; Appell, J. J. Phys. Chem. 1981, 85, 2511-2519. (8) Imae, T.; Kamiya, R.; Ikeda, S. J. Colloid Interface Sci. 1985, 108, 215-225. (9) Wong, M. In Hair and Hair Care; Johnson, D. H., Ed.; Cosmetic Science and Technology Series 17; Marcel Dekker: New York, 1997; Chapter 3, p 33.

10.1021/la030069v CCC: $25.00 © 2003 American Chemical Society Published on Web 10/11/2003

Effect of Counterion Binding on Micellar Solutions

undergo a sphere-to-rod transition upon addition of only 0.1 M NaBr, while micelles formed by its chloride analogue require adding over 1.0 M NaCl to undergo this shape transition.8,11 Multivalent counterions, such as Al3+ and Ca2+, are known to be much more effective promoters of micelle growth than monovalent counterions, such as Na+, at the same ionic strength.12-14 Furthermore, lipophilic counterions, such as the aromatic salicylates, methylsalicylic acid, chlorobenzoates, and toluic acid15-18 as well as alkyl sulfonates19 and quaternary ammonium ions,20 strongly promote the formation of elongated rodlike or wormlike micelles in ionic surfactant solutions, where the presence of these micellar structures imparts viscoelasticity to the system. Therefore, the development of a quantitative, molecularly based thermodynamic theory capable of predicting the micellization behavior of ionic surfactant-electrolyte systems, including the effect of the counterion type, would be extremely valuable for the selection of ionic surfactants that exhibit the desired bulk solution behavior. Moreover, the micellization properties predicted by such a theory can often be correlated with solution properties of practical relevance,21 including the solution solubilization capacity and the solution rheological behavior, thus reducing the time and resources associated with trial-and-error type experimentation. Theoretical studies of ionic micellar systems22-26 have typically utilized variations of the Poisson-BoltzmannStern theory,27,28 which describes the formation of the diffuse counterion cloud surrounding the charged micelle, residing beyond a Stern layer of steric exclusion adjacent to the micelle surface. However, unlike charged solid surfaces, the micelle surface is porous and nonuniform, and the micelles are highly fluctuating entities. Therefore, it is certainly possible for some counterions to penetrate into the Stern layer, where they are highly restricted from moving parallel to the micelle surface due to lateral electrostatic and steric interactions. In this respect, a selfconsistent field lattice model of ionic surfactant micelles in the presence of indifferent counterions indicated that the surfactant heads are rather widely distributed outside (10) Rosen, M. J. Surfactants and Interfacial Phenomena; Wiley: New York, 1989. (11) Dorshow, R. B.; Bunton, C. A.; Nicoli, D. F. J. Phys. Chem. 1983, 87, 1409-1416. (12) Alargova, R. G.; Ivanova, V. P.; Kralchevsky, P. A.; Mehreteab, A.; Broze, G. Colloids Surf., A 1998, 142, 201-218. (13) Alargova, R.; Petkov, J.; Petsev, D.; Ivanov, I. B.; Broze, G.; Mehreteab, A. Langmuir 1995, 11, 1530-1536. (14) Alargova, R. G.; Danov, K. D.; Kralchevsky, P. A.; Broze, G.; Mehreteab, A. Langmuir 1998, 14, 4036-4049. (15) Hoffmann, H.; Rehage, H.; Reizlein, K.; Thurn, H. In Macroand Microemulsions: Theory and Applications; Shah, D. O., Ed.; ACS Symposium Series 272; American Chemical Society: Washington, DC, 1985; Chapter 4, p 41. (16) Rehage, H.; Hoffmann, H. Mol. Phys. 1991, 74, 933-973. (17) Lin, Z.; Cai, J. J.; Scriven, L. E.; Davis, H. T. J. Phys. Chem. 1994, 98, 5984-5993. (18) Magid, L. J.; Han, Z.; Warr, G. G.; Cassidy, M. A.; Butler, P. D.; Hamilton, W. A. J. Phys. Chem. B 1997, 101, 7919-7927. (19) Oda, R.; Narayanan, J.; Hassan, P. A.; Manohar, C.; Salkar, R. A.; Kern, F.; Candau, S. J. Langmuir 1998, 14, 4364-4372. (20) Kumar, S.; Aswal, V. K.; Goyal, P. S.; KabirudDin J. Chem. Soc., Faraday Trans. 1 1998, 94, 761-764. (21) Shiloach, A.; Blankschtein, D. Langmuir 1998, 14, 1618-1636. (22) Nagarajan, R.; Ruckenstein, E. Langmuir 1991, 7, 2934-2969. (23) Gunnarsson, G.; Jo¨nsson, B.; Wennerstro¨m, H. J. Phys. Chem. 1980, 84, 3114-3121. (24) Jo¨nsson, B.; Wennerstro¨m, H. J. Colloid Interface Sci. 1981, 80, 482-496. (25) Evans, D. F.; Mitchell, D. J.; Ninham, B. W. J. Phys. Chem. 1984, 88, 6344-6348. (26) Hayter, J. B. Langmuir 1992, 8, 2873-2876. (27) Verwey, E. J. W.; Overbeek, J. T. G. Theory of the Stability of Lyophobic Colloids; Elsevier: New York, 1948. (28) Bockris, J. O.; Reddy, A. K. N. Modern Electrochemistry I; Plenum Press: New York, 1977.

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the micelle core and that about 50-75% of the counterions released by the surfactant heads accumulate between them causing partial neutralization of the micelle charge.29 A theoretical analysis of electrophoretic measurements in the sodium dodecyl sulfate (SDS)-NaCl system30 indicated that a model which considers the presence of the Na+ ions solely in the diffuse layer largely overestimates the diffuse-layer electrostatic potential and postulated that the Na+ ions could be penetrating between the sulfate heads at the micelle surface (a similar observation has been made in the case of ionic surfactants adsorbed at the air-water interface as well31). Furthermore, molecular dynamics simulations of SDS micelles32,33 have indicated the formation of contact ion pairs by about 12-25% of the released Na+ ions with the sulfate heads on the micelle surface, as well as the presence of bridged Na+ ions shared by multiple sulfate heads. Therefore, a molecular model of micellization that allows for the presence of bound (that is, localized) counterions within the Stern layer of ionic surfactant micelles should be able to capture the electrostatic effects associated with the charged micelle-aqueous solvent interface. Furthermore, the introduction of counterion binding in a molecular model of micellization should be able to account for the counterion specificity of micellar solution properties, which cannot be addressed adequately on the basis of a diffuse-layer model alone, as demonstrated in previous theoretical studies.34 Theoretical efforts aimed at modeling counterion binding onto the micelle-aqueous solvent interface are few, due primarily to the complexities involved in describing the micelle interfacial region as well as in modeling the interactions between the various ionic species involved, including the discrete surfactant heads and the bound counterions. In the detailed model of the micelle interfacial region developed by Stigter,35-37 the Stern layer is modeled as a highly concentrated electrolyte solution, composed of the immobile hydrated surfactant heads and some counterions having restricted mobility in the free volume between the surfactant heads, due to their lateral electrostatic and steric interactions. Rathman and Scamehorn developed two triple-layer models of the Stern layer, where the counterions are modeled as either being mobile within the Stern layer or being localized on certain ionic surfactant heads as described by binding isotherms.38,39 Heindl and Kohler developed a molecular model of micellization that utilizes Langmuir isotherms to describe the direct adsorption of the counterions in the Stern layer.34 In all these theoretical studies, experimental results for the micelle surface charge and/or for the micelle size are utilized, in conjunction with a model of the electrostatic effects, to infer the degree of counterion binding. A predictive theory of counterion binding onto a charged micelle-solution interface, which considers explicitly the counterions in the Stern layer as being bound to the (29) Bo¨hmer, M. R.; Koopal, L. K.; Lyklema, J. J. Phys. Chem. 1991, 95, 9569-9578. (30) Stigter, D.; Mysels, K. J. J. Phys. Chem. 1955, 59, 45-51. (31) Warszyn´ski, P.; Barzyk, W.; Lunkenheimer, K.; Fruhner, H. J. Phys. Chem. B 1998, 102, 10948-10957. (32) Shelley, J.; Watanabe, K.; Klein, M. Int. J. Quantum Chem. 1990, 17, 103-117. (33) Bruce, C. D.; Berkowitz, M. L.; Perera, L.; Forbes, M. D. E. J. Phys. Chem. B 2002, 106, 3788-3793. (34) Heindl, A.; Kohler, H.-H. Langmuir 1996, 12, 2464-2477. (35) Stigter, D. J. Phys. Chem. 1964, 68, 3603-3611. (36) Stigter, D. J. Phys. Chem. 1975, 79, 1008-1014. (37) Stigter, D. J. Phys. Chem. 1975, 79, 1015-1022. (38) Rathman, J. F.; Scamehorn, J. F. J. Phys. Chem. 1984, 88, 58075816. (39) Rathman, J. F.; Scamehorn, J. F. Langmuir 1987, 3, 372-377.

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charged surfactant heads, was developed by Beunen and Ruckenstein.40 The central aspect of this theory involves the derivation of an equation to predict the degree of counterion binding onto the micelle surface, based on a detailed evaluation of the work of formation of the charged micelle surface, composed of the discrete ionic entities, along with the diffuse counterion cloud. In addition, these authors developed a simplified model of the charged micelle-solution interface that neglects the discreteness of the constituent ions and their nonrandom organization on the micelle surface, which can be utilized in the context of a predictive molecular model of micellization.41 The resulting model was then utilized to describe the behavior of SDS with added NaCl, and the trends of the model predictions were rationalized at a fundamental level. An important conclusion of the theoretical study in ref 41 is that the micelle size distribution is quite sensitive to even small contributions to the standard work of micelle formation, which in turn demands that one very accurately models the various electrostatic and nonelectrostatic contributions to the work of micelle formation. With the previous theoretical work in mind, in this paper, we present a fully predictive, molecular-thermodynamic theory of micellization for ionic surfactantelectrolyte systems, in which counterion binding onto micelles is accounted for explicitly. Our theory builds on molecular-thermodynamic theories of micellization developed previously by our group for single nonionic surfactants42 and for binary nonionic surfactant mixtures43,44 and extended recently to binary surfactant mixtures of ionic, zwitterionic, and nonionic surfactants.21 Specifically, a thermodynamic framework describing the free energy of the micellar solution is combined with a detailed molecular model of micellization, which evaluates the various physicochemical contributions to the standard work of micelle formation, which we refer to hereafter as the free energy of micellization. Our theory is similar, in spirit, to the simplified model of the charged micellewater interface developed by Ruckenstein and Beunen,41 in that the degree of counterion binding onto micelles is predicted based on a molecular model of micellization. The key differences between our theoretical approach and that of ref 41 are as follows: (1) Our model of counterion binding assumes that the bound counterions adsorb at the same plane as the ionic surfactant heads within the Stern layer and remain as independent entities in the Stern layer. The main implications of the choice of our model for the micelle-water interfacial region on the free energy of micellization will be discussed in sections 3 and 6 below. (2) We have applied our theory to model the behavior of ionic surfactant systems containing multivalent counterions as well as lipophilic counterions (see paper 2 of this series). In the case of lipophilic counterions, the pendant lipophilic moiety of the counterion resides in the micelle hydrophobic core consisting of the surfactant tails and therefore contributes to the free-energy changes associated with the formation of the micelle core. (3) The free-energy contributions associated with the formation of the micelle core have been evaluated in a more rigorous manner, with the appreciation that the level of detail with which these nonelectrostatic free-energy contributions are (40) Beunen, J. A.; Ruckenstein, E. J. Colloid Interface Sci. 1983, 96, 469-487. (41) Ruckenstein, E.; Beunen, J. A. Langmuir 1988, 4, 77-90. (42) Puvvada, S.; Blankschtein, D. J. Chem. Phys. 1990, 92, 37103724. (43) Puvvada, S.; Blankschtein, D. J. Phys. Chem. 1992, 96, 55675579. (44) Puvvada, S.; Blankschtein, D. J. Phys. Chem. 1992, 96, 55795592.

Srinivasan and Blankschtein

modeled affects the actual quantitative agreement of the theoretical predictions with the experimental data, particularly in the case of micelle sizes. The remainder of this paper is organized as follows. The thermodynamic framework describing the micellar solution is similar to that developed in our previous theoretical studies of micellization of single surfactants and binary surfactant mixtures,21,42-44 and the salient features of this framework, as they apply in the case of ionic surfactant micelles with bound counterions, are presented in section 2. The essence of the molecular model of micellization consists of visualizing the formation of a micelle as a series of reversible steps, each accompanied by a free-energy change. In section 3, we discuss the evaluation of the various free-energy changes, which can be divided into those associated with the formation of the micelle hydrophobic core and those associated with the formation of the micelle-water interfacial shell region, surrounded by the diffuse counterion cloud. In section 4, we describe the evaluation of various micellar solution properties, including the cmc, the optimal degree of counterion binding, and the average micelle aggregation numbers, based on the molecular-thermodynamic theory of micellization developed in sections 2 and 3. In section 5, we validate our model of counterion binding by comparing our predictions of the optimal degree of counterion binding, the electrostatic potentials at the micelle surface of charge and at the Stern surface, and the cmc’s in the SDS-NaCl system with the relevant experimental data. In addition, we provide a physical rationalization of the qualitative trends resulting from our theoretical predictions. In section 6, we discuss the main assumptions and limitations of our micellization model. Finally, in section 7, we present our concluding remarks. In paper 2 of this series, we implement the molecularthermodynamic theory developed here to quantitatively predict micellization properties of various ionic surfactant-electrolyte systems, including the cmc, the micelle shape and average micelle sizes, and the optimal degree of binding of various counterion species onto the charged micelles. The ionic surfactants considered are all of the symmetric 1:1 type, and the different cases examined correspond to solutions containing multiple counterion species, including monovalent, multivalent, and/or organic counterions. These ionic surfactant-electrolyte systems have been selected with the specific aim of studying the influence of the nature of the counterion, including its hydrated size, valence, and lipophilic character, on the surfactant micellization process. It should be stressed here that for the different ionic surfactant-electrolyte systems considered in paper 2 of this series, we were not able to achieve qualitative and quantitative accuracy in the prediction of micellar solution properties (especially the micelle sizes) by utilizing the previously developed micellization models,21,42 which ignored the presence of bound counterions at the micelle surface. Therefore, the systems considered in paper 2 of this series serve to illustrate the broad range of applicability of the molecular-thermodynamic theory developed here. 2. Thermodynamic Framework The underlying thermodynamic framework to describe single and mixed surfactant systems has been described in detail previously.21,42-44 In our earlier treatments of ionic surfactant systems, however, the ionic surfactants were assumed to be fully dissociated, and the released counterions were assumed to be distributed only in the


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electrical diffuse double layer surrounding the micelle according to the Boltzmann distribution. In this section, we present the essential elements of our theoretical framework as it applies to ionic surfactants when counterions are allowed to bind onto the micelle surface and hence become integral constituents of the micellar entity. In our earlier treatments of micellization, ionic surfactants were considered only in the context of ionic-nonionic surfactant mixtures.21,45 In such mixtures, the degree of counterion binding is expected to decrease rapidly with decreasing micelle composition of the ionic surfactant, consistent with many experimental observations.38,46 Therefore, the neglect of counterion binding in our earlier treatments of mixed micellization appears justified. The ionic micellar solution under consideration contains surfactant ions (component s), counterions dissociated from the surfactant molecules, and counterions and coions released by any added electrolytes. In general, the counterions released from the added electrolyte(s) in solution may be different from those released from the surfactant molecules, in which case, the various counterion species in solution would tend to compete for adsorption onto the micelle surface. The number of counterions of species cj that remain associated with the micelle, per surfactant molecule in the micelle, is referred to hereafter as the degree of binding of counterion cj and is denoted by βj. Therefore, the ionic character of the micelle is determined by the number of surfactant molecules in the micelle, that is, by its micelle aggregation number, n, as well as by the degrees of counterion binding, {βj}. It then follows that at thermodynamic equilibrium, the chemical potential of the micelle (an n{βj}-mer), µn{βj}, can be related to the chemical potential of the surfactant monomers in the bulk aqueous solution, µs, and to that of the counterions in the bulk aqueous solution, µcj, as follows:

nµs +

∑

j)1,2,..

nβjµcj ) µn{βj}

(1)

Equation 1 represents the conventional micellemonomer chemical equilibrium condition,1,22,42 modified to include the counterions in the formation of the micelle. The chemical potentials of the individual species in the solution, µs and µcj, and of the micelle, µn{βj}, can be obtained given an expression for the total Gibbs free energy of the surfactant solution, as shown in our earlier treatments of mixed micellar solutions.21,43 By substituting the resulting expressions for the chemical potentials in eq 1, we obtain the following expression describing the population distribution of micelles of varying sizes and degrees of counterion binding:

Xn{βj} )

() 1 e

∏j (Xc e)nβ ×

(Xse)n

[ (

exp -

j

j

0 n µn{βj}(S, lc, {βj})

kT

n

- µ0s -

other words, µ0i includes the interactions of component i (i ) n{βj}-mer, s, and cj) with the aqueous “hypersolvent” containing the various ionic species. Interactions between the micelles, however, have been neglected here. In ionic surfactant systems, the intermicellar interactions, which include excluded-volume (steric) interactions, electrostatic interactions, and attractive interactions, typically become important only at high micelle concentrations and/or at low electrolyte concentrations.47-49 In dilute surfactant solutions having low micelle concentrations and high electrolyte concentrations, the average intermicellar spacing (which can be estimated by nmic-1/3, where nmic is the micelle number density) is much larger than the average size of the micelle along with its electrical double layer (which can be estimated by R + κ-1, where R is the micelle radius based on its excluded volume and κ-1 is the Debye-Hu¨ckel screening length of the solution, a measure of the length scale associated with the decay of the electrostatic interactions for the charged micelle). Under these conditions, the intermicellar interactions should be small,14 and the free-energy contributions associated with these interactions may therefore be neglected relative to the free-energy contributions associated with the intramicellar effects. The surfactant-electrolyte systems that we have modeled in paper 2 of this series can all be considered to be thermodynamically dilute solutions, and therefore, our neglect of intermicellar interactions is valid for these systems. It is illuminating and computationally useful to rewrite eq 2 such that the size (n) dependence of the micelle population distribution is separated from its dependence on the ionic character (as reflected in {βj}). Specifically, the size distribution can be expressed in terms of the translational entropy loss of the surfactant molecules and the free-energy gain per surfactant molecule as follows:

∑j βjµc0

j

)]

(2)

where Xi (i ) n{βj}-mer, s, and cj) is the mole fraction of component i, µ0i (i ) n{βj}-mer, s, and cj) is the standardstate chemical potential of component i, k is the Boltzmann constant, and T is the solution absolute temperature. The standard-state chemical potentials, µ0i , are defined for the various components in a state of infinite dilution in the aqueous solution at the prescribed ionic strength. In (45) Shiloach, A.; Blankschtein, D. Langmuir 1997, 13, 3968-3981. (46) Akisada, H. J. Colloid Interface Sci. 2001, 240, 323-334.

Xn{βj} ) where

g˜ mic )

(

0 µn{β j}

n

()

[

]

g˜ mic({βj, Xcj}) 1 n Xs exp -n e kT

- µ0s -

)

∑j βjµc0 ∑j βjkT ln(Xc ) - kT(1 + ∑j βj) j

(3)

j

(4)

In eq 3, the pre-exponential term represents the loss in translational entropy associated with localizing the n monomeric surfactant ions in the micelle, while in the exponential term, g˜ mic, defined as the free energy of micellization, reflects the net free-energy gain on a “per surfactant” molecule basis associated with transferring the surfactant monomers, s, and the counterions, {cj}, from the bulk aqueous solution, where the counterion concentrations are {Xcj}, to form the n{βj}-mer. Specifically, the first term in brackets in eq 4 for g˜ mic reflects the freeenergy advantage associated with forming the micelle,42,43 while the second term in eq 4 reflects the translational entropy loss incurred in binding each of the nβj counterions (cj) onto the micelle. Essentially, the translational entropy loss incurred due to the loss in mobility of the bound counterions opposes the binding of counterions onto the micelle, while the reduction in the total micelle surface (47) Odijk, T. J. Chem. Phys. 1990, 93, 5172-5176. (48) Mileva, E. J. Colloid Interface Sci. 1996, 178, 10-17. (49) Zoeller, N.; Lue, L.; Blankschtein, D. Langmuir 1997, 13, 52585275.

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charge resulting from the bound counterions promotes counterion binding. Note that the last term in eq 4 arises from the entropy of mixing the various species in the aqueous solution (s, c, and nβj-mers), which has been defined based on the number densities of the entities in solution42,43 rather than on their mole fractions. The free energy of micellization, g˜ mic, is a function of the micelle shape, S, its core minor radius, lc, and the ionic character of the micelle, as reflected in the degrees of counterion binding onto the micelle surface, {βj}. The calculation of the first term in brackets in eq 4 for g˜ mic, according to the molecular model of micellization developed here, is described in section 3 below. The mole fractions of the counterions in the bulk aqueous solution, {Xcj}, required to evaluate the second term in eq 4, can be obtained by performing a mass balance on each of the counterion species in the solution, in conjunction with the condition of electroneutrality in the bulk aqueous solution. Specifically, the bulk aqueous solution, which contains ionic surfactant monomers, counterions dissociated from the surfactant molecules, and counterions and inert coions dissociated from the added electrolytes, should be electroneutral (and consequently, the micelle along with its electrical double layer should be electroneutral), which implies that the various {Xcj} are related to Xs. Finally, to determine Xs, the following mass balance equation must be satisfied: ∞

Xsurf ) Xs +

∑ ∑nXn{β }

n)2 {βj}

j

(5)

where Xsurf is the total mole fraction of ionic surfactant in the solution, which is distributed between the monomeric and the micellar fractions. Equation 3 is then substituted in eq 5, along with the expressions for {Xcj} derived as indicated above, resulting in a single equation for Xs, which can then be solved numerically. The optimal degree of binding of a counterion, cj, onto a micelle of aggregation number n, denoted by β/j (n), can then be obtained by maximizing the micelle population distribution, Xn{βj} in eq 3, with respect to βj, or equivalently by minimizing g˜ mic in eq 4 with respect to βj for each counterion cj. It is reasonable to assume that all micelles having aggregation number n have the same degree of counterion binding given by {β/j (n)}, namely, that the micelle population distribution is sharply peaked around its maximum value, Xn{β*j (n)}. This approximation parallels the one made in mixed micellar systems21 regarding the composition (R) dependence of the micelle size distribution, XnR. The rationale behind this approximation is that the exponential term in eq 3 has the aggregation number, n, as a multiplicative factor to g˜ mic, and hence, small deviations of g˜ mic from its optimal value at {β/j } will result in Xn{βj} ≈ 0. This approximation simplifies the solution of eq 5, which now involves carrying out only a summation over n to obtain Xs and the micellar size distribution, Xn{β*j }. 3. Free Energy of Micellization In this section, we discuss briefly the salient features of the molecular model of micellization (described in detail previously21,42-44) and explain in detail the new physicochemical contributions to the free energy of micellization that arise due to the binding of counterions onto the micelle surface. The molecular model of micellization presented here applies to the generalized case when there are several

Figure 1. Schematic representation of the micelle-aqueous solvent interfacial region. The adsorbed counterions and the surfactant heads are located on the micelle surface of charge, positioned at a distance r ) Rch ) lc + dch from the center of the micelle. The Stern surface, located at a distance dst from the micelle surface of charge, marks the beginning of the diffuse layer, where the remaining counterions are distributed. Values of the dimensions, dch and dst, and of the dielectric constants, ηs and ηb, of the different regions are provided in the text.

counterion species in the solution capable of binding onto the micelle surface. 3.1. Assumptions of Our Model of a Charged Micelle. A schematic representation of our model of a charged micelle is shown in Figure 1. For spherical micelles, lc is the radius of the spherical hydrocarbon core, for cylindrical micelles, lc is the radius of the circular cross section of the cylindrical hydrocarbon core, and for discoidal micelles, lc is the half-thickness of the planar hydrocarbon core. The center of charge of the ionic surfactant heads lies on a surface (r ) Rch), referred to as the micelle surface of charge (where Rch ) lc + dch, and dch can be obtained from the molecular structure of the surfactant). The Stern surface (r ) Rs) marks the distance of closest approach of the counterions in the diffuse layer to the micelle surface of charge due to their finite size (where Rs ) Rch + dst ) lc + dch + dst, and the thickness of the Stern layer, dst, can be estimated from the molecular structures of the hydrated surfactant heads and the hydrated counterions). In our model, we assume that the bound counterions are able to penetrate into the Stern layer all the way to adsorb at the same Helmholtz plane as the ionic surfactant heads, namely, at the micelle surface of charge (see Figure 1). In addition, our model assumes that the counterions adsorbed onto the micelle surface of charge preserve their freedom, which implies that a counterion can be shared by several surfactant heads, and that no ion-pairing of a surfactant head and a counterion occurs. In essence, our model, in which the surfactant ions and the counterions are adsorbed independently on the same Helmholtz plane, is similar to the Stern layer model of Warszyn´ski et al.31,50 used to describe ionic surfactant adsorption at the airionic surfactant solution planar interface. The surfactant ions and the bound counterions are considered to be arranged randomly on the micelle surface of charge, and possible ordering effects resulting from the interactions between the surfactant ions and the bound counterions have been ignored (note that the consequences of assuming a random distribution are discussed in detail in section 6.2). In the diffuse ion cloud, all the ions are treated as point charges having no physical excluded volume except (50) Warszyn´ski, P.; Lunkenheimer, K.; Czichocki, G. Langmuir 2002, 18, 2506-2514.


for a minimum distance of closest approach to the charged micelle surface, as reflected in the context of the Stern layer model.28 Our model is different from most previously developed models of charged surfaces in micelles and monolayers38,41 that utilize Grahame’s concept of the triple layer, in which the counterions are considered to adsorb on an inner Helmholtz plane within the Stern layer, which is different from the plane on which the surfactant heads reside. We have chosen to utilize the same plane to describe the location of both ionic species, mainly in order to reduce the detailed structural information needed otherwise regarding the micelle interfacial region, including the dimensions of the different regions within the Stern layer, as well as the dielectric constant of the solvent in these different regions. There is considerable variability in the experimentally determined values of these structural parameters,28 and consequently, by adopting a simplified model of the Stern layer, we reduce the sensitivity of our model predictions to these structural parameters. To test if the proposed arrangement of the surfactant ions and the bound counterions on the micelle surface of charge is reasonable, we consider the case of a spherical SDS micelle. Based on known structural information about the SDS molecule1 (hydrophobic volume, vt ≈ 324 Å3, and extended tail length, lmax ≈ 15.5 Å), the average available interfacial area per sulfate head is estimated to be about 62 Å2 (see eq 9). Considering that the cross-sectional area of the hydrated sulfate head, ah,s, is equal to 25 Å2 (see section 5.1), we can conclude that there is sufficient available free area on the plane where the sulfate heads are located to accommodate the small Na+ counterions (ah,Na ) 15 Å2, see section 5.1). In addition, we have implemented our model to describe surfactant systems containing multivalent counterions and lipophilic counterions. For such counterions, our description of counterion binding onto the plane on which the surfactant heads reside is more applicable, since multivalent counterions can be shared by several surfactant heads, and lipophilic counterions adsorb such that their lipophilic moieties penetrate into the micelle core. With these considerations in mind, we have adopted this simplified model of the micelle interfacial region in which the surfactant ions and the bound counterions are located on the same Helmholtz plane. In section 6.1, we will further discuss experimental and simulation results concerning the proximity of counterions to the charged micelle surface and show that these results provide partial justification for the choice of our model of the Stern layer. Finally, we discuss the dielectric properties of the solvent in the regions surrounding the charged micelle. The discrete nature of the solvent molecules has been neglected throughout, and the solvent has been treated as a dielectric continuum. The dielectric constant of water outside of the Stern layer is taken to be that of pure water, denoted by ηb (see Figure 1). Within the Stern layer, the dielectric constant of the solvent, denoted by ηs, is lower due to the high degree of orientation of the water dipoles in that region under the influence of the high concentration of ions, including the surfactant ions and the bound counterions.28 Having discussed the model adopted to describe the micelle-solvent interfacial region, we describe next the molecular model of micellization developed to evaluate g˜ mic. The first term in brackets in eq 4 for g˜ mic, which represents the free-energy change involved in transferring the constituents of the micelle in their standard states in the aqueous solution to form the micellar entity, can be evaluated using a thought process which visualizes the

Langmuir, Vol. 19, No. 23, 2003 9937

formation of a micelle as a series of reversible steps, each associated with a physicochemical contribution to the freeenergy change upon micellization. The relevant freeenergy contributions can be divided into those associated with the formation of the micelle hydrophobic core, consisting of the surfactant tails and the hydrophobes of any bound lipophilic counterions, and those associated with the formation of the micelle interfacial shell region, consisting of the surfactant heads and the bound counterions. In sections 3.2 and 3.3 below, we describe our evaluation of the various free-energy contributions associated with the formation of the micelle core and the micelle interfacial shell regions, respectively, for the case of both inorganic and organic counterions. Note that inorganic counterions, such as Na+ and Al3+, adsorb at the plane of the surfactant heads (see Figure 1) and therefore influence only the free-energy changes associated with the formation of the micelle interfacial shell. On the other hand, a hydrophobic counterion behaves, in spirit, like a second, oppositely charged surfactant. In this case, the counterions contribute to the free-energy changes involved in the formation of both the hydrophobic core and the interfacial shell. 3.2. Formation of the Micelle Core. The micelle core is considered to be a “dry core” having a uniform density equal to that of bulk liquid hydrocarbon. In this section, we derive expressions for the free-energy changes associated with the formation of the micelle core, on a per surfactant molecule basis, for the general case when one of the counterion species, say, species ck (whose degree of binding onto the micelle is given by βk), is organic. We will mainly highlight the new aspects that arise as a consequence of counterion binding, since complete details have been provided in our previous articles.21,42-44 The following free-energy contributions are involved in the formation of the micelle core: 1. The transfer free energy, gtr, is the attractive freeenergy contribution per surfactant molecule associated with transferring the hydrocarbon tail of the surfactant and the lipophilic moiety of the organic counterion, ck, from the bulk aqueous solvent to form a bulk hydrocarbon phase, which can be viewed as the precursor of the micelle core. On a per surfactant molecule basis, this can be modeled generally as follows:

gtr ) gtr,s + βkgtr,ck

(6)

where gtr,i (i ) s and ck) is the free-energy change associated with transferring component i from the aqueous solution to a bulk hydrocarbon phase. Equation 6 reflects the fact that the organic counterion has a greater driving force to bind onto the micelle surface, as compared to an inorganic counterion. For the surfactant alkyl tails, gtr,s can be evaluated from experimental data on the aqueous solubility of linear alkanes.1 For the lipophilic portion of the organic counterion, ck, the term gtr,ck can also be evaluated using either experimental data for its aqueous solubility (similar to the case of the surfactant tails) or groupcontribution methods51 which empirically relate the aqueous solubility of a molecule to structural descriptors of that molecule. 2. The interfacial free energy, gint, represents the freeenergy change per surfactant molecule associated with forming an interface between the hydrocarbon micelle core and the surrounding aqueous solution. This contribution is evaluated using the concept of a macroscopic interfacial (51) Klopman, G.; Zhu, H. J. Chem. Inf. Comput. Sci. 2001, 41, 439445.

9938



free energy of a hydrocarbon-water interface, including its dependence on interfacial curvature, as follows:

gint ) σ(a - a0)

(7)

where σ is the curvature-dependent interfacial tension of the hydrocarbon core against the aqueous solvent, a is the interfacial area available per surfactant molecule, and a0 is the interfacial area per surfactant molecule screened from contact with the aqueous solvent due to the physical connection of the hydrophobic groups (such as the tail of the surfactant or the tail of the lipophilic counterion) present in the micelle core to the hydrophilic groups outside (such as the hydrophilic head of the surfactant or the head of the lipophilic counterion) present in the micelle interfacial shell. If the molecular volume of the hydrophobic moiety of counterion ck that penetrates into the micelle core is denoted by vck and that of the surfactant tail by vt, then the volume fraction of counterion ck in the micelle core, ηk, is given by

ηk )

βkvck vt + βkvck

(8)

The interfacial area, a, on a per surfactant molecule basis, can be obtained from geometrical considerations and is given by

a)

( )( ) Svt 1 lc 1 - η k

(9)

where S is a shape factor (3 for spheres, 2 for infinitesized cylinders, and 1 for infinite-sized disks or bilayers). For a micelle containing only inorganic counterions, for which βk ) 0, eq 9 yields the familiar expression: a ) Svt/lc.2 Simply put, eq 9 reflects the fact that the presence of the organic counterions pushes the surfactant heads apart, thereby increasing the average area at the interface accessible to each surfactant head along with the associated bound organic counterions. To estimate σ in eq 7, a volume-fraction-weighted expression is utilized to express σ in terms of σs and σck, the curvature-dependent interfacial tensions of the surfactant tail and the organic counterion hydrophobe against water, respectively. Specifically,

σ ) (1 - ηk)σs + ηkσck

(10)

The curvature dependence of the interfacial tension, σi (i ) s and ck), is approximated using the Gibbs-TolmanKoenig-Buff equation,52 which describes the decrease of σi with respect to its value for a planar interface (details are given in ref 42). Finally, the area shielded at the interface by the hydrophilic head in the interfacial shell, a0, is estimated using the following composition-weighted expression:

a0 ) a0,s + βka0,ck

(11)

where a0,s is the interfacial area screened from contact with the aqueous solvent by surfactant s and a0,ck is that corresponding to counterion ck. The molecular parameters of the surfactant tails, which are typically n-alkanes, utilized in eqs 8-11, are given by vt ) 27.4 + 26.9(nc 1) Å3, a0,s ) 21.0 Å2, and the fully extended (all-trans) tail length lmax ) 1.54 + 1.269(nc - 1) Å.1,42 The molecular parameters of the organic counterion, ck, are specific to (52) Tolman, R. C. J. Chem. Phys. 1949, 17, 333-337.

the counterion being modeled and will be computed in paper 2 of this series for the specific counterion considered (salicylate). 3. The packing free energy, gpack, represents the freeenergy change per surfactant molecule associated with the loss in conformational degrees of freedom of the surfactant tails and the hydrophobic groups of the organic counterions, ck, in the micelle core, as compared to the bulk hydrocarbon phase, since one end of each of these molecules is anchored at the micelle-water interface. The free energy of packing, gpack, is evaluated by implementing a single-chain mean-field model developed by Ben-Shaul, Szleifer, and Gelbart.53-55 In this approach, we first generate all the internal conformations of a hydrophobe (surfactant or counterion tail) utilizing a model for the torsional states of the bonds in the molecule. The excludedvolume interactions between this central surfactant (or counterion) tail and the neighboring tails are accounted for in the context of a radially varying mean field acting on this central hydrophobe. The strength of the mean field is evaluated self-consistently by imposing the constraint that the density inside the micelle core be uniform and liquid-hydrocarbon-like.53 The free energy of packing, gpack, which is a strong function of S, lc, and βk, is then obtained by evaluating the partition functions of the central surfactant and counterion tails, which are acted upon by this mean field. Structural parameters of the alkyl chain and of the counterion hydrophobe, such as the bond lengths and bond angles, the molecular volumes, and models for their internal configurational states (such as the rotational isomeric state model for alkyl chains56), are required to generate the conformations. The actual implementation of this computational approach has been explained in detail previously for the case of both single surfactant micelles42 and binary surfactant mixed micelles,44 and we shall therefore omit the details here for the sake of brevity. 3.3. Formation of the Micelle-Water Interfacial Shell. The following free-energy contributions are involved in the formation of the micelle-water interfacial shell: 1. The steric free energy, gst, accounts for steric interactions between the surfactant heads and the adsorbed counterions at the micelle-water interface. The molecules present at the micelle surface are treated as components of an ideal localized monolayer. The resulting steric free-energy contribution, gst, can be expressed as follows:

gst ) -kT(1 +

∑j

(

βj) ln 1 -

ah,s +

∑j βjah,c a

)

j

(12)

where ah,s and ah,cj are the effective cross-sectional areas of the surfactant head, s, and the counterion, cj, respectively. In deriving eq 12, the ions are modeled as hard spheres occupying areas ah,s and {ah,cj} at the micelle corewater interface. Equation 12 reflects the fact that the presence of bound counterions along with the surfactant heads at the micelle surface leads to an increase in the steric repulsions. The expression for gst in eq 12 was derived using the test-particle approach described in our earlier theoretical studies of micellization in binary (53) Ben-Shaul, A.; Szleifer, I.; Gelbart, W. M. J. Chem. Phys. 1985, 83, 3597-3611. (54) Szleifer, I.; Ben-Shaul, A.; Gelbart, W. M. J. Chem. Phys. 1986, 85, 5345-5358. (55) Szleifer, I.; Ben-Shaul, A.; Gelbart, W. M. J. Chem. Phys. 1987, 86, 7094-7109. (56) Flory, P. J. Statistical Mechanics of Chain Molecules; Interscience: New York, 1969.


Langmuir, Vol. 19, No. 23, 2003 9939

surfactant mixtures.44 The physical picture here is that (na - n(ah,s + ∑jβjah,cj)) represents the total available unoccupied area on the monolayer and is related to the number of ways in which a test particle can be introduced in the monolayer without occupying a region already occupied by another particle (a surfactant head or a bound counterion). The cross-sectional area of the surfactant head, ah,s, is obtained from the molecular structure of the surfactant head, and those of the counterions, {ah,cj}, are obtained from knowledge of the hydrated ion radii. 2. The reduction in the electrostatic repulsions between the ionic surfactant heads resulting from the adsorbed counterions is the main driving force promoting counterion binding. The electrostatic free energy, gelec, is calculated as the reversible work required to assemble this micelle starting from the monomeric ionic species in the bulk aqueous solution25,27 and depends on the micelle surface charge, through the degree of counterion binding, {βj}, as well as on the micelle structure, through S and lc. The evaluation of gelec for surfactant-electrolyte systems is explained in detail in section 3.4. 3. The entropic gain associated with mixing the surfactant heads and the bound counterions at the micelle surface is denoted by gent. As stressed in section 3.1, the surfactant ions and the bound counterions are considered to be arranged randomly on the micelle surface, and therefore, we utilize the following ideal mixing entropy expression to model gent:

gent ) kT ln

( ) 1

1+

∑j βj

+

∑j βjkT ln

( ) βj

1+

∑j βj

(13)

Upon evaluation of the various free-energy contributions in eqs 6-13 and the evaluation of gelec as described in section 3.4 below, we can compute g˜ mic and predict the optimal micellar solution properties as described in section 4 below. 3.4. Electrostatic Free Energy. In this section, we describe the evaluation of the electrostatic free energy, gelec, based on the structure of the micelle interfacial shell, as depicted in Figure 1 and described in section 3.1. The electrostatic effects associated with transferring the ionic surfactants and the counterions in their standard states in the bulk aqueous solution to form a charged monolayer at the micelle surface and the related distribution of the remaining counterions to form the electrical diffuse layer surrounding the micelle can be modeled using the following thought process. The n ionic surfactant heads (s) and each of the nβj counterions (cj) in their standard states, which will assemble into the n{βj}-mer, are first discharged in the bulk aqueous solution. The discharged surfactant heads and counterions are then assembled to form a neutral micelle surface. This neutral micelle surface can be modeled as a two-dimensional monolayer, where steric repulsions and entropic effects due to mixing exist between the discharged constituent ions, with the corresponding free energies, gst and gent, evaluated using eqs 12 and 13, respectively. The constituent ions are then recharged, bringing the micelle surface to its final charge density. The counterions remaining in the bulk solution, in response to the attractive electrostatic potential generated by the charged micelle, assemble to form the diffuse ion cloud, thus creating a neutral, “dressed” micelle. It then follows that the free energy associated with the electrostatic effects on a per surfactant molecule basis, gelec, in

the micelle-water interfacial shell is given by

gelec ) Welec + gdis

(14)

where gdis denotes the electrostatic self-energy of each ion comprising the micelle released during the discharging process and Welec denotes the reversible work done, on a per surfactant molecule basis, in charging the micelle surface to its final charge density against the electrostatic potential that it generates.25,27 The net release of the electrostatic self-energy during the discharging process, gdis, is given by gdis ) gdis,s + ∑jβjgdis,cj, where gdis,i represents the electrostatic self-energy released by species i (i ) s and {cj}). The expression for gdis,i, the electrostatic selfenergy released by an ionic species i, is given by28

gdis,i ) -

zi2e02 2brh,i(1 + κrh,i)

(15)

Equation 15 is the familiar Debye-Hu¨ckel expression for the self-energy of ionic species i having a valence zi, where e0 is the electronic charge, b ) 4πηb0 (where 0 is the dielectric permittivity of vacuum), and rh,i is the hydrated radius of ionic species i in the bulk aqueous solvent. The Debye screening length of this ionic aqueous solution, κ-1, is a measure of the length scale over which electrostatic effects decay in the solution (where κ ) (8πe02I/ bkT)1/2, and I denotes the solution ionic strength28). Note that the self-energy expression in eq 15 includes the Born energy associated with the ion (i)-solvent interactions, as well as the self-atmosphere energy associated with the interactions of ion i with other ions in the bulk aqueous solution (see pages 56 and 223 of ref 28). We next need to calculate the reversible work incurred in charging the micelle surface to its final surface charge, Welec. During this charging process, the charges are smeared uniformly over the plane of adsorption, such that the micelle surface is treated as a uniform, twodimensional charged surface. The work done in charging this uniformly charged surface is given by the Guntelberg charging process:27

Welec )

∫0q ψ0(q) dq f

(16)

where ψ0 is the electrostatic potential at the micelle surface of charge (r ) Rch ) lc + dch, see Figure 1), qf is the final average micelle charge on a per surfactant molecule basis, and q is the instantaneous micelle charge on a per surfactant molecule basis, which varies from zero to qf as the micelle evolves from being uncharged to being fully charged. The instantaneous and final micelle charges, q and qf, depend on the degree of binding of the counterions, and for a monovalent surfactant ion, |qf| ) (1 - ∑jβjzcj)e0. The electrostatic surface potential, ψ0, is determined by solving the Poisson equation separately in the Stern layer and in the diffuse region. In the Stern layer, there are no free ions present between the two bounding surfaces, the micelle surface of charge (r ) Rch ) lc + dch) and the Stern surface (r ) Rs ) Rch + dst), and therefore, the electrostatic potential in the Stern layer is governed by the well-known Laplace equation. By solving the Laplace equation, utilizing Gauss’ law for the charged micelle surface, and requiring the condition of continuity of the electrostatic potential at the Stern surface, one obtains Welec in eq 16 as21

Welec )

2πqf2 F (S) + sach

∫0q ψs(q) dq f

(17)

9940



where ach is the area available per surfactant molecule at the micelle surface of charge, ψs is the electrostatic potential at the Stern surface, and F (S) is a shapedependent function given by

F (S) ) dst

S ) 1 (planar micelle)

(18)

F (S) ) Rch ln(1 + dst/Rch) S ) 2 (cylindrical micelle) (19)

4. Evaluation of Various Optimal Micellar Solution Properties The free energy of micellization, g˜ mic(S, lc, {βj}), can be evaluated by summing the free-energy contributions described in sections 3.2 and 3.3, along with the translational entropy loss of the bound counterions. Specifically,

g˜ mic ) gtr + gint + gpack + gst + gelec + gent -

∑j βjkT ln(Xc ) - kT(1 + ∑j βj) j

F (S) ) dst/(1 + dst/Rch) S ) 3 (spherical micelle) (20)

(22)

where the dimensionless variables utilized for the instantaneous electrostatic potential, the position, and the instantaneous micelle surface charge density are y0 ) e0ψs/ kT, x0 ) κRs, and s ) (4πe0/bkTκ)q/as (where as is the interfacial area per surfactant molecule at the Stern surface, r ) Rs), respectively. For a solution containing only monovalent ions, eq 21 is solved numerically, using a Newton-Raphson scheme, to obtain y0 for a given value of the instantaneous micelle surface charge density, s. Finally, ψs is integrated numerically, according to eq 17, to evaluate Welec, the work of charging the micelle surface. Finally, it should be stressed here that in the evaluation of Welec, we have ignored the discrete nature of the surfactant ions and the bound counterions. The consequences of ignoring discrete-ion effects (and thereby, any nonrandomness in the arrangement of these ions on the micelle surface) is discussed in detail in section 6.2.

For each of the three regular micelle shapes (spheres, infinite cylinders, and infinite bilayers), the free energy of micellization, g˜ mic, is calculated and subsequently minimized with respect to the set, {βj}, to determine the optimal degrees of binding, {β/j }. In other words, differences between the degrees of binding of the various counterions arise due to differences in their effect on the free energy of micelle formation. For example, for an organic counterion, the hydrophobic driving force will promote higher values of β/j as compared to that for an inorganic counterion, as reflected in eq 6. The optimum micelle structural characteristics, l/c and S*, are then estimated by minimizing g˜ mic with respect to lc and S. The minimum value of g˜ mic, which corresponds to the free energy of forming the micelle with optimal structural and compositional characteristics, defined by S*, l/c , and {β/j }, is referred to hereafter as g˜ /mic. The cmc of the surfactant is, as in our previous work,42 estimated by the expression exp(g˜ /mic/kT). According to eq 3, the population distribution of micelles having the optimal characteristics becomes significant only when the surfactant monomer concentration, Xs, is comparable to the quantity exp(g˜ /mic/kT). Therefore, this quantity serves as a good quantitative indicator of the onset of micelle formation in solution, namely, of the surfactant cmc. The optimal micelle shape, S*, corresponds to the shape of the micelles that form predominantly in the solution. Specifically, if S* corresponds to a sphere, then finite-sized spherical micelles, whose aggregation number is related to l/c and whose surface charge is related to {β/j }, will be favored. If S* corresponds to an infinite-sized cylinder or bilayer, then a wide distribution of polydisperse micelles, described by eq 3, will form. Because the surfactant systems that we have considered in paper 2 of this series do not form discoidal micelles, we will only discuss the treatment of finite cylindrical micelles or spherocylinders. As shown in earlier treatments,5,22,41,42 finite cylindrical micelles can be modeled as consisting of a cylindrical body terminating in spherical sections on either end, and the average aggregation numbers can be evaluated from the moments of the micelle size distribution (this aspect is discussed in more detail in paper 2 of this series). The two regions of a spherocylindrical micelle, the two spherical sections and the cylindrical region, can have different values of {β/j }. This is similar to the case of binary surfactant mixtures, where the surfactant composition in the two regions in spherocylindrical micelles can be quite different. The optimal degree of binding, β/j , of counterion cj in a finite spherocylinder can be expressed as a linear interpolation between the optimal degrees of binding in the cylindrical body and in the spherical zones, β/j (cyl) and β/j (sph), respectively, as follows:

(57) Ohshima, H.; Healy, T. W.; White, L. R. J. Colloid Interface Sci. 1982, 90, 17-26.

β/j (n) ) β/j (cyl) + (nsph/n)(β/j (sph) - β/j (cyl)) (23)

In the diffuse region, the ions arrange under the balance of electrostatic forces and thermal diffusion, and their distribution is assumed to be described by the Boltzmann distribution.28 As a result, ψs is obtained by applying the Poisson-Boltzmann (PB) model for the electrostatic effects in this region. The PB equation, along with the appropriate boundary conditions, can be solved analytically only for a planar geometry. In the case of spherical and cylindrical geometries, several attempts have been made25,26,57 to simplify the PB equation to a form that can be solved numerically in a manner that is less computationally intensive. An analysis of the several approximations available for the PB equation was made by Shiloach and Blankschtein21 in the context of modeling the electrostatic effects associated with nonionic-ionic and cationicanionic mixed micelles, where the electrostatic surface potential varies with the micelle composition. The analytical approximation to the PB equation for symmetric 1:1 electrolytes derived by Ohshima, Healy, and White (OHW),57 which served to bridge the Debye-Hu¨ckel approximation suitable for the low-potential limit and the planar solution expansion suitable for the high-potential limit, was found to be most appropriate to describe the electrostatic effects in these systems. In our model of ionic surfactant micelles, the micelle charge can vary depending on the degree of counterion binding to the micelles, and therefore, we utilize the OHW approximation to obtain ψs for use in eq 17. For a solution containing only monovalent ions, the OHW approximation for a micelle of shape S yields the following relation:

s ) 2 sinh(y0/2) +

2(S - 1) tanh(y0/4) x0

(21)


Langmuir, Vol. 19, No. 23, 2003 9941

5. Model Validation In this section, we validate our molecular-thermodynamic theory of micellization by implementing it in the case of the well-studied and well-understood ionic surfactant system SDS with added NaCl. Our aim here is primarily to assess the accuracy with which the electrostatic effects have been captured in our model of counterion binding. Accordingly, we only present our theoretical predictions of the degree of Na+ binding onto the SDS micelle surface, βNa, the electrostatic potentials at the micelle surface of charge and at the Stern surface, ψ0 and ψs, respectively, and the cmc and compare these with the available experimental data. We stress that, although not reported here for the sake of brevity, our quantitative predictions of the average micelle aggregation numbers in the SDS-NaCl system were found to compare favorably with the experimental results of Missel et al.5,58 In paper 2 of this series, we will present our theoretical predictions of various micellar solution properties in several interesting surfactant systems containing multivalent and organic counterions, which are less well understood at a fundamental level. In section 5.1, we apply the molecular-thermodynamic theory of micellization to the SDS-NaCl system. In sections 5.2, 5.3, and 5.4, we present our theoretical predictions of the degree of Na+ binding, the electrostatic potentials ψ0 and ψs, and the cmc’s, respectively, and compare them with the available experimental results. 5.1. Application of the Molecular-Thermodynamic Theory of Micellization to the SDS-NaCl System. In the SDS-NaCl system, the surfactant and the electrolyte share the common counterion, Na+, which will tend to adsorb onto the negatively charged SDS micelle surface (therefore, in this system, c1 ) Na+). The molecular parameters of the SO4- head were estimated from knowledge of the bond angles and bond lengths and are given by ah,s ) 25 Å2 and dch ) 3.7 Å. The bound Na+ counterion lies on the micelle surface of charge at r ) Rch, and its cross-sectional area for steric effects is given by ah,Na ) πrh,Na2. The radius of the hydrated Na+ counterion, rh,Na, is 2.18 Å, based on its hydration number reported from ionic activity coefficient measurements.59 The Stern layer thickness, dst, was determined from the geometry of the sulfate head and the radius of the hydrated Na+ counterion and has a value of 3.58 Å. Finally, the dielectric constant of the solvent in the Stern layer was taken to be about half of its value in the bulk aqueous region, as suggested by Bockris and Reddy,28 and is given by ηs ) 40. The various free-energy contributions to g˜ mic in eq 22 were computed utilizing the expressions given in sections 3.2 and 3.3. To calculate the term in eq 22 accounting for the translational entropy lost by the bound counterions, the bulk counterion concentration, XNa, needs to be evaluated by applying the condition of electroneutrality in the bulk solution. Specifically,

XNa ) XCl + XDS ≈ XNaCl + cmc

(24)

To arrive at eq 24, the following assumptions were made. Since the Cl- ions are repelled from the negatively charged SDS micelle surface, their concentration in the bulk aqueous solution nearly equals the concentration of the added NaCl electrolyte, XNaCl. In addition, XDS can be (58) Missel, P. J.; Mazer, N. A.; Benedek, G. B.; Carey, M. C. J. Phys. Chem. 1983, 87, 1264-1277. (59) Marcus, Y. Ion Solvation; Wiley: Chichester, U.K., 1985; p 79.

Figure 2. Predicted optimal degree of binding of Na+ counterions onto SDS micelles, β/Na(S), at 25 °C as a function of NaCl concentration, CNaCl, for S ) spheres (solid line) and S ) infinite cylinders (dashed line).

approximated by the cmc of SDS, since the DS- monomer concentration remains approximately constant above the cmc. Once the translational entropy loss of the bound counterions is evaluated using eq 24 for XNa, the free energy of micellization, g˜ mic, is calculated for each of the three regular shapes and then minimized with respect to βNa and lc for each micelle shape, as discussed in section 4. For the inorganic Na+ counterion, the main driving force for counterion binding is the reduction in the electrostatic free energy, gelec, due to the decrease in the micelle surface charge density. In addition, the entropy of mixing, gent, is another factor favoring counterion binding. The unfavorable loss of translational entropy incurred due to the loss of mobility of the bound counterions, -βNa ln XNa, as well as the increased steric repulsions associated with the bound counterions at the micelle surface, reflected in gst, are the main factors opposing counterion binding. The optimal degree of Na+ binding, β/Na, is controlled by all these competing effects and will therefore vary with the counterion concentration, XNa (or equivalently, with the NaCl concentration), and with the micelle shape, S, as discussed next in section 5.2. 5.2. Comparison of the Predicted Degree of Na+ Binding in SDS Micelles with the Available Experimental Results. In Figure 2, we plot our theoretical predictions of β/Na(S) for the SDS + NaCl system, as a function of the NaCl concentration, CNaCl (where CNaCl has been defined in molar units), for spherical micelles (solid line) and for infinite cylindrical micelles (dashed line). The degree of Na+ binding for a finite-sized cylindrical micelle depends on the aggregation number of the micelle and, accordingly, will lie between the values for the sphere and the infinite cylinder, as shown in eq 23. In other words, as the SDS micelle shape changes progressively from spheres to cylinders upon addition of SDS or NaCl, the degree of Na+ binding shifts from that given by the solid line to that given by the dashed line in Figure 2. Figure 2 reveals that the degree of Na+ binding increases as the micelle curvature decreases, that is, β/Na(cyl) > β/Na(sph). This is because, as the micelle curvature decreases, the electric field generated by the micelle increases, requiring that a greater fraction of the Na+ counterions remain bound to the SDS micelle surface in order to minimize the electrostatic free energy. In other words, while the translational entropy loss is not shape-dependent, gelec depends on shape and increases as the micelle curvature

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decreases, thereby promoting higher values of β/Na in cylinders as compared to spheres. In addition, Figure 2 reveals that β/Na(S) increases with an increase in the NaCl concentration and that this increase is more pronounced at low NaCl concentrations. For example, β/Na(sph) increases from about 50% in the absence of any added NaCl to about 60% in a solution containing 1 M NaCl. A similar qualitative trend was predicted by Ruckenstein and Beunen,41 although their exact values for β/Na(S) were much higher (75-90% for spheres and over 90% for cylinders), and also by Konop and Colby60 using their model of counterion condensation onto micelles. The observed qualitative trend in the dependence of β/Na(S) on CNaCl can be explained by the fact that the translational entropy lost in binding the Na+ counterions decreases logarithmically as the NaCl concentration increases, thus promoting higher values of β/Na(S). On the other hand, a higher NaCl concentration brings about increased screening of the electrostatic repulsions by the diffuse counterion cloud (that is, an increase in κ), thereby reducing the incentive of the Na+ counterions to bind onto the SDS micelle surface. It is this opposing mechanism that causes the salt dependence of β/Na(S) to diminish at high NaCl concentrations. Our predicted trends agree well with the experimental findings of Romsted and co-workers,61,62 who developed a technique to estimate the counterion concentrations in the interfacial region of surfactant aggregates and applied it to the related surfactant system: cetyltrialkylammonium halide, CTAX, with an added electrolyte containing the common halide ion, X-. Their experimental results indicate that the interfacial counterion concentration increases steadily as the bulk electrolyte concentration increases. Interestingly, this increase is particularly rapid at low bulk electrolyte concentrations (

Effect of Counterion Binding on Micellar Solution Behavior: 1

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