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Langmuir 1997, 13, 5258-5275
Statistical-Thermodynamic Framework to Model Nonionic Micellar Solutions Nancy Zoeller, Leo Lue,† and Daniel Blankschtein* Department of Chemical Engineering and Center for Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 Received March 21, 1997X The McMillan-Mayer theory of multicomponent solutions is utilized to formulate a statisticalthermodynamic description of surfactant solution behavior from which quantitative predictions of micelle formation, micellar size distribution, and micellar solution phase separation can be made. Specifically, a model is constructed for the Gibbs free energy of the micellar solution, which is divided into ideal and excess contributions. The advantage of this approach is that it enables a systematic analysis of various models of intermicellar interactions. In this paper, we focus on micelles of nonionic surfactants which exhibit both repulsive and attractive interactions. The repulsive interactions are described using excludedvolume considerations, while the attractive ones are modeled using a mean-field description. Utilizing this statistical-thermodynamic framework, expressions for the chemical potentials of each of the solution components are obtained and used, along with the principle of multiple chemical equilibrium, to calculate the micellar size distribution and its moments. An analysis of the effect of excluded-volume interactions on the monomer and micelle concentrations and on the weight-average aggregation number of micelles which exhibit one-dimensional (cylindrical) growth indicates that these steric interactions promote micelle formation and growth. Interestingly, in the limit of extensive cylindrical micellar growth, we recover the well-known expressions for the micellar size distribution and its moments corresponding to the popular phenomenological “ladder model”, with modified “ladder model” parameters which are explicit functions of the excluded-volume parameters. In addition, quantitative predictions of the critical micellar concentration, the polydispersity of the micellar size distribution, and phase separation characteristics are presented and found to compare favorably with available experimental data for aqueous micellar solutions of alkyl poly(ethylene oxide) nonionic surfactants.
I. Introduction When surfactants are dissolved in water at concentrations which exceed the critical micellar concentration (cmc), these amphiphilic molecules self-assemble into microstructures known as micelles, with their polar groups in contact with water and their nonpolar groups shielded from water in the micellar interior. Micelles can form in a variety of shapes and display narrow or broad size distributions depending on solution conditions.1-3 Indeed, by tuning overall surfactant concentration, temperature, or ionic strength, it is possible to induce dramatic changes in micellar shape, size, and size distribution.2,4,5 In some cases, it is also possible to induce macroscopic phase separation of a micellar solution into a micelle-rich phase coexisting with a micelle-poor phase by varying solution conditions such as temperature.5 In view of the rich behavior exhibited by micellar solutions, it would be valuable to develop a theoretical description of their behavior which explicitly incorporates the unique chemical structure of the surfactant molecules. In the past, the modeling of micellar solution behavior has proceeded primarily along two seemingly unrelated * To whom correspondence should be addressed. † Present address: Department of Chemical Engineering, University of California at Berkeley, Berkeley, CA, 94720-1462. X Abstract published in Advance ACS Abstracts, September 15, 1997. (1) For an introduction to the field of micellar solutions see: Mittal, K. L., Ed. Micellization, Solubilization, and Microemulsions; Plenum: New York, 1977; Vols. 1 and 2. Gelbart, W. M., Ben-Shaul, A., Roux, D., Eds. Micelles, Membranes, Microemulsions, and Monolayers; Springer: Berlin, 1994. (2) Tanford, C. The Hydrophobic Effect, 2nd ed.; John Wiley and Sons: New York, 1980 (3) Israelachvili, J. Intermolecular and Surface Forces, 2nd ed.; Academic Press: New York, 1992. (4) Carale, T. R.; Blankschtein, D. J. Phys. Chem. 1992, 96, 459. (5) Blankschtein, D.; Thurston, G. M.; Benedek, G. B. J. Chem. Phys. 1986, 85, 7268 and references cited therein.
S0743-7463(97)00308-9 CCC: $14.00
fronts: (i) describing the micellization process itself and (ii) describing the overall micellar solution phase behavior. Along the first front, extensive work has been done on modeling micelle formation on the basis of physicochemical arguments.1,2,6-13 These models calculate the free energy of micellization, which is the free-energy change (per surfactant molecule) associated with transferring a surfactant molecule from bulk solution into a micelle. In this description, the micelle is assumed to be at infinite dilution, and intermicellar interactions can thus be neglected. The cmc and information regarding micellar shape and size can then be computed directly from the free energy of micellization.2,7-12 Many of these models also make use of various phenomenological parameters based on experimental data to obtain a high degree of predictive accuracy. For example, an empirical constant has been estimated on the basis of cmc data to quantify the repulsive contributions to the free energy of micellization,2,11 and geometric packing constraints have been utilized to determine optimal micellar shapes.13 Later models incorporate the unique molecular structure of the surfactant molecules into the formulation by explicitly describing the repulsive free-energy contributions associ(6) For a comprehensive account of theoretical developments in the field of micellization see: Mittal, K. L., Lindman, B., Eds. Surfactants in Solution; Plenum: New York, 1984; Vols. 1, 2, and 3. (7) Puvvada, S.; Blankschtein, D. J. Chem. Phys. 1990, 92, 3710 and references cited therein. (8) Puvvada, S.; Blankschtein, D. MRS Symp. Proc. 1990, 177, 129. Puvvada, S.; Blankschtein, D. In Proceedings of the 8th International Symposium on Surfactants in Solution; Mittal, K. L., Shah, D. O., Eds.; Plenum: New York, 1991; Vol. 11, p 95 and references cited therein. (9) Zoeller, N.; Blankschtein, D. Ind. Eng. Chem. Res. 1995, 34, 4150. (10) Zoeller, N.; Shiloach, A.; Blankschtein, D. CHEMTECH 1996, 26 (3), 24. (11) Nagarajan, R.; Ruckenstein, E. J. Colloid. Interface Sci. 1979, 71, 580. (12) Nagarajan, R.; Ruckenstein, E. Langmuir 1991, 7, 2934 and references cited therein. (13) Israelachvili, J. N.; Mitchell, D. J.; Ninham, B. W. J. Chem. Soc., Faraday Trans. 2 1976, 72, 1525.
© 1997 American Chemical Society
Nonionic Micellar Solutions
ated with micellization, through separate steric, interfacial, and electrostatic contributions.7,11,12,14 Along the second front, theoretical work has been driven by the need to both fundamentally rationalize and quantitatively predict micellar growth and micellar solution phase behavior.5,15-20 At low surfactant concentrations, micelles are somewhat globular and monodisperse. As the surfactant concentration is increased, two additional micellar solution characteristics become important: (i) micelles may change shape and size, becoming more polydisperse, and (ii) intermicellar interactions may become stronger. Early attempts to model micellar solution phase behavior assumed ideal solution behavior, where interactions between micelles are negligible, and concentrated mainly on the entropy of mixing polydisperse micelles in solution.11,21 However, intermicellar interactions must be treated in order to model certain micellar solution characteristics at higher surfactant concentrations, such as the stabilization of particular micellar structures,22-24 micellar diffusion coefficients,25 the micellar size distribution,26 and phase separation.5,15 Experimentally, the importance of light and neutron scattering in elucidating micellar structures has also led to a demand for detailed descriptions of intermicellar interactions. Specifically, an accurate model of intermicellar interactions is required to interpret scattering data and obtain a clear description of the micellar size distribution and the diffusion coefficients.25,27-32 Clearly, there is a need for a theoretical approach capable of unifying the previously disconnected treatments of micellization and overall micellar solution phase behavior. Several theoretical approaches have examined the coupling of intramicellar and intermicellar interactions as they affect micelle shape, average size, and size distribution.18,19,22,26,33,34 In addition, to describe dilute, isotropic micellar solution phase separation, a phenom(14) Eriksson, J. C.; Ljunggren, S. Prog. Colloid Polym. Sci. 1988, 76, 188. (15) Blankschtein, D.; Thurston, G. M.; Benedek, G. B. Phys. Rev. Lett. 1985, 54, 955. (16) Reatto, L.; Tau, M. Chem. Phys. Lett. 1984, 108, 292. Tau, M.; Reatto, L. J. Chem. Phys. 1985, 83, 1921. (17) Leng, C. A. J. Chem. Soc., Faraday Trans. 2 1985, 84, 145. Evans, H.; Tildesley, D. J.; Leng, C. A. J. Chem. Soc., Faraday Trans. 2 1987, 83, 1525. (18) Odijk, T. J. Phys. 1987, 48, 125. (19) Gelbart, W. M.; McMullen, W. E.; Ben-Shaul, A. J. Phys. 1985, 46, 1137. (20) See, for example: (a) Kjellander, R. J. Chem. Soc., Faraday Trans. 2 1982, 78, 2025. (b) Goldstein, R. E. J. Chem. Phys. 1986, 84, 3367. (21) Nagarajan, R. Colloids Surf., A: Physicochem. Eng. Aspects 1993, 71, 39. (22) Israelachvili, J. N.; Sornette, D. J. Phys. 1985, 46, 2125. (23) Ljunggren, S.; Eriksson, J. C. J. Chem. Soc., Faraday Trans. 2 1984, 80, 489. (24) Bergstro¨m, M. J. Colloid Interface Sci. 1996, 181, 208 and references cited therein. (25) Corti, M.; Degiorgio, V. J. Phys. Chem. 1981, 85, 711. (26) Gelbart, W. M.; Ben-Shaul, A.; McMullen, W. E.; Masters, A. J. Phys. Chem. 1984, 88, 861. (27) Dorshow, R.; Briggs, J.; Bunton, C. A.; Nicoli, D. F. J. Phys. Chem. 1982, 86, 2388. Dorshow, R.; Bunton, C. A.; Nicoli, D. F. J. Phys. Chem. 1983, 87, 1409. (28) Appell, J.; Porte, G. J. Colloid Interface Sci. 1981, 81, 85. Porte, G.; Appell, J. J. Phys. Chem. 1981, 85, 2511. (29) Kato, T.; Anzai, S.; Takano, S.; Seimiya, T. J. Chem. Soc., Faraday Trans. 1 1989, 85, 2499. (30) Sheu, E. Y.; Wu, C.-F.; Chen, S.-H. J. Phys. Chem. 1986, 90, 4179. Lin, T.-L.; Chen, S.-H.; Gabriel, N. E.; Roberts, M. F. J. Phys. Chem. 1990, 94, 855. (31) Zulauf, M.; Weckstrm ¨ , K.; Hayter, J. B.; Degiorgio, V.; Corti, M. J. Phys. Chem. 1985, 89, 3411. (32) Huang, J. S.; Safran, S. A.; Kim, M. W.; Grest, G. S.; Kotlarchyk, M.; Quirke, N. Phys. Rev. Lett. 1984, 53, 592. (33) Taylor, M. P.; Berger, A. E.; Herzfeld, J. Mol. Cryst. Liq. Cryst. 1988, 157, 489.
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enological model for the total solution Gibbs free energy, which incorporates intermicellar interactions at a meanfield level of description, was recently formulated.5,15 In this phenomenological model, the Gibbs free energy of the micellar solution is decomposed into three contributions: formation, mixing, and interactions. Although this model has been highly successful in describing many of the experimentally observed micellar solution phenomena, splitting the solution nonidealities into mixing and interaction free-energy contributions without a rigorous underlying statistical-mechanical basis can sometimes lead to ambiguities. Specifically, when the model is unable to describe some aspect of the experimentally observed micellar solution behavior, it is difficult to unambiguously determine if the source of the discrepancy lies in the mixing or the interaction contributions to the micellar solution Gibbs free energy. In view of the above, we have developed a theoretical framework for the calculation of the micellar solution Gibbs free energy which is based on rigorous statisticalmechanical principles in the context of the McMillanMayer theory of multicomponent solutions. An advantage of this theoretical framework is that the approximations made in constructing the solution Gibbs free energy model are clearly delineated, and therefore, in principle, it is possible to systematically improve upon the theory, if needed. In addition, the theoretical framework allows for the implementation of a variety of excess free energy models. Determining the optimal model for the excess free energy requires a tradeoff between accuracy and computational complexity. In other words, a complex model may have a high degree of accuracy for all types of surfactants but may be computationally difficult to handle. A more simple phenomenological model may be computationally fast but would have to be specialized for each type of surfactant. In this paper, we present an excess free energy model which is somewhat computationally complex and yields an accurate description of nonionic micellar solution behavior over a wide range of surfactant concentrations and temperatures. Indeed, a similar theoretical framework has already been succesfully applied to model solute partitioning in phase-separated surfactant solutions.35 In future work, we will generalize the theoretical framework to incorporate ionic surfactants. The remainder of the paper is organized as follows. In section II, we present a general description of the McMillan-Mayer theory and its application to multicomponent solutions. Specifically, we utilize the McMillan-Mayer theory to develop a general statisticalthermodynamic framework for the calculation of the solution Gibbs free energy. Note that this section, as well as Appendix A, is primarily aimed at those readers who may be less familiar with the thermodynamic principles underlying the McMillan-Mayer theory. In section III, we implement the statistical-thermodynamic framework developed in section II in the case of aqueous nonionic micellar solutions which exhibit one-dimensional (cylindrical) micellar growth in the presence of repulsive, excluded-volume interactions and attractive, mean-field type interactions. Interestingly, in the limit of extensive cylindrical micellar growth, we recover the well-known expressions for the micellar size distribution and its moments corresponding to the popular phenomenological “ladder model”, with modified “ladder model” parameters which are explicit functions of the excluded-volume parameters. In section IV, a qualitative analysis of the effect of excluded-volume interactions on the monomer (34) Safran, S. A.; Pincus, P. A.; Cates, M. E.; MacKintosh, F. C. J. Phys. (Paris) 1990, 51, 503. (35) Lue, L.; Blankschtein, D. Ind. Eng. Chem. Res. 1996, 35, 3032.
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and micelle concentrations and on the weight-average aggregation number of micelles which exhibit cylindrical growth indicates that these steric interactions promote micelle formation and growth. In addition, quantitative predictions of cmc’s, micellar size distribution polydispersity, critical surfactant concentrations signaling the onset of phase separation, and the osmotic compressibility of aqueous micellar solutions of alkyl poly(ethylene oxide) nonionic surfactants are presented and compared with experimental data. Finally, in section V, we summarize the main results of the paper. II. McMillan-Mayer Theory of Multicomponent Solutions For many multicomponent fluids of practical interest, such as dilute colloidal dispersions and micellar solutions, one of the components, referred to as the solvent, is present at a much higher concentration than those of the other components, referred to as the solutes. There are several levels of approximation with which to approach the statistical-mechanical problem of constructing a freeenergy model for such systems. A widely-used approximation involves treating the solvent as a background, or continuum, through which the solute molecules interact with each other. Properties of the solvent molecules are averaged and are therefore not accounted for explicitly in the statistical-mechanical analysis. As a result, the interactions between the solute molecules now include not only the “bare” solute-solute interactions, which are present when the solutes are located in vacuum, but also additional interactions due to the presence of the solvent. The resulting effective interaction potential between the solute molecules, W({Nσ}), where {Nσ} ) {N1, N2, ..., Nn} is a shorthand notation for the various numbers of solute molecules, is known as the potential of mean force36 and is a required input to the theory. W({Nσ}) can be quite complex, since it must reflect the properties of the solvent, such as its structure and the manner in which it restructures in the presence of the solutes, in addition to its dependence on the thermodynamic state of the system (for example, on temperature and pressure). Consequently, approximations are typically required to model W({Nσ}). Once an expression for the potential of mean force is available, the next step is to construct the free energy of a system consisting of molecules interacting via this potential. There are many approximate free-energy models for various molecular systems,3,37,38 such as hardsphere and Lennard-Jones fluids, but these models are strictly only applicable to particles interacting in vacuum rather than embedded in a solvent. In view of this, it would be extremely useful to be able to apply the models for particles interacting in vacuum to particles interacting in a solvent. The McMillan-Mayer theory establishes such a connection.36,39 In the remainder of this section, the basic results of the McMillan-Mayer theory are summarized, with details of the derivations left to Appendix A. In what follows, Greek indices (σ, R) refer only to solute species, the index w refers to the solvent (that is, water), and all other roman indices (i, j) refer to both solvent and solute species. In the McMillan-Mayer theory, the natural independent variables of the system are temperature, T, volume, (36) McMillan, W. G., Jr.; Mayer, J. E. J. Chem. Phys. 1945, 13, 276. (37) Mansoori, G. A.; Carnahan, N. F.; Starling, K. E.; Leland, T. W., Jr. J. Chem. Phys. 1971, 54, 1523. (38) Maitland, G. C.; Rigby, M.; Smith, E. B.; Wakeham, W. A. Intermolecular Forces: Their Origin and Determination; Clarendon Press: Oxford, 1981. (39) Friedman, H. L. J. Solution Chem. 1971, 1, 387.
Zoeller et al.
V†, {Nσ}, and the chemical potential of the pure solvent, µ°w. Note that V† ) V(T,p+Π,{Nσ},Nw) is the total volume of the solution at T, p + Π, {Nσ}, and Nw, where Π is the osmotic pressure of the solution and Nw is the number of solvent molecules. In other worlds, V† is the volume of the solution in osmotic equilibrium with pure solvent at T, p, and Nw. The reason for this selection of variables will become apparent later in the paper. The free energy associated with this set of independent variables is a Legendre transform40 of the Helmholtz free energy, A, whose natural variables are T, V†, {Nσ}, and Nw. If Nw is transformed to its conjugate variable, µ°w, then the Helmholtz free energy, A, is transformed into the McMillan-Mayer free energy, F; that is,
F ) A - Nwµ°w
(1)
In order to evaluate the McMillan-Mayer free energy, F, we will decompose it into IDEAL and EXCESS contributions, FID and FEX, where F ) FID + FEX. An IDEAL solution is defined as one in which the solute molecules do not interact with each other. The IDEAL contribution to the McMillan-Mayer free energy is given by35,41
FID(T,V†,{Nσ},µ°w) )
∑σ NσµQσ +
∑σ Nσ(ln(c†σ/cQσ ) - 1) - p(T,{0},µ°w)V†
kBT
(2)
where µQσ is the standard-state chemical potential of solute σ corresponding to a concentration cQσ , c†σ ) Nσ/V† is the concentration of solute σ, kB is the Boltzmann constant, T is the absolute temperature, and p(T, {0},µ°w) is the pressure of the pure solvent at T and µ°w, where {0} denotes zero solute concentrations. Note that the standard-state chemical potential of solute σ, µQσ , is chosen to be solute molecules dissolved in solvent molecules at concentration cQσ , usually chosen to be 1 mol/L, in a hypothetical standard state in which the solute molecules do not interact with each other. This standard state is frequently utilized to model electrolyte solutions.42 An EXCESS property is defined as the difference between the property of the actual system and that of an IDEAL system at the same T, V†, {Nσ}, and µ°w. In particular, the EXCESS McMillan-Mayer free energy, FEX, is given by
FEX(T,V†,{Nσ},µ°w) ) F(T,V†,{Nσ},µ°w) FID(T,V†,{Nσ},µ°w) (3) In the system of interest, referred to hereafter as the “solvent” system, the solute molecules interact with each other through a potential of mean force, W({Nσ}), which depends on the positions and orientations of all the solute molecules in the system. Now, let us consider a system in which the solute molecules are placed in vacuum but still interact with each other through the same potential, W({Nσ}). This system will be referred to hereafter as the “vacuum” system. In what follows, properties of the vacuum system will carry a tilde to distinguish them from (40) Tester, J. W.; Modell, M. Thermodynamics and Its Applications, 3rd ed.; Prentice-Hall: Upper Saddle River, NJ, 1996. (41) Note that eq 8 in ref 35 contains two typographical errors: (i) it is mising the -pV† term, which is the last term given in eq 2, and (ii) µw should be replaced by µ°w. (42) Atkins, P. W. Physical Chemistry, 3rd ed.; Freeman: New York, 1986.
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those of the solvent system. For example, the Helmholtz free energy of the vacuum system will be denoted by A ˜. The central result of the McMillan-Mayer theory is the following relation36
FEX(T,V†,{Nσ},µ°w) ) A ˜ res(T,V,{Nσ})
(4)
where the superscript res denotes the residual property of the vacuum system, that is, the difference between the property of the actual vacuum system and that of an ideal vacuum system at the same T, V, and {Nσ}. Note that the volume of the vacuum system is V(T,p,{Nσ}), evaluated at a different pressure than the volume of the solvent system, V†(T,p+Π,{Nσ},Nw). Note also that the “vacuum” system depends on one less variable than the solvent system because it contains one less component, namely, the solvent. Equation 4 establishes a relation between the residual Helmholtz free energy of a vacuum system and the EXCESS McMillan-Mayer free energy of a solvent system. Accordingly, a model for the residual Helmholtz free energy of solute molecules in vacuum can be utilized to calculate the EXCESS McMillan-Mayer free energy of solute molecules interacting through solvent. In theory, therefore, we have developed a complete model for the system of interest. Specifically, given the T, V†, {Nσ}, and µ°w of a system, eqs 2, 3, and 4 can be utilized to calculate F, from which other thermodynamic properties, such as the chemical potentials and the osmotic pressure, can then be obtained. The difficulty in implementing this approach in practice, however, is that µ°w is not a convenient, experimentally-accesible variable. Instead, in a typical experiment, the independent variables which are most easily manipulated are temperature, T, pressure, p, the numbers of solute molecules, {Nσ}, and the number of solvent molecules, Nw. The free energy associated with this set of independent variables is the Gibbs free energy, G, and therefore, in order to effectively model the thermodynamic behavior of the system, one needs a model for the Gibbs free energy, instead of one for the McMillanMayer free energy. Consequently, in order to make a connection with actual experimental measurements, a relation between the McMillan-Mayer free energy, F, and the Gibbs free energy, G, is required. As in the case of F, in order to derive an expression for G, we will separately compute its ideal and excess contributions. As before, an ideal solution is defined as one in which there are no interactions between the solute molecules. Note, however, that an “ideal” solution is different from the “IDEAL” solution defined earlier, in spite of the fact that in both systems the solutes do not interact with each other. This difference reflects the fact that the solvent molecules can still interact with themselves as well as with the solute molecules, and the effect of these interactions depends on the thermodynamic variables which are held constant. For the ideal solution, these variables are T, p, {Nσ}, and Nw, while for the IDEAL solution, the variables are T, V†, {Nσ}, and µ°w. The ideal Gibbs free energy, Gid, is given by35
Gid(T,p,{Nσ},Nw) ) Nwµid w + ) Nwµ°w +
∑σ Nσµidσ ∑σ Nσµ°σ + kBT∑Nσ(ln mσ - 1) σ
(5)
state chemical potential of component i (i ) w or σ), and mσ ) Nσ/Nw is the “molality” of solute σ. Note that eq 6 is obtained from eq 5 by expanding µid w to leading order in mσ. For the solvent, the standard state is chosen to be pure solvent at the system T and p. For a solute species, the standard state is chosen to be a solute molecule at infinite dilution in the solvent at the system T and p. Note that µQσ (at cQσ ) in eq 2 and µ°σ (at infinite dilution) in eq 6 are related through the following expression42
µ°σ - µQσ - kBT ln cQσ Vw
where Vw ) V(T,p,{0},Nw)/Nw is the volume per molecule in pure solvent (water). From Gid, all the other thermodynamic properties of the ideal system can be determined. For example, one finds that
µid σ )
id where µid w is the ideal chemical potential of water, µσ is the µ° is the standardideal chemical potential of solute σ, i
( ) ∂Gid ∂Nσ
T,p,{NR*σ},Nw
) µ°σ + kBT ln mσ
(8)
) µ°w - kBTm
(9)
and
µid w )
( ) ∂Gid ∂Nw
T,p,{Nσ}
where m ) ∑σ mσ is the total solute molality. In addition, an excess property is defined, denoted by a superscript ex, to differentiate it from the superscript EX utilized earlier, as the difference between the property of the actual system and the property of an ideal system at the same T, p, {Nσ}, and Nw. Specifically,
Gex(T,p,{Nσ},Nw) ) G(T,p,{Nσ},Nw) Gid(T,p,{Nσ},Nw) (10) The excess Gibbs free energy can be obtained from the EXCESS McMillan-Mayer free energy through the appropriate thermodynamic transformations39 (see Appendix A for a detailed derivation). This yields
Gex(T,p,{Nσ},Nw) ) FEX(T,V†,{Nσ},µ°w) †
∫pp+ΠV(T,p′,{Nσ},Nw) dp′ + ΠV† - NkBT ln VwVNw (11) The advantage of rewriting Gex in this form is that FEX can be related to the residual Helmholtz free energy of solute molecules interacting in vacuum with potentials equal to the potentials of mean force (see eq 4). Therefore, one can apply all the available models for dilute fluids directly to solute molecules interacting in a solvent. From the excess Gibbs free energy expression given in eq 11, one can obtain the following expression for the excess chemical potential of solute σ, µex σ (see Appendix A for a detailed derivation)
µex σ ) (6)
(7)
( ) ∂Gex ∂Nσ
) µEX σ -
T,p,{NR*σ},Nw †
∫pp+ΠVh σ(p′) dp′ - kBT ln VwVNw
(12)
where V h σ(p′) is the partial molar volume of solute σ at
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Zoeller et al.
pressure p′. Similarly, the excess chemical potential of the solvent, µex w , is given by (see Appendix A)
µex w )
( )
µn ) nµ1
∂G ∂Nw
T,p,{Nσ}
∫pp+ΠVh w(p′)dp′
(13)
where V h w(p′) is the partial molar volume of the solvent at pressure p′. The total chemical potentials of the solutes and the solvent include both the ideal contributions (from eqs 8 and 9) and the excess contributions (from eqs 12 and 13), as shown below for solute σ ex µσ ) µid σ + µσ ) µ° σ + kBT ln mσ + †
∫pp+ΠVh σ(p′) dp′ - kBT ln VwVNw
(14)
Equation 17 will be utilized later to derive an expression for the micellar size distribution. First, however, the chemical potentials of the various solution components are calculated, as described next. B. Calculation of Chemical Potentials. The chemical potential of an n-mer, µn, is obtained from eq 14 with σ ) n, where n ) 1 for the monomers and n > 1 for the micelles. The chemical potential of water is given in eq 15. h w are the partial molar volumes In eqs 14 and 15, V h n and V of an n-mer and water, respectively, which, in general, can be complicated functions of solute concentration, pressure, and temperature. Consequently, to evaluate the volume terms in eqs 14 and 15, a model for the volumetric behavior of the solution is required. In general, the total volume of the solution, V, is given by ∞
and for the solvent
µw )
µid w
+
(17)
ex
) kBTm -
µEX σ -
chemical equilibrium, that is,3
µex w
) µ°w -
∫p
p+Π
V ) NwV hw + V h w(p′) dp′
(15)
In summary, we have shown that the McMillan-Mayer theory allows one to use the thermodynamic properties of a system in vacuum to predict the thermodynamic properties of a system in a continuum solvent. In the next section, we discuss the inputs needed to utilize the results derived in this section to construct a Gibbs freeenergy model for an aqueous nonionic micellar solution. III. Application to Aqueous Nonionic Micellar Solutions In the previous section, we described a geneal statisticalthermodynamic framework for constructing a Gibbs freeenergy model of a solution of solute species dissolved in a solvent, given the potentials of mean force between the solute species. In this section, this theoretical framework is implemented in the case of aqueous nonionic micellar solutions. In particular, the chemical potentials of the solute species (micelles and monomers) are calculated and utilized, along with the principle of multiple chemical equilibrium, to obtain an expression for the micellar size distribution and its moments, as well as to predict micellar solution phase separation characteristics. A. Definition of System. The surfactant-water micellar system is modeled as a multicomponent solution containing (i) Nw water molecules, (ii) N1 surfactant monomers, and (iii) a distribution of {Nn} surfactant micelles of aggregation number n also referred to as n-mers, where {Nn} is a short-hand notation for the various numbers of n-mers, {N1, N2, ..., Nn}. Note that the subscript σ utilized in section II is now replaced by the subscript n. The main objective of this section is to calculate the chemical potentials of the surfactant monomers, µ1, the micelles of aggregation number n, µn, and the water, µw. Knowledge of these chemical potentials is equivalent to knowing the total micellar solution Gibbs free energy, G, since
∑ NnVh n
(18)
n)1
Since the micellar solutions considered in this paper are assumed to be quite dilute, we will neglect solute concentration effects on the partial molar volumes, V hw and V h n. In addition, we will neglect pressure effects on h n, since these should not be significant at the V h w and V atmospheric conditions typically encountered experimentally. With these assumptions in mind, V h w ) Vw, the volume occupied by a water molecule, which we take to be a constant, Vw ≈ Ωw ) 30 Å3. Similarly, the partial molar volume of a surfactant monomer, V h 1, is equal to Ωs, the volume occupied by a surfactant monomer, independent of surfactant concentration and pressure. Regarding a micelle of aggregation number n, we assume that (i) the partial molar volume, V h n, is equal to the sum of the partial molar volumes of its n constituent surfactant molecules and (ii) the partial molar volume of a surfactant molecule in a micelle is equal to that of a free surfactant monomer in solution. In other words, (i) and (ii) imply h 1 ) nΩs. that V h n ) nV Combining these volumetric approximations, the total volume of the micellar solution is given by
V(T,p,{Nn},Nw) ) V†(T,p+Π,{Nn},Nw) ) NwΩw + NsΩs (19) Note that the neglect of pressure effects (incompressibility assumption) in eq 19 also imply that the solute concentrations utilized in the McMillan-Mayer theory, c†n ) Nn/V†, are equal to the solute concentrations in the actual system, cn ) Nn/V. In the context of the model for the volumetric behavior of the micellar solution given in eq 19, the chemical potential expressions in eqs 14 and 15 can be simplified (the pressure integrals can be easily carried out). Specifically,
µn ) µ°n + kBT ln cnΩw + µEX n - nΠΩs
(20)
µw ) µ°w - ΠΩw
(21)
∞
G ) Nwµw +
∑ Nnµn
(16)
n)1
Recall that micelles are self-assembling aggregates which continually exchange surfactant molecules with each other and with the monomers in solution. These material exchanges must satisfy the principle of multiple
and
Note that the natural log terms in eq 14 have been combined in eq 20 to transform from molalities, mn ) Nn/Nw, to concentrations, cn ) Nn/V.
Nonionic Micellar Solutions
Langmuir, Vol. 13, No. 20, 1997 5263
Next, the micelle (n > 1) and monomer (n ) 1) chemical potential expressions in eq 20 can be utilized in the chemical equilibrium condition, µn ) nµ1, to obtain the following expression for the micellar size distribution
cyl from n ) n0 to infinity. In this case, gmic (n) is estimated by linearly interpolating between the gmic values corresponding to a sphere and an infinite cylinder, namely,7
cyl gmic (n) )
EX Ωwcn ) (Ωwc1)n exp{-β(µ°n - nµ°1) - β(µEX n - nµ1 )}
(22) Note that the osmotic pressure contribution in eq 20 cancels out in eq 22. Equation 22 defines the entire micellar size distribution in terms of the concentrations of n-mers for any n > 1. The first term, (Ωwc1)n, corresponds to the entropic cost associated with localizing n monomers at one position to form the n-mer, and the exponential term reflects the free-energy advantage associated with forming the n-mer. This last term includes both an ideal (infinite dilution) contribution and an EXCESS contribution. If there were no interactions among the various n-mers, then the EXCESS contribution would be zero, and we would recover the “traditional” expression for the micellar size distribution equation.3 Up to this point, the only approximation made involves the use of eq 19 for the volumetric behavior of the micellar solution. Consequently, eq 22 can be applied to any micellar solution which satisfies eq 19. To complete the calculation, models are required for the standard-state chemical potential difference, µ°n - nµ°1, and for the EXCESS chemical potential difference, µEX - nµEX n 1 , appearing in eq 22. The standard-state chemical potential calculation is discussed in the next section, section IIIC, and the EXCESS chemical potential calculation is discussed in section IIID. C. Standard-State Chemical Potential. Each nmer has a distinct standard-state chemical potential, µ°n, which is equal to the chemical potential of a micelle of aggregation number n at infinite dilution in water. In eq 22, the relevant quantity is the difference between the standard-state chemical potential of an n-mer and that of n monomers. This difference, per surfactant molecule, is referred to as the free energy of micellization, gmic.7 Specifically,
gmic(n) ) (µ°n - nµ°1)/n
(23)
gmic represents the free-energy change when a surfactant molecule is transferred from the aqueous solvent to a micelle of aggregation number n present at infinite dilution in the solvent. The magnitude of gmic can be evaluated using a thought process which describes the formation of a micelle from individual surfactant monomers as a series of reversible steps, each associated with a physicochemical contribution to the micellization process.7,8 The calculation7,8 of gmic depends on the micellar core radius, lc, and on the micelle shape, s, which can be a sphere (s ) 3) or an infinite cylinder (s ) 2). (Theoretically, the shape can also be an infinite bilayer (s ) 1), but this case will not be addressed here, since it is not relevant to the experimental systems examined in this paper.) gmic is then minimized with respect to lc and s to determine the optimal micellar core radius, l*c, and shape, s*, of the micelle. Note that the optimal l* c value for a sphere may be different from that corresponding to an infinite cylinder. If the optimal shape, s*, corresponds to a sphere, the spherical micelles are assumed to be monodisperse with an aggregation number, n0, derived from geometric 3 considerations. Specifically, n0 ) 4π(l* c) /3vtail, where vtail is the volume of the surfactant tail. If the optimal shape, s*, corresponds to an infinite cylinder, the micelles can be quite polydisperse, with aggregation numbers ranging
n0 (n - n0) gsph + gcyl n n
(24)
where gsph is the optimal micellization free energy, gmic (l*c,s* ) 3), of a sphere and gcyl is the optimal micellization free energy, gmic(l* c,s* ) 2), of an infinite cylinder. D. EXCESS Chemical Potential. The EXCESS chemical potential of an n-mer, µEX n , can be calculated from the EXCESS McMillan-Mayer free energy, FEX, as follows
βµEX n )
( ) ∂βFEX ∂Nn
(25)
T,V†,{Nm*n},µ°w
where β ) 1/kBT (recall that, according to eq 19, V† ) V). FEX can be obtained by integrating over the osmotic pressure of the micellar solution, Π, with respect to the total solute concentration, c; that is,35,39,40
βFEX ) N
dc′ - 1) ∫0c(βΠ c′ c′
(26)
where N ) ∑n)1 Nn and c ) ∑n)1 cn are the total number and total concentration of aggregates (micelles and monomers), respectively. In order to model Π, we utilize the virial equation of state, since it provides a reasonable, mathematically tractable representation of the nonidealities arising from solute-solute interactions in a dilute micellar solution. Specifically,35 ∞
βΠ ) c +
∞
∞
∞
∞
(2) (3) Bnm cncm + ∑ ∑ ∑ Bnmp cncmcp + ... ∑ ∑ n)1m)1 n)1m)1p)1
(27)
(2) where Bnm is the second-viral coefficient between ag(3) gregates of aggregation numbers n and m and Bnmp is the third-virial coefficient between aggregates of aggregation numbers n, m, and p. Note that the first term in eq 27 represents the IDEAL contribution to the osmotic pressure (βΠID ) c), while the additional terms represent the EXCESS contributions, βΠEX. As discussed above, the micellar solution is dilute, and therefore, the contribution of the third- and higher-order virial coefficients in eq 27 can be neglected. For convenience, hereafter, the superscript 2 is dropped from the second-virial coefficient; that (2) t Bnm. is, Bnm Using eq 27 for βΠ in eq 26 yields
∞
βFEX ) V
∞
∑ ∑ Bnmcncm
(28)
n)1m)1
The EXCESS chemical potential, µEX n , can then be obtained using eq 28 in eq 25 (recall that cn ) Nn/V). This yields ∞
βµEX n ) 2
∑ Bnmcm
(29)
m)1
In a typical micellar solution, the interaction potentials reflect three types of interactions: (i) hard-core, steric repulsions, (ii) electrostatic repulsions, and (iii) attractions. This paper focuses on nonionic surfactants, for which electrostatic interactions are negligible. Consequently,
5264 Langmuir, Vol. 13, No. 20, 1997
Zoeller et al.
the EXCESS chemical potential is divided into hard-core (HC) and attractive (att) contributions, each characterized by its own second-virial coefficient. Specifically,
βµEX n
)
βµEX,HC n
+
βµEX,att n
given by35,43,44 (see Appendix B for details)
BHC n1 )
∞
and
∑ m)1
HC ) Bnm
∞
HC Bnm cm + 2
att Bnm cm ∑ m)1
(31)
HC att and Bnm are the hard-core and attractive where Bnm contributions to the second-virial coefficient, respectively. att The attractive contribution, Bnm , in eq 31 is estimated utilizing a mean-field approximation. Specifically,
att Bnm )-
nmC(T)Ωs 2kBT
(32)
where the attractive interaction between an n-mer and an m-mer is assumed to be proportional to the number of pairwise interactions (nm/2) associated with the n-mer and the m-mer. Use of this simple form for the attractive intermicellar interactions implies averaging over all possible micellar configurations, a reasonable assumption for isotropic micellar solutions which lack both positional and orientational long-range order. C(T) in eq 32 is a phenomenological parameter reflecting the magnitude of the attraction between two surfactant molecules, in units of kBT. Note that C(T) in eq 32 is multiplied by the volume of a surfactant molecule, Ωs, in order to express the virial att in the coefficient in volumetric units. Using eq 32 for Bnm second summation of eq 31 yields the attractive contribution to the EXCESS chemical potential, that is,
) -n βµEX,att n
C(T)Ωs kBT
∞
∑ mcm
) 8Ωsc1 + Ωsγ2(cs - c1) + Ωsγ2n2/3 βµEX,HC 1 0 (c - c1) (37) ∞ ∑m)n 0
cm is the total surfactant concenwhere cs ) c1 + ∞ c is the total aggregate tration and c ) c1 + ∑m)n 0 m concentration. Note that, for the case of monodisperse spherical micelles, n0(c - c1) ) cs - c1, and the EXCESS chemical potential of a monomer becomes βµEX,HC (sph) ) 1 8Ωsc1 + Ωsγ3(cs - c1). Utilizing eqs 35 and 36 in eq 34, the following expression is obtained for the hard-core EXCESS chemical potential of a spherocylindrical micelle of aggregation number n g n0,
HC Bnm cm
(34)
m)n0 HC Bn1
∞
πd2
∞
where the second-virial coefficients,
where d ) 2R and γ ) (1 + 1/n1/3 0 ) is a geometric factor associated with the smallest micelle of aggregation number HC n0. Note that, for the case of spherical micelles, Bn1 HC (sph) can be obtained from eq 35 with n ) n0 and Bnm (sph) can be obtained from eq 36 with Ln ) Lm ) 0. HC Utilizing eq 35, along with the fact that B11 ) 4Ωs, in eq 34 with n ) 1, the following expression is obtained for the hard-core EXCESS chemical potential of a monomer (see Appendix B for details)
(33)
m)1
∑
2πd3 πd2 π + (Ln + Lm) + dLnLm, n, m g n0 3 2 4 (36)
) Ωsγ2(n + n2/3 βµEX,HC n 0 )c1 +
∞ mcm ) cs, the total surfactant concentration. where ∑m)1 The hard-core repulsive contribution is somewhat more difficult to quantify because nonionic micelles often exhibit growth from monodisperse spheres into flexible, polydisperse cylindrical aggregates. The model for the repulsive interactions should be applicable to the full range of shapes and sizes. Gelbart et al. have developed26,43 a description of the hard-core, repulsive interactions between cylindrical micelles by modeling them as rigid spherocylinders, consisting of a cylindrical body of length Ln and crosssectional radius R, terminating in two hemispherical endcaps of radius R. Note that R ) l*c + lhg, where lhg is the length of the surfactant head. In this model, the smallest micelle that can form is a sphere (Ln ) 0) having 3 3 aggregation number n0 ) 4π(l* c) /(3vtail) ≈ 4πR /(3Ωs). For the spherocylindrical micelle case, the excluded, in eq 31 can be written as volume contribution, βµEX,HC n
βµEX,HC ) 2BHC n n1 c1 + 2
(35)
(30)
Equations 29 and 30 indicate that
βµEX n ) 2
Ωsγ2 (n + n2/3 0 ), n g n0 2
and
HC Bnm ,
are
(43) Ben-Shaul, A.; Gelbart, W. M. J. Phys. Chem. 1982, 86, 316.
∑ m)n
4πd3 3
∞
∑
cm +
m)n0 ∞
π cm(Ln + Lm) + d cmLnLm (38) 2 m)n0 0
∑
Note that, for the case of monodisperse spherical micelles, βµEX,HC (sph) can be obtained from eq 38, with n ) n0 and n Ln ) Lm ) 0. E. Micellar Size Distribution and Its Moments. EX The EXCESS chemical potentials, µEX n and µ1 , are now fully defined through eqs 30, 33, 37, and 38 and can be utilized to evaluate the micellar size distribution through eq 22. The attractive contribution to the EXCESS chemical potential in eq 33 was computed in the context of a mean-field approximation and was shown to depend ∞ solely on the total surfactant concentration, cs ) ∑m)1 mcm. As a result, this contribution cancels out when used in eq 22. Specifically, ∞
µEX,att - nµEX,att ) -nC(T)Ωs n 1
∑ mcm + m)1 ∞
n[C(T)Ωs
∑ mcm] ) 0
(39)
m)1
In view of eq 39, only the hard-core, repulsive interactions contribute to the EXCESS chemical potentials appearing in the size-distribution equation. This hard-core contri(44) Onsager, L. Ann. N. Y. Acad. Sci. 1949, 51, 627.
Nonionic Micellar Solutions
Langmuir, Vol. 13, No. 20, 1997 5265
bution can be conveniently written as a linear function of the micelle aggregation number, n. Specifically, for the case of spherocylindrical micelles, one obtains EX EX,HC cyl β(µEX - nµEX,HC ) ) Acyl n - nµ1 ) ) β(µn 1 0 - nA1
(40) where 2 2/3 Acyl 0 ) Ωsγ n0 c1 +
8Ωs 2πd3 (c - c1) + (c - c1) 9 3 s
(41)
and
volume interactions with an alternative equation of state.46 However, they only considered the relatively high surfactant concentration limit and, hence, neglected the monomer excluded-volume contributions. In addition, the spherocylindrical micelles at these conditions were assumed to be sufficiently long such that the effect of the excluded-volume contributions associated with the hemispherical end caps could be neglected. Consequently, their cyl expressions for Acyl 0 and A1 are slightly different from those given in eqs 41 and 42. In order to better characterize the spherocylindrical micellar system, it is useful to introduce various moments of the micellar size distribution. In general, the kth moment of the micellar size distribution is given by
2/3 2 2 Acyl 1 ) [8 - γ (2 + n0 )]Ωsc1 + Ωsγ cs +
Ωsγ2n2/3 0 c -
8Ω2s (c 3 s
8Ωs (c - c1) 3 πd
∞
Mk ) - c1) (42)
and Asph can For the case of monodisperse spheres, Asph 0 1 be obtained directly from eqs 41 and 42 by making use of the fact that n ) n0 ) πd3/(6Ωs) and n0(c - c1) ) cs - c1. 2 2/3 sph Specifically, Asph 0 ) Ωsγ n0 c1 + 4Ωs(cs - c1) and A1 ) [12 2(1 + γ)]Ω c + Ω γ3c - 4Ω c. -γ s 1 s s s cyl Note that Acyl 0 and A1 in eqs 41 and 42 are dimensionless quantities which are independent of n. When very few micelles are present in the solution, Acyl 0 and Acyl are very small, since, as c f 0, c ≈ c ≈ c s 1 s, and 1 cyl therefore, Acyl 0 and A1 f 0. Utilizing eqs 23 and 40 in eq 22, the following expression is obtained for the micellar size distribution of spherocylindrical micelles cyl Ωwcn ) (Ωwc1)n exp{-βngmic(n) - (Acyl 0 - nA1 )} (43) cyl gmic (n)
given in eq 24. Note that eq 43 is with gmic(n) ) also applicable to monodisperse spherical micelles of cyl (n) replaced by gsph ) aggregation number n0, with gmic cyl sph gmic(l*c,s* ) 3) and with A0 and Acyl 1 replaced by A0 and sph A1 , respectively. cyl By substituting the expression for gmic (n) given in eq 24 in eq 43, the following convenient form is obtained for the micellar size distribution of spherocylindrical micelles
cn )
qn K
(44)
where
K ) Ωw exp[βn0(gsph - gcyl) + Acyl 0 ]
∑ nkcn n)1
From these moments, one can calculate various average characteristics of the micellar size distribution, which can be measured experimentally. For example,5
〈n〉n )
M1 M0
(48)
〈n〉w )
M2 M1
(49)
and
M3M1
σ)
M22
nkcn
(51)
Expressions for 〈n〉n, 〈n〉w, and σ corresponding to spherocylindrical micelles can be obtained by using eq 51 for Mcyl k (with k ) 0, 1, 2, and 3) in eqs 48-50, respectively. Utilizing eq 44 in eq 51 with k ) 0 and k ) 1, expressions for c and cs, respectively, can be derived. Specifically, ∞
∑
n)n0
(45) Missel, P. J.; Mazer, N. A.; Benedek, G. B.; Young, C. Y.; Carey, M. C. J. Phys. Chem. 1980, 84, 1044.
∑ n)n
0
c ) Mcyl 0 ) c1 +
Equations 44-46 describe the effect of excluded-volume interactions on the micellar size distribution of spheroand cylindrical micelles through the parameters Acyl 0 Acyl 1 . It is noteworthy that, in the absence of excludedcyl volume interactions (the “ideal case”), Acyl 0 ) A1 ) 0, and one recovers the well-known expressions for cn, q, and K.45 Interestingly, Gelbart et al. obtained26 a similar expression for the size distribution of spherocylindrical micelles by utilizing the same model for the excluded-
(50)
∞
Mcyl k ) c1 +
(45)
(46)
-1
where 〈n〉n is the number-average micellar aggregation number, 〈n〉w is the weight-average micellar aggregation number, and σ is the relative variance of the micellar size distribution, constituting a measure of micellar-size polydispersity. For spherocylindrical micelles, Mk in eq 47 is given by
and
q ) (Ωwc1) exp[-βgcyl + Acyl 1 ]
(47)
qn K
) c1 +
qn0
(52)
K(1 - q)
and ∞
cs ) Mcyl 1 ) c1 +
∑
n)n0
qn
n
K
) c1 + qn0
[n0(1 - q) + q] (53) K(1 - q)2 (46) Gelbart et al.26,43 utilized the “y-expansion”59 which rescales the virial expansion in terms of a new concentration variable, thus incorporating the effect of higher-order terms. For the relatively dilute surfactant solutions of interest in this paper, the two approaches yield similar results.
5266 Langmuir, Vol. 13, No. 20, 1997
Zoeller et al.
Equation 44, describing the size distribution of spherocylindrical micelles, can also be utilized to define a critical micellar concentration (or cmc). Specifically, by taking the natural log of cn in eq 44 and keeping only terms that are of order n, because n . 1, one can show7,45 that the cmc (in units of mole fraction) is given by
decreasing exponential function of n. Using eq 58 in eq 47, the following expression for the kth moment is obtained:
cmc ≈ exp(βgcyl - Acyl 1 )
In particular, the zeroth moment is given by Mcyl 0 ) c ) (cs/K)1/2. The parameter K provides a quantitative measure of the ability of the spherocylindrical micelles to grow.45 Recall that K is a function of the surfactant concentration, cs, through the excluded-volume contribution, Acyl 0 (see eqs 41 and 45). In the ideal case, when Acyl ) 0, K does 0 not depend on cs, and c increases as the square root of cs. However, when Acyl 0 > 0, as cs increases, K also increases, thus reducing the rate at which c increases with cs. In addition, in the limit of extensive micellar growth, the average micellar aggregation numbers are given by
(54)
where gcyl ) gmic(l*c,s* ) 2). In view of the fact that Acyl 1 depends on surfactant concentration (see eq 42), it is necessary to set cs ) cmc and solve eq 54 iteratively. Note that eqs 48-54 are also applicable to monodisperse spherical micelles of aggregation number n0, with gcyl cyl cyl replaced by gsph ) gmic(l* c,s* ) 3) and with A0 and A1 sph sph replaced by A0 and A1 , respectively. In summary, using eqs 45 and 46, along with eqs 41 and 42, in eq 53, one obtains an implicit equation relating c1 and cs. Note that solving eq 53 requires the simultaneous solution of eq 52, since K and q depend on Acyl 0 and Acyl 1 , respectively, which, in turn, are functions of c. Solving this set of equations numerically, one can, in principle, calculate c1 as a function of cs, which can then be inserted back in eq 44 to obtain the entire distribution of micellar sizes, {cn}, as a function of cs and T. Illustrative example calculations of (i) the variation of c1 with cs, (ii) the cmc, and (iii) the characteristics of the micellar size distribution, including c, 〈n〉w and σ, will be presented in section IV. F. Limit of Extensive Micellar Growth. It is instructive to consider the limiting case of extensive spherocylindrical micellar growth, in which 〈n〉w . n0. For this purpose, it is convenient to introduce a concentration, cb, which corresponds to the maximum monomer concentration attainable in this limit. Therefore, Ωwcb is equivalent to the cmc defined in eq 54, that is,5,45
Ωwcb ) exp(βgcyl - Acyl 1 )
(55)
In view of eqs 46 and 55, it follows that q ) c1/cb. Note that cb is a function of the total surfactant concentration, cs, through the excluded-volume parameter, Acyl 1 (see eq 42). In particular, as cs increases, the excluded-volume term, Acyl 1 , increases, and cb decreases. At high cs values, the monomer concentration, c1, approaches its maximum value, cb, and hence q ≈ 1. In order to explore this limiting behavior, it is useful to introduce a parameter f ) 1 - q. Substituting q ) 1 - f in eq 53, cs can be expressed in terms of this new parameter as follows
cs ) c1 +
(
(1 - n0f) n0 1 - f + K f f2
)
(56)
In the limit of extensive micellar growth, q ≈ 1, and f , 1. Expanding the right-hand side of eq 56 in powers of f (to leading order in f) and rearranging terms gives the following expression for f 45
f ) [K(cs - c1)]-1/2
(57)
At high surfactant concentrations, (cs - c1) ≈ cs, and by utilizing eq 57 in eq 44, one obtains the following remarkably simple expression for cn45
1 ccyl exp[-n(Kcs)-1/2], n . n0 n ) K
(58)
Consequently, in the limit of extensive micellar growth, the micellar size distribution, {cn}, is a monotonically
Mcyl k )
k! (c K)(k+1)/2, n . n0 K s
〈n〉n )
〈n〉w )
Mcyl 1 Mcyl 0 Mcyl 2 Mcyl 1
(59)
) (csK)1/2
(60)
) 2(csK)1/2
(61)
Equations 60 and 61, along with eq 45, indicate that excluded-volume interactions affect these aggregation numbers through the growth parameter, K, which depends cyl on Acyl 0 . When A0 ) 0 (ideal case), both 〈n〉n and 〈n〉w increase as the square-root of the total surfactant concentration, cs. However, in the presence of excludedvolume interactions, Acyl 0 increases as cs increases (see eq 41), which, in turn, leads to an increase of K with cs. As a result, in the presence of excluded-volume interactions, 〈n〉w and 〈n〉n not only are larger but also increase at a rate faster than c1/2 with increasing cs. Interestingly, als though excluded-volume interactions separately affect both 〈n〉w and 〈n〉n, in the limit of extensive micellar growth, their ratio, 〈n〉w/〈n〉n, remains equal to 2 (see eqs 60 and 61), a well-known finding corresponding to the ideal case.5,45 In addition, the relative variance of the micellar size distribution, σ, has a value of 0.5 even in the presence of excluded-volume interactions, which is another wellknown finding corresponding to the ideal case.5,45 G. Micellar Solution Phase Separation. 1. Spinodal Curve and Critical Point. Nonionic surfactants, particulary those of the alkyl poly(ethylene oxide) variety, can phase separate into a micelle-rich phase coexisting with a micelle-poor phase by varying temperature and/or surfactant concentration. The phase separation behavior can be quantified by making use of thermodynamic stability requirements. The spinodal curve is obtained from the following requirement40
( ) ∂µ1 ∂cs
)0
(62)
T,p
The critical point is obtained from the following additional requirement40
( ) ∂2µ1 ∂c2s
)0
(63)
T,p
The derivatives of the monomer chemical potential in eqs 62 and 63 can be taken directly using eq 20, with n
Nonionic Micellar Solutions
Langmuir, Vol. 13, No. 20, 1997 5267
) 1. The following expression is then obtained for the spinodal curve
( ) ( ) ∂µ1 ∂cs
∂µEX 1 ∂cs
)
T,p
+ T,p
()
kBT ∂c1 c1 ∂cs
- Ωs
T,p
( ) ∂Π ∂cs
)0
T,p
(64)
Due to the Gibbs-Duhem equation, it can be shown that (∂Π/∂cs)T,p ) 0, and therefore, the last term in eq 64 can be set to zero.40,47 A second differentiation of eq 64, with (∂Π/∂cs)T,p ) 0 as explained above, with respect to cs yields the additional expression required to evaluate the critical point, that is,
( ) ( ) ∂2µEX 1
∂2µ1 ∂c2s
)
∂c2s
T,p
T,p
-
()
kBT ∂c1 c2 ∂cs 1
2
+
T,p
( )
kBT ∂2c1 c1 ∂c2 s
T,p
ccrit s , by simultaneously solving eqs 67 and 68, utilizing the expressions for the derivatives given in Appendix C. Alternatively, given the critical temperature, Tc, solving eq 68 yields the critical surfactant concentration, ccrit s . Then, eq 67 can be utilized to solve for the attraction parameter at the critical point, Ccrit(Tc). Illustrative calculations of the critical surfactant concentration are presented in section IVB. 2. Coexistence Curve. The coexistence curve can be obtained through the requirements of thermodynamic equilibrium. Specifically, the temperature, the pressure, and the chemical potential of each component present in the two coexisting micellar phases A and B should be equal, namely,
)0 (65)
For the micellar solutions exhibiting phase separation considered in this paper, the micelles are cylindrical in shape.48,49 Therefore, in what follows, we model the micelles as spherocylinders and utilize the results derived in sections IIID-IIIE for this micellar shape. Recall that the monomer EXCESS chemical potential includes both attractive (eq 33 with n ) 1) and repulsive (eq 37) contributions, that is,
µAn ) µBn , n g 1
(69)
µAw ) µBw
(70)
and
Recall that due to the multiple chemical equilibrium condition (see eq 17), requiring µA1 ) µB1 is equivalent to requiring µAn ) nµA1 ) nµB1 ) µBn for any n. Hence, eq 69 can be replaced by the following requirement
EX,att + µEX,HC µEX 1 ) µ1 1
cs + kBTΩsγ2n2/3 0 c + 2 kBTΩs[8 - γ (1 + n2/3 0 )]c1
µA1 ) µB1
2
)kBTΩsγ
- C(T)Ωscs (66)
in eq 64 gives the following Utilizing eq 66 for µEX 1 expression for the spinodal curve
(
Ωs γ2 -
)
( )
C(T) ∂c + Ωsγ2n2/3 0 kBT ∂cs
{
ΠA ) ΠB T,p
}( )
1 ∂c1 c1 ∂cs
) 0 (67)
T,p
The critical point is then obtained from the following additional requirement, obtained by utilizing Eq 66 for µEX 1 in eq 65
() { ∂2c ∂c2s
+ Ωs[8 - γ2(1 + n2/3 0 )] +
T,p
In addition, specifying the water chemical potentials in phases A and B is equivalent to specifying the osmotic pressures of phases A and B (see eq 21). Hence, eq 70 can be replaced by
(72)
+
Ωs[8 - γ2(1 + n2/3 0 )] +
Ωsγ2n2/3 0
(71)
( ) () ∂2c1
1 ∂c1 - 2 2 ∂cs T,p c1 ∂cs
}
1 × c1 2
) 0 (68)
T,p
Note that c and c1, as well as their derivatives with respect to cs, which appear in eqs 67 and 68, can be calculated from eqs 52 and 53, respectifely (see Appendix C for details). Typically, both c and c1 depend on solution conditions such as temperature through the free energy of micellization, gmic. Given models for gmic(T) and the attraction parameter, C(T), one can predict both the critical temperature, Tc, and the critical surfactant concentration, (47) At constant temperature and pressure, the Gibbs-Duhem equation requires that (∂µw/∂cs)T,p,Nw also vanishes at the spinodal. Since the standard-state chemical potential, µ°w(T,p), depends only on T and p, eq 21 indicates that (∂µw/∂cs)T,p,Nw ) -Ωw(∂Π/∂cs)T,p,Nw ) 0. (48) Balmbra, R. R.; Clunie, J. S.; Corkill, J. M.; Goodman, J. F. Trans Faraday Soc. 1962, 58, 1661. (49) Corti, M.; Minero, C.; Degiorgio, V. J. Phys. Chem. 1984, 88, 309. Corti, M.; Degiorgio, V. J. Phys. Chem. 1981, 85, 1442.
The monomer chemical potential, µ1, is given by eq 20 with n ) 1, with µEX given in eq 66. Utilizing the 1 resulting expressions in eq 71 yields A B 2 Ωsγ2(cAs - cBs ) + Ωsγ2n2/3 0 (c - c ) + Ωs(8 - γ (1 + A B n2/3 0 ))(c1 - c1 ) -
C(T)Ωs A cA1 (cs - cBs ) + ln B ) 0 (73) kBT c 1
where the superscripts A and B indicate the values of c1, c, and cs in phases A and B, rspectively. The osmotic pressure, Π, is obtained directly from the virial equation of state used in section IIID to model the attractive and repulsive intermicellar interactions. Utilizing the second-virial coefficients corresponding to the attractive interactions (see eq 32), along with those corresponding to the hard-core repulsive interactions (see eqs 35 and 36), in eq 27, truncated at quadratic order in concentration, results in the following expression for the osmotic pressure
C(T)Ωsc2s βΠ ) c + Ωsγ2csc1 + Ωsγ2n2/3 0 cc1 + 2kBT πd3 (c - c1)2 + 9 8Ωs 4Ω2s (c - c1)2 (74) (c - c1)(cs - c1) + 3 s 3 πd
2 Ωs[4 - γ2(1 + n2/3 0 )]c1 +
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Zoeller et al.
Utilizing eq 74 in eq 72 yields
cA - cB -
C(T)Ωs A 2 [(cs ) - (cBs )2] + Ωsγ2[cAs cA1 - cBs cB1 ] + 2kBT
2/3 A 2 A A B B 2 Ωsγ2n2/3 0 [c c1 - c c1 ] + Ωs(4 - γ (1 + n0 ))[(c1 ) -
(cB1 )2] +
πd3 A [(c - cA1 )2 - (cB - cB1 )2] + 9
8Ωs A [(c - cA1 )(cAs - cA1 ) - (cB - cB1 )(cBs - cB1 )] + 3 4Ω2s A [(cs - cA1 )2 - (cBs - cB1 )2] ) 0 (75) πd3 Accordingly, given a model for the attraction parameter, C(T), a simultaneous solution of eqs 73 and 75 at a fixed temperature and pressure yields the two coexisting concentrations, cAs and cBs . Repeating this calculation for a range of temperatures generates the entire coexistence curve in the temperature-concentration plane. Note that eq 74 can also be utilized to derive an expression for the osmotic compressibility of the micellar -1 . By difsolution in the one-phase region, (∂Π/∂cs)T,p ferentiating eq 74 with respect to cs, we obtain the following result
( ) {( ) -1
∂Π ∂cs
)β
∂c ∂cs
[ ( )]
T,p
-
C(T)Ωs c + kBT s
( )]
[( ) ( ) [( ) ( ) ] [ ( )] [( ) ( )] [ ( ) ]} T,p
∂c1 ∂cs
∂c + ∂c T,p s T,p ∂c1 ∂c1 c + 2Ωs(4 - γ2(1 + n2/3 + 0 ))c1 ∂cs T,p ∂cs T,p ∂c1 2πd3 ∂c + (c - c1) 9 ∂cs T,p ∂cs T,p ∂c1 8Ωs 8Ωs ∂c + (c - c1) 1 (c - c1) 3 ∂cs T,p 3 s ∂cs T,p 2
Ωsγ c1 + cs
∂c1 ∂cs
+
T,p
Figure 1. Predicted monomer concentration, c1, as a function of total surfactant concentration, cs, for an aqueous solution of C12E6 at 20 °C. Predictions were made using the excludedvolume model (s) and the ideal solution model (- - -), which has zero excluded volume.
+ Ωsγ2n2/3 0 c1
∂c1 8Ω2s (c - c1) 1 3 s ∂cs πd
-1
(76)
T,p
As stated earlier, the derivatives of c and c1 with respect to cs are given in Appendix C. If the temperature dependence of C(T) is known in the one-phase region (that is, for T < Tc in the case of the alkyl poly(ethylene oxide) surfactants considered in this paper), then solving eq 76 at a given value of cs yields the osmotic compressibility for temperatures less than Tc. Illustrative calculations for both the coexistence curve and the osmotic compressibility are presented in section IVB. IV. Results In this section, we apply the theoretical results derived in sections II and III to aqueous micellar solutions of alkyl poly(ethylene oxide) surfactants, a representative, widelyused class of nonionic surfactants. First, a qualitative analysis of the effect of excluded-volume interactions is presented. Then, micellar solution properties, such as the cmc, the relative variance of the micellar size distribution, the critical surfactant concentration for phase separation, and the osmotic compressibility, are quantitatively predicted and compared to experimental data. A. Excluded-Volume Contribution. As a representative example, aqueous solutions of the nonionic surfactant C12E6, CH3(CH2)11(OCH2CH2)6-OH, were examined at 20 °C. Using eq 53, the monomer concentration,
Figure 2. Predicted micelle concentration, c - c1, as a function of total surfactant concentration, cs, for an aqueous solution of C12E6 at 20 °C. Predictions were made using the excludedvolume model (s) and the ideal solution model (- - -), which has zero excluded volume.
c1, was predicted as a function of the total surfactant concentration, cs, and plotted in Figure 1 (solid line). In addition, the same prediction was made assuming ideal behavior, that is, in the absence of excluded-volume interactions (dashed line in Figure 1). Two main conclusions can be drawn from these predictions. First, the monomer concentration is lower when excluded-volume interactions are accounted for, indicating that such interactions encourage free surfactant monomers to form micelles. Second, as the total surfactant concentration increases, the monomer concentration initially increases both in the presence and in the absence of excluded-volume interactions. However, beyond the cmc (signaled in Figure 1 by the abrupt changes in the slopes of each curve), the monomer concentration remains constant in the ideal case, while it decreases in the presence of excluded-volume interactions. In general, as micelle formation becomes more favorable from a free-energy perspective, larger micelles are formed.2 Indeed, this is also true in the present case. Figure 2 depicts the micelle concentration, c - c1, predicted using eq 52, as a function of cs for the ideal case (dashed line) and in the presence of excluded-volume interactions (solid line). In the ideal case, as more surfactant molecules are added, they aggregate to form new micelles, such that the overall micelle concentration increases. When excludedvolume interactions are considered, as cs increases, the micelle concentration increases at a slower rate. This indicates that, instead of forming new micelles, the additional surfactant molecules are incorporated into already existing micelles, thus inducing micellar growth.
Nonionic Micellar Solutions
Figure 3. Predicted weight-average micellar aggregation number, 〈n〉w, as a function of total surfactant concentration, cs, for an aqueous solution of C12E6 at 20 °C. Predictions were made using the excluded-volume model (s) and the ideal solution model (- - -), which has zero excluded volume.
This is further corroborated in Figure 3, which shows predictions of the weight-average micellar aggregation number, 〈n〉w, as a function of cs. Indeed, Figure 3 reveals that, for a given cs value, when excluded-volume interactions are accounted for (solid line), the micelles that form have larger aggregation numbers than those corresponding to the ideal case (dashed line).26 The results in Figures 1-3 indicate that the overall effect of excluded-volume interactions is to encourage micellar growth, since a solution containing fewer, larger micelles excludes less volume than one containing a larger number of smaller micelles. This is a general result which should be applicable to any micellar system in the presence of excluded-volume interactions. However, the reader should keep in mind that the predictions presented here were made using a particular excluded-volume model, namely, that corresponding to rigid spherocylinders treated at the second-virial expansion level of approximation. This derivation assumes that (i) third- and higher-order body interactions can be neglected and (ii) micelle flexibility can be neglected. In order to test the validity and range of applicability of assumptions (i) and (ii) corresponding to the excluded-volume description presented above, other excluded-volume models were analyzed. First, to examine assumption (i), the third-virial coefficient corresponding to rigid spherocylinders26 was utilized in eq 27 to account for the effect of three-body interactions. The various EXCESS chemical potentials were then obtained following a procedure similar to that described in section III, and predictions for the micellar size distribution, analogous to those presented in Figures 1-3, were made. As expected, at relatively low surfactant concentrations, no difference was observed between the predictions made in the context of the second-order and third-order virial equations. Only when the surfactant concentration exceeded about 200 mM, which is approximately 1000 times the cmc, did the predictions made using the third-order virial equation begin to deviate from those made using the second-order virial equation. This clearly indicates that, for the relatively dilute surfactant concentrations of interest in this paper (0-100 mM), it is reasonable to neglect three-body excluded-volume effects. Second, to examine assumption (ii), an equation of state for flexible, hard-sphere chains58 was utilized, instead of the virial equation of state, to investigate the effect of micellar flexibility on the theoretical predictions. Similar to our finding regarding the third-virial coefficient contribution, deviations from the rigid case were only observed at surfactant concentrations greater than ≈200 mM. At these relatively high surfactant concentrations, the flexible
Langmuir, Vol. 13, No. 20, 1997 5269
Figure 4. Predicted cmc as a function of the number of ethylene oxide (EO) groups, j, for aqueous solutions of C10Ej (s), C12Ej (- - -), and C16Ej (‚‚‚) at 20 °C. Experimental values7,50,51 are denoted by circles for C10Ej, diamonds for C12Ej, and stars for C16Ej.
model predicted slightly higher monomer concentrations and higher micellar aggregation numbers than the virial equation approach, indicating that the nonideal contributions were even stronger for this equation of state. However, for the relatively low surfactant concentrations of interest in this paper (0-100 mM), is is reasonable to neglect micellar flexibility. In summary, we verified that the virial equation of state truncated at quadratic order, in the context of an excludedvolume model for rigid spherocylinders, is adequate for the calculation of micellar solution characteristics at the solution conditions of interest in this paper. Carrying out these additional calculations was also valuable in demonstrating the versatility of the McMillan-Mayer theory and the relative ease with which alternative models of intermicellar interactions and micellar flexibility can be analyzed in the context of this statistical-thermodynamic framework. B. Comparison with Experiments. In this section, the excluded-volume model is utilized to make several quantitative predictions of micellar solution characteristics. The molecular parameters, Ωs, n0, and gmic, of the CiEj surfactants examined in this paper (i ) 10, 12, and 16; j ) 4-9) were determined using a previously developed molecular model of micellization.7-9 gmic has a strong dependence on temperature, while n0 and Ωs are approximately constant over the range of temperatures examined. Specifically, n0 ) 34, 48, and 84 for C10, C12, and C16, respectively, and Ωs ) vtail + vhead, where vtail ) 312, 366, and 473 Å3 for C10, C12, and C16, respectively, and vhead ) (42.3 + 63.5j) Å3. Note that the CH2 group adjacent to the poly(ethylene oxide) head has been included as part of the head.7 For details of this molecular model of micellization, including a description of the calculation of gmic(T), see refs 7-9. Figure 4 shows predicted cmc’s at 20 °C corresponding to aqueous solutions of CiEj surfactants as a function of the number of ethylene oxide (EO) groups, j, for i ) 10 (solid line), i ) 12 (dashed line), and i ) 16 (dotted line). The circles, diamonds, and stars denote experimental values for C10Ej, C12Ej, and C16Ej, respectively.7,50,51 Note that the theory consistently captures the trend of increasing cmc with increasing j. As stressed in section III, the relative variance of the micellar size distribution, σ, constitutes a quantitative (50) Mukherjee, P.; Mysels, K. J. Critical Micelle Concentration of Aqueous Surfactant Systems; National Standard Reference Data Series (U. S., National Bureau of Standards) No. 36; U. S. Department of Commerce: Washington, DC, 1971. (51) Becher, P. In Nonionic Surfactants; Shick, M. J., Ed.; Arnold: London, 1967; p 478.
5270 Langmuir, Vol. 13, No. 20, 1997
Figure 5. Predicted relative variance of the micellar size distribution of C12Ej (j ) 5, 6, 7, and 8) micelles in aqueous solution as a function of temperature (solid lines). The arrows denote the experimentally determined shape transition temperatures for j ) 6,52 7,53 and 8.54.
Figure 6. Predicted critical surfactant concentration for aqueous solutions of various CiEj surfactants. The left-hatched bars denote theoretical predictions, and the white bars denote experimental values.49,53,55-57
measure of polydispersity. In particular, elongated, polydisperse cylindrical micelles are characterized by σ ) 0.5, whereas small, monodisperse spherical micelles are characterized by σ ) 0. Figure 5 illustrates the predicted temperature variation of the relative variance of the micellar size distribution for C12Ej surfactants in aqueous solutions, where j ) 5, 6, 7, and 8. In particular, for j ) 6, 7, and 8, the narrow temperature range over which the relative variance changes rapidly from 0 to 0.5 corresponds to a sphere-to-cylinder micellar shape transition. The experimentally determined shape transition temperatures (indicated by the various arrows in Figure 5) are 16,52 34,53 and 50 °C54 for C12E6, C12E7, and C12E8, respectively. As can be seen, the theory is capable of predicting the micellar shape transition behavior quite accurately. The critical behavior of several aqueous solutions of CiEj surfactants was predicted by solving eqs 67 and 68. As discussed in section IIIG, the quantities c1 and c depend on temperature through gmic(T). Consequently, in order to make these predictions, experimental values of the critical temperatures, Tc, served as inputs to the theory. Then, eq 68 was used to solve for the critical surfactant concentration. Figure 6 illustrates the predictions of the critical surfactant concentration for several CiEj surfactants. The experimental critical temperature values are 21,55 44,55 and 58 °C56 for C10E4, C10E5, and C10E6, respectively, and 3.5,17 25,53 50,53 62,53 and 77 °C49 for (52) Brown, W.; Rymden, R. J. Phys. Chem. 1987, 91, 3565. (53) Fujimatsu, H.; Ogasawara, S.; Kuroiwa, S. Colloid Polym. Sci. 1988, 266, 594. (54) Zana, R.; Weill, C. J. Phys., Lett. 1985, 46, L953. (55) Lang, J. C.; Morgan, R. D. J. Chem. Phys. 1980, 73, 5849. (56) Mulley, B. A.; Metcalf, A. D. J. Colloid Sci. 1962, 17, 523.
Zoeller et al.
Figure 7. Predicted osmotic compressibility along the critical isochore, ccs ) 57 mM, as a function of temperature for an aqueous solution of C12E6 (s). Experimental values are denoted by stars49 and diamonds.48
C12E4, C12E5, C12E6, C12E7, and C12E8, respectively. The theoretical predictions are given by the left-hatched bars, and the experimental values are given by the white bars.49,53,55-57 As can be seen, the theory yields accurate predictions for the critical behavior of CiEj surfactants. It should also be noted that the predicted coexistence curves corresponding to the aqueous CiEj micellar solutions examined (not shown) were relatively flat, in accordance with experimental observations.49,53 In order to gain a quantitative understanding of the mean-field attraction parameter, C(T), experimental values48 for cAs (T) and cBs (T) were used in eq 73 to predict C(T) as a function of T (>Tc). The predictions close to the critical point (T - Tc < 2 °C) indicated a linear dependence on temperature (specifically, C(T)/kB ) 0.45T - 141, in units of K). Then, this expression for C(T) was assumed to be applicable for T < Tc as well and was utilized in eq 76 to predict the osmotic compressibility along the critical isochore, for which the surfactant concentration is equal to the critical concentration (cc ) 57 mM48), for temperatures in the range 15 °C < T < Tc. Predictions of the osmotic compressibility are shown in Figure 7 (solid line) and compared to experimental data from ref 49 (stars) and from ref 48 (diamonds). The expected divergence of the osmotic compressibility as the temperature approaches Tc is clearly observed. The predictions follow the experimental data closely, indicating that the linear dependence on temperature found for C(T) at T > Tc is a reasonable approximation, even when extrapolated to temperatures which are less than Tc. In addition, because the osmotic compressibility and the coexistence curve represent two independent characteristics of the micellar solution, we can interpret the agreement of the osmotic compressibility predictions with experiments as an independent validation of the present theory. V. Conclusions In summary, we have developed a statistical-thermodynamic framework to model nonionic micellar solutions based on the McMillan-Mayer theory of multicomponent solutions. The advantage of this approach is that it clearly delineates the ideal and excess contributions to the solution Gibbs free energy, thus allowing, in principle, for successive improvements of the theory as equations of state of increasing complexity and accuracy are imple(57) Strey, R.; Pakusch, A. In Surfactants in Solutions; Mittal, K. L., Bothorel, P., Eds.; Plenum: New York, 1986; p 465. (58) See, for example: Boublik, T. Mol. Phys. 1974, 27, 1415. Boublik, T.; Vega, C.; Diaz-Pena, M. J. Chem. Phys. 1990, 93, 730. (59) Barboy, B.; Gelbart, W. M. J. Chem. Phys. 1979, 71, 3053.
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Langmuir, Vol. 13, No. 20, 1997 5271
mented. In addition, this statistical-thermodynamic framework allows for the quantitative analysis of the effect of intermicellar interactions on micellar solution characteristics, such as micelle formation, micellar size distribution, and micellar solution phase separation. In the calculations presented in this paper, intermicellar interactions were modeled in the context of a virial equation of state, truncated at quadratic order. In particular, attractive interactions were modeled using a mean-field description, while repulsive interactions were described using a model for excluded-volume interactions between rigid spherocylinders. It was shown that the inclusion of excluded-volume interactions has a profound effect on the micellar size distribution. Specifically, excluded-volume interactions encourage micellar growth, resulting in fewer, larger micelles. The theory was compared to an earlier phenomenological model (the ladder model45), which was developed for an ideal system devoid of excluded-volume interactions. In particular, in the limit of extensive micellar growth, the expressions for the average micellar aggregation numbers and the relative variance of the micellar size distribution were found to have exactly the same mathematical form as those predicted by the ladder model. The only difference is that, in the presence of excluded-volume interactions, the growth parameter, K, depends on surfactant concentration and more readily promotes micellar growth. It is very interesting that a micellar solution model based on an entirely different thermodynamic framework leads to the same limiting behavior. In addition, several quantitative predictions of micellar solution characteristics were made and found to compare favorably with experimental data. These include (i) the cmc, (ii) the variance of the micellar size distribution, (iii) the critical surfactant concentration for phase separation, and (iv) the osmotic compressibility for aqueous solutions of alkyl poly(ethylene oxide) nonionic surfactants. In view of this, for the dilute micellar solutions examined in this paper, the model for intermicellar interactions presented here constitutes a reasonable approximation. However, if more concentrated micellar solutions are investigated, then alternative models of intermicellar interactions may need to be considered. As better descriptions of intermicellar interactions are developed, they can be incorporated into the statistical-thermodynamic framework presented here. For example, we are currently engaged in modeling electrostatic intermicellar interactions so that ionic micellar solutions can be modeled in the context of the current theoretical framework. Electrostatic interactions are, in general, longer-ranged than the attractive and excluded-volume interactions considered in this paper, making this a very challenging problem. However, due to its versatility and computational accuracy, the statistical-thermodynamic framework presented in this paper appears well suited to address these challenges. Appendix A. Excess Gibbs Free Energy and Excess Chemical Potentials In this appendix, we present a derivation of eq 11 in the main text and we derive expressions for the excess chemical potentials. We begin with the definition of the excess Gibbs free energy, Gex. Specifically,
G (T,p,{Nσ},Nw) ) G(T,p,{Nσ},Nw) ex
G (T,p,{Nσ},Nw) (A1) id
We would like to establish a connection between the Gibbs free energy at pressure p and the McMillan-Mayer free energy at pressure p + Π. Accordingly, the first step
involves transforming G(T,p,{Nσ},Nw) from pressure p to pressure p + Π. In view of the fact that
(∂G ∂p )
T,{Nσ},Nw
)V
(A2)
where V(T,p,{Nσ},Nw) is the system volume, integration of eq A2 from p to p + Π yields
G(T,p,{Nσ},Nw) ) G(T,p+Π,{Nσ},Nw) -
∫pp+ΠV(T,p′,{Nσ},Nw) dp′
(A3)
Inserting the expression for G given in eq A3, along with the expression for Gid given in eq 6, into eq A1 yields the following expression for Gex
Gex(T,p,{Nσ},Nw) ) G(T,p+Π,{Nσ},Nw) -
∫pp+ΠV(T,p′,{Nσ},Nw) dp′ - [Nwµ°w + ∑Nσµ°σ + ∑σ
kBT
σ
Nσ(ln mσ - 1)] (A4)
Next, a Legendre transformation40 is utilized to transform from the Gibbs free energy, G, to the McMillan-Mayer free energy, F. Specifically, in order to transform from G(T,p+Π,{Nσ},Nw) to F(T,V†,{Nσ},µ°w), the variables p + Π and Nw should be transformed to the variables V† and µ°w, respectively (recall that V† ) V(T,p+Π,{Nσ},Nw), as described in section II). This Legendre transformation yields
F(T,V†,{Nσ},µ°w) ) G(T,p+Π,{Nσ},Nw) - (p + Π)V† Nwµ°w (A5) Utilizing the relation between G and F, established in eq A5, in eq A4 yields
Gex(T,p,{Nσ},Nw) ) F(T,V†,{Nσ},µ°w) + (p + Π)V† -
∫pp+ΠV(T,p′,{Nσ},Nw) dp′ - ∑Nσµ°σ ∑σ
k BT
σ
Nσ(ln mσ - 1) (A6)
Next, the standard-state chemical potentials of the solutes in eq A6 are transformed from µ°σ to µQσ with the use of eq 7. Using this standard state will allow a connection with FID given in eq 2. In addition, the resulting expression for Gex is rewritten below utilizing number concentrations, c†σ, rather than molalities, mσ (recall that mσ ) c†σV†/Nw). Specifically,
Gex(T,p,{Nσ},Nw) ) F(T,V†,{Nσ},µ°w) + (p + Π)V† -
∑σ
[
Nσ µQσ
∫pp+ΠV(T,p′,{Nσ},Nw) dp′ -
+ kBT ln
1
]
cQσ Vw
[( ) ]
∑σ Nσ ln
- kBT
c†σV† Nw
-1
(A7)
where Vw ) V(T,p,{0},Nw)/Nw is the volume per molecule in pure solvent (water).
5272 Langmuir, Vol. 13, No. 20, 1997
Zoeller et al.
By rearranging the natural log terms in eq A7, the following expression for Gex is obtained
Gex(T,p,{Nσ},Nw) ) F(T,V†,{Nσ},µ°w) + (p + Π)V† -
∫p
∑σ
p+Π
V(T,p′,{Nσ},Nw) dp′ -
[() ] c†σ
∑σ Nσ ln
kBT
- 1 - NkBT ln
cQσ
NσµQσ -
V
( )
∑σ NσµQσ + kBT∑σ Nσ ln
-1 -
cQσ
pV† + FEX(T,V†,{Nσ},µ°w) (A9) Utilizing the expression for F given in eq A9 in eq A8 and canceling the appropriate terms yields the following expression for Gex
Gex(T,p,{Nσ},Nw) ) FEX(T,V†,{Nσ},µ°w) + ΠV† -
∫p
p+Π
( ) ( ) ( ) ∂G ∂Nσ
∂Π ∂Nσ
+
T,p,{NR*σ},Nw
∫p
ΠV h †σ
p+Π
T,p,{NR*σ},Nw
( )
∂Π +V ∂Nσ †
-
V†
†
∫pp+ΠVh σ(p′) dp′ - kBT ln VwVNw
(A13)
The osmotic pressure, Π, can be split into IDEAL and EXCESS contributions. Specifically,
Π ) ΠID + ΠEX ) c†kBT + ΠEX
(A14)
where c† ) N/V† is the total solute concentration. In addition, the derivative of FEX in eq A13 can be expressed in terms of its natural variables, T, V†, {Nσ}, and µ°w. Specifically,
( ) ∂FEX ∂Nσ
( ) ( ) ( )
)
T,p,{NR*σ},Nw
∂FEX ∂Nσ
∂FEX ∂V†
+
T,V†,{NR*σ},µ°w
∂V† ∂Nσ
T,{Nσ},µ°w
( ) ∂FEX ∂Nσ
(A15)
T,p,{NR*σ},Nw
T,V†,{NR*σ},µ°w
) µEX σ
(A16)
and
( ) ∂FEX ∂V†
T,{Nσ},µ°w
) ΠEX
(A17)
eq A15 can be rewritten as follows
( ) ∂FEX ∂Nσ
V h σ(p′) dp′ - kBT ln
V† VwNw
h †σ is the partial molar volume of solute σ at where V pressure p + Π, that is,
( ) ∂V† ∂Nσ
EX µex σ ) µσ -
T,p,{NR*σ},Nw
NkBT † V h σ (A11) V†
V h †σ )
]
NkBT
T,p,{NR*σ},Nw
EX † ) µEX hσ σ - Π V
(A18)
Using eq A18 for the derivative of FEX, along with eq A14 for the osmotic pressure, Π, in eq A13 and canceling terms, one obtains the following expression for µex σ
T,p,{NR*σ},Nw
-
[
+V h †σ Π -
In view of the fact that
ex
∂FEX ) ∂Nσ
V†
T,p,{NR*σ},Nw
V† V(T,p′,{Nσ},Nw) dp′ - NkBT ln (A10) VwNw
Equation A10 is the Gex expression given in eq 11 of the main text. The advantage of utilizing this approach is that the McMillan-Mayer free energy can be related to a model for solute molecules interacting through vacuum. Equation A10 provides the necessary relation to transform the McMillan-Mayer free energy, FEX, to the Gibbs free energy, Gex, for solute molecules interacting through solvent. The excess contributions to the chemical potentials of the solutes and the solvent can now be obtained directly by differentiation of Gex with respect to Nσ and Nw, respectively. Specifically, for solute σ, one obtains
µex σ )
∂FEX ∂Nσ
(A8)
where N ) ∑σ Nσ. Recall that the McMillan-Mayer free energy, F, can be divided into IDEAL and EXCESS contributions (see eq 3). Utilizing eq 2 for FID in eq 3, the following expression for F is obtained
F(T,V†,{Nσ},µ°w) )
( )
µex σ )
†
VwNw
c†σ
Canceling terms in eq A11 and rearranging, one obtains the following expression for µex σ
(A12)
(A19)
Equation A19 is the µex σ expression given in eq 12 of the main text. For the solvent, the excess chemical potential is also obtained by differentiation of the Gex expression given in eq A10. Specifically,
µex w )
)
T,p,{NR*σ},Nw
and V h σ(p′) is the partial molar volume of solute σ at pressure p′. Note that, in taking the derivative of the integral term in eq A10, the limits of integration must also be differentiated. This yields the additional -V†(∂Π/∂Nσ)T,p,{NR*σ},Nw term in eq A11.
†
∫pp+ΠVh σ(p′) dp′ - kBT ln VwVNw
V†
( ) ( ) ( ) ∂Gex ∂Nw
T,p,{Nσ}
∂FEX ∂Nw
∂Π ∂Nw
T,p,{Nσ}
T,p,{Nσ}
+ ΠV h †w + V†
-
( ) ∂Π ∂Nw
-
T,p,{Nσ}
∫pp+ΠVh w(p′) dp′ -
[
NkBT V
†
V h †w -
]
V† Nw
(A20) h †w is the partial molar volume of the solvent at where V
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Langmuir, Vol. 13, No. 20, 1997 5273
pressure p + Π, that is,
V h †w )
( ) ∂V† ∂Nw
(A21) T,p,{Nσ}
and V h w(p′) is the partial molar volume of the solvent at pressure p′. The derivative of FEX in eq A20 can be expressed in terms of its natural variables, T, V†, {Nσ}, and µ°w. This yields
( ) ∂FEX ∂Nw
( )
∂FEX ) ∂V† T,p,{Nσ}
T,{Nσ},µ°w
T,p,{Nσ}
V h †w
EX
) -Π
∫pp+ΠVh w(p′) dp′
Appendix B. Hard-Core Contribution to the Excess Chemical Potentials of Spherocylindrical Micelles As described in eq 34 with n ) 1, the hard-core contribution to the monomer EXCESS chemical potential is given by ∞
∑ m)n
HC B1m cm
(B1)
0
2π π (R + R)3 + (R1 + R)2Ln 3 1 2
(B3)
(n - n0)Ωs )
2
(B4)
πR2
πd
Since 4πR3/3 ) πd3/6 ) n0Ωs, eq B4 indicates that Ln ) 0 when n ) n0. From the definitions of R1 and R given above, it follows that
R1 )
( ) 3Ωs 4π
1/3
)
R n1/3 0
(B5)
Using eqs B4 and B5 in eq B2, the following expression HC is obtained for Bn1
BHC n1
(
Ωs 1 ) 1 + 1/3 2 n 0
)
2
(n +
n2/3 0 )
Ωsγ2 ) (n + n2/3 0 ), 2 n g n0 (B6)
where γ ) (1 + 1/n1/3 0 ). HC HC Using Bn1 from eq B6, along with B11 ) 4Ωs, in eq B1 yields
) 8Ωsc1 + Ωsγ2(cs - c1) + Ωsγ2n2/3 βµEX,HC 1 0 (c - c1)
The contribution of the excluded-volume interactions HC can be estimated by between two monomers to B11 modeling each monomer as a sphere of radius R1, such that 4πR31/3 ) Ωs, the volume of a surfactant molecule. In that case, the monomer-monomer excluded volume is HC represents the given by 8(4πR31/3) ) 8Ωs. Since B11 HC excluded volume per monomer, it follows that B11 ) 8Ωs/2 ) 4Ωs. The contribution of the excluded-volume interactions between a spherocylindrical micelle of aggregation number HC HC ) B1n is given by44,35 n g n0 and a monomer to Bn1
BHC n1 )
4nΩs - (2/3)πd3
(A23)
Equation A23 is the expression for µex w given in eq 13 of the main text.
HC ) 2B11 c1 + 2 βµEX,HC 1
Ln )
(A22)
Utilizing eqs A14 and A22 in eq A20 and canceling and rearranging terms yields the following expression for µex w (recall that N/Nw ) m)
µex w ) kBTm -
πd3 d2 Ln + ) nΩs 2 6
()
Vn ) π
where d ) 2R. Using eq B3, Ln can be written in terms of the aggregation number, n, as follows
( ) ∂V† ∂Nw
head. We have chosen to use l* c(s ) 2) rather than l* c(s ) 3) because the systems described in this paper are typically elongated rods, where the majority of the surfactant molecules reside in the cylindrical body of the micelle. Ln can be calculated by using the total volume of the spherocylindrical n-mer, Vn, which is given by
(B7) where cs ) c1 + ∑mgn0 mcm and c ) c1 + ∑mgn0 cm. Equation B7 is the expression for βµEX,HC given in eq 37 of the main 1 text. The hard-core contribution to the n-mer EXCESS , was given in eq 38 in the main chemical potential, µEX,HC n text, which we repeat below for clarity
βµEX,HC n
2
) Ωsγ (n + ∞
(B2)
where Ln is the length of the cylindrical body of a spherocylindrical micelle of aggregation number n and R is the radius of the two hemispherical end caps having aggregation number n0 ) 4π(l*c)3/3vtail ≈ 4πR3/3Ωs. Note that we have made the reasonable simplifying assumption that the radius of the hemispherical end caps, Rsph, is equal to the cross-sectional radius of the cylindrical body, Rcyl. In other words, we have assumed that Rsph ) Rcyl t R ) (l*c(s ) 2) + lhg), where lhg is the length of the polar
πd2
n2/3 0 )c1
4πd3 + 3
∞
∑ m)n
cm + 0
∞ π cm(Ln + Lm) + d cmLnLm (B8) 2 m)n0 0
∑ m)n
∑
The expression for Ln in eq B4 can be utilized in eq B8 to obtain the following relation between µEX,HC and n n
8Ωs 2πd3 (c - c1) + (c - c1) + 9 3 s 8Ω2s 8Ωs (cs - c1) (B9) n Ωsγ2c1 + (c - c1) + 3 πd3
βµEX,HC ) Ωsγ2n2/3 n 0 c1 +
(
)
5274 Langmuir, Vol. 13, No. 20, 1997
Zoeller et al.
Using eqs B9 and B7, one can compute β(µEX,HC n nµEX,HC ) for spherocylindrical micelles. Specifically, 1
where (see eqs 41 and 42)
a1 )
2πd3 9
(C6)
a2 )
8Ωs 3
(C7)
cyl β(µEX,HC - nµEX,HC ) ) Acyl n 1 0 - nA1 )
Ωsγ2n2/3 0 c1 +
[
8Ωs 2πd3 (c - c1) + (c - c1) 9 3 s
2 2 2/3 n [8 - γ2(2 + n2/3 0 )]Ωsc1 + Ωsγ cs + Ωsγ n0 c -
a3 )
]
8Ω2s 8Ωs (cs - c1) (B10) (c - c1) 3 πd3 Equation B10, corresponding to eq 40 of the main text, cyl permits the identification of Acyl 0 and A1 , which are given in eqs 41 and 42 of the main text, respectively.
8Ω2s
(C8)
πd3
a4 ) Ωsγ2
(C9)
a5 ) Ωsγ2n2/3 0
(C10)
The first derivative of c1 in eq C1 with respect to cs is given by
( )
∂c1 X1 ) ∂cs Y1
Appendix C. Derivatives of the Monomer and Micelle Concentrations
(C11)
where The derivatives presented in this Appendix apply to the case of micellar solutions consisting of cylindrical micelles which are modeled as rigid spherocylinders. As explained in the main text, this is the optimal micellar shape of experimental interest in this paper. The first and second derivatives of the monomer concentration, c1, and of the zeroth moment, c, with respect to the total surfactant concentration, cs, can be obtained by utilizing eqs 52 and 53. The resulting derivative expressions are very long and complicated, due to the dependence of the excluded-volume contributions on the cyl size distribution. Specifically, since Acyl 0 and A1 depend on c, in order to calculate (∂c1/∂cs), it is also necessary to determine (∂c/∂cs).
(a22 - a1a3 + a1a4 - a2a5)(D21 - D2D3) (C12) and
Y1 ) 1 + 2(a2 - a5)D1 + (3a2 + a3 - 2a4)D2 + a1D3 -
a1 - a22 + a1a3 + 3a1a2 - 2a1a4 + c1
)
2a2a5 - a25 (D21 - D2D3) (C13)
D1 )
(C1) D2 )
where
K ) Ωw exp[βn0(gsph - gcyl) + Acyl 0 ]
(
qn0 [n0 + (1 - n0)q] K(1 - q)2
(C2)
and
q ) (Ωwc1) exp[-βgcyl +
(C14)
qn0 [n20 + (1 + 2n0 - 2n20)q + K(1 - q)3 (1 - 2n0 + n20)q2] (C15) D3 )
Acyl 1 ]
D2 + c1
with Di (i ) 1, 2, and 3) given by
We begin by rewriting eq 53 as follows
qn0 [n0(1 - q) + q] c1 ) cs K(1 - q)2
X1 ) 1 + (2a2 - a5)D1 + (a3 - a4)D2 + a1D3 +
(C3)
qn0 K(1 - q)
(C16)
The first derivative of c with respect to cs is obtained by differentiating eq 52 with respect to cs. This yields
( )
X2 ∂c ) ∂cs Y2
cyl with the excluded-volume parameters, Acyl 0 and A1 , given
by
(C17)
where
Acyl 0 ) a2cs + a1c + (a5 - a1 - a2)c1
(C4)
Acyl 1 ) (a4 - a3)cs + (a5 - a2)c + (4a2 + a3 - 2a4 - a5)c1 (C5)
X2 )
( )(
∂c1 D1 1+ + (4a2 + a3 - 2a4 - a5)D1 + ∂cs c1
)
(a1 + a2 - a5)D3 + (a4 - a3)D1 - a2D3 (C18)
Nonionic Micellar Solutions
Langmuir, Vol. 13, No. 20, 1997 5275
The D′i’s (i ) 1, 2, and 3) in eqs C21-C25 are given by
and
Y2 ) 1 + (a2 - a5)D1 + a1D3
(C19)
A second differentiation of eqs C11 and C17 with respect to cs yields the corresponding second derivatives. The algebra is tedious but straightforward. The resulting expression for the second derivative of the monomer concentration is given by
( ) ( ()) ∂2c1 ∂c2s
∂c1 1 ) X′1 Y′ Y1 ∂cs 1
(C20)
where
D′1 )
1
( )
)
(
( )
( )
a1 ∂c1 (C22) c2 ∂cs 1
( ) ( ())
( )(
( )
) ( )(
(a1 + a2 - a5)D′3 +
∂c2s
1+
( ) [ ( ) ( )] ∂Acyl ∂q 1 ∂c1 1 )q + ∂cs c1 ∂cs ∂cs
2a4 - a5)D1 + (a1 + a2 - a5)D3 + (a4 - a3)D′1 a2D′3 (C24)
and
Y′2 ) (a2 - a5)D′1 + a1D′3
(C29)
and
() ( ) ∂Acyl ∂K 0 )K ∂cs ∂cs
( ) ( )
(C30)
()
()
(C25)
()
(C31)
∂Acyl 1 ∂c ) a4 - a3 + (a5 - a2) + (4a2 + a3 - 2a4 ∂cs ∂cs
D1 + (4a2 + a3 c1
)
(C28)
where
∂Acyl ∂c1 0 ∂c ) a2 + a1 + (a5 - a1 - a2) ∂cs ∂cs ∂cs
∂c1 D′1 D1 ∂c1 - 2 + (4a2 + a3 - 2a4 - a5)D′1 + ∂cs c1 c1 ∂cs ∂2c1
]( ) ( )}
with
(C23)
where
X′2 )
(C26)
]( ) ( )}
∂D3 n0 1 ∂K ∂q 1 ) D3 + ∂cs q K ∂cs (1 - q) ∂cs
In these and the following equations, the ′ indicates differentiation with respect to cs. The expression for the second derivative of the micelle concentration is given by
1 ∂2c ∂c ) X′ Y′ Y2 2 ∂cs 2 ∂c2s
∂q 1 ∂K ∂cs K ∂cs
( ) {[
( ) {[
2a1a4 + 2a2a5 - a25 (2D1D′1 - D′2D3 - D2D′3) + (D21 - D2D3)
{[
∂q + ∂cs
(C27)
D′3 )
Y′1 ) 2(a2 - a5)D′1 + (3a2 + a3 - 2a4)D′2 + D′2 D2 ∂c1 a1 - 2 + a1D′3 - a22 + a1a3 + 3a1a2 c1 ∂c c1 c s
( ) ]( ) ( )}
∂D2 qn0 ) {(1 + 2n0 - 2n20) + 2(1 ∂cs K(1 - q)3 n0 1 ∂K ∂q ∂q 3 + D2 2n0 + n20)q} + ∂cs q K ∂cs (1 - q) ∂cs
D′2 )
X′1 ) (2a2 - a5)D′1 + (a3 - a4)D′2 + a1D′3 + (a22 a1a3 + a1a4 - a2a5)(2D1D′1 - D′2D3 - D2D′3) (C21) and
( )
∂D1 qn0 ) (1 - n0) ∂cs K(1 - q)2 n0 2 D1 + q (1 - q)
a5)
()
∂c1 (C32) ∂cs
Acknowledgment. L.L. thanks the Miller Institute for Basic Research for the award of a Postdoctoral Fellowship and J. M. Prausnitz for his hospitality during the author’s stay at the University of California at Berkeley. N.Z. and D.B. are grateful to Kodak, Unilever, and Witco for partial support of this work. LA970308C
