Molecular-Thermodynamic Theory of Micellization of Multicomponent

Apr 20, 2007 - In article 1 of this series, we developed a molecular-thermodynamic (MT) theory to model the micellization of mixtures containing an ar...
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Langmuir 2007, 23, 5953-5962

5953

Molecular-Thermodynamic Theory of Micellization of Multicomponent Surfactant Mixtures: 2. pH-Sensitive Surfactants Arthur Goldsipe and Daniel Blankschtein* Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 ReceiVed October 19, 2006. In Final Form: February 22, 2007 In article 1 of this series, we developed a molecular-thermodynamic (MT) theory to model the micellization of mixtures containing an arbitrary number of conventional (pH-insensitive) surfactants. In this article, we extend the MT theory to model mixtures containing a pH-sensitive surfactant. The MT theory was validated by examining mixtures containing both a pH-sensitive surfactant and a conventional surfactant, which effectively behave like ternary surfactant mixtures. We first compared the predicted micellar titration data to experimental micellar titration data that we obtained for varying compositions of mixed micelles containing the pH-sensitive surfactant dodecyldimethylamine oxide (C12DAO) mixed with either a cationic surfactant (dodecyltrimethylammonium bromide, C12TAB), a nonionic surfactant (dodecyl octa(ethylene oxide), C12E8), or an anionic surfactant (sodium dodecyl sulfate, SDS) surfactant. The MT theory accurately modeled the titration behavior of C12DAO mixed with C12E8. However, C12DAO was observed to interact more favorably with SDS and with C12TAB than was predicted by the MT theory. We also compared predictions to data from the literature for mixtures of C12DAO and SDS. Although the pH values of solutions with no added acid were modeled with only qualitative accuracy, the MT theory resulted in quantitatively accurate predictions of solution pH for mixtures containing added acid. In addition, the predicted degree of counterion binding yielded a lower bound to the experimentally measured value. Finally, we predicted the critical micelle concentration (cmc) of solutions of two pH-sensitive surfactants, tetradecyldimethylamine oxide (C14DAO) and hexadecyldimethyl betaine (C16Bet), at varying solution pH and surfactant composition. However, at the pH values considered, the pH sensitivity of C16Bet could be neglected, and it was equivalently modeled as a zwitterionic surfactant. The cmc’s predicted using the MT theory agreed well with the experimental cmc’s and were found to be comparable to and sometimes better than the cmc’s determined using the regular solution theory (RST), even though the empirical RST utilizes experimentally measured cmc’s as an input. The MT theory presented here represents the first molecular-based quantitative description of the micellization behavior of mixtures of pH-sensitive surfactants and conventional surfactants, and allows qualitative and quantitative predictions of the micellization behavior of a variety of surfactant systems.

1. Introduction series,1

As discussed in article 1 of this typical detergent formulations2-4 consist of two or more types of surfactants. The major surfactant component is usually a conventional, pHinsensitive surfactant, typically anionic but sometimes nonionic. Small amounts of amphoteric (that is, pH-sensitive) surfactants are often added because of their performance-boosting properties. In fact, the unique properties of amphoteric surfactants (for example, solubility over large pH ranges, low potential for skin and eye irritation, and good foaming characteristics) have led to their use in a wide variety of products, including personal care formulations, disinfectants, detergents, and anticorrosion coatings.5 Furthermore, pH-sensitive surfactants are becoming increasingly important in novel applications, including controlled drug release, targeted gene delivery, and * To whom correspondence should be addressed. Mailing address: Department of Chemical Engineering, Room 66-444, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139. Tel: (617) 253-4594. Fax: (617) 252-1651. E-mail: [email protected]. (1) Goldsipe, A.; Blankschtein, D. Langmuir 2007, 23, 5942-5952. (2) Hargreaves, T. Chemical Formulation: An OVerView of Surfactant-Based Preparations Used in EVeryday Life; Royal Society of Chemistry: Cambridge, U.K., 2003. (3) Liquid Detergents, 2nd ed.; Lai, K.-Y., Ed.; Surfactant Science Series 129; Taylor & Francis: Boca Raton, FL 2006. (4) Showell, M. S. In Powdered Detergents; Schowell, M. S., Ed.; Surfactant Science Series 71; Marcel Dekker: New York, 1998; Chapter 1, pp 1-19. (5) Amphoteric Surfactants, 2nd ed.; Lomax, E. G., Ed.; Surfactant Science Series 59; Marcel Dekker: New York, 1996.

the fabrication of nanoscale materials for optics, electronics, and sensors.6-21 In particular, alkylamine oxide surfactants represent a promising and well-studied class of pH-sensitive surfactants.22,23 (6) Imae, T.; Trend, B. Langmuir 1991, 7, 643-646. (7) Briscoe, P.; Caniggia, I.; Graves, A.; Benson, B.; Huang, L.; Tanswell, A. K.; Freeman, B. A. Am. J. Physiol. 1995, 268, L374-L380. (8) Liang, C.-W. Use of Biodegradable pH-Sensitive Surfactants in Liposome Mediated Oligonucleotide Delivery. Thesis, University of Florida, Gainesville, FL, 1998. (9) Liang, E.; Hughes, J. A. J. Membr. Biol. 1998, 166, 37-49. (10) Liang, E.; Rosenblatt, M. N.; Ajmani, P. S.; Hughes, J. A. Eur. J. Pharm. Sci. 2000, 11, 199-205. (11) Wang, S.; Bui, V.; Hughes, J. A.; King, M. A.; Meyer, E. M. Neurochem. Int. 2000, 37, 1-6. (12) Hafez, I. M.; Ansell, S.; Cullis, P. R. Biophys. J. 2000, 79, 1438-1446. (13) Matsui, H.; Gologan, B. J. Phys. Chem. B 2000, 104, 3383-3386. (14) Fielden, M. L.; Perrin, C.; Kremer, A.; Bergsma, M.; Stuart, M. C.; Camilleri, P.; Engberts, J. B. F. N. Eur. J. Biochem. 2001, 268, 1269-1279. (15) Bergsma, M.; Fielden, M. L.; Engberts, J. B. F. N. J. Colloid Interface Sci. 2001, 243, 491-495. (16) Guo, X. Ortho Ester-Based Surfactants for pH-Triggered Release in Drug and Gene Delivery. Thesis, University of California, San Francisco, CA, 2001. (17) Kawasaki, H.; Souda, M.; Tanaka, S.; Nemoto, N.; Karlsson, G.; Almgren, M.; Maeda, H. J. Phys. Chem. B 2002, 106, 1524-1527. (18) Johnsson, M.; Wagenaar, A.; Stuart, M. C. A.; Engberts, J. B. F. N. Langmuir 2003, 19, 4609-4618. (19) Chen, F.-J.; Asokan, A.; Cho, M. J. Biochim. Biophys. Acta. 2003, 1611, 140-150. (20) Chung, M.-H.; Chung, Y.-C.; Chun, B. C. Colloids Surf., B 2003, 29, 75-80. (21) Chung, M.-H.; Park, C.; Chun, B. C.; Chung, Y.-C. Colloids Surf., B 2004, 34, 179-184. (22) Maeda, H. Colloids Surf., A 1996, 109, 263-271 and references therein. (23) Maeda, H.; Kakehashi, R. AdV. Colloid Interface Sci. 2000, 88, 275293 and references therein.

10.1021/la063082z CCC: $37.00 © 2007 American Chemical Society Published on Web 04/20/2007

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In addition, pH-sensitive molecules, including surfactants, dyes, and drugs, have been used to probe micelles and, in particular, to infer the electrostatic potential at the micelle surface (ψ0).24-36 Probe molecules are utilized by solubilizing them in micelles and using hydrogen ion titration to determine the equilibrium deprotonation parameter of the probe in the micelle (pKm). The pKm is assumed to vary with ψ0 according to the following equation:37-39

e0ψ0 pKm ) pK0m + (log e) kBT

(1)

where pK0m is the probe pKm in a neutral micelle (at ψ0 ) 0), e0 is the charge of a proton, kB is the Boltzmann constant, and T is the absolute temperature. pK0m is typically determined by titrating the probe solubilized in a nonionic micelle because pK0m can differ significantly from the deprotonation constant of the probe (pKa) in bulk aqueous solutions.25,26,28-31,33 In spite of the practical and technological relevance of mixtures containing pH-sensitive surfactants, to our knowledge very little attention has been devoted to the theoretical modeling of these surfactants. The pKm of lauric acid in nonionic, cationic, and anionic micelles was recently predicted theoretically by da Silva et al. using Poisson-Boltzmann calculations and Monte Carlo simulations.33 However, these authors limited their studies to dilute concentrations of lauric acid, with approximately one lauric acid molecule per micelle. Zimmerman and Schnaare utilized regular solution theory (RST) and the pseudophase approximation to predict the pH dependence of the critical micelle concentration (cmc) of mixtures of tetradecyldimethylamine oxide (C14DAO) and hexadecyldimethyl betaine (C16Bet), which are ternary mixtures of the deprotonated C14DAO, the protonated C14DAO, and the deprotonated C16Bet at the pH values that these authors considered.40 Although RST provided reasonable estimates of the cmc, a significant experimental effort was required to obtain the required inputs, including the three β parameters that characterize the pairwise surfactant interactions. In addition, we recently utilized RST to develop a model of the pKm of a pHsensitive surfactant in mixed micelles.39 Although the model agreed well with the experimental data,39 the empirical nature of RST limits its ability to provide physical insight at the molecular level. With the above in mind, we have developed a molecularthermodynamic (MT) theory to overcome the shortcomings of (24) Mukerjee, P.; Banerjee, K. J. Phys. Chem. 1964, 68, 3567-3574. (25) Ferna´ndez, M. S.; Fromherz, P. J. Phys. Chem. 1977, 81, 1755-1761. (26) Garcı´a-Soto, J.; Ferna´ndez, M. S. Biochim. Biophys. Acta 1983, 731, 275-281. (27) Drummond, C. J.; Warr, G. G.; Grieser, F.; Ninham, B. W.; Evans, D. F. J. Phys. Chem. 1985, 89, 2103-2109. (28) Lovelock, B.; Grieser, F.; Healy, T. W. J. Phys. Chem. 1985, 89, 501507. (29) Drummond, C. J.; Grieser, F. Photochem. Photobiol. 1987, 45, 19-34. (30) Drummond, C. J.; Grieser, F.; Healy, T. W. J. Phys. Chem. 1988, 92, 2604-2613. (31) Grieser, F.; Drummond, C. J. J. Phys. Chem. 1988, 92, 5580-5593. (32) Cassidy, M. A.; Warr, G. G. J. Phys. Chem. 1996, 100, 3237-3240. (33) da Silva, F. L. B.; Bogren, D.; So¨derman, O.; Åkesson, T.; Jo¨nsson, B. J. Phys. Chem. B 2002, 106, 3515-3522. (34) Whiddon, C. R.; Bunton, C. A.; So¨derman, O. J. Phys. Chem. B 2003, 107, 1001-1005. (35) Chakraborty, H.; Banerjee, R.; Sarkar, M. Biophys. Chem. 2003, 104, 315-325. (36) Chakraborty, H.; Sarkar, M. J. Colloid Interface Sci. 2005, 292, 265270. (37) Maeda, H. J. Colloid Interface Sci. 2003, 263, 277-287. (38) Goldsipe, A.; Blankschtein, D. Langmuir 2006, 22, 3547-3559. (39) Goldsipe, A.; Blankschtein, D. Langmuir 2006, 22, 9894-9904. (40) Zimmerman, J. A.; Schnaare, R. L. J. Colloid Interface Sci. 1999, 220, 75-80.

the limited theoretical descriptions that are currently available to model surfactant mixtures containing pH-sensitive surfactants. The MT approach is fully predictive in that it does not require any experimental data on the micellization properties of the surfactants of interest. Instead, the MT theory models the micellization process molecularly and requires only knowledge of the molecular structures of the surfactants as well as of the solution conditions, including the temperature, the total surfactant concentration and composition, and the type and concentration of additives (salt, acid, and/or base). The pH-dependent behavior of the micelles is captured by modeling the equilibrium of the surfactant monomers and requires knowledge of the deprotonation constant of the pH-sensitive surfactant monomers (pK1). The remainder of this article is organized as follows. The materials and methods are briefly described in section 2. In section 3, the thermodynamic and the molecular components of the theory are discussed. The prediction of various micellar solution properties in the context of the MT theory is discussed in section 4. In section 5, theoretical predictions are compared to experimental data, and the theory is utilized to gain physical insight into the micellization behavior of various mixtures of conventional and pH-sensitive surfactants. Finally, concluding remarks are presented in section 6. 2. Materials and Experimental Methods Although the theory will be validated in section 5 using titration data that we collected recently,39 that study was limited to mixtures of pH-sensitive surfactant dodecyldimethylamine oxide (C12DAO) and either cationic surfactant dodecyltrimethylammonium bromide (C12TAB) or nonionic surfactant dodecyl octa(ethylene oxide) (C12E8). In this article, we used the same experimental procedure to conduct titrations on 51 mM solutions of binary mixtures of C12DAO (lot no. 436579/1 31402) obtained from Sigma-Aldrich (St. Louis, MO) and anionic surfactant sodium dodecyl sulfate (SDS) (lot no. 7718X07624) obtained from Mallinckrodt Chemicals (Paris, KY). Hydrochloric acid (HCl) (0.1 N lot no. X25515 and 0.5 N lot no. X10505) and sodium hydroxide (NaOH) (0.1 N lot no. X38505 and 0.5 N lot no. X27503) solutions were obtained from J. T. Baker (Phillipsburg, NJ). All of these materials were used as received. Note that the 51 mM surfactant concentration was selected in order for the concentration of the micellized C12DAO surfactant to be much higher than that of the C12DAO surfactant monomers, thereby allowing us to neglect the contribution of the C12DAO surfactant monomers during the titrations. Sodium chloride (NaCl) was added at a concentration of 0.1 M to maintain a constant solution ionic strength. Smirnova et al. reported that the dissolution temperature of the C12DAO/SDS solutions reaches a maximum of approximately 48 °C at equimolar compositions and low pH.41 Consequently, all of the titrations were conducted at a temperature of 50 °C to avoid precipitation. Three different mixtures were examined, having C12DAO compositions of 1.00, 0.90, and 0.75. Solutions containing both C12DAO and SDS became turbid when more than approximately 25% of C12DAO was in the protonated state. Note that Smirnova et al. also reported that some C12DAO/ SDS solutions were not transparent below pH 5.5.41 As a result, solutions having lower C12DAO compositions were not examined. Although the exact nature of these solutions is unknown, they are probably not isotropic micellar solutions, which are usually transparent. (Some possible phases that could be forming include vesicles, ordered cylindrical phases, and lamellae.42,43) Because the theory developed in this article is applicable only in the case of (41) Smirnova, N. A.; Murch, B.; Pukinsky, I. B.; Churjusova, T. G.; Alexeeva, M. V.; Vlasov, A. Y.; Mokrushina, L. V. Langmuir 2002, 18, 3446-3453. (42) Hoffmann, H.; Bergmeier, M.; Gradzielski, M.; Thunig, C. Prog. Colloid Polym. Sci. 1998, 109, 13-20. (43) Kawasaki, H.; Imahayashi, R.; Tanaka, S.; Almgren, M.; Karlsson, G.; Maeda, H. J. Phys. Chem. B 2003, 107, 8661-8668.

Micellization of pH-SensitiVe Surfactant Mixtures isotropic micellar solutions, experimental titration data are reported only for the conditions at which the micellar solution was transparent.

3. Theory The MT theory is developed here for a pH-sensitive surfactant mixed with an arbitrary number of conventional surfactants. However, the experimental data used to validate the theory (section 5) is limited to mixtures of a pH-sensitive surfactant and a single conventional surfactant. The basic thermodynamic framework is based on the theory developed in article 1 to model the micellization of mixtures of conventional surfactants.1 pH equilibrium is incorporated into the theory following the approach that we developed recently to model solutions containing only pH-sensitive surfactants.38 Although a theoretical framework was developed to incorporate the effect of intermicellar interactions in the MT approach,44 we have not considered these interactions here because the micellar solutions under consideration are quite dilute. This neglect of intermicellar interactions is discussed in greater detail in ref 45. 3.1. Notation. The deprotonated and the protonated forms of the pH-sensitive surfactant are denoted as S and HS, respectively, reflecting the reaction H+ + S f HS+. The total number of conventional surfactants is denoted as nsurf. The subscript i is used to refer generally to any of these (nsurf + 2) pH-sensitive and conventional surfactant species. The solution may also contain any number of counterions, which may result from the addition of ionic surfactants, acid, base, or salt. The total number of counterions is denoted as nion, and the individual ions are denoted by the subscript j, which varies from 1 to nion. Note that using this notation allows many of the thermodynamic conditions to be written in exactly the same form as in the case of the multicomponent surfactant mixtures discussed in article 1 of this series.1 A micelle is described in terms of several characteristics. The number of surfactant molecules in a micelle (or the aggregation number) is denoted as n. The micelle composition, denoting the mole fractions of the various surfactant types in the micelle, is RHS, RS, and Ri for the protonated surfactant, the deprotonated surfactant, and the conventional surfactant i, respectively. Counterions are explicitly allowed to bind to the micelle. The number of counterions j bound to the micelle is nβˆ j, where βˆ j is referred to as the degree of counterion binding. The coreminor radius of the micelle is denoted as lc. The shape of the micelle is captured through the shape factor S, which is 1 for planar bilayers, 2 for cylindrical micelles, and 3 for spherical micelles. The mole fraction of micelles of aggregation number n, composition {Ri}, degree of counterion binding { βˆ j}, coreminor radius lc, and shape S is denoted as Xn{Ri,βˆ j}. Note that S and lc are not explicitly included in the subscript to simplify the notation. The total mole fraction of surfactant in micelles of any type is denoted as Xmic. The total monomer concentration is denoted as X1. The monomer composition is denoted as R1HS, R1S, and R1i for the protonated surfactant monomers, the deprotonated surfactant monomers, and the conventional surfactant monomers i, respectively. The concentration of free ions of type j is denoted as X1j. The bulk concentrations of H+ and OH- are denoted as X1H and X1OH, respectively. The experimentally controlled concentrations are the total concentrations of the various species. The total concentration of total surfactant is denoted as Xsurf , and the composition of added total total for the protonated surfactant is denoted as RHS , Rtotal S , and Ri (44) Puvvada, S.; Blankschtein, D. J. Phys. Chem. 1992, 96, 5567-5579.

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surfactant, the deprotonated surfactant, and the conventional surfactant i, respectively. The total concentration of ion j, which may result from any added ionic surfactant, acid, base, . Finally, the total concentrations of or salt, is denoted as Xtotal j H+ and OH- (added to the solution from acid and protonated surfactant and from base, respectively) are denoted as Xtotal H and . Xtotal OH 3.2. Thermodynamic Framework. Using the notation presented in section 3.1, the condition governing the thermodynamic equilibrium between the micelles, the surfactant monomers, and the free counterions is identical to that given in ref 1:

Xn{Ri,βˆ j} )

1 e

∏i (eX1i)nR ∏j (eX1j)nβˆ exp × i

j

{

-

n kBT

[gmic(S, lc, {Ri}, {βˆ j})]

}

(2)

where gmic is the free energy of micellization representing the free-energy gain per surfactant molecule associated with forming a micelle. To facilitate calculations, eq 2 can be expressed in terms of the total monomer concentration, X1, as follows:

[

]

1 n Xn{Ri,βˆ j} ) Xn1 exp g e kBT m

(3)

where gm is the modified free energy of micellization, defined as

gm ) gmic - kBT[

∑i Ri ln R1i + ∑j βˆ j ln X1j + ∑j βˆ j + 1]

(4)

Note that eqs 2-4 are identical in form to eqs 2-4 in article 1 of this series.1 However, subscript i in eqs 2-4 here denotes the nsurf conventional surfactants and the protonated and the deprotonated forms of the pH-sensitive surfactant. In addition, the monomer composition of the pH-sensitive surfactant (R1S and R1HS) is subject to the pH equilibrium constraint, which can be written in terms of the Henderson-Hasselbalch equation: 23,38,39,46

pK1 ) pH - log

X1S R1HS ) pH + log X1HS R1S

(5)

Each species is also subject to a mass balance constraint. For the conventional surfactants, the mass balance is given by the following expression:

Xtotal Rtotal i surf ) R1iX1 +

∑

S,n,lc,{Ri,βˆ j}

nRiXn{Ri,βˆ j} + Edl,i

(6)

where i denotes only the conventional surfactant components (1 to nsurf) and Edl,i is a correction due to the enhancement or the depletion of surfactant i in the diffuse double layer (Appendix). In the case of the pH-sensitive surfactant, the appropriate mass balance is for the total concentration of both the protonated and deprotonated forms38 (45) Srinivasan, V.; Blankschtein, D. Langmuir 2003, 19, 9932-9945. (46) Stryer, L. Biochemistry, 3rd ed.; W. H. Freeman: New York, 1988. (47) CRC Handbook of Chemistry and Physics, 84th ed.; Lide, D. R., Ed.; CRC Press: Boca Raton, FL, 2003.

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Goldsipe and Blankschtein

total total (Rtotal HS + RS )Xsurf ) (R1HS + R1S)X1 +

∑

S,n,lc,{Ri,βˆ j}

n(RHS + RS)Xn{Ri,βˆ j} + Edl,S (7)

where Edl,S is a correction due to the enhancement or the depletion of the pH-sensitive surfactant in the diffuse double layer (Appendix). The mass balance for ion j is identical to that in eq 8 in article 1 of this series1

) X1j + Xtotal j

∑

S,n,lc,{Ri,βˆ j}

nβˆ jXn{Ri,βˆ j} + Edl,j

(8)

where Edl,j is a correction due to the enhancement or the depletion of ion j in the diffuse double layer (Appendix). Similar mass balances can be obtained for the H+ and OH- ions present in the solution. However, as discussed in ref 38, the following combined mass balance is more useful total (Xtotal H - XOH ) ) (X1H - X1OH) + R1HSX1 +

∑

S,n,lc,{Ri,βˆ j}

nRHSXn{Ri,βˆ j} + (Edl,H - Edl,OH) (9)

where Edl,H and Edl,OH are corrections due to the enhancement or the depletion of the H+ and the OH- ions in the diffuse double layer (Appendix). Finally, X1OH is determined by utilizing the equilibrium constant for the dissociation of water (Kw, obtained from ref 47):

X1HX1OH ) Kw

(10)

As in ref 38, we have not included activity coefficients in eqs 5 and 10, which implies that the activity coefficients are 1 or, alternatively, that they can be accounted for in terms of effective values of the equilibrium constants. total In summary, the experimentally controlled variables (Xsurf , total total total total Ri , Xj , XH , and XOH ) are related to the concentrations of the various species in the bulk (X1, R1i, Xj, X1H, and X1OH) and to the micelle concentration (Xn{Ri,βˆ j}) by simultaneously solving eqs 3-10. Note that a model is also required for gmic, which we describe next. 3.3. Molecular Model for the Free Energy of Micellization, gmic. Because of our choice of notation, the equations describing the gmic model for mixtures of pH-sensitive and conventional surfactants are identical to those obtained for multicomponent surfactant mixtures. (See the Appendix of article 1 of this series.1) For completeness, the main physical contributions to the gmic model are summarized briefly below. The free energy of micellization, gmic, is modeled as the sum of six contributions:48

gmic ) gtr + gint + gpack + gst + gelec + gent

(11)

The first three contributions are associated with the formation of the micelle core: The transfer contribution, gtr, accounts for the transfer of the surfactant hydrophobic tails from an aqueous to an oil-like environment (eq A1 of article 1). The interfacial contribution, gint, accounts for the free-energy penalty associated with the formation of an interface between the oil-like micelle core and the surrounding aqueous environment (eq A2 of article 1). The packing contribution, gpack, accounts for the additional entropic constraints associated with anchoring one end of each surfactant hydrophobic tail at the micelle core-water interface (48) Goldsipe, A.; Blankschtein, D. Langmuir 2005, 21, 9850-9865.

(eq A3 of article 1). The fourth and fifth contributions are associated with the formation of the micelle interfacial region: The steric contribution, gst, accounts for the steric interactions between the surfactant hydrophilic heads and the bound counterions in the micelle interfacial region (eq A4 of article 1). The electrostatic contribution, gelec, accounts for the electrostatic interactions at the micelle interface that operate in the case of ionic surfactants and bound counterions and of zwitterionic surfactants (eqs A6-A10 of article 1). The last contribution, gent, accounts for the entropy of mixing the various components in a micelle (eq A5 of article 1).

4. Determination of Useful Micellar Solution Properties As described in article 1 of this series,1 the optimal micelle characteristics are determined by minimizing gm with respect to {Ri}, { βˆ j}, lc, and S. Denoting the minimum value of gm as g/m, the cmc is approximated as exp(g/m/kBT). The minimization is performed explicitly for spherical and infinite cylindrical micelles, and the corresponding gm of finite cylinders is determined by interpolation (section 3.1 of article 1 and refs 48 and 38). In the case of micelles containing pH-sensitive surfactants, an additional property of interest is pKm, which is a micellar deprotonation parameter analogous to pK1 of the surfactant monomers. pKm is defined as23

pKm ) pH + log

( )

(

RHS xHS ) pH + log RS 1 - xHS

)

(12)

where xHS ) RHS/(RHS + RS) is the degree of protonation of the micellized pH-sensitive surfactant. Note that defining the pKm of mixed micelles in terms of xHS, which always varies from 0 to 1, is more convenient than using the micelle composition because (RHS + RS) is generally less than unity.

5. Results and Discussion The theory presented in section 3 is validated below using titration data that we obtained (section 2 and ref 39) and experimental data from the literature for systems containing a pH-sensitive surfactant and a conventional surfactant. In most cases, the pH-sensitive surfactant is C12DAO. A longer-chain homologue, tetradecyldimethylamine oxide (C14DAO), is used to validate the cmc predictions because of the availability of experimental (cmc) data.40 Note that pK1 of the C12DAO and the C14DAO monomers is approximately 4.9,49-51 indicating that half of the surfactant monomers are protonated at a pH of 4.9 (eq 5). The conventional surfactants considered include anionic, cationic, and nonionic surfactants. We also consider a pH-sensitive surfactant that, at the pH values considered, can be accurately modeled as a pH-insensitive, zwitterionic surfactant. Estimated values of the molecular characteristics of the surfactants considered are taken from refs 38 and 48 and are also provided as Supporting Information. 5.1. Prediction of Micellar Titration Behavior. We first use the MT theory to predict the titration behavior of micelles of C12DAO and a conventional surfactant. Three conventional surfactants are examined: the nonionic surfactant C12E8 and the cationic surfactant C12TAB, which were also analyzed recently using a RST-based theory,39 and the anionic surfactant SDS. The (49) Tokiwa, F.; Ohki, K. J. Phys. Chem. 1966, 70, 3437-3441. (50) Maeda, H.; Tsunoda, M.-a.; Ikeda, S. J. Phys. Chem. 1974, 78, 10861090. (51) Kakehashi, R.; Yamamura, S.; Tokai, N.; Takeda, T.; Kaneda, K.; Yoshinaga, K.; Maeda, H. J. Colloid Interface Sci. 2001, 243, 233-240.

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Figure 1. Comparison of the predicted (-) and the experimental (b) ∆pKm values of C12DAO/C12E8 micelles as a function of the degree of protonation of C12DAO (xC12DAOH). The composition of C12E8 (RCtotal ) is (a) 0.90, (b) 0.75, (c) 0.50, and (d) 0.29. Error bars denote the 12E8 experimental uncertainty as estimated using eq 22 of ref 39.

C12DAO/SDS titrations are reported only up to 25% protonation of C12DAO because solutions with greater protonation became turbid. Because isotropic micellar phases are usually transparent, the turbidity suggests the formation of other phases (such as vesicles, ordered cylindrical phases, and lamellae42,43) that are not modeled within the context of the MT theory presented here, which assumes that the micellar solution is isotropic. Note that the complete set of C12DAO/SDS titration data is included as Supporting Information, along with additional predictions (micelle aggregation numbers, monomer concentrations, and counterion binding). Just as the deprotonation constant (pKa) is commonly used to characterize the pH equilibrium of a simple acid, the pH equilibrium of a micellized surfactant can be modeled in terms of the pKm (section 4). In ref 39, the predicted pKm of C12DAO was consistently lower than the experimental pKm, a discrepancy that was tentatively attributed to hydrogen bonding and/or to additional electrostatic synergy. In this article, we are largely interested in the ability of the MT theory to capture the additional effect of the added conventional surfactant on pKm. Therefore, rather than comparing absolute pKm values, which largely reflect the ability to model the self-interaction of C12DAO, we compare relative pKm values (∆pKm), which are defined as follows:

∆pKm ) pKm - pKm(RC12DAO f 1)

(13)

where pKm(RC12DAO f 1) denotes the pKm in the limit of a micelle containing only deprotonated C12DAO. The ∆pKm should better reflect the ability of the MT theory to model the specific effect of the added conventional surfactant. Experimentally, pKm(RC12DAO f 1) was measured to be 5.88 at room temperature (approximately 20 °C) and 5.79 at 50 °C. The predicted value of pKm(RC12DAO f 1) was found to be 5.48 ( 0.01 over the temperature range of 20-50 °C. 5.1.1. Mixtures of C12DAO and Nonionic Surfactant C12E8. In Figure 1, the predicted (lines) and experimental (circles) ∆pKm values are compared as a function of the protonated fraction of C12DAO [xC12DAOH ≡ RC12DAOH/(RC12DAO + RC12DAOH)] for

mixtures of C12DAO/C12E8 with RCtotal ) 0.90 (Figure 1a), 0.75 12E8 (Figure 1b), 0.50 (Figure 1c), and 0.29 (Figure 1d). Error bars denote the experimental uncertainty as estimated using eq 22 of ref 39. Excellent agreement is observed at RCtotal ) 0.90 for 12E8 which the ∆pKm is slightly negative (0 to -0.5), indicating that it is slightly more difficult to incorporate C12DAOH+ into C12E8rich micelles than into C12DAO-rich micelles. The ∆pKm shows only a weak dependence on xC12DAOH when RC12E8total is high, as indicated by the small slopes of the curves in Figure 1a,b. As decreases, the ∆pKm curves have a more negative slope, RCtotal 12E8 which is accurately captured by the MT predictions. However, ) 0.50 and 0.29, the MT theory slightly underpredicts at RCtotal 12E8 the ∆pKm by approximately 0.5. The lower ∆pKm values correspond to MT predictions “favoring” the deprotonated form of C12DAO too much. One possible explanation that is consistent with the observations in pure C12DAO micelles38 is that the gelec contribution to gm may be too high for ionic surfactants and/or too low for zwitterionic surfactants. As indicated in ref 39, the most important free-energy contribution for the determination of the pKm is gelec. Consequently, the observed behavior can be understood qualitatively by examining the effects of RC12E8 and xC12DAOH on gelec. As shown in ref 39, the pKm decreases as gelec increases. In particular, gelec increases with the average valence per micellized surfactant molecule, zmic ) xC12DAOH(1 - RC12E8), as shown in eq 22 of ref 48. Therefore, as xC12DAOH increases, zmic and gelec increase, leading to a decrease in pKm. However, as RC12E8 approaches unity, zmic and gelec increase only slightly with xC12DAOH, and the pKm decreases only slightly as xC12DAOH increases. 5.1.2. Mixtures of C12DAO and Cationic Surfactant C12TAB. Interesting titration behavior was found in mixtures of C12DAO and C12TAB. In Figure 2, the predicted (lines) and the experimental (circles) ∆pKm values are compared as a function of xC12DAOH for mixtures with RCtotal ) 0.90 (Figure 2a), 0.76 12TAB (Figure 2b), 0.50 (Figure 2c), and 0.24 (Figure 2d). Error bars denote the experimental uncertainty as estimated using eq 22 of ref 39. Although the predicted ∆pKm curves capture the slope

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Figure 2. Comparison of the predicted (-) and the experimental (b) pKm values of C12DAO/C12TAB micelles as a function of the degree of protonation of C12DAO (xC12DAOH). The composition of C12TAB (RCtotal ) is (a) 0.90, (b) 0.76, (c) 0.50, and (d) 0.24. Error bars denote 12TAB the experimental uncertainty as estimated using eq 22 of ref 39.

of the experimental ∆pKm curves reasonably well, the predicted ∆pKm values are consistently too low by 1 to 1.5. The ∆pKm values are all negative because the presence of the cationic C12TAB surfactant increases gelec and favors the micelle incorporation of the zwitterionic C12DAO surfactant rather than that of the cationic C12DAOH+ surfactant. However, the experimental ∆pKm data are much less negative than the predicted values. In fact, ) 0.90, the experimental ∆pKm is approximately at RCtotal 12TAB -0.8 to -0.9, which is only slightly more negative than the value of -0.5 observed in the C12DAO/C12E8 mixtures at RCtotal 12E8 ) 0.90. Interestingly, this indicates that the incorporation of C12DAOH+ into a highly charged C12TAB-rich micelle is only slightly more “difficult’’ (that is, it produces a slightly greater increase in gm) than the incorporation of C12DAOH+ into a nearly neutral C12E8-rich micelle. This result is surprising because electrostatic interactions would suggest a relatively large repulsion between C12DAOH+ and the cationic C12TAB surfactant. Although we previously suggested that gelec may be the source of the observed discrepancy in the C12DAO/C12E8 mixture, the success of the MT theory for other predictions suggests that gelec is sufficiently accurate that it cannot be solely responsible for the discrepancy observed in the C12DAO/C12TAB mixture. Therefore, we previously conjectured that part of the observed discrepancy may arise from ion-specific interactions and, in particular, from the interactions of the bromide ions with the micelle.39 In fact, micelles of hexadecyltrimethylammonium are markedly more sensitive to bromide than to chloride, requiring only 0.1 M sodium bromide (NaBr) but 1.0 M NaCl to undergo a sphere-to-rod micelle shape transition.52-54 However, the pKm of C12DAO is lower in 0.06 M NaBr55 than in 0.05 M NaCl,50 suggesting that the bromide ions do not have any particular affinity for C12DAOH+ . Instead, the bromide ions could have (52) Dorshow, R. B.; Bunton, C. A.; Nicoli, D. F. J. Phys. Chem. 1983, 87, 1409-1416. (53) Ozeki, S.; Ikeda, S. Colloid Polym. Sci. 1984, 262, 1435-1536. (54) Imae, T.; Kamiya, R.; Ikeda, S. J. Colloid Interface Sci. 1985, 108, 215225. (55) Rathman, J. F.; Christian, S. D. Langmuir 1990, 6, 391-395.

an indirect effect on the pKm, for example, by favoring the formation of cylindrical micelles through their interactions with C12TAB. Such an effect is consistent with experimental observations that the pKm of C14DAO cylindrical micelles is higher than the pKm of spherical C12DAO micelles.23 Whether this effect is sufficient to explain the observed discrepancy between the predicted and experimental ∆pKm values in the C12DAO/C12TAB mixture is presently unclear. Therefore, further experimental studies are warranted on C12DAO/C12TAB and related mixtures. For example, it would be instructive to conduct titrations in mixtures of C12DAO and the chloride analogue of C12TAB. In addition, micellar titrations and lightscattering studies at varying concentrations of NaCl and NaBr may help to identify the specific roles of counterion type and micelle shape on pKm. 5.1.3. Mixtures of C12DAO and Anionic Surfactant SDS. Finally, the predicted (lines) and experimental (circles) ∆pKmvalues are compared in Figure 3 for mixtures of C12DAO and SDS. Because of the tendency of this system to become turbid, the only SDS total ) studied were 0.25 (Figure 3a) and 0.10 compositions (RSDS (Figure 3b), and the maximum xC12DAOH value is 0.25. Because xC12DAOH is relatively low, the error bars denoting the experimental uncertainty (estimated using eq 22 of ref 39) are larger than typical error bars in Figures 1 and 2. Because the anionic SDS surfactant interacts more favorably with C12DAOH+ than with C12DAO, the ∆pKm values are positive at nearly all of the conditions examined. Only after the amount of C12DAOH+ in the micelle exceeds the amount of SDS does ∆pKm become negative. Because of the limited range of experimental conditions examined, such negative ∆pKm values are observed only for total ) 0.1 when xC12DAOH is greater than 0.15 (predicted) or 0.20 RSDS (experimental). The predicted ∆pKm values agree well with the experimental total ) 0.1. Although the ∆pKm values are correctly values for RSDS total ) 0.25, the predicted predicted to be more positive for RSDS ∆pKm values are 0.5 to 1.0 units lower than the experimental ∆pKm values. Perhaps more surprising, the experimental ∆pKm

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Langmuir, Vol. 23, No. 11, 2007 5959

Figure 4. Comparison of the predicted (-) and experimental57 (b) degrees of sodium (Na) binding (βˆ Na) to C12DAO/SDS micelles as total ). Experimental uncertainties a function of SDS composition (RSDS were not reported by the authors.

Figure 3. Comparison of the predicted (-) and the experimental (b) pKm values of C12DAO/SDS micelles as a function of the degree of protonation of C12DAO (xC12DAOH). The composition of SDS total ) is (a) 0.25 and (b) 0.10. Error bars denote the experimental (RSDS uncertainty as estimated using eq 22 of ref 39.

values are nearly constant, except for the final data point at xC12DAOH ) 0.23, which is very close to the point where the solution became turbid. However, ∆pKm is predicted to be total f 1. These independent of xC12DAOH only in the limit where RSDS results indicate that the MT theory underestimates the synergy between SDS and C12DAOH+ . It may be possible to improve the ∆pKm predictions by investigating additional sources of synergy, such as the formation of SDS/C12DAOH+ ion pairs. Indeed, there is some evidence that SDS/C12DAO mixtures form a complex with a stoichiometric ratio of 1:4.41,56 Although the ∆pKm values predicted using the MT theory are underpredicted for mixtures of C12DAO with both C12TAB and SDS, the observed discrepancy provides useful information at the molecular level because it indicates that additional physical interactions may need to be accounted for, and it also provides useful guidance for future experimental studies. 5.2. Prediction of the Degree of Counterion Binding and the Solution pH. We next validate the MT theory using data from the literature for various mixtures of pH-sensitive surfactant C12DAO and anionic surfactant SDS. We first examine the degree of sodium (Na) binding to micelles (βˆ Na), as measured by Imae and Kakitani using Na-selective electrodes.57 The predicted total (circles) and experimental (line) βˆ Na values as a function of RSDS are compared in Figure 4. As we previously noted,48 the predicted degree of counterion binding is expected to be a lower bound for the experimentally measured β values because the model considers counterions to be bound only if they are adsorbed directly onto the surface of a micelle. In fact, the predictions reproduce the qualitative behavior of βˆ Na, and they are slightly lower than the experimental βˆ Na values. Because the pK1 of C12DAO is approximately 4.9,49-51 C12DAO is typically assumed to be only in its deprotonated, zwitterionic state at neutral pH or higher (pH >7). However, as we will show below, the presence of small amounts of protonated C12DAO (denoted as C12DAOH+) can have a significant effect (56) Weers, J. G.; Rathman, J. F.; Scheuing, D. R. Colloid Polym. Sci. 1990, 268, 832-846. (57) Imae, T.; Kakitani, M. Colloid Polym. Sci. 1996, 274, 1170-1175.

Figure 5. Comparison of the predicted (various lines) and the experimental57 (various symbols) solution pH values in 80 mM total solutions of C12DAO/SDS as a function of SDS composition (RSDS ) for solutions containing no salt (-, b) and 50 mM NaCl (---, 2). Experimental uncertainties were not reported by the authors.

on the solution pH when there is no added acid. In particular, C12DAOH+ is expected to interact very strongly with anionic surfactants. Therefore, we examine aqueous solutions containing C12DAO and SDS, which were studied by Imae and Kakitani.57 The pH values of 80 mM solutions of C12DAO/SDS and varying total compositions of SDS (RSDS ) were measured at 25 °C. The predicted (various lines) and the experimental (various symbols) total are compared in Figure solution pH values as a function of RSDS 5. The experiments were conducted either with no added salt (solid line and circles) or with 50 mM NaCl (dashed line and triangles). Note that the fraction of C12DAO monomers that are protonated under these experimental conditions (eq 5) is extremely small (less than 0.001 at a pH of 8 and less than 3 × 10-7 at a pH of 11.5). Because the vast majority of the pH-sensitive surfactant monomers are deprotonated at the conditions considered, one might assume that C12DAOH+ could be neglected. However, because the system contains no strong acid or base and no buffer, the solution pH is very sensitive to the degree of protonation of the pH-sensitive surfactant and to the presence of trace amounts of contaminants. Therefore, we consider the predicted and experimental pH values presented in Figure 5 to be in generally good agreement. We discuss the quantitative behavior in more detail below, along with one qualitative discrepancy. Although the solutions considered contain no added acid or total base, the solution pH varies considerably with RSDS and with salt concentration. The experimental points (circles and triangles) in total Figure 5 show that the solution pH increases steadily with RSDS total and eventually plateaus at 0.4 j RSDS j 0.6, whereas the total predicted solution pH plateaus over a much wider range of RSDS values (see below). The solution pH changes because the addition of SDS favors the formation and incorporation of C12DAOH+ into the micelles (see below).

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Goldsipe and Blankschtein Table 1. Compositions of Mixtures of C14DAO and C16Bet Studied by Zimmerman and Schnaare40 component compositions

Figure 6. Comparison of the predicted (-) and the experimental41 (b) solution pH values in 56 mM equimolar mixtures of C12DAO/ SDS as a function the relative amount of added HCl (X′H). Experimental uncertainties were not reported by the authors.

As the amount of C12DAOH+ in the micelles increases, the bulk concentration of H+ ions must decrease in order to satisfy the H+ and OH- mass balance (eq 9), which corresponds to an increase in the solution pH. In the case of no added salt, the experimental pH (the circles in Figure 5) is always higher than the predicted pH (the solid line in Figure 5), suggesting that the theory underestimates the total amount of protonated surfactant (C12DAOH+) in the micelles, primarily as a result of the electrostatic interactions. The electrostatic model (captured in gelec) may be partially responsible. For example, if the ionic strength of the solution is overestimated, then gelec will be too small, thereby reducing the driving force for the incorporation of C12DAOH+ into the micelles. This effect of increasing ionic strength on the solution pH is observed in both the predicted and the experimental data because the solutions containing 50 mM NaCl have lower pH values than those with no added salt. The one significant qualitative deviation between the predictotal tions and the experiments is the solution pH at RSDS ) 0. Specifically, the experimental solution pH decreases upon adding total ) 0 in Figure 5), and the salt (circles and triangles at RSDS predicted solution pH increases upon adding salt, corresponding to the region in Figure 5 where the dashed line lies above the solid line. The predicted results are consistent with a decrease in gelec for the slightly cationic micelles that exist when no SDS is present, which leads to a larger amount of C12DAOH+ in the micelles at a given pH. Therefore, the solution pH is predicted to rise with salt concentration when no SDS is added. However, the experimental pH actually decreases slightly upon addition of NaCl. Because the solution pH under these conditions can be very sensitive to the presence of trace contaminants, the experimental trends should be verified. If the observed discrepancy is real, then it may indicate that the electrostatic model requires greater accuracy to capture the observed trends quantitatively at extremely low ionic strengths. Smirnova et al.41 also measured the pH of equimolar solutions of C12DAO and SDS. The experimental conditions used were 56 total ) 0.5, no added salt, and a temperature mM surfactant with RSDS of 50 °C. The predicted and the experimentally measured solution pH values are compared in Figure 6. We have followed the authors’ convention and reported the results as a function of X′H total total ≡ Xtotal H /(XH + XC12DAO). The experimental conditions at X′H ) 0 (no added acid) are similar to those used by Imae and Kakitani total at RSDS ) 0.5 (Figure 5). However, the reported pH values differ markedly (11.5 in Figure 5 and 8.5 in Figure 6). As mentioned earlier, when there is no added acid or base, the solution pH can be very sensitive to the presence of trace contaminants. In particular, the presence of dissolved CO2 can significantly decrease the solution pH. Although Smirnova et al. report that they took special precautions to limit the absorption of CO2, the

mixture

C16Bet

C14DAO

C14DAOH+

pH

A B C D E F G

0.50 0.90 0.75 0.50 0.25 0.10 0.50

0.10 0.04 0.11 0.22 0.33 0.40 0.36

0.40 0.06 0.14 0.28 0.42 0.50 0.14

4.3 4.8 4.8 4.8 4.8 4.8 5.3

significant discrepancy between their results and those of Imae and Kakitani suggests that one or both groups may not have removed all of the contaminants from the solution. As a result, we consider the overprediction of the solution pH at X′H < 0.2 in Figure 6 to be within the experimental uncertainty under these conditions. At X′H > 0.2, the solution is much less sensitive to the presence of trace contaminants, and excellent agreement is observed between the predicted and the experimental pH values. 5.3. Prediction of cmc’s of a pH-Sensitive Surfactant Mixture. Zimmerman and Schnaare40 determined the cmc’s of various solutions of C14DAO and the zwitterionic surfactant hexadecyldimethyl betaine (C16Bet) at 30 °C. The counterion concentration was maintained at 0.02 M through the addition of NaCl. Although C16Bet is also a pH-sensitive surfactant, the authors argue that only the zwitterionic (deprotonated) form of C16Bet needs to be accounted for at the pH conditions considered. To confirm that this approximation was reasonable, preliminary cmc calculations were made in which C16Bet was modeled as a pH-sensitive surfactant with a pK1 of 1.83.40 These calculations indicated that the composition of protonated C16Bet in the micelle is negligible (less than 0.015) when the solution pH is above 4 (data not shown). Because the minimum pH value considered below is 4.3, the neglect of the protonated C16Bet surfactant constitutes a reasonable approximation. Treating C16Bet as a conventional surfactant mixed with C14DAO effectively results in a ternary surfactant mixture. Therefore, Zimmerman and Schnaare implemented the multicomponent extension of regular solution theory (RST) in the context of the pseudophase model58 to predict the cmc’s of the ternary surfactant mixtures using cmc data obtained from binary surfactant mixtures. Note that the binary surfactant mixtures include (1) protonated and deprotonated C14DAO, (2) protonated C14DAO and C16Bet, and (3) deprotonated C14DAO and C16Bet. Unfortunately, these authors appear to have implemented the Henderson-Hasselbalch equation (eq 5) incorrectly when calculating the compositions of the protonated and the deprotonated C14DAO monomers. In particular, the reported compositions of the protonated and the deprotonated C14DAO are inconsistent with the reported pH values in Table 1 of ref 40. For example, the degree of protonation of C14DAO is incorrectly reported to increase with pH. Therefore, we have assumed that the authors correctly reported the solution pH because the pH is the quantity that they measured experimentally, and we have recalculated the compositions of the protonated and the deprotonated C14DAO surfactant. In addition, we have recalculated the cmc’s obtained using RST and inputting the correct compositions. The cmc’s in these mixtures are determined largely by the composition of the C16Bet surfactant because it has the longest tail. Consequently, most of the corrected cmc’s differ from those reported in ref 40 by less than 1%, but two of the cmc’s were modified by 20-30%. With this in mind, we have (58) Holland, P. M.; Rubingh, D. N. J. Phys. Chem. 1983, 87, 1984-1990.

Micellization of pH-SensitiVe Surfactant Mixtures

Figure 7. Comparison of experimental40 (white bars), RST calculations (gray bars), and MT predictions (black bars) of the cmc’s of mixtures of C14DAO and C16Bet. The compositions of the mixtures are listed in Table 1. The error bars correspond to the reported experimental uncertainties.

utilized the corrected RST calculations of the cmc in the comparisons that follow. In Figure 7, cmc’s obtained experimentally (white bars), using RST (gray bars), and using our MT theory (black bars) are compared for the ternary surfactant mixtures examined. The pH and the corresponding compositions of these mixtures (denoted as mixtures A-G) are listed in Table 1. Both the RST and the MT predictions compare favorably to the experimental cmc’s. Even though the empirical RST approach makes use of some experimental cmc data, the MT predictions are better than those of the RST for mixtures A and D. In addition, the cmc’s predicted by both the MT theory and the RST are within the experimental uncertainty for mixtures C and F (error bars in Figure 7). Mixtures B-F correspond to decreasing amounts of C16Bet at a pH of 4.8. Because C16Bet has a much lower cmc than either form of the C14DAO surfactant, it is not surprising that the cmc’s predicted by the MT theory and by the RST tend to increase with decreasing C16Bet composition (that is, as we proceed from left to right from mixture B to mixture F in Figure 7). However, the experimental cmc’s do not show such a trend, and both the RST and the MT theory show significant deviations from the experimental cmc’s for mixtures B and E. In fact, both theories would agree well if the experimental cmc’s for mixtures B and E were swapped. Therefore, we suggest that these experiments be repeated to determine whether the observed experimental cmc trend is real.

6. Conclusions A molecular-thermodynamic (MT) theory was developed and validated for mixtures of pH-sensitive and conventional surfactants. We first examined experimental pKm data that we obtained for mixtures of C12DAO with C12E8, C12TAB, and SDS. Although the C12DAO/C12E8 mixtures were modeled reasonably well using the MT theory, both C12TAB and SDS exhibited more favorable interactions with the protonated C12DAOH+ surfactant than were predicted. The results for C12TAB are particularly surprising because they indicate an unusually strong interaction between two positively charged surfactants (C12DAOH+ and C12TAB). The observed discrepancy suggests that the current model may somewhat overestimate the electrostatic repulsions between the charged surfactant heads in a micelle. In addition, we suggest that counterions may play a particularly important role in the titration behavior of C12DAO/C12TAB mixtures, and we propose additional experimental studies that could elucidate this behavior. Other solution properties were also examined. In particular, the theory accurately predicted the pH of C12DAO/SDS mixtures

Langmuir, Vol. 23, No. 11, 2007 5961

upon the addition of HCl but only qualitatively captured the behavior of the solution pH without added acid. The MT theory explicitly accounts for counterion binding, and predictions of sodium binding in C12DAO/SDS mixtures compared favorably with the experimental values. Finally, the MT theory was used to predict the cmc’s of mixtures of C14DAO and C16Bet, which behave effectively like ternary surfactant mixtures under the solution conditions considered. In most cases, good agreement was found between the predicted and the experimental cmc’s. In addition, the MT theory cmc predictions were comparable to and sometimes better than the empirical RST cmc predictions. However, the completely predictive MT theory offers significant advantages over the empirical RST, which requires a minimum of six cmc measurements to predict cmc’s of ternary surfactant mixtures. The MT theory often provides quantitatively accurate predictions without any experimental effort. In addition, when significant discrepancies are observed between the MT theory predictions and the experimental data, the theory can be useful in guiding further experimental studies as well as in suggesting directions for future modeling efforts on the molecular level. Therefore, the MT theory may prove useful for understanding and designing surfactant mixtures of both fundamental and practical interest. In particular, the theory may assist in formulating commercial detergents, which typically utilize pHsensitive surfactants as secondary components. Furthermore, the relationship between the pKm and the type and amount of conventional surfactant may lead to new or improved pHresponsive systems. Acknowledgment. A.G. gratefully acknowledges the Fannie and John Hertz Foundation for financial support.

Appendix: Diffuse-Layer Contribution to Mass Balances at High Micelle Concentrations and at Low Ionic Strengths Several of the experimental conditions considered in this article are at a micelle concentration that is much higher than the ionic strength of the solution. Unfortunately, the model that we developed in ref 48 for the diffuse-layer contribution to the mass balances is inappropriate under such conditions because it can lead to negative concentrations of the co-ions in the solution. Although the model for the diffuse-layer contribution developed in ref 59 is somewhat less rigorous, it does not lead to negative co-ion concentrations (because this model essentially assumes that the distribution of the co-ions is unaffected by the presence of the micelles). Therefore, for the calculations carried out in this article, we have used the diffuse-layer model developed in ref 59, which is summarized here for completeness. In practice, diffuse-layer behavior lies somewhere between the extremes represented by these two models. Specifically, the model in ref 59 overestimates the enhancement of the counterions in the diffuse layer, and the model in ref 48 overestimates the depletion of the co-ions in the diffuse layer. Until a better model of the diffuse layer is developed, the best choice of diffuse-layer models will largely depend on the experimental conditions: (1) at low micelle concentrations, either model is appropriate because they both lead to negligible enhancements or depletions; (2) at high ionic strengths and high micelle concentrations, the model developed in ref 48 may be more appropriate; and (3) at low ionic strengths and high micelle concentrations, the model developed in ref 59 (eq A1) may be more appropriate. (59) Srinivasan, V.; Blankschtein, D. Langmuir 2003, 19, 9946-9961.

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The model for the diffuse-layer enhancement of the counterions presented in ref 59 is written in our notation as follows:

Edl,j ) -zjX1j

[

]

zmicXmic 2µc

(A1)

where X1j and Xmic are the bulk mole fractions of ions of species j and of micelles, respectively, zmic is the average valence per micellized surfactant molecule, and µc ) 1/2∑jz2j X1j is the contribution of the counterions to the solution ionic strength in mole fraction units. In this model, note that j refers only to the counterion species. For every other species k, the model assumes that their enhancement or depletion in the diffuse layer is negligible (that is, Edl,k ) 0). Also, in contrast to eq A8 in ref 48, the denominator of eq A1 includes only counterion mole fractions.

Perhaps in the future a more rigorous model for the diffuse layer can be developed. In particular, an improved model must avoid the linearization of the Boltzmann distribution of ion concentrations that is inherent in the development of both eq A8 in ref 48 and of eq A1 in this Appendix. If such a model can be developed, it may allow more accurate predictions of the activities of ions in micellar solutions, particularly at high micelle concentrations. Supporting Information Available: Complete set of titration data for mixtures of C12DAO/SDS, estimated values of the molecular characteristics of the surfactants considered, and additional predictions (micelle aggregation numbers, monomer concentrations, and counterion binding) for C12DAO/C12TAB, C12DAO/C12E8, and C12DAO/SDS mixtures. This material is available free of charge via the Internet at http://pubs.acs.org. LA063082Z