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Molecular-Thermodynamic Prediction of Critical Micelle Concentrations of Commercial Surfactants Isaac Reif, Michael Mulqueen, and Daniel Blankschtein* Department of Chemical Engineering, Center for Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 Received April 16, 2001. In Final Form: July 5, 2001 Commercial surfactants have widespread use in many practical applications, including detergents, cosmetics, pharmaceuticals, enhanced oil recovery, and surfactant-based separation processes. However, despite their practical relevance, the solution behavior of commercial surfactants is not well understood at a fundamental molecular level. With this in mind, a new computational approach, based on a recently developed molecular-thermodynamic theory of mixed micellization, was developed to predict the critical micelle concentrations of commercial surfactants containing any number of surfactant components. The new computational approach was then implemented, in the context of two user-friendly computer programs, PREDICT and MIX2, to molecularly predict the critical micelle concentrations of several commercial surfactants of known chemical composition, including their binary mixtures. The commercial surfactants examined include the cationic BTC-8358, the nonionics GENAPOL UD-079, GENAPOL UD-110, GENAPOL 26-L-98, and GENAPOL 26-L-50, and the anionic STEOL CS-330. The accuracy of the predicted critical micelle concentrations was found to be comparable to that attained in the case of single (pure) surfactants and their binary mixtures, thus demonstrating the practical utility of the computer-assisted molecularthermodynamic modeling as a predictive tool in commercial surfactant characterization.
1. Introduction Commercial surfactants are commonly used in almost all practical applications of surfactants. They consist of multicomponent surfactant mixtures which usually result from a synthesis procedure which produces a homologous series of surfactants whose purification process may be either too difficult or expensive to undertake. The widespread use of commercial surfactants for industrial purposes has stimulated research on polydisperse surfactant mixtures, and as a result, a number of papers have been published dealing mainly with experimental measurements of the solution properties of these complex systems, including the interaction, stability, and microenvironmental properties of mixed micelles of Triton X100 and n-alkyltrimethylammonium bromides, and the partitioning of nonionic polydisperse ethoxylated octyl phenol and anionic alkyl benzene sulfonate surfactant mixtures between oil/microemulsion/water phases.1-9 However, despite these efforts, the solution behavior of commercial surfactants is not well understood at a fundamental molecular level. Indeed, the selection of a suitable commercial surfactant for a specific application * To whom correspondence should be addressed. (1) Carnero Ruiz, C.; Aguiar, J. Langmuir 2000, 16, 7946. (2) Holland, P. M. In Mixed Surfactant Systems; Holland, P. M., Rubingh, D. N., Eds.; ACS Symposium Series 501; American Chemical Society: Washington, DC, 1992; p 114. (3) Warr, G. G.; Grieser, F.; Healy, T. W. J. Phys. Chem. 1983, 87, 1220. (4) Scamehorn, J. F.; Schechter, R. S.; Wade, W. H. J. Dispersion Sci. Technol. 1982, 3, 261. (5) Osborne-Lee, I. W.; Schechter, R. S.; Wade, W. H.; Barakat, Y. J. J. Colloid Interface Sci. 1985, 108, 60. (6) Osborne-Lee, I. W.; Schechter, R. S. In Phenomena in Mixed Surfactant Systems; Scamehorn, J. F., Ed.; ACS Symposium Series 311; American Chemical Society: Washington, DC, 1986; p 30. (7) Graciaa, A.; Lachaise, J.; Bourrel, M.; Osborne-Lee, I.; Schechter, R. S.; Wade, W. H. Soc. Pet. Eng. J. 1987, 2, 305. (8) Yoesting, O. E.; Scamehorn, J. F. Colloid Polym. Sci. 1986, 264, 148. (9) Graciaa, A.; Ben Ghoulam, M.; Mendiboure, B.; Lachaise, J.; Schechter, R. S. In Mixed Surfactant Systems; Holland, P. M., Rubingh, D. N., Eds.; ACS Symposium Series 501; American Chemical Society: Washington, DC, 1992; p 165.
is typically made based only on experience, empirical evidence, or time-consuming trial-and-error research. To the best of our knowledge, no predictive, molecularly based theory of surfactant solution behavior has been developed to model commercial surfactants. Clearly, the availability of such a molecular-thermodynamic theory to predict the solution behavior of commercial surfactants would help clarify how their solution properties, including the critical micelle concentration (cmcmix), micelle shape, and micelle size, vary with the molecular structure of each of the surfactant components composing the commercial surfactants as well as with the solution conditions, including temperature, total surfactant concentration, and salt type and concentration. In particular, the prediction of critical micelle concentrations of commercial surfactants is extremely valuable, since this property not only represents a fundamental indicator of surfactant micelleforming capacity but can also be correlated with applied surfactant performance characteristics, such as solubilization, dispersion, emulsification, and skin irritation.10-12 Binary surfactant mixtures have traditionally been modeled using the pseudophase separation approach, in which the micelles are treated as a separate, infinite phase in equilibrium with the monomer phase.13-16 When mixed micelle formation is ideal, the critical micelle concentration of the surfactant mixture can be determined solely from knowledge of the critical micelle concentrations of each single surfactant component composing the mixture. If mixed micelle formation is nonideal, then additional information is needed on the interaction between the two surfactant components composing the mixed micelle. (10) Jost, F.; Leiter, H.; Schwuger, M. J. Colloid Polym. Sci. 1988, 226, 554. (11) Garcia, M. T.; Ribosa, I.; Sanches Leal, J.; Comelles, F. J. Am. Oil Chem. Soc. 1992, 69, 25. (12) Rhein, L. D.; Simion, F. A.; Hill, R. L.; Cagan, R. H.; Mattai, J.; Maibach, H. I. Dermatologica 1990, 180, 18. (13) Shinoda, K. J. Phys. Chem. 1954, 58, 541. (14) Lange, H.; Beck, K. H. Kolloid Z. Z. Polym. 1973, 251, 424. (15) Clint, J. J. Chem. Soc. 1975, 71, 1327. (16) Rubingh, D. N. In Solution Chemistry of Surfactants; Mittal, K., Ed.; Plenum: New York, 1979; Vol. 1, p 337.
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Nonideal mixed micelle formation is most commonly modeled using regular solution theory.16 For a binary mixture of surfactants A and B, in the context of regular solution theory, the nonideality is represented by a term of the form βABRA(1 - RA), where βAB, the pairwise interaction parameter, is considered to be a constant independent of RA, the micelle composition of surfactant A, with RB ) (1 - RA). A negative value of βAB indicates synergism in mixed micelle formation, a positive value indicates antagonism, and if βAB ) 0, then mixed micelle formation is ideal.17 The larger the absolute value of βAB, the stronger the mixing nonideality. The regular solution theory approach has been extended to multicomponent surfactant mixtures,18,19 by decomposing all the surfactant interactions into additive interactions between every possible surfactant pair. In this way, cmcmix of surfactant mixtures, including commercial surfactants, can be calculated once the critical micelle concentration of each single surfactant component and the values of the various interaction parameters, βAB, corresponding to each surfactant pair are known.2,18 However, a major disadvantage of this approach is that it is not predictive, since the critical micelle concentration of each single surfactant component composing the mixture needs to be measured, and the various βAB interaction parameters are typically obtained by fitting to experimental data, specifically to the measured critical micelle concentrations of binary surfactant mixtures at various solution compositions.17,18 As will be shown in the next section, a rather large number of experimental critical micelle concentration measurements are then required to obtain all the necessary βAB values corresponding to the various surfactant pairs composing the surfactant mixture. This is particularly problematic in the case of commercial surfactants, since they generally contain a large number of surfactant components. For example, the number of surfactant components in the commercial surfactants considered in this paper ranges from 3 to 45 (see section 3). The challenges that need to be addressed when considering the quantitative prediction of the solution properties of commercial surfactants are considerable. Nevertheless, the significant progress that has been made in recent years in the theoretical modeling of the solution behavior of pure and mixed surfactant solutions provides a solid foundation for the understanding of the solution behavior of commercial surfactants and for the development of theoretical approaches to predict their micellar solution properties.20-32 The useful, practical value of such predictive molecular-thermodynamic treatments, which require no experimental measurements, cannot be overemphasized. (17) Holland, P. M. In Mixed Surfactant Systems; Holland, P. M., Rubingh, D. N., Eds.; ACS Symposium Series 501; American Chemical Society: Washington, DC, 1992; p 31. (18) Holland, P. M.; Rubingh, D. N. J. Phys. Chem. 1983, 87, 1984. (19) Graciaa, A.; Ben Ghoulam, M.; Marion, G.; Lachaise, J. J. Phys. Chem. 1989, 93, 4167. (20) Nagarajan, R. Langmuir 1985, 1, 331. (21) Nagarajan, R. Adv. Colloid Interface Sci. 1986, 26, 205. (22) Nagarajan, R.; Ruckenstein, E. Langmuir 1991, 7, 2934. (23) Nagarajan, R. In Mixed Surfactant Systems; Holland, P. M., Rubingh, D. N., Eds.; ACS Symposium Series 501; American Chemical Society: Washington, DC, 1992; p 54. (24) Eriksson, J. C.; Ljunggren, S. Langmuir 1990, 6, 895. (25) Bergstrom, M. J. Colloid Interface Sci. 1996, 181, 208. (26) Bergstrom, M.; Eriksson, J. C. Langmuir 2000, 16, 7173. (27) Bergstrom, M. Langmuir 2001, 17, 993. (28) Puvvada, S.; Blankschtein, D. J. Chem. Phys. 1990, 92, 3710. (29) Puvvada, S.; Blankschtein, D. J. Phys. Chem. 1992, 96, 5567. (30) Puvvada, S.; Blankschtein, D. J. Phys. Chem. 1992, 96, 5579. (31) Shiloach, A.; Blankschtein, D. Langmuir 1997, 13, 3968. (32) Shiloach, A.; Blankschtein, D. Langmuir 1998, 14, 1618.
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The main goal of this paper is to present a new computational approach for the prediction of cmcmix of commercial surfactants. A recently developed molecularthermodynamic theory of micellization,28-32 capable of predicting a broad spectrum of solution properties of single surfactants and their binary mixtures, is applied to commercial surfactants by treating them as a polydisperse mixture of surfactants. A complete theory for multicomponent surfactant mixtures would predict not only cmcmix but also other important surfactant mixture solution properties such as micelle shape and size. As a first step in this direction, in this paper we focus on the prediction of cmcmix. To accomplish this goal, we make the approximation that the interactions between the various surfactant components composing the mixed micelle can be expressed as a sum of pairwise interactions between all possible surfactant pairs. It has recently been shown that in this case, both the molecular-thermodynamic theory and the pseudophase separation theory (within the regular solution approximation) yield the same thermodynamic relationships if the micelle composition is considered to be equal to its optimal value and the mixed micelle aggregation number is sufficiently large.29,33,34 In particular, the molecular-thermodynamic approach yields the same equation for cmcmix as that obtained in the context of the pseudophase separation approach. Specifically, this equation relates cmcmix to the critical micelle concentrations of each single surfactant component composing the mixture and to the βAB interaction parameters corresponding to each surfactant pair composing the mixed micelle.13-19 However, in contrast to the pseudophase separation approach, the molecular-thermodynamic approach presented here enables the molecular prediction of the critical micelle concentrations of each single surfactant component as well as of the various βAB parameters characterizing each surfactant-pair interaction. In this way, using the molecular-thermodynamic approach, it becomes possible to predict cmcmix for a commercial surfactant without the need to perform a single experimental measurement. The new computational approach described above was implemented, in the context of two user-friendly computer programs, PREDICT and MIX2, to molecularly predict the critical micelle concentrations of several commercial surfactants of known chemical composition. The commercial surfactants examined include the cationic BTC8358, the nonionics GENAPOL UD-079, GENAPOL UD110, GENAPOL 26-L-98, and GENAPOL 26-L-50, and the anionic STEOL CS-330. The same approach can be used to molecularly predict the critical micelle concentrations of mixtures of commercial surfactants. This fact is illustrated here by predicting cmcmix for the binary mixture of the commercial cationic surfactant BTC-8358 and the commercial nonionic surfactant GENAPOL 26-L-98 over the entire range of solution compositions. The accuracy of the predicted critical micelle concentrations was found to be comparable to that attained in the case of single (pure) surfactants and their binary mixtures, thus demonstrating the practical utility of the computer-assisted molecular-thermodynamic modeling as a predictive tool in commercial surfactant characterization. The remainder of the paper is organized as follows. In section 2, we present the basic theoretical framework by summarizing the important theoretical concepts used in the molecular-thermodynamic approach, followed by a description of the two user-friendly computer programs, (33) Shiloach, A.; Blankschtein, D. Langmuir 1998, 14, 4105. (34) Reif, I.; Somasundaran, P. Langmuir 1999, 15, 3411.
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PREDICT and MIX2, which were used to make the required molecular predictions of (i) the critical micelle concentrations of each single surfactant component involved, (ii) the various βAB interaction parameters, and (iii) cmcmix of the various commercial surfactants examined. In section 3, useful information about the cationic, nonionic, and anionic commercial surfactants examined in this paper will be given, including their chemical structure and composition. A detailed description of the experimental method employed to measure the critical micelle concentrations of the various commercial surfactants will also be given. In section 4, a comparison between the theoretically predicted and the experimentally measured cmcmix of these commercial surfactants, including the binary mixture of BTC-8358 and GENAPOL 26-L-98, will be presented. Finally, in section 5, we present concluding remarks.
2.1. Molecular-Thermodynamic Theory of Micellization. In a molecular-thermodynamic treatment of mixed micellization, the mixed micelles are characterized by a size and composition distribution. Each mixed micelle of a given size (aggregation number) and composition is viewed as a distinct chemical species, and the entire surfactant solution is therefore described as a multicomponent solution.28-30 The micelle size and composition distribution is then derived through the use of a multiple chemical equilibrium condition between surfactant monomers in the bulk solution and surfactant molecules in the mixed micelles. For example, in a binary surfactant mixture of surfactants A and B at thermodynamic equilibrium, the chemical potential of a mixed micelle of aggregation number n and composition RA, µnRA, can be related to the chemical potentials, µ1A and µ1B, of the nRA surfactant A monomers and n(1 - RA) surfactant B monomers composing the mixed micelle as follows:27-31
(1)
By use of appropriate expressions for the chemical potentials in eq 1, the following expression is obtained for the size and composition distribution of mixed micelles:29,30,32
XnRA ) (1/e)(X1A)nRA(X1B)n(1-RA)e-n[gmic(RA)/kT-1]
XnRA ) (1/e)(X1)ne-n[gm/kT] ) (1/e)(X1/egm/kT)n
(3)
where
2. Theoretical Framework
µnRA ) nRA µ1A + n(1 - RA)µ1B
micelle distribution is then obtained by maximizing the mixed micelle distribution given by eq 2 with respect to the micelle core minor radius and the micelle composition.29,30,32 To determine X1A and X1B, two mass balance equations related to the total number of surfactant A and B molecules present in the solution need to be solved.29,30,32 Once the mixed micelle size and composition distribution given in eq 2 is known, all the equilibrium properties of the mixed micellar solution associated with this distribution can be predicted molecularly. Of particular relevance to this paper are the molecular prediction of cmcmix and the βAB interaction parameters.29,30,32 To obtain cmcmix for the binary surfactant mixture, eq 2 can be rewritten as follows:32
(2)
where XnRA is the mole fraction of mixed micelles of aggregation number n and composition RA, X1A and X1B are the mole fractions of surfactant monomers of type A and B, respectively, and gmic(RA) is the free energy of mixed micellization. More specifically, gmic(RA) is the free-energy change per surfactant monomer required to form an isolated mixed micelle in the infinitely dilute state from the singly dispersed nRA surfactant A monomers and n(1 - RA) surfactant B monomers in the bulk solution. To actually calculate the mixed micelle size and composition distribution given in eq 2, the free energy of mixed micellization, gmic(RA), and the monomer mole fractions, X1A and X1B, must be known. A molecular model of mixed micellization, including a detailed micellization thought process, is used to calculate the free energy of mixed micellization from the known chemical structures of surfactants A and B and the solution conditions, including the temperature, the total surfactant concentration and composition, and the type and concentration of any additives (for example, salts).29,30,32 For every shape (sphere, cylinder, disk or bilayer), the optimal mixed
X1 ) X1A + X1B
(4)
and
gm/kT ) gmic(R/A)/kT - 1 [R/A lnR1A + (1 - R/A) ln(1 - R1A)] (5) is a modified free energy of mixed micellization. In eq 5, R/A is the optimal micelle composition and R1A is the composition of surfactant A monomers in the bulk solution. Since the mixed micelle aggregation number n is typically large and appears as an exponent in eq 3, the value of X1 must be very close to
X1 ) egm/kT
(6)
since otherwise, XnRA will be either too large (.1) or too small (f0). It then follows that X1 ) X1A + X1B ) exp[gm/kT] in eq 6 corresponds approximately to the binary surfactant mixture critical micelle concentration. Note that the free energy of mixed micellization, gmic, is calculated molecularly as the sum of several free-energy contributions corresponding to the process of forming a mixed micelle out of free surfactant monomers in solution. The following micellization steps are accounted for:28-31 (i) the transfer of the surfactant tails of each surfactant component from the aqueous solution to the core of the mixed micelle, (ii) the formation of the interface between the micellar core and the aqueous solution, (iii) the packing of the surfactant tails in the micellar core, (iv) the steric interactions between the surfactant heads at the micellar core-aqueous solution interface, (v) the electrostatic interactions (in the case of ionic and zwitterionic surfactants) between the surfactant heads at the micellar coreaqueous solution interface, and (vi) the entropy of mixing the two surfactant components in the mixed micelle. To molecularly predict βAB, it is convenient to define the excess free energy of mixed micellization, gexcess mic , as follows:32 excess gmic ) gideal mic + gmic
(7)
where gideal mic , the free energy associated with forming a mixed micelle in which the various surfactant components mix ideally, is given by / A / B gideal mic ) RA gmic + (1 - RA)gmic +
kT[R/A ln R/A + (1 - R/A) ln(1 - R/A)] (8)
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In eq 8, gAmic and gBmic are the free energies of micellization of the single surfactants A and B, respectively, and the last term accounts for the entropy of mixing the two surfactant components within the mixed micelle. Note that gAmic and gBmic are related to the critical micelle concentrations of the single surfactants A, cmcA, and B, cmcB, by the following expressions:28-30
cmcA ) exp[(gAmic - 1)/kT]
(9)
cmcB ) exp[(gBmic - 1)/kT]
(10)
and
In this way, the excess free energy of mixed micellization can be determined as a function of micelle composition in terms of the difference between the calculated values of gmic and gideal mic (see eq 7). Although the resulting actual on micelle composition can be rather dependence of gexcess mic complex, a simple approximation can be used which works well in many cases. This approximation is given by32
) βABR/A(1 - R/A) gexcess mic
(11)
where the interaction parameter βAB is considered to be a constant, independent of the mixed micelle composition. Note that eq 11 should be viewed as providing a formal definition of the interaction parameter βAB. The value of βAB can then be estimated by performing a least-squares fit between the calculated values of the excess free energy of mixed micellization and the expression given in eq 11.32 More generally, eqs 7 and 8, along with the molecularly predicted gmic, may be utilized to evaluate the various contributions to the excess free energy of mixed micellization, gexcess mic , including its dependence on micelle composition, beyond the simple quadratic approximation given provides a quantitative measure of in eq 11. Since gexcess mic the deviations from ideal mixed micelle formation, its detailed molecular evaluation may be used to determine the magnitudes, and hence the relative importance, of the various sources of surfactant synergism (packing, interfacial, steric, electrostatic) contributing to the nonidealities in mixed micelle formation. However, since our main goal here is to develop a useful computational approach to predict cmcmix of commercial surfactants, in expression given in the context of the approximate gexcess mic eq 11, a detailed discussion of synergism is beyond the scope of the paper. The mixed micelle distribution is sharply peaked in composition at its optimal value, R/A, when the micelle aggregation number is sufficiently large, as is typically the case.32 It can then be shown that at this optimal mixed micelle composition, R/A, the excess free energy of mixed micellization given by eq 11 results in the following relationships at the mixture critical micelle concentration:30,33
R/A fAcmcA ) R1Acmcmix
(12)
R/B fBcmcB ) R1Bcmcmix
(13)
and
where fA and fB are the activity coefficients of surfactants A and B in the mixed micelle, respectively, and are given by
fA ) exp[ βAB(1 - R/A)2/kT ]
(14)
fB ) exp[ βAB(R/A)2/kT ]
(15)
Note that as emphasized above, eqs 12-15, obtained here in the context of the molecular-thermodynamic approach, are identical to those obtained using the pseudophase separation approach within the regular solution approximation.13-19 Eliminating the optimal micelle compositions, R/A and R/B, from eqs 12 and 13, using the fact that (R/A + R/B) ) 1, yields the following well-known expression for cmcmix of a binary surfactant mixture:
R1A R1B 1 ) + cmcmix fAcmcA fBcmcB
(16)
Note that close to the mixture critical micelle concentration, cmcmix, R1A ) RsA, and R1B ) RsB, the solution compositions of surfactants A and B, respectively, since the amount of surfactant present in the form of mixed micelles is rather small. Equations 12-16 can be generalized for surfactant mixtures containing any number of surfactant components, as long as the excess free energy of mixed micellization is expressed as a linear combination of pairwiseinteraction terms of the form given in eq 11. In particular, for a surfactant mixture containing m components, the following expression is obtained for cmcmix:13-19 m
1
)
cmcmix
∑
Rsp
p)1 fpcmcp
(17)
In eq 17, cmcp is the critical micelle concentration of surfactant component p, Rsp is the solution composition of surfactant component p, and fp is the activity coefficient of surfactant component p given by
ln fp )
m
m q-1
q*p
q)1 k)1 p*q*k
∑ βpq(R/q)2 + ∑ ∑ (βpq + βpk - βqk)R/qR/k q)1
(18)
Equation 18 indicates that in the pairwise-interaction approximation, the activity coefficient of each surfactant component in the mixed micelle, fp, depends on the value of the interaction parameters corresponding to all the possible surfactant pairs, βpq, composing the surfactant mixture. Equation 17 has proven to be very useful in describing the micellization behavior of a wide variety of surfactant mixtures, including those that contain ionic surfactants.17,18 However, to predict cmcmix for a commercial surfactant using eq 17, it is necessary to know the critical micelle concentration, cmcp, the solution composition, Rsp, and the activity coefficient, fp, of each one of the m surfactant components present in the mixture. Equation 18 shows that the determination of the activity coefficient of each surfactant component, fp, requires knowledge of the interaction parameters, βpq, corresponding to all the possible surfactant pairs, pq, composing the surfactant mixture. The various βpq interaction parameters have usually been determined empirically by measuring cmcp, cmcq, and cmcmix at at least one solution composition for each binary surfactant p and surfactant q mixture and then using these data to iteratively solve for the micelle composition, R/p, in the following equation which has been derived using eqs 12-15:17-33
Prediction of Critical Micelle Concentration
(
(1 - R/p)-2 ln
)
Rspcmcmix R/pcmcp
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)
(
(R/p)-2 ln
)
(1 - Rsp)cmcmix (1 - R/p)cmcq
) βpq (19)
Once R/p is known, the value of βpq is obtained directly from eq 19. However, the accurate determination of each interaction parameter requires the measurement of several critical micelle concentrations at different solution compositions for every binary surfactant mixture. Since several surface tension measurements are involved in the determination of each required critical micelle concentration, it is clear that a considerable amount of experimentation is necessary to obtain the value of cmcmix of a multicomponent surfactant mixture. This is particularly problematic for commercial surfactants, since they generally contain a large number of surfactant components and hence a large number of surfactant pairs. The molecular-thermodynamic approach presented above provides a very convenient way to predict all the information required for the prediction of cmcmix of commercial surfactants without the need to perform a single experiment. Specifically, cmcp of each individual surfactant component p as well as the interaction parameters, βpq, for all the possible surfactant pairs, pq, in the mixture can be predicted molecularly. The practical benefit of such a predictive, molecularly-based approach cannot be overemphasized. The validity and range of applicability of the computational approach presented above are determined primarily by the validity and range of applicability of the utilized here simple, approximate expression for gexcess mic (see eq 11, also referred to as the regular solution approximation).16-19 As stressed earlier, a complete mobeyond eq 11 is possible, lecular evaluation of gexcess mic including an explicit treatment of counterions in the case of ionic surfactants, and would not only enable more accurate predictions of cmcmix but also allow the prediction of other important micellization characteristics of commercial surfactants, including micelle shape and size. However, as will be shown in section 4, the simpler computational approach presented here, which makes use on of eq 11 to approximate the dependence of gexcess mic micelle composition through the interaction parameter βAB, does a reasonable job at molecularly predicting cmcmix of the commercial surfactants examined. Section 2.2 provides a brief description of the two userfriendly computer programs, PREDICT and MIX2, used in the predictions presented in section 4. For a more detailed discussion of computer programs PREDICT and MIX2, the interested reader is referred to refs 35 and 36. Note, however, that the interested reader can utilize the information available in the literature (see refs 28-33) to make predictions similar to those made in this paper without using these computer programs. 2.2. Computer Programs PREDICT and MIX2. Two user-friendly computer programs have been developed to facilitate the implementation of the molecular-thermodynamic theory of micellization discussed above. These include computer program PREDICT for single surfactants and computer program MIX2 for binary surfactant mixtures. (35) Zoeller, N. J.; Blankschtein, D. Ind. Eng. Chem. Res. 1995, 34, 4150. (36) Zoeller, N. J.; Shiloach, A.; Blankschtein, D. CHEMTECH 1996, 26, 24.
Figure 1. Required surfactant molecular parameters for programs PREDICT and MIX2 (see Table 1). Table 1. Required Surfactant Molecular Parameters for Programs PREDICT and MIX2a nonionic
ionic
zwitterionic
nc ah
nc ah dcharge z
nc ah dsep
a n is the number of carbon atoms in the surfactant tail, a is c h the cross-sectional area of the surfactant head, dcharge is the distance from the location of the charge in the head of an ionic surfactant to the beginning of the tail, z is the valence of an ionic surfactant, and dsep is the distance between the two charges in the head of a zwitterionic surfactant.
To operate programs PREDICT and MIX2, the user must input information about the molecular structures of the polar head and the nonpolar tail of the various surfactants under consideration as well as about the solution conditions.35,36 All the required structural information can be deduced from the known chemical structures of the surfactant polar head and nonpolar tail. To describe the surfactant tail molecular structure of a linear hydrocarbon-based surfactant, the total number of carbon atoms in the tail, nc (current capabilities allow values between 4 and 18), is required. To describe the surfactant head molecular structure, the effective cross-sectional area of the head, ah, is needed. Furthermore, if the head is ionic, two additional inputs are required: the distance from the position of the charge in the head to the beginning of the tail, denoted as dcharge, and the charge or valence, z, of the head. Finally, if the head is zwitterionic (or dipolar), the distance between the two charges composing the dipole in the head, denoted as dsep, must be specified. For a summary of the surfactant molecular parameters which need to be input to programs PREDICT and MIX2 in the case of nonionic, ionic, and zwitterionic (not examined in this paper) surfactants, see Figure 1 and Table 1. The solution conditions are described by the temperature, the total surfactant concentration, the solution composition (in the case of surfactant mixtures), and the type and concentration of any added salt. Figures 2 and 3 summarize the inputs required and the predictions which can be made with the use of programs PREDICT and MIX2, respectively. As an illustration of the predictive capabilities of program PREDICT and of particular relevance to this paper, Figure 4 shows a comparison between the predicted critical micelle concentrations and the experimental critical micelle concen-
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Figure 2. Flow diagram of program PREDICT. The inputs required include information about the surfactant molecular structure and the solution conditions. The program molecularly predicts a variety of micellar solution properties of single surfactants. Figure 4. Comparison between predicted and experimentally measured critical micelle concentrations of almost 100 single anionic (9), cationic (b), zwitterionic (2), and nonionic ([) surfactants.
Figure 3. Flow diagram of program MIX2. The surfactant molecular structures and the solution conditions are required as inputs. The program predicts a variety of micellar solution properties of binary surfactant mixtures.
trations of almost a hundred single surfactants taken from the well-known book by Rosen.37 Since reported experimental values of critical micelle concentrations of a given surfactant, deduced using different experimental techniques, can differ by as much as a factor of 2, the fact that (37) Rosen, M. J. Surfactants and Interfacial Phenomena; Wiley: New York, 1989; Chapter 3.
the majority of the predicted critical micelle concentrations are within a factor of 5 from the experimental values indicates that reasonable agreement is obtained between the cmc’s predicted using program PREDICT and the measured cmc values. The use of program PREDICT allows the molecular prediction of the cmc for each one of the m surfactant components composing the commercial surfactant. Program MIX2 can then be used to molecularly predict the various βpq interaction parameters corresponding to all the possible surfactant pairs, pq, in the surfactant mixture. With this information in hand, the activity coefficient, fp, of each surfactant component can be predicted using eqs 12, 13, and 18.18 In this way, all the information required for the molecular prediction of cmcmix of a commercial surfactant through the use of eq 17 can be obtained without the need to perform a single experimental measurement. Programs PREDICT and MIX2 are used in section 4 to predict cmcmix of a number of cationic, nonionic, and anionic commercial surfactants as well as for a representative binary mixture of commercial surfactants. As we will see, reasonable agreement (within a factor of 5) is obtained between the theoretically predicted cmcmix and the experimentally measured cmcmix deduced using the surface tension method. In the next section, we provide useful information about the chemical composition of the commercial surfactants examined as well as a description of the surface tension method used to determine cmcmix experimentally. 3. Materials and Methods Table 2 lists all the commercial surfactants examined in this paper, along with the names of the manufacturers, the lot numbers, and a description of the chemical compositions. All the
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Table 2. Chemical Composition of the Commercial Cationic, Nonionic, and Anionic Surfactants Examined in This Paper and Their Manufacturers and Lot Numbers commercial surfactant
manufacturer (lot number)
BTC-8358 (cationic)
Stepan (8-16095)
GENAPOL UD-079 (nonionic)
Clariant (62560)
GENAPOL UD-110 (nonionic)
Clariant (TB11333)
GENAPOL 26-L-98 (nonionic)
Clariant (TB08521)
GENAPOL 26-L-50 (nonionic)
Clariant (T005520)
STEOL CS-330 (anionic)
Stepan (0-30620)
chemical composition polydisperse alkyl dimethyl benzyl ammonium chloride (CiDBAC) 40% Ci)12 + 50% Ci)14 + 10% Ci)16 polydisperse ethoxylated alcohol (CiEj), j is Poisson distributed about 〈j〉 ) 7, and i ) 11 polydisperse ethoxylated alcohol (CiEj), j is Poisson distributed about 〈j〉 ) 11, and i ) 11 polydisperse ethoxylated alcohol (CiEj), j is Poisson distributed about 〈j〉 ) 11, and 65% Ci)12 + 30% Ci)14 + 5% Ci)16 polydisperse ethoxylated alcohol (CiEj), j is Poisson distributed about 〈j〉 ) 7, and 65% Ci)12 + 30% Ci)14 + 5% Ci)16 polydisperse sodium ethoxy sulfate (CiEjS04Na), j is distributed about 〈j〉 ) 2, and 65% Ci)12 + 30% Ci)14 + 5% Ci)16
commercial surfactants were used as received. The commercial cationic surfactant BTC-8358 is a polydisperse mixture of three surfactant components, all having the same cationic head, dimethyl benzylammonium chloride, but having either a C12, C14, or C16 alkyl tail. Values of ah ) 35 Å2, dcharge ) 2.5 Å, and z ) 1 were used as inputs.38 The commercial nonionic surfactants from the GENAPOL series are all polydisperse mixtures of ethoxylated alcohol surfactants of the CiEj type, where i denotes the number of carbon atoms in the alkyl tail and j denotes the number of ethylene oxide groups in the polar head. The GENAPOL UD-079 is made of about 15 surfactant components, all having the same C11 alkyl tail but having a Poisson distribution of ethoxylated alcohol heads with an average value of 〈j〉 ) 7. Figure 5 presents an example of a calculated Poisson distribution with an average value of 〈j〉 ) 7. The GENAPOL UD-110 also consists of about 15 surfactant components, all having the same C11 alkyl tail but having a Poisson distribution of ethoxylated alcohol heads with an average value of 〈j〉 ) 11. The GENAPOL 26-L-98 consists of about 45 surfactant components, each one having either a C12, C14, or C16 alkyl tail and having a Poisson distribution of ethoxylated alcohol heads for each tail with an average value of 〈j〉 ) 11. The GENAPOL 26-L-50 is made of
about 45 surfactant components, each one having either a C12, C14, or C16 alkyl tail and having a Poisson distribution of ethoxylated alcohol heads for each tail with an average value of 〈j〉 ) 7. To estimate the value of ah for these poly(ethylene oxide) surfactant heads, we have used the scaling relation ah(j) ) ah(j)6) (j/6)0.8 with ah(j)6) ) 42.3 Å2 (see ref 28 for a detailed derivation). Finally, the commercial anionic surfactant STEOL CS-330 is a polydisperse mixture of about 30 surfactant components, each one having either a C12, C14, or C16 alkyl tail and with a non-Poisson distribution of sodium ethoxy sulfate heads for each tail, provided by the manufacturer and shown in Figure 6, with an average value of 〈j〉 ) 2. A recent study on sodium dodecyl hexa(ethylene oxide) sulfate indicated that the predicted critical micelle concentration agrees best with the experimental value when the ethylene oxide chain is assumed to be fully extended.39 The same assumption was made here, leading to ah ) 25 Å2 and dcharge(j) ) (3.7 + 3.8j) Å, with z ) -1. All the solutions were prepared using deionized water that was purified using a Milli-Q filtration system. Final surfactant solution concentrations were obtained using a series of two dilutions on a mass basis, and all the solutions were used within 24 h after preparation. All glassware was carefully cleaned by soaking in a 1 M NaOH-ethanol bath for at least 8 h, followed by soaking in a 1 M nitric acid bath for at least another 8 h. The
Figure 5. Calculated Poisson probability distribution of j values having an average value of 〈j〉 ) 7, where j refers to the number of ethylene oxide groups in the surfactant head. The arrow denotes the mean value.
Figure 6. Weight distribution of j values for the anionic STEOL CS-330 commercial surfactant as reported by the manufacturer, where j refers to the number of ethylene oxide groups in the surfactant head. The arrow denotes the mean value of 〈j〉 ) 2.
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Reif et al. Table 3. Critical Micelle Concentration Predictions for the Commercial Cationic Surfactant BTC-8358 Surfactant: BTC-8358, Alkyl Dimethyl Benzyl Ammonium Chloride, (CiDBAC), 40% Ci)12 + 50% Ci)14 + 10% Ci)16 predicted cmc using program PREDICT (mM) C12DBAC C14DBAC C16DBAC
10.2 2.40 0.56 predicted βAB using program MIX2 -1.33 -1.46 -4.92
C12DBAC/C14DBAC C14DBAC/C16DBAC C12DBAC/C16DBAC
cmcmix (mM)
predicted
expt
1.15
1.30
Table 4. Number of Pairwise Interactions between m Surfactant Components
Figure 7. Critical micelle concentration determinations for the two commercial nonionic surfactants (b) GENAPOL UD079 and (9) GENAPOL UD-110 (see text). glassware was then rinsed with Milli-Q water and dried in an oven overnight. The critical micelle concentrations were determined at 25 °C by measuring the surface tension as a function of total bulk surfactant concentration, at a constant total bulk surfactant composition in the case of the binary mixture of BTC-8358 and GENAPOL 26-L-98, using a Kruss K-10 tensiometer with the Du-Nou¨y ring method.22 Each surface tension measurement was repeated three times, and the experimental error was estimated to be less than 1%. The cmcmix was then determined by the observed dramatic change in the slope of the surface tension versus the log of the surfactant concentration curve at the cmcmix. Specifically, this break point in the curve was precisely determined by the intersection of two lines: (i) a linear, least-squares fit to the three measurements just prior to this break point and (ii) a linear, least-squares fit to the three measurements just after this break point. The typical error in the cmcmix determination was less than 5%. As an illustration, the experimentally measured surface tension versus total bulk surfactant concentration as well as the fitted lines used in the cmcmix determination are shown in Figure 7 for the solutions of two of the commercial surfactants examined: GENAPOL UD-079 and GENAPOL UD110. The surface tension behaviors of the other commercial surfactant solutions were similar and, in the interest of brevity, are not shown here.
4. Molecularly Predicted and Experimentally Measured cmcmix Values: A Quantitative Comparison The commercial surfactants examined in this paper can be classified into three classes of homologous series (see Table 2). In the first class, which includes BTC-8358, all the surfactant components have the same cationic head (DBAC+) but differ in the lengths of their hydrocarbon tails (Ci)12, Ci)14, and Ci)16). The second class, which includes GENAPOL UD-079 and GENAPOL UD-110, (38) Wells, A. Structural Inorganic Chemistry, 2nd ed.; Oxford University Press: New York, 1984. (39) Shiloach, A.; Blankschtein, D. Langmuir 1998, 14, 7166.
number of surfactant components, m
number of pairwise interactions, m!/[(2!(m - 2)!]
3 4 5 10 20 30
3 6 10 45 190 435
consists of surfactant components having the same hydrocarbon tail lengths (Ci)11) but having a Poisson distribution of ethylene oxide groups in the head. Finally, the third class, which includes GENAPOL-26-L-98, GENAPOL 26-L-50, and STEOL CS-330, consists of surfactant components having a distribution of ethylene oxide groups in the head and a distribution of hydrocarbon tail lengths (Ci)12, Ci)14, and Ci)16). Table 3 shows the results obtained for the commercial cationic surfactant BTC-8358. Since this mixture consists of three (m ) 3) surfactant components, only predictions for their three cmc’s and for the interaction parameters corresponding to the three possible surfactant pairs (see Table 2) are needed. As expected, the predicted critical micelle concentration of each surfactant component decreases as its hydrocarbon tail length increases, due to an increase in the free-energy change per surfactant molecule associated with the transfer of the hydrocarbon tail from the aqueous solution to the hydrocarbon core of the mixed micelle.28 The predicted values of the interaction parameters show the experimentally known increase in synergism between two ionic surfactants having identical heads as the difference in hydrocarbon chain lengths between the two surfactant tails increases.40,41 The predicted cmcmix is 1.15 mM which is very close to the experimentally measured value of 1.30 mM. As the number of surfactant components, m, in a commercial surfactant increases, the number of pairwise interactions grows as m!/[2!(m - 2)!]. Table 4 shows that even for surfactant mixtures consisting of only 10 components, the number of pairwise interactions is equal to 45, which clearly indicates the convenience of using simplifying approximations in the calculation of cmcmix. The simplifying approximations to treat multicomponent (40) Garcia-Mateos, I.; Velazquez, M. M.; Rodriguez, L. J. Langmuir 1990, 6, 1078. (41) Lopez-Fontan, J. L.; Suarez, M. J.; Mosquera, V.; Sarmiento, F. Phys. Chem. Chem. Phys. 1999, 1, 3583.
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surfactant mixtures with m > 3 adopted in this paper are explained below. The components of the two commercial nonionic surfactants, GENAPOL UD-079 and GENAPOL UD-110, all have the same hydrocarbon tail length (Ci)11) but have a Poisson distribution of ethylene oxide groups in the head with different average values (〈j〉 ) 7 and 〈j〉 ) 11). In general, for mixtures of CiEj nonionic surfactants, cmcmix given in eq 17 can be rewritten as follows:
1
)
cmcmix
Rsij
∑i ∑j f cmc ij
(20)
ij
where Rsij, cmcij, and fij refer to the solution composition, critical micelle concentration, and activity coefficient of the CiEj surfactant component, respectively. When all the CiEj surfactant components have the same i value, eq 20 simplifies to
1
)
cmci,mix
Rsij
∑j f cmc ij
(21) ij
where cmci,mix is the critical micelle concentration of a mixture of CiEj surfactants having the same Ci tail. To calculate cmc11,mix with 〈j〉 ) 7 for GENAPOL UD079 and cmc11,mix with 〈j〉 ) 11 for GENAPOL UD-110, the following considerations were made. First, since the chemical ethoxylation process generates a Poisson distribution of ethylene oxide chain lengths, j, about a mean value 〈j〉, for any i value,3 it follows that
〈j〉 j Rsij ) Rsj ) e-〈j〉 j!
(22)
Second, we assume that interactions in the mixed micelle between CiEj nonionic surfactants having the same tail length (Ci) are close to ideal. Therefore, we make the approximation that
fij ) 1
(23)
for all the CiEj surfactant components having the same i. Finally, we follow the procedure of Warr et al.3 to simplify the mathematical summation over the Poisson distribution in eq 22, by using an approximate linear relationship obtained using program PREDICT between ln(cmcij) and the number of ethylene oxide units, j, in the head for a given value of i (surfactant tail length). Specifically, program PREDICT was used to molecularly predict cmcij for a series of CiEj surfactants having the same i but different j values. These predictions were made for i ) 11, 12, 14, and 16. Similar results have been reported previously using program PREDICT.28 The numerical results were then fitted to the following linear relationship:
ln(cmcij) ) ai + jbi
(24)
Figure 8 shows the predicted linear dependence of ln(cmcij) on j, for i ) 11, 12, 14, and 16, obtained using program PREDICT. Table 5 lists the calculated values of the coefficients, ai and bi, in eq 24. Figure 9 and Table 5 indicate that the coefficients ai show a linear dependence on the length of the hydrocarbon tail, while the bi coefficients are essentially constant, independent of either i or j. To understand this behavior, it is necessary to realize
Figure 8. Predicted dependence of ln(cmcij) on j, the number of ethylene oxide groups in the surfactant head, and on i, the number of carbon atoms in the surfactant tail, of CiEj nonionic surfactants: ([) C11Ej surfactants, (9) C12Ej surfactants, (2) C14Ej surfactants, and (b) C16Ej surfactants. Table 5. Values of ai and bi (See Equation 24) Predicted Molecularly Using Program PREDICT for Various CiEj Nonionic Surfactants i
ai
bi
i
ai
bi
11 12
-3.823 -5.316
0.489 0.487
14 16
-8.135 -10.946
0.483 0.472
that ln(cmcij) is given by the free energy of micellization of the CiEj surfactant (see eqs 9 and 10). As stated above, one of the contributions to the free energy of micellization involves the transfer of the surfactant hydrocarbon tails from the bulk solution to the micellar core. This transfer free energy depends only on the number of carbon atoms in the surfactant tail (i), and its dependence on i is linear.28-30 This is, in fact, the source of the observed linear behavior with respect to the surfactant tail length (reflected in the i value) displayed by the ai coefficients. On the other hand, the approximate linear dependence on the j value shown in eq 24 is the result of a balance of the interfacial, packing, and steric contributions to the free energy of micellization, where each one of these contributions exhibits a complex dependence on the size of the surfactant head (reflected in the j value).28-30 Substituting eqs 22, 23, and 24 in eq 21, it follows that
1 cmci,mix
∞
)
( )( )
e ∑ j)1
〈j〉j
1
j!
eai+jbi
-〈j〉
cmci,mix ) exp[ai + 〈j〉(1 - e-bi)]
(25) (26)
Note that only when |bi| , 1, one obtains the following relationship:
cmci,mix ) eai+bi〈j〉 ) cmci,j)〈j〉
(27)
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surfactant components. All the surfactant components belonging to each group are expected to behave ideally within that group as well as to have very similar nonideal interactions with the surfactant components belonging to the other groups. It can be shown that for such systems, their cmcmix is given by2
1
)
cmcmix
Rsg
∑g f cmc g
(28) g
where the summation index g runs over all the surfactant groups, and
Rsg )
∑t Rst
(29)
is the solution composition of surfactant group g. In eq 29, the summation index t runs over all the surfactant components which belong to group g. In addition, cmcg is the critical micelle concentration of surfactant group g and is given by
cmcg )
(∑ ) Rst
t
Figure 9. Linear dependence of the coefficients ai in eq 24 on the length of the hydrocarbon tail, i, as molecularly predicted using program PREDICT. The line is drawn to guide the eye. Table 6. Critical Micelle Concentration Predictions for the Commercial Nonionic Surfactants GENAPOL UD-079 and GENAPOL UD-110 Surfactant: GENAPOL UD-079, Polydisperse Ethoxylated Alcohol (CiEj), i ) 11 and j is Poisson Distributed about 〈j〉 ) 7 cmcmix (mM) predicted using program PREDICT (see eq 24) experimental
0.33 0.28
Surfactant: GENAPOL UD-110, Polydisperse Ethoxylated Alcohol (CiEj), i ) 11 and j is Poisson Distributed about 〈j〉 ) 11 cmcmix (mM) predicted using program PREDICT (see eq 24) experimental
1.54 0.41
It is important to stress that in general, it is not true that in a mixture of CiEj nonionic surfactants, the CiEj component having the average molecular structure, CiEj)〈j〉, determines the critical micelle concentration of the CiEj mixture. In this particular case of a Poisson distribution of head sizes, only when |bi| , 1, eq 27 shows that the mixture cmc is equal to the cmc of the CiEj component having the average molecular structure of the mixture, namely, CiEj)〈j〉.3 Table 6 shows the predicted and experimentally measured cmcmix values obtained for the two commercial nonionic surfactants, GENAPOL UD-079 and GENAPOL UD-110. The predicted higher critical micelle concentration of GENAPOL UD-110 is due to the larger surfactant head sizes of the various components composing the commercial surfactant. Reasonable agreement between theory and experiment is obtained in both cases, since the ratios between the predicted and the experimental cmcmix values are 1.2 and 3.8, respectively. To simplify the cmcmix calculations in the case of commercial surfactants where the interactions between all the surfactant components cannot be considered ideal, we will divide the commercial surfactant into groups of
-1
(30)
cmct
since within each group, all the interactions are assumed to be ideal. Finally, fg is the activity coefficient of surfactant group g, which we have taken to be the activity coefficient given by the surfactant component having the average molecular structure within each group. We will utilize eqs 28, 29, and 30 to predict cmcmix for the two commercial nonionic surfactants GENAPOL 26L-98 and GENAPOL 26-L-50. We assume that all the CiEj surfactants having the same tail length have ideal interactions with each other and similar nonideal interactions with the rest of the surfactants in the mixed micelle. In other words, all the CiEj surfactants having the same Ci can be considered to form a group, with g ) i. Therefore, the two commercial surfactants GENAPOL 26-L-98 and GENAPOL 26-L-50 each have three groups (g ) i ) 12, 14, and 16) of surfactant components, with cmcmix given by
1
)
cmcmix
Rsg
∑g f cmc g
g
Rsi
∑
)
i)12,14,16 ficmci,mix
(31)
where
Rsg ) Rsi )
∑j Rsij
(32)
and
1 cmcg
)
1 cmci,mix
)
Rsij
∑j f cmc ij
(33)
ij
with fij ) 1 since the CiEj (nonionic) surfactant components having the same tail length (Ci) are considered to interact ideally. The group activity coefficient, fg ) fi, was calculated using the βgg′ ) βii′ values predicted molecularly using program MIX2 for the interaction between the average surfactant, Ci)gEj)〈j〉, characterizing group g ) i and the average surfactant, Ci′)g′Ej′)〈j′〉, characterizing group g′ ) i′. Due to the fact that j is Poisson distributed about 〈j〉
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Table 7. Critical Micelle Concentration Predictions for the Commercial Nonionic Surfactant GENAPOL 26-L-98
Table 9. Critical Micelle Concentration Predictions for the Commercial Anionic Surfactant STEOL CS-330
Surfactant: GENAPOL 26-L-98, Polydisperse Ethoxylated Alcohol (CiEj), j is Poisson Distributed about 〈j〉 ) 11, 65% Ci)12 + 30% Ci)14 + 5% Ci)16
Surfactant: STEOL CS-330, Polydisperse Sodium EthoxySulfate (CiEjSO4Na), j is distributed about 〈j〉)2, 65% Ci)12+30% Ci)14+5% Ci)16
predicted cmcg (mM) using program PREDICT (see eq 24) Ci)12E〈11〉 Ci)14E〈11〉 Ci)16E〈11〉
0.34 0.020 0.0011 predicted βgg′ using program MIX2
Ci)12Ej)〈j〉)11/Ci)14Ej)〈j〉)11 Ci)14Ej)〈j〉)11/Ci)16Ej)〈j〉)11 Ci)12Ej)〈j〉)11/Ci)16Ej)〈j〉)11
cmcmix (mM)
0.31 0.25 0.50
predicted cmcg using program PREDICT (mM) Ci)12E〈2〉SO4Na Ci)14E〈2〉SO4Na Ci)16E〈2〉SO4Na
predicted βgg′ using program MIX2 Ci)12Ej)〈j〉)2SO4Na/Ci)14Ej)〈j〉)2SO4Na Ci)14Ej)〈j〉)2SO4Na/Ci)16Ej)〈j〉)2SO4Na Ci)12Ej)〈j〉)2SO4Na/Ci)16Ej)〈j〉)2SO4Na
predicted
expt
0.017
0.020 cmcmix(mM)
Table 8. Critical Micelle Concentration Predictions for the Commercial Nonionic Surfactant GENAPOL 26-L-50 Surfactant: GENAPOL 26-L-50, Polydisperse Ethoxylated Alcohol (CiEj), j is Poisson Distributed about 〈j〉)7, 65% Ci)12 + 30% Ci)14 + 5% Ci)16 predicted cmcg (mM) using program PREDICT (see eq 24) Ci)12E〈7〉 Ci)14E〈7〉 Ci)16E〈7〉
0.073 0.0043 0.00025 predicted βgg′ using program MIX2 -0.31 -0.28 -0.75
Ci)12Ej)〈j〉)7/Ci)14Ej)〈j〉)7 Ci)14Ej)〈j〉)7/Ci)16Ej)〈j〉)7 Ci)12Ej)〈j〉)7/Ci)16Ej)〈j〉)7
cmcmix(mM)
2.66 0.62 0.15
predicted
expt
0.0033
0.018
within each group, it follows that cmci,mix will be given by eq 24 for each group. Tables 7 and 8 show the results obtained for the two commercial nonionic surfactants GENAPOL 26-L-98 and GENAPOL 26-L-50. In all cases, as expected, the predicted critical micelle concentration of the mixture is larger for the larger surfactant head size and the shorter surfactant tail length. The predicted interaction parameter values are close to zero in all cases, indicating a behavior close to ideal in these systems. Reasonable agreement between the predicted and the experimental critical micelle concentrations is again obtained for both surfactants, since the ratios between the predicted and the experimental cmcmix values are 0.9 and 0.2, respectively. The cmcmix predictions for the commercial anionic surfactant STEOL CS-330 were done following the same procedure as for the commercial surfactants GENAPOL 26-L-98 and GENAPOL 26-L-50. All the surfactant components having the same hydrocarbon tail length were considered to form a group, and therefore three cmcg and three βgg′ values had to be determined for the i ) 12, 14, and 16 tail lengths. The results of these calculations are shown in Table 9. Again, longer surfactant tails have lower predicted critical micelle concentrations, and the predicted values of the interaction parameters exhibit the experimentally known increase in their absolute values when the differences in the hydrocarbon chain lengths are larger. Once again, reasonable agreement is obtained between
-0.81 -0.15 -2.71
predicted
expt
0.67
0.27
the predicted and the experimentally measured cmcmix values, since the ratio between the predicted and experimental cmcmix values is equal to 2.5. Figure 10 is a plot of the predicted cmcmix versus the experimentally measured cmcmix for the six commercial surfactants examined in this paper. As can be seen, all the points lie close to the diagonal line, with the largest deviation between the predicted and the experimental cmcmix values equal to a factor of 5, indicating reasonable agreement between theory and experiment for all the six commercial surfactants considered which include cationic, nonionic, and anionic commercial surfactants. The same general computational framework can be implemented for the molecular prediction of cmcmix for mixtures of commercial surfactants at different solution compositions. As an illustration, we present results for a
Figure 10. Comparison between predicted and experimentally measured critical micelle concentrations for all the commercial surfactants examined in this paper: (2) GENAPOL UD-110, (b) BTC-8358, (9) STEOL CS-330, (0) GENAPOL UD-079, ([) GENAPOL 26-L-98, and (O) GENAPOL 26-L-50.
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Reif et al. Table 10. Critical Micelle Concentration Predictions for the Binary Mixture of the Commercial Cationic BTC-8358 Surfactant and the Commercial Nonionic GENAPOL 26-L-98 Surfactant commercial surfactant
surfactant groups
predicted cmc (mM) using program PREDICT
BTC-8358
C12DBAC C14DBAC C16DBAC
10.2 2.40 0.56 predicted cmc (mM) using program PREDICT (see eq 24)
GENAPOL 26-L-98
Ci)12E〈11〉 Ci)14E〈11〉 Ci)16E〈11〉
0.34 0.020 0.0011
The Following 15 βgg′ Values Were Predicted Using Program MIX2 C12DBAC/C14DBAC -1.33 C14DBAC/C16DBAC -1.46 C12DBAC/C16DBAC -4.92 Ci)12Ej)〈j〉)11/Ci)14Ej)〈j〉)11 0.31 Ci)14Ej)〈j〉)11/Ci)16Ej)〈j〉)11 0.25 Ci)12Ej)〈j〉)11/Ci)16Ej)〈j〉)11 0.50
Figure 11. Comparison between predicted (line) and experimentally measured (b) critical micelle concentrations of the binary mixture of the commercial cationic BTC-8358 surfactant and the commercial nonionic GENAPOL 26-L-98 surfactant over the entire range of solution compositions.
binary mixture of the commercial cationic BTC-8358 surfactant and the commercial nonionic GENAPOL 26L-98 surfactant. The mixture is made up of 48 surfactant components (BTC-8358 has 3 surfactant components, and GENAPOL 26-L-98 has 45 surfactant components), which were modeled considering them to form six surfactant groups, as shown in Table 10. Each surfactant group, which belongs to a given commercial surfactant, consists of all the surfactant components having the same tail length. Therefore, the cationic BTC-8358 is described by three groups (each group made of one surfactant having either i ) 12, 14, or 16), and the nonionic GENAPOL 26L-98 provides the other three groups (each group made of about 15 surfactant components having either i ) 12, 14, or 16). Therefore six cmcg values are needed, and [6!/ (2!4!)] ) 15 βgg′ interaction parameters are required (see Table 10). The predicted βgg′ values for the monovalent cationic and nonionic surfactant mixtures fall within the range -5 < β < -1, as expected.2 Figure 10 shows a plot of the molecularly predicted and the experimental cmcmix as a function of solution composition for this binary mixture of commercial surfactants. As before, reasonable agreement is obtained over the entire range of solution compositions. 5. Conclusions A new computational approach, which combines a molecular-thermodynamic theory of micellization with a simple approximate expression for the excess free energy of mixed micellization, was developed to molecularly predict the critical micelle concentration of surfactant mixtures containing any number of surfactant components. This computational approach was then implemented, through the use of two user-friendly computer programs, PREDICT and MIX2, to molecularly predict
Ci)12Ej)〈j〉)11 Ci)14Ej)〈j〉)11 Ci)16Ej)〈j〉)11
C12DBAC
C14DBAC
C16DBAC
-3.02 -2.75 -2.30
-3.98 -4.18 -3.70
-5.17 -5.27 -5.43
the critical micelle concentrations of several commercial surfactants of known chemical composition, including one representative binary mixture of commercial surfactants. The commercial surfactants examined include the cationic BTC-8358, the nonionics GENAPOL UD-079, GENAPOL UD-110, GENAPOL 26-L-98, and GENAPOL 26-L-50, and the anionic STEOL CS-330. The accuracy of the predicted critical micelle concentration values is comparable to that attained in the case of single (pure) surfactants and their binary mixtures, thus demonstrating the practical utility of the computer-assisted molecular-thermodynamic modeling as a predictive tool in commercial surfactant characterization. Any future improvements in the ability to predict critical micelle concentrations of single surfactants and their binary mixtures will also improve the ability to predict critical micelle concentrations of commercial surfactants using the computational approach presented in this paper. The validity and range of applicability of the computational approach presented in this paper are determined primarily by the validity and range of applicability of the simple, approximate expression for the excess free energy of mixed micellization, gexcess mix , utilized here. A complete beyond the approximation molecular evaluation of gexcess mix used here, including an explicit treatment of counterions in the case of ionic surfactants, would not only enable the improvement of the prediction of cmcmix but also allow the prediction of other important micellization characteristics of commercial surfactants, including micelle shape and size. Work along these lines is in progress. Acknowledgment. We are grateful to Dow Chemical, Kodak, Schlumberger-Dowell, and Unilever USA for partial support of this work. LA0105578
