Langmuir 1999, 15, 2661-2668
2661
NMR Studies of Binary Surfactant Mixture Thermodynamics: Molecular Size Model for Asymmetric Activity Coefficients Charles D. Eads* and Lora C. Robosky Miami Valley Laboratories, The Procter & Gamble Company, P.O. Box 538707, Cincinnati, Ohio 45253-8707 Received May 5, 1998. In Final Form: January 13, 1999 Pulsed-field-gradient NMR methods are used for determining the partitioning of surfactants between unimeric and micellar forms in mixed surfactant systems. The method allows determination of the composition dependence of activity coefficients in binary surfactant mixtures. Application of the regular solution approach to NMR data for several mixtures gives β parameters which are dependent on concentration and which differ systematically among the surfactants within the same binary mixtures. The β parameter summarizes interactions among different pairs of surfactants and, in principle, should not be composition dependent. The observed behavior is therefore not consistent with expectations from the regular solution model. Use of the van Laar expressions, on the other hand, accounts well for the composition dependence of the activity coefficients. The van Laar expressions also account for the often-observed composition dependence of regular solution β parameters determined from CMC measurements. Though similar to the symmetric regular solution model, the van Laar expressions contain an additional parameter which reflects differences in the sizes of the mixture components. The results therefore suggest that headgroup size and headgroup packing are important contributors to nonideal surfactant behavior. Computer algorithms are described for extracting the van Laar interaction energy- and size-related parameters from NMR-derived results, from mixed critical micelle concentrations, or from heats of mixing. Data for several binary surfactant mixtures are presented and discussed. The results emphasize that when accurate data are available, the single-parameter regular solution model will not always fully account for nonideal surfactant mixing.
Introduction Aqueous mixed surfactant solutions often exhibit properties quite different from those of the constituent materials.1-3 For example, critical micelle concentrations for mixtures often deviate substantially from expectations based on ideal mixing considerations. Micelle composition and surface activity also become complex functions of both the relative amounts of the constituents and the total surfactant concentration. Following the original work of Rubingh4,5 and unpublished work by J. M. Corkill, experimental characterization of intercomponent interactions in nonideal binary mixed surfactant systems is frequently carried out using the regular solution approach. The regular solution model describes deviations from ideal mixing6 using a single parameter, β, which summarizes the pairwise interaction energies of the various molecules.7 For surfactant systems, the regular solution β parameter is usually determined by fitting to mixed CMC (critical micelle concentration) data for a series of surfactant compositions.4,5 The wide-spread utility of this approach stems in part from the ability to characterize and catalog interactions using a single parameter which has a simple physical interpretation. This is not a predictive approach because the β parameter must be determined experimentally for each surfactant pair. Current literature activity
in this area includes efforts to predict various micellar properties including the β parameter for surfactant mixtures using molecular descriptors.8-11 The use of NMR pulsed gradient spin-echo diffusion methods12-16 for studying mixed surfactant systems offers advantages over the more commonly used method of measuring CMCs. NMR data measure partitioning between micellar and unimeric forms of each individual surfactant via analysis of self-diffusion coefficients. This in turn allows calculation of activity coefficients directly for each component in individual solutions rather than indirectly through fitting CMCs for a series of solutions. The NMR approach also gives the micellar composition directly, offering advantages in understanding mixed micelle composition at concentrations near the CMC where micelle composition differs significantly from the solution composition. The initial plan for interpretation of the NMR data presented in this report was to use simple algebraic manipulation of the regular solution equations to directly determine the regular solution β parameters for various binary surfactant mixture compositions based on NMRderived activity coefficients. However, certain deviations from the expected behavior were observed. In particular, the NMR-derived β parameters were dependent on concentration and differed systematically among the
(1) Holland, P. M.; Rubingh, D. N., Eds. Mixed Surfactant Systems; ACS Washington, DC, 1992. (2) Kronberg, B. Curr. Opin. Colloid Interface Sci. 1997, 2, 456. (3) Holland, P. M.; Rubingh, D. N. In Catioinic Surfactants: Physical Chemistry; Rugingh, D. N., Holland, P. M., Eds.; Surface Science Series 37; Marcel Dekker: New York, 1990; p 141. (4) Rubingh, D. N. In Solution Chemistry of Surfactants; Mittal, K. L., Ed., Plenum Press: New York, 1979; Vol. 1, p 337. (5) Holland, P. M.; Rubingh, D. N. J. Phys. Chem. 1983, 87, 1984. (6) Clint, J. H. J. Chem. Soc., Farady Trans. 1 1975, 71, 1327. (7) Hill, T. L. An Introduction to Statistical Thermodynamics; 1960; Dover: New York, 1986.
(8) Shiloach, A.; Blankschtein, D. Langmuir 1997, 13, 3968. (9) Blankschtein, D.; Shiloach, A.; Zoeller, N Curr. Opin. Colloid Interface Sci. 1997, 2, 294. (10) Nagarajan, R. Langmuir 1985, 1, 331. (11) Sarmoria, C.; Puvvada, S.; Blankschtein, D. Langmuir 1992, 8, 2690. (12) Faucompre´, B.; Lindman, B. J. Phys. Chem. 1987, 91, 383. (13) Carlfors, J.; Stilbs, P. J. Phys. Chem. 1984, 88, 4410. (14) Stilbs, P. J. Colloid Interface Sci. 1982, 87, 385. (15) So¨derman, O.; Stilbs, P. Prog. Nucl. Magn. Reson. Spectrosc. 1994, 26, 445. (16) Stilbs, P. Prog. Nucl. Magn. Reson. Spectrosc. 1987, 19, 1.
10.1021/la980525t CCC: $18.00 © 1999 American Chemical Society Published on Web 03/20/1999
2662 Langmuir, Vol. 15, No. 8, 1999
Eads and Robosky
surfactants within the same binary mixtures. This behavior is not expected based on the regular solution approach. It was suggested (D. Rubingh, personal communication) that this behavior could be a result of differing headgroup sizes. This paper therefore explores the use of a modified regular solution approach, which includes a size-related parameter, to describe the composition dependence of activity coefficients determined by NMR in mixed surfactant systems. In particular, the van Laar expressions for activity coefficients are used to account for the data. Simple theoretical arguments show that the van Laar expressions are valid for interactions among headgroups which operate over long distances (e.g., electrostatic interactions) or short distances (e.g., contact interactions). Two-parameter modifications of the regular solution approach have been used previously for characterizing mixed surfactant systems.17-19 The present treatment maintains the advantage of the regular solution approachs a small number of parameters having a simple physical interpretationswhile extending the model to handle the asymmetry commonly found in surfactant mixtures. Materials and Methods Diffusion Data Acquisition and Analysis. Experiments were carried out on a GE Omega 500 MHz NMR spectrometer and on a Varian Unity-Plus 300 MHz NMR spectrometer. All experiments were performed at 25 °C using commercially available surfactants. Gradient strength was calibrated by scaling to the known diffusion coefficient of residual hydrogen atoms in deuterium oxide, 1.902 × 10-5 cm2 s-1.20 Diffusion data were collected using the LED modification of the stimulated echo sequence.21 All interpulse delays were held constant while the gradient strength was systematically varied. Data were fit to the equation of Stejskal and Tanner22 using software supplied with the spectrometers. It was particularly important in this application to acquire diffusion data as accurately as possible. Therefore many (>10) gradient strengths spaced over a wide range were used, and spectra were processed carefully using matched apodization filters and polynomial baseline corrections. Pentaoxyethylene glycol monooctyl ether (C8E5), pentaoxyethylene glycol monododecyl ether (C12E5), and sodium dodecyl sulfate (C12S) were obtained from Sigma. N,N-Dimethyldecylamine N-oxide (C10AO) was obtained from Aldrich. All materials were obtained in the highest available purity and used without further purification. Determination of Unimer-Micelle Partitioning. Surfactant unimer-micelle partitioning can be studied by NMR diffusion techniques because the diffusion coefficient D is the average of the diffusion coefficients for the unimeric and micellar forms, weighted according to the mole fractions of material in each form:12-16
D ) Duχu + Dmχm
(1)
or
χu )
D - Dm Du - Dm
(2)
In these equations D is the diffusion coefficient for the sample of interest, Du is the diffusion coefficient for the unimeric form of the surfactant, Dm is the diffusion coefficient of the micelle, (17) Maeda, H. J. Colloid Interface Sci., 1995, 172, 98. (18) Georgiev, G. S. Colloid Polym. Sci. 1996, 274, 49. (19) Kameyama, K.; Muroya, A.; Takagi, T. J. Colloid Interface Sci. 1997, 196, 48. (20) Holz, M.; Weinga¨rtner, H. J. Magn. Reson. 1991, 92, 115. (21) Gibbs, S. J.; Johnson, C. S., Jr. J. Magn. Reson. 1991, 92, 395. (22) Stejskal, E. O.; Tanner, J. E. J. Chem. Phys. 1965, 42, 288.
χu is the mole fraction of material in the unimeric form, and χm ) 1 - χu is the mole fraction of material in the micellar form. Mole fractions can therefore be determined if values for D, Du, and Dm are available. D is measured for each surfactant in each sample of interest. Du is measured for each surfactant in solutions having concentrations below the CMC. Dm is determined by measuring the diffusion coefficient of a hydrophobic probe molecule placed in the system of interest. The hydrophobic probe resides completely in the micelles, and therefore the diffusion coefficient of the probe reflects the diffusion coefficient of the micelles. We find that L-limonene makes a useful probe because it has unique chemical shifts, it is highly water-insoluble, it is easy to handle, and it has a relatively low molecular weight. It appears to have a negligible effect on partitioning between unimeric and micellar forms when present at low concentrations (unpublished observation). Sample Preparation. For systems in which the CMCs of both surfactants are comparable, stock solutions of both surfactants were prepared at concentrations approximately twice the CMC, so that about half of each surfactant resides in the micellar form. These solutions were then mixed in systematically varied proportions. Under these conditions both surfactants will partition roughly equally between the two forms unless the system is strongly nonideal. For systems in which the CMCs of the surfactants are highly disparate, solutions were prepared in which the low CMC surfactant was present far above it’s CMC, and the appropriate parameters were extracted from the behavior of the high-CMC component. A probe molecule is not necessary in this case because the vast majority of the low-CMC surfactant resides in the micellar form and its diffusion coefficient represents that of the micelles.
Formulas for Asymmetric Activity Coefficients van Laar Equations. The van Laar expressions for activity coefficients are given by
[ [
ln(γA) ) βAB
nBF nA + nBF
ln(γB) ) βBA
nA nA + nBF
] ]
2
(3)
2
(4)
In these expressions, γA and γB are the activity coefficients for components A and B, nA and nB are the numbers of moles of each material, F is an asymmetry parameter related to differences in the sizes of the molecules, and βAB and βBA ) FβAB summarize interaction energy differences among various types of molecules. Scatchard’s23 and Hildebrand’s24 derivations of these expressions imply that F ) vB/vA, where vB and vA are the molecular volumes of B and A, respectively. When the molecules have equal sizes (F ) 1), eqs 3 and 4 reduce to the regular solution model commonly used to treat mixed surfactant systems. As will be shown below, eqs 3 and 4 account very well for the activity coefficients and other measures of nonideality determined for mixed surfactant systems involving ionic and nonionic surfactants, even though these expressions were initially intended to apply to mixtures of nonpolar liquids. It is therefore worth considering how these relations apply to surfactant mixtures involving ionic headgroups. Use of van Laar Expressions for Long-Range Interactions. Consider the interaction between the headgroups in a mixed micelle containing A and B headgroups, either or both of which can be ionic. For the present treatment it will be assumed that the main (23) Scatchard, G. Chem. Rev. 1931, 8, 321. (24) Hildebrand, J. H.; Wood, S. E. J. Chem. Phys. 1993, 1, 817.
Asymmetric Surfactant Activity Coefficients
Langmuir, Vol. 15, No. 8, 1999 2663
contribution to nonideality arises from interactions among the headgroups. To calculate the energy of interaction among headgroups, each surfactant headgroup in the micelle will be “switched on” one at a time and in a random order. In this process, all interactions of the surfactant headgroup with the headgroups which have already been switched on will be added to the energy. For example, the total energy of interaction among A headgroups, eAA, is given by:
eAA )
∫0n UAA(dA) dn A
(5)
where UAA(dA) is the energy of interaction of a single A headgroup with all other switched on headgroups, expressed as a function of the density of headgroups which have already been switched on, dA. It is assumed as a first approximation that UAA(dA) is directly proportional to the number density of the switched on A headgroups.
n UAA(dA) ) uAA Vh
(6)
In this expression, Vh is the volume available to the headgroups, n is the number of headgroups which have been switched on, and uAA is a composition-independent proportionality constant relating the energy of interaction to the number density. This is a low-order approximation, though the use of a composition-independent proportionality constant to summarize electrostatic energies has been used successfully in a predictive model of mixed micelles.11 This gives for the total energy among A headgroups
eAA )
∫0n uAAVnh dn ) 21 A
uAA 2 n Vh A
(7)
A similar relation holds for the total interaction energy among B headgroups
eBB )
∫0n uBBVnh dn ) 21 B
uBB 2 n Vh B
(8)
To evaluate the interaction energy due to A-B interactions, eAB, assume that all the B headgroups have been switched on and integrate the interaction of each A headgroup with the B groups already present as each A headgroup is switched on. For this calculation it is assumed that the interaction of an A headgroup with the B headgroups is proportional to the number density of the B headgroups.
eAB )
n
nAnB Vh
∫0n uABVBh dn ) uAB A
(9)
It is also assumed that the headgroup volumes are additive, Vh ) nAvA + nBvB, where vA and vB are the headgroup volumes. The total energy of interaction among headgroups v is therefore given by
nA2 1 + Uh ) eAA + eBB + eAB ) uAA 2 nAvA + nBvB nB2 nAnB 1 uBB + uAB (10) 2 nAvA + nBvB nAvA + nBvB
Given this expression, it is straightforward to derive the headgroup contribution to the concentrated standard state activity coefficients by taking derivatives of eq 10 with respect to nA or nB, holding all other parameters constant. In determining the activity coefficient it is also assumed that the entropy of the mixture is given by the ideal expression, ∆S/R ) nA ln(χA) + nB ln(χB), where χA and χB are the mole fractions of the components. This is the defining assumption of regular solution theories. This gives eqs 3 and 4 for the activity coefficients, in which the parameters are given by
βAB )
ωAB vBkT
βBA ) FβAB )
ωAB ) uAB -
(
(11)
ωAB vAkT
(12)
)
vA 1 vB uAA + uBB 2 vA vB
F)
vB vA
(13)
(14)
Use of van Laar Expressions for Short-Range Interactions. One may also derive eqs 3 and 4 using a nearest-neighbor contact model. One advantage of doing so is that it provides a different perspective for interpretation of the adjustable parameters. Rather than molecular volumes, the contact number derivation ascribes mixture asymmetry to differences in contact numbers experienced by the mixture components. Since the derivation applies to lattice mixtures, the considerable computational convenience provided by such models becomes available for interpreting experimental parameters. One expects this perspective to be valid when nearest-neighbor interactions dominate, for example in mixtures of nonionic surfactants, or when long-range electrostatic interactions are screened by high concentrations of counterions. According to some arguments, electrostatic interactions in micelles may have a fairly short range due to the very high local ionic strength near an ionic micelle surface.25 Details of the derivation for short-range interactions will be presented in a separate publication (J. Phys. Chem. B, submitted). The main assumption in using contact numbers for the derivation of eqs 3 and 4 is that in the calculation of the number of contacts among particles, the particle numbers are weighted according to the number of contacts each particle makes when surrounded by particles of the other type. Using this approach, one can calculate, for example, that the probability pAB that a B particle occupies a site next to a given A particle is given by pAB ) ZBAnB/(ZABnA + ZBAnB), and that the total number of AB contacts is mAB ) pABZABnA. In these expressions, the symbols Zij represent the maximum number of contacts a particle of type i can make with other particles when completely surrounded by particles of type j. This is illustrated in Figure 1 for ZAB and ZBA. Lattice sampling calculations show that these expressions accurately predict interparticle contact numbers. The internal energy is calculated by summing the contributions of AB, AA, and BB contacts, wAB, wAA, and wBB. This again leads to the van Laar form for activity (25) Shelley, J. C.; Sprik, M.; Klein, M. L. Prog. Colloid Polym. Sci. 1997, 103, 146.
2664 Langmuir, Vol. 15, No. 8, 1999
Eads and Robosky
of surfactant A in the micelle, χA,m, combined with measurements of aqueous surfactant unimer concentrations for two systems. One of these unimer concentrations is that which is in equilibrium with the standard state, C0A,w Since the standard state in the pseudophase model is taken as the pure micelles, the unimer concentration for this case is given by the critical micelle concentration, C*A. The other unimer concentration, CA,w, is that measured for the system consisting of micelles with the appropriate composition.
γA,m ) Figure 1. Effect of Unequal Molecular Size on Contact Numbers.
coefficients, in which the adjustable parameters have the following interpretations:
ωAB βAB ) ZAB kT
(15)
ωAB ) FβAB βBA ) ZBA kT
(16)
ωAB ) wAB -
(
)
ZBB 1 ZAA w + w 2 ZAB AA ZBA BB F)
ZBA ZAB
(17)
(18)
The functional form for the activity coefficients resulting from the assumption appropriate for long-range interactions (interaction energies are proportional to number densities of headgroups) is the same as that resulting based on the assumption appropriate for short-range interactions (contact numbers and internal energy depend on weighting the molecule numbers by the number of contacts each headgroup can make with headgroups of the other type). This result indicates that the van Laar form of the activity coefficients should be robust and generally applicable for both short-range and long-range interactions, so long as the interaction energies are small enough that the regular solution ideal entropy assumption holds. When used with predictive molecular thermodynamic theories, the parameters can, in principle, be calculated based on geometric considerations and on appropriate models for the interactions among headgroups. Numerical Analysis. Using regular solution theory, the β parameter can be determined by least-squares fitting to CMC data using a functional relation between the pure surfactant CMCs, the β parameter, and the experimental mixed CMCs.4 The NMR diffusion approach extracts different information and therefore requires a different approach to data analysis. For the NMR experiments, solutions of various compositions whose total surfactant concentration exceeds the mixed CMC are prepared. The NMR data provide a measure of the amount of each surfactant in the unimeric and in the micellar form. Thus, the unimer concentrations and micelle composition are measured rather than the mixed CMC. Procedures for extracting parameters from data in this form are not available in the literature, so such methods are presented in this section. It may be readily shown4 that the activity coefficient γA,m can be determined from knowledge of the mole fraction
1
( ) ( ) CA,w
χA,m C0 A,w
)
1
χA,m
CA,w C*A
(19)
All of these parameters are available from the NMR experiments as described above. The critical micelle concentration is taken as the surfactant unimer concentration in a solution of the pure surfactant. Thus, the NMR method allows direct calculation of activity coefficients for the individual surfactants. Using the regular solution formalism, the β parameter can be determined from the activity coefficients and the micellar mole fractions by solution of eqs 3 and 4 with the assumption that F ) 1. Using the two-parameter formalism, βAB and F must be determined by simultaneous solution of eqs 3 and 4 using the experimentally determined values of γA,m, γB,m, nA, and nB from a single solution. Fitting to Unimer Concentrations. A more accurate technique for extracting parameters from NMR data is to fit these parameters to measured unimer concentrations for a series of solutions in which the composition is systematically varied. In the present work, the leastsquares fitting was carried out using a Simplex optimization routine.26 This approach requires computation of the expected distribution of surfactants between micellar and unimeric forms given an overall solution composition and guesses at the parameters. For a given micelle composition χA, the concentration of unimeric surfactants A and B, CA,w and CB,w can be determined using a rearrangement of eq 19
CA,w ) γA,mχA,mC*A
(20)
CB,w ) γB,mχB,mC*B
(21)
The activity coefficients themselves can also be expressed in terms of mole fraction because the quantities nA and nB in Equations 3 and 4 can be replaced with χA and χB, respectively, without affecting the value of the equations. In this way, the unimer concentrations can be expressed entirely as functions of the micelle composition. For specific values of βAB and F, and for specific total (micellar plus unimeric) surfactant concentrations CAt and CBt, there is a unique micelle composition χA which makes eqs 20 and 21 self-consistent. That is, there is a micellar composition χA which predicts unimer concentrations CA,w and CB,w, which, when subtracted from the total surfactant concentrations CAt and CBt, gives back finally the original micellar composition. The appropriate value of χA which satisfies eqs 20 and 21 can be rapidly located using numerical methods such as the false position modification of the iterative secant algorithm.26 This algorithm finds the value of a variable parameter which causes an objective function Y to equal (26) Press: W. H.; Flannery, B. P.; Teukolsky, S. A.; Vetterling, W. T. Numerical Recipes in C; Cambridge University Press: Cambridge, England, 1988.
Asymmetric Surfactant Activity Coefficients
Langmuir, Vol. 15, No. 8, 1999 2665
zero. In this case, the variable is χA and the objective function Y is calculated using the following procedure: 1. The unimer concentrations are predicted using eqs 20 and 21 for the current trial value of χA. 2. For surfactant A, subtract the predicted unimer concentration CA,w from the total concentration CAt and assign the remaining material to the amount of A in the micelle, nA. 3. For surfactant B, assign enough of the total concentration CBt to the amount of B in the micelle, nB, to produce the starting value of χA. 4. Compare the remaining amount of unimeric B to the unimer concentration predicted based on χA. The difference gives the objective function Y. The value of χA which serves as a root for this objective function is the correct micelle composition for the current solution composition and the current values of βAB and F. In the fitting routine, the unimer concentrations CA,w and CB,w which result from these parameters can be compared to the experimentally observed unimer concentrations. The optimum least-squares fit to the experimental data arises from the values of βAB and F which best predict the measured unimer concentrations using this procedure. The algorithm requires bounds to be placed on the values of χA. One bound is taken to be the value of χA in the overall solution, and the other bound is taken to be the micelle composition at the mixed CMC. It is therefore necessary to compute the mixed CMC for the solution as well. Determining Mixed Critical Micelle Concentrations. In addition to setting bounds for the data fitting routine, it is necessary to calculate the mixed CMC and the composition of micelles infinitesimally above the mixed CMC to establish if the total surfactant concentration is above or below the minimum necessary to form micelles. Furthermore, it is necessary to calculate the mixed CMC to use the two-parameter approach to fit CMC data. To determine the mixed CMC, we define R, the overall composition of the solution, by
CA,t R) CA,t + CB,t
(22)
(23)
where
F)
CA,w CA,w + CB,w
Figure 3. Log of Activity Coefficient of Surfactant i vs Square of Contact Number Fraction of Surfactant j in C8E5-C12S Mixture. Circles show activity coefficients for C12S; squares show activity coefficients for C8E5. The lines are based on a fit of the two-parameter asymmetric model to the unimer concentrations. This model predicts different behavior for the two surfactants as indicated by the lines.
Results and Discussion
Infinitesimally above the mixed CMC, since the micelles will hold negligible surfactant, this value of R gives the composition of the unimeric part of the system which would be in equilibrium with the micelles. Hence, the mixed CMC is equal to the sum of the unimer concentrations which result from finding the micelle composition which would equilibrate with a unimer composition reflecting the total composition of the system, R. As above, we may use the iterative secant algorithm to rapidly find the CMC and micelle composition. In this case, the objective function Y is the difference between the solution composition R and a function F of predicted unimer concentrations.
Y)R-F
Figure 2. Log of Activity Coefficient of Surfactant i vs Square of Mole Fraction of Surfactant j in C8E5-C12S Mixture. Circles show activity coefficients for C12S; squares show activity coefficients for C8E5. A single straight line is expected for a symmetrical system.
(24)
and CA,w and CB,w are determined using eqs 20 and 21 with the trial value of χA. The value of χA giving Y ) 0 in eq 23 corresponds to the micelle composition at the mixed CMC. The sum of the corresponding values CA,w and CB,w gives the mixed CMC itself.
Demonstration of Asymmetry in Activity Coefficients. The symmetric regular solution model predicts that a plot of ln(γA) vs (χB)2 should give a straight line with slope of β. Similarly, a plot of ln(γB) vs (χA)2 should give a straight line with an identical slope. Figure 2 shows such a plot for a system containing C8E5 plus C12S in which the micellar compositions and activity coefficients were determined from NMR measurements of the amounts of surfactant in the unimeric and micellar forms. Neither of the two surfactants gives a straight line, nor do the two curves coincide. The slopes differ most at low values of the mole fractions, that is, when there is a high concentration of the surfactant whose activity coefficient is being plotted. Attempts to determine β parameters from these data give composition-dependent results and very different values for the two different surfactants. This is evidence for an asymmetry in the system which cannot be described using the regular solution treatment. Figure 3 shows a plot of the same data but with the mole fractions replaced by asymmetry parameter-weighted fractions (eqs 3 and 4). The values of βAB and F used to generate the lines through the data are based on a leastsquares fit to the unimer concentrations. Except for the lowest activity coefficients (which have a high uncertainty), the two-parameter model accounts well for the activity coefficients, correctly predicting the different behavior of the two surfactants. Thus, Figure 3 demonstrates that
2666 Langmuir, Vol. 15, No. 8, 1999
Eads and Robosky
Figure 4. Measured and Best Fit Unimer Concentrations for C8E5-C12S Mixture. Circles show the values for C8E5, squares show the values for C12S, and diamonds show the values for the sum of the unimer concentrations. Solid lines show the values using the best fit of the two-parameter model. Broken lines show the best fit of the single-parameter regular solution model. Table 1. Results of Least-Squares Fitting of the Two-Parameter Asymmetric Regular Solution Model to NMR-Derived Surfactant Unimer Concentrations composition
CMC of A
CMC of B
βAB
F
C8E5 (A) C12S (B) C12E5 (A) C12S (B) C8E5 (A) C10AO (B) pH 10 C8E5 (A) C10AO (B) pH 3
7.29
5.54
-2.67
2.28
0.06a
6.61b
-1.47
2.93
8.40
19.06
8.28
32.6
0.00c -0.54
1.00c 1.49
a The CMC for this material was taken from the literature. The value is too low and too different from the CMC of C12S to be studied in the same sample using the NMR technique. The parameters are therefore determined from a fit to C12S unimer concentrations only. b Since no L-limonene was used in this system, no measurement of the micelle diffusion coefficient in the absence of C12S was possible. The reported value is from the fitting procedure. c This sample behaved ideally, so no attempt to fit the parameters was made.
the two-parameter model correctly accounts for the observed asymmetric behavior. Figure 4 compares fits of the two-parameter treatment and the one-parameter regular solution treatment to the surfactant unimer concentrations determined from NMR diffusion measurements. The two-parameter fit can reproduce the data within experimental uncertainty, while the regular solution approach cannot. Thus, the twoparameter asymmetric treatment provides a better description of surfactant unimer concentrations compared to the regular solution model, as well as a better description of activity coefficients determined from the unimer concentrations. The data set shown in Figure 4 is different from that used in Figures 2 and 3. The two data sets were independently acquired by the two authors. The values of βAB and F determined from these data were -2.67 and +2.28 for the data in Figure 3, and -2.73 and +2.28 for the data in Figure 4. Thus, the values of these parameters are quite reproducible. Results for Several Binary Mixtures. Table 1 shows the results of fitting the two-parameter model to NMRderived unimer concentrations for several alkyl ethoxylate containing binary mixtures. Several interesting aspects of mixed surfactant systems are apparent from these results. For the two systems consisting of C8E5 mixed
Figure 5. Critical Micelle Concentrations for an Asymmetric System. The heavy line shows the critical micelle concentrations predicted using the two-parameter model using, βAB ) -2.5, F ) 3, CMCA ) 3 mM, and CMCB ) 6 mM. The thin line shows the best fit of the regular solution model to the six values listed in Table 2.
with C12S or C12E5 mixed with C12S, the values of βAB and F are different though the headgroups are identical. This emphasizes the importance of the effect of the surfactant tails on the interactions among the headgroups. It is likely that the two different mixtures have different geometries resulting from differences in the volume available for mixing the headgroups. This will lead to differences in the distance of closest approach, changes in the effect of one surfactant on the distances between headgroups of the other surfactant, etc. and should influence both βAB and F. For the amine oxide containing systems, the high-pH mixture behaves ideally, indicating that interaction energies among all surfactant headgroup pairs have similar values. At low pH, the behavior becomes nonideal. This is apparently due to the protonation of the oxygen atom and consequent development of a positive charge on the amine oxide headgroups. Under these conditions, interactions among amine oxide headgroups are energetically unfavorable (positive sign), leading to a negative value of ωAB via eq 13. This illustrates the general result, true also of the alkyl sulfate headgroups, that ionic surfactants should show synergistic interactions with nonionic cosurfactants because such cosurfactants reduce the number of unfavorable interactions among like charges. Identifying Asymmetry Using CMC and Heat of Mixing Measurements. Since a great deal of published and ongoing research on mixed surfactant systems addresses mixed CMCs as a function of composition, the ability to detect size asymmetry using CMC data instead of unimer concentrations will be explored. Figure 5 shows predicted CMCs as a function of composition for a system characterized by the parameters βAB ) -2.5, F ) 3, CA* ) 3 mM, and CB* ) 6 mM (heavy line). The thin line gives the results of a fit to six evenly spaced values using the symmetric regular solution model (F ) 1). Fitting to the symmetric model gives a root-mean-square-average difference between the fit and the simulated CMCs of 0.12 mM. This is probably within the uncertainty in CMC determinations for many systems. Thus, for systems with moderate asymmetry, the regular solution model can do an acceptable job of fitting CMC data unless the CMCs are determined very precisely. However, note that the symmetric regular solution fit gives the result β ) -4.06, which is significantly different from either βAB or FβAB ) βBA. The value of the symmetric regular solution β parameter derived from fitting to CMC data will differ from the more precise descriptors βAB and F.
Asymmetric Surfactant Activity Coefficients
Langmuir, Vol. 15, No. 8, 1999 2667
Table 2. Critical Micelle Concentrations for an Asymmetric Surfactant Mixturea RB
CMC
β
1.0 0.8 0.6 0.4 0.2 0.0
6.0 2.07 1.58 1.40 1.40 3.00
-3.68 -4.02 -4.33 -4.68
a The mixture is characterized by the parameters β ΑΒ ) -2.5, F ) 3, CMC A ) 3.0 mM, CMC B ) 6.0 mM. The β parameters were determined assuming the symmetric regular solution model. Note that the values vary systematically with the solution composition.
β parameters may also be determined from individual CMC measurements of mixed systems given the CMCs of the pure surfactants.4 For an asymmetric system such as that described by Figure 5, one finds a systematic variation in the regular solution β parameter determined for such a system as a function of composition as shown in Table 2. Thus, one diagnostic of asymmetric systems available from CMC measurements in a systematic, monotonic variation in the β parameter. Recent literature has many examples (e.g., refs 27 and 28) of carefully determined CMC measurements in which this trend is evident. Thus, carefully obtained CMC measurements giving composition-dependent β parameters can indicate asymmetry in the mixture, and inspection of existing literature gives many examples of this type of behavior. The symmetric regular solution approach and the asymmetric modification described here both ascribe nonideal behavior to the heat of mixing, assuming that the entropy of mixing is ideal. To test the validity of the models, one may compare heats of mixing determined calorimetrically to parameters determined from fitting to CMCs or to unimer concentrations.29-31 Holland29,30 noted discrepancies between the excess heat of mixing and the expected values obtained from fitting CMCs to the regular solution model. This discrepancy is evident from the asymmetric composition dependence of the excess heats of mixing for C10E5 with several anionic surfactants. The present model predicts such asymmetric behavior, so we revisited the data of Holland29 using the van Laar model. Assuming that the enthalpy and internal energy are interchangeable at the present level of approximation, eq 10 predicts that the enthalpy of mixing should be given by ∆H ) (nAnB/Vh)ωAB ) kTβABnAnBF/(nA + nBF). We fit published enthalpies of mixing for mixtures of C12E2S with C10E5 using this formula and obtained βAB ) -1.9, F ) 2.47. The enthalpies were fit to within about 4% of their values using this model. Holland obtained a value of β ) -1.6 from CMC measurements on the same system. Thus, although the β parameters determined from the different models are not strictly comparable, the agreement between the methods is satisfactory given the differing conditions for the two types of measurements. The values are also comparable to parameters determined for similar systems in the present work. The asymmetric model presented here appears satisfactory for fitting heats of mixing of ionic and nonionic surfactants. (27) Hines, J. D.; Thomas, R. K.; Garrett, P. R.; Rennie, G. K.; Penfold, J. J. Phys. Chem. B 1997, 101, 9215. (28) Haque, M. E., Das, A. R., Rakshit, A. K., Moulik, S. P. Langmuir 1996, 12, 4084. (29) Holland, P. M. In Structure/Performance Relationships in Surfactants; Rosen, M. J., Ed.; ACS Symposium Series 253; American Chemical Society: Washington, DC, 1984; p 141. (30) Holland, P. M. Adv. Colloid Interface Sci. 1986, 26, 111. (31) Hey, M. J.; MacTaggart, J. W. J. Chem. Soc., Faraday Trans. 1 1985, 81, 207.
Conclusions Results presented here indicate that asymmetry in mixed surfactant systems may be very common, though the use of CMC measurements to characterize nonideal surfactant interactions may sometimes mask the asymmetry because of the small effect of the asymmetry on CMCs and because of the intrinsic uncertainty of CMC measurements. The ability to experimentally demonstrate asymmetry in surfactant mixtures depends on the ability to measure either unimer concentrations, CMCs, or heats of mixing sufficiently accurately. In cases where accurate data are available and a composition dependence of the β parameters or activity coefficients is evident, one should consider a more detailed description such as that provided by the van Laar model. The NMR approach for studying mixed surfactant systems offers several unique and useful features. First, the NMR method gives two experimental values for each mixture, namely the unimer concentrations (or equivalently the micellar concentrations) for both surfactants. In contrast the CMC approach gives a single CMC value for each solution composition. Second, the NMR approach allows the determination of activity coefficients for each surfactant in the mixture for each composition studied. By examining and comparing the dependence of activity coefficients on micelle composition for each surfactant, one can identify systematic differences in the behavior of each surfactant. As shown in Figure 3, asymmetry can be convincingly demonstrated using this approach. Finally, the NMR approach avoids the necessity of determining the critical micelle concentration using techniques which can occasionally be somewhat ambiguous. For example, the use of surface tension measurements to determine CMCs in mixed surfactants does not always yield two linear branches and a clean break in the plot of the surface tension vs the logarithm of the concentration. Several cautions regarding the use of the NMR technique and the asymmetric regular solution model are in order. A disadvantage of the NMR approach is that when the CMCs of the two surfactants are widely different, it is difficult to design experimental conditions for which unimer concentrations of both surfactants can be measured. Also, the NMR approach requires use of a hydrophobic probe if both surfactants have significant unimer concentrations. One must exercise care since the probe may perturb the properties of interest. The asymmetric regular solution parameters may be somewhat dependent on parameters such as total surfactant concentration. For example, it is known that aggregation number and micelle geometry differ at high concentrations compared to concentrations near the CMC. Such geometric differences will undoubtedly influence the headgroup packing geometry and interaction energies. Headgroup packing can be influenced by the surfactant tails, as illustrated for the data for the C8E5-C12S mixture compared to the C12E5C12S mixture. The asymmetric regular solution parameters are therefore functions of both the headgroups and the tails. These cautions apply equally to the widely used symmetric one-parameter regular solution model. Compared to the symmetric regular solution model, the two-parameter asymmetric model advocated here gives improved fits to key parameters used to characterize mixed surfactant systems, including activity coefficients from NMR data, composition-dependent CMC data, and heats of mixing. It also predicts observed qualitative features of binary surfactant mixtures, especially asymmetric activity coefficients and composition-dependent regular
2668 Langmuir, Vol. 15, No. 8, 1999
solution β parameters. These results all support the value of the asymmetric model when high-precision data are available. Though the present treatment is not intended to be predictive, the resulting parametrization lends itself to analysis by predictive models. For example, the proportionality constants such as uAA can, in principle, be calculated using electrostatic theories for mixtures involving ionic surfactants.11 A wide range of experimental data support the asymmetric nature of intercomponent interactions in binary surfactant mixtures. Unless size effects (or other sources of asymmetry) are properly incorporated in both theory and analysis of experimental data, models which aspire to predict the β parameter may
Eads and Robosky
disagree with measured values, even if electrostatics or other intermolecular interactions are handled completely correctly by the theory. It is therefore hoped that the asymmetric two-parameter treatment advocated here may also contribute to the development of predictive models. Acknowledgment. We thank our colleagues Donn Rubingh, Robert Laughlin, and John Shelley, for their support and for many helpful discussions and suggestions. We also thank Gerard Stijntjes for providing the motivation and many important applications of this work. LA980525T
