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Demixing in Ternary Mixed Micelles
JOURNAL OF COLLOID AND INTERFACE SCIENCE ARTICLE NO.
200, 74–80 (1998)
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Demixing in Ternary Mixed Micelles M. Ben Ghoulam,* N. Moatadid,* A. Graciaa,† J. Lachaise,† G. Marion, and R. S. Schechter‡ ,1 *Faculty of Sciences, University Moulay Ismail, Mekne`s, Morocco; †LTEMPM, Universite´ de Pau et des Pays de l’Adour, Pau, France; and ‡Department of Chemical Engineering, University of Texas, Austin, Texas 78712 Received July 8, 1997; accepted November 17, 1997
bon moiety mixed with a fluorocarbon surfactant indicated that hydrocarbon-rich micelles and fluorocarbon-rich aggregates coexisted over some range of mixture compositions. Other investigators have confirmed their conclusions (2– 21). Micellar demixing in this surfactant system has been predicted based on both a group contribution method and a regular solution model (2, 6, 11, 14, 22–24). Mixed micelles composed of surfactants, all having hydrocarbon moieties, have been reported to form conjugate micellar phases (25–27) in binary surfactant mixtures. These binary systems each contain one surfactant having a complex hydrophile exhibiting both nonionic and ionic character and a nonionic surfactant. Apart from these special systems, few other examples of micelle demixing are known. Since aqueous solutions containing conjugate micellar phases may serve dual functions, i.e., detergency and wetting or solubilization of two different apolar compounds, then it is of practical interest to determine under what conditions, if any, micellar demixing occurs in systems composed of mixed hydrocarbon surfactants. The purpose of this paper is to propose a systematic approach to help identify the conditions for micellar demixing in ternary surfactant mixtures and once having established them verify the predictions. The system studied here is a mixture of cationic, anionic, and nonionic hydrocarbon surfactants. Blends of these three types of surfactants are used as detergents (16). The unique feature of this paper is that micellar demixing is shown to occur when certain proportions of these three different types of surfactant are blended. This may account for the advantages derived from detergent formulations that contain all three surfactant types.
Micellar demixing yielding conjugate micellar phases that coexist in an aqueous solution is known for a very few surfactant systems even though the presence of two distinct types of micelles in solution has practical application. Here, on the basis of a pseudophase regular solution model for surfactant chemical potentials, we predict micellar demixing to be widely prevalent in ternary surfactant mixtures that contain anionic, cationic, and nonionic surfactants. While demixing of micelles in these ternary systems has not previously been reported, the prediction is confirmed here on the basis of surface tension measurements. q 1998 Academic Press Key Words: mixed micelles; demixing.
INTRODUCTION
When mixtures of surfactants are dissolved in an aqueous solution, mixed micelles often form. A mixed micelle is an aggregate of surfactant molecules composed of the different types of surfactant present in the aqueous solution. In such cases, there exists a distribution of aggregation numbers as well as micellar compositions. However, if the aggregates are of sufficient size, then the deviation of sizes and compositions about their mean values will be small, permitting one to treat an individual micelle as a separate phase having a distinct composition. This ‘‘pseudophase’’ approximation has been successfully used to model mixed micelle behavior. It is used here as the basis for understanding some interesting and perhaps quite useful phenomena. The pseudophase model greatly simplifies thermodynamic considerations, but information relating to the distribution of aggregation numbers and compositions is not developed. Thus, we shall refer to the composition of certain micelles as though all the micelles of that type have precisely the same composition. In terms of the pseudophase model, it is reasonable to ask whether conditions exist such that micelles of two distinct compositions coexist at equilibrium. Such systems may be thought to contain conjugate micellar phases. As far as we know, Mukerjee and Yang (1) were the first to suggest this possibility. Their studies of a surfactant having a hydrocar1
DEMIXING IN TERNARY SYSTEMS
Micellar Pseudophases Lange and Beck (28) were apparently the first to apply this model to mixed micelles. The composition of a mixed micelle is defined by the mole fraction, Xi , of surfactant i in the micelle. In this simple model, the chemical potential of a surfactant in a micelle is given by mmic Å mV mic / RT ln gi Xi . i i
To whom correspondence should be addressed. 74
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is a standard chemical potential which in this Here mV mic i model does not depend on the aggregation number, the micellar shape, nor the arrangement of the various surfactants in the palisade layer. The activity coefficient, gi , takes into account nonidealities. Mixed micelles have been considered to be regular solutions with some success. For a regular solution, the following expression for the activity coefficients in terms of the dimensionless binary interaction parameters, bi , j ( Åai , j /RT ) has been used for multicomponent systems (29): N
N
i0 1
ln gj Å ∑ bi , j X 2i / ∑ ∑ [ bi , j / bk , j 0 bi , k ]Xi Xk . iÅ1 i xj
[2]
iÅ1 kÅ1 j xk xi
A few binary interaction coefficients for mixed micelles have been tabulated (30). These have been determined based on the mixture CMC. If for a binary regular solution b1,2 ú 2, then demixing may occur. Since for most (in fact, all reported in (29–32)) binary combinations of surfactants having hydrocarbon moieties b1,2 is negative, micellar demixing in binary mixtures of such surfactants is not expected. Insufficient data were given in (25–27) to evaluate the binary interaction coefficients of these unusual systems that have been reported to demix. Thus, it would seem that micellar demixing is a phenomenon restricted to mixtures of surfactant having disparate hydrophobes or complex multifunctional hydrophiles. It is known, however, that, in regular solutions, ternary mixtures may demix even if all of the binary interaction coefficients are negative. Thus, we may inquire as to the possibility of micellar phase separation in ternary mixtures of surfactants having similar hydrocarbon hydrophobes. The chemical potential of surfactant i dispersed in the aqueous phase may be written as miw Å mV iw / RT ln Ci Å mV iw / RT ln ai Cijk .
[3]
inventory is present as monomer. Thus, at the mixed micelle binary CMC, a1C12 Å X1C 01 exp[ b1,2 X 22 ] É z1CT
[5]
a2C12 Å X2C 02 exp[ b1,2 X 21 ] É z2CT ,
[6]
and
where C12 is the mixture CMC ( ÅC1 / C2 ) and CT is the total surfactant concentration. Equations [5] and [6] may be used in conjunction with measurements of the binary CMC as a function of z1 to determine the binary interaction parameter. This method is used here. Phase Separation in Ternary Regular Solutions We briefly review the conditions for thermodynamic stability of ternary systems. At certain micellar compositions (expressed by the Xi ) a ternary phase may become unstable with the resultant formation of conjugate phases. The conditions that ensure stability of a ternary phase are (33) m1,1 ú 0; m2,2 ú 0; ( m1,1m2,2 0 m21,2 ) ¢ 0,
where m i,j Å Ìmi / Ìnj . A phase becomes unstable at points in composition space where one or more of these strict inequalities is violated. The spinodal boundary separating stable from unstable states in composition space is determined by Eq. [8] (33): m1,1m2,2 0 m21,2 Å 0.
ai Cijk Å gi Xi C 0i ,
[4]
where C 0i is the CMC of surfactant i at the system counterion concentration, ai is the fraction of dispersed surfactant that is component i, and Cijk is the aqueous monomer (dispersed) surfactant concentration. We also let zi be the fraction of component i in the total surfactant inventory. It is the mole fraction in the original surfactant mixture. Note that this fraction, in general, differs from ai since a portion of the surfactant inventory is present in the form of micelles. For a binary system (i Å 1, 2) at a surfactant concentration such that micelles first form virtually the entire surfactant
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[8]
It can be shown that the following equation defines the spinodal surface of a ternary regular solution at constant temperature and pressure: LX1 X2 X3 / 2[ b1,2 X1 X2 / b1,3 X1 X3 / b2,3 X2 X3 ] Å 1,
For equilibrium between the surfactant in micelles with the dispersed surfactant, we must have (29)
[7]
[9]
where L Å b 21,2 / b 21,3 / b 22,3 0 2b1,2b1,3 0 2b1,2b2,3 0 2b1,3b2,3 . A solution of this equation may be written as [34] q
H { H 2 / LX1 X2 X3 Å 1,
[10]
where H Å b1,2 X1 X2 / b1,3 X1 X3 / b2,3 X2 X3 . The systems considered here are those for which bi , j õ 0. For these systems, the conditions yielding surfaces sepa-
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rating stable from unstable phases are those for which L ú 0. An examination of the function L reveals that L is positive when negative binary interaction coefficients satisfy the following inequality: q
q
q
0 b1,2 ú 0 b1,3 / 0 b2,3 .
[11]
This inequality is a necessary condition. There may or may not exist certain permitted values of Xi (0 ° Xi ° 1) that satisfy Eq. [10] with the consequent phase separation. However, demixing is assured by the inequality q
q
q
0 b1,2 ú 2 0 b1,3 / 2 0 b2,3 .
[12]
This is an important and useful result because it represents a necessary and sufficient condition. THE TERNARY SYSTEM
Chemicals Used The ternary system studied here is a mixture of anionic, cationic, and nonionic surfactants. The nonionic surfactant was a monoisomeric hexaethoxylate of dodecyl alcohol (C12H25 (OCH2CH2 )6OH) of high purity manufactured by Nikko Chemical Co. (designated NI). It was used as received. There was no minimum in the surface tension curve to indicate the presence of impurities. The anionic surfactant was a sodium dodecyl benzenesulfonate C12H25C6H4SO 30 Na / denoted here as SDBS. This surfactant was synthesized at The University of Texas at Austin and was purified by procedures previously reported (32). The cationic surfactant was tetradocyltrimethylammonium bromide C14H29N(CH3 ) 3/ Br 0 denoted here as TTAB. This surfactant was provided by the Aldrich Chemical Co. and was purified by recrystallization several times in an acetone–water mixture. The CMC’s were determined at 257C based on surface tension measurements using the Whilhelmy plate method (Kruss digital tensiometer). The CMC’s of these surfactants were determined by surface tension measurements to be Ca Å 2.4 1 10 03 M, Cn Å 6.8 1 10 05 M, and Cc Å 3.68 1 10 03 M, where the subscript refers to the surfactant type. Dimensionless Binary Interaction Coefficients The binary interaction coefficients for the three binary systems (anionic–cationic, anionic–nonionic, and cationic– nonionic) have been determined based on measurement of the critical micelle concentration of binary mixtures measured as a function of the overall solution composition. The mixture CMC values were determined by the position of a sharp break in the surface tension plotted as a function of the total surfactant concentration. These CMC values are
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FIG. 1. The mixture CMC shown as a function of the mole fraction ( Xn in the anionic – nonionic mixture; Xc in the mixtures with cationics ) of one of the surfactants in the binary mixtures. The data show the measured critical micelle concentrations. The lines show the values computed using the dimensionless binary interaction coefficients selected to give the best fit.
plotted in Fig. 1 as a function of mole fraction of one of the surfactants in the binary mixture. The binary interaction coefficients are determined by fitting Eqs. [5] and [6] to the data. The binary interaction coefficients best fitting the data are as follows: bac Å 025.5, ban Å 03.5, and bcn Å 02.5. Figure 1 shows a comparison of the experimental data with the calculated curves. These interaction coefficients satisfy inequality [11]; therefore, this system is expected to exhibit conjugate micellar phases. One dominant interaction coefficient is needed to satisfy inequality [11]. In this case, it is the cationic–anionic interaction. It should be noted that the influence of the counterion concentration has been ignored in determining the interaction coefficients, and these may, therefore, vary somewhat with the total surfactant concentration. However, the variation is expected to be modest since the cationic and anionic surfactants are virtually ion-paired, behaving as a nonionic entity which is not sensitive to the counterion concentrations. Calculation of the Phase Diagram The three binary interaction parameters determined above may be used to construct a phase diagram in micellar compo-
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FIG. 2. The micellar composition ternary phase diagram showing the calculated binodal curve, the spinodal curve, and the two critical points.
sition space. The spinodal curve is determined by Eq. [9]. A convenient display for the calculated results is a triangular diagram. Figure 2 shows the calculated ternary micellar demixing phase diagram complete with the spinodal, binodal curves, and critical points. Also shown are tie lines that define the conjugate micellar compositions. The binodal curve corresponds to two types of micelles in equilibrium and the tie lines are, therefore, found by equating the chemical potentials of surfactants in the two conjugate micellar phases designated as Xi and X i* . This leads to the following equations to be solved simultaneously (35): ln
Xa X * n Xn X * a
Å bn,a[(Xa 0 X * a ) 0 (Xn 0 X * n )] / ( bn,c 0 ba,c )(Xc 0 X * c ),
ln
Xc X * n Xn X * c
[13]
c Å 1 0 2( bn,a Xn Xa / bn,c Xn Xc / ba,c Xc Xa ) 0 LXn Xa Xc Å 0,
Å bn,c[(Xc 0 X * c ) 0 (Xn 0 X * n )]
[16]
and / ( bn,a 0 ba,c )(Xa 0 X * a ) 1 (Xn / X * n )ln / (Xc / X * c )ln
Xn X* n Xc
[14]
/ (Xa 0 X * a )ln Å0
Ìc Ì ln Cacn Ìc Ì ln Cacn 0 Å 0. ÌXa ÌXc ÌXc ÌXa
Xa X* a [15]
X* c
The tie lines are not quite vertical since ban x bcn . However, their essentially vertical orientation may be anticipated
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by visualizing cationic–anionic ion pairs to be a single surfactant that is nonionic in character. Thought of in this way, a given overall micellar composition represented by a point in the interior of the ternary diagram, Fig. 2, represents conjugate nonionic-rich micelles with the surplus ionic surfactant, those molecules not ion-paired, partitioning in roughly equal proportions between the two ‘‘nonionic’’ micelles. These conjugate micelles, although primarily nonionic in character, have different hydrophobes and, therefore, may serve a dual function such as solubilizing different compounds. The binodal curve shown in Fig. 2 encloses the spinodal curve except at the two critical points where the two curves have a common tangent. The critical points are found by simultaneous solution of the equations (36),
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[17]
Equation [16] is the same as Eq. [9], and Eq. [17] expresses the tangency of the spinodal and the binodal curves. The two critical points shown in Fig. 2 are at (Xa Å 0.64, Xc Å 0.08, Xn Å 0.28) and (0.07, 0.65, and 0.28). As the temperature increases, at a critical value Tc , these
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FIG. 3. The calculated monomer concentration as a function of the total surfactant concentration for the special ternary system (zn Å 0.3; za Å zc Å 0.35).
two critical points merge into one and then disappear. Beyond this critical point (called the second-order critical point) the system is always stable (see (33)). The coordinates of this critical point are Tc Å 231.337C, Xn Å 0.5, Xa Å 0.24, and Xc Å 0.26. The Dispersed Surfactant Concentration It is important to note that the phase diagram depicted by Fig. 2 does not present a complete picture of the physical chemistry of our system or all of its potential applications because the composition of the dispersed surfactant ( monomer ) in equilibrium with the conjugate micellar phases is not shown. The dispersed or monomer phase must be in equilibrium with the demixed micelles. The composition of the aqueous solution is, therefore, related to the composition of the micelles. Thus, associated with a point in the interior of the triangular diagram, Fig. 2 is a well-defined aqueous solution in equilibrium with the conjugate micellar phases. It is not a straightforward matter for an experimentalist increasing the overall surfactant concentration to anticipate the composition of the monomer as well as the concentration and compositions of the micelles. To illustrate this complexity, Fig. 3 shows the monomer concentrations as a function of the total surfactant concentration which is increased maintaining constant proportions of each surfactant type ( i.e., za Å zc Å 0.35; zn Å 0.3 ) . The first CMC ( first breakpoint ) which occurs essentially when the solution concentrations of the cationic and anionic surfactants are sufficient to form micelles. This critical micelle concentration is represented by the binary diagram, Fig. 1. The cationic – anionic micelles formed will
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contain only a small fraction of nonionic surfactant ( Xn ! 1 ) , and the nonionic surfactant monomer concentration will continue to increase steadily as the total surfactant concentration is increased. When the nonionic solution concentration has increased sufficiently, nonionic micelles then form. This is the second CMC shown by Fig. 3. To a practitioner, the monomer concentration may be as important as the composition of the micelles since it is the monomer composition that determines the surface tension or the adsorption ( 37, 38 ) . Figure 4 shows the surface tensions measured as a function of the surfactant concentration for a mixture of the three surfactants corresponding to the conditions used in calculating Fig. 3. It is possible to match the surface tension results to the calculated monomer concentrations. Figure 5 shows an entire phase map for the case in which the SDBS / TTAB ratio is maintained at unity ( za / zc Å 1 ) . To understand this map, it is helpful to refer to Fig. 3. The micellar phase map shown by Fig. 5 is divided into four regions delineated by the curves f, g, and h. In Region I, below f, there are no micelles. In Region III between f and g there are cationic – anionic-rich micelles whereas in Region IV there are nonionic-rich micelles. In Region II, both types of micelles coexist. Point E corresponds to the monomeric composition at which demixing occurs at the CMC. Curve g begins at the CMC of the pure nonionic surfactant and then approaches a vertical line at zn Å 0.04, which is at the intersection of the binodal curve in Fig. 2 with a line connecting the point ( xa Å xc Å 0.5 ) to the nonionic vertex. The data points shown on the phase map in Fig. 5 are determined by the break points in the surface tension curve as illustrated by Fig. 4. The open circles are points ob-
FIG. 4. The surface tension shown as a function of the surfactant concentration for conditions corresponding to Fig. 3 (zn Å 0.3; za Å zc Å 0.35).
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CONCLUSIONS
Based upon a relationship between the binary interaction coefficients that is obtained by considering the micelles to be a regular solution, micellar demixing is predicted to occur in a ternary mixture of anionic, cationic, and nonionic surfactants in aqueous solutions. Micellar demixing in ternary systems has not been previously reported in prior work, although ternary surfactant systems have found application. The predicted micellar demixing was confirmed experimentally from surface tension measurements. Even though the predicted values of the solution concentrations at which micellar demixing was expected to occur do not closely correspond to the measured values, the fundamental inequality [12] does provide guidance in the search for surfactant systems that exhibit micellar demixing. REFERENCES 1. 2. 3. 4. 5.
FIG. 5. Ternary phase map delineating various regions of micellar compositions as a function of the total surfactant concentration and the proportion of nonionic surfactant (for za /zc Å 1.0). Region I, no micelles; Region II, conjugate micellar phases; Region III, cationic–nonionic micelles; and Region IV, nonionic micelles. The curves are calculated. Points represent experimental results.
tained from surface tension measurements that herald the formation of conjugate micellar phases. The second CMC is shown by Fig. 4. We expected these points to lie on curve g as predicted by regular solution theory. A discrepancy is seen. This discrepancy is neither surprising nor particular disturbing because the calculated second CMC hinges on the validity of the model represented by Eq. [ 4 ] applied to surfactant aggregates. Clearly, further work to refine the thermodynamic model is warranted. It is, however, important to stress that the existence of the second CMC, and, therefore, existence of conjugate micellar phases has been predicted entirely based upon knowledge of the binary interaction coefficients determined from the surface tension of binary surfactant mixtures. Application of regular solution model seems to be an appropriate method for evaluating the potential for micellar demixing in systems not known to exhibit such phenomena. This demonstrates our primary contention that micellar demixing may occur when the surfactants in a mixture all have hydrocarbon hydrophobes and when the binary surfactant pairs are miscible in all proportions.
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6. 7. 8. 9. 10. 11. 12. 13. 14. 15.
16.
17. 18. 19. 20. 21. 22. 23. 24. 25.
26.
Mukerjee, P., and Yang, A. Y., J. Phys. Chem. 80, 1388 (1976). Funasaki, N., and Hada, S., Chem. Lett. 717 (1979). Funasaki, N., and Hada, S., J. Phys. Chem. 84, 736 (1980). Funasaki, N., and Hada, S., J. Colloid Interface Sci. 73, 425 (1980). Ben Ghoulam, M., Moatadid, N., Graciaa, A., Marion, G., and Lachaise, J., Langmuir 12, 5048 (1996). Shinoda, K., and Nomura, T., J. Phys. Chem. 84, 365 (1980). Funasaki, N., and Hada, S., J. Colloid Interface Sci. 78, 376 (1980). Funasaki, N., and Hada, S., J. Phys. Chem. 86, 2504 (1982). Mukerjee, P., J. Am. Oil Chem. Soc. 59, 573 (1982). Meguro, K., Ueno, M., and Suzjki, T., Yukagaku 31, 909 (1982). Funasaki, N., and Hada, S., J. Phys. Chem. 87, 342 (1983). Funasaki, N., Hada, S., and Neya, S., Bull. Chem. Soc. Jpn. 56, 3839 (1983). Harada, S., and Sahara, H., Chem. Lett. 1199 (1984). Asakawa, T., Johten, K., Miyagishi, S., and Nishida, M., Langmuir 1, 347 (1985). Zhao, G. X., and Zhu, B. Y., in ‘‘Phenomena in Mixed Surfactant Systems’’ (J. F. Scamehorn, Ed.), ACS Symposium Series 311, Chap. 14. American Chemical Society, Washington, DC, 1986. Scamehorn, J. F., in ‘‘Phenomena in Mixed Surfactant Systems’’ (J. F. Scamehorn, Ed.), ACS Symposium Series 311, Chap. 1. American Chemical Society, Washington, DC, 1986. Asakawa, T., Miyagishi, S., and Nishida, M., Langmuir 3, 821 (1987). Asakawa, T., Johten, K., Miyagishi, S., and Nishida, M., Langmuir 4, 136 (1988). Sugihara, G., Yamamoto, M., Wada, Y., Murata, Y., and Ikawa, Y., J. Solution Chem. 17, 225 (1988). Kalyanasundram, K., Langmuir 4, 942 (1988). Asakawa, T., Mouri, M., Miyagishi, S., and Nishida, M., Langmuir 5, 343 (1989). Kamrath, R. F., and Franses, E. I., Ind. Eng. Chem. Fund. 22, 230 (1983). Kamrath, R. F., and Franses, E. I., J. Phys. Chem. 88, 1642 (1984). Asakawa, T., Johten, K., Miyagishi, S., and Nishida, M., J. Colloid Interface Sci. 104, 279 (1985). Xia, J., Dubin, P. L., and Zhang, H., in ‘‘Mixed Surfactant Systems’’ (P. H. Holland and D. N. Rubingh, Eds.), ACS Symposium Series 501, Chap. 14. American Chemical Society, Washington, DC, 1992. Ogino, K., and Abe, M., in ‘‘Mixed Surfactant Systems’’ (P. H. Holland and D. N. Rubingh, Eds.), ACS Symposium Series 501, Chap. 8. American Chemical Society, Washington, DC, 1992.
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27. Abe, M., and Ogino, K., in ‘‘Mixed Surfactant Systems’’ (K. Ogino and M. Abe, Eds.). Surfactant Science Series 46, Chap. 13. Dekker, New York, 1993. 28. Lange, H., and Beck, K. H., Kolloid Z. Z. Polym. 251, 424 (1973). 29. Holland, P. M., and Rubingh, D. N., J. Phys. Chem. 87, 1984 ( 1983 ) . 30. Holland, P. M., in ‘‘Mixed Surfactant Systems’’ (P. H. Holland and D. N. Rubingh, Eds.), ACS Symposium Series 501, Chap. 2. American Chemical Society, Washington, DC, 1992. 31. Graciaa, A., Ben Ghoulam, M., Marion, G., and Lachaise, J., J. Phys. Chem. 93, 4167 (1989).
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32. Graciaa, A., Ben Ghoulam, M., Marion, G., and Lachaise, J., Prog. Colloid Polym. Sci. 89, 288 (1992). 33. Prigogine, I., and Defay, R., ‘‘Chemical Thermodynamics.’’ Longmans, New York, 1967. 34. Meijiring, J. L., Philips Res. Rep. 5, 333 (1950). 35. Meijiring, J. L., Phillips Res. Rep. 6, 183 (1951). 36. Lupis, C. H. P., in ‘‘Chemical Thermodynamics of Materials,’’ Chap. 11. North-Holland, Amsterdam, 1983. 37. Clint, J. H., J. Chem. Soc., Faraday Trans. 71, 1327 (1975). 38. Trogus, F. J., Schechter, R. S., and Wade, W. H., J. Colloid Interface Sci. 70, 293 (1979).
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CS975318
Demixing in Ternary Mixed Micelles M. Ben Ghoulam,* N. Moatadid,* A. Graciaa,† J. Lachaise,† G. Marion, and R. S. Schechter‡ ,1 *Faculty of Sciences, University Moulay Ismail, Mekne`s, Morocco; †LTEMPM, Universite´ de Pau et des Pays de l’Adour, Pau, France; and ‡Department of Chemical Engineering, University of Texas, Austin, Texas 78712 Received July 8, 1997; accepted November 17, 1997
bon moiety mixed with a fluorocarbon surfactant indicated that hydrocarbon-rich micelles and fluorocarbon-rich aggregates coexisted over some range of mixture compositions. Other investigators have confirmed their conclusions (2– 21). Micellar demixing in this surfactant system has been predicted based on both a group contribution method and a regular solution model (2, 6, 11, 14, 22–24). Mixed micelles composed of surfactants, all having hydrocarbon moieties, have been reported to form conjugate micellar phases (25–27) in binary surfactant mixtures. These binary systems each contain one surfactant having a complex hydrophile exhibiting both nonionic and ionic character and a nonionic surfactant. Apart from these special systems, few other examples of micelle demixing are known. Since aqueous solutions containing conjugate micellar phases may serve dual functions, i.e., detergency and wetting or solubilization of two different apolar compounds, then it is of practical interest to determine under what conditions, if any, micellar demixing occurs in systems composed of mixed hydrocarbon surfactants. The purpose of this paper is to propose a systematic approach to help identify the conditions for micellar demixing in ternary surfactant mixtures and once having established them verify the predictions. The system studied here is a mixture of cationic, anionic, and nonionic hydrocarbon surfactants. Blends of these three types of surfactants are used as detergents (16). The unique feature of this paper is that micellar demixing is shown to occur when certain proportions of these three different types of surfactant are blended. This may account for the advantages derived from detergent formulations that contain all three surfactant types.
Micellar demixing yielding conjugate micellar phases that coexist in an aqueous solution is known for a very few surfactant systems even though the presence of two distinct types of micelles in solution has practical application. Here, on the basis of a pseudophase regular solution model for surfactant chemical potentials, we predict micellar demixing to be widely prevalent in ternary surfactant mixtures that contain anionic, cationic, and nonionic surfactants. While demixing of micelles in these ternary systems has not previously been reported, the prediction is confirmed here on the basis of surface tension measurements. q 1998 Academic Press Key Words: mixed micelles; demixing.
INTRODUCTION
When mixtures of surfactants are dissolved in an aqueous solution, mixed micelles often form. A mixed micelle is an aggregate of surfactant molecules composed of the different types of surfactant present in the aqueous solution. In such cases, there exists a distribution of aggregation numbers as well as micellar compositions. However, if the aggregates are of sufficient size, then the deviation of sizes and compositions about their mean values will be small, permitting one to treat an individual micelle as a separate phase having a distinct composition. This ‘‘pseudophase’’ approximation has been successfully used to model mixed micelle behavior. It is used here as the basis for understanding some interesting and perhaps quite useful phenomena. The pseudophase model greatly simplifies thermodynamic considerations, but information relating to the distribution of aggregation numbers and compositions is not developed. Thus, we shall refer to the composition of certain micelles as though all the micelles of that type have precisely the same composition. In terms of the pseudophase model, it is reasonable to ask whether conditions exist such that micelles of two distinct compositions coexist at equilibrium. Such systems may be thought to contain conjugate micellar phases. As far as we know, Mukerjee and Yang (1) were the first to suggest this possibility. Their studies of a surfactant having a hydrocar1
DEMIXING IN TERNARY SYSTEMS
Micellar Pseudophases Lange and Beck (28) were apparently the first to apply this model to mixed micelles. The composition of a mixed micelle is defined by the mole fraction, Xi , of surfactant i in the micelle. In this simple model, the chemical potential of a surfactant in a micelle is given by mmic Å mV mic / RT ln gi Xi . i i
To whom correspondence should be addressed. 74
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is a standard chemical potential which in this Here mV mic i model does not depend on the aggregation number, the micellar shape, nor the arrangement of the various surfactants in the palisade layer. The activity coefficient, gi , takes into account nonidealities. Mixed micelles have been considered to be regular solutions with some success. For a regular solution, the following expression for the activity coefficients in terms of the dimensionless binary interaction parameters, bi , j ( Åai , j /RT ) has been used for multicomponent systems (29): N
N
i0 1
ln gj Å ∑ bi , j X 2i / ∑ ∑ [ bi , j / bk , j 0 bi , k ]Xi Xk . iÅ1 i xj
[2]
iÅ1 kÅ1 j xk xi
A few binary interaction coefficients for mixed micelles have been tabulated (30). These have been determined based on the mixture CMC. If for a binary regular solution b1,2 ú 2, then demixing may occur. Since for most (in fact, all reported in (29–32)) binary combinations of surfactants having hydrocarbon moieties b1,2 is negative, micellar demixing in binary mixtures of such surfactants is not expected. Insufficient data were given in (25–27) to evaluate the binary interaction coefficients of these unusual systems that have been reported to demix. Thus, it would seem that micellar demixing is a phenomenon restricted to mixtures of surfactant having disparate hydrophobes or complex multifunctional hydrophiles. It is known, however, that, in regular solutions, ternary mixtures may demix even if all of the binary interaction coefficients are negative. Thus, we may inquire as to the possibility of micellar phase separation in ternary mixtures of surfactants having similar hydrocarbon hydrophobes. The chemical potential of surfactant i dispersed in the aqueous phase may be written as miw Å mV iw / RT ln Ci Å mV iw / RT ln ai Cijk .
[3]
inventory is present as monomer. Thus, at the mixed micelle binary CMC, a1C12 Å X1C 01 exp[ b1,2 X 22 ] É z1CT
[5]
a2C12 Å X2C 02 exp[ b1,2 X 21 ] É z2CT ,
[6]
and
where C12 is the mixture CMC ( ÅC1 / C2 ) and CT is the total surfactant concentration. Equations [5] and [6] may be used in conjunction with measurements of the binary CMC as a function of z1 to determine the binary interaction parameter. This method is used here. Phase Separation in Ternary Regular Solutions We briefly review the conditions for thermodynamic stability of ternary systems. At certain micellar compositions (expressed by the Xi ) a ternary phase may become unstable with the resultant formation of conjugate phases. The conditions that ensure stability of a ternary phase are (33) m1,1 ú 0; m2,2 ú 0; ( m1,1m2,2 0 m21,2 ) ¢ 0,
where m i,j Å Ìmi / Ìnj . A phase becomes unstable at points in composition space where one or more of these strict inequalities is violated. The spinodal boundary separating stable from unstable states in composition space is determined by Eq. [8] (33): m1,1m2,2 0 m21,2 Å 0.
ai Cijk Å gi Xi C 0i ,
[4]
where C 0i is the CMC of surfactant i at the system counterion concentration, ai is the fraction of dispersed surfactant that is component i, and Cijk is the aqueous monomer (dispersed) surfactant concentration. We also let zi be the fraction of component i in the total surfactant inventory. It is the mole fraction in the original surfactant mixture. Note that this fraction, in general, differs from ai since a portion of the surfactant inventory is present in the form of micelles. For a binary system (i Å 1, 2) at a surfactant concentration such that micelles first form virtually the entire surfactant
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[8]
It can be shown that the following equation defines the spinodal surface of a ternary regular solution at constant temperature and pressure: LX1 X2 X3 / 2[ b1,2 X1 X2 / b1,3 X1 X3 / b2,3 X2 X3 ] Å 1,
For equilibrium between the surfactant in micelles with the dispersed surfactant, we must have (29)
[7]
[9]
where L Å b 21,2 / b 21,3 / b 22,3 0 2b1,2b1,3 0 2b1,2b2,3 0 2b1,3b2,3 . A solution of this equation may be written as [34] q
H { H 2 / LX1 X2 X3 Å 1,
[10]
where H Å b1,2 X1 X2 / b1,3 X1 X3 / b2,3 X2 X3 . The systems considered here are those for which bi , j õ 0. For these systems, the conditions yielding surfaces sepa-
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rating stable from unstable phases are those for which L ú 0. An examination of the function L reveals that L is positive when negative binary interaction coefficients satisfy the following inequality: q
q
q
0 b1,2 ú 0 b1,3 / 0 b2,3 .
[11]
This inequality is a necessary condition. There may or may not exist certain permitted values of Xi (0 ° Xi ° 1) that satisfy Eq. [10] with the consequent phase separation. However, demixing is assured by the inequality q
q
q
0 b1,2 ú 2 0 b1,3 / 2 0 b2,3 .
[12]
This is an important and useful result because it represents a necessary and sufficient condition. THE TERNARY SYSTEM
Chemicals Used The ternary system studied here is a mixture of anionic, cationic, and nonionic surfactants. The nonionic surfactant was a monoisomeric hexaethoxylate of dodecyl alcohol (C12H25 (OCH2CH2 )6OH) of high purity manufactured by Nikko Chemical Co. (designated NI). It was used as received. There was no minimum in the surface tension curve to indicate the presence of impurities. The anionic surfactant was a sodium dodecyl benzenesulfonate C12H25C6H4SO 30 Na / denoted here as SDBS. This surfactant was synthesized at The University of Texas at Austin and was purified by procedures previously reported (32). The cationic surfactant was tetradocyltrimethylammonium bromide C14H29N(CH3 ) 3/ Br 0 denoted here as TTAB. This surfactant was provided by the Aldrich Chemical Co. and was purified by recrystallization several times in an acetone–water mixture. The CMC’s were determined at 257C based on surface tension measurements using the Whilhelmy plate method (Kruss digital tensiometer). The CMC’s of these surfactants were determined by surface tension measurements to be Ca Å 2.4 1 10 03 M, Cn Å 6.8 1 10 05 M, and Cc Å 3.68 1 10 03 M, where the subscript refers to the surfactant type. Dimensionless Binary Interaction Coefficients The binary interaction coefficients for the three binary systems (anionic–cationic, anionic–nonionic, and cationic– nonionic) have been determined based on measurement of the critical micelle concentration of binary mixtures measured as a function of the overall solution composition. The mixture CMC values were determined by the position of a sharp break in the surface tension plotted as a function of the total surfactant concentration. These CMC values are
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FIG. 1. The mixture CMC shown as a function of the mole fraction ( Xn in the anionic – nonionic mixture; Xc in the mixtures with cationics ) of one of the surfactants in the binary mixtures. The data show the measured critical micelle concentrations. The lines show the values computed using the dimensionless binary interaction coefficients selected to give the best fit.
plotted in Fig. 1 as a function of mole fraction of one of the surfactants in the binary mixture. The binary interaction coefficients are determined by fitting Eqs. [5] and [6] to the data. The binary interaction coefficients best fitting the data are as follows: bac Å 025.5, ban Å 03.5, and bcn Å 02.5. Figure 1 shows a comparison of the experimental data with the calculated curves. These interaction coefficients satisfy inequality [11]; therefore, this system is expected to exhibit conjugate micellar phases. One dominant interaction coefficient is needed to satisfy inequality [11]. In this case, it is the cationic–anionic interaction. It should be noted that the influence of the counterion concentration has been ignored in determining the interaction coefficients, and these may, therefore, vary somewhat with the total surfactant concentration. However, the variation is expected to be modest since the cationic and anionic surfactants are virtually ion-paired, behaving as a nonionic entity which is not sensitive to the counterion concentrations. Calculation of the Phase Diagram The three binary interaction parameters determined above may be used to construct a phase diagram in micellar compo-
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FIG. 2. The micellar composition ternary phase diagram showing the calculated binodal curve, the spinodal curve, and the two critical points.
sition space. The spinodal curve is determined by Eq. [9]. A convenient display for the calculated results is a triangular diagram. Figure 2 shows the calculated ternary micellar demixing phase diagram complete with the spinodal, binodal curves, and critical points. Also shown are tie lines that define the conjugate micellar compositions. The binodal curve corresponds to two types of micelles in equilibrium and the tie lines are, therefore, found by equating the chemical potentials of surfactants in the two conjugate micellar phases designated as Xi and X i* . This leads to the following equations to be solved simultaneously (35): ln
Xa X * n Xn X * a
Å bn,a[(Xa 0 X * a ) 0 (Xn 0 X * n )] / ( bn,c 0 ba,c )(Xc 0 X * c ),
ln
Xc X * n Xn X * c
[13]
c Å 1 0 2( bn,a Xn Xa / bn,c Xn Xc / ba,c Xc Xa ) 0 LXn Xa Xc Å 0,
Å bn,c[(Xc 0 X * c ) 0 (Xn 0 X * n )]
[16]
and / ( bn,a 0 ba,c )(Xa 0 X * a ) 1 (Xn / X * n )ln / (Xc / X * c )ln
Xn X* n Xc
[14]
/ (Xa 0 X * a )ln Å0
Ìc Ì ln Cacn Ìc Ì ln Cacn 0 Å 0. ÌXa ÌXc ÌXc ÌXa
Xa X* a [15]
X* c
The tie lines are not quite vertical since ban x bcn . However, their essentially vertical orientation may be anticipated
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by visualizing cationic–anionic ion pairs to be a single surfactant that is nonionic in character. Thought of in this way, a given overall micellar composition represented by a point in the interior of the ternary diagram, Fig. 2, represents conjugate nonionic-rich micelles with the surplus ionic surfactant, those molecules not ion-paired, partitioning in roughly equal proportions between the two ‘‘nonionic’’ micelles. These conjugate micelles, although primarily nonionic in character, have different hydrophobes and, therefore, may serve a dual function such as solubilizing different compounds. The binodal curve shown in Fig. 2 encloses the spinodal curve except at the two critical points where the two curves have a common tangent. The critical points are found by simultaneous solution of the equations (36),
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[17]
Equation [16] is the same as Eq. [9], and Eq. [17] expresses the tangency of the spinodal and the binodal curves. The two critical points shown in Fig. 2 are at (Xa Å 0.64, Xc Å 0.08, Xn Å 0.28) and (0.07, 0.65, and 0.28). As the temperature increases, at a critical value Tc , these
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BEN GHOULAM ET AL.
FIG. 3. The calculated monomer concentration as a function of the total surfactant concentration for the special ternary system (zn Å 0.3; za Å zc Å 0.35).
two critical points merge into one and then disappear. Beyond this critical point (called the second-order critical point) the system is always stable (see (33)). The coordinates of this critical point are Tc Å 231.337C, Xn Å 0.5, Xa Å 0.24, and Xc Å 0.26. The Dispersed Surfactant Concentration It is important to note that the phase diagram depicted by Fig. 2 does not present a complete picture of the physical chemistry of our system or all of its potential applications because the composition of the dispersed surfactant ( monomer ) in equilibrium with the conjugate micellar phases is not shown. The dispersed or monomer phase must be in equilibrium with the demixed micelles. The composition of the aqueous solution is, therefore, related to the composition of the micelles. Thus, associated with a point in the interior of the triangular diagram, Fig. 2 is a well-defined aqueous solution in equilibrium with the conjugate micellar phases. It is not a straightforward matter for an experimentalist increasing the overall surfactant concentration to anticipate the composition of the monomer as well as the concentration and compositions of the micelles. To illustrate this complexity, Fig. 3 shows the monomer concentrations as a function of the total surfactant concentration which is increased maintaining constant proportions of each surfactant type ( i.e., za Å zc Å 0.35; zn Å 0.3 ) . The first CMC ( first breakpoint ) which occurs essentially when the solution concentrations of the cationic and anionic surfactants are sufficient to form micelles. This critical micelle concentration is represented by the binary diagram, Fig. 1. The cationic – anionic micelles formed will
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contain only a small fraction of nonionic surfactant ( Xn ! 1 ) , and the nonionic surfactant monomer concentration will continue to increase steadily as the total surfactant concentration is increased. When the nonionic solution concentration has increased sufficiently, nonionic micelles then form. This is the second CMC shown by Fig. 3. To a practitioner, the monomer concentration may be as important as the composition of the micelles since it is the monomer composition that determines the surface tension or the adsorption ( 37, 38 ) . Figure 4 shows the surface tensions measured as a function of the surfactant concentration for a mixture of the three surfactants corresponding to the conditions used in calculating Fig. 3. It is possible to match the surface tension results to the calculated monomer concentrations. Figure 5 shows an entire phase map for the case in which the SDBS / TTAB ratio is maintained at unity ( za / zc Å 1 ) . To understand this map, it is helpful to refer to Fig. 3. The micellar phase map shown by Fig. 5 is divided into four regions delineated by the curves f, g, and h. In Region I, below f, there are no micelles. In Region III between f and g there are cationic – anionic-rich micelles whereas in Region IV there are nonionic-rich micelles. In Region II, both types of micelles coexist. Point E corresponds to the monomeric composition at which demixing occurs at the CMC. Curve g begins at the CMC of the pure nonionic surfactant and then approaches a vertical line at zn Å 0.04, which is at the intersection of the binodal curve in Fig. 2 with a line connecting the point ( xa Å xc Å 0.5 ) to the nonionic vertex. The data points shown on the phase map in Fig. 5 are determined by the break points in the surface tension curve as illustrated by Fig. 4. The open circles are points ob-
FIG. 4. The surface tension shown as a function of the surfactant concentration for conditions corresponding to Fig. 3 (zn Å 0.3; za Å zc Å 0.35).
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CONCLUSIONS
Based upon a relationship between the binary interaction coefficients that is obtained by considering the micelles to be a regular solution, micellar demixing is predicted to occur in a ternary mixture of anionic, cationic, and nonionic surfactants in aqueous solutions. Micellar demixing in ternary systems has not been previously reported in prior work, although ternary surfactant systems have found application. The predicted micellar demixing was confirmed experimentally from surface tension measurements. Even though the predicted values of the solution concentrations at which micellar demixing was expected to occur do not closely correspond to the measured values, the fundamental inequality [12] does provide guidance in the search for surfactant systems that exhibit micellar demixing. REFERENCES 1. 2. 3. 4. 5.
FIG. 5. Ternary phase map delineating various regions of micellar compositions as a function of the total surfactant concentration and the proportion of nonionic surfactant (for za /zc Å 1.0). Region I, no micelles; Region II, conjugate micellar phases; Region III, cationic–nonionic micelles; and Region IV, nonionic micelles. The curves are calculated. Points represent experimental results.
tained from surface tension measurements that herald the formation of conjugate micellar phases. The second CMC is shown by Fig. 4. We expected these points to lie on curve g as predicted by regular solution theory. A discrepancy is seen. This discrepancy is neither surprising nor particular disturbing because the calculated second CMC hinges on the validity of the model represented by Eq. [ 4 ] applied to surfactant aggregates. Clearly, further work to refine the thermodynamic model is warranted. It is, however, important to stress that the existence of the second CMC, and, therefore, existence of conjugate micellar phases has been predicted entirely based upon knowledge of the binary interaction coefficients determined from the surface tension of binary surfactant mixtures. Application of regular solution model seems to be an appropriate method for evaluating the potential for micellar demixing in systems not known to exhibit such phenomena. This demonstrates our primary contention that micellar demixing may occur when the surfactants in a mixture all have hydrocarbon hydrophobes and when the binary surfactant pairs are miscible in all proportions.
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6. 7. 8. 9. 10. 11. 12. 13. 14. 15.
16.
17. 18. 19. 20. 21. 22. 23. 24. 25.
26.
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27. Abe, M., and Ogino, K., in ‘‘Mixed Surfactant Systems’’ (K. Ogino and M. Abe, Eds.). Surfactant Science Series 46, Chap. 13. Dekker, New York, 1993. 28. Lange, H., and Beck, K. H., Kolloid Z. Z. Polym. 251, 424 (1973). 29. Holland, P. M., and Rubingh, D. N., J. Phys. Chem. 87, 1984 ( 1983 ) . 30. Holland, P. M., in ‘‘Mixed Surfactant Systems’’ (P. H. Holland and D. N. Rubingh, Eds.), ACS Symposium Series 501, Chap. 2. American Chemical Society, Washington, DC, 1992. 31. Graciaa, A., Ben Ghoulam, M., Marion, G., and Lachaise, J., J. Phys. Chem. 93, 4167 (1989).
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32. Graciaa, A., Ben Ghoulam, M., Marion, G., and Lachaise, J., Prog. Colloid Polym. Sci. 89, 288 (1992). 33. Prigogine, I., and Defay, R., ‘‘Chemical Thermodynamics.’’ Longmans, New York, 1967. 34. Meijiring, J. L., Philips Res. Rep. 5, 333 (1950). 35. Meijiring, J. L., Phillips Res. Rep. 6, 183 (1951). 36. Lupis, C. H. P., in ‘‘Chemical Thermodynamics of Materials,’’ Chap. 11. North-Holland, Amsterdam, 1983. 37. Clint, J. H., J. Chem. Soc., Faraday Trans. 71, 1327 (1975). 38. Trogus, F. J., Schechter, R. S., and Wade, W. H., J. Colloid Interface Sci. 70, 293 (1979).
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