Micellization of Ionic Surfactants in Nonaqueous Solvents (24) M. L. Williams, R. F. Landel, and J. D. Ferry, J. Amer. Chem. Soc., 77, 3701 (1955). (25) J. D. Ferry, ”Viscoelastic Properties of Polymers,” Wiley, New York, N.Y., 1970, p 292. (26) M. H. Cohen and D. Turnbull, J. Chem. fhys., 31, 1164 (1959). (27) D. Turnbull and M. H. Cohen, J. Chem. fhys., 34, 120 (1961). (28) A. J. Kovacs, Fortschr. Hochpolym. Forsch., Ed. 3, 394 (1963). (29) G. Adam and J. H. Gibbs, J. Chem. fhys., 43, 139 (1965). (30) J. H. Gibbs and E. A. Di Marzio, J. Chem. fhys., 28, 373 (1958).
287 (31) I. M. Ward, “Mechanical Properties of Solid Polymers,” Wiley, New York, N.Y., 1971, p 156. (32) N. E. Hill, W. E. Vaughan, A. H. Price, and M. Davies, “Dielectric Properties and Molecular Behaviour,” Van Nostrand-Reinhold, London, 1969, p 69. (33) J. P. Soulier, B. Chabert, J. Chauchard, P. Berticat, and J. F. May, to be submitted for publication. (34) R. F. Boyer, Rubber Chem. Techno/., 36, 1303 (1963); J. folym. Sci. C, 14, 267 (1966).
A Multiple-Equilibrium Model for the Micellization of Ionic Surf actants in Nonaqueous Solvents’ Norbert Muller Department of Chemistry, Purdue University, West Lafayette, lndiana 47907 (Received March 18, 1974; Revised Manuscript Received October 29, 7974)
It is demonstrated that the reported concentration dependence of nmr chemical shifts for solutions of alkylammonium carboxylates in organic solvents can be reproduced equally well using either a single-equilibrium model where the micelle size is a disposable parameter or a multiple-equilibrium model where it is assumed that dimers and all higher species with even aggregation numbers coexist in each solution. The latter model is suggested by simple electrostatic energy calculations which show that trimers and other small odd-order aggregates should disproportionate to give the more stable even-order species. Although the available data do not suffice to determine which scheme is better, the multiple-equilibrium model seems more acceptable on physical grounds and avoids some difficulties encountered with the single-equilibrium interpretation. A major point of difference is that the single-equilibrium approach appears to show that only small micelles are present, but with the multiple-equilibrium model it is seen that the data are consistent with the presence, in the more concentrated solutions, of significant numbers of micelles with aggregation numbers of 20 or more.
Introduction Nuclear magnetic resonance spectroscopy has recently been used to study the formation of “inverted” micelles of alkylammonium carboxylate surfactants in organic solv e n t ~ . ~For - ~ each surfactant-solvent pair, the chemical shift, 6, of a selected probe nucleus depends in a characteristic way on the total surfactant concentration, [SO],and the data can be rationalized on the assumption that monomers, s, are in equilibrium with micelles, s,, having a fixed aggregation number, m, that is mS
e s,
K =
[s,l/[slm
and
Here 6(S) and 6(S,) are the chemical shifts for the monomeric and the micellar forms, evaluated by extrapolating the data to zero concentration, for 6(S), or to infinite conl/[So] 0) for 6(S,). The equilibrium concentration (k, stant K, defined subject to the assumption of unit activity coefficients, and the aggregation number are then evaluated by constructing a plot6 of log m [S,] against log [SI, which should yield a straight line with slope m and intercept log mK. Once the parameters are thus fixed, this sin-
-
gle-equilibrium model leads to calculated values of 6 as a function of [SO]which are in excellent accord with observations, The derived values of m are always small, ranging between about 3 and 8, but for a given surfactant m is by no means independent of solvent4or t e m p e r a t ~ r e . ~ Several considerations prompt one to question whether or not the characterization of each surfactant-solvent pair at a given temperature by means of the four parameters, 6(S), 6(S,), m, and K , represents the optimum procedure for interpreting these data. One disturbing feature is the occurrence4 of nonintegral values of m. Another is the enigmatic variation of m with experimental conditions, since no attempt to visualize the aggregates in terms of structural models (see below) accounts in a reasonable way for the fact that small changes in temperature or solvent composition can apparently change the m value of the most stable micelle from 3 to 4 or 5. Moreover, it is inconvenient to compare results for a surfactant in different solvents, or in the same solvent at different temperatures, when each set of data refers to a different aggregation process. In particular, the thermodynamic relation d In K / d ( l / T ) = -AHO/R (31 cannot be used to find the enthalpy of micellization when the micelle size is strongly temperature dependent. The Journal of Physical Chemistry, Voi. 79, No. 3, 1975
288
Norbert Muller
It seems likely that these difficulties may be avoided if one turns to a multiple-equilibrium model of micellization,7-9 explicitly recognizing that micelles of several different sizes coexist in each solution and that the derived values of m represent average aggregation numbers. Although a number of properties of such multiple-equilibrium systems have been derived and presented, the usefulness of these models has been restricted by the fact that far more parameters are needed to define the equilibria than can be evaluated experimentally. It is therefore necessary to make sweeping simplifications. Restricting considerations to micelles of a single size, as outlined above, is one way of doing this. The purpose of this work is to explore two simple alternatives.
The Multiple-Equilibrium Model It is generally agreed that aggregation probably occurs as a stepwise process involving addition of single monomers to already existing micelles, as suggested by the scheme
s + s = s, s, + s = s3 S,.l + s = s,
K, = K3 =
[s,]/[sl~ [s,l/[s,l[sl
(4)
Ki = [S,l/[S,-*l[~I
When all of these equilibrium conditions are satisfied simultaneously, the total surfactant concentration is given by
and the total concentration of surfactant in aggregated form is [AI =
[sol - [SI
*
=
msi1
(6)
f .2
If the chemical shift for species Si is 6 ( S i ) the observed shift is
while the concentration of the nth micellar species is [S,] = Kin-l[SIn. Since the average chemical shift still depends on the whole set of micellar shift values, a further simplification is needed. An appealing possibility8 is to assume that all aggregates with i 1 2 have the same chemical shift, 6(A), so that this shift and the monomer shift are the only additional variables to be evaluated. It then follows that 6 = 6(A)
+
so that the dilution-shift curve is readily calculated. Apart from any specific objections which might be raised against the plausibility of this model (see below), it suffers from the serious defect that the number of disposable parameters has been reduced to only 3, and thus it has one less degree of freedom than the single-equilibrium model. Available dilution-shift curves present such a variety of shapes that attempts to fit them using this model are not successful except in a few instances. It appears that no scheme will have sufficient flexibility to fit all of the data unless it provides a t least four adjustable parameters. Case 2. Dimers and Even-Order Polymers. When the stabilities of small aggregates of alkylammonium carboxylates are estimated on the basis of simple electrostatics it is quickly apparent that the assumption of equal association constants is far from realistic. Each monomer consists of an ion pair with the charges separated by a distance d , and its electrostatic potential energy in a medium of dielectric constant t is - e 2 / t d . For a dimer the most stable form is a square planar one, and if the distance between adjacent charges is still d the energy is V = ( - e 2 / d ( 4 - 2/v%
= -2.586e2/
