An analytical expression for the approximate dependence of the degree of counterion binding on the composition of ionic-nonionic mixed micelles

A simple analytical expression is presented to describe the dependence of the degree of counterion binding beta of ionic-nonionic mixed micelles on the composition x(I) (the micelle mole fraction of the ionic species): 1/(1-beta)=1-x(I)+x(I)/1-beta(x(I)=1)]. In the application of the relation, the value of beta for the pure ionic micelle, beta(x(I)=1), should be known in advance. An equivalent expression was first proposed by Hall et al. on purely empirical grounds but in the present study the relation is derived on the basis of the Poisson-Boltzmann equation of the plate model of micelles in a salt-free medium. The insensitive nature of beta toward a change in micelle concentration as well as to the addition of a salt is also derived under reasonable approximations. When the relation was applied to 12 mixed micelles both with and without added salt, it described quite well the observed composition dependencies of all examples examined. Two other simple analytical expressions are also derived, but they are much less satisfactory in describing the experimental data.