The McMillan−Mayer theory of multicomponent solutions is
utilized to formulate a statistical-thermodynamic description of surfactant solution behavior from which
quantitative predictions of micelle
formation, micellar size distribution, and micellar solution phase
separation can be made. Specifically,
a model is constructed for the Gibbs free energy of the micellar
solution, which is divided into ideal and
excess contributions. The advantage of this approach is that it
enables a systematic analysis of various
models of intermicellar interactions. In this paper, we focus on
micelles of nonionic surfactants which
exhibit both repulsive and attractive interactions. The repulsive
interactions are described using excluded-volume considerations, while the attractive ones are modeled using a
mean-field description. Utilizing
this statistical-thermodynamic framework, expressions for the chemical
potentials of each of the solution
components are obtained and used, along with the principle of multiple
chemical equilibrium, to calculate
the micellar size distribution and its moments. An analysis of the
effect of excluded-volume interactions
on the monomer and micelle concentrations and on the weight-average
aggregation number of micelles
which exhibit one-dimensional (cylindrical) growth indicates that these
steric interactions promote micelle
formation and growth. Interestingly, in the limit of extensive
cylindrical micellar growth, we recover the
well-known expressions for the micellar size distribution and its
moments corresponding to the popular
phenomenological “ladder model”, with modified “ladder model”
parameters which are explicit functions
of the excluded-volume parameters. In addition, quantitative
predictions of the critical micellar
concentration, the polydispersity of the micellar size distribution,
and phase separation characteristics are
presented and found to compare favorably with available experimental
data for aqueous micellar solutions
of alkyl poly(ethylene oxide) nonionic surfactants.