Analytical description of molecular mechanism of fast relaxation of spherical micelles with the extended Becker–Döring differential equation

Abstract

To improve and expand kinetic analysis of fast relaxation (via attachments and detachments of surfactant molecules) in an ensemble of spherical micelles in surfactant solutions, a general scheme for reducing the linearized difference Becker–Döring equations to the differential equation of an arbitrary order with respect to the aggregation number is proposed. A perturbation theory is formulated for any model of spherical micelles, where the main approximation corresponds to the kinetic Aniansson equation for the case of a symmetrical potential well of the aggregation work and the perturbation operator is written in the Hermitian form. The latter allows one to use standard perturbation techniques to find the fast relaxation times with the help of extended differential kinetic equation. The calculations were carried out in the second order of the perturbation theory, and the longest fast relaxation times were found as a function of the surfactant concentration for the droplet and the quasi-droplet models of direct spherical micelles and the star model of diblock polymeric spherical micelles. In the case of the droplet model, inclusion of corrections gives the concentration dependence of the longest fast relaxation time that virtually coincides with results of numerical solution of the system of linearized Becker–Döring difference equations. For the quasi-droplet model, the fast relaxation time found in the main approximation considerably deviate from the numerical result (up to 50%). Addition of corrections allows us to reduce these deviations to a considerably smaller value (to 10%). For the star micelle model, a fine agreement between the analytical and numerical solutions is obtained.