Kinetics of micellisation and relaxation of cylindrical micelles described by the difference Becker-Döring equation

A numerical description of micellisation and relaxation to an aggregate equilibrium in a nonionic surfactant solution with spherical premicellar aggregates and stable polydisperse cylindrical micelles is presented for a wide interval of total surfactant concentrations and initial conditions. The Smoluchowsky-type model for the attachment-detachment rates of surfactant monomers to and from surfactant aggregates with matching rates for small spherical premicellar aggregates and the rates for larger cylindrical micelles have been used. The full discrete spectrum of characteristic times of micellar relaxation and the first three relaxation modes in their dependence on the equilibrium monomer concentration have been computed using the linearized form of the Becker-Döring difference equations. The overall time behavior of the surfactant monomer and aggregate concentrations in micellisation and relaxation at large initial deviations from the final equilibrium has been studied with the help of nonlinearized discrete Becker-Döring kinetic equations. The studies demonstrate a possibility for non-monotonic evolution of the monomer concentration at the initial stages. A comparison of the computed results with the analytical ones known from solutions of the linearized and nonlinearized differential Becker-Döring kinetic equations demonstrates a general agreement at higher concentrations of the surfactant above the critical micellar concentration.