Localization and transformation of physically significant modes in relaxation of ensembles of spherical and cylindrical micelles

A general picture of relaxation in micellar solution of nonionic one-component surfactant on the basis of numerical solution of linearized set of the Becker–Döring equations for spherical and for cylindrical micelles has been analyzed in the form of a series in eigenvectors of the matrix of the kinetic coefficients. Two general characteristic cases have been considered as the initial conditions, the addition of monomers to the equilibrium system and the dilution of the equilibrium system. In both cases, the significant eigenvectors (relaxation modes) have been localized in the space of the aggregation numbers, that are responsible for the stages of ultrafast, fast, and slow relaxation, and corresponding eigenvalues (inverse relaxation times) have been selected from the huge number of all eigenvalues of the matrix of the kinetic coefficients. The analytical methods for finding the relaxation times of the ultrafast, fast and slow relaxation, recently developed and new ones proposed in this article, have been considered. The accuracy of the analytical calculations was controlled by comparison with much more resource-intensive computations using the matrix of the linearized equation. The analytical determination of the fast relaxation modes was based on the transition to the continual boundary-value problem with the potential for the distribution function of micelles over the aggregation numbers and using the perturbation theory. The spectrum was found numerically using the Runge–Kutta method. A new analytical solution for fast relaxation of the ensemble of cylindrical micelles with physically sound coefficients of monomer attachment to cylindrical micelles has been found with reducing the boundary value problem for the differential Becker–Döring–Frenkel equation to the equation for the Airy function.