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.

The relaxation kinetics of polydisperse cylindrical micelles has been numerically studied under different initial conditions corresponding to fast concentration and dilution of a surfactant solution. The kinetic description has been based on a set of difference Smoluchowski equations, which takes into account the capture and release of surfactant monomers by micelles, as well as the fusion and fission of micelles. The dependences of fusion coefficients of cylindrical aggregates on aggregation numbers have been plotted on the basis of the Burgers−Oseen equations for translational motion of spherocylindrical particles in a viscous liquid. The solution of the generalized Smoluchowski kinetic equations obtained for a nonequilibrium distribution of cylindrical micelles over aggregation numbers has been compared with the numerical solution of the Becker−Döring kinetic equations for cylindrical micelles under the same initial conditions given the mechanism of molecular aggregation via the attachment and detachment of surfactant monomers.

On the basis of the linearized analytical and numerical kinetic description of stepwise aggregation of surfactant aggregates, the hierarchical relaxation times have been found for a polydisperse micellar system close and above the critical micellar concentration. The description was based on the difference and differential Becker–Döring kinetic equations with using a specific boundary condition and improved models for the attachment rates of surfactant monomers to cylindrical aggregates. Two models have been considered: the linear model for cylindrical aggregates and the attachment rate to elongated spheroidal aggregates. The rate of attachment of monomers to an elongated spheroidal aggregate was found explicitly as a function of the aggregation number. With applying the truncation techniques, the analytical solution of differential kinetic equations for fast relaxation of polydisperse micellar systems has been obtained for a linear model of the aggregation rate. In the case of the attachment rate for an elongated spheroidal aggregate, the semi-analytical solution has been found.

Theoretical results published in the last 17 years on the kinetics of aggregation and relaxation in micellar surfactant solutions have been reviewed. The results obtained by the analytical and direct numerical solution of the Becker–Döring kinetic equations and the Smoluchowski generalized equations, which describe different possible mechanisms of aggregation and relaxation on all time scales from ultrafast relaxation while reaching the quasi-equilibrium in the region of subcritical molecular aggregates to the last stage of slow relaxation of micelles to the final aggregated state, have been considered in detail. The droplet model and the model linear with respect to aggregation numbers have been used for the work of aggregation to describe the dynamics of the rearrangement of micellar systems consisting of only spherical, only cylindrical, and coexisting spherical and cylindrical aggregates, with the dynamics being both linear and nonlinear with respect to deviations from equilibrium. The results of molecular simulation of the rearrangement kinetics of micellar systems subjected to initial disturbance have been reviewed.

An analytical expression has been derived for the quasi-stationary size distribution of surfactant aggregates in a micellar system approaching the final equilibrium state. In contrast to previously known relations, the derived expression takes into account variations in the concentration of monomers during the slow relaxation and enables one to determine the previously unknown fine structure of the linearized mode of slow relaxation, i.e., its dependence on the aggregation numbers in the range between the maximum and minimum of the work of aggregation. This dependence has been reliably confirmed by the numerical solution of the set of linearized Becker–Döering difference equations, which describe the molecular mechanism of the kinetics of micellization and micellar relaxation. In turn, the expression found for the relaxation mode makes it possible to refine the analogous “fine structure” of aggregation rates at different points of the same range between the maximum and minimum of the work of aggregation, in which the aggregation rates appear to be low but exhibit a nonmonotonic behavior. This behavior is also confirmed by the numerical solution of the Becker–Döering difference kinetic equations.

Relaxation of micellar systems can be described with the help of the Becker–Döring kinetic difference equations for aggregate concentrations. Passing in these equations to continual description, when the aggregation number is considered as continuous variable and the concentration difference is replaced by the concentration differential, allows one to find analytically the eigenvalues (to whom the inverse times of micellar relaxation are related) and eigenfunctions (or the modes of fast relaxation) of the linearized differential operator of the kinetic equation corresponding to the Fokker–Planck approximation. At this the spectrum of eigenvalues appears to be degenerated at some surfactant concentrations. However, as has been recently found by us, there is no such a degeneracy at numerical determination of the eigenvalues of the matrix of coefficients for the linearized difference Becker–Döring equations. It is shown in this work in the frameworks of the perturbation theory, that taking into account the corrections to the kinetic equation produced by second derivatives at transition from differences to differentials and by deviation of the aggregation work from a parabolic form in the vicinity of the work minimum, lifts the degeneracy of eigenvalues and improves markedly the agreement of concentration-dependent fast relaxation time with the results of the numerical solution of the linearized Becker–Döring difference equations.

Full-time kinetics of self-assembly and disassembly of spherical micelles with their fusion and fission in non-ionic micellar solutions has been considered in detail on the basis of direct numerical solutions of the generalized Smoluchowski equations describing the evolution of the time-dependent concentrations of molecular aggregates for every aggregation number. The cases of instant increase of the monomer concentration up or dilution of a surfactant solution below the critical micelle concentration at large initial deviations from the final equilibrium state have been studied. Different stages in assembly or disassembly of micelles have been described and compared with the results of the stepwise mechanism of monomer attachment-detachment described by the Becker-Döring kinetic equations. A relation of the full-time kinetics to micellar relaxation at small deviations from the equilibrium state has been checked.

The whole picture of evolution of coexisting spherical and cylindrical micelles has been described for initial states far from equilibrium.

The eigenvalues and eigenvectors of the matrix of coefficients of the linearized kinetic equations applied to aggregation in surfactant solution determine the full spectrum of characteristic times and specific modes of micellar relaxation. The dependence of these relaxation times and modes on the total surfactant concentration has been analyzed for concentrations in the vicinity and well above the second critical micelle concentration (cmc2) for systems with coexisting spherical and cylindrical micelles. The analysis has been done on the basis of a discrete form of the Becker-Döring kinetic equations employing the Smoluchowsky diffusion model for the attachment rates of surfactant monomers to surfactant aggregates with matching the rates for spherical aggregates and the rates for large cylindrical micelles. The equilibrium distribution of surfactant aggregates in solution has been modeled as having one maximum for monomers, another maximum for spherical micelles and wide slowly descending branch for cylindrical micelles. The results of computations have been compared with the analytical ones known in the limiting cases from solutions of the continuous Becker-Döring kinetic equation. They demonstrated a fair agreement even in the vicinity of the cmc2 where the analytical theory looses formally its applicability.

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.

A numerical description of micellization and relaxation to an aggregate equilibrium in surfactant solution with nonionic spherical micelles has been developed on the basis of a discrete form of the Becker-Döring kinetic equations. Two different models for the monomer-aggregate attachment-detachment rates have been used, and it has been shown that the results are qualitatively the same. The full discrete spectrum of characteristic times of micellar relaxation and first relaxation modes in their dependence on equilibrium monomer concentration have been found with using the linearized form of the Becker-Döring kinetic equations. Overall time behavior of surfactant monomer and aggregate concentrations in micellization and relaxation at large initial deviations from final equilibrium has been studied with the help of nonlinearized discrete Becker-Döring kinetic equations. Comparison of the computed results with the analytical ones known in the limiting cases from solutions of the linearized and nonlinearized continuous Becker-Döring kinetic equation demonstrates general agreement.