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.
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.
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 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 (cmc 2 ) for systems with coexisting spherical and cylindrical micelles. The analysis has been done on the basis of a discrete form of the BeckerDöring 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 cmc 2 where the analytical theory looses formally its applicability.
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