Kinetic description of the relaxation of surfactant solutions in the absence of activation barrier between spherical and cylindrical micelles

Abstract

The realistic situation, where there is no potential barrier of aggregation work between the spherical and cylindrical micelles, is analyzed within the framework of the kinetic description of the relaxation of nonionic surfactant solution containing spherical and cylindrical micelles. The variations in the sum of the total concentrations of spherical and cylindrical micelles caused by the total flux of molecular aggregates over the potential barrier of aggregation work between the monomers and spherical micelles is taken into account; it is also accounted for that, in the absence of the potential barrier of aggregation work between the spherical and cylindrical micelles, they are present in the mutual quasi-equilibrium. The closed linearized relaxation equation determining the variations (with time) in the total concentrations of spherical and cylindrical micelles in a materially isolated surfactant solution is derived and solved. The variations (with time) of the total concentrations of spherical and cylindrical micelles, the concentration of surfactant monomers, as well as the total amount of surfactant in cylindrical micelles, in the process of approach of a materially isolated micellar solution to the final equilibrium state are described analytically. It is disclosed that, at the mutual quasi-equilibrium of spherical and cylindrical micelles, the opposite-sign deviations of the total amount of substance in cylindrical micelles and the total amount of cylindrical micelles from their values in the final equilibrium state of a materially isolated solution is attributed to the relatively large cylindrical micelles. The agreement between the results obtained and analogous results of the description of micellar relaxation in the presence of the potential barrier of aggregation work between spherical and cylindrical micelles whose relative height is at least slightly lower than that between monomers and spherical micelles is demonstrated. Comparison is performed for the cases of the absence of either spherical or cylindrical micelles. Analytical expressions, which are accessible for experimental verification, for the ratios of relaxation times of surfactant solutions containing spherical and cylindrical micelles to those of surfactant solutions containing either spherical or cylindrical micelles alone are derived.

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

“…Depending on the initial conditions, we have found that there is a possibility for non-monotonic evolution of the monomer concentration and the total concentration of small pre-critical aggregates. A comparison of the computed results with the analytical ones, [14][15][16][17][18][19][20][21][22][23] known in the limiting cases from solutions of the linearized and nonlinearized differential Becker-Döring kinetic equation, demonstrates quantitative agreement at higher concentrations of the surfactant above the critical micellar concentration. Surprisingly, there is also qualitative agreement with the formulas [14][15][16][17][18][19][20][21][22][23] even at premicellar states of surfactant solutions, where the analytical theory cannot formally be applied.…”

“…A comparison of the computed results with the analytical ones, [14][15][16][17][18][19][20][21][22][23] known in the limiting cases from solutions of the linearized and nonlinearized differential Becker-Döring kinetic equation, demonstrates quantitative agreement at higher concentrations of the surfactant above the critical micellar concentration. Surprisingly, there is also qualitative agreement with the formulas [14][15][16][17][18][19][20][21][22][23] even at premicellar states of surfactant solutions, where the analytical theory cannot formally be applied. Unfortunately, we do not know the direct experimental or numerical simulation results on the kinetics of cylindrical micelles to compare with the predictions obtained from our work, but we are planning to perform such simulations in the near future.…”

“…The situation of a large initial excess of the surfactant in aggregates can be achieved by setting A ¼ 3000 in eqn (17). Direct computations with the help of eqn ( 7)-( 9), ( 2)-( 4) and ( 6 As is seen from Fig.…”

“…The direct J 0 and back J 00 uxes of the aggregates can be determined with the help of eqn (12e), ( 21) and ( 22) as [14][15][16][17][18]23…”