Concentrations of Monomers and Cylindrical Micelles above the Second CMC

Abstract

Based on thermodynamically substantiated linear dependence of the work of cylindrical micelle formation on the aggregation number within a wide range of aggregation numbers where the cylindrical micelles are accumulated in a surfactant solution, the second critical micellization concentration (CMC) is introduced as an overall surfactant concentration at which the ratio of the total amount of substance in cylindri- cal micelles to the amount of substance in monomers is equal to 0.1, i.e., it is already noticeable. It is shown that this ratio increases rather rapidly with a monomer concentration. The coefficient of the linear dependence of the work of cylindrical micelle formation on the aggregation number in the important practical situation where the ratios of the total concentration of cylindrical micelles and total amount of substance in these micelles to the monomer concentration are equal by the order of magnitude to 1 and 10 5 , respectively, while disc micelles and extended bilayers are still not appeared. In the same situation, the ratios of the total concentration of spher- ical micelles and total amount of substance in these micelles to the monomer concentration are equal by the order of magnitude to 1 and 10 2 , respectively. The relationship between the overall surfactant concentration and monomer concentration is found. It is shown that the second CMC exceeds by two orders of magnitude the first CMC corresponding to the onset of the noticeable accumulation of surfactant in spherical micelles. The distri- bution of cylindrical micelles over the aggregation numbers is analyzed. It is demonstrated that, in agreement with the experiment, the distribution is almost uniform in the considerable part of the wide range of aggregation numbers and drops exponentially in the remaining (right-hand) part of this range. Experimental result is con- firmed that the total concentration of cylindrical micelles, the mean value, and the mean statistical scatter of aggregation numbers in a cylindrical micelle is proportional to the square root of the overall surfactant concen- tration. The balance equation of surfactant amount in the vicinity of the final equilibrium state of a materially isolated solution is linearized. This linearization makes it possible to express the deviations of monomer and aggregate concentrations from their equilibrium values at the lower boundary of the region of the linear depen- dence of the work of cylindrical micelle formation on the aggregation numbers via the deviations of experimen- tally observed total concentrations of spherical and cylindrical micelles from their equilibrium values. The case of the solutions of such surfactants, for which spherical shape appeared to be unrealizable due to their molecular structure and packing conditions, is considered separately.