POPULATION-DYNAMICS OF SMALL SYSTEMS .1. INSTANTANEOUS AND IRREVERSIBLE REACTIONS IN REVERSED MICELLES

Molecules solubilized in the water pools of reversed micelles are redistributed over these micelles when they coalesce to form temporary dimers and then break apart again. The manner in which the solutes are redistributed will determine the form of the probability distribution function for the solutes over the entire micellar population. A general, self-consistent mathematical framework based on population balances is developed for the description of the population dynamics of these interacting colloidal aggregates, allowing for cooperative, repulsive, and random redistribution of solutes over any two interacting micelles during the coalescence/redispersion process. Analytical solutions to the general steady-state population balance equations are obtained in terms of these binary redistribution functions; the Poisson, binomial, and geometric probability distribution functions are derived as special cases. The dynamic evolution of the probability distribution functions for the micellar population following the mixing of two different reactive or nonreactive micellar solutions is calculated, and the reaction rates predicted are slower than those based on simple second-order rate equations using the same kinetic parameters. This points to the importance of accounting for the statistical distribution of solutes when analyzing experimental rate data.