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Kinetics of Surfactant Micellization: a Free Energy Approach

We present a new theoretical approach to the kinetics of micelle formation in surfactant solutions, in which the various stages of aggregation are treated as constrained paths on a single free-energy landscape. Three stages of well-separated time scales are distinguished. The first and longest stage involves homogeneous nucleation of micelles, for which we derive the size of the critical nuclei, their concentration, and the nucleation rate. Subsequently, a much faster growth stage takes place, which is found to be diffusion-limited for surfactant concentrations slightly above the critical micellar concentration ({\it cmc}), and either diffusion-limited or kinetically limited for higher concentrations. The time evolution of the growth is derived for both cases. At the end of the growth stage the micelle size may be either larger or smaller than its equilibrium value, depending on concentration. A final stage of equilibration follows, during which the micelles relax to their equilibrium size through fission or fusion. Both cases of fixed surfactant concentration (closed system) and contact with a reservoir of surfactant monomers (open system) are addressed and found to exhibit very different kinetics. In particular, we find that micelle formation in an open system should be kinetically suppressed over macroscopic times and involve two stages of micelle nucleation rather than one.