Critical micelle concentration and the size distribution of surfactant aggregates

Abstract

The critical micelle concentration (cmc) exhibited by many surfactants in water represents an abrupt change in the solution from a state with mostly unassociated molecules to a state with large molecular aggregates or micelles. This transition can be monitored by corresponding abrupt changes in any of a wide variety of properties, including osmotic pressure, electrical conductance, light scattering, viscosity, and dye solubilization. Micelles typically have sizes ranging from twenty to several hundred surfactant molecules.‘ Recently Ruckenstein and NagarajanZ4 proposed an intriguing connection between the occurrence of the cmc on the one hand, and a qualitative change in the size distribution of aggregates on the other hand. At very low total surfactant concentration the aggregate size distribution decreases monotonically with increasing size, i.e., monomers are the most frequently encountered species, dimers are less frequent, trimers even rarer, etc. However, as the total surfactant concentration increases a point is reached at which this monotonicity disappears. Subsequently, the aggregate size distribution continues to decline with size only for small sizes, .passes through a subsequent minimum and maximum, and then reverts to monotonicity. Ruckenstein and Nagarajan identify the transition between these two regimes (the concentration at which the size distribution displays a horizontal point of inflection) with the cmc. It is our aim in this paper to test the Ruckenstein-Nagarajan proposal for an elementary aggregation model. For present purposes it is sufficient to suppose that the surfactant solution is ideal with respect to all aggregation species (this assumption is implicit in ref 2-4). Then the concentration pi of i-molecule aggregates can be written