Role of condensed counterions in the thermodynamics of surfactant micelle formation with and without oppositely charged polyelectrolytes

We present a simple model of micelle formation that includes any effects of condensed counterions. The critical micelle concentration (cmc) of ionic surfactants has a much weaker dependence on surfactant tail length than nonionic surfactants because condensed counterions on the ionic surfactant micelle lose their translational entropy, as predicted by Murray and Hartley in 1935. The critical aggregation concentration (cac) of ionic surfactants in the presence of oppositely charged polyelectrolytes is much smaller than the cmc, and has a strong dependence on surfactant tail length, similar to the cmc of nonionic surfactants. The oppositely charged polyelectrolyte has a strong electrostatic interaction with the micelle surface, partially accounting for the low cac. The binding of micelle and polyelectrolyte also means that no condensed counterions are necessary on the micelle surface, which also lowers the cac and changes the dependence of the cac on surfactant tail length. For strongly charged polyelectrolytes, the cac is lowered further still because counterions are released from the polyelectrolyte when the surfactant binds to it. Our model correctly predicts that the cac increases as salt is added while the cmc decreases with added salt, owing to the different roles of counterion condensation. The experimental increases in the cac as either the charge density or concentration of the polyelectrolyte increases are also explained by the model. We experimentally find that the number of counterions released from the polyelectrolyte per bound surfactant is roughly equal to the fraction of neutralized monomers with bound counterions.