Model for phase equilibria in micellar solutions of nonionic surfactants

The formation of spherical micelles in aqueous solutions of nonionic surfactants and the equilibria between two such micellar phases are studied with a phenomenological model incorporating hydrophobic interactions and the configurational entropy of the amphiphiles. The distribution of micelle sizes is determined over the entire temperature-composition phase diagram, and moments of that distribution function determine the consolute point parameters. In a generalization of an analysis given by Stillinger and Ben-Naim, the mathematical properties of various thermodynamic functions in the neighborhood of the critical micelle concentration are related to the location of branch points of the osmotic pressure in the complex concentration plane. The model attributes the experimentally observed lower critical solution points in these systems to surfactant–water hydrogen bonding, whose temperature dependence is described with a mean field approximation. Calculated phase diagrams are in qualitative agreement with those from experiments, in particular, exhibiting closed solubility loops with quite distinct upper and lower critical compositions, and values for the lower critical composition on the order of several percent volume fraction. The relevance of certain aspects of the model to the understanding of microemulsions is discussed.