Evolution with temperature of the middle phase from micellar to bicontinuous microemulsion

Abstract

We study the evolution of a middle phase microemulsion, from micellar solution to bicontinuous microemulsion to inverse micellar solution, as temperature is increased in a lattice model of a ternary mixture of oil, water, and nonionic amphiphile. Our model, which is studied within the Bethe approximation, includes orientational degrees of freedom of water as well as of amphiphile. It yields a two-phase to three-phase to two-phase progression with changing temperature, i.e., a triple line bounded by lower and upper critical endpoints. The cmc surfaces of normal and inverted micelles are calculated as a function of temperature, and we find that at both critical endpoints, the middle phase emerges from a critical micellar phase. Osmotic pressures are calculated to display the manner in which they reflect the cmcs. The connectivities of oil and water regions are obtained so that the region in which the middle phase is bicontinuous can be delineated. We find that when the temperature is increased from the lower critical endpoint, the micelle concentration falls sharply as the middle phase becomes bicontinuous; when the temperature is decreased from the upper critical endpoint, the much smaller population of inverse micelles decreases less sharply as the system becomes bicontinuous. The correlation functions of the middle phase are calculated, and we determine the region in which they decay nonmonotonically at large distances. Comparison shows that while the regions in which the middle phase displays its structure to connectivity and to scattering experiments do overlap, they do not coincide. Most of our results compare well to experiments on ternary mixtures which contain a weak, nonionic amphiphile.