DESCRIPTION OF THE GEOMETRICAL AND TOPOLOGICAL-STRUCTURE IN AMPHIPHILIC SYSTEMS

A unified description of internal interfaces in oil-water-surfactant mixtures is proposed. Surfactant degrees of freedom are explicitly taken into account in the form of a vector field. A general definition of average curvatures in terms of the vector field is given. They are averages of the mean and Gaussian curvatures and characterize globally the geometrical and topological structure of the internal interface. It is argued that this definition can be applied to both sharp and diffuse oil-water interfaces, in ordered phases and in disordered microemulsions. A few examples concerning ordered phases are considered, and the results for the average curvatures are compared with the standard approach, in which the interface is modeled by an infinitely thin mathematical surface. It is also shown that the approach reduces to the standard one in the case of well-defined surfactant monolayers. Finally, the definition of average curvatures is extended to the case of microscopic Hamiltonians.