Elastic models of amphiphilic membranes

Membranes consisting of molecular layers of amphiphiles are the subject of extensive experimental and theoretical studies stimulated by their relevance to the biological membranes and the modern physico-chemical systems. The first theoretical approaches were developed to describe the simplest cases of weakly bent membranes with homogeneous curvatures. Recent successful applications of advanced experimental techniques revealed a broad spectrum of shapes of membranes formed spontaneously by various classes of amphiphiles in the process of their self-assembly, and made it possible to study quantitatively the elastic properties of the membranes of different configurations. Attempts to understand the physics of these systems require development of more sophisticated theoretical models treating the elastic behavior of amphiphile membranes characterized by high and inhomogeneous curvatures. In the present review we give a short phenomenological description of structures formed by the amphiphile molecules, survey the main theoretical ideas involved in the analysis of these structures and give a motivation for the future theoretical research.