Theory of curved interfaces and membranes: Mechanical and thermodynamical approaches

The mechanical and thermodynamical approaches to the theory of the general curved interfaces are presented and compared. In the mechanical approach a curved interface or membrane is characterized by the tensors of surface stresses and moments. They are connected by the surface balances of the linear and angular momentum. On the other hand, in the thermodynamical approach the surface is characterized by the scalar dilation and shear tensions as well as by the bending and torsion moments. In this review we investigate the problem about the relationships connecting the mechanical and thermodynamical approaches. We find that these two approaches are in a good agreement, that they are complementary to each other and represent the two parts of a self-consistent theory. The latter can be applied to any system where curved interfaces, thin films or membranes are present: microemulsions, lamellar and sponge phases, lipid vesicles and cell membranes, capillary waves at interfaces, undulation and peristaltic surface forces, lateral capillary forces between particles in thin liquid films, etc.