Curvature elasticity of thin films

A tutorial review of the theory of curvature elasticity of thin films is pr esented with an emphasis on the physical origins of the bending energy. We begin with a discussion of surface curvature and focus on the role of speci al surfaces of curvature to show how such surfaces can be defined to elimin ate either the coupling of the compressibility and bending terms (neutral s urface) or the saddle-splay (Gaussian curvature) modulus. Next, we consider phenomenological models for curvature elasticity and discuss the coupling of the curvature degrees of freedom with other properties of the system suc h as the packing area and the number of molecules at the interface. The pre ssure distribution in the film is related to the bending moduli. We then co nnect the elastic moduli to the physical properties of both solid and liqui d thin films with a detailed discussion of the role of solid elasticity (in cluding defects), electrostatic interactions (applicable to polar head grou ps and chain packing (using a block copolymer model of amphiphilic molecule s). Finally, we demonstrate the effects of fluctuations and inhomogeneities in these systems in a discussion of the role of thermal undulations in ren ormalizing the bending moduli and of mixtures of amphiphiles of different c hain lengths in fluid films. The article is concluded with a brief review o f experimental characterizations of curvature elasticity in self-assembling systems.