Calculation of the rigidity constant in a Landau model for microemulsions

A theoretical investigation into the validity of different mean-field formulas for the rigidity constant of a microemulsion system is presented. As a theoretical model for the microemulsion system we use a Landau-theory-like expression for the free-energy in which, besides the usual gradient of the density squared term, a term proportional to the square of the second derivative is present, while the coefficient of the squared gradient term is taken to be negative in the microemulsion phase. Gompper and Zschocke (Phys. Rev. A 46, 4386 (1992)) used this model and derived mean-field expressions for the surface tension, spontaneous curvature and rigidity constants. The formula for the rigidity constant of bending features the curvature dependent density profile. It was observed that approximating the curvature dependent density profile with the density profile of the planar interface, yields approximate values for the rigidify constant which are in good agreement with a numerical calculation of the free energy of the curved surface. This is somewhat surprising since the same approximation for the rigidity constant of a simple liquid-vapour interface was shown to be incorrect. In this paper we investigate the differences and similarities between the two systems by calculating the approximate and exact values of the rigidity constant. We investigate when the approximate value of the rigidity constant is a good approximation to the exact one.