Derivation of microscopic expressions for the rigidity constants of a simple liquid—vapor interface

We derive equations for the spontaneous curvature and rigidity constants of a simple liquid-vapor interface analogous to the virial expression for the surface tension given by Kirkwood and Buff [J. Chem. Phys. 17 (1949) 338]. Using an approximation for the pair density of a curved interface we give expressions for these quantities in terms of an integral over the interaction potential and the pair correlation function in the liquid region. Near the critical point the rigidity constants are explicitely calculated to yield k = 0.631kBTc and k̵ = 0.239kBTc.