Van der Waals theory of curved surfaces

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Abstract
Van der Waals theory of a planar liquid-vapour interface, extended with a squared Laplacian term in the expression for the free energy, is used to study curved interfaces. Expressions, closely related to previously derived expressions by Fisher, M. P. A., and Wortis, M., 1984, Phys. Rev. B, 29, 6252, and Gompper, G., and Zschocke, S., 1992, Phys. Rev. A, 46, 4836 are given for the Tolman length (or, equivalently, spontaneous curvature) and rigidity constants of bending and Gaussian curvature. It is shown that these expressions follow also from a statistical mechanical (or virial) approach in which a local, mean field approximation is made for the pair density. The density profile of the curved surface is calculated to first order in the curvature which allows the derivation of explicit expressions for the various coefficients.