Triezenberg-Zwanzig expression for the surface tension of a liquid drop

Formulas, analogous to the Triezenberg-Zwanzig expression for the surface tension of a planar interface, are presented for the Tolman length, the bending rigidity, and the rigidity constant associated with Gaussian curvature. These expressions feature the Ornstein-Zernike direct correlation function and are derived from considering the deformation of a liquid drop in the presence of an external field. This approach is in line with the original analysis by Yvon in 1948. It is shown that our expressions reduce to previous results from density functional theory when a mean-field approximation is made for the direct correlation function. We stress the importance of the form of the external field used and show how the values of the rigidity constants depend on it.