The generalized Laplace equation of capillarity II. Hydrostatic and thermodynamic derivations of the Laplace equation for high curvatures

Two derivations are given of the generalized Laplace equation of capillarity, which assume constant surface tensions and constant curvature potentials over the surface for highly curved interfaces a priori: one based on hydrostatics and the other on thermodynamics. The excess hydrostatic equation is integrated across the interfacial zone, leading to the Boruvka and Neumann (BN) generalized Laplace equation. Based on the minimum free energy principle, a straightforward derivation is presented, also resulting in the BN generalized Laplace equation. The agreement between the two independent approaches provides a confirmation of the BN generalized Laplace equation for high curvature systems.