The generalized Laplace equation of capillarity I. Thermodynamic and hydrostatic considerations of the fundamental equation for interfaces

Abstract

Theories of capillarity are reviewed. The limitations of the classical theory as developed by Gibbs are demonstrated. The generalized theories of capillarity initiated by Buff, and later by Murphy, Kondo, Kralchevsky, and Boruvka and Neumann are scrutinized by considering the basic requirements of formulating thermodynamic fundamental equations. The different generalized Laplace equations of different theories stem from the different setups of the fundamental equation. It is concluded that only Boruvka and Neumann's (BN) generalized theory satisfies all the requirements of thermodynamics and mathematics. Further, to test the BN theory, a hydrostatic treatment of a two phase capillary system is presented. This non-thermodynamic approach is based on the concept of virtual work as the condition for equilibrium of a capillary system, and on the concept of parallel surfaces for evaluating the stress tensor field and excess properties within the interfacial region. Following a straight-forward procedure, it is shown that the hydrostatic results for surface tension γ and two bending moments, C 1, and C 2, agree with the results of the BN generalized thermodynamic theory of capillarity. This agreement indicates that the form of the BN fundamental equation for surfaces with the extensive geometric curvatures ( J and K total mean and total Gaussian curvatures, respectively) is the proper expression required to generalize the theory of capillarity.