A Theoretical Study of the Liquid Bridge Forces between Two Rigid Spherical Bodies

Abstract

This paper is concerned with liquid bridges between two spherical rigid bodies of equal radii under conditions where the effects of gravity are negligible. Previous work on the necessary condition for the stability of such bridges is examined and the minimum free-energy formulation applicable to any contact angle is proven. It is shown that this formulation is a more fundamental criterion for specifying the stable numerical solutions of the Laplace-Young equation, although equivalent to previous conjectures based on the liquid bridge neck diameter and filling angle. At relatively low contact angles, say <40°, the critical separation for rupture is given by the cube root of the liquid bridge volume to a good approximation. The toroidal approximation provides a simple method of estimating the total liquid bridge force. The "gorge method" of evaluation leads to errors of <10% for all stable separations and a wide range of bridge volumes. The accuracy is independent of contact angle because of geometrical self-similarity. Simple scaling coefficients can be introduced into the toroidal approximation to allow force estimations to be made with relatively high accuracy.