The capillary bridge between two spheres: New closed-form equations in a two century old problem

We discuss progress in obtaining explicit equations for the capillary force between nano and micron sized solid spheres. Early approaches to this two-century old problem adopted approximations to the geometry. With the toroidal approximation, the meridian profile is approximated by an arc, and the approach leads to the capillary force being dependent on the location at which the force is evaluated. The Derjaguin approximation further assumes that the meridian radius is orders of magnitude smaller than the azimuth radius. An explicit expression for the capillary force is obtained, but the equation is limited to sufficiently small liquid volumes and separation distances. Significant progress has been made in recent years in using numerical solutions to derive analytical expressions for capillary bridges. Early numerical investigation established that the maximum separation for stable capillary bridges before rupture scales to the cubic root of the liquid volume. We report new progress in using numerical solutions to obtain more accurate and more general closed-form expressions for capillary bridges. Simple explicit algebraic equations have been observed to fit the numerical results well, leading to a closed-form solution applicable to capillary bridges between equal and unequal spheres and with zero or finite solid-liquid contact angles. The newly derived closed-form equation is more accurate and reduces to the Derjaguin equation when the liquid volume (or half-filling angle) and separation distance are both sufficiently small.