Capillary bridges between unequal-sized spherical particles: rupture distances and capillary forces

This study focuses on capillary bridges between unequal-sized spherical particles in the pendular regime where the capillary bridge surface is axisymmetric. An analytical theory as well as closed-form expressions have been developed for the rupture distance and the capillary force. These expressions have been validated with a very large dataset of numerical solutions of the governing Young-Laplace equation, generated through a high-resolution integration method. For small capillary bridge volumes and contact angles smaller than 20 ^∘ , it has been shown that the meridional profile of the capillary bridge between unequal-sized particles can be accurately described by part of ellipses and that the contact radii for the large and the small particle are approximately equal. Contrary to the widely used toroidal approximation, the developed analytical theory takes into account the governing Young-Laplace equation, according to which the capillary force is constant along the capillary bridge. The analytical theory rigorously shows that expressions developed for cases with equal-sized particles can be directly employed in cases with unequal-sized particles by the use of the Derjaguin radius when the capillary bridge volume is small. Expressions for the rupture distance and the capillary force have also been derived analytically. For large capillary bridge volumes the use of the Derjaguin radius is not sufficient to accurately describe the properties of capillary bridges between particles with unequal sizes. By curve-fitting to the large dataset of numerical solutions of the Young-Laplace equation, a closed-form expression for the rupture distance has been developed, that accounts for the influence of the particle size ratio and that is accurate over a wide range of capillary bridge volumes. Expressions in the literature for the capillary force between unequal-sized spherical particles have been rigorously evaluated. With the new expression for the rupture distance, an improved closed-form expression for the capillary force has been formulated that is accurate for a wide range of small and large capillary bridge volumes (more specifically, when the ratio of the capillary bridge volume to the cubic root of the Derjaguin radius is smaller than 0.5) and separation distances, for contact angles smaller than 40 ^∘ .