Capillary adhesion between solid surfaces: Small-volume liquid rings and bridges

A small amount of wetting liquid is placed about the point of contact between a solid sphere and a planar solid wall. It is well known in the colloid science literature that the leading-order approximation for the adhesive force between the two surfaces is independent of the liquid volume. This independence breaks down when the sum of the contact angles on the two surfaces is close to 180°. We identify the pertinent distinguished limit in this scenario, linking the proximity to 180° to the liquid volume. We address the adhesion problem at that limit, where the meridional curvature becomes comparable to the azimuthal curvature. Our key result is a closed-form approximation to the adhesive force, which does depend upon the liquid volume. We quantify the condition of negligible gravity in the context of the distinguished limit.