Line tension and morphology of a droplet and a bubble attached to the inner wall of a spherical cavity

Abstract

The effects of line tension on the morphology of a lens-shaped droplet and bubble placed on the inner wall of a spherical cavity are studied. The contact angle between the lens-shaped droplet and the concave spherical substrate is expressed by the generalized Young's formula. The equator of the spherical substrate is found to play a crucial role. Neither a droplet with its contact line on the upper hemisphere of the substrate nor one with its contact line on the lower hemisphere can transform into each other continuously. On a hydrophobic substrate, the contact angle jumps discontinuously to $180^{\circ}$, and the droplet is detached from the substrate to form a spherical droplet when the line tension is positive and large. This is similar to the drying transition on a flat substrate. On the other hand, on a hydrophilic substrate, the contact angle jumps discontinuously to $0^{\circ}$ when the line tension is positive and large. Then, the droplet spread over the whole inner wall to leave a spherical bubble. Therefore, not only the drying transition but also the wetting transition is induced by positive line tension on a concave spherical substrate. There also exist stable as well as metastable droplets, whose phase diagrams can be complex. When the line tension is negative and its magnitude increases, the contact line approaches the equator infinitesimally from either above or below. However, it cannot cross the equator of a spherical cavity continuously. The droplet with a contact line that coincides with the equator is a singular droplet. The contact line is pinned and cannot move, irrespective of the magnitude of the line tension.