Line tension and morphology of a sessile droplet on a spherical substrate

TL;DR

This paper investigates how line tension influences the shape and contact angle of a droplet on a spherical substrate, revealing possible drying and wetting transitions depending on the line tension's magnitude and sign.

Contribution

It provides a theoretical analysis of droplet morphology on spherical substrates considering line tension effects, extending understanding of wetting transitions beyond flat surfaces.

Findings

Positive line tension causes the droplet to detach or spread depending on surface hydrophobicity.

Negative line tension leads to contact line pinning near the equator of the sphere.

Transitions between different droplet morphologies depend on line tension magnitude and substrate wettability.

Abstract

The effects of line tension on the morphology of a sessile droplet placed on top of a convex spherical substrate are studied. The morphology of the droplet is determined from the global minimum of the Helmholtz free energy. The contact angle between the droplet and the spherical substrate is expressed by the generalized Young's formula. When the line tension is positive and large, the contact angle jumps discontinuously to $180^{\circ}$, the circular contact line shrinks towards the top of the substrate, and the droplet detaches from the substrate, forming a spherical droplet if the substrate is hydrophobic (i.e., the Young's contact angle is large). This finding is consistent with that predicted by Widom [J. Phys. Chem. {\bf 99}, 2803 (1995)]; the line tension induces a drying transition on a flat substrate. On the other hand, the contact angle jumps to $0^{\circ}$, the circular contact line shrinks towards the bottom of the substrate, and the droplet spreads over the substrate to form a wrapped spherical droplet if the substrate is hydrophilic (i.e., the Young's contact angle is small). Therefore, not only the drying transition of a cap-shaped to a detached spherical droplet but also the wetting transition of a cap-shaped to a wrapped spherical droplet could occur on a spherical substrate as the surface area of the substrate is finite. When the line tension is negative and its magnitude increases, the contact line asymptotically approaches the equator from either above or below. The droplet with a contact line that coincides with the equator is an isolated, singular solution of the first variational problem. In this instance, the contact line is pinned and cannot move as far as the line tension is smaller than the critical magnitude, where the wetting transition occurs.