Spreading equilibria under mildly singular potentials: pancakes versus droplets

TL;DR

This paper analyzes the shape and support of equilibrium states of thin liquid layers influenced by various potentials, revealing conditions for droplet or pancake shapes and rigorously confirming de Gennes' spreading theory.

Contribution

It provides a rigorous mathematical analysis of spreading equilibria with mildly singular potentials, extending and completing de Gennes' formal discussion.

Findings

Global minimizers are compactly supported with a microscopic contact angle of π/2.

The macroscopic shape can be droplet-like or pancake-like depending on the potential.

A transition profile exists at zero spreading coefficient.

Abstract

We study global minimizers of a functional modeling the free energy of thin liquid layers over a solid substrate under the combined effect of surface, gravitational, and intermolecular potentials. When the latter ones have a mild repulsive singularity at short ranges, global minimizers are compactly supported and display a microscopic contact angle of $\pi/2$. Depending on the form of the potential, the macroscopic shape can either be droplet-like or pancake-like, with a transition profile between the two at zero spreading coefficient. These results generalize, complete, and give mathematical rigor to de Gennes' formal discussion of spreading equilibria. Uniqueness and non-uniqueness phenomena are also discussed.