Spreading equilibria under mildly singular potentials: pancakes versus droplets

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 π/ 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 for purely repulsive potentials. These results generalize, complete, and give mathematical rigor to de Gennes’ formal discussion of spreading equilibria. Uniqueness and non-uniqueness phenomena are also discussed.