We consider the statics of compound droplets made of two immiscible fluids on a
rigid substrate, in the limit when gravity is dominated by capillarity. In particular,
we show that the merging of four phases along a single contact line is a persistent
and robust phenomenon from a mechanical and thermodynamic perspective; it can
and does occur for a range of interfacial energies and droplet volumes. We give an
interpretation for this in the context of the macroscopic Young–Laplace law and
its microscopic counterpart due to van der Waals, and show that the topological
transitions that result can be of either a continuous or discontinuous type depending
on the interfacial energies in question.