Droplet shapes on structured substrates and conformalinvariance

Abstract

We consider the finite-size scaling of equilibrium droplet shapesfor fluid adsorption (at bulk two-phase coexistence) onheterogeneous substrates and also in wedge geometries in which onlya finite domain ΛA of the substrate is completely wet.For three-dimensional systems with short-ranged forces we userenormalization group ideas to establish that both the shape of thedroplet height and the height-height correlations can be understoodfrom the conformal invariance of an appropriate operator. Thisallows us to predict the explicit scaling form of the droplet heightfor a number of different domain shapes. For systems withlong-ranged forces, conformal invariance is not obeyed but thedroplet shape is still shown to exhibit strong scaling behaviour.We argue that droplet formation in heterogeneous wedge geometriesalso shows a number of different scaling regimes depending on therange of the forces. The conformal invariance of the wedge dropletshape for short-ranged forces is shown explicitly.