Droplet shapes on structured substrates and conformal invariance

We consider the finite-size scaling of equilibrium droplet shapes for fluid adsorption (at bulk two-phase coexistence) on heterogeneous substrates and also in wedge geometries in which only a finite domain Lambda (A) of the s ubstrate is completely wet. For three-dimensional systems with short-ranged forces we use renormalization group ideas to establish that both the shape of the droplet height and the height-height correlations can be understood from the conformal invariance of an appropriate operator. This allows us t o predict the explicit scaling form of the droplet height for a number of d ifferent domain shapes. For systems with long-ranged forces, conformal inva riance is not obeyed but the droplet shape is still shown to exhibit strong scaling behaviour. We argue that droplet formation in heterogeneous wedge geometries also shows a number of different scaling regimes depending on th e range of the forces. The conformal invariance of the wedge droplet shape for short-ranged forces is shown explicitly.