Critical wetting in power-law wedge geometries

We investigate critical wetting transitions for fluids adsorbed in wedge-like geometries where the substrate height varies as a power-law, $z(x,y) \sim |x| ^\gamma$, in one direction. As $\gamma$ is increased from 0 to 1, the substrate shape is smoothly changed from a planar-wall to a linear wedge. The continuous wetting and filling transitions pertinent to these limiting geometries are known to have distinct phase boundaries and critical singularities. We predict that the intermediate critical wetting behaviour occurring for $0<\gamma< 1$ falls into one of {\it{three}} possible regimes depending on the values of $\gamma$, p and q. The unbinding behaviour is characterised by a high degree of non-universality, strongly anisotropic correlations and enhanced interfacial roughness. The shift in phase boundary and emergence of universal critical behaviour in the linear wedge limit is discussed in detail.