Phase transitions and droplet shapes above a wetting stripe

•We determine the finite-size scaling shift of the wetting temperature.•We derive a Kelvin-like equation for the finite-size shift of the pre-wetting line.•We highlight the scaling and conformally invariant properties of liquid drops adsorbed on a wetting stripe.•We present new density-functional-theory results for adsorption of liquid on nano-patterned wall. We consider fluid adsorption on a planar wall patterned by a single stripe of width L of a wet material imbedded within a partially wet surface. If L were infinite we suppose that the wall would exhibit a first-order wetting transition at temperature Tw, and an associated line of pre-wetting transitions extending off bulk-coexistence. For finite widths L, we a) determine the finite-size scaling shift of the wetting temperature Tw(L), b) derive an equation, analogous to the Kelvin equation for capillary condensation, for the shift of the pre-wetting line but involving boundary tensions and a boundary contact angle and c) highlight the scaling and conformally invariant properties of condensed liquid drops lying above the stripe at bulk coexistence. We point out analogies between these predictions and those for capillary condensation and criticality in a parallel plate geometry and test them against numerical results obtained from a microscopic non-local classical density functional theory.