Wetting and drying at a curved substrate: Long-ranged forces

We examine the interfacial properties of a hard spherical cavity, radius R, immersed in a solvent in which the fluid-fluid interaction potential contains both a hard-sphere repulsive part and an attractive -r ^-6 component. Near to liquid-gas coexistence where the chemical potential deviation δμ≡μ-μ _co(T)→0 ⁺ complete wetting by the gas (drying) occurs and a coarse-grained effective Hamiltonian approach shows that the wall/liquid surface tension has a term in R ^-2/3, i.e., a leading-order power-law nonanalyticity in the curvature (R ^-1) in the large cavity limit. For states sufficiently well removed from coexistence the surface tension can be expanded in integer powers of the curvature R ^-1, provided R>R _c with the length scale given by R _c=2γ _gl(∞)/(Δρδμ), where γ _gl(∞) is the planar liquid/gas surface tension and Δρ is the difference between the coexisting densities. However, even in these circumstances there are additional R ^-2 In R contributions to the surface tension arising from the dispersion forces. An exact statistical mechanical sum rule is used to relate the density of the fluid at the point of contact with the cavity, ρ(R ⁺,μ), to the pressure of the reservoir and the surface tension. This predicts that ρ(R ⁺,μ) acquires a term in R ^-5/3 in the regime R<R _c. Numerical results obtained by applying classical density functional theory to this model confirm all the predictions from the coarse-grained approach for both the surface tension and the contact density. We argue that our results for leading-order nonanalytic contributions are exact, i.e., they should remain valid in the presence of interface fluctuations, and we discuss briefly the repercussions for solvation phenomena and for other wetting situations.