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When does Wenzel's extension of Young's equation for the contact angle of droplets apply? A density functional study

he contact angle of a liquid droplet on a surface under partial wetting conditions differs for a nanoscopically rough or periodically corrugated surface from its value for a perfectly flat surface. Wenzel's relation attributes this difference simply to the geometric magnification of the surface area (by a factor $r_{\rm w}$), but the validity of this idea is controversial. We elucidate this problem by model calculations for a sinusoidal corrugation of the form $z_{\rm wall}(y) = Δ\cos(2πy/λ)$ , for a potential of short range $σ_{\rm w}$ acting from the wall on the fluid particles. When the vapor phase is an ideal gas, the change of the wall-vapor surface tension can be computed exactly, and corrections to Wenzel's equation are typically of order $σ_{\rm w}Δ/λ^2$. For fixed $r_{\rm w}$ and fixed $σ_{\rm w}$ the approach to Wenzel's result with increasing $λ$ may be nonmonotonic and this limit often is only reached for $λ/σ_{\rm w}>30$. For a non-additive binary mixture, density functional theory is used to work out the density profiles of both coexisting phases both for planar and corrugated walls, as well as the corresponding surface tensions. Again, deviations from Wenzel's results of similar magnitude as in the above ideal gas case are predicted. Finally, a crudely simplified description based on the interface Hamiltonian concept is used to interpret corresponding simulation results along similar lines. Wenzel's approach is found to generally hold when $λ/σ_{\rm w}\gg 1$, $Δ/λ<1$, and conditions avoiding proximity of wetting or filling transitions.