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

The 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_w), but the validity of this idea is controversial. We elucidate this problem by model calculations for a sinusoidal corrugation of the form z_wall(y) = Δ cos(2πy/λ), for a potential of short range σ_w acting from the wall on the fluid particles. When the vapor phase is an ideal gas, the change in the wall-vapor surface tension can be computed exactly, and corrections to Wenzel's equation are typically of the order σ_wΔ/λ². For fixed r_w and fixed σ_w, the approach to Wenzel's result with increasing λ may be nonmonotonic and this limit often is only reached for λ/σ_w > 30. For a non-additive binary mixture, density functional theory is used to work out the density profiles of both coexisting phases 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 the corresponding simulation results along similar lines. Wenzel's approach is found to generally hold when λ/σ_w ≫ 1 and Δ/λ < 1 and under conditions avoiding proximity of wetting or filling transitions.