Gibbsian Thermodynamics of Wenzel Wetting (Was Wenzel Wrong? Revisited)

When a drop is in contact with a rough surface, it can rest on top of the rough features (the Cassie-Baxter state) or it can completely fill the rough structure (the Wenzel state). The contact angle (theta) of a drop in these states is commonly predicted by the Cassie-Baxter or Wenzel equations, respectively, but the accuracy of these equations has been debated. Previously, we used fundamental Gibbsian composite-system thermodynamics to rigorously derive the Cassie-Baxter equation, and we found that the contact line determined the macroscopic contact angle, not the contact area that was originally proposed. Herein, to address the various perspectives on the Wenzel equation, we apply Gibbsian composite-system thermodynamics to derive the complete set of equilibrium conditions (thermal, chemical, and mechanical) for a liquid drop resting on a homogeneous rough solid substrate in the Wenzel mode of wetting. Through this derivation, we show that the roughness must be evaluated at the contact line, not over the whole interfacial area, and we propose a new Wenzel equation for a surface with pillars of equal height. We define a new dimensionless number H = h(1 - lambda(solid))/R to quantify when the drop's radius of curvature (R) is large enough compared to the size of the pillars for the new Wenzel equation to be simplified (h is the pillar height; lambda(solid) is the line fraction of the spherical cap's circumference that is on the pillars). Our new line-roughness Wenzel equation can be simplified to cos theta(w) = rho cos theta(Y) when H << 1, where rho is the line roughness. We also perform a thermodynamic free-energy analysis to determine the stability of the equilibrium states that are predicted by our new Wenzel equation.