Hydrophobic properties of a wavy rough substrate

The wetting/non-wetting properties of a liquid drop in contact with a chemically hydrophobic rough surface (thermodynamic contact angle theta(e)>pi/2) are studied for the case of an extremely idealized rough profile: the liquid drop is considered to lie on a simple sinusoidal profile. Depending on surface geometry and pressure values, it is found that the Cassie and Wenzel states can coexist. But if the amplitude h of the substrate is sufficiently large the only possible stable state is the Cassie one, whereas if h is below a certain critical value hcr a transition to the Wenzel state occurs. Since in many potential applications of such super-hydrophobic surfaces, liquid drops often collide with the substrate (e.g. vehicle windscreens), in the paper the critical drop pressure pW is calculated at which the Cassie state is no longer stable and the liquid jumps into full contact with the substrate (Wenzel state). By analyzing the asymptotic behavior of the systems in the limiting case of a large substrate corrugation, a simple criterion is also proposed to calculate the minimum height asperity h necessary to prevent the Wenzel state from being formed, to preserve the super-hydrophobic properties of the substrate, and, hence, to design a robust super-hydrophobic surface.