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.

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In this paper we discuss the adhesion of a thin elastic plate to a randomly rough hard substrate. It is shown that at small magnification (long length scales) the plate, because of its higher compliance, is able to adhere in apparent full contact to the long wavelength corrugation of the underlying surface. That is, at length scales longer than the plate thickness, the gain in the adhesion energy upon the contact with the substrate overcomes the repulsive elastic energy produced by the elastic deformations, and the plate is able to fill out the large cavities of the rigid substrate. This produces a larger area of contact and an enhanced capability to adhere to a rough surface in comparison to the semi-infinite elastic solid case. However, at large enough magnification (small length scales) the plate behaves as a semi-infinite solid, and, depending on the roughness statistical properties, the area of true atomic contact may be much smaller than the nominal contact area.