Wetting transitions and apparent contact angles on smoothly textured surfaces

We analyze the dewetting transition on a hydrophobic solid substrate of sinusoidal shape. For a given Young angle, this shape is compatible with air entrapment only for a sufficiently high amplitude (or, equivalently, for a sufficiently small period) of the shape. Entrapment may then take place in the form of either deep or shallow air pockets. Regardless of the amplitude, the interfacial energy associated with the formation of deep pockets is lower than that associated with the formation of shallow pockets. Compared with the nominal wetted state, the deep-pocket state is energetically beneficial only beyond a threshold amplitude, which is a strictly decreasing function of the contact angle. The apparent contact angle associated with that superhydrophobic state is obtained using the principle of virtual work.