Superhydrophobicity: Localized Parameters And Gradient Surfaces

The use of Cassie and Baxter's equation and that of Wenzel has been subject to some criticism of late. It has been suggested that researchers use these equations without always considering the assumptions that have been made and sometimes apply them to cases that are not suitable. This debate has prompted a reconsideration of the derivation of these equations using the concept of parameters for the Wenzel roughness and Cassie-Baxter solid surface fractions that are local to the three-phase contact lines. In such circumstances, we show the roughness and Cassie-Baxter solid fractions depend not only on the substrate material, but also on which part of the substrate is being sampled by the three-phase contact lines of a given droplet. We show that this is not simply a theoretical debate, but is one which has direct consequences for experiments on surfaces where the roughness or spatial pattern varies across the surface. We use the approach to derive formulae for the contact angle observed on a double length scale surface under the assumption that the small-scale features on the peaks of larger scale features are either wetted or non-wetted. We also discuss the case of curved and re-entrant surface features and how these bring the Young's law contact angle into the formula for roughness and the condition for suspending droplets without penetration into the surface. To illustrate the use of local parameters, we consider the case of a variation in Cassie-Baxter fraction across a surface possessing a homogeneous hydrophobic surface chemistry and discuss the conditions (droplet volume, surface hydrophobicity, gradient in superhydrophobicity and contact angle hysteresis) under which a droplet may be set into motion. We show that different contact angles on each side of a droplet of water placed on such a surface can generate sufficient lateral force for the droplet to move towards the region of the surface with the lowest contact angle. Using an electrodeposited copper surface with a radial gradient in superhydrophobicity we exemplify these ideas by showing experimentally that droplets enter into self-actuated motion and accumulate in the centre of the surface where the wettability is higher. In principle, paths can be defined and water droplets can be collected by creating such gradients in superhydrophobicity through changes in the lateral topography of the surface.