Drops down the hill: theoretical study of limiting contact angles and the hysteresis range on a tilted plate

The limiting inclination angle (slip angle), for which a two-dimensional water drop may be at equilibrium on a chemically heterogeneous surface, is exactly calculated for a variety of cases. The main conclusion is that, in the cases studied, the contact angles at the upper and lower contact line do not always simultaneously equal the receding and advancing contact angles, respectively. On a hydrophobic surface, the lowest contact angle (at the upper contact line) tends to be approximately equal to the receding contact angle, while the highest contact angle (at the lower contact line) may be much lower than the advancing contact angle. For hydrophilic surfaces, the opposite is true. These conclusions imply that the hysteresis range cannot in general be measured by analyzing the shape of a drop on an inclined plane. Also, the limiting inclination angle cannot in general be calculated from the classical equation based only on the advancing and receding contact angles.