Migration and spreading of a droplet driven by a chemical step

The chemical step is an elementary pattern in chemically heterogeneous substrates, featuring two regions of different wettability separated by a sharp border. Within the framework of lubrication theory, we investigate droplet motion and the contact-line dynamics driven by a chemical step, with the contact-line singularity addressed by the Navier slip condition. For both two-dimensional (2-D) and three-dimensional (3-D) droplets, two successive stages are identified: the migration stage, when the droplet traverses both regions, and the asymmetric spreading stage, when the droplet spreads on the hydrophilic region while being constrained by the border. For 2-D droplets, we present a matched asymptotic analysis that agrees with numerical solutions. In the migration stage, a 2-D droplet can exhibit translational motion with a constant speed. In the asymmetric spreading stage, the contact line at the droplet rear is pinned at the border. We show that a boundary layer still exists near the pinned contact line, across which the slope is approximately constant, whereas the curvature would diverge in the absence of slip. For 3-D droplets, our numerical simulations show that the evolution is qualitatively analogous to the 2-D case, while being significantly affected by the lateral flow. The droplet length and width exhibit non-monotonic variations due to the lateral flow. Eventually, the droplet detaches from the border and approaches equilibrium at the hydrophilic substrate. Additionally, we demonstrate that the variation of the apparent contact angle at the instant of border contact only affects the early stage of droplet migration.