Migration and spreading of a droplet driven by a chemical step

Abstract

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 (2D) and three-dimensional (3D) 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 2D droplets, we present a matched asymptotic analysis which agrees with numerical solutions. In the migration stage, a 2D 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 3D droplets, our numerical simulations show that the evolution is qualitatively analogous to the 2D case, while being significantly affected by the lateral flow. At early times, the contact line on the hydrophilic region advances linearly and spreads transversely according to a power law $t^{1/2}$. The droplet length and width exhibit non-monotonic variations due to the lateral flow. Eventually, the droplet detaches from the border and reaches equilibrium at the hydrophilic substrate.