Droplet spreading on chemically heterogeneous substrates

Consider the spreading dynamics of a two-dimensional droplet over chemically heterogeneous substrates. Assuming small slopes and strong surface tension effects, a long-wave expansion of the Stokes equations yields a single evolution equation for the droplet thickness. The contact line singularity is removed by assuming slip at the liquid-solid interface. The chemical nature of the substrate is incorporated by local variations in the microscopic contact angle, which appear as boundary conditions in the governing equation. By asymptotically matching the flow in the bulk of the droplet with the flow in the vicinity of the contact lines, we obtain a set of coupled ordinary differential equations for the locations of the two droplet fronts. We verify the validity of our matching procedure by comparing the solutions of the ordinary differential equations with solutions of the full governing equation. The droplet dynamics is examined in detail via a phase-plane analysis. A number of interesting features that are not present in chemically homogeneous substrates are found, such as the existence of multiple equilibria, the pinning of the droplet fronts at localized chemical features, and the possibility for the droplet fronts to exhibit a stick-slip behavior.