Droplet evaporation on inclined patterned substrates

The evaporation of droplets on inclined substrates is important for a broad range of applications, including ink-jet printing, surface cooling, and micro-structure assembly. Despite its apparent simplicity, the precise configuration of an evaporating droplet on a solid surface has proven notoriously difficult to predict and control. Here, we study the effect of smooth patterns on a droplet evaporating on an inclined plane. The evaporation is assumed to be quasi-static so that the dynamics of the droplet can be quantified by the equilibrium properties of the system. We first examine the limiting case of small gravity (small Bond number) and perform asymptotic analysis to obtain approximate solutions. For higher Bond numbers we derive exact solutions written in terms of elliptic integrals that can be solved numerically. By studying the stability of these solutions, and how it depends with the droplet size, we perform a bifurcation analysis that shows the emergence of a hierarchy of bifurcations that strongly depends on the particular underlying chemical pattern. We show in turn that gravity affects the topology of the bifurcation diagrams, reducing the stability regions of the phase space. To understand the droplet dynamics upon evaporation on inclined surfaces, we perform numerical simulations of the Cahn-Hilliard and Navier-Stokes system of equations. We focus on the quasi-static regime, where droplet evaporation is dominated by diffusion into the gas phase, and the density contrast between fluids is taken into account via a Boussinesq approximation. We observe very good agreement between the numerical results and the behaviour predicted by the quasi-static analysis. Our results show that the interplay between a phase change and surface wettability can be exploited to control the motion of droplets on inclined patterned solid surfaces.