A modified phase-field method for the investigation of wetting transitions of droplets on patterned surfaces

A variant of the phase-field method, suitably modified to enable equilibrium (static) computations, is proposed for the calculations of the equilibrium wetting states of a droplet on patterned surfaces. Complemented by a parameter continuation method and stability analysis, the method is capable of mapping the solution space of the wetting states and computing the energy barriers of the wetting transitions. The wetting behavior of a droplet on a patterned surface, e.g. an array of pillars, is generally affected by the orientation, surface density (distance), size, and the shape of the pillars. The focus is on the effect of the shape (fine features) of the pillar, isolated from the effect of the pillar array; a simple system of a droplet lying on a single pillar is considered. The wetting on technologically feasible shapes of pillars is studied and the pillars are evaluated individually as single units with respect to the energy barrier of the transition from the Cassie-Baxter (CB) state to the Wenzel (W) state. The study brings out the intricate solution space of the wetting states on a single pin-shaped or rippled pillar. For example, 15 in total stable (impregnating) and unstable equilibrium states are calculated for the rippled pillar at a contact angle of 60°, which makes the calculation of the energy barrier more complex as it increases the number of potential transition paths. The evaluation verifies that the energy barrier of the CB to W transition is quite higher for pillars with re-entrant geometric features, in contrast to pillars with sharp protruding edges. Pin-shaped and inverted conical frustum pillars demonstrate the greater energy barrier of CB to W transition. The inherent flexibility of the phase-field method (handling of droplet breakup and coalescence) in combination with the mapping of the solution space, can facilitate the design of surface patterns with desired wetting behavior.