Precise expression for the reactive energy of a sessile spherical-cap droplet in the electrostatic surrogate model

Derive an exact analytical expression for the reaction-induced long-range interaction energy (the reactive energy in the electrostatic surrogate model) of a single sessile chemically active droplet with spherical-cap geometry and contact angle θ on a planar wall, under linear bulk reactions mapped to electrostatics with Neumann boundary conditions. The expression should quantify the reactive energy as a function of droplet size (area/volume) and θ, replacing the current reliance on approximations based on a neutral-wall (half-spherical) droplet, to enable accurate evaluation of nucleation barriers and shape deformations in the presence of walls.

Background

The paper maps linear bulk reactions in an active Cahn–Hilliard system to an equilibrium surrogate with a nonlocal energy that admits an electrostatic interpretation, where deviations c(r)−c0 act as effective charges and the reactive energy is an electrostatic self-energy. For quantitative predictions of nucleation barriers and shape deformations of sessile droplets, one needs the reactive energy for the spherical-cap geometry at a wall.

Because an exact expression is not available, the authors approximate the reactive energy by using the energy of one half of a spherical bulk droplet (neutral wall) as an upper bound for a spherical cap, neglecting explicit dependence on the contact angle. This approximation limits their ability to capture the coupling between reaction strength and wall affinity. A closed-form expression for the spherical-cap case would remove this limitation and improve predictive accuracy.