Static wetting of a liquid droplet on a soft elastic substrate

A refined spherical cap model, combined with an elastic foundation model for the elastic substrate, is proposed to study static wetting of a liquid droplet on a soft elastic substrate. The strain energy of the substrate is evaluated by the JKR (Johnson-Kendall-Roberts) model, and the increase of the surface energy of the substrate outside the contact zone is calculated based on the elastic foundation model. The total potential energy of the droplet-substrate system is given in terms of four geometrical parameters: the contact radius, the contact angle of the droplet, the deflection angle inside the contact zone, and the maximum downward displacement of the substrate surface at the contact zone center. The equilibrium state is determined based on the stationary condition of total potential energy. The present model reduces to the Young's equation for a rigid substrate and to the Neumann's triangle for a liquid-like substrate. Three equations are given to determine the liquid droplet shape in terms of surface energies and substrate's elastic modulus. Reasonable agreement with existing experimental data and simulation results shows that the present model with derived formulas has the potential to catch the role of substrate's elastic deformation on static wetting and fill the gap between the Young's equation and the Neumann's triangle for a soft elastic substrate.