Spreading of a thin droplet on a soft substrate

A thin liquid droplet spreads on a soft viscoelastic substrate with arbitrary rheology. Lubrication theory is applied to the governing field equations in the liquid and solid domains, which are coupled through the free boundary at the solid-liquid interface, to derive a set of reduced equations that describe the spreading dynamics. Fourier transform techniques and the finite difference method are used to construct a solution for the dynamic liquid-gas and solid-liquid interface shapes, as well as the macroscopic contact angle. Substrate properties affect the spreading dynamics through the contact angle and internal droplet flow fields, and these mechanisms are revealed. Increased substrate softness increases the spreading rate, whereas increased viscoelasticity decreases the spreading rate. For the case of a purely elastic substrate, the spreading power-law exponent recovers Tanner's law in the rigid limit and increases with substrate softness.