Surfactant-dependent contact line dynamics and droplet spreading on textured substrates: derivations and computations

Abstract

We study the adhesion of a droplet with insoluble surfactant laid on its capillary surface to a textured substrate. In this process, the surfactant-dependent surface tension dominates the behaviors of the whole dynamics, particularly the moving contact lines. This allows us to derive the full dynamics of the droplets laid by the insoluble surfactant: (i) the moving contact lines, (ii) the evolution of the capillary surface, and (iii) the surfactant dynamics on this moving surface with a boundary condition at the contact lines. Our derivations base on Onsager's principle with Rayleigh dissipation functionals for either the viscous flow inside droplets or the motion by mean curvature of the capillary surface. We also prove the Rayleigh dissipation functional for the viscous flow case is stronger than the one for the motion by mean curvature. After incorporating the textured substrate profile, we design numerical schemes based on unconditionally stable explicit boundary updates and moving grids, which enable efficient computations for many challenging examples showing the significant contributions of the surfactant.