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

We study spreading of a droplet, with insoluble surfactant covering its capillary surface, on 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, (iii) the surfactant dynamics on this moving surface with a boundary condition at the contact lines and (iv) the incompressible viscous fluids inside the droplet. 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 viscous flow case is stronger than the one for the motion by mean curvature. After incorporating the textured substrate profile, we design a numerical scheme based on unconditionally stable explicit boundary updates and moving grids, which enable efficient computations for many challenging examples showing significant impacts of the surfactant to the deformation of droplets.