Primal-dual splitting methods for phase-field surfactant model with moving contact lines

Surfactants have important effects on the dynamics of droplets on solid surfaces, which has inspired many industrial applications. Phase-field surfactant model with moving contact lines (PFS-MCL) has been employed to investigate the complex droplet dynamics with surfactants, while its numerical simulation remains challenging due to the coupling of gradient flows with respect to transport distances involving nonlinear and degenerate mobilities. We propose a novel structure-preserving variational scheme for PFS-MCL model with the dynamic boundary condition based on the minimizing movement scheme and optimal transport theory for Wasserstein gradient flows. The proposed scheme consists of a series of convex minimization problems and can be efficiently solved by our proposed primal-dual splitting method and its accelerated versions. By respecting the underlying PDE's variational structure with respect to the transport distance, the proposed scheme is proved to inherits the desirable properties including original energy dissipation, bound-preserving, and mass conservation. Through a suite of numerical simulations, we validate the performance of the proposed scheme and investigate the effects of surfactants on the droplet dynamics.