Modified multi-phase diffuse-interface model for compound droplets in contact with solid

In this study, a novel diffuse-interface (phase-field) model is developed to efficiently describe the dynamics of compound droplets in contact with a solid object. Based on a classical four-component Cahn–Hilliard-type system, we propose modified governing equations, in which the solid is represented by an initially fixed phase. By considering Young's equality between surface tensions and microscale contact angles, equilibrium profiles of diffuse interfaces, and horizontal force balance between contact and interfacial angles, a correction term is derived and added into the phase-field equations to reflect the accurate contact line property for each component. The proposed model can be implemented on Eulerian grids in the absence of complicated treatment on the liquid-solid boundary. The standard finite difference method (FDM) is adopted to perform discretization in space. The linear second-order time-accurate method based on the two-step backward differentiation formula (BDF2) and a stabilization technique are adopted to update the phase-field variables. To accelerate convergence in solving the resulting fully discrete system, we use the linear multigrid method. At each time step, the calculations are completely decoupled. The numerical experiments not only indicate the desired accuracy but also show superior capability in complex geometries. Furthermore, the numerical and analytical results for the compound droplets on a flat solid are in good agreement with each other.