Energetic variational formulation for electrohydrodynamics of surfactant-laden droplets

Abstract

The coupling between surfactant-laden droplet dynamics and electric fields plays an important role in liquid-handling technologies such as digital microfluidics. We develop an energetic variational framework for the coupled electrohydrodynamics of surfactant-laden droplets, incorporating two-phase Stokes flow, insoluble surfactant transport on a moving interface, and electrostatic effects. Based on Onsager's principle, the governing equations are derived by minimizing the Rayleighian, defined as the sum of the rate of change of the free energy and the dissipation functional, subject to the incompressibility constraint. This formulation simultaneously yields the Stokes equations in each bulk phase, the interfacial stress-balance condition incorporating Marangoni and Maxwell stresses, the electrostatic equation, the surface transport equation for the surfactant concentration, and the moving contact-line dynamics. By replacing the viscous dissipation functional with Rayleigh dissipation, we further derive a reduced model describing the evolution of surfactant-laden droplets driven by motion by mean curvature. For sessile droplets represented as graph surfaces, the model reduces to a one-dimensional coupled electrohydrodynamic model for the liquid height, surfactant concentration, and electrostatic potential. A first-order implicit-explicit numerical scheme is proposed for the reduced system, and numerical results illustrate the coupled effects of surfactant transport and electric fields on driving droplet deformation and migration.