Droplet dynamics: A phase-field model of mobile charges, polarization, and its leaky dielectric approximation

This paper presents a Poisson–Nernst–Planck–Navier–Stokes–Cahn–Hillard (PNP–NS–CH) model for an electrically charged droplet suspended in a viscous fluid under an external electric field. Our model incorporates spatial variations in electric permittivity and diffusion constants, as well as interfacial capacitance. Based on a time scale analysis, we derive two approximations of the original model: a dynamic model for the net charge (assuming unchanged conductance) and a leaky-dielectric model (assuming unchanged conductance and net charge). For the leaky-dielectric model, we perform a detailed asymptotic analysis to demonstrate the convergence of the diffusive-interface leaky-dielectric model to the sharp interface model as the interface thickness approaches zero. Numerical computations are conducted to validate the asymptotic analysis and demonstrate the model's effectiveness in handling topology changes, such as electro-coalescence. Our numerical results from these two approximation models reveal that the polarization force, induced by the spatial variation in electric permittivity perpendicular to the external electric field, consistently dominates the Lorentz force arising from the net charge. The equilibrium shape of droplets is determined by the interplay between these two forces along the direction of the electric field. Moreover, in the presence of interfacial capacitance, a local variation in effective permittivity results in the accumulation of counter-ions near the interface, leading to a reduction in droplet deformation. Our numerical solutions also confirm that the leaky-dielectric model is a reasonable approximation of the original PNP–NS–CH model when the electric relaxation time is sufficiently short. Both the Lorentz force and droplet deformation decrease significantly when the diffusion of net charge increases.