  In this paper, we propose a Poisson-Nernst-Planck-Navier-Stokes-Cahn-Hillard
(PNP-NS-CH)model for an electrically charged droplet suspended in a viscous
fluid subjected to an external electric field. Our model incorporates spatial
variations of electric permittivity and diffusion constants, as well as
interfacial capacitance. Based on a time scale analysis, we derive two
approximations of the original model, namely a dynamic model for the net charge
and a leaky-dielectric model. For the leaky-dielectric model, we conduct 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 performed to
validate the asymptotic analysis and demonstrate the model's effectiveness in
handling topology changes, such as electrocoalescence. Our numerical results of
these two approximation models reveal that the polarization force, which is
induced by the spatial variation of electric permittivity in the direction
perpendicular to the external electric field, consistently dominates the
Lorentz force, which arises 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. Furthermore, in the presence of the
interfacial capacitance, a local variation of effective permittivity leads to
an accumulation of counter-ions near the interface, resulting in a reduction in
droplet deformation. Our numerical solutions also confirm that the leaky
dielectric model serves as a reasonable approximation of the original PNP-NS-CH
model when the electric relaxation time is sufficiently short. The Lorentz
force and droplet deformation both decrease when the diffusion of net charge is
significant.
