Electrophoretic motion of a liquid droplet with Brinkman-screened internal hydrodynamics

We develop a theory for the electrophoresis of a spherical porous liquid droplet with prescribed uniform surface charge. The exterior electrokinetics is governed by the Poisson-Nernst-Planck-Stokes equations, while the internal liquid motion is described by the Brinkman-Debye-Bueche equation. A regular perturbation expansion in the applied electric field reduces the governing equations to coupled radial ordinary differential equations. In the Debye-H\"uckel regime, we derive a closed-form mobility expression valid for arbitrary Debye layer thickness. The analysis shows that the porous interior modifies clean-droplet electrophoresis through a single Brinkman-screened hydrodynamic resistance, yielding a continuous transition between clean-droplet and rigid-particle limits. Numerical solutions beyond the low-potential regime reveal a non-universal role of permeability: increasing the Darcy number can either suppress or enhance the mobility. This reversal is determined by the sign of the interfacial-velocity mode, which is governed by the competition between tangential Maxwell traction and hydrodynamic shear generated by electric-double-layer distortion. Dielectric polarization, surface charge and double-layer thickness can reverse the internal circulation, while the Darcy number controls how strongly this circulation is transmitted through the porous interior. This permeability sensitivity is especially pronounced for highly polarizable droplets in the thin-double-layer regime. The theory provides a basis for tuning electrokinetic transport of soft porous droplets in microfluidic and biomedical technologies.