Macro-scale description of transient electro-kinetic phenomena over polarizable dielectric solids

We have studied the temporal evolution of electro-kinetic flows in the vicinity of polarizable dielectric solids following the application of a ‘weak’ transient electric field. To obtain a macro-scale description in the limit of narrow electric double layers (EDLs), we have derived a pair of effective transient boundary conditions directly connecting the electric potentials across the EDL. Within the framework of the above assumptions, these conditions apply to a general transient electro-kinetic problem involving dielectric solids of arbitrary geometry and relative permittivity. Furthermore, the newly derived scheme is applicable to general transient and spatially non-uniform external fields. We examine the details of the physical mechanisms involved in the relaxation of the induced-charging process of the EDL adjacent to polarizable dielectric solids. It is thus established that the time scale characterizing the electrostatic relaxation increases with the dielectric constant of the solid from the Debye time (for the diffusion across the EDL) through the ‘intermediate’ scale (proportional to the product of the respective Debye- and geometric-length scales). Thus, the present rigorous analysis substantiates earlier results largely obtained by heuristic use of equivalent RC-circuit models. Furthermore, for typical values of ionic diffusivity and kinematic viscosity of the electrolyte solution, the latter time scale is comparable to the time scale of viscous relaxation in problems concerning microfluidic applications or micro-particle dynamics. The analysis is illustrated for spherical micro-particles. Explicit results are thus presented for the temporal evolution of electro-osmosis around a dielectric sphere immersed in unbounded electrolyte solution under the action of a suddenly applied uniform field, combining both induced charge and ‘equilibrium’ (fixed charge) contributions to the zeta potential. It is demonstrated that, owing to the time delay of the induced-EDL charging, the ‘equilibrium’ contribution to fluid motion (which is linear in the electric field) initially dominates the (quadratic) ‘induced’ contribution.