Nonlinear Smoluchowski slip velocity and micro-vortex generation

When an electric field is applied across a conducting and ion-selective porous granule in an electrolyte solution, a polarized surface layer with excess counter-ions is created. The depth of this layer and the overpotential V across this layer are functions of the normal electric field j on the granule surface. By transforming the ionic flux equations and the Poisson equation into the Painleve equation of the second type and by analysing the latter's asymptotic solutions, we derive a linear universal j-V correlation at large flux with an electrokinetic slip length beta. The flux-induced surface polarization produces a nonlinear Smoluchowski slip velocity that can couple with the granule curvature to produce micro-vortices in micro-devices. Such vortices are impossible in irrotational electrokinetic flow with a constant zeta-potential and a linear slip velocity.