Migration of ion-exchange particles driven by a uniform electric field

A cation-selective conducting particle is suspended in an electrolyte solution and is exposed to a uniformly applied electric field. The electrokinetic transport processes are described in a closed mathematical model, consisting of differential equations, representing the physical transport in the electrolyte, and boundary conditions, representing the physicochemical conditions on the particle boundary and at large distances away from it. Solving this mathematical problem would in principle provide the electrokinetic flow about the particle and its concomitant velocity relative to the otherwise quiescent fluid. Using matched asymptotic expansions, this problem is analysed in the thin-Debye-layer limit. A macroscale description is extracted, whereby effective boundary conditions represent appropriate asymptotic matching with the Debye-scale fields. This description significantly differs from that corresponding to a chemically inert particle. Thus, ion selectivity on the particle surface results in a macroscale salt concentration polarization, whereby the electric potential is rendered non-harmonic. Moreover, the uniform Dirichlet condition governing this potential on the particle surface is transformed into a non-uniform Dirichlet condition on the macroscale particle boundary. The Dukhin–Derjaguin slip formula still holds, but with a non-uniform zeta potential that depends, through the cation-exchange kinetics, upon the salt concentration and electric field distributions. For weak fields, an approximate solution is obtained as a perturbation to a reference state. The linearized solution corresponds to a uniform zeta potential; it predicts a particle velocity which is proportional to the applied field. The associated electrokinetic flow is driven by two different agents, electric field and salinity gradients, which are of comparable magnitude. Accordingly, this flow differs significantly from that occurring in electrophoresis of chemically inert particles.