Surprising consequences of ion conservation in electro-osmosis over a surface charge discontinuity

A variety of microfluidic technologies utilise electrokinetic transport over rigid surfaces possessing rapid variations in charge. Here, as a paradigmatic model system for such situations, we consider electro-osmosis past a flat plate possessing a discontinuous jump in surface charge. Although the problem is relatively simple to pose, our analysis highlights a number of interesting and somewhat surprising features. Notably, the standard assumption that the electric field outside the diffuse screening layer is equal to the uniform applied field leads to a violation of ion conservation, since the applied field drives an ionic Surface current along the diffuse layer downstream of the jump, whereas there is zero surface current upstream. Instead, at the surface charge discontinuity, field lines are drawn into the diffuse layer to supply ions from the bulk electrolyte, thereby ensuring ion conservation. A simple charge conservation argument reveals that the length-scale over which this process occurs is of the order of the ratio of surface-to-bulk electrolyte conductivities, L-H = sigma(s)/sigma(b). For a highly charged surface, LH can be several orders of magnitude greater than the Debye screening length lambda(D), which is typically nanometres in size. Remarkably, therefore, nano-scale surface conduction may cause micrometre-scale gradients in the bulk electric field. After a distance O(L-H) downstream, the bulk field 'heals' and is once again equal to the applied field. Scaling all distances with the 'healing length' L-H yields a universal set of equations for the bulk field and fluid flow, which are solved numerically. Finally, we discuss the role of surface conduction in driving a non-uniform ion distribution, or concentration polarization, in the bulk electrolyte.