Dynamic Stern layers in charge-regulating electrokinetic systems: three regimes from an analytical approach

We present analytical solutions for the electrokinetic equations at a charged surface with both non-zero Stern-layer conductance and finite chemical reaction rates. We have recently studied the same system numerically [B.L. Werkhoven et al., Phys. Rev. Lett. 120, 264502 (2018)], and have shown that an applied pressure drop across the surface leads to a non-trivial, laterally heterogeneous surface charge distribution at steady state. In this work, we linearise the governing electrokinetic equations to find closed expressions for the surface charge profile and the generated streaming electric field. The main results of our calculations are the identification of three important length and time scales that govern the charge distribution, and consequently the classification of electrokinetic systems into three distinct regimes. The three governing time scales can be associated to (i) the chemical reaction, (ii) diffusion in the Stern layer, and (iii) conduction in the Stern layer, where the dominating (smallest) time scale characterises the regime. In the reaction-dominated regime, we find a constant surface charge with an edge effect and recover the Helmholtz-Smoluchowski equation. In the other two regimes, we find that the surface charge heterogeneity extends over the entire surface, either linearly (diffusion-dominated regime) or nonlinearly (conduction-dominated regime).