Electric conduction from an electrolyte solution into a charge selective solid, such as ion exchange membrane or electrode, becomes unstable when the electrolyte concentration near the interface approaches zero owing to diffusion limitation. The sequence of events leading to instability is as follows: upon the decrease of the interface concentration, the electric double layer at the interface transforms from its common quasi-equilibrium structure to a different, non-equilibrium one. The key feature of this new structure is an extended space charge added to the usual one of the quasi-equilibrium electric double layer. The non-equilibrium electro-osmotic slip related to this extended space charge renders the quiescent conductance unstable. A unified asymptotic picture of the electric double-layer undercurrent, encompassing all regimes from quasi-equilibrium to the extreme non-equilibrium one, is developed and employed for derivation of a universal electro-osmotic slip formula. This formula is used for a linear stability study of quiescent electric conduction, yielding the precise parameter range of instability, compared with that in the full electroconvective formulation. The physical mechanism of instability is traced both kinematically, in terms of non-equilibrium electro-osmotic slip, and dynamically, in terms of forces acting in the electric double layer.
