Faradaic and capacitive charging of an electrolyte-filled pore in response to a small applied potential

Electrochemical devices often charge both through Faradaic reactions and electric double layer formation. Here, we study these coupled processes in a model system of a long electrolyte-filled pore subject to a small suddenly applied potential, close to the equilibrium potential Ψeq at which there is no net Faradaic charge transfer. Specifically, we solve the coupled Poisson-Nernst-Planck and Frumkin-Butler-Volmer equations by asymptotic approximations, using the pore's small inverse aspect ratio as the small parameter. In the early time limit, the reaction-diffusion equations yield an extended Faradaic transmission line model that includes a voltage source, Ψeq, biasing the Faradaic reactions, captured by the resistance RF. In the long-time limit, the pore exhibits a nontrivial potential of zero total charge, Ψpztc=Ψeq1-Ẑ(0)/RF, where Ẑ(0) is the experimentally accessible zero-frequency impedance of the system. This expression provides a new means to measure the Faradaic contribution to Ψpztc experimentally.