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

TL;DR

The paper develops a first-principles FBV–PNP model for charging a long, electrolyte-filled pore under a small applied potential near the equilibrium, and shows that early-time dynamics map onto a biased Faradaic transmission-line circuit with parameters determined from microscopic pore properties. In the long-time limit, the model yields a nontrivial potential of zero charge given by $\Psi^{pzc}=\Psi^{eq}\left[1-\hat{Z}(0)/R_F\right]$, linking PZC to the zero-frequency impedance and Faradaic resistance. The work provides a transparent framework to quantify the Faradaic contribution to PZC and offers spatiotemporal insights into pore charging that extend traditional TL and macrohomogeneous models. This approach enables experimental extraction of $R_F$ and improves understanding of charge transfer in porous electrodes with coupled EDL formation. Potential extensions include unequal diffusivities, multi-step reactions, and networks of pores for more realistic porous electrodes.

Abstract

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 $Ψ^\text{eq}$ 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, $Ψ_\text{eq}$, biasing the Faradaic reactions, captured by the resistance $R_F$. In the long-time limit, the model exhibits a nontrivial potential of zero charge, $Ψ_\text{pzc} = Ψ_\text{eq}[1 - \hat{Z}(0)/R_F]$, where $\hat{Z}(0)$ is the experimentally accessible zero-frequency impedance of the system. This expression provides a new means to experimentally measure the Faradaic contribution to $Ψ_\text{pzc}$.

Figures (5)

• Figure 1: (a) Schematic of a cylindrical pore connected with a bulk reservoir. The pore surface is chemically active with oxidation reaction \ref{['eq:redox']}. (b) Schematic illustration of the ions (red and blue discs are cations and anions, respectively) in a two-dimensional cut of the cylindrical setup colored gray in (a). The pore obtains positive charge while the reservoir remains neutral. (c) Faradaic TL circuit.
• Figure 2: The centerline potential ${\psi}_c(t,z)$\ref{['eq:phi_c-sol']} for $\text{Bi}=R_p/R_r=10$, and $\text{Da}=R_p/R_F=1$. The curves correspond to $t/(R_p C)=10^{-4}, 10^{-3}, 10^{-2}, 10^{-1}, 1, 10$ (purple to yellow). The black dotted line shows the steady state value $\psi_c^\text{ss}(z)$ [\ref{['eq:psi_c-ss']}].
• Figure 3: The spatial distribution of the potential $\psi(t,r,z)$ at times $t/(R_p C)=0.001,0.01,0.1,1$ from (a) to (d), respectively. The heatmap is calculated by \ref{['eq:phi-distr']} for $\text{Bi}=R_p/R_r=10$, $\text{Da}=R_p/R_F=1$, $\lambda_D=\lambda_S=0.1 \varrho_p$, $\Lambda = R_p C D/\ell_p^2=0.1$, $e\Psi/(kT) = - 0.2$, and $e\Psi^\text{eq}/(kT) = - 0.1$.
• Figure 4: The scaled steady-state centerline potential $\psi_c^\text{ss}/\delta\Psi$ [\ref{['eq:psi_c-ss']}] for $\text{Bi}= 5$ and $\text{Da}$ from $0$ to $2.5$ with the step $0.25$ (solid curves from purple to yellow).
• Figure 5: The steady-state ratio $Q^\text{ss}/Q^\text{nr}$ calculated in the plane of $\Psi_\text{eq}$ and $\Psi$ for $\text{Bi}=1$ and $\text{Da}=2$. The red line lies at $Q^\text{ss} = 0$ and illustrates \ref{['eq:PZC']}.