Nonlinear Dynamics of Capacitive Charging and Desalination by Porous Electrodes

TL;DR

This paper develops a unified mean-field theory for ion exchange in porous electrodes, covering capacitive charging and desalination, with simplified models for different regimes and numerical solutions for practical parameters.

Contribution

It introduces a comprehensive mean-field framework for ion dynamics in porous electrodes, unifying capacitive charging and desalination processes with analytical and numerical insights.

Findings

The theory captures two regimes: super-capacitor and desalination.

Numerical solutions validate the models for realistic parameters.

Simplified models provide insights into different time scales of ion exchange.

Abstract

The rapid and efficient exchange of ions between porous electrodes and aqueous solutions is important in many applications, such as electrical energy storage by super-capacitors, water desalination and purification by capacitive deionization (or desalination), and capacitive extraction of renewable energy from a salinity difference. Here, we present a unified mean-field theory for capacitive charging and desalination by ideally polarizable porous electrodes (without Faradaic reactions or specific adsorption of ions) in the limit of thin double layers (compared to typical pore dimensions). We illustrate the theory in the case of a dilute, symmetric, binary electrolyte using the Gouy-Chapman-Stern (GCS) model of the double layer, for which simple formulae are available for salt adsorption and capacitive charging of the diffuse part of the double layer. We solve the full GCS mean-field theory numerically for realistic parameters in capacitive deionization, and we derive reduced models for two limiting regimes with different time scales: (i) In the "super-capacitor regime" of small voltages and/or early times where the porous electrode acts like a transmission line, governed by a linear diffusion equation for the electrostatic potential, scaled to the RC time of a single pore. (ii) In the "desalination regime" of large voltages and long times, the porous electrode slowly adsorbs neutral salt, governed by coupled, nonlinear diffusion equations for the pore-averaged potential and salt concentration.