A framework to investigate the immittance responses for finite length-situations: fractional diffusion equation, reaction term, and boundary conditions

TL;DR

This paper develops a comprehensive framework for analyzing the impedance response of finite-length electrolytic cells, incorporating fractional diffusion, reaction terms, and boundary effects, with exact solutions and experimental validation.

Contribution

It introduces a novel formalism extending the PNP model to include fractional diffusion, reactions, and boundary conditions, providing exact impedance solutions for finite systems.

Findings

Model accurately predicts impedance spectra of electrolytic solutions.

Good agreement between theoretical predictions and experimental data.

Framework captures surface effects like charge transfer and adsorption-desorption.

Abstract

The Poisson-Nernst-Planck (PNP) diffusional model for the immittance or impedance spectroscopy response of an electrolytic cell in a finite-length situation is extended to a general framework. In this new formalism, the bulk behavior of the mobile charges is governed by a fractional diffusion equation in the presence of a reaction term. The solutions have to satisfy a general boundary condition embodying, in a single expression, most of the surface effects commonly encountered in experimental situations. Among these effects, we specifically consider the charge transfer process from an electrolytic cell to the external circuit and the adsorption-desorption phenomenon at the interfaces. The equations are exactly solved in the small AC signal approximation and are used to obtain an exact expression for the electrical impedance as a funcion of the frequency. The predictions of the model are compared to and found to be in good agreement with the experimental data obtained for an electrolytic solution of CdCl2H20.