Analytical distribution function of relaxation times for porous electrodes and analysis of the distributions of time constants

Abstract

Based on the work by Boukamp (Boukamp, 2017), the method of Fuoss and Kirkwood (Fuoss and Kirkwood, 1941) is applied to derive an analytical distribution function of relaxation times for physics based porous electrode impedance cases. These impedance models are typically described by transcendental transfer functions. The porous electrode impedance treated here reflects a balance of the effective ionic and electronic impedances inside a porous electrode consisting of particles. Therefore, first the DFRT of the single particle interface impedance is derived. This includes treatment of charge transfer, double layer charging, solid state diffusion inside the particles, open-circuit voltage variations due to solid-state concentration, and insulating layers surrounding the particles. The resulting single particle DFRT relations are then incorporated into a mathematical description of the porous electrode DFRT. The results show that the DFRT of the porous electrode can be clearly separated into distributions of time constants corresponding to charge transfer, solid state diffusion and in case of intercalating particles, like in lithium-ion batteries, a third distribution of time constants is identified. A novelty of this work is the explicit treatment of the low-frequency capacitance and the resulting distribution of time constants in porous electrode systems. Analytical relations for the individual time constants are derived and reported. Since the ideal distribution of time constants can be represented by a series of R||C circuit elements, validation is performed by reconstruction of the impedance spectra, based on the analytical results.