An Asymptotic Analysis of Space Charge Layers in a Mathematical Model of a Solid Electrolyte

TL;DR

This paper provides an asymptotic analysis of space-charge layers in a solid electrolyte model, identifying distinct layer behaviors and introducing a novel pseudo-matching method to connect different asymptotic regimes, validated by numerical simulations.

Contribution

It introduces a detailed asymptotic framework for analyzing space-charge layers in solid electrolytes, including a new pseudo-matching approach for different regimes.

Findings

Identification of strong and weak space-charge layers with distinct behaviors

Development of a pseudo-matching method for asymptotic regimes

Excellent agreement between asymptotics and numerical simulations

Abstract

We review a model for a solid electrolyte derived under thermodynamics principles. We non-dimensionalise and scale the model to identify small parameters, where we identify a scaling that controls the width of the space-charge layer in the electrolyte. We present asymptotic analyses and numerical solutions for the one dimensional zero charge flux equilibrium problem. We introduce an auxiliary variable to remove singularities from the domain in order to facilitate robust numerical simulations. From the asymptotics we identify three distinct regions: the bulk, boundary layers, and intermediate layers. The boundary and intermediate layers form the space charge layer of the solid electrolyte, which we can further distinguish as strong and weak space-charge-layers respectively. The weak space-charge-layer is characterised by a length, $\lambda$, which is equivalent to the Debye length of a standard liquid electrolyte. The strong space-charge-layer is characterised by a scaled Debye length, which is larger than $\lambda$. We find that both layers exhibit distinct behaviour, we see quadratic behaviour in the strong space-charge-layer and exponential behaviour in the weak space-charge-layer. We find that matching between these two asymptotic regimes is not standard and we implement a pseudo-matching approach to facilitate the transition between the quadratic and exponential behaviours. We demonstrate excellent agreement between asymptotics and simulation.