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

SIAM Journal on Applied Mathematics, Volume 84, Issue 4, Page 1413-1438, August 2024.
Abstract. We review a model for a solid electrolyte derived under thermodynamics principles. We nondimensionalize 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: bulk, boundary, 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 characterized by a length, [math], which is equivalent to the Debye length of a standard liquid electrolyte. The strong space charge layer is characterized by a scaled Debye length, which is larger than [math]. We find that both layers exhibit distinct behavior; we see quadratic behavior in the strong space charge layer and exponential behavior in the weak space charge layer. We find that matching between these two asymptotic regimes is not standard, and we implement a pseudomatching approach to facilitate the transition between the quadratic and exponential behaviors. We demonstrate excellent agreement between asymptotics and simulation.