Mathematical Modelling Of Electric Double Layers In Electrolytes For Lithium-Ion Batteries

Abstract

In this thesis we explore electric double layers (EDLs) in electrolytes for lithium-ion batteries using mathematical modelling tools. We review three standard continuum modelling approaches applied to model electrolytes: dilute theory, moderately concentrated theory, and thermodynamically consistent theory. We implement the thermodynamically consistent formulation to model a solid electrolyte whereby we investigate the structure of the EDLs both from numerical and asymptotic perspectives. We introduce an auxiliary variable to remove singularities from the domain, allowing for standard numerical methods and robust numerical simulations. In our non-dimensionalisation of the model we uncover a length scale representing the true width of these double charge layers. This informs an asymptotic reduction of the model whereby we reveal that the EDL is composed of two distinct regions: a boundary layer and an intermediate layer. The boundary layer exhibits polynomial behaviour while the intermediate layer exhibits exponential behaviour. We refer to the boundary layer as the strong space charge layer, and the intermediate layer as the weak space charge layer. Asymptotic matching between these two layers is non-standard, therefore we introduce a pseudo matching technique to complete the asymptotic solutions. We observe excellent agreement between our numerical simulations and asymptotics. Motivated by these results we apply the thermodynamic formulation to a liquid electrolyte to investigate the differences between the two electrolytes; noting that throughout the literature it is posited that these double charge layers in solid electrolytes are wider than those of the liquid, and that the liquid exhibits exponential behaviour in these layers, without any reference to a polynomial region. Through our numerics we confirm that the layers are wider in the solid, however, via our asymptotics we determine that the structure of these layers in the liquid also displays both polynomial and exponential behaviour. We introduce a parameter into the model to reconcile this thermodynamic model with the standard Poisson-Nernst-Planck (PNP) model, which is widely associated to the observation of exponential behaviour in the double layers. We find that the PNP model becomes ill-posed under the prescribed boundary conditions and suggest ways to rectify that.