Size effects in nanostructured Li-ion battery cathode particles

Sundararajan Natarajan,Katerina Aifantis

doi:10.1515/jmbm-2020-0004

Abstract

Abstract Cathode materials for Li-ion batteries exhibit volume expansions on the order of 10% upon maximum lithium insertion. As a result internal stresses are produced and after continuous electrochemical cycling damage accumulates, which contributes to their failure. Battery developers resort to using smaller particle sizes in order to limit damage and some models have been developed to capture the effect of particle size on damage. In this paper, we present a gradient elasticity framework,which couples the mechanical equilibrium equations with Li-ion diffusion and allows the Young’s modulus to be a function of Li-ion concentration. As the constitutive equation involves higher order gradient terms, the conventional finite element method is not suitable, while, the two-way coupling necessitates the need for higher order shape functions. In this study, we employ B-spline functions with the framework of the iso-geometric analysis for the spatial discretization. The effect of the internal characteristic length on the concentration evolution and the hydrostatic stresses is studied. It is observed that the stress amplitude is significantly affected by the internal length, however, using either a constant Young’s modulus or a concentration dependent one yields similar results.

Highlights

Li-ion batteries are the most promising energy storage devices for portable devices and electric vehicles
Battery developers resort to using smaller particle sizes in order to limit damage and some models have been developed to capture the effect of particle size on damage
This work is licensed under the Creative Commons classical linear elasticity with only one extra material pa- Here u is the displacement vector, σ is the Cauchy stress rameter that multiplies the tensor, c represents the concentration and J represents the Laplacian of the Hookean stress, which is added to diffusive flux

Summary

Introduction

Li-ion batteries are the most promising energy storage devices for portable devices and electric vehicles. This work is licensed under the Creative Commons classical linear elasticity with only one extra material pa- Here u is the displacement vector, σ is the Cauchy stress rameter (the internal length scale, lo) that multiplies the tensor, c represents the concentration and J represents the Laplacian of the Hookean stress, which is added to diffusive flux. The diffusive flux is given by the gradient of the classical linear elasticity expression [6]. The paper is organized as follows: Section 2 presents the governing differential equations for the diffusion of Liions and the corresponding gradient elasticity equations. Where M is the mobility of the lithium ions and the chemical potential in an ideal solid solution is expressed as:. The influence of the concentration on the stress field can be modelled by modifying the stress field within the gradient elasticity [6] by thermal analogy

Theoretical formulation

Weak form

Spatial and temporal discretization

Results