Size effects in nanostructured Li-ion battery cathode particles

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.