Transport-Mechanical Model for Deformation and Stress Calculation in Phase-Change Active Materials for Electrodes of Lithium-Ion Batteries
Stress and consequent crack propagation are one of the main causes of damage in the electrodes’ microstructure of lithium-ion batteries. In this kind of problems, two physical fields are involved: The concentration of lithium ions, which are driven by diffusion, and the mechanical field, which arises as a consequence of the inhomogeneous lithium ions concentration. A transport-mechanical model is developed in COMSOL Multiphysics® using the Transport of Diluted Species and Solid Mechanics interfaces, with the aim of accurately computing concentration distribution, stress and strain in the particles of phase-change active materials (such as graphite and LFP) used in lithium-ion batteries’ electrodes.
Firstly, the simplest case is considered: The Transport of Diluted Species interface is used as it is to model a Fickian diffusion of lithium ions in active material particles, and hygroscopic swelling is implemented in the Solid Mechanics interface, to model the deformation induced by the lithium-ions concentration. A concentration-dependent partial molar volume, obtained from X-ray diffraction measurements, is considered as the proportionality coefficient between lithium ion concentration and deformation.
Secondly, the influence of the mechanical stress on lithium-ions diffusion is taken into account by modifying the built-in equation of the Transport of Diluted Species interface. In this case, diffusion is no longer Fickian, but depends both on the gradient of concentration and hydrostatic stress.
Thirdly, mechanical coupling and the phase-change behavior of some active materials are considered. Indeed, materials such as graphite - the state of the art of anodic material - do not show a smooth diffusion, but some lithium concentration levels, corresponding to the different “stages” are favored. This peculiar behavior, which is similar to the phase change behavior of LFP and other active materials, causes a steep concentration gradient between the “favored” concentration levels of the stages, causing localized stress peaks which are preferred spots for crack nucleation and growth. This particular transport mechanism is derived theoretically with a thermodynamic consistent approach and is numerically implemented in COMSOL Multiphysics® by modifying further the equation of the Transport of Diluted Species interface, incorporating the activity coefficient. This coefficient affects the overall diffusion causing the localized concentration jumps between the different phases. The activity coefficient depends on the open circuit voltage of graphite with respect to lithium, which is a function of the lithium ions concentration in turn.
Graphite is considered as a case study. The results of the latter model incorporating the activity coefficient show that the concentration distribution in the graphite particle is in agreement with the experimental measurements, capturing the alternating stages. Concerning stress, the difference with respect to the traditional Fickian diffusion model is substantial, as the stress is up to 85% higher.
The results confirm the significant influence of the phase-change behavior of active materials, usually neglected by most of the works in literature computing stress in the electrode microstructure, and show that these assumptions lead to more serious loading conditions, and thus faster battery degradation.