Modeling and simulation are essential to study the generation and effects of stress inside batteries. Treating the intercalation-induced stress analogously to thermal stress, a model has been developed to study the stress and concentration inside a particle. ^{1} This model has been extended to study various problems at both particle and cell levels. Nevertheless, this model and its extensions are based on the assumption of solid particles. This assumption does not apply for active materials with an agglomerate structure, such as LiNi_{0.8}Co_{0.15}Al_{0.05}O_{2} (NCA). ^{2} In these materials, many nanometer-scale primary particles agglomerate to a micrometer-scale secondary particle by the adhesion of binder. The secondary particle is porous rather than a compact solid, as the electrolyte is found to be soaked into the agglomerate.^{ 2 }Therefore, charge transfer reactions are expected to occur between the primary particle surface and the electrolyte inside a secondary particle.

Although electrochemical models have been developed to investigate the characteristic of an agglomerate accounting for the effects of its internal structure, ^{3} no mechanical model has been developed to study the stress in agglomerates for lithium-ion batteries. Meanwhile, multiple experiments have reported observations of fracture of agglomerates after cycling, which is a major mechanism of capacity degradation. This calls for a fully understanding of the mechanical behaviors at the agglomerate level.

This work presents a coupled mechanical and electrochemical model to predict the intercalation-induced stress in a secondary particle with an agglomerate structure, as shown in** Fig. 1**. In this model, the electrochemical and transport processes are accounted for at both the secondary and primary particle levels. The porous electrode theory is applied at the secondary particle level, and the solid diffusion is incorporated at the primary particle level. Simulation results from the electrochemical model revealed that a major concentration gradient exists along the radius of the secondary particle, while the concentration is fairly uniform in each primary particle.

Based on this finding, the mechanical model focused on the stress generation at the secondary particle level. The secondary particle is assumed to be mechanically homogeneous with effective properties, which can be calculated from the porosity and properties of bulk materials. Because the primary particle is much smaller than the secondary particle, the secondary particle is regarded as a continuum. Each spatial point in the secondary particle is composed of many primary particles at that location. Therefore the stress at each spatial point represents the loading stress exerted on the primary particles at that location. This loading stress is important to know since it is the cause of separation of primary particles, i.e. fracture in the secondary particle. The intercalation-induced stress is calculated using the analogy to thermal stress.

The developed model has been applied to investigate factors affecting the stress generation behaviors. The results are summarized as follows: 1) A strong dependence of OCP on the solid lithium concentration leads to a more uniform current density in the secondary particle, which reduces the stress level. 2) A large magnitude of over-potential at the secondary particle surface causes severely non-uniform current density, and thus larger stresses. 3) The primary particle size shows a significant effect on the current density, concentration and stress profiles. A larger primary particle size results in a smaller active surface area per volume, which reduces the impact of non-uniform current density and thus reduces the stress level in the secondary particle. However, the concentration gradient inside the primary particle becomes pronounced with the increase of the primary particle size, which may generate stress inside the primary particle. 4) The comparison between a porous secondary particle and a solid particle of the same size shows that the stress is greatly alleviated in the porous secondary particle. This is attributed to the lower Young’s modulus of the porous particle, and more importantly, to the smaller concentration gradient in the porous secondary particle.

References

1. X. Zhang, W. Shyy, and A. Marie Sastry, *J Electrochem Soc*, **154**, A910 (2007).

2. D. Abraham, D. Dees, J. Knuth, and E. Reynolds, *ARGONNE National Laboratory (ANL-05/21)*, 196 (2005).

3. S. Lueth, U. S. Sauter, and W. G. Bessler, *J Electrochem Soc*, **163**, A210 (2016).