IntroductionIn lithium-ion batteries, electrode active particles are used to store and release lithium ions. The insertion and extraction of lithium ions are accompanied by a local deformation of the electrode active material, which can lead to mechanical degradation. Studies on the chemo-mechanical behavior of electrode particles can favor to improve the capacity and cycle life of batteries. Silicon has a super high capacity as an anode and is considered as a promising electrode material. However, its excessively high volume change rate hinders the further development of silicon electrodes. Two main reasons lead to mechanical failure of silicon electrode particles, i.e., the non-uniform volume change due to Li-ion concentration gradient, and the contact between particles upon lithium insertion induced volume expansion. Contacts between elastic deformable parts with simple morphologies are investigated based on the Hertz theory, which mainly analyzes contacts via calculating the contact area and indentation depth. Simulations of solids are mainly based on the Lagrangian description, which requires to create multiple systems and inter-system connections that are computationally complex. A phase-field model is developed with Eulerian description that can conveniently identify and distinguish the inside and outside of an object by the value of a variable “order parameter”. In addition, some examples for calculation the contact between electrode particles with different morphologies are also calculated to explore the ability of the proposed method.MethodsThe calculation model is based on 3 equations, i.e., the equilibrium equation, Cahn-Hilliard equation and the reference map equation. After splitting the Cahn-Hilliard equation into two parts, 4 variables v,ϕ, μ, ξ need to be calculated. The constitutive law is calculated through ξ and contact force is calculated through the ϕ of the contact bodies. Deformation gradient is decomposed into elastic part and lithium-intercalation-induced part. The elastic part is constructed on neo-Hookean theory, and lithium-intercalation-induced part is calculated through the concentration of Li. Elastic constants are also functions of the concentration as Si and Li both contribute to it. 4 PDEs are generated, including the Cahn-Hilliard equation, the equilibrium equation and the evolution equation of ξ. After applying boundary conditions, the PDEs are transformed to weak forms. The simulations are performed by the finite element method.Results and discussionA simple elastic contact problem is simulated to verify the feasibility and accuracy of the method. The result is compared with the Hertz theory. The lithiation of electrode particles is then simulated. The electrode particles are initialized as cylinders with regular and irregular profiles. The figure of von Mises stress shows that the contact between particles with regular profile is equivalent to the contact between a nanowire and rigid plane. And particles show different behaviors when they start lithiation at different durations. Particles with irregular circle profile shows a higher stress when the contact with rigid plane while the stress is lower when contacting with other particles. That is because particles rotate when the irregular profiles contact, which releases part of the stress and expands the contact area. To exclude the effects of rotation, considering that friction and binder can impede rotation, an example of 1/4 electrode particles is simulated. Particle rotation has a significant effect on stress relief, especially in the final stages of expansion, where the maximum stress can be reduced by approximately 1/3. Finally, simulations are performed with a reference to the actual electrode particle morphology. The internal stress in the particles becomes stable at a certain point and no longer increases significantly with deeper lithiation.ConclusionsThe simulation modeling of the contact between solids was implemented based on the phase-field method, and its feasibility and accuracy were verified through a simple example. Subsequently, the simulations of the contact problem of the lithiation of electrode particles were carried out to investigate the deformation and stress state of electrode particles with different shapes when the contact occurs to show the region where mechanical degradation might occur. The simulation of electrode particles’ lithiation showed different behaviors when the profile changed. This was because particles with irregular profile rotate when contacts occured so that concave surface and convex surface coupled. However, smaller contact area resulted in higher stress when concave surface contacted with rigid plane. The final simulations were performed with a reference to the morphology of the electrode particles in the real situation. These simulations could provide a reference for the study of the lithiation behavior of electrode particles in Li-ion batteries, showing the possible sites of mechanical damage or fracture of the electrode particles when they squeezed against each other, which in turn could be expected to contribute to the optimal design of Li-ion batteries in the future.