Explicit formulae for the internal stress in spherical particles of active material within lithium ion battery cathodes during charging and discharging

• Closed form expression proposed for Li ion concentration in a spherical particle.
• This solution for fast (dis)charging depends explicitly on time and radius.
• Explicit internal stress solution derived further is useful in design calculations.

The fundamental process underlying the operation of lithium ion batteries is the diffusion of charge-carrying ions through the electrolyte that separates the anode from the cathode, and the reversible insertion of lithium ions into the host material’s crystal structure (intercalation and de-intercalation). Alongside the principal electrochemical consequences of this process, mechanical phenomena that accompany (de)intercalation are of fundamental significance for deformation and fragmentation of the active materials, since it is these phenomena that ultimately determine the battery structural integrity and durability.This article presents a sequentially coupled analytical treatment of the transient diffusion and stress analysis (eigenstrain) problem related to the lithiation and de-lithiation processes at the level of an individual spherical secondary particle of active material. Explicit closed form approximate solutions are derived for the stresses that arise within the particles during fast charging. They provide a firm basis for the assessment of the charging conditions influence on the internal stress states and the effects on battery damage, mechanical integrity and durability.