Modeling analysis for the growth of a Li sphere and Li whisker in a solid-state lithium metal battery

One of the challenges when using lithium metal as the anode in rechargeable lithium batteries is the formation and growth of lithium dendrites. The recent observation by He et al. (Nat. Nanotechnol., 2019, 14, 1042-1047) and Zhang et al. (Nat. Nanotechnol., 2020, 15, 94-98) confirm the presence of the root-growth mode for the growth of lithium dendrites (whiskers, spheres and hillocks). In this work, we introduced a non-Newtonian flow model to describe the flow of lithium in lithium metal and incorporate the contributions of viscous dissipation, surface energy, kinetic energy and strain energy in the analysis of the stress relaxation and the growth of a Li-sphere and a Li-whisker. Nonlinear second-order differential equations are derived for the growth of the Li-sphere and the Li-whisker. Closed-form solutions for the temporal evolution of the Li-sphere and the Li-whisker are obtained under the conditions such that the contributions of the surface energy, kinetic energy and strain energy stored in the cantilever beam to the stress relaxation in the lithium metal are negligible. Using the Lippmann relation in specific surface energy, we demonstrated that increasing the electric potential reduces the resistance to the flow of lithium into the Li-sphere and the Li-whisker. The results reveal the need to suppress the cycling-induced strain energy (misfit strain and biaxial modulus) in order to mitigate the growth of Li dendrites (whiskers, spheres and hillocks).