Geometry-dependent elastic stability map of solid and hollow alloy particles for lithium-ion battery anodes

Alloy-type anodes such as silicon (Si), germanium (Ge), and tin (Sn) offer substantially higher theoretical capacities than conventional graphite anodes, but their practical implementation is hindered by severe volumetric expansion and diffusion-induced mechanical degradation during lithiation. Although nanoscale particles and hollow architectures have been widely explored to alleviate stress accumulation, quantitative design criteria that consistently account for material properties, particle geometry, and operating conditions remain lacking. Here, we implement a coupled chemo-mechanical framework to investigate lithium transport, deformation, and stress evolution in both solid and hollow alloy particles. Based on this framework, we establish geometry-dependent elastic stability maps that define the maximum allowable lithiation rate as a function of particle size and shell geometry. The proposed maps enable a systematic comparison of representative alloy systems (Si, Ge, and Sn) under identical electrochemical loading conditions and reveal distinct material-dependent design trends. Hollow architectures substantially expand the elastic operating window by mitigating diffusion-induced stress, particularly in Si and Ge, whereas the benefit of geometric optimization is comparatively limited in Sn. Among the materials studied, Ge exhibits the broadest elastic stability regime owing to its favorable balance of diffusivity, volumetric expansion, and mechanical strength. Furthermore, the results demonstrate that elastic operating windows provide a more practical design criterion than peak stress alone when evaluating mechanically robust alloy anodes. The resulting design maps offer predictive guidelines for selecting material-geometry combinations under application-relevant charging conditions, bridging the gap between nanoscale mechanical stability and microscale practical implementation in high-capacity lithium-ion battery anodes.