The chemo-mechanics of initiation of crack advance in Niobium Tungsten Oxide single crystals due to delithiation

Niobium tungsten oxide (NWO) micron-sized single crystals show promise as the active particles in the anodes of lithium ion batteries due to their fast charge/discharge and high storage capabilities. Within the "block" phases such as Nb16W5O55, Nb14W3O44 and the H-Nb2O5 polymorph, Li ion diffusion occurs in a unidirectional manner along a single axis of the crystal and lithiation results in significant length changes in this direction. A chemo-mechanics analysis is developed for delithiation of an NWO "block" phase single crystal that contains a pre-existing flaw in the form of a through-thickness edge crack. Galvanostatic delithiation occurs by the flux of Li ions through the free ends of the crystal and also through the faces of the crack, with concomitant shrinkage of the surface layer over a depth equal to that of the crack. The induced tensile stress in the surface layer can be of sufficient magnitude to advance the crack in the NWO crystal, consistent with recent experimental observations for Nb14W3O44. A 2D full-field finite element version of the chemo-mechanics model is used to explore the sensitivity of cracking to crack length, delithiation rate of the anode, diffusivity of Li ions within the NWO crystal and to the kinetics of Li ion transfer across its surface. Additionally, a simplified, lumped parameter model is developed by assuming that the Li ion occupancy is spatially uniform in a surface layer that extends to a depth of the pre-existing crack, and has a spatially uniform but different value in the underlying core of the particle. This simplified model can be expressed as a set of first order differential equations in time and gives the main features of delithiation and the associated transient stress state in the single crystal. Both the full model and the lumped parameter version predict that crack advance from a pre-existing defect is triggered by a high delithiation rate, consistent with the experimental literature.