In the theoretical treatment, we consider a general intercalation system in two dimensions in which ion intercalation proceeds through a first-order phase transformation with isotropic misfit strain and elasticity. The phase boundary between the Li-rich and Li-poor phases is assumed to maintain (partial) coherency. We consider interface-limited phase boundary migration, i.e. the interface velocity is proportional to the Li chemical potential difference between the Li-rich and Li-poor phases at the interface, which includes contribution from coherency stress. Linear stability analysis is carried out to determine the stability of a small periodic perturbation applied to the originally flat interface, Figure 1(b). The analysis results in the following main findings:
1) The phase boundary is unstable against perturbations with wave vector below a critical value kc, or equivalently, when perturbation wavelength is larger than the critical wave length λc = 2*π/kc. As shown in Figure 1(c), there exists a fastest growing wave vector km, at which the perturbation amplitude has the maximum growth rate. Both kc and km are proportional to A/(E*e02), where A is interfacial energy, E is the Young’s modulus and e0 is the misfit strain. With typical values of these properties for intercalation compounds (A ~ 0.1 J/m^2, E ~ 100 GPa, e0~ 0.01), the critical and fastest growing wave lengths of interface perturbation are in the range of 10 – 100 nm. Interface stability is hence expected in nano- and micro-sized particles.
2) Interface is most susceptible to instability development at the beginning stage of the (de)intercalation process. As shown in Figure 1(d), the fastest growing wave vector of interface perturbation depends on the depth of the interface from the particle surface, z0. While km and kcare the largest when interface is near the electrode surface, they decay with interface moving into the interior of the particle.
3) Interface instability is suppressed in the interior of intercalation compounds below a critical particle size. Figure 1(d) illustrates that the fastest growing wave vector drops to 0, i.e. no perturbation with finite wave length can grow, when the system dimension in the main intercalation direction is smaller than a critical thickness L*, which is on the order of 100 nm. Therefore, nanoscaling electrodes not only benefits intercalation kinetics but also improves interface stability.
Complementing the linear stability analysis, we also performed phase-field simulation of the phase boundary evolution during delithiation of LiFePO4 cathode. Consistent with the theoretical prediction, interface instability is observed in the simulation (Figure 1(e)), which reveals the details of the interface morphological evolution at later stage. Notably, similar interface morphology is observed in a recent operando hard x-ray microscopy experiment .
Based on the findings from this study, we identify two useful approaches to stabilize phase boundary during ion (de)intercalation:
i) Reduce one dimension of the electrode particles to the sub-micron regime, i.e. nanoplate morphology is desirable.
ii) Apply large current pulses at the beginning of discharge/charge to have interface rapidly traverse the near-surface region where instability is easiest to develop.
 Asaro and Tiller, Metall. Trans. 3 1789 (1972); Grinfeld, Sov. Phys. Dokl. 31, 831 (1986).