Kinetics of non-equilibrium lithium incorporation in

. In particular, once transport limitations at the electrode level are removed through carbon addition and particle size reduction, the innate rate capability of LiFePO 4 is revealed to be very high. We demonstrate that the reason LiFePO 4 functions as a cathode at reasonable rate is the availability of a single-phase transformation path at very low overpotential, allowing the system to bypass nucleation and growth of a second phase. The Li x FePO 4 system is an example where the kinetic transformation path between LiFePO 4 and FePO 4 is fundamentally different from the path deduced from its equilibrium phase diagram. In the considerable volume of literature on the subject, the lithiation mechanism in (Li)FePO 4 is conventionally described and interpreted as a two-phase growth process 12-18 involving the coexistence of both phases (LiFePO 4 and FePO 4 ), initiated by a nucleation event. According to classical nucleation theory, on lithiation of FePO 4 the inserted Li will pool together to form clusters, which only grow once a critical size is reached. The critical size, below which the new phase dissolves again, exists because the driving force for transformation scales with volume, and the interfacial energy which hinders the transformation scales with area. These concepts lead to the well-known expressions for the critical radius (r * ) and critical nucleation barrier (G r * ), r * = 2γ ·v/(|φ| − g s ) and LiFePO 4 particle in a composite electrode, an underpotential in excess of 50 mV must be applied, and G r * is at least several hundred thousand kT (thermal energy) at room temperature per cluster. This very large energy required to form the critical nucleus makes nucleation very unlikely. Even if one were to ignore the coherency strain energy altogether, and reduce the interfacial energy by a factor of two (for example, to account for heterogeneous nucleation), G r * for a 50 nm diameter nucleus exceeds 300,000 kT per cluster. Considering this very large G r * , and the fact that the voltage hysteresis between charge and discharge in very slow discharging experiments (C/1,000) approaches only ∼20 mV (ref. 22), the lithiation mechanism in LiFePO 4 is clearly inadequately described by classical nucleation and growth. An identical analysis performed here applies to the charging procedure. If nucleation does not proceed then the Li x FePO 4 and Li 1−x FePO 4 solid solutions can be highly oversaturated. To investigate the energetics of metastable solid solutions in Li x FePO 4 as an alternative transformation path, we have used the cluster expansion approach combined with ab initio density functional theory calculations 23 . The cluster expansion allows determination of the energy of any Li/vacancy and Fe 2+ /Fe 3+ configuration in the system, and was parametrized from the calculated energies of 245 different Li/vacancy and electron/hole configurations in Li x FePO 4 (0 ≤ x ≤ 1) shown in To better quantify the free energy and voltage along a metastable transformation path, we performed canonical Monte Carlo simulations with the LiFePO 4 cluster expansion 7 , with appropriate constraints to avoid phase separation, to obtain the free energy of non-equilibrium states. Specifically, we used small NATURE MATERIALS | ADVANCE ONLINE PUBLICATION | www.nature.com/naturematerials