Intercalate Diffusion in Multiphase Electrode Materials and Application to Lithiated Graphite

Phase transformations in electrode materials used in lithium batteries can be characterized in a plot of the open-circuit voltage U ( x ) vs. state of charge x as regions that are almost constant. Some models represent these phase transitions as moving boundary points where jump discontinuities in concentration occur. The question of when it is necessary to include these effects in a transport model is difficult to resolve, because the measured values of U ( x ) for real materials are never completely flat and always exhibit a small dependence on state of charge. We show that the jump discontinuities arise as a singular limit of a nonlinear diffusion equation when chemical potentials are used as the driving force for transport. The chemical potential can be expressed in terms of U ( x ), and the discontinuities occur in regions where dU / dx vanishes. Published measurements of lithium diffusion in graphite illustrate these facts. The use of chemical potentials simplifies the tabulation of diffusion coefficients and eliminates the difficult numerical task of solving a moving boundary problem. A Kirchhoff transformation is suggested to solve the resulting near-singular diffusion equation. A previously developed method of measuring diffusion coefficients based on GITT (galvanostatic intermittent titration technique) is proposed for use on such materials.