We study the phase-field model with the elasticity energy arising from the coherency stress across the two-phase interface for lithium-ion batteries by using olivine LiFePO4 as a model system. The existence of global solutions about this model which is a coupled system of elliptic and parabolic equations with the singular points in the case of a logarithmic nonlinear term is proved. We demonstrate the model satisfies the second law of thermodynamics, and numerical simulations are given to show that the evolution of lithium concentration and the motion of interfaces are consistent with the experiments results of Laffont et al.
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