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 $p(c)$ 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.

中文翻译：

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锂离子电池具有对数电势的相场模型的整体解决方案

我们使用橄榄石LiFePO ₄作为模型系统，研究了锂离子电池两相界面上相干应力产生的弹性能的相场模型。关于该模型的整体解的存在，在对数非线性项的情况下，该模型是具有奇异点的椭圆和抛物方程的耦合系统$p（C）$被证明。我们证明了该模型满足热力学第二定律，并通过数值模拟表明锂浓度的演变和界面的运动与Laffont等人的实验结果一致。