Lithium forms ordered stages when it reacts with graphite. These stages have distinct colors; therefore, optical microscopy gives direct information about the lithium concentration in the graphite. Here we present in situ optical images during charging and discharging of a graphite electrode. Stages are observed to coexist with each other even after extended rest. There is considerable spatial nonuniformity on the microscale. To predict this concentration distribution, we employ a model which combines porous-electrode theory and Cahn-Hilliard phase-field theory to describe the flux of lithium within the graphite. The model closely matches the experimental voltage and concentration distribution. The spatial nonuniformity can be approximated with a relatively simple model of distributed resistances. Finally, we discuss the implications of using the phase-field model instead of a solid-solution model for prediction of lithium plating. The two models give similar predictions of cell voltage and risk of lithium plating under many operating conditions, with the main difference being the relaxation of concentration gradients within particles during rest. The distributed-resistance model shows a higher risk of lithium plating because well-connected particles are overworked as their more-resistive neighbors require a higher driving force for passage of current.

中文翻译：

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石墨中锂分布的原位观察和数学建模

锂与石墨反应时会形成有序的阶段。这些阶段具有不同的颜色。因此，光学显微镜可提供有关石墨中锂浓度的直接信息。在这里，我们展示了石墨电极充电和放电过程中的原位光学图像。即使长时间休息，也可以观察到阶段彼此共存。在微观尺度上存在相当大的空间不均匀性。为了预测这种浓度分布，我们采用了结合多孔电极理论和Cahn-Hilliard相场理论的模型来描述锂在石墨中的通量。该模型与实验电压和浓度分布紧密匹配。可以使用相对简单的分布电阻模型来近似空间不均匀性。最后，我们讨论了使用相场模型而不是固溶模型来预测锂电镀的含义。两种模型对电池电压和在许多操作条件下镀锂的风险做出了相似的预测，主要区别在于在静止过程中颗粒内浓度梯度的松弛。分布电阻模型显示出较高的锂电镀风险，因为连接良好的粒子会过度劳累，因为它们的高电阻邻居需要更高的驱动力以通过电流。主要区别在于在休息期间颗粒内浓度梯度的松弛。分布电阻模型显示出较高的锂电镀风险，因为连接良好的粒子会过度劳累，因为它们的高电阻邻居需要更高的驱动力以通过电流。主要区别在于在休息期间颗粒内浓度梯度的松弛。分布电阻模型显示出较高的锂电镀风险，因为连接良好的粒子会过度劳累，因为它们的高电阻邻居需要更高的驱动力以通过电流。