Next-generation lithium-ion batteries with silicon anodes have positive characteristics due to higher energy densities compared to state-of-the-art graphite anodes. However, the large volume expansion of silicon anodes can cause high mechanical stresses, especially if the battery active particle cannot expand freely. In this article, a thermodynamically consistent continuum model for coupling chemical and mechanical effects of electrode particles is extended by a change in the boundary condition for the displacement via a variational inequality. This switch represents a limited enlargement of the particle swelling or shrinking due to lithium intercalation or deintercalation in the host material, respectively. For inequality constraints as boundary condition a smaller time step size is need as well as a locally finer mesh. The combination of a primal-dual active set algorithm, interpreted as semismooth Newton method, and a spatial and temporal adaptive algorithm allows the efficient numerical investigation based on a finite element method. Using the example of silicon, the chemical and mechanical behavior of one- and two-dimensional representative geometries for a charge-discharge cycle is investigated. Furthermore, the efficiency of the adaptive algorithm is demonstrated. It turns out that the size of the gap has a significant influence on the maximal stress value and the slope of the increase. Especially in two dimension, the obstacle can cause an additional region with a lithium-poor phase.

翻译：下一代采用硅负极的锂离子电池，相较于当前先进的石墨负极，具有更高能量密度的积极特性。然而，硅负极的大体积膨胀会引发高机械应力，尤其当电池活性颗粒无法自由膨胀时更为显著。本文通过变分不等式对位移边界条件进行切换，扩展了电极颗粒化学-力学耦合效应的热力学一致连续介质模型。该切换机制分别对应主体材料中锂嵌入/脱出导致的颗粒有限溶胀或收缩。针对不等式约束边界条件，需要采用更小的时间步长及局部加密的网格。结合可解释为半光滑牛顿法的原始-对偶活动集算法，与时空自适应算法，实现了基于有限元方法的高效数值研究。以硅材料为例，对单维及二维代表性几何构型在充放电循环中的化学-力学行为进行了分析，并论证了自适应算法的有效性。结果表明：空隙尺寸对最大应力值及其增长斜率具有显著影响，特别是在二维情形下，障碍物可能导致额外贫锂相区域的形成。