FRG: Collaborative Research: Mathematical Modeling of Rechargeable Batteries

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。该项目将为可充电电池的数学建模开发一个新的框架，考虑到统计热力学、浓溶液反应速率、弹性、晶体各向异性、随机效应和复合微观结构。现有的工程模型简单地根据经验拟合开路电压，并通过插层锂的线性扩散来假设动力学，但最近的实验与相分离材料的这一情况相矛盾。相比之下，该团队将开发强大的数学模型来预测整个操作条件下的电压和电流响应。在单晶水平上建模的基础将是具有非线性边界条件的Cahn-Hilliard偏微分方程，表示依赖化学势的反应。目标将是提供新兴的高速率材料的第一个数学描述，其中相变通过非线性插入波，耦合各向异性扩散和电化学反应发生。这项工作还将提出线性和非线性稳定性、简并波解和数值方法方面的基本数学问题。尽管在过去的几十年里进行了大量的工程设计，但可充电电池的性能只是逐步提高。对于电动汽车和可再生能源存储等应用，功率密度（单位质量的充电/放电率）和循环寿命仍必须大幅提高，这将需要对如何从多孔电极中插入和提取离子有更好的基本理解。为了满足这一需求，该项目创建了一个由数学、化学工程和材料科学组成的重点研究小组，以开发锂离子电池的新理论范式。该小组将指导新型超快锂离子电池的工程设计，这种电池能够在几秒钟而不是几小时内充放电，同时为应用数学开辟了富有成效的方向。该小组将培训研究生、本科生和博士后，组织年度研讨会，并开发一门关于电化学能量系统数学建模的课程。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).The project will develop a new framework for mathematical modeling of rechargeable batteries, taking into account statistical thermodynamics, concentrated-solution reaction rates, elasticity, crystal anisotropy, stochastic effects, and composite microstructures. Existing engineering models simply fit the open circuit voltage empirically and postulate dynamics by linear diffusion of intercalated lithium, but recent experiments contradict this picture for phase-separating materials. In contrast, the team will develop robust mathematical models to predict the voltage and current response over the full range of operating conditions. The basis for modeling at the single-crystal level will be Cahn-Hilliard partial differential equations with nonlinear boundary conditions, expressing chemical-potential dependent reaction reactions. The goal will be to provide the first mathematical description of emerging high-rate materials, where phase transformations occur via nonlinear intercalation waves, coupling anisotropic diffusion and electrochemical reactions. This effort will also raise basic mathematical questions in linear and nonlinear stability, degenerate wave solutions, and numerical methods.In spite of extensive engineering over the past few decades, the performance of rechargeable batteries has improved only incrementally. Power density (charge/discharge rate per unit mass) and cycle life must still improve drastically for applications such as electric vehicles and renewable energy storage, and this will require a better fundamental understanding of how ions are inserted and extracted from porous electrodes. To meet this need, the project creates a Focused Research Group from mathematics, chemical engineering, and materials science to develop a new theoretical paradigm for Li-ion batteries. The group will guide the engineering of new ultrafast Li-ion batteries, capable of charging and discharging in seconds rather than hours, while opening fruitful directions for applied mathematics. The group will train graduate and undergraduate students and postdocs, organize annual workshops, and develop a course on mathematical modeling of electrochemical energy systems.