FRG: Collaborative Research: Mathematical Modeling of Rechargeable Batteries

该奖项由2009年美国复苏和再投资法案(公法111-5)资助。该项目将开发一个新的框架，用于可充电电池的数学建模，其中考虑了统计热力学、浓溶液反应速率、弹性、晶体各向异性、随机效应和复合微结构。现有的工程模型简单地符合开路电压的经验和假设的动力学插层锂的线性扩散，但最近的实验与相分离材料的情况相矛盾。相比之下，该团队将开发强大的数学模型，以预测整个运行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偏微分方程组，该方程表示化学势相关的反应。我们的目标将是为新兴的高速率材料提供第一个数学描述，其中相变通过非线性插层波、耦合各向异性扩散和电化学反应发生。这一努力还将在线性和非线性稳定性、简并波解和数值方法方面提出基本的数学问题。尽管在过去几十年中进行了广泛的工程设计，但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(单位质量的充放电速率)和循环寿命仍必须大幅提高，这将需要更好地从根本上理解离子是如何插入和从多孔电极中提取的。为了满足这一需求，该项目成立了一个专注于数学、化学工程和材料科学的研究小组，以开发一种新的锂离子电池理论范式。该小组将指导新的超高速锂离子电池的工程设计，能够在几秒钟而不是几个小时内充电和放电，同时为应用数学开辟富有成效的方向。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源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.