Some Mesoscale Issues for Applied Mathematics

应用数学的一些介尺度问题

• 批准号: 0072194
• 负责人: David Kinderlehrer
• 金额: $ 19.77万
• 依托单位: Carnegie-Mellon University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-08-01 至 2004-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0072194&HistoricalAwards=false
• 关键词: Some; Mesoscale; Issues; Applied; Mathematics

Mesoscale is a term intended to convey an intermediate level description ofa physical system. Its function is to capture the interactions of thesystem at finer scales with interactions or influences from larger coarserscales, and the mesoscale level itself. Such systems, active acrossdisparate length and time scales, are inherently metastable. This featureis often revealed by hysteretic behavior or by a reluctance to evolvequickly to equilibrium. The focus in this proposal is on severalprototypes of these systems that occur in materials sceince. One is the role of interfaces, or grain boundaries, in determining or limiting the behavior of polycrystalline materials. The energy and mobility of grain boundaries depends on crystallography and geometry, according to established thermodynamic principles. Innovative new ways to determine these functions explicitly for important materials are the objective of the Mesoscale Interface Mapping Project. This involves developing automated microscopy to harvest large amounts of data from samples and then formulating and solving a complex inverse problem. One way to approach determination of mobility consists in the development of large scale simulations of grain boundary evolution. The second focus is the coarse grained descriptions of mesoscale systems, the functional analytic limit processes in the microstructure of solids or the averaging to distribution functions in systems with stochastic behavior. The new methods allow study of situations where kinetics arise directly in terms of thermodynamic state functions and naturally carry with them an appropriate topology. The issue of metastability has been under investigation in this context. There is now the opportunity to improve understanding, for example, of microstructural evolution in shape-memory materials and diffusion mediated transport in certain liquid crystal systems and in protein motors. This will include diagnostics for these systems.The challenge of the mesoscale in materials science is to understand how it constrains finer scale systems (at the molecular scale) and determines larger scale systems (at the scale of entire devices). This is accomplishedthrough coarse graining procedures. For example, many technologicallyuseful materials are polycrystalline, or granular, in nature. The aluminumskin of an aircraft and the copper or copper-aluminum interconnects incomputer chips are but two examples at vastly different size scales of suchgranular materials. It is widely understood that many aspects of thesematerials depend on the interfaces they contain, or their grain boundaries.Properties of grain boundaries determine the reliability as well as the mechanical strength. This project will exploit the exciting opportunities and challenges for mathematical science in this field and beyond. Specifically, coarse graining methods will be developed in order to better understand phenomena that occur in protein motors and in liquid crystals. A related problem of coarse graining arises when information is to be extracted from the immense amounts of data that can be produced with simulations ofcomplex systems, such as polycrystalline materials. This problem of coarse graining at an information scale rather than a physical scale will also be addressed in this project.

中尺度是一个术语，旨在传达一个物理系统的中间水平的描述。 它的功能是捕捉的相互作用，在更细的尺度与相互作用或影响，从更大的coarserscales，和中尺度水平本身。 这样的系统，活跃在不同的长度和时间尺度上，本质上是亚稳态的。 这一特征通常表现为滞后行为或不愿迅速发展到平衡状态。 本提案的重点是材料科学中出现的这些系统的几个原型。 一个是界面或晶界在决定或限制多晶材料行为中的作用。根据已建立的热力学原理，晶界的能量和流动性取决于晶体学和几何形状。 中尺度界面测绘项目的目标是明确确定重要材料的这些功能的创新方法。 这涉及开发自动显微镜，从样品中收集大量数据，然后制定和解决复杂的逆问题。 一种方法来接近确定的流动性包括在大规模模拟晶界演变的发展。 第二个重点是粗粒度的描述中尺度系统，功能分析的极限过程中的微观结构的固体或平均系统中的随机行为的分布函数。新的方法允许研究的情况下，动力学直接出现在热力学状态函数和自然进行与他们适当的拓扑结构。 亚稳态问题一直在这方面的调查。 现在有机会提高理解，例如，在形状记忆材料和扩散介导的运输在某些液晶系统和蛋白质马达的微观结构的演变。 这将包括这些系统的诊断。材料科学中的中尺度的挑战是了解它如何约束更精细的尺度系统（在分子尺度上）和确定更大尺度的系统（在整个设备的尺度上）。 这是通过粗粒化程序进行的。 例如，许多技术上有用的材料本质上是多晶的或颗粒状的。 飞机的铝外壳和计算机芯片中的铜或铜-铝互连只是这种颗粒材料在尺寸尺度上差异很大的两个例子。 人们普遍认为，这些材料的许多方面都取决于它们所包含的界面或晶界，晶界的性质决定了材料的可靠性和机械强度。该项目将利用数学科学在这一领域和超越令人兴奋的机遇和挑战。 具体而言，将开发粗粒化方法，以更好地理解蛋白质马达和液晶中发生的现象。 一个相关的粗粒化问题出现时，信息是从大量的数据，可以产生与模拟的复杂系统，如多晶材料。这个问题的粗粒化在一个信息规模，而不是一个物理规模也将在这个项目中解决。