Mathematical Sciences: Percolation in Composite Materials

9622367我们建议开发新的分析和数值技术，以研究高对比度复合材料研究中出现的一些渗流模型的行为。一些发展方向包括：(I)根据与伊辛模型直接对应的新的方法，探索随机电阻网络的临界行为；(Ii)分析多孔介质和基质颗粒复合材料的连续介质渗流模型；(Iii)发展一种非破坏性地判断复合材料中某一相是否渗漏的诊断方法，该方法可应用于复合材料制造中的材料失效检测或质量控制；以及(Iv)模拟热和卤水在多孔介质海冰中的传输。%材料科学中的广泛问题涉及复合材料，其特点是部件的性能具有很高的对比度。对于这类介质，有效行为通常由特定相的连通性或渗流性质决定。例子包括多孔介质、掺杂半导体、压敏电阻和热敏电阻等智能材料、金属薄膜(用于太阳能生产)、含油砂岩、岩石和金属的断裂模式、人体中的许多结构以及海冰(它在气候中起着重要作用，是全球变化的指标)。在拟议的工作中，我们将研究此类材料的数学模型。我们预计，该项目将进一步提高我们预测自然发生或人工制造的复杂随机材料的有效行为的能力，并帮助开发旨在实现新的或最佳性能的复合材料。***

9622367 Golden We propose to develop new analytical and numerical techniques for investigating the behavior of a number of percolation models arising in the study of high contrast composite materials. Some of the directions for development include: (i) exploring the critical behavior of the random resistor network in light of a new, direct correspondence with the Ising model, (ii) analysis of continuum percolation models for porous media and matrix-particle composites, (iii) developing a diagnostic method for non-destructively telling whether one phase in a composite percolates or not, which could have applications to the detection of material failure or quality control for composite manufacturing, and (iv) modeling the transport of heat and brine through sea ice, a porous medium. %%% A broad range of problems in materials science involve composites which can be characterized by a high contrast in the properties of the components. For such media, the effective behavior is often dominated by the connectedness, or percolation properties of a particular phase. Examples include porous media, doped semiconductors, smart materials such as piezoresistors and thermistors, thin metal films (used in solar energy production), oil bearing sandstones, fracture patterns in rocks and metals, many structures in the human body, and sea ice (which plays a major role in climate and is an indicator of global change). In the proposed work we will study mathematical models of such materials. We anticipate that this project will further our capabilities to predict the effective behavior of naturally occurring or artificially made complex random materials, as well as aid the development of composites designed to achieve novel or optimal properties. ***