Mathematics of Anisotropic Electrical and Dielectric Properties of Nanocomposites

郑DMS-08-7954研究了纳米复合材料中的两个重要领域：(1)纳米复合材料的均匀有效介电常数；(2)纳米复合材料的渗流主导电导率。该项目主要研究具有取向有序纳米颗粒的各向异性体系，基于对流动处理数据库中纳米颗粒取向概率分布函数的精确描述，然后通过体积平均和渗流现象将其映射到有效的性质。在第一部分中，有效介电张量被考虑在两个显式体系中：高介电常数纳米粒子分散在聚合物中和导电纳米粒子分散在聚合物中。观察到这两个系统都能提高有效介电性能。这需要仔细的准备，因为这些特性对组成和加工的细节极其敏感。在第二部分中，考虑了显著电学性质的关键方面，如机械加载的渗流阈值修正、多尺度均匀化方法和分子尺度电子跳跃模型。需要开发数值和数学均化和非平衡渗流技术来应对这些挑战。拟议的项目修改、应用、测试和扩展了现有的两相(或三相，如果考虑界面)复合材料的数学理论和数值方法，其中颗粒的数量和新的表面积完全超过了现有的工具。通过将设计的纳米颗粒混合到传统材料中来实现极端性能增强的愿望是基于原型实验。然而，纳米复合材料存在着独特的新挑战，没有可预测的数学或计算工具。这个项目是对这些需求的回应，通过数学分析和科学计算的协调组合，范围从单个颗粒到整体性质。今天，粒子数量的限制只允许测量非粒子分布的低阶矩，而材料对实验中观察到的分布细节的内在敏感性是未知的。该项目深入了解极端财产收益背后的机制，从而为设计和控制策略提供关键输入。

ZhengDMS-08-7954 The investigator studies two important areas innanocomposites: (1) homogenized effective dielectric constant ofnanocomposites; and (2) percolation-dominated electricalconductivity of nanocomposites. This project is focused onanisotropic systems with orientationally ordered nanoparticles,and is based on an accurate description of the nanoparticleorientational probability distribution function fromflow-processing databases, which we then map to effectiveproperties through volume averaging and percolation phenomena. In the first part, the effective dielectric tensor is consideredin two explicit systems of current interests: high dielectricconstant nanoparticles dispersed in polymers, and conductingnanoparticles dispersed in polymers. Both systems are observedto lead to enhanced effective dielectric properties. Thisrequires careful preparation, because the properties areextremely sensitive to details of composition and processing. Inthe second part, key aspects of salient electric properties areconsidered, such as percolation threshold modification bymechanical loading, multiscale homogenization method, and amolecular-scale electron hopping model. Numerical andmathematical homogenization and non-equilibrium percolationtechniques need to be developed to address these challenges. Theproposed projects modify, apply, test, and extend existingmathematical theory and numerical approaches for two-phase (orthree phase, if interphase is considered) composites, where thenumber of particles and new surface area simply overwhelmexisting tools. The aspirations for extreme property enhancements throughdesigner nanoparticles mixed into traditional materials are basedon prototype experiments. However, there are new challengesunique to nanocomposites for which there are no predictivemathematical or computational tools. This project is a responseto these needs, through a coordinated combination of mathematicalanalysis and scientific computation, ranging in scale fromindividual particles to bulk properties. Today, limitationsimposed by particle numbers only allow low moments of thenanoparticle distributions to be measured, and the inherentmaterial sensitivity to the details of the distribution observedin experiments is unknown. This project gives insight into themechanism underlying extreme property gains, and thus provideskey input into design and control strategies.