Bulk-Interface Coupled Response in Novel Materials: Pattern Formation and Interactive Migration

形状记忆合金和含有表面活性剂的微流控块体是所谓的新型材料的例子，它们是由单分子厚的界面层连接的多相块体的复合材料。在外延生长制造工艺中，位错和晶界等材料缺陷是界面层，它们的迁移可能导致致命的塑性变形。本项目将集中于对新材料中界面层和微流体流的迁移模式进行数学分析，以了解它们的机制并预测它们的模式。例如，对它们的发展提供准确的理论预测将促进微流控设备的设计和更低的制造成本。本科生和高中生的培训将被整合到这一跨学科研究项目中。该项目的第一个科学目标是开发一种新的体界面交互动力学分解，并用它来预测多层相变剖面的模式形成。基于具有动态边界条件的偏微分方程解的全局稳定性和多层膜慢运动存留的约化不变流形的定量估计，主要目的是证明耦合的体动力学不破坏材料缺陷慢运动的主导阶。第二个目标是系统地推广Onsager对具有拓扑变化的交互动力学的障碍问题的线性响应原理。研究人员的目标是建立一种新的微流控流几何运动相场模型，该模型在发生拓扑变化时自动加入界面层的迁移机制。这个新的模型，加上数学验证，将揭示隐藏的变分原理和选择标准。这一变分原理将结合固体-流体、固体-固体相互作用机制，以及针对一般复杂系统的连贯的微观-宏观结构。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Shape-memory alloys and microfluidic bulks with surfactants are examples of so-called novel materials, which are composites of multi-phase bulks connected by single-molecule-thick interfacial layers. In epitaxial growth manufacturing processes, material defects such as dislocations and grain boundaries are interfacial layers, whose migration may lead to fatal plastic deformation. This project will focus on the mathematical analysis of migration patterns of interfacial layers and microfluidic flows in novel materials to understand their mechanisms and predict their patterns. Providing accurate theoretical predictions of their evolution will advance, for instance, the design of microfluidic devices and lower manufacturing costs. The training of undergraduate students and high school students will be integrated into this interdisciplinary research project.The first scientific goal of this project is to develop a novel decomposition for bulk-interface interactive dynamics and use it to predict the pattern formation of a multilayer transition profile. Using quantitative estimates based on the global stability of partial differential equations with dynamic boundary conditions and a reduced invariant manifold for the slow-motion persistence of the multilayer pattern, the main intent is to show that the coupled bulk dynamics does not destroy the slow-motion pattern of material defects at the leading order. The second goal is to systematically generalize Onsager’s linear response principle to obstacle problems for interactive dynamics with topological changes. The investigator aims to derive a new phase field model for geometric motions of microfluidic flows, which automatically builds in migration mechanisms of interfacial layers whenever topological changes happen. This new model, together with mathematical validations, will unveil the hidden variational principle and a selection criterion. This variational principle will incorporate solid-fluid, solid-solid interactive mechanisms, and coherent micro-macro structures for general complex systems.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.