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

形状记忆合金和带有表面活性剂的微流体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。在外延生长制造过程中，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。本项目将重点对新型材料中界面层和微流体流动的迁移模式进行数学分析，以了解其机制并预测其模式。提供精确的理论预测它们的演变将推进，例如，微流体装置的设计和降低制造成本。本科生和高中生的培养将被纳入这个跨学科的研究项目。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方法，并用它来预测多层过渡剖面的模式形成。利用基于具有动态边界条件的偏微分方程的全局稳定性的定量估计和减少不变流形的多层图案的慢动作持久性，主要目的是表明耦合体动力学不会破坏材料缺陷在领先阶的慢动作图案。第二个目标是系统地将Onsager的线性响应原理推广到具有拓扑变化的交互动力学障碍问题。本研究旨在建立一种新的微流体几何运动相场模型，该模型可在拓扑结构发生变化时自动建立界面层的迁移机制。这个新模型，连同数学验证，将揭示隐变分原理和选择标准。这种变分原理将结合固体-流体，固体-固体相互作用机制，以及一般复杂系统的连贯微观宏观结构。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.