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

具有表面活性剂的形状记忆合金和微流体块体是所谓的新型材料的例子，其是由单分子厚的界面层连接的多相块体的复合物。在外延生长制造工艺中，诸如位错和晶界的材料缺陷是界面层，其迁移可能导致致命的塑性变形。 该项目将侧重于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。对它们的演变提供准确的理论预测将促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训将被整合到这个跨学科的研究项目中。这个项目的第一个科学目标是开发一种新的体-界面相互作用动力学分解，并使用它来预测多层过渡剖面的图案形成。使用定量估计的基础上的全局稳定性的偏微分方程与动态边界条件和减少不变流形的慢动作持久性的多层图案，主要目的是表明，耦合的散装动力学不破坏慢动作图案的材料缺陷的领导秩序。第二个目标是将Onsager的线性响应原理系统地推广到拓扑变化的相互作用动力学障碍问题。研究者的目标是推导出一种新的微流体流动的几何运动相场模型，该模型在拓扑变化发生时自动建立界面层的迁移机制。这个新的模型，连同数学验证，将揭示隐藏的变分原理和选择标准。该变分原理将包含固体-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NSF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.