Computational Problems For Interfaces With Bending Stiffness In Strongly Anisotropic Thin Films And Inhomogeneous Biomembranes

该项目的主要目标是通过(1)开发和应用最先进的自适应数值方法进行大规模计算，以及(2)对重要组成过程进行分析，数值和建模研究，来研究数学，材料科学和生物学之间的界面问题。在数学和材料科学的交叉点上，我们将专注于纳米级晶体薄膜的基础研究，在那里我们将研究强各向异性系统中纳米级结构的动力学和粗化。我们的研究不仅对于获得对薄膜演化的基本理解很重要，而且对于阐明可用于促进空间有序纳米结构的机制也很重要，因此有可能影响新型电子器件的设计，如高效激光器和量子元胞自动机架构。在数学，材料科学和生物学之间的界面，我们将专注于非均质生物膜的基础研究，在那里我们将研究组成生物膜的多种脂质组分的形状变形，流体流动和表面相分离之间的耦合。我们的基础研究有可能影响对许多细胞和亚细胞生物学过程的理解，因为生物膜在细胞运动、粘附、信号转导等细胞功能中起着积极和关键的作用。尽管我们要研究的问题的物理起源非常不同，但数学控制方程有许多相似之处，因此可以使用常见的分析和计算技术来处理。例如，我们研究的一个关键组成部分是研究表面张力和弯曲力在两种应用的动力学中所起的核心作用。作为这项研究的一部分，我们将开发新的数学和自适应数值技术，这些技术也将在当前背景之外得到应用；例子包括合金微结构、乳液和聚合物混合物、血流、组织和实体肿瘤生长。事实上，数学分析的能力通过利用数学描述中的共性来阐明、预测和控制看似不同的物理和生物系统中的基本过程，这正是数学如此有价值的原因。最后，加州大学欧文分校的加州州立数学与科学暑期学校（COSMOS）将为有天赋的高中生开设一门关于晶体和薄膜生长的课程。这门课程也将有助于招收新的数学和科学专业的学生，并提高高中学生对研究的参与。两名博士生和一名博士后将在开展工作的同时接受跨学科培训。拟议的工作还扩大了代表性不足群体的参与，因为其中一名学生是女性。

The main objective of this project is to investigate problems at the interface between mathematics, materials science, and biology, by (1) developing and applying state-of-the-art adaptive numerical methods to large-scale computation, and (2) performing analytical, numerical, and modeling studies of important constituent processes. At the interface between mathematics and materials science, we will focus on fundamental studies of nanoscale, crystalline thin films, where we will investigate the dynamics and coarsening of nanometer-scale structures in strongly anisotropic systems. Our study is important not only for gaining a fundamental understanding of thin-film evolution, but also for elucidating mechanisms that can be used to promote spatially ordered nanostructures, and thus has the potential to impact the design of novel electronic devices such as high-efficiency lasers, and quantum cellular automata architectures. At the interface between mathematics, materials science, and biology, we will focus on fundamental studies of inhomogeneous biomembranes, where we will investigate the coupling between shape deformation, fluid flow, and surface phase segregation of the multiple lipid components that comprise the biomembrane. Our fundamental study has the potential to impact the understanding of many cellular and subcellular biological processes, since biomembranes play an active and critical role in cell functions such as cell locomotion, adhesion, signal transduction, etc. In spite of the very different physical origins of the problems we propose to study, the mathematical governing equations have many similarities and thus can be treated using common analytical and computational techniques. For example, a key component of our study is to investigate the central role that the surface tension and bending forces play in the dynamics of both applications. As part of this study, we will develop new mathematical and adaptive numerical techniques that will also have application beyond the present context; examples include alloy microstructures, emulsions and polymer blends, blood flow, tissue and solid tumor growth. Indeed, the ability of mathematical analyses to elucidate, predict, and control fundamental processes in seemingly disparate physical and biological systems, by exploiting the commonalities in their mathematical descriptions, is what makes mathematics so valuable. Finally, a course on crystal and thin-film growth for gifted high-school students will be developed as part of the Calif. State Summer School for Mathematics and Science (COSMOS) at UC Irvine. This course will also help to recruit new math and science majors and enhance the participation of high-school students in research. Two Ph.D students and one postdoctoral researcher will receive interdisciplinary training while performing the proposed work. The proposed work also broadens the participation of underrepresented groups, as one of the students is a woman.