Collaborative Research: Computational Problems in Heterogeneous Nanomaterials

在从能源到生物医学的许多应用中，纳米晶体材料，如量子点和纳米线，有望产生革命性的新技术。这一前景的实现受到可再生制造具有可控形貌和成分的纳米晶体材料所固有的挑战的阻碍。这些纳米材料通常是不均匀的，由多种成分的合金组成。虽然已经有很多工作在制定条件下，几乎均匀形状和尺寸的空间有序纳米晶体可以产生，确定合金成分的空间分布的机制的定量描述，这是至关重要的器件性能，仍然知之甚少。研究人员及其合作者在本提案中解决了这个问题。他们通过(1)开发和应用最先进的自适应数值方法进行大规模计算，(2)对重要组成过程进行分析、数值和建模研究，研究非均质、应变的纳米晶体材料的非线性动力学。研究人员关注的动态，非线性耦合之间的形状，弹性应力和组成的背景下(i)薄膜合金和量子点的动力学在远离平衡的加工条件下，可能存在不同成分的体积和表面输运，以及相分解；封顶纳米晶体的粗化动力学和稳定性。在应用中需要帽材料来提供电荷载流子的限制电位以及对外部环境的钝化。这些问题的特点是存在多个组成部分，体-表面相互作用，复杂的模式形成和/或奇点（即空间复杂性）。数学模型涉及高阶空间导数（例如，高达六阶），不断发展的自由边界和高度非线性的相互作用，使分析和模拟变得困难，特别是在3D中。这些问题的高度非线性性质使得快速、准确和鲁棒的数值方法对它们的研究至关重要。纳米晶合金材料的物理特性使其非常适合广泛的潜在应用，包括先进的电子和磁性设备以及生物和化学传感器。非均质纳米晶元件的空间组成和纳米晶的几何形状决定了纳米器件的性能。实验技术的最新进展使表征纳米晶体的纳米级组成变化成为可能。然而，对这些变化的定量理解仍然难以捉摸，但对设备性能至关重要。研究人员和他们的合作者通过开发新的数学模型、理论和计算方法来解决这个问题，这些方法可以表征和量化纳米晶体形状、弹性应力和成分之间的相互作用。研究人员还考虑了帽状纳米结构，其中帽状材料提供了许多应用所需的环境保护，包括在硅基电子电路中使用纳米晶体。纳米晶体和封盖材料之间的相互作用带来了额外的复杂性。这些问题是多学科的，进展需要材料科学和应用与计算数学研究者的综合专业知识。通过这项研究，研究人员为纳米晶体成分变化的实验测量提供了定量解释指导，并提出了优化的加工条件，以实现所需的器件形状、成分和性能。该项目在两所院校之间建立了新的合作关系，并为两名博士生和一名博士后提供跨学科培训。此外，研究人员以他们最近的成功为基础，继续开发和教授一门关于晶体和外延生长的课程，作为加州大学欧文分校加州州立数学与科学暑期学校（COSMOS）的一部分。这门课程也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。

In many applications ranging from energy to biomedicine, nanocrystalline materials, such as quantum dots and nanowires, promise to yield revolutionary new technologies. The realization of this promise is hindered by the challenges inherent in reproducibly fabricating nanocrystalline materials with controlled morphologies and compositions. These nanomaterials are typically heterogeneous and consist of alloys with multiple constituents. While there has been much work on formulating conditions under which spatially ordered nanocrystals with nearly uniform shapes and sizes may be produced, a quantitative description of the mechanisms that determine the spatial distribution of the alloy components, which is crucial to device performance, is still poorly understood. The investigators and their collaborators address this issue in this proposal. They study the nonlinear dynamics of heterogeneous, strained strained nanocrystalline materials by (1) developing and applying state-of-the-art adaptive numerical methods to large-scale computation and (2) performing analytical, numerical and modelling studies of important constituent processes. The investigators focus on the dynamic, nonlinear coupling among shape, elastic stress and composition in the context of (i) the dynamics of thin film alloys and quantum dots under far-from-equilibrium processing conditions where there may be bulk and surface transport of the different constituents, as well as phase decomposition; and (ii) the coarsening dynamics and stability of capped nanocrystals. The cap material is needed in applications to provide the confinement potential for charge carriers as well as passivation against the external environment. These problems are characterized by the presence of multiple constitutive components, bulk-surface interactions, complex pattern formation and/or singularities (i.e. spatial complexity). The mathematical models involve high-order spatial derivatives (e.g. up to sixth-order), evolving free boundaries and highly nonlinear interactions that make analysis and simulation difficult, particularly in 3D. The highly nonlinear nature of these problems makes fast, accurate and robust numerical methods essential to their study. Nanocrystalline alloy materials have physical properties that make them ideally suited for a wide range of potential applications including advanced electronic and magnetic devices as well as biological and chemical sensors. The properties of nanoscale devices are determined both by the spatial composition of the heterogeneous nanocrystal components and the nanocrystal geometry. Recent advances in experimental techniques have enabled the characterization of nanoscale composition variation in nanocrystals. However, a quantitative understanding of these variations remains elusive and yet is critical to device performance. The investigators and their collaborators address this issue by developing new mathematical models, theory and computational methods that make it possible to characterize and quantify the interactions among nanocrystal shape, elastic stress and composition. The investigators also consider capped nanostructures where the cap material provides protection from the environment that is needed in many applications including the use of nanocrystals in silicon-based electronic circuits. The interaction among the nanocrystalline and capping materials introduces additional complexity. These problems are multidisciplinary and progress requires the combined expertise of the investigators in materials science and applied and computational mathematics. Through this study, the investigators provide guidance in the quantitative interpretation of experimental measurements of composition variation in nanocrystals and suggest optimized processing conditions to achieve desired device shape, composition and performance. The project establishes a new collaboration between two institutions and provides interdisciplinary training of two Ph.D students and one postdoctoral researcher. In addition, the investigators build on their recent success and continue to develop and teach a course on crystal and epitaxial growth for gifted high school students as part of the Calif. State Summer School for Mathematics and Science (COSMOS) at UC Irvine. This course also helps to recruit new math and science majors and enhance the participation of high school students in research.