Collaborative Research: Computational problems in heterogeneous nanomaterials

在从能源到生物医学的许多应用中，纳米晶材料，如量子点和纳米线，有望产生革命性的新技术。这一承诺的实现受到可重复制造具有受控形态和组成的纳米晶材料所固有的挑战的阻碍。 这些纳米材料通常是异质的，并且由具有多种成分的合金组成。虽然已经有很多工作的制定条件下，空间有序的纳米晶体几乎均匀的形状和尺寸可以产生，定量描述的机制，确定合金成分的空间分布，这是至关重要的设备性能，仍然知之甚少。研究人员及其合作者在本提案中解决了这一问题。他们通过以下方式研究非均质应变纳米晶材料的非线性动力学：（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。研究人员专注于形状，弹性应力和成分之间的动态，非线性耦合，（i）在远离平衡处理条件下薄膜合金和量子点的动力学，其中可能存在不同成分的体和表面传输，以及相分解;和（ii）覆盖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势以及针对外部环境的钝化。这些问题的特征在于存在多个组成成分、体-表面相互作用、复杂图案形成和/或奇异性（即空间复杂性）。数学模型涉及高阶空间导数（例如高达六阶），不断变化的自由边界和高度非线性的相互作用，这使得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。纳米晶合金材料具有的物理特性使它们非常适合于广泛的潜在应用，包括先进的电子和磁性器件以及生物和化学传感器。纳米器件的性质由异质异质结组分的空间组成和异质结的几何形状决定。近年来实验技术的进步使得对纳米晶体中纳米级成分变化的表征成为可能，然而，对这些变化的定量理解仍然是难以捉摸的，但对器件性能至关重要。研究人员及其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，这些模型，理论和计算方法使得能够表征和量化形状，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，其中帽材料提供了许多应用中所需的环境保护，包括在硅基电子电路中使用纳米晶体。纳米晶和覆盖材料之间的相互作用引入了额外的复杂性。这些问题是多学科的，需要材料科学、应用数学和计算数学研究人员的综合专业知识。通过这项研究，研究人员在定量解释纳米晶体组成变化的实验测量方面提供了指导，并提出了优化的加工条件，以实现所需的器件形状，组成和性能。该项目在两个机构之间建立了新的合作关系，并为两名博士生和一名博士后研究人员提供跨学科培训。此外，研究人员建立在他们最近的成功，并继续开发和教授晶体和外延生长的课程，为有天赋的高中学生作为加州的一部分。加州大学欧文分校数学与科学暑期学校（COSMOS）这门课程还有助于招收新的数学和科学专业的学生，并提高高中生在研究中的参与。

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.