Corner regularizations for nanoscale crystal growth

纳米级晶体生长的角正则化

• 批准号: 0505497
• 负责人: Brian Spencer
• 金额: $ 21.12万
• 依托单位: SUNY at Buffalo
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-08-15 至 2009-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0505497&HistoricalAwards=false
• 关键词: Corner; regularizations; nanoscale; crystal; growth

The investigator focuses on a mathematical issue in thedescription of nanocrystal growth: how to properly resolve theill-posedness inherent in dynamic models involving crystallinesurfaces with strong anisotropy. Strong anisotropy in the surfaceenergy manifests itself physically in the formation of corners ona crystal. Traditional mathematical models applied to theformation of corners are mathematically ill-posed and thusintractable. Because the formation of corners during crystalgrowth is ubiquitous, the ill-posedness of corner formation is aproblem inherent in simulations of both industrial and naturallyoccurring crystal growth. Moreover, it is of critical importanceto modeling crystal growth of nanoscale materials because of thedominant role of surface effects at small length scales. Theinvestigator characterizes and evaluates different methodsproposed to remove or regularize the ill-posedness. A widelyemployed regularization is a singular perturbation. A significantmathematical challenge is to characterize the behavior of thesingularly perturbed corner. The investigator studies separatelythe role of the regularization in three dimensions and its effectin the presence of elastic stress. Another important scientificissue is to determine which of many regularization procedures istrue to the atomic-scale behavior of different materials. Theinvestigator also considers this question by studying the dynamicbehavior of regularizations in relation to experimentalobservations and the relation of regularizations to atomic-scalemodels. Overall, the project has the potential for significantimpact on the understanding of a key mathematical issue regardingregularization of ill-posedness in a classic moving boundaryproblem, and the impact of the work in a broader scientificcontext is that it contributes to the understanding of how tomodel the growth of crystalline solids in materials science. In the growth of crystals for nanotechnology and othermaterials applications, the formation of structures with corners(as on a grain of salt) is a natural occurrence. The physicaleffects responsible for the existence of a corner are wellunderstood and a mathematical description of an existing cornercan be accomplished with a classical mathematical model. However,the classical model is incapable of describing the actual dynamicsof corner formation. This problem is present in all mathematicalsimulations of crystal growth in which corners form. Moreover, itis of magnified importance in the simulation of the growth ofnanoscale structures: when the crystal decreases in size, cornersbecome an increasingly dominant part of the overall structure. Thus, to correctly describe the growth of nanostructuredmaterials, it is essential to have a correct model for cornerformation. To obtain tractable models for corner formation,different "regularization" ideas have been proposed to make themathematical problem of corner formation solvable, but there aremany different approaches and no universally accepted procedure. One aspect of this project is a critical comparison of thedifferent regularization approaches and how they behave inrelation to actual material systems. A second aspect of the workrelates to the fact that some of these models are "singularperturbations," which means that the results obtained when theregularization effect approaches zero can be different than if theregularization is not present at all. This type of unexpectedbehavior can mean that a small regularization that is added toallow for corner formation might give a different corner shape insimulations than should be present from the accepted classicalmodel. Thus, understanding such singular perturbation behavior isan important part of validating such regularization methods toensure that they give the correct overall behavior when used inlarge-scale crystal growth simulations. Taken as a whole, theproject has the potential for significant impact as a buildingblock in our ability to simulate the fabrication of nanomaterials,and by extension could contribute to the creation ofpurpose-specific materials, especially those with nanoscalefeatures, in electronics and other applications. In addition, theproject involves the training of a graduate student and includestwo undergraduate students, for whom the experience may serve asstimulus to pursue graduate degrees in the mathematical sciences.

研究人员关注描述纳米晶体生长的一个数学问题：如何正确地解决包含强各向异性晶体表面的动力学模型中固有的不适定性。表面能的强各向异性在物理上表现在晶体拐角的形成上。应用于角点形成的传统数学模型在数学上是不适定的，因此难以处理。由于在晶体生长过程中拐角的形成是普遍存在的，拐角形成的不适性是工业和自然发生的晶体生长模拟中固有的问题。此外，由于表面效应在小尺寸尺度上的主导作用，模拟纳米材料的晶体生长是至关重要的。研究者对不同的方法进行了描述和评价，这些方法被用来消除或正规化病态。广泛使用的正则化是一种奇异摄动。一个重大的数学挑战是描述这些奇异扰动的角落的行为。研究人员分别研究了正则化在三维空间中的作用及其在弹性应力存在时的影响。另一个重要的科学问题是确定许多正则化程序中哪一个更适用于不同材料的原子尺度行为。研究人员还通过研究与实验观测有关的正则化的动态行为以及正则化与原子尺度模型的关系来考虑这个问题。总体而言，该项目有可能对理解一个关键的数学问题产生重大影响，该问题涉及经典移动边界问题中不适定性的正则化，而这项工作在更广泛的科学背景下的影响是，它有助于理解如何在材料科学中模拟晶体固体的生长。在用于纳米技术和其他材料应用的晶体生长中，具有角的结构(如在盐粒上)的形成是自然发生的。角点存在的物理效应是很好理解的，现有角点的数学描述可以用经典的数学模型来完成。然而，经典模型不能描述拐角形成的实际动态。这个问题存在于所有形成拐角的晶体生长的数学模拟中。此外，它在模拟纳米结构的生长过程中具有放大的重要性：当晶体尺寸减小时，角点成为整个结构中日益占主导地位的部分。因此，要正确地描述纳米结构材料的生长，就必须有一个正确的角化模型。为了获得易于处理的拐角形成模型，人们提出了不同的“正则化”思想，使拐角形成的数学问题可解，但有许多不同的方法，也没有普遍接受的方法。这个项目的一个方面是对不同的正则化方法以及它们如何与实际材料系统相关的行为进行关键的比较。这项工作的第二个方面涉及这样一个事实，即这些模型中的一些是“奇异摄动”，这意味着当规则化效应接近于零时，得到的结果可能与根本不存在规则化时的结果不同。这种类型的意外行为可能意味着，添加一个允许拐角形成的小正则化可能会在模拟中给出与公认的经典模型中应该呈现的不同的拐角形状。因此，了解这种奇异扰动行为是验证这种正则化方法的重要组成部分，以确保它们在用于大规模晶体生长模拟时给出正确的整体行为。作为一个整体，该项目有可能对我们模拟纳米材料制造的能力产生重大影响，进而可能有助于创造特定用途的材料，特别是那些具有纳米级特征的材料，在电子和其他应用中。此外，该项目涉及对一名研究生和两名本科生的培训，对他们来说，这一经历可能会刺激他们攻读数学科学研究生学位。