Corner regularizations for nanoscale crystal growth

研究者关注的是在描述非线性增长过程中的一个数学问题：如何正确地解决包含具有强各向异性的堆叠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为晶体角的形成。 传统的数学模型应用于角的形成是数学不适定的，因此难以处理。 由于在晶体生长过程中角的形成是普遍存在的，角形成的不适定性是工业和自然发生的晶体生长模拟中固有的问题。 此外，由于表面效应在小长度尺度上的主导作用，它对模拟纳米材料的晶体生长至关重要。 研究人员的特点和评价提出了不同的方法来消除或规范的不适定性。 一个广泛使用的正则化是奇异摄动。 一个重要的数学挑战是描述奇异扰动角点的行为。 研究者分别研究了正则化在三维空间中的作用和在弹性应力存在下的作用。 另一个重要的科学问题是确定许多正则化过程中的哪一个是不同材料的原子尺度行为的原因。 研究者还考虑了这个问题，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。 总的来说，该项目对理解经典移动边界问题中的不适定性正则化这一关键数学问题具有潜在的重大影响，并且该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。 在纳米技术和其他材料应用的晶体生长中，形成具有角的结构（如在盐粒上）是自然发生的。 角点存在的物理效应是很好理解的，并且可以用经典的数学模型来完成对现有角点的数学描述。 然而，经典模型无法描述拐角形成的实际动力学。 这个问题存在于所有的晶体生长的角形模拟中。 此外，它在模拟纳米结构的生长中具有放大的重要性：当晶体的尺寸减小时，角成为整体结构中越来越占主导地位的部分。因此，要正确描述纳米结构材料的生长，必须有一个正确的角形成模型。 为了得到易于处理的角点形成模型，人们提出了不同的"正则化"思想来解决角点形成的数学问题，但方法很多，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在实际物质系统中的表现进行了重要的比较。 工作的第二个方面涉及到这样一个事实，即这些模型中的一些是"奇异扰动"，这意味着当正则化效应接近零时所获得的结果可能不同于正则化根本不存在时的结果。 这种类型的意外行为可能意味着，一个小的正则化，增加了允许角的形成可能会给一个不同的角形状在模拟比应该从公认的经典模型。 因此，理解这种奇异摄动行为是验证这种正则化方法的重要部分，以确保它们在大规模晶体生长模拟中使用时给出正确的整体行为。 作为一个整体，该项目有可能作为我们模拟纳米材料制造能力的基石产生重大影响，并通过扩展可能有助于创造特定用途的材料，特别是那些具有纳米尺度特征的材料，在电子和其他应用中。 此外，该项目还包括一名研究生和两名本科生的培训，对他们来说，这些经验可能有助于他们攻读数学科学的研究生学位。

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.