Variational Problems Associated with Models for Both Orthodox and Unorthodox Materials

与正统和非正统材料模型相关的变分问题

• 批准号: 0072816
• 负责人: Victor Mizel
• 金额: $ 7.5万
• 依托单位: Carnegie-Mellon University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-08-01 至 2004-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0072816&HistoricalAwards=false
• 关键词: Variational; Problems; Associated; Models; Both

0072816MizelThe goal of the part of the present proposal involving unorthodox materials is to develop an understanding of step structure and step motion in the limit of small step free energy for crystals with surface defects consisting of monatomic height steps separated by terraces. Such defects are fundamental structures since steps--as sinks and sources for atoms--influence mass transport at surfaces and therefore influence fundamental multidimensional surface processes such as nucleation and growth, facet formation and thermal roughening in crystals and their applications. Consequently these defects have been studied since the 1950's and fairly successful macroscopic models to describe behavior in most materials have been developed. These models express the energy of an isothermally wandering step as the integral of the (generally orientation dependent) step energy per unit length. However in the limit of vanishing step free energy, new physics [e.g. higher-order elastic interactions and step-to-step interactions] enters the picture. These higher-order effects can lead to the spontaneous development of highly periodic step structures. Such novel step morphologies are members of an increasingly important class of self-organizing systems: systems that spontaneously form atomic scale periodic patterns. A recent (1997) pathbreaking experiment on Boron-doped Silicon in which the isothermal wanderings in question, contrary to the simple models cited, are periodic and increase with a decrease of the steady state temperature in an interval presents a significant challenge to the development of appropriate macroscopic models for this branch of crystal theory. It reveals that there is a major gap in the current technical understanding of defects in crystals of this nature, since detailed analysis of the dynamics of these fluctuating step edges is incompatible with current theories of step energetics. Resolving this issue is important for near term attainment of non-routine applications involving crystals with small step free energy, since it has direct impact on understanding the atomic-scale structures they develop. In turn, this understanding is needed for the development of materials with uniformly closely spaced and nearly congruent self-organized [quantum dot] defects--such self-organizing materials being crucial to the development of monochromatic lasers and quantum wires, for example, as device scales shrink. The goal of that part of the present proposal involving orthodox materials is focussed on clarifying an as yet unresolved issue in the theory of nonlinearly elastic materials. Equilibria for such materials subjected to specified boundary displacements correspond to deformations which minimize a stored energy integral whose integrand associates to each deformation a nonnegative real valued function. Such stored energy integrands are subject to various constraints in order to correctly represent possible physical materials. In the last decade a previously unnoticed issue has been raised. Namely in view of a little known one-dimensional variational phenomenon (originally established in 1926) that for certain variational integrands the minimizing functions can differ depending on the smoothness of the class of deformations under consideration -- even though the smoother class of functions is dense in the larger class. In fact there can be a nonzero difference between the infima of the integrals associated with such classes [Lavrentiev's gap phenomenon]. A corresponding gap between classes of continuous deformations in three dimensional nonlinear elasticity would imply that the global equilibrium deformation in the larger class would be energy minimizing and more singular than the energy minimizing global deformation in the smaller class--whereby structures devised on the basis of the less singular deformations could develop flaws [fractures], contrary to the evidence provided by calculations based on standard methods. To summarize, the PI proposes in the case of the unorthodox type of crystalline material exemplified by Boron doped Silicon to devise variational models that will shed light on the entirely nonstandard physical behavior of such materials. Such models will involve devising what are known as Landau-de Gennes type order parameter terms for the free energy of such materials to reflect the particularly delicate atomic interactions governing the nanatomic structure of such crystals. On the other hand, in the case of orthodox materials the PI intends to clarify whether in the well-established theory of nonlinear elasticity there can occur materials which for certain classes of boundary conditions can exhibit an energy gap between the minimum energy on one class of continuous smooth deformations and the minimum energy on a smaller class. The occurrence of such an energy gap could lead to cases in which structures devised on the basis of computations associated with the smaller class could develop flaws because the actual energy minimizing deformation is more singular than the computations suggest.

0072816 Mizel本提案中涉及非正统材料的部分的目标是，对于具有由台阶分隔的单原子高度台阶组成的表面缺陷的晶体，在小台阶自由能的限制下，发展对台阶结构和台阶运动的理解。这种缺陷是基本结构，因为步骤-作为原子的汇和源-影响表面的质量传输，因此影响基本的多维表面过程，如晶体中的成核和生长，小面形成和热粗糙化及其应用。因此，自20世纪50年代以来，人们一直在研究这些缺陷，并开发出了相当成功的宏观模型来描述大多数材料的行为。这些模型将等温漂移步骤的能量表示为每单位长度的（通常依赖于取向的）步骤能量的积分。然而，在消失的步骤自由能的限制，新的物理[例如，高阶弹性相互作用和步骤到步骤的相互作用]进入画面。这些高阶效应可以导致高度周期性台阶结构的自发发展。这种新颖的台阶形态是一类越来越重要的自组织系统的成员：自发形成原子尺度周期性图案的系统。 最近（1997年）对掺硼硅进行的一项开创性实验表明，与所引用的简单模型相反，所讨论的等温漂移是周期性的，并且在一定间隔内随着稳态温度的降低而增加，这对为晶体理论的这一分支发展适当的宏观模型提出了重大挑战。它表明，有一个主要的差距，在目前的技术理解缺陷的晶体的这种性质，因为详细分析这些波动的步骤边缘的动力学是不符合目前的理论步骤能量学。 解决这个问题对于短期实现涉及具有小步阶自由能的晶体的非常规应用是重要的，因为它对理解它们开发的原子尺度结构有直接影响。反过来，需要这种理解来开发具有均匀紧密间隔和几乎全等的自组织[量子点]缺陷的材料-这种自组织材料对于单色激光器和量子线的开发至关重要，例如，随着设备规模缩小。本提案中涉及正统材料的部分的目标集中于澄清非线性弹性材料理论中尚未解决的问题。 这种材料在特定的边界位移下的平衡对应于使储能积分最小化的变形，该储能积分的被积函数与每个变形的非负真实的值函数相关联。 这样的储能被积体受到各种约束，以便正确地表示可能的物理材料。 在过去的十年里，一个以前没有注意到的问题被提出来了。即鉴于一种鲜为人知的一维变分现象（最初成立于1926年），对于某些变分被积函数，最小化函数可能会根据所考虑的变形类的平滑度而有所不同--尽管更平滑的函数类在更大的类中是稠密的。事实上，可以有一个非零的差异之间的infima积分与这些类[拉夫连季耶夫的差距现象]。 三维非线性弹性中连续变形类别之间的相应间隙意味着较大类别中的全局平衡变形将是能量最小化的，并且比较小类别中的能量最小化全局变形更奇异--由此，基于较不奇异变形设计的结构可能会产生缺陷[裂缝]，这与基于标准方法的计算所提供的证据相反。总而言之，PI建议在以硼掺杂硅为例的非正统类型的晶体材料的情况下设计变分模型，以揭示此类材料的完全非标准物理行为。 这种模型将涉及设计所谓的Landau-de Gennes型有序参数项，用于这些材料的自由能，以反映控制这种晶体的纳米原子结构的特别微妙的原子相互作用。另一方面，在正统材料的情况下，PI旨在澄清在非线性弹性的成熟理论中是否会出现这样的材料，即对于某些类别的边界条件，可以在一类连续光滑变形的最小能量和较小类别的最小能量之间表现出能量差。 这种能隙的出现可能会导致这样的情况，即根据与较小类别相关的计算设计的结构可能会产生缺陷，因为实际的能量最小化变形比计算所建议的更奇异。