Analysis of Stationary Solutions and the Pinning Phenomena for the Modeling of Phase Boundary Motions in Materials Science

建议：DMS-0406033 PI：Nung Kwan Yip机构：普渡大学题目：材料科学中相边界运动建模的定态解和钉扎现象分析摘要本提案旨在理解材料相边界所经历的能量景观。在材料科学的应用分析和建模领域，这是一个非常有趣和重要的问题。这是因为相界在形成过程中不可避免地会遇到来自环境中的缺陷和杂质的复杂微观结构。研究人员将研究相界的稳定和不稳定模式。这些对应于材料能量泛函上的局部极小点和鞍点。该建议的主要重点是研究能量景观上相边界的时间演化，特别是从一种稳定构型到另一种稳定构型的过渡，以及响应于某些外部强迫的钉扎和脱钉现象。科学活动的成果将为理解材料对基本空间环境和外部参数的动态响应提供坚实的数学框架。在这个提议中涉及的数学技术包括变分方法，渐近分析，和分歧theory.The理解的材料性能在现代材料科学中变得越来越有用，特别是在当前追求的小尺度（纳米）材料。一个关键因素是将材料的性能与其微观结构联系起来。所提出的活动的结果可以帮助描述材料相界的稳定和不稳定的图案形成和它们的动力学行为时，外部条件的变化。这使人们更好地了解材料对外部环境的反应，从而更好地设计和制造材料。这项调查的主题是高度跨学科的，这可以促进数学家和材料科学家之间的密切合作。它还为人力资源和教育发展提供了良好的机会。

Proposal: DMS-0406033PI: Nung Kwan YipInstitution: Purdue UniversityTitle: Analysis of Stationary Solutions and the Pinning Phenomena for the Modeling of Phase Boundary Motions in Materials ScienceABSTRACTThis proposal strives to understand the energy landscape experienced by materials phase boundaries. This is a tremendously interesting and important problem in the regimes of applied analysis and modeling in materials science. It is due to the fact that phase boundaries, during their formations, will inevitably encounter complicated microstructures coming from the defects and impurities embedded in the environment. The investigator will study the stable and unstable patterns of the phase boundaries. These correspond to the local minimizers and saddle points on the materials energy functionals. The main emphasis of this proposal is to study the time evolution of the phase boundaries on the energy landscape, in particular, the transition from one stable configuration to another and from the pinning and de-pinning phenomena in response to some external forcing. The outcome of the scientific activity will provide a solid mathematical framework for the understanding of the dynamical response of the materials with respect to the underlying spatial environment and external parameters. The mathematical techniques involved in this proposal include variational methods, asymptotic analysis, and bifurcation theory.The understanding of material properties is becoming more and more instrumental in modern materials science, especially in the current pursuit of small scale (nano-)materials. A crucial factor is to relate the materials properties to their microstructures. The outcome of the proposed activity can help the description of the stable and unstable pattern formations of the materials phase boundaries and their dynamical behavior when external conditions changes. This gives a better understanding of the materials response to the external environment, which in turn gives better design and manufacture of the materials. The topic of this investigation is highly interdisciplinary, which can lead to close collaboration between mathematicians and materials scientists. It also provides good opportunities for human resources and educational development.