Mathematical Problems from Materials Science

提案：0313744PI: Robert V. Kohn [kohn@cims.nyu.edu]机构：纽约大学标题：材料科学的数学问题摘要该项目涉及数学，物理和材料科学之间的界面主题，特别是(a)晶体晶界中能量驱动的图案形成，以及(b)能量驱动的相混合物和缺陷的粗化。对于前者，PI将使用基于非相干能量最小化的新观点来研究扭曲晶界中的图案形成。关于后者：PI将使用基于耗散和等周不等式的新观点，从相分离，晶粒生长和涡旋湮灭等方面研究问题。能量驱动的模式形成发生在许多不同的物理系统中；例子包括ii型超导体中的涡流晶格，晶界中的位错模式，以及由相分离产生的迷宫结构。这个项目寻求一种理解——对于特定的、广泛研究的系统——为什么形成某些模式而不是其他模式。这样的理解对于解释实验观察是至关重要的，即通过观察系统形成的模式来推断系统的物理特性。

Proposal: 0313744PI: Robert V. Kohn [kohn@cims.nyu.edu]Institution: New York UniversityTitle: Mathematical Problems from Materials ScienceABSTRACTThis project addresses topics at the interface between mathematics, physics, and materials science, particularly (a) energy-driven pattern formation in crystalline grain boundaries, and (b) energy-driven coarsening of phase mixtures and defects. Concerning the former: the PI will investigate pattern formation in twist grain boundaries, using a new viewpoint based on minimization of incoherence energy. Concerning the latter: the PI will investigate problems from phase separation, grain growth, and vortex annihilation, using a new viewpoint based on dissipation and isoperimetric inequalities.Energy-driven pattern formation occurs in many different physical systems; examples include vortex lattices in type-II superconductors, dislocation patterns in grain boundaries, and labyrinthine structures created by phase separation. This project seeks an understanding -- for specific, widely studied systems -- why certain patterns form rather than others. Such understanding is crucial for interpreting experimental observations, i.e. for drawing inferences about the physics of a system from observations of the patterns it forms.