Mathematical Sciences Issues in Phase Transformations

固体和液体物质可以以不同的“状态”存在，无论状态的概念是通常的液/固相，还是其他一些条件，如晶体的变体，等等。引起和伴随状态变化的过程，虽然通常在广义上是已知的，但往往也有神秘的方面。对这些过程的理解是当今材料科学的一个基本要点。面对鲜为人知的基础物理学，数学模型不断被提出，希望能揭示一些洞见。本文主要讨论了相变时空变化的连续介质模型。在典型情况下，这种变化包括两个相之间或更一般类型的状态之间的界面运动。不同的模型并不总是预测相同的事情，例如各向异性介质中界面运动的详细性质。提出的工作的一个目的是更好地阐明重要模型类别的不同假设，并将这些不同的假设更清楚地与它们的预测联系起来。在比较这些模型时，将特别注意非本地模型。最后，一个神秘的特定类型的相变固体二元合金被称为不连续沉淀或胞状沉淀将进一步建模和分析。新型高性能材料的开发对经济和其他国家利益至关重要。材料的许多特性与其性能相关，其中包括渐进相变的可能性以及交替固相或晶体变体之间界面的发展。有必要对这些界面形成和移动的过程有更基本的了解。这项研究将通过发展、比较和分析基于物理定律的这些现象的数学模型来促进这种理解。

9703483 Fife Solid and liquid materials can exist in different ``states'', whether the concept of state is that of the usual liquid/solid phase, or some other condition like a variant of a crystal, etc. The processes which cause and accompany changes of state, while usually known in broad terms, often also have mysterious aspects. The understanding of these processes is a basic thrust of present day materials science. In the face of little known basic physics, mathematical models are continually being proposed in the hopes of shedding insight. The present proposal focusses attention on certain classes of continuum models for the spatio-temporal variation of phase transformations. In typical cases, this variation consists of motion of an interface between two phases or between more general types of states. Different models do not always predict the same thing, an example being the detailed nature of the motion of the interface in anisotropic media. One aim in the proposed work will be to better elucidate the differing assumptions of important classes of models, and to tie these different assumptions more clearly with their predictions. Special attention, in comparing these models, will be paid to nonlocal ones. Finally, a mysterious specific type of phase transformation for solid binary alloys known as discontinuous precipitation or cellular precipitation will be further modeled and analyzed. The development of new high performance materials is vital to economic and other national interests. Among the many properties of materials which are relevant to their performance is the possibility of progressive phase change and the development of interfaces between alternate solid phases or crystalline variants. There is need for a great deal more basic understanding of the processes by which such interfaces are formed and move. This research will contribute to this understanding by the development, comparison, and analysis of mathematical models of these phen omena based on physical laws.