Mathematical Sciences Issues in Phase Transformations

小行星9703483 固体和液体材料可以以不同的“状态”存在，无论状态的概念是通常的液体/固体相，还是其他一些条件，如晶体的变体等。 对这些过程的理解是当今材料科学的基本推动力。 面对鲜为人知的基础物理学，数学模型不断被提出，希望能带来深刻的见解。 本建议集中注意力在某些类别的连续模型的时空变化的相变。 在典型的情况下，这种变化包括两相之间或更一般类型的状态之间的界面的运动。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。 拟议工作的一个目标是更好地阐明重要类别模型的不同假设，并将这些不同的假设与它们的预测更清楚地联系起来。在比较这些模型时，将特别注意非局部模型。最后，一个神秘的特定类型的相变固体二元合金称为不连续沉淀或细胞沉淀将进一步建模和分析。 开发新型高性能材料对经济和其他国家利益至关重要。 在与材料性能相关的许多性质中，有可能发生渐进相变，并在交替的固相或结晶变体之间形成界面。我们需要对这些界面的形成和移动过程有更多的基本了解。 本研究将有助于这一理解的发展，比较和分析的数学模型，这些现象的基础上的物理定律。

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.