Collaborative Research: Computational and theoretical approaches for the morphological control of material microstructures

该奖项根据 2009 年美国复苏和再投资法案（公法 111-5）提供资金。研究人员和同事研究了固/液和固/固扩散相变中两相微观结构的预测和形态控制。该领域的大部分研究都涉及复杂模式（例如树突生长形状）的详细和广泛的研究。在许多应用（例如铸件）中，需要控制枝晶的形成并生长致密的形状，然而，对此的研究却少得多。该项目有助于填补这一空白，旨在制定可生长具有所需形状的微结构的指南。研究小组计划（1）发展一种合适的致密析出物生长的非线性理论，包括自相似形状的存在性、唯一性和稳定性； (2) 开发并采用最先进的自适应 3D 数值方法来测试理论的有效性和局限性； (3) 将理论和数值结果与现有实验进行比较，以检验数学假设的有效性并验证理论和模拟得出的预测的准确性。扩散相变涉及熔体转变为沉淀物（反之亦然）以及固体（例如金属合金）分离成不同的相。这些现象对于包括铸造、焊接和锡焊、晶体生长以及有关蛋白质和大分子结晶的相关问题在内的各种工艺具有重要意义。例如，用于技术应用的晶体生长工艺始于 19 世纪末，并构成了当今几乎所有现代半导体电子学和光子学的基石。研究活动将提供新的数学理论和数值模拟，可用于制定控制某些固化材料形态的指南。该项目开发的新数学理论和自适应数值方法可应用于更广泛的相关问题，包括多相流、生物结构和实体瘤的生长。此外，该项目将为年轻研究人员提供宝贵的跨学科培训机会。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). The investigator and colleagues study the prediction and morphological control of two-phase microstructures in solid/liquid and solid/solid diffusional phase transitions. Much of the research in this area is concerned with detailed and extensive studies of complex patterns such as dendritic growing shapes. In many applications (e.g. castings), it is desirable to control the formation of dendrites and grow compact shapes, which, however, has been much less studied. This project helps to fill the gap and aims to develop guidelines by which microstructures with desired shapes may be grown. The research team plans to (1) develop a suitable nonlinear theory of compact precipitate growth including existence, uniqueness, and stability of self-similar shapes; (2) develop and employ state-of-the-art adaptive 3D numerical methods to test the validity and limitations of theory; (3) compare theoretical and numerical results with existing experiments to test the validity of the mathematical assumptions and to verify the accuracy of predictions derived from the theory and simulations. Diffusional phase transformations deal with transformations of melts into precipitates (and vice-versa) as well as the separation of solids (e.g. metal alloys) into distinct phases. These phenomena have importance for a variety of processes including casting, welding and soldering, crystal growth, and related problems concerning protein and macromolecular crystallization. For example, crystal growth processes for technological applications began in the late 19th century, and form the cornerstone of virtually all modern semiconductor electronics and photonics today. The research activities will provide new mathematical theory and numerical simulations that can be used to develop guidelines for controlling the morphology of certain solidified materials. The new mathematical theory and adaptive numerical methods developed in the project have applications to a broader set of related problems including multiphase flows, biostructures and growth of solid tumors. In addition, this project will provide valuable interdisciplinary training opportunities for young researchers.