Complex Transient Patterns in Phase-Field Models

论文题目：相场模型中的复杂瞬态模式【摘要】凝固和熔化过程会产生复杂而不稳定的材料微观组织，这些组织会随着时间的推移而演变。该项目涉及复杂微观结构的数学研究，可以在凝固和熔化过程的基本相场模型中观察到。除了经典和热力学一致的模型外，该研究还考虑了最近的随机和非局部扩展。后者纳入了热噪声和远程相互作用的影响，从而解决了经典相场模型的两个主要缺点。PI获得了微观结构动力学的详细数学描述，解释了微观结构的形成及其复杂几何形状的关键特征。他研究噪声和远距离相互作用如何影响图案几何。此外，与一名研究生合作，PI开发了数值模拟工具来处理随机和非局部效应。近年来，相场建模已被证明是模拟和预测材料科学中各种过程的有价值的工具。提出的研究提供了可用于精确预测微观结构变化的工具，因此对材料设计和小规模生物和物理过程的理解感兴趣。由于现代材料的微观结构几何形状直接影响材料的宏观性能，因此本项目的研究结果可以为开发新型高性能材料提供帮助，这对经济和其他国家利益至关重要。特别是由于包括热噪声和远程相互作用，该研究允许处理更现实的模型。虽然提议的研究集中在一个特定的情况下，其影响远远超出了材料科学的应用。该项目可直接应用于数学生物学领域。细菌内部复杂蛋白质模式的形成和神经模式形成的各个方面都可以用相场型模型来描述。此外，拟议的活动提供了一个很好的框架，可以在几个层次上，包括本科生和研究生的研究，把研究和教育结合起来。

Proposal: DMS-0406231PI: Thomas WannerInstitution: George Mason UniversityTitle: Complex Transient Patterns in Phase-Field ModelsABSTRACTSolidification and melting processes can generate complicated and unstable material microstructures which evolve with time. This project is concerned with the mathematical study of complex microstructures that can be observed in fundamental phase-field models for solidification and melting processes. In addition to classical and thermodynamically consistent models, the study considers recent stochastic and nonlocal extensions. The latter incorporate the effects of thermal noise and long-range interactions, thereby addressing two key shortcomings of the classical phase-field models. The PI obtains a detailed mathematical description of the microstructure dynamics, which explains both the formation of the microstructures and key characteristics of their complex geometry. He studies how noise and long-range interactions influence the pattern geometry. Furthermore, in collaboration with a graduate student, the PI develops numerical simulation tools to treat stochastic and nonlocal effects.In recent years, phase-field modeling has proved to be a valuable tool for simulating and predicting a variety of processes in materials science. The proposed research provides tools that can be used for precisely predicting microstructure changes and therefore is of interest for materials design and the understanding of small-scale biological and physical processes. Since the microstructure geometry of modern materials has direct implications for macroscopic material properties, the results of this project can offer help in the development of new high performance materials, which is vital to economic and other national interests. Especially due to the inclusion of thermal noise and long-range interactions, the study allows for the treatment of more realistic models. While the proposed research is focused on a specific situation, its impact reaches well beyond the materials science application. Immediate applications of the project can be expected in the field of mathematical biology. Both the formation of complicated protein patterns inside bacteria and aspects of neural pattern formation are described by phase-field-type models. In addition, the proposed activities offer an excellent framework for integrating research and education at several levels, including undergraduate and graduate research.