Pattern and wavenumber selection in the wake of fronts

在各种化学、生物和制造过程中，图案都是在界面之后出现的。这一建议特别受到反复降水中的列色岗模式、细菌菌落模式和离子轰击表面下自组织纳米波纹形成的启发。在所有这些情况下，在实验和直接模拟中，当它们在移动锋的尾迹中形成时，已经观察到大量规则和复杂的模式。从数学的角度来看，模式形成锋面的理解很少，但它们为系统地理解相分离过程、表面演化和生物生长中的模式提供了主要途径。一个系统的，独立于模型的理解在锋面之后的模式，结合定量预测的计算工具，有可能解释经典的百年历史的化学实验，并为生物聚集机制提供新的亮点。锋面后连贯的模式允许对模式进行清晰的定性和定量预测。在这方面，提出的研究将为定量的、确定性的模型验证提供新的工具。对图案表面不稳定性的预测在制造过程中具有潜在的高影响。拟议研究的教育方面涉及两个夏季reu的本科生和四个研究生。

Pattern and wavenumber selection in the wake of frontsIn a variety of chemical, biological, and manufacturing processes, patterns emerge in the wake of interfaces. This proposal is particularly motivated by Liesegang patterns in recurrent precipitation, bacterial colony patterning, and self-organized nano-ripple formation under ion-bombardment of surfaces. In all those contexts, a plethora of regular and complex patterns has been observed as they form in the wake of moving fronts, both experimentally and in direct simulations. Pattern-forming fronts have been poorly understood from a mathematical view point, yet they provide a prime avenue to a systematic understanding of patterning in phase separation processes, surface evolution, and biological growth. A systematic, model-independent understanding of patterning in the wake of fronts, combined with computational tools for quantitative predictions, has the potential to both explain classic 100-year old chemical experiments and shed new lights on biological aggregation mechanisms. Coherent patterns in the wake of fronts allow for clean qualitative and quantitative predictions of patterns. In this respect, the proposed research will give new tools for quantitative, deterministic model validation. Predictions on patterned surface instabilities have potentially high impact in manufacturing processes. The educational aspect of the proposed research involves undergraduate students during two summer REUs and four graduate students.