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Patterns, Geometry, and Growth

图案、几何形状和生长

基本信息

批准号：
1907391
负责人：
Arnd Scheel
金额：
$ 47.01万
依托单位：
University of Minnesota-Twin Cities
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-08-15 至 2022-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1907391&HistoricalAwards=false
关键词：
Patterns Geometry Growth

项目摘要

Many physical systems spontaneously self-organize in regular patterns. Without external control, they evolve into almost crystalline states showing stripes or spots aligned in somewhat regular fashions. Examples range from the self-organizational processes in early developmental states of the embryo to phase separation dynamics in manufacturing processes such as dip-coating. In order to control those processes and possibly harvest the self-organizational capabilities for the manufacturing of micro-structured materials, one needs to understand how the patterning is influenced by parameters and system geometry. The investigator focuses on a particularly relevant situation in which patterns arise through a directional quenching process where the patterned region expands in space, either in an externally controlled fashion, or in a self-organized growth process. It turns out that the result of patterning is very rigid, across many physical and biological contexts, in the sense that the final pattern is robust against imperfections and reproducible from many different initial states of the experiment. The final pattern does however depend sensitively on growth rates and quenching geometry. The investigator and his collaborators develop analytic and numerical tools that enable systematic prediction and control of resulting patterns, with the ultimate goal of designing processes that result in a desired, pre-specified pattern. Graduate students are engaged in the research of the project.The investigator and his collaborators analyze prototypical systems such as the Swift-Hohenberg or the Cahn-Hilliard equation in situations where the pattern-forming region expands in time at a prescribed rate. In the simplest case, these systems develop striped patterns with a fixed wavelength and orientation relative to the direction of growth. Numerical tools developed for this scenario allow for a systematic exploration of the relation between parameters and resulting orientations and wavelengths. Analytic tools can guide the numerics by exhibiting universal mechanisms such as pinning and detachment of structures. Analysis also complements numerical studies in limiting regimes where computational cost is prohibitively high. Both analysis and numerics focus on the study of coherent structures, which are the simplest pattern-forming dynamic states of the system. They are typically stationary or time-periodic in a frame moving with the quenching interface, and asymptotic to a selected pattern in the pattern-forming region. The first part of the project focuses on the formation of stripes, or lamellar crystals, in simple model problems, where the growth process roughly selects an orientation angle of stripes relative to the boundary. The second part broadens the scope by including more realistic models, different growth laws, and different preferred crystalline states. Graduate students are engaged in the research of the project.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

许多物理系统自发地以规则的模式自组织。在没有外部控制的情况下，它们进化成几乎是晶体的状态，显示出以某种规则的方式排列的条纹或斑点。例子范围从胚胎早期发育状态的自组织过程到生产过程中的相分离动力学，如浸渍涂层。为了控制这些过程并可能获得制造微结构材料的自组织能力，需要了解图案如何受到参数和系统几何形状的影响。研究者专注于一种特别相关的情况，在这种情况下，图案通过定向淬火过程产生，其中图案区域在空间中扩展，要么以外部控制的方式，要么以自组织的生长过程。事实证明，在许多物理和生物环境中，模式的结果是非常严格的，因为最终的模式是坚固的，可以抵抗缺陷，并且可以从许多不同的实验初始状态中重现。然而，最终的图案确实敏感地取决于生长速率和淬火几何形状。研究者和他的合作者开发了分析和数值工具，能够系统地预测和控制产生的模式，最终目标是设计产生期望的、预先指定的模式的过程。研究生从事该项目的研究。研究者和他的合作者分析了典型的系统，如斯威夫特-霍恩伯格方程或卡恩-希利亚德方程，在这种情况下，模式形成区域以规定的速度在时间上膨胀。在最简单的情况下，这些系统形成具有固定波长和相对于生长方向的方向的条纹图案。为这种情况开发的数值工具允许系统地探索参数与结果方向和波长之间的关系。分析工具可以通过展示结构的钉住和分离等普遍机制来指导数值。分析还补充了在计算成本过高的限制条件下的数值研究。分析和数值都集中在相干结构的研究上，这是系统最简单的形成模式的动态状态。它们在随淬灭界面移动的框架中通常是静止的或时间周期性的，并且在模式形成区域中渐近于选定的模式。该项目的第一部分侧重于简单模型问题中条纹或片层晶体的形成，其中生长过程大致选择条纹相对于边界的方向角。第二部分通过纳入更现实的模型、不同的生长规律和不同的优选晶态来扩大范围。研究生从事该项目的研究。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。