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常规模式进行自我组织。如果没有外部控制，它们就会演变成几乎结晶状态，显示了条纹或斑点，以某种规则的时尚对齐。示例范围从胚胎的早期发育状态的自组织过程到制造过程中的相位分离动态，例如浸入涂层。为了控制这些过程，并可能收集制造微型结构材料的自组织能力，人们需要了解图案如何受参数和系统几何形状影响。研究者专注于一种特别相关的情况，在这种情况下，模式是通过方向性淬火过程在空间中以外部控制的方式或自组织增长过程在太空中扩展的。事实证明，在许多物理和生物学环境中，图案的结果非常僵硬，从某种意义上说，最终模式是可靠的，从实验的许多初始状态中可以重现，并且可以重现。然而，最终模式确实取决于增长率和淬火几何形状。研究者及其合作者开发了分析和数值工具，可以实现系统的预测和控制模式，其最终目标是设计过程，从而导致所需的预先指定的模式。研究生从事该项目的研究。调查员及其合作者分析了原型系统，例如Swift-Hohenberg或Cahn-Hilliard方程，在以规定的速度及时扩展的情况下。在最简单的情况下，这些系统会相对于生长方向发展具有固定波长和方向的条纹模式。针对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诸如固定和脱离结构的通用机制来指导数字。分析还补充了计算成本高度高的限制制度中的数值研究。分析和数字都集中在相干结构的研究上，这是系统中最简单的动态状态。它们通常是静止的或与淬火界面移动的框架中的时间周期性，并渐近地归于模式形成区域中选定的模式。项目的第一部分着重于在简单模型问题中的条纹或层状晶体的形成，其中生长过程大致选择了条纹的方向角度相对于边界。第二部分通过包括更现实的模型，不同的增长定律和不同的首选结晶状态来扩大范围。研究生从事该项目的研究。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点评估和更广泛影响的评论标准来支持的。