Patterns, Geometry, and Growth

许多物理系统自发地以规则的模式自组织。 如果没有外部控制，它们会演变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制造过程中的相分离动力学，如浸涂。 为了控制这些过程，并可能收获微结构材料制造的自组织能力，人们需要了解图案化如何受到参数和系统几何形状的影响。 研究人员专注于一个特别相关的情况下，图案出现通过定向淬火过程中的图案化区域在空间中扩大，无论是在外部控制的方式，或在一个自组织的增长过程。 事实证明，在许多物理和生物背景下，图案化的结果是非常严格的，从某种意义上说，最终的图案对缺陷是鲁棒的，并且可以从实验的许多不同初始状态中重现。 然而，最终的图案敏感地依赖于生长速率和淬火几何形状。 研究人员和他的合作者开发分析和数值工具，使系统的预测和控制产生的模式，最终目标是设计过程，导致一个理想的，预先指定的模式。 研究生参与了该项目的研究。研究员和他的合作者分析了原型系统，如斯威夫特-霍恩伯格或卡恩-希利亚德方程的情况下，模式形成区域在时间上以规定的速度扩展。 在最简单的情况下，这些系统发展出相对于生长方向具有固定波长和取向的条纹图案。 为这种情况下开发的数值工具，允许系统的探索参数之间的关系和由此产生的取向和波长。 分析工具可以通过展示结构的钉扎和分离等普遍机制来指导数值计算。 分析还补充了有限的制度，计算成本高得令人望而却步的数值研究。 分析和数值都集中在相干结构的研究上，相干结构是系统中最简单的模式形成动态状态。 它们通常是静止的或时间周期性的，在一个框架中移动的淬火界面，并渐近选定的图案中的图案形成区域。 该项目的第一部分侧重于简单模型问题中条纹或层状晶体的形成，其中生长过程大致选择条纹相对于边界的取向角。 第二部分通过包括更现实的模型、不同的生长定律和不同的优选结晶状态来拓宽范围。 该奖项反映了NSF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。

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.