Topological Analysis of Pattern-Forming Systems

自然界和实验室中的各种现象会产生涟漪、正方形或六边形等图案。例子从实验室气候模型到负责空间记忆的大脑区域的六角形放电神经元阵列，其中流体从下到下被加热。两类图案形成系统推动了本项目的工作。第一个涉及用宽束离子束轰击固体表面产生的纳米级图案。这可以产生各种各样的自组织纳米级图案。当固体受到辐射时，纳米级图案的自组装不仅令人着迷：在未来，离子轰击可能会被证明是制造纳米结构的重要工具。人们普遍认为，新兴的纳米技术领域将带来进步，这些进步将改变能源、电子和医学等截然不同的领域。第二类图案涉及化学体系中的颜色变化，在化学体系中，蒸汽与固体或液体发生反应。例如，一种名为花青素的重要植物色素的溶液中可能会出现彩色图案，因为它暴露在常见的大气污染物中。这些模式是针对小学生和高中生的赞助外展活动的一部分。在这两个系统中，图案从具有少量缺陷的高度有序的波纹或晶格到由缺陷主导的图案，以至于晶格结构可能不容易被识别。这些缺陷限制了离子轰击产生的纳米结构的应用。它们还指示了模式形成的潜在化学或物理机制，因此有助于研究人员理解这些机制。在这项工作中，研究人员开发了数学工具来理解缺陷的形成。这些工具还被用于帮助提出实验方法，以消除离子轰击产生的纳米材料中的缺陷。这些工具还提供了对驱动模式形成的机制的洞察。这项研究涉及本科生和研究生，他们从事的是理论和实验相结合的工作。在自然界和实验室生产的图案中，缺陷往往很常见，因此图案远不是理想的波纹或六角晶格。这些缺陷可以解释为具有拓扑特征的数据集。在这个项目中，研究人员将拓扑数据分析(TDA)的方法应用于由非线性偏微分方程建模的模式。特别是，研究人员和他们的同事开发了使用各种TDA方法的输出来量化模式中的顺序的方法。实验和模拟表明，长波形变(即零模)对缺陷在发展中的持续起着重要作用。研究人员使用TDA方法的多维扩展来检验这一假设。通过推导花样的振幅和相位方程，探讨了缺陷的稳定性。使用TDA开发了预测当图案演变时缺陷将在哪里形成的方法。最后，将TDA与机器学习工具相结合，该团队从实验数据中确定模式形成模型中的参数。该方法适用于任何图案形成系统。然而，有两类系统为这项工作提供了重点。第一种是当固体表面受到宽束离子束轰击时形成的纳米级图案。利用TDA，研究人员确定了当二元材料表面受到轰击时导致形成六角纳米点阵列的不稳定性的性质，这是一个相当有争议的主题。第二类是一组反应扩散系统，我们称之为汽相变色实验。在这些实验中，水蒸气与含有聚合物或颜料的溶液相互作用，聚合物或颜料在与水蒸气相互作用时会改变颜色。该团队开发了用于汽色图案形成的反应-扩散-对流的数学模型。使用TDA分析模式的方法是由实验和偏微分方程模型的数值模拟产生的模式激发并在其上进行测试的。研究生和REU本科生参与了这项研究。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Diverse phenomena in nature and the laboratory give rise to patterns such as ripples, squares, or hexagons. Examples range from laboratory models of climate in which a fluid is heated from below to hexagonal arrays of firing neurons in the region of the brain responsible for spatial memory. Two classes of pattern-forming systems motivate the work of this project. The first involves nanoscale patterns produced by bombarding a solid surface with a broad ion beam. This can produce a wide variety of self-organized nanoscale patterns. The self-assembly of nanoscale patterns that occurs when solids are irradiated is not just fascinating: in the future, ion bombardment may prove to be an important tool in the fabrication of nanostructures. It is widely believed that the burgeoning field of nanotechnology will lead to advances that will transform fields as disparate as energy, electronics, and medicine. The second class of patterns involves color changes in chemical systems in which a vapor reacts with a solid or liquid. For example, a colored pattern may appear in a solution of an important class of plant pigments called anthocyanins as it is exposed to common atmospheric pollutants. These patterns are a part of the sponsored outreach to elementary and high school students. In both of these systems, the patterns vary from highly ordered ripples or lattices with a few defects to patterns so dominated by defects that a lattice structure may not be easily recognized. These defects limit the utility of nanostructures produced by ion bombardment. They are also indicative of the underlying chemical or physical mechanism by which the patterns form, and therefore help the investigators to understand those mechanisms. In this work, the investigators develop mathematical tools to understand the formation of defects. The tools are also applied to help propose experimental methods to eliminate defects in nanopatterns produced by ion bombardment. The tools also provide insight into the mechanisms driving pattern formation. The research involves undergraduate and graduate students in integrated theoretical and experimental work.Defects are often prevalent in patterns produced in nature and the laboratory, so that the patterns are far from ideal ripples or hexagonal lattices. These defects can be interpreted as data sets that have topological characteristics. In this project, the investigators apply methods of topological data analysis (TDA) to patterns modeled by nonlinear partial differential equations. In particular, the investigators and their colleagues develop methods to quantify the order in a pattern using the output of various TDA methods. Experiments and simulations suggest that long-wavelength deformations (i.e., the zero mode) can play a significant role in the persistence of defects in a developing pattern. The investigators test this hypothesis using a multidimensional extension of TDA methods. The stability of defects is probed by deriving equations for the amplitude and phase of the patterns. Methods of predicting where defects will form as a pattern evolves are developed using TDA. Finally, combining TDA with machine learning tools, the team determines parameters in models of pattern formation from experimental data. The methods are applicable to any pattern-forming system. However, two classes of systems provide focus for this work. The first is the formation of nanoscale patterns when a solid surface is bombarded by a broad ion beam. Using TDA, the investigators determine the nature of the instability that leads to the formation of a hexagonal array of nanodots when the surface of a binary material is bombarded, a subject of considerable debate. The second class is a set of reaction-diffusion systems that we call vaporchromatic experiments. In these experiments, vapor interacts with a solution containing a polymer or pigment that changes color upon interaction with the vapor. The team develops mathematical reaction-diffusion-convection models for the formation of vaporchromatic patterns. The methods of analyzing patterns using TDA are motivated by and tested on patterns produced both by experiments and by numerical simulations of partial differential equation models. Graduate and REU undergraduate students participate in the research.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.