LEAPS-MPS: Network Statistics of Rupturing Foams

当气泡被困在液体或固体材料中时，泡沫就形成了。几个世纪以来，泡沫中发现的结构和对称性一直吸引着纯粹的数学家的注意，但泡沫的物理性质，无论是令人满意的还是不希望的，在工业和制造业中也是高度相关的。这个项目将通过在两个透明平板之间压缩肥皂泡沫并加热来诱导边缘断裂，在受控环境下研究二维泡沫的统计特性。然后通过图像处理算法捕获网络动态。还将严格分析用于模拟破裂的基本微分方程式，即描述粘性颗粒聚集的所谓凝聚方程的变体。由于泡沫塑料的消除在洗涤剂生产等多个工业过程中至关重要，该项目旨在产生与泡沫塑料减少量直接相关的数据，以阐明如何优化这一过程。该项目包括本科生研究培训机会，以及面向当地高中的数据科学推广活动。对材料微观结构或小尺度成分的研究，对于在宏观层面理解材料属性至关重要。虽然已经有很多关于气体通过胞壁扩散导致微结构粗化的文章，但本项目将重点放在加热导致边缘破裂的二维泡沫的实验生成和理论模拟上。使用形态学中常见的细胞跟踪算法，该项目将检查不断演变的网络统计数据，如拓扑频率、粗化率以及大量细胞的创建和生长。由于破裂导致的细胞合并的二级反应可以通过Smoluchowski凝血方程等过程进行建模，因此本项目将考虑受这种合并启发的一类此类方程，并将重点放在严格推导得到的星团统计数据的性质上。长期目标是使用本研究中开发的实验、计算和理论方法来构建粗化过程的一般理论。该奖项反映了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。

Foams form when pockets of air are trapped inside a liquid or solid material. The structures and symmetries found in foams have drawn the attention of pure mathematicians for centuries, but physical properties of foams, both desirable and undesirable, are also highly relevant in industry and manufacturing. This project will study statistical properties of two-dimensional foams in a controlled setting through compressing a soap foam between two transparent plates and heating to induce edge breaking. The network dynamics will be then captured through image processing algorithms. The underlying differential equations for modeling ruptures, a variation of the so-called coagulation equations describing sticky particle aggregation, will also be rigorously analyzed. Because the elimination of foam is vital in multiple industrial processes such as in the production of detergents, the project aims to generate data directly related to rates of foam reduction to shed light on how to optimize this process. The project includes undergraduate research training opportunities and outreach activities to local high schools focused on data science.The study of microstructure, or composition of materials at small scales, is critical for understanding material properties at the macroscopic level. While much has been written about the coarsening of microstructure via gas diffusion across cell walls, this project will focus on the experimental generation and theoretical modeling of two-dimensional foams with edge rupture induced by heating. Using cell-tracking algorithms commonly found in morphology, the project will examine the evolving network statistics such as topological frequencies, coarsening rates, and the creation and growth of massive cells. Because the second-order reaction of cell merging resulting from rupture can be modeled through processes like the Smoluchowski coagulation equation, this project will consider a class of these equations inspired by this merging and will focus on rigorously deriving properties of the resulting cluster statistics. The long-term goal being to use the experimental, computational, and theoretical methods developed in this study for constructing a general theory of coarsening processes.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.