Montjoy Douglas G, Hou Harrison, Vecchio Drew A, Wilson Elizabeth A K, Bahng Joong Hwan, Eskafi Aydin, Kotov Nicholas A
Department of Chemical Engineering, University of Michigan, Ann Arbor, Michigan 48109, United States.
PPG Industries, Pittsburgh, Pennsylvania 15272, United States.
Chem Mater. 2025 Jul 7;37(16):6161-6172. doi: 10.1021/acs.chemmater.5c00651. eCollection 2025 Aug 26.
Graph-theoretical (GT) representations, conceptually analogous to chemical formulas, offer a powerful and versatile framework for describing the structure of nanomaterialsincluding complex assemblies with nano-, meso-, and microscale organization. GT formulas of nanostructures can capture repetitive structural patterns that combine both order and disorder needed to attain the desired combination of properties. These repetitive structural patterns are extracted from microscopy, spectroscopy, and diffractometry. However, methods for constructing GT models of complex particles with diverse geometries and architectures remain underdeveloped, as do approaches linking graph features to material properties. In this work, we address these gaps using hedgehog particles (HPs)nanostructured colloids characterized by halos of rigid nanospikesas model complex particles. HPs were synthesized with a spectrum of solid/hollow cores and spikes and multiple materials, which enabled systematic variation of their structural patterns. We detail the process of building the GT models, accounting for multiple structural elements, chemical phases, and interfaces between them. Consistent GT elements (subgraphs) are assigned to one-, two-, and three-dimensional structural elements of HPs identified from electron microscopy images. Solid and hollow cores carrying solid and hollow spikes find unique representations in GT 'formulas' of the nanostructures. Analysis of complexity metrics for HPs indicated that the key aspects of structural patterns contributing to complexity are (1) dimensionality of the building blocks, (2) levels of hierarchy, and (3) variety of structural components. All studied HPs consistently display enhanced dispersibility and strong Mie scattering. These findings point to the relations between the graph models and chemical properties, whose expected limitations are also discussed. The GT descriptions can be utilized to engineer hierarchical particles toward hard-to-reach functionalities.
图论（GT）表示法在概念上类似于化学分子式，为描述纳米材料的结构提供了一个强大且通用的框架，其中包括具有纳米、介观和微观尺度组织的复杂聚集体。纳米结构的GT分子式能够捕捉重复的结构模式，这些模式结合了获得所需性能组合所需的有序和无序。这些重复的结构模式是从显微镜、光谱学和衍射测量中提取的。然而，构建具有不同几何形状和结构的复杂粒子的GT模型的方法仍然不够完善，将图形特征与材料属性联系起来的方法也是如此。在这项工作中，我们使用刺猬粒子（HPs）——以刚性纳米尖刺晕为特征的纳米结构胶体——作为模型复杂粒子来填补这些空白。合成了具有一系列实心/空心核和尖刺以及多种材料的HPs，这使得它们的结构模式能够系统地变化。我们详细介绍了构建GT模型的过程，考虑了多个结构元素、化学相以及它们之间的界面。将一致的GT元素（子图）分配给从电子显微镜图像中识别出的HPs的一维、二维和三维结构元素。带有实心和空心尖刺的实心和空心核在纳米结构的GT“分子式”中有独特的表示。对HPs的复杂性指标分析表明，导致复杂性的结构模式的关键方面是：（1）构建块的维度，（2）层次级别，以及（3）结构组件的多样性。所有研究的HPs都始终表现出增强的分散性和强烈的米氏散射。这些发现指出了图形模型与化学性质之间的关系，同时也讨论了其预期的局限性。GT描述可用于设计具有难以实现的功能的分级粒子。
