Graph Representations of Nanostructured Hedgehog Particles with Variable Complexity

Graph-theoretical (GT) representations, conceptually analogous to chemical formulas, offer a powerful and versatile framework for describing the structure of nanomaterialsincluding 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 nanospikesas 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.