Graph curvature measures quantify disorder and complexity for self-assembled particle systems.
NP assemblies can be viewed as a series of graphs or networks in which nodes represent NPs and edges represent strong interparticle interactions. ORC and AFRC can be found for every edge in these graphs, quantifying the structural order and complexity in the local area around those edges. Scale bars, 200 nm.
Researchers from the Center for Complex Particle Systems (COMPASS) at the University of Michigan, UIUC and USC, have published a breakthrough study in Science that provides a new mathematical framework for understanding and optimizing complex nanoparticle assemblies.
The study, titled “Decoding collective dynamics and complexity in nanoparticle assemblies using graph theory,” addresses a long-standing challenge in materials science: how to quantify and predict the properties of “messy” or partially disordered structures that fall between perfectly ordered crystals and completely random clusters.
A New Language for Nano-Materials
While traditional symmetry-based descriptors work well for highly ordered crystals, they fail to capture the nuances of nanoparticle (NP) gels and other low-density states. By applying Graph Theory (GT) the COMPASS team developed a method to track the interactions and structural transitions of thousands of nanoparticles in real-time.
“One cannot design, optimize, or reproduce materials with structures that one cannot quantify,” the researchers noted. Their new GT framework allows for the continuous assessment of structural organization from fully ordered to fully disordered states.
Key Discoveries: The Curvature of Complexity
The team utilized advanced metrics, specifically Forman-Ricci curvature (AFRC) and Ollivier-Ricci curvature (ORC), to analyze the “bending” and connectivity of the nanoparticle networks:
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AFRC was found to reflect the energetic state and reconfigurability of the assembly.
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ORC quantified structural complexity and revealed a “Goldilocks” regime. This specific intermediate state, composed of interconnected mesocrystals with low symmetry, maximizes the material’s plasmonic response, which is critical for the development of highly sensitive biosensors like those used in COVID and pregnancy tests.
Experimental and Computational Synergy
The researchers validated their findings across three distinct material systems, gold nanocubes, gold nanoprisms, and indium tin oxide nanospheres. They combined wide-frame time-resolved liquid-phase transmission electron microscopy (LPTEM) to capture real-time particle dynamics with large-scale molecular dynamics simulations to confirm the universality of their graph-based metrics.
Broad Impact
This unified framework offers a roadmap for engineering high-entropy materials with “correlated disorder”. Beyond optics, the approach is generalizable to a variety of other systems, including molecular solids, slurries, and dispersions, potentially impacting fields from electronics to drug delivery.
The study was a multidisciplinary effort, first co-authored by COMPASS students Jonas Hallstrom, Puquan Pan, and Jayson Sia and led by Nicholas A. Kotov (University of Michigan), along with corresponding authors Thomas M. Truskett, Delia J. Milliron, Qian Chen, Xiaoming Mao, and Paul Bogdan.