5/18/2026 University of Michigan
Professor Qian Chen and graduate student Puquan Pen worked with the University of Michigan and the University of Southern California to develop a graph theory-based mathematical framework that can quantify and predict the structural properties of nanoparticle assemblies across the full spectrum from ordered crystals to disordered clusters — a longstanding challenge in materials science. This breakthrough, published in Science, has broad implications for engineering advanced materials.
Written by University of Michigan
Researchers from the Center for Complex Particle Systems (COMPASS) at the University of Michigan, the University of Illinois Urbana-Champaign and University of Southern California 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," involves contributions from Professor Qian Chen of the Illinois Department of Materials Science and Engineering at The Grainger College of Engineering, along with her graduate student, Puquan Pan, a co-first author. It 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.
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.
"Watching thousands of nanoparticles reorganize in real time through LPTEM gives us an unprecedented window into how these assemblies evolve," said Chen. "What's exciting about this framework is that it finally gives us the mathematical vocabulary to describe what we're seeing — moving from qualitative observation to quantitative, reproducible structural analysis."
Broad Impact
This unified framework offers a road map 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.
"What drew me to this problem is that so many of the most interesting materials — biological systems, photonic gels, sensing platforms — live in this messy intermediate space between order and disorder. Having a rigorous way to quantify that complexity opens doors for engineering materials we couldn't rationally design before," said Pan.
The study was a multidisciplinary effort first-coauthored from COMPASS students Pan, Jonas Hallstrom and Jayson Sia and led by Nicholas A. Kotov (University of Michigan), along with corresponding authors Chen, Thomas M. Truskett, Delia J. Milliron, Xiaoming Mao and Paul Bogdan.
- AFRC was found to reflect the energetic state and reconfigurability of the assembly.
- 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.
Illinois Grainger Engineering Affiliations
Qian Chen is an Illinois Grainger Engineering professor in the Department of Materials Science and Engineering and is affiliated with the Materials Research Laboratory, the Beckman Institute for Advanced Science and Technology, Carl R. Woese Institute for Genomic Biology, the Department of Chemistry, the Department of Chemical and Biomolecular Engineering and the Carle Illinois College of Medicine. She serves as an investigator for Chan Zuckerberg Biohub Chicago and holds the Racheff Faculty Scholar appointment.