Scientists have cracked a problem that's stumped engineers for decades: how to measure complexity in a way that actually predicts what a material can do. Researchers from the University of Michigan Engineering, USC Viterbi School of Engineering, and the University of Illinois Urbana-Champaign have developed the first quantitative complexity metric for nanomaterials, published in Science, that transforms nanomaterials engineering from a matter of luck to one of precise design.
The breakthrough matters because it opens a new frontier in material science. For years, engineers have had only two options: randomly scattered nanoparticles or perfectly ordered crystals with uniform patterns. Both have limitations. But the real magic, it turns out, lives somewhere in between—in structures that blend order and randomness in just the right proportions to unlock properties that don't exist in nature or in any material made so far.
Nick Kotov, the Irving Langmuir Distinguished University Professor of Chemical Sciences and Engineering at Michigan and principal investigator of the Center for Complex Particle Systems, describes the insight simply: "It's like a structure that has clusters and some bridges that connect these clusters throughout the system, and these interconnected communities of particles give you something new." Those bridges connecting communities are key. They're what creates the sweet spot where complexity becomes a tool rather than an obstacle.
The team proved the concept works by assembling gold nanoparticles into loose networks of crystal clusters that strongly reflect infrared light. Gold nanoparticle suspensions can't do this. Gold nanoparticle crystals do it poorly. But the hybrid—mixing clusters with connecting bridges—achieves what neither can alone. It's the combinatorial effect of order and disorder working in concert that produces the novel property.
The idea that complexity relates to a material's capabilities traces back to physics Nobel laureate Murray Gell-Mann, who theorized that truly complex structures contain both order and randomness, while simple ones have only one or the other. But Gell-Mann never developed the mathematical tools to actually measure it. He left behind a rough conceptual curve linking complexity to functionality, but no numbers, no equations, no way for engineers to design toward it.
This team changed that. By mapping the interactions between nanoparticles as a network graph—using gray lines for ordered crystalline patterns, red lines for disordered structures, and blue lines for bridges between clusters—they created a system to calculate complexity precisely. More importantly, they showed that graph-based measures strongly correlate with real material properties, turning an abstract principle into an engineering tool.
"It allows engineers to harness complexity as a powerful design parameter, opening pathways to capabilities that cannot be achieved with simple materials," said Xiaoming Mao, a U-M professor of physics and mechanical engineering and co-corresponding author of the study. The research provides a framework other scientists can use to quantify order and disorder and predict what properties will emerge.
The implications reach far beyond gold nanoparticles. Bones, for example, demonstrate nature's mastery of this principle—curved nanocrystals merge into twisted plates, which combine into larger structures woven through collagen filaments. That cascading complexity from nanoscale to macroscale makes bone both hard and flexible, a combination that's been notoriously difficult to replicate synthetically. With a way to measure and design complexity, engineers can now approach materials like nature does.