New insights into the softness of amorphous materials
For decades, scientists have puzzled over why glass and other amorphous materials deform more easily in some regions than in others. A collaborative study from major Japanese institutions—The University of Osaka, the National Institute of Advanced Industrial Science and Technology (AIST), Okayama University, and the University of Tokyo—has shed light on this mystery. By applying a mathematical framework known as persistent homology, the researchers reveal that soft regions are sculpted by hidden hierarchical structures where ordered and disordered atomic arrangements intertwine.
Unlike crystalline solids, which boast long-range order in their atomic layouts, amorphous materials lack this perfect repetition. Yet they are not completely random. They exhibit medium-range order (MRO), subtle patterns that span a few nanometers and influence how the material responds to stress. The link between MRO and mechanical behavior has long been suspected, but the complexity of the atomic networks has made it difficult to connect the dots—until now.
Persistent homology uncovers hierarchical rings
The team focused on amorphous silicon, a prototypical covalent glass widely used in solar cells and electronics. Using persistent homology, they mapped the structural landscape across multiple scales and found a striking feature: hierarchical ring structures. Small rings with irregular edges nest inside larger rings, creating a nested, multi-layered organization that blends order with local disorder.
This coexistence of order and disorder means that mechanical softness does not result from randomness alone. It emerges from the constraints imposed by MRO as it interweaves with disordered regions. In other words, soft spots appear where the hierarchy of rings constrains motion, guiding where atoms can rearrange with less energy cost.
Softness, vibrations, and the boson peak
The researchers also connected these hierarchical structures to vibrational properties of glasses. Low-energy localized vibrations—often referred to as the boson peak—are a universal feature of amorphous solids. The study shows a strong correlation between the identified hierarchical rings and these soft vibrational modes. In practical terms, regions governed by this topology tend to accommodate localized, low-energy vibrations, which facilitate deformation under stress.
Lead author Emi Minamitani from The University of Osaka emphasizes the broader significance: “This work provides a new route to link the atomic structure of amorphous materials with their mechanical responses. We believe these insights will accelerate the design of durable glass and other advanced amorphous materials.”
Implications for material design and applications
The central finding—that mechanically soft regions arise where disorder is embedded within medium-range order—offers a concrete design principle. Engineers can leverage this principle to tailor amorphous solids that balance flexibility with strength. Potential applications span display technologies, protective coatings, and energy devices where precise mechanical properties are crucial.
The study’s framework also invites further exploration across different amorphous systems beyond silicon, potentially guiding the creation of more robust glasses, polymers, and related materials. By translating topological features into tangible mechanical behavior, researchers are moving toward a future where atomic-scale architecture dictates macroscopic performance.
What this means for the field
In a field often challenged by the lack of long-range order, topology offers a powerful lens. The use of persistent homology to reveal hierarchical structures demonstrates that insights from mathematics can directly inform materials science. As computational tools and experimental techniques advance, the ability to predict and engineer the mechanical properties of amorphous solids will only improve.
The study, titled “Persistent homology elucidates hierarchical structures responsible for mechanical properties in covalent amorphous solids,” appears in Nature Communications (DOI: 10.1038/s41467-025-63424-z). The work stands as a landmark step toward a topology-guided design paradigm for next-generation amorphous materials.
