For decades, the origins of “soft spots” in amorphous materials like glass remained a mystery. A new study shows that hierarchical atomic structures, uncovered through mathematical topology, are responsible for their mechanical behavior.

Osaka, Japan — Why do glass and other amorphous materials deform more easily in some regions than in others? A research team from The University of Osaka, the National Institute of Advanced Industrial Science and Technology (AIST), Okayama University, and the University of Tokyo has uncovered the answer. By applying a mathematical method known as persistent homology, the team demonstrated that these soft regions are governed by hidden hierarchical structures, where ordered and disordered atomic arrangements coexist.

Crystalline solids, such as salt or ice, have atoms neatly arranged in repeating patterns. Amorphous materials, including glass, rubber, and certain plastics, lack this long-range order. However, they are not completely random: they possess medium-range order (MRO), subtle atomic patterns that extend over a few nanometers.

MRO has long been suspected to play a critical role in determining the physical properties of amorphous materials, particularly their mechanical responses. Yet, because of the complexity of atomic networks, conventional analysis methods have been unable to clarify how MRO relates to regions that deform more easily than their surroundings. The structural origins of mechanical softness in amorphous solids have therefore remained elusive.

The research team applied persistent homology, a branch of topological data analysis that captures structural features across multiple scales. In amorphous silicon—a prototypical covalent amorphous material widely used in solar cells and electronic devices—they discovered hierarchical ring structures: smaller rings with irregular edge lengths are nested inside larger rings.

This coexistence of order and disorder means that softness emerges not from randomness alone, but from constraints imposed by medium-range order interwoven with local disorder. The study also revealed that these hierarchical structures strongly correlate with low-energy localized vibrations, a universal feature of glasses known as the “boson peak.”

“This work provides a new route to link the atomic structure of amorphous materials with their mechanical responses,” says Emi Minamitani of The University of Osaka, the lead author of the study. “We believe these insights will accelerate the design of durable glass and other advanced amorphous materials.”

The discovery establishes a clear structural principle: mechanically soft regions arise where disorder is embedded within medium-range order. This counterintuitive finding provides a practical guideline for developing amorphous solids that are both flexible and strong—benefiting applications from displays and coatings to energy devices.

###

The article “Persistent homology elucidates hierarchical structures responsible for mechanical properties in covalent amorphous solids” will be published in Nature Communications at DOI: https://doi.org/10.1038/s41467-025-63424-z.