A recent study by researchers from the University of Osaka and other institutions used a special mathematical tool called persistent homology to find out what’s happening at the atomic level.
Persistent Homology: How Math Reveals Hidden Atomic Structures
Scientists have long wondered why glass and other amorphous materials don’t deform evenly. Some areas seem softer and easier to bend, while others stay hard. A recent study by researchers from the University of Osaka and other top Japanese institutes has found an exciting answer using a smart math tool called persistent homology. This discovery helps us understand the hidden structures inside these materials that control their softness.
The Mystery of Disordered Structures
Amorphous materials such as amorphous silicon (a-Si) challenge scientists because their atomic structures lack the ordered patterns found in crystalline materials. Unlike crystals, which respond to forces through well-defined defects called dislocations, amorphous materials exhibit complex mechanical responses due to their disordered and irregular atomic arrangements. This makes it difficult to predict how these materials behave under stress.
Researchers have used various models to explain these behaviors. But many have struggled to connect microscopic atomic arrangements directly with mechanical properties like elasticity and hardness. A key difficulty is related to how atoms move unevenly when the material is subjected to strain. Some move predictably (affine displacement), while others shift unpredictably (nonaffine displacement). These different movements affect the material’s stiffness and strength.
Medium-range order (MRO) describes how atoms in amorphous materials form network structures beyond immediate neighbors, usually at distances from 5 to 20 Angstroms. In covalent glasses like a-Si, MRO involves networks of polyhedral units linked together by shared corners or edges. These larger structures influence how regions in the material respond mechanically.
Persistent homology, a mathematical tool that tracks how rings and voids form and disappear as spheres around atoms grow, helps scientists analyze these networks on multiple scales simultaneously. This approach reveals hierarchical structural patterns within the material, linking them with properties such as softness and vibration behaviors.
The new study used mathematical topology techniques, specifically persistent homology, to map out the atomic rings inside amorphous silicon—a type widely used in solar cells and electronics. The results showed that irregular small rings are nested inside larger rings, creating hierarchical ring structures. This combination leads to certain areas becoming softer than others.
Soft Regions Explained by Atomic Rings
The study finds that areas surrounding atoms with small Born terms, which relate to basic elastic responses, consist mainly of smaller rings without internal child rings. These show disorder in bond lengths and angles—signs of disruptions at short scales (short-range order).
Larger Rings Influence Nonaffine Displacements
By contrast, regions exhibiting large nonaffine atomic displacements contain bigger rings that include child cycles forming complex hierarchical structures reflecting multi-scale MRO. Such areas are softer because they allow atoms more freedom to move irregularly under strain.
MRO Links With Low-Energy Vibrations
The research shows that these soft regions also display low-energy localized vibrations. This suggests that constraints imposed by MRO govern not only static structure but also dynamic mechanical behavior—crucial for designing better amorphous materials with desirable properties.
A New Link Between Atomic Structure and Mechanical Behavior
The researchers found that these hierarchical structures relate closely to what’s called the “boson peak.” This feature describes low-energy vibrations common to glasses worldwide. These vibrations impact how glasses absorb energy when bent or hit. Knowing this link opens new work paths on material innovation.
The Power of Persistent Homology
Persistent homology is a branch of mathematics that helps spot multiple structural features at different scales by measuring changes continuously. Applying this method allowed scientists to connect tiny atomic arrangements with larger soft regions directly—something traditional methods couldn’t achieve.
Reference
- Minamitani, E., Nakamura, T., Obayashi, I., & Mizuno, H. (2025). Persistent homology elucidates hierarchical structures responsible for mechanical properties in covalent amorphous solids. Nature Communications, 16(1). https://doi.org/10.1038/s41467-025-63424-z
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Neha Adikane holds a Master’s degree in Environmental Science from Pune University, bringing a unique blend of scientific expertise and creative writing skills to her craft. She specializes in creating engaging, insightful, and impactful content across various niches. With a passion for translating complex ideas into simple, relatable narratives, she excels at crafting blogs, articles, website content, and social media posts that captivate readers and drive engagement. Neha’s background in environmental science adds depth to her writing, allowing her to effectively cover topics related to sustainability, eco-awareness, and scientific innovations.
