The formation of glass states involves a rapid increase in viscosity upon cooling from the liquid state. The process is highly dependent on the cooling rate, with slower cooling rates resulting in more stable glass states. However, distinguishing structural differences between liquid and glass, or between glasses formed at different cooling rates, based on their static structures is challenging due to subtle differences and disordered atomic arrangements. This necessitates the development of novel analytical methods to reveal hidden structural features. Molecular dynamics (MD) simulations have been extensively used to study the cooling rate dependency, employing techniques such as Voronoi polyhedral and bond-orientational order analyses. These methods, however, often fail to capture essential structural features beyond short ranges or are sensitive to the specific glass system being studied. Computational persistent homology, a method for analyzing topological features of 3D objects, offers a powerful alternative. It can visualize hierarchical features via persistence diagrams by identifying holes created and annihilated as atomic size changes. Combining persistent homology with machine learning provides a systematic approach for identifying important geometrical patterns from the complex data.

Previous research on the structure and dynamics of glass formation often relied on molecular dynamics (MD) simulations and analyses like Voronoi polyhedral analysis and bond-orientational order analysis. While these approaches provided some insight into local structural variations with cooling rates, they struggled to elucidate the essential structural components beyond short-range order, and their findings were often system-specific. The introduction of computational persistent homology provided a new avenue to explore the topological features of glass structures, identifying holes and their evolution with changes in atomic size, but the interpretation of these diagrams alone remained challenging. The current study builds upon this foundation, adding machine learning techniques to systematically extract relevant structural information from the complex topological data.

This research uses molecular dynamics (MD) simulations to create structural models of Pd80Si20 metallic glass at various cooling rates (1.7 × 10¹⁰ to 1.7 × 10¹⁴ K s⁻¹). Two sets of models were generated: one for linear regression analysis (LRA) with one model per cooling rate and another for principal component analysis (PCA) with 30 models per five selected cooling rates. Each model contains 12,000 atoms. The structure factors were validated against experimental diffraction data. Persistent homology analysis was performed on these models to compute persistence diagrams (PDs), with PD2 representing closed holes (voids) for Pd atoms and PD1 representing open holes (rings) for Si atoms. These diagrams capture the hierarchical arrangement of atoms. To analyze the cooling-rate dependence, LRA was first applied to the PDs from the 37 MD configurations, transforming the diagrams into vectors and modeling the relationship between birth-death pair counts and cooling rates. PCA was then employed on 30 MD models for each of the five cooling rates to further reduce dimensionality and extract the most significant structural variations linked to cooling rate. Finally, inverse analysis is conducted to reconstruct the local atomic structures corresponding to specific regions in the reconstructed persistence diagrams that show the strongest cooling rate dependence. This process involved identifying birth-death pairs in the characteristic regions of the persistence diagrams and reconstructing the associated local atomic structures from the original MD data.

Linear regression analysis (LRA) and principal component analysis (PCA) were used to determine the cooling rate dependence of the features observed in the persistence diagrams. The analyses revealed that significant structural changes are associated with cooling rates. The reconstructed persistence diagrams, generated using the vectors learned by both LRA and PCA, highlight regions (in blue and red) corresponding to slower and faster cooling rates, respectively. Inverse analysis based on PCA, which provides better statistical significance, revealed the atomic structures responsible for these cooling-rate-dependent changes. In slower-cooled samples, the key structures identified were ordered arrangements of 3-4 prism-type clusters (superclusters), creating smaller holes in the Si network. Faster cooling yielded superclusters comprised of 5-9 prism-type clusters, resulting in larger holes. The size distribution of these prism-type clusters shows that slower cooling leads to less distorted, more compact clusters. Local structure factor calculations confirm that the superclusters found in slowly cooled samples are more aligned with the experimentally observed second peak splitting in the structure factor, a characteristic feature of glassy structures. Conventional methods like pair distribution function (PDF) and Voronoi polyhedral analyses showed limited ability to detect these extended-range structural variations compared to the persistence diagram analyses.

This study demonstrates that the combined approach of persistent homology and machine learning effectively identifies extended-range structural changes during glass formation, changes not easily discernible using traditional techniques. The findings suggest that the formation of superclusters, particularly the smaller, more compact ones, significantly contributes to the development of the characteristic glassy structural order observed experimentally. The size and organization of these superclusters are strongly influenced by the cooling rate. The results support the hypothesis that glass formation is not solely determined by changes in individual atomic clusters but rather by the evolution of their spatial arrangement and connectivity over extended ranges. Future research could explore the existence of an ideal glass structure, potentially characterized by a specific arrangement of less-distorted prism-type clusters without long-range order, as suggested by the analysis.

This research successfully combined computational persistent homology with machine learning techniques (LRA and PCA) to analyze the cooling rate dependence of structural changes in Pd-Si metallic glass. The study identified a crucial role of extended-range structures, particularly superclusters of prism-type atomic clusters, in glass formation. The size and arrangement of these superclusters vary drastically with cooling rate, influencing the final glassy structure. This approach provides a powerful new tool for studying glass formation in various systems and could lead to a deeper understanding of the glass transition.

The study focuses on a specific metallic glass system (Pd80Si20). While the methods are potentially applicable to other glass systems, further research is needed to confirm the generalizability of the findings. The cooling rates achievable by molecular dynamics simulations are still limited compared to the extremely rapid cooling rates used in experimental glass formation. Future studies should explore methods to overcome this limitation to better represent real-world conditions.