Hierarchical structures of amorphous solids characterized by persistent homology
This article proposes a topological method that extracts hierarchical structures of various amorphous solids. The method is based on the persistence diagram (PD), a mathematical tool for capturing shapes of multiscale data. The input to the PDs is given by an atomic configuration and the output is e...
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| Published in: | Proceedings of the National Academy of Sciences - PNAS Vol. 113; no. 26; p. 7035 |
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| Main Authors: | , , , , , |
| Format: | Journal Article |
| Language: | English |
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United States
28.06.2016
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| ISSN: | 1091-6490, 1091-6490 |
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| Abstract | This article proposes a topological method that extracts hierarchical structures of various amorphous solids. The method is based on the persistence diagram (PD), a mathematical tool for capturing shapes of multiscale data. The input to the PDs is given by an atomic configuration and the output is expressed as 2D histograms. Then, specific distributions such as curves and islands in the PDs identify meaningful shape characteristics of the atomic configuration. Although the method can be applied to a wide variety of disordered systems, it is applied here to silica glass, the Lennard-Jones system, and Cu-Zr metallic glass as standard examples of continuous random network and random packing structures. In silica glass, the method classified the atomic rings as short-range and medium-range orders and unveiled hierarchical ring structures among them. These detailed geometric characterizations clarified a real space origin of the first sharp diffraction peak and also indicated that PDs contain information on elastic response. Even in the Lennard-Jones system and Cu-Zr metallic glass, the hierarchical structures in the atomic configurations were derived in a similar way using PDs, although the glass structures and properties substantially differ from silica glass. These results suggest that the PDs provide a unified method that extracts greater depth of geometric information in amorphous solids than conventional methods. |
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| AbstractList | This article proposes a topological method that extracts hierarchical structures of various amorphous solids. The method is based on the persistence diagram (PD), a mathematical tool for capturing shapes of multiscale data. The input to the PDs is given by an atomic configuration and the output is expressed as 2D histograms. Then, specific distributions such as curves and islands in the PDs identify meaningful shape characteristics of the atomic configuration. Although the method can be applied to a wide variety of disordered systems, it is applied here to silica glass, the Lennard-Jones system, and Cu-Zr metallic glass as standard examples of continuous random network and random packing structures. In silica glass, the method classified the atomic rings as short-range and medium-range orders and unveiled hierarchical ring structures among them. These detailed geometric characterizations clarified a real space origin of the first sharp diffraction peak and also indicated that PDs contain information on elastic response. Even in the Lennard-Jones system and Cu-Zr metallic glass, the hierarchical structures in the atomic configurations were derived in a similar way using PDs, although the glass structures and properties substantially differ from silica glass. These results suggest that the PDs provide a unified method that extracts greater depth of geometric information in amorphous solids than conventional methods. This article proposes a topological method that extracts hierarchical structures of various amorphous solids. The method is based on the persistence diagram (PD), a mathematical tool for capturing shapes of multiscale data. The input to the PDs is given by an atomic configuration and the output is expressed as 2D histograms. Then, specific distributions such as curves and islands in the PDs identify meaningful shape characteristics of the atomic configuration. Although the method can be applied to a wide variety of disordered systems, it is applied here to silica glass, the Lennard-Jones system, and Cu-Zr metallic glass as standard examples of continuous random network and random packing structures. In silica glass, the method classified the atomic rings as short-range and medium-range orders and unveiled hierarchical ring structures among them. These detailed geometric characterizations clarified a real space origin of the first sharp diffraction peak and also indicated that PDs contain information on elastic response. Even in the Lennard-Jones system and Cu-Zr metallic glass, the hierarchical structures in the atomic configurations were derived in a similar way using PDs, although the glass structures and properties substantially differ from silica glass. These results suggest that the PDs provide a unified method that extracts greater depth of geometric information in amorphous solids than conventional methods.This article proposes a topological method that extracts hierarchical structures of various amorphous solids. The method is based on the persistence diagram (PD), a mathematical tool for capturing shapes of multiscale data. The input to the PDs is given by an atomic configuration and the output is expressed as 2D histograms. Then, specific distributions such as curves and islands in the PDs identify meaningful shape characteristics of the atomic configuration. Although the method can be applied to a wide variety of disordered systems, it is applied here to silica glass, the Lennard-Jones system, and Cu-Zr metallic glass as standard examples of continuous random network and random packing structures. In silica glass, the method classified the atomic rings as short-range and medium-range orders and unveiled hierarchical ring structures among them. These detailed geometric characterizations clarified a real space origin of the first sharp diffraction peak and also indicated that PDs contain information on elastic response. Even in the Lennard-Jones system and Cu-Zr metallic glass, the hierarchical structures in the atomic configurations were derived in a similar way using PDs, although the glass structures and properties substantially differ from silica glass. These results suggest that the PDs provide a unified method that extracts greater depth of geometric information in amorphous solids than conventional methods. |
| Author | Hiraoka, Yasuaki Escolar, Emerson G Nishiura, Yasumasa Nakamura, Takenobu Hirata, Akihiko Matsue, Kaname |
| Author_xml | – sequence: 1 givenname: Yasuaki surname: Hiraoka fullname: Hiraoka, Yasuaki email: hiraoka@wpi-aimr.tohoku.ac.jp organization: World Premier International Research Center Initiative-Advanced Institute for Materials Research, Tohoku University, Sendai, Miyagi 980-8577, Japan; hiraoka@wpi-aimr.tohoku.ac.jp – sequence: 2 givenname: Takenobu surname: Nakamura fullname: Nakamura, Takenobu organization: World Premier International Research Center Initiative-Advanced Institute for Materials Research, Tohoku University, Sendai, Miyagi 980-8577, Japan – sequence: 3 givenname: Akihiko surname: Hirata fullname: Hirata, Akihiko organization: World Premier International Research Center Initiative-Advanced Institute for Materials Research, Tohoku University, Sendai, Miyagi 980-8577, Japan – sequence: 4 givenname: Emerson G surname: Escolar fullname: Escolar, Emerson G organization: World Premier International Research Center Initiative-Advanced Institute for Materials Research, Tohoku University, Sendai, Miyagi 980-8577, Japan – sequence: 5 givenname: Kaname surname: Matsue fullname: Matsue, Kaname organization: The Institute of Statistical Mathematics, Tachikawa, Tokyo 190-8562, Japan – sequence: 6 givenname: Yasumasa surname: Nishiura fullname: Nishiura, Yasumasa organization: World Premier International Research Center Initiative-Advanced Institute for Materials Research, Tohoku University, Sendai, Miyagi 980-8577, Japan |
| BackLink | https://www.ncbi.nlm.nih.gov/pubmed/27298351$$D View this record in MEDLINE/PubMed |
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| References | 10044969 - Phys Rev Lett. 1991 Aug 5;67(6):711-714 25990900 - Nat Mater. 2015 Jun;14(6):547-52 9985651 - Phys Rev B Condens Matter. 1996 Dec 1;54(22):15808-15827 21838366 - Phys Rev Lett. 2011 Jul 15;107(3):034503 26150288 - Nanotechnology. 2015 Jul 31;26(30):304001 10041537 - Phys Rev Lett. 1990 Apr 16;64(16):1955-1958 21482760 - Proc Natl Acad Sci U S A. 2011 Apr 26;108(17):7265-70 19805115 - Proc Natl Acad Sci U S A. 2009 Oct 6;106(40):16907-12 23845945 - Science. 2013 Jul 26;341(6144):376-9 9996322 - Phys Rev B Condens Matter. 1991 Jan 1;43(1):1194-1197 |
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