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Research Article

Topological framework for local structure analysis in condensed matter

View ORCID ProfileEmanuel A. Lazar, Jian Han, and David J. Srolovitz
PNAS October 27, 2015 112 (43) E5769-E5776; first published October 12, 2015; https://doi.org/10.1073/pnas.1505788112
Emanuel A. Lazar
aDepartment of Materials Science and Engineering, University of Pennsylvania, Philadelphia, PA 19104;
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  • ORCID record for Emanuel A. Lazar
  • For correspondence: mLazar@seas.upenn.edu
Jian Han
aDepartment of Materials Science and Engineering, University of Pennsylvania, Philadelphia, PA 19104;
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David J. Srolovitz
aDepartment of Materials Science and Engineering, University of Pennsylvania, Philadelphia, PA 19104;
bDepartment of Mechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia, PA 19104
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  1. Edited by Pablo G. Debenedetti, Princeton University, Princeton, NJ, and approved September 10, 2015 (received for review March 25, 2015)

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Significance

Richard Feynman famously described the hypothesis “All things are made of atoms” as among the most significant of all scientific knowledge. How atoms are arranged in “things” is an interesting and natural question. However, aside from perfect crystals and ideal gases, understanding these arrangements in an insightful yet tractable manner is challenging. We introduce a unified mathematical framework for classifying and identifying local structure in imperfect condensed matter systems using Voronoi topology. This versatile approach enables visualization and analysis of a wide range of complex atomic systems, including highly defected solids and glass-forming liquids. The proposed framework presents a new perspective into the structure of discrete systems of particles, ordered and disordered alike.

Abstract

Physical systems are frequently modeled as sets of points in space, each representing the position of an atom, molecule, or mesoscale particle. As many properties of such systems depend on the underlying ordering of their constituent particles, understanding that structure is a primary objective of condensed matter research. Although perfect crystals are fully described by a set of translation and basis vectors, real-world materials are never perfect, as thermal vibrations and defects introduce significant deviation from ideal order. Meanwhile, liquids and glasses present yet more complexity. A complete understanding of structure thus remains a central, open problem. Here we propose a unified mathematical framework, based on the topology of the Voronoi cell of a particle, for classifying local structure in ordered and disordered systems that is powerful and practical. We explain the underlying reason why this topological description of local structure is better suited for structural analysis than continuous descriptions. We demonstrate the connection of this approach to the behavior of physical systems and explore how crystalline structure is compromised at elevated temperatures. We also illustrate potential applications to identifying defects in plastically deformed polycrystals at high temperatures, automating analysis of complex structures, and characterizing general disordered systems.

  • structure classification
  • Voronoi topology
  • atomic systems visualization

Footnotes

  • ↵1To whom correspondence should be addressed. Email: mLazar{at}seas.upenn.edu.
  • Author contributions: E.A.L., J.H., and D.J.S. designed research; E.A.L. and J.H. performed research; E.A.L. and J.H. contributed new reagents/analytic tools; E.A.L., J.H., and D.J.S. analyzed data; and E.A.L., J.H., and D.J.S. wrote the paper.

  • The authors declare no conflict of interest.

  • This article is a PNAS Direct Submission.

  • This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1505788112/-/DCSupplemental.

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Topological framework for local structure analysis
Emanuel A. Lazar, Jian Han, David J. Srolovitz
Proceedings of the National Academy of Sciences Oct 2015, 112 (43) E5769-E5776; DOI: 10.1073/pnas.1505788112

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Topological framework for local structure analysis
Emanuel A. Lazar, Jian Han, David J. Srolovitz
Proceedings of the National Academy of Sciences Oct 2015, 112 (43) E5769-E5776; DOI: 10.1073/pnas.1505788112
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Proceedings of the National Academy of Sciences: 112 (43)
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  • Physical Sciences
  • Applied Physical Sciences

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    • Abstract
    • Theory
    • Finite-Temperature Crystals
    • Defect Visualization
    • Comparison with Other Methods
    • Grain Boundary Characterization and Analysis
    • Disordered Structures
    • Conclusions and Discussion
    • Materials and Methods
    • Enumerating Primary and Secondary Types
    • Resolving Indeterminate Types
    • Quantitative Comparison at High Temperatures
    • Quenched Systems
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