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Three-dimensional atomic packing in amorphous solids with liquid-like structure

Nature Materials volume 21, pages 95–102 (2022)Cite this article

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Abstract

Liquids and solids are two fundamental states of matter. However, our understanding of their three-dimensional atomic structure is mostly based on physical models. Here we use atomic electron tomography to experimentally determine the three-dimensional atomic positions of monatomic amorphous solids, namely a Ta thin film and two Pd nanoparticles. We observe that pentagonal bipyramids are the most abundant atomic motifs in these amorphous materials. Instead of forming icosahedra, the majority of pentagonal bipyramids arrange into pentagonal bipyramid networks with medium-range order. Molecular dynamics simulations further reveal that pentagonal bipyramid networks are prevalent in monatomic metallic liquids, which rapidly grow in size and form more icosahedra during the quench from the liquid to the glass state. These results expand our understanding of the atomic structures of amorphous solids and will encourage future studies on amorphous–crystalline phase and glass transitions in non-crystalline materials with three-dimensional atomic resolution.

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Fig. 1: Determination of the 3D atomic structure of monatomic amorphous materials.
Fig. 2: Polytetrahedral packing in the amorphous Ta film and two Pd nanoparticles.
Fig. 3: Correlation of 3D local mass density heterogeneity and polytetrahedral packing.
Fig. 4: Quantitative characterization of 3D atomic packing of pentagonal bipyramids.
Fig. 5: Direct observation of PBNs in monatomic amorphous materials.
Fig. 6: PBNs in the MD-simulated Ta liquid and metallic glass.

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Data availability

The raw and processed experimental data are available at https://github.com/AET-AmorphousMaterials/Supplementary-Data-Codes. The 3D atomic coordinates of the Ta thin film and Pd1 and Pd2 nanoparticles have been deposited in the Materials Data Bank (www.materialsdatabank.org) with Materials Data Bank identification numbers TaXX00001, PdXX00001 and PdXX00002, respectively.

Code availability

The MATLAB source codes for the RESIRE reconstruction and data analysis used in this work are available at https://github.com/AET-AmorphousMaterials/Supplementary-Data-Codes.

References

  1. Frank, F. C. Supercooling of liquids. Proc. R. Soc. Lond. A 215, 43–46 (1952).

    Article  CAS  Google Scholar 

  2. Bernal, J. D. A geometrical approach to the structure of liquids. Nature 183, 141–147 (1959).

    Article  CAS  Google Scholar 

  3. Scott, G. D. Packing of spheres: packing of equal spheres. Nature 188, 908–909 (1960).

    Article  Google Scholar 

  4. Finney, J. L. Random packings and the structure of simple liquids. I. The geometry of random close packing. Proc. R. Soc. Lond. A 319, 479–493 (1970).

    Article  CAS  Google Scholar 

  5. Nelson, D. R. Order, frustration, and defects in liquids and glasses. Phys. Rev. B 28, 5515–5535 (1983).

    Article  CAS  Google Scholar 

  6. Jonsson, H. & Andersen, H. C. Icosahedral ordering in the Lennard-Jones liquid and glass. Phys. Rev. Lett. 60, 2295–2298 (1988).

    Article  CAS  Google Scholar 

  7. Chen, L. C. & Spaepen, F. Calorimetric evidence for the micro-quasicrystalline structure of ‘amorphous’ Al/transition metal alloys. Nature 336, 366–368 (1988).

    Article  CAS  Google Scholar 

  8. Reichert, H. et al. Observation of five-fold local symmetry in liquid lead. Nature 408, 839–841 (2000).

    Article  CAS  Google Scholar 

  9. Spaepen, F. Five-fold symmetry in liquids. Nature 408, 781–782 (2000).

    Article  CAS  Google Scholar 

  10. Kelton, K. F. et al. First X-ray scattering studies on electrostatically levitated metallic liquids: demonstrated influence of local icosahedral order on the nucleation barrier. Phys. Rev. Lett. 90, 195504 (2003).

    Article  CAS  Google Scholar 

  11. Schenk, T., Holland-Moritz, D., Simonet, V., Bellissent, R. & Herlach, D. M. Icosahedral short-range order in deeply undercooled metallic melts. Phys. Rev. Lett. 89, 075507 (2002).

    Article  CAS  Google Scholar 

  12. Jakse, N. & Pasturel, A. Local order of liquid and supercooled zirconium by ab initio molecular dynamics. Phys. Rev. Lett. 91, 195501 (2003).

    Article  Google Scholar 

  13. Di Cicco, A., Trapananti, A., Faggioni, S. & Filipponi, A. Is there icosahedral ordering in liquid and undercooled metals? Phys. Rev. Lett. 91, 135505 (2003).

    Article  Google Scholar 

  14. Stratton, W. G. et al. Aluminum nanoscale order in amorphous Al92Sm8 measured by fluctuation electron microscopy. Appl. Phys. Lett. 86, 141910 (2005).

    Article  Google Scholar 

  15. Anikeenko, A. V. & Medvedev, N. N. Polytetrahedral nature of the dense disordered packings of hard spheres. Phys. Rev. Lett. 98, 235504 (2007).

    Article  CAS  Google Scholar 

  16. Ma, D., Stoica, A. D. & Wang, X. L. Power-law scaling and fractal nature of medium-range order in metallic glasses. Nat. Mater. 8, 30–34 (2009).

    Article  CAS  Google Scholar 

  17. Dmowski, W., Iwashita, T., Chuang, C. P., Almer, J. & Egami, T. Elastic heterogeneity in metallic glasses. Phys. Rev. Lett. 105, 205502 (2010).

    Article  CAS  Google Scholar 

  18. Hirata, A. et al. Direct observation of local atomic order in a metallic glass. Nat. Mater. 10, 28–33 (2011).

    Article  CAS  Google Scholar 

  19. Zhong, L., Wang, J., Sheng, H., Zhang, Z. & Mao, S. X. Formation of monatomic metallic glasses through ultrafast liquid quenching. Nature 512, 177–180 (2014).

    Article  CAS  Google Scholar 

  20. Ding, J. & Ma, E. Computational modeling sheds light on structural evolution in metallic glasses and supercooled liquids. npj Comput. Mater. 3, 9 (2017).

    Article  Google Scholar 

  21. Tanaka, H., Tong, H., Shi, R. & Russo, J. Revealing key structural features hidden in liquids and glasses. Nat. Rev. Phys. 1, 333–348 (2019).

    Article  Google Scholar 

  22. Khmicha, A., Sbiaaib, K. & Hasnaouib, A. Structural behavior of tantalum monatomic metallic glass. J. Non-Cryst. Solids 510, 81–92 (2019).

    Article  Google Scholar 

  23. Gemma, R., Baben, M., Pundt, A., Kapaklis, V. & Hjörvarsson, B. The impact of nanoscale compositional variation on the properties of amorphous alloys. Sci. Rep. 10, 11410 (2020).

    Article  Google Scholar 

  24. Nelson, D. R. & Spaepen, F. Polytetrahedral order in condensed matter. Solid Stat. Phys. 42, 1–90 (1989).

    Article  CAS  Google Scholar 

  25. Sheng, H. W., Luo, W. K., Alamgir, F. M., Bai, J. M. & Ma, E. Atomic packing and short-to-medium-range order in metallic glasses. Nature 439, 419–425 (2006).

    Article  CAS  Google Scholar 

  26. Cheng, Y. Q. & Ma, E. Atomic-level structure and structure–property relationship in metallic glasses. Prog. Mater. Sci. 56, 379–473 (2011).

    Article  CAS  Google Scholar 

  27. Chen, M. W. A brief overview of bulk metallic glasses. NPG Asia Mater. 3, 82–90 (2011).

    Article  Google Scholar 

  28. Luo, W. K. et al. Icosahedral short-range order in amorphous alloys. Phys. Rev. Lett. 92, 145502 (2004).

    Article  CAS  Google Scholar 

  29. Hirata, A. et al. Geometric frustration of icosahedron in metallic glasses. Science 341, 376–379 (2013).

    Article  CAS  Google Scholar 

  30. Shechtman, D., Blech, I., Gratias, D. & Cahn, J. Metallic phase with long-range orientational order and no translational symmetry. Phys. Rev. Lett. 53, 1951–1953 (1984).

    Article  CAS  Google Scholar 

  31. Levine, D. & Steinhardt, P. J. Quasicrystals: a new class of ordered structures. Phys. Rev. Lett. 53, 2477–2480 (1984).

    Article  CAS  Google Scholar 

  32. Miao, J., Ercius, P. & Billinge, S. J. L. Atomic electron tomography: 3D structures without crystals. Science 353, aaf2157 (2016).

    Article  Google Scholar 

  33. Scott, M. C. et al. Electron tomography at 2.4-ångström resolution. Nature 483, 444–447 (2012).

    Article  CAS  Google Scholar 

  34. Chen, C. C. et al. Three-dimensional imaging of dislocations in a nanoparticle at atomic resolution. Nature 496, 74–77 (2013).

    Article  CAS  Google Scholar 

  35. Goris, B. et al. Three-dimensional elemental mapping at the atomic scale in bimetallic nanocrystals. Nano Lett. 13, 4236–4241 (2013).

    Article  CAS  Google Scholar 

  36. Xu, R. et al. Three-dimensional coordinates of individual atoms in materials revealed by electron tomography. Nat. Mater. 14, 1099–1103 (2015).

    Article  CAS  Google Scholar 

  37. Haberfehlner, G. et al. Formation of bimetallic clusters in superfluid helium nanodroplets analysed by atomic resolution electron tomography. Nat. Commun. 6, 8779 (2015).

    Article  CAS  Google Scholar 

  38. Yang, Y. et al. Deciphering chemical order/disorder and material properties at the single-atom level. Nature 542, 75–79 (2017).

    Article  CAS  Google Scholar 

  39. Zhou, J. et al. Observing crystal nucleation in four dimensions using atomic electron tomography. Nature 570, 500–503 (2019).

    Article  CAS  Google Scholar 

  40. Tian, X. et al. Correlating the three-dimensional atomic defects and electronic properties of two-dimensional transition metal dichalcogenides. Nat. Mater. 19, 867–873 (2020).

    Article  CAS  Google Scholar 

  41. Kim, B. H. et al. Critical differences in 3D atomic structure of individual ligand-protected nanocrystals in solution. Science 368, 60–67 (2020).

    Article  CAS  Google Scholar 

  42. Yang, Y. et al. Determining the three-dimensional atomic structure of an amorphous solid. Nature 592, 60–64 (2021).

    Article  CAS  Google Scholar 

  43. Egami, T. & Billinge, S. J. L. Underneath the Bragg Peaks: Structural Analysis of Complex Materials 2nd edn, Vol. 16 (Pergamon Materials Series, Pergamon-Elsevier Science, 2012).

  44. Lechner, W. & Dellago, C. Accurate determination of crystal structures based on averaged local bond order parameters. J. Chem. Phys. 129, 114707 (2008).

    Article  Google Scholar 

  45. Lee, G. W. et al. Difference in icosahedral short-range order in early and late transition metal liquids. Phys. Rev. Lett. 93, 037802 (2004).

    Article  CAS  Google Scholar 

  46. Hales, T. C. A proof of the Kepler conjecture. Ann. Math. 162, 1065–1185 (2005).

    Article  Google Scholar 

  47. Plimpton, S. Fast parallel algorithms for short-range molecular dynamics. J. Comput. Phys. 117, 1–19 (1995).

    Article  CAS  Google Scholar 

  48. Tang, C. & Harrowell, P. Anomalously slow crystal growth of the glass-forming alloy CuZr. Nat. Mater. 12, 507–511 (2013).

    Article  CAS  Google Scholar 

  49. Ohring, M. Materials Science of Thin Films 2nd edn (Academic Press, 2001).

  50. Liu, A. J. & Nagel, S. R. Jamming is not just cool any more. Nature 396, 21–22 (1998).

    Article  CAS  Google Scholar 

  51. Kelton, K. F. & Greer, A. L. Nucleation in Condensed Matter: Applications in Materials and Biology (Pergamon, 2010).

  52. Debenedetti, P. G. & Stillinger, F. H. Supercooled liquids and the glass transition. Nature 410, 259–267 (2001).

    Article  CAS  Google Scholar 

  53. Liu, Y. et al. Surfactant-induced postsynthetic modulation of Pd nanoparticle crystallinity. Nano Lett. 11, 1614–1617 (2011).

    Article  CAS  Google Scholar 

  54. Kim, S.-W. et al. Synthesis of monodisperse palladium nanoparticles. Nano Lett. 3, 1289–1291 (2003).

  55. Nag, A. et al. Metal-free inorganic ligands for colloidal nanocrystals: S2–, HS–, Se2–, HSe–, Te2–, HTe–, TeS32–, OH–, and NH2– as surface ligands. J. Am. Chem. Soc. 133, 10612–10620 (2011).

    Article  CAS  Google Scholar 

  56. Zheng, K. et al. Electron-beam-assisted superplastic shaping of nanoscale amorphous silica. Nat. Commun. 1, 24 (2010).

    Article  Google Scholar 

  57. Cao, C. R. et al. Liquid-like behaviours of metallic glassy nanoparticles at room temperature. Nat. Commun. 10, 1966 (2019).

    Article  CAS  Google Scholar 

  58. Tran, D. T., Svensson, G. & Tai, C. W. SUePDF: a program to obtain quantitative pair distribution functions from electron diffraction data. J. Appl. Crystallogr. 50, 304–312 (2017).

    Article  CAS  Google Scholar 

  59. Billinge, S. J. L. The rise of the X-ray atomic pair distribution function method: a series of fortunate events. Phil. Trans. R. Soc. A 377, 20180413 (2019).

    Article  CAS  Google Scholar 

  60. Dabov, K., Foi, A., Katkovnik, V. & Egiazarian, K. Image denoising by sparse 3-D transform-domain collaborative filtering. IEEE Trans. Image Process. 16, 2080–2095 (2007).

    Article  Google Scholar 

  61. Rogers, S. S., Waigh, T. A., Zhao, X. & Lu, J. R. Precise particle tracking against a complicated background: polynomial ftting with Gaussian weight. Phys. Biol. 4, 220–227 (2007).

    Article  CAS  Google Scholar 

  62. Lloyd, S. Least squares quantization in PCM. IEEE Trans. Inf. Theory 28, 129–137 (1982).

    Article  Google Scholar 

  63. Brünger, A. T. et al. Crystallography & NMR System: a new software suite for macromolecular structure determination. Acta Crystallogr. D 54, 905–921 (1998).

    Article  Google Scholar 

  64. Ophus, C. A fast image simulation algorithm for scanning transmission electron microscopy. Adv. Struct. Chem. Imag. 3, 13 (2017).

    Article  Google Scholar 

  65. Pryor, A., Ophus, C. & Miao, J. A streaming multi-GPU implementation of image simulation algorithms for scanning transmission electron microscopy. Adv. Struct. Chem. Imag. 3, 15 (2017).

    Article  Google Scholar 

  66. Steinhardt, P. J., Nelson, D. R. & Ronchetti, M. Bond-orientational order in liquids and glasses. Phys. Rev. B 28, 784–805 (1983).

    Article  CAS  Google Scholar 

  67. Ravelo, R., Germann, T. C., Guerrero, O., An, Q. & Holian, B. L. Shock-induced plasticity in tantalum single crystals: interatomic potentials and large-scale molecular-dynamics simulations. Phys. Rev. B 88, 134101 (2013).

    Article  Google Scholar 

  68. Purja Pun, G. P., Darling, K. A., Kecskes, L. J. & Mishin, Y. Angular-dependent interatomic potential for the Cu–Ta system and its application to structural stability of nano-crystalline alloys. Acta Mater. 100, 377–391 (2015).

    Article  CAS  Google Scholar 

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Acknowledgements

We thank J. Ciston for help with data acquisition and X. Tian for help with data analysis. This work was primarily supported by the US Department of Energy, Office of Science, Basic Energy Sciences, Division of Materials Sciences and Engineering under award no. DE-SC0010378. We also thank the support by STROBE: a National Science Foundation Science and Technology Center under award no. DMR-1548924. Some of the data analysis was partially supported by the National Science Foundation Designing Materials to Revolutionize and Engineer our Future (DMREF) programme under award no. DMR-1437263 and the Army Research Office Multidisciplinary University Research Initiative (MURI) programme under grant no. W911NF-18-1-0431. The ADF-STEM imaging with TEAM I was performed at the Molecular Foundry, which is supported by the Office of Science, Office of Basic Energy Sciences of the US Department of Energy under contract no. DE-AC02-05CH11231.

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Author notes

  1. These authors contributed equally: Yakun Yuan, Dennis S. Kim, Jihan Zhou.

Authors and Affiliations

  1. Department of Physics and Astronomy and California NanoSystems Institute, University of California, Los Angeles, Los Angeles, CA, USA

    Yakun Yuan, Dennis S. Kim, Jihan Zhou, Dillan J. Chang, Fan Zhu, Yao Yang & Jianwei Miao

  2. Department of Chemistry, Brown University, Providence, RI, USA

    Yasutaka Nagaoka & Ou Chen

  3. Department of Mathematics, University of California, Los Angeles, Los Angeles, CA, USA

    Minh Pham & Stanley J. Osher

  4. National Center for Electron Microscopy, Molecular Foundry, Lawrence Berkeley National Laboratory, Berkeley, CA, USA

    Peter Ercius & Andreas K. Schmid

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Contributions

J.M. conceived the idea and directed the project; A.K.S. synthesized the amorphous Ta thin film; Y.N. and O.C. synthesized the amorphous Pd nanoparticles; J.Z., A.K.S., P.E. and J.M. discussed and performed the AET experiments of the amorphous materials; and M.P., Y. Yuan, S.J.O. and J.M. developed the 3D reconstruction algorithm. Y. Yuan, D.S.K., D.J.C., F.Z., Y. Yang and J.M. performed image reconstruction and atom tracing, analysed the data and interpreted the results; D.S.K. and J.M. discussed and performed the MD simulations. Y. Yuan, D.S.K. and J.M. wrote the manuscript. All authors commented on the manuscript.

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Correspondence to Jianwei Miao.

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The authors declare no competing interests.

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Peer review information Nature Materials thanks Matthew Kramer and Yong Yang for their contribution to the peer review of this work.

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Supplementary information

Supplementary Information

Supplementary Table 1 and Figs. 1–22.

Supplementary Video 1

Experimental 3D atomic model of the amorphous Ta thin film. The reconstructed volume of the thin film consists of 6,615 disordered atoms (in blue) with several crystal nuclei of 1,669 atoms on the surface (in grey).

Supplementary Video 2

Experimental 3D atomic model of the amorphous Pd1 nanoparticle. The nanoparticle consists of 51,170 disordered atoms (in green) with several small crystal nuclei of 1,138 atoms on the surface (in grey).

Supplementary Video 3

Experimental 3D atomic model of the amorphous Pd2 nanoparticle. The nanoparticle consists of 74,893 disordered atoms (in green) with several small crystal nuclei of 1,345 atoms on the surface (in grey).

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Yuan, Y., Kim, D.S., Zhou, J. et al. Three-dimensional atomic packing in amorphous solids with liquid-like structure. Nat. Mater. 21, 95–102 (2022). https://doi.org/10.1038/s41563-021-01114-z

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