SCIENCE CHINA Physics, Mechanics & Astronomy, Volume 63 , Issue 7 : 276111(2020) https://doi.org/10.1007/s11433-020-1539-4

Revisiting the breakdown of Stokes-Einstein relation in glass-forming liquids with machine learning

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  • ReceivedJan 16, 2020
  • AcceptedMar 5, 2020
  • PublishedMar 25, 2020
PACS numbers


The Stokes-Einstein (SE) relation has been considered as one of the hallmarks of dynamics in liquids. It describes that the diffusion constant D is proportional to (τ/T)–1, where τ is the structural relaxation time and T is the temperature. In many glass-forming liquids, the breakdown of SE relation often occurred when the dynamics of the liquids becomes glassy, and its origin is still debated among many scientists. Using molecular dynamics simulations and support-vector machine method, it is found that the scaling between diffusion and relaxation fails when the total population of solid-like clusters shrinks at the maximal rate with decreasing temperature, which implies a dramatic unification of clusters into an extensive dominant one occurs at the time of breakdown of the SE relation. Our data leads to an interpretation that the SE violation in metallic glass-forming liquids can be attributed to a specific change in the atomic structures.

Funded by

the National Natural Science Foundation of China(Grant,Nos.,11804027,11525520)

the National Basic Research Program of China 973 Program(Grant,No.,2015CB856801)

and the Fundamental Research Funds for the Central Universities(Grant,No.,2018NTST24)


We thank all members of the Beijing Metallic Glass Club for the long-term useful discussions. This work was supported by the National Natural Science Foundation of China (Grant Nos. 11804027, and 11525520), the National Basic Research Program of China (Grant No. 2015CB856801), and the Fundamental Research Funds for the Central Universities (Grant No. 2018NTST24).

Interest statement

These authors contributed equally to this work.


Supporting Information

The supporting information is available online at phys.scichina.com and link.springer.com. The supporting materials are published as submitted, without typesetting or editing. The responsibility for scientific accuracy and content remains entirely with the authors.


[1] Debenedetti P. G., Stillinger F. H.. Nature, 2001, 410: 259 CrossRef PubMed ADS Google Scholar

[2] Wang W. H.. Prog. Mater. Sci., 2019, 106: 100561 CrossRef Google Scholar

[3] Qiao J. C., Wang Q., Pelletier J. M., Kato H., Casalini R., Crespo D., Pineda E., Yao Y., Yang Y.. Prog. Mater. Sci., 2019, 104: 250 CrossRef Google Scholar

[4] Wu Z. W., Kob W., Wang W. H., Xu L.. Nat. Commun., 2018, 9: 5334 CrossRef PubMed ADS arXiv Google Scholar

[5] Luo P., Li Y. Z., Bai H. Y., Wen P., Wang W. H.. Phys. Rev. Lett., 2016, 116: 175901 CrossRef PubMed ADS Google Scholar

[6] Scopigno T., Ruocco G., Sette F., Monaco G.. Science, 2003, 302: 849 CrossRef PubMed ADS arXiv Google Scholar

[7] Wang L., Ninarello A., Guan P., Berthier L., Szamel G., Flenner E.. Nat. Commun., 2019, 10: 26 CrossRef PubMed ADS arXiv Google Scholar

[8] Kawasaki T., Kim K.. Sci. Adv., 2017, 3: e1700399 CrossRef PubMed ADS arXiv Google Scholar

[9] Hu Y. C., Li F. X., Li M. Z., Bai H. Y., Wang W. H.. J. Appl. Phys., 2016, 119: 205108 CrossRef ADS Google Scholar

[10] Soklaski R., Tran V., Nussinov Z., Kelton K. F., Yang L.. Philos. Mag., 2016, 96: 1212 CrossRef ADS arXiv Google Scholar

[11] Xu L., Mallamace F., Yan Z., Starr F. W., Buldyrev S. V., Eugene Stanley H.. Nat. Phys., 2009, 5: 565 CrossRef ADS Google Scholar

[12] Sastry S., Austen Angell C.. Nat. Mater., 2003, 2: 739 CrossRef PubMed ADS Google Scholar

[13] Stillinger F. H., Hodgdon J. A.. Phys. Rev. E, 1994, 50: 2064 CrossRef PubMed ADS Google Scholar

[14] Tarjus G., Kivelson D.. J. Chem. Phys., 1995, 103: 3071 CrossRef ADS Google Scholar

[15] Becker S. R., Poole P. H., Starr F. W.. Phys. Rev. Lett., 2006, 97: 055901 CrossRef ADS arXiv Google Scholar

[16] Pan S., Wu Z. W., Wang W. H., Li M. Z., Xu L.. Sci. Rep., 2017, 7: 39938 CrossRef PubMed ADS Google Scholar

[17] Schoenholz S. S., Cubuk E. D., Sussman D. M., Kaxiras E., Liu A. J.. Nat. Phys., 2016, 12: 469 CrossRef ADS Google Scholar

[18] Sun Y. T., Bai H. Y., Li M. Z., Wang W. H.. J. Phys. Chem. Lett., 2017, 8: 3434 CrossRef PubMed Google Scholar

[19] Plimpton S.. J. Comput. Phys., 1995, 117: 1 CrossRef ADS Google Scholar

[20] Mendelev M. I., Sordelet D. J., Kramer M. J.. J. Appl. Phys., 2007, 102: 043501 CrossRef ADS Google Scholar

[21] Kob W., Andersen H. C.. Phys. Rev. E, 1995, 52: 4134 CrossRef PubMed ADS arXiv Google Scholar

  • Figure 1

    (Color online) (a) Mean-squared displacements and (b) self-intermediate scattering functions at temperatures ranged from 1500 to 1000 K in steps of 100 K. (c) Self-diffusion coefficient D as a function of structural relaxation time τ scaled by temperature T. The system follows the SE relation till to 1150 K and violates the SE relation below 1150 K.

  • Figure 2

    Total number of clusters formed by connected solid-like atoms and its derivative with respect to temperature. (a) The population has a maximum in number occurring at 1330 K; (b) the main peak of the cluster number derivative at 1140 K, which roughly coincides with the temperature where the SE relation breaks down.

  • Figure 3

    Comparison of the total size of the solid-like atoms in the system with respect to the largest and second largest formed clusters. The size of the largest cluster almost coincides with the total number of solid-like atoms below 1140 K, and the size of the second largest cluster (inset) peaks at 1330 K.

  • Figure 4

    Growth of clusters comprising solid-like atoms in the system. Typical simulation snapshots depicting the growth of clusters at (a) 1500, (b) 1330, (c) 1140, and (d) 860 K (Tg). The percentages show the proportion of solid-like atoms at different temperatures. Different color spheres represent different disconnected clusters.

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