Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Article
  • Published:

Visualizing slow internal relaxations in a two-dimensional glassy system

Nature Physics volume 19, pages 969–977 (2023)Cite this article

Subjects

Abstract

Unlike a crystal, a glassy solid state displays slow internal relaxation processes besides vibrational modes even when structural (α) relaxation is frozen. The precise nature of such residual local relaxation modes remains poorly understood due to the lack of real-space information. Here we directly visualize the internal relaxations in a glass via a long-time observation of the dynamics of a mechanically driven two-dimensional granular system that shows a pinning-induced transition. This allows directly visualizing the internal relaxations in a glass. On approaching the glass transition, vanishing cage-breaking motion is observed, accompanied by the emergence of a restricted dynamic mode characterizing slow β relaxation. The emergence of bond rigidity freezes the structure relaxation and leads to slow β motion. Our findings indicate that unlike crystallization, where all the bonds are constrained, vitrification in a disordered system freezes α relaxation and accompanies elastic percolation, but the remaining non-constrained bonds provide room for slow β relaxation.

This is a preview of subscription content, access via your institution

Access options

Buy this article

  • Purchase on Springer Link
  • Instant access to full article PDF
Buy now

Prices may be subject to local taxes which are calculated during checkout

Fig. 1: Experimental details and reduction in internal structure relaxation across a pinning-induced glass transition.
Fig. 2: Pinning-fraction dependence of the internal relaxation dynamics profiles.
Fig. 3: Four-particle tetragonal representation of the relaxation mode.
Fig. 4: Double signature of glass transition seen in internal relaxations.

Similar content being viewed by others

Data availability

The data that support the findings of this study are available from the corresponding authors upon reasonable request.

Code availability

The codes used in this study are available from the corresponding authors upon reasonable request.

References

  1. Stevenson, J. D. & Wolynes, P. G. A universal origin for secondary relaxations in supercooled liquids and structural glasses. Nat. Phys. 6, 62–68 (2010).

    Google Scholar 

  2. Zhang, Z., Ispas, S. & Kob, W. Roughness and scaling properties of oxide glass surfaces at the nanoscale. Phys. Rev. Lett. 126, 066101 (2021).

    ADS  Google Scholar 

  3. Biroli, G. & Urbani, P. Breakdown of elasticity in amorphous solids. Nat. Phys. 12, 1130–1133 (2016).

    Google Scholar 

  4. Ozawa, M., Ikeda, A., Miyazaki, K. & Kob, W. Ideal glass states are not purely vibrational: insight from randomly pinned glasses. Phys. Rev. Lett. 121, 205501 (2018).

    ADS  Google Scholar 

  5. Yeh, W.-T., Ozawa, M., Miyazaki, K., Kawasaki, T. & Berthier, L. Glass stability changes the nature of yielding under oscillatory shear. Phys. Rev. Lett. 124, 225502 (2020).

    ADS  Google Scholar 

  6. Scalliet, C., Berthier, L. & Zamponi, F. Nature of excitations and defects in structural glasses. Nat. Commun. 10, 5102 (2019).

    ADS  Google Scholar 

  7. Saito, M. et al. Slow processes in supercooled o-terphenyl: relaxation and decoupling. Phys. Rev. Lett. 109, 115705 (2012).

    ADS  Google Scholar 

  8. Ngai, K. L. & Paluch, M. Classification of secondary relaxation in glass-formers based on dynamic properties. J. Chem. Phys. 120, 857–873 (2004).

    ADS  Google Scholar 

  9. Boyer, R. F. Dependence of mechanical properties on molecular motion in polymers. Polym. Eng. Sci. 8, 161–185 (1968).

    Google Scholar 

  10. Johari, G. P. Glass transition and secondary relaxations in molecular liquids and crystals. Ann. N. Y. Acad. Sci. 279, 117–140 (1976).

    ADS  Google Scholar 

  11. Tanaka, H. Origin of the excess wing and slow β relaxation of glass formers: a unified picture of local orientational fluctuations. Phys. Rev. E 69, 021502 (2004).

    ADS  Google Scholar 

  12. Zuriaga, M. et al. New microscopic mechanism for secondary relaxation in glasses. Phys. Rev. Lett. 103, 075701 (2009).

    ADS  Google Scholar 

  13. Zhao, Z. F., Wen, P., Shek, C. H. & Wang, W. H. Measurements of slow β-relaxations in metallic glasses and supercooled liquids. Phys. Rev. B. 75, 174201 (2007).

    ADS  Google Scholar 

  14. Wang, Q. et al. Unusual fast secondary relaxation in metallic glass. Nat. Commun. 6, 7876 (2015).

    ADS  Google Scholar 

  15. Yu, H.-B., Richert, R. & Samwer, K. Structural rearrangements governing Johari-Goldstein relaxations in metallic glasses. Sci. Adv. 3, e1701577 (2017).

    Google Scholar 

  16. Zhang, H., Wang, X., Yu, H.-B. & Douglas, J. F. Dynamic heterogeneity, cooperative motion, and Johari–Goldstein β-relaxation in a metallic glass-forming material exhibiting a fragile-to-strong transition. Eur. Phys. J. E 44, 56 (2021).

    ADS  Google Scholar 

  17. Franz, S., Parisi, G., Urbani, P. & Zamponi, F. Universal spectrum of normal modes in low-temperature glasses. Proc. Natl Acad. Sci. USA 112, 14539–14544 (2015).

    ADS  Google Scholar 

  18. Bi, D., Lopez, J. H., Schwarz, J. M. & Manning, M. L. A density-independent rigidity transition in biological tissues. Nat. Phys. 11, 1074–1079 (2015).

    Google Scholar 

  19. Bi, D., Yang, X., Marchetti, M. C. & Manning, M. L. Motility-driven glass and jamming transitions in biological tissues. Phys. Rev. X 6, 021011 (2016).

    Google Scholar 

  20. Ngai, K. L. Relaxation and Diffusion in Complex Systems (Springer, 2011).

  21. Scalliet, C., Guiselin, B. & Berthier, L. Excess wings and asymmetric relaxation spectra in a facilitated trap model. J. Chem. Phys. 155, 064505 (2021).

    ADS  Google Scholar 

  22. Guiselin, B., Scalliet, C. & Berthier, L. Microscopic origin of excess wings in relaxation spectra of supercooled liquids. Nat. Phys. 18, 468–472 (2022).

    Google Scholar 

  23. Dennis, R. C. & Corwin, E. I. Jamming energy landscape is hierarchical and ultrametric. Phys. Rev. Lett. 124, 078002 (2020).

    ADS  Google Scholar 

  24. Charbonneau, P. et al. Numerical detection of the Gardner transition in a mean-field glass former. Phys. Rev. E 92, 012316 (2015).

    ADS  Google Scholar 

  25. Berthier, L. et al. Growing timescales and lengthscales characterizing vibrations of amorphous solids. Proc. Natl Acad. Sci. USA 113, 8397–8401 (2016).

    ADS  Google Scholar 

  26. Kob, W. & Berthier, L. Probing a liquid to glass transition in equilibrium. Phys. Rev. Lett. 110, 245702 (2013).

    ADS  Google Scholar 

  27. Cammarota, C. & Biroli, G. Ideal glass transitions by random pinning. Proc. Natl Acad. Sci. USA 109, 8850–8855 (2012).

    ADS  Google Scholar 

  28. Ozawa, M., Kob, W., Ikeda, A. & Miyazaki, K. Equilibrium phase diagram of a randomly pinned glass-former. Proc. Natl Acad. Sci. USA 112, 6914–6919 (2015).

    ADS  Google Scholar 

  29. Cammarota, C. & Biroli, G. Aging and relaxation near random pinning glass transitions. Europhys. Lett. 98, 16011 (2012).

    ADS  Google Scholar 

  30. Gokhale, S., Nagamanasa, K. H., Ganapathy, R. & Sood, A. K. Growing dynamical facilitation on approaching the random pinning colloidal glass transition. Nat. Commun. 5, 4685 (2014).

    ADS  Google Scholar 

  31. Nagamanasa, K. H., Gokhale, S., Sood, A. K. & Ganapathy, R. Direct measurements of growing amorphous order and non-monotonic dynamic correlations in a colloidal glass-former. Nat. Phys. 11, 403–408 (2015).

    Google Scholar 

  32. Angelani, L., Paoluzzi, M., Parisi, G. & Ruocco, G. Probing the non-Debye low-frequency excitations in glasses through random pinning. Proc. Natl Acad. Sci. USA 115, 8700–8704 (2018).

    ADS  Google Scholar 

  33. Tong, H. & Tanaka, H. Revealing hidden structural order controlling both fast and slow glassy dynamics in supercooled liquids. Phys. Rev. X 8, 011041 (2018).

    Google Scholar 

  34. Li, B., Lou, K., Kob, W. & Granick, S. Anatomy of cage formation in a two-dimensional glass-forming liquid. Nature 587, 225–229 (2020).

    ADS  Google Scholar 

  35. Watanabe, K., Kawasaki, T. & Tanaka, H. Structural origin of enhanced slow dynamics near a wall in glass-forming systems. Nat. Mater. 10, 512–520 (2011).

    ADS  Google Scholar 

  36. Stevenson, J. D., Schmalian, J. & Wolynes, P. G. The shapes of cooperatively rearranging regions in glass-forming liquids. Nat. Phys. 2, 268–274 (2006).

    Google Scholar 

  37. Scalliet, C., Berthier, L. & Zamponi, F. Absence of marginal stability in a structural glass. Phys. Rev. Lett. 119, 205501 (2017).

    ADS  Google Scholar 

  38. Tong, H., Sengupta, S. & Tanaka, H. Emergent solidity of amorphous materials as a consequence of mechanical self-organisation. Nat. Commun. 11, 4863 (2020).

    ADS  Google Scholar 

  39. Yip, C.-T. et al. Direct evidence of void-induced structural relaxations in colloidal glass formers. Phys. Rev. Lett. 125, 258001 (2020).

    ADS  Google Scholar 

  40. Chandler, D. & Garrahan, J. P. Dynamics on the way to forming glass: bubbles in space-time. Annu. Rev. Phys. Chem. 61, 191–217 (2010).

    Google Scholar 

  41. Vogel, M. & Glotzer, S. C. Spatially heterogeneous dynamics and dynamic facilitation in a model of viscous silica. Phys. Rev. Lett. 92, 255901 (2004).

    ADS  Google Scholar 

  42. Keys, A. S., Hedges, L. O., Garrahan, J. P., Glotzer, S. C. & Chandler, D. Excitations are localized and relaxation is hierarchical in glass-forming liquids. Phys. Rev. X 1, 021013 (2011).

    Google Scholar 

  43. Mishra, C. K., Nagamanasa, K. H., Ganapathy, R., Sood, A. K. & Gokhale, S. Dynamical facilitation governs glassy dynamics in suspensions of colloidal ellipsoids. Proc. Natl Acad. Sci. USA 111, 15362–15367 (2014).

    ADS  Google Scholar 

  44. Cheng, Y. et al. Highly tunable β-relaxation enables the tailoring of crystallization in phase-change materials. Nat. Commun. 13, 7352 (2022).

    ADS  Google Scholar 

  45. Peng, S.-X. et al. Uncovering β-relaxations in amorphous phase-change materials. Sci. Adv. 6, eaay6726 (2020).

    ADS  Google Scholar 

  46. Ji, W., de Geus, T. W. J., Popović, M., Agoritsas, E. & Wyart, M. Thermal origin of quasilocalized excitations in glasses. Phys. Rev. E 102, 062110 (2020).

    ADS  Google Scholar 

  47. Tanaka, H. Bond orientational order in liquids: towards a unified description of water-like anomalies, liquid-liquid transition, glass transition, and crystallization. Eur. Phys. J. E 35, 113 (2012).

    Google Scholar 

  48. Tong, H. & Tanaka, H. Structural order as a genuine control parameter of dynamics in simple glass formers. Nat. Commun. 10, 5596 (2019).

    ADS  Google Scholar 

  49. Cubuk, E. D. et al. Structure-property relationships from universal signatures of plasticity in disordered solids. Science 358, 1033–1037 (2017).

    ADS  Google Scholar 

  50. Widmer-Cooper, A. & Harrowell, P. Predicting the long-time dynamic heterogeneity in a supercooled liquid on the basis of short-time heterogeneities. Phys. Rev. Lett. 96, 185701 (2006).

    ADS  Google Scholar 

  51. Kawasaki, T., Araki, T. & Tanaka, H. Correlation between dynamic heterogeneity and medium-range order in two-dimensional glass-forming liquids. Phys. Rev. Lett. 99, 215701 (2007).

    ADS  Google Scholar 

  52. Scalliet, C. & Berthier, L. Rejuvenation and memory effects in a structural glass. Phys. Rev. Lett. 122, 255502 (2019).

    ADS  Google Scholar 

  53. Albert, S. et al. Fifth-order susceptibility unveils growth of thermodynamic amorphous order in glass-formers. Science 352, 1308–1311 (2016).

    ADS  Google Scholar 

  54. Weeks, E. R., Crocker, J. C., Levitt, A. C., Schofield, A. & Weitz, D. A. Three-dimensional direct imaging of structural relaxation near the colloidal glass transition. Science 287, 627–631 (2000).

    ADS  Google Scholar 

  55. Zheng, Z., Wang, F. & Han, Y. et al. Glass transitions in quasi-two-dimensional suspensions of colloidal ellipsoids. Phys. Rev. Lett. 107, 065702 (2011).

    ADS  Google Scholar 

  56. Scalliet, C., Gnoli, A., Puglisi, A. & Vulpiani, A. Cages and anomalous diffusion in vibrated dense granular media. Phys. Rev. Lett. 114, 198001 (2015).

    ADS  Google Scholar 

  57. Lin, J., Jorjadze, I., Pontani, L.-L., Wyart, M. & Brujic, J. Evidence for marginal stability in emulsions. Phys. Rev. Lett. 117, 208001 (2016).

    ADS  Google Scholar 

  58. Hammond, A. P. & Corwin, E. I. Experimental observation of the marginal glass phase in a colloidal glass. Proc. Natl Acad. Sci. USA 117, 5714–5718 (2020).

    ADS  Google Scholar 

  59. Chen, K. et al. Measurement of correlations between low-frequency vibrational modes and particle rearrangements in quasi-two-dimensional colloidal glasses. Phys. Rev. Lett. 107, 108301 (2011).

    ADS  Google Scholar 

  60. Lozano, C., Gomez-Solano, J. R. & Bechinger, C. Active particles sense micromechanical properties of glasses. Nat. Mater. 18, 1118–1123 (2019).

    ADS  Google Scholar 

Download references

Acknowledgements

P.T. acknowledges the National Natural Science Foundation of China (nos. 12174071, 12105050, 12035004, 11734014 and 11725521), the Innovation Program of Shanghai Municipal Education Commission (no. 2023ZKZD06) and the Science and Technology Commission of Shanghai Municipality (no. 20JC1414700). Y.C. acknowledges the China Postdoc Science Foundation (nos. BX2021081 and 2021M690707). H. Tanaka acknowledges the Grants-in-Aid for Specially Promoted Research (JP20H05619) from the Japan Society for the Promotion of Science (JSPS).

Author information

Authors and Affiliations

  1. State Key Laboratory of Surface Physics and Department of Physics, Fudan University, Shanghai, China

    Yanshuang Chen, Zefang Ye, Kexin Wang, Jiping Huang & Peng Tan

  2. Institute for Nanoelectronic Devices and Quantum Computing, Fudan University, Shanghai, China

    Yanshuang Chen & Peng Tan

  3. Department of Physics, University of Science and Technology of China, Hefei, China

    Hua Tong

  4. Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing, China

    Yuliang Jin

  5. Institute of Physics, Chinese Academy of Sciences, Beijing, China

    Ke Chen

  6. Research Center for Advanced Science and Technology, University of Tokyo, Meguro-ku, Japan

    Hajime Tanaka

  7. Department of Fundamental Engineering, Institute of Industrial Science, University of Tokyo, Meguro-ku, Japan

    Hajime Tanaka

Authors
  1. Yanshuang Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zefang Ye
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Kexin Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Jiping Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Hua Tong
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Yuliang Jin
    View author publications

    You can also search for this author in PubMed Google Scholar

  7. Ke Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  8. Hajime Tanaka
    View author publications

    You can also search for this author in PubMed Google Scholar

  9. Peng Tan
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

P.T. and H. Tanaka conceived and supervised the research, performed the data analysis and modelling, and wrote the paper. Y.C. performed the experiments, data analysis and modelling. Z.Y. performed the experiments and data analysis. K.W. and J.H. performed the data analysis. H. Tong and Y.J. performed the data analysis and modelling, and K.C. performed the colloidal experiments.

Corresponding authors

Correspondence to Hajime Tanaka or Peng Tan.

Ethics declarations

Competing interests

The authors declare no competing interests.

Peer review

Peer review information

Nature Physics thanks Jana Shanelova and Shuai Wei for their contribution to the peer review of this work.

Additional information

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Extended data

Extended Data Fig. 1 Our 2D binary magnetic disk system under mechanical perturbation.

a, A typical raw image of our sample. b, MSD of our pinning samples over τs at a high perturbation strength. Within τs, we can see a significant rise from the MSD plateau, which means that the system behaves as a viscous fluid. In this case, the long-time MSD becomes superdiffusive under our mechanical perturbation. With increasing the pinning fraction c, we can see that the system’s response gradually becomes solid-like. Thus, the mechanical perturbation strength should be small enough to guarantee a solid-like response within τs. c-e, Relaxation dynamics illustrated by the particle-level Van Hove correlation function at Δt = 20τs(c), Δt = 100τs (d), and Δt = 500τs (e). The self part of the particle-level time-dependent Van Hove correlation function, Gs,i(r, t), is calculated for a group of particles i that have a narrow range of 〈Δr2(t)〉 (Δt = 500τs). The long-time motion of the fastest particles forms a second peak or a shoulder at a characteristic moving distance, whereas the motion of the slowest particles exhibits a Gaussian distribution.

Source data

Extended Data Fig. 2 Wall-induced effects on the relaxation dynamics in our systems.

a, The particle number N counted with respect to the distance RC/a from the centre. The smooth circular wall induces the layering of particles up to a distance around 2a(a grey area) from the wall. b, c, Particles’ mean-square displacements (MSD), 〈Δr2(t)〉/a2, with respect to RC/a at Δt = 10τs (b) and Δt = 200τs (c). The wall slows down the particle motion near the wall (a grey area). d, e, 〈Δr2(t)〉/a2 of bulk particles (d) and boundary particles (e). Although particles near the wall have slower dynamics than the bulk particles, we can observe the same threshold value of c*. It indicates that the wall does not change the glass transition point but makes the system less fragile, that is, stronger.

Source data

Extended Data Fig. 3 Characterization of internal-relaxation profiles.

a-d, The unified ls,l − ΔR relation for T1 events. a, c ~ 0.012, b, c ~ 0.024, c, c ~ 0.036, d, c ~ 0.048. Here, ΔR is the total moving distance of the four particles in each tetragon. We can see a similar curve as in Fig. 2a-d. Note that each tetragon has four values of Nb(i). e-g, Cluster analysis of large-T1 and small-T1 events for all samples as a function of 〈Δr2(t = 200τs)〉/a2, where c* is located around 〈Δr2(t)〉/a2 ~ 10−3. e, The number of T1-tetragon, NT1, in each sample. f, The average particle number, 〈NP〉, contained in the T1-cluster. g, The coordination number, \(\overline{n_c}\), of T1-tetragon in the T1 cluster. Large-T1 cluster is string-like cooperative excitation at \(c\le {c}^{* }(\langle {N}_{P}\rangle \sim 10,\overline{{n}_{c}} \sim 2)\), whereas small-T1 cluster is sparse, localized excitation at all \(c(\langle {N}_{P}\rangle \sim 4,\overline{{n}_{c}} \sim 0)\). h, i, The number of T1-tetragon, \(\overline{N}\), excited with respect to t at c < c* (h) and c > c* (i). Large-T1 and small-T1 events are excited at different time scales, suggesting that small-T1 event contributes to slow-β relaxation.

Source data

Extended Data Fig. 4 Spatio-temporal illustration of internal-relaxation dynamics with increasing the pinning fraction.

Relaxation dynamics is illustrated by ΔR(i) (indicated by the arrow length and direction), lsl (indicated by black and grey tetragons representing large-ratio and small-ratio T1 events, respectively), and Nb(i) (indicated by the particle and displacement colours). a, Sequential accumulation of large-ratio T1 events forms a compact cluster with Nb(i)≥1 for a typical sample of α-type relaxation (c ~ 1.2%). Note that the motion that forms the cluster’s core is also string-like. b, A collective excitation of spatially extended large-ratio T1 events forms a string-like cage-breaking excitation for a typical sample at c ~ 2.4%. c and d correspond to the cases of a glass state (c ~ 3.6% and c ~ 4.8% respectively), where large-ratio T1 events vanishes, but localized small-ratio T1 events can persist. The displacement fields in c and d are magnified 4 times. e, Sequential accumulation of large-T1 events that develop into a string-like larger cluster at a pinning fraction similar to a.

Source data

Extended Data Fig. 5 Spatio-temporal illustration of internal-relaxation dynamics in a supercooled colloidal liquid and a granular, crystalline system with dislocations.

We use bidisperse Nipam colloidal particles to form a 2D glassy system. Relaxation dynamics is illustrated by Δr(i) (indicated by the arrow length and direction), lsl (indicated by black and grey tetragons representing large-ratio and small-ratio T1 events, respectively), and Nb(i) (indicated by the particle and the displacement colours). a-c, Reversible bond-breaking process occurring at the fast-β regime (t < τ0). d-f, Formation of string-like excitation after the fast-β regime. See the red and orange strings of Δr in f. g-i, Irreversible cage-breaking events serving as precursors of relaxation. See the recovery of Δr in g for the orange string in f. j, k, Spatial illustration of structure relaxations in a crystalline system with dislocations. We use large disk particles (10 mm diameter) in our magnetic glass system to form the 2D hexagonal crystalline structure with dislocations. Dislocation dynamics is responsible for large-ratio T1 events. We rarely find small-ratio T1 events, indicating the absence of the slow-β mode.

Source data

Extended Data Fig. 6 Comparison of the relaxation dynamics for two glass systems and crystals, using the four-particle tetragon model.

a, A colloidal supercooled colloidal liquid system with short-range interactions that represents fast-β and α modes. We use bidisperse Nipam colloidal particles to form 2D glassy system. Note that the MSD plateau value is around 2.0 × 10−2. Small-ratio T1 events occur within τ0, whereas large-ratio T1 events typically occur around a time of 30 τ0 (left panel). RM is absent for the relaxation dynamics examined at Δt = 0.2τ0 (middle panel) and Δt = 100τ0 (right panel). b, The deeply supercooled granular liquid sample with long-range soft interaction that represents slow-β and α modes (c ~ 1.2%). Note that the MSD plateau value is around 2.0 × 10−4. Small-ratio T1 events typically occur around a time of 2.0 × 104τ0, whereas large-ratio T1 events typically occur around a time of 1.0 × 105τ0 (left panel). We find a weak signature of RM at the time scale of slow-β relaxation (middle panel), whereas RM is absent at the timescale of early-α relaxation (right panel). c, The granular glass sample with long-range soft interaction that represents slow-β mode (c = 4.8%). Note that the MSD plateau value is around 4.0 × 10−4. Small-ratio T1 events typically occur around a time of 2.0 × 105τ0, whereas large-ratio T1 events are absent (left panel). We find a strong signature of RM at the time scale shorter than (middle panel) and comparable (right panel) to the slow-β relaxation time. d, The crystalline system with dislocations. We use large disk particles (10 mm diameter) used in our glass system to form a 2D hexagonal crystalline structure with dislocations. Both RM and small-T1 events are absent.

Source data

Extended Data Fig. 7 Spontaneous fluctuations of ‘rigid’ bonds in glass samples.

a, c ~ 3.6%, b, c ~ 4.8%. For a specific bond, we can define its rigidity through the time percentage that its dynamics belongs to the RM mode and not to the FM mode. The bonds with more than 7.5%, 10%, and 15% of their time in a ‘rigid’ state are shown. We can see spontaneous fluctuations of ‘rigid’ bonds that present RM when approaching a glass state.

Source data

Extended Data Fig. 8 Probability distribution of tan−1(ΔLm,n/ΔLh,k) for the two glass and crystalline systems with dislocations.

a, A strong signature of FM is observed, whereas RM is absent in the supercooled colloidal liquid with short-range interactions that represents fast-β (left panel) and α (right panel) modes. b, The deeply supercooled granular liquid sample with long-range soft interaction that represents slow-β and α modes (c ~ 1.2%). We find a weak signature of RM for slow-β relaxation (middle panel), whereas RM is absent for early-α relaxation (right panel). c, The granular glass sample with long-range soft interaction that represents slow-β mode (c ~ 1.2%). We find a strong signature of RM when early-α relaxation is absent. d, The granular, crystalline system with dislocations. Both RM and small-T1 events are absent. We find a strong signature of \(\Delta {L}_{m,n}/\Delta {L}_{h,k} \sim \sqrt{3}\), indicating the elastic nature. The results shown here correspond to those in Extended Data Fig. 6.

Source data

Source data

Source Data Fig. 1

Source data for Fig. 1.

Source Data Fig. 2

Time-sequential configurations.

Source Data Fig. 3

Time-sequential configurations.

Source Data Fig. 4

Source data for Fig.4.

Source Data Extended Data Fig. 1

Source data for Extended Data Fig. 1.

Source Data Extended Data Fig. 2

Source data for Extended Data Fig. 2.

Source Data Extended Data Fig. 3

Source data for Extended Data Fig. 3.

Source Data Extended Data Fig. 4

Time-sequential configurations.

Source Data Extended Data Fig. 5

Time-sequential configurations.

Source Data Extended Data Fig. 6

Time-sequential configurations.

Source Data Extended Data Fig. 7

Time-sequential configurations.

Source Data Extended Data Fig. 8

Source data for Extended Data Fig. 8.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chen, Y., Ye, Z., Wang, K. et al. Visualizing slow internal relaxations in a two-dimensional glassy system. Nat. Phys. 19, 969–977 (2023). https://doi.org/10.1038/s41567-023-02016-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s41567-023-02016-4

This article is cited by

Search

Advanced search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing