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Microscopic origin of excess wings in relaxation spectra of supercooled liquids


Glass formation is encountered in diverse materials. Experiments have revealed that the dynamic relaxation spectra of supercooled liquids generically become asymmetric near the glass transition temperature Tg, where an extended power law emerges at high frequencies. The microscopic origin of this ‘wing’ remains unknown, and has so far been inaccessible to simulations. Here we develop a novel computational approach and study the equilibrium dynamics of model supercooled liquids near Tg. We demonstrate the emergence of a power-law wing in the numerical spectra, which originates from relaxation at rare, localized regions over broadly distributed timescales. We rationalize the asymmetric shape of the relaxation spectra by constructing an empirical model associating heterogeneous activated dynamics with dynamic facilitation, which are the two minimal physical ingredients revealed by our simulations. Our work offers a glimpse into the molecular motion responsible for glass formation at relevant experimental conditions.

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Fig. 1: Emergence of excess wings in a 3D glass former near the glass transition temperature.
Fig. 2: Visualization of spatially heterogeneous and facilitated dynamics.
Fig. 3: Microscopic origin of excess wings.
Fig. 4: Facilitated trap model generically predicts asymmetric winged relaxation spectra.

Data availability

Source data are provided with this paper. The data that support the findings of this study are available from the corresponding author upon reasonable request.

Code availability

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


  1. Berthier, L. & Biroli, G. Theoretical perspective on the glass transition and amorphous materials. Rev. Mod. Phys. 83, 587 (2011).

    ADS  Article  Google Scholar 

  2. Schmidtke, B., Petzold, N., Kahlau, R., Hofmann, M. & Rössler, E. A. From boiling point to glass transition temperature: transport coefficients in molecular liquids follow three-parameter scaling. Phys. Rev. E 86, 041507 (2012).

    ADS  Article  Google Scholar 

  3. Lunkenheimer, P., Schneider, U., Brand, R. & Loid, A. Glassy dynamics. Contemp. Phys. 41, 15–36 (2000).

    ADS  Article  Google Scholar 

  4. Körber, T., Stäglich, R., Gainaru, C., Böhmer, R. & Rössler, E. A. Systematic differences in the relaxation stretching of polar molecular liquids probed by dielectric vs magnetic resonance and photon correlation spectroscopy. J. Chem. Phys. 153, 124510 (2020).

    ADS  Article  Google Scholar 

  5. Schmidtke, B., Petzold, N., Kahlau, R. & Rössler, E. A. Reorientational dynamics in molecular liquids as revealed by dynamic light scattering: from boiling point to glass transition temperature. J. Chem. Phys. 139, 084504 (2013).

    ADS  Article  Google Scholar 

  6. Gainaru, C., Kahlau, R., Rössler, E. A. & Böhmer, R. Evolution of excess wing and β-process in simple glass formers. J. Chem. Phys. 131, 184510 (2009).

    ADS  Article  Google Scholar 

  7. Flämig, M., Hofmann, M., Fatkullin, N. & Rössler, E. A. NMR relaxometry: the canonical case glycerol. J. Phys. Chem. B 124, 1557–1570 (2020).

    Article  Google Scholar 

  8. Hecksher, T. et al. Toward broadband mechanical spectroscopy. Proc. Natl Acad. Sci. USA 114, 8710–8715 (2017).

    Article  Google Scholar 

  9. Schneider, U., Brand, R., Lunkenheimer, P. & Loidl, A. Excess wing in the dielectric loss of glass formers: a Johari-Goldstein β relaxation? Phys. Rev. Lett. 84, 5560–5563 (2000).

    ADS  Article  Google Scholar 

  10. Lunkenheimer, P., Wehn, R., Riegger, T. & Loidl, A. Excess wing in the dielectric loss of glass formers: further evidence for a β-relaxation. J. Non Cryst. Solids 307-310, 336–344 (2002).

    ADS  Article  Google Scholar 

  11. Dixon, P. K., Wu, L., Nagel, S. R., Williams, B. D. & Carini, J. P. Scaling in the relaxation of supercooled liquids. Phys. Rev. Lett. 65, 1108–1111 (1990).

    ADS  Article  Google Scholar 

  12. Leheny, R. L. & Nagel, S. R. Dielectric susceptibility studies of the high-frequency shape of the primary relaxation in supercooled liquids. J. Non Cryst. Solids 235-237, 278–285 (1998).

    ADS  Article  Google Scholar 

  13. Menon, N. et al. Wide-frequency dielectric susceptibility measurements in glycerol. J. Non Cryst. Solids 141, 61–65 (1992).

    ADS  Article  Google Scholar 

  14. Menon, N. & Nagel, S. R. Evidence for a divergent susceptibility at the glass transition. Phys. Rev. Lett. 74, 1230–1233 (1995).

    ADS  Article  Google Scholar 

  15. Wu, L. Relaxation mechanisms in a benzyl chloride–toluene glass. Phys. Rev. B 43, 9906–9915 (1991).

    ADS  Article  Google Scholar 

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

    ADS  Article  Google Scholar 

  17. Blochowicz, T., Tschirwitz, C., Benkhof, S. & Rössler, E. A. Susceptibility functions for slow relaxation processes in supercooled liquids and the search for universal relaxation patterns. J. Chem. Phys. 118, 7544–7555 (2003).

    ADS  Article  Google Scholar 

  18. Bauer, T., Lunkenheimer, P., Kastner, S. & Loidl, A. Nonlinear dielectric response at the excess wing of glass-forming liquids. Phys. Rev. Lett. 110, 107603 (2013).

    ADS  Article  Google Scholar 

  19. Duvvuri, K. & Richert, R. Dielectric hole burning in the high frequency wing of supercooled glycerol. J. Chem. Phys. 118, 1356–1363 (2003).

    ADS  Article  Google Scholar 

  20. Lunkenheimer, P., Wehn, R., Schneider, U. & Loidl, A. Glassy aging dynamics. Phys. Rev. Lett. 95, 055702 (2005).

    ADS  Article  Google Scholar 

  21. Diezemann, G., Mohanty, U. & Oppenheim, I. Slow secondary relaxation in a free-energy landscape model for relaxation in glass-forming liquids. Phys. Rev. E 59, 2067–2083 (1999).

    ADS  Article  Google Scholar 

  22. Domschke, M., Marsilius, M., Blochowicz, T. & Voigtmann, T. Glassy relaxation and excess wing in mode-coupling theory: the dynamic susceptibility of propylene carbonate above and below Tc. Phys. Rev. E 84, 031506 (2011).

    ADS  Article  Google Scholar 

  23. Sethna, J. P., Shore, J. D. & Huang, M. Scaling theory for the glass transition. Phys. Rev. B 44, 4943–4959 (1991).

    ADS  Article  Google Scholar 

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

    Article  Google Scholar 

  25. Viot, P., Tarjus, G. & Kivelson, D. A heterogeneous picture of α relaxation for fragile supercooled liquids. J. Chem. Phys. 112, 10368–10378 (2000).

    ADS  Article  Google Scholar 

  26. Chamberlin, R. V. Mesoscopic mean-field theory for supercooled liquids and the glass transition. Phys. Rev. Lett. 82, 2520–2523 (1999).

    ADS  Article  Google Scholar 

  27. Dyre, J. C. & Schrøder, T. B. Universality of a.c. conduction in disordered solids. Rev. Mod. Phys. 72, 873–892 (2000).

    ADS  Article  Google Scholar 

  28. Berthier, L. & Garrahan, J. P. Numerical study of a fragile three-dimensional kinetically constrained model. J. Phys. Chem. B 109, 3578–3585 (2005).

    Article  Google Scholar 

  29. Ninarello, A., Berthier, L. & Coslovich, D. Models and algorithms for the next generation of glass transition studies. Phys. Rev. X 7, 021039 (2017).

    Google Scholar 

  30. Berthier, L., Charbonneau, P., Ninarello, A., Ozawa, M. & Yaida, S. Zero-temperature glass transition in two dimensions. Nat. Commun. 10, 1508 (2019).

    ADS  Article  Google Scholar 

  31. Berthier, L. et al. Configurational entropy measurements in extremely supercooled liquids that break the glass ceiling. Proc. Natl Acad. Sci. USA 114, 11356–11361 (2017).

    ADS  Article  Google Scholar 

  32. Berthier, L., Flenner, E., Fullerton, C. J., Scalliet, C. & Singh, M. Efficient swap algorithms for molecular dynamics simulations of equilibrium supercooled liquids. J. Stat. Mech. 2019, 064004 (2019).

    MathSciNet  Article  Google Scholar 

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

    ADS  Article  Google Scholar 

  34. Illing, B. et al. Mermin–Wagner fluctuations in 2D amorphous solids. Proc. Natl Acad. Sci. USA 114, 1856–1861 (2017).

    ADS  Article  Google Scholar 

  35. Flenner, E. & Szamel, G. Fundamental differences between glassy dynamics in two and three dimensions. Nat. Commun. 6, 7392 (2015).

    ADS  Article  Google Scholar 

  36. Berthier, L. & Ediger, M. D. How to ‘measure’ a structural relaxation time that is too long to be measured? J. Chem. Phys. 153, 044501 (2020).

    Article  Google Scholar 

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

    Article  Google Scholar 

  38. 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 

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

    ADS  Article  Google Scholar 

  40. Dyre, J. C. Master-equation appoach to the glass transition. Phys. Rev. Lett. 58, 792–795 (1987).

    ADS  Article  Google Scholar 

  41. Bouchaud, J.-P. Weak ergodicity breaking and aging in disordered systems. J. Phys. I 2, 1705–1713 (1992).

    Google Scholar 

  42. Rehwald, C., Rubner, O. & Heuer, A. From coupled elementary units to the complexity of the glass transition. Phys. Rev. Lett. 105, 117801 (2010).

    ADS  Article  Google Scholar 

  43. Rehwald, C. & Heuer, A. How coupled elementary units determine the dynamics of macroscopic glass-forming systems. Phys. Rev. E 86, 051504 (2012).

    ADS  Article  Google Scholar 

  44. Arkhipov, V. I. & Baessler, H. Random-walk approach to dynamic and thermodynamic properties of supercooled melts. 1. Viscosity and average relaxation times in strong and fragile liquids. J. Phys. Chem. 98, 662–669 (1994).

    Article  Google Scholar 

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

    ADS  Article  Google Scholar 

  46. Xia, X. & Wolynes, P. G. Microscopic theory of heterogeneity and nonexponential relaxations in supercooled liquids. Phys. Rev. Lett. 86, 5526–5529 (2001).

    ADS  Article  Google Scholar 

  47. van Meel, J. A., Filion, L., Valeriani, C. & Frenkel, D. A parameter-free, solid-angle based, nearest-neighbor algorithm. J. Chem. Phys. 136, 234107 (2012).

    ADS  Article  Google Scholar 

  48. Schrøder, T. B., Sastry, S., Dyre, J. C. & Glotzer, S. C. Crossover to potential energy landscape dominated dynamics in a model glass-forming liquid. J. Chem. Phys. 112, 9834–9840 (2000).

    ADS  Article  Google Scholar 

  49. Götze, W. Complex Dynamics of Glass-Forming Liquids: A Mode-Coupling Theory Vol. 143 (Oxford Univ. Press, 2009).

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We thank G. Biroli, M. Ediger and J. Kurchan for discussions, and S. Nagel for detailed explanations about the experiments. Some simulations were performed at MESO@LR platform at the University of Montpellier. This work was supported by a grant from the Simons Foundation (no. 454933; L.B.), the European Research Council under the EU’s Horizon 2020 programme via grant no. 740269 (C.S.), a Herchel Smith Postdoctoral Research Fellowship (C.S.), a Ramon Jenkins Research Fellowship from Sidney Sussex College, Cambridge (C.S.), and Capital Fund Management - Fondation pour la Recherche (B.G.).

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Authors and Affiliations



B.G., C.S. and L.B. designed the research. B.G. and C.S. carried out the simulations. B.G, C.S. and L.B. analysed the data and wrote the paper.

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Correspondence to Ludovic Berthier.

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Nature Physics thanks Reiner Zorn, Thomas Voigtmann and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Supplementary information

Supplementary Information

Supplementary Sections I and II and Figs. 1–4.

Source data

Source Data Fig. 1

The {x, y} data for Fig. 1a–c.

Source Data Fig. 3

The {x, y} data for Fig. 3.

Source Data Fig. 4

The {x, y} data for Fig. 4c.

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

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