Thermal vestige of the zero-temperature jamming transition


When the packing fraction is increased sufficiently, loose particulates jam to form a rigid solid in which the constituents are no longer free to move. In typical granular materials and foams, the thermal energy is too small to produce structural rearrangements. In this zero-temperature (T = 0) limit, multiple diverging1,2,3,4,5,6,7,8 and vanishing2,9,10 length scales characterize the approach to a sharp jamming transition. However, because thermal motion becomes relevant when the particles are small enough, it is imperative to understand how these length scales evolve as the temperature is increased. Here we used both colloidal experiments and computer simulations to progress beyond the zero-temperature limit to track one of the key parameters—the overlap distance between neighbouring particles—which vanishes at the T = 0 jamming transition. We find that this structural feature retains a vestige of its T = 0 behaviour and evolves in an unusual manner, which has masked its appearance until now. It is evident as a function of packing fraction at fixed temperature, but not as a function of temperature at fixed packing fraction or pressure. Our results conclusively demonstrate that length scales associated with the T = 0 jamming transition persist in thermal systems, not only in simulations but also in laboratory experiments.

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Figure 1: Schematic jamming phase diagram.
Figure 2: Pair-correlation function g(r ) for the large particles at all experimental packing fractions.
Figure 3: Peak value of g(r), g1, measured from simulations.
Figure 4: Dynamics approaching the structural maximum.


  1. 1

    O'Hern, C. S., Langer, S. A., Liu, A. J. & Nagel, S. R. Random packings of frictionless particles. Phys. Rev. Lett. 88, 075507 (2002)

    ADS  Article  Google Scholar 

  2. 2

    O'Hern, C. S., Silbert, L. E., Liu, A. J. & Nagel, S. R. Jamming at zero temperature and zero applied stress: The epitome of disorder. Phys. Rev. E 68, 011306 (2003)

    ADS  Article  Google Scholar 

  3. 3

    Drocco, J. A., Hastings, M. B., Reichhardt, C. J. O. & Reichhardt, C. Multiscaling at point J: Jamming is a critical phenomenon. Phys. Rev. Lett. 95, 088001 (2005)

    ADS  CAS  Article  Google Scholar 

  4. 4

    Ellenbroek, W. G., Somfai, E., van Hecke, M. & van Saarloos, W. Critical scaling in linear response of frictionless granular packings near jamming. Phys. Rev. Lett. 97, 258001 (2006)

    ADS  Article  Google Scholar 

  5. 5

    Olsson, P. & Teitel, S. Critical scaling of shear viscosity at the jamming transition. Phys. Rev. Lett. 99, 178001 (2007)

    ADS  Article  Google Scholar 

  6. 6

    Silbert, L. E., Liu, A. J. & Nagel, S. R. Vibrations and diverging length scales near the unjamming transition. Phys. Rev. Lett. 95, 098301 (2005)

    ADS  Article  Google Scholar 

  7. 7

    Wyart, M., Silbert, L. E., Nagel, S. R. & Witten, T. A. Effects of compression on the vibrational modes of marginally jammed solids. Phys. Rev. E 72, 051306 (2005)

    ADS  Article  Google Scholar 

  8. 8

    Xu, N., Vitelli, V., Wyart, M., Liu, A. J. & Nagel, S. R. Energy transport in jammed sphere packings. Phys. Rev. Lett. 102, 038001 (2009)

    ADS  Article  Google Scholar 

  9. 9

    Silbert, L. E., Liu, A. J. & Nagel, S. R. Structural signatures of the unjamming transition at zero temperature. Phys. Rev. E 73, 041304 (2006)

    ADS  Article  Google Scholar 

  10. 10

    Donev, A., Torquato, S. & Stillinger, F. H. Pair correlation function characteristics of nearly jammed disordered and ordered hard-sphere packings. Phys. Rev. E 71, 011105 (2005)

    ADS  MathSciNet  Article  Google Scholar 

  11. 11

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

    ADS  CAS  Article  Google Scholar 

  12. 12

    Trappe, V., Prasad, V., Cipelletti, L., Segre, P. N. & Weitz, D. A. Jamming phase diagram for attractive particles. Nature 411, 772–775 (2001)

    ADS  CAS  Article  Google Scholar 

  13. 13

    Durian, D. J. Foam mechanics at the bubble scale. Phys. Rev. Lett. 75, 4780–4783 (1995)

    ADS  CAS  Article  Google Scholar 

  14. 14

    Pelton, R. Temperature-sensitive aqueous microgels. Adv. Colloid Interf. Sci. 85, 1–33 (2000)

    ADS  CAS  Article  Google Scholar 

  15. 15

    Saunders, B. R. & Vincent, B. Microgel particles as model colloids: theory, properties and applications. Adv. Colloid Interf. Sci. 80, 1–25 (1999)

    CAS  Article  Google Scholar 

  16. 16

    Senff, H. & Richtering, W. Temperature sensitive microgel suspensions: Colloidal phase behavior and rheology of soft spheres. J. Chem. Phys. 111, 1705–1711 (1999)

    ADS  CAS  Article  Google Scholar 

  17. 17

    Wu, J. Z., Zhou, B. & Hu, Z. B. Phase behavior of thermally responsive microgel colloids. Phys. Rev. Lett. 90, 048304 (2003)

    ADS  Article  Google Scholar 

  18. 18

    Lyon, L. A. et al. Microgel colloidal crystals. J. Phys. Chem. B 108, 19099–19108 (2004)

    CAS  Article  Google Scholar 

  19. 19

    Alsayed, A. M., Islam, M. F., Zhang, J., Collings, P. J. & Yodh, A. G. Premelting at defects within bulk colloidal crystals. Science 309, 1207–1210 (2005)

    ADS  CAS  Article  Google Scholar 

  20. 20

    Han, Y., Ha, N. Y., Alsayed, A. M. & Yodh, A. G. Melting of two-dimensional tunable-diameter colloidal crystals. Phys. Rev. E 77, 041406 (2008)

    ADS  CAS  Article  Google Scholar 

  21. 21

    Han, Y. L. et al. Geometric frustration in buckled colloidal monolayers. Nature 456, 898–903 (2008)

    ADS  CAS  Article  Google Scholar 

  22. 22

    Kob, W. & Andersen, H. C. Testing mode-coupling theory for a supercooled binary Lennard-Jones mixture—The Van Hove correlation-function. Phys. Rev. E 51, 4626–4641 (1995)

    ADS  CAS  Article  Google Scholar 

  23. 23

    Perera, D. N. & Harrowell, P. Relaxation dynamics and their spatial distribution in a two-dimensional glass-forming mixture. J. Chem. Phys. 111, 5441–5454 (1999)

    ADS  CAS  Article  Google Scholar 

  24. 24

    Murray, C. A. & Grier, D. G. Video microscopy of monodisperse colloidal systems. Annu. Rev. Phys. Chem. 47, 421–462 (1996)

    ADS  CAS  Article  Google Scholar 

  25. 25

    Crocker, J. C. & Grier, D. G. Methods of digital video microscopy for colloidal studies. J. Colloid Interf. Sci. 179, 298–310 (1996)

    ADS  CAS  Article  Google Scholar 

  26. 26

    Kegel, W. K. & van Blaaderen, A. Direct observation of dynamical heterogeneities in colloidal hard-sphere suspensions. Science 287, 290–293 (2000)

    ADS  CAS  Article  Google Scholar 

  27. 27

    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  CAS  Article  Google Scholar 

  28. 28

    Abate, A. R. & Durian, D. J. Approach to jamming in an air-fluidized granular bed. Phys. Rev. E 74, 031308 (2006)

    ADS  CAS  Article  Google Scholar 

  29. 29

    Busse, L. E. & Nagel, S. R. Temperature-dependence of the structure factor of As2Se3 glass up to the glass-transition. Phys. Rev. Lett. 47, 1848–1851 (1981)

    ADS  CAS  Article  Google Scholar 

  30. 30

    Lacevic, N., Starr, F. W., Schroder, T. B., Novikov, V. N. & Glotzer, S. C. Growing correlation length on cooling below the onset of caging in a simulated glass-forming liquid. Phys. Rev. E 66, 030101 (2002)

    ADS  CAS  Article  Google Scholar 

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We thank T. Lubensky, D. Durian and K. Chen for discussions and a critical reading of the manuscript. We acknowledge the financial support of the Department of Energy and the National Science Foundation: DE-FG02-05ER46199 (A.J.L., N.X.), DE-FG02-03ER46088 (S.R.N., N.X.), the University of Chicago MRSEC DMR-0820054 (S.R.N., N.X.), DMR-080488 (A.G.Y.), and the PENN MRSEC DMR-0520020 (A.G.Y., A.J.L., Z.Z.). Z.Z. gratefully acknowledges partial support from Rhodia. Finally, we acknowledge the support of the Teraport computer cluster at the University of Chicago.

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Correspondence to Zexin Zhang or Ning Xu.

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Zhang, Z., Xu, N., Chen, D. et al. Thermal vestige of the zero-temperature jamming transition. Nature 459, 230–233 (2009).

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