A phase diagram for jammed matter


The problem of finding the most efficient way to pack spheres has a long history, dating back to the crystalline arrays conjectured1 by Kepler and the random geometries explored2 by Bernal. Apart from its mathematical interest, the problem has practical relevance3 in a wide range of fields, from granular processing to fruit packing. There are currently numerous experiments showing that the loosest way to pack spheres (random loose packing) gives a density of 55 per cent4,5,6. On the other hand, the most compact way to pack spheres (random close packing) results in a maximum density of 64 per cent2,4,6. Although these values seem to be robust, there is as yet no physical interpretation for them. Here we present a statistical description of jammed states7 in which random close packing can be interpreted as the ground state of the ensemble of jammed matter. Our approach demonstrates that random packings of hard spheres in three dimensions cannot exceed a density limit of 63.4 per cent. We construct a phase diagram that provides a unified view of the hard-sphere packing problem and illuminates various data, including the random-loose-packed state.

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Figure 1: Phase diagram of jamming: theory.
Figure 2: Representation of the volume landscape of jammed matter ( ξ, W).
Figure 3: Predictions of the equation of state of jammed matter in the X φ S space.
Figure 4: Phase diagram of jamming: simulations.


  1. 1

    Hales, T. C. The Kepler conjecture. Preprint at 〈http://arxiv.org/abs/math/9811078v2〉 (2002)

  2. 2

    Bernal, J. D. & Mason, J. Packing of spheres: co-ordination of randomly packed spheres. Nature 188, 910–911 (1960)

    Article  ADS  Google Scholar 

  3. 3

    Behringer, R. P. & Jenkins, J. T. (eds) Powders & Grains 97 (Balkema, Rotterdam, 1997)

    Google Scholar 

  4. 4

    Scott, G. D. & Kilgour, D. M. The density of random close packing of spheres. J. Phys. D 2, 863–866 (1969)

    Article  ADS  Google Scholar 

  5. 5

    Onoda, G. Y. & Liniger, E. G. Random loose packings of uniform spheres and the dilatancy effect. Phys. Rev. Lett. 64, 2727–2730 (1990)

    CAS  Article  ADS  Google Scholar 

  6. 6

    Berryman, J. D. Random close packing of hard spheres and disks. Phys. Rev. A 27, 1053–1061 (1983)

    CAS  Article  ADS  Google Scholar 

  7. 7

    Edwards, S. F. & Oakeshott, R. B. S. Theory of powders. Physica A 157, 1080–1090 (1989)

    MathSciNet  Article  ADS  Google Scholar 

  8. 8

    Nowak, E. R., Knight, J. B., Ben-Naim, E., Jaeger, H. M. & Nagel, S. R. Density fluctuations in vibrated granular materials. Phys. Rev. E 57, 1971–1982 (1998)

    CAS  Article  ADS  Google Scholar 

  9. 9

    Blumenfeld, R. & Edwards, S. F. Granular entropy: explicit calculations for planar assemblies. Phys. Rev. Lett. 90, 114303 (2002)

    Article  ADS  Google Scholar 

  10. 10

    Ball, R. C. & Blumenfeld, R. Stress field in granular systems: loop forces and potential formulation. Phys. Rev. Lett. 88, 115505 (2002)

    Article  ADS  Google Scholar 

  11. 11

    Schröter, M., Goldman, D. I. & Swinney, H. L. Stationary state volume fluctuations in a granular medium. Phys. Rev. E 71, 030301(R) (2005)

    Article  ADS  Google Scholar 

  12. 12

    Fierro, A., Nicodemi, M., Tarzia, M., de Candia, A. & Coniglio, A. Jamming transition in granular media: A mean-field approximation and numerical simulations. Phys. Rev. E 71, 061305 (2005)

    CAS  Article  ADS  Google Scholar 

  13. 13

    Aste, T., Saadatfar, M. & Senden, T. J. Local and global relations between the number of contacts and density in monodisperse sphere packs. J. Stat. Mech. P07010 (2006)

  14. 14

    da Cruz, F., Lechenault, F., Dauchot, O. & Bertin, E. Free volume distributions inside a bidimensional granular medium, in Powders and Grains 2005 (eds García-Rojo, R., Herrmann, H. J. & McNamara, S.) (Balkema, Rotterdam, 2005)

    Google Scholar 

  15. 15

    Bertin, E., Dauchot, O. & Droz, M. Definition and relevance of nonequilibrium intensive thermodynamic parameters. Phys. Rev. Lett. 96, 120601 (2006)

    Article  ADS  Google Scholar 

  16. 16

    Ciamarra, M. P., Coniglio, A. & Nicodemi, M. Thermodynamics and statistical mechanics of dense granular media. Phys. Rev. Lett. 97, 158001 (2006)

    Article  ADS  Google Scholar 

  17. 17

    Brujić, J., Edwards, S. F., Hopkinson, I. & Makse, H. A. Measuring distribution of interdroplet forces in a compressed emulsion system. Physica A 327, 201–212 (2003)

    Article  ADS  Google Scholar 

  18. 18

    Torquato, S., Truskett, T. M. & Debenedetti, P. G. Is random close packing of spheres well defined? Phys. Rev. Lett. 84, 2064–2067 (2000)

    CAS  Article  ADS  Google Scholar 

  19. 19

    Alexander, S. Amorphous solids: their structure, lattice dynamics and elasticity. Phys. Rep. 296, 65–236 (1998)

    CAS  Article  ADS  Google Scholar 

  20. 20

    Edwards, S. F. & Grinev, D. V. Statistical mechanics of stress transmission in disordered granular arrays. Phys. Rev. Lett. 82, 5397–5400 (1999)

    CAS  Article  ADS  Google Scholar 

  21. 21

    Makse, H. A., Johnson, D. L. & Schwartz, L. M. Packing of compressible granular materials. Phys. Rev. Lett. 84, 4160–4163 (2000)

    CAS  Article  ADS  Google Scholar 

  22. 22

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

    Article  ADS  Google Scholar 

  23. 23

    Silbert, L. E., Ertas, D., Grest, G. S., Halsey, T. C. & Levine, D. Geometry of frictionless and frictional sphere packings. Phys. Rev. E 65, 031304 (2002)

    MathSciNet  Article  ADS  Google Scholar 

  24. 24

    Torquato, S. & Stillinger, F. H. Multiplicity of generation, selection, and classification procedures for jammed hard-particle packings. J. Phys. Chem. B 105, 11849–11853 (2001)

    CAS  Article  Google Scholar 

  25. 25

    Brujić, J. et al. Granular dynamics in compaction and stress relaxation. Phys. Rev. Lett. 95, 128001 (2005)

    Article  ADS  Google Scholar 

  26. 26

    Stillinger, F. H. A topographic view of supercooled liquids and glass formation. Science 267, 1935–1939 (1995)

    CAS  Article  ADS  Google Scholar 

  27. 27

    Zhang, H. P. & Makse, H. A. Jamming transition in emulsions and granular materials. Phys. Rev. E 72, 011301 (2005)

    CAS  Article  ADS  Google Scholar 

  28. 28

    Parisi, G. & Zamponi, F. The ideal glass transition of hard spheres. J. Chem. Phys. 123, 144501 (2005)

    Article  ADS  Google Scholar 

  29. 29

    Krzakala, F. & Kurchan, J. Landscape analysis of constraint satisfaction problems. Phys. Rev. E 76, 021122 (2007)

    MathSciNet  Article  ADS  Google Scholar 

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This work is supported by the National Science Foundation, CMMT Division and the US Department of Energy, Office of Basic Energy Sciences, Geosciences Division. We are grateful to J. Brujić, A. Yupanqui and M. Makse for stimulating discussions.

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Corresponding author

Correspondence to Hernán A. Makse.

Supplementary information

Supplementray Information

The file contains Supplementary Notes which describe the details of the calculations leading to the phase diagram of jammed matter. The main theoretical result is the deffinition of the volume function based on the Voronoi volume of a particle explained in Section I. Section II describes the isostatic condition that deffines the ensemble of jammed matter through the W function. Section III explains the difference between the geometrical, z, and mechanical, Z, coordination number, which is important to deffine the canonical partition function. Section IV deffines the partition function for the ensemble of jammed states under the isostatic condition, Qiso. Section V explains how to solve the partition function Qiso which leads to the equations of state (4) and (5), and the phase diagram of Fig. 1. Finally, Section VI explains the numerical studies to test the theoretical predictions. The file also includes Supplementary Figures 1-14 with Legends. (PDF 930 kb)

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Song, C., Wang, P. & Makse, H. A phase diagram for jammed matter. Nature 453, 629–632 (2008). https://doi.org/10.1038/nature06981

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