Glass and Ceramics

, Volume 75, Issue 9–10, pp 345–351 | Cite as

Hard-Sphere Close-Packing Models: Possible Applications for Developing Promising Ceramic and Refractory Materials (Review)

  • A. V. SmirnovEmail author
  • S. G. Ponomarev
  • V. P. Tarasovskii
  • V. V. Rybal’chenko
  • A. A. Vasin
  • V. V. Belov

The current status of theoretical research on close-packed systems of hard spheres is reviewed. The basic models of regular and random close-packings of hard spheres are described. Examples are presented of the application of modeling of the close packing of hard spheres for solving applied problems in the development of promising materials made from ceramics, including refractories.

Key words

hard-sphere model random close-packing regular closest-packing modeling of hard-sphere packing powder materials 


Financial support for this work was provided by the Ministry of Education and Science of the Russian Federation as part of government task No. 11.5987.2017/VU for performing the work ‘Organization of scientific research’ (Publication No. 11.5987.2017/6.7) using equipment from the Center for Collective Use ‘Science intensive technologies in machine engineering’ at Moscow Polytechnic University.


  1. 1.
    G. Parisi and F. Zamponi, “Mean-field theory of hard sphere glasses and jamming,” Rev. Mod. Phys., 82(1), 789 – 845 (2011).CrossRefGoogle Scholar
  2. 2.
    A. R. Kansal, T. M. Truskett, and S. Torquato, “Nonequilibrium hard-disk packings with controlled orientational order,” J. Chem. Phys., 113(12), 4844 – 4851 (2000).CrossRefGoogle Scholar
  3. 3.
    L. Burtseva, B. V. Salas, F. Werner, and V. Petranovskii, Modeling of Monosized Sphere Packings into Cylinders, Technical Report, January 2015; DOI:
  4. 4.
    T. P. Bondareva, “Computer simulation of the structure of random packing of systems of spherical particles,” Nauch. Vedom. Belgorod. Gos. Univ., Ser. Ékonomika, Informatika, 25(1-1), 78 – 85 (2013).Google Scholar
  5. 5.
    T. Hales, M. Adams, G. Bauer, et al., “A formal proof of the Kepler conjecture,” Forum of Mathematics, Pi 5; DOI:
  6. 6.
    K. Jeffery, J. Wilson, G. Casali, and R. Hayman, “Neural encoding of large-scale three-dimensional space properties and constraints,” Frontiers in Psychology, No. 6 (2015);
  7. 7.
    A. L. Mackay, “A dense noncrystallographic packing of equal spheres,” Acta Cryst., 15(9), 916 – 918 (1962).CrossRefGoogle Scholar
  8. 8.
    D. Hilbert and S. Cohn-Vossen, Geometry and the Imagination, Chelsea, New York (1999), pp. 45 – 53.Google Scholar
  9. 9.
    H. Steinhaus, Mathematical Snapshots, Dover, New York (1999), pp. 202 – 203.Google Scholar
  10. 10.
    D. Wells, The Penguin Dictionary of Curious and Interesting Geometry, Penguin, London (1991), pp. 237 – 238.Google Scholar
  11. 11.
    T. Aste, “Circle, sphere, and drop packings,” Phys. Rev. E, 53, 2571 (1996).CrossRefGoogle Scholar
  12. 12.
    G. D. Scott and D. M. Kilgour, “The density of random close packing of spheres,” Brit. J. Appl. Phys., 2(6), 863 (1969).Google Scholar
  13. 13.
    O. Pouliquen, M. Nicolas, and P. D. Weidman, “Crystallization of non-Brownian spheres under horizontal shaking,” Phys. Rev. Lett., 79, 3640 – 3643 (1997).CrossRefGoogle Scholar
  14. 14.
    E. E. Lord, A. L. Mackay, and S. Ranganathan, New Geometry for New Materials [Russian translation], Fizmatlit, Moscow (2010).Google Scholar
  15. 15.
    J. D. Bernal, “A geometrical approach to the structure of monatomic liquids,” Nature, 183, 141 – 147 (1959).CrossRefGoogle Scholar
  16. 16.
    J. D. Bernal, “Geometry of the structure of monatomic liquids,” Nature A, 185, 68 – 70 (1960).CrossRefGoogle Scholar
  17. 17.
    J. D. Bernal, “The structure of liquids,” Sci. Am. B, 201, 124 – (1)31 (1960).Google Scholar
  18. 18.
    J. D. Bernal, “The structure of liquids,” Proc. Roy. Soc. London A, 208, 299 – 322 (1964a).Google Scholar
  19. 19.
    J. D. Bernal, “The structure of liquids” New Sci. B, No. 8, 453 – 435 (1964).Google Scholar
  20. 20.
    S. Torquato, T. M. Truskett, and P. G. Debenedetti, “Is random close packing of spheres well defined?,” Phys. Rev. Lett., 20(5), 20 (2000).Google Scholar
  21. 21.
    J. M. Wills, “A quasicrystalline sphere-packing with unexpected high density,” J. Phys. France, 51, 860 – 864 (1990).CrossRefGoogle Scholar
  22. 22.
    K. Gotoh and J. L. Finney, “Statistical geometrical approach to random packing density of equal spheres,” Nature, 252, 202 – 205 (1974).CrossRefGoogle Scholar
  23. 23.
    D. R. Hudson, “Density and packing in an aggregate of mixed spheres,” J. Appl. Phys., 20, 154 (1949); doi: Scholar
  24. 24.
    P. I. O’Toole and T. S. Hudson, “New high-density packings of similarly sized binary spheres,” J. Phys. Chem. C, 115(39), 19037 (2011).CrossRefGoogle Scholar
  25. 25.
    L. V. Korolev, A. P. Lupanov, and Yu, M. Pridatko, “Dense packing of polydisperse particles in composite building materials,” Sovr. Probl. Nauki Obraz., No. 6-1 (2007); URL: (appeal date: 07.11.2017).
  26. 26.
    M. Borkovec, W. De Paris, and R. Peikert, “The fractal dimension of the Apollonian sphere packing,” Fractals, 2(4), 521 – 526 (1994).CrossRefGoogle Scholar
  27. 27.
    R. Blaak, “Optimal packing of polydisperse hard-sphere fluids, II,” J. Chem. Phys., 112, 9041 (2000).CrossRefGoogle Scholar
  28. 28.
    J. Zhang, R. Blaak, E. Trizac, et al., “Optimal packing of polydisperse hard-sphere fluids,” J. Chem. Phys., 110, 5318 (1999).CrossRefGoogle Scholar
  29. 29.
    V. Baranau, D. Hlushkou, S. Khirevich, and U. Tallarek, “Pore-size entropy of random hard-sphere packings,” Soft Matter, No. 9, 3361 – 3372 (2013).Google Scholar
  30. 30.
    V. Baranau and U. Tallare, “Random-close packing limits for monodisperse and polydisperse hard spheres,” Soft Matter, No. 10, 3826 – 3841 (2014).Google Scholar
  31. 31.
    V. V. Belov and M. A. Smirnov, Building Composites of Optimized Mineral Mixtures [in Russian], TvGTU, Tver (2012): URL: (appeal date: 11/17/2017).
  32. 32.
    A. R. Kansai, S. Torquato, and F. H. Stillinger, “Computer generation of dense polydisperse sphere packings,” J. Chem. Phys., ll7, 8212 (2002).CrossRefGoogle Scholar
  33. 33.
    R. M. Baram and H. J. Herrmann, “Self-similar space – filling packings in three dimensions,” Fractals, 12(3), 293 (2004).CrossRefGoogle Scholar
  34. 34.
    R. G. Eromasov, Composite Ceramic Materials Based on Coarse-Grained Technogenic Filler, Author’s Abstract of Candidate’s Thesis [in Russian], Krasnoyarsk (2014).Google Scholar
  35. 35.
    V. V. Belov, M. A. Smirnov, and I. V. Obraztsov, “Theoretical foundations of the method of optimizing the particle size composition of compositions for producing non-firing construction conglomerates,” Stroit. Mater., Oborud., Tekhnol. XXI Veka, No. 6 (2012); URL: (appeal date: 11/17/2017).
  36. 36.
    I. I. Loktev, K. Yu. Vergazov, V. A. Vlasov, and I. A. Tikhomirov, “On modeling some technological properties of dispersed materials,” Izv. TPU, 308(6), 85 – 89 (2005).Google Scholar
  37. 37.
    V. V. Belov, I. V. Obraztsov, V. K. Ivanov, and E. N. Konoplev, Computer Implementation of the Solution of Scientific, Technical and Educational Problems [in Russian], TvGTU, Tver (2015); URL: (appeal date: 11/17/2017).
  38. 38.
    Yu. E. Pivinskii, Theoretical Aspects of the Technology of Ceramics and Refractories: Selected Works, Vol. 1 [in Russian], St. Petersburg (2003).Google Scholar
  39. 39.
    G. D. Scott, “Radial distribution of the random close packing of equal spheres,” Nature, 192, 956 – 957 (1962).CrossRefGoogle Scholar
  40. 40.
    A. G. Aslamazov and A. A. Varlamov, Amazing Physics, Dobrosvet, Moscow (2002).Google Scholar
  41. 41.
    G. A. Tingate, “Some geometrical properties of packings of equal spheres in cylindrical vessels,” Nuclear Eng. Design, 24, 153 – 179 (1973).CrossRefGoogle Scholar
  42. 42.
    G. E. Mueller, “Radial void fraction distributions in randomly packed fixed beds of uniformly sized spheres in cylindrical containers,” Powder Technol., 72, 269 – 275 (1992).CrossRefGoogle Scholar
  43. 43.
    Yi. Gan, M. Kamlah, and J. Reimann, “Computer simulation of packing structure in pebble beds,” Fusion Eng. Design, 85, 1782 – 1787 (2010).CrossRefGoogle Scholar
  44. 44.
    R. P. Zou and A. B. Yu, “The packing of spheres in a cylindrical container: The thickness effect,” Chem. Eng. Sci., 50, 1504 – 1507 (1995).CrossRefGoogle Scholar
  45. 45.
    R. M. German, Particle Packing Characteristics, Metal Powder Industries Federation, Princeton, New Jersey (1989).Google Scholar
  46. 46.
    R. K. McGeary, “Mechanical packing of spherical particles,” J. Am. Ceram. Soc., 44, 513 – 522 (1961).CrossRefGoogle Scholar
  47. 47.
    R. Lakes, “Materials with structural hierarchy,” Nature, 361, 511 – 515 (1993).CrossRefGoogle Scholar
  48. 48.
    R. Fratzl and R. Weinkamer, “Nature’s hierarchical materials,” Progr. Mater. Sci., 52, 1263 – 1334 (2007).CrossRefGoogle Scholar
  49. 49.
    E. Olevsky, “Theory of sintering: from discrete to continuum,” Mater. Sci. Eng., R23, 41 – 100 (1998).CrossRefGoogle Scholar
  50. 50.
    V. Tikare, M. Braginsky, and E. A. Olevsky, “Numerical simulation of solid-state sintering: I, Sintering of three particles,” J. Am. Ceram. Soc., 86(1), 49 – 53 (2003).CrossRefGoogle Scholar
  51. 51.
    M. Braginsky, V. Tikare, and E. Olevsky, “Numerical simulation of solid state sintering,” Int. J. Solids Struct., 42, 621 – 636 (2005).CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • A. V. Smirnov
    • 1
    Email author
  • S. G. Ponomarev
    • 1
  • V. P. Tarasovskii
    • 1
  • V. V. Rybal’chenko
    • 1
  • A. A. Vasin
    • 1
  • V. V. Belov
    • 2
  1. 1.Moscow Polytechnic UniversityMoscowRussia
  2. 2.Tver State Technical UniversityTverRussia

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