- Physical Review Journals
All JournalsPhysics Magazine

Physical Review E

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Random packing fraction of binary hyperspheres with small or large size difference: A geometric approach

H. J. H. Brouwers*

  • *Contact author: jos.brouwers@tue.nl
PDF

Phys. Rev. E 112, 025411 – Published 8 August, 2025

DOI: https://doi.org/10.1103/xhhh-hvhg

Abstract

The random packing fraction of binary particles in D-dimensional Euclidean space RD is studied using a geometric approach. First, the binary packing fraction of assemblies with small size difference are studied, using a recently developed model that has its foundations in the excluded volume model by Onsager for cylinders and spherocylinders (D=3). According to this model the packing increase by bidispersity is proportional to (1−f)(uD−1)2, with f as monosized packing fraction, u as size ratio and D as space dimension. The model predictions are compared with computational results for disks in two dimensions (D=2) and hyperspheres in the large-dimension limit (D→∞), yielding very good agreement. Subsequently, the packing of hyperspheres with large size difference is modeled, employing the classic theory of Furnas. This theory, developed for three dimensions, starts from an infinite size ratio of larger and smaller particles (u→∞). Here, the pertaining equations are applied to hyperspheres, and successfully compared with computational results for hyperspheres in the large-dimension limit. Furthermore, an asymptotic approximation of the binary packing fraction for large size ratio is derived, which shows that the first order variation of the Furnas packing fraction (u−1=0) is proportional to (2−f)u−1. Finally, a normalized D-dimensional binary packing graph is presented, governing a simplified phase diagram that borders the binary random packing fraction of amorphous assemblies. To summarize, basic space-filling and geometric (“athermal”) theories on “simple” hard spheres appear to be a valuable tool for the study of hyperspheres’ random packing and amorphization.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (47)

  1. S. Torquato and F. H. Stillinger, Rev. Mod. Phys. 82, 2633 (2010).
  2. G. Parisi and F. Zamponi, Rev. Mod. Phys. 82, 789 (2010).
  3. P. Charbonneau, A. Ikeda, G. Parisi, and F. Zamponi, Phys. Rev. Lett. 107, 185702 (2011).
  4. C. C. Furnas, Department of Commerce, Bureau of Mines, Report of Investigation Serial No. 2894 (1928).
  5. C. C. Furnas, Bull. U.S. Bureau Mines 307, 74 (1929).
  6. A. E. R. Westman and H. R. Hugill, J. Am. Ceram. Soc. 13, 767 (1930).
  7. C. C. Furnas, Ind. Eng. Chem. 23, 1052 (1931).
  8. A. E. R. Westman, J. Am. Ceram. Soc. 19, 127 (1936).
  9. M. A. Caquot, Mémoires de la Société des Ingénieurs Civils de France, Mémoire 562 (1937).
  10. P. C. Mangelsdorf and E. L. Washington, Nature (London) 187, 930 (1960).
  11. R. K. McGeary, J. Am. Ceram. Soc. 44, 513 (1961).
  12. R. Jeschar, W. Pötke, V. Petersen, and K. Polthier, in Proceedings of the Symposium on Blast Furnace Aerodynamics, Wollongong, Australia, 25–27 September 1975, edited by N. Standish (Illawara branch of the Australasian Institute of Mining and Metallurgy (Aus. I.M.M.), 1975), pp. 136–147.
  13. A. S. Clarke and J. D. Wiley, Phys. Rev. B 35, 7350 (1987).
  14. A. P. Shapiro and R. F. Probstein, Phys. Rev. Lett. 68, 1422 (1992).
  15. A. Yang, C. T. Miller, and L. D. Turcoliver, Phys. Rev. E 53, 1516 (1996).
  16. D. He, N. N. Ekere, and L. Cai, Phys. Rev. E 60, 7098 (1999).
  17. H. J. H. Brouwers, Phys. Rev. E 74, 031309 (2006); 74, 069901(E) (2006); 84, 059905(E) (2011).
  18. H. J. H. Brouwers, Phys. Rev. E 76, 041304 (2007).
  19. Y. Shi and Y. Zhang, Appl. Phys. A 92, 621 (2008).
  20. H. J. H. Brouwers, Phys. Rev. E 78, 011303 (2008).
  21. I. Biazzo, F. Caltagirone, G. Parisi, and F. Zamponi, Phys. Rev. Lett. 102, 195701 (2009).
  22. R. S. Farr and R. D. Groot, J. Chem. Phys. 131, 244104 (2009).
  23. M. Clusel, E. I. Corwin, A. O. N. Siemens, and J. Brujić, Nature (London) 460, 611 (2009).
  24. M. Danisch, Y. Jin, and H. A. Makse, Phys. Rev. E 81, 051303 (2010).
  25. A. V. Kyrylyuk, A. Wouterse, and A. P. Philipse, Progr. Colloid Polym. Sci. 137, 29 (2010).
  26. K. A. Newhall, I. Jorjadze, E. Vanden-Eijnden, and J. Brujić, Soft Matter 7, 11518 (2011).
  27. A. B. Hopkins, F. H. Stillinger, and S. Torquato, Phys. Rev. E 88, 022205 (2013).
  28. H. J. H. Brouwers, Phys. Rev. E 87, 032202 (2013).
  29. K. W. Desmond and E. R. Weeks, Phys. Rev. E 90, 022204 (2014).
  30. H. Ikeda, K. Miyazaki, H. Yoshino, and A. Ikeda, Phys. Rev. E 103, 022613 (2021).
  31. C. Anzivino, M. Casiulis, T. Zhang, A. S. Moussa, S. Martiniani, and A. Zaccone, J. Chem. Phys. 158, 044901 (2023).
  32. H. J. H. Brouwers, Physics–Uspekhi 67, 510 (2024).
  33. L. Onsager, Ann. N.Y. Acad. Sci. 51, 627 (1949).
  34. Y. H. Yang and D. D. Wan (private communications, 2025).
  35. D. Y. Chen, Y. Jiao, and S. Torquato, J. Phys, Chem. B 118, 7981 (2014).
  36. L. Meng, P. Lu, S. Li, J. Zhao, and T. Li, Powder Technol. 228, 284 (2012).
  37. J. A. Anderson, J. Glaser, and S. C. Glotzer, Comput. Mater. Sci. 173, 109363 (2020).
  38. K. W. Desmond (private communications, 2025).
  39. K. W. Desmond and E. R. Weeks, Phys. Rev. E 80, 051305 (2009).
  40. H. J. H. Brouwers, Soft Matter 19, 8465 (2023).
  41. A. P. Philipse, Colloids Surf. A 213, 167 (2003).
  42. G. Parisi and F. Zamponi, J. Stat. Mech. (2006) P03017.
  43. M. Skoge, A. Donev, F. H. Stillinger, and S. Torquato, Phys. Rev. E 74, 041127 (2006).
  44. B. Schmid and R. Schilling, Phys. Rev. E 81, 041502 (2010).
  45. H. J. H. Brouwers, Phys. Rev. E 89, 052211 (2014).
  46. H. Y. Sohn and C. Moreland, Can. J. Chem. Eng. 46, 162 (1968).
  47. G. D. Scott, Nature (London) 188, 908 (1960).

Outline

Information

Phys. Rev. E 112, 025411– Published 8 August, 2025
  • Received 29 January 2025
  • Revised 21 March 2025
  • Accepted 26 June 2025
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation