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Maximum packing densities of basic 3D objects

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  • Mechanics
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Chinese Science Bulletin

Abstract

Numerical simulation results show that the upper bound order of random packing densities of basic 3D objects is cube (0.78) > ellipsoid (0.74) > cylinder (0.72) > spherocylinder (0.69) > tetrahedron (0.68) > cone (0.67) > sphere (0.64), while the upper bound order of ordered packing densities of basic 3D objects is cube (1.0) > cylinder and spherocylinder (0.9069) > cone (0.7854) > tetrahedron (0.7820) > ellipsoid (0.7707) > sphere (0.7405); these two orders are significantly different. The random packing densities of ellipsoid, cylinder, spherocylinder, tetrahedron and cone are closely related to their shapes. The optimal aspect ratios of these objects which give the highest packing densities are ellipsoid (axes ratio = 0.8:1:1.25), cylinder (height/diameter = 0.9), spherocylinder (height of cylinder part/diameter = 0.35), tetrahedron (regular tetrahedron) and cone (height/bottom diameter = 0.8).

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References

  1. Aste T, Weaire D. The Pursuit of Perfect packing. Bristol: Institute of Physics Publishing, 2000

    Book  Google Scholar 

  2. Hales T C. A proof of the Kepler conjecture. Ann Math, 2005, 162: 1065–1185

    Article  Google Scholar 

  3. Bernal J D. A geometrical approach to the structure of liquids. Nature, 1959, 183: 141–147

    Article  Google Scholar 

  4. Zhao L, Li S X, Liu Y W. Numerical simulation of sphere packing with arbitrary diameter distribution. Chin J Comput Phys, 2007, 24: 625–630

    Google Scholar 

  5. Cumberland D J, Crawford R J. The Packing of Particles. Amsterdam: Elsevier, 1987

    Google Scholar 

  6. Williams S R, Philipse A P. Random packings of spheres and spherocylinders simulated by mechanical contraction. Phys Rev E, 2003, 67: 051301

    Article  Google Scholar 

  7. Ye D N. Problem of particle packing. Geol Sci Tech Inform, 1988, 7: 15–17

    Google Scholar 

  8. Donev A, Cisse I, Sachs D, et al. Improving the density of jammed disordered packings using ellipsoids. Science, 2004, 303: 990–993

    Article  Google Scholar 

  9. Dong Q, Ye D N. Specific gravity, shape and haphazard packing of particles. Chinese Sci Bull, 1993, 38: 54–57

    Google Scholar 

  10. Li S X, Zhao J. Sphere assembly model and relaxation algorithm for packing of non-spherical particles. Chin J Comput Phys, 2009, 26: 454–460

    Google Scholar 

  11. Li S X, Zhao L, Liu Y W. Computer simulation of random sphere packing in an arbitrarily shaped container. CMC-Comput Mater Con, 2008, 7: 109–118

    Google Scholar 

  12. Rosen D L, Georgiev G, Goldman T, et al. Random thoughts on densest packing. Phys Today, 2008, 61: 12–15

    Article  Google Scholar 

  13. Weitz D A. Packing in the spheres. Science, 2004, 303: 968–969

    Article  Google Scholar 

  14. Zhao J, Li S X. Numerical simulation of random close packings in particle deformation from spheres to cubes. Chin Phys Lett, 2008, 25: 4034–4037

    Article  Google Scholar 

  15. Trovato A, Hoang X T, Banavar J R, et al. Symmetry, shape, and order. Proc Natl Acad Sci USA, 2007, 104: 19187–19192

    Article  Google Scholar 

  16. Chen T, Zhang Z L, Glotzer S C. A precise packing sequence for self-assembled convex structures. Proc Natl Acad Sci USA, 2007, 104: 717–722

    Article  Google Scholar 

  17. Conway J H, Torquato S. Packing, tiling, and covering with tetrahedra. Proc Natl Acad Sci USA, 2006, 103: 10612–10617

    Article  Google Scholar 

  18. Chen E R. A dense packing of regular tetrahedra. Discrete Comput Geom, 2008, 40: 214–240

    Article  Google Scholar 

  19. Torquato S, Jiao Y. Dense packings of the Platonic and Archimedean solids. Nature, 2009, 460: 876–879

    Article  Google Scholar 

  20. Chaikin P, Wang S, Jaoshvili A. packing of tetrahedral and other dice. In: American Physical Society, APS March Meeting, March 5–9, 2007

  21. Latham J P, Lu Y, Munjiza A. A random method for simulating loose packs of angular particles using tetrahedra. Geotechnique, 2001, 51: 871–879

    Google Scholar 

  22. Li S X, Zhao J, Zhou X. Numerical simulation of random close packing with tetrahedra. Chin Phys Lett, 2008, 25: 1724–1726

    Article  Google Scholar 

  23. Charlles R A, Frederico W T, Marcelo C. Influence of particle shape on the packing and on the segregation of spherocylinders via Monte Carlo simulations. Powder Tech, 2003, 134: 167–180

    Article  Google Scholar 

  24. Zou R P, Yu A B. Wall effect on the packing of cylindrical particles. Chem Eng Sci, 1996, 51: 1177–1180

    Article  Google Scholar 

  25. Zhang W L, Thompson K E, Reed A H, et al. Relationship between packing structure and porosity in fixed beds of equilateral cylindrical particles. Chem Eng Sci, 2006, 61: 8060–8074

    Article  Google Scholar 

  26. Donev A, Stillinger F H, Chaikin P M, et al. Unusually dense crystal packings of ellipsoids. Phys Rev Lett, 2004, 92: 255506

    Article  Google Scholar 

  27. Donev A, Connelly R, Stillinger F H, et al. Underconstrained jammed packings of nonspherical hard particles: Ellipse and ellipsoids. Phys Rev E, 2007, 75: 051304

    Article  Google Scholar 

  28. Man W N, Donev A, Stillinger F H, et al. Experiments on random packings of ellipsoids. Phys Rev Lett, 2005, 94: 198001

    Article  Google Scholar 

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

    Article  Google Scholar 

  30. Yu A B, Standish N, Mclean A. Porosity calculation of binary mixture of nonspherical particles. J Am Ceram Soc, 1993, 76: 2813–2816

    Article  Google Scholar 

  31. Fraige F Y, Langston P A, Chen G Z. Distinct element modelling of cubic particle packing and flow. Powder Tech, 2008, 186: 224–240

    Article  Google Scholar 

  32. Zou R P, Yu A B. Evaluation of the packing characteristics of mono-sized non-spherical particles. Powder Tech, 1996, 88: 71–79

    Article  Google Scholar 

  33. Wouterse A, Williams S R, Philipse A P. Effect of particle shape on the density and microstructure of random packings. J Phys: Condens Matter, 2007, 19: 406215

    Article  Google Scholar 

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Authors and Affiliations

  1. State Key Laboratory for Turbulence and Complex Systems, College of Engineering, Peking University, Beijing, 100871, China

    ShuiXiang Li, Jian Zhao, Peng Lu & Yu Xie

Authors
  1. ShuiXiang Li
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  2. Jian Zhao
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  3. Peng Lu
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  4. Yu Xie
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Correspondence to ShuiXiang Li.

Additional information

We acknowledge Professor Pu Chen (Peking University) for his earnest help and support to this work. We thank Professor S. Torquato (Princeton University, U.S.), Dr. A. Kyrylyuk (Utrecht University, Holland), Dr. Xiaodong Jia (University of Leeds, U.K.) and Dr. Haiping Zhu (University of New South Wales, Australia) for their valuable discussion and help. This work was supported by the National Natural Science Foundation of China (Grant No. 10772005) and National Basic Research Program of China (Grant No. 2007CB714603).

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Li, S., Zhao, J., Lu, P. et al. Maximum packing densities of basic 3D objects. Chin. Sci. Bull. 55, 114–119 (2010). https://doi.org/10.1007/s11434-009-0650-0

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  • DOI: https://doi.org/10.1007/s11434-009-0650-0

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