Random close pack
Random close packing (RCP) is an empirical parameter used to characterize the maximum volume fraction of solid objects obtained when they are packed randomly. For example, when a solid container is filled with grain, shaking the container will reduce the volume taken up by the objects, thus allowing more grain to be added to the container. In other words, shaking increases the density of packed objects. But shaking cannot increase the density indefinitely, a limit is reached, and if this is reached without obvious packing into a regular crystal lattice, this is the empirical random close-packed density.
Experiments and computer simulations have shown that the most compact way to pack hard perfect spheres randomly gives a maximum volume fraction of about 64%, i.e., approximately 64% of the volume of a container is occupied by the spheres. It seems as if because it is not possible to precisely define 'random' in this sense it is not possible to give an exact value.[1] The random close packing value is significantly below the maximum possible close-packing of (equal sized) hard spheres into a regular crystalline arrangements, which is 74.04%.[2] Both the face-centred cubic (fcc) and hexagonal close packed (hcp) crystal lattices have maximum densities equal to this upper limit, which can occur through the process of granular crystallisation.
Definition
Random close packing does not have a precise geometric definition. It is defined statistically, and results are empirical. A container is randomly filled with objects, and then the container is shaken or tapped until the objects do not compact any further, at this point the packing state is RCP. The definition of packing fraction can be given as: "the volume taken by number of particles in a given space of volume". In other words, packing fraction defines the packing density. It has been shown that the filling fraction increases with the number of taps until the saturation density is reached.[3][4] Also, the saturation density increases as the tapping amplitude decreases. Thus RCP is the packing fraction given by the limit as the tapping amplitude goes to zero, and the limit as the number of taps goes to infinity.
Effect of object shape
The particle volume fraction at RCP depends on the objects being packed. If the objects are polydispersed then the volume fraction depends non-trivially on the size-distribution and can be arbitrarily close to 1. Still for (relatively) monodisperse objects the value for RCP depends on the object shape; for spheres it is 0.64, for M&M's candy it is 0.68.[5]
For spheres
| Model | Description | Void fraction | Packing density |
|---|---|---|---|
| Thinnest regular packing | cubic lattice (Coordination number 6) | 0.4764 | 0.5236 |
| Very loose random packing | E.g., spheres slowly settled | 0.44 | 0.56 |
| Loose random packing | E.g., dropped into bed or packed by hand | 0.40 to 0.41 | 0.59 to 0.60 |
| Poured random packing | Spheres poured into bed | 0.375 to 0.391 | 0.609 to 0.625 |
| Close random packing | E.g., the bed vibrated | 0.359 to 0.375 | 0.625 to 0.641 |
| Densest regular packing | fcc or hcp lattice (Coordination number 12) | 0.2595 | 0.7405 |
Example
Products containing loosely packed items are often labeled with this message: 'Contents May Settle During Shipping'. Usually during shipping, the container will be bumped numerous times, which will increase the packing density. The message is added to assure the consumer that the container is full on a mass basis, even though the container appears slightly empty. Systems of packed particles are also used as a basic model of porous media.
References
- Torquato, S.; Truskett, T.M.; Debenedetti, P.G. (2000). "Is Random Close Packing of Spheres Well Defined?". Physical Review Letters. 84 (10): 2064–2067. arXiv:cond-mat/0003416. Bibcode:2000PhRvL..84.2064T. doi:10.1103/PhysRevLett.84.2064. PMID 11017210.
- Modes of wall induced granular crystallisation in vibrational packing.Granular Matter, 21(2), 26
- Rosato, Anthony D.; Dybenko, Oleksandr; Horntrop, David J.; Ratnaswamy, Vishagan; Kondic, Lou (2010). "Microstructure Evolution in Density Relaxation by Tapping". Physical Review E. 81: 061301. doi:10.1103/physreve.81.061301.
- Ratnaswamy, V.; Rosato, A.D.; Blackmore, D.; Tricoche, X.; Ching, Luo; Zuo, L. (2012). "Evolution of Solids Fraction Surfaces in Tapping: Simulation and Dynamical Systems Analysis". Granular Matter. 14 (2): 163–68. doi:10.1007/s10035-012-0343-2.
- Donev, A.; Cisse, I.; Sachs, D.; Variano, E. A.; Stillinger, F. H.; Connelly, R.; Torquato, S.; Chaikin, P. M. (2004). "Improving the Density of Jammed Disordered Packings Using Ellipsoids". Science. 303 (5660): 990–993. Bibcode:2004Sci...303..990D. CiteSeerX 10.1.1.220.1156. doi:10.1126/science.1093010. PMID 14963324.
- Dullien, F. A. L. (1992). Porous Media: Fluid Transport and Pore Structure (2nd ed.). Academic Press. ISBN 978-0-12-223651-8.
- Jaeger, H. M.; Nagel, S. R. (1992). "Physics of Granular States". Science. 255 (5051): 1523–31. Bibcode:1992Sci...255.1523J. doi:10.1126/science.255.5051.1523. PMID 17820163.
- Donev, A.; Cisse, Ibrahim; Sachs, David; Variano, Evan A.; Stillinger, Frank H.; Connelly, Robert; Torquato, Salvatore; Chaikin, P. M. (2004). "Improving the Density of Jammed Disordered Packings using Ellipsoids". Science. 303 (5660): 990–993. Bibcode:2004Sci...303..990D. CiteSeerX 10.1.1.220.1156. doi:10.1126/science.1093010. PMID 14963324.