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.

Experiments have shown that the most compact way to pack spheres randomly gives a maximum density of about 64%. Most recent research predicts analytically that the volume fraction filled by the solid objects in random close packing cannot exceed a density limit of 63.4% for (monodisperse) spherical objects.^[1] This is significantly smaller than the maximum theoretical filling fraction of 0.74048 that results from hexagonal close pack (HCP – also known as close-packing). This discrepancy demonstrates that the "randomness" of RCP is vital to the definition.

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. 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.^[2]

For spheres

Comparison of various models of close sphere packing (monodispersed)^[3]

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

The permeability of beds of close packed spheres has been extensively studied because it is one of the basic models of porous media. Multiple formulas have been proposed to express permeability of such a bed a function of bed porosity. The Carman-Kozeny model predicts that:^[4]

$K=\frac {\epsilon^3}{36k(1-\epsilon)^2}d^2$

where:

• $K$ – permeability (m²)
• $\epsilon$ – porosity (dimensionless)
• $d$ – sphere diameter (m)
• $k$ – Kozeny-Carman constant, $k = 5$ for beds packed with spherical particles^[5]

Rumpf and Gupte gave the following equation, which may offer better fit with experimental data:^[4][6]

$K=\frac {\epsilon^{5.5}}{5.6}d^2$

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.

See also

• Sphere packing
• Close-packing of equal spheres

References

1. Song, C.; Wang, P.; Makse,H.A. (29 May 2008). "A phase diagram for jammed matter". Nature 453 (7195): 629–632. arXiv:0808.2196. Bibcode:2008Natur.453..629S. doi:10.1038/nature06981. PMID 18509438.
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. doi:10.1126/science.1093010. PMID 14963324.
3. F.A.L. Dullien, "Porous Media. Fluid Transport and Pore Structure", 2nd edition, Academic Press Inc., 1992.
4. M.Kaviany, "Principles of Heat Transfer in Porous Media", Springer-Verlag, 1991
5. Kaviany, M (1995). Principles of heat transfer in porous media (2nd Ed.). New York: Springer.
6. H.Rumpf and A.R.Gupte, "The influence of porosity and grain size distribution on the permeability equation of porous flow", Chemie Ing. Techn. (Weinheim), v. 43, no. 6, p 367-375, 1975

• "Physics of Granular States." Science 255, 1523, 1992.
• "Improving the Density of Jammed Disordered Packings using Ellipsoids." Science, 303, 990–993, 2004.