Random Close Packing

Random close packing is a mathematically ill-defined notion that is better replaced by maximally random jammed packing (Torquato et al. 2000).

Zaccone's (2022a) model estimates a packing density of approximately 0.8865 for random close packing of circles in two dimensions.

Random close packing of spheres in three dimensions typically gives a packing density of (Bernal and Mason 1960, Jaeger and Nagel 1992, Zaccone 2022a), significantly smaller than the optimal packing density for cubic close packing or hexagonal close packing of 0.74048. Zaccone's (2022a) model gives the estimate

			(1)
			(2)

using the Percus-Yevick approximation, or

(3)

using the Carnahan-Starling approximation. These values are model-dependent estimates rather than exact packing densities, and the derivation has been disputed (Zaccone 2022b, Chen and Ni 2022).

Donev et al. (2004) showed that a maximally random jammed state of M&Ms chocolate candies has a packing density of about 68%, or 4% greater than spheres. Furthermore, Donev et al. (2004) also showed by computer simulations other ellipsoid packings resulted in random packing densities approaching that of the densest sphere packings, i.e., filling nearly 74% of space.