Random close packing (RCP) of spheres 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 an ordered structure, such as a regular crystal lattice, this is the empirical random close-packed density for this particular procedure of packing. The random close packing is the highest possible volume fraction out of all possible packing procedures.
Experiments and computer simulations have shown that the most compact way to pack hard perfect same-size 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. The problem of predicting theoretically the random close pack of spheres is difficult mainly because of the absence of a unique definition of randomness or disorder.[1] The random close packing value is significantly below the maximum possible close-packing of same-size 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.
The random close packing fraction of discs in the plane has also been considered a theoretically unsolved problem because of similar difficulties. An analytical, though not in closed form, solution to this problem was found in 2021 by R. Blumenfeld.[3] The solution was found by limiting the probability of growth of ordered clusters to be exponentially small and relating it to the distribution of `cells', which are the smallest voids surrounded by connected discs. The derived maximum volume fraction is 85.3542%, if only hexagonal lattice clusters are disallowed, and 85.2514% if one disallows also deformed square lattice clusters.
An analytical and closed-form solution for both 2D and 3D, mechanically stable, random packings of spheres has been found by A. Zaccone in 2022 using the assumption that the most random branch of jammed states (maximally random jammed packings, extending up to the fcc closest packing) undergo crowding in a way qualitatively similar to an equilibrium liquid.[4][5] The reasons for the effectiveness of this solution are the object of ongoing debate.[6]