Packing of Regular (Spherical) and Irregular Particles

Abstract

RECENT correspondence^1,2 in Nature cites a theorem on the ordered packing of equal spheres to prove that packings of graded sizes of coal have the same voidage (fraction of free space) for all sizes. This theorem can be stated in its most general form as follows: the overall voidage of any one of the four possible ordered arrangements of equal spheres packed into a container is independent of the relative size of the container provided its size and shape are such that it will contain an integral number of ‘unit’ cells of the particular ordered arrangement; in all other cases an ordered packing cannot be obtained. Graton and Fraser³ have given the size and shape of the ‘unit’ cells for the four arrangements, where one sphere touches six (cubic), eight (orthorhombic), ten (tetragonal) and twelve (rhombohedral) neighbours respectively. The same authors give the correct voidages 0·4764, 0·3954, 0·3019 and 0·2595 for the four cases and discuss also the case of ordered packings of spheres of different sizes. (Mr. Ackermann's figures for the case of eight contacts on each sphere appear to be incorrect.)