On the caging number of two- and three-dimensional hard spheres

Local structural arrest in random packings of colloidal or granular spheres is quantified by a caging number, defined as the average minimum number of randomly placed spheres on a single sphere that immobilise all its translations. We present an analytic solution for the caging number for two-dimensional hard disks immobilised by neighbour disks for which the random positioning of disks is constrained by a non-overlap condition. Immobilization of a disk by disks of arbitrary size is solved analytically for a size ratio larger than one, for disks with size ratio lower than one it can be evaluated accurately with an approximate excluded volume model that also applies to spheres in higher dimension. Comparison of our exact two-dimensional caging number with studies on random disk packing indicates that it relates to the average co-ordination number of random loose packing, whereas the parking number is more indicative for coordination in random dense packing of disks.