Caging of ad-dimensional sphere and its relevance for the random dense sphere packing

Abstract

We analyze the caging of a hard sphere (i.e., the complete arrest of all translational motions) by randomly distributed static contact points on the sphere surface for arbitrary dimension d>/=1, and prove that the average number of uncorrelated contacts required to cage a sphere is <N>(d)=2d+1. Computer simulations, which confirm this analytical result, are also used to model the effect of correlations between contacts that occur in real hard-sphere systems. Our analysis predicts an average coordination number of 4.79 (+/-0.02) for caged spheres, which agrees surprisingly well with the experimental coordination number for random sphere packings reported by Mason [Nature 217, 733 (1968)]. This result supports the physical picture that the coordination number in random dense sphere packings is primarily determined by caging effects. It also suggests that it should be possible to construct such packings from a local caging rule.

Simulation of random packing of binary sphere mixtures by mechanical contraction

“…A sphere is caged if it is surrounded by contacting spheres such that the sphere is unable to move [21]. A surprisingly high percentage of the spheres is non-caged in the range 0ox 2 o0:4, Fig.…”

Geometrical cluster ensemble analysis of random sphere packings

“…Collectively jammed and strictly jammed are more stringent conditions 18 where a collection of spheres or all spheres cannot translate or rotate. Peters et al 19 analyzed a specific case of local jamming, namely, the caging of a sphere with the corresponding caging number defined as the average minimum number of spheres that needs to be placed at random on the surface of sphere S to block all translational degrees of freedom of S with the condition of nonoverlap for spheres. 19,20 In a disorded sphere packing it is expected that as a first approximation the contacts on each sphere are distributed randomly over the sphere surface, constrained by the nonoverlap condition.…”

“…Peters et al 19 analyzed a specific case of local jamming, namely, the caging of a sphere with the corresponding caging number defined as the average minimum number of spheres that needs to be placed at random on the surface of sphere S to block all translational degrees of freedom of S with the condition of nonoverlap for spheres. 19,20 In a disorded sphere packing it is expected that as a first approximation the contacts on each sphere are distributed randomly over the sphere surface, constrained by the nonoverlap condition. Thus in our approach, for spheres in a random packing to be locally jammed, the average number of contacts at least equals the caging number if contacts are distributed randomly on the surfaces of spheres.…”

“…The mathematical criterion for a sphere S to be noncaged, namely, that a hemisphere on S can be chosen such that all contacts are part of that hemisphere, 19,20 can be cast into a problem of contact normal forces to give a more physical picture. For a noncaged sphere S all vector sums of nonzero normal forces applied at the contact points are nonzero.…”

On contact numbers in random rod packings

“…2.1 we describe the simulation method and the evaluation of contact numbers employing expansion of particles that interact via a spring-dashpot model. The caging problem, i.e., finding the average minimal number of uncorrelated contacts needed to arrest a particle, has only been solved for spheres [20,21] and 2-dimensional discs [22]. In Sect.…”