Marginal stability in jammed packings: Quasicontacts and weak contacts

Maximally random jammed (MRJ) sphere packing is a prototypical example of a system naturally poised at the margin between underconstraint and overconstraint. This marginal stability has traditionally been understood in terms of isostaticity, the equality of the number of mechanical contacts and the number of degrees of freedom. Quasicontacts, pairs of spheres on the verge of coming in contact, are irrelevant for static stability, but they come into play when considering dynamic stability, as does the distribution of contact forces. We show that the effects of marginal dynamic stability, as manifested in the distributions of quasicontacts and weak contacts, are consequential and nontrivial. We study these ideas first in the context of MRJ packing of $d$-dimensional spheres, where we show that the abundance of quasicontacts grows at a faster rate than that of contacts. We reexamine a calculation of Jin et al. [Phys. Rev. E 82, 051126 (2010)], where quasicontacts were originally neglected, and we explore the effect of their inclusion in the calculation. This analysis yields an estimate of the asymptotic behavior of the packing density in high dimensions. We argue that this estimate should be reinterpreted as a lower bound. The latter part of the paper is devoted to Bravais lattice packings that possess the minimum number of contacts to maintain mechanical stability. We show that quasicontacts play an even more important role in these packings. We also show that jammed lattices are a useful setting for studying the Edwards ensemble, which weights each mechanically stable configuration equally and does not account for dynamics. This ansatz fails to predict the power-law distribution of near-zero contact forces, $P\left(f\right)\sim f^{\theta }$.

Figure 1

Calculation of the (expected) volume of the Voronoi region around an arbitrary sphere. The Voronoi region (medium gray) is the set of all points that are closer to the center of a given sphere than to the center of any other (left). Point $A$ is in the Voronoi region of the sphere with center $O$ if and only if the (light gray) spherical region centered at $A$ and such that $O$ is on its boundary has no sphere centers in its interior (right). Since the probability that this region is empty depends only on the distance $c=\left|AO\right|$, we denote the region by $\Omega \left(c\right)$. The fraction of a spherical surface $S$ (dotted line) of radius $r$ centered at $O$ that intersects $\Omega \left(c\right)$ (thick line) is denoted by $f\left(\frac{r}{2c}\right)$.

Figure 2

Red circles: densities of disordered jammed packings as a function of dimension from Ref. [9]. Blue crosses: densities of jammed Bravais lattice packings from Ref. [20]. The black lines illustrate the scaling exponent predicted as a lower bound by the calculation of Appendix pp2.

Figure 3

The distribution of contact forces over all contacts of all minimally extreme eight-dimensional lattices (green triangles) and in the 200 densest lattices (purple circles). The line shows the best-fit truncated Gaussian as a guide for the eye.