Edge Effects in Jammed Systems

Packings of spheres at zero temperature and shear stress exhibit a jamming/unjamming transition as a function of density. For spheres that repel when they overlap and do not otherwise interact, packings are jammed with a nonzero static shear modulus at high densities. As density decreases towards the unjamming transition, the number of interacting neighbors per particle, $z$, decreases towards a critical value $z_c$, so that at the unjamming transition the system just has the minimum number of interacting neighbors to be mechanically stable. In 2005, Wyart, et al. [1] proposed that there is a diverging length scale, $l^*$, associated with this transition, that can be understood from a ``cutting argument." Thus, if one cuts a cluster of linear dimension $L$, the cluster will have zero-frequency vibrational modes (soft modes) only for $L