On the rigidity of a hard-sphere glass near random close packing

Abstract

In fig. 4, the abscissa has an incorrect unit: the force must be divided by a factor 5.65. Obviously, this does not change the observed power law.

Force distribution affects vibrational properties in hard-sphere glasses

“…This directly leads to an effective force law f ðhÞ ≈ k B T=h and allows one to map a hard-sphere system near the random close packing ϕ c to a zero-temperature elastic network. These two sets of results yield a stability constraint on the microscopic structure of hard-sphere glasses, which in practice appears to lie very close to saturation (6,7,12). Such marginal stability implies the abundance of very soft elastic modes, as confirmed empirically (6,7,(12)(13)(14)(15)(16), and fixes the scaling behavior of elasticity as jamming is approached (7).…”

“…These two sets of results yield a stability constraint on the microscopic structure of hard-sphere glasses, which in practice appears to lie very close to saturation (6,7,12). Such marginal stability implies the abundance of very soft elastic modes, as confirmed empirically (6,7,(12)(13)(14)(15)(16), and fixes the scaling behavior of elasticity as jamming is approached (7). In particular the particles' mean-squared displacement was predicted to follow hδR 2 i ∼ ðϕ c − ϕÞ κ with κ = 1:5 (7) instead of the naive κ = 2, which would hold in a crystal: Particles in the glass fluctuate much more than the size of their cage (defined as the typical distance between particles), due to the presence of collective soft modes.…”

“…Using [4], this becomes δz J e ð1+αÞ=ð2+αÞ , extending the previous result δz J ffiffi ffi e p (10) to the case α < 0. In packings of particles, e ∝ jϕ − ϕ c j, and the latter bound was argued to be saturated, based on dynamical considerations (6,7,10).…”

“…This scenario is presumably what mode-coupling theory captures (2,3) and can be rationalized via density functional theory (4) and via the replica method (5). Recently a real-space description of mechanical stability and elasticity in hard-sphere glasses has been proposed (6,7), which is most easily tested at large pressure, deep in the glass phase. It is based on two results.…”

“…Second, within a long-lived metastable state the vibrational free energy of a hard-sphere system can be approximated as a sum of local interaction terms between pairs of colliding particles, which are said to be in contact. On a time scale that contains many collisions, at high packing fraction the interaction follows approximately V ðhÞ ≈ −k B T log h, where h is the time-averaged distance between two adjacent particles (6,7). This directly leads to an effective force law f ðhÞ ≈ k B T=h and allows one to map a hard-sphere system near the random close packing ϕ c to a zero-temperature elastic network.…”