Critical and noncritical jamming of frictional grains

We probe the nature of the jamming transition of frictional granular media by studying their vibrational properties as a function of the applied pressure p and friction coefficient μ. The density of vibrational states exhibits a crossover from a plateau at frequencies ω≳ω^*(p,μ) to a linear growth for ω≲ω^*(p,μ). We show that ω^* is proportional to Δz, the excess number of contacts per grain relative to the minimally allowed, isostatic value. For zero and infinitely large friction, typical packings at the jamming threshold have Δz→0, and then exhibit critical scaling. We study the nature of the soft modes in these two limits, and find that the ratio of elastic moduli is governed by the distance from isostaticity.