Emergence of a critical scale in jamming of frictional grains

We probe the characteristic scale of two-dimensional frictional granular media close to the jamming transition by studying their vibrational properties as function of the applied pressure $P$ and friction coefficient $\mu$. The density of vibrational states exhibits a crossover from a plateau at frequencies $\omega > \omega^*(P,\mu)$ to a linear growth for $\omega < \omega^*(P,\mu)$. Both for large and for zero friction, this characteristic frequency $\omega^*$ vanishes when $P$ is lowered towards zero. For moderate friction, however, $\omega^*$ saturates at a finite value for $P\downarrow0$. We show that $\omega^*$ is proportional with $\Delta z$, the excess number of contacts per grains relative to the minimally allowed, isostatic value, and that only for zero and infinitely large friction, systems at the jamming threshold have $\Delta z \rightarrow 0$ and hence are critical.