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 $\mu$. The density of vibrational states exhibits a crossover from a plateau at frequencies $\omega \gtrsim {\omega }^{*}(p,\mu )$ to a linear growth for $\omega \lesssim {\omega }^{*}(p,\mu )$. We show that ${\omega }^{*}$ is proportional to $\mathrm{\Delta }z$, 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 $\mathrm{\Delta }z\to 0$, 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.

Figure 1

(a) Average contact number $z$ as a function of $p^{1∕3}$ for various $\mu$ as indicated. (b)–(e) Vibrational DOS for granular packings for friction coefficients as indicated, and for pressures approximately $5\times {10}^{-6}$, $5\times {10}^{-5}$, $5\times {10}^{-4}$, $4\times {10}^{-3}$, and $3\times {10}^{-2}$. For decreasing $p$ the DOS becomes steeper for small $\omega$, and the crossover frequency ${\omega }^{*}$, indicated in (e), decreases with $p$. The packing with $\mu =0^{+}$ is obtained by first making a frictionless packing and then turning on the tangential frictional forces in the DOS calculation. As noted in the text, all frequencies are scaled by a factor $p^{1∕6}$.

Figure 2

(a) ${\omega }^{*}$ for $\mu =10$ as a function of ${\omega }_{\mathrm{overlap}}$—the regime $(1<{\omega }_{\mathrm{overlap}}<3)$ corresponds to the crossover regime in the DOS that we focus on here [see text and (b)]. (b) The rescaled DOS for $\mu =10$ exhibit good data collapse in the crossover regime. Here 20 rescaled DOS are shown with $p$ ranging from $9\times {10}^{-7}$ to $3\times {10}^{-2}$.

Figure 3

(a) ${\omega }^{*}$ as a function of pressure $p$ for a range of friction coefficients $\mu$—error bars are similar to or smaller than symbol sizes. (b) Deviation from isostaticity for the same range of parameters. (c) ${\omega }^{*}$ scales linearly with the distance to isostaticity for frictional packings. (d), (e) ${\omega }^{*}$ for frictionless packings scales with both $p$ and $z-4$. Dashed lines indicate power laws with exponents as indicated. For details, see text.

Figure 4

Scaling of bulk modulus $K$ and shear modulus $G$ as function of $p$ and $\mu$—as for the data for ${\omega }^{*}$, the trivial $p^{1∕3}$ dependence has been divided out. (a) The rescaled bulk modulus $K$ (curve) essentially levels off for small $p$, while the shear modulus $G$ (symbols as in Fig. 3) varies strongly with both $p$ and $\mu$. (b) $G∕K$ scales like the excess contact number for small $\mathrm{\Delta }z$.