Universal Microstructure and Mechanical Stability of Jammed Packings

The mechanical properties of jammed packings depend sensitively on their detailed local structure. Here we provide a complete characterization of the pair correlation close to contact and of the force distribution of jammed frictionless spheres. In particular we discover a set of new scaling relations that connect the behavior of particles bearing small forces and those bearing no force but that are almost in contact. By performing systematic investigations for spatial dimensions $d=3–10$, in a wide density range and using different preparation protocols, we show that these scalings are indeed universal. We therefore establish clear milestones for the emergence of a complete microscopic theory of jamming. This description is also crucial for high-precision force experiments in granular systems.

Figure 1

The extrapolated jamming density ${\phi }_p^{\gamma \to 0}$ following the protocol described in Ref. 28 is extended to higher $d$ (solid line and crosses), and compared with the G-RT prediction for ${\phi }_{\mathrm{GCP}}$ (dashed line). (top inset) The range of jamming densities ${\phi }_p^{\gamma }$ (squares) is compared to ${\phi }_e^{\max }$ (circles) and ${\phi }_e^{\min }$ (triangles). Note that ${\phi }_e^{\max }\sim {\phi }_p^{\gamma =3\times {10}^{-4}}$ and ${\phi }_e^{\min }\sim {\phi }_p^{\gamma =3\times {10}^{-2}}$. (bottom inset) The $d=3$ increase of ${\phi }_e^{\downarrow }$ with ${\sigma }_{+}$, in terms of the initial effective packing fraction.

Figure 2

Schematic of $Z(r)$ when approaching jamming from above (a) with protocol (ii), and from below (b) with protocol (i). Three distinct scaling regimes can be identified. The first regime is related to the growth of $Z(r)$ from 0 to $\overline{z}$. It corresponds to interparticle gaps $h=|r-\sigma |\sim \Delta \phi$, and hence to particles that are in contact when $\Delta \phi \to 0$. The last regime corresponds to gaps $h$ that remain finite for $\Delta \phi \to 0$. These particles remain separated at jamming, but these small gaps, $Z(r)-\overline{z}\propto h^{1-\alpha }$, form a “quasicontact” regime. The intermediate regime corresponds to gaps $h\sim \Delta {\phi }^{\mu }$. It matches the two other regimes and disappears when $\Delta \phi \to 0$.

Figure 3

Scaling of the cumulative structure function $Z(r)$ and the cumulative force distribution $G(f)$ in $d=3$ upon approaching jamming from above (a) by SS energy minimization (where $e\propto |\Delta \phi {|}^2\to 0$ and $\epsilon =\sqrt{ed/2}$), and from below (b) by HS compression (where $p\propto |\Delta \phi {|}^{-1}\to \infty$). (1a) For diminishing $e$, the height of the plateau (inset) converges to the isostatic value with $\zeta =0.53(3)$. (2a) The small $r<\sigma$ regime shows the “contact” scaling function ${\mathcal{Z}}_{+}(x)$, which agrees well with the G-RT prediction (red line). (3a) Rescaling $Z(r)$ using $\mu =(1+\theta )/(2+\theta -\alpha )$ and $\nu =\alpha \mu$ highlights the behavior of the scaling function $|{\mathcal{H}}_{-}(x)-{\mathcal{H}}_{-}(1)|\sim 1.2x$ (brown lines—intermediate gray) along with the $\theta =0.42(2)$ (red line—light gray) and the $\alpha =0.39(1)$ (blue line—dark gray) power-law regimes. (4a) $G(f)$, with power-law tail exponent $\theta =0.42(2)$ (dashed line). (1b) For increasing $p$, $Z(r)$ grows on an earlier scale $r-\sigma \sim p^{-1}$ to a plateau at the isostatic value, whose height (inset) decays with $\zeta =0.36(1)$. (2b) The small $r-\sigma$ regime shows the “contacts” scaling function ${\mathcal{Z}}_{-}(x)$, which agrees well with the G-RT prediction (red line). (3b) Rescaling $Z(r)$ using $\mu =(1+\theta )/(2+\theta -\alpha )$ and $\nu =\alpha \mu$ highlights the behavior of the scaling function $|{\mathcal{H}}_{-}(x)-{\mathcal{H}}_{-}(1)|\sim 6x$ (brown lines—intermediate gray) along with the $\theta =0.28(3)$ (red line—light gray) and $\alpha =0.42(2)$ (blue line—dark gray) power-law regimes. (4b) $G(f)$, with power-law tail exponent $\theta =0.28(3)$ (dashed line), compared with the G-RT prediction (solid line).