Jamming Transitions in Amorphous Packings of Frictionless Spheres Occur over a Continuous Range of Volume Fractions

We numerically produce fully amorphous assemblies of frictionless spheres in three dimensions and study the jamming transition these packings undergo at large volume fractions. We specify four protocols yielding a critical value for the jamming volume fraction which is sharply defined in the limit of large system size, but is different for each protocol. Thus, we directly establish the existence of a continuous range of volume fractions where nonequilibrium jamming transitions occur. However, these jamming transitions share the same critical behavior. Our results suggest that, even in the absence of partial crystalline ordering, a unique location of a random close packing does not exist.

Figure 1

(a) The final part of a compression-decompression cycle across the jamming transition for $N={10}^3$ and ${\varphi }_{\mathrm{init}}=0.3572$, showing the practical determination of ${\varphi }_J$. (b) Evolution of the energy density during compressions of several runs with $N=1000$. Different curves with the same color correspond to independent realizations at constant ${\varphi }_{\mathrm{init}}$, while different ${\varphi }_{\mathrm{init}}$ are shown with different colors: ${\varphi }_{\mathrm{init}}=0.3572$ (red), 0.5397 (dark green), 0.5672 (blue), 0.5744 (purple), 0.5816 (black), 0.5860 (orange), 0.5890 (light green), 0.5935 (maroon). (c) Same as (b) for the evolution of the number of contacts per particle. (d) All data in (b) and (c) collapse when $\varphi$ is eliminated, showing that the jamming transition is better defined by its properties than by its location.

Figure 2

Scaling behavior along the continuous line of $J$ points. The pressure vanishes as (a) $P\sim (\varphi -{\varphi }_J)$ while (b) the number of contacts is discontinuous, $z\sim z_c+\sqrt{\varphi -{\varphi }_J}$ with $z_c\simeq 6$. We show data for 4 representative points $J$ with 4 different ${\varphi }_J$.

Figure 3

(a) Statistical distribution of the number of contacts per particle, $z$, for 4 points $J$ as in Fig. 2. (b) Power law singularity near contact for $g_{11}(r)\sim (r-{\sigma }_{11}{)}^{\gamma }$ for the four points $J$, with $\gamma =-\frac{1}{2}$ shown as a dashed line. (c) $g_{11}(r)$ near second and third peaks for 4 points $J$. (d) As in (c) for the equilibrated fluid configurations at ${\varphi }_{\mathrm{init}}$.