Structural signatures of the unjamming transition at zero temperature

We study the pair correlation function $g\left(r\right)$ for zero-temperature, disordered, soft-sphere packings just above the onset of jamming. We find distinct signatures of the transition in both the first and split second peaks of this function. As the transition is approached from the jammed side (at higher packing fraction) the first peak diverges and narrows on the small-$r$ side to a $\delta$ function. On the high-$r$ side of this peak, $g\left(r\right)$ decays as a power law. In the split second peak, the two subpeaks are both singular at the transition, with power-law behavior on their low-$r$ sides and step-function drop-offs on their high-$r$ sides. These singularities at the transition are reminiscent of empirical criteria that have previously been used to distinguish glassy structures from liquid ones.

Figure 1

The pair correlation function $g\left(r\right)$ vs $r$ at two extreme values of $\Delta \varphi \equiv \varphi -{\varphi }_c={10}^{-6}$ (dotted line) and ${10}^{-1}$ (solid line). The maximum value of the first peak height is higher and its width narrower for the lower value of $\Delta \varphi$. For $\Delta \varphi ={10}^{-6}$, the first-peak maximum is approximately ${10}^6$, far beyond the scale of the graph.

Figure 2

The height $g\left(r_{\text{peak}}\right)$ (top panels) and the left-hand width $w_L$ (bottom panels) of the nearest-neighbor peak of $g\left(r\right)$, over several orders of magnitude of $\Delta \varphi$, for monodisperse spheres with purely repulsive (a) harmonic spring and (b) Hertzian interactions.

Figure 3

Pair distribution function $g\left(r\right)$ up to contact $0<r<1$ on a linear-logarithmic scale, for $\Delta \varphi =$ (a) ${10}^{-6}$, (b) ${10}^{-4}$, (c) ${10}^{-2}$, and (d) ${10}^{-1}$.

Figure 4

Evolution of the parameters ${\alpha }_1$ and ${\alpha }_2$ (top) and $g_0$ (bottom) in Eq. (3) with $\Delta \varphi$, for harmonic repulsions. The parameters ${\alpha }_{1,2}$ are clearly related to the pressure, while $g_0$ is associated with $g\left(r_{\text{peak}}\right)$.

Figure 5

The distribution of normal contact forces, $P\left(F\right)$, for the purely repulsive, harmonic potential, at different compressions $\Delta \varphi$.

Figure 6

Behavior just above the first-neighbor peak for a system at $\Delta \varphi ={10}^{-8}$. (a) $g(r>1)$ versus $r-1$. A power law with exponent of $-0.5$ is indicated. (b) $Z\left(\ell \right)$ vs $\ell \equiv r-1$, computed by numerically integrating $g\left(r\right)>1$. A power law with exponent of 0.5 is indicated.

Figure 7

Pair distribution function $g\left(r\right)$, just above the first peak for $\Delta \varphi ={10}^{-8}$ (×), ${10}^{-6}$ (circles), ${10}^{-4}$ (squares), ${10}^{-3}$ (diamonds), ${10}^{-2}$ (+). (a) $g\left(r\right)$ vs $r-1$, showing the power-law behavior in $(r-1)$. (b) The exponent $\eta$ characterizing the power-law behavior of (a), as a function of $\Delta \varphi$. (c) $g\left(r\right)$ from the nearest-neighbor, first peak at $r_{\text{peak}}$ out to the first minimum at $r\approx 1.4$ as a function of $r-r_{\text{peak}}$.

Figure 8

Pair distribution function $g\left(r\right)$, from the nearest-neighbor, first peak at $r_{\text{peak}}$ out to the first minimum at $r\approx 1.4$, at $\Delta \varphi ={10}^{-4}$. Different symbols represent different configuration protocols with and without temperature: conjugate gradient minimization to $T=0$ (open circles), quenched molecular dynamics at $T=0$ (open squares), molecular dynamics at a very low temperature (solid diamonds), and molecular dynamics at a higher temperature (+). The jump in $g\left(r\right)$ persists to small but nonzero temperature, and is smoothed out at high enough temperatures.

Figure 9

The distribution $P\left(Z\right)$ of the number of overlapping neighbors per particle, $Z$, at several different compressions. Key same as Fig. 5.

Figure 10

$g\left(r\right)$ region around the split second peak for $\Delta \varphi =\left(\mathrm{a}\right)$ ${10}^{-6}$, (b) ${10}^{-3}$, and (c) ${10}^{-1}$.

Figure 11

Functional form of the subpeaks of the split second peak in $g\left(r\right)$ for $\Delta \varphi ={10}^{-6}$. Left-hand subpeak at $r=\sqrt{3}$; right-hand subpeak at $r=2$. The solid line corresponds to a power-law fit, the dashed line to a logarithmic fit, and the gray line to a shifted power-law fit.

Figure 12

Left: The particles at the left and at the right share two common neighbors and are separated by a distance $r=\sqrt{3}$. Right: The particles at the left and the right share one common neighbor and are separated by $r=2$.

Figure 13

(a) The three-particle angular correlation $P\left({\theta }_{i,j}\right)$ versus ${\theta }_{i,j}$ for $\Delta \varphi ={10}^{-6}$. Bottom four panels are fits to the regions around $\theta =\left(\mathrm{b}\right)$ 60°, (c) 120° left, (d) 120° right, and (e) 180°. The power-law exponents are $\approx 0.5$, 0.17, 0.17, and 0.75, respectively.