Glassiness, rigidity, and jamming of frictionless soft core disks

The jamming of bidisperse soft core disks is considered, using a variety of different protocols to produce the jammed state. In agreement with other works, we find that cooling and compression can lead to a broad range of jamming packing fractions ${\varphi }_J$, depending on cooling rate and initial configuration; the larger the degree of big particle clustering in the initial configuration, the larger will be the value of ${\varphi }_J$. In contrast, we find that shearing disrupts particle clustering, leading to a much narrower range of ${\varphi }_J$ as the shear strain rate varies. In the limit of vanishingly small shear strain rate, we find a unique nontrivial value for the jamming density that is independent of the initial system configuration. We conclude that shear driven jamming is a unique and well-defined critical point in the space of shear driven steady states. We clarify the relation between glassy behavior, rigidity, and jamming in such systems and relate our results to recent experiments.

Figure 1

Proposed jamming phase diagram, adapted from Ref. [2]. Note that the axis along which packing fraction varies is $1/\varphi$ rather than $\varphi$.

Figure 2

Jammed fraction $f$ vs packing fraction $\varphi$ for systems of different numbers of particles $N$. (a) and (b) are for the RAND and QS ensembles, respectively. Vertical dashed lines indicate the limiting $N\to \infty$ value of the jamming density ${\varphi }_J$ in each case.

Figure 3

(a) Jammed fraction $f$ vs packing fraction $\varphi$ comparing RAND with ensembles quenched after thermal equilibration at a fixed temperature $T$. Open symbols (solid lines) are for $N=256$ particles, while closed symbols (dashed lines) are for $N=512$ particles. (b) Clustering strength parameter $n_6$ vs $\varphi$ for the cases shown in (a) as well as for QS. Representative error bars are shown at select data points at the lowest $T$’s.

Figure 4

Jammed fraction $f$ vs packing fraction $\varphi$ for a system of $N=256$ particles, comparing RAND and QS ensembles with ensembles quenched from a fixed finite shear strain rate $\mathrm{\gamma ̇}$.

Figure 5

(a) Relaxation time $\tau$ and (b) clustering strength parameter $n_6$ vs $\varphi$ for $N=1024$ diffusing hard core particles. Representative error bars are shown.

Figure 6

Initial unjammed configurations A and B at $\varphi =0.80$. Configuration B is seen to have a greater degree of big (black) particle clustering than A. $N=1024$.

Figure 7

(a) Energy $E/N$ vs $\varphi$ for the two configurations A and B of Fig. 6 undergoing uniform compression. (b) $E/N$ vs strain $\gamma$ for the same configurations undergoing uniform shear starting from stress-free states at $\varphi =0.85$. (c) Clustering strength $n_6$ vs $\gamma$ as the two configurations are sheared at $\varphi =0.85$. $N=1024$.

Figure 8

(a) Strain averaged cluster strength $\langle n_6\rangle {}_{\gamma }$, (b) energy per particle $\langle E/N\rangle {}_{\gamma }$, and (c) jammed fraction $\langle f\rangle {}_{\gamma }$, vs total strain $\gamma$ at $\varphi =0.85>{\varphi }_J^{\mathrm{QS}}$; plots (d), (e), and (f) are the same quantities at the lower density $\varphi =0.84<{\varphi }_J^{\mathrm{QS}}$. The three curves correspond to simulations starting from three different initial configurations: A is the moderately clustered configuration and B is the highly clustered configuration of Figs. 6 and 7, while C is a configuration with random initial particle positions. The system has $N=1024$ particles. Note, symbols are displayed on every $2000\mathrm{th}$ data point for $\varphi =0.85$ and every $1000\mathrm{th}$ data point for $\varphi =0.84$, to help with curve identification.

Figure 9

Scatter plot of energy per particle $E/N$ vs the number of conjugate gradient iteration steps per particle $M_{\mathrm{iter}}/N$ needed to achieve minimization at $\varphi \approx {\varphi }_J$. (a) is for RAND with $N=16384$ particles at $\varphi =0.8415$; (b) is for QS with $N=4096$ particles at $\varphi =0.8430$. The horizontal line at $E/N={10}^{-16}$ shows the unjammed states, where the minimization has stopped upon reaching our cutoff threshold $e_{\mathrm{cut}}$.

Figure 10

Average number of conjugate gradient iteration steps per particle $M_{\mathrm{iter}}/N$ needed to energy minimize, vs packing fraction $\varphi$. (a) is for RAND; (b) is for QS. Results are shown for a few different systems sizes with particle numbers $N$.

Figure 11

(a) Dependence of jammed fraction $f$ and (b) energy per particle $E/N$ on shear strain step increment $\Delta \gamma$, within quasistatic shearing simulations, at fixed packing fraction $\varphi =0.8430\approx {\varphi }_J^{\mathrm{QS}}$. Results are shown for several different system sizes with particle numbers $N$.