Protocol dependence of the jamming transition

We propose a theoretical framework for predicting the protocol dependence of the jamming transition for frictionless spherical particles that interact via repulsive contact forces. We study isostatic jammed disk packings obtained via two protocols: isotropic compression and simple shear. We show that for frictionless systems, all jammed packings can be obtained via either protocol. However, the probability to obtain a particular jammed packing depends on the packing-generation protocol. We predict the average shear strain required to jam initially unjammed isotropically compressed packings from the density of jammed packings, shape of their basins of attraction, and path traversed in configuration space. We compare our predictions to simulations of shear strain-induced jamming and find quantitative agreement. We also show that the packing fraction range, over which shear strain-induced jamming occurs, tends to zero in the large system limit for frictionless packings with overdamped dynamics.

Figure 1

Fraction of jammed packings (or cumulative distribution) $\mathcal{C}(\varphi ,\gamma )$ along different paths in the packing fraction $\varphi$ and shear strain $\gamma$ plane for $N=32$. (a) ${\mathcal{C}}_1(\varphi ,\gamma )$ for protocol 1 (isotropic compression at fixed $\gamma$; solid line) and ${\mathcal{C}}_2(\varphi ,\gamma )$ for protocol 2, i.e., compression to $\varphi$ followed by shear strain to $\gamma =0.1$ (dotted line), 0.3 (dot-dashed line), and 0.5 (dashed line). I, II, and III indicate the packing fractions in (b). (b) We show ${\mathcal{C}}_1(\varphi ,\gamma )$ (dashed lines) and ${\mathcal{C}}_2(\varphi ,\gamma )$ (solid lines) at fixed $\varphi =0.815$, 0.824, and 0.832 indicated by I–III. Protocol dependence can be seen in the difference between ${\mathcal{C}}_1$ and ${\mathcal{C}}_2$ evaluated at the same $\varphi$ and $\gamma$, e.g., at $\varphi =0.824$ and $\gamma =0.67$ as highlighted by the dashed double arrow. Right and left solid arrows indicate protocols 1 and 2, respectively.

Figure 2

(a) Packing fraction $\varphi$ versus shear strain $\gamma$ for all isostatic jammed $N=6$ disk packings. The solid black line obeys $\varphi =A{(\gamma -{\gamma }_0)}^2+{\varphi }_0$ with $A=0.776, {\varphi }_0=0.665$, and ${\gamma }_0=0.35$. Filled circles (downward triangles) indicate packings with positive (negative) local slope. The solid vertical arrow indicates protocol 1, and the dashed vertical arrow followed by the dashed horizontal arrow indicates protocol 2 that was used to reach a jammed packing at $\gamma =0.8$ and $\varphi =0.725$. (b) Jammed packing fraction $\varphi \left(\gamma \right)$ using protocol 2 at fixed $\varphi$ in the range $0.64<\varphi <0.77$ for $N=6$. (c) Number of $N=32$ jammed packings at each $\varphi$ and $\gamma$ (increasing from dark to light) from protocol 1.

Figure 3

(a) Natural logarithm of the fraction of unjammed packings [normalized by the $\varphi$-dependent decay factor, $\mathcal{F}\left(\varphi \right)S_2\left(\varphi \right){\ell }_2\left(\varphi \right)]$, during protocol 2 at fixed $\varphi$ in the range $0.798<\varphi <0.844$ (solid lines) for $N=32$. The dashed line has slope $-1$. (b) Comparison of $-dln[M_2(\varphi ,\gamma )/M_0]/d\gamma$ (open circles) from protocol 2 and $-dln[M_1\left(\varphi \right)/M_0]/d\varphi$ from protocol 1 with $S_2\left(\varphi \right)\propto S_1\left(\varphi \right){\ell }_1\left(\varphi \right)$ (solid line) or $S_2\left(\varphi \right)\propto S_1\left(\varphi \right)$ (dashed line) for $N=32$. The inset shows the distances ${\ell }_1\left(\varphi \right)$ and $\alpha {\ell }_2\left(\varphi \right)$ (with $\alpha \approx 5.5$) traversed in configuration space during protocols 1 (solid line) and 2 (dashed line) for $N=32$.

Figure 4

(a) Average shear strain ${\overline{\gamma }}_j\left(\varphi \right)$ to jam an initially unjammed configuration at $\varphi$ and $\gamma =0$ (protocol 2) for $N=32$ (circles), 128 (triangles), and 512 (squares). We compare ${\overline{\gamma }}_j\left(\varphi \right)$ from protocol 2 to $1/\left(\mathcal{F}\left(\varphi \right)S_1\left(\varphi \right){\ell }_1^2\left(\varphi \right)\right)$ [Eqs. (3) and (4)] from protocol 1 for the same system sizes: $N=32$ (solid line), 128 (dashed line), and 512 (dotted line). (b) Distribution of jammed packing fractions ${\mathcal{P}}_1\left(\varphi \right)$ from protocol 1 for $N=32$ (solid line), 128 (dashed line), and 512 (dotted line) compared to predictions obtained from ${\mathcal{P}}_1\left(\varphi \right)=-M_0^{-1}d{M}_1\left(\varphi \right)/d\varphi$ with $M_1\left(\varphi \right)$ given by Eq. (2) and $\mathcal{F}\left(\varphi \right)S_1\left(\varphi \right)l_1\left(\varphi \right)$ given by Eq. (4) using the measured value of ${\overline{\gamma }}_j$ for $N=32$ (circles), 128 (triangles), and 512 (squares).

Figure 5

Distribution of jammed packing fractions ${\mathcal{P}}_1\left(\varphi \right)$ from protocol 1 for $N=64$ bidisperse spheres (solid line), compared to predictions obtained from ${\mathcal{P}}_1\left(\varphi \right)=-M_0^{-1}d{M}_1\left(\varphi \right)/d\varphi$ with $M_1\left(\varphi \right)$ given by Eq. (2) and $\mathcal{F}\left(\varphi \right)S_1\left(\varphi \right)l_1\left(\varphi \right)$ given by Eq. (4) using the measured value of ${\overline{\gamma }}_j$ for $N=64$ bidisperse spheres (circles).

Figure 6

(a) Packing fraction $\varphi$ and (b) ratio of the stress anisotropy $\tau$ to the pressure $P$ versus shear strain $\gamma$ for all isostatic jammed $N=6$ bidisperse disk packings. The solid black line in (a) obeys $\varphi =A{(\gamma -{\gamma }_0)}^2+{\varphi }_0$ with $A=0.776, {\varphi }_0=0.665$, and ${\gamma }_0=0.35$. Filled circles (downward triangles) indicate packings with positive (negative) local slope of $\varphi$ versus $\gamma$.

Figure 7

Average shear strain ${\overline{\gamma }}_j\left(\varphi \right)$ required to jam an initially unjammed configuration at $\varphi$ using simple shear (light circles) and pure shear strain (dark squares) for $N=32$ (blue symbols), 128 (green symbols), and 512 (red symbols).