Jamming in systems with quenched disorder

C. J. Olson Reichhardt, E. Groopman, Z. Nussinov, and C. Reichhardt

Phys. Rev. E 86, 061301 – Published 3 December 2012

Abstract

We numerically study the effect of adding quenched disorder in the form of randomly placed pinning sites on jamming transitions in a disk packing that jams at a well-defined point $J$ in the clean limit. Quenched disorder decreases the jamming density and introduces a depinning threshold. The onset of a finite threshold coincides with point $J$ at the lowest pinning densities, but for higher pinning densities there is always a finite depinning threshold even well below jamming. We find that proximity to point $J$ strongly affects the transport curves and noise fluctuations, and we observe a change from plastic behavior below jamming, where the system is highly heterogeneous, to elastic depinning above jamming. Many of the general features we find are related to other systems containing quenched disorder, including the peak effect observed in vortex systems.

Received 27 April 2012

DOI:https://doi.org/10.1103/PhysRevE.86.061301

Authors & Affiliations

C. J. Olson Reichhardt¹, E. Groopman^1,2, Z. Nussinov², and C. Reichhardt¹

¹Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
²Department of Physics, Washington University, St. Louis, Missouri 63160, USA

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Vol. 86, Iss. 6 — December 2012

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Images

Figure 1
(a,b) $〈 V_{x} 〉 / N$ vs driving force $F_{D}$ in samples with pinning strength $F_{p} = 2.0$ . (a) Samples with pin density $N_{p} / N_{J} = 0.415$ . Right (red) curve: at disk density $ϕ / ϕ_{J} = 0.014$ , the critical depinning force $F_{c} = 2.0$ . Left (black) curve: at $ϕ / ϕ_{J} = 0.761$ , $F_{c}$ is much lower. (b) Samples with pin density $N_{p} / N_{J} = 0.092 67$ . Lower (black) curve: at $ϕ / ϕ_{J} = 0.947$ , $F_{c} = 0$ . Upper (red) curve: at $ϕ / ϕ_{J} = 0.99$ , $F_{c}$ is finite. (c) Disk positions in the pinned state for the system with $N_{p} / N_{J} = 0.415$ and $ϕ / ϕ_{J} = 0.761$ in panel (a). (d) Disk trajectories over a period of time for the system in (c) at $F_{D} = 0.25 \approx 1.09 F_{c}$ showing plastic flow above the depinning threshold. (e) Disk positions in the pinned state for a system with $N_{p} / N_{J} = 0.415$ and $ϕ / ϕ_{J} = 1.038 78$ . (f) Disk trajectories over a period of time for the system in (e) at $F_{D} = 1.1 F_{c}$ showing elastic flow.Reuse & Permissions
Figure 2
$F_{c}$ vs $ϕ / ϕ_{J}$ for $N_{p} / N_{J} = 0.828$ (black $◯$ ), $0.415$ (red $▪$ ), $0.277$ (green $◊$ ), $0.138$ (blue $▴$ ), $0.092 67$ (black $▾$ ), $0.0346$ (red $▹$ ), $0.006 92$ (green $+$ ), and $0.001 38$ (blue $\times$ ). (a) Log-linear plot. Curves with $N_{p} / N_{J} ⩾ 0.277$ have $F_{c} > 0$ for all $ϕ / ϕ_{J}$ while curves with $N_{p} / N_{J} ⩽ 0.138$ have a pinned regime at low $ϕ / ϕ_{J}$ and another regime of finite $F_{c}$ at high $ϕ / ϕ_{J}$ where jamming occurs. (b) Blowup of the region near $ϕ / ϕ_{J} = 1.0$ . At jamming, $F_{c}$ drops for higher pinning densities and peaks for lower pinning densities.Reuse & Permissions
Figure 3
Black circles: the value of $N_{p} / N_{J}$ at which a finite $F_{c}$ first appears vs $ϕ$ . For small $ϕ$ , all disks are directly pinned in pinning sites, while for $ϕ$ near $ϕ_{J} = 0.844$ , the disks are pinned due to jamming. Dotted line: $N = N_{p}$ . Dashed line: the highest measured value of $N_{p} / N_{J}$ at which an $F_{c} = 0$ region still appears. Red squares: the value of $N_{p} / N_{J}$ at which $F_{c}$ reaches its peak value vs $ϕ$ , marking the onset of elastic depinning. Inset: the data in the main panel plotted against $ϕ_{J} - ϕ$ on a log-log scale in the region near $ϕ_{J}$ . Upper dashed line: a fit with a scaling exponent of $2 ν = 1.0$ ; lower dashed line: a fit with $2 ν = 1.2$ .Reuse & Permissions
Figure 4
(a,b) $S_{0}$ vs $ϕ / ϕ_{J}$ for $F_{D} = 1.1 F_{c}$ . (a) $N_{p} / N_{J} = 0.277$ . $S_{0}$ drops at the onset of jamming and at low $ϕ / ϕ_{J}$ . (b) $N_{p} / N_{J} = 0.0346$ . $S_{0}$ peaks just below jamming. (c) $S (f)$ vs $f$ from the system in (a) for (upper red curve) $ϕ / ϕ_{J} = 0.968$ where the depinning is plastic and (lower blue curve) $ϕ / ϕ_{J} = 1.141$ where the depinning is elastic and a narrowband noise signal appears. Solid black line: a fit to $1 / f^{0.9}$ . (d) $S (f)$ vs $f$ for the system in (b) for (upper red curve) $ϕ / ϕ_{J} = 0.989$ and (lower blue curve) $ϕ / ϕ_{J} = 1.141$ showing narrowband noise in the jammed phase. Solid black line: a fit to $1 / f^{2}$ . (e) The data from Fig. 2 plotted as $F_{c} {(N_{p} / N_{J})}^{- γ}$ vs $ϕ / ϕ_{J}$ with $γ = 0.43$ for $N_{p} / N_{J} = 0.828$ (black $◯$ ), $0.415$ (red $▪$ ), $0.277$ (green $◊$ ), $0.138$ (blue $▴$ ), $0.092 67$ (black $▾$ ), $0.0346$ (red $▹$ ), $0.006 92$ (green $+$ ), and $0.001 38$ (blue $\times$ ), showing noncritical scaling of $F_{c}$ in the jammed region. (f) The data from Fig. 4b plotted as $S_{0}$ vs $1 - ϕ / ϕ_{J}$ showing scaling below the jamming transition. The dotted line is a fit to $S_{0} \propto {(1 - ϕ / ϕ_{J})}^{- β}$ with $β = 2.09$ .Reuse & Permissions

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