Dynamic phases of active matter systems with quenched disorder

Depinning and nonequilibrium transitions within sliding states in systems driven over quenched disorder arise across a wide spectrum of size scales ranging from atomic friction at the nanoscale, flux motion in type II superconductors at the mesoscale, colloidal motion in disordered media at the microscale, and plate tectonics at geological length scales. Here we show that active matter or self-propelled particles interacting with quenched disorder under an external drive represents a class of system that can also exhibit pinning-depinning phenomena, plastic flow phases, and nonequilibrium sliding transitions that are correlated with distinct morphologies and velocity-force curve signatures. When interactions with the substrate are strong, a homogeneous pinned liquid phase forms that depins plastically into a uniform disordered phase and then dynamically transitions first into a moving stripe coexisting with a pinned liquid and then into a moving phase-separated state at higher drives. We numerically map the resulting dynamical phase diagrams as a function of external drive, substrate interaction strength, and self-propulsion correlation length. These phases can be observed for active matter moving through random disorder. Our results indicate that intrinsically nonequilibrium systems can exhibit additional nonequilibrium transitions when subjected to an external drive.

Figure 1

Images of disk locations (dots) in a system with $\varphi =0.55,r_l=300,N/N_p=2.0$, and $F_p=5.0$. Disks are colored according to the cluster to which they belong, with the largest cluster shown in blue, the second-largest in red, the third-largest in green, the fourth-largest in purple, and the fifth-largest in orange. All remaining clusters, including isolated disks, are colored black. White areas contain no disks. The disks are driven in the positive $x$ direction with a driving force of $F_D$. (a) At $F_D=0.0$, the system forms a uniform disordered state. (b) Disk trajectories (lines) in a portion of the sample at $F_D=0.5$, where the system is in a plastic flow state. (c) At $F_D=3.25$, a dense stripe coexists with a pinned liquid. The stripe structure is moving in the positive $x$ direction. (d) At $F_D=6.0$, a fully phase-separated state appears with a large cluster that translates in the positive $x$ direction.

Figure 2

(a) The velocity $\langle V\rangle$ vs $F_D$ for the system in Fig. 1 with $F_p=5.0,r_l=300, N_p=8000$, and varied $\varphi$ ranging from $\varphi =0.1745$ to $\varphi =0.8375$, corresponding to $N=5000$ to $N=24000$ in the figure legend. The letters a to d indicate the values of $F_D$ at which the images in Fig. 1 were obtained for the $\varphi =0.55$, which corresponds to the $16k$ curve. (b) The corresponding $d\langle V\rangle /d{F}_D$ vs $F_D$ curves. (c) The size of the largest cluster ${\tilde{C}}_L$ vs $F_D$. The dashed line indicates the transition into the moving stripe state, and the solid line denotes the transition to the moving phase-separated state. (d) The number of sixfold coordinated disks ${\tilde{P}}_6$ vs $F_D$. The error bars in this and all remaining figures are smaller than the size of the symbols.

Figure 3

Image of disk locations (dots) in a sample with $r_l=300,N_p=8000$, and $F_p=5.0$. Disks are colored according to the cluster to which they belong, with the largest cluster shown in blue. White regions contain no disks. (a) The uniform disordered state for $\varphi =0.8375$ and $F_D=0$. (b) The moving stripe state for $\varphi =0.8375$ and $F_D=1.5$. The stripe structure moves in the positive $x$ direction. (c) The pinned liquid state for $\varphi =0.35$ and $F_D=0$. (d) The moving phase-separated state for $\varphi =0.35$ and $F_D=5.0$, where all disks move in the positive $x$ direction.

Figure 4

(a) The dynamic phase diagram as a function of $F_D$ vs $\varphi$ constructed from the features in Fig. 2 for a sample with $F_p=5.0,N_p=8000$, and $r_l=300$. I, pinned phase; II, plastic flow phase; III, moving stripe phase; IV, moving fully phase-separated state; V, moving liquid phase. The dashed line in phase I indicates the separation between a pinned liquid and a pinned labyrinth state. (b) The dynamic phase diagram as a function of $F_D$ vs $F_p$ for a sample with $\varphi =0.55, N/N_p=2.0$, and $r_l=300$. The phases I through V are marked as above. Phase VI is the moving phase-separated state in which some disks can be temporarily pinned.

Figure 5

(a) $d\langle V\rangle /d{F}_D$ vs $F_D$ in samples with $\varphi =0.55, N/N_p=2.0$, and $r_l=300$ from the system in Fig. 4 for different values of $F_p$ ranging from $F_p=0$ to $F_p=9.0$. As $F_p$ increases, a second peak appears and shifts to higher $F_D$. (b) The corresponding $P_6$ vs $F_D$.

Figure 6

(a) $d\langle V\rangle /d{F}_D$ vs $F_D$ for samples with $\varphi =0.55,F_p=5.0$, and $N/N_p=2.0$ for varied $r_l$ ranging from $r_l=1$ to $r_l=1200$. The magnitude of the second peak in $d\langle V\rangle /d{F}_D$ decreases with decreasing $r_l$. (b) The corresponding $\langle V\rangle$ vs $F_D$ curves. (c) The corresponding $P_6$ vs $F_D$ curves, indicating that phase IV disappears for $r_l<60$.

Figure 7

(a) The dynamic phase diagram as a function of $F_D$ vs $r_l$ constructed from the features in Fig. 6 for a sample with $\varphi =0.55,N_p=8000$, and $F_p=5.0$. I, pinned phase; II, plastic flow phase; III, moving stripe phase; IV, moving fully phase separated state; V, moving liquid phase. (b) The dynamic phase diagram as a function of $F_D$ vs the number of pinning sites $N_p$ in samples with $\varphi =0.55,F_p=5.0$, and $r_l=300$.

Figure 8

(a) $\langle V\rangle$ vs $F_D$ for systems with $\varphi =0.55$ and $r_l=300$ for varied numbers of obstacles $N_o=2000$ to $N_o=12000$. The inset shows the normalized cluster size $C_L$ vs the number of obstacles $N_o$ for $\varphi =0.55$ and $r_l=300$ at $F_D=0.0$, where there is a transition from a phase-separated state at low $F_D$ to a uniform state at high $F_D$. (b) $d\langle V\rangle /d{F}_D$ vs $F_D$, corresponding to the system in panel (a), lacks any peak features. (c) The corresponding normalized $P_6$ vs $F_D$ has a smooth monotonic behavior.

Figure 9

Images of disk locations (dots) in samples with $\varphi =0.55$ and $r_l=300$ containing obstacles consisting of fixed disks (not shown). (a) At $F_D=0$ and $N_o=8000$ obstacles, a disordered state appears. (b) At $F_D=6.0$ and $N_o=8000$, the system is still disordered. (c) $F_D=0$ and $N_o=2000$. (d) At $F_D=2.0$ and $N_o=2000$, a moving stripe state forms.