Physical Review E 97, 052613 (2018)

Clogging and Depinning of Ballistic Active Matter Systems in Disordered Media

C. Reichhardt and C.J.O. Reichhardt

Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA

(Received 23 March 2018; published 21 May 2018)

We numerically examine ballistic active disks driven through a random obstacle array. Formation of a pinned or clogged state occurs at much lower obstacle densities for the active disks than for passive disks. As a function of obstacle density we identify several distinct phases including a depinned fluctuating cluster state, a pinned single cluster or jammed state, a pinned multicluster state, a pinned gel state, and a pinned disordered state. At lower active disk densities, a drifting uniform liquid forms in the absence of obstacles, but when even a small number of obstacles are introduced, the disks organize into a pinned phase-separated cluster state in which clusters nucleate around the obstacles, similar to a wetting phenomenon. We examine how the depinning threshold changes as a function of disk or obstacle density, and find a crossover from a collectively pinned cluster state to a disordered plastic depinning transition as a function of increasing obstacle density. We compare this to the behavior of nonballistic active particles and show that as we vary the activity from completely passive to completely ballistic, a clogged phase-separated state appears in both the active and passive limits, while for intermediate activity, a readily flowing liquid state appears and there is an optimal activity level that maximizes the flux through the sample.
I. INTRODUCTION
II. SIMULATION
III. PHASES OF ACTIVE BALLISTIC DISKS
IV. DEPINNING AND DRIVE DEPENDENCE
V. FINITE RUN TIME ACTIVE DISKS
VI. SUMMARY
REFERENCES


I.  INTRODUCTION

There is a wide range of soft and hard matter systems that can be modeled as collectively interacting particles which, when driven over quenched disorder, exhibit pinning-depinning behavior as well as transitions between different types of sliding regimes [1,2]. Such dynamics occur for vortex motion in type-II superconductors [3,4], sliding charge density waves [5], depinning of classical Wigner crystals [6,7], current driven motion of skyrmions in chiral magnets [8,9], colloids interacting with random [10,11,12,13,14,15] or ordered substrates [16], sliding in frictional systems [17], magnetic domain wall motion [18,19], erosion [20], granular matter [21,22], driven pattern forming systems [23], geophysical models of plate tectonics [24], and the motion of dislocations in crystalline materials [25]. The substrate may be random, ordered, or partially ordered, and it can be modeled as localized pinning sites with a finite trapping strength or as impenetrable obstacles. Under an applied drive, the particles exhibit a variety of pinned and moving order-disorder transitions that can be characterized by the moving structure, pattern formation features, changes in the velocity force curves, and fluctuation phenomena [1,2]. In the systems listed above, the particles themselves are passive or experience only thermal fluctuations, so the driving is strictly externally applied; however, recently a growing number of studies have focused on what are called active matter systems containing self-driven particles with an activity that is often modeled as arising from driven diffusive or run-and-tumble dynamics [26,27]. In the absence of a substrate, active disks exhibit a transition from a uniform gas or liquid state to a phase-separated or cluster state consisting of a high density solid coexisting with a low density active gas [28,29,30,31,32,33,34]. This transition occurs for fixed activity as a function of increasing density or for fixed density with increasing activity level. Active matter systems have also been been studied in the context of particle shape effects [27,35], active rotators [36], passive and active mixtures [37,38,39], boundary effects [40,41,42,43,44], and ratchet effects [45,46,47].
Several studies have examined different types of active matter systems coupled with ordered [48,49,50] or disordered substrates [33,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66]. Numerical simulations show that when a drift force is applied to run-and-tumble disks moving through a random obstacle array, the flux through the system is non-monotonic as a function of activity level, indicating that there is an optimal run length or run correlation time that maximizes the flux of disks through the obstacles [33]. At small run lengths, the disks behave thermally and easily become trapped behind the obstacles, giving a low disk flux. As the activity level or run length increases, the disks can more readily move around the obstacles, increasing the flux; however, when the run length is too large, a self-pinning effect occurs in which the disks self-cluster around the obstacles, reducing the flux [33,65]. As a result, the flux is maximized when the disks are active and the self-clustering is weak, while it is reduced when the activity becomes high enough for significant clustering or self-trapping to occur. When the run lengths are very long, the average flux under an applied drive is strongly reduced, but it never reaches zero since there are still long-time dynamical rearrangements that produce avalanches or intermittent flow of disks in the direction of the drift force [65]. Analytic and theoretical studies of active systems without a drift force also indicate that there is an optimal activity level that maximizes the diffusion in the system [64]. If the obstacles are replaced by pinning sites, the onset of clustering can have the opposite effect of increasing rather than decreasing the flux, since the clusters act like large rigid objects that are poorly trapped by individual pinning sites [60]. When the activity is low, a uniform liquid state appears and individual disks can be trapped readily by individual pinning sites [60]. These works also showed that within the phase separated state, increasing the number of obstacles or pinning sites produces a disorder-induced transition from a phase-separated state to a uniform disordered state [58,60].
In addition to the run-and-tumble models, several simulation studies of swarming or flocking active systems with quenched disorder show that there is an optimal noise level for the appearance of flocking [51,54], and that increasing the disorder strength can induce a transition from a flocking to a non-flocking state [53]. Experiments on colloidal rollers that exhibit flocking behavior have also revealed a transition from a drifting flocking state to a non-flocking state as a function of increasing obstacle density, where the flow becomes increasingly filamentary as the disordered state is approached [57].
In these systems, the activity is stochastic in nature. For run-and-tumble particles, after each run interval the particle randomly reorients and runs in a new direction, while in driven diffusive models, there is a noise term controlling the rate of rotational diffusion [27]. The deterministic limit of active matter is purely ballistic flow where the swimming direction of each particle is permanently fixed. This can be achieved in run-and-tumble disk systems by setting the running time to infinity, or in driven diffusive systems by setting the rotational diffusion coefficient to zero [52]. In the deterministic regime, the particles can form a cluster state even at very low densities, and Bruss et al. [67] argue that phase separation occurs when the mean time between collisions is smaller than the mean duration of an individual collision. Previous studies [52] of an active ballistic system showed that the disks can form a frozen cluster state where almost all the fluctuations are lost. This frozen state arises due to the lack of stochastic behavior in the system, and the resulting cluster can be regarded as an absorbed state. In contrast, at long but finite run times, the cluster can gradually evolve over time due to the possibility of rare stochastic reorganizations [65].
Previous studies of the active ballistic limit did not include an applied drift force, so no pinning-depinning phenomena occurred. In this work we consider active ballistic disks driven through an obstacle array, and we measure the average drift mobility 〈V〉 of the disks in the direction of the driving force FD. We find that the system can evolve toward a completely pinned or clogged state with 〈V〉 ≈ 0. We describe our simulation in Sec. II. In Sec. III we compare the active pinning or clogging to that observed in the zero activity or passive limit. The critical density of obstacles needed to induce the formation of a clogged state is much higher in the passive system than in the active ballistic system, and we show that in general the active ballistic disks are much more susceptible to forming clogs than the passive disks. In the active system, clogging is associated with the formation of a cluster state. As a function of increasing obstacle density, we identify four phases: a sliding state with either a fluctuating cluster or a liquid structure; a pinned single cluster or jammed state consisting of a large cluster held in place by a small number of obstacles; a pinned multicluster state containing several distinct pinned clusters; and a pinned disordered state in which the disk density remains spatially uniform. As the active disk density increases, we also find what we term a pinned gel state where the disks form a percolating labyrinth structure. For low active disk densities where a flowing uniform liquid appears in the absence of obstacles, we find that introduction of a small number of obstacles causes a transition to a pinned cluster state, which we compare to the active wetting of clusters around an obstacle. In Sec. IV we examine the velocity-force relations and the behavior of the critical depinning force Fc. The depinning threshold for a cluster is finite, and there is a pronounced increase in Fc at the transition from the pinned single cluster state to the pinned disordered state, reminiscent of the "peak effect" observed in superconducting vortex systems at a transition from a collectively pinned ordered or quasiordered vortex crystal to a vortex glass state [2,3]. In Sec. V we show how passive clogging can be connected to the ballistic clogging limit by considering finite but increasing run times for run-and-tumble active disks in obstacle arrays. In both the passive and ballistic clogged states, the disks form a cluster, while between these two limits a flowing liquid structure appears. There is an optimal run time at which the disk flux is maximized, and the flux decreases with increasing run length until the disks form a completely clogged state at infinite run times. We discuss how these results can be related to granular jamming transitions [68,69,70] and jamming in systems with quenched disorder [71,72], where the activity or the obstacle density represent an additional set of parameters that can be used to induce a jammed state. In Sec. VI we summarize our results.

II.  SIMULATION

We consider a two-dimensional system with periodic boundary conditions in the x and y-directions containing Na active or mobile disks that interact through a stiff repulsive harmonic spring, Fs = k(d−2R)Θ(d − 2R)∧d, where d is the distance between two disks, d is the displacement vector, k=100 is the spring constant, and the disk radius is R = 0.5. For this value of k the disk-disk overlaps remain very small, allowing us to define the active disk density in terms of the area covered by the disks, ϕa = NaπR2/L2, where the system size is L = 100. In the limit of no activity and no obstacles, the disks form a hexagonal lattice at ϕa = 0.9. In addition to the mobile active disks, we introduce Nobs obstacles that are identical to the active disks but are permanently fixed in place. The obstacles are initially placed in a hexagonal lattice and are randomly diluted until we reach the desired obstacle density of ϕobs = NobsπR2/L2. Placing the obstacles in an initial hexagonal lattice ensures that there is a fixed minimum distance between any two obstacles, thereby avoiding the rare obstacle density fluctuations such as large gaps or tight obstacle clustering that can arise from a completely random obstacle placement and dominate the dynamics. The total disk coverage of both active and fixed disks is ϕtot = ϕobs + ϕa. The dynamics of the active disks is obtained from the following overdamped equation of motion:
η dri
dt
 = Fs,ai + Fmi + Fs, obsi + FD.
(1)
Here η = 1.0 is the damping constant and the interaction with other active disks Fs,a has the form of the stiff spring repulsion Fs. Each active disk has a motor force Fmi that is constant in magnitude and oriented in a randomly chosen direction. In run-and-tumble systems, the motor force orientation is held fixed during the run time τr, after which a new random orientation is chosen for the next running time. In the ballistic limit, we set τr to infinity so that the running direction never changes. The forces from the obstacles Fs, obs are also given by Fs, and the external driving force FD=FD∧x is applied uniformly to all active disks. We also consider the passive particle limit in which Fm = 0 and the only driving force is the externally applied FD. To initialize the system, after establishing the location of the obstacles we place mobile disks with artificially reduced radii in nonoverlapping locations and allow the mobile disks to rearrange while gradually expanding the radii to the final value of R = 0.5. With this method we can reach disk densities up to ϕtot = 0.86.
To characterize the system we measure the average drift velocity of the disks 〈V〉 = 〈Na−1∑Nai=0 v i·∧x〉 in the direction of the drive. We wait 1×106 simulation time steps for the system to settle into a steady state and then average over an additional 9×106 simulation time steps to obtain 〈V〉. We have found that increasing the waiting time produces negligible changes in the results. In this work, unless otherwise noted we set FD = 0.05, |Fm| = 0.5, and τr=∞; however, we also consider varied FD and finite τr.
Fig1.png
Figure 1: (a,b) An active ballistic system with an area coverage of ϕa=0.3495 active disks and ϕobs=0.00393 obstacles for a total ϕtot = 0.3534. An external drift force of FD = 0.05 is applied in the positive x-direction. (a) The fraction of disks in the largest cluster, Cmax/Na, versus time in simulation time steps. (b) The drift velocity V of the active disks vs time. There is a transition from a fluctuating cluster state that is drifting in the direction of drive, illustrated in Fig. 2(a), to a pinned single cluster, shown in Fig. 2(b). At the transition, Cmax/Na abruptly increases to a value close to one and V simultaneously drops nearly to zero. The letters a and b indicate the times corresponding to the images in Fig. 2(a,b). (c) Cmax/Na and (d) V vs time for the same system in the passive |Fm|=0 limit at ϕtot = 0.3534 and ϕobs = 0.1178. For this value of ϕtot, the system can reach a pinned or clogged state only when ϕobs ≥ 0.098. The system evolves over time into a pinned state, with a gradual drop in V accompanied by a gradual increase in Cmax/Na. The initial unclogged state at the time marked c is illustrated in Fig. 2(c), while the V=0 pinned state at the time marked d is shown in Fig. 2(d).
Fig2.png
Figure 2: (a,b) The active ballistic disk positions (red circles) and obstacle locations (blue circles) for the system in Fig. 1(a,b) with ϕtot = 0.3534, ϕobs = 0.00393, and a drift force FD=0.05 applied in the positive x direction. (a) Depinned fluctuating clusters appear at time 1×106, marked a in Fig. 1(a). (b) The pinned single cluster state at time 3×106, marked b in Fig. 1(a). (c,d) Passive disk positions (green circles) and obstacle locations (blue circles) for the system from Fig. 1(c,d) with ϕtot = 0.3534 and ϕobs = 0.1178. (c) The initial flowing state at the time marked c in Fig. 1(c). (d) The clogged or pinned state at time 3×106, marked d in Fig. 1(c).

III.  PHASES OF ACTIVE BALLISTIC DISKS

In Fig. 1(a) we plot the fraction Cmax/Na of active disks in the largest cluster versus time in simulation time steps for a system with FD = 0.05, Na = 4450, and Nobs=50, giving an active disk density of ϕa = 0.3495, an obstacle density of ϕobs = 0.00393, and an overall density of ϕtot = 0.3534. The largest cluster size Cmax is defined as the largest number of disks in direct contact with each other as determined using the cluster identification algorithm described in Ref. [73]. Figure 1(b) shows the corresponding disk velocity V=Na−1∑i=0Navi ·∧x versus time. Both Cmax/Na and V exhibit strong fluctuations during the first 2.5×106 time steps, indicating that the system is in a dynamic unpinned fluctuating state. The disks then become trapped in a single pinned cluster, as shown by the sudden jump in Cmax/Na to Cmax/Na ≈ 1.0 which is accompanied by a drop in V to nearly zero. There are still some small fluctuations in both Cmax/Na and V due to the presence of a small number of freely running disks that do not join the cluster. In Fig. 2(a) we show a snapshot of the depinned fluctuating clusters at time 1.0×106, where the active disks form temporary clusters. Here V = 0.046, which is close to the expected obstacle-free value of V = 0.05. A pinned single cluster state appears at time 3×106, as illustrated in Fig. 2(b), where the active disks form a single large immobile cluster. We find that even a small number of obstacles (Nobs/Na = 0.011) can produce a pinned state for active ballistic disks. In contrast, for passive disks at the same total density of ϕtot = 0.3534, the system does not reach a pinned or clogged state until Nobs/Na ≥ 0.384, indicating that nearly 35 times more obstacles are required to pin the passive disks compared to the active ballistic disks.
In Fig. 1(c,d) we plot Cmax/Na and V versus time for a passive |Fm|=0 system with ϕtot = 0.3534 and ϕobs = 0.1178, where the disks reach a pinned or clogged state. In contrast to the active ballistic system in Fig. 1(a,b), the pinned state does not appear abruptly; instead, the passive disks continuously evolve toward the pinned state over time, with a growing number of pinned clusters gradually emerging as indicated by the steady increase in Cmax/Na and the gradual decrease in V. In Fig. 2(c) we illustrate the initial uniform spatial distribution of the passive disks, while in Fig. 2(d) we show a snapshot of the disk positions in the clogged state at a time of 2×106. Unlike the active ballistic disks in Fig. 2(b), the passive disks in Fig. 2(d) do not form a single clump but instead assemble into a number of smaller clumps. This indicates that the passive and active clogged or pinned states are very different in nature.
Fig3.png
Figure 3: (a) CL and (b) 〈V〉 vs obstacle density ϕobs for active ballistic disks with FD = 0.05 and ϕtot = 0.3534. For ϕobs < 0.0039, 〈V〉 is finite and the system is in phase Ifc, the depinned fluctuating cluster state, while at the transition to phase II, the pinned single cluster state, 〈V〉 drops to zero. Phase II extends from 0.0039 ≤ ϕobs < 0.025 and is illustrated in Fig. 4(a). For 0.025 ≤ ϕobs < 0.065, the system is in phase III, a pinned multicluster state, illustrated in Fig. 4(b), while for ϕobs ≥ 0.065 the system is in phase IV, the pinned disordered phase illustrated in Fig. 4(c,d) at ϕobs = 0.0942 and ϕobs=0.1963. The labels a to d in (a) indicate the values of ϕobs at which the images in Fig. 4 are obtained. (c) CL and (d) 〈V〉 vs ϕobs for passive disks at the same FD and ϕtot. Here there are only two phases: a plastic flow state ϕobs < 0.098, and a completely pinned or clogged state for ϕobs ≥ 0.098. There is a peak in CL at ϕobs=0.098 where 〈V〉 drops to zero. The labels a to d in (c) indicate the values of ϕobs at which the images in Fig. 5 are obtained.
Fig4.png
Figure 4: The active ballistic disk positions (red circles) and obstacle locations (blue circles) for the system in Fig. 3(a,b) with FD=0.05 and ϕtot=0.3534 obtained at the values of ϕobs marked by the letters a to d in Fig. 3(a). (a) The pinned single cluster phase II at ϕobs = 0.01178. (b) The pinned multicluster phase III at ϕobs = 0.039. (c) The pinned disordered phase IV at ϕobs=0.0942 consists of a group of small clusters. (d) The pinned disordered phase IV at ϕobs = 0.1963 is composed of even smaller clusters.
In Fig. 3(a,b) we plot CL=〈Cmax/Na〉 and 〈V〉 versus ϕobs for the active ballistic system from Fig. 1(a) with ϕtot = 0.3534 at FD = 0.05. For ϕobs < 0.0039 the system is in a depinned fluctuating cluster state, labeled phase Ifc, where 〈V〉 is finite and the long-time average of Cmax/Na is CL ≈ 0.5. For 0.0039 ≤ ϕobs < 0.025, we find a pinned single cluster state, denoted phase II, with CL > 0.9 and 〈V〉 ≈ 0.0. In Fig. 4(a) we illustrate phase II at ϕobs = 0.01178. Phase II can be viewed as a pinned jammed state in which a single clump has nucleated around the obstacles and acts as a rigid solid. Active disks that are not adjacent to obstacles are pinned or prevented from moving by other active disks through contact interactions, so the collective pinning of the clump is dominated by disk-disk interactions rather than by direct disk-obstacle interactions. Since we are using monodisperse disks rather than the bidisperse disk mixture commonly studied in jammed systems, the particles forming the cluster have a substantial amount of hexagonal or crystalline ordering, whereas typical 2D jammed systems form amorphous rather than polycrystalline packings [69,71]; however, in both our system and the jamming systems, it is the disk-disk contact interactions that cause the system to act like a solid that can be pinned by a small number of obstacles. Previous work [52] on active ballistic systems revealed similar large-scale frozen cluster states, but did not include an external drift force. In the present study, the formation of the single frozen cluster in the presence of a drift force results in a pinned state.
For 0.025 ≤ ϕobs < 0.065 in Fig. 3(a,b), we observe a pinned multicluster state termed phase III in which 〈V〉 = 0 where CL decreases from CL=0.9 to CL=0.12 with increasing ϕobs. A snapshot of the disk positions in phase III at ϕobs=0.039 appears in Fig. 4(b). For ϕobs ≥ 0.065, the system is in phase IV, a pinned disordered state with CL < 0.15 in which the disks form numerous small clumps that gradually decrease in size with increasing ϕobs, as illustrated in Fig. 4(c) at ϕobs = 0.0942 and in Fig. 4(d) at ϕobs = 0.1963.
Fig5.png
Figure 5: Passive disk positions (green circles) and obstacle locations (blue circles) for the system in Fig. 4(c,d) with FD=0.05 and ϕtot = 0.3534 obtained at the values of ϕobs marked by the letters a to d in Fig. 3(c). (a) The flowing state at ϕobs = 0.01178. (b) At ϕobs = 0.055, clusters begin to form. (c) The clogged state at ϕobs = 0.14137. (d) The clogged state at ϕobs = 0.1963.
We plot CL and 〈V〉 versus ϕobs for passive disks with ϕtot=0.3534 and FD=0.05 in Fig. 3(c,d). The passive disks do not reach a pinned state with 〈V〉 ≈ 0 until ϕobs > ϕc=0.098. Figure 3(c) shows that CL is small for low ϕobs and increases to a peak value of CL=0.5 just below ϕc. For ϕobs > ϕc, CL decreases with increasing obstacle density. In Fig. 5(a) we show a snapshot of the flowing state at ϕobs = 0.01178. Although no clusters appear, the disks tend to form one-dimensional (1D) flowing chains. In the flowing state at ϕobs=0.055, illustrated in Fig. 5(b), small clusters are beginning to appear. The pinned cluster state near the peak value of CL at ϕobs = 0.1178 is shown in Fig. 2(d). Above the peak in CL, the size of the clusters decreases with increasing obstacle density and a clogged state forms as shown in Fig. 5(c) for ϕobs = 0.14137 and in Fig. 5(d) for ϕobs = 0.1963, where the clusters have become quite small.
Fig6.png
Figure 6: The scaling of V ∝ |ϕc − ϕobs|β for the passive disks in Fig. 3(d) with ϕc=0.098 and β = 0.35.
The peak or divergence in CL for the passive disk system in Fig. 3(c) suggests that the onset of complete clogging at ϕc occurs at a critical point. We have tried performing a power law fit CL ∝ |ϕc − ϕobs|ν on either side of the divergence; however, we find only a limited range for the fit resulting in strong variations in the exponent. We find more consistent scaling of the average drift velocity as ϕc is approached, with 〈V〉 ∝ |ϕc − ϕobs|β, as shown in Fig. 6 where β = 0.35. We have studied the critical clogging behavior of passive disks in more depth in Ref. [14], where we find a robust power law divergence in the transient times near ϕc consistent with an absorbing phase transition. The focus of the present work is active ballistic jamming and we measure the passive disks for comparison. Our results indicate that the onset of pinned or clogged states for the active ballistic disks is very different in nature from the clogging of passive disks. In particular, we find only two phases for the passive disks and four phases for the active ballistic disks. The Ifc-II transition marking the onset of a pinned state for the active ballistic disks produces discontinuities in both 〈V〉 and CL, consistent with a first order phase transition, while for the passive disks, the onset of pinning or clogging has the character of a second order phase transition or a crossover phenomenon.
Fig7.png
Figure 7: (a) 〈Cmax〉, the average number of disks in the largest cluster, vs ϕobs at FD=0.05 for ϕtot=0.668 (Ntot = 8500, turquoise), 0.589 (7500, light green), 0.511 (6500, red), 0.432 (5500, dark green), 0.354 (4500, blue), 0.275 (3500, maroon), 0.196 (2500, violet), 0.118 (1500, orange), and 0.0786 (1000, magenta). (b) The corresponding CL vs ϕobs curves. The I-II transition is associated with a jump or increase in 〈Cmax〉 and CL, while at large ϕobs, the system enters a pinned disordered phase as indicated by the drop in Cmax and CL to nearly zero. The labels a to d in (a) indicate the values of ϕobs at which the images in Fig. 8 were obtained.
Fig8.png
Figure 8: The active ballistic disk positions (red circles) and obstacle locations (blue circles) for the system in Fig. 7 obtained at the values of ϕtot and ϕobs marked by the letters a to d in Fig. 7. (a) The drifting liquid for ϕtot = 0.196 and ϕobs = 0.003926. (b) The pinned gel phase containing a large-scale percolating cluster at ϕtot = 0.589 and ϕobs = 0.1256. (c) The pinned disordered phase at ϕtot = 0.589 and ϕobs = 0.283. (d) The high density depinned fluctuating cluster state at ϕtot = 0.668 and ϕobs = 0.000156.
In Fig. 7(a) we plot 〈Cmax〉, the average number of disks in the largest cluster, versus ϕobs for the active ballistic system at varied ϕtot to highlight the evolution of phases I through IV. Figure 7(b) shows the collapse of the curves when the same results are plotted in terms of CL, the average fraction of disks in the largest cluster, versus ϕobs. For ϕtot ≤ 0.275, phase I is a uniform drifting liquid with CL < 0.04, as illustrated in Fig. 8(a) for ϕtot = 0.196 at ϕobs = 0.003926. As ϕobs increases, there is a transition from the drifting liquid phase I to the pinned single cluster phase II, as indicated by the large increase in CL to a value of CL=0.8 or higher. The obstacle density ϕobs at which the I-II transition occurs shifts upward as ϕtot decreases, and at the lowest values of ϕtot that we consider, the system always remains in phase I, as shown for ϕtot = 0.0786 in Fig. 7. We note that the maximum allowed value of ϕobs decreases with decreasing ϕtot since it is bounded by the total disk density. For ϕtot ≥ 0.35, instead of the drifting liquid phase I, we find the depinned fluctuating cluster state Ifc as described previously, and in all cases the I-II and Ifc-II transitions are associated with a jump or increase in CL. The II-III transition also shifts to higher values of ϕobs with increasing ϕtot.
Within the pinned disordered phase IV in Fig. 7, an additional feature emerges for ϕtot = 0.432 in the form of a peak in 〈Cmax〉 and CL near ϕobs = 0.115. This peak grows in both height and extent with increasing ϕtot. At the onset of the pinned disordered phase IV, CL is low and the disks form a small number of isolated clumps. As ϕobs increases, these clumps break apart and the disks are spread more evenly over the substrate. At higher overall disk densities ϕtot ≥ 0.432, this produces a percolation transition in which the broken clumps merge to form a pinned gel or labyrinth state of the type illustrated in Fig. 8(b) at the peak in 〈Cmax〉 and CL for the ϕtot = 0.589 and ϕobs = 0.1256 system in Fig. 7. For higher ϕobs, the active ballistic disks spread further apart and the gel transforms to a pinned disordered state, as shown in Fig. 8(c) for the ϕtot = 0.589 system from Fig. 7 at ϕobs = 0.283. The emergence of the intermediate pinned gel state is responsible for the additional peak in 〈Cmax〉 and CL. At the highest value ϕtot=0.668 in Fig. 7, the II-III transition is lost and the 〈Cmax〉 and CL curves become nearly featureless below the transition to the pinned disordered phase IV. When the total disk density is high, motion in the depinned fluctuating cluster phase Ifc becomes less intermittent and the steady state value of CL increases to CL > 0.95. The high density depinned fluctuating cluster state is illustrated in Fig. 8(d) for ϕtot = 0.668 and ϕobs = 0.00156.
Fig9.png
Figure 9: The locations of the phases in Fig. 7 as a function of ϕobs vs ϕtot. Phase I (magenta) is the drifting liquid state; phase Ifc (pink) is the depinned fluctuating clump state; phase II (yellow) is the pinned single cluster state; phase III (dark green) is the pinned multicluster state; phase IIIpc (light green) is the pinned gel state; and phase IV (blue) is the pinned disordered state.
Using the results in Fig. 7, we can construct a phase diagram showing the evolution of the different phases as a function of ϕobs versus ϕtot, as shown in Fig. 9. The unpinned state is a drifting liquid (phase I) for low ϕtot and a depinned fluctuating clump state (phase Ifc) for high ϕtot. Phase II is the pinned single clump state and phase III is the pinned multiclump state. For large ϕtot we find a window of phase IIIpc, the pinned gel state, at ϕobs values above phase III. Phase IV is the pinned disordered state. The features in the CL and 〈V〉 curves indicate that the I-II and Ifc-II transitions are first order in nature, while the II-III and III-IV transitions are continuous or show crossover behavior. In Ref. [14] we perform a detailed study of the behavior of passive disks as a function of ϕobs versus ϕtot, where we find that the pinned phase appears at much higher values of ϕobs than in the active ballistic system. We note that there may be additional phases in the active ballistic system at values of ϕtot higher than those shown in Fig. 9, particularly upon approaching ϕtot=0.9 where the system crystallizes into a close-packed lattice.
Fig10.png
Figure 10: (a) CL vs ϕtot for active ballistic disks with FD=0.05. Light green circles: In the obstacle-free system ϕobs=0, clustering begins near ϕtot = 0.28. Dark green squares: A system with ϕobs = 0.00157, showing that inclusion of a small number of obstacles can stabilize a cluster state at much lower densities ϕtot ≈ 0.1. (c) Cmax/Na and (d) instantaneous velocity V versus time for a system with ϕtot=0.196 and ϕobs=0.00785 that remains in phase I but is near the I-II transition. Drops in V indicate the temporary formation of a pinned cluster as shown by the corresponding jumps in Cmax/Na.
In nonballistic active particle systems that undergo driven diffusion or have run-and-tumble motion with finite τr, a phase separated state appears in the absence of quenched disorder as a function of disk density ϕtot and activity. Typically there is a density ϕtotmin below which phase separation does not occur. For the ballistic active matter system we consider, in the absence of obstacles a phase separated state appears only for ϕtot ≥ 0.35. The introduction of obstacles makes it possible for a cluster state to nucleate at much smaller values of ϕtot. For FD = 0.05, we find clustering at densities as small as ϕtot ≈ 0.1. To illustrate this more clearly, in Fig. 10 we plot CL versus ϕtot for systems with ϕobs = 0 and ϕobs = 0.00157. In the obstacle-free system, the cluster size begins to increase near ϕtot = 0.28, reaching a value of CL = 0.8 at ϕtot = 0.47. In contrast, adding a small number of obstacles shifts the onset of clustering down to ϕtot = 0.095, and CL reaches a value of CL = 0.8 at ϕtot = 0.12. This indicates that the obstacles are responsible for nucleating stable clusters over the range 0.12 < ϕtot < 0.28. The disorder-induced cluster state can be viewed in terms of a wetting phenomenon where the active particles accumulate not along walls [74] but next to the obstacles.
The ability of a cluster to form at low obstacle densities also depends on FD. As FD decreases, the transition from the drifting liquid phase I to the pinned single cluster phase II occurs at lower ϕobs. In general, even when FD is too large to stabilize phase II for a given value of ϕobs, a transient pinned cluster can still form on a temporary basis. An example of this appears in Fig. 10(b,c) where we plot Cmax/Na and instantaneous velocity V versus time for a system with ϕtot = 0.196 and ϕobs = 0.00785, which is close to the I-II transition on the phase I side. There are a series of dips in V correlated with jumps in Cmax/Na which arise when a pinned cluster forms, reducing the velocity temporarily. The cluster quickly breaks apart, restoring V and Cmax/Na to their steady state average values.

IV.  DEPINNING AND DRIVE DEPENDENCE

Fig11.png
Figure 11: (a) 〈V〉 and (b) CL vs FD for active ballistic disks with ϕtot = 0.3534 and ϕobs = 0.00178. At FD=0.05 the system is in phase II, the pinned single cluster state, and at FD=Fc=0.1 it depins and enters phase Ifc. (c) 〈V〉 and (d) CL vs FD for passive ballistic disks at the same values of ϕtot and ϕobs. There is no depinning threshold and clustering does not occur.
We next consider the effect of changing FD in order to construct velocity-force (v−f) curves and measure the depinning threshold Fc. We compare the depinning of the active ballistic disks to that of passive disks. In Fig. 11(a,b) we plot 〈V〉 and CL versus FD for an active ballistic system with ϕtot = 0.3534 and ϕobs = 0.00178, which is in the pinned single cluster phase II at FD = 0.05. We find a drive-induced depinning transition at FD=Fc=0.1 from phase II to phase Ifc, as shown by the sharp drop in CL that coincides with the onset of a linear increase in 〈V〉 with increasing FD. In Fig. 11(c,d) we show 〈V〉 and CL versus FD for a passive disk system with the same values of ϕtot and ϕobs. There is no depinning threshold and CL < 0.005 for all values of FD, indicating a complete lack of clustering.
Fig12.png
Figure 12: (a) 〈V〉 and (b) CL vs FD for active ballistic disks with ϕtot = 0.3534 at ϕobs = 0.00235 (light blue), 0.01178 (red), 0.03927 (green), 0.0628 (dark blue), 0.0785 (violet), 0.0942 (brown), and 0.1256 (magenta). (c) 〈V〉 and (d) CL vs FD for passive disks with ϕtot = 0.3534 at ϕobs = 0.00235 (light blue), 0.01178 (red), 0.03927 (green), 0.0628 (dark blue), 0.094274 (violet), and 0.1267 (brown), showing that a finite depinning threshold does not appear until ϕobs > 0.094274.
In Fig. 12(a,b) we plot 〈V〉 and CL versus FD for the active ballistic disks with ϕtot = 0.3534 at ϕobs = 0.00235 to 0.1256. We find that the II-Ifc transition, corresponding to the depinning threshold, drops to lower values of FD as ϕobs decreases, as indicated most clearly by the ϕobs=0.00235 curve in Fig. 12(a) which has Fc=0.035. The II-Ifc depinning transition is generally quite sharp, and the system goes directly from the pinned state to a fully flowing state without passing through a regime in which moving and pinned active particles coexist. In contrast, the depinning transition separating phases III and IV for ϕobs ≥ 0.03927 is smooth or continuous and is plastic in nature, so that above depinning only a portion of the active disks are flowing while the other portion remains pinned. In studies of depinning in other systems such as superconducting vortices or colloidal particles moving over quenched disorder, elastic depinning is associated with a sharp transition in the v−f curve and a scaling of V ∝ (FD −Fc)α with α < 1.0. Plastic depinning is accompanied by an extensive nonlinear regime in the v−f curves with α > 1.0. In our active ballistic disk system, the resolution of the v−f curves is not high enough to perform a scaling analysis; however, the qualitative change in the v−f curve at the Ifc-II transition is consistent with an elastic or collective depinning at low ϕobs crossing over to plastic depinning for higher ϕobs. For the III-IV depinning transition, there is generally a small peak in CL at Fc produced when the disks start to accumulate behind the obstacles, and in all cases there is an overall drop in CL at higher values of FD in the moving phase. For ϕobs = 0.1256, depinning does not occur until FD=Fc = 1.35.
In Fig. 12(c,d) we plot 〈V〉 and CL versus FD for the passive disks with ϕtot = 0.3534 at ϕobs = 0.00235 to 0.1267. There is no finite depinning threshold for ϕobs ≤ 0.094274. We find an extended regime of nonlinear flow for ϕobs > 0.0628 associated with a coexistence of flowing and clogged disks. In Fig. 12(d), CL < 0.01 at all FD for ϕobs = 0.00235 and ϕobs=0.01178, while for higher ϕobs there is generally a decrease in CL with increasing FD for FD > 0.3. The maximum value of CL occurs for ϕobs = 0.094274, which is just below the obstacle density at which a finite depinning threshold first appears. These results show that for varied FD, the active ballistic disks have a much higher susceptibility to becoming pinned than the passive disks.
Fig13.png
Figure 13: (a) Fc (red squares) and the value of CL at FD=0.05 (green circles) vs ϕobs for the active ballistic disks at ϕtot = 0.3534. The depinning threshold remains finite down to ϕobs = 0.0008, and there is an increase in Fc at the onset of phase IV. (b) Fc and (c) the value of CL at FD=0.05 vs ϕtot for ϕobs = 0.09427. The behavior of Fc is nonmonotonic, and Fc increases with increasing ϕtot for ϕtot > 0.6 when CL=1, indicating the formation of a jammed state. The labels a to d in (c) indicate the values of ϕtot at which the images in Fig. 14 were obtained.
In Fig. 13(a) we plot the evolution of the depinning threshold Fc along with the value of CL at FD=0.05 versus ϕobs for the active ballistic disks at ϕtot = 0.3534. There is a sharp increase from Fc=0 at ϕobs=0 to Fc=0.007 at ϕobs=0.0008, the lowest nonzero obstacle density we considered, for which the ratio of obstacles to active particles is Na/Nobs = 440. For 0.0045 < ϕobs < 0.039, there is a more gradual linear increase of Fc with increasing ϕobs over the range of phase II depinning through half of the phase III depinning. This is followed by a regime of roughly constant Fc for 0.039 < ϕobs < 0.065 in the second half of phase III depinning, while in phase IV for ϕobs > 0.65, there is a rapid increase in Fc which coincides with a drop in CL. Plastic depinning appears in phase IV, and the rapid increase in Fc in this regime is reminiscent of the "peak effect" phenomenon observed for the depinning of superconducting vortices [75,3,2]. At low disorder strength, the vortices form a crystal that that can be collectively pinned by a small number of pinning sites; however, when the pinning strength is increased, the crystalline structure breaks apart, the vortex structure becomes amorphous, and there is a pronounced increase in the depinning force that is much larger than what would be expected from the increase in the pinning strength [2,3]. In the active ballistic disk system, the clusters have considerable crystalline order, and when the amount of disorder is increased by raising the number of obstacles, the large clusters break up into smaller clusters that can be pinned more easily, leading to the increase in Fc.
Fig14.png
Figure 14: The active ballistic disk positions (red circles) and obstacle locations (blue circles) for the system in Fig. 13(b,c) with ϕobs = 0.09427 obtained at the values of ϕtot marked by the letters a to d in Fig. 13(c). (a) The pinned liquid at ϕtot = 0.1963. (b) A pinned weakly clustered state at ϕobs = 0.275 (phase IV). (c) A pinned gel state at ϕtot = 0.51 (phase IIIpc). (d) A jammed solid state at ϕtot = 0.8246.
In Fig. 13(b,c) we plot Fc and the value of CL at FD=0.05 versus ϕtot for the active ballistic disks at ϕobs = 0.09427. For low ϕtot < 0.35, a disordered but strongly pinned phase appears, as indicated by the large Fc and the low CL < 0.1. In Fig. 14(a) we illustrate the disk configuration at ϕtot = 0.1963 where a pinned liquid with small local disk clusters forms. As ϕtot increases, Fc decreases and a pinned weakly clustered state emerges, as shown in Fig. 14(b) at ϕtot = 0.275. Since the number of obstacles is fixed, as ϕtot increases, each obstacle must restrain a larger number of mobile disks, causing Fc to decrease with increasing ϕtot. In Fig. 14(c) we plot the disk configuration at ϕtot = 0.51, where the system forms a pinned gel phase with low Fc. Figure 13(c,d) shows a local minimum in Fc near ϕtot = 0.628, where CL = 0.92. For ϕtot > 0.628, Fc begins increasing with increasing ϕtot and CL approaches CL=1 as the disks assemble into a single jammed solid packing, as shown in Fig. 14(d) at ϕtot = 0.8246.
Fig15.png
Figure 15: (a) Fc and (b) the value of CL at FD=0.05 vs ϕobs at ϕtot=0.3534 for the active ballistic disks (circles) and passive disks (squares). (c) Fc and (d) the value of CL at FD=0.05 vs ϕtot at ϕobs=0.094 for the active ballistic disks (circles) and passive disks (squares). At high ϕtot, both the active and passive disks undergo a transition into a jammed state.
Fig16.png
Figure 16: The passive disk positions (green circles), obstacle locations (blue circles), and disk trajectories for the system in Fig. 15(c,d) (a) At ϕtot = 0.196, 1D flow channels coexist with pinned disks. (b) At ϕtot = 0.4712, collective interactions between disks cause a larger fraction of the disks to flow.
In Fig. 15(a,b) we plot Fc and the value of CL at FD=0.05 versus ϕobs at ϕtot=0.3534 for active ballistic disks and passive disks. The depinning threshold remains finite in the active system for ϕobs ≥ 0.0008, whereas in the passive system the depinning threshold drops to zero when ϕobs ≤ 0.094. This indicates that 100 times fewer obstacles are needed to pin the active system compared to the passive system. In addition, the depinning threshold for the active system is always higher than that of the passive system. In Fig. 15(c,d) we show Fc and the value of CL at FD=0.05 versus ϕtot at ϕobs = 0.094 for the active ballistic and passive disks. Here the depinning threshold for the passive disks does not become finite until ϕtot > 0.7, the density above which CL increases to CL=1.0. The high density active ballistic and passive disk states are similar in nature and are dominated by the formation of a jammed solid state. Even when the depinning threshold in the passive disk system is zero, the v−f curves can show strongly nonlinear behavior, and a large fraction of the passive disks remain pinned or immobile for ϕtot < 0.35, the same total disk density at which the active ballistic disks exhibit a large increase in Fc. In Fig. 16(a) we show the passive disk locations and trajectories at ϕtot = 0.196 and FD = 0.05. A portion of the disks move in 1D filamentary channels while the remaining disks are pinned. These filamentary channels persist down to FD = 0.0 in the passive disks, but are generally absent for active ballistic disks. In the passive disk system, as ϕtot increases, cooperative interactions between the mobile disks reduce the overall trapping and lead to a higher fraction of flowing disks, as illustrated in Fig. 16(b) at FD = 0.05 and ϕtot = 0.4712. As ϕtot further increases, the system approaches a jammed solid with a finite depinning threshold.

V.  FINITE RUN TIME ACTIVE DISKS

Fig17.png
Figure 17: (a) The disk mobility M and (b) CL vs run length lr for finite run time active disks at FD=0.05, ϕtot = 0.51 and ϕobs = 0.14137. The low lr behavior is similar to that found in the passive disk limit while the high lr behavior is similar to that found in the active ballistic limit. Between these two limits the disks form a uniform density, highly mobile liquid state. The labels a to d in (b) indicate the values of lr at which the images in Fig. 18 were obtained. (c) M and (d) CL vs lr for finite run time active disks at ϕtot = 0.667 and ϕobs=0.14137.
Fig18.png
Figure 18: The finite run time active disk locations (orange circles) and obstacle locations (blue circles) for a system with ϕtot = 0.51 and ϕobs = 0.14137 obtained at the values of lr marked by the letters a to d in Fig. 17(b). (a) The low mobility clogged state at lr = 0.001. (b) The high mobility uniform liquid at lr = 2.0. (c) The actively clogged state at lr = 10. (d) The actively clogged state at lr = 200.
We next show how the passive disk pinning and active ballistic disk pinning limits can be connected to each other by considering run-and-tumble particles where we gradually increase the running time from close to zero, which is the passive limit, to large values which approach the ballistic limit. In Fig. 17(a,b) we plot the mobility M = 〈V〉/〈V0〉 per disk and CL versus run length lr for a finite run time active disk system at ϕtot = 0.51, ϕobs = 0.14137, and FD = 0.05. Here 〈V0〉 is the average drift velocity of an individual disk in the absence of obstacles or other disks, so for FD=0.05, 〈V0〉 = 0.05, and in the obstacle-free limit, M = 1.0. The disk dynamics are the same as those described in Eq. 1 except the running time τr is now finite. For convenience, we characterize the activity level in the system at a fixed |Fm| in terms of the run length lr = |Fm|τr, so that large lr corresponds to large τr. In the passive limit, lr=0, while in the ballistic limit, lr is infinite. For the parameters we consider in this section, the system reaches a completely pinned state in both the passive and ballistic limits. In Fig. 17(a) at low lr, 〈M〉 is small, indicating that a clogged state has formed that is similar to the passive disk clogged state which appears at lr=0. At the same time, CL > 0.8 in Fig. 17(b), indicating the formation of a large cluster. In Fig. 18(a) we plot the disk configurations for lr = 0.001, where the clogged state has highly heterogeneous local disk density. For 0.01 < lr < 5, an easily flowing liquid appears, as indicated by the increase in M and the drop in CL to CL < 0.01. We illustrate the flowing liquid at lr=2.0 in Fig. 18(b), where the disk density is uniform and clustering behavior is lost. As lr increases, M decreases once self-clustering of the disks begins to occur, as shown in Fig. 18(c) for lr=10. At large lr, CL approaches CL=0.9 and M drops to a low value. In this regime, the image of the lr=200 system in Fig. 18(d) indicates that clustering similar to that found in the active ballistic pinned state is present. We note that M > 0 for any finite lr since there is always a chance that the activity can unpin a fraction of the disks even when lr becomes large. The motion in the finite but large lr regime becomes highly intermittent and exhibits avalanche-like fluctuations, as has been described in detail elsewhere [65]. In Fig. 17(c,d) we plot M and CL versus lr for the run-and-tumble disks at a higher ϕtot = 0.667. We find the same trend in which clustering occurs at both small and large lr, with a correspondingly low value of M, while for intermediate lr, the clustering disappears, CL is low, and M is high. The maximum value of M is lower at ϕtot=0.667 than at ϕtot=0.51 due to the crowding effect that appears at higher disk densities.
Fig19.png
Figure 19: (a) The disk mobility M and (b) CL vs run length lr at FD = 0.05, ϕtot = 0.3543 and ϕobs = 0.14137 for finite run time active disks. (c) M and (d) CL vs lr at ϕtot = 0.2356 and ϕobs = 0.14137 for finite run time active disks, where the system is always in the disordered regime.
In Fig. 19(a,b) we plot M and CL for finite run time active disks at ϕtot=0.3534 and ϕobs = 0.1413. We find the same trend as in Fig. 17 where high values of CL are associated with low values of M; however, at this lower total disk density ϕtot, the maximum value of CL is reduced. A peak in CL near lr=0.005 indicates the appearance of additional clustering just before the activity becomes large enough to liquefy the system and increase M. This suggests that the activity becomes strong enough to cause untrapping of individual disks at lower lr but that the untrapped disks then pool into clusters that are too large to be broken apart by individual disks until lr increases, at which point the clumps break apart and the system flows in a liquid state. In Fig. 19(c,d) we plot M and CL versus lr for finite run length disks at ϕtot = 0.2356 and ϕobs = 0.1413. At this low total disk density, little clustering occurs and CL is always small. We still observe a peak in M at intermediate lr values; however, M is relatively low overall, reaching a maximum value of only M=0.1. This result emphasizes the role of collective disk-disk interactions in liquefying the system and increasing the mobility for intermediate lr.
Although we consider only monodisperse disks in this work, the introduction of obstacles causes the clogged states to exhibit a considerable amount of structural disorder, similar to that found in jammed states for bidisperse or completely amorphous systems [68,70,71]. For passive disks at low obstacle densities, previous studies have shown that jamming occurs near the obstacle-free jamming density of ϕj or point J, but that the jamming transition shifts to lower disk densities as the obstacle density increases [70,72]. When the obstacle density is high enough, the behavior of the system changes and instead of a uniform jammed state, the passive disks assemble into a clogged state with strongly heterogeneous local disk density [14]. For active disks we find three limiting regimes of behavior. At high disk densities, the effect of the activity becomes negligible and the behavior is similar to that found in the passive high disk density limit, which is controlled by jamming near point J in amorphous systems or crystallization at a density ϕtot=0.9 for monodisperse disks. The second limiting regime appears for low activity and intermediate disk densities, where the active disks form a clogged configuration similar to that found for the clogging of passive disks. The third limiting regime, consisting of the actively pinned or clogged states that appear for active ballistic disks or for disks with finite but large lr, is unique to the active disks and does not appear in passive disk systems. This regime extends over a wide range of obstacle densities and can assume the form of phase II, III, or IV, as described in this work. Our results suggest that in addition to the density, temperature, and load axes on the jamming phase diagram [68,69], there could be two additional axes, the activity level and the obstacle density or disorder level, that produce jammed states.
In future work for both the passive and active disk systems, it would be interesting to investigate how different the pinned phases are. For example, at high disk densities near the jamming transition, it is likely that the system is highly stable to perturbations since there are fewer available degrees of freedom. Similarly, in a clogged state with high obstacle densities, a perturbed system is likely to fall back into the same or a similar clogged state. On the other hand, in phases II or III, a small perturbation could readily break up one or more of the clusters, permitting the system to flow again, so that although the active systems are more susceptible to clogging, the clogged state they reach may be more fragile than that formed by passive disks.

VI.  SUMMARY

We have examined the pinning and clogging behaviors of active run-and-tumble disks in the ballistic limit driven through an array of obstacles. As a function of increasing obstacle density, we find four generic phases: an unpinned fluctuating cluster phase, a pinned single cluster phase in which a small number of obstacles can pin a large number of active disks, a pinned multiclump phase, and a pinned disordered phase. We find that in contrast to passive disks, the active ballistic disks can reach a pinned state at relatively low obstacle densities. Within the pinned disordered phase, as the density of active ballistic disks increases, a pinned gel state or labyrinth pattern appears. By constructing velocity-force curves we find that the active ballistic disks exhibit a finite depinning threshold. As the obstacle density increases, there is a transition from collective depinning of the pinned clusters to plastic depinning of the disordered pinned states that is associated with a large increase in the depinning threshold. As a function of total disk density, the depinning threshold for the active ballistic disks is nonmonotonic, dropping at intermediate disk densities when collective disk-disk interactions reduce the threshold, but rising at high disk densities as the system approaches a jammed or crystalline state. In contrast, for passive disks the depinning threshold is only finite at high disk densities, while at lower disk densities filamentary channels of disk flow form that are absent in the active disk system. For both the active and passive disks, the pinned or clogged states are phase separated; however, the phase separation appears at a much lower obstacle density for the active disks. Finite run time active disks provide a connection between the active ballistic and passive disk systems, and exhibit a low mobility phase separated state for short run times in the passive limit as well as for long run times in the active ballistic limit. At intermediate run times, the active disks form an easily flowing uniform liquid with reduced clustering, and there is an optimal level of activity that maximizes the flux through the system. We describe how our results can be related to systems that exhibit jamming. Since a self-jamming state appears in the limit of high activity or sufficiently large obstacle density, both activity and disorder strength act as two additional parameters that can produce jamming in particle assemblies.

ACKNOWLEDGMENTS

This work was carried out under the auspices of the NNSA of the U.S. DoE at LANL under Contract No. DE-AC52-06NA25396.

REFERENCES

[1]
D.S. Fisher, Collective transport in random media: From superconductors to earthquakes, Phys. Rep. 301, 113 (1998).
[2]
C. Reichhardt and C. J. Olson Reichhardt, Depinning and nonequilibrium dynamic phases of particle assemblies driven over random and ordered substrates: A review, Rep. Prog. Phys. 80, 026501 (2017).
[3]
S. Bhattacharya and M. J. Higgins, Dynamics of a disordered flux line lattice, Phys. Rev. Lett. 70, 2617 (1993).
[4]
C. J. Olson, C. Reichhardt, and F. Nori, Nonequilibrium dynamic phase diagram for vortex lattices, Phys. Rev. Lett. 81, 3757 (1998).
[5]
G. Grüner, The dynamics of charge-density waves, Rev. Mod. Phys. 60, 1129 (1988).
[6]
F.I.B. Williams, P. A. Wright, R. G. Clark, E. Y. Andrei, G. Deville, D. C. Glattli, O. Probst, B. Etienne, C. Dorin, C. T. Foxon, and J. J. Harris, Conduction threshold and pinning frequency of magnetically induced Wigner solid, Phys. Rev. Lett. 66, 3285 (1991).
[7]
C. Reichhardt, C. J. Olson, N. Grønbech-Jensen, and F. Nori, Moving Wigner glasses and smectics: Dynamics of disordered Wigner crystals, Phys. Rev. Lett. 86, 4354 (2001).
[8]
J. Iwasaki, M. Mochizuki, and N. Nagaosa, Universal current-velocity relation of skyrmion motion in chiral magnets, Nat. Commun. 4, 1463 (2013).
[9]
C. Reichhardt, D. Ray, and C. J. Olson Reichhardt, Collective transport properties of driven skyrmions with random disorder, Phys. Rev. Lett. 114, 217202 (2015).
[10]
A. Pertsinidis and X. S. Ling, Statics and dynamics of 2D colloidal crystals in a random pinning potential, Phys. Rev. Lett. 100, 028303 (2008).
[11]
P. Tierno, Depinning and collective dynamics of magnetically driven colloidal monolayers, Phys. Rev. Lett. 109, 198304 (2012).
[12]
S. Deutschländer, T. Horn, H. Löwen, G. Maret, and P. Keim, Two-dimensional melting under quenched disorder, Phys. Rev. Lett. 111, 098301 (2013).
[13]
Y. Yang, D. McDermott, C.J.O. Reichhardt, and C. Reichhardt, Dynamic phases, clustering, and chain formation for driven disk systems in the presence of quenched disorder, Phys. Rev. E 95, 042902 (2017).
[14]
H. Peter, A. Libal, C. Reichhardt, and C.J.O. Reichhardt, Crossover from jamming to clogging behaviors in heterogeneous environments, Sci. Rep., in press (2018).
[15]
R.L. Stoop and P. Tierno, Clogging and jamming of colloidal monolayers driven across a disordered landscape, arXiv:1712.05321.
[16]
T. Bohlein, J. Mikhael, and C. Bechinger, Observation of kinks and antikinks in colloidal monolayers driven across ordered surfaces, Nature Mater. 11, 126 (2012).
[17]
A. Vanossi, N. Manini, M. Urbakh, S. Zapperi, and E. Tosatti, Modeling friction: From nanoscale to mesoscale, Rev. Mod. Phys. 85, 529 (2013).
[18]
S. Zapperi, P. Cizeau, G. Durin, and H.E. Stanley, Dynamics of a ferromagnetic domain wall: Avalanches, depinning transition, and the Barkhausen effect. Phys. Rev. B 58, 6353 (1998).
[19]
J.P. Sethna, K. Dahmen, and C.R. Myers, Crackling noise, Nature 410, 242 (2001).
[20]
P. Aussillous, Z. Zou, É. Guazzelli, L. Yan, and M. Wyart, Scale-free channeling patterns near the onset of erosion of sheared granular beds, Proc. Natl. Acad. Sci. (USA) 113, 11788 (2016).
[21]
J.A. Drocco, M.B. Hastings, C.J.O. Reichhardt, and C. Reichhardt, Multiscaling at Point J: Jamming is a critical phenomenon, Phys. Rev. Lett. 95, 088001 (2005).
[22]
R. Candelier and O. Dauchot, Journey of an intruder through the fluidization and jamming transitions of a dense granular media, Phys. Rev. E 81, 011304 (2010).
[23]
C. Reichhardt, C.J.O. Reichhardt, I. Martin, and A.R. Bishop, Dynamical ordering of driven stripe phases in quenched disorder, Phys. Rev. Lett. 90, 026401 (2003).
[24]
J.M. Carlson and J.S. Langer, Properties of earthquakes generated by fault dynamics, Phys. Rev. Lett. 62, 2632 (1989).
[25]
M.-C. Miguel, A. Vespignani, S. Zapperi, J. Weiss, and J.-R. Grasso, Intermittent dislocation flow in viscoplastic deformation, Nature 410, 667 (2001).
[26]
M.C. Marchetti, J.F. Joanny, S. Ramaswamy, T.B. Liverpool, J. Prost, M. Rao, and R.A. Simha, Hydrodynamics of soft active matter, Rev. Mod. Phys. 85, 1143 (2013).
[27]
C. Bechinger, R. Di Leonardo, H. Löwen, C. Reichhardt, G. Volpe, and G. Volpe, Active Brownian particles in complex and crowded environments, Rev. Mod. Phys. 88, 045006 (2016).
[28]
Y. Fily and M. C. Marchetti, Athermal phase separation of self-propelled particles with no alignment, Phys. Rev. Lett. 108, 235702 (2012).
[29]
G. S. Redner, M. F. Hagan, and A. Baskaran, Structure and dynamics of a phase-separating active colloidal fluid, Phys. Rev. Lett. 110, 055701 (2013).
[30]
M.E. Cates and J. Tailleur, Motility-induced phase separation, Annu. Rev. Condens. Mat. Phys. 6, 219 (2015).
[31]
J. Palacci, S. Sacanna, A.P. Steinberg, D.J. Pine, and P.M. Chaikin, Living crystals of light-activated colloidal surfers, Science 339, 936 (2013).
[32]
I. Buttinoni, J. Bialké, F. Kümmel, H. Löwen, C. Bechinger, and T. Speck, Dynamical clustering and phase separation in suspensions of self-propelled colloidal particles, Phys. Rev. Lett. 110, 238301 (2013).
[33]
C. Reichhardt and C.J.O. Reichhardt, Active matter transport and jamming on disordered landscapes, Phys. Rev. E 90, 012701 (2014).
[34]
C. Reichhardt and C.J.O. Reichhardt, Active microrheology in active matter systems: Mobility, intermittency, and avalanches, Phys. Rev. E 91, 032313 (2015).
[35]
V. Prymidis, S. Samin, and L. Filion, State behaviour and dynamics of self-propelled Brownian squares: a simulation study, Soft Matter, 12, 4309 (2016).
[36]
N.H.P. Nguyen, D. Klotsa, M. Engel, and S.C. Glotzer, Emergent collective phenomena in a mixture of hard shapes through active rotation, Phys. Rev. Lett. 112, 075701 (2014).
[37]
Z. Ma , Q. Lei, and R. Ni, Driving dynamic colloidal assembly using eccentric self-propelled colloids, Soft Matter 13, 8940 (2017)
[38]
R. Wittkowski, J. Stenhammar, and M.E. Cates, Nonequilibrium dynamics of mixtures of active and passive colloidal particles, New. J. Phys. 19, 105003 (2017).
[39]
F. Vega-Reyes and A. Lasanta, Hydrodynamics of a granular gas in a heterogeneous environment, Entropy 19(10), 536 (2017).
[40]
X. Yang, M.L. Manning, and M.C. Marchetti, Aggregation and segregation of confined active particles, Soft Matter 10, 6477 (2014).
[41]
D. Ray, C. Reichhardt, and C.J.O. Reichhardt, Casimir effect in active matter systems, Phys. Rev. E 90, 013019 (2014).
[42]
R. Ni, M.A. Cohen Stuart, and P.G. Bolhuis, Tunable long range forces mediated by self-propelled colloidal hard spheres, Phys. Rev. Lett. 114, 018302 (2015).
[43]
Y. Fily, A. Baskaran, and M.F. Hagan, Dynamics and density distribution of strongly confined noninteracting nonaligning self-propelled particles in a nonconvex boundary, Phys. Rev. E 91, 012125 (2015).
[44]
A. P. Solon, Y. Fily, A. Baskaran, M.E. Cates, Y. Kafri, M. Kardar, and J. Tailleur, Pressure is not a state function for generic active fluids,” Nat. Phys. 11, 673 (2015).
[45]
S.A. Mallory, C. Valeriani, and A. Cacciuto, Curvature-induced activation of a passive tracer in an active bath, Phys. Rev. E 90, 032309 (2014).
[46]
N. Nikola, A.P. Solon, Y. Kafri, M. Kardar, J. Tailleur, and R. Voituriez, Active particles with soft and curved walls: equation of state, ratchets, and instabilities, Phys. Rev. Lett. 117, 098001 (2016).
[47]
C.J.O. Reichhardt and C. Reichhardt, Ratchet effects in active matter systems, Annl. Rev. Condens. Mat. Phys. 8, 51 (2017).
[48]
G. Volpe, I. Buttinoni, D. Vogt, H.-J. Kummerer, and C. Bechinger, Microswimmers in patterned environments, Soft Matter 7, 8810 (2011).
[49]
C. Lozano, B. ten Hagen, H. Löwen, and C. Bechinger, Phototaxis of synthetic microswimmers in optical landscapes, Nat. Commun. 7, 12828 (2016).
[50]
Cs. Sándor, A. Libál, C. Reichhardt, and C.J.O. Reichhardt, Collective transport for active matter run-and-tumble disk systems on a traveling-wave substrate, Phys. Rev. E 95, 012607 (2017).
[51]
O. Chepizhko, E.G. Altmann, and F. Peruani, Optimal noise maximizes collective motion in heterogeneous media, Phys. Rev. Lett. 110, 238101 (2013).
[52]
C. Reichhardt and C.J.O. Reichhardt, Absorbing phase transitions and dynamic freezing in running active matter systems, Soft Matter 10, 7502 (2014).
[53]
D. Quint and A. Gopinathan, Topologically induced swarming phase transition on a 2D percolated lattice, Phys. Biol. 12, 046008 (2015).
[54]
O. Chepizhko and F. Peruani, Active particles in heterogeneous media display new physics, Eur. Phys. J. Spec. Top. 224, 1287 (2015).
[55]
E. Pince, S.K.P. Velu, A. Callegari, P. Elahi, S. Gigan, G. Volpe, and G. Volpe, Disorder-mediated crowd control in an active matter system, Nat. Commun. 7, 10907 (2016).
[56]
M. Khatami, K. Wolff, O. Pohl, M.R. Ejtehadi, and H. Stark, Active Brownian particles and run-and-tumble particles separate inside a maze, Sci. Rep. 6, 37670 (2016).
[57]
A. Morin, N. Desreumaux, J.-B. Caussin, and D. Bartolo, Distortion and destruction of colloidal flocks in disordered environments, Nat. Phys. 13, 63 (2017).
[58]
Cs. Sándor, A. Libál, C. Reichhardt, and C.J.O. Reichhardt, Dewetting and spreading transitions for active matter on random pinning substrates, J. Chem. Phys. 146, 204903 (2017).
[59]
M. Zeitz, K. Wolff, and H. Stark, Active Brownian particles moving in a random Lorentz gas, Eur. Phys. J. E 40, 23 (2017).
[60]
Cs. Sándor, A. Libál, C. Reichhardt, and C.J.O. Reichhardt, Dynamic phases of active matter systems with quenched disorder, Phys. Rev. E 95, 032606 (2017).
[61]
J. Wang, D. Zhang, B. Xia, and W. Yu, Spatial heterogeneity can facilitate the target search of self-propelled particles, Soft Matter 13, 758 (2017).
[62]
A. Morin, D.L. Cardozo, V. Chikkadi, and D. Bartolo, Diffusion, subdiffusion and localisation of active colloids in random post lattices, Phys. Rev. E 96, 042611 (2017).
[63]
Z. Mokhtari, T. Aspelmeier, and A. Zippelius, Collective rotations of active particles interacting with obstacles, EPL 120, 1400 (2017).
[64]
T. Bertrand, Y. Zhao, O. Bénichou, J. Tailleur, and R. Voituriez, Optimized diffusion of run-and-tumble particles in crowded environments, arXiv:1711.05209.
[65]
C.J.O. Reichhardt and C. Reichhardt, Avalanche dynamics for active matter in heterogeneous media, New J. Phys. 20, 025002 (2018).
[66]
C. Reichhardt and C.J.O. Reichhardt, Negative differential mobility and trapping in active matter systems, J. Phys.: Condens. Matter 30, 015404 (2018).
[67]
I.R. Bruss and S.C. Glotzer, Phase separation of self-propelled ballistic particles, arXiv:1712.01983.
[68]
A.J. Liu and S.R. Nagel, Jamming is not just cool any more, Nature 396, 21 (1998).
[69]
C.S. O'Hern, L.E. Silbert, A.J. Liu, and S.R. Nagel, Jamming at zero temperature and zero applied stress: The epitome of disorder, Phys. Rev. E 68, 011306 (2003).
[70]
C.J.O. Reichhardt, E. Groopman, Z. Nussinov, and C. Reichhardt, Jamming in systems with quenched disorder, Phys. Rev. E 86, 061301 (2012).
[71]
C. Reichhardt and C.J.O. Reichhardt, Aspects of jamming in two-dimensional athermal frictionless systems, Soft Matter 10, 2932 (2014).
[72]
A.L. Graves, S. Nashed, E. Padgett, C.P. Goodrich, A.J. Liu, and J.P. Sethna, Pinning susceptibility: the effect of dilute, quenched disorder on jamming, Phys. Rev. Lett. 116, 235501 (2016).
[73]
S. Luding and H. J. Herrmann, Cluster-growth in freely cooling granular media, Chaos 9, 673 (1999).
[74]
N. Sepúlveda and R. Soto, Wetting transitions displayed by persistent active particles, Phys. Rev. Lett. 119, 078001 (2017).
[75]
A.B. Pippard, A possible mechanism for the peak effect in type II superconductors, Phil. Mag. 19, 217 (1969).


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