Physical Review E 90, 012701 (2014)

Active Matter Transport and Jamming on Disordered Landscapes

C. Reichhardt and C. J. Olson Reichhardt

Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA

(Received 13 February 2014; revised manuscript received 10 June 2014; published 3 July 2014)

We numerically examine the transport of active run-and-tumble particles with steric particle-particle interactions driven with a drift force over random disordered landscapes comprised of fixed obstacles. For increasing run lengths, the net particle transport initially increases before reaching a maximum and decreasing at larger run lengths. The transport reduction is associated with the formation of cluster or living crystal states that become locally jammed or clogged by the obstacles. We also find that the system dynamically jams at lower particle densities when the run length is increased. Our results indicate that there is an optimal activity level for transport of run-and-tumble type active matter through quenched disorder, and could be important for understanding biological transport in complex environments or for applications of active matter particles in random media.
I. INTRODUCTION
II. SIMULATION
III. RESULTS
IV. DISCUSSION
V. SUMMARY
References

I.  INTRODUCTION

There has been tremendous growth in interest in what are termed active matter systems, where the individual units comprising the system undergo self-propulsion [1,2,3]. Biological examples of such systems include swimming bacteria [4,5], crawling cells [3], and flocking or swarming particles or agents [6]. Non-biological active matter, such as self-driven colloids [7,8,9,10], artificial swimmers [11], and assemblies of self-motile mechanical devices such as bristle bots [12], has been the focus of an increasing number of studies. Active matter systems exhibit a rich variety of behaviors that disappear when the fluctuations experienced by the particles are only equilibrium or thermal in nature. There are different classes of dynamical rules for the motion of self-driven particles. In run-and-tumble dynamics of the type exhibited by some bacteria [4,5], the particle moves in a fixed direction for a period of time before undergoing a tumbling event and changing directions. Active Brownian particles are driven by a motor in a direction that slowly rotates due to a noise term, a dynamics that appears in many self-driven colloidal systems [2,7,10,13,14,15]. Flocking particles select their direction of motion based on the average swimming direction of their neighbors in a manner captured by the Vicsek model [6]. It was recently shown that active Brownian particle dynamics and the run-and-tumble (RT) dynamics can be mapped into each other and can be considered equivalent [16], whereas Vicsek type flocking models form a separate class.
When collections of sterically interacting particles such as hard disks undergo thermal fluctuations, they form a uniformly dense state; however, when the particles are active and obey RT dynamics there can be a transition to a phase-separated state consisting of high density clusters residing in a low density active gas [10,13,14,15,17,18,16,19,20,21]. In two-dimensional (2D) systems of monodisperse active disks, the clusters have internal hexagonal or crystalline ordering [10,18,19]. Recent experiments on light-activated self-driven colloidal particles that form such crystalline cluster states revealed that the clusters are highly dynamical, frequently breaking up and reforming, so that the cluster state has been termed a 'living crystal' [10]. Similar living crystals have been experimentally observed in other self-driven colloidal systems [19]. The onset of the cluster phase depends both on the particle density and on the activity level, defined as the distance Rl the particles move in a fixed direction during each run in a dilute system.
Studies of active matter have generally focused on systems with a smooth substrate. For example, jamming and glassy effects have been used to connect activity with an effective temperature [14,22,23,24]. Understanding how active matter particles move in random or complex landscapes is an open question that is relevant to biological systems operating in a complex environment. Possible active matter applications could also require the particles to interact with disordered landscapes. In studies of RT particles interacting with arrays of asymmetric barriers, ratchet effects arise when the activity causes a breaking of detailed balance in the particle-barrier interactions [5,25,26,27,28,29]. Recent studies of particles obeying Vicsek flocking rules without steric particle-particle interactions showed that there is an optimum noise level that maximizes the amount of collective motion, defined as the number of particles that are all swimming in the same direction [30]. Studies of swarming models in disordered environments indicate that disorder can destroy the collective motion phase [31], while a recent study showed that disordered substrates can localize or trap flocking active particles [32]. Since the flocking models are a different class of system from RT models, it would be interesting to understand if an optimal activity level can occur in the RT system.
Here we consider RT particles with steric interactions which undergo a clustering transition at longer Rl when local jamming effects occur. We study particle transport over a disordered substrate in the presence of an external drift force, which could be produced experimentally for active colloids through fluid flow or an electrophoretic current, or for bacteria by a chemotactic drift. In the absence of a substrate, the net particle velocity 〈Vx〉 is determined only by the drift force FD=Fd∧x and is independent of Rl. Particle transport over disordered substrates under an external drift is a very general problem that has been studied in a variety of condensed matter systems including colloids moving over random substrates [33], vortices moving in dirty type-II superconductors [34], sliding charge density waves [35], classical charge transport [36] and the motion of magnetic domain walls [37]. In these systems, the particle transport in the direction of the external drift increases with increasing thermal fluctuations, which diminish the effectiveness of pinning by the substrate. RT particles might be expected to experience significantly reduced pinning by the substrate since, as Rl is increased, the particles would be better able to escape from effective trapping regions; however, in this work we show that this is generally not the case. We find that as the obstacle density increases, 〈Vx〉 monotonically decreases, and that for fixed obstacle density, 〈Vx〉 initially increases with Rl before reaching a maximum and then decreasing to zero. This decrease in transport coincides with the formation of large clusters or living crystals of particles due to local jamming effects, which permit an entire cluster to be pinned by a small number of obstacles. Our results show that there is an optimal run length or activity at which maximum transport of the RT particles through disordered media can be achieved. We also demonstrate that the jamming transitions observed in 2D assemblies of non-active disks [38,39,40,41] can be connected to active matter systems and that activity can form a new axis of the jamming phase diagram.

II.  SIMULATION

We consider 2D systems of size L ×L with periodic boundary conditions in the x− and y−directions containing an assembly of Na active disks with radius rd that interact with each other via a harmonic repulsion. The dynamics of disk i located at Ri is obtained by integrating the following equation of motion:
η d Ri
dt
 = Fmi + Fsi + Fbi + FD,
(1)
where η = 1.0 is the damping constant. Under the motor force Fmi, a particle moves in a fixed direction with a constant velocity Fm/η for a fixed run time τr, and at the end of each running time a new running direction is chosen at random. We select Fm such that the particles can never pass through one another or through the obstacles. A single particle in the absence of other particles or obstacles moves a distance Rl = Fmτr in one running time. The steric disk-disk repulsion is modeled as Fsi = ∑Nai ≠ j(2rd − |rij|)Θ(2rd −|rij|)∧rij, where rij = Ri − Rj, ∧rij = rij/|rij|, and we take rd=0.5. The term Fbi represents the particle-obstacle interactions, where we model the obstacles as Np immobile disks that have the same steric interactions as the active disks. The drift force FD=Fd∧x with Fd=0.05. The disk density ϕ is determined by the area covered by all the disks, ϕ = Nπrd2/L2, where N=Na+Np, and the pin density is ϕp=Npπrd2/L2; here, we consider systems with L=100. In the absence of any obstacles or thermal fluctuations, the disks form a hexagonal solid at ϕs ≈ 0.9. To measure transport we calculate the average drift velocity 〈Vx〉 = (1/N)∑Ni = 1vi· ∧x. In the absence of a drift force, 〈Vx 〉 = 0, and in the absence of any obstacles, all the particles move with 〈Vx〉 = Fd.
Fig1.png
Figure 1: (a,b) Snapshots of disk positions in an obstacle-free system with ϕ = 0.667 and Np=0. (a) Uniform liquid at Rl = 0.4. (b) Phase separated state at Rl = 100 with a large cluster (red) of local density ϕ ≈ 0.9 surrounded by a low density gas of particles. (c) Cl/N, the fraction of particles in the largest cluster, vs ϕ in the Np=0 system for Rl = 40, 20, 1.4, 0.2, and 0.04, from left to right. The onset of clustering occurs at higher ϕ for decreasing Rl. (d,e,f) Positions of pinned disks (light blue circles), active disks (dark red circles), and disk trajectories over a period of time (lines) in small sections of samples with Np=300 and ϕp=0.0235. (d) ϕ = 0.667 and Rl=20. (e) ϕ = 0.188 and Rl=0.004. (f) ϕ = 0.188 and Rl=0.8.

III.  RESULTS

We first characterize the onset of clustering in the absence of quenched disorder. To quantify the amount of clustering present, we utilize a cluster-identification algorithm described in Ref. [42]. In Fig. 1(a,b) we show the onset of cluster formation at a fixed density of ϕ = 0.667 when Rl is increased. At Rl = 0.4 in Fig. 1(a), there is a uniform liquid phase, while at Rl=100 in Fig. 1(b) the system phase separates into a crystalline hexagonally ordered solid phase with a local density of ϕ = 0.9 surrounded by a low density gas phase. This behavior is identical to that found in simulations of active Brownian particles at a fixed density when the persistence length is increased [18]. Cluster formation also occurs when ϕ is increased for fixed Rl. In Fig. 1(c) we plot Cl/N, the average fraction of particles in the largest cluster, versus ϕ for systems with Rl = 0.04 to 40. At the smallest run length of Rl = 0.04, clustering does not occur until the density is well above ϕ = 0.8, while for large Rl the onset of clustering shifts to much lower values of ϕ.
Fig2.png
Figure 2: (a) 〈Vx〉, the average velocity per particle in the drift direction, vs Rl for a system with ϕ = 0.667 and obstacle density ϕp = 0.00039, 0.00157, 0.00472, 0.0785, 0.0157, 0.02356, 0.055, 0.0942, 0.01413, and 0.188, from top to bottom. When Rl > 5.0, 〈Vx〉 decreases with increasing Rl. (b) 〈Vx〉 vs Rl passes through a maximum for all the curves, as shown more clearly in this blow-up of panel (a) for ϕp = 0.055, 0.094, 0.1413, and 0.188, from top to bottom.
We next consider the effect of adding a drift force and Np obstacles in the form of immobile disks, as illustrated in Fig. 1(d). In Fig. 2(a) we plot a series of transport curves showing 〈Vx〉 versus Rl at fixed ϕ = 0.667 and ϕp ranging from 0.00039 to 0.188. For fixed ϕ, Fig. 2(a) shows that increasing ϕp monotonically decreases the transport. For any given ϕp, the transport initially increases with increasing Rl, reaches an optimal value, and then decreases at larger Rl. In Fig. 2(b) we highlight the transport curves for ϕp = 0.055, 0.094, 0.1413, and 0.188, where the maximum in 〈Vx〉 occurs near Rl = 1.0. The decrease in 〈Vx〉 at both large and small Rl is correlated with a heterogeneous distribution of particles in the system, due to clogging behind obstacles for small Rl and due to living crystal formation for large Rl. At ϕp=0.188 in Fig. 2(a), the system completely jams at higher Rl with 〈Vx〉 ≈ 0.
In driven systems with quenched disorder, such as superconducting vortices driven over random landscapes [34], increasing the temperature T generally monotonically increases the transport up to a saturation value 〈Vx〉s. In our RT system, increasing Rl for small Rl is is similar to increasing the temperature; however, once Rl is large enough for clustering to occur, the particle density becomes inhomogeneous and 〈Vx〉 drops. We have examined a variety of values of ϕ and ϕp and find that in general, for ϕ > 0.4, when quenched disorder is present the drift velocity is maximized at an optimal value of Rl.
Fig3.png
Figure 3: (a) Cl/N (dark curve) and 〈Vx〉 (light curve) vs Rl for a system with ϕ = 0.667 and ϕp = 0.094. The drop in 〈Vx〉 is correlated with the formation of cluster states. (b) The same for a system with ϕ = 0.337 and ϕp = 0.02356.
Fig4.png
Figure 4: (a) A log-log plot of 〈Vx〉 vs Rl for systems with ϕ = 0.667 at ϕp = 0.00039, 0.00157, 0.00472, 0.0785, 0.0157, 0.02356, 0.055, 0.0942, 0.01413, and 0.188, from top to bottom. The dashed line is a fit to a power law with Vx(Rl) ∝ R−αl, with α = 1.31 ±0.02. (b) 〈Vx〉 vs ϕp at a fixed ϕ = 0.667 for Rl = 1.0, 4.0, 20.0, and 80.0, from top to bottom. At small Rl, 〈Vx〉 decreases linearly with increasing obstacle density, while for large Rl the decrease is much more rapid and approximately follows 〈Vx〉 ∝ 1/ϕp.
To show that the onset of clustering correlates with the decrease in transport, in Fig. 3(a) we plot 〈Vx〉 and Cl/N versus Rl for a system with ϕ = 0.667 and ϕp = 0.094. The decrease in 〈Vx〉 coincides with an increase in Cl/N, indicating cluster formation. When Rl is small, all of the mobile particles are in the gas phase, and if a particle encounters an obstacle, it has a high probability of quickly changing its swimming direction and moving away from the obstacle. As Rl increases and the system enters the cluster phase, even a single obstacle can pin a cluster since the cluster has transient rigidity. As a result, the moving clusters can become clogged by the obstacles, reducing the transport. Fig. 1(d) illustrates the trapping of a high density cluster by obstacles, while the particles in lower density regions remain more mobile. The particles that belong to a cluster become unpinned once the cluster breaks apart. The lifetime of individual clusters increases with increasing Rl and the longer-lived clusters remain clogged for longer periods of time, resulting in a gradual reduction in 〈Vx〉 with increasing Rl as shown in Fig. 2. For ϕp > 0.0785, in the cluster regime where 〈Vx〉 decreases with increasing Rl we find Vx(Rl) ∝ R−αl, with α = 1.31 ±0.02, as illustrated in Fig. 4(a). If we fix Rl and vary ϕp, Fig. 4(b) shows that for small Rl, 〈Vx〉 decreases linearly with increasing ϕp, while for large Rl the decrease is much more rapid and approximately follows 〈Vx〉 ∝ 1/ϕp. In Fig. 3(b) we plot Cl/N and 〈Vx〉 vs Rl for a system with ϕ = 0.337 and ϕp = 0.02356. The transport is almost constant at the values of Rl where clustering is absent; however, at the onset of clustering where Cl/N increases, 〈Vx〉 begins to drop rapidly.
At low Rl, the drift force can cause individual particles to become trapped behind the obstacles and form locally jammed states even when ϕ is well below the jamming density in a clean system, as illustrated in Fig. 1(e) at ϕ = 0.188, ϕp = 0.0235 and Rl = 0.004 where there is one freely flowing channel but a considerable portion of the particles are trapped. Due to the finite Rl, trapped particles can occasionally escape, as shown in the upper part of Fig. 1(e). As Rl increases, particles can more easily escape from behind the obstacles by moving in directions opposite and transverse to the drift force, and the mobility of the system is enhanced, as illustrated in Fig. 1(f) at Rl = 0.8. Thus, increasing the activity by increasing Rl can increase the mobility by reducing the trapping of individual particles; however, at larger Rl, the particles begin clustering into living crystal structures which become locally clogged, causing the mobility to fall at higher Rl.
Fig5.png
Figure 5: (a) 〈Vx〉 vs Rl for systems with ϕp = 0.0942 at ϕ = 0.667, 0.746, 0.785, 0.8246, and 0.864 (from top to bottom) showing the decrease in the transport for increasing ϕ. (b) Cl/N (circles) and 〈Vx〉 (squares) vs ϕ for samples with ϕp = 0.0235 at Rl = 0.04 (open symbols) and Rl = 20 (filled symbols) showing the correlation between the onset of clustering and the drop in the transport.
We next show how the transport is affected by jamming at higher particle density. The jamming concept was originally applied to non-active particles in the zero-temperature limit, where it was shown that as a function of increasing density the system becomes jammed at ϕj, termed point J [38]. In 2D assemblies of bidisperse disks, point J falls at ϕj = 0.844 [39], while for monodisperse disk assemblies the system forms a hexagonal solid at ϕ = 0.9. Recently it was shown that the addition of quenched disorder to a jamming system decreases the jamming density, where jamming was defined to occur when the drift velocity 〈Vx〉 = 0 under an external applied drive [40]. In the RT system for the non-active limit of Rl = 0, we can obtain a jammed state with 〈Vx〉 = 0 by increasing ϕ for fixed ϕp or increasing ϕp for fixed ϕ. In Fig. 5(a) we plot 〈Vx〉 vs Rl for samples with ϕp = 0.0942 at ϕ = 0.667, 0.746, 0.785, 0.8246, and 0.864. Each curve has an a optimal Rl for transport, and the magnitude of 〈Vx〉 decreases with increasing ϕ. For ϕ > 0.746, 〈Vx〉 = 0 at Rl = 0, corresponding to the jammed phase. For nonzero Rl, the decrease in 〈Vx〉 with increasing ϕ is due to the formation of clusters which increase in size with increasing ϕ. For moderate to large values of ϕp, we find that the system reaches the 〈Vx〉 = 0 jammed state if infinite run times are used, so that complete jamming can occur at the limits Rl=0 and Rl=∞. In Fig. 5(b) we plot Cl/N and 〈Vx〉 versus ϕ for samples with ϕp=0.0235 at Rl = 0.04 and Rl = 20. For Rl = 0.04, 〈Vx〉 is constant over most of the range of ϕ but rapidly decreases to a value near zero at ϕ = 0.9, where we also find a rapid increase in Cl/N. For Rl = 20, the onset of clustering occurs at a lower value of ϕ and is also correlated with a drop in 〈Vx〉. Our results suggest the possibility of adding a new axis to the jamming phase diagram [38] corresponding to the activity Rl, where for large activity there could be another critical jamming point Aj similar to point J.

IV.  DISCUSSION

We have shown that an RT active matter model with quenched disorder has an optimal activity level at which transport under a drift force is maximized. In Ref. [30] it was demonstrated that a flocking model with quenched disorder and without steric particle-particle interactions has an optimum noise level at which collective or flocking motion is maximized. The flocking model falls in a very different class of behavior from the RT model, and it is interesting to compare the two results. We measure a flux through our system while in Ref. [30] the tendency for particles to swim in the same direction is measured. These are two very distinct quantities that cannot be mapped onto each other. The decrease in the transport in our system at larger Rl is due to local jamming effects caused by steric interactions between the particles. These effects are absent in the flocking model, which instead has a transition from coordinated to uncoordinated swimming as a function of noise. Thus, the optimal activity level and optimum noise level observations are distinct and are produced by quite different physical effects.
We find that increasing the run length or particle density can substantially decrease the mobility of the RT particles through a substrate due to cluster formation and local clogging. In order to avoid such clogging effects, new types of motion rules could be devised for the RT particles, such as having the particles reverse or randomize their swimming direction if they become immobilized by an obstacle or trapped in a cluster. It would be interesting to explore whether biologically relevant active matter such as swimming organisms have evolved methods to avoid self-clogging.

V.  SUMMARY

We have examined the transport of run-and-tumble active matter particles driven through quenched disordered environments. Unlike thermalized particles driven over random disorder, where previous studies have shown that the transport is generally increased when the thermal fluctuations increase, we find that for active matter the transport first increases and then decreases as the activity level or running length increases due to the formation of living crystals that can be locally jammed or clogged. The system becomes effectively jammed at lower densities as the run length increases, suggesting that activity could form a new axis of the jamming phase diagram. Since recent theoretical work has shown that run-and-tumble dynamics and active Brownian motion can be mapped to each other [38], our results can be generalized to other active systems driven over quenched disorder.
We thank L. Lopatina for useful discussions. This work was carried out under the auspices of the NNSA of the U.S. DoE at LANL under Contract No. DE-AC52-06NA25396.

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