Proc. SPIE 8458, Optical Trapping and Optical Micromanipulation IX, 84581I (2012)

Dynamics of Self-Driven and Flocking Particles on Periodic Arrays

J. Drocco, L.M. Lopatina, C. Reichhardt, and C.J. Olson Reichhardt
Center for Nonlinear Studies and Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico, USA 87545

ABSTRACT
Recently there has been growing interest in what is called active matter, or collections of particles that are self driven rather than driven with an external field. Examples of such systems include swimming bacteria, flocks of birds or fish, and pedestrian flow. There have also been recent experimental realizations of self-driven systems using colloidal particles undergoing self-catalytic interactions. One example of this is light-induced catalysis where the colloids become self-driven in the presence of light. Almost all of these studies have been performed in the absence of a substrate. Here we examine how a substrate can be used to direct the motion of the particles. We demonstrate a self-induced ratchet effect that occurs in the presence of disorder as well as the direction of the particle along symmetry directions of the substrate. The type of substrate we consider may be created using various optical techniques, and studies of this system could lead to insights into the nonequilibrium behavior of active matter as well as to applications such as sorting of different active particle species or of active and non-active particles.

Keywords: Active matter, ratchet effect, optical traps

1. INTRODUCTION
2. COMPUTATIONAL MODEL
3. FLOCKING MODELS ON ASYMMETRIC SUBSTRATES
4. ACTIVE MATTER ON 2D PERIODIC POTENTIALS
5. SUMMARY
REFERENCES

1.  INTRODUCTION

Assemblies of particles that exhibit self-motility have been termed active matter, and appear in a range of systems including swimming bacteria, motile cells, flocks of birds and fish, pedestrians, and traffic systems [1]. Artificial active matter has also been created in the form of swimmers [2] and self-driven colloids [3,4]. These systems have generally been divided into two classes. The first class contains individual swimmers where particles obey some rule of motion such as the run and tumble dynamics found for bacteria [5]. In this class, the particle-particle interactions are small or negligible. The second class contains the flocking models where the particle dynamics are governed by rules associated with the motion of the surrounding particles [1]. One of the earliest proposed flocking models is the Vicsek model where particles choose a direction of motion based on the average velocity of the surrounding particles within some interaction radius [6]. There are also individual swimmer models using only simple repulsive interactions that produce flocks or clumps when the particle density is high enough [7]. In most models that have been studied, the effects of an underlying substrate consisting of a periodic potential or random quenched disorder have not been considered.
For non-active matter, it is known that a periodic or random substrate can strongly affect the ground state orderings of the system. For example, for particles interacting with a periodic substrate, there can be commensuration effects when the number of particles is some integer or rational fraction of the number of substrate minima [8,9]. These commensuration effects can strongly enhance the melting transition in these systems and can even change the universality class of the transition [10]. Under an applied drive, particles moving over periodic or random substrates can exhibit a rich variety of dynamic phases, including dynamic reordering transitions [11,12,13,14,15], directional locking [16,17,18,19,20,21], phase locking [22], friction effects [23], and pattern formation [24]. If the substrate contains some asymmetry, then an externally applied ac drive can induce a ratchet effect where a net dc motion of the particles occurs [25,26,27,28]. The direction of the resulting dc output can often show reversals as the number of particles per substrate minima is varied [28,27]. For active matter, one could expect a similar richness in the dynamics; however, there are important differences between active and non-active systems. Typically in active matter there is a net direction to the motion of the particles. For example, particles obeying run and tumble dynamics will move away from their starting point over time in a random direction. For flocking models such as the Vicsek model, it is possible for symmetry breaking to occur where all the particles move in a single direction; however, this direction is not fixed and changes from one simulation realization to another [6].
Fig1.png
Figure 1: Illustration of the system, consisting of hard walls with a periodic array of asymmetric funnels. Blue dots are individual swimmers that obey run and tumble dynamics with alignment rules for interactions with the walls and funnels. (a) The initial configuration has equal numbers of particles on each side of the funnel array. (b) The configuration at later time shows a build-up of particles in the upper part of the sample.
Active matter experiments that have been conducted with ordered substrates include swimming bacteria in a periodic sawtooth array [29]. If non-active matter particles are placed into such a system, and undergo only thermal motion, no net current or flux of particles moves through the array; however, when active or swimming particles are placed in the same geometry, a net flux of particles occurs in the easy funnel direction as shown in Fig. 1. This directed motion has been shown to result from two factors: the fact that the particles move in a single direction over a fixed length due to the motor force, and the fact that the particles align with and run along the walls, breaking detailed balance in the wall interactions [30,31]. When other wall interactions, such as reflection or scattering, are used instead, no directed motion of the particles occurs. Based on the initial studies, it was later shown that the same basic mechanism can be used to create gears turned by bacteria, and that an asymmetric sawtooth gear placed in a bath of bacteria undergoes a net rotation in one preferred direction [32,33,34,35]. Studies of individual swimmers have only focused on relatively low particle densities and steric particle-particle repulsion, where the directed motion only occurs in the easy flow direction as illustrated in Fig. 1. Studies of eukaryotic cells moving over periodic asymmetric arrays have shown that depending on the type of cell, some species move in the easy flow direction while others move against the easy flow direction [36]. Due to the complexity of the locomotive mechanism for eukaryotic cells, it is difficult to determine in such systems how these flow reversals can occur; however, it is clear that active matter can exhibit ratchet reversals. Recent flocking simulations using a modified Vicsek model showed that the ratchet effect in the easy flow direction can be strongly enhanced when the particles flock, and that when the flocks become increasingly incompressible, the direction of motion through the asymmetric funnels can be reversed [37].
For self-driven colloid systems, it has been shown that if the particles move over a triangular substrate, the motion is not simply random but occurs preferentially along certain directions determined by the symmetry of the underlying substrate [38]. This effect is similar to directional locking effects found for non-active matter under an external drive in the presence of a periodic substrate [16,17,18,19,20,21]. In directional locking, the particle motion locks to a symmetry direction of the underlying substrate such as at 60° angles for triangular substrates and 90° angles for square substrates. In addition to the main symmetry directions, certain fractional driving angles also produce similar locking effects such as 45° angles for square arrays and 30° angles for triangular arrays [16,17]. The particle motion locks to a symmetry direction of the substrate even when the external drive is not aligned exactly with that direction. For active matter in the absence of a substrate, a particle moves in some fixed direction for a certain time until a tumbling event occurs. If a substrate is present, the particle locks to a symmetry direction of the substrate that is close to the angle along which the particle was originally moving.
In this work we present results for both individual swimmers and flocking particles moving over asymmetric funnel arrays and over square periodic substrates. For the flocking particles we find both forward and reverse ratchet effects. For the square substrate array, we find guided motion of the particles along the symmetry direction of the substrate; however, we observe almost no locking effects for the higher order symmetry directions, unlike the case of non-active particles driven over a periodic substrate. We also show that in general for directional locking, a muffin tin type potential produces more higher order locking effects than the egg carton type potential we examine in this work. We demonstrate that the substrate affects the diffusion of the particles and produces multiple diffusive regimes, and that as the running length of the particles is decreased, the motion becomes subdiffusive.

2.  COMPUTATIONAL MODEL

We consider several two-dimensional (2D) model systems of individual swimmers or flocking particles. In the individual swimmer model, the motion of a single particle i is obtained by integrating the following overdamped equation of motion:
η dRi
dt
 = Fmi(t) + FiB + Fppi+ FTi.
(1)
Here η = 1 is a phenomenological damping term. The first term Fmi(t) represents the motor force propelling the particle. The force remains oriented in a randomly chosen direction for a fixed time duration τl, and after this time the force is applied in a new randomly chosen direction for another interval τl. Since the dynamics is overdamped, the particle moves at a constant velocity determined by the magnitude of the motor force Fm, so that during the time period τl the particle moves a distance l, termed the run length. The resulting motion obeys run and tumble dynamics of the type observed in real bacteria. For infinitesimal τl, the motor force resembles a thermal force. The second term FiB corresponds to interactions with barriers or walls. We consider a simple rule in which the motion of the particle aligns with the wall so that the particle moves along the wall until, after the run time τl has elapsed, the particle has a chance to move away from the wall depending on the new randomly chosen direction of motion. The third term Fppi is the particle-particle interaction which we take to be simple steric repulsion, and the last term FiT is the thermal force which is generally weak on the size scale of bacteria. The thermal force is modeled as Langevin kicks employed during the simulated annealing that have the properties 〈FTi〉 = 0 and 〈FTi(t)FTj(t′)〉 = 2ηkBTδijδ(t − t′) where kB is the Boltzmann constant.
We also consider swimmers moving over a simple periodic substrate with periodic boundary conditions in the x and y-directions. In this case the barrier force term is
FBi = Asin(2πx/10)
^
x
 
 + Asin(2πy/10)
^
y
 
 .
(2)
This produces a square egg carton potential with a maximum force amplitude of A.
For the flocking model we consider a variant on the Vicsek model using the overdamped equation of motion
vi = FiVC(t) + Fir(t) + FiB(t) .
(3)
Here the consensus force FiVC(t) is determined by the velocities of the surrounding particles within a fixed radius rf. This model contains a noise term ξ, and it has been shown previously that when the noise is large enough, there can be a transition to a disordered state. The density of the system is ρ = N/L2, where N is the number of particles and L is the system size, taken as L = 66. The particle-particle interactions are modeled as stiff springs of the form Fir = A(2re − rij)Θ(2re − rij)∧rij. Here Θ is the Heaviside step function, re is the particle exclusion radius, and we set Ar = 200. We use a similar form for the barrier interactions. Unlike the individual swimmers, in the flocking model there is no rule for aligning the motion of the particle with a barrier; instead, the particle simply experiences a repulsion from the barrier.
Fig2.png
Figure 2: The particle positions (dots) for the same geometry in Fig. 1 but for a flocking model. Here the particle exclusion radius is small, re = 0.07. (a) An early time configuration with equal numbers of particles on both sides of the funnel barrier. (b) At a later time, particles start to accumulate in the upper chamber. Particles enter the upper chamber in the form of flocks that squeeze through the funnel tips. (c) In steady state, most of the particles are in the upper chamber. (d) The density ρ of particles in the upper (top blue curve) and lower (bottom red curve) chambers vs time. The letters indicate the times illustrated in panels a, b, and c.

3.  FLOCKING MODELS ON ASYMMETRIC SUBSTRATES

In Fig. 1 we illustrate the geometry of the system with asymmetric barriers. In our previous work we showed that individual swimmers interacting with the geometry in Fig. 1 become concentrated in the upper part of the system provided that the particle motion is aligned or at least partially aligned by the walls. Figure 2 shows the same system but for flocking particles. The particles are initialized in random positions with a uniform density throughout the sample, as shown in Fig. 2(a). As time progresses, Fig. 2(b) indicates that the particles begin to concentrate in the upper chamber as larger flocks form and move as a unit through the funnel barriers, forming tendril-like shapes. At long times when the system has reached steady state, the flocks are primarily found in the upper chamber as shown in Fig. 2(c). We note that the concentration of particles in the upper chamber occurs much more quickly for the flocking particles than it did for non-flocking particles in the same geometry. Figure 2(d) shows the particle density in the upper and lower chambers as a function of time for the flocking system. In steady state, the upper chamber contains nearly 10 times more particles than the lower chamber, whereas for individual swimmers the upper chamber typically contained only 4 times more particles than the lower chamber [30]. The letters in Fig. 2(d) denote the times at which the images in Fig. 2(a,b,c) were taken. We also find a clear jump in ρ as the system approaches the steady state. This jump occurs since the particles generally do not pass individually through the funnel barrier but instead cross into the upper chamber as flocks.
Fig3.png
Figure 3: The particle positions (dots) for the flocking system in Fig. 2 with a larger exclusion radius re = 0.23. (a) An early time configuration with equal numbers of particles on both sides of the funnel barrier. Here the flocks are larger than those in Fig. 2(a) due to the larger re. (b) At a later time, flocks in the lower chamber form crystalline like states at the tips of the funnels that block the flow though the funnel. (c) In steady state there is a buildup of particles in the lower chamber. (d) The density ρ of particles in the upper (bottom blue curve) and lower (upper red curve) chambers vs time. The letters indicate the times at which the images from panels (a,b,c) were taken. Here the change in density is much slower than for the forward ratchet effect in Fig. 2(d).
We observe a forward rectification, or concentration of the particles in the upper chamber, in the limit of small exclusion radius when the flocks are very compressible. In this limit, entire flocks can squeeze through the funnel tips as illustrated in Fig. 2(b). If the exclusion radius is larger than the funnel tip aperture, particles cannot cross from one side of the funnel barrier to the other; however, when the exclusion radius is smaller than the funnel tip aperture but still large, we find a reverse ratchet effect in which the particles accumulate in the lower part of the chamber so that the net particle flux is against the easy flow direction of the funnels. In Fig. 3(a) we illustrate an early time configuration with equal numbers of particles on either side of the funnel barrier for a system with a larger exclusion radius of re=0.23. Figure 3(b) shows that when a large flock approaches the funnels from below, the particles crystallize in the shape of the funnel due to the large steric repulsion of the particles. The sawtooth shape imprinted on the flock remains stable for some time before breaking apart. The particles jam in the funnel tip and are prevented from crossing into the upper chamber due to the incompressibility of the flock. In contrast, when a flock approaches the funnels from above, it can be split into two flocks by the tip of the funnel, and one or more particles may escape the flocks and cross into the lower chamber. After a long time, particles slowly accumulate in the lower chamber due to this process as shown in Fig. 3(c). The reverse rectification proceeds much more slowly than the forward rectification found for the compressible flocks, as demonstrated in Fig. 3(d) where we plot the particle density in each chamber as a function of time. We find a nearly linear change in particle density with time. The letters in Fig. 3(d) denote the times at which the images in Fig. 3(a,b,c) were taken.
Fig4.png
Figure 4: Model of self-driven ring particles to mimic cell motion in a funnel geometry. Left: Early time configuration. Right: At later time, clogging begins to occur leading to a drop in the flux of particles.
It is also possible to consider active matter that mimics moving cells. In Fig. 4 we show an example of a system of particles confined into flexible rings. A randomly chosen bead in each ring undergoes run and tumble dynamics of the type used in the individual swimmer model, and this bead drags the remainder of the particles in the ring along with it. We place the self-propelled rings in a hopper geometry and find that a clogging effect can arise as the rings accumulate near the tip of the hopper. In future we will investigate different regimes of the motion of the rings to explore whether a reversal can occur due to ring shape or different types of ring motion.
Fig5.png
Figure 5: Individual swimmers obeying run and tumble dynamics with motor force Fm=2.0 in a system with a square egg carton substrate potential with amplitude A=3.0. (a) Sequential stroboscopic particle positions for a run length of l=1.5 where the particles are mostly confined to the substrate minima. (b) Sequential stroboscopic particle positions for a run length of l=5.0 showing more motion between neighboring substrate minima. (c) Sequential stroboscopic particle positions for a run length of l=15.0 showing increased motion across the system. The dominant motion occurs along the symmetry directions of the substrate. (d) The mean square particle displacement 〈r2〉 vs time. Red curve: l=1.5; green curve: l=5.0; blue curve: l=15.0. For short run lengths the system shows caging effects at longer times.

4.  ACTIVE MATTER ON 2D PERIODIC POTENTIALS

We next consider the case of nonflocking swimmers with repulsive interactions moving over a periodic egg carton potential. We simulate a box of side length L = 99. After placing the particles randomly on the substrate, we measure their mean square displacement 〈r2〉. We fix the substrate strength to A = 3.0 and the swimming force to Fm=2.0, and consider the effect of different run lengths l. Figure 5(a) shows sequential stroboscopic particle positions for a run length of l=1.5. In this case the particles are almost completely confined within the potential minima with a small number of hopping events. Individual particles cannot be seen since they are moving rapidly inside the traps producing a large density of stroboscopic points. In Fig. 5(d) we plot 〈r2〉 for this system as a red curve. At short times, 〈r2〉 increases rapidly since the particles can move freely inside the traps; however, at longer times, the particles remain mostly confined in the traps, producing a plateau in 〈r2〉. There is a slight rise in 〈r2〉 at longer times due to occasional hopping of particles out of the wells. We note that for noninteracting particles, there is no motion out of the wells since the maximum attractive force of the substrate minima is greater than the motor force. Hopping from well to well can only occur when groups of particles push against one another. For run lengths of l=5.0, Fig. 5(b) shows that there is more hopping between substrate minima, while the green curve in Fig. 5(d) indicates a similar rapid increase 〈r2〉 at short times followed by a plateau at longer times. At still longer times, however, 〈r2〉 begins to increase again. This behavior is similar to a subdiffusive caging effect where at long times the particle motion is diffusive but at short times it is subdiffusive. For a run length of l=15, Fig. 5(c) shows that the particle motions are more rapid, while the blue curve in Fig. 5(d) shows that 〈r2〉 has a change in slope at long times but no longer has a plateau feature. Additionally, Fig. 5(c) indicates that the particle motion between minima is confined to move in one-dimensional paths along the symmetry directions of the substrate. This is similar to the guidance effects observed for self-driven colloids moving over triangular substrate arrays. We do not find additional directionally locked motion at other symmetry angles such as 45°, which are observed for driven non-active particles moving over periodic substrates. This may be due to the nature of the substrate. Most studies of directional locking systems consider a muffin tin type of potential or an array of obstacles, either of which permit certain trajectories to flow along the higher order symmetry directions. The egg carton potential we consider here tends to suppress such motion. We will investigate this effect further in the future.

5  SUMMARY

We have shown that active matter on ordered and asymmetric substrates exhibits a rich variety of behaviors. Non-active matter particles have previously been shown to exhibit a ratchet effect on asymmetric substrates when there is some form of external forcing such as an ac drive or a flashing substrate. We find that the active matter particles can exhibit a self-induced ratchet effect without any external drive. For individual swimmers the rectification occurs due to the non-thermal motion of the particle and the breaking of detailed balance in the particle-wall interactions. For individual swimmers we have not observed any ratchet reversals; however, when we consider a flocking model, we observe both an enhanced forward ratchet effect as well as a reversed ratchet effect. We have studied the effects of more complex geometries for individual swimmers and find a jamming type effect in a hopper geometry for a moving loop model of swimming cells. For repulsively interacting swimmers moving on a two-dimensional square periodic substrate, we observe directional locking effects where the particle motions are confined to certain symmetry directions of the substrate. This is similar to the directional locking observed in experiments for self-driven colloids on triangular substrates. We also find that as the running time of the particles is varied, different diffusive regimes appear including a short time superdiffusive regime and a long time subdiffusive regime.
Up until this point, experiments on active matter systems have been performed with non-optical substrates; however, from recent studies of colloidal particles we expect that a variety of substrates could be created using optical techniques. An advantage of optical traps is that new types of dynamic substrates could be considered. For example, if an asymmetric substrate is flashed on and off, it may be possible to realize a flashing ratchet for active matter. Other possibilities include using optical means to drag a single trap through a bath of active particles to test nonequilibrium fluctuation theorems and to study how drag effects change with different types of swimming or flocking models. There are also many types of magnetic bacteria so it should be possible to create a magnetic substrate with arrays of nanomagnets.
This work was carried out under the auspices of the NNSA of the U.S. Department of Energy at Los Alamos National Laboratory under Contract No. DE-AC52-06NA25396.

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