Active Phases for Particles on Resource Landscapes

We introduce an active matter model composed of sterically interacting particles which absorb resources from a substrate and move in response to resource gradients. For varied ratios of absorption rate to substrate recovery rate, we find a variety of phases including periodic waves, partial clustering, stochastic motion, and a frozen state. If passive particles are added, they can form crystalline clusters in an active fluid. This model could be implemented using colloidal systems on feedback landscapes and can provide a soft matter realization of excitable media and ecological systems.

Active matter exhibits self-mobility [1][2][3], which can appear in biological [4,5], social [6,7], robotic [8,9], and soft matter systems [10,11].For active matter composed of particles, the motility can be modeled as a motor force providing run-and-tumble or driven diffusive propulsion [1][2][3], and additional dynamics can be included which induce different types of flocking behaviors [12,13].Active particle assemblies exhibit a variety of phenomena that are absent in Brownian systems, such as the motilityinduced phase separation or clustering that can arise even when all the pairwise particle-particle interactions are repulsive [2,10,11,14,15].In certain biological active matter systems, the motion can be affected by local or global gradients in the environment, producing effects such as chemotaxis or drift of particles in certain directions [4,16].There are also numerous examples of active particles that are coupled to complex environments [2], leading to phenomena such as active matter ratchets [17,18], trapping or substrate-induced clustering [2,[19][20][21], or topotaxis, where spatial gradients in the landscape generate directed motion [22,23].
It is also possible for activity to arise not from a motor force carried by each particle, but instead from forces induced by a substrate.Such situations can arise in time dependent environments [7,24], particles coupled to excitable media [25], and colloids on feedback substrates [26].Recently, Wang et al. [9] introduced an ecologyinspired active matter system of robots interacting with a resource substrate where the robots consume the resources and are attracted to regions with the highest resource concentration.This system exhibited numerous phases such as crystalline, liquid, glass, and jammed states.
Here we propose a new type of soft matter active system for an assembly of sterically repulsive particles that couple to a resource landscape.The particle motion is governed by local gradients of the resource concentration.When a single particle sits over a group of resource sites, it experiences a net force directed towards the sites containing the highest resource levels.Collisions between adjacent particles cause localized pinning-depinning transitions that make the behavior distinct from that found in a reaction-diffusion system.Resources are depleted from sites occupied by the particle at a fixed rate, while all sites in the system recover resource levels at a different fixed rate up to a maximum resource value.
We find that for high absorption rates and low recovery rates, particle motion occurs in periodic bursts of activity via the propagation of waves through the system, similar to the reaction diffusion patterns found in excitable media [25,27,28].As the recovery rate is increased, the pulse frequency increases until a transition occurs to a continuously fluctuating or chaotic fluid state.When the recovery rate is larger than the absorption rate, the particles become frozen into place since a resource gradient never develops.For some rate combinations, the particles form a partially clustered phase due to collisions of oppositely propagating waves.When we add a second species of passive particles that do not couple to the substrate but only interact sterically with the active particles, we find a crystallization of the passive particles within the fluctuating fluid phase similar to the motility induced phase separation observed for active particles with motor forces.
Our system could be realized using colloids interacting with an optical substrate in the presence of feedback mechanisms [26,29] or colloids coupled to an excitable medium where the presence of a colloid locally alters the diffusion rate in the medium.Our results indicate that soft active matter could be used to explore excitable media and ecologically inspired systems.
Simulation and System-We consider a twodimensional system of size L x × L y with L x = 100 and L y = 200 containing N p = 3500 particles of diameter d = 1.0 with steric repulsive interactions.The particle area coverage is φ = πd 2 /(4L x L y ) = 0.549, well below the jamming density of φ J = 0.9.The particles also couple to a substrate consisting of a fine mesh of N g = 80000 grid sites, each of size l g × l g with l g = 0.5.A given grid site interacts with at most one particle at a time.Defining g i c (t) to be the grid site closest to the center of particle i at time t, the "occluded sites" which exert an attractive force on the particle are defined to be the eight nearest neighbors of g i c (t) (including diagonal nearest neighbors) and the four linear next-nearest neighbors of g i c (t).The overdamped equation of motion of particle i is given by where v i = dr i /dt is the velocity of particle i and η is the damping constant which is set to unity.The particleparticle interaction force If all sites occluded by a particle have the same value of S g , the attractive forces cancel and the particle does not move.The evolution of the resource level at grid site l is S l g (t + ∆t) = S l g (t) − O l g r abs + r rec , where r abs is the rate at which a particle absorbs resources from the grid site and O l g = 1 (0) if the grid site is occluded (unoccupied).All grid sites recover resources at rate r rec , but the value of S l g is not allowed to exceed S l g = 1.For initialization, S l g (0) for each site is set randomly to values between 0 and 1, while the particles are placed at randomly chosen locations with particle overlap forbidden.To characterize the behavior, we measure the fraction of particles that are moving at each instant, M = N −1 p Np i Θ(|F tot i | − 0.01), its time average M , and its standard deviation σ M .We also measure C l , the fraction of particles belonging to the largest cluster in the system, where particles that are in contact with each other are defined as belong to a given cluster.
Results-In Fig. 1 we show snapshots of a portion of the system in different dynamic phases.At high absorption rates and low recovery rates, illustrated in Fig. 1 at r abs = 8.5 × 10 −3 and r rec = 5 × 10 −5 , long intervals of no motion are interspersed with bursts of motion in the form of a front or wave which traverses the system and leaves behind a region of depleted resource sites.After each wave dissipates, the system returns to a nonmoving state until the grid sites have recovered enough resources to reactivate the particles.If we increase r rec while holding r abs fixed, the time between pulses of motion is reduced, the propagating fronts become wider, and we find a partial clustering effect as shown in Fig. 1(b) at r abs = 8.5 × 10 −3 and r rec = 5 × 10 −4 .As the recovery rate is further increased, the pulse frequency increases until a transition occurs to a state in which most particles are moving most of the time, creating a fluctuating fluid as illustrated in Fig. 1(c) for r abs = 2.5 × 10 −3 and r rec = 7 × 10 −4 .For high recovery and low absorption rates, the system becomes trapped in a frozen phase, as shown in Fig. 1(d r rec = 8 × 10 −4 .The overall behavior can be described in ecological terminology by allowing the resources to represent food.Oscillations occur when there is not enough food to support continuous motion; instead, a food accumulation period is required to induce movement.Once the food has recovered, at least one particle moves and collides with another particle, triggering the wave of motion.After the wave has propagated through the system, the food is depleted again and motion does not resume until the food has been replenished.In the fluctuating state of Fig. 1(c), when food is depleted at a local grid site, the particle can immediately find sufficient food at neighboring grid sites, permitting the particle to maintain nearly constant motion.When the food is extremely plentiful, a particle always finds more than enough food at its current position, so it does not need to move.
In Fig. 2(a) we plot the fraction of moving particles M versus time for a system with propagating waves at r rec = 2.5 × 10 −5 and r abs = 0.01, where long intervals of no activity are interspersed with sharp bursts of motion.For the same system with a lower r abs = 0.001 in Fig. 2(b), M still has windows of close to zero activity, but the interval between activity bursts is almost 3.5 times shorter.In Fig. 2(c), where r rec = 3 × 10 −4 and r abs = 0.006, we find a more rapid oscillating behavior associated with the propagation of multiple fronts though the system with slightly different periods, producing beat-like patterns in M .For r rec = 8 × 10 −4 and r abs = 0.004 in Fig. 2(d), the motion is continuous and chaotic, similar to what is shown in Fig. 1(c).In the frozen phase, such as that illustrated in Fig. 1(d), we find M = 0 apart from a brief initial transient.
In Fig. 2(e-h) we plot the time averaged fraction of moving particles M , its standard deviation σ M , and the fraction of time P 0 during which M = 0 versus r rec at r abs = 0.01, 0.001, 0.006, and 0.004, respectively.We use these measures to delineate the dynamic phases of the system.In phase I, intermittent or periodic burst behavior of the type shown in Figs.2(a,b) appears.Here, a single moving front of particles activates adjacent particles by pushing them forward, creating a wave-like excitation.The motion depletes the resources of the substrate, so the particles become motionless after the wave passes, and a finite recovery time is necessary before the substrate can support particle motion again.A long recovery time appears as large values of P 0 for low r rec .Throughout phase I, M increases with increasing r rec , but it exhibits large variations as indicated by the increase in σ M with increasing r rec .Phase II is defined to occur when P 0 < 0.5 but the motion is still periodic.In this regime the waiting time for the substrate recovery is reduced, and motion initiates at multiple points, producing multiple simultaneously propagating fronts.The result is very rapid oscillations in M and the appearance of a peak in σ M .At even higher r rec , there is a transition to phase III, consisting of continuous or chaotic motion.Here, σ M decreases with increasing r rec as the intermittent nature of the motion disappears, while P 0 drops and M becomes large.When r rec is increased further, there is a sudden transition to the frozen state, termed phase IV, where the recovery is sufficiently rapid that all grid sites remain saturated to S g = 1 and gradients in S g , required to produce motion of the particles, do not develop.Figure 2(f) shows that in a sample with r abs = 0.001, the frozen phase IV appears at r rec = 0.001, where M drops abruptly to zero.The oscillating motion in phases I and II and the transition to the chaotic phase III motion are similar to what is observed in reaction-diffusion systems [25,27,28].In our system, the fronts are correlated with translating motion of the particles rather than with reaction activity.By conducting a series of simulations in which we measure M and P 0 , we construct a dynamic phase diagram as a function of r abs versus r rec as shown in Fig. 3(a).The transition from I to II appears at r abs = 75r rec , the II-III crossover occurs when r abs ≈ 7.5r rec , and the tran- At later times, the passive particles are pushed together into crystalline clusters.Movies of these states appear in the supplemental material [30].
sition from III to IV falls at r abs = r rec .In Fig. 3(b), we plot a heat map of C l , the fraction of particles in the largest cluster, as a function of r abs versus r rec for the same system.The partial clustering state appears in phase II close to the peak value of σ M , indicated by a black line in Fig. 3(a).In general, clustering occurs when the pulses of motion begin to overlap.In phase III, the motion becomes too rapid for cluster formation to be possible.
We have also considered a system in which half of the particles are passive and interact only with other particles but not with the substrate.As shown in the dynamic phase diagram of Fig. 3(c), we find the same four phases which appeared in the system containing only active particles; however, the range of parameters over which phase III extends is expanded.In phase II there is no partial clustering; instead, a phase separated crystallization state appears in region III, as illustrated in the heat map of C l in Fig. 3(d).In the crystallized state, the passive particles form dense clusters with triangular ordering.The behavior at early and later times is illustrated in Fig. 4(a,b) for the system in Fig. 3(d) at r abs = 0.006 and r rec = 0.001.At early times, the particles are in a uniform fluid state, but at later times, dense crystalline clusters of passive particles coexist with a fluid of active particles, similar to what is observed in motility induced phase separation systems [2,3,10,11,15].The crystallization occurs when r rec is high and r abs is at intermediate values.For lower r abs , the active particles move rapidly from site to site with a mostly Brownian characteristic, while for high r abs , the active particles move only in bursts separated by long time intervals, so there is not enough activity to generate the phase separation.At intermediate values of r abs , the substrate gradients are maximized, generating the longest persistent intervals of active particle motion, and producing behavior similar to that found in run-and-tumble or driven diffusive systems.
Summary-We have proposed a new type of active matter system in which particles interact with a resource landscape of grid sites which impart forces to the particles.Particles are attracted to sites with the most resources, and simultaneously experience steric repulsion from other particles.Sites underneath particles have their resources depleted at a fixed absorption rate, while all sites recover their resources at a different fixed recovery rate up to a maximum value.When the absorption rate is much larger than the recovery rate, we find periodic pulses of propagating waves of translating particles.As the recovery rate increases, the time between pulses decreases until there is a transition to a fluid-like state with continuous motion.When the recovery rate is greater than or equal to the absorption rate, the system becomes trapped in a frozen state.When we introduce a second species of passive particles that do not couple to the substrate but interact sterically with all other particles, crystalline clusters of passive particles can form within the fluid state.Our system could be realized using colloidal particles interacting with an optical feedback light substrate, reactive excitable media, or robotic assemblies with an active substrate.This system could also be used to create soft matter versions of ecological models and excitable media.
r ij arises from harmonic repulsion with spring constant k = 20, r ij = |r i −r j |, and rij = (r i −r j )/r ij , where Θ is the Heaviside step function.The attractive substrate force is summed over the 12 grid sites occluded by particle i, F g i = − 12 k=1 S k g rik , where rik = (r i − r k )/r ik , r ik = |r i − r k |, and r k is the center of grid site k.The attraction is proportional to the resource level S k g (t) available at grid site k at time t, which ranges from the completely depleted value S k g = 0 to the saturated maximum value S k g = 1.

FIG. 2 .
FIG. 2. Left column: The fraction of moving particles M versus time in simulation steps.(a) In phase I at rrec = 2.5×10 −5 and r abs = 0.01, there are slow periodic oscillations in M .(b) At rrec = 2.5 × 10 −5 and r abs = 0.001, we find phase I motion but with shorter intervals between bursts.(c) At rrec = 3 × 10 −4 and r abs = 0.006, the system is in phase II with rapid periodic motion.(d) For rrec = 8×10 −4 and r abs = 0.004, we find phase III continuous motion.Right column: The time averaged fraction of moving particles M (blue), its standard deviation σM (orange), and the fraction of time P0 during which the motion is zero (green) versus recovery rate rrec at absorption rates of r abs = (e) 0.01, (f) 0.001, (g) 0.006, and (g) 0.004 matching the rates shown in panels (a) through (d).Dashed lines in the left column indicate the value of rrec illustrated in the right column.

FIG. 3 .
FIG. 3. (a) Dynamic phase diagram as a function of r abs versus rrec showing phases I (periodic bursts, green), II (oscillations, orange), III (continuous motion, purple), and IV (frozen, red).The black line indicates the location of the peak in σM .(b) Heat map of the fraction C l of particles in the largest cluster for the same system.Cluster size is maximized in phase II near the location of the peak in σM .(c) Dynamic phase diagram as a function of r abs versus rrec in a system where half of the particles are passive and do not interact with the substrate.(d) The corresponding heat map of C l showing that crystallization now occurs in phase III, where the passive particles form triangular crystallites as illustrated in Fig. 4.

FIG. 4 .
FIG. 4. Simulation snapshots of the entire sample showing moving (orange) and stationary (blue) active particles along with passive (red) particles on a resource field with values ranging from Sg = 0 (black) to Sg = 1 (white) in a sample with r abs = 0.006 and rrec = 0.001.(a) At early times, the system forms a phase III uniform fluctuating state.(b) At later times, the passive particles are pushed together into crystalline clusters.Movies of these states appear in the supplemental material [30].