Taming Polar Active Matter with Moving Substrates: Directed Transport and Counterpropagating Macrobands

Following the goal of using active particles as targeted cargo carriers aimed, for example, to deliver drugs towards cancer cells, the quest for the control of individual active particles with external fields is among the most explored topics in active matter. Here, we provide a scheme allowing to control collective behaviour in active matter, focusing on the fluctuating band patterns naturally occurring e.g. in the Vicsek model. We show that exposing these patterns to a travelling wave potential tames them, yet in a remarkably nontrivial way: the bands, which initially pin to the potential and comove with it, upon subsequent collisions, self-organize into a macroband, featuring a predictable transport against the direction of motion of the travelling potential. Our results provide a route to simultaneously control transport and structure, i.e. micro- versus macrophase separation, in polar active matter.

Much of what we know about active systems and the patterns they form roots in explorations of minimal models which to some extend represent broader classes of active systems showing the same symmetries. The pioneering example of such a minimal model is the Vicsek model describing polar self-propelled particles such as actin-fibres mixed with motor proteins [20,21], certain microorganisms [22], self-propelled rods [23,24] or "birds" [22,25] which only see their neighbors and have a tendency to align with them, in competition with noise. While forbidden in equilibrium [26] the Vicsek model shows true long-range order in two dimensions [25], meaning that activity makes orientational correlations robust against noise over arbitrarily long distances. The phase transition from the disordered phase which occurs for strong noise to the long-range ordered Toner-Tu phase is now known to be discontinuous [27] and features a remarkably large coexistence region [28,29] where high-density bands of comoving polarized particles spontaneously emerge and traverse through a background of a low-density disordered gas-like phase. These bands be-have highly randomly; they merge when colliding with each other but also split up frequently, rendering an irregular pattern of sharply localized and strongly polarized moving bands. The latter choose their direction of motion spontaneously depending on initial state and fluctuating molecular environment (noise realization); thus, when averaging over many realizations, there is no net motion. This randomness is unfortunate in view of potential key applications of active matter, e.g. for targeted drug delivery, crucially requiring schemes to control active particles. Here, while single particle guidance with external fields is among the most explored problems in active matter [5,[30][31][32][33][34][35][36][37] and there is a moderate knowledge on interacting particles in external fields (complex environments) [38][39][40][41][42][43] and their control [44][45][46][47][48], surprisingly little is known about the controllability of polar active particles and band patterns they naturally form.
In the present work we ask for a scheme to tame band patterns, i.e. if we can force the bands in the Vicsek model to settle down into a pattern featuring a predictable and externally controllable direction of motion. To achieve this, we apply a "traveling wave potential" (also called travelling potential ratchet [49]) to the Vicsek model. We find that such an external field does in fact allow to control the late time direction of motion of particle ensembles in polar active matter, yet, in a remarkably nontrivial way. In our simulations, for appropriate parameter regimes, we see the formation of bands that at early times pin to the minima of the travelling potential and comove with it. When time proceeds, one of the bands suddenly unpins and starts counterpropagating in the travelling potential. Upon subsequent collisions the band swells towards a macroband containing most particles in the system. This macroband emerges representatively in a large parameter window and shows a predictable motion against the travelling direction of the potential. Our results show that a moving (or tilted, see Fig. 1) substrate tames the collective behaviour of polar active particles and can be used to control the transition from microphase separation (band patterns) to a macrophase separated state which does show predictable transport. In the following, we specify these results and analyze the mechanism underlying the emergence of a counterpropagating macroband.

II. MODEL
We consider N = 5000 active overdamped particles in a quasi-1D-potential V (x, y, t), which is uniform in ydirection and represents a traveling wave in x-direction; i.e. it is periodically modulated and moves with constant where ω, k are the frequency and wave vector of the travelling potential ( Fig.  1 (b)). Such a potential has previously been considered for active point particles [42] and disks [50] and can be realized e.g. by a micropatterned ferrite garnet film substrate [51], by optical lattices traversing at speeds of a few µ/s, or effectively (see Fig. 1 (c)), simply by a tilted washboard potential [52][53][54]. Note here that in the comoving frame (moving with a constant velocity v L ) the dynamics translates into motion of a particle in a static tilted washboard potential (see Fig. 1 (b),(c) and Sec. IV). Besides experiencing the external potential, the active particles also self-propel, a fact effectively described by a self-propulsion force γv 0 p i where p i = cos θ i e x + sin θ i e y ; i = 1, .., N are the self-propulsion directions of the particles and γ is the Stokes drag coefficient. In bulk, the particles would move with a constant speed v 0 . As in the Vicsek model, we assume that the particles align with each other. We define the dynamics of the particles by the following equations of motion [55]: Here, g controls the strength of alignment of a particle with its neighbors within a range R and the sum is performed over all these neighbors (see Fig. 1 (a)). Alignment competes with rotational Brownian diffusion, occurring with a rate D r ; η i represents Gaussian white noise of zero mean and unit variance. The force due to the substrate reads where u 0 is the strength of the external force. Here, in all of our results we express lengths and times in units of µm and s respectively, i.e. we introduce parameters D r = D r · s, g = g · s/µm 2 etc. and omit primes for simplicity, allowing thus for a straightforward comparison with potential experiments. However, for readers interested in the actual dimensionless control parameters, see [56]. Here vi = v0pi is the self-propulsion velocity of the i-th particle, aligning with adjacent partices (red circle). A Galilei transformation to the comoving frame turns the travelling wave potential (panel (b)) into a static, tilted periodic potential (panel (c)). Thus, motion in the travelling wave potential (b) is equivalent to motion in a static tilted lattice in the comoving frame which displaces through space (relative to the laboratory frame) with a constant speed vL (panel (c)). The pinned state, where particles comove with the travelling wave corresponds to particles resting around a minimum of the tilted lattice in (c). We find that in the lab frame, for certain values of the parameters, polar active particles can move faster down the tilted lattice (to the left) than the lattice displaces through space (see Sec IV). The dynamical pathway to achieve this sliding state involves a controllable transition from microphase separation (patterns) to macrophase separation (counterpropagating macroband).
We now study the dynamics of the described model using Brownian dynamics simulations and an elongated simulation box of size L x ×L y = 500×5, fixing the density to ρ = 2, as well as random but uniformly distributed initial particle positions and orientations. Using more quadratic boxes leads to qualitatively similar phenomena.

III. COUNTERPROPAGATING MACROBAND
In the absence of a lattice, our simulations reveal the usual phenomenology of the Vicsek model [27][28][29]: For a given alignment strength (g = 0.07) and comparatively strong noise D r > D c r ≈ 0.15 (or high temperatures), we find a disordered uniform phase ( Fig. 2 (a),(b)), whereas noise values D r < D c r lead to a polarized phase (Fig. 2 (c),(d)). In the latter phase, particles self-organize into polarized bands of high density which move with a speed ∼ v 0 and coexist with gas-like unpolarized regions in between the bands. The bands occur at seemingly irregular distances to each other. As time proceeds, they occasionally split up (for D r = 0) and typically merge when they collide with each other; overall, the number and size of the bands changes dynamically.
In the presence of the traveling wave (moving lattice) and at weak noise (D r = 3 × 10 −4 < D c r ) the behaviour of the bands may change dramatically. While very steep lattices of course pin the particles permanently to the lat-tice minima, leading to a state where all particles comove with the lattice, shallow lattices have little impact on the behaviour of the system and its tendency to form bands. In this latter regime, the lattice exerts a periodic force which essentially averages out before the particles move much. Thus, we here focus on moderate lattice depth (u 0 = 0.3) and lattice speeds comparable to that of the particles (v 0 = v L = 0.2), so that the particles can occasionally overcome the potential maxima. In this regime, for sufficiently weak noise (here D r < D c r = 3 × 10 −4 ), particles form quickly bands, most of which are pinned to the lattice and thus co-move with it ( Fig. 2 (e),(g),(h)). Note that the polarization of such bands is almost unity even for small times and maintains this very high value during the time evolution ( Fig. 2 (g),(h)). Occasionally, we observe that a band, assisted by the existing noise, changes direction and counterpropagates; it then soon collides with another band (see Fig. 2 (g),(h) insets for such collision events). Here, the two bands merge and form one larger band which in some cases becomes pinned and in other cases slides, still against the direction of motion of the lattice. In the latter case, the band soon encounters further bands and can in each case, either stop moving (get pinned to the lattice) or continue sliding. One might expect that this seemingly random result of the collision processes should ultimately lead back to a pinned state. Strikingly, however, in many simulations we observe cases where a band counterpropagates through the entire lattice and systematically consumes all other bands. The result is one macroband which contains most of the N particles and counterpropagates against the direction of lattice motion (Fig. 2 (f),(h)). Since the particles counterpropagate, even when viewed from the laboratory frame, with respect to the forces acting on a pinned particle in a minimum of the lattice, they feature an absolute negative mobility. Thus, we observe a spontaneous reversal from a comoving state where most particles have followed the lattice to a counterpropagating state.
The striking difference between a finally pinned ( Fig.  2 (e),(g)) and a finally sliding state (Fig. 2 (f),(h)) featuring a current reversal is further illustrated in Fig. 2 (i),(j),(k). Here we observe that the mean polarization (averaged over all particles) P = cos θ 2 + sin θ 2 increases from the pinned state at short times to a value of almost one for the sliding macroband ( Fig. 2 (i)). It turns out (Fig. 2 (j)) that p x = cos θ ≈ −1, meaning that the particles collective self-propel against the direction of the lattice motion (still in the laboratory frame), i.e. along −e x . The average velocity of the particles is ẋ ≈ v L = 0.2 (see also Eq. (1a)) for the pinned state ( Fig. 2 (k)) and acquires a negative value oscillating in time for the case of the sliding macroband.

IV. PINNED AND SLIDING SOLUTIONS
We can get some first insight into the mechanism underlying the surprising counterpropagation of the bands by examining the single-particle dynamics in the zero noise limit. When projected to the x-axis Eq. (1a) reduces toẋ where the Galilean transformation to the comov- This equation is well known as the overdamped limit of the equations of motion of e.g. the forced nonlinear pendulum [57], the driven Frenkel-Kontorova (FK) model [58] and the resistively shunted junction (RSJ) model of Josephson junctions [59]. It is known to attain two different kinds of solutions depending on the value ofṽ x u0 . For ṽx u0 ≤ 1 the system is in the so-called pinned phase where the particle cannot overcome the potential barrier u 0 and remains therefore trapped within one of its wells ( x ∞ t → 0), yielding an asymptotic time averaged velocity ẋ ∞ t = v L = ω/k. In the opposite case ṽx u0 > 1, the particle is fast enough to overcome the potential barrier separating the wells (or in the example of the pendulum to lead to a rotation) and thus the system exhibits a sliding phase where the particle permanently moves (slides or rotates) in one and the same direction with an oscillating velocityẋ [60] of period T = For the N -particle system (Eqs.(1a),(1b)) the particles' self-propulsion directions p i change due to alignment interactions (Eq. (1b)) and noise (Eq.(1a)). Hence, the projection of the particle speed onto the x-axis changes in time, so that the sliding condition ṽx u0 > 1 subsequently may and may not be fulfilled. In terms of p i , the sliding condition reads In the present parameter regime (v 0 = v L = 0.2, u 0 = 0.3) sliding occurs for p xi < − 1 2 , or θ i ∈ 2π 3 , 4π 3 . Thus roughly 1/3 of particles will initially be in the sliding phase. Importantly, all of these particles which can in principle slide, move against the direction of lattice motion (negative p xi ), i.e. sliding can only occur against the direction of lattice motion, as observed in Figs. 2 (f),(h),(j),(k). The main effect of rotational diffusion (noise in the particle orientations) consists in the smoothening of the pinned-to-sliding transition at ṽx u0 = 1.

V. COLLISIONS OF VICSEK BANDS
We now exploit these considerations regarding pinned and sliding states for single particles to understand the dynamics of the polarized bands in the lattice. In our simulations, shortly after their formation, the magnitude of the polarization of the individual bands quickly approaches a value close to one (Fig. 2 (g),(h)), i.e. most bands are moving with almost constant individual velocities relative to the lattice. Thus, in the absence of collisions, the bands essentially behave like single particles and are either pinned or slide through the lattice. To study band collisions, it is useful to assign effective "masses" m n to the bands representing the number of particles contained in the band. When two bands with polarization angles θ 1 , θ 2 and masses m 1 , m 2 collide, they usually merge into a larger band of total mass m 1,2 = m 1 + m 2 (Fig. 3(a)) and average their polarizations.
(Formally, there are two fixpoints of the orientational dynamics when two bands merge: one reads θ 1,2 = Θ 0 with Θ 0 = m1θ1+m2θ2 m1+m2 , the other one θ 1,2 = Θ 0 − 2π m1 m1+m2 ; here the one lying within the smaller arc between θ 1 and θ 2 is stable and thus observed, the other one is unstable.) In our simulations, the polarization direction of an isolated band can freely rotate (Goldstone mode); noise therefore creates a random dynamics of the band polarization direction. Once the polarization angle of an initially pinned band reaches a value θ 1 ∈ ( 2π 3 , 4π 3 ) the band will move over a lattice barrier in the direction opposite to the lattice motion (Figs. 3(d),(e),(g),(h)). Since the motion of the band towards an adjacent lattice site occurs on timescales which are short compared to the time noise needs to significantly change θ 1 , the band will typically feature an angle close to 2π/3 or 4π/3 when it encounters another pinned band. Depending on its relative orientation to the band it encounters, after the collision, its angle may either be out of the sliding interval ( Fig. 3(b)), or, similarly likely, may be deeper in the sliding interval (Fig. 3(c)). In the latter case, the band continues counterpropagating through the lattice. Statistically, further collisions with other bands can be essentially viewed as a random walk of the band's polarization direction. Here, however, the effective mass of the band increases within each collision, corresponding to a decrease of the stepsize after each step. Hence, when the polarization of a band after a first few collisions is deeply in the sliding regime, i.e. θ ≈ π, the sliding of the band is highly robust against further collisions. This is why we have observed the emergence of a counterpropagating macroband consuming all other bands (Fig. 2 (f),(h)).
To understand the broad width of the counterpropagating macroband, it is instructive to resolve the collisions slightly further. When sliding bands collide, the positions of the contained particles do not fully mix up; rather, the resulting band features a substructure of microbands stacked one behind the other (Fig. 3 (i)). This fact is responsible for the large width of the observed macroband ( Fig. 2 (f)) in the case of finally sliding states. This is because successive collisions typically result in a macroband with an average polarization close to π, which is the centre of the sliding interval 2π 3 , 4π 3 . Thus after the collisions the involved particles move in the negative x direction with p i ≈ −e x (see Fig. 3 (i)), a fact that prohibits them from mixing along the y direction. Conversely, band collisions leading to pinning do not induce a pronounced substructure. Here, the involved particles move significantly in the y-direction and therefore tend to mix in the course of the dynamics (Fig. 3 (f)).

VI. EFFECT OF THE PARTICLE SPEED
Having explored the mechanism leading to the dynamical reversal of the direction of motion of the particles in the moving lattice, we now ask how representative this scenario is. Here, we stay in the low noise regime (D r = 3 × 10 −5 ) and with our previous values for the lattice velocity v L = 0.2 and height u 0 = 0.3 but vary the self-propulsion velocity of the particles. For v 0 ≤ 0.1, we have u 0 − v L > v 0 and hence sliding is not possible, whereas for v 0 > 0.5, where v 0 > u 0 + v L , sliding can be achieved also in the positive direction (see Eq. (6)). In the complete interval 0.1 < v 0 ≤ 0.5 sliding is possible only against the lattice motion (negative direction) as discussed above. Within this interval, larger values of v 0 yield a larger interval of polarization angles leading to sliding (Figs. 4 (a),(b)). To specify this, we simulate 50 particle ensembles for each value of v 0 and count the number (ratio) of ensembles R s which have reached a sliding and counterpropagating macroband and the corresponding ratio of ensembles R p which have settled in an overall pinned configuration. Fig. 4 (b) shows that R s increases monotonically in v 0 , crossing from a regime where most of the bands are pinned, even at late times Complementary information about the finally sliding states, featuring a current reversal, is provided by Figs. 4 (c),(d). The final average polarization of these states P s is for all v 0 very close to 1 (Fig. 4 (c)), owing to the low noise which results in particles clustering to a macroband with a certain alignment. In contrast, the direction of this alignment, quantified by p x s = cos θ s , is affected strongly by v 0 (Fig. 4 (d)). For 0.1 < v 0 ≤ 0.5 we have p x s ≈ −0.95, indicating that within this interval the particles' velocities are all approximately aligned towards −e x (as in Fig. 3 (i)) and thus the particles counterpropagate at roughly their maximum velocity. This picture changes when v 0 > 0.5 where a sliding also in the forward direction becomes possible. Different realizations result in finally sliding states with a different alignment p i and thus their ensemble average p x s → 0 as v 0 increases, recovering the isotropy in the direction of sliding bands of the Vicsek model in the absence of a lattice.

VII. EFFECT OF THE NOISE AMPLITUDE
Rotational diffusion, whose strength is controlled by D r , is crucial to initiate the emergence of sliding states; on the other hand, there is an upper critical noise strength D c r above which the Vicsek model does not show polar order, but is in the isotropic phase. We now systematically explore how the transition to the counterpropagating macroband is affected when changing D r . For u 0 = 0.3, v 0 = v L = 0.2 where sliding is possible only in the negative x direction, the ratio of states which are pinned at the end of our simulations R p decreases as D r increases ( Fig. 5 (a)) and finally approaches zero for D r 0.1D c r . The reason for this behaviour is probably that larger noise turns the orientation of the polarization of initially pinned bands faster and therefore initiates sliding earlier (and more often). The respective ratio of finally sliding states R s (Fig. 5 (b)) increases only slightly as D r increases from zero for D r 0.05D c r and afterwards decreases tending towards zero. Physically, when D r is too large, the polarization of a band may significantly change between each subsequent collisions with other bands and may therefore leave the sliding regime before encountering another collision. Thus, for too strong noise, the emergence of a transition to a counterpropagating macroband is rather unlikely. The generic behaviour of the system for such high noise values is that of a mixture of individual particles, both in the pinned and in the sliding phase, whose polarization orientation changes fast in time, providing the picture of an overall disordered phase modulated by the existing potential wells (Figs. 6 (a),(b)). There is however still a possibility of obtaining a finally sliding state even in the high noise regime (Fig. 5 (b)). Such states are significantly less polarized than the ones in the low noise regime (Figs. 5 (c), 6 (c)) featuring also a larger variety in the direction of alignment, quantified by p x s (Figs. 5 (d), 6 (d)). Furthermore, for such cases of high noise (e.g. D r = 0.3, D r = 0.95) the time evolution of both p x and P is much slower (Figs. 6 (e),(f)) than the ones observed for lower noise values (e.g. D r = 0.0003, D r = 0.003), indicating the diffusive character of the dynamics expected for highly noisy systems.

VIII. CONCLUSIONS
The present results provide a scheme allowing to control the typically highly irregular collective dynamics of polar active particles. In particular, we have seen that the bands occurring in the Vicsek model, which normally move in unpredictable directions and irregularly merge and split up can be tamed by applying a traveling wave-shaped potential, as can be realized e.g. using a micropatterned moving substrate or a traversing optical lattice. We find that while most particles in the system self-organize into polarized bands which comove with the lattice at early times, they can later experience a remarkable reversal, initiated by the counterpropagation of a single band which subsequently consumes all other bands in the system. The asymptotic state is a strongly polarized macroband which predictably moves opposite to the direction of the motion of the external substrate. This behaviour is representative in a large parameter window and can be controlled e.g. by tuning the relative speed of the active particles and the lattice.
These results may inspire further research of the interface between nonlinear dynamics and active matter and perhaps also applications regarding collective targeted cargo delivery using polar active matter.