Flowing active liquids in a pipe: Hysteretic response of polar flocks to external fields

We investigate the response of colloidal flocks to external fields. We first show that individual colloidal rollers align with external flows as would a classical spin with magnetic fields. Assembling polar active liquids from colloidal rollers, we experimentally demonstrate their hysteretic response: confined colloidal flocks can proceed against external flows. We theoretically explain this collective robustness, using an active hydrodynamic description, and show how orientational elasticity and confinement protect the direction of collective motion. Finally, we exploit the intrinsic bistability of confined active flows to devise self-sustained microfluidic oscillators.

For centuries, applying an external pressure difference has remained the only solution to flow a liquid in a pipe. Over the last ten years, by engineering soft materials from self-propelled units we have learned how to drive fluids from within [1][2][3][4][5][6][7][8][9][10]. The generic strategy consists in assembling orientationally ordered liquids from selfpropelled particles [11][12][13]. From a fundamental perspective, significant efforts have been devoted to explaining the emergence of collective motion in ensembles of interacting motile bodies, and the flow patterns of the resulting polar and nematic phases [10][11][12][13]. However, we still lack basic understanding of these non-equilibrium materials. One of the major questions that remains to be elucidated is the response of active phases to external fields [14,15]. The situation is all the more unsatisfactory because, from an applied perspective, the potential of active fluids as smart materials will be chiefly determined by their ability to sustain their spontaneous flows against external perturbations.
Here, combining experiments and theory, we elucidate how confined active fluids with broken rotational symmetry respond to external fields. Our experiments are based on colloidal rollers self-assembled into polar flocks, i.e. active liquids with orientational order akin to that of a ferromagnet [4,16,17]. We first demonstrate that isolated colloidal rollers align their direction of motion with an external flow as would classical spins with a magnetic field. In contrast, we establish that the response of polar liquids is intrinsically non-linear. When confined in channels, transverse confinement and bending elasticity act together to protect the direction of collective motion against external flows. We close this paper showing how the resulting hysteretic relation between the flock velocity and external flows results in the spontaneous oscillations of confined polar-liquid droplets.

RESPONSE OF MOTILE COLLOIDS TO EXTERNAL FLOWS: ALIGNMENT WITH EXTERNAL FIELDS
In our experiments, we exploit the so-called Quincke mechanism to motorize inert colloidal particles and turn them into self-propelled particles [4]. We recall the motorization principle in Appendix A, and provide details about the experimental set-up in Appendix B. In brief, we observe the 2D motion of colloidal rollers of diameter 2a = 4.8 µm confined in 50 mm × 16 mm × 0.1 mm channels filled with hexadecane oil, as illustrated in Fig. 1a. They behave as persistant random walkers moving at a constant speed v 0 = 0.98 ± 0.1 mm/s, and having a rotational diffusivity D = 1.5 s −1 [18]. As a result, the distribution of the roller velocities is isotropic and narrowly peaked on a circle of radius v 0 , see Fig. 1b.
We investigate their individual response to external flows by injecting a fresh hexadecane solution at constant flow rate. Given the aspect ratio of the fluidic channel, the flow varies only in the z direction along which it has a Poiseuille profile, Fig. 1a. We denote h the magnitude of the hexadecane flow evaluated at z = a along thex direction. Over a wide range of h, the speed of the rollers is virtually unchanged, see Figs. 1b and 1c. Their orientational distribution is however strongly biased. As seen in Fig. 1b, the average roller velocity points in the direction of the flow and the angular fluctuations are reduced upon increasing h. More quantitatively this behavior is very well captured by the following equations of motion: where the r i (t) andv i ≡ (cos θ i (t), sin θ i (t)) are respectively the positions and velocity orientations of the colloids. µ is a constant mobility coefficient and ξ(t) is a Gaussian white noise of unit variance. Eq. (2) corresponds to the over-damped Langevin dynamics of a classical spin coupled to a constant magnetic field. We henceforth use this magnetic analogy and define the average roller magnetization as V = v i i . Eq. (2) is readily solved and used to measure the rotational mobility µ = 0.08 µm −1 from the magnetization curve in Fig. 1d. This value is in excellent agreement with the estimate derived from first principles in [4] and has a sign opposite to that of the colloidal surfers studied in [19]. Isolated rollers align with a flow field as would uncoupled XY spins with a magnetic field.

ROBUSTNESS TO EXTERNAL FIELDS: HYSTERETIC RESPONSE OF POLAR LIQUIDS Experiments
In the absence of external flows, ferromagnetic orientational order emerges over system-spanning length scales when increasing the roller packing fraction ρ above 10 −2 [4]. A homogeneous polar liquid then forms and spontaneously flows through the microchannels as illustrated in Fig. 2a and Supplementary video 1. We now address the response of this active ferromagnet to external fields taking advantage of the coupling between the roller velocity and the surrounding fluid flows. To do so, we assemble a polar liquid confining hundreds of thousands of rollers in a race-track pattern of width W = 0.5 mm (the area fraction is set to ρ = 0.1). Once a homogeneous and stationary polar order is established, we study its longitudinal response by applying a uniform flow along one of the two straight parts of the channel as sketched in Fig. 2a, and detailed in Appendix B. For the sake of clarity, we henceforth refer to the hexadecane flow field evaluated at z = a as the external field h = hx. Fig. 2b shows that applying a field h along the direction of V reduces the transverse velocity fluctuations of collective motion.
In stark contrast to the individual response, Fig. 2b also shows that collective motion can occur in the direction opposite to the external field with a high level of ordering. But this robustness has a limit. Increasing |h| above h c = 63 µm/s, the rollers abruptly change their direction of motion and align with h. As quantitatively demonstrated in Fig. 2c, this behavior translates into a strong hysteresis of the magnetization curve V(h) upon cycling the magnitude of the external field.
We now elucidate the origin of this collective robustness, or more precisely, what sets the magnitude of the coercive field h c in this active ferromagnet. We thus focus on the regime where V · h < 0, and leave the discussion of the case where V · h > 0 to Appendix C. In an infinite system h c should vanish as the polar-liquid flow stems from the spontaneous breaking of a continuous rotational symmetry. In this active system, however, orientation is intimately coupled to mass transport. Therefore, the homogeneous rotation of the roller velocity is forbidden by the confining boundaries: reversing the direction of the flock requires finite wave-length distortions. As shown in Fig. 2d, this picture is supported by the very sharp increase of h c measured when decreasing the channel width.
To gain a more quantitative insight, we inspect the inner structure of the roller flow field. Supplementary video 2 and Fig. 3a both show that applying a field in the upstream direction causes finite wave-length distortions dominated by bending deformations. We introduce the Fourier transform of the roller flow field as v(q, t) = v i (t) exp[iq · r i (t)] i , and plot in Fig. 3b the time-averaged spectrum of the bending modes along the q = (q x , 0) direction: B(q x ) = |v y (q x , t)| 2 t . The bending deformations are dominated by spatial oscillations at a well-defined wave-length λ B . Fig. 3c indicates that confinement sets λ B = 2W , see also Supplementary video 3. Increasing the magnitude of the field strongly increases the amplitude of the bending oscillations until |h| = h c , Fig. 3d. As exemplified in Supplementary video 4, the bending waves are then destabilized into vortices leading to flow reversal. Subsequently, the external field stabilizes a strongly polarized homogeneous polar liquid flowing in the direction of h. We stress that the reversal of the flow is completed without resorting to local melting. Orientational order is locally preserved regardless of the direction and magnitude of the external field. The weak decrease of the magnetization curve seen in Fig. 2c in the negative V · h region chiefly stems from the constrained oscillations of the spontaneous flow.

Theory
We use a hydrodynamic description of the polar liquid to theoretically account for the bistability of the spontaneous flows and for the variations of the coercive field h c (W ) with confinement. The Toner-Tu equations are the equivalents of the Navier-Stokes equations for polar active liquids. They describe the dynamics of the velocity v(r, t) and density fields ρ(r, t). In the presence of an external driving field h(r, t), they take the generic form: and These phenomenological equations involve a number of hydrodynamic coefficients which we do not describe here (see e.g. [13] for a comprehensive discussion). The only two parameters relevant to the following discussion are: (i) the convective coefficient λ 1 which translates the lack of translational invariance of the system: the rollers drag on a fixed substrate defining a specific frame. (ii) D T and D 2 which are two elastic constants of the broken symmetry fluid. D T and D 2 both hinder the orientational distortions of the velocity field.
Describing the flow-reversal dynamics and the underlying spatiotemporal patterns would require solving the strongly non-linear system given by Eqs. (3) and (4). This task goes far beyond the scope of this article. Here, we instead exploit our experimental observations to construct a minimal model. As Fig. 3 indicates that a single bending mode of wavevector q = π/Wx dominates the deformations of the velocity and density fields, we make a simplifying ansatz writing: v(x, t) = v x (t)x + v y (t) cos(qx − ωt)ŷ, and neglect all contributions from spatial frequencies higher than π/W . We also ignore density fluctuations and restrain our analysis to longitudinal external fields h = −hx. Within this two-mode approximation, Eq. (3) is always satisfied and Eq. (4) reduces to the dynamical system: for the amplitudes of the two coupled velocity modes: To gain more intuition on the physical meaning of the dynamical system, we plot the force field F(v x , v y ) in Fig. 4 for four values of h. Anticipating on the comparison with our measurements, we use the parameter values estimated in [4] and recalled in Appendix D. Looking for fixed points in the absence of external field, F(V, h = 0) = 0, we find five solutions for V when α > D T q 2 , see Fig. 4a [31]. The trivial solution (0, 0) is obviously unstable and corresponds to a fluid at rest. The four other solutions are given by Fig. 4a. The former are saddle points while the latter are linearly stable fixed points corresponding to the two possible homogeneous flows at speed v 0 = α/β along the ±x directions. It is worth noting that the force field focuses the position of the dynamical system along a closed line connecting the four fixed points thereby making the dynamics of V almost one dimensional, see Fig. 4a.
The bistability of the flow at finite h, viz the existence of a finite coercive field, is understood from the dynamics of the fixed points in phase space. Upon increasing h the dynamical system, and therefore the active flow, explore three different states labeled with a star symbol in Figs. 4b, 4c and 4d: (i) We start from a polar liquid flowing in the +x direction, and an opposing external field h = −hx, with h > 0. As sketched in Fig. 4b, this state where V = (v 0 − h/(2α), 0) corresponds to a polar fluid uniformly flowing against the external field. This situation re- Fig. 4b. Above this value, the homogeneous flow is unstable to buckling, and the stable point V = (v 0 −h/(2α), 0) becomes a saddle point.
(ii) Yet, the flow is not reversed. The system indeed reaches one of the two new stable fixed points with v y = 0. They both correspond to homogeneously buckled conformations, see Fig. 4c. This prediction is consistent with the buckled patterns observed prior to flow reversal shown in Fig. 3 and Supplementary videos 2 and 3.
(iii) Further increasing h, the buckled state approaches the topmost saddle point. The two points eventually merge at a critical value h c corresponding to Fig. 4d. The only stable conformation then corresponds to a situation where V = (−v x , 0). The flow is reversed and aligns along the direction prescribed by h. h c defines the value of the coercive field.
The value of h c is determined analytically by the merging condition between the saddle and the fixed point, see Fig. 4d.
We find that h c stems from the competition between the external field and all the velocity-alignment terms (α and D T ): predicts that the stability of the flows opposing an external field is enhanced when further confining the polar liquid, i.e increasing q = π/W . Remarkably, this simplified picture also provides a reasonable estimate of the magnitude of the coercive field, see Figs. 2d, and 2e.
In summary, we have established the bistability of polar active fluids. Their hysteretic response originates from buckled flow patterns stabilized by orientational elasticity. We expect this phenomenology to apply to all confined active fluids with uniaxial orientational order. Our theory builds on the observation of a single set of buckling modes. Explaining the pattern-selection process remains, however, a significant technical challenge.

APPLICATION: SPONTANEOUS OSCILLATIONS OF POLAR-LIQUID DROPLETS
We close this article exploiting the intrinsic multistability of polar-liquid flows and demonstrating emergent functionalities in active microfluidics [22][23][24]. The existence of a hysteresis loop in the response function provides a very natural design strategy for spontaneous oscillators via the relaxation-oscillation mechanism [25]. Simply put, and having mechanical devices in mind, relaxation oscillations stem from the coupling between a system with a hysteretic "force-velocity" relation and a harmonic spring. This minimal design rule is transposed to active fluids by confining them in curved containers, and applying a constant and homogeneous external field h = hx. As illustrated in Supplementary video 5, and in the image sequence of Fig. 5a, the colloids form a polarliquid droplet that spontaneously glides along the confining boundary in an oscillatory fashion. We denote α the polar angle defining the position along the confining disc, and v(α, t) the azimuthal component of the velocity field averaged over the radial direction. Fig. 5b shows the variations of the velocity field v(α, t). The oscillatory dynamics of the polar-liquid droplets is clearly periodic with well-defined period and amplitude both decreasing with the magnitude of the stationary external field, Figs. 5c, and 5d.
We now explain these collective oscillations as the periodic exploration of the four states (i), (ii), (iii) and (iv) along the hysteresis loop established in Fig. 2c and sketched in 2e . The key observation is that the droplet follows the boundary of the circular chamber. The droplet hence experiences a longitudinal field of magnitude h (α) = h sin α which either favors or hinders its motion. The periodic exploration of the hysteresis loop is supported by Fig. 5e. Fig. 5e shows the distribution P (v CoM , h ), where v CoM is the polar-liquid velocity evaluated at the center of mass of the droplet α CoM . The support of this distribution corresponds to the rectangular shape of the velocity-field relation measured in Fig. 2 for a straight channel. The droplet spends most of its time exploring the stable horizontal branches and quickly jumps from one stable conformation to the other along the vertical ones. We can gain more intuition on this oscillatory dynamics describing the four states one at a time: In state (i), the head of the flock is located at α < 0 and v(α) · h < 0. The flock proceeds in the direction opposite to the azimutal component of the external field. The system moves toward the left of the bottom branch of the hysteresis loop, Fig. 5e. As the flock moves toward the negative α direction, the field strength |h | increases and reaches the maximal value h c at an angle −α c . The system then reaches the left vertical branch of the response curve and hence becomes unstable, state (ii). The flock bends and reverses its direction to reach the upper branch of the response curve, state (iii). When the flock proceeds in the positive α region, it experiences an increasingly high field in the direction opposite to its motion. As h = h c the flock reaches the right vertical branch of the response curve at the maximal angle +α c (state iv), thereby leading the system back to state (i). The hysteresis loop is periodically explored.
This oscillatory motion relates to the conventional relaxation-oscillation picture as follows: the response curve (h, V) plays the role of the force-velocity relation in a mechanical system, and the angle-dependent longitudinal flow plays the role of the harmonic spring.

CONCLUSION
In conclusion, we have established that colloidal rollers respond to external flows as classical spins to magnetic fields. Assembling active fluids with broken orientational symmetry from these elementary units, we have experimentally demonstrated, and theoretically explained, the hysteretic response of polar-active-fluid flows. We have shown how confinement and bending elasticity act together to protect emergent flows against external fields. Finally, we have effectively exploited the bistability of active flows to engineer active-fluid oscillators with frequency and amplitude set by the geometry of the container. Together with the virtually unlimited geometries accessible to microfabrication, the intrinsic nonlinearity of active flows offer an effective framework for the design of emergent microfluidic functions [22][23][24]. Our experiments are based on colloidal rollers, see [4]. We turn inert polystyrene colloids of diameter a = 4.8 µm into self-propelled bodies by taking advantage of the socalled Quincke electro-rotation mechanism [26,27]. Applying an electric field to an insulating body immersed in a conducting fluid results in a dipolar distribution of its surface charges. Increasing the magnitude of the electric field, E 0 , above the Quincke threshold E Q destabilizes the dipole orientation, which in turn makes a finite angle with the electric field. A net electric torque builds up and competes with viscous friction to power the spontaneous rotation of the colloids at constant angular velocity. As sketched in Fig. 1a, the colloids are let to sediment on a flat electrode, rotation is then readily converted into translational motion at constant speed v 0 in the direction opposite to the charge dipole. The direction of motion is randomly chosen and freely diffuses as a result of the spontaneous symmetry breaking of the surface-charge distribution.

Appendix B: Methods
We disperse commercial polystyrene colloids (Thermo Scientific G0500) in a mixture of hexadecane and AOT with concentration [AOT ] = 0.13 mol/L. We inject this solution into microfluidic chambers made of two electrodes spaced by a 110 µm-thick scotch tape. The electrodes are glass slides, coated with indium tin oxide (Solems, ITOSOL30, thickness: 80 nm). A voltage amplifier (TREK 609E-6) applies a DC electric field between the two electrodes. We image the system with a Nikon AZ100 microscope with a 3.6X magnification and record videos with a CMOS camera (Basler Ace) at framerate up to 380 Hz. We use conventional techniques to detect and track all particles [28][29][30]. To confined the colloidal rollers inside racetracks, we pattern the bottom electrode by mean of photolithography using a 2 µm-thick layer of UV photoresist (Microposit S1818) as in [17]. The geometry of the microfluidic device is detailed in Fig. 6. We study the response of rollers to external field, by injecting a fresh hexadecane solution at a controlled flow-rate using a high-precision syringe pump (Cetoni neMESYS). Each measurement was done at least 60 seconds after the relaxation of the flow pattern in the main branch of the racetrack. The construction of the hysteresis loop in Fig. 2b corresponds to a seven-hour long experiment.
Appendix C: Non-linear response of the ordered phase: V · h > 0 We discuss here the strengthening of orientational order when V · h > 0. We plot in Fig. 7a the variations of δV ≡ |V(h) − V(h = 0)| in this regime. At small h, V responds linearly to the external field. However, the increase of δV(h) becomes sub-linear for field amplitudes as small as h = 3 × 10 −2 v 0 . The simplest possible explanation of this anomalous attenuation is that V is a bounded quantity which is maximal and equals 1 when all the rollers move along the very same direction. A second and more involved explanation was put forward in [15]. For finite systems, a crossover from linear response at small h to the anomalous scaling law δV(h) ∼ h 1/3 was predicted from renormalization group analysis [15]. As shown in Fig. 7a, this scaling law is consistent with our experiments for system sizes ranging from W = 0.175 mm to W = 0.5 mm. However, the finite size scaling shown in Fig. 7b fails to ascertain this explanation. Disentangling the two effects would require operating closer to the transition toward collective motion where the fluctuations of V are more prominent. Such a regime cannot be achieved in our experiment due to the strongly first-order nature of the transition toward collective motion.

Appendix D: Hydrodynamic parameters of the roller fluid
We recall the estimates of the hydrodynamic parameters relevant to the computation of the coercive field h c .
Starting from the Stokes and Maxwell equations describing the microscopic dynamics of the colloids, we established in [4] the hydrodynamics of colloidal-roller liquids. The results of this kinetic theory are summarized in Table I. We determined the value of µ h following the same procedure, and found µ h = 1 2 µv 0 , where µ is the rotational mobility measured in Fig. 1.

ACKNOWLEDGMENTS
We acknowledge support from ANR program MiTra and Institut Universitaire de France. We thank M.C. Marchetti for valuable discussions. |h| (µm/s) 0    . Unlike what would be expected from the RG theory of [15], all plots do not collapse on the same master curve.