Freezing a Flock: Motility-Induced Phase Separation in Polar Active Liquids
Geyer Delphine, Martin David, Tailleur Julien, Denis Bartolo
FFreezing a Flock:Motility-Induced Phase Separation in Polar Active Liquids
Delphine Geyer, David Martin, Julien Tailleur, and Denis Bartolo Univ. Lyon, ENS de Lyon, Univ. Claude Bernard,CNRS, Laboratoire de Physique, F-69342, Lyon, France Laboratoire Matiere et Systemes Complexes, UMR 7057 CNRS/P7, Universit´e Paris Diderot,10 rue Alice Domon et Leonie Duquet, 75205 Paris cedex 13, France (Dated: March 5, 2019)Combining model experiments and theory, we investigate the dense phases of polar active matterbeyond the conventional flocking picture. We show that above a critical density flocks assembledfrom self-propelled colloids arrest their collective motion, lose their orientational order and formsolids that actively rearrange their local structure while continuously melting and freezing at theirboundaries. We establish that active solidification is a first-order dynamical transition: activesolids nucleate, grow, and slowly coarsen until complete phase separation with the polar liquidsthey coexists with. We then theoretically elucidate this phase behaviour by introducing a minimalhydrodynamic description of dense polar flocks and show that the active solids originate from aMotility-Induced Phase Separation. We argue that the suppression of collective motion in the formof solid jams is a generic feature of flocks assembled from motile units that reduce their speed asdensity increases, a feature common to a broad class of active bodies, from synthetic colloids toliving creatures.
I. INTRODUCTION
The emergence of collective motion in groups of liv-ing creatures or synthetic motile units is now a wellestablished physical process [1–6]: Self-propelled parti-cles move coherently along the same direction whenevervelocity-alignment interactions overcome orientationalperturbations favoring isotropic random motion. Thisminimal picture goes back to Vicsek’s seminal work [7]and made it possible to elucidate the flocking dynamicsof systems as diverse as bird groups, polymers surfing onmotility assays, shaken grains, active colloidal fluids anddrone fleets [6, 8–13]. From a theoretical perspective,flocks are described as flying ferromagnets where point-wise spins move at constant speed along their spin direc-tion [1, 3, 7, 14, 15]. However, this simplified descriptionis inapt to capture the dynamics of dense populationswhere contact interactions interfere with self-propulsionand ultimately arrest the particle dynamics. Until now,aside from rare theoretical exceptions [14, 16–19], theconsequences of motility reduction in dense flocks hasremained virtually uncharted despite its relevance to aspectrum of active bodies ranging from marching animalsto robot fleets and active colloids.In this article, combining quantitative experiments andtheory, we investigate the suppression of collective mo-tion in high-density flocks. We show and explain howpolar assemblies of motile colloids turn into lively solidphases that actively rearrange their amorphous structure,but do not support any directed motion. We establishthat active solidification of polar liquids is a first-orderdynamical transition: active solids nucleate, grow, andslowly coarsen until complete phase separation. Eventhough they are mostly formed of particles at rest, weshow that active solids steadily propagate through thepolar liquids they coexist with. Using numerical simula- tions and analytical theory, we elucidate all our experi-mental findings and demonstrate that the solidification ofcolloidal flocks provides a realization of the long sought-after complete Motility-Induced Phase Separation [5, 20–25].
II. SOLIDIFICATION OF COLLOIDAL FLOCKS
Our experiments are based on colloidal rollers. Inbrief, taking advantage of the so-called Quincke insta-bility [10, 26], we motorize inert colloidal beads andturn them into self-propelled rollers all moving at thesame constant speed ν = 1040 µ m / s when isolated,see also Appendix B. We observe 5 µ m colloidal rollerspropelling and interacting in microfluidic racetracks oflength L = 9 . ρ ( r , t ) is homogeneous and the local particlecurrent W ( r , t ) vanishes. Increasing the average pack-ing fraction ρ above ρ F = 0 .
02, Fig. 1c, the rollers un-dergo a flocking transition. The transition is first-order,and polar liquid bands, where all colloids propel on av-erage along the same direction, coexist with an isotropicgas [10, 30]. Further increasing ρ , the ordered phase fillsthe entire system and forms a homogeneous polar liquidwhich flows steadily and uniformly as illustrated in Sup-plementary Movie 1. In polar liquids both W ( r , t ) and ρ ( r , t ) display small (yet anomalous) fluctuations, andorientational order almost saturates, see Fig. 1d and [31].This low-density behavior provides a prototypical exam-ple of flocking physics. a r X i v : . [ c ond - m a t . s o f t ] M a r . t/ T W ( m/s) - . - . . . t/T j ( m / s ) -0.500.5 t/T j ( m / s ) -0.2-0.100.10.2 t/T j ( m / s ) -0.500.5 a. . t/ T j( m/s) - . - . . . . t/ T j( m/s) - . - . . . . t/ T j( m/s) - . - . . . . t/ T j( m/s) - . - . . . . t/ T j( m/s) - . - . . . b. Gas c. Vicsek Bands e. Solid Jam f. Active Solidd. Polar Liquid . t/ T W ( m/s) - . - . . . . t/ T W ( m/s) - . - . . . . t/ T W ( m/s) - . - . . . . t/ T W ( m/s) - . - . . . t/T W ( m / s ) -0.2-0.100.10.2 t/T j ( m / s ) -0.2-0.100.10.2 FIG. 1.
The dynamical phases of Quincke rollers. a . Picture of a microfluidic racetrack where ∼ × Quincke rollersinteract. An active solid (dark grey) propagates though a polar liquid (light grey). Scale bar: 2 mm. b–f.
Top panel: close-uppictures of Quincke rollers in the racetrack. Scale bars: 250 µ m. Bottom panel: longitudinal component of the particle current W and density plotted as a function of a normalized time. Both W and ρ are averaged over an observation window of size56 µ m × µ m. T is arbitrarily chosen to be the time taken by an active solid to circle around the race track. b. Gas phase( ρ = 0 . c. Coexistence betweenan active gas and a denser polar band ( ρ = 0 . d. Polar-liquid phase ( ρ = 0 . e. Coexistence between a polar liquid and an amorphous active solid ( ρ = 0 . f. Homogeneous active solid phase( ρ = 0 . However, when ρ exceeds ρ S (cid:39) .
55 collective motionis locally suppressed and flocking physics fails in explain-ing our experimental observations. Particles stop theircollective motion and jam as exemplified in Supplemen-tary Movies 2 and 3. The jams are active solids thatcontinuously melt at one end while growing at the otherend. This lively dynamics hence preserves the shape andlength ( L S ) of the solid which propagates at constantspeed upstream the polar-liquid flow, see the kymographof Fig. 2a. Further increasing ρ , the solid region growsand eventually spans the entire system, Fig. 1f.Active solids form an amorphous phase. The pair cor-relation function shown in Figs. 2b and 2c indicate thatactive solids are more spatially ordered than the polarliquid they coexist with, but do not display any sign oflong-range translational order. As clearly seen in Sup-plementary Movie 2, the colloid dynamics are howevermarkedly different in the two phases. While they contin-uously move at constant speed in the polar liquid, in theactive solid, the rollers spend most of their time at restthereby suppressing any form of collective motion andorientational order, Figs. 2d and 2e.We stress that the onset of active solidification corre-sponds to an area fraction ρ S (cid:39) . ρ = 0 .
84) and thanthe crystalization point of self-propelled hard disks re-ported by Briand and Dauchot ( ρ ∼ .
7) in [32]. Thismarked difference hints towards different physics which we characterize and elucidate below.
III. THE EMERGENCE OF AMORPHOUSACTIVE SOLIDS IS A FIRST-ORDER PHASESEPARATION
We now establish that the formation of active solids oc-curs according to a first order phase-separation scenario.Firstly, Figs. 3a and b indicate that, upon increasing ρ ,the extent of the solid phase has a lower bound: start-ing from a homogeneous polar liquid, the solid length L S discontinuously jumps to a finite value before increasinglinearly with ρ − ρ b L , where ρ b L = 0 .
53, see Fig.3b. Thesmallest solid observed in a stationary state is as largeas L S ∼ . ρ , its value saturates as it co-exists with an active solid, Fig. 3c. At coexistence, thelocal densities in the bulk of the liquid and solid phasesare independent of the average density ρ : ρ b S = 0 . ρ b L = 0 .
53, which again supports a nucleation andgrowth scenario. Increasing ρ leaves the inner structureof both phases unchanged and solely increases the frac- -1 0 1 v (mm/s) -1-0.500.51 v ( mm / s ) t(s) x / L -2 0 2 v (mm/s) -2-1012 v ( mm / s ) r /a g S (r) r /a g L (r) abd ec FIG. 2.
Structure and dynamics of active jams. a.
Thekymograph of the measured light intensity averaged over theracetrack width shows how an active solid (dark region) prop-agates at constant speed through a homogeneous polar liquid(light region). b and c.
Pair-correlation functions measuredin the polar liquid ( g L ) and in the coexisting active-solid phase( g S ). ρ = 0 .
58. Both pair-correlation functions are plottedversus the interparticle distance r ⊥ in the direction transverseto the mean polar-liquid flow. g S displays more peaks than g L revealing a more ordered structure, but translational ordermerely persists over a few particle diameters. d . Probabilitydensity functions of the roller velocities in the polar liquidregion. The distribution is peaked around ν ˆ x (cid:107) , where ˆ x (cid:107) isthe vector tangent to the racetrack centerline. e . Probabil-ity density functions of the roller velocities in the active jamregion. The distribution is peaked around 0. The rollers re-main mostly at rest. b , c , d , and e : average area fraction ρ = 0 . tion of solid L S /L in the racetrack. We find that, as inequilibrium phase separation, the length of the solid re-gion is accurately predicted using a lever rule constructedfrom the stationary bulk densities ρ bS and ρ bL , see Fig. 3b.Thirdly, we stress that when multiple jams nucleate inthe device, they propagate nearly at the same speed, seeSupplementary Movie 4. Therefore, they cannot catchup and coalesce. The system in fact reaches a stationarystate thanks to a slow coarsening dynamics illustratedin Fig. 3d where we show the temporal evolution of thelength of two macroscopic active solids (red symbols) andof the overall solid fraction (dark line). One solid jamgrows at the expense of the other and coarsening operatesleaving the overall fraction of solid constant. All of our experiments end with complete phase separation: a sin-gle macroscopic active solid coexists with a single activeliquid phase. The final state of the system is thereforeuniquely determined by two macroscopic control param-eters: the average density ρ and the magnitude E ofthe electric field used to power the rollers.Finally, the most compelling argument in favour of agenuine first-order phase separation is the bistability ofthe two phases. Fig. 3b shows that at the onset of so-lidification, depending on the (uncontrolled) initial con-ditions, the system is either observed in a homogeneouspolar liquid or at liquid-solid coexistence. The bistabil-ity of the active material is even better evidenced whencycling the magnitude of E (cycling the average densityis not experimentally feasible). Fig. 3e shows the tem-poral variations of the active-solid fraction upon triangu-lar modulation of E , see also Supplementary Movie 5.When E increases an active-solid nucleates and quicklygrows. When E decreases, the solid continuously shrinksand eventually vanishes at a field value smaller than thenucleation point. The asymmetric dynamics of L S /L demonstrates the existence of a metastable region in thephase diagram. As shown in Fig. 3f, the metastabilityof the active solid results in the hysteretic response of L S , the hallmark of a first-order phase transition. Wealso note that the continuous interfacial melting observedwhen E smoothly decreases contrasts with the responseto a rapid field quench, see Supplementary Movie 6.Starting with a stationary active solid, a rapid quenchresults in a destabilization of the solid bulk akin to aspinodal decomposition dynamics.Altogether these measurements firmly establish thatthe emergence of active solids results from a first-orderphase separation, which we theoretically elucidate below. IV. MOTILITY-INDUCED PHASESEPARATION IN HIGH-DENSITY POLARFLOCKSA. Nonlinear hydrodynamic theory
As a last experimental result, we show in Fig. 4a howthe roller speed ν ( ρ ) varies with the local density ρ ( r , t )evaluated in square regions of size 12 a ∼ µ m. Thesemeasurements correspond to an experiment where a solidjam coexists with a homogeneous polar liquid. ν ( ρ )hardly varies at the smallest densities and sharply dropstowards ν ( ρ ) = 0 when ρ ( r , t ) exceeds ¯ ρ ∼ .
35. Al-though we cannot positively identify the microscopic ori-gin of this abrupt slowing down, we detail a possible ex-planation in Appendix A. Simply put, the lubrication in-teractions between nearby colloids with colinear polariza-tions result in the reduction of their rotation rate whichultimately vanishes at contact: near-field hydrodynamicinteractions frustrate self-propulsion. Instead of elabo-rating a microscopic theory specific to colloidal rollers as t(s) L S / L L S / L E (V/ m) L S / L t(s) L S / L E ( V / m ) L P , S LPS
123 12 3 a b cd e f L S / L lever-ruleexperiments L P , S LPS
FIG. 3.
Active solidification is a first-order phase separation. a.
Solid jams in a racetrack at ρ = 0 . , . , . b. Solid fraction plotted versus the averagedensity ρ . Note the discontinuous jump and the two possible states at the onset of solidification. c. Density of the polar-liquidphase (blue circles) and of the active-solid phase (red circles) plotted versus ρ . In steady state, the liquid density increaseswith ρ until an active solid forms, the density in both phases then remains constant. In b . and c . the shaded regions indicatethe coexistence between the polar liquid and an active solid phases. d. Coarsening dynamics. Two solid jams coexist onlyover a finite time period. The larger jam (filled symbols) shrinks and eventually vanishes in favor of the smaller one (opensymbols). The total fraction of solid phase in the racetrack (black line) remains constant over time. e. Solid fraction (blueline) and magnitude of the electric field (red line) plotted versus time. Over a range of E values, the active-solid fraction isdifferent when increasing or decreasing the electric field. f. The extent of the traffic jam follows a hysteresis loop when cyclingthe field amplitude. in [10], we instead adopt a generic hydrodynamic descrip-tion to account for all our experimental findings.We start with a minimal version of the Toner-Tu equa-tions which proved to correctly capture the coexistence ofactive gas and polar-liquid drops at the onset of collectivemotion [2, 33, 34]. For sake of simplicity we ignore fluctu-ations transverse to the mean-flow direction and write thehydrodynamic equations for the one-dimensional density ρ ( x, t ) and longitudinal current W ( x, t ): ∂ t ρ + ∂ x W = D ρ ∂ xx ρ, (1) ∂ t W + λW ∂ x W = D W ∂ xx W − ∂ x [ (cid:15) ( ρ ) ρ ]+ [ ρ(cid:15) ( ρ ) − φ ] W − a W . (2)We modify the Toner-Tu hydrodynamics to account forthe slowing down of the rollers when ρ exceeds ¯ ρ , Fig. 4a.Two terms must be modified to capture this additionalphysics: the so-called pressure term ρ(cid:15) ( ρ ), and thedensity-dependent alignment term ρ(cid:15) ( ρ ) responsible forthe emergence of orientational order and collective mo-tion.Coarse-graining microscopic flocking models typicallyleads to a pressure term proportional to the particle speed [1, 28]. We therefore expect (cid:15) ( ρ ) to sharply de-crease when ρ ( x, t ) > ¯ ρ . At even higher densities, wealso expect the repulsion and contact interactions be-tween the particles to result in a pressure increase withthe particle density [10, 35]. We henceforth disregardthis second regime which is not essential to the nucle-ation and propagation of active solids. The functionalforms of (cid:15) ( ρ ) is phenomenologically deduced from theloss of orientational order in the solid phase reported inFigs. 2e. This property is modelled by a function (cid:15) ( ρ )which decreases from a constant positive value in thelow-density phases to a vanishing value deep in the solidphase. In all that follows, we conveniently assume (cid:15) ( ρ )and (cid:15) ( ρ ) to be proportional. This assumption is sup-ported by the similar variations observed for the rollerspeed and local current in Fig. 4a. In practice, we take (cid:15) i = σ i [1 − tanh(( ρ − ¯ ρ ) /ξ )], where σ and σ are constant.Numerical resolutions of Eqs. (1) and (2) at increas-ing densities faithfully account for the five successivephases observed in our experiments, see Fig. 4 and Ap-pendix B. At low densities, we first observe the standard .
510 200 400 600 800
W ( m/s) ( ) ( m/s) W ( m / s ) () ( m / s ) b c e fd x/L W -2-1012 ; x/L W -2-1012 x/L W -505 x/L W -2-1012 x/L W -505 . t/ T j( m/s) - . - . . . . t/ T j( m/s) - . - . . . . t/ T j( m/s) - . - . . . . t/ T j( m/s) - . - . . . . t/ T j( m/s) - . - . . . . t/ T W ( m/s) - . - . . . . t/ T W ( m/s) - . - . . . . t/ T W ( m/s) - . - . . . . t/ T W ( m/s) - . - . . . . t/ T W ( m/s) - . - . . . a ¯ ⇢
Nonlinear hydrodynamics of polar active matter. a.
Average velocity ν ( r , t ) of the colloids and local magnitudeof the longitudinal current W ( r , t ) plotted as a function of the local density ρ ( r , t ). b-e. Successive phases observed in theresolution of Eqs. (1) and (2) at increasing densities ρ = (0 . , . , . , . , . x is normalized by the systemsize L . Simulation parameters: D ρ = 0 . D W = 1, λ = 1, a = 0 . σ = 0 .
2, ¯ ρ = 2, ξ = 0 . σ = 1, φ = 0 . L = 200, dx = 0 . dt = 0 . Vicsek transition: a disordered gas phase is separatedfrom a homogeneous polar liquid phase by a coexistenceregion where ordered bands propagate through a disor-dered background [15]. This phase transition occurs atvery low area fraction ( ρ ∼ φ (cid:28) ¯ ρ ), in a regime wherethe colloidal rollers experience no form of kinetic hin-drance as they interact, therefore (cid:15) i ( ρ ) (cid:39) σ i . In agree-ment with our experiments, a second transition leads tothe coexistence between a polar liquid of constant den-sity ρ b L and an apolar dense phase of constant density ρ b S .This jammed phase propagates backwards with respectto the flow of the polar liquid as does the active solidswe observe in our experiments. This second transitionshares all the signatures of the first-order phase separa-tion reported in Fig. 3. Figs. 5a, 5b and 5c indicate thatthe jammed phase obeys a lever rule, its width increaseslinearly with ρ − ρ b L , while the velocity c and the shapeof the fronts remains unchanged upon increasing ρ . Thefirst order nature of the transition is further supportedby Supplementary Movie 7 which shows the existence ofa hysteresis loop when ramping up and down the averagedensity. B. Spinodal instability of polar liquids and domainwall propagation
Having established the predictive power of our hydro-dynamic model, we now use it to gain physical insightinto the origin of active solidification. We focus on the ex-perimentally relevant situation where ¯ ρ (cid:29) φ g , i.e. wherethe slowing down of the particle occurs at area fractionsmuch larger than the onset of collective motion. Giventhis hierarchy, at low densities, when ρ (cid:28) ¯ ρ , the hy-drodynamic equations Eqs. (1)-(2) correspond to thatthoroughly studied in [33, 34, 36]. They correctly pre-dict a first order transition from an isotropic gas to apolar-liquid phase, see also Appendix C.The phase separation between a polar liquid and ajammed phase becomes also clear when performing a linear stability analysis of the homogeneous solutions ofEqs. (1)-(2), see Appendix C where the stability of thevarious phases is carefully discussed. At high density, thestability of polar liquids where ρ = ρ and W = W (cid:54) = 0 islimited by a phenomenon that is not captured by classicalflocking models. When ρ (cid:29) φ + (cid:15) / (2 λ(cid:15) + 4 a (cid:15) ), polarliquids are stable with respect to the spinodal decompo-sition into Vicsek bands, however another instability setsin whenever (cid:15) (cid:48) i ( ρ ) ρ + (cid:15) i ( ρ ) is sufficiently large and neg-ative, which occurs when ρ approaches ¯ ρ . This instabilityis responsible for the formation of active-solid jams. Welearn from the stability analysis that it ultimately relieson the decrease of the effective pressure with density inEq. (2) as a result of the slowing down of the colloidsin dense environments. This criterion is exactly analo-gous to the spinodal decomposition condition in MIPSphysics: the formation of active-solid jams results froma complete motility induced phase separation [20]. C. Discussion
Two comments are in order. Firstly, our results pro-vide a novel microscopic mechanism leading to MIPS.In classical systems such as Active Brownian Particles,repulsive interactions and persistent motion conspire toreduce the local current when active particles undergohead-on collisions [21, 23, 24]. Here we show that thismicroscopic dynamics is however not necessary to ob-serve phase separation and MIPS transitions solely relyon the reduction of the active particle current as densitybecomes sufficiently high, irrespective of its microscopicorigin. In the case of colloidal rollers, particle indeed donot experience any frontal collision when an active solidnucleate in a polar liquid, phase separation is howevermade possible by the slowing down of their rolling mo-tion.Secondly, another marked difference with the densephases of conventional MIPS system is the steady propa-gation of the active solids through the dilute polar liquid. x/L W L S / L -1.15-1.14-1.13-1.12 c D c . t/ T j( m/s) - . - . . . . t/ T W ( m/s) - . - . . . a b c d L S / L lever-ruleexperiments simulationslever-rule FIG. 5.
Shape and dynamics of active-solid jams. a.
Density and velocity profiles computed for different value of ρ : from 1.85 (light colors) to 2.25 (dark colors). Numerical resolution of Eqs. (1) and (2) with the hydrodynamic parameters: D ρ = 5, D W = 10, λ = 1, a = 1, σ = 1, ¯ ρ = 2, ξ = 0 . σ = 1, phi = 0 . L = 200, dx = 0 . dt = 0 . b. Length ofthe solid jam plotted versus ρ (symbols) and lever rule (solid line). Same parameters as in a . c. The speed of the solid jam c does not depend on the average density ρ . d. Variations of the propagation speed c as a function of the effective diffusivity D ρ . Numerical parameters: D W = 5, xi = 1, a = 0 . σ = 0 .
2, ¯ ρ = 2, ξ = 0 . σ = 1, phi = 0 . L = 200, dx = 0 . dt = 0 . ρ = 1 . This dynamics can be accounted for by our model. Thetwo boundaries of the active solid are two domain wallsthat propagate at the same speed. The propagation ofthe domain wall at the front of the solid jam relies on amechanisms akin to actual traffic jam propagation: thedirected motion of the particles incoming from the polarliquid cause an accumulation at the boundary with thearrested phase, in the direction opposing the spontaneousflow. By contrast, the propagation of the second do-main wall, at the back of the solid jam, requires arrestedparticles to resume their motion. The formation of thissmooth front originates from the mass diffusion termsin Eq. (1), which allows particles to escape the arrestedsolid phase and progressively resume their collective mo-tion when reaching a region of sufficiently low densityin the polar liquid. This diffusive spreading, however,does not rely on thermal diffusion. The roller difusivity D m ∼ − m / s is indeed negligible on the timescale ofthe experiments. Fortunately, other microscopic mech-anisms, and in particular anisotropic interactions, leadto diffusive contributions to the density current [37]. Asimple way to model this effect is to consider a velocity-density relation of the form ν ( ρ )[1 − r u · ∇ ρ ] where u isthe orientation of the particle and r , which could be den-sity dependent, quantifies the anisotropic slowing downof particles ascending density gradients. This anisotropicform is consistent with the polar symmetry of the flowand electric field induced by the Quincke rotation of thecolloids. Coarse-graining the dynamics of self-propelledparticles interacting via such a nonlocal quorum sensingrule was done in [37] and leads to an effective Fickian con-tribution ∼ − ρν r ∇ ρ to the density current in Eq. (1).The ratio between the magnitude of this effective Fickianflux and that of thermal diffusion is readily estimated as( ρν r ) /D m ∼ , assuming that r is of the order of the a colloid diameter. Anisotropic interactions are there-fore expected to strongly amplify the magnitude of D ρ .In order to confirm the prominent role of this diffusionterm in the active solid dynamics, we numerically mea-sure the propagation speed c of the jammed region as afunction of D ρ . In agreement with the above discussion c is found to vanish as D ρ →
0, Fig. 5d thereby confirmingthe requirement of a finite diffusivity to observe stableactive-solid jams. Simply put, the steady propagation ofactive solids relies on the balance between two distinctmacroscopic phenomena: motility reduction at high den-sity which results in the formation of sharp interfaceswith the polar liquid, and the diffusive smoothing of theinterfaces that enables particles trapped in the arrestedsolid phase to resume their motion by rejoining the polar-liquid flock.
V. CONCLUSION
In summary, combining expriments on Quincke rollersand active-matter theory, we have shown that the phasebehavior of polar active units is controled by a seriesof two dynamical transitions: a Flocking transition thattransforms active gases into spontaneously flowing liq-uids, and a Motility-Induced Phase Separation that re-sults in the freezing of these polar fluids and the for-mation of active solids. Although most of their con-stituents are immobile, active solid jams steadily prop-agate through the active liquid they coexist with dueto their continuous melting and freezing at their bound-aries. Remarkably, Quincke rollers provide a rare exam-ple of an unhindered MIPS dynamics that is not boundto form only finite-size clusters, see [38] and referencestherein. Beyond the specifics of active rollers, we haveshown that the freezing of flocking motion and the emer-gence of active solids is a generic feature that solely relieson polar ordering and speed reduction in dense environe-ments. A natural question to ask is wether suitably tai-lored polar and quorum-sensing interactions could yieldordered active solids. More generally understanding theinner structure and dynamic of active solids is an openchallenge to active matter phycisists.
ACKNOWLEDGMENTS
D. G. and D. M. have equal contributions. We thankA. Morin for invaluable help with the experiments.
Appendix A: Quincke rollers.1. Motorization
Our experiments are based on colloidal rollers [10]. Wemotorize inert polystyrene colloids of radius a = 2 . µ mby taking advantage of the so-called Quincke electro-rotation instability [26, 39]. Applying a DC electric fieldto an insulating body immersed in a conducting fluidresults in a surface-charge dipole P . Increasing the mag-nitude of the electric field E above the Quincke thresh-old E Q destabilizes the dipole orientation, which in turnmakes a finite angle with the electric field. A net elec-tric torque T E ∼ P × E builds up and competes with aviscous frictional torque T V ∼ η Ω where Ω is the colloidrotation rate and η is the fluid shear viscosity. In steadystate, the two torques balance and the colloids rotate atconstant angular velocity. As sketched in Fig. 6a and6b, when the colloids are let to sediment on a flat elec-trode, rotation is readily converted into translationnalmotion at constant speed v (in the direction opposite tothe charge dipole). We stress that the direction of mo-tion is randomly chosen and freely diffuses as a result ofthe spontaneous symmetry breaking of the surface-chargedistribution.
2. Arresting Quincke rotation
We conjecture a possible microscopic mechanism to ex-plain the arrest of the Quincke rotation at high area frac-tion: the frustration of rolling motion by lubrication in-teractions, Fig. 6c. The viscous torque T V acting on twonearby colloids rolling along the same direction is chieflyset by the lubricated flow in the contact region separatingthe two spheres. T V therefore increases logarithmicallywith d − a where d in the interparticle distance [40].As there exists an upper bound to the magnitude of theelectric torque T E , torque balance requires the rolling mo-tion to become vanishingly slow as d − a goes to zero:lubricated contacts frustrate collective motion. E ( V µ m − ) V ( mm s − ) E V a cb -2 -1 0 1 2-2-1012 -3 V // (mm s − ) V ⊥ ( mm s − ) E ( V µ m − ) V ( mm s − ) E V a cb -2 -1 0 1 2-2-1012 -3 V // (mm s − ) V ⊥ ( mm s − ) E ( V µ m − ) V ( mm s − ) E V a cb -2 -1 0 1 2-2-1012 -3 V // (mm s − ) V ⊥ ( mm s − ) E ( V µ m − ) V ( mm s − ) E V a cb -2 -1 0 1 2-2-1012 -3 V // (mm s − ) V ⊥ ( mm s − ) d
Quincke rollers. a.
When applying a DC electricfield E to an insulating sphere immersed in a conductingfluid, a charge dipole forms at the sphere surface. When E >E Q , the electric dipole makes a finite angle with the electricfield causing the steady rotation of the sphere at constantangular speed Ω. b. The rotation is converted into translationby letting the sphere to sediment on one electrode. Whenisolated, the resulting Quincke rotor rolls without sliding atconstant speed: ν (0) = a Ω. c. When two colloids rollingin the same direction are close to each other the lubricationtoque acting on the two spheres separated by a distance dscales as ln d and hinders their rolling motion. Appendix B: Experimental and Numerical Methods1. Experiments
The experimental setup is identical to that describedin [31]. We disperse polystyrene colloids of radius a =2 . µ m (Thermo Scientific G0500) in a solution of hex-adecane including 0 .
055 wt% of AOT salt. We inject thesolution in microfluidic chambers made of two electrodesspaced by a 110 µ m-thick scotch tape. The electrodesare glass slides, coated with indium tin oxide (Solems,ITOSOL30, thickness: 80 nm). We apply a DC electricfield between the two electrodes ranging from 1 . /µ mto 2 . /µ m using a voltage amplifier. If not specifiedotherwise, the data correspond to experiments performedat 1 . /µ m, i.e. at E /E Q = 2. We confine the rollersinside racetrack by coating the bottom electrode with aninsulating pattern preventing the electric current to flowoutside of the racetrack. To do so, we apply 2 µ m-thicklayer of insulating photoresist resin (Microposit S1818)and pattern it by means of conventional UV-Lithographyas explained in [31].In order to keep track of the individual-colloid posi-tion and velocity, we image the system with a NikonAZ100 microscope with a 4.8X magnification and recordvideos with a CMOS camera (Basler Ace) at framerateup to 900 Hz. We use conventional techniques to detectand track all particles [41–43]. When performing large-scale observations we use a different setup composed ofa 60 mm macro lens (Nikkor f/2.8G, Nikon) mounted ona 8 Mpxls, 14bit CCD camera (Prosilica GX3300).
2. Numerics
The numerical resolution of the Toner-Tu equationsEqs. (1) and (2) were done using a semi-spectral methodwith semi-implicit Euler scheme.
Appendix C: Linear Stability of the generalizedToner-Tu equations
In this appendix we show how the succession of insta-bilities of the homogeneous solutions of Eqs. (1) and (2)correctly predict the full phase behavior observed in ourexperiments and numerical simulations.
1. Stability of the disordered gas
We start by considering a homogneous gas phase where ρ = ρ , W = 0. The linearized dynamics of a small perturbation δX ≡ ( δρ, δW ) is given, in Fourier space,by δ ˙ X k = M k δX k , where the dynamical matrix M k isgiven by M k = (cid:18) − D ρ k − ik − ( (cid:15) + (cid:15) (cid:48) ρ ) ik ( ρ (cid:15) − φ ) − D W k (cid:19) (C1)The eigenvalues λ ± of M k determines the stability of thegas phase. We find: λ ± = − ( D ρ k + D W k − ρ (cid:15) + φ ) ± (cid:112) ( D ρ k + D W k − ρ (cid:15) + φ ) − k ( (cid:15) + (cid:15) (cid:48) ) − D W k − ρ(cid:15) + φ ) D ρ k ρ (cid:15) − φ > (cid:15) + ρ (cid:15) (cid:48) ) < − ( D W k − ρ(cid:15) + φ ) D ρ (C4)The first condition is the standard spinodal instabilityleading to the emergence of collective motion and localorientational order [1, 36]. Beyond this classical results,we find a second instability akin to the MIPS criterion inthe presence of translationnal diffusion [20]. This secondcondition arise from the decrease of the pressure termin Eq. (2) when ρ increases. This condition is not metin our experiments where we find that the emergence ofpolar order occurs before the onset of solidification.
2. Stability of polar liquids
Let us now consider a polar liquid where ρ = ρ , and W = W with a W = ( ρ (cid:15) − φ ). Following the sameprocedure as in the previous section, the linearized dy-namics of a small perturbation δX = ( δρ, δW ) is definedby the dynamical matrix: M k = (cid:18) M ρρ M ρW M W ρ M W W (cid:19) (C5)where M ρρ = − D ρ k , (C6) M ρW = − ik, (C7) M W ρ = − ( (cid:15) + (cid:15) (cid:48) ρ ) ik + ( (cid:15) + (cid:15) (cid:48) ρ ) W (C8) M W W = ( ρ (cid:15) − φ ) − D W k − λikW − a W (C9) Looking for the condition under which an eigenvalueadmits a positive real part, so that the homogeneous so-lution becomes unstable, leads to: − α k − α k − α k − α > α = D W D ρ ( D ρ + D W ) /W (C11) α = λ D ρ D W + 2 D ρ D W ( D ρ + D W )2 a W + ( D ρ + D W ) (cid:2) D ρ a + ( (cid:15) + (cid:15) (cid:48) ρ ) /W (cid:3) (C12) α = [ D ρ a + [( D ρ + D W ) ( ρ (cid:15) (cid:48) + (cid:15) )]+ 2 ( D ρ + D W ) 2 a (cid:0) D ρ a W + ( ρ (cid:15) (cid:48) + (cid:15) ) (cid:1) + D ρ D W (2 a W ) − λD ρ ( ρ (cid:15) (cid:48) + (cid:15) ) (C13) α = (2 a ) (cid:2) D ρ a W + ( ρ (cid:15) (cid:48) + (cid:15) ) (cid:3) + ( ρ (cid:15) (cid:48) + (cid:15) ) 2 a W − ( ρ (cid:15) (cid:48) + W (cid:15) ) (C14)The analysis of the polynomial equation Eq.(C10) leadsto a cumbersome instability criterion, that however sim-plifies in the context of our experiments where φ (cid:28) ¯ ρ .At low density, ρ (cid:28) ¯ ρ , so that ρ (cid:15) (cid:48) + (cid:15) = σ and ρ (cid:15) (cid:48) + (cid:15) = σ , and only the constant term α can bepositive, which yields a simplified instability criterion: σ / ( σ ρ − φ ) > a D ρ ( σ ρ − φ ) + 4 a σ + 2 λσ (C15)In the limit of small D ρ , relevant for our experiments [31],one recovers the spinodal instability criterion of a homo-geneous polar liquid [34] φ + σ λσ + 4 a σ (cid:38) σ ρ (C16)For larger densities, deep in the polar-liquid phasewhen ρ ∼ ¯ ρ , the instability criterion (C16) cannot besatisfied. However, (cid:15) (cid:48) i ( ρ ) is not negligible anymore, anda novel instability dictates the dynamics. All the α , α and α terms can be negative when (cid:15) (cid:48) i ρ + (cid:15) i is sufficientlylarge and negative (provided that D W > D ρ , which holdsin our experimental conditions [31]). Again, one recoversa standard MIPS instability, which occurs in the vicinityof ¯ ρ provided that the decrease of (cid:15) i ( ρ ) is sharp enough.We note that α could also change and become negativefor (cid:15) (cid:48) i ρ + (cid:15) i sufficiently large and positive. This situa-tion is however not relevant for our experimental systembut hints towards a possible additional instability of theordered polar liquid, which will be discussed elsewhere. Appendix D: Description of the Movies a. Movie 1.
Movie 1 shows a polar liquid flowingalong a microfluidic racetrack. Dimensions of the obser-vation window: 1 . mm × . mm . The movie is sloweddown by a factor 3.8. b. Movie 2. Movie 2 Close-up in the microfluidicracetrack. We first see the polar liquid phase and thenthe compact active solid forming at one end and meltingat the other end. Dimensions of the observation window:1 . mm × . mm . wide. The movie is slowed down by afactor 3.8. c. Movie 3. Movie 3 shows a typical experimentin a racetrack where an active solid steadily propagatethrough the homogenous polar liquid it coexists with.The movie is sped up by a factor 30. d. Movie 4.
Movie 4 shows the coarsening dynamicsof multiple active solids. The movie is sped up by a factor30. e. Movie 5.
Hysteresis dynamics upon cycling themagnitude of the electric-field. This movie correspond tothe experiments of Figs. 3e and 3f in the main document.The movie is sped up by a factor 30. f. Movie 6.
Response to an electric-field quench Thebulk of the active solid is destabilized at all scales. Thisphenomenon is reminiscent to a spinodal decompositionscenario. The movie is sped up by a factor 30. g. Movie 7.
Numerical simulations of Eqs (1-2) withthe same parameters as in Fig 5a-c, but with dx = 0 . dt = 0 .
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