Downloaded by guest on October 1, 2021 oeulbimcletv yaiso neatn cierotors. active interacting of dynamics collective nonequilibrium the to critical are behavior. pat- particles collective active of vortex and the emergence and into interactions rollers boundary hydrodynamic the self-organize the colloidal between situations, walls these and In side (27–29). (26), terns by crystals confined living bacteria substrate clus- the rotating on into surfers For form colloidal collec- aggregate absent. (23–25), microorganisms substrate to otherwise the and rise near is ters cells which gives particles, sperm also active but instance, not of cases, system motion equilibrium active tive surface in an introduces as in and relevant escaping tension, wall A from becomes particles confinement systems. the systems prevents the active only that passive in is important in example and ingredients subtle trivial extremely is originally particles by active of behavior (16–22). collective intriguing especially emergent among the active phenomena, nonequilibrium which The exotic (10–15). exhibits rotation often motility: matter or particle indi- (7–9) into energy particles translation ambient self-propelled constituent or the stored convert materials systems, vidually smart active-matter and In microdevices (1–6). in applications potential and A confinement account. into taken properly be should and behavior matter collective active on of influences significant have and compressibility structure, of dynamic occurrence inhomogeneity, the confinement-induced that show the or identi- thus regions results well Our solid-like rearrangement. are particle of defect-induced origins modes percolation different structural the the The by fied between varies. transitions fraction the packing of the differ- as three has modes behavior ent collective a the drive properties, active oscillation then on the chiral depending Moreover, can boundary. the confinement which of the along motion particles, matter collective the oscillating the of spatially couples translation striking and fric- the friction interrotor position-dependent to from spin a results to that stress leads that tional find inhomogeneity We density theory. and the self-spinning experiment, of composed simulation, matter through this active rotors chiral investigate minimal we effect a Here, in whose unclear. effect remains correlations, behavior density collective particle-position inhomogeneous on to an due induces also distribution confinement single-particle How- the the behavior. at level, collective couplings play of external introducing emergence boundary besides the ever, and in particles role important active or an hydrodynamic between that interactions indicated in steric have matter studies active Extensive unconfined in 2019) systems. nature, absent 23, behavior December out-of-equilibrium collective review exhibit for may inherent (received confinement 2020 its 15, April to approved and Due MA, Cambridge, University, Harvard Weitz, A. David by Edited China 523808, Guangdong Dongguan, China; Laboratory, 621000, China; Mianyang 325001, College, Zhejiang Teachers’ Wenzhou, Mianyang Physics, and Mathematics of China; 100049, Beijing Sciences, Lu Ying atrPyis nttt fPyis hns cdm fSine,Biig109,China; 100190, Beijing Sciences, of Academy Chinese Physics, of Institute Physics, Matter a Liu Peng confinement rotors in active of motion collective Oscillating www.pnas.org/cgi/doi/10.1073/pnas.1922633117 ejn ainlLbrtr o odne atrPyis nttt fPyis hns cdm fSine,Biig109,China; 100190, Beijing Sciences, of Academy Chinese Physics, of Institute Physics, Matter Condensed for Laboratory National Beijing nti ril,w td o ofieetmyiflec the influence may confinement how study we article, this In played role the particles, active of motility the to Thanks w eae u oisitiscnnqiiru nature nonequilibrium intrinsic its to past the due in decades interest considerable two attracted has matter ctive a,b,c,h a,b,c,1 | olciebehavior collective igZheng Ning , oge Zhu Hongwei , d d,2 colo hsc,BiigIsiueo ehooy ejn 001 China 100081, Beijing Technology, of Institute Beijing Physics, of School | aguYe Fangfu , g d,1 hrlatv matter active chiral olg fPyisadEetois hnogNra nvriy ia 504 hn;and China; 250014, Jinan University, Normal Shandong Electronics, and Physics of College igZeng Ying , a,b,c,f,h,2 e,1 | rotor n ighn Yang Mingcheng and , unl Du Guangle , a,b,c,f doi:10.1073/pnas.1922633117/-/DCSupplemental at online information supporting contains article This 2 1 oeet a iers oasailyoclaigeg flow, antisymmetric edge density-dependent continuum oscillating a a spatially through with clarified a theory inho- is hydrodynamic density to mechanism underlying the rise its that and give show can We a the matter. mogeneity in active of behavior chiral collective effects emergent confined the the on we investigate inhomogeneity Here, density experimentally behavior. collective and and on numerically systems effects many-body inhomogene- unexpected interacting have density confined could The in ubiquitous 36). is distribution (35, ity nonuniform particles, density spatially the particle-number a with of cause interactions also steric can momentum. the confinement angular from “orbital” apart conversion into However, the momentum inter- allows the spin-angular that particle–boundary condition of to topologically steric boundary even the a parallel be case, impose to actions flow this proven In rotors edge is (34). interacting which protected collective of 33), fluid 32, a (12, a interest boundary yields that reported theoretical confinement been fundamental in has of It par- 31). subject have both (30, systems the breaks active-rotor been that the recently matter symmetries, time-reversal active and chiral Nevertheless, ity explored. representative less a com- much remain as systems rotors counterparts, active of translational posed their with Compared is ulse a 9 2020. 19, May published First the under Published Submission.y Direct PNAS a is article This interest.y competing no declare paper. authors the The wrote M.Y. M.Y. F.Y., and and F.Y., N.Z., N.Z., Y.L., P.L., K.C., and D.W., data; L.N., G.D., analyzed Y.Z., F.Y., H.Z., N.Z., P.L., G.D., research; Y.Z., performed H.Z., M.Y. P.L., and research; designed M.Y. F.Y., and N.Z., contributions: Author owo orsodnemyb drse.Eal [email protected] ningzheng@bit. [email protected], Email: addressed. [email protected] be or edu.cn, may correspondence whom work.y To this to equally contributed Y.Z. and H.Z., P.L., uu Ning Luhui , nesadn fcletv eairo hrlatv atrin matter active chiral collec- confinement. our of different advance behavior collective significantly varies, of findings media understanding These active emerge. the modes Moreover, of tive pattern. structure the oscillatory the spatially influences as a greatly causing flow, spinners edge of distribution structureless confinement-induced nonuniform and that been show incompressible we robust generally Here, of fluid. for homogeneous has context the feature collec- potentials in unique with unidirectional discussed This confinement, spontaneous transport. in a material flow of A edge matter area. emergence tive active developing the chiral rapidly of is and phenomenon self- interesting exciting of particularly time-reversal an composed and is parity matter both symmetries, breaks active which chiral objects, spinning of exploration The Significance f ezo nttt,Uiest fCieeAaeyo Sciences, of Academy Chinese of University Institute, Wenzhou a,b,c,2 c colo hsclSine,Uiest fCieeAaeyof Academy Chinese of University Sciences, Physical of School a,b,c PNAS NSlicense.y PNAS uyuWang Dunyou , | e eerhCne fCmuainlPyis School Physics, Computational of Center Research ue2 2020 2, June | g . y o.117 vol. eChen Ke , https://www.pnas.org/lookup/suppl/ h oghnLk Materials Lake Songshan | y b o 22 no. aoaoyo Soft of Laboratory a,b,c,h | , 11901–11907

APPLIED PHYSICAL SCIENCES frictional stress. Furthermore, we identify three different collec- boundary wall can induce a spatially oscillating particle distribu- tive modes of motion and their respective structural origins. tion in passive fluids to minimize the system free energy (35, 36). The number density distribution, n(r), of the active rotors is also Results plotted in Fig. 2B and exhibits a behavior similar to that of the The simulation system consists of N spinning disks of diame- passive system (SI Appendix), implying that the structural prop- ter σs confined in a circular boundary of radius R, as shown erties of the chiral active system are insensitive to the spin. Based in Fig. 1A. Each constituent disk spins counterclockwise, driven on the spatial inhomogeneity and the active spinning, the driving by a constant torque, and different rotors interact via a repul- force for the edge flow can be easily identified. As illustrated in sive potential with a surface friction. To focus our study on Fig. 2A, the particles in the outermost layer, i.e., rotor 1, experi- confinement-induced inhomogeneity, we only allow a radial ence a tangential force from rotor 2, F21, generated due to the repulsive interaction between the wall and the particles. The friction between the spinning particles. Because the confinement particle dynamics is described by the underdamped Langevin wall is smooth and applies no tangential force on rotor 1, F21 equation. In experiments, a circular vessel containing a mono- drives rotor 1 to move counterclockwise. Nevertheless, for the layer of gear-like granular rotors (Fig. 1B) is mounted on an particles in other layers, e.g., rotor 2, the outer-layer rotor 1 and electromagnetic shaker, and the rotors’ spinning is driven by the inner-layer rotor 3 contribute opposite frictional forces F12 vertical vibration. To compare the simulations with the experi- and F32 on it. If the number density of the inner layer is higher ments, we use a dimensionless number, ωs /Dr , to characterize than that of the outer layer, the inner layer will, on average, apply the motility of an isolated rotor for both systems. Here, ωs and a larger tangential friction, and, hence, rotor 2 will move coun- Dr refer to the spin velocity and the rotational diffusion coeffi- terclockwise; otherwise, it will move clockwise. Therefore, the cient of the isolated rotor, respectively. The experimental ωs /Dr oscillating number density distribution, which has an equilibrium is measured as 6.2 (SI Appendix), and we choose ωs /Dr = 6.0 in structural origin, can give rise to a position-dependent (spatially simulations. oscillatory) frictional stress, which then drives an oscillating edge flow in space. Far away from the boundary, the system density as Oscillating Collective Edge Flow. We first consider a fluid system well as the frictional stress tends to be homogeneous, and, hence, with the packing fraction ρ = 0.6 in simulation (Movie S1). Fol- the macroscopic flow vanishes. Such a scenario is formulated via lowing previous work (12, 32), we measure the steady-state mean a continuum hydrodynamic theory in Theoretical Description for tangential velocity of the particles in different concentric annuli, the Oscillating Edge Flow. vt (parallel to the boundary), to quantify the collective motion. The corresponding experimental results are given in Fig. 2C, Note that the mean radial velocity normal to the wall, vr , van- which plots the orbital angular velocity of the granular spinners ishes due to the confinement. Fig. 2B displays the orbital angular (Fig. 1B) with a low packing fraction ρ = 0.65 (the ratio of the velocity of the rotor fluid, vt (r)/r, as a function of the distance area occupied by the particles to that of the vessel). The results from the system center, r. Throughout the paper, the orbital also show a spatially oscillating edge flow (Movie S2), with the angular velocity is normalized by the spin velocity of the isolated period being around the spinner diameter. Although the gran- rotor, ωs . Indeed, there exists an edge flow near the bound- ular spinners are macroscopic and dissipative, confinement can ary. Interestingly, the vt (r)/r varies nonmonotonously with the still lead to a spatially inhomogeneous density distribution sim- distance and exhibits a significant oscillation in space and even ilar to the simulation. Consequently, the essential requirements changes the sign. The oscillation period of the collective motion for the emergence of oscillating collective motion (i.e., nonuni- is equal to the rotor diameter. The magnitude of the oscilla- formity, spin, and interparticle friction) are properly satisfied. tion decays substantially as r decreases and vanishes far away The experimental results thus provide strong support for our from the boundary, indicating that the edge flow is localized near theoretical predictions. the boundary. Previous studies on the confined spinners also Nevertheless, two apparent distinctions exist between the sim- reveal the emergence of the edge flow, but with no oscillation ulation and the experiment. One is that vt (r)/r of the simulation (12, 32, 33). oscillates around zero (Fig. 2B), while in the experiment, vt (r)/r To understand the microscopic mechanism of the spatially oscillates around a reference value that decays substantially with oscillating edge flow, we note that the existence of the con- decreasing r (Fig. 2C). The other is that the oscillation mag- finement breaks the spatial uniformity of the system. Thus, the nitude of the simulation vt (r)/r is stronger than that of the environment felt by the particles close to the boundary is consid- experimental vt (r)/r. We speculate that these discrepancies erably different from that far from the boundary. Particularly, the come from the following facts. In the experiment, the shaken

Fig. 1. (A) Simulation snapshot of 1,000 spinning disks in confinement, with the packing fraction ρ = 0.6, where Inset shows a zoomed-in image. (B) Experimental snapshot of the gear-like spinners in a circular vessel with ρ = 0.65. Lower Inset is the sketch (side view) of a 3D-printed active rotor, and Upper Inset is the top view of the rotor with D1 = 15.50 ± 0.06 mm and D2 = 21.26 ± 0.06 mm.

11902 | www.pnas.org/cgi/doi/10.1073/pnas.1922633117 Liu et al. Downloaded by guest on October 1, 2021 Downloaded by guest on October 1, 2021 lhuhtecia ciefli ne td scompressible. is study under fluid active chiral the although imply symmetry i tal. et Liu state, steady the In vectors. with system, coordinate polar a In spinners in I of flow density density edge mass tum oscillatory the the are of variables emergence of the which terms rotors, for the fluid. essential of active distribution is stress inhomogeneous chiral frictional the position-dependent by the induced a continuum describe consider oscillating (2D) we to two-dimensional spatially Importantly, (12) a the theory use of hydrodynamic we Flow. mechanism motion, Edge underlying collective Oscillating the the clarify for Description Theoretical good in 2C is Fig. speculation. which in our results 2D, experimental Fig. the in with displayed agreement are model modified results simulation the The of lengths). coupling ratio multiple size the the mimicking (with sizes different transla- of disks the spinning decrease first coefficient, we frictional simulation, tional the in aspects account these To lengths, distances. for summit–cleft characteristic and transla- summit–summit multiple the their e.g., possess that the interactions Besides, so weakened. greatly gear–gear baseplate is substrate the the with friction from tional separate values often peak the gears to fit exponential an are lines dashed the and distribution, density simulation number center: the system to velocity. the refer angular to axis orbital distance vertical the the right of of the function and a lines as magenta density) the number velocity, particle mean the by with (normalized results distribution density number particle the and 2. Fig. n nua oetm h ascniut qainreads equation mass-continuity The momentum, momentum. mass, angular of and conservations the obey respectively, which, (r, h osraino oetmtksteform, the takes momentum of conservation The t )ω r ˆ h ria nua velocity angular orbital The (B–D) system. rotor active confined the in motion collective the of mechanism microscopic the for diagram Schematic (A) (r, and h hydrodynamic The Flow. Edge Collective Oscillating γ t with ), = φ ˆ %(r, C ( results experimental (B), 100 eaaeybigterda n agnilunit tangential and radial the being separately v t %(∂ ∇ · )v(r, I (r, (∂ % t t + 0 = t t + ) n h nua oetmdensity momentum angular the and ), v h pne oeto nri density, inertia of moment spinner the v ∇ · uhta Eq. that such , ∇ · · ∇ γ v )v ∇ n hnueabnr itr of mixture binary a use then and , r + v i 0 = (r) ˆ = = 0, = · ∇ r ∂ ∂ j v)% r σ + ij oehrwt h system the with together 0 = − φ ˆ ,adsmlto eut with results simulation and ), 1 1 r Γv becomes ∂ . %(r, φ i , and t h momen- the ), ' v n supports and ˆ = roughly 1.3, obetter To rv r + φv [1] [3] [2] ˆ t γ ˆ eae otefitoa ofceto igesinrby spinner single of 4γρ/πσ coefficient frictional the to related with becomes hrlatv udi o tecnrptlacceleration centripetal (the low is fluid active chiral oscillatory spatially a η in results fraction particle-packing local the prefactor the with results A simulation independent fits well which have we Thus, contact. at function Aρ correlation pair the z where g as approx- be can colli- estimated frequency the collision imately the particles. and theory, surrounding Enskog its fraction with the spinner packing From tagged spinner a of the frequency sion to proportional is inho- the on Since focus rotational here and We mogeneous fraction. shear confinement-induced packing of the The inhomogeneity is via stress. pressure position-dependent term the tangential are viscosity in viscosities the absorbed bulk to be stress the can contribute frictional flow; it as antisymmetric the Eq. ignored, the to to owing to spin Eq. absent the refers In couples term friction. that last interrotor from the arises 4, that viscosity rotational = R (σ · ' ,weetebu ie n h etvria xsrfrt h angular the to refer axis vertical left the and lines blue the where (D), 2 steeg o swa n h enlsnme fthe of number Reynolds the and weak is flow edge the As × (∇ 2 Fg 3, (Fig. s g 2.07 = ) (σ Γ η η R s s 2 = v) h rcinlcefiin rmevrnet hc is which environment, from coefficient frictional the 1−7ρ/16 ) n h testensor, stress the And . σ steserviscosity, shear the is .Eq. Appendix). (SI (1−ρ) with eut rmteitratcefitoa olsos it collisions, frictional interparticle the from results ij n,hne rcinlstress. frictional hence, and, Inset) η = η  R R ij 2 −p A = (ρ) ∂ hti h rgno h siltr deflow. edge oscillatory the of origin the is that i − nukonpeatr iilepninyields expansion virial A prefactor. unknown an v δ j ij 8(1−ρ) PNAS ρ h otct fteflwfil,and field, flow the of vorticity the Aρ + 3 n h d icst em(0 1 is 31) (30, term viscosity odd the and 2, /16 η 8ρg 2 (∂ 4  | i 1 3) uhta h oainlviscosity rotational the that such (38), (σ v 5 ue2 2020 2, June (1 − j en htteoclaoypol of profile oscillatory the that means s + − ) 7ρ/16 p  ∂ σ ij ρ j k ij ) v 2 B h eiCvt symbol, Levi–Civita the sepesdas expressed is , i T + ) − /π |  8(1 m ij o.117 vol. ρ η σ 3 R − s 2 /16 (2ω ρ) 3) with (37), p 4 | −  n osnot does and o 22 no. , Ω), = Γ η %v | R g γ η 11903 (σ ∇ · = Ω n R the [5] [4] s = = v )

APPLIED PHYSICAL SCIENCES from the rapid decay of the edge flow, and the frictionless boundary condition at the confined sidewall, σφr (R − σs ) =  vt
vt
 ηR ∂r vt + r − 2ω + η ∂r vt − r r=R−σs = 0. Fig. 3 plots the theoretical orbital angular velocity that reproduces all of the features of the oscillatory edge flow obtained from the simu- lation. Nevertheless, the theoretical calculation underestimates the oscillation magnitude by a factor of around two, which may be attributed to the approximations employed in the theoretical derivation. In addition, there is apparently a phase shift between the theoretical calculation and the simulation, which originates from the following fact. The continuum theory neglects the finite size of the rotor, so that the fluid flow is driven by the local gradi- ent of the frictional stress, while in the simulations, the fluid flow is produced by the variation of the frictional stress on the length scale of the rotor size. Except for the quantitative differences, the theoretical calculation compares well to the simulation measure- ment, confirming the microscopic mechanism of the oscillatory edge flow proposed in Oscillating Collective Edge Flow. With the obtained vt , the spin angular velocity is determined via Eq. 9, which also agrees well with that measured in the simulation (SI Appendix). Further verification of the continuum theory is Fig. 3. Comparison of the orbital angular velocities obtained from the sim- given in SI Appendix by exploring wider parameter spaces. ulation (blue square) and the continuum theory with the nonuniform ηR (red solid line). The system is the same as that of Fig. 2B. In the theoretical Collective Motion Modes of Higher Densities. The nonuniform par- calculation, the position-dependent ηR (Inset) is introduced by substituting the packing fraction profile from the simulation measurement into Eq. 5. For ticle distribution has been shown to be critical for the emergence comparison, we also plot the theoretically calculated vt /r for an incompress- of the oscillating edge flow, which highlights the importance of ible fluid of rotors (i.e., constant ηR; black dashed line), and the calculation the compressibility of the chiral active system. To further study details are provided in SI Appendix. the effect of the compressibility on the collective motion, we consider the simulation systems with a wide range of packing fraction from ρ = 0.50 to 0.82. Interestingly, we find three dif- is negligible), from Eqs. 3 and 4, the steady-state equation of ferent modes of collective motion as ρ increases, as shown in momentum conservation thus is Fig. 4A. For low ρ = 0.6, the collective flow oscillates around zero, and the magnitude decays to zero as r decreases. The 2 0 = −∂i p + η∇ vi + ij ∂j [ηR(2ω − Ω)] − Γvi . [6] decaying vt (r)/r is reminiscent of the circular Couette flow of a viscous fluid confined in two concentric cylinders (39), in which Here, for simplicity, η is regarded as a constant and equals to its the outer cylinder rotates at constant angular velocity, while the value in the bulk, η ' 25, which can be determined from inde- inner cylinder remains fixed. This implies that the chiral active pendent simulations by externally imposing a shear flow in an system of low ρ is in the fluid regime. For high ρ = 0.8, the unconfined active fluid (SI Appendix). system rotates as a rigid body at a constant angular velocity with- The angular momentum conservation is written as out any periodic oscillation (only with fluctuation). In this case, the system is an elastic solid. At moderate ρ = 0.7, the vt (r)/r 2 I (∂t + v · ∇)ω = −Γr ω − 2ηR(2ω − Ω) + Dω∇ ω + τ, [7] oscillates around a constant nonzero value (without decay), indi- cating that different domains of the active system can slide over where Γr refers to the rotational friction coefficient from envi- each other and, at the same time, are constrained in an elastic ronment, Dω to the diffusion coefficient, and τ to the torque background. density field. Γr and τ are, respectively, related to their single- For comparison, we also plot the orbital angular velocity of the 2 particle counterparts, γr and Td , by Γr = 4γr ρ/πσs and τ = experimental spinners at low (ρ = 0.65) and moderate (ρ = 0.78) 2 4Td ρ/πσs . In the steady state, v · ∇ω = 0, and thus Eq. 7 packing fractions, as shown in Fig. 4B. The experimental results reduces to are consistent with the simulation in Fig. 4A. Here, an experi-

2 mental system of high ρ like a rigid body cannot be achieved due 0 = −Γr ω − 2ηR(2ω − Ω) + Dω∇ ω + τ. [8] to the gear-like structure of the particles, so the corresponding results are not provided. In the present packing fraction (ρ = 0.6), the diffusion coefficient Dω is small such that the diffusion term in Eq. 8 can be safely Structural Origins of the Transitions between Different Modes. To neglected. The spin angular velocity thus approximately reads elucidate the structural origin of different modes of the col- lective motion, we measure the orientational order parameter τ + 2ηRΩ P 6iθjk ω = . [9] ψ6j = ( e )/Nj in the simulations (Fig. 5A), which Γr + 4ηR k∈Nj characterizes local crystalline order (40, 41). Here, the sum is Inserting Eq. 9 into Eq. 6, we have for the tangential component taken over the Nj nearest neighbors of particle j , and θjk is the of the momentum equation in polar coordinates, angle between rk − rj and a fixed arbitrary axis. At ρ = 0.60, the mean ψ6j is small, and the system is in a viscous fluid ∂ (βr) (β − r∂ β + r 2Γ) regime. In this case, the particles can easily change their posi- β∂2v + r ∂ v − r v − 2∂ β0 = 0, [10] r t r r t r 2 t r tions and cannot sustain a rigid rotation (Movie S1). At ρ = 0.80, most areas (except for the boundary region) have quite large −1 0 −1 where β = η + Γr ηR(Γr + 4ηR) and β = βτ(Γr + 4ηR) are ψ6j , and the inner-layer particles form a defect-free crystal. defined. Thus, the edge flow can be obtained by solving Eq. Thus, the active particles cannot change their relative positions, 10, with the boundary condition vt (r = 0) = 0, which arises only allowing a whole rotation (Movie S3). Nevertheless, for

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APPLIED PHYSICAL SCIENCES Transitions between various modes must arise from the struc- with m = 1 being the particle mass, γ = 100 the translational friction coeffi- tural changes of the chiral active matter. According to the cient, and Fr and Fw the interparticle and particle-wall steric forces, respec- previous discussions, the structural changes correspond to the tively. Here, the stochastic force η is Gaussian distributed with hη(t)i = 0 0 0 percolation of the solid-like regions or the formation of a defect- and η(t)η(t ) = 2kBTγδ(t − t ), with the temperature kBT = . Similarly, free crystal. Fig. 5D plots the probability of the percolation as a the spin-angular velocity of the particles evolves according to ρ function of . Here, the percolation is thought to occur once the I ω˙ = T + ξ − γ ω, [12] x or y dimension of the connected solid-like region [i.e., region s d r with ψ6j ≥ 0.84 (43)] is greater than a threshold size 2TS R, with 1 2 1 2 where Is = 8 mσs refers to the rotor momenta of inertia, γr = 3 σs γ the TS an imposed number; and the percolation probability is esti- rotational friction coefficient, Td = 6 the driving torque, and ξ the Gaussian 0 mated as the average percolation frequency per frame. Clearly, distributed stochastic torque with zero mean and ξ(t)ξ(t ) = 2kBTγr δ(t − the percolation probability becomes nonzero at ρ = 0.64, which t0). In addition, the bounce-back collision (44) that generates the friction agrees quantitatively with the first transition point of the motion between two rotors in contact (say, i and j) can be realized by instanta- neously updating v and ω, according to δv = δp /m and δω = − 1 r × mode given in Fig. 5B. (Note that the percolation here does not i i i 2 ij necessarily indicate a liquid-to-hexatic phase transition, given δpi/I. Here, the impulse δpi is determined by conservation laws, that our system is too small to distinguish whether there exists  κ  δp = −m ˜vk + ˜v⊥ , [13] a long-range correlation.) Fig. 5E plots the mean number of i ij 1 + κ ij defects, ND , per frame in the bulk as a function of ρ. Here, a defect is defined as a region with ψ6j less than a prescribed value 2 ˜k ˜⊥ with the parameter κ = 4Is/mσs , and vij and vij the components of the TD . The results show that the defects occur only when ρ ≤ 0.8, 1 relative velocity at collision point, (vi − vj) − 2 (ωi + ωj) × (ri − rj), parallel which is perfectly consistent with the second transition point of and perpendicular to ri − rj, respectively. The velocity Verlet algorithm is the motion mode, as determined in Fig. 5C. Thus, we clarify used to integrate the equations of motion with the time step ∆t = 10−3 × p 2 the microscopic structural origins of the transitions between the mσs /. different modes of collective motion. The rotors are initially randomly distributed; 105 steps are performed to eliminate the effects of the initial configuration, and 6.4 × 109 steps are per- Discussion formed to compute the physical quantities. To measure the orbital angular The emergent collective motion of the confined active rotors velocity vt /r and the number density distribution n of the rotors, we divide the system into concentric annuli, with the width ∆r = 1 σ . has been studied numerically, theoretically, and experimen- 4 s tally. Remarkably, this minimal chiral active matter exhibits Experiment. Gear-like rotors are put in a circular vessel mounted on an elec- rich collective behavior, resulting from an inhomogeneous den- tromagnetic shaker. An acrylic cover placed on the top of the vessel can sity distribution induced by confinement boundary. In partic- suppress vertical motion of the particles, and thus ensure that the parti- ular, the collective motion has a significant spatial oscillation cles move horizontally on the 2D baseplate. The shaker provides a vertical and experiences three different modes as the packing fraction vibration Z = Asin(2πft), with f the vibration frequency and A the vibration changes. The microscopic mechanisms underlying the collec- amplitude. The vibration strength is characterized by Γ = A(2πf)2/g, with g tive behaviors have been elucidated: The position-dependent the gravitational acceleration. Experiments are performed with f = 50 Hz frictional stress drives an oscillating edge flow; the percolation and Γ = 1.7. A high-resolution camera system is used to track the particle of solid-like regions induces the penetration of the oscilla- trajectories. The granular rotors used in our experiment are similar to those in tion into the bulk; and the vanishing of defects disables the previous work (45–48). The particles rest on circularly aligned tilted legs rearrangement of particle positions, resulting in a rigid rota- (Fig. 1B), which are manufactured from polylactide by using a three- tion of the bulk. Our findings highlight the importance of the dimensional (3D) printer. Owing to geometric asymmetry, the tilted legs compressibility and the confinement-induced inhomogeneity in act as elastic springs and transfer vibrational energy from the shaker the oscillatory collective motion of chiral active system and into a unidirectional rotation of the rotor without active translation. The also the influence of nonequilibrium structure on dynamics of distribution of translational displacement of a single rotor is symmet- active matter. rical with respect to the origin, suggesting that the rotor indeed per- forms an unbiased random walk (SI Appendix). The interactions between Methods rotors are short-ranged repulsive. The protruding teeth of the gear-like rotors can significantly enhance the interparticle friction. Nevertheless, disc- Simulation. Different rotors interact via a repulsive Lennard–Jones (LJ) shaped rotors made of a material with a large friction coefficient are h σs
2l σs
li type of potential, U(r) = 4 r − r + , with r being the distance expected to exhibit similar collective behavior. The effect of interrotor between rotor centers. Here, we set the disk diameter σs = 2, the interaction friction on the collective motion is studied by simulation and theory (SI intensity  = 1, and the potential stiffness l = 12. Besides the radial poten- Appendix). tial interaction, different disks also couple tangentially through a surface friction realized by the bounce-back collision (44). The interaction between Data Availability. All data discussed in the paper are available in the main the boundary wall and the rotors is chosen as the repulsive LJ potential with text and SI Appendix. l = 24 and the interaction length σs, without any friction. In simulations, N = 1,000 is fixed, and the packing fraction is adjusted by changing R. ACKNOWLEDGMENTS. We thank T. C. Lubensky, D. Frenkel, R. Podgornik, The translational degree of freedom of the active particles satisfies the and R. Blumenfeld for helpful discussions. This work was supported by underdamped Langevin equation (44), National Natural Science Foundation of China Grants 11874397, 11674365, and 11774394; Key Research Program of Frontier Sciences of Chinese Academy of Sciences Grant QYZDB-SSW-SYS003; and the K. C. Wong mv˙ = Fr + Fw + η − γv, [11] Education Foundation.

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