Elasticity-induced force reversal between active spinning particles in dense passive media

The self-organization of active particles is governed by their dynamic effective interactions. Such interactions are controlled by the medium in which such active agents reside. Here we study the interactions between active agents in a dense non-active medium. Our system consists of actuated, spinning, active particles embedded in a dense monolayer of passive, or non-active, particles. We demonstrate that the presence of the passive monolayer alters markedly the properties of the system and results in a reversal of the forces between active spinning particles from repulsive to attractive. The origin of such reversal is due to the coupling between the active stresses and elasticity of the system. This discovery provides a mechanism for the interaction between active agents in complex and structured media, opening up opportunities to tune the interaction range and directionality via the mechanical properties of the medium.

L ife occurs out of equilibrium. Living organisms are continuously generating and consuming energy to achieve self-generated motion. In addition, equilibrium conditions are barely found in nature or industrial material processing. Thus, due to their out-of-equilibrium nature, active systems have attracted much attention in recent years. These systems exhibit exotic behaviours not possible under equilibrium constraints such as emergent collective motion 1,2 , pattern formation [3][4][5] or even phase segregation in the absence of attractive interactions [6][7][8][9] . The most studied active agents are those which convert some sort of energy into translational motion. Such systems resemble how bacteria swim, and are known as self-propelled agents 6,7,[10][11] . Energy conversion into rotational motion is also common in nature; important examples are the motor adenosine triphosphate synthase 12 , certain cilia 13 , the vortex array formation of sperm cells 3 and the dancing Volvox 14 . Furthermore, experiments and several numerical and theoretical studies have focused on this type of active system in viscous media [15][16][17][18][19][20][21][22][23][24][25] . In addition to the type of activity, the medium can have a great influence on the effective interaction between different particles, which can be particularly important for active systems. Mixtures of active and passive particles can be used as a model system where active particles are embedded in a complex passive system, a scenario that is prevalent in many biological systems or processes. For example, bacterial biofilms, where live and dead bacteria phase segregate 26,27 , cell migration through tissues 28,29 or sperm swimming through the viscoelastic cervical mucus 30 . Although these systems are ubiquitous, very few works have investigated these hybrid active-passive matter systems 2,31-36 . McCandlish et al. 31 reported phase segregation of active rods in the presence of passive rods, where they point to a dynamical instability as the origin of activity-induced phase segregation; this instability originates from the differential parallel and transversal diffusion coefficients coming from the anisotropy of the rods. Ni et al. focused on the behaviour of a passive particle suspension in a glassy state doped with active agents. They observed that the presence of active particles shift the glass transition toward higher packing fractions 32 and promote the crystallization of hard-sphere glasses 37 . Stenhammar et al. 34 showed that mixtures of self-propelled and passive particles phase separate into a dense and a dilute phase, between which the interfacial tension is negative 38 . However, despite all these efforts, the origin of the emergent interactions between active agents in mixtures with passive agents remains unclear.
To shed light on the emergent interactions that govern the self-organization of non-Brownian active rotating particles, henceforth referred to as spinners, in systems composed of mixtures of active and passive particles, we use both experiments and simulations. We focus on the behaviour of pairs of co-rotating and counter-rotating spinners suspended in a viscous fluid or embedded in dense monolayers of passive particles. Importantly, we show a force reversal between spinners as the concentration of passive particles increases above a threshold. In particular, we observe that in a viscous fluid at small but finite Reynold numbers (Re), the fluid flows generated by co-rotating spinners produce a repulsion between spinners (Fig. 1a), whereas for counter-rotating spinners the resulting forces are attractive (Fig. 1b). By contrast, two co-rotating spinners in a dense passive monolayer attract each other (Fig. 1c), whereas counter-rotating spinners repel (Fig. 1d). We demonstrate that this force reversal is induced by the change in the mechanical properties of the matrix, from a viscous medium, if suspended in the fluid, to a solid-like viscoelastic medium, in the presence of passive particles. We anticipate that this mechanical attraction between co-rotating spinners is responsible for the phase separation between active and passive particles in macroscopic systems.

Results
Spinners in a viscous fluid. Our experimental system is composed of spherical ferromagnetic particles of diameter s coupled to an external rotating magnetic field of frequency o, see Supplementary Fig. 1. For the experimental conditions, the rotational frequency of the particles always coincides with the rotational frequency of the field. Spinners suspended in an incompressible fluid (ru ¼ 0) of viscosity Z and density r generate fluid flows that can be described by the Navier-Stokes equation: where we have assumed no-slip boundary conditions at the particle surface, u ¼ U þ x Â r being U the translational velocity of the particle, o the angular velocity and r the vector pointing from the center of mass of the particle to the surface of the particle. In equation (1), we have chosen a translating reference frame at the center of the spinner (that is, such as the flow is steady) and scaled the velocities and lengths by os and s, respectively. u corresponds to the fluid velocity field, Re is the Reynold number (Re ¼ os 2 r/Z), p is the pressure and f is the force density exerted by the particle on the fluid. Therefore, in the absence of any other particle, and in the limit of Re ¼ 0 (that is, where the left-hand terms in equation (1) are 0), a rotating spherical particle generates a velocity field given by ref. 39.
where r is the position of the fluid from the center of the particle, t is the torque acting on the particle, and t pZs 3 corresponds to the angular rotational frequency (o) of the spinner, which is constant in our system. This rotating field decays as 1/r 2 from the center of the spinner, as shown in Supplementary Figs 2-4 and Supplementary Discussion.
First, we study the effective interaction between two spinners in a dilute system of co-rotating spinners in the absence of passive particles, see Supplemetary Methods. In this system, the interaction between spinners is controlled by the fluid flows generated by the rotation of the spinners and the magnetic dipole-dipole attraction between them. Experimentally, we observe that spinners initially positioned further than four particle diameters do not feel either the fluid flow created by other spinners or the permanent dipole of the other spinners. Therefore, they rotate in place without experiencing any translation, as shown in Fig. 2. By contrast, spinners closer than 4s attract due to the magnetic dipole-dipole interaction, thereby forming a doublet and rotating around its center of mass. To get a deeper insight in the behaviour of the system, we also perform hybrid molecular dynamic simulations of the spinners sedimented onto a wall within a channel of height h ¼ 30 Dx and coupled to a Lattice-Boltzmann (LB) fluid 40 . These simulations, which lack the dipole-dipole interaction, show that co-rotating spinners closer than 3s experience a hydrodynamic repulsion, while rotating around the center of mass of the repulsive pair. We hypothesize that in our experiments, the hydrodynamic repulsion is hidden by the strong dipole-dipole interaction between ferromagnetic particles. To test this, we increase the rotation frequency of the applied magnetic field up to 50 Hz. At this frequency the hydrodynamic repulsion overcomes the dipoledipole attraction, and spinners separate up to 3s, blue circles in Fig. 2. The fact that the hydrodynamic repulsion increases with Re proves the inertial nature of the interaction 15 , and it is in good agreement with previous observations on millimeter-sized rotating magnetic disks adsorbed at the air-water interface 15,16 . Although our experimental set-up does not provide us with control over the direction of rotation of individual spinners, our simulations allow us to explore the case of spinners rotating on opposite directions. In this case, we find that two counter-rotating spinners closer than 3s attract each other until the separation distance between them becomes about 1.15s (see Supplementary  Fig. 5), and simultaneously translate as a doublet in the direction orthogonal to the vector joining both centers 41 . The equilibrium distance between counter-rotating spinners does not depend on the Re, but the strength of the interaction and translational velocity of the center of mass does (see Supplementary Fig. 5 and Supplementary Discussion), which indicates that the observed attraction between counter-rotating spinners is also inertial in nature.
Inertial contributions to the fluid velocity field, left-hand terms in equation (1), are the origin of the repulsion between corotating spinners 17,42 and the attraction between counter-rotating spinners. At a finite Re, inertial terms generate additional forces on the particles due to the momentum of the fluid. These type of forces, known as lift forces, originate from the relative translation of a rotating particle with respect to the fluid 43 , known as Magnus forces, or by the translation of the rotating particle with a shear flow 44 . Both lift forces depend on the translational velocity of the rotating particle. Under these conditions, the fluid velocity profile generated by a rotating sphere, equation (2), needs to be corrected to include these inertial terms, which generate a so-called secondary flow. Perturbation methods have been used to calculate the secondary flow around a rotating sphere due to small inertial effects 45 ; these studies have shown that the secondary flow produces no correction in the azimuthal part of the fluid velocity profile, equation (2). However, because of the centrifugal force effect, the fluid is pulled in towards the poles and expelled from the equator, which generates a secondary flow on the zx-plane (see Supplementary Fig. 6). The presence of a second rotating sphere breaks the symmetry of the secondary flow; around the equator of the spheres the fluid velocity between the spinners decreases for co-rotating spinners and increases for counter-rotating spinners, as shown in Fig. 3a,b. We compute the forces exerted by the fluid on co-and counter-rotating spinner pairs along trajectories of repulsion and attraction, as shown in For two co-rotating spinners, hydrodynamic forces generate a net repulsion between them, while a hydrodynamic attraction is generated for the case of counterrotating spinners. The fluid flows generated by two co-rotating spinners cause the spinners to rotate around their center of mass. This translation of the spinners generate a lift force that results in a repulsion of co-rotating spinners 17,18 . Previous studies have shown that both co-rotating and counter-rotating spinners, repel each other as a consequence of the lift forces 42 ; however, in the presence of a channel the hydrodynamic attraction between two counter-rotating spinners overcome the lift force, resulting in an effective attraction 25  Spinners in a dense passive media. The behaviour of spinners embedded in monolayers of passive particles is completely different. At finite Re, the fluid flow generated by a spinner repels neighbouring passive particles, and generates the rotation of the first shell of particles around it (see Supplementary Movies 1 and 2). The distance of this first shell of passive particles with respect to the spinner depends on the area fraction of the monolayer, f A , and the angular rotational frequency of the spinner, o. Therefore, the spinner produces a local increase of the mobility of neighbouring passive particles and compresses the monolayer ( Supplementary Fig. 8). When more than one spinner are present in the monolayer, we observe that two co-rotating spinners attract each other; this behaviour is opposite to that observed in the absence of passive particles, as shown in both experiments and simulations in Fig. 4. The experimental trajectories of two co-rotating spinners in a monolayer with an area fraction of f A ¼ 0.7±0.1 show two well-differentiated regimes: (i) If the distance between the spinners is smaller than 4s, then the slope of the trajectory is sharp; this indicates that the attraction between spinners in this regime is governed by the strong magnetic dipole-dipole interaction. (ii) At distances larger than 4s, the slope is small, which indicates that the attraction between spinners must be of a different nature. By contrast, in the simulations the trajectories exhibit a single regime of slow attraction due to the lack of dipole-dipole interactions in our model. Furthermore, once the spinners squeeze out all the passive particles initially positioned between them, they remain as a doublet at a distance of about 2s for monolayers of f A ¼ 0.8.
In our experiments, we can only study co-rotating spinners, a limitation absent in our simulations. Thus, using simulations we find that two counter-rotating spinners repel each other at a distance of about 5s within a monolayer of f A ¼ 0.8. Therefore, we also observe a reversal of the interaction force between two counter-rotating spinners with respect to the pure viscous media in the presence of a passive matrix.
To investigate the nature of the interaction between two spinners in the presence of the passive matrix, we define four different regions in the system and label the particles within these regions accordingly. These four regions are: the first shell of particles around the spinners, named as corona, the region between the two spinners, referred as bridge, the region besides the spinners on the opposite side of the bridge, denoted as the surroundings, and the bulk, as illustrated in Fig. 5a,b. Particles located in the corona rotate coherently around the spinners, and collide against neighbouring particles transferring their momentum. These particles rarely escape from this region and the number of particles in the corona remains almost constant until the coronas of the two spinners starts to collide with each other. Therefore, we count these particles as a part of the spinner for every calculation (green shade region in Fig. 5a,b). The bridge and the surroundings are very dynamic; particles in these regions are in continuous motion. The stresses generated by the spinners through the corona are released in these regions. To relax the stresses they need to yield.
To study the evolution of the passive particles in the bridge, surroundings and bulk decoupled from the spinners rearrangement, we perform numerical simulations freezing the distance between the spinners at different values. These constrained   systems are just able to relax the stress coming from the activity of the spinners through the displacement of passive particles, and thus they never reach a steady state. However, the initial time evolution of these systems allows us to study the process of loading and yielding of the monolayer without the relaxation of the system through the displacement of the spinners. In Fig. 5c,d, the initial time evolution of the particle area fraction in the different defined regions are presented for co-rotating and counter-rotating spinners separated 6s and 4s, respectively. We observe that for co-rotating spinners, the f A of particles in the bridge is significantly reduced as compared with the f A in the surroundings and bulk, as shown in Fig. 5c and Supplementary  Fig. 9. The compression and shear stresses produced by the spinners in the bridge, through the corona, result in a f A reduction within this region as it is constantly yielding due to shear stresses induced by the spinners. Thus, the higher mobility of passive particles initially located in this region allows them to migrate to less-stressed regions. On the contrary, for counterrotating spinners, the f A of passive particles in the bridge is significantly increased compared with the bulk and surroundings, as shown in Fig. 5d and Supplementary Fig. 9. Therefore, the mechanism by which the passive matrix mediates the interaction between spinners is related to the type of stresses that the spinners exert on their vicinity. Co-rotating spinners compress and shear the bridge, as schematically illustrated in Fig. 5a. To alleviate the stress, the system prefers to yield by transporting particles from the bridge into the other regions. This occurs through avalanches and single-particle hopping, as will be shown later. Clearly, this migration reduces the density on the bridge. This imbalance repositions the spinners closer to each other, thereby restoring the temporal mechanical equilibrium. This process is continuously occurring, which slowly degrades the bridge until the active particles are able to come together ( Supplementary Movies 1 and 2). During this process the monolayer is annealed, inducing the defects to migrate and concentrate around the spinners, as depicted in Supplementary   Fig. 10 and Supplementary Discussion. We discard the migration and coalescence of defects as the driving force for the co-rotating spinners attraction by performing simulations in which the initial configuration is a perfect hexagonal close packing lattice. In this case, co-rotating spinners aggregation is still observed, as shown in Supplementary Fig. 11. By contrast, counter-rotating spinners produce compression and dilation stresses in the bridge. Both spinners move passive particles into the bridge, which increases the pressure in this region; this pushes both spinners away, thereby resulting in a repulsion between counter-rotating spinners (Supplementary Movie 3).
To further investigate this emergent interaction between spinners embedded in passive matrixes, we estimate the mean spinner-spinner interaction potential in monolayers of f A ¼ 0.8, as shown in Fig. 6. By means of the application of harmonic springs, F ¼ k(r À r 0 ), to restrain the separation distance between the spinner pair at r 0 , we calculate a potential of mean force between co-and counter-rotating spinners, as presented in Fig. 6a,b, respectively. The mean interaction potential, black circles, is obtained by averaging over five independent initial configurations at each separation distance r 0 . The emergent interaction between two co-rotating spinners within a passive monolayer is repulsive at distances smaller than 2s. At larger distances, shallow attractive minima are separated by small energy barriers, as shown in Fig. 6a. The local energy input coming from the spinners' activity is enough to overcome these small energy barriers. The presence of these energy barriers at distances larger than 2s agree with the observation that the loading of the bridge plays a key role for this interaction. Moreover, the shallow minima would then correspond with the state of the system after an yielding event. In the case of counterrotating spinners, the interaction is repulsive with a strong repulsive peak at 1.5s; it also presents a small energy minimum at 3s, as can be seen in Fig. 6b. This shallow energy minimum explains that along the repulsive trajectories, the spinners spend long times at this distance, as it can be seen in Supplementary  Fig. 12B. We also use this methodology to evaluate the interaction potential between co-and counter-rotating spinners in the absence of passive particles, represented by the magenta solid circles in Fig. 6a,b, respectively. In agreement with our previous calculations, Fig. 6 clearly shows that the change in the mechanical properties of the media produces not only a reversal of the forces acting on the spinners but also significantly increases the range of the interaction. Interestingly, the potential of mean force at the different separation distances strongly depends on the initial configuration of the monolayer, as shown in Fig. 6; the average s.d. between the different initial configurations is about 10%. Therefore, in spite of the general trend captured by these mean spinner-spinner interaction potentials, this interaction is of a stochastic nature and thus, the strength and range of the interaction between spinners in passive matrixes is determined by the instantaneous and previous configurations of the monolayers, that is memory effects. Another proof of that is shown in Supplementary Fig. 13, where we compute, by means of harmonics springs that restrain the separation distance between co-rotating spinners, the attractive force between the spinners for 100 configurations selected from the unconstrained trajectory presented in Fig. 4 of spinners initially positioned 6s apart, as shown in Supplementary Fig. 13A. The different configurations present significantly different values of the force, as it can be seen in Supplementary Fig. 13B; however, the average force as a function of the separation distance between the spinners is in good agreement with the one estimated by averaging over five independent configurations at each separation distance. We would like to point out that the direction of the interaction is not determined for a given distance (Supplementary Discussion; Supplementary Fig. 14). For example, in Fig. 6a at r ij ¼ 4.5s the interaction can be either attractive or repulsive depending on the configuration (different seeds). Thus, for any spinner pair it is not possible to predict the exact local evolution, but only that they will attract or repel over the length of the experiment. We have also found in both experiments and simulations that the aggregation process of co-rotating spinners embedded in a monolayer is governed by the elasticity of the medium and the ability of the spinners to increase the elastic energy of the system. This can be directly confirmed by measuring the storage and loss moduli of the system in the absence of active particles, and observing at what particle area fraction the attractive interaction between co-rotating spinners is lost (Supplementary Discussion; Supplementary Figs 15 and 16). In Fig. 7, the initial distance between two co-rotating spinners is represented against the distance reached after a long run. For passive monolayers at f A ¼ 0.8 and 0.7, spinners initially separated up to 6s are attracted to each other up to a distance of about 2s and 2.5s, respectively. However, for particle area fractions smaller than 0.7, passive-mediated interactions are no longer effective and only the hydrodynamic repulsion is observed. According to the mechanical properties of the passive monolayer, the elastic response of the system dominates at long times for monolayers of f A ¼ 0.8 and 0.7, as depicted in Supplementary Fig. 16A,B; for the latter, elastic and viscous responses are almost equivalent at long times. On the contrary, for area fractions of 0.6 and 0.5 the viscous response dominates for the entire frequency range, Supplementary Fig. 16C,D. This demonstrates that an elastic response of the passive media is a necessary condition for co-rotating spinners to interact. As mentioned before, this elasticity-mediated attraction between active spinning particles is inherently stochastic. Thus, the initial configuration of the system determines the length of interaction and the time required for spinners to aggregate, as shown by the different experimental and simulation trajectories in Fig. 4. Hence, one can imagine a rough and dynamic energy landscape in which the spinners must traverse to form a dimer, and where multiple paths exist when moving from state to state, as schematically illustrated in Fig. 8a. Erosion of the bridge, which happens when passive particles are removed from it, occurs through three main mechanisms: (i) Single-particle removal from the bridge to the corona and then to the bulk, as shown in Fig. 8b; (ii) multiple particle removal, where two or three particles in the bridge, due to the shear stresses, are moved into the bulk or intermittently moved to the corona and then ejected into the bulk, as seen in Fig. 8c; and (iii) avalanches where entire lines of particles in the bridge are pushed into the bulk by the shear exerted by the spinners, as depicted in Fig. 8d. We use the name avalanche to invoke the instantaneous and dramatic nature of the particle removal process. Whereas the bridge erosion produced by the two first mechanisms is slow and depends on stochastic collisions, the removal of particles in avalanche happens almost instantaneously. In fact, it has been previously shown that the devitrification of hard-sphere glasses is mediated by large rearrangements of particles, so-called avalanches 46 . This interaction opens up possibilities to study the mechanism of elasticity-mediated interaction between active particles in these systems governed by a glassy dynamics.

Discussion
In summary, the forces exerted by an incompressible fluid at small but finite Re on a pair of non-Brownian active rotating particles depends on the relative sense of rotation of each particle, resulting repulsive for co-rotating spinners and attractive for counter-rotating spinners when confined in a channel. The presence of a dense passive matrix modify the mechanical properties of the system from a viscous media to a viscoelastic material. In this latter case, the interaction between spinners becomes controlled by elastic effects, which act in the opposite direction than inertial effects [47][48][49] . Hence, the switch between inertial and elastic stresses derives in a reversal of the interaction between spinners, resulting in an attraction of co-rotating spinners and a repulsion of counter-rotating spinners. In fact, the structure of the passive dense medium cannot be treated at the mean field level. For example, assuming a pure viscous scenario where the passive matrix would be an homogeneous continuum with higher viscosity than the system in the absence of passive particles, the stress would be dissipated by the viscous media, and one would just observe the repulsion between co-rotating spinners, but the strength of the secondary flows would be smaller due to the viscosity increment. Therefore, the change of the mechanical properties of the matrix from a viscous material to a solid-like material is the responsible for the force reversal. Furthermore, the interaction between spinners in dense monolayers of passive particles is of stochastic nature; it depends on the configuration of the passive monolayer. Thus, it is the instantaneous configuration of the monolayer that determines the strength and range of the interaction, and the dynamics of attraction is intimately related to the timescale for the rearrangement of the monolayer. This can be better understood by looking at the oscillation between periods of well-defined distances and periods of fast attraction along the trajectory between two co-rotating spinners, Figure 4 and Supplementary  Fig. 12A. The level of stress put into the bridge by the spinners must reach a configuration dependent threshold value, and when this level is reached the system yields by removing entire groups of particles from the bridge, as depicted in Fig. 8. Then, the spinners approach each other and start stress loading the bridge again. Therefore, this effective interaction mediated by the passive medium cannot be seen as a position-dependent interaction potential (U(r ij )). Interestingly, the dynamic trajectories of the distance between the spinners initially positioned at different distances show an almost linear regime of attraction, as it can be seen in Fig. 4 and Supplementary Fig. 11. This means that the spinners approach each other on average at a constant speed.
Assuming a Stokes' scenario, F ¼ 6pms/2U, for the translation of the spinners through the monolayer along their attractive trajectory, the strength would be a constant and independent on the distance between them. Moreover, this elasticity-mediated interaction is of a very long range. For example, in our simulations we observe that spinners separated up to sixparticle diameters still interact, whereas the hydrodynamic interaction reach only three-particle diameters. Remarkably, our experimental measurements show that the interaction threshold shifts even at longer distances, and spinners separated by up to 17s attract each other 50 , while the dipole-dipole interaction reach 4s (Figs 2 and 4). Our results resemble other elastic media such as lipid membranes, where elasticity-mediated forces between transmembrane embedded proteins show logarithmic decays 51,52 . However, the origin of the stresses in our system are different.
In conclusion, we have shown that the interaction between active spinning particles depends on the properties of the medium and the dynamics of the active particles. Therefore, this cooperative interaction between the mechanical and dynamical properties of the system offers a variety of possibilities to tune this type of interaction between active agents. Remarkably, we have also observed that the spinners produce an annealing of the passive matrix structure ( Supplementary Fig. 10), in agreement with previous simulation results in hybrid passive-active systems but with a different type of active particles 33 . We anticipate that this mechanical attractive force between co-rotating spinners is responsible for the phase separation observed in systems with higher concentrations of spinners. Moreover, in ternary hybrid active-passive systems, composed of mixtures of spinners rotating either clockwise or counter clockwise, we observe phase segregation in three different phases: one composed of the passive particles, other of co-rotating spinners, and the last one of counter-rotating spinners. In principle, this elasticityinduced interaction between active spinning particles is general for other active agents such as self-propelled particles. Therefore, this study opens up routes to control the range and direction of the interaction between active units in passive and structured environments. This interaction between active spinning particles in passive matrixes is different from the emergent interactions observed between passive objects within active fluids. objects depend on the mobility of the passive objects 53,54 and their shape 55 . Therefore, it would be very interesting to investigate these effects in the opposite scenario between active particles in passive matrixes ( Supplementary Fig. 17). Even more, this type of interaction could play an important role in overcoming diffusive limitations that active biological molecules encounter if interacting within dense viscoelastic materials such as the highly viscoelastic nucleus of the cell 56 or cells in extracellular polysaccharide matrix secreted by biofilm-forming bacteria 57 .

Methods
Experimental set-up. To study the behaviour of active rotating particles in pure viscous and in dense passive environments, we have designed a hybrid active-passive system composed by ferromagnetic particles (the active units or spinners) and polystyrene particles (the passive units). To measure pairwise interactions between spinners we mix an extremely dilute solution of active ferromagnetic particles with a solution containing passive polystyrene particles. Both particles have a diameter of 5 mm. We first study the case of pure spinners, in the absence of any passive particle. We then study the interactions between spinners in a monolayer of passive particles at a particle area fraction of f A ¼ 0.7 ± 0.1. These solutions, dilute spinners and dense mixtures, are placed between a cover slip and a slide, and placed in our experimental set-up, which is composed by four magnetic coils ( Supplementary Fig. 1); this allows us to generate a rotating magnetic field that is parallel to the substrate and rotates around the z axis (see Supplementary Information for more details). The frequency of rotation used in this study was 5 Hz. On actuation of the external rotating magnetic field the magnetic dipole moment of the spinner couples to the applied field, and the particle spins in place around the z axis.
Simulation. To gain a more detailed insight into the non-equilibrium nature of this system we carry out numerical simulations using hybrid molecular dynamic simulations of the colloidal particles coupled to a LB fluid 40 . The simulation box is discretized in three-dimensional grids with resolution N x Â N y Â N z ¼ 214 Â 214 Â 30 bounded in the z direction by no-slip walls and periodic boundary conditions in the x and y directions. The LB fluid is described by the fluctuating LB equation 58 , which properly describes the dissipative and fluctuating hydrodynamic interactions. We implement the discrete 19-velocity model (D3Q19). The LB fluid parameters are density r ¼ 1, kinematic viscosity n ¼ 1/6 and temperature k B T ¼ 2 Â 10 À 5 . For simplicity, we set the grid spacing Dx and the LB time step Dt equal to unity. Interactions between the LB fluid and the particles are described by the bounce-back rule 59 , and enforcing no-slip boundary conditions at the surface of the particles. Specifically, we implement the Aidun, Lu, Ding (ALD) method 60 , where particles are treated as real solid objects. Therefore, we do not taken into account lubrication forces; however, we tested that including lubrication forces only shifts towards smaller Re numbers (that is, smaller rotational frequencies) the behaviour observed here. In our simulation model, colloidal particles are considered as hard spheres 61 of diameter s ¼ 12Dx; thus, we are just considering excluded volume interactions. The spinner activity is generated by imposing an external torque, t, about the z axis. To form the monolayer we also include a gravity force F G ¼ 0.005. Since we are interested in the hydrodynamic interactions that occur between spinners, and not their magnetic interaction, we do not include dipole-dipole interactions in our simulation model to more clearly delineate the origin of the effective interactions. The typical simulation length is of about 0:2t B , where t B is the characteristic Brownian time.