Emergence of reconfigurable wires and spinners via dynamic self-assembly

Dissipative colloidal materials use energy to generate and maintain structural complexity. The energy injection rate, and properties of the environment are important control parameters that influence the outcome of dynamic self-assembly. Here we demonstrate that dispersions of magnetic microparticles confined at the air-liquid interface, and energized by a uniaxial in-plane alternating magnetic field, self-assemble into a variety of structures that range from pulsating clusters and single-particle-thick wires to dynamic arrays of spinners (self-assembled short chains) rotating in either direction. The spinners emerge via spontaneous breaking of the uniaxial symmetry of the energizing magnetic field. Demonstration of the formation and disaggregation of particle assemblies suggests strategies to form new meso-scale structures with the potential to perform functions such as mixing and sensing.

T he use of self-assembly to generate functional structures that have sizes or properties not otherwise accessible has been a largely unrealized objective of materials science for decades. Examples of partial successes include nanoparticles and aggregates in biosensing 1,2 , ferrofluids and magneto-rheological liquids for clutches and bearings, and optical colloids 3,4 . Most of these systems are at equilibrium or steady state.
Out-of-equilibrium (active) colloidal suspensions [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19] are much more challenging. Model colloids have helped to understand the principles that guide dynamic self-assembly: those systems offer a variety of properties due to their high structural controllability, and the range of possible interactions among them 20,21 . External electromagnetic fields [22][23][24][25] , UV radiation 26 or chemical reactions 27 have been used to supply energy for dynamic colloidal self-assembly. The resulting dynamic structures 8,9,19,28,29 are usually unique, and not accessible under equilibrium conditions. Complex collective motion and hierarchical ordering in these out-of-equilibrium systems reflect the balance between many types of interactions among particles, ranging from short-range steric to long-range hydrodynamic and electromagnetic ones. Here we report on the emergent dynamics, and resulting self-assembly of suspensions of ferromagnetic microparticles confined at a liquid-air interface, and energized by a uniform inplane uniaxial alternating magnetic field (see Methods). Reported self-assembled structures range from dynamic wires and clusters to spinners. The alternating magnetic field maintains the system out of equilibrium, and controls both structural morphology and time-dependent behavior of the structures.

Results
Surface tension confines the ferromagnetic microparticles (,90 mm) with intrinsically pinned magnetic moments (see Methods) to the surface of the water-air interface. An alternating magnetic field applied parallel to the interface (Fig. 1a) exerts torque on them. The torque is dissipated locally in the liquid and generates hydrodynamic flows around each particle. Consequently, the particles interact by two temporally related, but physically distinct types of forces: magnetic (dipole-dipole interactions) and hydrodynamic. Dynamic selfassembly reflects the interplay between magnetic interactions and hydrodynamic flows. The relative contributions of these two primary interactions are modulated by the parameters of the energizing alternating magnetic field. As a result, the system exhibits a remarkable diversity of quasi-stable dynamic states. The phase diagram in Fig. 1b delineates these various self-assembled phases as a function of the frequency and amplitude of the applied magnetic field. All the states are dynamic, and exist only while energy is supplied by the a.c. magnetic field. Fig. 1cg illustrates major distinctive phases.
Loose clusters extended along the a.c. magnetic field are formed at low frequencies. The clusters exhibit periodic changes in shape (pulsations): over time the cluster extends and contracts with a fraction of the driving magnetic field frequency (see Supplementary Video 1).
At elevated frequencies of the applied field, the cluster transforms into a cloud of continuously rearranging short chains (Fig. 1e). In striking contrast with a loose cluster, the cloud switches the axis of elongation and extends perpendicular to the a.c. magnetic field. Further increase in the frequency of the applied magnetic field yields a new dynamic phase: spinners (see Fig. 1f,g). In this phase the particles self-assemble into short chains, and rotate in the plain of the air-water interface at the frequency of the applied a.c. magnetic field. This rotation creates strong in-plane hydrodynamic flows. The spinners exhibit complex dynamic behavior: they move across the surface, collide, disintegrate, and re-assemble (see Supplementary Video 2 and 3). The multi-spinner state (gas) has no apparent preferred direction and covers the entire area of the container uniformly. At higher frequencies of the applied field the spinners give way to dynamic wires (see Fig. 1d). All phases form and disassemble reversibly (see Supplementary Video 4), and their behavior is controlled by the parameters of the applied a.c. magnetic field.  Dynamic self-assembled wires. The formation of dynamic wires proceeds through an elongation of the initial cluster of particles along the axis of the applied magnetic field (see Fig. 2a). During its growth, the wire shows imperfections (e.g. kinks and perforations). These are eventually eliminated by the rearrangements of near-by particles. The extension of a wire continues until it becomes oneparticle thick (see Supplementary Video 5). Since all dynamic phases are reversible and controlled by the parameters of the applied a.c. field, the formation of a wire is also possible from the initial spinner phase by adjustment of the a.c. field frequency. In this case, growth of the wire proceeds by successive attachments of individual particles and merging of linear chain fragments (see Supplementary Video 6).
To characterize the dynamics of growth of the wires, we monitored the evolution of the length of a wire as a function of the frequency and amplitude of the driving magnetic field within a parameter space of a wire phase. Fig. 2a shows the stages of formation of a typical wire. Initially all particles aggregate in a static compact cluster due to ferromagnetic interactions among particles: strong magnetic dipoledipole inter-particle interactions exist even in the absence of the external magnetic field. Upon application of the alternating magnetic field, the static cluster transforms into a thick non-uniform bundle aligned along the field axis and continues to extend until it reaches a single-particle thickness. This process contrasts with a common mode of aggregation 30 where growth proceeds through a succession of attachments (aggregation) of individual particles from the surrounding surface. The dynamics of formation of wires can be appro- Here L ' is the established asymptotic length of the wire at time t R '; t 0 is the time when the a.c. field was applied; and t is a characteristic time for wire growth. The saturation of growth is caused by the depletion of particles in the vicinity of the wire. Fig. 2b shows a typical evolution of the length of the wire with time.
From this type of analysis, we have extracted the characteristic growth times, t, for different parameters of the applied magnetic field. Increases in the magnitude of the applied field significantly decrease the characteristic formation time (see inset to Fig. 2b). In contrast, variations in frequency of the applied field (within the boundaries of the dynamic wire phase) do not produce detectable changes in the characteristic formation time.
Reversible assembly of straight wires from ferromagnetic particles is a challenging task. Under equilibrium conditions, a static magnetic field will produce only a deformed compact cluster of ferromagnetic particles. Out-of-equilibrium dynamics enables assembly of a more complex wire pattern in this system: parallel arrays of single-particlethick wires (Fig. 2c). For this case, an additional small static (d.c.) magnetic field (20 Oe) applied perpendicular to the interface was added to stabilize the pattern. The spacing between wires in the array is determined by the magnitude of the d.c. field and the meniscus: the spacing is slightly smaller in the middle of the array. Upon removal of the stabilizing field the array of wires evolves towards formation of a single wire by successive coalescence of separate wires (seen as ''Y''shaped defects in Fig. 2c).
Spinners and spontaneous symmetry breaking. Surprisingly, in a certain range of parameters for the applied field (region 2 in Fig. 1b) a striking transformation takes place: the uniaxial symmetry of the field breaks spontaneously, and an ensemble of spinners rotating both clockwise and counterclockwise emerges, see Fig. 1f,g and Supplementary Video 3. Dynamic spinners are composed of short self-assembled chains of micro-particles ferromagnetically ordered due to the prevailing magnetic dipole-dipole inter-particle interactions. Spinners rotate clockwise or counterclockwise with the frequency of the applied field. The onset of spinning is related to the following mechanism. Individual ferromagnetic particles aggregate into short chains with the resulting magnetic moment oriented along  the chains. The applied a.c. magnetic field forces the chain to align with the field orientation, resulting in periodic (with the field frequency) reversals of the direction. If the chain's rotational inertia is large compared to the viscous drag, the chain will preferably rotate in one of the directions. Since for the case of uniaxial magnetic field the clock/counterclockwise senses of rotation are equally probable, the initial direction is selected by a variety of factors, like interactions with neighboring particles and flows.
The rotating chains exert viscous torques on the liquid which trigger strong long-range hydrodynamic vortical flows at the interface. In the bulk, the spinners create swirling flow, which decays roughly as the square of the depth. Since our experiments are performed in deep containers (the ration of the depth of the container to the length of the spinners is of the order of 100), the effects of the container's bottom on the vortical flow at the interface are negligible.
Spinners move (seemingly randomly) due to magnetic interactions and the flows generated by other spinners. Their collective motion creates an overall gas-like appearance of the phase (see Supplementary Video 2). The structure of each spinner is not fixed. Spinners collide, disintegrate and reassemble, and thus create complex time-dependent hydrodynamic patterns at the interface. The hydrodynamic surface flows generated by spinners have been characterized by particle-image velocimetry (Methods). Fig. 3a,b shows snapshots of the velocity field for the same system at two different times. Fig. 3c shows streamlines of a typical velocity field generated by the spinners at the air-liquid interface. The pattern of surface flow exhibits complex spatio-temporal dynamics, manifested by abrupt local bursts and subsequent cessation of activity (Fig. 3d). The characteristic size reached by the spinners depends on the frequency of the applied a.c. field. An increase in the frequency of the field shortens the length of the spinning chains. The length is determined by a balance between magnetic and viscous torques acting on a chain at the liquid interface. Balancing the torques, we find (in the slender body approximation) 31 Here, L s is a  characteristic length of a spinner, a is a particle diameter, and A is a factor that depends on magnetic moment of the particles, their radius, the amplitude of the applied magnetic field and the viscosity of the liquid. The expression provides an adequate description of the experimentally observed behavior (Fig. 3e).
Simulations. We numerically solved the equations of motion for N particles on a plane. Particles are energized by an in-plane uni-axial magnetic field, and interact due to their magnetic dipole moments, self-induced hydrodynamic flows, and steric repulsion. In simulations, the length was scaled on the particle size, time was scaled by 1/f, where f is the frequency of the applied magnetic field, and field amplitude was scaled by m/(4pa 3 ), where m is a magnetic moment per particle. The parameters in the equations of motion were calculated from actual properties of the particles and fluid (size, magnetic susceptibility, viscosity). See Methods for the details of the model and simulations. The simulations successfully reproduced the morphology of experimental phase diagram (Fig. 4a) and captured most of the observed phases (see Fig. 4).
Synchronization of spinners. Strong hydrodynamic and magnetic couplings between the spinners hint at possible synchronization (perhaps similar to synchronization of applause after musical performances 32 , or synchronization of weakly coupled pendulum clocks 33 ). While clock/counterclockwise spinners are equally probable, synchronization can lead to the emergence of phases with a dominant direction of rotation.
To quantify this effect we introduce an order parameter I related to the imbalance of spinners with different sense of rotation: I 5 (N 1 2 N 2 )/(N 1 1 N 2 ). Here N 1 and Nare the number of clockwise and counterclockwise spinners respectively. The emergence of complete global synchronization of the sense of rotation of spinners rotation would correspond to the order parameter values I 5 61; the completely random non-synchronized state would correspond to I 5 0. The results of experimental and computational studies are illustrated in Fig. 5. Smaller number of particles in a simulation window (see Methods) than those used in experiments expectedly produce smaller amount of spinners per simulation frame (,10). Consequently, asymmetry in the number of spinners with different rotational direction results in a high magnitude of imbalance compared to the larger experimental system (,10 2 spinners).
While both experiment and simulation revealed significant fluctuations in the order parameter (see Fig. 5a,c), no global synchronization was detected. Analysis of the fluctuation of the order  parameter resulted in Gaussian distribution peaking at zero, and suggested no long-lived synchronization events (see Fig. 5d). To better understand the collective dynamics of spinners, we performed a Fourier analysis of the fluctuations in the order parameter. The results are shown in Fig. 5e,f. The spectra reveal pronounced peaks at certain characteristic frequencies (Fig. 5e), suggesting deterministic origin of the seemingly random order parameter fluctuations. Simulations reproduced similar phenomenon (Fig. 5f). The origin of spontaneous symmetry breaking and rotation is associated with the non-negligible inertia of the colloidal particles and the fluid their motion entrains. Consider for simplicity an individual particle in a uniaxial field H 5 H 0 sin(vt). The evolution of a particle's orientation w is described by the balance of viscous and magnetic torques and the particle's inertia, where I r , a r , m are the moment of inertia (including the inertia of the entrained liquid) of the particle, the rotational drag, and the magnetic moment. The right part of Eq (1) can be decomposed into clockwise and counter-clockwise rotating waves This equation is well-known in the context of deterministic chaos 34 .
If we neglect one of the waves in Eq (2), then a fixed point of this equation will correspond to a steady rotation of the particle with frequency v. The rotation is possible as long as the viscous friction is below a certain threshold, a r v , m H 0 /2. However, increasing the magnitude of the field H 0 results in an interaction with a counterrotating wave. It eventually leads to the rotation reversal and the onset of the overall chaotic behavior. The so-called ''resonance overlap criterion'' 35 for the onset of chaotic behavior is given approximately by ffiffiffiffiffiffiffiffiffiffiffiffi 2m H 0 I r r wv Typical trajectories of Eq (1) are characterized by chaotic reversal of direction of rotation with well-pronounced peaks in the spectrum at the fractions of the driving frequency v. Corresponding spectral peaks are captured in both experimental data and simulations (see Fig. 5e,f). Quantitative differences between experimental and simulation spectra originate from two major sources: particle size inhomogeneity in the experiment (75-100 mm) and approximations of the model used in the simulations (see Methods).
In conclusion, we have demonstrated that a dispersion of ferromagnetic microparticles, confined at the air-liquid interface and energized by a uniaxial in-plane alternating magnetic field, form a rich variety of nontrivial, dynamically self-assembled states, ranging from pulsating clusters and tunable, single-particle-thick wires, to self-assembled spinners. The morphology and complex collective behavior of these out-of-equilibrium self-assembled structures could be tuned by the parameters of the applied magnetic field. Our study did not reveal global synchronization in the gas of spinners. However, the propensity towards synchronization can be enhanced by the use of bi-axial fields 8,11 , e.g. chiral rotational field that will break the clock/anticlockwise rotational symmetry. In this case, one sense of rotation will be more preferred than the opposite.
Our work provides new insights into the engineering of tunable complex dynamic architectures by the means of out-of-equilibrium self-assembly. All reported dynamic structures are fully reversible. Possible applications might include design of wire networks via dynamic self-assembly with a topology controlled by a magnetic field, e.g. for flexible electronics and microrobotics, or non-contact mixing at the interface. Another intriguing extension of our work would be a dynamic self-assembly of ferromagnetic particles suspended in the bulk of the fluid rather than at the interface. This technique, complimentary to self-assembly of diamagnetic particles via magnetic levitation 37,38 , has the potential to lead to a variety of novel, three-dimensional, dynamic structures.

Methods
Ferromagnetically ordered nickel spherical particles (Alfa Aesar Company) with an average diameter of ,90 mm (75-100 mm uniform size distribution) were used in the experiments. Magnetic saturation moment per particle is ,200 memu at 4 kOe field. Because of intrinsic defects in particles their magnetic moments are strongly pinned, and the particles behave as magnetically ''hard'' microspheres. The presence of ''pinned'' magnetic moments in the particles is important prerequisite for observation of reported phenomena. Similar dynamic self-assembly has been observed with smaller particles down to ,30 mm. However, since the spinner phase exists due to inertia of the particles, we anticipate that this phase will disappear for too small particles (below ,10 mm).
Particles (about 10 3 particles) were dispersed at the interface between deionized water and air. The suspension was energized by a uniform uni-axial alternating magnetic field, H ac 5 H 0 sin(2pft), applied parallel (in-plane) to the interface. The amplitude of the ac magnetic field, H 0 , was in the range of 10-60 Oe and the frequency, f, was in the range of 5-300 Hz. The alternating in-plane magnetic field was created by a pair of custom-made precision Helmholtz coils.
Optical microscopy and particle image velocimetry. A glass container (5 cm in diameter) containing the particles was mounted on a microscope stage (Leica MZ9.5). The trajectories of the particles were monitored by a fast CCD camera (RedLake MotionPro). The particle image velocimetry (PIV) was performed at the water-air interface. Nonmagnetic tracer particles (copper 10-20 mm spheres) were added at the water interface and were supported by surface tension. We recorded image sequences (1280 3 1024 resolution) at a frame-rate of 400 fps. Image and data analysis of the time sequences were carried out using ImageJ, MatPIV and custom scripts. where m is the mass of particle, m is its magnetic moment, I is moment of inertia, and a t,r are, respectively, translation and rotation drag coefficients, F i and T i are forces and torques due to magnetic dipole-dipole interaction and hard-core repulsions (see for detail Refs. 18,36). Hydrodynamic interactions between the particles are described by the advection with velocity v, where v is the hydrodynamic velocity created due to rotation of all particles (in the Stokes limit, a rotating particle produces azimuthal flow with the velocity v~d w dt a 3 =r 2 , where a is the radius of the particle and r is the distance from the center). For simplicity we neglected the secondary flow generated by a nonsteady rotating particle and realignment of the particles in shear flow. The last term in Eq. (4) describes the torque from the applied magnetic field H 5 H 0 sin(vt) on a permanent magnetic dipole. The algorithm was implemented on a GPU card. 100-200 particles were simulated over more than 20,000 periods of the applied field at different parameters of the driving magnetic field.