Abstract
Assemblies of selfrotating particles are gaining interest as a novel realization of active matter with unique collective behaviors such as edge currents and nontrivial dynamic states. Here, we develop a continuum model for a system of fluidembedded spinners by coarsegraining the equations of motion of the discrete particles. We apply the model to explore mixtures of clockwise and counterclockwise rotating spinners. We find that the dynamics is sensitive to fluid inertia; in the inertialess system, after transient turbulentlike motion the spinners segregate and form steady traffic lanes. At small but finite Reynolds number instead, the turbulentlike motion persists and the system exhibits a chirality breaking transition leading to a single rotation sense state. Our results shed light on the dynamic behavior of nonequilibrium materials exemplified by active spinners.
Similar content being viewed by others
Introduction
Active matter has gained much attention as a field rich with unique and potentially useful nonequilibrium phenomena^{1}. These systems consist of selfdriven units that have relatively simple individual dynamics but collectively exhibit macroscopic coherent motions^{2}. While flocks of birds and schools of fish are the commonly given examples of such emergent collective behavior, similar phenomena have been observed on much smaller scales with active microagents such as bacteria^{3,4,5,6,7,8,9,10}, chemically activated motile colloids^{11,12,13,14,15}, microtubule–kinesin bundles (active nematics)^{16}, living liquid crystals (suspensions of motile bacteria in liquid crystals)^{17,18}, and fielddriven colloids^{19,20,21,22,23,24,25,26,27}.
Most studies of active suspensions have focused on translational particles such as bacteria and their inanimate mimics^{28,29,30,31,32,33,34,35}. However, there has been rising interest in systems containing selfrotating particles^{36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51}. An isolated microrotor has no means of selfmotility. But when a rotor is coupled with other rotors, motility becomes possible. Theoretical studies showed that coupling mediated by contact in dry systems^{36} or hydrodynamic interactions^{37,52} can induce propulsion and selforganization. While biological examples of rotors are limited, e.g., the bacterium Thiovulum majus^{53}, there has been a growing number of synthetic designs of rotors fueled by chemical reaction^{54} or external forcing such as magnetic fields^{39,44,46,51,55}, electric fields^{56,57,58}, oscillating platform^{59}, ultrasound^{60}, and applied air flow^{61,62}.
Simulations based on agentbased models, without hydrodynamics, for gearshaped spinners^{36,63} and rotating dimers^{64} showed that a mixture of clockwise (CW) and counterclockwise (CCW) rotors segregate into samespin phases. This behavior was actualized in an experiment^{65} using a large number of small robots spinning on a table. The collective dynamics of fluidsuspended rotors, however, can be quite different from that observed in the dry systems due to hydrodynamic interactions. For example, unlike the frozen state of gearshaped spinners at low density^{36}, fluidembedded spinners exhibit a gaslike phase, where the particles move randomly in the stirred fluid^{38}. Furthermore, while both the dry and the fluid systems form similar largescale dynamic patterns, such as lanes and vortices, their quantitative characteristics are different. Much insight has been gained by studying these systems using discretespinner models, however, this approach becomes computationally expensive when considering large systems with a high spinner density. In such cases, it becomes favorable to utilize continuum models that replace the discrete individual particles with a continuous distribution. Such continuum approach allows for the computationally feasible study of the collective dynamics of large systems^{27,66,67,68,69}.
The existing continuum models of selfrotating systems are phenomenological and typically based on the simplified hydrodynamic theory that includes either chiral components in the stress tensor or additional chiral forcing terms in the Navier–Stokes equation^{45,46,70}. These models capture phenomena such as the surface waves in active spinner systems^{46}, active coarsening, and the emergence of selfpropelled vortex doublets^{70}, etc. However, the phenomenological implementation of active rotation makes it challenging to assess if the observed dynamics are physical or dependent on the postulated continuum formulation. It also becomes difficult to directly compare experiment and theory.
Here we derive a continuum model for a monolayer of spherical spinners with CW and CCW spins suspended in liquid in a threedimensional domain. The model is motivated by experimental realizations such as spinners trapped at a fluid interface^{39,71} or on a solid substrate^{46}. We present numerical simulations assuming rotation due to an external DC (direct current) electric field, known as Quincke rotation. We explore the mixtures of CW and CCW spinners and study samespin phase separation. We find that the dynamics is sensitive to fluid inertia: while in the inertialess system after a long transient turbulentlike motion the spinners settle and form steady trafficlike lanes, at small but finite Reynolds number the turbulentlike motion persists. For even larger Reynolds numbers, a population initially composed of an equal amount of CW/CCW spinners exhibits a chiralitybreaking transition and collapses into a single rotation sense state. While our results are obtained for Quincke spinners, a similar continuum approach can be applied to model a monolayer of magnetic spinners energized by an applied AC magnetic field^{39,71}. Our results advance a fundamental understanding of active systems, where largescale chiral states emerge as a result of spontaneous symmetry breaking, as for example spinners, rotating bacteria, microtubules assays, etc.
Results and discussion
Summary of the continuum model
In the “Methods” section, we derive a continuum model for a twodimensional system of N active spinners of mass m and radius a suspended in a fluid with viscosity μ and density ρ confined to a plane of area A, see Fig. 1a. The spinner concentration, c = N/A, is assumed to be uniform. The system is governed by the following equations for the spinner angular velocity ω(r, t), fluid velocity u(r, t), and pressure p(r, t):
where \({I}_{p}=\frac{2}{5}m{a}^{2}\) and \(D=\frac{{k}_{B}T}{6\pi \mu a}\). The thermal energy of the system is given by k_{B}T. Details of the model derivation are provided in the “Methods” section.
Equation (1) is the spin angular momentum balance of the spinners; the three righthand side terms account for the thermal noise, the torque on the spinners due to fluid drag, and the externally applied torque τ. Equation (2) is the linear momentum conservation for the fluid (the Navier–Stokes equations) with the addition of the last term on the righthand side, which drives the fluid flow due to the rotation of the spinners. Equation (3) is the incompressibility condition since we assume constant fluid density ρ.
It is worth noting that Eqs. (1)–(3) are generic for any spherical spinners. The specifics of the particular physical system are introduced in the form of the externally applied torque τ, which is what causes the particles to spin. In this paper we consider the case of Quincke spinners, studied experimentally in ref. ^{72}, τ = c^{−1}(P×E)_{z}, where P is the polarization vector of the individual spinner and E is the externally applied constant (DC) electric field, see Fig. 1b. The governing equations for the polarization vector P are provided in the “Methods” section, see Eqs. (28)–(29).
The complete model is provided in nondimensional form in the Methods section, see Eqs. (31)–(36). All equations are nondimensionalized using the spinner radius a as the characteristic length scale and the Maxwell–Wagner polarization time t_{Q}, which sets the magnitude of the rotation rate, as the characteristic time scale. The scaling of the physical variables and the definitions of the model parameters are summarized in Table 1. The scaling analysis shows that the most important parameter governing the behavior of the system is the Reynolds number for the fluid, \({\rm{Re}}=\rho {a}^{2}/\mu {t}_{Q}\). For \({\rm{Re}}\ll 1\), the effect of inertial terms in Eq. (2) of is small; fluid inertia becomes important for \({\rm{Re}}\gg 1\). In the rest of the paper, we explore the system behavior as Re increases. Unless otherwise noted, all simulations are performed with α = 0.25, γ = 1.1, κ = 0.1, D^{*} = 1, \({D}_{P}^{* }=0.1\), box size L = 480a, N = 1024, and time step dt = 0.00025t_{Q}.
Spinners stir largescale fluid flow
Figure 2a–c and Supplementary Movie 1 show that the spinner fluid undergoes a phase separation into clusters of CCW (yellow) and CW spinners (blue), in agreement with the simulations using a discrete spinner model^{38}. These clusters themselves are rotating, growing, separating, and reconnecting as also observed in the active coarsening continuum model in ref. ^{70}.
The flow is localized the interface between CW and CCW clusters, see Fig. 2b and Supplementary Movie 2. Physically, the fluid flow is canceled out in the region between the same spin spinners. Hence, even though the spinners might rotate very fast, the flow might be very weak, especially in the interior of the same spin regions; accordingly, while inertia might be very important in the angular conservation equation, it might be less so in the linear momentum conservation (Navier–Stokes equation for the fluid flow). At the boundary between clusters, opposite spin spinners pump fluid in the same direction causing bands of flow or edge currents, see an illustration of this in Fig. 2d.
Emergence of lanes depends on Reynolds number
An important question concerns the longterm behavior of a system composed of spinners that are initially randomly 50 CW : 50 CCW. Previous theoretical studies^{38,41} of discrete microspinners have shown that the system exhibits separation into samespin clusters that eventually evolve into emergent patterns, such as lanes or vortices. This work however was restricted to Stokes flow (\({\rm{Re}}=0\)), while recent colloidal experiments^{39,44} suggest a nonnegligible Reynolds number.
Using the continuum model, if we consider \({\rm{Re}}=0\) and an initial 50 CW : 50 CCW random configuration, lanes always eventually form, see Fig. 3a and Supplementary Movie 3, regardless of the initial configuration. Therefore, the continuum model is in agreement with the discrete spinner model. The formation of lanes can be qualitatively interpreted as a result of the fact that the Stokes flow corresponds to least dissipation. Segregation into the same spin regions is thus favored since the flow vanishes in the interior of the same spin region and the only remaining flow is confined to the interface between opposite spin regions. The flow (and thus the dissipation) is reduced if the interface between these same spin domains is minimized, which in the case of the periodic boundary conditions considered in our simulations, is a straight line. Conservation of mass further requires two interfaces with opposite flow directions resulting in a lane configuration.
A small, nonzero Reynolds number, such as \({\rm{Re}}=0.01\), leads to distorted lanes, see Fig. 3b and Supplementary Movie 4. These quasistable lanes have turbulentlike flow near the interfaces between the two oppositespin clusters. The lanes can also spontaneously dissolve into to a transient turbulentlike state and then reappear.
For small but finite Reynolds number, e.g., \({\rm{Re}}=0.1\) and greater, the system only exhibits a turbulentlike motion with no emergent collective largescale structure, as shown in Fig. 3c and Supplementary Movie 1. If the initial state is lanes, instead of a random configuration, the lanes become unstable and dissolve into turbulentlike motion, see Supplementary Movie 5.
For an even larger Reynolds number, such as \({\rm{Re}}=1\), the ratio of CW to CCW spinners is no longer conserved and the system exhibits a chiralitybreaking transition. As the inertial terms of the Navier–Stokes equation become more significant, the fluid vorticity is not solely determined by the spinner angular velocity distribution. Instead, the inertialinduced component of the fluid vorticity can cause the spinners to flip spin orientation. For this case, the fluid flow initially exhibits turbulentlike motion, but eventually begins to favor a single spin orientation, see Supplementary Movie 6. Clearly this behavior is not present in dry system of spinners.
We can quantitatively distinguish between the lane and the turbulentlike cases by studying the probability distribution function (PDF) of the fluid velocity, u. As shown in Fig. 4a, the PDF for lanes exhibits a clear bimodal distribution due to the two sides of the lane. For the turbulentlike case, Fig. 4b, the PDF is Gaussian unlike the nonGaussian behavior observed in experiments with colloidal spinners^{71,73}. Overly populated tails in our simulation were found to be due to underresolved simulations. Velocity and vorticity correlation functions for turbulentlike regimes are displayed in Fig. 4c, d.
Finally, it is worth mentioning that the discrete spinner model^{38,41} also observed the formation of vortices in the case of larger spinner concentrations, which is not captured by our continuum model. This deviation is likely because steric repulsion is neglected in the continuum model but becomes nonnegligible at higher spinner concentrations. Steric repulsion can be included in the continuum model by introducing a particle stress tensor^{27}.
Turbulentlike flow has a powerlaw energy spectrum
In many quasitwodimensional active systems exhibiting turbulentlike behavior, such as magnetic rollers^{39}, or swimming bacteria^{74}, the energy is injected at the microscopic scale. For example, rotation of bacterial flagella results in the onset of largescale collective behavior and socalled "active turbulence”^{6}, characterized by the inverse cascade^{75}. It is tempting to assume that in the Quincke system, the inverse cascade should be present as well due to the microscopic rotation of individual spinners. However, the situation is less clear in our model because the microscopic energy injection scale is coarsegrained in the continuum approximation and replaced by the effective driving torque τ in Eq. (1). This torque may slowly vary in space and does not produce the inverse cascade.
As the simulations show turbulentlike behavior, it is of interest to compute the energy spectrum, E(k), and the energy flux in kspace, Π_{E}. The energy spectrum E(k) is obtained by taking the Fourier transform of the spatial autocorrelation of u:
such that \({{\bf{r}}}_{0}={x}_{0}\hat{{\bf{x}}}+{y}_{0}\hat{{\bf{y}}}\) and 〈. . . 〉_{∣k∣=k} is the average over all wavenumbers k such that ∣k∣ = k. Correspondingly, the energy flux is obtained as follows
Here u^{<k} is the lowpass filtered velocity field with the wave numbers outside ∣k∣ < k set to zero:
and \(\tilde{{\bf{u}}}({\bf{k}},t)\) is the Fourier transform of u(r, t)^{76}.
A hallmark of the inverse cascade in 2D turbulence is the energy scaling E(k) ~ k^{−5/3} and negative value of the energy flux Π_{E}(k) for small wavenumbers k. As shown in Fig. 5, the energy cascade scales as k^{−5/3}. However, the energy flux is negative only in a very narrow range, Π_{E}(k) < 0 for k < k_{0} ≈ 0.016, which is not consistent with the energy cascade observed in 2D classical turbulence^{76,77,78} and colloidal experiments^{44,71}. We suspect that this behavior is due to the relatively small system size in the computations and coarsegaining of the microscopic energy infection scale. In the cases where lanes form at long times, the fluid flow initially exhibits transient turbulencelike motion with the same k^{−5/3} scaling, prior to settling into lanes.
The turbulence inertial range is characterized by two lengthscales, the Taylor microscale^{76}, and the integral scale^{79}. The Taylor microscale falls in between the largescale eddies and the smallscale eddies. The integral scale L_{I} can be extracted from the velocity autocorrelation function in Fig. 4c. It gives L_{I} ≈ 100 for particle Reynolds number Re = 0.1. The corresponding r.m.s. (root mean square) velocity u_{rms} can be determined from Fig. 4b and is about 5, resulting in the integral scale Reynolds number of the order of 50. The Taylor microscale λ is several times smaller, and is determined from the fluctuations of velocity and velocity gradients, \({({\partial }_{{\bf{r}}}\langle {\bf{u}}\rangle )}^{2}={{\bf{u}}}_{{\rm{rms}}}^{2}/{\lambda }^{2}\). It roughly can be estimated from the velocity gradients (vorticity) autocorrelation function, Fig. 4d, which gives λ ≈ 15.
Initially, the spinners begin to phase separate into larger and larger same spin clusters. However, the energy spectrum exhibits a peak that occurs at the length scale of the domain, ~L. This demonstrates the cluster size becomes limited by the size of the domain. The spatial correlation functions for the fluid flow^{73} and fluid vorticity are another useful metric for studying turbulent flows. As shown in Fig. 4, the turbulent flow case exhibits no characteristic length scale as seen in mesoscale turbulence. This behavior is consistent with colloidal spinner experiments^{71,73}.
Activity enhances transport
The capability of the spinner system for active transport can be quantified by tracking the position of point particles s_{i} that are being passively advected by the fluid flow, ∂s_{i}/∂t = u(s_{i}, t) and computing the meansquaredisplacement (MSD), \(\left\langle  {{\bf{s}}}_{i}(t+{t}_{0}){{\bf{s}}}_{i}({t}_{0}){ }^{2}\right\rangle \). In addition, the mean square velocity (MSV) for the fluid flow, \(\left\langle  {\bf{u}}{ }^{2}\right\rangle \), is another useful metric for the studying the active transport of the system.
As shown in Fig. 6a, the MSD initially exhibits a transient ballistic regime for all shown \({\rm{Re}}\) values. Counterintuitively, the MSD and MSV decrease as \({\rm{Re}}\) increases, Fig. 6b. For increasing \({\rm{Re}}\), the spinnerdriven flow begins to compete with the effects from the inertial terms. Also, the force term in Eq. (2), accounting for the spinners disturbing the fluid, is proportional to viscosity. Therefore, the disturbance in the fluid generated by the spinners does not get damped down with higher fluid viscosity.
The curves for different \({\rm{Re}}\) values begin to qualitatively diverge in the long term. The lane formation cases, \({\rm{Re}}=0\) and \({\rm{Re}}=0.01\), show sustained ballistic motion in MSD (~t^{2}) and a relatively constant MSV, as all the motion is confined to the interface of the lanes. Yet for sustained turbulentlike behavior, \({\rm{Re}}=0.1\) and \({\rm{Re}}=1\), the MSD shifts away from a ballistic regime closer towards a diffusive regime (~t). However, a reliable evaluation of the diffusion coefficient would require a much longer integration time. For Re = 1, the MSV also exhibits an eventual decrease. This is due to the chirality breaking of the spinner orientation.
Conclusions
To summarize, we develop a continuum coarsegrained model for a suspension of discrete microspinners. Assuming Quincke rotation, we then present numerical simulations that demonstrate the emergence of lanes of samespin fluid and turbulentlike behavior depending on the Reynolds number. Our work sheds light on active spinner materials and makes testable predictions for experiments on the effects of fluid inertia and chirality symmetry breaking.
For the case of the Stokes flow, the formation of lanes of spinners is favored as it minimizes the interfaces between counterspin clusters, thus minimizing the energy of the system. This behavior is in agreement with the existing discrete spinner models^{36,38}. Yet, even small inertia causes turbulentlike behavior, which becomes more pronounced with increasing Re: turbulent lanes for \({\rm{Re}} \sim 0.01\) and sustained turbulentlike flow for \({\rm{Re}}\ge 0.1\). We also quantitatively characterize lane formation and the turbulentlike flow through the mean square displacement, MSV, and the energy spectrum statistics. Furthermore, for nonzero Reynolds numbers we observe that initially 50:50 CW/CCW population of spinners exhibit a chirality symmetry breaking transition and consequent condensation to a single sense of rotation state.
While this work used periodic boundaries, future work could apply this model with both fixed, impenetrable boundaries and soft, deformable boundaries, as well as arrays of artificial obstacles. Another possible avenue is the case of nonuniform spinner densities. As the spinner densities, in experiments, are often nonconstant in space and time, it would be of interest to see how this would affect the collective and turbulent behavior seen in this study. The model can be modified to analyze other experimental systems, e.g., magnetic spinners at waterair interface^{25,39} and could also be easily generalized for a threedimensional suspension of neutrally buoyant microrotors.
Methods
Derivation of the continuum model
Let us consider N spherical spinners of radius a and mass m confined to a 2D plane of area A. The translational motion of a spinner is the result of being passively advected by the surrounding fluid flow, u, and Brownian diffusion. The translational velocity of an individual spinner located at position r is then:
The thermal noise of the system is modeled by a random variable ξ(t) that is Wiener process governed by the standard normal distribution. In the lowReynolds number limit, the diffusion constant is D = k_{B}T/6πμa, where k_{B}T is the thermal energy and μ is the fluid viscosity. We also assume low spinner concentration such that the effect of steric repulsion is negligible.
We introduce a probability density, Ψ(r, Ω, t), of finding a spinner at position r and with angular velocity Ω. Since there are N total spinners, the normalization for Ψ is given as
Using Eqs. (7)–(14) and the Fokker–Planck equation^{80}, it can be shown that the governing equation for Ψ is given by the following Smoluchowski equation
assuming an incompressible fluid (∇ ⋅ u).
From this point on the derivation of our continuum model follows ref. ^{67}. It is useful to define a function \(\left\langle \right..{\rangle }_{{{\Omega }}}\) such that
We will therefore define the local spinner density as c(r, t) = 〈1〉_{Ω}, the local angular velocity per unit area as \(\overline{\omega }({\bf{r}},t)={\langle {{\Omega }}\rangle }_{{{\Omega }}}\), and the local external torque per unit area as \(\overline{\tau }({\bf{r}},t)={\langle {\tau }_{e}\rangle }_{{{\Omega }}}\).
Integrating Eq. (9) leads to
We can ignore the boundary term in Eq. (11) as spinners with ∣Ω∣ → ∞ are nonphysical and inherently have zero probability, Ψ = 0.
Therefore Eq. (12) gives the evolution equation for the spinner density of the system. It is worth noting that, with appropriate boundary conditions, c is conserved and ∫_{A}cdA = N according to Eq. (8).
Spin angular momentum equation
If we multiply Eq. (9) by Ω and integrate with respect to Ω, it is straightforward to show that
We can again ignore the boundary term as spinners with ∣Ω∣ → ∞ are nonphysical and have zero probability.
The conservation of spin angular momentum of a single spinner is
where \({I}_{p}=\frac{2}{5}m{a}^{2}\) is moment of inertia and ζ_{p} = 8πμa^{3} is the friction coefficient for a sphere and the externally applied torque on the spinner is τ_{e}(Ω, t). The effect of thermal noise is ignored in Eq. (14) because the spinners rotation rate is very high and thus the rotational Peclet number Ω/D_{r} ≫ 1, where D_{r} = k_{B}T/(8πμa^{3}).
Using Eq. (14), Eq. (13) simplifies to
Furthermore, if we assume a constant spinner density c and define \(\omega =\overline{\omega }/c\), \(\tau =\overline{\tau }/c\), Eq. (15) can be simplified to
The above derivation is based on the assumption that a spinner is not selfpropelled. In other words, \({\bf{u}}({\bf{r}}^{\prime} ,t)\) is dependent only on spinners located at \({\bf{r}}\ne {\bf{r}}^{\prime} \).
Fluid flow
Assuming Stokes flow, the fluid flow generated by a single spinner can be approximated as a point rotlet^{81}, with torque \(8\pi \mu {a}^{3}{{\Omega }}\hat{{\bf{z}}}\). The resulting fluid flow is given by the solution to the following forced Stokes equation:
For distribution of N spinners with center of mass position r_{0i} and angular velocity Ω_{i}, the fluid flow can be written as
The pressure distribution, p(r, z, t) can be computed similarly. By multiplying Eq. (17) by Ψ(r_{0}) and integrating, the Stokes equation for the spinner distribution can be shown to be
For flows with nonnegligible inertia, Eq. (20) can be approximated for low Reynolds flow as
Since the majority of experiments and previous theoretical work involve an effectively twodimensional system^{25,39,60,61}, we will therefore consider a twodimensional fluid flow in the plane of the spinners. For the case of twodimensional flow, a frictional drag term, −βu, can be added to the right side of Eq. (21). For small β values, this term does not qualitatively change the observed behaviors. The effect of larger β is more extensively studied in ref. ^{70}. The form of Eqs. (1)–(3) is consistent with the continuum model derived for ferrofluids^{82} and the model used to analytically study the hydrodynamics for a system of microrotors^{83,84}.
Spinner’s torque
A common spinner system consists of ferromagnetic colloids in either a rotating or oscillating magnetic field^{42}. The external torque is then given by
where μ_{0} permeability of vacuum, magnetization is M, and the magnetic field is H. An oscillating magnetic field allows for spontaneoussymmetry breaking in the direction of rotation. Therefore, spinners can have either a CW or CCW spin^{44,85}. However, a rotating magnetic field predetermines the spinner’s direction of rotation, causing the spinners to be either all CW or all CCW^{46}.
We, instead, consider a system of Quincke spinners driven by a DC electric field. Quincke rotation has the advantage of being well studied^{86,87,88,89} and it allows for simultaneous populations of CW and CCW spinners.
Quincke rotation
For a fluid with conductivity and permittivity (σ_{1},ϵ_{1}) and a suspended spherical particle with conductivity and permittivity (σ_{2},ϵ_{2}), Quincke rotation occurs when
and the external electric field ∣E∣ is above a threshold given by
where
is the Maxwell–Wagner relaxation time. The characteristic angular velocity for a Quincke spinner is
Neglecting the higherorder electromagnetic interactions^{89} between spinners, we instead focus on the spinners’ hydrodynamic interactions. If R is the distance between two rotors, the hydrodynamic forces scale as ~R^{−2} while the electromagnetic forces^{86} scale as ~R^{−4}.
The external torque from the electric field is then given as
with electric polarization density P(r, t) (dipole moment per unit area) governed by
where the equilibrium polarization density is P_{eq} is
and
Eq. (28)–Eq. (29) are provided in ref. ^{83} with the inclusion an artificial diffusion term for numerical stability.
Continuum model of a Quincke spinner fluid
We will assume a constant, uniform electric field E parallel to x–y plane. Without loss of generality, we can define \({\bf{E}}=E\hat{{\bf{x}}}\).
We nondimensionalize the system of equations in order to reduce the number of parameters. Equations (1)–(3) and Eqs. (26)–(29) can be nondimensionalized as Eqs. (31)–(36) using the scalings and dimensionless parameters shown in Table 1. X^{*} is the corresponding dimensionless value of a variable X.
The system involves seven dimensionless parameters: the dimensionless spinner diffusion coefficient (D^{*}), the particle Reynolds number (κ^{−1}), the strength of the electric field relative to critical field (γ), the Reynolds number (Re) for the fluid, and the percentage of area covered by spinners (α), and a small artificial diffusion constant (\({D}_{P}^{* }\)) for the polarization evolution equation. Since Quincke rotation only occurs for E > E_{c}, we will operate for γ > 1. For the dimensionless form of the equations see Supplementary Note 1.
The parameters γ and α control how fast the spinners rotate and the corresponding effect on the fluid flow. D^{*} determines the characteristic length for the interface between two oppositespinning clusters. Therefore, the only parameters that could significantly change the qualitative behavior of the system are Re and κ.
In the initialization of ω^{*}, each grid point is randomly assigned a small positive value (CW) or a small negative value (CCW).
Scaling and dimensionless parameters
We scale the length by the particle radius a, time by the t_{Q} is the Maxwell–Wagner relaxation time t_{Q}. The electric field E is normalized by the critical electric field E_{c}. Here N is the number of spinners, A is the area covered by spinners, and μ is the fluid dynamic viscosity. Correspondingly, the polarization P is scaled by Nϵ/A, where N is the number of spinners, A is the area covered by spinners, and ϵ is given by Eq. (30). Finally, the pressure p is normalized by μ/t_{Q}, where μ is the fluid dynamic viscosity. Definitions of the model dimensionless parameters are given in Table 1.
Numerical methods
The reported results are numerical simulations of the continuum model using a square domain of length L, double periodic boundary conditions. The code is highly parallelized using CUDA to run on a NVIDIA graphics card. Spatial derivatives are computed using a pseudospectral method^{90}. All Fourier and inverse transforms are computed using the fast Fourier transform (FFT). Equations (31)–(34) are then timestepped using firstorder exponential time differences^{91}. For Stokes flow, the fluid velocity is solved using the streamfunction formulation^{92}. For nonStokes flow, the fluid velocity is timestepped using the method of Chorin Projection^{93}. Details about the numerical implementation can be found in Supplementary Note 2.
Data availability
The data that support the findings of this study are available in the main text and supplementary information. Additional information is available from the corresponding author upon request.
Code availability
The code is available from the corresponding author upon reasonable request.
References
Gompper, G. et al. The 2020 motile active matter roadmap. J. Phys.: Condens. Matter 32, 193001 (2020).
Vicsek, T. & Zafeiris, A. Collective motion. Phys. Rep. 517, 71 – 140 (2012).
Dombrowski, C., Cisneros, L., Chatkaew, S., Goldstein, R. & Kessler, J. Selfconcentration and largescale coherence in bacterial dynamics. Phys. Rev. Lett. 93, 098103 (2004).
Zhang, H. P., Be’er, A., Florin, E.L. & Swinney, H. L. Collective motion and density fluctuations in bacterial colonies. Proc. Natl Acad. Sci. USA 107, 13626–13630 (2010).
Sokolov, A. & Aranson, I. S. Physical properties of collective motion in suspensions of bacteria. Phys. Rev. Lett. 109, 189–5 (2012).
Dunkel, J. et al. Fluid dynamics of bacterial turbulence. Phys. Rev. Lett. 110, 228102 (2013).
Wioland, H., Woodhouse, F. G., Dunkel, J., Kessler, J. O. & Goldstein, R. E. Confinement stabilizes a bacterial suspension into a spiral vortex. Phys. Rev. Lett. 110, 268102 (2013).
Wioland, H., Woodhouse, F. G., Dunkel, J. & Goldstein, R. E. Ferromagnetic and antiferromagnetic order in bacterial vortex lattices. Nat. Phys. 12, 341–U177 (2016).
Nishiguchi, D., Aranson, I. S., Snezhko, A. & Sokolov, A. Engineering bacterial vortex lattice via direct laser lithography. Nat. Commun. 9, 1–8 (2018).
Reinken, H. et al. Organizing bacterial vortex lattices by periodic obstacle arrays. Commun. Phys. 3, 1–9 (2020).
Howse, J. R. et al. Selfmotile colloidal particles: From directed propulsion to random walk. Phys. Rev. Lett. 99, 048102 (2007).
Ibele, M., Mallouk, T. & Sen, A. Schooling behavior of lightpowered autonomous micromotors in water. Angew. Chem. Int. Ed. 48, 3308–3312 (2009).
Palacci, J., CottinBizonne, C., Ybert, C. & Bocquet, L. Sedimentation and effective temperature of active colloidal suspensions. Phys. Rev. Lett. 105, 088304 (2010).
Baker, R. et al. Fight the flow: the role of shear in artificial rheotaxis for individual and collective motion. Nanoscale 11, 10944–10951 (2019).
Baker, R. D. et al. Shapeprogrammed 3d printed swimming microtori for the transport of passive and active agents. Nat. Commun. 10, 1–10 (2019).
Sanchez, T., Chen, D. N., DeCamp, S., Heymann, M. & Dogic, Z. Spontaneous motion in hierarchically assembled active matter. Nature 491, 431+ (2012).
Zhou, S., Sokolov, A., Lavrentovich, O. D. & Aranson, I. S. Living liquid crystals. Proc. Natl Acad. Sci. USA 111, 1265–1270 (2014).
Genkin, M. M., Sokolov, A., Lavrentovich, O. D. & Aranson, I. S. Topological defects in a living nematic ensnare swimming bacteria. Phys. Rev. X 7, 011029 (2017).
Gangwal, S., Cayre, O. J., Bazant, M. Z. & Velev, O. D. Inducedcharge electrophoresis of metallodielectric particles. Phys. Rev. Lett. 100, 058302 (2008).
Bricard, A., Caussin, J.B., Desreumaux, N., Dauchot, O. & Bartolo, D. Emergence of macroscopic directed motion in populations of motile colloids. Nature 503, 95–98 (2013).
Bricard, A. et al. Emergent vortices in populations of colloidal rollers. Nat. Commun. 6, 378–8 (2015).
Ma, F., Yang, X., Zhao, H. & Wu, N. Inducing propulsion of colloidal dimers by breaking the symmetry in electrohydrodynamic flow. Phys. Rev. Lett. 115, 208302 (2015).
Driscoll, M. et al. Unstable fronts and motile structures formed by microrollers. Nat. Phys. 13, 375–379 (2016).
Kaiser, A., Snezhko, A. & Aranson, I. S. Flocking ferromagnetic colloids. Sci. Adv. https://advances.sciencemag.org/content/3/2/e1601469 (2017).
Kokot, G. & Snezhko, A. Manipulation of emergent vortices in swarms of magnetic rollers. Nat. Commun. 9, 1–7 (2018).
Karani, H., Pradillo, G. E. & Vlahovska, P. M. Tuning the random walk of active colloids. Phys. Rev. Lett. 123, 208002 (2019).
Han, K. et al. Emergence of selforganized multivortex states in flocks of active rollers. Proc. Natl Acad. Sci. USA 117, 9706–9711 (2020).
Aranson, I. S. Active colloids. Phys.Uspekhi 56, 79–92 (2013).
Elgeti, J., Winkler, R. G. & Gompper, G. Physics of microswimmers – single particle motion and collective behavior: a review. Rep. Prog. Phys. 78, 056601 (2015).
Zhang, J., Luijten, E., Grzybowski, B. A. & Granick, S. Active colloids with collective mobility status and research opportunities. Chem. Soc. Rev. 46, 5551–5569 (2017).
Zoettl, A. & Stark, H. Emergent behavior in active colloids. J. Phys.: Condens. Matter 28, 253001 (2016).
Illien, P., Golestanian, R. & Sen, A. Fuelled motion: phoretic motility and collective behaviour of active colloids. Chem. Soc. Rev. 46, 5508–5518 (2017).
Aubret, A., Ramananarivo, S. & Palacci, J. Eppur si muove, and yet it moves: Patchy (phoretic) swimmers. Curr. Opin. Colloid Interface Sci. 30, 81–89 (2017).
Saintillan, D. Rheology of active fluids. Annu. Rev. Fluid Mech. 50, 563–592 (2018).
Driscoll, M. & Delmotte, B. Leveraging collective effects in externally driven colloidal suspensions: experiments and simulations. Curr. Opin. Colloid Interface Sci. 40, 42–57 (2019).
Nguyen, N. H. P., Klotsa, D., Engel, M. & Glotzer, S. C. Emergent collective phenomena in a mixture of hard shapes through active rotation. Phys. Rev. Lett. 112, 1–5 (2014).
Lushi, E. & Vlahovska, P. M. Periodic and chaotic orbits of planeconfined microrotors in creeping flows. J. Nonlinear Sci. 25, 1–13 (2015).
Yeo, K., Lushi, E. & Vlahovska, P. M. Collective dynamics in a binary mixture of hydrodynamically coupled microrotors. Phys. Rev. Lett. 114, 1–5 (2015).
Kokot, G., Piet, D., Whitesides, G. M., Aranson, I. S. & Snezhko, A. Emergence of reconfigurable wires and spinners via dynamic selfassembly. Sci. Rep. 5, 9528 (2015).
Goto, Y. & Tanaka, H. Purely hydrodynamic ordering of rotating disks at a finite Reynolds number. Nat. Comm. 6, 5994 (2015).
Yeo, K., Lushi, E. & Vlahovska, P. M. Dynamics of inert spheres in active suspensions of microrotors. Soft Matter 12, 5645–5652 (2016).
Snezhko, A. Complex collective dynamics of active torquedriven colloids at interfaces. Curr. Opin. Colloid Interface Sci. 21, 65–75 (2016).
Steimel, J. P., Aragones, J. L., Hu, H., Qureshi, N. & AlexanderKatz, A. Emergent ultra–longrange interactions between active particles in hybrid active–inactive systems. Proc. Natl Acad. Sci. USA 113, 4652–4657 (2016).
Kokot, G. et al. Active turbulence in a gas of selfassembled spinners. Proc. Natl Acad. Sci. USA 114, 1–13 (2017).
Banerjee, D., Souslov, A., Abanov, A. G. & Vitelli, V. Odd viscosity in chiral active fluids. Nat. Commun. 8, 1573 (2017).
Soni, V. et al. The odd free surface flows of a colloidal chiral fluid. Nat. Phys. 15, 1188–1194 (2019).
Shen, Z., Würger, A. & Lintuvuori, J. S. Hydrodynamic selfassembly of active colloids: chiral spinners and dynamic crystals. Soft Matter 15, 1508–1521 (2019).
Scheibner, C. et al. Odd elasticity. Nat. Phys. 16, 475+ (2020).
Oppenheimer, N., Stein, D. B. & Shelley, M. J. Rotating membrane inclusions crystallize through hydrodynamic and steric interactions. Phys. Rev. Lett. 123, 148101 (2019).
Shen, Z. & Lintuvuori, J. S. Twophase crystallization in a carpet of inertial spinners. Phys. Rev. Lett. 125, 228002 (2020).
Zhang, B., Sokolov, A. & Snezhko, A. Reconfigurable emergent patterns in active chiral fluids. Nat. Commun. 11, 4401 (2020).
Fily, Y., Baskaran, A. & Marchetti, M. Cooperative selfpropulsion of active and passive rotors. Soft Matter 8, 3002 (2012).
Petroff, A. P., Wu, X.L. & Libchaber, A. Fastmoving bacteria selforganize into active twodimensional crystals of rotating cells. Phys. Rev. Lett. 114, 3474–6 (2015).
Wang, Y. et al. Dynamic interactions between fast microscale rotors. J. Am. Chem. Soc. 131, 9926+ (2009).
Han, K. et al. Reconfigurable structure and tunable transport in synchronized active spinner materials. Sci. Adv. https://advances.sciencemag.org/content/6/12/eaaz8535 (2020).
Sapozhnikov, M., Tolmachev, Y. V., Aranson, I. & Kwok, W.K. Dynamic selfassembly and patterns in electrostatically driven granular media. Phys. Rev. Lett. 90, 114301 (2003).
Shields IV, C. W. et al. Supercolloidal spinners: complex active particles for electrically powered and switchable rotation. Adv. Funct. Mater. 28, 1–7 (2018).
Pradillo, G. E., Karani, H. & Vlahovska, P. M. Quincke rotor dynamics in confinement: rolling and hovering. Soft Matter 15, 6564–6570 (2019).
Tsai, J. C., Ye, F., Rodriguez, J., Gollub, J. P. & Lubensky, T. C. A chiral granular gas. Phys. Rev. Lett. 94, 241–4 (2005).
Sabrina, S. et al. Shapedirected microspinners powered by ultrasound. ACS Nano 12, 2939–2947 (2018).
Workamp, M., Ramirez, G., Daniels, K. E. & Dijksman, J. A. Symmetryreversals in chiral active matter. Soft Matter 14, 5572–5580 (2018).
Farhadi, S. et al. Dynamics and thermodynamics of airdriven active spinners. Soft Matter 14, 5588–5594 (2018).
Spellings, M. et al. Shape control and compartmentalization in active colloidal cells. Proc. Natl Acad. Sci. USA 112, E4642–E4650 (2015).
van Zuiden, B. C., Paulose, J., Irvine, W. T. M., Bartolo, D. & Vitelli, V. Spatiotemporal order and emergent edge currents in active spinner materials. Proc. Natl Acad. Sci. USA 113, 12919–12924 (2016).
Scholz, C., Engel, M. & Pöschel, T. Rotating robots move collectively and selforganize. Nat. Commun. 9, 1–8 (2018).
Saintillan, D. & Shelley, M. J. Instabilities and pattern formation in active particle suspensions: kinetic theory and continuum simulations. Phys. Rev. Lett. 100, 600–604 (2008).
Saintillan, D. & Shelley M.J. Theory of Active Suspensions. In Complex Fluids in Biological Systems. Biological and Medical Physics, Biomedical Engineering. (ed Spagnolie, S.) (Springer, New York, NY, 2015).
Rapp, L., Bergmann, F. & Zimmermann, W. Systematic extension of the CahnHilliard model for motilityinduced phase separation. Eur. Phys. J. E 42, 71–10 (2019).
James, M., Bos, W. J. T. & Wilczek, M. Turbulence and turbulent pattern formation in a minimal model for active fluids. Phys. Rev. Fluids 3, 1–9 (2018).
Sabrina, S., Spellings, M., Glotzer, S. C. & Bishop, K. J. M. Coarsening dynamics of binary liquids with active rotation. Soft Matter 11, 8409–8416 (2015).
Kokot, G. et al. Active turbulence in a gas of selfassembled spinners. Proc. Natl Acad. Sci. USA 114, 12870–12875 (2017).
Pradillo, G. E., Karani, H. & Vlahovska, P. M. Quincke rotor dynamics in confinement: rolling and hovering. Soft Matter 15, 6564–6570 (2019).
Snezhko, A. & Aranson, I. S. Velocity statistics of dynamic spinners in outofequilibrium magnetic suspensions. Soft Matter 11, 6055–6061 (2015).
Sokolov, A., Aranson, I. S., Kessler, J. O. & Goldstein, R. E. Concentration dependence of the collective dynamics of swimming bacteria. Phys. Rev. Lett. 98, 158102 (2007).
Słomka, J. & Dunkel, J. Spontaneous mirrorsymmetry breaking induces inverse energy cascade in 3d active fluids. Proc. Natl Acad. Sci. USA 114, 2119–2124 (2017).
Alexakis, A. & Biferale, L. Cascades and transitions in turbulent flows. Phys. Rep. 767, 1–101 (2018).
Boffetta, G. & Ecke, R. E. Twodimensional turbulence. Annu. Rev. Fluid Mech. 44, 427–451 (2012).
Chen, S. et al. Physical mechanism of the twodimensional inverse energy cascade. Phys. Rev. Lett. 96, 084502 (2006).
O’Neill, P. L. et al. Autocorrelation functions and the determination of integral length with reference to experimental and numerical data. In 15th Australasian Fluid Mechanics Conference, vol. 1, 1–4 (Univ. of Sydney Sydney, NSW, Australia, 2004).
Risken, H. & Haken, H.The FokkerPlanck Equation: Methods of Solution and Applications Second Edition (Springer, 1989).
Chwang, A. & Wu, T. Hydromechanics of lowReynoldsnumber flow. Part 2. Singularity method for Stokes flows. J. Fluid Mech. 67, 787–815 (1975).
Rosensweig, R. E. Continuum equations for magnetic and dielectric fluids with internal rotations. J. Chem. Phys. 121, 1228–1242 (2004).
Huang, H.F., Zahn, M. & Lemaire, E. Continuum modeling of microparticle electrorotation in Couette and Poiseuille flows—The zero spin viscosity limit. J. Electrost. 68, 345–359 (2010).
Huang, H.F., Zahn, M. & Lemaire, E. Negative electrorheological responses of micropolar fluids in the finite spin viscosity small spin velocity limit. I. Couette flow geometries. J. Electrost. 69, 442–455 (2011).
Matsunaga, D. et al. Controlling collective rotational patterns of magnetic rotors. Nat. Commun. 10, 1–9 (2019).
Das, D. & Saintillan, D. Electrohydrodynamic interaction of spherical particles under Quincke rotation. Phys. Rev. E 87, 194–14 (2013).
Brosseau, Q., Hickey, G. & Vlahovska, P. M. Electrohydrodynamic Quincke rotation of a prolate ellipsoid. Phys. Rev. Fluids 2, 251–11 (2017).
Pradillo, G. E., Karani, H. & Vlahovska, P. M. Quincke rotor dynamics in confinement: rolling and hovering. Soft Matter 15, 6564–6570 (2019).
Hu, Y., Vlahovska, P. M. & Miksis, M. J. Colloidal particle electrorotation in a nonuniform electric field. Phys. Rev. E 97, 1–14 (2018).
Fornberg, B. A Practical Guide to Pseudospectral Methods. (Cambridge University Press, 1998).
Cox, S. M. & Matthews, P. C. Exponential time differencing for stiff systems. J. Comput. Phys. 176, 430–455 (2001).
Lamb, H. Hydrodynamics. (Cambridge University Press, Cambridge, 1916).
Chorin, A. J. Numerical solution of the NavierStokes equations. Math. Comput. 22, 745–762 (1968).
Lemaire, E. & Lobry, L. Chaotic behavior in electrorotation. Phys. A 314, 663–671 (2002).
Acknowledgements
Research of I.S.A. was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, under Award DESC0020964. P.V. was supported by NSF award CBET1704996.
Author information
Authors and Affiliations
Contributions
P.M.V. and I.S.A. designed the research. C.J.R. developed numerical tools and performed numerical studies. All authors analyzed the data, discussed the results, and wrote the paper.
Corresponding authors
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Reeves, C.J., Aranson, I.S. & Vlahovska, P.M. Emergence of lanes and turbulentlike motion in active spinner fluid. Commun Phys 4, 92 (2021). https://doi.org/10.1038/s42005021005962
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/s42005021005962
This article is cited by

Shaping active matter from crystalline solids to active turbulence
Nature Communications (2024)

Simultaneous emergence of active turbulence and odd viscosity in a colloidal chiral active system
Communications Physics (2023)

Diffusive regimes in a twodimensional chiral fluid
Communications Physics (2022)

Fluctuating hydrodynamics of chiral active fluids
Nature Physics (2021)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.