Mesoscopic turbulence and local order in Janus particles self-propelling under an ac electric field

To elucidate mechanisms of mesoscopic turbulence exhibited by active particles, we experimentally study turbulent states of nonliving self-propelled particles. We realize an experimental system with dense suspensions of asymmetrical colloidal particles (Janus particles) self-propelling on a two-dimensional surface under an ac electric field. Velocity fields of the Janus particles in the crowded situation can be regarded as a sort of turbulence because it contains many vortices and their velocities change abruptly. Correlation functions of their velocity field reveal the coexistence of polar alignment and antiparallel alignment interactions, which is considered to trigger mesoscopic turbulence. Probability distributions of local order parameters for polar and nematic orders indicate the formation of local clusters with particles moving in the same direction. A broad peak in the energy spectrum of the velocity field appears at the spatial scales where the polar alignment and the cluster formation are observed. Energy is injected at the particle scale and conserved quantities such as energy could be cascading toward the larger clusters.

Figure 1

(a) Janus particles used in our experiments are polystyrene beads whose hemispheres are covered with chromium. (b) The suspension of the Janus particles was sandwiched between two horizontal transparent ITO electrodes. After sedimentation of the Janus particles, an ac electric field was applied in the $z$ direction and the Janus particles moved in a quasi-two-dimensional horizontal plane ($xy$ plane). Images were captured through the bottom electrode using a microscope and a CCD camera.

Figure 2

A trajectory of a Janus particle for 15 seconds overlaid on the snapshot at the last moment. Due to rotational Brownian motion and frequent collisions of the particles, their trajectories are not straight but curved. The polarities of the particles are invisible with this magnification. The scale bar corresponds to 10 $\mu \mathrm{m}$ (see movie 1 of the Supplemental Material [33] ).

Figure 3

Log-log plot of the power spectral density of the velocity field of the Janus particles. The power spectrum was calculated according to Eq. (2) using the experimentally obtained velocity correlation function defined as Eq. (1). The velocity correlation function $C\left(R\right)$ is shown in the inset. The blue and green lines are slopes with the exponents 0.5 and 1 respectively to guide the eye. Wave number is normalized by the wave number corresponding to the diameter of the Janus particle. The vertical axis is not normalized.

Figure 4

(a) Color map of the velocity correlation function $C^{\prime }(R,\psi )$ defined in Eq. (3). The coordinate is $x^{\prime }=Rcos\psi$ and $y^{\prime }=Rsin\psi$. Note that $x^{\prime }$ and $y^{\prime }$ are different from the axes defined in Fig. 1. The filled gray circle represents a reference Janus particle, whose direction of motion (the red arrow inside) is rearranged to the right ($x^{\prime }$ direction). The dashed circle corresponds to twice the radius, which the other particle cannot enter due to exclusive interaction. In reality, repulsive electrostatic interactions, surfactants absorbed on the colloidal surface, and the suspending fluid dominated by Stokesian dynamics keep the particles apart and they cannot get close to this dashed circle. No particles are detected in the white region. (b) Slices of (a) at $y^{\prime }=0$ ($x^{\prime }$ direction, parallel to the velocity, red circles) and $x^{\prime }=0$ ($y^{\prime }$ direction, perpendicular to the velocity, blue squares). Negative correlation in the $y^{\prime }$ direction shows antiparallel alignment, and positive correlation in the $x^{\prime }$ direction indicates that the particles are following other particles in front of them.

Figure 5

The local average values of the orientational order parameter $\left|\langle e^{i\theta }\rangle \right|$ and the nematic order parameter $\left|\langle e^{2i\theta }\rangle \right|$ can distinguish local states of collective behaviors.

Figure 6

The probability distributions of the order parameters (a) $\left|\langle e^{i\theta }\rangle \right|$ and (b) $\left|\langle e^{i\theta }\rangle \right|$ defined as Eq. (4) are calculated at the scale of $L=9,10,12,15\mu \mathrm{m}$ respectively. When $L$ is smaller than 12 $\mu \mathrm{m}$, both order parameters have nonzero peaks meaning that the particles are locally moving in the same direction but opposing lanes are not formed. These peaks gradually grow as $L$ is decreased. In the calculation of order parameters, the regions in the grid including fewer than three particles are neglected in order to extract the results coming from the collective dynamics.

Figure 7

(a) In a snapshot of the experimental movie, each particle is colored depending on its direction of instantaneous motion. Spaces between the particles are also colored according to average velocities of the nearby particles with Gaussian weight (see Appendix pp1 for detail). We can see some regions connected with same colors (see Supplemental Material movies 3 and 4 [33]). (b)Trajectories of the particles moving in the same direction are overlaid on the snapshot at the last moment. Different colors indicate different clusters (see Supplemental Material movie 2 [33]). The scale bars correspond to 10 $\mu \mathrm{m}$.

Figure 8

Color map of the vorticity. Various scales of vortices with positive and negative vorticity coexist.

Figure 9

The spherical coordinates used for calculation. The origin is at the center of the Janus particle. The right hemisphere ($x>0$ or $\left|\theta \right|>\pi /2$) is covered with metal. Note that the axes defined in this Appendix pp3 are different from those used in Figs. 1, 4, and 8.

Figure 10

Values of the coefficients of spherical harmonics (a) $B_{l,m}$ in a linear plot and (b) $\left|B_{l,m}\right|$ in a log-log plot. $B_{l,m}$ has nonzero values only when $m=0,\pm 2$. For any $l, B_{l,+2}=B_{l,-2}$ holds due to symmetry in the sign of $\varphi$.

Figure 11

Streamlines of the flow field around a Janus particle. (a),(c),(e) Flow parallel to the electric field. (b),(d),(f) Flow perpendicular to the electric field. (a) and (b) are flow fields in the particle frame, and (c)–(f) are those in the laboratory frame. The far-field flow shown in (e) is reminiscent of that of pusher-type microswimmers. The radius of the particle is set to 1. The color indicates the amplitude of the flow field. The speed of the flow is faster in the red region.

Figure 12

The parallel configuration of the two Janus particles moving in the same direction is unstable. Because the observed plane in the experiment is the plane perpendicular to the applied electric field, the flow shown in this figure is accountable for the hydrodynamic interaction. The hydrodynamic flow between the two particles causes them to repel each other. The radius of the particle is set to 1.

Figure 13

Pair correlation function $g\left(r\right)$ has the same form as simple liquids. There is no positional order in the crowd of the Janus particles. Error bars are smaller than symbols.