Algebraic correlations and anomalous fluctuations in ordered flocks of Janus particles fueled by an AC electric field

We study the polar collective dynamics of Janus colloidal particles fueled by an AC electric field. When the density is high enough, the polar interactions between the particles induce a polar orientationally ordered state, which exhibits features reminiscent of the Vicsek model such as true long-range order and giant number fluctuations. Independent measurements of the polarity and velocity at the single particle level allowed us to investigate the single particle dynamics within the ordered state. We discovered theoretically unaddressed statistical properties of the ordered state such as the asymmetric relation of polarity and velocity, enhanced rotational diffusion stronger than in the disordered state, and an algebraic autocorrelation of the polarity. Our experimental findings, at the crossroads of the Vicsek physics and the active Brownian particles physics, shed light on the so-far-unexplored physics arising from the interplay between the polarity and the velocity.

Figure 1

Schematic image of the experimental system. (a) A silica-based Janus particle with a metallic titanium hemisphere. (b) The suspension of Janus particles were sandwiched by two ITO coated electrodes. The particles sediment to the surface of the ITO electrode, which leads to quasi-two-dimensional motion on the $xy$ plane.

Figure 2

Dynamics of Janus particles in the disordered and ordered states. (a), (b) Trajectories of the particles in the disordered state (3 s) and the ordered state (2 s), respectively. The bright and dark colors indicate the early and late times of each trajectory, respectively. (c), (d) Velocity distribution of the particles in the disordered (9909 particles) and ordered state (40205 particles), respectively [46]. The mean speeds for the disordered and ordered state were ${\overline{v}}_d=14.6\mu \mathrm{m}$/s and ${\overline{v}}_o=8.5\mu \mathrm{m}$/s, respectively. (e) Binarized experimental snapshot of the Janus particles in the ordered state. Each particle is colored based on the deviation of its polarity from the global order $|\theta -\langle \theta \rangle |$ (radian). The black arrow shows the direction of global order $\langle \theta \rangle$.

Figure 3

Statistics of binary interactions of Janus particles. (a) Schematic picture describing the relative angle of particle pairs. (b) The relative polarity angle $\mathrm{\Delta }\theta$ distribution of Janus particles in a low-density situation. $R$ denotes the distance of two isolated Janus particles. (c) Typical trajectories of two Janus particles when they are in close distance.

Figure 4

The polar order parameter for the ordered and disordered states. (a) Log-log plot of the polar order parameter $C$ vs $S$, the area of the ROI. The blue and red points show the order parameter for the ordered and disordered state, respectively. (b) The same data as in (a) for the ordered state in a magnified range (log-log). (c) The same data as in (b) with the asymptotic value $C_{\infty }=0.74$ subtracted (log-log). The solid black line shows the nonlinear fit: $C=C_{\infty }+k{S}^{-\gamma /2}$ with $C_{\infty }=0.74$, $k=3.1$, and $\gamma /2=0.53$. The data for ROIs with $S>{10}^5\mu {\mathrm{m}}^2$ have been excluded from the nonlinear fitting due to the existence of counterflows in the system. Error bars in (a), (b), (c): standard error.

Figure 5

Autocorrelation functions. (a) Polarity autocorrelation function in the ordered (red) and disordered state (blue) on a semilog scale. $\tau$ denotes the time delay. The dashed line shows $exp(-2D_r\tau )$ where $D_r\simeq 1/24.2{\mathrm{s}}^{-1}$ is the rotational diffusion constant for a $3.17\mu \text{m}$ diameter particle derived from the Einstein-Stokes relation. (b) The same data as in (a) in a magnified view, which is shown as a gray box in (a) (semilog). Error bars show the standard error. (c) The connected autocorrelation functions for polarity:$f_{p,C}\left(\tau \right)$ and velocity: $f_{v,C}\left(\tau \right)$ in the ordered state. The blue dashed line with a slope: $-0.7$ is a guide for the eye.

Figure 6

Fluctuations of the mean polarity in the ordered region. (a) Time series of the fluctuations of global order ${\langle \theta -\langle \theta \rangle \rangle }_{\mathrm{ordered}}$ where ${\langle ·\rangle }_{\text{ordered}}$ takes the average within the highly ordered region ($S<{10}^5\mu \mathrm{m}$). (b) Autocorrelation of the polarity fluctuations from global order $f_{\mathrm{global},C}\left(\tau \right)=\langle \mathrm{\Delta }{\tilde{\mathit{n}}}_p\left(t\right)·\mathrm{\Delta }{\tilde{\mathit{n}}}_p(t+\tau )\rangle$, where $\mathrm{\Delta }{\tilde{\mathit{n}}}_p\left(t\right)={\langle {\mathit{n}}_p-{\langle {\mathit{n}}_p\rangle }_{t,\text{ordered}}\rangle }_{\text{ordered}}$. The blue dashed line with a slope: $-0.85$ is a guide for the eye.

Figure 7

Polarity-velocity cross-correlation functions. (a) The cross-correlation function $g_{pv}\left(\tau \right)$ for Janus particles in the ordered (blue) and disordered state (red). (b) The same data as in (a) in a magnified view for the gray box shown in (a). Error bars: standard error. The difference in error bar size is caused by the sparsity in the disordered state.

Figure 8

Scaling of number fluctuations. (a) Number fluctuations in the ordered (blue) and disordered (red) state. (b) Same data as in (a) with the number fluctuations normalized by ${\langle N\rangle }^{0.5}$. The cyan and magenta dashed lines show $\mathrm{\Delta }N\propto {\langle N\rangle }^{0.8}$ and $\mathrm{\Delta }N\propto {\langle N\rangle }^{0.5}$, respectively, as a guide to the eye. The gray dashed line indicates the scale of counterflows up to which the true long-range order persists.

Figure 9

Orientation fluctuation correlation function in the ordered state. (a) Color map of the correlation function $C_C\left(\mathit{R}\right)$. The global mean director is set aligned with the vertical axis ($R_{\parallel }$). The region of $\left|\mathit{R}\right|<6\mu \text{m}$ were excluded from calculations due to the small statistics. (b) Log-log plot of $C_C\left(\mathit{R}\right)$ both in the transverse ($R_{\perp }$, purple) and longitudinal ($R_{\parallel }$, green) direction. The purple and green solid lines are a guide for the eye, which show a slope of $-0.52$. Note, the exponents from the Toner-Tu conjecture predict an anisotropic decay of $C_C\left(\mathit{R}\right)\propto R_{\perp }^{-2/5},R_{\parallel }^{-2/3}$ [16, 17, 23] and recent numerical simulations suggested $C_C\left(\mathit{R}\right)\propto R_{\perp }^{-0.62},R_{\parallel }^{-0.65}$ [24].