Collective Motion and Pattern Formation in Phase-Synchronizing Active Fluids

Many active particles, such as swimming micro-organisms or motor proteins, do work on their environment by going though a periodic sequence of shapes. Interactions between particles can lead to synchronization of their duty cycles. Here, we study the collective dynamics of a suspension of active particles coupled through hydrodynamics. We find that at high enough density the system transitions to a state of collective motion by a mechanism that is distinct from other instabilities in active matter systems. Second, we demonstrate that the emergent nonequilibrium states feature stationary chimera patterns in which synchronized and phase-isotropic regions coexist. Third, we show that in confinement, oscillatory flows and robust unidirectional pumping states exist, and can be selected by choice of alignment boundary conditions. These results point toward a new route to collective motion and pattern formation and could guide the design of new active materials.

Figure 1

(a) The real part of the growth rate $\sigma$ as a function of wave numbers $\{k_{\parallel },k_{\perp }\}$ in a periodic box, where $k_{\parallel }$ and $k_{\perp }$ are wave vectors parallel and normal to the fixed particle orientation $\mathbf{n}$, respectively. The unstable region is within the red curve. (b) Growth rate in a channel, computed from the linear stability as a function of the wave number $k_x$ for different particle alignments. The dominant wave number $k_c$ for $\theta =0$ is indicated. A snapshot of the Kuramoto order parameter $|Z_1(\mathbf{x})|$ (top) and instantaneous streamlines (bottom) from the emergent travelling wave state is shown for a periodic box (c),(d) and in a channel for $\theta =0$ (e),(f), and $\theta =\pi /4$ (g),(h). Parameters: $\eta =2$, $\gamma =1$, $c_0={\omega }_0=8$, $X(\phi )=\cos \phi +2\sin \phi$. Channel height, $H=5\pi$. Illustration of the active particles indicates their orientation.

Figure 2

(a) Phase diagram of the two emergent states: spontaneous pumping and oscillations. (b) Mean amplitude of the active stress $⟨\mathcal{M}⟩$ as a function of $\theta$ for varying channel height $H$. The asterisk on (a) indicates the state shown in Fig. 3.

Figure 3

(a) Variation of $\mathcal{M}(\mathbf{x})$ in the channel demonstrates that through synchronization the suspension behaves as an extensile (or contractile) fluid. Also shown is the direction of pumping results from the spatial distribution of the stress. (b) Variation of the phase along the channel width as a function of time illustrates the existence of metachronal phase waves. Parameters: $H=5\pi$, $L=3H$, $\theta =\pi /8$, $\gamma =1$, $\eta =2$, ${\omega }_0=8$, $d_{\phi }=1$, $D=1$, $X(\phi )=\cos \phi +2\sin \phi$.

Figure 4

Snapshots of various field variables for two different boundary conditions of particles. (a),(d) The Kuramoto order parameter from the final state. (b),(e) Associated snapshot of the velocity field and the streamlines highlight the formation of vortical structures and fluid jets. (c),(f) $\theta (\mathbf{x})$ showcases the variation in the orientation and the associated director field. Prameters: (top row) $H=4\pi$, $L=8\pi$, $\theta {|}_{y=\pm H/2}=\pi /5$ and (bottom row) $H=6\pi$, $L=12\pi$, ${\partial }_y\theta {|}_{y=\pm H/2}=0$.