Freezing a Flock: Motility-Induced Phase Separation in Polar Active Liquids

Combining model experiments and theory, we investigate the dense phases of polar active matter beyond the conventional flocking picture. We show that above a critical density flocks assembled from self-propelled colloids arrest their collective motion, lose their orientational order, and form solids that actively rearrange their local structure while continuously melting and freezing at their boundaries. We establish that active solidification is a first-order dynamical transition: active solids nucleate, grow, and slowly coarsen until complete phase separation with the polar liquids with which they coexist. We then theoretically elucidate this phase behavior by introducing a minimal hydrodynamic description of dense polar flocks and show that the active solids originate from a motility-induced phase separation. We argue that the suppression of collective motion in the form of solid jams is a generic feature of flocks assembled from motile units that reduce their speed as density increases, a feature common to a broad class of active bodies, from synthetic colloids to living creatures.

Popular Summary

Using microscopic synthetic particles capable of self-propulsion, researchers have created a new generation of materials assembled from mobile units: active matter. In one remarkable example, micrometer-size particles move about and probe the orientation of their neighbors. When interacting, these active particles self-assemble into synthetic flocks to form spontaneously flowing liquids. Here, we show that upon increasing the local density of particles, liquid flocks collectively turn into arrested solids through a freezing transition akin to that observed when cooling water below $0°\mathrm{C}$.

By combining experiments and theory, we demonstrate that polar liquids assembled from motile colloids undergo a “motility-induced solidification,” a transition generic to any flocking group in which the individuals slow down as the distance to their neighbors decreases. This transition results in the formation of a novel state of active matter: amorphous active solids, the first experimental confirmation of a theoretical prediction put forward more than a decade ago.

Amorphous active solids have a lively inner structure and continuously melt and reform at their boundaries. Although composed of particles that spend most of their time at rest, they steadily propagate through the active liquids with which they coexist, much like traffic jams along crowded highways.

Beyond the specifics of synthetic active matter, these findings should be relevant to understanding motion of robot fleets, bacteria swarms, and animal flocks in nature.

Figure 1

The dynamical phases of Quincke rollers. (a) Picture of a microfluidic racetrack where $\sim 3\times {10}^6$ Quincke rollers interact. An active solid (dark gray) propagates through a polar liquid (light gray). Scale bar is 2 mm. (b)–(f) Top: Close-up pictures of Quincke rollers in the racetrack. Scale bars are $250\mu \mathrm{m}$. Bottom: Longitudinal component of the particle current $W$ and density plotted as a function of a normalized time. Both $W$ and $\varphi$ are averaged over an observation window of size $56\times 1120\mu {\mathrm{m}}^2$. $T_0$ is arbitrarily chosen to be the time taken by an active solid to circle around the race track. (b) Gas phase (${\varphi }_0=0.002$). The density is homogeneous and the system does not support any net particle current. (c) Coexistence between an active gas and a denser polar band (${\varphi }_0=0.033$). A heterogeneous polar-liquid drop propagates at constant speed through a homogeneous isotropic gas in the form of a so-called Vicsek band. (d) Polar-liquid phase (${\varphi }_0=0.096$). The homogeneous active fluid supports a net flow. (e) Coexistence between a polar liquid and an amorphous active solid (${\varphi }_0=0.49$). The high-density solid is homogeneous but, unlike the polar liquid, does not support any net particle current. (f) Homogeneous active-solid phase (${\varphi }_0=0.70$).

Figure 2

Structure and dynamics of active jams. (a) The kymograph of the measured light intensity averaged over the racetrack width shows how an active solid (dark region) propagates at constant speed through a homogeneous polar liquid (light region). (b),(c) Pair-correlation functions measured in the polar liquid ($g_L$) and in the coexisting active-solid phase ($g_S$). ${\varphi }_0=0.58$. Both pair-correlation functions are plotted versus the interparticle distance $r_{\perp }$ in the direction transverse to the mean polar-liquid flow. $g_S$ displays more peaks than $g_L$, revealing a more ordered structure, but translational order merely persists over a few particle diameters. The dashed lines indicate the distance corresponding to one particle diameter. (d) Probability density functions of the roller velocities in the polar-liquid region. The distribution is peaked around ${\nu }_0{\widehat{\mathbf{x}}}_{\parallel }$, where ${\widehat{\mathbf{x}}}_{\parallel }$ is the vector tangent to the racetrack centerline. (e) Probability density functions of the roller velocities in the active jam region. The distribution is peaked around 0. The rollers remain mostly at rest. (b)–(e) Average area fraction ${\varphi }_0=0.58$.

Figure 3

Active solidification is a first-order phase separation. (a) Solid jams in a racetrack at ${\varphi }_0=0.38$, 0.58, 0.65. Increasing the average packing fraction, the extent of the active solid increases. (b) Solid fraction plotted versus the average packing fraction ${\varphi }_0$. Note the discontinuous jump and the two possible states at the onset of solidification. (c) Density of the polar-liquid phase (blue circles) and of the active-solid phase (red circles) plotted versus ${\varphi }_0$. In steady state, the liquid density increases with ${\varphi }_0$ until an active solid forms, the density in both phases then remains constant. In (b) and (c) the shaded regions indicate the coexistence between the polar-liquid and active-solid phases. (d) Coarsening dynamics. Two solid jams coexist only over a finite time period. The larger jam (filled symbols) shrinks and eventually vanishes in favor of the smaller one (open symbols). The total fraction of solid phase in the racetrack (black line) remains constant over time. (e) Solid fraction (blue line) and magnitude of the electric field (red line) plotted versus time. Over a range of $E_0$ values, the active-solid fraction is different when increasing or decreasing the electric field. (f) The extent of the traffic jam follows a hysteresis loop when cycling the field amplitude.

Figure 4

Nonlinear hydrodynamics of polar active matter. (a) Average velocity ${\nu }_0(\mathbf{r},t)$ of the colloids and local magnitude of the longitudinal current $W(\mathbf{r},t)$ plotted as a function of the local packing fraction $\varphi (\mathbf{r},t)$. (b)–(e) Successive phases observed in the resolution of Eqs. (1) and (2) at increasing average densities ${\rho }_0=(0.10,0.18,1.7,1.96,6.5)$. The position $x$ is normalized by the system size $L$. Simulation parameters are $D_{\rho }=0.2$, $D_W=1$, $\lambda =1$, $a_4=0.45$, $\sigma =0.2$, $\overline{\rho }=2$, $\xi =0.01$, $\alpha =1$, ${\rho }_c=0.5$, $L=200$, $dx=0.05$, $dt=0.005$.

Figure 5

Shape and dynamics of active-solid jams. (a) Density and velocity profiles computed for different value of ${\rho }_0$: from 1.85 (light colors) to 2.25 (dark colors). Numerical resolution of Eqs. (1) and (2) with the hydrodynamic parameters $D_{\rho }=5$, $D_W=10$, $\lambda =1$, $a_4=1$, $\sigma =1$, $\overline{\rho }=2$, $\xi =0.1$, $\alpha =1$, ${\rho }_c=0.5$, $L=200$, $dx=0.01$, $dt=0.001$. (b) Length of the solid jam plotted versus ${\rho }_0$ (symbols) and lever rule (solid line). Same parameters as in (a). (c) The speed of the solid jam $c$ does not depend on the average density ${\rho }_0$. (d) Variations of the propagation speed $c$ as a function of the effective diffusivity $D_{\rho }$. Numerical parameters are $D_W=5$, $xi=1$, $a_4=0.45$, $\sigma =0.2$, $\overline{\rho }=2$, $\xi =0.01$, $\alpha =1$, ${\rho }_c=0.5$, $L=200$, $dx=0.05$, $dt=0.005$, ${\rho }_0=1.96$.

Figure 6

Quincke rollers. (a) When applying a dc electric field ${\mathbf{E}}_{\mathbf{0}}$ to an insulating sphere immersed in a conducting fluid, a charge dipole forms at the sphere surface. When $E_0>E_Q$, the electric dipole makes a finite angle with the electric field causing the steady rotation of the sphere at constant angular speed $\mathrm{\Omega }$. (b) The rotation is converted into translation by allowing the sphere to sediment on one electrode. When isolated, the resulting Quincke rotor rolls without sliding at constant speed: ${\nu }_0(0)=a\mathrm{\Omega }$. (c) When two colloids rolling in the same direction are close to each other, the lubrication torque acting on the two spheres separated by a distance $d$ scales as $\log (d-2a)$ and hinders their rolling motion.