Motility-Induced Microphase and Macrophase Separation in a Two-Dimensional Active Brownian Particle System

As a result of nonequilibrium forces, purely repulsive self-propelled particles undergo macrophase separation between a dense and a dilute phase. We present a thorough study of the ordering kinetics of such motility-induced phase separation (MIPS) in active Brownian particles in two dimensions, and we show that it is generically accompanied by microphase separation. The growth of the dense phase follows a law akin to the one of liquid-gas phase separation. However, it is made of a mosaic of hexatic microdomains whose size does not coarsen indefinitely, leaving behind a network of extended topological defects from which microscopic dilute bubbles arise. The characteristic length of these finite-size structures increases with activity, independently of the choice of initial conditions.

Figure 1

Hexatic domains and bubbles in a MIPS droplet. A steady-state snapshot of ${512}^2$ ABPs at moderate density ($\varphi =0.25$) and high activity ($\mathrm{Pe}=200$) showing a macrodroplet made of a mosaic of hexatic microdomains, delimited by clusters of topological defects from which microbubbles arise. Colors indicate the projection of the particles’ hexatic parameter ${\psi }_{6,j}$ (see the definition in the text) onto the direction of its global average ${\mathrm{\Psi }}_6=(1/N){\sum }_j{\psi }_{6,j}$. Insert: stationary distribution of hexatic domain radii ${\tilde{R}}_H$ for three Pe and equal fraction of dense and dilute coexisting phases.

Figure 2

Dense phase formation. (a) Typical dense phase size from $S(k,t)$ and (b) number of clusters, both as a function of time, after a quench into the MIPS coexistence region, in a system with $N=1024^2$ particles at different Pe on the curve with equal surface fraction of dense and dilute phases. Three time regimes are discerned: nucleation, condensation and aggregation, and scaling.

Figure 3

Macrodroplet against hexatic microdomains. Growth of the dense phase radius of gyration, $R_G$, normalized by the system size $L$ (a) and the hexatic length $R_H$ (b) for $\mathrm{Pe}=100$ and different $N$ on the $50:50$ curve. In (a) $R_G/L$ saturates at $\sqrt{5/48}$ (a droplet that occupies half the available surface). In (b) we represent the slow growth $t^{0.13}$, and $t^{1/3}$ for comparison. The vertical dotted lines mark the crossover between the three regimes in the largest system. (c)–(f) A zoom over a sector of these systems with linear size $160{\sigma }_d$ taken at $t=15$ (scaling regime) is shown.

Figure 4

Parameter dependencies of the dense phase and hexatic growing lengths. $N=128^2$ at fixed $\varphi =0.5$ and varying Pe (a), (b); at fixed $\mathrm{Pe}=100$ and varying $\varphi$ (c) and the asymptotic $R_H^s$ for $\mathrm{Pe}=50$, 100, 200 (in MIPS) and $\mathrm{Pe}=10$ (before MIPS) as a function of $\varphi$ (d). An exponential growth $a\exp [b({\varphi }^{-1}-{\tilde{\varphi }}^{-1}{)}^{-1/2}]$ is shown in dotted lines (for $\mathrm{Pe}=10$: $a=0.778$, $b=0.214$, and $\tilde{\varphi }=0.797$; for $\mathrm{Pe}=50$: $a=3.172$, $b=0.232$, and $\tilde{\varphi }=0.868$). Inset in (b): dependence of the asymptotic value $R_H^s$ on Pe; the continuous line is $R_H^s\propto \ln [\mathrm{Pe}-a]+b$ with $a=29.01$ and $b=-1.79$ (vanishing at the transition point).

Figure 5

Gas microbubbles. Snapshot at $\varphi =0.75$, $\mathrm{Pe}=200$ (a). The color scale (below) is associated with the local density. In (b) a zoom over a sector of (a) showing hexatic domains delimited by clusters of defects (represented by black dots), where bubbles (dilute regions) are likely to emerge. (c) Evolution of $R_B$ and $R_H$ from an initial hexatically ordered disk or slab at $\mathrm{Pe}=100$ and $\varphi =0.5$ (see movies 2 and 3 [21]). (d) Growth of the bubble typical radius $R_B$ (empty symbols) confronted with the one of the hexatic order $R_H$ (filled symbols) for two Pe. (e) Steady-state distribution of bubble radii ${\tilde{R}}_B$ (parameters given in the key).