Continuum Theory of Phase Separation Kinetics for Active Brownian Particles

Active Brownian particles (ABPs), when subject to purely repulsive interactions, are known to undergo activity-induced phase separation broadly resembling an equilibrium (attraction-induced) gas-liquid coexistence. Here we present an accurate continuum theory for the dynamics of phase-separating ABPs, derived by direct coarse graining, capturing leading-order density gradient terms alongside an effective bulk free energy. Such gradient terms do not obey detailed balance; yet we find coarsening dynamics closely resembling that of equilibrium phase separation. Our continuum theory is numerically compared to large-scale direct simulations of ABPs and accurately accounts for domain growth kinetics, domain topologies, and coexistence densities.

Figure 1

ABP (left) and continuum (right) simulations with particle area fractions $\varphi$ as indicated. Snapshots are taken at equal time $t=500{\tau }_r$ and shown at the same spatial scale. Inset numbers indicate the area fraction of the dense phase, measured through numerical integration of $P(\varphi )$ (Fig. 4). The color bars run from $\varphi =0$ (black) to $\varphi =1$ (yellow or light gray).

Figure 2

Density-dependent swim speed $v(\varphi )$ (black symbols) and diffusivity $D(\varphi )$ (red or light gray symbols), and the ratio $D/v^2$ (blue or dark gray curve and symbols) obtained from an ABP simulation at $\mathrm{Pe}=40$. The black and red curves show the best fits to the functions $v_0(1-a\varphi )$ and $D_0(1-b\varphi {)}^2$, respectively, with the optimized values $a=1.05$ and $b=1.04$. Dashed lines show the predicted zero-density values. Plotted quantities are in Lennard-Jones units $\sigma$ and ${\tau }_{\mathrm{LJ}}$, as defined in 30.

Figure 3

Time-dependent coarsening length $L(t)$ obtained from the inverse first moment of the structure factor at $\varphi =0.5$. The dashed lines indicate the fitted exponents $\alpha =0.27(9)$ (ABPs), $\alpha =0.28(7)$ (continuum model), and $\alpha =0.27(9)$ [continuum model with detailed balance (DB) term]. The latter curve has been vertically shifted for clarity.

Figure 4

Probability distribution $P(\varphi )$ of the local area fraction $\varphi$ obtained from ABP simulations (black curve), from the continuum model as written (red or light gray curve) and with detailed balance restored (blue or dark gray curve). $P(\varphi )$ was sampled over quadratic coarse graining areas of side length $0.8\lambda$ at $\varphi =0.5$ and averaged over the time window $500{\tau }_r\le t\le 3500{\tau }_r$.