Effective Cahn-Hilliard Equation for the Phase Separation of Active Brownian Particles

The kinetic separation of repulsive active Brownian particles into a dense and a dilute phase is analyzed using a systematic coarse-graining strategy. We derive an effective Cahn-Hilliard equation on large length and time scales, which implies that the separation process can be mapped onto that of passive particles. A lower density threshold for clustering is found, and using our approach we demonstrate that clustering first proceeds via a hysteretic nucleation scenario and above a higher threshold changes into a spinodal-like instability. Our results are in agreement with particle-resolved computer simulations and can be verified in experiments of artificial or biological microswimmers.

Figure 1

(a) Instability diagram for repulsive self-propelled disks as a function of area fraction $\varphi$ and propulsion speed $v_0$. The symbol color indicates the fraction of particles that are part of the largest cluster. The open symbols correspond to the homogeneous suspension ($P<0.1$) and closed squares to the cluster phase. The vertical dotted line marks the freezing density for the passive ($v_0=0$) suspension, closed triangles indicate solid. The instability line is calculated from the analytical result Eq. (13). A change from a continuous (solid) to a discontinuous (dashed) transition at the density ${\varphi }_0\simeq 0.32$ occurs. (b) Illustration of the double tangent construction. We also sketch the bulk free energy density $f(c)$ for ${\sigma }_1<0$ (dashed line). (c) Derivation of Eq. (13): Given a point (${\varphi }_1$, $v_1$) on the instability line, at the slightly larger speed $v_2=v_1(1+ϵ)$ phase separation into the dense phase and the dilute gas phase with area fraction ${\varphi }_2={\varphi }_{-}$ will occur.

Figure 2

(a) Hysteresis at low area fraction $\varphi$ and vanishing at higher values of $\varphi$. Shown is the mean fraction $P$ of particles in the largest cluster after the (metastable) steady state has been reached for two initial conditions: starting from the homogeneous disordered passive suspension (solid lines, open symbols) and already containing an ordered cluster (dashed lines, closed symbols). (b) Effective free energy density $f(c)$ for ${\sigma }_1>0$ containing an additional $c^4$ term that stabilizes the high density phase. The dashed line indicates $g=g_{*}$. For $g>g_{*}$, the energy density develops a nonconvex part indicating coexistence of regions of two different particle densities, which promotes hysteresis.