Flow-Induced Phase Separation for Active Brownian Particles in Four-Roll-Mill Flow

TL;DR

This work investigates how background flow and crowding affect active Brownian particles (ABPs) in a two-dimensional four-roll-mill flow, revealing a flow-induced phase separation (FIPS) that emerges for packing fractions $φ≥0.6$. Using numerical simulations, it characterizes transport via MSD, drift, and diffusivity (with $D_e = D_0 (1 - λ φ)^2$ and $λ≈0.8$), fluctuations via number statistics, and clustering via cluster-size distributions and the Okubo-Weiss parameter. FIPS manifests as density inhomogeneities anchored in strain-dominated flow regions, with four-lobed morphologies at high density and a flow-dominated transport regime distinct from classical motility-induced phase separation (MIPS). The results highlight a new mechanism for controlled clustering in driven active matter and motivate future work on hydrodynamic coupling, three-dimensional flows, and experimental realizations in microfluidic systems.

Abstract

We investigate the collective dynamics of active Brownian particles (ABPs) subjected to a steady two-dimensional four-roll-mill flow using numerical simulations. By varying the packing fraction ($φ$), we uncover a novel flow-induced phase separation (FIPS) that emerges beyond a critical density ($φ\geq 0.6$). The mean-square displacement (MSD) exhibits an intermediate bump between ballistic and diffusive regimes, indicating transient trapping and flow-guided clustering. The effective diffusivity decreases quadratically with $φ$, while the drift velocity remains nearly constant, demonstrating that large-scale transport is primarily dictated by the background flow. Number fluctuations show a crossover from normal to giant scaling, signaling the onset of long-range density inhomogeneities in the FIPS regime. Our findings provide new insights into the coupling between activity, crowding, and flow, offering a unified framework for understanding phase behavior in driven active matter systems.

Figures (7)

• Figure 1: Representative snapshots of ABPs without (top panel) and with (bottom panel) imposed flow. The left, center, and right panels correspond to $\phi = 0.2, 0.4$, and $0.7$, respectively. The color bar represents the normalized cluster size.
• Figure 2: Mean square displacement (MSD) of ABPs as a function of time for various packing fractions $\phi \in [0.1, 0.9]$. The black solid (dashed) lines represent slopes of 2 (1), corresponding to ballistic (diffusive) regimes, respectively, and serve as visual guides in the log-log plot. At higher packing fractions, the emergence of FIPS alters the crossover behavior, leading to deviations from simple ballistic-diffusive dynamics due to persistent clustering. (Top-left inset) normalized Drift velocity $\rm \overline{v}_d$ and (bottom-right inset) normalized diffusivity $\rm D_e / D_0$ as functions of the packing fraction $\rm \phi$. The green curve shows a fit of the form $\rm D_e = D_0 (1 - \lambda \phi)^2$, with fitting parameter $\rm \lambda = 0.8$, indicating a quadratic decay of diffusivity with packing fraction.
• Figure 3: Number fluctuations in ABPs with and without flow, color code is same as Fig.2. The standard deviation $\rm \Delta N_\ell$ of particle number is plotted against the mean number $\rm N_\ell$ in subregions of size $\rm \ell^2$ of ABPs with size $\rm a = 0.02$ and scaled speed $\rm V = 0.4$, for packing fractions $\rm \phi \in [0.1, 0.9]$. (a) In the absence of flow, a transition from normal ($\rm \Delta N_\ell \sim N_\ell^{1/2}$) to giant fluctuations ($\rm \Delta N_\ell \sim N_\ell$) is observed as $\rm \phi$ increases beyond 0.4, consistent with the onset of MIPS. (b) In the presence of background flow, a similar crossover is observed near $\rm \phi \approx 0.5$, indicative of FIPS. Dashed and dash-dotted lines serve as visual guides for $\rm \alpha = 0.5$ and $\rm \alpha = 1$, respectively. (c) Scaling exponent $\rm \alpha$ of the number fluctuations $\rm \Delta N_\ell \sim N_\ell^\alpha$ plotted against packing fraction $\rm \phi$ without (red triangles) and with (blue squares) background flow.
• Figure 4: Normalized cluster size distribution $\rm P(n)/P(1)$ as a function of ratio of cluster size to maximum number of particle at corresponding packing fractions $\rm n/N_{max}$ for ABPs with size $\rm a = 0.02$ at various packing fractions $\rm \phi \in [0.1, 0.9]$: (a) without background flow exhibiting MIPS behavior, and (b) with background flow exhibiting FIPS behavior.
• Figure 5: Plot of (a)the exponent $\beta$ and (b) maximum cluster size ($\rm n_{max}$) as a function of packing fraction $\phi$ for ABPs with particle size $a = 0.02$. Red triangles (blue diamonds) represent simulations without (with) background flow.