Two-field theory for phase coexistence of active Brownian particles

TL;DR

The study addresses phase coexistence and wall wetting in highly persistent active Brownian particles (ABPs) by developing a two-field continuum model for density $\rho$ and polarization $\mathbf{q}$. Starting from microscopic ABP dynamics, it combines the Dean coarse-graining with phenomenological closures for the interparticle flux $G^{\alpha}$ and the stress tensor $T^{\alpha\beta}$, and introduces an adiabatic slaving limit for $q$ and a non-variational active stress to sustain phase separation. The resulting PDEs reproduce the observed transition from a homogeneous to an interfacial polarization regime and capture the formation and stabilization of wetting films, aligning well with quasi-one-dimensional ABP simulations. The framework clarifies how persistent polarization drives phase coexistence without a pre-imposed double-well, and it provides a platform for extending to higher dimensions and exploring transient contributions and other free-energy forms.

Abstract

Active Brownian particles (ABPs) serve as a minimal model of active matter systems. When ABPs are sufficiently persistent, they undergo a liquid-gas phase separation and, in the presence of obstacles, accumulate around them, forming a wetting layer. Here, we perform simulations of ABPs in a quasi-one-dimensional domain in the presence of a wall, studying the dynamics of the polarization field. On the course of time, we observe a transition from a homogeneous (where all particles are aligned) to a heterogeneous (where particles align only at the interface) polarization regime. We propose coarse-grained equations for the density and polarization fields based on microscopic and phenomenological arguments that correctly account for the observed phenomena.

Figures (6)

• Figure 1: Snapshots of one of the simulations for $t = 0$ (bottom), 500 (middle), and 5000 (top). The color indicates the particle orientation angle $\theta$. Only the central part in $x$ of the system is shown.
• Figure 2: Top: Spatiotemporal diagrams obtained from simulations for the (a) density, (b) $x$ component of polarization, $q^x$, and (c) $yy$ component of the nematic tensor, $Q^{yy}$. Spatiotemporal diagrams for density (d) and polarization (e) fields obtained from the numerical solutions of Eqs. \ref{['eq:rho_approx']} and \ref{['eq:qx_approx']}. Bottom: Steady state profiles of the (f) density, (g) $x$ component of polarization, $q^x$, and (h) $yy$ component of the nematic tensor, $Q^{yy}$ of one interface, centered at $x_0$, with the dilute phase at the left and the dense at the right (see text for details). The blue lines are the results obtained from simulations and the green lines are obtained by solving numerically Eqs. \ref{['eq:rho_approx']} and \ref{['eq:qx_approx']}.
• Figure 3: Spatiotemporal diagrams obtained from a simulation with $D_1=0.01$, with the rest of the parameters equal to the other simulations. (a) shows the density and (b) the $x$ component of polarization, $q^x$.
• Figure 4: Panel (a) shows the stationary profile of $G^{x}$ with the fit given by Eq. \ref{['eq: G_approx']}. The black (red) dashed line is for the approximation with (without) the thermodynamic term. For the free energy density it is used $f(\rho) = (\rho - \rho_l / 2)^{4}$. Panel (b) shows $(T^{xx} + T^{yy})/2$ in blue and $T^{xx} - T^{yy}$ in green as a function of density. Panels (c) and (d) show in blue the separate components $T^{xx}$ and $T^{yy}$ as a function of density. The dashed black lines show the fits given by Eq. \ref{['eq: T_approx']}.
• Figure 5: (a) Wetting layer width evolution for simulations and model. (b) Polarization decay near the wall. The dotted line represents the function $\sim\exp(-t / \tau_q)$.