Active Cahn--Hilliard theory for non-equilibrium phase separation: quantitative macroscopic predictions and a microscopic derivation

TL;DR

This work develops a quantitative active Cahn–Hilliard framework at fourth order in spatial gradients, avoiding density Taylor expansions and introducing five density-dependent coefficient functions that capture non-equilibrium interfacial physics. By systematically coarse-graining a microscopic model of thermal quorum-sensing active particles (tQSAPs) with a novel multiple-scale analysis, the authors derive the full $O(\nabla^4)$ theory and connect its coefficient functions to microscopic parameters. They show how to compute binodal densities, three interfacial tensions (two for Ostwald dynamics and one for capillary waves), and interface profiles, revealing nontrivial phenomena such as a curvature-coupled binodal shift, Ostwald-reversal regimes, capillary-wave instabilities, and a re-entrant phase behavior driven by non-Fickian currents. The MS-based theory improves upon previous continuum models by matching particle simulations across broader parameter ranges, including cases where the diffusion-drift (DD) approximation fails, and demonstrates a clear dependence of phase behavior on the QS length scale $\gamma$. The results provide a systematic route from microscopic active-particle models to a predictive continuum theory for non-equilibrium phase separation with broad potential applications in active matter and beyond.

Abstract

Phase-separating active systems can display phenomenology that is impossible in equilibrium. The binodal densities are not solely determined by a bulk (effective) free energy, but also affected by gradient terms, while capillary waves and Ostwald processes are determined by three distinct interfacial tensions. These and related phenomena were so far explained at continuum level using a top-down minimal theory (Active Model B+). This theory, by Taylor-expanding in the scalar order parameter (or density), effectively assumes that phase separation is weak, which is not true across most of the phase diagram. Here we develop a quantitative account of active phase separation, by introducing an active counterpart of Cahn-Hilliard theory, constructing the density current from all possible terms with up to four spatial derivatives without Taylor-expanding in the density. From this O(grad^4) theory, we show how to compute binodals and interfacial tensions for arbitrary choices of the five density-dependent 'coefficient functions' that specify the theory (replacing the four constant coefficients of Active Model B+). We further consider a particle model composed of thermal quorum-sensing active particles (tQSAPs) yielding a fully specified example of the O(grad^4) theory upon coarse-graining. We find that to coarse-grain consistently at O(grad^4) requires a novel procedure, based on multiple-scale analysis, to systematically eliminate fast-evolving orientational moments. Using this, we calculate from microscopic physics all five coefficient functions of the active Cahn-Hilliard theory for tQSAPs. We identify contributions that were missed in previous continuum theories, and show how neglecting them becomes justified only in the limit of large quorum-sensing range parameter. Comparison with particle simulations of tQSAPs shows that our O(grad^4) theory improves on previous continuum models [...]

Figures (15)

• Figure 1: Stepwise abolition of common tangent construction in $\mathcal{O}(\nabla^4)$ theory, for a bulk (effective) free energy density defined via $f'(\rho)=\mu_0(\rho)$. a) Passive case (common tangent construction): $\Delta P = \Delta\mu_0 = 0$; b) AMB+ (uncommon tangent construction): $\Delta P \neq 0$, $\Delta\mu_0 = 0$; c) $\mathcal{O}(\nabla^4)$ theory (no tangent construction): $\Delta P \neq 0$, $\Delta\mu_0 \neq 0$.
• Figure 2: Binodals in the constant-coefficient $\mathcal{O}(\nabla^4)$ theory (with free energy parameters $a=b=1$). a) Binodal width $\Delta\rho=\rho_\mathrm{L}-\rho_\mathrm{V}$ as a function of $\nu$ and $\bar{\lambda}$. b) Binodal center $\bar{\rho}=(\rho_\mathrm{L}+\rho_\mathrm{V})/2$ as a function of $\nu$ and $\bar{\lambda}$. The gray area is the forbidden region (F.R.) where no binodal solutions exist. Orange, red, and pink lines as in Fig. \ref{['fig:constantcoeff_forbiddenregion']}.
• Figure 3: Forbidden region in the constant-coefficient $\mathcal{O}(\nabla^4)$ theory (with free energy parameters $a=b=1$). a) The $\nu$ term acts as a density pump across an interface. The arrow shows the direction of the current for positive $\nu$. b) Characterization of the forbidden region. The orange vertical line here corresponds to AMB+ ($\nu = 0$). The red line indicates the boundary $\nu_\mathrm{F}$ of the forbidden region; it crosses the pink line, to the right of which the pseudodensity becomes oscillatory, at $\nu = 3Kb/a$. As $\nu\uparrow\nu_\mathrm{F}$, depending on whether the pseudodensities are oscillatory or not, either one of the binodals diverges (dotted red line, panel c) or both end at finite values, annihilating with another, unphysical solution of Eq. \ref{['eq:binodals']} (solid red line, panel d). The green and purple line segments in panel b) indicate the range of values of $\nu$ for which binodals are shown in panels c,d).
• Figure 4: Interfacial tensions in the constant-coefficient $\mathcal{O}(\nabla^4)$ theory (with free energy parameters $a=b=1$), as functions of $\nu/K$ and $\bar{\lambda}/K$ for different $\zeta$. Orange, pink and red curves as in Figs. \ref{['fig:constantcoeff_binodals']} and \ref{['fig:constantcoeff_forbiddenregion']}; the dashed black line indicates a sign change in $\sigma$. a--c) Ostwald interfacial tension for vapor bubbles. The liquid droplet interfacial tension can be found by setting $\bar{\lambda}, \zeta, \nu \to -\bar{\lambda}, -\zeta, \nu$. d--f) Capillary wave interfacial tension.
• Figure 5: Predictions of binodal curves of DD and MS-$\mathcal{O}(\nabla^4)$ theory for tQSAPs with various parameter values (see legends), as functions of $\eta_T$. The difference between the two theories becomes larger upon increasing $A$, $S$ or $\alpha_\gamma$. The two theories exactly agree on the location of the critical point, and at $\eta_T=0$. Here and in subsequent figures, $\tilde{\alpha}_\gamma = c_\gamma^{-1} \approx 2.77$ denotes the value of $\alpha_\gamma$ obtained for $v_1\tau=1$, $r_\mathrm{cut}=1$.