Non-equilibrium phase coexistence in conserved chemically active mixtures

Abstract

Chemical activity is known to affect phase coexistence and coarsening in liquid mixtures, most commonly through reaction-induced changes of intermolecular interactions. Here, we analyze a scenario in which chemical reactions regulate particle transport while leaving thermodynamic interactions unchanged. We study an incompressible mixture of thermodynamically identical solutes with unequal diffusivities that interconvert through driven chemical reactions. Using linear stability analysis and finite-element simulations, we show that the system can phase-separate into solute-rich and solute-poor domains via two qualitatively different pathways. When interactions are too weak to induce phase separation, patterns arise through a generalized mass-redistribution instability and coarsen uninterruptedly. When interactions favor phase separation, coarsening can be arrested if chemical activity locally enriches faster-diffusing solutes within dense domains. In the limit of fast chemical turnover, the system always coarsens, and phase coexistence is governed by an effective free energy that explicitly depends on kinetic parameters. Beyond this limit, we develop a sharp-interface theory that predicts the onset of arrested coarsening, stationary droplet sizes, and nucleation conditions under chemical driving. Taken together, our results establish kinetic regulation as a minimal and robust mechanism to control phase coexistence and coarsening in chemically active mixtures.

Figures (11)

• Figure 1: Chemically active mixture. A mixture consisting of two solutes, A (red) and B (blue), and an inert solvent (not shown) phase separates into solute-rich (gray) and solute-poor domains (white). The solutes are thermodynamically identical but differ in their diffusivities (${D_\mathrm{A}>D_\mathrm{B}}$). Density-dependent chemical reactions convert A and B (small arrows), leading to preferential enrichment of the faster-diffusing species A within solute-rich domains. Bottom panel: Since chemical conversion competes with diffusive transport, the relative composition of A and B inside a droplet depends on its size, with smaller droplets remaining closer to the background composition. The resulting composition gradients generate non-equilibrium mass fluxes that redistribute solute from regions of high to low effective mobility, i.e., from larger to smaller droplets.
• Figure 2: Linear stability analysis. (a--c) Linear spectrum $\sigma(q)$ of the homogeneous steady state ${(\bar{\rho},\bar{s})}$ for a ternary mixture in a two-dimensional system (${d=2}$), with ${\ell=20}$, ${\bar{\rho}=0.4}$, and ${(\chi, D)=}$ (a) ${(2.35,0)}$, (b) ${(2.35,0.9)}$, (c) ${(1.8,-0.8)}$. Panels (a,b) display both branches $\sigma_\pm(q)$; in all shown cases, the spectrum is real-valued.
• Figure 3: Bifurcation diagram. The homogeneous steady state is stable (white) or displays a type-I (red/light gray) or type-II (blue/gray) instability. (a) Bifurcation diagram as a function of $D$ and $\chi$ for fixed ${\bar{\rho}=0.4}$, ${d=2}$, and ${\ell\in\{5,10,20\}}$ (light gray, gray, and black lines). In the presence of driven chemical reactions, the boundary of the type-II regime (black line) is determined by Eq. \ref{['app:instabilityCon2']}. In the equilibrium limit (${\ell \to \infty}$), the homogeneous state is linearly unstable above the (dashed) spinodal line ${\chi=1/[2\bar{\rho}(1 - \bar{\rho})]}$, corresponding to ${f^{\prime\prime}(\bar{\rho})<0}$. (b, c) Upper branch of the dispersion relation, $\sigma_+(q)$, for parameters corresponding to the different markers in subpanel (a) and ${\ell=20}$. The figure is adapted from Ref. prl.
• Figure 4: Oscillatory modes. Spectrum $\sigma(q)$ of the homogeneous steady state ${(\bar{\rho},\bar{s})}$ obtained from linear stability analysis of a ternary mixture in a two-dimensional system (${d=2}$) with ${\ell=20}$, ${\bar{\rho}=0.85}$, ${\chi=2.35}$, and ${D=0.35}$. Solid and dashed lines indicate the real [Re$(\sigma)$] and imaginary [Im$(\sigma)$] part of $\sigma$, respectively.
• Figure 5: Non-equilibrium phase separation. Snapshots of the solute volume fraction $\rho$ from numerical simulations of a ternary mixture [Eq. \ref{['eq:RescaledDynamics']}] in a square domain of size ${L=200}$ with periodic boundary conditions for ${D=-0.3}$ (top) and ${D=0.3}$ (bottom). The system is initialized with small perturbations around the homogeneous steady state at average solute volume fraction ${\bar{\rho}=0.4}$. Color indicates the local solute volume fraction $\rho$ (colorbar); in grayscale renderings, interfaces appear white. The remaining parameters are ${\ell=20}$, and ${\chi=2.4}$.