Local composition controls pattern formation in conserved active emulsions

Abstract

Phase separation in passive systems leads to uncontrolled droplet growth, limiting structural control in soft materials and cells. We identify a generic mechanism to arrest coarsening based on chemical interconversion between molecular species with different diffusivities. Sharp-interface theory and simulations show that when the faster-diffusing species becomes enriched inside droplets, composition gradients emerge that oppose mass influx. This transport asymmetry stabilizes droplet sizes even without interaction asymmetries, offering a minimal route to regulate structure formation in active emulsions.

Figures (4)

• Figure 1: Linear stability analysis. The homogeneous steady state is stable (white) or displays a type-I (red/light gray) or type-II (blue/gray) instability. (a) Stability 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{['eq:massRedistributionInstability']}. In the equilibrium limit, a homogeneous state is linearly unstable above the (dashed) spinodal line ${\chi=1/[2\bar{\rho}(1 - \bar{\rho})]}$. (b, c) Upper branch of the dispersion relation, $\sigma_+(q)$, for parameters corresponding to the different markers in subpanel (a) and ${\ell=20}$. (d, e) Composition weighted diffusivity $D\, s$ (blue/gray) for a weakly perturbed homogeneous density $\rho$ (black). Horizontal arrows indicate composition- (blue/gray) and chemical-potential-driven (black) fluxes, respectively.
• Figure 2: Phase diagram. Snapshots of the conserved density $\rho$ at time ${t=10^8/(1+D)}$, obtained from FEM simulations in a square domain of size ${L=200}$ with periodic boundary conditions. The system is initialized with small perturbations around the homogeneous steady state. Color indicates local density (colorbar); in grayscale renderings, interfaces appear white. The inset shows snapshots for ${D\leq0.1}$ (green box). Dashed and solid lines mark the boundaries of the type-I and type-II unstable regimes, respectively (Fig. \ref{['fig:LSAFigure']}). All simulations use ${\ell=20}$ and the other parameters as in Fig. \ref{['fig:LSAFigure']}.
• Figure 3: Sharp-interface limit for an isolated droplet. An infinitely sharp interface at ${r=R}$ separates the high-density interior (gray) from the low-density exterior (white). Within each domain, the solute density $\rho$ (black) and chemical potential $\mu$ (red/light gray) remain close to their equilibrium values $\rho_\text{in/out}$ and $\mu(R)$. The composition $s$ (blue) varies on a characteristic length scale $\ell$, and relaxes towards chemical equilibrium in the far field [${s \to \bar{s}(\rho_\infty)}$].
• Figure 4: Single droplet steady states in a supersaturated solution. Stable (solid) and unstable (dashed) stationary radii obtained from sharp-interface theory and FEM simulations (bullets) in a three-dimensional (${d=3}$) spherically symmetric system (${L=15\,\ell}$) SI for ${\chi=2.4}$ (${\rho_-=0.17}$), and ${\{\rho_\infty,\ell\}=}$${\{0.185,200\}}$ (black), ${\{0.18,200\}}$ (blue/gray), ${\{0.18,100\}}$ (red/light gray). The inset shows the asymptotic behavior near the onset $D_c$ of arrested coarsening [Eq. \ref{['eq:criticalD']}].