Theory of Reversed Ripening in Active Phase Separating Systems

TL;DR

The paper addresses reversed ripening in chemically active emulsions, proposing that activity can arrest Ostwald-like ripening and even drive monodisperse emulsions. It develops a minimal ternary model with a single reaction $A <-> B$, introducing a conserved density $ψ = (c_A+c_B)/2$ and a non-conserved $ξ = (c_A-c_B)/2$, all derived from a Flory-Huggins free energy to describe droplet phase separation. A key result is the emergence of a stable fixed point in single-droplet dynamics, leading to arrested growth or arrested ripening, and a late-time scaling regime where the size distribution narrows and collapses onto a stationary radius $R_s$ via a scaling form $n(R,t)= (1/R_s) f(x) e^{λ t}$ with $x=((R-R_s)/R_s) e^{λ t}$. The framework extends to multi-component active emulsions and has implications for understanding biomolecular condensates and engineered active systems, while pointing to open issues such as fluctuations and division dynamics.

Abstract

The ripening dynamics in passive systems is governed by the theory of Lifshitz-Slyozov-Wagner (LSW). Here, we present an analog theory for reversed ripening in active systems. To derive the dynamic theory for the droplet size distribution, we consider a minimal ternary emulsion with one active reaction, leading to one conserved quantity. Even for cases where single droplets constantly grow, coupling many droplets via the conserved density in the far field leads to a self-organized reversal of ripening and, thus, a monodisperse emulsion. For late times, we find a scaling ansatz leading to the collapse of the rescaled size distributions, different from the LSW theory. This scaling behavior arises from a stable fixed point in the single droplet dynamics and may capture the late-time behavior of many active matter systems exhibiting reversed ripening.

Figures (4)

• Figure 1: Passive and active emulsion. (a,c): phase diagram of ternary emulsions with passive or active chemical reaction A $\rightleftharpoons$ B. The binodal (thick blue), tie-lines (thin blue), and the reaction nullcline for passive (yellow) and active (red) reactions are shown. Lines of constant $\psi = 0.07, 0.13, 0.19$ (a) and $\psi = 0.03, 0.08, 0.16$ (b) are highlighted (green). (b,d): Interface velocities $\dot{R}$ as a function of radius $R$ for single droplets are shown for these values of $\psi$. Phase diagram and interface velocities were obtained using a Flory-Huggins mean field model of mixtures; see the End Matter for details.
• Figure 2: Dynamics of droplet size distributions in chemically active emulsions. (a) Distribution $n(R)$ of droplet sizes for different times $t$ for an arrested growth scenario (thick blue lines). The shading indicates distributions at intermediate times, revealing the arrest in a monodisperse distribution at late times. The inset shows late-time distributions. (b) Interface velocities $\dot{R}(R)$ for selected time points. The average radii at different times are indicated (dashed vertical lines) (c) Droplet number $N$, average radius $\langle R \rangle$, and the phase fraction $\phi$ as a function of time. (d-f) Same plots as (a-c) but for a scenario of arrested ripening. Parameters: Initial droplet number $N=200$ in reference volume $V_\text{ref}$, Gaussian initial distribution $\mathcal{N}(\bar{R}_0, \sigma_R)$, $b=1$, $c_0=1$, $D_{\psi/\xi}^\text{i/o} = 1$, $\gamma = 1/6$, $V_\text{ref} = 4 \pi R_\text{ref}^3/3$, $R_\text{ref}=600$, $k_{AB}^\text{i/o}/k_{BA}^\text{i/o}=2.5$; arrested growth: $\alpha = 0.15 \pi$, $\Psi_\text{tot}=-0.5$, $k_{AB}^\text{i/o}=0.0013$, $\bar{R}_0= 20$, $\sigma_R = 20$, arrested ripening: $\alpha = 0.23\pi$, $\Psi_\text{tot}=-0.7$, $k_{AB}^\text{i/o}=0.0009$, $\bar{R}_0 = 15$, $\sigma_R = 10$.
• Figure 3: Shape of droplet size distributions at late times. (a) Dynamics of the droplet size distribution $n(R)$ for the same system as shown in Fig. \ref{['fig:2example']}(a) but with an initial shape $g(x)$. The inset shows the distribution at late times. (b) Late time distribution as shown in (a)(inset), but scaled and shown as a function of scaled radius $x$. (c,d) Scaled plot as in (b) but showing late time distributions of arrested growth (Fig. \ref{['fig:2example']}(a)) and arrested ripening (Fig. \ref{['fig:2example']}(d)).
• Figure 4: Effect of diffusivities inside ($D^\text{o}$) and outside ($D^\text{i}$) of the droplet on the droplet size distribution $n(R,t)$. Early times (a) and late times (b) of the same dynamics shown in Fig. \ref{['fig:2example']}(a-c) (blue) in comparison to the same system with a ten-times faster diffusion for both species $A$ and $B$ in the phase outside the droplet ($D^\text{o}=10 \, D^\text{i}$). For simplicity, different species have equal diffusivity in each phase. For a movie of these dynamics, see the supplementary information. We see that the enhanced diffusivity outside speeds up the relaxation of the droplet size distribution to the state of arrested growth, while the final stationary distribution is less affected by this change of diffusivity.