Size control guidelines for chemically active droplets

TL;DR

This work tackles the problem of controlling droplet size in phase-separating systems by introducing a binary-fluid model driven out of equilibrium with chemical reactions. The authors develop a thermodynamically consistent framework with both passive and active reaction channels, connecting droplet dynamics to reaction-diffusion lengths and effective parameters $l_\alpha$ and $ε_\alpha$ that couple chemistry to diffusion. They derive a thin-interface description yielding an interface-driven radius dynamics and identify two distinct regimes—volume-limited and interface-limited—each with simple scaling laws for the steady-state radius $R$ in terms of the internal/external reaction-diffusion lengths $l_{\mathrm{in}}$, $l_{\mathrm{out}}$ and energy scales $ε_{\mathrm{in}}$, $ε_{\mathrm{out}}$, plus a critical point that bounds viable finite droplets. The results provide actionable design guidelines: to obtain small or volume-limited droplets one must balance internal vs external production/degradation energetics and kinetics, with clear tradeoffs depending on whether the goal is minimal interior size or overall tiny droplets. This framework advances understanding of condensate size control and suggests extensions to multi-droplet emulsions, multi-component systems, and fluctuating/elastic environments relevant to biology and synthetic microreactors.

Abstract

Biological cells and synthetic analogues use liquid-liquid phase separation to dynamically compartmentalize their environment for various applications. In many cases, multiple droplets need to coexist, and their size needs to be controlled, which is challenging because large droplets tend to grow at the expense of smaller ones. Chemical reactions can, in principle, control droplet sizes, but there are no clear guidelines on how to robustly achieve size control. To provide guidelines, we consider a binary fluid model driven out of equilibrium by chemical reactions. We reveal two different classes of size-controlled droplets, depending on the ratio of droplet radius to the reaction-diffusion length. Moreover, we determine parameter regimes in which droplets become small. Taken together, our theory allows us to separately predict the chemical reactions necessary for maintaining droplets of a given class or size.

Figures (7)

• Figure 1: Conceptual model. (a) Schematic representation of the model, in which components A and B segregate from each other and inter-convert through chemical reactions. The internal energy difference between the components, $\varepsilon_0$, drives a passive reaction, while an active reaction is facilitated by an additional input of external energy, $\Delta\mu$. Concentration-dependent reaction rates can lead to differential production in the two phases. (b) Concentration field $\varphi(\vec{r})$ at three time points illustrating effective droplet size control. Simulations of Eqs. \ref{['eq:continuity']}-\ref{['eq:TST']} were done on a two-dimensional periodic grid of size $100w\times100w$ and resolution $0.5w$ using py-pdeZwicker2020 for model parameters $f\left(\varphi\right)=\frac{B}{2}\varphi^2(1-\varphi)^2-0.05\,B\varphi$, $\Delta\mu=0.1\,B\nu$, $\Lambda_p(\varphi)=0.07[1-\tanh(\frac{\varphi-0.5}{0.01})]M$, $\Lambda_a(\varphi)=0.09[1+\tanh(\frac{\varphi-0.5}{0.01})]M$.
• Figure 2: Volume-limited and interface-limited reactions. Quasi-static reaction flux $s$ as a function of the radial coordinate $r$ for different values of $l_\mathrm{in}$. The shaded blue and teal areas correspond to degradation and production of droplet material, respectively. (a) Model parameters are $\sigma=0.15\,Bw$, $\Delta\varphi=1$, $\varepsilon_0=0.05\,B\nu$, $\varepsilon_\mathrm{in}=0.05\,B\nu$, $\varepsilon_\mathrm{out}=-0.05\,B\nu$, $l_\mathrm{in}=10^4\,w$, $l_\mathrm{out}=1.11\cdot10^4\,w$, and $R=2\cdot 10^5\,w$. (b) Model parameters are $\sigma=0.15\,Bw$, $\Delta\varphi=1$, $\varepsilon_0=0.01\,B\nu$, $\varepsilon_\mathrm{in}=0.09\,B\nu$, $\varepsilon_\mathrm{out}=-0.01\,B\nu$, $l_\mathrm{in}=500\,w$, $l_\mathrm{out}=1000\,w$, and $R=455\,w$.
• Figure 3: Thin-interface model captures stationary state radii. Steady state droplet radius $R$ as a function of the scaled reaction-diffusion lengths $w/l_\mathrm{in}$ and $w/l_\mathrm{out}$. The system is homogeneously enriched in component A (white region), homogeneously enriched in component B (gray region), or exhibits an isolated droplet whose radius is indicated by color. (a) Data based on the stable fixed points of Eq. \ref{['eq:Req']} for model parameters $\varepsilon_\mathrm{in}=0.09B\nu$, $\varepsilon_\mathrm{out}=-0.01B\nu$, $\sigma=0.15B$, and $M_\mathrm{in}=M_\mathrm{out}=M$. (b) Data based on simulations of Eqs. \ref{['eq:continuity']}-\ref{['eq:TST']} on a three-dimensional, spherically symmetric grid of radius $1000w$ and resolution $0.5w$ using py-pdeZwicker2020 for the equivalent model parameters $f\left(\varphi\right)=\frac{B}{2}\varphi^2(1-\varphi)^2-0.01B\varphi$, $\Delta\mu=0.1\,B\nu$, $\Lambda_p(\varphi)=l_\mathrm{out}^{-2}[1-\tanh(\frac{\varphi-0.5}{0.01})]M$, $\Lambda_a(\varphi)=l_\mathrm{in}^{-2}[1+\tanh(\frac{\varphi-0.5}{0.01})]M$.
• Figure 4: Volume-limited and interface-limited droplets obey scaling laws. (a) Droplet radius $R$ normalized by the external reaction-diffusion length $l_\mathrm{out}$ as a function of the control parameter expected for volume-limited droplets in comparison to the scaling law given by Eq. \ref{['eq:limiting_cases']} (dashed blue line). (b) $R$ normalized to internal reaction-diffusion length $l_\mathrm{in}$ as a function of the control parameter expected for interface-limited droplets in comparison to the scaling law given by Eq. \ref{['eq:limiting_cases2']} (dashed red line). (a, b) The colored data points represent stable fixed points of Eq. \ref{['eq:Req']} for a range of parameter values. The color-coding is prescribed by the colorbar. Model parameters are $\sigma=0.15\,Bw$.
• Figure 5: Parameter regions leading to volume- vs. interface-limited droplets. Relative steady state droplet radius $R/l_\mathrm{in}$ as a function of the scaled reaction-diffusion lengths $w/l_\mathrm{in}$ and $w/l_\mathrm{out}$. Each panel corresponds to a different combination of energy scales $\varepsilon_\mathrm{in}/B\nu$ and $\varepsilon_\mathrm{out}/B\nu$. The system is homogeneously enriched in component A (white region), homogeneously enriched in component B (gray region), or exhibits an isolated droplet whose radius is indicated by color. Model parameters are $\sigma=0.15\,Bw$.