Advection selects pattern in multi-stable emulsions of active droplets

Abstract

Controlling the size of droplets, for example in biological cells, is challenging because large droplets typically outcompete smaller droplets due to surface tension. This coarsening is generally accelerated by hydrodynamic effects, but active chemical reactions can suppress it. We show that the interplay of these processes leads to three different dynamical regimes: (i) Advection dominates the coalescence of small droplets, (ii) diffusion leads to Ostwald ripening for intermediate sizes, and (iii) reactions finally suppress coarsening. Interestingly, a range of final droplet sizes is stable, of which one is selected depending on initial conditions. Our analysis demonstrates that hydrodynamic effects control initial droplet sizes, but they do not affect the later dynamics, in contrast to passive emulsions.

Figures (10)

• Figure 1: Turnover causes hexagonal droplet pattern. (A) Schematic of our model involving two species that inter-convert and phase separate from each. (B) Simulation snapshot showing $c(\boldsymbol{r})$ at stationarity for $\mathrm{P\space e} = 0$, $c_0 = 0.35 \, c_\mathrm{in}$, $k = 0.02/ \tau$, $\tau=w^2/D$, and $w = \sqrt{\kappa/a}$.
• Figure 2: Reaction rate controls stationary pattern size. (A) Pattern size $L_\mathrm{eq}$, determined numerically (symbols) and predicted by Eq. \ref{['eq:AnalyticalLengthScale']} (lines), as a function of reaction rate $k$ for various average compositions $c_0$. Inset shows corresponding radii. (B) Concentration profile $c(\boldsymbol{r})$ for $k=10^{-3}/\tau$ in a unit cell with adjustable length $L$. (C) Average pattern size $\bar{L}$ as a function of time for various $k$, where $\bar{L}= 2\pi^{-1/2} (V_\mathrm{sys}/N)^{1/2}$ from droplet count $N$ and system size $V_\mathrm{sys}$. Numerical data is compared to $L_\mathrm{eq}$ from Eq. \ref{['eq:AnalyticalLengthScale']} (gray dotted lines) and a $t^{1/3}$ power law (black dashed line). (D) Distribution of droplet radii $R$ (left) and the average radius $\bar{R}$ (right) for various time points for $k = 10^{-5}/\tau$. Numerical data is compared to the prediction from spinodal decomposition (gray dashed line) and $R_\mathrm{eq} = \sqrt{c_0/4 c_\mathrm{in}} \, L_\mathrm{eq}$ (black dashed line). (A--D) Model parameters are $c_0 = 0.25\,c_\mathrm{in}$, $\tau = w^2/D$, and $w = \sqrt{\kappa/a}$.
• Figure 3: Many stationary states are stable. (A) Pattern size $\bar{L}$ as a function of time $t$ for three initial conditions with $c_0 = 0.4 \, c_\mathrm{in}$ and differing droplet size (see snapshots and Appendix \ref{['app:effsim']}). (B) Final pattern size $\bar{L} = 2/\sqrt{\pi} (V_\mathrm{sys}/N)^{1/2}$ as a function of the initial pattern size $\bar{L}_\mathrm{i}$ for various $c_0$. Colored dashed lines indicate minimal stable size $L_\mathrm{min}$ obtained from stability analysis. Open circles indicate shape instabilities. (A--B) Gray dotted lines mark free energy minimum for $c_0 = 0.4 \, c_\mathrm{in}$ given by Eq. \ref{['eq:AnalyticalLengthScale']}. Model parameters are $k = 10^{-3}/\tau$, $\tau = w^2/D$, and $w = \sqrt{\kappa/a}$.
• Figure 4: Advection accelerates coarsening. Pattern size $\bar{L}$ (shaded area indicates SD, $n=20$) as a function of time $t$ for various Péclet numbers $\mathrm{P\space e} = w^2 \, a \, c_\mathrm{in}^2/(D \eta)$ and reaction rates $k$ (across panels). Black dashed lines show power laws $\bar{L} \propto t$ (left) and $\bar{L} \propto t^{1/3}$ (right). Gray dotted lines show $L_\mathrm{eq}$ calculated from Eq. \ref{['eq:AnalyticalLengthScale']}. Inset in (B) shows the velocity field generated by two colliding droplets (gray arrows) for $k = 10^{-4}/\tau$ and $\mathrm{P\space e} = 100$. (D--F) Volume averaged viscous (blue) and diffusive (orange) dissipation corresponding to (A--C) for $\mathrm{P\space e} = 100$ (Appendix \ref{['app:dissipation']}). The curves were smoothed by convolution with a Gaussian kernel with a standard deviation of 2 data-points to make the plot more readable. Black dashed line in (D) shows a power law $\propto t^{-2/3}$. (A--F) Model parameters are $c_0 = 0.4\,c_\mathrm{in}$, $\tau = w^2/D$, and $w = \sqrt{\kappa/a}$.
• Figure S1: Pattern size $\bar{L}$ (for $n=20$ simulations per line) as a function of time $t$ for various Péclet numbers $\mathrm{P\space e} = w^2 \, a \, c_\mathrm{in}^2/(D \eta)$ and reaction rates $k$ (across panels). Black dashed line shows power laws $\bar{L} \propto t^{1/3}$ (A). (A--B) Model parameters are $c_0 = 0.25\,c_\mathrm{in}$, $\tau = w^2/D$, and $w = \sqrt{\kappa/a}$.