Suppression of Ostwald ripening in active emulsions

Emulsions consisting of droplets immersed in a fluid are typically unstable since they coarsen over time. One important coarsening process is Ostwald ripening, which is driven by the surface tension of the droplets. Stability of emulsions is relevant not only in complex fluids but also in biological cells, which contain liquidlike compartments, e.g., germ granules, Cajal bodies, and centrosomes. Such cellular systems are driven away from equilibrium, e.g., by chemical reactions, and thus can be called active emulsions. In this paper, we study such active emulsions by developing a coarse-grained description of the droplet dynamics, which we analyze for two different chemical reaction schemes. We first consider the simple case of first-order reactions, which leads to stable, monodisperse emulsions in which Ostwald ripening is suppressed within a range of chemical reaction rates. We then consider autocatalytic droplets, which catalyze the production of their own droplet material. Spontaneous nucleation of autocatalytic droplets is strongly suppressed and their emulsions are typically unstable. We show that autocatalytic droplets can be nucleated reliably and their emulsions stabilized by the help of chemically active cores, which catalyze the production of droplet material. In summary, different reaction schemes and catalytic cores can be used to stabilize emulsions and to control their properties.

Figure 1

Schematic representations of emulsions with droplets of enriched $B$ components (dark orange) in a background of $A$ components (light blue). (a) Full spatiotemporal description of the volume fractions ${\varphi }^A$ and ${\varphi }^B$ including diffuse interfaces. (b) Simplified description in terms of the droplet radii $R_i$ and the average volume fractions ${\varphi }_0^A$ and ${\varphi }_0^B$ in the background fluid.

Figure 2

Chemical reaction schemes of building blocks $A$ (light blue) and droplet material $B$ (medium orange) considered in this paper. (a) First-order reactions. (b) Autocatalytic droplets where $B$ is produced by the reaction $A+B\to 2B$ and at the catalytic cores (dark green), such that the production is effectively restricted to inside the droplets. Droplet material $B$ is converted back to $A$ by a simple first-order reaction.

Figure 3

Behavior of a single droplet with first-order reactions. (a) Growth rate $\dot{R}=dR/dt$ given in Eq. (13) as a function of the droplet radius $R$ in an infinite system. The passive system (light orange, $k_{\mathrm{f}}=k_{\mathrm{b}}=0,{\overline{\varphi }}^B=1/11)$ is compared to droplets with first-order kinetics (dark blue, $k_{\mathrm{f}}/k_0=3.6\times {10}^{-5},k_{\mathrm{b}}/k_0=3.6\times {10}^{-4})$. The critical radius $R_{\mathrm{crit}}$ and the stable radius $\overline{R}$ are indicated. (b) Bifurcation diagram of the droplet radius as a function of the rate constant $k_{\mathrm{f}}$ with $k_{\mathrm{b}}=10k_{\mathrm{f}}$. $\overline{R}$ (solid line) and $R_{\mathrm{crit}}$ (dotted line) were determined by solving Eq. (13) numerically. The saddle-node bifurcation point given in Eq. (17) is marked by a blue dot. The triangle indicates the scaling predicted by Eq. (16). [(c) and (d)] Same plots as in (a) and (b) for a finite system with volume $V_{\mathrm{s}}=1.8\times {10}^6{w}^3$. The maximal droplet volume is $V_{\mathrm{s}}{\overline{\varphi }}^B$ in the passive system. The gray area in (d) denotes the region where systems are relatively small, $V_{\mathrm{s}}\ll (4\pi /3){[D_B/(k_{\mathrm{f}}+k_{\mathrm{b}})]}^{3/2}$. In all panels, $\psi =1$ and quantities are normalized to the length scale $w=6\gamma \beta$ generated by the surface tension $\gamma$ and the associated rate $k_0=D_B/w^2$.

Figure 4

Behavior of two droplets as a function of their radii $R_1$ and $R_2$. The black arrows indicate the temporal evolution of the state variables $R_1$ and $R_2$ following from Eq. (13), with ${\varphi }_0^B$ given by Eq. (14) in steady state. The medium blue and light orange lines are the nullclines, which indicate where the growth rate of droplets 1 and 2 vanish, respectively. Their intersections are stable (disks) or unstable fixed points (open circles). (a) Passive system, $k_{\mathrm{f}}=k_{\mathrm{b}}=0$, with ${\overline{\varphi }}^B=1/11$. (b) First-order reactions, $k_{\mathrm{f}}/k_0=1.4\times {10}^{-4}$ and $k_{\mathrm{b}}=10k_{\mathrm{f}}$. Model parameters are $\psi =1$ and $V_{\mathrm{s}}/w^3=5.8\times {10}^4$, where $w=6\gamma \beta$ is the length scale generated by the surface tension $\gamma$.

Figure 5

Numerical simulations of the dynamics of two droplets. (a) Illustration of the numerical procedure for a passive system $\left(k_{\mathrm{f}}=k_{\mathrm{b}}=0,{\overline{\varphi }}^B=1/11\right)$. We numerically solve Eq. (3) in a cylindrical geometry as described in Appendix pp5. Snapshots (top panel) are used to measure the droplet volumes $V_i$ as a function of time $t$ (lower left panel). The rate $\lambda$ is determined by a linear fit to the logarithm of the volume difference $\mathrm{\Delta }V$ (lower right panel). (b) Coarsening rate $\lambda$ as a function of the average droplet volume $\overline{V}$ for $k_{\mathrm{f}}=k_{\mathrm{b}}=0$ with ${\overline{\varphi }}^B=1/11$. The analytical prediction (solid line) given by Eq. (18) is compared to numerical results (symbols) for $V_{\mathrm{s}}/w^3=1.5\times {10}^5$ (discs) and $V_{\mathrm{s}}/w^3=1.2\times {10}^6$ (squares). (c) Same as (a) for $k_{\mathrm{b}}/k_0=5\times {10}^{-3}$ and $k_{\mathrm{f}}/k_0=5\times {10}^{-4}$. (d) Rate $\lambda$ as a function of the backward rate constant $k_{\mathrm{b}}$ with $k_{\mathrm{f}}=0.1k_{\mathrm{b}}$. The analytical prediction given by combining Eqs. (18) and (20) (solid line) is compared to numerical results (orange disks) for $V_{\mathrm{s}}/w^3=1.5\times {10}^5$.

Figure 6

Stability of multiple droplets with first-order reactions. (a) The maximal number $N_{\max }$ of stable droplets (blue line) as a function of the forward reaction rate constant $k_{\mathrm{f}}$ with $k_{\mathrm{b}}=10k_{\mathrm{f}}$ in a finite system of volume $V_{\mathrm{s}}=2.3\times {10}^5{w}^3$. $N_{\max }$ is obtained from a numerical linear stability analysis of Eq. (13). The theoretical stability boundaries $\chi =1$ (left black line) given by Eq. (21) and $k_{\mathrm{f}}=k_{\mathrm{f}}^{\mathrm{c}}$ (right black line) given by Eq. (17) are indicated. (b) Stability of possible states in an infinite system as a function of the reaction rate constants $k_{\mathrm{f}}$ and $k_{\mathrm{b}}$. The homogeneous state ${\varphi }^B\left(\mathit{r}\right)={\varphi }_0^B$ with ${\varphi }_0^B=k_{\mathrm{f}}/(k_{\mathrm{f}}+k_{\mathrm{b}})$ is unstable and bicontinous structures typically form in the orange hatched region bounded by the stability condition given in Appendix pp6 (dashed line). Droplets enriched in $B$ components are stable in the upper blue region, which is defined by $3k_{\mathrm{f}}<k_{\mathrm{b}}<16k_0{\left({\varphi }_0^B\right)}^3$, following from Eq. (17). The color indicates the maximal droplet density $n_{\mathrm{c}}$, which follows from Eq. (21) for $\chi =1$. Correspondingly, $A$-rich droplets are stable in the lower blue region. The homogeneous state ${\varphi }^B\left(\mathit{r}\right)={\varphi }_0^B$ is the only stable state in the white region.

Figure 7

Behavior of a single autocatalytic droplet in an infinite system (orange; $Q_1=0)$, in a finite system (blue; $V_{\mathrm{s}}/w^3=6.9\times {10}^7,Q_1=0)$, and in a finite system with a catalytically active core (green; $V_{\mathrm{s}}/w^3=6.9\times {10}^7,Q_1=300w^3{k}_0)$. (a) Growth rate $\dot{R}=dR/dt$ derived from Eq. (22) as a function of the droplet radius $R$ for $\overline{\varphi }=0.02$. The critical radius $R_{\mathrm{crit}}$ and the stable radius $\overline{R}$ derived from Eq. (23) are indicated. (b) Bifurcation diagram of the droplet radius as a function of the total volume fraction $\overline{\varphi }$ of material. Dotted lines indicate unstable steady states. Model parameters are $k/k_0=0.02,k_{\mathrm{b}}/k_0=2\times {10}^{-4}$, and $\psi =0.1$, with $k_0=D_B/w^2$.

Figure 8

Behavior of two autocatalytic droplets as a function of their radii $R_1$ and $R_2$. The black arrows indicate the temporal evolution of the state variables $R_1$ and $R_2$ following from Eq. (22). The medium blue and light orange lines are the nullclines, which indicate where the growth rate of droplets 1 and 2 vanish, respectively. Their intersections are stable (disks) or unstable fixed points (open circles). (a) No cores, $Q_i=0$. (b) Catalytically active cores, $Q_i=300w^3{k}_0$. The volume fractions ${\varphi }_0^A$ and ${\varphi }_0^B$ are fixed to their stationary state value and model parameters are $k_{\mathrm{f}}=0,k_{\mathrm{b}}/k_0=2\times {10}^{-4},k/k_0=0.02,\psi =0.1,\overline{\varphi }=0.02$, and $V_{\mathrm{s}}/w^3=6.9\times {10}^7$.

Figure 9

Behavior of emulsions of autocatalytic droplets. (a) Rate $\lambda$ as a function of the average droplet volume $\overline{V}$. The analytical prediction given by Eq. (24) for small systems $({\varphi }_0^B={\varphi }_{+}^B$, lower blue line) and large systems $({\varphi }_0^B=0$, upper blue line) is compared to numerical results for small (orange disks, $V_{\mathrm{s}}/w^3=1.5\times {10}^5)$ and large (orange squares, $V_{\mathrm{s}}/w^3=8.8\times {10}^5)$ systems, which were obtained as described in Fig. 5. Model parameters are $\psi =0.1,k/k_0=100,k_{\mathrm{b}}/k_0=0.01$. (b) Maximal number $N_{\max }$ of stable droplets as a function of the total volume fraction $\overline{\varphi }$ of material and the catalytic activity $Q_i$ of cores within the droplets. The red dashed line denotes the approximate threshold value of $Q_i$ given in Eq. (25) with ${\varphi }_0^B\approx 0$ valid for large systems. Model parameters are $\psi =0.1,k/k_0=0.02,k_{\mathrm{b}}/k_0=2\times {10}^{-4},V_{\mathrm{s}}/w^3=6.9\times {10}^7$.

Figure 10

Behavior of two autocatalytic droplets with cores of unequal activity $Q_i$. (a) Temporal evolution of the droplet radii $R_1$ and $R_2$ (black lines) following from Eq. (22) for $Q_1=1000w^3{k}_0$ and $Q_2=300w^3{k}_0$. The medium blue and light orange lines are the nullclines, which indicate where the growth rate of droplets 1 and 2 vanish, respectively. Their intersection (disk) is the only fixed point, which is stable. (b) Droplet volumes $V_i$ in steady state as a function of the catalytic activity $Q_1$ of one core for $Q_2=300w^3{k}_0$. In both panels, ${\varphi }_0^A$ and ${\varphi }_0^B$ are fixed to their stationary state values and model parameters are $k_{\mathrm{f}}=0,k_{\mathrm{b}}/k_0=2\times {10}^{-4},k/k_0=0.02,\psi =0.1,\overline{\varphi }=0.02$, and $V_{\mathrm{s}}/w^3=6.9\times {10}^7$.