Cluster Phases and Bubbly Phase Separation in Active Fluids: Reversal of the Ostwald Process

It is known that purely repulsive self-propelled colloids can undergo bulk liquid-vapor phase separation. In experiments and large-scale simulations, however, more complex steady states are also seen, comprising a dynamic population of dense clusters in a sea of vapor, or dilute bubbles in a liquid. Here, we show that these microphase-separated states should emerge generically in active matter, without any need to invoke system-specific details. We give a coarse-grained description of them and predict transitions between regimes of bulk phase separation and microphase separation. We achieve these results by extending the ${\varphi }^4$ field theory of passive phase separation to allow for all local currents that break detailed balance at leading order in the gradient expansion. These local active currents, whose form we show to emerge from coarse graining of microscopic models, include a mixture of irrotational and rotational contributions and can be viewed as arising from an effective nonlocal chemical potential. Such contributions influence, and in some parameter ranges reverse, the classical Ostwald process that would normally drive bulk phase separation to completion.

Popular Summary

Active colloids are micrometer-sized particles that can move under their own power, a feat that has potential applications in environmental cleanup, drug delivery, and microfluidics. Systems composed of purely repulsive active colloids are capable of self-separating dense liquidlike phases of matter from dilute gaslike ones. Moreover, researchers often see dynamic populations of dense clusters in a sea of vapor or phase separation between a gas phase and seemingly boiling liquid. These are typically explained using system-specific details. Here, we show that such microphase separation does not require system-specific explanations but stems generically from the nonequilibrium nature of active systems.

We study the simplest fully generic field theory describing phase separation when time-reversal symmetry is locally broken, and we show that it arises from explicit coarse graining of particle models. We then discover a new and generic explanation of active microphase separation. At the heart of our results is the fact that, due to activity, the “Ostwald ripening process”—a phenomenon that describes how passive systems undergo full phase separation—can go into reverse.

By rationalizing the occurrence of microphase separation in active systems, and providing both the minimal description and particle model expected to show such phenomenology, we hope that our work will trigger computational and experimental studies to investigate this new phase of matter and control it in the future.

Figure 1

(a) Bubbly phase separation as described in our field theory, a phase coexistence of boiling liquid (yellow) with a vapor phase (blue). Vapor bubbles are continuously created inside the bulk liquid phase, which are then expelled to the exterior vapor phase (indicated by a blue arrow in the figure, also see Movie 4 in Ref. [36]). The vapor phase can then diffuse back inside the liquid, and the process continues indefinitely. Time-reversal symmetry is manifestly broken in this steady state. (b) Arrested phase separation: Initially, the system phase separates into a dense phase and a dilute phase. However, the average cluster size saturates to a finite value at a steady state. Additional steady states, representing (a) a cluster phase in coexistence with excess dense liquid and (b) a strict bubble phase, are found by inverting the order parameter scale shown the top right, as it follows from the invariance of the model under the duality mapping explained in Sec. 2. The parameters we use are $\zeta =4$, $\lambda =1$, ${\varphi }_0=-0.4$ in (a) and $\zeta =-4$, $\lambda =-1$, ${\varphi }_0=-0.6$ in (b).

Figure 2

The binodals ${\varphi }_1$ and ${\varphi }_2$ (coexisting densities for a flat interface) for $K=1$ and any value of $A$ in the mean-field limit ($D=0$). The binodals depend on a combination of the two activity parameters ($\zeta -2\lambda$) and not on them separately. Dashed lines indicate spinodal density defined by $f^{\prime \prime }({\varphi }_s)=0$.

Figure 3

Stationary values of the density for a spherical dense droplet in a dilute environment in the mean-field approximation ($D=0$). We plot in (a) the outer density ${\varphi }_{-}(R)$ and in (b) the inner density ${\varphi }_{+}(R)$ as a function of droplet radius $R$ for two different sets of activity parameters: $\zeta =-1$ and $\lambda =0.5$ [red, corresponding to region A of Fig. 4] and $\zeta =-4$ and $\lambda =-1$ [blue, corresponding to region B of Fig. 4]. Points indicate results from the numerical simulations, and lines indicate analytical results. When the pseudosurface tension $\sigma$ is positive (red line), ${\varphi }_{\pm }(R)$ are both decreasing functions of $R$. Instead, when $\sigma <0$ (blue line), ${\varphi }_{-}(R)$ is increases when $R$ increases.

Figure 4

(a) Mean-field phase diagram of AMB+ with $K=1$, $K_1=0$, and any value of $A>0$, in $d=2$, 3. Ostwald ripening is reversed where $\sigma <0$. In region A, $\sigma >0$, and we have normal or forward Ostwald ripening for both bubbles and droplets. In region B (resp. C), Ostwald ripening is reversed for dense droplets dispersed in a dilute environment (resp. dilute bubbles in a dense environment). Points are results from simulations of two droplets (red squares for reverse and blue circles for forward Ostwald ripening). (b) Evolution of two droplets with initial radii $R_1>R_2$ for $(\zeta ,\lambda )=(-1,0.5)$ (red), corresponding to region A, and $(-4,-1)$ (blue), corresponding to region B. Solid lines are direct numerical simulations of Eqs. (5) and (6) with $D=0$, and dashed lines are predictions from our theory (see Appendix pp5). We observe forward Ostwald ripening in the first case and reverse Ostwald ripening for the second, as predicted.

Figure 5

(a) Steady-state phase diagram as a function of global density ${\varphi }_0=(1/V)\int d\mathbf{r}\varphi$ for two different sets of activity parameters: $\zeta =1$, $\lambda =-0.5$ corresponding to region A in Fig. 4, and $\zeta =4$, $\lambda =1$ corresponding to region C. In region A, the steady-state phase diagram resembles equilibrium Model B with full phase separation into dense (yellow) and dilute (blue) phases. In region C, besides the low- and high-density uniform phases, we observe either bubbly phase separation or a homogeneous bubble phase. Since AMB+ is symmetric under $(\varphi ,\lambda ,\zeta )\to -(\varphi ,\lambda ,\zeta )$, the phase diagram in region B is obtained by exchanging dense and dilute regions. (b) Probability distribution of bubble radius for parameters $\zeta =4$, $\lambda =1$ as a function of time, starting from an initial uniform density $\varphi (\mathbf{r},t=0)={\varphi }_0=0.6$. The system reaches a steady state at $t\simeq 32000$. (c) Average radius of bubbles $\overline{R}$ and number density of bubbles $\overline{n}$ in the homogeneous bubble phase as a function of $D$. Parameters are $\zeta =4$, $\lambda =1$, and ${\varphi }_0=0.6$.

Figure 6

Snapshots of the total current $\mathbf{J}$ for the two droplet simulations shown in Fig. 4 at short time $t=1000$. The top half corresponds to regime A or normal Ostwald ($\zeta =-0.8$, $\lambda =-0.4$), and the bottom half corresponds to regime B or reverse Ostwald ($\zeta =-1.6$, $\lambda =-0.8$). (Note that the system is symmetric in $y$. The arrows are magnified for better visibility; the typical magnitude of $\mathbf{J}$ is around ${10}^{-4}$.)