Activity-Suppressed Phase Separation

We use a continuum model to examine the effect of activity on a phase-separating mixture of an extensile active nematic and a passive fluid. We highlight the distinct role of (i) previously considered interfacial active stresses and (ii) bulk active stresses that couple to liquid crystalline degrees of freedom. Interfacial active stresses can arrest phase separation, as previously demonstrated. Bulk extensile active stresses can additionally strongly suppress phase separation by sustained self-stirring of the fluid, substantially reducing the size of the coexistence region in the temperature-concentration plane relative to that of the passive system. The phase-separated state is a dynamical emulsion of continuously splitting and merging droplets, as suggested by recent experiments. Using scaling analysis and simulations, we identify various regimes for the dependence of droplet size on activity. These results can provide a criterion for identifying the mechanisms responsible for arresting phase separation in experiments.

Figure 1

Phase diagram in the $({\varphi }_0,\alpha )$ plane. The black lines are the binodal (solid) and spinodal (dashed) curves for the mean field theory in equilibrium. The numerical data for three values of activity $\alpha$ clearly show the shift of the critical point and its increase with activity. The lines through the data are parabolic fits and the thick dots mark the critical point. The vertical line at ${\varphi }_0=0$ highlights the shift of the critical point to values of ${\varphi }_0$ that correspond to more than 50% of passive fluid. The error bars are the precision with which we can visually differentiate uniform and phase-separated states.

Figure 2

Top: time evolution of the width $\mathrm{\Delta }\varphi$ of the coexistence region for $a=-0.2$ (blue curve) and $a=-1.0$ (red curve) corresponding to states above and below the critical point $a_c^{*}\approx -0.3$ of the active mixture, respectively. Here, $|\alpha |=0.8$. Bottom: snapshots of the simulations at times $t=0$, 3, 10, and 90 marked by vertical lines in the plot. The color of the frame of the snapshots matches the color of the corresponding $\mathrm{\Delta }\varphi$ curves.

Figure 3

Finite-size scaling of $\mathrm{\Delta }\varphi (a,{\varphi }_0^c)$ for $|\alpha |=1.2$. The critical point $a_c$ is defined as the value of $a$ where $\mathrm{\Delta }\varphi =0$. The black line is the equilibrium mean field line of the passive system, with $a_c=0$. For active mixtures, $a_c$ is shifted to lower (negative) values and approaches $a_c\approx -0.4$ for the largest system sizes considered. $d\mathrm{\Delta }\varphi /da$ diverges at the active critical point. The inset shows an enlarged portion of the data in the black rectangle.

Figure 4

Droplet size as a function of the time for an initially uniform state and various activity for two sets of parameters. Left: $l_D\approx 10l_{\alpha }$ (left), i.e., regime (i) where coarsening is controlled by diffusion and we estimate $L^{*}\sim |\alpha {|}^{-1/3}$. Right: $l_{\alpha }\approx 10l_D$ (right), i.e., regime (ii) where coarsening is controlled by elasticity and we estimate $L^{*}\sim |\alpha {|}^{-1/2}$. The insets show the scaling of the saturation value $L^{*}$ with $|\alpha |$.