Capillary Interfacial Tension in Active Phase Separation

In passive fluid-fluid phase separation, a single interfacial tension sets both the capillary fluctuations of the interface and the rate of Ostwald ripening. We show that these phenomena are governed by two different tensions in active systems, and compute the capillary tension ${\sigma }_{\mathrm{cw}}$ which sets the relaxation rate of interfacial fluctuations in accordance with capillary wave theory. We discover that strong enough activity can cause negative ${\sigma }_{\mathrm{cw}}$. In this regime, depending on the global composition, the system self-organizes, either into a microphase-separated state in which coalescence is highly inhibited, or into an “active foam” state. Our results are obtained for Active Model $\mathrm{B}+$, a minimal continuum model which, although generic, admits significant analytical progress.

Figure 1

Mean-field phase diagram for $\zeta >0$, showing sign regimes of interfacial tensions $\sigma$ and ${\sigma }_{\mathrm{cw}}$. When ${\sigma }_{\mathrm{cw}}>0$, the interface is stable and unstable otherwise. Orange circles and blue squares respectively denote the results of direct simulations of $\mathrm{AMB}+$ where the instability of the interface is or is not observed. Right: interfacial instability ($\zeta =1.5$, $\lambda =2$).

Figure 2

(Left) liquid-vapor interface for ${\sigma }_{\mathrm{cw}}>0$ ($L_x=L_y=256$), $D=0,\zeta =2\lambda$ and its relaxation compared to the theoretical predictions for initial perturbations at wave number $2\pi n/L_x$. Dashed lines are predictions obtained using $\tau (q)$, converging to the $q\to 0$ prediction (continuous line). (Right) snapshot in steady state for $D=5\times {10}^{-3}$ and scaled structure factor $q^{2}S(q)/D$ vs $q$ compared to the $q\to 0$ analytical prediction; results are averaged over 30 realisations of duration ${10}^6$ after equilibration.

Figure 3

Instability at ${\sigma }_{\mathrm{cw}}<0$. The densities on the two sides of the interface adjust quasistatically at values that depend on its local curvature. The ensuing diffusive density fluxes on the vapor side is always stabilising (white arrow); that in the liquid is stabilising when $\sigma >0$ (arrow 1) and becomes destabilizing when $\sigma <0$ (arrow 2). This (one-sided) reverse-Ostwald current does not trigger an instability unless the current in the liquid outweighs that in the vapor, which requires ${\sigma }_{\mathrm{cw}}<0$.

Figure 4

(Left) phase diagram when ${\sigma }_{\mathrm{cw}}<0$ as a function of the global density ${\varphi }_0=-1,-0.4$, 0.4, 1.2 at $D=0.05$, $L_x=256$, $L_y=512$, and $\lambda =1.75$, $\zeta =2$, for which ${\varphi }_1=-0.9$, ${\varphi }_2=1.08$. At high and low ${\varphi }_0$, the system is homogeneous (liquid or vapor states). Within the binodals, when ${\varphi }_0>({\varphi }_1+{\varphi }_1)/2$, the system shows microphase-separated vapor bubbles whose coalescence is highly inhibited. At lower ${\varphi }_0$, the system forms a continuously evolving active foam state. (Middle and right): area distribution of vapor regions for the active foam state (${\varphi }_0=-0.4$) and in the microphase-separated state (${\varphi }_0=0.2$, noise values in the legend).