Thermodynamic coarsening arrested by viscous fingering in partially miscible binary mixtures

We study the evolution of binary mixtures far from equilibrium, and show that the interplay between phase separation and hydrodynamic instability can arrest the Ostwald ripening process characteristic of nonflowing mixtures. We describe a model binary system in a Hele-Shaw cell using a phase-field approach with explicit dependence of both phase fraction and mass concentration. When the viscosity contrast between phases is large (as is the case for gas and liquid phases), an imposed background flow leads to viscous fingering, phase branching, and pinch off. This dynamic flow disorder limits phase growth and arrests thermodynamic coarsening. As a result, the system reaches a regime of statistical steady state in which the binary mixture is permanently driven away from equilibrium.

Figure 1

(Top) The common tangent construction (blue line) on the bulk free energy of pure phases yields the equilibrium concentrations (green circles) within gas and liquid. For parameters ${\alpha }_l=1, {\beta }_l=500, {\alpha }_g=200, {\beta }_g=2\times {10}^{-4}$, the equilibrium compositions are $c_l^{\text{eq}}\approx 0.304$ and $c_g^{\text{eq}}\approx 0.828$. (Bottom) Zoomed-in snapshots of $c$ illustrate the process of spinodal decomposition of a domain that is initially filled with supersaturated liquid ($t=0$). The progression shows vapor bubbles (high $c$, red) that nucleate and coarsen out of the liquid phase (low $c$, pink).

Figure 2

Snapshots of $c$ (top) and $\varphi$ (bottom) illustrating the Ostwald ripening process in a system with a small and a large vapor bubble, in a liquid bath that is initially at local equilibrium.

Figure 3

Snapshots of $c$ at $t=40,150,350$, and 500, under no flow (top) and with periodic left-to-right flow imposed at $t>40$ (bottom). See Video 1 [50].

Figure 4

(a) ${\langle r\rangle }^3$ vs $t$ for simulations without flow (dashed red line) and with background flow after $t=40$ (solid black line), emphasizing arrest of thermodynamic coarsening in the presence of flow. Inset: $\langle r\rangle$ vs $t$ in log-log scale, emphasizing algebraic growth of spinodal decomposition ($\langle r\rangle \sim t^{1/3}$) in the absence of flow. (b)–(c) Normalized distribution of $r/\langle r\rangle$ at sampling times for simulations without flow (b) and with flow introduced at $t=40$ (c). (d) $\overline{\langle r\rangle }$ vs Ca for different Pe. (e) $\overline{\langle r\rangle }$ rescaled with ${\mathrm{Pe}}^{-0.078}$ vs Ca. Inset: $\overline{\langle r\rangle }$ vs Pe for different Ca.

Figure 5

Evolution of the averaged liquid-phase concentration $\langle c_l\rangle$ for systems without background flow (red solid line) and with background flow after $t=40$ (black solid line). The gray dashed line indicates equilibrium liquid-phase concentration from the common tangent construction [Fig. 1 (top)]. Insets: zoomed-in snapshots of $c$. The circle highlights pinch off of a small bubble that quickly dissolves into the liquid. The color map range is $(0.3,0.35)$ to emphasize concentration variations around the $c_l^{\text{eq}}$; see Video 2 [50].