Pattern formation arising from condensation of a homogeneous gas into a binary, phase-separating liquid

We examine the nucleated growth of a binary, immiscible liquid drop within a homogeneous gas. The system couples the growth of the liquid drop with the phase separation of the immiscible components and, thus, can potentially reveal novel pattern formation. To carry out this study, we first characterize the thermodynamic properties of the system in terms of an appropriate Ginzburg-Landau free energy density. By minimizing this free energy, we construct the equilibrium phase diagram for the system. We then use a lattice Boltzmann algorithm to solve the hydrodynamic equations describing the dynamical evolution of the fluid. We observe intriguing tentaclelike structures within the nucleation and growth regime and explore how the formation of these structures depends on the thermodynamic and transport properties of the system. We give scaling laws describing domain growth in both the diffusion- and flow-limited regimes. The results highlight the novel physics that can emerge when there is interplay between the ordering of a density and a concentration field.

Figure 1

The phase diagrams for two different values of the interspecies interaction strength $\lambda$. They show the behavior of a system with an average density $\overline{\rho }$ and temperature $\theta$. If the temperature is below the critical point, and $\overline{\rho }$ lies between the two solid curves, then the system lowers its free energy by phase separating into a liquid and a gas phase. The lambda line divides binary states to the right from homogeneous states to the left. The spinodal line encloses states that spontaneously phase separate. (a) A weaker interspecies interaction, $\lambda =0.008$. (b) A stronger interspecies interaction, $\lambda =0.014$. The following symbols mark these specific cases: $\times$, simulation results of the coexistence curve; 엯, nucleated tentacle structures; ▵, no tentacles form; and ◻, spinodal decomposition.

Figure 2

The time evolution of the density difference profile $\phi$ across the system. The time is measured in lattice units. Black and white areas denote the A-rich and B-rich liquid, respectively, and gray areas represent gas. (a) High viscosity $\nu =1$. (b) Low viscosity $\nu =0.125$.

Figure 3

(a) The variation in time of the tip radius (solid lines) and the core radius (dashed lines). The viscosity was $\nu =1$ and the mobility was $M=10$. The time is measured in lattice units. By taking a linear regression, the tip and core radius growth rates were found to be 0.020 and 0.0068, respectively. (b) The variation of tip and core radius growth rates as a function of kinematic viscosity $\nu$, with fixed mobility $M=10$. The dotted line shows the transition between the droplet forming/not forming a tentacle structure. (c) The variation of tip and core radius growth rates as a function of mobility $M$. The viscosity was $\nu =0.5$.

Figure 4

Snapshots, shown in perspective, of nucleated droplets in 3D (the gas phase is transparent). Black and white areas are the A-rich and B-rich liquid, respectively. The mobility was $M=5$. (a) Snapshot at $t=9000$ with high viscosity $\nu =2$. (b) Snapshot at $t=7800$ with low viscosity $\nu =0.1$.

Figure 5

The core and tip radii of a 3-D droplet, as a function of time. The low viscosity was $\nu =0.1$ and the high viscosity was $\nu =2$. The mobility was $M=5$.

Figure 6

A contour plot of the free energy density $\psi$ (see Eq. (2)) [15]. The three minima show the coexistence between the gas and liquid A-rich and B-rich phases.