Dynamics of phase separation of sheared binary mixtures after a nonisothermal quenching

When a symmetric regular binary mixture, subjected to a constant shear, is quenched into the unsteady region of its phase diagram under a temperature gradient, it phase separates following very complicated patterns. The phase separation process is simulated using a thermodynamics-based phase-field model where the fundamental balance equations are coupled with the constitutive equations for the diffusive fluxes of chemical species, momentum, and energy by applying the rules of nonequilibrium thermodynamics. The evolution of phase separation shows distinct features, being more complex than a simple superposition of patterns emerging in a shear flow and under a thermal gradient when taken individually. The imposed temperature gradient causes a preferential nucleation at the cooler wall, so that the emerging droplets drift towards the center of the domain while following the imposed flow field, causing a change in droplet movement as they cross the domain centerline and enhancing coalescence. The imposed temperature gradient breaks the symmetry compared to instantaneous quenching, with stable droplets which remain attached to the cooler wall and move coherently with it. The capillary number $\left(N_{\mathrm{Ca}}\right)$ determines breakup and phase separation evolving as stripes for $N_{\mathrm{Ca}}\gg 1$, while droplets nucleate and grow for $N_{\mathrm{Ca}}\ll 1$. The Lewis number $\left(N_{\mathrm{Le}}\right)$ affects the pace and propagation of phase separation: for $N_{\mathrm{Le}}>10$ phase separation takes place rather uniformly, being similar to instantaneous quenching, while for $N_{\mathrm{Le}}<0.1$ the mixture cools slowly and a phase separation front proceeds from the cooler wall. A similar behavior is induced by a composition-dependent thermal conductivity. The mixture dimensionless heat capacity $\left(N_c\right)$ has a significant effect on phase separation because, for $N_c\ll 1$, heat dissipation counterbalances the effect of the applied temperature quench, thus retarding or even reversing the process of phase separation. These results and the variety of patterns reproduced by the model highlight the necessity of integrating a consistent thermodynamic description to the hydrodynamics and heat transport in phase-field modeling.

Figure 1

Phase separation of a binary mixture at different dimensionless times under a thermal gradient in the limit case of no imposed shear (i.e., $\tilde{\gamma }=0$) at $N_{\alpha }={10}^{-2}$ and $N_c^{-1}=0$ for two values of the Lewis number. In these simulations a square domain with $\tilde{W}=150$ is used. The other parameters, kept constant throughout this study if not otherwise specified (see Table 1), are $N_{\mathrm{Sc}}={10}^2$, ${\mathrm{\Psi }}^{+}=2.6$, ${\mathrm{\Psi }}^{\text{–}}=2.1$, ${\mathrm{\Psi }}_0=1.9$, ${\varphi }_0=0.4$, $\tilde{D}=\tilde{\eta }=\tilde{c}=\tilde{k}=1$.

Figure 2

Phase separation of a binary mixture having a high capillary number ($N_{\mathrm{Ca}}=50>N_{\mathrm{Ca}}^{cr}$) under a thermal gradient for an imposed dimensionless shear rate $\tilde{\gamma }={10}^{-2}$ for two values of the Lewis number (left and right panels). The fluidity number is set to $N_{\alpha }={10}^{-2}$ while $N_c^{-1}=0$. The first snapshot reports the imposed velocity profile with gray arrows. In all the other snapshots, the black arrows are used to guide the eye to schematically indicate the direction of the red phase while it moves.

Figure 3

Phase separation of a binary mixture having a small capillary number ($N_{\mathrm{Ca}}=5\times {10}^{-3}<N_{\mathrm{Ca}}^{cr}$) under a thermal gradient for an imposed dimensionless shear rate $\tilde{\gamma }={10}^{-2}$ for $N_{\mathrm{Le}}=0.1$. The fluidity number is set to $N_{\alpha }={10}^2$ while $N_c^{-1}=0$. The two panels report two different evolutions depending on whether phase separation starts with nucleation at the top of the minority phase (a) or of the majority phase (b).

Figure 4

Phase separation of a binary mixture having a small capillary number ($N_{\mathrm{Ca}}=5\times {10}^{-3}<N_{\mathrm{Ca}}^{\mathrm{cr}}$) under a thermal gradient for an imposed dimensionless shear rate $\tilde{\gamma }={10}^{-2}$ for $N_{\mathrm{Le}}=10$. The fluidity number is set to $N_{\alpha }={10}^2$ while $N_c^{-1}=0$.

Figure 5

Phase separation of a binary mixture under a thermal gradient for different values of the dimensionless heat capacity $N_c$ at two representative times. The imposed dimensionless shear rate is $\tilde{\gamma }={10}^{-2}$ while $N_{\mathrm{Le}}=10$ and $N_{\alpha }={10}^2$, corresponding to $N_{\mathrm{Ca}}=5\times {10}^{-3}<N_{\mathrm{Ca}}^{\mathrm{cr}}$.

Figure 6

(a) Distribution of the Margules parameter Ψ for two different values of the dimensionless heat capacity $N_c$ at two representative times for $\tilde{\gamma }={10}^{-2}$, $N_{\mathrm{Le}}=10$ and $N_{\alpha }={10}^2$, which are the same conditions as in Fig. 5; (b) dimensionless entropy production term and chemical potential difference, with superimposed velocity streamlines, for the case $N_c=1$ at $\tilde{t}=600$.

Figure 7

Phase separation of a binary mixture having the composition-dependent thermal conductivity reported in Eq. (28), under a thermal gradient and for an imposed dimensionless shear rate of $\tilde{\gamma }={10}^{-2}$ with $N_{\mathrm{Le}}=10$, $N_{\alpha }={10}^2$, and $N_c^{-1}=0$. In this simulation round droplets nucleate at the top boundary.

Figure 8

Phase separation of a binary mixture having the composition-dependent thermal conductivity reported in Eq. (28), under a thermal gradient and for an imposed dimensionless shear rate of $\tilde{\gamma }={10}^{-2}$ with $N_{\mathrm{Le}}=10$, $N_{\alpha }={10}^2$, and $N_c^{-1}=0$. In this simulation an elongated stripe nucleates at the top boundary.

Figure 9

Main results of the study summarized in a diagram which highlights the principal effects of capillary number, Lewis number, and dimensionless heat capacity on the regimes of nonisothermal phase separation of a sheared binary mixture.