Spinodal decomposition of chemically reactive binary mixtures

We simulate the influence of a reversible isomerization reaction on the phase segregation process occurring after spinodal decomposition of a deeply quenched regular binary mixture, restricting attention to systems wherein material transport occurs solely by diffusion. Our theoretical approach follows a diffuse-interface model of partially miscible binary mixtures wherein the coupling between reaction and diffusion is addressed within the frame of nonequilibrium thermodynamics, leading to a linear dependence of the reaction rate on the chemical affinity. Ultimately, the rate for an elementary reaction depends on the local part of the chemical potential difference since reaction is an inherently local phenomenon. Based on two-dimensional simulation results, we express the competition between segregation and reaction as a function of the Damköhler number. For a phase-separating mixture with components having different physical properties, a skewed phase diagram leads, at large times, to a system converging to a single-phase equilibrium state, corresponding to the absolute minimum of the Gibbs free energy. This conclusion continues to hold for the critical phase separation of an ideally perfectly symmetric binary mixture, where the choice of final equilibrium state at large times depends on the initial mean concentration being slightly larger or less than the critical concentration.

Figure 1

Dimensionless reaction rates as a function of concentration from Eq. (13) (solid) and Eq. (15) (dashed).

Figure 2

Thermodynamic Gibbs free energy for a binary mixture with ${\mathrm{\Psi }}^{+}=2.1, {\mathrm{\Psi }}^{-}=0$, and ${\mu }_0=-0.01$. The absolute minimum of the free energy is at ${\varphi }_{\text{eq}}^{*}=0.7046$.

Figure 3

Mass fraction snapshots at different (nondimensional) times $t=5\times {10}^{-2},5\times {10}^{-1},\text{and}2$ (left to right) from phase-field simulations of an asymmetric binary mixture with ${\mathrm{\Psi }}^{+}=2.1, {\mathrm{\Psi }}^{-}=0$, and ${\mu }_0=-0.01$ on a ${512}^2$ grid with Damköhler numbers ${\text{Da}}_{{}_L}=0.531,5.31,53.1,\text{and}531$ (top to bottom).

Figure 4

Temporal evolution of the characteristic length scale of single-phase microdomains from phase-field simulations of an asymmetric binary mixture with ${\mathrm{\Psi }}^{+}=2.1, {\mathrm{\Psi }}^{-}=0$, and ${\mu }_0=-0.01$ on a ${512}^2$ grid and different Damköhler numbers.

Figure 5

Normalized mean concentration vs time (reaction time units) from phase-field simulations of an asymmetric binary mixture with ${\mathrm{\Psi }}^{+}=2.1, {\mathrm{\Psi }}^{-}=0$, and ${\mu }_0=-0.01$ at different Damköhler numbers.

Figure 6

Mass fraction snapshots at different (nondimensional) times $t=4\times {10}^{-3},2.4,3.1,12,17$, and 28 (left to right and top to bottom) from phase-field simulations of a perfectly symmetric binary mixture with ${\mathrm{\Psi }}^{+}=2.1, {\mathrm{\Psi }}^{-}=0, {\mu }_0=0$, and $\text{Da}=\frac{k{a}^2}{D\left({\varphi }_0\right)\psi \left({\varphi }_0\right)}=4.76$ on a ${256}^2$ grid.

Figure 7

Mass fraction snapshots at different (nondimensional) times $t=0.01,1,\text{and}2.5$ (left to right) from phase-field simulations of a perfectly symmetric binary mixture consisting of a monodisperse array of $\alpha$-phase drops embedded in a continuous $\beta$ phase (at the initial time) with ${\mathrm{\Psi }}^{+}=2.1, {\mathrm{\Psi }}^{-}=0, {\mu }_0=0$, and $\text{Da}=\frac{k{a}^2}{D\left({\varphi }_0\right)\psi \left({\varphi }_0\right)}=0.62$ on a ${512}^2$ grid with $\langle \varphi \rangle \left(0\right)=0.4864$ (top) and $\langle \varphi \rangle \left(0\right)=0.5150$ (bottom).

Figure 8

Mass fraction snapshots at different (nondimensional) times $t=0.02,1,\text{and}80$ (left to right) from phase-field simulations of a perfectly symmetric binary mixture at critical concentration (with random noise superimposed at the initial time) with ${\mathrm{\Psi }}^{+}=2.1, {\mathrm{\Psi }}^{-}=0, {\mu }_0=0$, and $\text{Da}=\frac{k{a}^2}{D\left({\varphi }_0\right)\psi \left({\varphi }_0\right)}=2.44$ on a ${256}^2$ grid with $\langle \varphi \rangle \left(0\right)=0.50017$ (top) and $\langle \varphi \rangle \left(0\right)=0.49994$ (bottom).