Phase amplification in spinodal decomposition of immiscible fluids with interconversion of species

A fluid composed of two molecular species may undergo phase segregation via spinodal decomposition. However, if the two molecular species can interconvert, e.g., change their chirality, then a phenomenon of phase amplification, which has not been studied so far to our best knowledge, emerges. As a result, eventually, one phase will completely eliminate the other one. We model this phenomenon on an Ising system which relaxes to equilibrium through a hybrid of Kawasaki-diffusion and Glauber-interconversion dynamics. By introducing a probability of Glauber-interconversion dynamics, we show that the particle conservation law is broken, thus resulting in phase amplification. We characterize the speed of phase amplification through scaling laws based on the probability of Glauber dynamics, system size, and distance to the critical temperature of demixing.

Figure 1

(a) The spontaneous equilibrium order parameter ($\varphi ={\varphi }_0$) in the lattice-gas or binary-lattice mixture along the liquid-vapor phase coexistence (red domain). One of the two alternative magnetizations (${\varphi }_0>0$ and ${\varphi }_0<0$) in the Ising ferromagnet in zero field is shown in the red domain with blue arrows. The solid curve is the crossover from mean-field behavior (dashed) to the asymptotic scaling power law $\varphi \propto \mathrm{\Delta }{\widehat{T}}^{\beta }$ with $\beta =0.326$ [12, 13], while the crosses are our simulation data for $\ell =100$ and averaged over 1000 realizations. (b) The growth rate solution to Eq. (1), known as the amplification factor, for three probabilities: $p_r=0$ (pure swapping dynamics, $L=0$), $p_r=1$ (pure flipping dynamics, $M=0$), and $p_r=L/(M+L)=0.01$ (mixed dynamics) after quenching the system into the unstable region at $\mathrm{\Delta }\widehat{T}=0.1$.

Figure 2

Phase amplification: the growth of the order parameter for different probabilities of Glauber dynamics at a system size $\ell =100$. (a) Full-time behavior, $T=4.4$ and 100 realizations, $p_r=1$; (b–d) initial time behavior, $T=4.0$ and 1000 realizations: (b) $p_r=1.0$, (c) $p_r=0.1$, and (d) $p_r=1.0\times {10}^{-7}$. The solid horizontal line, $\varphi =0$, corresponds to Kawasaki dynamics, $p_r=0$.

Figure 3

The evolution of the order parameter during phase amplification. (a) The RMS of the distribution of the growth rates for different probabilities captured at the same time, $t=300$. The solid curve is the crossover between $\sigma \propto \sqrt{p_r}$ and $\sigma \propto p_r$, approximated as $\sigma =a\sqrt{p_r}(1+b{p}_r)/(1+\sqrt{p_r})$. (b–d) The growth of the order parameter for different (b) probabilities at $\mathrm{\Delta }\widehat{T}=0.11$ and $\ell =100$, (c) system sizes at $p_r=1.0$ and $\mathrm{\Delta }\widehat{T}=0.11$, and (d) distances to the critical point at $p_r=1.0$ and $\ell =100$; the colored values in (b) and (d) correspond to the colored curves of different $p_r$ and $\mathrm{\Delta }\widehat{T}$, respectively. The inset of (d) shows the power law for the initial growth of the reduced order parameter, $\varphi /{\varphi }_0\propto t^{3/4}$.

Figure 4

Scaling properties of the growth of the reduced order parameter, $\widehat{\varphi }=\langle \left|\varphi \right|\rangle /{\varphi }_0$. (a) The order parameter growth with time rescaled by system size. The size dependence of the rescaling parameter, $\tau \left(\ell \right)$, is shown in the inset, in the log-log scale with a slope of 1. The colors are the same as in Fig. 3. (b) The order parameter growth with time rescaled by probability; the rescaling parameter ${\tau }_0\left(p_r\right)$, inversely proportional to the probability, is shown in the inset. The colors are the same as in Fig. 3.

Figure 5

Topological characteristics of the time dependence of phase amplification for $\ell =100$. (a) For spherical domains, the reduced deviation from the equilibrium order parameter, $\mathrm{\Delta }\widehat{\varphi }=1-\varphi /{\varphi }_0$ scales as $t^{3/2}$. This is shown for temperatures, $\mathrm{\Delta }\widehat{T}$, as 0.11 (cyan), 0.025 (purple), 0.014 (orange), and 0.009 (brown). The inset shows the effect for a cylindrical domain at $\mathrm{\Delta }\widehat{T}=0.11$. (b) A zero curvature Schwarz-P interface is initially formed by simulating a system with Kawasaki dynamics ($p_r=0$) for a long time. At $t-t_0=0$, the system obtains Glauber dynamics ($p_r=1$), and the collapse of one of the phases is shown. One can see a transitions from random-walk behavior, $\sqrt{t}$, at short times (see inset) to exponential behavior (straight line) before saturation.