Geometry of phase separation

We study the domain geometry during spinodal decomposition of a 50:50 binary mixture in two dimensions. Extending arguments developed to treat nonconserved coarsening, we obtain approximate analytic results for the distribution of domain areas and perimeters during the dynamics. The main approximation is to regard the interfaces separating domains as moving independently. While this is true in the nonconserved case, it is not in the conserved one. Our results can therefore be considered as a “first-order” approximation for the distributions. In contrast to the celebrated Lifshitz-Slyozov-Wagner distribution of structures of the minority phase in the limit of very small concentration, the distribution of domain areas in the 50:50 case does not have a cutoff. Large structures (areas or perimeters) retain the morphology of a percolative or critical initial condition, for quenches from high temperatures or the critical point, respectively. The corresponding distributions are described by a $c{A}^{-\tau }$ tail, where $c$ and $\tau$ are exactly known. With increasing time, small structures tend to have a spherical shape with a smooth surface before evaporating by diffusion. In this regime, the number density of domains with area $A$ scales as $A^{1/2}$, as in the Lifshitz-Slyozov-Wagner theory. The threshold between the small and large regimes is determined by the characteristic area $A\sim t^{2/3}$. Finally, we study the relation between perimeters and areas and the distribution of boundary lengths, finding results that are consistent with the ones summarized above. We test our predictions with Monte Carlo simulations of the two-dimensional Ising model.

Figure 1

Snapshots of the 2D Ising model evolving with a locally conserved order parameter at $t=1000$ and $t=16000$ equivalent MC steps using the accelerated bulk algorithm explained in Sec. 3. The initial condition is a random configuration of $\pm 1$ spins taken with probability one half, that is to say, an infinite-temperature state. Evolution occurs at $T=1.0$.

Figure 2

(Color online) Left panel: number density of domains areas per unit area for the 2DIM evolving at $T=1.0$ after a quench from $T_0\to \infty$ using the accelerated algorithm. The dotted line represents the analytical prediction. Right panel: rescaled data using the typical domain area time dependence $A\sim t^{2/3}$. The dotted line is the theoretical prediction in Eq. (11) with $\tau$ replaced by ${\tau }^{\prime }$, appropriate to a disordered initial state.

Figure 3

(Color online) Simulations with the Kawasaki algorithm, without and with disorder. The solid lines represent the analytic prediction with $c_d=0.025$, ${\tau }^{\prime }=2.055$, and ${\lambda }_d=0.3$. For the disordered case, the growth law $R\left(t\right)$ is extracted from the spatial correlation function. Even if there are deviations from the theoretical curve, the data seem to show superuniversality.

Figure 4

(Color online) Number density of domain areas per unit area for the 2DIM evolving at $T=1.0$ after a quench from $T_0=T_c$ using the accelerated bulk algorithm. The lines represent Eq. (11) with constant $c_d$ and $2c_d$.

Figure 5

(Color online) Number density of domains areas per unit area for the 2DIM evolving at $T=1.5$ after a quench from $T_0\to \infty$ using the accelerated bulk algorithm. The prefactor ${\lambda }_d\left(T\right)$ in the growing length is chosen to be ${\lambda }_d\left(T\right)=0.0060$. Compare to the results shown in Fig. 2 obtained for a lower-working temperature.

Figure 6

(Color online) The time-dependent relation between the area and the domain boundary evolving at $T=1.0$ after a quench from $T_0\to \infty$ using conserved and nonconserved order-parameter dynamics. The lines are fits to the data points. For the large structures, the fit yields ${\alpha }^{\prime >}\simeq 1$ in both cases, while for the small structures it yields ${\alpha }^{\prime >}\simeq 2$ for conserved order-parameter dynamics (circular domains) and ${\alpha }^{\prime >}\simeq 1.8$ for nonconserved order-parameter dynamics. Here $z=2$ and 3 for nonconserved and conserved dynamics, respectively.

Figure 7

(Color online) The time-dependent number density of perimeters evolving at $T=1.0$ from an initial condition at $T_0\to \infty$. Left panel: raw data; note that the time dependence is visible in the whole range of values of $p$ (while in the area number densities, the large area tails were very weakly dependent on time). Right panel: scaled data and analytic predictions for the small and large area regimes.

Figure 8

(Color online) Comparison between the number density of domain areas in the clean and disordered 2D Ising model with nonconserved dynamics and conserved dynamics in a zoom over the small area regime. The data points are the result of the numerical simulations, while the straight lines are the analytic predictions.