Kinetics of first-order phase transitions with correlated nuclei

We demonstrate that the time evolution of a first-order phase transition may be described quite generally in terms of the statistics of point processes, thereby providing an intuitive framework for visualizing transition kinetics. A number of attractive and repulsive nucleation scenarios is examined followed by isotropic domain growth at a constant rate This description holds for both uncorrelated and correlated nuclei, and may be employed to calculate the nonequilibrium, $n\text{−}\mathrm{point}$ spatiotemporal correlations that characterize the transition. Furthermore, it is shown that the interpretation of the one-point function in terms of a stretched-exponential, Kolmogorov-Johnson-Mehl-Avrami result is problematic in the case of correlated nuclei, but that the calculation of higher-order correlation functions permits one to distinguish among various nucleation scenarios.

Figure 1

The transformed fraction $\varphi \left(t\right)$ as a function of scaled time $n^{1/2}Gt$ for correlation lengths $\xi ={10}^{-4}$ (circles), 1.0 (diamonds), and 2.0 (squares) [Eq. (10)] for nucleation scenario one and $\ell =8$ (triangles) for scenario two as obtained from simulation. The results are averaged over approximately ${10}^3$ initial conditions. Also shown (lines) are the values obtained for $\varphi \left(t\right)$ using the point-process model [see Eq. (7)]. It should be noted that the point-process model accurately represents the simulation data.

Figure 2

A plot of $ln\left[-ln(C_1\left(t\right))\right]$ versus $ln\left(t\right)$ for scenario one. The corresponding lines are the best fits to the data. Thus, even for correlated nuclei, a stretched exponential fit can be a very good approximation.

Figure 3

(a) The correction function ${\gamma }_2\left(s\right)$ versus time $s=|{\vec{r}}_1-{\vec{r}}_2|/2Gt$ from the simulation of spatially uncorrelated nuclei (points), as given in Eq. (3). The dashed line is the predicted functional form obtained by simple geometry [Eq. (4)] for uncorrelated nuclei. (b) The same as (a) except for correlated nuclei with $\xi =1.0$ (circles) and $\xi =2.0$ (squares) for $Gt=2.0$. The dashed line again is the predicted functional form for uncorrelated nuclei.