Competitive nucleation and the Ostwald rule in a generalized Potts model with multiple metastable phases

We introduce a simple nearest-neighbor spin model with multiple metastable phases, the number and decay pathways of which are explicitly controlled by the parameters of the system. With this model, we can construct, for example, a system which evolves through an arbitrarily long succession of metastable phases. We also construct systems in which different phases may nucleate competitively from a single initial phase. For such a system, we present a general method to extract from numerical simulations the individual nucleation rates of the nucleating phases. The results show that the Ostwald rule, which predicts which phase will nucleate, must be modified probabilistically when the new phases are almost equally stable. Finally, we show that the nucleation rate of a phase depends, among other things, on the number of other phases accessible from it.

Figure 1

Metastable transition graphs: (a) kinetic Ising model, (b) succession of three phases, (c) single metastable phase decaying to two competing phases, (d) as in (c), but such that one phase can decay further, and (e) three competing phases.

Figure 2

(Color online) Time dependence of magnetization per site $M_{\alpha }∕N$ of each phase $\alpha$, in a single run of the model with transition graph $1\to 2\to 3\to 4\to 5$ (a succession of phases). Parameters are $L=50$, $\beta =1.25$, $h_{\alpha }=0.1(\alpha -1)$, $K_1=0.1$, and $K_2=1.0$. Labels denote the dominant phase. Configuration snapshots depict postcritical nuclei of each new phase embedded in the previous phase. These grow to fill the system, producing the next phase in sequence.

Figure 3

(Color online) Probability $p_2\left(h_3\right)$ that phase 2 nucleates before phase 3 as a function of $\Delta h∕h_2$, with $h_2=0.1$, $K_1=0.1$, $K_2=1$, $\beta =1.25$, and $L=50$ unless otherwise noted. Up to $n={10}^4$ trials were used for each data point; statistical errors are of the order of the symbol size. Inset: system-size dependence of $p_2$ for two values of $h_3$ with fixed parameter values.

Figure 4

(Color online) (a) Comparison of nucleation rates ${\lambda }_i\left(h_3\right)$ of phases $i=2,3$ from phase 1 in the model of Fig. 1c with $h_2$ fixed, calculated directly from simulations and using forward-flux sampling. (b) Nucleation probability $p_2\left(h_2\right)$ for $h_3=0.1$ and $h_4=0.2$ varying $h_2$, with and without phase 4, in the model of Fig. 1d. Dashed lines show positions of equal nucleation probability and equal field of the two phases. In both figures, parameters are as in Fig. 3, with $L=50$ and $\beta =1.25$.

Figure 5

(Color online) Configuration snapshots of the multidroplet regime in the model with transition graph Fig. 1e, with $L=200$, $\beta =1.25$, and $h_1=0$, $h_2=h_3=h_4=0.15$. Nucleation of droplets of three equally stable phases from a single metastable phase is followed by droplet growth and then domain coarsening. The “coating” of the initial phase visible between domains in the final snapshot is due to repulsive interactions between the new phases.