Traversing the nucleation-growth landscape through heterogeneous random walks

The nucleation-growth process is a crucial component of crystallization. While previous theoretical models have focused on nucleation events and postnucleation growth, such as the classical nucleation theory and Lifshitz-Slyozov-Wagner model, recent advancements in experiments and simulations have highlighted the inability of classical models to explain the transient dynamics during the early development of nanocrystals. To address these shortcomings, we present a model that describes the nucleation-growth dynamics of individual nanocrystals as a series of reversible chain reactions, with the free energy landscape extended to include activation-adsorption-relaxation reaction pathways. By using the Monte Carlo method based on the transition state theory, we simulate the crystallization dynamics. We derive a Fokker-Planck formalism from the master equation to describe the nucleation-growth process as a heterogeneous random walk on the extended free energy landscape with activated states. Our results reveal the transient quasiequilibrium of the prenucleation stage before nucleation starts, and we identify a postnucleation crossover regime where the dynamic growth exponents asymptotically converge towards classical limits. Additionally, we generalize the power laws to address the dimension and scale effects for the growth of large crystals.

Figure 1

Reaction mechanism and free energy profiles of activation energies. (a) The scheme of the chain reactions and the activation-adsorption-relaxation pathway of crystallization. $n$ denotes the normalized crystal volume. (b) The free energy profile along the reaction pathway, when $n<n_c$. $\mathrm{\Delta }{G}_{\pm }^{*}$ is the forward or backward activation energy. The activated conformations are in the shapes of lune and gourd. (c) The size-dependent activation energy. At $n_c, \mathrm{\Delta }{G}_{+}^{*}=\mathrm{\Delta }{G}_{-}^{*}$.

Figure 2

Time-resolved evolution of the nucleation-growth dynamics. The curves denote the numerical solutions of ME and FPE. The markers show the results of MCS. (a) Mean crystal volume and mean crystal radius versus time. The horizontal line denotes ${\overline{n}}_c$ and ${\overline{r}}_c$ at quasiequilibrium. (b) Dynamic growth exponents based on mean volume and mean radius versus time. The upper and lower horizontal lines respectively denote the asymptotic limits as ${\alpha }_n\to 1$ and ${\alpha }_r\to 1/3$.

Figure 3

The probability density functions at different crystallization stages. (a) Prenucleation stage. (b) Transition from prenucleation to early postnucleation stage. (c) Exponential decay of small clusters during nucleation. The solid lines denote the numerical solutions of ME and FPE. The markers show the results of MCS. (d) Late postnucleation stage.