Overcoming the Time Limitation in Molecular Dynamics Simulation of Crystal Nucleation: A Persistent-Embryo Approach

The crystal nucleation from liquid in most cases is too rare to be accessed within the limited time scales of the conventional molecular dynamics (MD) simulation. Here, we developed a “persistent embryo” method to facilitate crystal nucleation in MD simulations by preventing small crystal embryos from melting using external spring forces. We applied this method to the pure Ni case for a moderate undercooling where no nucleation can be observed in the conventional MD simulation, and obtained nucleation rate in good agreement with the experimental data. Moreover, the method is applied to simulate an even more sluggish event: the nucleation of the $B2$ phase in a strong glass-forming Cu-Zr alloy. The nucleation rate was found to be 8 orders of magnitude smaller than Ni at the same undercooling, which well explains the good glass formability of the alloy. Thus, our work opens a new avenue to study solidification under realistic experimental conditions via atomistic computer simulation.

Figure 1

Using the persistent-embryo method to reach the critical nucleus. (a) The excess free energy (black) and spring constant (red) as a function of the crystalline cluster size $N$. $N_0$ is the number of atoms in the constrained embryo. The red curve shows that the strength of the spring constant decreases with the increasing cluster size. The spring is completely removed when the cluster size reaches the threshold value $N_{\mathrm{sc}}$. $N^{*}$ is the critical size. (b) A cross section of the as-grown crystalline cluster around embryo. The yellow atoms with spring icon are the persistent embryo. The red are the as-grown atoms, showing the crystalline packing. The gray are the liquid atoms.

Figure 2

The persistent-embryo MD simulation of the crystal nucleation in the undercooled liquid Ni. (a) The nucleus size versus time of one Ni nucleation trajectory at 1480 K. The blue dashed line shows the atom number $N_0$ in the persistent embryo. The green dashed line indicates the threshold to remove the spring and the red solid line indicates the critical size $N^{*}$. Two insets enlarge two plateaus at the critical size. (b) The critical size as a function of the temperature. (c) The upper panel shows the nucleus size versus time for the isoconfigurational ensemble with 30 MD runs. Each color indicates an independent MD trajectory. The bottom panel shows the ensemble average of ${|\mathrm{\Delta }{N}^{*}(t)|}^2={|N(t)-N^{*}|}^2$. The dashed line indicates the linear fitting to the first 5 ps to derive the attachment rate. (d) The nucleation rate as a function of the temperature for Ni. The simulation results are connected to guide the eye. The experimental data are from Ref. [26] (upward triangle) and Ref. [27] (downward triangle).

Figure 3

The persistent-embryo MD for $B2$ nucleation in ${\mathrm{Cu}}_{50}{\mathrm{Zr}}_{50}$ undercooled liquid at 1097 K. (a) Nucleus size as a function of time. The inset shows the $B2$ critical nucleus. (b) 30 MD runs starting from the configuration with critical nucleus are performed. Each color indicates an independent MD trajectory. The dashed line shows the linear fitting of the ensemble average to derive the attachment rate.