Beyond Finite-Size Scaling in Solidification Simulations

Although computer simulation has played a central role in the study of nucleation and growth since the earliest molecular dynamics simulations almost 50 years ago, confusion surrounding the effect of finite size on such simulations has limited their applicability. Modeling solidification in molten tantalum on the Blue Gene/L computer, we report here on the first atomistic simulation of solidification that verifies independence from finite-size effects during the entire nucleation and growth process, up to the onset of coarsening. We show that finite-size scaling theory explains the observed maximal grain sizes for systems up to about 8 000 000 atoms. For larger simulations, a crossover from finite-size scaling to more physical size-independent behavior is observed.

Figure 1

Percentage of the simulation cell that has solidified as function of time (solid line), along with the compression ratio (dashed line). The melt curve for Ta is shown in the inset, which also depicts the isothermal compression path (black arrow). The labeled points refer to (*) the crossing of the equilibrium melt curve, (a) the onset of nucleation, (b) the period of explosive growth, and (c) the onset of coalescence.

Figure 2

Cross sections of the 16M-atom (a)–(c) and 64k-atom (d)–(f) simulation taken at different stages during the solidification. Labels (a)–(c) refer to points shown in Fig. 1, while labels (d)–(f) refer to corresponding times for the 64k-atom sample. [Animation of the solidification is available online [26].]

Figure 3

The average size of the largest clusters in the system as a function of time, plotted for simulation cells containing 64 000 (blue), 250k (magenta), 2M (red), 8M (green), and 16M (black) atoms. The marked points represent the average size of the largest clusters at the onset of coalescence.

Figure 4

Average maximum cluster size at the start of coarsening as a function of simulation size. The solid line represents the result of finite-size scaling theory. We find that the maximum cluster size is independent of system size for simulations larger than about 8M atoms. Inset: log-log plot of the results, showing the linear (power-law) dependence. The 1M-atom simulation (square) was run in an $NPT$ ensemble.