Large-scale atomistic simulation of diffusion in refractory metals and alloys

The equilibrium vacancy concentration and atomic diffusion coefficients in dilute and complex refractory alloys have been calculated using various computational methods. The most productive technique has been large-scale atomistic simulation in the form of a numerical experiment in which a crystal with free surfaces was simulated for a relatively long time. This method is based on the concept that the free surface acts as a source of point defects and provides a natural way to achieve an equilibrium concentration of the defects within the bulk after an initial annealing stage. For complex concentrated alloys (CCAs), this numerical experiment offers the possibility to study diffusion processes where standard analytical approaches are difficult to apply due to the large variety of microscopic states. As the simulation results, we found that the transition from dilute alloys to CCA is accompanied by a significant increase in atomic diffusion rate due to both a substantial increase in the vacancy concentration and their mobility.

Figure 1

Calculated migration energies for the alloying atom moving by exchange with the vacancy. Available DFT data [5, 6, 43, 44] are compared with the results obtained from the ADP [47], tabGAP [55], SNAP [53], and MTP [54] models.

Figure 2

Nine vacancy jumps distinguished according to the Le Claire model [58]. The green spheres represent the matrix atoms. A vacancy and an impurity atom in a nearest-neighbor configuration are shown by an empty box and a violet sphere, respectively. ${\omega }_2$ corresponds to the vacancy–impurity atom exchange, ${\omega }_3/{\omega }_4, {\omega }_3^{\prime }/{\omega }_4^{\prime }, {\omega }_5/{\omega }_6$, and ${\omega }_3^{″}/{\omega }_4^{″}$ represent the back/forth vacancy jumps to the second, third, fourth, and fifth neighbors of the impurity atom, respectively, and ${\omega }_0$ is the jump frequency of all other possible vacancy jumps between nearest-neighbor positions (also, ${\omega }_6={\omega }_0$ according to this model).

Figure 3

Calculated temperature dependences of vacancy diffusion coefficients in pure metals and ternary equiatomic alloy. The results are given for absolute (left panel) and homologous temperatures (right panel). For absolute temperature, MD simulation results (large symbols) are compared with predictions based on the static calculations (small symbols). For homologous temperature, the green dashed line indicates GM of the values in the pure metals.

Figure 4

Calculated temperature dependence of the vacancy formation energies in pure metals: solid lines and dashed lines correspond to energies $E_v^f$ and $G_v^f$, respectively. The obtained results for W are compared with the prediction based on the ab initio ML approach [16].

Figure 5

Vacancy formation energy depending on alloy composition: the calculation results are given for two binary alloys (W-Nb and Mo-Nb) and ternary alloy (${\mathrm{WMo})}_{(1-x)}{\mathrm{Nb}}_x$. Open symbols correspond to $E_{v0}^f$ calculated with the DFT method for pure metals [79]: triangle—W; diamond—Mo; circle—Nb.

Figure 6

(a)–(d) Snapshots of large-scale MD simulations of Mo at $T=2850$ K during the first 1.8 ns of the modeling (the simulation time is shown in the snapshots). Panel (a) shows all simulated atoms. Panels (b)–(d) show the evolution of defects and their trajectories during the MD run: only atoms with the highest displacements ($\mathrm{\Delta }r>2.0$ Å) are shown. (e) The calculated dependence of $C_v$ (left axis) and the number of vacancies $N$ (right axis) in the diffusion region on the simulation time for pure Mo and ternary equiatomic alloy. The results are shown for different temperatures.

Figure 7

Calculated temperature dependence of the equilibrium vacancy concentration. The symbols show the results of the direct large-scale MD simulations performed for dilute alloys and the ternary equiatomic alloy. The solid lines show the analytical function $C_v=exp[-G_v^f/k_{B}T]$ for pure metals: black, blue, and red colors correspond to Nb, Mo, and W, respectively. The dashed line is a guide to the eye for the values obtained in the equiatomic alloy. The obtained results are given for homologous (left panel) and absolute temperatures (right panel).

Figure 8

Correlation plot between the simulated melting temperature and the vacancy concentration in this state. The figure contains the simulation results obtained with different interatomic potentials and the vacancy concentrations obtained from the experimental data for W [86, 87, 88, 89], Mo [87, 90], and Nb [87, 90]. The three shaded regions indicate a scatter in the available experimental values. The black solid line corresponds to the relationship $C_{\mathrm{fit}}exp(T_m/T_{\mathrm{fit}})$ discussed in the text.

Figure 9

Temperature dependence of the atomic diffusion coefficients in dilute alloys and ternary equiatomic alloy. Solid lines correspond to the self-diffusion $D=C_v·D_v$ based on the MD results; dotted lines with small solid symbols indicate the impurity diffusion (and self-diffusion) obtained with the nine-jump model; the large solid symbols are the results of the large-scale MD simulation; large open symbols correspond to the measured coefficients [7, 8, 98, 99, 100, 101, 102, 103]. For comparison, the panel with the results for the WMN alloy also contains the calculated $D$ for pure metals.

Figure 10

Comparison of the calculated diffusion coefficients of a single metal in different matrices: open symbols—the results of the large-scale MD simulation in the dilute alloys; solid line—the geometric mean of the values in the dilute alloys; solid squares—the diffusion coefficients in the ternary equiatomic alloy calculated in the large-scale MD simulation.