Interaction models and configurational entropies of binary MoTa and the MoNbTaW high entropy alloy

We introduce a simplified method to model the interatomic interactions of high entropy alloys based on a lookup table of cluster energies. These interactions are employed in replica exchange Monte Carlo simulations with histogram analysis to obtain thermodynamic properties across a broad temperature range. Kikuchi's cluster variation method entropy formalism is applied to directly calculate entropy from statistics on short- and long-range chemical order, and we discuss the convergence of the entropy as clusters of differing size are included. A high temperature series expansion aids in our understanding of the convergence. Computer codes implementing these methods, and supporting data, are freely available on the internet.

Figure 1

bcc unit cell (left) and bcc tetrahedron (right). Nearest-neighbor bonds are in yellow and next-nearest-neighbor bonds are in blue. Tetrahedron sites $a$ and $b$ are “even” (cube vertex), while $c$ and $d$ are “odd” (body center) sites.

Figure 2

Parity plot for tetrahedron model energy/atom vs full (unrelaxed) DFT energy/atom for (a) MoTa and (b) MoNbTaW. All structures are equiatomic; black shows random 16-atom structures, red shows random 128-atom structures, and green shows MC-generated 128-atom structures. See the Appendix for a discussion of the skew.

Figure 3

Illustration of the quaternary MoNbTaW replica exchange simulation in an $L\times L\times L$ supercell with $L=4$. (a) $T\left(t\right)$ graph illustrating the replica exchange. Each color represents the time evolution of a single initial configuration whose temperature is repeatedly swapped with its neighbors; (b) energy histograms at selected temperatures.

Figure 4

Binary MoTa of size $L=8$ (1024 atoms). (a) Occupation $x_{\alpha }$ of even sites $a$ and $b$ (dotted, short-dashed, and long-dashed lines are for sizes 2, 4, and 6, respectively). (b) $ac$ site pair frequencies $y_{\alpha \gamma }$ normalized by global mean concentrations ${\overline{x}}_{\alpha }$ and ${\overline{x}}_{\gamma }$. (c) Simulated histogram and CVM-predicted entropies (inset: residuals with respect to the histogram). The dashed line shows $S/k_{\mathrm{B}}=ln2$.

Figure 5

Binary MoTa. (a) Specific heat $c\left(T\right)$ and (b) susceptibility $\chi \left(T\right)$ vs temperature $T$ for system sizes $L=2$ (black), 4 (red), 6 (green), and 8 (blue). Insets show scaling functions as defined in text.

Figure 6

Quaternary MoNbTaW of size $L=8$ (1024 atoms). (a) Occupation $x_{\alpha }$ of the $a$ site. (b) $ac$ site pair frequencies $y_{\alpha \gamma }$ normalized by global mean concentrations ${\overline{x}}_{\alpha }$ and ${\overline{x}}_{\gamma }$. (c) Simulated histogram and CVM-predicted entropies (inset: residuals with respect to the histogram). The dashed line shows $S/k_{\mathrm{B}}=ln4$.

Figure 7

Quaternary MoNbTaW. (a) Specific heat $c\left(T\right)$ and (b) susceptibility $\chi \left(T\right)$ vs temperature $T$ for system sizes $L=2$ (black), 4 (red), 6 (green), and 8 (blue). Insets show scaling functions as defined in text.

Figure 8

$L=8$ quaternary MoNbTaW at $T=390$ K (top) showing the B2 phase and at $T=100$ K (bottom) showing phase separation. The color scheme is chosen so that purple (Mo) and magenta (W) alternate with cyan (Nb) and blue (Ta) in the B2 phase. Lighter colors (NbW in the B32 structure) separate from dark colors (MoTa in the B2 structure) at low $T$.

Figure 9

Parity plot for tetrahedron plus polarization model energy/atom vs full DFT energy/atom for (a) MoTa and (b) MoNbTaW. All structures are equiatomic; black shows random 16-atom structures, red shows random 128-atom structures, and green shows MC-generated 128-atom structures.