Interstitial carbon in bcc HfNbTiVZr high-entropy alloy from first principles

The remarkable mechanical properties of high-entropy alloys can be further improved by interstitial alloying. In this work we employ density functional theory calculations to study the solution energies of dilute carbon interstitial atoms in tetrahedral and octahedral sites in bcc HfNbTiVZr. Our results indicate that carbon interstitials in tetrahedral sites are unstable, and the preferred octahedral sites present a large spread in the energy of solution. The inclusion of carbon interstitials induces large structural relaxations with long-range effects. The effect of local chemical environment on the energy of solution is investigated by performing a local cluster expansion including studies of its correlation with the carbon atomic Voronoi volume. However, the spread in solution energetics cannot be explained with a local environment analysis only pointing towards a complex, long-range influence of interstitial carbon in this alloy.

Figure 1

Tetrahedral and octahedral interstitial sites in the bcc structure.

Figure 2

Schematic comparison of the perfect and distorted nearest neighbor polyhedra, and the positions probed to find the most suitable position to place an interstitial atom.

Figure 3

Radical Voronoi volume distribution of tetrahedral and octahedral sites in the relaxed HEA.

Figure 4

Pair distribution function of the DFT relaxed HfNbTiVZr and compared with the experimental neutron total scattering results in Ref. [6]. The function from DFT is scaled so that the baseline and the first peak are at the same level as in the experimental function.

Figure 5

Total energies of supercells modeling HfNbTiVZr HEA with C interstitials in octahedral and tetrahedral sites. The initial energy corresponds to the total energy of supercells with interstitial carbon placed in the SQS modeling of a carbon-free relaxed HEA, while the final energy is calculated for the fully relaxed supercell with the carbon atom in the system. The $x$ axis shows each of the interstitial positions, which have been sorted in ascending radical Voronoi volume.

Figure 6

Energy of solution for octahedral sites.

Figure 7

Distribution of the solution energies of carbon atoms in octahedral sites for different $M_i$ rich, as well as the HEA-like local environments.

Figure 8

Atomic displacements of metal atoms upon carbon inclusion as a function of the distance to the carbon interstitial.

Figure 9

(a) Comparison of displacements of metal atoms in pure Nb, V and average displacement in HfNbTiVZr as a function of distance to the carbon interstitial. (b) Schematic illustration of atomic displacements in bcc Nb.

Figure 10

$\mathrm{\Delta }{E}_{\mathrm{sol}}$ vs Voronoi volume, radical Voronoi volume, nearest neighbor polyhedral volume, and valence electron concentration.

Figure 11

Calculated vs predicted energies using on-site interactions obtained by a local cluster expansion.