Interplay between Lattice Distortions, Vibrations and Phase Stability in NbMoTaW High Entropy Alloys

Abstract

Refractory high entropy alloys (HEA), such as BCC NbMoTaW, represent a promising materials class for next-generation high-temperature applications, due to their extraordinary mechanical properties. A characteristic feature of HEAs is the formation of single-phase solid solutions. For BCC NbMoTaW, recent computational studies revealed, however, a B2(Mo,W;Nb,Ta)-ordering at ambient temperature. This ordering could impact many materials properties, such as thermodynamic, mechanical, or diffusion properties, and hence be of relevance for practical applications. In this work, we theoretically address how the B2-ordering impacts thermodynamic properties of BCC NbMoTaW and how the predicted ordering temperature itself is affected by vibrations, electronic excitations, lattice distortions, and relaxation energies.

1. Introduction

Refractory high entropy alloys (HEAs) such as BCC NbMoTaW are of great interest due to their extraordinary mechanical properties [1,2,3,4,5,6]. Characteristic of HEAs is the lack of ordering commonly attributed to the large configurational entropy [7,8,9]. For BCC NbMoTaW, no signature of ordering has been experimentally found in the annealed state or at room temperature [3]. However, thermodynamic equilibrium at experimental conditions is limited due to slow diffusion inherent in refractory alloys. Recent theoretical works have reported a B2 ordering between mixed (Mo,W) and (Nb,Ta) sites at ambient temperatures [10,11,12,13,14]. A recent first-principles study combining chemical interactions from the generalized perturbation method with Monte Carlo simulations revealed an ordering temperature of about 750 K [15]. However, lattice distortions as well as vibrations (phonons) have not been taken into account within this approach, and their impact on the order–disorder temperature remains unknown. Furthermore, the B2 ordering could also impact lattice excitations, and hence thermodynamic properties such as the lattice expansion and the heat capacity.

In this work, we close the previous simulation gap. We theoretically address the question of how local relaxations as well as vibrational and electronic entropies impact the theoretical B2 ordering temperature. We also report how the B2 ordering affects the heat capacity, thermal expansion, and bulk modulus of the alloy.

2. Materials and Methods

Density functional theory calculations have been carried out employing the vasp code [16,17] and the projector-augmented wave (PAW) method [18] within the generalized gradient approximation [19]. The semi core p states were treated as valence within the PAW potentials. The planewave cutoff was set to 400 eV. Chemical disorder was simulated utilizing the concept of special quasi-random structures (SQS) [20] for 128 atom supercells ($4\times 4\times 4$ unit cells). For B2, Mo and W (Nb and Ta) are randomly distributed on the first (second) sublattice. The employed SQS have been constructed by minimizing the correlation functions of the first two nearest-neighbor shells. The employed 128 atom cells provide a fair compromise between mimicking chemical disorder within a limited supercell size and computational efficiency. For a recent discussion on the application of SQS for HEAs, we refer to [21]. A $4\times 4\times 4$ Γ-centered k-point mesh and the Methfessel–Paxton method (order 1) with a smearing parameter of 0.1 eV were chosen for the total energy calculations. To study the impact of lattice expansion, all calculations were performed at nine different lattice constants around the theoretical equilibrium. The cubic cell shape was fixed, while the internal atom positions have been fully relaxed.

Ground state volume, bulk modulus, and its derivative were obtained by parametrizing the computed ground state energies, employing Vinet’s equation of state [22]. The electronic free energy is derived by occupying the electronic density of states with the Fermi–Dirac distribution. The Gibbs energy due to vibrations is computed by utilizing the Debye–Grüneisen model [23], as explained in detail in [24].

3. Results and Discussion

Based on the previous findings, we constructed two supercells (shown in Figure 1) for the A2 (solid solution) in (a) and (b), as well as for the B2 phase (d) and (e). For both structures, we computed the total energies

$$E_{\text{tot}}^{\sigma }(V)=E_{\text{static}}^{\sigma }(V)+\Delta E_{\text{relax}}^{\sigma }(V)$$

(1)

where σ indicates the A2 and B2 phase. As we are interested in the impact of internal atomic relaxations, the total energy above is split into the contributions arising from the static bcc lattice, $E_{\text{static}}^{\sigma }$, and the energy contribution due to internal relaxations, denoted as $\Delta E_{\text{relax}}^{\sigma }$. Indeed, lattice distortions constitute one of the main features in HEAs [7,8,9], and have been investigated, for example, for the FeCoNiCrMn (Cantor) HEA recently [25]. For binary alloys, the importance of relaxation energies to phase stability computations have been recognized some time ago (e.g., [26,27,28]). For the here-considered refractory HEA, a similar analysis is, however, lacking so far.

The results for $E_{\text{tot}}^{\sigma }(V)$ for the A2 and B2 phases are shown in Figure 1c,f, respectively. We first notice that the B2 phase is lower in energy at zero K, compared to the A2 solid solution. This is consistent with the B2 ordering at ambient temperatures found in previous works [10,11,12,13,14,15]. Considering the impact of atomic relaxations, we find that the A2 total energy is lowered by ≈ 12 meV per atom (difference between red line/circles and black line/squares in Figure 1c) whereas there is only very little impact of atomic relaxations on the B2 total energy (< 1 meV) shown in Figure 1f. The relaxation energies are also summarized in

Table 1. Ground state properties, Debye temperatures, and relaxation energies of the NbMoTaW A2 and B2(NbTa;MoW) phase.

Phase	Method	V0 (A3)	B0 (GPa)	B′	ΔErelax (meV)	ΘD (K)
B2(MoW;NbTa)	PAW + SQS	16.98	233	4	-	296
	PAW + SQS + relax	16.97	233	4	0.6	296
A2 solid solution	PAW + SQS	16.97	232	4	-	296
	PAW + SQS + relax	16.99	231	4	12.5	295
	PAW + SQS + relax [30]	17.08	228	n/a	n/a	n/a
	Experiment [4]	16.70	220	-	-	-

. The rather small impact of atomic relaxations and relaxation energies can be intuited by examining the B2 ordered phase in Figure 1e. The two sublattices are occupied by larger (Nb,Ta) and smaller (Mo,W) atoms, shown in blue and red, respectively. Note that the atomic sizes of the larger Nb and Ta atoms (dark and light blue) are exaggerated to facilitate the visualization. From the atomic arrangement of larger and smaller atoms occupying distinct sublattices, only very little relaxations are indeed expected for the B2 phase. In contrast to this, the A2 phase shown in Figure 1a,b features a random distribution of larger (Nb, Ta) and smaller (Mo,W) atoms, causing a significantly larger amount of local distortions, and hence relaxation energies. From the different $\Delta E_{\text{relax}}^{\sigma }$ contributions for A2 and B2, we can conclude that local atomic relaxations will have an impact on the ordering temperature, which we will discuss below.

Before discussing the ordering temperature, we first investigate the impact of B2 ordering on ground state and thermodynamic properties. In Table 1, we summarize the ground state volume, $V_0$, bulk modulus, $B_0$, and its derivative, $B^{\prime }$, derived from the ground state energies in Figure 1. Although the relaxation energy for A2 is not negligible, the impact on the ground state properties is rather small. This indicates an almost constant and volume-independent $\Delta E_{\text{relax}}^{\sigma }$. We note that our results are also in good agreement with previous calculations [30]. In order to evaluate the impact of B2 ordering on the thermodynamic properties, we compute the Gibbs energies of both phases, $G^{\sigma }(T)$, from which thermodynamic properties such as the volume expansion or heat capacity can be derived.

The Gibbs energies of the B2 and A2 solid solution at ambient pressure are given in the adiabatic approximation as

$$G^{\sigma }(T)=H_{\text{tot}}^{\sigma }+G_{\text{el}}^{\sigma }(T)+G_{\text{vib}}^{\sigma }(T)-T{S}_{\text{conf}}^{\sigma }$$

(2)

where $H_{\text{tot}}^{\sigma }$ is the enthalpy, $G_{\text{el}}^{\sigma }(T)$ is the electronic Gibbs energy, $G_{\text{vib}}^{\sigma }(T)$ is the vibration Gibbs energy, and $S_{\text{conf}}^{\sigma }$ denotes the configuration entropy. The ideal configuration entropy for B2 (with disordered Nb,Ta and Mo,W on each sublattice) is $S_{\text{conf}}^{\mathrm{B}2}=log(2)$, and for the A2 solid solution $S_{\text{conf}}^{\mathrm{A}2}=log(4)$. Note that configuration entropy is, in general, temperature dependent (e.g., [15,31]), as discussed below.

The vibration contribution, $G_{\text{vib}}^{\sigma }(T)$, is modeled within the quasi-harmonic Debye–Grüneisen model [23], as described in detail in [24]. A similar approach has recently been employed to study thermodynamic properties of other refractory alloys (HfNbZr, HfNbTiZr, and HfNbTaTiZr) [32]. Note that despite recent progress in the computation of phonons for disordered binary alloys [33,34], these methods have not yet been advanced to the stage where they are applicable to multi-componen alloys.

The key ingredient for our Debye model is the Debye temperature, ${\mathrm{\Theta }}_D$, which can be derived from the ground state properties. The values for ${\mathrm{\Theta }}_D$ are listed in Table 1. For B2 and A2, the same ${\mathrm{\Theta }}_D$ of 296 K is found. Only a marginal reduction of ≈ 1K is found for ${\mathrm{\Theta }}_D$ of the A2 when the effects of local relaxations are included.

From Equation (2) we compute for both phases (B2 and A2) the specific heat capacity, $C_P(T)$, the volume expansion, $V(T)$, as well as the bulk modulus, $B(T)$. The results for B2 and A2 are shown in Figure 2 as black and red solid lines, respectively. The dashed lines show the results without taking electronic excitations into account. For all considered temperatures, the impact of B2 ordering turns out to be negligible. This is consistent with the very similar ground state properties and ${\mathrm{\Theta }}_D$ listed in Table 1 derived for both phases. The impact of B2 ordering on the electronic contribution is also negligible, although $G_{\text{el}}^{\sigma }$ is of course important for accurate thermodynamic predictions—e.g., for the specific heat capacity in Figure 2a.

We can therefore conclude that the B2 ordering has practically no impact on the lattice parameter or the bulk modulus, and probably not on the elastic properties either. The heat capacities of B2 and A2 do not differ significantly from one another. Note, however, that Figure 2 does not contain the contributions of chemical short-range order, which gives a non-vanishing contribution to the heat capacity around the phase transition.

From the computed Gibbs energies (Equation 2), we can relate the ordering temperature to the Gibbs energy difference between the A2 and B2 phase; i.e.,

$$\Delta G^{\mathrm{A}2-\mathrm{B}2}(T)=G^{\mathrm{A}2}(T)-G^{\mathrm{B}2}(T)=\Delta H_{\text{tot}}+\Delta G_{\text{el}}(T)+\Delta G_{\text{vib}}(T)-T\Delta S_{\text{conf}}$$

(3)

From above equation, we estimate the ordering temperature via $\Delta G^{\mathrm{A}2-\mathrm{B}2}(T_{\text{ord}})\equiv 0$. An advantage of our theoretical approach is that it allows us to “switch on” and “off” different contributions entering Equation (3), and thus to estimate their individual importance on $T_{\text{ord}}$.

We first assume an ideal bcc lattice without taking local atomic relaxations into account; i.e., $E_{\text{tot}}^{\sigma }\equiv E_{\text{static}}^{\sigma }$ and $\Delta E_{\text{relax}}^{\sigma }=0$. If only configurational entropy in Equation (3) is taken into account, a transition temperature of 717 K is obtained (see also

Table 2. Enthalpy difference at T = 0 K between B2(NbTa;MoW) and A2(NbMoTaW) solid solution, as well as predicted B2-ordering temperatures if configuration entropy, lattice excitations, and electronic contributions are taken into account.

Method	$\mathbf{\Delta }{\mathit{H}}_{\mathbf{\text{tot}}}^{\mathbf{A}\mathbf{2}\mathbf{-}\mathbf{B}\mathbf{2}}$ (meV)	${\mathit{T}}_{\mathbf{\text{ord}}}^{\mathbf{\text{conf}}}$ (K)	${\mathit{T}}_{\mathbf{\text{ord}}}^{\mathbf{\text{conf}}\mathbf{+}\mathbf{\text{vib}}}$ (K)	${\mathit{T}}_{\mathbf{\text{ord}}}^{\mathbf{\text{conf}}\mathbf{+}\mathbf{\text{vib}}\mathbf{+}\mathbf{\text{el}}}$
PAW + SQS	42.8	717	717	717
PAW + SQS + relax	30.9	517	508	508

), being in good agreement with our previous result of ≈ 750 K based on the generalized perturbation method derived on the static lattice and Monte Carlo simulations [15]. The slightly lower ordering temperature derived here compared to [15] can be partially attributed to the overestimated ideal configurational entropy $S_{\text{conf}}^{\mathrm{A}2}$ in the present work. Neither the inclusion of vibrations nor of electronic contributions change the transition temperature. This is consistent with the very similar ground state properties (in particular, bulk modulus and Debye temperature) of the A2 and B2 phase in Table 1 and the negligible impact of B2 ordering on thermodynamic properties in Figure 2.

We now include atomic relaxations, which, as discussed above, lowers the A2 energy by about 12 meV compared to the B2 phase, which has been neglected in [15]. Indeed, if the relaxation energies, $\Delta E_{\text{relax}}^{\sigma }$, are included, the ordering temperature is reduced by almost 30% down to 517 K. Similar to the unrelaxed scenario, vibrations do not significantly change the ordering temperature (< 10 K), revealing a final predicted ordering temperature of 508 K if relaxations, vibrations, electronic excitations, and configurational entropy are taken into account. We can therefore conclude that vibrations and electronic excitations are not critical for the phase stability analysis in NbMoTaW HEAs, while local atomic relaxations and relaxation energies turn out to be significant.

4. Conclusions

We studied the A2–B2 ordering in NbMoTaW refractory alloys, in particular the impact of vibrations, electronic excitations, and lattice distortions on the ordering temperature, as well as the impact of B2 ordering on thermodynamic properties. We find that relaxations and B2 ordering have only marginal impact on the ground state volume and bulk modulus. The heat capacity of B2 and A2 differ little, although there is of course a significant contribution due to long and short-range order around the order–disorder temperature. The volume expansion and bulk modulus do not appear to be affected by the B2 ordering.

Based on total energy calculations and ideal configurational entropy, the A2–B2 ordering temperature is found to be in good agreement with previous works. Vibrations are found to play a minor role in determining the ordering temperature. In contrast to the A2 phase, the B2 phase is only slightly affected by atomic relaxations, due to the similar size of Nb and Ta as well as Mo and W. Regarding the competition of A2 and B2 Gibbs energies, the lattice relaxations turn out to significantly reduce the predicted ordering temperature by almost 30%.

Our results suggest that although thermodynamic properties of the refractory NbMoTaW HEA can be efficiently modeled based on static lattice calculations, accurate ordering temperature computations require the inclusion of local atomic relaxations.