A. DFT total energy calculations

The DFT calculations at zero temperature were performed using the VASP (Vienna Ab Initio Simulation Package, Vienna, Austria), ^{Reference Kresse and Hafner40,Reference Kresse and Furthmüller41} a plane-wave pseudo-potential software package. Projector augmented-wave potentials ^{Reference Blochl42} were used as supplied with VASP and the Perdew–Burke–Ernzerhof ^{Reference Perdew, Burke and Ernzerhof43} gradient approximation for the exchange-correlation functional. The semi-core 3p/4p/5p electrons of Cr, Fe, Mn, Mo, Nb, Ni, Os, Re, Ru, Ta, and W are explicitly treated as valence. The special quasi-random structure (SQS) method ^{Reference Wei, Ferreira, Bernard and Zunger44,Reference Zunger, Wei, Ferreira and Bernard45} was used to mimic the disordered atomic structures for equimolar HEAs as previously reported by Gao et al. ^{Reference Gao, Niu, Jiang, Irving, Gao, Yeh, Liaw and Zhang36} The cell size of the SQS models used in this work is: 64 atoms for quaternary HEAs, and 125, 125, and 160 atoms for quinary FCC, BCC, and HCP HEAs, respectively. For alloys that contain magnetic elements Co, Cr, Fe, Mn or Ni, collinear spin polarization (i.e., using ferromagnetic and anti-ferromagnetic spin configurations as initial input) were considered. The Brillouin-zone integrations were performed, using the Monkhorst–Pack k-point meshes. ^{Reference Monkhorst and Pack46} The energy convergence with respect to k-points mesh is about 1–2 meV/atom. The plane-wave energy cutoff was held constant at 500 eV, 264 eV, and 322 eV for CoCrFeNi, MoNbTaW, and CoOsReRu, respectively. All SQS models first underwent a volumetric relaxation only (i.e., fixing the lattice shape and ionic positions), followed by full relaxation under zero pressure until the energy convergence reached 1 meV/atom. For simplicity, the zero-point energy was neglected in this study. The enthalpy of formation was calculated by subtracting the composition–weighted total energies of the constituent elements in their ground state from the total energy of the alloy.

Lattice phonon calculations were done using the harmonic approximation. The vibrational entropy (S ^vib) is calculated by:

(1) $${S^{{\rm{vib}}}}\left( {V,T} \right) = 3{k_{\rm{B}}}\int_0^\infty {{n^{{\rm{vib}}}}<$$><$$>\left[ {\left( {{f_{{\rm{BE}}}} + 1} \right)\ln \left( {{f_{{\rm{BE}}}} + 1} \right) - {f_{{\rm{BE}}}}\ln {f_{{\rm{BE}}}}} \right]{\rm{d}}\varepsilon \quad ,}$$

where n ^vib is the phonon DOS and f _BE is the Bose–Einstein distribution function.

Electronic excitation across the Fermi level by migrating electrons from the valance band to the conduction band gives rise to electronic entropy (S ^elec), which can be determined by:

(2) $${S^{{\rm{elec}}}}\left( {V,T} \right) = - 2{k_{\rm{B}}}\int_{ - \infty }^\infty {{n^{{\rm{elec}}}}<$$><$$>\left[ {{f_{{\rm{FD}}}}\ln {f_{{\rm{FD}}}} + \left( {1 - {f_{{\rm{FD}}}}} \right)\ln \left( {1 - {f_{{\rm{FD}}}}} \right)} \right]} {\rm{d}}\varepsilon \quad ,$$

where n ^elec is the electron DOS and f _FD is Fermi–Dirac distribution function.

B. MC/MD hybrid simulations

The hybrid MC/MD method proposed by Widom et al. ^{Reference Widom, Huhn, Maiti and Steurer32,Reference Widom, Gao, Yeh, Liaw and Zhang33} was used to calculate the configurational entropy. Supercells of 108, 128, and 96 atoms were used for FCC CoCrFeNi, BCC MoNbTaW, and HCP CoOsReRu, respectively. The simulations were done by alternating the molecular dynamics at each temperature with MC swaps, each performed from first principles using VASP. The canonical ensemble (i.e., constant amount of substance, volume, and temperature) was adopted for the FCC and BCC structures and the isothermal-isobaric ensemble (i.e., constant amount of substance, pressure and temperature) for the HCP structure. A minimum of 1000 MC steps with MD time steps of 10 or 20 fs were carried out for these three alloys, resulting in a total MD simulation time of greater than 10 ps. More details on the procedures used are provided elsewhere. ^{Reference Widom, Huhn, Maiti and Steurer32,Reference Widom, Gao, Yeh, Liaw and Zhang33,Reference Huhn and Widom47,Reference Gao, Zhang, Gao, Zhang, Ouyang, Widom and Hawk48} The reduction in configurational entropy due to a short-range chemical order can be calculated using the pairwise truncation of the Kikuchi cluster variation method ^{Reference Ackermann, Inden and Kikuchi49} :

(3) $$I = \sum\limits_{i,\,j = 1}^N {{y_{ij}}\ln \left( {{{{y_{ij}}} \mathord{\left/ {\vphantom {{{y_{ij}}} {{c_i}{c_j}}}} \right. \kern-\nulldelimiterspace} {{c_i}{c_j}}}} \right)} \quad ,$$

where y _ij is the near neighbor correlations and c _i and c _j are the mole fractions of elements i and j, respectively.

C. CALPHAD modeling

The CALPHAD calculations were carried out using a TCNI8 thermodynamic database supplied by ThermoCalc™ (Solna, Sweden). ^{Reference Sundman, Jansson and Andersson50} For equilibrium calculations, all phases are restored. To calculate the entropy of mixing of the FCC and BCC phases, the ordered L1₂ and B2 phases were rejected. The TCNI8 database covers the complete constituent binaries and limited ternaries. ^{Reference Sundman, Jansson and Andersson50} Recent research ^{Reference Zhang, Zhang, Chen and Cao24–Reference Zhang, Gao, Gao, Yeh, Liaw and Zhang28} details the database development and applications of the CALPHAD modeling for HEAs.

A. Re-examining those predicted single-phase compositions ^{Reference Troparevsky, Morris, Kent, Lupini and Stocks14,Reference Gao, Gao, Yeh, Liaw and Zhang30,Reference Bei38,Reference Senkov, Miller, Miracle and Woodward39}

The new single-phase HEAs that were collected in Ref. Reference Gao, Gao, Yeh, Liaw and Zhang30 were mainly predicted by Bei, ^{Reference Bei38} Troparevsky, ^{Reference Troparevsky, Morris, Kent, Lupini and Stocks14} Senkov, ^{Reference Senkov, Miller, Miracle and Woodward39} and Gao. ^{Reference Gao, Gao, Yeh, Liaw and Zhang30} Note that Troparevsky ^{Reference Troparevsky, Morris, Kent, Lupini and Stocks14} did not specify the crystal structures of their predicted single-phase HEAs. The thermodynamic properties of most of these compositions are unknown and whether they would truly form a single-phase solid solution in the as-cast condition awaits experimental verification. To begin, those reassessed empirical rules ^{Reference Gao, Zhang, Gao, Zhang, Ouyang, Widom and Hawk48} will be applied to these alloys.

The equations to calculate the empirical parameters are provided in the section of the Supplementary Material. The widely used parameters include the ideal configurational entropy $\left( {S_{{\rm{ideal}}}^{{\rm{conf}}}} \right)$ , enthalpy of mixing in the liquid phase $\left( {\Delta H_{{\rm{mix}}}^{{\rm{liq}}}} \right)$ , ^{Reference Zhang, Zhou, Lin, Chen and Liaw6} atomic size difference (δ), ^{Reference Zhang, Zhou, Lin, Chen and Liaw6} the Ω-parameter, ^{Reference Zhang, Lu, Ma, Liaw, Tang, Cheng and Gao7} average valence electron concentration $\left( {\overline {{\rm{VEC}}} } \right)$ , ^{Reference Guo, Ng, Lu and Liu8} electronegativity difference (Δχ), ^{Reference Fang, Xiao, Xia, Li and Dong9} the ϕ-parameter, ^{Reference Ye, Wang, Lu, Liu and Yang10} elastic residual strain root mean square (ε_R.M.S., or $\sqrt {\left\langle {{\varepsilon ^2}} \right\rangle }$ ), ^{Reference Ye, Liu and Yang11} the atomic size-related α₂-parameter, ^{Reference Wang, Qiu, Yang and Liu12} intrinsic elastic strain energy (E ₂/E ₀), ^{Reference Wang, Qiu, Yang and Liu12} the Φ-parameter, ^{Reference King, Middleburgh, McGregor and Cortie13} the η-parameter, ^{Reference Troparevsky, Morris, Kent, Lupini and Stocks14,Reference Gao, Zhang, Gao, Zhang, Ouyang, Widom and Hawk48} and the $\kappa _1^{{\rm{cr}}}$ parameter. ^{Reference Senkov and Miracle15}

Very recently Gao et al. ^{Reference Gao, Zhang, Gao, Zhang, Ouyang, Widom and Hawk48} assessed the most-up-to-date reported HEAs (single-phase, multiphase, and amorphous) and re-evaluated the effectiveness of those empirical rules. In general, the rules are useful since they are able to separate single-phase compositions from all amorphous compositions and many multiphase compositions. However, the threshold values are sensitive to the experimental data assessed, the atomic radii used, and whether or not two FCC or two BCC phases are counted as single-phase HEAs. They ^{Reference Gao, Zhang, Gao, Zhang, Ouyang, Widom and Hawk48} found that the probably necessary requirements needed to form single-phase solid solutions are: −16.25 kJ/mol ≤ $\Delta H_{{\rm{mix}}}^{{\rm{liq}}}$ ≤ +5 kJ/mol, δ ≤ 6%, Ω ≥ 1, η ≥ 0.19, ϕ ≥ 7.0, ε_R.M.S. ≤ 0.061, and E ₂/E ₀ ≤ 13.6 × 10⁻⁴. However, these are not sufficient conditions since many multiphase compositions also satisfy these criteria.

The select empirical parameters of those predicted single-phase HEA compositions ^{Reference Gao, Gao, Yeh, Liaw and Zhang30} are shown in Fig. 1, and all parameters evaluated in this work are summarized in Table SI (Supplementary Material). Clearly, the vast majority of the predicted compositions satisfy those reevaluated criteria by Gao et al. ^{Reference Gao, Zhang, Gao, Zhang, Ouyang, Widom and Hawk48} The calculated Δχ values using the Pauling electronegativity are less than 0.36. Many compositions do not satisfy the $\Delta {H_{{\rm{IM}}}}/\Delta H_{{\rm{mix}}}^{{\rm{liq}}} \lt \kappa _1^{{\rm{cr}}}$ criterion. ^{Reference Senkov and Miracle15} Those compositions recommended by Troparevsky et al. ^{Reference Troparevsky, Morris, Kent, Lupini and Stocks14} have a $\overline {{\rm{VEC}}}$ greater than 7.43, suggesting that most of them either prefer the FCC or HCP structure as the main phase in their microstructures [Fig. 1(e)]. The average bulk and shear moduli were estimated using the ROM:

(4) $$\bar P = \sum\limits_i {{c_i}{P_i}} \quad ,$$

where P _i refers to the properties of the ith element [lattice parameter (a), elastic constants (C _ij ), bulk modulus (B), and shear modulus (G)], and $\bar P$ refers to the corresponding average value for the alloy. The result is shown in Fig. 1(f). The HEAs that comprise of alkali earth and/or rare earth elements are very soft since their shear and bulk moduli are very small, while other HEAs that contain high-melting transition metals have a significantly higher shear and bulk moduli.

FIG. 1.

Calculated empirical parameters for those predicted single-phase HEA compositions ^{Reference Troparevsky, Morris, Kent, Lupini and Stocks14,Reference Gao, Gao, Yeh, Liaw and Zhang30,Reference Senkov, Miller, Miracle and Woodward39} : (a) $\Delta H_{{\rm{mix}}}^{{\rm{liq}}}$ versus δ, (b) η-parameter versus ϕ-parameter, (c) Ω-parameter versus Δχ, (d) $\Delta {H_{{\rm{IM}}}}/\Delta H_{{\rm{mix}}}^{{\rm{liq}}}$ versus $\kappa _1^{{\rm{cr}}}$ , (e) $\overline {{\rm{VEC}}}$ versus $\Delta S_{{\rm{ideal}}}^{{\rm{conf}}}$ , and (f) $\bar B$ versus $\bar G$ . The marked threshold values are obtained from the work by Gao et al. ^{Reference Gao, Zhang, Gao, Zhang, Ouyang, Widom and Hawk48}

B. CALPHAD predictions: new septenary, octonary, and ennead BCC HEAs

Figure 2 shows the calculated equilibrium phase mole fraction as a function of temperature for septenary CrMoNbTaTiVW, octonary CrMoNbReTaTiVW, and ennead CrFeMoNbReRuTaVW, using the TCNI8 database. ^{Reference Sundman, Jansson and Andersson50} The insets are the nonequilibrium solidification using the Scheil–Gulliver models, ^{Reference Gulliver51,Reference Scheil52} which assume equilibrium mixing in the liquid state and no diffusion in the solid state. Both the equilibrium and Scheil simulations predict the formation of a single BCC solid solution in CrMoNbTaTiVW and CrMoNbReTaTiVW, while formation of C14 and HCP phases in very small fractions toward the end of the Scheil solidification is predicted for CrFeMoNbReRuTaVW.

FIG. 2.

Calculated equilibrium phase mole fraction as a function of temperature for (a) CrMoNbTaTiVW, (b) CrMoNbReTaTiVW, and (c) CrFeMoNbReRuTaVW, using the TCNI8 database. The insets are the corresponding Scheil simulations.

It has been proposed ^{Reference Gao, Gao, Yeh, Liaw and Zhang30} that a stable single-phase solid solution in as-cast state is promoted when there is a wide temperature range (T _rd) between the solidus temperature (T _sol) and decomposition temperature (T _dec):

(5) $${T_{{\rm{rd}}}} = \left( {{T_{{\rm{sol}}}} - {T_{{\rm{dec}}}}} \right)/{T_{{\rm{sol}}}} \ge 0.3\quad .$$

The calculated T _rd parameters from the equilibrium thermodynamic calculation are 0.40, 0.43, and 0.29 for CrMoNbTaTiVW, CrMoNbReTaTiVW, and CrFeMoNbReRuTaVW, respectively. Both the Scheil simulations and the calculated T _rd values suggest that CrFeMoNbReRuTaVW may be on the verge of forming a single BCC phase in the as-cast condition.

A. Entropy of mixing calculated from CALPHAD

The compositional dependences of the entropy of mixing for the FCC phase in the Co–Cr–Fe–Mn–Ni system and the BCC phase in the Mo–Nb–Ta–V–W system at T = 1273 K are illustrated in Fig. 3 using the CALPHAD method. The ideal configurational entropies are also shown as a reference in dashed lines. Due to the presence of a short-range order, or segregation, and the contribution from the lattice phonon vibration, or magnetic contribution, the total entropy of mixing $\left( {\Delta S_{{\rm{mix}}}^\varphi } \right)$ for HEAs may not always follow ideal mixing and can cause positive or negative deviation from $- R\sum\limits_i {{c_i}\ln {c_i}}$ (i.e., excess entropy). The excess entropy $\left( {^{{\rm{ex}}}S_{{\rm{mix}}}^\varphi } \right)$ of a solution phase (φ) is calculated by subtracting the ideal configurational entropy from the total entropy of the alloy $\left( {\Delta S_{{\rm{mix}}}^\varphi } \right)$ ^{Reference Zhang, Gao, Gao, Yeh, Liaw and Zhang28} :

(6) $$^{{\rm{ex}}}S_{{\rm{mix}}}^\varphi = \Delta S_{{\rm{mix}}}^\varphi + R\sum\limits_i {{c_i}\ln {c_i}} \quad .$$

FIG. 3.

Calculated total entropy of mixing at T = 1273 K using the TCNI8 database for: (a) the FCC phase in the Co–Cr–Fe–Ni system; (b) the FCC phase in the CoCr _x , CoFeCr _x , CoFeNiCr _x , and CoFeMnNiCr _x systems; (c) the BCC phase in the Mo–Nb–Ta–W system; and (d) the BCC phase in the MoV _x , MoNbV _x , MoNbTaV _x , and MoNbTaWV _x systems. The star symbols in (b) and (d) mark the equimolar compositions. The dashed curves represent the ideal mixing.

The FCC phase in the Co–Cr–Fe–Mn–Ni system [Figs. 3(a) and 3(b)] exhibits a positive deviation from the ideal mixing and thus positive excess entropy for equal and near-equimolar compositions. The maximum excess entropy becomes smaller with increasing the number of components [Fig. 3(b)]. The compositions with the highest $\Delta S_{{\rm{mix}}}^{{\rm{FCC}}}$ also do not necessarily occur at the equimolar compositions. In contrast, the BCC phase in the Mo–Nb–Ta–V–W system [Figs. 3(c) and 3(d)] shows a negative deviation from the ideal mixing, and thus the negative excess entropy at equal and near-equimolar compositions except that the Mo–V binary shows an ideal mixing behavior [Fig. 3(d)]. At equimolar compositions, the calculated entropy of mixing for both FCC and BCC phases increases with increasing the number of components, suggesting the dominance of configurational entropy over other entropy sources at T = 1273 K. However, the contrasting behavior in the excess entropy between FCC and BCC structures indicates that the other entropy sources such as the vibrational entropy of mixing may also be important, as revealed from DFT calculations (see Sec. IV.B).

B. DFT calculations: enthalpy of formation and entropy sources

In general, SQS models tend to maximize random distributions of atoms in the lattice, but the permutation in the atomic position between elements may alter the local atomic environment and accordingly the fluctuation in the energy. The total number of atomic configurations for an HEA SQS model that result from atomic permutation scales up almost exponentially with increasing the number of equimolar components, and it is 24 for an equimolar quaternary HEA (4! = 24) and 120 for an equimolar quinary HEA (5! = 120). To examine the effect of element permutation on the enthalpy of formation (ΔH _SS), DFT calculations were carried out on all 24 configurations of FCC CoCrFeNi, BCC MoNbTaW, and HCP CoOsReRu, and the results are shown in Fig. 4(a). The averaged ΔH _SS are 8.354 ± 0.266 kJ/mol, −7.407 ± 0.069 kJ/mol, and 2.724 ± 0.49 kJ/mol for CoCrFeNi, MoNbTaW, and CoOsReRu, respectively. Relatively speaking, CoOsReRu is a bit more sensitive to the atomic configuration than the other two structures, presumably due to the anisotropy of the HCP structure compared to the cubic structure. Other factors that may also contribute to the sensitivity with respect to the atomic configuration in SQS models are electronegativity, magnetism, and atomic size of the constituent elements. It can be assumed that this sensitivity may decrease with increasing the SQS cell size which, however, may increase the computing time substantially.

FIG. 4.

(a) Enthalpy of formation (ΔH _SS) versus SQS configurations for FCC CoCrFeNi, BCC MoNbTaW, and HCP CoOsReRu. (b) DFT-calculated enthalpy of formation versus atomic size difference for various single-phase HEAs with the FCC, BCC, and HCP structures at zero temperature.

The calculated ΔH _SS for widely studied single-phase HEAs are shown in Fig. 4(b). Note that only one randomly chosen atomic configuration of the SQS model was used in the calculations (except CoCrFeNi, MoNbTaW, and CoOsReRu whose averaged ΔH _SS was used), so a small fluctuation in ΔH _SS is expected upon atomic permutation tests. For FCC and HCP HEAs, the calculated ΔH _SS is positive or close to zero. BCC AlNbTiV has the most negative ΔH _SS at −10.3 kJ/mol, followed by BCC MoNbTaW. Addition of HCP metals such as Hf, Ti, and Zr to BCC refectory metals causes a near-zero, or even positive, ΔH _SS.

The electronic DOS for FCC CoCrFeNi, BCC MoNbTaW, and HCP CoOsReRu is shown in Fig. 5. Integration of the difference between spin-up and spin-down electronic DOS gives rise to the magnetic moment, which is 0.62 ± 0.03 μ_B/atom for CoCrFeNi and 0.11 ± 0.02 μ_B/atom for CoOsReRu (here μ_B = Bohr magneton), averaged from total 24 atomic configurations. A clear pseudo-gap exists only for MoNbTaW, and the Fermi level sits on the left edge to the gap. Existence of a deep pseudo-gap of an alloy usually suggests thermodynamic stability, and this is consistent with the negative ΔH _SS of −7.407 kJ/mol for BCC MoNbTaW. Conversely, the Fermi levels of CoCrFeNi and CoOsReRu sit on (relatively speaking) a plateau with a low DOS.

FIG. 5.

DFT-calculated electronic DOS for (a) FCC CoCrFeNi, (b) BCC MoNbTaW, and (c) HCP CoOsReRu. The vertical dashed lines mark the Fermi level.

The entropy sources of a solid solution phase may comprise contributions from the lattice vibration (S ^vib), configuration (S ^conf), electronic excitation (S ^elec), and magnetic spin fluctuations (S ^mag):

(7) $${S^{{\rm{total}}}} = {S^{{\rm{vib}}}} + {S^{{\rm{conf}}}} + {S^{{\rm{elec}}}} + {S^{{\rm{mag}}}}\quad .$$

Since the critical magnetic ordering temperatures for FCC CoCrFeNi and CoCrFeMnNi are well below room temperature, ^{Reference Lucas, Mauger, Munoz, Xiao, Sheets, Semiatin, Horwath and Turgut53,Reference Lucas, Belyea, Bauer, Bryant, Michel, Turgut, Leontsev, Horwath, Semiatin, McHenry and Miller54} it is sensible to assume that their magnetic entropy term should be temperature independent and thus can be ignored at T ≥ 293 K for simplicity. The HCP CoOsReRu HEA was first suggested by Gao and Alman, ^{Reference Gao and Alman25} and there are no experiments reported yet.

Figure 6(a) presents the phonon DOS of FCC CoCrFeNi, BCC MoNbTaW, and HCP CoOsReRu. Excellent agreement in the phonon DOS for CoCrFeNi with the experiment ^{Reference Lucas, Mauger, Munoz, Xiao, Sheets, Semiatin, Horwath and Turgut53} is evident. At low energies (less than 10 meV), the phonon DOS is comparable for CoCrFeNi and CoOsReRu, but both are smaller than MoNbTaW. In contrast, the calculated vibrational heat capacity [C _V, see Fig. 6(b)] is very similar for all three alloys, although it is slightly larger for CoOsReRu and MoNbTaW than that for CoCrFeNi at T ≤ 400 K. At higher temperatures, the vibrational C _V approaches the theoretical limit 3R.

FIG. 6.

(a) DFT-calculated vibrational DOS using the harmonic approximation. The experimental data for FCC CoCrFeNi was measured by Lucas et al. ^{Reference Lucas, Mauger, Munoz, Xiao, Sheets, Semiatin, Horwath and Turgut53} (b) DFT-calculated vibrational heat capacity at a constant volume. (c) DFT- and MC/MD-predicted entropies of mixing from lattice configurations (S ^conf), phonon vibrations $\left( {\Delta S_{{\rm{mix}}}^{{\rm{vib}}}} \right)$ , and electron excitations $\left( {\Delta S_{{\rm{mix}}}^{{\rm{elec}}}} \right)$ . (d) Total entropy of mixing based on (c).

Alloy entropies of mixing (ΔS _mix) are calculated by subtracting the composition–weighted total entropies of the constituent elements from the entropy of the alloy for total entropy, vibrational entropy, and electronic entropy, respectively. DFT-calculated total vibrational and electronic entropies for FCC CoCrFeNi, BCC MoNbTaW, and HCP CoOsReRu were reported elsewhere. ^{Reference Gao, Niu, Jiang, Irving, Gao, Yeh, Liaw and Zhang36,Reference Gao, Zhang, Gao, Zhang, Ouyang, Widom and Hawk48} The vibrational entropies of pure elements of FCC Co, BCC Cr, antiferromagnetic FCC Fe, and FCC Ni were used for calculating the ΔS _mix of CoCrFeNi. FCC Cr was discarded because of the presence of 64 imaginary vibrational modes (out of a total of 324 modes) and its elastic instability.

The calculated ΔS _mix from various entropy sources for these HEAs are shown in Fig. 6(c). The calculated entropies of mixing for these three HEAs in descending order are: S ^conf ≫ $\left| {\Delta S_{{\rm{mix}}}^{{\rm{vib}}}} \right|$ ≫ $\left| {\Delta S_{{\rm{mix}}}^{{\rm{elec}}}} \right|$ . At a temperature close to the solidus temperatures, the calculated S ^conf is fairly close to their ideal value (R ln 4) and start to decrease very gradually until the temperature reaches 1100 K. At lower temperatures, S ^conf decreases rapidly, signaling the development of chemical short-range order in the alloys. Note that S ^conf at room temperature is much lower for MoNbTaW than for the others due to the tendency in forming the ordered BCC (i.e., B2) as revealed by Widom et al. ^{Reference Widom, Huhn, Maiti and Steurer32,Reference Huhn and Widom47} The vibrational entropies of mixing (ΔS ^vib) approach constant values at temperatures above the Debye temperature (i.e., T ≥ ∼400 K), namely, +2.8 J/K/mol, −3.6 J/K/mol, and −0.4 J/K/mol for CoCrFeNi, MoNbTaW, and CoOsReRu, respectively. This is because the heat capacities approach their classical limit at 3R. In contrast, the calculated $\Delta S_{{\rm{mix}}}^{{\rm{elec}}}$ values are close to zero. The sum of all entropy sources shows that the total entropy of the mixing in descending order for these alloys is: CoCrFeNi > CoOsReRu > MoNbTaW [Fig. 6(d)].