# Designing Quaternary and Quinary Refractory-Based High-Entropy Alloys: Statistical Analysis of Their Lattice Distortion, Mechanical, and Thermal Properties

^{1}

^{2}

^{*}

## Abstract

**:**

_{D}), thermal conductivity (κ), Grüneisen parameter (γ

_{α}), and dominant phonon wavelength (λ

_{dom}). The NbTaTiV(M9) and TiVNbHf(M10) models have significantly reduced lattice thermal conductivities (κ

_{L}). This reduction is due to the mass increase and strain fluctuations, which in turn signify lattice distortion. The findings not only provide an understanding of these promising materials but also offer guidance for the design of next-generation HEAs with properties tailored for potential specific applications.

## 1. Introduction

_{B}is the Boltzmann constant, ω is the number of ways of mixing, R is the gas constant, and n is the number of elements within the HEA [4]. Materials are considered low-entropy when ΔS

_{mix}is less than 0.69 R, medium-entropy when ΔS

_{mix}is between 0.69 R and 1.59 R, and high-entropy when ΔS

_{mix}is larger than 1.60 R [5]. HEAs come with very unique effects such as significant lattice distortion and the cocktail effect [6,7] which result in some unique properties such as high hardness, high resistance to corrosion, high thermoelectric (TE) performance [8,9], oxidation resistance [10], magnetic properties [11], and improved radiation performance [12,13,14,15]. The significant lattice distortion in HEAs results from the mismatch of mass, size, and bond state of different elements or ions in the structure of the HEA.

_{0.5}Ta

_{0.5}[25], CrNbTiVZr [26,27], and TiZrNbMoV

_{x}(x = 0–1.5) [28]. Ab initio molecular dynamics (AIMD) simulation was used to study the phase stability of HfNbTaTiZr HEA [29]. The ab initio method combined with virtual crystal approximation (VCA) was used to predict the tensile strength and shear strength of TiVNbMo HEA [30]. They found that alloying does not result in a remarkable change in tensile strength and shear strength of TiVNbMo. The microstructure and mechanical properties of TiZrHfNbV and TiZrHfNbCr HEAs were investigated using X-ray diffractometry and scanning electron microscopy (SEM) [31] The SEM study showed that strength and hardness were enhanced for these HEAs. The phase stability of MoNbTaW HEA was tested by using first principles calculations [32]. The distribution of interatomic distances and the lattice distortion (LD) in BCC MoNbTaVW HEA and its five sub-quaternary systems at different temperatures were studied by Toda et al. [33] Experimental (including X-ray diffraction and SEM) and computational study was carried out to investigate the microstructure and thermodynamic properties of BCC HfNbTaTiVZr HEA [34]. The effect of alloying on the elastic properties of ZrNbHf, ZrVTiNb, ZrVNbHf, and ZrVTiNbHf refractory-based HEAs was investigated via ab initio calculations by Li et al. [35]. A theoretical study was carried out to investigate the B2-ordering impacts on the thermodynamic properties of BCC NbMoTaW [36]. Elastic properties of TiZrVNb, TiZrNbMo, and TiZrVNbMo HEAs were investigated by using ab initio calculation combined with a special quasi-random structure (SQS) approach and coherent potential approximation (CPA) [37]. Hybrid Monte Carlo/molecular dynamics simulation was carried out to study the elastic properties and LD of NbTiVZr, CrMoNbV, HfNbTaZr, and MoNbTaW [38]. A theoretical study [39] showed that the effect of local lattice distortion is larger in refractory HEAs than in 3d transition metal HEAs. Another theoretical study [40] was carried out to investigate the elastic and thermal properties of single-phase ternary and quaternary Al-Ti-V-Cr-Nb-Mo refractory-based HEAs.

^{2}σT/κ, where σ is the electrical conductivity, S is the Seebeck coefficient, T is the absolute temperature, and κ is the thermal conductivity. The considerable mass difference in HEAs causes chemical disorder, which results in significant phonon scattering and reduces κ. In addition, the distinct chemical environments in HEAs can produce force constant variations that modify the phonon spectral distribution, where the phonon dispersion can be expressed as [42]:

_{c}are the force constant, atomic mass, wave vector, and cut-off wave vector, respectively. The speed of phonons ($\omega $) can be changed by changes in these parameters. Large M and small F (weak chemical bonds) indicate low $\omega $ and low lattice thermal conductivity (κ

_{L}). The variations in mass and force constant induce phonon-scattering processes, which are used to enhance the TE performance of HEAs. Thus, HEAs are good candidates for TE applications [43,44] and heat shield materials [45]. Studies on using chemical disorder to induce phonon scattering in HEAs are so far lacking. Some previous studies addressing phonon scattering in disordered alloys were limited to binaries [46,47,48,49,50,51]. The manipulation of mass and force constant in HEAs to induce phonon scattering and reduce κ

_{L}is not yet fully explored.

^{*}) of every atom in each model, and the calculated mechanical parameters are discussed in detail. Moreover, interatomic bonding and local lattice distortion (LD) are discussed.

## 2. Computational Modeling and Method

#### 2.1. Optimization and Mechanical Properties

^{−5}eV, and the force convergence set at −1.0 × 10

^{−3}eV/Å for ionic relaxation.

#### 2.2. Electronic Structure

## 3. Results and Discussion

#### 3.1. Electronic Structure

_{F}). None of the investigated DOS is minimum at E

_{F}. By comparison, in the case of FCC, the minimum can be seen at E

_{F,}which means that FCC HEAs are more stable and less deformed [73]. Detailed total and partial DOS of the 12 models are shown in Figure S1 in Supplementary Materials (SM). Notably, the 5d elements had lower DOS at E

_{F}. For instance, M5, M6, and M7 are 4-component models, containing, e.g., W; and thus, this influences a lower total DOS for the models. Although M1 and M2 contain W and their total DOS are also low compared to the non-W models, it is not as low as M5 and M6 (see Figure S1). This is due to the higher number of components in their structure. Moreover, the partial DOS of elements contributing to the 12 models almost have same trend regardless of their different local chemical environments in the supercell. On the other hand, the partial DOS of 3d elements are usually higher, which is consistent with FCC HEAs such as the Cantor alloy [2,74].

#### 3.2. Interatomic Bonding and Lattice Distortion

#### 3.3. Mechanical Properties

_{11}, C

_{22}, and C

_{33}are strongly correlated with unidirectional compression along the principal x, y, and z directions [76] and have the same value in cubic structures. Synonymously, C

_{11}, C

_{22}, and C

_{33}can describe the resistance of a material against the deformation along the [100], [010], and [001] directions, respectively. C

_{44}measures the resistance against shear deformation in the (100) plane. A large value of C

_{11}indicates incompressibility under uniaxial stress along the x-axis. The C

_{11}of M5 and M6 are much larger than the C

_{11}of all remaining models, indicating that M5 and M6 are much less compressible under uniaxial stress along the x, y, and z directions. It also means that the bonding strength in M5 and M6 along the x, y, and z axes is much stronger than the bonding strength in all remaining models. The C

_{44}of M5 and M6 are larger than the C

_{44}of all remaining models. Larger C

_{11}and stronger bonding characteristics can result in higher values of K, G, and E. Larger C

_{11}and C

_{44}also indicate higher transverse (shear) velocity (v

_{s}) and longitudinal sound velocity (v

_{l}). The pure elements W and Ta have much larger densities and higher melting temperatures than the other elements in these 12 models. Thus, alloying with W and Ta in M5 and M6 results in harder materials for many mechanical applications. However, this also results in higher lattice thermal conductivity, which makes M5 and M6 much less applicable as TE materials. On the other hand, the pure elements Ti, V, and Zr have the lowest density among the studied elements. Thus, alloying with Ti and V in M10 and M11, and Ti, Vi, and Zr in M12, may result in softer materials with smaller C

_{11}and K for TE applications. A low value of C

_{44}indicates high shearability. Due to having the lowest C

_{44}value, M9 and M10 have the highest shearability among all solid solutions. The results in Figure 7a show the model M6 has the highest values for K, G, and E, while M9 exhibits the lowest values for both G and E. However, M12 has the lowest K value among the models. It should be noted that the G and E have the same trend from M6 (highest) to M9 (lowest). It can be observed that the model consisting of Ta has higher bulk modulus, especially the models with 4 components, because their number of atoms in the 4-element model is 125 atoms each. By contrast, models containing Hf possess lower bulk modulus. Vickers hardness (Hv) was calculated using the formula of Tian et al. [77]:

_{44}> 0, C

_{11}> |C

_{12}|, and C

_{11}+ 2C

_{12}> 0 [81]. From Table 3, the calculated elastic constants satisfy these criteria, thus these alloys are expected to be mechanically stable. Frantsevich’s rule of Poisson’s ratio [82] is used to characterize material’s brittleness or ductility. It suggests that if Poisson’s ratio (η) is less than 0.26, the material tends to be brittle, otherwise it is ductile in nature. From Table 3, we notice that all refractory-based HEAs under study have an η much higher than 0.26. Hence, these HEAs are ductile and both Frantsevich’s rule and Pugh’s criterion are equivalent for these HEAs. Cauchy pressure (CP), which is given by: (C

_{12}− C

_{44}) [83], can be used to characterize materials’ bonding nature. Generally speaking, a positive value of CP indicates metallic bonding dominating, while a negative value of CP suggests that the material is dominated by covalent bonding. The calculated positive values of CP in Table 3 show that these HEAs have a metallic character. M3, M5, M6, and M9 have the highest metallic bonding character, which may indicate that alloying with Ta increases the metallic character of the bonding. The Zener ratio (A

_{Z}= 2C

_{44}/(C

_{11}− C

_{12})) determines the elastic anisotropy of materials [40]. A

_{Z}is a unity for isotropic materials. From Table 3, we notice that these refractory-based HEAs are elastically anisotropic. We can associate the electronic structure and chemical properties with the mechanical properties. For instance, the correlation between TBOD and bulk modulus (see Figure 7b) does not exhibit a perfect linear relationship but does display a closer-to-linear nature, with the exception of a few outliers (M5 and M6). This observation suggests the potential future prediction of bulk modulus based on TBOD. Another example is effective charge versus bulk modulus, with their coefficient of determination (R

^{2}) equaling 0.97, as shown in Figure 3a. The figure illustrates that an increase in $VE{C}_{av}$ within the model leads to an increase in the elastic moduli, particularly the bulk modulus. These insights offer valuable guidance for the design of HEAs.

#### 3.4. Thermal Properties

_{D}), which originates from the theory of thermal vibration of atoms. Θ

_{D}is an important parameter for high temperature applications and correlates strongly to thermal conductivity (κ). Lower Θ

_{D}indicates softer materials with lower melting temperatures (T

_{melt}), while higher Θ

_{D}indicates harder materials with stronger interatomic bonds and higher T

_{melt}[84,85]. Θ

_{D}is calculated here using Anderson’s method, shown in Equation (S1) in the SM. Average sound velocity (v

_{m}), transverse (shear) velocity (v

_{s}), and longitudinal sound velocity (v

_{l}) are calculated using the equations shown in Equations (S2)–(S4) respectively. The calculated density(ρ), ν

_{s}, ν

_{l}, ν

_{m}, and Θ

_{D}for the 12 BCC HEAs are listed in Table 4 and plotted in Figure 8a–c. Figure 8a shows that M5 and M6 have the highest densities, while M12 has the lowest density. M1, M7, M9, M10, M11, and M12 have the lowest Θ

_{D}and ν

_{s}while M3, M5, and M6 have the highest Θ

_{D}and ν

_{s}. This indicates that alloying with Ti, V, and Hf may suppress transverse phonon velocity and Θ

_{D}in HEAs, while alloying with W, Mo, Cr, and Ta may increase transverse phonon velocity and Θ

_{D}. As individual elements, Ti, Hf, and Zr have the smallest lattice thermal conductivity(κ

_{L}) [86], whereas W, Mo, and Ta, Nb, and Cr have the largest κ

_{L}at room temperature [86], among the elements constituting these 12 HEAs. This indicates that the κ

_{L}value of the individual elements constituting these 12 HEAs also counts in determining the value of κ

_{L}for refractory-based HEA models. For example, M6 has the largest κ

_{L}because it consists of W, Mo, Nb, and Ta, alongside M5, which contains the three elements W, Ta, and Cr. It is important to identify the thermal limits or melting temperature (T

_{melt}) of a material. Low T

_{melt}indicates lower Θ

_{D}and higher thermal expansion. T

_{melt}is calculated using Equation (S5). Models M1, M7, M9, M10, M11, and M12 also have the lowest T

_{melt}, while M3, M5, and M6 have the highest T

_{melt}among the models. This indicates that alloying with Ta increases T

_{melt}and makes the alloys much harder. Significantly dampened transverse phonon modes (ν

_{s}) would strengthen the scattering of phonons [87], which in turn results in reduced lattice thermal conductivity(κ

_{L}).

_{min}) and lattice thermal conductivity (κ

_{L}) at 300 K are estimated using Clarke’s model [89], Cahill’s model [90], Slack’s model, and mixed model [91]. κ

_{min}and κ

_{L}were calculated using Equations (S6)–(S8) and (S10) and are listed in Table 5. κ

_{L}for the 12 BCC HEAs is also shown in Figure 8d. M5 and M6 have the largest values of κ

_{L}while M9 and M10 have the smallest κ

_{L}. Phonon velocities or sound velocities and κ

_{L}are correlated through Equation (6) [92]:

_{L}requires suppressing sound velocity, particularly shear velocity (v

_{s}). v

_{s}and v

_{l}are directly correlated to the shear elastic constant (C

_{44}), C

_{11}, and density (ρ) by Equations (7) and (8) below [93]:

_{L}values when compared to other models.

_{α}indicates strong anharmonic vibrations, which also indicate higher phonon scattering and thus low κ

_{L}(depressed and temperature-independent lattice thermal conductivity). Element substitution in HEAs creates disorder, leading to weak displacements of the atoms and bonds resulting in higher bond anharmonicity and higher γ

_{α}. Sound or phonon velocity and the strength of interatomic interactions are positively correlated. Weaker interatomic interactions between atoms indicate a lower sound velocity and thus larger γ

_{α}[95]. In summary, alloying induces internal strain fields, which reduces the speed of sound.

_{L}. This can be fully understood by following formula [96], which correlates the phonon frequency (ω), Grüneisen parameter tensor (γ

_{ij}), and strain tensor (ε

_{ij}):

_{L}. In this study, γ

_{α}is calculated using Equation (S11). The calculated γ

_{α}for the 12 BCC HEA models are summarized in Table 5 and Figure 9. The four-component alloys M9 and M10 have the largest γ

_{α}(weaker chemical bonds) and thus the lowest κ

_{L}(see Figure 8d), which indicates strong anharmonic vibrations due to higher mass and force constant. The four-component alloys M5, M6, and M7 have the lowest γ

_{α}and largest κ

_{L}(see Figure 8d), which indicates weak anharmonic vibrations that result from the lower mass and force constant.

_{dom}). λ

_{dom}is defined as the wavelength at which the phonon energy distribution curve strikes its maximum value. λ

_{dom}and mean free path (MFP) are positively correlated and both play a significant role in controlling κ

_{L}. MFP is the average distance that a phonon travels between two successive inelastic collisions. Shortening MFP increases inelastic collisions between phonons, which means increasing the scattering of phonons and reducing κ

_{L}[97]. This requires shifting the heat phonon spectra towards shorter wavelengths (smaller λ

_{dom}). λ

_{dom}can be roughly estimated at 300 K by using Equation (S13). The calculated λ

_{dom}for the 12 BCC refractory-based HEAs are shown in Table S5. M5 and M6 have the largest λ

_{dom}, whereas M9 and M10 have the smallest λ

_{dom}among the HEAs.

## 4. Conclusions

_{0.38}V

_{0.15}Nb

_{0.23}Hf

_{0.24}(M11), and TiZrHfVNb(M12)—were investigated using first-principles calculations. The random solid solution model (RSSM) was used for alloying these solid solutions with large supercells of 500 atoms. We highlight the significance of TBOD as a valuable parameter in understanding the bonding of HEAs. Our calculations showed that TBOD is positively correlated with the mechanical properties, especially with bulk modulus. The average partial charge ${Q}_{av}^{*}$ is positively correlated with the bulk modulus, which is a new important finding from our current calculations. This feature of these refractory-based HEAs can be used to design new HEAs. Based on our calculations, all these 12 HEA models are mechanically stable. M3, M5 and M6 have the largest density and largest Young’s, bulk and shear moduli, while M9 and M10 have the lowest Young’s and shear moduli. Alloying with both W and Ta elements in M5 and M6 or both Mo and Ta in M3 results in very large elastic constants (C

_{11}and C

_{44}) compared with other models, indicating higher hardness with higher fracture toughness and melting temperature. This feature can be useful for many mechanical and high temperature applications. Also, due to the high strength and ductility of M3, M5, and M6, they can be used as joint surrogate metals instead of the traditional stainless steels and titanium alloys, especially because they consist of refractory elements that are mostly non-toxic and hypoallergenic. However, this in turn results in high sound velocities or high phonon speeds, indicating larger lattice thermal conductivity. This feature makes the M5 and M6 models less applicable for TE applications. Considering that Ta has a higher density than Hf, and Mo has a higher density than V (see Figure 8a), replacing Hf in M10 with Ta in M9 and replacing V in M1 with Mo in M2 leads to a significant increase in the values of C

_{11}and bulk modulus. Compared to M1, a significant reduction in the values of C

_{11}and bulk modulus of M12 is observed when W is replaced with Nb, since W has a much higher density than Nb (see Figure 8a). M9 and M10 have the smallest v

_{s}, v

_{m}, Θ

_{D}, and κ

_{L}, whereas M5 and M6 have the largest v

_{s}, v

_{m}, Θ

_{D}, and κ

_{L}. M9 and M10 have the largest γ

_{α}and thus the highest anharmonic vibrations. Thus, M9 and M10 are more suitable for TE applications. It is difficult to determine the main factor that caused the significant reduction in κ

_{L}in M9 and M10. In general, the models from M9 to M12 have smaller κ

_{L}than the other models. All these models (from M9 to M12) contain the elements Ti and V, which are the lightest of the remaining elements. This can cause a larger mismatch between size and mass, which may lead to larger lattice distortion in these models and thus smaller κ

_{L}. These 12 HE models were investigated for their local lattice distortion (LD). M1, M7, and M12 have the highest LD while models M3, M5, M6, and M9 have the lowest LD. It is known that alloying with heavy elements, such as W and Ta in M5 and M6, may result in high LD. However, with the lower LD of M5 and M6, we conclude that heavy elements are not the only factor making the lattice more distorted. LD is correlated with lattice thermal conductivity (κ

_{L}). A high LD indicates higher phonon scattering and thus low κ

_{L}, while a small LD indicates higher κ

_{L}. This correlation is revealed for most models, whereas it is not clear for the model M9. The higher LD and lower κ

_{L}that some of these HE models have may not make them perfect for TE applications, since they are all metals with a zero energy band gap that indicates a very small Seebeck coefficient. Thus, more research work is required for enhancing the value of Seebeck coefficient and figure of merit. The promising current results encourage and inspire us to continue research in this direction for more complex and interesting high-entropy materials. Our DFT calculations could be improved by using better options, such as using either hybrid potential or Becke–Johnson potential. Overall, we believe our results can facilitate the design of new high-entropy materials with wider applications.

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Yeh, J.W.; Chen, S.K.; Lin, S.J.; Gan, J.Y.; Chin, T.S.; Shun, T.T.; Tsau, C.H.; Chang, S.Y. Nanostructured High-Entropy Alloys with Multiple Principal Elements: Novel Alloy Design Concepts and Outcomes. Adv. Eng. Mater.
**2004**, 6, 299–303. [Google Scholar] [CrossRef] - Cantor, B.; Chang, I.T.H.; Knight, P.; Vincent, A.J.B. Microstructural Development in Equiatomic Multicomponent Alloys. Mater. Sci. Eng. A
**2004**, 375–377, 213–218. [Google Scholar] [CrossRef] - Yeh, J.W.; Chen, Y.L.; Lin, S.J.; Chen, S.K. High-Entropy Alloys—A New Era of Exploitation. Mater. Sci. Forum
**2007**, 560, 1–9. [Google Scholar] [CrossRef] - Zhu, H.; Zhao, T.; Zhang, B.; An, Z.; Mao, S.; Wang, G.; Han, X.; Lu, X.; Zhang, J.; Zhou, X. Entropy Engineered Cubic N-Type AgBiSe2 Alloy with High Thermoelectric Performance in Fully Extended Operating Temperature Range. Adv. Energy Mater.
**2021**, 11, 2003304. [Google Scholar] [CrossRef] - Sun, L.; Cava, R.J. High-Entropy Alloy Superconductors: Status, Opportunities, and Challenges. Phys. Rev. Mater.
**2019**, 3, 090301. [Google Scholar] [CrossRef] - Yeh, J.W. Alloy Design Strategies and Future Trends in High-Entropy Alloys. JOM
**2013**, 65, 1759–1771. [Google Scholar] [CrossRef] - Dąbrowa, J.; Zajusz, M.; Kucza, W.; Cieślak, G.; Berent, K.; Czeppe, T.; Kulik, T.; Danielewski, M. Demystifying the Sluggish Diffusion Effect in High Entropy Alloys. J. Alloys Compd.
**2019**, 783, 193–207. [Google Scholar] [CrossRef] - Jiang, B.; Yu, Y.; Cui, J.; Liu, X.; Xie, L.; Liao, J.; Zhang, Q.; Huang, Y.; Ning, S.; Jia, B.; et al. High-Entropy-Stabilized Chalcogenides with High Thermoelectric Performance. Science
**2021**, 371, 830–834. [Google Scholar] [CrossRef] - Jiang, B.; Wang, W.; Liu, S.; Wang, Y.; Wang, C.; Chen, Y.; Xie, L.; Huang, M.; He, J. High Figure-of-Merit and Power Generation in High-Entropy GeTe-Based Thermoelectrics. Science
**2022**, 377, 208–213. [Google Scholar] [CrossRef] - Waseem, O.A.; Ryu, H.J. Combinatorial Synthesis and Analysis of AlxTayVz-Cr20Mo20Nb20Ti20Zr10 and Al10CrMoxNbTiZr10 Refractory High-Entropy Alloys: Oxidation Behavior. J. Alloys Compd.
**2020**, 828, 154427. [Google Scholar] [CrossRef] - Chen, C.; Zhang, H.; Fan, Y.; Zhang, W.; Wei, R.; Wang, T.; Zhang, T.; Li, F. A Novel Ultrafine-Grained High Entropy Alloy with Excellent Combination of Mechanical and Soft Magnetic Properties. J. Magn. Magn. Mater.
**2020**, 502, 166513. [Google Scholar] [CrossRef] - Granberg, F.; Nordlund, K.; Ullah, M.W.; Jin, K.; Lu, C.; Bei, H.; Wang, L.M.; Djurabekova, F.; Weber, W.J.; Zhang, Y. Mechanism of Radiation Damage Reduction in Equiatomic Multicomponent Single Phase Alloys. Phys. Rev. Lett.
**2016**, 116, 135504. [Google Scholar] [CrossRef] [PubMed] - Zhang, Y.; Egami, T.; Weber, W.J. Dissipation of Radiation Energy in Concentrated Solid-Solution Alloys: Unique Defect Properties and Microstructural Evolution. MRS Bull.
**2019**, 44, 798–811. [Google Scholar] [CrossRef] - Zhang, Y.; Stocks, G.M.; Jin, K.; Lu, C.; Bei, H.; Sales, B.C.; Wang, L.; Béland, L.K.; Stoller, R.E.; Samolyuk, G.D.; et al. Influence of Chemical Disorder on Energy Dissipation and Defect Evolution in Concentrated Solid Solution Alloys. Nat. Commun.
**2015**, 6, 8736. [Google Scholar] [CrossRef] [PubMed] - Lu, C.; Niu, L.; Chen, N.; Jin, K.; Yang, T.; Xiu, P.; Zhang, Y.; Gao, F.; Bei, H.; Shi, S.; et al. Enhancing Radiation Tolerance by Controlling Defect Mobility and Migration Pathways in Multicomponent Single-Phase Alloys. Nat. Commun.
**2016**, 7, 13564. [Google Scholar] [CrossRef] - Ikeda, Y.; Grabowski, B.; Körmann, F. Ab Initio Phase Stabilities and Mechanical Properties of Multicomponent Alloys: A Comprehensive Review for High Entropy Alloys and Compositionally Complex Alloys. Mater. Charact.
**2019**, 147, 464–511. [Google Scholar] [CrossRef] - El-Atwani, O.; Li, N.; Li, M.; Devaraj, A.; Baldwin, J.K.S.; Schneider, M.M.; Sobieraj, D.; Wróbel, J.S.; Nguyen-Manh, D.; Maloy, S.A.; et al. Outstanding Radiation Resistance of Tungsten-Based High-Entropy Alloys. Sci. Adv.
**2019**, 5, eaav2002. [Google Scholar] [CrossRef] - Zhang, Y.; Zhou, Y.J.; Lin, J.P.; Chen, G.L.; Liaw, P.K. Solid-Solution Phase Formation Rules for Multi-Component Alloys. Adv. Eng. Mater.
**2008**, 10, 534–538. [Google Scholar] [CrossRef] - Takeuchi, A.; Inoue, A. Quantitative Evaluation of Critical Cooling Rate for Metallic Glasses. Mater. Sci. Eng. A
**2001**, 304–306, 446–451. [Google Scholar] [CrossRef] - Guo, S.; Ng, C.; Lu, J.; Liu, C.T. Effect of Valence Electron Concentration on Stability of Fcc or Bcc Phase in High Entropy Alloys. J. Appl. Phys.
**2011**, 109, 103505. [Google Scholar] [CrossRef] - Wu, Z.; Bei, H.; Otto, F.; Pharr, G.M.; George, E.P. Recovery, Recrystallization, Grain Growth and Phase Stability of a Family of FCC-Structured Multi-Component Equiatomic Solid Solution Alloys. Intermetallics
**2014**, 46, 131–140. [Google Scholar] [CrossRef] - Senkov, O.N.; Wilks, G.B.; Miracle, D.B.; Chuang, C.P.; Liaw, P.K. Refractory High-Entropy Alloys. Intermetallics
**2010**, 18, 1758–1765. [Google Scholar] [CrossRef] - Senkov, O.N.; Scott, J.M.; Senkova, S.V.; Miracle, D.B.; Woodward, C.F. Microstructure and Room Temperature Properties of a High-Entropy TaNbHfZrTi Alloy. J. Alloys Compd.
**2011**, 509, 6043–6048. [Google Scholar] [CrossRef] - Senkov, O.N.; Scott, J.M.; Senkova, S.V.; Meisenkothen, F.; Miracle, D.B.; Woodward, C.F. Microstructure and Elevated Temperature Properties of a Refractory TaNbHfZrTi Alloy. J. Mater. Sci.
**2012**, 47, 4062–4074. [Google Scholar] [CrossRef] - Senkov, O.N.; Woodward, C.F. Microstructure and properties of a refractory NbCrMo
_{0.5}Ta_{0.5}TiZr alloy. Mater. Sci. Eng. A**2011**, 529, 311–320. [Google Scholar] [CrossRef] - Senkov, O.N.; Senkova, S.V.; Woodward, C.; Miracle, D.B. Low-Density, Refractory Multi-Principal Element Alloys of the Cr-Nb-Ti-V-Zr System: Microstructure and Phase Analysis. Acta Mater.
**2013**, 61, 1545–1557. [Google Scholar] [CrossRef] - Senkov, O.N.; Senkova, S.V.; Miracle, D.B.; Woodward, C. Mechanical Properties of Low-Density, Refractory Multi-Principal Element Alloys of the Cr-Nb-Ti-V-Zr System. Mater. Sci. Eng. A
**2013**, 565, 51–62. [Google Scholar] [CrossRef] - Zhang, Y.; Yang, X.; Liaw, P.K. Alloy Design and Properties Optimization of High-Entropy Alloys. Jom
**2012**, 64, 830–838. [Google Scholar] [CrossRef] - Gao, M.C.; Alman, D.E. Searching for Next Single-Phase High-Entropy Alloy Compositions. Entropy
**2013**, 15, 4504–4519. [Google Scholar] [CrossRef] - Tian, F.; Wang, D.; Shen, J.; Wang, Y. An Ab Initio Investgation of Ideal Tensile and Shear Strength of TiVNbMo High-Entropy Alloy. Mater. Lett.
**2016**, 166, 271–275. [Google Scholar] [CrossRef] - Fazakas, E.; Zadorozhnyy, V.; Varga, L.K.; Inoue, A.; Louzguine-Luzgin, D.V.; Tian, F.; Vitos, L. Experimental and Theoretical Study of Ti20Zr20Hf 20Nb20X20 (X = v or Cr) Refractory High-Entropy Alloys. Int. J. Refract. Met. Hard Mater.
**2014**, 47, 131–138. [Google Scholar] [CrossRef] - Huhn, W. Thermodynamics from First Principles: Prediction of Phase Diagrams and Materials Properties Using Density Functional Theory. Ph.D. Thesis, Carnegie Mellon University, Pittsburgh, PA, USA, 2014. [Google Scholar]
- Toda-Caraballo, I.; Wróbel, J.S.; Dudarev, S.L.; Nguyen-Manh, D.; Rivera-Díaz-Del-Castillo, P.E.J. Interatomic Spacing Distribution in Multicomponent Alloys. Acta Mater.
**2015**, 97, 156–169. [Google Scholar] [CrossRef] - Gao, M.C.; Zhang, B.; Yang, S.; Guo, S.M. Senary Refractory High-Entropy Alloy HfNbTaTiVZr. Metall. Mater. Trans. A Phys. Metall. Mater. Sci.
**2016**, 47, 3333–3345. [Google Scholar] [CrossRef] - Li, X.; Tian, F.; Schönecker, S.; Zhao, J.; Vitos, L. Ab Initio-Predicted Micro-Mechanical Performance of Refractory High-Entropy Alloys. Sci. Rep.
**2015**, 5, 12334. [Google Scholar] [CrossRef] [PubMed] - Körmann, F.; Sluiter, M.H.F. Interplay between Lattice Distortions, Vibrations and Phase Stability in NbMoTaW High Entropy Alloys. Entropy
**2016**, 18, 403. [Google Scholar] [CrossRef] - Tian, L.Y.; Wang, G.; Harris, J.S.; Irving, D.L.; Zhao, J.; Vitos, L. Alloying Effect on the Elastic Properties of Refractory High-Entropy Alloys. Mater. Des.
**2017**, 114, 243–252. [Google Scholar] [CrossRef] - Feng, B.; Widom, M. Elastic Stability and Lattice Distortion of Refractory High Entropy Alloys. Mater. Chem. Phys.
**2018**, 210, 309–314. [Google Scholar] [CrossRef] - Song, H.; Tian, F.; Hu, Q.M.; Vitos, L.; Wang, Y.; Shen, J.; Chen, N. Local Lattice Distortion in High-Entropy Alloys. Phys. Rev. Mater.
**2017**, 1, 023404. [Google Scholar] [CrossRef] - Ge, H.; Tian, F.; Wang, Y. Elastic and Thermal Properties of Refractory High-Entropy Alloys from First-Principles Calculations. Comput. Mater. Sci.
**2017**, 128, 185–190. [Google Scholar] [CrossRef] - He, J.; Tritt, T.M. Advances in Thermoelectric Materials Research: Looking Back and Moving Forward. Science
**2017**, 357, eaak9997. [Google Scholar] [CrossRef] - Wu, Y.; Chen, Z.; Nan, P.; Chen, L.; Wu, Y.; Chen, Z.; Nan, P.; Xiong, F.; Lin, S.; Zhang, X.; et al. Lattice Strain Advances Thermoelectrics Lattice Strain Advances Thermoelectrics. Joule
**2019**, 3, 1276–1288. [Google Scholar] [CrossRef] - Shafeie, S.; Guo, S.; Hu, Q.; Fahlquist, H.; Erhart, P.; Palmqvist, A. High-Entropy Alloys as High-Temperature Thermoelectric Materials. J. Appl. Phys.
**2015**, 118, 184905. [Google Scholar] [CrossRef] - Fan, Z.; Wang, H.; Wu, Y.; Liu, X.J.; Lu, Z.P. Thermoelectric High-Entropy Alloys with Low Lattice Thermal Conductivity. RSC Adv.
**2016**, 6, 52164–52170. [Google Scholar] [CrossRef] - Lee, J.I.; Oh, H.S.; Park, E.S. Manipulation of σ
_{y}/κ Ratio in Single Phase FCC Solid-Solutions. Appl. Phys. Lett.**2016**, 109, 061906. [Google Scholar] [CrossRef] - Dutta, B.; Ghosh, S. Vibrational Properties of NixPt
_{1−x}Alloys: An Understanding from Ab Initio Calculations. J. Appl. Phys.**2011**, 109, 053714. [Google Scholar] [CrossRef] - Grånäs, O.; Dutta, B.; Ghosh, S.; Sanyal, B. A New First Principles Approach to Calculate Phonon Spectra of Disordered Alloys. J. Phys. Condens. Matter
**2012**, 24, 015402. [Google Scholar] [CrossRef] [PubMed] - Dutta, B.; Bisht, K.; Ghosh, S. Ab Initio Calculation of Phonon Dispersions in Size-Mismatched Disordered Alloys. Phys. Rev. B-Condens. Matter Mater. Phys.
**2010**, 82, 134207. [Google Scholar] [CrossRef] - Alam, A.; Ghosh, S.; Mookerjee, A. Phonons in Disordered Alloys: Comparison between Augmented-Space-Based Approximations for Configuration Averaging to Integration from First Principles. Phys. Rev. B-Condens. Matter Mater. Phys.
**2007**, 75, 134202. [Google Scholar] [CrossRef] - Chouhan, R.K.; Alam, A.; Ghosh, S.; Mookerjee, A. Interplay of Force Constants in the Lattice Dynamics of Disordered Alloys: An Ab Initio Study. Phys. Rev. B-Condens. Matter Mater. Phys.
**2014**, 89, 060201. [Google Scholar] [CrossRef] - Wang, Y.; Zacherl, C.L.; Shang, S.; Chen, L.Q.; Liu, Z.K. Phonon Dispersions in Random Alloys: A Method Based on Special Quasi-Random Structure Force Constants. J. Phys. Condens. Matter
**2011**, 23, 485403. [Google Scholar] [CrossRef] - Tian, F. A Review of Solid-Solution Models of High-Entropy Alloys Based on Ab Initio Calculations. Front. Mater.
**2017**, 4, 36. [Google Scholar] [CrossRef] - Zunger, A.; Wei, S.H.; Ferreira, L.G.; Bernard, J.E. Special Quasirandom Structures. Phys. Rev. Lett.
**1990**, 65, 353. [Google Scholar] [CrossRef] - King, D.J.M.; Burr, P.A.; Obbard, E.G.; Middleburgh, S.C. DFT Study of the Hexagonal High-Entropy Alloy Fission Product System. J. Nucl. Mater.
**2017**, 488, 70–74. [Google Scholar] [CrossRef] - Soven, P. Coherent-Potential Model of Substitutional Disordered Alloys. Phys. Rev.
**1967**, 156, 809–813. [Google Scholar] [CrossRef] - Bellaiche, L.; Vanderbilt, D. Virtual Crystal Approximation Revisited: Application to Dielectric and Piezoelectric Properties of Perovskites. Phys. Rev. B-Condens. Matter Mater. Phys.
**2000**, 61, 7877–7882. [Google Scholar] [CrossRef] - Ching, W.Y.; San, S.; Brechtl, J.; Sakidja, R.; Zhang, M.; Liaw, P.K. Fundamental Electronic Structure and Multiatomic Bonding in 13 Biocompatible High-Entropy Alloys. npj Comput. Mater.
**2020**, 6, 45. [Google Scholar] [CrossRef] - Dharmawardhana, C.C.; Misra, A.; Ching, W.Y. Quantum Mechanical Metric for Internal Cohesion in Cement Crystals. Sci. Rep.
**2014**, 4, 7332. [Google Scholar] [CrossRef] [PubMed] - Kresse, G. Efficient Iterative Schemes for Ab Initio Total-Energy Calculations Using a Plane-Wave Basis Se. Phys. Rev. B
**1996**, 54, 11169. [Google Scholar] [CrossRef] [PubMed] - Kresse, G. Ab Initio Molecular Dynamics for Liquid Metals. Phys. Rev. B
**1993**, 47, 558. [Google Scholar] [CrossRef] - Reuss, A. Berechnung Der Fließgrenze von Mischkristallen Auf Grund Der Plastizitätsbedingung Für Einkristalle. ZAMM-J. Appl. Math. Mech./Zeitschrift für Angewandte Mathematik und Mechanik
**1929**, 9, 49–58. [Google Scholar] [CrossRef] - Hill, R. The Elastic Behaviour of a Crystalline Aggregate. Proc. Phys. Soc.
**1952**, 65, 349. [Google Scholar] [CrossRef] - Bao, L.; Kong, Z.; Qu, D.; Duan, Y. Revealing the Elastic Properties and Anisotropies of Mg2X(X = Si, Ge and Sn) with Different Structures from a First-Principles Calculation. Mater. Today Commun.
**2020**, 24, 101337. [Google Scholar] [CrossRef] - Mo, Y.; Rulis, P.; Ching, W.Y. Electronic Structure and Optical Conductivities of 20 MAX-Phase Compounds. Phys. Rev. B-Condens. Matter Mater. Phys.
**2012**, 86, 165122. [Google Scholar] [CrossRef] - Ching, W.Y.; Yoshiya, M.; Adhikari, P.; Rulis, P.; Ikuhara, Y.; Tanaka, I. First-Principles Study in an Inter-Granular Glassy Film Model of Silicon Nitride. J. Am. Ceram. Soc.
**2018**, 101, 2673–2688. [Google Scholar] [CrossRef] - Mulliken, R.S. Electronic Population Analysis on LCAO-MO Molecular Wave Functions. I. J. Chem. Phys.
**1955**, 23, 1833–1840. [Google Scholar] [CrossRef] - Hasan, S.; Adhikari, P.; Baral, K.; Ching, W.Y. Conspicuous Interatomic Bonding in Chalcogenide Crystals and Implications on Electronic, Optical, and Elastic Properties. AIP Adv.
**2020**, 10, 075216. [Google Scholar] [CrossRef] - Hasan, S.; Baral, K.; Li, N.; Ching, W.Y. Structural and Physical Properties of 99 Complex Bulk Chalcogenides Crystals Using First-Principles Calculations. Sci. Rep.
**2021**, 11, 9921. [Google Scholar] [CrossRef] [PubMed] - Hasan, S.; Rulis, P.; Ching, W. First-Principles Calculations of the Structural, Electronic, Optical, and Mechanical Properties of 21 Pyrophosphate Crystals. Crystals
**2022**, 12, 1139. [Google Scholar] [CrossRef] - Hunca, B.; Dharmawardhana, C.; Sakidja, R.; Ching, W.Y. Ab Initio Calculations of Thermomechanical Properties and Electronic Structure of Vitreloy Zr41.2Ti13.8Cu12.5Ni10Be22.5. Phys. Rev. B
**2016**, 94, 144207. [Google Scholar] [CrossRef] - Poudel, L.; Twarock, R.; Steinmetz, N.F.; Podgornik, R.; Ching, W.Y. Impact of Hydrogen Bonding in the Binding Site between Capsid Protein and MS2 Bacteriophage SsRNA. J. Phys. Chem. B
**2017**, 121, 6321–6330. [Google Scholar] [CrossRef] - Adhikari, P.; Li, N.; Shin, M.; Steinmetz, N.F.; Twarock, R.; Podgornik, R.; Ching, W.Y. Intra- And Intermolecular Atomic-Scale Interactions in the Receptor Binding Domain of SARS-CoV-2 Spike Protein: Implication for ACE2 Receptor Binding. Phys. Chem. Chem. Phys.
**2020**, 22, 18272–18283. [Google Scholar] [CrossRef] [PubMed] - San, S.; Tong, Y.; Bei, H.; Kombaiah, B.; Zhang, Y.; Ching, W.Y. First-Principles Calculation of Lattice Distortions in Four Single Phase High Entropy Alloys with Experimental Validation. Mater. Des.
**2021**, 209, 110071. [Google Scholar] [CrossRef] - Bérardan, D.; Franger, S.; Dragoe, D.; Meena, A.K.; Dragoe, N. Colossal Dielectric Constant in High Entropy Oxides. Phys. Status Solidi-Rapid Res. Lett.
**2016**, 10, 328–333. [Google Scholar] [CrossRef] - Yao, H.; Ouyang, L.; Ching, W.Y. Ab Initio Calculation of Elastic Constants of Ceramic Crystals. J. Am. Ceram. Soc.
**2007**, 90, 3194–3204. [Google Scholar] [CrossRef] - Deng, H. Theoretical Prediction of the Structural, Electronic, Mechanical and Thermodynamic Properties of the Binary α-As2Te3 and β-As2Te3. J. Alloys Compd.
**2015**, 656, 695–701. [Google Scholar] [CrossRef] - Tian, Y.; Xu, B.; Zhao, Z. Microscopic Theory of Hardness and Design of Novel Superhard Crystals. Int. J. Refract. Met. Hard Mater.
**2012**, 33, 93–106. [Google Scholar] [CrossRef] - Pugh, S.F. Relations between the Elastic Moduli and the Plastic Properties of Polycrystalline Pure Metals. Lond. Edinb. Dublin Philos. Mag. J. Sci.
**1954**, 45, 823–843. [Google Scholar] [CrossRef] - Ali, M.A.; Hossain, M.A.; Rayhan, M.A.; Hossain, M.M.; Uddin, M.M.; Roknuzzaman, M.; Ostrikov, K.; Islam, A.K.; Naqib, S.H. First-Principles Study of Elastic, Electronic, Optical and Thermoelectric Properties of Newly Synthesized K2Cu2GeS4 Chalcogenide. J. Alloys Compd.
**2019**, 781, 37–46. [Google Scholar] [CrossRef] - Heciri, D.; Belkhir, H.; Belghit, R.; Bouhafs, B.; Khenata, R.; Ahmed, R.; Bouhemadou, A.; Ouahrani, T.; Wang, X.; Bin Omran, S. Insight into the Structural, Elastic and Electronic Properties of Tetragonal Inter-Alkali Metal Chalcogenides CsNaX (X = S, Se, and Te) from First-Principles Calculations. Mater. Chem. Phys.
**2019**, 221, 125–137. [Google Scholar] [CrossRef] - Mouhat, F.; Coudert, F.X. Necessary and Sufficient Elastic Stability Conditions in Various Crystal Systems. Phys. Rev. B-Condens. Matter Mater. Phys.
**2014**, 90, 224104. [Google Scholar] [CrossRef] - Thakur, V.; Pagare, G. Theoretical Calculations of Elastic, Mechanical and Thermal Properties of REPt3 (RE = Sc, Y and Lu) Intermetallic Compounds Based on DFT. Indian J. Phys.
**2018**, 92, 1503–1513. [Google Scholar] [CrossRef] - Yang, Y.; Wang, W.; Gan, G.-Y.; Shi, X.-F.; Tang, B.-Y. Structural, Mechanical and Electronic Properties of (TaNbHfTiZr)C High Entropy Carbide under Pressure: Ab Initio Investigation. Phys. B Phys. Condens. Matter
**2018**, 550, 163–170. [Google Scholar] [CrossRef] - Yang, A.; Bao, L.; Peng, M.; Duan, Y. Explorations of Elastic Anisotropies and Thermal Properties of the Hexagonal TMSi2 (TM = Cr, Mo, W) Silicides from First-Principles Calculations. Mater. Today Commun.
**2021**, 27, 102474. [Google Scholar] [CrossRef] - Ma, H.; Zhang, X.; Wang, F. First-Principles Study of the Lattice Vibration, Elastic Anisotropy and Thermodynamical Properties of Tantalum Silicide with the Different Crystal Structures. Vacuum
**2021**, 191, 110410. [Google Scholar] [CrossRef] - Senkov, O.N.; Miracle, D.B.; Chaput, K.J.; Couzinie, J.P. Development and Exploration of Refractory High Entropy Alloys—A Review. J. Mater. Res.
**2018**, 33, 3092–3128. [Google Scholar] [CrossRef] - Li, X.Y.; Zhang, H.P.; Lan, S.; Abernathy, D.L.; Otomo, T.; Wang, F.W.; Ren, Y.; Li, M.Z.; Wang, X.L. Observation of High-Frequency Transverse Phonons in Metallic Glasses. Phys. Rev. Lett.
**2020**, 124, 225902. [Google Scholar] [CrossRef] [PubMed] - Shen, Y.; Clarke, D.R.; Fuierer, P.A. Anisotropic Thermal Conductivity of the Aurivillus Phase, Bismuth Titanate (Bi
_{4}Ti_{3}O_{12}): A Natural Nanostructured Superlattice. Appl. Phys. Lett.**2008**, 93, 102907. [Google Scholar] [CrossRef] - Clarke, D.R. Materials Selections Guidelines for Low Thermal Conductivity Thermal Barrier Coatings. Surf. Coatings Technol. [CrossRef]
- Cahill, D.G.; Watson, S.K.; Pohl, R.O. Lower Limit to the Thermal Conductivity of Disordered Crystals. Phys. Rev. B
**1992**, 46, 6131–6140. [Google Scholar] [CrossRef] - Morelli, D.T.; Slack, G.A. High Lattice Thermal Conductivity Solids. In High Thermal Conductivity Materials; Springer: New York, NY, USA, 2006; pp. 37–68. [Google Scholar] [CrossRef]
- Murphy, K.F. Strain Effects on Thermal Conductivity of Nanostructured Silicon by Raman Piezothermography. Doctoral Dissertation, University of Pennsylvania, Philadelphia, PA, USA, 2014. [Google Scholar]
- Bhowmick, S. Effect of Strain on the Thermal Conductivity of Solids. J. Chem. Phys.
**2006**, 125, 164513. [Google Scholar] [CrossRef] - Knura, R.; Parashchuk, T. Origins of Low Lattice Thermal Conductivity of Pb
_{1 − x}Sn_{x}Te Alloys for Thermoelectric Applications. Dalton Trans.**2021**, 50, 4323–4334. [Google Scholar] [CrossRef] - Jia, T.; Chen, G.; Zhang, Y. Lattice Thermal Conductivity Evaluated Using Elastic Properties. Phys. Rev. B
**2017**, 95, 155206. [Google Scholar] [CrossRef] - Brugger, K. Generalized Griineisen Parameters in the Anisotropic Debye Model. Phys. Rev.
**1965**, 137, A1826. [Google Scholar] [CrossRef] - Malhotra, A.; Maldovan, M. Impact of Phonon Surface Scattering on Thermal Energy Distribution of Si and SiGe Nanowires. Sci. Rep.
**2016**, 6, 25818. [Google Scholar] [CrossRef] [PubMed] - Jiang, S.; Shao, L.; Fan, T.W.; Duan, J.M.; Chen, X.T.; Tang, B.Y. Elastic and thermodynamic properties of high entropy carbide (HfTaZrTi)C and (HfTaZrNb)C from ab initio investigation. Ceram. Int.
**2020**, 46, 15104–15112. [Google Scholar] [CrossRef] - Boudiaf, K.; Bouhemadou, A.; Boudrifa, O.; Haddadi, K.; Saoud, F.S.; Khenata, R.; Al-Douri, Y.; Bin-Omran, S.; Ghebouli, M.A. Structural, Elastic, Electronic and Optical Properties of LaOAgS-Type Silver Fluoride Chalcogenides: First-Principles Study. J. Electron. Mater.
**2017**, 46, 4539–4556. [Google Scholar] [CrossRef] - Guo, F.; Zhou, X.; Li, G.; Huang, X.; Xue, L.; Liu, D.; Jiang, D. Structural, mechanical, electronic and thermodynamic properties of cubic TiC compounds under different pressures: A first-principles study. Solid State Commun.
**2020**, 311, 113856. [Google Scholar] [CrossRef] - Anderson, O.L. A simplified method for calculating the Debye temperature from elastic constants. J. Phys. Chem. Solids
**1963**, 24, 909–917. [Google Scholar] [CrossRef] - Wachter, P.; Filzmoser, M.; Rebizant, J. Electronic and elastic properties of the light actinide tellurides. Phys. B Condens. Matter
**2001**, 293, 199–223. [Google Scholar] [CrossRef] - Fine, M.E.; Brown, L.D.; Marcus, H.L. Elastic constants versus melting temperature in metals. Scr. Metall.
**1984**, 18, 951–956. [Google Scholar] [CrossRef] - Benkaddour, K.; Chahed, A.; Amar, A.; Rozale, H.; Lakdja, A.; Benhelal, O.; Sayede, A. First-principles study of structural, elastic, thermodynamic, electronic and magnetic properties for the quaternary Heusler alloys CoRuFeZ (Z = Si, Ge, Sn). J. Alloys Compd.
**2016**, 687, 211–220. [Google Scholar] [CrossRef] - Liu, S.Y.; Zhang, S.; Liu, S.; Li, D.J.; Li, Y.; Wang, S. Phase stability, mechanical properties and melting points of high-entropy quaternary metal carbides from first-principles. J. Eur. Ceram. Soc.
**2021**, 41, 267–6274. [Google Scholar] [CrossRef] - Clarke, D.R.; Levi, C.G. Materials design for the next generation thermal barrier coatings. Annu. Rev. Mater. Res.
**2003**, 33, 383–417. [Google Scholar] [CrossRef] - Cahill, D.G.; Ford, W.K.; Goodson, K.E.; Mahan, G.D.; Majumdar, A.; Maris, H.J.; Merlin, R.; Phillpot, S.R. Nanoscale thermal transport. J. Appl. Phys.
**2003**, 93, 793–818. [Google Scholar] [CrossRef] - Morelli, D.T.; Heremans, J.P. Thermal conductivity of germanium, silicon, and carbon nitrides. Appl. Phys. Lett.
**2002**, 81, 5126–5128. [Google Scholar] [CrossRef] - Toberer, E.S.; Zevalkink, A.; Snyder, G.J. Phonon engineering through crystal chemistry. J. Mater. Chem.
**2011**, 21, 15843–15852. [Google Scholar] [CrossRef] - Arab, F.; Sahraoui, F.A.; Haddadi, K.; Bouhemadou, A.; Louail, L. Phase stability, mechanical and thermodynamic properties of orthorhombic and trigonal MgSiN 2: An ab initio study. Phase Transitions
**2016**, 89, 480–513. [Google Scholar] [CrossRef] - Naher, M.I.; Afzal, M.A.; Naqib, S.H. A comprehensive DFT based insights into the physical properties of tetragonal superconducting Mo5PB2. Results Phys.
**2021**, 28, 104612. [Google Scholar] [CrossRef]

**Figure 3.**(

**a**) Calculated Q_av^* versus bulk modulus K, (

**b**) Q_av^* versus total electronic energy for the 12 BCC HEAs investigated. The dashed lines denote linear fit.

**Figure 6.**Lattice distortion (LD) for the 12 BCC HEA models. FWHM of the Gaussian curve fitted to the histogram distribution of the bimodal peaks. The two peaks denote the NN and SNN.

**Figure 7.**(

**a**) The distribution of Young’s modulus (E), bulk modulus (K), and shear modulus (G) for the 12 BCC HEAs. (

**b**) Illustrates the relationship between total bond order density (TBOD) and the bulk modulus (K) for the same set of 12 BCC HEAs.

**Figure 8.**(

**a**) The theoretical density (ρ), (

**b**) longitudinal ν

_{l}, transverse ν

_{s}, and average sound velocities ν

_{m}, (

**c**) Debye temperature (Θ

_{D}), (

**d**) thermal conductivities (κ) at 300 K for each model of the 12 BCC HEAs investigated.

**Table 1.**The optimized structure parameters along with first and second nearest neighbors of the 12 BCC HEA models.

Models | a(Å) | b(Å) | c(Å) | $\mathit{\alpha}$ | $\mathit{\beta}$ | $\mathit{\gamma}$ | Vol (Å^{3}) | NN (Å) | SNN (Å) | |
---|---|---|---|---|---|---|---|---|---|---|

M1 | WTiVZrHf ^{1} | 19.871 | 19.915 | 19.794 | 90.327 | 90.329 | 89.885 | 7832.808 | 2.867 | 3.310 |

M2 | HfMoTiWZr ^{1} | 19.91 | 19.987 | 19.889 | 90.174 | 90.232 | 89.945 | 7914.625 | 2.876 | 3.321 |

M3 | MoTaTiV ^{2} | 19.015 | 19.015 | 18.988 | 89.928 | 90.14 | 90.056 | 6865.245 | 2.743 | 3.168 |

M4 | MoTaTiVZr ^{1} | 19.589 | 19.507 | 19.487 | 90.164 | 90.355 | 89.943 | 7446.129 | 2.819 | 3.255 |

M5 | WTaCrV ^{2} | 18.53 | 18.542 | 18.543 | 89.74 | 89.823 | 89.903 | 6370.888 | 2.676 | 3.09 |

M6 | MoNbTaW ^{2} | 19.387 | 19.388 | 19.388 | 89.997 | 90.065 | 89.901 | 7287.275 | 2.742 | 3.878 |

M7 | TiZrHfW ^{2} | 20.252 | 20.338 | 20.269 | 90.129 | 90.067 | 90.175 | 8348.226 | 2.928 | 3.381 |

M8 | TiZrNbMoTa ^{1} | 19.844 | 19.857 | 19.875 | 89.903 | 89.918 | 90.103 | 7831.877 | 2.866 | 3.31 |

M9 | NbTaTiV ^{2} | 19.288 | 19.325 | 19.290 | 90.060 | 90.058 | 90.16 | 7189.958 | 2.786 | 3.217 |

M10 | TiVNbHf ^{2} | 19.711 | 19.72 | 19.628 | 89.843 | 89.583 | 90.268 | 7628.582 | 2.841 | 3.281 |

M11 | Ti_{0.38}V_{0.15}Nb_{0.23}Hf_{0.24} ^{2} | 19.821 | 19.788 | 19.792 | 90.06 | 89.941 | 90.29 | 7762.676 | 2.858 | 3.300 |

M12 | TiZrHfVNb ^{1} | 20.136 | 20.113 | 20.014 | 89.94 | 90.011 | 89.869 | 8105.873 | 2.899 | 3.348 |

^{1}5-component model;

^{2}4-component model. NN and SNN stand for average distances of nearest neighbors (NN) and second nearest neighbors (SNN).

**Table 2.**List of partial charge (PC) and effective charge (Q*) for each atom in the 12 BCC HEA models.

Models | M1 | M2 | M3 | M4 | M5 | M6 | M7 | M8 | M9 | M10 | M11 | M12 | VEC | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Ti | PC | −0.26 | −0.16 | 0.07 | −0.03 | -- | -- | −0.32 | −0.09 | −0.12 | −0.26 | −0.24 | −0.31 | 4 |

Q* | 4.26 | 4.16 | 3.93 | 4.03 | -- | -- | 4.32 | 4.09 | 4.12 | 4.26 | 4.24 | 4.31 | ||

V | PC | −0.24 | -- | 0.01 | −0.06 | 0.08 | -- | -- | -- | −0.12 | −0.25 | −0.23 | −0.28 | 5 |

Q* | 5.24 | -- | 4.99 | 5.06 | 4.92 | -- | -- | -- | 5.12 | 5.25 | 5.23 | 5.28 | ||

Cr | PC | -- | -- | -- | -- | −0.71 | -- | -- | -- | -- | -- | -- | -- | 6 |

Q* | -- | -- | -- | -- | 6.71 | -- | -- | -- | -- | -- | -- | -- | ||

Zr | PC | 0.24 | 0.31 | 0.38 | -- | -- | 0.19 | 0.32 | -- | -- | -- | 0.18 | 4 | |

Q* | 3.76 | 3.69 | 3.62 | -- | -- | 3.81 | 3.68 | -- | -- | -- | 3.82 | |||

Nb | PC | -- | -- | -- | -- | -- | 0.32 | -- | 0.21 | 0.22 | 0.09 | 0.09 | 0.02 | 5 |

Q* | -- | -- | -- | -- | -- | 4.68 | -- | 4.79 | 4.78 | 4.91 | 4.91 | 4.98 | ||

Mo | PC | -- | −0.55 | −0.31 | −0.41 | −0.43 | -- | −0.46 | -- | -- | -- | -- | 6 | |

Q* | -- | 6.55 | 6.31 | 6.41 | 6.43 | -- | 6.46 | -- | -- | -- | -- | |||

Hf | PC | 0.45 | 0.53 | -- | -- | -- | -- | 0.41 | -- | -- | 0.41 | 0.44 | 0.39 | 4 |

Q* | 3.55 | 3.47 | -- | -- | -- | -- | 3.59 | -- | -- | 3.59 | 3.56 | 3.61 | ||

Ta | PC | -- | -- | 0.23 | 0.12 | 0.35 | 0.08 | -- | 0.02 | 0.03 | -- | -- | -- | 5 |

Q* | -- | -- | 4.77 | 4.88 | 4.65 | 4.92 | -- | 4.98 | 4.97 | -- | -- | -- | ||

W | PC | −0.2 | −0.12 | -- | -- | 0.28 | 0.02 | −0.28 | -- | -- | -- | -- | -- | 6 |

Q* | 6.2 | 6.12 | -- | -- | 5.72 | 5.98 | 6.28 | -- | -- | -- | -- | -- | ||

${\mathit{Q}}_{\mathit{a}\mathit{v}}^{\mathit{*}}$ or $\mathit{V}\mathit{E}{\mathit{C}}_{\mathit{a}\mathit{v}}$ | 4.6 | 4.8 | 5.0 | 4.8 | 5.5 | 5.5 | 4.5 | 4.8 | 4.75 | 4.5 | 4.5 | 4.4 |

**Table 3.**The calculated elastic coefficients (C11, C12, C44), Young’s modulus (E), bulk modulus (K), shear modulus (G), Vicker’s hardness (HV), Poisson’s ratio (η), Pugh’s ratio (G/K), Cauchy pressure (CP), total bond order density (TBOD), and Zener ratio (AZ) for the 12 BCC HEAs.

Model | C_{11} | C_{12} | C_{44} | K | G | E | Hv | CP: C_{12}–C_{44} | η | G/K | TBOD | A_{Z} |
---|---|---|---|---|---|---|---|---|---|---|---|---|

M1 | 170.71 | 107.04 | 37.11 | 128.24 | 34.86 | 95.88 | 2.59 | 69.93 | 0.375 | 0.272 | 0.041 | 1.166 |

M2 | 201.82 | 114.19 | 38.84 | 143.37 | 40.73 | 111.63 | 3.04 | 75.35 | 0.370 | 0.284 | 0.042 | 0.887 |

M3 | 258.38 | 139.11 | 31.43 | 178.46 | 40.72 | 113.51 | 2.37 | 107.67 | 0.394 | 0.228 | 0.055 | 0.527 |

M4 | 201.89 | 116.36 | 32.02 | 144.84 | 35.92 | 99.53 | 2.38 | 84.34 | 0.386 | 0.248 | 0.047 | 0.749 |

M5 | 376.99 | 159.09 | 48.63 | 230.47 | 67.53 | 184.57 | 4.50 | 110.47 | 0.367 | 0.293 | 0.057 | 0.446 |

M6 | 389.28 | 159.94 | 49.61 | 234.94 | 69.88 | 190.73 | 4.69 | 110.33 | 0.365 | 0.297 | 0.051 | 0.433 |

M7 | 164.35 | 102.72 | 44.12 | 123.22 | 38.20 | 103.88 | 3.20 | 58.60 | 0.360 | 0.310 | 0.038 | 1.432 |

M8 | 201.18 | 118.75 | 33.52 | 146.21 | 36.38 | 100.77 | 2.41 | 85.23 | 0.385 | 0.249 | 0.042 | 0.813 |

M9 | 200.66 | 136.46 | 24.30 | 157.83 | 27.15 | 77.03 | 1.29 | 112.16 | 0.419 | 0.172 | 0.048 | 0.757 |

M10 | 165.92 | 110.54 | 28.56 | 128.99 | 28.17 | 78.78 | 1.73 | 81.98 | 0.398 | 0.218 | 0.040 | 1.031 |

M11 | 159.86 | 105.79 | 32.62 | 123.81 | 30.24 | 83.89 | 2.07 | 73.17 | 0.387 | 0.244 | 0.040 | 1.207 |

M12 | 151.23 | 99.48 | 30.62 | 116.72 | 28.61 | 79.34 | 2.00 | 68.86 | 0.387 | 0.245 | 0.036 | 1.183 |

**Table 4.**The theoretical density (ρ), calculated sound velocity (longitudinal ν

_{l}, transverse ν

_{s}, and average ν

_{m}), Debye temperature (Θ

_{D}), and melting temperature (T

_{melt}) of 12 BCC HEAs.

Model | ρ (Kg/m^{3}) | ν_{l} (m/s) | ν_{s} (m/s) | ν_{m} (m/s) | Θ_{D} (K) | T_{melt} (K) |
---|---|---|---|---|---|---|

M1 | 11,709.54 | 3862.68 | 1725.29 | 1946.48 | 231.60 | 1561.88 |

M2 | 12,533.18 | 3971.51 | 1802.80 | 2032.49 | 241.00 | 1745.76 |

M3 | 11,358.89 | 4526.66 | 1893.26 | 2141.44 | 266.25 | 2080.01 |

M4 | 10,412.46 | 4302.30 | 1857.34 | 2098.35 | 253.92 | 1746.15 |

M5 | 15,238.36 | 4586.22 | 2105.19 | 2372.21 | 302.38 | 2781.03 |

M6 | 15,769.33 | 4561.45 | 2105.10 | 2371.51 | 289.05 | 2853.66 |

M7 | 12,466.75 | 3737.64 | 1750.56 | 1970.71 | 229.56 | 1524.32 |

M8 | 10,790.15 | 4247.93 | 1836.06 | 2074.22 | 246.82 | 1741.94 |

M9 | 10,758.12 | 4246.85 | 1588.55 | 1802.85 | 220.72 | 1738.88 |

M10 | 10,071.86 | 4066.47 | 1672.39 | 1892.72 | 227.19 | 1533.59 |

M11 | 9629.90 | 4128.41 | 1772.07 | 2002.46 | 238.98 | 1497.77 |

M12 | 9453.01 | 4047.57 | 1739.70 | 1965.78 | 231.23 | 1446.77 |

**Table 5.**Calculated minimum thermal conductivities (κ

_{min}) (W⋅m

^{−1}·K

^{−1}) at 300 K, lattice thermal conductivities (κ

_{L}) (W⋅m

^{−1}·K

^{−1}) at 300 K, and Grüneisen parameter (γ

_{a}), thermal expansion coefficient (α), and dominant phonon wavelength (λ

_{dom}) at 300 K for each model of the 12 BCC HEAs investigated.

Model | Clarke Model κ _{min} (W·m^{−1}·K^{−1}) | Cahill Model κ _{min} (W·m^{−1}·K^{−1}) | Slack Model κ _{L} (W·m^{−1}·K^{−1}) | Mixed Model κ _{L} (W·m^{−1}·K^{−1}) | γ_{α} | α (×10^{−5}) | λ_{dom} (Å) |
---|---|---|---|---|---|---|---|

M1 | 0.54897 | 0.49659 | 0.69331 | 0.73710 | 2.361 | 4.59 | 0.815 |

M2 | 0.56862 | 0.50962 | 0.88163 | 0.91413 | 2.311 | 3.93 | 0.851 |

M3 | 0.66220 | 0.62294 | 0.76401 | 0.79356 | 2.556 | 3.93 | 0.897 |

M4 | 0.61351 | 0.56618 | 0.70941 | 0.75364 | 2.464 | 4.45 | 0.879 |

M5 | 0.76628 | 0.68246 | 1.75668 | 1.63648 | 2.276 | 2.37 | 0.994 |

M6 | 0.70012 | 0.62166 | 1.84346 | 1.76763 | 2.260 | 2.29 | 0.993 |

M7 | 0.53075 | 0.46744 | 0.87195 | 0.91217 | 2.213 | 4.19 | 0.825 |

M8 | 0.58636 | 0.54074 | 0.71212 | 0.76133 | 2.460 | 4.40 | 0.869 |

M9 | 0.54351 | 0.54905 | 0.34487 | 0.41132 | 2.860 | 5.89 | 0.755 |

M10 | 0.54606 | 0.51904 | 0.45064 | 0.51310 | 2.604 | 5.68 | 0.793 |

M11 | 0.56966 | 0.52752 | 0.56240 | 0.62115 | 2.481 | 5.29 | 0.839 |

M12 | 0.54328 | 0.50271 | 0.52396 | 0.58808 | 2.477 | 5.59 | 0.823 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

San, S.; Hasan, S.; Adhikari, P.; Ching, W.-Y.
Designing Quaternary and Quinary Refractory-Based High-Entropy Alloys: Statistical Analysis of Their Lattice Distortion, Mechanical, and Thermal Properties. *Metals* **2023**, *13*, 1953.
https://doi.org/10.3390/met13121953

**AMA Style**

San S, Hasan S, Adhikari P, Ching W-Y.
Designing Quaternary and Quinary Refractory-Based High-Entropy Alloys: Statistical Analysis of Their Lattice Distortion, Mechanical, and Thermal Properties. *Metals*. 2023; 13(12):1953.
https://doi.org/10.3390/met13121953

**Chicago/Turabian Style**

San, Saro, Sahib Hasan, Puja Adhikari, and Wai-Yim Ching.
2023. "Designing Quaternary and Quinary Refractory-Based High-Entropy Alloys: Statistical Analysis of Their Lattice Distortion, Mechanical, and Thermal Properties" *Metals* 13, no. 12: 1953.
https://doi.org/10.3390/met13121953