First-principles prediction of yield strength and hardness in body-centered cubic refractory-based medium, high, and super high entropy alloys

Abstract

Refractory-based high-entropy alloys (RHEAs) are compelling materials for super-high-temperature (1500–2000 K) applications, such as jet-engines, gas turbines, and nuclear powerplants. However, only a small portion of their vast compositional design space has been explored, while many potentially interesting RHEAs remain to be discovered. In this work, a large region of 9-refractory elements: Ti, V, Cr, Zr, Nb, Mo, Hf, Ta, and W design space is computationally explored to identify 18 RHEAs with simultaneously high yield strength and body-centered cubic (BCC) solid solution stability. We provide a comprehensive analysis of the impact of elemental variations on yield strength, lattice distortion, and thermodynamic properties within 18 RHEAs. We show that these specific elemental variations in composition lead to significant changes in the calculated properties, such as a significant large yield strength of 2693.618 MPa for the alloy Ti₁₁₄V₁₁₄Cr₁₇₁Zr₅₇Nb₁₁₄Mo₁₇₁Hf₅₇Ta₁₁₃W₁₁₃, thereby guiding the experimental efforts. This alloy with significantly high yield strength is revealed for the first time in this work. Our findings reveal significant insights into how these elemental variations in composition contribute to the unique high yield strength of the 18 RHEAs at room and high temperatures, potentially opening new avenues for the design of these materials across various industrial sectors.

Refractory-based high-entropy alloys (RHEAs) are compelling materials for super-high-temperature (1500–2000 K) applications, such as jet-engines, gas turbines, and nuclear powerplants.

1. Introduction

High-Entropy Alloys (HEAs) are a fascinating class of materials that have attracted the attention of the materials science community due to their unique properties, such as exceptional strength,¹ ductility,² corrosion resistance,³ high-temperature stability,^4,5 fatigue resistance,^6,7 high wear resistance,⁸ dielectric,⁹ and superconducting properties.¹⁰ Multi-principal elements alloys with equimolar or near-equimolar atomic ratios are the original definition of the HEAs concept.¹¹ Senkov et al.^12,13 extended HEAs to include refractory-based high entropy alloys (RHEAs) which are based on refractory elements with high melting points: titanium (Ti), vanadium (V), chromium (Cr), zirconium (Zr), niobium (Nb), molybdenum (Mo), hafnium (Hf), tantalum (Ta), and tungsten (W). RHEAs possess a body-centered cubic (BCC) crystal structure and have a large number of promising applications in the aero-space,^14,15 nuclear,^16,17 and bio-medical industries as potential candidates for biomedical implants.^18,19 RHEAs exhibit an excellent combination of strength and ductility under both compressive and tensile loading at ambient temperatures²⁰ along with outstanding mechanical properties arising from their remarkable solid solution strengthening and thermal stability at high temperatures.^21–23 The slow diffusion of refractory elements and the remarkable solid solution strengthening result in the strength of RHEAs at elevated temperatures.^11,24,25 Due to their very high melting points, RHEAs are in high demand for ultra-high temperature (∼1800 °C) applications such as hypersonic vehicles,²⁶ aero-engines,²⁷ and gas turbines, where traditional nickel-based super alloys fail.^21,28–30 RHEAs can form single-phase solid solutions with very high solid-solution strengthening.³¹ Single-phase RHEAs tend to be more ductile than multi-phase RHEAs with phase stability even at high temperatures.³²

Regular gas turbines are made from Ni-based super-alloys with melting points near 1350 °C. The regular gas turbines are defined as internal combustion engines that convert the chemical energy from different types of fuels into a mechanical power using a continuous flow of air and gases, operating based on the Brayton cycle.³³ These gas turbines operate at 1100 °C, and they require thermal-barrier coatings and internal cooling to sustain the working gas temperature as hot as 1600 °C.² Modern gas turbines have begun using hydrogen, which burns much hotter than the natural gases.³ Thus, these modern turbines must be made from the next-generation materials capable of withstanding temperatures above 1600 °C, while maintaining phase stability and high yield strength at such elevated temperatures. These materials allow the use of hotter gas and reduce the need for internal cooling, thereby, improving fuel efficiency by up to 7% and lowering environmentally-harmful emissions. The yield strength of typical Ni-based super-alloys, such as Inconel 718 and Haynes 230, drops to near zero at 1200 °C and 110 MPa at 1000 °C for these two alloys respectively.^24,34 RHEAs are considered the next-generation materials with high melting points and high yield strengths, achieved via solid-solution strengthening well above 1000 °C. For example, HfMoNbTaTi RHEA is a promising alloy that maintains its yield strength maintained up to 699 MPa at 1200 °C, which decreases to 367 MPa at 1400 °C.³⁵

Extensive work has been done to improve the yield strength and mechanical properties for a large number of RHEAs. The first two RHEAs: MoNbTaW and MoNbTaVW were synthesized and reported for the first time in 2010 by Senkov et al.¹² with yield strengths (Δσ_y) of 1058 MPa and 1246 MPa at room temperature for these two alloys respectively, but Δσ_y decreases to 405 MPa and 477 MPa respectively at 1600 °C. HfNbTaTiZr with a BCC structure has shown excellent tensile ductility and Δσ_y of 929 MPa at room temperature, but a poor Δσ_y of 295 MPa at elevated temperature (∼1000 °C).³⁶ Juan et al.³⁷ reported a large Δσ_y of 1600 MPa at room temperature and 855 MPa at higher temperature (∼1000 °C) for HfMoTaTiZr. Other RHEAs such as TiZrNbTaMo, AlNbTiVZr_x, and MoNbTaTiW were studied for their excellent mechanical properties.^38–40 The microstructure and mechanical properties of TiZrHfNbV and TiZrHfNbCr RHEAs were investigated experimentally.⁴¹ These two RHEAs have Δσ_y of 1170 MPa and 1375 MPa at room temperature, respectively. Another study by Wang et al.⁴² found that VNbMoTa exhibits a large Δσ_y of 811 MPa at high temperature (∼1000 °C). An experimental study⁴³ found that the fine-grained NbMoTaWVCr alloy sintered at 1500 °C possesses a very high Δσ_y of 3416 MPa. Another experimental study performed by Yang et al.⁴⁴ reported a high Δσ_y of 851 MPa at 1200 °C due to the high precipitation strengthening, and a high Δσ_y of 1713 MPa at room temperature due to the combined effect of solid-solution strengthening and precipitation strengthening for HfMoNbTaTi alloy. A high Δσ_y of 775 MPa at 1200 °C for CrMoNbTiW RHEA was reported by Raman et al.⁴⁵ The Δσ_y of TiNbMoTaW and Ti₁(NbMoTa)₂W_0.5 alloys were calculated to be about 1300 MPa and 1100 MPa at room temperature, respectively.⁴⁶ An experimental study showed that NbMoTaWVTi has a compressive Δσ_y of 2709 MPa at room temperature.⁴⁷ The TiHfNbTa alloy demonstrated a high Δσ_y of 1280 MPa.⁴⁸ Another study showed that TiZrNbTaMo exhibits Δσ_y of 1390 MPa.³⁸ Matsuzaka et al.⁴⁹ recently calculated the Δσ_y of the super-high entropy alloy TiNbTaZrMoHfWCr and RHEA TiNbTaZrMo to be 2292 MPa and 1604 MPa respectively. Bai et al.⁵⁰ recently found that the Δσ_y of HfMoTaTiZrCr_0.5 reaches about 1800 MPa at room temperature. The Δσ_y of CrMoNbV RHEA has been reported recently to be 629 MPa at 1250 °C.⁵¹ A nitride-reinforced NbMoTaWHfN RHEA was reported with Δσ_y of 792 MPa at 1400 °C.⁵² An experimental study performed by Senkov et al.⁵³ showed that HfNbTaTiZr RHEA maintains high Δσ_y of 1114 MPa at 1100 °C. Recently, an experimental study⁵⁴ reported that Ti₃Zr₁.₅NbVAl₀.₂₅ RHEA has Δσ_y of 876 MPa at 1200 °C. Another experimental study done by Lee et al.² showed that NbTaTiV RHEA exhibits Δσ_y of 1273 MPa at room temperature and 688 MPa at 900 °C. Xu et al.⁵⁵ showed that the room temperature Δσ_y of HfMoTaTiZrCr and HfMoTaTiZr RHEAs reaches 2003 MPa and 1543.1 MPa, respectively. Yurchenko et al.⁵⁶ have reported recently a large Δσ_y of 2430 MPa and 1110 MPa at room temperature and 1200 °C respectively for MoNbTaTiSi. Large number of RHEAs was reported with high Δσ_y at room temperature, but most of them cannot retain this high Δσ_y at temperatures higher than 1500 K. To provide a clear overview of RHEAs yield strength and operating temperature, we have included Table S1 in the SI. This table shows that there are less RHEAs that have high yield strength at temperature above 1500 °C. This table also contains large number of recent theoretical calculations of yield strength for large number of RHEAs.

Recently, computational materials science has begun to play a crucial role in RHEAs development, including CALPHAD-based high-throughput computational methods,^57,58 machine learning assisted design coupled with modeling/experimental data,^59–61 and density functional theory (DFT).⁶² Recently, RHEAs composition field has expanded significantly, thus it demands an extensive computational work, which can be performed using the numerically efficient DFT methods.⁶³ Large number of computational studies has been performed on RHEAs (as displayed in Table S1). The massive calculations in these computational studies aimed to find and design stable RHEAs with improved mechanical properties and higher yield strength at high temperatures. Understanding the mechanisms behind the enhancement of yield strength is the key driver to improve the mechanical properties of RHEAs, especially at high temperatures. That is why we focus on yield strength calculations in our current study. On the other hand, ab initio DFT calculation is an effective tool that has been widely used in studying HEAs to understand and enrich their systems. We use DFT theoretical modeling in our current study to enable in-depth analysis of the electronic structures and interatomic bonding of the current designed RHEAs.

In this work, we aim to fill the gap and find RHEAs with large yield strength at room and higher temperatures. We use DFT and focus extensively on the effect of composition on yield strength calculations for 18 BCC RHEAs, listed in Table S2 providing additional information on their lattice parameters and components.^64,65 These 18 RHEAs are composed of nine refractory elements: Ti, V, Cr, Zr, Nb, Mo, Hf, Ta, and W. The diverse atomic sizes, atomic masses, elastic properties, valencies, and electron configurations of these nine elements contribute significantly to the microstructure and yield strength of the resulting alloys. We explore the complex balance between lattice distortion, mechanical, and electronic structure contributions to the yield strength of 18 RHEAs by manipulating the ratios of the nine refractory elements. First, we study the phase formation rules of these 18 alloys by calculating several crucial affecting parameters such as entropy of mixing (ΔS_mix), enthalpy of mixing (ΔH_mix), change in Gibbs free energy (ΔG_mix), valence electron concentration (VEC), the atomic size difference (δ), and electronegativity difference (Δχ). Second, the yield strength and hardness of these complex alloys are calculated to provide insights into the role of atomic sizes, atomic masses, and elastic behaviour in determining overall yield strength. The correlation between the yield strength and total bond order density (TBOD), a measure of the bond strength, is revealed. TBOD is a single quantum mechanical metric to describe the internal cohesion of the RHEA. Third, the temperature-dependent yield strength is calculated and discussed in detail. By systematically exploring these aspects, the current study aims to significantly advance the fundamental understanding of RHEAs, thereby guiding the development of new materials with tailored properties for advanced applications in aerospace, automotive, and energy sectors, among others.

2. Computational methods

2.1. Optimization and bonding properties

These 18 RHEAs were modeled initially from a simple Nb BCC unit cell with space group 229. (In materials science, a space group is the symmetry group of a repeating pattern in space, usually in three dimensions. In crystallography, the space group is also called the crystallographic group, and represents a description of the symmetry of the crystal⁶⁶). 5 × 5 × 5 super-cells of BCC unit cell were constructed for the four and five component alloys (labeled as Mx, where x = 1 to 12 and listed in Tables S2 and 1), resulting in random solid solution RHEAs with a total of 500 atoms each. Whereas for the nine and seven component alloys (labeled as M_x, where x = 13 to 18 and listed in Tables S2 and 1), a larger super-cell was needed. Thus a super-cell of 8 × 8 × 8 was created for each model from M13 to M18 to obtain six RHEAs random solid solutions containing 1024 atoms each. While super-cell approaches involving 500 to 1024 atoms are generally considered computational demanding for DFT methods, this scale proved efficient for our specific framework. Each model underwent random alloying with refractory elements (Ti, V, Cr, Zr, Nb, Mo, Hf, Ta, and W), using random ratios of compositions including equiatomic and non-equiatomic. The specific values of the composition (the atomic fractions) were chosen randomly in these 18 RHEAs, considering changing (reducing or increasing) the composition of the elements with the smallest atomic radius (Cr, Mo, V), the elements with the largest atomic radius (Zr and Hf), and the lightest element (Ti). In addition, Mo and Cr have BCC structure and the highest elemental VEC (VEC = 6), whereas Ti, Zr, and Hf have hexagonal close packed (HCP) structure and the lowest elemental VEC (VEC = 4). Thus, the elemental variation in composition of these six elements was chosen to change (reducing or increasing) to produce different alloys with different lattice distortion, yield strength, and mechanical properties.

Table 1. Enthalpy of mixing (ΔH_mix), entropy of mixing (ΔS_mix), valence electron concentration (VEC), the parameter Ω, atomic size difference (δ), the combination parameter (Λ), Pauling electronegativity difference (Δχ_Pauling), and Allen electronegativity difference (Δχ_Allen) in the 18 RHEAs systems (note: the data for the models from M13 to M18 are taken from our previous published work⁶⁴).

Model	Composition	ΔHmix (kJ mol−1)	ΔSmix (J K−1 mol−1)	VEC	Ω	δ%	Λ (J K−1 mol−1)	ΔχPauling	ΔχAllen%
M1	W100Ti100V100Zr100Hf100	−4.80	13.382	4.60	6.942	7.686	0.227	0.385	9.350
M2	Hf100Mo100Ti100W100Zr100	−5.60	13.382	4.80	6.291	6.867	0.284	0.439	8.461
M3	Mo125Ta125Ti125V125	−2.75	11.526	5.00	10.803	4.080	0.692	0.266	5.210
M4	Mo100Ta100Ti100V100Zr100	−2.88	13.382	4.80	11.556	6.843	0.286	0.281	5.671
M5	W125Ta125Cr125V125	−4.25	11.526	6.30	7.694	4.958	0.469	0.336	7.457
M6	Mo125Nb125Ta125W125	−6.50	11.526	5.50	5.600	2.318	2.145	0.564	3.765
M7	Ti125Zr125Hf125W125	−5.25	11.526	4.50	5.635	6.286	0.292	0.430	8.480
M8	Ti100Zr100Nb100Mo100Ta100	−1.76	13.382	4.80	21.749	5.457	0.449	0.426	3.840
M9	Nb125Ta125Ti125V125	−0.25	11.526	4.75	117.150	3.927	0.747	0.394	5.010
M10	Ti125V125Nb125Hf125	0.25	11.526	4.50	108.114	6.443	0.278	0.130	9.752
M11	Ti190V75Nb115Hf120	0.70	11.081	4.38	36.393	5.584	0.355	0.122	8.922
M12	Ti100Zr100Hf100V100Nb100	0.16	13.382	4.40	96.225	7.055	0.269	0.138	8.948
M13	Ti114V114Cr114Zr114Nb114Mo114Hf114Ta114W114	−4.79	18.267	5.00	9.9851	7.638	0.3131	0.3368	9.2908
M14	Ti114V114Cr114Zr114Nb114Mo114Hf114Ta114W114	−4.79	18.267	5.00	9.9851	7.638	0.3131	0.3368	9.2908
M15	Ti114V114Cr138Zr90Nb114Mo138Hf90Ta113W113	−5.06	18.199	5.10	9.4638	7.056	0.3655	0.3349	9.2505
M16	Ti164V65Cr118Zr118Nb99Mo118Hf104Ta119W119	−5.00	18.080	4.97	9.4442	7.485	0.3227	0.3416	9.0476
M17	Ti114V114Cr171Zr57Nb114Mo171Hf57Ta113W113	−5.19	17.801	5.23	9.0787	7.341	0.3303	0.3283	9.1897
M18	Ti114V114Cr228Nb114Mo228Ta113W113	−4.80	15.712	5.45	8.7423	7.121	0.3098	0.3053	9.1287

A Python script was used to determine the lattice parameters, accounting for the inclusion of the nine refractory metals within their specific cells. This Python script is available in the SI. For TiVCrZrNbMoHfTaW, two configurations (M13 and M14) were constructed, while one structure was constructed for each of the remaining models (compositions). These 18 distinct RHEA models (listed in Table S2) were subsequently optimized using the Vienna Ab initio Simulation Package (VASP).⁶⁷ In VASP's optimizations, the Projector Augmented Wave (PAW) of Perdew, Burke, and Ernzerhof (PBE)⁶⁸ method was used within the Generalized Gradient Approximation (GGA) to treat the exchange–correlation potential. The kinetic energy cutoff of the plane wave expansion was set to be 500 eV. The convergence thresholds of the electronic and ionic relaxations were set to be 1.0 × 10⁻⁶ eV and −1.0 × 10⁻⁴ eV Å⁻¹, respectively. The energy cutoff and the convergence thresholds of the electronic and ionic relaxations were tested very well in our previous published works.^69–72 The spin–orbit coupling (SOC) effect was not included in our DFT calculations. The Monkhorst scheme⁷³ was used with a 1 × 1 × 1 k-point mesh for all these RHEA solid solutions. The optimized structures from VASP were used to calculate the elastic tensors by employing a stress–strain response approach. Note that all elastic properties of these 18 RHEAs were published in our previous works.^64,65

To investigate the electronic structure and bonding properties, the Orthogonalized Linear Combination of Atomic Orbitals (OLCAO)⁷⁴ package was used. The structures optimized by VASP were used as the starting point. The local density approximation (LDA) as the exchange and correlation potential is used in OLCAO with Gaussian-type orbitals (GTO) as the atomic orbitals that are part of the basis set for expanding the Bloch functions. OLCAO was successfully applied in DFT calculations for crystalline,^{71,72,75–79} and non-crystalline materials,^80,81 as well as HEAs.^70,82 A 1 × 1 × 1 Monkhorst k-points mesh was used for density of states (DOS) calculations within OLCAO. To calculate the TBOD, the OLCAO first calculates the bond order (BO) based on Mulliken's analysis.⁸³ The BO, indicative of bond strength, is calculated as the overlap population ρ_α,β between atom pairs (α,β) at a given bond length:

1

The formula for ρ_αβ involves summing over occupied bands (n, occ) and integrating the product of the complex conjugate of the coefficient the coefficient Cⁿ_jβ, and the overlap integral S_iα,jβ across the orbital indices i and j. The total bond order (TBO) yields from the aggregation of all BOs within a model which when normalized by the model's volume, provides the TBOD. TBOD serves as a quantum mechanical measure of the alloy's structural integrity.⁸⁴ The combination of the products of eigenvector coefficients and the overlap integrals over all bands and orbitals gives the effective charge the charge distribution deviation from a neutral state:

2

The average overall effective charge of each RHEA can be obtained from the formula: where c_i is the composition of the components. Lattice distortion (LD) is obtained from the BO vs. bond length (BL) data. Providing a precise description of LD is challenging without a baseline reference for the undistorted lattices of our 18 RHEAs. To determine these values, one must calculate the LD for individual BCC constituent elements; however, these do not qualify as HEAs by definition, as the latter requires a multi-element composition. In this work, LD is obtained by plotting the histogram distribution of BL data of interatomic pairs for all RHEAs models. These figures are published in our previous works.^64,65 The two peaks in these figures basically belong to the first nearest neighbor (NN) and second nearest neighbor (SNN) interatomic pairs. These data were fitted with two Gaussians exhibiting bimodal distributions for the NN and SNN. The full-width at half maximum (FWHM) values obtained from BO values are presented in Table 6.

Table 6. The FWHM of the two Gaussians peaks and their summation and average as a measure for lattice distortion, and the density of states at the Fermi energy N(E_F) in the 18 RHEAs models.^64,65.

Models		FWHM 1	FWHM 2	Sum	Ave	N(EF)
M1	W100Ti100V100Zr100Hf100	0.221	0.697	0.918	0.459	530
M2	Hf100Mo100Ti100W100Zr100	0.273	0.404	0.677	0.339	494
M3	Mo125Ta125Ti125V125	0.167	0.197	0.364	0.182	570
M4	Mo100Ta100Ti100V100Zr100	0.266	0.337	0.603	0.302	545
M5	W125Ta125Cr125V125	0.152	0.169	0.321	0.161	300
M6	Mo125Nb125Ta125W125	0.086	0.092	0.178	0.089	289
M7	Ti125Zr125Hf125W125	0.267	0.704	0.971	0.486	520
M8	Ti100Zr100Nb100Mo100Ta100	0.225	0.275	0.500	0.250	552
M9	Nb125Ta125Ti125V125	0.167	0.241	0.408	0.204	657
M10	Ti125V125Nb125Hf125	0.314	0.437	0.751	0.376	588
M11	Ti190V75Nb115Hf120	0.245	0.683	0.928	0.464	589
M12	Ti100Zr100Hf100V100Nb100	0.229	0.721	0.950	0.475	561
M13	Ti114V114Cr114Zr114Nb114Mo114Hf114Ta114W114	0.258	0.334	0.592	0.296	1235
M14	Ti114V114Cr114Zr114Nb114Mo114Hf114Ta114W114	0.260	0.303	0.563	0.282	1222
M15	Ti114V114Cr138Zr90Nb114Mo138Hf90Ta113W113	0.254	0.283	0.537	0.269	1197
M16	Ti164V65Cr118Zr118Nb99Mo118Hf104Ta119W119	0.252	0.327	0.579	0.289	1140
M17	Ti114V114Cr171Zr57Nb114Mo171Hf57Ta113W113	0.211	0.241	0.452	0.226	1119
M18	Ti114V114Cr228Nb114Mo228Ta113W113	0.151	0.168	0.319	0.159	732

2.2. Super-cells construction

Several random solid solution model (RSSM) approaches are used to simulate the solid solution phases,⁸⁵ such as special quasi-random structure (SQS),⁸⁶ super-cell (SC) method,⁸⁷ coherent potential approximation (CPA),⁸⁸ and virtual crystal approximation (VCA).⁸⁹ HEAs have very high chemical disorder, which requires using large super-cells when using the SC method. In the current study, the cubic super-cells for the 18 RHEAs in the BCC lattice are constructed using SC based on the RSSM. For the models from M1 to M12, the size of the super-cell, or the number of atoms (N) in the BCC super-cell is determined by the simple formula: N = 4 × (n)³ with n equals to 5. So the super-cell has 500 atoms in the cubic cell of length equal to na, where a is the lattice constant of the single atom BCC crystal of a typical transition metal. We believe that a 500-atom super-cell for the four and five component alloys (the alloys from M1 to M12), may be the minimal size to justify the use of the RSSM with sufficient confidence. It should be pointed out that our super-cell is different from those used in the SQS structures in the simulations by other researchers, which are usually much smaller. For the models from M13 to M18, the size of the super-cell, or the number of atoms (N) in the BCC super-cell is determined by the simple formula: N = 2 × (n)³ with n equals to 8, so the super-cell has 1024 atoms in the cubic cell of length equal to na. We believe that a 1024-atom super-cell for the nine and seven component alloys (the alloys from M13 to M18) is the minimal size to justify the use the RSSM with sufficient confidence. To account for the different possible structural configurations, many such so-called “super-cells” have to be used. In the present study, we assert that the statistical distribution of random distribution of refractory metals is sufficient, since the super-cell is sufficiently large and with periodical boundary conditions and can account for the random distribution of the nearest neighbor (NN), second NN and also the third NN and beyond for each atom in the model. The use of large super-cells for many HEAs is efficient, tractable, and conducive for detailed bond analysis and other calculation. Four, five, seven, and nine atomic species of equal and unequal percentage are chosen from the following nine refractory elements: Ti, V, Zr, Nb, Mo, Hf, W, Cr, and Ta, and are distributed randomly in the lattice sites of the super-cell with periodic boundaries. The initial lattice constant for the super-cell is obtained from the scaled average of the crystal lattice constant for each atom. To fully comply with the spirit of RSSM, a python script was used such that the atomic occupation of each site is completely random with no restriction to their NN atoms and beyond (Fig. 1).

Fig. 1. Ball and stick structure of TiVCrZrNbMoHfTaW (M13) solid solution.

3. Results and discussion

3.1. Phase formation stability

The Hume-Rothery rules^90,91 used to predict phase formation in HEAs are based on the enthalpy of mixing, atomic-size differences, electronegativity differences, and valence electron concentration. The entropy of mixing (ΔS_mix), enthalpy of mixing (ΔH_mix), and Gibbs free energy (ΔG_mix)⁹² are the first three principal components that affect the phase formation:

3

4

ΔG_mix = ΔH_mix − TΔS_mix 5

where R is the molar gas constant, [R = 8.314 J (mol⁻¹. K)], n is the number of components of the multi-component material, c_i and c_j are the content or atomic fraction of the ith and jth components (at%) (as listed in Table S3 for the 18 RHEAs). ΔH^AB_mix is the mixing enthalpy of binary liquid A₅₀B₅₀ alloys calculated based on Miedema's model⁹³ and T is the absolute temperature in kelvins. Single-phase solid solutions favor a low magnitude of ΔH_mix (a slightly positive or negative) and a high ΔS_mix. ΔS_mix increases with the increasing number of components, thereby reducing ΔG_mix of the solid solution, and leading to more thermodynamically stable phases. However, secondary phases such as intermetallic compounds can be formed in HEAs even at high ΔS_mix.^94,95 In general, a single-phase HEA requires ΔH_mix and ΔS_mix in the following ranges: −22.0 kJ mol⁻¹ ≤ ΔH_mix ≤ 7.0 kJ mol⁻¹ and 11.0 J (K mol)⁻¹ ≤ ΔS_mix ≤ 19.5 J (K mol)⁻¹, respectively.⁹⁶ Amorphous and intermetallic phases tend to form at more negative ΔH_mix (−49.0 kJ mol⁻¹ ≤ ΔH_mix ≤ −5.5 kJ mol⁻¹).⁹⁶ From Table 1, these 18 RHEAs meet these two criteria for ΔH_mix and ΔS_mix. It is worth to mention that a large negative ΔH^AB_mix value between two elements leads to the formation of stable intermetallic phases. Thereby, the near-zero ΔH_mix (positive or negative) was chosen as a condition of stable solid solution formation. However, small negative (near-zero) ΔH_mix values do not guarantee that intermetallic phases will not form. By quantifying the dominance of entropy over enthalpy, Yang et al. proposed the parameter Ω to predict whether an alloy is a solid solution. Ω is defined as a ratio of the product of ΔS_mix and the melting temperature to the absolute value of ΔH_mix of HEAs:⁹⁷

6

where T_m is the average melting temperature in kelvin of the alloy. Using the rule of mixtures, T_m is given by:

7

where (T_m)_i is the melting point for the ith element. Ω should be roughly larger than 1.1, so the contribution from TΔS_mix in eqn (4) is larger than the contribution from ΔH_mix. Thus, the RHEA is mainly composed of the solid solutions. If Ω < 1.0, ΔH_mix value dominates ΔG_mix and thus, intermetallic compounds and segregations are formed prior to solid-solution phases.⁹⁷ Our calculation show that Ω > 1.1 for all the 18 RHEAs, thus, the 18 RHEAs are mainly composed of solid solutions, as indicated by the Ω data in Table 1.

Zhang et al.⁹² suggested the atomic size difference parameter (δ), is

8

where n is the total number of elements in the solid solution, c_i is the atomic fraction of the ith element, r_i is the atomic radius of element i, and (where N is the total number of components in the alloy). The parameter δ is widely used to predict the formation of HEAs. The Hume Rothery rules state that binary solid solutions will be frustrated if the δ value between two elements is larger than 15%.^98,99 Metallic glasses tend to form at larger δ values (δ > 11%) while HEA formation is favored at smaller δ values. Yang et al.⁹⁷ and Guo et al.¹⁰⁰ separately calculated the values of δ for large number of HEAs that were prepared experimentally in previous works. According to these calculations, they proposed a critical condition for disordered solid solution formation phase: (δ ≤ 6.6), while the intermetallic phases and amorphous phases form at larger values of δ. However, their calculations for δ indicated that some HEAs deviated from this critical condition and still form solid solutions even at larger values of δ. For example, CoCrFeNiCuAl_0.5B and AlCoCrCuFeMoNiTiVZr⁹⁶ still form solid solutions at δ of 9.52 and 8.54, respectively. On the other hand, intermetallic phases still can form even at small values of δ (δ < 6.6) along with disordered solid solution in some HEAs.¹⁰¹ The δ values listed in Table 1 range from 2.0 to 7.6, which indicates formation of solid solutions in these 18 RHEAs. However, even we considered these values of δ are small, small δ does not guarantee the absence of intermetallic phases. Singh et al.¹⁰² suggested the Λ parameter (named the combination parameter) to predict the formation of disordered solid solution as follows:

9

High ΔS_mix and low δ stabilize the single-phase solid solution. Single-phase solid solution requires Λ > 0.96, two-phase solid solution form when 0.24 < Λ < 0.96, while intermetallic compounds form at Λ < 0.24. The calculated values of Λ for all 18 RHEAs range from 0.269 to 0.747 (except for M1 and M6), which may indicate the formation of intermetallic phases in addition to the solid solution (see Table 1). M6 has Λ value of 2.145 which may indicate the formation of single-phase solid solution, whereas M1 has Λ value of 0.227 which may indicate the formation of intermetallic phases in addition to the solid solution. However, the value of the parameter Λ can only be used to make a rough prediction of the phase formation, and it is not sufficient to make a final determination of the phase in the current 18 RHEAs. VEC is the total number of valence electrons per atom. VEC can be calculated as follows:⁹⁶

10

where (VEC)_i is the VEC for the ith element. The parameter VEC parameter for phase formation selection has been used recently in large number of research in HEAs. It can be calculated easily even before the alloys are experimentally prepared, to predict of the phase formation in RHEAs. In general, high VEC values favor FCC structures, but low VEC values prefer BCC phases. When VEC ≥ 8.0, a single FCC solid solution phase exists; for 6.87 ≤ VEC < 8.0, mixed FCC and BCC solid solution phases co-exist; and for VEC < 6.87, a single BCC solid solution phase exists.^4,103–110

From Table 1, the VECs for all 18 RHEAs are less than 6.87, which indicates the formation of single BCC solid solution phases. Pauling electronegativity difference (Δχ_Pauling) and Allen electronegativity difference (Δχ_Allen) parameters can be used to assess the phase formation of HEAs, and they are given by:^{96,103,111–113}

11

12

The atomic radius,¹¹⁴ elemental VEC,^115,116 elemental Pauling electronegativity(χ_Pauling),^115,116 elemental Allen electronegativity(χ_Allen),¹¹⁵ and elemental melting point (T_m) in kelvin¹¹⁶ of Ti,V, Cr, Zr, Nb, Mo, Hf, Ta, and W elements are listed in Table S4. Electronegativity is the ability of an atom to attract electrons toward itself.¹¹⁷ Electronegativity difference provides insights into the bonding characteristics, structure stability, and other potential properties of the materials. A lower value of Δχ indicates more metallic bonding (a tendency for ductility), promoting better miscibility and formation of homogeneous and stable solid solutions. When the electronegativity difference is low (Δχ_Pauling < 0.117), the atoms in a HEA can attract electrons uniformly, forming stable solid solutions, whereas, the formation of amorphous structures is favored at high electronegativity difference.¹¹⁸ The values of Δχ_Pauling and Δχ_Allen for all 18 models are listed in Table 1. Laves phase may form when Δχ_Allen exceeds 7%.¹¹⁹ Laves phases are intermetallic phases that have the composition AB₂ formula and tetrahedral close-packed crystal structures. These phases are generally known for being hard and brittle at room temperature, while offering high strength and corrosion resistance.^120,121 The calculated values of Δχ_Pauling for all 18 alloys are relatively high (see Table 1), which may indicate the formation of intermetallic phases in addition to the solid solution.

In testing the phase stability, we intended to perform a rough prediction of the phase formation based on different selection rules and using several empirical parameters, such as ΔS_mix, ΔH_mix, ΔG_mix, VEC, δ, and Δχ. These phase selection rules were proposed based on known experimental data for HEAs, applying them for new alloys might not always work because they do not take into account the competing phases in the system of new alloys. The empirical parameters themselves can be calculated even before the alloys are experimentally prepared, and they cannot guarantee 100% the formation of single-phase solid solution. So the final determination of the phase selection can be made after the experiment is carried out (such as X-ray diffraction (XRD) tests), which is out of scope in our current study. It is crucial to mention that experimental validation techniques, such as XRD and other microstructural characterization methods are needed to verify the accurate phase formation for our current 18 RHEAs.

3.2. Short-range order (SRO) parameter

One of the issues frequently discussed in the HEAs community is the possible existence of the short-range order (SRO), and if this SRO parameter can be quantified. SRO is potentially important, if enigmatic, factor in the engineering of this class of HE materials. The formation of SRO is used to explain experimental anomalies. SRO is expected in HEAs because of the chemical complexity, and was predicted by the DFT.^122,123 The RHEA system does not consist of completely randomly distributed atoms due to the different mixing energy between pairs of atoms among the multiple metals in the alloy (four, five, seven or nine elements).¹²⁴ Instead, some pairs of metals usually tend to attract each other, while some tend to repulse each other. Thus, we need to investigate the SRO value and effect on these 18 RHEAs, and to quantify the chemical ordering around an atomic species. A commonsense rule is that a single atom has no SRO, or SRO = 0, while a diatomic molecule has the maximum SRO (SRO = 1). In our current work, the SRO in RHEAs can be calculated from the root-mean-square (RMS) deviation of [PBOD/TBOD], where PBOD is the partial bond order density. PBOD can identify the contributions of bond order (BO) from various types of bonds, and it is obtained from TBOD decomposition. The data for PBOD, TBOD, and BO is published in our previous works^64,65 and put here in the SI (Fig. S1 and S2). First, the ratio of PBOD/TBOD for each of the 18 RHEAs models are calculated from the data listed in the Table S5, Fig. S1, and S2 in the SI. Next the RMS value for each model is calculated and listed in SI Table S5. As can be seen in Table S5, SRO mainly depends on the number of atomic components in the RHEA model. The 4-component alloys have higher SRO than the 5-component alloy; and the 5-component alloys have higher SRO than the 7- and 9- component alloy. The larger the atomic components, the lower the SRO.

We thus conclude that SRO in RHEAs under the RSSM is very small. Since the TBOD and PBOD are obtained from our quantum mechanics calculations of individual BO values, therefore they provide the vitally important details regarding the nature of the phase formation of RHEAs. It is important to point out that the above calculations for SRO is purely based on the homogenized RSSM used in our super-cells, thus they may not apply to different cases where other factors may involve under different assumptions.^125,126 The actual experimental sample is much larger than our samples and may contain multiphase structures, and the impurities or dislocations that may induce some form of SRO beyond the simple notion of number of atomic components. However, our current method of quantifying SRO could still be applied in such a case as long as the super-cell itself contains far more complicated non-homogeneous structures beyond the RSSM. More discussion about SRO will be in section 3.5.

3.3. Yield strength from solid solution strengthening

The total yield strength (Δσ_y, in MPa) of HEAs¹²⁷ can be expressed as:

Δσ_y = σ^mix_y + Δσ_ss + Δσ_ps + Δσ_gb + Δσ_df 13

Here, σ^mix_y is the mixing yield strength from the pure elements, which can be defined as the composition (c_i) average of the intrinsic yield strength (σ_yi, listed in Table S4) of the pure metals of the constituent elements:

14

In eqn (13), Δσ_ss, Δσ_ps, Δσ_gb, and Δσ_df are the contributions from the solid solution strengthening (SSS), precipitate strengthening, grain boundary strengthening, and dislocation-forest strengthening, respectively. Previous studies have shown that solid solution strengthening is the predominant factor in maintaining high strength in RHEAs and Δσ_ps can be negligible.^128–131 Moreover, Δσ_df of an undeformed alloy is negligible because of the low dislocation density.¹²⁷ Experiments have reported that the Δσ_gb rarely contributes to the Δσ_y of many RHEAs, such as HfNbTaTiZr,⁵³ Ti₃Zr_1.5NbV,¹³² TiZrNbMo,¹³³ and VNbTaTi¹²⁹ due to their large average grain sizes, hence it is also ignored in the current work. Zhao et al.¹³⁴ calculated Δσ_gb to be about 27 MPa for HfNbTaTiZr, which is very small compared to Δσ_ss (933 MPa). In recent study by Kim et al.'s,⁴⁶ the Δσ_gb was calculated as 44.7 MPa and 43.4 MPa in TiNbMoTaW and Ti₁(NbMoTa)₂W_0.5 alloys, respectively, indicating SSS is the dominant strengthening mechanism. Another study⁴⁵ showed that SSS is the main contribution to the Δσ_y of CrMoNbTiW RHEA. Wang et al.¹³⁵ found that the experimental Δσ_y value of Zr_1.2V_0.8NbTi_xAl_y alloys mostly comes from the SSS.

It is worth mentioning that Δσ_gb can make a significant contribution to Δσ_y in some mechanically alloyed RHEAs, such as NbMoTaWVCr (Δσ_gb = 598 MPa at 1400 °C) which was sintered by spark plasma sintering (SPS) and exhibited extremely fine-grained (grain boundary size = 1.24 µm).⁴³ The strengthening of RHEAs at most comes from the SSS. The SSS provides about 80–90% of the total strength in BCC Senkov alloys: TiZrNbHfTa, TiZrNbTa, and TiZrNbMo.^13,132,133 Apart from experimental observations,^{1,13,132,133,136–138} another first-principles calculation also confirmed that the SSS effect governs the strengthening mechanism of RHEAs.¹³⁹

As mentioned-above, the contributions from Δσ_gb, Δσ_ps, and Δσ_df are neglected, and the focus is placed on the SSS effect on Δσ_y for the current 18 RHEAs. Thus, Δσ_y of BCC RHEAs can be simplified as:^140,141

Δσ_y = σ^mix_y + Δσ_ss 15

The SSS effect originates from the elastic interactions between the local stress fields generated by solute atoms and those associated with dislocations. In this work, we assume that we have an ideal situation where each atom acts as a solute atom in the simplest case. Each atom randomly occupies a crystal lattice site without possible clustering. The misfit whether in atomic sizes or elastic moduli induces mutual interactions among the solute atoms, causing elastic distortions in the lattice and generating local stress fields around the solutes. Hence, solute atoms can be effectively represented as point defects that distort the crystal lattice. Then, the interactions between these local elastic stress fields and the stress fields of dislocations during the deformation can significantly hinder dislocations motion, thereby increasing the force required for dislocation to move, and contributing to the overall strengthening of the alloy.¹⁴² This process is defined as an SSS effect.¹⁴³

In a single-phase BCC alloy, the SSS is the major strengthening mechanism. Based on Gypen and Deruyttere method,^144–146 the Δσ_ss summarizes the SSS effect caused by the ith element, which can be written as:

16

where the Δσ_i is the strength increase of the ith element and is written as:

Δσ_i = AGδ_i^4/3c_i^2/3 17

The above formula is known as Labush's approach which has been successfully and widely used to investigate the SSS effect in BCC RHEAs.^{127,140,141,147} The constant A is a dimensionless parameter, empirically determined to be about 0.04 based on previous analyses of various RHEAs.^129,136,148G is the shear modulus of the alloy computed via our DFT calculations (listed in Table 2), rather than using the rule of mixtures (ROM). When the shear modulus of an alloy under a study is missing or not available, researchers usually rely on the rule of mixtures to roughly calculate the shear modulus of that alloy. In our case, we have used DFT to calculate the shear modulus for all 18 RHEAs under our current study. We want to completely rely on our DFT calculation, known for accuracy, for shear moduli instead of using the rough rule of mixtures. The c_i represents the concentration of the ith element. Fleischer parameter or the total misfit parameter (difference parameter) (δ_i) for the ith element arises from the mutual interaction of solute atoms caused by the shear modulus misfit (difference) (δ_Gi) and the atomic misfit (δ_ai) between the solute and the solvent and can be given as:

18

Table 2. Density functional theory (DFT)-computed elastic constant C₄₄, bulk modulus (K), Young's modulus (E), shear modulus (G), Poisson's ratio (η), Pugh's ratio (G/K), and theoretical density (ρ) for the 18 BCC RHEAs. (Note: these data for the elastic properties are taken from our previously published works^64,65).

Model	Composition	C44 (GPa)	K (GPa)	E (GPa)	G (GPa)	η	G/K	ρ (g cm−3)
M1	W100Ti100V100Zr100Hf100	37.11	128.24	95.88	34.86	0.375	0.272	11.71
M2	Hf100Mo100Ti100W100Zr100	38.84	143.37	111.63	40.73	0.370	0.284	12.53
M3	Mo125Ta125Ti125V125	31.43	178.46	113.51	40.72	0.394	0.228	11.36
M4	Mo100Ta100Ti100V100Zr100	32.02	144.84	99.53	35.92	0.386	0.248	10.41
M5	W125Ta125Cr125V125	48.63	230.47	184.57	67.53	0.367	0.293	15.24
M6	Mo125Nb125Ta125W125	49.61	234.94	190.73	69.88	0.365	0.297	15.77
M7	Ti125Zr125Hf125W125	44.12	123.22	103.88	38.20	0.360	0.310	12.47
M8	Ti100Zr100Nb100Mo100Ta100	33.52	146.21	100.77	36.38	0.385	0.249	10.79
M9	Nb125Ta125Ti125V125	24.30	157.83	77.03	27.15	0.419	0.172	10.76
M10	Ti125V125Nb125Hf125	28.56	128.99	78.78	28.17	0.398	0.218	10.07
M11	Ti190V75Nb115Hf120	32.62	123.81	83.89	30.24	0.387	0.244	9.63
M12	Ti100Zr100Hf100V100Nb100	30.62	116.72	79.34	28.61	0.387	0.245	9.45
M13	Ti114V114Cr114Zr114Nb114Mo114Hf114Ta114W114	32.48	160.22	106.27	38.24	0.390	0.239	10.57
M14	Ti114V114Cr114Zr114Nb114Mo114Hf114Ta114W114	34.46	160.09	111.51	40.29	0.384	0.252	10.57
M15	Ti114V114Cr138Zr90Nb114Mo138Hf90Ta113W113	35.48	170.38	118.95	42.99	0.384	0.252	10.55
M16	Ti164V65Cr118Zr118Nb99Mo118Hf104Ta119W119	38.75	157.72	117.24	42.60	0.376	0.270	10.48
M17	Ti114V114Cr171Zr57Nb114Mo171Hf57Ta113W113	39.56	184.82	135.95	49.35	0.377	0.267	10.52
M18	Ti114V114Cr228Nb114Mo228Ta113W113	50.35	213.68	179.94	66.17	0.360	0.310	10.42

The coefficient β embodies the type of dislocations and relates to the interaction force between the dislocations and solute atoms. Its value depends on the type of mobile dislocation. Generally speaking, β ranges from 2 to 4 for screw dislocations and is typically 16 or higher for edge dislocations.^149–151 The dislocation is described as a linear defect or irregularity within the crystal structure that involves an unexpected change in the arrangement of atoms in the lattice. These dislocations control the material deformation. Edge dislocations involve an extra half-plane of atoms. That means they are caused by the termination of a plane of atoms in the middle of the crystal.¹⁵² In such a case, the surrounding atomic planes are not straight, but instead they bend around the edge of the terminating plane so that the crystal structure is perfectly ordered on either side. Whereas screw dislocations create a helical, spiral distortion in the atomic lattice. Screw dislocations are formed by cutting halfway through a crystal and sliding those regions on each side of the cut parallel to the cut to create spiraling atomic planes.¹⁵³ The screw dislocation line would be located in the central axis of the spiral. Both of these two types of dislocations (edge and screw) are essential in understanding ductility and material strength.¹⁵⁴ Pinning points in a material halt the dislocation's movement. Thus, pinning of dislocations is a strengthening mechanism where moving dislocation is halted by obstacles in the material's crystalline lattice, such as solutes, precipitates, or other dislocations.¹⁵⁵

Regarding the value of coefficient β, in conventional BCC alloys, the pinning of edge dislocations makes them less able to move compared to screw dislocations. In addition, screw dislocations are less susceptible to pinning by different alloying atoms. Thereby, the edge dislocations mainly determine the SSS in BCC solid solutions. We set β = 9 for BCC metals due to the mixed nature of edge and screw dislocations in most RHEAs, and β = 16 for HCP metals. Senkov¹³ suggested that the atomic size misfit parameter (δ_ai) and the shear modulus misfit parameter (δ_Gi) of element i of BCC HEAs cause local lattice distortion and can be expressed as:^{13,141,156–158}

19

20

The constant (9/8) originates from the fact that for a single-phase HEA with BCC structure, the crystal lattice has 8 nearest-neighbor atoms, thus forming a 9-atom cluster. In this case, the atomic size and modulus mismatches are the function between one element and all its nearest neighbors. δ_aij and δ_Gij are the atomic size difference and shear modulus difference between the atoms i and j, respectively, and they have the following expressions:

21

22

Here, r_i and G_i are the atomic radius and shear modulus of the ith pure element, and the c_j, r_j, and G_j are the concentration, atomic radius, and shear modulus of the neighbor j elements, respectively.¹³ The calculated shear modulus (G), Poisson's ratio (η), and theoretical density (ρ) for the 18 BCC RHEAs are listed in Table 2. The calculated values of δ_aij and δ_Gij for each element are listed in Table 3. To analyze the contribution of the constituent elements to SSS, the calculated δ_Gi and δ_ai between each pair of atoms in these 18 RHEAs are listed in Table 4 and plotted in Fig. S3 in the SI. From eqn (17) and (18), Δσ_ss mainly depends on the values of δ_ai and δ_Gi. The increase in the values of δ_ai and δ_Gi, (and consequently δ_i) leads to an increase in Δσ_ss. The alloys containing V and Cr elements exhibit large δ_ai value except in M5, M17, and M18. Alloys containing Zr and Hf elements also exhibit a large δ_ai value. The reason behind this large δ_ai is that V and Cr have the smallest atomic radius in comparison to other refractory elements, whereas Zr and Hf have the largest atomic radius. The alloy involving V or Cr together with Zr or Hf or both Zr and Hf resulted in very large δ_ai value. In M5, M6, and M18, Zr and Hf are not involved, thus V has a small δ_ai value. δ_ai has been significantly reduced in M18 due to the absence of Hf and Zr. Mixing HCP elements (Ti, Zr, Hf) that have a small shear modulus with elements that have a large shear modulus (W and Mo) in M2 and in the alloys from M13 to M18 resulted in very large δ_Gi (see Fig. S3(c and d)). The data for δ_i are listed in Table 5.

Table 3. Relative atomic size difference (δ_aij, underlined numbers) and shear modulus difference (δ_Gij, bold numbers) of different alloying element pairs in BCC Ti, V, Cr, Zr, Nb, Mo, Hf, Ta, and W.

Element i/j δaij/δGij	Ti	Zr	Hf	V	Nb	Ta	Mo	W	Cr
Ti	0.0	0.232	0.380	−0.044	0.178	−0.440	−0.9586	−1.141	−0.893
Zr	0.090	0.0	0.100	−0.710	−0.067	−0.710	−1.1283	−0.782	−1.108
Hf	0.083	−0.006	0.0	−0.4211	−0.240	−0.325	−0.628	−0.709	−1.172
V	−0.080	−0.170	−0.163	0.0	0.197	−0.325	−0.890	−0.709	−0.740
Nb	0.008	−0.082	−0.075	0.088	0.0	−0.5794	−1.0732	−1.2362	−1.007
Ta	0.008	−0.081	−0.075	0.088	0.001	0.0	−0.5846	−0.800	−0.500
Mo	−0.040	−0.129	−0.123	0.040	−0.047	−0.048	0.0	−0.2439	0.083
W	−0.034	−0.123	−0.117	0.046	−0.041	−0.042	0.006	0.0	0.333
Cr	−0.118	−0.248	−0.233	−0.050	−0.126	−0.135	−0.078	−0.084	0.0

Table 4. The total atomic size misfit parameter (δ_ai, underlined numbers) and the total shear modulus misfit parameter (δ_Gi, bold numbers) of different alloying elements pairs: Ti, V, Cr, Zr, Nb, Mo, Hf, Ta, and W in the 18 RHEAs systems.

Model		Ti	V	Cr	Zr	Nb	Mo	Hf	Ta	W
M1	δ ai	−0.0133	−0.1033	—	0.0875	—	—	0.0803	—	−0.0513
	δ Gi	−0.1244	0.1004	—	−0.3654	—	—	−0.3622	—	0.7517
M2	δ ai	−0.0223	—	—	0.0783	—	0.0671	0.0713	—	−0.0603
	δ Gi	−0.2624	—	—	−0.4595	—	0.4836	−0.4088	—	0.6471
M3	δ ai	0.0315	−0.0585	—	—	—	−0.0135	—	0.0405	—
	δ Gi	−0.4001	−0.3349	—	—	—	0.6462	—	0.0889	—
M4	δ ai	0.0050	−0.08505	—	0.1058	—	−0.0398	—	0.0142	—
	δ Gi	−0.2560	−0.21128	—	−0.4386	—	0.5479	—	0.1600	—
M5	δ ai	—	−0.0236	0.0757	—	—	—	—	0.0745	0.0248
	δ Gi	—	−0.4989	0.2551	—	—	—	—	−0.2093	0.4531
M6	δ ai	—	—	—	—	0.0245	−0.0284	—	0.0256	−0.0250
	δ Gi	—	—	—	—	−0.8125	0.3977		−0.2265	0.6413
M7	δ ai	−0.0391	—	—	0.0616	—	—	0.0546	—	−0.0771
	δ Gi	−0.1488	—	—	−0.2571	—	—	−0.3344	—	0.7403
M8	δ ai	−0.0149	—	—	0.0860	−0.0063	−0.0594	—	−0.0054	—
	δ Gi	−0.2191	—	—	−0.4809	−0.3883	0.8067	—	0.2816	—
M9	δ ai	0.0180	−0.0720	—	—	0.0267	—	—	0.0273	—
	δ Gi	−0.0841	−0.0256	—	—	−0.2732	—	—	0.3781	—
M10	δ ai	−0.0031	−0.0031	—	—	0.0059	—	0.0903	—	—
	δ Gi	0.1465	0.1465	—	—	−0.0380	—	−0.1578	—	—
M11	δ ai	−0.0115	−0.0115	—	—	−0.0018	—	0.0732	—	—
	δ Gi	0.1424	0.1424	—	—	−0.0396	—	−0.2835	—	—
M12	δ ai	−0.0425	−0.0425	—	0.1998	−0.0533	—	0.0709	—	—
	δ Gi	0.0790	0.0790	—	−0.3517	0.1382	—	−0.2568	—	—
M13	δ ai	0.0104	0.0104	−0.1340	0.1161	0.0190	−0.0344	0.1079	0.0208	−0.0276
	δ Gi	−0.3359	−0.3359	0.5005	−0.5797	−0.4955	0.6378	−0.4969	0.1869	0.7443
M14	δ ai	0.0104	0.0104	−0.1340	0.1161	0.0190	−0.0344	0.1079	0.0208	−0.0276
	δ Gi	−0.3359	−0.3359	0.5005	−0.5797	−0.4955	0.6378	−0.4969	0.1869	0.7443
M15	δ ai	0.0191	0.0191	−0.1236	0.1261	0.0277	−0.0257	0.1176	0.0297	−0.0190
	δ Gi	−0.4013	−0.4013	0.4391	−0.6422	−0.5592	0.5943	−0.5425	0.1578	0.7209
M16	δ ai	0.0076	0.0076	−0.1361	0.1135	0.0162	−0.0371	0.1052	0.0180	−0.0304
	δ Gi	−0.3560	−0.3560	0.4827	−0.5729	−0.5162	0.6274	−0.5058	0.1798	0.7534
M17	δ ai	0.0310	0.0310	−0.1091	0.1395	0.0396	−0.0138	0.1306	0.0419	−0.0071
	δ Gi	−0.4900	−0.4900	0.3541	−0.7263	−0.6451	0.5340	−0.6037	0.1175	0.6880
M18	δ ai	0.0519	0.0519	−0.0837	—	0.0604	0.0070	—	0.0633	0.0137
	δ Gi	−0.6452	−0.6452	0.2052		−0.7955	0.4286		0.0469	0.6304

Table 5. The total misfit (difference) parameter (δ_i), solid solution strengthening (Δσ_ss), the mixing yield strength from the pure metallic elements (σ_y^mix), the total yield strength (Δσ_y), and the hardness (H_V) in the 18 RHEAs systems.

Models		δ i	Δσss (MPa)	σ y mix (MPa)	Δσy (MPa)	H V = 3Δσy (GPa)
M1	W100Ti100V100Zr100Hf100	4.976	1740.429	355.0	2095.429	6.286
M2	Hf100Mo100Ti100W100Zr100	4.700	1811.221	380.6	2191.821	6.576
M3	Mo125Ta125Ti125V125	2.392	942.802	257.6	1200.402	3.601
M4	Mo100Ta100Ti100V100Zr100	3.783	1484.238	313.6	1797.838	5.394
M5	W125Ta125Cr125V125	2.641	1789.812	416.3	2206.062	6.618
M6	Mo125Nb125Ta125W125	2.367	1671.034	443.3	2114.280	6.343
M7	Ti125Zr125Hf125W125	3.665	1562.525	366.3	1928.775	5.786
M8	Ti100Zr100Nb100Mo100Ta100	3.462	1282.600	299.6	1582.200	4.747
M9	Nb125Ta125Ti125V125	2.434	722.330	282.5	1004.830	3.015
M10	Ti125V125Nb125Hf125	2.627	1055.918	246.3	1302.168	3.907
M11	Ti190V75Nb115Hf120	2.664	989.950	233.4	1223.350	3.670
M12	Ti100Zr100Hf100V100Nb100	4.805	1355.733	253.0	1608.733	4.826
M13	Ti114V114Cr114Zr114Nb114Mo114Hf114Ta114W114	8.674	2009.008	339.8	2348.786	7.046
M14	Ti114V114Cr114Zr114Nb114Mo114Hf114Ta114W114	8.674	2116.709	339.8	2456.486	7.370
M15	Ti114V114Cr138Zr90Nb114Mo138Hf90Ta113W113	8.975	2215.602	344.3	2559.915	7.680
M16	Ti164V65Cr118Zr118Nb99Mo118Hf104Ta119W119	8.648	2179.637	338.4	2518.005	7.554
M17	Ti114V114Cr171Zr57Nb114Mo171Hf57Ta113W113	9.461	2343.488	350.1	2693.618	8.081
M18	Ti114V114Cr228Nb114Mo228Ta113W113	5.244	2058.114	360.0	2418.091	7.254

As shown in Table 5, manipulating atomic ratios or elemental variations is the key to enhancing the mechanical properties and yield strength in RHEAs. Among four-component alloys (M3, M5, M6, M7, M9, M10, M11), M7 has the largest δ_i whereas M6 has the lowest δ_i. Based on these results, we hypothesized that the higher concentration of HCP elements in M7 (Ti, Zr, Hf) causes a significant change in δ_i and lattice distortion (LD) in comparison to the other four-component alloys. This hypothesis can be supported by the LD data in Table 6 (ref. 64 and 65) where M7 has the highest LD while M6 has the lowest LD among the other four-component alloys. The same hypothesis is valid for the five-component alloys (M1, M2, M4, M8, and M12) where M12 has the largest δ_i and LD whereas M8 has the lowest δ_i and LD (see Tables 5 and 6). It is worthwhile to mention that M12 has a very large δ_i in comparison to the other five-component and four-component alloys, and this causes very large Δσ_ss. HCP elements can produce a higher degree of local atomic shuffling, which causes larger LD.¹⁵⁹ Alloys with a high concentration of HCP elements tend to have phase instability, which also causes larger LD. LD has a considerable effect on the properties of RHEAs,² including microstructure and mechanical properties. LD induces large elastic-strains which hinder the dislocation motion during deformation.¹⁴³ LD can significantly modify the dislocation core structure¹⁶⁰ suggesting a correlation between LD and strength/ductility in HEAs.¹⁶¹ Among nine-component alloys (from M13 to M18), M18 has the lowest δ_i and LD, and the third largest shear modulus due to the absence of the HCP elements (Zr and Hf). V, Cr, and Mo have the smallest atomic volume among other refractory elements; hence, increasing the V, Cr, and Mo compositions in this class of RHEAs could be the key factor in improving the SS yield strength. For example, M17 has the largest δ_i and the second largest Δσ_ss after M12 (see Table 5) due to the increase in Cr and Mo content from 114 atoms to 171 atoms and a reduction in the content of Zr and Hf from 114 atoms to 57 atoms.

The elastic constant: C₄₄ determines the resistance against the shear deformation in the (100) plane. The C₄₄ of M5, M6, and M18 is larger than the C₄₄ of all remaining models. Fig. S4 shows a positive correlation between C₄₄ and Δσ_y (higher C₄₄ leads to larger Δσ_y). Fig. 2 represents the δ_i of different alloying elements pairs in BCC Ti, V, Cr, Zr, Nb, Mo, Hf, Ta, and W in the 18 RHEAs systems. In Fig. 3(a), Δσ_ss increases with the increase of δ_i. The yield strength of an alloy can also be controlled by the concentration of HCP elements. Our first-principles calculations show that a higher concentration of HCP elements in the alloy can contribute to the electronic structure and phase instability. The shear modulus of an alloy increases or decreases depending on the concentration of HCP elements. Among the alloys from M1 to M12, M5 and M6 have the lowest total density of states at the Fermi energy (N(E_F)) (see Table 6) which indicates that these two alloys are more stable than the other ones due to the absence of HCP elements in their compositions. Among the alloys from M13 to M18, M18 has the lowest N(E_F) (see Table 6) which indicates that this alloy is more stable than the other ones due to the absence of HCP elements (Zr and Hf) in its composition. Hence, M5, M6, and M18 have the largest shear modulus among the remaining alloys. The HCP structure of an alloy can be stabilized (destabilized BCC structure) by increasing the concentration of HCP elements (Ti, Zr, Hf) and reducing BCC elements (V) within a BCC structure of a RHEA, which results in a significant decrease in shear modulus. As mentioned before, the only two alloys that have the same composition but different atomic configurations are M13 and M14. Changing the atomic configuration of an alloy results in a significant different properties. For instance, the N(E_F) of M14 is lower than N(E_F) in M13 indicating more structure stability, higher Young's and shear moduli, and larger Δσ_ss for M14 (see Tables 2, 5, and 6). Thus, manipulating the atomic configuration is an effective strategy to enhance the mechanical properties and Δσ_y. Similarly, changing atomic configurations could be carried out for each of the current alloys (from M1 to M18) to produce other new different models of alloys with different physical properties. However, this process would be very computationally expensive in the current study.

Fig. 2. The total misfit parameter (δ_i) of different alloying elements pairs: Ti, V, Cr, Zr, Nb, Mo, Hf, Ta, and W in the 18 RHEAs systems, (a) for the alloys from M1 to M9, and (b) for the alloys from M10 to M18.

Fig. 3. (a) Solid solution strengthening (Δσ_ss) versus the total misfit parameter (δ_i), (b) shear modulus versus the density of states at the Fermi energy N(E_F), and (c) the total yield strength (Δσ_y) in the 18 RHEAs systems.

From Tables 2 and 6, among the four-element alloys (M3, M5, M6, M7, M9, M10, and M11), M5 and M6 have the lowest N(E_F), and the largest shear modulus among all the 18 RHEAs. This is because M5 and M6 contain higher percentage of 3d and 4d elements (Cr and Mo). Whereas M9 has the highest N(E_F) among the four-element alloys, and the smallest shear modulus among all the 18 RHEAs. Among the five-element alloys (M1, M2, M4, M8, and M12), M2 has the lowest N(E_F) and the largest shear modulus. Among the seven- and nine-element alloys (the alloys from M13 to M18), M18 has the lowest N(E_F) and the largest shear modulus. A higher N(E_F) indicates the availability of more electronic states for electrons to occupy at this energy position, which means stronger metallicity and less covalency. The electrons will be more localized between the atoms to form strong covalent bonds between atoms in the alloy. The covalent bonds make the electrons bound between specific pairs of atoms, which reduces the number of electronic states that can move freely near the Fermi level, resulting in a decrease in the N(E_F). The shift in the density of states plotted in Fig. S5 towards the high-energy levels in M5 and M18 alloys that have lower N(E_F) than M3 and M13 alloys, indicates that the types of the chemical bonds between the atoms in the alloy and their strengths have changed. This often implies to enhancement of covalency. Electrons are more strongly bound between the atoms when stronger covalent bonds are formed between atoms, resulting in a higher energy of electrons and shift in the density of states of M5 and M18 plotted in Fig. S5 towards the high-energy levels (see also Table S6).

This can lead to a larger shear modulus because the covalent bonds are generally stronger than the metallic bonds.

Lowering the concentrations of Zr and Hf and increasing the concentrations of Cr and Mo (BCC elements) stabilizes the BCC structure and results in a larger shear modulus. As can be seen in Fig. 3(b), the shear modulus decreases as the N(E_F) increases. In Fig. 3(b), there are two distinguished groups of alloys: the one circled by the red dashed circle (the alloys from M1 to M12) and the other circled by the blue dashed circle (the alloys from M13 to M18). In both of these groups, shear modulus and N(E_F) have a negative correlation. The current findings are promising and can guide further experiment efforts to improve the Δσ_y and mechanical properties of RHEAs. The hardness (H_V) can be estimated using:¹⁵¹

H_V = 3Δσ_y 23

From Table 5, M5 has the largest Δσ_y and H_V among the four-component alloys due to the largest shear modulus. M12 has the largest Δσ_y and H_V among five-component alloys due to the largest δ_i, whereas M17 has the largest Δσ_y and H_V among nine-component alloys due to the largest δ_i. The interesting fact is that most of these 18 RHEAs have incredibly large hardness, which has not been reported for many other RHEAs. M13 and M14 have different atomic configurations and this can result in different properties.

3.4. Temperature-dependent yield strength

In this section, we apply the well-known Curtin's solid solution strengthening model for edge dislocations¹⁶² in a random BCC alloy to predict the yield strength and understand the strengths in the 18 RHEAs. Lee et al.¹⁶² found that the edge dislocations are the main effecting factor in the yield strengths prediction of NbTaTiV and CrMoNbV alloys. The strengthening theory for the shear yield stress developed recently by Maresca and Curtin^163,164 is a reduced analytic model that assumes individual solutes interact elastically with dislocations in the average alloy matrix. In a simplified picture of this theory, it depends only on elastic moduli and atomic misfit volumes.¹⁶² This theory was used to design a large number of different compositions in the Al–Cr–Hf–Mo–Nb–Ta–Ti–W–Zr family.¹⁶² For a BCC alloy containing n types of alloying components with concentrations c_n, the shear yield strength and energy barrier (ΔE_b) for thermally activated flow at zero-temperature can be estimated as:

24

25

Here, G, and η represent the shear modulus and Poisson's ratio, respectively. α = 1/12 is a weighting factor for the dislocation line tension in BCC lattices. The average Burgers vector b is given as with a denotes the lattice constant of BCC RHEA. ΔV_n is the atomic misfit volume for the type-n element, which can be determined using Vegard's Law, as:¹⁶⁵

ΔV^Vegard_n = V_0n − V̄^Vegard 26

where V_0n are the elemental BCC atomic volumes and the alloy volume:

27

Then, the predicted yield stress σ_y at finite temperature T and finite strain rate ε̇ can be expressed as:

28

Here, ε̇ = 10⁴ s⁻¹ is a reference strain rate and 3.067 is the Taylor factor for polycrystal strengths controlled by edge dislocations. The finite strain rate ε̇ here is set to 5 × 10⁻³ s ⁻¹. The atomic volumes (V_0n) used to make predictions viaeqn (26) and (27) are shown in Table S4 in the SI.

In the current study, we focus on the yield strength calculations above 1200 °C (or 1500 K). The calculated σ_y0 and Δσ_y are listed in Table 7 and plotted in Fig. 4. Due to the combination of large shear modulus and large energy barrier (ΔE_b) (ΔE_b = 3.096 eV), M17 has the largest Δσ_y of 951.021 MPa at 1500 K among all the 18 RHEAs, which is larger than many reported RHEAs (see Fig. 4 and Table 7). M5 has the largest Δσ_y of 890.69 MPa at 1500 K among four-component alloys due to the largest ΔE_b (ΔE_b = 3.177 eV) and large shear modulus. A significantly higher shear modulus of M5, M6, M17, and M18 indicates significant edge dislocation strengthening.

Table 7. The yield strength at zero K (Δσ₀), the yield strength at 1500 K (Δσ_y), the total misfit volume of the alloy the zero-stress energy barrier (ΔE_b), and the melting temperature (T_melt) in the 18 RHEAs systems.

Models		Δσy0 (MPa)	Δσy (MPa) at 1500 K	∑ciΔVi2	ΔEb(eV)	T m (K)
M1	W100Ti100V100Zr100Hf100	2254.771	567.322	12.947	2.540	1561.88
M2	Hf100Mo100Ti100W100Zr100	2193.153	611.528	10.517	2.765	1745.76
M3	Mo125Ta125Ti125V125	1253.169	173.408	2.560	1.710	2080.01
M4	Mo100Ta100Ti100V100Zr100	2088.732	471.155	9.180	2.336	1746.15
M5	W125Ta125Cr125V125	2745.387	890.690	4.443	3.177	2781.03
M6	Mo125Nb125Ta125W125	1591.982	402.583	1.710	2.474	2853.66
M7	Ti125Zr125Hf125W125	1581.989	358.698	7.789	2.345	1524.32
M8	Ti100Zr100Nb100Mo100Ta100	1486.972	284.240	5.979	2.080	1741.94
M9	Nb125Ta125Ti125V125	910.663	84.400	2.777	1.396	1738.88
M10	Ti125V125Nb125Hf125	1645.421	280.441	9.130	1.932	1533.59
M11	Ti190V75Nb115Hf120	1675.265	301.295	9.218	1.999	1497.77
M12	Ti100Zr100Hf100V100Nb100	1727.503	346.319	11.217	2.103	1446.77
M13	Ti114V114Cr114Zr114Nb114Mo114Hf114Ta114W114	2635.645	703.891	11.274	2.666	2618.42
M14	Ti114V114Cr114Zr114Nb114Mo114Hf114Ta114W114	2725.169	765.457	11.274	2.783	2618.42
M15	Ti114V114Cr138Zr90Nb114Mo138Hf90Ta113W113	2841.176	829.956	10.487	2.880	2631.89
M16	Ti164V65Cr118Zr118Nb99Mo118Hf104Ta119W119	2691.049	781.557	10.728	2.865	2611.79
M17	Ti114V114Cr171Zr57Nb114Mo171Hf57Ta113W113	2968.456	951.021	8.818	3.054	2646.13
M18	Ti114V114Cr228Nb114Mo228Ta113W113	2553.855	806.914	4.371	3.096	2671.05

Fig. 4. The yield strength (Δσ_y) as a function of temperature (a) for the models from M1 to M9, and (b) for the models from M10 to M18 in the 18 RHEAs systems.

Hf and Zr have much larger atomic volumes than V and W, thus M1 has the largest misfit volume among all 18 RHEAs, leading to a significant strengthening of 2254.77 MPa and 567.32 MPa at 0 K and 1500 K, respectively. V has a smaller atomic volume compared to other elements in M12, thus the misfit volume is large (see Table 7). Ti, Ta, and Nb have very close values of atomic volume. At the same time, V, Mo, and W have very close values of atomic volume, so the misfit volume in M3, M6, and M9 is small. Hence, M3 and M9 have small Δσ_y of 173.41 MPa and 84.40 MPa at 1500 K, respectively. As mentioned before, M5 and M6 have more structure stability than the remaining 16 RHEAs, and they have the largest shear modulus of 67.53 MPa and 69.88 MPa respectively among the remaining 16 RHEAs due to the absence of HCP elements in their compositions. Thus, M5 and M6 have a larger Δσ_y of 890.69 and 402.58 MPa at 1500 K respectively, compared to M3 and M9. V has a smaller atomic volume comparing to other elements in M10 and M11, so the misfit volume is the largest among four-component alloys. Fig. 5(a and b) illustrates the correlation between ΔE_b and Δσ_y at 300 K and 1500 K. This correlation exhibits a closer-to-linear nature with their coefficient of determination (R²) equalling 0.89 and 0.93 for Δσ_y at 300 K and 1500 K, respectively. From Fig. 5, it is clear that the large values of ΔE_b lead to a significant strength retention with increasing temperature. From Table 7, most of these 18 RHEAs have very large zero-stress ΔE_b of 2.0–3.0 eV. These values are significant and much larger than the ΔE_b values reported for many RHEAs, and this indicates high stresses are needed to overcome these barriers, even at high temperatures, and thereby causing plastic flow in the alloys. Plastic flow is a permanent or an irreversible deformation of the materials when they are subjected to an applied stress that exceeds their yield strength, causing them to behave like highly viscous fluids. Large ΔE_b requires large shear modulus or large misfit volume or large shear modulus and misfit volume at the same time (see eqn (25)), which is difficult to achieve. High concentration of HCP elements in an alloy reduces the shear modulus but leads to a large misfit volume (and consequently larger ΔE_b), such as in M1, M2, and the alloys from M12 to M16. On the other hand, a low concentration of HCP elements reduces misfit volume but leads to a large shear modulus (and consequently larger ΔE_b), such as in M5, M6, and M18. In a comparison between M1 and M4, Hf was replaced by Ta which has a smaller atomic volume; and W was replaced by Mo which has an atomic volume very close to that of W. The resulting misfit volume is lower compared to M1 and M2. However, M4 still has a large Δσ_y of 471.16 MPa at 1500 K. Melting properties are critical for designing novel materials with high performance at high temperatures. Alloys with high melting points (T_m) are expected to retain strength up to higher temperatures. The T_m for M5 and the alloys from M13 to M18 have been calculated using eqn (7) to be in the range of 2600–2800 K (see Table 7). These seven alloys sustain high strength in the range of 700–950 MPa up to 1500 K (see Table 7). The high strength of these seven alloys can be related to their high T_m. To prove this analysis, we plotted the correlation between Δσ_y at two different temperatures (300 K and 1500 K) and T_m in Fig. 5(c and d), and noticed a positive correlation, especially at 1500 K.

Fig. 5. (a and b) The yield strength (Δσ) as a function of energy barriers: (a) at 300 K and (b) at 1500 K, (c and d) Δσ as a function of melting point (T_m): (c) at 300 K and (d) at 1500 K in the 18 RHEAs systems.

3.5. The correlation between negative enthalpy of mixing and the yield strength

A relationship between Δσ_y and ΔH_mix has been established in recent studies.^166,167 The interactions between constituent elements can be presented by ΔH_mix. If these interactions are strong enough, local atomic ordering and short-range ordered structures can be promoted. The magnitude of ΔH_mix (negative or positive) strongly affects the formation of short-range atomic clusters or long-range ordered intermetallic compounds.^{92,100,168,169} As discussed in section 3.1, ΔH_mix is a fundamental thermodynamic parameter widely used for phase stability prediction.^{167,170–172} Negative ΔH_mix has not been used to design HEAs with high strength and ductility until Han et al. recently reported that negative ΔH_mix in (HfNbTiV)₉₀Al₁₀ RHEA induces Local Chemical Ordering (LCO), which affects the dislocation interactions and improves hardening.^24,173 LCO in HEAs indicates a non-random arrangement of atoms where certain atoms tend to cluster or segregate into specific local regions rather than being uniformly distributed. LCO is driven by negative ΔH_mix, and it involves SRO, medium-range order (MRO), and long-range order (LRO) domains. Enhancing LCO is used to improve the strength and ductility of HEAs, as it has a significant influence on their mechanical properties. However, very large negative ΔH_mix (ΔH_mix < −10 kJ mol⁻¹) can promote strong chemical interactions and large binding energies between different elements, leading to the formation of LRO atomic arrangements. Thus, brittle intermetallic phases are promoted and the formation of single-phase solid solutions is suppressed, leading to reduced toughness and ductility of HEAs.^24,92,173 He et al.¹³⁷ showed that the increase of Al content in Al_x(CoCrCuFeNi)_100−x HEA led to an increase in negative ΔH_mix from −3.2 to −4.78 kJ mol⁻¹, resulting in a structural transition from FCC to BCC. The V₁₆(CoNi)₇₄Mo₁₀ HEA was reported by Xu et al.¹⁷⁴ with high negative ΔH_mix (ΔH_mix = −9.5 kJ mol⁻¹), which caused an increase in yield and tensile strengths from 504 to 923 MPa and 1027 to 1189 MPa, respectively. Negative ΔH_mix emerges as a crucial parameter in achieving high Δσ_y and ductility by driving local chemical fluctuations, SRO, and second-phase formation, thereby optimizing the microstructure. The correlations between negative ΔH_mix and Δσ_y are plotted in Fig. 6 for the 18 RHEAs (a is for Δσ_y calculated using eqn (15), b and c are for Δσ_y0 and Δσ_y calculated using eqn (28)). M1, M2, M5, M6, M7, and the alloys from M13 to M18 have the highest negative ΔH_mix. From Fig. 6, there is an increase in Δσ_y with increase of negative ΔH_mix which is evidence that the Δσ_y can also be controlled by the value of negative ΔH_mix. M10 and M11 alloys have interestingly positive ΔH_mix and small Δσ_y. W and Cr result in large negative values of ΔH_mix (from −6 to −12 kJ mol⁻¹) when interacting with other refractory elements (see Table S7). Thus W and Cr have the largest contribution to the negative ΔH_mix, indicating that the more atomic fractions of W and Cr in the alloy, the higher negative ΔH_mix becomes.

Fig. 6. Yield strength (Δσ_y) versus the enthalpy of mixing (ΔH_mix) in the 18 RHEAs systems ((a) is for Δσ_y calculated using eqn (15), (b and c) are for Δσ_y0 and Δσ_y calculated using eqn (24) and (28)).

3.6. The correlation between atomic size difference, Allen electronegativity difference and yield strength

The atomic size difference (δ), calculated using eqn (8) in section 3.1, is an indicator of a distorted lattice. From Fig. 7(a), the correlation between δ and Δσ_ss is positive, which means there is a clear trend suggesting that a higher δ value correlates with higher SSS. These interesting results indicate that alloys in the current study exhibiting the highest δ values also demonstrated the highest SSS values. It is worth discussing the reasons behind large or small δ due to the strong correlation with SSS. Interestingly, even though V has a slightly smaller r than Mo, replacing Mo in M2 by V in M1 resulted in a noticeable increase in the value of δ (from 6.867 to 7.686) (see Table 1). The element Zr has the largest r among other refractory elements; so adding Zr into the composition of M3 resulted in larger lattice distortion and δ in M4 (from 4.08 to 6.843) (see Table 1).

Fig. 7. (a) Solid solution strengthening (Δσ_ss) versus the atomic size difference (δ) and (b) Δσ_ssversus Allen electronegativity difference in the 18 RHEAs systems.

Interestingly, adding just one more element (Zr) caused this large change (increase) in δ. Cr and V have the smallest r among other refractory elements; so that replacing Mo and Nb elements in M6 by V and Cr elements in M5 resulted in a larger δ (from 2.318 to 4.958) (see Table 1). Replacing the Ta element in M9 by the Hf element in M10 resulted in a much larger δ (from 3.927 to 6.443) (see Table 1). However, a positive relationship between the increase in lattice distortion (δ) and the increase in the SSS component was not observed across all alloys, as can be noticed for M6 (see Fig. 7(a)). On the other hand, Oh et al.¹⁷⁵ suggested a new model to roughly estimate the SSS effect from the electronegativity difference among the constituent elements (χ) as follows:

Δσ_ss α Δχ_Allen 29

This model suggests that a large electronegativity difference should lead to a high SSS. In Fig. 7(b), the data for Allen electronegativity difference are plotted versus the Δσ_ss. The figure indicates a positive relationship between Δχ_Allen and Δσ_ss for the 18 RHEAs under study.

It is important to discuss the effect of the VEC on the mechanical properties and the melting point (T_m) of RHEAs under the current study. High or low T_m, Young's modulus (E), bulk (K) and shear (G) moduli can be controlled by the value of VEC. In Fig. 8(a–d), the data for VEC from Table 1 are plotted versus K, G, E, and T_m. As shown in these figures, VEC is positively correlated to the three elastic moduli and T_m. This suggests that increasing the composition of elements with low VEC values such as Ti, Zr, and Hf lowers the VEC of the RHEA, thereby reducing K, G, and E moduli (as observed in M1, M7, M10, M11, M12, and M16). Conversely, decreasing the composition of Ti, Zr, and Hf while increasing the proportion of elements with high VEC values, such as Cr, Mo, and W, raises the VEC of the RHEA, resulting in a significantly larger K, G, and E (as seen in M5, M6, M17, and M18). VEC values also control the T_m of RHEAs (see Fig. 8(d)). Fig. 8(d) shows that there are two groups of alloys, the one circled by the red dashed circle and the other circled by the blue dashed circle. In both groups, T_m shows a positive linear correlation with VEC. The correlation between VEC and Δσ_y at 1500 K is plotted in Fig. 9(a).

Fig. 8. (a) Bulk modulus versus VEC, (b) shear modulus versus VEC, (c) Young's modulus versus VEC, and (d) melting point versus VEC in the 18 RHEAs systems.

Fig. 9. (a) The yield strength at 1500 K versus VEC, and (b) yield strength at 1500 K versus total bond order density (TBOD) in the 18 RHEAs systems.

The correlation between VEC and Δσ_y is roughly positive, indicating an effect of VEC on the Δσ_y. VEC and the type of crystalline structure can be used to manipulate the mechanical and thermal properties of RHEAs. This analysis opens new avenues for designing more RHEAs for multiple applications.

We identify the relation between the underlying electronic structure/bonding characteristics and Δσ_y by plotting the correlation between TBOD and Δσ_y at 1500 K for these 18 RHEAs in Fig. 9(b). Most of these 18 RHEAs show a positive correlation, in which Δσ_y steadily increases with TBOD, which is a significant new finding. M3 and M9 deviate from this positive correlation.

We plotted the correlation between and Δσ_y at 0 K and 1500 K for these 18 RHEAs in Fig. 10(a and b). We notice that there is a positive correlation in which Δσ_y steadily increases with This indicates that is extremely important for designing RHEAs because it is positively correlated with Δσ_y. We discuss here the relationship between the Debye temperature (Θ_D) and yield strength. High Θ_D indicates strong bonding, large density, and higher T_m, while lower Θ_D indicates softer materials with lower density and T_m. In Anderson's method, Θ_D depends on the sound velocity: the average (v_m), the transverse (shear) (vs.), and the longitudinal sound velocities (v_l). Materials with stronger bonds and higher density tend to have higher sound velocities, which results in a higher Θ_D. The values of Θ_D, v_m, and v_s, are calculated and published in our previous works.^64,65 From Fig. S6 and 11, we find that the yield strength of the 18 RHEAs is positively correlated with the Θ_D, v_m, and v_s. From Fig. S6 and 11, M5, M17, and M18 have the strongest interatomic bonds at 1500 K, thus they have the largest Θ_D, v_m, v_s, and yield strength. In Fig. S6(a) and Fig. 11(a and c), having high Δσ_ss and small Θ_D, v_m, and v_s, M12 deviates from the rest of the alloys. M3 also behaves differently from the rest of the alloys.

Fig. 10. (a) The yield strength at 0 K (Δσ_y0) versus the average effective charge and (b) yield strength at 1500 K versus in the 18 RHEAs systems.

Fig. 11. (a) Solid solution strengthening (Δσ_ss) versus shear sound velocity (v_s), (b) yield strength at 1500 K versus v_s, (c) Δσ_ssversus average sound velocity (v_m), and (d) yield strength at 1500 K versus v_m in the 18 RHEAs systems.

4. Conclusion

In current the study, the yield strength (Δσ_y) of 18 BCC refractory-based HEAs (the alloys from M1 to M18) is calculated at 0 K and 1500 K. The random solid solution method was used for alloying these 18 solid solutions using large super-cells. The phase stability of these 18 RHEAs has been assessed by calculating thermodynamic parameters and other parameters, such as VEC, Ω, δ, Λ, Δχ_Pauling, and Δχ_Allen. These empirical parameters were calculated to provide a theoretical and preliminary prediction of the phase formation, rather than providing a definitive confirmation of the phase formation in these 18 RHEAs. The atomic size of the refractory element and the type of crystal structure (BCC or HCP) are responsible for large or small lattice distortion and shear modulus, and thus they determine the value of Δσ_ss and Δσ_y. M12 (Ti₁₀₀Zr₁₀₀Hf₁₀₀V₁₀₀Nb₁₀₀) has significantly larger Δσ_ss and Δσ_y of 2595.849 MPa and 2848.849 MPa respectively whereas M9 (Nb₁₂₅Ta₁₂₅Ti₁₂₅V₁₂₅) has the smallest Δσ_ss and Δσ_y among the 18 RHEAs of 722.330 MPa and 1004.830 MPa respectively. The alloy M5 and the alloys from M13 to M18 retain extraordinary high Δσ_y at 1500 K (∼900 MPa), which makes them suitable for high-temperature applications. In general, the seven- and nine-element alloys (M13–M18) have larger lattice distortion δ_i than the four- and five-element alloys (M1–M12), except for M12 which has a larger δ_i than M18. Reducing the fraction of the HCP elements (Ti, Zr, Hf) in the alloys composition leads to a large increase in the shear modulus and energy barrier, and thus Δσ_y as can be seen in M5, M6, M17, and M18. M17 (Ti₁₁₄V₁₁₄Cr₁₇₁Zr₅₇Nb₁₁₄Mo₁₇₁Hf₅₇Ta₁₁₃W₁₁₃) has the largest Δσ_y at 1500 K of 951.021 MPa. The new findings in our current study can be used to design new RHEAs for multiple uses. The promising current results inspire us to do more research in this direction for more complex refractory–based high-entropy materials. We believe our results can facilitate the design of new high-entropy materials with broader applications. Despite a large number of complex calculations on mechanical properties of very large super-cells of RHEAs is done for the first time in the current work, there are obviously some drawbacks, such as ignoring the temperature effect on shear modulus at high temperatures and the spin–orbit coupling (SOC) calculations. However, we are encouraged by the current promising results and aspire to continue research in this direction for more complex and interesting RHEAs, considering using more desirable options, such as using temperature-dependent shear modulus model and using either hybrid potential or Becke–Johnson potential to improve the DFT calculations. We should point out that there are cases^176,177 where the neglect of SOC may not result in too large a difference due to unpredictable fluctuations. Our results without SOC are within the limit of these fluctuations and will still be useful as a first step to more accurate calculations in the RHEAs field.

Author contributions

Wai-Yim Ching (W. C.) conceived and directed the project. W. C., Sahib Hasan (S. H.) performed the calculations. S. H. made all figures. S. H., Puja Adhikari, Khagendra Baral, and W. C. drafted the paper. All authors participated in the discussion and interpretation of the results. All authors edited and proofread the final manuscript.

Conflicts of interest

The authors declare no competing or conflicts of interests.

Data availability

The authors declare that the main data supporting the findings of this study are available within the article and its supplementary information (SI). Any additional information, such as codes used in this study, is available either in the SI or upon request via contacting the first author at: sahib.hasan@mu.edu.iq. Supplementary information is available. See DOI: https://doi.org/10.1039/d6ra01515h.