Ultra-High Mixing Entropy Alloys with Single bcc, hcp, or fcc Structure in Co–Cr–V–Fe–X (X = Al, Ru, or Ni) Systems Designed with Structure-Dependent Mixing Entropy and Mixing Enthalpy of Constituent Binary Equiatomic Alloys

Abstract

Non-equiatomic high-entropy alloys (HEAs) for which the mixing (S_mix), configuration (S_config), and equivalent ideal (S_ideal) entropies satisfy S_mix > S_config = S_ideal were reported for Co–Cr–V–Fe–(Al, Ru, or Ni) systems. Three Co₂₀Cr₂₀Fe₂₀V₁₀X₃₀ (X = Al, Ru, or Ni) alloys (referred to as Al₃₀, Ru₃₀, and Ni₃₀ alloys) were studied here using conventional arc melting and subsequent annealing. The X-ray diffraction profiles revealed that the Al₃₀, Ru₃₀, and Ni₃₀ alloys annealed at 1600 K for 1 h exhibited B2 ordered, hcp, and fcc structures, respectively. A single structure was verified by scanning electron microscopy observations combined with elemental mapping via energy-dispersive X-ray spectroscopy. Thermodynamic calculations of S_mix normalized by the gas constant (S_mix/R) revealed that Al₃₀, Ru₃₀, and Ni₃₀ alloys at 1600 K had S_mix/R = 0.833, 1.640, and 1.618, respectively, where the latter two alloys exceeded S_config/R = 1.557. A compositionally optimized Al-containing HEA for S_mix with a single bcc structure was computationally predicted and verified experimentally for the Al₆Co₂₇Cr₃₄Fe₁₉V₁₄ alloy (Al₆ alloy). The non-equiatomic Al₆ alloy with S_config/R = 1.480 exhibited S_mix/R of 1.703 at 1600 K, surpassing S_config/R = ln 5 = 1.609 for the exact equiatomic (EE) quinary alloy. The bcc Al₆, hcp Ru₃₀, and fcc Ni₃₀ alloys were regarded as ultra-high mixing entropy alloys (UMHEAs) according to S_mix > S_config. Structure-dependent S_mix and the mixing enthalpy of constituent binary EE alloys are useful for future UHMEAs as a subset of HEAs.

1. Introduction

The present study addresses the systematic development of high-entropy alloys (HEAs)^1,2) by utilizing precise thermodynamic calculations based on the CALculation of PHase Diagram (CALPHAD) scheme. The specific objective is to develop non-equiatomic HEAs, with the mixing (S_mix), configuration (S_config), and its equivalent ideal entropies (S_ideal) satisfying S_mix > S_config (= S_ideal). The variation of S_mix from S_config is frequently discussed in some literature.^3–6) Followed by these early studies, the present authors adopted a thermodynamic approach and calculated the value of S_mix using Thermo-Calc^7,8) and a specially designated database for HEAs (TCHEA4). The present study was motivated by the authors’ previous precise thermodynamic calculations,⁹⁾ indicating that S_mix < S_config (= S_ideal) holds for most of the exact equiatomic (EE) HEAs with a single bcc structure; in particular, for Al-containing bcc EE HEAs. Only one exception was reported,⁹⁾ for the bcc CrMoNbTaTiVZr HEA¹⁰⁾ at 1300 K, with S_mix/S_config = 1.004. In addition to bcc EE HEAs, six of the nine fcc EE HEAs at 1600 K exhibited S_mix < S_config, except the fcc CoCrFeMnNi HEA²⁾ with S_mix/S_config = 1.087, the fcc CoCuFeMnNi HEA¹¹⁾ with S_mix/S_config = 1.003, and the fcc CuIrNiPdPtRh HEA¹²⁾ with S_mix/S_config = 1.032. In addition, it was reported⁹⁾ that hcp EE HEAs^13–15) comprising heavy lanthanides with and without Y behaved as an ideal solution, exhibiting S_mix/S_config = 1. Among the above exceptions, the authors have come to recognize that those HEAs that satisfy S_mix > S_config are ultra-high mixing entropy alloys (UHMEAs), for which the solid solution is more stable than for conventional HEAs, owing to S_mix > S_config. The significance of UHMEAs lies in the fact that S_mix > S_config differs from N-element EE HEAs that merely exhibit large S_config (= S_ideal) = ln N owing to high N. The previous study⁹⁾ was performed for EE HEAs only; thus, it is worth exploring UHMEAs by extending the category of HEAs to non-equiatomic types.

In the present study, single-phase non-equiatomic HEAs with bcc, hcp, and fcc structures and with S_mix/S_config > 1 were fabricated as UHMEAs.

2. Methods

2.1 Alloy design

The reference alloy for UHMEAs was the fcc Co₂₀Cr₂₀Fe₂₀Mn₂₀Ni₂₀ HEA (the Cantor alloy);²⁾ the motivation was to fabricate bcc, hcp, and fcc UHMEAs by optimizing the components and compositions of the reference alloy. The authors’ strategy for the development of UHMEAs was to determine a prototypical alloy consisting of main/common and supplemental inclusions. The main inclusion was to be composed of a set of common components and compositions, whereas the supplemental one was a structure-designated element (forming element, former, or structure determinant) intended to form a single bcc, hcp, or fcc structure. Such alloy design, controlling the structure with forming elements, such as Al, Ru, and Ni for bcc, hcp, and fcc structures, was somewhat different from the conventional ones, focusing on the similarities between elements with respect to their chemical features and/or geometrical factors in terms of their atomic differences. Taking HEAs as example metallic materials, Cu can be substituted by Ni and vice versa, owing to exchangeability.¹⁶⁾ The exchangeability works for Ti, Zr, and Hf in the same group in the periodic table. The role of exchangeability in metallic materials can be the same as isomorphous substitution¹⁷⁾ between Si⁴⁺ and Al³⁺ in oxides and clay minerals¹⁸⁾ in high-entropy ceramics and oxides.

The following section describes two essential aspects of determining a prototypical alloy. The first aspect is based on the authors’ previous analysis⁹⁾ of S_mix, revealing that Cr–Mn and Co–Cr atomic pairs, respectively, in the Co₂₀Cr₂₀Fe₂₀Mn₂₀Ni₂₀ HEA significantly decreased and increased S_mix relative to S_config. This led to the selection of Cr and omission of Mn as constituent elements. The second aspect, based on a preliminary study, was the use of a private database for interaction parameters Ω_ij(c_i, T) of a sub-regular solution model. Specifically, the values of Ω_ij(c_i, T) were stored in a private database from a previous database for solid solutions (SSOL4) where its latest version is SSOL8.¹⁹⁾ As for the extraction procedure, the authors referred to the contents described in the Thermo-Calc user’s manual. The values of Ω_ij(c_i, T) were stored as a function of the content of the i-th element (c_i) and absolute temperature (T). The analysis of Ω_ij(c_i, T) suggested that Co–Cr, Cr–Fe, Cr–Ni, Co–V, and Fe–Ni atomic pairs in the bcc structure tended to increase S_mix owing to the negative sign⁹⁾ of the temperature coefficient of Ω_ij(c_i, T). On the other hand, Mn-containing atomic pairs also exhibited a similar tendency, but atomic pairs were limited to Mn–Cu and Mn–Zr. Hence, the constituent elements of Mn with the ability to increase S_mix were not included in the constituent elements of the reference Co₂₀Cr₂₀Fe₂₀Mn₂₀Ni₂₀ HEA. These aspects led to the substitution of Mn with V while retaining Cr, leading to the EE Co₂₀Cr₂₀Fe₂₀Ni₂₀V₂₀ alloy as a prototypical alloy.

Subsequently, the composition of the prototypical EE Co₂₀Cr₂₀Fe₂₀Ni₂₀V₂₀ alloy was optimized in terms of the appearance of the fcc structure, resulting in the Ni₃₀ alloy (Co₂₀Cr₂₀Fe₂₀Ni₃₀V₁₀) as the first candidate for UHMEAs. Preliminary thermodynamic calculations revealed that stabilizing the Ni₃₀ alloy as an fcc HEA was achieved by suppressing the precipitation of the sigma (σ) phase. Then, the Al₃₀ alloy was considered as the second candidate for UHMEAs by replacing Ni from the fcc-former with Al from the bcc-former.¹⁾ Finally, the Co₂₀Cr₂₀Fe₂₀Ru₃₀V₁₀ alloy was considered as the third candidate for UHMEAs in the same replacing scheme by referring to the literature.²⁰⁾ In deciding Ru as an hcp-former, the authors focused on the ratio of the compositions of c_Fe:c_Ru = 2:3. This ratio matches the formation of an hcp structure from the Fe–Ru binary phase diagram²¹⁾ by referring to the authors’ previous work²⁰⁾ for the hcp Fe₁₂Ir₂₀Re₂₀Rh₂₀Ru₂₈ HEA. Hence, it was expected that Ru would affect the formation of hcp UHMEAs with the help of Fe. The features of the three UHMEA candidates all had in common that the Co₂₀Cr₂₀Fe₂₀V₁₀ as the main/common inclusion was accompanied by structural determinants of Al, Ru, and Ni, from bcc-, hcp-, and fcc-former.

2.2 Thermo-Calc and TC-Python

Thermodynamic calculations were performed using Thermo-Calc^7,8) ver. 2021a, using the TCHEA4 database. The details of the calculations of S_mix and other thermodynamic quantities were the same as those in the authors’ previous studies.⁹⁾ An important aspect of calculating S_mix was to set it to zero for pure constituent elements, which was achieved by selecting the designated structure as the standard element reference (SER).⁹⁾ A compositionally optimized alloy for S_mix/R was modeled using TC-Python²²⁾ which is a Python™ language-based software development kit available with Thermo-Calc. TC-Python allows for easy and flexible coupling of Thermo-Calc calculations with other software programs. The optimization eventually resulted in the Al₆Co₂₇Cr₃₄Fe₁₉V₁₄ alloy (Al₆ alloy).

2.3 Experiments

Alloy ingots (weight, 5 g) with nominal compositions of Al₃₀Co₂₀Cr₂₀Fe₂₀V₁₀, Co₂₀Cr₂₀Fe₂₀Ru₃₀V₁₀, and Co₂₀Cr₂₀Fe₂₀Ni₃₀V₁₀ (at%) alloys (Al₃₀, Ru₃₀, and Ni₃₀ alloys) were prepared via arc-melting from raw metals with industrial purity of 99.99%Co, 99.9%V, 99.995%Ni, 99.9%Cr, 99.99%Al, 99.95%Fe, and 99.9%Ru. The raw materials in the form of fragments and Ru powders were weighed at the designed molar ratio and mixed uniformly in a mixer. The 5-g samples were formed into button-shaped ingots (diameter, ∼10 mm; height, ∼5 mm). After arc melting, the samples were annealed at a high temperature to confirm the equilibrium phases. The as-prepared ingots were annealed for 1 h with a magnesium oxide crucible as the contacting material in an electric furnace. The annealing temperature was fixed at 1600 K, which was determined by preliminary thermodynamic calculations using Thermo-Calc. The chamber of the furnace was vacuumed (∼10⁻² Pa) in advance and then filled with high-purity Ar gas at ambient pressure. The samples were homogenized by annealing, followed by cooling in a furnace or quenching in water. The cross-sections of these alloys, which were cut into two pieces perpendicular to the base, were examined for their structure using X-ray diffraction (XRD). Co radiation with a wavelength (λ) of 0.1788965 nm (Co-Kα₁) was used for the XRD analysis. The samples’ morphologies were studied using scanning electron microscopy (SEM) as secondary electron (SE) images, and the distributions of the elements in the same area as the SE images were analyzed using energy-dispersive X-ray spectroscopy (EDX) equipped with SEM.

2.4 Numerical calculation of thermodynamic and physical quantities

In the authors’ previous studies, the following thermodynamic and physical quantities were computed using conventional procedures:^23,24) configuration entropy normalized by the gas constant (S_config/R), mixing enthalpy (ΔH_mix) calculated using Miedema’s empirical model, the delta parameter (δ), the valence electron concentration (VEC),²⁵⁾ and S_mix. The present study intentionally distinguished between ΔH_mix and H_mix, with the former corresponding to the mixing enthalpy calculated using the Miedema model, while the latter was computed using Thermo-Calc. Based on simple deductions of these quantities and on previous studies, S_config was equivalent to the ideal entropy (S_ideal), ΔH_mix was calculated by referring to literature,^26–28) and δ corresponded to the difference between the constituent atomic sizes.²⁹⁾ By contrast, the VECs of transition metals (TMs) and alloys containing TMs were calculated by averaging the intrinsic VEC of the i-th element (VEC)_i. Then, (VEC)_i corresponded exactly to the number of groups in the periodic table of the constituent i-th element, except for Al, for which (VEC)_Al was 3 by referring to the original literature.²⁵⁾ A detailed description of the S_mix calculation scheme can be found in the authors’ previous work.⁹⁾

3. Results

Figure 1 shows that the (a) Al₃₀, (b) Al₆, (c) Re₃₀, and (d) Ni₃₀ alloys exhibited a single phase for temperatures around 1600 K. As a reference, Fig. 1(e) shows that the common inclusion, Co₂₀Cr₂₀Fe₂₀V₁₀ (the Co_28.5714Cr_28.5714Fe_28.5714V_14.2857 alloy), tended to exhibit a mixture of bcc, hcp, and fcc structures in the intermediate range of temperatures (800–1600 K), while it exhibited a single bcc region within 80 K just above 1500 K. This narrow temperature interval was not advantageous for the reference case in Fig. 1(e) (Co₂₀Cr₂₀Fe₂₀V₁₀, the Co_28.5714Cr_28.5714Fe_28.5714V_14.2857 alloy) to form a single bcc phase. Figure 1 shows that a chaotic state in a mixture structure of the common inclusion in Fig. 1(e) was eliminated in Figs. 1(a), (c), (d) by adding structure-forming elements. The predicted stable structure at ∼1600 K was a B2-ordered structure for (a) the Al₃₀ alloy, hcp for (c) the Ru₃₀ alloy, and fcc for (d) the Ni₃₀ alloy, as well as bcc for (b) the Al₆ alloy as an optimized alloy for S_mix. These predicted structures were confirmed by preliminary experiments for as-prepared ingots without being subjected to annealing (XRD data for ingots are not shown in the present paper).

Fig. 1

Property diagrams, calculated using Thermo-Calc 2021a and the TCHEA4 database, for (a) Al₃₀, (b) Al₆, (c) Re₃₀ and (d) Ni₃₀ alloys, together with (e) Co₂₀Cr₂₀Fe₂₀V₁₀ (Co_28.5714Cr_28.5714Fe_28.5714V_14.2857 alloy) as the common inclusion for comparison.

Figure 2 shows the XRD patterns for the (a) Al₃₀, (b) Al₆, (c) Ru₃₀, and (d) Ni₃₀ alloys annealed at 1600 K for 1 h. Annealing of the Al₃₀, Ru₃₀, and Ni₃₀ alloy samples was followed by furnace cooling, whereas annealing of the Al₆ alloy sample was followed by quenching in water. A preliminary experiment revealed that water quenching, instead of furnace cooling, was necessary for the annealed Al₆ alloy sample, to suppress a mixture of complicated compounds. The XRD profiles shown in Fig. 2 demonstrate that the alloys exhibited distinct peaks corresponding to the presence of crystalline grains. From the XRD patterns, the structures of the alloys were identified as chemically ordered B2 for the Al₃₀ alloy, bcc for the Al₆ alloy, hcp for the Re₃₀ alloy, and fcc for the Ni₃₀ alloy. As shown in Fig. 2(a), the Al₃₀ alloy sample exhibited a (100) chemically ordered reflection peak at 2θ ∼ 36°, accompanied by faint-ordered reflections from (111) and (210) at 2θ ∼ 65° and ∼ 88°, respectively. In addition, Fig. 2(a) reveals weak shoulder peaks at each right side (high 2θ side) of the reflection peak, except for the (111) and (210) reflection peaks. These shoulder peaks were attributed to the Co-Kα₂ radiation with λ = 0.179285 nm, owing to the high crystallinity (fine crystalline grains) of the Al₃₀ alloy. The appearance of the shoulder peaks was preliminary confirmed experimentally for standard pure Si powders.

Fig. 2

XRD patterns measured using the Co-Kα radiation, for Al₃₀, Al₆, Re₃₀, and Ni₃₀ alloys annealed at 1600 K for 1 h.

Specifically, λ = 0.179285 nm of the Co-Kα₂ radiation was approximately 0.217% longer than that of the Co-Kα₁ radiation with λ = 0.1788965 nm, explaining the presence of the shoulder peaks at the high 2θ side owing to the Co-Kα₂ radiation. The differences between the ordered B2 and disordered bcc structures of the Al₃₀ and Al₆ alloys, respectively, were supported by the thermodynamic calculation results, as shown in Fig. 1.

Further investigations were performed using the SEM and EDX methods. Figure 3 shows an SE image and EDX element mapping images, for the Al₃₀ alloy annealed at 1600 K for 1 h. The SE image reveals different-brightness grains, with sizes ranging from a few to several hundred micrometers. The EDX mapping images at the same location as the SE image revealed a homogeneous distribution of constituent elements both inside the grains and over other grains, including the grain boundary under the current observation conditions of a few hundred micrometer scale or more. The experimental results in Figs. 1–3 indicate that the Al₃₀ alloy annealed at 1600 K for 1 h had a single B2 structure. Similar SE and EDX images were observed for the Al₆, Ru₃₀, and Ni₃₀ alloy samples (The data are not shown in the present paper).

Fig. 3

SE image obtained using SEM, and element-mapping images obtained using EDX, for the Al₃₀ alloy annealed at 1600 K for 1 h.

The estimated lattice constants of the Al₃₀, Al₆, and Ni₃₀ alloys with the B2, bcc, and fcc structures were a_B2 = 0.2884 nm, a_bcc = 0.2890 nm, and a_fcc = 0.3590 nm, respectively. The a_B2 and a_bcc values were close to a_Cr,bcc = 0.28846 nm, whereas the a_fcc value was near a_Co,fcc = 0.3544 nm. On the other hand, the lattice constants for the Ru₃₀ alloy were a_hcp = 0.2617 nm and c_hcp = 0.4201 nm, and the resultant ratio was c/a = 1.605. These data were not close to a_Ru,hcp = 0.2706 nm, c_Ru,hcp = 0.4458 nm, and (c/a)_Ru,hcp = 1.5824, although the Ru₃₀ alloy contained as much as 30 at%Ru as the major component.

The above results were analyzed based on the thermodynamic and physical properties of the alloys, as listed in Table 1. The Al₃₀, Al₆, Ru₃₀, and Ni₃₀ alloys exhibited VECs of 6.0, 6.87, 7.5, and 8.1, respectively, while the value of S_config/R was the same (1.557) for the Al₃₀, Ru₃₀, and Ni₃₀ alloys, and was 1.480 for the Al₆ alloy. The values of ΔH_mix/kJ mol⁻¹, computed using the Miedema model and based on the original literature,^26,30) ranged from approximately −7.1 to −14, somewhat larger and more negative than the values for conventional HEAs,²⁹⁾ for which ΔH_mix/kJ mol⁻¹ ranges from 0 to −5. The value of δ was in the 2.5–3.7 for the Ni₃₀, Ru₃₀, and Al₆ alloys with the bcc, hcp, and fcc structures, respectively; these values were within the δ values range (0.4–4.6) for disordered HEAs.²⁹⁾ On the other hand, the value δ = 5.5 for the Al₃₀ alloy with the B2 structure was also in the range for ordered HEAs (4.6 < δ < 6.4).²⁹⁾

Table 1 Values of the configuration entropy normalized by the gas constant (S_config/R), mixing enthalpy (ΔH_mix/kJ mol⁻¹) calculated using Miedema’s empirical model for equiatomic binary alloys in a liquid state, delta parameter (δ), valence electron concentration (VEC), and mixing entropy normalized by the gas constant (S_mix/R), calculated using Thermo-Calc and the TCHEA4 database, for the Al₃₀, Al₆, Re₃₀, and Ni₃₀ alloys with the compositions of Al₃₀Co₂₀Cr₂₀Fe₂₀V₁₀, Al₆Co₂₇Cr₃₄Fe₁₉V₁₄, Co₂₀Cr₂₀V₁₀Re₃₀Ru₂₀, and Co₂₀Cr₂₀Fe₂₀Ni₃₀V₁₀, respectively.

Table 1 contains further precise calculation results for S_mix/R, obtained using Thermo-Calc. The S_mix/R values for the Al₃₀, Al₆, Ru₃₀, and Ni₃₀ alloys were 0.833, 1.703, 1.640, and 1.618, respectively. These results indicate that the Ru₃₀ and Ni₃₀ HEAs, respectively, have larger values of S_mix/R = (1.640, 1.618) than S_config/R = 1.557, and that the Al₆ HEA also has a higher value of S_mix/R = 1.703 than S_config/R = 1.480. Hence, Al₆, Ru₃₀, and Ni₃₀ alloys were regarded as a new class of UHMEAs, because they satisfy S_mix > S_config, together with the CoCrFeMnNi HEA reported based on the authors’ previous precise thermodynamic calculations.⁹⁾ The ratio of S_mix (at 1600 K)/S_config was as high as 1.15 for the Al₆ alloy, the highest reported value.

4. Discussion

4.1 Reversible and irreversible processes affecting S_mix

In the HEA literature, S_mix is often presented as S_config or its equivalent S_ideal. This misrepresentation can be avoided by recognizing that S_mix is the sum of S_config and the excess entropy S_excess, as expressed by eq. (1)   

\begin{equation} S_{\text{mix}}=S_{\text{ideal(config)}}+S_{\text{excess}} \end{equation} (1)

The two terms on the right-hand side of eq. (1) are important in that S_config is the entropy that is generated in an irreversible process from the second law of thermodynamics, whereas conventional S (including S_excess) is generated in reversible processes, within the framework of the first law of thermodynamics. S_config can increase without heat exchange,³¹⁾ which can be understood as a phenomenon of the free expansion of an ideal gas to vacuum, via an adiabatic process. On the other hand, S_excess is necessarily accompanied by heat exchange as a reversible quasi-static process, according to the first law of thermodynamics. Inevitably, S_config is positive, whereas S_excess can be either positive or negative. Thus, a condition for yielding positive S_excess leads to the development of UHMEAs.

The following considers only S_excess in the reversible process based on the authors’ previous achievements.⁹⁾ In the case of EE alloys as the simplest example, S_mix(T) of the multi-component alloy of interest can be obtained by averaging the quantity of the constituent binary alloys over the contents of the alloy. Other alloys with non-equiatomicity can be dealt with using eq. (2) for S_excess, similar to eq. (3) for H_mix, which is the formulation of a sub-regular solution model. $\varOmega_{\text{ij}(\text{A}_{50}\text{B}_{50})}$ on the right-hand side of eq. (3) is a constant at the binary equiatomic A–B compositions of A₅₀B₅₀, and is a simple case of Ω_ij(c_i, T). Replacing $\varOmega_{\text{ij}(\text{A}_{50}\text{B}_{50})}$ in eq. (3) with $S_{\text{excess}(\text{A}_{50}\text{B}_{50})}$ yields eq. (2).   

\begin{equation} S_{\text{excess}} = 4\sum_{j = 1}^{N}\sum_{i > j}^{N}S_{\text{excess(A${_{50}}$B${_{50}}$)}}c_{i}c_{j} \end{equation} (2)

  

\begin{equation} H_{\text{mix}} = 4\sum_{j = 1}^{N}\sum_{i > j}^{N}\Omega_{\text{${ij}$(A${_{50}}$B${_{50}}$)}}c_{i}c_{j} \end{equation} (3)

The actual procedure for calculating S_excess is described in eq. (4).   

\begin{align} &S_{\text{mix(alloy),approx}}/R \\ &\quad = (S_{\text{ideal(=config)}} + S_{\text{excess}})R = -\sum_{i = 1}^{N}c_{i}\ln c_{i} \\ &\qquad + 4\sum_{j = 1}^{N}\sum_{i > j}^{N}\{S_{\text{mix,i-j}}/(R\ln 2) - 1\}c_{i}c_{j} \end{align} (4)

An important aspect of eq. (4) is that the S_mix value of an alloy, which surely is an S term, can be evaluated in a scheme of sub-regular solution models as H_mix. Specifically, the S_mix value of an alloy can be approximated by extending the S_mix value of the binary equiatomic alloy.

4.2 Structure-dependent S_mix and H_mix and their temperature dependence

4.2.1 Structure-dependent S_mix

An example of the structure dependence of S_mix is presented in Table 2. A feature of the Co₂₀Cr₂₀Fe₂₀V₁₀ main inclusion lies in the high value of $S_{\text{mix}(\text{A}_{50}\text{B}_{50})}/(R\ln 2)$ ∼ unity or more for most of the constituent atomic pairs consisting of Co, Cr, Fe, and V for the yellow-hatched columns. Only a set of exceptions was observed in the Co–V atomic pair for the hcp and fcc structures. Table 2 indicates that the efficient selection of atomic pairs with $S_{\text{mix}(\text{A}_{50}\text{B}_{50})}/(R\ln 2) > 1$ leads to UHMEAs. In addition, large and negative values of $S_{\text{mix}(\text{A}_{50}\text{B}_{50})}/(R\ln 2)$ are shown in red letters in Table 2 for atomic pairs with Al mainly, indicating a decrease in S_mix for the alloys with Al as a constituent element.

Table 2 Structure-dependent mixing entropy (S_mix) of binary equiatomic A–B compositions of A₅₀B₅₀, $S_{\text{mix}(\text{A}_{50}\text{B}_{50})}$, normalized by the product of the gas constant (R) and ln 2. In each upper right column, the values at the top and bottom, respectively, are $S_{\text{mix}(\text{A}_{50}\text{B}_{50})}/(R\ln 2)$ for the liquid phase at 3000 K and for the bcc at 1600 K. Lower left columns at the top and bottom show the results for hcp and fcc, respectively, at 1600 K. The underlined values, consisting of the Al, Re, and Ni atomic pairs, were not investigated in the present study.

The S_mix values of EE-HEAs can be calculated even for a multi-component alloy with N elements, just by averaging the values of $S_{\text{excess,i-j}} \equiv S_{\text{excess}(\text{A}_{50}\text{B}_{50})}$ over the constituent atomic pairs. Equation (4) explains the actual approximation procedure for estimating S_mix of an alloy, S_{mix(alloy),approx}, using the values of $S_{\text{mix}(\text{A}_{50}\text{B}_{50})}/(R\ln 2)$ listed in Table 2. The term, S_mix,i-j/(R ln 2) − 1, in eq. (4) corresponds to S_excess, and is zero for ideal solutions. The first term on the right-hand side of eq. (4), $ - \sum_{i = 1}^{N}c_{i}\ln c_{i}$, corresponds to S_config, where the coefficient four on the right side of eq. (4) is required to compensate for the c_ic_j term at the equiatomic composition (c_i = c_j = 0.5). The estimation using eq. (4) and the data from Table 2 yielded S_mix/R = 1.506 for the Al₆ UHMEA, 0.737 for the Al₃₀ HEA, 1.647 for the Ru₃₀ UHMEA, and 1.658 for the Ni₃₀ UHMEA. Comparisons of S_mix/R in Table 2 indicate that the evaluations based on eq. (4) and Table 2 were valid at the rates of 88.5%, 96.7%, 99.6%, and 102.5% for the Al₆ UHMEA, Al₃₀ HEA, Ru₃₀ UHMEA, and Ni₃₀ UHMEA, respectively. The relative error between the value of S_mix using eq. (4) combined with the data from Table 2 and the thermodynamical S_mix computed using Thermo-Calc was at most 10%. The parabolic approximation as a function of c_i for the second term on the right-hand side of eq. (4) mainly caused errors. Specifically, the inclusion of 10 at%V and 30 at%X (X = Al, Ru, or Ni) degraded the estimation accuracy for the Al₃₀ HEA, Ru₃₀ UHMEA, and Ni₃₀ UHMEA, whereas as small as 6 at% inclusion of Al as well as non-equiatomicity of Al₆ UHMEA yielded the worst accuracy. However, the estimation of S_mix using eq. (4) provided acceptable reliability when considering the simple formulae in eq. (4), based on the empirical method for estimating S_mix. In contrast, the values of $S_{\text{mix}(\text{A}_{50}\text{B}_{50})}/(R\ln 2)$ listed in Table 2 provide a useful guide for judging whether the alloys of interest are suitable for forming UHMEAs by their magnitude of $S_{\text{mix}(\text{A}_{50}\text{B}_{50})}/(R\ln 2) > 1$.

4.2.2 Structure-dependent H_mix

In Table 2, large and negative values of $S_{\text{mix}(\text{A}_{50}\text{B}_{50})}/(R\ln 2)$ are listed in red letters for atomic pairs with Al mainly, decreasing S_mix of the alloy when S_mix/R is estimated by eq. (4). A similar large and negative value of $H_{\text{mix}(\text{A}_{50}\text{B}_{50})}$ for atomic pairs with Al is listed in Table 3. This similarity between $S_{\text{mix}(\text{A}_{50}\text{B}_{50})}$ and $H_{\text{mix}(\text{A}_{50}\text{B}_{50})}$ indicates that the inclusion of Al in an alloy efficiently leads to the formation of a single bcc (or B2) HEAs, owing to large and negative H_mix, but the resultant alloy does not exhibit a high S_mix value for HEAs. This decrease in S_mix should be compensated by the decrease in H_mix in the framework of the formula G_mix = H_mix − TS_mix, for maintaining the G_mix stabilization by decreasing G_mix. Hence, it is understood that Al-containing HEAs can be of the H_mix-HEA type, rather than of the S_mix-HEA type. This proves that the bcc and B2 structures were formed in the present study for the A₆ and A₃₀ alloys, respectively, suggesting that a large amount of Al inclusions enhanced the chemical ordering owing to the large and negative H_mix value. The above suggestion was supported by the ΔH_mix value calculated using the Miedema model (Table 1) where ΔH_mix/kJ mol⁻¹ = −8.3 for the Al₆ alloy and −14.3 for the Al₃₀ alloy.

Table 3 Structure-dependent mixing enthalpy (H_mix) of binary equiatomic A–B compositions of A₅₀B₅₀, $H_{\text{mix}(\text{A}_{50}\text{B}_{50})}$. In each upper right column, the values at the top and bottom, respectively, are $H_{\text{mix}(\text{A}_{50}\text{B}_{50})}$ for the liquid phase at 3000 K and for the bcc at 1600 K. Lower left column plots (top and bottom) show the values for the hcp and fcc, respectively, at 1600 K. The underlined values, consisting of the Al, Re, and Ni atomic pairs, were not investigated in the present study.

The above values of ΔH_mix were not sufficiently valid because of the following two aspects. First, the ΔH_mix value used for the evaluation was originally derived for the EE-binary liquid phase,^26,30) to use ΔH_mix for evaluating the forming ability of amorphous and glassy alloys. Second, the atomic pairs of TM and non-TM (NTM) (where NTM includes Al) were not precisely treated, owing to the limitation on ΔH_mix for the liquid phase. Specifically, as has been reported,³²⁾ ΔH_mix of a solid solution including Al from NTM should be modified by considering the R term (≠ gas constant R) in addition to the P and Q terms.²⁷⁾ The alternatively calculated H_mix values using Thermo-Calc and the TCHEA4 database (and assuming the temperature of 1600 K) instead of ΔH_mix calculations using the Miedema model for the TM-NTM in the state of solids for the Al₆ and Al₃₀ alloys were H_mix/kJ mol⁻¹ = −3.043 and −25.980, respectively. These values of H_mix, computed using Thermo-Calc and the TCHEA4 database, and assuming the temperature to be 1600 K, also supported the tendency of the Al₃₀ alloy to exhibit considerably larger and more negative H_mix than the Al₆ alloy, indicating the H_mix-HEA type of the Al₃₀ alloy.

The values of H_mix can be approximated using eq. (5), using $H_{\text{mix}(\text{A}_{50}\text{B}_{50})}$, and the results are shown in Table 3.   

\begin{equation} H_{\text{mix(alloy),approx}} = 4\sum_{j = 1}^{N}\sum_{i > j}^{N}H_{\text{mix,i-j}}c_{i}c_{j} \end{equation} (5)

The approximated values for the Al₆ and Al₃₀ alloys were H_mix/kJ mol⁻¹ = −6.462 and −29.132, respectively. These results did not provide precise values of H_mix/kJ mol⁻¹ = −3.043 and −25.980, respectively, for the Al₆ and Al₃₀ alloys, that were calculated using Thermo-Calc and the TCHEA4 database, assuming the temperature at 1600 K. However, the approximations yielded almost the same ranges for H_mix. This indicates that $H_{\text{mix}(\text{A}_{50}\text{B}_{50})}$ (Table 3) is also useful for the further development of HEAs.

4.2.3 Temperature dependence of structure-dependent S_mix and H_mix

In the present study, experiments were performed by fixing the annealing temperature at 1600 K; this specific temperature was selected by referring to a single structure temperature range based on thermodynamic calculations, as shown in Fig. 1. The structure-dependent S_mix and H_mix of binary EE alloys are listed in Tables 2 and 3, respectively, for solids at 1600 K for a single phase, allowing to calculate those of a multicomponent alloy at 1600 K. In reality, however, S_mix and H_mix are not constant with respect to T, and rather should be considered as functions of T. The calculation results for S_mix and H_mix performed using Thermo-Calc and the TCHEA4 database are shown in Fig. 4. Figure 4(a) shows the S_mix/R ratio, computed using Thermo-Calc 2021a and the TCHEA4 database; the results are shown for the Al₆, Al₃₀, Ru₃₀, and Ni₃₀ alloys. The solid curves show the S_mix/R ratios for the designated single structures as thermodynamically stable phases, whereas the dotted curves indicate the designated single structures of the alloys in metastable states. The solid curves for the Al₆, Ru₃₀, and Ni₃₀ alloys with single bcc, hcp, and fcc structures, respectively, exhibit a slight increase in S_mix/R with decreasing T, which characterizes UHMEAs. The increase in the S_mix/R ratio for the Ru₃₀ and Ni₃₀ alloys at T < 1396 K (Curie temperature, T_c, of pure Co) was attributed to the change in the standard element reference (SER)⁹⁾ of pure Co for ferro- and para-magnetism, which affected the Al₆ alloy at T < 1450 K. This tendency of Co, T_c(fcc) ≤ T_c(hcp) < T_c(bcc), was qualitatively supported by the first-principles prediction of high T_c for ferromagnetic bcc-Co, calculated as reported in the literature³³⁾ using the Korringa–Kohn–Rostoker (KKR) Green function and full-potential linearized augmented plane-wave (FLAPW). The reported values³³⁾ were $T_{\text{c}}^{\text{KKR}}(\text{fcc-Co}) = 1280$ K, $T_{\text{c}}^{\text{KKR}}(\text{hcp-Co}) = 1300$ K, and $T_{\text{c}}^{\text{KKR}}\text{(bcc-Co)} = 1370$ or 1420 K, as $T_{\text{c}}^{\text{FLAPW}}(\text{fcc-Co}) = 1200$ K, $T_{\text{c}}^{\text{FLAPW}}(\text{hcp-Co}) = 1350$ K, and $T_{\text{c}}^{\text{FLAPW}}(\text{bcc-Co}) = 1670$ K.

Fig. 4

(a) Normalized S_mix/R calculated using Thermo-Calc 2021a and the TCHEA4 database, for the Al₆, Al₃₀, Ru₃₀, and Ni₃₀ alloys. Solid curves denote S_mix/R of the designated single structure, whereas dotted curves indicate the designated single structure of the alloys being in a mixture or metastable states. B2 ordering takes place for T < 1810.9 K for the Al₃₀ alloy, whereas a bcc disordered structure is a stable single structure for T > 1343.3 K for the Al₆ alloy. Increase in the S_mix/R ratio for the Ru₃₀ and Ni₃₀ alloys at T < 1396 K (= Curie temperature, T_c of pure Co) was owing to the change in the standard element reference (SER)⁹⁾ of pure Co for ferro- and para-magnetism, which presumably affected the Al₆ alloy at T < 1450 K. Specific heat at constant pressure (C_p) of Co in the (b) bcc, and (c) hcp and fcc structures.

The T_c values of 1396 K and 1450 K, obtained in the present study, were verified using the calculation results of the specific heat at constant pressure (C_p) of Co for the bcc, hcp, and fcc structures, and the results are shown in Fig. 4(b). Significantly, the ferromagnetic elements (Co, Fe, Ni) increased S_mix/R owing to the magnetic specific heat at constant pressure ($C_{p}^{\textit{magnetic}}$) at T < T_c when an alloy of interest and the pure ferromagnetic element possessed the same structure. Specifically, the magnetic second-order transition from ferromagnetism to paramagnetism, which was accompanied by $C_{p}^{\textit{magnetic}}$, contributed to S_mix/R as magnetic entropy (S_mag) by a factor of $S = \int (C_{p}^{\textit{magnetic}}/T) dT$ around T_c. In Fig. 4(b), S_mag tends to correspond to the hatched area surrounded by the λ-shaped C_p curve and the baseline for each structure. This specific increase in S_mix/R was supported by the following two results.⁹⁾ First, as described in Section 1 in the present study, the fcc CoCrFeMnNi HEA²⁾ and the fcc CoCuFeMnNi HEA,¹¹⁾ both containing Co, exhibited S_mix > S_config at 1600 K. Second, S_mix > S_config was not observed in the authors’ previous work⁹⁾ for bcc HEAs without Co. By contrast, the Al₃₀ alloy with a single B2 structure exhibited a considerable decrease in S_mix/R with decreasing T. The Al₆ and Al₃₀ alloys exhibited critical temperatures for chemical order/disorder (B2/bcc) at T_B2/bcc = 1343.3 and 1810.9 K, respectively, where the former for the Al₆ alloy was the transition between dual phases (B2 + bcc) and a single bcc structure. Figure 4 suggests that the Al₃₀ alloy formed in a single B2 structure exhibited a small S_mix/R < 1, but the Al₃₀ alloy formed in a metastable bcc structure in the range of T > 1810.9 K possessed a rather high S_mix/R > 1.41, which was 90% of S_config/R = 1.557. The decrease in S_mix/R began at T = 1810.9 K with decreasing T, indicating that chemical ordering decreased S_mix/R.

The decrease in S_mix/R for the Al₃₀ alloy can be explained by considering the site fraction (f) of the sublattice elements. The change in the value of f is shown in Fig. 5. Considering the fractional composition of the Al₃₀ alloy (Al₃₀Co₂₀Cr₂₀Fe₂₀V₁₀ = Al_0.3Co_0.2Cr_0.2Fe_0.2V_0.1), the above f values indicate that the B2 structure was an approximately Al–(Fe, Co) ordered structure in which the constituent elements were mainly occupied in sublattices 1–2, respectively. Meanwhile, V and Cr exhibited weak affinity to Al and (Fe, Co). These elemental distributions indicate that the B2 structure of the Al₃₀ alloy as a single-phase exhibits compositional homogeneity on a macroscopic scale but includes an atomistic inhomogeneity owing to the B2 ordering, because of the development of a short-range order. As shown in Fig. 5, the f values for both sublattices of the B2 structure exactly correspond to the alloy content of the Al₃₀ alloy for T ≥ 1810.9 K, although the stable phase was a liquid structure in this temperature range. Figures 4 and 5 and Tables 1 and 2 indicate that the Al₃₀ alloy was classified as neither an UHMEA nor a conventional HEA. More generally, the present results for the Al₃₀ alloy suggest the need to exclude B2 ordered single-phase alloys from the class of HEAs.

Fig. 5

Site fraction of sublattice elements for the Al₃₀ alloy, for single stable and metastable B2 ordered structures, and for a stable liquid structure.

There is a shortcoming in Table 2 in that the S_mix/(R ln 2) value is listed only for 1600 K, and its temperature dependence is not considered. The disadvantage of Table 2 can be compensated by deriving the molar Gibbs mixing energy (G_mix) of the alloys as a function of T. Below, we show that G_mix can be fitted with eq. (6).^9,34)   

\begin{equation} G_{\text{mix}} = a + bT + cT\ln T \end{equation} (6)

Applying G = H − TS and S = −(∂G/∂T)_P to eq. (6) yields eqs. (7) and (8), respectively:   

\begin{equation} S_{\text{mix}} = -b - c(\ln T + 1) \end{equation} (7)

  

\begin{equation} H_{\text{mix}} = a - cT \end{equation} (8)

Specifically, the constants a, b, and c were derived by fitting the G_mix data, which were calculated using Thermo-Calc and the TCHEA4 database, as in eq. (6). Table 4 summarizes the coefficients a, b, and c in eq. (6) for solids, and the lowest and highest temperatures (T_low and T_high) for forming a designated structure, as well as S_mix/R at T_low and T_high, as a result of the competition between the designated structure and liquid structure only. Table 4 indicates that the maximum of S_mix/R = 1.8259 (S_mix/S_config = 1.17) was obtained for the Ru₃₀ alloy at 680 K. In addition, Table 4 indicates that H_mix at 0 K can be estimated using eq. (8) by extrapolating T → 0. The value of H_mix at 0 K was approximately −50 kJ mol⁻¹ for the Al₃₀ alloy, whereas the absolute value of H_mix at 0 K was within 1.5 kJ mol⁻¹ for the other alloys. The large and negative H_mix at 0 K for the Al₃₀ alloy indicated a strong tendency to form a chemically ordered structure, where the H_mix at 0 K near the origin supported the formation of a disordered structure.

Table 4 The coefficients of a, b, and c of eq. (6) for solids, lowest and highest temperatures (T_low and T_high) for forming a designated structure, and S_mix/R at T_low and T_high where R² is the determination coefficient indicating the reliability of the fitting using the least-squares method for fitting the dependent variable (= G_mix) by independent variables (T and T ln T) in accordance with Y = a + bT + cT ln T; the fitting was performed in the range T_low ≤ T ≤ T_high, which includes a single range of a designated structure. The T_low and T_high values were determined by selecting two structures: a designated and a liquid structure as a counterpart.

At present, there are two ways to predict S_mix/R for multicomponent alloys. The first approach utilizes $S_{\text{mix}(\text{A}_{50}\text{B}_{50})}/(R\ln 2)$, as shown in Table 2. This method is convenient for judging the tendency of $S_{\text{mix}(\text{A}_{50}\text{B}_{50})}/(R\ln 2)$ to decrease or increase from $S_{\text{mix}(\text{A}_{50}\text{B}_{50})}/(R\ln 2)$. The evaluation of S_mix/R for a multicomponent alloy is valid for UHMEAs because of the slight increase in S_mix/R with decreasing T, as shown in Fig. 4. The second method uses the coefficients a, b, and c of eq. (6), to derive S_mix/R for a multicomponent alloy. The second method estimates S_mix/R more accurately than the first method, but the coefficients a, b, and c should be derived in advance. The combination of the first and second methods will contribute to the development of HEAs in the near future.

5. Conclusions

Experiments revealed that the Al₆Co₂₇Cr₃₄Fe₁₉V₁₄, Al₃₀Co₂₀Cr₂₀Fe₂₀V₁₀, Co₂₀Cr₂₀Fe₂₀Ru₃₀V₁₀, and Co₂₀Cr₂₀Fe₂₀Ni₃₀V₁₀ alloys (Al₆, Al₃₀, Ru₃₀, and Ni₃₀ alloys) annealed at 1600 K for 1 h were formed into single bcc, B2 ordered, hcp, and fcc structures. The bcc Al₆ alloy with S_config/R = 1.480 exhibited S_mix/R = 1.703 at 1600 K as an UHMEA, as well as Re₃₀ and Ni₃₀ UHMEAs. The hcp Ru₃₀ UMHEA contained as many 3d transition metals as 70 at%, at the highest amount among the hcp HEAs fabricated via conventional solidification from a melt to date. The inclusion of Co as a ferromagnetic element contributed to increasing S_mix/R. The chemical ordering of B2 from the disordered bcc structure decreased S_mix/R in the B2 Al₃₀ alloy down to S_mix/S_config ∼ 0.5, at 1600 K. Evaluations of S_mix and H_mix, both with structure dependency, were first performed based on $S_{\text{mix}(\text{A}_{50}\text{B}_{50})}/(R\ln 2)$ and $H_{\text{mix}(\text{A}_{50}\text{B}_{50})}$, respectively, of the binary exact equiatomic constituents. Structure-dependent S_mix and H_mix contribute greatly to the further development of UHMEAs and HEAs through a strategy of structure-dependent S_mix and H_mix.

Acknowledgments

This study was supported by JSPS KAKENHI (grant number JP17H03375).

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