Mixing Entropy of Exact Equiatomic High-Entropy Alloys Formed into a Single Phase

Abstract

Exact equiatomic high-entropy alloys (EE-HEAs) comprising N elements (N ≥ 5) formed into a single phase with either bcc, fcc or hcp structure were investigated based on sub-regular solution model. The analysis was performed by utilizing relationships among Gibbs energy (G), enthalpy (H), entropy (S), absolute temperature (T) and pressure (P), G = H − TS and S = −(∂G/∂T)_P, for representative EE-HEAs, such as bcc-MoNbTaVW, fcc-CoCrFeMnNi and hcp-EE-HEAs comprising heavy lanthanides with and without Y. Mixing entropy (S^mix) was evaluated as the sum of excess entropy (S^excess) and ideal entropy (S^ideal), the latter of which is equivalent to configuration entropy (S^config). Calculation tools contained commercial software (Thermo-Calc 2020a) using a database for HEAs (TCHEA4) mainly for the bcc- and fcc-EE-HEAs and that for solid solutions (SSOL5) for the hcp-EE-HEAs. The analysis revealed that the bcc-MoNbTaVW and NbTaTiVW HEAs exhibited the greatest decrease in S^mix normalized with gas constant (R) down to approximately 87% of S^ideal/R = ln N due to a positive T dependence of interaction parameter, Ω_i−j(T), of i-j atomic pairs in mixing enthalpy (H^mix). In contrast, S^mix/R of the fcc-CoCrFeMnNi HEA was approximately 9% greater than ln N. The hcp-EE-HEAs from a class of athermal solutions behaved as ideal solutions in practice. The results revealed that a relationship of S^mix/R = ln N does not always hold in EE-HEAs.

Relationships between $\text{X} = H_{\text{m}}^{\text{excess}}$ and $\text{Y} = - S_{\text{m}}^{\text{excess}}/(\ln N/\ln 2)$ of EE-HEAs from $H_{\text{m}}^{\text{excess}}$ and $S_{\text{m}}^{\text{excess}}$ consisting of $H_{\text{m}}^{\text{excess(Sub-Regular Solution Model)}}$, together with supplemental Y₁ and Y₂ axes.

1. Introduction

Recently, increasing scientific attention has drawn to high-entropy alloys (HEAs)^1–3) as advanced metallic materials because of their unique properties. The terminology of HEAs originates from a large magnitude of entropy (S) when alloying or mixing in a multi-component system with equiatomic or near equiatomic ranging 5 to 35 at%. The large magnitude of S contributes to decreasing Gibbs energy (G) of the system of HEAs due to the thermodynamic formula of G = H − TS where H and T are enthalpy and absolute temperature, respectively, leading to stabilizing G of a solid solution. The latest definitions of HEAs have allowed us to contain multiple phases including intermetallic compounds as well as a single phase. However, HEAs formed into a single phase is of great importance to analyze the fundamental features of HEAs.

Under the circumstances, the present paper describes thermodynamic analysis and calculations of exact equiatomic (EE) HEAs^4,5) formed into a single phase for their mixing entropy (S^mix) by focusing on excess entropy (S^excess).⁶⁾ Specifically, thermodynamic relationships among G, H, S, T and pressure (P), G = H − TS and S = −(∂G/∂T)_P, were utilized to evaluate S^excess. In analyzing, types of solid solutions were considered to characterize the thermodynamic features of EE-HEAs for a set of mixing enthalpy (H^mix) and S^mix. Since EE-HEAs, as well as conventional HEAs, are a class of solid solutions, they should be classified into either

1. (1)    ideal solution: (H^mix, S^mix) = (0, S^ideal),
2. (2)    athermal solution: (H^mix, S^mix) = (0, ≠S^ideal),
3. (3)    regular solution: (H^mix, S^mix) = (≠0, S^ideal), or
4. (4)    real solution: (H^mix, S^mix) = (≠0, ≠S^ideal).

Here, S^mix = S^ideal holds for ideal and regular solutions where S^mix exactly corresponds to configuration entropy (S^config) described below. Features of the regular solution are that S is supposed to be composed of only a term S^ideal and that the other excess terms regarding S (S^excess), even if they exist, deviated from an ideal state are included in H^mix as an excess G (G^excess) in alloying or mixing. This inclusion can be dealt with a sub-regular solution model with an interaction parameter, Ω_i−j(x_i,x_j,T), of the i-j atomic pairs as functions of the contents of x_i and x_j (0 ≤ x_i, x_j ≤ 1) and T. However, this inclusion is never considered with a sub-regular solution model with a constant interaction parameter (Ω_i−j). In reality, T dependence of H^mix and S^excess cannot be recognized even with a sub-regular solution model at a glance without performing thermodynamic calculations in practice. This can simply be exemplified by a case that an additional bT term in H affects G by a formula of G = H − TS. Specifically, an additional bT term with b being a positive value that resulted from H as (H + bT) − TS in the sub-regular solution model can also be interpreted as H − T(S − b) as an additional term of S. This suggests⁶⁾ that the temperature dependence of H^mix, bT, can be regarded as S^excess with the ability to reduce its magnitude by b when b is a positive value. This two-aspects of the bT term between H and S motivated the present study to evaluate S^mix precisely based on the sub-regular solution model.

The present study also submits a problem with mixture usage of S^mix and S^ideal. By assuming HEAs as regular solutions, S^mix is often described incorrectly in many literature as eqs. (1) with a content of the i-th element (x_i) and resultant x_i = N⁻¹ in EE-EHAs where R is gas constant.   

\begin{equation} S^{\text{mix}} = - R\sum_{i = 1}^{N}\mathrm{x}_{\text{i}}\ln \mathrm{x}_{\text{i}} = R\ln N \end{equation} (1)

Originally, eqs. (1) should be described for S^config that exactly corresponds to S^ideal as eqs. (2). Relationships in eqs. (2) always hold regardless of the four types of solid solutions. In strong contrast, eqs. (1) can be valid only for the alloys of interest are regarded as ideal and regular solutions.   

\begin{equation} S^{\text{config}} = S^{\text{ideal}} = -R\sum_{i = 1}^{N}\mathrm{x}_{\text{i}}\ln \mathrm{x}_{\text{i}} = R\ln N \end{equation} (2)

In other words, eqs. (1) does not necessarily valid for athermal and real solutions with a relationship S^mix ≠ S^ideal. Thus, it is worth analyzing S^mix for EE-HEAs to characterize thermodynamic nature.

Based on the above consideration, the present study regards S^mix of experimentally observed EE-HEAs formed into a single-phase (as an arbitral α-phase: α = bcc, fcc or hcp) could be described as eq. (3) by considering additional term of S^excess.   

\begin{equation} S^{\text{mix}} = S^{\text{config}} + S^{\text{excess}} \end{equation} (3)

Besides, allotropic entropy (ΔS^{allotropic,β-≠α-phase}) due to the difference in the phase of interest of constituent elements might be necessary to take into account in evaluating S^mix. However, preliminary investigation revealed that ΔS^{allotropic,β-≠α-phase} affects in the form of linear function and shifts S^mix by the content of the i-th element of the alloys in a composition–G diagram by the value of x_i · ΔS^{allotropic,β-≠α-phase} for each content. In reality, ΔS^{allotropic,β-≠α-phase} term works only when the phase of interest of alloy, α-phase that is acquired from experiments, differs from that of the constituent element, β-phase ≠ α-phase. In other words, ΔS^{allotropic,β-≠α-phase} can be ignored when the phases of the alloy and constituent elements are the same at given T and P. Thus, ΔS^{allotropic,β-≠α-phase} does not affect the formulae of eq. (3) essentially.

The present paper aims to examine whether the formula, S^mix = R ln N, holds for EE-HEAs actually by performing thermodynamic evaluations and to clarify the contribution of S^excess to S^mix in forming in N-component EE-HEAs formed into a single phase.

2. Methods

2.1 Thermodynamic analysis

We dealt with a set of thermodynamic quantities of G, H, and S of an N-element alloy (N ≥ 5) with x_i, which satisfies $\sum\limits_{\text{i} = 1}^{\text{N}}\text{x}_{\text{i}} = 1$. Only a single phase of alloys under a constant P was considered to simplify the subsequent analysis. In the actual analysis, molar thermodynamic quantities were considered, such as, molar G, H, and S (G, H, and S per one mole of atoms) by giving a symbol (m) at subscript of each G, H, and S below. In short, molar mixing entropy $(S_{\text{m}}^{\text{mix}})$ was analyzed as a function of T in a process of evaluating molar mixing Gibbs energy $(G_{\text{m}}^{\text{mix}})$ with a single phase of the EE-HEAs as a function of T. In practice, the present study defined that $G_{\text{m}}^{\text{mix}}$ is a quantity with values of zero at the pure constituent elements as eq. (4).   

\begin{equation} G_{\text{m}}^{\text{mix}} = G_{\text{m}}^{\alpha} - \sum_{\text{i=1}}^{\text{N}}\mathrm{x}_{\text{i}}{}^{0}G_{\text{m}}^{\text{${\alpha}$-i}} \end{equation} (4)

Here, ${}^{0}G_{\text{m}}^{\alpha \text{-i}}$ is a molar G of an α-phase (α = bcc, fcc or hcp) of the i-th element at a given T and P where superscript zero indicates the pure state of the element. The α-phase was set to be the same as that of HEA experimentally observed. When the α-phase did not correspond to a standard element reference (SER) that was defined by CALPHAD (CALculation of PHAase Diagrams) scheme of the i-th element that resulted from the stable crystallographic phase at a given T and P, additional G term of allotropic transformation (ΔG^allotropic) took place, but the present study did not deal with ΔG^allotropic furtherly as described in the former Section. Besides, there is another definition of SER in Thermo-Calc software⁷⁾ that stands for selected element reference. The SER’s from CALPHAD and Thermo-Calc software are equivalent in practice.

2.2 Alloys

Among 100 or more HEAs^2,3) found to date, the EE-HEAs that are formed into a single-phase either a simple crystallographic structure of bcc, fcc, or hcp phase as solid solutions were considered for evaluations. In reality, the EE-HEAs were acquired form a book³⁾ and their original literature by limiting their preparation methods of either arc-melting or induction melting, and subsequent annealing in need. These production methods matched the concept of HEAs that are thermodynamically stable solid solutions due to large S value, reducing G of systems due to G = H − TS. The actual alloys investigated were 18 bcc-EE-HEAs consisting of HfMoNbTiZr,⁸⁾ MoNbTaVW,^9,10) MoNbTaTiVW,¹¹⁾ CrMoNbReTaVW¹²⁾ and others,^13–21) 9 fcc-EE-HEAs of CoCrFeMnNi^22–24) and others,^25–32) and 4 hcp-EE-HEAs of DyGdHoTbY,³³⁾ DyGdLuTbY and DyGdLuTbTm³⁴⁾ and GdHoLaTbY.³⁵⁾

2.3 Interaction parameter, Ω_i−j(T), of $H_{\text{m}}^{\text{mix}}$ in a formula of sub-regular solution model and relationships between $G_{\text{m}}^{\text{mix}}$ and $G_{\text{m}}^{\text{excess}}$

First, $G_{\text{m}}^{\text{mix}}$ of EE-HEAs were calculated with Thermo-Calc as a function of T. Then, $G_{\text{m}}^{\text{mix}}$ was fitted with coefficients a, b, c as described in eqs. (5).   

\begin{align} G_{\text{m}}^{\text{mix}} & = a + bT + cT\ln T\\ & = a + (b + c\ln T)T \end{align} (5)

It might be necessary to consider the succeeding terms to cT ln T at the first line on the right side of eqs. (5), such as, by adding Tⁿ (1 < n ≤ 4 or 7) and T^−m, (m = 1 or 9) as can be seen in unary SGTE (Scientific Group Thermodata Europe) database.³⁶⁾ However, preliminary investigation revealed that most of the alloys including the EE-HEAs investigated in the present study can be described $G_{\text{m}}^{\text{mix}}$ as well as $G_{\text{m}}^{\alpha }$ in eq. (4) up to the three terms as the first line on the right side of eqs. (5). In case of requiring further additional terms, see details in the Appendix.

On the other hand, in a framework of a sub-regular solution model, $H_{\text{m}}^{\text{mix}}$ can be described with Ω_i−j(T) as eqs. (6) and (7) when the fitting $G_{\text{m}}^{\text{mix}}$ was thoroughly made up to the T ln T term.   

\begin{align} H_{\text{m}}^{\text{mix}}& = \sum_{\text{j} \neq \text{i}}^{N}\sum_{\text{i} = 1}^{N}\Omega_{\text{i-j}}(T)\mathrm{x}_{\text{i}}\mathrm{x}_{\text{j}} \\ &= 4\sum_{\text{j} \neq \text{i}}^{N}\sum_{\text{i=1}}^{N}\left\{\sum_{v = 0}^{v}L_{\text{i-j}}^{(\text{${v}$-th})}(T)(\mathrm{x}_{\text{i}} - \mathrm{x}_{\text{j}})^{v}\right\}\mathrm{x}_{\text{i}}\mathrm{x}_{\text{j}} \end{align} (6)

  

\begin{equation} L_{\text{i-j}}^{(\text{${v}$-th})}(T) = a_{\text{i-j}}^{(\text{${v}$-th})} + b_{\text{i-j}}^{(\text{${v}$-th})}T + c_{\text{i-j}}^{(\text{${v}$-th})}T\ln T \end{equation} (7)

Here, a part of the right side of eqs. (6) described in a formula of Redlich-Kister (R-K) polynomial functions of (x_i − x_j)^v can be expressed as eqs. (8) for EE-HEAs with v = 0 due to a relationship of x_i = x_j = N⁻¹ for every i-j atomic pair.   

\begin{align} &\sum_{\text{j} \neq \text{i}}^{N}\sum_{\text{i} = 1}^{N}\left\{\sum_{\text{v}}L_{\text{ij}}^{(v)}(T)(x_{\text{i}} - x_{\text{i}})^{v}\right\}x_{\text{i}}x_{\text{i}} \\ &\quad = \sum_{\text{j} \neq \text{i}}^{N}\sum_{\text{i} = 1}^{N}\left\{\sum_{\text{v}}L_{\text{ij}}^{(v)}(T)\left(\frac{1}{N} - \frac{1}{N}\right)^{v}\right\}\frac{1}{N^{2}}\\ &\quad = \frac{1}{N^{2}}\sum_{\text{j} \neq \text{i}}^{N}\sum_{\text{i} = 1}^{N}\left\{\sum_{\text{v}}L_{\text{ij}}^{(0)}(T)\right\} \end{align} (8)

Eventually, eqs. (6) and (7) can be simplified as eqs. (9).   

\begin{align} H_{\text{m, EE-HEA}}^{\text{mix}} & = 4N^{-2}\sum_{\text{j} \neq \text{i}}^{N}\sum_{\text{i} = 1}^{N}(a_{\text{i-j}}^{(0)} + b_{\text{i-j}}^{(0)}T + c_{\text{i-j}}^{(0)}T\ln T)\\ & = a^{(0)} + b^{(0)}T + c^{(0)}T\ln T\\ & = L_{\text{EE-HEA}}^{(0)}(T) \end{align} (9)

Here, Ω_i−j(T) in eqs. (6) originated from the sub-regular solution model, which enables us to deal with asymmetric composition dependence of $H_{\text{m}}^{\text{mix}}$ against equiatomic composition (x_A = x_B = 0.5) in an A–B binary system due to the presence of v from odd numbers. On the other hand, coefficient four on the right side of eqs. (6) is a modifier to x_ix_j term so that $L_{\text{i-j}}^{(v\text{-th})}(T)$ corresponds to the actual value of $L_{\text{i-j}}^{(v\text{-th})}(T)$ at x_i = x_j = 0.5 in each binary system. A significant feature of EE-HEAs is that equiatomicity (x_i = x_j) holds for all atomic pairs, which greatly reduced the necessary terms to be considered to only one set of coefficients of the 0-th order (a⁽⁰⁾, b⁽⁰⁾, c⁽⁰⁾, …) of $L_{\text{EE-HEA}}^{(0)}(T)$ in eqs. (9) due to x_i = x_j in eqs. (6). This feature allows us to use a conventional regular solution model in practice as the 0-th order approximation of the sub-regular solution model at x_i = x_j = N⁻¹.

Next, we considered deriving the formula of the sub-regular solution model from eqs. (5). According to the sub-regular as well as the regular solution model, which defines S as $S_{\text{m}}^{\text{config}} = S_{\text{m}}^{\text{ideal}}$, one can rewrite eqs. (5) as eqs. (10) for EE-HEAs where S-RSM stands for the sub-regular solution model. In reality, the coefficients (a, b and c) without superscript (0) at the first line in eqs. (9) originated from a conventional description by eqs. (5) and those with superscript (0) at the second and third lines in eqs. (9) resulted from the 0-th order approximation of the S-RSM. The coefficients (a, b and c) with and without superscript (0) are equivalent to each other for EE-HEAs.   

\begin{align} G_{\text{m}}^{\text{mix}} & = \{a + (b + c\ln T + S_{\text{m}}^{\text{ideal}})T\} - \mathit{TS}_{\text{m}}^{\text{ideal}}\\ & = \{a^{(0)} + (b^{(0)} + c^{(0)}\ln T + S_{\text{m}}^{\text{ideal}})T\} - \mathit{TS}_{\text{m}}^{\text{ideal}}\\ & = \{a^{(0)} + (b^{(0)} + c^{(0)}\ln T + R\ln N)T\} - \mathit{TS}_{\text{m}}^{\text{ideal}}\\ & = H_{\text{m}}^{\text{mix (S-RSM)}} - \mathit{TS}_{\text{m}}^{\text{ideal}}\\ & = G_{\text{m}}^{\text{excess (S-RSM)}} - \mathit{TS}_{\text{m}}^{\text{ideal}} \end{align} (10)

At the first line on the right side of eqs. (10), a term of $ - TS_{\text{m}}^{\text{ideal}}$ was intentionally added at the last term, which was compensated for by adding $TS_{\text{m}}^{\text{ideal}}$ in the brace at the second term on the first line. Besides, x_i = x_j for EE-HEAs made it possible to consider v = 0-th terms only described in eq. (7) and at the second and third lines of eqs. (10). The fourth line on the right side of eqs. (10) is the conventional description of the sub-regular solution model where mixing enthalpy is intentionally described as $H_{\text{m}}^{\text{mix (S-RSM)}}$. Note that a term RT ln N is added at the third line in eqs. (10) as $H_{\text{m}}^{\text{mix (S-RSM)}}$, which differs from $H_{\text{m, EE-HEA}}^{\text{mix}} = a^{(0)} + b^{(0)}T + c^{(0)}T\ln T$ where RT ln N is absent in eqs. (9). This was due to the two-aspect of the bT term between H and S in the sub-regular solution model as described in Section 1. The RT ln N term originates from S^config, which is equivalent to S^ideal from eqs. (2), and thus, the term $H_{\text{m}}^{\text{mix (S-RSM)}}$ containing S^ideal unconsciously in its formulation can be rewritten as G consisting of H and S instead of H solely. Consequently, the fourth line of eqs. (10) can be rewritten as the last line by regarding $H_{\text{m}}^{\text{mix (S-RSM)}} = a^{(0)} + (b^{(0)} + c^{(0)}\ln T + R\ln N)T$ for EE-HEAs as $G_{\text{m}}^{\text{excess(S-RSM)}}$. The significance of the appearance of $G_{\text{m}}^{\text{excess(S-RSM)}}$ in eqs. (10) will be explained in the next paragraph.

In contrast to eqs. (10), the present study provides another formula for describing $G_{\text{m}}^{\text{mix}}$ as shown in eqs. (11) with keeping the relationships of G = H − TS and S = −(∂G/∂T)_P.   

\begin{align} G_{\text{m}}^{\text{mix}} & = a + bT + cT\ln T\\ & = (a^{(0)} - c^{(0)}T) - \{-b^{(0)} - c^{(0)}(\ln T + 1)\}T\\ & = (a^{(0)} - c^{(0)}T) - \{-b^{(0)} - c^{(0)}(\ln T + 1) - R\ln N\}T \\ &\quad - \mathit{TS}_{\text{m}}^{\text{ideal}}\\ & = H_{\text{m}}^{\text{excess}} - \mathit{TS}_{\text{m}}^{\text{excess}} - \mathit{TS}_{\text{m}}^{\text{ideal}}\\ & = H_{\text{m}}^{\text{excess}} - \mathit{TS}_{\text{m}}^{\text{mix}}(=G_{\text{m}}^{\text{excess(S-RSM)}} - \mathit{TS}_{\text{m}}^{\text{ideal}}\\ & = H_{\text{m}}^{\text{excess(S-RSM)}} - \mathit{TS}_{\text{m}}^{\text{ideal}}) \end{align} (11)

Note that mixing enthalpy at the fourth and fifth lines in eqs. (11) is intentionally expressed as $H_{\text{m}}^{\text{excess}}$ to distinguish it from $H_{\text{m}}^{\text{mix (S-RSM)}}$ in eqs. (10), and thus, $H_{\text{m}}^{\text{excess}} = a^{(0)} - c^{(0)}T$ at the last line in eqs. (11) does not agree with $H_{\text{m}}^{\text{mix (S-RSM)}} = \{ a^{(0)} + (b^{(0)} + c^{(0)}\ln T + R\ln N)T\} $ for EE-HEAs in eqs. (10). This difference again indicated the two-aspect of the bT term between H and S in sub-regular solution model as described in Section 1 in that only = a⁽⁰⁾ − c⁽⁰⁾T from $H_{\text{m}}^{\text{excess}}$ should be considered as H in reality instead of {a⁽⁰⁾ + (b⁽⁰⁾ + c⁽⁰⁾ ln T + R ln N)T} from $H_{\text{m}}^{\text{mix (S-RSM)}}$ in frameworks of thermodynamic quantities: G = H − TS and S = −(∂G/∂T)_P. In short, the two-aspect allows us to describe $H_{\text{m}}^{\text{mix (S-RSM)}}$ in eqs. (9) from $H_{\text{m}}^{\text{excess}}$ in eqs. (11) by converting $TS_{\text{m}}^{\text{excess}}$ as an H term as shown at the fourth line in eqs. (11). Hence, neither $H_{\text{m}}^{\text{excess}} = a^{(0)} - c^{(0)}T$ nor $S_{\text{m}}^{\text{excess}} = \{ - b^{(0)} - c^{(0)}(\ln T + 1)\} $ for EE-HEAs in eqs. (11) correlated with $G_{\text{m}}^{\text{excess (S-RSM)}} = \{ a^{(0)} + (b^{(0)} + c^{(0)}\ln T + R\ln N)T\} $ for EE-HEAs in eqs. (10) in that the former ($H_{\text{m}}^{\text{excess}}$ and $S_{\text{m}}^{\text{excess}}$) was from the thermodynamic excess quantities investigated in the present study and the latter $(G_{\text{m}}^{\text{excess (S-RSM)}})$ was due to the sub-regular solution model. Further attention should be paid to that the last line of eqs. (10), $G_{\text{m}}^{\text{mix}} = G_{\text{m}}^{\text{excess(S-RSM)}} - TS_{\text{m}}^{\text{ideal}}$, was also derived in eqs. (11) by considering $S_{\text{m}}^{\text{excess}} = - b^{(0)} - c^{(0)}(\ln T + 1) - R\ln N$ for EE-HEAs.

3. Results and Discussion

3.1 bcc-EE-HEAs

The analysis revealed that 14 in 18 alloys^8–21) exhibited a single bcc phase from calculations with Thermo-Calc. Specifically, $S_{\text{m}}^{\text{mix}}$ was computed for 14 alloys for both bcc and liquid phases whereas $S_{\text{m}}^{\text{mix}}$ of liquid phase only was calculated for the other 4 alloys (AlCoFeNiTi, AlCrMoNbTi, CoCuHfPdTiZr, and CrMoNbTaTiVWZr). As a set of examples, amounts of phases, and $S_{\text{m}}^{\text{mix}}/R$ of the bcc and liquid phases of the bcc-MoNbTaVW EE-HEA are shown in Fig. 1. Figure 1(a) shows that a single bcc phase was stable at a wide temperature range of 682.9 ∼ 2839.3 K (T^{bcc, single, low} ∼ T^{bcc, single, high}), whereas a single liquid phase was stable at T ≥ 2976.5 K (T^{liquid, single, low}). At a range of T lower than 682.9 K, this alloy exhibited dual bcc-phase of bcc+bcc#2 from calculation results. Figure 1(b) demonstrates $S_{\text{m}}^{\text{mix}}/R$ of the single bcc and liquid phases drawn with thick solid curves, accompanied by those of the other states with dotted curves. Each $S_{\text{m}}^{\text{mix}}/R$ of a single bcc and liquid phase is almost constant, which are slightly smaller than ln 5 ∼ 1.609, whereas $S_{\text{m}}^{\text{mix}}/R$ of the single liquid phase is slightly larger than that of the single bcc phase. Figure 1(b) indicates that bcc-MoNbTaVW EE-HEA in both bcc and liquid single-phase dose not possess $S_{\text{m}}^{\text{mix}}/R = \ln 5$. Figure 1(b) artificially demonstrates the change in $S_{\text{m}}^{\text{mix}}/R$ due to the phase transitions: phase separation at T < 682.9 K (T^{bcc, single, low}) and a mixture of bcc and liquid phases at T = 2839.3∼2976.5 K (T^{bcc, single, high}∼T^{liquid, single, low}). The latter case corresponds to the increase/decrease in $S_{\text{m}}^{\text{mix}}/R$ due to the melt/solidification of the alloy. The values shown in dotted curves were calculated artificially, but the values of $S_{\text{m}}^{\text{mix}}/R$ were in principle valid only for the calculation results for a single-phase drawn in thick solid curves.

Fig. 1

(a) Stable phases and their amounts in the unit of one mole of the bcc-MoNbTaVW EE-HEA calculated with Thermo-calc 2020a and the TCHEA4 database. A single bcc phase was stable at a wide temperature range of 682.9 ∼ 2839.3 K (T^{bcc, single, low} ∼ T^{bcc, single, high}), whereas a single liquid phase was stable at T ≥ 2976.5 K (T^{liquid, single, low}). (b) The values of $S_{\text{m}}^{\text{mix}}/R$ of the bcc-MoNbTaVW EE-HEA at bcc and liquid single phases (solid thick curves), respectively, and those at other states (dotted curves).

Calculation results of the $S_{\text{m}}^{\text{mix}}$ and their ratio to $S_{\text{m}}^{\text{ideal}}$ of the all alloys including the bcc-MoNbTaVW EE-HEA are depicted in Fig. 2. In Fig. 2, the values of $S_{\text{m}}^{\text{mix}}/R$ were calculated for single bcc and liquid phases at 1600 and 4000 K, respectively. These temperatures were determined by the calculation results that all the 14 alloys in bcc-phase and all the 18 alloys in liquid phase exhibited a single phase at these temperatures and that $S_{\text{m}}^{\text{mix}}/R$ were almost unchanged at each temperature range forming a single phase as depicted in Fig. 1(b). Figure 2 shows that the values of the ratio, $S_{\text{m}}^{\text{mix}}/S_{\text{m}}^{\text{ideal}}$, of bcc phase (given at the second horizontal bar in each column numbered and drawn in a light-green bar in online version) are approximately in the range of 0.9 to 1, excepting for alloys with Nos. 7, 9 and 10 that tended to exhibit lower ratios than $S_{\text{m}}^{\text{mix}}/S_{\text{m}}^{\text{ideal}} \sim 0.9$. Only the CrMoNbTaTiVZr EE-HEA (No. 17) of a single bcc phase exhibits slightly greater than $S_{\text{m}}^{\text{mix}}/S_{\text{m}}^{\text{ideal}} = 1$. As for the ratios of $S_{\text{m}}^{\text{mix}}/S_{\text{m}}^{\text{ideal}}$ of liquid phase (given at the fourth bar in each column numbered and light-blue in green bar in the online version), alloys with No. 1, 2, 10 and 11 had a tendency exhibiting lower $S_{\text{m}}^{\text{mix}}/S_{\text{m}}^{\text{ideal}}$ than approximately 0.8. Here, No. 11 may be excluded from this tendency, since SSOL5 database was used for calculating $S_{\text{m}}^{\text{mix}}/R$ due to the inclusion of Pd that is out of the applicability of TCHEA4. Only the HfNbTiVZr EE-HEA (No. 5) and HfNbTaTiVZr EE-HEA (No. 13) of a single liquid phase exhibit slightly greater than $S_{\text{m}}^{\text{mix}}/S_{\text{m}}^{\text{ideal}} = 1$. As a whole, it is found that alloys with V and W simultaneously (Nos. 7 and 9) and with Al (Nos. 1, 2 and 10), in particular with atomic pairs of Al–Fe or Al–Ti, seemed to cause smaller $S_{\text{m}}^{\text{mix}}/S_{\text{m}}^{\text{ideal}}$. However, the effect of decreasing tendency of $S_{\text{m}}^{\text{mix}}/S_{\text{m}}^{\text{ideal}}$ of alloys with V and W simultaneously (Nos. 7 and 9) was weakened in further multicomponent alloying with N > 5, which can be seen in the values of $S_{\text{m}}^{\text{mix}}/S_{\text{m}}^{\text{ideal}} \sim 0.9$ or more in alloys in Nos. 14, 16 and 18. The reason for decreasing in $S_{\text{m}}^{\text{mix}}/R$ with the addition of V and W simultaneously in Nos. 7 and 9 and with Al–Fe or Al–Ti atomic pairs will be explained in Section 3.4.

Fig. 2

Values of $S_{\text{m}}^{\text{mix}}/S_{\text{m}}^{\text{ideal}}$ and $S_{\text{m}}^{\text{mix}}/R$ of bcc-EE-HEAs in single bcc calculated with Thermo-calc at 1600 K and liquid phases at 4000 K, together with original references and database for calculations. Each horizontal columns have four bars at the maximum as shown in the explanatory notes. Bars of bcc single phase of Nos. 1, 2, 11 and 18 were absent, since the single bcc phase was not achieved from calculations. Temperature ranges of a single bcc and liquid phase are supplementary shown in Fig. 5.

3.2 fcc-EE-HEAs

The calculation results of the nine fcc-EE-HEAs are summarized in Table 1 where these alloys were reported^22–32) as a single fcc-EE-HEAs through either arc-melting or induction melting and subsequent annealing in need. Of nine alloys, five alloys of AuCuNiPdPt, CoCrFeMnNi, CoCuFeMnNi, CuFeMnNiPt and CuIrNiPdPtRh with Nos. 1, 3, 5, 8 and 9, respectively, exhibited a single fcc phase from calculations with Thermo-Calc. Specifically, calculations were performed with TCHEA4 database for two alloys of CoCrFeMnNi and CoCuFeMnNi EE-HEAs (Nos. 3 and 5) whereas the other three alloys containing Pd, Pt, and Rh (Nos. 1, 8 and 9) were with SSOL5 database. The calculation results summarized in Table 1 show that the ratios of $S_{\text{m}}^{\text{mix}}/S_{\text{m}}^{\text{ideal}}$ of fcc phase were in the range of approximately 0.85 (No. 1) to 1.09 (No. 3) whereas those of liquid phase was nearly 0.68 (No. 6) to 1.01 (No. 5). Here, it should be noted that the CoCrFeMnNi (No. 3), CoCuFeMnNi (No. 5) and CuIrNiPdPtRh (No. 9) in fcc phase as well as CoCrCuFeNi (No. 2) and CoCuFeMnNi (No. 5) in liquid phase exhibit greater $S_{\text{m}}^{\text{mix}}/S_{\text{m}}^{\text{ideal}} = 1$. The greatest value of the ratios of $S_{\text{m}}^{\text{mix}}/S_{\text{m}}^{\text{ideal}} = 1.087$ was calculated in the CoCrFeMnNi (No. 3) for its fcc phase. On the other hand, alloys containing Ti at a liquid phase appear to exhibit low $S_{\text{m}}^{\text{mix}}/S_{\text{m}}^{\text{ideal}} < 0.7$ as shown in alloys CoCrFeNiTi (No. 4) and CoCuFeNiTi (No. 6). Furthermore, the CrCuFeMoNi (No. 7) in a liquid phase also exhibit small $S_{\text{m}}^{\text{mix}}/S_{\text{m}}^{\text{ideal}} \sim 0.8$. The reason for these decreases in $S_{\text{m}}^{\text{mix}}/R$ of alloys in liquid phases with Nos. 4, 6 and 7 containing Fe and Ni together and with Ti or Mo will be explained in Section 3.4.

Table 1 Values of $S_{\text{m}}^{\text{mix}}/R$ of the fcc-EE-HEAs calculated with Thermo-Calc. Calculations were performed for a single fcc and liquid phase at 1300 and 4000 K, respectively, which are common for all alloys exhibiting a single phase.

3.3 hcp-EE-HEAs

The calculations performed for four hcp-EE-HEAs with SSOL5 database are summarized in Table 2. Table 2 revealed that the DyGdHoTbY, DyGdLuTbY and DyGdLuTbTm alloys were formed into a single hcp phase as experimental results.^33,34) On the other hand, the GdHoLaTbY exhibited a single bcc phase from calculations as a stable phase instead of the hcp phase from experiment.³⁵⁾ The results clearly show that hcp-EE-HEAs possesses ideal $S_{\text{m}}^{\text{mix}}/S_{\text{m}}^{\text{ideal}} = 1$ in solid and liquid states. This coincidence is due to the chemical similarity between the constituents that are selected from heavy lanthanides mainly. The chemical similarity of Y to heavy lanthanides also contributes to the $S_{\text{m}}^{\text{mix}}/S_{\text{m}}^{\text{ideal}} = 1$. These results indicate that the hcp-EE-HEAs, which were classified into athermal solutions, behaved as ideal solutions in practice.

Table 2 Values of $S_{\text{m}}^{\text{mix}}/R$ of the hcp-EE-HEAs calculated with Thermo-Calc. Calculations were performed for a single hcp and liquid phase at 1600 and 4000 K, respectively, which are common for all alloys exhibiting a single phase. The GdHoLaTbY exhibited a single bcc phase instead of the hcp phase from calculation.

3.4 Temperature dependence of Ω_i−j(T) and its effect on $S_{\text{m}}^{\text{mix}}/R$

The calculation results as shown in Fig. 2 demonstrated that relatively large decrease in $S_{\text{m}}^{\text{mix}}/R$ from $S_{\text{m}}^{\text{ideal}}/R = \ln N$ were observed in bcc-EE-HEAs with simultaneous inclusions of V and W bcc-EE-HEAs in a bcc phase (Nos. 7, 9 in Fig. 2) and Al-containing bcc-EE-HEAs in a liquid phase as well as bcc phases (Nos. 1, 2 and 10 in Fig. 2). In contrast, Table 1 declared that $S_{\text{m}}^{\text{mix}}/R > S_{\text{m}}^{\text{ideal}}/R = \ln N$ were observed in the CoCrFeMnNi and CoCuFeMnNi fcc-EE-HEAs (Nos. 3 and 5 in Table 1) in an fcc phase. This Sub-Section discloses the reason for these peculiar deviations of $S_{\text{m}}^{\text{mix}}/R$ from $S_{\text{m}}^{\text{ideal}}/R = \ln N$.

First, the composition dependence of $S_{\text{m}}^{\text{mix}}/R$ was preliminary calculated with Thermo-Calc at T = 1600 K for bcc phase for V–W and Al–Ti binary systems and V–Nb binary system for comparison. Figure 3 shows that $S_{\text{m}}^{\text{mix}}/R$ at x_B = 0.5 of the V–W binary alloy is considerably smaller than that of V–Nb alloy that exactly exhibit $S_{\text{m}}^{\text{ideal}}/R$. Furthermore, the Al–Ti binary alloy exhibits negative $S_{\text{m}}^{\text{mix}}/R$ smaller than −0.5 at a central composition range over equiatomicity. These different tendencies between V–Nb and other binary alloys were analyzed furtherly to explain with Ω_i−j(T) of an i–j atomic pair as a result of calculating $G_{\text{m}}^{\text{mix}}$. Here, B2 ordered phase was stable in Al–Ti binary system at 1600 K at x_B ∼ 0.5–0.7 in Fig. 3, which was not investigated furtherly in the analysis of 0-th order approximation as shown later.

Fig. 3

$S_{\text{m}}^{\text{mix}}/R$ of the V–Nb, V–W sub-binary systems, together with that of the Al–Ti binary system computed with Thermo-Calc at 1600 K and $S_{\text{m}}^{\text{ideal}}/R$ for comparison. These values were calculated with Thermo-Calc using TCHEA4 database at a temperature range exhibiting a single bcc phase: 546.0–2192.3 K in V–Nb, 298.15–2632.9 K in V–W and 298.15–1666.8 K in Al–Ti sub-binary systems. The Al–Ti sub-binary system exhibited hcp and L1₀-AlTi as stable solid phases, and thus, bcc as well as B2 ordered and liquid phases only were intentionally selected for calculations. The $G_{\text{m}}^{\text{mix}}/R$ of the V–Nb, V–W and Al–Ti sub-binary systems can be approximated up to b⁽⁰⁾T term.

The above tendencies of T dependence of Ω_i−j(T) were investigated for the V–Nb, V–W and Al–Ti sub-binary systems summarized in Table 3. Table 3 shows that the values of $S_{\text{m}}^{\text{mix}}/R$ of the V–Nb and V–W, which are previously shown in Fig. 3 as 0.69319 and 0.11518, respectively, was evaluated from Table 3 as −b⁽⁰⁾/R. Similarly, $S_{\text{m}}^{\text{mix}}/R$ of the exact equiatomic Al–Ti at 1600 K as a disordered bcc phase was calculated to be −0.50842, which is enough as fist approximation to $S_{\text{m}}^{\text{mix}}/R = - 0.508$ in Fig. 3. The results of $S_{\text{m}}^{\text{mix}}/R$ of the V–Nb and V–W indicate that $S_{\text{m}}^{\text{mix}}$ cannot be $S_{\text{m}}^{\text{ideal}}$ when $H_{\text{m}}^{\text{mix}}$ has T dependence. In addition to the V–Nb sub-binary system, preliminary calculations, which are not shown in the current figures and tables, revealed that V–Mo, Nb–Ta, Mo–W sub-binary systems in a single bcc and liquid phase at 1600 and 4000 K, respectively, and V–Ta and Mo–Ta sub-binary system at 4000 K in a single liquid phase exhibited $S_{\text{m}}^{\text{mix}}/R \sim \ln 2$. As a whole, the minimum $S_{\text{m}}^{\text{mix}}/R$ was observed in the V–W sub-binary system and the maximum $S_{\text{m}}^{\text{mix}}/R \sim \ln 2$ was in V–Nb and some other sub-binary systems in the MoNbTaVW EE-HEA. These tendencies explained approximately a 13% decrease in $S_{\text{m}}^{\text{mix}}/R$ of a single bcc phase of the MoNbTaVW EE-HEA (No. 7) at 1600 K as shown in Fig. 2. These tendencies also explained the CrMoNbTaTiVZr EE-HEA (No. 17) of a single bcc phase exhibiting $S_{\text{m}}^{\text{mix}}/S_{\text{m}}^{\text{ideal}} \sim 1$ as the maximum for bcc-EE-HEAs in Fig. 2 The analysis above for bcc phase in Table 3 was also performed for liquid phase of Ni–Ti, Fe–Ti and Mo–Ni sub-binary alloy systems obtained by approximating $G_{\text{m}}^{\text{mix}}$. The results for liquid phases are summarized in Table 4 shows that these alloys in a liquid phase exhibit positive T dependence of Ω_i−j(T), resulting in decreasing $S_{\text{m}}^{\text{mix}}/R$ from its ideal value of ln 2 = 0.693.

Table 3 Coefficients (a⁽⁰⁾ and b⁽⁰⁾) of the 0th order of the interaction parameter, Ω_i−j(T), of a single bcc phase of the V–Nb, V–W and Al–Ti binary alloy systems obtained by approximating $G_{\text{m}}^{\text{mix}}$. A single bcc phase was computed at a temperature range of 546.0–2192.3 K in V–Nb, 298.15–2632.9 K in V–W and 298.15–1666.8 K in Al–Ti sub-binary systems. The Al–Ti sub-binary system exhibited hcp and L1₀-AlTi as stable solid phases, and thus, bcc and liquid phases only were intentionally selected for calculations. The coefficients obtained by approximating $G_{\text{m}}^{\text{mix}}$ up to T term with Ω_i−j(T) = 4{a⁽⁰⁾ + (b⁽⁰⁾ + R ln 2)T} from the sub-regular solution model where $S_{\text{m}}^{\text{mix}}/R$ was able to be computed as −b⁽⁰⁾/R.

Table 4 Coefficients (a⁽⁰⁾ and b⁽⁰⁾) of the 0th order of the interaction parameter, Ω_i−j(T), of a single liquid phase of the of Ni–Ti, Fe–Ti and Mo–Ni binary alloy systems. A single liquid phase was obtained at a temperature range of 1583.7–4000 K in Ni–Ti, 1585.8–4000 K in Fe–Ti and 2018.3–4000 K in Mo–Ni sub-binary systems. The coefficients obtained by approximating $G_{\text{m}}^{\text{mix}}$ up to T term with Ω_i−j(T) = 4{a⁽⁰⁾ + (b⁽⁰⁾ + R ln 2)T} from the sub-regular solution model where $S_{\text{m}}^{\text{mix}}/R$ was able to be computed as −b⁽⁰⁾/R.

A similar analysis was performed for all the sub-binary systems of the CoCrFeMnNi EE-HEA as shown in Fig. 4. Features of the sub-binary systems of the CoCrFeMnNi EE-HEA were that the values of the $S_{\text{m}}^{\text{mix}}/R$ at each equiatomic composition (at x_B = 0.5) are smaller than $S_{\text{m}}^{\text{mix}}/R = \ln 2$ in only three sub-binary systems of Fe–Ni, Fe–Mn and Cr–Mn. This would considerably contribute to increase the $S_{\text{m}}^{\text{mix}}/R$ of the fcc-CoCrFeMnNi HEA (No. 3) by 9% than ln N as demonstrated in Table 1. In general, T dependence of Ω_i−j(T) of the EE-HEAs necessarily became averaged in magnitude as shown in eqs. (8) due to alloying within the maximum and minimum tendencies of T dependences of Ω_i−j(T)’s of the sub-binary systems. This averaging can be confirmed in the second line of eqs. (9) in a process of summation for j ≠ i and i. Figure 4 differs from Fig. 3 in that most of $S_{\text{m}}^{\text{mix}}/R$’s excepting for Cr–Fe, Co–Ni, Fe–Mn sub-binary systems in Fig. 4 exhibit asymmetric profiles against x_B = 0.5. These asymmetric characteristics of the plots never affect the present approximation for EE-HEAs that hold exact equiatomicity. However, they might affect the applicability of the present approximation technique to near-equiatomic- (NE-) HEAs. These asymmetric characteristics could be dealt with a sub-regular solution model where Ω_i−j(T) in eqs. (6) allows v from odd numbers.

Fig. 4

$S_{\text{m}}^{\text{mix}}/R$ of the sub-binary systems comprising CoCrFeMnNi fcc-EE-HEA calculated at 1300 K with Thermo-Calc using TCHEA4 database. The calculations were performed based on a sub-regular solution model, which can deal with asymmetry against x_B = 0.5.

3.5 Advantageous features of $G_{\text{m}}^{\text{excess(S-RSM)}}\unicode{x2013}T$ and $G_{\text{m}}^{\text{mix}}\unicode{x2013}T$ charts

The above tendencies of $S_{\text{m}}^{\text{mix}}/R$ were investigated for all the bcc- and fcc-EE-HEAs in a form of $G_{\text{m}}^{\text{excess(S-RSM)}}\unicode{x2013}T$ chart³⁷⁾ based on the sub-regular solution model. Figure 5 shows that most of the plots, excepting for three fcc-EE-HEAs of CoCrFeMnNi (No. 3), CoCuFeMnNi (No. 5) and CuIrNiPdPtRh (No. 8) alloys in each single fcc phase in Fig. 5(b), can be approximated as linear functions. This suggests that most of the EE-HEAs possess a simple linear T dependence of Ω_i−j(T), indicating the presence of b⁽⁰⁾ and absence of c⁽⁰⁾ term as shown in eqs. (10) and Tables 3 and 4. The $G_{\text{m}}^{\text{excess(S-RSM)}}\unicode{x2013}T$ chart indicates that the positive or negative sign of the slope of the plots indicates the decrease or increase in $S_{\text{m}}^{\text{mix}}/R$ from its ideal value of $S_{\text{m}}^{\text{mix}}/R = \ln N$. Figure 5(a) demonstrates that almost all the bcc-EE-HEAs in single bcc and liquid phases (solid and dotted lines, respectively) exhibited positive slopes over a T range of 1300 K (bcc phase) and 4000 K (liquid phase), suggesting $S_{\text{m}}^{\text{mix}}/R < \ln N$. Minor exceptions were the CrMoNbTaTiVZr EE-HEA (No. 17) of a single bcc phase and the HfNbTiVZr (No. 5) and HfNbTaTiVZr (No. 13) alloys formed in a liquid phase. These three are plotted almost flat in Fig. 5(a), corresponding to $S_{\text{m}}^{\text{mix}}/S_{\text{m}}^{\text{ideal}} \sim 1$ in Fig. 2. The AlCoFeNiTi alloy (No. 1) in the liquid state exhibits the largest positive slope in Fig. 5(a), explaining the smallest value of the ratio of $S_{\text{m}}^{\text{mix}}/S_{\text{m}}^{\text{ideal}} = 0.354$ as shown in Fig. 2. In strong contrast, Fig. 5(b) indicate that some of the fcc-EE-HEAs formed in an fcc solid solutions over a T range of 1600 K exhibited weak negative slopes, such as CoCrFeMnNi (No. 3) and CoCuFeMnNi (No. 5) and CuIrNiPdPtRh (No. 9) systems. The negative slopes of these three alloys in Fig. 5(b) explained $S_{\text{m}}^{\text{mix}}/R > \ln N$. Here, as supplementals, T dependency of Ω_i−j(T) of the fcc-CoCrFeMnNi EE-HEA exhibiting $S_{\text{m}}^{\text{mix}}/R > \ln N$ does not agree with an early report³⁷⁾ where the plots in $G_{\text{m}}^{\text{excess(S-RSM)}}\unicode{x2013}T$ chart indicates the positive slope. This disagreement with the early³⁷⁾ and the present study would be due to the different databased for the analysis. Further investigation will be performed shortly regarding this difference.

Fig. 5

$G_{\text{m}}^{\text{excess(S-RSM)}}\unicode{x2013}T$ chart based on a sub-regular solution model of (a) bcc- and (b) fcc-EE-HEAs and their liquid phases calculated with Thermo-Calc. The positive or negative sign of the slope of the plots indicates the decrease or increase, respectively, in $S_{\text{m}}^{\text{mix}}/R$ from its ideal value of $S_{\text{m}}^{\text{mix}}/R = \ln N$. $G_{\text{m}}^{\text{excess(S-RSM)}}\unicode{x2013}T$ chart can be converted to $G_{\text{m}}^{\text{mix}}\unicode{x2013}T$ chart by subtracting $TS_{\text{m}}^{\text{ideal}} = RT\ln N$ due to $G_{\text{m}}^{\text{mix}} = G_{\text{m}}^{\text{excess(S-RSM)}} - TS_{\text{m}}^{\text{ideal}}$ where $G_{\text{m}}^{\text{mix}}\unicode{x2013}T$ chart directly express $S_{\text{m}}^{\text{mix}}$ by the slopes of the plots.

The significance of the present analysis was to focus on the T dependence of Ω_i−j(T). The present results indicate that $S_{\text{m}}^{\text{mix}}/R < \ln N$ when T dependence of Ω_i−j(T) > 0 and $S_{\text{m}}^{\text{mix}}/R > \ln N$ when T dependence of Ω_i−j(T) < 0. The above results suggested that selecting binary systems with negative temperature dependence of Ω_i−j(T) efficiently from the database would lead to developing novel super EE-HEAs. This partially takes place in the fcc-CoCrFeMnNi EE-HEA where seven sub-binary systems apart from only the three Fe–Ni, Fe–Mn and Cr–Mn sub-binary systems tended to increase $S_{\text{m}}^{\text{mix}}/R$. This example of fcc-CoCrFeMnNi EE-HEA implies that one could find out truly HEAs that possess considerably high S values than $S_{\text{m}}^{\text{mix}}/R = \ln N$ by utilizing the negative T dependence of H^mix.

The $G_{\text{m}}^{\text{mix}}\unicode{x2013}T$ chart can be drawn by subtracting RT ln N due to a relationship of $G_{\text{m}}^{\text{mix}} = G_{\text{m}}^{\text{excess}} - TS_{\text{m}}^{\text{ideal}}$. An advantageous aspect of the $G_{\text{m}}^{\text{mix}}\unicode{x2013}T$ chart is that a fundamental thermodynamic relationships, G = H − TS and S = −(∂G/∂T)_P hold between $G_{\text{m}}^{\text{mix}}$ and T. The latter relationship helps to evaluate $S_{\text{m}}^{\text{mix}}$ directly from the slope of the $G_{\text{m}}^{\text{mix}}\unicode{x2013}T$ chart, which can be regarded as the significant aspect. In contrast, the $G_{\text{m}}^{\text{mix}}\unicode{x2013}T$ chart does not provide the universal trends among the EE-HEAs with different N, such as, $S_{\text{m}}^{\text{mix}}/R < \ln N$ when T dependence of Ω_i−j(T) > 0 and $S_{\text{m}}^{\text{mix}}/R > \ln N$ when T dependence of Ω_i−j(T) < 0 from the $G_{\text{m}}^{\text{excess(S-RSM)}}\unicode{x2013}T$ chart. However, the $G_{\text{m}}^{\text{mix}}\unicode{x2013}T$ chart is of great importance in a process of approximating $G_{\text{m}}^{\text{mix}}$ and subsequent obtaining coefficients of the v-th order, in particular, a⁽⁰⁾, b⁽⁰⁾ and c⁽⁰⁾ of $L_{\text{EE-HEA}}^{(0)}(T)$ in eqs. (9) for EE-HEAs. For instance, the representative $L_{\text{EE-HEA}}^{(0)}(T)$ of bcc-MoNbTaVW and fcc-CoCrFeMnNi with a single solid phase summarized in Table 5 shows that the sign of Judgment Factor, (b⁽⁰⁾ + R ln N) + c⁽⁰⁾(ln T + 1), whether positive or negative determines $R\ln N - S_{\text{m}}^{\text{mix}} > 0$ ($S_{\text{m}}^{\text{mix}} < R\ln N$) or $R\ln N - S_{\text{m}}^{\text{mix}} < 0$ ($S_{\text{m}}^{\text{mix}}/R > R\ln N$), respectively. The wide applicability of $G_{\text{m}}^{\text{excess(S-RSM)}}\unicode{x2013}T$ chart necessarily results from $G_{\text{m}}^{\text{mix}}\unicode{x2013}T$ charts where the latter chart has a process of calculating $G_{\text{m}}^{\text{mix}}$ by eqs. (5) and subsequent fitting as eqs. (6) and (7) to obtain coefficients. It is expected that researches of EE- and NE-HEAs will proceed by utilizing advantageous aspects and by compensating for each disadvantageous point between $G_{\text{m}}^{\text{excess(S-RSM)}}\unicode{x2013}T$ and $G_{\text{m}}^{\text{mix}}\unicode{x2013}T$ charts.

Table 5 Coefficients (a⁽⁰⁾, b⁽⁰⁾ and c⁽⁰⁾) of the 0th order of the interaction parameter, $L_{\text{EE-HEA}}^{(0)}(T)$, of bcc-MoNbTaVW and fcc-CoCrFeMnNi a single solid phase. The coefficients were obtained by approximating $G_{\text{m}}^{\text{mix}}$ up to T term or T ln T term for the latter cases from eqs. (9) as $H_{\text{m,EE-HEA}}^{\text{mix}} = L_{\text{EE-HEA}}^{(0)}(T) = a^{(0)} + b^{(0)}T + c^{(0)}T\ln T$ where $S_{\text{m}}^{\text{mix}}/R$ was able to be computed as −b⁽⁰⁾/R for the former and −{b⁽⁰⁾ + c⁽⁰⁾(ln T + 1)}/R for the latter EE-HEA. The magnitude of $S_{\text{m}}^{\text{mix}}$ smaller or larger than R ln N has a relationship to the judgment factor (JF), $R\ln N - S_{\text{m}}^{\text{mix}} = (b^{(0)} + R\ln N) + c^{(0)}(\ln T + 1)$, which corresponds with the slope of the $G_{\text{m}}^{\text{excess(S-RSM)}}\unicode{x2013}T$ chart by its sign of whether positive or negative, resulting in $R\ln N - S_{\text{m}}^{\text{mix}} > 0$ ($S_{\text{m}}^{\text{mix}} < R\ln N$) or $R\ln N - S_{\text{m}}^{\text{mix}} < 0$ ($S_{\text{m}}^{\text{mix}}/R > R\ln N$), respectively.

3.6 Estimation of ranges of $S_{\text{m}}^{\text{mix}}/R$ and $S_{\text{m}}^{\text{mix}}/S_{\text{m}}^{\text{ideal}}$ of EE-HEAs

In a framework of the sub-regular solution model, the ranges of $S_{\text{m}}^{\text{mix}}/R$ and $S_{\text{m}}^{\text{mix}}/S_{\text{m}}^{\text{ideal}}$ of the EE-HEAs were estimated by plotting the alloys in an X-Y chart with $H_{\text{m}}^{\text{excess}}$ and $S_{\text{m}}^{\text{excess}}$, respectively, which consist of $H_{\text{m}}^{\text{excess(S-RSM)}}$ in a part of eqs. (11). Figure 6 depict the plots of the EE-HEAs for X = a⁽⁰⁾ − c⁽⁰⁾T and Y = {b⁽⁰⁾ + R ln 2 + c⁽⁰⁾(ln T + 1)}/(ln N/ln 2) as well as two additional perpendicular axes (Y₁ and Y₂) for comparison. The ln N/ln 2 in Y-axis was the conversion factor to adjust the magnitude of the quantities of the N-component EE-HEAs to that of binary alloys (N = 2). In Fig. 6, the ideal solution is demonstrated by Y = 0, Y₁ = ln N, and Y₂ = 1. The analysis with Fig. 6 revealed that the EE-HEAs tended to be plotted along a zone (Zone 1) that distributes through the second and fourth quadrants of Fig. 6 in the X-Y axes. The Zone 1 consists of a direct proportional approximation function of Y = −0.15X drawn with a dash-dotted line and two parallel lines of different intersections of ±0.85.

Fig. 6

Relationships between $\text{X} = H_{\text{m}}^{\text{excess}}$ and $\text{Y} = - S_{\text{m}}^{\text{excess}}/(\ln N/\ln 2)$ where $H_{\text{m}}^{\text{excess}}$ and $S_{\text{m}}^{\text{excess}}$ consist of $H_{\text{m}}^{\text{excess(S-RSM)}}$ in a framework of the sub-regular solution model. The factor of ln N/ln 2 is a denominator for the N-component EE-HEAs to convert their magnitude to those of binary alloy (N = 2). The X-Y chart is accompanied by additional perpendicular axes (Y₁ and Y₂) with reverse direction to Y-axis. The values of Y’s were calculated after determining T for the fcc-EE-HEAs with Nos. 3, 5 and 8 because of the presence of c⁽⁰⁾ term. Besides, the values of the EE-HEAs with hcp structure needed to use coefficients up to the g⁽⁰⁾ term by referring to eqs. (A.2) and (A.3) in the Appendix because of a rather small increase in $G_{\text{m}}^{\text{mix}}$ at a low-temperature range of T ≤ 1000 K. However, the magnitude of the X of the EE-HEAs with hcp structure can be evaluated to be ∼0 in the units of kJ mol⁻¹ and that of the Y ∼ 0 Jmol⁻¹K⁻¹.

The ranges of $S_{\text{m}}^{\text{mix}}/R$ and $S_{\text{m}}^{\text{mix}}/S_{\text{m}}^{\text{ideal}}$ of EE-HEAs were estimated from conventional $H_{\text{m}}^{\text{mix}}$ in X-axis. Figure 6 demonstrates that all the EE-HEAs exhibited H^mix in a range of −20 ≤ H^mix/kJ mol⁻¹ ≤ 5, which exactly agreed with the data by Zhang et al.³⁸⁾ and almost corresponded with −22 ≤ H^mix/kJ mol⁻¹ ≤ 7 by Guo and Liu.³⁹⁾ On the other hand, the present H^mix in Fig. 6 in a range of −12 ≤ H^mix/kJ mol⁻¹ ≤ 5 by excepting the bcc-HEA with No. 16 roughly agreed with −11.6 ≤ H^mix/kJ mol⁻¹ ≤ 3.2 reported by Guo et al.⁴⁰⁾ In this exceptional case of −12 ≤ H^mix/kJ mol⁻¹ ≤ 5, a rectangular area was formed with −0.5 ≤ Y ≤ 1.15 from approximate maximum and minimum values of $S_{\text{m}}^{\text{mix}}/S_{\text{m}}^{\text{ideal}}$, which were given from the fcc-EE-HEAs with Nos. 3 and bcc-EE-HEAs with Nos. 10, respectively. Then, considering the overlap between the rectangular and Zone 1 led to the truncated polygon hatched in Fig. 6, which includes almost all the plots of the EE-HEAs exception for the bcc-EE-HEA with No. 16 (CrMoNbReTaVW). Thus, it was found that Zone 1 worked effectively to screening the EE-HEAs with the support of the limits of conventional $H_{\text{m}}^{\text{mix}}$ of the EE-HEAs.

The exceptional plot of the bcc-EE-HEA with No. 16 in Fig. 6 was included in another zone (Zone 2) that contains candidates of EE-HEA as well as X-axis with Y = 0, Y₁ = ln N, and Y₂ = 1 as defined by the ideal solid solution. Although the range (width) of Zone 2 is unclear, Zone 2 given in Fig. 6 also contains the bcc-EE-HEA with Nos. 5, 6, 13, and 17, the fcc-EE-HEA with Nos. 5 and 9 as well as hcp-EE-HEAs with Nos. 1–3 in addition to bcc-EE-HEA with No. 16. Thus, Zone 2 based on the characteristics of the ideal- and near-ideal solid solutions is significant as well as Zone 1 for screening the plots of EE-HEAs.

The results above suggested that the EE-HEAs would possess $S_{\text{m}}^{\text{mix}}/R$ and $S_{\text{m}}^{\text{mix}}/S_{\text{m}}^{\text{ideal}}$ in an approximate range of 20% lower to 10% greater than the ideal solid solutions. It is expected that novel EE-HEAs can be developed by utilizing the above results from Fig. 6.

4. Conclusions

Exact equiatomic high-entropy alloys (EE-HEAs) with N-component (N ≥ 5) formed into a single-phase experimentally confirmed were analyzed thermodynamically for such representative alloys as bcc-NbMoTaVW, fcc-CoCrFeMnNi, and hcp-heavy lanthanides with and without Y. The analysis revealed that mixing entropy (S^mix) does not correspond with either ideal entropy (S^ideal) nor configuration entropy (S^config) in case of presenting temperature dependence of Ω_i−j(T) of i-j atomic pairs in mixing enthalpy (H^mix) of a sub-binary system. The calculations with Thermo-Calc using TCHEA4 as well as SSOL5 database indicate that bcc-MoNbTaVW and NbTaTiVW EE-HEAs exhibited S^mix normalized with gas constant (R), S^mix/R, smaller by 13% to S^ideal/R = ln N. In contrast, CoCrFeMnNi, CoCuFeMnNi fcc-EE-HEAs exhibited greater S^mix/R than ln N. These different tendencies can be explained by the T dependence of Ω_i−j(T) in H^mix. As an exceptional case, the hcp-EE-HEAs comprising heavy lanthanides mainly from athermal solutions behave as ideal solutions. The $G_{\text{m}}^{\text{excess(S-RSM)}}\unicode{x2013}T$ chart from the sub-regular solution model declared that the positive or negative signs of the slope of plots directly represent S^mix/R < ln N or S^mix/R > ln N, respectively. The present results from the sub-regular solution model have clarified that EE-HEAs do not always exhibit S^mix/R = ln N in case of the presence of T dependence of Ω_i−j(T) in H^mix that affects S^mix. The present results with the help of H^mix analysis revealed that the EE-HEAs would possess S^mix/R approximately 20% smaller or 10% greater than S^ideal/R as their minimum or maximum limits.

Acknowledgment

This work was supported by JSPS KAKENHI Grant Number JP17H03375.

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Appendices

Appendix

The following describes a case when $G_{\text{m}}^{\text{mix}}$ is necessary to describe with the succeeding terms to cT ln T in eqs. (5), such as eqs. (A.1). According to unary SGTE (Scientific Group Thermodata Europe) database,³⁶⁾ ${}^{0}G_{\text{m}}^{\alpha } - {}^{0}H_{m}^{298.15}$ could be described with T ln T and polynomials of Tⁿ (0 ≤ n ≤ 4 or 7), T^−m, (m = 1 or 9) where ${}^{0}H_{m}^{298.15}$ is an enthalpy at 298.15 K from standard element reference stage. Below, $G_{\text{m}}^{\text{mix}}$ was supposed to be described with T ln T and polynomials as eqs. (A.1).   

\begin{align} G_{\text{m}}^{\text{mix}} &= a + bT + cT\ln T + dT^{2} + eT^{3} \\ &\quad + fT^{-1} + gT^{4} + hT^{7} + iT^{-9}+ \end{align} (A.1)

Applying G = H − TS and S = −(∂G/∂T)_P to eqs. (A.1) yields eqs. (A.2) and (A.3).   

\begin{align} S_{\text{m}}^{\text{mix}} &= -b - c(\ln T + 1) - 2dT - 3eT^{2} \\ &\quad + fT^{-2} - 4gT^{3} - 7hT^{6} + 9iT^{-10} \end{align} (A.2)

  

\begin{align} H_{\text{m}}^{\text{mix}} &= a - cT - dT^{2} - 2eT^{3} + 2fT^{-1} \\ &\quad - 3gT^{4} - 6hT^{7} + 10iT^{-9} \end{align} (A.3)

In case of necessity of using the succeeding terms to cT ln T in eqs. (5), eqs. (A.1) to (A.3) could be used for analysis. In this case, a criteria to judge (Judgement Factor) whether the system of interest has S^mix/R greater or smaller than ln N is expressed as eqs. (A.4)/R instead of {−b⁽⁰⁾ − c⁽⁰⁾(ln T + 1)}/R in Table 3.   

\begin{align} &\{-b^{(0)} - c^{(0)}(\ln T + 1) - 2d^{(0)}T - 3e^{(0)}T^{2} \\ &\quad + f^{(0)}T^{-2} - 4g^{(0)}T^{3} - 7h^{(0)}T^{6} + 9i^{(0)}T^{-10}\} \end{align} (A.4)

In practice, eqs. (A.1) and subsequent eqs. (A.2) to (A.4) do not necessary in approximating $G_{\text{m}}^{\text{mix}}$ instead of eqs. (5), since even a complicated formulation of eqs. (A.1) could be divided into simple formulations, such as eqs. (5) by selecting appropriate temperature ranges. Equations (A.2) and (A.3) demonstrate that coefficients of a and b in eqs. (5) as well as relevant a⁽⁰⁾ and b⁽⁰⁾ from 0-th order approximation in eq. (7) are inherent coefficients to $H_{\text{m}}^{\text{mix}}$ and $S_{\text{m}}^{\text{mix}}$, respectively. On the other hand, coefficients c and c⁽⁰⁾ in eqs. (5) and subsequent coefficients, d to i in eqs. (A.2) and (A.3) and relevant d⁽⁰⁾ to i⁽⁰⁾ in the 0-th order approximation, have a characteristic of affecting both $H_{\text{m}}^{\text{mix}}$ and $S_{\text{m}}^{\text{mix}}$.