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[Taichi Abe](https://orcid.org/0000-0002-5065-0939), [Kwangsik Han](https://orcid.org/0000-0002-5701-5348), Yumi Goto, [Ikuo Ohnuma](https://orcid.org/0000-0003-4874-4941), Toshiyuki Koyama

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[Stabilization of Equiatomic Solutions Due to High-Entropy Effect](https://mdr.nims.go.jp/datasets/0845e3c0-4e55-4a71-b5ed-3129cbe2974e)

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Stabilization of Equiatomic Solutions Due to High-Entropy EffectStabilization of Equiatomic Solutions Due to High-Entropy EffectTaichi Abe1,+, Kwangsik Han1, Yumi Goto1, Ikuo Ohnuma1 and Toshiyuki Koyama21National Institute for Materials Science, Tsukuba 305-0047, Japan2Nagoya University, Nagoya 464-8601, JapanThe stability of solid-solution phases in FCC and BCC lattices was examined in multi-component alloys based on the CALPHADtechnique using the compound energy formalism and regular solution model. From the thermodynamic calculations, it was found in ternarysystems that the single solid-solution phase became stable around the equiatomic composition where the configurational entropy was the largestvalue. The transition temperature from the disordered phase to ordered phase(s) or miscibility gap(s) decreased with the increasing number ofelements in the system. The order-disorder transition temperature on the FCC lattice was affected by the end member of the ordered phasesexisting at the equiatomic composition, whereas it was not significant for the order-disorder transition in the BCC lattice. The single solid-solution phase region at equiatomic compositions was affected by variations in the interaction parameters. In multi-component systems, thevariations were averaged with increasing the number of elements in the system. This suggests that high-entropy alloys can afford a variety ofelements. This study shows that the disordered state can be formed in multicomponent systems around the equiatomic composition and suggestsclearly that due to the high-entropy effect, the solution phases are stabilized. [doi:10.2320/matertrans.MT-M2022167](Received October 19, 2022; Accepted January 16, 2023; Published February 3, 2023)Keywords: regular solution, order-disorder transition, phase stability, miscibility gap1. IntroductionHigh-entropy alloys (HEAs)1) have been intensivelyinvestigated and synthesized in various alloy systems withface-centered cubic (FCC),2­5) body-centered cubic(BCC),6,7) and hexagonal close-packed (HCP) structures.8)HEAs are defined based on the configurational entropy, Sconf,in the Bragg-Williams-Gorsky (B-W-G) approximation,which is given by Sconf ¼ �RPxi ln xi, where R is the gasconstant and xi is the mole fraction of element i. By mixingmultiple elements, Sconf increases and becomes “high” inHEAs. The Sconf of equiatomic alloys with five elements isR ln 5 ’ 1:61R, which is used as a guiding value for HEAs.This concept has been expanded to lower entropy alloys,such as medium-entropy alloys (MEAs) with 0.69R <Sconf < 1.61R and low-entropy alloys with Sconf < 0.69R =R ln 2. According to the narrow definition of HEAs, themicrostructure of these alloys consists of a single solid-solution phase. However, in many HEAs, microstructuresinclude second phases such as intermetallic compounds orother solution phases. As the co-existence of the secondphases changes the Sconf of the matrix phase, they may affectthe mechanical properties of HEAs.9) For the formation ofHEAs, Yang and Zhang10) proposed two parameters: +,which is related to the mixing quantity and ¤, which is relatedto the size misfits between atoms. They found that HEAscan form in the region where + ² 1.1 and ¤ ² 6.6%. Basedon phase diagrams, the existence of a single solid-solutionphase region is prevented by the formation of i) intermetalliccompounds, ii) miscibility gaps, or iii) other solid-solutionphases. Although Takeuchi et al.8,11­13) examined thestability of solution phases based on the valence electronconcentration (VEC), the stability of solution phases withrespect to intermetallic compounds and miscibility gaps hasnot yet been discussed. Therefore, the purpose of this studywas to examine the relative stability of single-phase HEAswith FCC or BCC structures in the formation of compoundsand miscibility gaps based on the CALPHAD (CALculationsof PHAse Diagrams) technique.14)2. Gibbs Energy Functions2.1 Gibbs energy of solution phasesThe Gibbs energy of the solution phases, Gdism , is describedby the substitutional solution model14) as follows:Gdis ¼Xxi0Gdisi þ RTXxi ln xi þGexcess ð1Þwhere 0Gdisi and T are the Gibbs energy of a pure element iand the temperature in Kelvin, respectively. The third term onthe right-hand side, Gexcess, is the excess Gibbs energy givenby the Redlich-Kister (R-K) polynomial as follows:Gexcess ¼Xi;jxixjX¯¼0LðvÞi;j ðxi � xjÞ¯ ð2Þwhere LðvÞi;j is the v-th R-K parameter between elements i andj. When LðvÞi;j ¼ 0 (except Lð0Þi;j 6¼ 0), this solution is calledthe regular solution. When Lð0Þi;j < 0 (attractive interactionbetween i and j), the solution phases can have miscibilitygaps15) at low temperatures, while when Lð0Þi;j > 0, orderedphases are formed instead of miscibility gaps. For the idealsolution, the excess Gibbs energy term of eq. (1) is zero(Gexcess = 0). For equiatomic n-component alloys, wherexi = 1/n, eq. (2) can be simply written as:Gexcess ¼ 1n2Xi;jLð0Þi;j : ð3ÞBy using a pair interaction energy, wi, j, eq. (3) can be writtenas:Gexcess ¼ zn2Xi;jwi;j ð4Þwhere z is the coordination number in the nearest-neighborshell, and is equal to 12 and 8 for the FCC and BCC phases,respectively. A negative value of the parameter (wi, j < 0)indicates an attractive interaction between the i and j atoms,whereas a positive value indicates a repulsive interaction+Corresponding author, E-mail: abe.taichi@nims.go.jpMaterials Transactions, Vol. 64, No. 4 (2023) pp. 877 to 884©2023 The Japan Institute of Metals and Materialshttps://doi.org/10.2320/matertrans.MT-M2022167between the i and j atoms. In the following calculations, thePxi0Gdisi term in eq. (1) is zero because the Gibbs energyof pure elements is considered as the reference state. Onlybinary interactions were considered for the excess Gibbsenergy term.2.2 Gibbs energy of BCC-based phases with twosublatticesThe B2 ordered phase and A2 disordered phase weremodeled using the two-sublattice model. The sublatticeconfiguration of the B2 phase is illustrated in Fig. 1(a). Itis selected in such a way that all sublattices are equivalentand have an equal number of sites and bonds to the othersublattices. The Gibbs energy equation for the two-sublatticephase in an i- j binary system using the compound energyformalism (CEF)16) isG2sl ¼Xi;jyð1Þi yð2Þj Gi:j þ RT12X2m¼1XiyðmÞi ln yðmÞi ð5Þwhere Gi: j is the Gibbs energy of the i: j compound. A colonin the suffix of the Gibbs energies separates elements ondifferent sublattices and a comma separates elements on thesame sublattice. In the two-sublattice model, the sublatticesare numbered as #1 and #2 from left to right in the suffix.For example, i: j stands for i on the sublattice#1 and j on thesublattice#2. By counting the number of the unlike pairs, theGibbs energies of the end members are given by Gi: j =Gj:i = 4wi, j and Gi:i = Gj:j = 0.To simplify the integration of an ordered BCC phase with adatabase where many systems have BCC phases without anyorderings, it is advantageous to partition the Gibbs energy ofan ordered phase into two parts (eq. (6)), as follows:Gord ¼ GdisðfxigÞ þ�GordðfyðmÞi gÞ ð6Þwhere {xi} and fyðmÞi g denote the mole fraction and the sitefraction of all independent components. When the orderedphase is disordered, the second term on the right side ofeq. (6) becomes zero, i.e., �GordðfyðmÞi gÞ ¼ 0; Gdis({xi})contains all the parameters needed to describe the disorderedphase and is given by eq. (1). �GordðfyðmÞi gÞ is calculatedusing the sublattices, and one way to ensure that it is zerowhen the phase is disordered is to calculate eq. (5) twice,once with the original site fractions, yðmÞi , and once with themreplaced by the mole fractions, xi, as follows:�Gord ¼ G2slðfyðmÞi gÞ �G2slðfyðmÞi ¼ xigÞ: ð7Þ2.3 Gibbs energy of FCC-based phases with foursublatticesTo describe ordered structures in the FCC lattice, the foursublattices were selected in the way that all sublattices wereequivalent, had an equal number of sites, and bonds tothe other sublattices. The Gibbs energy equation for a four-sublattice phase in an i- j binary system using the CEF is:G4sl ¼Xi;j;k;lyð1Þi yð2Þj yð3Þk yð4Þl Gi:j:k:lþ RT14X4m¼1XiyðmÞi ln yðmÞi ð8Þwhere Gi:j:k:l is the Gibbs energy of the i: j:k:l compound. Inthis model, as presented in Fig. 1(b), the FCC lattice isdivided into four simple cubic sublattices which areequivalent to each other. In the four-sublattice model, thesublattices are numbered as #1, #2, #3, and #4 from left toright in the suffix of the Gibbs energy of the compounds.Therefore, L10 and L12 ordered phases are written as#1=#2º#3=#4 (L10), and #1=#2=#3º#4 (L12). Becauseof the symmetry of the sublattice configurations, the Gibbsenergies of the end members in the i-j binary system haveto fulfill the relations:i atomj atom122222222(a) B2 phaseSublattice #4Sublattice #1 Sublattice #2 Sublattice #3(b) L10 phase43124444444132Sublattice #2Sublattice #1Fig. 1 (a) Four equivalent sublattices for the FCC lattice in the L10 phase, where #1=#2º#3=#4, and (b) two equivalent sublattices forthe BCC lattice in the B2 phase, where #1º#2 in the i-j binary system.T. Abe, K. Han, Y. Goto, I. Ohnuma and T. Koyama878Gi:i:i:j ¼ Gi:i:j:i ¼ Gi:j:i:i ¼ Gj:i:i:i ¼ 3wi;j;Gi:i:j:j ¼ Gi:j:i:j ¼ Gi:j:j:i ¼ Gj:i:i:j ¼ Gj:i:j:i¼ Gj:j:i:i ¼ 4wi;j;Gj:j:j:i ¼ Gj:j:i:j ¼ Gj:i:j:j ¼ Gi:j:j:j ¼ 3wi;j:ð9ÞUsing the same manner in eq. (9), the constraints due tothe symmetry are applied for the i- j-k ternary and i- j-k-lquaternary end members. They can be given by countingunlike pairs, asGi:i:j:k ¼ 2wi;j þ 2wi;k þwj;k;Gi:j:k:l ¼ wi;j þ wi;k þ wi;l þwj;k þ wj;l þwk;l:ð10ÞThe Gibbs energy of FCC phases with four sublattices isgiven by the CEF using eq. (7), where �Gord ¼G4slðfyðmÞi gÞ �G4slðfyðmÞi ¼ xigÞ.To calculate the phase equilibria, isothermal sections, andtransition temperatures, the Gibbs energy functions werewritten in a TDB format17) for Pandat,18) CaTCalc,19) andThermo-Calc20) software packages used in the present work.3. Results and Discussions3.1 Stability range of the single solid-solution phase inbinary systemsThe A-B binary phase diagrams are calculated using theGibbs energies defined in Section 2. When the interactionparameter between A and B is repulsive, i.e., wA,B = +1kJmol¹1 > 0, the solid-solution phase decomposes into twophases at low temperatures, as presented in Fig. 2(a). Whenthe interaction parameter between A and B is attractive, i.e.,wA,B = ¹1 kJmol¹1 < 0, the long-range orderings take placeat low temperatures, as presented in Figs. 2(b) and (c).For the miscibility gap in the A-B binary system, the peaktemperature of the two-phase region can be obtained byputting the second derivative of the Gibbs energy in eq. (1)with respect to the mole fraction to zero,d2Gdisdx2B¼ 0: ð11ÞAt the equiatomic composition, the peak temperature TC ofthe miscibility gap is +zwA,B/2R (+6wA,B/R in the FCClattice and +4wA,B/R in the BCC lattice). For the order-disorder transition in the A-B binary system, TC can beobtained from the second derivative of the Gibbs energy ineq. (5) for A2/B2 and in eq. (8) for A1/L10 with respectto the site fraction to zero. The peak temperature of the A2/B2 transition is ¹4wA,B/R at the equiatomic composition.The A1/L10 transition at the equiatomic composition is¹2wA,B/R. Using thermodynamic software packages, theserelations were confirmed numerically as presented inFigs. 3(a) and (b).It should be noted that the purpose of this study is todiscuss the relative stability of the disordered state, whereasthe thermodynamic modeling of the order-disorder tran-sitions. Based on the B-W-G approximation, it is known that(a)123－RT/wABA1L12L10L120 0.2 0.4 0.6 0.8 1Mole fraction of B00 0.2 0.4 0.6 0.8 11234A2B2Mole fraction of B－RT/wAB(b)(c)02460 0.2 0.4 0.6 0.8 1Mole fraction of BRT/wABA1A1+A1Fig. 2 A-B binary phase diagrams: (a) miscibility gap with the interaction parameter wA,B = +1 kJmol¹1 and the coordination numberz = 12, (b) order-disorder transition in the BCC lattice with wA,B = ¹1 kJmol¹1, and (c) order-disorder transitions in the FCC latticewith wA,B = ¹1 kJmol¹1. The dotted lines indicate the second-order transitions.Stabilization of Equiatomic Solutions Due to High-Entropy Effect 879the Al/L10 transition at the equiatomic composition is thesecond order21) as presented in Fig. 2(c), while it is the firstorder transition in the real alloy systems.22,23) For detaileddiscussions of the order-disorder transitions, it may requiremore precise thermodynamic models such as the clustervariation method.3.2 Stability range of the single solid-solution phase internary systemsIn this section, the stability of the solid-solution phasesis examined in the A-B-C ternary system. The interactionparameters used in the calculations are as follows:wA,B ¼ wA,C ¼ wB,C ¼ w: ð12ÞThe isothermal sections of the ternary system were calculatedfor the FCC lattice, and are presented in Figs. 4(a), (b), and(c). Since the interaction parameter is positive (repulsivebetween A, B, and C), i.e., w = +1 kJmol¹1, the FCC solidsolution has miscibility gaps. At 600K in Fig. 4(a), there arethree two-phase regions. At 500K in Fig. 4(c), one three-phase triangle appears in the middle of the ternary isothermand the single solid-solution phase is only stable at thecorners of the pure elements. At an intermediate temperature(530K), the two-phase regions in Fig. 4(a) merge and formthree three-phase triangles (Fig. 4(b)). It is worth emphasiz-ing that the single solid-solution region remains around theequiatomic composition. This is because the configurationalentropy Sconf has the largest value at the equiatomiccomposition, which stabilizes the solution phase. Detailedexaminations of the stability of solution phases to thespinodal decompositions in ternary systems can be found inthe literature.24)When the interaction parameter is negative in the BCClattice (attractive between A, B, and C), i.e., w =¹1 kJmol¹1, the calculated isothermal sections are presentedin Figs. 5(a), (b), and (c). B2-type ordering (#1º#2) at 500Koccurs around the binary stoichiometric compositions (AB,AC, and BC), as presented in Fig. 5(a). As the temperaturedecreases to 100K (Fig. 5(c)), the single solid-solutionregions are limited to the corners of pure elements A, B,and C. At an intermediate temperature (350K) in Fig. 5(b),the single solid-solution region remains as stable around theequiatomic composition.For the orderings of the FCC lattice in the ternary system,the calculated isothermal sections are presented in Figs. 6(a)and (b), where the interaction parameter w = ¹1 kJmol¹1was used for the calculations on CaTCalc software.18) Inbinary systems, the A1, L10, and L12 phases can be describedby equivalence in sublattices such as (1) #1=#2=#3=#4, (2)#1=#2º#3=#4, and (3) #1=#2=#3º#4, respectively. TheTransition temperature, TC/K Normalized interaction parameter, wA,BR−1/K Miscibility gap(a)B2⁄C = −4 A,B ⁄C = +4 A,BA2+A2−400A2−60005001000150020002500−200 0 200 400 600Transition temperature, TC/K Normalized interaction parameter, wA,BR−1/K (b)⁄C = −2 A,B⁄C = +6 A,BL10 Miscibility gapA1A1+A10500100015002000250030003500−400−600 −200 0 200 400 600Fig. 3 Calculated transition temperatures, TC, of the equiatomic alloy in the A-B binary system as a function of the normalized interactionparameter wA,BR¹1 in the (a) BCC lattice and (b) FCC lattice.(a) 600K0 0.2 0.4 0.6 0.8 100.20.40.60.81Bmole fraction of CA1A C(c) 500K0 0.2 0.4 0.6 0.8 100.20.40.60.81A CBmole fraction of CA1A1A1(b) 530K0 0.2 0.4 0.6 0.8 100.20.40.60.81A CBmole fraction of CA1A1A1A1Fig. 4 Isothermal sections with the miscibility gaps in the A-B-C ternary system at (a) 600K, (b) 530K, and (c) 500K. The interactionparameters used in the calculations are wA,B = wA,C = wB,C = w = 1 kJmol¹1. The coordination number is z = 12 (FCC lattice).T. Abe, K. Han, Y. Goto, I. Ohnuma and T. Koyama880ordering behaviors are more complex in the ternary systems,as higher-order phases can be stable in the central part of theisothermal section in Fig. 6(b). The indicated configurationsof the ternary ordered phases are (4) #1=#2º#3º#4, and(5) #1=#2º#3º#4, yð3ÞA ¼ yð4ÞC , yð3ÞC ¼ yð4ÞA , and yð3ÞB ¼ yð4ÞB . Inaddition, there are two more configurations: (6) #1º#2º#3º#4, and (7) #1º#2º#3º#4, yð1ÞA ¼ yð2ÞC , yð1ÞC ¼ yð2ÞA ,yð1ÞB ¼ yð2ÞB , yð3ÞA ¼ yð4ÞC , yð3ÞC ¼ yð4ÞA , and yð3ÞB ¼ yð4ÞB , which arenot stable in Fig. 6(b). Consequently, the phase equilibria onthe isothermal sections become complex with these orderedphases in the four-sublattice model using the CEF.25) Atlow temperatures, where the disordered state is not stable inany of the compositions, an isothermal section cannot befully calculated with any of the thermodynamic calculationpackages adopted in the present work and is hence omittedin Fig. 6.In this section, the stability of the solid-solution phaseswas examined in the A-B-C ternary system. The results canbe summarized that the single solid-solution phases in allthree configurations in Figs. 4­6 become stable around theequiatomic composition, where Sconf has the largest value.Consistently, around the equiatomic compositions, HEAs canbe effectively synthesized.3.3 Transition temperature (TC) in higher-order systemsIn Section 3.2, it was demonstrated that the single solid-solution phase became stable around the equiatomiccomposition in ternary systems in the BCC and FCC lattices.In this section, we investigate the change in the transitiontemperature of equiatomic alloys with the number ofelements in the system. In Figs. 7(a) and (b), the transitiontemperatures of the equiatomic alloys are numericallycalculated using thermodynamic software packages and arepresented as a function of the normalized interactionparameter, wR¹1. With an increase in the number of elementsin the system (n), the transition temperature decreases. Thehorizontal bars indicate the ranges where the HEAs areobserved.10) Previously reported HEAs were in the range ofthe single solid-solution phase.In Fig. 8, the transition temperature is presented as afunction of the number of elements in the system, where theinteraction parameters are w = +1 kJmol¹1 for the misci-0 0.2 0.4 0.6 0.8 100.20.40.60.81A CBmole fraction of C(c) 100KA2A2 A2B2+B2+B2B2(a) 500K0 0.2 0.4 0.6 0.8 100.20.40.60.81A CBmole fraction of CA2B2 B2B20 0.2 0.4 0.6 0.8 100.20.40.60.81A CBmole fraction of C(b) 350KA2A2A2 A2B2Fig. 5 Isothermal sections with the order-disorder transitions in the A-B-C ternary system at (a) 500K, (b) 350K, and (c) 100K. Theinteraction parameters used in these calculations are wA,B = wA,C = wB,C = w = ¹1 kJmol¹1. The dotted lines indicate the second-ordertransitions.(a) 200Kmole fraction of C00.20.40.60.81A CB0 0.2 0.4 0.6 0.8 1(b) 175K100.20.40.60.80 0.2 0.4 0.6 0.8mole fraction of C1A CBFig. 6 Isothermal sections with the order-disorder transitions in the FCC lattice with four sublattices at (a) 200K and (b) 175K. Theinteraction parameters used in these calculations are wA,B = wA,C = wB,C = w = ¹1 kJmol¹1. The numbers in the figures indicatephases; 1: A1 disordered phase (#1=#2=#3=#4), 2: L10-type ordered phase (#1=#2º#3=#4), 3: L12-type ordered phase (#1=#2=#3º#4), 4: an ordered phase (#1=#2º#3º#4), and 5: an ordered phase (#1=#2º#3º#4, yð3ÞA ¼ yð4ÞC , yð3ÞC ¼ yð4ÞA , yð3ÞB ¼ yð4ÞB ).Stabilization of Equiatomic Solutions Due to High-Entropy Effect 881bility gap and w = ¹1 kJmol¹1 for the orderings in the FCCand BCC lattices. Due to the increase in Sconf, the transitiontemperatures decrease with increasing n in the system.Notably, the order-disorder transition temperature curve doesnot smoothly decrease with increasing n. This is possibly dueto the stoichiometric compositions of the ordered phases atthe equiatomic composition, where A0.5B0.5 in the A-B binarysystem (n = 2) and A0.25B0.25C0.25D0.25 in the A-B-C-Dquaternary system (n = 4) are in equilibrium with thedisordered phase at the transition temperature. Consequently,the transition temperature becomes slightly higher due tothe stoichiometry of the compounds in equilibrium with thesolution phase. To avoid stoichiometric compounds at theequiatomic composition, adding more than four elementsmay be effective because quaternary stoichiometric com-pounds of transition elements are rarely found in Pearson’scrystal structure database.26) For the A2 and B2 phasesdescribed by the CEF with the two sublattices, althoughthe stoichiometric compound is formed at the equiatomiccomposition in the binary system (n = 2, indicated by anarrow in Fig. 8), the curve seems to be smoothly changing.This may suggest that this effect is not significant in the BCClattice with the present sublattice configurations.3.4 Variations in the interaction parameters in ternarysystemsIn the previous sections, the phase equilibria werecalculated for the symmetric cases where all the interactionparameters were the same as in eq. (12). However, binaryinteractions are different in actual multi-component systems.Therefore, in the present section, we examine the change inthe transition temperature with small changes in theinteraction parameters. When the interaction parameters havethe same values, the A1 phase becomes stable at 530Karound the equiatomic composition, as presented in Fig. 4(b).With a weaker repulsive interaction between B and C(wB,C = +0.9 kJmol¹1) than others (wA,B = wA,C = +1kJmol¹1), the A1 single-phase region at the center of theisothermal section disappears (Fig. 9(a)) and becomeswider. With an even more repulsive interaction (wB,C =+1.1 kJmol¹1) than the others, the single solid-solutionphase at the equiatomic composition also disappears, and thethree-phase triangle becomes stable (Fig. 9(b)). This isqualitatively the same for the ordering cases, as presentedin Figs. 10(a) and (b) for the BCC lattice and Figs. 11(a)and (b) for the FCC lattice. The ordered states are less stablewith less attractive wB,C = ¹0.9 kJmol¹1, and more stablewith more attractive wB,C = ¹1.1 kJmol¹1 interactions, whilethe other interactions are the same, wA,B = wA,C = ¹1kJmol¹1. Because the ordering behaviors are complex inthe present four-sublattice model, as mentioned in Section3.2, thermodynamic calculation packages cannot completelysolve isothermal sections at low temperatures. Figure 11(b) isa typical example where the phase equilibria at the centralpart of the isothermal section are not well calculated. Hence,the stable phases were confirmed based on point calculationson CaTCalc software. The results revealed that the A1 phasewas not stable in the central part of the diagram. Theseslight changes in the interactions between B and C resultin different isothermal sections. The change of «0.1 kJmol¹1in the interaction parameter approximately corresponds toa change of «2 kJmol¹1 in the mixing enthalpy at the05001000150020002500−600 −400 −200 0 200 400 600Transition temperature, TC/K Normalized interaction parameter, wR−1/K 0500100015002000250030003500−600 −400 −200 0 200 400 600Transition temperature, TC/K Normalized interaction parameter, wR−1/K Miscibility gap(a) (b)A2Ordering Ordering Miscibility gapA1n=2n=3n=5n=10n=2n=3n=5HEAs10)HEAs10)Fig. 7 Calculated transition temperatures, TC, of the equiatomic alloy in the n-element system as a function of the normalized interactionparameter wR¹1 in the (a) BCC lattice and (b) FCC lattice. The horizontal bars indicate the range of HEAs, as previously reported.10)05001000150020002500300035000 2 4 6 8 10 12Transition temperature, TC/K Number of elements, nMiscibility gap (z=12)Order-disorder transition in the FCC lattice11 kJmolw11 kJmolwOrder-disorder transition in the BCC lattice11 kJmolwFig. 8 The calculated transition temperatures, TC, of the equiatomic alloysas a function of the number of elements in the system, n. The interactionparameters used in the calculations are given in the figure. The arrowsindicate the equiatomic alloys where the end member exists in the BCClattice.T. Abe, K. Han, Y. Goto, I. Ohnuma and T. Koyama882A10 0.2 0.4 0.6 0.8 100.20.40.60.81A CBmole fraction of CA1 A1(a) (b) 0 0.2 0.4 0.6 0.8 100.20.40.60.81A CBmole fraction of CA1 A1Fig. 9 Miscibility gaps in the FCC lattice at 530K where the interaction parameters used in the calculations are wA,C = wA,C = +1kJmol¹1, and (a) wB,C = +0.9 kJmol¹1 and (b) wB,C = +1 kJmol¹1.(a) (b) 0 0.2 0.4 0.6 0.8 100.20.40.60.81A CBmole fraction of CA2B2A2B20 0.2 0.4 0.6 0.8 100.20.40.60.81A CBmole fraction of CB2A2A2A2Fig. 10 Order-disorder transitions in the BCC lattice at 350K, where the interaction parameters used in the calculations arewA,C = wA,C = ¹1 kJmol¹1, and (a) wB,C = ¹0.9 kJmol¹1 and (b) wB,C = ¹1.1 kJmol¹1. The dotted lines indicate the second-ordertransitions.0 0.2 0.4 0.6 0.8 100.20.40.60.81A CBmole fraction of C(a) 0 0.2 0.4 0.6 0.8 1mole fraction of CA CB00.20.40.60.81(b) Fig. 11 Order-disorder transitions in the FCC lattice at 175K where the interaction parameters used in the calculations arewA,C = wA,C = ¹1 kJmol¹1, and (a) wB,C = ¹0.9 kJmol¹1 and (b) wB,C = ¹1.1 kJmol¹1. The numbers in the figures indicate thephases; 1: A1 disordered phase (#1=#2=#3=#4), 2: L10-type ordered phase (#1=#2º#3=#4), and 3: L12-type ordered phase(#1=#2=#3º#4). The phase areas in the central part are not resolved since the calculation is not completed.Stabilization of Equiatomic Solutions Due to High-Entropy Effect 883equiatomic composition. With this variation, the stability ofthe solid-solution phase is affected and can decrease. Thisvariation is smaller than the variations in the mixing enthalpyin the binary subsystems of the Cantor alloy,27) where theyare in the range of +3³¹7 kJmol¹1. This contradicts the factthat Cantor alloys and many HEAs have been synthesized.10)One of the reasons is that the mixing enthalpies of equiatomicalloys converge to an ideal solution with the increasingnumber of elements in the system. The convergence in theCantor alloy and the BCC-based HEA is demonstrated inFig. 12.27­29) This implies that the characteristics of thebinary interactions are averaged in multi-component systems.Consequently, HEAs can accommodate a variety of elementswith different interaction parameters.4. ConclusionsIn the present study, using the regular solution model andCEF with two and four sublattices, the stability of the solid-solution phases was examined in multi-component alloysbased on the CALPHAD technique. The following resultswere obtained:(1) The isothermal sections of the ternary systems withmiscibility gaps and orderings on the BCC and FCClattices were calculated using the thermodynamicmodels. The results exhibited that the single solid-solution region remained stable around the equiatomiccomposition, where Sconf had the largest value in thesystem.(2) The transition temperatures from the disordered stateto the ordered state or miscibility gaps decreased withthe increasing number of elements in the system. Theorder-disorder transition temperature in the FCC latticewas affected by the end member of the ordered phaseexisting at the equiatomic composition, whereas it wasnot significant for the order-disorder transition in theBCC lattice.(3) The stability of the single solid-solution phases atequiatomic compositions was affected by smallvariations in the interaction parameters. By addingmultiple elements, the characteristics of the binaryinteractions were averaged; this resulted in HEAscomprising a variety of elements whose interactionparameters were different.AcknowledgmentsThis work was supported by the Grant-in-Aid for ScientificResearch on Innovative Area, “High Entropy Alloys” (No.18H05454).REFERENCES1) J.-W. Yeh, S.-K. Chen, S.-J. Lin, J.-Y. Gan, T.-S. Chin, T.-T. Shun,C.-H. Tsau and S.-Y. Chang: Adv. Eng. Mater. 6 (2004) 299­303.2) B. Cantor, I.T.H. Chang, P. Knight and A.J.B. Vincent: Mater. Sci. Eng.A 375­377 (2004) 213­218.3) J.-I. Lee, K. Tsuchiya, W. Tasaki, H.-S. Oh, T. Sawaguchi, H.Murakami, T. Hiroto, Y. Matsushita and E.-S. Park: Sci. Rep. 9 (2019)13140.4) M. Kawamura, M. Asakura, N.L. Okamoto, K. Kishida, H. Inui andE.P. George: Acta Mater. 203 (2021) 116454.5) D. Zhou, Z. Chen, K. Ehara, K. Nitsu, K. Tanaka and H. Inui: Scr.Mater. 191 (2021) 173­178.6) O.N. Senkov, D.B. Miracle, K.J. Chaput and J.-P. Couzinie: J. 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