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Prediction of Face-Centered Cubic Single-Phase Formation for Non-EquiatomicCr­Mn­Fe­Co­Ni High-Entropy Alloys Using Valence Electron Concentration andMean-Square Atomic Displacement

Kodai Niitsu1,2,+, Makoto Asakura1, Koretaka Yuge1 and Haruyuki Inui1,2

1Department of Materials Science and Engineering, Kyoto University, Kyoto 606-8501, Japan2Center for Elements Strategy Initiative for Structure Materials (ESISM), Kyoto University, Kyoto 606-8501, Japan

We have investigated the face-centered cubic (FCC) single-phase formability of non-equiatomic Cr­Mn­Fe­Co­Ni HEAs as well asequiatomic derivative medium/high-entropy alloys (M/HEAs) considering their valence electron concentration (VEC) and mean-square atomicdisplacement (MSAD). While VEC remains the most decisive parameter to predict phase formation, MSAD can be a complementary parameterthat modifies the VEC boundary. Multiplicity of constituent elements was beneficial to accommodate a larger MSAD, which resulted in adownward shift of the VEC boundary for the FCC single phase. This offers information about the correlations between the phase formationpreference, VEC, and MSAD of M/HEAs with various compositions. [doi:10.2320/matertrans.MT-M2020144]

(Received May 7, 2020; Accepted June 9, 2020; Published July 28, 2020)

Keywords: high-entropy alloys, superalloys, atomic displacement, phase diagram, phase transformations

1. Introduction

Medium/high-entropy alloys (M/HEAs) are receivingincreasing interest as a new class of materials with promisingmechanical properties such as the combination of highstrength and ductility at low temperature.1­7) HEAs wereoriginally defined as being composed of more than fivemetallic elements with (nearly) equiatomic compositions withan entropy of mixing Sconf of no less than 1.5R (R is the gasconstant) and MEAs were defined as being composed ofthree or four metallic elements with 1.0R ¯ Sconf ¯ 1.5R.5)

M/HEAs were initially believed to exist as a stable single-phase solid solution because of their large Sconf.1,2) However,recent studies have elucidated that Sconf is not the dominantfactor regulating the formation of a single-phase solidsolution. Thereby, the definition of M/HEAs has now beenexpanded to cover multiphase alloys.8­10) The concept ofM/HEAs allows vast degrees of freedom in their chemicaldesign in terms of the number of elements and theirconcentrations, as well as in their microstructural designsuch as the degree/range of atomic ordering, texturing, andgrain size. In particular, reliable guidance for stabilizingsingle-phase solid solutions is required to help screen andoptimize the mechanical properties of M/HEAs.

There have been several attempts to tackle this challengein terms of the Hume­Rothery rules. Zhang et al.11,12)

delineated the conditions under which a single-phase solidsolution formed by constructing a matrix of the mixingenthalpy of a solid solution and the atomic-size misfit ofthe constituent elements ¤ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPNi¼1 xið1� ri=

PNj¼1 xjrjÞ2

p,

where xi and xj are the compositions of elements i and j, riand rj are the atomic radii of elements i and j, and N is thenumber of constituent elements. The atomic radii wereobtained from Ref. 13), which are now widely used todescribe those of elements in HEAs. Takeuchi et al.14)

presented candidates for face-centered cubic (FCC) HEAsby evaluating mixing enthalpy and ¤. Guo and co-workers

proposed that the valence electron concentration (VEC) is themost promising parameter that dictates the formationfeasibility of FCC, body-centered cubic (BCC), andFCC+BCC phases.15) However, recent studies have raisedtwo concerns about these parameterizations. One is thedescriptions of effective atomic displacement and atomic-sizemisfit. Okamoto et al.16) proposed that mean-square atomicdisplacement (MSAD) can be a more comprehensive atomicdisplacement to describe the static lattice distortion in solid-solution alloys including M/HEAs and that the effectiveatomic size derived from MSAD calculations is quitedifferent from that commonly used. Another is the scalabilityof the above-mentioned parameterizations for M/HEAs withcompositions far from equiatomicity. Considering that thisresearch community has shown recent interest in multi-component alloys well away from equiatomicity,10) theapplicability of the above-mentioned parameterizationsshould be revisited for non-equiatomic multicomponentHEAs as well as equiatomic M/HEAs. In this study, weinvestigate the phase formation preferences of various non-equiatomic quinary Cr­Mn­Fe­Co­Ni alloys as well asequiatomic derivative M/HEAs and discuss how the phaseformation of these alloys can be predicted using VEC andMSAD.

2. Methodology

To obtain experimental information about the possiblecorrelations between the phase formation preference, VEC,and MSAD, we prepared quinary alloys with the generalformula AxB25¹(x/4)C25¹(x/4)D25¹(x/4)E25¹(x/4) (A­E: Cr, Mn,Fe, Co, and Ni) in addition to equiatomic Cr­Mn­Fe­Co­Ni,as listed in Table 1. Each alloy is hereafter abbreviated withthe non-equiatomic element and its composition, such as

“10Cr alloy”. VEC is defined as VEC ¼ PiciVECi, where ci

and VECi are the atomic ratio and the number of valenceelectrons of element i, respectively. VECi was simply set to 6,7, 8, 9, and 10 for i = Cr, Mn, Fe, Co, and Ni, respectively,+Corresponding author, E-mail: niitsu.koudai.8z@kyoto-u.ac.jp

Materials Transactions©2020 The Japan Institute of Metals and Materials

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in line with Mizutani’s approximation.17) The alloys werefabricated by arc melting and then cold-rolled to 60%reduction. Cold-rolled plates were recrystallized and homo-genized at 1473K for 8 days, followed by water quenching.The phase formation and chemical compositions of the alloyswere investigated by scanning electron microscopy (SEM);energy-dispersive spectroscopy (EDS), which was attached tothe SEM; and X-ray diffraction (XRD). All calculations for

the derivation of MSAD were performed based on the densityfunctional theory (DFT) formulated within the generalizedgradient approximation (GGA) by Perdew, Burke, andErnzerhof (PBE).18) The Kohn­Sham equations were solvedwith the Vienna Ab initio simulation package (VASP)code.19­21) We used the special quasirandom structure(SQS) package22,23) to model the atomic configurations witha 3 © 3 © 3 extension of the FCC unit cell for the ternary,

Table 1 Calculated lattice constant acalc, ¤, VEC, MSADi, and averaged MSAD for equiatomic and non-equiatomic ternary, quaternary,and quinary M/HEAs.

Continued on next page.

K. Niitsu, M. Asakura, K. Yuge and H. Inui2

Advance Viewquaternary, and quinary systems listed in Table 1. The first-and second-nearest neighbors were taken into account forpair correlation functions. The plane-wave energy cutoffwas set to 400 eV. The Methfessel­Paxton technique24) with asmearing value of 0.1 eV and k-point grids of 4 © 4 © 4 withthe Monkhorst­Pack scheme25) was used. The SQS structureswere relaxed until the residual forces became less than10¹3 eV/¡ with fixed global FCC symmetry. The calcu-lations were iterated with independently relaxed cell volumeand atomic positions and terminated when convergence wasreached. MSAD was derived as MSAD ¼ P

iciMSADi using

the MSAD of element i (MSADi) that can be obtained as anaverage of atomic displacements for i atoms from regularlattice points. For equiatomic ternary, quaternary, and quinaryM/HEAs, we derived the effective atomic radius rieff tominimize � ¼ P

j

� Qi6¼j

reffi � ðravei 6¼jÞX�( j = Cr, Mn, Fe, Co,

Ni, and X = 2, 3, and 4 for ternary, quaternary, and quinary

systems, respectively), where ravei6¼j ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia3i6¼j=16

ffiffiffi2

p3p

16) withlattice constant aiºj for the system not including element i.A combination of rieff conceptually gives the most plausibleset of atomic sizes for all derivative subsystems when atomswould be placed at the rigid lattice points. Together with thewell-known parameter ¤, we herein evaluated the scalability

using ¤eff ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPN

i¼1 xið1� reffi =PN

j¼1 xjreffi Þ2

q. All derived

numerical values are listed in Tables 1 and 2.

3. Results and Discussions

Typical SEM images of selected alloy samples are shownin Fig. 1. We confirmed that most of the prepared alloysformed a single FCC phase, but there were some exceptionslike 40Cr and 5Ni alloys, which contained additional phases.To identify the crystal structure of the additional phases,SEM-EDS and powder XRD were performed for these

Continued.

Prediction of Face-Centered Cubic Single-Phase Formation for Non-Equiatomic Cr­Mn­Fe­Co­Ni High-Entropy Alloys 3

Advance Viewalloys. As shown in Fig. 2(a), peaks in the XRD profile forthe 40Cr alloy were unambiguously indexed as those of theFCC, BCC, and · phases. Meanwhile, the 5Ni alloy onlyshowed FCC peaks, which was presumably caused by thesmall fraction of the secondary phase. Considering theferromagnetic nature of the 5Ni alloy, the crystal structureof the secondary phase was deduced to be BCC. The FCC

lattice constants determined by XRD experiments, aexp, andfirst-principles calculations, acalc, were shown in Fig. 2(b).While there exists a certain level of offset, they show a goodlinearity for alloys forming FCC single-phase state. Thissupports that the calculations provide qualitatively plausiblelattice models of M/HEAs involving their lattice constantsand also atomic displacements.

The relationship between the constituent phases and VECfor the studied alloys is depicted in Fig. 3. For comparison,those of equiatomic ternary and quaternary MEAs26) andother reported non-equiatomic quinary Cr­Mn­Fe­Co­Nialloys27­29) are also included. It is clear that the alloysforming the FCC single phase almost consistently possesseda certain range of VEC. According to Guo et al.,15) the VECranges in which the BCC and FCC phases form can beroughly classified as follows: the FCC single phase formswhen 8.0 ¯ VEC, the FCC and BCC phases coexist when6.87 ¯ VEC < 8.0, and the BCC single phase forms whenVEC < 6.87. Tsai et al.29) also suggested that the ·-proneregime for Cr-containing M/HEAs is 6.88 < VEC < 7.84.

Chemical compositions of the FCC phase coexisting withother phases provide information about the phase boundaryconditions. The VEC values of the FCC phase of the 40Crand 5Ni alloys were 7.78 and 7.61, respectively. In view ofthe fact that Cr-rich alloys tend to form the · phase, theformer and latter values are responsible for the FCC/· andFCC/BCC boundary conditions, respectively. The criterionfor the FCC/· boundary is in good agreement with the valueof 7.84 reported by Tsai and colleagues.29) In contrast, thatfor the FCC/BCC boundary is considerably lower than thevalue of 8.0 reported in Ref. 15).

To better understand the phase formation preferences ofthe alloys, we constructed several matrices that allow us toelicit correlations between the phase formation preference,lattice constant acalc, MSAD, ¤, and VEC. First, Fig. 4(a)presents the relationship between acalc and MSAD, both ofwhich were derived by first-principles calculations. It wasclearly observed that a larger lattice favors a larger MSAD,which indicates that the derived MSAD is a good measure toquantify the room to displace atoms from the regular latticepositions. Notably, the slope of the linear dependency seemsto increase slightly with the number of constituent elementsfor alloys forming an FCC single phase. This trend is likelyto indicate that the multiplicity of the constituent elementshelps to increase the capacity for local atomic displacement.Phase separation seems to be preferred for alloys with larger

Fig. 1 Back-scattered electron images of typical alloy samples after homogenization at 1473K.

Table 2 Effective radius reff, and its size misfit ¤eff for equiatomic ternary,quaternary, and quinary M/HEAs.

K. Niitsu, M. Asakura, K. Yuge and H. Inui4

Advance Viewacalc and MSAD, but some exceptions were observed(CrFeCo (#4), CrMnFeCo (#11), and 5Ni (#21)), whichsuggests that acalc or MSAD alone is not a decisive parameter.

When revisiting the classic Hume­Rothery rule,30) onemay think the classical atomic-size misfit ¤ would becomplementary to MSAD or ¤eff; their relationship is shownin Fig. 4(b). No correlations between the phase formationpreference, the atomic-size-misfit parameters (¤ and ¤eff), andMSAD were identified. The lack of an apparent correlationbetween MSAD and the atomic-size-misfit parameterssuggests that MSAD is a more descriptive parameter thanthe misfit parameters because it can incorporate the proximitysize effect unique to the multicomponent random config-urations. Figure 4(c) presents the effects of ¤ and ¤eff on thephase formation preference based on the most decisiveparameter, VEC (horizontal axis). The wide variations of ¤and ¤eff did not provide any additional information about thephase formation tendencies of the alloys. Instead of ¤ and ¤eff,

the effect of MSAD on the phase formation preference basedon VEC is shown in Fig. 4(d). As mentioned in Fig. 4(a),multiplicity of the constituent elements is likely to helpaccommodate a larger MSAD, which results in an upwardexpansion of

ffiffiffiffiffiffiffiffiffiffiffiffiffiMSAD

p=acalc with increasing number of

constituent elements. With the global trend of largerffiffiffiffiffiffiffiffiffiffiffiffiffiMSAD

p=acalc for lower VEC alloys, this leads to the

expansion of the FCC single-phase region as highlightedwith masked regions in Fig. 4(d). Indeed, the lowest VEC forthe FCC single phase for the ternary and quaternary systemswas 8.00, whereas that for the quinary systems shifted to7.75, which is close to the experimentally determined valueof 7.61. MSAD is therefore considered to be a comple-mentary parameter to VEC that tunes the lower end of VECfor the FCC single-phase formation preference.

Lastly, the detailed distributions of MSAD values ofconstituent elements (MSADi) and their averages aredisplayed in Fig. 4(e). Although there are some exceptions,

Fig. 3 Relationship between the VEC and phase(s) of the HEAs prepared in this study and reported M/HEAs.14,25­28)

Fig. 2 (a) XRD profiles and analytical compositions of phases in the 40Cr and 5Ni alloy samples. (b) Relationship between experimental(aexp) and calculated (acalc) lattice constants for some of the studied quinary alloys. Filled and open plots represent the alloys forming aFCC single phase and the other cases at 1473K, respectively.

Prediction of Face-Centered Cubic Single-Phase Formation for Non-Equiatomic Cr­Mn­Fe­Co­Ni High-Entropy Alloys 5

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elements with smaller atomic number tend to possess largerMSAD. In particular, MSAD of Cr often deviatesconsiderably from those of the other elements. As reportedin several studies,31,32) Cr is considered to be a key elementto induce short-range chemical ordering in equiatomic M/HEAs. We think that the obtained MSAD trend does notcontradict this because the deviated MSAD of Cr might drivelocal strain relaxation by coordinating to selected atoms withsmaller MSADi from the perspective of the elastic proximityeffect. In turn, alloys with constituent elements with a narrowrange of MSAD values, such as MnFeCo (#7), FeCoNi (#10),MnFeCoNi (#15), 5Cr (#17), and 40Mn (#33), may beexpected to involve statistically more random configurations.

4. Conclusion

The preference for FCC single-phase formation for non-equiatomic Cr­Mn­Fe­Co­Ni HEAs and equiatomic deriv-ative M/HEAs has been investigated in terms of VEC andMSAD. The existing VEC criterion was found to remain themost decisive to dictate the single-phase formation preferenceof the studied multicomponent alloy systems but failed toexplain a downshift of the VEC boundary between the FCCsingle-phase and BCC+FCC two-phase regimes as thenumber of constituent elements increases. MSAD calcu-lations elucidated that alloys with smaller VEC values tend tohave larger

ffiffiffiffiffiffiffiffiffiffiffiffiffiMSAD

p=acalc values and that the multiplicity of

(d)

Fig. 4 FCC formation tendency plotted on matrices between (a)ffiffiffiffiffiffiffiffiffiffiffiffiffiMSAD

pand acalc, (b)

ffiffiffiffiffiffiffiffiffiffiffiffiffiMSAD

pand atomic-size misfit (¤, ¤eff), (c) atomic-

size misfit (¤, ¤eff) and VEC, and (d)ffiffiffiffiffiffiffiffiffiffiffiffiffiMSAD

p=acalc and VEC. (e) Distributions of MSADi and average values for ternary, quaternary, and

quinary M/HEAs. Filled and open plots represent the alloys forming a FCC single phase and the other cases at 1473K, respectively.

K. Niitsu, M. Asakura, K. Yuge and H. Inui6

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the constituent elements offers an additional capacity ofMSAD, which result in a downshift of the VEC boundary forthe FCC single phase. Namely, we found that MSAD is apromising parameter that tunes the VEC lower boundary forFCC single-phase formation. For most of the alloys studied,the MSAD value of Cr is found to be notably larger thanthose of other constituent elements. This tendency canpromote the atomic rearrangement in the vicinity of Cr atomsto lower the local strain, which does not contradict with thereported Cr-driven short-range ordering of M/HEAs. Theconcept of MSAD provides rich implications for the phaseformation preference and short-range ordering tendency ofM/HEAs.

Acknowledgment

This work was supported by Grant-in-Aids for ScientificResearch on Innovative Areas on High Entropy Alloys (thegrant number JP18H05450 and JP18H05451), the JapanSociety for the Promotion of Science (JSPS) KAKENHI(grant number 19K22053 and 19H00824), the ElementsStrategy Initiative for Structural Materials (ESISM) fromthe Ministry of Education, Culture, Sports, Science andTechnology (MEXT) of Japan, and Advanced Low CarbonTechnology Research and Development Program sponsoredby Japan Science and Technology Agency (JST-ALCA)(grant number JPMJAL1004).

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Prediction of Face-Centered Cubic Single-Phase Formation for Non-Equiatomic Cr­Mn­Fe­Co­Ni High-Entropy Alloys 7