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Current Opinion in Solid State and Materials Science 21 (2017) 238–251

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Current Opinion in Solid State and Materials Science

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Thermodynamics of concentrated solid solution alloys

http://dx.doi.org/10.1016/j.cossms.2017.08.0011359-0286/� 2017 Elsevier Ltd. All rights reserved.

⇑ Corresponding author at: National Energy Technology Laboratory, Albany, OR97321, USA.

E-mail address: [email protected] (M.C. Gao).

M.C. Gao a,b,⇑, C. Zhang c, P. Gao d, F. Zhang c, L.Z. Ouyang d, M. Widom e, J.A. Hawk a

aNational Energy Technology Laboratory, Albany, OR 97321, USAbAECOM, P.O. Box 1959, Albany, OR 97321, USAcCompuTherm LLC, Middleton, WI 53562, USAd Tennessee State University, Nashville, TN 37209, USAeCarnegie Mellon University, Pittsburgh, PA 15213, USA

a r t i c l e i n f o

Article history:Received 18 December 2016Revised 15 August 2017Accepted 20 August 2017

a b s t r a c t

This paper reviews the three main approaches for predicting the formation of concentrated solid solutionalloys (CSSA) and for modeling their thermodynamic properties, in particular, utilizing the methodologiesof empirical thermo-physical parameters, CALPHAD method, and first-principles calculations combinedwith hybrid Monte Carlo/Molecular Dynamics (MC/MD) simulations. In order to speed up CSSA develop-ment, a variety of empirical parameters based on Hume-Rothery rules have been developed. Herein, theseparameters have been systematically and critically evaluated for their efficiency in predicting solid solu-tion formation. The phase stability of representative CSSA systems is then illustrated from the perspec-tives of phase diagrams and nucleation driving force plots of the r phase using CALPHAD method. Thetemperature-dependent total entropies of the FCC, BCC, HCP, and r phases in equimolar compositionsof various systems are presented next, followed by the thermodynamic properties of mixing of theBCC phase in Al-containing and Ti-containing refractory metal systems. First-principles calculations onmodel FCC, BCC and HCP CSSA reveal the presence of both positive and negative vibrational entropiesof mixing, while the calculated electronic entropies of mixing are negligible. Temperature dependentconfigurational entropy is determined from the atomic structures obtained from MC/MD simulations.Current status and challenges in using these methodologies as they pertain to thermodynamic propertyanalysis and CSSA design are discussed.

� 2017 Elsevier Ltd. All rights reserved.

1. Introduction

Traditional physical metallurgy has paid relatively little atten-tion to multi-component concentrated solid solution alloys (CSSA)until Yeh [1] and Cantor [2] independently published their firstpapers on the subject in 2004. (Note: Yeh has named such alloysas ‘‘high-entropy alloys”, but for the sake of consistency only theterm ‘‘CSSA” will be used in this paper.) Comprehensive reviewson the structures and properties of CSSA are provided in Refs. [3–5]. One important fundamental property of CSSA is the thermody-namics that governs their phase stability, which impacts themicrostructure and resulting mechanical, physical and environ-mental properties. The main objective of this paper is to providea critical review on the prediction of CSSA formation and the mod-eling of thermodynamic properties of single-phase CSSA systemsusing three representative methodologies: (1) empirical thermo-physical parameters, (2) CALPHAD (acronym of CALculation of

PHAse Diagrams) method, and (3) first-principles density func-tional theory (DFT) calculations combined with hybrid MC/MDsimulations. Current status and the challenges in using thesemethodologies as they pertain to the thermodynamic evaluationand alloy design are also reviewed. Efficient strategies to acceleratethe development of new CSSA by combining these methodologieswill be discussed.

One important topic for thermodynamics of CSSA is predictingsolid solution formation, given an arbitrary combination of metal-lic elements in the periodic table. Since the maximum ideal config-urational entropy occurs at equimolar compositions (i.e.,

Sconfmax ¼ R lnN, where N is the total number of components in thesolution and R is the ideal gas constant), most studies on CSSAfocus on equimolar or near equimolar compositions. Althoughincreasing the number of principal components in the system hasbeen claimed to potentially stabilize the solid solution againstordered intermetallic compounds, to date the vast majority of mul-ticomponent CSSA published in the literature contain more thanone phase in the microstructure (i.e., in the as-cast state or afterannealing) [3,6–8]. The presence of additional phases in themicrostructure usually cause elemental partitioning among these

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phases, and thus, reduces the configurational entropy of the solidsolution. The total number of single-phase CSSA reported in the lit-erature is still very limited, and one possible reason may be lack ofeffective searching guidelines [9,10]. Another reason for this couldbe that the entropy does not always dominate over enthalpy. Forhitherto reported multicomponent CSSA in the literature, thealloys with the face-centered cubic (FCC) structure are mostlybased on CoCrFeMnNi [2] and its derivatives. The alloys with thebody-centered cubic (BCC) structure are mostly based on refrac-tory metals [11,12], while the alloys with the hexagonalclosed-packed (HCP) structure are mostly based on either rareearth elements or late transition metals [10].

Various empirical thermo-physical parameters [7,13–23] havebeen proposed in the literature to predict solid solution variantsof CSSA, which originated from the Hume-Rothery rules. Due tolack of reliable phase diagrams for multicomponent CSSA systems,composition screening using this approach can be fast but at thecost of reliability. Ambiguity in defining single-phase solid solutionalloys has cast doubt on some of empirical parameters. For exam-ple, many alloys that contain two FCC phases (e.g., coexistence ofCu-rich FCC phase with another FCC phase), or two BCC phases(e. g., coexistence of a BCC phase with another BCC phase or B2phase), are counted as single-phase compositions [6,13,22–24].Furthermore, some empirical rules have been determined basedon limited number of compositions [16,18,21,22]. In order to betterunderstand the strengths and weaknesses of each set of empiricalparameters, this study collects and assesses the most up-to-dateexperimentally reported CSSA compositions with the FCC, BCC,HCP, multiphase, and amorphous structures (see Table S1), andevaluates the efficiency in predicting single-phase solid solutions.Alloys with two FCC or BCC phases in the microstructure are trea-ted as multi-phase compositions. In addition, discrepancies in theatomic radii used by various research groups are also addressed.For consistency the atomic radii that corresponds to coordinationnumber 12 are used here for metallic elements while the radii oftetrahedral covalent bonds are used for interstitial elements. Thesevalues are taken from Ref. [25]. Assessment of experimentalreports in the literature and subsequent evaluation of the widelyused empirical thermo-physical parameters used in predictingsingle-phase CSSA are presented in Section 2.

Contrasted to the use of empirical parameters is CALPHADmethod [26,27]. It is the most robust approach for predicting phasestability and it is the most effective way in calculating multi-component phase diagrams as long as a reliable thermodynamicdatabase for the system under investigation is available. As amature methodology, CALPHAD method has been applied to abroad range of materials science and engineering problems beyondjust phase diagrams, including solidification, coating, joining, andphase transformation. In particular, applications of CALPHADmethod to CSSA systems, using PanHEA [28], TCNI8 [29], and TTNI8[29] databases, for materials design have been reported in Refs.[12,30–35] with reasonable success. The applications of CALPHADmodeling to CSSA are detailed in Section 3. Phase diagrams forthe Co-Cr-Fe-Mn-Ni system with varying number of componentsare developed, followed by nucleation driving force contour plotsfor the competing r phase in the temperature-composition spaceof the various systems. Then the total entropies for the FCC, BCC,HCP and r phases, as a function of composition and temperature,are presented. Finally, the entropies of mixing and enthalpies ofmixing of BCC phases in Al-containing and Ti-containing CSSA sys-tems are calculated to illustrate the concept of excess entropy.

First-principles DFT calculations prove to be very useful in pro-viding physics-based thermo-chemistry data and for predictingphase equilibria without experimental input. The atomic structure,the phase stability, elasticity, electronic structure, magnetic prop-erty, and vibrational properties of multicomponent CSSA have been

modeled using DFT [36–42]. Applications of DFT methods andhybrid Monte Carlo/Molecular Dynamics (MC/MD) simulations topredict the thermodynamic properties of CSSA are presented inSection 4. Enthalpies of formation of single-phase CSSA with theFCC, BCC and HCP structures at zero temperature are presented,followed by the entropy sources, in particular, from lattice vibra-tion, electronic excitation and solid solution configuration. Suchcalculations are supplementary to CALPHAD modeling, and canalso be used as input to CALPHAD database development.

Each of these approaches used in tackling the formation andthermodynamics of CSSA has its own strengths and weaknesses,and identifying the underlying hypotheses and potential pitfallsare important to rationally interpreting the results. This is coveredin Section 5. In particular the roadblocks to developing reliablethermodynamic databases for multi-component CSSA systems,and accordingly the solution to them, are discussed in detail. Forexample, the lack of experimental data on phase equilibria andthermo-chemistry of multicomponent CSSA systems is addressedas is the importance in assigning physically meaningful energydata to those hypothetical phases (partially for the purpose ofdatabase compatibility), and integrating DFT energetics intoCALPHAD as input.

2. Empirical thermo-physical parameters

Following the Hume–Rothery rules, various empirical thermo-physical parameters are proposed to predict CSSA formation. These

parameters include ideal configurational entropy (Sconfideal), enthalpy

of mixing in the liquid phase (DHliqmix) [13], atomic size difference

(d) [13], the X-parameter that is determined by TmSconfideal=jDHliq

mixj(where Tm is the composition-weighted average melting point)[7], valence electron concentration (VEC) [14], electronegativitydifference (Dv) [15], the /-parameter [16] that defines the ratio

of the difference between Sconfideal and the entropy at Tm over the phe-nomenological ‘‘excess entropy” that is calculated using atomicsize and packing efficiency, elastic residual strain root mean square(eR:M:S: ; or

ffiffiffiffiffiffiffiffiffihe2i

p) [17], the atomic size-related a2 -parameter [18],

and intrinsic elastic strain energy (E2=E0) [18]. The equations tocalculate these parameters are compiled in Supplementary Materi-als. Commonly accepted criteria based on these parameters to form

single-phase CSSA are: �15 kJ/mol � DHliqmix � +5 kJ/mol, d � 6.6%,

X � 1.1, eR:M:S: � 0.05, and / � 20.More sophisticated empirical models that compare the free

energies (enthalpy and entropy) of both the solid solution phaseand the intermetallic compound are also proposed, which includethe U-parameter [19], g-parameter [20], and the jcr

1 parameter[21]. These oversimplified models only consider one hypotheticalsolid solution phase and one hypothetical intermetallic compoundat the nominal composition. The hypothesis for these models isthat, in order to form a solid solution, the free energy of the hypo-thetical solid solution must be more negative than that of thehypothetical intermetallic compound at any given temperature.However, this hypothesis is not true for multicomponent alloysbecause solid-state phase equilibria for multi-component systemsusually involve multiple phases (potentially three or more) at agiven temperature. In fact, phase stability in CSSA systems at con-stant temperature and pressure is determined by minimizing theGibbs free energy of the system (Gu ¼ Hu � TSu for single phase

u, where H is enthalpy and S is entropy; G ¼ f iPN

i Gi for multiphase

systems, where f i is the molar fraction of phase i).The proposed criteria to form a single-phase solid solution using

the free energy model are U P 1 [19], gP 1 [20], and

DHIM=DHliqmix < jcr

1 [21]. Here DHIM refers to the composition-

240 M.C. Gao et al. / Current Opinion in Solid State and Materials Science 21 (2017) 238–251

weighted average enthalpy of formation of the most stable con-stituent binary intermetallic compounds (DHIM

ij ) that are takenfrom DFT data in the literature. The U-parameter is defined asthe absolute value of the ratio of the Gibbs free energy of solidsolution phase over that of the intermetallic compound. Theenthalpy for both the solid solution and the compound phasesare derived using the Miedema model [43] while ignoring theentropy of the compound. The g-parameter [20] is determined by

TannSconfideal=jDHIM

ij jmax. Here an annealing temperature (Tann) definesthe temperature where annealing experiments are usually per-formed to reach homogeneous microstructure. The jcr

1 parameter

is determined by 1� TannðSconfideal � DSIMÞ=DHliqmix (here DSIM represents

the entropy of an intermetallic compound). Although the entropyof intermetallic compounds can vary in a wide range dependingon its crystal structure and constituent elements, Senkov and Mir-

acle [21] assigned it to be a fraction (e.g., 0.6) of Sconfideal, for the sake ofsimplicity.

Fig. 1 compares the effectiveness of various empirical rules. The

DHliqmix � d relation (Fig. 1a) appears to be very powerful. All single-

phase compositions are located within the region d � 6.1% and

�16 kJ/mol � DHliqmix < +5 kJ/mol, while all the amorphous and a

fraction of the multiphase compositions are located outside theregion. However, this criterion is not a sufficient requirement sincethe region also contains most of the multiphase compositions. Sim-ilarly, theX-parameter (Fig. 1b) is useful in separating single-phasecompositions from amorphous compositions with the thresholdvalue of 1.0–1.1, but most multiphase compositions also possessX � 1.1. In contrast, the Dv parameter (Fig. 1b) is not quite as effec-tive since for alloys below 0.44 significant scatter is observed.Fig. 1c shows a threshold value of 0.19 rather than 1.0 for the g-parameter derived from Ref. [20], and a threshold value of 7.0rather than 20 for /-parameter [16]. Note that neither g � 0.19nor / � 7.0 are sufficient conditions for forming solid solutions.However, combining these parameters proves to be much moreeffective than using them alone (Fig. 1c). Alternatively, the

DHIM=DHliqmix < jcr

1 relation (Fig. 1d) separates the vast majority ofmultiphase compositions from single-phase compositions. How-ever, exceptions do occur. Many amorphous compositions arelocated in the same region as most single-phase alloys, and abouta dozen single-phase and multiphase compositions fail to meetthe criteria of these relations as well. The elastic strain energyvs. residual strain (Fig. 1e) indicates that eR:M:S: � 6.1% orE2=E0 � 13.6 � 10�4 promotes the formation of a single-phase solidsolution, but considerable overlap exists between multiphase andsingle-phase compositions, suggesting that both are necessary,but not sufficient, conditions in forming a solid solution. Note thatthe eR:M:S: and d values are almost equivalent (see Table S1) andindeed both involve only atomic sizes.

In order to predict the crystal structure type of a solid solution,Guo et al. [14] first proposed the VEC criterion. They claimed thatFCC phases form at high VEC (> 8), BCC phases form at low VEC

(< 6.87), and FCC + BCC phases form at intermediate VEC

(6.87–8). As Fig. 1f shows, the threshold VEC identified in this studydiffers slightly from Ref. [14]. That is, FCC alloys are found to haveVEC � 7.8, and BCC alloys have VEC < 6. Rare-earth HCP alloys haveVEC of 3, while transition-metal HCP alloys have VEC about 7.5–8.5.Although the VEC criterion can serve as a useful guide for predict-ing the structure type of a solid solution, it cannot be used to pre-dict whether or not solid solution will form. For example, nothreshold VEC could be established for multiphase and amorphous

alloys. Note that most multiphase alloys have fairly high DSconfideal (�14 J/K/mol, Fig. 1f), and no pattern can be identified between the

phases and DSconfideal. This suggests that high configurational entropydoes not necessarily, at least not always, lead to formation ofsingle-phase solid solution because entropy does not always dom-inate the contribution to the Gibbs free energy of a system. Thesolid solution entropy may be less than ideal, and additionallythe coexisting phases themselves may lead to the minimum Gibbsenergy of the system at a given temperature, pressure and bulkcomposition.

3. CALPHAD modeling: Phase diagrams, driving forces, andthermodynamic properties

The highly studied FCC CoCrFeMnNi alloy was discovered byCantor et al. [2] who investigated two equi-molar alloys with 20and 16 components. These alloys contained multiple phases andwere brittle in the as-cast and as-melt-spun states. Since then,many reports studied the phase stability and microstructure ofthe CoCrFeMnNi alloy [44,45] and its sub-system alloys [46,47].For example, Otto et al. [44] confirmed that a single FCC solid solu-tion formed only in CoCrFeMnNi while five other alloys (i.e., CoCr-CuFeMn, CoCrMnNiV, CoFeMnMoNi, CoFeMnNiV and CrFeMnNiTi)consisted of more than one phase after annealing at 1273 K or at1123 K for 3 days. It is noteworthy that the experimental observa-tion by Otto et al. [44] is reasonably well reproduced by CALPHADmodeling [8]. Many reported CSSAs with the major FCC phase inthe microstructure are derivatives of the Co-Cr-Fe-Mn-Ni system,and were developed by adding (or replacing) elements, such as,AlxCoCrCuFeNi [48,49], AlxCo1.5CrFeNi1.5Tiy [50], AlxCoCrFeNi[34,51,52], CoCrCuFeNi [53], CoCrFeNiNbx [54], CoCrFeNiMox [55]and CoCrFeMnNiVx [56]. Single phase FCC CSSA with four or moreprincipal components are very limited [8,57]. For example, addingCu causes formation of a Cu-rich FCC phase [48,49,53], while add-ing Al promotes the formation of BCC and B2 phases[34,48,49,51,52]. In addition, if Ti is added in large amount to thisCSSA, precipitation of other intermetallic compounds occurs[50,58]. The FCC phase in CoCrFeMnNi and its sub-systems are veryductile. Precipitation of second phases usually enhance thestrength of the alloy but leads to reduced ductility.

The phase stability of the Co-Cr-Fe-Mn-Ni, Al-Co-Cr-Fe-Ni andCo-Cr-Cu-Fe-Ni systems were modeled using the PanHEA databasevia the PandatTM software [28], and selected isopleths are shown inFig. 2. The database covers the complete descriptions of the con-stituent binary and ternary systems of the Al-Co-Cr-Cu-Fe-Mn-Ni-Si-Ti-Zr system [33,34]. While Cr still acts as a strong BCC stabi-lizer in this system (Fig. 2a), the FCC phase has unusually large Crsolubility (close to �30 at.%). This is primarily due to the entropystabilization effect. Similar to its role in steels Ni behaves as potentFCC stabilizer in this system (Fig. 2b). Conversely, the FCC phasehas very small solubility for Al (Fig. 2c) due to the large elasticstrain energy of hosting Al atoms in the FCC lattice of 3d transitionmetals (TM) and the tendency to form extremely stable Al-TMintermetallic compounds [59]. Although Cu has an FCC crystalstructure and has very similar atomic radius to that of other 3dTM, the mutual solubility between Cu and TM (except Ni andMn) is very small due to their inherent repulsive interaction. As aresult, Cu solubility in CoCrCuxFeNi is limited to �8% (Fig. 2d).The calculated phase diagrams (Fig. 2) agree well with experiments[34,51–53] at least at higher temperatures.

In CALPHAD phase diagrams are calculated by minimizing theGibbs free energy of the system at given temperature and pressure.Typical phase diagrams show the most stable phases at varyingtemperature, pressure, and composition, while the driving forcediagram shows the nucleation driving force of forming one ‘‘child”phase from the ‘‘parent” phase in the temperature-compositionspace. One main intermetallic compound competing against the

Fig. 1. Comparison of various empirical parameters in separating single-phase alloys from multiphase alloys and amorphous alloys.


FCC solid solution in the Co-Cr-Fe-Mn-Ni system is the r phasethat has Pearson symbol tP30 and space group P42/mnm with pro-totype Cr0.49Fe0.51. Precipitation of the r phase in the CoCrFeMnNialloy after annealing at 973 K for 500 days is reported in Ref. [45],which has a composition: 18.1% Co, 46.8% Cr, 16.9% Fe, 12.5% Mn,and 5.7% Ni.

Fig. 3 shows the nucleation driving force contour lines of form-ingr phase from the FCC phase in the binary Co-Cr system, ternaryCoFe-Cr section, quaternary CoFeMn-Cr section, and quinaryCoFeMnNi-Cr section. During these calculations, only the FCCphase is set ‘‘active” while the r phase is set ‘‘dormant”; all otherphases are suspended. The area represented by green lines hasnegative driving force, i.e., r phase will not form. The area repre-sented by red lines has positive driving force, i.e., r phase can

form. The larger the positive value, the higher the driving forceto form the r phase from the FCC phase. The calculations indicatethat driving force to form r phase from FCC is strong in the Co-Crbinary (Fig. 3a), but decreases with increasing configurationalentropy of the systems, namely, CoFe-Cr, CoFeMn-Cr, andCoFeMnNi-Cr sections (Fig. 3b–d). This demonstrates that increas-ing entropy can decrease the nucleation driving force of inter-metallic compounds. Driving force contour lines can also beplotted in a phase diagram to show the driving force of forminga certain phase (referred to as the stable phases), as shown inFig. S1 (Supplementary Materials).

The total entropies of the FCC, BCC, HCP and r phases areshown in Fig. 4 for equimolar compositions of CoCr, CoCrFe, CoCr-FeMn, and CoCrFeMnNi. The default stable structures at 1 atm and

Fig. 2. Calculated isopleths of (a) CoFeMnNiCrx, (b) CoCrFeMnNix, (c) CoCrFeNiAlx, and (d) CoCrFeNiCux, using the PanHEA database.


298.15 K are used as reference (i.e., Co�HCP, Cr�BCC, Fe�BCC,Mn�CBCC_A12, and Ni�FCC). The total entropy of each phaseincreases monotonically with increasing the number of compo-nents, reflecting the contribution from configurational entropy.They also increase with increasing temperature, indicating domi-nant contribution from vibrational entropy. Keep in mind thatthe ideal configurational entropies for equimolar binary, ternary,quaternary and quinary solid solution alloys are 5.8, 9.1, 11.5,and 13.4 J/K/mol, respectively, and they only constitute a smallfraction of the total entropy. This agrees with other reports [41,60].

Due to presence of short range order, or segregation, and thecontribution from lattice phonon vibration, or magnetic contri-bution, the total entropy of mixing (DS/mix) for a CSSA may notalways follow ideal mixing and can cause positive or negativedeviation from �R

Pici ln ci (i.e., excess entropy). The excess

entropy (exS/mix) of a solution phase (u) is calculated by subtract-ing the ideal configurational entropy from the total entropy ofthe alloy (DS/mix) [33]: exS/mix ¼ DS/mix þ R

Pici ln ci. The CALPHAD

analysis on the thermodynamic properties of mixing for theFCC phase in the Co-Cr-Fe-Mn-Ni system at 1273 K by Zhangand Gao [33] using TCNI8 database [29] reveals positive excessentropy for binary Co-Cr and pseudo-binary CoFe-Cr, CoFeNi-Cr,and CoFeMnNi-Cr systems.

The thermodynamic properties of mixing for the BCC phase at1273 K for Al-containing systems (Fig. 5a and b) andTi-containing systems (Fig. 5c and d) were calculated by Ruiet al. [61] using TCNI8 database. The results reveal significant neg-ative excess entropy for those equimolar compositions (Fig. 5a).The calculated DSBCCmix at 1273 K are �4, �2, +2, +3, +6, and +10 J/(mol�K) for equimolar AlTi, AlNb, AlNbTi, AlTiV, AlNbTiV, andAlCrNbTiV, respectively. The ideal configurational entropies are+6, +9, +12 and +13 J/(mol�K) for equimolar binary, ternary, quater-nary and quinary compositions, respectively. In contrast, theexcess entropy is predicted to be negligible in refractory CSSA sys-tems when Al is absent (i.e., CrTix, CrVTix, CrNbVTix and CrNbVZrTixsystems; see Fig. 5c). The strong attractive interatomic interactionbetween Al and transition metals [61] also causes very large nega-tive values of enthalpy of mixing (DHBCC

mix ), as shown in Fig. 5b, and

reducing Al contents results in much less negative DHBCCmix . Con-

versely, absence of Al in the refractory metal systems correspondsto greatly reduced jDHBCC

mix j values (Fig. 5d). The DHBCCmix values are

positive for CrTix, CrVTix, CrNbVTix and CrNbVZrTix systems exceptTi-poor CrVTix compositions, implying overall repulsive interac-tions for these compositions. Note that all these predictionsobtained from CALPHAD method are subject to experimental veri-fication and/or confirmation from DFT calculations in the future,

Fig. 3. Calculated driving force contours [J/mol] of the r phase nucleating from the FCC phase within the (a) CoCrx, (b) CoFeCrx, (c) CoFeMnCrx, and (d) CoFeMnNiCrx systemsusing the PanHEA database, overlaid with the corresponding phase diagrams.


and improvement to the database will enhance the reliability ofthese and subsequent calculations.

4. DFT calculations and hybrid MC/MD simulations: Entropysources

For a long time the CSSA community has approximated the totalentropy of mixing for a multicomponent CSSA to be the ideal con-figurational entropy. The exact values are not known for both thetrue configurational entropy and total entropy of mixing except afew recent theoretical reports [40,41]. As such this section mainlyfocuses on predicting enthalpy of formation of solid solution(DHSS), and more importantly, predicting the entropy sources ofexample CSSA compositions using DFT and MC/MD simulations.The DFT calculations were performed using VASP (Vienna Ab InitioSimulation Package) [62,63], and details on the VASP settings areprovided elsewhere [40,41]. The input atomic structures forequimolar quaternary CSSA are taken from the special quasi-random structure (SQS) reported in Ref. [41]: the 64-atom SQSmodels for quaternary compositions, the 160-atom SQS modelfor quinary HCP alloys, and the 125-atom SQS models for quinaryFCC and BCC alloys. The calculated DHSS of widely studied single-phase CSSA are shown in Fig. 6, which are obtained from oneSQS configuration per alloy composition. Prior study by Gao et al.

[41] shows that the fluctuation in DHSS due to the variation inthe atomic configuration are relatively small, at least for the CoOs-ReRu, CoCrFeNi, and MoNbTaW alloys, and the averaged DHSS are2.724 ± 0.49, 8.354 ± 0.266, and �7.407 ± 0.069 kJ/mol,respectively.

In general, the total entropy of a solid solution phase may con-

sist of the contributions from lattice vibration (Svib), configuration

(Sconf ), electron excitation (Selec), and magnetic spin fluctuations(Smag):

Stotal ¼ Sv ib þ Sconf þ Selec þ Smag : ð1ÞThe entropy sources are temperature and volume dependent. It

is assumed that the magnetic entropy term to FCC CoCrFeNi andCoCrFeMnNi should be small at T � 293 K because their criticalmagnetic ordering temperatures are well below room temperature[64,65]. No magnetism is reported for those refractory BCC multi-

component CSSAs. The vibrational entropy (Svib) is calculated by:

SvibðV ; TÞ ¼ 3kBZ 1

0nv ib½ðf BE þ 1Þlnðf BE þ 1Þ � f BElnf BE�de; ð2Þ

where nvib is the phonon density of states, and f BE is the Bose-Einstein distribution function. Phonon frequencies of the CSSA werecalculated using the harmonic approximation by diagonalizing the

Fig. 4. Calculated total entropy for the BCC, FCC, HCP, and r phases for different alloys using the PanHEA database: (a) CoCr; (b) CoCrFe; (c) CoCrFeMn; (d) CoCrFeMnNi.


dynamical matrix based on the interatomic force constants reportedby VASP. The quasi-harmonic approximation can improve the accu-racy at high temperatures, and more importantly, can predict thecoefficient of thermal expansion. For a comprehensive review onthe lattice vibration of crystalline solids, readers can refer to thework by Fultz [60].

Electronic excitation across the Fermi level by migrating fromvalence band to conduction band gives rise to electronic entropy

(Selec), which can be determined by:

SelecðV ; TÞ ¼ �2kBZ 1

�1nelec f FDlnf FD þ ð1� f FDÞlnð1� f FDÞ½ �de; ð3Þ

where nelec is the electron density of states, and f FD is Fermi-Diracdistribution function.

For a solid solution as the temperature decreases, chemicalordering or phase separation sets in, which reduces the configura-tional entropy. At zero temperature, all entropy sources shouldvanish according to the Third Law of Thermodynamics. Such exam-ples are shown in refractory CSSA by Widom et al. [37,40] usinghybrid MC/MD simulations. The canonical ensemble (NVT) (i.e.,constant amount of substance, volume and temperature) wasadopted for the FCC and BCC structures, and the isothermal-isobaric ensemble (NPT) (i.e., constant amount of substance, pres-sure and temperature) for the HCP structure. The supercells used inthe present study contain 108, 128, and 96 atoms for FCC CoCrFeNi,BCC MoNbTaW, and HCP CoOsReRu, respectively. The simulationswere done by alternating molecular dynamics at each temperaturewith Monte Carlo swaps, each performed from first principlesusing VASP. Monte Carlo swaps of atomic species on different sites

are always accepted if the swap reduces the energy(DE ¼ Eswap � Einitial < 0; Eswap and Einitial are the energies in theswapped and initial configurations, respectively). For those swapsthat increase the energy (DE > 0), the acceptance is determinedby assessing the probability, P ¼ expð�DE=kBTÞ. The energy barrierseparating the swapped and initial states are not relevant forMonte Carlo simulations and thus not considered. More detailson the procedures are provided elsewhere [37,40,66]. The reduc-tion in configurational entropy due to short-range chemical ordercan be calculated using the Guggenheim quasichemical model[67,68], Bethe lattice [69], or Kikuchi cluster variation method(CVM) [70]. Specifically, the reduction in entropy due to nearneighbor correlations (yij) using Kikuchi CVM is calculated by:

I ¼XNi;j¼1

yijlnðyij=cicjÞ ð4Þ

where ci and cj are the mole fraction of elements i and j,respectively.

Fig. 7 compares the vibrational, electronic, and configurationalentropies for three model CSSAs: (1) FCC CoCrFeNi, (2) BCC MoN-bTaW, and (3) HCP CoOsReRu. Note that the vibrational entropyof BCC Cr was used here because of presence of 64 imaginary vibra-tional modes and the elastic instability for FCC Cr. The vibrationalentropy is very similar for MoNbTaW and CoOsReRu (as expecteddue to their similar average masses), but it is slightly smaller forCoCrFeNi (also as expected due to its lower atomic mass). This isshown in Fig. 7a. Vibrational entropy is sensitive to temperature.It vanishes at zero temperature and increases rapidly with

Fig. 5. Calculated entropy of mixing (DSBCCmix ) and enthalpy of mixing (DHBCCmix ) of the BCC solid-solution phase at 1273 K using the TCNI8 database: (a, b) AlTix, AlVTix, AlNbVTix

and AlCrNbVTix; (c, d) CrTix, CrVTix, CrNbVTix and CrNbVZrTix systems. The reference state is the BCC phase at 1273 K and 1 atm. The dashed curves in (a) and (c) refer to theideal configurational entropies.

Fig. 6. DFT-calculated enthalpy of formation of various single-phase CSSA in the FCC (blue), BCC (green) and HCP (red) structures. (For interpretation of the references to colorin this figure legend, the reader is referred to the web version of this article.)


increasing temperature. Note that the vibrational entropies for thethree alloys are already �2.5 times the ideal configurationalentropy at T = 300 K (Fig. 7a).

In contrast, the electronic entropy is small for all three compo-sitions even at high temperatures (Fig. 7b). Configurational entro-

pies (Sconf ) for these alloys are fairly close to their ideal value

Fig. 7. Calculated (a) vibrational entropy, (b) electronic entropy, (c) configurational entropy, and (d) total entropy of FCC CoCrFeNi, BCC MoNbTaW, and HCP CoOsReRu, usingDFT and MC/MD methods.


(R ln 4) at temperatures close to the solidus temperatures, and startto decrease very gradually until the temperature reaches 1100 K

(Fig. 7c). At this point Sconf decreases rapidly, signaling the develop-

ment of chemical short-range order in the alloys. Note that Sconf atroom temperature is much lower for MoNbTaW than the othersdue to the tendency in forming the ordered BCC (i.e., B2) asrevealed by Widom et al. [37,66]. The total entropies (i.e., thesum of vibrational, electronic, and configurational entropy) of

these alloys are shown in Fig. 7d. Stotal is equivalent for MoNbTaVand CoOsReRu, and is greater than CoCrFeNi, following the sametrend as seen for vibrational entropy (Fig. 7a).

Alloy entropies of mixing are determined as:

DSmix ¼ Salloy �XNi¼1

ciSi ð5Þ

where Si is the entropy (total, vibrational, etc.) of the ith element, ciis the molar composition of the ith element, and N is the total num-ber of elements in the CSSA. Note that the pure elements lack con-

figurational entropy, and hence, DSconfmix ¼ Sconfalloy.Fig. 8 compares the mixing vibrational and electronic entropies

for the three model alloys. The calculations predict that latticevibration increases the total entropy of CoCrFeNi, while vibration

decreases the total entropy of MoNbTaW (Fig. 8a). The DSvibmix values

for the three cases are small relative to the individual Sviballoy becausethe alloy vibrational frequencies lie close to the average of the indi-

vidual elements. Indeed, often DSvibmix < Sconf , and DSvibmix is near zerofor CoOsReRu. At temperatures above the Debye temperature(i.e., T � �400 K), the vibrational entropies of mixing approach

constant values, namely, +2.8 J/K/mol, �3.6 J/K/mol and �0.4 J/K/mol for CoCrFeNi, MoNbTaW and CoOsReRu, respectively,because the heat capacities approach their classical limit at 3R.

Note that the predicted DSvibmix of +2.8 J/K/mol for CoCrFeNi is inexcellent agreement with the calculated excess entropy of +2.5

J/K/mol [33] at 1273 K using CALPHAD method. In contrast, DSelecmix

values are close to zero (Fig. 8b). The total entropies of mixing

(i.e., DSvibmix þ DSelecmix þ Sconf ) are shown in Fig. 8c and correspond to,in descending order: CoCrFeNi > CoOsReRu > MoNbTaW.

The positive contribution of phonon vibration to the solid solu-tion stabilization was also reported in the literature by Fultz andco-workers [60,71,72]. For example, Fultz et al. [71] measured

DSvibmix to be +1.17, +1.67 and +1.78 J/K/mol for disordered BCC solidsolution alloys Fe70Cr30, Fe53Cr47 and Fe30Cr70, respectively. Nagelet al. [72] found that the disordered FCC Cu3Au has a higher vibra-tional entropy than the ordered L12 Cu3Au by approximately+1.16 J/K/mol at high temperature. However, generally speakingit is not obvious whether chemical disordering causes positive ornegative vibrational entropy of mixing, or whether a disorderedalloy will exhibit greater vibrational entropy than its ordered form.The vibrational entropy of mixing of an alloy may depend on thecrystal structure, bonding and enthalpy of the alloy, as well asthe molar volumes and structures of the constituent elements.

5. Discussion and outlook

The three major approaches that are widely used to predictCSSA formation and analyze the underlying thermodynamicproperties have been reviewed. Due to inherent hypotheses or

Fig. 8. Calculated (a) vibrational entropy of mixing, (b) electronic entropy of mixing, and (c) total entropy of mixing for FCC CoCrFeNi, BCC MoNbTaW, and HCP CoOsReRu,using DFT and MC/MD methods.


limitations in each approach, they possess strengths and weakness.Accordingly, it is prudent not to overstate their predictive capabil-ity, especially when using empirical thermo-physical parametersalone. In the following section the reasons that may lead to inaccu-racies in predictions when using these empirical parameters willbe discussed. Additionally, ways to overcome common issues indeveloping reliable physics-based thermodynamic databases formulti-component CSSA systems will also be explored.

5.1. Empirical parameters: Flaws and limitations

This critical re-evaluation of empirical thermo-physical parame-ters demonstrates that most parameters are efficient in separatingsingle-phase compositions from amorphous compositions. In gen-eral, they fail to separate single-phase compositions from multi-phase compositions. Considerable overlap between single-phase

compositions and multiphase compositions requires that newparameters, or identification methodologies, be developed that arestricter and more effective. The necessary requirements needed(but not sufficient) to form single-phase solid solution identified in

this work are: �16.25 kJ/mol � DHliqmix � +5 kJ/mol, d � 6%, X � 1,

g � 0.19, / � 7.0, eR:M:S: � 0.061, and E2=E0 � 13.6 � 10-4. AlthoughVEC by itself cannot be used to predict solid solution formation,VEC can be effective in separating BCC from FCC compositions. Forexample, the FCC phase forms when VEC � 7:8, while BCC phaseforms when VEC � 6:0. Note that the threshold values identified inthisworkdiffer somewhat from those reported in the literature. Thisappears to be due to the different compositional spaces assessed aswell as different values of atomic radii. Asmore reliable and verifiedexperimental data emerge in the literature, the threshold valuesmay become a range rather than a fixed value [61].


Besides bias due to the availability of experimental composi-tions, the empirical rules also suffer in other ways. Most empiricalrules assign the enthalpy of mixing of the liquid as determinedfrom the Miedema model to the solid solution phase, primarilydue to lack of reliable enthalpy data for the solid solution phasefor CSSA systems. It is known that the enthalpy values from Mie-dema models can differ significantly from experimental data forcertain systems. Gao et al. [12] studied these empirical parametersand thermodynamic properties for sixteen refractory BCC CCSAand found that both the sign and absolute value of DHBCC

mix are not

necessarily in accord with DHliqmix. In that work nine alloys exhibit

opposite sign while seven alloys show significant contrast in theirabsolute values.

Secondly, the true configurational entropy of solid solution maynot always follow ideal mixing behavior, and the deviation can besubstantial. Positive excess entropies were predicted for the FCCphase at 1273 K in Co-Cr-Fe-Mn-Ni system [33], and negativeexcess entropies for the BCC phase at 1273 K in Al-containing CSSAsuch as Al-Cr-Nb-Ti-V [61]. Similarly, the enthalpy of mixing forthe solid solution phase cannot be ignored, especially for systemsthat exhibit strong tendency to form extremely stable intermetalliccompounds (e.g., Al-containing CSSA) [61].

Lastly, the configurational entropy of intermetallic compoundscan be very small if the compositional homogeneity range remainsextremely small in the multicomponent systems. Otherwise, theconfigurational entropies, in general, cannot be ignored, as arguedby Senkov and Miracle [21]. Therefore, future work is needed todifferentiate compositions that will likely show appreciable com-positional homogeneity so that the empirical rules can be modifiedaccordingly.

Regardless of the subtle differences among those empiricalthermo-physical rules, they share one common thermodynamicflaw: The phase stability of a multicomponent system is not deter-mined by the simple comparison of the Gibbs free energies of a hypo-thetical solid solution with a hypothetical compound phase. Rather, itis determined by the equilibrium condition that the chemical potentialof each component should be equal in all phases (or equivalently, thealloy free energy should touch the convex hull of free energies of allcompeting phases). Consequently, these rules cannot predict theequilibrium phases and their crystal structures of arbitrary alloysreliably, nor can they predict the temperature and pressure depen-dence of phase stability. In fact, the simple comparison of the Gibbsfree energy at one composition to determine phase equilibriaamong phases is only valid for the unary system.

5.2. CALPHAD for CSSA systems: Issues and solutions

To overcome those defects to which there are no easy solutionswithin the framework of the empirical thermo-physical parame-ters, CALPHAD method proves to be the most direct solution tophase stability of multicomponent systems. In terms of CSSA, pre-dicted solid solution formation is confirmed through experiments[73–77]. Screening is carried out to identify low-density equimolarcompositions in single-phase solid solution [32,78]. Furthermore,non-equilibrium solidification using Scheil model [79] is used topredict segregation behavior in the as-cast state [33,73,75–77,80]. Quantitative analyses in enthalpy, entropy, and Gibbs freeenergy of CSSA are also presented in Refs. [12,44,75,76]. However,all these CALPHADmodeling predictions are susceptible to the reli-ability of the thermodynamic databases.

The essence of CALPHAD approach for a multicomponent sys-tem is to develop reliable self-consistent thermodynamic descrip-tions of all constituent binary and ternary systems, by fittingexperimentally determined thermochemistry and phase equilibriaas well as DFT-predicted data. The description of a quaternary or

higher-order system can be obtained via an extrapolation methodfrom the binary and ternary systems [81], subject to further adjust-ment of the database in comparison to experiments [33]. However,extrapolation of binaries and ternaries into higher order systemsmay not always guarantee satisfactory agreement with experi-ments. For example, if there are new stable quaternary or higher-order phases, or if the descriptions for those unstable or hypothet-ical phases/endmembers are so unphysical that cause their unde-sired over-stabilization or underestimated stability, thenextrapolation may not work. Since CALPHAD itself is not a predic-tive methodology like DFT, the thermodynamic descriptions for allphases in the system must be pre-defined. In case there exist newquaternary or higher-order compound phases in the system, thentheir Gibbs free energy descriptions in CALPHAD framework mustbe determined first.

Developing thermodynamic databases for multicomponentCSSA systems requires that the database be valid for the entirecomposition range, whereas traditional alloys are only concernedabout the compositions in the corner of a principal element. Forexample, development of 10-component CSSA system requiresoptimization over the entire composition ranges for 45 constituentbinaries and 120 constituent ternaries. Moreover, to date completeternary phase diagrams are still very limited in extent, and evensome binary phase diagrams require further experimental studyto better define them. For example, a recent DFT study by Widom[40] predicts the existence of stable C14 NbV2 Laves phase atT � 1100 K in the binary Nb-V system. This prediction has yet tobe confirmed by experiment. Element Tc has the HCP crystal struc-ture, and it is intuitively expected that there will be large mutualsolubility between Tc and several transition elements (such asRe, Ru, and Os). However, most of the binary phase diagrams per-taining to Tc are still unknown. Therefore, a complete databaseincluding 10 or more components is rarely available. For example,even though commercial thermodynamic databases developed fortraditional alloys may contain greater than 20 components, theyare not intended for CSSA use where compositions are located faraway from the principal element corner. Individuals doing CCSAthermodynamic research should be aware that using the thermo-dynamic databases developed for traditional alloys with one prin-cipal element may lead to erroneous results.

Another main challenge in developing databases for multicom-ponent CSSA systems is the lack of reliable energy data for hypo-thetical phases and endmembers, as discussed in Ref. [33]. Oneexample is the r phase in the Al-Co-Cr-Cu-Fe-Mn-Ni-Ti system.For example, the r phase is stable in the Co-Cr and Cr-Fe systems,but it is not stable for Al- or Cu-containing binaries. Their energies,however, should be physically meaningful, and need to beaccounted for in the database for the sake of compatibility. Theadopted sublattice model and the assigned energies for the corre-sponding endmembers have profound influence on the stability ofthe r phase. For instance, both TCNI8 and PanHEA databasesunderestimate the precipitation temperature of the r phase forCoCrFeMnNi (i.e., 867 K by the former and 624 K by the latter).Experiment has shown that r phase precipitates at 973 K [45].The predicted composition of r phase also differs greatly fromexperiments [45,82]. TCNI8 predicts a composition of r phase atT = 773 K to be 5.3% Co - 37.2% Cr - 30.0% Fe - 26.3% Mn - 1.2%Ni; PanHEA on the other hand predicts a composition atT = 623 K to be 2.1% Co - 48.5% Cr - 20.5% Fe - 28.5% Mn - 0.4% Ni.

Another example is Laves phase (i.e., C14, C15 and C36) inrefractory metal system (e.g., Al-Cr-Hf-Mo-Nb-Ta-Ti-V-W-Zr).Laves phase is stable in Cr-M binaries (where M = Hf, Nb, Ta, Ti,Zr), but is not stable in many other systems (e.g., Cr-M (M = V,Mo, W), Ti-M (M = Mo, Nb, Ta, V, W) and Al-containing binaries).How a physically meaningful energy is assigned to these hypothet-ical or unstable phases remains an open question in CALPHAD


community. Ideally, their composition and temperature depen-dence should be provided as well. With recent progress in first-principles DFT calculations, the enthalpy of formation [83–89]and even Gibbs free energy for relevant compounds [40] can bepredicted without experimental input. The application of thesedata to the hypothetical compounds in binary and ternary systemsis particularly important for developing physically meaningfuldatabases for CSSA systems.

Lastly, very few experiments are dedicated to phase equilibria[34,45] or thermo-chemistry data of multicomponent CSSAs. Themajority experiments on CSSA in the literature report only on themicrostructure and properties of the alloys in the as-cast state. Inthis condition, alloys possess severe compositional and structuralinhomogeneity in their microstructures. Proper homogenization/annealing is necessary to achieve an ‘‘equilibrium” state [90]. Forexample, Zhang et al. [34] annealed the Al0.3CoCrFeNi at 973 Kfor 500 h and the Al0.7CoCrFeNi alloy at 1523 K for 1000 h, and theyobtained good agreement between CALPHAD prediction and exper-iments. The experiment by Otto et al. [45] that annealed the CoCr-FeMnNi alloy at 1173 K, 973 K, and 773 K for 500 days providesvaluable first-hand experimental data to show that a multicompo-nent CSSA is thermodynamically stable within a certain tempera-ture range, and decomposes to multiphase structures atsufficiently low temperatures, similar to binary and ternary CSSA.

In terms of predicting solid solution formation in multicompo-nent systems, one feasible and effective approach would be to usea combination of phase diagram inspection [9,12], empiricalparameter screening [7,13–21], CALPHAD screening [9,10,12,30–32,78,91], first-principles DFT calculations [40], and ab initiomolecular dynamics (AIMD) simulations [3,8–10]. In fact, usingthis strategy, Gao [8] predicted hundreds of quaternary andhigher-order equimolar compositions in the Dy-Er-Gd-Ho-Lu-Sc-Sm-Tb-Tm-Y, Ba-Ca-Eu-Sr-Yb, Mo-Nb-Ta-Ti-V-W, and Mo-Nb-Re-Ta-Ti-V-W systems. Although CALPHAD method is powerful in cal-culating phase diagrams, it relies on pre-determination of allphases in the system and their thermodynamic descriptions.Future work to accelerate developing physics-based thermo-chemistry data for numerous hypothetical, or unstable phases/endmembers, will contribute substantially in developing reliablethermodynamic databases for CSSA systems. Reliable experimentson phase equilibria and phase thermo-chemistry ternaries, andhigher-order systems, are also critically important.

6. Conclusions

This work reviews the present status in predicting the forma-tion of CSSA and modeling their thermodynamic properties, fromthe perspectives of empirical thermo-physical parameters,CALPHAD method, and first-principles calculations combined withhybrid MC/MD simulations. The effectiveness of most empiricalrules to predict solid solution formation has been critically evalu-ated in a systematic way. Applications of CALPHAD method to pre-dict phase stability and visualize phase diagrams of CSSA systemswere illustrated. Identifying entropy sources of three model CSSAusing first-principles DFT were also presented. Challenges associ-ated with using these methodologies as they pertained to thermo-dynamics property development and alloy design were reviewed,and the efficient strategies they engendered in designing new CSSAwere discussed. The following conclusions were developed:

1. Most proposed empirical thermo-physical parametersexcept Dv are efficient in separating single-phase composi-tions from amorphous compositions, but they fail to sepa-rate single-phase compositions from multiphasecompositions. Considerable overlapping of single-phase

compositions with multiphase compositions requires thedevelopment of new empirical parameters or methodologiesthat are stricter and more effective.

2. The necessary requirements needed to form single-phasesolid solutions were identified in the present work are:

�16.25 kJ/mol � DHliqmix � +5 kJ/mol,

d � 6%, X � 1, g � 0.19, / � 7.0,eR:M:S: � 0.061, and E2=E0 � 13.6 � 10�4.However, they are not sufficient conditions.

3. VEC is effective in separating BCC from FCC compositions.The FCC phase forms when VEC � 7.8, while the BCC phaseforms when VEC � 6:0. However, VEC cannot solely be usedto predict solid solution formation.

4. Using the PanHEA database, calculated isopleths of CoCrx-FeMnNi, CoCrFeMnNix, AlxCoCrFeMn, and CoCrCuxFeMnagree with experiments at higher temperatures. The drivingforce for r phase nucleation from the FCC phase in Co-Crbinary, CoFe-Cr, CoFeNi-Cr, and CoFeMnNi-Cr vertical sec-tions were shown, and the driving force decreased withincreasing the number of principal components.

5. The PanHEA database predicts: For solid solution phases(FCC, BCC, and HCP) and r compound, (1) the total entropyincreased with increasing the number of principal compo-nents; and (2) configurational entropy constituted a verysmall fraction of the total entropy, as determined fromequimolar CoCr, CoCrFe, CoCrFeMn, and CoCrFeMnNi alloys.

6. TCNI8 database predicts noticeable negative excess entropy(exSBCCmix ) together with large negative DHBCC

mix for the BCC phasein Al-containing systems (namely, AlTix, AlVTix, AlNbVTixand AlCrNbVTix), in sharp contrast to non-Al-containing sys-tems (namely, CrTix, CrVTix, CrNbVTix and CrNbVZrTix).

7. DFT calculated enthalpy of formation of select single-phasemulticomponent CSSA are: �7.3 kJ/mol < DHf < +8.4 kJ/mol.

8. Vibrational entropy of FCC CoCrFeNi, BCC MoNbTaW andHCP CoOsReRu is �2.5 times the ideal configurationalentropy at room temperature. The contribution from elec-tronic entropy is truly negligible. The total entropies areequivalent for MoNbTaV and CoOsReRu, and greater thanCoCrFeNi, and follow the same trend as was seen for vibra-tional entropy.

9. Vibrational entropy of mixing is small compared to the idealconfigurational entropy. At T > 400 K, the calculated vibra-tional entropy of mixing is +2.8, �3.6 and �0.3 J/K/mol forFCC CoCrFeNi, BCC MoNbTaW and HCP CoOsReRu,respectively. The total entropies of mixing (i.e.,

DSvibmix þ DSelecmix þ Sconf ) were observed to trend in the followingmanner: CoCrFeNi > CoOsReRu > MoNbTaW.

10. DFT predicted DSvibmix equal to +2.8 J/K/mol for FCC CoCrFeNi,which is in excellent agreement with the calculated excessentropy of +2.5 J/K/mol [33] at 1273 K using CALPHADmethod.

11. Calculated configurational entropies (Sconf ) for CoCrFeNi,MoNbTaW and CoOsReRu alloys were fairly equivalent totheir ideal value (R ln 4) at temperatures close to their soli-dus temperatures. The calculated entropies then start to

decrease very gradually till 1100 K. Rapid decrease in Sconf

becomes more apparent at T < 1100 K, signaling develop-ment of chemical short-range order in the alloys.

Acknowledgements

This work was carried out in support of the Cross-Cutting Tech-nologies Program at the National Energy Technology Laboratory


(NETL) – Strategic Center for Coal, managed by Robert Romanosky(Technology Manager) and Charles Miller (Technology Monitor).The Research was executed through NETL’s Office of Researchand Development’s Innovative Process Technologies (IPT) FieldWork Proposal. Research performed by AECOM Staff was con-ducted under the RES contract DE-FE-0004000. L.Z. Ouyangacknowledges support by DE-FE-0011549 and DE-NA0002630.Work at Carnegie Mellon was supported under grant DE-SC0014506. M.C. Gao thanks Yifan Ye for kindly sharing his Matlabcodes to calculate /-parameter and eR:M:S:.

Disclaimer

The computational modeling work presented in the paper is pro-ject was funded by the Department of Energy, National EnergyTechnology Laboratory, an agency of the United States Govern-ment, through a support contract with AECOM. Neither the UnitedStates Government nor any agency thereof, nor any of theiremployees, nor AECOM, nor any of their employees, makes anywarranty, expressed or implied, or assumes any legal liability orresponsibility for the accuracy, completeness, or usefulness ofany information, apparatus, product, or process disclosed, or repre-sents that its use would not infringe privately owned rights. Refer-ence herein to any specific commercial product, process, or serviceby trade name, trademark, manufacturer, or otherwise, does notnecessarily constitute or imply its endorsement, recommendation,or favoring by the United States Government or any agencythereof. The views and opinions of authors expressed herein donot necessarily state or reflect those of the United States Govern-ment or any agency thereof.

Appendix A. Supplementary material

Supplementary data associated with this article can be found, inthe online version, at http://dx.doi.org/10.1016/j.cossms.2017.08.001.

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