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 ⇑ 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 a National Energy Technology Laboratory, Albany, OR 97321, USA b AECOM, P.O. Box 1959, Albany, OR 97321, USA c CompuTherm LLC, Middleton, WI 53562, USA d Tennessee State University, Nashville, TN 37209, USA e Carnegie Mellon University, Pittsburgh, PA 15213, USA article info abstract

Article history: This paper reviews the three main approaches for predicting the formation of concentrated solid solution Received 18 December 2016 alloys (CSSA) and for modeling their thermodynamic properties, in particular, utilizing the methodologies Revised 15 August 2017 of empirical thermo-physical parameters, CALPHAD method, and first-principles calculations combined Accepted 20 August 2017 with 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, these parameters 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. The temperature-dependent total of the FCC, BCC, HCP, and r phases in equimolar compositions of various systems are presented next, followed by the thermodynamic properties of mixing of the BCC phase in Al-containing and Ti-containing refractory metal systems. First-principles calculations on model FCC, BCC and HCP CSSA reveal the presence of both positive and negative vibrational entropies of mixing, while the calculated electronic entropies of mixing are negligible. Temperature dependent configurational is determined from the atomic structures obtained from MC/MD simulations. Current status and challenges in using these methodologies as they pertain to thermodynamic property analysis and CSSA design are discussed. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction PHAse Diagrams) method, and (3) first-principles density func- tional theory (DFT) calculations combined with hybrid MC/MD Traditional physical metallurgy has paid relatively little atten- simulations. Current status and the challenges in using these tion to multi-component concentrated solid solution alloys (CSSA) methodologies as they pertain to the thermodynamic evaluation until Yeh [1] and Cantor [2] independently published their first and alloy design are also reviewed. Efficient strategies to accelerate papers on the subject in 2004. (Note: Yeh has named such alloys the development of new CSSA by combining these methodologies as ‘‘high-entropy alloys”, but for the sake of consistency only the will be discussed. term ‘‘CSSA” will be used in this paper.) Comprehensive reviews One important topic for thermodynamics of CSSA is predicting on the structures and properties of CSSA are provided in Refs. [3– solid solution formation, given an arbitrary combination of metal- 5]. One important fundamental property of CSSA is the thermody- lic elements in the periodic table. Since the maximum ideal config- namics that governs their phase stability, which impacts the urational entropy occurs at equimolar compositions (i.e., microstructure and resulting mechanical, physical and environ- conf ¼ Smax R ln N, where N is the total number of components in the mental properties. The main objective of this paper is to provide solution and R is the ideal gas constant), most studies on CSSA a critical review on the prediction of CSSA formation and the mod- focus on equimolar or near equimolar compositions. Although eling of thermodynamic properties of single-phase CSSA systems increasing the number of principal components in the system has using three representative methodologies: (1) empirical thermo- been claimed to potentially stabilize the solid solution against physical parameters, (2) CALPHAD (acronym of CALculation of ordered intermetallic compounds, to date the vast majority of mul- ticomponent CSSA published in the literature contain more than ⇑ one phase in the microstructure (i.e., in the as-cast state or after Corresponding author at: National Energy Technology Laboratory, Albany, OR annealing) [3,6–8]. The presence of additional phases in the 97321, USA. E-mail address: [email protected] (M.C. Gao). microstructure usually cause elemental partitioning among these http://dx.doi.org/10.1016/j.cossms.2017.08.001 1359-0286/Ó 2017 Elsevier Ltd. All rights reserved. M.C. Gao et al. / Current Opinion in Solid State and Materials Science 21 (2017) 238–251 239 phases, and thus, reduces the configurational entropy of the solid modeled using DFT [36–42]. Applications of DFT methods and solution. The total number of single-phase CSSA reported in the lit- hybrid Monte Carlo/Molecular Dynamics (MC/MD) simulations to erature is still very limited, and one possible reason may be lack of predict the thermodynamic properties of CSSA are presented in effective searching guidelines [9,10]. Another reason for this could Section 4. of formation of single-phase CSSA with the be that the entropy does not always dominate over . For FCC, BCC and HCP structures at zero temperature are presented, hitherto reported multicomponent CSSA in the literature, the followed by the entropy sources, in particular, from lattice vibra- alloys with the face-centered cubic (FCC) structure are mostly tion, electronic excitation and solid solution configuration. Such based on CoCrFeMnNi [2] and its derivatives. The alloys with the calculations are supplementary to CALPHAD modeling, and can body-centered cubic (BCC) structure are mostly based on refrac- also be used as input to CALPHAD database development. tory metals [11,12], while the alloys with the hexagonal Each of these approaches used in tackling the formation and closed-packed (HCP) structure are mostly based on either rare thermodynamics of CSSA has its own strengths and weaknesses, earth elements or late transition metals [10]. and identifying the underlying hypotheses and potential pitfalls Various empirical thermo-physical parameters [7,13–23] have are important to rationally interpreting the results. This is covered been proposed in the literature to predict solid solution variants in Section 5. In particular the roadblocks to developing reliable of CSSA, which originated from the Hume-Rothery rules. Due to thermodynamic databases for multi-component CSSA systems, lack of reliable phase diagrams for multicomponent CSSA systems, and accordingly the solution to them, are discussed in detail. For composition screening using this approach can be fast but at the example, the lack of experimental data on phase equilibria and cost of reliability. Ambiguity in defining single-phase solid solution thermo-chemistry of multicomponent CSSA systems is addressed alloys has cast doubt on some of empirical parameters. For exam- as is the importance in assigning physically meaningful energy ple, many alloys that contain two FCC phases (e.g., coexistence of data to those hypothetical phases (partially for the purpose of Cu-rich FCC phase with another FCC phase), or two BCC phases database compatibility), and integrating DFT energetics into (e. g., coexistence of a BCC phase with another BCC phase or B2 CALPHAD as input. phase), are counted as single-phase compositions [6,13,22–24]. Furthermore, some empirical rules have been determined based on limited number of compositions [16,18,21,22]. In order to better 2. Empirical thermo-physical parameters understand the strengths and weaknesses of each set of empirical parameters, this study collects and assesses the most up-to-date Following the Hume–Rothery rules, various empirical thermo- experimentally reported CSSA compositions with the FCC, BCC, physical parameters are proposed to predict CSSA formation. These HCP, multiphase, and amorphous structures (see Table S1), and conf parameters include ideal configurational entropy (Sideal), enthalpy evaluates the efficiency in predicting single-phase solid solutions. of mixing in the liquid phase (DHliq ) [13], atomic size difference Alloys with two FCC or BCC phases in the microstructure are trea- mix (d) [13], the X-parameter that is determined by T Sconf =jDHliq j ted as multi-phase compositions. In addition, discrepancies in the m ideal mix atomic radii used by various research groups are also addressed. (where Tm is the composition-weighted average melting point) For consistency the atomic radii that corresponds to coordination [7], valence electron concentration (VEC) [14], electronegativity number 12 are used here for metallic elements while the radii of difference (Dv) [15], the /-parameter [16] that defines the ratio conf tetrahedral covalent bonds are used for interstitial elements. These of the difference between Sideal and the entropy at Tm over the phe- values are taken from Ref. [25]. Assessment of experimental nomenological ‘‘excess entropy” that is calculated using atomic reports in the literature and subsequent evaluation of the widely size and packingpffiffiffiffiffiffiffiffiffi efficiency, elastic residual strain root mean square used empirical thermo-physical parameters used in predicting 2 (eR:M:S:; or he i) [17], the atomic size-related a2 -parameter [18], single-phase CSSA are presented in Section 2. and intrinsic elastic strain energy (E2=E0) [18]. The equations to Contrasted to the use of empirical parameters is CALPHAD calculate these parameters are compiled in Supplementary Materi- method [26,27]. It is the most robust approach for predicting phase als. Commonly accepted criteria based on these parameters to form stability and it is the most effective way in calculating multi- single-phase CSSA are: 15 kJ/mol DHliq +5 kJ/mol, d 6.6%, component phase diagrams as long as a reliable thermodynamic mix X 1.1, e : : : 0.05, and / 20. database for the system under investigation is available. As a R M S More sophisticated empirical models that compare the free mature methodology, CALPHAD method has been applied to a energies (enthalpy and entropy) of both the solid solution phase broad range of materials science and engineering problems beyond and the intermetallic compound are also proposed, which include just phase diagrams, including solidification, coating, joining, and the U-parameter [19], g-parameter [20], and the jcr parameter phase transformation. In particular, applications of CALPHAD 1 [21]. These oversimplified models only consider one hypothetical method to CSSA systems, using PanHEA [28], TCNI8 [29], and TTNI8 solid solution phase and one hypothetical intermetallic compound [29] databases, for materials design have been reported in Refs. at the nominal composition. The hypothesis for these models is [12,30–35] with reasonable success. The applications of CALPHAD that, in order to form a solid solution, the free energy of the hypo- modeling to CSSA are detailed in Section 3. Phase diagrams for thetical solid solution must be more negative than that of the the Co-Cr-Fe-Mn-Ni system with varying number of components hypothetical intermetallic compound at any given temperature. are developed, followed by nucleation driving force contour plots However, this hypothesis is not true for multicomponent alloys for the competing r phase in the temperature-composition space because solid-state phase equilibria for multi-component systems of the various systems. Then the total entropies for the FCC, BCC, usually involve multiple phases (potentially three or more) at a HCP and r phases, as a function of composition and temperature, given temperature. In fact, phase stability in CSSA systems at con- are presented. Finally, the entropies of mixing and enthalpies of stant temperature and pressure is determined by minimizing the mixing of BCC phases in Al-containing and Ti-containing CSSA sys- of the system (Gu ¼ Hu TSu for single phase tems are calculated to illustrate the concept of excess entropy. P ¼ N i First-principles DFT calculations prove to be very useful in pro- u, where H is enthalpy and S is entropy; G f i i G for multiphase viding physics-based thermo-chemistry data and for predicting systems, where f i is the molar fraction of phase i). phase equilibria without experimental input. The atomic structure, The proposed criteria to form a single-phase solid solution using the phase stability, elasticity, electronic structure, magnetic prop- the free energy model are U P 1 [19], g P 1 [20], and = liq < cr erty, and vibrational properties of multicomponent CSSA have been DHIM DHmix j1 [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- conf phases and DSideal. This suggests that high configurational entropy IM stituent binary intermetallic compounds (DHij ) that are taken does not necessarily, at least not always, lead to formation of from DFT data in the literature. The U-parameter is defined as single-phase solid solution because entropy does not always dom- the absolute value of the ratio of the Gibbs free energy of solid inate the contribution to the Gibbs free energy of a system. The solution phase over that of the intermetallic compound. The solid solution entropy may be less than ideal, and additionally enthalpy for both the solid solution and the compound phases the coexisting phases themselves may lead to the minimum Gibbs are derived using the Miedema model [43] while ignoring the energy of the system at a given temperature, pressure and bulk entropy of the compound. The g-parameter [20] is determined by composition. conf =j IMjmax TannSideal DHij . Here an annealing temperature (Tann) defines the temperature where annealing experiments are usually per- 3. CALPHAD modeling: Phase diagrams, driving forces, and cr thermodynamic properties formed to reach homogeneous microstructure. The j1 parameter is determined by 1 T ðSconf DS Þ=DHliq (here DS represents ann ideal IM mix IM The highly studied FCC CoCrFeMnNi alloy was discovered by the entropy of an intermetallic compound). Although the entropy Cantor et al. [2] who investigated two equi-molar alloys with 20 of intermetallic compounds can vary in a wide range depending and 16 components. These alloys contained multiple phases and on its crystal structure and constituent elements, Senkov and Mir- were brittle in the as-cast and as-melt-spun states. Since then, conf acle [21] assigned it to be a fraction (e.g., 0.6) of Sideal, for the sake of many reports studied the phase stability and microstructure of simplicity. the CoCrFeMnNi alloy [44,45] and its sub-system alloys [46,47]. Fig. 1 compares the effectiveness of various empirical rules. The For example, Otto et al. [44] confirmed that a single FCC solid solu- liq DHmix d relation (Fig. 1a) appears to be very powerful. All single- tion formed only in CoCrFeMnNi while five other alloys (i.e., CoCr- phase compositions are located within the region d 6.1% and CuFeMn, CoCrMnNiV, CoFeMnMoNi, CoFeMnNiV and CrFeMnNiTi) liq consisted of more than one phase after annealing at 1273 K or at 16 kJ/mol DHmix < +5 kJ/mol, while all the amorphous and a fraction of the multiphase compositions are located outside the 1123 K for 3 days. It is noteworthy that the experimental observa- region. However, this criterion is not a sufficient requirement since tion by Otto et al. [44] is reasonably well reproduced by CALPHAD the region also contains most of the multiphase compositions. Sim- modeling [8]. Many reported CSSAs with the major FCC phase in ilarly, the X-parameter (Fig. 1b) is useful in separating single-phase the microstructure are derivatives of the Co-Cr-Fe-Mn-Ni system, compositions from amorphous compositions with the threshold and were developed by adding (or replacing) elements, such as, value of 1.0–1.1, but most multiphase compositions also possess AlxCoCrCuFeNi [48,49],AlxCo1.5CrFeNi1.5Tiy [50],AlxCoCrFeNi X 1.1. In contrast, the Dv parameter (Fig. 1b) is not quite as effec- [34,51,52], CoCrCuFeNi [53], CoCrFeNiNbx [54], CoCrFeNiMox [55] tive since for alloys below 0.44 significant scatter is observed. and CoCrFeMnNiVx [56]. Single phase FCC CSSA with four or more Fig. 1c shows a threshold value of 0.19 rather than 1.0 for the g- principal components are very limited [8,57]. For example, adding parameter derived from Ref. [20], and a threshold value of 7.0 Cu causes formation of a Cu-rich FCC phase [48,49,53], while add- rather than 20 for /-parameter [16]. Note that neither g 0.19 ing Al promotes the formation of BCC and B2 phases nor / 7.0 are sufficient conditions for forming solid solutions. [34,48,49,51,52]. In addition, if Ti is added in large amount to this However, combining these parameters proves to be much more CSSA, precipitation of other intermetallic compounds occurs effective than using them alone (Fig. 1c). Alternatively, the [50,58]. The FCC phase in CoCrFeMnNi and its sub-systems are very liq cr ductile. Precipitation of second phases usually enhance the DHIM=DH < j relation (Fig. 1d) separates the vast majority of mix 1 strength of the alloy but leads to reduced ductility. multiphase compositions from single-phase compositions. How- The phase stability of the Co-Cr-Fe-Mn-Ni, Al-Co-Cr-Fe-Ni and ever, exceptions do occur. Many amorphous compositions are Co-Cr-Cu-Fe-Ni systems were modeled using the PanHEA database located in the same region as most single-phase alloys, and about via the PandatTM software [28], and selected isopleths are shown in a dozen single-phase and multiphase compositions fail to meet Fig. 2. The database covers the complete descriptions of the con- the criteria of these relations as well. The elastic strain energy stituent binary and ternary systems of the Al-Co-Cr-Cu-Fe-Mn- vs. residual strain (Fig. 1e) indicates that eR:M:S: 6.1% or Ni-Si-Ti-Zr system [33,34]. While Cr still acts as a strong BCC stabi- E =E 13.6 10 4 promotes the formation of a single-phase solid 2 0 lizer in this system (Fig. 2a), the FCC phase has unusually large Cr solution, but considerable overlap exists between multiphase and solubility (close to 30 at.%). This is primarily due to the entropy single-phase compositions, suggesting that both are necessary, stabilization effect. Similar to its role in steels Ni behaves as potent but not sufficient, conditions in forming a solid solution. Note that FCC stabilizer in this system (Fig. 2b). Conversely, the FCC phase the e : : : and d values are almost equivalent (see Table S1) and R M S has very small solubility for Al (Fig. 2c) due to the large elastic indeed both involve only atomic sizes. strain energy of hosting Al atoms in the FCC lattice of 3d transition In order to predict the crystal structure type of a solid solution, metals (TM) and the tendency to form extremely stable Al-TM Guo et al. [14] first proposed the VEC criterion. They claimed that intermetallic compounds [59]. Although Cu has an FCC crystal FCC phases form at high VEC (> 8), BCC phases form at low VEC structure and has very similar atomic radius to that of other 3d (< 6.87), and FCC + BCC phases form at intermediate VEC TM, the mutual solubility between Cu and TM (except Ni and (6.87–8). As Fig. 1f shows, the threshold VEC identified in this study Mn) is very small due to their inherent repulsive interaction. As a differs slightly from Ref. [14]. That is, FCC alloys are found to have result, Cu solubility in CoCrCuxFeNi is limited to 8% (Fig. 2d). VEC 7.8, and BCC alloys have VEC < 6. Rare-earth HCP alloys have The calculated phase diagrams (Fig. 2) agree well with experiments VEC of 3, while transition-metal HCP alloys have VEC about 7.5–8.5. [34,51–53] at least at higher temperatures. In CALPHAD phase diagrams are calculated by minimizing the Although the VEC criterion can serve as a useful guide for predict- Gibbs free energy of the system at given temperature and pressure. ing the structure type of a solid solution, it cannot be used to pre- Typical phase diagrams show the most stable phases at varying dict whether or not solid solution will form. For example, no temperature, pressure, and composition, while the driving force threshold VEC could be established for multiphase and amorphous diagram shows the nucleation driving force of forming one ‘‘child” conf alloys. Note that most multiphase alloys have fairly high DSideal ( phase from the ‘‘parent” phase in the temperature-composition 14 J/K/mol, Fig. 1f), and no pattern can be identified between the space. One main intermetallic compound competing against the M.C. Gao et al. / Current Opinion in Solid State and Materials Science 21 (2017) 238–251 241

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 phase form. The larger the positive value, the higher the driving force that has Pearson symbol tP30 and space group P42/mnm with pro- to form the r phase from the FCC phase. The calculations indicate totype Cr0.49Fe0.51. Precipitation of the r phase in the CoCrFeMnNi that driving force to form r phase from FCC is strong in the Co-Cr alloy after annealing at 973 K for 500 days is reported in Ref. [45], binary (Fig. 3a), but decreases with increasing configurational which has a composition: 18.1% Co, 46.8% Cr, 16.9% Fe, 12.5% Mn, entropy of the systems, namely, CoFe-Cr, CoFeMn-Cr, and and 5.7% Ni. CoFeMnNi-Cr sections (Fig. 3b–d). This demonstrates that increas- Fig. 3 shows the nucleation driving force contour lines of form- ing entropy can decrease the nucleation driving force of inter- ing r phase from the FCC phase in the binary Co-Cr system, ternary metallic compounds. Driving force contour lines can also be CoFe-Cr section, quaternary CoFeMn-Cr section, and quinary plotted in a phase diagram to show the driving force of forming CoFeMnNi-Cr section. During these calculations, only the FCC a certain phase (referred to as the stable phases), as shown in phase is set ‘‘active” while the r phase is set ‘‘dormant”; all other Fig. S1 (Supplementary Materials). phases are suspended. The area represented by green lines has The total entropies of the FCC, BCC, HCP and r phases are negative driving force, i.e., r phase will not form. The area repre- shown in Fig. 4 for equimolar compositions of CoCr, CoCrFe, CoCr- sented by red lines has positive driving force, i.e., r phase can FeMn, and CoCrFeMnNi. The default stable structures at 1 atm and 242 M.C. Gao et al. / Current Opinion in Solid State and Materials Science 21 (2017) 238–251

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., CoHCP, CrBCC, FeBCC, The thermodynamic properties of mixing for the BCC phase at MnCBCC_A12, and NiFCC). The total entropy of each phase 1273 K for Al-containing systems (Fig. 5a and b) and increases monotonically with increasing the number of compo- Ti-containing systems (Fig. 5c and d) were calculated by Rui nents, reflecting the contribution from configurational entropy. et al. [61] using TCNI8 database. The results reveal significant neg- They also increase with increasing temperature, indicating domi- ative excess entropy for those equimolar compositions (Fig. 5a). nant contribution from vibrational entropy. Keep in mind that BCC The calculated DSmix at 1273 K are 4, 2, +2, +3, +6, and +10 J/ the ideal configurational entropies for equimolar binary, ternary, (molK) for equimolar AlTi, AlNb, AlNbTi, AlTiV, AlNbTiV, and quaternary and quinary solid solution alloys are 5.8, 9.1, 11.5, AlCrNbTiV, respectively. The ideal configurational entropies are and 13.4 J/K/mol, respectively, and they only constitute a small +6, +9, +12 and +13 J/(molK) for equimolar binary, ternary, quater- fraction of the total entropy. This agrees with other reports [41,60]. nary and quinary compositions, respectively. In contrast, the Due to presence of short range order, or segregation, and the excess entropy is predicted to be negligible in refractory CSSA sys- contribution from lattice phonon vibration, or magnetic contri- tems when Al is absent (i.e., CrTix, CrVTix, CrNbVTix and CrNbVZrTix / bution, the total entropy of mixing (DSmix) for a CSSA may not systems; see Fig. 5c). The strong attractive interatomic interaction always follow idealP mixing and can cause positive or negative between Al and transition metals [61] also causes very large nega- deviation from R c ln c (i.e., excess entropy). The excess BCC i i i tive values of enthalpy of mixing (DHmix ), as shown in Fig. 5b, and entropy (exS/ ) of a solution phase (u) is calculated by subtract- BCC mix reducing Al contents results in much less negative DHmix . Con- ing the ideal configurational entropy from the total entropy of P versely, absence of Al in the refractory metal systems corresponds the alloy (DS/ ) [33]: exS/ ¼ DS/ þ R c ln c . The CALPHAD j BCC j BCC mix mix mix i i i to greatly reduced DHmix values (Fig. 5d). The DHmix values are analysis on the thermodynamic properties of mixing for the positive for CrTix, CrVTix, CrNbVTix and CrNbVZrTix systems except FCC phase in the Co-Cr-Fe-Mn-Ni system at 1273 K by Zhang Ti-poor CrVTix compositions, implying overall repulsive interac- and Gao [33] using TCNI8 database [29] reveals positive excess tions for these compositions. Note that all these predictions entropy for binary Co-Cr and pseudo-binary CoFe-Cr, CoFeNi-Cr, obtained from CALPHAD method are subject to experimental veri- and CoFeMnNi-Cr systems. fication and/or confirmation from DFT calculations in the future, M.C. Gao et al. / Current Opinion in Solid State and Materials Science 21 (2017) 238–251 243

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 systems using the PanHEA database, overlaid with the corresponding phase diagrams.

and improvement to the database will enhance the reliability of [41] shows that the fluctuation in DHSS due to the variation in these and subsequent calculations. the atomic configuration are relatively small, at least for the CoOs-

ReRu, CoCrFeNi, and MoNbTaW alloys, and the averaged DHSS are 2.724 ± 0.49, 8.354 ± 0.266, and 7.407 ± 0.069 kJ/mol, 4. DFT calculations and hybrid MC/MD simulations: Entropy respectively. sources In general, the total entropy of a solid solution phase may con- v sist of the contributions from lattice vibration (S ib), configuration For a long time the CSSA community has approximated the total conf elec entropy of mixing for a multicomponent CSSA to be the ideal con- (S ), electron excitation (S ), and magnetic spin fluctuations mag figurational entropy. The exact values are not known for both the (S ): true configurational entropy and total entropy of mixing except a total ¼ vib þ conf þ elec þ mag : ð Þ few recent theoretical reports [40,41]. As such this section mainly S S S S S 1 focuses on predicting enthalpy of formation of solid solution The entropy sources are temperature and volume dependent. It (DHSS), and more importantly, predicting the entropy sources of is assumed that the magnetic entropy term to FCC CoCrFeNi and example CSSA compositions using DFT and MC/MD simulations. CoCrFeMnNi should be small at T 293 K because their critical The DFT calculations were performed using VASP (Vienna Ab Initio magnetic ordering temperatures are well below room temperature Simulation Package) [62,63], and details on the VASP settings are [64,65]. No magnetism is reported for those refractory BCC multi- provided elsewhere [40,41]. The input atomic structures for vib component CSSAs. The vibrational entropy (S ) is calculated by: equimolar quaternary CSSA are taken from the special quasi- Z 1 random structure (SQS) reported in Ref. [41]: the 64-atom SQS v v S ibðV; TÞ¼3k n ib½ðf þ 1Þlnðf þ 1Þf lnf d ; ð2Þ models for quaternary compositions, the 160-atom SQS model B BE BE BE BE e 0 for quinary HCP alloys, and the 125-atom SQS models for quinary vib FCC and BCC alloys. The calculated DHSS of widely studied single- where n is the phonon , and f BE is the Bose- phase CSSA are shown in Fig. 6, which are obtained from one Einstein distribution function. Phonon frequencies of the CSSA were SQS configuration per alloy composition. Prior study by Gao et al. calculated using the harmonic approximation by diagonalizing the 244 M.C. Gao et al. / Current Opinion in Solid State and Materials Science 21 (2017) 238–251

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 reported are always accepted if the swap reduces the energy by VASP. The quasi-harmonic approximation can improve the accu- (DE ¼ Eswap Einitial < 0; Eswap and Einitial are the energies in the racy at high temperatures, and more importantly, can predict the swapped and initial configurations, respectively). For those swaps coefficient of thermal expansion. For a comprehensive review on that increase the energy (DE > 0), the acceptance is determined the lattice vibration of crystalline solids, readers can refer to the by assessing the probability, P ¼ expðDE=kBTÞ. The energy barrier work by Fultz [60]. separating the swapped and initial states are not relevant for Electronic excitation across the by migrating from Monte Carlo simulations and thus not considered. More details valence band to conduction band gives rise to electronic entropy on the procedures are provided elsewhere [37,40,66]. The reduc- (Selec), which can be determined by: tion in configurational entropy due to short-range chemical order

Z 1 can be calculated using the Guggenheim quasichemical model elecð ; Þ¼ elec½þð Þ ð Þ ; ð Þ [67,68], Bethe lattice [69], or Kikuchi cluster variation method S V T 2kB n f FDlnf FD 1 f FD ln 1 f FD de 3 1 (CVM) [70]. Specifically, the reduction in entropy due to near

elec neighbor correlations (yij) using Kikuchi CVM is calculated by: where n is the electron density of states, and f FD is Fermi-Dirac distribution function. XN For a solid solution as the temperature decreases, chemical ¼ ð = ÞðÞ I yijln yij cicj 4 ordering or phase separation sets in, which reduces the configura- i;j¼1 tional entropy. At zero temperature, all entropy sources should vanish according to the Third Law of Thermodynamics. Such exam- where ci and cj are the mole fraction of elements i and j, ples are shown in refractory CSSA by Widom et al. [37,40] using respectively. hybrid MC/MD simulations. The canonical ensemble (NVT) (i.e., Fig. 7 compares the vibrational, electronic, and configurational constant amount of substance, volume and temperature) was entropies for three model CSSAs: (1) FCC CoCrFeNi, (2) BCC MoN- adopted for the FCC and BCC structures, and the isothermal- bTaW, and (3) HCP CoOsReRu. Note that the vibrational entropy isobaric ensemble (NPT) (i.e., constant amount of substance, pres- of BCC Cr was used here because of presence of 64 imaginary vibra- sure and temperature) for the HCP structure. The supercells used in tional modes and the elastic instability for FCC Cr. The vibrational the present study contain 108, 128, and 96 atoms for FCC CoCrFeNi, entropy is very similar for MoNbTaW and CoOsReRu (as expected BCC MoNbTaW, and HCP CoOsReRu, respectively. The simulations due to their similar average masses), but it is slightly smaller for were done by alternating molecular dynamics at each temperature CoCrFeNi (also as expected due to its lower atomic mass). This is with Monte Carlo swaps, each performed from first principles shown in Fig. 7a. Vibrational entropy is sensitive to temperature. using VASP. Monte Carlo swaps of atomic species on different sites It vanishes at zero temperature and increases rapidly with M.C. Gao et al. / Current Opinion in Solid State and Materials Science 21 (2017) 238–251 245

BCC BCC Fig. 5. Calculated entropy of mixing (DSmix ) and enthalpy of mixing (DHmix ) 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 the ideal 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 color in this figure legend, the reader is referred to the web version of this article.) increasing temperature. Note that the vibrational entropies for the In contrast, the electronic entropy is small for all three compo- three alloys are already 2.5 times the ideal configurational sitions even at high temperatures (Fig. 7b). Configurational entro- entropy at T = 300 K (Fig. 7a). pies (Sconf ) for these alloys are fairly close to their ideal value 246 M.C. Gao et al. / Current Opinion in Solid State and Materials Science 21 (2017) 238–251

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

(R ln 4) at temperatures close to the solidus temperatures, and start constant values, namely, +2.8 J/K/mol, 3.6 J/K/mol and 0.4 J/ to decrease very gradually until the temperature reaches 1100 K K/mol for CoCrFeNi, MoNbTaW and CoOsReRu, respectively, (Fig. 7c). At this point Sconf decreases rapidly, signaling the develop- because the heat capacities approach their classical limit at 3R. v conf ib ment of chemical short-range order in the alloys. Note that S at Note that the predicted DSmix of +2.8 J/K/mol for CoCrFeNi is in room temperature is much lower for MoNbTaW than the others excellent agreement with the calculated excess entropy of +2.5 elec due to the tendency in forming the ordered BCC (i.e., B2) as J/K/mol [33] at 1273 K using CALPHAD method. In contrast, DSmix revealed by Widom et al. [37,66]. The total entropies (i.e., the values are close to zero (Fig. 8b). The total entropies of mixing sum of vibrational, electronic, and configurational entropy) of vib þ elec þ conf (i.e., DSmix DSmix S ) are shown in Fig. 8c and correspond to, these alloys are shown in Fig. 7d. Stotal is equivalent for MoNbTaV in descending order: CoCrFeNi > CoOsReRu > MoNbTaW. and CoOsReRu, and is greater than CoCrFeNi, following the same The positive contribution of phonon vibration to the solid solu- trend as seen for vibrational entropy (Fig. 7a). tion stabilization was also reported in the literature by Fultz and Alloy entropies of mixing are determined as: co-workers [60,71,72]. For example, Fultz et al. [71] measured vib XN DSmix to be +1.17, +1.67 and +1.78 J/K/mol for disordered BCC solid DSmix ¼ Salloy ciSi ð5Þ solution alloys Fe70Cr30,Fe53Cr47 and Fe30Cr70, respectively. Nagel i¼1 et al. [72] found that the disordered FCC Cu3Au has a higher vibra- tional entropy than the ordered L1 Cu Au by approximately where S is the entropy (total, vibrational, etc.) of the ith element, c 2 3 i i +1.16 J/K/mol at high temperature. However, generally speaking is the molar composition of the ith element, and N is the total num- it is not obvious whether chemical disordering causes positive or ber of elements in the CSSA. Note that the pure elements lack con- negative vibrational entropy of mixing, or whether a disordered conf ¼ conf figurational entropy, and hence, DSmix Salloy. alloy will exhibit greater vibrational entropy than its ordered form. Fig. 8 compares the mixing vibrational and electronic entropies The vibrational entropy of mixing of an alloy may depend on the for the three model alloys. The calculations predict that lattice crystal structure, bonding and enthalpy of the alloy, as well as vibration increases the total entropy of CoCrFeNi, while vibration the molar volumes and structures of the constituent elements. vib decreases the total entropy of MoNbTaW (Fig. 8a). The DSmix values vib for the three cases are small relative to the individual Salloy because 5. Discussion and outlook the alloy vibrational frequencies lie close to the average of the indi- vib < conf vib vidual elements. Indeed, often DSmix S , and DSmix is near zero The three major approaches that are widely used to predict for CoOsReRu. At temperatures above the Debye temperature CSSA formation and analyze the underlying thermodynamic (i.e., T 400 K), the vibrational entropies of mixing approach properties have been reviewed. Due to inherent hypotheses or M.C. Gao et al. / Current Opinion in Solid State and Materials Science 21 (2017) 238–251 247

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. compositions and multiphase compositions requires that new Accordingly, it is prudent not to overstate their predictive capabil- parameters, or identification methodologies, be developed that are ity, especially when using empirical thermo-physical parameters stricter and more effective. The necessary requirements needed alone. In the following section the reasons that may lead to inaccu- (but not sufficient) to form single-phase solid solution identified in racies in predictions when using these empirical parameters will liq this work are: 16.25 kJ/mol DHmix +5 kJ/mol, d 6%, X 1, be discussed. Additionally, ways to overcome common issues in -4 g 0.19, / 7.0, eR:M:S: 0.061, and E2=E0 13.6 10 . Although developing reliable physics-based thermodynamic databases for VEC by itself cannot be used to predict solid solution formation, multi-component CSSA systems will also be explored. VEC can be effective in separating BCC from FCC compositions. For 5.1. Empirical parameters: Flaws and limitations example, the FCC phase forms when VEC 7:8, while BCC phase forms when VEC 6:0. Note that the threshold values identified in This critical re-evaluation of empirical thermo-physical parame- this work differ somewhat from those reported in the literature. This ters demonstrates that most parameters are efficient in separating appears to be due to the different compositional spaces assessed as single-phase compositions from amorphous compositions. In gen- well as different values of atomic radii. As more reliable and verified eral, they fail to separate single-phase compositions from multi- experimental data emerge in the literature, the threshold values phase compositions. Considerable overlap between single-phase may become a range rather than a fixed value [61]. 248 M.C. Gao et al. / Current Opinion in Solid State and Materials Science 21 (2017) 238–251

Besides bias due to the availability of experimental composi- higher-order system can be obtained via an extrapolation method tions, the empirical rules also suffer in other ways. Most empirical from the binary and ternary systems [81], subject to further adjust- rules assign the enthalpy of mixing of the liquid as determined ment of the database in comparison to experiments [33]. However, from the Miedema model to the solid solution phase, primarily extrapolation of binaries and ternaries into higher order systems due to lack of reliable enthalpy data for the solid solution phase may not always guarantee satisfactory agreement with experi- for CSSA systems. It is known that the enthalpy values from Mie- ments. For example, if there are new stable quaternary or higher- dema models can differ significantly from experimental data for order phases, or if the descriptions for those unstable or hypothet- certain systems. Gao et al. [12] studied these empirical parameters ical phases/endmembers are so unphysical that cause their unde- and thermodynamic properties for sixteen refractory BCC CCSA sired over-stabilization or underestimated stability, then BCC extrapolation may not work. Since CALPHAD itself is not a predic- and found that both the sign and absolute value of DHmix are not necessarily in accord with DHliq . In that work nine alloys exhibit tive methodology like DFT, the thermodynamic descriptions for all mix phases in the system must be pre-defined. In case there exist new opposite sign while seven alloys show significant contrast in their quaternary or higher-order compound phases in the system, then absolute values. their Gibbs free energy descriptions in CALPHAD framework must Secondly, the true configurational entropy of solid solution may be determined first. not always follow ideal mixing behavior, and the deviation can be Developing thermodynamic databases for multicomponent substantial. Positive excess entropies were predicted for the FCC CSSA systems requires that the database be valid for the entire phase at 1273 K in Co-Cr-Fe-Mn-Ni system [33], and negative composition range, whereas traditional alloys are only concerned excess entropies for the BCC phase at 1273 K in Al-containing CSSA about the compositions in the corner of a principal element. For such as Al-Cr-Nb-Ti-V [61]. Similarly, the enthalpy of mixing for example, development of 10-component CSSA system requires the solid solution phase cannot be ignored, especially for systems optimization over the entire composition ranges for 45 constituent that exhibit strong tendency to form extremely stable intermetallic binaries and 120 constituent ternaries. Moreover, to date complete compounds (e.g., Al-containing CSSA) [61]. ternary phase diagrams are still very limited in extent, and even Lastly, the configurational entropy of intermetallic compounds some binary phase diagrams require further experimental study can be very small if the compositional homogeneity range remains to better define them. For example, a recent DFT study by Widom extremely small in the multicomponent systems. Otherwise, the [40] predicts the existence of stable C14 NbV Laves phase at configurational entropies, in general, cannot be ignored, as argued 2 T 1100 K in the binary Nb-V system. This prediction has yet to by Senkov and Miracle [21]. Therefore, future work is needed to be confirmed by experiment. Element Tc has the HCP crystal struc- differentiate compositions that will likely show appreciable com- ture, and it is intuitively expected that there will be large mutual positional homogeneity so that the empirical rules can be modified solubility between Tc and several transition elements (such as accordingly. Re, Ru, and Os). However, most of the binary phase diagrams per- Regardless of the subtle differences among those empirical taining to Tc are still unknown. Therefore, a complete database thermo-physical rules, they share one common thermodynamic including 10 or more components is rarely available. For example, flaw: The phase stability of a multicomponent system is not deter- even though commercial thermodynamic databases developed for mined by the simple comparison of the Gibbs free energies of a hypo- traditional alloys may contain greater than 20 components, they thetical solid solution with a hypothetical compound phase. Rather, it are not intended for CSSA use where compositions are located far is determined by the equilibrium condition that the chemical potential away from the principal element corner. Individuals doing CCSA of each component should be equal in all phases (or equivalently, the thermodynamic research should be aware that using the thermo- alloy free energy should touch the convex hull of free energies of all dynamic databases developed for traditional alloys with one prin- competing phases). Consequently, these rules cannot predict the cipal element may lead to erroneous results. equilibrium phases and their crystal structures of arbitrary alloys Another main challenge in developing databases for multicom- reliably, nor can they predict the temperature and pressure depen- ponent CSSA systems is the lack of reliable energy data for hypo- dence of phase stability. In fact, the simple comparison of the Gibbs thetical phases and endmembers, as discussed in Ref. [33]. One free energy at one composition to determine phase equilibria example is the phase in the Al-Co-Cr-Cu-Fe-Mn-Ni-Ti system. among phases is only valid for the unary system. r 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, 5.2. CALPHAD for CSSA systems: Issues and solutions however, should be physically meaningful, and need to be accounted for in the database for the sake of compatibility. The To overcome those defects to which there are no easy solutions adopted sublattice model and the assigned energies for the corre- within the framework of the empirical thermo-physical parame- sponding endmembers have profound influence on the stability of ters, CALPHAD method proves to be the most direct solution to the r phase. For instance, both TCNI8 and PanHEA databases phase stability of multicomponent systems. In terms of CSSA, pre- underestimate the precipitation temperature of the r phase for dicted solid solution formation is confirmed through experiments CoCrFeMnNi (i.e., 867 K by the former and 624 K by the latter). [73–77]. Screening is carried out to identify low-density equimolar Experiment has shown that r phase precipitates at 973 K [45]. compositions in single-phase solid solution [32,78]. Furthermore, The predicted composition of r phase also differs greatly from non-equilibrium solidification using Scheil model [79] is used to experiments [45,82]. TCNI8 predicts a composition of r phase at predict segregation behavior in the as-cast state [33,73,75– T = 773 K to be 5.3% Co - 37.2% Cr - 30.0% Fe - 26.3% Mn - 1.2% 77,80]. Quantitative analyses in enthalpy, entropy, and Gibbs free Ni; PanHEA on the other hand predicts a composition at energy of CSSA are also presented in Refs. [12,44,75,76]. However, T = 623 K to be 2.1% Co - 48.5% Cr - 20.5% Fe - 28.5% Mn - 0.4% Ni. all these CALPHAD modeling predictions are susceptible to the reli- Another example is Laves phase (i.e., C14, C15 and C36) in ability of the thermodynamic databases. refractory metal system (e.g., Al-Cr-Hf-Mo-Nb-Ta-Ti-V-W-Zr). The essence of CALPHAD approach for a multicomponent sys- Laves phase is stable in Cr-M binaries (where M = Hf, Nb, Ta, Ti, tem is to develop reliable self-consistent thermodynamic descrip- Zr), but is not stable in many other systems (e.g., Cr-M (M = V, tions of all constituent binary and ternary systems, by fitting Mo, W), Ti-M (M = Mo, Nb, Ta, V, W) and Al-containing binaries). experimentally determined thermochemistry and phase equilibria How a physically meaningful energy is assigned to these hypothet- as well as DFT-predicted data. The description of a quaternary or ical or unstable phases remains an open question in CALPHAD M.C. Gao et al. / Current Opinion in Solid State and Materials Science 21 (2017) 238–251 249 community. Ideally, their composition and temperature depen- compositions with multiphase compositions requires the dence should be provided as well. With recent progress in first- development of new empirical parameters or methodologies principles DFT calculations, the enthalpy of formation [83–89] that are stricter and more effective. and even Gibbs free energy for relevant compounds [40] can be 2. The necessary requirements needed to form single-phase predicted without experimental input. The application of these solid solutions were identified in the present work are: data to the hypothetical compounds in binary and ternary systems liq 16.25 kJ/mol DHmix +5 kJ/mol, is particularly important for developing physically meaningful d 6%, X 1, g 0.19, / 7.0, databases for CSSA systems. 4 eR:M:S: 0.061, and E2=E0 13.6 10 . Lastly, very few experiments are dedicated to phase equilibria However, they are not sufficient conditions. [34,45] or thermo-chemistry data of multicomponent CSSAs. The 3. VEC is effective in separating BCC from FCC compositions. majority experiments on CSSA in the literature report only on the The FCC phase forms when VEC 7.8, while the BCC phase microstructure and properties of the alloys in the as-cast state. In : this condition, alloys possess severe compositional and structural forms when VEC 6 0. However, VEC cannot solely be used inhomogeneity in their microstructures. Proper homogenization/ to predict solid solution formation. annealing is necessary to achieve an ‘‘equilibrium” state [90]. For 4. Using the PanHEA database, calculated isopleths of CoCrx- FeMnNi, CoCrFeMnNix,AlxCoCrFeMn, and CoCrCuxFeMn example, Zhang et al. [34] annealed the Al0.3CoCrFeNi at 973 K agree with experiments at higher temperatures. The driving for 500 h and the Al0.7CoCrFeNi alloy at 1523 K for 1000 h, and they obtained good agreement between CALPHAD prediction and exper- force for r phase nucleation from the FCC phase in Co-Cr iments. The experiment by Otto et al. [45] that annealed the CoCr- binary, CoFe-Cr, CoFeNi-Cr, and CoFeMnNi-Cr vertical sec- FeMnNi alloy at 1173 K, 973 K, and 773 K for 500 days provides tions were shown, and the driving force decreased with valuable first-hand experimental data to show that a multicompo- increasing the number of principal components. nent CSSA is thermodynamically stable within a certain tempera- 5. The PanHEA database predicts: For solid solution phases ture range, and decomposes to multiphase structures at (FCC, BCC, and HCP) and r compound, (1) the total entropy sufficiently low temperatures, similar to binary and ternary CSSA. increased with increasing the number of principal compo- In terms of predicting solid solution formation in multicompo- nents; and (2) configurational entropy constituted a very nent systems, one feasible and effective approach would be to use small fraction of the total entropy, as determined from a combination of phase diagram inspection [9,12], empirical equimolar CoCr, CoCrFe, CoCrFeMn, and CoCrFeMnNi alloys. parameter screening [7,13–21], CALPHAD screening [9,10,12,30– 6. TCNI8 database predicts noticeable negative excess entropy ex BCC BCC 32,78,91], first-principles DFT calculations [40], and ab initio ( Smix ) together with large negative DHmix for the BCC phase molecular dynamics (AIMD) simulations [3,8–10]. In fact, using in Al-containing systems (namely, AlTix, AlVTix, AlNbVTix this strategy, Gao [8] predicted hundreds of quaternary and and AlCrNbVTix), in sharp contrast to non-Al-containing sys- higher-order equimolar compositions in the Dy-Er-Gd-Ho-Lu-Sc- tems (namely, CrTix, CrVTix, CrNbVTix and CrNbVZrTix). Sm-Tb-Tm-Y, Ba-Ca-Eu-Sr-Yb, Mo-Nb-Ta-Ti-V-W, and Mo-Nb-Re- 7. DFT calculated enthalpy of formation of select single-phase Ta-Ti-V-W systems. Although CALPHAD method is powerful in cal- multicomponent CSSA are: 7.3 kJ/mol < DHf < +8.4 kJ/mol. culating phase diagrams, it relies on pre-determination of all 8. Vibrational entropy of FCC CoCrFeNi, BCC MoNbTaW and phases in the system and their thermodynamic descriptions. HCP CoOsReRu is 2.5 times the ideal configurational Future work to accelerate developing physics-based thermo- entropy at room temperature. The contribution from elec- chemistry data for numerous hypothetical, or unstable phases/ tronic entropy is truly negligible. The total entropies are endmembers, will contribute substantially in developing reliable equivalent for MoNbTaV and CoOsReRu, and greater than thermodynamic databases for CSSA systems. Reliable experiments CoCrFeNi, and follow the same trend as was seen for vibra- on phase equilibria and phase thermo-chemistry ternaries, and tional entropy. higher-order systems, are also critically important. 9. Vibrational entropy of mixing is small compared to the ideal configurational entropy. At T > 400 K, the calculated vibra- 6. Conclusions tional entropy of mixing is +2.8, 3.6 and 0.3 J/K/mol for FCC CoCrFeNi, BCC MoNbTaW and HCP CoOsReRu, This work reviews the present status in predicting the forma- respectively. The total entropies of mixing (i.e., vib þ elec þ conf tion of CSSA and modeling their thermodynamic properties, from DSmix DSmix S ) were observed to trend in the following the perspectives of empirical thermo-physical parameters, manner: CoCrFeNi > CoOsReRu > MoNbTaW. CALPHAD method, and first-principles calculations combined with vib 10. DFT predicted DSmix equal to +2.8 J/K/mol for FCC CoCrFeNi, hybrid MC/MD simulations. The effectiveness of most empirical which is in excellent agreement with the calculated excess rules to predict solid solution formation has been critically evalu- entropy of +2.5 J/K/mol [33] at 1273 K using CALPHAD ated in a systematic way. Applications of CALPHAD method to pre- method. dict phase stability and visualize phase diagrams of CSSA systems 11. Calculated configurational entropies (Sconf ) for CoCrFeNi, were illustrated. Identifying entropy sources of three model CSSA MoNbTaW and CoOsReRu alloys were fairly equivalent to using first-principles DFT were also presented. Challenges associ- their ideal value (R ln 4) at temperatures close to their soli- ated with using these methodologies as they pertained to thermo- dus temperatures. The calculated entropies then start to dynamics property development and alloy design were reviewed, decrease very gradually till 1100 K. Rapid decrease in Sconf and the efficient strategies they engendered in designing new CSSA becomes more apparent at T < 1100 K, signaling develop- were discussed. The following conclusions were developed: ment of chemical short-range order in the alloys. 1. Most proposed empirical thermo-physical parameters except Dv are efficient in separating single-phase composi- Acknowledgements tions from amorphous compositions, but they fail to sepa- rate single-phase compositions from multiphase This work was carried out in support of the Cross-Cutting Tech- compositions. Considerable overlapping of single-phase nologies Program at the National Energy Technology Laboratory 250 M.C. Gao et al. / Current Opinion in Solid State and Materials Science 21 (2017) 238–251

(NETL) – Strategic Center for Coal, managed by Robert Romanosky [14] S. Guo, C. Ng, J. Lu, C.T. Liu, Effect of valence electron concentration on stability (Technology Manager) and Charles Miller (Technology Monitor). of fcc or bcc phase in high entropy alloys, J. Appl. Phys. 109 (2011) 103505. [15] S. Fang, X. Xiao, L. Xia, W. Li, Y. Dong, Relationship between the widths of The Research was executed through NETL’s Office of Research supercooled liquid regions and bond parameters of Mg-based bulk metallic and Development’s Innovative Process Technologies (IPT) Field glasses, J. Non-Cryst. Solids 321 (2003) 120–125. Work Proposal. Research performed by AECOM Staff was con- [16] Y.F. Ye, Q. Wang, J. Lu, C.T. Liu, Y. Yang, Design of high entropy alloys: a single- parameter thermodynamic rule, Scripta Mater. 104 (2015) 53–55. ducted under the RES contract DE-FE-0004000. L.Z. Ouyang [17] Y.F. Ye, C.T. Liu, Y. Yang, A geometric model for intrinsic residual strain and acknowledges support by DE-FE-0011549 and DE-NA0002630. phase stability in high entropy alloys, Acta Mater. 94 (2015) 152–161. Work at Carnegie Mellon was supported under grant DE- [18] Z.J. Wang, W.F. Qiu, Y. Yang, C.T. Liu, Atomic-size and lattice-distortion effects in newly developed high-entropy alloys with multiple principal elements, SC0014506. M.C. Gao thanks Yifan Ye for kindly sharing his Matlab Intermetallics 64 (2015) 63–69. codes to calculate /-parameter and eR:M:S:. [19] D.J.M. King, S.C. Middleburgh, A.G. McGregor, M.B. Cortie, Predicting the formation and stability of single phase high-entropy alloys, Acta Mater. 104 (2016) 172–179. Disclaimer [20] M.C. Troparevsky, J.R. Morris, P.R.C. Kent, A.R. Lupini, G.M. Stocks, Criteria for predicting the formation of single-phase high-entropy alloys, Phys. Rev. X 5 (2015) 011041. The computational modeling work presented in the paper is pro- [21] O.N. Senkov, D.B. Miracle, A new thermodynamic parameter to predict ject was funded by the Department of Energy, National Energy formation of solid solution or intermetallic phases in high entropy alloys, J. Technology Laboratory, an agency of the United States Govern- Alloys Compd. 658 (2016) 603–607. ment, through a support contract with AECOM. Neither the United [22] M.G. Poletti, L. Battezzati, Electronic and thermodynamic criteria for the occurrence of high entropy alloys in metallic systems, Acta Mater. 75 (2014) States Government nor any agency thereof, nor any of their 297–306. employees, nor AECOM, nor any of their employees, makes any [23] I. Toda-Caraballo, P.E.J. Rivera-Diaz-del-Castillo, A criterion for the formation warranty, expressed or implied, or assumes any legal liability or of high entropy alloys based on lattice distortion, Intermetallics 71 (2016) 76– 87. responsibility for the accuracy, completeness, or usefulness of [24] Y. Zhang, S. Guo, C.T. Liu, X. Yang, Phase formation rules, in: M.C. Gao, J.W. Yeh, any information, apparatus, product, or process disclosed, or repre- P.K. Liaw, Y. Zhang (Eds.), High-Entropy Alloys: Fundamentals and sents that its use would not infringe privately owned rights. Refer- Applications, first ed., Springer International Publishing, Cham, Switzerland, 2016, pp. 21–50. ence herein to any specific commercial product, process, or service [25] W.B. Pearson, Crystal chemistry and physics of metals and alloys, Phys. Today by trade name, trademark, manufacturer, or otherwise, does not 26 (1972) 59. necessarily constitute or imply its endorsement, recommendation, [26] L. Kaufman, Computer Calculation of Phase Diagrams, Academic Press, New York, 1970. or favoring by the United States Government or any agency [27] Y.A. Chang, S.L. Chen, F. Zhang, X.Y. Yan, F.Y. Xie, R. Schmid-Fetzer, et al., Phase thereof. The views and opinions of authors expressed herein do diagram calculation: past, present and future, Prog. Mater. Sci. 49 (2004) 313– not necessarily state or reflect those of the United States Govern- 345. [28] PandatTM. Thermodynamic Calculations and Kinetic Simulations. Madison, WI ment or any agency thereof. 53562, USA: CompuTherm LLC. [29] ThermoCalc. www.thermocalc.com. [30] C. Zhang, F. Zhang, S.L. Chen, W.S. Cao, Computational thermodynamics aided Appendix A. Supplementary material high-entropy alloy design, JOM 64 (2012) 839–845. [31] F. Zhang, C. Zhang, S.L. Chen, J. Zhu, W.S. Cao, U.R. Kattner, An understanding of Supplementary data associated with this article can be found, in high entropy alloys from phase diagram calculations, CALPHAD 45 (2014) 1– 10. the online version, at http://dx.doi.org/10.1016/j.cossms.2017.08. [32] O.N. Senkov, J.D. Miller, D.B. Miracles, C. Woodward, Accelerated exploration 001. of multi-principal element alloys for structural applications, CALPHAD 50 (2015) 32–48. [33] C. Zhang, M.C. Gao, CALPHAD modeling of high-entropy alloys, in: M.C. Gao, J. 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