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Acta Materialia 55 (2007) 3281–3303

Integrated design of Nb-based superalloys:Ab initio calculations, computational thermodynamics

and kinetics, and experimental results

G. Ghosh *, G.B. Olson

Department of Materials Science and Engineering, Robert R. McCormick School of Engineering and Applied Science, Northwestern

University, 2220 Campus Drive, Evanston, IL 60208-3108, USA

Received 29 July 2006; received in revised form 22 January 2007; accepted 22 January 2007Available online 23 March 2007

Abstract

An optimal integration of modern computational tools and efficient experimentation is presented for the accelerated design of Nb-based superalloys. Integrated within a systems engineering framework, we have used ab initio methods along with alloy theory toolsto predict phase stability of solid solutions and intermetallics to accelerate assessment of thermodynamic and kinetic databases enablingcomprehensive predictive design of multicomponent multiphase microstructures as dynamic systems. Such an approach is also applicablefor the accelerated design and development of other high performance materials. Based on established principles underlying Ni-basedsuperalloys, the central microstructural concept is a precipitation strengthened system in which coherent cubic aluminide phase(s) pro-vide both creep strengthening and a source of Al for Al2O3 passivation enabled by a Nb-based alloy matrix with required ductile-to-brit-tle transition temperature, atomic transport kinetics and oxygen solubility behaviors. Ultrasoft and PAW pseudopotentials, asimplemented in VASP, are used to calculate total energy, density of states and bonding charge densities of aluminides with B2 andL21 structures relevant to this research. Characterization of prototype alloys by transmission and analytical electron microscopy demon-strates the precipitation of B2 or L21 aluminide in a (Nb) matrix. Employing Thermo-Calc and DICTRA software systems, thermody-namic and kinetic databases are developed for substitutional alloying elements and interstitial oxygen to enhance the diffusivity ratio ofAl to O for promotion of Al2O3 passivation. However, the oxidation study of a Nb–Hf–Al alloy, with enhanced solubility of Al in (Nb)than in binary Nb–Al alloys, at 1300 �C shows the presence of a mixed oxide layer of NbAlO4 and HfO2 exhibiting parabolic growth.� 2007 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Ab initio electron theory; Analytical electron microscopy; Calphad; Elastic properties; Kinetics

1. Introduction

The efficiency of turbine engines is primarily limited bythe operating temperature, which in turn is determinedby several temperature-dependent properties of materialsused in turbine engines. A great deal of research is under-way to design new materials capable of operating at tem-peratures higher than current capability, as the efficiencyof a Carnot engine is directly related to the homologous

1359-6454/$30.00 � 2007 Acta Materialia Inc. Published by Elsevier Ltd. All

doi:10.1016/j.actamat.2007.01.036

* Corresponding author. Tel.: +1 847 467 2595; fax: +1 847 491 7820.E-mail address: [email protected] (G. Ghosh).

operating temperature. Currently, Ni-base alloys are thematerials of choice for high temperature turbine bladeapplications as they satisfy a unique combination ofdesired properties. Fig. 1 shows the evolution of Ni-basedsuperalloys over the years as a function of their tempera-ture capability [1].

The operating temperature capability of Ni-based super-alloys is less than 1150 �C as the most advanced Ni-basedsingle crystal superalloys melt below 1350 �C [2], thus trig-gering the need for new alloys that can operate around1300 �C. Ideally, such an alloy should have a substantiallyhigher melting point (or low homologous temperature(<0.5Tm) at 1300 �C), and at operating temperature it should

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Fig. 1. Evolution of temperature capability of Ni-base superalloys [1].

3282 G. Ghosh, G.B. Olson / Acta Materialia 55 (2007) 3281–3303

have high creep strength and high oxidation resistance. Foroxidation resistance, the material should be capable of pas-sive oxidation, making it capable of forming a protectiveoxide scale. The refractory metal niobium has a meltingtemperature of 2467 �C and also has a low density, thusmaking it an attractive candidate for the replacement ofnickel. However, Nb has a poor oxidation resistance [3]and only moderate strength at high temperature [4].

A considerable amount of research has been done on thedesign of composite microstructures consisting of (Nb),Nb5Si3 and other intermetallics, such as the Laves phaseCr2Nb to improve oxidation, strength, creep at elevatedtemperatures, as well as ductility and damage tolerance atambient temperatures [5–9]. However, as the intermetallicphases are not present as fine scale precipitates, creep insuch composites is not controlled by dislocation climb.Rather, creep strength is imparted by Nb5Si3, and is con-trolled by diffusion of Nb in it [2]. Furthermore, it has beenfound that such multiphase Nb-based composites formnonprotective oxide scales above 1200 �C, and the scalesconsist of several oxides, including Nb2O5, CrNbO4 andNb2TiO7 [10]. Here, we present a systems-based approach[11] to design a new precipitation strengthened Nb-basedsuperalloy having desired oxidation resistance, creep andductile-to-brittle transition temperature (DBTT) for aero-turbine applications operating at 1300 �C or above.

The remainder of the paper is organized as follows. InSection 2, we present the conceptual design principles,and the integration of computational tools and experimen-tal methods. In Section 3, we present the computationalmethodology employed in this study. In Section 4, we pre-sent alloy preparation and various experimental methodsused to characterize the prototype alloys. In Section 5,we present computational and experimental results. In Sec-tion 6, we discuss the results of ab initio total energy andelectronic structure calculations. Conclusions are summar-ized in Section 7.

2. Integrated design: principles, computational and

experimental tools, and relevance to prior research on Nb-

based alloys

To design precipitation strengthened Nb-base superal-loys, we employ a science-based materials design approachby integrating a number of computational capabilitieswithin the framework of systems engineering [11]. A sys-tems approach integrates a hierarchy of computationaland experimental tools to quantify and to predict the pro-cessing–microstructure–properties–performance links. Fig.2 summarizes the conceived processing–structure–prop-erty–performance links governing the behavior of a multi-level-structured coated Nb-based superalloy system. Thethree main properties of interest in this research are creepresistance, oxidation resistance, and low-temperature duc-tility. The relevant research tools, both computationaland experimental, and their acronyms are indicated in therectangular box at the bottom of Fig. 2.

The principles of strengthening in classical c/c 0 Ni-basedsuperalloys [12] are extended to design a Nb-based superal-loy strengthened by an ordered aluminide phase. In body-centered cubic (bcc) alloys, such strengthening can beachieved by precipitating either B2 (nearest-neighborordered structure) and/or L21 (next-nearest-neighborordered structure) phases. The ordered intermetallic needsto have high thermodynamic stability at high temperature.Furthermore, to maintain coherency over a long period oftime and to obtain a uniform distribution of precipitates,the lattice mismatch should be very small (preferably<0.1%). In this work, we consider three aluminide phases,PdAl with B2 structure, and Pd2HfAl and Ru2NbAl withL21 (Heusler) structure, for the conceptual design of aNb-based alloy where one or more aluminide may be pre-sent as precipitates.

The consideration of PdAl and Ru2NbAl as candidateprecipitates is based on the experimental study of phaserelations in Nb–Pd–Al and Nb–Ru–Al systems that indi-cate the presence of (Nb) + PdAl and (Nb) + Ru2NbAlphase field, respectively, at 1100 �C [13]. While L21-Pd2HfAl was reported long ago [14], our recent experimen-tal study of phase relations in Nb-Pd-Hf-Al system revealsthe presence of (Nb) + PdAl, (Nb) + Pd2HfAl and(Nb) + PdAl + Pd2HfAl phase fields at 1200 �C [15,16].Prior to our phase relations studies, quantum mechanicaltotal energy calculations [17] were carried out using the fullpotential-linear muffin-tin orbital (FP-LMTO) method [18]in conjunction with the local density approximation (LDA)[19] to obtain the heat of formation and lattice parameterof six Heusler phases, Ni2TiAl, Ni2VAl, Ni2ZrAl, Ni2N-bAl, Ni2HfAl and Ni2TaAl. The calculated results showthat Ni2HfAl provides the lowest lattice misfit with Nb(�7.6%) and has the highest thermodynamic stability (for-mation energy DEf = �91.5 kJ mol�1 of atom). In order todecrease the lattice misfit further, the Ni was replaced byPd, which has the same electronic configuration but a lar-ger atomic radius. Quantum mechanical calculations of

Fig. 2. A materials systems design chart showing processing–microstructure–properties links and how they influence the performance of precipitationstrengthened Nb-base superalloys. The computational methods and the experimental techniques for the design and prototype evaluation of such alloys areshown at the bottom of the processing, microstructure and properties systems: electronic density functional theory (DFT); diffusion-controlledtransformation (DICTRA); differential thermal analysis (DTA); Thermo-Calc (TC); high-resolution electron microscopy (HR-AEM); thermogravimetricanalysis (TGA); transmission electron microscopy (TEM); scanning electron microscopy (SEM); X-ray diffraction (XRD).

G. Ghosh, G.B. Olson / Acta Materialia 55 (2007) 3281–3303 3283

Pd2HfAl [20] using the full potential-linearized augmentedplane wave method (FLAPW) [21] within LDA show thatindeed there is a decrease in lattice mismatch between Nband Pd2HfAl (�3.8%), and also the latter has a reasonablyhigh thermodynamic stability (formation energy DEf =�79.9 kJ mol�1 of atom).

The oxidation resistance, which can be extrinsic andintrinsic, is the life-limiting property of Nb-based alloys.The intrinsic oxidation resistance may be obtained by theformation of a protective oxide scale governed by Wahl’smodification [22] of Wagner’s criterion [23] for externaloxidation. The extrinsic oxidation resistance can beobtained by integration of thermal barrier coatings andoxygen barrier coatings to the underlying base alloy. Thiswould necessarily require a suitable bond coat material toincrease adherence to the external coating.

A self-protective oxide scale usually exhibits parabolicoxidation kinetics, and the adherence of the oxide scalewith the base alloy is governed by a low value of Pilling–Bedworth (PB) ratio [24], typically close to or less than 1.Analysis of available thermodynamic and kinetic data ofvarious oxides indicates that the slowest growing oxidescale is Al2O3 up to about 1300 �C, above which SiO2 isthe slowest growing oxide scale [1]. Cr2O3 also exhibits arelatively slow growth kinetics. Considering thermody-namic stability and PB ratio among candidate oxides

(Al2O3, Cr2O3, HfO2, Nb2O5, NbO, SiO2, TiO2, ZrO2),Al2O3 has the highest stability and the lowest PB ratio[1]. Therefore, Al2O3 is the preferred candidate oxide scalein the precipitation strengthened Nb-based superalloy. Thisalso further justifies the choice of aluminides as strengthen-ing precipitates, as they can act as a source for Al.

The critical amount of Al needed in solid solution toform external alumina scale may be predicted from Wahl’smodification [22] of Wagner’s theory [23] from transitionfrom internal to external oxidation. The modified theorypredicts that the critical amount of Al can be minimizedby minimizing the oxygen solubility (N ðSÞO ) in the matrixand by minimizing the ratio of oxygen diffusivity (DO) tothe aluminum diffusivity (DAl) in the matrix. In a multi-component Nb-based solid solution, these three parameterscan be greatly influenced by the presence of other alloyingelements.

A systematic study of oxidation behavior of bcc Nb–X(X = Al, B, Be, Cr, Co, Fe, Mn, Mo, Ni, Si, Ti, V, W orZr) between 600 and 1000 �C was reported by Sims et al.[25]. This study showed that Ti is most effective in reducingthe oxidation rate of Nb, but oxide scale formed wasNb2O5 even in an alloy containing 25 at.% Ti. Niobiumdissolves about 8 at.% Al at 1300 �C, and when oxidizedit also forms Nb2O5. Oxidation studies of intermetallicNbAl3 [26–28] show the formation of Al2O3 scale, but it


cannot sustain this growth over a prolonged time andNbAl3 decomposes to form NbAl2. The growth of a pro-tective Al2O3 scale at 1400 �C has also been demonstratedin a 30Nb–28Ti–42Al (at.%) alloy with B2 structure [26].The addition of Cr and V reduces the Al required to about37 at.%. The B2 structure of the alloy favors rapid diffusionof Al to the surface, but the B2 phase decomposes, into c-TiAl and Nb2Al, on cooling to room temperature.

The dissolved solutes in the Nb matrix play severalimportant roles: (i) provide solid solution strengthening;(ii) control the lattice parameter of the bcc matrix, hencelattice mismatch with the precipitate; (iii) control atomictransport kinetics affecting both coarsening of precipitatesand oxidation behavior; and (iv) influence dislocationmobility affecting the ductility and fracture toughness.The alloying element(s) needed to control one or more ofthese properties often has a conflicting influence on theother(s). For example, a systematic study by Begley [29]shows that alloying elements such as Al, Cr, Mo, Re, V,W and Zr cause embrittlement and raise DBTT, while Hfand Ti do not increase DBTT of Nb. Recent results haveindicated that Ti enhances both tensile ductility and frac-ture toughness of Nb solid solution [30], while Al and Crexert the opposite effect [31,32]. However, Al and Cr alloy-ing additions are needed to enhance the oxidation resis-tance, which is the life-limiting property of Nb-basedcomposites [5,6,8,9].

Rice [33] has suggested that the ratio of the surfaceenergy (cs) and unstable stacking energy (cus) is a key mea-sure of the propensity for brittle fracture, and also a criter-ion for DBTT. Using these concepts as a guideline, Chan[34] calculated the surface energy and the Peierls–Nabarrobarrier energy as a function of composition of bcc solidsolutions in a quaternary system Nb–Al–Cr–Ti. Consistentwith the experimental data, Chan [34] showed that Tireduces the Peierls–Nabarro barrier energy, promotes dis-location mobility, and enhances the tensile ductility andfracture toughness of Nb-based solid solutions [34]. Onthe other hand, Cr and Al exert opposite effects. The criter-ion of cs/cus has also been used by Waghmare et al. [35] asthe basis for selecting ternary alloying elements to improvethe ductility of MoSi2 via first-principles calculations of cs

and cus.In summary, Fig. 2 illustrates the complex interactions

of structure on the primary property objectives of creepresistance, oxidation resistance, and low-temperaturetoughness and ductility, as well as the processing variableswhich control them. These interactions allow us to de-emphasize the weak interaction(s) while explore and quan-tify the strong interaction(s) in order to optimize a complexsystem.

3. Computational methodology

Due to the multicomponent and multiphase nature ofthe Nb-based alloys of interest in this research, the directapplication of accurate first-principles methods (i.e., meth-

ods using quantum-mechanical calculations requiring onlythe atomic numbers of the constituent elements and theirarrangement as input) to the direct modeling of phase sta-bility in these systems is intractable. By contrast, computa-tional thermodynamic methods based upon the calphadapproach [36] can be employed readily in the modeling ofmulti-component alloy phase stability. The predictivepower of this approach, however, is limited by the avail-ability of sufficient experimental data to generate accuratefree energy functions. In the absence of adequate experi-mental data, first-principles calculations offer the best strat-egy for supplementing thermodynamic databases, therebyincreasing the range of applicability of computational ther-modynamic methods while limiting the reliance uponextensive experimental measurements.

First-principles calculations are performed to calculateformation energies and atomic volumes of intermetalliccompounds for use in developing reliable thermodynamicand lattice-parameter databases for required stable and vir-tual ordered phases. Stable phases are those which are pre-sent in the equilibrium phase diagram, irrespective oftemperature or composition ranges of stability. The con-cept of virtual phase is a mathematical one in the contextof calphad modeling of intermetallics having a finite homo-geneity range, using a sublattice model [37]. In cases whereexperimental data are available, these calculations providea basis for determining the overall accuracy of the first-principles methods, while in cases where such data arelacking, they can be used to augment the thermodynamicdatabases that are employed in the integrated design ofB2 and/or L21 strengthened Nb-base alloys.

3.1. Ab initio total energy calculations

The ab initio calculations presented here are based onelectronic density-functional theory (DFT), and have beencarried out using the ab initio program VASP (Vienna abinitio simulation package) [38–40]. Most current calcula-tions make use of Vanderbilt-type ultrasoft pseudopoten-tials (US-PP) [41], as implemented in VASP. Electronicwavefunctions are expanded in plane waves with akinetic-energy cutoff of 281 eV, which is at least 1.38 timesthe default cutoff value for Al, Hf, Nb, Pd and Ru. TheUS-PPs employed in this work explicitly treat three valenceelectrons for Al (3s2p1), four valence electrons for Hf(5d36s1), five valence electrons for Nb (4d45s1), 10 valenceelectrons for Pd (4d95s1) and eight valence electrons forRu (4d75s1). All calculated results were derived employingthe generalized gradient approximation (GGA) forexchange-correlation energy due to Perdew and Wang[42]. Brillouin zone integrations were performed usingMonkhorst–Pack [43] k-point meshes, and the Methfes-sel–Paxton [44] technique with the smearing parameter of0.1 eV.

The B2 and L21 phases considered in this study are listedin Table 1. The k-point meshes used for Brillouin zone inte-gration for total energy calculations were 24 · 24 · 24 and

Table 1Crystallographic data of intermetallics used in the ab initio calculations

Phase Pearsonsymbol

Structurberichtdesignation

Spacegroup (#)

Prototype

Stable

PdAl cP2 B2 Pm�3m (221) CsClRuAl cP2 B2 Pm�3m (221) CsClRuNb cP2 B2 Pm�3m (221) CsClPd2HfAl cF16 L21 Fm�3m (225) Cu2AlMnRu2NbAl cF16 L21 Fm�3m (225) Cu2AlMn

Virtual

HfAl cP2 B2 Pm�3m (221) CsClNbAl cP2 B2 Pm�3m (221) CsClPdHf cP2 B2 Pm�3m (221) CsClPdHfAl2 cF16 L21 Fm�3m (225) Cu2AlMnPdHf2Al cF16 L21 Fm�3m (225) Cu2AlMnRuNbAl2 cF16 L21 Fm�3m (225) Cu2AlMnRuNb2Al cF16 L21 Fm�3m (225) Cu2AlMn


12 · 12 · 12 for B2 and L21 phases, respectively. The corre-sponding k-points in the irreducible Brillouin zone are 364and 56, respectively. All calculations are performed usingthe ‘‘accurate’’ setting within VASP to avoid wrap-arounderrors. With the chosen plane-wave cutoff and k-point sam-pling, the reported formation energies are estimated to beconverged to a precision of better than 2 meV atom�1

(�0.2 kJ mol�1 of atom).We have also calculated the formation energies of pseu-

dobinary alloys of Pd(HfxAl1�x) and Ru(NbxAl1�x),0 6 x 6 1, with B2 structure from first principles using asupercell (SC) method. This approach involves calculatingtotal energy of structures where several crystallographicallyequivalent sites are created by repeating the B2 unit cellalong appropriate directions; the energies of compoundswith site-substituted species are thus calculated at discretecompositions. Specifically, we have used 32-atom super-cells, with 16 sites (equivalent to 1b position in B2 struc-ture) occupied by Pd or Ru atoms and the remaining 16sites (equivalent to 1a position in B2 structure) occupiedby either Al + Hf or Al + Nb atoms. The lattice vectorsof the supercell are defined as aSC

X ¼ ½2; 2; 0�a0,aSC

Y ¼ ½0; 2; 2�a0 and aSCZ ¼ ½2; 0; 2�a0, with a0 being the lat-

tice constant of B2 unit cell. We have used a k-point meshof 7 · 7 · 7 for the total energy calculations usingsupercells.

In addition to total energy calculations, to obtain elec-tronic structures and to understand the bonding character-istics of three B2 (PdAl, RuAl, RuNb) and two Heuslerphases (Ru2NbAl, Pd2HfAl) we have also calculated theelectronic density of states and charge densities. Such cal-culations are performed using potentials constructed bythe projector-augmented wave (PAW) method [45], whichretains the all-electron character but the all-electron wavefunction is decomposed into a smooth pseudo-wave func-tion and a rapidly varying contribution localized with thecore region. In its current implementation in VASP, thePAW method freezes the core orbital to those in a referenceconfiguration, although it is not strictly necessary. In fact,

very recently a relaxed core PAW method has been pro-posed [46] that is shown to yield results with an accuracycomparable to the FLAPW method. The PAW method isfree of any shape approximation for both charge densityand electronic potential. Therefore, PAW potentials arean improvement over Vanderbilt-type US-PP [47], as theycombine the elegance and computational efficiency of planewaves with the chemically appealing concept of localizedfunctions. The valence configurations of PAW potentialsfor Al, Hf and Ru were the same as in US-PP; however,for Nb and Pd the PAW potentials were constructed treat-ing the occupied semicore 4p electronic states as valencestates. Once again, the exchange-correlation energy dueto Perdew and Wang [42] was used. A kinetic-energy cutoffof 525 eV was used for the expansion of the electronicwavefunctions in plane waves. For the calculation ofcharge densities, we have used k-point meshes of35 · 35 · 35 for the B2 phases and 19 · 19 · 19 for theL21 phases.

The total energies of the intermetallics listed in Table 1,and also relevant pure elements, are calculated as a func-tion of volume, and the results are fit to the equation ofstate (EOS) due to Vinet et al. [48]. The zero-temperatureequations of state (EOS) define pressure–volume relation-ships. Vinet et al. [48] assumed that the interatomic interac-tion-versus-distance relation in solids can be expressed interms of a relatively few material constants. Specifically,the pressure P is expressed in terms of isothermal bulkmodulus (B0), its pressure derivative (B00) and a scaledquantity (x):

P ¼ 3B0x�2ð1� xÞ exp½vð1� xÞ� ð1Þwith x = (V/V0)1/3 and v ¼ 3=2ðB00 � 1), where V0 is theequilibrium volume. Based on Eq. (1) and the relations be-tween pressure and energy, the total energy (E) and vo-lume-dependence of the bulk modulus can be expressed as

EðV Þ � EðV 0Þ ¼9B0V 0

v2f1� ½vð1� xÞ� exp½vð1� xÞ�g; ð2Þ

BðV ÞB0

¼ x�2½1þ ðvxþ 1Þð1� xÞ� exp½vð1� xÞ�: ð3Þ

Fig. 3a and b shows the E–V plots defining zero-tempera-ture EOS parameters for Pd2HfAl and Ru2NbAl,respectively.

3.2. Computational thermodynamics and kinetics

The principal integrated computational design toolsused are the Thermo-Calc [49] and DICTRA (diffusioncontrolled transformation) [50] software systems. Thermo-Calc employs the calphad [36] approach to predict phasediagrams and phase stability as a function of compositionand temperature in multicomponent alloys. Calphademploys a bottom-up approach, whereby the Gibbs freeenergies of lower-order systems (unary, binary, ternary,etc.) are modeled using analytical functions, which in turn

Fig. 3. Calculated total energy, at zero-temperature and without zero-point motion, as a function of volume for (a) L21-Pd2HfAl, and (b) L21-Ru2NbAl. The filled circles represent calculated point, and the line is a fitto EOS in Eq. (2).


are used to predict the phase stability and phase diagramsof higher-order systems, where the number of componentsmay be as high as 10. Past experiences have demonstratedthat modeling ternary systems usually lead to satisfactorypredictions of phase stability and phase diagrams of multi-component alloys. Besides calculation of multicomponentequilibrium phase diagrams [49], at Northwestern Univer-sity the applications of computational thermodynamics(using Thermo-Calc software) have been extended toinclude the prediction of glass forming ability of alloysby rapid solidification [51], the calculation of kineticallyconstrained solid–solid phase equilibria [52], and modelingand prediction of the kinetics of diffusionless martensiticstructural transformations [53–56].

DICTRA is a software package for simulating diffusion-controlled transformations in multicomponent systemsinvolving multiple phases having simple geometries. DIC-TRA solves one-dimensional diffusion equations in a

volume-fixed frame of reference. Besides single-phase andmulti-phase diffusional simulations [50], at NorthwesternUniversity the applications of computational kinetics(using DICTRA software) have been extended to includesimulation of growth kinetics under kinetically constrainedconditions [52,57] during a solid–solid phase transforma-tion, and the dissolution of solid metal in liquid solder dur-ing electronic package assembly [58]. A DICTRAsimulation requires both thermodynamic and atomic mobi-lity data bases. While Thermo-Calc is used widely, due tothe availability of relevant thermochemical databases, theapplications of DICTRA are limited by the availability ofmobility databases. As a part of our research program, asynergistic approach for thermodynamic and kinetic mod-eling relevant to the design of Nb-based superalloys.

4. Experimental procedure

Three prototype alloys with aluminide precipitates areused for microstructural investigation. The nominal com-positions (in at.%) of these alloys are: 84Nb–8Al–8Pd,82Nb–8Al–10Ru and 81.6Nb–6.3Al–2.8Hf–9.3Pd. Anotherprototype alloy, 45Nb–34Hf–21Al (in at.%), is used for theoxidation study. Elements (all obtained from Alfa Aesar,Ward Hill, MA) of the following purity (in mass%) are usedin preparing the alloys: Al: 99.999%, Hf: 99.9%, Nb: 99.8%,Pd: 99.95%, Ru: 99.95%. About 10–15 g of each alloy ismade by arc-melting in an inert argon atmosphere. Duringarc-melting, each alloy button is flipped and re-melted 10times in order to achieve homogeneity. Samples are cutfrom the ingots, and then heat-treated at 1000 and1200 �C for 100–500 h. All heat treatments are carried outafter vacuum encapsulation of specimens in a quartz tubewith tantalum foil to getter residual oxygen.

The oxidation studies are carried at 1300 �C in air [1].Semi-elliptical discs with thickness 1–6 mm are cut fromthe as-cast alloys, and the flat surfaces are polished to 800grit. The samples are placed in alumina boats (so as to collectall spalled oxides), which are then placed in a tube furnace at1300 �C for air oxidation. The weight of the boat + sampleassembly is recorded before and after oxidation treatmentto measure the weight gain due to oxidation. Conventionalmetallography techniques are employed to reveal the cross-section microstructure of oxidized specimens.

Both scanning electron microscopy (SEM) and trans-mission electron microscopy (TEM) are performed formicrostructual characterization at various length scales.A Hitachi 3500 scanning electron microscope is used formicrostructural characterization of oxidized specimens.Thin foils for transmission and analytical electron micro-scopy are prepared by a standard dual jet electropolishingmethod. An electrolyte containing 5% H2SO4 and 2% HFin methanol is maintained at �60 to �70 �C, and the thinfoils are prepared by applying a voltage of 80 V and a cur-rent of 30–40 mA. Conventional TEM is carried out in aHitachi 8100 microscope operating at 200 kV. The high-resolution electron microscopy (HREM) and analytical


characterization are performed in a cold field emission gunhigh-resolution AEM (Hitachi HF-2000) equipped with aGatan 666 parallel electron energy loss spectrometry detec-tor, an ultrathin window Link EDX detector and data pro-cessor (QX2000) and a Gatan CCD camera for HREMimaging. The AEM is operated at 200 kV. The take-offangle for the X-ray detector is 68�. The X-ray collectiontime is 100–200 s, and the electron probe size is about10 nm. Care is taken to insure that the particle being ana-lyzed is not in a two-beam condition in order to minimizeelectron-channeling effects.

5. Results

5.1. Ab initio results: pure elements

The calculated zero-temperature cohesive properties orpure elements are compared with available experimentaldata in Table 2. The lattice parameter of Al [59] and Nb[60] are taken from the measured values at 4.2 K. The latticeparameters of Hf [61], Pd [62] and Ru [63,64] at 0 K areobtained by extrapolating the experimental data at highertemperatures to 0 K. We find that, in general, the calculatedand experimental (measured or extrapolated) lattice para-meters agree within ±1%, except for Pd, where the discre-pancy is 2.1%. The bulk moduli values of Al [60,65,66], Hf[67], Nb [68], Pd [69] and Ru [70] correspond to 4.2 K orextrapolated value at 0 K. Our calculated bulk moduli ofHf and Nb agree within ±2% of the experimental value,while the calculated bulk moduli of Pd and Ru are underes-timated by about 19%. The experimental data for Al show alarge scatter. In Table 2, we also note that the lattice para-meters of pure elements calculated using US-PP and PAWare very similar.

Experimental data of B00 are available primarily at ambi-ent temperature. Once again, the values of Al [71–73] showsome scatter. Other B00 values are taken from Steinberg [74].In general, the agreement between the calculated andexperimental values should be considered good. It has beenpointed out [74] that, depending on the measurement tech-nique, ultrasonic resonance, vs. the initial slope of the locusof Hugoniot states in shock-velocity particle-velocity coor-dinates, the values of B00 may differ even though ideally theyshould be the same. It is not uncommon that the B00 pre-dicted by ab initio techniques differs from the experimentalvalue by as much as 30%.

5.2. Ab initio results: intermetallics

We take the formation energy at zero-temperature andzero-pressure and without zero-point motion (DE/

f ) of anintermetallic ApBqCr, where p, q and r are integers, as akey measure of the relative stability. The formation energyper atom is evaluated relative to the composition-averagedenergies of the pure elements in their equilibrium crystalstructures. For example, the DE/

f of a ternary intermetallicis defined as

DE/f ðApBqCrÞ ¼

1

pþ qþ rE/

ApBqCr

� ppþ qþ r

EhAþ

qpþ qþ r

EuB þ

rpþ qþ r

EwC

� �

ð4Þ

where E/ApBqCr

is the total energy of intermetallic ApBqCr

with structure /, EhA, Eu

B and EwC are the total energy per

atom of A, B and C with structure h, u and w, respectively.The results of ab initio calculations for the B2 and L21

phases are summarized in Table 3. There have been severalstudies of phase stability and electronic structure of someof the intermetallics considered here using ab initio meth-ods. The previous studies can be summarized as follows:FLAPW [75], FP-LMTO [76] and full-potential linearizedaugmented Slater-type orbital [77] (FLASTO) [78] forPdAl; FP-LMTO [79,76], LAPW [80] and FLASTO [78]for RuAl; FLAPW [20] for Pd2HfAl; FLASTO for Ru2N-bAl [81]. Except for Nguyen-Manh and Pettifor [76], all theaforementioned studies made use of Hedin–Lundqvist [19]parametrization of LDA. On the other hand, Nguyen-Manh and Pettifor [76] made use of Perdew–Zunger [82]parametrization of LDA.

Among the stable intermetallics of interest to us, theheat of formation has been measured only for PdAl[83,84] and RuAl [85] by direct reaction calorimetry.

Our calculated value of DEf for PdAl using both US-PP(GGA) and PAW (GGA) underestimates the experimentalvalue [83,84] by about 7–10 kJ mol�1 of atom. On the otherhand, DEf values from both FLAPW- [75] and FP-LMTO(LDA) [76] calculations agree with the experimental datawithin a few kJ mol�1 of atom. In contrast, our result ofDEf for RuAl using US-PP (GGA) and PAW (GGA) agreewithin a few kJ mol�1 of atom of experimental value [85],while DEf by FLASTO (LDA) [78] method shows an excel-lent agreement. Mehl et al. [80] calculated only the elasticproperties of RuAl by ab initio methods and did not reportthe formation energy. Unlike PdAl and RuAl, in Table 3,we note that the DEf for Pd2HfAl and Ru2NbAl calculatedby US-PP (GGA) and various all-electron methods such asPAW (GGA), FLAPW (LDA) [20] and FLASTO (LDA)[81] are very similar; the theoretical result is thus one thatis not sensitive to DFT computational procedure.

It is important to note that both B2-PdAl and B2-RuNbare high-temperature phases in the respective equilibriumphase diagram [86]; B2-PdAl is stable between 850 and1645 �C and B2-RuNb is stable between 900 and1942 �C. As a result, there is no measured lattice parameterat ambient and/or at a low temperature for these twophases. The zero-temperature lattice parameter of B2-PdAlobtained by linear extrapolation of high-temperature data[87] is listed in Table 3. The lattice parameter of B2-RuNbat 298 K is obtained by linear extrapolation of Nb-rich B2lattice parameter data [88] to the equiatomic composition.The lattice parameter of B2-RuAl [89–92], L21-Pd2HfAl[14] and L21-Ru2NbAl [13] has been measured at ambient

Table 2A comparison of selected ab initio structural and elastic properties of pure elements at 0 K (this study) with the available experimental data

Element Structure Lattice parameter (nm) B0 (·1010 N m�2) B00

Ab initio Expt. Ab initio Expt. Ab initio Expt.

Al FCC (cF4) a = 0.40436a a = 0.40322 [59] 7.42 8.82 [65] 4.11 4.0 [71]a = 0.40469b 7.94 [66] 5.19 [72]

8.20 [60] 4.42 [73]

Hf HCP (hP2) a = 0.31804a a = 0.31930 [61] 11.03 11.06 [67] 3.43 3.95, 4.28 [74]c = 0.50208 c = 0.50395a = 0.31976b

c = 0.50488

Nb BCC (cI2) a = 0.32923a a = 0.33026 [60] 17.05 17.33 [68] 3.58 4.06, 3.80 [74]a = 0.33229b

Pd FCC (cF4) a = 0.39617a a = 0.38780 [62] 16.43 19.55 [69] 5.34 5.42, 5.20 [74]a = 0.39575b

Ru HCP (hP2) a = 0.27335a a = 0.27030 [63,64] 29.88 35.26 [70] 4.24 –c = 0.43055 c = 0.42719a = 0.27295b

c = 0.43021

Some properties are calculated using both ultrasoft and PAW pseudopotentials.a US-PP (GGA).b PAW (GGA).


temperature. It may be noted in Table 3 that our calculatedlattice parameter agrees within ±1% except for B2-PdAl,where the discrepancy is about 2.7%.

As for the elastic properties, the experimental bulk mod-ulus is available only for RuAl at ambient temperature [93].As seen in Table 3, our calculated value of B0 agrees withthe experimental data to within 5%.

In addition to the thermodynamic stability of the stablealuminides in Table 1, we also investigated their mechani-cal stability with respect to tetragonal deformation byapplying an appropriate strain of up to 5%. We found thatonly B2-RuNb is mechanically unstable, as the tetragonalshear modulus (C 0 = (C11 � C12)/2) is found to be negative.This is consistent with the experimental observation thatB2-RuNb undergoes a martensitic transformation sponta-neously upon cooling from high temperature to about900 �C to form a tetragonal phase [94,95], which in turnundergoes further structural transformations at lower tem-peratures. Unlike B2-RuNb, we found that B2-PdAl ismechanically stable at 0 K even though it is stable above850 �C and undergoes complex (yet unknown) orderingat lower temperatures, as indicated in the equilibriumphase diagram [86]. Among the virtual phases, the mechan-ical stability of B2-PdHf was also checked with respect totetragonal strain and was found to be mechanically stable.

5.3. Microstructure of prototype alloys: aluminide

precipitates in a (Nb) matrix

Figs. 4–6 show the results of microstructure characteri-zation of following prototype alloys: 84Nb–8Al–8Pd (inat.%) heat treated at 1000 �C for 500 h, 81.6Nb–6.3Al–2.8Hf–9.3Pd (in at.%) heat treated at 1000 �C for 200 h,and 82Nb–8Al–10Ru (in at.%) heat treated at 1200 �C

for 100 h, respectively. They were heat treated after arc-melting. In all three alloys, based on the morphology anddistribution of the second phase, they are clearly the resultof solid-state precipitation by nucleation and growth pro-cesses from supersaturated solid solutions.

Fig. 4a and b shows the bright-field (BF) and dark-field(DF) micrographs, but different areas, of 84Nb–8Al–8Pdalloy. The DF micrograph is taken using {100}B2-typesuperlattice reflection. The precipitates have diameters inthe range of 100–200 nm, but the presence of interfacialdislocations implies the semi-coherent nature of thematrix/precipitate interface. Fig. 4c and d shows the EDSspectra obtained from the matrix and the precipitate,respectively. The matrix EDS spectrum indicates smallamounts of dissolved Al and Pd in (Nb). The precipitateEDS spectrum indicates a small amount of Nb dissolvedin PdAl. This result is consistent with the experimental iso-thermal section at 1100 �C [13] that also indicates a fewpercent of Nb solubility in PdAl.

Fig. 5a and b shows the BF and the corresponding DFmicrographs of 81.6Nb–6.3Al–2.8Hf–9.3Pd alloy, the lat-ter taken using f111gL21

-type superlattice reflection. Preci-pitates with diameters in the range of 20–70 nm may beseen, and all appear to be fully coherent with the matrix.Fig. 5c and d shows the EDS spectra obtained from thematrix and the precipitate, respectively. Like the Nb–Pd–Al alloy above, once again the matrix EDS spectrum indi-cates small amounts of dissolved Al and Pd in (Nb) and anegligible amount of Hf. On the other hand, the precipitateEDS spectrum indicates a substantial amount of Nb dis-solved in Pd2HfAl. An important point to note is that inthe matrix spectrum (Fig. 5c) the peak height of Nb-L ishigher than Nb-Ka; however, in the precipitate spectrum(Fig. 5d) the peak height of Nb-L is much smaller than

Table 3A comparison of ab initio and experimental cohesive properties of the intermetallics

Phase EOS parameters Lattice const. (nm) DEf, (kJ mol�1 of atom) Misfit (da)

V0 B0 B00 Ab initio Expt. Ab initio Expt.

Stable

PdAl 0.014682 15.14 4.77 0.30851a 0.30167b �83.31a �93.6c �6.3%0.30799d �84.55d �90.6 ± 2.2e �7.3%0.30200f �90.57f

0.014055g 20.1g �93.58g

0.30196h �87.68h

RuAl 0.013635 19.91 4.30 0.30099a 0.30300i �65.29a �62.05 ± 1.65j �8.6%0.30086d �64.35d �9.4%

22.0k 0.30200k 0.30360l �74.93k

0.013095g 22.3g �58.15g

23.0m 4.5m 0.29600m

20.8n 0.30300o

0.29635h �61.67h

RuNb 0.015921 23.01 4.38 0.31696a 0.31500p �15.72a – �3.7%0.31857d �14.68d �4.1%

Pd2HfAl 0.016536 15.48 4.60 0.64197a 0.63670q �79.36a – �2.5%0.64239d �80.59d �3.3%0.63500r 0.63680s �79.94r

Ru2NbAl 0.014772 22.44 4.32 0.61828a 0.61350t �63.96a – �6.1%0.62042d �62.25d – �6.6%0.61254u �63.0u

Virtual

HfAl 0.018687 10.92 3.33 0.33434a – �20.13a –NbAl 0.016765 14.17 4.09 0.32246a – �2.71a –PdHf 0.018234 15.30 6.46 0.33162a – �59.36a –PdHfAl2 0.016808 11.53 4.32 0.64547a – �38.02a –PdHf2Al 0.018675 12.94 4.07 0.66854a – �34.60a –RuNbAl2 0.015196 16.31 4.23 0.62414a – �22.57a –RuNb2Al 0.016305 18.10 4.17 0.63897a – �5.32a –

The units of V0 and B0 are nm3 atom�1 and 1010 N m�2, respectively. The lattice misfit (da) is based on our calculated lattice parameter of pure Nb and theintermetallic at 0 K. The reference states for DEf are fcc-Al, hcp-Hf, bcc-Nb, fcc-Pd and hcp-Ru.

a US-PP (GGA) [this study].b Data of Ref. [87] are extrapolated 0 K.c Direct reaction calorimetry [83].d PAW (GGA) [this study].e Direct reaction calorimetry [84].f FLAPW (LDA) [75].g FP-LMTO (LDA) [76].h FLASTO (LDA) [78].i At 298 K [89].j Direct reaction calorimetry [85].

k LMTO (LDA) [79].l At 298 K [91].

m LAPW (LDA) [80].n At 298 K [93].o At 298 K [92].p At 298 K and extrapolated to equiatomic composition [88].q At 298 K [14].r FLAPW (LDA) [20].s At 298 K [15].t At 298 K [13].u FLASTO (LDA) [81].


Nb-Ka. This could be due to absorption and/or fluores-cence effects.

Fig. 6a and b shows the BF and the corresponding DFmicrographs of 82Nb–8Al–10Ru alloy, the latter takenusing f111gL21

-type superlattice reflection. Precipitates

with diameters in the range of 50–200 nm may be seen. Afew interfacial dislocations around some of the precipitates(see Fig. 6a) are seen while some precipitates exhibit theremnants of a coherency-induced morphological effect,i.e. cuboidal shape. In addition, some precipitates are seen

Fig. 4. Transmission electron micrographs showing the two-phase microstructure of a prototype alloy 84Nb–8Al–8Pd (in at.%) heat treated at 1000 �C for500 h. The presence of interfacial dislocations, in both (a) BF and (b) DF (different area) micrographs, imply the semi-coherent nature of the Nb/B2-PdAlinterface. Also shown are the EDS spectra from (c) (Nb) matrix and (d) B2-PdAl precipitate obtained in a high-resolution analytical electron microscope.


to have undergone physical coalescence by a process com-monly observed during sintering of powders. Fig. 6c and dshows the EDS spectra obtained from the matrix and theprecipitate, respectively. The matrix EDS spectrum indi-cates small amounts of dissolved Al and Ru in (Nb). Onthe other hand, the precipitate EDS spectrum indicates asubstantial amount of Nb dissolved in Pd2HfAl. Onceagain, like the Nb–Pd–Hf–Al case above, in the matrixspectrum (Fig. 6c) the peak height of Nb-L is higher thanNb-Ka; however, in the precipitate spectrum (Fig. 6d) thepeak height of Nb-L is smaller than Nb-Ka. Once again,this is most likely to have originated from absorptionand/or fluorescence effects.

Transmission electron diffraction patterns are used toevaluate the lattice mismatch, da (= (aprecipitate � amatrix)/amatrix), between the matrix and precipitate. The diffractionpatterns shown in Fig. 7a–c represent [110]bcc, [110]bcc and[100]bcc zone axes taken from 84Nb–8Al–8Pd, 81.6Nb–6.3Al–2.8Hf–9.3Pd and 82Nb–8Al–10Ru alloy, respec-tively. In the first two cases, we have used (110)A2/B2 (orð220ÞL21

), (002)A2/B2 (or ð004ÞL21) and (112)A2/B2 (or

ð224ÞL21), while in the third case we have used (020)A2

(or ð040ÞL21), (002)A2 (or ð004ÞL21

) and (022)A2 (orð044ÞL21

) reflections to estimate a mean value of da. Thesereflections are labelled and clearly distinguishable in the

respective diffraction pattern, as shown in Fig. 7a–c. Themeasured values of da are �8.6%, �3.9% and �6%, respec-tively, in the above three alloys. These values comparefavorably with the calculated values of �6.3%, �2.5%and �6.1% in the corresponding two-phase systems (seeTable 3). It should be noted that the da values listed inTable 3 are based on the lattice parameters of pure Nband stoichiometric intermetallic calculated at 0 K. How-ever, the measured values may be influenced by severalintrinsic and extrinsic factors, such as (i) the ambient tem-perature effect, (ii) the dissolved solutes in (Nb), (iii) thedeviation from ideal stoichiometry of the intermetallicsand (iv) possibly some instrumental/measurement artifactsdue to the finite size of the diffraction spots. Taking thesefactors into account, the observed agreement should beconsidered as good.

5.4. Computational thermodynamics: modeling and

calculation of the Al–Hf–Nb phase diagram

Here, we demonstrate the modeling of phase stabilityin Al–Hf–Nb, which represents an important ternary sub-system in one of the prototype alloys discussed above.Since the experimental phase diagram and thermody-namic data for this system were very limited, ab initio

Fig. 5. Transmission electron micrographs showing two-phase microstructure of a prototype alloy 81.6Nb–6.3Al–2.8Hf–9.3Pd (in at.%) heat treated at1000 �C for 200 h. (a) The BF and (b) the corresponding DF micrograph show fully coherent L21-Pd2HfAl precipitates in a (Nb) matrix. Also shown arethe EDS spectra from (c) (Nb) matrix and (d) L21-Pd2HfAl precipitate obtained in a high-resolution analytical electron microscope.


methods were exploited to accelerate the development ofthe databases required to employ computational thermo-dynamics methods, based on the calphad approach [36],as a predictive framework for guiding the alloy designprocess. In this work, the application of the methods isfacilitated by employing the Alloy Theoretic AutomatedToolkit [96,97] for calculating the heat of mixing of solidsolutions by cluster expansion method [98] and also thevibrational entropy of formation of several relevant inter-metallics. In the calphad method, the parameters inmodel free energy functions are generally fit to availableexperimental data for measured phase boundaries andthermodynamic properties. However, in the absence ofsufficient measured data to uniquely fit the model para-meters, ab initio methods can be employed as a ‘‘virtualcalorimeter’’ to augment thermodynamic databases withcomputed values of enthalpies and entropies of formationfor solid solutions and intermetallic compounds. Thedetails of such an integrated approach have been dis-cussed elsewhere [99]. In the following, we describe theprocedure very briefly.

In the calphad modelling of Al–Nb intermetallics, a two-sublattice model for Al3Nb and AlNb3 and a three-sublat-tice model for the r phase were used. An advantage pro-vided by integrating ab initio methods with calphad isthat the energy parameters of all the virtual phases,required in the construction of the sublattice models, can

be calculated directly. For example, in a three-sublatticedescription of the r phase, (Al,Nb)10(Nb)4(Al,Nb)16,values are required for the formation free energies of thethree virtual phases Al26Nb4, Al16Nb14 and Nb30 havingthe structure of the r phase. Similarly, a two-sublatticedescription of the A15 phase, (Al,Nb)(Al,Nb)3, gives riseto three virtual A15 phases, Al4, Al3Nb and Nb4. For theconstruction of the calphad free-energy functions we havecalculated the energy of formation of all virtual phases inthe Al–Nb system. Similarly, in the Al–Hf system we calcu-lated the heats of formation of stable and several virtualAl–Hf intermetallics [100]. For calphad modelling of phasediagrams, both enthalpies and entropies of formation ofphases are required. The calculation of vibrational entro-pies of formation is in general computationally demanding.In particular, the Al3Hf2, Al2Hf3 and r phases have orthor-hombic and tetragonal crystal structures, respectively, with40, 20 and 30 atoms per unit cell. In such cases, we haveestimated the entropy of formation using a correlationbetween intermetallic enthalpies, and entropies of forma-tion were parametrized from extensive ab initio calcula-tions involving relatively simpler phases [99]. Theintegration ab initio phase stability of the Hf–Nb systemwithin calphad formalism has been discussed elsewhere[101].

In modeling the Al–Hf–Nb phase diagram, all Al–Hfintermetallics were treated as stoichiometric binary phases,

Fig. 6. Transmission electron micrographs showing two-phase microstructure of a prototype alloy 82Nb–8Al–10Ru (in at.%) heat treated at 1200 �C for100 h. (a) The BF micrograph shows only a few dislocations around L21-Ru2NbAl precipitates, and the corresponding DF micrograph is shown in (b).Also shown are the EDS spectra from (c) (Nb) matrix and (d) L21-Ru2NbAl precipitate obtained in a high-resolution analytical electron microscope.


i.e. without allowing any Nb solubility, with the singleexception of Al3Hf. Since both Al3Nb and Al3Hf have theDO22 structure at high temperature, they were treated asone phase, i.e. Al3(Hf,Nb), allowing random mixing ofNb and Hf on one sublattice. The sublattice models forAlNb3 (A15) and AlNb2 (r) were extended to includeHf, i.e. (Al,Hf,Nb)(Al,Hf,Nb)3 and (Al,Hf,Nb)10(Nb,Hf)4

(Al,Hf,Nb)16, respectively. This introduces additional virtualphases, the energy parameters of which were all calculated byab initio methods. Modeling of ternary solid-solution phaseswas facilitated by cluster-expansion calculations of the ther-modynamic properties face-centered cubic (fcc) and hexago-nal close-packed (hcp) solid solution in the Hf–Nb and theAl–Nb system, respectively. In a more traditional calphadapproach, relying on measured data entirely, such data wouldnot be accessible due to the fact that fcc and hcp solid-solutions phases are absent in the respective equilibriumphase diagrams. No ternary interaction was introduced forthe solution phases (liquid, bcc, fcc and hcp).

Fig. 8 shows the calculated isothermal section of the Al–Hf–Nb system at 1300 �C. In the context of the design ofNb-base superalloy, it turns out to be a very important pre-diction as the solubility of Al in (Nb) can be increased sig-nificantly by adding Hf. For example, in the binary Al–Nbsystem (Nb) 8 at.% Al is dissolved at 1300 �C. However,

our calculated results show that by adding 30 at.% Hf thesolubility of Al can be increased by more than a factor oftwo, to 17 at.%. As will be shown below, based on theresults of kinetics simulations that were later verifiedexperimentally [1], this increase in Al solubility proves tobe highly beneficial for significantly reducing oxygen trans-port kinetics in the (Nb) solid-solution phase. Anotherimportant feature in Fig. 8 is the presence of (Nb) + rphase field in the ternary regime, which is absent in the bin-ary Al–Nb system. Usually the formation of r phase is det-rimental to the mechanical properties; therefore thecalculated phase diagram helps greatly in the alloydesign/selection so as to avoid r phase.

5.5. Oxidation behavior of a prototype Al–Hf–Nb alloy

As mentioned in Section 2, the critical amount of Alneeded to form a self-protective Al3O3 scale can be mini-mized by minimizing the oxygen solubility (N ðSÞO ) in thematrix and by minimizing the ratio of oxygen diffusivity(DO) to the aluminum diffusivity (DAl) in the matrix. There-fore, it is desirable to add elements that would increase thesolubility of Al in (Nb). As shown above, Hf is one ofthem, and the available experimental phase diagrams[102] show that Cr, Ti and V also exert a similar effect.

Fig. 7. Transmission electron diffraction patterns used to estimate the lattice mismatch in prototype alloys: (a) along the [110] zone axis in 84Nb–8Al–8Pd(in at.%) heat treated at 1000 �C for 500 h, (b) along the [110] zone axis in 81.6Nb–6.3Al–2.8Hf–9.3Pd (in at.%) heat treated at 1000 �C for 200 h, and (c)along the [100] zone axis in 82Nb–8Al–10Ru (in at.%) heat treated at 1200 �C for 100 h. In (b) and (c), H refers to the Heusler phase.

0

0.1

0.2

0.4

0.6

0.8

0.9

1.0

Mol

e Fr

actio

n H

f

0 0.2 0.4 0.6 0.8 1.0

Mole Fraction NbAlAl NbNb

HfHf

sσ

AlNbAlNb22AlNbAlNb33

AlAl33NbNb

AlAl33HfHf

AlAl22HfHf

AlAl33HfHf22

AlHfAlHf

AlHfAlHf22

(Nb)(Nb)

((aα-Hf)

Liquid+Liquid+AlAl33(Hf,Nb)(Hf,Nb)

Al33HfHf22+AlNb+AlNb22++AlAl33(Hf,Nb)(Hf,Nb)

AlAl

33 HfHf

22 +AlHf+AlNb

+AlHf+AlNb22

0.7

0.5

0.3

Al

Fig. 8. Calculated isothermal section of the Al–Hf–Nb system at 1300 �C.


Based on our calculated isothermal section at 1300 �Cshown in Fig. 8, 45Nb–34Hf–21Al (in at.%) was chosenfor the oxidation study. Static oxidation was carried outat 1300 �C using several specimens, for up to 50 h. Theresults are summarized in Fig. 9. The weight gain behaviorclearly exhibits a parabolic kinetics, and the oxide scalethickness is also found to exhibit a parabolic kinetics [1].

The oxidized specimens were characterized by a combina-tion of X-ray diffraction and SEM to ascertain the nature ofthe oxide scale(s). Fig. 10 shows the SEM micrographs of theoxide scale formed on the alloy after 5 and 25 h of oxidation.In both cases, the oxide scale was adherent, and did not spalloff after cooling. As seen in Fig. 10, the oxide scale has twodistinct layers: (i) the outermost layer is a transient Nb2O5

phase with some dissolved Al and Hf; and (ii) underneaththe Nb2O5 layer is a two-phase oxide layer consisting ofNbAlO4, and HfO2 with dissolved Al and Nb. The transientnature of the outermost Nb2O5 layer is characterized by itsnearly constant thickness even after varying exposure times.The mixed oxide layer NbAlO4 and HfO2 exhibits parabolic

0

50

100

150

200

0 10 20 30 40 50 60

Time of Oxidation (hrs)

Wt

Gai

n (

mg

/cm

2 )

1300 oC

Fig. 9. Static oxidation behavior of a 45Nb–34Hf–21Al (in at.%) alloy at1300 �C [1].


growth, implying that the oxidation kinetics is a diffusioncontrolled process.

5.6. Computational kinetics: a comparison of diffusion of

oxygen in pure Nb, and bcc alloys of Al–Nb and Al–Hf–Nb

One of the most important challenges in successfuldesign of Nb-base superalloys for 1300 �C application is

Fig. 10. SEM micrographs of oxide layer formed on alloy 45Nb–34Hf–21Alinteraction zone at a low magnification; (b) a higher magnification of (a), showfor 25 h at low magnification; and (d) a higher magnification of (c) [1].

to improve its intrinsic oxidation resistance significantly.This requires the formation of a self-protective externaloxide scale, and a decrease in oxygen solubility and oxygentransport kinetics, both by several orders in magnitudecompared with pure Nb. Within the framework of compu-tational thermodynamics and kinetics, we demonstrate thatAl–Hf–Nb solid solutions help achieve these goals.

The modelling and simulation of oxygen transport wascarried out using the DICTRA package [50]. DICTRAuses Thermo-Calc to calculate the thermodynamic factorof the phases to convert mobility into diffusivity and alsoto compute the local equilibrium between the phases. Inother words, to use DICTRA successfully a complete ther-modynamic description of the participating phase(s) isneeded first, and then the kinetic description of the corre-sponding phase(s). In parallel to the development of ther-modynamic database described above, a multicomponentmobility database relevant to Nb-base alloys was alsodeveloped [1].

Fig. 11 compares the simulated oxygen profiles in pureNb and two Nb-base alloys at 1300 �C, clearly demonstrat-ing the effect of alloy chemistry on the intrinsic transportkinetics of oxygen in solid solution. An assumption madein these simulations is that the oxygen content at the sur-face of the bulk alloy is 5 at.%, which is based on theNb–O phase diagram. This is equivalent to the assumption

(in at.%) at 1300 �C: (a) for 5 h, showing the complete oxide layer anding the transient outermost layer and the steady-state layer underneath; (c)

0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0 A

tom

ic P

erce

nt O

xyge

n

0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Distance, mm

1h1h 5h5h

1h1h 5h5h

11 5h5h

Pure NbPure Nb92Nb8Al Alloy92Nb8Al Alloy45Nb34Hf21Al Alloy45Nb34Hf21Al Alloy

Fig. 11. Intrinsic oxygen transport kinetics in pure Nb and Nb-base alloysat 1300 �C, demonstrating the effect of alloy chemistry.


that, irrespective of the type of oxide, the oxygen potentialat the oxide/base metal interface is constant. The simulatedoxygen penetration depths after 1 and 5 h at 1300 �C are3.21 and 6.01 mm in pure Nb, 1.26 and 2.51 mm in92Nb8Al (at.%) alloy, and 0.14 and 0.35 mm in45Nb34Hf21Al (at.%) alloy. The values for Nb–Al andNb–Al–Hf alloys are in excellent agreement with theexperimental data as verified by the hardness profile andalso by direct observation of microstructures [1]. For exam-ple, in the ternary alloy the measured oxygen penetrationdepths were 0.13 and 0.34 mm after 1 and 25 h exposure,respectively, at 1300 �C. Compared to pure Nb, the oxygenpenetration depth is reduced by a factor of more than 20 inNb–Al–Hf alloy. Mechanistically, this is most likely due tostrong binding energies of Al–O and Hf–O, and possiblyHf–O–Nb and Al–Hf–O in the ternary alloy.

6. Discussion

6.1. An estimation of the energy for B2! L21 congruent

ordering

In this section, we present an estimate of B2! L21 order-ing energy at Pd2HfAl and Ru2NbAl compositions from abinitio calculations. We are motivated by the fact that L21-Ru2NbAl is more stable than the pseuodobinary alloys ofRu(NbxAl1�x) with B2 structure. This is implied by the pre-sence of two two-phase fields, RuAl + Ru2NbAl and Ru2N-bAl + RuNb, in the isothermal section of Ru–Nb–Al at1100 �C [13]. Once again, it should be noted that both B2-RuAl and B2-RuNb are stable phases at 1100 �C in therespective binary phase diagram [86]. However, in the caseof PdAl–PdHf pseudobinary alloys, B2-PdAl is the stablephase while B2-PdHf is a virtual phase. In an L21 structureof composition A2BC, atoms A, B and C occupy three dis-tinct sublattices, designated as I, II and III. In a pseuodobin-

ary alloy of A2BC with B2 structure, atoms of A occupysublattice I, while atoms B and C remain disordered on sub-lattices II and III. In a bcc solid solution, atoms A, B and Cremain disordered on all three sublattices. To estimate theenergy associated with congruent ordering (B2! L21) atA2BC, it is necessary to model compositional disorder inA(BxC1�x) alloys of with B2 structure.

Within the ab initio framework, three approaches maybe employed to investigate the effect of compositionaldisorder on the formation energy, namely (i) the conven-tional supercell method; (ii) the SQS (special quasi-randomstructure) supercell method [103]; and (iii) the sublatticecluster expansion method [104] that has been widely usedto model anion and/or cation disorder in ceramic systems[105]. As mentioned in Section 3.1, we have employed theconventional supercell method only. Specifically, we havecalculated the formation energy of Pd(HfxAl1�x) andRu(NbxAl1�x) with B2 structure at x = 0.125, x = 0.25,x = 0.375, x = 0.5, x = 0.625, x = 0.75 and x = 0.875. Itis important to note that, for a particular composition,there are many ways to arrange Al + Hf or Al + Nb atomson 16 sites in 32-atom supercells. Due to limited computa-tional resources, the total energies of all possible arrange-ments could not be performed. Instead, at a particularcomposition we have calculated the total energies for 2–6different arrangements (chosen arbitrarily). Then, the mix-ing energy, as an example for Pd(HfxAl1�x), is defined asDEmix ¼ DEPdðHfxAl1�xÞ

f � ½ð1� xÞDEPdAlf þ xDEPdHf

f �.The results of ab initio calculations are summarized in

Tables 4 and 5. In the supercell method, total energiesare calculated allowing volume relaxation only, and alsoallowing full (volume, shape, ionic) relaxation. The meanvalue of mixing energies at each composition (see Tables4 and 5) is plotted in Fig. 12. In both Pd(HfxAl1�x) andRu(NbxAl1�x) alloys the difference between volume relaxedand fully relaxed DEmix is only a few kJ mol�1 of atom,with the latter being more negative as expected. In the caseof Pd(HfxAl1�x) alloys the DEmix is weakly positive, whileit is moderately negative in Ru(NbxAl1�x) alloys. We havenoted that the magnitude of relaxations (volume, shape,ionic) is larger in Pd(HfxAl1�x) than in Ru(NbxAl1�x); thisis believed to be due to the larger size mismatch betweenB2-PdAl and B2-PdHf than between B2-RuAl and B2-RuNb (see Table 3). This, along with a small number ofconfigurations used for total energy calculations at aparticular composition, has contributed to the oscillatorybehavior, though within 5 kJ mol�1 of atom, of DEmix ofPd(HfxAl1�x) alloys. Nonetheless, as indicated, Fig. 12aand b defines the energy for B2! L21 congruent ordering,which is about �8 and �4.5 kJ mol�1 of atom at Pd2HfAland Ru2NbAl compositions, respectively.

6.2. Electronic structure and bonding mechanism in B2 and

L21 phases

Large negative formation energies of the stable phases inTable 1 are indicative of strong bonding tendencies

Table 4Calculated formation (DEf) and mixing (DEmix) energies (in kJ mol�1 ofatom) of Pd(HfxAl1�x) alloys with B2 structure

x DEf DEmix

Volume relaxed Fully relaxed Volume relaxed Fully relaxed

0.125 �78.273 �78.938 2.045 1.380�79.178 �79.583 1.139 0.918�77.945 �78.292 2.373 2.026

(�78.465) (�78.938) (1.852) (1.441)

0.250 �77.358 �78.051 �0.038 �0.727�74.718 �75.412 2.606 1.913

(�76.038) (�76.732) (1.284) (0.593)

0.375 �70.383 �70.845 3.948 3.486�71.269 �72.454 3.061 1.876�70.903 �72.762 3.428 1.568

(�70.852) (�72.021) (3.479) (2.310)

0.500 �65.585 �66.009 5.752 5.328�79.412 �79.363 �8.074 �8.026�68.871 �70.663 2.467 0.674�74.276 �74.931 �2.939 �3.594�69.824 �71.048 1.512 0.289�67.704 �69.786 3.632 1.551

(�70.945) (�71.967) (0.392) (�0.629)

0.625 �63.331 �64.381 2.045 1.380�64.381 �68.698 1.139 0.918�64.352 �64.496 2.373 2.026

(�64.021) (�65.858) (4.322) (2.485)

0.750 �63.938 �66.444 1.412 �1.093�63.988 �66.521 1.363 �1.171�61.144 �65.384 4.206 �0.034

(�63.023) (�66.116) (2.327) (�0.766)

0.875 �60.663 �61.694 1.693 0.662�60.307 �61.219 2.050 1.067�60.123 �60.991 2.233 1.375

(�60.364) (�61.301) (1.992) (1.035)

The energies of formation at a particular composition are calculated usingdifferent configurations of Al and Hf in 32-atom supercells, and theaverage value is given in parenthesis. The reference states for DEf are fcc-Al, hcp-Hf, and fcc-Pd.

Table 5Calculated formation (DEf) and mixing (DEmix) energies (in kJ mol�1 ofatom) of Ru(NbxAl1�x) alloys with B2 structure

x DEf DEmix

Volume relaxed Fully relaxed Volume relaxed Fully relaxed

0.125 �65.089 �65.677 �5.992 �6.584�65.793 �66.091 �6.690 �6.999�65.012 �65.089 �5.919 �5.997

(�65.298) (�65.619) (�6.202) (�6.527)

0.250 �65.844 �66.306 �12.947 �13.409�65.419 �65.931 �12.523 �13.034

(�65.632) (�66.119) (�12.735) (�13.222)

0.375 �62.715 �63.053 �16.015 �16.352�63.496 �63.862 �16.795 �17.162�62.321 �63.545 �15.620 �16.824

(�62.844) (�63.487) (�16.143) (�16.780)

0.500 �57.313 �57.987 �16.809 �17.483�63.730 �63.730 �23.226 �23.226�54.760 �57.602 �14.255 �17.098�60.493 �61.321 �19.988 �20.817�57.862 �58.276 �17.358 �17.772�56.330 �57.188 �15.826 �16.683

(�58.415) (�59.351) (�17.911) (�18.846)

0.625 �48.326 �48.721 �14.018 �14.413�47.758 �48.682 �13.449 �14.374�47.324 �48.355 �13.016 �14.047

(�47.803) (�48.586) (�13.494) (�14.278)

0.750 �38.453 �40.091 �10.341 �11.979�37.017 �38.954 �8.905 �10.842

(�37.735) (�39.523) (�9.623) (�11.411)

0.875 �26.440 �28.319 �4.525 �6.404�26.624 �28.108 �4.708 �6.192�26.335 �27.462 �4.419 �5.546

(�26.466) (�27.963) (�4.551) (�6.047)

The energies of formation at a particular composition are calculated usingdifferent configurations of Al and Nb in 32-atom supercells, and theaverage value is given in parenthesis. The reference states for DEf are fcc-Al, bcc-Nb, and hcp-Ru.


between the constituent atoms. Also, as discussed above,there is a driving energy for B2! L21 ordering at Pd2HfAland Ru2NbAl compositions. To obtain further insight intothe nature of the bonding, we present the electronic densi-ties of states (DOS) and bonding charge densities of threeB2 (PdAl, RuAl, RuNb) and two L21 (Pd2HfAl and Ru2N-bAl) phases. In the following, we present both total andpartial DOSs, where the latter quantities were computedusing projections into site-centered atomic spheres, eachwith radii equal to half the nearest-neighbor spacing. Wealso present the bonding charge density (sometimes calledthe deformation charge density), defined as the difference(point-by-point using identical FFT grid) between theself-consistent charge density qð x!Þ of the intermetallicand a reference charge density constructed from the super-position of non-interacting atomic charge density at thecrystal sites. For example, the bonding charge density of

B2-PdAl is defined as dq = q(PdAl) � q(Al atom) � q(Pdatom). Similarly, the bonding charge density of L21-Ru2N-bAl is defined as dq = q(Ru2NbAl) � q(Al atoms) � q(Nbatoms) � q(Ru atoms).

Fig. 13a and b shows the calculated the electronic DOSand the bonding charge-densities of B2-PdAl, respectively.In Fig. 13a, different Y-scales between s, p and d compo-nents of the DOS may be noted. The total DOS is in agree-ment with the previous report [75,78]. The Fermi level liesin the pseudogap minimum separating the bonding andanti-bonding states. It is seen that below the Fermi levelpronounced peaks of Pd-d strongly hybridize with Al-p.The Al-s contribution becomes visible at about 5 eV belowthe Fermi level, and a corresponding Pd-d peak may alsobe seen. Fig. 13b shows the bonding charge density plotin the (11 0) plane where Al and Pd are the nearest neigh-bors. Delocalization of the bonding charge density in the

Fig. 12. Calculated mixing energy of (a) PdHfxAl1�x and (b) RuNbxAl1�x

alloys with B2 structure. These also define the B2! L21 ordering energyat Pd2HfAl and Ru2NbAl composition, respectively.

a

b

Fig. 13. Electronic structure of B2-PdAl calculated using all-electronPAW potentials: (a) angular momentum and site decomposed electronicdensity of states, n(E), with the Fermi level marked by a dotted line; (b) thedistribution of bonding (or deformation) charge density in the (110)plane, with selected contour lines drawn at a constant interval of0.003 e A�3. In the color scale bar, the bonding (or deformation) chargedensity ranges from �0.116 (depleted region: dq(�)) to 0.032 (enhancedregion: dq(+)) e A�3.


interstitial region is seen. There is also a directional build-up of bonding charge, with a maximum halfway betweenand Al and Pd atoms, which, according to Fu [75], maynot be the covalent type. These features, along with theangular momentum-resolved DOS, imply that the bondingmechanism may be described as Al-sp–Pd-d hybridization.Fu [75] proposed that the bonding character in B2-PdAl isa combination of metallic bonding and charge-transfercomponents. In Fig. 13b, it is interesting to note that thebonding charge density (dq) isocontour lobes are orientedperpendicular to the Al–Pd bonding direction.

Fig. 14a and b shows the calculated the electronic DOSand the bonding charge-densities of L21-Pd2HfAl, respec-tively. Once again, in Fig. 14a, different Y-scales betweens, p and d components of the DOS may be noted. TheFermi level lies about 2 eV to the right of the pseudogapminimum, i.e. the bonding states are completely occupied.As seen in Fig. 14a, the bonding states are dominated byHf-d and Pd-d, while the anti-bonding states are domi-

nated by Hf-d. Below the Fermi level, pronounced peaksof Hf-d and Pd-d strongly hybridize, while both of themalso hybridize with Al-p in the entire energy region. UnlikeB2-PdAl, a strong Al-s peak is promoted in this structureat about 6.6 eV below the Fermi level which hybridizeswith Pd-d. The Al-d and Hf-p contributions are negligible,while Hf-s, Pd-s and Pd-p make only a small contribution,with the latter being spread out on both sides of the Fermilevel. Fig. 14b shows the bonding charge density plot in the(110) plane. A significant redistribution of bonding chargein the interstitial region is seen. The build-up of bondingcharge along the Al–Hf and Al–Pd bond directions mayonly be described as moderate; however, it is very strongalong the Hf–Pd bond direction. We also notice a maxi-mum in bonding charge density along the direction ofHf–Pd bonding, but its distribution is skewed towardsthe Pd site and the dq isocontour lobes are oriented perpen-

b

a

Fig. 14. Electronic structure of L21-Pd2HfAl calculated using all-electronPAW potentials: (a) angular momentum and site decomposed electronicdensity of states, n(E), with the Fermi level marked by a dotted line; (b) thedistribution of bonding (or deformation) charge density in the (110)plane, with selected contour lines drawn at a constant interval of0.006 e A�3. In the color scale bar, the bonding (or deformation) chargedensity ranges from �0.164 (depleted region: dq(�)) to 0.065 (enhancedregion: dq(+)) e A�3.

b

a

Fig. 15. Electronic structure of B2-RuAl calculated using all-electronPAW potentials: (a) angular momentum and site decomposed electronicdensity of states, n(E), with the Fermi level marked by a dotted line; (b) thedistribution of bonding (or deformation) charge density in the (110)plane, with selected contour lines drawn at a constant interval of0.0046 e A�3. In the color scale bar, the bonding (or deformation) chargedensity ranges from �0.223 (depleted region: dq(�)) to 0.047 (enhancedregion: dq(+)) e A�3.


dicular to the Hf–Pd bonding direction. The angularmomentum-resolved DOS in conjunction with the bondingcharge density plot implies that the bonding mechanismconsists of short-range band mixing between the d statesof Hf and Pd, and also sp states of Al and d states of Hfand Pd along with the long-range charge transfer (electro-static) effect.

Fig. 15a and b shows the calculated the electronic DOSand the bonding charge densities of B2-RuAl, respectively.Here, the Fermi level lies about 1 eV to the left of pseudo-gap minimum, i.e. the bonding states are incompletelyoccupied. The partial DOS in Fig. 15a provide direct evi-dence that hybridization is present between the Ru-d andAl-p states, which reflects a strong directional bondingbetween the Ru and Al atoms. The Al-s states become sig-nificant only at about 5 eV below the Fermi level. The totalDOS at the Fermi level is rather high and, due to insuffi-cient number of valence electrons, a lot of bonding states

are unoccupied. All features of DOS in Fig. 15a are in verygood agreement with the previous results calculated byLMTO–LDA [17]. Fig. 15b shows the bonding charge den-sity plot in the (110) plane. While there is delocalization ofbonding charge in the interstitial region that resemblesmetallic bonding, there is also evidence of directionalityof electron density along Æ111æ direction which may contri-bute to the covalent character. The distribution of bondingcharge density is clearly asymmetric, being skewed towardsthe Ru site. The presence of a covalent character is furthersupported by the fact that the energy interval for bondingis much larger than for anti-bonding, as seen in Fig. 15a.Unlike B2-PdAl and L21-Pd2HfAl, it is seen that manydq contour lobes are oriented along the Al–Ru bondingdirection. Combining angular momentum-resolved DOSand the bonding charge density plot, it may be concludedthat the mechanism of cohesion is dominated by the


short-range band mixing between Al-sp and Ru-d statesalong with the long-range charge transfer (electrostatic)effect.

Fig. 16a and b shows the calculated the electronic DOSand the bonding charge-densities of B2-RuNb, respec-tively. Like B2-RuAl, a pseudogap separating bondingand anti-bonding states in total DOS is seen, but at about1 eV below the Fermi level. The anti-bonding states aredominated by both Ru-d and Nb-d states. It is seen that,at about 1.7 eV below the Fermi level, pronounced peaksof Ru-d and strongly hybridize with Nb-d. The sp statesof both Nb and Ru also exhibit hybridization, but theircontributions are much smaller compared with the d–dcontribution. Fig. 16b shows the bonding charge densityplot in (110) plane. It is seen that there is significant delo-calization of bonding charge in the entire interstitial region,suggestive of metallic bonding. Meanwhile, the nature of

b

a

Fig. 16. Electronic structure of B2-NbRu calculated using all-electronPAW potentials: (a) angular momentum and site decomposed electronicdensity of states, n(E), with the Fermi level marked by a dotted line; (b) thedistribution of bonding (or deformation) charge density in the (110)plane, with selected contour lines drawn at a constant interval of0.0059 e A�3. In the color scale bar, the bonding (or deformation) chargedensity ranges from �0.237 (depleted region: dq(�)) to 0.059 (enhancedregion: dq(+)) e A�3.

band mixing between the spd states of Nb and Ru alsoimplies some covalent character. In fact, nonspherical ani-sotropy of bonding charge density is observed along thedirection between Ru and Nb atoms. Like B2-RuAl inFig. 15b, the distribution of bonding charge density inB2-RuNb also shows asymmetry, being skewed towardsthe Ru site. Once again, combining angular momentum-resolved DOS and the bonding charge density plot, itmay be concluded that the bonding mechanism is primarilydominated by the short-range band mixing involving dstates of Ru and Nb.

Fig. 17a and b shows the calculated the electronic DOSand the bonding charge-densities of L21-Ru2NbAl, respec-tively. Like B2-RuAl and B2-RuNb, a pronounced deep ora pseudogap separating bonding and anti-bonding states intotal DOS is seen, but here the Fermi level lies in the pseu-dogap. It is interesting to note that RuAl has 5.5 valence

b

a

Fig. 17. Electronic structure of L21-Ru2NbAl calculated using all-electronPAW potentials: (a) angular momentum and site decomposed electronicdensity of states, n(E), with the Fermi level marked by a dotted line; (b) thedistribution of bonding (or deformation) charge density in the (110)plane, with selected contour lines drawn at a constant interval of0.01 e A�3. In the color scale bar, the bonding (or deformation) chargedensity ranges from �0.289 (depleted region: dq(�)) to 0.1 (enhancedregion: dq(+)) e A�3.


electrons per atom with the pseudogap 1 eV above theFermi level, and RuNb has 6.5 valence electrons per atomwith the pseudogap 1 eV below the Fermi level; therefore, itis not surprising that Ru2NbAl with 6 valence electrons peratom has the Fermi level in the pseudogap. The total DOSin Fig. 17a is in very good agreement with the previousresults of Weinert and Watson [106]. A very low densityof states at Fermi level implies that L21-Ru2NbAl is a verystable compound. These results (total DOS vis-a-vis DEf ofB2-RuAl, B2-RuNb and L21-Ru2NbAl) are consistent withthe idea that the most stable phase is the one which opti-mizes the filling of bonding states (increasing cohesion)within a rigid band model. As seen in Fig. 17a, the anti-bonding states are dominated by Ru-d and Nb-d. Belowthe Fermi level, the hybridization is dominated by Al-p,Nb-d and Ru-d. Although Al-p does not make a significantcontribution, it is found to participate in the hybridizationin the entire energy region. Like L21-Pd2HfAl, we note thata strong Al-s peak is promoted in this structure at about6.6 eV below the Fermi level which hybridizes with Ru-d.Fig. 17b shows the bonding charge density plot in the(110) plane. It shows that a depletion of electron densityat the Al and Nb sites is accompanied by a build-up ofcharge density around the Ru sites. The build-up of bond-ing charge along the Al–Nb and Al–Ru bond directionsmay only be described as moderate; however, along theNb–Ru bond direction the build-up is rather strong. LikeB2-RuAl in Fig. 15b and B2-RuNb in Fig. 16b, the distri-bution of bonding charge density in L21-Ru2NbAl alsoshows asymmetry, being skewed towards the Ru atom.We also notice that the dq isocontour lobes are orientedalong the Nb–Ru bonding direction. Like B2-RuAl, inFig. 17a we see that the energy interval for bonding is muchlarger than for anti-bonding, suggestive of the presence of acovalent character associated with the bonding between theatoms. Consistent with the partial DOS in Fig. 17b, thebonding between Ru and Nb dominates over both Ru–Aland Nb–Al bonding. Based on these results, it is concludedthat the bonding mechanism consists of the short-rangeband mixing between the d states of Nb and Ru, and alsosp states of Al and d states of Nb and Ru, and the long-range charge transfer (electrostatic) effect.

6.3. Integrated materials design: current trend and

limitations

A major focus of our current research has been an opti-mal and efficient integration of modern computationaland experimental tools to facilitate quantitative evaluationof processing–microstructure–property–performance linksand rapid prototyping. The present study demonstratesthe efficacy of this approach by modeling Nb-based super-alloy as an integrated microstructure. This representation,shown in Fig. 2, is then used to identify and prioritize thekey process–structure and structure–property links to bequantified as a part of the design exercise. The multilevelstructure and hierarchy of computational models demands

that a hierarchy of experimental tools be employed to cre-ate the requisite databases, which are not yet available, andto validate the predictions of the modeling. Due to conflict-ing nature of the effect of alloying elements on many of therelevant properties, as discussed in Section 2, the optimiza-tion of this materials system can only be achieved by themethod of systems design employing different computa-tional tools. The potential economic impact of thisapproach is significant as it can reduce the time to invent,produce and test a new alloy. Obviously there is a clear eco-nomic driving force for adopting such an integrated sys-tems approach. Specific to the conceived Nb-basedsuperalloy, while the total system integrates compatiblethermal barrier and bond coat subsystems, here we havepresented results on predictive design of the underlyingprecipitation strengthened alloy by integrating the resultsof ab initio calculations, computational thermodynamicsand kinetics. Following the initial computational modelingand design exercise, selected experiments are performed tovalidate the precipitation strengthened microstructures andoxidation behavior of prototype alloys.

Notwithstanding tremendous advances in computerhardware, algorithm, software, alloy theory and user-inter-face, in terms of both availability and affordability, it isimportant to underscore the limitations of ab initio meth-ods in the design of new engineering alloys. Here, we willmention only a few important ones. First, due to the multi-component (where the number of components may be ashigh as 10) and multiphase nature of engineering alloys,the direct application of first-principles methods (i.e. whereatomic number and atomic arrangement are the only inputparameters) to the direct modeling of phase stability isintractable. A convenient way to overcome this drawbackis to employ calphad-based approaches of computationalthermodynamic and kinetics where multicomponent, mul-tiphase equilibria and diffusion problems can be solvedrather rapidly requiring minimal computational resources.Second, it may not be possible to calculate all relevantproperties directly by ab initio methods. Instead, the physi-cal parameters/properties calculable by ab initio methodsmay be correlated with the relevant engineering properties.For example, it is not possible to calculate the rafting beha-vior, the directional coarsening of the precipitates intoplates or rafts under applied load and the coarsening rateof precipitates directly by ab initio methods even in rela-tively simple alloys. However, it is possible to calculatethe physical parameters/properties that control them, suchas the lattice misfit, precipitate/matrix interfacial energy,solute diffusivity and solubility. These parameters may thenbe correlated with the creep property of an alloy. Similarly,it is not possible to calculate the DBTT of an alloy directlyby ab initio methods. However, it is possible to calculatethe surface and unstable stacking fault energies, the ratioof which has been shown to correlate well with the propen-sity for brittle fracture and thus may be used as a criterionfor DBTT. Third, sometimes the physical laws that governone or more properties of an alloy may not be available at


the time of alloy design. Consequently, new physical law(s)need to be established prior to a decision being made as towhat physical parameters/property should be calculated byab initio methods.

Successful application of computational thermodynamicand kinetics tools in materials design also suffer from lim-itations. In recent years, these tools based on the calphadframework have become widely used as the basis for mod-eling phase stability and phase transformation kinetics incomplex alloy systems; however, it is important to recog-nize that the accuracy of predictions derived from thesetools depends critically on the accuracy of relevant data-bases. This in turn depends on the accuracy of experimen-tal data and the assessment methods that define theaccuracy of thermodynamic and kinetics model para-meters. Furthermore, for new, relatively unexplored sys-tems, design and modeling efforts are often hindered bythe need for extensive experimental measurements neededin the development of robust thermodynamic and kineticdatabases. Some of these drawbacks can be overcome byperforming ab initio calculations, thus significantly limitingthe extent of costly experimental measurements required inthermodynamic and kinetics database development, butmay require significant computational resources.

7. Conclusions

An optimal integration of ab initio total energy andalloy theory, and selective experimental determination ofhigh temperature phase relations allows accelerated devel-opment of multicomponent thermodynamic and kineticdatabases for computational materials design. In this con-text, the principles of integrated design of Nb-based super-alloys are discussed. The conceptual design of Nb-basedsuperalloys exploits precipitation strengthening by alumi-nide phase(s), having nearest-neighbor (B2) and next-near-est-neighbor (L21) ordered structures based on a bcclattice, that also offer a high potential for combining oxida-tion resistance and creep strength for application at1300 �C and above. The following conclusions are drawn:

(i) We have used US-PP–GGA as implemented in VASPto determine the zero-temperature equation of state,electronic density of states, and charge densities ofselected aluminides with B2 and L21 structures rele-vant to the design of precipitation strengthened Nb-based superalloys.

(ii) We find that, in general, the calculated zero-tempera-ture lattice parameters of Al, Hf, Nb, Pd and Ruagree within ±1%, and the calculated B0 agreeswithin ±5% when the calculated are compared withthe corresponding experimental values (either mea-sured or extrapolated).

(iii) The calculated zero-temperature formation energy ofB2-PdAl is underestimated by about 7–10 kJ mol�1

of atom, while that of B2-RuAl agrees within a fewkJ mol�1 of atom when compared with the calori-

metric data. The zero-temperature formation energyof L21-Pd2HfAl and L21-Ru2NbAl calculated byUS-PP–GGA (VASP), PAW–GGA (VASP),FLAPW–LDA and FLASTO–LDA give almostidentical results.

(iv) For the stable intermetallics considered, the zero-tem-perature lattice parameter agrees within ±1%, exceptfor B2-PdAl, when compared the experimental data(either measured or extrapolated) at low tempera-tures. The calculated bulk modulus of B2-RuAl isfound to agree with 5% of the experimental value atambient temperature.

(v) The B2! L21 ordering energy at Pd2HfAl andRu2NbAl compositions estimated from ab initio cal-culations to be about �8 and �4.5 kJ mol�1 of atom,respectively.

(vi) The phase stability and bonding mechanism in rele-vant B2 (PdAl, RuAl, RuNb) and L21 (Pd2HfAl,Ru2NbAl) phases are discussed in terms of their elec-tronic density of states and bonding (or deformation)charge density.

(vii) The following two-phase microstructures are observedin prototype alloys: (Nb) + B2-PdAl, (Nb) + L21-Pd2HfAl and (Nb) + L21-Ru2NbAl. The structureand composition (qualitative) of matrix and precipi-tates are confirmed by transmission electron diffrac-tion, and the corresponding EDS spectra obtainedin a high-resolution analytical electron microscope,respectively.

(viii) Integration of ab initio alloy phase stability and com-putational thermodynamics predict that the solubilityof Al in (Nb) at 1300 �C is significantly increased byadding Hf. This is shown to be highly beneficial forimproving oxidation resistance of Nb-based alloys.

(xi) Guided by the results of computational thermody-namics, the oxidation study of a prototype alloy of45Nb–34Hf–21Al (in at.%) at 1300 �C was carriedout. Both weight gain and thickening kinetics of theoxide layer exhibited the parabolic behavior, andthe layer was found to be a mixture of the NbAlO4

and HfO2 phases.

Acknowledgments

This work was sponsored by the Air Force Office ofScientific Research, USAF, under Grant No. F49620-01-1-0529. Computational resources, the Itanium clustersas a part of the teragrid facility at the University of Illi-nois at Urbana-Champaign and at the San Diego Super-computing Center, provided by NPACI (NationalPartnership for Advanced Computational Infrastructure),are gratefully acknowledged. The views and conclusionscontained herein are those of the authors and shouldnot be interpreted as necessarily representing the officialpolicies or endorsements, either expressed or implied, ofthe Air Force Office of Scientific Research or the U.S.Government.


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