Integrated design of Nb-based superalloys: Ab initio calculations, computational thermodynamics and kinetics, and experimental results

Acta Materialia 55 (2007) 3281–3303 www.elsevier.com/locate/actamat

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 2007 Available online 23 March 2007

Abstract An optimal integration of modern computational tools and efficient experimentation is presented for the accelerated design of Nbbased superalloys. Integrated within a systems engineering framework, we have used ab initio methods along with alloy theory tools to predict phase stability of solid solutions and intermetallics to accelerate assessment of thermodynamic and kinetic databases enabling comprehensive predictive design of multicomponent multiphase microstructures as dynamic systems. Such an approach is also applicable for the accelerated design and development of other high performance materials. Based on established principles underlying Ni-based superalloys, the central microstructural concept is a precipitation strengthened system in which coherent cubic aluminide phase(s) provide both creep strengthening and a source of Al for Al2O3 passivation enabled by a Nb-based alloy matrix with required ductile-to-brittle transition temperature, atomic transport kinetics and oxygen solubility behaviors. Ultrasoft and PAW pseudopotentials, as implemented in VASP, are used to calculate total energy, density of states and bonding charge densities of aluminides with B2 and L21 structures relevant to this research. Characterization of prototype alloys by transmission and analytical electron microscopy demonstrates the precipitation of B2 or L21 aluminide in a (Nb) matrix. Employing Thermo-Calc and DICTRA software systems, thermodynamic and kinetic databases are developed for substitutional alloying elements and interstitial oxygen to enhance the diffusivity ratio of Al 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 by the operating temperature, which in turn is determined by several temperature-dependent properties of materials used in turbine engines. A great deal of research is underway to design new materials capable of operating at temperatures higher than current capability, as the efficiency of a Carnot engine is directly related to the homologous

*

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 the materials of choice for high temperature turbine blade applications as they satisfy a unique combination of desired properties. Fig. 1 shows the evolution of Ni-based superalloys over the years as a function of their temperature capability [1]. The operating temperature capability of Ni-based superalloys is less than 1150 C as the most advanced Ni-based single crystal superalloys melt below 1350 C [2], thus triggering the need for new alloys that can operate around 1300 C. Ideally, such an alloy should have a substantially higher melting point (or low homologous temperature (<0.5Tm) at 1300 C), and at operating temperature it should

1359-6454/$30.00  2007 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2007.01.036

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2. Integrated design: principles, computational and experimental tools, and relevance to prior research on Nbbased alloys

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

have high creep strength and high oxidation resistance. For oxidation resistance, the material should be capable of passive oxidation, making it capable of forming a protective oxide scale. The refractory metal niobium has a melting temperature of 2467 C and also has a low density, thus making it an attractive candidate for the replacement of nickel. 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 the design of composite microstructures consisting of (Nb), Nb5Si3 and other intermetallics, such as the Laves phase Cr2Nb to improve oxidation, strength, creep at elevated temperatures, as well as ductility and damage tolerance at ambient temperatures [5–9]. However, as the intermetallic phases are not present as fine scale precipitates, creep in such composites is not controlled by dislocation climb. Rather, creep strength is imparted by Nb5Si3, and is controlled by diffusion of Nb in it [2]. Furthermore, it has been found that such multiphase Nb-based composites form nonprotective oxide scales above 1200 C, and the scales consist of several oxides, including Nb2O5, CrNbO4 and Nb2TiO7 [10]. Here, we present a systems-based approach [11] to design a new precipitation strengthened Nb-based superalloy having desired oxidation resistance, creep and ductile-to-brittle transition temperature (DBTT) for aeroturbine applications operating at 1300 C or above. The remainder of the paper is organized as follows. In Section 2, we present the conceptual design principles, and the integration of computational tools and experimental methods. In Section 3, we present the computational methodology employed in this study. In Section 4, we present alloy preparation and various experimental methods used to characterize the prototype alloys. In Section 5, we present computational and experimental results. In Section 6, we discuss the results of ab initio total energy and electronic structure calculations. Conclusions are summarized in Section 7.

To design precipitation strengthened Nb-base superalloys, we employ a science-based materials design approach by integrating a number of computational capabilities within the framework of systems engineering [11]. A systems approach integrates a hierarchy of computational and experimental tools to quantify and to predict the processing–microstructure–properties–performance links. Fig. 2 summarizes the conceived processing–structure–property–performance links governing the behavior of a multilevel-structured coated Nb-based superalloy system. The three main properties of interest in this research are creep resistance, oxidation resistance, and low-temperature ductility. The relevant research tools, both computational and experimental, and their acronyms are indicated in the rectangular box at the bottom of Fig. 2. The principles of strengthening in classical c/c 0 Ni-based superalloys [12] are extended to design a Nb-based superalloy strengthened by an ordered aluminide phase. In bodycentered cubic (bcc) alloys, such strengthening can be achieved by precipitating either B2 (nearest-neighbor ordered structure) and/or L21 (next-nearest-neighbor ordered structure) phases. The ordered intermetallic needs to have high thermodynamic stability at high temperature. Furthermore, to maintain coherency over a long period of time 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 with L21 (Heusler) structure, for the conceptual design of a Nb-based alloy where one or more aluminide may be present as precipitates. The consideration of PdAl and Ru2NbAl as candidate precipitates is based on the experimental study of phase relations in Nb–Pd–Al and Nb–Ru–Al systems that indicate the presence of (Nb) + PdAl and (Nb) + Ru2NbAl phase field, respectively, at 1100 C [13]. While L21Pd2HfAl was reported long ago [14], our recent experimental study of phase relations in Nb-Pd-Hf-Al system reveals the presence of (Nb) + PdAl, (Nb) + Pd2HfAl and (Nb) + PdAl + Pd2HfAl phase fields at 1200 C [15,16]. Prior to our phase relations studies, quantum mechanical total energy calculations [17] were carried out using the full potential-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 parameter of six Heusler phases, Ni2TiAl, Ni2VAl, Ni2ZrAl, Ni2NbAl, Ni2HfAl and Ni2TaAl. The calculated results show that Ni2HfAl provides the lowest lattice misfit with Nb (7.6%) and has the highest thermodynamic stability (formation energy DEf = 91.5 kJ mol1 of atom). In order to decrease the lattice misfit further, the Ni was replaced by Pd, which has the same electronic configuration but a larger atomic radius. Quantum mechanical calculations of

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

Pd2HfAl [20] using the full potential-linearized augmented plane wave method (FLAPW) [21] within LDA show that indeed there is a decrease in lattice mismatch between Nb and Pd2HfAl (3.8%), and also the latter has a reasonably high thermodynamic stability (formation energy DEf = 79.9 kJ mol1 of atom). The oxidation resistance, which can be extrinsic and intrinsic, is the life-limiting property of Nb-based alloys. The intrinsic oxidation resistance may be obtained by the formation of a protective oxide scale governed by Wahl’s modification [22] of Wagner’s criterion [23] for external oxidation. The extrinsic oxidation resistance can be obtained by integration of thermal barrier coatings and oxygen barrier coatings to the underlying base alloy. This would necessarily require a suitable bond coat material to increase adherence to the external coating. A self-protective oxide scale usually exhibits parabolic oxidation kinetics, and the adherence of the oxide scale with 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 of various oxides indicates that the slowest growing oxide scale is Al2O3 up to about 1300 C, above which SiO2 is the slowest growing oxide scale [1]. Cr2O3 also exhibits a relatively slow growth kinetics. Considering thermodynamic 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 scale in the precipitation strengthened Nb-based superalloy. This also further justifies the choice of aluminides as strengthening precipitates, as they can act as a source for Al. The critical amount of Al needed in solid solution to form external alumina scale may be predicted from Wahl’s modification [22] of Wagner’s theory [23] from transition from internal to external oxidation. The modified theory predicts that the critical amount of Al can be minimized ðSÞ by minimizing the oxygen solubility (N O ) in the matrix and by minimizing the ratio of oxygen diffusivity (DO) to the aluminum diffusivity (DAl) in the matrix. In a multicomponent Nb-based solid solution, these three parameters can be greatly influenced by the presence of other alloying elements. A systematic study of oxidation behavior of bcc Nb–X (X = Al, B, Be, Cr, Co, Fe, Mn, Mo, Ni, Si, Ti, V, W or Zr) between 600 and 1000 C was reported by Sims et al. [25]. This study showed that Ti is most effective in reducing the oxidation rate of Nb, but oxide scale formed was Nb2O5 even in an alloy containing 25 at.% Ti. Niobium dissolves about 8 at.% Al at 1300 C, and when oxidized it also forms Nb2O5. Oxidation studies of intermetallic NbAl3 [26–28] show the formation of Al2O3 scale, but it

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cannot sustain this growth over a prolonged time and NbAl3 decomposes to form NbAl2. The growth of a protective Al2O3 scale at 1400 C has also been demonstrated in a 30Nb–28Ti–42Al (at.%) alloy with B2 structure [26]. The addition of Cr and V reduces the Al required to about 37 at.%. The B2 structure of the alloy favors rapid diffusion of Al to the surface, but the B2 phase decomposes, into cTiAl and Nb2Al, on cooling to room temperature. The dissolved solutes in the Nb matrix play several important roles: (i) provide solid solution strengthening; (ii) control the lattice parameter of the bcc matrix, hence lattice mismatch with the precipitate; (iii) control atomic transport kinetics affecting both coarsening of precipitates and oxidation behavior; and (iv) influence dislocation mobility affecting the ductility and fracture toughness. The alloying element(s) needed to control one or more of these properties often has a conflicting influence on the other(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 Hf and Ti do not increase DBTT of Nb. Recent results have indicated that Ti enhances both tensile ductility and fracture toughness of Nb solid solution [30], while Al and Cr exert the opposite effect [31,32]. However, Al and Cr alloying additions are needed to enhance the oxidation resistance, which is the life-limiting property of Nb-based composites [5,6,8,9]. Rice [33] has suggested that the ratio of the surface energy (cs) and unstable stacking energy (cus) is a key measure of the propensity for brittle fracture, and also a criterion for DBTT. Using these concepts as a guideline, Chan [34] calculated the surface energy and the Peierls–Nabarro barrier energy as a function of composition of bcc solid solutions in a quaternary system Nb–Al–Cr–Ti. Consistent with the experimental data, Chan [34] showed that Ti reduces the Peierls–Nabarro barrier energy, promotes dislocation mobility, and enhances the tensile ductility and fracture toughness of Nb-based solid solutions [34]. On the other hand, Cr and Al exert opposite effects. The criterion of cs/cus has also been used by Waghmare et al. [35] as the basis for selecting ternary alloying elements to improve the 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 creep resistance, oxidation resistance, and low-temperature toughness and ductility, as well as the processing variables which control them. These interactions allow us to deemphasize the weak interaction(s) while explore and quantify the strong interaction(s) in order to optimize a complex system. 3. Computational methodology Due to the multicomponent and multiphase nature of the Nb-based alloys of interest in this research, the direct application of accurate first-principles methods (i.e., meth-

ods using quantum-mechanical calculations requiring only the atomic numbers of the constituent elements and their arrangement as input) to the direct modeling of phase stability in these systems is intractable. By contrast, computational thermodynamic methods based upon the calphad approach [36] can be employed readily in the modeling of multi-component alloy phase stability. The predictive power of this approach, however, is limited by the availability of sufficient experimental data to generate accurate free energy functions. In the absence of adequate experimental data, first-principles calculations offer the best strategy for supplementing thermodynamic databases, thereby increasing the range of applicability of computational thermodynamic methods while limiting the reliance upon extensive experimental measurements. First-principles calculations are performed to calculate formation energies and atomic volumes of intermetallic compounds for use in developing reliable thermodynamic and lattice-parameter databases for required stable and virtual ordered phases. Stable phases are those which are present in the equilibrium phase diagram, irrespective of temperature or composition ranges of stability. The concept of virtual phase is a mathematical one in the context of calphad modeling of intermetallics having a finite homogeneity range, using a sublattice model [37]. In cases where experimental data are available, these calculations provide a basis for determining the overall accuracy of the firstprinciples methods, while in cases where such data are lacking, they can be used to augment the thermodynamic databases that are employed in the integrated design of B2 and/or L21 strengthened Nb-base alloys. 3.1. Ab initio total energy calculations The ab initio calculations presented here are based on electronic density-functional theory (DFT), and have been carried out using the ab initio program VASP (Vienna ab initio simulation package) [38–40]. Most current calculations make use of Vanderbilt-type ultrasoft pseudopotentials (US-PP) [41], as implemented in VASP. Electronic wavefunctions are expanded in plane waves with a kinetic-energy cutoff of 281 eV, which is at least 1.38 times the default cutoff value for Al, Hf, Nb, Pd and Ru. The US-PPs employed in this work explicitly treat three valence electrons for Al (3s2p1), four valence electrons for Hf (5d36s1), five valence electrons for Nb (4d45s1), 10 valence electrons for Pd (4d95s1) and eight valence electrons for Ru (4d75s1). All calculated results were derived employing the generalized gradient approximation (GGA) for exchange-correlation energy due to Perdew and Wang [42]. Brillouin zone integrations were performed using Monkhorst–Pack [43] k-point meshes, and the Methfessel–Paxton [44] technique with the smearing parameter of 0.1 eV. The B2 and L21 phases considered in this study are listed in Table 1. The k-point meshes used for Brillouin zone integration for total energy calculations were 24 · 24 · 24 and

G. Ghosh, G.B. Olson / Acta Materialia 55 (2007) 3281–3303 Table 1 Crystallographic data of intermetallics used in the ab initio calculations Phase

Pearson symbol

Structurbericht designation

Space group (#)

Prototype

Stable PdAl RuAl RuNb Pd2HfAl Ru2NbAl

cP2 cP2 cP2 cF16 cF16

B2 B2 B2 L21 L21

Pm3m  Pm3m Pm3m Fm3m Fm3m

(2 2 1) (2 2 1) (2 2 1) (2 2 5) (2 2 5)

CsCl CsCl CsCl Cu2AlMn Cu2AlMn

Virtual HfAl NbAl PdHf PdHfAl2 PdHf2Al RuNbAl2 RuNb2Al

cP2 cP2 cP2 cF16 cF16 cF16 cF16

B2 B2 B2 L21 L21 L21 L21

Pm3m Pm3m Pm3m Fm3m Fm3m Fm3m Fm3m

(2 2 1) (2 2 1) (2 2 1) (2 2 5) (2 2 5) (2 2 5) (2 2 5)

CsCl CsCl CsCl Cu2AlMn Cu2AlMn Cu2AlMn Cu2AlMn

12 · 12 · 12 for B2 and L21 phases, respectively. The corresponding k-points in the irreducible Brillouin zone are 364 and 56, respectively. All calculations are performed using the ‘‘accurate’’ setting within VASP to avoid wrap-around errors. With the chosen plane-wave cutoff and k-point sampling, the reported formation energies are estimated to be converged to a precision of better than 2 meV atom1 (0.2 kJ mol1 of atom). We have also calculated the formation energies of pseudobinary alloys of Pd(HfxAl1x) and Ru(NbxAl1x), 0 6 x 6 1, with B2 structure from first principles using a supercell (SC) method. This approach involves calculating total energy of structures where several crystallographically equivalent sites are created by repeating the B2 unit cell along appropriate directions; the energies of compounds with site-substituted species are thus calculated at discrete compositions. Specifically, we have used 32-atom supercells, with 16 sites (equivalent to 1b position in B2 structure) occupied by Pd or Ru atoms and the remaining 16 sites (equivalent to 1a position in B2 structure) occupied by either Al + Hf or Al + Nb atoms. The lattice vectors of the supercell are defined as aSC X ¼ ½2; 2; 0a0 , SC aSC ¼ ½0; 2; 2a and a ¼ ½2; 0; 2a , with a 0 0 0 being the latY Z tice constant of B2 unit cell. We have used a k-point mesh of 7 · 7 · 7 for the total energy calculations using supercells. In addition to total energy calculations, to obtain electronic structures and to understand the bonding characteristics of three B2 (PdAl, RuAl, RuNb) and two Heusler phases (Ru2NbAl, Pd2HfAl) we have also calculated the electronic density of states and charge densities. Such calculations are performed using potentials constructed by the projector-augmented wave (PAW) method [45], which retains the all-electron character but the all-electron wave function is decomposed into a smooth pseudo-wave function and a rapidly varying contribution localized with the core region. In its current implementation in VASP, the PAW method freezes the core orbital to those in a reference configuration, although it is not strictly necessary. In fact,

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very recently a relaxed core PAW method has been proposed [46] that is shown to yield results with an accuracy comparable to the FLAPW method. The PAW method is free of any shape approximation for both charge density and electronic potential. Therefore, PAW potentials are an improvement over Vanderbilt-type US-PP [47], as they combine the elegance and computational efficiency of plane waves with the chemically appealing concept of localized functions. The valence configurations of PAW potentials for Al, Hf and Ru were the same as in US-PP; however, for Nb and Pd the PAW potentials were constructed treating the occupied semicore 4p electronic states as valence states. Once again, the exchange-correlation energy due to Perdew and Wang [42] was used. A kinetic-energy cutoff of 525 eV was used for the expansion of the electronic wavefunctions in plane waves. For the calculation of charge densities, we have used k-point meshes of 35 · 35 · 35 for the B2 phases and 19 · 19 · 19 for the L21 phases. The total energies of the intermetallics listed in Table 1, and also relevant pure elements, are calculated as a function of volume, and the results are fit to the equation of state (EOS) due to Vinet et al. [48]. The zero-temperature equations of state (EOS) define pressure–volume relationships. Vinet et al. [48] assumed that the interatomic interaction-versus-distance relation in solids can be expressed in terms of a relatively few material constants. Specifically, the pressure P is expressed in terms of isothermal bulk modulus (B0), its pressure derivative (B00 ) and a scaled quantity (x): P ¼ 3B0 x2 ð1  xÞ exp½vð1  xÞ 1/3

ð1Þ

3=2ðB00

and v ¼  1), where V0 is the with x = (V/V0) equilibrium volume. Based on Eq. (1) and the relations between pressure and energy, the total energy (E) and volume-dependence of the bulk modulus can be expressed as EðV Þ  EðV 0 Þ ¼

9B0 V 0 f1  ½vð1  xÞ exp½vð1  xÞg; v2

BðV Þ ¼ x2 ½1 þ ðvx þ 1Þð1  xÞ exp½vð1  xÞ: B0

ð2Þ ð3Þ

Fig. 3a and b shows the E–V plots defining zero-temperature EOS parameters for Pd2HfAl and Ru2NbAl, respectively. 3.2. Computational thermodynamics and kinetics The principal integrated computational design tools used are the Thermo-Calc [49] and DICTRA (diffusion controlled transformation) [50] software systems. ThermoCalc employs the calphad [36] approach to predict phase diagrams and phase stability as a function of composition and temperature in multicomponent alloys. Calphad employs a bottom-up approach, whereby the Gibbs free energies of lower-order systems (unary, binary, ternary, etc.) are modeled using analytical functions, which in turn

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volume-fixed frame of reference. Besides single-phase and multi-phase diffusional simulations [50], at Northwestern University the applications of computational kinetics (using DICTRA software) have been extended to include simulation of growth kinetics under kinetically constrained conditions [52,57] during a solid–solid phase transformation, and the dissolution of solid metal in liquid solder during electronic package assembly [58]. A DICTRA simulation requires both thermodynamic and atomic mobility data bases. While Thermo-Calc is used widely, due to the availability of relevant thermochemical databases, the applications of DICTRA are limited by the availability of mobility databases. As a part of our research program, a synergistic approach for thermodynamic and kinetic modeling relevant to the design of Nb-based superalloys. 4. Experimental procedure

Fig. 3. Calculated total energy, at zero-temperature and without zeropoint motion, as a function of volume for (a) L21-Pd2HfAl, and (b) L21Ru2NbAl. The filled circles represent calculated point, and the line is a fit to EOS in Eq. (2).

are used to predict the phase stability and phase diagrams of higher-order systems, where the number of components may be as high as 10. Past experiences have demonstrated that modeling ternary systems usually lead to satisfactory predictions of phase stability and phase diagrams of multicomponent alloys. Besides calculation of multicomponent equilibrium phase diagrams [49], at Northwestern University the applications of computational thermodynamics (using Thermo-Calc software) have been extended to include the prediction of glass forming ability of alloys by rapid solidification [51], the calculation of kinetically constrained solid–solid phase equilibria [52], and modeling and prediction of the kinetics of diffusionless martensitic structural transformations [53–56]. DICTRA is a software package for simulating diffusioncontrolled transformations in multicomponent systems involving multiple phases having simple geometries. DICTRA solves one-dimensional diffusion equations in a

Three prototype alloys with aluminide precipitates are used for microstructural investigation. The nominal compositions (in at.%) of these alloys are: 84Nb–8Al–8Pd, 82Nb–8Al–10Ru and 81.6Nb–6.3Al–2.8Hf–9.3Pd. Another prototype alloy, 45Nb–34Hf–21Al (in at.%), is used for the oxidation study. Elements (all obtained from Alfa Aesar, Ward Hill, MA) of the following purity (in mass%) are used in 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 is made by arc-melting in an inert argon atmosphere. During arc-melting, each alloy button is flipped and re-melted 10 times in order to achieve homogeneity. Samples are cut from the ingots, and then heat-treated at 1000 and 1200 C for 100–500 h. All heat treatments are carried out after vacuum encapsulation of specimens in a quartz tube with 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 from the as-cast alloys, and the flat surfaces are polished to 800 grit. The samples are placed in alumina boats (so as to collect all spalled oxides), which are then placed in a tube furnace at 1300 C for air oxidation. The weight of the boat + sample assembly is recorded before and after oxidation treatment to measure the weight gain due to oxidation. Conventional metallography techniques are employed to reveal the crosssection microstructure of oxidized specimens. Both scanning electron microscopy (SEM) and transmission electron microscopy (TEM) are performed for microstructual characterization at various length scales. A Hitachi 3500 scanning electron microscope is used for microstructural characterization of oxidized specimens. Thin foils for transmission and analytical electron microscopy are prepared by a standard dual jet electropolishing method. An electrolyte containing 5% H2SO4 and 2% HF in methanol is maintained at 60 to 70 C, and the thin foils are prepared by applying a voltage of 80 V and a current of 30–40 mA. Conventional TEM is carried out in a Hitachi 8100 microscope operating at 200 kV. The highresolution electron microscopy (HREM) and analytical

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characterization are performed in a cold field emission gun high-resolution AEM (Hitachi HF-2000) equipped with a Gatan 666 parallel electron energy loss spectrometry detector, an ultrathin window Link EDX detector and data processor (QX2000) and a Gatan CCD camera for HREM imaging. The AEM is operated at 200 kV. The take-off angle for the X-ray detector is 68. The X-ray collection time is 100–200 s, and the electron probe size is about 10 nm. Care is taken to insure that the particle being analyzed is not in a two-beam condition in order to minimize electron-channeling effects. 5. Results 5.1. Ab initio results: pure elements The calculated zero-temperature cohesive properties or pure elements are compared with available experimental data in Table 2. The lattice parameter of Al [59] and Nb [60] are taken from the measured values at 4.2 K. The lattice parameters of Hf [61], Pd [62] and Ru [63,64] at 0 K are obtained by extrapolating the experimental data at higher temperatures to 0 K. We find that, in general, the calculated and experimental (measured or extrapolated) lattice parameters agree within ±1%, except for Pd, where the discrepancy 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 or extrapolated value at 0 K. Our calculated bulk moduli of Hf and Nb agree within ±2% of the experimental value, while the calculated bulk moduli of Pd and Ru are underestimated by about 19%. The experimental data for Al show a large scatter. In Table 2, we also note that the lattice parameters of pure elements calculated using US-PP and PAW are very similar. Experimental data of B00 are available primarily at ambient temperature. Once again, the values of Al [71–73] show some scatter. Other B00 values are taken from Steinberg [74]. In general, the agreement between the calculated and experimental values should be considered good. It has been pointed out [74] that, depending on the measurement technique, ultrasonic resonance, vs. the initial slope of the locus of Hugoniot states in shock-velocity particle-velocity coordinates, the values of B00 may differ even though ideally they should be the same. It is not uncommon that the B00 predicted by ab initio techniques differs from the experimental value by as much as 30%. 5.2. Ab initio results: intermetallics We take the formation energy at zero-temperature and zero-pressure and without zero-point motion (DE/f ) of an intermetallic ApBqCr, where p, q and r are integers, as a key measure of the relative stability. The formation energy per atom is evaluated relative to the composition-averaged energies of the pure elements in their equilibrium crystal structures. For example, the DE/f of a ternary intermetallic is defined as

DE/f ðAp Bq C r Þ ¼

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1 E/ p þ q þ r Ap Bq Cr   p q r EhA þ EuB þ EwC  pþqþr pþqþr pþqþr ð4Þ

where E/Ap Bq Cr is the total energy of intermetallic ApBqCr with structure /, EhA , EuB 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 several studies of phase stability and electronic structure of some of the intermetallics considered here using ab initio methods. The previous studies can be summarized as follows: FLAPW [75], FP-LMTO [76] and full-potential linearized augmented Slater-type orbital [77] (FLASTO) [78] for PdAl; FP-LMTO [79,76], LAPW [80] and FLASTO [78] for RuAl; FLAPW [20] for Pd2HfAl; FLASTO for Ru2NbAl [81]. Except for Nguyen-Manh and Pettifor [76], all the aforementioned studies made use of Hedin–Lundqvist [19] parametrization of LDA. On the other hand, NguyenManh and Pettifor [76] made use of Perdew–Zunger [82] parametrization of LDA. Among the stable intermetallics of interest to us, the heat 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 experimental value [83,84] by about 7–10 kJ mol1 of atom. On the other hand, DEf values from both FLAPW- [75] and FP-LMTO (LDA) [76] calculations agree with the experimental data within a few kJ mol1 of atom. In contrast, our result of DEf for RuAl using US-PP (GGA) and PAW (GGA) agree within a few kJ mol1 of atom of experimental value [85], while DEf by FLASTO (LDA) [78] method shows an excellent agreement. Mehl et al. [80] calculated only the elastic properties of RuAl by ab initio methods and did not report the formation energy. Unlike PdAl and RuAl, in Table 3, we note that the DEf for Pd2HfAl and Ru2NbAl calculated by US-PP (GGA) and various all-electron methods such as PAW (GGA), FLAPW (LDA) [20] and FLASTO (LDA) [81] are very similar; the theoretical result is thus one that is not sensitive to DFT computational procedure. It is important to note that both B2-PdAl and B2-RuNb are high-temperature phases in the respective equilibrium phase diagram [86]; B2-PdAl is stable between 850 and 1645 C and B2-RuNb is stable between 900 and 1942 C. As a result, there is no measured lattice parameter at ambient and/or at a low temperature for these two phases. The zero-temperature lattice parameter of B2-PdAl obtained by linear extrapolation of high-temperature data [87] is listed in Table 3. The lattice parameter of B2-RuNb at 298 K is obtained by linear extrapolation of Nb-rich B2 lattice 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

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

B0 (·1010 N m2)

B00

Ab initio

Expt.

Ab initio

Expt.

Ab initio

Expt.

a = 0.40436a a = 0.40469b

a = 0.40322 [59]

7.42

8.82 [65] 7.94 [66] 8.20 [60]

4.11

4.0 [71] 5.19 [72] 4.42 [73]

HCP (hP2)

a = 0.31804a c = 0.50208 a = 0.31976b c = 0.50488

a = 0.31930 [61] c = 0.50395

11.03

11.06 [67]

3.43

3.95, 4.28 [74]

Nb

BCC (cI2)

a = 0.32923a a = 0.33229b

a = 0.33026 [60]

17.05

17.33 [68]

3.58

4.06, 3.80 [74]

Pd

FCC (cF4)

a = 0.39617a a = 0.39575b

a = 0.38780 [62]

16.43

19.55 [69]

5.34

5.42, 5.20 [74]

Ru

HCP (hP2)

a = 0.27335a c = 0.43055 a = 0.27295b c = 0.43021

a = 0.27030 [63,64] c = 0.42719

29.88

35.26 [70]

4.24

–

Element

Structure

Al

FCC (cF4)

Hf

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 calculated lattice parameter agrees within ±1% except for B2-PdAl, where the discrepancy is about 2.7%. As for the elastic properties, the experimental bulk modulus is available only for RuAl at ambient temperature [93]. As seen in Table 3, our calculated value of B0 agrees with the experimental data to within 5%. In addition to the thermodynamic stability of the stable aluminides in Table 1, we also investigated their mechanical stability with respect to tetragonal deformation by applying an appropriate strain of up to 5%. We found that only B2-RuNb is mechanically unstable, as the tetragonal shear modulus (C 0 = (C11  C12)/2) is found to be negative. This is consistent with the experimental observation that B2-RuNb undergoes a martensitic transformation spontaneously upon cooling from high temperature to about 900 C to form a tetragonal phase [94,95], which in turn undergoes further structural transformations at lower temperatures. Unlike B2-RuNb, we found that B2-PdAl is mechanically stable at 0 K even though it is stable above 850 C and undergoes complex (yet unknown) ordering at lower temperatures, as indicated in the equilibrium phase diagram [86]. Among the virtual phases, the mechanical stability of B2-PdHf was also checked with respect to tetragonal 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 characterization of following prototype alloys: 84Nb–8Al–8Pd (in at.%) 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 arcmelting. In all three alloys, based on the morphology and distribution of the second phase, they are clearly the result of solid-state precipitation by nucleation and growth processes from supersaturated solid solutions. Fig. 4a and b shows the bright-field (BF) and dark-field (DF) micrographs, but different areas, of 84Nb–8Al–8Pd alloy. The DF micrograph is taken using {1 0 0}B2-type superlattice reflection. The precipitates have diameters in the range of 100–200 nm, but the presence of interfacial dislocations implies the semi-coherent nature of the matrix/precipitate interface. Fig. 4c and d shows the EDS spectra obtained from the matrix and the precipitate, respectively. The matrix EDS spectrum indicates small amounts of dissolved Al and Pd in (Nb). The precipitate EDS spectrum indicates a small amount of Nb dissolved in PdAl. This result is consistent with the experimental isothermal section at 1100 C [13] that also indicates a few percent of Nb solubility in PdAl. Fig. 5a and b shows the BF and the corresponding DF micrographs of 81.6Nb–6.3Al–2.8Hf–9.3Pd alloy, the latter taken using f1 1 1gL21 -type superlattice reflection. Precipitates with diameters in the range of 20–70 nm may be seen, and all appear to be fully coherent with the matrix. Fig. 5c and d shows the EDS spectra obtained from the matrix and the precipitate, respectively. Like the Nb–Pd– Al alloy above, once again the matrix EDS spectrum indicates small amounts of dissolved Al and Pd in (Nb) and a negligible amount of Hf. On the other hand, the precipitate EDS spectrum indicates a substantial amount of Nb dissolved in Pd2HfAl. An important point to note is that in the matrix spectrum (Fig. 5c) the peak height of Nb-L is higher than Nb-Ka; however, in the precipitate spectrum (Fig. 5d) the peak height of Nb-L is much smaller than

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

Stable PdAl

EOS parameters V0

B0

B00

0.014682

15.14

4.77

0.014055g

20.1g

Lattice const. (nm)

DEf, (kJ mol1 of atom)

Ab initio

Expt.

Ab initio

Expt.

0.30851a 0.30799d 0.30200f

0.30167b

83.31a 84.55d 90.57f 93.58g 87.68h

93.6c 90.6 ± 2.2e

6.3% 7.3%

0.30300i

65.29a 64.35d 74.93k 58.15g

62.05 ± 1.65j

8.6% 9.4%

0.30196h RuAl

0.013635

0.013095g

19.91 22.0k 22.3g 23.0m 20.8n

4.30

4.5m

0.30099a 0.30086d 0.30200k

0.30360l

Misfit (da)

0.29600m 0.30300o 0.29635h a

p

61.67h

RuNb

0.015921

23.01

4.38

0.31696 0.31857d

0.31500

15.72a 14.68d

–

3.7% 4.1%

Pd2HfAl

0.016536

15.48

4.60

0.64197a 0.64239d 0.63500r

0.63670q

79.36a 80.59d 79.94r

–

2.5% 3.3%

a

6.1% 6.6%

0.63680s

Ru2NbAl

0.014772

22.44

4.32

0.61828 0.62042d 0.61254u

0.61350t

63.96a 62.25d 63.0u

– –

Virtual HfAl NbAl PdHf PdHfAl2 PdHf2Al RuNbAl2 RuNb2Al

0.018687 0.016765 0.018234 0.016808 0.018675 0.015196 0.016305

10.92 14.17 15.30 11.53 12.94 16.31 18.10

3.33 4.09 6.46 4.32 4.07 4.23 4.17

0.33434a 0.32246a 0.33162a 0.64547a 0.66854a 0.62414a 0.63897a

– – – – – – –

20.13a 2.71a 59.36a 38.02a 34.60a 22.57a 5.32a

– – – – – – –

The units of V0 and B0 are nm3 atom1 and 1010 N m2, respectively. The lattice misfit (da) is based on our calculated lattice parameter of pure Nb and the intermetallic 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 fluorescence effects. Fig. 6a and b shows the BF and the corresponding DF micrographs of 82Nb–8Al–10Ru alloy, the latter taken using f1 1 1gL21 -type superlattice reflection. Precipitates

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

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Fig. 4. Transmission electron micrographs showing the two-phase microstructure of a prototype alloy 84Nb–8Al–8Pd (in at.%) heat treated at 1000 C for 500 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-PdAl interface. 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 commonly observed during sintering of powders. Fig. 6c and d shows the EDS spectra obtained from the matrix and the precipitate, respectively. The matrix EDS spectrum indicates small amounts of dissolved Al and Ru in (Nb). On the other hand, the precipitate EDS spectrum indicates a substantial amount of Nb dissolved in Pd2HfAl. Once again, like the Nb–Pd–Hf–Al case above, in the matrix spectrum (Fig. 6c) the peak height of Nb-L is higher than Nb-Ka; however, in the precipitate spectrum (Fig. 6d) the peak height of Nb-L is smaller than Nb-Ka. Once again, this is most likely to have originated from absorption and/or fluorescence effects. Transmission electron diffraction patterns are used to evaluate the lattice mismatch, da (= (aprecipitate  amatrix)/ amatrix), between the matrix and precipitate. The diffraction patterns shown in Fig. 7a–c represent [1 1 0]bcc, [1 1 0]bcc and [1 0 0]bcc zone axes taken from 84Nb–8Al–8Pd, 81.6Nb– 6.3Al–2.8Hf–9.3Pd and 82Nb–8Al–10Ru alloy, respectively. In the first two cases, we have used (1 1 0)A2/B2 (or ð2 2 0ÞL21 ), (002)A2/B2 (or ð0 0 4ÞL21 ) and (112)A2/B2 (or ð2 2 4ÞL21 ), while in the third case we have used (0 2 0)A2 (or ð0 4 0ÞL21 ), (0 0 2)A2 (or ð0 0 4ÞL21 ) and (0 2 2)A2 (or ð0 4 4ÞL21 ) reflections to estimate a mean value of da. These reflections are labelled and clearly distinguishable in the

respective diffraction pattern, as shown in Fig. 7a–c. The measured values of da are 8.6%, 3.9% and 6%, respectively, in the above three alloys. These values compare favorably with the calculated values of 6.3%, 2.5% and 6.1% in the corresponding two-phase systems (see Table 3). It should be noted that the da values listed in Table 3 are based on the lattice parameters of pure Nb and stoichiometric intermetallic calculated at 0 K. However, the measured values may be influenced by several intrinsic and extrinsic factors, such as (i) the ambient temperature effect, (ii) the dissolved solutes in (Nb), (iii) the deviation from ideal stoichiometry of the intermetallics and (iv) possibly some instrumental/measurement artifacts due to the finite size of the diffraction spots. Taking these factors into account, the observed agreement should be considered as good. 5.4. Computational thermodynamics: modeling and calculation of the Al–Hf–Nb phase diagram Here, we demonstrate the modeling of phase stability in Al–Hf–Nb, which represents an important ternary subsystem in one of the prototype alloys discussed above. Since the experimental phase diagram and thermodynamic data for this system were very limited, ab initio

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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 at 1000 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 are the 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 of the databases required to employ computational thermodynamics methods, based on the calphad approach [36], as a predictive framework for guiding the alloy design process. In this work, the application of the methods is facilitated by employing the Alloy Theoretic Automated Toolkit [96,97] for calculating the heat of mixing of solid solutions by cluster expansion method [98] and also the vibrational entropy of formation of several relevant intermetallics. In the calphad method, the parameters in model free energy functions are generally fit to available experimental data for measured phase boundaries and thermodynamic properties. However, in the absence of sufficient measured data to uniquely fit the model parameters, ab initio methods can be employed as a ‘‘virtual calorimeter’’ to augment thermodynamic databases with computed values of enthalpies and entropies of formation for solid solutions and intermetallic compounds. The details of such an integrated approach have been discussed elsewhere [99]. In the following, we describe the procedure very briefly. In the calphad modelling of Al–Nb intermetallics, a twosublattice model for Al3Nb and AlNb3 and a three-sublattice model for the r phase were used. An advantage provided by integrating ab initio methods with calphad is that 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-sublattice description of the r phase, (Al,Nb)10(Nb)4(Al,Nb)16, values are required for the formation free energies of the three virtual phases Al26Nb4, Al16Nb14 and Nb30 having the structure of the r phase. Similarly, a two-sublattice description of the A15 phase, (Al,Nb)(Al,Nb)3, gives rise to three virtual A15 phases, Al4, Al3Nb and Nb4. For the construction of the calphad free-energy functions we have calculated the energy of formation of all virtual phases in the Al–Nb system. Similarly, in the Al–Hf system we calculated the heats of formation of stable and several virtual Al–Hf intermetallics [100]. For calphad modelling of phase diagrams, both enthalpies and entropies of formation of phases are required. The calculation of vibrational entropies of formation is in general computationally demanding. In particular, the Al3Hf2, Al2Hf3 and r phases have orthorhombic and tetragonal crystal structures, respectively, with 40, 20 and 30 atoms per unit cell. In such cases, we have estimated the entropy of formation using a correlation between intermetallic enthalpies, and entropies of formation were parametrized from extensive ab initio calculations involving relatively simpler phases [99]. The integration ab initio phase stability of the Hf–Nb system within calphad formalism has been discussed elsewhere [101]. In modeling the Al–Hf–Nb phase diagram, all Al–Hf intermetallics were treated as stoichiometric binary phases,

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Fig. 6. Transmission electron micrographs showing two-phase microstructure of a prototype alloy 82Nb–8Al–10Ru (in at.%) heat treated at 1200 C for 100 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 single exception of Al3Hf. Since both Al3Nb and Al3Hf have the DO22 structure at high temperature, they were treated as one phase, i.e. Al3(Hf,Nb), allowing random mixing of Nb and Hf on one sublattice. The sublattice models for AlNb3 (A15) and AlNb2 (r) were extended to include Hf, 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 virtual phases, the energy parameters of which were all calculated by ab initio methods. Modeling of ternary solid-solution phases was facilitated by cluster-expansion calculations of the thermodynamic properties face-centered cubic (fcc) and hexagonal close-packed (hcp) solid solution in the Hf–Nb and the Al–Nb system, respectively. In a more traditional calphad approach, relying on measured data entirely, such data would not be accessible due to the fact that fcc and hcp solidsolutions phases are absent in the respective equilibrium phase diagrams. No ternary interaction was introduced for the 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 of Nb-base superalloy, it turns out to be a very important prediction as the solubility of Al in (Nb) can be increased significantly by adding Hf. For example, in the binary Al–Nb system (Nb) 8 at.% Al is dissolved at 1300 C. However,

our calculated results show that by adding 30 at.% Hf the solubility of Al can be increased by more than a factor of two, to 17 at.%. As will be shown below, based on the results of kinetics simulations that were later verified experimentally [1], this increase in Al solubility proves to be highly beneficial for significantly reducing oxygen transport kinetics in the (Nb) solid-solution phase. Another important feature in Fig. 8 is the presence of (Nb) + r phase field in the ternary regime, which is absent in the binary Al–Nb system. Usually the formation of r phase is detrimental to the mechanical properties; therefore the calculated phase diagram helps greatly in the alloy design/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 Al needed to form a self-protective Al3O3 scale can be miniðSÞ mized by minimizing the oxygen solubility (N O ) in the matrix and by minimizing the ratio of oxygen diffusivity (DO) to the aluminum diffusivity (DAl) in the matrix. Therefore, it is desirable to add elements that would increase the solubility of Al in (Nb). As shown above, Hf is one of them, and the available experimental phase diagrams [102] show that Cr, Ti and V also exert a similar effect.

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Fig. 7. Transmission electron diffraction patterns used to estimate the lattice mismatch in prototype alloys: (a) along the [1 1 0] zone axis in 84Nb–8Al–8Pd (in at.%) heat treated at 1000 C for 500 h, (b) along the [1 1 0] 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 [1 0 0] 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.

1.0

Hf (a α -Hf)

0.9 0.8 f

0.7

Mo le

0.6 AlHf

0.5

Al3Hf2

0.4 Al2Hf

0.3

Al3Hf

0.2 0.1 0 0 Al

bb 22 NlN AAl f+f+ lHlH +AA ff 2+2 l H3H AAl 3

Fra cti

on H

AlHf2

(Nb)

Al3Hf2+AlNb2+ Al3(Hf,Nb)

Liquid+ Al3(Hf,Nb)

0.2

sσ Al3Nb

0.4 0.6 AlNb2 0.8 AlNb3 1.0 Nb Mole Fraction Nb

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

Based on our calculated isothermal section at 1300 C shown in Fig. 8, 45Nb–34Hf–21Al (in at.%) was chosen for the oxidation study. Static oxidation was carried out at 1300 C using several specimens, for up to 50 h. The results are summarized in Fig. 9. The weight gain behavior clearly exhibits a parabolic kinetics, and the oxide scale thickness is also found to exhibit a parabolic kinetics [1]. The oxidized specimens were characterized by a combination of X-ray diffraction and SEM to ascertain the nature of the oxide scale(s). Fig. 10 shows the SEM micrographs of the oxide scale formed on the alloy after 5 and 25 h of oxidation. In both cases, the oxide scale was adherent, and did not spall off after cooling. As seen in Fig. 10, the oxide scale has two distinct layers: (i) the outermost layer is a transient Nb2O5 phase with some dissolved Al and Hf; and (ii) underneath the Nb2O5 layer is a two-phase oxide layer consisting of NbAlO4, and HfO2 with dissolved Al and Nb. The transient nature of the outermost Nb2O5 layer is characterized by its nearly constant thickness even after varying exposure times. The mixed oxide layer NbAlO4 and HfO2 exhibits parabolic

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1300 C

Wt Gain (mg/cm2)

150

100

50

0 0

10

20 30 40 Time of Oxidation (hrs)

50

60

Fig. 9. Static oxidation behavior of a 45Nb–34Hf–21Al (in at.%) alloy at 1300 C [1].

growth, implying that the oxidation kinetics is a diffusion controlled 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 successful design of Nb-base superalloys for 1300 C application is

to improve its intrinsic oxidation resistance significantly. This requires the formation of a self-protective external oxide scale, and a decrease in oxygen solubility and oxygen transport kinetics, both by several orders in magnitude compared with pure Nb. Within the framework of computational thermodynamics and kinetics, we demonstrate that Al–Hf–Nb solid solutions help achieve these goals. The modelling and simulation of oxygen transport was carried out using the DICTRA package [50]. DICTRA uses Thermo-Calc to calculate the thermodynamic factor of the phases to convert mobility into diffusivity and also to compute the local equilibrium between the phases. In other words, to use DICTRA successfully a complete thermodynamic description of the participating phase(s) is needed first, and then the kinetic description of the corresponding phase(s). In parallel to the development of thermodynamic database described above, a multicomponent mobility database relevant to Nb-base alloys was also developed [1]. Fig. 11 compares the simulated oxygen profiles in pure Nb and two Nb-base alloys at 1300 C, clearly demonstrating the effect of alloy chemistry on the intrinsic transport kinetics of oxygen in solid solution. An assumption made in these simulations is that the oxygen content at the surface of the bulk alloy is 5 at.%, which is based on the Nb–O phase diagram. This is equivalent to the assumption

Fig. 10. SEM micrographs of oxide layer formed on alloy 45Nb–34Hf–21Al (in at.%) at 1300 C: (a) for 5 h, showing the complete oxide layer and interaction zone at a low magnification; (b) a higher magnification of (a), showing the transient outermost layer and the steady-state layer underneath; (c) for 25 h at low magnification; and (d) a higher magnification of (c) [1].

G. Ghosh, G.B. Olson / Acta Materialia 55 (2007) 3281–3303 5.0

Pure Nb Pure Nb 92Nb8Al Alloy 92Nb8Al Alloy 45Nb34Hf21Al Alloy 45Nb34Hf21Al Alloy

4.5

Atomic Percent Oxygen

4.0 3.5 3.0 2.5

1h 1h

2.0

5h 5h

1h 5h 1h 5h

1.5 1.0

1 5h

0.5 0

0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Distance, mm Fig. 11. Intrinsic oxygen transport kinetics in pure Nb and Nb-base alloys at 1300 C, demonstrating the effect of alloy chemistry.

that, irrespective of the type of oxide, the oxygen potential at the oxide/base metal interface is constant. The simulated oxygen penetration depths after 1 and 5 h at 1300 C are 3.21 and 6.01 mm in pure Nb, 1.26 and 2.51 mm in 92Nb8Al (at.%) alloy, and 0.14 and 0.35 mm in 45Nb34Hf21Al (at.%) alloy. The values for Nb–Al and Nb–Al–Hf alloys are in excellent agreement with the experimental data as verified by the hardness profile and also by direct observation of microstructures [1]. For example, in the ternary alloy the measured oxygen penetration depths were 0.13 and 0.34 mm after 1 and 25 h exposure, respectively, at 1300 C. Compared to pure Nb, the oxygen penetration depth is reduced by a factor of more than 20 in Nb–Al–Hf alloy. Mechanistically, this is most likely due to strong binding energies of Al–O and Hf–O, and possibly Hf–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 ordering energy at Pd2HfAl and Ru2NbAl compositions from ab initio calculations. We are motivated by the fact that L21Ru2NbAl is more stable than the pseuodobinary alloys of Ru(NbxAl1x) with B2 structure. This is implied by the presence of two two-phase fields, RuAl + Ru2NbAl and Ru2NbAl + RuNb, in the isothermal section of Ru–Nb–Al at 1100 C [13]. Once again, it should be noted that both B2RuAl and B2-RuNb are stable phases at 1100 C in the respective binary phase diagram [86]. However, in the case of PdAl–PdHf pseudobinary alloys, B2-PdAl is the stable phase while B2-PdHf is a virtual phase. In an L21 structure of composition A2BC, atoms A, B and C occupy three distinct sublattices, designated as I, II and III. In a pseuodobin-

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ary alloy of A2BC with B2 structure, atoms of A occupy sublattice I, while atoms B and C remain disordered on sublattices II and III. In a bcc solid solution, atoms A, B and C remain disordered on all three sublattices. To estimate the energy associated with congruent ordering (B2 ! L21) at A2BC, it is necessary to model compositional disorder in A(BxC1x) alloys of with B2 structure. Within the ab initio framework, three approaches may be employed to investigate the effect of compositional disorder on the formation energy, namely (i) the conventional supercell method; (ii) the SQS (special quasi-random structure) supercell method [103]; and (iii) the sublattice cluster expansion method [104] that has been widely used to model anion and/or cation disorder in ceramic systems [105]. As mentioned in Section 3.1, we have employed the conventional supercell method only. Specifically, we have calculated the formation energy of Pd(HfxAl1x) and Ru(NbxAl1x) 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. It is important to note that, for a particular composition, there are many ways to arrange Al + Hf or Al + Nb atoms on 16 sites in 32-atom supercells. Due to limited computational resources, the total energies of all possible arrangements could not be performed. Instead, at a particular composition we have calculated the total energies for 2–6 different arrangements (chosen arbitrarily). Then, the mixing energy, as an example for Pd(HfxAl1x), is defined as PdðHf x Al1x Þ DEmix ¼ DEf  ½ð1  xÞDEPdAl þ xDEPdHf . f f The results of ab initio calculations are summarized in Tables 4 and 5. In the supercell method, total energies are calculated allowing volume relaxation only, and also allowing full (volume, shape, ionic) relaxation. The mean value of mixing energies at each composition (see Tables 4 and 5) is plotted in Fig. 12. In both Pd(HfxAl1x) and Ru(NbxAl1x) alloys the difference between volume relaxed and fully relaxed DEmix is only a few kJ mol1 of atom, with the latter being more negative as expected. In the case of Pd(HfxAl1x) alloys the DEmix is weakly positive, while it is moderately negative in Ru(NbxAl1x) alloys. We have noted that the magnitude of relaxations (volume, shape, ionic) is larger in Pd(HfxAl1x) than in Ru(NbxAl1x); this is believed to be due to the larger size mismatch between B2-PdAl and B2-PdHf than between B2-RuAl and B2RuNb (see Table 3). This, along with a small number of configurations used for total energy calculations at a particular composition, has contributed to the oscillatory behavior, though within 5 kJ mol1 of atom, of DEmix of Pd(HfxAl1x) alloys. Nonetheless, as indicated, Fig. 12a and b defines the energy for B2 ! L21 congruent ordering, which is about 8 and 4.5 kJ mol1 of atom at Pd2HfAl and Ru2NbAl compositions, respectively. 6.2. Electronic structure and bonding mechanism in B2 and L21 phases Large negative formation energies of the stable phases in Table 1 are indicative of strong bonding tendencies

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Table 4 Calculated formation (DEf) and mixing (DEmix) energies (in kJ mol1 of atom) of Pd(HfxAl1x) alloys with B2 structure

Table 5 Calculated formation (DEf) and mixing (DEmix) energies (in kJ mol1 of atom) of Ru(NbxAl1x) alloys with B2 structure

x

x

DEf

DEmix

Volume relaxed

Fully relaxed

Volume relaxed

Fully relaxed

0.125

78.273 79.178 77.945 (78.465)

78.938 79.583 78.292 (78.938)

2.045 1.139 2.373 (1.852)

1.380 0.918 2.026 (1.441)

0.250

77.358 74.718 (76.038)

78.051 75.412 (76.732)

0.038 2.606 (1.284)

0.375

70.383 71.269 70.903 (70.852)

70.845 72.454 72.762 (72.021)

0.500

65.585 79.412 68.871 74.276 69.824 67.704 (70.945)

0.625

DEf

DEmix

Volume relaxed

Fully relaxed

0.125

65.089 65.793 65.012 (65.298)

65.677 66.091 65.089 (65.619)

5.992 6.690 5.919 (6.202)

6.584 6.999 5.997 (6.527)

0.727 1.913 (0.593)

0.250

65.844 65.419 (65.632)

66.306 65.931 (66.119)

12.947 12.523 (12.735)

13.409 13.034 (13.222)

3.948 3.061 3.428 (3.479)

3.486 1.876 1.568 (2.310)

0.375

62.715 63.496 62.321 (62.844)

63.053 63.862 63.545 (63.487)

16.015 16.795 15.620 (16.143)

16.352 17.162 16.824 (16.780)

66.009 79.363 70.663 74.931 71.048 69.786 (71.967)

5.752 8.074 2.467 2.939 1.512 3.632 (0.392)

5.328 8.026 0.674 3.594 0.289 1.551 (0.629)

0.500

57.313 63.730 54.760 60.493 57.862 56.330 (58.415)

57.987 63.730 57.602 61.321 58.276 57.188 (59.351)

16.809 23.226 14.255 19.988 17.358 15.826 (17.911)

17.483 23.226 17.098 20.817 17.772 16.683 (18.846)

63.331 64.381 64.352 (64.021)

64.381 68.698 64.496 (65.858)

2.045 1.139 2.373 (4.322)

1.380 0.918 2.026 (2.485)

0.625

48.326 47.758 47.324 (47.803)

48.721 48.682 48.355 (48.586)

14.018 13.449 13.016 (13.494)

14.413 14.374 14.047 (14.278)

0.750

63.938 63.988 61.144 (63.023)

66.444 66.521 65.384 (66.116)

1.412 1.363 4.206 (2.327)

1.093 1.171 0.034 (0.766)

0.750

38.453 37.017 (37.735)

40.091 38.954 (39.523)

10.341 8.905 (9.623)

11.979 10.842 (11.411)

0.875 0.875

60.663 60.307 60.123 (60.364)

61.694 61.219 60.991 (61.301)

1.693 2.050 2.233 (1.992)

0.662 1.067 1.375 (1.035)

26.440 26.624 26.335 (26.466)

28.319 28.108 27.462 (27.963)

4.525 4.708 4.419 (4.551)

6.404 6.192 5.546 (6.047)

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

between the constituent atoms. Also, as discussed above, there is a driving energy for B2 ! L21 ordering at Pd2HfAl and Ru2NbAl compositions. To obtain further insight into the nature of the bonding, we present the electronic densities of states (DOS) and bonding charge densities of three B2 (PdAl, RuAl, RuNb) and two L21 (Pd2HfAl and Ru2NbAl) phases. In the following, we present both total and partial DOSs, where the latter quantities were computed using projections into site-centered atomic spheres, each with radii equal to half the nearest-neighbor spacing. We also present the bonding charge density (sometimes called the deformation charge density), defined as the difference (point-by-point using identical FFT grid) between the self-consistent charge density qð! x Þ of the intermetallic and a reference charge density constructed from the superposition of non-interacting atomic charge density at the crystal sites. For example, the bonding charge density of

Volume relaxed

Fully relaxed

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

B2-PdAl is defined as dq = q(PdAl)  q(Al atom)  q(Pd atom). Similarly, the bonding charge density of L21-Ru2NbAl is defined as dq = q(Ru2NbAl)  q(Al atoms)  q(Nb atoms)  q(Ru atoms). Fig. 13a and b shows the calculated the electronic DOS and the bonding charge-densities of B2-PdAl, respectively. In Fig. 13a, different Y-scales between s, p and d components of the DOS may be noted. The total DOS is in agreement with the previous report [75,78]. The Fermi level lies in the pseudogap minimum separating the bonding and anti-bonding states. It is seen that below the Fermi level pronounced peaks of Pd-d strongly hybridize with Al-p. The Al-s contribution becomes visible at about 5 eV below the Fermi level, and a corresponding Pd-d peak may also be seen. Fig. 13b shows the bonding charge density plot in the (1 1 0) plane where Al and Pd are the nearest neighbors. Delocalization of the bonding charge density in the

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a

b

Fig. 12. Calculated mixing energy of (a) PdHfxAl1x and (b) RuNbxAl1x alloys with B2 structure. These also define the B2 ! L21 ordering energy at Pd2HfAl and Ru2NbAl composition, respectively.

interstitial region is seen. There is also a directional buildup of bonding charge, with a maximum halfway between and Al and Pd atoms, which, according to Fu [75], may not be the covalent type. These features, along with the angular momentum-resolved DOS, imply that the bonding mechanism may be described as Al-sp–Pd-d hybridization. Fu [75] proposed that the bonding character in B2-PdAl is a combination of metallic bonding and charge-transfer components. In Fig. 13b, it is interesting to note that the bonding charge density (dq) isocontour lobes are oriented perpendicular to the Al–Pd bonding direction. Fig. 14a and b shows the calculated the electronic DOS and the bonding charge-densities of L21-Pd2HfAl, respectively. Once again, in Fig. 14a, different Y-scales between s, p and d components of the DOS may be noted. The Fermi level lies about 2 eV to the right of the pseudogap minimum, i.e. the bonding states are completely occupied. As seen in Fig. 14a, the bonding states are dominated by Hf-d and Pd-d, while the anti-bonding states are domi-

Fig. 13. Electronic structure of B2-PdAl calculated using all-electron PAW potentials: (a) angular momentum and site decomposed electronic density of states, n(E), with the Fermi level marked by a dotted line; (b) the distribution of bonding (or deformation) charge density in the (1 1 0) plane, with selected contour lines drawn at a constant interval of ˚ 3. In the color scale bar, the bonding (or deformation) charge 0.003 e A density ranges from 0.116 (depleted region: dq()) to 0.032 (enhanced ˚ 3. region: dq(+)) e A

nated by Hf-d. Below the Fermi level, pronounced peaks of Hf-d and Pd-d strongly hybridize, while both of them also hybridize with Al-p in the entire energy region. Unlike B2-PdAl, a strong Al-s peak is promoted in this structure at about 6.6 eV below the Fermi level which hybridizes with 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 Fermi level. Fig. 14b shows the bonding charge density plot in the (1 1 0) plane. A significant redistribution of bonding charge in the interstitial region is seen. The build-up of bonding charge along the Al–Hf and Al–Pd bond directions may only be described as moderate; however, it is very strong along the Hf–Pd bond direction. We also notice a maximum in bonding charge density along the direction of Hf–Pd bonding, but its distribution is skewed towards the Pd site and the dq isocontour lobes are oriented perpen-

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a

a

b b

Fig. 14. Electronic structure of L21-Pd2HfAl calculated using all-electron PAW potentials: (a) angular momentum and site decomposed electronic density of states, n(E), with the Fermi level marked by a dotted line; (b) the distribution of bonding (or deformation) charge density in the (1 1 0) plane, with selected contour lines drawn at a constant interval of ˚ 3. In the color scale bar, the bonding (or deformation) charge 0.006 e A density ranges from 0.164 (depleted region: dq()) to 0.065 (enhanced ˚ 3. region: dq(+)) e A

dicular to the Hf–Pd bonding direction. The angular momentum-resolved DOS in conjunction with the bonding charge density plot implies that the bonding mechanism consists of short-range band mixing between the d states of Hf and Pd, and also sp states of Al and d states of Hf and Pd along with the long-range charge transfer (electrostatic) effect. Fig. 15a and b shows the calculated the electronic DOS and the bonding charge densities of B2-RuAl, respectively. Here, the Fermi level lies about 1 eV to the left of pseudogap minimum, i.e. the bonding states are incompletely occupied. The partial DOS in Fig. 15a provide direct evidence that hybridization is present between the Ru-d and Al-p states, which reflects a strong directional bonding between the Ru and Al atoms. The Al-s states become significant only at about 5 eV below the Fermi level. The total DOS at the Fermi level is rather high and, due to insufficient number of valence electrons, a lot of bonding states

Fig. 15. Electronic structure of B2-RuAl calculated using all-electron PAW potentials: (a) angular momentum and site decomposed electronic density of states, n(E), with the Fermi level marked by a dotted line; (b) the distribution of bonding (or deformation) charge density in the (1 1 0) plane, with selected contour lines drawn at a constant interval of ˚ 3. In the color scale bar, the bonding (or deformation) charge 0.0046 e A density ranges from 0.223 (depleted region: dq()) to 0.047 (enhanced ˚ 3. region: dq(+)) e A

are unoccupied. All features of DOS in Fig. 15a are in very good agreement with the previous results calculated by LMTO–LDA [17]. Fig. 15b shows the bonding charge density plot in the (1 1 0) plane. While there is delocalization of bonding charge in the interstitial region that resembles metallic bonding, there is also evidence of directionality of electron density along Æ1 1 1æ direction which may contribute to the covalent character. The distribution of bonding charge density is clearly asymmetric, being skewed towards the Ru site. The presence of a covalent character is further supported by the fact that the energy interval for bonding is much larger than for anti-bonding, as seen in Fig. 15a. Unlike B2-PdAl and L21-Pd2HfAl, it is seen that many dq contour lobes are oriented along the Al–Ru bonding direction. Combining angular momentum-resolved DOS and the bonding charge density plot, it may be concluded that the mechanism of cohesion is dominated by the

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short-range band mixing between Al-sp and Ru-d states along with the long-range charge transfer (electrostatic) effect. Fig. 16a and b shows the calculated the electronic DOS and the bonding charge-densities of B2-RuNb, respectively. Like B2-RuAl, a pseudogap separating bonding and anti-bonding states in total DOS is seen, but at about 1 eV below the Fermi level. The anti-bonding states are dominated by both Ru-d and Nb-d states. It is seen that, at about 1.7 eV below the Fermi level, pronounced peaks of Ru-d and strongly hybridize with Nb-d. The sp states of both Nb and Ru also exhibit hybridization, but their contributions are much smaller compared with the d–d contribution. Fig. 16b shows the bonding charge density plot in (1 1 0) plane. It is seen that there is significant delocalization of bonding charge in the entire interstitial region, suggestive of metallic bonding. Meanwhile, the nature of

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band mixing between the spd states of Nb and Ru also implies some covalent character. In fact, nonspherical anisotropy of bonding charge density is observed along the direction between Ru and Nb atoms. Like B2-RuAl in Fig. 15b, the distribution of bonding charge density in B2-RuNb also shows asymmetry, being skewed towards the Ru site. Once again, combining angular momentumresolved DOS and the bonding charge density plot, it may be concluded that the bonding mechanism is primarily dominated by the short-range band mixing involving d states of Ru and Nb. Fig. 17a and b shows the calculated the electronic DOS and the bonding charge-densities of L21-Ru2NbAl, respectively. Like B2-RuAl and B2-RuNb, a pronounced deep or a pseudogap separating bonding and anti-bonding states in total DOS is seen, but here the Fermi level lies in the pseudogap. It is interesting to note that RuAl has 5.5 valence

a a

b

Fig. 16. Electronic structure of B2-NbRu calculated using all-electron PAW potentials: (a) angular momentum and site decomposed electronic density of states, n(E), with the Fermi level marked by a dotted line; (b) the distribution of bonding (or deformation) charge density in the (1 1 0) plane, with selected contour lines drawn at a constant interval of ˚ 3. In the color scale bar, the bonding (or deformation) charge 0.0059 e A density ranges from 0.237 (depleted region: dq()) to 0.059 (enhanced ˚ 3. region: dq(+)) e A

b

Fig. 17. Electronic structure of L21-Ru2NbAl calculated using all-electron PAW potentials: (a) angular momentum and site decomposed electronic density of states, n(E), with the Fermi level marked by a dotted line; (b) the distribution of bonding (or deformation) charge density in the (1 1 0) plane, with selected contour lines drawn at a constant interval of ˚ 3. In the color scale bar, the bonding (or deformation) charge 0.01 e A density ranges from 0.289 (depleted region: dq()) to 0.1 (enhanced ˚ 3. region: dq(+)) e A

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electrons per atom with the pseudogap 1 eV above the Fermi level, and RuNb has 6.5 valence electrons per atom with the pseudogap 1 eV below the Fermi level; therefore, it is not surprising that Ru2NbAl with 6 valence electrons per atom has the Fermi level in the pseudogap. The total DOS in Fig. 17a is in very good agreement with the previous results of Weinert and Watson [106]. A very low density of states at Fermi level implies that L21-Ru2NbAl is a very stable compound. These results (total DOS vis-a´-vis DEf of B2-RuAl, B2-RuNb and L21-Ru2NbAl) are consistent with the idea that the most stable phase is the one which optimizes the filling of bonding states (increasing cohesion) within a rigid band model. As seen in Fig. 17a, the antibonding states are dominated by Ru-d and Nb-d. Below the Fermi level, the hybridization is dominated by Al-p, Nb-d and Ru-d. Although Al-p does not make a significant contribution, it is found to participate in the hybridization in the entire energy region. Like L21-Pd2HfAl, we note that a strong Al-s peak is promoted in this structure at about 6.6 eV below the Fermi level which hybridizes with Ru-d. Fig. 17b shows the bonding charge density plot in the (1 1 0) plane. It shows that a depletion of electron density at the Al and Nb sites is accompanied by a build-up of charge density around the Ru sites. The build-up of bonding charge along the Al–Nb and Al–Ru bond directions may only be described as moderate; however, along the Nb–Ru bond direction the build-up is rather strong. Like B2-RuAl in Fig. 15b and B2-RuNb in Fig. 16b, the distribution of bonding charge density in L21-Ru2NbAl also shows asymmetry, being skewed towards the Ru atom. We also notice that the dq isocontour lobes are oriented along the Nb–Ru bonding direction. Like B2-RuAl, in Fig. 17a we see that the energy interval for bonding is much larger than for anti-bonding, suggestive of the presence of a covalent character associated with the bonding between the atoms. Consistent with the partial DOS in Fig. 17b, the bonding between Ru and Nb dominates over both Ru–Al and Nb–Al bonding. Based on these results, it is concluded that the bonding mechanism consists of the short-range band mixing between the d states of Nb and Ru, and also sp states of Al and d states of Nb and Ru, and the longrange charge transfer (electrostatic) effect. 6.3. Integrated materials design: current trend and limitations A major focus of our current research has been an optimal and efficient integration of modern computational and experimental tools to facilitate quantitative evaluation of processing–microstructure–property–performance links and rapid prototyping. The present study demonstrates the efficacy of this approach by modeling Nb-based superalloy as an integrated microstructure. This representation, shown in Fig. 2, is then used to identify and prioritize the key process–structure and structure–property links to be quantified as a part of the design exercise. The multilevel structure and hierarchy of computational models demands

that a hierarchy of experimental tools be employed to create the requisite databases, which are not yet available, and to validate the predictions of the modeling. Due to conflicting nature of the effect of alloying elements on many of the relevant properties, as discussed in Section 2, the optimization of this materials system can only be achieved by the method of systems design employing different computational tools. The potential economic impact of this approach is significant as it can reduce the time to invent, produce and test a new alloy. Obviously there is a clear economic driving force for adopting such an integrated systems approach. Specific to the conceived Nb-based superalloy, while the total system integrates compatible thermal barrier and bond coat subsystems, here we have presented results on predictive design of the underlying precipitation strengthened alloy by integrating the results of ab initio calculations, computational thermodynamics and kinetics. Following the initial computational modeling and design exercise, selected experiments are performed to validate the precipitation strengthened microstructures and oxidation behavior of prototype alloys. Notwithstanding tremendous advances in computer hardware, algorithm, software, alloy theory and user-interface, in terms of both availability and affordability, it is important to underscore the limitations of ab initio methods in the design of new engineering alloys. Here, we will mention only a few important ones. First, due to the multicomponent (where the number of components may be as high as 10) and multiphase nature of engineering alloys, the direct application of first-principles methods (i.e. where atomic number and atomic arrangement are the only input parameters) to the direct modeling of phase stability is intractable. A convenient way to overcome this drawback is to employ calphad-based approaches of computational thermodynamic and kinetics where multicomponent, multiphase equilibria and diffusion problems can be solved rather rapidly requiring minimal computational resources. Second, it may not be possible to calculate all relevant properties directly by ab initio methods. Instead, the physical parameters/properties calculable by ab initio methods may be correlated with the relevant engineering properties. For example, it is not possible to calculate the rafting behavior, the directional coarsening of the precipitates into plates or rafts under applied load and the coarsening rate of precipitates directly by ab initio methods even in relatively simple alloys. However, it is possible to calculate the physical parameters/properties that control them, such as the lattice misfit, precipitate/matrix interfacial energy, solute diffusivity and solubility. These parameters may then be correlated with the creep property of an alloy. Similarly, it is not possible to calculate the DBTT of an alloy directly by ab initio methods. However, it is possible to calculate the surface and unstable stacking fault energies, the ratio of which has been shown to correlate well with the propensity for brittle fracture and thus may be used as a criterion for DBTT. Third, sometimes the physical laws that govern one or more properties of an alloy may not be available at

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the time of alloy design. Consequently, new physical law(s) need to be established prior to a decision being made as to what physical parameters/property should be calculated by ab initio methods. Successful application of computational thermodynamic and kinetics tools in materials design also suffer from limitations. In recent years, these tools based on the calphad framework have become widely used as the basis for modeling phase stability and phase transformation kinetics in complex alloy systems; however, it is important to recognize that the accuracy of predictions derived from these tools depends critically on the accuracy of relevant databases. This in turn depends on the accuracy of experimental data and the assessment methods that define the accuracy of thermodynamic and kinetics model parameters. Furthermore, for new, relatively unexplored systems, design and modeling efforts are often hindered by the need for extensive experimental measurements needed in the development of robust thermodynamic and kinetic databases. Some of these drawbacks can be overcome by performing ab initio calculations, thus significantly limiting the extent of costly experimental measurements required in thermodynamic and kinetics database development, but may require significant computational resources.

(iv)

(v)

(vi)

(vii)

7. Conclusions An optimal integration of ab initio total energy and alloy theory, and selective experimental determination of high temperature phase relations allows accelerated development of multicomponent thermodynamic and kinetic databases for computational materials design. In this context, the principles of integrated design of Nb-based superalloys are discussed. The conceptual design of Nb-based superalloys exploits precipitation strengthening by aluminide phase(s), having nearest-neighbor (B2) and next-nearest-neighbor (L21) ordered structures based on a bcc lattice, that also offer a high potential for combining oxidation resistance and creep strength for application at 1300 C and above. The following conclusions are drawn: (i) We have used US-PP–GGA as implemented in VASP to determine the zero-temperature equation of state, electronic density of states, and charge densities of selected aluminides with B2 and L21 structures relevant to the design of precipitation strengthened Nbbased superalloys. (ii) We find that, in general, the calculated zero-temperature lattice parameters of Al, Hf, Nb, Pd and Ru agree within ±1%, and the calculated B0 agrees within ±5% when the calculated are compared with the corresponding experimental values (either measured or extrapolated). (iii) The calculated zero-temperature formation energy of B2-PdAl is underestimated by about 7–10 kJ mol1 of atom, while that of B2-RuAl agrees within a few kJ mol1 of atom when compared with the calori-

(viii)

(xi)

3301

metric data. The zero-temperature formation energy of L21-Pd2HfAl and L21-Ru2NbAl calculated by US-PP–GGA (VASP), PAW–GGA (VASP), FLAPW–LDA and FLASTO–LDA give almost identical results. For the stable intermetallics considered, the zero-temperature lattice parameter agrees within ±1%, except for B2-PdAl, when compared the experimental data (either measured or extrapolated) at low temperatures. The calculated bulk modulus of B2-RuAl is found to agree with 5% of the experimental value at ambient temperature. The B2 ! L21 ordering energy at Pd2HfAl and Ru2NbAl compositions estimated from ab initio calculations to be about 8 and 4.5 kJ mol1 of atom, respectively. The phase stability and bonding mechanism in relevant B2 (PdAl, RuAl, RuNb) and L21 (Pd2HfAl, Ru2NbAl) phases are discussed in terms of their electronic density of states and bonding (or deformation) charge density. The following two-phase microstructures are observed in prototype alloys: (Nb) + B2-PdAl, (Nb) + L21Pd2HfAl and (Nb) + L21-Ru2NbAl. The structure and composition (qualitative) of matrix and precipitates are confirmed by transmission electron diffraction, and the corresponding EDS spectra obtained in a high-resolution analytical electron microscope, respectively. Integration of ab initio alloy phase stability and computational thermodynamics predict that the solubility of Al in (Nb) at 1300 C is significantly increased by adding Hf. This is shown to be highly beneficial for improving oxidation resistance of Nb-based alloys. Guided by the results of computational thermodynamics, the oxidation study of a prototype alloy of 45Nb–34Hf–21Al (in at.%) at 1300 C was carried out. Both weight gain and thickening kinetics of the oxide layer exhibited the parabolic behavior, and the layer was found to be a mixture of the NbAlO4 and HfO2 phases.

Acknowledgments This work was sponsored by the Air Force Office of Scientific Research, USAF, under Grant No. F4962001-1-0529. Computational resources, the Itanium clusters as a part of the teragrid facility at the University of Illinois at Urbana-Champaign and at the San Diego Supercomputing Center, provided by NPACI (National Partnership for Advanced Computational Infrastructure), are gratefully acknowledged. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the Air Force Office of Scientific Research or the U.S. Government.

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