Journal of Nuclear Materials 419 (2011) 177–185

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Journal of Nuclear Materials

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The U–Ti system: Strengths and weaknesses of the CALPHAD method

Saurabh Bajaj a, Alexander Landa b, Per Söderlind b, Patrice E.A. Turchi b, Raymundo Arróyave a,c,⇑a Department of Mechanical Engineering, Texas A&M University, College Station, TX 77843-3123, United Statesb Lawrence Livermore National Laboratory, Livermore, CA 94551-0808, United Statesc Materials Science and Engineering Program, Texas A&M University, College Station, TX 77843-3123, United States

a r t i c l e i n f o a b s t r a c t

Article history:Received 29 April 2011Accepted 29 August 2011Available online 12 September 2011

0022-3115/$ - see front matter � 2011 Elsevier B.V. Adoi:10.1016/j.jnucmat.2011.08.050

⇑ Corresponding author at: 119 ENPH, Mail Stop 3College Station, TX 77843-3123, United States. Tel.: +845 3081.

E-mail address: rarroyave@tamu.edu (R. Arróyave)

Input from Density Functional Theory (DFT) calculations is used to understand phase equilibria in abinary metallic alloy fuel system: U–Ti. The CALPHAD approach is employed to calculate a U–Ti phasediagram that is consistent not only with experimental data but also—more importantly—with thermody-namic data from DFT calculations: heat of formation of c(bcc)-U–Ti alloys as a function of composition,and formation enthalpy of the d-U2Ti compound. Three DFT-based electronic structure methods are uti-lized: SR-KKR-ASA-CPA, SR-EMTO-CPA, and FPLMTO-SQS, and the use of derived ab initio data avoids themanifestation of unreasonable or inaccurate phase stabilities that result from an otherwise uncon-strained Gibbs energy minimization within the CALPHAD approach. We also investigate phase formationof the d-U2Ti phase in the U–Ti system, that stabilizes in the same C32 structure as other binary metallicfuel alloys such as U–Zr and Np–Zr.

� 2011 Elsevier B.V. All rights reserved.

1. Introduction

This work is part of our continuing effort to develop a thermo-dynamic database for candidate metallic fuel alloys that couldpotentially be utilized in Gen-IV type fast breeder reactors. Thesereactors aim to achieve high burn-up rates (upwards of 20–50%)that lead to fissioning and transmutation of transuranic elementssuch as Np, Pu, Am, and Cm, thus recycling spent fuel and closingthe fuel cycle [1]. Our focus in this work is to determine phaseequilibria in the U–Ti system by adopting an ab initio aidedCALPHAD approach that we previously employed in the analysisof the Np–Zr system [2]. A recently published thermodynamicmodel of the U–Ti system [3] shows good agreement with experi-mental information. However, as it will be shown later in thiswork, this model results in unreasonable ‘‘hidden’’ phase stabilitiesthat are manifested in metastable phase diagram calculations. As aword of caution to CALPHAD practitioners, it must be stressed herethat the process of parameter optimization and error minimizationinvolved in the CALPHAD method is an inverse problem withinfinite degrees of freedom. Thus, many possible combinations ofvalues of user-defined parameters could produce phase boundariesand diagrams that overlap each other. The use of ab initio data inCALPHAD modeling imposes a restriction in this optimizationprocess, thus avoiding the appearance of ‘‘artifacts’’. Such amethodology (which is now being used by CALPHAD practitioners

ll rights reserved.

123, Texas A&M University,1 979 845 5416; fax: +1 979

.

especially in cases when experimental information is sparse, suchas in the present system) also provides a fundamental insight intophase behavior. In particular, we use ab initio calculations to inves-tigate the thermodynamic properties of the high-temperature bccphase and the d-U2Ti intermetallic compound, which constitutethe primary defining characteristics of the U–Ti phase diagram.

The study of phase equilibria in the U–Ti system is particularlyinteresting from the point of view of its comparison to phase equi-libria in the U–Zr system. These systems are isoelectronic, i.e. Tiand Zr have the same number of valence electrons (electronic con-figurations: Ti–[Ar]3d24s2, Zr–[Kr]4d25s2), and when alloyed withU, it is expected that the phase stability in these two alloy systemsshould be somewhat similar. However, atomic size difference, andoccupation of additional higher states in Zr cause a few majorchanges between these two phase diagrams: firstly, the suppres-sion of a stable miscibility gap in bcc U–Zr alloys to a low-temper-ature metastable bcc miscibility gap in the U–Ti system, secondly,a shift in stoichiometry of the d-phase from UZr2 to U2Ti, and lastly,a change from solid solution behavior of d-UZr2 to a stoichiometricd-U2Ti compound.

The purpose of this work is three fold: (i) to present a self-consistent thermodynamic description of the U–Ti system byincorporating reliable experimental and ab initio data, (ii) providea physical understanding of the phase stability of this systemthrough the use of ab initio methods, and (iii) show that CALPHADdescriptions with insufficient data may lead to artifacts whenGibbs energy functions are extrapolated to conditions far fromthose used during optimization. Experimental information avail-able on the U–Ti phase diagram is discussed in Section 2. TheCALculation of PHAse Diagrams (CALPHAD) method and phase

178 S. Bajaj et al. / Journal of Nuclear Materials 419 (2011) 177–185

models employed to develop the thermodynamic model are de-scribed in Section 3. Ab initio calculations in this work are per-formed by employing three state-of-the-art computationaltechniques: (i) the scalar-relativistic (SR) Green’s function tech-nique based on the Korringa–Kohn–Rostoker (KKR) method withinthe Atomic Sphere Approximation (ASA), (ii) the scalar-relativisticexact muffin-tin orbital (EMTO) method, and (iii) the all-electronfull potential linear muffin-tin orbital method (FPLMTO) that ac-counts for all relativistic effects. Details of these calculations arementioned in Section 4. The optimization process followed forthe user-defined parameters in the thermodynamic models is de-scribed in Section 5. Results obtained using these methodologiesare presented and discussed in Section 6. Finally, concluding re-marks are presented in Section 7.

2. Experimental data

2.1. Phase diagram

The most comprehensive experimental data on the U–Ti phasediagram are discussed here. Other information found for this sys-tem will be mentioned in successive sections as and when needed.

Buzzard et al. [4] analyzed alloys in the U-rich region (70 at.% U)of the phase diagram by means of thermal, microscopy, and X-rayanalysis. However, as the melting procedures that were utilized ledto metal contamination and loss that increased in severity with anincrease in content of Ti, several of the results obtained in thiswork differ significantly from those of other published articles[5,6] on this system. Some of which include: a steep rise in liquidusupon adding Ti to U, and the presence of a new ‘‘delta’’ (d) phase(not to be confused with U2Ti that is called the ‘‘epsilon’’ (�) phasein this work, i.e. Ref. [4]) for which the stability region expandedwith an increase in concentration of Ti, thus explaining its origindue to impurities [7].

Udy and Boulger [5] fabricated and heat treated 22 U–Ti alloysamples of different compositions, and subjected them to metal-lographic and X-ray examination. It was confirmed in this workthat the ‘‘delta’’ phase observed in Ref. [4] was an artifact causedby high levels of contamination. However, the lack of identifica-tion of b-U in this study is merely because no samples wereprepared with compositions in this region of the phase diagram.The liquidus and solidus determined are in good agreement withRef. [8].

Knapton [6] also investigated the U–Ti phase diagram using 25alloy samples at well-spaced compositions. The uranium firstused was in the form of bars which contained several oxide par-ticles. However, coarse powder uranium, that was considerablycleaner than the bars, was later used for analysis on the U-richside. As no major contaminations were observed in the samplesafter treatment, the phase diagram predicted in this work isbelieved to be most reliable, and data points of equilibrium linesin this article are used for phase diagram assessment in thepresent work.

2.2. d-U2Ti

It was found in Refs. [5,9] that d-U2Ti decomposes at 1163 K to asolid solution containing bcc (c)-U, and bcc (b)-Ti. This phase waspredicted to be a disordered C32 structure (space group:P6/mmm)with a 10 at.% solubility. However, in Ref. [6], it was confirmed thatthis phase stabilizes as a stoichiometric compound rather than asolid solution, and it is also speculated that metastability may bethe reason causing the non-stoichiometry in Ref. [5]. The c

a ratioof this C32 structure is confirmed to be equal to 0.589 in Refs.[10,11].

3. CALPHAD models

To investigate global and metastable phase equilibria in theU–Ti system, the CALPHAD method [12] is utilized to develop an‘‘ab initio consistent’’ thermodynamic model. The Gibbs energiesG/

m of each phase / taking part in the equilibria are defined usingthe models described below. In these expressions, xi is the molarfraction of component i and, oG/

i is the Gibbs energy of the phasecontaining only the pure component i that is obtained from theScientific Group Thermodata Europe (SGTE) database [13]. Thisdatabase contains various types of information developed using amixture of experimental data, and phenomenology. The use ofCALPHAD methodology is particularly appropriate to examinethermodynamic properties as: (i) extrapolation of phase equilibriato regions in the temperature-composition space that are difficultto access using experiments is made possible (Note: in this work,CALPHAD calculations are performed at atmospheric pressure:101.325 kPa), (ii) metastable equilibria can be investigated, (iii)analysis may be preformed to determine sensitivities of variousparameters that are critical to the phase diagram, (iv) thermody-namic information may be provided to experiments for verificationpurposes as, for this class of materials, they are costly, and special-ized training is required for handling radioactive materials, and (v)CALPHAD (or similar mean-field) models are necessary to modelthe range of solution phases, particularly when attempting toextrapolate to higher order systems.

3.1. Random solution

A random solution model is one where components randomlyoccupy any of the spatial positions/sites in the lattice. In this work,the liquid, bcc, hcp, orthorhombic, and tetragonal phases are mod-eled with this assumption. The Gibbs energy of such a solutionphase is given by:

G/m ¼

X

i¼U;Ti

xioG/

i þ RTX

i¼U;Ti

xilogexi þ xUxTiL/U;Ti ð1Þ

where L/U;Ti is an interaction parameter that includes the effect of

non-ideality. It is further expanded using the Redlich–Kister formal-ism [14]:

L/U;Ti ¼

X

v

vL/U;TiðxU � xTiÞv ð2Þ

When v = 0, the phase is a regular solution, and when v > 1, thephase is a sub-regular solution. To include temperature dependencyin this parameter, it is further expanded as:

vL/U;Ti ¼

vAþ vBT ð3Þ

where vA and vB are model parameters that need to be optimized.

3.2. Sublattice model

In this model, preferential, rather than random, occupation ofspecific lattice sites is the manner in which components are dis-tributed within the structure. The phase model can be visualizedas containing two inter-locking sublattices. The Gibbs energyexpression for this model is described using [12]:

G/m ¼

X

I0

PI0ðYÞoG/I0 þ RT

X

s

NsX

i

ysi logeys

i þX

Z>0

X

IZ

PIZðYÞL/IZ ð4Þ

where PI0 is a product of sublattice site fractions when each of themis occupied by only one component, Ns is the number of sites onsublattice s, PIZ is also a product of sublattice site fractions but whenone or more sublattices contains Z components and the remaining

S. Bajaj et al. / Journal of Nuclear Materials 419 (2011) 177–185 179

are occupied by one component. ysi is called the site fraction defined

by:

ysi ¼

nsi

Ns ð5Þ

where nsi is the number of atoms of component i on sublattice s.

When Z = 0, the phase is a regular solution in which mixing on eachsublattice is independent of site occupations in other sublattices.However, in sub-regular solutions, this is not the case. The inter-dependence of sublattice site fractions is then included in the inter-action parameters of Eq. (4) above as:

L/a;b:c ¼ y1

ay1by1

c

X

v

vL/a;b:c y1

a � y1b

� �v ð6Þ

where there are four different combinations of mixing possibilitiesin this two-sublattice system (a,b)1(c,d)1, one of which is shownabove.

As determined from experimental measurements in Ref. [6],and from first-principles calculations discussed later in Section6.2, the d-U2Ti phase is a stoichiometric compound. On the otherhand, experiments suggest that the d-phases in the U–Zr [15]and Np–Zr [16] systems show significant degrees of non-stoichiometry. Calculations [2,17] by some of the authors of thepresent paper indicate that configurational disorder is energeti-cally favored in the U–Zr and Np–Zr systems, confirming the exper-imental observations. Thus, in order to be consistent with thesublattice solution models used for similar d-C32 phases in theseother systems, and to be able to easily incorporate solution behav-ior when extrapolating to higher-order multi-component systems,the d-U2Ti phase is modeled as a special case of the sublatticemodel (instead of a stoichiometric line compound) in which thecompositions on both sublattices of the C32 structure are fixed,i.e., the first sublattice containing two sites is assumed to only beoccupied by U atoms, and the second sublattice containing one siteby Ti atoms. This distribution thus results in a fixed stoichiometriccomposition of 66.67 at.% U, and 33.33 at.% Ti.

4. First-principles techniques

The calculations we have referred to as SR-KKR-ASA are per-formed using the SR Green function technique based on the KKRmethod within the ASA [18–21]. For the present study this approx-imation is improved by including higher multipoles of the chargedensity [20] and the so-called muffin-tin correction [21] to theelectrostatic energy. The calculations are performed for the basisset including valence spdf orbitals, and the semi-core 6p statesfor uranium, where the core states are recalculated at every itera-tion (soft-core approximation). For the electron exchange and cor-relation energy functional, the generalized gradient approximation(GGA) [22] is considered. Integration over the Brillouin zone is per-formed using the special k-point technique [23] with 285 points inthe irreducible wedge of the Brillouin zone for the bcc structures ofc-U, b-Ti, and c-U–Ti alloys. The moments of the density of states,needed for the kinetic energy and valence charge density, are cal-culated by integrating the Green function over a complex energycontour (with a 3.0 Ry diameter) using a Gaussian integration tech-nique with 40 points on a semi-circle enclosing the occupiedstates.

To treat compositional disorder the SR-KKR-ASA method iscombined with the Coherent Potential Approximation (CPA) [24].The ground state properties of the chemically random U–Ti alloysare obtained from SR-KKR-ASA-CPA calculations that include theCoulomb screening potential and energy [25–27]. The screeningconstants are determined from supercell calculations using the lo-cally self-consistent Green function (LSGF) method [28] for a 1024atoms supercell that models the random equiatomic alloy. The a

and b screening constants (see Refs. [25,26] for details) are foundto be 0.7583 and 1.1099, respectively, for U–Ti alloys.

Though the SR-KKR-ASA-CPA formalism is well suited to treatclose-packed structure, it would produce a significant error whenbeing applied to less compact structures (such as the C32 structureof d-U2Ti). That is why we use another Green function technique,based on the SR-EMTO formalism, that is not limited by the geo-metrical restrictions imposed by the ASA.

The SR-EMTO calculations are performed using the scalar-rela-tivistic Green function technique based on the improved screenedKKR method, where the one-electron potential is represented byoptimized overlapping muffin-tin (OOMT) potential spheres[29,30]. The potential is spherically symmetric inside the spheres,and is constant between the spheres. Radii of these potentialspheres, and constant value in the interstitial region are deter-mined by minimizing (i) deviation between the exact and overlap-ping potentials and (ii) errors coming from the overlap betweenspheres. Within the SR-EMTO formalism, one-electron states arecalculated exactly for the OOMT potentials. As an output ofSR-EMTO calculations, one can determine a self-consistent Greenfunction for the system, and the complete non-sphericallysymmetric charged density. Finally, the total energy is calculatedusing the full charge-density technique [31]. Like in the case ofSR-KKR-ASA calculations, the GGA is used for the electronexchange-correlation approximation, and SR-EMTO is combinedwith the CPA for calculation of the total energy of chemicallyrandom alloys [32]. Integrations over the Brillouin zone and com-plex energy contour, and the choice of screening constants areidentical to those in the SR-KKR-ASA-CPA method. The numberof k-points in the irreducible wedge of the Brillouin zone used inSR-EMTO-CPA calculations are 285, 350, 576, and 81 for the bcc,orthorhombic, hcp, and C32 structures, respectively.

For elemental metals, the most accurate calculations areperformed using a full-potential (no geometrical approximations)approach. These are fully-relativistic in the sense that spin-orbitinteraction is accounted for through the conventional perturbativescheme [33] that has the accuracy of solving the Dirac equation forlight actinides [34]. Although unable to model disorder in the CPAsense, it provides important information for the metals, and alsoserves to confirm the CPA calculations mentioned above. For thispurpose, we use a version of FPLMTO [35], where full-potential(FP) in FPLMTO refers to the use of non-spherical contributionsto the electron charge density and potential. This is accomplishedby expanding charge density and potential in cubic harmonics in-side non-overlapping muffin-tin spheres, and in a Fourier series inthe interstitial region. We use two energy tails associated witheach basis orbital. For U, semi-core 6s, 6p states, and valence states(7s, 7p, 6d, and 5f) pairs are different. With this ‘‘double basis’’approach we use a total of six energy tail parameters, and a totalof 12 basis functions per atom. Spherical harmonic expansionsare carried out up to lmax = 6 for the basis, potential, and chargedensity. As in the case of SR-EMTO method, the GGA is used forthe electron exchange-correlation approximation. Finally, a specialquasi-random structure (SQS) method, utilizing a 16-atom super-cell, was used to treat compositional disorder within the FPLMTOformalism [36].

Equilibrium values of Wigner–Seitz (W–S) radii, atomic vol-umes, energies, etc., are determined by fitting to the Murnaghanequation of state [37].

5. Parameter optimization

After defining the Gibbs energy expressions for all the phases inthe system, the next step in the thermodynamic assessment in-volves the evaluation and optimization of user-defined parameters.

180 S. Bajaj et al. / Journal of Nuclear Materials 419 (2011) 177–185

For this, the PARROT module of Thermo-Calc software [38] is uti-lized. These parameters are optimized against input experimentaldata on phase boundaries, invariant points, etc., and results fromDFT calculations that are discussed in subsequent sections (seeSection 6.3). In the present work, the parameter optimization pro-cess was carried out by introducing one phase at a time, in the or-der: bcc, d-U2Ti, hcp, liquid, tetragonal, and orthorhombic. Thisorder was chosen according to the conceived importance of phasesin defining the major thermodynamical and phase stability charac-teristics of the phase diagram. Such a procedure ensures the avoid-ance of unwanted artifacts in phase equilibria (such as miscibilitygaps) especially at high-temperatures up to 4000 K. Another con-straint used for this purpose was the driving force that is definedas an affinity between reacting chemical species. Specifically, thedriving force for formation of the bcc phase was set to be negativein the high temperature region of the phase diagram. Conversely,the driving force for the formation of other phases (in the lowtemperature region) was set to be positive whenever experimentalevidence suggested their presence.

The data from Ref. [6], as it is the most conclusive informationavailable on the U–Ti phase diagram, is used for the optimizationof bcc-d, bcc-hcp, liquidus, bcc-tetragonal, and tetragonal-orthorhombic equilibrium lines, as well as for the invariantreactions. Appropriate statistical weight values were assigned toequilibrium points according to their perceived importance andreliability. During the initial assessment, the temperature indepen-dent parts of the interactions parameters (see Eq. (3)) for the bccphase were kept fixed at values that corresponded to SR-KKR-ASA-CPA results in Fig. 1. Finally, after assessing each phase,these parameters were allowed to ‘‘relax’’ together with all otherparameters. Also, the interaction parameters of the tetragonaland orthorhombic phases were first chosen, and then optimizedto obtain solubilities that conformed with results in Ref. [6].

6. Results and discussion

6.1. Ground-state properties of c (bcc) U1�x–Tix alloys

DFT calculations were carried out for c-U1�x–Tix alloys to deter-mine equilibrium values of Wigner–Seitz radius, and energies atselected alloy compositions. The heat of formation of these alloyswere then calculated with respect to the ground states of bcc (c)-U and bcc (b)-Ti, and are plotted in Fig. 1 for all three methods uti-lized as a function of composition. It is interesting to note that the

0 0.2 0.4 0.6 0.8 1

Mole Fraction of Ti

-6

-4

-2

0

2

4

6

Hea

t of

form

atio

n (k

J/m

ol-a

tom

)

SR-EMTO-CPASR-KKR-ASA-CPAFPLMTO-SQS

U Ti

BCC

Fig. 1. Formation enthalpy of c(bcc)-U–Ti alloys with respect to bcc-U and bcc-Ti asa function of composition. Full lines that join the points serve only as guides to theeye.

SR-KKR-ASA-CPA calculations suggest a qualitative change in thenature of the bcc phase solution behavior: below x (Ti) � 0.6, thesolution is endothermic (phase-separating), while it is exothermicat higher Ti composition. This result is corroborated by FPLMTO-SQS calculations that show an increase in the negativity of heatof formation for c-U0.25–Ti0.75 alloy in comparison with c-U0.50–Ti0.50 alloy. Note, however, that the SR-EMTO-CPA calculations re-veal the endothermic nature of c-U1�x–Tix solutions over the entirecomposition range.

Equilibrium atomic volumes of c-U–Ti alloys is plotted in Fig. 2as a function of composition within SR-KKR-ASA-CPA and SR-EMTO-CPA methods. A change from positive to negative deviationfrom Vegard’s law in the case of SR-KKR-ASA-CPA calculations isconsistent with formation enthalpy data from Fig. 1. Conversely,SR-EMTO-CPA calculations result in only a positive deviation fromVegard’s law for c-U–Ti alloys within the whole composition rangethat is consistent with the endothermic behavior of these alloysshown in Fig. 1. Agreement between the results of Figs. 1 and 2suggests that the electronic structure calculations are wellconverged.

6.2. Ground-state properties of d-U2Ti

To determine the equilibrium ca ratio of d-U2Ti, first, using the

experimental value of ca = 0.589, the W–S radius was optimized to

0.164 nm using the SR-EMTO-CPA formulation. Then, using this va-lue for W–S radius, a sweep of c

a ratios was performed. This proce-dure was repeated until the difference in energy was less than0.3 kJ/mole, until finally resulting in c

a = 0.60, and W–S radiusrU2Ti-C32

opt = 0.165 nm.The primitive cell of the C32 structure is described by three

atomic positions: (0,0,0), 23 ;

13 ;

12

� �, and 1

3 ;23 ;

12

� �. The occupation

of these three sites by 2 U atoms and 1 Ti atom can be accom-plished by three possible atomic configurations as shown by Landaet al. [17]: (i) ‘‘random ordering’’ for which U and Ti atoms arerandomly distributed over the three sites, hence correspondingto a U2

3Ti1

3disordered alloy, (ii) ‘‘complete ordering’’ in which Ti

atoms occupy the (0,0,0) position, and U atoms occupy the23 ;

13 ;

12

� �; 1

3 ;23 ;

12

� �positions, and (iii) ‘‘partial ordering’’ configura-

tion, where U atoms occupy the (0,0,0) position, but a randommixture of U and Ti atoms occupies the 2

3 ;13 ;

12

� �; 1

3 ;23 ;

12

� �posi-

tions. SR-EMTO-CPA calculations were performed for all threeconfigurations, and Fig. 3 displays the total energies obtained asa function of W–S radius. It is found that the ‘‘complete ordering’’configuration is the lowest energy configuration by �50 J/mol

0 0.2 0.4 0.6 0.8 1

Mole Fraction of Ti

0.017

0.018

0.019

0.02

0.021

Ato

mic

vol

ume

(nm

3 )

SR-EMTO-CPASR-KKR-ASA-CPA

U Ti

BCC

Fig. 2. Equilibrium atomic volumes of c(bcc)-U–Ti alloys as a function of compo-sition. Dotted lines represent atomic volumes that follow Vegard’s law. Full linesthat join the points serve only as guides to the eye.

0.1600 0.1625 0.1650 0.1675

Wigner-Seitz radius (nm)

0

0.02

0.04

0.06

0.08

Ene

rgy

(kJ/

mol

-ato

m)

Partial orderingComplete orderingRandom ordering

δ-U2Ti (C32)

Fig. 3. Total energy of d-U2Ti as a function of Wigner–Seitz radius in three possibleatomic configurations from SR-EMTO-CPA calculations. Reference point is taken atthe minimum energy point rU2 Ti-C32

opt = 0.165 nm of the ‘‘complete’’ ordering config-uration as determined from the optimization process. Full lines that join the pointsserve only as guides to the eye.

-0.4 -0.3 -0.2 -0.1 0.0 E

F

0.1

Energy (Ry)

0

10

20

30

40

50

60

Tot

al D

OS

(sta

tes/

Ry.

atom

)

A2 (bcc)δ-C32

U2Ti

Fig. 4. Total electronic DOS for U2Ti in the bcc-A2 and hcp-C32 structures. Fermienergy, EF = 0 Ry.

-0.4 -0.3 -0.2 -0.1 0.0 E

F

0.1

Energy (Ry)

0

10

20

30

40

50

60

DO

S (s

tate

s/R

y.at

om)

U ordered (C32)U disordered (A2)Ti ordered (C32)Ti disordered (A2)Total (C32)Total (A2)

U2Ti

Fig. 5. Total, and partial electronic DOS for U and Ti in the bcc-A2 and hcp-C32structures. Fermi energy, EF = 0 Ry.

S. Bajaj et al. / Journal of Nuclear Materials 419 (2011) 177–185 181

atom, that is corroborated by X-ray diffraction experiments in Ref.[10]. This result confirms the measurements made in Ref. [6] thatpoint to d-U2Ti in being a stoichiometric compound. The agree-ment between experiments and calculations illustrates the useful-ness of these electronic structure calculation techniques todetermine the likely atomic configuration in intermetallic phases.In contrast to the behavior of the d phase in the U–Ti system, thed-UZr2 phase actually has a wide range of non-stoichiometryaccording to experiments [15]. This was confirmed in Ref. [17],where it was found that the ‘‘partial ordering’’ configuration is infact the lowest energy configuration. The different stoichiometriesas well as the degrees of stoichiometries in the d phase for the iso-electronic U–Ti and U–Zr systems seems to suggest an importantdifference in the contribution of 3d vs 4d electrons, respectively,to the stabilization of the C32 structure. It is also worth noting thatZr has a larger atomic size than U and the reverse is true for Ti.Hence the variation in electronic character combined with the dif-ference in atomic sizes provides an explanation for the formationof the d phase at high (low) U composition in U–Ti (U–Zr).

The formation enthalpy of this compound is then found to beequal to �38.806 kJ/mol atom with respect to the bcc-U andbcc-Ti phases, and �33.884 kJ/mol atom with respect to theroom-temperature structures ort-U and hcp-Ti from SR-EMTO-CPA calculations. FPLMTO calculations resulted in a formationenthalpy for d-U2Ti of �25.865 kJ/mol atom (using the relaxed va-lue for c

a) relative to bcc-U and bcc-Ti. It must also be mentionedhere that VASP calculations, reported in Ref. [3], also found a veryhigh negative value (�26.160 kJ/mol atom) for the formationenthalpy of d-U2Ti compound, although, with respect to bcc-Uand hcp-Ti. For a comparison with SR-EMTO-CPA results, similarcalculations from VASP (using GGA, and non-spin polarized)resulted in E (bcc-Ti) � E (hcp-Ti) = 9.323 kJ/mol atom, resultingin a formation enthalpy of �35.483 kJ/mol atom referenced tobcc-U and bcc-Ti. Using experimental lattice parameters of thed-U2Ti C32 structure, a = 0.483 nm, c = 0.285 nm, and the expres-sion that relates volume of a structure to the total atomic volume,i.e.,

ffiffi3p

2 a2c ¼ N � 43 � p � r3, where N = 3 is the number of atoms

in a C32 unit cell, and r is the W-S radius, we obtain an experimentalequivalent of W–S radius, rU2Ti-C32

exp = 0.166 nm that is in excellentagreement with our optimized value of 0.165 nm.

In Fig. 4, we plotted the total electronic density of states (DOS)for both the disordered bcc-A2 structure (with composition x(Ti) � 0.33, and at an optimized volume), and the ordered C32

structure. It is observed that there is a significant drop in the totalDOS in the vicinity, as well as at the Fermi level (EF), signaling highstability of the d-U2Ti–C32 compound in comparison with theU2Ti-bcc structure, and a tendency toward ordering from a bcc toa C32 structure.

In Fig. 5, the partial contributions of U and Ti to the totalelectronic DOS are respectively plotted and also show a localizeddrop in DOS near the Fermi energy. To determine which type ofelectrons play a major role in the stability of these structures, inFig. 6a and b the contributions to the partial DOS for both U andTi, respectively, are displayed. As expected, f-electrons in U, andd-electrons in Ti play a major role in the formation of U2Ti with aC32 structure. Due to negligible contributions, s- and p- electronsare not shown in these plots.

6.3. Thermodynamic model

As mentioned in Sections 1 and 5, results from the above DFTcalculations were entered in the optimization process to obtain athermodynamically-consistent model. This data includes heat offormation of bcc alloys as a function of composition, and formationenthalpy of d-U2Ti. For the former, SR-KKR-ASA-CPA results werechosen as, from Fig. 1, they seem to be closely matched byFPLMTO-SQS calculations. In the case of the latter, the value of

-0.4 -0.3 -0.2 -0.1 0.0 E

F

0.10

10

20

30

40

50

Con

trib

utio

ns to

DO

S (s

tate

s/R

y.at

om) (a)d-ordered (C32)

f-ordered (C32)d-disordered (A2)f-disordered (A2)

U

-0.4 -0.3 -0.2 -0.1 0.0 E

F

0.1

Energy (Ry)

0

2

4

6

8

Con

trib

utio

ns to

DO

S (s

tate

s/R

y.at

om) (b)d-ordered (C32)

f-ordered (C32)d-disordered (A2)f-disordered (A2)

Ti

Fig. 6. Contributions to partial electronic DOS for (a) U and, (b) Ti in the bcc-A2 andhcp-C32 structures. Fermi energy, EF = 0 Ry. DOS lines are thickened for (a)-electrons in case of U, and (b) d-electrons in case of Ti.

Table 1Model description and assessed parameters for the phases in the U–Ti system (withtemperature T in Kelvin). Parameters not shown are taken from the SGTE database[13].

Phase Model (Va = Vacancy) Evaluated parameters (J/mole)

Liquid Random solution (U,Ti) 0LLiqU;Ti ¼ þ138;191� 88�T

1LLiqU;Ti ¼ þ20065

bcc Random solution (U,Ti) 0LbccU;Ti:Va ¼ þ4749� 9�T

1LbccU;Ti:Va ¼ �14698þ 9�T

hcp Random solution (U,Ti) 0GhcpU:Va ¼ 0Gbcc

U þ 50000Lhcp

U;Ti:Va ¼ �7379þ 6�T

Ortho Random solution (U,Ti) 0GortTi � 0Ghcp

Ti ¼ þ18;000Tetra Random solution (U,Ti) 0Gtet

Ti � 0GhcpTi ¼ þ10;000

d-U2Ti Sublattice model (U)2(Ti)1 0GU2TiU:Ti � 2�0Gort

U � 0GhcpTi

¼ �94;819þ 61�T

Fig. 7. Calculated U–Ti phase diagram, compared with results from Ref. [6] (not allexperimental points used for optimization are shown – see Section 5). Dashed linerepresents a bcc metastable miscibility gap as explained in Section 6.3.

Table 2Invariant reactions in the U–Ti system (a-U: orthorhombic, b-U: tetragonal, a-Ti:hcp, c-U

Reaction Ref. [6]

x (Ti) Temp (K)

b-Ti ? a-Ti + d-U2Ti 0.85 928(c-U,b-Ti) ? d-U2Ti 0.33 1171c-U ? b-U + d-U2Ti 0.07 993b-U ? a-U + d-U2Ti 0.02 941

182 S. Bajaj et al. / Journal of Nuclear Materials 419 (2011) 177–185

formation enthalpy of d-U2Ti was allowed to relax between thelimits of SR-EMTO-CPA calculations, i.e. ��39 kJ/mol atom, andFPLMTO calculations, i.e. ��25 kJ/mol atom (see Section 6.2).

The set of model parameters that have resulted from the opti-mization process are listed in Table 1 along with the phase modelsused. The phase diagram generated using these parameter values isshown in Fig. 7, and the invariant reactions are listed in Table 2. Asevident from this phase diagram, not all the phase equilibriumlines calculated align perfectly with those determined by Knapton[6]. But, the results of Knapton [6] cannot be considered error-freeas, except for a small region on the U-rich side, the samples usedfor analysis of the phase diagram were in the form of bars, andwere found to contain oxide impurities. Also, as is the case withsome phase diagram calculations using CALPHAD, the focus wasnot placed on obtaining perfect agreement between experimentaland calculated phase boundaries, but on obtaining reasonableresults for parameter values that make sense from the point of viewof thermodynamics and electronic structure-based results. Theheat of formation of c-U–Ti alloys calculated from this thermody-namic model is plotted, and compared with SR-KKR-ASA-CPAresults in Fig. 8. Additionally, in Fig. 9, we display the formationenthalpies of d-U2Ti, both from CALPHAD and DFT calculations,and as shown, the resulting formation enthalpy of d-U2Ti fromthe CALPHAD model is equal to �35.864 kJ/mol atom with respectto bcc-U and bcc-Ti. Both these sets of data that were used in thecalculation of the phase diagram, are in good agreement withDFT results. In Fig. 10a, we show a metastable bcc phase diagramthat is calculated from our model by suspending all other phasesexcept bcc, and which reflects the effect of CALPHAD-calculatedbcc heat of formation in Fig. 8. It can be seen from this figure that

and b-Ti: bcc).

CALPHAD (this work) Reaction type

x (Ti) Temp (K)

0.87 911 Eutectoid0.33 1173 Congruent0.02 1025 Eutectoid0.002 938 Eutectoid

0 0.2 0.4 0.6 0.8 1

Mole Fraction of Ti

-4

-2

0

2

4

Hea

t of

form

atio

n (k

J/m

ol-a

tom

)

CALPHADSR-KKR-ASA-CPA

U Ti

BCC

Fig. 8. Heat of formation of c(bcc)-U–Ti alloys as a function of compositionobtained from the U–Ti CALPHAD model (at T = 0 K), and compared with SR-KKR-ASA-CPA calculations. Reference states are bcc-U and bcc-Ti. Full lines that join thepoints serve only as guides to the eye.

0 0.2 0.4 0.6 0.8 1

Mole Fraction of Ti

-40

-36

-32

-28

-24

-20

-16

-12

-8

-4

0

4

Hea

t of

form

atio

n (k

J/m

ol-a

tom

)

ΔH (bcc) -CALPHADΔH (bcc) - SR-KKR-ASA-CPAΔH (δ-U

2Ti) - CALPHAD

ΔH (δ-U2Ti) - SR-EMTO-CPA

ΔH (δ-U2Ti) - FPLMTO

ΔH (δ−U2Ti) - Ref. [3]

U Ti

Fig. 9. Formation enthalpy of the d-U2Ti compound obtained from the U–TiCALPHAD model, and compared with DFT results. Heat of formation of the bccphase is also shown for comparison. Reference states are bcc-U and bcc-Ti. In case ofbcc-CALPHAD and bcc-SR-KKR-ASA-CPA calculations, full lines that join the pointsserve only as guides to the eye.

Fig. 10. Metastable bcc phase diagram produced from the U–Ti phase diagrams in(a) this work which shows a miscibility gap in the bcc phase up to T = 612 K, (b) inRef. [3] which shows a high-temperature inverse miscibility gap above � 2700 K,and (c) in Ref. [40] which shows a miscibility gap up to T = 1128 K.

Fig. 11. Heat of formation of the bcc phase as a function of composition calculatedfrom the U–Ti thermodynamic model in (a) this work, (b) Ref. [3], and (c) in Ref.[40]. Reference states are bcc-U and bcc-Ti.

S. Bajaj et al. / Journal of Nuclear Materials 419 (2011) 177–185 183

there is a metastable miscibility gap in the bcc phase up toT = 612 K. However, due to a relatively higher formation energy,and thus, stability of the d-U2Ti compound over the bcc phase, thismiscibility gap remains metastable, and does not appear in theequilibrium phase diagram shown in Fig. 7 (as also evident fromthe DOS calculations in Fig. 4).

Recently, a U–Ti phase diagram was published by Berche et al.[3] that seems to be in good agreement with experimental data[4–6]. Using the model parameters of this work, we calculated ametastable bcc phase diagram, similar to the one generated withour model parameters in Fig. 10a. The result of this calculation isplotted in Fig. 10b, and shows complete solubility in the bcc phaseup to �2600 K, but an inverse miscibility gap above this tempera-ture. The appearance of an inverse miscibility gap is not forbiddenfrom the thermodynamics point of view. However, such a feature isusually the result of a decrease in entropy of a mixture as temper-ature increases. In this relatively simple system, we thus considerthe inverted miscibility gap to be an artifact that can and must be

avoided by using ab initio data in CALPHAD optimizations.Additionally, the manifestation of such unrealistic phase stabilitiescould also be prevented by introducing special phase stabilityconstraints (ensuring positive curvature of the Gibbs energy curvesfor the bcc phase at high temperatures, for example) such as phasestability (QF), or phase driving force (DGM) functions in Thermo-Calc, or by employing Kaptay’s function for the excess Gibbs energy[39]. In our work, the former approach was utilized. The heat offormation of the bcc phase resulting from the U–Ti model in Ref.[3] is shown in Fig. 11b, and is another demonstration of a disparitywith DFT data in Fig. 1. Other deficiencies of this model include the

0

1

2

3

4

5

6

7

8

9

10

Tem

pera

ture

, T*1

000

K

0 0.2 0.4 0.6 0.8 1.0Mole Fraction, TiU Ti

Fig. 12. U–Ti phase diagram calculated from the present work, and plotted up toT = 10000 K.

184 S. Bajaj et al. / Journal of Nuclear Materials 419 (2011) 177–185

lack of use of formation enthalpy of d-U2Ti in the CALPHAD calcu-lation, and the assumption of no solubility of Ti in the a-U phase.

Another thermodynamic assessment of the U–Ti system waspublished by Murray in Ref. [40]. From this study, the calculatedenthalpy and entropy of formation of the d-U2Ti phase are�4.104 kJ/mol atom and �2.054 J/mol atom K, respectively (withreference to bcc-U and bcc-Ti), compared to the relatively muchhigher values of �35.864 kJ/mol atom and �24.506 J/mol atom K,respectively found in this work. The results from Ref. [40] also dif-fer by an order of magnitude from those obtained using varioustheoretical methods discussed earlier. The much higher enthalpyof formation calculated and assessed in this work necessitates anentropy of formation roughly ten times as large as that assessedby Murray in order to have the d phase decompose at the temper-ature suggested by experiments. Despite the large difference in en-ergy scales of both models (Murray’s and this work), the ratioDHformation

DSformationis of the same order of magnitude—�2 in case of Murray

[40] and �1.5 in the present work, which suggests that both mod-els are at least similar in the relative importance of the enthalpicand entropic contributions to the stability of the delta phase. Themetastable bcc phase diagram, and heat of formation of bcc alloysare calculated from this model, and plotted in Figs. 10c and 11c,respectively. The metastable bcc miscibility gap in Ref. [40] is as-sumed to occur up to a much higher temperature, close to thedecomposition temperature of d-U2Ti at T = 1171 K. This assump-tion may not necessarily be the case as the metastable bcc miscibil-ity gap could occur up to a much lower temperature, such asT = 612 K that is found from our result in Fig. 10a, and be ‘‘masked’’by the high stability of the d-U2Ti compound.

-50

0

50

100

150

Mol

ar E

ntha

lpy

(kJ/

mol

e)

500 1000 1500 2000 2500Temperature, K

liquid

x(Ti)=0.30

x(Ti)=0.85

ort

bcc

Fig. 13. Molar enthalpy of U–Ti versus temperature at two alloy compositions, x(Ti) = 0.30, 0.85. Reference states are ort-U and hcp-Ti.

7. Conclusions

In this work, we have performed DFT calculations on the bcc,and d-phases in the U–Ti system, and used these results to obtaina thermodynamically consistent phase diagram. Such an approach,the importance of which has been stressed before [41], has helpedus avoid unwanted phase equilibria, such as a high-temperatureinverted bcc miscibility gap, or an unnecessary assumption of coin-cidence of bcc miscibility gap with the decomposition temperatureof d-U2Ti compound, which are produced from other published U–Ti models. The resulting heat of formation of the bcc phase fromCALPHAD also shows a change from positive to negative valueswith an increase concentration of Ti, and is in good agreement withSR-KKR-ASA-CPA and FPLMTO-SQS calculations. The formation en-thalpy of d-U2Ti was allowed to relax during the parameter optimi-zation process of phase diagram calculation between the twolimits imposed by SR-EMTO-CPA and FPLMTO calculations. Theresulting value from CALPHAD is DHd�U2Ti ¼ �35:864 kJ/mol atom,with respect to bcc-U and bcc-Ti.

To investigate the relatively high formation enthalpy of d-U2Ti,the electronic DOS of both the d-C32 and bcc structure in a U2Tistoichiometry were calculated, and it was found that there was asudden drop in DOS at and in the vicinity of the Fermi level inthe case of the ordered C32 structure, thus indicating a highstability for this compound. The formation energy of thedisordered bcc-U2Ti structure was found to be equal toDHbcc�U2Ti ¼ þ5:69 kJ/mol atom, and +2.86 kJ/mol atom from SR-EMTO-CPA and SR-KKR-ASA-CPA, respectively, which also signalsinstability of this structure with respect to the C32-U2Ticompound. When comparing the d-phases of the U–Zr and U–Tisystems (isoelectronic counterparts), it is established that a changein phase behavior from a d-UZr2 solid solution, to a stoichiometricd-U2Ti compound is caused due to a change in atomic distributions.

Another observation made during this work that we want tobring to attention to practitioners of the CALPHAD method is the

re-stabilization of low-temperature phases at very high tempera-tures. The U–Ti phase diagram in Fig. 7 is re-calculated with the li-quid phase suspended up to T = 10,000 K, and the result is shown inFig. 12. As seen from this phase diagram, the Gibbs energies of bothlow-temperature phases of pure metals U and Ti, orthorhombicand hcp, when extrapolated to high temperatures, become lowerthan those of their high-temperature bcc phase counterparts at acertain temperature. It is noted here that even though most ofthe Gibbs energy functions in the SGTE database are undefinedfor temperatures outside the range of 298–4000 K, and that suchhigh temperatures are not important for practical purposes, theuser must be aware of such shortcomings when calculating alloyphase diagrams. It is thus stressed here that the SGTE databasemust be ‘‘revisited’’ by the CALPHAD community to avoid suchinconsistencies in phase stabilities at high temperatures.

Similar to our work in Ref. [2], in order to provide assistance inthe form of thermodynamic data to future experiments on such

S. Bajaj et al. / Journal of Nuclear Materials 419 (2011) 177–185 185

systems (such as differential scanning calorimetry (DSC), and dif-ferential thermal analysis (DTA)), we have calculated and plottedin Fig. 13, the molar enthalpies of two alloys of compositions x(Ti) = 0.30, and x (Ti) = 0.85 as functions of temperature (with ref-erence states taken as a-U (ort) and a-Ti (hcp)). These alloys arechosen as they are believed to be critical in verifying the thermo-dynamics of the U–Ti system. As expected, there are sudden jumpsin molar enthalpies across phase transitions in Fig. 7.

Acknowledgements

This work has been performed under the auspices of the US DOEby the Lawrence Livermore National Laboratory under contract No.DE-AC52-07NA27344. S.B. would like to acknowledge the supportprovided by the Computational Chemistry and Materials Science(CCMS) Summer Institute 2010 held at Lawrence Livermore Na-tional Laboratory, where a part of this work was performed. S.B.also thanks R.A. for the computational resources provided duringthe entirety of this work. R. A. would like to acknowledge the TexasA&M Supercomputing Facility as well as the Texas Advanced Com-puting Center for computational resources provided.

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