Thermodynamic evaluation of the Nb-O system

Thermodynamic evaluation of the Nb-O system

Ali R. Massih and Rosa Jerlerud Pérez

QuantumTechnologiesAB,UppsalaSciencePark,SE75183Uppsala

KTH,Materialvetenskap,SE10044Stockholm

E-mail:alma@quantumtech.se

10th April2006

Abstract

Thephasediagramofthebinarysystemniobium-oxygenhasbeen

evaluatedbymeansofaCALPHAD(CALculationofPHAseDiagram)

method. The experimental data on solubility, melting temperatures

and thefreeenergy of formationof niobium oxides aresurveyed and

the thermodynamic models based on the previous assessment of the

Nb-Osystemare delineated.Theresultsofour separateindependent

computations and comparison with experimental data indicate that

themodelsdescribetheNb-Ophasediagramadequately.

1 Introduction

Niobium and oxygen are alloying elements in the zirconium alloys used in

thecoreofthecurrentpressurizedwaterpowerreactors[1].Forexample,the

alloys ZIRLO[2]andM5[3 ] areincreasinglyusedasfuelcladding materials

in pressurized water reactors, Zr-2.5Nb has been employed as a standard

material for the pressure tubes in pressurized heavy water reactors [4, 5],

and E110 [6] and E635 [7] alloys are utilized in the Russian built VVER

andRMBKcores.Inordertoassess thethethermodynamicsof theZ-Nb-O

basealloys,thethermodynamicsofthebinarycomponentsarenecessary.For

the Nb-Zr and O-Zr, we have already reported such evaluations [8, 9]. For

theNb-O system, presently thereis no publishedthermodynamic modeling

assessment;thereare,however,asetofcorrelationsandanoutlineofmodels

for the Gibbs free energies of the Nb-O system listed in the ZIRCOBASE

web site [10]and abrief private communication [11] thatcite thereferences

justifyingthecorrelations'experimentalbasis.

The aim of this note is to review of the literature on the Nb-O system

and evaluate the thermodynamic quantities and the phase diagram of the

2 Survey of experimental data

Niobium (-Nb) has a body-centered cubic (bcc) crystal structure with a

melting point 2741 K and boiling point 5015 K. Introduction of oxygen

into Nb lowers the melting point to 2188 K at around 10 at%O. Oxygen

residesasan interstitial inthesolidsolutionof -Nblattice.Oxygen's

boil-ing point is at 90.188 K and melting point at 54.8 K. In solid state at low

temperatures, oxygen has three allotropic forms, namely, -O

2

(rhombic),

-O

2

(rhombohedral) and -O

2

(cubic). Thetransitiontemperaturesbetween

thesephasesareT(!)=23:8KandT( ! )=43:8K.Here, weonly

considerthehightemperature (T 298 K)properties.

Besidesthatoxygenbeingininterstitialsolidsolutionwithinthebcc

nio-biumstructure,numerousoxidesandsub-oxidescanbeformeddependingon

the temperature and oxygen pressure or concentration. Briey,Nb-O solid

solution (bcc), NbO

x

(tetragonal), Æ-Nb-O (hexagonal), -Nb

2 O 5 (mono-clinic), -Nb 2 O 5

(orthorhombic) areformed inair at eitheratmospheric or

reduced pressure, whereas, NbO

2

(tetragonal), NbO (cubic, NaCltype)are

formedinairat reducedpressure.Structural relationsbetween theoxidesof

niobiumarediscussed byTerao [13 ].A classical work onthepolymorphism

ofNb

2 O

5

isHoltzbergetal. [14].

Thesolidsolubilityofoxygeninniobiumhasbeendeterminedbyav

ari-etyofmethodssuchasX-rayanalysis, thermaltechniques, internal friction,

micro-hardness, electric resistivityand solidstate electrolytic cell technique

(EMFmeasurements).Seybolt[15 ]byX-raydiractionandmicroscopic

ex-amination reported the solid solubility of oxygen in thetemperature range

775 Æ

C to 1000 Æ

C. He found that thesolubility is a function oftemperature

andvariesinthattemperature rangefrom 0.25to 1 wt%,respectively.

Elliott [16] studied the Nb-O phase equilibria bymeans metallographic

examination.Theexaminedsampleswereas-castandannealedarc-cast

niobium-oxygen alloys. The empirically constructed phase diagram is shown in Fig.

1.Salient featuresofthis diagramareasfollows:

 There are three oxides of niobium: NbO, NbO

2 and Nb 2 O 5 . These

oxides melt congruently at 1925 Æ C (2198 K), 1915 Æ C (2188 K) and 1595 Æ

C(1768K).Figure2displaysmeasureddataonincipientmelting

oftheNb-O specimens asafunction ofoxygenconcentration.

 Aeutecticreactionoccursat1915 Æ

C:L(10.5wt%O)!Nb(0.7wt%O)+NbO,

where Lstands for liquid.

 Aeutectic reactionoccursat 1810 Æ

C: L(21wt%O)!NbO+NbO

2 .

 Aperitecticreactionoccursat1510 Æ C:L(29.5wt%O)+NbO 2 !Nb 2 O 5 .

 The solid solubility of oxygen varies between 0.25 wt% at 500 Æ

Elliott by X-ray investigation, à la Debye-Scherrer [17], mapped out the

powder patterns of NbO, NbO

2

and Nb

2 O

5

and conrmed that thecrystal

structure of NbO is simple cubic with a lattice parameter of 4.210 Å. In

particular, he found that Nb atoms lie at (00 1 2 ),(0 1 2 0) and ( 1 2 00); O atoms lieat ( 1 2 1 2 0), ( 1 2 0 1 2 ) and (0 1 2 1 2

);and lattice sites(000)and ( 1 2 1 2 1 2 ) arevacant. Moreover, the Nb 2 O 5

patterns revealed arutile [18] (TiO

2

) structure.

Bryant[19 ]determinedthesolubilityofoxygeninniobiuminthe

temper-ature range 700 Æ

C to 1550 Æ

C by establishing equilibrium between an oxide

lmonthesurfaceofthespecimenandasaturatedsolidsolutionofoxygenin

metal. The solubility wasmeasured byremoving the oxide scale after

equi-librium had been reached and analyzing the underlying metal for oxygen.

The time required to establish equilibrium was determined by measuring

specimen hardiness as a function of time. Since oxygen hardens niobium

appreciably, any change in oxygen content is accompanied by a change in

hardness.Oxygenequilibriumthereforeisestablishedwhennofurther

hard-ness increase was observed. The results of Bryant's data on solubility are

depicted in Fig. 3. Figure 4 shows the weight to atom% unit conversion in

theNb-Osystemfor reader's convenience.

Gebhardt and Rothenbacher [20 ] using a combination of X-ray

dirac-tion,microhardnessandelectricresistivitymeasurementsplusmetallography

studied thesolubilityof oxygeninniobium between temperatures of500 Æ

C

and1900 Æ

C. Theirresults show thatthesolubilityincreases from 1.1at%O

at750 Æ

Cto5.5at%Oat1540 Æ

C.Inthistemperaturerange,thetemperature

dependenceofsolubilityfollowed anArrhenius description(Fig. 3).

Taylor and Doyle [21 ] investigated the solid solubilityof oxygen in

nio-biumbymeansofX-raydiraction,metallographic andthermaltechniques.

TheresultsofTaylorandDoylefallmidwaybetweenthoseofSeybolt[15]and

Gebhardtand Rothenbacher [20],Fig.3. Thesolubilitylimit dataobtained

byElliott[16 ]indicatemuchloweroxygenvaluesintheregionof1800 Æ

Cthan

otherdata.

Fromm [22] reported some data for the Nb-O system in temperatures

between 1000and2000Kbymeansofsolidstateelectromotiveforce(EMF)

measurements. He found that at the temperature of 1276 K the solubility

limit is2.42at%O(Fig. 3).

Theresultsobtainedbytheaforementionedinvestigationsarenotalways

inagreement.Inorder tore-solve the issue, Nickersonand Altstetter [23 ]in

temperature range of 973 to 1773 K employed accurate solid state

electro-motive force measurements. Nickerson and Altstetter results of the EMF

measurements oer quantitative information on the enthalpy and entropy

change.Thesedataareingoodagreementwith thecorrespondingvalues

ex-tracted from Gebhardt and Rothenbacher's work [20 ] and theearlier study

pro-vided anArrhenius formexpression forthe oxygensolidsolubilitylimit, C 1 O =49:4exp  8000 R T  ; (1) whereC 1 O

isthesolidsolubilityofoxygeninat%O, R=1:987calK 1

mol 1

isthegasconstantandT isthetemperatureinkelvin.Equation(1)isplotted

inFig. 3 togetherwith the corresponding correlation based on thework of

Gebhardtand Rothenbacher[20] andthe aforementioned data.

Matsui and Naito [24 ] measured the vapor pressure over the

niobium-oxygen solid solution by mass-spectrometric method in the temperature

range 2091 to 2379 K. The solidus line of the Nb-O solid solution and the

solubilitylimitofoxygeninniobiummetalathightemperatureswere

deter-mined from the vaporization behavior. The phase transition temperatures

obtained by Matsui and Naito are depicted in Fig. 5. In an earlier

sim-ilar study, Matsui and Naito [25] measured the vapor pressure over

non-stoichiometric NbO

2x

(with O/Nb in the range of 1.972 to 2.037) in the

temperature range 1958-2326 K. Thephase transitiontemperatures around

NbO

2x

athightemperaturesweredeterminedfromthevaporizationstudy.

Table1liststhese(melting)dataplussimilardatafromotherstudies.InT

a-ble1,T

r1

correspondstothetransitionfromsolidNbO

2x

tothemixture of

solidNbO

2x

and liquidussolution; while T

r2

,around 2188 K, corresponds

to the transitionfromthemixture of liquidussolutionand solidNbO

2x to

theliquidussolution.

Table 1:Phase transitionboundariesaround NbO

2x

at hightemperatures

[25].The transitiontemperatures T

r1 and T

r2

aredened inthetext.

Oxygencontent T r1 T r2 Method Reference atom fraction K K 0.6667 - 2188 Mass Spectrometry [25 ] 0.6679 2163 2188 0.6693 2123 2188 0.6707 2028 2188 0.6667 - 2188 Metallography [16 ] 0.6711 - 2183 0.6676 - 2173 Mass Spectrometry [26 ] 0.6657 2063 2188 Mass Spectrometry [25 ] 0.6647 2103 2188 0.6635 2103 2188 0.6466 2083 - Metallography [16 ] 0.5595 2108

-Table 2: Free energy of formation of niobium oxides determined by EMF

measurementsasa functionof temperature [27 ].

Oxide Freeenergy of formation, cal/mol T,K

NbO
G Æ = 99500+20:7T (500) 1177-1388 NbO 2
G Æ = 184500+38:7T (500) 1100-1400 Nb 2 O 5
G Æ = 440200+94:1T (500) 1000-1400

Table3:Crystalstructuresandlatticeconstantsofelementsandcompounds

inthe Nb-O system.

Phase Structure Spacegroup Lattice constant, Å

-Nb bcc Im3m a=3:300 NbO cubic Pm3m a=4:210 NbO 2 tetragonal I4 1 =a a=4:837 c=2:988 -Nb 2 O 5 monoclinic P2 a=21:20 b=3:824 c=19:39 -Nb 2 O 5 orthorhombic Pban a=7:317 b=15:728 c=10:749 -O 2 rhombic C2/m a=5:403 b=3:429 c=5:086 -O 2 rhombohedral R  3m a=3:307 c=1:126 -O 2 cubic Pm3n a=6:83

Hiraoka et al. [27] usinga solidstate electrolytic cell technique (EMF

mea-surements)inthetemperaturerangeof1000to1400K.Theyalsocompared

their results with those of other investigators (see references therein). The

results of their measurements are systematized in terms of linear relations

between the standard Gibbs free energy of formation and temperature for

NbO, NbO 2 and Nb 2 O 5 (Table2).

The phase equilibria databriey surveyed above form theexperimental

basis for the models and the phase diagram presented in the subsequent

sections. The crystal structure and the phase description of the elements

and compound in O-Nb system are summarized in Table 3, [13 , 14]. The

3 Thermodynamic modeling

Alistofmodels proposedforthis workissummarizedinTable 4,where the

square symbol  stands for vacancies. Mathematical descriptions for these

modelsare relegatedto Appendix A.All thesemodelsareformulated using

thecompoundenergyformalismforsolutionphaseswheretheconceptof

sub-latticeisutilized[28].TheGibbsfreeenergyexpressionsareinputparameters

tothemodelforcalculationofphaseequilibria/diagramsofcompounds.The

thermodynamicparametersfortheNb-Osystemusedinourcalculationsare

outlinedinAppendix B.

The basic idea of the computational method is to dene theGibbs free

energy of each element with respect to its stable magnetically disordered

state at 298.15 K and 101 325 Pa, referred to as the stable element

refer-ence (SER), as recommended by the Scientic Group Thermodata Europe

(SGTE). In theSGTE database [29],the Gibbs free energy isexpressed in

termsofa power seriesexpansion intemperature intheform:

G(T)=a+bT +cTln(T)+ X n d n T n ; (2)

where n takes the values 2;3; 1;:::,and a;b;c;d

n

areempirical constants

determinedbyexperimentaldata.The referencestatesfor thepureniobium

andoxygenarebcc-Nband gaseous oxygen.

Table 4: List of models selected for the Nb-O system basedon the work of

Dupinand Ansara [11,10]. Thesquare symbolstands for vacancies.

Phase Model Constitution

Gas Ideal mixture (O,O

2 ,O 3 ,Nb,NbO,NbO 2 )

Liquid Ionic melt (Nb

+2 ) P (O 2 ; 2 ;NbO 2 ;NbO 5=2 ) Q

bcc Interstitial solution (Nb)(O,)

3

Cubic NbO Stoichiometric (Nb)(O)

Tetragonal NbO 2 Stoichiometric (Nb)(O) 2 Monoclinic Nb 2 O 3 Stoichiometric (Nb) 2 (O) 5 4 Results of calculations

The binary phase diagram for the Nb-O system calculated by using the

Thermo-Calcprogram [30],armedwiththemodels delineatedinthe

forego-ingsection, is presented inFig.6. Thisgure alsodepicts theexperimental

high-experimentaldata,respectively.Thecompletephasediagramwithout

exper-imental data is shown in Fig. 9. The presented diagrams rest on the data

surveyed insection2 and theunpublishedassessmentof Dupin and Ansara

[11,10].Thisassessmentshowsthatthemaximumsolubilityofoxygenatthe

eutectictemperatureof2192Kis0.9at%.Thecalculatedcongruentmelting

points ofthe three typesofniobium oxides arelistedinTable 5.

Table 5: The congruent melting points of the stoichiometric oxides in the

Nb-Osystem[11 ], cf.Fig.9.

Oxide Oxygen content (at.%) Meltingtemperature (K)

NbO 50 2218 NbO 2 66.7 2182 Nb 2 O 5 71.4 1923

Thecalculated invariant reactions andpointsinthesystemareoutlined

in Table 6. The calculated values of the compositions and temperatures in

thistable aredue to Dupinand Ansara [11 ].

Table6:Invariant reactions inthe Nb-O system(x

O

=mole fraction)[11 ].

Reaction Type Composition, x

O Temperature, K Liquid +gas Nb 2 O 5 ... 0.713 1.0000.714 1919 Liquid Nb 2 O 5 +NbO 2 Eutectic 0.709 0.7140.667 1782.3 Liquid NbO+NbO 2 Eutectic 0.616 0.5000.667 2099.2

Liquid -Nb+NbO Eutectic 0.393 0.0090.500 2191.9

5 Verication and discussion

5.1 Gibbs free energy calculations

TheGibbsfreeenergyofthe-Nb-Ointerstitialsolutionisevaluated

accord-ing to formulae of section A.3. In particular, we write the the total Gibbs

freeenergy ofthe bcc phase, insimpliednotation, as

G inter (y;T) = (1+3y) 1  y
G Nb;O +(1 y)
G Nb + +3R T  ylogy+(y 1)log(1 y)  + +y(y 1)  L 0 +L 1 (2y 1)   ; (3) where y  y O

is the site fraction in the sub-lattice model for oxygen and

L 0

;L 1

de-pendent.Wehaveusedthetemperature-dependent correlationsfor
G Nb;O ,
G Nb ,L 0 ;L 1

listedinAppendixBtoevaluateG

inter

(y;T).Figure10shows

the plots of G

inter

as a function of y, while Fig. 11 illustrates the relation

between the mole fraction of oxygen x  x

O

and y; and Fig. 12 shows

the plots of G

inter

as a function the mole fraction of oxygen x at dierent

temperatures. Using theThermo-Calcprogram [30 ] we have found identical

results.

5.2 Free energy of formation

The Gibbsenergy of formation for thestoichiometric niobium oxidesNbO,

NbO 2 andNb 2 O 5

arecalculatedaccordingtoEq.(A.17).Letusrstevaluate

the temperature dependence of the term G bcc

Nb

a O

b

(T) using the correlations

listed in Appendix B. The results in thetemperature range of 300 to 2900

K are plotted in Fig. 13, where the values of the free energies are divided

bythenumberofatoms peroxide.Thecorrespondingresults fortheenergy

offormation,
G bcc A a B b

(T),arecalculated inthetemperature rangeof900to

1500 K and the results are depicted inFig. 14. In the same gure we have

plotted the experimental data of Hiraoka et al. [27 ] as symbols. We note

thatthe agreement between modelcalculations and Hiraoka et al. dataare

excellent. Utilizing the Thermo-Calc program [30], we have found identical

results.Moreover, at room temperature (298.16 K), we have calculated the

enthalpy of formation
H

f

for NbO, NbO

2 and Nb 2 O 5 compounds and

compared our results withsome of experimental datareportedin literature

inorderto checkthestabilityoftheseoxides(Table7).Ascanbeseenfrom

Table7,thecalculationsbasedontheDupin-AnsaraassessmentoftheNb-O

system[31 ] show good agreement with theexperiments. It should be noted

that, in the thermodynamic assessment by Dupin and Ansara [11 , 10 ], all

theoxideswere considered to be stoichiometric andtheir Gibbsfreeenergy

expressions were chosen from the SGTE Substance Database [31], which is

available intheThermo-Calcprogram [30].

6 Closure

In this note we rst reviewed the experimental data available in literature

concerningthe thermodynamic properties of theNb-O binarysystem; then

we summarized the thermodynamic models that are developed to describe

this system. These models had been assessed in a previous study for the

Nb-O systemandhere theywere employed in both in-house MATLABles

and inthe Thermo-Calc program for theevaluation of phasediagram. Our

calculations agreebycomparing theoutput of separatecomputer programs

Table 7: The enthalpy of formation
H

f

(kJ/mole) for NbO, NbO

2 and Nb 2 O 5 at 298.15 K. Compound
H f

uncertainty Method Reference

::: kJ/mol kJ/mol ... ... NbO -426.2 5.1 Calorimetry [32 ] -419.7 12.6 ::: [33 ] -455.22 2.51 Calorimetry [34 ] -419.65 ::: Assessed [31 ] NbO 2 -792.7 4.2 Calorimetry [32 ] -795.0 8.4 ::: [33 ] -833.87 2.5 Calorimetry [34 ] -794.96 ::: Assessed [31 ] NbO 2:42 -930.2 1.4 Calorimetry [32 ] -930.5 ::: Calorimetry [35 ] NbO 2:47 -942.9 0.6 Calorimetry [32 ] NbO 2:485 -945.6 0.3 Calorimetry [32 ] NbO 2:5 -949.8 4.2 ::: [33 ] -988.68 ::: Calorimetry [34 ] -949.78 ::: Assessed [31 ]

References

[1] C. Lemaignan and A. T. Motta. Zirconium alloys in nuclear

applica-tions. In R.W. Cahn, P. Haasen, and E. J. Kramer, editors, Nuclear

Materials, volume 10B of Material Science and Technology, chapter 7.

VCH, Weinheim, Germany,1994. Volume editor B.R.T.Frost.

[2] R.J.Comstock,G.Schoenberger,andG.P.Sabol.Inuenceof

process-ing variables and alloy chemistry on the corrosion behavior of ZIRLO

nuclear fuelcladding. InE. R.Bradleyand G.P.Sabol,editors,

Zirco-nium in Nuclear Industry: Eleventh International Symposium, volume

ASTMSTP1295,pages710725,WestConshohocken,PA,USA,1996.

American Societyfor Testing andMaterials.

[3] J-P. Mardon, D. Charquet, and J. Senevat. Inuence of composition

andfabricationprocessonout-of-pileandin-pilepropertiesofM5alloy.

InG.P.SabolandG.D.Moan,editors,ZirconiuminNuclear Industry:

TwelfthInternationalSymposium,volumeASTMSTP1354,pages505

522,WestConshohocken,PA,USA,2000.AmericanSocietyfor Testing

and Materials.

[4] D.O.Northwood,X.Meng-Burany,andB.W.Warr. Microstructureof

Zr-2.5nballoypressuretubing.InC.M.EukenandA.M.Garde,editors,

ZirconiuminNuclearIndustry:NinthInternationalSymposium,volume

ASTM STP 1132, pages 156176, Philadelphia, USA, 1991. American

Societyfor Testingand Materials.

[5] D.Srivastava,G.K. Dey,and S.Banerjee. Evolutionof microstructure

duringfabricationofZr2.5wtpctNballoypressuretubes. Metallurgical

Transactions A,26A:27072718, 1995.

[6] P.V.Shebaldovandetal.E110alloycladdingtubepropertiesandtheir

interrelationwithalloystructurephaseconditionandimpuritycontent.

InG.P.SabolandG.D.Moan,editors,ZirconiuminNuclear Industry:

TwelfthInternationalSymposium,volumeASTMSTP1354,pages545

559,WestConshohocken,PA,USA,2000.AmericanSocietyfor Testing

and Materials.

[7] A. V. Nikulina and et al. Zirconium alloy E635 as a material for fuel

rod cladding and other components of VVER and RMBK core. In

E. R. Bradley and G. P. Sabol, editors, Zirconium in Nuclear

Indus-try:EleventhInternationalSymposium,volumeASTMSTP1295,pages

785804, West Conshohocken, PA, USA, 1996. American Society for

[8] A.R.Massih. Thermodynamiccomputationsofthestablephasesinthe

Zr-Nb system. Technical Report PM 04-006, Quantum Technologies,

2004.

[9] A.R.Massih. Phaseboundariesof the O-Zrsystem. Technical Report

PM 05-001, QuantumTechnologies, 2005.

[10] See: www.inpg.fr/ltpcm/base/zircobase, 1999.

[11] N. Dupin and I. Ansara. Systèm Nb-O. Unpublished communication,

1999.

[12] R.Jerlerud-Pérez andA.R.Massih. Thermodynamicevaluationofthe

Nb-O-Z r system. Technical Report PM05-003, Quantum Technologies

AB, 2005.

[13] N. Terao. Structures of niobium oxides. Japanese Journal of Applied

Physics, 2:156174, 1963.

[14] F. Holtzberg, A. Reisman, M. Berry, and M. Berkenblit. Chemistry of

the groupVB pentoxides. VI the polymorphism of Nb

2 O

5

. Journal of

American Chemical Society, 79:20392043, 1957.

[15] A. U. Seybolt. Solid solubility in columbium. Transactions of AIME

Journal of Metals, 200:774776,1954.

[16] R. P. Elliott. Columbium-oxygen system. Transactions of the ASM,

52:9001014, 1960.

[17] A.Guinier. X-ray Diraction. DoverPublications,Mineola,NewYork,

1994.

[18] D. E. Sands. Introduction to Crystallography. Dover, New York,1993.

[19] R.T.Bryant.Thesolubilityofoxygenintransitionmetalalloys.Journal

of LessCommon Metals,4:6268, 1962.

[20] E. Gebhardt and R.Rothenbacher. Untersuchungen im S ystem

Niob-Sauersto. Z. Metallkunde,54:623630,1963.

[21] A.TaylorandN.J.Doyle. Thesolid-solubilityofoxygeninNb and

N-richNb-MoandNb-Walloys. Journalof LessCommonMetals,13:313

330, 1967.

[22] E.Fromm.Emfmeasurementsofniobium-oxygenandtantalum-oxygen

solidsolutions. Journalof LessCommon Metals, 22:139140,1970.

[23] W. Nickerson and C. Altstetter. A study of nobium-oxygen solid

so-lution utilizingsolid electrolytic cells. Scripta Metallurgica, 7:229232,

[24] T. Matsui and K. Naito. Vaporization study of the niobium-oxygen

solid solution by by mass-spectrometric method. Journal of Nuclear

Materials, 115:178186, 1983.

[25] T. Matsui and K. Naito. Vaporization study of nonstoichiometric

NbO

2x

by mass-spectrometricmethod. Journal of Nuclear Materials,

102:227234, 1981.

[26] S.A.Shchukarev,G.A.Semenov,andK.E.Frantseva.RussianJournal

of Inorganic Chemistry,11:129, 1966.

[27] T.Hiraoka,N.Sano,andY.Matsushita. Electricalmeasurementofthe

standard free energies of formation of niobiumoxides. Transactions of

the Iron Institute of Japan, 11:102106,1971.

[28] M. Hillert. The compound energy formalism. Journal of Alloys and

Compounds,320:161176, 2001.

[29] A.T.Dinsdale.SGTE dataforpureelements. CALPHAD,15:317425,

1991.

[30] Thermo-Calc Program User's Guide. Stockholm, Sweden, 2003.

www.thermocalc.se.

[31] SGTE substance database. AvailableinThermo-Calc software.

[32] H.Inaba,T.Mima,andK.Naito. Measurementofenthalpiesof

forma-tionofniobiumoxidesat920kinatian-calvet-typecalorimetr. Journal

of Chemical Thermodynamics,16:411418,1984.

[33] JANAFthermochemical tables,2nd edition. Supplement 1976.

[34] M. P.Marozavaand L.L.Gestkina. Enthalpyof formationof niobium

oxides. Z.Obeshei kximij,29:10491052, 1959. In Russian.

[35] J.F.Marucco,R.Tetot,P.Gerdanian,andC.Picard. Etude

thermody-namique dudioxyde de niobiumahautetemperature. Journal of Solid

A Thermodynamic relations

A.1 Gas phase

Thegasphase isconsidered asamixture ofideal gases consistingof O,O

2 ,

O

3

,Nb, NbO and NbO

2

. The Gibbs energy per mole of species in the gas

phaseisgiven by
G gas (T;P)= X j x j
G gas j (T;P)+R T  X j x j logx j +log P P 0  ; (A.1) and
G gas =G gas (T;P) X i X j x j a j i H SER i (298:5) (A.2) where
G gas j

(T;P)istheGibbsfreeenergy ofpurespeciesjattemperature

T and pressure P, H SER

i

(298:5) is the reference enthalpy of element i at

298.15 K and 0.1 MPa, so called Stable Element Reference, x

j

is themole

fractionof speciesj in thegasphase, a j

i

is the numberof atoms of element

iinthe gaseousspeciesjand Risthegasconstant.Here, thesums areover

allthe involved species.

A.2 Liquid phase

Inbinarysystems,whenoneoftheelementsisnotmetallic,theliquidphase

is described by the ionic melt sub-lattice model [28] witha genericformula

as (A v A ) P (B v B ; v  ;B;AB;AB x ) Q

. Here, A denotes the metallic element,

occupyingtherstsub-lattice,Bthenon-metallicone(chargedandneutral)

andthevacancies,denotedby,whichoccupythesecondsub-lattice;v

A ,v

B

andv



arethevalancechargesofthespeciesA,Bandvacancies,respectively;

and P and Q are the number of sites of each sub-lattice. The Gibbs free

energy permole isexpressedas

G l iq =G l iq ref +G l iq id +G l iq ex : (A.3)

Herethe referencefree energy is

G l iq ref =Q y 
G l iq A +y B
G l iq B
+y B v B
G l iq A:B ; (A.4)

thecongurational (ideal) freeenergy

G l iq id =R TQ X i y i logy i ; (A.5)

andthe interaction freeenergy

G l iq ex = X i6=j y i y j L l iq A:i;j ; (A.6)

where the sums over i and j cover elements , B v B and B. In relations (A.4)-(A.6),y  ;y B Q and y B

arethe site fractions of thevacancies, B ions

and neutral B atoms, respectively on theanion lattice. Onthe cation

sub-lattice,A is theonly residing species,hence y

A

=1.Moreover, we have the

following constraints: y  +y B v B +y B =1 (A.7) and P = v B y B v B +Qy  and Q= v  y A = v  : (A.8) In Eq.(A.4),
G l iq A and
G l iq B

represent theGibbsenergies permole of

atomsofliquidmetalAandactitiouspureliquidnon-metalB,respectively.

G l iq

A:B

represents the molar Gibbs energy of the ideal hypothetical liquid

A v B B v A . In Eq. (A.6), L l iq A:i;j

accounts for the interaction energies between

speciesi,jandA;itisexpressedgenerically bytheso-calledRedlich-Klister

polynomial oftheform:

L l iq A:i;j = n X k=0 L k;l iq A (y i y j ) k : (A.9)

Alltheaforementionedfreeenergies,includingL k;l iq

A

aretemperature

depen-dent and areobtained by evaluationofexperimentaldata.

Finally,theconcentration ofthe oxygeninthe solutionis relatedto the

sub-latticesite fractions accordingto

x B v B = Qy B v B P +Q(1 y  ) (A.10)

Weshouldnotethatthethreesitefractionsintheanionsub-lattice:y

 ;y B v B and y B

, are determined at a given solute concentration by minimising the

GibbsfreeenergyG l iq =G l iq (y  ;y B v B ;y B

)withrespecttothesite fractions

subjectto theconstraintsdened byEqs.(A.7),(A.8) and (A.10).

Applying theabove formalismto theNb-O system,A=Nb, B=O, v

A = +2,v  = 2andv B = 2;hencewehave(Nb +2 ) P (O 2 ; 2 ;NbO 2 ;NbO 5=2 ) Q withP =2y O 2 +Qy  andQ=2.Furthermore,
G l iq Nb =G l iq Nb H SER Nb and
G l iq O =G l iq O H SER O

arethemolarGibbsfreeenergiesofpureNbandO,

re-spectively,whicharelistedinref.[29] and
G l iq Nb;O =G l iq Nb;O H SER Nb H SER O

isthe molarGibbsenergyofthecompoundNbO;itisdeterminedby

assess-ingexperimental data [10].Also, L l iq

A;B; =L

l iq

Nb;O;

denotes theinteraction

between the oxygen atoms and vacancies in the interstitial sub-lattice; it

is determined by assessingexperimental data. Finally, theconcentration of

oxygeninthe liquid solution, Eq. (A.10)becomes

x = Qy O 2 = y O 2 : (A.11)

A.3 Interstitial solution

Oxygen is aninterstitial atom insolid solutions -Nb (bcc). Interstitial

so-lutions can be described by the sub-lattice model. In general, for a binary

system A-B, one of the sub-lattices is assumed to be entirely occupied by

theelement A (y

A

=1), whilethe second sub-lattice contains Band

vacan-cies with the formula: (A)

p (B,)

q

. The ratio p=q is related to the crystal

structureand thetype ofsites occupied by B.

Thethree componentsoftheGibbsfreeenergy inphase, G  ref ,G  id and G  ex ,areexpressedas G  ref = 1 p+qy B  y 
G  (A;) +y B
G  A;B  ; (A.12) G  id = q p+qy B R T X i=B; y i logy i ; (A.13) G  ex = y B y  p+qy B L  A;B; : (A.14)

For application totheNb-O system,A=Nb,B=O,hence(Nb)

p (O,) q with p = 1 and q = 3. Furthermore,
G  (Nb;) = G  Nb H SER Nb is the molar

Gibbs free energy of pure Nb, listed in ref. [29 ] and
G  Nb;O = G  Nb;O H SER Nb qH SER O

isthemolarGibbsenergyofthecompoundwithallinterstitial

siteslled; it isdetermined byassessing experimental data. Also, L  A;B; = L  Nb;O;

denotes the interaction between the oxygenatoms andvacanciesin

the interstitial sub-lattice; it is determined by assessment of experimental

data.

Similar to theionic melt model, y

Nb

=1 inthe metallic sub-lattice and

inthe interstitial sub-lattice, we have theconstraint:

y

O +y



=1: (A.15)

The compositions of the phase is related to the sub-lattice site fraction y

O accordingto: x O = qy O 1+qy O ; x Nb =1 x O : (A.16)

Note that the variables y

O and y



are determined at once by Eqs. (A.15)

and (A.16); hence no minimisation of the free energy with respect to y

i is

neededhere.

A.4 Niobium oxide phases

Thecubic NbO, tetragonal NbO

2 and monoclinic Nb 2 O 3 are stoichiometric

compounds, i.e., they have xed composition at all theapplicable

A a B b ) iscalculated accordingto
G  A a B b (T)=G  A a B b (T) aG  A (T) bG  B (T): (A.17)

For example for NbO

2 ,G  NbO 2 , G  Nb and G  O

are the Gibbs freeenergies of

NbO

2

,pureniobiumandoxygen,foragivenphase(herebcc),respectively;

andsimilarly for the othertwo oxides.

B A summary of thermodynamic parameters for

the Nb-O system

Table 8 lists the thermodynamic parameters, the Gibbs free energies as a

function of temperature, describing the Nb-O system with respect to the

stableelement reference H SER

(attemperature T =298:14 Kand pressure

P = 101325 Pa). The interaction parameters are denoted by L. These

re-sults,which areinputto Thermo-Calc, arebasedonthework ofDupin and

Ansara [11, 10 ] presented here with some minor misprint corrections and

Table8:Thermodynamic properties ofthedierentphases oftheNb-O sys-tem. Parameter(J/mol) Range (K) G NbO = 434220:337+246:876204T 42:99897Tln(T) 298:14<T <2210 0:0044367135T 2 +7:61348510 10 T 3 +201346:65T 1 G NbO = 456057:083+399:171599T 62:76Tln(T) 2210 <T <6000 G NbO2 = 817191:531+381:593377T 64:17126Tln(T) 298:14<T <700 9:08246E 04T 2 4:02243510 6 T 3 +418142:7T 1 G NbO2 = 811038:971+277:505525T 47:77082Tln(T) 700<T <1000 0:01985076T 2 G NbO 2 = 7995222:39+66544:4364T 9449:355Tln(T) 1000 <T <1300 +5:400225T 2 5:8015910 4 T 3 +1:021178510 9 T 1 G NbO 2 = 825146:769+515:555749T 83:0524Tln(T) 1300 <T <2175 G NbO 2 = 849262:299+611:84916T 94:14Tln(T) 2175 <T <6000 G Nb 2 O 5 = 1942063:25+674:379374T 115:742Tln(T) 298:14<T <700 0:0547895T 2 +8:2494533310 6 T 3 +534527T 1 G Nb2O5 = 1967843:43+1010:19042T 166:3182Tln(T) 700<T <1500 0:010714595T 2 +1:0469763310 6 T 3 +2995953T 1 G Nb 2 O 5 = 1970867:16+1053:49937T 172:7281Tln(T) 1500 <T <1785 0:006198805T 2 +5:18174510 7 T 3 +2995953T 1 G Nb 2 O 5 = 2077756:73+1625:45741T 242:2536Tln(T) 1785 <T <6000 Liquidphase G L (Nb +2 ;O 2 )=2G NbO +310681:92 135:66551T 298:14<T <6000 G L (NbO 2 )=G NbO2 +62301 16:9083T 298:14<T <6000 G L (NbO 5=2 )=0:5G Nb 2 O 5 +19682:659 10T 298:14<T <6000 L 0;L (Nb +2 ;O 2 ;)=56277:338 298:14<T <6000 bcc-A2 phase G bcc (Nb:O)=G Nb 2 O 5 +G NbO +250000 298:14<T <6000 L 0;bcc (Nb:O;)= 670149+76:4T 298:14<T <6000 L 1;bcc (Nb:O;)= 354266 298:14<T <6000

hcp-A3metastable phase

G hcp

(Nb:O;)=G

NbO

C Figures

Figure 1: The equilibrium phase diagram of niobium (Cb)-oxygen system

0

5

10

15

20

25

30

35

1600

1700

1800

1900

2000

2100

2200

2300

Oxygen content (wt%)

Temperature (K)

Incipient melting data in Nb−O system, Elliott (1960)

melting points

NbO

NbO

2

Nb

2

O

5

Figure2: Melting datafor niobium-oxygen systemaccording to the

800

1000

1200

1400

1600

1800

2000

2200

10

−1

10

0

10

1

Temperature (K)

C

O

∞

(at%)

Oxygen solubility in niobium

Nickerson & Altstetter

Gebhardt & Rothenbacher

Seybolt

Elliott

Bryant

Taylor & Doyle

Fromm

Cost

Figure3:Solubilitydatafor niobium-oxygensystemfrom various

investiga-tions[15,16,19,21,22,20,23].TheworkofCostwascitedin[23]asprivate

0

20

40

60

80

100

0

10

20

30

40

50

60

70

80

90

100

Oxygen (wt%)

Oxygen (at%)

Oxygen in niobium

0

2

4

6

8

10

12

2000

2100

2200

2300

2400

2500

2600

2700

2800

Oxygen content (atom%)

Temperature (K)

Nb (solid)

Liquid + Nb (solid)

Nb (solid) + NbO (solid)

Phase transition data around Nb−O solid solution, Matsui & Naito (1983)

Figure 5: Phase diagram aroundniobium-oxygen solid solution determined

Elliott 1960

NbO

NbO2

Nb2O5

Seybolt 1954

Matsui and Naito 1981

Matsui and Naito 1983

Bryant 1962

Taylor and Doyle 1967

Cost 1973

Fromm 1970

Gebhardt and Rothenbacher 1963

Nickerson and Altstetter 1973

300

600

900

1200

1500

1800

2100

2400

2700

3000

Temperature K

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Mole Fraction O

BCC+ NbO

NbO

2

Nb

2

O

5

_Gas

Gas

Liquid

Figure 6: Calculated phase diagram for the binary Nb-O system using the

models outlined in Table 4 and the experimental data reviewed in section

2 [15, 16, 19, 21, 22, 20, 23 , 24 , 25 ]. The work of Cost was cited in [23] as

600

900

1200

1500

1800

2100

2400

2700

Temperature K

0

0.05

0.10

0.15

0.20

Mole Fraction O

BCC+NbO

BCC+Liquid

BCC

Figure 7: Calculated Nb-rich portion of the Nb-O system phase diagram

togetherwiththeexperimentaldatareviewedinsection2.Thesamesymbols

1500

1800

2100

2400

2700

Temperature K

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Mole Fraction O

BCC+NbO

NbO

2

Nb

2

O

5

Liquid

Gas

Gas

Figure 8: Calculated high-temperature portion of the Nb-O system phase

diagramwiththeexperimentaldatareviewedinsection2.Thesamesymbols

300

600

900

1200

1500

1800

2100

2400

2700

3000

TEMPERATURE_KELVIN

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

M

OLE_FRACT

ION

_O

BCC

NbO

NbO

2

Nb

2

O

5

Gas

Gas

Liquid

Figure 9: Calculated phase diagram for the binary Nb-O system using the

0

0.2

0.4

0.6

0.8

1

−400

−350

−300

−250

−200

−150

−100

−50

0

y

O

(−)

G

inter

(kJmol

−

1

)

The Nb−O System, bcc Phase

T = 1000 K

T = 1500 K

T = 2000 K

T = 2500 K

Figure10: The calculated Gibbs freeenergy of -Nb(in bcc phase) vs.the

0

0.2

0.4

0.6

0.8

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

y

O

(−)

X

O

(

−

)

The Nb−O System, bcc Phase

Figure 11: Relation between the sub-lattice site fraction y

O

and the mole

fractionofoxygenin-Nblattice, x

O .

0

0.2

0.4

0.6

0.8

−400

−350

−300

−250

−200

−150

−100

−50

0

X

O

(−)

G

inter

(kJmol

−

1

)

The Nb−O System, bcc Phase

T = 1000 K

T = 1500 K

T = 2000 K

T = 2500 K

Figure12: The calculated Gibbs freeenergy of -Nb(in bcc phase) vs.the

mole fraction of interstitial oxygen at dierent temperatures. The symbols

indicate the calculated Gibbs free energy of the stoichiometric compound

0

500

1000

1500

2000

2500

3000

−450

−400

−350

−300

−250

−200

T (K)

G

m

(kJmol

−

1

)

NbO

2

Nb

2

O

5

NbO

Figure13:ThecalculatedGibbsfreeenergyofstoichiometricniobiumoxides

asa function of temperature; note that the freeenergy is evaluated as per

900

1000

1100

1200

1300

1400

1500

−220

−210

−200

−190

−180

−170

−160

−150

−140

T (K)

G

m

(kJmol

−

1

)

NbO

2

Nb

2

O

5

NbO

Figure14: Thecalculated Gibbsfree energy offormation of niobiumoxides

as a function of temperature, as in Fig. 13, the free energy is evaluated as