0012-5016/00/0012- $25.00 © 2000

åÄIä “Nauka

/Interperiodica”0249

Doklady Physical Chemistry, Vol. 375, Nos. 4–6, 2000, pp. 249–253. Translated from Doklady Akademii Nauk, Vol. 375, No. 4, 2000, pp. 491–495.Original Russian Text Copyright © 2000 by Gusev, Rempel’.

Phase equilibria in the Zr–Nb–C ternary system orin partial sections of it are poorly studied [1–5]. Mostinformation concerns the temperature span from 1273to 2773 K. Low-temperature phase equilibria and theposition of the liquidus surface for ternary systems arepractically unstudied. However, at temperatures below1300 K, the carbon sublattice of nonstoichiometric car-bides

ZrC

y

and

NbC

y

, which exist in the Zr–Nb–Csystem, experiences disorder–order transitions to var-ious ordered phases [6]. Along with the ordering ofthe carbon sublattice of the aforementioned carbides,ordering or exsolution is possible in the metal sublat-tice of

ZrC

y

–Nb

solid solutions that exist in theZr–Nb–C system.

An experimental study of phase equilibria in the Zr–Nb–C system at 1973 [2] and 1773 K [3] and a theoret-ical analysis of equilibria suggested to occur in this sys-tem at 1273 K [4] showed that cubic zirconium and nio-bium carbides, whatever their carbon content, form acontinuous solid solution series. Note that cubic

NbC

y

(having the

B

1

structure) has a homogeneity region of

NbC

0.70

–NbC

1.00

and that

Zr

C

y

has a homogeneityregion of

ZrC

0.60

–ZrC

0.98

[7]. The solidus temperaturesof alloys have been determined, and isothermal sec-tions through the solidus surface of the Zr–Nb–C sys-tem for carbon percentages from 0 to 68 at. % havebeen obtained [5]. No experimental evidence concern-ing phase equilibria in the Zr–Nb–C or

ZrC

y

–Nb

system for temperatures below 1273 K were availableuntil 1999. However, the theoretical simulations of aphase diagram for the pseudobinary ZrC–NbC system,which is constituted of near-stoichiometric carbides,imply that an extensive exsolution region exists in thissystem at equilibrium at

T

< 800 K [8–10]. After homo-geneous dilute solid solutions

(ZrC

0.98

)

1 –

x

(NbC

1.00

)

x

containing 1 mol %

ZrC

0.98

had been annealed, zirco-nium carbide precipitated on their surface [9, 10]; theanneal was carried out at 2300 K with the temperatureslowly decreasing to 1000 K. This is experimental evi-dence of demixing of carbide solid solutions.

Cy'

Cy'

Structural ordering has been studied only in nonsto-ichiometric carbides

NbC

y

and

ZrC

y

[6, 7]. In particu-lar, a monoclinic (space group

C

2/

m

) ordered phase of

Nb

6

C

5

and a cubic (space group

Fd m

) or trigonal

(space group

R m

) ordered phase of

Zr

2

C

have beendiscovered. According to [6, 7], niobium and zirconiumcarbides undergo ordering at temperatures below 1300and 1170 K, respectively.

To design multicomponent materials with refractorycarbides, a knowledge of equilibrium phase diagramsof carbide-forming systems for both high temperatures(where equilibration is quite rapid) and low tempera-tures (where the diffusion mobility of atoms is low andfull equilibrium is unachievable) is pertinent. The liter-ature shows no successful attempts to map a phase dia-gram for the Zr–Nb–C system from low to high temper-atures on the basis of experimental data.

Here, we report on the first calculation of an equilib-rium phase diagram for the Zr–Nb–C system that takesinto account ordering in nonstoichiometric cubic

ZrC

y

and

NbC

y

phases (having the

B

1

structure) and the sub-solidus exsolution of

ZrC

y

–

NbC

y

'

carbide solid solutions.Our calculations have shown that cubic zirconium andniobium carbides form unlimited solid solutions at tem-peratures above 1210 K and that (

ZrC

y

)

1 –

x

(Nb

)

x

carbide solutions demix at

T

< 1200 K in the subsolidusregion.

It has been shown [7, 8] that systems involving com-pounds with gross nonstoichiometry are poorlydescribed by the regular solutions model. Therefore, asubregular solutions model was suggested for calculat-ing the phase diagrams of such systems [7, 8, 11]. Thedetails of this model, which takes into account varia-tions in the energy of exchange in a phase as a functionof composition and temperature, are described in [7].Its application to calculate phase diagrams of ternarysystems is illustrated in [12] for the Ti–B–C system.Therefore, here, we only mention the basic features ofthe model and some of its details that are essential tothe calculation of the subsolidus exsolution region inthe phase diagram of the Zr–Nb–C system.

3

3

Cy'

PHYSICALCHEMISTRY

Phase Diagram for the Zr–Nb–C System

A. I. Gusev and S. V. Rempel’

Presented by Academician G.P. Shveikin August 7, 2000

Received August 16, 2000

Institute of Solid-State Chemistry, Ural Division, Russian Academy of Sciences, Pervomaiskaya ul. 91, Yekaterinburg, 620219 Russia

250

DOKLADY PHYSICAL CHEMISTRY

Vol. 375

Nos. 4–6

2000

GUSEV, REMPEL’

For a fixed pressure, the Gibbs free energy of the

j

thphase in a multicomponent system can be written as

(1)

where

x

i

is the concentration of the

i

th component in the

j

th phase,

∆

G

j

(

x

i

= 1,

T

)

is the Gibbs free energy of a

component of the system, and

({

x

i

},

T

)

is the excessGibbs free energy of mixing of the

j

th phase. In the sub-regular solutions model, the deviation of the systemfrom ideality is governed by the value of the excess

Gibbs free energy of mixing

=

B

j

({

x

i

},

T

)

,where

B

j

({

x

i

},

T

)

is the energy of exchange in the

j

thphase, that is, the energy parameter that characterizesthe interaction of the components in the

j

th phase andthat is a function of composition and temperature.

Let us address a binary system in which the concen-tration of the second component in the liquid or solid is

x

. According to [7], for systems of nonstoichiometriccompounds, the energy of exchange is

B

j

(

x

) =

B

0

j

+

xB

1

j

. The energy of exchange for a liquid,

B

l

(

x

) =

e

0

+

e

p

, is the sum of the electron interaction parameter

e

0

and the internal pressure parameter

e

p

. Apart from

e

0

and

e

p

, the energy of exchange of a solid includes theelectron interaction parameter in the solid

e

1

and theparameter of elastic crystal-lattice distortion

e

2

; that is,

B

s

(

x

) =

e0 + ep + e1 + e2. By representing the Gibbs freeenergies of all phases of the system in the form of (1),taking into account the explicit form of the energy ofexchange for each phase, and solving the known set ofequations describing the conditions for phase equilib-ria, we can find the position of the liquidus and solidusfor the system in question. The calculation formulas fore0, ep, e1, and e2 are found in [7, 8, 12].

In terms of the subregular model, the excess Gibbs

free energy of mixing for a solid is = x(1 – x)(B0s +xB1s) and the Gibbs free energy of mixing for a solidGs(x) is

(2)

For theoretical analysis of the Gibbs free energy of asolid as a function of B0 and B1, a more convenient formof expression (2) is

(3)

where (x) = Gs(x)/RT*, b0s = B0/RT*, b1s = B1/RT*,and Tr = T/T* are dimensionless quantities.

∆G j xi{ } T,( ) xi∆G j xi 1 T,=( )i

∑=

+ RT xi xi G je xi{ } T,( ),+ln

i

∑

G je

G je xi

j( )∏

Gse

Gs x( ) = x 1 x–( ) B0s xB1s+( )+ RT x x 1 x–( ) 1 x–( )ln+ln[ ] .

Gs* x( ) x 1 x–( ) b0s xb1s+( )=

+ Tr x x 1 x–( ) 1 x–( )ln+ln[ ] ,

Gs*

Given a high enough temperature (Tr > 0.6), theGibbs free energy curve has one minimum [10], whichsuggests a single-phase state of the binary system, i.e.,the unlimited solid miscibility of the components. Thesecond minimum in the (x) curve for Tr < 0.6 meansthat the homogeneous solid solution demix into a pairof phases of different compositions—phases 1 and 2—in this range of temperatures. The conditions forphases 1 and 2 to be in equilibrium are given by the setof equations

(4)

Transforming set (4), the equilibrium conditions arewritten as

(5)

(6)

Here, x1 and x2 are the amounts of the second compo-nent in exsolution phases 1 and 2. Solving the set ofEqs. (5) and (6) yields the boundaries of the exsolutionregion.

The conditions for critical miscibility are the nullvalues of the second and third derivatives of the Gibbsfree energy of mixing with respect to composition:

(7)

From this, , the critical exsolution point, i.e., themiscibility-gap temperature, can be found; this is thetemperature above which the components have unlim-ited solid miscibility and below which an immiscibilityregion appears. The same expressions yield xmax, thecomposition of the solid solution corresponding to theupper critical exsolution point. Solving set (7) of equa-tions with referencing to Eq. (2) gives

(8)

(9)

Only those solutions to Eqs. (8) and (9) for which

≥ 0 and 1 ≥ xmax ≥ 0 are physically meaningful.

The liquidus surface of the Zr–Nb–C system and thesurface of the subsolidus exsolution region of ZrCy–Nb solid solutions were calculated in terms of thesubregular solutions model. The B0s and B1s exchangeparameters of the solid for selected ZrCy–Nbpseudobinary sections that were used to calculate theexsolution region are found in Table 1. The phase dia-

Gs*

∂G1 x( )/∂x 0, ∂G2 x( )/∂x 0.= =

RT 1 x1–( ) x12 B0s x1B1s+( ) x1

2 1 x1–( )B1s–+ln

= RT 1 x2–( )ln x22

B0s x2B1s+( ) x22 1 x2–( )B1s,–+

RT x1 1 x1–( )2 B0s x1B1s+( )+ln x1 1 x1–( )2B1s+

= RT x2 1 x2–( )2 B0s x2B1s+( ) x2 1 x2–( )2B1s.+ +ln

∂G2 x( )/∂x2 0, ∂G3 x( )/∂x3 0.= =

Tdecompmax

Tdecompmax 6B1sxmax

21 xmax–( )2/ 2xmax 1–( )R[ ] ,=

xmax19--- 4 B0s/B1s( )–{=

± 7 B0s/B1s( ) B0s/B1s( )2+ +[ ]1/2 } .

Tdecompmax

Cy'

Cy'

DOKLADY PHYSICAL CHEMISTRY Vol. 375 Nos. 4–6 2000

PHASE DIAGRAM FOR THE Zr–Nb–C SYSTEM 251

gram for the Zr–Nb system was mapped using experi-mental data from [13]. The phase diagrams for the Zr–C and Nb–C systems were calculated taking intoaccount ordering in the nonstoichiometric carbidesZrCy and NbCy and using the ordering parameter func-tional [6, 7].

The general view of the simulated phase diagram forthe Zr–Nb–C system over the range of 300–3900 K ismapped in Fig. 1. Figure 2 depicts the liquidus surfaceof the Zr–Nb–C system divided into four fields by thebinary eutectic curves that radiate from the invariantpoints of the binary systems. Each field corresponds tothe primary separation of one of the following phases:carbon C, a cubic carbide phase MCy (M = Zr1 – xNbx),hexagonal carbide phase Nb2Cy, and body-centeredcubic metal phase M (M = Zr1 – xNbx). The curves thatradiate from the invariant points of the Nb–C system,one associated with peritectic reaction p1 (L + NbCy ⇔Nb2Cy) and the other with eutectic reaction e2 (L ⇔ Nb +Nb2Cy), meet the curve that radiates from the invariantpoint of the Zr–C system and is associated with eutecticreaction e4 (L ⇔ β-Zr + ZrCy). This results in eutecticreaction E(L ⇔ M + Nb2Cy + MCy (M = Zr1 – xNbx))at 2503 K (Fig. 2). In other words, at ternary eutecticpoint E, melt L is in equilibrium with the Zr1 – xNbx

metal solid solution, Zr1 – xNbxCy carbide solid solution,and lowest hexagonal niobium carbide Nb2Cy. Thecurve radiating from point e4 passes through a mini-mum at 1995 K, which is near the azeotropic minimum(lying at T = 2013 K) of the liquidus curve of the Zr–Nb system, and then through a weak maximum withT = 2520 K, which is located near eutectic compositionE (Fig. 2).

Figure 3 maps isothermal sections through the surfaceof the exsolution region of the (ZrCy)1 – x(Nb )x carbidesolid solutions of the Zr–Nb–C system. Clearly, the upper

critical exsolution temperature rises as the

Cy'

Tdecompmax

amount of carbon in the carbide solid solution decreasesto reach the highest value for the (ZrC0.60)1 – x(NbC0.70)x

composition, which corresponds to the lower bound ofthe homogeneity region of the cubic carbide phase.

Thus, our simulation shows that for T > 1200 K.whatever the carbon content within the homogeneityregion of the cubic carbide phases, ZrCy and NbCy form

38503800

37003600

3400

30002800

3870

27422610

2800

1850

1594

900

12101100

800

Nb6C5Nb3C2Nb2Cy

1258

893

Nb

Zr

C

T, ä

3000

2000

1000

400

11601129

Zr6ë5Zr3ë2Zr2ë

38003570

36943600

3400

3125

210024002600

2128

600

800

3200

Fig. 1. Spatial phase diagram for the Zr–Nb–C ternary sys-tem for temperatures from 300 to 3900 K and isothermalsections of the liquidus surface and the surface of the sub-solidus exsolution region. Notations: Zr2C, Zr3C2, andZr6C5 are ordered phases of zirconium carbide ZrCy; Nb3C2and Nb6C5 are ordered phases of niobium carbide NbCy;and Nb2Cy is the lowest hexagonal niobium carbide. Thefigures in the diagram denote temperatures in Kelvindegrees.

Parameters B0s and B1s of the energy of exchange Bs = B0s + xB1s of the solid for selected pseudobinary joins of carbide solidsolutions (ZrCy)1–x(NbCy')x

Pseudobinary join B0s, kJ/mol B1s, kJ/mol

ZrC0.60 – NbC0.70 21.272 – 3.117 × 10–3T + 0.101 × 10–6T2 4.943 – 0.776 × 10–3T + 0.035 × 10–6T2

ZrC0.70 – NbC0.70 18.158 – 3.678 × 10–3T + 0.145 × 10–6T2 3.340 – 0.538 × 10–3T + 0.033 × 10–6T2

ZrC0.75 – NbC0.75 15.844 – 3.470 × 10–3T + 0.130 × 10–6T2 5.471 – 0.810 × 10–3T + 0.042 × 10–6T2

ZrC0.80 – NbC0.80 14.544 – 3.310 × 10–3T + 0.124 × 10–6T2 6.880 – 1.011 × 10–3T + 0.048 × 10–6T2

ZrC0.82 – NbC0.83 13.577 – 3.215 × 10–3T + 0.124 × 10–6T2 7.075 – 1.053 × 10–3T + 0.050 × 10–6T2

ZrC0.85 – NbC0.85 12.049 – 3.080 × 10–3T + 0.125 × 10–6T2 7.258 – 1.097 × 10–3T + 0.053 × 10–6T2

ZrC0.90 – NbC0.90 11.225 – 3.226 × 10–3T + 0.143 × 10–6T2 8.430 – 1.337 × 10–3T + 0.068 × 10–6T2

ZrC0.95 – NbC0.95 10.580 – 3.060 × 10–3T + 0.132 × 10–6T2 8.863 – 1.383 × 10–3T + 0.0.68 × 10–6T2

ZrC0.98 – NbC1.00 10.142 – 2.964 × 10–3T + 0.123 × 10–6T2 9.197 – 1.399 × 10–3T + 0.066 × 10–6T2

252

DOKLADY PHYSICAL CHEMISTRY Vol. 375 Nos. 4–6 2000

GUSEV, REMPEL’

a continuous series of solid solutions. However, thesubsolidus exsolution region exists at lower tempera-tures in this system. The upper critical exsolution tem-perature increases from 844 K for the ZrC0.98–NbC1.00

section (for 66.9 mol % NbC1.00) to 1210 K for theZrC0.60–NbC0.70 section (for 57.2 mol % NbC0.70). Theasymmetry of the exsolution region and the shift of theupper critical point toward NbCy in ZrCy–Nb sec-tions (66.9 mol % NbC1.0 for the ZrC0.98–NbC1.0 sectionand 56.3 mol % NbC0.70 for the ZrC0.70–NbC0.70 sec-tion) mean that the ZrCy solubility in niobium carbidefor T < Tdecomp is several times lower than the NbCy sol-ubility in ZrCy.

In terms of [13, 14], the Zr–Nb system is character-ized by the unlimited miscibility of β-Zr and β-Nb attemperatures above 1260 K; at T < 1260 K, a miscibil-ity gap appears to give rise to the subsolidus exsolutionregion. The exsolution region is very extensive. Even at770 K, its boundaries are Zr1 – xNbx solid solutions with1 and 95 at. % Nb. Therefore, there are grounds to sug-gest that the demixing of Zr1 – xNbxCy carbide solid solu-tions in the Zr–Nb–C system is genetically related tothe demixing of Zr1 – xNbx metal solid solutions.

The results of our simulation are consistent withexperimental evidence for the demixing of the carbidesolid solutions [10], as a result of which even a dilutesolution of zirconium carbide in niobium carbide,(ZrC)1 – x(NbC)x, precipitates single-crystal grains ofZrCy on its surface. Intriguingly, the surface segregationof zirconium atoms has been observed in a similar metalsystem Zr–Nb, which does not contain carbon [15].As already mentioned, a solid miscibility gap alsoexists in the Zr–Nb system.

ACKNOWLEDGMENTS

This work was supported by the Russian Foundationfor Basic Research, project no. 98-03-32856a.

REFERENCES

1. Koval’skii, A.E. and Umanskii, Ya.S., Zh. Fiz. Khim.,1946, vol. 20, no. 8, pp. 769–782.

2. Stecher, P., Benesovsky, F., Neckel, A., and Nowotny, H.,Monatsh. Chem., 1964, vol. 95, no. 6, pp. 1630–1645.

3. Fedorov, T.F., Popova, N.M., and Gladyshevskii, Yu.I.,Izv. Akad. Nauk SSSR, Met., 1965, no. 3, pp. 158–163.

4. Fedorov, T.F. and Gladyshevskii, Yu.I., Poroshk. Metall.(Kiev), 1965, no. 1, pp. 37–40.

5. Emel’yanov, V.S., Godin, Yu.G., Suchkov, I.I., and Yas-trebkov, A.A., Metallurgiya i metallovedenie chistykhmetallov (Physical Metallurgy of Pure Metals), Moscow:Atomizdat, 1967, no. 6, pp. 92–100.

6. Gusev, A.I., Usp. Fiz. Nauk, 2000, vol. 170, no. 1, pp. 3–40.

Cy'

Nb

ZrC

80

60

40

2080

60

40

20

20 40 60 80

700 K

900940

1100

1100

940900700

400500

600700

800

840

800 K

Nb, at. %C, at. %

Zr, at. %

ZrCy

2600

2520

2400

1995

2200

Nb, at. %60

80

20

C, at. %

Zr, at. %

Nb

ZrC

2600 K2800

32003400

3600

38503800

37003600

3700

3700

3000

3800

40

40

80

20 40 60 80

35003400

3200

36002600

2400

2200ZrC0.82

(3694 K)

NbC0.83

(3870 K)

e2

e4

e1

e3

p1 E

NbCy

Fig. 2. Liquidus surface for the Zr–Nb–C system. The tem-perature of ternary eutectic E(L ⇔ M + Nb2Cy + MCy (M =Zr1 – xNbx)) is 2503 K; the temperature of peritectic p1(L +NbCy ⇔ Nb2Cy) is 2800 K; the temperature of eutecticse1(L ⇔ NbCy + C), e2(L ⇔ Nb + Nb2Cy), e3(L ⇔ ZrCy +C), and e4(L ⇔ β-Zr + ZrCy) are 3570, 2610, 3125, and2100 K, respectively.

Fig. 3. Positions of carbide solid solutions Zr1 – xNbxCy(0 ≤ x ≤ 1.0, 0.6 ≤ y ≤ 1.0) with the B1 structure in the Zr–Nb–C system and isothermal sections of the surface of theZr1 – xNbxCy exsolution region.

DOKLADY PHYSICAL CHEMISTRY Vol. 375 Nos. 4–6 2000

PHASE DIAGRAM FOR THE Zr–Nb–C SYSTEM 253

7. Gusev, A.I., Fizicheskaya khimiya nestekhiometriches-kikh tugoplavkikh soedinenii (Physical Chemistry ofRefractory Nonstoichiometric Compounds), Moscow:Nauka, 1991.

8. Gusev, A.I., Zh. Fiz. Khim., 1985, vol. 59, no. 3, pp. 579–584.

9. Rempel’, S.V. and Gusev, A.I., Dokl. Akad. Nauk, 1999,vol. 367, no. 1, pp. 85–89 [Dokl. Phys. Chem. (Engl.Transl.), vol. 367, nos. 1–3, pp. 187–191].

10. Rempel’, S.V., Rempel’, A.A., and Gusev, A.I., Zh. Fiz.Khim., 2000, vol. 74, no. 3, pp. 412–417.

11. Gusev, A.I., Izv. Akad. Nauk SSSR, Neorg. Mater., 1984,vol. 20, no. 7, pp. 1132–1137.

12. Gusev, A.I., J. Solid State Chem., 1997, vol. 133, no. 1,pp. 205–210.

13. Flewitt, P.E.J., Acta Metall., 1974, vol. 22, no. 1, pp. 47–79.

14. Abriata, J.P. and Bolcich, J.C., Bull. Alloy Phase Dia-grams, 1982, vol. 3, no. 1, pp. 34–44.

15. Buribaev, I., Talipov, N., and Nurmanov, N.A., Izv. Akad.Nauk SSSR, Ser. Fiz., 1991, vol. 55, no. 12, pp. 2340–2343.