5
0012-5016/03/0009- $25.00 © 2003 åÄIä “Nauka /Interperiodica” 0235 Doklady Physical Chemistry, Vol. 392, Nos. 1–3, 2003, pp. 235–239. Translated from Doklady Akademii Nauk, Vol. 392, No. 3, 2003, pp. 356–361. Original Russian Text Copyright © 2003 by Gusev. Spontaneous surface segregation of zirconium car- bide ZrC has recently been found in carbide solid solu- tions (ZrC y ) 1 – x (NbC y' ) x with x 0.95, which are formed in the ZrC y –NbC y' pseudobinary system [1–3]. Nobody has heretofore observed surface segregation in carbide solid solutions, although it is rather common in differ- ent substitutional systems, for example, in the metal alloys Au–Ni [4] and Zr–Nb [5] and the oxide systems CaO–MgO [6], Ca–Y 3 Al 5 O 12 , and Ca–M–Y 3 Al 5 O 12 (M = Sr, Nd, Cr) [7]. It is likely that information on sur- face segregation in other cubic carbide solid solutions (M (1) C y ) 1 – x (M (2) C y' ) x (M (1) , M (2) = Ti, Zr, Hf, V, Nb, Ta) is lacking because little is known about the phase equi- libria in the range of low concentrations of one of the constituent carbides. A necessary condition for segregation is the exist- ence of the solid-phase decomposition region [3], since the surface segregation of one of the components of a solid solution is possible if its content exceeds the sol- ubility limit [3, 7]. This allows one to specify pseudo- binary carbide systems promising for a search for sur- face segregation. The calculations of the phase diagrams of M (1) C– M (2) C systems formed by cubic (B1) carbides of Group IV and V transition metals [1, 8–10] showed that the pseudobinary systems ZrC 0.98 –TiC, TiC–HfC, VC 0.88 NbC, VC 0.88 –TaC, ZrC 0.98 –TaC, HfC–TaC, ZrC y NbC y' (0.60 y 0.98, 0.70 y' 1.00), and HfC–NbC under equilibrium conditions are characterized by infi- nite mutual solubility of the components in a definite temperature range, whereas solid-phase decomposition regions appear at a lower temperature T < . The phase diagrams were calculated using the subregular solution model [8–10], which implies the interchange energy in different phases to be a function of composi- tion and temperature. Limited experimental data on the location of decomposition regions are available only for the VC 0.88 –NbC and VC 0.88 –TaC systems [11, 12]. T decomp max In the subregular solution model, the deviation of a system from the ideal behavior is determined by the excess energy of mixing = B j ({x i },T ) , where B j ({x i },T ) is the interchange energy in the j th phase (an energy parameter that characterizes the inter- action of the components in the j th phase and is a func- tion of composition and temperature). Physically, the interchange energy B in regular or subregular solution models is the difference between the energies of pair- wise interactions of unlike (ε AB ) and like (ε AA , ε BB ) atoms; i.e., B = N A [2ε AB – (ε AA + ε BB )]. For a binary system in the subregular approximation, = x A x B B j ({x i }, T ) x(1 – x)B j (x, T ). In the regular approx- imation, the interchange energy is a nonzero constant independent of composition and temperature (B j = const); thus, = x A x B B j x(1 – x)B j . Let us consider a system in which the concentration of the second component is x. According to [8, 9], the interchange energy is B j (x) = B 0j + xB 1j in carbide, nitride, or boride systems. The interchange energy of a liquid phase is the sum of the electron interaction parameter e 0 and the internal pressure parameter e p , B l (x) = e 0 + e p . The interchange energy of a solid phase includes, in addition to e 0 and e p , the parameter of elec- tron interaction in the solid phase e 1 and the parameter of elastic distortions of the crystal lattice e 2 , i.e., B s (x) = e 0 + e p + e 1 + e 2 . The e 0 , e p , e 1 , and e 2 parameters were calculated by the formulas in [8, 9]. Formulating the free energy of all phases in the system as (1) with regard to the explicit form of the phase inter- change energies and solving a set of equations describ- ing the equilibrium conditions, the authors of [9] found the positions of the liquidus, solidus, and decomposi- tion regions in the above carbide systems. The phase equilibrium conditions are invariant with respect to the addition of an arbitrary linear function of composition to the energy of mixing G j (x) = x(1 – x)(B 0j + xB 1j ) + RT[x ln x + (1 – x)ln(1 – x)]. Thus, the phase equilibria G j e x i j (29 G j e G j e G j xT , ( 29 1 x ( 29 G A j T ( 29 xG B j T ( 29 + = + x 1 x ( 29 B j xT , ( 29 RT x x ln 1 x ( 29 1 x ( 29 ln + [ ] + PHYSICAL CHEMISTRY Analysis of Surface Segregation and Solid-Phase Decomposition of Substitutional Solid Solutions A. I. Gusev Presented by Academician G.P. Shveikin April 2, 2003 Received April 9, 2003 Institute of Solid-State Chemistry, Ural Division, Russian Academy of Sciences, ul. Pervomaiskaya 91, Yekaterinburg, 620219 Russia

Analysis of Surface Segregation and Solid-Phase Decomposition of Substitutional Solid Solutions

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0012-5016/03/0009- $25.00 © 2003

åÄIä “Nauka

/Interperiodica”0235

Doklady Physical Chemistry, Vol. 392, Nos. 1–3, 2003, pp. 235–239. Translated from Doklady Akademii Nauk, Vol. 392, No. 3, 2003, pp. 356–361.Original Russian Text Copyright © 2003 by Gusev.

Spontaneous surface segregation of zirconium car-bide ZrC has recently been found in carbide solid solu-tions

(ZrC

y

)

1 –

x

(NbC

y

'

)

x

with

x

0.95

, which are formedin the

ZrC

y

–NbC

y

'

pseudobinary system [1–3]. Nobodyhas heretofore observed surface segregation in carbidesolid solutions, although it is rather common in differ-ent substitutional systems, for example, in the metalalloys Au–Ni [4] and Zr–Nb [5] and the oxide systemsCaO–MgO [6],

Ca–Y

3

Al

5

O

12

,

and

Ca–M–Y

3

Al

5

O

12

(M = Sr, Nd, Cr) [7]. It is likely that information on sur-face segregation in other cubic carbide solid solutions(

M

(1)

C

y

)

1 –

x

(M

(2)

C

y

'

)

x

(M

(1)

, M

(2)

= Ti, Zr, Hf, V, Nb, Ta)is lacking because little is known about the phase equi-libria in the range of low concentrations of one of theconstituent carbides.

A necessary condition for segregation is the exist-ence of the solid-phase decomposition region [3], sincethe surface segregation of one of the components of asolid solution is possible if its content exceeds the sol-ubility limit [3, 7]. This allows one to specify pseudo-binary carbide systems promising for a search for sur-face segregation.

The calculations of the phase diagrams of

M

(1)

C–M

(2)

C

systems formed by cubic (

B

1

) carbides of GroupIV and V transition metals [1, 8–10] showed that thepseudobinary systems

ZrC

0.98

–TiC, TiC–HfC, VC

0.88

–NbC, VC

0.88

–TaC, ZrC

0.98

–TaC, HfC–TaC

,

ZrC

y

–NbC

y

'

(0.60

y

0.98

,

0.70

y

'

1.00

), and HfC–NbCunder equilibrium conditions are characterized by infi-nite mutual solubility of the components in a definitetemperature range, whereas solid-phase decomposition

regions appear at a lower temperature

T

<

. Thephase diagrams were calculated using the subregularsolution model [8–10], which implies the interchangeenergy in different phases to be a function of composi-tion and temperature. Limited experimental data on thelocation of decomposition regions are available onlyfor the

VC

0.88

–NbC

and

VC

0.88

–TaC

systems [11, 12].

Tdecompmax

In the subregular solution model, the deviation of asystem from the ideal behavior is determined by the

excess energy of mixing

=

B

j

({

x

i

},

T

)

,where

B

j

({

x

i

},

T

)

is the interchange energy in the

j

thphase (an energy parameter that characterizes the inter-action of the components in the

j

th phase and is a func-tion of composition and temperature). Physically, theinterchange energy

B

in regular or subregular solutionmodels is the difference between the energies of pair-wise interactions of unlike (

ε

AB

) and like (

ε

AA

,

ε

BB

)atoms; i.e.,

B

=

N

A

[2

ε

AB

– (

ε

AA

+

ε

BB

)]

. For a binary

system in the subregular approximation,

=

x

A

x

B

B

j

({

x

i

},

T

)

x

(1 –

x

)

B

j

(

x

,

T

)

. In the regular approx-imation, the interchange energy is a nonzero constantindependent of composition and temperature (

B

j

=

const); thus,

=

x

A

x

B

B

j

x

(1 –

x

)

B

j

.

Let us consider a system in which the concentrationof the second component is

x

. According to [8, 9], theinterchange energy is

B

j

(

x

) =

B

0

j

+

xB

1

j

in carbide,nitride, or boride systems. The interchange energy of aliquid phase is the sum of the electron interactionparameter

e

0

and the internal pressure parameter

e

p

,Bl(x) = e0 + ep. The interchange energy of a solid phaseincludes, in addition to e0 and ep, the parameter of elec-tron interaction in the solid phase e1 and the parameterof elastic distortions of the crystal lattice e2, i.e., Bs(x) =e0 + ep + e1 + e2. The e0, ep, e1, and e2 parameters werecalculated by the formulas in [8, 9]. Formulating thefree energy of all phases in the system as

(1)

with regard to the explicit form of the phase inter-change energies and solving a set of equations describ-ing the equilibrium conditions, the authors of [9] foundthe positions of the liquidus, solidus, and decomposi-tion regions in the above carbide systems. The phaseequilibrium conditions are invariant with respect to theaddition of an arbitrary linear function of compositionto the energy of mixing Gj(x) = x(1 – x)(B0j + xB1j) +RT[xlnx + (1 – x)ln(1 – x)]. Thus, the phase equilibria

G je xi

j( )∏

G je

G je

∆G j x T,( ) 1 x–( )GA j T( ) xGB j T( )+=

+ x 1 x–( )B j x T,( ) RT x xln 1 x–( ) 1 x–( )ln+[ ]+

PHYSICALCHEMISTRY

Analysis of Surface Segregation and Solid-Phase Decomposition of Substitutional Solid Solutions

A. I. GusevPresented by Academician G.P. Shveikin April 2, 2003

Received April 9, 2003

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

236

DOKLADY PHYSICAL CHEMISTRY Vol. 392 Nos. 1–3 2003

GUSEV

can be described by the relationship between the freeenergies of mixing of phases at equilibrium.

The existence of solid-phase decomposition regions

in the carbide systems means that, at T < , theymeet the necessary condition for segregation. Sufficientconditions for surface segregation are associated withthe segregation energy and diffusion.

The diffusion decomposition of solid solutions canfollow two mechanisms [13]. One of them—spinodaldecomposition—occurs everywhere over the volume ofa solution without nucleation of a new phase and with acontinuous (without a jump) decrease in the free energyof the system. The other mechanism involves fluctua-tion nucleation of phases and their subsequent growth.As shown in [3], the first decomposition mechanism isnot realized in carbide solid solutions, since, at about1500 K and below, the diffusion mobility of atoms in acrystal is too low for spatial separation of phases withdifferent carbide contents. In fluctuation nucleation, thegrowth of grains of a new phase in the surface layer isfacilitated due to the positive role of the interfaceenergy [13]; thus, segregation of one of the phases tothe surface is possible even at a relatively low tempera-ture.

Regular solution models [4, 6, 14] of the equilib-rium state of the surface of very dilute solid solutionsimply that solids under equilibrium conditions com-prise volume and surface phases. In other words, it isassumed that surface segregation already exists and,thus, the equilibrium state model merely describes seg-regation rather than predicts it.

According to [4, 6, 14], the segregation energy—∆Hseg = ∆Hint + ∆Hbin + ∆Hdef—is contributed to by theinterface energy ∆Hint , the pairwise interaction energy∆Hbin, and the deformation energy ∆Hdef . All contribu-tions and the segregation energy as their superpositionare independent of the composition of a solid solutionand temperature, which is contrary to fact. Inclusion of∆Hbin and ∆Hdef in the segregation energy is actually anattempt to allow for the possibility of solid-phasedecomposition, which is the necessary condition forsegregation. However, in the subregular solutionmodel, the contributions that have the same physicalmeaning (the electron interaction parameter e1 and theelastic lattice distortion parameter e2) are already com-ponents of the interchange energy and the free energyof the system. Moreover, these energy contributions, aswell as the others, are functions of composition andtemperature. In effect, the only contribution that shouldbe additionally considered in the subregular approxi-mation is the interface energy. In [4, 6], the interfaceenergy is defined as

∆Hint = (γA – γΒ)SBNA, (2)

where γA and γB are the specific (per unit surface) inter-face energies of solute A and solvent B, respectively; SBis the surface area per solvent B molecule; and NA isAvogadro’s number. However, the interface energy

Tdecompmax

∆Hint (2) is a constant independent of both the compo-sition of a solid solution and the grain size of a precip-itated phase. This is a very rough approximation. In thiscontext, let us survey how to allow for the dependenceof ∆Hint on the composition of a solid solution.

The decomposition of a solid solution leads to theappearance of interfaces, which makes an additionalpositive contribution to the free energy of the system.Segregation of any phase to the surface decreases theinterface area and is accompanied by a decrease in freeenergy, i.e., by the transition of the system into a morestable state. Present the interface energy as

∆Hint(x) = |(γA – γΒ)|(2x – 1)S(x), (3)

where γA and γB are the specific surface energies ofcomponents A and B of the A–B binary solid solution(for uniqueness of our speculations, we assume thatγB > γA); x ≡ xB is the relative (in mole fractions) contentof the second component, i.e., component B with ahigher specific surface energy; and S(x) is the interfacearea between two phases per mole of the solid solution.Only the component with a lower specific surfaceenergy, on the one hand, and the impurity component,i.e., the component with a lower content in the solidsolution, on the other hand, can segregate to the surface.To take into account these physical limitations, the nor-malizing factor (2x – 1) is introduced, which deter-mines the sign of the interface energy as a contributionto the free energy of the system.

Let us determine the interface area between twophases upon segregation of component B from a cubicsolid solution A1 – xBx; its lattice constant follows Veg-

ard’s rule, i.e., a(x) = aA(1 + kx), where k = . If

the volume concentration of component B in the A1 – xBx

solid solution is c, the molar volume is V(c) =

NA[a(x)]3 = (1 + kx)3. At the same time, V(c) =

NA[(1 – c)VA + cVB], where VA = and VB = .

Thus, the volume concentration c can be written as c =

. Let the particles of two phases fill the

entire space, without free volumes among them. In thiscase, the crystal can be represented by a set of particlesshaped as Voronoi polyhedra, i.e., as a distortedWigner–Seitz cell. If the volume of one particle is v,then the total number of particles per molar volume

V(c) is N = . If a particle has z faces and the area

of each face is s, the number of faces shared by neigh-boring particles of different phases is n = zc(1 – c)N =

and the interface area is S = sn =

aB aA–aA

-----------------

14--- 1

4---NAaA

3

aA3

4-----

aB3

4-----

1 kx+( )3 1–

1 k+( )3 1–------------------------------

V c( )v

-----------

zc 1 c–( )V c( )v

----------------------------------

DOKLADY PHYSICAL CHEMISTRY Vol. 392 Nos. 1–3 2003

ANALYSIS OF SURFACE SEGREGATION AND SOLID-PHASE DECOMPOSITION 237

. For carbide solid solutions with sub-

stitution of metal atoms, we should consider theWigner–Seitz cell in the metal face-centered cubic (fcc)sublattice. The Wigner–Seitz cell in the fcc lattice is aregular rhombododecahedron [9] with twelve faces (z =12), the centers of the cells being the sites of the crystallattice. If the characteristic size of a rhombododecahe-dral particle (the distance between the centers of neigh-

boring particles) is D, then s = , v = , and

= . On this basis and using z = 12, V(c) =

(1 + kx)3, and the volume concentration c of the

second component expressed in terms of x, we obtainthe interface area in the decomposing solid solutionA1 – xBx as a function of the composition x:

(4)

If the average size of particles is independent of com-position, the interface energy ∆Hint is

(5)

Dependence of the size of precipitated particles onthe composition of the solid solution can be taken intoaccount as follows: The minimal size of the precipitated

particle of the impurity component is Dmin = , i.e.,

DminB(x = 0) = and DminA(x = 1) = . In the

range 0.5 ≤ x ≤ 1, when x decreases from 1 to xb (thecontent of the impurity component A increases from 0to (1 – x) ≤ (1 – xb)), the size of the precipitated particlesDA increases to some limiting value and does notchange with a further decrease in x from xb to 0.5. Thus,to a first approximation, the change in the size of pre-cipitated particles with changing composition in therange 0 ≤ (1 – x) ≤ 0.5 (or, which is the same, 0.5 ≤ x ≤1.0) can be represented as

(6)

zc 1 c–( )V c( )sv

------------------------------------

24

-------D2 22

-------D3

sv---- 1

2D-------

14--- NAaA

3

S x( ) 32---NAaA

3 1 kx+( )3

D---------------------=

× 1 kx+( )3 1–

1 k+( )3 1–------------------------------

1 1 kx+( )3 1–

1 k+( )3 1–------------------------------–

.

∆H int x( ) 32---NAaA

3 γA γB– 2x 1–( ) 1 kx+( )3

D---------------------=

× 1 kx+( )3 1–

1 k+( )3 1–------------------------------

1 1 kx+( )3 1–

1 k+( )3 1–------------------------------–

.

22

-------a

22

-------aB2

2-------aA

DA x( ) Dmin A 1 m 1 x–( )+[ ] 1 f H x xb–( )–[ ]{=

+ 1 m 1 xb–( )+[ ] f H x xb–( ) } ,

where xb = const is the boundary content of the secondcomponent, corresponding to the limiting size of theparticles of the precipitated phase;

where fH is the Heaviside function. Analogously, for therange 0 ≤ x ≤ 0.5, in which component B is an impurity,we can obtain the symmetric expression for the size DBof phase B:

(7)

where

Inasmuch as component B is an impurity at x < 0.5and component A is an impurity at x > 0.5, the generalequation for D(x) can be written as

(8)

where DA(x) and DB(x) are determined by Eqs. (6) and(7), respectively;

It is easily seen that the use of relationships (6)–(8)provides the fulfillment of all normalization conditions:at x → 0, phase B particles with the size DB(x → 0) →DminB are precipitated, while at x → 1, phase A particleswith the size DA(x → 1) → DminA are precipitated. Withallowance for the dependence of the size of precipitatedparticles on the composition of the solid phase, theinterface energy takes the form

(9)

where D(x) is determined by formula (8). Substitutingthe change in the lattice constant with temperature intoEq. (5) or (9), we find the interface energy as a functionof composition and temperature, ∆Hint(x, T).

To examine the dependence of the interface energyon the composition x of the A1 – xBx solid solution,

f H x xb–( )1, if x xb≤0, if x xb,>

=

DB x( ) Dmin B 1 mx+( ) 1 f H 1 x– xb–( )–[ ]{=

+ 1 m 1 xb–( )+[ ] f H 1 x– xb–( ) } ,

f H 1 x–( ) xb–[ ]1, if 1 x–( ) xb≤0, if 1 x–( ) xb.>

=

D x( ) DA x( ) 1 f H x 0.5–( )–[ ]=

+ DB x( ) f H x 0.5–( ) Dmin A B,( ) f D x( ),≡

f H x 0.5–( )1, if x 0.5≤0, if x 0.5.>

=

∆H int x( ) 32---NAaA

3 γA γB– 2x 1–( ) 1 kx+( )3

D x( )---------------------=

× 1 kx+( )3 1–

1 k+( )3 1–------------------------------

1 1 kx+( )3 1–

1 k+( )3 1–------------------------------–

,

238

DOKLADY PHYSICAL CHEMISTRY Vol. 392 Nos. 1–3 2003

GUSEV

∆Hint(x) is conveniently represented by the dimension-

less quantity :

(10)

Here, fD(x) ≡ 1 when the particle size D = Dmin = const

and is independent of x. Figure 1 shows the plot. If the size of precipitated particles is independentof the composition of a solid solution, is negative

at x < 0.5 and positive at x > 0.5. The largest value is attained at x ≈ 0.8. Surface segregation of theprecipitated impurity phase leads to a decrease in thecontent of the corresponding component in the solidsolution and is thermodynamically possible if thischange in the solid solution composition is accompa-nied by a decrease in free energy. Figure 1 shows thatthis condition is met only in the range 0.8 ≤ x < 1.0. Thismeans that only component A with the lower surfaceenergy can be a segregating component. In real sys-tems, surface segregation is observed in the narrow

∆H int* x( )

∆H int* x( )2∆H int x( )Dmin

3NAaA3 γA γB–( )

------------------------------------------=

= 2x 1–( ) 1 kx+( )3

f D x( )--------------------- 1 kx+( )3 1–

1 k+( )3 1–------------------------------

1 1 kx+( )3 1–

1 k+( )3 1–------------------------------–

.

∆H int* x( )

∆H int*

∆H int*

range 0.95–0.98 ≤ x < 1.0. Taking into account thedependence of the particle size on the composition of asolid solution noticeably narrows the segregationregion (Fig. 1, curves 2, 3). This supports the assump-tion that D is a function of x. The growth of particles ofthe precipitated phases (increase in m) is accompaniedby a decrease in magnitude and by narrowing ofthe concentration range where surface segregation ispossible.

In view of Eq. (9) for ∆Hint(x), the free energy ofmixing takes the form

(11)

The specific surface energies γ [15] and the latticeconstants a [9] of MCy carbides used in calculation ofthe interface energy (9) and the free energy of mixing(11) of carbide solid solutions are listed in the table.The interchange energies Bs(x, T ) of carbide solidsolutions required for this calculation were takenfrom [1, 9, 10].

As an example, Fig. 2 shows the plots of ∆Hint(x) (9)and Gs(x) (11) vs. concentration in decomposing car-bide solid solutions of the ZrC0.82–NbC0.83 system cal-culated for T = 600 K. D(x) was found using yb = 0.8 and

∆H int*

Gs x( ) x 1 x–( ) B0s xB1s+( )=

+ ∆H int x( ) RT x xln 1 x–( ) 1 x–( )ln+[ ] .+

0.2 0.4 0.6 0.8 1.00

–0.10

–0.05

0.05

0

0.10

x

∆Hint*

3

2

1

Fig. 1. Model interface energy (x) =

as a function of the composition x of the

A1 – xBx solid solution (calculated for k = –0.078): (1) thesize of the particles of the precipitated phase is independentof composition (D = Dmin = const); (2) the dependence ofthe particle size on composition is described by function (8)with yb = 0.8 and m = 20; (3) the dependence of the particlesize on composition is described by function (8) with yb =0.8 and m = 50. Segregation of component A is possibleonly in the region to the right of the dashed line, where thedecrease in the content of A is accompanied by a decreasein interface energy.

∆Hint*

2∆Hint x( )Dmin

3NAaA3 γA γB–

----------------------------------------

20 40 60 80 NbC0.83ZrC0.82mol %

–2

–1

0

1

∆Gs(

x), k

J m

ol–

1

∆Hin

t(x),

kJ

mol

–1

T = 600 K

1

2

3

Fig. 2. Isotherms of Gs(x) and ∆Hint(x) at T = 600 K forsolid solutions of the ZrC0.82–NbC0.83 system: (1) Gs(x) =x(1 – x)(B0s + xB1s) + RT[xlnx + (1 – x)ln(1 – x)] is theenergy of mixing without regard to the interface energy; (2)interface energy ∆Hint(x); (3) Gs(x) = x(1 – x)(B0s + xB1s) +RT[xlnx + (1 – x)ln(1 – x)] + ∆Hint(x) is the energy of mix-ing with allowance for ∆Hint(x). Surface segregation ofZrC0.82 is possible in the region to the right of the dotted-and-dashed line (at a content of niobium carbide exceeding91.5 mol %). In this region, the decrease in the content ofZrC0.82 in the solid solution is accompanied by a decreasein the free energy of mixing. The tangent to curve (3) deter-mines the compositions of the phases that appear, withregard to surface segregation, upon the decomposition ofthe solid solution.

DOKLADY PHYSICAL CHEMISTRY Vol. 392 Nos. 1–3 2003

ANALYSIS OF SURFACE SEGREGATION AND SOLID-PHASE DECOMPOSITION 239

m = 50. For comparison, Fig. 2 also shows the isothermof the free energy of mixing calculated without regardto the ∆Hint(x) energy. As can be seen, when ∆Hint(x) istaken into account, the curve of the free energy of mix-ing has, in addition to two local minima pointing to thedecomposition of the solid solution into two phases, alocal maximum in the range of low zirconium carbidecontents. If zirconium carbide segregates to the surface,its content in the solid solution and the free energyGs(x) decrease; i.e., the system transforms into a ther-modynamically stable state. The calculation shows thatsurface segregation is possible if the relative molar con-tent of ZrC0.82 is no more than 0.085 (or the niobiumcontent is x ≥ 0.915). This value is close to that experi-mentally found x ≥ 0.95, at which surface segregationof zirconium carbide from the solid solution wasobserved in [1–3]. The tangent to the curve of the freeenergy of mixing (3) determines the compositions ofthe phases that appear (with allowance for surface seg-regation) upon the decomposition of the solid solution.

Analogous calculations were performed for thesolid solution of the ZrC0.98–TiC, TiC–HfC, VC0.88–NbC, VC0.88–TaC, ZrC0.98–TaC, HfC–TaC, and HfC–NbC systems, in which decomposition regions existand surface segregation is possible. Our estimates showthat segregating components are VC0.88 in the VC0.88–NbC and VC0.88–TaC systems, zirconium carbide in theZrC0.98–TiC and ZrC0.98–TaC systems, HfC in the HfC–TaC and HfC–NbC systems, and titanium carbide in theTiC–HfC system.

ACKNOWLEDGMENTS

I am grateful to A.A. Rempel’ for valuable advice onestimation of the interface surface area.

This work was supported by the Russian Foundationfor Basic Research, project no. 03–03–32031.

REFERENCES1. Rempel’, S.V. and Gusev, A.I., Zh. Fiz. Khim., 2001,

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Lattice constant a and specific surface energy γ in MCy carbides, used for calculation of the interface energy ∆Hint(x) of solidsolutions

Parameter TiC1.0 ZrC0.98 ZrC0.82 HfC1.0 VC0.88 NbC1.0 NbC0.83 TaC1.0

a, nm [9] 0.4326 0.46967 0.47011 0.4641 0.4166 0.44695 0.44509 0.4456

γ, J m–2 [15] 2.22 2.13 2.13 2.29 2.44 2.60 2.60 2.73