ISSN 0021-3640, JETP Letters, 2008, Vol. 88, No. 7, pp. 435–440. © Pleiades Publishing, Ltd., 2008.Original Russian Text © S.V. Rempel’, A.I. Gusev, 2008, published in Pis’ma v Zhurnal Éksperimental’no
œ
i Teoretichesko
œ
Fiziki, 2008, Vol. 88, No. 7, pp. 508–513.
435
The separation of one of the components of a solid-phase system on the surface as a separate phase isobserved in multicomponent systems that are diluted(in one of the components) solid solutions. The surfacesegregation appears in various substitution systemssuch as metal alloys Fe–Sn, Au–Ni [1], Zr–Hf, and Zr–Nb [2] and oxide systems CaO–MgO [3], Ca–Y
3
Al
5
O
12
, and Ca–M–Y
3
Al
5
O
12
(M = Sr, Nd, Cr) [4].
The separation of the second phase from the homo-geneous solid solution is possible if its content exceedsthe solubility limit. In this case, the separation canoccur in the bulk or on the surface. According to [2], thesurface segregation is observed at a noticeable differ-ence between the surface tensions of the system com-ponents; the component with the lower surface energyis separated. In turn, the surface energy of a componentcorrelates with its sublimation (evaporation) heat andmelting temperature. In view of this circumstance, thesurface segregation of the component with the lowermelting temperature is more probable.
A decrease in the surface energy was initially con-sidered as the driving force of segregation. However, toexplain grain-boundary segregation, a decrease in thelattice strain energy associated primarily with the dif-ference between the sizes of mutually substituted atomsof the solid solution was treated as the driving force.Considering the surface and deformation energies sep-arately, one can arrive at qualitatively different predic-tions of segregation in the same diluted solid solution.In particular, the minimization of the surface freeenergy of the ideal solid solution of the A–B system,where the surface energy of component A is lower thanthat of component B, indicates that component A con-centrates on the surface of the solid solution of anycomposition. If only the strain energy is taken intoaccount, the component whose content in the solidsolution bulk is lower should be concentrated on the
surface. The simultaneous effect of the surface andstrain energies is more probable; in this case, theyenhance and weaken each other in solid solutionswhere
c
B
>
c
A
and
c
B
<
c
A
respectively.
In this work, the surface segregation of zirconiumcarbide from the (ZrC
y
)
1 –
x
(NbC
y
'
)
x
solid solution witha low content of zirconium carbide 0.001 < (1 –
x
)
≤
0.05, i.e., from the diluted ZrC solid solution in nio-bium carbide is discussed. The surface segregation hasnot yet been observed in carbide solid solutions andthere are no published data on this item; however, infor-mation on the zirconium segregation from a dilutedsolid solution of the related Zr–Nb system exists.According to the experimental [5–8] and theoretical [9–11] data, the Zr–Nb binary system in the solid state ischaracterized by the unlimited mutual solubility of
β
-Zr and
β
-Nb above 1261 K, whereas a break in mis-cibility and the decomposition region appear at
T
≤
1261 K. The decomposition region is wide: its bound-aries at 770 K already correspond to the Zr
1 –
x
Nb
x
solidsolutions with 1 and 95 at % Nb. The investigation [2]of the Zr–Nb solid solutions with a low Zr content (nomore than 1 at %) shows that the separation (segrega-tion) of zirconium atoms is observed on the surface ofa solid solution after 20-h heating in a vacuum of (3–4)
×
10
–8
Pa at temperatures of 1300–2100 K. The seg-regation of Zr atoms on the surface begins at a temper-ature of about 900–1000 K, and the zirconium concen-tration in the surface layer at 1300 K reaches 70–80 at %.In addition to zirconium, carbon also segregates on thesurface of Zr–Nb alloys with an admixture of 0.16 at %C in the temperature range 1000–1300 K: the carbonconcentration on the surface is 10–15 at %. The inves-tigation of the
β
Zr–Nb solid solution with 25 at % Nbshows that long-term aging at 600–800 K gives rise todecomposition with the formation of the
α
-Zr-enrichedphase [12].
Surface Segregation in Decomposing Carbide Solid Solutions
S. V. Rempel’ and A. I. Gusev
Institute of Solid State Chemistry, Ural Division, Russian Academy of Sciences, Yekaterinburg, 620041 Russiae-mail: [email protected]
Received July 17, 2008
The surface segregation of ZrC carbide from the ZrC
0.82
–NbC
0.83
solid solutions has been investigated usingthe X-ray diffraction, electron microscopy, X-ray microanalysis, and laser mass analysis. A method for includ-ing the energy of the interfaces in the free energy of mixing, which ensures the determination of the composi-tions of the solid solution in which the surface segregation can be observed, has been proposed for the carbidesolid solution.
PACS numbers: 64.70.Kb, 64.80.Eb, 64.80.Gd, 81.40.Cd, 82.69.Lf
DOI:
10.1134/S0021364008190065
436
JETP LETTERS
Vol. 88
No. 7
2008
REMPEL’, GUSEV
The (ZrC)
1 –
x
(NbC)
x
solid solutions with 0.001 <(1 –
x
)
≤
0.05 were synthesized using the solid-phasesintering of the NbC and ZrC carbides at a temperatureof 2300 K in a vacuum of 0.001 Pa or vacuum sinteringof Nb, Zr, and C at a temperature of 2500 K. The syn-thesized samples were additionally annealed with adecrease in the temperature from 1300 to 300 K with arate of 4 K/min.
The phase composition and lattice parameters ofvarious phases were determined using X-ray diffractionon a DRON-UM1 autodiffractometer. The diffractionmeasurements were performed using the Bragg–Bren-tano method in Cu
K
α
1, 2
radiation in the 2
θ
angularrange from 20
°
to 120
°
with a step of
∆
2
θ
= 0.02
°
. Inorder to determine the homogeneity degree of the syn-thesized solid solutions and to analyze their decompo-sition, X-ray diffraction patterns were obtained on aSiemens D-500 autodiffractometer in Cu
K
α
1, 2
radia-tion in the 2
θ
angular range from 10
°
to 158
°
in thestep-by-step scanning with a step of
∆
2
θ
= 0.02
°
and anexposition of 12 s at each point. The surface of the sam-
ples was analyzed on a JEOL-SuperProbe 733 electronprobe X-ray microanalyzer, which can operate simulta-neously in the regime of a scanning electron micro-scope, providing an image of the surface in back-reflected electrons and in the regime of an X-raymicroanalyzer spectrometer, ensuring control over thecomposition of segregating crystal grains in the charac-teristic X-ray radiation. The semiquantitative elementanalysis of the solid solution samples was performedwith an EMAL-2 laser energy–mass analyzer.
All of the synthesized solid solutions with a low ZrCcontent are single phase and have the
B
1 cubic struc-ture; the lattice period
a
B
1
increases from 0.4468 to0.4477 nm within a measurement error of 0.00002 nmwhen the ZrC content increases from 0.01 to 0.05. Afterannealing, the surface of the samples has the dark silvertint, whereas it is dark before annealing. Figure 1 showsthe X-ray diffraction patterns from the surface and bulkof the synthesized sample of the (ZrC)
0.02
(Nb)
0.98
solidsolution and of the same sample after annealing. The X-ray diffraction pattern (see Fig. 1) of the surface showsthat very intense lines of a new phase with the
B
1 struc-ture and with a lattice period of
a
B
1
= 0.46986 nm,which is close to the lattice period
a
B
1
= 0.4699 nm ofZrC
0.98
carbide [13]. In addition, traces (less than 1 mol %)of the (ZrC)
1 –
x
(NbC)
x
solid solutions with a high ZrCcontent and a low NbC content are present on the sur-face of the annealed samples. This follows from thepresence of a nearly constant intensity above the back-ground in a narrow angular range to the right from thezirconium carbide lines (see inset in Fig. 1). The latticeperiod of these solid solutions ranges from 0.4698 to0.4670 nm, which corresponds to the niobium carbidecontent
x
from 0.004 to 0.120. The synthesized(ZrC)
0.02
(Nb)
0.98
solid solution has the same latticeperiod
a
B
1
= 0.4469 nm both in the bulk and on the sur-face. The lattice period of the solid solution in the bulkslightly decreases (to
a
B
1
= 0.4466 nm) after annealing,because the most of the ZrC carbide passes from thebulk to the surface of the sample.
Electron microscopy reveals well-faceted grains ofthe second phase on the surface of the annealed(ZrC)
1
−
x
(NbC)
x
samples with (1 –
x
)
≤
0.05. The grainsizes are 3–10
µ
m (some grains up to 20
µ
m), and thesize of the grains of the main phase is about 1
µ
m. Thethree- and six-face shapes of the segregating particles(see Fig. 2) are characteristic of the [111] section ofcubic crystals.
Surface scanning with the detection of the charac-teristic radiation shows that the sample matrix containsniobium and the segregating grains of the new phasecontain zirconium and almost do not contain niobium(see Fig. 2). The semiquantitative element analysis withthe EMAL-2 laser energy–mass analyzer confirms thatthe matrix contains niobium and carbon, whereas themass analysis of the substance evaporating from largegrains of the segregating phase shows that its maincomponents are zirconium and carbon.
Fig. 1.
X-ray diffraction patterns from the bulk and surfaceof the synthesized sample of the (ZrC)
0.02
(NbC)
0.98
solidsolution and of the same sample after annealing: the bulk ofthe annealed sample contains only the solid solution withthe
B
1 structure and a lattice period of
a
B
1
= 0.4466 nm,which is slightly smaller than the period
a
B
1
= 0.44687 nmof the synthesized sample; the cubic phase with a latticeperiod of
a
B
1
= 0.46986 nm corresponding to zirconiumcarbide is observed on the surface of the annealed sample.The presence of a nearly constant intensity above the back-ground in a narrow angular range to the right from the zir-conium carbide lines (see inset) implies that traces (lessthan 1 mol %) of the (ZrC)
1 –
x
(NbC)
x
solid solutions witha low NbC content (0.004 <
x
< 0.120) are present on thesurface.
JETP LETTERS
Vol. 88
No. 7
2008
SURFACE SEGREGATION IN DECOMPOSING CARBIDE SOLID SOLUTIONS 437
The segregation of the second phase—ZrC grains—on the surface of the samples can be due to the decom-position or the initial inhomogeneity of the solid solu-tions. The diffraction measurement shows that thehomogeneity degree of the initial solid solutions is~0.994–0.997, i.e., is close to unity. Therefore, theappearance of the second phase after annealing can bedue to the decomposition of the solid solutions ratherthan to their composition inhomogeneity.
A necessary condition of segregation is the presenceof the solid-phase decomposition region, because thesurface segregation of one of the components of a solidsolution is possible if its content is higher than the sol-ubility limit [4].
The previous calculation [14] of the phase diagramof the ZrC
y
–NbC
y
'
system shows that zirconium, ZrC
y
,and niobium, NbC
y
'
, carbides with any carbon contentin the homogeneity regions of the cubic phases form acontinuous series of solid solutions at
T
> 1200 K; how-ever, the decomposition region in the solid state existsin this system at lower temperatures. The maximumtemperature of the decomposition of solid solutions,
, increases from 843 K for the ZrC
1.0
–NbC
1.0
section to 1210 K for the ZrC
0.60
–NbC
0.70
section. Thedecomposition region is asymmetric and its vertex isshifted towards NbC
y
niobium carbide. This means thatthe solubility of ZrC
y
in niobium carbide at
T < Tdecompis several times lower than the solubility of NbCy inZrCy. Thus, the necessary condition of segregation—the presence of the decomposition region—is satisfiedin carbide solid solutions. The sufficient conditions ofthe surface segregation are associated with the segrega-tion energy.
According to [15], the fluctuation mechanism ofdecomposition occurs in carbide solid solutions. In thiscase, the growth of grains of the new phase in the sur-face layer is facilitated due to the positive role of theenergy of the interfaces [16]; for this reason, the segre-gation of one of the phases on the surface is possibleeven at a relatively low temperature.
In the models [1, 3, 17] of the equilibrium state ofthe surface of strongly diluted solid solutions, whichare based on the regular approximation, it is assumedthat the bulk and surface phases coexist in a solid underthe equilibrium conditions. In other words, it isassumed that segregation on the surface already existsand, therefore, the model of the equilibrium state onlydescribes the segregation phenomenon rather than pre-dicts it.
According to [1, 3], the segregation energy is∆Hseg = ∆Hint + ∆Hbin + ∆Hdef, where ∆Hint is the energyof the interfaces, ∆Hbin is the energy of the binary inter-atomic interactions, and ∆Hdef is the deformationenergy. All of the contributions and segregation energy,which is their sum, are independent of the compositionof a solid solution and temperature in contradiction toreality. The inclusion of the contributions ∆Hbin and
Tdecompmax
∆Hdef is an attempt to take into account the possibilityof the solid-phase decomposition, i.e., the necessarycondition of segregation. The model of subregular solu-tions [13] was used in [14] to calculate the decomposi-tion region. In that model, the contributions having thesame physical meaning as ∆Hbin and ∆Hdef, namely, theelectron interaction parameter and the parameter of theelastic distortions of the lattice, are already included inthe mutual exchange energy and free energy of the sys-tem. Moreover, these contributions are functions of thecomposition and temperature. For this reason, todescribe segregation in the subregular approximation,the only contribution that should be additionally takeninto account is the energy of the interfaces ∆Hint. In [1,3], the energy of the interfaces was taken as ∆Hint =(γA – γB)SBNA, where γA and γB are the specific (per unitarea) energies of the interfaces of solute A and solventB, respectively; SB is the surface area per molecule ofsolvent B; and NA is the Avogadro number. This is avery rough approximation, because ∆Hint appears to beindependent of the composition of the solid solution.Let us discuss how this dependence can be taken intoaccount.
The decomposition of the solid solution leads to theappearance of the interfaces; this introduces an addi-tional positive contribution to the free energy of the sys-tem. The segregation of a certain phase on the surfacereduces the area of the interfaces and is accompaniedby a decrease in the free energy, i.e., by the transition ofthe system to a more stable state. Let us represent theenergy of the interfaces as
(1)∆H int x( ) γ A γ B–( ) 2x 1–( )S x( ),=
10 µm
Zr
Nb
Fig. 2. Distributions of the intensities of the characteristicX-ray radiation of Zr and Nb from the surface scanning ofthe annealed solid solution (ZrC)0.02(NbC)0.98 along thehorizontal line. The maxima of the X-ray radiation of Zrcorrespond to the grains of the segregating ZrC zirconiumcarbide and the maxima of the X-ray radiation of Nb corre-spond to the matrix solid solution with 98 mol % NbC.
438
JETP LETTERS Vol. 88 No. 7 2008
REMPEL’, GUSEV
and take that γB > γA for certainty. In Eq. (1), x ≡ xB isthe relative (in mol %) content of the second compo-nent, i.e., component B with a higher specific surfaceenergy and S(x) is the area of the interface between twophases per mole of the solid solution. Only the compo-nent with a lower specific surface energy can segregateon one side of the interface and only the impurity com-ponent, i.e., the component with the lower content inthe solid solution can segregate on the other side of theinterface. To take into account these physical con-straints, the normalizing factor 2x – 1 is introduced; itdetermines the sign of the energy of the interfaces as acontribution to the free energy of the system.
Let the lattice period of the A1 – xBx cubic solid solu-tion additively vary as a function of the composition,i.e., a(x) = aA(1 + kx), where k = (aB – aA)/aA. In thiscase, the solid solution with the volume concentrationc = [(1 + kx)3 – 1]/[(1 + k)3 – 1] of the B component has
the molar volume V(c) = NA (1 + kx)3/4. If the parti-cles of two phases fill the space without voids betweenthem, the crystal can be represented as a set of particlesin the form of the Voronoi polygons, i.e., the distortedWigner–Seitz cell. The total number of particles in themolar volume V(c) is N = V(c)/ν, where ν is the volumeof one particle. For a particle with z faces each with thearea s, the area of the surface of the interfaces is S =zc(1 – c)V(c)s/ν. In carbide solid solutions under dis-cussion with the substitution of metal atoms in the fccsublattice, the Wigner–Seitz cell is a rhombic dodeca-hedron [15] with z = 12 faces and the centers of the cellscoincide with the sites of the crystal lattice. If the char-acteristic size of the particle (distance between the cen-ters of the neighboring particles) in the form of the
rhombic dodecahedron is D, then s = ( /4)D2, ν =
( /2)D3, and s/ν = 1/2D. Therefore, the area of theinterfaces in the decomposing A1 – xBx solid solution asa function of the content x is given by the expression
(2)
If the mean size of the particles is independent of thecontent, the energy of the interfaces is expressed as
(3)
The dependence of the size of segregating particles onthe composition of the solid solution can be taken intoaccount as follows. The minimum size of the segregat-
aA3
2
2
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 γ A γ B– 2x 1–( )=
× 1 kx+( )3
D--------------------- 1 kx+( )3 1–
1 k+( )3 1–------------------------------
⎩ ⎭⎨ ⎬⎧ ⎫
1 1 kx+( )3 1–
1 k+( )3 1–------------------------------–
⎩ ⎭⎨ ⎬⎧ ⎫
.
ing particle of the impurity component is Dmin =
( /2)a; hence, Dmin B(x = 0) = ( /2)aB and
Dmin A(x = 1) = ( /2)aA. As x decreases from 1 to acertain bound value xb in the range 0.5 ≤ x ≤ 1 (or as thecontent of impurity component A increases from 0 to(1 – x) ≤ (1 – xb), the size of segregating particles DAincreases to a certain limiting value; with a furtherdecrease in x from xb to 0.5, the size remainsunchanged. Thus, a change in the size of segregatingparticles as a function of the composition in the range0 ≤ (1 – x) ≤ 0.5 (i.e., 0.5 ≤ x ≤ 1.0) in the first approx-imation can be represented as
(4)
Here, xb = const is the bound content of the secondcomponent, which corresponds to the limiting size ofthe particles of the segregating phase and
fH(x – xb) =
is the Heaviside step function. Similarly, for the range0 ≤ x ≤ 0.5, where component B is an impurity, one canobtain the following symmetric expression for the sizeDB of the B phase:
(5)
Since the B and A components are impurities in theranges x < 0.5 and x > 0.5, respectively, the generalexpression for D(x) has the form
(6)
where DA(x) and DB(x) are specified by Eqs. (4) and (5),respectively. It is easy to see that all of the normalizationconditions—the particles of the B phase with the sizeDB(x 0) Dmin B segregate at x 1 and theparticles of the A phase with the size DA(x 1) Dmin A segregate at x 1—are satisfied when usingEqs. (4)–(6). In view of Eq. (6), the energy of the inter-faces between two phases has the form
(7)
where D(x) is specified by Eq. (6).To analyze the dependence of the energy of the
interfaces on content x of the B component in the
2 2
2
DA x( ) Dmin A 1 m 1 x – ( ) + [ ] 1 f H x x b – ( ) – [ ]{ =
+ 1
m
1
x
b
–
( )
+
[ ]
f
H
x x
b
–
( ) }
.
1, if x xb≤0, if x xb>⎩
⎨⎧
DB x( ) Dmin B 1 mx + ( ) 1 f H 1 x – x b – ( ) – [ ]{ =
+ 1
m
1
x
b
–
( )
+
[ ]
f
H
1
x
–
x
b
–
( ) }
.
D x( ) DA x( ) 1 f H x 0.5–( )–[ ]=
+ DB x( ) f H x 0.5–( ) Dmin A B ,( ) f D x ( ) , ≡
∆H int x( ) 32---NAaA
3 γ A γ B– 2x 1–( )=
× 1 kx+( )3
D x( )--------------------- 1 kx+( )3 1–
1 k+( )3 1–------------------------------
⎩ ⎭⎨ ⎬⎧ ⎫
1 1 kx+( )3 1–
1 k+( )3 1–------------------------------–
⎩ ⎭⎨ ⎬⎧ ⎫
,
JETP LETTERS Vol. 88 No. 7 2008
SURFACE SEGREGATION IN DECOMPOSING CARBIDE SOLID SOLUTIONS 439
A1 − xBx solid solution, it is convenient to represent∆Hint(x) as the dimensionless quantity
(8)
where fD(x) ≡ 1 when the size of the particles is D =Dmin = const and is independent of x. The ∆ (x)dependence is shown in Fig. 3. If the size D of segregat-ing particles is independent of the composition of thesolid solution, then ∆ is negative and positive forx < 0.5 and x > 0.5, respectively. The maximum of∆ is reached at x ≈ 0.8. The surface segregation ofthe segregating impurity phase reduces the content ofthe corresponding component in the solid solution andis thermodynamically possible if such a change in thecomposition of the solid solution is accompanied by adecrease in the free energy. As seen in Fig. 3, this con-dition is satisfied only for the range 0.8 ≤ x < 1.0. Thismeans that only the A component with a lower specificsurface energy can segregate. Surface segregation in
∆H int* x( )2∆H int x( )Dmin
3NAaA3 γ A γ B–
-------------------------------------=
= 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*
H int*
H int*
actual systems is observed in the narrow range 0.95–0.98 ≤ x < 1.0. Allowance for the dependence of the sizeof the particles on the composition of the solid solutionsignificantly narrows the segregation region (lines 2and 3 in Fig. 3). This confirms the assumption of theexistence of the D(x) dependence. The increase in sizeD of the particles of the segregating phase (increase inm) is accompanied by a decrease in the absolute valueof ∆ and by the narrowing of the concentrationinterval where surface segregation is possible.
With the inclusion of the energy ∆Hint(x) given byEq. (7), the free energy of the mixing of the A1 – xBx
solid solution in the model of subregular solutions [13]has the form
(9)
where Bs(x, T) is the energy of mutual exchange of thesolid solution.
Figure 4 shows the concentration dependences cal-culated by Eqs. (7) and (9) at T = 600 K for the energyof the interfaces ∆Hint(x) and mixing free energy Gs(x)of decomposing solid solutions ZrC0.82–NbC0.83. Themutual exchange energy Bs(x, T) of the solid solutions(ZrC0.82)1 – x(NbC0.83)x is taken from [13], and the latticeperiods a and specific surface energies γ of ZrCy andNbCy carbides are taken from [13] and [18], respec-
H int*
Gs x( ) x 1 x–( )Bs x T,( ) ∆H int x( )+=
+ RT x xln 1 x–( ) 1 x–( )ln+[ ],
Fig. 3. Model energy of the interfaces ∆ (x) =
2∆Hint(x)Dmin/(3NA |γA – γB |) versus the content x of the
B component in the A1 – xBx solid solution (the calculationwas performed for k = –0.078): (1) the size of the particlesof the segregating phase is independent of the composition(D = Dmin = const), (2) the composition dependence of theparticle size is described by function (6) with yb = 0.8 andm = 20, and (3) the content dependence of the particle sizeis described by function (6) with yb = 0.8 and m = 50. Thesegregation of the A component is possible only to the rightof the dashed line, where the decrease in the content of theA component in the solid solution is accompanied by adecrease in the energy of the interfaces.
Hint*
aA3
Fig. 4. Isotherms of the mixing free energy Gs(x) and theenergy of the interfaces ∆Hint(x) of the ZrC0.82–NbC0.83solid solutions at a temperature of T = 600 K: (1) the mixingenergy Gs(x) disregarding the energy of the interfaces,(2) the energy of the interfaces ∆Hint(x), and (3) the mixingenergy Gs(x) given by Eq. (9) with the inclusion of ∆Hint(x).The surface segregation of ZrC0.82 carbide is possible in theregion to the right of the dash–dotted line (at a niobium car-bide content higher than 91.5 mol %). In this region, thedecrease in the content of ZrC0.82 in the solid solution isaccompanied by a decrease in the mixing free energy. Thetangent to line 3 of the mixing free energy determines thecomposition of the phases into which the solid solutiondecomposes allowing for the surface segregation.
3
2
1
b
b
s
440
JETP LETTERS Vol. 88 No. 7 2008
REMPEL’, GUSEV
tively. The quantity D(x) is calculated at yb = 0.8 andm = 50. For comparison, we present the isotherm of themixing free energy calculated disregarding the energy∆Hint(x). As seen in Fig. 4, the inclusion of ∆Hint(x)leads to the appearance of a local maximum on the mix-ing free energy in the region of a low content of zirco-nium carbide in addition to two local minima indicatingthe decomposition of the solid solution into two phases.In the case of the surface segregation of zirconium car-bide, its content in the solid solution decreases and thefree energy Gs(x) decreases; i.e., the system passes to amore stable state. The calculation indicates that surfacesegregation is possible if the relative molar content ofZrC0.82 carbide is ≤0.085 (or the content of niobium car-bide x ≥ 0.915). This is sufficiently close to the experi-mental data according to which the surface segregationof zirconium carbide from the solid solution wasobserved at x ≥ 0.95. The tangent to line 3 of the mixingfree energy determines the composition of the phasesinto which the solid solution decomposes allowing forsurface segregation.
The proposed method for estimating the energy ofthe interfaces and determining the composition regionof the solid solution in which surface segregation ispossible is applicable to any substitution solid solu-tions.
This work was supported by the Russian Foundationfor Basic Research, project no. 06-03-32047.
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Translated by R. Tyapaev
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