Surface segregation in decomposing carbide solid solutions

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  • ISSN 0021-3640, JETP Letters, 2008, Vol. 88, No. 7, pp. 435440. Pleiades Publishing, Ltd., 2008.Original Russian Text S.V. Rempel, A.I. Gusev, 2008, published in Pisma v Zhurnal ksperimentalno

    i Teoretichesko

    Fiziki, 2008, Vol. 88, No. 7, pp. 508513.

    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 FeSn, AuNi [1], ZrHf, and ZrNb [2] and oxide systems CaOMgO [3], CaY

    3

    Al

    5

    O

    12

    , and CaMY

    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 AB 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 ZrNb system exists.According to the experimental [58] and theoretical [911] data, the ZrNb binary system in the solid state ischaracterized by the unlimited mutual solubility of

    -Zr and

    -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 ZrNb 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 (34)

    10

    8

    Pa at temperatures of 13002100 K. The seg-regation of Zr atoms on the surface begins at a temper-ature of about 9001000 K, and the zirconium concen-tration in the surface layer at 1300 K reaches 7080 at %.In addition to zirconium, carbon also segregates on thesurface of ZrNb alloys with an admixture of 0.16 at %C in the temperature range 10001300 K: the carbonconcentration on the surface is 1015 at %. The inves-tigation of the

    ZrNb solid solution with 25 at % Nbshows that long-term aging at 600800 K gives rise todecomposition with the formation of the

    -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: gusev@ihim.uran.ru

    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 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 segregationthe presence of the decomposition regionis 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 isHseg = 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 =(A B)SBNA, where A and B 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( ) A B( ) 2x 1( )S x( ),=

    10 m

    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 B > A 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 hasthe 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 distortedWignerSeitz cell. The total number of particles in themolar volume V(c) is N = V(c)/, where 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/. In carbide solid solutions under dis-cussion with the substitution of metal atoms in the fccsublattice, the WignerSeitz 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 therhombic dodecahedron is D, then s = ( /4)D2, =( /2)D3, and s/ = 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

    22

    S x( ) 32---NAaA3 1 kx+( )3

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

    1 kx+( )3 11 k+( )3 1

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

    1 1 kx( )3 1

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

    .

    H int x( ) 32---NAaA3 A B 2x 1( )=

    1 kx+( )3

    D---------------------

    1 kx+( )3 11 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 andDmin 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 normalizationconditionsthe 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 1are 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 22

    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 xb0, 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---NAaA3 A B 2x 1( )=

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

    1 kx+( )3 11 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 representHint(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( )2H int x( )Dmin3NAaA

    3 A B-------------------------------------=

    = 2x 1( ) 1 kx+( )3

    f D x( )---------------------

    1 kx+(...

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