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    STATE KEY LABORATORY OF SOLIDIFICATION PROCESSING

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    Grain growth in nanocrystalline materials

    Feng Liu, Ke Zhang, Mingming Gong

    30 Nov 2010

    The Saudi International Nanotechnology Conference 2010

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    Contents

    Experiments

    Thermo-kinetics of grain growth

    Theoretical background of thermal stability

    Introduction

    Conclusions

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    1 Introduction

    1.1 Character and application of nano-material

    1.2 Importance of stability of nano-material

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    Character Application

    Nano-

    material

    Surface-effect

    Size-effect

    Quantum-effect

    Structure material

    Function material

    device

    1.1 Character and application of nano-material

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    1.2 Importance of stability of nano-material

    Stability

    grain sizeimportant factor

    reflecting properties ofnanocrystalline materials

    actual use

    grain growth

    performancedecreasing

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    2 Theoretical background ofthermal stability

    2.1 Thermodynamic models of grain growth

    2.2 Kinetic models of grain growth

    2.3 Advantage and disadvantage of previous models2.4 Development of current models

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    2.1 Thermodynamic models of grain growth

    1. Weissmllers model

    ---A concept of stabilization of NC solids was first presented.

    ( ) ( ) { } { } { }0

    , , ln

    M

    dilute M sol sol mix

    in M in GB g P T P T M

    N

    N H H T S N R N

    = +

    specific excess, , integral molar heats of solution in matrix and GB,

    excess entropy of mixing, number of atoms of component in the matrix.N

    sol

    MinHsol

    GBinH

    { }mixS MNNanostruct. Mater., 3(1993) 261

    Acta Mater., 50 (2002) 413

    2. Kirchheims model

    ---From a general thermodynamic consideration, an additional analytical

    model was derived.

    with is the GB energy for pure solvent, the saturated solute excess, the

    content of matrix and the enthalpy change.

    ( )0 0 lnb b g seg R T X H = +

    0 0b

    segH

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    3. Liu and Kirchheims model

    Liu and Kirchheims model

    Gibbs adsorptionequation

    McLeans GBsegregation model

    =i

    iidSdTd

    0 0 03ln b mb b g seg

    V R T X H D

    = +

    Metastable equilibrium grain size D*

    ( )0 0 0 03

    exp ( ) /

    b M

    b seg b g

    VD

    X H R T

    =

    F.Liu et al. J.Cryst.Growth, 264 (2004) 385

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    Validation in Ni-P and Pd-Zr alloys [Acta Mater., 48 (2000) 789; Z. Metallkd., 96 (2005) 1134]

    as prepared

    heat treated, 573K

    fits

    Relation of grain size and P-contents in NCNi-P alloys at 573K

    Relation of grain size and temperature in NCPd-Zr alloys at 573K

    It is shown that this model fits well with grain growth in NC Ni-P and Pd-Zr alloys. This further provides evidence

    that reduces to zero once grain growth is suppressed with saturated GBs.b

    0

    (J/ m2)

    Gseg

    (kJ/ mol)

    b

    (mol/ m2)

    b0

    (mol/ m2)

    Error

    (%)

    0.51 55.2 1.751.8510-5 1.8810-5 5

    Solute content 0

    (J/ m2

    )

    Gseg

    (kJ/ mol)

    b0

    (mol/ m2

    )

    Error

    (%)Pd90Zr10

    Pd85Zr15

    Pd80Zr20 (6731400 K)

    Pd80Zr20 (11731400K)

    0.7

    0.7

    0.7

    0.7

    55.9

    42.5

    59.2

    59.2

    2.0710-5

    3.0710-5

    1.7810-5

    1.7810-5

    2

    16

    7

    2

    Fitting parameters and errors of fits are given in Table 1 and 2

    Table 1

    Table 2

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    2.2 Kinetic models of grain growth

    1. Burke and Turnbulls model [Prog Met Phys., 3 (1952) 220]

    --- A so-called parabolic law applicable to grain growth in highly pure and coarse-grained polycrystalline materials was proposed.

    tMDDD

    M

    dt

    dDb

    b

    == 202

    2

    with Mas the GB mobility, the initial mean grain size at annealing time t=0 and

    the GB energy.

    2. Burkes solute drag model [Trans Metall Soc AIME., 175 (1949) 73]

    --- Regarding the effect of the solute, the case where a drag term is independent of

    the grain size is first considered.

    0D b

    tD

    k

    DD

    DD

    D

    DD

    max2

    max

    0max

    max

    0ln =

    +

    with as the maximum size and kinetic parameter.bk M=maxD

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    3. Michelss model [Acta Mater., 47 (1999) 2143 ]

    --- A grain-size-dependent drag term is introduced to restrain the grain growth andto stabilize the grain size of nano-material.

    ( )2

    1

    max2

    2

    0max2

    max2 2exp

    =D

    ktDDDD

    4. Rabkins model [Scripta Mater., 42 (2000) 1199 ]

    --- From Cahn, the drag term acts as function of both GB concentration and V,Rabkin then shows that,

    ( ) ( ) tMDDMDD b

    =+3

    0

    32

    0

    2

    32

    1

    with , =const.

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    2.3 Advantage and disadvantage of models

    Thermodynamic model favors a state or tendency,

    whereas, cannot describe the evolution of grain size

    Weissmllersmodel

    Liu et alsmodel

    Kirchheimsmodel

    Thermodynamics

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    Constant GB energy b is assumed and stabilized grain

    size cannot be determined. It points to a real process.

    Burke andTurnbulls

    RabkinsBurkes Michelss

    Kinetics

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    2.4 Development of current models

    Advantage and

    disadvantage of

    thermodynamic

    and kinetic models

    Establish an intact nano-graingrowth model considering mixed

    effect of thermodynamicsand kinetics

    Factors affecting

    grain growth

    Until now, thermodynamics and kinetics of grain growth have been studied onlyindependently of one another for NC materials.

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    3 Thermo-kinetics of grain growth

    3.1 Validity of thermo-kinetic models

    3.2 State criterion of initial GB segregation3.2.1 Thermodynamic state with saturated GB segregation

    3.2.2 Evolution of thermodynamic factors upon grain growth with unsaturated

    GB segregation

    3.3 Thermo-kinetic models of grain growth3.3.1 Thermo-kinetic models with saturated GB segregation

    3.3.2 Thermo-kinetic models with unsaturated GB segregation

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    3.1 Validity of thermo-kinetic models

    Kinetic process Thermodynamicstate

    Lius GB energymodel

    Both Kinetic process and Thermodynamic state can lead to the same GB energy model.

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    The coupling of GB diffusion, bulk diffusion (kinetics) and GB energy (thermodynamics) is reasonable.

    Kinetic process

    2

    0 0

    ln ln

    b

    l

    b a a

    DkT Dm ma a

    =

    ( )

    ( )

    ( )

    ( )

    ( )

    ( )

    ( )

    0 0

    0 0 0 0

    0 0 0

    0 0

    0 0 0

    b b b

    fb mb bb g l b l b

    fb fb f f l l l

    fb mb bb g l l l b

    b b g mb mb m mb b b

    f m b g l b l b

    bb bb b b

    l l l f m b g

    S S S R H H H H

    S S S R R T H H H H

    S S S R H H H H

    S S S R

    + + + + + = + + + + + + + + + +

    ln X segH

    0 0 lnb b g seg R T X H = +

    b l bG G =

    Borisov semi-empirical equation[Fiz. Metal. Metall., 17 (1964) 80]

    The term ofentropy change

    The term ofenthalpy change

    Lius GB energy model

    F.Liu et al. Acta Mater., 57 (2009) 1466

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    Both kinetic process and thermodynamic state lead to the same GB energy model. This further proves that

    incorporation of GB energy decreasing with GB segregation into the parabolic kinetics is physically practicable.

    R. Kirchheim, Acta Mater., 50 (2002) 413

    F.Liu et al. J.Cryst.Growth, 264 (2004) 385

    Thermodynamic state

    Kirchheims model:The segregation enthalpy movesolute atoms from the GBs into thegrains. The changes of theconfigurational entropy is( ). Assuming further

    that the GBs are always saturatedwith solute atoms ( ) leads to Lius GB energy model

    +=

    seg

    g

    bbH

    XTR

    ln00

    McLeans GB segregation model

    = TR

    H

    X

    X

    XX

    X

    g

    seg

    GBGB

    GB

    exp10

    ==

    dX

    d

    TR

    X

    d

    d

    g

    b

    Gibbs adsorption equation

    0 0ln

    b b g seg R T X H = +

    segH

    ( ) lnb gda R T X

    0b b =

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    3.2 State criterion of initial GB segregation

    Upon grain growth, the evolution of GB energy and GB excess amount with grain size must depend on whetherGB segregation is saturated or not. So it is necessary to analyze the initial GB excess amount and evaluate thestate of initial GB segregation.

    0

    exp1

    segGB

    GB GB g

    HX X

    X X X R T

    =

    McLeans GBsegregation modelConservationlaw Conservationlaw

    0 3 /b M X V D = b GBX =

    ( )03

    0

    1

    exp1

    seg

    g

    b M

    b

    Hb R T

    V

    DX

    +

    ( ) ( )0

    0

    0

    3

    00

    1 1

    exp exp1 1

    seg seg

    g g

    b M

    b

    H Hb R T R T

    V

    DXX

    < +++

    >>+

    =

    1112211

    121211

    11

    2

    0

    2

    ... nnnn

    t

    tttttkttktk

    tttttktk

    tttk

    DDLL

    1 Lius thermo-kinetic model

    with k1>k2>>kn and n can be chosen as 2 or 3.

    F.Liu et al. Thermochimica Acta, 443 (2006) 212

    Lius model

    ---Liu et al introduced variable activation energy Q and into the parabolic growthlaw.

    b

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    Validation in nano-RuAl alloy [Acta Mater., 49 (2001) 395]

    The grain size, D, of nano-RuAl produced bymechanical alloying, was evaluated as a

    function of annealing time, t

    n = 2

    n = 3

    The present model can describe the experiment results well (n=2 and n=3).

    ( )12

    2 2 2

    max max 0 2

    max

    2exp

    kt D D D D

    D

    =

    Michels,

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    2 Thermo-kinetic numerical model

    Lius modelParabolic

    equation

    Numerical model

    +=

    seg

    g

    bbH

    XTR

    ln00

    Numerical model of thermo-kinetics

    0 0 0

    3ln b mb g seg

    V R T X H

    DdDM

    dt D

    + =

    ( )

    ( )

    33

    0 3

    330

    0 3

    exp

    1 1

    b

    segb

    b b gb

    D DXHD

    R TD DX

    D

    = +

    McLeans GB

    segregation model

    =

    TR

    H

    X

    X

    XX

    X

    g

    seg

    GBGB

    GB exp10

    Conservation

    law

    ( )fX

    fXx GB

    +=

    1

    0

    F.Liu et al. J. Alloys Compd., 475 (2009) 893

    Two-stage (i.e., thermo-kinetic) fittingis applied,

    1. Determination of thermodynamicparameters: , , and

    2. Determination of kineticparameters: M

    0 b 0b segH

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    Validation in nano-CGO [Acta Mater., 54 (2006) 1721]

    Average grain size of the CGO spray pyrolysis films as afunction of annealing time and temperature

    The generalized parabolic growth model cannot predict well, whereas a good fit to the experimental data is

    obtained using numerical model.

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    Effect of GB energy

    obviously tends to its saturated value , a metastable equilibrium, i.e. GB energy tends to zero, results and

    grain growth stops.

    b 0b

    The GB energy vs. grain size with the temperaturesA plot of solute excess vs. grain size fordifferent annealing temperatures

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    Regarding the effect of the solute segregation, Q=2.00.3 eV>Qsurface=1.30.1 eV.

    The consumed annealing time before grain growth stops should be determined by the GB mobility which is

    described by annealing temperature and atom diffusion activation energy.

    = TR

    QMM

    g

    exp0

    Effect of interface mobility

    The GB energy vs. annealing time with the temperatures Arrhenius plot of the GB mobility Magainst

    the reciprocal annealing temperature

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    0

    0 0

    bGB

    V V X X X V V

    = + ( ) XXGBGBb =

    Conservationequation

    ( ) 06

    b GB

    D X X X

    = +

    ( )00

    6

    6

    seg

    b

    H X D

    +=

    GB excess

    bsegb H = 0

    Krills model

    Simplified GB energy model

    It is consistent with the phase-field approach in which the GB energy reduces linearly;

    Only one parameter is used.seg

    3 Thermo-kinetic analytical model

    F.Liu et al. Acta Mater., 57 (2009) 1466

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    In comparison with Rabkins and Kirchheims models, the present model is a relative intact one in that it couples

    the effects due to thermodynamics and kinetics.

    ( )00

    6

    6

    seg

    b

    H X D

    +=

    Simplified GB energymodel

    Kinetic equation withsolute drag

    with1 0 0

    2 0

    ,

    6

    seg

    seg

    H X

    H X

    = =

    ( ) ( )[ ] ( )

    t

    D

    D

    M

    DDM

    DD

    =

    +

    +

    ++

    021

    21

    3

    2

    2

    1

    2

    2

    1

    02

    2

    1

    2

    2

    021

    2

    213

    2

    ln

    21

    2

    Analytical thermo-kinetic model of grain growth

    Analytical thermo-kinetic model

    F.Liu et al. Acta Mater., 57 (2009) 1466

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    Analytical thermo-

    kinetic model of

    grain growth

    b

    Pure thermodynamic model

    (is zero)

    ( ) ( ) tMDDMDD b

    =+ 3032

    0

    2

    32

    1

    MtDD

    D

    D=

    +

    2

    0

    021

    21

    2

    2

    1 ln

    ( ) ( )[ ]

    ( ) MtD

    DDD

    DD

    =

    ++

    ++

    021

    21

    2

    2

    21

    2

    2

    10

    2

    1

    2

    2

    021

    2

    212

    2

    ln

    21

    2

    1

    Pure kinetic model( is constant)

    Mixed model (= /M)2

    NC Ni-O alloys annealed at 673K

    Validation in NC Ni-O alloys [Scripta Mater., 44 (2001) 2321]

    Pure thermodynamic model and mixed model can describe the experimental data well, whereas, the pure kinetic

    model cannot bring into agreement.

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    4 Experiments

    4.1 Preparation and stability of NC Fe-C alloys

    4.2 Thermal stability of as-spun nano-Fe-B alloys

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    XRD patterns of Fe-C alloy powders and grain size

    in them corresponding to different milling time

    SEM pictures of Fe-C alloy powders

    subjected to different milling time

    4.1 Preparation and stability of NC Fe-C alloys

    XRD

    Nano-crystalline Fe-1at%C alloy powdershave been synthesized by high energyball milling.

    The morphologies of powders have beenrevealed by SEM.

    The average grain size evolution withmilling time have been obtained from XRDpatterns by applying Scherrer formula.

    Final grain size is about 8nm.

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    Isothermal and isochronal DSC curves of NC Fe-1at%C solid solution

    The nano-scale growth (stability) can be studied experimentally by DSC method. The grainsize evolution with annealing time can be indirectly reflected by the obtained DSC curves.

    Isochronal DSC experiments subjected to different heating rates have been performed to reveal the temperaturerange where grain growth happens.

    Isothermal DSC experiments at annealing temperatures of 250, 270, 320, 350 and 450 oC have been conducted.

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    The GB energy (Krills GB energy model), the total GB enthalpy of the nano-multicrystalline system are

    incorporated into the kinetic equation for grain growth. And the equations used to describe the total energy

    change during grain growth are obtained.

    and

    ( )211n gb gb

    gb mn n nm

    dH MH

    H gV dt g V = +

    1

    2

    m

    gb m

    gVD

    H gV

    =

    +

    1 0 0

    2 0

    ,

    6seg

    seg

    H XH X

    = =

    with the enthalpy of GB,

    the shape factor,

    the molar volume

    gbH

    g

    mV

    1n

    dD M

    dt D

    =

    0

    0

    0

    1 2

    ( 6 )

    6

    b seg

    seg

    H

    H x D

    D

    = +

    =

    =

    mgb

    g VH

    D

    =

    Krills GB energy model

    Kinetic equation

    Total GB enthalpy

    The Isothermal DSC curves can be analytically fitted by the following model which

    describes the total energy change during isothermal grain growth.

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    3.115.02.003.710-18142450

    2.512.12.177.310-20165350

    4.411.02.221.910-20176320

    Error %D*(nm)nMHseg (kJ mol-1)Temperature()

    Fitting parameters and errors of fits are given in the following table

    The reasons for the much larger values ofHsegare:

    1. All the other factors that stabilize the grain size (e.g. entropy) have been incorporated into the thermodynamic effect (i.e. GB

    energy decreasing with grain growth);2. The present alloy system is a little bit different from an ideal dilute solution, which is the basis of the thermodynamic model used

    here.

    Fitted results of DSC curves for isothermal anneal of NC Fe-C powders

    With increasing annealing temperature

    T, GB mobility Mincreases due to the

    improvement of atom activity.The metastable equilibrium grain size

    D* increases with increasing T.

    Manuscript is in preparation

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    4.2 Thermal stability of as-spun nano-Fe-B alloys

    XRD profile for the melt spun Fe-10at.%B alloy

    (7000rpm) annealed at 700

    for different time

    Bright field TEM image for Fe-10at.%B nano-grain(7000rpm) annealed at 700 oC for 0.5 h

    For Fe-8at.%B, Fe-10at.%B, Fe-12at.%Band Fe-14at.%B alloys, the melt spinningwith 7000 rpm was performed, andsubsequently, isothermal annealing of theas-spun nano-grain at 700oC wasconducted for different time.

    Typical bright field morphologies (a) anddiffraction spot (b) of the melt spun Fe-10at.%B

    SSSS with rotational speed as 7000 rpm.

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    Pure kinetic, pure thermodynamic, and thermo-kinetic models have been adopted to fit the experimental data.Obviously, pure kinetic model with solute drag cannot be brought into agreement with the experimental data.However, the thermo-kinetic model can be adopted to describe well the experimental data due to the introduction

    of GB energy effect.

    Evolution of the average grain size with the annealing timefor as-quenched Fe-B nano-grain (7000rpm) annealed at

    700 oC, The symbols are the experimental data; the dotted,dashed, solid lines are calculated using pure kinetic, purethermodynamic and thermo-kinetic models, respectively.

    Pure thermodynamic model(is zero)

    ( ) ( ) tMDDMDD b

    =+ 3032

    0

    2

    32

    1

    MtDD

    D

    D=

    +

    2

    0

    021

    21

    2

    2

    1 ln

    ( ) ( )[ ]( ) Mt

    D

    DDD

    DD

    =

    ++

    ++

    021

    21

    2

    2

    2

    1

    2

    2

    10

    2

    1

    2

    2

    021

    2

    2122

    ln

    21

    2

    1

    Pure kinetic model

    ( is constant)

    thermo-kinetic model (= /M)

    Thermo-kinetic

    model at

    unsaturated GB

    segregation

    b

    2

    F.Liu et al. J. Crystal Growth, 2010 (313): 81-93

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    Effect of GB energy

    decreases with grain growth, but it approaches only infinitely zero, indicating that the effect of GB energy plays

    an dominated role in inhibiting grain growth in Fe-B alloys.b

    GB energy equationMtb

    b 2

    20

    0

    1 ln

    =+

    F.Liu et al. J. Crystal Growth, 2010 (313): 81-93

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    Conclusions

    1. Based on Borisovs equation, a qualitative description between self-diffusionin the lattice and along the GBs and the GB energy is provided. It is shownthat the incorporation of decreasing GB energy into the grain growth kineticsis physically practicable.

    2. A state criterion of initial GB segregation has been proposed, which can beused to evaluate whether the GBs are saturated or not at the initial stage.

    3. Based on the state criterion for initial GB segregation, the stop of graingrowth corresponds to the saturated GB, but the saturated GB does notnecessarily correspond to the stop of grain growth.

    4. Thermo-kinetic model with saturated GB segregation has been proposed andsuccessfully applied in NC Ni-O alloy with oxygen content as 6039ppm.

    5. Thermo-kinetic models with unsaturated GB segregation have also beenderived and successfully applied in NC RuAl, NC CGO and NC Ni-O alloyswith oxygen contents as 956 and 1805 ppm.

    6. NC Fe-1at%C alloy powders have been prepared by high energy ball millingand their thermal stability has been investigated by DSC. Isothermalannealing of as-spun Fe-B NC grain at 700oC was conducted. It is due to thereduction of GB energy, but not the kinetic factor, e.g. solute drag, thatdominantly controls the thermal stability of NC materials.

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    3.2.1 Thermodynamic state with saturated GB segregation

    For the condition that the

    initial reaches its

    saturated value , the

    bulk concentration will

    approach the ideal bulkconcentration at

    metastable equilibrium

    b

    0b

    X

    0X

    3 2 2 Evolution of thermodynamic factors upon grain growth with

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    3.2.2 Evolution of thermodynamic factors upon grain growth with

    unsaturated GB segregation

    Grain growth is a kinetic

    process controlled by

    thermodynamic factor.

    , withD and finally GB energy

    approaches 0, grain growth

    stops

    0b b

    0X X

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    ( ) ( )0 00

    0

    0

    ln

    3

    ln

    1 ln

    gb

    segm

    g seg

    g seg

    D X X R T XHV

    R T X H

    R T X H

    = +

    +

    = +

    For strong segregated alloys, ,and decrease upon growth

    For weak segregated alloys, ,and increases upon growth

    Large makes close to 0;

    while small ( ) leadsto deviating substantially from zeroat the state of

    0X X >X

    0X