Atomic Transport Phase Transformations - uni Transport Phase Transformations Lecture 9 PD Dr....

download Atomic Transport Phase Transformations - uni   Transport  Phase Transformations Lecture 9 PD Dr. Nikolay Zotov zotov@imw.uni

of 37

  • date post

    20-Mar-2018
  • Category

    Documents

  • view

    223
  • download

    3

Embed Size (px)

Transcript of Atomic Transport Phase Transformations - uni Transport Phase Transformations Lecture 9 PD Dr....

  • Atomic Transport

    &

    Phase Transformations

    Lecture 9

    PD Dr. Nikolay Zotov

    zotov@imw.uni-stuttgart.de

  • 2

    Part I Alloy Thermodynamics

    Lecture Short Description

    1 Introduction; Review of classical thermodynamics

    2

    3 Phase equilibria, Classification of phase transitions

    4 Thermodynamics of solutions I

    5 Thermodynamics of solutions II

    6 Binary Phase Diagrams I

    7 Binary Phase Diagrams II

    8 Binary Phase Diagrams III

    9 Order Disorder Phase Transitions

    Atomic Transport & Phase Transformations

  • Lecture I-9 Outline

    Degree of order

    Order parameters (General)

    Examples of order-disorder structures

    Short-range order (SRO) parameters

    Determination of SRO parameters

    Long-range order (LRO) parameters

    Determination of the LRO parameters

  • Degree of Order in the Solid State

    Interaction parameter WS < 0

    2eAB eAA eBB = eAB (eAA + eBB )< 0 Tendency for formation of A-B bonds

    For random distribution:

    Number of A-B bonds: PABRan = NozXAXB;

    Degree of ordering:

    No (random solution) PAB = PABRan = NozXAXB;

    Partial ordering

    Short-range order (SRO) PABPar > PAB

    Ran

    0,0 0,2 0,4 0,6 0,8 1,0

    0,00

    0,05

    0,10

    0,15

    0,20

    0,25

    XB

    XAXB;

  • Degree of Order in the Solid StateThermodynamic description

    DGmix = DHmix TDSmix;

    DHmix = WSXAXB < 0

    Non-random configurational entropy

    DSmix = DSmixid + DSmix

    Ex = - R[XAln(XA) + XBln(XB)] + DSmixEx ;

    Quasi-Chemical model (Gugenheim 1952) for weak SRO:

    DSmixEx = -R[yAAln(yAA/XA

    2) + yBBln(yBB/XB2) + 2yABln(yAB/XAXB)]

    yAA - fraction of A-A bonds (for random distribution XA2)

    yBB - fraction of B-B bonds (for random distribution XB2)

    yAB - fraction of A-B bonds (for random distribution XAXB)

    In the presence of SRO yAA < XA2, yBB < XB

    2 but yAB > XAXB ; SRO reduces the configurational entropy

  • Order Parameters

    Order Parameter (h)

    G = G(T,p,ni,h) h = 1 Ordered phase with LRO

    h > 0 SRO

    h = 0 random distribution

    Positional order-disorder

    Orientational (magnetic) order-disorder

  • Phase Diagrams with Order- Disorder Phase Transitions

    A

    B

    A Homogeneous solid

    solution

    Disordered (Cu,Au) phase

    B - Ordered CuAuI phase

  • Phase Diagrams with Order- Disorder Phase Transitions

    High-temperature

    disordered structure

    Low-temperature

    ordered structure

  • Order Disorder Structures

    T > Tc = 410oC T < Tc= 410

    oC

    S.G F m -3 m

    Strukturbericht symbol A1

    Crystall class: cubic

    Point Group: m -3 m

    S.G P 4/mmm

    Strukturbericht symbol L10Crystall class: tetragonal

    Point Group: 4/mmm

    Cu

    Au

    CuAuI

  • Order Disorder Structures

    Cu

    Zn

    S.G I m -3 m

    Strukturbericht symbol A2

    Crystall class: cubic

    Point Group: m -3 m

    S.G P m -3 m

    Strukturbericht symbol B2

    Crystall class: cubic

    Point Group: m -3 m

    T > TC ~ 727 KT < TC ~ 727 K

    CuZn CuZn

  • Order Disorder Structures

    S.G P m -3 m

    Strukturbericht symbol L12Crystall class: cubic

    Point Group: m -3 m

    Cu3Au

    S.G F m -3 m

    Strukturbericht symbol A1 (fcc)

    Crystall class: cubic

    Point Group: m -3 m

    T < Tc = 390oC T > Tc= 390

    oC

  • Disordered phase the different types of atoms occupy all crystallographic sites in the

    unit cell randomly with probabilities corresponding to the overall composition

    All sites have the same site symmetry

    Fcc (F m -3 m) 0 0 0 00 0

    Stoichiometric CuAuI

    pCu = pAu =

    Site Symmetry 4a m -3 m

    Order Disorder Structures

    SRO, AmBn Clusters

  • Atom Site Site Symmetry_Coordinates______No of Atoms

    Au 2e mmm 0 4x1/2 = 2

    Cu 1a 4/mmm 0 0 0 8x1/8 = 1

    Cu 1c 4/mmm 0 2x1/2 = 1

    Ordered structure (LRO) Different atoms occupy different set of

    equivalent atomic positions (different sites)

    Each site generates a sub-lattice.

    Substitutional Disorder in ordered structures

    Random substitution of a given sublattice

    Space group symmetry is preserved

    Ordered non-stoichiometric compounds

    Au

    Cu

    Order Disorder Structures

    Cu

    Au Cu

    Cu Au

    CuAuI

    4-fold axis

  • Order Disorder Structures

    Disordered liquid crystal Orientationally-ordered liquid crystal

  • Order Disorder Structures

    LRO Substitutional Disorder SRO (Random) Disordered structure

    G G G G

    WS

  • Short-range order parameter (1):

    hSRO = (PABSol PAB

    Ran) / (PABMax PAB

    Ran) ; Bethe SRO parameter

    Short-range order parameter (2):

    a = 1 P1/XA ; P1 probability to find an atom A as a

    nearest-neighbour of an atom B.

    a Warren Cowley SRO parameter

    = 0 random distribution (P1 = XA)

    < 0 SRO (P1 > XA)

    > 0 Clustering (P1 < XA)

    Short-range Order Parameters

    random

    hSRO ~ 0.3

    Easterling and Potter (2009)

    AB4 clusters

  • Short-range Order Parameters

    Warren Cowley parameters

    an = 1 Pn/XA, Pn Probability to find atom A on the

    n-th shell, if an atom B is at (0,0,0)

    a0 = a000 = 1 R0 = ( 0 0 0)

    a1 = a 0 R1 = ( 0 )

    a2 = a 010 R2 = ( 010 )

    a (lmn)

    No SRO Pn = XA and all an 0

    Clustering aj > 0 and aj 0 at large distances

    SRO aj oscilate between positive and negative values

    (0,0,0)

    R1

    R2

  • Short-range Order Parameters

    Determination using Diffuse X-ray and Neutron Scattering

    Basics of Diffuse Scattering:

    A = FT(r) A Amplitude of scattered X-ray/neutrons

    r Scattering density

    In the presence of defects (and/or SRO) r = + Dr - Scattering density of the average lattice

    Dr (local) variations due to presence

    of SRO and/or defects

    A = FT() + FT(Dr)

    Itot = |A|2 ~ IBragg

    Av + Idiff; Small concentration of defects or degree of SRO

    IBraggAv ~ |A()|2

    IDiff = ISRO + IStrain +

  • Short-range Order Parameters

    Determination using Diffuse X-ray and Neutron Scattering

    SRO Diffuse Scattering:

    (C = 2 components, each atom has the same number of nearest-neighbours)

    ISRO = NXAXB(fA fB)2 + NXAXB(fA-fB)

    2Sj Skj ajk exp(2piQ.Rjk)

    = NXAXB(fA fB)2 Sl SmSnalmn exp[2pi(lh + mk + nl]}; l,m,n - +, a000 = 1

    fA, fB Atomic scattering factors for the A and B atoms

    Rjk = Rk Rj ; difference of radius vectors of atoms j and k

    la + mb + nc; a, b c lattice vectors

    Q = 4psin(Q)/l Scattering vector length

    Q = ha* + kb* + lc*; a* b* c* lattice parameters of the

    reciprocal lattice

    almn = aj = IFT{ISRO/ NXAXB(fA fB)2 }

    a(-l m n ) = a (lmn)

  • Short-range Order Parameters

    Determination using Diffuse X-ray and Neutron Scattering

    Q ~ 0 fCu ~ ZCu = 29

    fAu ~ ZAu = 79

    (fCu fAu)2 ~ 2500

    Cowley (1950) ; Cu Ka radiation

    Cu3Au cubic; P m -3 m (L12) type

    Cu3Au single crystal

    at each temperature 450 measurements

    in reciprocal space for diffrenet h,k,l

  • Short-range Order Parameters

    Determination using Diffuse X-ray and Neutron Scattering

    h

    k

    (100) Plane 405 oC

    1 2 3 4 5 6 7

    -0,4

    -0,2

    0,0

    0,2

    0,4

    0,6

    0,8

    1,0

    SR

    O P

    ara

    mete

    rs

    Shell

    405

    460

    550

    Perfect order

    (200) (400)

    (220) (420)

    Cu3Au

    Cowley (1950)

    (110) (200) (211)

    nA(i) = NiPi = NiXA(1 ai) Crystal

    i=1 12 x 0.75 x 1.152 = 10.4 (12)

    i=2 6 x 0.75 x 0.814 = 3.7 (0)

  • Short-range Order Parameters

    Determination using Diffuse X-ray and Neutron Scattering

    1 2 3 4 5 6 7

    -0,2

    -0,1

    0,0

    0,1

    0,2

    0,3

    SR

    O P

    ara

    mete

    rs

    Shell

    426 oC CuAuI

  • Long-range Order Parameters

    Au Cu

    Cu Au

    a

    b

    Ideally, A atoms ordered on a sites

    B atoms ordered on b sites

    Definitions:

    ra fraction of a sites occupied by A atoms

    rb fraction of b sites occupied by B atoms

    wa fraction of wrong (B) atoms occupying a sites

    wa = 1 ra ;

    wb fraction of wrong (A) atoms occupying b sites

    wb = 1 rb ;

    hLRO = (ra XA)/(1 XA)

    hLRO = (rb XB)/(1 XB)hLRO = 1 ideal LRO

    hLRO < 1 Increasing degree of substitutional disorder

  • Long-range