Phase Transformation of Materials - Seoul National · PDF file 2018. 1. 30. · 1...

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Transcript of Phase Transformation of Materials - Seoul National · PDF file 2018. 1. 30. · 1...

  • 11

    Phase Transformation of MaterialsPhase Transformation of Materials

    EunEun SooSoo ParkPark

    Office: 33-316 Telephone: 880-7221 Email: espark@snu.ac.kr Office hours: by an appointment

    2009 fall

    09.22.2009

  • 2

    Contents for previous class

    1) Simple Phase Diagrams

    2) Systems with miscibility gap

    4) Simple Eutectic Systems

    3) Ordered Alloys

    5) Phase diagrams containing intermediate phases

    합금의 평형조성 주어진 온도에서 얻은 자유에너지 곡선으로 얻음

    평형은 온도변화에 따라 어떻게 변화되어 가는가?

    - Gibbs Phase Rule

    - Binary phase diagrams

    F = C − P + 2 (from T, P)

  • 3

    • Effect of Temperature on Solid Solubility

    • Equilibrium Vacancy Concentration

    • Influence of Interfaces on Equilibrium

    • Ternary Equilibrium: Ternary Phase Diagram

    Contents for today’s class

  • 4

    Effect of T on solid solubility

    BBBB o

    BB o

    BB o

    B o

    BB

    BBB o

    B

    XRTXG

    GGGGG

    XRTXG

    ln)1(

    ln)1(

    2

    0

    2

    −−Ω−=−

    −=−=−=Δ

    +−Ω+= →

    αα

    ααβαβααβ

    αα

    μ

    μμ

    μ

    )exp(

    ln

    )1,(

    )1(ln

    ln)1( 2

    2

    RT GX

    GXRT

    Xhere

    XGXRT

    XRTXG

    Be B

    B e B

    e B

    BBB

    BBB

    Ω+Δ −=>>

    Ω−Δ−=

  • 5

    • Vacancies increase the internal energy of crystalline metal due to broken bonds formation.

    • Vacancies increase entropy because they change the thermal vibration frequency and also the configurational entropy.

    • Total entropy change is thus

    V VH H XΔ ≅Δ

    = + Δ = + Δ − Δ + + − −A A V V V V V V V VG G G G H X T S X RT{X ln X (1 X )ln(1 X )}

    The molar free energy of the crystal containing Xv mol of vacancies

    Δ = Δ − + − −V V V V V VS S X R{X ln X (1 X )ln(1 X )}

    With this information, estimate the equilibrium vacancy concentration.

    Equilibrium Vacancy Concentration STHG Δ−Δ=Δ

  • 6

    • In practice, ∆HV is of the order of 1 eV per atom and XV

    e

    reaches a value of about 10-4~10-3

    at the melting point of the solid

    =

    ⎛ ⎞ =⎜ ⎟

    ⎝ ⎠ eV VV X X

    dG 0 dX

    Δ − Δ + =eV V VH T S RTln X 0

    Δ −Δ = ⋅

    Δ = Δ − Δ −Δ

    =

    e V V V

    V V V

    e V V

    S HX exp exp R RT

    putting G H T S GX exp

    RT

    at equilibrium

  • 7

    Interface (α/β)=γ r VG mγ2=Δ 의 effect

    )2exp(

    )/2exp(

    )exp(

    )exp(

    RTr VX

    RT rVGX

    RT GX

    RT GX

    mr B

    mBrr B

    Br B

    Be B

    γ

    γ

    ∞=

    =

    ∞=

    =

    −Ω+Δ −=

    Ω+Δ −=

    Ω+Δ −=

    Ex) γ=200mJ/m2, Vm=10-5 m3,T=500K

    )( 11 nmrX

    X r += ∞

    r=10nm 이면 10% 증가

    RTr V

    RTr V

    X X mm

    r B

    rr B γγ 21)2exp( +≈=∞= =

    r P γ2=Δ

    Gibbs-Thomson effect: 계면에너지로 인해 자유에너지가 증가하는 현상

  • 8

    β formation in α

    β Nucleation & growth in α

    Interface (α/β) : size barrier composition barrier

    ↓↑Δ

    Δ =

    Δ =

    *,

    22*

    rT

    TL T

    G r m

    V

    γγ

    Undercooling이클수록 r*가작다 Nucleation ↑β 상의수 size barrier (r*)

    r VG mγ2=Δ

    mT TLG Δ≅Δ

  • 9

    Additional Thermodynamic Relationships for Binary Solutions

    1 )(d

    X d

    X d AB

    A

    B

    B

    A μμμμ −==− 1

    AB

    BdX dG μμ −

    =

    2

    2A A B B A B B d GX d X d X X dX dX

    μ μ− = =

    0A A B BX d X dμ μ+ =

    조성변화로인한화학퍼텐셜의변화계산: Gibbs-Duhem식

  • 10

    The Gibbs-Duhem Equation be able to calculate the change in chemical potential (dμ) that result from a change in alloy composition (dX).

    = + +Ω + +A A B B A B A A B BG X G X G X X RT(X lnX X lnX ) 2

    2 2 A B

    d G RT dX X X

    = − Ω

    For a ideal solution, Ω = 0,

    2

    2 A B

    d G RT dX X X

    =

    μ = + = + γB B B B B BG RTlna G RTln X

    ln1 1 ln

    B B B B

    B B B B B B

    d RT X d RT d dX X dX X d X μ γ γ

    γ ⎧ ⎫ ⎧ ⎫

    = + = +⎨ ⎬ ⎨ ⎬ ⎩ ⎭ ⎩ ⎭

    For a regular solution,

  • 11

    a similar relationship can be derived for dμA/dXB

    ln ln1 1 ln ln

    A B A A B B B B

    A B

    d dX d X d RT dX RT dX d X d X

    γ γμ μ ⎧ ⎫ ⎧ ⎫

    − = = + = +⎨ ⎬ ⎨ ⎬ ⎩ ⎭ ⎩ ⎭

    2

    2A A B B A B B d GX d X d X X dX dX

    μ μ− = =

    2

    2

    ln ln1 1 ln ln

    A B A B

    A B

    d G d dX X RT RT dX d X d X

    γ γ⎧ ⎫ ⎧ ⎫ = + = +⎨ ⎬ ⎨ ⎬

    ⎩ ⎭ ⎩ ⎭

  • 12

    Driving force: precipitation

    * Consider the chemical potential of component B in phase alpha compared to B in beta. This difference, labeled as ∆Gn on the right of the lower diagram is the driving force (expressed as energy per mole, in this case).

    * To convert to energy/volume, divide by the molar volume for beta: ∆GV = ∆Gn/Vm.

    R S

    T

    U

    N

    P

    Q

    Te

    Driving force for the reaction : ∆G0

    Driving force for nucleation : ∆Gn Because the first nuclei of beta to appear do not ignificantly change the composition of the parent material

  • 13

  • 14

    What are ternary phase diagram?

    www.sjsu.edu/faculty/selvaduray/page/phase/ternary_p_d.pdf

  • 15

    Gibbs Phase Rule for 3-component Systems

  • 16

    Gibbs Triangle An Equilateral triangle on which the pure components are represented by each corner.

    Concentration can be expressed as

    either “wt. %” or “at.% = molar %”.

    XA+XB+XC = 1

    Used to determine

    the overall composition

  • 17

    Overall Composition

  • 18

    Overall Composition

  • 19

    Ternary Isomorphous System

  • 20

    Ternary Isomorphous System

  • 21

    Ternary Isomorphous System

  • 22

    Ternary Isomorphous System

  • 23

    Ternary Isomorphous System

  • 24

    Ternary Isomorphous System

    Isothermal section → F = C - P

  • 25

    Ternary Isomorphous System

    Isothermal section

  • 26

    Ternary Isomorphous System

    Isothermal section → F = C - P

  • 27

    Ternary Isomorphous System Locate overall composition using Gibbs triangle

  • 28

  • 29

    Ternary Eutectic System (No Solid Solubility)

  • 30

    Ternary Eutectic System

    Liquidus projection

    (No Solid Solubility)

  • 31

    Ternary Eutectic System (No Solid Solubility)

  • 32

    Ternary Eutectic System (No Solid Solubility)

  • 33

    Ternary Eutectic System (No Solid Solubility)

  • 34

    Ternary Eutectic System (No Solid Solubility)

  • 35

    Ternary Eutectic System (No Solid Solubility)

  • 36

    Ternary Eutectic System (No Solid Solubility)

  • 37

    Ternary Eutectic System (No Solid Solubility)

    T= ternary eutectic temp.

    A C

    B

    L+A+C

    L+A+B L+B+C

  • 38

    Ternary Eutectic System (with Solid Solubility)

  • 39

    Ternary Eutectic System (with Solid Solubility)

  • 40

    Ternary Eutectic System (with Solid Solubility)

  • 41

    Ternary Eutectic System (with Solid Solubility)

  • 42

    Ternary Eutectic System (with Solid Solubility)

  • 43

    Ternary Eutectic System (with Solid Solubility)

  • 44

    Ternary Eutectic System (with Solid Solubility)

  • 45

    Ternary Eutectic System (with Solid Solubility)

  • 46

    Ternary Eutectic System (with Solid Solubility)

    T= ternary eutectic temp.

    A C L+β+γ

    L+α+γL+α+β

  • 47

    Ternary Eutectic System (with Solid Solubility)

    정해솔 학생 제공 자료 참조: 실제 isothermal section의 온도에 따른 변화

    http://www.youtube.com/watch?v=yzhVomAdetM

  • 48

    Ternary Eutectic System Solidification Sequence

  • 49

    Ternary Eutectic System Solidification Sequence

    2 상영역에서 수직 단면이 tie line과 불일 치하므로