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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

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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)

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• 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−=

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• 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

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• 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

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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

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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

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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

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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,

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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⎧ ⎫ ⎧ ⎫ = + = +⎨ ⎬ ⎨ ⎬

⎩ ⎭ ⎩ ⎭

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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

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What are ternary phase diagram?

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

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Gibbs Phase Rule for 3-component Systems

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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

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Overall Composition

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Overall Composition

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Ternary Isomorphous System

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Ternary Isomorphous System

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Ternary Isomorphous System

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Ternary Isomorphous System

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Ternary Isomorphous System

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Ternary Isomorphous System

Isothermal section → F = C - P

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Ternary Isomorphous System

Isothermal section

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Ternary Isomorphous System

Isothermal section → F = C - P

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Ternary Isomorphous System Locate overall composition using Gibbs triangle

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Ternary Eutectic System (No Solid Solubility)

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Ternary Eutectic System

Liquidus projection

(No Solid Solubility)

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Ternary Eutectic System (No Solid Solubility)

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Ternary Eutectic System (No Solid Solubility)

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Ternary Eutectic System (No Solid Solubility)

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Ternary Eutectic System (No Solid Solubility)

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Ternary Eutectic System (No Solid Solubility)

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Ternary Eutectic System (No Solid Solubility)

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Ternary Eutectic System (No Solid Solubility)

T= ternary eutectic temp.

A C

B

L+A+C

L+A+B L+B+C

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Ternary Eutectic System (with Solid Solubility)

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Ternary Eutectic System (with Solid Solubility)

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Ternary Eutectic System (with Solid Solubility)

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Ternary Eutectic System (with Solid Solubility)

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Ternary Eutectic System (with Solid Solubility)

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Ternary Eutectic System (with Solid Solubility)

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Ternary Eutectic System (with Solid Solubility)

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Ternary Eutectic System (with Solid Solubility)

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Ternary Eutectic System (with Solid Solubility)

T= ternary eutectic temp.

A C L+β+γ

L+α+γL+α+β

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Ternary Eutectic System (with Solid Solubility)

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

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

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Ternary Eutectic System Solidification Sequence

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Ternary Eutectic System Solidification Sequence

2 상영역에서 수직 단면이 tie line과 불일 치하므로