phase transformation of materials - seoul national...

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Phase Transformation of MaterialsPhase Transformation of Materials

EunEun SooSoo ParkPark

Office: 33-316 Telephone: 880-7221Email: [email protected] 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

BBBBo

BBo

BBo

Bo

BB

BBBo

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

RTGX

GXRT

Xhere

XGXRT

XRTXG

BeB

BeB

eB

BBB

BBB

Ω+Δ−=>>

Ω−Δ−=

<<

−Ω−Δ−=

−−Ω−=Δ

→

→

→

→

αβ

αβ

αβ

αβ

)exp()exp(

RTH

RSX

STHG

BBeB

BBB

Ω+Δ−

Δ=

Δ−Δ=Δ→→

→→→

αβαβ

αβαβαβ 이므로

Q : heat absorbed (enthalpy) when 1 mole of β dissolves in A rich α as a dilute solution.

↑↑ eBXT

⎭⎬⎫

⎩⎨⎧−=

RTQAX e

B exp

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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 V

V X X

dG 0dX

Δ − Δ + =eV V VH T S RTln X 0

Δ −Δ= ⋅

Δ = Δ − Δ−Δ

=

e V VV

V V V

e VV

S HX exp expR RT

putting G H T SGX exp

RT

at equilibrium

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Interface (α/β)=γrVG mγ2

=Δ 의 effect

)2exp(

)/2exp(

)exp(

)exp(

RTrVX

RTrVGX

RTGX

RTGX

mrB

mBrrB

BrB

BeB

γ

γ

∞=

=

∞=

=

−Ω+Δ−=

Ω+Δ−=

Ω+Δ−=

Ex) γ=200mJ/m2, Vm=10-5 m3,T=500K

)(11nmrX

X r +=∞

r=10nm 이면 10% 증가

RTrV

RTrV

XX mm

rB

rrB γγ 21)2exp( +≈=∞=

=

rP 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 barriercomposition barrier

↓↑Δ

Δ=

Δ=

*,

22*

rT

TLT

Gr m

V

γγ

Undercooling이클수록 r*가작다Nucleation ↑β 상의수size barrier (r*)

rVG mγ2

=ΔmTTLG Δ

≅Δ

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Additional Thermodynamic Relationships for Binary Solutions

1)(d

Xd

Xd AB

A

B

B

A μμμμ −==−

1AB

BdXdG μμ −

=

2

2A A B B A B Bd GX d X d X X dXdX

μ μ− = =

0A A B BX d X dμ μ+ =

조성변화로인한화학퍼텐셜의변화계산: Gibbs-Duhem식

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The Gibbs-Duhem Equationbe 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 2A B

d G RTdX X X

= − Ω

For a ideal solution, Ω = 0,

2

2A B

d G RTdX X X

=

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

ln1 1ln

B B B B

B B B B B B

d RT X d RT ddX X dX X d Xμ γ γ

γ⎧ ⎫ ⎧ ⎫

= + = +⎨ ⎬ ⎨ ⎬⎩ ⎭ ⎩ ⎭

For a regular solution,

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a similar relationship can be derived for dμA/dXB

ln ln1 1ln ln

A BA A B B B B

A B

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

γ γμ μ⎧ ⎫ ⎧ ⎫

− = = + = +⎨ ⎬ ⎨ ⎬⎩ ⎭ ⎩ ⎭

2

2A A B B A B Bd GX d X d X X dXdX

μ μ− = =

2

2

ln ln1 1ln ln

A BA B

A B

d G d dX X RT RTdX d X d X

γ γ⎧ ⎫ ⎧ ⎫= + = +⎨ ⎬ ⎨ ⎬

⎩ ⎭ ⎩ ⎭

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Driving force: precipitation

* Consider the chemical potential of component B in phase alphacompared 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.

RS

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 TriangleAn 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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21


22


23


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Isothermal section → F = C - P

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Isothermal section

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Isothermal section → F = C - P

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Ternary Isomorphous SystemLocate 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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32


33


34


35


36


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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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40


41


42


43


44


45


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T= ternary eutectic temp.

A CL+β+γ

L+α+γL+α+β

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

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

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Ternary Eutectic SystemSolidification Sequence

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Ternary Eutectic SystemSolidification Sequence

2 상영역에서 수직 단면이 tie line과 불일치하므로 다른 온도에서 평형상만 나타내고 조성은 표시할 수 없음.

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4성분원소들 가운데서 임의의 3성분의농도가 독립적으로 변할 수 있는 함수이므로 여러가지 조성의 Quar-ternary alloy은 공간적으로 표시된다. 3원계의조성은 정4면체의 면상에, 그리고 4원계합금의 조성은 정4면체의 내부공간에표시된다. 합금의 조성은 정4면체의 기하학적성질에 의하여 결정된다. 4원계에서 상조성을 결정하기 위하여 lever rule을 이용한다. 4원합금의 변태과정을고찰할 때 정4면체안의 추상적인 4차원투영을 이용한다.

Quarternary의 평형상태를 기하학적으로 표시한 그림

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