Phase Transformation of Materials

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1

재 료 상 변 태

2008. 11. 25.

박 은 수

서울대학교 재료공학부

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Contents in Phase Transformation

(Ch1)

열역학과 상태도

: Thermodynamics (Ch2)

확 산 론

: Kinetics

(Ch3)

결정계면과 미세조직

(Ch4)

응 고: Liquid →

Solid

(Ch5)

고체에서의 확산 변태

: Solid → Solid (Diffusional) (Ch6)고체에서의 무확산 변태: Solid → Solid (Diffusionless)

상변태를 상변태를

이해하는데 이해하는데 필요한 필요한 배경 배경

대표적인 대표적인 상변태 상변태

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< Phase Transformation in Solids >

1) Diffusional Transformation

2) Non-diffusional Transformation (Athermal Transformation)

From what we’ve learned, what can we say roughly about diffusional transformation in solid?

1) Thermally-activated process: rate ∝ exp (-ΔG

^*

/kT) 2) Misfit strain energy lattice distortion

3) Kinetic path for low nucleation barrier

→

coherent or semi-coherent interfaces

→

heterogeneous nucleation

4) Local equilibrium for incoherent (rough) interfaces

→

diffusion-controlled growth

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(b) Eutectoid Transformation (a) Precipitation

Composition of product phases differs from that of a parent phase.

5. Diffusion Transformations in solid

→

long-range diffusion

Which transformation proceeds

by short-range diffusion?

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(d) Massive Transformation (e) Polymorphic Transformation (c) Order-Disorder

Transformation

5. Diffusion Transformations in solid

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Homogeneous Nucleation in Solids

1) Volume Free Energy : − Δ V G

_V

2) Interface Energy : A γ

3) Misfit Strain Energy :

V G Δ

S

V S

G V G A γ V G Δ = − Δ + + Δ

for spherical nucleation

3 2

4 ( ) 4

3

^V ^S

G π r G G π γ r

Δ = − Δ − Δ +

Free Energy Change Associated with the Nucleation

Plot of ΔG vs r?

r* = ? ΔG* = ?

Negative and Positive Contributions to ΔG?

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* 2

(

_V _S

)

r G G

= γ

Δ − Δ

3

2

* 16

3(

_V _S

)

G G G

Δ = πγ

Δ − Δ

Homogeneous Nucleation in Solids

Fig. 5.2 The variation of

Δ

G with r for a homogeneous nucleus.

There is a activation energy barrier ΔG*.

(8)

8

*

0

exp( * / )

C = C −Δ G kT

Nucleation Rate

hom 0

exp G

^m

exp G *

N C

kT kT

ω ⎛ Δ ⎞ ⎛ Δ ⎞

= ⎜ ⎝ − ⎟ ⎠ ⎜ ⎝ − ⎟ ⎠

The Concentration of Critical Size Nuclei

C₀ : number of atoms

per unit volume in the phase

hom

*

N = f C

If each nucleus can be made supercritical at a rate of f per second,

( G kT )

exp

f = ω − Δ

_m

vibration frequency, area of critical nucleus ω ∝

m

:

G activation energy for atomic migration Δ

Homogeneous Nucleation in Solids

3 2

* 16

3( _V _S)

G G G

Δ = πγ

Δ − Δ

ΔG* ∞ ΔG

_V

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hom 0

exp G

^m

exp G *

N C

kT kT

ω ⎛ Δ ⎞ ⎛ Δ ⎞

= ⎜ ⎝ − ⎟ ⎠ ⎜ ⎝ − ⎟ ⎠

3

2

* 16

3(

_V _S

)

G G G

Δ =

πγ

Δ − Δ

ΔG* ∞ ΔG

_V

(화학적 구동력)

Liquid → solid α → α + β : B

성분이 과포화되어 β 석출

1) For X₀, solution treatment at T₁

2) For X₀, quenching down to T₂

ΔG_V

(화학적 구동력) 는 조성에 따라 변함

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P A A B B

G X X

per mol removed

α β α β

μ μ

= β +

Q A A B B

G X X

per mol formed

β β β β

μ μ

β

= +

n V

m

G G per unit volume of

V β

Δ = Δ

0

V e

G X where X X X

Δ ∝ Δ Δ = −

Total Free Energy Decrease per Mole of Nuclei Driving Force for Precipitate Nucleation

ΔG

₀

β α

β

α

μ

_A

X

_A _B

X

_B

G = +

Δ

₁

β β β

β

μ

_A

X

_A _B

X

_B

G = +

Δ

₂

1

2

G

_n

= Δ − Δ Δ

For dilute solutions,

T X

G

_V

∝ Δ ∝ Δ

Δ

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Rate of Homogeneous Nucleation Varies with Undercooling

hom 0

exp G^m exp G*

N C

kT kT

ω ^⎛ ^Δ ^⎞ ^⎛ ^Δ ^⎞

= ⎜⎝− ⎟⎠ ⎜⎝− ⎟⎠

undercooling

3 2

* 16

3( _V _S)

G G G

Δ = πγ

Δ − Δ T X

G_V ∝ Δ ∝ Δ Δ

임계 핵의 증가확률

원자 이동도: T↓시 ↓

구동력 ΔGv너무 작으므로 N ↓

확산이 너무 느리기 때문에N ↓

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The Effect of Δ T on ∆ G*

_het

& ∆ G*

_hom

?

Fig. 4.9 (a) Variation of ∆G* with undercooling (∆T) for homogeneous and heterogeneous nucleation.

(b) The corresponding nucleation rates assuming the same critical value of ∆G*

-3 1 hom

1 cm

N ≈ s

⁻

Plot ∆G*_het & ∆G* _hom vs ΔT and N vs ΔT.

임계 과냉도

ΔT_c^hom

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Rate of Homogeneous Nucleation Varies with Undercooling

hom 0

exp G^m exp G*

N C

kT kT

ω ^⎛ ^Δ ^⎞ ^⎛ ^Δ ^⎞

= ⎜⎝− ⎟⎠ ⎜⎝− ⎟⎠

undercooling

3 2

* 16

3( _V _S)

G G G

Δ = πγ

Δ − Δ T X

G_V ∝ Δ ∝ Δ Δ

임계 핵의 증가확률

원자 이동도: T↓시 ↓

구동력 ΔGv너무 작으므로 N ↓

확산이 너무 느리기 때문에N ↓ 중간정도 과냉에서 N 최대

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The Effect of Alloy Composition on the Nucleation Rate Compare the two plots of T vs N(1) and T vs N(2).

Homogeneous Nucleation in Solids

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Heterogeneous Nucleation in Solids

* 2 /

_V

r = γ

_αβ

Δ G

hom hom

* *

* * ( )

het het

G V

G V S θ

Δ = =

Δ

1 2

( ) (2 cos )(1 cos ) S θ = 2 + θ − θ

( )

het V S d

G V G G A γ G Δ = − Δ − Δ + − Δ

Nucleation on Grain Boundaries When ΔG

_S

(불일치 변형 에너지) = 0,

대부분의 핵생성이 해당, 적합한 위치는 격자결함

(

핵생성이 격자결함 제거 역할

)

αβ αα

γ γ

θ / 2

cos =

G V G

V

A

_{αβ αβ}

γ A

_{αα αα}

γ

Δ = − Δ + −

핵의 모양 전체 계면 자유에너지를 최소로 하는 상태

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V

A

V

B

Barrier of Heterogeneous Nucleation

4

) cos cos

3 2 ( 3

) 16 3 (

* 16

3 2

3 θ πγ θ θ

πγ ⋅ − +

= Δ Δ ⋅

= Δ

V SL V

SL

S G G G

θ θ

⎛ − + ⎞

Δ = Δ ⎜ ⎟

⎝ ⎠

3

*

2 3 cos cos

4

*

sub homo

G G

2 3 cos cos

3

4 ( )

A

A B

V S

V V

θ θ θ

− +

= =

+

het S v SL SL SM SM SM ML

G V G A γ A γ A γ

Δ = − Δ + + −

S(θ) has a numerical value ≤ 1 dependent only on θ (the shape of the nucleus)

* *

( )

hom

G

het

S θ G

Δ = Δ

₍ ₎

3

* 16

* 2 ₂

3 θ

πγ

γ S

G G G and

r

V SL V

SL ⋅

= Δ Δ Δ

=

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How can V* and ΔG* be reduced even further?

→ By nucleation on a grain edge or a grain corner.

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Activation Energy Barrier

* *

hom

cos

, .

het

Compare the plots of

G G vs

for grain boundaries edges and corners

θ

Δ Δ

Heterogeneous Nucleation in Solids

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< Nucleus with Coherent Interface >

High-angle grain boundaries are particularly effective nucleation sites for incoherent precipitates with high γ

_αβ

. If the matrix and precipitate make a coherent interface, V* and ΔG* can be further reduced.

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Decreasing order of ΔG*

1) homogeneous sites 2) vacancies

3) dislocations 4) stacking faults

5) grain boundaries and interphase boundaries 6) free surfaces

Rate of Heterogeneous Nucleation

3 1 1

exp

^m

exp *

het

G G

N C nuclei m s

kT kT

ω ⎛ Δ ⎞ ⎛ Δ ⎞

⁻ ⁻

= ⎜ − ⎟ ⎜ − ⎟

⎝ ⎠ ⎝ ⎠

C₁ : concentration of heterogeneous nucleation sites per unit volume

hom 0

exp G^m exp G *

N C

kT kT

ω ^⎛ ^Δ ^⎞ ^⎛ ^Δ ^⎞

= ⎜⎝− ⎟⎠ ⎜⎝− ⎟⎠

(21)

21 hom

1

hom 0

* *

het

exp

het

N C G G

N C kT

Δ −Δ

⎛ ⎞

= ⎜ ⎝ ⎟ ⎠

1 0

( )

C GB thickness C D grain size

= δ

C

₁

/C

₀

for GB nucleation?

The Rate of Heterogeneous Nucleation during Precipitation

> 0

For D = 50 μm, δ = 0.5 nm

1 5 0

C 10

C D

δ

₋

= ≈

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C

₁

/C

₀

for Various Heterogeneous Nucleation Sites

2 1

0

C for nucleation on grain edge

C D

⎛ ⎞ δ

= ⎜ ⎟ →

⎝ ⎠

3 1

0

C for nucleation on grain corner

C D

⎛ ⎞ δ

= ⎜ ⎟ ⎝ ⎠ →

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In order to make nucleation occur exclusively

on the grain corner, how should the alloy be cooled?

At small driving forces ( Δ G

_v

), when activation energy barriers for nucleation are high, the highest nucleation rates will be produced by grain-corner nucleation.

At very high driving forces it may be possible for

the (C

₁

/C

₀

) term to dominate and then homogeneous nucleation provides the highest nucleation rates.

At very high cooling rate?

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Precipitate Growth

→

Origin of the Widmanstätten morphology

If the nucleus consists of semi-coherent

and incoherent interfaces, what would

be the growth shape?

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Growth behind Planar Incoherent Interfaces

Diffusion-Controlled Thickening

Incoherent → similar to rough interface → local equilibrium

→ diffusion-controlled

From mass conservation,

( )

( dC dx ) dt

D J

B of mole dx

C C

B

=

e

=

β

−

D: interdiffusion coefficient or interstitial diffusion coeff.

e

dx D dC

v = dt = C

_β

C ⋅ dx

−

→ v = f (ΔT or ΔX, t)

평형값

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Simplification of concentration profile

/

0

/

dC dx = Δ C L

0 0

(C_β −C x) = ΔL C / 2 Q

e

dx D dC

v = dt = C

_β

C ⋅ dx

−

2 0

0

( )

2(

_e

)( )

D C

v C

_β

C C

_β

C x

= Δ

− −

0

( )

e

x X Dt

X

_β

X

= Δ

−

0 e

C

_β

− C ≅ C

_β

− C

0

2(

_e

)

X D

v X

_β

X t

= Δ

− Δ X

0

= X

0

− X

e

0 0

2( )

L C

_β

C x C

← = − Δ

CV

m

X =

( )

2 0

2

_e 2

D X

xdx dt

X

_β

X

= Δ

− x ∝ ( Dt )

v ∝ Δ X

0

( / ) v ∝ D t

planar

→

no curvature

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Fig. 5.16 The effect of temperature and position on growth rate, v.

( )

x ∝ Dt v ∝ Δ X 0 v ∝ ( / ) D t

Growth behind Planar Incoherent Interfaces

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The Effect of Overlap of Separate Precipitates

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Diffusion Controlled lengthening of Plates or Needles

r

D C

v C

_β

C kr

= ⋅ Δ

−

0

1

r

*

X X

r

⎛ ⎞

Δ = Δ ⎜⎝ − ⎟⎠

0

0 0

r e

X X X X X X Δ = − Δ = −

0

1 *

(

_r

) 1

D X r

v k X

_β

X r r

Δ ⎛ ⎞

= − ⋅ ⎜ ⎝ − ⎟ ⎠

V

→

constant x ∝ t

Needle →

Gibbs-Thomson increase in G = 2γV_m/r instead of γV_m/r

→

the same equation but the different value of r^*

e

dx D dC

v = dt = C

_β

C ⋅ dx

Plate Precipitate of constant thickness −

(30)

30 0

1 r *

X X

r

⎛ ⎞

Δ = Δ ⎜ ⎝ − ⎟ ⎠

0

0 0

r e

X X X

Δ = −

The Gobs-Thomson Effect

Diffusion Controlled lengthening of Plates or Needles

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Thickening of Plate-like Precipitates

v uh

= λ

0

,

0

(

_e

) (

_e

)

D X D X

u v

k X

_β

X h k X

_β

X λ

Δ Δ

= =

− −

u : rate of lateral migration

Thickening of

Plate-like Precipitates by Ledge Mechanism

Half Thickness Increase

For the diffusion-controlled growth, a monatomic-height ledge

should be supplied constantly.

sources of monatomic-height ledge

→

spiral growth, 2-D nucleation, nucleation at the precipitate edges, or from intersections with other precipitates (heterogeneous 2-D)

Assuming the diffusion-controlled growth,

diffusion-control or interface-control?

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Thickening of γ Plate in the Al-Ag system

Ledge nucleation is rate controlling.

What does this data mean?

Thickening of Plate-like Precipitates