Phase Transformation of Materials Phase Transformation of Materials

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Phase Transformation of Materials Phase Transformation of Materials

Eun Eun Soo Soo Park Park

Office: 33-316

Telephone: 880-7221 Email: espark@snu.ac.kr

Office hours: by an appointment

2009 fall

11. 19. 2009

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“Alloy solidification” ^- Solidification of single-phase alloys

1) Equilibrium Solidification

: perfect mixing in solid and liquid

• Three limiting cases Planar S/L interface

→ unidirectional solidification

Contents for previous class

2) No Diffusion in Solid, Perfect Mixing in Liquid

- Sufficient time for diffusion in solid & liquid (low cooling rate) - Relative amount of solid and liquid : lever rule

: high cooling rate, efficient stirring

- Separate layers of solid retain their original compositions

mean comp. of the solid ( ) < X_s XS

Scheil equation

: non-equilibrium lever rule

(k 1)

L O L

X  X f

X

 kX

 ¹  f



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“Alloy solidification” ^- Solidification of single-phase alloys

Contents for previous class

3) No Diffusion on Solid, Diffusional Mixing in the Liquid

D/v

) )]

/ exp( (

1 1

[ D 

x k

X k

X

 





4

실제의 농도분포

두 가지 경우의 중간형태

Zone Refining

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“Alloy solidification”

Contents for previous class

At the interface,

T_L = T_e (not T_E ) = T₃

Constitutional Supercooling

Diffusional Mixing in the Liquid Steady State

* Temperature gradient in Liquid

T

’

* equilibrium solidification temp. change

T

T

' /v < (T

-T

)/D

Fig. 4.24 The breakdown of an initially planar solidification front into cells

온도구배 임계값 이하로 감소시 평면계면 불안정

돌출부 성장

Cellular Solidification

Heat flow Heat flow

용질축적 Te 감소

Local melting

T

' /v < (T

-T

)/D

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Fig. 4.25

Note that solute enrichment in the liquid between the cells, and coring in the cells with eutectic in the cell walls.

용질 농축 → 공정응고 → 제 2 상 형성

X₀ << X_max

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Fig. 4.26 Cellular microstructures.

(a) A decanted interface of a cellularly solidified Pb-Sn alloy (x 120)

(after J.W. Rutterin Liquid Metals and Solidification, American Society for Metals, 1958, p. 243).

(b) Longitudinal view of cells in carbon tetrabromide (x 100) (after K.A. Jackson and J.D. Hunt, ActaMetallurgica13 (1965) 1212).

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Fig. 4.27 Cellular dendrites in carbon tetrabromide.

( After L.R. Morris and W.C. Winegard, Journal of Crystal Growth 6 (1969) 61.) 형태 변화

세포상 조직 과 수지상 조직의 천이 온도구배가 온도구배 ↓ 면 특정한 범위로 수지상 형성 유지될 때 안정

Fig. 4.28 Columnar dendrites in a transparent organic alloy.

1차 가지 성장 방향 변화

열전도 방향 → 결정학적 우선 방향

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4.3 Alloy solidification

- Solidification of single-phase alloys - Eutectic solidification

- Off-eutectic alloys

- Peritectic solidification

4.4 Solidification of ingots and castings

- Ingot structure

- Segregation in ingot and castings - Continuous casting

Contents for today’s class

4.6 Solidification during quenching from the melt

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4.3.2 Eutectic Solidification

Normal eutectic Anomalous eutectic

Fig. 4.29 Al-CuAl₂ lamellar eutectic normal to the growth direction (x 680).

The microstructure of the Pb-61.9%Sn (eutectic) alloy presented a coupled growth of the (Pb)/bSn eutectic.

There is a remarkable change in morphology increasing the degree of undercooling with transition

from regular lamellar to anomalous eutectic.

http://www.matter.org.uk/solidification/eutectic/anomalous_eutectics.htm

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4.3.2 Eutectic Solidification

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4.3.2 Eutectic Solidification (Thermodynamics)

Plot the diagram of Gibbs free energy vs. composition at T

and T

.

What is the driving force for nucleation of  and ?

What is the driving force for the eutectic reaction (L → + ) at T

at C

?

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Eutectic Solidification (Kinetics)

If  is nucleated from liquid and starts to grow, what would be the composition at the interface of /L determined?

→ rough interface (diffusion interface) & local equilibrium How about at /L? Nature’s choice?

What would be a role of the curvature at the tip?

→ Gibbs-Thomson Effect interlamellar

spacing →

1) λ ↓→ 성장속도 ↑

2) λ ↓→ γ_αβ↑로 계의 계면 E↑

최소 λ 존재

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( ) 2 V

G 

    

( )

E

H T G  ^{ } T

    

Critical spacing, 

:  G ( 

)  0

*

0

2 T

V

H T

   

 

What would be the minimum ?

For an interlamellar spacing, , there is a total of (2/ ) m

of / interface per m

of eutectic.

( ) ? G 

 

How many / interfaces per unit length?  1   2

2

G   V

      

( ) 2 V

G 

    ⁾

:

SL SL m

V V

v

2 2 T 1

cf r G L T

L latent heat per unit volume

 ^  ^

       

m

G L T

 T ^

   

Eutectic Solidification

Driving force for nucleation

λ → ∞ ,

L = ΔH =H^L - H^S

*

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*

0

2 T V

identical to critical radius H T

    

 

Gibbs-Thomson effect in a G-composition diagram?

β

α

<

3 상 모두 평형 상태

G 증가의 원인은 α/β/L의 3중점에서 계면장력이 균형을 유지하기 위하여 α/L 계면과 β/L 계면이 곡률을 갖기 때문

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Fig. 4.33 (a) Molar free energy diagram at (T_E - ∆T₀) for the case

λ * < λ

< ∞

공정의 성장속도

v

→ α/L와 β/L 계면의 이동도가 커서 액상을 통한 용질이동과 비례

→ 성장 확산 제어

) (

 X

X

dl

D dC  



) (

 X

X

dl

D dC  



  k

D   ^X

0

*

,

0 ,

X X

X























0 0

*

0

( 1 )

T X

X X













 



) 1

(

* 0

2





 ^

  T D k v

Maximum growth rate at a fixed T₀

   2 

계면 과냉 변화시킴에 따라 성장속도와 간격 서로 독립적으로 변화시킬 수 있음.

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S S L L V

K T   K T   vL

From

L L V

v K T L

 



2

r

V

T T

L r

  

S

0, If T  

C i

T T T

  

Gibbs-Thomson effect:

melting point depression

r

T 2 T

G L

m

V











C L

V

T K

L r

  

Thermodynamics at the tip?

Closer look at the tip of a growing dendrite

different from a planar interface because heat can be conducted away from the tip in three dimensions.

Assume the solid is isothermal  T 

 0 

A solution to the heat-flow equation for a hemispherical tip:

'

( )

L

T negative T

r

 

L L V

v K T L

 

 v

 r 1

However, T also depends on r.

How?

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*

2

v o

r T

L T

 



r o

T r T

  r



* m

0 r

min

: T T T T r

r     







c

L L L

V V V

T T

T T

K K K r

v L r L r L r r

    

 

        

 

*

0 0

1

v as r r due to Gibbs-Thomson effect as r due to slower heat condution

 

  0

Minimum possible radius ( r)?

r o

Express T by r r and T  ,

 .

Maximum velocity?  r  2r

The crit.nucl.radius

2 _m

r

V

T T

L r

  

21 D

r

0

T T

T    



total r D

G G G

    

2

r

G V

free energy dissipated in forming / interfaces





 

 



G

free energy dissipated in diffusion

  Corresponding location at phase diagram?

Fig. 4.34 Eutectic phase diagram showing the relationship between ∆X and ∆X₀ (exaggerated for clarity)

curvature composition gradient

22

) 1

(

* 0

2





 ^

  T D k

v

   2 

* 0

2

0

k D T / 4 

v  

*

0

2 T

V

H T

   

 

T

 1 / 



 

2 4 0 0

3 2

0 0

)

( k

T v

k v

 

 

D

r

T

T

T    



계면 곡률효과 극복 과냉도

확산 위한 충분한 조 성차주기 위한 과냉



 T

const T 



T

