5931.pdf

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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

10.08.2009

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• Interstitial Diffusion / Substitution Diffusion

1. Self diffusion in pure material 2. Vacancy diffusion

3. Diffusion in substitutional alloys

• Atomic Mobility

• Tracer Diffusion in Binary Alloys

• High-Diffusivity Paths

1. Diffusion along Grain Boundaries and Free Surface 2. Diffusion Along Dislocation

• Diffusion in Multiphase Binary Systems

Contents for previous class

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• Diffusion in substitutional alloys

J_A

J_B J_V

At t = 0

,

At t = t₁

,

J_A

J_V^A J_A

’

J_B J_V^B

J_B

’

x D C vC

J J

J

J_{A A v}^A _{A A} ^A

∂

− ∂

= +

= +

′ = ~

x D C vC

J J

J

J_{B B B}^v _{B B} ^B

∂

− ∂

= +

= +

′ = ~

B A A

BD X D

X

D~ = +

⎟⎠

⎜ ⎞

⎝

⎛

∂

∂

∂

= ∂

∂

∂

x D C x t

C_A ~ _A

A

B

J

J ′ = − ′

농도 구배에 의한 속도 + 격자면 이동에 의한 속도

∴

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Atomic mobility

A diffusion flux is a combined

quantity of a drift velocity and random jumping motion.

M x

v

_{B B} ^B

∂

− ∂

= μ

M_B : mobility of B atoms

: chemical force causing atom to migrate x

B

∂

− ∂μ

C x M

J

_{B B B} ^B

∂

− ∂

= μ

B B

B

v c

J =

How the mobility of an atom is related to its diffusion coefficient?

The problem of atom migration can be solved by considering the thermodynamic condition for equilibrium; namely that the chemical potential of an atom must be the same everywhere. In general the flux of atoms at any point in the lattice is proportional to the chemical potential gradient: diffusion occurs down the slope of the chemical potential.

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Relationship between M

_B

and D

_B

For ideal or dilute solutions,

near X_B ≈ 0, γ_B = const. with respect to X_B

x X RT X

x X RT x

X RT

x G x

B B

B

B B

B B B

B

∂

∂

∂ + ∂

=

∂ + ∂

∂

= ∂

∂ +

= ∂

∂

∂

) ln ln

1 ln (

ln ) ( ln

) ln

( ⁰

γ γ μ γ

x D C

x RTF C

M

x X X

X RT V

M X J

B B

B B

B B

B B

m B B B

∂

− ∂

∂ =

− ∂

=

∂

∂

∂ + ∂

−

=

∴

ln ) 1 ln

( γ

ln ) 1 ln

(

B B

X d

F = + d γ

= 1

∴ F D

_B

= M

_B

RT

C x M

J

_{B B B} ^B

∂

− ∂

= μ μ

^B

= G

^B

+ RT ln a

^B

= G

^B

+ RT ln γ

^B

X

^B

m B m

B A

B B

B

V

X V

n n

n V

C n =

= +

= ( )

RTF M

D

_B

=

_B

For non-ideal concentrated solutions,

thermodynamic factor must be included.

Related to the curvature of the molar free energy-composition curve.

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Contents for today’s class

• Interstitial Diffusion / Substitution Diffusion

• Atomic Mobility

• Tracer Diffusion in Binary Alloys

• High-Diffusivity Paths

1. Diffusion along Grain Boundaries and Free Surface 2. Diffusion Along Dislocation

• Diffusion in Multiphase Binary Systems

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2.5 Tracer diffusion in binary alloys

It is possible to use radioactive tracers (D^*_Au) to determine the intrinsic diffusion coefficients (D_Au) of the components in an alloy .

*

Au Au

How does D differ from D ?

* *

0 ,

mix Au Au Ni Ni

H D D D D

Δ > → < <

D*_Au gives the rate at which Au* (or Au) atoms diffuse in a chemically homogeneous alloy, whereas D_Au gives the diffusion rate of Au when concentration gradient is present.

1) Au* in Au

2) Au* in Au-Ni

D^*_Au = D_Au(self diffusivity)

If concentration gradient exhibit,

the rate of homogenization will there fore be slower.

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Since the chemical potential gradient is the driving force for diffusion in both types of experiment, it is reasonable to suppose that the atomic

mobility are not affected by the concentration gradient.

What would be the relation between the intrinsic chemical diffusivities D_B and tracer diffusivities D*_B?

In the tracer diffusion experiment the tracer essentially forms a dilute solution in the alloy.

B B B

D

^*

= M RT

^*

= M RT

B B ^B B B

D M RT d FM RT

d X 1 ln

ln γ

⎧ ⎫

= ⎨ + ⎬ =

⎩ ⎭

A A

B B

D FD D FD

*

*

=

= D % = X D

^{B A}

+ X D

^{A B}

= F X D (

^{B A*}

+ X D

^{A B*}

)

A B A B

A B

d d X X d G

F d X d X RT dX

2 2

ln ln

1 1

ln ln

γ γ

⎧ ⎫ ⎧ ⎫

= ⎨ + ⎬ ⎨ = + ⎬ =

⎩ ⎭ ⎩ ⎭

A B

A B

A B

Eq

d d

X X d G RT RT

dX d X d X

2 2

.(1.71)

ln ln

1 1

ln ln

γ γ

⎧ ⎫ ⎧ ⎫

= ⎨ + ⎬= ⎨ + ⎬

⎩ ⎭ ⎩ ⎭

F : Thermodynamic Factor

Additional Thermodynamic Relationships for Binary Solution:

Variation of chemical potential (dμ) by change of alloy compositions (dX)

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B A A B

D % = X D + X D

(

^{B A A B}

)

D % = F X D

^*

+ X D

^*

2.5 Tracer diffusion in binary alloys

D*_Au

> D*

_Ni

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2.6 Diffusion in ternary alloys:

Additional Effects

Example)

Fe-Si-C system (Fe-3.8%Si-0.48%C) vs. (Fe-0.44%C) at 1050

℃

① Si raises the μ_C in solution.

(chemical potential of carbon)

② M_Si (sub.) ≪ M_C (int.),

(M : mobility)

at t = 0

A C B

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How do the compositions of A and B change with time?

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Grain boundary dislocation surface

Real materials contain defects.

= more open structure fast diffusion path.

2.7.1 High-diffusivity paths

Diff. along lattice

Diff. along grain boundary Diff. along free surface

) exp( RT D Q

D

_l

=

_lo

−

^l

) exp( RT D Q

D

_b

=

_bo

−

^b

) exp( RT D Q

D

_s

=

_so

−

^s

s b l

D > D > D

But area fraction

→ lattice > grain boundary

> surface

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Atoms diffusing along the boundary will be able to penetrate much deeper than atoms which only diffuse through the lattice.

In addition, as the concentration of solute builds up in the boundaries, atoms will also diffuse from the boundary into the lattice.

Diffusion along grain boundaries

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Combined diffusion of grain boundary and lattice?

Thus, grain boundary diffusion makes a significant contribution

only when D_bδ > D_ld.

D d D

D_{app = l + b} δ

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The relative magnitudes of D

_bδ

and D

_l

d are most sensitive to temperature.

Due to

Q

_b

< Q

_l

,

the curves for D_l and D_bδ/d cross

in the coordinate system of lnD versus 1/T.

D

_b

> D

_l

at all temp.

Therefore, the grain boundary diffusion becomes predominant at temperatures lower than the crossing temperature.

(T < 0.75~0.8 T_m)

-Q/2.3R

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2.7.2 Diffusion along dislocations

app p

l l

D D

D = + ⋅ 1 g D

app l p

D = D + ⋅ g D

hint) ‘g’ is the cross-sectional area of ‘pipe’ per unit area of matrix.

D

_app

= ?

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ex) annealed metal ~ 10

⁵

disl/mm

²

; one dislocation(

┴

) accommodates 10 atoms in the cross-section; matrix contains 10

¹³

atoms/mm

²

.

g

5 6

7

13 13

10 * 10 10 10 10 10

= = =

−

At high temperatures,

diffusion through the lattice is rapid and gD

_p

/D

_l

is very small

so that the dislocation contribution to the total flux of atoms is very small.

At low temperatures,

gD

_p

/D

_l

can become so large that the apparent diffusivity is entirely due to diffusion along dislocation.

g = cross-sectional area of ‘pipe’ per unit area of matrix

Due to

Q

_p

< Q

_l

,

the curves for D_l and gD_p/D_l cross in the coordinate system of lnD versus 1/T.

(T < ~0.5 T_m)

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2.8 Diffusion in multiple binary system

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2.8 Diffusion in multiple binary system

A diffusion couple made by welding together pure A and pure B

Draw a phase distribution and composition profile in the plot of distance vs. X

_B

after annealing at T

₁

.

Draw a profile of activity of B atom,

in the plot of distance vs. a

_B

after annealing at T

₁

.

→ a layered structure containing α, β & γ.

What would be the microstructure evolved after annealing at T₁ ?

A B

α β γ

A or B atom → easy to jump interface (local equil.)

→ μ_A^α= μ_A^β, μ_A^β= μ_A^γ at interface (a_A^α= a_A^β, a_A^β= a_A^γ)

β

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If unit area of the interface moves a distance dx, a volume (dx⋅1) will be converted

from α containing C

_B^α

atoms/m

³

to β containing C

_B^β

atoms/m

³

. ( C

^Bβ

− C

^Bα

) dx → shaded area

How can we formulate the interface (α/β, β/γ) velocity?

( C

^B

C

^B

) dx equal to which quantity?

β

−

α

α→β

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( ) ( ) = ( )

b a

b a

B B

B B

C C

D D dt C C dx

x x

β α

⎧ ⎛ ∂ ⎞ ⎛ ∂ ⎞ ⎫

⎪ − − − ⎪ −

⎨ ⎜ ∂ ⎟ ⎜ ∂ ⎟ ⎬

⎪ ⎝ ⎠ ⎝ ⎠ ⎪

⎩ % % ⎭

In a time dt, there will be an accumulation of B atoms given by

( )

b B B

J D C

x

β

= − β ∂

% ∂

B

( )

^Ba

J D C

x

α

= − α ∂

% ∂

1 ( ) ( )

( )

(velocity of the / interface)

a b

B B

b a

B B

C C

v dx D D

dt C C α x β x

α β

⎧ ∂ ∂ ⎫

= = − ⎨ ⎩ % ∂ − % ∂ ⎬ ⎭

A flux of B towards the interface from the β phase

A flux of B away from the interface into the α phase

Local equilibrium is assumed.

x J t

C

∂

− ∂

∂ =

∂

2 2

x D C t

C

∂

− ∂

∂ =

∂

- [J_B-J_A] dt dCdx

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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) 상변태를상변태를

이해하는데 이해하는데 필요한필요한 배경배경

대표적인대표적인 상변태상변태

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• Types of Interface

1. Free surface (solid/vapor interface)

2. Grain boundary (

α/ α

interfaces)

> same composition, same crystal structure

> different orientation

3. inter-phase boundary (

α

/

β

interfaces)

> different composition &

crystal structure

3. Crystal interfaces and microstructure

solid vapor

β

α defect

energy ↑