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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

09.29.2009

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β = A

₁₅

B

₇₅

C

₁₀

= A

₃

B

₁₅

C

₂

L = A

₃₅

B

₅₀

C

₁₅

= A

₇

B

₁₀

C

₃

β = 40%, L=60%

Homework1 website 참고

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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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Contents for previous class

• Diffusion

• Interstitial Diffusion / Substitution Diffusion

- Steady-state diffusion– Fick’s First Law

- Nonsteady-state diffusion – Fick’s Second Law

-

For random walk in 3 dimensions,

after n steps of length α

•

Effect of Temperature on Diffusivity

Movement of atoms to reduce its chemical potential μ.

driving force: Reduction of G

Down-hill diffusion

movement of atoms from a high C_B region to low C_B region.

Up-hill diffusion

movement of atoms from a low C_B region to high C_B region.

(atoms m^-2 s^-1)

( ) ⎛ α ⎞ ∂ ∂

= Γ − = − ⎜ ⎝ Γ ⎟ ∂ ⎠ = − ∂

2

1 2

1 1

6 6

B B

B B B B

C C

J n n D

x x

Concentration varies with position.

∂ ∂

∂ = ∂

2 2

B B

B

C C

t D x

Concentration varies with time and position

.

⎟⎠

⎜ ⎞

⎝

− ⎛

= . R T

D Q log D

log

1

3

0

2

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Contents for today’s class

• Solutions to the diffusion equations

• Interstitial Diffusion / Substitution Diffusion

1. Self diffusion in pure material 2. Vacancy diffusion

3. Diffusion in substitutional alloys

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Solutions to the diffusion equations (Fick’s 2

^nd

law의 응용)

Ex1. Homogenization

of sinusoidal varying composition

in the elimination of segregation in casting

l : half wavelength

l C x

C = + β

₀

sin π

: the mean composition C 　

0 : the amplitude of the initial concentration profile β

Initial or Boundary Cond.?

at t=0

2 0

2 >

∂

∂ x C_B

2 0

2 <

∂

∂ x C_B

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Rigorous solution of for

2 2

x D C t

C

∂

= ∂

∂

∂ ( )

l C x

x

C ,0 = + β₀sinπ

( )

( )

D l

t n

n

n n

n n

Dt

l e C x

C

A B

C A

l C x

C t

x B

x A

A C

e x B

x A

C

2 2

2

/ 0

0 1

0

0 1

0

sin , 0 ,

;

sin 0

cos sin

sin cos

π

λ

β π

β β π

λ λ

λ λ

−

∞

=

−

+

≡

∴

=

=

=

+

≡

→

=

+ +

=

∴

+

=

∴

∑

(A_n = 0 for all others)

τ : relaxation time

⎟ ⎠

⎜ ⎞

⎝ + ⎛ −

= τ

β π t

l C x

C

₀

sin exp

D l

2 2

τ = π

) 2 /

0

exp(

x l at

t

=

−

=

β τ

β

2 2

2 2 2

1 1

Let

λ

−

≡

=

=

=

dx X d X dt

dT DT

dx X DT d

dt X dT

XT C

x B

x A

X dx X

X d

λ λ

λ

sin cos

2 0

2 2

+ ′

= ′

= +

변수분리법이용

l C β sin π x X(x,0) ≡ +

₀

e Dt

T T

dt D T d

dt D dT T

2

0

2 2

ln 1

λ

λ λ

=

−

=

−

=

8

Solutions to the diffusion equations

Ex1. Homogenization

of sinusoidal varying composition in the elimination of segregation in casting

l C x

C = + β

₀

sin π

^{at t=0}

2 0

2 >

∂

∂ x C_B

2 0

2 <

∂

∂ x C_B

τ : relaxation time

⎟ ⎠

⎜ ⎞

⎝ + ⎛ −

= τ

β π t

l C x

C

₀

sin exp

D l

2 2

τ = π

) 2 /

0

exp(

x l at

t

=

−

=

β τ

β

농도분포 진폭(β)는

시간에 따라 지수적으로 감소

균질화 속도 결정

실제의 경우, 초기 농도 분포 = 파장과 진폭이 서로다른 무한히 많은 sine wave의 합 각 파는 각각의 τ에 의해서 결정되는 속도에 따라 소멸됨

결국 균질화는 가장 긴 파장의 성분에 대한 τ에 의해서 최종적으로 결정됨.

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Solutions to the diffusion equations

Ex2. Carburization of Steel

The aim of carburization is to increase the carbon concentration in the surface layers of a steel product in order to achiever a harder wear-resistant surface.

1. Holding the steel in CH₄ and/or Co at an austenitic temperature.

2. By controlling gases the concentration of C at the surface of the steel can be maintained at a suitable constant value.

3. At te same time carbon continually diffuses from the surface into the steel.

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Carburizing of steel

Boundary conditions

The error function solution:

( ) ( ) 0 , , t C

₀

C

C t

C

B

s B

=

∞

=

( )

( )

^{Ζ =}

∫

^{Ζ −}

⎟⎠

⎜ ⎞

⎝

− ⎛

−

=

0 0

2 2

2 dy e

erf

Dt erf x

C C

C C

y s

s

π

Depth of Carburization?

• Since erf(0.5)

≈

0.5, the depth at which the carbon concentration is midway between C

_s

and C

₀

is given

that is

( / 2 x Dt ) ≅ 0.5 x ≅ Dt

erf(0.5) ≈ 0.5

⎟ ⎠

⎜ ⎞

⎝

= ⎛

−

−

Dt erf x

C C

C C

s s

0

2

0

2 C

s

C C = +

Fick’s 2^nd law의 해

이용

시편 무한히 길다/

확산계수 농도의 평균값인 경우로 가정

→

Depth of Carburization

* 농도분포는

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Error function

In mathematics, the error function (also called the Gauss error

function) is a non-elementary function which occurs in probability, statistics and partial differential equations.

It is defined as:

By expanding the right-hand side in a Taylor series and integrating, one can express it in the form

for every real number x. (

From Wikipedia, the free encyclopedia

)

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Error function = −

π ∫OZ 2

erf(z) 2 exp( y )dy

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Carburizing of steel

Boundary conditions

The error function solution:

( ) ( ) 0 , , t C

₀

C

C t

C

B

s B

=

∞

=

( )

( )

^{Ζ =}

∫

^{Ζ −}

⎟⎠

⎜ ⎞

⎝

− ⎛

−

=

0 0

2 2

2 dy e

erf

Dt erf x

C C

C C

y s

s

π

Depth of Carburization?

• Since erf(0.5)

≈

0.5, the depth at which the carbon concentration is midway between C

_s

and C

₀

is given

that is

( / 2 x Dt ) ≅ 0.5 x ≅ Dt

erf(0.5) ≈ 0.5

⎟ ⎠

⎜ ⎞

⎝

= ⎛

−

−

Dt erf x

C C

C C

s s

0

2

0

2 C

s

C C = +

Fick’s 2^nd law의 해

이용

시편 무한히 길다/

확산계수 농도의 평균값인 경우로 가정

→

Depth of Carburization

* 농도분포는

•

14

Carburizing of steel

⎟ ⎠

⎜ ⎞

⎝

= ⎛

Dt erf x

C

C

⁰

2

≅ Dt

Dt

Thus the thickness of the carburized layer is .

Note also that the depth of any is concentration line is directly proportion to , i.e. to obtain a twofold increase

in penetration requires a fourfold increase in time.

Decarburization of Steel?

15

= = −∞ < <

= = < < ∞

= = −∞ < < ∞

= = ∞ < < ∞

1 2 1 2

0, 0

0, 0 , 0 , 0

C C t x

C C t x

C C x t

C C x t

Joining of two semi-infinite specimens of compositions C

₁

and C

₂ ^(C₁ ^{> C}₂ ⁾

Ex3. Diffusion Couple

Draw C vs. x with time t = 0 and t > 0. Boundary conditions?

Solutions to the diffusion equations

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The section is completed with 4 example solutions to Fick's 2nd law:

carburisation, decarburisation, diffusion across a couple and homogenisation.

The solutions given are as follows:

Process Solution

Carburisation

C_S = Surface concentration C₀ = Initial bulk concentration

Decarburisation

C₀ = Initial bulk concentration

Diffusion Couple

C₁ = Concentration of steel 1 C₂ = Concentration of steel 2

Homogenisation

C_mean = Mean concentration

b₀ = Initial concentration amplitude l = half-wavelength of cells

t = relaxation time

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Substitutional Diffusion

1. Self diffusion in pure material 2. Vacancy diffusion

3. Diffusion in substitutional alloys

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Substitutional diffusion

1. Self diffusion in pure material

What would be the jump frequency in substitutional diffusion?

An atom next to a vacancy can make a jump provided it has enough thermal energy to overcome ΔG_m.

The probability that an adjacent site is vacant

→ zX

_v

→ exp(-ΔG

_m

/kT) Probability of vacancy x probability of jump

→ Γ = ν

_V

−Δ G

^m

zX exp

RT

_V

=

^e_V

= −Δ G

^V

X X exp

RT

1

2

A

6

A

D = Γ α

_A

= α ν 1

²

− Δ ( G

^m

+ Δ G )

^V

D z exp

6 RT

For most metals: ν ~ 10

¹³

, fcc metals : z = 12, α = a 2

Gm

cf ) z exp for int erstitials RT

Γ = ν −Δ

Z=최인접 원자수/ ν= 진동수

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Experimental Determination of D

∂ = ∂

∂ ∂

2 2

B B

B

C C

t D x

Solution for

the infinite boundary condition

M : quantity·m^-2/ C : quantity·m^-3

- Deposit a known quantity (M) of a radioactive isotope A*

(방사성동위원소)

4 ) 2 exp(

2

Dt x Dt

C = M −

π

4 )

2 exp(

²

Dt x Dt

C = M −

π

Solution for the semi-infinite B.C.

20

0

exp

^ID

B B

D D Q

RT

= −

m m m

G H T S

Δ = Δ − Δ 1

²

B

6

B

D = Γ α

B

? jump frequency Γ

Thermally activated process

Ζ : nearest neighbor sites ν : vibration frequency

ΔG_m : activation energy for moving

( ) ( )

ID m

m m

B

Q H

RT H

R S

D

≡ Δ

Δ

⎥⎦ −

⎢⎣ ⎤

⎡ Ζ Δ

= exp / exp /

6

1 α

₂

ν

( G

_m

RT )

B

= Ζ exp − Δ /

Γ ν

previous class: interstitial diffusion

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⎟ ⎠

⎜ ⎞

⎝

− ⎛

= . R T

D Q log D

log 1

3

0

2

Temperature Dependence of Diffusion

=

₀

exp −

^ID

B B

D D Q

RT

Therefore, from the slope of the D-curve in an log D vs 1/T coordinate, the activation energy may be found.

How to determine Q

_ID

experimentally?

previous class: interstitial diffusion

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)

0

exp(

RT D Q

D

_A

= −

^SD

Substitutional diffusion

1. Self diffusion in pure material

− Δ + Δ

= α ν

^{2 m V}

A

1 ( G G )

D z exp

6 RT

For most metals: ν ~ 10

¹³

, fcc metals : z = 12, α = a 2

방사성동위원소

Z=최인접 원자수/ ν= 진동수

S T H

G = Δ − Δ Δ

) exp(

6 exp 1

₂

RT

H H

R S z S

D

_A ^{m V}

Δ

^m

+ Δ

^V

Δ − +

= α ν Δ

R S z S

D = exp Δ

^m

+ Δ

^V

6

1

₂

0

α ν

V m

SD

H H

Q = Δ + Δ

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For a given structure and bond type, Q/RT

_m

is roughly constant;

Q is roughly proportional to T_m

.

ex) for fcc and hcp, Q/RT_m ~ 18 and D(T_m) ~ 1 μm²s^-1 (10^-12 m²s^-1)

For a given structure and bond type, D(T/T

_m

) ~ constant

T/T_m : homologous temperature

Within each class, D(T

_m

) and D

₀

are approximately constants.

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25

26

ex) At 800

^o

C, D

_Cu

= 5 × 10

^-9

mm

²

s

^-1

, α = 0.25 nm

2

6

1

_B

α

D

B

= Γ Γ

_Cu

= 5 × 10

⁵

jumps s

⁻¹

After an hour, diffusion distance (x)? Dt ~ 4 μ m

Cu

: ? Γ

At 20

^o

C, D

_Cu

~ 10

^-34

mm

²

s

^-1

Γ

_Cu

~ 10

⁻²⁰

jumps s

⁻¹

→ Each atom would make one jump every 10

¹²

years!

Hint) From the data in Table 2.2, how do we estimate D_Cu at 20^oC?

How do we determine D

_Cu

at low temperature such as 20

^o

C?

Cu에서 자기 확산에 대한 온도의 영향:

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All the surrounding atoms are possible

jump sites of a vacancy, which is analogous to interstitial diffusion.

) /

exp(

) / 6 exp(

1 6 1

2 2

RT H

R S

zv D

m m

v v

Δ

− Δ

=

Γ

=

α α

Comparing D_v with the self-diffusion coefficient of A, D_A,

2. Vacancy diffusion

e v A

v

D X

D = /

This shows in fact that the diffusivity of vacancy (D_v) is many orders of magnitude greater than the diffusivity of substitutional atoms (D_A).

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3. 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 J

J

J_{A A A}^{v A}

∂

− ∂

= +

′ = ~

x D C J

J

J_{B B B}^{v B}

∂

− ∂

= +

′ = ~

B A A

BD X D

X

D~ = +

⎟⎠

⎜ ⎞

⎝

⎛

∂

∂

∂

= ∂

∂

∂

x D C x t

C_A ~ _A

A

B

J

J ′ = − ′

∴

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