PowerPoint 프레젠테이션

재 료 상 변 태

Phase Transformation of Materials

2008.09.30.

박 은 수

서울대학교 재료공학부

Contents for previous class

• Ternary Equilibrium: Ternary Phase Diagram

< Chapter 2 >

• Diffusion

• Interstitial Diffusion – Fick’s First Law

• Effect of Temperature on Diffusivity

* Simple Eutectic Systems Δ H

_mix^L

= 0

Tie line: binary system

>> 0 Δ H

_mix^S

α L+α L L+β β

α L

β

Tie line: ternary system

Diffusion

Diffusion

Diffusion : Mechanism by which matter transported through matter

What is the driving force for diffusion?

⇒ a concentration gradient (x)

⇒ a chemical potential (o)

But this chapter will explain with concentration

gradients for a convenience.

Substitutional diffusion Interstitial diffusion

Diffusion: THE PHENOMENON

Assume that there is no lattice distortion and also that there are always six vacant sites around the diffusion atom.

J_B : Flux of B atom

Γ_B : Average jump rate of B atoms n₁ : # of atoms per unit area of plane 1 n₂ : # of atoms per unit area of plane 2

2

1

6

1 6

1 n n

J

_B

= Γ

_B

− Γ

_B

Random jump of solute B atoms in a dilute solid solution of a simple cubic lattice

Interstitial diffusion

α α

= =

B B

C (1) n

₁

, C (2) n

₂

(atoms m^-2 s^-1)

( n

₁

− n

₂

) = α ( C

B

(1) − C

B

(2) )

D_B: Intrinsic diffusivity or Diffusion coefficient of B

⇒ depends on

microstructure of materials

Fick’s First Law of Diffusion

(

_{1 2}

)

2

1

6

1 6

1 6

1 n n n n

J

_B

= Γ

_B

− Γ

_B

= Γ

_B

−

( ) ⎛ α ⎞ ∂ ∂

= Γ − = − ⎜ ⎝ Γ ⎟ ∂ ⎠ = − ∂

2

1 2

1 1

6 6

B B

B B B B

C C

J n n D

x x

( )

α

− = − ∂ ∂

B B B

C (1) C (2) C x

α: Jump distance

Magnitude of D in various media Gas : D ≈ 10^-1 cm²/s

Liquid : D ≈ 10^-4 ~ 10^-5 cm²/s

Solid : Materials near melting temp. D ≈ 10^-8 cm²/s Elemental semiconductor (Si, Ge) D ≈ 10^-12 cm²/s

Concentration varies with position.

EFFECT OF TEMPERATURE on Diffusivity

Thermal Activation

Temperature Γ D, Diffusivity

(Interstitial atom jumps times per second)

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

⎟ ⎠

⎜ ⎞

⎝

− ⎛

= . 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?

Contents for today’s class

• Steady-state diffustion – Fick’s First Law

• Nonsteady-state diffusion – Fick’s Second Law

• Solutions to the diffusion Equations

Steady-state diffusion

The simplest type of diffusion to deal with is when the concentration at every point does not change with time.

Steady-state diffusion

The simplest type of diffusion to deal with is when the concentration at every point does not change with time.

Nonsteady-state diffusion

In most practical situations steady-state conditions are not established, i.e. concentration varies with both distance and time, and Fick’s 1st law can no longer be used.

How do we know the variation of C

_B

with time?

→ Fick’s 2nd law

The number of interstitial B atoms that diffuse into the slice across plane (1) in a small time interval dt :

→ J

₁

A dt Likewise : J

₂

A dt

Sine J₂ < J₁, the concentration of B within the slice will have increased by

( ) δ

δ δ

=

¹

−

²

B

J J A t

C A x

Due to mass conservation

( J

1

− J

2

) A t δ = = ? δ C A x

_B

δ

Nonsteady-state diffusion

B B

C J

t x

∂ ∂

∂ = − ∂

Fick’s Second Law

substituting Fick's 1st law gives

⎟ ⎠

⎜ ⎞

⎝

⎛

∂

∂

∂

= ∂

∂

∂

x D C

x t

C

_B

B B

∂ ∂

∂ = ∂

2 2

B B

B

C C

t D x

x D C

J

_B ^B

∂

− ∂

=

If variation of D with concentration can be ignored,

x ? C

_B

∂ =

∂

2 2

∂ = ∂

∂ ∂

2 2

B B

B

C C

t D x

Fick’s Second Law

Concentration varies with time and position.

Note that is the curvature.

₂

2

x C_B

∂

∂

Diffusion

Solutions to the diffusion equations

Ex1. Homogenization

of sinusoidal varying composition

in the elimination of segregation in casting

τ : relaxation time

l : half wavelength

l C x

C = +

β

₀sin

π

⎟ ⎠

⎜ ⎞

⎝ + ⎛ −

= τ

β π t

l C x

C

₀

sin exp

D l

2 2

τ = π

sin 2 x T

⎛ π ⎞

⎜ ⎟

⎝ ⎠

: the mean composition C 　

0 : the amplitude of the initial concentration profile β

Initial or Boundary Cond.?

at t=0

) 2 /

0

exp(

x l at

t =

−

= β τ

β

Rigorous solution of for

2 2

x D C t

C

∂

= ∂

∂

∂

( )

l C x

x

C ,0 = + β₀sinπ

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

e Dt

T T

dt D T d

dt D dT T

2

0

2 2

ln 1

λ

λ λ

=

−

=

−

=

⎟⎠

⎜ ⎞

⎝ + ⎛ −

=

τ

β π t

l C x

C

₀

sin exp

( )

( )

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)

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.

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 = +

→ Depth of Carburization

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 isoconcentration line is

directly proportion to , i.e. to obtain a twofold increase in penetration requires a fourfold increase in time.

Decarburization of Steel?

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

)

Error function = − π ∫OZ 2

erf(z) 2 exp( y )dy

= = −∞ < <

= = < < ∞

= = −∞ < < ∞

= = ∞ < < ∞

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

(C

₁

> C

₂

) Ex3. Diffusion Couple

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

Solutions to the diffusion equations

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