ema5001 lecture 19 spinodal decomposition...ema5001 lecture 19 spinodal decomposition ema 5001...

© 2016 by Zhe Cheng

EMA5001 Lecture 19

Spinodal Decomposition

EMA 5001 Physical Properties of Materials Zhe Cheng (2016) 19 Spinodal Decomposition

Introduction

Certain solid-state transformation does not have (significant) barrier to

nucleation

Spinodal decomposition

Topics

Basics and unique features

Driving force

Features

Simplicity and more precise mathematical description

Transformation kinetics determined purely by diffusion

2


Composition & Temperature Range for


Binary system with a miscibility

gap

Gibbs free energy – composition

curve at a given temperature

below Tc

Cooling from T1 (> Tc ) to T2 (<Tc)

For certain composition X0

Initial free energy G0 higher than

equilibrium

Slight composition variation near X0

leads to lowering of system energy

− Spontaneous decomposition

− “Uphill” diffusion

− Will always reach equilibrium

Condition for such transformation or

unstable region

3

A B

G

eq

A

X1 X2

eq

B

XB

02

2

dX

Gd

A B

T

X1 X2 XB

α1 + α2

α

T2

T1

At T2

X0

G0

Tc

02

2

dX

Gd

Geq


Composition Change with Time for


4

A B

G

eq

A

X1 X2

eq

B

XB

02

2

dX

Gd

A B

T

X1 X2 XB

α1 + α2

α

T2

T1

At T2

X0

G0

Tc

Geq

Phase Transformations in Metals &

Alloys, Porter, 3rd Ed, 2008, p. 305


Transformation Outside Spinodal

Decomposition Range

Outside (chemical) spinodal

Same system as before

Cooling from T1 (> Tc ) to T2 (<Tc)

For certain composition X’0

Initial free energy higher than

equilibrium

Small composition fluctuation near

original composition (X’0) leads to

increase in system free energy

− Need nucleation process with nuclei

composition very different from matrix

(> Xc)

− “Down-hill” diffusion

− May stay at meta-stable condition

Condition for such transformation or

unstable region

5

A B

G

eq

A

X1 X2

eq

B

XB

A B

T

X1 X2 XB

α1 + α2

α

T2

T1

At T2

X’0

G0

Tc

Geq

02

2

dX

GdSXXX '01

Xc

XS

XS


Transformation Outside Spinodal

Decomposition Range

6

A B

G

eq

A

X1 X2

eq

B

XB

A B

T

X1 X2 XB

α1 + α2

α

T2

T1

At T2

X’0

G0

Tc

Geq

Phase Transformations in Metals &

Alloys, Porter, 3rd Ed, 2008, p. 306

XS


Spinodal Decomposition vs.

Nucleation & Growth

Type of precipitation Nucleation & Growth Spinodal Decomposition

Free energy – composition curve

Condition Critical supercooling Composition fluctuation

Nucleus formation? Yes No

Composition change from matrix to new phase

Very large Small

Structure change from matrix to new phase

Typically large Essentially no change

Interface Distinct Diffuse and not well defined

Direction of diffusion Down-hill Uphill

Rate of transformation Slow Fast

Precipitate size and distribution

Small number of precipitates with large size

Large number of precipitates with small size

7

02

2

dX

Gd0

2

2

dX

Gd


Driving Force for Spinodal

Decomposition (1)

Assuming initial matrix

composition of X0

For simplicity, assume it locally

experiences compositional

fluctuation of δX

Local sys. free energy before fluctuation

Local sys. free energy after fluctuation

(omitting 3rd order and higher terms)

Local system free energy change:

8

A B

G

eq

A

X1 X2

eq

B

XB

02

2

dX

Gd

A B

T

X1 X2 XB

α1 + α2

α

T2

T1

At T2

X0

G0

Tc

Geq

0G

2

2

2

0

2

2

2

00

2

1

2

1

2

1

2

1'

XdX

GdX

dX

dGG

XdX

GdX

dX

dGGG

00 ' GGGv


Driving Force for Spinodal

Decomposition (2)

Continue from p.8

Assuming system interface energy and straining energy can be neglected

System free energy change for such local composition fluctuation is

Since

We have

Therefore, only when

we have

9

00 ' GGGv

2

2

2

0

2

2

2

002

1

2

1

2

1

2

1' X

dX

GdX

dX

dGGX

dX

GdX

dX

dGGG

22

2

2

1X

dX

GdGv

02

2

dX

Gd

02

1 2

2

2

XdX

GdGv

The phase separation/transformation will be spontaneous


Up-Hill Diffusion in Spinodal

Decomposition (1)

Describing diffusive flux of atoms from chemical potential gradient

Net drift velocity v superimposed on random jumping due to a driving force/potential

Intrinsic diffusive flux of B (with respect to a lattice plane) is related to drift velocity as

Drift velocity is proportional to local gradient of chemical potential, i.e.,

Where MB is the atomic mobility

Therefore, flux of B due to the driving force is

10

BBB CvJ

xM

xMv B

BB

BB

'

xCMJ B

BBB



Decomposition (2)

Continue from p. 10

Consider:

We have

For binary alloy,

Under constant temperature and pressure, Gibbs-Duham equation gives

On the other hand,

We have

11

xCMJ B

BBB

mBB VXC /

x

C

XXM

x

C

VXVXMJ B

B

BBB

B

mB

BmBBB

//

BBAA XXG

0 VdPSdTdXdX BBAA

BBAABBAA dXdXdXdXdG

BBAA dXdXdG



Decomposition (3)

Continue from p.11

Therefore,

For regular solution

Similarly,

12

BBAA dXdXdG

ABB

B

AA

B dX

dX

dX

dG

)ln(ln 00

BBBBBB XRTaRT

A

A

B

B

B

A

A

A

B

B

B dX

d

dX

d

dX

dX

dX

d

dX

d

dX

Gd

2

2

B

B

BB

B

BB

B

Xd

d

X

RT

dX

d

XRT

dX

d

ln

ln1

ln1

A

A

AA

A

Xd

d

X

RT

dX

d

ln

ln1



Decomposition (4)

Continue from p.12

Again, remember and

We have

Or

Therefore,

Therefore,

13

B

B

BAB

B

BAA

A

B

B

B Xd

d

XX

RT

Xd

d

XXRT

dX

d

dX

d

dX

Gd

ln

ln1

ln

ln1

112

2

A

A

AA

A

Xd

d

X

RT

dX

d

ln

ln1

B

B

BB

B

Xd

d

X

RT

dX

d

ln

ln1

0 BBAA dXdX

0ln

ln1

ln

ln1

B

B

B

A

BA

A

A

A

A dXXd

d

X

RTXdX

Xd

d

X

RTX

0ln

ln1

ln

ln1

B

B

BB

A

A dXXd

ddX

Xd

d

B

B

A

A

Xd

d

Xd

d

ln

ln1

ln

ln1

1 BA XX



Decomposition (5)

Continue from p.13

On the other hand, and

Therefore,

Diffusion coefficient

Therefore, diffusion coefficient

When

14

B

B

BAB Xd

d

XX

RT

dX

Gd

ln

ln1

2

2

x

C

XXMJ B

B

BBBB

2

2

B

BAB

B

BBB

dX

GdXXM

XXMD

B

B

BB

B

Xd

d

X

RT

dX

d

ln

ln1

A

A

AA

A

Xd

d

X

RT

dX

d

ln

ln1

A

AA

B

BB

B

B

B

BAdX

dX

dX

dX

Xd

dRT

dX

GdXX

ln

ln1

2

2

02

2

dX

Gd0D


Interfacial Energy & Strain Energy Effects

on Spinodal Decomposition

If interfacial energy and strain energy could NOT be neglected

Interfacial energy / gradient energy term

− Origin from bonding energy difference between A-B versus A-A and B-B types of bonding

− Value

K proportional constant

Wavelength of composition variation

Strain energy term

− Origin from size difference in atoms

− Value

Defined as

v Poisson’s ratio

15

2

XKG

mms VEXVv

EXG '

1

2222

dX

da

a

1


Interfacial Energy & Strain Energy Effects

on Spinodal Decomposition

Total free energy change has three contributions

We have

For Spinodal decomposition to occur

At a given temperature, the limiting composition for Spinodal decomposition to

occur (i.e., when ) satisfies

Minimal wavelength given by

16

ms Vv

EXG

1

22

2

XKG 2

2

2

2

1X

dX

GdGv

2

'22

2

2

22

2 XVE

K

dX

GdG m

mVEdX

Gd'2 2

2

2

0'22 2

22

2

mVEK

dX

Gd

mVEdX

Gd

K

'2

2

2

2

2

2

ema5001 lecture 19 spinodal decomposition...ema5001 lecture 19 spinodal decomposition ema 5001...

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