© 2016 by Zhe Cheng

EMA5001 Lecture 7

Short-Circuit Diffusion, &

Reaction Diffusion

EMA 5001 Physical Properties of Materials Zhe Cheng (2016) 7 Short Circuit Diff & Reaction Diff

Diffusion Along Grain Boundaries (1)

Diffusion along grain boundaries or surface also follow Arrhenius

Equation

Relationship of lattice (bulk), grain

boundaries, and surface diffusion

Example

Steady state diffusion through a thin

polycrystalline sheet

Grain boundary perpendicular to sheet

Concentration gradient identical in

grain boundaries and in lattice

2

RT

QDD b

bb exp0

RT

QDD s

ss exp0

lbs DDD

(1) (2) (1)

Grain boundary

0 x

C

Grain Jl

Jb

d

EMA 5001 Physical Properties of Materials Zhe Cheng (2016) 7 Short Circuit Diff & Reaction Diff

Diffusion Along Grain Boundaries (2)

Continue from p.10

Flux through lattice:

Flux through the grain boundary

If grain size of d,

If grain boundaries has effective thickness of , and

Total flux

Therefore,

For diffusion through such a polycrystalline film, the apparent diffusion coefficient

3

dx

dCDJ ll

dx

dCDJ bb

d

dx

dCDd

dx

dCD

dd

JdJ

d

JdJJ bl

blbl 1

dx

dCD

dDJ bl

blapp Dd

DD

EMA 5001 Physical Properties of Materials Zhe Cheng (2016) 7 Short Circuit Diff & Reaction Diff

1/T

lgD

Db

Dl

(/d)Db

Diffusion Along Grain Boundaries (3)

Continue from p.11

Or

The relative importance of lattice and grain

diffusion depends on ratio of

Normally

− Db > Dl

− Qb 0.5 Ql

Temperature effect

− Grain boundary diffusion dominates at low T;

− Lattice diffusion dominates at high T

− Transition temperature of 0.75-0.80 Tm

More pronounced for material w/ small grains

Grain boundary diffusion depends on specific boundary and even direction

4

l

b

l

app

dD

D

D

D 1

blapp Dd

DD

lb dDD /

Dapp

0.75-0.80Tm

EMA 5001 Physical Properties of Materials Zhe Cheng (2016) 7 Short Circuit Diff & Reaction Diff

Diffusion Along Dislocations

Diffusion along dislocation is often also

faster than lattice diffusion

Qd 0.5 Ql

Flux through lattice:

Flux through the dislocation

If area fraction of total dislocation is g, (g ~ 10-7)

Total flux

5

Dislocation

Grain

Jl

Jd

Unit area

dx

dCDJ ll

dx

dCDJ dd

dx

dCgDD

dx

dCgD

dx

dCDgJJJ dldldl

dlapp gDDD

l

d

l

app

D

Dg

D

D1

Area fraction of

dislocation is g

EMA 5001 Physical Properties of Materials Zhe Cheng (2016) 7 Short Circuit Diff & Reaction Diff

Diffusion Involving Two Phases -

with Interfacial Reactions

In binary system involving 2 phases, if

Slow diffusion process

Fast interfacial process local equilibrium

Fast phase transformation at the phase

boundary

Phase analysis

Binary alloy, fixed surface concentration

of CS The number of phases can

be present:

At constant temperature of T1,

Single phase region: composition can

change

At two phase boundary, composition

will be fixed

6

B

T

A

α β

T1

x

CB

0 CS

BC

BC

2 pcf

α

t+dt

t

x

aB

0 aS

t+dt

t

/

Ba

dx

t

CS

BC

BC

α

CS

BC

BC

EMA 5001 Physical Properties of Materials Zhe Cheng (2016) 7 Short Circuit Diff & Reaction Diff

Rate of Interface Movement if

Controlled by Diffusion

For reaction diffusion

In a very short time dt, interface between

α and moves from x to x + dx.

The change in B within dx is

From flux point of view, the change in B is

Therefore,

Introduce

7

x

CB 0

CS

BC

BC

α

t+dt

t dx

CS

BC

BC

dxCC BBB

dtx

CD

x

CDdt

x

CDdt

x

CD

BBBB CCCC

B

~~~~

BB CCBB x

CD

x

CD

CCdt

dx ~~1

t

x

d

dC

tx

t

x

d

dC

xd

dC

x

C

1

EMA 5001 Physical Properties of Materials Zhe Cheng (2016) 7 Short Circuit Diff & Reaction Diff

Rate of Interface Movement if

Controlled by Diffusion

Continue from p. 15

Because the concentration in each phase at the interface are constant, is a

constant depending on concentration

Therefore,

Integrate on both sides, we have

8

d

dC

tx

C

1

d

dC

BB CCBB x

CD

x

CD

CCdt

dx ~~1

tCA

dt

dx 1)('

tCBx )('

tCBx )(2

t

x

EMA 5001 Physical Properties of Materials Zhe Cheng (2016) 7 Short Circuit Diff & Reaction Diff

Concentration Profile Change for

Diffusion involving Single Phase

Two piece of Ni-Cu alloys with

different initial compositions put

in close contact at elevated

temperature, plot the

change of concentration profile

for i) t1 = 0, ii) t2 finite, iii) t3 ∞

9

C

x

C

x C

x

t1=0

t2 finite

t3 →∞

EMA 5001 Physical Properties of Materials Zhe Cheng (2016) 7 Short Circuit Diff & Reaction Diff

Concentration Profile Change for

Diffusion involving Two-Phases (1)

Two pieces of similar sized alloys

with different composition and

phases (one is α with pure A and

the other is γ) are put in close

contact at elevated temperature,

as below.

Assuming slow diffusion and

local equilibrium at interface

Please plot concentration profile

for i) t1 = 0, ii) t2 finite, iii) t3 ∞

10

B

T

A

α β

T1

C

x C

x C

x

BC

BC

t1=0

t2 finite

t3 →∞ BC

BC 0

BC

0BC

0BC

0BC

BC

BC

EMA 5001 Physical Properties of Materials Zhe Cheng (2016) 7 Short Circuit Diff & Reaction Diff

Concentration Profile Change for

Diffusion involving Two-Phases (2)

A very large pieces of one is α

with pure A and a very small

piece of γ are put in close

contact at elevated temperature,

as below.

Assuming slow diffusion and

local equilibrium at interface

Please plot concentration profile

for i) t1 = 0, ii) t2 finite, iii) t3 ∞

11

B

T

A

α β

T1

C

x C

x C

x

t1=0

t2 finite

t3 →∞

BC

BC 0

BC

0BC

0BC

0BC

BC

BC

EMA 5001 Physical Properties of Materials Zhe Cheng (2016) 7 Short Circuit Diff & Reaction Diff

Homework

Porter 3rd Ed, Exercise 2.7

Due Feb 29 class

12