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© 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