V. Diffusion in Solids
MECE 3345 Materials Science
VI . Diffusion in Solids
copyright © 2008 by Li Sun
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V. Diffusion in Solids
MECE 3345 Materials Science
Examples
• Biology
• Ink in water
blood
air
Transpiration
• Smell
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V. Diffusion in Solids
MECE 3345 Materials Science
Definition
Diffusion: The dispersal of material by random atom or molecular
movement from regions of high concentration to those of lower
concentration.
The result of diffusion is to decrease the concentration non-uniformity.
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V. Diffusion in Solids
MECE 3345 Materials Science
Ni-Cu
Cu
Ni
Ni
Cu
Cu in Ni Ni in Cu
NiCu alloy
x=0
concentration
100
0
t=0
t = t1
t=∞
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V. Diffusion in Solids
MECE 3345 Materials Science
Diffusion Mechanism
Vacancy diffusion
Interstitial diffusion
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V. Diffusion in Solids
MECE 3345 Materials Science
Activation Energy
Energy
G  H  TS
G
Vibration frequency: 
Nearest neighbor sites: z
Atom position
Interstitial diffusion
G m  H m  TS m
Jump Frequency
:
Vacancy diffusion
 Gm 
  zX V v exp  

RT 

 H m  T  S m

 H m 
D  D 0 exp  

RT 

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V. Diffusion in Solids
MECE 3345 Materials Science
Vacancy Diffusion
• Applies to substitutional impurities
• Atoms must hop to open vacancy
• Rate depends on probability of overcoming migration activation
energy, Qm
pQ = Bexp(-Hm/kT)
And
• Probability of a vacancy
pv = Nv/N = exp(-Hv/kT)
• Here is Diffusion Coefficient is:
D = D0 exp(-Hd/kT)
where Hd = Hm + Hv
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V. Diffusion in Solids
MECE 3345 Materials Science
Interstitial Diffusion
•
•
•
•
Applies to interstitial impurities (tend to be low concentration)
Most interstitial sites are not occupied
More rapid than vacancy diffusion
Activation energy involves overcoming the migration activation
energy, Hm.
• Rate is defined by the Diffusion Coefficient
D = D0exp(-Hd/kT)
where D0 is a pre-exponential
factor (m2/s) and Hd = Hm
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V. Diffusion in Solids
MECE 3345 Materials Science
Data
Interstitial diffusion in Fe
Self-diffusion data for metals
solute
Hm(kJmol-1)
D0(10-6m2s-1)
C
84
2.0
N
76
0.3
H
13
0.1
metal
Tm(K
)
Q
(kJmol-1)
D0
(10-6m2s-1)
D(Tm)
(10-12m2s-1)
Al
933
142
170
1.9
Cu
1356
200
31
0.59
Ni
1726
279
190
0.65
-Fe
1805
284
49
0.29
isotope
x=0
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V. Diffusion in Solids
MECE 3345 Materials Science
Other Diffusion Mechanisms
Direct exchange
Zener ring
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V. Diffusion in Solids
MECE 3345 Materials Science
Fick’s First Law
n 2  n1  a
n2
n1
 
1
dn
v : Jump Frequency
dx
: average tune if atome stay at the lattice site
v
Jump Frequency
x
a
along x direction :
J  J 1 2  J 2 1
1
6
1
 1

J  a
n1 
n2 
6
 6

 a
2
1 dn
6 dx
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V. Diffusion in Solids
MECE 3345 Materials Science
Geometric Factor and Diffusion Coefficient
J  D
dn
J  J 1 2  J 2 1
dx
D 
Diffusion constant
Einstein formula
Simple cubic
 a
 
a

Average stay time at interstitial site
Geometric factor
D 
1 1
6
FCC
 
2
2
1
6
Lattice parameter
2
 
1
12

2
 : jump distance
BCC
a
 
3
a
2
 
1
8
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V. Diffusion in Solids
MECE 3345 Materials Science
Steady State Diffusion
concentration
c
Steady state: the state where the concentration profile does not change with time,
which means at any time the concentration at certain spot in space is a constant.
Fick’s first law:
A
B
J  D
dn
J
dx
Position
Position
n: number density
x
x
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V. Diffusion in Solids
MECE 3345 Materials Science
Fick’s Second Law
c
concentration
c
concentration
t1
A
B
x
A
t2
A
B
x
x
x
J
Position
n2
n1
Position
J 1 2
x
J 2 1
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V. Diffusion in Solids
MECE 3345 Materials Science
Diffusion into a Semi-infinite Solid
t > 0, diffusion start
n (x=0) = ns , n (x = ∞) = n0
A
B
n
t ≤ 0, before diffusion start
n(x) = n0, 0≤ x≤ ∞
A
B
ns
n0
0
concentration
ns
ns
x=0
n0
0
x=∞
x=0
Solution to the Fick’s second law
n x (t )  n 0
n s  n0
x=∞
t increases
n0
t=0
0
Position
x
Fixed surface concentration.

x
 1  erf 
 2 Dt




,
erf: Gaussian error function
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V. Diffusion in Solids
MECE 3345 Materials Science
Error function
erf  x  
2

x  2
0 e
d
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V. Diffusion in Solids
MECE 3345 Materials Science
Examples
Steady state diffusion : Gas purification
Hydrogen gas purification can be achieved
by pass mixed gas through palladium sheet.
Pump
Pump
Assume the there is a hydrogen purification
system with cross-sectional area of 0.2m2, the
palladium film thickness is 10mm hydrogen
concentration on the two sides are 3kg/m3 and
0.5kg/m3. If the room temperature hydrogen
diffusion coefficient in Pd is 1.0×10-15m2/s and
the activation energy is 27.8kJ/mol. Calculate
the pure hydrogen generation speed (kg/hour)
when the system works at 500oC.
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MECE 3345 Materials Science
I.
What we know:
C1= 3kg/m3, C2 = 0.5kg/m3
V. Diffusion in Solids
Calculate how much hydrogen can diffuse
through in 1hour at 500oC.
Palladium film thickness
d=10mm = 0.01m
Diffusion area A=0.2m2
D @ 300K = 1.0×10-11m2/s
Qd = 27.8kJ/mol
diffusion direction
C2
C1
C2
C1
0
xA xB
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V. Diffusion in Solids
MECE 3345 Materials Science
Diffusion into a Semi-infinite Solid
II. Calculate the carburizing time to achieve 0.45 wt% C at a position of 2mm into a 0.2
weight percent steel. Sample was maintained at 1000oC with 1.3 wt% carbon atmosphere.
C0
0.2 wt %
c
concentration
Cs
1.3 wt %
Cs
t = t1 t2 > t1
C0
0
t=0
Position
x
1 .0
0 .8
e rf(z)
0 .6
0 .4
0 .2
0 .0
0 .0
0 .5
1 .0
1 .5
2 .0
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V. Diffusion in Solids
MECE 3345 Materials Science
Diffusion Coefficient and Thermally Activated Process
Thermally activated processes: The processes have exponentially
increasing rates versus temperature.
Vacancy formation, electrical conductivity, creep and
diffusion are thermally activated processes.
 Q 
D  D 0 exp   d 
 RT 
D0: temperature independent parameter (m2/s).
Qd: activation energy for diffusion (J/mol).
R: gas constant 8.31J/molK
T: absolute temperature
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V. Diffusion in Solids
MECE 3345 Materials Science
Arrhenius Plot
 Q 
rate  C exp  

RT


log( rate )  log( C ) 
Q 1
RT
Arrhenius Plot: log (rate) vs.
1
T
From the slope the thermal activation energy can be calculated
10-8
D (m2/s)
10-12
10-15
10-18
10-21
10-24
0
0.4
0.8
1.2
1000
T
1.6
2.0
2.4
(1/K)
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V. Diffusion in Solids
MECE 3345 Materials Science
Diffusion and Temperature
10-8
300
600
1000
1500
D has exponential dependence on T
D (m2/s)
T(C)
Dinterstitial
C in -Fe
C in -Fe
10-14
10-20
0.5
1.0
1.5
>> D substitutional
Al in Al
Fe in a-Fe
Fe in g-Fe
1000K/T
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