Diffusion in Solids
DIFFUSION
Atom movement from a region of high
concentration to a region of low concentration
Driving Force = Concentration Gradient
Importance of Diffusion
Many reactions and processes rely on
Diffusion:
• nucleation and growth
• recrystallization
• phase transformations
• creep
• carburization and nitridation
Case Study 1: Carburizing
• Case Hardening:
--Diffuse carbon atoms
into the host iron atoms
at the surface.
--Example of interstitial
diffusion is a case
hardened gear.
Fig. 5.0,
Callister 6e.
(Fig. 5.0 is
courtesy of
Surface
Division,
MidlandRoss.)
8
Case Study 2: Doping
• Doping Silicon with P for n-type semiconductors:
• Process:
1. Deposit P rich
layers on surface.
silicon
2. Heat it.
3. Result: Doped
semiconductor
regions.
Fig. 18.0,
Callister 6e.
silicon
9
Types of Diffusion
Self-diffusion
- atoms of same
material moving
within lattice
Interdiffusion or
Impurity Diffusion
- one type of material
diffuses into another
type of material
Prerequisites of Diffusion
For atoms to diffuse:
a) There must be an
empty adjacent site
b) The atom must
have sufficient energy
to break bonds with its
neighbor and then
cause lattice distortion
during displacement
Diffusion Mechanisms
Vacancy Diffusion
occurs at high temperature
since more vacancies are
formed
  Qv 
N v  N exp

 kT 
Diffusion Mechanisms
Interstitial Diffusion
• involves interdiffusion of impurities such
as H, C, N or O
Mathematical Models of Diffusion


Two Models
Steady-State Diffusion
Non-Steady State Diffusion
Diffusion Flux, J = quantitative measure
of diffusion
Units = kg/m2-s
Diffusion Case I
Steel
0.2% carbon
Con’c changes
linearly over
distance
Con’c
Profile doesn’t
change over
time
Distance, x
Steady-State Diffusion
• state of diffusion where diffusion flux
(mass/area-time) does not change with time
and position
dC
J  D
dx
J = diffusion flux (kg/m2-s)
D = diffusion coefficient (m2/s)
C = concentration (kg/m3)
x = distance (m)
Fick’s First Law
• not observed in practical applications
Diffusion Case II
Steel
0.2% carbon
Con’c gradient
changes over
time and
position
Distance, x
Nonsteady-State Diffusion
• occurs when diffusion flux and concentration gradient
vary with time
• most practical diffusion situations are nonsteady
ones
• governed by Fick’s Second Law:
C
C
D 2
t
x
2
Nonsteady-State Diffusion
Solution of this differential equation depends on the initial
boundary conditions.
One solution maybe:
C x  Co
 x 
 1  erf 

Cs  Co
 2 Dt 
Cx = concentration at depth x
Cs = concentration at the surface
Co = initial concentration
X`
erf(x)
erfc(x)
x
erf(x)
erfc(x)
0.00
0.0000000
1.0000000
1.30
0.9340079
0.0659921
0.05
0.0563720
0.9436280
1.40
0.9522851
0.0477149
0.10
0.1124629
0.8875371
1.50
0.9661051
0.0338949
0.15
0.1679960
0.8320040
1.60
0.9763484
0.0236516
0.20
0.2227026
0.7772974
1.70
0.9837905
0.0162095
0.25
0.2763264
0.7236736
1.80
0.9890905
0.0109095
0.30
0.3286268
0.6713732
1.90
0.9927904
0.0072096
0.35
0.3793821
0.6206179
2.00
0.9953223
0.0046777
0.40
0.4283924
0.5716076
2.10
0.9970205
0.0029795
0.45
0.4754817
0.5245183
2.20
0.9981372
0.0018628
0.50
0.5204999
0.4795001
2.30
0.9988568
0.0011432
0.55
0.5633234
0.4366766
2.40
0.9993115
0.0006885
0.60
0.6038561
0.3961439
2.50
0.9995930
0.0004070
0.65
0.6420293
0.3579707
2.60
0.9997640
0.0002360
0.70
0.6778012
0.3221988
2.70
0.9998657
0.0001343
0.75
0.7111556
0.2888444
2.80
0.9999250
0.0000750
0.80
0.7421010
0.2578990
2.90
0.9999589
0.0000411
0.85
0.7706681
0.2293319
3.00
0.9999779
0.0000221
0.90
0.7969082
0.2030918
3.10
0.9999884
0.0000116
0.95
0.8208908
0.1791092
3.20
0.9999940
0.0000060
1.00
0.8427008
0.1572992
3.30
0.9999969
0.0000031
1.10
0.8802051
0.1197949
3.40
0.9999985
0.0000015
1.20
0.9103140
0.0896860
3.50
0.9999993
0.0000007
Factors Affecting Diffusion
Solute
Solvent
D(500 C)
D(1000 C)
Carbon
FCC Iron
5 x 10-15
3 x 10-11
Carbon
BCC Iron
10-12
2 x 10-9
Iron
FCCAPF
Iron
Iron
BCC Iron
10-20
3 x 10-14
Carbon
HCP Ti
3 x 10-16
2 x 10-11
Silver
Silver(crystal)
10-17
10-12
Silver
Silver(grain
boundary)
10-11
-
TEMPERATURE
2 x 10-23
2 x 10-16
Factors Affecting Diffusion
Solute
Solvent
D(500 C)
D(1000 C)
Carbon
FCC Iron
5 x 10-15
3 x 10-11
Carbon
BCC Iron
10-12
2 x 10-9
Iron
FCC Iron
2 x 10-23
2 x 10-16
Iron
BCC Iron
10-20
3 x 10-14
Carbon
Size Ti
of
HCP
3 x 10-16
2 x 10-11
10-17
10-12
10-11
-
Silver
Silver
Diffusing
Silver(
crystal)
Specie
Silver(grain
boundary)
Factors Affecting Diffusion
Solute
Solvent
D(500 C)
D(1000 C)
Carbon
FCC Iron
5 x 10-15
3 x 10-11
Carbon
BCC Iron
10-12
Iron
FCC Iron
-9
2
x
10
Bond
2 x 10-23 Strength
2 x 10-16
of Host
Iron
BCC Iron
10-20
3 x 10-14
Carbon
HCP Ti
3 x 10-16
2 x 10-11
Silver
Silver(crystal)
10-17
10-12
Silver
Silver(grain
boundary)
10-11
-
Factors Affecting Diffusion
Solute
Solvent
D(500 C)
D(1000 C)
Carbon
FCC Iron
5 x 10-15
3 x 10-11
Carbon
BCC Iron
10-12
2 x 10-9
Iron
FCC Iron
2 x 10-23
2 x 10-16
Iron
BCC Iron
10-20
3 x 10-14
Carbon
HCP Ti
3 x 10-16
2 x 10-11
Silver
Silver(crystal)
10-17
10-12
Silver(grain
boundary)
10-11
-
Silver
Defects
Factors Affecting Diffusion
1. Temperature
- increasing temperature increases diffusion rates
  Qd 
D  Do exp

 RT 
Do = a temp-independent pre-exponential
Qd = activation energy
R = gas constant
Factors Affecting Diffusion
2. Diffusing Species
- the smaller the diffusing atom, the faster diffusion is
3. Atomic Packing Factor
- the lower the APF, the faster diffusion is
4. Bonds of Structure
- the weaker the bond, the faster diffusion is
5. Presence of other diffusion paths
- dislocations and grain boundaries hastens diffusion
Applications of Diffusion
Carburizing
• technique of case hardening of steel by increasing
carbon content of the surface
• involves diffusing carbon into the steel from a gas, liquid
or solid source
Nitriding
• involves diffusing nitrogen into steel for added strength
CARBURIZING
KINKERDALL VOIDS
QUIZ

In which case will diffusion occur faster?
Con’c
B
A
Distance, x
QUIZ

a.
b.
c.
d.
Diffusion is inversely proportional to the
following except:
Temperature
Size of diffusing specie
Bond strength
APF
QUIZ
In which metal, fine-grain or coarsegrain metal will diffusion occur faster?
Why?
QUIZ
1. The diffusion coefficients for nickel in iron are given at two
different temperatures:
T(K)
D(m2/s)
1473
2.2 x 10-15
1673
4.8 x 10-14
Determine the values of Do and the activation energy Qd.
(R = 8.31 J/mol-K)
2. If a condition of steady state is achieved during the carburization
of an iron plate done at 700C, calculate the diffusion flux of
carbon through the plate if the concentrations of carbon at
positions of 5 and 10 mm beneath the carburizing surface are 1.2
and 0.8 kg/m3, respectively. Assume a diffusion coefficient of 3 x
10-11 m2/s at this temperature.
QUIZ

In which alloy,
Steel A = 0.8% Carbon or
Steel B = 0.2% Carbon will diffusion of
carbon occur faster? Why?
In which metal, fine-grain or coarsegrain metal will diffusion occur faster?
Why?
Thanks for Listening!
1. The diffusion coefficients for nickel in iron are given
at two different temperatures:
T(K)
D(m2/s)
1473
2.2 x 10-15
1673
4.8 x 10-14
Determine the values of Do and the activation energy
Qd.
(R = 8.31 J/mol-K)
2. If a condition of steady state is achieved during the
carburization of an iron plate done at 700C, calculate
the diffusion flux of carbon through the plate if the
concentrations of carbon at positions of 5 and 10 mm
beneath the carburizing surface are 1.2 and 0.8
kg/m3, respectively. Assume a diffusion coefficient of 3
x 10-11 m2/s at this temperature.