1
Materials Science
CL 409
Diffusion
Instructor: Jhumpa Adhikari
Department of Chemical Engineering
Indian Institute Of Technology, Bombay
2
Diffusion
 Phenomenon of material transport by atomic motion
• Mass transport by atomic motion
 Diffusion is the net flux of any species (ions/atoms/electrons/holes,
etc.), the magnitude of which depends on the concentration gradient
and temperature
 Atomic Mechanisms by which Diffusion occurs
• Gases & Liquids – random (Brownian) motion
Atomic scale diffusion is relatively rapid in liquids
Relatively easy to visualize
• Solids – vacancy diffusion or interstitial diffusion
Chemical composition of engineering materials is changed as a result of movement of
atoms during production and application
Possible Cases:Case 1: Atoms are redistributed within the microstructure of the material
Case 2: Atoms are added from the materials’ environment
Case 3: Atoms from the material may be discharged into the environment
Diffusion
• Interdiffusion or Impurity diffusion: In an alloy, atoms tend to
migrate from regions of high conc. to regions of low conc.
• Example: Cu - Ni Diffusion Couple
Initially
After heating & then cooling to room temp
From Figs. 6.1 and 6.2
Callister’s Materials Science and Engineering,
Adapted Version.
Chapter 6 -
Diffusion
• Self-diffusion: In an elemental solid, atoms also migrate.
Label some atoms
C
A
D
B
After some time
C
D
A
B
• Elemental solids like metals
• All atoms exchanging positions are of the same type
• Not subject to observation based on compositional changes
Chapter 6 -
Diffusion
 Atomic Perspective
• Stepwise migration of atoms from lattice site to lattice site
• Atoms in solids are in constant motion
Rapidly changing positions
 For an atom to make such a move, conditions are
• There must be an empty adjacent site
• Atom must have sufficient energy to break bonds with its neigbhouring
atoms
Some lattice distortion during displacement
Energy is vibrational in nature
 Fraction of total # of atoms capable of diffusive motion
• Small at some specific temperature
By virtue of the magnitudes of their vibrational energies
• Increases with increasing temperature
• Dominating mechanisms for metallic solids – vacancy diffusion or
interstitial diffusion
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Diffusion Mechanisms
Vacancy Diffusion:
• diffusing atoms exchange positions with vacancies
• self-diffusion & interdiffusion occur by this mechanism
• interdiffusion: applies to substitutional impurities atoms
--- impurity atoms substitute for host atoms
• rate depends on:
--number of vacancies
--activation energy to exchange.
increasing elapsed time
Chapter 6 -
Diffusion Mechanisms
• Interstitial diffusion – smaller atoms can diffuse between
atoms.
• Atoms migrate from an interstitial position to a neighbouring one that is
empty : More rapid than vacancy diffusion
– Interstitial atoms smaller and more mobile
– More empty interstitial positions than vacancies: Probability of
interstitial atom movement > vacancy diffusion
• Interdiffusion of impurities e.g. hydrogen, carbon, nitrogen, oxygen:
atoms small enough to fit into interstitial positions
• Host or substitutional impurity atoms rarely form interstitials
– Do not diffuse via this mechanism, generally
More rapid than vacancy diffusion
From Fig. 6.3 (b)
Callister’s Materials Science and Engineering, Adapted
Version.
Chapter
6-
8
 Temperature increases the ability of atoms/ions to diffuse.
 The relationship of the movement rate to the temperature is
given by the Arrhenius equation:
Q
 −Q 
Rate = c0 exp 
or
ln
Rate
=
ln
c
−
( ) ( 0)

RT
 RT 
9
 Example: Interstitial atoms move from one site to another at
the rates 5 x 108 jumps/s at 500oC and 8 x 1010 jumps/s at
800oC. Find the activation energy Q for the process.
 The problem can be solved graphically or by writing 2
simultaneous equations.
Rate = co exp ( −Q / RT )


jumps
−Q
5 10
= c0 exp 
 = c0 exp ( −0.0001556Q )
s
 (8.314 )( 500 + 273) 
8


jumps
−Q
8 10
= c0 exp 
 = c0 exp ( −0.0001121Q )
s
 (8.314 )(800 + 273) 
10
10
 Since
5 10 jump/s
= c0  8 1010
exp ( −0.0001556Q )
8
 We can solve for Q,
510 ) exp ( −0.0001121Q )
(
=
8
exp ( −0.0001556Q )
Q = 1.167 105 J/mol
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Mechanisms for Diffusion
 Interdiffusion can occur in 2 ways:
• Vacancy diffusion: When an atom leaves a lattice site to fill
another one, it creates a vacancy. Thus diffusion involves
counterflows of atoms and vacancies. High temperature
increases vacancies.
• Interstitial diffusion: This occurs when a small atom/ion moves
from one interstitial site to another. No vacancies are required
for this kind of diffusion, so it occurs more easily. Generally,
smaller interstitial species diffuse faster.
Activation Energy for Diffusion
 A diffusing atom must squeeze past other atoms to reach a
new position, which requires activation energy.
 A low activation energy indicates easy diffusion.
 Generally, interstitial atoms require a lower activation energy
than substitutional atoms.
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Processing Using Diffusion
• Case Hardening:
--Diffuse carbon atoms
into the host iron atoms
at the surface.
--Example of interstitial
diffusion is a case
hardened gear.
From chapteropening
photograph,
Chapter 6,
Callister’s
Materials Science
and Engineering,
Adapted Version.
(Courtesy of
Surface Division,
Midland-Ross.)
• Result: The presence of C
atoms makes iron (steel) harder.
Chapter 6 -
Processing Using Diffusion
• Doping silicon with phosphorus for n-type semiconductors:
0.5 mm
• Process:
1. Deposit P rich
layers on surface.
magnified image of a computer chip
silicon
2. Heat it.
3. Result: Doped
semiconductor
regions.
silicon
light regions: Si atoms
light regions: Al atoms
From chapter-opening photograph, Chapter 17
Callister’s Materials Science and Engineering,
Adapted Version.
Chapter 6 -
15
Rate of Diffusion (Fick’s 1st Law)
 Fick’s 1st Law relates the rate of diffusion (written in terms of flux
J) to the diffusivity D of the diffusion couple, and the
concentration gradient.
dc
J = −D
dx
 The concentration gradient acts as the thermodynamic driving
force for diffusion
Diffusion
• How do we quantify the amount or rate of diffusion?
moles (or mass) diffusing mol
kg
J  Flux 
=
or 2
2
(surface area)(time)
cm s m s
• Measured empirically
– Make thin film (membrane) of known surface area
– Impose concentration gradient
– Measure how fast atoms or molecules diffuse through the
membrane
M l dM
J=
=
At A dt
M=
mass
diffused
J  slope
time
Chapter 6 -
Steady-State Diffusion
Rate of diffusion independent of time
dC
Flux proportional to concentration gradient =
dx
Fick’s first law of diffusion
C1 C1
dC
J = −D
dx
C2
x1
if linear
x
C2
D  diffusion coefficient
x2
dC C C2 − C1

=
dx x x2 − x1
Chapter 6 -
Diffusion and Temperature
• Diffusion coefficient increases with increasing T.
 Qd 
D = Do exp−
 RT 
D = diffusion coefficient [m2/s]
Do = pre-exponential [m2/s]
Qd = activation energy [J/mol or eV/atom]
R = gas constant [8.314 J/mol-K]
T = absolute temperature [K]
Chapter 6 -
Diffusion and Temperature
300
600
1000
10-8
1500
D has exponential dependence on T
T(C)
D (m2/s)
Dinterstitial >> Dsubstitutional
C in a-Fe
C in g-Fe
10-14
10-20
0.5
1.0
1.5
Al in Al
Fe in a-Fe
Fe in g-Fe
1000 K/T
From Fig. 6.7
Callister’s Materials Science and Engineering, Adapted Version.
(Date for Fig. 6.7 taken from E.A. Brandes and G.B. Brook (Ed.)
Smithells Metals Reference Book, 7th ed., Butterworth-Heinemann,
Oxford, 1992.)
Chapter 6 -
Example: At 300ºC the diffusion coefficient and
activation energy for Cu in Si are
D(300ºC) = 7.8 x 10-11 m2/s
Qd = 41.5 kJ/mol
What is the diffusion coefficient at 350ºC?
transform
data
D
Temp = T
1
  and
 T2 
Q
D
 lnD2 − lnD1 = ln 2 = − d
D1
R
Qd
lnD2 = lnD0 −
R
ln D
1/T
Qd
lnD1 = lnD0 −
R
 1 1
 − 
 T2 T1 
 1
 
 T1 
Chapter 6 -
 Qd
D2 = D1 exp−
 R
 1 1 
 − 
 T2 T1 
T1 = 273 + 300 = 573 K
T2 = 273 + 350 = 623 K
 − 41,500 J/mol  1
1 
D2 = (7.8 x 10−11 m2/s) exp 
−


 8.314 J/mol - K  623 K 573 K 
D2 = 15.7 x 10-11 m2/s
Chapter 6 -
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Rate of Diffusion
 Example. A 0.05 cm layer of MgO is deposited between
layers of nickel and tantalum. At 1400oC, Ni ions diffuse
through the MgO to the tantalum layer. Determine the
number of Ni ions passing through the MgO per second. At
1400oC, the Ni lattice parameter is 3.6 x 10-8 cm and the
diffusion coefficient is 9 x 10-12 cm2/s.
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Rate of Diffusion
 Example. A 0.05 cm layer of MgO is deposited between
layers of nickel and tantalum. At 1400oC, Ni ions diffuse
through the MgO to the tantalum layer. Determine the
number of Ni ions passing through the MgO per second. At
1400oC, the Ni lattice parameter is 3.6 x 10-8 cm and the
diffusion coefficient is 9 x 10-12 cm2/s.
 The Ni composition is 100% Ni at Ni/MgO interface
cNi/MgO =
4 Ni atoms
(3.6 10
−8
cm )
3
Ni atoms
= 8.573 10
cm3
22
24
 The composition at the MgO/Ta interface is 0% Ni. Hence
c 0 − 8.573 1022
24 Ni atoms
=
= −1.715 10
x
0.05
cm4
 The flux of Ni atoms through the MgO layer is
c
−12
24
13 Ni atoms
J = −D
= − ( 9 10 )( −1.71510 ) = 1.543 10
x
cm2  s
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 The total number of Ni atoms crossing the 2 cm x 2 cm
interface per second is
# of Ni atoms
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13 Ni atoms
= J  Area = (1.543 10 ) ( 2 )( 2 ) = 6.17 10
s
s
 In 1 second, the volume of Ni atoms removed from the
Ni/MgO interface is
3
6.17 1013
cm
−10
=
7.2

10
8.573 1022
s
 The thickness by which the Ni layer is reduced each second
is
7.2 10−10 cm3 /s
−10 cm
=
1.8

10
4 cm2
s
Non-steady State Diffusion
• The concentration of diffucing species is a function of
both time and position C = C(x,t)
• In this case Fick’s Second Law is used
Fick’s Second Law
C
 2C
=D 2
t
x
Chapter 6 -
Non-steady State Diffusion
• Copper diffuses into a bar of aluminum.
Surface conc.,
Cs of Cu atoms
bar
pre-existing conc., Co of copper atoms
Cs
Adapted from
Fig. 5.5,
Callister 7e.
B.C.
at t = 0, C = Co for 0  x  
at t > 0, C = CS for x = 0 (const. surf. conc.)
C = Co for x = 
Chapter 6 -
Solution:
C (x ,t ) − Co
 x 
= 1 − erf 

Cs − Co
 2 Dt 
C(x,t) = Conc. at point x at
time t
erf (z) = error function
2 z −y 2
=
e dy

0

CS
C(x,t)
erf(z) values are given in
Table 5.4 in the next slide
Co
Chapter 6 -
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Error Function Values
Non-steady State Diffusion
• Example: An FCC iron-carbon alloy initially
containing 0.20 wt% C is carburized at an elevated
temperature and in an atmosphere that gives a
surface carbon concentration constant at 1.0 wt%. If
after 49.5 h the concentration of carbon is 0.35 wt%
at a position 4.0 mm below the surface, determine
the temperature at which the treatment was carried
out.
• Solution:
C( x, t ) − Co
 x 
= 1 − erf 

Cs − Co
 2 Dt 
Chapter 6 -
C( x ,t ) − Co
 x 
= 1 − erf 

Cs − Co
 2 Dt 
– t = 49.5 h
– Cx = 0.35 wt%
– Co = 0.20 wt%
x = 4 x 10-3 m
Cs = 1.0 wt%
C( x, t ) − Co 0.35 − 0.20
 x 
=
= 1 − erf 
 = 1 − erf (z )
Cs − Co
1.0 − 0.20
 2 Dt 
 erf(z) = 0.8125
Chapter 6 -
We must now determine the value of z for which the error function
is 0.8125. An interpolation is necessary as follows
z
erf(z)
0.90
z
0.95
0.7970
0.8125
0.8209
Now solve for D
z − 0.90
0.8125 − 0.7970
=
0.95 − 0.90 0.8209 − 0.7970
z = 0.93
x
z=
2 Dt
D=
x2
4z2t
−3 2
 x2 
(
4
x
10
m)
1h
=
D = 
= 2.6 x 10−11 m2 /s
 4z 2t  ( 4)(0.93)2 ( 49.5 h) 3600 s


Chapter 6 -
• To solve for the temperature at
which D has above value, we
use a rearranged form of
Equation shown
Qd
T=
R(lnDo − lnD)
For diffusion of C in FCC Fe
Do = 2.3 x 10-5 m2/s Qd = 148,000 J/mol

T=
148,000 J/mol
(8.314 J/mol - K)(ln 2.3x10−5 m2/s − ln 2.6x10−11 m2/s)
T = 1300 K = 1027°C
Chapter 6 -
34
Factors Affecting Diffusion
 Temperature & Diffusion Coefficient: Diffusion kinetics are
strongly temperature dependent.
D = D0 exp ( −Q / RT )
 In covalently bonded materials, the D value is low because Q
(activation energy) is high.
 Similarly, ionic materials have high Q because ions can only
diffuse into sites having same charge.
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Factors Affecting Diffusion
 Types of Diffusion:
• Volume diffusion: Slow, high activation energy (Q)
• Grain boundary diffusion: Faster, lower Q
• Surface diffusion: Fastest, lowest Q
 Time: Diffusion requires time; controlling diffusion time
during processing can be the key to producing uniform or
unique (non-equilibrium) properties & microstructures.
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Factors Affecting Diffusion
 Dependence on Bonding & Crystal Structure:
• Interstitial diffusion occurs faster than vacancy diffusion.
• Close-packed crystal structures have higher Q than open
crystal structures.
• Cations diffuse faster than anions due to size.
• Ionic diffusion also transports charge/enables conductivity.
 The diffusion coefficient also depends on the concentration
of the diffusing species and composition of the matrix.
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Diffusion & Materials Processing
 Diffusional processes become very important when treating
processing materials at elevated temperatures.
 Sintering:
• Sintering is the high-temperature treatment that joins particles
into a solid mass by reducing pore space between them.
• Diffusion from the bulk into the neck region causes
densification.
• E.g. manufacture of ceramics & superalloys
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Diffusion & Materials Processing
 Grain growth:
• Polycrystalline materials tend to undergo grain
growth/reduction in grain boundary areas at high
temperatures due to diffusion.
• Processing at high temperatures is carefully monitored for
grain growth.
• Grain growth reduces material strength, and may affect
optical, magnetic & electric properties.
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OPTICAL
• Transmittance:
--Aluminum oxide may be transparent, translucent, or
opaque depending on the material structure.
single crystal
polycrystal:
low porosity
polycrystal:
high porosity
Adapted from Fig. 1.2,
Callister’s Materials Science
and Engineering,
Adapted Version.
(Specimen preparation,
P.A. Lessing; photo by S.
Tanner.)
40
Diffusion & Materials Processing
 Diffusion Bonding:
• It is a 3 step process used to join reactive or dissimilar metals
and to join ceramics.
Summary
Diffusion FASTER for...
Diffusion SLOWER for...
• open crystal structures
• close-packed structures
• smaller diffusing atoms
• larger diffusing atoms
• lower density materials
• higher density materials
Chapter 6 -