2. 3rd Sem-Physical Metallurgy-
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Learning Objective...
What is Diffusion?
How does diffusion occur?
Why is it an important part of processing?
How can the rate of diffusion be predicted for some simple
cases?
How does diffusion depend on structure and temperature?
3. Selection of steel for gears
Wear resistance
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5. Cementite is hard as well as brittle
Hardness or strength is desirable.
But brittleness is not.
Silica Glass is also hard and brittle
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6. Selection of steel for gears
High Carbon Steel: Good wear resistance
but
Btittle
Low Carbon steel: Good ductility
but
poor wear resistance
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7. Wear resistance is required only at the
surface
High C steel on the surface
Mild steel inside
Q: How do you achieve this?
Ans: By case carburization
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8. Case carburization
Pack a mild steel gear in carbon and heat at a
high temperature in the austenite phase field
for some time.
Carbon will enter into the mild steel to give a
high-carbon wear resistant surface layer
called case.
How do carbon enter into solid steel?
At what temperature and how long should we do the
carburization?
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9. The process why which Carbon enters into solid
steel during Case carburization is an example of
DIFFUSION
Diffusion is relative movement of atoms inside a
solid
We can find appropriate time and tempearture
for case carburization by solution of Fick’s
second law
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10. How do we create an n-p junction in silicon chip?
Ans: by DIFFUSION
Deposit n
type element
Deposit p
type element Heat
Si substrate Si substrate
Si substrate
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11. 3rd Sem-Physical Metallurgy- MME, NITRR 11
Oxidation
Roles of Diffusion
Creep
Aging
Sintering
Doping Carburizing
Metals
Precipitates
Steels
Semiconductors
Many more…
Many mechanisms
Material Joining Diffusion bonding
Diffusion is relative flow of one material into another
Mass flow process by which species change their position relative to their neighbours.
Diffusion of a species occurs from a region of high concentration to low concentration
(usually). More accurately, diffusion occurs down the chemical potential (µ) gradient.
To comprehend many materials related phenomenon (as in the figure below) one must
understand Diffusion.
The focus of the current chapter is solid state diffusion in crystalline materials.
In the current context, diffusion should be differentiated with flow (of usually fluids and
sometime solids).
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Mass flow process by which species change their position relative to their neighbours.
Diffusion is driven by thermal energy and a ‘gradient’ (usually in chemical potential).
Gradients in other physical quantities can also lead to diffusion (as in the figure below).
In this chapter we will essentially restrict ourselves to concentration gradients.
Usually, concentration gradients imply chemical potential gradients; but there are
exceptions to this rule. Hence, sometimes diffusion occurs ‘uphill’ in concentration
gradients, but downhill in chemical potential gradients.
Thermal energy leads to thermal vibrations of atoms, leading to atomic jumps.
In the absence of a gradient, atoms will still randomly ‘jump about’, without any net flow
of matter.
Diffusion
Chemical potential
Electric
Gradient
Magnetic
Stress
First we will consider a continuum picture of
diffusion and later consider the atomic basis for the
same in crystalline solids. The continuum picture is
applicable to heat transfer (i.e., is closely related to
mathematical equations of heat transfer).
13. Diffusion: flow of matter
Heat: flow of thermal energy
Electric current: flow of electric charge
Different kinds of flows in
material
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14. Heat: flow of thermal energy
Fourier’s law of heat
conduction (1811)
x
T
q
q: heat flux (J m-2 s-1)
x
T
Gradient?
Temperature gradient
Thermal conductivity
Joseph Fourier
(1768-1830)
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15. Electric current: flow of charge
Ohm’s law of electrical conduction
(1827)
x
V
E
j
j : charge flux (C m-2 s-1), current density
x
V
Gradient?
Electric potential gradient,
electric field E
electrical conductivity
Georg Simon Ohm
(1787-1854)
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16. Diffusion: flow of mass
Fick’s first law of diffusion
1855
x
c
D
j
j : mass flux (kg m-2 s-1, moles m-2 s-1)
x
c
Gradient?
concentration gradient, kg m-4
D: Diffusivity, m2 s-1
1829-1901
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17. 1 dn
J
A dt
Flow direction
Area (A)
Concentration gradient. Concentration can be designated in many ways (e.g. moles per unit
volume). Concentration gradient is the difference in concentration between two points
(usually close by).
We can use a restricted definition of flux (J) as flow per unit area per unit time:
→ mass flow / area / time [Atoms / m2 / s].
Steady state. The properties at a single point in the system does not change with time. These
properties in the case of fluid flow are pressure, temperature, velocity and mass flow rate.
In the context of diffusion, steady state usually implies that, concentration of a given
species at a given point in space, does not change with time.
Important terms
2
mass atoms
J
area time m s
In diffusion problems, we would typically like to address one of the following problems.
(i) What is the composition profile after a contain time (i.e. determine c(x,t))?
18. Fick’s* I law
Assume that only species ‘S’ is moving across an area ‘A’. Concentration gradient for
species ‘S’ exists across the plane.
The concentration gradient (dc/dx) drives the flux (J) of atoms.
Flux (J) is assumed to be proportional to concentration gradient.
The constant of proportionality is the Diffusivity or Diffusion Coefficient (D).
‘D’ is assumed to be independent of the concentration gradient.
Diffusivity is a material property. It is a function of the composition of the material
and the temperature.
In crystals with cubic symmetry the diffusivity is isotropic (i.e. does not depend on direction).
Even if steady state conditions do not exist (concentration at a point is changing with
time, there is accumulation/depletion of matter), Fick’s I-equation is still valid (but not
easy to use).
dx
dc
DA
dt
dn
dx
dc
J
dx
dc
D
J
dx
dc
D
dt
dn
A
J
1
Fick’s first law (equation)
As we shall see the ‘law’is actually an equation
Area
Flow direction
The negative sign implies that
diffusion occurs down the
concentration gradient
* Adolf Fick in 1855
A material property
19. 0
c J
t x
Fick’s law Equation of continuity
x
c
D
J
dx
2
2
c c
D
x t
J
C(x,t)
20. dx
dc
DA
dt
dn
No. of atoms
crossing area A
per unit time
Cross-sectional area
Concentration gradient
ve sign implies matter transport is down the concentration gradient
Diffusion coefficient/Diffusivity
A
Flow direction
As a first approximation assume D f(t)
Let us emphasize the terms in the equation
Note the strange unit of D: [m2/s]
2 3
1
[ ]
dc number number
J D D
dx m s m m
Let us look at the units of Diffusivity
2
[ ]
m
D
s
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Fick’s II law
The equation as below is often refered to as the Fick’s II law (though clearly
this is an equation and not a law).
This equation is derived using Fick’s I-equation and mass balance.
The equation is a second order PDE requiring one initial condition and two
boundary conditions to solve.
2
2
x
c
D
t
c
c c
D
t x x
If ‘D’ is not a function of the position, then it can be ‘pulled out’.
Derivation of this equation will taken up next.
Another equation
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Let us consider a 1D diffusion problem.
Let us consider a small element of width x in
the body.
Let the volume of the element be the control
volume (V) = 1.1. x = x. (Unit height and thickness).
Let the concentration profile of a species ‘S’ be
as in the figure.
The slope of the c-x curve is related to the flux
via the Fick’s I-equation.
In the figure the flux is decreasing linearly.
The flux entering the element is Jx and that
leaving the element is Jx+x.
Since the flux at x1 is not equal to flux leaving
that leaving at x2 and since J(x1) > J(x2), there is
an accumulation of species ‘S’ in the region x.
The increase in the matter (species ‘S’) in the
control volume (V) = (c/t).V = (c/t). x.
24. x x+x
c c+c
j j + j
Mass in at x: min = A t j
Mass out at x+ x: mout = A t (j + j)
Mass accumulation between x and x+ x
m = min-mout
= A t ( j – j - j ) = -A t j
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25. Change in concentration in a volume V = A x
and time interval t :
m = -A t j
x
A
m
c
x x+x
c c+c
j j + j
Average rate of change of concentration
between x and x + x in time interval t:
x
j
t
c
x
A
j
t
A
x
j
t
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26. x
j
t
c
Instantaneous change in concentration
at a time t, at a point x:
x
j
t
c
t
x
t
x
0
0
0
0 lim
lim
x
j
t
c
Fick’s 2nd Law
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28. 28
Jx Jx+x
x
x
x
x J
J
on
Accumulati
x
x
J
J
J
on
Accumulati x
x
x
x
J
J
J
x
t
c
x
x
J
s
m
Atoms
m
s
m
Atoms
2
3
.
1
x
x
J
x
t
c
x
c
D
x
t
c
Fick’s first law
x
c
D
x
t
c D f(x)
2
2
x
c
D
t
c
A B
Calculation of units
If Jx is the flux arriving at plane A and Jx+x is the flux leaving plane B. Then the
Accumulation of matter is given by: (Jx Jx+x).
c
J
t
In 3D
Arises from mass conservation
(hence not valid for vacancies)
2
c
D c
t
In 3D
29. Solution to Fick’s 2nd law:
2
2
x
c
D
t
c
Solution depends on the boundary condition.
t
D
x
B
A
t
x
c
2
erf
)
,
(
A and B : constants depending on the boundary
conditions
erf (z) : Gaussian error function
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30. The Gaussian Error Function
z
d
z
erf
0
2
)
(
exp
2
)
(
0.2
0.4
0.6
0.8
1
exp (-2)
-3 -2 -1 1 2 3
0 z
Hatched area (2/)
= erf (z)
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32. Carburisation of steel
cs
Distance in steel from
surface
c (wt% C)
x
c0
Surface
concentration
initial
concentration
Concentration profile after
carburization for time t at a
temperature T
Boundary conditions:
1. c=c0 at x>0 , t=0
2. c=cs at x=0 , t>0
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33. Carburisation of steel
t
D
x
B
A
t
x
c
2
erf
)
,
(
Boundary conditions:
2. c(x,t) = c0 at x > 0, t = 0
1. c(x,t) = cs at x = 0, t > 0
B.C. 1 cs = A – B erf(0) = A
B.C. 2 c0 = A – B erf(+) = A-B
A = cs
B = cs – c0
t
D
x
c
c
c
t
x
c s
s
2
erf
)
(
)
,
( 0
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34. 3rd Sem-Physical Metallurgy-
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kT
Q
e
D
D 0
Temperature dependence of diffusivity
Arrhenius type
Diffusion is an activated process and hence the Diffusivity depends exponentially on
temperature (as in the Arrhenius type equation below).
‘Q’ is the activation energy for diffusion. ‘Q’ depends on the kind of atomic processes (i.e.
mechanism) involved in diffusion (e.g. substitutional diffusion, interstitial diffusion, grain
boundary diffusion, etc.).
This dependence has important consequences with regard to material behaviour at elevated
temperatures. Processes like precipitate coarsening, oxidation, creep etc. occur at very high
rates at elevated temperatures.
35. Case carburization of steel Problem 8.4
Initial concentration c0 = 0.2 wt% C
Surface concentration cs = 1.4 wt % C
Temperature = 900 ºC = 1173 K
Desired concentration c = 1.0 wt% C at x = 0.2 mm
At 900 ºC the equilibrium phase of steel is austenite ()
Diffusivity data for C in austenite:
D0 = 0.7 x 10-4 m2s-1
Q = 157 kJ mol-1
RT
Q
D
D exp
0 = 7.13688 x 10-12 m2s-1
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36. Carburization of steels
0.1 0.2 0.3 0.4 0.5
0.2
0.4
0.6
0.8
1.0
1.2
1.4
100 s
1000 s
10000 s
Distance in steel from surface, mm
wt% C
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39. Solved
Example
A 0.2% carbon steel needs to be surface carburized such that the concentration of
carbon at 0.2mm depth is 1%. The carburizing medium imposes a surface
concentration of carbon of 1.4% and the process is carried out at 900C (where, Fe
is in FCC form).
Data: 4 2
0
D (C in -Fe) 0.7 10 m / s
157 /
Q kJ mole
Given: T = 900° C, C0 = C(x, 0) = C(, t) = 0.2 % C,
Cf = C(0.2 mm, t1) = 1% C (at x = 0.2 mm), Cs = C(0, t) = 1.4% C
The solution to the Fick’ second law:
( , )
2
x
C x t A B erf
Dt
The constants A & B are determined from boundary and initial conditions:
(0, ) 0.014
S
C t A C
, 0
( , ) 0.002
C t A B C
or 0
( ,0) 0.002
C x A B C
S 0
B C C 0.012
, ( , ) 0.014 0.012
2
x
C x t erf
Dt
-4
4
1
1
2 10
(2 10 , ) 0.01 0.014 0.012
2
C m t erf
Dt
S S 0
( , ) C (C C )
2
x
C x t erf
Dt
0
( , )
=
2
S
S
C x t C x
erf
C C Dt
(2)
(1)
4
1
1 2 10
3 2
erf
Dt
40. x (in mm from surface)
%
C
0.2
0.4
0.6
0.8
1.0
1.2
1.4
0.2
t =0
t = t1 = 14580s
t = 1000s
t = 7000s
t
0.4 0.6 0.8 1.0 1.2 1.4
x (in mm from surface)
%
C
0.2
0.4
0.6
0.8
1.0
1.2
1.4
0.2
t =0
t = t1 = 14580s
t = 1000s
t = 7000s
t
0.4 0.6 0.8 1.0 1.2 1.4
The following points are to be noted:
The mechanism of C diffusion is interstitial diffusion
The diffusivity ‘D’ has to be evaluated at 900C using: 0 exp
Q
D D
RT
0 exp
Q
D D
RT
3
4 157 10
(0.7 10 )exp
8.314 1173
2
12
7.14 10
m
s
-4
1
12
1
2 10
(0.3333) 0.309
2 7.14 10
erf
t
2
4
1 12
1 10
14580
0.33
7.14 10
t s
From equation (2)
-4
1
1 2 10
3 2
erf
Dt
41. 1 2
Vacant site
c = atoms / volume
c = 1 / 3
concentration gradient dc/dx = (1 / 3)/ = 1 / 4
Flux = No of atoms / area / time = ’/ area = ’/ 2
2
4
2
'
'
)
/
(
dx
dc
J
D
kT
Hm
e
D 2
2
0
D
kT
Q
e
D
D 0
On comparison
with
42. Temperature dependence of Diffusivity
RT
Q
D
D exp
0
D0 = preexponential factor
Q = activation energy
Empirical
constants
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43. Self-Diffusion in Amorphous Se (Problem 8.3)
0.00305 0.00315 0.00325
-34
-32
-30
-28
ln D
1/T
T (ºC) D (m2s-1)
35 7.7 x 10-16
40 2.4 x 10-15
46 3.2 x 10-14
56 3.2 x 10-13
D0 = 2 x 1027 m2 s-1
Q = 250 kJ mol-1
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44. Atomic Mechanism of Diffusion
How does C enter into solid steel?
is INTERSTITIAL solid solution of C in FCC Fe
C occupies octahedral voids in FCC Fe
Maximum solubility of C in austenite () is
2.14 wt% at 1150 ºC.
C
at%
9
85
.
55
98
12
2
12
2
C
wt%
2
Thus 9 out of 91
OH voids are
occupied.
90 % of OH
voids are empty
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45. C atoms can jump from one interstitial site to
another vacant interstitial site.
This is interstitial diffusion.
For OH voids the void size is 0.414 R but the
window through which C atoms can jump outside is
only 0.155 R.
Thus to jump out of an interstitial OH site the C
atoms will have to displace neighbouring Fe atoms.
This will increase the energy of the system
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46. OH void
OH void
Potential
energy
Hm
A carbon atom can jump
to a neighbouring site if
it has sufficient energy
Hm.
It can gain this energy
only through random
thermal vibration.
If thermal vibration
frequency is then it
makes attempts per
second.
Only a fraction
RT
Hm
exp of these attempts
will have an energy Hm and will be successful.
46
47. 1 2
C1 C2
No. of successful jumps per
second from plane 1 to plane 2,
n1->2 = A c1 exp(- Hm/RT) p
No. of successful jumps per
second from plane 2 to plane 1,
n2->1 = A c2 exp(- Hm/RT) p
Net jumps per second from plane 1
to plane 2
n = n1->2 - n2->1 = A (c1-c2) exp(- Hm/RT) p
1
2
2
exp
c
c
RT
H
p
A m
Flux:
x
c
RT
H
p
A
n
j m
exp
2
47
48. x
c
RT
H
p
j m
exp
2
Fick’s 1st Law
RT
H
p
D m
exp
2
2
0
p
D m
H
Q
An atom making a successful jump may remain in
plane 1, go to the back plane or jump to forward
plane 2. Thus only a fraction p of successful jumps
are from plane 1 to plane 2. This factor has been
omitted in the textbook.
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Mechanisms of Diffusion
1. Self-diffusion: there is no gradient of chemical potential:
Diffusion is the stepwise migration of atoms from lattice site to lattice
site. For an atom to make such a move, two conditions must be met:
1. there must be an empty adjacent site, and
2. the atom must have sufficient energy to break bonds with its
neighbor atoms and then cause some lattice distortion during the
displacement.
a) Interstitial b) Vacancy b) Kick-out
50. 50
Atomic Models of Diffusion
1) Interstitial Diffusion
Usually the solubility of interstitial atoms (e.g. carbon in steel) is small. This implies that most
of the interstitial sites are vacant. Hence, if an interstitial species (like carbon) wants to jump,
this is ‘most likely’ possible as the the neighbouring site will be vacant.
Light interstitial atoms like hydrogen can diffuse very fast. For a correct description of
diffusion of hydrogen anharmonic and quantum (under barrier) effects may be very important
(especially at low temperatures).
The diffusion of two important types of species needs to be distinguished:
(i) species in a interstitial void (interstitial diffusion)
(ii) species ‘sitting’ in a lattice site (substitutional diffusion).
1 2
1 2
Hm
At T > 0 K vibration of the atoms provides
the energy to overcome the energy barrier
Hm (enthalpy of motion).
→ frequency of vibrations,
’ → number of successful jumps / time.
kT
Hm
e
'
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2) Substitutional diffusion via Vacancy Mechanism
For an atom in a lattice site (or a large atom associated with the motif), a jump to a neighbouring site
can take place only if it is vacant. Hence, vacancy concentration plays an important role in the diffusion
of species at lattice sites via the vacancy mechanism.
Vacancy clusters and defect complexes can alter this simple picture of diffusion involving vacancies.
Probability for an atomic jump
(probability that the site is vacant) (probability that the atom has sufficient energy)
'
f m
H
kT
H
kT
e e
kT
H
H m
f
e
'
Where, is the jump distance
Hm → enthalpy of motion of atom
across the barrier.
’ → frequency of successful jumps.
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kT
Hm
e
D 2
For Substitutional Diffusion
kT
H
H m
f
e
D 2
D (C in FCC Fe at 1000ºC) = 3 1011 m2/s
D (Ni in FCC Fe at 1000ºC) = 2 1016 m2/s
0
f m
H H
kT
D D e
0
m
H
kT
D D e
which is of the form
A comparison of the value of diffusivity for interstitial diffusion and substitutional diffusion is
given below. The comparison is made for C in -Fe and Ni in -Fe (both at 1000C).
It is seen that Dinterstitial is orders of magnitude greater than Dsubstitutional.
This is because the “barrier” (in the exponent) for substitutional diffusion has two ‘opposing’
terms: Hf and Hm (as compared to interstitial diffusion, which has only one term).
For Interstitial Diffusion
which is of the form
2
D
Hence, ’ is of the form:
( )
Enthalpy
kT
e
If is the jump distance then the diffusivity can be written as:
( )
2
Enthalpy
kT
D e
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Important.
During self-diffusion there is no change of chemical potential.
Realization of each of the mechanisms depends on
Type of intrinsic defects that prevails in the solid
Activation energy for each of the mechanisms, if more than one may
be realized.
Presence of other defects (vacancies).
Realization of vacancy or kick-out diffusion is possible only at the temperatures
with sufficient concentration of vacancies. Therefore, prevailing mechanism
may change with temperature.
In general, EVERY component in solid undergoes self-diffusion, however, if a
solid contains more than one component, the ratio between self-diffusion
coefficient depends on the type of bonding:
Solids with covalent bonding typically have very low self-diffusion
coefficients.
Solids with ionic bonding may have very different self-diffusion
coefficients for anion and cation.
Metals and metal alloys usually show fast self-diffusion.
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Diffusion of impurities.
a) Interstitial b) Vacancy b) Kick-out
Important.
The diffusion mechanism of an impurity depends on many factors:
type of the solution: interstitial or substitutional;
size of the diffusant and size of the host sites;
temperature;
presence of other impurities;
electronic structure of the host: metal, dielectric or
semiconductor.
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Diffusion Paths with Lesser Resistance
Qsurface < Qgrain boundary < Qpipe < Qlattice
Experimentally determined activation energies for diffusion
The diffusion considered so far (both substitutional and interstitial) is ‘through’ the
lattice.
In a microstructure there are many features, which can provide ‘easier’ paths for
diffusion. These paths have a lower activation barrier for atomic jumps.
The ‘features’ to be considered include grain boundaries, surfaces, dislocation cores, etc.
Residual stress can also play a major role in diffusion.
The order for activation energies (Q) for various paths is as listed below. A lower
activation energy implies a higher diffusivity.
However, the flux of matter will be determined not only by the diffusivity, but also by
the cross-section available for the path.
The diffusion through the core of a dislocation (especially so for edge dislocations) is
termed as Pipe Diffusion.
56. 3
• Interdiffusion: In an alloy, atoms tend to migrate
from regions of large concentration.
Initially After some time
100%
Concentration Profiles
0
Adapted from
Figs. 5.1 and
5.2, Callister
6e.
DIFFUSION: THE PHENOMENA (1)
57. 4
• Self-diffusion: In an elemental solid, atoms
also migrate.
Label some atoms After some time
A
B
C
D
DIFFUSION: THE PHENOMENA (2)
58. How is diffusion
taking place in a
substitutional
solid solution ?
Mechanism of substitutional diffusion
Vacancy mechanism of substitutional diffusion
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MME, NITRR
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59. However, only a very small fraction of the order
of 10-4 to 10-30 are vacant.
A jump can only be successful if the neighbouring
site is vacant.
Probability of finding a vacant site
= fraction of vacant site
RT
H
N
n f
exp
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MME, NITRR
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60. x
c
RT
H
RT
H
p
j
f
m
exp
exp
2
f
m
subs H
H
Q
2
,
0
p
D subs
Sbstitutional diffusion is usually slower than
interstitial diffusion due to difficulty of
finding a vacant site.
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MME, NITRR
60
