Phase Transformation Lecture 3

MM-501 Phase Transformation in
Solids
Fall Semester-2015
Lecture No: 03
Diffusion :
How do atoms move through
solids?

What is diffusion?
• Material/Mass transport by atomic motion is called
Diffusion. OR
• It is a transport phenomenon caused by the motion
of chemical species (molecules, atoms or ions), heat
or similar properties of a medium (gas, liquid or solid)
as a consequence of concentration (or, strictly,
chemical potential) differences.
• In general, the species move from high
concentration areas to low concentrations areas until
uniform concentration is achieved in the medium.
3

4
• Diffusion is easy in liquids and gases where atoms are
relatively free to move around:
• In solids, atoms are not fixed at its position but
constantly moves (oscillates) . So, Diffusion is difficult in
solids due to bonding and requires, most of the time,
external energy to mobilize the atoms.

5
Mass transport can generally involve:
1. fluid flow – dominant in gases or liquids
2. viscous flow – flow of a viscous material, generally amorphous or
semi crystalline (e.g. glasses and polymers) due to the forces
acting on it at that moment;
3. atomic diffusion – principal mechanism in solids and in static
liquids (as occurs in solidification).
Atomic diffusion occurs during important processes such as:
1. solidification of materials .
2. precipitation hardening, e.g. Al-Cu alloys
3. annealing of metals to reduce excess vacancies & dislocations
formed during working
4. manufacture of doped silicon, e.g. as used in many electronic
devices

• For an active diffusion to occur,
the temperature should be high
enough to overcome energy
barriers to atomic motion
• for atom to jump into a vacancy
site, it needs enough energy
(thermal energy) to break the
bonds and squeeze through its
neighbors and take the new
position. The energy necessary
for motion is Em called the
activation energy for vacancy
motion.
• At activation energy Em has to be
supplied to the atom so that it
could break inter-atomic bonds
and to move into the new
position. 6
Figure: Schematic representation of the
diffusion of an atom from its original
position into a vacant lattice site
Inhomogeneous materials
can become homogeneous by
diffusion.

12
Diffusion Mechanisms
How do atoms move between atomic sites?
For diffusion to occur:
1. Adjacent site needs to be empty (vacancy or
interstitial).
2. Sufficient energy must be available to break
bonds and overcome lattice distortion.
There are many diffusion mechanism to be observed but
two possible mechanisms are considered:
1. Vacancy diffusion.
2. Interstitial diffusion.

13
1. Substitutional Diffusion
a) Direct Exchange
b)Ring
c) Vacancy
2. Interstitial Diffusion

15
Vacancy Mechanism
Atoms can move from one site to another if there is
sufficient energy present for the atoms to overcome a
local activation energy barrier and
if there are vacancies present for the atoms to move into.
The activation energy for diffusion is the sum of the
energy required to form a vacancy and the energy to move
the vacancy.

16
Vacancy diffusion
- An atom adjacent to a vacant lattice site moves into it.
Essentially looks like
an interstitial atom:
lattice distortion
First, bonds with the neighboring
atoms need to be broken
From Callister 6e resource CD.
• To jump from lattice site to lattice
site, atoms need energy to break
bonds with neighbors, and to cause
the necessary lattice distortions
during jump. This energy comes
from the thermal energy of atomic
vibrations (Eav ~ o CT)
• Materials flow (the atom) is
opposite the vacancy flow
direction.

17
Interstitial atoms like hydrogen, helium, carbon, nitrogen,
etc) must squeeze through openings between interstitial
sites to diffuse around in a crystal.
The activation energy for diffusion is the energy required
for these atoms to squeeze through the small openings
between the host lattice atoms.
Interstitial Mechanism

18
Interstitial Diffusion
Migration from one interstitial site to another (mostly for small atoms that can be
interstitial impurities: (e.g. H, C, N, and O) to fit into interstices in host.
Carbon atom in Ferrite
Interstitial diffusion is generally faster than
vacancy diffusion because bonding of interstitials to
the surrounding atoms is normally weaker and there are
many more interstitial sites than vacancy sites to jump
to.

Interstitial Diffusion-Animation
19

20
• How do we quantify the amount or rate of diffusion?
• Flux (J): No of atom s diffusing through unit area per
unit time OR Materials diffusion through unit area per
unit time.
• Measured empirically
– Make thin film (membrane) of known surface area
– Impose concentration gradient
– Measure how fast atoms or molecules diffuse through the membrane
   sm
kg
or
scm
mol
timeareasurface
diffusingmass)(ormoles
Flux
22
J
M =
mass
diffused
time
J  slope

21
Temperature Dependence of the
Diffusion Coefficient :
• Diffusion coefficient increases with increasing T.
= pre-exponential [m2/s]
= diffusion coefficient [m2/s]
= activation energy [J/mol or eV/atom]
= gas constant [8.314 J/mol-K]
= absolute temperature [K]
D
Do
Qd
R
T
With conc. gradient fixed, higher D means higher flux of mass transport.
TR
Q
-D=D d
olnln





TR
Q
-D=D d
o exp

22
• Diffusivity increases with T.
• Experimental Data:
D has exp. dependence on T
Recall: Vacancy does also!
Dinterstitial >> Dsubstitutional
C in -Fe
C in -Fe Al in Al
Cu in Cu
Zn in Cu
Fe in -Fe
Fe in -Fe
Diffusion and Temperature
ln D  ln D0 
Qd
R
1
T






log D  log D0  Qd
2.3R
1
T






Note:
pre-exponential [m2/s]
activation energy
gas constant [8.31J/mol-K]
D DoExp 
Q
d
RT
diffusivity
[J/mol],[eV/mol]

23
Steady-State Diffusion
dx
dC
DJ 
Fick’s first law of diffusionC1
C2
x
C1
C2
x1 x2
D  diffusion coefficient
(be careful of its unit)
Rate of diffusion independent of time
Flux proportional to concentration gradient =
dx
dC
12
12linearif
xx
CC
x
C
dx
dC







24
Diffusivity -- depends on:
1. Diffusion mechanism. Substitutional vs interstitial.
2. Temperature.
3. Type of crystal structure of the host lattice.
4. Type of crystal imperfections.
(a) Diffusion takes place faster along grain boundaries
than elsewhere in a crystal.
(b) Diffusion is faster along dislocation lines than
through bulk crystal.
(c) Excess vacancies will enhance diffusion.
5. Concentration of diffusing species.

Microstructural Effect on Diffusion:
• If a material contains grains, the grains will act as diffusion
pathways, along which diffusion is faster than in the bulk
material.
25

26
Physical Aspect of D
1. D is the indicator of how fast atom moves.
2. In liquid state, D reaches similar level regardless of structure.
3. In solid state, D shows high sensitivity to temperature and
structure.
4. Absolute temperature and Tm are what we should care about.

27
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?













1
01
2
02
1
lnlnand
1
lnln
TR
Q
DD
TR
Q
DD dd







121
2
12
11
lnlnln
TTR
Q
D
D
DD d
transform
data
D
Temp = T
ln D
1/T

28
Example (cont.)














 
K573
1
K623
1
K-J/mol314.8
J/mol500,41
exp/s)m10x8.7( 211
2D













12
12
11
exp
TTR
Q
DD d
T1 = 273 + 300 = 573K
T2 = 273 + 350 = 623K
D2 = 15.7 x 10-11 m2/s

29
• Steel plate at 7000C with geometry shown:
• Q: In steady-state, how much carbon transfers
from the rich to the deficient side?
Adapted from Fig.
5.4, Callister 6e.
Example: Steady-state Diffusion
Knowns:
C1= 1.2 kg/m3 at 5mm
(5 x 10–3 m) below surface.
C2 = 0.8 kg/m3 at 10mm
(1 x 10–2 m) below surface.
D = 3 x10-11 m2/s at 700 C.
700 C

30
• Concentration profile,C(x),
changes with time.
14
• To conserve matter: • Fick's First Law:
• Governing Eqn.:
Non-Steady-State Diffusion
In most real situations the concentration profile and the
concentration gradient are changing with time. The
changes of the concentration profile is given in this case
by a differential equation, Fick’s second law.
Called Fick’s second law

31
Fick's Second Law of Diffusion
In words, the rate of change of composition at position x with
time, t, is equal to the rate of change of the product of the
diffusivity, D, times the rate of change of the concentration
gradient, dCx/dx, with respect to distance, x.






xd
Cd
D
xd
d
=
td
Cd xx
Non-Steady-State Diffusion
Co
Cs
position, x
C(x,t)
to
t1
t2
t3

32
• Copper diffuses into a bar of aluminum.
15
• General solution:
"error function"
Values calibrated in Table 5.1, Callister 6e.
Co
Cs
position, x
C(x,t)
to
t1
t2
t3 Adapted from Fig.
5.5, Callister 6e.
Example: Non Steady-State Diffusion
t3>t2>t1
Fig. 6.5: Concentration profiles nonsteady-state diffusion taken at three different times
C0=Before diffusion
For t=0, C=C0 at 0x 

33
Non-steady State Diffusion
• Sample Problem: 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 tip: use Eqn.









Dt
x
CC
CtxC
os
o
2
erf1
),(

34
Solution (cont.):
– t = 49.5 h x = 4 x 10-3 m
– Cx = 0.35 wt% Cs = 1.0 wt%
– Co = 0.20 wt%









Dt
x
CC
C)t,x(C
os
o
2
erf1
)(erf1
2
erf1
20.00.1
20.035.0),(
z
Dt
x
CC
CtxC
os
o 











 erf(z) = 0.8125

35
Solution (cont.):
We must now determine from Table 5.1 the value of z for which the error
function is 0.8125. An interpolation is necessary as follows
z erf(z)
0.90 0.7970
z 0.8125
0.95 0.8209
7970.08209.0
7970.08125.0
90.095.0
90.0




z
z  0.93
Now solve for D
Dt
x
z
2

tz
x
D
2
2
4

/sm10x6.2
s3600
h1
h)5.49()93.0()4(
m)10x4(
4
211
2
23
2
2












tz
x
D

36
• To solve for the temperature at
which D has above value, we use a
rearranged form of Equation
(5.9a);
)lnln( DDR
Q
T
o
d


from Table 5.2, for diffusion of C in FCC Fe
Do = 2.3 x 10-5 m2/s Qd = 148,000 J/mol
/s)m10x6.2ln/sm10x3.2K)(ln-J/mol314.8(
J/mol000,148
21125 

T
Solution (cont.):
T = 1300 K = 1027°C

37






TR
Q
-D=D d
o exp
TR
Q
-D=D d
olnln
Where
D is the Diffusivity or Diffusion Coefficient ( m2 / sec )
Do is the prexponential factor ( m2 / sec )
Qd is the activation energy for diffusion ( joules / mole )
R is the gas constant ( joules / (mole deg) )
T is the absolute temperature ( K )
Temperature Dependence of the
Diffusion Coefficient
OR

39
Experimental Determination of Diffusion
Coefficient
Tracer method
• Radioisotopic tracer atoms are deposited at surface of solid by e.g. electro deposition
• isothermal diffusion is performed for a given time t, often quartz ampoules are used
(T<1600°C)
• Sample is then divided in small slices either mechanically, chemically or by sputtering
techniques
• Mechanically: for diffusion length of > 10 µm; D>10-11 cm2/s
• Sputtering of surface: for small diffusion length (at low temperatures) 2nm …10µm, for
the range D = 10-21 …10-12 cm2/s

40
Experimental Determination of Diffusion
Coefficient
• Example:
Diffusion of Fe in Fe3Si
• From those figures the
diffusion constant can be
determined with an accuracy
of a few percent
• Stable isotopes can be used as
well, when high resolution
SIMS is used
• This technique is more
difficult

42
• 10 hours processed at 600 C gives desired C(x).
• How many hours needed to get the same C(x) at 500 C?
16
• Result: Dt should be held constant.
• Answer:
Note: values of D are
provided.
Key point 1: C(x,t500C) = C(x,t600C).
Key point 2: Both cases have the same Co and Cs.
Processing Question

4317
• The experiment: we recorded combinations of
t and x that kept C constant.
• Diffusion depth given by:
C(xi,ti)  Co
Cs  Co
 1 erf
xi
2 Dti







 = (constant here)
Diffusion Analysis

44
Non steady-state diffusion
From Fick’s 1st Law:
dx
dc
DJ 
Take the first derivative w.r.t. x: 






dx
dc
D
dx
d
dx
dJ
Conservation of mass:
i.e. flux to left and to right has to correspond to concentration change.
dx
dJ
dx
JJ
dt
dc lr



Sub into the first derivative: 






dx
dc
D
dx
d
dt
dc Fick’s 2nd law
JrJl
dx c = conc.
inside box
Partial differential equation. We’ll need boundary conditions to solve…
In most practical cases steady-state conditions are
not established, i.e. concentration gradient is not
uniform and varies with both distance and time. Let’s
derive the equation that describes non steady-state
diffusion along the direction x.

45
EX: NON STEADY-STATE DIFFUSION
• Copper diffuses into a bar of aluminum (semi infinite solid).
• General solution:
"error function"
Values calibrated in Table 5.1, Callister 6e.
Adapted from
Fig. 5.5,
Callister 6e.
From Callister 6e resource CD.
Co
Cs
position, x
C(x,t)
At to, C = Co inside the Al bar
to
At t > 0, C(x=0) = Cs and C(x=∞) = Co
t1
t2
t3

46
If it is desired to achieve a specific concentration C1
i.e.






os
o
os
o
CC
CC
CC
CtxC 1),(
constant
which leads to:

Dt
x
2
constant
Known for given system
Specified with C1
1
1

48
PROCESSING QUESTION
• 10 hours at 600C gives desired C(x).
• How many hours would it take to get the same C(x)
if we processed at 500C?
• Result: Dt should be held constant.
• Answer:
Note: values
of D are
provided here.
Key point 1: C(x,t500C) = C(x,t600C).
Key point 2: Both cases have the same Co and Cs.
Adapted from Callister 6e resource CD.
Dt2

49
Diffusion: Design Example
During a steel carburization process at 1000oC, there is a drop in
carbon concentration from 0.5 at% to 0.4 at% between 1 mm and
2 mm from the surface (g-Fe at 1000oC).
– Estimate the flux of carbon atoms at the surface.
Do = 2.3 x 10-5 m2/s for C diffusion in -Fe.
Qd = 148 kJ/mol
r-Fe = 7.63 g/cm3
AFe = 55.85 g/mol
– If we start with Co = 0.2 wt% and Cs = 1.0 wt% how long does it take to
reach 0.6 wt% at 0.75 mm from the surface for different processing
temperatures?

50
T (oC) t (s) t (h)
300 8.5 x 1011 2.4x108
900 106,400 29.6
950 57,200 15.9
1000 32,300 9.0
1050 19,000 5.3
Need to consider factors such as cost of maintaining furnace at different T for
corresponding times.
27782 yrs!
Diffusion: Design Example Cont’d

51
A Look at Diffusion Bonding

52
Introduction
• Diffusion bonding is a method of creating a joint between
similar or dissimilar metals, alloys, and nonmetals.
• Two materials are pressed together (typically in a vacuum) at a
specific bonding pressure with a bonding temperature for a
specific holding time.
• Bonding temperature
– Typically 50%-70% of the melting temperature of the most
fusible metal in the composition
– Raising the temperature aids in the interdiffusion of atoms
across the face of the joint.

53
How does diffusion bonding work?
• Bonding pressure
– Forces close contact between the edges of the
two materials being joined.
– Deforms the surface asperities to fill all of the
voids within the weld zone .
– Disperses oxide films on the materials, leaving
clean surfaces, which aids the diffusion and
coalescence of the joint.

54
• Holding Time
– Always minimized
• Minimizing the time reduces the physical force on the
machinery.
• Reduces cost of diffusion bonding process.
• Too long of a holding time might leave voids in the weld
zone or possibly change the chemical composition of
the metal or lead to the formation of brittle
intermetallic phases when dissimilar metals or alloys
are being joined.

55
• Sequence for diffusion bonding
a ceramic to a metal
– a) Hard ceramic and soft metal
edges come into contact.
– b) Metal surface begins to yield
under high local stresses.
– c) Deformation continues mainly
in the metal, leading to void
shrinkage.
– d) The bond is formed

56
Advantages of diffusion bonding
• Properties of parent materials are generally unchanged.
• Diffusion bonding can bond similar or dissimilar metals and
nonmetals.
• The joints formed by diffusion bonding are generally of very
high quality.
• The process naturally lends itself to automation.
• Does not produce harmful gases, ultraviolet radiation, metal
spatter or fine dusts.
• Does not require expensive solders, special grades of wires or
electrodes, fluxes or shielding gases.

58
Summary II
1. Diffusion is just one of many mechanisms for
mass transport.
2. Electrical field can produce mass transport.
3. Magnetic field can produce mass transport.
4. Combination of fields can produce mass
transport such as electrochemical transport.

60
Application: Homogenization time
Solidification usually results in chemical heterogeneities
– Represent it with a sinusoid of wavelength, λ
– Composition should homogenize when, x > λ/2
– The approximate time necessary is:
Homogenization time
- increases with λ2
- decreases exponentially with T

61
Application:
Service Life of a Microelectronic Device
•Microelectronic devices
– have built-in heterogeneities
– Can function only as long as these doped regions survive
• To estimate the limit on service life, ts
– Let doped island have dimension, λ
– Device is dead when, x ~ λ/2, hence
Service life
- decreases with miniaturization (λ2)
- decreases exponentially with T

62
Influence of Microstructure on Diffusivity
Interstitial species
– Usually no effect from microstructure
– Stress may enhance diffusion
Substitutional species
– Raising vacancy concentration increases D
• Quenching from high T
• Solutes
• Irradiation
– Defects provide “short-circuit” paths
• Grain boundary diffusion
• Dislocation “core diffusion”

63
Adding Vacancies Increases D
• Quench from high T
– Rapid cooling freezes in high cv
– D decreases as cv evolves to equilibrium
• Add solutes that promote vacancies
– High-valence solutes in ionic solids
• Mg++ increases vacancy content in Na+Cl-
• Ionic conductivity increases with cMg
– Large solutes in metals
– Interstitials in metals
• Processes that introduce vacancies directly
– Irradiation
– Plastic deformation

64
Grain Boundary Diffusion
• Grain boundaries have high defect densities
– Effectively, vacancies are already present
– QD ~ Qm
• Grain boundaries have low cross-section
– Effective width = δ
– Areal fraction of cross-section:

67
Diffusion – Thermally Activated
Process (I)
In order for atom to jump into a
vacancy site, it needs to posses
enough energy (thermal energy) to
break the bonds and squeeze
through its neighbors. The energy
necessary for motion, Em, is called
the activation energy for vacancy
motion.
At activation energy Em has to be
supplied to the atom so that it could
break inter-atomic bonds and to
move into the new position.
Schematic representation of the
diffusion of an atom from its original
position into a vacant lattice site.

68
Diffusion – Thermally Activated
Process (II)
The average thermal energy of an atom (kBT = 0.026 eV for room
temperature) is usually much smaller that the activation energy
Em (~ 1 eV/atom) and a large fluctuation in energy (when the
energy is “pooled together” in a small volume) is needed for a
jump.
The probability of such fluctuation or frequency of jumps, Rj,
depends exponentially from temperature and can be described by
equation that is attributed to Swedish chemist
Arrhenius :
where R0 is an attempt frequency proportional to the frequency of
atomic vibrations.

69
Diffusion – Thermally Activated Process (III)
For the vacancy diffusion mechanism the probability for any atom in a solid to
move is the product of the probability of finding a vacancy in an adjacent
lattice site (see Chapter 4):
and the probability of thermal fluctuation needed to overcome the
energy barrier for vacancy motion
The diffusion coefficient, therefore, can be estimated as
Temperature dependence of the diffusion coefficient, follows the Arrhenius
dependence.

70
Diffusion – Temperature Dependence (I)
Diffusion coefficient is the measure of
mobility of diffusing species.
D0 – temperature-independent preexponential (m2/s)
Qd – the activation energy for diffusion (J/mol or eV/atom)
R – the gas constant (8.31 J/mol-K or 8.62x105 /atom-K
T – absolute temperature (K)
The above equation can be rewritten as
The activation energy Qd and preexponential D0, therefore, can be estimated by
plotting lnD versus 1/T or logD versus 1/T. Such plots are Arrhenius plots.

71
Diffusion – Temperature Dependence (II)
Graph of log D vs. 1/T has
slop of –Qd/2.3R, intercept
of ln Do

72
Diffusion – Temperature Dependence (III)
Arrhenius plot of diffusivity data for some metallic systems

73
Diffusion of different species
Smaller atoms diffuse more readily than big ones, and diffusion is
faster in open lattices or in open directions

74
Diffusion: Role of the microstructure (I)
Self-diffusion coefficients
for Ag depend on the
diffusion path.
In general, the diffusivity is
greater through less
restrictive structural regions
– grain boundaries,
dislocation cores, external
surfaces.

75
Diffusion: Role of the microstructure
(II)
The plots (opposite) are from the computer
simulation by T. Kwok, P. S. Ho, and S. Yip.
Initial atomic positions are shown by the
circles, trajectories of atoms are shown by
lines.
We can see the difference between atomic
mobility in the bulk crystal and in the grain
boundary region.

76
Exercise
1. A thick slab of graphite is in contact with a 1mm thick sheet of
steel. Carbon steadily diffuses through the steel at 925C. The carbon
reaching the free surface reacts with CO2 gas to form CO, which is
then rapidly pumped away.
Determine the carbon concentration, C2, adjacent to the free surface,
and the find the carbon flux in the steel, given that the reaction
velocity for C+CO22CO is =3.010-6cm/sec.
At 925C, the solubility of carbon in the steel in contact with
graphite is 1.5wt% and the diffusivity of carbon through steel is
D=1.710-7cm2/sec. The equilibrium solubility of carbon in steel,
Ceq, is 0.1wt% for the CO/CO2 ratio established at the surface of the
steel.

77
Exercise
Pe 
l
D
 1.76The Péclet number is
Note: The value of the Péclet number suggests mixed kinetic behavior is expected.
C2  Ceq 
C0  Ceq
1
 l
D
 0.1
1.5  0.1
11.76
, [wt%]
C2  0.61wt%.
The carbon concentration in
the steel at the free surface,
C2, is
The steady-state flux is Jss = 1.51  10-6 [wt% C  cm/s]
Jss = 1.18  10-7 [g/ cm2-s]
Divide by the density of steel, =12.8 cm3/100g to obtain the steady-state flux of
carbon

78
Exercise
2. Two steel billets—a slab and a solid cylinder—contain 5000ppm
residual H2 gas. These billets are vacuum annealed in a furnace at
725C for 24 hours to reduce the gas content. Vacuum annealing is
capable of maintaining a surface concentration in the steel of
10ppm H2 at the annealing temperature.
Estimate the average residual concentration of H2 in each billet
after vacuum annealing, given that the diffusivity of H in steel at
725C is DH=2.2510- 4 cm2/sec.

79
Exercise
15 cm
2h=10cm
2h=10 cm
15 cm
10 cm
2h = 10 cm
2h=10cm
Rectangular and cylindrical slabs of steel
10 cm

80
Given:
t=24 hr=86400 s
Co= Initial Concentration= 5000 ppm
Cs= Surface concentration= 10 ppm
DH= 2.25x10-4 cm2/s
C1= average residual concentration=?
We know that:
(C1-Co) / (Cs-Co) = Constant(z) and also
X  (Dt) or x = Constant x (Dt)
or Constant(z) = (Dt) / x2
Now we can write:
(C1-Co) / (Cs-Co) = (Dt) / x2 or C1= (Co-Cs) x (f) + Cs
Therefore,
For slab:
C1= (Co-Cs) x (flong x fshort x fshort) + Cs
and
For Cylinder:
C1= (Co-Cs) x (flong x fshort) + Cs

83
STRUCTURE & DIFFUSION
Diffusion FASTER for...
• open crystal structures
• lower melting T materials
• materials with secondary
bonding
• smaller diffusing atoms
• lower density materials
Diffusion SLOWER for...
• close-packed structures
• higher melting T materials
• materials with covalent
bonding
• larger diffusing atoms
• higher density materials

84
Factors that Influence Diffusion:
Summary
" Temperature - diffusion rate increases very rapidly with
increasing temperature
" Diffusion mechanism - interstitial is usually faster
than vacancy
" Diffusing and host species - Do, Qd is different for
every solute, solvent pair
" Microstructure - diffusion faster in polycrystalline vs.
single crystal materials because of the accelerated diffusion
along grain boundaries and dislocation cores.

85
Concepts to remember
• Diffusion mechanisms and phenomena.
– Vacancy diffusion.
– Interstitial diffusion.
• Importance/usefulness of understanding diffusion
(especially in processing).
• Steady-state diffusion.
• Non steady-state diffusion.
• Temperature dependence.
• Structural dependence (e.g. size of the diffusing
atoms, bonding type, crystal structure etc.).

Phase Transformation Lecture 3

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Phase Transformation Lecture 3