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Materials of all types are often heat treated to improve
their properties.
The phenomena that occur during a heat treatment almost
always involve atomic diffusion.
Often an enhancement of diffusion rate is desired; on
occasion measures are taken to reduce it.
Heat-treating temperatures and times, and/or cooling rates
are often predictable using the mathematics of diffusion
and appropriate diffusion constants.
The steel gear
shown on this page
has been case
hardened; that is,
its hardness and
resistance to failure
by fatigue have
been enhanced by
diffusing excess
carbon or nitrogen
into the outer
surface layer.
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Many reactions and processes that are important in the
treatment of materials rely on the transfer of mass either
within a specific solid (ordinarily on a microscopic level) or
from a liquid, a gas, or another solid phase.
This is necessarily accomplished by Diffusion, the
phenomenon of material transport by atomic motion.
The phenomenon of diffusion may be demonstrated with the
use of a diffusion couple, which is formed by joining bars of
two different metals together so that there is intimate contact
between the two faces; this is illustrated for copper and
nickel in Figure 7.1, which includes schematic
representations of atom positions and composition across
the interface.
This couple is heated for an extended period at an elevated
temperature (but below the melting for both metals), and
cooled to room temperature.
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Interdiffusion: In an alloy, atoms tend to migrate from regions of
high conc. to regions of low conc.
Initially
After some time
Figs. 7.1 & 7.2,
Callister &
Rethwisch 9e.
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Chemical analysis will reveal a condition similar to that
represented in Figure 7.2 namely, pure copper and nickel at the
two extremities of the couple, separated by an alloyed region.
Concentrations of both metals vary with position as shown in
Figure 7.2.
This result indicates that copper atoms have migrated or
diffused into the nickel, and that nickel has diffused into copper.
This process, whereby atoms of one metal diffuse into another,
is termed interdiffusion, or impurity diffusion.
Interdiffusion may be detected from a macroscopic
perspective by changes in concentration which occur over time,
as in the example for the Cu–Ni diffusion couple.
There is a net drift or transport of atoms from high- to lowconcentration regions.
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Diffusion also occurs for pure metals, but all atoms exchanging
positions are of the same type; this is termed self-diffusion.
Of course, self-diffusion is not normally subject to observation
by noting compositional changes.
Self-diffusion: In an elemental solid, atoms also migrate.
Label some atoms
C
A
D
B
After some time
C
D
A
B
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From an atomic perspective, diffusion is just the stepwise
migration of atoms from lattice site to lattice site.
In fact, the atoms in solid materials are in constant motion,
rapidly changing positions.
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.
This energy is vibrational in nature. At a specific temperature
some small fraction of the total number of atoms is capable of
diffusive motion, by virtue of the magnitudes of their vibrational
energies. This fraction increases with rising temperature.
Several different models for this atomic motion have been
proposed; of these possibilities, the following two dominate for
metallic diffusion.
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Vacancy Diffusion:
One mechanism involves the interchange of an atom from a normal
lattice position to an adjacent vacant lattice site or vacancy (self-diffusion)
This mechanism also applies to substitutional impurities atoms
(interdiffusion)
Diffusion rate depends on:
 number of vacancies
 activation energy to exchange.
increasing elapsed time
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Interstitial diffusion
Another diffusion mechanism involves smaller atoms can
diffuse into spaces between atoms.
Fig. 7.3 (b), Callister & Rethwisch 9e.
More rapid than vacancy diffusion
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Case Hardening:
-- Diffuse carbon atoms into the
host iron atoms at the surface.
-- Example of interstitial diffusion
is a case hardened gear.
Chapter-opening
photograph,
Chapter 7,
Callister &
Rethwisch 9e.
(Courtesy of
Surface Division,
Midland-Ross.)
Result: The presence of C atoms makes iron (steel) harder.
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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
Adapted from Figure 19.27, Callister &
Rethwisch 9e.
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How do we quantify the amount (or rate) of diffusion?
Measured empirically
– Make thin film (membrane) of known cross-sectional area
– Impose concentration gradient
– Measure how fast atoms or molecules diffuse through the membrane
J 
M
1 dM

At A dt
M=
mass
diffused
J  slope
time
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Rate of diffusion independent of time
Flux proportional to concentration gradient =
C1 C1
Fick’s first law of diffusion
C2
C2
x1 x x2
D  diffusion coefficient
 One practical example of steady-state diffusion is found in the
purification of hydrogen gas.
 One side of a thin sheet of palladium metal is exposed to the impure gas
composed of hydrogen and other gaseous species.
 The hydrogen selectively diffuses through the sheet to the opposite side,
which is maintained at a constant and lower hydrogen pressure.
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Methylene chloride is a common ingredient of paint
removers. Besides being an irritant, it also may be
absorbed through skin. When using this paint remover,
protective gloves should be worn.
If butyl rubber gloves (0.04 cm thick) are used, what is the
diffusive flux of methylene chloride through the glove?
Data:
◦ diffusion coefficient in butyl rubber:
D = 110 x10-8 cm2/s
◦ surface concentrations:
C1 = 0.44 g/cm3
C2 = 0.02 g/cm3
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Solution – assuming linear conc. gradient
glove
C1
paint
remover
tb 
2
6D
skin
C2
x1 x2
Data:
D = 110 x 10-8 cm2/s
C1 = 0.44 g/cm3
C2 = 0.02 g/cm3
x2 – x1 = 0.04 cm
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Most practical diffusion situations are nonsteady-state ones.
That is, the diffusion flux and the concentration gradient at
some particular point in a solid vary with time, with a net
accumulation or depletion of the diffusing species resulting.
This is illustrated in Figure 7.5, which shows concentration
profiles at three different diffusion times.
Under conditions of nonsteady state, use of Equation 7.3 is no
longer convenient; instead, the partial differential equation
(7.4b)
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known as Fick’s second law, is used. If the diffusion
coefficient is independent of composition (which should be
verified for each particular diffusion situation)
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Copper diffuses into a bar of aluminum.
Surface conc.,
Cs of Cu atoms
bar
pre-existing conc., Co of copper atoms
Cs
Fig. 7.5,
Callister &
Rethwisch 9e.
I.C.
at t = 0, C = Co for 0  x  ∞
B.C
at t > 0, C = CS for x = 0 (constant surface conc.)
C = Co for x = ∞
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Application of these boundary conditions to Equation 7.4b yields
the solution
(7.5)
C(x,t) = Conc. at point x at
time t
erf(z) = error function
CS
C(x,t)
erf(z) values are given in
Table 7.1
Co
Fig. 7.5, Callister & Rethwisch 9e.
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Where Cx represents the concentration at depth x after time t.
The expression erf( ) is the Gaussian error function, values of
which are given in mathematical tables for various values; a
partial listing is given in Table 7.1.
The concentration parameters that appear in Equation 7.5 are
noted in Figure 7.6, a concentration profile taken at a specific
time.
Equation 7.5 thus demonstrates the relationship between
concentration, position, and time namely, that being a function of
the dimensionless parameter may be determined at any time and
position if the parameters and D are known.
Suppose that it is desired to achieve some specific concentration
of solute, in an alloy; the left-hand side of Equation 7.5 now
becomes
,and the right-hand side
Hence
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Fig_07_05
Fig_07_06
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Fig. 7.5, Callister & Rethwisch 9e.
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7.4 NONSTEADY-STATE DIFFUSION
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7.4 NONSTEADY-STATE DIFFUSION
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Diffusing Species
 The magnitude of the diffusion coefficient D is indicative of the rate at
which atoms diffuse.
 Coefficients, both self- and interdiffusion, for several metallic systems
are listed in Table 7.2.
 The diffusing species as well as the host material influence the
diffusion coefficient.
 For example, there is a significant difference in magnitude between
self-diffusion and carbon interdiffusion in iron at 500 C, the D value
being greater for the carbon interdiffusion (3.0x10-21 vs. 1.4x10-12
m2/s).
 This comparison also provides a contrast between rates of diffusion
via vacancy and interstitial modes as discussed above.
 Self-diffusion occurs by a vacancy mechanism, whereas carbon
diffusion in iron is interstitial.
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Temperature
Temperature has a most profound influence on the
coefficients and diffusion rates.
For example, for the self-diffusion of Fe in α-Fe, the
diffusion coefficient increases approximately six orders
of magnitude (from 3.0x10-21 to 1.8x10-15 ) in rising
temperature from 500 to 900o C (Table 7.2).
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Temperature
The temperature dependence of the diffusion coefficients is
D  Do exp 
Qd
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]
Diffusion coefficient increases with increasing T
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Table 7.2 A Tabulation of Diffusion Data
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Table 7.2 A Tabulation of Diffusion Data
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Temperature
300
600
1000
10-8
1500
D has exponential dependence on T
T(°C)
D (m2/s)
Dinterstitial >> Dsubstitutional
C in α-Fe
C in γ-Fe
10-14
10-20
0.5
1.0
1.5
Al in Al
Fe in α-Fe
Fe in γ-Fe
1000 K/T
Adapted from Fig. 7.7, Callister & Rethwisch 9e.
(Data for Fig. 7.7 taken from E.A. Brandes and G.B. Brook (Ed.) Smithells Metals
Reference Book, 7th ed., Butterworth-Heinemann, Oxford, 1992.)
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Example: At 300°C the diffusion coefficient and
activation energy for Cu in Si are
D(300°C) = 7.8 x 10-11 m2/s
Qd = 41.5 kJ/mol
What is the diffusion coefficient at 350°C?
transform
data
D
Temp = T
ln D
1/T
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T1 = 273 + 300 = 573 K
T2 = 273 + 350 = 623 K
D2 = 15.7 x 10-11 m2/s
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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: use Eqn. 7.5
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◦ t = 49.5 h
◦ Cx = 0.35 wt%
◦ Co = 0.20 wt%
x = 4 x 10-3 m
Cs = 1.0 wt%
 erf(z) = 0.8125
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We must now determine from Table 7.1 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
z  0.93
Now solve for D
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Solution (cont.):
To solve for the temperature at which D
has the above value, we use a
rearranged form of Equation (8.9a);
from Table 8.2, for diffusion of C in FCC Fe
Do = 2.3 x 10-5 m2/s Qd = 148,000 J/mol

T = 1300 K = 1027°C
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Diffusion FASTER for...
Diffusion SLOWER for...
• open crystal structures
• close-packed structures
• materials w/secondary
bonding
• materials w/covalent
bonding
• smaller diffusing atoms
• larger diffusing atoms
• lower density materials
• higher density materials
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