CHAPTER 5:
DIFFUSION IN SOLIDS
ISSUES TO ADDRESS...
• 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?
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DIFFUSION DEMO
• Glass tube filled with water.
• At time t = 0, add some drops of ink to one end
of the tube.
• Measure the diffusion distance, x, over some time.
• Compare the results with theory.
to
x (mm)
t1
t2
t3
xo
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x1
time (s)
x2 x3
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DIFFUSION: THE PHENOMENA (1)
• Interdiffusion: In an alloy, atoms tend to migrate
from regions of high concentration to regions of low
concentration.
Initially
After some time
Adapted
from Figs.
5.1 and 5.2,
Callister 6e.
100%
0
Cu
Ni
100%
Concentration Profiles
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0
Concentration Profiles
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DIFFUSION: THE PHENOMENA (2)
• Self-diffusion: In an elemental solid, atoms
also migrate.
Label some atoms
After some time
C
C
A
A
D
B
B
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D
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Diffusion: The Mechanism
• All atoms are constantly vibrating around their lattice
positions (as long as temperature is above absolute zero).
• The atoms vibrate with a distribution of frequencies and
amplitudes, i.e. there is a distribution of vibrational
energies
• For one atom, the vibrational energy will vary over time
• As temperature increases the average vibrational energy
increases
Diffusion is the movement of an atom from one lattice
position to another. An atom can diffuse if (1) there is
an adjacent space and (2) the atom has sufficient energy
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Substitutional Diffusion
• applies to substitutional impurities
• atoms exchange with vacancies
• rate depends on:
--number of vacancies
--activation energy to exchange.
increasing elapsed time
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INTERSTITIAL DIFFUSION
• Applies to interstitial impurities.
• More rapid than vacancy diffusion.
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PROCESSING USING DIFFUSION (1)
• 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.)
• Result: The "Case" is
--hard to deform: C atoms
"lock" planes from shearing.
--hard to crack: C atoms put
the surface in compression.
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MODELING DIFFUSION: FLUX
• Flux:
Flux is
perpendicular
to this plane
⎡ atoms⎤
1 dM ⎡ kg ⎤
J=
⇒⎢
⎥ or ⎢ 2 ⎥
2
A dt
⎣m s ⎦
⎣ m s ⎦
• Directional Quantity
y J
y
Jz
Metal A
Jx
x
z
• Flux can be measured for:
--vacancies
--host (A) atoms
--impurity (B) atoms
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Metal B
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CONCENTRATION PROFILES & FLUX
• Concentration Profile, C(x): [kg/m3]
Cu flux Ni flux
Concentration
of Cu [kg/m3]
Concentration
of Ni [kg/m3]
Adapted
from Fig.
5.2(c),
Callister 6e.
• Fick's First Law:
flux in x-dir.
[kg/m2-s]
Position, x
Diffusion coefficient [m2/s]
dC
Jx = − D
dx
concentration
gradient [kg/m4]
• The steeper the concentration profile,
the greater the flux!
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STEADY STATE DIFFUSION
• Steady State: the concentration profile doesn't
change with time.
Steady State:
Jx(left)
Jx(right)
Jx(left) = Jx(right)
x
Concentration, C, in the box doesn’t change w/time.
dC
• Apply Fick's First Law: J x = −D
dx
⎛ dC ⎞
⎛ dC ⎞
=⎜ ⎟
• If Jx)left = Jx)right , then ⎜ ⎟
⎝ dx ⎠ left ⎝ dx ⎠ right
• Result: the slope, dC/dx, must be constant
(i.e., slope doesn't vary with position)!
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Diffusion Coefficient, D
•
•
•
•
•
Also called diffusivity
Units: m2/s
Depends on diffusing species and host
Depends on temperature
Higher diffusivity means a higher flux for
the same concentration gradient
dC
Fick's First Law: J x = −D
dx
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EX: STEADY STATE DIFFUSION
• Steel plate at
700°C with
Carbon
geometry
rich
shown:
gas
3
m
g/
k
2
3
.
m
=1
/
kg
C1
8
.
=0
C2
Steady State =
straight line! Adapted
Carbon
deficient
gas
from Fig.
5.4,
Callister 6e.
D=3x10-11m2/s
5m
m
m
m
10
0 x1 x2
• Q: What is the flux of carbon from the left to the
right?
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NON STEADY STATE DIFFUSION
dx
• Concentration profile,
C(x), changes
J(left)
w/ time.
• To conserve matter:
J(right) − J(left)
= − dC
dx
dt
dJ = dC
−
dt
dx
• Governing Eqn.:
J(right)
Concentration,
C, in the box
• Fick's First Law:
dC
=
−
J
D
or
dx
d2 C (if D does
dJ =
−D
not vary
dx
dx2 with x)
equate
dC
d2C
=D
dt
dx 2
Fick’s 2nd Law
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Non-steady state diffusion
dC
d2C
=D
dt
dx 2
Fick’s 2nd Law
• In order to analyze non-steady state
diffusion behavior we must solve this
partial differential equation.
• The solution depends on the initial
conditions and the boundary conditions.
• We will consider one type of non-steady
state diffusion only.
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Diffusion through a semi infinite solid
with constant surface concentration
• We can use this approximation if L > 10 Dt
• Before diffusion begins the diffusing atoms are
uniformly distributed in the solid at a concentration C0.
• At t=0 the concentration at the surface of the solid is
suddenly changed to CS.
t<0, C=C0
x
x=0
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t=0, C=C0
CS
x=0
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x
CS
t=t1, C=C(x,t)
x
x=0
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Diffusion through a semi infinite solid
with constant surface concentration
CS
Blue line is the
concentration
profile at t=0.
C0
Solution to
Fick’s 2nd Law
for this case:
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⎛ x ⎞
C(x, t) − Co = −
⎟
1 erf ⎜⎝
2 Dt ⎠
Cs − Co
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Error Function Table
x
Note: z is the independent variable, i.e. Z =
2 Dt
and erf(z) is the dependent variable.
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Example 1
Consider the impurity diffusion of gallium into a silicon wafer
at 1100°C. Prior to commencement of diffusion the wafer was
free of gallium. At time = 0 the surface concentration is
changed to 1024 atoms/m3. Find the depth below the surface at
which the concentration will be 1022 atoms/m3 after 3 hours.
Given: D=7.0 x 10-17 m2/s
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Example 2
Consider the gas carburizing of steel at 927ºC. The diffusion
coefficient of carbon in steel at this temperature is 1.28 x 10-11
m2/s. The carbon content at the surface is 0.90% and the initial
carbon content is 0.20%. Calculate the carbon content 0.50
mm below the surface after 5 hours of carburizing time.
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Example 3
Consider the surface treatment of steel with C at 950°C.
The initial concentration of C in the steel is 0.25 wt% and
the surface concentration during treatment is 1.2 wt%. How
long will it take to reach 0.8 wt% C at a position of 0.5 mm
below the surface?
Given: D=1.6 x 10-11 m2/s
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DIFFUSION AND TEMPERATURE
• Diffusivity increases with T.
D = Do
diffusivity
pre-exponential [m2/s] (see Table 5.2, Callister 6e)
activation energy
⎛ Q ⎞ [J/mol],[eV/mol]
exp ⎜− d ⎟ (see Table 5.2, Callister 6e )
⎝ RT ⎠
γFe
Ci
300
nα
-Fe
Al
in
Al
Cu
Cu in Fe
in Cun αei e
Zn
F
γ-F
in
Fe
0.5
in
10-14
10-20
600
C
10-8
D (m2/s)
1000
1500
• Experimental Data:
1.0
1.5
gas constant [8.31J/mol-K]
T(C)
D has exp. dependence on T
Recall: Vacancy does also!
Dinterstitial >> Dsubstitutional
Cu in Cu
C in α-Fe
Al in Al
C in γ -Fe
Fe in α-Fe
Fe in γ -Fe
Zn in Cu
2.0 1000K/T
Adapted from Fig. 5.7, Callister 6e. (Date for Fig. 5.7 taken from
E.A. Brandes and G.B. Brook (Ed.) Smithells Metals Reference
Book, 7th ed., Butterworth-Heinemann, Oxford, 1992.)
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Diffusion Coefficient
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PROCESSING QUESTION
• Copper diffuses into a bar of aluminum.
• 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?
Key point 1: C(x,t500C) = C(x,t600C).
Key point 2: Both cases have the same Co and Cs.
• Result: Dt should be held constant.
⎛ x ⎞
C(x,t) − Co
(Dt)500ºC =(Dt)600ºC
⎟
= 1 − erf ⎜⎝
⎠
Cs − C
2
Dt
o
5.3x10-13m2/s
10hrs
• Answer:
(Dt)600
= 110 hr
t 500 =
D500
4.8x10-14m2/s
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Note: values
of D are
provided here.
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SUMMARY:
STRUCTURE & DIFFUSION
Diffusion FASTER for...
Diffusion SLOWER for...
• open crystal structures
• close-packed structures
• lower melting T materials
• higher melting T materials
• materials w/secondary
bonding
• materials w/covalent
bonding
• smaller diffusing atoms
• larger diffusing atoms
• cations
• anions
• lower density materials
• higher density materials
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