mse 280 chap 6
TRANSCRIPT
MSE 280: Engineering Materials ©Lane W. Martin Spring 2011 1
Chapter 6: Diffusion in Solids
MSE 280: Engineering Materials
Professor Lane W. Martin
MSE 280: Engineering Materials ©Lane W. Martin Spring 2011 2
Overview
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?
MSE 280: Engineering Materials ©Lane W. Martin Spring 2011 3
Mechanisms
• Gases & Liquids – random (Brownian) motion
• Solids – vacancy diffusion or interstitial diffusion
Importance of diffusion?
• All materials processing
• Energy storage (e.g. fuel cells, batteries…)
• Filtration
• Chemical reactions
• Semiconductor devices
• And more…
Diffusion
MSE 280: Engineering Materials ©Lane W. Martin Spring 2011 4
• Diffuse carbon atoms into the host
iron atoms at the surface
• Example of interstitial diffusion is a
case hardened gear
• Result
• The “case” is:
• Hard to deform C atoms
“lock” planes from shearing
• Hard to crack C atoms
put surface in compression
From Callister 6e resource CD.
Processing Using Diffusion (1)
Callister, Fig. 6.0
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• Silicon with P for n-type semiconductors:
• Process:
1. Deposit P rich
layers on surface.
2. Heat it.
3. Result: Doped
semiconductor regions.
silicon
silicon
Fig. 18.0, Callister 6e.
Processing Using Diffusion (2)
MSE 280: Engineering Materials ©Lane W. Martin Spring 2011 6
: Atoms of one material diffusing into another and
vice versa (e.g. In an alloy, atoms tend to migrate from regions of
large concentration)
Initially After some time
100%
Concentration Profiles0
Adapted from
Figs. 5.1 and
5.2, Callister 6e.
Diffusion: The Phenomenon
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: Atoms within one material exchanging
positions. (i.e. In an elemental solid, atoms also migrate)
Label some atoms at t = 0 Observe at t > 0
A
B
C
D
Diffusion: The Phenomenon
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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 different possible mechanisms but let’s
consider the simple/more common cases:
1. Vacancy diffusion
2. Interstitial diffusion
Diffusion Mechanisms
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Lattice
distortion
First, bonds with the neighboring
atoms need to be broken
1) Vacancy Diffusion
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Typically more rapid than vacancy diffusion
SMALLER species diffuse FASTER
Callister Fig. 6.3
2) Interstitial Diffusion
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Consider atoms (or mass, M) going through a plane
t = 0
t = t’
2 atoms passed from left to
right (+ direction)
1 atom passed from right
to left (- direction)
Net result: '
1
tarea
atom
or
'At
M
Modeling Diffusion: Flux
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• :
• Directional Quantity
• Flux can be measured for: --vacancies
--host (A) atoms
--impurity (B) atoms
From Callister 6e resource CD.
In general: diffusion flux may or may not be
the same over time
Modeling Diffusion: Flux
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What causes net flow of atoms?
• Concentration Profile, C(x): [kg/m3]
• Fick's First Law:
Concentration
of Cu [kg/m3]
Concentration
of Ni [kg/m3]
Position, x
Cu flux Ni flux
• The steeper the concentration profile, the greater the flux!
Adapted from
Fig. 5.2(c),
Callister 6e.
From Callister 6e resource CD.
Derivation of Fick’s First Law (1)
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Derivation of Fick’s First Law (1)
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Derivation of Fick’s First Law (2)
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: the concentration profile doesn't
change with time
• Apply Fick's First Law:
• Result: the slope, dC/dx, must be constant
(i.e., slope doesn't vary with position)!
Jx D
dC
dx
dC
dx
left
dC
dx
right
• If Jx,left = Jx,right , then
Steady State Diffusion
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dx
dCDJ
Fick’s first law of diffusion C1
C2
x
C1
C2
x1 x2
D diffusion coefficient
Rate of diffusion independent of time
Flux proportional to concentration gradient = dx
dC
12
12 linear ifxx
CC
x
C
dx
dC
Steady State Diffusion
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• Steel plate at 700°C
with geometry shown:
• Q: How much carbon
transfers from the rich to
the deficient side?
= 2.4x10-10 g/cm2s
Steady State Diffusion: Example
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• Methylene chloride is a common ingredient of paint
removers. Besides being an irritant, it also may be
absorbed through the 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:
C2 = 0.02 g/cm3
C1 = 0.44 g/cm3
Example: Personal Protective Equipment
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scm
g 10 x 16.1
cm) 04.0(
)g/cm 44.0g/cm 02.0(/s)cm 10 x 110(
25-
3328-
J
12
12- xx
CCD
dx
dCDJ
Dtb
6
2
glove
C1
C2
skin paint
remover
x1 x2
• Solution – assuming linear conc. Gradient (i.e. steady-state)
D = 110 x 10-8 cm2/s
C2 = 0.02 g/cm3
C1 = 0.44 g/cm3
x2 – x1 = 0.04 cm
Data:
Compare this result to J = 2.4x10-10 g/cm2s calculated in the previous
example of C diffusion in Fe (at higher T and smaller diffusing species).
FASTER diffusion through materials with weak (secondary) bonds
Example: Personal Protective Equipment
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Example – Fick’s First Law
We can purify hydrogen gas by diffusing it through a palladium
sheet. Compute the number of kilograms of hydrogen that
pass / hour through a 6 mm thick sheet of palladium with an
area of 0.25 m2 at 600°C.
Assume:
• D = 1.7x10-8 m2 / s
• CH = 2.0 kg / m3
• CL = 0.4 kg / m3
• That steady state conditions have been achieved.
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D Do exp
Qd
R 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
Note: Same type of T-dependence as
vacancies and interstitials
Temperature Dependence of Diffusion
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Adapted from Fig. 6.7, Callister & Rethwisch 3e. (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.)
D interstitial >> D substitutional
C in α-Fe C in γ-Fe
Al in Al Fe in α-Fe Fe in γ-Fe
1000 K/T
D (m2/s)
0.5 1.0 1.5 10-20
10-14
10-8
T(C) 1
50
0
10
00
600
300
Again, SMALLER
species diffuse FASTER
Temperature Dependence of Diffusion
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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
1lnln and
1lnln
TR
QDD
TR
QDD dd
121
212
11lnlnln
TTR
Q
D
DDD d
transform data D
Temp = T
ln D
1/T
Temperature Dependence of Diffusion
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K 573
1
K 623
1
K-J/mol 314.8
J/mol 500,41exp /s)m 10 x 8.7( 211
2D
12
12
11exp
TTR
QDD d
T1 = 273 + 300 = 573 K
T2 = 273 + 350 = 623 K
D2 = 15.7 x 10-11 m2/s
Temperature Dependence of Diffusion
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1000 K/T
D (m2/s)
0.5 1.0 1.5 10-20
10-14
10-8
T(C) 1
50
0
10
00
600
30
0
C diffusion in γ-Fe
(FCC) is slower than
in α-Fe (BCC) even at
higher T!
More OPEN STRUCTURES allow FASTER diffusion.
Crystal Structure and Diffusion
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Note that we’d have to remove carbon from the right
side and add to the left side to keep a constant flux.
Possible in the previous example of processing using
gas source that can be added and removed but in many
cases this may not be possible….
Non steady-state diffusion
And now for the complication…
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• The concentration of diffusing species is a function of both time and position C = C(x,t)
• In this case is used
2
2
x
CD
t
C
Non-Steady State Diffusion
MSE 280: Engineering Materials ©Lane W. Martin Spring 2011 29
From Fick’s 1st Law:
dx
dcDJ
Take the first derivative w.r.t. x:
dx
dcD
dx
d
dx
dJ
Conservation of mass:
Jr Jl
i.e. fluxes to left and to right have to correspond to concentration change.
dx
dJ
dx
JJ
dt
dc lr
dx
Sub into the first derivative:
dx
dcD
dx
d
dt
dcFick’s 2nd law
c = conc.
inside box
Partial differential equation. We’ll need boundary conditions to solve…
Non-Steady State Diffusion
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• Copper diffuses into a bar of aluminum (semi infinite solid).
C o
C s
position, x
C( x , t ) At to, C = Co inside the Al bar
t o
At t > 0, C(x=0) = Cs and C(x=∞) = Co
t 1 t 2
t 3
Example Non-Steady State Diffusion
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C(x,t) = Conc. at point x at time t
erf (z) = error function
erf(z) values are given in Table 6.1 in text
CS
Co
C(x,t)
Dt
x
CC
Ct,xC
os
o
2 erf1
dye yz 2
0
2
Solution
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Suppose it is desired to achieve a specific concentration
(C1) in an alloy, therefore:
os
o
os
o
CC
CC
CC
CtxC 1),(constant
which leads to:
Dt
x
2constant
Known for given system
Specified with C1
Non-Steady State Diffusion Cont.
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• Copper diffuses into a bar of aluminum
• 10 hours at 600°C 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.
Dt2
Example Processing Question
MSE 280: Engineering Materials ©Lane W. Martin Spring 2011 36
• 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 (γ-Fe at 1000oC).
• Estimate the flux of carbon atoms at the surface given:
Do = 2.3 x 10-5 m2/s for C diffusion in γ-Fe.
Qd = 148 kJ/mol
ργ-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?
Diffusion Design Example
MSE 280: Engineering Materials ©Lane W. Martin Spring 2011 37
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
27,782 yrs!
0.1
10
1000
105
107
109
1011
200 400 600 800 1000 1200
tim
e (
hours
)
T (K)
Diffusion Design Example
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Fick’s 2nd Law Example
Determine the carburizing time necessary to achieve a carbon
concentration of 0.30 wt% at a position 4 mm into an iron-
carbon alloy that initially contains 0.10 wt% C. The surface
concentration is to be maintained at 0.90 wt% C and the
treatment is to be conducted at 1100°C.
Assume: D = 5.3x10-11 m2 / s
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Need to consider coulomb interactions between ions…
+ - - - - - - + + + +
+ - - - - - + + + + +
+ - - - - - - + + + +
+ - - - - - + + + +
+ - - - - - - + + + +
+ - - - - - + + + + +
Diffusion in Ionic Solids
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Which do you expect to diffuse faster, cations
or anions?
•Smaller cations will usually diffuse faster
•But analogous to charge neutrality required for
defect formation, each ion will need counter
charge to move with it (e.g. vacancy, impurity or
free electrons or holes).
Diffusion in Ionic Solids
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• Open crystal structures
• Lower melting T materials
• Materials w/secondary
bonding
• Smaller diffusing atoms
• Lower density materials
• Close-packed structures
• Higher melting T materials
• Materials w/covalent
bonding
• Larger diffusing atoms
• Higher density materials
Structure and Diffusion
MSE 280: Engineering Materials ©Lane W. Martin Spring 2011 42
• Diffusion mechanisms and phenomena – Vacancy diffusion
– Interstitial diffusion
• Importance/usefulness of understanding diffusion (especially in processing)
• Steady-state diffusion
• Temperature dependence – similarity to vacancies and interstitials
• Non steady-state diffusion
• Structural dependence (e.g. size of the diffusing atoms, bonding type, openness of medium)
Concepts to Remember