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17 March 2015
Diffusion in Solids
ELEC3215 Fluids and Mechanical Materials
Pt 3 Electromechanical Engineering
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• C
changes as we solidify.• Cu-Ni case:
• Fast rate of cooling:Cored structure
• Slow rate of cooling:Equilibrium structure
First
to solidify has C
= 46 wt% Ni.Last
to solidify has C
= 35 wt% Ni.
Cored vs Equilibrium Phases
First to solidify: 46 wt% Ni
Uniform C:35 wt% Ni
Last to solidify: < 35 wt% Ni
Diffusion and Kinetics of Phase Transformation are important!
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ISSUES TO ADDRESS...
• How does diffusion occur?
• Why is it an important part of processing?
• How can the rate of diffusion be predicted forsome simple cases?
• How does diffusion depend on structureand temperature?
Diffusion in Solids
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Diffusion
Diffusion - Mass transport by atomic motion
Mechanisms
Gases & Liquids – random (Brownian) motion
Solids – vacancy diffusion or interstitial diffusion
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• Interdiffusion: In an alloy, atoms tend to migrate from regions of high concentration to regions of low concentration
Initially
Diffusion
After some time
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• Self-diffusion: In an elemental solid, atoms also migrate.
Label some atoms After some time
Diffusion
A
B
C
DA
B
C
D
Mechanisms:
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Diffusion Mechanisms
Vacancy Diffusion:• atoms exchange with vacancies• applies to substitutional impurities atoms • rate depends on:
--number of vacancies--activation energy to exchange.
increasing elapsed time
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Diffusion Mechanisms
Interstitial diffusion – smaller atoms can diffuse between atoms.
More rapid than vacancy diffusion
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• Case Hardening:--Diffuse carbon atoms
into the host iron atomsat the surface.
--Example of interstitialdiffusion is a casehardened gear.
• Result: The presence of C atoms makes iron (steel) harder.
Processing Using Diffusion
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• Doping silicon with phosphorus for n-type semiconductors:• Process:
3. Result: Dopedsemiconductorregions.
silicon
Processing Using Diffusion
magnified image of a computer chip
0.5mm
light regions: Si atoms
light regions: Al atoms
2. Heat it.
1. Deposit P richlayers on surface.
silicon
Doped region need to be interconnected !
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• What material to choose? Cu, Ag, Au, Al ?• Process involves further heat treatments.
Conducting circuit paths
Al is the best choice in spite of lower electric conductivity
Cu can be use but thin barrier layer of Ta has to be deposited first ( cost ! )
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Diffusion
How do we quantify the amount or rate of diffusion?
Measured empirically
Make thin film (membrane) of known surface area
Impose concentration gradient
Measure how fast atoms or molecules diffuse through the membrane
smkgor
scmmol
timearea surfacediffusing mass) (or molesFlux 22J
1M dMJAt A dt
M =mass
diffusedtime
J
slope
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Geometry of Fick’s first law
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Steady-State Diffusion
dxdCDJ
Fick’s first law of diffusionC1
C2
x
C1
C2
x1 x2
D
diffusion coefficient
Rate of diffusion independent of timeFlux proportional to concentration gradient =
dxdC
12
12 linear ifxxCC
xC
dxdC
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Example: Chemical Protective Clothing (CPC)
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 = 110x10-8 cm2/s surface concentrations:
C2 = 0.02 g/cm3
C1 = 0.44 g/cm3
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scmg 10 x 16.1
cm) 04.0()g/cm 44.0g/cm 02.0(/s)cm 10 x 110( 2
5-33
28-
J
Example (cont).
12
12- xxCCD
dxdCDJ
D
tb 6
2
gloveC1
C2
skinpaintremover
x1 x2
• Solution – assuming linear conc. gradient
D = 110x10-8 cm2/s
C2 = 0.02 g/cm3
C1 = 0.44 g/cm3
x2 – x1 = 0.04 cm
Data:
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Thermal Activation
Process path showing how an atom must overcome an activation energy, q, to move from one stable position to a similar adjacent position
/( )"jump" rate * Bq k Tconst e
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Diffusion and Temperature
• Diffusion coefficient increases with increasing T.
D Do exp
Qd
RT
= 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]
DDo
Qd
RT
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Diffusion and Temperature
D has exponential dependence on T
Dinterstitial >> DsubstitutionalC in -FeC in -Fe
Al in AlFe in -FeFe in -Fe
1000K/T
D (m2/s) C in -Fe
C in -Fe
Al in Al
Fe in -Fe
Fe in -Fe
0.5 1.0 1.510-20
10-14
10-8T(C)15
00
1000
600
300
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Alternative Diffusion Paths
Self-diffusion coefficients for silver depend on the diffusion path
In general, diffusivity is greater through less- restrictive structural regionsSurfacesGrain boundaries
(Lattice)
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Diffusion at Micro scale
Schematic illustration of how a coating of impurity B can penetrate more deeply into grain boundaries and even further along a free surface of polycrystalline A, consistent with the relative values of diffusion coefficients
(Dvolume < Dgrain boundary < Dsurface )
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Effect of Grain Boundaries
Represent a grain by a cylinder: diameter d with grain boundary width .
Area of grain =
Area of grain boundary =
Hence ratio of areas is
2lattice gbD D D
d
“Top view”
22d
22 2d 2
d
lattice lattice gb gbJ A J A J A
2lattice gbJ J J
d
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Effect of Grain Boundaries
Grain boundaries have a more open structure, so Qgb < Qlattice
Should result in faster diffusion in polycrystalline samples than in single crystals.
However, the grain boundary area forms such a small fraction of the total area
The effect is only significant at low temperatures, when Dlattice << Dgb .
The effect is dependent on grain size.
For smaller grains the effect of grain boundary diffusion is greater.
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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/sQd = 41.5 kJ/mol
What is the diffusion coefficient at 350ºC?
101
202
1lnln and 1lnlnTR
QDDTR
QDD dd
121
212
11lnlnln TTR
QDDDD d
transform data
D
Temp = T
ln D
1/T
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Example (cont.)
K 5731
K 6231
K-J/mol 314.8J/mol 500,41exp /s)m 10 x 8.7( 211
2D
1212
11exp TTR
QDD d
T1 = 273 + 300 = 573KT2 = 273 + 350 = 623K
D2 = 15.7 x 10-11 m2/s
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Non-steady State Diffusion
The concentration of diffucing species is a function of both time and position C = C(x,t)
In this case Fick’s Second Law is used
2
2
xCD
tC
Fick’s Second Law
Profile changes in time!Steady State: Fixed profile
v.s.
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Non-steady State Diffusion
B.C. at t = 0, C = Co for 0
x
at t > 0, C = CS for x = 0 (const. surf. conc.)
C = Co for x =
• Copper diffuses into a bar of aluminum.
pre-existing conc., Co of copper atoms
Surface conc., C of Cu atoms bars
Cs
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Solution:
CS
Co
C(x,t)
Dtx
CCCt,xC
os
o
2 erf1
dye yz 2
0
2
C(x,t) = Conc. at point x at time t
erf (z) = error function
erf(z) values are given in tables
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Error FunctionERF
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Approximation
0
01
2s
c c xc c Dt
0
00.25
s
c cc c
If
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Example: 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: use equation given:
Dtx
CCCtxC
os
o
2erf1),(
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Solution (cont.)
t = 49.5 h x = 4 x 10-3 mCx = 0.35 wt% Cs = 1.0 wt%Co = 0.20 wt%
Dtx
CCC)t,x(C
os
o
2erf1
)(erf12
erf120.00.120.035.0),( z
Dtx
CCCtxC
os
o
erf(z) = 0.81250
00.19 0.25
s
c cc c
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Solution (cont.)
We must now determine from Table ERF the value of z for which the error function is 0.8125. An interpolation is necessary as follows
z erf(z)0.90 0.7970z 0.81250.95 0.8209
7970.08209.07970.08125.0
90.095.090.0
z
z
0.93
Now solve for D
Dtxz
2
tzxD 2
2
4
/sm 10 x 6.2s 3600
h 1h) 5.49()93.0()4(
m)10 x 4(4
2112
23
2
2
tzxD
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To solve for the temperature at which D has above value, we use a rearranged form of Arrhenius equation )lnln( DDR
QTo
d
from tables, for diffusion of C in FCC Fe
Do = 2.3 x 10-5 m2/s Qd = 148,000 J/mol
/s)m 10x6.2 ln/sm 10x3.2 K)(ln-J/mol 314.8(J/mol 000,148
21125 T
Solution (cont.)
T = 1300 K = 1027°C
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Diffusion FASTER for...
• open crystal structures
• materials with secondarybonding
• smaller diffusing atoms
• lower density materials
Diffusion SLOWER for...
• close-packed structures
• materials with covalentbonding
• larger diffusing atoms
• higher density materials
Summary