Chapter 5: Atom and Ion Movements in Materials
Chapter 5:
Atom and Ion
Movements
in Materials
© 2011 Cengage Learning Engineering. All Rights Reserved.
Chapter 5 -5 - 1
Course Outcomes
CO-1:
Characterize the structure-property-performance
relationship for engineering materials.
CO-2:
Identify the structure and basic properties of
different types of materials including metals,
polymers, ceramics, and composites.
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Engineering. All Rights Reserved.
Chapter 5 -1 - 2
Chapter 5: Atom and Ion Movements in Materials
Learning Objectives
1.
Applications of diffusion
2.
Stability of atoms and ions
3.
Mechanisms for diffusion: How does diffusion occur?
4.
Activation energy for diffusion
5.
Rate of diffusion [Fick’s first law]: How can Diffusion rate be
predicted for simple cases?
6.
Factors affecting diffusion [How does diffusion depend on
structure and temperature?]
7.
Permeability of polymers
8.
Composition profile [Fick’s second law]
9.
Diffusion and materials processing [Why is an important part
of Processing?
© 2011 Cengage Learning Engineering. All Rights Reserved.
Chapter 5 -
5-3
Chapter 5: Atom and Ion Movements in Materials
Applications of Diffusion
 Diffusion
 Net flux of any species, such as ions, atoms, electrons,
holes, and molecules.
 Carburization/Nitriding for surface hardening of
steels –
 A source of carbon is diffused into steel components.
 In nitriding, nitrogen is introduced into the surface of a
metallic material.
 Dopant diffusion for semiconductor devices
 A p-n junction is a region of the semiconductor, one
side of which is doped with n-type dopants and the
other side is doped with p-type dopants.
© 2011 Cengage Learning Engineering. All Rights Reserved.
Chapter 5 -
5-4
Diffusion
• Self-diffusion: In an elemental solid, atoms
also migrate.
Label some atoms
C
A
D
B
After some time
C
D
A
B
Chapter 5 - 5
Diffusion Mechanisms: Vacancy type
Vacancy Diffusion:
• atoms exchange with vacancies
• applies to substitutional impurities atoms
• rate depends on:
-- number of vacancies
-- activation energy to exchange.
increasing elapsed time
Chapter 5 - 6
Diffusion Mechanisms: Interstitial type
• Interstitial diffusion – smaller atoms can
diffuse between atoms.
Adapted from Fig. 5.3(b), Callister & Rethwisch 8e.
More rapid than vacancy diffusion
Chapter 5 - 7
Diffusion Simulation
• Simulation of
interdiffusion
across an interface:
This slide contains an animation that requires Quicktime
and a Cinepak decompressor. Click on the message or
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• Rate of substitutional
diffusion depends on:
-- vacancy concentration
-- frequency of jumping.
(Courtesy P.M. Anderson)
Chapter 5 - 8
Processing Using Diffusion
• Case Hardening:
-- Diffuse carbon atoms
into the host iron atoms
at the surface.
-- Example of interstitial
diffusion is a case
hardened gear.
Adapted from
chapter-opening
photograph,
Chapter 5,
Callister &
Rethwisch 8e.
(Courtesy of
Surface Division,
Midland-Ross.)
• Result: The presence of C
atoms makes iron (steel) harder.
In the same manner, N, can be released from NH3, and diffused
into iron to make steels, and/or harden steels.
Carbo – Nitriding can be carried out by using CH4 and NH3
Chapter 5 - 9
Surface Hardening by Carburization / Nitriding
Based on the Cracking of methane gas in carburization or the
decomposition of ammonia (NH3)gas in a controlled chamber
CH4 + H2
2NH3 + H2
3H2 + C
4H2 + 2N
CH4 + H2 = 3H2 + C
Carburizing chamber
can be modified for
optimizing the entire
process
Chapter 5 -
Processing Using Diffusion
• 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
Boron can be used to get P-type Semiconductor
Adapted from Figure 18.27,
Callister
Chapter
5 - 11&
Rethwisch 8e.
Diffusion Modeling: Steady State Case
Diffusion:
Material transport predicated upon the establishment of
Atomic/Molecular Flux
Without Flux, material transport would be non-diffusional as in the
case of Brownian motion of gaseous atoms.
Under Steady State Case:
Flux is assumed proportional to Concentration gradient
Under Non-Steady Case:
Flux gradient is assumed equal to rate of change of Concentration
Chapter 5 -
MODELING DIFFUSION: FLUX
• Flux:
• Directional Quantity
• Flux can be measured for:
--vacancies
--host (A) atoms
--impurity (B) atoms
Chapter 5 - 10
Chapter 5: Atom and Ion Movements in Materials
Figure 5.8
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Chapter 5 - 5 - 14
Diffusion
• How do we quantify the amount or rate of diffusion?
moles (or mass) diffusing
mol
kg
J  Flux 

or
2
surface area time 
cm s m2s
• Measured empirically
– Make thin film (membrane) of known surface area
– Impose concentration gradient
– Measure how fast atoms or molecules diffuse through the
membrane
M l dM
J

At A dt
M=
mass
diffused
J  slope
time
Chapter 5 - 15
Chapter 5: Atom and Ion Movements in Materials
Rate of Diffusion [Fick’s First Law]
 Fick’s first law explains the net flux of atoms:
where
J
D
dc
dx
J = -D dc
dx
flux
diffusivity or diffusion coefficient (cm2/s)
concentration gradient (atoms/(cm3∙cm))
 The negative sign in the equation indicates that the flux of
diffusing species is from higher to lower concentrations.
© 2011 Cengage Learning Engineering. All Rights Reserved.
Chapter 5 - 5 - 16
Diffusion
• Interdiffusion: In an alloy, atoms tend to migrate
from regions of high conc. to regions of low conc.
Initially
After some time
Adapted from
Figs. 5.1 and
5.2, Callister &
Rethwisch 8e.
Chapter 5 - 17
CONCENTRATION PROFILES & FLUX
• Concentration Profile, C(x): [kg/m3]
Cu flux Ni flux
Concentration
of Cu [kg/m3]
• Fick's First Law:
Concentration
of Ni [kg/m3]
Adapted
from Fig.
5.2(c),
Callister 6e.
Position, x
• The steeper the concentration profile,
the greater the flux!
Chapter 5 - 11
Steady-State Diffusion
Rate of diffusion independent of time
dC
Flux proportional to concentration gradient =
dx
Fick’s first law of diffusion
C 1 C1
C2
x1
if linear
x
C2
dC
J  D
dx
x2
dC C C2  C1


dx
x x2  x1
D  diffusion coefficient
Chapter 5 - 19
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 = 110 x10-8 cm2/s
– surface concentrations: C1 = 0.44 g/cm3
C2 = 0.02 g/cm3
Chapter 5 - 20
Example (cont).
• Solution – assuming linear conc. gradient
glove
C1
2
tb 
6D
paint
remover
skin
Data:
D = 110 x 10-8 cm2/s
C1 = 0.44 g/cm3
C2 = 0.02 g/cm3
x2 – x1 = 0.04 cm
C2
x1 x2
J   (110 x 10
-8
dC
C2  C1
J -D
 D
dx
x2  x1
(0.02 g/cm3  0.44 g/cm3 )
g
cm /s)
 1.16 x 10 -5
(0.04 cm)
cm2s
2
Chapter 5 - 21
Diffusion and Temperature
• Diffusion coefficient increases with increasing T.
 Qd 

D  Do exp
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]
Chapter 5 - 22
Diffusion and Temperature
300
600
1000
10-8
1500
D has exponential dependence on T
T(C)
D (m2/s)
Dinterstitial >> Dsubstitutional
C in a-Fe
C in g-Fe
10-14
10-20
0.5
1.0
1.5
Al in Al
Fe in a-Fe
Fe in g-Fe
1000 K/T
Adapted from Fig. 5.7, Callister & Rethwisch 8e. (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.)
Chapter 5 - 23
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?
transform
data
D
Temp = T
1
  and
 T2 
Q
D
 lnD2  lnD1  ln 2   d
D1
R
Qd
lnD2  lnD0 
R
ln D
1/T
Qd
lnD1  lnD0 
R
 1 1
  
 T2 T1 
 1
 
 T1 
Chapter 5 - 24
Example (cont.)
 Qd
D2  D1 exp 
 R
 1 1 
  
 T2 T1 
T1 = 273 + 300 = 573 K
T2 = 273 + 350 = 623 K
D2  (7.8 x 10
11
  41,500 J/mol  1
1 
m /s) exp 



 8.314 J/mol - K  623 K 573 K 
2
D2 = 15.7 x 10-11 m2/s
Chapter 5 - 25
NON STEADY STATE DIFFUSION
• Concentration profile,
C(x), changes
w/ time.
• To conserve matter:
• Fick's First Law:
• Governing Eqn.:
Chapter 5 - 14
Chapter 5: Atom and Ion Movements in Materials
Composition Profile [Fick’s Second Law]
• Fick’s second law
If D is not a function of location, x, and the
concentration (c) of diffusing species
The concentration of diffusing species is a function of both time
and position (or location) C = C(x,t)
In this case Fick’s Second Law is used
© 2011 Cengage Learning Engineering. All Rights Reserved.
Chapter 5 - 5 - 27
Chapter 5: Atom and Ion Movements in Materials
Composition Profile [Fick’s Second Law]
 The mathematical definition of the error function
 Fick’s second law is the technique behind carburization
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Chapter 5 - 5 - 28
Chapter 5: Atom and Ion Movements in Materials
Figure 5.19
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Chapter 5 - 5 - 29
Solutions of Non-Steady State Diffusion
Cx = C0/{2√(πDt)}exp-[x2/4Dt]
Case II: Open Planar THIN-FLIM
Case III: Open Plate Surface Concentration C0 constant
C0
[Cx - C0] / [Cs -C0]={1- erf[ x/2√(Dt)] }
Case IV: Sandwich Plate
C0
[Cx - C0] / [Cs -C0]=1/2{1- erf[ x/2√(Dt)] }
Chapter 5 -
VMSE: Student Companion Site
Diffusion Computations & Data Plots
Chapter 5 - 31
Non-steady State Diffusion
• Copper diffuses into a bar of aluminum.
Surface conc.,
Cs of Cu atoms
bar
pre-existing conc., Co of copper atoms
Cs
Adapted from
Fig. 5.5,
Callister &
Rethwisch 8e.
B.C.
at t = 0, C = Co for 0  x  
at t > 0, C = CS for x = 0 (constant surface conc.)
C = Co for x = 
Chapter 5 - 32
Solution:
C x , t   Co
 1  erf
Cs  Co
C(x,t) = Conc. at point x at
time t
erf (z) = error function
2



z
0
e
y 2
dy
erf(z) values are given in
Table 5.1
 x 


 2 Dt 
CS
C(x,t)
Co
Adapted from Fig. 5.5,
Callister & Rethwisch 8e.
Chapter 5 - 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: use Eqn. 5.5
C( x, t )  Co
 x 
 1  erf 

Cs  Co
 2 Dt 
Chapter 5 - 34
Solution (cont.):
– t = 49.5 h
– Cx = 0.35 wt%
– Co = 0.20 wt%
C( x , t )  Co
 x 
 1  erf 

Cs  Co
 2 Dt 
x = 4 x 10-3 m
Cs = 1.0 wt%
C( x, t )  Co 0.35  0.20
 x 

 1  erf 
  1  erf ( z )
Cs  Co
1.0  0.20
 2 Dt 
 erf(z) = 0.8125
Chapter 5 - 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
z
0.95
0.7970
0.8125
0.8209
Now solve for D
z  0.90
0.8125  0.7970

0.95  0.90 0.8209  0.7970
z  0.93
x
z
2 Dt
D
x2
4 z 2t
3 2
 x2 
(
4
x
10
m)
1h


D 

 2.6 x 10 11 m2 /s
 4z 2t  ( 4)(0.93 )2 ( 49.5 h) 3600 s


Chapter 5 - 36
Solution (cont.):
• To solve for the temperature at
which D has the above value,
we use a rearranged form of
Equation (5.9a);
Qd
T
R(lnDo  lnD)
from Table 5.2, for diffusion of C in FCC Fe
Do = 2.3 x 10-5 m2/s Qd = 148,000 J/mol

T
148,000 J/mol
(8.314 J/mol - K)(ln 2.3 x10 5 m2 /s  ln 2.6 x10 11 m2 /s)
T = 1300 K = 1027ºC
Chapter 5 - 37
Chapter 5: Atom and Ion Movements in Materials
Composition Profile [Fick’s Second Law]
 Limitations to applying the error-function solution
 It is assumed that D is independent of the
concentration of the diffusing species.
 The surface concentration of the diffusing species
(cs) is always constant.
 There are situations under which these conditions
may not be met and hence the concentration profile
evolution will not be predicted by the errorfunction solution.
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Chapter 5 - 5 - 38
Chapter 5: Atom and Ion Movements in Materials
Further Applications of Diffusion
 Conductive ceramics
 Used in products such as oxygen sensors in cars,
touch-screen displays, fuel cells, and batteries.
 Creation of plastic beverage bottles - limit the
occurrence of diffusion for certain species
 For instance, the diffusion of CO2 must be minimized.
 Oxidation of aluminum
 Al2O3 forms a thin oxide coating. The coating does not
have a color (making it invisible) and hinders further
oxidation of the metal.
© 2011 Cengage Learning Engineering. All Rights Reserved.
Chapter 5 - 5 - 39
Chapter 5: Atom and Ion Movements in Materials
Further Applications of Diffusion continued

Thermal barrier coatings for turbine blades
 Ceramic coatings are applied to protect the underlying
metallic alloy from high temperatures.
 Optical fibers and microelectronic components
 Optical fibers are coated with polymeric materials to
prevent diffusion of water molecules.
Coatings and thin films
Used to limit the diffusion of water vapor, oxygen,
or other chemicals.
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Chapter 5 - 5 - 40
Chapter 5: Atom and Ion Movements in Materials
Diffusion during Sintering process:
Figure 5.20
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Chapter 5 - 5 - 41
Chapter 5: Atom and Ion Movements in Materials
Figure 5.23
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Chapter 5 - 5 - 42
Chapter 5: Atom and Ion Movements in Materials
Diffusion and Materials Processing: Sintering
 Grain growth: Movement of grain boundaries,
permitting larger grains to grow at the expense of
smaller grains (Figure 5.23).
 Driving force for grain growth is the reduction in grain
boundary area.
 In normal grain growth, the average grain size
increases steadily and the width of the grain size
distribution is not affected severely.
 In abnormal grain growth, the grain size distribution
tends to become bi-modal.
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Chapter 5 - 5 - 43
Chapter 5: Atom and Ion Movements in Materials
Figure 5.24: In Diffusion Bonding (grain growth)
Chapter 5 - 5 - 44
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Summary
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
Chapter 5 - 45