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?
Chapter 5 - 1
Diffusion
Diffusion - Mass transport by atomic motion
Migration of atoms from lattice site to lattice site.
Two conditions must be met:
1. There must be an empty adjacent site.
2. Atoms must have sufficient energy to break bonds
with its neighbor atoms.
Mechanisms
• Gases & Liquids – random (Brownian) motion
• Solids – vacancy diffusion or interstitial diffusion
Chapter 5 - 2
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
7e.
Chapter 5 - 3
Inter diffusion
A process
whereby atoms of
one metal difuse
into another.
f01_05_pg110
f02_05_pg111
Chapter 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 Diffusion:
• atoms exchange with vacancies
• applies to substitutional impurities atoms
• rate depends on:
--number of vacancies (vacancy diffusion is a function of the
number of these defects that are present)
--activation energy to exchange.
increasing elapsed time
Chapter 5 - 6
Diffusion Mechanisms
• Interstitial diffusion – smaller atoms can
diffuse between atoms.
Adapted from Fig. 5.3 (b), Callister 7e.
More rapid than vacancy diffusion
Chapter 5 - 7
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 7e.
(Courtesy of
Surface Division,
Midland-Ross.)
• Result: The presence of C
atoms makes iron (steel) harder.
Chapter 5 - 8
Diffusion
• How do we quantify the amount or rate of diffusion?
• Diffusion flux (J): mass or equivalently number of
atoms M diffusing through and perpendicular a unit
cross sectional area per unit of time
moles (or mass) diffusing
atoms
kg
J  Flux 

or 2
2
surface area time 
ms
ms
• 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
Chapter 5 - 9
Steady-State Diffusion
Rate of diffusion independent of time
dC
Flux proportional to concentration gradient =
dx
Fick’s first law of diffusion
C1 C1
C2
x1
x
C2
dC
J  D
dx
x2
dC C C2  C1
if linear


dx
x
x2  x1
D  diffusion coefficient
Chapter 5 - 10
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
11
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 - 12
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/cm 3  0.44 g/cm 3 )
g
cm /s)
 1.16 x 10-5
(0.04 cm)
cm2s
2
Chapter 5 - 13
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 - 14
Diffusion and Temperature
300
600
1000
10-8
1500
D has exponential dependence on T
D (m2/s)
T(C)
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 7e. (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 - 15
t02_05_pg119
Chapter 5 - 16
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 - 17
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 - 18