Material Science topic

1. Chapter 5 - 1
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: Diffusion

2. Chapter 5 - 2
After studying this chapter you should be able to do the following:
1. Name and describe the two atomic mechanisms of diffusion.
2. Distinguish between steady-state and non steady state diffusion.
3.(a) Write Fick’s first and second laws in equation form and define all
parameters.
(b) Note the kind of diffusion for which each of these equations is normally
applied.
4. Write the solution to Fick’s second law for diffusion into a semi-infinite solid
when the concentration of diffusing species at the surface is held constant. Define
all parameters in this equation.
Learning Objectives

3. Chapter 5 - 3
Diffusion
Diffusion - Mass transport by atomic motion
Many reactions and processes that are important in the
treatment of materials rely on the transfer of mass either
within a specific solid (ordinarily on a microscopic level) or
from a liquid, a gas, or another solid phase.This is necessarily
accomplished by diffusion.
Mechanisms
Gases & Liquids – random
(Brownian) motion
Solids – vacancy diffusion or
interstitial diffusion

4. Chapter 5 - 4
Ni
Cu
Ni
Cu
Cu Ni

5. Chapter 5 - 5
• Interdiffusion: In an alloy, atoms tend to migrate
from regions of high conc. to regions of low conc.
Initially
Adapted from
Figs. 5.1 and
5.2, Callister &
Rethwisch 8e.
Diffusion
After some time
This process, whereby atoms of one metal diffuse into another, is termed inter diffusion,
or impurity diffusion.

6. Chapter 5 - 6
• Self-diffusion: In an elemental solid, atoms
also migrate.
Label some atoms
Diffusion
A
B
C
D
After some time
A
B
C
D
Diffusion also occurs for pure metals, but all atoms exchanging
positions are of the same type; this is termed self-diffusion.

7. Chapter 5 - 7
DIFFUSION MECHANISMS
From an atomic perspective, diffusion is just the stepwise migration
of atoms from lattice site to lattice site. In fact, the atoms in solid
materials are in constant motion, rapidly changing positions. For an
atom to make such a move, two conditions must be met:
(1) there must be an empty adjacent site, and
(2) the atom must have sufficient energy to break bonds with its
neighbor atoms and then cause some lattice distortion during the
displacement.This energy is vibrational in nature.
At a specific temperature some small fraction of the total number
of atoms is capable of diffusive motion, by virtue of the
magnitudes of their vibrational energies. This fraction increases
with rising temperature.

8. Chapter 5 - 8
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

9. Chapter 5 - 9
Diffusion Mechanisms
• Interstitial diffusion – smaller atoms can
diffuse between atoms.
More rapid than vacancy diffusion
Adapted from Fig. 5.3(b), Callister & Rethwisch 8e.

10. Chapter 5 - 10
Diffusion
• How do we quantify the amount or rate of diffusion?
   s
m
kg
or
s
cm
mol
time
area
surface
diffusing
mass)
(or
moles
Flux
2
2



J
J  slope
dt
dM
A
l
At
M
J 

M =
mass
diffused
time
• Measured empirically
– Make thin film (membrane) of known surface area
– Impose concentration gradient
– Measure how fast atoms or molecules diffuse through the
membrane

11. Chapter 5 - 11
Steady-State Diffusion
dx
dC
D
J 

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
1
2
1
2
linear
if
x
x
C
C
x
C
dx
dC






The negative sign in this expression indicates that the direction of diffusion is down the concentration
gradient, from a high to a low concentration.

12. Chapter 5 - 12
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

13. Chapter 5 - 13
s
cm
g
10
x
16
.
1
cm)
04
.
0
(
)
g/cm
44
.
0
g/cm
02
.
0
(
/s)
cm
10
x
110
(
2
5
-
3
3
2
8
-




J
Example (cont).
1
2
1
2
-
x
x
C
C
D
dx
dC
D
J





D
tb
6
2


glove
C1
C2
skin
paint
remover
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:

14. Chapter 5 - 14
Non-steady State Diffusion
• The concentration of diffusing species is a function of
both time and position C = C(x,t)
• In this case Fick’s Second Law is used
2
2
x
C
D
t
C





Fick’s Second Law

15. Chapter 5 - 15
Non-steady State Diffusion
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 = 
• Copper diffuses into a bar of aluminum.
pre-existing conc., Co of copper atoms
Surface conc.,
C of Cu atoms bar
s
Cs

16. Chapter 5 - 16
Solution:
C(x,t) = Conc. at point x at
time t
erf (z) = error function
erf(z) values are given in
Table 5.1
CS
Co
C(x,t)
 










Dt
x
C
C
C
t
,
x
C
o
s
o
2
erf
1
dy
e y
z 2
0
2 



Adapted from Fig. 5.5,
Callister & Rethwisch 8e.
Gaussian error
function,

17. Chapter 5 -
FACTORS THAT INFLUENCE DIFFUSION
• Diffusing Species
• The magnitude of the diffusion coefficient D is
indicative of the rate at which atoms diffuse.
• Temperature
Diffusion coefficient increases with increasing T.
17

18. Chapter 5 - 18
Diffusion and Temperature
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.)
D has exponential dependence on T
Dinterstitial >> Dsubstitutional
C in a-Fe
C in g-Fe
Al in Al
Fe in a-Fe
Fe in g-Fe
1000K/T
D (m2/s)
0.5 1.0 1.5
10-20
10-14
10-8
T(C)
1500
1000
600
300

19. Chapter 5 - 19
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]
D
Do
Qd
R
T

20. Chapter 5 -
• The activation energy may be thought of as that energy required
to produce the diffusive motion of one mole of atoms.
20
D  Do exp







Qd
RT

21. Chapter 5 -
DIFFUSION IN SEMICONDUCTING
MATERIALS
• One technology that applies solid-state diffusion is the
fabrication of semiconductor integrated circuits (ICs).Each
integrated circuit chip is a thin square wafer having dimensions
on the order of 6 mm by 6 mm by 0.4 mm; furthermore, millions
of interconnected electronic devices and circuits are embedded
in one of the chip faces. Single-crystal silicon is the base
material for most ICs. In order for these IC devices to function
satisfactorily, very precise concentrations of an impurity (or
impurities) must be incorporated into minute spatial regions in a
very intricate and detailed pattern on the silicon chip; one way
this is accomplished is by atomic diffusion.
21

22. Chapter 5 -
OTHER DIFFUSION PATHS
• Atomic migration may also occur along
dislocations, grain boundaries, and external
surfaces.These are sometimes called “short-
circuit” diffusion paths inasmuch as rates are
much faster than for bulk diffusion. However,
in most situations short-circuit contributions to
the overall diffusion flux are insignificant
because the cross sectional areas of these
paths are extremely small.
22

23. Chapter 5 - 23
Diffusion FASTER for...
• open crystal structures
• materials w/secondary
bonding
• smaller diffusing atoms
• lower density materials
Diffusion SLOWER for...
• close-packed structures
• materials w/covalent
bonding
• larger diffusing atoms
• higher density materials
Summary