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
• The phenomenon of material transport by
which atomic diffusion occurs.
Chapter 5 -2
DIFFUSION DEMO
• Glass tube filled with water.
• At time t = 0, add some drops of ink to one end
of the tube.
• Measure the diffusion distance, x, over some time.
• Compare the results with theory.
Chapter 5 -3
DIFFUSION: THE PHENOMENA (1)
Interdiffusion:
•Atoms of one metal diffuse into another.
•In an alloy, atoms tend to migrate from regions of large
concentration.
Initially
After some time
Adapted
from Figs.
5.1 and 5.2,
Callister 6e.
100%
0
Concentration Profiles
Chapter 5 -4
DIFFUSION: THE PHENOMENA (2)
Self-diffusion:
•All atoms exchanging positions are of the same type.
•In an elemental solid, atoms also migrate.
Label some atoms
After some time
C
A
D
B
Chapter 5 -5
DIFFUSION MECHANISMS
For an atom to make a move, two conditions must
be met
1. there must be and empty adjacent site
2. the atom must have sufficient energy to break
bonds with its neighbor atoms and then cause
some lattice distortion during the displacement
Chapter 5 -6
DIFFUSION SIMULATION
• Simulation of
interdiffusion
across an interface:
• Rate of substitutional
diffusion depends on:
--vacancy concentration
--frequency of jumping.
(Courtesy P.M. Anderson)
Chapter 5 -7
DIFFUSION MECHANISMS
Substitutional Diffusion:
• applies to substitutional impurities
• atoms exchange with vacancies
• rate depends on:
--number of vacancies
--activation energy to exchange.
Chapter 5 -8
DIFFUSION MECHANISMS
Vacancy Diffusion:
• involves the interchange of an atom from a normal
lattice position
Chapter 5 -9
DIFFUSION MECHANISMS
Interstitial Diffusion:
• involves the atoms that migrate from an interstitial
position to a neighboring one that is empty
Position of interstitial
atom before diffusion
Position of interstitial
atom after diffusion
Chapter 5 -10
INTERSTITIAL SIMULATION
• Applies to interstitial
impurities.
• More rapid than
vacancy diffusion.
• Simulation:
--shows the jumping of a
smaller atom (gray) from
one interstitial site to
another in a BCC
structure. The
interstitial sites
considered here are
at midpoints along the
unit cell edges.
(Courtesy P.M. Anderson)
Chapter 5 -11
PROCESSING USING DIFFUSION (1)
• Case Hardening:
--Diffuse carbon atoms
into the host iron atoms
at the surface.
--Example of interstitial
diffusion is a case
hardened gear.
Fig. 5.0,
Callister 6e.
(Fig. 5.0 is
courtesy of
Surface
Division,
MidlandRoss.)
• Result: The "Case" is
--hard to deform: C atoms
"lock" planes from shearing.
--hard to crack: C atoms put
the surface in compression.
Chapter 5 -12
PROCESSING USING DIFFUSION (2)
• Doping Silicon with P for n-type semiconductors:
• Process:
1. Deposit P rich
layers on surface.
silicon
Fig. 18.0,
2. Heat it.
Callister 6e.
3. Result: Doped
semiconductor
regions.
silicon
Chapter 5 -13
DOPING
• What is Doping?
• Doping is the process of adding some
impurity atoms in the semi conductor.
These impurity atoms are known as
dopants. After addition of these dopants
some of the properties of the conductors can
be changed according to our need.
Chapter 5 -14
DOPING
Basic conditions that are required for the doping
process are given below:
1. The atom which is to be doped in the crystal
must be placed at the position same as that of the
position of the semiconductor atom.
2. There should be no distortion in the crystal after
the insertion of the dopants.
3. The size of the dopants should be exactly same as
that of the size of the atom of the crystal.
4. In a crystal the percentage of doping should not
be more than one percent.
Chapter 5 -15
DOPING
Some basic doping techniques of doping are
shown below:
1. First we have to heat up the semiconductor
crystal. The heating must be at the place where
the dopants are present in the atmosphere. After
heating the diffusion of the dopants will take
place in the lattice site of the crystal.
2. In the second method of doping, the
semiconductor is bombarded with the ions of the
semiconductor itself. Then the dopants are
embedded.
Chapter 5 -16
MODELING DIFFUSION: FLUX
• Flux:
• Directional Quantity
• Flux can be measured for:
--vacancies
--host (A) atoms
--impurity (B) atoms
Chapter 5 -17
CONCENTRATION PROFILES & FLUX
• Concentration Profile, C(x): [kg/m3]
Cu flux Ni flux
Concentration
of Cu [kg/m3]
Concentration
of Ni [kg/m3]
Adapted
from Fig.
5.2(c),
Callister 6e.
• Fick's First Law:
Position, x
• The steeper the concentration profile,
the greater the flux!
Chapter 5 -18
STEADY STATE DIFFUSION
• Steady State: the concentration profile doesn't
change with time.
dC
• Apply Fick's First Law: J x = -D
dx
æ dC ö
æ dC ö
=ç ÷
• If Jx)left = Jx)right , then ç ÷
è dx ø left è dx ø right
• Result: the slope, dC/dx, must be constant
(i.e., slope doesn't vary with position)!
Chapter 5 -19
SAMPLE PROBLEM S1
A plate of iron is exposed to a carburizing (carbonrich) atmosphere on one side and decarburizing
(carbon-deficient) atmosphere on the other side at
700oC (1300oF). If a condition of steady state is
achieved, calculate the diffusion flux of carbon
through the plate if the concentration of carbon at
positions of 5 and 10 mm (5 x10-3 and 10-2 m)
beneath the carburizing surface are 1.2 and 0.8
kg/m3, respectively. Assume a diffusion
coefficient of 3 x 10-11 m2/s at this temperature
Chapter 5 -20
EX: STEADY STATE DIFFUSION
• Steel plate at
700C with
geometry
shown:
Adapted
from Fig.
5.4,
Callister 6e.
• Q: How much
carbon transfers
from the rich to
the deficient side?
Chapter 5 -21
NON STEADY STATE DIFFUSION
• Concentration profile,
C(x), changes
w/ time.
• To conserve matter:
• Fick's First Law:
• Governing Eqn.:
Chapter 5 -22
TABULATION OF ERROR FUNCTION VALUES
z
erf (z)
z
erf (z)
z
erf (z)
0
0
0.55
0.5633
1.3
0.9340
0.025
0.0282
0.60
0.6039
1.4
0.9523
0.05
0.0564
0.65
0.6420
1.5
0.9661
0.10
0.1125
0.70
0.6778
1.6
0.9763
0.15
0.1680
0.75
0.7112
1.7
0.9838
0.20
0.2227
0.80
0.7421
1.8
0.9891
0.25
0.2763
0.85
0.7707
1.9
0.9928
0.30
0.3286
0.90
0.7970
2.0
0.9953
0.35
0.3794
0.95
0.8209
2.2
0.9981
0.40
0.4284
1.0
0.8427
2.4
0.9993
0.45
0.1125
1.1
0.8802
2.6
0.9998
0.50
0.5205
1.2
0.9103
2.8
0.9999
Chapter 5 -23
SAMPLE PROBLEM U1
Consider one such alloy that initially has a uniform
carbon concentration of 0.25 wt% and is to be
treated at 950oC (1750oF). If the concentration of
carbon at the surface is suddenly brought to and
maintained at 1.20 wt%, how long will it take to
achieve a carbon content of 0.80 wt% at a position
of 0.5 mm below the surface? The diffusion
coefficient for carbon in iron at this temperature is
1.6 x 10-11 m2/s; assume that the steel piece is
semi-infinite.
Chapter 5 -24
SAMPLE PROBLEM U2
The diffusion coefficients for copper in
aluminum at 500 and 600oC are 4.8 x10-14
and 5.3 x 10-13 m2/s, respectively.
Determine the approximate time at 500oC
that will produce the same diffusion result
(in terms of concentration of Cu at some
specific point in Al) as a 10-h heat
treatment at 600oC.
Chapter 5 -25
EX: NON STEADY STATE DIFFUSION
• Copper diffuses into a bar of aluminum.
Cs
C(x,t)
to t1
Co
t3
t2
Adapted from
Fig. 5.5,
Callister 6e.
position, x
• General solution:
"error function"
Values calibrated in Table 5.1, Callister 6e.
Chapter 5 -26
PROCESSING QUESTION
• Copper diffuses into a bar of aluminum.
• 10 hours at 600C gives desired C(x).
• How many hours would it take to get the same C(x)
if we processed at 500C?
Key point 1: C(x,t500C) = C(x,t600C).
Key point 2: Both cases have the same Co and Cs.
• Result: Dt should be held constant.
• Answer:
Note: values
of D are
provided here.
Chapter 5 -27
DIFFUSION DEMO: ANALYSIS
• The experiment: we recorded combinations of
t and x that kept C constant.
æ
ö
C(xi , ti )- C o
x
i ÷
= 1- erf çç
= (constant here)
÷
Cs - Co
è 2 Dti ø
• Diffusion depth given by:
Chapter 5 -28
DATA FROM DIFFUSION DEMO
• Experimental result: x ~ t0.58
• Theory predicts x ~ t0.50
• Reasonable agreement!
Chapter 5 -29
DIFFUSION AND TEMPERATURE
• Diffusivity increases with T.
• Experimental Data:
D has exp. dependence on T
Recall: Vacancy does also!
Dinterstitial >> Dsubstitutional
Cu in Cu
C in a-Fe
Al in Al
C in g-Fe
Fe in a-Fe
Fe in g-Fe
Zn in Cu
Adapted from Fig. 5.7, Callister 6e. (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 -30
SUMMARY:
STRUCTURE & DIFFUSION
Diffusion FASTER for...
Diffusion SLOWER for...
• open crystal structures
• close-packed structures
• lower melting T materials
• higher melting T materials
• materials w/secondary
bonding
• materials w/covalent
bonding
• smaller diffusing atoms
• larger diffusing atoms
• cations
• anions
• lower density materials
• higher density materials
Chapter 5 -31