Diffusion is the mass transport of atoms in solids by atomic motion. There are two main mechanisms: vacancy diffusion, where atoms exchange with vacancies in the lattice, and interstitial diffusion, where smaller atoms diffuse between lattice sites. The rate of diffusion depends on factors like temperature, activation energy, and the concentration gradient. Fick's laws can be used to calculate the flux of diffusing atoms and model diffusion processes. Controlling diffusion is important for applications like alloy processing and semiconductor doping.

1. Chapter 5 -
1
ISSUES TO ADDRESS...
• How does diffusion occur?
• Why is it an important part of processing?
• How can estimate the rate of diffusion ?
• How does diffusion depend on:
Structure and temperature ?
Chapter 5
Diffusion in Solids
• What is diffusion ?

2. Chapter 5 -
2
Diffusion
Diffusion: Mass transport by atomic motion
From regions of High Concentration to regions of Low Conc.
Mechanisms
• Gases & Liquids – random molecular motion
• In Solids:
• Every Atom is vibrating  Vibrational Energy
 Atoms are in constant motion: migration from lattice site to another
• Types in Solids:
1) Vacancy diffusion
- Substitutional
- Self
2) Interstitial diffusion (more rapid)

3. Chapter 5 -
3
Diffusion Mechanisms
Vacancy Diffusion:
• atoms exchange with vacancies
• applies to substitutional impurities atoms
• Rate depends on:
--number of vacancies
--activation energy to exchange.
With time

4. Chapter 5 -
4
• Interdiffusion: e.g. in an alloy:
atoms tend to migrate from regions of high conc. to regions of low conc.
Types of Diffusion
After some timeInitially

5. Chapter 5 -
5
• Self-diffusion: In an elemental solid:
Atoms also migrate.
Label some atoms After some time
Types of Diffusion
A
B
C
D
A
B
C
D
Random Atomic Motion

6. Chapter 5 -
6
Types of Diffusion
Interstitial diffusion:
smaller atoms can diffuse between atoms.
Interstitial diffusion is faster than vacancy diffusion

7. Chapter 5 -
7
Process: Case Hardening
Starting with low carbon gear (iron)
Subject it to carbon medium
Diffuse carbon atoms into the host iron atoms at the surface.
This is an example of interstitial diffusion (case hardened gear).
Result
The presence of C atoms
makes iron to become steel
i.e. harder.
Application - Processing Using Diffusion

8. Chapter 5 -
8
• Doping silicon with phosphorus for n-type semiconductors:
• Process:
3. Result: Doped
semiconductor
regions.
silicon
Another Application
magnified image of a computer chip
0.5mm
light regions: Si atoms
light regions: Al atoms
2. Heat it.
1. Deposit P rich
layers on surface.
silicon

9. Chapter 5 -
9
Diffusion Calculations
( )( ) sm
kg
or
scm
mol
timeareasurface
diffusingmass)(ormoles
Flux 22
=≡≡J
Estimation of the rate of diffusion?
Steady State Diffusion: Rate is independent of time
Flux is proportional to concentration gradientgradient =
dx
dC
C1
C2
x
C1
C2
x1 x2
Fick’s first law of diffusion
dx
dC
DJ −=
12
12linearif
xx
CC
x
C
dx
dC
−
−
=
∆
∆
≅
D ≡ diffusion coefficientcoefficient

10. Chapter 5 -
10
Diffusion Experiments
Experiment:
– Make a thin film (membrane) of known surface area
– Impose concentration gradient (dc/dx)
– Measure how fast atoms or molecules diffuse through the membrane:
i.e. measure moles or mass (M) passes versus time
Then, calculate:
( )( ) sm
kg
or
scm
mol
timeareasurface
diffusingmass)(ormoles
Flux: 22
=≡≡JAgain
dt
dM
A
l
At
M
J ==
M =
mass
diffused
time
J ∝ slope
For obtaining rate of diffusion Empirically

11. Chapter 5 -
11
Application and Example
Chemical Protective Clothing (CPC)
Background of the Example:
• 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.
Example: Diffusion of Methylene chloride through Gloves
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

12. Chapter 5 -
12
scm
g
10x16.1
cm)04.0(
)g/cm44.0g/cm02.0(
/s)cm10x110(
2
5-
33
28-
=
−
−=J
Example-Solution
12
12-
xx
CC
D
dx
dC
DJ
−
−
−≅=
D
tb
6
2

=
glove
C1
C2
skinpaint
remover
x1 x2
Assuming linear concentration gradient – Steady State
D = 110x10-8
cm2
/s
C2 = 0.02 g/cm3
C1 = 0.44 g/cm3
x2 – x1 = 0.04 cm
Data:

13. Chapter 5 -
13
Parameters affecting Diffusion Coefficient
Temperature
Diffusion Coefficient depends on
1) Type and size of diffusing species (& host materials)
2) 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
A plot ln D versus 1/T ?
Slope = -Qd/R
ln D
1/T

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

15. Chapter 5 -
15






−=





−=
1
01
2
02
1
lnlnand
1
lnln
TR
Q
DD
TR
Q
DD dd






−−==−∴
121
2
12
11
lnlnln
TTR
Q
D
D
DD d
transform
data
D
Temp = T
ln D
1/T
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?

16. Chapter 5 -
16
Example (cont.)












−
−
= −
K573
1
K623
1
K-J/mol314.8
J/mol500,41
exp/s)m10x8.7( 211
2D












−−=
12
12
11
exp
TTR
Q
DD d
T1 = 273 + 300 = 573K
T2 = 273 + 350 = 623K
D2 = 15.7 x 10-11
m2
/s

17. Chapter 5 -
17
Non-steady State Diffusion
Optional
• 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
I.C. at t = 0, C = Co for 0 ≤ x ≤ ∞
B.C. at t > 0, C = CS for x = 0 (const. surf. conc.)
C = Co for x = ∞
For solving: we need boundary and initial conditions

18. Chapter 5 -
18
Solution of Partial Differential Equation
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
CC
Ct,xC
os
o
2
erf1
dye y
z 2
0
2 −
∫π
=

19. Chapter 5 -
19
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