1. DIFFUSION IN SOLIDS ISSUES TO ADDRESS... • How does diffusion occur?
Gear from case-hardened
steel (C diffusion)
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?

2. Mass transport by atomic motion
Diffusion
Mass transport by atomic motion
Mechanisms
Gases & Liquids – random (Brownian) motion
Solids – vacancy diffusion or interstitial diffusion
Cause (Driving force)
Concentration gradient

3. Diffusion: gases and liquids
• 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.
time (s)
x (mm)
Possible to measure concentration

4. Interdiffusion • Interdiffusion: in alloys, atoms tend to migrate
from regions of large concentration.
Initially
After some time

5. Selfdiffusion
• Selfdiffusion: In an elemental solid, atoms also migrate.
Label some atoms
After some time
This can be observed on metal surface under UHV conditions
with scanning tunneling microscopy

6. Concentration of Vacancies at temp T
Vacancy diffusion or interstitial diffusion
• applies to substitutional impurities
• atoms exchange with vacancies
• rate depends on
(1) number of vacancies
(2) activation energy to exchange.
Concentration of Vacancies at temp T
Vacancy
The two processes have
an activation energy
(eV/atom).
Interstitial

7. Interstitial diffusion
Concentration gradient
More rapid than vacancy diffusion

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.
• Result: The "Case" is
-hard to deform: C atoms
"lock" planes from shearing.
-hard to crack: C atoms put
the surface in compression.

9. Processing using Diffusion
• Doping Silicon with P for n-type semiconductors:
• Process:
1. Deposit P rich
layers on surface.
2. Heat it.
3. Result: Doped
semiconductor
regions.

10. Modeling rate of diffusion: flux
• Flux can be measured for:
- vacancies
- host (A) atoms
- impurity (B) atoms
Empirically determined:
– Make thin membrane of known surface area
– Impose concentration gradient
– Measure how fast atoms or molecules diffuse through the membrane
A = Area of flow

11. Steady-state Diffusion
• Concentration Profile, C(x): [kg/m3]
• Fick's First Law:
D = m2/s
D=diffusion coefficient
• The steeper the concentration profile, the greater the flux!

12. Steady-State Diffusion
• Steady State: concentration profile not changing with time.
Qm,r = Qm,l
Qm,r
Qm,l
C(x)
• Apply Fick's First Law:
• since Qm,l = Qm,r then
The slope, dC/dx, must be constant (i.e., doesn't vary with position)

13. Steady-State Diffusion
Rate of diffusion independent of time C1 C2 x x1 x2

14. Example: C Diffusion in steel plate
• Steel plate at 7000C with geometry shown:
700 C
Knowns:
C1= 1.2 kg/m3 at 5mm
(5 x 10–3 m) below surface.
C2 = 0.8 kg/m3 at 10mm
(1 x 10–2 m) below surface.
D = 3 x10-11 m2/s at 700 C.
In steady-state, how much carbon transfers from the rich to the deficient side?
1 amu = Kg
J= at/m2s = at/cm2s

15. Example: Chemical Protection Clothing
Methylene chloride is a common ingredient of paint removers. Besides being an irritant, it also may be absorbed through skin. When using, 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?
glove C1 C2
skin
paint
remover x1 x2
Data:
D in butyl rubber: D = 110 x10-8 cm2/s
surface concentrations: C1 = 0.44 g/cm3 C2 = 0.02 g/cm3
Diffusion distance: x2 – x1 = 0.04 cm

16. Non-Steady-State Diffusion
Concentration profile
C(x) changes with time
Qm,r
Qm,l
C(x)
• To conserve matter:
• Fick's First Law:
• Governing Eqn.:
• Fick's Second Law

17. Diffusion and Temperature
• Diffusivity increases with T exponentially (so does Vacancy conc.). D = Do
exp æ è ç ö ø ÷ -
EACT R T
= 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]

18. • Experimental Data: T(C) 10-8 D (m2/s) 10-14 10-20 1000 K/T
C in a-Fe
C in g-Fe
Al in Al
Fe in a-Fe
Fe in g-Fe
0.5
1.0
1.5
10-20
10-14
10-8
T(C)
1500
1000
600
300
Callister & Rethwisch 3e.
from E.A. Brandes and G.B. Brook (Ed.) Smithells Metals Reference Book, 7th ed., Butterworth-Heinemann, Oxford, 1992.)

19. Example: Comparing Diffusion in Fe
Is C in fcc Fe diffusing faster than C in bcc Fe?
1000 K/T
D (m2/s)
C in a-Fe
C in g-Fe
Al in Al
Fe in a-Fe
Fe in g-Fe
0.5
1.0
1.5
10-20
10-14
10-8
T(C)
1500
1000
600
300
fcc-Fe: D0=2.3x10–5 (m2/s)
EACT=1.53 eV/atom
T = 900 C D=5.9x10–12(m2/s)
bcc-Fe: D0=6.2x10–7(m2/s)
EACT=0.83 eV/atom
T = 900 C D=1.7x10–10(m2/s)
• FCC Fe has both higher activation energy EACT and D0 (holes larger in FCC).
BCC and FCC phase exist over limited range of T (at varying %C).
Hence, at same T, BCC diffuses faster due to lower EACT.
Cannot always have the phase that you want at the %C and T you want!
which is why this is all important.

20. Vacuum cases 1) Gas permeation through chamber’s walls
The pressure gradient depends on the vacuum inside the chamber
and is a steady state situation, after initial pumpdown
1° Fick’s law
2) Gas permeation from a solid located inside the chamber
The initial concentration of gas si decreasing with time since
the gas molecules are pumped away. (outgassing)
2° Fick’s law
Relative concentration for a sheet of thickness d
and different values of Dt/d2

21. Material outgassing The role for UHV chambers example
Not all materials are good for UHV
At RT, the flux stabilizes after few hours
example
After 20 h, the flux from untreated 3mm thick inox is 1 x 10-7 mbar L/(s cm2)
So for surface area of cm2, the flux is 2 x 10-3 mbar L/s
Pumping speed S = Q/p’ = 250 l/s
p’ = 2x10-3/250 = 8 x10-6 mbar
Inox machined and degreased after 24 h, flux 5 x mbar L/(s cm2)
So for surface area of cm2, the flux is 1 x 10-6 mbar L/s
p’ = 1x10-6/250 = 4 x10-9 mbar
Not included all evaporators, manipulators, etc
In my lab 3 x 10-8 after two days

22. Material outgassing Flux composition: H20 40% CO 20% CO2 12%
So to remove H2O and decrease the outgassing flux from material you need to
BAKE OUT
The entire chamber and instruments for 48 h