1. Chapter 6
Diffusion in
Solids

2. Diffusion in solids In general Diffusion in solids
Example of diffusion…
1. Movement of smoke particles in air:
-- very fast.
2. Movement of dye in water:
-- relatively slow.
3. Movement of atom in solid (solid state reactions):
-- very restricted movement due to bonding.
In general
Diffusion is a process by which:
-- a matter is transported through
another matter. OR
-- atoms are moved through solids.
Occurs at an elevated temperature, where
atoms tend to move from high conc. region
to low conc. region. .
Atoms diffuse in solids if:
1. vacancies or other crystal defects are
present.
2. there is enough activation energy.
- Atoms move into the vacancies present.
- Normally, at higher temperature;
-- more vacancies are created.
-- diffusion rate is higher.
Diffusion in solids
type (classification)
mechanism (method)
state (condition)
industrial application
impurity diffusion
substitutional
steady state diffusion
non-steady state diffusion
self-diffusion
interstitial
case hardening
doping silicon
Example:
If atom ‘A’ has sufficient activation energy, it moves into the vacancy.
diffusing species
solid
heat
Diffusion process…
Activation energy of diffusion
Activation energy to form a vacancy
Activation energy to move a atom = +

3. Diffusion in solids type (classification) Mechanism (method)
- atoms migrate itself in solid.
-- change its position.
diffusion occurs in 2 different solids.
atoms migrate between 2 solids.
Initially
After some time
Initially
After some time
self-diffusion A B C D A B C D
inter-diffusion
atoms move from one interstitial
site to another.
the diffusing atoms that move
must be much smaller than the
host atom.
i.e: carbon interstitially diffuses
into α-iron -iron (FCC).
Example: Cu-Ni alloy
1. Join 2 metals (as diffusion couple).
2. Heat (below Tm) & cool.
3. After heat-treated, the couple:
-- contains alloyed diffusion region.
-- occurs diffusion process.
interstitial diffusion
Interstitial diffusion more rapid than vacancy diffusion
- atoms exchange with vacancies.
applies to substitutional impurities
atoms.
- Rate of diffusion depends on:
-- number of vacancies (vacancy
concentration).
. -- activation energy to exchange
(frequency of jumping).
Mechanism (method)
vacancy diffusion

4. Diffusion in solids  dC J  dx @ dC J = -D dx
state (condition)
Steady state diffusion
rate of diffusion independent of time
does not change with time).
concentration of diffusing species
depends with position.
Example: Chemical Protective Clothing (CPC)
Rate of flux, J
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?
Given: D in butyl rubber = 110 x10-8 cm2/s
surface concentrations:
How do we quantify the amount or rate of diffusion?
concentration gradient
rate of diffusion 
J = moles (or mass) diffusing J dC dx 
surface area x time C1 C2 x x1 x2
C1 = 0.44 g/cm3
C2 = 0.02 g/cm3
mol
cm2s @ kg
m2s
Answer:
unit:
assuming linear conc. gradient
glove C1 C2
skin
paint
remover x1 x2
Measured empirically:
make thin film (membrane) of known surface area.
impose concentration gradient.
measure how fast atoms or molecules diffuse through the membrane.
apply Fick’s 1st law J dC dx
= -D
D  diffusion coefficient
C  surface concentration
x  position J dC dx
= -D
C2 – C1
x2 – x1
(0.02 g/cm3 – 0.44 g/cm3)
(0.04 cm – 0)
= -(110x10-8 cm2/s) M t
J  slope
J = 1.16 x 10-5 g/cm2s

5. Diffusion in solids @ J = moles (or mass) diffusing state (condition)
Non-steady state diffusion
occur for most practical diffusion
situations.
rate of diffusion & concentration
gradient vary with time & position.
concentration of diffusing species is
a function of both time and position.
C = C(x,t)
- apply Fick’s 2nd law
- boundary conditions (B. C)
Rate of flux, J
Position, x Co
C(x,t) Cs
time = t2
time= t1
time = to x
How do we quantify the amount or rate of diffusion?
J = moles (or mass) diffusing
surface area x time
at t = 0, C = Co for 0  x  
at t > 0, C = CS for x = 0 (constant surface conc.)
C = C(x,t) for x = xt = x2
C = Co for x = 
Example:
mol
cm2s @ kg
m2s
unit:
Copper diffuses into a bar of aluminum. Co
Al bar
before diffusion CS
Measured empirically:
make thin film (membrane) of known surface area.
impose concentration gradient.
measure how fast atoms or molecules diffuse through the membrane.
- B. C are related to equation:
during diffusion
C(x,t)
after some time x1 x2
where; M t
J  slope
Co = pre-existing concentration
CS = surface concentration
erf (z) = error function
Note: erf(z) values are given in Table 6.1
C(x,t) = concentration at depth x after
time t

6. Diffusion in solids state (condition) Non-steady state diffusion
Table 6.1: Tabulation of Error Function Values z
erf (z)
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
0.4755
1.1
0.8802
2.6
0.9998
0.50
0.5205
1.2
0.9103
2.8
0.9999

7. Diffusion in solids state (condition) Non-steady state diffusion
Example:
Consider one such alloy that initially has a uniform carbon concentration of 0.25 wt% and is to be treated at 950oC. If the concentration of carbon at the surface is suddenly brought to & maintained at 1.20 wt%, how long will it take to achieve a carbon content of 0.80 wt% at a position 0.5 mm below the surface? The D for carbon in iron at this temperature is 1.6 x m2/s.
Answer:
Given;
An interpolation is necessary as follows
x = 0.5 mm = 5 x 10-4 m
C(x,t) = 0.80 wt% C
Cs = 1.20 wt% C
Co = 0.25 wt% C
D = 1.6 x m2/s
z – = –
0.40 – –
z = 0.392
Therefore,
x = z
2 (Dt)1/2
C(x,t) – Co = 1 – erf (z)
Cs – Co
5 x 10-4 m = 0.392
2 [(1.6 x m2/s)(t)]1/2
0.80 – 0.25 = 1 – erf (z)
t = s = 7.1 h
1.20 – 0.25
 erf (z) =
use table 6.1 to find z value z
erf (z)
0.35
0.40
0.3794
0.4210
0.4284

8. Factors that influence diffusion
Diffusion in solids
Factors that influence diffusion
Temperature
Diffusing species
Smaller diffusing species, diffusion faster.
diffusion coefficient, D increases with increasing T.
apply the formula: æ - Qd R T ö
D vs T for several metals… D = Do
exp ç è ø
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
= 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
- rearrange;
ln D = ln Do – Qd
R (T) D
interstitial
>> D
substitutional
C in a-Fe
C in g-Fe
Al in Al
Fe in a-Fe
Fe in g-Fe
The above formula can be used for steady state & non-steady state diffusion
Note: Diffusion data values are given in Table 6.2 8

9. Factors that influence diffusion
Diffusion in solids
Factors that influence diffusion
Temperature
Diffusing species
Table 6.2: Tabulation of diffusion data
Diffusing Species
Host Metal
Do (m2/s)
Activation Energy Qd
Calculated Values
kJ/mol
eV/atom
T (oC)
D (m2/s) Fe
α-Fe (BCC)
2.8 x 10-4
251
2.60
500
3.0 x 10-21
900
1.8 x 10-15
-Fe (FCC)
5.0 x 10-5
284
2.94
1.1 x 10-17
1100
7.8 x 10-16 C
α-Fe
6.2 x 10-7 80
0.83
2.4 x 10-12
1.7 x 10-10
-Fe
2.3 x 10-5
148
1.53
5.9 x 10-12
5.3 x 10-11 Cu
7.8 x 10-5
211
2.19
4.2 x 10-19 Zn
2.4 x 10-5
189
1.96
4.0 x 10-18 Al
2.3 x 10-4
144
1.49
4.2 x 10-14
6.5 x 10-5
136
1.41
4.1 x 10-14 Mg
1.2 x 10-4
131
1.35
1.9 x 10-13 Ni
2.7 x 10-5
256
2.65
1.3 x 10-22 9

10. Factors that influence diffusion
Diffusion in solids
Factors that influence diffusion
Temperature
Diffusing species
Example 1:
At 300ºC the diffusion coefficient and activation energy for Cu in Si are 7.8 x m2/s and 41.5 kJ/mol. What is the diffusion coefficient at 350ºC?
Answer: æ - Qd R T ö
transform data D
Temp = T
ln D
1/T D = Do
exp ç è ø
T1 = = 573 K
T2 = = 623 K
D2 = 15.7 x m2/s 10

11. Factors that influence diffusion
Diffusion in solids
Factors that influence diffusion
Temperature
Diffusing species
Example 2: (for non-steady state diffusion application)
An FCC iron-carbon alloy initially containing 0.20 wt% C is carburized at an elevated temperature and in an atmosphere that gives a surface carbon concentration constant at 1.0 wt%. If after 49.5 h the concentration of carbon is 0.35 wt% at a position 4.0 mm below the surface, determine the temperature at which the treatment was carried out.
Answer:
z = 0.93
Given;
t = 49.5 h
x = 4 x 10-3 m
C(x,t) = 0.35 wt%
Cs = 1.0 wt%
Co = 0.20 wt%
Now solve for D
from Table 6.2, for diffusion of Carbon in FCC Fe
Do = 2.3 x 10-5 m2/s Qd = 148,000 J/mol
 erf(z) =
We must now determine from Table 6.1 the value of z for which the error function is An interpolation is necessary as follows  z
erf(z) z
T = 1300 K = 1027°C

12. Factors that influence diffusion
Diffusion in solids
Factors that influence diffusion
Temperature
Diffusing species
Example 3:
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 breakthrough time (tb), i.e., how long could the gloves be used before methylene chloride reaches the hand?
Given;
D in butyl rubber = 110 x10-8 cm2/s
Answer:
assuming linear conc. gradient
Breakthrough time = tb
glove C1 C2
skin
paint
remover x1 x2
 Time required for breakthrough ca. 4 min

13. Industrial application
Diffusion in solids
sliding and rotating parts needs to have hard
surfaces.
- these parts are usually machined with low
carbon steel as they are easy to machine.
- their surface is then hardened by carburizing.
steel parts are placed at elevated temperature
(927oC) in an atmosphere of hydrocarbon gas
(CH4).
carbon diffuses into iron (steel) surface and fills
interstitial space to make it harder.
Case hardening
Low carbon
steel part
Diffusing carbon
atoms
Industrial application
Doping silicon with phosphorus, P for n-type semiconductors.
impurities (P) are made to diffuse into silicon wafer to change
its electrical characteristics.
- used in integrated circuits.
silicon wafer is exposed to vapor of impurity at 1100oC in a
quartz tube furnace.
the concentration of impurity at any point depends on
depth and time of exposure.
Doping silicon wafer
Doping process…
3. Result: Doped semiconductor
regions.
1. Deposit P rich
layers on surface.
2. heat
silicon
silicon

14. Summary Diffusion FASTER for... Diffusion SLOWER 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 14

15. End of Chapter 6 15