Chapter 5: Thermally Activated
Processes & Diffusion
ME 2105
Dr. R. Lindeke
DIFFUSION is observed to occur:
IN LIQUIDS: ink in water, etc.
IN GASES: swamp gas in air, exhaust fumes
into Smog, etc.
Carburization
and IN SOLIDS
Surface coating
Processing Using Diffusion
• Case Hardening:
• Diffuse carbon atoms
into the host iron atoms
at the surface.
• Example of interstitial
diffusion to produce a
surface (case) hardened
gear.
Courtesy of
Surface Division,
Midland-Ross.
The carbon atoms (interstitially) diffuse from a carbon rich
atmosphere into the steel thru the surface.
Result: The presence of C atoms makes the iron (steel)
surface harder.
Typical Arrhenius plot of data compared with Equation 5.2.
This behavior controls most molecular movement driven
behavior (like vacancy formation or diffusion). The slope
equals −Q/R, and the intercept (at 1/T = 0) is ln(C)
rate  Ce
where:
Q
RT
C is a constant (rate w/o temperature)
Q is activation energy
T is absolute temperature
Note: this is a “Semi-log”
plot
Process path showing how an atom must overcome an
activation energy, q, to move from one stable position to
a similar adjacent position.
And it is this “Activation” Energy barrier – which we can model as in Ex. 5.1
– that determines the “Rate Limiting Step” in any process …
The overall thermal expansion (ΔL/L) of aluminum is measurably greater than
the lattice parameter expansion (Δa/a) at high temperatures because vacancies
are produced by thermal agitation(a). A semilog (Arrhenius-type) plot of
ln(vacancy concentration)
(b) versus 1/T based on the data of part (a). The slope of the plot (−Ev/k)
indicates that 0.76 eV of energy is required to create a single vacancy in
the aluminum crystal structure. (From P. G. Shewmon, Diffusion in Solids,
McGraw-Hill Book Company, New York, 1963.)
Atomic migration (“Diffusion”) occurs by a mechanism of
vacancy migration. Note that the overall direction of material
flow (the atom) is opposite to the direction of vacancy flow.
So diffusion
is faster at
higher
temperature
since more
vacancies
will exist in
the lattice!
Diffusion by an ‘interstitialcy’ mechanism illustrates the randomwalk nature of atomic migration (which is quicker as temperature
increases)
Diffusion of importance to material engineers is observed to
occur by both mechanisms – vacancy migration and random
moving interstitials
The interdiffusion of materials A
and B. Although any given A or B
atom is equally likely to “walk” in
any random direction (see Figure
5.6), the concentration gradients of
the two materials can result in a net
flow of A atoms into the B material,
and vice versa. (From W. D.
Kingery, H. K. Bowen, and D. R.
Uhlmann, Introduction to Ceramics,
2nd ed., John Wiley & Sons, Inc.,
New York, 1976.)
The interdiffusion of materials on an atomic scale was
illustrated in the previous Figure --This figure shows
interdiffusion of copper and nickel in a comparable example
on the microscopic scale:
Quantifying Diffusion: Fick’s First Law (Equation 5.3) is a
statement of Material Flux across a ‘Barrier’
moles (or mass) diffusing
mol
kg
J  Flux 

or
2
surface area time 
cm s m2s
We will consider this model as a Steady State Diffusion system
Quantifying Diffusion: Fick’s Second Law (Equation 5.4)
is a statement of Concentration Variation over time
across a ‘Barrier’
 cx    cx 

D
 t  x   x 
in the usual case where D is
independent of Concentration
 cx
 2 cx
D 2
t
u
Solution to Fick’s second law for the case of a semi-infinite solid, constant surface
concentration of the diffusing species cs , initial bulk concentration c0, and a
constant diffusion coefficient, D.
We will consider this model as a Non-Steady State (transient) Diffusion system
A practically important solution is for a semiinfinite solid in which the surface
concentration is held constant. Frequently the
source of the diffusing species is a gas phase,
which is maintained at a constant pressure.
A bar of length l is considered to be semi-infinite when:
l  10 Dt
The following assumptions are implied for a good solution:
1.
Before diffusion, any of the diffusing solute atoms in the solid are
uniformly distributed with concentration of C0.
2.
The value of x (position in the solid) is taken as zero at the surface
and increases with distance into the solid.
3.
The time is taken to be zero the instant before the diffusion process
begins.
Non-steady State Diffusion
• Copper diffuses into a bar of aluminum.
Surface conc.,
C s of Cu atoms
bar
pre-existing conc., Co of copper atoms
Cs
Notice: the concentration
decreases at increasing x
(from surface) while it
increases at a given x as
time increases!
Boundary Conditions:
at t = 0, C = Co for 0  x  
at t > 0, C = CS for x = 0 (const. surf. conc.)
C = Co for x = 
Mathematical Solution:
C x , t   Co
 x 
 1  erf 

Cs  Co
 2 Dt 
CS
C(x,t)
C(x,t) = Conc. at point x at time t
erf (z) = error function
2 z y 2

e dy

0

erf(z) values are given in Table 5.1
(see next slide!)
Co
Similar F.S.L. Diffusion Studies have been
documented for other than Semi-Infinite Solids:
The parameter cm is the average concentration of diffusing species within the sample. Again, the surface concentration,
cs , and diffusion coefficient, D, are assumed to be constant. (From W. D. Kingery, H. K. Bowen, and D. R. Uhlmann,
Introduction to Ceramics, 2nd ed., John Wiley & Sons, Inc., New York, 1976.)
Non-steady State Diffusion
• Sample Problem: 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 (Cs ) 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.
• Solution: use Eqn. 5.5
C( x, t )  Co
 x 
 1  erf 

Cs  Co
 2 Dt 
The solution requires the use of the erf function which was
developed to model conduction along a semi-infinite rod
Solution (cont.):
C( x , t )  Co
 x 
 1  erf 

Cs  Co
 2 Dt 
– t = 49.5 h
– Cx = 0.35 wt%
– Co = 0.20 wt%
x = 4 x 10-3 m
Cs = 1.0 wt%
C ( x, t )  Co 0.35  0.20
 x 

 1  erf 
  1  erf ( z )
C s  Co
1.0  0.20
 2 Dt 
0.1875  1  erf ( z )
erf ( z )  0.8125
 erf(z) = 0.8125
Solution (cont.):
We must now determine from Table 5.1 the value of z for which the error
function is 0.8125. An interpolation is necessary as follows:
z
erf(z)
0.90
z
0.95
0.7970
0.8125
0.8209
Now By LINEAR Interpolation:
z  0.90
0.8125  0.7970

0.95  0.90 0.8209  0.7970
z  0.93
And finally solve for D:
x
z
2 Dt
D
x2
4 z 2t
3
2
 x2 
(
4
x
10
m)
1h

D  
 2.6 x 10 11 m2 /s
 4z 2t  ( 4)(0.93)2 ( 49.5 h) 3600 s


Diffusion and Temperature
The Diffusion coefficient seen in Fick’s Laws increases with
increasing T is a “Classic” Arrhenius Model:
 Qd 

D  Do exp 
R
T


D = diffusion coefficient [m2/s]
Do = pre-exponential [m2/s]
Qd = activation energy for diffusion [J/mol, Kcal/mol, or eV/atom]
R
= gas constant [8.314 J/mol-K]
T
= absolute temperature [K]
So, using this model, we should be able to “back out” the
temperature at which this process took place!
Arrhenius plot of the diffusivity of carbon in α-iron over a range of
temperatures:
Arrhenius plot of diffusivity data for a number of metallic systems. (From
L. H. Van Vlack, Elements of Materials Science and Engineering, 4th
ed., Addison-Wesley Publishing Co., Inc., Reading, MA, 1980.)
In a computationally simpler form (to read!):
Similar Data also exists for Ionic (and Organic) Compounds:
From P. Kofstad, Nonstoichiometry, Diffusion, and Electrical Conductivity in Binary Metal Oxides, John Wiley & Sons, Inc.,
NY, 1972; and S. M. Hu in Atomic Diffusion in Semiconductors, D. Shaw, Ed., Plenum Press, New York, 1973
.
And in a Tabular Form:
Now, Returning to the Solution to our Carburizing problem:
•
To solve for the temperature at which D has
above value, we use a rearranged the D
(Arrhenius) Equation:
Qd
T
R(lnDo  lnD)
142,000 J/mol
 T
(8.314 J/mol - K)(ln 2.0x10 5  ln 2.6x10 11 )
142,000 J/mol
142,000 J/mol
T

(8.314 J/mol - K)(13.553) 112.681 J/mol - K
T  1260.2 K  987 C
Following Up:
• In industry one may wish to speed up this
process
– This can be accomplished by increasing:
• Temperature of the process
• Surface concentration of the diffusing species
• If we choose to increase the temperature,
determine how long it will take to reach the
same concentration at the same depth as in the
previous study?
Diffusion time calculation:
• Given target X (depth of ‘case’) and concentration are equal:
– Here we known that D*t is a constant for the diffusion process (where D is a
function of temperature)
– D1260 was 2.6x10-11m2/s at 987C while the process took 49.5 hours
– How long will it take if the temperature is increased to 1250 ˚C?
D1250C  D0 e
 QD
5
142000
RT
 2.0 10 e
5
5
8.311523
D1250C  2.0  10 1.34110  2.68 10
 D1250C t1250C  D1027 C t1027 C  t1250C 
t1250C
2.6 1011  49.5hr

 4.8 hr
10
2.68 10
10
m2
s
D987 C t987 C
D1250C
Considering a “First Law” or Steady-State Diffusion
Case
Here, The Rate of diffusion is independent of time
dC
 Flux is proportional to concentration gradient =
dx
Note, steady state diffusion concentration gradient ( dC/dx) is linear
C1 C1
dC
J  D
dx
C2
x1
x
C2
D is the diffusion coefficient
x2
if Behavior is linear
dC C C2  C1


dx x
x2  x1
F.F.L. Example: Chemical Protective Clothing
(CPC)
• Methylene chloride is a common ingredient in 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 of MeChl in butyl rubber: D = 110 x10-8 cm2/s
– surface concentrations:
C1 = 0.44 g/cm3 (outside surface)
C2 = 0.02 g/cm3(inside surface)
Example (cont).
J -D
glove
C1
paint
remover
tb 
2
6D
skin
Data:
C2
x1 x2
J   (110 x 10
dC
C  C1
 D 2
dx
x2  x1
D = 110 x 10-8 cm2/s
C1 = 0.44 g/cm3
C2 = 0.02 g/cm3
x2 – x1 = X = 0.04 cm
-8
(0.02 g/cm 3  0.44 g/cm 3 )
g
cm /s)
 1.16 x 10-5
(0.04 cm)
cm2s
2
What happens to a Worker?
• If a person is in contact with the irritant and more than about 0.5 gm of the
irritant is deposited on their skin they need to take a wash break
• If 25 cm2 of glove is in the paint thinner can, How Long will it take before
they must take a wash break?
Flux  1.16 105 g
2
cm -s
if the exposed area of the gloves are 25 cm
2
how long will it take to get 0.5g of M-C onto the hands?
Flux  25cm  1.16  10
2
Exposure time= 0.5g
5
g
2
cm -s
 2.9 104 g
2.9 10 g
s
 1724s  0.48 hr
4
s
Self-diffusion coefficients for silver (and other materials in other
metals) depend on the diffusion path. In general, diffusivity is greater
through less-restrictive structural regions. (From J. H. Brophy, R. M.
Rose, and J. Wulff, The Structure and Properties of Materials, Vol. 2:
Thermodynamics of Structure, John Wiley & Sons, Inc., New York,
1964.)
While shown for
“self-diffusion” –
this type of
diffusing behavior
is typical – Areas
high in vacancies
are ones where
diffusion occurs at
a faster rate!
Schematic illustration of how a coating of impurity B can penetrate more deeply into
grain boundaries and even further along a free surface of polycrystalline A, consistent
with the relative values of diffusion coefficients (Dvolume < Dgrain boundary < Dsurface).
Summary:
Diffusion FASTER for...
Diffusion SLOWER for...
• open crystal structures
• close-packed structures
• materials w/secondary
bonding
• materials w/covalent
bonding
• smaller diffusing atoms
• larger diffusing atoms
• lower density materials
• higher density materials