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?
1
1
Chapter 5: DIFFUSION
z Why
study Diffusion?
z
z
Heat-treated to improve their properties.
z
z
Heat-treatment almost always involve
atomic diffusion.
z
z desired results depends on diffusion
rate
z
z
Heat-treatment temperature, time,
and/or rate of heating/cooling can be
predicted by the mathematics of
diffusion
z
z
Steel gear Î Case hardened to improve
hardness and resistance to fatigue Î
diffusing excess carbon or nitrogen into
outer surface layer.
2
5.1 Introduction
Diffusion: The phenomenon of material transport by
atomic motion.
Many reactions and processes that are important in the material
treatment rely on the mass transfer:
z
z Either with a specific solid ( at microscopic level )
z
z Or from a liquid, a gas, or another solid phase.
z
z
z
z
This chapter covers:
z
z Atomic mechanism
z
z Mathematics of diffusion
z
z Influence of temperature and diffusing species of the
diffusion rate
3
5.1 Introduction (Contd.)
z
z
Phenomenon of diffusion
z
z Explained using diffusion couple,
formed by joining bars of two
different materials having intimate
contact
z
z
Copper and Nickel diffusion couple
z
z Figure 5.1 shows as formed
z
z Atom locations and concentration
z
z
Heated for an extended period at an
elevated temperature ( but below
melting temperature of both ) and
cooled to room temperature.
4
DIFFUSION
• Interdiffusion: In an alloy, atoms tend to migrate
from regions of large concentration.
After some time
Initially
100%
0
Cu
Ni
Concentration Profiles
100%
0
Concentration Profiles5
3
5.1 Introduction (Contd.)
z
z
Chemical analysis reveals
z Alloy region
z Variation of concentration
z Atoms migrated or diffused into one
another
Interdiffusion or impurity diffusion
z Atoms of one metal diffuses into another
z Net drift of atoms from high to lower
concentration
6
DIFFUSION
• Self-diffusion: In an elemental solid, atoms
also migrate. Self-diffusion
All atoms exchanging positions are of same type
No compositional Diffusion in pure metal
changes
Label some atoms
C
A
D
B
After some time
C
D
A
B
4
7
5.2 Diffusion Mechanism
z
z
Atoms in solids are in constant motion rapidly changing positions.
Diffusion is just the stepwise migration of atoms from a lattice site
to other lattice site.
z
z Two conditions for movement:
1. There must be an empty adjacent site
2. Atom must have sufficient energy to break bonds with neighbor
atoms
Atomic vibration (Section 4.7):
z
z Every atom is vibrating very rapidly about its lattice position within
the crystal
z
z At any instant, not all vibrate with same frequency and amplitude.
z
z Not all atoms have same energy
z
z Same atom may have different level of energy at different time
8
z
z Energy increases with temperature
z
z
5.2 Diffusion Mechanism (Contd.)
z Several
z Two
different models for atomic motion
dominate for metallic diffusion
z VACANCY
z Involves
DIFFUSION
interchange of an atom from a normal
lattice position to an adjacent vacant lattice site or
vacancy
z Necessitates presence of vacancies
z Diffusing atoms and vacancies exchange positions
Î they move in opposite directions
z Both self- and inter-diffusion occurs by this
9
mechanism
DIFFUSION MECHANISMS
Vacancy Diffusion:
• applies to substitutional impurities
• atoms exchange with vacancies
• rate depends on:
--number of vacancies
--activation energy to exchange.
increasing elapsed time
5
10
5.2 Diffusion Mechanism (Contd.)
11
5.2 Diffusion Mechanism (Contd.)
z
z
INTERSTITIAL DIFFUSION
z
z Atoms migrate from an interstitial position to a neighboring
one that is empty
z
z Found for interdiffusion of impuries such as hydrogen,
carbon, nitrogen, and oxygen Î atoms small enough to fit
into interstitial positions.
z
z Host or substitutional impurity atoms rarely have
insterstitial diffusion
z
z
Interstitial atoms are smaller and thus more mobile Î
interstitial diffusion occurs much more rapidly then by
vacancy mode
z
z
There are more empty interstitial positions than vacancies Î
12
interstitial atomic movement have greater probability
INTERSTITIAL DIFFUSION
• 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)
7
13
5.3 Steady-State Diffusion
z
z
The quantity of an element that is transported within another is a
function of time Î diffusion is a time-dependent process.
z
z
Diffusion flux (J)
z
z
Rate of diffusion or mass transfer
z
z
Defined as “mass or number of atoms (M) diffusing through and
perpendicular to a unit cross-sectional area of solid per unit time.
z
z
Mathematically, J = M / (At)
z
z
In differential form: J = (1/A)(dM/dt)
A: area across which diffusion is occurring
t: elapsed diffusion time
14
Diffusion
z
How do we quantify the amount or rate of
diffusion?
kg
moles (or mass) diffusing
mol
or
J ≡ Flux ≡
=
2
(surface area )(time )
cm s m 2 s
z
Measured empirically
z
z
z
z
z
z
Make thin film (membrane) of known surface area
Impose concentration gradient
Measure how fast atoms or molecules diffuse through the
membrane
M=
J ∝ slope
mass
M
l dM
diffused
J=
=
At A dt
time
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Steady-State Diffusion
Rate of diffusion independent of time
dC
Flux proportional to concentration gradient =
dx
C1 C1
Fick’s first law of diffusion
C2
x1
x
C2
dC
J = −D
dx
x2
dC
∆ C C 2 − C1
if linear
≅
=
dx
∆x
x 2 − x1
D ≡ diffusion coefficient
16
5.3 Steady-State Diffusion (Contd.)
z
z
z
z
If the diffusion flux does not change with time Î steady-state
diffusion
Example:
z
z Diffusion of a gas through a plate of metal
z
z Concentration (or pressure) of diffusing species on both side are
held constant
z
z Concentration profile: Concentration versus position
z
z Assumed linear concentration profile as shown in figure (b)
17
5.3 Steady-State Diffusion (Contd.)
z
z
Concentration gradient
z
z Slope at a particular point on the concentration profile
curve
z
z Concentration gradient = dC / dx
z
z
For linear concentration shown in figure 5.4b:
Conc. Gradient = ∆C/∆x = (CAA – CBB) / (xAA – xBB)
z
z
Fick’s first law: For steady-state diffusion, the flux is
proportional to the concentration gradient
J = -D(dC/dx)
D: diffusion coefficient (sq. m per second )
-ve sign: direction of diffusion from a high to a low
concentration
18
5.3 Steady-State Diffusion (Contd.)
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Example: Chemical Protective Clothing (CPC)
z
z
z
z
z
z
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?
Data:
z
z diffusion coefficient in butyl rubber:
-8 cm22/s
D = 110 x10-8
z
z surface concentrations:
C1 = 0.44 g/cm3
C2 = 0.02 g/cm3
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Example (cont).
• Solution – assuming linear conc. gradient
glove
dC
C 2 − C1
≅ −D
J = -D
x 2 − x1
dx
C1
paint
remover
skin
l2
tb =
C2
6D
x1 x2
J = − (110 x 10
Data:
D = 110 x 10-8 cm2/s
C1 = 0.44 g/cm3
C2 = 0.02 g/cm3
x2 – x1 = 0.04 cm
-8
g
( 0 .02 g/cm 3 − 0 .44 g/cm 3 )
= 1 .16 x 10 - 5
cm /s)
( 0 .04 cm)
cm 2 s
2
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5.4 Nonsteady-State Diffusion
z
z
z
z
Most practical diffusion situations
are non-steady
Non-steady
z
z Diffusion flux and the
concentration flux at some
particular point of solid vary
with time
z
z Net accumulation or depletion
of the diffusing species
z
z Figure shown concentration
profile at three different times
22
NON STEADY STATE DIFFUSION
• Concentration profile,
C(x), changes
w/ time.
dx
J(left)
• To conserve matter:
J (right) − J (left)
= − dC
dx
dt
dJ = dC
−
dt
dx
• Governing Eqn.:
J(right)
Concentration,
C, in the box
• Fick's First Law:
dC
J = −D
or
dx
d2 C (if D does
dJ =
−D
not vary
dx
dx 2 with x)
equate
dC
d 2C
=D
dt
dx 2
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23
z
z
z
z
z
z
z
z
Solution for Semi-infinite Solid with constant surface
concentration
Assumptions
z
z Initial concentration C00
z
z X = 0 at the surface and increases with distance into the
solid
z
z Initial time = 0
Boundary conditions
z
C = Coo at 0 ≤ x ≤ ∝
z For t = 0,
z
C = Css (Constant surface concentration) at
z For t > 0,
x=0
C x − C0
⎛ x
C = C00 at x = ∝
= 1 − erf ⎜
C s − C0
⎝ 2 Dt
Solution
z
z erf ( ) : Gaussian error function
24
z
z Values given in Table 5.1
⎞
⎟
⎠
NON STEADY STATE DIFFUSION
• Copper diffuses into a bar of aluminum.
Surface conc.,
C s of Cu atoms
Cs
C( x,t)
C o to
t1
t2
bar
pre-existing conc., C o of copper atoms
t3
position, x
• General solution:
⎛ x ⎞
C(x, t) − C o = −
⎟
1 erf ⎜⎝
2 Dt ⎠
Cs − Co
"error function"
Values calibrated in Table 5.1, Callister 6e.
15
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EXAMPLE PROBLEM
• 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.
• Dt should be held constant.
⎛ x
C (x ,t) − Co
= 1 − erf ⎜⎝
Cs − C
2Dt
o
5.3 x10 -13 m 2 /s
• Answer:
⎞
⎟
⎠
(Dt) 500ºC =(Dt) 600ºC
10hrs
(Dt) 600
= 110 hr
t 500 =
D 500
4.8x10 -14 m 2 /s
Note: values
of D are
Given here.
16
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Factors That Influence Diffusion
29
Factors That Influence Diffusion (Contd.)
z
z
DIFFUSING SPECIES
z
z
Magnitude of diffusion coefficient (D) Î indicative of the
rate at which atoms diffuse
z
z
D depends on both the diffusing species as well as the host
atomic structure
z
z
Self-diffusion
Fe in α-Fe
3.0E(-21) m22/s
Vacancy Diffusion
Inter-diffusion C in α-Fe
2.4E(-12) m22/s
Interstitial Diffusion
z
z
Interstitial is faster than vacancy diffusion
30
Factors That Influence Diffusion (Contd.)
TEMPERATURE
z
Temperature has a most profound influence
on the coefficients and diffusion rate
z
Example: Fe in α-Fe (Table 5.2)
500ooC D=3.0E(-21) m22/s
Î approximately
900ooC D=1.8E(-15) m22/s
six orders
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DIFFUSION AND TEMPERATURE
• Diffusivity increases with T.
D = Do
diffusivity
pre-exponential [m 2 /s] (see Table 5.2, Callister 6e )
activation energy
⎛ Q ⎞ [J/mol],[eV/mol]
exp ⎜− d ⎟ (see Table 5.2, Callister 6e )
⎝ RT ⎠
300
600
1000
log D = log D0 −
γFe
Ci
nα
-Fe
Al
in
Al
Cu
Cu in -Fe
in Cun α
ei e
Zn
F
γ-F
in
Fe
0.5
in
10 -14
10 -20
gas constant [8.31J/mol-K]
Q
ln D = ln D0 − d
R
T(C)
C
10 -8
D (m 2 /s)
1500
• Experimental Data:
1.0
1.5
⎛1⎞
⎜ ⎟
⎝T ⎠
Qd ⎛ 1 ⎞
⎜ ⎟
2.3R ⎝ T ⎠
D has exp. dependence on T
Recall: Vacancy does also!
D interstitial >> D substitutional
C in α-Fe
C in γ-Fe
2.0 1000K/T
Cu in Cu
Al in Al
Fe in α-Fe
Fe in γ-Fe
Zn in Cu
19
32