Diffusion
Atomic Motion in Solids
Material Sciences and Engineering
MatE271
Week6
1
Diffusion in Materials
Q. How do changes in microstructure and chemical
composition actually occur?
A. Atoms must be able to move around (diffusion)
Diffusion occurs in solids, liquids and gases:
• Redistribution of non-uniform chemical species
(impurity diffusion or interdiffusion)
• Random atomic movement can also occur in
chemically uniform materials (self diffusion)
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1
Diffusion is “driven” by Nonuniformity
B
B
A
Temperature
Concentration of “A”
Concentration of “A”
A
Time
Distance (x)
Distance (x)
Concentration Profile
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Interdiffusion forming a solid solution
Initial
Intermediate time
Much longer time
What is this time scale?
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2
Diffusion?
Diffusion is necessary for:
Ø Redistribution of chemical species
Ø Physical changes in microstructure
Ø Densification of powder compacts
Ø Deformation at high temperature (creep)
Ø Formation of solid state reaction products
Ø One kind of conductivity in ceramics (ionic)
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Diffusion:
Perfect crystal:
Atoms would not move around because
there would be no places for them to move to
(all sites would be occupied)-- “locked in place”
Ø Point defects must be present in a crystal to permit atomic
movement (diffusion)
In a way, atomic diffusion is actually the movement of
defects.
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3
Diffusion Mechanisms
I. Vacancy diffusion
o Only adjacent atoms can move into a vacancy
o Vacancy moves in opposite direction of atomic
motion
o Rate depends on concentration of vacancies
Vacancy
flux
Atomic
flux
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Diffusion Mechanisms
II. Interstitial Diffusion
o Atom can move into any adjacent empty
interstitial position (usually smaller atoms)
o Rate depends on concentration of interstitial
atoms
Ø (Usually faster than vacancy diffusion)
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4
Diffusion Mechanisms
o Would you expect vacancy or interstitial
diffusion to be faster?
o Why?
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Diffusion occurs by random jumps
Ø After many random jumps by an atom, it’s displacment”
can be calculated by the theory of “random walks”
Net migration
after n jump
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5
Quantitative Description of Diffusion
- The rate of diffusion is characterized by describing atomic
fluxes at particular locations in the material
- Critical quantities
area
c
J = atomic flux (atoms/m2-s, kg/ m2-s)
(dc/dx) = concentration gradient
J
(atoms/m4)
D = diffusion coefficient (m2/s)
dc/dx
x
Fick’s first law: J = - D (dc/dx)
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Interrelating the quantities
• Fick’s first law: J = - D (dc/dx)
(negative sign indicates that the direction of diffusion flux
is “down” the concentration gradient from high to low
concentration)
• For steady state diffusion (local flux doesn’t change
with time), Fick’s First Law can be solved directly
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6
Example
Hydrogen (H) gas can be purified by passing atomic
hydrogen through a thin sheet of palladium (Pa) at 700oC in
a paladium diffusion cell. If the impure hydrogen gas is
maintained at 1 atm on one side of a 1 mm thick Pd sheet
(A=1 m2), and the pressure on the purified side is
maintained at 0.1 atm by pumping, what is the mass of the
hydrogen purified in 1 hr? Assume steady state conditions.
The concentration of H2 at 1 atm is 9.0x10-3 gm/cm3 and
D(H) in Pa is 1.2x10-6 cm2/sec.
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Non-steady state diffusion
The diffusion flux at a particular point
varies with time
Ø • (There is a net accumulation or depletion
of the diffusing species at a given location)
Ø • i.e., local concentration of diffusing
species changes with time as diffusion
proceeds
Ø • This is the most common situation
What is this time scale?
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Non-steady state diffusion
• Fick’s Second Law governs
C(x, t)=Cx
C(0,t)=Cs
∂C = D ∂2C
∂t
∂x2
?
C(∞,t)=Co
x
C(x,0)=Co
• Many solutions exist for particular geometries
(initial and boundary conditions)
• Diffusion from a constant source into an semi-infinite solid
BC-1:
For t = 0, C = C0 at 0 ≤ x ≤ ∞
BC-2:
t > 0, C = Cs at x = 0
BC-3:
C = C0 at x = ∞
(Cx - C0) = 1 - erf x
(Cs-C0)
2√Dt
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Non-steady state diffusion
(Cx - C0) = 1 - erf x
(Cs-C0)
2√Dt
?
C(x, t)=Cx
C(∞,t)=Co
C(0,t)=Cs
C(x,0)=Co
to< t1 < t2 < t3
Cs
C
to
C0
x=0
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t1
t2
t3
Cx
x
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8
Example
Surface Treatment of Steel:
For some applications (e.g. gears), it is necessary to harden
the surface of a steel (Fe-C alloy) above that of its interior.
One way of accomplishing this is by increasing the surface
concentration of carbon in the steel (as we will see later)
using a process termed carburizing. In carburizing the steel
piece is exposed, at elevated temperature, to an atmosphere
rich in a hydrocarbon gas, such as methane (CH4).
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Example: Surface Treatment of Steel:
Consider on such alloy that initially has a uniform carbon
concentration of 0.25 wt% and is to be treated at 950° C.
If the concentration of carbon at the surface is suddenly
brought to and 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 diffusion coefficient for C
in Fe at this temperature is 1.6 x 10-11 m2/sec. Assume
piece is semi-infinite.
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9
Factors that Influence Diffusion
I. Diffusing Species
• magnitude of diffusion coefficient, D - indicates
the rate at which atoms diffuse
• both diffusing species and host material
influence the coefficient
• Relative sizes of atoms
• “Openess” of lattice
• Ionic charges
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Factors that Influence Diffusion
Ø
Atomic Size/Mechanism
For example:
• For the host species of iron:
- Self diffusion at 500°C (Fe moving in Fe)
D = 1.1 x 10-20 m2/s (vacancy diffusion)
- Carbon interdiffusion at 500°C (C moving in Fe
D = 2.3 x 10-12 m2/s (interstitial diffusion)
This shows the contrast between rates of
vacancy and interstitial diffusion
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Factors that Influence Diffusion
II. Temperature
• very strong effect on the diffusion coefficient:
æ −Q d ö
D = Do exp
è RT ø
Do = T independent preexponential
Qd = the activation energy for diffusion (J/mol, or eV/atom)
R = the gas constant, 8.31 J/mol - K or 8.662 x 10 -5 eV/ atom − K
T = absolute temperature, (K)
• A large activation energy results in a small D
ln D = ln D o −
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−Q d æ 1 ö
R è Tø
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Plot ln D vs 1/T - get straight line
(to measure activation energy and Do)
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Example:
o Using data from Table 5.2, compute the
diffusion coefficient of C in α−Fe and γ−Fe
at 900ºC.
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Solution:
æ −Q d ö
D = Do exp
è RT ø
Do = T independent preexponential
Q = the activation energy for diffusion (J/mol, or eV/atom)
d
o C in α−Fe
(BCC) at 900ºC
R = the gas constant, 8.31 J/mol - K or 8.662 x 10 -5 eV/ atom − K
D = 6.2x10-7 m2/sec exp (-0.83 eV/atom / (1173K)(8.62x10-5 eV/atom-K)
T = absolute temperature, (K)
D = 1.7x10-10 m2/sec
o C in γ−Fe (FCC) at 900ºC
D = 2.3x10-5 m2/sec exp (-1.53 eV/atom / (1173K)(8.62x10-5 eV/atom-K)
D = 5.9x10-12 m2/sec
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What does this tell you about interstitial sites
in BCC and FCC?….
• BCC more open than FCC for interstitial
diffusion…i.e. it is easier to move from one
interstitial site to another in BCC
• But it does not say anything about the size
or number of interstitial sites in
each….actually, as you will see, FCC has
bigger (and more) interstitial sites
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12
Other Diffusion Paths
(Besides through volume of the crystal)
• Atomic migration often occurs more rapidly along socalled “short circuiting paths”
• Dislocations
• Grain boundaries
• External surfaces
• However, there is usually small total area
for this to occur - so not always important
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Week 6
Volume, grain boundary and surface diffusion
surface
Grain boundary
Ag in Ag
volume
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Diffusion and Materials Processing
o Properties of materials are altered through
diffusion
• steelmaking
• sintering
• semiconductor doping
o “Heat treatment” is used to allow these to
occur over a reasonable time frame.
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Summary
• Recognize various imperfections in crystals
• Point imperfections
• Impurities
• Line imperfections (dislocations)
• Bulk imperfections
• Define various diffusion mechanisms
• Identify factors controlling diffusion processes
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Reading Assignment
Shackelford 2001(5th Ed)
– Read Chapter 5, pp 158-181
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