DIFFUSION IN SOLIDS

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DIFFUSION IN SOLIDS

• FICKS LAWS
• KIRKENDALL EFFECT
• ATOMIC MECHANISMS

Diffusion in Solids P.G. Shewmon McGraw-Hill,
New York (1963)
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H2 diffusion direction
Ar
H2
Movable piston with an orifice
Piston motion
Ar diffusion direction
Piston moves in thedirection of the
slowermoving species
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Kirkendall effect

• Materials A and B welded together with Inert
  marker and given a diffusion anneal
• Usually the lower melting component diffuses
  faster (say B)

A
B
Marker motion
Inert Marker thin rod of a high melting
material which is basically insoluble in A B
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Diffusion

• Mass flow process by which species change their
  position relative to their neighbours
• Driven by thermal energy and a gradient
• Thermal energy ? thermal vibrations ? Atomic
  jumps

Concentration / chemical potential
Electric
Gradient
Magnetic
Stress
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• Flux (J) (restricted definition) ? Flow / area
  / time Atoms / m2 / s

• Assume that only B is moving into A
• Assume steady state conditions ? J ? f(x,t) (No
  accumulation of matter)

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Ficks I law
Diffusion coefficient/ diffusivity
No. of atoms crossing area Aper unit time
Cross-sectional area
Concentration gradient
Matter transport is down the concentration
gradient
Flow direction
A

• As a first approximation assume D ? f(t)

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Ficks first law
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• Diffusivity (D) ? f(A, B, T)

Steady state diffusion
D ? f(c)
C1
Concentration ?
C2
D f(c)
x ?
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D ? f(c)
Steady state J ? f(x,t)
D f(c)
Diffusion
D ? f(c)
Non-steady stateJ f(x,t)
D f(c)
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Ficks II law
?x
Jx
Jx?x
Ficks first law
D ? f(x)
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RHS is the curvature of the c vs x curve
LHS is the change is concentration with time
ve curvature ? c ? as t ?
?ve curvature ? c ? as t ?
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Solution to 2o de with 2 constantsdetermined
from Boundary Conditions and Initial Condition

• Erf (?) 1
• Erf (-?) -1
• Erf (0) 0
• Erf (-x) -Erf (x)

Area
Exp(? u2) ?
0
?
u ?
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Applications based on Ficks II law
Determination of Diffusivity
A B welded together and heated to high
temperature (kept constant ? T0)
t2 gt t1 c(x,t1)
t1 gt 0 c(x,t1)
t 0 c(x,0)
f(x)t
C2
Non-steadystate
Flux
f(t)x
Cavg

• If D f(c) ? c(x,t) ? c(-x,t) i.e.
  asymmetry about y-axis

Concentration ?
? t
A
B
C1
x ?

• C(x, 0) C1
• C(?x, 0) C2

• A (C1 C2)/2
• B (C2 C1)/2

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Temperature dependence of diffusivity
Arrhenius type
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Applications based on Ficks II law
Carburization of steel

• Surface is often the most important part of the
  component, which is prone to degradation
• Surface hardenting of steel components like
  gears is done by carburizing or nitriding
• Pack carburizing ? solid carbon powder used as C
  source
• Gas carburizing ? Methane gas CH4 (g) ? 2H2 (g)
  C (diffuses into steel)

CS
C1
x ?
0

• C(x, 0) C1
• C(0, t) CS

• A CS
• B CS C1

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Approximate formula for depth of penetration
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ATOMIC MODELS OF DIFFUSION
1. Interstitial Mechanism
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2. Vacancy Mechanism
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3. Interstitialcy Mechanism
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4. Direct Interchange and Ring
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Interstitial Diffusion
1
2

• At T gt 0 K vibration of the atoms provides the
  energy to overcome the energy barrier ?Hm
  (enthalpy of motion)
• ? ? frequency of vibrations, ? ? number of
  successful jumps / time

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• c atoms / volume
• c 1 / ? 3
• concentration gradient dc/dx (?1 / ? 3)/? ?
  1 / ? 4
• Flux No of atoms / area / time ? / area
  ? / ? 2

On comparisonwith
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Substitutional Diffusion

• Probability for a jump ? (probability that the
  site is vacant) . (probability that the atom has
  sufficient energy)
• ?Hm ? enthalpy of motion of atom
• ? ? frequency of successful jumps

As derived for interstitial diffusion
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Calculated and experimental activation energies
for vacancy Diffusion
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Interstitial Diffusion

• D (C in FCC Fe at 1000ºC) 3 ? 10?11 m2/s

Substitutional Diffusion

• D (Ni in FCC Fe at 1000ºC) 2 ? 10?16 m2/s

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DIFFUSION PATHS WITH LESSER RESISTANCE
Experimentally determined activation energies for
diffusion
Qsurface lt Qgrain boundary lt Qlattice
Lower activation energy automatically implies
higher diffusivity

• Core of dislocation lines offer paths of lower
  resistance ? PIPE DIFFUSION

• Diffusivity for a given path along with the
  available cross-section for the path will
  determine the diffusion rate for that path

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Comparison of Diffusivity for self-diffusion of
Ag ? single crystal vs polycrystal
Schematic

• Qgrain boundary 110 kJ /mole
• QLattice 192 kJ /mole

Polycrystal
Log (D) ?
Singlecrystal
1/T ?
? Increasing Temperature
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