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DIFFUSION IN SOLIDS

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Title: DIFFUSION IN SOLIDS


1
DIFFUSION IN SOLIDS
  • FICKS LAWS
  • KIRKENDALL EFFECT
  • ATOMIC MECHANISMS

Diffusion in Solids P.G. Shewmon McGraw-Hill,
New York (1963)
2
H2 diffusion direction
Ar
H2
Movable piston with an orifice
Piston motion
Ar diffusion direction
Piston moves in thedirection of the
slowermoving species
3
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
4
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
5
  • 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)

6
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)

7
Ficks first law
8
  • Diffusivity (D) ? f(A, B, T)

Steady state diffusion
D ? f(c)
C1
Concentration ?
C2
D f(c)
x ?
9
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)
10
Ficks II law
?x
Jx
Jx?x
Ficks first law
D ? f(x)
11
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 ?
12
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 ?
13
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

14
Temperature dependence of diffusivity
Arrhenius type
15
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

16
Approximate formula for depth of penetration
17
ATOMIC MODELS OF DIFFUSION
1. Interstitial Mechanism
18
2. Vacancy Mechanism
19
3. Interstitialcy Mechanism
20
4. Direct Interchange and Ring
21
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

22
  • c atoms / volume
  • c 1 / ? 3
  • concentration gradient dc/dx (?1 / ? 3)/? ?
    1 / ? 4
  • Flux No of atoms / area / time ? / area
    ? / ? 2

On comparisonwith
23
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
24
Calculated and experimental activation energies
for vacancy Diffusion
25
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

26
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

27
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
28
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