Download Diffusion

Diffusion
• Diffusion is the relative movement of atoms in a solid, liquid or gas
concerned here with diffusion in solids.
• Is very important in
1. phase transformations
2. corrosion resistant coatings
3. separation of U235 by gaseous
diffusion
4. permeability
5. impurity transistors
6. metal joining by diffusion bonding
7. radiation damage / defects - defect
migration
Fick's 1st Law : (no time dependence)
dc
J = - D dx
we are
{ = - D ∇C in 3-dimensions}
[J] = #/cm2s [C] = #/cc [D] = cm2/s
• note - J along negative concentration
gradient
J1 - J 2 = (
∂C
∂J
) ∆x = ∆x
∂t
∂x
Fick's 2nd Law : C + f(x,t)
∂C
= D ∇ 2C
∂t
∂C
∂J
∂
∂C
∂ 2C
or = − = − (−D ) = D 2 ,
∂t
∂x
∂x
∂x
∂x
if D≠f(x) which is true for most cases:
∂C
= D ∇ 2C
∂t
⇒ diffusion eqn.
Find the general and particular solutions
{C(x,t)} to the diffusion equation using
initial and boundary conditions.
KL Murty
page 1
J2
J1
1
∆x
2
NE409/509
Examples
1. Thin-film solution {find C(x,t)}
α
-L
x
0
-x
L
α is the amount in the thin layer at x=0
and L is very large
∂C
∂ 2C
=D
∂t
∂x 2
x = 0 C → ∞ as t → 0
&
x > 0 C → 0 as t → 0
3. Carburizing / Decarburizing
Co
x
x2
note C = A exp(- 4Dt ) satisfies the
Cs
above equation
Solution
+∞
A is defined by ∫ C( x, t )dx = α
−∞
∴C =
Cs - Cx
x
=
erf(
)
Cs - Co
2 Dt
α
x2
)
exp(−
4Dt
2 πDt
2. Pair of semi-infinite solids
Cu
-L
Ni
-x
0
x
L
x
2
2
⌠
where erf(x) =
⌡exp(-ξ ) dξ
πo
C'
α
0
-x
C(x,t) =
 C 0 = initial concentration
 Cs = surface concentration
for t = 0, C = Co at o ≤ x ≤ ∞
for t > o, C = Cs (constant) at x = 0
and C = Co at x = ∞
0
erf (0) = 0
erf (∞) = 1
erf (x) = erf(-x)
0
x
C' ∞
( x − α) 2
)dα
∫ exp(−
4Dt
2 πDt 0
C'
x
C(x,t) = 2 [ 1 + erf(
]
2 Dt
KL Murty
page 2
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4. Diffusion Experiment (self diffusion)
α
α is the radioisotope of the metal A
A
x
0
L
x2
α
)
C=
exp(−
4Dt
2 πDt
• T1
t o T1
ln C
ln D(T)
-Q/R
t o T2
-1/4Dt o
x
2
• T2
1/T
QD
D = Do exp(- RT)
Mechanism (?)
KL Murty
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Atomic Theory of Diffusion
{Ref. Paul Shewmon, Diffusion in Solids , 2nd edition, TMS, 1989}
D ⇒ atomic jump distance & frequency ⇔ random walk theory (text, Eq.7.25)
Another way to do this is (see notes) : (no specific micromechanism !)
Consider two adjacent crystal lattice planes 1 and 2 [Fig.] separated by α.
a
{if the planes are {110} type α = 2 since the direction x then is <110> type}
n1, n2 : no. of atoms / unit area
n1
|
|
|
n2
|
|
|
1
n Γ δt is the number jumping from
2 1
|
______|____ X
plane 1 to plane 2
1 ←α→ 2
Γ = number of jumps they make per sec.
The atoms cannot stay in-between since there is no lattice plane between 1 and 2
The flux of atoms crossing a plane between 1 and 2 [J] :
number of atoms
1
J = 2 (n1 - n2) Γ = (area) (time) .
In terms of the concentrations [c, per unit volume] : c1α = n1 and c2α = n2
dc
1
1
J = 2 (c1 - c2) α Γ or J = 2 {-αdx } α Γ
dc
1
1
Recalling Fick's first law {J = - D dx }, D = 2 α2 Γ or D = 6 α2 Γ
This equation is similar to the one derived in the text using random-walk theory,
1
r2 = Γ t α2 (Eq.7.17) and also r2 = 6 D t (Eq.7.24), so that D = 6 α2 Γ
• note : no specific mechanisms of atom jumping assumed •
KL Murty
page 4
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Diffusion Mechanisms
Vacancy
Interstitial
Interstitialcy
Ring
Self-diffusion via Vacancy Mechanism
Γ is proportional to the number of
nearest neighbors [β] and the probability
of finding the lattice point vacant, i.e.,
the vacancy concentration [Cv] so that
Γ = β Cv ω,
where ω is the atomic jump frequency, ω
= νD e- (Em / kT)
1
1
∴ D = 6 α2 Γ = 6 α2 β Cv νD e- (Em/kT)
Cv = e- (Ef / kT)
β
D = 6 α2 νD e- (Em/ kT) e- (Ef / kT)
• For diffusion of vacancies, there is no
need for Cv and only Em appears in the
expression for Dv •
z
L
Or DL = DSD = 6 α2 νD e- (ED/ kT) = Do e- (ED/ kT) , ED= Ef + Em
KL Murty
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i
i
⇒ similarly for interstitial diffusion : Di = Do e-(Em /kT)
Melting Point, ÞC
Short-Circuit Diffusion
• Surface (Qs < QD)
• Grain-Boundary (QGB < QD)
surface
lnD
• Dislocations (Pipe) (QGB ≈ Q⊥)
self
(lattice)
GB / Dislocation
QGB ≈ 0.35 QD
1/T
KL Murty
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Diffusion in Ionics
1
e.g. Schottky-type : ED = Em + 2 Es
NaCl + Ca
NaCl with Ca++
lnD
at low temperatures, where extrinsic
vacancies dominate, ED = Em ⇐
vacancy concentration is independent of
temperature (athermal vacancies)
⇒ thus can determine both Es and Em
QD
Qm
1/T
In metals, it is not easy to determine Ef and Em separately 1. Can get Ef from
(a) Simmons-Balluffi experiment,
or
(b) Quenched in Resistivity
2. Annealing of quenched-in vacancies ⇒ Em
Quench from high temperature ,
determine ∆ρο ; heat to T for t ; quench and determine ∆ρ
∆ρ (t) = ∆ρo e-t/τ where τ(Τ) is the time for a vacancy to anneal out to sink
l2
τ ∝ D where l is the sink distance and Dv ∝ e-Em/kT
v
β
{note : Dv = 6 α2 e-Em/kT no Cv appears since T is low enough thta the
quenched-in vacancies from ‘high’ temperature is far larger}
40ÞC
Example : Bauerle & Koehler in Au
60ÞC
Em 1 1
τ1 Dv(T2)
= D (T ) = exp{- k (T - T ) }
τ2
v 1
2
1
ln(∆ρ/∆ρο)
Tq=700ÞC
Em = 0.82 eV (18.9 kCal/mole)
t, hours
KL Murty
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Group Work
(solutions)
For the example below, determine
(a) β and α (in cm), and (b) lattice diffusivity, DL in cm2/s at 500 ˚C.*
QD
β
DL = 2 D α2 νD exp(),
RT
β = no. of positions an atom can jump to,
and D in the denominator is the dimension (1, 2 or 3)
for Cu (fcc) along <110> : β = 12, α =
Group 1.
12
a
= 2.556x10-8 cm,
2
50000
DL = 6 (2.556x10-8) 2 1013 exp(- 1.987x773 ) = 9.547x10-17 cm2/sec
for Cu (fcc) along <100> : β = 6, α = a = 3.615x10-8 cm
Group 2.
6
50000
DL = 6 (3.615x10-8) 2 1013 exp(- 1.987x773 ) = 9.519x10-17 cm2/sec
3
for Cu (fcc) along <111> : β = 8, α = 2 a = 3.131x10-8 cm
Group 3.
8
50000
DL = 6 (3.131x10-8) 2 1013 exp(- 1.987x773 ) = 9.521x10-17 cm2/sec
Group 4.
for Zr (hcp) along c-axis :
β = 2, α = c = 1.593x3.231 Å = 5.147x10-8 cm
2
70000
DL = 2 (5.147x10-8) 2 1013 exp(- 1.987x773 ) = 4.270x10-22 cm2/sec
note ‘2’ in the denominator - since the atom jumps are 1-D (!)
*
Cu : fcc, a = 3.615 Å, QD = 50 kCal/mole
c
νD = 1013 per sec
a
Zr : hcp, a = 3.231Å, c/a = 1.593, QD = 70 kCal/mole, QD = 60 kCal/mole
KL Murty
page 8
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