재 료 상 변 태
Phase Transformation of Materials
2008.10.09.
박 은 수
서울대학교 재료공학부
Contents for previous class
• Diffusion in substitutional alloys 1) Interdiffusion
2) Velocity of the movement of the lattice plane 3) Fick’s 2
ndlaw for substitutional alloys
4) Diffusion in dilute substitutional alloys
• Atomic mobility: Relationship between M
Band D
B• Diffusion in substitutional alloys
JA
JB JV
At t = 0
,
At t = t1
,
JA
JVA JA
’
JB JVB
JB
’
x D C JJ
JA A Av A
∂
− ∂
= +
′ = ~
x D C J
J
JB B Bv B
∂
− ∂
= +
′ = ~
B A A
BD X D
X
D~ = +
⎟⎠
⎜ ⎞
⎝
⎛
∂
∂
∂
= ∂
∂
∂
x D C x t
CA ~ A
A
B
J
J ′ = − ′
∴
Atomic mobility
A diffusion flux is a combined
quantity of a drift velocity and random jumping motion.
M x
v
B B B∂
− ∂
= μ
MB : mobility of B atoms
: chemical force causing atom to migrate x
B
∂
− ∂μ
C x M
J
B B B B∂
− ∂
= μ
B B
B
v c
J =
How the mobility of an atom is related to its diffusion coefficient?
The problem of atom migration can be solved by considering the thermodynamic condition for equilibrium; namely that the chemical potential of an atom must be the same everywhere. In general the flux of atoms at any point in the lattice is proportional to the chemical potential gradient: diffusion occurs down the slope of the chemical potential.
Relationship between M
Band D
BFor ideal or dilute solutions,
near XB ≈ 0, γB = const. with respect to XB
x X RT X
x X RT x
X RT
x G x
B B
B
B B
B B B
B
∂
∂
∂ + ∂
=
∂ + ∂
∂
= ∂
∂ +
= ∂
∂
∂
) ln ln
1 ln (
ln ) ( ln
) ln
( 0
γ γ μ γ
x D C
x RTF C
M
x X X
X RT V
M X J
B B
B B
B B
B B
m B B B
∂
− ∂
∂ =
− ∂
=
∂
∂
∂ + ∂
−
=
∴
ln ) 1 ln
( γ
ln ) 1 ln
(
B B
X d
F = + d γ
= 1
∴ F D
B= M
BRT
C x M
J
B B B B∂
− ∂
= μ μ
B= G
B+ RT ln a
B= G
B+ RT ln γ
BX
Bm B m
B A
B B
B
V
X V
n n
n V
C n =
= +
= ( )
RTF M
D
B=
BFor non-ideal concentrated solutions,
thermodynamic factor must be included.
Related to the curvature of the molar free energy-composition curve.
Contents for today’s class
• Tracer Diffusion in Binary Alloys
• Diffusion in Ternary Alloy
• High-Diffusivity Paths
1) Diffusion along Grain Boundaries and Free Surface 2) Diffusion Along Dislocation
• Diffusion in Multiphase Binary Systems
2.5 Tracer diffusion in binary alloys
It is possible to use radioactive tracers (D*Au) to determine the intrinsic diffusion coefficients (DAu) of the components in an alloy .
*
Au Au
How does D differ from D ?
* *
0 ,
mix Au Au Ni Ni
H D D D D
Δ > → < <
D*Au gives the rate at which Au* (or Au) atoms diffuse in a chemically homogeneous alloy, whereas DAu gives the diffusion rate of Au when concentration gradient is present.
1) Au* in Au
2) Au* in Au-Ni
D*Au = DAu(self diffusivity)
If concentration gradient exhibit,
obtained by maker’s movement
Since the chemical potential gradient is the driving force for diffusion in both types of experiment it is reasonable to suppose that the atomic mobility are not affected by the concentration gradient.
What would be the relation between the intrinsic chemical diffusivities DB and tracer diffusivities D*B?
In the tracer diffusion experiment the tracer essentially forms a dilute solution in the alloy.
B B B
D
*= M RT
*= M RT
B B B B BD M RT d FM RT
d X 1 ln
ln γ
⎧ ⎫
= ⎨ + ⎬ =
⎩ ⎭
A A
B B
D FD D FD
*
*
=
= D % = X D
B A+ X D
A B= F X D (
B A*+ X D
A B*)
A B A B
A B
d d X X d G
F d X d X RT dX
2 2
ln ln
1 1
ln ln
γ γ
⎧ ⎫ ⎧ ⎫
= ⎨ + ⎬ ⎨ = + ⎬ =
⎩ ⎭ ⎩ ⎭
A B
A B
A B
Eq
d d
X X d G RT RT
dX d X d X
2 2
.(1.71)
ln ln
1 1
ln ln
γ γ
⎧ ⎫ ⎧ ⎫
= ⎨ + ⎬= ⎨ + ⎬
⎩ ⎭ ⎩ ⎭
F : Thermodynamic Factor
Additional Thermodynamic Relationships for Binary Solution:
Variation of chemical potential (dμ) by change of alloy compositions (dX)
B A A B
D % = X D + X D
(
B A A B)
D % = F X D
*+ X D
*2.5 Tracer diffusion in binary alloys
D*Au
> D*
Ni2.6 Diffusion in ternary alloys:
Additional EffectsExample)
Fe-Si-C system (Fe-3.8%Si-0.48%C) vs. (Fe-0.44%C) at 1050
℃① Si raises the μC in solution.
(chemical potential of carbon)
② MSi (sub.) ≪ MC (int.),
(M : mobility)
at t = 0
A C B
How do the compositions of A and B change with time?
Grain boundary dislocation surface Real materials contain defects.
= more open structure fast diffusion path.
2.7.1 High-diffusivity paths
Diff. along lattice
Diff. along grain boundary Diff. along free surface
) exp( RT D Q
D
l=
lo−
l) exp( RT D Q
D
b=
bo−
b) exp( RT D Q
D
s=
so−
ss b l
D > D > D
But area fraction
→ lattice > grain boundary> surface
Atoms diffusing along the boundary will be able to penetrate much deeper than atoms which only diffuse through the lattice.
In addition, as the concentration of solute builds up in the boundaries, atoms will also diffuse from the boundary into the lattice.
Diffusion along grain boundaries
Combined diffusion of grain boundary and lattice?
Thus, grain boundary diffusion makes a significant contribution
only when Dbδ > Dld.
The relative magnitudes of D
bδand D
ld are most sensitive to temperature.
Due to
Q
b< Q
l,
the curves for Dl and Dbδ/d cross
in the coordinate system of lnD versus 1/T.
D
b> D
lat all temp.
Therefore, the grain boundary diffusion becomes predominant at temperatures lower than the crossing temperature.
(T < 0.75~0.8 Tm)
-Q/2.3R
2.7.2 Diffusion along dislocations
app p
l l
D D
D = + ⋅ 1 g D
app l p
D = D + ⋅ g D
hint) ‘g’ is the cross-sectional area of ‘pipe’ per unit area of matrix.
D
app= ?
ex) annealed metal ~ 10
5disl/mm
2; one dislocation(
┴) accommodates 10 atoms in the cross-section; matrix contains 10
13atoms/mm
2.
g
5 6
7
13 13
10 * 10 10 10 10 10
= = =
−At high temperatures,
diffusion through the lattice is rapid and gD
p/D
lis very small
so that the dislocation contribution to the total flux of atoms is very small.
At low temperatures,
gD
p/D
lcan become so large that the apparent diffusivity is entirely due to diffusion along dislocation.
g = cross-sectional area of ‘pipe’ per unit area of matrix
Due to
Q
p< Q
l,
the curves for Dl and gDp/Dl cross in the coordinate system of lnD versus 1/T.
(T < ~0.5 Tm)