ema5001 lecture 3 steady state & nonsteady state diffusion...ema 5001 physical properties of...
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EMA5001 Lecture 3
Steady State &
Nonsteady State Diffusion -
Fick’s 2nd Law & Solutions
EMA 5001 Physical Properties of Materials Zhe Cheng (2016) 3 Nonsteady State Diff – Fick’s 2nd Law
Steady State
Steady State = Equilibrium?
− Similarity: State function (e.g., , C) does NOT change with time
− Difference: Net flux (or net reaction rate)
• Zero (0) net flux for equilibrium state vs. non-zero net flux for steady state
Fick’s 1st Law
If steady state diffusion
CB does NOT change with time
JB does NOT change with time
For 1-D, if DB constant, what is the
concentration profile under steady state?
Steady-State Diffusion
2
No!
x
CDJ B
BB
dxD
JxC
B
BB )(
xD
JxCxC
B
BBB )0()( Linear concentration profile x
CB
CB (x=0)
Slope = B
B
D
J
0
dx
dCDJ B
BB
EMA 5001 Physical Properties of Materials Zhe Cheng (2016) 3 Nonsteady State Diff – Fick’s 2nd Law
Nonsteady-State Diffusion
Nonsteady State
Concentration changes with both
− Location (x, y, z)
− Time (t)
Take a small slice at location x
− δx : Thickness of the slice
− J1 : Flux into the slice
− J2 : Flux out from the slice
In small time period δt, the change
of concentration in that slice
We have
3
xA
tAJtAJC B
21
x
C
0
x
J
0
x x+δx
x+δx x
J2
J1
J1 J2 Area A
δx x
JJ
t
C B
21
EMA 5001 Physical Properties of Materials Zhe Cheng (2016) 3 Nonsteady State Diff – Fick’s 2nd Law
Fick’s 2nd Law
Continued from p. 3
As δx and δt 0, we have
Therefore
Invoking Fick’s 1st Law
We have
i.e.,
If DB constant, simply to
4
x
J
x
JJ
t
CB
B
21
x
CDJ B
B
B
x
CD
xx
CD
xt
C BB
BB
B
x
J
t
CB
B
x
CD
xt
C BB
B Fick’s 2nd Law
2
2
x
CD
t
C BB
B
EMA 5001 Physical Properties of Materials Zhe Cheng (2016) 3 Nonsteady State Diff – Fick’s 2nd Law
Implications of Fick’s 2nd Law
Two concentration profiles
Does the concentration in the specified region increase or decrease with time?
5
x
CB
0 x
CB
0
02
2
x
CB 02
2
x
CB
CB increases with time CB decreases with time
02
2
x
CD
t
C BB
B0
2
2
x
CD
t
C BB
B
EMA 5001 Physical Properties of Materials Zhe Cheng (2016) 3 Nonsteady State Diff – Fick’s 2nd Law
Special Case – Homogenization
Diffusion to eliminate to local
concentration variation
Simplest case: t = 0, CB varies sinusoidally
Assumption:
− DB constant
Solution to Fick’s 2nd Law takes the form
in which is relaxation time
The amplitude of the variation
6
x
C
0
t = 0
t = τ
High CB
Low CB
Low CB
High CB
l
02
2
x
CB02
2
x
CB
l
xCCB
sin0
C
β0
t
l
xCtxCB expsin, 0
BD
l2
2
t
exp0
2
2
x
CD
t
C BB
B
EMA 5001 Physical Properties of Materials Zhe Cheng (2016) 3 Nonsteady State Diff – Fick’s 2nd Law
Special Case – Spin-on Dopant for
Silicon Wafer
Diffusion in Semi-Infinite Bar w/ Fixed Amount of Total Dopant
Doping silicon surface with boron or
phosphorous spin-on dopants and
diffuse at 800-1000 oC
Fick’s 2nd Law
Assumption:
− DB constant
Initial condition
− CB (x =0, t = 0) = ∞; CB (x >0, t = 0) = 0
Boundary condition
− Zero concentration far away from surface
CB (x ∞) = 0
If the total amount of dopant is fixed of N, then
7
x
CD
xt
C BB
B
Dt
x
Dt
NtxCB
4exp,
2
x
C
0
t1 t2 t3
t1 < t2 < t3
EMA 5001 Physical Properties of Materials Zhe Cheng (2016) 3 Nonsteady State Diff – Fick’s 2nd Law
Special Case – Infinite Diffusion Couple
Diffusion in Infinite Diffusion Couple
Two dilute alloys of B in A welded together
Fick’s 2nd Law
Assumption:
− DB constant
Initial condition
− CB (x > 0, t = 0) = C2; CB (x < 0, t = 0) = C1
Boundary condition:
− CB (x ∞, t > 0) = C2 ; CB (x -∞, t > 0) = C1
Solution is
in which error function, erf is given by
8
(1) (2) C1 C2
x
CD
xt
C BB
B
Dt
xerf
CCCCtxC
222, 2121
dyyzerfz
0
2 )exp(2
)(
0
1
erf(x)
x
-1
C
0
t1 t2
x
C1
C2
t0
t0 < t1 < t2
EMA 5001 Physical Properties of Materials Zhe Cheng (2016) 3 Nonsteady State Diff – Fick’s 2nd Law
Special Case – Carburization &
Decarburization of Steel
Diffusion in Semi-Infinite Bar w/ Constant Surface Concentration
Increase/decrease carbon concentration in surface
− Carburization: CH4/CO atmosphere
at elevated temperature (for FCC γ-Fe)
− Decarburization: vacuum at elevated
temperature
Boundary/Initial conditions:
− Carburization:
CB (x = 0) = CS; CB (x ∞) = C0
− Decarburization:
CB (x = 0) = 0; CB (x ∞) = C0
Solutions are
− Carburization
− Decarburization
9
x
C
0
C0
CS
t1 t2 t3
Dt
xerfCCCtxC SS
2)(, 0
Dt
xerfCtxC
2, 0 x
C
0
C0
t1 t2 t3
Carburization of steel
Decarburization of steel
t1 < t2 < t3
t1 < t2 < t3
EMA 5001 Physical Properties of Materials Zhe Cheng (2016) 3 Nonsteady State Diff – Fick’s 2nd Law
Diffusion Length
Example: Carburization of steel
For error function, if erf (z) = 0.5, z ≈ 0.5
Therefore, for concentration profile
When
Indicating
Therefore,
10
x
C
0
C0
CS
t1 t2 t3
Carburization of steel
5.02
Dt
xerf
2
, 0CCtxC S
5.02
Dt
xz
Dtx
Dt
xerfCCCtxC SS
2)(, 0
2
0CCS
x1 x2 x3
If 32
321
xxx
What is the relationship between
t1, t2, and t3 assuming D constant?
32
32
1
DtDtDt
2
3
2
21
32
ttt
Diffusion Length - characteristic length in a material
within which it experiences significant change due to diffusion
EMA 5001 Physical Properties of Materials Zhe Cheng (2016) 3 Nonsteady State Diff – Fick’s 2nd Law
Microscopic View of Diffusion Length (1)
For an interstitial atom
Each random jump with displacement of
After n jumps, total displacement vector is
To obtain absolute displacement length after n jumps, we have
11
Original
position
Position after
n jumps
Rn
n
i
inn rrrrrR1
321 ...
n
i
i
n
i
inn rrRR11
1
1
)1(1
2
1
)2(2
2
1
22
1
1
11
1
22...22i
inn
i
inn
n
i
i
n
i
ii
n
i
i rrrrrrrrrr
ir
1
1 11
2 2n
j
ij
jn
i
j
n
i
i rrr
r1
r4
r5 r3 r2
r1 rn
r6
EMA 5001 Physical Properties of Materials Zhe Cheng (2016) 3 Nonsteady State Diff – Fick’s 2nd Law
Microscopic View of Diffusion Length (2)
Continue from p. 11
Successful jumping occurs only to the nearest neighbor, then for i = 1, 2, … n,
in which is the jumping distance for an (interstitial) atom to its nearest neighbor.
Consider
in which is the angle between and
The displacement after n jumps will be
Now consider random jumping of a large amount of atoms:
Each atoms jumps for n times, and the “average” displacement among all atoms
12
ir
1
1 1
,
2222
n
j
jn
i
ijjn CosnR
1
1 11
2 2n
j
ij
jn
i
j
n
i
inn rrrRR
ijjijj Cosrr ,
2
ijj ,jr ijr
1
1 1
,
2222
n
j
jn
i
ijjn CosnR
EMA 5001 Physical Properties of Materials Zhe Cheng (2016) 3 Nonsteady State Diff – Fick’s 2nd Law
Microscopic View of Diffusion Length (3)
Continued from p. 12
For average over a large amount of atoms, we have
Therefore,
If is the successful jump frequency, and if the n jumps take time t,
From earlier derivation about diffusion of interstitial atoms,
Therefore,
The “average” (root mean square) displacement after time t for random walk is
13
01
1 1
,
n
j
jn
i
ijjCos
tn
ttnRn
2222)(
2
6
1BBD
tDtR Bn 6)( 22
tDR Bn 4.22
21
1 1
,
2222 nCosnR
n
j
jn
i
ijjn
BB D62
EMA 5001 Physical Properties of Materials Zhe Cheng (2016) 3 Nonsteady State Diff – Fick’s 2nd Law
Homework
Porter 3rd Ed, Exercise 2.1, 2.3, 2.6
Due Feb 3 class
14