Post on 30-Apr-2020
transcript
© 2016 by Zhe Cheng
EMA5001 Lecture 8
Review for Diffusion
EMA 5001 Physical Properties of Materials Zhe Cheng (2016) 8 Diffusion – Review
Expectations about Diffusion (1)
Can name major categories and mechanisms of diffusion
Can give practical examples of applications of diffusion
Understand the similarity & difference between down-hill and up-hill
diffusion
Remember Fick’s 1st and 2nd Law and can derive them based on 1-D
geometry
Can derive thermo activation of diffusion and calculate activation
energy and frequency factor, if given diffusion data at different
temperatures
Can derive concentration profile for steady state diffusion in 1-D
Can name major example solutions Fick’s 2nd Law and assumptions
(no need to remember the math)
Remember expression of diffusion length and can explain how its
significance
2
EMA 5001 Physical Properties of Materials Zhe Cheng (2016) 8 Diffusion – Review
Expectations about Diffusion (2)
Understand the relationship between self diffusion and vacancy
diffusion coefficients
Understand the significance of Kirkendall effect
Understand Darken’s equations
Can derive the relationship between diffusion coefficient and mobility
Can explain the relationship between tracer diffusion, intrinsic
diffusion, and interdiffusion coefficients
Understand why Matano analysis is needed and can explain the
significance of Matano interface
Can describe how diffusion coefficients are obtained in experiments
Can derive relationship between apparent diffusion coefficient for
diffusion through multi-crystalline or single crystalline materials with
dislocations
Can draw the concentration profile for reaction diffusion involving
multi-phases
3
EMA 5001 Physical Properties of Materials Zhe Cheng (2016) 8 Diffusion – Review
Diffusion coefficient of Carbon
in α-Fe (BCC) is ~100 times higher
than in -Fe (FCC) at 910 oC
“Closer packing” for -Fe is cited
“The reason for this huge difference lies in
the fact that the bcc structure is more open
and the diffusion process requires less lattice
distortion (enthalpy or thermodynamically
favored).” http://www.eng.utah.edu/~lzang/images/lecture-7-experimental-
measurement-interdiffusion-coefficient-kirkendall-effect.pdf
Diffusion Coefficient of Carbon in
α-Fe vs. -Fe (1)
4
Deformation-Mechanism Maps, The Plasticity and Creep of Metals and Ceramics, by Harold J Frost, Dartmouth College, USA, and Michael F Ashby, Cambridge University, UK. Chapter 8 http://engineering.dartmouth.edu/defmech/
Crystal structure D0 (mm2/s) Q (kJ/mol)
C in α-Fe (BCC) 2 84.1
C in -Fe (FCC) 20 142
James P Schaffer, et al. The Science & Design of Engineering Materials, 2nd Ed, McGraw-Hill, (1999), p. 131
EMA 5001 Physical Properties of Materials Zhe Cheng (2016) 8 Diffusion – Review
Diffusion Coefficient of Carbon in
α-Fe vs. -Fe (2)
Size & Number of Interstitials in BCC & FCC structures
Carbon occupy octahedral sites in both α-Fe (BCC) and -Fe (FCC)
despite the size of octahedral site is much smaller in α-Fe (BCC)
“at least 85% of the carbon atoms occupy octahedral interstices” G.K. Williamson, “X-ray evidence for the interstitial position of carbon in α- iron,” Acta Cryst. (1953), p.361
The number (density) of octahedral interstitials per unit cell (or per
atom) is much larger for α-Fe (BCC) than for -Fe (FCC)
5
Crystal structure
Packing density
Ratio of radius of largest
tetrahedral atom to host
Ratio of radius of largest
octahedral atom to host
Number of tetrahedral site
per unit cell (per host atoms)
Number of Octahedral site
per unit cell (per host atoms)
BCC 0.68 0.291 0.155 12 (6) 6 (3)
FCC 0.74 0.225 0.414 8 (2) 4 (1)
James P Schaffer, etc. The Science & Design of Engineering Materials, 2nd Ed, McGraw-Hill, (1999), p. 82 & 87
EMA 5001 Physical Properties of Materials Zhe Cheng (2016) 8 Diffusion – Review
Diffusion Coefficient of Carbon in
α-Fe vs. -Fe (3)
The jumping distance in α-Fe (BCC) is much shorter than in -Fe (FCC)
FCC:
BCC:
6
http://www.tf.uni-kiel.de/matwis/amat/def_en/kap_1/illustr/t1_3_3.html
CFeFeFeC rra3
1
3
2
3
4
2
1
2
1
FeFeFeC rra 22
4
2
1
2
1
FCC BCC
EMA 5001 Physical Properties of Materials Zhe Cheng (2016) 8 Diffusion – Review
Successful Jump Frequency,
Vibration Frequency
Thermal vibration frequency
- Thermal vibration frequency
(Successful) Jump frequency
In text book, p. 72-73, (successful) jump frequency was written slightly
differently:
’ is for “in the x direction it makes ’ attempts per second to jump…” (i.e.,
vibration frequency in x direction)
z is the number of such sites (e.g., acceptable interstitials) in 3D
Naturally,
It does not help further understanding, and we did not adopt it here
7
RT
GmB exp
RT
Gz m
B exp'
x
G
Gm
' z
EMA 5001 Physical Properties of Materials Zhe Cheng (2016) 8 Diffusion – Review
Concentration Profile & Darken’s Equation
http://www.matsceng.ohio-state.edu/~gupta/mse732/quiz2.pdf
8
EMA 5001 Physical Properties of Materials Zhe Cheng (2016) 8 Diffusion – Review
Movement for Kirkendall Interface
Displacement of Kirkendall Interface l satisfy l2 = At, if at that time t, the
interface has moved l, what is the velocity of the Kirkendal interface ?
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Atl 2
t
l
t
l
t
At
t
A
dt
Atd
dt
dlu
222
Adtldl 2
l
A
dt
dlu
2
EMA 5001 Physical Properties of Materials Zhe Cheng (2016) 8 Diffusion – Review
Darken’s Equation & Flux due to
Brownian Motion & Interface Movement
The problem and the original solution given is awkward and, most
likely, wrong.
I would use
JA’ = 100 mol/(m2sec); JB’ = -60 mol/(m2sec)
C0 = 5 mol/m3
CB = 5 x 60% = 3 mol/m3 ; CA = 5 x 40% = 2 mol/m3
JA = JA’ + uCA ; JB = JB’ + uCB
JA + JB = 0
Therefore,
JA = 84 mol/(m2sec)
u = -8 m/sec http://www.matsceng.ohio-state.edu/~gupta/mse732/quiz1.pdf
10
EMA 5001 Physical Properties of Materials Zhe Cheng (2016) 8 Diffusion – Review
(a) Steady state constant flux
Assuming constant diffusion
coefficient Linear profile
Changing diffusion coefficient
Curve
(b) D changes with concentration
Constant flux
Porter 3rd Exercise 2.1
11
Surface 1 for
carburization
Surface 2 for
decarburization
x
C
x
CDJ B
BB
21
)()( 21
xx
BB
xx
BBB
x
CxxD
x
CxxDJ
32.07.7
5.2
)(
)(
1
2
2
1
xxD
xxD
x
C
x
C
B
B
xx
B
xx
B
Larger D, Smaller dC/dx
Smaller D, Larger dC/dx
EMA 5001 Physical Properties of Materials Zhe Cheng (2016) 8 Diffusion – Review
Porter 3rd Exercise 2.1
Using average diffusion coefficient D = 5.1x10-11 m2/sec
Flux
The answer given in the book is too complicated and end up with the
same answer
12
m
mkgsm
x
CDJ B
BB002.0
/60%8.0/%)15.0%4.1(/101.5
3211
126104.2 skgmJ B
EMA 5001 Physical Properties of Materials Zhe Cheng (2016) 8 Diffusion – Review
Porter 3rd Exercise 2.3
Concentration profile for “spin on dopant” type
At constant time, we have
Plotting lnCB versus x2,
slope will be -1/(4Dt)
Plotting the data, we have
13
Dt
x
Dt
NtxCB
4exp,
2
Dt
x
Dt
NxCB
4lnln
2
81035.94
1
Dt
smD /101.31035.93600244
1 215
8
y = -9.35E+08x + 4.57E+00 R² = 9.96E-01
2.0
2.5
3.0
3.5
4.0
4.5
5.0
0 1E-09 2E-09 3E-09
EMA 5001 Physical Properties of Materials Zhe Cheng (2016) 8 Diffusion – Review
Porter 3rd Exercise 2.6
Free energy Chemical potential
14
A B
G
(1) (2)
G1
G2
Gsys_eq
1
A
2
A
2
B
1
B
Gα Gβ
(1’) (2’)
eq
A
eq
BGsys_i
A rich B rich
A
B
(1) (2)
eq
AAA '2'1
eq
BBB '2'1
EMA 5001 Physical Properties of Materials Zhe Cheng (2016) 8 Diffusion – Review
Porter 3rd Exercise 2.7
Interdiffusion coefficient
As Zn concentration increases, all three diffusion coefficients increase due to lower
melting point of Zn
15
sec/105.4~ 213 mDXDXD CuZnZnCu
sec/106.2 8 mmx
XDDu Zn
CuZn
22.0ZnX 78.0CuX mmx
X Zn /089.0
sec/105.422.078.0 213 mDD CuZn
sec/1092.2/089.0
sec/106.2 2138
mmm
mmDD CuZn
sec/101.5 213 mDZn
sec/102.2 213 mDCu