材料科学基础
Fundamental of Materials
Prof,Tian Min Bo
Tel,62795426, 62772851
E-mail,tmb@mail.tsinghua.edu.cn
Department of Material Science and Engineering
Tsinghua University,Beijing 100084
Chapter Ⅵ
Diffusion in Solids
Lesson twenty-six
§ 6.1 Introduction
? Ⅰ,Atom movement in solids
mechanical movement
thermal vibration
(short-range migration)
Thermal motion
diffusion (long-range migration)
evaporation &
solidification
surface diffusion
volume diffusion
? Ⅱ.Classification of diffusion in
solids
1,According to,
Steady state diffusion
and Non-steady state diffusion
)( ???C
2., self-diffusion,both in pure metal and in
homogeneous alloy.
,inter-diffusion,down-hill
up-hill
0)( ??? xC
0)( ???xC
3,According to diffusion path
① Volume diffusion (with grain,or lattice)
② Grain boundary diffusion
③ Surface diffusion short-cut diffusion
④ diffusion along dislocation
4,According to structure
Diffusion in single phase
Diffusion in multiphase
? Ⅲ.Significance:
heat treatment,welding,sintering,corrosion
? Ⅳ.Content
phenomenological theory
microscopic theory
§ 6.2 Fick’s Laws
? Ⅰ.Fick ’s first law
For binary,single-phase system tA
x
Cm ?
?
???
f l u xdi f f u s i on
d
d
f l owdi f f u s i onlim
d
d
0
—
—
x
C
D
tA
m
J
A
x
C
D
t
m
t
m
t
?
?
???
?
?
??
?
?
?
??
界面
x
C2 C1 C
2 > C1，均为固定值
For 3 dimensional diffusion:
CDCDJ ????? g r a d
?????? ?? RTQDD e x p0
Discussion:
1) It is an empirical rule
2) minus sign
3) D-material constant.
4) dimension
? Ⅱ.Fick ’s second law
t
C
x
J
x
J
x
JJ
xx
?
?
?
?
?
??
?
?
??
?
?
???? 0
21
0
limlim
?x
x x+?x
x
Jx Jx+?x
2
2
)(
x
C
D
t
C
x
x
C
t
C
?
?
?
?
?
?
?
?
??
??
?
?
if D=const,→
CDC
y
C
x
CD
t
C 2
2
2
2
2
2
2
z
????
?
?
???
?
?
??
?
??
?
??
?
?
For 3 dimensional diffusion:
1,In Cartesian coordination system
)z(z)()( ????????????????? CDyCDyxCDxtC
)
z
(c o n s t,if 2
2
2
2
2
2
?
??
?
??
?
??
?
?? C
y
C
x
CD
t
CD
2,For cylindrically symmetrical diffusion.
3,For spherical symmetrical diffusion.
r
C
r
D
r
CD
r
Cr
rr
D
t
C
?
??
?
??
?
?
?
??
?
?
2
2
)(
r
C
r
D
r
CD
r
Cr
rr
D
t
C
?
??
?
??
?
?
?
??
?
? 2)(
2
2
2
2
Relation between Fick’s first law and Fick’s second law
For spherically symmetrically
21
12
12
2
2
1
1
2
2
2
)(
0
0)(
rr
rr
CC
a
b
r
a
cb
r
a
cb
r
a
c
r
C
r
r
C
r
rr
D
t
C
?
?
??
??????????
?
?
?
?
?
?
?
?
?
?
?
21
12
122 )(4
d
d4
d
d
d
d rr
rr
CCD
r
CrD
r
CDA
t
m
?
??????? ??
§ 6.3 Steady State Diffusion
C
??C
0C
??C
? 0
1r 2r
2
10,
t
tt
r
t 增加
12 rr ??
? ?
???
T
?
??? ???
A B
??C 0C
0T
??C
??
Application to crystallization at initial stage.
)(π4)(π4dd
1
L2
121
12
L
r
CCDrrr
rr
CCD
t
m oo ?? ???
?
???
t
CC
CC
Dr
t
r
CC
DrCC
t
r
CC
DrCCrrm
)(2
d)(d)(
d)(4))(d4(d
LL
L2
1
1
L
1LL
1
L2
1LL1
2
1
??
?
?
??
?
??
??
?
?
??
?
???
?
????
?
?
?
Example
A 0.05cm layer of MgO is deposited between layers of Ni and
Ta to provide a diffusion barrier that prevents reactions
between the two metals,At 1400℃,Ni ions are created and
diffuse through the MgO ceramic to the Ta,Determine the
number of nickel ions that pass through the MgO per second,
The diffusion coefficient of Ni ions in MgO is 9× 10-12cm2/s,
and the lattice parameter of nickel at 1400℃ is 3.6× 10-8cm.
Solution
3
22
38M g O/Ni cm
a t o m s1057.8
cm106.3
c e l lu n i t
a t o m sNi4
??
?
? ?C
cmcm
a t o m s1071.1
05.0
1057.80
3
2422
????
???
?
?
x
C
scm
a t o m sNi
1054.1
)1071.1)(109(
2
13
2412
?
??
????
?
?
?? ?
x
C
DJ
s/a t o m sNi1016.6
221054.1
13
13
??
????Total Ni atoms per second
Examples and Discussions
Exercise
Thank you !
26