Transcript lecture 12 powerpoint

Defects in Solids
• 0-D or point defects
– vacancies, interstitials, etc. concentrations 
– control mass diffusion
• 1-D or linear defects
– dislocations
– control deformation processes
• 2-D or planar defects
– grain boundaries, surfaces, interfaces
• 3-D or volume defects
– voids, secondary components (phases)
Diffusional Processes
diffusion
coefficient
Pd
flux
H2
c
x
c
0
x
J 0
response  driving force
cH
+ CO
J  D
concentration gradient
Fick’s first law (similar to Ohm’s law)
phenomenological
+ CO2
H
x
Applies under steady state conditions
c(x)  f(t)
Units of D
hydrogen separation membrane
c
#
J  D
area  time
x
length 2
time
#
vol
length
Diffusional Processes
Non steady-state: c(x) = f(t)
A
B
Continuity requirements
  c 
c
J
 D 

x  x 
t
x
t=0
CA
Fick’s second law
CB
c   c 
 D 
t x  x 
t>0
CA
if D  f(x)
CB
# /vol
time
c
 2c
D 2
t
x
# /vol
length 2
length 2
D~
time
Atomistics of diffusion
Random walk
(1)
x
(2)
ao
planes of atoms
tracer species with a concentration gradient
c = concentration #/cm3
n = #/cm2 # density in the plane
n = cao
n1 = on plane (1)
n2 = on plane (2)
Flux from plane (1) to plane (2)
J12  ½ n1
(½ jump to the left)
Flux from plane (2) to plane (1)
J 21  ½ n2
Net from plane (1) to plane (2)
J12  J 21  ½   n1  n2 
c2  c1 c
ao

J net 
  c1  c2 
ao
x
2
J net
 c  3-D
 ½
 
1
 x 
D  ao2 
6
D
ao2
Mechanisms of Diffusion
Vacancy
net transport
vacancy to right
atom to left
Interstitial (self or impurity)
Atomistics from Mechanism
jump distance
D 
geometric constant
~ 1/(# nearest neighbor sites)
2
 ao  P 
crystallographic sites
probability that an atom will
jump into an available site
atom vibrates at frequency
nD = Debye frequency
 DGm 
success rate
exp 

of jumping
k
T
 b 
G
DGm
occupied
 N 

probability that a nearest neighbor site
is vacant (available) for jumping into
Evaluate terms

fraction of atoms that participate
position
empty
 DGm 
   n D exp 

k
T
 b 
Atomistics from Mechanism
fraction of atoms that participate
P and [N] differ depending on mechanism
probability that a nearest neighbor site
is vacant (available) for jumping into
A. Vacancy
•
P = concentration
of vacancies
•
[N] = 1 – P  1
B. Interstitial
•
P = 1 – [N]  1
•
[N] = concentration
of interstitial atoms
C. Substitutional impurity
•
P = concentration
of vacancies
•
[N] = fixed, < 1
D. Interstitial impurity
•
P=1
•
[N] = fixed, < 1
 DG f 
defect concentrations  exp 

k
T
 b 
 Q 
 (DGm  DG f ) 

D
exp
 D  Do exp 


o

k
T
kbT
 b 


Classic Diffusion Problem
Expose a solid material to a gas phase and observe diffusion into the solid
surface concentration
Gas
cs
Solid
c
c
 2c
D 2
t
x
t
t=0
co
initial concentration
Boundary conditions:
at t = 0, c(x) = co
0x
t > 0, c(x=0) = cs
}
x
0 1
c( x)  co
 x 

1

erf


Solution: c  c
2
Dt


s
o
t    erf(0)  0  c(x)  cs
Characteristic diffusion distance & time: set argument = 1, 1-erf(1) = 0.157
l  2 D