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Transport

Zuoan Li

NorFERM-2008
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Outline
 Diffusion basis
▲ Diffusion mechanism
▲ Mathematics of diffusion
▲ D vs. T
 Diffusion in electric gradient (conductivity)
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Diffusion
Cu-Ni diffusion couple

Before

heat treatment
After
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Diffusion Mechanism

Vacancy diffusion
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Interstitial diffusion
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Interstitialcy diffusion

Collinear jump
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Proton diffusion

H 2 O  O  VO  2OH 
O             O

Vehicle mechanism

Free transport mechanism (Grotthuss mechanism)
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BaCeO3

Kreur, Annu. Rev. Mater. Res. 2003
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Mathematics of diffusion
Fick’s 1st law
c

A

J: flux of particles across
plane with area A

x

 dc 
J   D 
 dx 
 
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c     j   c        2c
Fick’s 2nd law
t
    
x x
D
  x   D  x2


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Solution to Fick’s 2nd law
Thin film
 Thin layer of radioactive isotopes is located at x=0 of a semi-
infinite sample (self-exhausting source).

c,t   0
 Boundary conditions:

0
cdx  c0

 Concentration after time t:
 x2 
cx,t  
c0
 4 Dt 
exp     
2  Dt          
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Deposition   Annealing   Cutting
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Error function
 The concentration of tracer at x=0 is kept constant after
diffusion (non-exhausting source).

c,t   0
 Boundary conditions:

c (0, t )  cs

 Concentration after time t:

cx,t   c0            x 
 1  erf       
cs  c0                2 Dt 
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O18 depth profile
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Activation energy of diffusion coefficient
 E 
D  D0 exp    
 kT 

Random diffusion:               D r   s 2

frequency of successful jumps

  

: probability that jump can overcome energy barrier
: probability of site being ready for jump
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Hm

 Gm         S m      H m
   exp         exp      exp
RT           R         RT
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: probability of site for jump

Defects
Vacancy
 1           Interstitial solute
Interstitial

Constituents         Nd / N

T, defect structure, and pO2
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  Nd / N                     Dr N  Dd N d

ci                Di
Defects    small & variable   large & constant

Constituents   large & constant   small & variable
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Vacancy diffusion in elemental solids

S d  S m      (H d  H m )
Dr  s  exp
2
exp
R               RT

E  H d  H m

or      E  H m
Constant or frozen vacancy
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Vacancy diffusion in oxides
Predominant oxygen vacancy (MaOb-d)
1
OO  vO  2e'  O2 ( g )
x    ..

2

 
1/ 3
1    1         
vO  n   K v..  PO21/ 6
..

2    4 O 

Sv. .                       H v. .
1/ 3         O
 S m         (         O
 H m )
1
Dr  s    PO21/ 6 exp 3
2         
exp          3
4               R                              RT

Constant oxygen vacancy

S m      H m
Dr  s  exp 2
exp
R         RT
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Protons

 1
Rotation is easy

Erotation  0.1ev

Jump depends on oxygen-oxygen distance

Large soft lattices
2 3
EH  (  ) EV
3 4
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H/D isotope effects

Classical effects:         (mO  mH ) / mO mH

DH / DD  2
0    0

1
 DH / DD  1
0    0

2

Non-classical effects:      ED  EH  0.05ev
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Transport

d
ji  ci B i F  ci B i ( zi e )
dx

I i  zi eji                     i  zi eci ui

ui  zi eBi
Di  kTBi
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Nernst-Einstein relationship

kT
Di  σ i     2 2
ci zi e

ln( iT )  1 / T             E   Activation energy
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Conductivity

σtot  enun  epu p   ci zi eui

ti   i /  t        t   i   1

Solid ionic conductor            ci  n  p
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Electronic conductivity

 el   n   p  enun  epu p
Intrinsic semiconductor
Intrinsic ionization         0  e  h   

  Eg   
K i  np  N C NV exp
 kT     

        
n p

  Eg / 2 
 el  e N v N C (un  u p ) exp
 kT      
          
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Extrinsic semiconductor (n-type)

Low temperature
  Ed / 2 
 el  e N D N C un exp                       Eg
 kT                  2k

n  ND
Intermediate temperature

Log n
 el  eN Dun
 Ed
2k
High temperature
intrinsic
1/T, K
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Non-stoichiometric semiconductor

1
O  v  2e  O2 ( g )
x
O
..
O
'

2

  
n  2 vO  2Kv..
..
O

1/ 3     
PO21/ 6

 tot  2evO uv  enun
..
..
O
Sv..          H v..
 tot   n  eun 2 P
1/ 3    1/ 6
exp       O
exp          O

un  uv..                                                        O2
3R            3RT
O

Defect equilibrium, T, P
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un & up
Magnitude
3 / 2
Non-polar solids            un,lattice  const  T

un,imp  const  T           3/ 2

Polar oxides
ularge,pol  const  T 1/ 2
e                       E 
usm all,pol       D  const  T 1 exp   u 
kT                       kT 
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Transport
ci Di di       d
ji  ci B i F        (     zi e )
kT dx          dx
 i di       d
               2
(         zi e        )
( z i e)           dx            dx

itot   zi e ji
i

d    itot    t i di
      
dx     tot i zi e dx
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Voltage over a sample

S  S z  ze 
d    itot    t n d n 1 d  e
                                          Neutral form
dx     tot n z n e dx e dx
II                  II
itot             tn
U II  I  II  I                dx         dn
I
 tot        n I zn e

itot= 0 → transport number (EMF)

itot≠ 0 → fuel cell
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Flux of a specific species

ti itot    i di        t k d k
ji              2
(   zi           )
zi e ( zi e) dx       k z k dx

ti itot
II
i                  tk
ji X          X          2
(di  zi  dk )
zi e      I
( zi e)             k zk
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Some diffusion terms
Self diffusion            Ds  D r

Tracer diffusion         D *  fD r

Defect diffusion          NDr  N d Dd

Chemical diffusion         Dchem  Dr

Ambipolar diffusion
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Summary
 H m 
D  D0 exp      
 kT 

kT
Di  σ i
ci zi2 e 2
Intrinsic (n=p)

 el  enun  epu p    Extrinsic (n≠p)

Nonstoichiometric (defects)
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Organization committee