Modeling Diffusion, Facilitated Diffusion, and Active Transport.
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1
1
Diffusion
• Diffusion occurs from a concentration gradient• The difference between diffusion in metals and
in ceramics is that the diffusing species inceramics is often charged (vacancies,interstitials)
• The movement of a charged ion results in acurrent
• Thus diffusion and ionic conductivity arelinked
2
• Frenkel Defect --a cation is out of place.• Shottky Defect --a paired set of cation and anion vacancies.
• Equilibrium concentration of defects kT/QDe~!
Defects in ceramic structures
Shottky Defect:
Frenkel Defect
2
3
Schottky and Frenkel defectsSchottky defects in NaCl Both cation and anion are
missing from their regularlattice sites
At room temperature, 1 in1015 sites are vacant
200 kJ/mole (2.7 eV) creationenergy
Cation Frenkel defects inAgCl
Cation displaced fromregular lattice site ontointerstitial site
150 kJ/mole (1.6 eV)creation energy
4
• Impurities must also satisfy charge balance = Electroneutrality
• Ex: NaCl
• Substitutional cation impurity
Impurities
Na+ Cl-
initial geometry Ca2+ impurity resulting geometry
Ca2+
Na+
Na+Ca2+
cation vacancy
• Substitutional anion impurity
initial geometry O2- impurity
O2-
Cl-
anion vacancy
Cl-
resulting geometry
3
5
Crystalline point defects• If cationic impurities are introduced into a solid and the
dopant does not have the same valence as the cationit is replacing, extrinsic defects will be introduced– Ca2+, Y3+ in ZrO2 have anion vacancies– Ca2+ or Cd2+ in NaCl creates cation vacancies
• Real crystals contain both intrinsic and extrinsicdefects
• The dominate defect type depends on temperatureand doping level– Typically
• High temperatures– intrinsic
• Low temperatures– extrinsic defects
6
Kröger-Vink notation• Standard notation for defects in ionic crystals• Composed of 3 parts• Main body identifies the defect
– V = vacancy– M = metal– X = non-metal
• Subscript denotes site the defect occupies– i = interstitial– x = non-metal– M= cation site
• Superscript identifies the effective charge– • = positive charge– ' = negative charge
4
7
ExamplesConsider MgO
V
Mg
'' A vacancy on the Mg siteIt has a double negative charge since Mg is 2+
V
O
•• A vacancy on the O siteIt has a double positive charge since O is 2-
Al
i
••• An Al interstitialIt has a triple positive charge since it is 3+
Al
Mg
• An Al on a Mg siteIt has a positive charge since Al is 3+ and Mg is 2+
Li
Mg
' An L on a Mg siteIt has a negative charge since Li is 1+ and Mg is 2+
8
Defection associations andconcentrations
[V
Mg
'' ]
[Al
Mg
• ]
Concentrations given in brackets:
[e '] = n
[ !h] = p= concentration of electrons= concentration of holes
5
9
Defect reactionsReactions occur for defects, just like other chemicalspecies in the lattice
Consider Shottky defects in MX (M=cation, X=anion)
There is a random distribution of cation and anion vacancies
V
M
'+V
X
•! null
K
S= equilibrium constant = [V
M
' ][VX
• ]
From the definition of the equilibrium constant:
Ks= exp
!"gs
kT
#
$%&
'(= exp
!"hs+T"s
s
kT
#
$%&
'(= exp
"ss
k
#
$%&
'(exp
!"hs
kT
#
$%&
'() exp
!"hs
kT
#
$%&
'(
1
10
Defect reactions
K
S= equilibrium constant = [V
M
' ][VX
• ]
= exp!"g
s
kT
#
$%&
'(
When these are the only defects present, then
[VM
' ] = [VX
• ] = exp!"g
s
2kT
#
$%&
'(
Frenkel defects
M
M!V
M
'+M
i
•
K
F= [V
M
' ][Mi
• ]
Electronic defects
e '+ !h ! null K
e= [e '][ !h] = np
6
11
Rules for defect reactions
• These rules must be satisfied:– Mass conservation
• Not creating or destroying matter!– Electroneutrality
• The + and - charges must be balanced on each side ofreaction equation
– Site ratio conservation• Different crystal structures are not created
aiA
i! b
iB
i
k =
"i
"i
[Bi]b
i
[Ai]a
i
= exp#$G
kT
%&'
()*
12
Oxidization and reduction
M O M O M OO M O M O M
M O M O M O
O M O M O M
1/2O2 (g)2e
M (g)
2h Oxidation - generate holesReduction - generate electrons
V
M
' V
M
''
V
O
••
V
O
•
e
e
CB
VB
7
13
Oxidation and reduction
O
O!V
O
••+ 2e '+
1
2O
2(g)
KR= [V
O
•• ]n2pO2
1/2= K
R
o exp!"g
R
kT
#
$%&
'(
1
2O
2(g) +V
O
••!O
O+ 2 !h
KO=
p2
[VO
•• ]pO2
1/2= K
O
o exp!"g
O
kT
#
$%&
'(
14
Examples
1. Sodium tungstate bronzeNaxWO3 x: 0.32-0.93 perovskite with
V
Na
'
n-type for x < 0.25Metallic conductivity for x > 0.25
2. Ce3S4Ce2.67S4 ρ ~ 10-3 Ω-cmCe3S4 ρ ~ 109 Ω-cm
3. BaTiO3 heated and quenched in H2BaTiO3-x good semiconductorTi4+→Ti3+ + h•
4. ZnO sintering rate increases as pO2 decreasesformation of Zni
8
15
Impurity induced, ion compensated
There is no such thing as a 'pure' materialCan get 99.9999% pure (4 9's, Alfa Aesar, e.g.)
Concentration of impurities is 100 ppm or 10-4
Consider adding CdCl2 to NaClAssume Cd sits on Na site (not interstitial, too large)
Cd is 2+, for charge balance must form Na vacancies orCl interstitials (unlikely)
CdCl
2
2NaCl! "!! Cd
Na
•+V
Na
'+ 2Cl
Cl Na1-2xCdxCl
CaO
ZrO2! "!! Ca
Zr
''+V
O
••+O
OZr1-xCaxO2(1-x)
16
Frenkel defects
M X M X M
X M X M X
M X M X M
X M X M X
M
AgBr, CaF2
N = number of normal sitesN* = number of interstitial sites
nF= (NN
* )1/2 exp!Q
F
2kT
"
#$%
&'= number of Frenkel painr
9
17
Assumptions
• Only have one type of predominate defect– Schottky or Frenkel
• Assume a dilute solution– Neglect interactions between defects
• Constant volume• Energy for defect formation independent
of T
18
Diffusion in lightly doped NaClConsider adding CdCl2 to NaCl
CdCl
2
2NaCl! "!! Cd
Na
•+V
Na
'+ 2Cl
Cl
The Na diffusion coefficient is
DNa
= [VNa
' ]!"2 exp#$G
VNa
*
kT
%
&''
(
)**
ΔGVNa* is the energy for migration of free vacancies
DNa
= [CdCl2]!"2 exp
#SNa
*
k
$
%&
'
() exp
*#HNa
*
kT
$
%&
'
()
At low temperatures, extrinsic behavior observed
10
19
At high temperatures, there are additional vacancies fromSchottky defects that swamp the effect of the impurity
DNa
= [VNa
' ]!"2 exp#$G
Na
*
kT
%
&'
(
)* = !"2 exp
#$sS
2k
%
&'(
)*exp
#$SNa
*
k
%
&'
(
)* exp
#$HS
2kT
%
&'(
)*exp
#$HNa
*
kT
%
&'
(
)*
-1/k(ΔHNa* + 1/2Δhs)
-1/k(ΔHNa*)
1/T
ln D
(cm
2 /se
c)
low Textrinsic
high Tintrinsic
20
Diffusion in cation-deficient oxides
The transition metal oxides are typically cation deficient
Ni1-xO, Co1-xO, Mn1-xO, Fe1-xOx↑ 3 x 10-4 10-2 at 1300˚C
Consider Co1-xO
1
2O
2(g) = O
O+V
Co K
1= [V
Co]a
O2
!1/2
VCo
=VCo
'+ !h K
2=
[VCo
' ]p
[VCo
]
VCo
'=V
Co
''+ !h K
3=
[VCo
'' ]p
[VCo
' ]
x = [V
Co]+ [V
Co
' ]+ [VCo
'' ]
Can get up to x = 0.15, thenform F2O3
electrical conductivity is p-type
11
21
Diffusion in highly doped oxide - cubicstabilized ZrO2
CaO
ZrO2! "!! Ca
Zr
''+V
O
••+O
O
Usually 8-15% added
[Ca
Zr
'' ] = [VO
•• ]
Brouwer approximation:
Large concentrationof oxygen vacanciescompared with mostoxides
ΔG* = 1 eV (small)
Ca1-xZrxO2(1-x) fast ionconductor
single cubicphase
cubi
c +
tetra
gona
ldefectclustering
22
Electrical conductivity• Conductivity values range over 25 orders of magnitude
– Most insulating LiF (band gap > 12 eV)– Superconductors (no band gap)
• Electrical conductivity arises from– Movement of charged ions
• Ionic conductivity– Sensors, electrochemical pumps, solid electrolytes in fuel cells, high T battery systems
– Movement of electrons• Measured electrical conductivity
– From both ions and electrons– σtotal = σelec + σion– ti = transference number = σi/σtotal
• If telec > tion electronic conductor• If tion > telec ionic conductor
• Oxides that are easily reduced are n-type semiconductors– e.g. TiO2, SnO2, ZnO, BaTiO3
• Oxides that are easily oxidized are p-type semiconductors– e.g. transition metal monoxides (NiO, FeO, CoO)
12
2310-18
10-12
10-6
100
106 RuO2 (thick films)
SrTiO3 (photoelectrode)
TiO
V2O3•P2O3 (glass)
TiO2-x (oxygen sensor)
Al2O3 (substrate)
ZnO (varistor)
TiO2
SnO2•In2O3 (transparent elect.)LaNiO3 (fuel cell electrode)
fast ionconductor
solidelectrolyte
insulator
metallic
semiconducting
insulating
Na β-Al2O3 Na/S battery
ZrO2-Y2O3 (1000˚C) Oxygen sensor Li2O-LiCl-B2O3 (glass, 300˚C) KxPb1-xF1.75Primary battery
LaF3, EuF2fluorine ionspecific electrode
NaCl
SiO2
passivation onSi devices
IONIC CONDUCTORS ELECTRON CONDUCTORSYBa2Cu3O7-x
!
(Ω-cm)-1
24
Mobility
Mobility = velocity
driving force chemical, electric field, mechanical
In a chemical gradient, the absolute mobility given by
(ergs/mole) !µ
i
The chemical mobility = Bi' = Bi/NA
note: this is the chemical potential, notthe electrical mobility
Bi=
velocity (cm / sec)
force (ergs / cm)=
vi
Fi
=v
i
!1
NA
" !µi
"x
#
$%
&
'(
)
*++
,
-..
13
25
Mobility and diffusivity
Ji = civi = ciBiFi
Ji= !
1
NA
" !µi
"x
#
$%
&
'( B
ic
i
For an ideal solution,
!µi= µ
o+ RT lna
i= µ
o+ RT ln!
ic
i" µ
o+ RT lnc
i
d !µi
dx= RT
1
ci
#
$%&
'(dc
i
dx
Ji= )
1
NA
RT
ci
dci
dx
#
$%&
'(B
ic
i= )
RT
NA
Bi
dci
dx= )D
i
dci
dx
Di = kTBiNernst-Einstein relation
26
Fi(electrical) = z
ie
d!dx
= zieE
Ji= c
iB
iF
i= c
i
Di
kT
"
#$%
&'(z
ieE) =
ziec
iD
iE
kT
Ji= c
iv
i=
ziec
iD
iE
kT
vi=
zieD
i
kTE
µi=
vi
E=
zieD
i
kT= z
ieB
i=
ziFB
i
NA
= ziFB
i
' F = Faraday's constant =96,500C/mole = eNA
Instead of using a chemical potential (hard to measure), putthese expressions in terms of an electric field
relating electronic mobility with chemical mobility
14
27
Ionic conductivity
!i=
zi
2e
2D
ic
i
kT
µi=
zieD
i
kT !
i= z
ieµ
ic
i
Usually written in (Ω-cm)-1 or S-m-1
where S = Ω-1
e = 1.6 x 10-19 CD in cm2/secc in #/cm3
k in ergs (107ergs = J)
Conductivity depends oncarrier concentrationmobility of carriertemperature
At room temperaturenot many defectsmobility low
28
Diffusion and electrical conductivitymeasurements• Diffusion of a radioactive tracer element Na
was measured• The electrical conductivity was measured
D
1/T
tracer
conductivity
Difference is ~ 2 x 1011
cm2/sec
15
29
The electrochemical potential
• Gradients in chemical potential (concentration)and electric field mobilize defects
• Even in the absence of an external field,internal electric fields are present– Non uniform distribution of space charge
• Driving force for mass transport is theelectrochemical potential (η) instead of just thechemical potential
!
i= µ
i+ z
i"F
F = Faraday's constant = eNA = 96,500 C/mole
30
The force on the particle, Fi, is the negative gradient of ηi
Fi= !
1
NA
d"i
dx
#
$%&
'(
Ji=!c
iB
i
NA
d"i
dx
#
$%&
'(=!c
iB
i
NA
d !µi
dx+ z
iF
d)dx
#
$%
&
'(
Even a modest electrical field can offset the effectof the concentration gradient in the oppositedirection
16
31
Ambipolar diffusion
• Coupled transport of different charged species• Ionic crystals must maintain charge neutrality
– Long range charge separation must be avoided– Charge species are coupled
• Effect of slowing down faster diffusing species andspeeding up slower diffusing species
• Both diffuse with a common diffusivity– Chemical or ambipolar diffusion coefficient
!D
32
Consider MgO
VMg
''+V
O
••! null K
S= [V
Mg
'' ][VO
•• ]
e '+ h•! null K
i= np
OO!V
O
••+ 2e '+
1
2O
2(g) K
O=V
O
••n2pO
2
1/2
The flux of oxygen vacancies must be matched by anequivalent charge flux of electrons outward, holes inward
µe > µh
2JVO = Je
Using the ambipolar diffusion coefficient:
JV
O
••= ! !D
dcV
O
dx
"
#$$
%
&''
and Je '= ! !D
dn
dx
"#$
%&'
17
33
How does D depend on DVO and De?~
Rewrite Fick's first law in terms of ηi acting on the 2 defectsseparately, equating the fluxes to solve for the internal field.
2JV
O
•• = !2c
VO
DV
O
RT
" !µV
O
"x+ 2F
"#"x
$
%&&
'
())
flux is raised by internal field
Je '
= !nD
e
RT
" !µe
"x! F
"#"x
$
%&'
() flux is lowered by internal field
Then, rewriting the 2 expressions to get ∂φ/∂x
!"!x
=RT
F
De# D
VO
( )!c
VO
!x
De+ 2D
VO
JV
O
••= #
3DeD
VO
De+ 2D
VO
$
%&&
'
())
!cV
O
!x
!D =
3DeD
VO
De+ 2D
VO
the concentration gradients, !n
!x= 2
!cV
O
!x
n = 2cV
O
"
#
$$$
%
&
'''
!µ = !µ
o+ RT lnc( )
34
If De >> DVO, the D = 3DVO
Ambipolar diffusion rate is controlled by the slower speciesAmbipolar coupling causes rate to be enhanced by 3X
If DVO>>De, then D = 1.5 D
Slower species is rate controllingAmbipolar coupling increases effective diffusion coefficient
The ambipolar diffusion coefficient is greater than thatof the slower defect, due to charge-coupling to thefaster one
~
~