Skjalg Erdal1
Dept. of Chemistry, Centre for Materials Science and Nanotechnology, University of Oslo,
FERMIO, Gaustadalleen 21, NO-0349 Oslo, [email protected]
Mixed conduction
Outline• Defects• Derivation of flux equations• Flux of a particular species• Fluxes in a mixed proton, oxygen ion, electron conductor• Fluxes in a mixed proton, electron conductor
– Various defect situations• Fluxes in a mixed proton, oxygen ion conductor• Some materials of interest• Potential issues
Defects
Defects are crucial to the functional properties of materials
Provide paths for transport
Provide traps for other species
Alters the solubility of alien species
Act as donors/acceptors
Become dominating migrating species themselves
Defects
Defects typically dealt with:
Oxygen vacancies: Vo¨
Oxygen interstitials: Oi´´
Hydroxide ion on oxygen site: OHo˙
Metal vacancies: VM´´
Metal interstitials: Mi¨
Addition of aliovalent element to structure: CaLa´
Electronic defects: e´ and h˙
Equilibria
K = Equilibrium constant
ΔH0 = Stand. Enthalpy Change
ΔS0 = Stand. Entropy Change
pX = Partial pressure of gas X
R = Ideal gas constant
T = Absolute temperature
RT
H
R
S
pp
pK
gg
OHOH
H
OHOH
O
00
2/1
222
22
22
2
2expexp
;gOHO2/1H
RT
H
R
S
pvOK
Ovg
OHOH
OHOXO
OH
XOO
002O
O2
expexp]][[
][OH
;OH2OH
2
2
Transport of charged species
xFz
xcB
xcB
x
PcBj i
iii
iii
iiii d
d
d
d
d
d
d
d
Nernst-Einstein:
RT
DcFzBcFzFcz iiiiiiiiii
22
xFz
xFzj i
i
i
ii d
d
d
d2
xFz
xFzFjzi i
i
i
iiii d
d
d
d
ji = flux density
Bi = mech. mobility
ci = concentration
zi = elem. charge
F = Faradays const.
μi = chem. potential
φ = el. potential
σi = conductivity
Di = diff. coefficient
ii = current density
Transport of charged species
),0(d
d
d
dcircuitopenat
xFz
xFzFjzi
kk
k
k
k
kkktot
Then, use the definition of total conductivity and transport number to find an expression for the electrical potential gradient in terms of transport numbers and chemical potential gradient of all charge carriers:
k
ktot
kk
k
tot
kkt
k
k
k
k
dx
d
Fz
t
dx
d
and
Transport of charged species
We now need to represent the chemical potential of charged species as the chemical potential of their neutral counterparts.
Assume chemical equilibrium between neutral and charged species and electrons, the electrochemical red-ox reaction:
zeEE z E ~ neutral chemical entity, z ~ + or -
Equilibrium expression is in terms of products and reactants:
eEE
zddd Z Substitute into expression for dx
d
The chemical potential for the neutral species can be expressed via activities, or partial pressures.
dx
d
Fdx
d
Fz
t
dx
d e
n
n
n
n
1 1k
kt,
Voltage over the sample: assume no gradient in electron chemical potential last term becomes zero.
Transport of charged species
xFz
xFzj i
i
i
ii d
d
d
d2
k
k
k
k
dx
d
Fz
t
dx
d
k
k
k
ki
i
i
ii xz
tz
xFzj
d
d
d
d2
Calculate flux density of species i, in the company of other species
Represents the flux at a particular point in the membrane.
II
I2
II
I
ddk
kk
kii
i
iii z
tz
FzLjdxj
We need the steady state condition constant flux everywhere in membrane
II
I2 dd
1
kk
k
kiii
i
i z
tz
FzLj
Transport of charged species
II
I2 dd
1
kk
k
kiii
i
i z
tz
FzLj
Expression for SS flux.
Still with chemical potentials of charged species substitute in expressions for neutrals
nd Corresponding neutral species
II
I2 dd
1
nn
n
niini
i
i z
tz
FzLj
kd All charged species
II
I2 dd
1
nn
n
niini
i
i z
tz
FzLj
Fluxes in mixed H+, O2-, el-conductors
Possible contributions to proton transport:
Hnd eHt
Hnd2 OHtAmbipolar with oxygen ions:
Ambipolar with electrons:
Hd - driven
If conduction of oxygen ions in an
oxygen gradient, with charge
compensating flow of protons:
Ond2 OHt Od - dep.
II
IOeO2
dt2
1d t t
1-2--2 OHHH LF
j
Fluxes in mixed H+, O2-, el-conductors
Equilibrium between hydrogen, oxygen and water:
22dlnd OO pRT
22dlnd HH pRT and
II
I
)(O)(eO2 2-2
2--2 dlntdln t t2
4 gOgHHHpp
LF
RTj
II
I
)(O)(e2 2-2
2- dlntdln t
2 gOHgHHHpp
LF
RTj
Fluxes in mixed H+, O2-, el-conductor
II
I
)(H)(eH2 22-22 dlnt2dln t t
8 gHgOOOpp
LF
RTj
Equilibrium between hydrogen, oxygen and water:
II
I
)(H)(e2 22-22 dlnt2dln t
8 gOHgOOOpp
LF
RTj
Ambipolar proton-electron conduction
II
I
)(O)(eO2 2-2
2--2 dlntdln t t2
4 gOgHHHpp
LF
RTj
Assume: 0t -2O
II
I
)(e2 2- dlnt
2 gHHHp
LF
RTj
Need to know how the proton conductivity and the electron transport number vary with in order to integrate the expression.)(2 gHp
Ambipolar proton-electron conduction
Examples of partial pressure dependencies with varying defect situations
I2/1)(
II2/1)(2
0,II
I
)(2/1
)(2
0,
II
I
)(2/1
)(2
0,
2222
22
d2
dln2
gHgHH
gHgHH
gHgHH
H
ppLF
RTpp
LF
RT
ppLF
RTj
Assuming: Protons minority defects
Electronic transport no. ~1
II
I
)(4/3
)(2
0,
II
I
)(4/1
)(2
0,
22
22
d2
dln2
gHgHH
gHgHH
H
ppLF
RT
ppLF
RTj
Assuming: Protons majority defects,
compensated by electrons
Electronic transport no. ~1
Examples of partial pressure dependencies with varying defect situations
II
I
)()(
2
II
I
)(2
2
2
2
d1
2
dln2
gHgH
H
gHH
H
ppLF
RT
pLF
RTj
Assuming: Protons majority defects, compensated by acceptor dopants
Electronic transport no. ~1
Ambipolar proton-electron conduction
Proton concentration and conductivity independent of pH2
II
I
)(2
II
I
)(2
II
I
)(2
2
22
dln2
dln2
dln2
gHe
gHHegHeHH
pLF
RT
ptLF
RTpt
LF
RTj
Ambipolar proton-electron conduction
If the transport number of protons ~1:
If protons charge compensate acceptors, the electronic conductivities have dependencies:
2/1)(2 gHe
p 2/1)(2
gHhp
Ambipolar proton-electron conduction
Examples of partial pressure dependencies with varying defect situations
II
I
)(2/1
)(2
0,
II
I
)(2/1
)(2
0,
22
22
d2
dln2
gHgHn
gHgHn
H
ppLF
RT
ppLF
RTj
Assuming: Protons majority defects, compensated by acceptors
Protonic transport no. ~1
Limiting n-type conductivity
II
I
)(2/3
)(2
0,
II
I
)(2/1
)(2
0,
22
22
d2
dln2
gHgHp
gHgHp
H
ppLF
RT
ppLF
RTj
Assuming: Protons majority defects, compensated by acceptors
Protonic transport no. ~1
Limiting p-type conductivity
Ambipolar proton-oxygen ion conduction
If the material is a mixed proton-oxygen ion conductor with negligible electronic transport number:
II
I
)(H2
II
I
)(O2 22
2-2 dlnt
2dlnt
2 gOHOgOHHHp
LF
RTp
LF
RTj
Water vapor
pressure ~ driving force
If material is acceptor doped, and protons or oxygen vacancies can be majority (compensating) defects:
If vO¨ compensating: 2/1
)(2 gOHHp
If H˙ compensating: 1
)(22
gOHOp
TITANATES
TUNGSTATES
NIOBATES
TANTALATES
10-6
10-5
10-4
10-3
10-2
10-1T
ota
l co
ndu
ctiv
ity,
tot /
S
/cm
10-40 10-30 10-20 10-10 100
Oxygen pressure, pO2 / atm
1200°C 1000°C 800°C 600°C 400°C
To
tal c
on
duct
ivity
/
S/c
m
2
3
4
5
6
7
0.001
2
3
4
5
10-20
10-15
10-10
10-5
100
La0.99Ca0.01NbO4
Oxygen pressure, pO 2 / atm
1200
800
1000
LaTaO4
LaNbO4
La6WO12
La2Ti2O7
Oxygen pressure, pO2 / atm
10-20
10-15
10-10
10-5
100
La0.99Ca0.01TaO4
To
tal c
on
duct
ivity
/
S/c
m
2
3
4
5
67
0.001
2
3
4
5
1200
1000
800
2
4
68
2
4
68
2
4
10-30 10-20 10-10 100
Oxygen pressure, / atm pO2
600°C
800°C
1000°C
To
tal c
on
duct
ivity
,
to
t /
S
/cm
10-2
10-4
10-3
41
Oh 2p 4
1
Oe 2p
)g(O2
1e2vO 2O
XO
OO OHv2cAc
Material examples
R. Haugsrud,2007
elH
elHambH2
J
0.01
2
4
6
80.1
2
4
6
81
2
4
6
810
Hyd
roge
n pe
rmea
nce,
JH
2 /
mL
N /
min
cm2
1.21.11.00.90.80.7
1000K/T
1200 1000 800 600°C
LaW1/6O2
SrCeO3
ErW1/6O2
assuming 1% acceptor doping
Material examples
(pH2)1/2-dep
ln (pH2)-dep.
10-5
10-4
10-3
10-2
Con
duct
ivity
,
/ S
/cm
1.81.61.41.21.00.8
1000K/T
1000 800 600 400°C
tot
H
O
el
ramp(10 kHz)
Material examples
Partial conductivities modeled under reducing conditions
La6WO12
k
ktot
Protons dominate until ~ 800 °C
All conductivities rise with rising T, until ~ 800 °C
Total conductivity rise with rising T
,2O e
increase in entire T-window
We have a small T-region where 2O dominates
Potential Issues
Neutral H diffusion could lead to ambipolar conductivity measurements not telling the whole tale about hydrogen transport
Wet H2 Wet Ar
H2
H+
e -H2= 2H++2e- H2
H2O= H2+1/2O2
O2-
Water splitting and oxygen conduction giving hydrogen on wrong side difficult to measure correct hydrogen flux
•Dry sweep?
H2= 2HH
H2
II
I
)(e2 2- dlnt
2 gHHHp
LF
RTj
Potential Issues
What if the electronic transport number is dependent on oxygen partial pressure gradients?
How do we integrate the expression for the flux density in such a case?
II
I
)(e
1)(1
II
I
)(e)(1
II
I
)(e 2-
22-
22- dtdlntdlnt gH
XgHgH
XgHgHHH
ppKppKpKj
Integration by parts over a beer, anyone?
Sources
Norby, T. and Haugsrud, R., 2007, Membrane Technology Vol. 2: Membranes for energy conversion, Weinheim: WILEY-VCH
Kofstad, P. and Norby. T, 2006, Defects and transport in crystalline solids, University of Oslo
Haugsrud, R. 2007, New High-Temperature Proton Conductors (HTPC)-Applications in Future Energy Technology, New Materials for Membranes, GKSS
Serra, E., Bini, A.C., Cosoli, G. and Pilloni, L., 2005, Journal of the American Ceramic Society, 88, 15-18
Cheng, S., Gupta, V.K. and Lin, J.Y.S., 2005, Solid State Ionics, 176, 2653-2663
Hamakawa, S., Li, L., Li, A. and Iglesia, E., 2002, Solid State Ionics, 148, 71-83