CACHE Modules on Energy in the Curriculum
Fuel Cells
Module Title: Ion and Electrical Conduction in a Solid Oxide Fuel Cell
Module Author: Michael D. Gross
Module Affiliation: Department of Chemical Engineering
Bucknell University, Lewisburg, PA 17837
Course:
Science of Materials (Introduction to Materials Science)
Text Reference:
Callister and Rethwisch, Fundamentals of Materials Science and
Engineering: An Integrated Approach, 3rd edition (2008)
Sections 12.1-12.4, 12.8, 12.16
Concept Illustrated: Ionic and electronic conduction as a function of temperature
Problem Motivation: Fuel cells are a promising alternative energy conversion technology.
There are numerous types of fuel cells, as described in Module 0, which are typically
distinguished (and named) by either 1) the ion conducted across the electrolyte or 2) the
electrolyte material. In this module, the Solid Oxide Fuel Cell (SOFC) will be explored with
focus on the materials that conduct ions in the electrolyte and conduct electrons in the electrodes.
A general schematic of an SOFC operating on H2 fuel is shown in Figure 1. Oxygen from air
supplied to the cathode is reduced to O2- at the cathode, transported across the electrolyte, and
reacted with H2 fuel at the anode releasing electrons. The electrons released at the anode are
transported through an external circuit where electrical power can be drawn.
H2 + O2- → H2O + 2eFuel
SOFC reactions:
eAnode
O2-
Electrolyte
O2-
_______________________________________________________________________________
Cathode
eAir
½O2 + 2e- → O2-
Figure 1. General schematic of SOFC.
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Anode: H2 + O-2  H2O + 2 eCathode: ½ O2 + 2 e-  O-2
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Overall: H2 + ½ O2  H2O
Figure 2 is a schematic of the processes occurring in a SOFC near the electrode/electrolyte
interfaces, which provides better insight into understanding the requirement for ion and electron
conducting materials in a SOFC. Clearly, oxygen ion conduction is required for transport of O2from the cathode to the anode. In addition, electron conduction is required in the anode for
transport of e- to an external circuit and in the cathode for transport of e- from the external circuit
to the cathode.
a)
H2O
b)
H2
O2
ee-
eeO2O2-
e-
eO2catalyst
O2-
O2-
O2-
O2-
O2-
catalyst
O2O2-
dense electrolyte membrane
dense electrolyte membrane
Figure 2. Schematic of SOFC near a) anode/electrolyte interface and b) cathode/electrolyte interface.
Conducting Materials for SOFC
The most common electrolyte material for SOFC is a dense yttria-stabilized zirconia (YSZ)
membrane, with a composition of ZrO2 doped with 8 mol% Y2O3. For every replacement of 2
Zr4+ ions with 2 Y3+ ions, a total charge of +2 is lost. This loss of +2 charge becomes balanced
by the creation of one O2- vacancy (loss of a -2 charge). With a sufficient concentration of O2vacancies, O2- can be conducted through YSZ via a vacancy hopping mechanism. Ionic
conduction in YSZ is a thermally activated process; that is, conductivity increases with
temperature and can be described by the Arrhenius equation.
The most common e- conducting material for a SOFC cathode is strontium-doped lanthanum
manganate (La1-xSrxMnO3, LSM), which has the cubic perovskite structure. LSM exhibits p-type
semiconducting behavior; e- conductivity increases with increasing temperature. Typically, the
cathode consists of a porous LSM-YSZ composite.
The most common e- conducting material for a SOFC anode is nickel (Ni). Ni exhibits metallic
conduction behavior; e- conductivity decreases with increasing temperature due to increased escattering. Typically, the anode consists of a porous Ni-YSZ composite.
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Arrhenius Plots
Many phenomena in materials science and engineering are classified as thermally activated
processes–they require thermal energy to take place. In a thermally activated process, some
energy barrier needs to be overcome in order for the process to occur. This barrier is referred to
as an activation energy. Increasing the temperature provides extra thermal energy so that more
particles (these could be atoms, molecules, electrons, etc.) can surpass the activation energy.
This ultimately causes the process to occur faster or more easily.
The temperature dependence of all thermally activated processes can be represented by a general
equation of the form:
  EA 
z  Ao exp 

 RT 
Eqn. 1
where
z = the phenomenon that is thermally activated
EA = the activation energy for the process
Ao = a constant, sometimes called the "pre-exponential constant"
R = universal gas constant (8.314 J·mol-1.K-1)
T
= absolute temperature (Kelvin)
In order to easily fit data to an Arrhenius relationship, equation 1 can be manipulated. By taking
the log of both sides, we obtain:
  E A  1 
ln( z )  
   ln( Ao )
 R  T 
Eqn. 2
Equation x is now in the familiar form of a straight line,
y = mx + b
Eqn. 3
where
y = ln(z)
x = (1/T)
m = the slope of the line, which is equal to (-EA/R)
b = the y-intercept of the line, which is equal to ln(Ao)
A plot of data according to Eqn. 2 [ln(z) vs. (1/T)], is called an Arrhenius plot, from which the activation
energy and pre-exponential constant can be determined from the slope and intercept of the best-fit line
through the data.
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Calculating Conduction Resistance in a SOFC
Conductivity and resistance are related according to Eqn. 4.
σ
σ
l
R
A
=
=
=
=
l
RA
conductivity
thickness
resistance
area
Eqn. 4
units: S·cm-1 or Ω-1·cm-1
units: cm
units: Ω
units: cm2
In a fuel cell, the resistance to e- conduction in the anode and cathode and the resistance to O2conduction in the electrolyte can be modeled as resistances in series (Figure 3).
e-
Ranode
O2-
Relectrolyte
e-
Rcathode
Figure 3. Schematic of modeling anode, cathode, and electrolyte as resistances in series.
Considering resistances in series, the total fuel cell resistance to conduction can be calculated
with Eqn. 5.
Rtotal = Ranode + Relectrolyte + Rcathode
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Eqn. 5
Problem Information
Example Problem Statement:
a. Use the YSZ O2- conductivity data below to generate an Arrhenius plot for the
electrolyte.
T (K)
σYSZ (S·cm-1)
873
973
1073
1173
0.0054
0.0167
0.0421
0.0903
b. Calculate the pre-exponential constant and activation energy for the conduction of O2- in
YSZ.
c. Generate a plot of normalized resistance (RA) with units of Ω·cm2 as a function of
temperature for the electrolyte, cathode, anode, and total (electrolyte + cathode + anode),
given the σYSZ expression you developed in part (a) and the information provided below.
 1000 
log σ LSM  YSZ cathode   - 0.41
  0.06
 T 
σ Ni YSZ anode  - 0.39T  1087
electrolyte thickness = lelectrolyte = 20 µm
cathode thickness = lcathode = 50 µm
anode thickness = lanode = 500 µm
d. Assuming the total normalized resistance for the fuel cell cannot exceed 0.1 Ω·cm2, what
temperature range could you operate the fuel cell?
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Example Problem Solution:
Part a.
Step 1. An Arrhenius plot is generated by plotting ln(σ) as a function of T-1.
T (K)
σYSZ (S·cm-1)
T-1 (K-1)
ln(σYSZ)
873
973
1073
1173
0.0054
0.0167
0.0421
0.0903
1.15E-03
1.03E-03
9.32E-04
8.53E-04
-5.22
-4.09
-3.17
-2.40
6.0
ln(σ) [ln(S·cm-1)]
4.0
2.0
0.0
-2.0
-4.0
ln(σ) = -9622T-1 + 5.80
-6.0
0.0E+00 2.0E-04 4.0E-04 6.0E-04 8.0E-04 1.0E-03 1.2E-03 1.4E-03
T-1 [K-1]
Part b.
Step 1. The activation energy for O2- conduction can be calculated with the slope of the
Arrhenius plot.
 EA
 9622 K
R
J 
J
kJ

E A    9622 K   8.314
 80
  80000
mol  K 
mol
mol

slope 
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Step 2. The pre-exponential constant can be calculated with the y-intercept of the Arrhenius
plot.
y  intercept  ln(  o )  5.80
 o  exp( 5.80) S  cm 1  330S  cm 1
Part c.
The normalized resistance, RA, can be calculated with conductivity and thickness. Example
calculations are shown for a temperature of 873 K.
Step 1. Electrolyte
RAelectrolyte, 873 K =
l

20m
20 x10 4 cm


 0.370cm 2  S 1  0.370  cm 2
1
1
0.0054S  cm
0.0054S  cm
Step 2. Cathode
 1000 
log σ LSM  YSZ cathode   - 0.41
  0.06
 T 
σ LSM  YSZ cathode  10
RAcathode, 873 K =



 1000 
 - 0.41
  0.06 
 873 


 0.295S  cm 1
l

50m
50 x10 4 cm

 0.017cm 2  S 1  0.017  cm 2
1
1
0.295S  cm
0.295S  cm
Step 3. Anode
σ NiYSZ anode  - 0.39T  1087   0.39873  1087  747S  cm 1
RAanode, 873 K =
l

500m
500 x10 4 cm


 6.7 x10 5 cm 2  S 1  6.7 x10 5   cm 2
1
1
747 S  cm
747 S  cm
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Step 4. Total RA at 873 K
RAtotal = RAelectrolyte + RAcathode + RAanode = 0.370 Ω·cm2 + 0.017 Ω·cm2 + 6.7x10-5 Ω·cm2
= 0.387 Ω·cm2
Step 5. Generate a plot of RA values as a function of T.
The values to be plotted (and the corresponding conductivities) are as follows:
σ (S·cm-1)
RA (Ω·cm2)
T (K) electrolyte cathode anode electrolyte cathode anode
873
0.0054
0.295
747
0.370
0.017 6.7E-05
973
0.0167
0.330
708
0.120
0.015 7.1E-05
1073
0.0421
0.361
669
0.048
0.014 7.5E-05
1173
0.0903
0.389
630
0.022
0.013 7.9E-05
0.45
electrolyte
cathode
anode
total
0.40
0.35
RA (Ω·cm2)
total
0.387
0.135
0.061
0.035
0.30
0.25
0.20
0.15
0.10
0.05
0.00
850
900
950
1000
1050
1100
1150
1200
Temperature (K)
Another way to view RA as a function of temperature would be to use a log scale on the yaxis. This more easily distinguishes RA values for the electrolyte, cathode, and anode by
order of magnitude.
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1.E+00
electrolyte
cathode
anode
total
RA (Ω·cm2)
1.E-01
1.E-02
1.E-03
1.E-04
1.E-05
850
950
1050
1150
Temperature (K)
Part d.
Operating temperatures of 1007 K and greater would result in RAtotal < 0.1 Ω·cm2 as shown
in the table below. The calculations were completed in the same manner as Part c Step 5.
σ (S·cm-1)
RA (Ω·cm2)
T (K) electrolyte cathode anode electrolyte cathode anode
1006
0.0232
0.341
695
0.086
0.015 7.2E-05
1007
0.0234
0.341
694
0.085
0.015 7.2E-05
1008
0.0236
0.341
694
0.085
0.015 7.2E-05
total
0.101
0.100
0.099
Analysis
It is clear that the resistance to O2- conduction in the electrolyte is the limiting factor for
SOFC, particularly at lower operating temperatures. Two ways to reduce resistance in the
electrolyte would be to make the electrolyte thinner or replace YSZ with a material that has a
higher O2- conductivity.
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Home Problem Statement:
La0.8Sr0.2Ga0.8Mg0.2O3-δ (LSGM) is a potential alternative electrolyte material to YSZ for
operation at lower temperatures due to higher ionic conductivities than YSZ.
a. Generate a plot of LSGM thickness as a function of temperature (from 873 K to 1273 K)
that would result in a normalized resistance (RA) of 0.05 Ω·cm2, given the following values
related to O2- conduction in LSGM.
EA = 67.6 kJ·mol-1
σo = 267 S·cm-1
b. On the same graph generated in part a, plot YSZ thickness as a function of temperature
(from 873 K to 1273 K) that would result in a normalized resistance (RA) of 0.05 Ω·cm2.
Use information provided in the Example problem. How do the results compare?
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