CHAPTER 4
FUEL CELL CHARGE TRANSPORT
The previous chapter on reaction kinetics detailed one of the most pivotal steps in the electrochemical generation of electricity: the production and consumption of charge via electrochemical half reactions. In this chapter, we address an equally important step in the
electrochemical generation of electricity: charge transport. Charge transport “completes
the circuit” in an electrochemical system, moving charges from the electrode where they
are produced to the electrode where they are consumed.
There are two major types of charged species: electrons and ions. Since both electrons
and ions are involved in electrochemical reactions, both types of charge must be transported.
The transport of electrons versus ions is fundamentally different, primarily due to the large
difference in mass between the two. In most fuel cells, ion charge transport is far more difficult
than electron charge transport; therefore, we are mainly concerned with ionic conductivity.
As you will discover, resistance to charge transport results in a voltage loss for fuel
cells. Because this voltage loss obeys Ohm’s law, it is called an ohmic, or IR, loss. Ohmic
fuel cell losses are minimized by making electrolytes as thin as possible and employing
high-conductivity materials. The search for high-ionic-conductivity materials will lead to
a discussion of the fundamental mechanisms of ionic charge transport and a review of the
most important electrolyte material classes.
4.1
CHARGES MOVE IN RESPONSE TO FORCES
The rate at which charges move through a material is quantified in terms of flux (denoted
with the symbol J ). Flux measures how much of a given quantity flows through a material
per unit area per unit time. Figure 4.1 illustrates the concept of flux: Imagine water flowing down this tube at a volumetric flow rate of 10 L/s. If we divide the flow rate by the
Fuel Cell Fundamentals: Third Edition. Ryan O’Hayre, Suk-Won Cha, Whitney G. Colella and Fritz B. Prinz
© 2016 by John Wiley & Sons, Inc. Published by John Wiley & Sons, Inc.
117
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FUEL CELL CHARGE TRANSPORT
A
JA
A
Figure 4.1. Schematic of flux. Imagine water flowing down this tube at a volumetric flow rate of
10 L/s. Dividing this flow rate by the cross-sectional area of the tube (A) gives the flux JA of water
moving down the tube. Generally, flux is measured in molar rather than volumetric quantities, so in
this example the liters of water should be converted to moles.
cross-sectional area of the tube (A), we get the volumetric flux JA of water moving down
the tube. In other words, JA gives the per-unit-area flow rate of water through the tube. Be
careful! Remember that flux and flow rate are not the same thing. By computing a flux, we
are normalizing the flow rate by a cross-sectional area.
The most common type of flux is a molar flux (typical units are mol/cm2 ⋅ s). Charge flux
is a special type of flux that measures the amount of charge that flows through a material
per unit area per unit time. Typical units for charge flux are C/cm2 ⋅ s = A∕cm2 . From these
units, you may recognize that charge flux is the same thing as current density. To denote
that charge flux represents a current density and carries different units than molar flux, we
give it the symbol j. The quantity zi F is required to convert from molar flux J to charge flux
j, where zi is the charge number for the charge-carrying species (e.g., zi is +1 for Na+ , –2
for O2– , etc.) and F is Faraday’s constant:
j = zi FJ
(4.1)
ELIMINATE CONFUSION BETWEEN zi AND n
As we move from the discussion of electrochemical kinetics (Chapter 3) to a discussion
of charge transport (Chapter 4), it is important to recognize the difference between the
quantities zi and n. The quantity n, which we have used throughout the book, refers to
the number of electrons transferred during an electrochemical reaction. For example, in
the electrochemical half reaction
H2 → 2H+ + 2e−
two electrons are transferred per mole of H2 gas reacted, and therefore n = 2. In contrast,
the quantity zi , which we introduce here in Chapter 4, refers to the amount of charge
carried by a charged species. For the charged species H+ , as an example, zi = +1, while
for the charged species e– , zi = −1.
CHARGES MOVE IN RESPONSE TO FORCES
In all materials, a force must be acting on the charge carriers (i.e., the mobile electrons
or ions in the material) for charge transport to occur. If there is no force acting on the charge
carriers, there is no reason for them to move! The governing equation for transport can be
generalized (in one dimension) as
∑
Ji =
Mik Fk
(4.2)
k
Where Ji represents a flux of species i, the Fk ’s represent the k different forces acting on i,
and the Mik ’s are the coupling coefficients between force and flux. The coupling coefficients
reflect the relative ability of a species to respond to a given force with movement as well
as the effective strength of the driving force itself. The coupling coefficients are therefore a
property both of the species that is moving and the material through which it is moving. This
general equation is valid for any type of transport (charge, heat, mass, etc.). In fuel cells,
there are three major driving forces that give rise to charge transport: electrical driving
forces (as represented by an electrical potential gradient dV∕dx), chemical driving forces
(as represented by a chemical potential gradient d𝜇∕dx), and mechanical driving forces (as
represented by a pressure gradient dP∕dx).
As an example of how these forces give rise to charge transport in a fuel cell, consider
our familiar hydrogen–oxygen PEMFC (see Figure 4.2). As hydrogen reacts in this fuel
e–
–
+
e– H+
–+
e– H+
–+
+
H2 e– H+
Anode
H
– + O2
e– H+
–+
e– H+
–+
Electrolyte
Cathode
Figure 4.2. In a H2 –O2 fuel cell, accumulation of protons/electrons at the anode and depletion of
protons/electrons at the cathode lead to voltage gradients which drive charge transport. The electrons move from the negatively charged anode electrode to the positively charged cathode electrode.
The protons move from the (relatively) positively charged anode side of the electrolyte to the (relatively) negatively charged cathode side of the electrolyte. The relative charge in the electrolyte at the
anode versus the cathode arises due to differences in the concentration of protons. This concentration
difference can also contribute to proton transport between the anode and cathode.
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FUEL CELL CHARGE TRANSPORT
cell, protons and electrons accumulate at the anode, while protons and electrons are consumed at the cathode. The accumulation/depletion of electrons at the two electrodes creates
a voltage gradient, which drives the transport of electrons from the anode to the cathode.
In the electrolyte, accumulation/depletion of protons creates both a voltage gradient and a
concentration gradient. These coupled gradients then drive the transport of protons from
the anode to the cathode.
In the metal electrodes, only a voltage gradient drives electron charge transport. However, in the electrolyte, both a concentration (chemical potential) gradient and a voltage
(electrical potential) gradient drive ion transport. How do we know which of these two
driving forces is more important? In almost all situations, the electrical driving force dominates fuel cell ion transport. In other words, the electrical effect of the accumulated/depleted
protons is far more important for charge transport than the chemical concentration effect
of the accumulated/depleted protons. The underlying reasons why electrical driving forces
dominate fuel cell charge transport are explained for the interested reader in an optional
section near the end of this chapter (see Section 4.7).
For the case where charge transport is dominated by electrical driving forces,
Equation 4.2 can be rewritten as
dV
(4.3)
j=𝜎
dx
where j represents the charge flux (not molar flux), dV∕dx is the electric field providing the
driving force for charge transport, and 𝜎 is the conductivity, which measures the propensity
of a material to permit charge flow in response to an electric field. This important application
of Equation 4.2 simplifies the terms of fuel cell charge transport. In certain rare situations,
both the concentration effects and electric potential effects may become important; in these
cases, the charge transport equations become considerably more difficult.
Comparing Equation 4.3 to Equation 4.2, it is apparent that conductivity 𝜎 is nothing
more than the name of the coupling coefficient that describes how flux and electrical driving forces are related. The relevant coupling coefficient that describes transport due to a
chemical potential (concentration) gradient is called diffusivity. For transport due to a pressure gradient, the relevant coupling coefficient is called viscosity. These transport processes
are summarized in Table 4.1 using molar flux quantities.
TABLE 4.1. Summary of Transport Processes Relevant to Charge Transport
Transport Process
Driving Force
Coupling Coefficient
Equation
Conduction
Electrical potential gradient,
dV∕dx
Conductivity 𝜎
J=
Diffusion
Concentration gradient, dc∕dx
Diffusivity D
Convection
Pressure gradient, dp∕dx
Viscosity 𝜇
dc
dx
Gc dp
J=
𝜇 dx
𝜎 dV
|zi |F dx
J = −D
Note: The transport equation for convection in this table is based on Poiseuille’s law, where G is a geometric
constant and c is the concentration of the transported species. Convection flux is often calculated simply as
J = 𝑣ci , where v is the transport velocity.
CHARGE TRANSPORT RESULTS IN A VOLTAGE LOSS
4.2
CHARGE TRANSPORT RESULTS IN A VOLTAGE LOSS
Unfortunately, charge transport is not a lossless process. It occurs at a cost. For fuel cells,
the penalty for charge transport is a loss in cell voltage. Why does charge transport result
in a voltage loss? The answer is because fuel cell conductors are not perfect—they have an
intrinsic resistance to charge flow.
Consider the uniform conductor pictured in Figure 4.3. This conductor has a constant
cross-sectional area A and length L. Applying this example conductor geometry to our
charge transport equation 4.3 produces
V
L
(4.4)
( )
L
𝜎
(4.5)
j=𝜎
Solving for V yields
V=j
You might recognize that this equation is similar to Ohm’s law: V = iR. In fact,
since charge flux (current density) and current are related by i = jA, we can rewrite
Equation 4.5 as
( )
L
= iR
(4.6)
V=i
A𝜎
where we identify the quantity L∕A𝜎 as the resistance R of our conductor. The voltage V in
this equation represents the voltage which must be applied in order to transport charge at
a rate given by i. Thus, this voltage represents a loss: It is the voltage that is expended,
or sacrificed, in order to accomplish charge transport. This voltage loss arises due to our
conductor’s intrinsic resistance to charge transport, as embodied by 1/𝜎.
Length = L
Area = A
j
j
R = L/Aσ
V
V
0
V = jL/σ = iR
0
x
L
Figure 4.3. Illustration of charge transport along a uniform conductor of cross-sectional area A,
length L, and conductivity 𝜎. A voltage gradient dV/dx drives the transport of charge down the conductor. From the charge transport equation j = 𝜎(dV∕dx) and the conductor geometry, we can derive
Ohm’s law: V = iR. The resistance of the conductor is dependent on the conductor’s geometry and
conductivity: R = L∕𝜎A.
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FUEL CELL CHARGE TRANSPORT
Voltage (V)
Because this voltage loss obey’s Ohm’s law, we call it an “ohmic” loss. Like the activation overvoltage loss (𝜂act ) introduced in the previous chapter, we give this voltage loss the
symbol η. Specifically, we label it 𝜂ohmic to distinguish it from 𝜂act . Rewriting Equation 4.6
to reflect our nomenclature and explicitly including both the electronic (Relec ) and ionic
Eo
Anode
Electrolyte
Cathode
Distance (x)
Voltage (V)
(a)
η act,C
η act,A
Anode
V
Electrolyte
Eo
Cathode
Distance (x)
(b)
Voltage (V)
122
η ohmic
Anode
Electrolyte
o
V E
Cathode
Distance (x)
(c)
Figure 4.4. (a) Hypothetical voltage profile of a fuel cell at thermodynamic equilibrium (recall
Figure 3.7). The thermodynamic voltage of the fuel cell is given by E0 . (b) Effect of anode and cathode
activation losses on the fuel cell voltage profile (recall Figure 3.9). (c) Effect of ohmic losses on fuel
cell voltage profile. Although the overall fuel cell voltage increases from the anode to the cathode,
the cell voltage must decrease between the anode side of the electrolyte and the cathode side of the
electrolyte to provide a driving force for charge transport.
CHARGE TRANSPORT RESULTS IN A VOLTAGE LOSS
(Rionic ) contributions to fuel cell resistance gives
𝜂ohmic = iRohmic = i(Relec + Rionic )
(4.7)
Because ionic charge transport tends to be more difficult than electronic charge transport,
the ionic contribution to Rohmic tends to dominate.
The direction of the voltage gradient in an operating fuel cell electrolyte can often seem
nonintuitive. As Figure 4.4c illustrates, although overall fuel cell voltage increases from
the anode to the cathode, the cell voltage must decrease between the anode side of the
electrolyte and the cathode side of the electrolyte to provide a driving force for charge
transport.
Example 4.1 A 10-cm2 PEMFC employs an electrolyte membrane with a conductivity of 0.10 Ω−1 ⋅ cm−1 . For this fuel cell, Relec has been determined to be 0.005 Ω.
Assuming the only other contribution to cell resistance comes from the electrolyte
membrane, determine the ohmic voltage loss (𝜂ohmic ) for the fuel cell at a current
density of 1 A∕cm2 in the following cases: (a) the electrolyte membrane is 100 𝜇m
thick; (b) the electrolyte membrane is 50 𝜇m thick.
Solution: We need to calculate Rionic based on the electrolyte dimensions and then
use Equation 4.7 to calculate 𝜂ohmic . Since the fuel cell has an area of 10 cm2 , the
current i of the fuel cell is 10 A:
i = jA = 1 A∕cm2 × 10 cm2 = 10 A
(4.8)
From Equation 4.6 we can calculate Rionic for the two cases (a), (b) given in this
problem:
L
0.01 cm
=
= 0.01 Ω
−1
𝜎A (0.10 Ω ⋅ cm−1 )(10 cm2 )
0.005 cm
=
= 0.005 Ω
(0.10 Ω−1 ⋅ cm−1 )(10 cm2 )
Case (a): Rionic =
Case (b): Rionic
(4.9)
Inserting these values into Equation 4.7 and using i = 10 A gives the following
values for 𝜂ohmic :
Case (a): 𝜂ohmic = i(Relec + Rionic ) = 10 A(0.005 Ω + 0.01 Ω) = 0.15 V
Case (b): 𝜂ohmic = 10 A(0.005 Ω + 0.005 Ω) = 0.10 V
(4.10)
With everything else equal, making the membrane thinner reduces the ohmic
loss! However, note that the payoff does not scale directly with membrane thickness.
Although the membrane thickness was cut in half in this example, the ohmic loss was
only reduced by one-third. This occurs because not all of the fuel cell’s resistance
contributions come from the electrolyte.
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FUEL CELL CHARGE TRANSPORT
4.3 CHARACTERISTICS OF FUEL CELL CHARGE TRANSPORT
RESISTANCE
As Equation 4.7 implies, charge transport linearly decreases fuel cell operating voltage
as current increases. Figure 4.5 illustrates this effect. Obviously, if fuel cell resistance is
decreased, fuel cell performance will improve.
Fuel cell resistance exhibits several important properties. First, resistance is geometry
dependent, as Equation 4.6 clearly implies. Fuel cell resistance scales with area: To normalize out this effect, area-specific resistances are used to compare fuel cells of different
sizes. Fuel cell resistance also scales with thickness; for this reason, fuel cell electrolytes
are generally made as thin as possible. Additionally, fuel cell resistances are additive; resistance losses occurring at different locations within a fuel cell can be summed together in
series. An investigation of the various contributions to fuel cell resistance reveals that the
ionic (electrolyte) component to fuel cell resistance usually dominates. Thus, performance
improvements may be won by the development of better ion conductors. Each of these
important points will now be addressed.
4.3.1
Resistance Scales with Area
Since fuel cells are generally compared on a per-unit-area basis using current density instead
of current, it is generally necessary to use area-normalized fuel cell resistances when discussing ohmic losses. Area-normalized resistance, also known as area-specific resistance
(ASR), carries units of Ω ⋅ cm2 . By using ASR, ohmic losses can be calculated from current
density via
(4.11)
𝜂ohmic = j(ASRohmic )
1.2
Cell voltage (V)
124
Theoretical EMF or ideal voltage
Ohmic loss:
ηohmic = iRohmic
Rohmic =
0.50 Ω
Rohmic =
0.75 Ω
0.5
Rohmic =
1.0 Ω
Current (A)
1.0
Figure 4.5. Effect of ohmic loss on fuel cell performance. Charge transport resistance contributes
a linear decrease in fuel cell operating voltage as determined by Ohm’s law (Equation 4.7). The
magnitude of this loss is determined by the size of Rohmic . (Curves calculated for Rohmic equal 0.50 Ω,
0.75 Ω, and 1.0 Ω, respectively.)
CHARACTERISTICS OF FUEL CELL CHARGE TRANSPORT RESISTANCE
where ASRohmic is the ASR of the fuel cell. Area-specific resistance accounts for the fact
that fuel cell resistance scales with area, thus allowing fuel cells of different sizes to be
compared. It is calculated by multiplying a fuel cell’s ohmic resistance Rohmic by its area:
ASRohmic = Afuel cell Rohmic
(4.12)
Be careful, you must multiply resistance by area to get ASR, not divide! This calculation
will probably seem unintuitive at first. Because a large fuel cell has so much more area
to flow current through than a small fuel cell, its resistance is far lower. However, on a
per-unit-area basis, their resistances should be about the same; therefore, the resistance of
the large fuel cell must be multiplied by its area. This concept may be more understandable
if you recall the original definition of resistance in Equation 4.6:
R=
L
A𝜎
(4.13)
Since resistance is inversely proportional to area, multiplication by area is necessary to
get area-independent resistances. This point is reinforced by Example 4.2.
Example 4.2 Consider the two fuel cells illustrated in Figure 4.6. At a current density
of 1 A∕cm2 , calculate the ohmic voltage losses for both fuel cells. Which fuel cell
incurs the larger ohmic voltage loss?
Fuel cell 1
A 1 = 1 cm 2
R1 = 0.1 Ω
Fuel cell 2
A 2 = 10 cm2
R2 = 0.02 Ω
Fuel cell 1 ASR
R1A1 = 0.1 Ω . cm2
Fuel cell 2 ASR
R2A2 = 0.2 Ω . cm2
Figure 4.6. The importance of ASR is illustrated by these two fuel cells. Fuel cell 2 has lower
total resistance than fuel cell 1 but yields a larger ohmic loss for a given current density. Fuel cell
resistance is best compared using ASR rather than R.
Solution: There are two ways to solve this problem. To calculate voltage loss based
on current density, we can either convert the resistances of the fuel cells to ASRs and
then use Equation 4.11 (solution 1) or convert the current densities into currents and
use Equation 4.6 (solution 2).
Solution 1: Calculating the ASRs for the two fuel cells gives
ASR1 = R1 A1 = (0.1 Ω)(1 cm2 ) = 0.1 Ω ⋅ cm2
ASR2 = R2 A2 = (0.02 Ω)(10 cm2 ) = 0.2 Ω ⋅ cm2
(4.14)
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FUEL CELL CHARGE TRANSPORT
Then, the ohmic voltage losses for the two cells can be calculated using
Equation 4.11:
𝜂1,ohmic = j(ASR1 ) = (1 A∕cm2 )(0.1 Ω ⋅ cm2 ) = 0.1 V
𝜂2,ohmic = j(ASR2 ) = (1 A∕cm2 )(0.2 Ω ⋅ cm2 ) = 0.2 V
(4.15)
Solution 2: Converting current densities for the two fuel cells into currents gives
i1 = jA1 = (1 A∕cm2 )(1 cm2 ) = 1 A
i2 = jA2 = (1 A∕cm2 )(10 cm2 ) = 10 A
(4.16)
Then, the ohmic voltage losses for the two cells can be calculated using
Equation 4.6:
𝜂1,ohmic = i1 (R1 ) = (1 A)(0.1 Ω) = 0.1 V
(4.17)
𝜂2,ohmic = i2 (R2 ) = (10 A)(0.02 Ω) = 0.2 V
In both solutions, the same answer is obtained; cell 2 incurs a greater voltage loss.
Although the total resistance of cell 2 is lower than cell 1 (0.02 Ω versus 0.1 Ω), the
ASR of cell 2 is higher than that of cell 1. Thus, on an area-normalized basis, cell 2
is actually more “resistive” than cell 1 and leads to poorer fuel cell performance.
4.3.2
Resistance Scales with Thickness
Referring again to Equation 4.6, it is apparent that resistance scales not only with the
cross-sectional area of the conductor but also with the length (thickness) of the conductor.
If we normalize resistance by using ASR, then
ASR =
L
𝜎
(4.18)
The shorter the conductor length L, the lower the resistance. It is intuitive that a shorter
path results in less resistance.
Ionic conductivity is orders of magnitude lower than the electronic conductivity of metals, so minimizing the resistance of the fuel cell electrolyte is essential. Hence, we want the
shortest path possible for ions between the anode and the cathode. Fuel cell electrolytes,
therefore, are designed to be as thin as possible. Although reducing electrolyte thickness
improves fuel cell performance, there are several practical issues that limit how thin the
electrolyte can be made. The most important limitations are as follows:
• Mechanical Integrity. For solid electrolytes, the membrane cannot be made so thin
that it risks breaking or develops pinholes. Membrane failure can result in catastrophic
mixing of the fuel and oxidant!
CHARACTERISTICS OF FUEL CELL CHARGE TRANSPORT RESISTANCE
• Nonuniformities. Even mechanically sound, pinhole-free electrolytes may fail if the
thickness varies considerably across the fuel cell. Thin electrolyte areas may become
“hot spots” that are subject to rapid deterioration and failure.
• Shorting. Extremely thin electrolytes (solid or liquid) risk electrical shorting, especially when the electrolyte thickness is on the same order of magnitude as the electrode
roughness.
• Fuel Crossover. As the electrolyte thickness is reduced, the crossover of reactants may
increase. This leads to an undesirable parasitic loss, which can eventually become so
large that further thickness decreases are counterproductive.
• Contact Resistance. Part of the electrolyte resistance is associated with the interface
between the electrolyte and the electrode. This “contact” resistance is independent of
electrolyte thickness.
• Dielectric Breakdown. The ultimate physical limit to solid-electrolyte thickness is
given by the electrolyte’s dielectric breakdown properties. This limit is reached when
the electrolyte is made so thin that the electric field across the membrane exceeds the
dielectric breakdown field for the material.
For most solid-electrolyte materials, the ultimate limit on thickness, as predicted by
the dielectric breakdown field, is on the order of several nanometers. However, the other
practical limitations listed above currently limit achievable thickness to about 10–100 𝜇m,
depending on the electrolyte.
4.3.3
Fuel Cell Resistances Are Additive
As Figure 4.7 illustrates, the total ohmic resistance presented by a fuel cell is actually a
combination of resistances coming from different components of the device. Depending on
how much precision is needed, it is possible to assign individual resistances to the electrical
interconnections, anode electrode, cathode electrode, anode catalyst layer, cathode catalyst
layer, electrolyte, and so on. It is also possible to ascribe contact resistances associated with
the interfaces between the various layers in the fuel cell (e.g., a flow structure/electrode contact resistance). Because the current produced by the fuel cell must flow serially through all
of these regions, the total fuel cell resistance is simply the sum of all the individual resistance contributions. Unfortunately, it is experimentally very difficult to distinguish between
all the various sources of resistance loss.
You might think that it should be a relatively easy experimental task to measure the
resistance of each component in a fuel cell (e.g., the electrodes, the flow structures, the
interconnections, the membrane) before assembling them together into a device. However,
such measurements never completely reflect the true total resistance of a fuel cell device.
Variations in contact resistances, assembly processes, and operating conditions make
total fuel cell resistance difficult to predict. These factors make fuel cell characterization
extremely challenging, as discussed in Chapter 7, and emphasize the necessity of in situ
fuel cell characterization. Despite the experimental difficulties involved in pinpointing all
the sources of fuel cell resistance loss, the electrolyte yields the biggest resistance loss for
most fuel cell devices.
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FUEL CELL CHARGE TRANSPORT
Rinterconnect
Anode
Ranode
Relectrolyte Rcathode
Electrolyte
Rinterconnect
Cathode
Figure 4.7. The total ohmic resistance presented by a fuel cell is actually a combination of resistances, each attributed to different components of the fuel cell. In this diagram, fuel cell resistance
is divided into interconnect, anode, electrolyte, and cathode components. Since current flows serially
through all components, total fuel cell resistance is given by the series sum of the individual resistance
components.
4.3.4
lonic (Electrolyte) Resistance Usually Dominates
The best electrolytes employed in fuel cells have ionic conductivities of around 0.10 Ω−1 ⋅
cm−1 . Even at a thickness of 50 𝜇m (very thin), this produces an ASR of 0.05 Ω ⋅ cm2 .
In contrast, a 50-𝜇m-thick porous carbon cloth electrode would have an ASR of less than
5 × 10−6 Ω ⋅ cm2 . This example illustrates how electrolyte resistance usually dominates
fuel cells.
Well-designed fuel cells have a total ASR in the range of 0.05–0.10 Ω ⋅ cm2 , and electrolyte resistance usually accounts for most of the total. If electrolyte thickness cannot
be reduced, decreasing ohmic loss depends on finding high-𝜎 ionic conductors. Unfortunately, developing satisfactory ionic conductors is challenging. The three most widely used
electrolyte classes, discussed in Sections 4.5.1– 4.5.3, are aqueous, polymer, and ceramic
electrolytes. The conductivity mechanisms and materials properties of these three electrolyte classes are quite different. Before we get to that discussion, however, it is helpful to
develop a clear physical picture of conductivity in general terms.
4.4
PHYSICAL MEANING OF CONDUCTIVITY
Conductivity quantifies the ability of a material to permit the flow of charge when driven by
an electric field. In other words, conductivity is a measure of how well a material accommodates charge transport. A material’s conductivity is influenced by two major factors: how
many carriers are available to transport charge and the mobility of those carriers within the
material. The following equation defines 𝜎 in those terms:
𝜎i = (|zi |F)ci ui
(4.19)
PHYSICAL MEANING OF CONDUCTIVITY
where ci represents the molar concentration of charge carriers (how many moles of carrier
are available per unit volume) and ui is the mobility of the charge carriers within the material. The quantity |zi |F is necessary to convert charge carrier concentration from units of
moles to units of coulombs. Here, zi is the charge number for the carrier (e.g., zi = +2 for
Cu2+ , zi = −1 for e– , etc.), the absolute-value function ensures that conductivity is always
a positive number, and F is Faraday’s constant.
A material’s conductivity is therefore determined by the product of carrier concentration
ci and carrier mobility ui . These properties are, in turn, set by the structure and conduction mechanisms within the material. Up to this point, the charge transport equations we
have learned apply equally well to both electronic and ionic conduction. Now, however,
their paths will diverge. Because electronic and ionic conduction mechanisms are vastly
different, electronic and ionic conductivities are also quite different.
CONDUCTIVITY AND MOBILITY
The difference between conductivity and mobility can be understood by an analogy. Pretend that we are studying the transport of people (in cars) down an interstate highway.
Mobility describes how fast the cars are driving down the highway. Conductivity, however, would also include information about how many cars are on the highway and how
many people each car can hold. This analogy is not perfect but may help keep the two
terms straight.
4.4.1
Electronic versus Ionic Conductors
Differences in the fundamental nature of electrons versus ions lead to differences in the
mechanisms for electronic versus ionic conduction. Figure 4.8 schematically contrasts a
typical electronic conductor (a metal) and a typical ionic conductor (a solid electrolyte).
Figure 4.8a illustrates the free-electron model of a metallic electron conductor. In this
model, the valence electrons associated with the atoms of the metal become detached from
the atomic lattice and are free to move about the metal. Meanwhile, the metal ions remain
intact and immobile. The free valence electrons constitute a “sea” of mobile charges, which
are able to move in response to an applied field.
By contrast, Figure 4.8b illustrates the hopping model of a solid-state ionic conductor.
The crystalline lattice of this ion conductor consists of both positive and negative ions, all
of which are fixed to specific crystallographic positions. Occasionally, defects such as missing atoms (“vacancies”) or extra atoms (“interstitials”) will occur in the material. Charge
transport is accomplished by the site-to-site “hopping” of these defects through the material.
The structural differences between the two kinds of conductors lead to dramatic differences in carrier concentrations. In a metal, free electrons are populous, while carriers in a
crystalline solid electrolyte are rare. The differences in the charge transport mechanisms,
as illustrated in Figure 4.8, also lead to dramatic differences in carrier mobility. Combined,
the differences in carrier concentration and carrier mobility lead to a very different picture
for electron conductivity in a metal versus ion conductivity in a solid electrolyte. Let us
take a closer look.
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FUEL CELL CHARGE TRANSPORT
e–
M+
e–
e–
M+
M+
e–
M+
M+
M+
e–
e–
M+
e–
M+
e–
M+
e–
e–
M+
e–
e–
M+
M+
e–
e–
e–
M+
e–
M+
e–
M+
M+
e–
M+
e–
M+
e–
e–
(a)
A–
C+
A–
C+
A–
C+
A–
C+
A–
C+
A–
C+
C+
A–
A–
C+
A–
C+
A–
C+
A–
C+
A–
C+
A–
C+
A–
C+
A–
A–
C+
C+
A–
C+
Vacancy
A–
C+
Interstitial
(b)
Figure 4.8. Illustration of charge transport mechanisms. (a) Electron transport in a free-electron
metal. Valence electrons detach from immobile metal atom cores and move freely in response to
an applied field. Their velocity is limited by scattering from the lattice. (b) Charge transport in this
crystalline ionic conductor is accomplished by mobile anions, which “hop” from position to position within the lattice. The hopping process only occurs where lattice defects such as vacancies or
interstitials are present.
4.4.2
Electron Conductivity in a Metal
For a simple electron conductor, such as a metal, the Drude model predicts that the mobility
of free electrons in the metal will be limited by scattering (from phonons, lattice imperfections, impurities, etc.):
q𝜏
u=
(4.20)
m
where 𝜏 gives the mean free time between scattering events, m is the mass of the electron
(m = 9.11 × 10−31 kg), and q is the elementary electron charge in coulombs (q = 1.602 ×
10−19 C).
Inserting the results for electron mobility (Equation 4.20) into the expression for conductivity (Equation 4.19) gives
|z F|c q𝜏
𝜎= e e
(4.21)
m
Carrier concentration in a metal may be calculated from the density of free electrons. In
general, each metal atom will contribute approximately one free electron. Atomic packing
PHYSICAL MEANING OF CONDUCTIVITY
densities are generally on the order of 1028 atoms/m3 , which yields molar carrier concentrations on the order of 104 mol/m3 .
Inserting typical numbers into Equation 4.21 allows us to calculate ballpark electronic
conductivity values. The charge number on an electron is, of course, –1(|ze | = 1). Typical scattering times (in relatively pure metals) are 10−12 –10–14 s. Using ce ≈ 104 mol∕m3
yields typical electron conductivities for metals in the range of 106 –108 Ω–1 ⋅ cm–1 ).
4.4.3
Ion Conductivity in a Crystalline Solid Electrolyte
The conduction hopping process illustrated in Figure 4.8b for a solid ion conductor leads
to a very different expression for mobility than that used for a metallic electron conductor.
Ion mobility for the material in Figure 4.8b is dependent on the rate at which ions can hop
from position to position within the lattice. This hopping rate, like the reaction rates studied
in the previous chapter, is exponentially activated. The effectiveness of the hopping process
is characterized by the material’s diffusivity D:
D = Do e−ΔGact ∕(RT)
(4.22)
where Do is a constant reflecting the attempt frequency of the hopping process, ΔGact is the
activation barrier for the hopping process, R is the gas constant, and T is the temperature (K).
The overall mobility of ions in the solid electrolyte is then given by
u=
|zi |FD
RT
(4.23)
Where |zi | is the charge number on the ion, F is Faraday’s constant, R is the gas constant,
and T is the temperature (K).
Inserting the expression for ion mobility (Equation 4.23) into the equation for conductivity (Equation 4.19) gives
c(zi F)2 D
(4.24)
𝜎=
RT
Carrier concentration in a crystalline electrolyte is controlled by the density of the
mobile defect species. Most crystalline electrolytes conduct via a vacancy mechanism.
These vacancies are intentionally introduced into the lattice by doping. Maximum
effective vacancy doping levels are around 8–10%, leading to carrier concentrations of
102 –103 mol∕m3 .
Typical ion diffusivities are on the order of 10–8 m2 ∕s for liquid and polymer electrolytes at room temperature, and on the order of 10–11 m2 ∕s for ceramic electrolytes at
700–1000∘ C. Typical ion carrier concentrations are 103 –104 mol∕m3 for liquid electrolytes,
102 –103 mol∕m3 for polymer electrolytes, and 102 –103 mol∕m3 for ceramic electrolytes at
700–1000∘ C. Inserting these values into Equation 4.24 yields ionic conductivity values of
10−4 –102 Ω–1 ⋅ m−1 (10−6 − 100 Ω–1 ⋅ cm−1 ).
Note that solid-electrolyte ionic conductivity values are well below electronic conductivity values for metals. As has been previously stated, ionic charge transport tends to be
far more difficult than electronic charge transport. Therefore, much of the focus in fuel cell
research is placed on finding better electrolytes.
131
132
FUEL CELL CHARGE TRANSPORT
4.5
REVIEW OF FUEL CELL ELECTROLYTE CLASSES
The search for better electrolytes has led to the development of three major candidate materials classes for fuel cells: aqueous, polymer, and ceramic electrolytes. Regardless of the
class, however, any fuel cell electrolyte must meet the following requirements:
•
•
•
•
•
•
High ionic conductivity
Low electronic conductivity
High stability (in both oxidizing and reducing environments)
Low fuel crossover
Reasonable mechanical strength (if solid)
Ease of manufacturability
Other than the high-conductivity requirement, the electrolyte stability requirement is
often the hardest to fulfill. It is difficult to find an electrolyte that is stable in both the highly
reducing environment of the anode and the highly oxidizing environment of the cathode.
4.5.1
Ionic Conduction in Aqueous Electrolytes/Ionic Liquids
In this section, we discuss ionic conduction in aqueous electrolytes and ionic liquids. An
aqueous electrolyte is a water-based solution containing dissolved ions that can transport
charge. An ionic liquid is a material which is itself simultaneously liquid and ionic. Sodium
chloride dissolved in water is an example of an aqueous electrolyte. Upon dissolution in
water, the NaCl separates into mobile Na+ ions and mobile Cl– ions, which can transport
charge by moving through the water solvent. Molten NaCl (when heated to high temperature) is an example of an ionic liquid. Pure H3 PO4 at 50∘ C is another example of an ionic
liquid. At room temperature, H3 PO4 is a somewhat waxy, white crystalline solid. However,
when heated above 42∘ C it becomes a viscous ionic liquid consisting of H+ ions, PO4 3–
ions, and H3 PO4 molecules.
Almost all aqueous/liquid electrolyte fuel cells use a matrix material to support or immobilize the electrolyte. The matrix generally accomplishes three tasks:
1. Provides mechanical strength to the electrolyte
2. Minimizes the distance between the electrodes while preventing shorts
3. Prevents crossover of reactant gases through the electrolyte
Reactant crossover, the last task on this list, is a particular problem for aqueous/liquid
electrolytes (much more so than for solid electrolytes). In an unsupported liquid electrolyte, reactant gas crossover can be severe; in these situations, unbalanced-pressure or
high-pressure operation is impossible. The use of a matrix material provides mechanical
integrity and reduces gas crossover problems, while still permitting thin (0.1–1.0-mm)
electrolytes.
Alkaline fuel cells use concentrated aqueous KOH electrolytes, while phosphoric acid
fuel cells use either concentrated aqueous H3 PO4 electrolytes or pure H3 PO4 (an ionic liquid). Molten carbonate fuel cells use molten (K/Li)2 CO3 immobilized in a supporting
REVIEW OF FUEL CELL ELECTROLYTE CLASSES
matrix. The (K/Li)2 CO3 material melts at around 450∘ C to become a liquid (“molten”)
electrolyte. (MCFCs must therefore obviously be operated above 450∘ C.)
Ionic conductivity in aqueous/liquid environments can best be approached using a driving force/frictional force balance model. In liquids, an ion will accelerate under the force of
an electric field until frictional drag exactly counteracts the electric field force. The balance
between the electric field and frictional drag determines the terminal velocity of the ion.
The electric field force, FE , is given by
FE = zi q
dV
dx
(4.25)
where zi is the charge number of the ion and q is the fundamental electron charge (1.6 ×
10–19 C). Although we do not show the derivation here, the frictional drag force FD may be
approximated from Stokes’s law as
FD = 6𝜋𝜇rv
(4.26)
where 𝜇 is the viscosity of the liquid, r is the radius of the ion, and v is the velocity of the
ion. Equating the two forces allows us to determine the mobility, ui , which is defined as the
ratio between the applied electric field and the resulting ion velocity (because mobility is
defined as a positive quantity, inclusion of the absolute value is again required):
| 𝑣 |
| = |zi |q
ui = ||
| 6𝜋𝜇r
dV∕dx
|
|
(4.27)
Thus, mobility is determined by the ion size and the liquid viscosity. Intuitively, this
expression makes sense: Bulky ions or highly viscous liquids should lead to lower mobilities, while nonviscous liquids and small ions should yield higher mobilities. The mobilities
of a variety of ions in aqueous solution are given in Table 4.2. Note that in aqueous solutions the H+ ion tends to be hydrated by one or more water molecules. This ionic species
is therefore better thought of as H3 O+ or H ⋅ (H2 O)x + , where x represents the number of
water molecules “hydrating” the proton.
Recall our expression for conductivity (Equation 4.19), which is repeated here for clarity:
(4.28)
𝜎i = (|zi |F)ci ui
If the values of ion mobilities in Table 4.2 are inserted into this expression, the ionic
conductivity of various aqueous electrolytes may be calculated. Unfortunately, these
TABLE 4.2. Selected Ionic Mobilities at Infinite Dilution in Aqueous Solutions at 25∘ C
Cation
Mobility, u (cm2 /V ⋅ s)
Anion
Mobility, u (cm2 /V ⋅ s)
H+ (H3 O+ )
3.63 × 10−3
OH−
2.05 × 10−3
K+
7.62 × 10−4
Br−
8.13 × 10−4
Ag+
6.40 × 10−4
I−
7.96 × 10−4
Na+
5.19 × 10−4
Cl−
7.91 × 10−4
Li+
4.01 × 10−4
HCO3 −
4.61 × 10−4
Source: From Ref. [6a].
133
134
FUEL CELL CHARGE TRANSPORT
calculations are only accurate for dilute aqueous solutions when the ion concentration is
low. At high ion concentration (or for ionic liquids) strong electrical interactions between
the ions make conductivity far more difficult to calculate. In general, the conductivity
of highly concentrated aqueous solutions or pure ionic liquids will be much lower than
that predicted by Equation 4.28. For example, the conductivity of pure H3 PO4 is experimentally determined to be 0.1–1.0 Ω−1 ⋅ cm−1 (depending on the temperature), whereas
Equation 4.28 predicts that the conductivity of pure H3 PO4 should be approximately
18 Ω−1 ⋅ cm−1 .
Table 4.2 does offer some other useful insights. For example, it explains why KOH is
the electrolyte of choice in alkaline fuel cells. Besides being extremely inexpensive, KOH
exhibits the highest ionic conductivity of any of the hydroxide compounds. (Compare the
u value for K+ to other candidate hydroxide cations such as Na+ or Li+ .) In alkaline fuel
cells, fairly concentrated (30–65%) solutions of KOH are used, resulting in conductivities
on the order of 0.1–0.5 Ω−1 ⋅ cm−1 . How much would the conductivity be reduced if a far
more dilute electrolyte was used? To get an answer, refer to Example 4.3, where the approximate conductivity of a 0.1 M KOH electrolyte solution is calculated using Equation 4.28.
Example 4.3 Calculate the approximate conductivity of a 0.1 M aqueous solution of
KOH.
Solution: We use Equation 4.28 as our guide. Assuming that 0.1 M KOH completely
dissolves into K+ ions and OH– ions (it does), the concentration of K+ and OH– will
also be 0.1 M. Converting these concentrations to units of moles per cubic centimeter
gives
cK+ = (0.1 mol∕L)(1 L∕1000 cm3 ) = 1 × 10−4 mol∕cm3
(4.29)
cOH− = (0.1 mol∕L)(1 L∕1000 cm3 ) = 1 × 10−4 mol∕cm3
The mobilities of K+ and OH– are given in Table 4.2. Inserting these numbers into
Equation 4.28 yields
𝜎K+ = (1)(96, 485)(1 × 10−4 mol∕cm3 )(7.62 × 10−4 cm2 ∕V ⋅ s)
= 0.0073 Ω−1 ⋅ cm−1
𝜎OH− = (1)(96, 485)(1 × 10−4 mol∕cm3 )(2.05 × 10−3 cm2 ∕V ⋅ s)
= 0.0198 Ω−1 ⋅ cm−1
(4.30)
The total ionic conductivity of the electrolyte is then given by the sum of the cation
and anion conductivities:
𝜎total = 𝜎K+ + 𝜎OH− = 0.0073 + 0.0198 = 0.0271 Ω−1 ⋅ cm−1
(4.31)
In reality, the conductivity of the 0.1 M KOH solution will likely be a little lower
than this predicted value. Note that most of the conductivity is provided by the OH–
ion, rather than the K+ ion. This is due to the higher mobility of the OH– ion.
REVIEW OF FUEL CELL ELECTROLYTE CLASSES
4.5.2
Ionic Conduction in Polymer Electrolytes
In general, ionic transport in polymer electrolytes follows the exponential relationship
described by Equations 4.22 and 4.24. By combining these two equations, we can obtain
(see problem 4.11)
(4.32)
𝜎T = APEM e−Ea ∕kT
where APEM is a preexponential factor and Ea represents the activation energy (eV/atom)
(Ea = ΔGact ∕F, where F is Faraday’s constant). As this equation indicates, conductivity
increases exponentially with increasing temperature. Most polymer and crystalline ion conductors obey this model quite well.
For a polymer to be a good ion conductor, at a minimum it should possess the following
structural properties:
1. The presence of fixed charge sites
2. The presence of free volume (“open space”)
The fixed charge sites should be of opposite charge compared to the moving ions, ensuring that the net charge balance across the polymer is maintained. The fixed charge sites
provide temporary centers where the moving ions can be accepted or released. In a polymer structure, maximizing the concentration of these charge sites is critical to ensuring high
conductivity. However, excessive addition of ionically charged side chains will significantly
degrade the mechanical stability of the polymer, making it unsuitable for fuel cell use.
Free volume correlates with the spatial organization of the polymer. In general, a typical
polymer structure is not fully dense. Small-pore structures (or free volumes) will almost
always exist. Free volume improves the ability of ions to move across the polymer. Increasing the polymer free volume increases the range of small-scale structural vibrations and
motions within the polymer. These motions can result in the physical transfer of ions from
site to site across the polymer. (See Figure 4.9.)
Because of these free-volume effects, polymer membranes exhibit relatively high ionic
conductivities compared to other solid-state ion-conducting materials (such as ceramics).
Polymer free volume also leads to another well-known transport mechanism, known as
the vehicle mechanism. In the vehicle mechanism, ions are transported through free-volume
–
–
+
–
–
– Charged site
–
–
– –
–
–
––
–
–
+ Ion
+
–
–
Polymer chain
Figure 4.9. Schematic of ion transport between polymer chains. Polymer segments can move or
vibrate in the free volume, thus inducing physical transfer of ions from one charged site to another.
135
136
FUEL CELL CHARGE TRANSPORT
spaces by hitching a ride on certain free species (the “vehicles”) as these vehicles pass by.
Water is a common vehicular species; as water molecules move through the free volumes in
a polymer membrane, ions can go along for the ride. In this case, the conduction behavior
of the ions in the polymer electrolyte is much like that in an aqueous electrolyte. Persulfonated polytetrafluoroethylene (PTFE)—more commonly known as Nafion—exhibits
extremely high proton conductivity based on the vehicle mechanism. Since Nafion is the
most popular and important electrolyte for PEMFC applications, we review its properties
in the next section.
Ionic Transport in Nafion. Nafion has a backbone structure similar to polytetrafluoroethylene (Teflon). However, unlike Teflon, Nafion includes sulfonic acid (SO3 – H+ ) functional groups. The Teflon backbone provides mechanical strength while the sulfonic acid
(SO3 – H+ ) chains provide charge sites for proton transport. Figure 4.10 illustrates the structure of Nafion.
It is believed that Nafion free volumes aggregate into interconnected nanometer-sized
pores whose walls are lined by sulfonic acid (SO3 – H+ ) groups. In the presence of water, the
protons (H+ ) in the pores form hydronium complexes (H3 O+ ) and detach from the sulfonic
acid side chains. When sufficient water exists in the pores, the hydronium ions can transport
in the aqueous phase. Under these circumstances, ionic conduction in Nafion is similar to
conduction in liquid electrolytes (Section 4.5.1). As a bonus, the hydrophobic nature of
the Teflon backbone further accelerates water transport through the membrane, since the
hydrophobic pore surfaces tend to repel water. Because of these factors, Nafion exhibits
proton conductivity comparable to that of a liquid electrolyte. To maintain this extraordinary
conductivity, Nafion must be fully hydrated with liquid water. Usually, hydration is achieved
by humidifying the fuel and oxidant gases provisioned to the fuel cell. In the following
paragraphs, we review the key properties of Nafion in more detail.1
Nafion Absorbs Significant Amounts of Water. The pore structure in Nafion can
hold significant amounts of water. In fact, Nafion can accommodate so much water that
its volume will increase up to 22% when fully hydrated. (Strongly polar liquids, such as
alcohols, can cause Nafion to swell up to 88%!) Since conductivity and water content are
strongly related, determining water content is essential to determining the conductivity of
the membrane. The water content λ in Nafion is defined as the ratio of the number of water
molecules to the number of charged (SO3 – H+ ) sites. Experimental results suggest that λ
can vary from almost 0 (for completely dehydrated Nafion) to 22 (for full saturation, under
certain conditions). For fuel cells, experimental measurements have related the water content in Nafion to the humidity condition of the fuel cell, as shown in Figure 4.11. Thus, if
the humidity condition of the fuel cell is known, the water content in the membrane can
be estimated. Humidity in Figure 4.11 is quantified by water vapor activity a𝑤 (essentially
relative humidity):
p
a𝑤 = 𝑤
(4.33)
pSAT
1 The
Nafion model reviewed here was suggested by Springer et al. [8]
REVIEW OF FUEL CELL ELECTROLYTE CLASSES
Nafion
Polytetraflouroethylene (PTFE)
F
F
F
F
F
F
F
F
C
C
C
C
C
C
C
C
F
F
F
F
F
O
F
F
F
C
F
F
C
CF3
n
n
O
F
C
F
C
=
O= S
n
m
F
F
O–
H+
O
(a)
H2O
+
H 3O
–
SO3
1nm
(b)
Figure 4.10. (a) Chemical structure of Nafion. Nafion has a PTFE backbone for mechanical stability
with sulfonic groups to promote proton conduction. (b) Schematic microscopic view of proton conduction in Nafion. When hydrated, nanometer-sized pores swell and become largely interconnected.
Protons bind with water molecules to form hydronium complexes. Sulfonic groups near the pore walls
enable hydronium conduction.
where p𝑤 represents the actual partial pressure of water vapor in the system and pSAT represents the saturation water vapor pressure for the system at the temperature of operation.
The data in Figure 4.11 can be represented mathematically as
{
λ =
0.043 + 17.18a𝑤 − 39.85a2𝑤 + 36.0a3𝑤
)
(
14 + 4 a𝑤 − 1
for 0 < a𝑤 ≤ 1
for 1 < a𝑤 ≤ 3
(4.34)
137
FUEL CELL CHARGE TRANSPORT
14
12
10
−
λ = H2O/SO3
138
8
6
4
2
0
0
0.2
0.4
0.6
0.8
1
Water vapor activity ( pw /pSAT)
Figure 4.11. Water content versus water activity for Nafion 117 at 303 K (30∘ C) according to
Equation 4.34. Water vapor activity is defined as the ratio of the actual water vapor pressure (p𝑤 )
for the system compared to the saturation water vapor pressure (pSAT ) for the system at the temperature of interest. Reprinted with permission from Ref. [8], Journal of the Electrochemical Society,
138: 2334, 1991. Copyright 1991 by the Electrochemical Society.
Equation 4.34 does not consider the effects of temperature; however, it is reasonably
accurate for PEMFCs operating near 80∘ C.
WATER VAPOR SATURATION PRESSURE
When the partial pressure of water vapor (p𝑤 ) within a gas stream reaches the water
vapor saturation pressure pSAT for a given temperature, the water vapor will start to
condense, generating water droplets. In other words, relative humidity is 100% when
p𝑤 = pSAT . Importantly, pSAT is a strong function of temperature:
log10 pSAT = −2.1794 + 0.02953T − 9.1837 × 10−5 T 2 + 1.4454 × 10−7 T 3
(4.35)
where pSAT is given in bars (1 bar = 100,000 Pa) and T is the temperature in degrees
Celsius. For example, if fully humidified air at 80∘ C and 3 atm is provided to a fuel cell,
the water vapor pressure is [9]
−5 ×802 +1.4454×10−7 ×803
pSAT = 10−2.1794+0.02953×80−9.1837×10
= 0.4669 bar
(4.36)
This gives the mole fraction of water in fully humidified air at 80∘ C and 3 atm as
0.4669 bar/3 atm = 0.4669 bar/(3 × 1.0132501 bar) = 0.154 assuming an ideal gas.
REVIEW OF FUEL CELL ELECTROLYTE CLASSES
Under these same conditions, if the air is instead only partially humidified, such that
the water mole fraction is 0.1, then the water vapor activity (or relative humidity) would
be (again assuming an ideal gas)
a𝑤 =
pH2 O𝑤
pSAT
=
xH2 O × ptotal
xH2 O,SAT × ptotal
=
0.1
= 0.65
0.154
(4.37)
Nafion Conductivity Is Highly Dependent on Water Content. As previously mentioned, conductivity and water content are strongly related in Nafion. Conductivity and
temperature are also strongly related. In general, the proton conductivity of Nafion increases
linearly with increasing water content and exponentially with increasing temperature, as
shown by the experimental data in Figures 4.12 and 4.13. In equation form, these experimentally determined relationships may be summarized as
)]
[
(
1
1
𝜎(T, λ) = 𝜎303K (λ) exp 1268
−
303 T
(4.38)
𝜎303K (λ) = 0.005193λ − 0.00326
(4.39)
where
where 𝜎 represents the conductivity (S/cm) of the membrane and T (K) is the temperature.
Since the conductivity of Nafion can change locally depending on water content, the total
area-specific resistance of a membrane is found by integrating the local resistivity over the
0.12
0.1
σ (S/cm)
0.08
0.06
0.04
0.02
0
0
5
10
15
20
25
λ = H O/SO
2
3
Figure 4.12. Ionic conductivity of Nafion versus water content λ according to Equations 4.38 and
4.39 at 303 K.
139
FUEL CELL CHARGE TRANSPORT
100˚C
50˚C
0˚C
–0.6
–0.7
log(σ) [log(S/cm)]
140
–0.8
–0.9
–1
–1.1
–1.2
–1.3
2.6
2.8
3
3.2
3.4
3.6
3.8
3
1/T (x10 K)
Figure 4.13. Ionic conductivity of Nafion versus temperature according to Equation 4.38
when λ = 22.
membrane thickness (tm ) as
ASRm =
∫0
tm
𝜌(z)dz =
∫0
tm
dz
𝜎[λ(z)]
(4.40)
Protons Drag Water with Them. Since conductivity in Nafion is dependent on water
content, it is essential to know how water content varies across a Nafion membrane. During
fuel cell operation, the water content across a Nafion membrane is generally not uniform.
Water content varies across a Nafion membrane because of several factors. Perhaps most
important is the fact that protons2 traveling through the pores of Nafion generally drag
one or more water molecules along with them. This well-known phenomenon is called
electro-osmotic drag. The degree to which proton movement causes water movement is
quantified by the electro-osmotic drag coefficient ndrag , which is defined as the number of
water molecules accompanying the movement of each proton (ndrag = nH2 O ∕H+ ). Obviously, how much water is dragged per proton depends on how much water exists in the
Nafion membrane in the first place. It has been measured that ndrag = 2.5 ± 0.2 (between
30 and 50∘ C) in fully hydrated Nafion (when λ = 22). When λ = 11, ndrag = ∼ 0.9. Commonly, it is assumed that ndrag changes linearly with λ as
ndrag = nSAT
drag
λ
22
for 0 ≤ λ ≤ 22
(4.41)
2 Actually, protons travel in the form of hydronium complexes as explained in the text. For simplicity, however,
we use the term “proton” in these discussions. Also, it is more straightforward to define the electro-osmotic drag
coefficient in terms of the number of water molecules per proton (rather than per hydronium, which contains a
water molecule already).
REVIEW OF FUEL CELL ELECTROLYTE CLASSES
where nSAT
≈ 2.5. Knowledge of the electro-osmotic drag coefficient allows us to estimate
drag
the water drag flux from anode to cathode when a net current j flows through the PEMFC:
JH2 O,drag = 2ndrag
j
2F
(4.42)
where J is the molar flux of water due to electro-osmotic drag (mol/cm2 ), j is the operating
current density of the fuel cell (A/cm2 ), and the quantity 2F converts from current density to
hydrogen flux. The factor of 2 in the front of the equation then converts from hydrogen flux
to proton flux. As you will see in Chapter 6, the drag coefficient becomes very important
in modeling the behavior of Nafion membranes in PEMFCs.
Back Diffusion of Water. In a PEMFC, electro-osmotic water drag moves water from
the anode to the cathode. As this water builds up at the cathode, however, back diffusion occurs, resulting in the transport of water from the cathode back to the anode. This
back-diffusion phenomenon occurs because the concentration of water at the cathode is
generally far higher than the concentration of water at the anode (exacerbated by the fact
that water is produced at the cathode by the electrochemical reaction). Back diffusion
counterbalances the effects of electro-osmotic drag. Driven by the anode/cathode water
concentration gradient, the water back-diffusion flux can be determined by
JH2 O,back
diffusion
=−
𝜌dry
Mm
Dλ
dλ
dz
(4.43)
where 𝜌dry is the dry density (kg/m3 ) of Nafion, Mm is the Nafion equivalent weight
(kg/mol), and z is the direction through the membrane thickness.
The key factor in this equation is the diffusivity of water in the Nafion membrane (Dλ ).
Unfortunately, Dλ is not constant but is a function of water content λ. Since the total water
flux in Nafion is simply the addition of electro-osmotic drag and back diffusion, we have
JH2 O = 2nSAT
drag
𝜌dry
j λ
dλ
D (λ)
−
2F 22 Mm λ dz
(4.44)
This combined expression makes it explicitly clear that the water flux in Nafion is a
complex function of λ. [We state the water diffusivity as Dλ (λ) in this equation to emphasize
its dependency on water content.]
Summary. Based on the fuel cell operating conditions (humidity and current density), we
can estimate the water content profile (λ(z)) in the membrane by using Equations 4.34 and
4.44. Once we have the water content profile, we can then calculate the ion conductivity
of the membrane by using Equation 4.38. In this fashion, the ohmic losses in a PEMFC
may be quantified. This procedure is demonstrated in Example 4.4. In Chapter 6 we will
combine these equations with the other fuel cell loss terms to create a complete PEMFC
model.
141
142
FUEL CELL CHARGE TRANSPORT
Example 4.4 Consider a hydrogen PEMFC powering an external load at 0.7 A/cm2 .
The activities of water vapor on the anode and cathode sides of the membrane are
measured to be 0.8 and 1.0, respectively. The temperature of the fuel cell is 80∘ C.
If the Nafion membrane thickness is 0.125 mm, estimate the ohmic overvoltage loss
across the membrane.
Solution: We can convert the water activity on the Nafion surfaces to water contents
using Equation 4.34:
λA = 0.043 + 17.18 × 0.8 − 39.85 × 0.82 + 36.0 × 0.83 = 7.2
λC = 0.043 + 17.18 × 1.0 − 39.85 × 1.02 + 36.0 × 1.03 = 14.0
(4.45)
With these values as boundary conditions, we then solve Equation 4.44. In this
equation, we have two unknowns, JH2 O and λ. For convenience, we will set JH2 O =
𝛼NH2 = 𝛼(j∕2F), where 𝛼 is an unknown that denotes the ratio of water flux to hydrogen flux. After rearrangement, Equation 4.44 becomes
(
) jM
λ
dλ
m
= 2nSAT
−
𝛼
drag 22
dz
2F𝜌dry Dλ
(4.46)
EQUIVALENT WEIGHT
The equivalent weight of a species is defined by its atomic weight or formula weight
divided by its valence:
Equivalent weight =
atomic (formula) weight
valence
(4.47)
Valence is defined by the number of electrons that the species can donate or accept.
For example, hydrogen has a valence of 1 (H+ ). Oxygen has a valence of 2 (O2– ). Thus,
hydrogen has an equivalent weight of 1.008 g∕mol∕1 = 1.008 g∕mol and oxygen has an
equivalent weight of 15.9994 g∕mol∕2 = 7.9997 g∕mol. In the case of sulfate radicals
(SO4 2– ), the formula weight is (1 × 32.06) + (4 × 15.9994) = 96.058 g∕mol. Thus, the
equivalent weight is (96.058 g∕mol)∕2 = 48.029 g∕mol.
The sulfonic group (SO3 – H+ ) in Nafion has a valence of 1, since it can accept only
one proton. Thus, the equivalent weight of Nafion is equal to the average weight of the
polymer chain structure that can accept one proton. This number is very useful since it
facilitates the calculation of sulfonic charge (SO3 – ) concentration in Nafion as
CSO− (mol∕m3 ) =
3
𝜌dry (kg∕m3 )
Mm (kg∕mol)
(4.48)
where 𝜌dry is the dry density of Nafion (kg/m3 ) and Mm is the Nafion equivalent weight
(kg/mol).
REVIEW OF FUEL CELL ELECTROLYTE CLASSES
In a similar fashion, water content, λ (H2 O∕SO3 – ), can be converted to water concentration in Nafion as
CH2 O (mol∕m3 ) = λ
𝜌dry (kg∕m3 )
(4.49)
M m (kg∕mol)
Typically, Nafion has an equivalent weight of around ∼ 1–1.1 kg∕mol and a dry density of ∼ 1970 kg∕m3 . Thus, the estimated charge density for Nafion would be
CSO− (mol∕m3 ) =
3
1970 kg∕m3
= 1970 mol∕m3
1 kg∕mol
(4.50)
WATER DIFFUSIVITY IN NAFION
As emphasized above, water diffusivity in Nafion (Dλ ) is a function of water content λ.
Experimentally (using magnetic resonance techniques), this dependence has been measured as
)]
[
(
1
1
−
Dλ = exp 2416
303 T
× (2.563 − 0.33λ + 0.0264λ2 − 0.000671λ3 ) × 10−6
for λ > 4 (cm2 ∕s)
(4.51)
The exponential part describes the temperature dependence, while the polynomial
portion describes the λ dependence at the reference temperature of 303 K. This equation
is only valid for λ > 4. For λ < 4, values extrapolated from Figure 4.14 (dotted line)
should be used instead.
Water diffusivity, Dλ (cm2/s)
4
x 10 −6
3.5
3
2.5
2
1.5
1
0.5
0
0
5
10
λ (H2O/SO3-)
15
Figure 4.14. Water diffusivity Dλ in Nafion versus water content λ at 303 K.
143
144
FUEL CELL CHARGE TRANSPORT
Even though this is an ordinary differential equation on λ, we may not solve it analytically since Dλ is a function of λ. However, if we assume λ in the membrane changes
from 7.2 to 14.0 according to the boundary conditions, we can see from Figure 4.14
that the water diffusivity is fairly constant over this range. If we assume an average
value of λ = 10, we can estimate Dλ from Equation 4.51 as
)]
[
(
1
1
−
Dλ = 10−6 exp 2416
303 353
× (2.563 − 0.33 × 10 + 0.0264 × 102 − 0.000671 × 103 )
= 3.81 × 10−6 cm2 ∕s
(4.52)
Now we can evaluate Equation 4.46, yielding the analytical solution
[
]
jMm nSAT
drag
11𝛼
11𝛼
λ(z) SAT + C exp
z =
22 F 𝜌dry Dλ
2.5
ndrag
[
]
(
)
0.7 A∕cm2 × (1.0 kg∕mol) × 2.5
+ C exp
z
(22 × 96, 485 C∕mol) × (0.00197 kg∕cm3 ) × (3.81 cm2 ∕s)
= 4.4𝛼 + C exp(109.8z)
(4.53)
where z is in centimeters and C is a constant to be determined from the boundary
conditions. If we set the anode side as z = 0, we have λ(0) = 7.2 and λ(0.0125) = 14
from Equation 4.45. Accordingly, Equation 4.53 becomes
λ(z) = 4.4𝛼 + 2.30 exp(109.8z)
where 𝛼 = 1.12
(4.54)
Now we know that about 1.12 water molecules are dragged per each hydrogen (or
in other words, about 0.56 water molecules per proton). Figure 4.15a shows the result
of how 𝜆 varies across the membrane in this example. At the start of the problem,
we assumed a constant Dλ for λ in the range of 7.2–14. We can confirm that this
assumption is reasonable from the results of Figure 4.15.
From Equations 4.38 and 4.54, we can determine the conductivity profile of the
membrane:
𝜎(z) = {0.005193[4.4𝛼 + 2.30 exp(109.8z)] − 0.00326}
)]
[
(
1
1
−
× exp 1268
303 353
= 0.0404 + 0.0216 exp(109.8z)
(4.55)
REVIEW OF FUEL CELL ELECTROLYTE CLASSES
Figure 4.15b shows the result. Finally, we can determine the area-specific resistance of the membrane using Equation 4.40:
∫0
tm
dz
=
𝜎[λ(z)] ∫0
0.0125
dz
= 0.15 Ω ⋅ cm2
0.0404 + 0.0216 exp(109.8z)
(4.56)
Thus, the ohmic overvoltage due to the membrane resistance in this PEMFC is
approximately
ASRm =
Vohm = j × ASRm = (0.7 A∕cm2 ) × (0.15 Ω ⋅ cm2 ) = 0.105 V
(4.57)
This section has focused exclusively on the details of Nafion. However, the conduction properties and characteristics of other polymer electrolyte alternatives are discussed in
Chapter 9 for the interested reader.
4.5.3
Ionic Conduction in Ceramic Electrolytes
This section explains the underlying physics of ion transport in SOFC electrolytes. As their
name implies, SOFC electrolytes are solid, crystalline oxide materials that can conduct ions.
The most popular SOFC electrolyte material is yttria-stabilized zirconia (YSZ). A typical
YSZ electrolyte contains 8% yttria mixed with zirconia. What is the meaning of zirconia and
yttria? Zirconia is related to the metal zirconium, and yttria derives its name from another
metal, yttrium. Zirconia has the chemical composition ZrO2 ; it is the oxide of zirconium.
By analogy, yttria, or Y2 O3 , is the oxide of yttrium. A mixture of zirconia and yttria is
called yttria-stabilized zirconia because the yttria stabilizes the zirconia crystal structure in
the cubic phase (where it is most conductive). Even more importantly, however, the yttria
introduces high concentrations of oxygen vacancies into the zirconia crystal structure. This
high oxygen vacancy concentration allows YSZ to exhibit high ion conductivity.
Adding yttria to zirconia introduces oxygen vacancies due to charge compensation
effects. Pure ZrO2 forms an ionic lattice consisting of Zr4+ ions and O2– ions, as shown
in Figure 4.16a. Addition of Y3+ ions to this lattice upsets the charge balance. As shown in
Figure 4.16b, for every two Y3+ ions taking the place of Zr4+ ions, one oxygen vacancy is
created to maintain overall charge neutrality. The addition of 8% (molar) yttria to zirconia
causes about 4% of the oxygen sites to be vacant. At elevated temperatures, these oxygen
vacancies facilitate the transport of oxygen ions in the lattice, as shown in Figure 4.8b.
As discussed in Section 4.4, a material’s conductivity is determined by the combination
of carrier concentration (c) and carrier mobility (u):
𝜎 = (|z|F)cu
(4.58)
In the case of YSZ, carrier concentration is determined by the strength of the yttria
doping. Because a vacancy is required for ionic motion to occur within the YSZ lattice, the
145
FUEL CELL CHARGE TRANSPORT
15
14
−)
Water content λ (HO/SO
2
3
13
12
11
10
9
8
7
0
0.002
Anode
0.004
0.006
0.008
Membrane thickness(cm)
0.01
0.012
Cathode
(a)
0.13
0.12
Local conductivity (S/cm)
146
0.11
0.1
0.09
0.08
0.07
0.06
0
0.002
Anode
0.004
0.006
0.008
Membrane thickness(cm)
0.01
0.012
Cathode
(b)
Figure 4.15. Calculated properties of Nafion membrane for Example 4.4. (a) Water content profile
across Nafion membrane. (b) Local conductivity profile across Nafion membrane.
REVIEW OF FUEL CELL ELECTROLYTE CLASSES
Vacancy
Zr 4+
Zr 4+
Zr 4+
O 2–
O2–
O 2–
Zr 4+
O2–
O2–
Zr 4+
O2–
Zr 4+
Zr 4+
Zr 4+
O2–
O2–
O2–
Zr 4+
Zr 4+
Zr 4+
Y 3+
O2–
O2–
O2–
O2–
O2–
O2–
˚ Zr 4+
Zr 4+
O2–
O2–
O2–
Zr 4+
Zr 4+
Zr 4+
Zr 4+
O 2–
Y 3+
(a)
O2–
Zr 4+
O2–
O2–
Zr 4+
Zr 4+
Zr 4+
O2–
Zr 4+
(b)
Figure 4.16. View of the (110) plane in (a) pure ZrO2 and (b) YSZ. Charge compensation effects
in YSZ lead to creation of oxygen vacancies. One oxygen vacancy is created for every two yttrium
atoms doped into the lattice.
oxygen vacancies can be considered to be the ionic charge “carriers.” Increasing the yttria
content will result in increased oxygen vacancy concentration, improving the conductivity.
Unfortunately, however, there is an upper limit to doping. Above a certain dopant or vacancy
concentration, defects start to interact with each other, reducing their ability to move. Above
this concentration, further doping is counterproductive and conductivity actually decreases.
Plots of conductivity versus dopant concentration show a maximum at the point where
defect interaction or “association” commences. For YSZ, this maximum occurs at about
8% molar yttria concentration. (See Figure 4.17.)
log(σT ) (Ω–1 · cm–1 K)
2.4
2.3
2.2
2.1
2
1.9
1.8
1.7
1.6
6
7
8
9
10
11
12
13
14
15
%Y2O3
Figure 4.17. YSZ conductivity versus %Y2 O3 (molar basis) [10]; YSZ conductivity is displayed as
σ(Ω–1 ⋅ cm–1 )times T (K). In the next section, Figure 4.18 will clarify why it is convenient to multiply
𝜎 with T.
147
148
FUEL CELL CHARGE TRANSPORT
The complete expression for conductivity combines carrier concentration and carrier
mobility, as described in Section 4.4.3:
𝜎=
c(zF)2 D
RT
(4.59)
where carrier mobility is described by D, the diffusivity of the carrier in the crystal lattice.
Diffusivity describes the ability of a carrier to move, or diffuse, from site to site within
a crystal lattice. High diffusivities translate into high conductivities because the carriers
are able to move quickly through the crystal. The atomic origins and physical explanation behind diffusivity will be detailed in forthcoming sections. For now, however, it is
sufficient to know that carrier diffusivity in SOFC electrolytes is exponentially temperature dependent:
(4.60)
D = D0 e−ΔGact ∕(RT)
where D0 is a constant (cm2 /s), ΔGact is the activation barrier for the diffusion process
(J/mol), R is the gas constant, and T is the temperature (K). Combining Equations 4.59 and
4.61 provides a complete expression for conductivity in SOFC electrolytes:
𝜎=
c(zF)2 D0 e−ΔGact ∕(RT)
RT
(4.61)
INTRINSIC CARRIERS VERSUS EXTRINSIC CARRIERS
In YSZ and most other SOFC electrolytes, dopants are used to intentionally create high
vacancy (or other charge carrier) concentrations. These carriers are known as extrinsic
carriers because their presence is extrinsically created by intentional doping. However,
any crystal, even an undoped one, will have at least some natural carrier population.
These natural charge carriers are referred to as intrinsic carriers because they occur
intrinsically due to the natural energetics of the crystal. Intrinsic carriers exist because no
crystal is perfect (unless it is at absolute zero). All crystals will contain “mistakes” such
as vacancies that can act as charge carriers for conduction. These mistakes are actually
energetically favorable, because they increase the entropy of the crystal. (Recall Section
2.1.4.) For the case of vacancies, an energy balance may be developed that considers the
enthalpy cost to create the vacancies versus the entropy benefit they deliver. Solving for
this balance results in the following expression for intrinsic vacancy concentration as a
function of temperature in an ionic crystal:
xV ≈ e−Δh𝑣 ∕(2kT)
(4.62)
where xV represents the fractional vacancy concentration (expressed as the fraction of
lattice sites of the species of interest that are vacant), Δh𝑣 is the formation enthalpy for
REVIEW OF FUEL CELL ELECTROLYTE CLASSES
the vacancy in electron-volts (in other words, the enthalpy cost to “create” a vacancy),
k is Boltzmann’s constant, and T is the temperature in Kelvin. This expression states that
the intrinsic concentration of vacancies within a crystal increases exponentially with
temperature. However, since Δh𝑣 is typically on the order of 1 eV or larger, intrinsic
vacancy concentrations are generally quite low, even at high temperatures. At 800∘ C,
the intrinsic vacancy concentration in pure ZrO2 is around 0.001, or about one vacancy
per 1000 sites. Compare this to extrinsically doped crystal structures, which can attain
vacancy concentrations as high as 0.1, or about one vacancy per 10 sites.
This equation can be further refined depending on whether the charge carriers are extrinsic or intrinsic:
• For extrinsic carriers, c is determined by the doping chemistry of the electrolyte. In
this case, c is a constant and Equation 4.62 can be used as is.
• For intrinsic carriers, c is exponentially dependent on temperature, and Equation 4.62
must be modified as follows:
𝜎=
csites (zF)2 D0 e−Δh𝑣 ∕(2kT) e−ΔGact ∕(RT)
RT
(4.63)
where csites stands for the concentration of lattice sites for the species of interest in
the material (moles of sites/cm3 ).
Almost all useful fuel cell electrolyte materials are purposely doped to increase the
number of charge carriers, and therefore the concentration of intrinsic carriers is usually
insignificant compared to the concentration of extrinsic carriers (see text box on previous
page). Thus, Equation 4.62 is far more important than Equation 4.63 for describing ionic
conduction in practical electrolytes. Equation 4.62 is often simplified to a pseudo-empirical
expression by lumping the various preexponential terms into a single factor, yielding
𝜎T = ASOFC e−ΔGact ∕RT
(4.64)
Similarly to Equation 4.32, the term ΔGact ∕RT can instead be written as Ea ∕kT, yielding
𝜎T = ASOFC e−Ea ∕kT
(4.65)
Experimental observations confirm the relationship described by Equation 4.64
(or 4.65).
Figure 4.18 shows experimental plots of log(𝜎T) versus 1∕T for both YSZ and
gadolinia-doped ceria (GDC, another candidate SOFC electrolyte). The multiplication of
𝜎 with T ensures that the slopes in these plots are indicative of the activation energy for
ion migration, ΔGact . The size of ΔGact is often critical for determining the conductivity
149
FUEL CELL CHARGE TRANSPORT
4
3
log(σT ) (Ω–1 · cm–1 K)
150
2
1
ΔGact=0.60eV
0
–1
–2
Gd-doped ceria
Y-stabilized zirconia
–3
–4
0.6
0.8
1.0
1.2
ΔGact=0.89eV
1.4
1000/T K
1.6
1.8
2.0
–1
Figure 4.18. Conductivity of YSZ and GDC electrolytes versus temperature.
of SOFC electrolytes. Typically, its value ranges between about 50,000 and 120,000 J/mol
(0.5–1.2 eV).
Further details on specific fuel cell electrolyte materials properties, including a more
in-depth discussion on YSZ and GDC, are provided in Chapter 9.
CALCULATING EXTRINSIC DEFECT CONCENTRATIONS IN
CRYSTALLINE CERAMIC MATERIALS
As was pointed out earlier in this chapter, almost all useful ceramic fuel cell electrolyte
materials are purposely doped to increase the number of charge carriers, and therefore
extrinsically created carriers dominate the conduction process. In order to calculate
the concentration of the extrinsically created charge carriers (c), which is needed in
Equation 4.62, information about the material composition, the doping concentration,
and the crystal structure or density is required.
As an example, consider the classic case of 8YSZ, which is zirconia doped with
8 mol% yttria. As shown in Figure 4.16, for every 2 Y that are substituted into the
ZrO2 lattice, one oxygen vacancy is created. These extrinsically created oxygen vacancies become the source of ionic conduction in this material. To create 8YSZ, 8 mol %
Y2 O3 is combined with 92 mol % ZrO2 . The chemical formula of 8YSZ can therefore
be represented as 0.92(ZrO2 ) + 0.08(Y2 O3 ) = Zr0.92 Y0.16 O2.08 . Because of the 2-to-1
relationship between Y dopants and the created oxygen vacancies, the number of oxygen
vacancies can be explicitly shown by writing the formula as Zr0.92 Y0.16 O2.08 V0.08 . One
REVIEW OF FUEL CELL ELECTROLYTE CLASSES
mole of this material will therefore contain 0.08 mol of oxygen vacancies. The fraction
of oxygen sites that are vacant, xv , is 0.08∕2.16 = 0.037. This vacancy fraction can be
converted into a vacancy concentration (cv , units of vacancies/cm3 ) by applying knowledge about the molecular weight and density of the material or by applying knowledge
about the molar volume of the material.
If the density of the material is known, this information can be used to convert molar
vacancy fraction to vacancy concentration as follows:
no
(4.66)
V
where co is the concentration of oxygen sites in the material (mol/cm3 ), no is the moles of
oxygen atoms per mole of material, and V is the molar volume of the material (cm3 /mol).
The molar volume can be calculated from the molecular weight (M, g/mol) and the density (𝜌, g∕cm3 ) as
M
V=
(4.67)
𝜌
c𝑣 = x𝑣 co = x𝑣
For 8YSZ, 𝜌 = 6.15 g∕cm3 and M = (91.22 g∕mol × 0.92 + 88.9 g∕mol × 0.16 +
16 g∕mol × 2.08) = 131.4 g∕mol. Thus
V=
131.4 g∕mol
= 21.4 cm3
6.15 g∕cm3
(
c𝑣 = 0.037 vacancies∕O site
2.16 mol O sites∕mol YSZ
(
)
21.4 cm3 ∕mol YSZ
(4.68)
)
(4.69)
3
= 0.0037 mol vacancies∕cm
If the lattice constant and crystal structure of the material are known, this information
can be used to convert vacancy fraction to vacancy concentration in an analogous fashion. In this case, the molar volume can be calculated from the unit cell information. For
example, 8YSZ has the cubic (fluorite-type) structure with a lattice constant a = 5.15 Å
and a total of four ZrO2 formula units per unit cell (e.g., four cations and eight anions).
Based on this information the molar volume can be estimated as
V=
(5.15A)3 × (6.022 × 1023 )
= 20.5 cm3
4
(4.70)
which is reasonably close to the density-based value calculated from Equation 4.68.
From this point, the vacancy concentration, cv , can be calculated as before using
Equation 4.69.
151
152
FUEL CELL CHARGE TRANSPORT
4.5.4
Mixed Ionic–Electronic Conductors
So far, this chapter has focused almost exclusively on pure ionic conductors. These are
materials that conduct charged ionic species but do not conduct electrons. Beyond the traditional classes of pure ionic conductors and pure electronic conductors, however, there are
also interesting classes of materials that can conduct both ions and electrons. These materials are known as “mixed ionic–electronic conductors” (MIECs) or, more simply, “mixed
conductors.”
Many doped metal oxide ceramic materials exhibit both electronic and ionic conductivity. This is because doping can introduce both ionic defects (like oxygen vacancies)
and electronic defects (like free electrons or free holes). Both the ionic and electronic
defects can then “wander” through the material, leading to simultaneous ionic and electronic conductivity. If an oxide material is a mixed conductor, it is unsuitable for use as a fuel
cell electrolyte (since the electronic conductivity would essentially “short” the fuel cell).
However, MIECs are extremely attractive for SOFC electrode structures, because they can
dramatically increase electrochemical reactivity and thereby improve fuel cell performance.
Why do MIECs increase electrochemical activity? As you may recall from Chapter 3
(Section 3.11), fuel cell reactions can only occur where the electrolyte, electrode, and gas
phases are all in contact. This requirement is expressed by the concept of the “triple-phase
zone,” which refers to regions or points where the gas pores, electrode, and electrolyte
phases converge (see Figure 3.14). In order to maximize the number of these three-phase
zones, most fuel cell electrode–electrolyte interfaces employ a highly nanostructured geometry with significant intermixing, or blending, of the electrode and electrolyte phases (along
with gas porosity). However, another strategy to increase the number of reaction zones is
to employ a mixed-conductor electrode. Because a MIEC conducts both ions and electrons, it can simultaneously provide both the ionic species and the electrons needed for an
electrochemical reaction. In this case, only one additional phase (the gas phase) is needed
for electrochemical reaction. Thus, fuel cell reactions can occur anywhere along the entire
surface of the MIEC where it is in contact with the gas phase. Figure 4.19 schematically
illustrates the difference between a standard fuel cell electrode (Figure 4.19a) and a MIEC
electrode (Figure 4.19b).
As you can imagine, MIECs are scientifically fascinating materials. Most MIECs are
ceramic materials and are therefore employed in SOFC electrodes—particularly as cathode
electrode materials. In contrast, there is very little research on MIECs for low-temperature
PEMFCs, but perhaps this will be an interesting area for future work. The prototypical
MIEC is (La,Sr)MnO3 (LSM). LSM is used as the cathode electrode in many SOFC designs.
In LSM, Sr2+ is substituted for La3+ as a dopant in order to create oxygen vacancies and
holes. Due to the charge difference between La3+ and Sr2+ , either oxygen vacancies or
electron holes must be created to maintain charge neutrality, as illustrated by the following
defect reactions:
Oxygen vacancy formation:
Electron hole formation:
′
2Oxo → 2SrLa
+ Vo−
′
null → 2SrLa
+ 2h⋅
MORE ON DIFFUSIVITY AND CONDUCTIVITY (OPTIONAL)
Standard Electrode: Only
TPBs are active for reaction
MIEC Electrode: Entire
surface is active for reaction
e–
e–
O2
O2
O2
O2
O2–
O2–
Electrolyte
O2–
Electrolyte
(a)
(b)
Figure 4.19. A standard SOFC cathode electrode (a) versus a mixed ionic–electronic conducting
(MIEC) SOFC cathode electrode (b).
In the first reaction, one oxygen vacancy (Vo⋅⋅ ) is formed for every two Sr2+ dopant substitutions. This process is identical to the vacancy creation process in YSZ (see Section 4.5.3).
In the second reaction, two holes (h⋅ ) are formed for every two Sr2+ dopant substitutions.
Under typical SOFC conditions, hole conduction in LSM is dominant compared to oxygen
vacancy conduction. Therefore, LSM is only a marginal MIEC (i.e., for all intents and purposes it is almost exclusively a p-type electronic conductor). Nevertheless, its remarkable
stability and compatibility with other SOFC materials make it a popular choice in many
SOFC designs.
Significant recent research has been conducted to develop better MIEC materials,
and there are several other La-based perovskites that show increased ionic conductivity,
and therefore better mixed-conduction behavior, compared to LSM. These materials
include (La,Sr)(Co,Mn)O3 , (La,Sr)FeO3 , and (La,Sr)CoO3 . These materials tend to
provide much higher ionic conductivity compared to LSM and therefore function as
true mixed ionic–electronic conductors. Unfortunately, these materials also tend to be
less stable than LSM and have therefore proven difficult to deploy in functional SOFC
designs. Nevertheless, the electrochemical benefits of MIEC electrodes are substantial, and
therefore MIEC development remains an extremely intriguing area of research. Further
details on these materials are provided for the interested reader in Chapter 9.
4.6
MORE ON DIFFUSIVITY AND CONDUCTIVITY (OPTIONAL)
In this optional section, we develop an atomistic picture to explore conductivity and
diffusivity in more detail. We find that for conductors where charge transport involves a
“hopping”-type mechanism, conductivity and diffusivity are intimately related. Diffusivity
measures the intrinsic rate of this hopping process. Conductivity incorporates how this
153
FUEL CELL CHARGE TRANSPORT
hopping process is modified by the presence of an electric field driving force. Diffusivity
is therefore actually the more fundamental parameter.
Diffusivity is a more fundamental parameter of atomic motion because even in the
absence of any driving force, hopping of ions from site to site within the lattice still occurs
at a rate that is characterized by the diffusivity. Of course, without a driving force, the net
movement of ions is zero, but they are still exchanging lattice sites with one another. This
is another example of a dynamic equilibrium; compare it to the exchange current density
phenomenon that we learned about in Chapter 3.
4.6.1
Atomistic Origins of Diffusivity
Using the schematic in Figure 4.20b, we can derive an atomistic picture of diffusivity. The
atoms in this figure are arranged in a series of parallel atomic planes. We would like to
calculate the net flux (net movement) of gray atoms from left to right across the imaginary
plane labeled A in Figure 4.20 (which lies between two real atomic planes in the material).
Examining atomic plane 1 in the figure, we assume that the flux of gray atoms hopping in
the forward direction (and therefore through plane A) is simply determined by the number
Concentration of
gray atoms
154
Jnet
Distance (x)
(a)
ΔX
A
JA+
JA–
(c1) (c2)
A
(b)
Figure 4.20. (a) Macroscopic picture of diffusion. (b) Atomistic view of diffusion. The net flux of
gray atoms across an imaginary plane A in this crystalline lattice is given by the flux of gray atoms
hopping from plane 1 to plane 2 minus the flux of gray atoms hopping from plane 2 to plane 1. Since
there are more gray atoms on plane 1 than plane 2, there is a net flux of gray atoms from plane 1 to
plane 2. This net flux will be proportional to the concentration difference of gray atoms between the
two planes.
MORE ON DIFFUSIVITY AND CONDUCTIVITY (OPTIONAL)
(concentration) of gray atoms available to hop times the hopping rate:
JA+ = 21 𝑣c1 Δx
(4.71)
where JA+ is the forward flux through plane A (mol/cm2 ⋅ s), v is the hopping rate (s–1 ),
c1 is the volume concentration (mol/cm3 ) of gray atoms in plane 1, Δx (cm) is the atomic
spacing required to convert volume concentration to planar concentration (mol/cm2 ), and
the 1/2 accounts for the fact that on average only half of the jumps will be “forward” jumps.
(On average, half of the jumps will be to the left, half of the jumps will be to the right.)
Similarly, the flux of gray atoms hopping from plane 2 backward through plane A will
be given by
JA− = 21 𝑣c2 Δx
(4.72)
where JA− is the backward flux through plane A and c2 is the volume concentration
(mol/cm3 ) of gray atoms in plane 2. The net flux of gray atoms across plane A is therefore
given by the difference between the forward and backward fluxes through plane A:
Jnet = 21 𝑣Δx(c1 − c2 )
(4.73)
We would like to make this expression look like our familiar equation for diffusion:
J = −D(dc∕dx) We can express Equation 4.73 in terms of a concentration gradient as
(c1 − c2 )
Δx
1
2 Δc
= − 2 𝑣(Δx)
Δx
1
2 dc
= − 2 𝑣(Δx)
(for small x)
dx
Jnet = − 12 𝑣(Δx)2
(4.74)
Comparison with the traditional diffusion equation J = −D(dc∕dx) allows us to identify
what we call the diffusivity as
(4.75)
D = 12 𝑣(Δx)2
We therefore recognize that the diffusivity embodies information about the intrinsic hopping rate for atoms in the material (v) and information about the atomic length scale (jump
distance) associated with the material.
As mentioned previously, the hopping rate embodied by v is exponentially activated.
Consider Figure 4.21b, which shows the free-energy curve encountered by an atom as it
hops from one lattice site to a neighboring lattice site. Because the two lattice sites are
essentially equivalent, in the absence of a driving force a hopping atom will possess the
same free energy in its initial and final positions. However, an activation barrier impedes the
motion of the atom as it hops between positions. We might associate this energy barrier with
the displacements that the atom causes as it squeezes through the crystal lattice between
lattice sites. (See Figure 4.21a, which shows a physical picture of the hopping process.)
155
FUEL CELL CHARGE TRANSPORT
(a)
C+
C+
C+
C+
C+
–
A
+
C
(b)
Free energy
156
∆Gact
Distance
Figure 4.21. Atomistic view of hopping process. (a) Physical picture of the hopping process. As the
anion (A− ) hops from its original lattice site to an adjacent, vacant lattice site, it must squeeze through
a tight spot in the crystal lattice. (b) Free-energy picture of the hopping process. The tight spot in the
crystal lattice represents an energy barrier for the hopping process.
In a treatment analogous to the reaction rate theory developed in the previous chapter,
we can write the hopping rate as
𝑣 = 𝑣0 e−ΔGact ∕(RT)
(4.76)
where ΔGact is the activation barrier for the hopping process and v0 is the jump attempt
frequency.
Based on this activated model for diffusion, we can then write a complete expression for
the diffusivity as
(4.77)
D = 12 (Δx)2 𝑣0 e−ΔGact ∕(RT)
or, lumping all the preexponential constants into a D0 term.
D = D0 e−ΔGact ∕(RT)
4.6.2
(4.78)
Relationship between Conductivity and Diffusivity (1)
To understand how conductivity relates to diffusivity, we take a look at how an applied
electric field will affect the hopping probabilities for diffusion. Consider Figure 4.22,
which shows the effect of a linear voltage gradient on the activation barrier for the hopping
process. From this picture, it is clear that the activation barrier for a “forward” hop is
MORE ON DIFFUSIVITY AND CONDUCTIVITY (OPTIONAL)
Free energy
dV
1
zF∆x
dx
2
Voltage gradient
dV
zF
dx
∆G’act
zF ∆x
1
∆x
2
dV
dx
∆x
Distance
Figure 4.22. Effect of linear voltage gradient on activation barrier for hopping. The linear variation
in voltage with distance causes a linear drop in free energy with distance. This reduces the forward
activation barrier (ΔG′act < ΔGact ). Two adjacent lattice sites are separated by Δx; therefore, the total
free-energy drop between them is given by zFΔx(dV∕dx). If the activation barrier occurs halfway
between the two lattice sites, ΔGact will be decreased by 21 zF Δx(dV∕dx). [In other words, ΔG′act =
ΔGact − 21 zF Δx(dV∕dx).]
reduced by 21 zF Δx(dV∕dx) while the activation barrier for the “reverse” hop is increased
by 12 zF Δx(dV∕dx). (We are assuming that the activated state occurs exactly halfway
between the two lattice positions, or in other words that 𝛼 = 12 .) The forward-(𝑣1 ) and
reverse-(𝑣2 ) hopping-rate expressions are therefore
𝑣1 = 𝑣0 exp
]
[
− ΔGact − 21 zF Δx (dV∕dx)
[
𝑣2 = 𝑣0 exp
− ΔGact +
RT
1
zF Δx (dV∕dx)
2
]
(4.79)
RT
This voltage gradient modification to the activation barrier turns out to be small. In fact,
1
2
zF
dV
Δx
≪1
RT
dx
157
158
FUEL CELL CHARGE TRANSPORT
so we can use the approximation ex ≈ 1 + x for the second term in the exponentials. This
allows us to rewrite the hopping rate expressions as
)
(
zF
dV
Δx
𝑣1 ≈ 𝑣0 e−ΔGact ∕(RT) 1 + 21
RT
dx
)
(
dV
1 zF
−ΔGact ∕(RT)
𝑣2 ≈ 𝑣0 e
1− 2
Δx
RT
dx
(4.80)
Proceeding as before, we can then write the net flux across an imaginary plane A in a
material as
(4.81)
Jnet = JA+ − JA− = 21 Δx(c1 𝑣1 − c2 𝑣2 )
Since we are interested in conductivity this time, we would like to consider a flux that is
driven purely by the potential gradient. In other words, we want to get rid of any effects of
a concentration gradient by saying that c1 = c2 = c. Making this modification and inserting
the formulas for v1 and v2 give
)
czF dV
Δx
RT
dx
)
(
czF
dV
1
= 2 (Δx)2 𝑣0 e−ΔGact ∕(RT)
RT dx
Jnet = 12 Δx 𝑣0 e−ΔGact ∕(RT)
(
(4.82)
Recognizing the first group of terms as our diffusion coefficient D, we thus have
Jnet =
czFD dV
RT dx
(4.83)
Comparing this to the conduction equation
J=
𝜎 dV
zF dx
we see that 𝜎 and D are related by
𝜎=
c(zF)2 D
RT
(4.84)
For conductors that rely on a diffusive hopping-based charge transport mechanism, this
important result relates the observed conductivity of the material to the atomistic diffusivity
of the charge carriers. This equation is our key for understanding the atomistic underpinnings of ionic conductivity in crystalline materials.
4.6.3
Relationship between Diffusivity and Conductivity (2)
Recall from Section 2.4.4 that the introduction of the electrochemical potential gave us an
alternate way to understand the Nernst equation. In a similar fashion, looking at charge
MORE ON DIFFUSIVITY AND CONDUCTIVITY (OPTIONAL)
transport from the perspective of the electrochemical potential gives us an alternate way to
understand the relationship between conductivity and diffusivity. Recall the definition of
the electrochemical potential (Equation 2.99):
𝜇̃i = 𝜇i0 + RT ln ai + zi F𝜙i
If we assume that activity is purely related to concentration (ai = ci ∕c0 ), then the electrochemical potential can be written as
𝜇̃i = 𝜇i0 + RT ln
ci
+ zi F𝜙i
c0
(4.85)
The charge transport flux due to a gradient in the electrochemical potential will include
both the flux contributions due to the concentration gradient and the flux contributions due
to the potential gradient:
)
(
)]
[ (
d ln ci ∕c0
𝜕 𝜇̃
dV
= −Mi𝜇 RT
+ zi F
(4.86)
Ji = −Mi𝜇
𝜕x
dx
dx
The concentration term in the natural logarithm can be processed by remembering the
chain rule of differentiation:
d[ln(ci ∕c0 )] c0 d(ci ∕c0 )
1 dci
=
=
dx
ci
dx
ci dx
(4.87)
Therefore, the total charge transport flux due to an electrochemical potential gradient is
really made up of two fluxes, one driven by a concentration gradient and one driven by a
voltage gradient:
Mi𝜇 RT dci
dV
− Mi𝜇 zi F
(4.88)
Ji = −
ci dx
dx
Comparing the concentration gradient term in this equation to our previous expression
for diffusion allows us to identify Mi𝜇 in terms of diffusivity:
Mi𝜇 RT
ci
=D
Dci
Mi𝜇 =
RT
(4.89)
Comparing the voltage gradient term in this expression to our previous expression for
conduction allows us to identify 𝜎 in terms of diffusivity:
Mi𝜇 zF =
c (zF)2 D
𝜎
, where 𝜎 = i
|z|F
RT
(4.90)
159
160
FUEL CELL CHARGE TRANSPORT
By using the electrochemical potential, we arrive at the same result as before. Interestingly, we did not have to make any assumptions about the mechanism of the transport
process this time. Thus, we see that the relationship between diffusivity and conductivity
is completely general. (In other words, it does not just apply to hopping mechanisms.) The
conductivity and diffusivity of a material are related because the fundamental driving forces
for diffusion and conduction are related via the electrochemical potential.
4.7 WHY ELECTRICAL DRIVING FORCES DOMINATE CHARGE
TRANSPORT (OPTIONAL)
Our relationship between conductivity and diffusivity allows us to explain why electrical
driving forces dominate charge transport.
In metallic electron conductors, the extremely high background concentration of free
electrons means that electron concentration is basically invariant across the conductor.
This means that there are no gradients in electron chemical potential across the conductor.
Additionally, since metal conductors are solid materials, pressure gradients do not exist.
Therefore, we find that electron conduction in metals is driven only by voltage gradients.
What about for ion conductors? Like the metallic conductors, most fuel cell ion conductors are also solid state, therefore pressure gradients do not exist. (Even in fuel cells
that employ liquid electrolytes, the electrolyte is usually so thin that convection does not
contribute significantly). Similarly, the background concentration of ionic charge carriers is
also usually large, so that significant concentration gradients do not arise. However, even if
large concentration gradients were to arise, we find that the “effective strength” of a voltage
gradient driving force is far greater than the effective strength of a concentration gradient
driving force. To illustrate this point, let’s compare the charge flux generated by a concentration gradient to the charge flux generated by a voltage gradient. The charge flux generated
by a concentration gradient (jc ) is given by
jc = zFD
dc
dx
(4.91)
The charge flux generated by a voltage gradient (j𝑣 ) is given by
j𝑣 = 𝜎
dV
dx
(4.92)
Note that the quantity zF is required to convert moles in the diffusion equation into
charge in coulombs. As we have learned, 𝜎 and D are related by
𝜎=
c(zF)2 D
RT
(4.93)
The maximum possible sustainable charge flux due to a concentration gradient across a
material is
c
jc = zFD 0
(4.94)
L
QUANTUM MECHANICS–BASED SIMULATION OF ION CONDUCTION IN OXIDE ELECTROLYTES (OPTIONAL)
where L is the thickness of the material and c0 is the bulk concentration of charge carriers. The voltage, V, that would be required to produce an equivalent charge flux can be
calculated from
j 𝑣 = jc
c0
(zF)2 D
RT
c
V
= zFD 0
L
L
Solving for V gives
V=
RT
zF
(4.95)
(4.96)
At room temperature, for z = 1, RT∕zF = 0.0257 V. Therefore a voltage drop of
25.7 mV across the thickness of the material accomplishes the same thing as the maximum
possible chemical driving force available from concentration effects. Effectively, the
quantity RT∕zF sets the strength of the electric driving force relative to the chemical
(concentration) driving force. Because RT∕zF is small (for the fuel cell temperature range
of interest), fuel cell charge transport is dominated by electrical driving forces rather than
chemical potential driving forces.
4.8 QUANTUM MECHANICS–BASED SIMULATION OF ION CONDUCTION
IN OXIDE ELECTROLYTES (OPTIONAL)
In the previous sections, we have discussed the atomistic mechanisms of conduction and
diffusion. In particular, you have learned that diffusion (and hence conduction) in crystalline
oxide electrolytes occurs by a hopping process and that the rate of this hopping process is
determined by the size of the energy barrier for motion, ΔGact . In general, materials with
a lower barrier height will yield higher ionic diffusivities and hence higher ionic conductivities. This is exemplified in Figure 4.18 where GDC displays higher ionic conductivity
than YSZ (especially at lower temperatures) due to a smaller ΔGact . The quest for new
solid-oxide electrolyte materials has therefore focused on creating materials with higher
concentrations of mobile defects and lower activation barriers.
New electrolyte development, like new catalyst development, is largely a trial-and-error
process. Researchers first develop new candidate materials and then screen them for high
ionic conductivity and stability. Recently, however, the same quantum mechanics techniques that have been developed to help identify new catalyst materials (recall Chapter
3.12) are also being applied to identify new oxide electrolyte materials. The basic idea is
that quantum mechanics techniques can be used to directly calculate the size of activation
barriers associated with atomic motion through a crystalline lattice. Based on these calculated barrier heights, the conductivity of potential new electrolyte materials can then be
theoretically predicted.
Consider a quantum simulation approach applied to YSZ. In YSZ the diffusing species
are oxide ions, which must jump from an occupied site in the lattice to an adjacent (unoccupied) “vacancy.” The height of the barrier associated with this jump depends on the exact
nature and symmetry of all the other atoms in the nearby vicinity. The exact neighborhood
161
FUEL CELL CHARGE TRANSPORT
surrounding a single atom in the lattice can vary significantly—in fact, a detailed analysis
reveals that there are 42 different atomic configurations that an oxide ion may encounter
when jumping into a neighboring vacancy in YSZ [11]! (And this analysis considers only
nearest neighbors and next-nearest neighbors.) The barrier heights for each of these 42 different atomic configurations will be different because the local environment associated with
each of these configurations is different. These barrier heights can be calculated based on
approximations to the Schrödinger equation (as discussed in Appendix D), which allows
the determination of the energy “landscape” for a system of atoms at zero degrees Kelvin.
The barrier height associated with moving an atom into a vacancy is calculated by determining the energy of the entire atomic configuration in a step-by-step fashion as the oxide
ion moves into the vacancy. Figure 4.23 shows the concept of this barrier height calculation, performed step by step by considering atomic rearrangements, applied to one of the
42 possible configurations in YSZ. Once this process has been completed for the first configuration, it must then be repeated for the other 41 atomic configurations—a laborious and
time-consuming process!
After calculating each of the 42 possible barrier heights associated with moving an
atom from its lattice position to an open vacancy, the next step is to employ the methods
of statistical thermodynamics to calculate the overall macroscopic diffusivity. Statistical
thermodynamics teaches us that barriers with lower height can be more easily overcome
than those with a higher barrier height. Thus, the macroscopic diffusivity will largely be
dominated by the atomic configurations that occur most frequently and that have the lowest barrier heights. Diffusion processes are typically simulated using kinetic Monte Carlo
(KMC) techniques, which assume that all atoms move randomly, but that the probability of
a successful move depends exponentially on the barrier height as we discussed in Section
4.5.3. In KMC methods, the rate of successful atomic jumps is proportional to a random
number multiplied with an exponential Boltzmann factor that contains the barrier height
for diffusion. By simulating hundreds of thousands (if not millions) of individual atomic
jumps using this KMC technique, the averaged “macroscopic” diffusivity for a material can
be estimated. This diffusivity information can then be used to predict the performance of
new ion conductors or help in understanding the behavior of current ion conductors.
Relative energy
162
ΔE
m
Migration path
Figure 4.23. Illustration of the migration energy barrier. The middle point corresponds to the saddle
where the oxygen ion and two cations such as zirconia align in the same plane before the oxide ion
continues its path forward creating a vacancy in the location where it started.
CHAPTER SUMMARY
2.4
–4.2
Experiment
–4.3
log(σT ) (Ω –1 · cm–1 K)
KMC
2.2
–4.4
2.1
–4.5
2.0
–4.6
1.9
–4.7
1.8
–4.8
1.7
1.6
log D/D0
2.3
–4.9
6
8
10
12
mole % Y2O3
14
16
Figure 4.24. Logarithmic plot of conductivity times T versus mol% Y2 O3 in YSZ comparing experiment (open squares) and calculation (closed circles).
As an example of the power provided by this combined quantum–KMC technique,
Figure 4.24 compares experimental measurements and theoretical predictions for the
conductivity of YSZ as a function of yttria dopant concentration. As discussed in Section
4.5.3, adding excessive amounts of yttria to zirconia will actually decrease ionic conductivity because defects begin to interact with one another, reducing their ability to move.
This subtle effect is captured beautifully by the combined quantum–KMC simulation
approach.
4.9
CHAPTER SUMMARY
• Charge transport in fuel cells is predominantly driven by a voltage gradient. This
charge transport process is known as conduction.
• The voltage that is expended to drive conductive charge transport represents a loss
to fuel cell performance. Known as the ohmic overvoltage, this loss generally obeys
Ohm’s law of conduction, V = iR, where R is the ohmic resistance of the fuel cell.
• Fuel cell ohmic resistance includes the resistance from the electrodes, electrolyte,
interconnects, and so on. However, it is usually dominated by the electrolyte resistance.
• Resistance scales with conductor area A, thickness L, and conductivity σ: R = L∕𝜎A.
• Because resistance scales with area, area-specific fuel cell resistances (ASRs) are
computed to make comparisons between different-size fuel cells possible (ASR =
A × R).
• Because resistance scales with thickness, fuel cell electrolytes are made as thin as
possible.
163
164
FUEL CELL CHARGE TRANSPORT
• Because resistance scales with conductivity, developing high-conductivity electrode
and electrolyte materials is critical.
• Conductivity is determined by carrier concentration and carrier mobility:
𝜎i = (|zi |F)ci ui .
• Metals and ion conductors show vastly different structures and conduction mechanisms, leading to vastly different conductivities.
• Ion conductivity even in good electrolytes is generally four to eight orders of magnitude lower than electron conductivity in metals.
• In addition to having high ionic conductivity, electrolytes must be stable in both highly
reducing and highly oxidizing environments. This can be a significant challenge.
• The three major electrolyte classes employed in fuel cells are (1) liquid, (2) polymer,
and (3) ceramic electrolytes.
• Mobility (and hence conductivity) in aqueous electrolytes is determined by the balance between ion acceleration under an electric field and frictional drag due to fluid
viscosity. In general, the smaller the ion and the greater its charge, the higher the
mobility.
• Conductivity in Nafion (a polymer electrolyte) is dominated by water content. High
water content leads to high conductivity. Nafion conductivity may be determined by
modeling the water content in the membrane.
• Conductivity in ceramic electrolytes is controlled by defects (“mistakes”) in the crystal lattice. Natural (intrinsic) defect concentrations are generally low, so higher (extrinsic) defect concentrations are usually introduced into the lattice on purpose via doping.
• Mixed ionic and electronic conductors (MIECs) conduct both electrons and ions. They
are useful for SOFC electrodes, where simultaneous conduction of electrons and ions
enables improved reactivity by extending three-phase boundaries into two-phase reaction zones.
• (Optional section) At the atomistic level, we find that conductivity is determined by a
more basic parameter known as diffusivity D. Diffusivity expresses the intrinsic rate
of movement of atoms within a material.
• (Optional section) By examining an atomistic picture of diffusion and conduction, we
can explicitly relate diffusivity and conductivity: 𝜎 = c(zF)2 D∕(RT).
• (Optional section) Using the relationship between conductivity and diffusivity, we can
understand why voltage driving forces (conduction) dominate charge transport.
CHAPTER EXERCISES
Review Questions
4.1
Why does charge transport result in a voltage loss in fuel cells?
4.2
If a fuel cell’s area is increased 10-fold and its resistance is decreased 9-fold, will the
ohmic losses in the fuel cell increase or decrease (for a given current density, all else
being equal)?
CHAPTER EXERCISES
4.3
What are the two main factors that determine a material’s conductivity?
4.4
Why are the electron conductivities of metals so much larger than the ion conductivities of electrolytes?
4.5
List at least four important requirements for a candidate fuel cell electrolyte. Which
requirement (other than high conductivity) is often the hardest to fulfill?
Calculations
4.6
Redraw Figure 4.4c for a SOFC, where O2– is the mobile charge carrier in the electrolyte. Is there any change in the figure?
4.7
Draw a fuel cell voltage profile similar to those shown in Figure 4.4 that simultaneously shows the effects of both activation losses and ohmic losses.
4.8
Given that fuel cell voltages are typically around 1 V or less, what would be the absolute minimum possible functional electrolyte thickness for a SOFC if the dielectric
breakdown strength of the electrolyte is 108 V/m?
4.9
In Section 4.3.2, we discussed how fuel cell electrolyte resistance scales with thickness (in general as L∕𝜎). Several practical factors were listed that limit the useful range
of electrolyte thickness. Fuel crossover was stated to cause an undesirable parasitic
loss which can eventually become so large that further thickness decreases are counterproductive! In other words, at a given current density, an optimal electrolyte thickness may exist, and reducing the electrolyte thickness below this optimal value will
actually increase the total fuel cell losses. We would like to model this phenomenon.
Assume that the leak current jleak across an electrolyte gives rise to an additional fuel
cell loss of the following form: 𝜂leak = A ln jleak . Furthermore, assume that jleak varies
inversely with electrolyte thickness L as jleak = B∕L. For a given current density j
determine the optimal electrolyte thickness that minimizes 𝜂ohmic + 𝜂leak .
4.10 A 5-cm2 fuel cell has Relec = 0.01 Ω and 𝜎electrolyte = 0.10 Ω−1 ⋅ cm−1 . If the electrolyte is 100 𝜇m thick, predict the ohmic voltage losses for this fuel cell at j =
50 mA∕cm2 .
4.11 Derive Equation 4.32 using Equations 4.22 and 4.24.
4.12 Consider a PEMFC operating at 0.8 A/cm2 and 70∘ C. Hydrogen gas at 90∘ C and
80% relative humidity is provided to the fuel cell at the rate of 8 A. The fuel cell
area is 8 cm2 and the drag ratio of water molecules to hydrogen, α, is 0.8. Find the
water activity of the hydrogen exhaust. Assume that p = 1atm and that the hydrogen
exhaust exits at the fuel cell temperature, 70∘ C.
4.13 Consider two H2 –O2 PEMFCs powering an external load at 1 A/cm2 . The fuel cells
are running with differently humidified gases: (a) aW,anode = 1.0, aW,cathode = 0.5;
(b) aW,anode = 0.5, aW,cathode = 1.0. Estimate the ohmic overpotential for both fuel cells
if they are both running at 80∘ C. Assume that they both employ a 125-𝜇m-thick
Nafion electrolyte. Based on your results, discuss the relative effects of humidity at
the anode versus the cathode.
165
166
FUEL CELL CHARGE TRANSPORT
4.14 (a) Calculate the diffusion coefficient for oxygen ions in a pure ZrO2 electrolyte at
T = 1000∘ C given ΔGact = 100 kJ∕mol, 𝑣0 = 1013 Hz. ZrO2 has a cubic unit
cell with a lattice constant a = 5 Å and contains four Zr atoms and eight O atoms.
Assume that the oxygen–oxygen “jump”distance Δx = 12 a.
(b) Calculate the intrinsic carrier concentration in the electrolyte given Δh𝑣 = 1 eV.
(Assume vacancies are the dominant carrier.)
(c) From your answers in (a) and (b), calculate the intrinsic conductivity of this electrolyte at 1000∘ C.
4.15 You have determined the resistance of a 100-𝜇m-thick, 1.0-cm2 -area YSZ electrolyte
sample to be 47.7 Ω at T = 700 K and 0.680 Ω at T = 1000 K. Calculate D0 and
ΔGact for this electrolyte material given that the material is doped with 8% molar
Y2 O3 . Recall from problem 4.14 that pure ZrO2 has a cubic unit cell with a lattice
constant of 5 Å and contains four Zr atoms and eight O atoms. Assume that the lattice
constant does not change with doping.
4.16 Which of the following is a correct statement for the water behavior in a Nafion-based
PEMFC operating on dry H2 /dry air at room temperature:
(a) Both electro-osmotic drag and backdiffusion move water from the anode to the
cathode.
(b) Both electro-osmotic drag and backdiffusion move water from the cathode to the
anode
(c) Electro-osmotic drag moves water from the cathode to the anode while backdiffusion moves water from the anode to the cathode
(d) Electro-osmotic drag moves water from the anode to the cathode while backdiffusion moves water from the cathode to the anode
4.17 A solid-oxide fuel cell electrolyte has ASR = 0.20 Ω ⋅ cm2 at T = 726.85∘ C and
ASR = 0.05 Ω ⋅ cm2 at T = 926.85∘ C. What is the activation energy (ΔG ) for
conduction in this electrolyte material?
act