Chapter 4' Fuel Cell Charge Transport

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Chapter 4.Fuel Cell Charge Transport
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4.1 Charges move in response to forces
Flux (J) the rate at which charges move through
a material measures how much of a given quantity
flows through a material per unit area per unit
time
Charge flux Amount of charge that flows through
a material per unit area per unit time C/cm2s
A/cm2 gt charge flux is the same thing as current
density gt denoted as j
(1)
zi charge number for the carrier F Faradays
constant
JA flow rate / cross-sectional area
3
Governing equation for transport
Ji flux of species I Fk k different forces
acting on I Mik coupling coefficients between
force and flux (property both of the species that
is moving and the material through which it is
moving)
(2)
In fuel cells 3 major driving forces for charge
transport 1) Electrical driving forces
(electrical potential gradient dV/dx) 2)
Chemical driving forces (chemical potential
gradient dµ/dx) 3) Mechanical driving forces
(pressure gradient dP / dx)
4
Example H2-O2 PEMFC
Electrode Voltage gradient drives electron
charge transport Electrolyte Both
concentration gradient and voltage gradient
drive ion transport gt In fuel cell, electrical
driving force dominates
5
J charge flux dV/dx electric field s
conductivity
(3)
In comparing Eq. (2) and (3) gt coupling
coefficient is conductivity Coupling coefficient
that describes transport due to Chemical
potential (concentration) diffusivity Pressure
gradient viscosity
Summary of transport processes relevant to charge
transport
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4.2 Charge transport results in a voltage loss
For fuel cells, charge transport causes a loss in
cell voltage gt because conductors are not
perfect intrinsic resistance to charge flow
(4)
(5)
(6)
(7)
Ionic charge transport tends to be more difficult
than electronic charge transport gt ionic
contribution to Rohmic tends to dominate
7
(a) Hypothetical voltage profile of a fuel cell
at thermodynamics equilibrium. 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. (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.
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4.3 Characteristics of fuel cell charge transport
resistance
Effect of ohmic loss on fuel cell performance.
Charge transport resistance contributes a linear
decrease in fuel cell operating voltage as
determined by Ohms law.
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• 4.3.1 Resistance scales with area

(8)
ASR area-specific resistance (area-normalized
resistance), Ocm2
(9)
gt Since resistance is inversely proportional to
area, multiplication by area is necessary to get
area-independent resistances.
(10)
10
4.3.2 Resistance scales with Thickness
From Eq. (9) and (10)
(10)
gt The shorter the conductor length L, the lower
the resistance. gt Reducing electrolyte thickness
improves fuel cell performance. Limitations to
prepare thin electrolytes 1) Mechanical
integrity (solid electrolytes) 2)
Nonuniformities (pinhole problems) 3) Shorting
(electrical shorting) 4) Fuel crossover 5)
Contact resistance 6) Dielectric breakdown
(electric field across the membrane
exceeds the dielectric breakdown field for the
material
11

• 4.3.3 Fuel cell resistances are additive

Total ohmic resistance presented by a fuel cell
is a combination of resistances In the diagram,
fuel cell resistance is divided into
interconnect, anode, electrolyte, and cathode
components. Current flows serially through all
components gt total fuel cell resistance is given
by the series sum of the individual resistance
components
12

• 4.3.4 Ionic (electrolyte) resistance usually
  dominates
• Best electrolytes employed in fuel cells
• conductivity of around 0.10 O-1cm-1
• even at a thickness of 50 µm (very thin) ASR is
  0.05 Ocm2.
• 50-µmthick porous carbon cloth electrode ASR
  is less than 510-6 Ocm2.
• gt example how electrolyte resistance usually
  dominates fuel cells

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4.4 Physical meaning of conductivity

• Conductivity the ability of a material to
  permit the flow of charge when driven by an
  electric field
• conductivity is influenced by
• 1) how many carriers are available to transport
  charge
• 2) the mobility of those carriers within the
  material

(11)
ci molar concentration of charge carriers ui
mobility of the charge carriers within the
material ziF convert charge carrier
concentration from units of moles to
units of coulombs zi charge number for the
carrier (zi 2 for Cu2, -1 for e-) F
Faradays constant
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Conductivity and mobility

• 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.

15

• 4.4.1 Electronic versus ionic conductors

a) 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.
16

• 4.4.2 Electron conductivity in a metal

Drude model for mobility of free electrons
t mean free time between scattering events m
mass of the electron (9.1110-31 kg) q
elementary electron charge in coulombs (1.68
10-19 C)
(12)
Inserting Eq. (12) into Eq. (11)
(13)
General atomic packing density order of 1028
atoms/m3 gt molar carrier concentration order of
104 mol/m3 Typical electron conductivity for
metals 106108 O-1m-1
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• 4.4.3 Ion conductivity in a crystalline solid
  electrolyte

Effectiveness of hopping process
(14)
D0 constant reflecting the attempt frequency of
hopping process ?Gact activation barrier for
hopping process R gas constant T temperature (K)
Overall mobility of ions in solid electrolyte
(15)
zi charge number on ion F Faradays constant
Conductivity by inserting Eq. (15) into Eq. (11)
(16)
Maximum effective vacancy doping 8-10 (carrier
concentration of 102-103 mol/m3) Typical ion
diffusivity 10-8 m2/s for liquid and polymer,
10-11 m2/s for ceramic (700-1000 oC) Typical
carrier concentration 103-104 mol/m3 for liquid,
102-103 mol/m3 for polymer and ceramic (700-1000
oC) gt Ionic conductivity of 10-4-102 O-1m-1
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4.5 Review of fuel cell electrolytes classes

• Any fuel cell electrolyte must meet the
  following requirements
• 1) High ionic conductivity
• 2) Low electronic conductivity
• 3) High stability
• 4) Low fuel crossover
• 5) Reasonable mechanical strength
• 6) Ease of manufacturability

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• 4.5.1 Ionic conduction in aqueous
  electrolytes/ionic liquids
• Example
• NaCl dissolved in water aqueous electrolyte
• molten NaCl ionic liquid
• Almost all aqueous/liquid electrolyte fuel cells
  use a matrix material to support or immobilize
  the electrolyte. The matrix generally
  accomplishes 3 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

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• Ionic conductivity in aqueous/liquid
  environments can best be approached using a
  driving force/frictional force balance model

Electric field force FE
ni charge number of ion q fundamental electron
charge (1.610-19 C)
(17)
Frictional drag force FD from Stokess law
µ viscosity of the liquid r radius of the
ion v velocity of the ion
(18)
gt mobility ui (ratio between the applied
electric field and the resulting ion velocity)
(19)
gt mobility is determined by the ion size and the
liquid viscosity
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Selected ionic mobilities at infinite dilution in
aqueous solutions at 25 oC
22

• 4.5.2 Ionic conduction in polymer electrolytes

By combining Eq. (14) and (16),
s0 conductivity at a reference state Ea
activation energy (eV/mol)
(20)
For a polymer to be a good ion conductor
(structural properties) 1. The presence of fixed
charge sites 2. The presence of free volume
(open space)
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.
23

• Vehicle mechanism
• Ions are transported through free-volume spaces
  by hitching a ride on certain free species as
  these vehicles pass by.
• Persulfonated polytetrafluoroethylene (PTFE)
  known as Nafion exhibits extremely high proton
  conductivity based on the vehicle mechanism.
• Ionic transport in Nafion

Chemical structure of Nafion. Nafion has a PTFE
backbone for mechanical stability with sulfonic
groups to promote proton conduction.
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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.
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Nafion absorbs significant amounts of water
(20)
pW actual partial pressure of water vapor pSAT
saturation water vapor pressure
Water content versus water activity for Nafion
117 at 303 K (30 oC) according to Eq. (20). 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.
for 0 lt aW lt1 for 1 lt aW lt3
(21)
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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. gt relative humidity
  is 100 when pWpSAT

pSAT in bars (1 bar 100,000 Pa) T temperature
in degrees Celsius
Example Fully himidified air at 80 oC and 3 atm
is provided to a fuel cell, then the water vapor
pressure is
gt mole fraction of water in fully humidified air
as 0.4669 bar/3 atm 0.154
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Nafion conductivity is highly dependent on water
content
(22)
where
s conductivity (S/cm) of the membrane T (K)
temperature
Total resistance of a membrane is found by
integrating the local resistances over the
membrane thickness (tm) as
(23)
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Ionic conductivity of Nafion versus water content
? according to Eq.(22) and 4.39 at 303 K.
Ionic conductivity of Nafion versus temperature
according to Eq. (22) when ?22.
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Protons drag water with them

• Conductivity in Nafion is dependent on water
  content
• Electro-osmotic drag protons traveling through
  the pores of Nafion drag one or more water
  molecules along with them
• Electro-osmotic drag coefficient ndrag number of
  water molecules accompanying the movement of each
  proton (ndrag nH2O/H)

(24)
Water drag flux from anode to cathode when a net
current j flows through the PEMFC
J molar flux of water due to electro-osmotic
drag (mol/scm2) j operating current density of
fuel cell (A/cm2) 2F convert from current
density to hydrogen flux (2 hydrogen to proton
flux)
(25)
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Back diffusion of water
In a PEMFC, electro-osmotic water drag moves
water from the anode to the cathode. gt Water
build up at the cathode gt Back diffusion occurs
from the cathode back to the anode The water
back-diffusion can be determined by
?dry dry density (kg/m3) of Nafion Mn Nafion
equivalent weight (kg/mol) z direction through
the membrane thickness
(26)
Total water flux in Nafion
D?(?) diffusivity of water in the Nafion
membrane, a function of water content ?
(27)
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Equivalent weight
Equivalent weight (atomic (formula) weight) /
(valence) Valence number of electrons that the
species can donate or accept Sulfonic group
(SO3-H) in Nafion has a valence of 1
(28)
(29)
Nafion has an equivalent weight of around 1 1.1
kg/mol and a dry density of 1970 kg/m3.
(30)
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Water diffusivity in Nafion

• Water diffusivity in Nafion (D?) is a function
  of water content ?. Experimentally, this
  dependence has been measured as

For ? gt 4 (cm2/s)
(31)
Water diffusivity D? in Nafion versus water
content ? at 303 K.
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• 4.5.3 Ionic conduction in ceramic electrolytes
• ion transport in SOFC
• YSZ yttria stabilized zirconia
• Adding yttria to zirconia introduces oxygen
  vacancies due to charge compensation effects

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 yttria atoms doped into
the lattice.
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Conductivity is determined by the combination of
carrier concentration c and carrier mobility u
(32)
In YSZ, carrier concentration is determined by
the strength of the yttria doping. Increasing
yttria contentgt increased oxygen vacancy
concentration, improving conductivity There is
an upper limit to doping
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Complete expression for conductivity
D carrier mobility diffusivity of the carrier
in the crystal lattice
(33)
High diffusivity gt high conductivity
D0 constant (cm2/s) ?gact activation barrier
for the diffusion process (J/mol) R gas
constant T temperature (K)
(34)
Complete expression for conductivity in SOFE
electrolytes
(35)
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Eq. (35) can be further refined depending on
whether the charge carriers are extrinsic or
intrinsic For extrinsic carrriers, c is
determined by the doping chemistry of the
electrolyte. In this case, c is a constant and
Eq. (35) can be used as is. For intrinsic
carriers, c is exponentially dependent on
temperature, and Eq. (35) must be modified as
follows
(36)
Where csites stands for the concentration of
lattice sites for the species of interest in the
material (moles of sites/cm3)
37
Conductivity of YSZ and GDC (Gd-doped ceria)
electrolytes versus temperature
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4.6 More on diffusivity and conductivity

• 4.6.1 Atomistic origins of diffusivity

Flux of grey atoms hopping in the forward
direction
(37)
JA forward flux through plane A v hopping
rate c1 volume concentration (mol/cm3) of grey
atoms in plane 1 ?x atomic spacing required to
convert volume concentration to planar
concentration (mol/cm2) ½ half of the jumps will
be forward jumps
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Flux of grey atoms hopping from plane 2 backward
through plane A
JA- backward flux through plane A c2 volume
concentration (mol/cm3) of grey atoms in plane 2
(38)
Net flux of grey atoms across plane A
(39)
Change to a familiar equation for diffusion J
-D(dc/dx)
(40)
(for small x)
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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.
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Hopping rate
?Gact activation barrier for the hopping
process v0 jump attempt frequency
(41)
Complete expression for diffusivity based on this
activated model for diffusion
(42)
(43)
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• 4.6.1 Relationship between conductivity and
  diffusivity
• 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(?Gactlt ?Gact). Two
  adjacent latticesites are separated by ?x
  therefore the total free-energy drop between them
  is given by zF ?x(dV/dx). If the activation
  barrier occurs halfway between the two lattice
  sites, ?Gact will be decreased by 1/2zF
  ?x(dV/dx). In other words, ?Gact ?Gact-1/2zF
  ?x(dV/dx).

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Forward hopping rate
(44)
Reverse hopping rate
(45)
gt
(47)
(46)
Can be used
44
We can rewrite the hopping rate as
(48)
Net flux across an imaginary plane A
(49)
Get rid of any effects of a concentration
gradient by c1c2c
(50)
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Recognizing the first group of terms as diffusion
coefficient D
(51)
Comparing to the conduction equation
(52)
We see that s and D are related by
(53)
gt Relation between observed conductivity of the
material and atomistic diffusivity of the charge
carriers
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Looking at charge transport from the perspective
of the electrochemical potential gives us an
alternate way to understand the relationship
between conductivity and diffusivity. From the
definition of the electrochemical pototential
(54)
Assume that activity is purely related to
concentration (ai ci/c0)
(55)
Charge transport flux due to a gradient in the
electrochemical potential will include flux
contributions by both the concentration and
potential gradient
(56)
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Concentration term in the natural logarithm
(57)
(58)
Concentration gradient
Voltage gradient
Identify Miµ in terms of diffusivity from
concentration gradient
(58)
Identify s in terms of diffusivity from voltage
gradient
(59)
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4.7 Why electrical driving forces dominate charge
transport

• In metallic electron conductors
• High concentration of free electrons
• gt there are no gradients in electron chemical
  potential across the conductor
• gt pressure gradients do not exist
• gt electron conduction in metals is driven only
  by voltage gradients
• In ion conductors (solid state in fuel cell)
• gt no pressure gradient no significant
  concentration gradient
• gt effective strength of voltage gradient gtgt
  effective strength of concentration gradient

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The charge flux generated by a concentration
gradient (jc) is given by
(60)
The charge flux generated by a voltage gradient
(jv) is given by
(61)
zF is required to convert moles in the diffusion
equation into charge in coulombs. D and s are
related by
(62)
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The maximum possible sustainable charge flux due
to a concentration gradient
L thickness of the material c0 bulk
concentration of charge carriers
(63)
Voltage (V) that would be required to produce an
equivalent charge flux
(64)
Solving for V
(65)
RT/zF sets the strength of the electric driving
force relative to the chemical driving force. In
fuel cell, RT/zF is small gt fuel cell transport
is dominated by electrical driving forces rather
than chemical potential driving forces.