ELECTROCHEMICAL IMPEDANCE SPECTROSCOPY.

1
ELECTROCHEMICAL IMPEDANCE SPECTROSCOPY.
Introduction. Electrochemical impedance
spectroscopy is a recent tool in corrosion and
solid state laboratories that is slowly making
its way into the service environment as units are
decreased in size and become portable. Impedance
Spectroscopy is also called AC Impedance or just
Impedance Spectroscopy. The usefulness of
impedance spectroscopy lies in the ability to
distinguish the dielectric and electric
properties of individual contributions of
components under investigation.
Most of the material displayed in this lecture is
taken from http//www.gamry.com/App_Notes/EIS_Pri
mer/EIS_Primer.htm
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ELECTROCHEMICAL IMPEDANCE SPECTROSCOPY
For example, if the behavior of a coating on a
metal when in salt water is required, by the
appropriate use of impedance spectroscopy, a
value of resistance and capacitance for the
coating can be determined through modeling of the
electrochemical data. The modeling procedure uses
electrical circuits built from components such as
resistors and capacitors to represent the
electrochemical behavior of the coating and the
metal substrate. Changes in the values for the
individual components indicate their behavior and
performance. Impedance spectroscopy is a
non-destructive technique and so can provide time
dependent information about the properties but
also about ongoing processes such as corrosion or
the discharge of batteries and e.g. the
electrochemical reactions in fuel cells,
batteries or any other electrochemical process.
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ELECTROCHEMICAL IMPEDANCE SPECTROSCOPY.

• Below is a listing of the advantages and
  disadvantages of the technique.
• Advantages.
• 1. Useful on high resistance materials such as
  paints and coatings.
• 2. Time dependent data is available
• 3. Non- destructive.
• 4. Quantitative data available.
• Use service environments.
• Disadvantages.
• 1. Expensive.
• 2. Complex data analysis for quantification.

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Five major topics are covered in this application
note

1. AC Circuit Theory and Representation of Complex
   Impedance Values.
2. Physical Electrochemistry and Circuit Elements.
3. Common Equivalent Circuit Models.
4. Extracting Model Parameters from Impedance Data.
5. Case studies

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AC Circuit Theory and Representation of Complex
Impedance Values
Impedance definition concept of complex impedance
Ohm's law defines resistance in terms of the
ratio between voltage E and current I.
The relationship is limited to only one circuit
element -- the ideal resistor. An ideal resistor
has several simplifying properties

• It follows Ohm's Law at all current and voltage
  levels
• It's resistance value is independent of
  frequency.
• AC current and voltage signals though a resistor
  are in phase with each other

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Real World

• Circuit elements that exhibit much more complex
  behavior. These elements force us to abandon the
  simple concept of resistance. In its place we use
  impedance, which is a more general circuit
  parameter.
• Like resistance, impedance is a measure of the
  ability of a circuit to resist the flow of
  electrical current. Unlike resistance, impedance
  is not limited by the simplifying properties
  listed above.
• Electrochemical impedance is usually measured by
  applying an AC potential to an electrochemical
  cell and measuring the current through the cell.
• Suppose that we apply a sinusoidal potential
  excitation. The response to this potential is an
  AC current signal, containing the excitation
  frequency and it's harmonics. This current signal
  can be analyzed as a sum of sinusoidal functions
  (a Fourier series).
• Electrochemical Impedance is normally measured
  using a small excitation signal. This is done so
  that the cell's response is pseudo-linear.
  Linearity is described in more detail in a
  following section. In a linear (or pseudo-linear)
  system, the current response to a sinusoidal
  potential will be a sinusoid at the same
  frequency but shifted in phase.

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Sinusoidal Current Response in a Linear System
The excitation signal, expressed as a function of
time, has the form of
E(t) is the potential at time tr Eo is the
amplitude of the signal, and ?? is the radial
frequency. The relationship between radial
frequency ? (expressed in radians/second) and
frequency f (expressed in Hertz (1/sec).
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Impedance as a Complex Number
Z(?,Vo) ?V(?) / ?I(?) The impedance at any
frequency ? is a complex number because ?I(?)
contains phase information as well as magnitude
- the AC current may have a phase lag ? with
respect to the AC voltage If we apply VVo ??V
? xcos(? t) and measure I Io ? ?I ?
cos(? t- ?) Then Z(?,Vo) (??V ? /? ?I ?)
xcos(?) i sin(?) where i2 -1 and both
magnitude and phase of the impedance, ? Z ? and ?
vary with ?
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Response in a linear System
In a linear system, the response signal, the
current I(t), is shifted in phase (?) and has a
different amplitude, I0
An expression analogous to Ohm's Law allows us to
calculate the admittance (the AC resistance) of
the system
The impedance is therefore expressed in terms of
a magnitude, Z0, and a phase shift, f. This
admittance may allso be written as complex
function
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Measure Z(?,Vbias)
The result will be Z(?,Vo) ?V(?) / ?I(?)
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Response dI of dE from the Current (I)/Field (E)
relation
Let us assume we have an electrical element to
which we apply an electric field E(t) and get the
response I(t), then we can disturb this system at
a certain field E with a small perturbation dE
and we will get at the current I a small response
perturbation dI. In the first approximation, as
the perturbation dE is small, the response dI
will be a linear response as well (mirror at the
tangent oft the I(E) curve!
If we plot the applied sinusoidal signal on the
X-axis of a graph and the sinusoidal response
signal I(t) on the Y-axis, an oval is plotted.
This oval is known as a "Lissajous figure".
Analysis of Lissajous figures on oscilloscope
screens was the accepted method of impedance
measurement prior to the availability of lock-in
amplifiers and frequency response analyzers.
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Complex writing
Using Eulers relationship it is possible to
express the impedance as a complex function. The
potential is described as, and the current
response as,   The impedance is then
represented as a complex number,    
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Data PresentationNyquist Plot with Impedance
Vector
Look at the Equation above. The expression for
Z(?) is composed of a real and an imaginary part.
If the real part is plotted on the X axis and the
imaginary part on the Y axis of a chart, we get a
"Nyquist plot". Notice that in this plot the
y-axis is negative and that each point on the
Nyquist plot is the impedance Z at one frequency.
On the Nyquist plot the impedance can be
represented as a vector of length Z. The angle
between this vector and the x-axis is f. Nyquist
plots have one major shortcoming. When you look
at any data point on the plot, you cannot tell
what frequency was used to record that point. Low
frequency data are on the right side of the plot
and higher frequencies are on the left. This is
true for EIS data where impedance usually falls
as frequency rises (this is not true of all
circuits).
The Nyquist plot the results from the RC circuit.
The semicircle is characteristic of a single
"time constant". Electrochemical Impedance plots
often contain several time constants. Often only
a portion of one or more of their semicircles is
seen.
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The Bode Plot
Another popular presentation method is the "Bode
plot". The impedance is plotted with log
frequency on the x-axis and both the absolute
value of the impedance (Z Z0 ) and phase-shift
on the y-axis. The Bode plot for the RC circuit
is shown below. Unlike the Nyquist plot, the Bode
plot explicitly shows frequency information.
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The different views on impedance data
The impedance data are the red points. Their
projection onto the Z-Z plane is called the
Nyquist plot The projection onto the Z-? plane
is called the Cole Cole diagram
Nyquist plot
Cole Cole diagram
Bode plot
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Electrochemistry - A Linear System?

• Electrical circuit theory distinguishes between
  linear and non-linear systems (circuits).
  Impedance analysis of linear circuits is much
  easier than analysis of non-linear ones.
• A linear system ... is one that possesses the
  important property of superposition If the input
  consists of the weighted sum of several signals,
  then the output is simply the superposition, that
  is, the weighted sum, of the responses of the
  system to each of the signals.
• Mathematically, let y1(t) be the response of a
  continuous time system to x1(t) ant let y2(t) be
  the output corresponding to the input x2(t).
• Then the system is linear if
• The response to x1(t) x2(t) is y1(t) y2(t)
• 2) The response to ax1(t) is ay1(t) ...

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Current versus Voltage Curve Showing
Pseudo-linearity
For a potentiostated electrochemical cell, the
input is the potential and the output is the
current. Electrochemical cells are not linear! 
Doubling the voltage will not necessarily double
the current.
However, we will show how electrochemical systems
can be pseudo-linear. When you look at a small
enough portion of a cell's current versus voltage
curve, it seems to be linear. In normal EIS
practice, a small (1 to 10 mV) AC signal is
applied to the cell. The signal is small enough
to confine you to a pseudo-linear segment of the
cell's current versus voltage curve. You do not
measure the cell's nonlinear response to the DC
potential because in EIS you only measure the
cell current at the excitation frequency.
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Steady State Systems
Measuring an EIS spectrum takes time (often many
hours). The system being measured must be at a
steady state throughout the time required to
measure the EIS spectrum. A common cause of
problems in EIS measurements and their analysis
is drift in the system being measured. In
practice a steady state can be difficult to
achieve. The cell can change through adsorption
of solution impurities, growth of an oxide layer,
build up of reaction products in solution,
coating degradation, temperature changes, to list
just a few factors. Standard EIS analysis tools
may give you wildly inaccurate results on a
system that is not at a steady state
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Time and Frequency Domains and Transforms
Signal processing theory refers to data domains.
The same data can be represented in different
domains. In EIS, we use two of these domains,
the time domain and the frequency domain. In the
time domain, signals are represented as signal
amplitude versus time. The Figure demonstrates
this for a signal consisting of two superimposed
sine waves.
Two Sine Waves in the Time Domain
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Time and frequency domain
The figures show the same data of the two sinus
waves in the time and the frequency domain.
Two Sine Waves in the Time Domain
Amplitude
Time
Two Sine Waves in the Frequency Domain
Amplitude
Frequency
Use the Fourier transform and inverse Fourier
transform to switch between the domains. The
common term, FFT, refers to a fast, computerized
implementation of the Fourier transform. Detailed
discussion of these transforms is beyond the
scope of this manual. Several of the references
given at the end of this chapter contain more
information on the Fourier transform and its use
in EIS. In modern EIS systems, lower frequency
data are usually measured in the time domain. The
controlling computer applies a digital
approximation to a sine wave to the cell by means
of a digital to analog converter. The current
response is measured using an analog to digital
computer. An FFT is used to convert the current
signal into the frequency domain.
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Electrical Circuit Elements
EIS data is commonly analyzed by fitting it to an
equivalent electrical circuit model. Most of the
circuit elements in the model are common
electrical elements such as resistors,
capacitors, and inductors. To be useful, the
elements in the model should have a basis in the
physical electrochemistry of the system. As an
example, most models contain a resistor that
models the cell's solution resistance. Some
knowledge of the impedance of the standard
circuit components is therefore quite useful. The
Table lists the common circuit elements, the
equation for their current versus voltage
relationship, and their impedance
Component Current Vs.Voltage Impedance
resistor E IR Z R
inductor E L di/dt Z i?L
capacitor I C dE/dt Z 1/i?C
Notice that the impedance of a resistor is
independent of frequency and has only a real
component. Because there is no imaginary
impedance, the current through a resistor is
always in phase with the voltage. The impedance
of an inductor increases as frequency increases.
Inductors have only an imaginary impedance
component. As a result, an inductor's current is
phase shifted 90 degrees with respect to the
voltage. The impedance versus frequency behavior
of a capacitor is opposite to that of an
inductor. A capacitor's impedance decreases as
the frequency is raised. Capacitors also have
only an imaginary impedance component. The
current through a capacitor is phase shifted -90
degrees with respect to the voltage.
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Serial and Parallel Combinations of Circuit
Elements
Very few electrochemical cells can be modeled
using a single equivalent circuit element.
Instead, EIS models usually consist of a number
of elements in a network. Both serial and
parallel combinations of elements occur.
Impedances in Series
Impedances in Parallel
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Serial and Parallel Combinations of Circuit
Elements
Suppose we have a 1? and a 4 ? resistor is
series. The impedance of a resistor is the same
as its resistance (see Table 2-1). We thus
calculate the total impedance Zeq
Resistance and impedance both go up when
resistors are combined in series. Now suppose
that we connect two 2 µF capacitors in series.
The total capacitance of the combined capacitors
is 1 µF
Impedance goes up, but capacitance goes down when
capacitors are connected in series. This is a
consequence of the inverse relationship between
capacitance and impedance.
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Example The Zn Air battery

• There are three reactions
• The reaction at the anode between metal ions and
  electrons
• The reaction at the cathode between water and
  electrons
• The reaction of the whole cell, i.e. the two
  half-cell reactions added together

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An electrochemical cell The Zn Air battery
For each of these reactions it is true that
where?G is the free energy change of the
reaction.?G0  is what the free energy change
would be if every component were in its standard
state. ax is the activity of reaction product X
 and  ay   is the activity of reactant Y. nx is
the stoichiometric coefficient of reaction
product X, and likewise for the reactants.(The
stoichiometric coefficient is the number of that
molecule that are involved in the reaction for
the whole-cell reaction written above, the
stoichiometric coefficient of water is 2, and of
oxygen gas is 1.)R is the ideal gas constant and
T  is the temperature.The symbol ? is the
multiplying equivalent of ?  all the terms
after it are multiplied together.
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An electrochemical cell The Zn Air battery
Equilibrium
If a reaction is at equilibrium, ?G0 , and the
free energy G of the system is at a minimum with
respect to how much of the reactants have been
converted to products. When this is the case, we
obtain where K is the equilibrium
constant. Thus we can deduce that ?G0  RTlnk 
this is true of a reaction whether it is at
equilibrium or not. (?G0 for a reaction is
determined by the energies of the bonds within
the molecules of the reactants and products, and
this is independent of how many such molecules
there are per unit volume.)                    
                                                  
             
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The Zn Air battery An electrochemical cell
The cathode reaction is at equilibrium if there
is no power supply connected to the circuit. It
can do this because each atom or ion has enough
energy to undergo the reaction in either
direction there is nothing stopping it being at
equilibrium. The anode reaction is also at its
own equilibrium. The reaction for the whole cell
is not at equilibrium. There is too much of an
energy barrier for it to be able to get there
the ions have to diffuse through the electrolyte
and the electrons have to go around through the
wires. (Or through a high impedance voltmeter,
which they almost certainly cannot do.) Thus for
the example given
?G ?0 and the quotient is not the equilibrium
constant but equal to the electric potential. We
can convert this into an expression for the
electrical potentials using the general rule
where z is the stoichiometric number of electrons
in the reaction. (This is due to Faradays law)
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An electrochemical cell The Zn Air battery
In this form we have the Nernst equation for the
cell and
The activities of Zn and water are one, because
Zn is in its standard state and the water is so
much more abundant than its solutes that it may
as well be in its standard state. Thus E0
is the equilibrium potential it is the
potential of the whole cell when the electrodes
are at equilibrium within themselves. It can be
worked out (easily, using algebra with a pen and
pencil) that                                 
                                                  
                                                  
  where K is the equilibrium constant i.e. if
we were at equilibrium over the whole
electrochemical cell, then E would be zero. E0
is a property of the system like that ?G0  , and
is still equal to the same number even when the
whole cell is not at equilibrium. If for some
reason it was required to find the value of E0  ,
we could use this expression. E0   is called the
standard electrode potential.
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Physical Electrochemistry and Equivalent Circuit
Elements
Electrolyte Resistance
Electrolyte resistance is often a significant
factor in the impedance of an electrochemical
cell. A modern 3 electrode potentiostat
compensates for the solution resistance between
the counter and reference electrodes. However,
any solution resistance between the reference
electrode and the working electrode must be
considered when you model your cell. The
resistance of an ionic solution depends on the
ionic concentration, type of ions, temperature
and the geometry of the area in which current is
carried. In a bounded area with area A and length
l carrying a uniform current the resistance is
defined as
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The electrolyte resisatnce
Standard chemical handbooks list ?? values for
specific solutions. For other solutions and solid
materials, you can calculate ? from specific ion
conductances. The units for ? are Siemens per
meter (S/m). The Siemens is the reciprocal of the
ohm, so 1 S 1/ohm
Unfortunately, most electrochemical cells do not
have uniform current distribution through a
definite electrolyte area. The major problem in
calculating solution resistance therefore
concerns determination of the current flow path
and the geometry of the electrolyte that carries
the current. A comprehensive discussion of the
approaches used to calculate practical
resistances from ionic conductances is well
beyond the scope of this manual. Fortunately, you
don't usually calculate solution resistance from
ionic conductances. Instead, it is found when you
fit a model to experimental EIS data.
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Double Layer Capacitance
A electrical double layer exists at the interface
between an electrode and its surrounding
electrolyte. This double layer is formed as
ions from the solution "stick on" the electrode
surface. Charges in the electrode are separated
from the charges of these ions. The separation is
very small, on the order of angstroms. Charges
separated by an insulator form a capacitor. On a
bare metal immersed in an electrolyte, you can
estimate that there will be approximately 30 µF
of capacitance for every cm2 of electrode
area. The value of the double layer capacitance
depends on many variables including electrode
potential, temperature, ionic concentrations,
types of ions, oxide layers, electrode roughness,
impurity adsorption, etc.
Principle of the Electric Double-Layer Here C
electrodes
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Polarization Resistance
Whenever the potential of an electrode is forced
away from it's value at open circuit, that is
referred to as polarizing the electrode. When
an electrode is polarized, it can cause current
to flow via electrochemical reactions that occur
at the electrode surface. The amount of current
is controlled by the kinetics of the reactions
and the diffusion of reactants both towards and
away from the electrode. In cells where an
electrode undergoes uniform corrosion at open
circuit, the open circuit potential is controlled
by the equilibrium between two different
electrochemical reactions. One of the reactions
generates cathodic current and the other anodic
current. The open circuit potential ends up at
the potential where the cathodic and anodic
currents are equal. It is referred to as a mixed
potential. The value of the current for either of
the reactions is known as the corrosion current.
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The Butler Volmer equation For the polarization
resistance of simple reactions at electrodes
When there are two simple, kinetically controlled
reactions occurring, the potential of the cell is
related to the current by the following (known as
the Butler-Volmer equation).
I is the measured cell current in amps,Icorr is
the corrosion current in amps,Eoc is the open
circuit potential in volts,?a is the anodic Beta
coefficient in volts/decade,?c is the cathodic
Beta coefficient in volts/decade
If we apply a small signal approximation (E-Eoc
is small) to the buler Volmer equation, we get
the following
which introduces a new parameter, Rp, the
polarization resistance. As you might guess from
its name, the polarization resistance behaves
like a resistor. If the Tafel constants ?i are
known, you can calculate the Icorr from Rp. The
Icorr in turn can be used to calculate a
corrosion rate. We will further discuss the Rp
parameter when we discuss cell models.
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Charge Transfer Resistance
A similar resistance is formed by a single
kinetically controlled electrochemical reaction.
In this case we do not have a mixed potential,
but rather a single reaction at
equilibrium. Consider a metal substrate in
contact with an electrolyte. The metal molecules
can electrolytically dissolve into the
electrolyte, according to or more generally
In the forward reaction in the first equation,
electrons enter the metal and metal ions diffuse
into the electrolyte. Charge is being
transferred. This charge transfer reaction has a
certain speed. The speed depends on the kind of
reaction, the temperature, the concentration of
the reaction products and the potential. The
general relation between the potential and the
current holds
io exchange current density Co concentration
of oxidant at the electrode surface Co
concentration of oxidant in the bulk CR
concentration of reductant at the electrode
surface
F Faradays constant T temperature R gas
constant a reaction order n number of
electrons involvedh overpotential ( E - E0 )
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Overvoltage potential
The overpotential, h, measures the degree of
polarization. It is the electrode potential minus
the equilibrium potential for the reaction. When
the concentration in the bulk is the same as at
the electrode surface, CoCo and CRCR. This
simplifies the last equation into This
equation is called the Butler-Volmer equation. It
is applicable when the polarization depends only
on the charge transfer kinetics. Stirring will
minimize diffusion effects and keep the
assumptions of CoCo and CRCR valid. When the
overpotential, h, is very small and the
electrochemical system is at equilibrium, the
expression for the charge transfer resistance
changes into From this equation the exchange
current i0 density can be calculated when Rct is
known.
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Diffusion Warburg impedance with infinite
thickness
Diffusion can create an impedance known as the
Warburg impedance. This impedance depends on the
frequency of the potential perturbation. At high
frequencies the Warburg impedance is small since
diffusing reactants don't have to move very far.
At low frequencies the reactants have to diffuse
farther, thereby increasing the Warburg
impedance. The equation for the "infinite"
Warburg impedance
On a Nyquist plot the infinite Warburg impedance
appears as a diagonal line with a slope of 0.5.
On a Bode plot, the Warburg impedance exhibits a
phase shift of 45. In the above equation, s is
the Warburg coefficient defined as
? radial frequency DO diffusion coefficient
of the oxidant DR diffusion coefficient of the
reductant A surface area of the electrode n
number of electrons transferred C bulk
concentration of the diffusing species (moles/cm3)
37
Diffusion Warburg impedance with finite thickness
The former equation of the Warburg impedance is
only valid if the diffusion layer has an infinite
thickness. Quite often this is not the case. If
the diffusion layer is bounded, the impedance at
lower frequencies no longer obeys the equation
above. Instead, we get the form          
with, ? Nernst diffusion layer thickness D
an average value of the diffusion coefficients of
the diffusing species This more general equation
is called the "finite" Warburg. For high
frequencies where ??? , or for an infinite
thickness of the diffusion layer where d ?? ,
this equation becomes the infinite Warburg
impedance.
38
Coating Capacitance
A capacitor is formed when two conducting plates
are separated by a non-conducting media, called
the dielectric. The value of the capacitance
depends on the size of the plates, the distance
between the plates and the properties of the
dielectric. The relationship is
With, ?o electrical permittivity ?r relative
electrical permittivity A surface of one
plate d distances between two plates Whereas
the electrical permittivity is a physical
constant, the relative electrical permittivity
depends on the material. Some useful ?r values
are given in the table
Material ?r
vacuum 1
water 80.1 ( 20 C )
organic coating 4 - 8
Notice the large difference between the
electrical permittivity of water and that of an
organic coating. The capacitance of a coated
substrate changes as it absorbs water. EIS can be
used to measure that change
39
Constant Phase Element (for double layer capacity
in real electrochemical cells)
Capacitors in EIS experiments often do not behave
ideally. Instead, they act like a constant phase
element (CPE) as defined below. When this
equation describes a capacitor, the constant A
1/C (the inverse of the capacitance) and the
exponent ? 1. For a constant phase element,
the exponent a is less than one. The "double
layer capacitor" on real cells often behaves like
a CPE instead of like a capacitor. Several
theories have been proposed to account for the
non-ideal behavior of the double layer but none
has been universally accepted. In most cases, you
can safely treat ? as an empirical constant and
not worry about its physical basis.
40
Common Equivalent Circuit Models
In the following section we show some common
equivalent circuits models. To elements used in
the following equivalent circuits are presented
in the Table. Equations for both the admittance
and impedance are given for each element.
Equivalent element Admittance Impedance
R 1/R R
C i?C 1/1/i?C
L 1/i?L i?L
W (infinite Warburg) Y0(i?)1/2 1/Y0(i?)1/2
O (finite Warburg)
Q (CPE) Y0(i?)? 1/Y0(i?)?
41
A Purely Capacitive Coating
A metal covered with an undamaged coating
generally has a very high impedance. The
equivalent circuit for such a situation is in the
Figure
The model includes a resistor (due primarily to
the electrolyte) and the coating capacitance in
series. A Nyquist plot for this model is shown in
the Figure. In making this plot, the following
values were assigned R 500 ? (a bit high but
realistic for a poorly conductive solution)C
200 pF (realistic for a 1 cm2 sample, a 25 µm
coating, and ?r 6 )fi 0.1 Hz (lowest scan
frequency -- a bit higher than typical) ff 100
kHz (highest scan frequency) 
The value of the capacitance cannot be determined
from the Nyquist plot. It can be determined by a
curve fit or from an examination of the data
points. Notice that the intercept of the curve
with the real axis gives an estimate of the
solution resistance. The highest impedance on
this graph is close to 1010 ? . This is close to
the limit of measurement of most EIS systems
42
A Purely Capacitive Coating in the Bode Plot
The same data are shown in a Bode plot in Figure
2-13. Notice that the capacitance can be
estimated from the graph but the solution
resistance value does not appear on the chart.
Even at 100 kHz, the impedance of the coating is
higher than the solution resistance
43
The Voigt network
An electrical layer of a device can often be
described by a resistor and capacitor in parallel
44
Cole-Cole Plots Impedance Plots in the Complex
Plane
When we plot the real and imaginary components of
impedance in the complex plane (Argand diagram),
we obtain a semicircle or partial semicircle for
each parallel RC Voigt network
The diameter corresponds to the resistance R.
The frequency at the 90 position corresponds to
1/t 1/RC
45
Analyzing Circuits
By using the various Cole-Cole plots we can
calculate values of the elements of the
equivalent circuit for any applied bias voltage
By doing this over a range of bias voltages, we
can obtain the field distribution in the layers
of the device (potential divider) and the
relative widths of the layers, since C 1/d
46
Randles Cell
The Randles cell is one of the simplest and most
common cell models. It includes a solution
resistance, a double layer capacitor and a charge
transfer or polarization resistance. In addition
to being a useful model in its own right, the
Randles cell model is often the starting point
for other more complex models. The equivalent
circuit for the Randles cell is shown in the
Figure. The double layer capacity is in parallel
with the impedance due to the charge transfer
reaction
The Nyquist plot for a Randles cell is always a
semicircle. The solution resistance can found by
reading the real axis value at the high frequency
intercept. This is the intercept near the origin
of the plot. Remember this plot was generated
assuming that Rs 20 ? and Rp 250 ? . The
real axis value at the other (low frequency)
intercept is the sum of the polarization
resistance and the solution resistance. The
diameter of the semicircle is therefore equal to
the polarization resistance (in this case 250?
).
47
Bode Plot oft Randalls cell
This Figure is the Bode plot for the same cell.
The solution resistance and the sum of the
solution resistance and the polarization
resistance can be read from the magnitude plot.
The phase angle does not reach 90 as it would
for a pure capacitive impedance. If the values
for Rs and Rp were more widely separated the
phase would approach 90.
Bode Plot for 1 mm/year Corrosion Rate
48
Mixed Kinetic and Diffusion Control
First consider a cell where semi-infinite
diffusion is the rate determining step, with a
series solution resistance as the only other cell
impedance. A Nyquist plot for this cell is shown
in Figure 2-17. Rs was assumed to be 20 W. The
Warburg coefficient calculated to be about 120
?sec-1/2 at room temperature for a two electron
transfer, diffusion of a single species with a
bulk concentration of 100 µM and a typical
diffusion coefficient of 1.6 x10-5 cm2/sec.    
Notice that the Warburg Impedance appears as a
straight line with a slope of 45.
49
Example Half a fuel cell
Adding to the previous example a double layer
with capacitance and a charge transfer impedance,
we get the equivalent circuit
This circuit models a cell where polarization is
due to a combination of kinetic and diffusion
processes. The Nyquist plot for this circuit is
shown in the Figure. As in the above example, the
Warbug coefficient is assumed to be  about 150 W
sec-1/2. Other assumptions Rs 20 ? , Rct
250 ? , and Cdl 40 µF.
50
Bode plot
The Bode plot for the same data is shown here.
The lower frequency limit was moved down to 1mHz
to better illustrate the differences in the slope
of the magnitude and in the phase between the
capacitor and the Warburg impedance. Note that
the phase approaches 45 at low frequency.  
51
Case studies

1. Relaxation Dispersion of O2- Ionic Conductivity
   in a ZrO0.85Ca0.15O1.85 Single Crystal
2. Effect of intergranular glass films on the
   electrical conductivityof 3Y-TZP

52
Relaxation Dispersion of O2- Ionic Conductivity
in a ZrO0.85Ca0.15O1.85 Single Crystal
The aim oft the study was To study the dynamic
behavior of the oxygen ion conductivity of a
cubic ZrO0.85Ca0.15O1.85 Single Crystal with AC
impedance spectroscopy and a dynamic pulse method
as a function of both, the frequency and
temperature in the range of 450 to1200 K and 20
to 108 Hz. We had the hypothesis that the oxygen
vacancies are clustered e.g. forming pairs with
the Ca dopant. Somewhen when heating up the
material we expected that the conductivity slope
in the Arrhenius plot would show two slopes One
for the O2- conductivity via clustered vacancies
and at higher temperature when the clusters are
broken up an lower activation energy.
The Method and Materials Single crytals of
dimensions of . 10 x 5 x 2 m3 were contacted in
four probe mode with Platinum . The electrodes
were painted on the specimen by applying a
conductive platinum paste (Delnetron 308A) from
Heraeus To minimize the stray capacitance of the
test leads, they were kept as short as possible.
The shields of the measurement terminals were
grounded.
The relaxation dispersion regions of the ionic
conductivity shift towards higher frequencies
with increasing temperature. This indicates that
these dispersions are thermally activated. At low
temperatures the intragrain relaxation process in
the zirconia lattice can be seen at high
frequencies, but the electrode effects are too
slow to be detected. In the temperature range
from 673 K to 873K both dispersions of the
electrodes and the bulk material are observed in
the frequency range between 20 and 106 Hz. At
higher temperatures the effect 0f the intragrain
processes disappears and only the dispersion of
the electrodes can be seen in the middle of the
frequency window.
53
Relaxation Dispersion of Ionic Conductivity in
aZrO0.85Ca0.15O1.85 Single Crystal
The results
The relaxation dispersion regions of the ionic
conductivity shift towards higher frequencies
with increasing temperature. This indicates that
these dispersions are thermally activated. At low
temperatures the intragrain relaxation process in
the zirconia lattice can be seen at high
frequencies, but the electrode effects are too
slow to be detected. In the temperature range
from 673 K to 873K both dispersions of the
electrodes and the bulk material are observed in
the frequency range between 20 and 106 Hz. At
higher temperatures the effect 0f the intragrain
processes disappears and only the dispersion of
the electrodes can be seen in the middle of the
frequency window.
54
Relaxation Dispersion of Ionic Conductivity in
aZrO0.85Ca0.15O1.85 Single Crystal
The temperature dependence of the intragrain bulk
ionic conductivity as determined from AC
impedance spectroscopy is shown as two Arrhenius
plots, log (J against l/T and log (J against
l/T,) in Fig. 5. No curvature in the Arrhenius
plots can be observed which would indicate that
some of the vacancy clusters would break up. The
slope of the straight line of the plot
corresponds to the activation energy of the ionic
conductivity.
55
Relaxation Dispersion of Ionic Conductivity in
aZrO0.85Ca0.15O1.85 Single Crystal

• Conclusion
• The determination of the relaxation frequency,
  ?r2?fr, corresponding to a mean jump frequency
  of oxygen vacancies, 1/? allows the
  determination of their mobility as weIl as the
  diffusion coefficient.
• A very narrow distribution of relaxation times
  shows that only one polarization mechanism
  exists.
• Activation energy of the ionic conductivity
  act. eng. of . relaxation frequency mobility of
  charge carriers
• It follows that the concentration of hopping
  charge carriers in calcia stabilized zirconia is
  invariant with temperature and no cluster break
  up was observed in the temperature range studied.

JA. Orliukas, P. Bohac, K. Sasaki L.
Gauckler Nichtmetallische Werkstoffe, ETH Zürich,
CH-8092 Zürich, Switzerlandournal of the European
Ceramic Society 12 (1993) 87-96
56
Effect of intergranular glass films on the
electrical conductivityof 3Y-TZP
The electrical conductivity of 3Y-TZP ceramics
containing Si02 and Al20 3 has been investigated
by complex impedance spectroscopy between 500 and
1270 K. At low temperatures, the total
electrical conductivity is suppressed by the
grain boundary glass films. The equilibrium
thickness of intergranular films is 1-2 nm, as
derived using the "brick-Iayer" model and
measured by HRTEM. A change in the slope of the
conductivity Arrhenius plots occurs at the
characteristic temperature Tb at which the
macroscopic grain boundary resistivity has the
same value as the resistivity of the grains. The
temperature dependence of the conductivity is
discussed in terms of a series combination of Re
elements.
TZP 3Y
Specimens were round pellets oft ca 1.5 cm in
diameter and 5 mm in height oft sintered ceramics
oft TZP with varous amounts oft SiO2 and Al2O3
additions . The coprecipitated powders were
calcined at 1050 C reground and pressed and
then sintered at 1500C to full density. The
pellets were carefully lapped to have planar
faces and contacted with sintered Pt paste
(without glass additive!!!)
57
Intergranular glass films on the electrical
conductivity of 3Y-TZP
Usually it is not possible to observe all three
dispersions simultaneously, due to a limited
frequency range used in this study (40 Hz-l MHz).
At temperatures below 500 K only the grain
dispersion can be seen at high frequencies.4o The
grain boundary and the electrode dispersion are
too slow to be detected at this temperature. In
the medium temperature range (500 K-800 K), we
can observe two dispersions, that of the grains
and that of the grain boundaries. Finally, above
800 K the intragrain dispersion shifts out of the
frequency window and the sluggish dispersion due
to the slower electrode processes becomes visible.
58
Intergranular glass films on the electrical
conductivity of 3Y-TZP
In Fig. 3 the frequency dependence of the
specific imaginary impedance contribution, p fI
Z" . LIA, is shown. From this figure the
individual dispersion regions of grains, grain
boundaries, and electrodes can be seen more
distinctly. The complex impedance data can be
displayed in the complex impedance plane with
real part ?' as the abscissa and the imaginary
part ?? as the ordinate (Cole Cole diagram). A
typical complex impedance spectrum of 3Y-TZP
(sample E-10) at a medium temperature of 596 K is
shown in Fig. 4.
59
Intergranular glass films on the electrical
conductivity of 3Y-TZP
Since the time constants (? RC) of individual
RC-elements differ by orders of magnitude,
individual semicircles of the grains and that of
the grain boundaries can clearly be distinguished
in this temperature range. The real specific
impedance sections between the distinct minima in
the imaginary part ?? reveal the macroscopic
specific resistivities of the grains (?? G) and
the grain boundaries (?'B), respectively. The
macroscopic specific resistivity of the grain
boundaries is equal to the difference between the
total (dc) specific resistivity of the sampie (p
T) and the macroscopic specific resistivity of
the grains ?GB ?T - ?G. Moreover, from the
maximum of imaginary impedance ? at the top of
each semicircle, the relaxation frequency ? of
the corresponding process can be determined from
the relation ??? 1, where ? 2?rfr, is the
angular frequency rad' s-1, fr the
corresponding frequency of the applied electrical
ac-field Hz, and ? RC the time constant of
the relaxation circuit.
60
Intergranular glass films on the electrical
conductivity of 3Y-TZP
61
Intergranular glass films on the electrical
conductivity of 3Y-TZP
Result 3 mol Y TZP zirconia specimens with a
ratio of 11 of SiO2 to Al2O3 impurities have
highest grain boundary resistances