EIS

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_Primer/EIS_Primer.htm

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.

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.
5.Use service environments.
Disadvantages.
1. Expensive.
2. Complex data analysis for quantification.

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

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
( )
( )
E t
R
I t


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.

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).
=2 f
 
0
( ) cos( )
E t E t



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 V=Vo + V  xcos( t) and measure I = Io +  I  cos( t- )
Then:
Z(,Vo) = (V  / I ) x{cos() + i sin()}
where i2 = -1
and both magnitude and phase of the impedance,  Z  and  vary with 

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:
0
( ) cos( )
I t I t
 
 
An expression analogous to Ohm's Law allows us to calculate the admittance (=the AC
resistance) of the system :
0
0
0
cos( )
( ) cos( )
( )
( ) cos( ) cos( )
E t
E t t
Z t Z
I t I t t
 
   
  
 
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:

Measure Z(,Vbias)
The result will be Z(,Vo) = V() / I()

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.

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,
exp( ) cos sin
i i
  
 
0
( ) exp( )
E t E j t


0
( ) exp( )
I t I i t i
 
 
0 0
exp( ) (cos sin )
E
Z Z i Z i
I
  
   
0
0
0
cos( )
( ) cos( )
( )
( ) cos( ) cos( )
E t
E t t
Z t Z
I t I t t
 
   
  
 

Data Presentation:
Nyquist 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.
0 0
exp( ) (cos sin )
E
Z Z i Z i
I
  
   
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.
1 1 1
Z R i C

 

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.
1 1 1
Z R i C

 

The different views on impedance data

Z´
Z”
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

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:
1) The response to x1(t) + x2(t) is y1(t) + y2(t)
2) The response to ax1(t) is ay1(t) ...

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.

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

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

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
Two Sine Waves in the
Frequency Domain
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.
Amplitude
Amplitude
Time
Frequency

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.

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: 1 2 3
eq
Z Z Z Z
  
Impedances in Parallel
1 2 3
1 1 1 1
eq
Z Z Z Z
  

Serial and Parallel Combinations of Circuit Elements
1 2 1 2 1 4 5
eq
Z Z Z R R
         
1 2
1 2
6 6 6
1 1 1
1 1 1
1 µF
(2 ) (2 ) (1 )
eq
e
Z Z
Z i C i C
i e i e i
 
  
  
   
   
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
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:
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.
C1 C2
R1 R2

Example: The Zn Air battery
There are three reactions:
1.The reaction at the anode between
metal ions and electrons :
2.The reaction at the cathode between
water and electrons
3.The reaction of the whole cell, i.e. the
two half-cell reactions added together:
2
2 2
2 4 2 2
Zn O H Zn H O
 
   
2
2 2 4
Zn Zn e

 
2 2
4 4 2
O H e H O

  

An electrochemical cell: The Zn Air battery
2
2 2
2 4 2 2
Zn O H Zn H O
 
   
2
2 2 4
Zn Zn e

 
2 2
4 4 2
O H e H O

  
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.
0 ln
x
y
n
X
X
n
Y
Y
a
G G RT
a
 
 
     
 
 



Equilibrium
If a reaction is at equilibrium, G=0 , 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.)
x
y
n
X
X
n
Y
Y
a
k
a
 
 

 
 
 



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:
2
2
2
2 1/2
0 ln
O
Zn H
Zn H O
a a p
G G RT
a a
 
 
     
 
 
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:
G zFE
  
where z is the stoichiometric number of electrons in the reaction. (This is due to
Faraday’s law)

In this form we have the Nernst equation for the cell and :
2
2
2
2 1/2
0
ln
O
Zn H
e
Zn H O
a a p
E E RT
a a
 
 
   
 
 
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.
 
2
2
0 2 1/2
ln
e O
Zn H
E E RT a a p
 
 
0
ln
RT
E k
zF


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:
where r is the solution resistivity.
The conductivity of the solution, , is more commonly used in solution resistance calculations.
Its relationship with solution resistance is:
1
R=
l
R
A
l l
A RA







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.

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

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.

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.

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 involved
h = overpotential ( E - E0 )

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, Co=Co*
and CR=CR*. 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 Co=Co* and CR=CR* 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.

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)

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:
0 r A
C
d
 

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

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.
( )
Z A i 
 


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)
0 coth( )
Y i B i
  0
tanh( )/
B i Y i
 
 
 

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:
C
R
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

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

The Voigt network
An electrical layer of a device can often be described by a resistor and capacitor in parallel

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

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

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 ).

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

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°.

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.

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.

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 conductivity
of 3Y-TZP

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.

Relaxation Dispersion of Ionic Conductivity in a
ZrO0.85Ca0.15O1.85 Single Crystal
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.
The results:

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.

Conclusion:
• The determination of the relaxation frequency, r==2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

Effect of intergranular glass films on the electrical conductivity
of 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.
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 1500°C to full density. The pellets
were carefully lapped to have planar faces and contacted with sintered Pt paste (without glass
additive!!!)
TZP 3Y

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.

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.

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.

Result:
3 mol Y TZP zirconia specimens with a ratio of 1:1 of SiO2 to Al2O3 impurities
have highest grain boundary resistances

EIS

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