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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

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

Five major topics are covered in this application

note

- AC Circuit Theory and Representation of Complex

Impedance Values. - Physical Electrochemistry and Circuit Elements.
- Common Equivalent Circuit Models.
- Extracting Model Parameters from Impedance Data.
- 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

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).

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 ?

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

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,

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.

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.

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

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) ...

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

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.

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

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

Impedances in Parallel

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.

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

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.

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.)

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)

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.

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

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 involvedh 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, 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.

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)

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.

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

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.

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.

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

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

- Relaxation Dispersion of O2- Ionic Conductivity

in a ZrO0.85Ca0.15O1.85 Single Crystal - Effect of intergranular glass films on the

electrical conductivityof 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

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.

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.

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

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!!!)

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.

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.

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.

Intergranular glass films on the electrical

conductivity of 3Y-TZP

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

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