Experimental Techniques

TUTORIAL 4TUTORIAL 4

► The interaction between electromagnetic waves and The interaction between electromagnetic waves and

dielectric materials can be determined by broadband dielectric materials can be determined by broadband

measurement techniques.measurement techniques.

► Dielectric relaxation spectroscopy allows the study of Dielectric relaxation spectroscopy allows the study of

molecular structure, through the orientation of dipoles molecular structure, through the orientation of dipoles

under the action of an electric field.under the action of an electric field.

► The experimental devices cover the frequency range The experimental devices cover the frequency range

1010-4-4 -10 -101111 Hz. Hz.

10-4 100 104 108 1012

Time-domain spectrometer

Frequency-response analyzer

AC-bridges

Reflectometers

Resonance circuits

Cavities and waveguides

MEASUREMENT SYSTEMS IN THE MEASUREMENT SYSTEMS IN THE TIME DOMAINTIME DOMAIN

► In linear systems the time-dependent response to a step In linear systems the time-dependent response to a step

function field and the frequency-dependent response to a function field and the frequency-dependent response to a

sinusoidal electric field are related through Fourier sinusoidal electric field are related through Fourier

transforms. transforms.

► For this reason, from a mathematical point of view, there For this reason, from a mathematical point of view, there

is no essential difference between these two types of is no essential difference between these two types of

measurement.measurement.

► Over a long period of time the equipment for Over a long period of time the equipment for

measurements in the time domain has been far less measurements in the time domain has been far less

developed than that used in the frequency domain. developed than that used in the frequency domain.

► As a result, available experimental data in the time As a result, available experimental data in the time

domain are much less abundant than those in the domain are much less abundant than those in the

frequency domain.frequency domain.

Time domain spectroscopyTime domain spectroscopy► To cover the lowest To cover the lowest

frequency range (from 10frequency range (from 10--

44 to 10 to 1011 Hz), time domain Hz), time domain

spectrometers have spectrometers have

recently been developed.recently been developed.

► In these devices, a In these devices, a

voltage step voltage step VoVo is applied is applied

to the sample placed to the sample placed

between the plates of a between the plates of a

plane parallel capacitor, plane parallel capacitor,

and the current and the current I(t)I(t) is is

recorded.recorded.

Vo

I(t)

2o o

o

e S e RC

d d

Rd

0

( ) ( )

( ) ( )

1( ) ( ') '

oo

o o

t

o o

I t d tC

V dt

d t I t

dt e S E

t I t dtC V

*

0

( )( ) exp( )

d ti t dt

dt

Complex Dielectric Function

Time DependentDielectric Function

*(ω)

*

0

( ) 1( ) exp( )

2

d ti t d

dt

(t)

The main item in the equipment is the The main item in the equipment is the

electrometer, which must be able to measure electrometer, which must be able to measure

currents as low as 10currents as low as 10-16-16A. A.

In many cases the applied voltage can be taken In many cases the applied voltage can be taken

from the internal voltage source of the from the internal voltage source of the

electrometer. electrometer.

Also low-noise cables with high insulation Also low-noise cables with high insulation

resistance must be used. resistance must be used.

MEASUREMENT SYSTEMS IN THE MEASUREMENT SYSTEMS IN THE FREQUENCY DOMAINFREQUENCY DOMAIN

► In the intermediate frequency range 10In the intermediate frequency range 10-1-1- 10- 1066

Hz, capacitance bridges have been the common Hz, capacitance bridges have been the common

tools used to measure dielectric permittivities.tools used to measure dielectric permittivities.

► The devices are based on the Wheatstone The devices are based on the Wheatstone

bridge principle where the arms are bridge principle where the arms are

capacitance-resistance networks.capacitance-resistance networks.

► The principle of measurement of capacitance The principle of measurement of capacitance

bridges is based on the balance of the bridge bridges is based on the balance of the bridge

placing the test sample in one of the arms. placing the test sample in one of the arms.

The sample is represented by an RC network in parallel or series.The sample is represented by an RC network in parallel or series.

When the null detector of the bridge is at its minimum value (as When the null detector of the bridge is at its minimum value (as

close as possible to zero), the equations of the balanced bridge close as possible to zero), the equations of the balanced bridge

provide the values of the capacitance and loss factor (or provide the values of the capacitance and loss factor (or

conductivity) for the test sampleconductivity) for the test sample

Frequency response analyzers have proved to be very useful in Frequency response analyzers have proved to be very useful in

measuring dielectric permittivities in the frequency range 10measuring dielectric permittivities in the frequency range 10-2-2 - 10 - 1066

Hz,.Hz,.

An a.c. voltage An a.c. voltage VV11 is applied to the sample, and then a resistor is applied to the sample, and then a resistor RR, ,

or alternatively a current-to-voltage converter for low frequencies, or alternatively a current-to-voltage converter for low frequencies,

converts the sample current converts the sample current IIss, into a voltage , into a voltage VV22 . .

By comparing the By comparing the amplitude amplitude and the and the phase anglephase angle between these between these

two voltages, the complex impedance of the sample two voltages, the complex impedance of the sample ZZss can be can be

calculated ascalculated as1 2 1 2

2s

s

V V V VZ R

I V

1 2 1 2

2s

s

V V V VZ R

I V

Conductivity

► Owing to parasitic inductances, the high-Owing to parasitic inductances, the high-

frequency limit is about 1 MHz, frequency limit is about 1 MHz,

► It is necessary to be very careful with the It is necessary to be very careful with the

temperature control, and for this purpose temperature control, and for this purpose

it is advisable to measure the it is advisable to measure the

temperature as close as possible to the temperature as close as possible to the

sample.sample.

► At frequencies ranging from 1 MHz to 10 At frequencies ranging from 1 MHz to 10

GHz, the inductance of the connecting GHz, the inductance of the connecting

cables contributes to the measured cables contributes to the measured

impedance.impedance.

► At frequencies above 1 GHz the At frequencies above 1 GHz the

technique often used to obtain dielectric technique often used to obtain dielectric

spectra is reflectometry. spectra is reflectometry.

► The technique is based on the reflection The technique is based on the reflection

of an electric wave, transported through of an electric wave, transported through

a coaxial line, in a dielectric sample cell a coaxial line, in a dielectric sample cell

attached at the end of the line.attached at the end of the line.

► In this case, the reflective coefficient is a In this case, the reflective coefficient is a

function of the complex permittivity of function of the complex permittivity of

the sample, and the electric and the sample, and the electric and

geometric cell lengths.geometric cell lengths.

*

*

( )*( )

( )refl

inc

U xr x

U x

**

0 *

1 ( )( )

1 ( )s

r lZ Z

r l

* * 2( ) (0)expr l r

i

2 " 2 '; =

n n

Incoming voltage

reflected voltage

Reflection coefficient

Reflection at the beginning of the line

Attenuation coefficient

Propagationcoefficient

IMMITTANCE ANALYSISIMMITTANCE ANALYSISBasic Immittance FunctionsBasic Immittance Functions

► In many cases, it is possible to reproduce the electric In many cases, it is possible to reproduce the electric

properties of a dipolar system by means of passive elements properties of a dipolar system by means of passive elements

such as resistors, capacitors or combined elements.such as resistors, capacitors or combined elements.

► One of the advantages of the models is that they often easily One of the advantages of the models is that they often easily

describe the response of a system to polarization processes.describe the response of a system to polarization processes.

► However, it is necessary to stress that the models in general However, it is necessary to stress that the models in general

only provide an approximate way to represent the actual only provide an approximate way to represent the actual

behavior of the system.behavior of the system.

► The analysis of dielectric materials is commonly made in The analysis of dielectric materials is commonly made in

terms of the complex permittivity terms of the complex permittivity functionfunction ** or its inverse, or its inverse,

the electric modulus the electric modulus M*M*

* * 1

1* *

* *

* *

( )

o

o

M

Y Z

Y i C

M i C Z

► electrical impedanceelectrical impedance and and

admittanceadmittance are the are the

appropriate functions to appropriate functions to

represent the response of represent the response of

the corresponding the corresponding

equivalent circuits.equivalent circuits.

► As a consequence, the As a consequence, the

four basic immittance four basic immittance

functions are functions are permittivitypermittivity, ,

electric moduluselectric modulus, ,

impedanceimpedance and and

admittanceadmittance..

► They are related by the They are related by the

following formulae:following formulae:

”

tan=’’/’

M’’

Mixed Circuit. Debye EquationsMixed Circuit. Debye Equations

= RC= RC22CC11 = =CCoo

CC22 = ( = (oo--) Co) Co

► As shown before, Debye equation can be obtained in As shown before, Debye equation can be obtained in three different ways: three different ways:

► (1)(1) on the grounds of some simplifying assumptions on the grounds of some simplifying assumptions concerning concerning rotational Brownian motionrotational Brownian motion, ,

► (2)(2) assuming time-dependent orientational assuming time-dependent orientational depolarization of a material governed by depolarization of a material governed by first order first order kineticskinetics, and , and

► (3)(3) from the linear response theory assuming the time from the linear response theory assuming the time dipole correlation function described by a simple dipole correlation function described by a simple decreasing exponentialdecreasing exponential..

► The actual expressions are given byThe actual expressions are given by

► Under certain circumstances, the admittance is Under certain circumstances, the admittance is increased on account of hopping conductivity increased on account of hopping conductivity processes. Then, a conductivity term must be includedprocesses. Then, a conductivity term must be included

oo is a d.c. conductivity. is a d.c. conductivity.

► However, the presence of interactions leads to the However, the presence of interactions leads to the inclusion of a frequency dependent term in the inclusion of a frequency dependent term in the conductivity in such a way thatconductivity in such a way that

EMPIRICAL MODELS TO REPRESENT EMPIRICAL MODELS TO REPRESENT DIELECTRIC DATA - DIELECTRIC DATA - Retardation Time SpectraRetardation Time Spectra

► The assumptions upon which the Debye equations are The assumptions upon which the Debye equations are

based imply, in practice, that very few systems display based imply, in practice, that very few systems display

Debye behaviorDebye behavior

► In fact, relaxations in complex and disordered systems In fact, relaxations in complex and disordered systems

deviate from this simple behavior.deviate from this simple behavior.

► An alternative way to extend the scope of the Debye An alternative way to extend the scope of the Debye

dispersion relations is to include more than one dispersion relations is to include more than one

relaxation time in the physical description of relaxation relaxation time in the physical description of relaxation

phenomena.phenomena.

► The term The term N(N()) represents the distribution of relaxation represents the distribution of relaxation

(or better retardation) times representing the fraction (or better retardation) times representing the fraction

of the total dispersion that has a retardation time of the total dispersion that has a retardation time

between between and and +d+d

► The real and imaginary parts of the complex The real and imaginary parts of the complex

permittivity are given in terms of the retardation times permittivity are given in terms of the retardation times

by:by:

► Alternatively, the retardation spectrum can be defined Alternatively, the retardation spectrum can be defined

asas

Retardation time spectraRetardation time spectra

► Advantages:Advantages:

► Better separation of Better separation of processesprocesses

► Processes are Processes are narrower than in narrower than in frequency domainfrequency domain

► Disadvantages:Disadvantages:

► Require numerical Require numerical evaluation of the evaluation of the spectrum.spectrum.

► No physical senseNo physical sense

10-2 100 102 104 106

2

3

4

10-2

10-1

"

'

f, Hz

-6 -4 -2 00,0

0,2

0,4

0,6

0,8

1,0

L

(ln

)

log

Cole - Cole EquationCole - Cole Equation

► Experimental data (Experimental data (’’ vs ’’ vs ’’)rarely fit to a Debye semicircle.)rarely fit to a Debye semicircle.

► Studying several organic crystalline compounds, Cole and Cole Studying several organic crystalline compounds, Cole and Cole

found that the centers of the experimental arcs were displaced found that the centers of the experimental arcs were displaced

below the real axis, the experimental data thus having the shape of below the real axis, the experimental data thus having the shape of

a depressed arc.a depressed arc.

1-={0,5 – 1}

Low frequencieshigh frequencies

► The corresponding The corresponding equivalent circuit is:equivalent circuit is:

► The admittance is given The admittance is given by:by:

► Note that the circuit Note that the circuit contains a new element, contains a new element, namely a constant phase namely a constant phase element element (CPE)(CPE), the , the admittance of which is admittance of which is given by given by

► The admittance reduces The admittance reduces to Rto R-1-1 when when = 0 = 0

When we can use Cole – Cole When we can use Cole – Cole equationequation

► Symmetric relaxations. Symmetric relaxations. ► In general all Secondary relaxations can In general all Secondary relaxations can

be fitted by Cole – Cole equation.be fitted by Cole – Cole equation.► The The (1-(1-)) parameter, give us an idea parameter, give us an idea

about how distributed is the relaxation about how distributed is the relaxation (how broad it is).(how broad it is).

► In general the In general the (1-(1-)) parameter, must parameter, must increase with the temperature. increase with the temperature.

Fuoss- Kirkwood EquationFuoss- Kirkwood Equation► 1941 - Fuoss and Kirkwood 1941 - Fuoss and Kirkwood

propose to extend the Debye propose to extend the Debye

equation, in order to fit equation, in order to fit

symmetric functions.symmetric functions.

► Assuming an Arrhenius Assuming an Arrhenius

dependency of the relaxation dependency of the relaxation

time with the temperature, it time with the temperature, it

is possible to express the FK is possible to express the FK

equation as a T function.equation as a T function.

max

max

max

"

"

"2

"( )sec ln

"( )sec ·ln

"( ) 21

o

o

m

om

o

h Debye

h m FK

When it’s possible to use the FK When it’s possible to use the FK eq. eq.

► Secondary relaxations – Symmetric Secondary relaxations – Symmetric relaxations.relaxations.

► Advantages: The temperature Advantages: The temperature dependencies of the loss factor have a dependencies of the loss factor have a very simple expression.very simple expression.

► There are some relation between the There are some relation between the mm parameter of the FK equation and parameter of the FK equation and the the (1- (1- )) parameter of the CC parameter of the CC equation.equation.

a=1-

Davison – Cole EquationDavison – Cole Equation

► The Cole – Cole and Fuoss – Kirkwood equations The Cole – Cole and Fuoss – Kirkwood equations

are very useful for symmetric relaxations. are very useful for symmetric relaxations.

► However, experimental data obtained from However, experimental data obtained from ” ”

vs. vs. ’ plots show skewness on the high ’ plots show skewness on the high

frequency side.frequency side.

► For this reason, Davison and Cole (1950) For this reason, Davison and Cole (1950)

proposed to fit the experimental data with the proposed to fit the experimental data with the

following equation:following equation:

Low frequencieshigh frequencies

” maximum

Characteristic maximum

max ≠ CD

Low frequencieshigh frequencies

Havriliak - Negami EquationHavriliak - Negami Equation

► The generalization of the Cole-Cole, and The generalization of the Cole-Cole, and Davison-Cole equation was proposed by Davison-Cole equation was proposed by Havriliak and Negami (1967).Havriliak and Negami (1967).

► The flexibility of the HN, five-parameter The flexibility of the HN, five-parameter equation, makes it one of the most widely equation, makes it one of the most widely used methods of representing dielectric used methods of representing dielectric relaxation data. relaxation data.

► The formal expression isThe formal expression is

Depressed(1-)

Asym

metric

Low frequencieshigh frequencies

When we can use HN eq.When we can use HN eq.

►For all dielectric processed, For all dielectric processed, ►We must use for the main relaxation We must use for the main relaxation

process (process ( - process) - process)►For secondary relaxation we can use, For secondary relaxation we can use,

taking taking = 1 = 1. .

Advantages: flexibilityAdvantages: flexibility

Disadvantages: number of parametersDisadvantages: number of parameters

KWW ModelKWW Model► Williams and Watt proposed to use a stretched Williams and Watt proposed to use a stretched

exponential for the decay function exponential for the decay function (t)(t), in a similar way , in a similar way

to Kohlrausch many years ago.to Kohlrausch many years ago.

► In this way, the normalized dielectric permittivity can In this way, the normalized dielectric permittivity can

be written asbe written as

KWW - ModelKWW - Model

► The resulting expression does not have a The resulting expression does not have a

closed form but can be expressed as a closed form but can be expressed as a

series expansionseries expansion

where where is the gamma function For is the gamma function For = 1= 1 the the

Debye equations are recovered.Debye equations are recovered.

► For low values of For low values of and and > 0.25> 0.25, the , the convergence of the series of the KWW eq. is convergence of the series of the KWW eq. is slow, and the following equation is proposedslow, and the following equation is proposed

► The KWW equations are nonsymmetrical in The KWW equations are nonsymmetrical in shape and for this reason it is particularly useful shape and for this reason it is particularly useful to describe the nonsymmetrical to describe the nonsymmetrical -relaxations.-relaxations.

Thermostimulated Depolarization andThermostimulated Depolarization andPolarizationPolarization

► Due to the fact that the charges are virtually immobile at low Due to the fact that the charges are virtually immobile at low

temperatures, it is possible to study the depolarization as a temperatures, it is possible to study the depolarization as a

temperature functiontemperature function

AETT

Tp,tp

E=Eo

To

h (ºC/min)

Eo

0

TfTp,tp

E=Eo

To

h (ºC/min)

Eo

0

TfTw,td

E=0

100 150 200 250 300 350

10-13

10-12

1E-14

1E-13

1E-12

I (A)

T, K

Poly 3 (Fluor) bencyl-methacrylate

► Thermostimulated depolarization currents is a Thermostimulated depolarization currents is a complementary technique for the evaluation of complementary technique for the evaluation of the dielectric properties.the dielectric properties.

► It’s also useful for the following of the chemical It’s also useful for the following of the chemical reaction in which the mobility of the dipoles reaction in which the mobility of the dipoles change due to structural changes.change due to structural changes.

► Could give information about the fine structure Could give information about the fine structure of the materialsof the materials

► It’s equivalent frequency is lower than the It’s equivalent frequency is lower than the dielectric spectroscopydielectric spectroscopy

10-1 100 101 102 103 104 105 106 107 108 109

0.01

0.1

"

f, Hz

*

*

O

O

n

F

F

n

DC

vacj

11 j

2

2

1

21 j

SummarySummary

► Experimental techniques:Experimental techniques: Time domainTime domain Frequency domain:Frequency domain:

►Frequency Response Analyzer (ac bridges)Frequency Response Analyzer (ac bridges)►RF Analyzer (reflectometry)RF Analyzer (reflectometry)

►Complex dielectric Function it is related Complex dielectric Function it is related with the Time Dependant dielectric with the Time Dependant dielectric function by means of the Fourier function by means of the Fourier TransformTransform

SummarySummary

► Immitance Immitance Functions:Functions:

► Electric ModulusElectric Modulus► PermittivityPermittivity► ImpedanceImpedance► AdmitanceAdmitance

* * 1

1* *

* *

* *

( )

o

o

M

Y Z

Y i C

M i C Z

SummarySummary

► Fitting of the Fitting of the experimental dataexperimental data

► Symmetric relaxation Symmetric relaxation broader than Debye broader than Debye relaxation:relaxation: Cole-Cole equationCole-Cole equation Fouss – KirkwoodFouss – Kirkwood

► Asymmetric relaxation:Asymmetric relaxation: Cole-DavisonCole-Davison

► Asymmetric and Asymmetric and broader relaxations:broader relaxations: Havriliak-NegamiHavriliak-Negami KWWKWW

SummarySummary

►Another fitting procedures: Another fitting procedures: Retardation time spectraRetardation time spectra

Equivalent circuitsEquivalent circuits

Wheaston bridge

D

a.c.signal generator

Z1=1/Y1 Z2=1/Y2

Z3=1/Y3 Z4=1/Y4

∫