Transcript Tutorial 4 – Experimental Techniques

Experimental
Techniques
TUTORIAL 4
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10
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The interaction between electromagnetic waves and dielectric
materials can be determined by broadband measurement
techniques.
Dielectric relaxation spectroscopy allows the study of molecular
structure, through the orientation of dipoles under the action of
an electric field. 0
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4
8
12
10
10
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The experimental devices cover the frequency range 10-4 -1011
Time-domain spectrometer
Hz.
Frequency-response analyzer
AC-bridges
Reflectometers
Resonance circuits
Cavities and waveguides
MEASUREMENT SYSTEMS IN THE TIME
DOMAIN
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In linear systems the time-dependent response to a step
function field and the frequency-dependent response to a
sinusoidal electric field are related through Fourier transforms.
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For this reason, from a mathematical point of view, there is no
essential difference between these two types of measurement.
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Over a long period of time the equipment for measurements in
the time domain has been far less developed than that used in
the frequency domain.
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As a result, available experimental data in the time domain are
much less abundant than those in the frequency domain.
Time domain spectroscopy
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To cover the lowest
Vo
frequency range (from 10-4 to
101 Hz), time domain
spectrometers have recently
been developed.
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In these devices, a voltage
step Vo is applied to the
I(t)
sample placed between the
plates of a plane parallel
capacitor, and the current I(t)
is recorded.
I (t )
d  (t )
 Co
Vo
dt
eo S eo R 2
Co 

d
d
R

d
d  (t )
I (t )

dt
eo  S  Eo

t
1
 (t ) 
I (t ')dt '

Co  Vo 0
d (t ) 1

dt
2

  ()    exp(it )d
*

0
*(ω)
(t)

d (t )
 ()     
exp(it )dt
dt
0
*
Complex Dielectric
Function
Time Dependent
Dielectric Function
 The main item in the equipment is the electrometer,
which must be able to measure currents as low as 1016A.
 In many cases the applied voltage can be taken from
the internal voltage source of the electrometer.
 Also low-noise cables with high insulation resistance
must be used.
MEASUREMENT SYSTEMS IN THE
FREQUENCY DOMAIN
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In the intermediate frequency range 10-1- 106 Hz,
capacitance bridges have been the common tools used
to measure dielectric permittivities.
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The devices are based on the Wheatstone bridge
principle where the arms are capacitance-resistance
networks.
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principle of measurement of capacitance bridges
is based on the balance of the bridge placing the test
sample in one of the arms.

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 close as
possible to zero), the equations of the balanced bridge provide the values of
the capacitance and loss factor (or conductivity) for the test sample

Frequency response analyzers have proved to be very useful in measuring
dielectric permittivities in the frequency range 10-2 - 106 Hz,.

An a.c. voltage V1 is applied to the sample, and then a resistor R, or
alternatively a current-to-voltage converter for low frequencies, converts the
sample current Is, into a voltage V2 .

By comparing the amplitude and the phase angle between these two
voltages, the complex impedance of the sample Zs can be calculated as
V1  V2 V1  V2
Zs 

R
Is
V2
V1  V2 V1  V2
Zs 

R
Is
V2
Conductivity
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Owing to parasitic inductances, the highfrequency limit is about 1 MHz,
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It is necessary to be very careful with the
temperature control, and for this purpose it is
advisable to measure the temperature as close
as possible to the sample.
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At frequencies ranging from 1 MHz to 10 GHz,
the inductance of the connecting cables
contributes to the measured impedance.
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At frequencies above 1 GHz the technique
often used to obtain dielectric spectra is
reflectometry.
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The technique is based on the reflection of an
electric wave, transported through a coaxial
line, in a dielectric sample cell attached at the
end of the line.
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In this case, the reflective coefficient is a
function of the complex permittivity of the
sample, and the electric and geometric cell
lengths.
Reflection coefficient
r *( x) 
reflected voltage
*
refl
*
inc
U ( x)
U ( x)
Incoming voltage
 2 
r (l )  r (0) exp 

   i 
*

*
Reflection
at the beginning
of the line
2 n "

; =
2 n '

Propagation
coefficient
Attenuation
coefficient
1  r (l )
Z ( )  Z 0
*
1  r (l )
*
*
s
IMMITTANCE ANALYSIS
Basic Immittance Functions
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In many cases, it is possible to reproduce the electric properties
of a dipolar system by means of passive elements such as
resistors, capacitors or combined elements.
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One of the advantages of the models is that they often easily
describe the response of a system to polarization processes.
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However, it is necessary to stress that the models in general
only provide an approximate way to represent the actual
behavior of the system.
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The analysis of dielectric materials is commonly made in terms
of the complex permittivity function * or its inverse, the
electric modulus M*
► electrical
impedance and
admittance are the
appropriate functions to
represent the response of
the corresponding
equivalent circuits.
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As a consequence, the four
basic immittance functions
are permittivity, electric
modulus, impedance and
admittance.
They are related by the
following formulae:
M  ( )
*
Y  Z
*
* 1

* 1
Y  i    Co  
*
*
M  i    Co  Z
*
*
” 
tan=’’/’
M’’
Mixed Circuit. Debye Equations
C1 =Co
= RC2
C2 = (o-) Co
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As shown before, Debye equation can be obtained in three
different ways:
(1) on the grounds of some simplifying assumptions concerning
rotational Brownian motion,
(2) assuming time-dependent orientational depolarization of a
material governed by first order kinetics, and
(3) from the linear response theory assuming the time dipole
correlation function described by a simple decreasing
exponential.
The actual expressions are given by
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Under certain circumstances, the admittance is increased on
account of hopping conductivity processes. Then, a conductivity
term must be included
o is a d.c. conductivity.
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However, the presence of interactions leads to the inclusion of a
frequency dependent term in the conductivity in such a way
that
EMPIRICAL MODELS TO REPRESENT DIELECTRIC
DATA - Retardation Time Spectra
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The assumptions upon which the Debye equations are based
imply, in practice, that very few systems display Debye behavior
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In fact, relaxations in complex and disordered systems deviate
from this simple behavior.
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An alternative way to extend the scope of the Debye dispersion
relations is to include more than one relaxation time in the
physical description of relaxation phenomena.
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The term N() represents the distribution of relaxation (or
better retardation) times representing the fraction of the total
dispersion that has a retardation time between  and +d
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The real and imaginary parts of the complex permittivity are
given in terms of the retardation times by:
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Alternatively, the retardation spectrum can be defined as
Retardation time spectra
► Advantages:
► Disadvantages:
► Better
► Require
separation of
processes
► Processes
are
narrower than in
frequency domain
numerical
evaluation of the
spectrum.
► No
physical sense
'
"
4
3
2
-2
10
0
10
2
10
4
10
10
-1
10
-2
6
10
f, Hz
1,0
0,8
L(ln )
0,6
0,4
0,2
0,0
-6
-4
-2
log 
0
Cole - Cole Equation
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Experimental data (’’ vs ’)rarely fit to a Debye semicircle.
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Studying several organic crystalline compounds, Cole and
Cole found that the centers of the experimental arcs were
displaced below the real axis, the experimental data thus
having the shape of a depressed arc.
1-={0,5 – 1}
high frequencies
Low frequencies
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The corresponding
equivalent circuit is:
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The admittance is given by:
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Note that the circuit contains
a new element, namely a
constant phase element
(CPE), the admittance of
which is given by
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The admittance reduces to
R-1 when = 0
When we can use Cole – Cole
equation
Symmetric relaxations.
► In general all Secondary relaxations can be
fitted by Cole – Cole equation.
► The (1-) parameter, give us an idea about
how distributed is the relaxation (how broad it
is).
► In general the (1-) parameter, must increase
with the temperature.
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Fuoss- Kirkwood Equation
 
 "( )
 sec h ln    Debye
"

 o 
►
1941 - Fuoss and Kirkwood
propose to extend the Debye
equation, in order to fit
symmetric functions.
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Assuming an Arrhenius
dependency of the relaxation
time with the temperature, it is
possible to express the FK
equation as a T function.
max

  
 "( )
 sec h  m·ln     FK
"

 o  

max
 "( )  2 "
max
  o m 

2m 
1   o  
When it’s possible to use the FK eq.
Secondary relaxations – Symmetric
relaxations.
► Advantages: The temperature
dependencies of the loss factor have a very
simple expression.
► There are some relation between the m
parameter of the FK equation and the (1- )
parameter of the CC equation.
►
a=1-
Davison – Cole Equation
► The
Cole – Cole and Fuoss – Kirkwood equations are
very useful for symmetric relaxations.
experimental data obtained from ” vs. ’
plots show skewness on the high frequency side.
► However,
► For
this reason, Davison and Cole (1950) proposed to
fit the experimental data with the following equation:
high frequencies
Low frequencies
Characteristic maximum
” maximum
max ≠ CD
high frequencies
Low frequencies
Havriliak - Negami Equation
► The
generalization of the Cole-Cole, and DavisonCole equation was proposed by Havriliak and
Negami (1967).
► The flexibility of the HN, five-parameter equation,
makes it one of the most widely used methods of
representing dielectric relaxation data.
► The formal expression is
Depressed
(1-)
high frequencies
Low frequencies
When we can use HN eq.
► For
all dielectric processed,
► We must use for the main relaxation
process ( - process)
► For secondary relaxation we can use, taking
 = 1.
Advantages: flexibility
Disadvantages: number of parameters
KWW Model
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Williams and Watt proposed to use a stretched exponential for
the decay function (t), in a similar way to Kohlrausch many
years ago.
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In this way, the normalized dielectric permittivity can be
written as
KWW - Model
► The
resulting expression does not have a closed
form but can be expressed as a series expansion
where  is the gamma function For = 1 the Debye
equations are recovered.
low values of  and > 0.25, the convergence of
the series of the KWW eq. is slow, and the following
equation is proposed
► For
► The
KWW equations are nonsymmetrical in shape and
for this reason it is particularly useful to describe the
nonsymmetrical -relaxations.
Thermostimulated Depolarization and
Polarization
T
T
►
E
A
Due to the fact that the charges are virtually immobile at low
temperatures, it is possible to study the depolarization as a
temperature function
Tp,tp
E=Eo
Tf
Tw,td
E=0
Tf
Tp,tp
E=Eo
h (ºC/min)
h (ºC/min)
To
To
Eo
Eo
0
0

I (A)
-12
10
1E-12


1E-13
-13
10
1E-14
100
150
200
Poly 3 (Fluor) bencyl-methacrylate
250
T, K
300
350
► Thermostimulated
depolarization currents is a
complementary technique for the evaluation of the
dielectric properties.
► It’s
also useful for the following of the chemical
reaction in which the mobility of the dipoles change
due to structural changes.
► Could
give information about the fine structure of the
materials
► It’s
equivalent frequency is lower than the dielectric
spectroscopy
*
*
n
O
"
O
F
0.1
F
0.01
10
-1
10
0
10
1
10
2
10
3
10
4
f, Hz
10
5
10
6
10
7
10
8
10
9

  DC  
 
1

1

j



j


 vac 
n





 2
1   j 2 
1 2

Summary
►
Experimental techniques:
 Time domain
 Frequency domain:
►Frequency
Response Analyzer (ac bridges)
►RF Analyzer (reflectometry)
► Complex
dielectric Function it is related with
the Time Dependant dielectric function by
means of the Fourier Transform
Summary
M  ( )
*
Y  Z
*
* 1
*

► Electric
*
M  i    Co  Z
*
Functions:
Modulus
► Permittivity
► Impedance
► Admitance
1
Y  i    Co  
*
► Immitance
*
Summary
►
►
Fitting of the experimental
data
Symmetric relaxation
broader than Debye
relaxation:
 Cole-Cole equation
 Fouss – Kirkwood
►
Asymmetric relaxation:
 Cole-Davison
►
Asymmetric and broader
relaxations:
 Havriliak-Negami
 KWW
Summary
► Another
fitting procedures:
 Retardation time spectra
 Equivalent circuits
Wheaston bridge
Z1=1/Y1
Z2=1/Y2
D
Z3=1/Y3
Z4=1/Y4
∫
a.c.signal generator