TUTORIAL 4

►
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
10 -4
► an electric field.
10 0 10 4 10 8 10 12
The experimental devices cover the frequency range 10 -4 -10 11
Frequency-response analyzer
AC-bridges
Reflectometers
Resonance circuits
Cavities and waveguides

MEASUREMENT SYSTEMS IN THE TIME
DOMAIN
►
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.
►
For this reason, from a mathematical point of view, there is no essential difference between these two types of measurement.
►
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.
►
As a result, available experimental data in the time domain are much less abundant than those in the frequency domain.

►
To cover the lowest frequency range (from 10 -4 to
10 1 Hz), time domain spectrometers have recently been developed.
Vo
►
In these devices, a voltage step Vo is applied to the sample placed between the plates of a plane parallel capacitor, and the current I(t) is recorded.
I(t)

C o
 e S o d
 e o
 d
R
2
R d
V o

C o d

( ) dt
 d
 dt
( )
 e S E o
  o



1
 o o
 t
0
( ') '

d
 dt
( )

2
1



0

( )

 exp(
 
 *(ω)
 (t)
Complex Dielectric
Function
   

 

0 d

( ) exp(
 
) dt
Time Dependent
Dielectric Function

 The main item in the equipment is the electrometer, which must be able to measure currents as low as 10 -
16 A.
 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
►
In the intermediate frequency range 10 -1 - 10 6 Hz, capacitance bridges have been the common tools used to measure dielectric permittivities.
►
The devices are based on the Wheatstone bridge principle where the arms are capacitance-resistance networks.
►
The 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 - 10 6 Hz,.
An a.c. voltage V
1 is applied to the sample, and then a resistor R , or alternatively a current-to-voltage converter for low frequencies, converts the sample current I s
, into a voltage V
2
.
By comparing the amplitude and the phase angle between these two voltages, the complex impedance of the sample Z s can be calculated as
Z s

V
1

V
2

V
1

V
2
I s
V
2
R

Z s

V
1

V
2

V
1

V
2
I s
V
2
R

Conductivity

►
Owing to parasitic inductances, the highfrequency limit is about 1 MHz,
►
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.
►
At frequencies ranging from 1 MHz to 10 GHz, the inductance of the connecting cables contributes to the measured impedance.

►
At frequencies above 1 GHz the technique often used to obtain dielectric spectra is reflectometry.
►
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.
►
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 * ( )

U
*
( ) refl
U
*
( ) inc
* r l
 r
*
( ) (0) exp


Reflection at the beginning of the line
2


Attenuation coefficient
Propagation coefficient
Z s
* reflected voltage
Incoming voltage
 
2
 n "


Z
0
2
 n '

1
 * r l
1
 * r l

IMMITTANCE ANALYSIS
Basic Immittance Functions
►
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.
►
One of the advantages of the models is that they often easily describe the response of a system to polarization processes.
►
However, it is necessary to stress that the models in general only provide an approximate way to represent the actual behavior of the system.
►
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.
►
As a consequence, the four basic immittance functions are permittivity , electric modulus , impedance and admittance .
►
They are related by the following formulae:
M
*
Y
* 
 
*

1
( )

1
Y
* i

C o

M
*

C o


*
Z
*

 ”  tan  =  ’’/  ’
M’’

Mixed Circuit. Debye Equations
C
1
= 

C o
 = RC
2
C
2
= (  o


) Co

►
►
►
►
►
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

►
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.
►
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
►
The assumptions upon which the Debye equations are based imply, in practice, that very few systems display Debye behavior
►
In fact, relaxations in complex and disordered systems deviate from this simple behavior.
►
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.

►
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 
►
The real and imaginary parts of the complex permittivity are given in terms of the retardation times by:
►
Alternatively, the retardation spectrum can be defined as

►
Advantages:
►
Disadvantages:
►
Better separation of processes
►
Processes are narrower than in frequency domain
►
Require numerical evaluation of the spectrum.
►
No physical sense


4
3

10
-1
10
-2
2
10
-2
10
0
10
2
10
4
10
6 f,

1,0
0,8
0,6
0,4
0,2
0,0
-6 -4 log 
-2 0

Cole - Cole Equation
►
Experimental data (  ’’ vs  ’ )rarely fit to a Debye semicircle.
►
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.

high frequencies
1 ={0,5 – 1}
Low frequencies

►
The corresponding equivalent circuit is:
►
The admittance is given by:
►
Note that the circuit contains a new element, namely a constant phase element
(CPE) , the admittance of which is given by
►
The admittance reduces to
R -1 when  = 0

►
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.


" max

" max





 o




Debye
 sec



·ln




 o







FK
  
2

" max 



1

  o
 m
  o

2 m




►
1941 - Fuoss and Kirkwood propose to extend the Debye equation, in order to fit symmetric functions.
►
Assuming an Arrhenius dependency of the relaxation time with the temperature, it is possible to express the FK equation as a T function.

►
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.
►
However, experimental data obtained from  ” vs.  ’ plots show skewness on the high frequency side.
►
For this reason, Davison and Cole (1950) proposed to fit the experimental data with the following equation:

high frequencies Low frequencies

 ” maximum
Characteristic maximum
 max
≠ 
CD

high frequencies Low frequencies

Havriliak - Negami Equation
►
The generalization of the Cole-Cole, and Davison-
Cole 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

high frequencies
Depressed
(1 )
Low frequencies

►
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

►
Williams and Watt proposed to use a stretched exponential for the decay function  (t) , in a similar way to Kohlrausch many years ago.
►
In this way, the normalized dielectric permittivity can be written as

►
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.

►
For low values of  and  > 0.25
, the convergence of the series of the KWW eq. is slow, and the following equation is proposed
►
The KWW equations are nonsymmetrical in shape and for this reason it is particularly useful to describe the nonsymmetrical  -relaxations.

Thermostimulated Depolarization and
Polarization
TT 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

Eo
0
T p, t p
E=E o
T w
,t d
E=0
To
Tf h (ºC/min)
T p, t p
E=E o
Eo
0
To h (ºC/min)
Tf


I (A)
10
-12

1E-12

1E-13
10
-13
100 150 200 250 300 350
T, K
Poly 3 (Fluor) bencyl-methacrylate
1E-14

►
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

 "
0.1
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
*
O
O n
*
F
F


 j
DC
  vac
 n



 

1
  j
 
1
 
 

 
2 
1
  j

2

1
 
2


►
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

M
*  
*

1
( )
Y
* 

1
►
Immitance Functions:
►
Electric Modulus
►
Permittivity
►
Impedance
►
Admitance
Y
*

C o
 
*
M
*

C o

Z
*

►
►
►
►
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

►
Another fitting procedures:
 Retardation time spectra
 Equivalent circuits

Wheaston bridge
Z
1
=1/Y
1 Z
2
=1/Y
2
D
Z
3
=1/Y
3 a.c.signal generator
Z
4
=1/Y
4