Experimental Techniques

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Experimental Techniques

• TUTORIAL 4

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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.
• The experimental devices cover the frequency
  range 10-4 -1011 Hz.

Time-domain spectrometer
Frequency-response analyzer
AC-bridges
Reflectometers
Resonance circuits
Cavities and waveguides
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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.

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Time domain spectroscopy

• To cover the lowest frequency range (from 10-4
  to 101 Hz), time domain spectrometers have
  recently been developed.
• 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.

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?(?)
?(t)
Time Dependent Dielectric Function
Complex Dielectric Function
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• The main item in the equipment is the
  electrometer, which must be able to measure
  currents as low as 10-16A.
• 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.

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MEASUREMENT SYSTEMS IN THE FREQUENCY DOMAIN

• In the intermediate frequency range 10-1- 106
  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-resistanc
  e 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.

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• 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

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Conductivity
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• Owing to parasitic inductances, the
  high-frequency 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.

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• 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.

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Reflection coefficient
reflected voltage
Incoming voltage
Propagation coefficient
Reflection at the beginning of the line
Attenuation coefficient
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IMMITTANCE ANALYSISBasic 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

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• 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

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?
??
tan??/?
M
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Mixed Circuit. Debye Equations
? RC2
C1 ??Co
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.
• However, the presence of interactions leads to
  the inclusion of a frequency dependent term in
  the conductivity in such a way that

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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.

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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?
• 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

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Retardation time spectra

• Advantages
• Better separation of processes
• Processes are narrower than in frequency domain

• Disadvantages
• Require numerical evaluation of the spectrum.
• No physical sense

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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.

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1-?0,5 1
Low frequencies
high frequencies
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• 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

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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

• 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.

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When its 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.

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a1-?
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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

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Low frequencies
high frequencies
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Characteristic maximum
? maximum
?max ? ?CD
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Low frequencies
high frequencies
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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

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Depressed (1-?)
Asymmetric ?
Low frequencies
high frequencies
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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

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KWW Model

• 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

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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.

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• For low values of ?? and ?gt 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.

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Thermostimulated Depolarization andPolarization
T
T

• Due to the fact that the charges are virtually
  immobile at low temperatures, it is possible to
  study the depolarization as a temperature function

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Poly 3 (Fluor) bencyl-methacrylate
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• Thermostimulated depolarization currents is a
  complementary technique for the evaluation of the
  dielectric properties.
• Its 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
• Its equivalent frequency is lower than the
  dielectric spectroscopy

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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

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Summary

• Immitance Functions
• Electric Modulus
• Permittivity
• Impedance
• Admitance

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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

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Summary

• Another fitting procedures
• Retardation time spectra
• Equivalent circuits

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• Wheaston bridge

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