Experimental Techniques

Experimental Techniques TUTORIAL 4

10-4 100 104 108 1012 • 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

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.

Vo I(t) 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.

*(ω) (t) Time Dependent Dielectric Function Complex Dielectric Function

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.

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-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 - 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

Conductivity

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.

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

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*

electrical impedance and admittanceare 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:

”  tan=’’/’ M’’

Mixed Circuit. Debye Equations = RC2 C1 =Co C2 = (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: Better separation of processes Processes are narrower than in frequency domain Disadvantages: Require numerical evaluation of the spectrum. No physical sense Retardation time spectra

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.

1-={0,5 – 1} high frequencies 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

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.

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.

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. • 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

Characteristic maximum ” 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

Depressed (1-) Asymmetric  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 • 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

Experimental Techniques