# Dielectric spectroscopy

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Dielectric spectroscopy (which falls in a subcategory of impedance spectroscopy) measures the dielectric properties of a medium as a function of frequency. [2] [3] [4] [5] It is based on the interaction of an external field with the electric dipole moment of the sample, often expressed by permittivity.

A dielectric is an electrical insulator that can be polarized by an applied electric field. When a dielectric is placed in an electric field, electric charges do not flow through the material as they do in an electrical conductor but only slightly shift from their average equilibrium positions causing dielectric polarization. Because of dielectric polarization, positive charges are displaced in the direction of the field and negative charges shift in the opposite direction. This creates an internal electric field that reduces the overall field within the dielectric itself. If a dielectric is composed of weakly bonded molecules, those molecules not only become polarized, but also reorient so that their symmetry axes align to the field.

Frequency is the number of occurrences of a repeating event per unit of time. It is also referred to as temporal frequency, which emphasizes the contrast to spatial frequency and angular frequency. The period is the duration of time of one cycle in a repeating event, so the period is the reciprocal of the frequency. For example: if a newborn baby's heart beats at a frequency of 120 times a minute, its period—the time interval between beats—is half a second. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio signals (sound), radio waves, and light.

The electric dipole moment is a measure of the separation of positive and negative electrical charges within a system, that is, a measure of the system's overall polarity. The SI units for electric dipole moment are coulomb-meter (C⋅m); however, a commonly used unit in atomic physics and chemistry is the debye (D).

## Contents

It is also an experimental method of characterizing electrochemical systems. This technique measures the impedance of a system over a range of frequencies, and therefore the frequency response of the system, including the energy storage and dissipation properties, is revealed. Often, data obtained by electrochemical impedance spectroscopy (EIS) is expressed graphically in a Bode plot or a Nyquist plot.

Electrical impedance is the measure of the opposition that a circuit presents to a current when a voltage is applied. The term complex impedance may be used interchangeably.

Spectroscopy is the study of the interaction between matter and electromagnetic radiation. Historically, spectroscopy originated through the study of visible light dispersed according to its wavelength, by a prism. Later the concept was expanded greatly to include any interaction with radiative energy as a function of its wavelength or frequency, predominantly in the electromagnetic spectrum, though matter waves and acoustic waves can also be considered forms of radiative energy; recently, with tremendous difficulty, even gravitational waves have been associated with a spectral signature in the context of the Laser Interferometer Gravitational-Wave Observatory (LIGO) and laser interferometry. Spectroscopic data are often represented by an emission spectrum, a plot of the response of interest, as a function of wavelength or frequency.

In electrical engineering and control theory, a Bode plot is a graph of the frequency response of a system. It is usually a combination of a Bode magnitude plot, expressing the magnitude of the frequency response, and a Bode phase plot, expressing the phase shift.

Impedance is the opposition to the flow of alternating current (AC) in a complex system. A passive complex electrical system comprises both energy dissipater (resistor) and energy storage (capacitor) elements. If the system is purely resistive, then the opposition to AC or direct current (DC) is simply resistance. Materials or systems exhibiting multiple phases (such as composites or heterogeneous materials) commonly show a universal dielectric response, whereby dielectric spectroscopy reveals a power law relationship between the impedance (or the inverse term, admittance) and the frequency, ω, of the applied AC field.

Alternating current (AC) is an electric current which periodically reverses direction, in contrast to direct current (DC) which flows only in one direction. Alternating current is the form in which electric power is delivered to businesses and residences, and it is the form of electrical energy that consumers typically use when they plug kitchen appliances, televisions, fans and electric lamps into a wall socket. A common source of DC power is a battery cell in a flashlight. The abbreviations AC and DC are often used to mean simply alternating and direct, as when they modify current or voltage.

A resistor is a passive two-terminal electrical component that implements electrical resistance as a circuit element. In electronic circuits, resistors are used to reduce current flow, adjust signal levels, to divide voltages, bias active elements, and terminate transmission lines, among other uses. High-power resistors that can dissipate many watts of electrical power as heat, may be used as part of motor controls, in power distribution systems, or as test loads for generators. Fixed resistors have resistances that only change slightly with temperature, time or operating voltage. Variable resistors can be used to adjust circuit elements, or as sensing devices for heat, light, humidity, force, or chemical activity.

A capacitor is a device that stores electrical energy in an electric field. It is a passive electronic component with two terminals.

Almost any physico-chemical system, such as electrochemical cells, mass-beam oscillators, and even biological tissue possesses energy storage and dissipation properties. EIS examines them.

An electrochemical cell is a device capable of either generating electrical energy from chemical reactions or using electrical energy to cause chemical reactions. The electrochemical cells which generate an electric current are called voltaic cells or galvanic cells and those that generate chemical reactions, via electrolysis for example, are called electrolytic cells. A common example of a galvanic cell is a standard 1.5 volt cell meant for consumer use. A battery consists of one or more cells, connected either in parallel, series or series-and-parallel pattern.

This technique has grown tremendously in stature over the past few years and is now being widely employed in a wide variety of scientific fields such as fuel cell testing, biomolecular interaction, and microstructural characterization. Often, EIS reveals information about the reaction mechanism of an electrochemical process: different reaction steps will dominate at certain frequencies, and the frequency response shown by EIS can help identify the rate limiting step.

A fuel cell is an electrochemical cell that converts the chemical energy of a fuel and an oxidizing agent into electricity through a pair of redox reactions. Fuel cells are different from most batteries in requiring a continuous source of fuel and oxygen to sustain the chemical reaction, whereas in a battery the chemical energy usually comes from metals and their ions or oxides that are commonly already present in the battery, except in flow batteries. Fuel cells can produce electricity continuously for as long as fuel and oxygen are supplied.

## Dielectric mechanisms

There are a number of different dielectric mechanisms, connected to the way a studied medium reacts to the applied field (see the figure illustration). Each dielectric mechanism is centered around its characteristic frequency, which is the reciprocal of the characteristic time of the process. In general, dielectric mechanisms can be divided into relaxation and resonance processes. The most common, starting from high frequencies, are:

The characteristic time is an estimate of the order of magnitude of the reaction time scale of a system. It can loosely be defined as the inverse of the reaction rate. In chemistry, the characteristic time is used to determine whether the problem needs to be solved as an equilibrium problem or a kinetic problem.

Resonance describes the phenomena of amplification that occurs when the frequency of a periodically applied force is in harmonic proportion to a natural frequency of the system on which it acts. When an oscillating force is applied at the resonant frequency of another system, the system will oscillate at a higher amplitude than when the same force is applied at other, non-resonant frequencies.

### Electronic polarization

This resonant process occurs in a neutral atom when the electric field displaces the electron density relative to the nucleus it surrounds.

This displacement occurs due to the equilibrium between restoration and electric forces. Electronic polarization may be understood by assuming an atom as a point nucleus surrounded by spherical electron cloud of uniform charge density.

### Atomic polarization

Atomic polarization is observed when the nucleus of the atom reorients in response to the electric field. This is a resonant process. Atomic polarization is intrinsic to the nature of the atom and is a consequence of an applied field. Electronic polarization refers to the electron density and is a consequence of an applied field. Atomic polarization is usually small compared to electronic polarization.

### Dipole relaxation

This originates from permanent and induced dipoles aligning to an electric field. Their orientation polarisation is disturbed by thermal noise (which mis-aligns the dipole vectors from the direction of the field), and the time needed for dipoles to relax is determined by the local viscosity. These two facts make dipole relaxation heavily dependent on temperature, pressure, [6] and chemical surrounding.

### Ionic relaxation

Ionic relaxation comprises ionic conductivity and interfacial and space charge relaxation. Ionic conductivity predominates at low frequencies and introduces only losses to the system. Interfacial relaxation occurs when charge carriers are trapped at interfaces of heterogeneous systems. A related effect is Maxwell-Wagner-Sillars polarization, where charge carriers blocked at inner dielectric boundary layers (on the mesoscopic scale) or external electrodes (on a macroscopic scale) lead to a separation of charges. The charges may be separated by a considerable distance and therefore make contributions to the dielectric loss that are orders of magnitude larger than the response due to molecular fluctuations. [2]

### Dielectric relaxation

Dielectric relaxation as a whole is the result of the movement of dipoles (dipole relaxation) and electric charges (ionic relaxation) due to an applied alternating field, and is usually observed in the frequency range 102-1010 Hz. Relaxation mechanisms are relatively slow compared to resonant electronic transitions or molecular vibrations, which usually have frequencies above 1012 Hz.

## Principles

For a redox reaction R ${\displaystyle \leftrightarrow }$ O + e, without mass-transfer limitation, the relationship between the current density and the electrode overpotential is given by the Butler–Volmer equation: [7]

${\displaystyle j_{\text{t}}=j_{0}\left(\exp(\alpha _{\text{o}}\,f\,\eta )-\exp(-\alpha _{\text{r}}\,f\,\eta )\right)}$

with

${\displaystyle \eta =E-E_{\text{eq}},\;f=F/(R\,T),\;\alpha _{\text{o}}+\alpha _{\text{r}}=1}$.
${\displaystyle j_{0}}$ is the exchange current density and ${\displaystyle \alpha _{\text{o}}}$ and ${\displaystyle \alpha _{\text{r}}}$ are the symmetry factors.

The curve ${\displaystyle j_{\text{t}}\;vs.\;E}$ is not a straight line (Fig. 1), therefore a redox reaction is not a linear system. [8]

### Dynamic behavior

In an electrochemical cell the faradaic impedance of an electrolyte-electrode interface is the joint electrical resistance and capacitance at that interface.

Let us suppose that the Butler-Volmer relationship correctly describes the dynamic behavior of the redox reaction:

${\displaystyle j_{\text{t}}(t)=j_{\text{t}}(\eta (t))=j_{0}\,\left(\exp(\alpha _{\text{o}}\,f\,\eta (t))-\exp(-\alpha _{\text{r}}\,f\,\eta (t))\right)}$

Dynamic behavior of the redox reaction is characterized by the so-called charge transfer resistance defined by:

${\displaystyle R_{\text{ct}}={\frac {1}{\partial j_{\text{t}}/\partial \eta }}={\frac {1}{f\,j_{0}\,\left(\alpha _{\text{o}}\,\exp(\alpha _{\text{o}}\,f\,\eta )+\alpha _{\text{r}}\,\exp(-\alpha _{\text{r}}\,f\,\eta )\right)}}}$

The value of the charge transfer resistance changes with the overpotential. For this simplest example the faradaic impedance is reduced to a resistance. It is worthwhile to notice that:

${\displaystyle R_{\text{ct}}={\frac {1}{f\,j_{0}}}}$

for ${\displaystyle \eta =0}$.

#### Double layer capacitance

An electrode ${\displaystyle |}$ electrolyte interface behaves like a capacitance called electrochemical double-layer capacitance ${\displaystyle C_{\text{dl}}}$. The equivalent circuit for the redox reaction in Fig. 2 includes the double-layer capacitance as well as the charge transfer resistance. Another analog circuit commonly used to model the electrochemical double-layer is called a constant phase element.

The electrical impedance of this circuit is easily obtained remembering the impedance of a capacitance which is given by:

${\displaystyle Z_{\text{dl}}(\omega )={\frac {1}{{\text{i}}\,\omega \,C_{\text{dl}}}}}$

where ${\displaystyle \omega }$ is the angular frequency of a sinusoidal signal (rad/s), and ${\displaystyle \scriptstyle {{\text{i}}^{2}=-1}}$.

It is obtained:

${\displaystyle Z(\omega )={\frac {R_{\text{t}}}{1+R_{\text{t}}\,C_{\text{dl}}\,{\text{i}}\,\omega }}}$

Nyquist diagram of the impedance of the circuit shown in Fig. 3 is a semicircle with a diameter ${\displaystyle \scriptstyle {R_{\text{t}}}}$ and an angular frequency at the apex equal to ${\displaystyle \scriptstyle {1/(R_{\text{t}}\,C_{\text{dc}})}}$ (Fig. 3). Other representations, Bode plots, or Black plans can be used. [9]

#### Ohmic resistance

The ohmic resistance ${\displaystyle R_{\Omega }}$ appears in series with the electrode impedance of the reaction and the Nyquist diagram is translated to the right.

### Universal dielectric response

Under AC conditions with varying frequency ω, heterogeneous systems and composite materials exhibit a universal dielectric response, in which overall admittance exhibits a region of power law scaling with frequency. ${\displaystyle Y\propto \omega ^{\alpha }}$. [10]

## Measurement of the impedance parameters

Plotting the Nyquist diagram with a potentiostat [11] and an impedance analyzer, most often included in modern potentiostats, allows the user to determine charge transfer resistance, double layer capacitance and ohmic resistance. The exchange current density ${\displaystyle j_{0}}$ can be easily determined measuring the impedance of a redox reaction for ${\displaystyle \eta =0}$.

Nyquist diagrams are made of several arcs for reactions more complex than redox reactions and with mass-transfer limitations.

## Applications

Electrochemical impedance spectroscopy is used in a wide range of applications. [12]

In the paint and coatings industry, it is a useful tool to investigate the quality of coatings [13] [14] and to detect the presence of corrosion. [15] [16]

It is used in many biosensor systems as a label-free technique to measure bacterial concentration [17] and to detect dangerous pathogens such as Escherichia Coli O157:H7 [18] and Salmonella, [19] and yeast cells. [20] [21]

Electrochemical impedance spectroscopy is also used to analyze and characterize different food products. Some examples are the assessment of food–package interactions, [22] the analysis of milk composition, [23] the characterization and the determination of the freezing end-point of ice-cream mixes, [24] [25] the measure of meat ageing, [26] the investigation of ripeness and quality in fruits [27] [28] [29] and the determination of free acidity in olive oil. [30]

In the field of human health monitoring is better known as bioelectrical impedance analysis (BIA) [31] and is used to estimate body composition [32] as well as different parameters such as total body water and free fat mass. [33]

Electrochemical impedance spectroscopy can be used to obtain the frequency response of batteries. [34] [35]

Biomedical sensors working in the microwave range relies on dielectric spectroscopy to detect changes in the dielectric properties over a frequency range. The IFAC database can be used as a resource to get the dielectric properties for human body tissues. [36]

## Related Research Articles

In physics and thermodynamics, an equation of state is a thermodynamic equation relating state variables which describe the state of matter under a given set of physical conditions, such as pressure, volume, temperature (PVT), or internal energy. Equations of state are useful in describing the properties of fluids, mixtures of fluids, solids, and the interior of stars.

The wave impedance of an electromagnetic wave is the ratio of the transverse components of the electric and magnetic fields. For a transverse-electric-magnetic (TEM) plane wave traveling through a homogeneous medium, the wave impedance is everywhere equal to the intrinsic impedance of the medium. In particular, for a plane wave travelling through empty space, the wave impedance is equal to the impedance of free space. The symbol Z is used to represent it and it is expressed in units of ohms. The symbol η (eta) may be used instead of Z for wave impedance to avoid confusion with electrical impedance.

In electromagnetism, absolute permittivity, often simply called permittivity, usually denoted by the Greek letter ε (epsilon), is the measure of capacitance that is encountered when forming an electric field in a particular medium. More specifically, permittivity describes the amount of charge needed to generate one unit of electric flux in a given medium. A charge will yield more electric flux in a medium with low permittivity than in a medium with high permittivity. Permittivity is the measure of a material's ability to store an electric field in the polarization of the medium.

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The Smith chart, invented by Phillip H. Smith (1905–1987), is a graphical aid or nomogram designed for electrical and electronics engineers specializing in radio frequency (RF) engineering to assist in solving problems with transmission lines and matching circuits. The Smith chart can be used to simultaneously display multiple parameters including impedances, admittances, reflection coefficients, scattering parameters, noise figure circles, constant gain contours and regions for unconditional stability, including mechanical vibrations analysis. The Smith chart is most frequently used at or within the unity radius region. However, the remainder is still mathematically relevant, being used, for example, in oscillator design and stability analysis.

In physics, the dissipation factor (DF) is a measure of loss-rate of energy of a mode of oscillation in a dissipative system. It is the reciprocal of quality factor, which represents the "quality" or durability of oscillation.

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The Havriliak–Negami relaxation is an empirical modification of the Debye relaxation model in electromagnetism. Unlike the Debye model, the Havriliak–Negami relaxation accounts for the asymmetry and broadness of the dielectric dispersion curve. The model was first used to describe the dielectric relaxation of some polymers, by adding two exponential parameters to the Debye equation:

The Cole–Cole equation is a relaxation model that is often used to describe dielectric relaxation in polymers.

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The Tafel equation is an equation in electrochemical kinetics relating the rate of an electrochemical reaction to the overpotential. The Tafel equation was first deduced experimentally and was later shown to have a theoretical justification. The equation is named after Swiss chemist Julius Tafel.

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The cone-shape distribution function, also known as the Zhao–Atlas–Marks time-frequency distribution,, is one of the members of Cohen's class distribution function. It was first proposed by Yunxin Zhao, Les E. Atlas, and Robert J. Marks II in 1990. The distribution's name stems from the twin cone shape of the distribution's kernel function on the plane. The advantage of the cone kernel function is that it can completely remove the cross-term between two components having the same center frequency. Cross-term results from components with the same time center, however, cannot be completely removed by the cone-shaped kernel.

The Butler–Volmer equation, also known as Erdey-Grúz–Volmer equation, is one of the most fundamental relationships in electrochemical kinetics. It describes how the electrical current on an electrode depends on the electrode potential, considering that both a cathodic and an anodic reaction occur on the same electrode:

In probability and statistics, the Gompertz distribution is a continuous probability distribution, named after Benjamin Gompertz. The Gompertz distribution is often applied to describe the distribution of adult lifespans by demographers and actuaries. Related fields of science such as biology and gerontology also considered the Gompertz distribution for the analysis of survival. More recently, computer scientists have also started to model the failure rates of computer codes by the Gompertz distribution. In Marketing Science, it has been used as an individual-level simulation for customer lifetime value modeling. In network theory, particularly the Erdős–Rényi model, the walk length of a random self-avoiding walk (SAW) is distributed according to the Gompertz distribution.

In dielectric spectroscopy, large frequency dependent contributions to the dielectric response, especially at low frequencies, may come from build-ups of charge. This Maxwell–Wagner–Sillars polarization, occurs either at inner dielectric boundary layers on a mesoscopic scale, or at the external electrode-sample interface on a macroscopic scale. In both cases this leads to a separation of charges. The charges are often separated over a considerable distance, and the contribution to dielectric loss can therefore be orders of magnitude larger than the dielectric response due to molecular fluctuations.

Acoustic attenuation is a measure of the energy loss of sound propagation in media. Most media have viscosity, and are therefore not ideal media. When sound propagates in such media, there is always thermal consumption of energy caused by viscosity. For inhomogeneous media, besides media viscosity, acoustic scattering is another main reason for removal of acoustic energy. Acoustic attenuation in a lossy medium plays an important role in many scientific researches and engineering fields, such as medical ultrasonography, vibration and noise reduction.

A biotransducer is the recognition-transduction component of a biosensor system. It consists of two intimately coupled parts; a bio-recognition layer and a physicochemical transducer, which acting together converts a biochemical signal to an electronic or optical signal. The bio-recognition layer typically contains an enzyme or another binding protein such as antibody. However, oligonucleotide sequences, sub-cellular fragments such as organelles and receptor carrying fragments, single whole cells, small numbers of cells on synthetic scaffolds, or thin slices of animal or plant tissues, may also comprise the bio-recognition layer. It gives the biosensor selectivity and specificity. The physicochemical transducer is typically in intimate and controlled contact with the recognition layer. As a result of the presence and biochemical action of the analyte, a physico-chemical change is produced within the biorecognition layer that is measured by the physicochemical transducer producing a signal that is proportionate to the concentration of the analyte. The physicochemical transducer may be electrochemical, optical, electronic, gravimetric, pyroelectric or piezoelectric. Based on the type of biotransducer, biosensors can be classified as shown to the right.

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