Materials, Relaxation and Non-Linear effects

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Materials, Relaxation and Non-Linear effects

Analysis of the electrical or magnetic attributes of materials, including examination of charge or molecular relaxation processes and higher harmonic phenomena

Material characterization through dielectric spectroscopy and impedance spectroscopy

The objective is to assess the electric or magnetic characteristics of materials using the complex impedance spectrum Z*(ω). This involves placing the material into a sample cell. The electrical characterization entails measuring the complex conductivity spectra σ*(ω) and the complex permittivity spectra ε*(ω), which are linked to the dynamics of charge transport and dipole reorientation within the material.

Materials can be generally classified as conductors or dielectrics. As in most cases, materials comprise both free charge carriers (such as electrons and ions) and charge carriers bound within molecules. When the electrical characteristics are primarily influenced by the presence of free charge carriers, the materials are classified as conductors; otherwise, they are categorized as dielectrics.

free charge carriers electrons and ions in Materials

charge carriers bound within molecules (Dipoles) in Materials

When an electric field E is applied to a material, the free charge carriers move within it, while polar molecules partially align themselves in the direction of the field. This phenomenon is typically described by the current density J or the dielectric polarization density P.

Current density: J = \frac{I}{A} = \frac{1}{A} \frac{\partial q_{free}}{\partial t} Dielectric polarization density: P = \frac{\partial q_{bound}}{\partial A}

Where:
q: Charge carriers whether free or bound
A: Area crossed/moved by the free/bound(oriented dipoles) charge carriers within time dt

A common simplification for the Electric current density assumes that the current is directly proportional to the electric field, which can be expressed as:

J = σ E \to σ = \frac{J}{E}

where:
$\mathbf {E}$ is the electric field
σ is the electrical conductivity

Assuming that the material is a linear dielectric, the electric susceptibility is defined as the constant of proportionality linking an electric field E to the induced dielectric polarization density P, such that:

P = ϵ_{0} x_{e} E

where:
$\mathbf {P}$ is the polarization density
$\varepsilon _{0}$ is the electric permittivity of free space (electric constant)
$\chi _{\text{e}}$ is the electric susceptibility
$\mathbf {E}$ is the electric field

And The susceptibility is related to its relative permittivity (dielectric constant) $\varepsilon _{\textrm {r}}$ by:

x_{e} = ϵ_{r} - 1 \to ϵ_{r} = \frac{P_{}}{ϵ_{0} E} + 1

The material in question will be in this case subjected to an Alternating current in the frequency domain. This would render our values from before to a complex frequency bound response as follows:

\begin{array}{lll} J = \frac{1}{A} \frac{\partial q_{free}}{\partial t} \to J^{*} (ω) = \frac{j ω q^{free *} (ω)}{A} \\ P = \frac{\partial q_{bound}}{\partial A} \to P^{*} (ω) = \frac{ω q^{bound *} (ω)}{A} \\ σ_{= \frac{J}{E} \to σ^{*} (ω) = \frac{J^{*} (ω)}{E^{*} (ω)}} \\ {ϵ_{r}}^{= \frac{P_{}}{ϵ_{0} E} + 1 \to ϵ^{*} (ω) = \frac{P^{*} (ω)}{ϵ_{0} E^{*} (ω)} + 1} \end{array}

Data measuring in the frequency domain provides significant supremacy over time domain beacuse:

The frequency representation consists of simpler algebraic terms with power laws in frequency.
Data can be measured more accurately and across a much broader range of conductance and permittivity.

Graph illustrating the dielectric constant as a function of frequency, depicting multiple resonances(relaxations) and plateaus.

The challenge is the fact that both dipoles and free charge carriers contribute frequency dependent charge components. while the response of the complex qfree*(ω) lags by 90° relative to the driving field, the dipole charge response Qbound*(ω) is in phase with the field. Thus, in principle, these contributions can be distinguished based on phase. However, in practice, this approach only works effectively at sufficiently low frequencies where the dynamic effects of free charge or dipole responses, related to their mass, can be disregarded. In such cases, the material’s electrical response is frequency-independent, and Real Part of ε*(ω) = ε, Real Part of σ*(ω) = σ. When dynamic effects become significant, dipole relaxation is invariably damped, resulting in a non-zero phase shift of Qbound*(ω) relative to E(ω). Additionally, the response of free charge carriers often deviates from the ideal 90° phase shift of Qfree*(ω) relative to E(ω). This deviation may become noticeable in the frequency range of dielectric or impedance spectroscopy, particularly for ions with a substantial mass and for molecular dipoles.

For materials that are purely conductive or purely dielectric, the relaxation process occurs rapidly, and the relaxation times are typically outside the measurement range of dielectric or impedance spectroscopy. However, in many other materials, such as those where the motion of mobile charge carriers or molecules is impeded by factors like the material’s viscosity, the relaxation times fall within a wide range, spanning from picoseconds to weeks. This corresponds to frequencies ranging from microhertz to several gigahertz, making them measurable using Eynocs Impedance Analyzers ( up to 100MHz). This observation is particularly relevant in the case of polymer materials, where the molecular dynamics are hindered by the entanglement of the molecular chains.

Non-Linear Material effects

Eynocs offers turnkey solutions that facilitates measurements of sample voltage and current, encompassing DC components, harmonic base waves, and higher harmonics up to the interface’s upper frequency limit. which means that complete support for nonlinear evaluation is provided by Eymea software. It read and visually represent all voltage and current base, as well as higher harmonic components. Additionally, various parameters such as DC material parameters, linear impedance, permittivity, conductivity, and their corresponding higher harmonic terms are processed. The higher harmonic current components are computed via complex Fourier Transform derived from the sampled current.