Effective Complex Permittivity Research Articles

Measurements of the complex permittivity and the complex permeability of low and medium loss isotropic and uniaxially anisotropic metamaterials at microwave frequencies

Split dielectric resonators operating in a quasi-TE011 mode were used for the measurement of the complex permittivity and the complex permeability of planar metamaterials deposited on low-loss dielectric substrates at microwave frequencies. Separation of the complex permittivity from the complex permeability for a specific metamaterial was achieved by performing measurements of the resonance frequencies and the Q-factors of a split post-dielectric resonator with two samples having different diameters but identical patterns. Measurements of the resonance frequencies and the Q-factors of empty substrates served as reference materials to determine the resonance frequency shifts and Q-factor changes due to the presence of the metamaterial only. The effective complex permittivity and permeability were determined on the basis of accurate electromagnetic modeling of the resonance structures containing the samples.

Effective complex permittivity and continuum percolation analysis of two-phase composite media

Using ab initio finite-element (FE) calculations we study the dielectric properties of the continuum (off-lattice)-percolation system consisting of two-dimensional equilibrium distributions of randomly distributed circular and partially penetrable disks (or parallel, infinitely long, identical, partially penetrable circular cylinders) throughout a host matrix. Theoretical investigations of the (relative) effective complex permittivity epsiv = epsiv' - iepsiv" were conducted a hybrid modeling that combine standard Metropolis Monte Carlo (MC) algorithm and continuum-electrostatics equations which are solved by finite element calculations. We present the details of the epsiv dependence on surface fraction Phi <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> of the disks, permittivity contrast between the two phases and arbitrary degree of impenetrability lambda (0 les lambda les 1), for wide ranges of these parameters. Careful evaluation of the critical exponents s and t governing the power-law behavior of epsiv' and epsiv" respectively, near the percolation threshold, are used to address controversial or unresolved issues, related to the underlying physics of the classical percolation model. Our results, corresponding to different values of lambda in the range 0 les lambda les 0.9 and for a wide range of phase's permittivity ratios, indicate that s and t can differ from the universal values, i.e. s = t cong 1.3 , characterizing the continuum percolation phenomena of statistically isotropic distributions of disks in a plane. As the distance to Phi <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2c</sub> is decreased, epsiv' and epsiv" display a smooth transition from a power-law dependence, which is well fit by the standard percolation expression, to a plateau regime. We associate the plateau with finitesize effects and the short-range multipolar interactions localized in disk clusters. The radial distribution function (RDF) results are consistent with the notion that larger area fractions lead to an increase in the distance over which one disk influences another via excluded volume effect. Furthermore, we perform a quantitative test of the McLachlan (TEPPE) equation by comparing its prediction of the effective permittivity to the simulation results obtained on systems with overlapping disks (0 les lambda les 0.9). We find that the analytic equation presented by McLachlan is consistent with FE-MC simulations only for Phi <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> < Phi <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2c</sub> . However, the failure of the TEPPE for Phi <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> > Phi <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> can be attributed to a poor representation of the various degrees of disk aggregation present in the equilibrium distributions where increased aggregation results in an enhanced multipolar interaction.

Electromagnetic Design and Realization of Innovative Fiber-Reinforced Broad-Band Absorbing Screens

This paper presents the design and the realization process for radar absorbing panels made of composite materials. It is demonstrated that the proper selection of the carbon fiber length and volume fraction allows reducing sensibly the overall thickness of the screen compared to a standard absorber, and obtaining at the same time broad-band absorption response in the X- and Ku-bands. To this end, the carbon fiber composite material realizing the lossy sheet of the absorbing screen is designed by simulation in order to have tailored complex effective permittivity. Two prototypes of absorbing screens are realized and tested experimentally. The first one is a three-layer panel, characterized by reflection coefficient lower than -10 dB in the frequency range from 8 to 14 GHz, and total thickness of 4.5 mm. The second one is a five-layer panel with reflection coefficient less than -20 dB in the range 9-18 GHz, and total thickness of 5.5 mm.

A Novel Method for Determining the R-Card Sheet Impedance Using the Transmission Coefficient Measured in Free-Space or Waveguide Systems

Free-space and rectangular waveguide techniques for determining the effective complex permittivity and, ultimately, the effective sheet impedance of an R-card using the forward transmission coefficient are presented. The advantage of using a transmission coefficient method instead of a more traditional reflection-based technique is discussed. The exact transcendental expressions relating the transmission coefficient and effective complex permittivity are derived and approximated using the Maclaurin series for sine and cosine. It is shown that the Maclaurin series expansion leads to simple closed-form solutions to the effective complex permittivity and avoids the use of sensitive and often unstable root search algorithms, which are necessary to solve transcendental equations. The accuracy of the approximations is directly related to the R-card's thickness and wavenumber. Free-space (4-16 GHz) and waveguide (8.2-12.4 GHz) measurements are made using two R-cards of differing thicknesses and impedances to demonstrate the method and regimes of validity. An uncertainty analysis is also performed to demonstrate the robustness of the technique.

Dielectric resonances at optical frequencies using metal nanoshells

A first-principles systematic investigation is reported to identify the dielectric (quasistatic) resonances properties of layered metal–dielectric particles. Specifically, we present remarkable results of finite-element simulations that model the effective permittivity of two-dimensional (or cross sections of infinite three-dimensional parallel, infinitely long, identical, circular cylinders, where the properties and characteristics are invariant along the cylinder axis) three-phase heterostructures made of coated discs embedded in a dielectric host. The derivation assumes that the quasistatic approximation holds valid even at relatively high angular frequencies of the harmonic external field. The coating (shell) material is taken as a conventional Drude noble metal and the core and shell phases are assumed to be dielectric, non-dispersive and lossless. One of the main results of this study is that the real part of the effective permittivity of an isotropic host matrix containing metallic nanoshells is tunable from negative through zero to positive values. The simulations reveal that a Drude–Lorentz-like dielectric function reproduces the overall shape of the effective complex permittivity of these nanostructures over an extended range of frequencies. The data nearly make a complete circle when all the (ε′, ε″) pairs for the various frequencies are included over the bandwidth about each resonance. For a given resonant frequency, the resonant features are determined by the complex permittivity of the metal (shell) phase, but also depend on the dielectric environment. Kramers–Kronig relations provide a consistency test for data of the complex effective permittivity. Numerical simulations are also used to investigate the sensitivity of field enhancement effects against frequency. The position and width of the resonance have the ability to be tailored by the selection of constituent metal phases and their nominal thicknesses.

Variable-temperature measurements of the dielectric relaxation in carbon black loaded epoxy composites

Technologically, an understanding of the temperature influence on the transport properties is essential to the study of many random conductor-insulator composites, while fundamentally it is related to a variety of questions in statistical physics, dielectrics, and materials science, to name a few. Variable-temperature measurements of the frequency dependent complex effective permittivity were performed on amine-cured epoxy resins loaded with carbon black (CB). Two series of prepercolative samples differing from the kind of CB particles (Raven 2000 and Raven 5000) mixed in an amine-cured epoxy matrix (diglycidylic ether of bisphenol F) were studied. In this effort to contribute to our understanding of the role of frequency (100 Hz–15 MHz) and temperature (from ambient temperature up to 90 °C) on the complex effective permittivity which describes the linear response of the system to an electromagnetic wave, we investigate these composites with CB loadings below the percolation threshold. Two features are observed. First, our observations cannot be understood in the typical framework of a simple Debye-like dipolar process. In this analysis, we argue that the appearance of the broad temperature and frequency dependent maximum loss can be understood within the heuristic framework proposed by Jonscher which applies to disordered heterogeneous systems. This theoretical framework is consistent with several aspects of the experiments, notably the power-law decays of the real and imaginary parts of the effective permittivity characterized by two fractional exponents m and n. These exponents are both positive and smaller than unity. We further quantified their different temperature variations: while m is strongly decreasing with increasing temperature, n takes a value close to 1. Second, the observed maximum loss frequency found for each CB volume fraction shifts to higher frequencies with increasing temperature and exhibits a non-Arrhenius temperature dependence well represented by a Vogel–Tammam–Fulcher (VTF) fit. Well below the percolation threshold, the associated activation energy and ordering temperature of the VTF fit are not significantly sensitive upon the CB concentration. Such results are compared to previous related work.

The role of the concentration and distribution of water in the complex permittivity of breast fat tissue

The structure of breast tissue is complicated and highly variable and presents a great challenge in the development of physical models that may be used to obtain its effective complex permittivity. Empirical models are commonly used by researchers to fit measured data and extrapolated to higher frequencies. However, these models have not been verified experimentally at higher frequencies. Theoretical models of tissue permittivity to explain the role of water are not available today. This communication is a systematic study of several models to estimate the complex permittivity of breast fat tissue based on volume content and distribution of water in the tissue. These models are implemented in (i) long wavelength, sparse concentration limit; (ii) full wave finite element simulation; and (iii) numerical implementation of dynamic multiple scattering theory. A comparison of the proposed models with experimental data is done at 3.2 GHz. Some of the measurement values are in fair agreement with the modeling. The results of the present study are useful for interpreting the large variability in experimentally measured values of the permittivity of breast fat tissue by taking the distribution of water into account.

Electrical impedance spectroscopy as a potential tool for recovering bone porosity

This paper deals with the recovery of porosity of bone from measurements of its effective electrical properties. The microstructural information is contained in the spectral measure in the Stieltjes representation of the bone effective complex permittivity or complex conductivity and can be recovered from the measurements over a range of frequencies. The problem of reconstruction of the spectral measure is very ill-posed and requires the use of regularization techniques. We apply the method to the effective electrical properties of cancellous bone numerically calculated using micro-CT images of human vertebrae. The presented method is based on an analytical approach and does not rely on correlation analysis nor on any a priori model of the bone micro-architecture. However the method requires a priori knowledge of the properties of the bone constituents (trabecular tissue and bone marrow). These properties vary from patient to patient. To address this issue, a sensitivity analysis of the technique was performed. Normally distributed random noise was added to the data to simulate uncertainty in the properties of the constituents and possible experimental errors in measurements of the effective properties. The values of porosity calculated from effective complex conductivity are in good agreement with the true values of bone porosity even assuming high level errors in the estimation of the bone components. These results prove the future potential of electrical impedance spectroscopy for in vivo monitoring of level and treatment of osteoporosis.

Reconstruction of spectral function from effective permittivity of a composite material using rational function approximations

The paper deals with the problem of reconstruction of microstructural information from known effective complex permittivity of a composite material. A numerical method for recovering geometric information from measurements of frequency dependent effective complex permittivity is developed based on Stieltjes analytic representation of the effective permittivity tensor of a two-component mixture. We derive the Stieltjes representation for the effective permittivity of the medium using the eigenfunction expansion of the solution of a boundary-value problem. The spectral function in this representation contains all information about the microgeometry of the mixture. A discrete approximation of the spectral measure is derived from a rational (Padé) approximation followed by its partial fractions decomposition. The approach is based on the least squares minimization with regularization constraints provided by the spectral properties of the operator. The method is applied to calculation of volume fractions of the components in a mixture of two materials in a Bruggeman effective medium analytic model which has a continuous spectral density and to analytical models of two-phase composites with coated cylindrical and ellipsoidal inclusions. The numerical results of reconstruction of spectral measure for a mixture of silver and silicon dioxide and a composite of magnesium and magnesium fluoride show good agreement between theoretical and predicted values. The approach is applicable to geological materials, biocomposites, porous media, etc.

Analysis of the complex effective permittivity of a heterogeneous sample by the finite-difference time-domain method

Numerical results of the complex effective permittivity and DC conductivity of heterogeneous samples with random constituents’ distribution are analyzed to search for a suitable mixture law. The calculation method consists of the simulation of a time-domain reflectometry experiment by means of the finite-difference time-domain (FDTD) algorithm. The permittivity and conductivity of several series of samples with their constituents distributed at random are calculated. Numerical results, as a function of the volume fractions of the mixture’s constituents, are compared with analytical mixture laws. For lossless binary media, it is found that both the Bruggeman law and a weighted geometric mean provide a good agreement. The last one can be generalized to composites with more than two constituents, with good results. On the other hand, for lossy binary mixtures it is shown that the weighted geometric mean of the constituents’ complex permittivities matches the numerical results obtained for the effective complex permittivity of the mixture.

Mathematical model for radiowave propagation in sparse forests: Rytov approximation

In this paper, we consider penetrating propagation of a plane monochromatic electromagnetic wave in woodlands where the concentration of randomly distributed forest elements is low. These elements lead to appearance of a randomly varying component in the effective complex permittivity of the forest. Assuming that the random perturbation of the refraction coefficient is small, we apply the Rytov approximation and find the normalized coherent and incoherent components of the electric field intensity at the receiving point.

Dielectric relaxation in particle-filled polymer: Influence of the filler particles and thermal treatments

Variable-temperature measurements of the frequency-dependent complex effective permittivity are performed on silicon dioxide loaded epoxy resin composites. Two relaxation peaks due to conductivity and α-relaxation processes are identified. The amplitude and relaxation frequency of each peak strongly vary according to filler and material treatments while the activation energy of the Arrhenius ionic conductivity remains unchanged. These trends are compared to previous works. We argue that the changes in amplitude and relaxation frequency can be attributed to ions trapped in silicon dioxide particles and to a significant change in the elasticity network of the polymer chains.

Microwave properties of surface modified Fe–Co–Zr alloy flakes with mechanochemically synthesized polystyrene

Polystyrene coated Fe–Co–Zr alloy flakes have been prepared from gas atomized powders and styrene monomer by high energy ball milling. The effective complex permittivity and complex permeability of flakes/paraffin composites within 3.95–12 GHz were measured to assess the application potential of these products used as microwave absorbers. It is found that both the real and imaginary parts of complex permittivity substantially decrease with the increasing amount of reactant styrene. For the complex permeability, the real part of polystyrene coated samples is larger than those of the as-milled sample in high frequencies, however, the imaginary part exhibits a decline while resonance frequency shifts to high values. The calculated reflection loss curve suggests that a proper thickness of coating can redound to improve the absorbing performance of polystyrene coated Fe–Co–Zr alloy flakes in high frequency range.

Simulation and analysis of the effective permittivity for two-phase composite medium

Based on the effective medium theory, we calculated the complex effective permittivity of the twophase composite medium, consisting of circular inclusions embedded in a surrounding host. Results show that the maximum dielectric loss of the composite medium can be obtained by tuning the dielectric properties of the inclusion. This was confirmed by the results in literature. The local high temperature, as a consequence of the local electric field enhancement phenomenon, is the main reason that microwave heating can lead to a dramatic increase in the chemical reaction rate.

Dehomogenization: reconstruction of moments of the spectral measure of the composite

This paper deals with the inverse homogenization or dehomogenization problem of recovering geometric information about the structure of a two-component composite medium from the effective complex permittivity of the composite. The approach is based on the reconstruction of moments of the spectral measure in the Stieltjes analytic representation of the effective property. The moments of the spectral measure are linked to n-point correlation functions of the structure of the composite and thus contain information about the microgeometry. We show that the moments can be uniquely recovered from the measurements of the effective property in a range of frequencies. Two methods of numerical reconstruction of the moments are developed and analyzed. One method, which is referred to as a direct method of moment reconstruction, is based on the solution of the Vandermonde system arising in series expansion of the Stieltjes integral. The second, indirect, method reformulates the problem and reduces it to the problem of reconstruction of the spectral function. This last problem is ill-posed and requires regularization. We show that even though the reconstructed spectral function can be quite sensitive to the choice of the regularization scheme, the moments of the spectral functions can be stably reconstructed. The applicability of these two methods in terms of the choice of data points is also discussed in this paper.

The application of finite-difference time-domain modelling for the assessment of GPR in magnetically lossy materials

Numerical modelling has recently established itself as an important, near-surface GPR interpretation tool with the finite-difference, time-domain (FDTD) method becoming one of the most popular techniques. Robust, flexible and accurate, the FDTD technique is capable of simulating GPR wave propagation in complex, three-dimensional, heterogeneous, lossy, subsurface environments to a high degree of realism. Unfortunately, many of the current FDTD methods still consider the subsurface materials as being ‘non magnetic’ and, as such, do not include the propagation and loss effects associated with magnetic materials (e.g., basic igneous rocks, iron-rich sands, corroded steel reinforced concrete, smelting wastes, etc). For magnetically lossy materials, the inclusion of a complex magnetic permeability into the FDTD scheme can result in smeared or ‘fuzzy’ interface problems, increased computational demand and equation-level coding changes. Therefore, it is prudent to describe the magnetically derived loss and propagation characteristics in a more generic manner where the ‘electric’ (e.g., permittivity and conductivity) properties of the material incorporate the magnetic loss effects explicitly. In this paper, we present a “generalised complex effective permittivity” approach to the FDTD material descriptors that allows for the true loss and propagation characteristics of the magnetic materials to modelled fully, regardless of their individual magnetic or electric field relaxation mechanisms. In doing so, we are able to incorporate the lossy, dispersive effects directly into existing FDTD schemes without modification, additional error or increased computational demand. To demonstrate its application, a three-dimensional, 450 MHz, near-surface model of GPR data simulation over a rusty pipe has been included that illustrates how the FDTD modelling can be used to evaluate subtle changes in the spectral nature of the reflected signals. The modelling results show that, for favourable conditions, GPR techniques could be used to provide important, practical information on the assessment of pipeline corrosion and pre-failure conditions.

Dielectric-Resonator-Based Composite Right/Left-Handed Transmission Lines and Their Application to Leaky Wave Antenna

Propagation and radiation characteristics of 1-D dielectric-resonator-based composite right/left-handed metamaterial structures with open mesh plates are investigated as a function of configuration parameters. The complex effective permittivity and permeability are estimated from the scattering parameters of the unit-cell structure. Based on this configuration, the single-mode transmission line is designed and the backfire-to-endfire frequency-scanned leaky wave antenna is implemented. It is found from numerical simulations that the 30-cell leaky wave antenna has a gain of 10 dBi with a scanning angle of over 90deg covering backfire-to-endfire directions. The measurement results show as wide beam-steering angles as numerically predicted.

Mechanical fatigue and dielectric relaxation of carbon black/polymer composites

Using a home-built experimental setup, we investigate the carbon black (CB) volume fraction-resolved, time-resolved, and frequency-resolved spectra of the room temperature (well above the glass transition temperature of the matrix) effective complex permittivity of well characterized CB filled ethylene butylacrylate copolymer samples that are submitted to a uniaxial tension. We focus here on three samples with CB volume fraction less, near, and well away the percolation threshold at about 8 vol %. Our primary observation is that the temporal evolution of the real and imaginary parts of the effective permittivity is distinctly different for samples containing a CB volume fraction below and above percolation threshold. For samples containing a CB volume fraction below and close to the percolation threshold, and at a given frequency, ε′ and ε″ remain constant over the time scale of our measurements. For the sample containing a CB volume fraction above the percolation threshold the evolution of ε′ and ε″ is different in the low and high elongation ratio regimes. At low strain, the temporal evolution of permittivity during aging under stress shows a logarithmic growth phase followed by a logarithmic decay phase, whereas for sufficiently large strain the permittivity behaviors coincide with those observed below the percolation threshold. To explain the issues involved, we argue that the phenomenology for physical aging in these materials is related to the change in the mesostructure, formed by the heterogeneous three-dimensional interconnected network of polymer and of aggregates (or agglomerates) of CB particles, as the composite is stretched. Indeed, it is possible to argue qualitatively that below the percolation threshold the time-independent permittivity behavior is reminiscent to the elasticity network properties of the polymeric matrix. The reorientation and breakdown of the CB aggregates are believed to be crucial for the physical aging understanding above the percolation threshold and low strain. An examination of the surface and volume morphological evolutions of these materials under the action of a mechanical stress at the microscale by scanning electron microscopy and atomic force microscopy indicates that aging during a few hundreds of hours, even at a moderate strain, generates voids and cracks that are aligned along the stretching direction. The overall behavior is compared to what occurs in another type of filled polymer system, i.e., plastoferrites, for which it was recently discovered that two characteristic time scales are required to describe physical aging.

Physical aging of plastoferrites under tensile stress and its effect on microwave properties

Aging phenomena in soft viscoelastic materials have been used as an important tool to investigate the physics of complexity for both scientific and practical purposes. At ambient conditions, microwave frequency-domain spectroscopy (gigahertz-FDS) is employed to investigate the electromagnetic properties at continuum length scales for two commercial plastoferrites (PFs) subjected to a uniaxial tension during a long time t of application. PFs are an attractive prototype for investigating the interactions between polarization and magnetization phenomena, and time. For both PFs we have measured simultaneously the time evolution of the effective complex permittivity ε=ε′−jε″ and magnetic permeability μ=μ′−jμ″ spectra during tensile-loading experiments. These changes allow a determination of how physical aging affects the microwave properties of PFs with tensile elongation in the range of 3%–17% on the time scale of hundreds of hours. At short times of strain application the ε and μ measurements under stress can be explained in terms of a Gaussian molecular network model (affine behavior) in the limit of low strain. However, the large-strain mechanical response does show nonaffine behavior. The key achievement is that we present observational evidence for a three-stage aging in which (“zone I”) ε′ and ε″ increase as the ln(t) and reach a peak value at tI, then (“zone II”) there is a slower decrease in ε′ and ε″∝−ln(t) as time moves from tI up to tII, and finally, (“zone III”) as time moves to higher values, ε′ and ε″ saturate for long times. We have provided experimental evidence that, while tI is found to be constant with the initial elongation ratio λ0 that is applied to the PF sample, the data revealed that tII∝λ0. In sharp contrast, there is no significant change in μ′ and μ″ as a function of the elongation ratio. It can be argued that the decoupling of the dielectric properties from the magnetic properties may be regarded as a consequence of rigidity of the ferrite grains. To probe the influence of aging that alters the initial morphology of the PF samples, scanning electron microscopy (SEM) was used. The results of the SEM studies indicate the existence of cracks at the PF surface and disagglomeration of the ferrite grains. A tentative phenomenological model is proposed to explain the three-stage evolution of the elasticity network which has been tracked by gigahertz-FDS during the isothermal application of tension. Because the structure of soft materials is scale dependent, the response to a mechanical load also depends on the length and time scales of the probing excitation. We argued that the effective permittivity increase during stage I can be attributed to the electric dipole reorientation with respect to the direction of the polarization under the tensile stress applied to the material. The physical mechanism driving the effective permittivity decrease in stage II is thought to be a consequence of the breakup of the adsorbed ferrite particle network due to the detachment of chains from the surface of the particles. Stage III reflects the eventual stable steady state for which local stresses are redistributed in the elasticity network of the filled polymer under strain. The understanding of the physical aging mechanisms is significant because it can offer a strategy for materials selection and PF-based device performance optimization.

Electromagnetic properties of FeNi alloy nanoparticles prepared by hydrogen-thermal reduction method

FeNi alloy nanoparticles were prepared by hydrogen-thermal reduction in nickel ferrite nanoparticles at 400 °C for 1 h. FeNi alloy nanoparticles with average particle size of 150 nm in diameter are composed of Fe, Ni, and FeNi solid solution. The effective complex permittivity and complex permeability of FeNi alloy nanoparticles in the frequency range of 2–18 GHz were measured, and the microwave reflection loss of FeNi/wax with different assumed coating thicknesses was calculated according to the transmission line theory. The hydrogen-thermal reduced FeNi alloy nanoparticles were found to possess high permeability and superior microwave absorption performance in the microwave band.