Maxwell-Garnett Mixing Formula Research Articles

High-Resolution L-Band TomoSAR Imaging on Forest Canopies with UAV Swarm to Detect Dielectric Constant Anomaly.

A rigorous TomoSAR imaging procedure is proposed to acquire high-resolution L-band images of a forest in a local area of interest. A focusing function is derived to relate the backscattered signals to the reflectivity function of the forest canopies without resorting to calibration. A forest voxel model is compiled to simulate different tree species, with the dielectric constant modeled with the Maxwell-Garnett mixing formula. Five different inverse methods are applied on two forest scenarios under three signal-to-noise ratios in the simulations to validate the efficacy of the proposed procedure. The dielectric-constant profile of trees can be used to monitor the moisture content of the forest. The use of a swarm of unmanned aerial vehicles (UAVs) is feasible to carry out TomoSAR imaging over a specific area to pinpoint potential spots of wildfire hazards.

Uniaxial Effective Permittivity of Anisotropic Bicontinuous Random Media Using NMM3D

In this letter, we generate anisotropic bicontinuous media with different vertical and horizontal correlation functions. With the computer-generated bicontinuous medium, we then use numerical solutions of Maxwell equations in 3-dimensions (NMM3D) to calculate the anisotropic effective permittivities and the effective propagation constants of V and H polarizations. The copolarization phase difference (CPD) of VV and HH is then derived. The CPDs have recently been applied to the retrieval of snow water equivalent, snow depth, and anisotropy. The NMM3D simulation results are also compared with the results of the strong permittivity fluctuations in the low frequency limit and compared against the Maxwell–Garnett mixing formula.

HOMOGENIZATION OF PERIODIC OBJECTS EMBEDDED IN LAYERED MEDIA

An effective medium modeling technique is proposed to homogenize the periodic objects embedded in layered media. The homogenization is based on the same scattering coefficients. An integral equation based approach is adopted to solve the scattering problem of original structures. Our modeling results are compared with Maxwell-Garnett mixing formula and published results. Good agreements have been observed. Periodic metal patches embedding in layered dielectric structure is fabricated and measured to validate the modeling technique. The difference between experiment results and proposed modeling results is less than 3%.

Modelling effective dielectric properties of materials containing diverse types of biological cells

An efficient and versatile numerical method for the generation of different realistically shaped biological cells is developed. This framework is used to calculate the dielectric spectra of materials containing specific types of biological cells. For the generation of the numerical models of the cells a flexible parametrization method based on the so-called superformula is applied including the option of obtaining non-axisymmetric shapes such as box-shaped cells and even shapes corresponding to echinocytes. The dielectric spectra of effective media containing various cell morphologies are calculated focusing on the dependence of the spectral features on the cell shape. The numerical method is validated by comparing a model of spherical inclusions at a low volume fraction with the analytical solution obtained by the Maxwell–Garnett mixing formula, resulting in good agreement. Our simulation data for different cell shapes suggest that around 1MHz the effective dielectric properties of different cell shapes at different volume fractions significantly deviate from the spherical case. The most pronounced change exhibits εeff between 0.1 and 1 MHz with a deviation of up to 35% for a box-shaped cell and 15% for an echinocyte compared with the sphere at a volume fraction of 0.4. This hampers the unique interpretation of changes in cellular features measured by dielectric spectroscopy when simplified material models are used.

Effects of structural and compositional factors on soil electromagnetic properties

Electromagnetic properties of clay soils can be affected by many factors, including clay mineralogy, clay percentage, flocculation of clay particles, pore fluid salt concentration, anisotropy, and water content. The influences of these factors on soil dielectric permittivity and electrical conductivity are analysed using a theoretical model developed for clay minerals and the modified Maxwell-Garnett mixing formula. The analysis shows that, at frequencies higher than 50 MHz, the real permittivity of an isotropic soil is affected primarily by water content, clay percentage and clay mineralogy and, to a lesser extent, by pore fluid salt concentration and flocculation. Three theoretical correlations are established for saturated isotropic clay–water and sand–water mixtures: (1) correlation between volumetric water content and the real permittivity at 1 GHz, (2) correlation between volumetric water content and the dielectric dispersion magnitude from 50 MHz to 1 GHz, and (3) correlation between volumetric water content and the effective electrical conductivity at 20 MHz. These correlations are validated by measured data for saturated sand, kaolinite, bentonite and mixtures of bentonite and silica flour. A new method to determine volumetric water content and specific surface area from electromagnetic measurements is proposed.

Microwave attenuation with composite of copper microwires

It is shown that copper microwires composite media attenuates microwave reflection of metallic surfaces. We show how the distance to the metallic surface, as well as the length and volume fraction of microwires, determine the frequency of maximum absorption and the return loss level. Furthermore, we were able to fit the experimental results with a theoretical model based on Maxwell-Garnett mixing formula.

Design and fabrication of nano-structured gradient index microlenses

We present a novel fabrication technology for nano-structured graded index micro-optical components, based on the stack-and-draw method used for photonic crystal fibres. These discrete structures can be described with an effective refractive index distribution. Furthermore we present spherical nano-structured microlenses with a flat facet fabricated with this method and designed using an algorithm based on the Maxwell-Garnett mixing formula. Finally we show theoretical verification by using FDTD simulations for a nano-structured lens as well as experimental data obtained in the microwave regime.

Weak polyelectrolytes tethered to surfaces: Effect of geometry, acid–base equilibrium and electrical permittivity

AbstractThe structural and thermodynamical properties of weak polyelectrolytes end‐tethered to surfaces of arbitrary geometry are studied using a molecular theory. The theory is based on writing down the free energy functional of the system including all the basic interactions and the explicit acid–base equilibrium for the chargeable groups of the polymer. The theory explicitly includes the size, shape, conformations, and charge distribution of all the molecular species. The electrostatic interactions include a density‐dependent dielectric function, modeled with the Maxwell–Garnett mixing formula, to account for the composition‐dependent permittivity. The minimization of the free energy leads to the distribution of all molecular species and their dependence on bulk pH and salt concentration. We apply the theory to polymer chains end‐tethered to planar, cylindrical, and spherical surfaces. The radius of the curved surfaces is small to enhance the curvature effect. We find that when the grafting surfaces are uncharged, the approximation of a constant dielectric function works very well for both structural and thermodynamic properties. The structure of weak polyelectrolytes tethered on cylindrical and spherical surfaces is different from that of polymers tethered on planar surfaces due to the available volume as a function of the distance from the surface. Specifically, the degree of dissociation increases with increasing curvature of the surface. This is a manifestation of the coupling between the local density of protons, counterions, and polymer segments. The results can be interpreted in terms of the local Le Chatelier principle for the acid–base equilibrium, with proper account of the three local contributions: counterions, protons, and chargeable groups. We find that one can achieve local changes of pH between one to two units within 1–2 nm. The thickness of the tethered layers as a function of bulk pH shows a large increase when the pH is equal to the bulk pK. However, the variation with salt concentration is different for the different geometries. The largest swelling is found for cylindrical surfaces. The predictions from scaling theories of a maximum in the thickness of the film as a function of salt concentration is found for planar films, but not for curved surfaces. Finally, the interactions between cylinders with tethered polyelectrolytes is very different from the equivalent planar surfaces. These results are important for the interpretation of force measurements with nanoscale AFM tips. The implications of the results for the rational design of responsive tethered polymer layers is discussed together with the limitations of the theoretical approach. © 2006 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 44: 2638–2662, 2006

Modeling of 3-D periodic multiphase composites by homogenization

An efficient technique is proposed for analyzing two- and three-dimensional lossy periodic composite materials, combining an asymptotic multiscale method with the unfolding method. The computed effective conductivities for square cylinders and cubes suspended in a host isotropic medium are compared to the Maxwell-Garnett mixing formula predictions. The electromagnetic field in a finite lattice of round cylinders is compared with the exact electromagnetic field calculated directly in the heterogeneous lattice

Analytical approximations in multiple scattering of electromagnetic waves by aligned dielectric spheroids.

In a dense medium, the failure to properly take into account multiple-scattering effects could lead to significant errors. This has been demonstrated in the past from extensive theoretical, numerical, and experimental studies of electromagnetic wave scattering by densely packed dielectric spheres. Here, electromagnetic wave scattering by densely packed dielectric spheroids with aligned orientation is studied analytically through quasicrystalline approximation (QCA) and QCA with coherent potential (QCA-CP). We assume that the spheroids are electrically small so that single-particle scattering is simple. Low-frequency QCA and QCA-CP solutions are obtained for the average Green's function and the effective permittivity tensor. For QCA-CP, the low-frequency expansion of the uniaxial dyadic Green's function is required. The real parts of the effective permittivities from QCA and QCA-CP are compared with the Maxwell-Garnett mixing formula. QCA gives results identical to those with the mixing formula, while QCA-CP gives slightly higher values. The extinction coefficients from QCA and QCA-CP are compared with results from Monte Carlo simulations. Both QCA and QCA-CP agree well with simulations, although qualitative disagreement is evident at higher fractional volumes.

Microphysical retrievals over stratiform rain using measurements from an airborne dual-wavelength radar-radiometer

The need to understand the complementarity of the radar and radiometer is important not only to the Tropical Rain Measuring Mission (TRMM) program but to a growing number of multi-instrumented airborne experiment that combine single or dual-frequency radars with multichannel radiometers. The method of analysis used in this study begins with the derivation of dual-wavelength radar equations for the estimation of a two-parameter drop size distribution (DSD). Defining a as the set of parameters that characterize snow density, cloud water, water vapor, and features of the melting layer, then to each storm model there will usually correspond a set of range-profiled drop size distributions that are approximate solutions of the radar equations. To test these solutions, a radiative transfer model is used to compute the brightness temperatures for the radiometric frequencies of interest. A storm model or class of storm models is considered optimum if it provides the best reproduction of the radar and radiometer measurements. Tests of the method are made for stratiform rain using simulated storm models as well as measured airborne data. Preliminary results show that the best correspondence between the measured and estimated radar profiles usually can be obtained by using a moderate snow density (0.1-0.2 g/cm/sup -3/), the Maxwell-Garnett mixing formula for partially melted hydrometeors (water matrix with snow inclusions), and low to moderate values of the integrated cloud liquid water (less than 1 kg/m/sup -/2). The storm-model parameters that yield the best reproductions of the measured radar reflectivity factors also provide brightness temperatures at 10 GHz that agree well with the measurements. On the other hand, the correspondence between the measured and modeled values usually worsens in going to the higher frequency channels at 19 and 34 GHz. In searching for possible reasons for the discrepancies, it is found that changes in the DSD parameter /spl mu/, the radar constants, or the path-integrated attenuation can affect the high frequency channels significantly. In particular, parameters that cause only modest increases in the median mass diameter of the snow, and which have a minor effect on the radar returns or the low frequency brightness temperature, can produce a strong cooling of the 31 GHz brightness temperature.

Multiple-scattering formalism for general discrete random composites.

A multiple-scattering formalism is developed for the determination of four effective parameters for bi-isotropic composites in the resonance range. Two models, the truncated quasicrystalline approximation and the dynamic Maxwell-Garnett model, are presented to calculate the effective parameters of general discrete random composites. All the results recover the Maxwell-Garnett mixing formula at low frequency. The degenerative two- and three-effective-parameter cases are also discussed in some detail due to the practical importance of these composites and the requirement of the inherent unity of the formalism.

Chiral Maxwell-Garnett mixing formula

The Maxwell-Garnett mixing formula, previously well known for dielectric mixtures, is generalised for the first time to cover chiral materials. The mixing formula expresses the permittivity, permeability and chirality of a two-component mixture with spherical inclusions as functions of the material parameters of the components. The result gives the classical Maxwell-Garnett formula as a special case. The mixing formula can be applied for the design of chiral composites that have attracted keen interest in recent years.