Debye Equation Research Articles

FD2TD ANALYSIS OF ELECTROMAGNETIC FIELD PROPAGATION IN MULTIPOLE DEBYE MEDIA WITH AND WITHOUT CONVOLUTION

This paper deals with the time-domain numerical calculation of electromagnetic (EM) fields in linearly dispersive media described by multipole Debye model. The frequency-dependent finite-difference time-domain (FD2TD) method is applied to solve Debye equations using convolution integrals or by direct integration. Original formulations of FD2TD methods are proposed using different approaches. In the first approach based on the solution of convolution equations, the exponential analytical behavior of the convolution integrand permits an efficient recursive FD2TD solution. In the second approach, derived by circuit theory, the transient equations are directly solved in time domain by the FD2TD method. A comparative analysis of several FD2TD methods in terms of stability, dispersion, computational time and memory is carried out.

Morphological determination of face-centered-cubic metallic nanoparticles by X-ray diffraction

X-ray diffraction studies of face-centered-cubic metallic nanoparticles with different morphologies are discussed based on the experimental data and the calculation of the Debye equation with the truncated and perfect strain models. At least four basic morphologies frequently observed in face-centered-cubic metallic nanoparticles, namely sphere, cube, decahedron, and icosahedron, can be clearly distinguished from their characteristic integrated intensity ratios of the first two X-ray diffraction peaks, that is, (200) to (111).

Electric-field-enhanced nutrient consumption in dielectric biomaterials that contain anchorage-dependent cells

This research contribution addresses electric-field stimulation of intra-tissue mass transfer and cell proliferation in viscoelastic biomaterials. The unsteady state reaction–diffusion equation is solved according to the von Kármán–Pohlhausen integral method of boundary layer analysis when nutrient consumption and tissue regeneration occur in response to harmonic electric potential differences across a parallel-plate capacitor in a dielectric-sandwich configuration. The partial differential mass balance with diffusion and electro-kinetic consumption contains the Damköhler (Λ 2) and Deborah (De) numbers. Zero-field and electric-field-sensitive Damköhler numbers affect nutrient boundary layer growth. Diagonal elements of the 2nd-rank diffusion tensor are enhanced in the presence of weak electric fields, in agreement with the formalism of equilibrium and nonequilibrium thermodynamics. Induced dipole polarization density within viscoelastic biomaterials is calculated via the real and imaginary components of the complex dielectric constant, according to the Debye equation, to quantify electro-kinetic stimulation. Rates of nutrient consumption under zero-field conditions are described by third-order kinetics that include local mass densities of nutrients, oxygen, and attached cells. Thinner nutrient boundary layers are stabilized at shorter dimensionless diffusion times when the zero-field intra-tissue Damköhler number increases above its initial-condition-sensitive critical value [i.e., {Λ 2 zero-field} critical ≥ 53, see Eq. (23)], such that the biomaterial core is starved of essential ingredients required for successful proliferation. When tissue regeneration occurs above the critical electric-field-sensitive intra-tissue Damköhler number, the electro-kinetic contribution to nutrient consumption cannot be neglected. The critical electric-field-sensitive intra-tissue Damköhler number is proportional to the Deborah number.

Dielectric, Ferroelectric, and Magnetic Characteristics of LuFeCuO4 Ceramics

The LuFeCuO4 ceramics with a rhombohedral crystal structure in space group were prepared by a solid‐state reaction process, and both the dielectric and magnetic characteristics were investigated. An obvious dielectric relaxation with activation energy of 0.37 eV was observed, which could be well fitted by the modified Debye equation and Arrhenius law. A P–E hysteresis loop was observed in the present ceramics at 150 K, with a remnant polarization of 5.78 × 10−2 μC/cm2 and a coercive field of 10.76 kV/cm. The LuFeCuO4 was antiferromagnetic with TN of 57 K, and a slim hysteresis loop observed at 5 K demonstrated the weak ferromagnetic behavior below Néel temperature. The room‐temperature magnetodielectric response in LuFeCuO4 suggested the existence of magnetic–dielectric coupling.

Microwave Dielectric Properties of Alfalfa Leaves From 0.3 to 18 GHz

Dielectric properties (i.e., permittivity) are essential in designing, simulating, and modeling microwave applications. The permittivity of stacked leaves of alfalfa (Medicago sativa) were measured with a network analyzer and a coaxial probe, and the effect of moisture content (MC: 12%-73% wet basis), frequency (300 MHz to 18 GHz), bound water (Cole-Cole dispersion equation), temperature (-15°C and 30°C), leaf-orientation, and pressure (0-11 kPa) were investigated. The measured permittivity increased with MC. A critical moisture level (CML) of 23% was reported, below which the permittivity decreased with increasing frequency at 22°C. Above CML and up to 5 GHz, the dielectric constants followed the Cole-Cole dispersion, and the dielectric loss factors consisted of ionic and bound water losses. Above 5 GHz, the behavior of the dielectric constant was similar to that of free water, and the polar losses became dominant. Above 0°C, the measured permittivity followed a trend similar to that of free saline water and was characterized by the Debye equation. Below 0°C, it was dominated by nonfreezing bound and unfrozen supercooled moistures. The relaxation parameters and the optimum pressure (9 kPa) for the leaf measurements were determined. The effects of variations among the samples, and their orientations had negligible effects on the measured permittivity.

The nature of x-ray scattering from geo-nanoparticles: Practical considerations of the use of the Debye equation and the pair distribution function for structure analysis

Interpretation of elastic scattering from nanocrystals is critical to building and testing models of the short and medium range atomic arrangements in environmentally relevant nanoparticles. Along with information about atomic arrangements, the diffraction pattern arising from a collection of nanoparticles contains information on the grain size, shape and defect structure and cannot simply be treated as a broadened collection of Bragg peaks. The Debye equation, a sum over all pairs of atoms in the particle, calculates the scattering at each value of Q, rather than at discrete values of hkl. Especially when atomic arrangements are well known, modeling using the Debye equation allows the refinement of particle shape, allowing the investigation of dominant growth axes. We show the example of 6-line ferrihydrite, fitting the diffraction pattern using a disc of 2.3(1)nm in the ab plane and 6(2)nm along the c axis. A powerful and intuitive way to help examine the diffraction pattern is to take the Fourier transform of the normalized total scattering (Bragg and diffuse), giving the pair distribution function (PDF). The PDF represents a bond length distribution of the material weighted by the respective scattering powers of the contributing atoms. Using examples from the literature, we show that by analyzing the medium range correlations (~5–15Å), the structure of a nanoparticle can be distinguished using PDFs generated from model structures.

Dielectric and electro-optical properties of solutions of chemically modified fullerene С60 in nematic liquid crystal

Solutions of fullerene molecules С60 with chemically attached molecules of diamine (С60D) in planar oriented nematic liquid crystal (NLC) were obtained by only heating and ultrasonic processing. The С60D concentration changes from 0 up to 3.0 wt.%. Within the ranges of frequencies 10 -1 - 10 6 Hz and temperatures 298-343 K, dielectric properties of solutions were investigated. It was shown that, at frequencies higher than 100 Hz, the frequency dispersion of the components of complex dielectric permittivity is absent. A value of conductivity of the solution was determined. It was also shown that the activation energy for the temperature dependence of the conductivity in nematic and isotropic phases does not depend on the concentration of molecules С60D. Obtained and explained were the reasons of the nonmonotonic conductivity dependence of solutions on the concentration of С60D molecules. For frequencies lower than 100 Hz, the dispersion of the components of complex dielectric permittivity is observed. It was shown that the dispersion can be described by the Debye equation. The temperature dependence of a value inverse to the relaxation time correlates with the temperature dependence of conductivity. Presence of С60D molecules in NLC tends to increasing the voltage for the Frederiksz transition. Made was the assumption that this effect may be explained by increase in viscosity of NLC as a consequence of aggregation of fullerene molecules.

Effect of ZnO Nano-Particles on The Dielectric Relaxation Behavior and Thermal Stability of Polycarbonate Host

The influence of ZnO nanoparticles on the dielectric properties, dielectric relaxation behavior, and thermal stability of PC host was investigated at different ZnO nanoparticles concentration. The dielectric study was carried out over a frequency range from 500 Hz up to 1 MHz as a function of ZnO concentration. Results obtained from the best fitting of relative permittivity data with Yan and Rhodes model reveal that the dielectric relaxation is not a single relaxation process and there are two values of relaxation time for each nanocomposite at room temperature and domain frequency range. Furthermore, addition of ZnO nanoparticles to PC host changes the dielectric properties of PC, mainly, increases relative permittivity, dielectric loss, and AC conductivity while decreases the impedance values of PC host. Moreover, effect of temperature on AC conductivity of ZnO-PC nanocomposites at 1 kHz was also considered. Besides, dielectric relaxation behavior of PC was investigated at 165°C (above the glass transition temperature of PC) in the domain frequency range. At this elevated temperature (165°C), dielectric loss data shows a single relaxation peak (α-relaxation) in the domain frequency range and was successfully fitted with Debye equation. Also, it was found that as the content of ZnO nanoparticles increases in the PC host the frequency of the peak maximum ( fmax) shifts toward higher frequency value and as a result decreases the value of relaxation time. Additionally, this study shows that the isothermal effect of ZnO nano particles on the α-relaxation of PC host is similar to the effect of temperature. Both of them have a linear dependence with ln( fmax). Furthermore, addition of ZnO nanoparticles to PC host will decrease the thermal stability and glass transition temperature of PC host. Finally, there is a strong evidence from the obtained dielectric and thermal results that addition of ZnO nanoparticles to PC host will highly enhance the chain mobility and also increase the polar character of PC host.

Heat capacity of methacetin in a temperature range of 6 to 300 K

Abstract Heat capacity of methacetin (N-(4-methoxyphenyl)-acetamide) has been measured in the temperature range 5.8–300 K. No anomalies in the Cp(T) dependence were observed. Thermodynamic functions were calculated. At 298.15 K, the values of entropy and enthalpy are equal to 243.1 J K−1 mol−1 and 36360 J mol−1, respectively. The heat capacity of methacetin in the temperature range 6–10 K is well fitted by Debye equation Cp = AT3. The thermodynamic data obtained for methacetin are compared with those for the monoclinic and orthorhombic polymorphs of paracetamol.

Microwave dielectric properties of aqueous potassium iodide solutions as a function of temperature

The complex dielectric permittivity of aqueous KI solutions was studied for molalities of 0.50–4.01 m and temperatures of 288–323 K in the region of water dielectric permittivity dispersion. The values of high-frequency of dielectric permittivity (e) and dielectric losses (e″) were obtained at seven frequencies ranging between 7.5 and 25 GHz. The low-frequency electrical conductivity of the aforementioned solutions was measured for calculating ionic losses. A single relaxation process is observed in these solutions, fitted by the Debye or Cole-Cole equation with small distribution parameters. The static dielectric constant and dielectric relaxation time were studied as functions of temperature and concentration; the activation enthalpy of dielectric relaxation was calculated. The temperature dependence of the static dielectric constant was found to disappear in highly concentrated solutions. The structure-breaking effect on water caused by K+ and I− ions was affirmed, this effect disappearing in going to elevated temperatures.

A FDTD formulation for high order dispersive media of muscle-equivalent

In this paper, shift operator finite-difference time-domain (SO-FDTD) method is applied for the calculation of the dispersive medium. The high efficiency and accuracy of this method is verified by calculating the reflection of the plane electromagnetic wave impinging on a muscle slab. For human tissues where multiterm Debye relaxation equations must generally be used. We describe a new differential equation approach, which can be used for general dispersive media. In this method D( t) is expressed in terms of E( t) by means of a differential equation involving D, E, and their time derivatives. The method is illustrated by means of example of media for which relative permittivity is given by a multiterm Debye equation, and for an approximate two-thirds muscle-equivalent model of the human body.

Estimation of critical grain size at different temperatures and thermodynamic stability of γ→α in NC Fe by QDA and EOS methods

Thermodynamic properties of Nanocrystalline (NC) materials are essentially different from the conventional coarse-grained materials (with the same chemical composition). The role of grain boundary is very important in the characterization of thermodynamic functions and thermal properties of NC materials when the grain size is less than 100 nm. Therefore, the traditional thermodynamics being applied for coarse-grained materials is not applicable for NC materials. In this study, Quasiharmonic Debye Approximation (QDA) and Equation of State (EOS) methods are used to calculate the Gibbs free energy in NC Fe. Since the Gibbs free energy for Fe, predicted by EOS and QDA methods, is inaccurate (especially at temperatures higher than the ambient temperature), a term called as Δ G Excess is proposed to modify the results. Thus, the Modified QDA (MQDA) and Modified EOS (MEOS) methods are introduced for this purpose. Thereafter, the change in the Gibbs free energy for γ-Fe to α-Fe phase transformation (Δ G γ→α ) via the grain size is calculated by MQDA and MEOS methods. The results obtained by the two methods are also compared and discussed. Finally, the critical grain size, at which Δ G γ→α =0, can be estimated at different temperatures, is found to increase with increasing temperature.

Electrical Properties of Rare Earth Double Perovskite Sr2CeSbO6

The double perovskite oxide strontium cerium antimonate, Sr2CeSbO6 (SCS) is synthesised by solid state reaction technique. Thermal behaviour of SCS are investigated using thermogravimetric analyser and differential scanning calorimeter in the temperature range from 303 to 1500 K. Dielectric relaxation spectroscopy is used to investigate the frequency dependent dielectric relaxation in SCS in the frequency range from 100 Hz to 1 MHz and in the temperature range from 303 to 703 K. The frequency dependent electrical data are analysed by impedance formalism. The relaxation mechanism of the sample is modelled by Davidson-Cole equation (modified Debye equation). The frequencies corresponding to the maxima of imaginary part of impedance at the various temperatures are found to obey the Arrhenius law with activation energy ∼ 0.15 eV. The scaling behaviour of imaginary part of impedance suggests that the relaxation describes the same mechanism at various temperatures.

Dielectric and magnetic characteristics of LuFeMgO4 ceramics

The magnetic and dielectric characteristics of LuFeMgO4 ceramics were investigated together with the crystal structure. The present ceramics were refined as a rhombohedral structure in space group R3¯m, and antiferromagnetic with a Néel temperature of 27 K. The slim hysteresis loop measured at 10 K demonstrated weak ferromagnetic ordering in LuFeMgO4 ceramics, which might be attributed to the canting angle in Fe moment. An obvious dielectric relaxation, which was well fitted by the modified Debye equation and Arrhenius law with an activation energy of 0.35 eV, was observed in the temperature range of 180–420 K. Moreover, the obtained dielectric constant was much larger than that reported previously. The dielectric relaxation could be strongly suppressed by oxygen annealing, indicating that oxygen vacancies played an important role in the dielectric response of LuFeMgO4 ceramics.

Translational and Rotational Diffusion during Radical Bulk Polymerization: A Comparative Investigation by Full Correlation Fluorescence Correlation Spectroscopy (fcFCS)

Diffusional changes during the radical bulk polymerization of styrene with and without cross-linker were investigated. Full correlation fluorescence correlation spectroscopy (fcFCS) allowed for the simultaneous detection of rotational and translational diffusion of single fluorescent perylene diimide probes. Small probes show a parallel decrease of both motions whereas for larger probes translation slows down faster than rotation. The latter behavior contradicts the relationship between the Stokes−Einstein and the Stokes−Einstein−Debye equation and is attributed to changes in the microviscosity and mesh size effects of the synthesized polymer chains. The introduction of permanent cross-links results in additional restrictions for translational diffusion whereas rotation is not significantly affected up to high monomer-to-polymer conversions.

Experimental and the Theoretical Studies of the Dielectric Properties of DMSO–H2O Mixtures

This paper focuses on the measurement of the permittivity of dimethyl sulfoxide (DMSO)–water (H2O) mixture solutions, at 2.45 GHz by using a resonant cavity perturbation method. A specific phenomenon was found, in that the imaginary part of the permittivity for the mixture solution was larger than the imaginary part for each component. Theoretical calculation indicated that the reason for that phenomenon was that the high frequency friction of the mixture was larger than that of each component. When comparing the theoretical results with the experimental data, it was found that the classical Debye equation must be modified in order to calculate the complex permittivity.

Incompressibility of osmium metal at ultrahigh pressures and temperatures

Osmium is one of the most incompressible elemental metals, and is used as a matrix material for synthesis of ultrahard materials. To examine the behavior of osmium metal under extreme conditions of high pressure and temperature, we measured the thermal equation of state of osmium metal at pressures up to 50 GPa and temperatures up to 3000 K. X-ray diffraction measurements were conducted in the laser heated diamond anvil cell at GeoSoilEnviroCARS and the High Pressure at the Advanced Photon Source and beamline 12.2.2 at the advanced light source. Ambient temperature data give a zero pressure bulk modulus of 421 (3) GPa with a first pressure derivative fixed at 4. Fitting to a high temperature Birch–Murnaghan equation of state gives a room pressure thermal expansion of 1.51(0.06)×10−5 K−1 with a first temperature derivative of 4.9(0.7)×10−9 K−2 and the first temperature derivative of bulk modulus of be dK0/dT=−0.055 (0.004). Fitting to a Mie–Grüneisen–Debye equation of state gives a Grüneisen parameter of 2.32 (0.08) with a q of 7.2 (1.4). A comparison of the high pressure, temperature behavior among Re, Pt, Os, shows that Os has the highest bulk modulus and lowest thermal expansion of the three, suggesting that Os-based ultrahard materials may be especially mechanically stable under extreme conditions.

Experimental and Theoretic Study of the Dielectric Properties of Ethanol + Methanol Mixtures

The main objective of this work is to measure the complex permittivity of ethanol + methanol liquid solutions over the full composition range at the temperatures 293, 303 and 313 K by the resonant cavity perturbation method. Dielectric parameters were derived using mixture formulae, and the values of excess permittivity and excess inverse relaxation time were determined using these parameters. By comparing the calculated results to the experiment values it was found that the classical Debye equation needs to be modified to calculate the complex permittivity.

Polarizability of red blood cells with an anisotropic membrane

We predict the complex polarizability of a realistic model of a red blood cell (RBC), with an inhomogeneous dispersive and anisotropic membrane. In this model, the frequency-dependent complex electrical parameters of the individual cell layers are described by the Debye equation while the dielectric anisotropy of the cell membrane is taken into account by the different permittivities along directions normal and tangential to the membrane surface. The realistic shape of the RBC is described in terms of the Jacobi elliptic functions. To calculate the polarizability, we evoke the effective dipole moment method to determine the cell internal electric field distribution, employing an adaptive finite-element numerical approach. We have furthermore investigated the influence of the anisotropic membrane and dispersive electrical parameters of each individual cell layer on the total complex polarizability. Our findings suggest that the individual layer contribution depends on two factors: the volume of the layer and the associated induced electric field, which in turn is influenced by other layers of the cell. These results further show that the average polarizability spectra of the cell are significantly impacted by the anisotropy and associated dispersion of the cellular compartments.

Electronic and ionic contributions to the constant-volume specific heat of carbon tetrachloride shocked at pressures up to 23 GPa

For carbon tetrachloride, a temperature Hugoniot at 7–23 GPa and a distribution of the constant-volume specific heat on the Hugoniot CV(T) at 1057–3275 K are simultaneously estimated from the Walsh–Christian (WC) equation such that the Hugoniot fits well to the existing measured data. The estimated CV(T) distribution reveals the significance of the contribution of electrons and ions to the specific heat. That is, in contrast to the almost uniform distribution of the specific heat predicted from the Debye equation, the CV(T) distribution increases significantly with an increase in the Hugoniot temperature due to thermal excitation of electrons at 1057–1500 K (7–10.1 GPa), additional activation of the dimerization reaction at 1500–2350 K (10.1–16 GPa), and also additional activation of the polymerization reaction at 2350–3275 K (16–23 GPa). As an example, evidence is presented that carbon tetrachloride is a semiconductor at 1500 K. The CV(T) distribution in each temperature range is formulated and a temperature Hugoniot is reevaluated from the WC equation using the specific heat equations formulated. It is confirmed by a good fit of the reevaluated Hugoniot to the existing measured data that the specific heat equations express the CV(T) distribution appropriately.