General Effective Media Equation Research Articles

Electrical conductivity of porous binary powder mixtures

Simultaneous data of the quasi-static compaction and electrical conductivity of porous, binary powder mixtures have been collected as a function of bulk density. The powder mixtures consist of a metal conductor, either titanium or iron, an insulator, and pores filled with ambient air. The data show a dependency of the conductivity in terms of relative bulk density and metal volume fraction on conductor type and conductor particle characteristics of size and shape. Finite element models using particle domains generated by discrete element method are used to simulate the bulk conductivity near its threshold while the general effective media equation is used to model the conductivity across the compression regime.

Numerical simulation of external sulphate attack in concrete considering coupled chemo-diffusion-mechanical effect

This paper proposes an extended numerical method based on Fick’s second law and second-order reaction kinetics for predicting the erosion process of concrete exposed to sodium sulphate solution. A three-phase concrete model was introduced for calculating the diffusivity of concrete to consider the effect of the interfacial transition zone (ITZ), and the diffusivities of diffusible phases were obtained based on the general effective medium equation. In addition, the effects of pore-filling and chemo–mechanical damage on the diffusivity of concrete were considered in the model, using the volume increase approach. Moreover, the expansive products, mainly gypsum and ettringite, were assumed to precipitate unevenly in the ITZ and cement paste owing to the different diffusivities and sulphate ion concentrations of these two phases. The distributions of the sulphate ions, gypsum, and ettringite in concrete, the expansion of concrete specimens, and the damage depth were determined through numerical calculations, and the simulation reliability was validated based on the comparisons with the available experimental data. Furthermore, the effects of the water–cement ratio and external solution concentration on the sulphate diffusion and specimen expansion were determined. The simulation results showed that the ITZ influenced the calculated results and should therefore be considered in the numerical model. The diffusivity of concrete decreased owing to the pore-filling effect but increased as the chemical–mechanical damage increased.

Influence of grain size and shape on volcanic ash electrical conductivity

Few studies have examined the electrical properties of volcanic ash or considered the effects of physical characteristics, such as grain size and shape on its electrification. This study measures the resistivity of eight volcanic ash samples, three milled-samples and five natural ashfall samples from Alaska, U.S.A., using a current amplifier and examines the influence of particle size and particle shape on calculated grain conductivity. Grain conductivities are calculated by applying spherical, oblate, and prolate ellipsoidal shape parameters within a general effective media equation. Volcanic ash is an electrically insulating material with average resistivity values ranging from 1.732E7 to 1.606E12 Ωm. Results show that grain size has a limited impact on resistivity. Calculated grain conductivities remained within the same order of magnitude between the three different shape calculations ranging between 1.367E−7 and 2.196E−12 S/m using the spherical shape parameter. Electrical measurements of heterogeneous powders are complex and challenging. This study aims to expand the community's dataset and understanding of the electrical properties of volcanic ash and the impact of different physical characteristics. This has implication for understanding ash hazards to electrical infrastructure and volcanic lightning.

Properties and mechanisms of self-sensing carbon nanofibers/epoxy composites for structural health monitoring

In this paper, carbon nanofibers (CNFs) with high aspect ratio were dispersed into epoxy matrix via mechanical stirring and ultrasonic treatment to fabricate self-sensing CNFs/epoxy composites. The mechanical, electrical and piezoresistive properties of the nanocomposites filled with different contents of CNFs were investigated. Based on the tunneling conduction and percolation conduction theories, the mechanisms of piezoresistive property of the nanocomposites were also explored. The experimental results show that adding CNFs can effectively enhance the compressive strengths and elastic moduli of the composites. The percolation threshold of the CNFs/epoxy composites is 0.186 vol% according to the modified General Effective Media Equation. Moreover, the stable and sensitive piezoresistive response of CNFs/epoxy composites was observed under monotonic and cyclic loadings. It can be demonstrated that adding CNFs into epoxy-based composites provides an innovative means of self-sensing, and the high sensitivity and stable piezoresistivity endow the CNFs/epoxy composites with considerable potentials as efficient compressive strain sensors for structural health monitoring of civil infrastructures.

Estimate the relative electrical conductivity of C–S–H gel from experimental results

A general effective media (GEM) equation-based scheme is proposed to estimate the relative electrical conductivity of C–S–H gel in this study. It is highlighted that the relative electrical conductivity of the solid phase in cement paste can be treated as a constant only on the iso-degree of hydration basis. On this basis, the relation between the electrical conductivity and capillary porosity of cement paste is fitted by GEM equation to deduce the relative conductivity of the solid phase, as a function of degree of hydration. The relative electrical conductivity of C–S–H gel is estimated from the relative conductivity of solid phase under fully hydrated state (an ideal state), and it equals 0.00775 as estimated.

Electrical resistivity of Au-ZnO nanocomposite films

The electrical resistivity of electron beam codeposited gold and zinc oxide (Au-ZnO) films was investigated over the full composition range. The electrical resistivity was shown to increase monotonically with increasing ZnO content, with three characteristic regimes of behavior associated primarily with (1) grain boundary electron scattering due to grain refinement at ZnO volume fractions below 0.3, (2) percolation theory for ZnO volume fractions at and above the percolation threshold (fc = 0.85), and (3) a transition region between these where it was proposed that resistivity was influenced by the formation of Au-Zn complexes due to an oxygen deficiency in the deposited ZnO. The electrical resistivity of the composite films remained below 100 μΩ cm for ZnO volume fractions below 0.5. A model combining the general effective media equation and Mayadas-Shatzkes grain boundary electron scattering model was shown to generally describe the composition dependence of electrical resistivity for the investigated oxide dispersion hardened metal-matrix composite thin films.

Electrical conductivity of CNT/polysulfonamide composites

Carbon nanotube/polysulfonamide (CNT/PSA) composite spinning solutions with different CNTs contents were prepared using the physical blending method. The corresponding composite fibers were extruded through a set of wet spinning unit and the composite membranes were prepared using a digital spin-coating technique. The distribution of CNTs in PSA matrix and the crystalline structure of composite fibers were investigated by scanning electron microscope (SEM) and X-ray diffraction, respectively. The influences of CNTs on mechanical properties as well as electrical conductivity of PSA composites were also analyzed. In addition, the general effective media equation was chosen to explain the electrical conductivity behavior of the CNT/PSA composites. The experimental results showed that CNTs with low contents can be distributed evenly in PSA; the crystallization in PSA was promoted at low CNTs contents; the mechanical properties of composites can be improved significantly; and the electrical conductivity of composites increased obviously and the percolation threshold of the blending system was at about 3 wt%.

Multi-scale modeling of the effective chloride ion diffusion coefficient in cement-based composite materials

N-layered spherical inclusions model was used to calculate the effective diffusion coefficient of chloride ion in cement-based materials by using multi-scale method and then to investigate the relationship between the diffusivity and the microstructure of cement-basted materials where the microstructure included the interfacial transition zone (ITZ) between the aggregates and the bulk cement pastes as well as the microstructure of the bulk cement paste itself. For the convenience of applications, the mortar and concrete were considered as a four-phase spherical model, consisting of cement continuous phase, dispersed aggregates phase, interface transition zone and their homogenized effective medium phase. A general effective medium equation was established to calculate the diffusion coefficient of the hardened cement paste by considering the microstructure. During calculation, the tortuosity (n) and constrictivity factors (Ds/D0) of pore in the hardened pastes are n≈3.2, Ds/D0=1.0×10−4 respectively from the test data. The calculated results using the n-layered spherical inclusions model are in good agreement with the experimental results; The effective diffusion coefficient of ITZ is 12 times that of the bulk cement for mortar and 17 times for concrete due to the difference between particle size distribution and the volume fraction of aggregates in mortar and concrete.

有限要素法を用いた複合熱電材料の特性解析

The addition of metal into thermoelectric materials has been used as one method to improve thermoelectric performance. Although the electrical resistivity of thermoelectric materials can be decreased largely by the metal composition, it is always accompanied with some negative effects such as the thermal conductivity increase. Therefore, it seems necessary to investigate the composite effects and their influence on thermoelectric properties. In the present work, analysis models of finite element method with series and random distribution for thermoelectric composites were proposed and established. Thermoelectric properties including electrical resistivity, Seebeck coefficient and thermal conductivity of thermoelectric composites were investigated by the proposed models. Thermoelectric properties calculated by the models were compared with those obtained by the experiments and theories of the rule of mixture and the general effective medium equation (GEM). The results revealed that the values of thermoelectric properties calculated from the proposed models were consistent with those from the experiments and the rule of mixture in the series model. In addition, the results calculated by the random distribution model showed a similar tendency with those from the experiments and GEM. The proposed analysis models are suitable to discuss the thermoelectric properties of thermoelectric composites.

Semi-empirical modeling of electrical conductivity for composite bipolar plate with multiple reinforcements

Carbon composite bipolar plates were developed by compression molding of novolac type phenol formaldehyde resin with natural graphite, carbon black, and carbon fiber. The General Effective Media equation was adapted to model the electrical conductivity of the bipolar plate. The experimental values of the electrical conductivity of the composites with different reinforcements were well predicted by the model. For resin–graphite system (2-component), the most effective in-plane and through-plane electrical conductivities for 70% graphite content were found to be 201.26 and 40.91 S cm−1, respectively. Similarly, for optimum resin–graphite–carbon black system (3-component), these values were found as 269.55 and 82.77 S cm−1, respectively. The most effective in-plane and through-plane electrical conductivities were found to be 285.54 and 91.79 S cm−1, respectively, for the composite with resin–graphite–carbon black–carbon fiber system (4-component). The predicted electrical conductivities for all the three systems were found to be in well agreement with the experimental values.

Computer simulation of electrical conductivity of colloidal dispersions during aggregation

The computation approach to the simulation of electrical conductivity of colloidal dispersions during aggregation is considered. We use the two-dimensional diffusion-limited aggregation model with multiple-seed growth. The particles execute a random walk, but lose their mobility after contact with the growing clusters or seeds. The two parameters that control the aggregation are the initial concentration of free particles in the system p and the concentration of seeds psi. The case of psi=1, when all the particles are the immobile seeds, corresponds with the usual random percolation problem. The other limiting case of psi=0, when all the particles walk randomly, corresponds to the dynamical percolation problem. The calculation of electrical conductivity and cluster analysis were done with the help of the algorithms of Frank-Lobb and Hoshen-Kopelman. It is shown that the percolation concentration phi c decreases from 0.5927 at psi=1 to 0 at psi --> 0. Scaling analysis was applied to study exponents of correlation length v and of conductivity t. For all psi>0 this model shows universal behavior of classical 2d random percolation with v approximately t approximately 4/3. The electrical conductivity sigma of the system increases during aggregation reaching up to a maximum at the final stage. The concentration dependence of conductivity sigma(phi) obeys the general effective medium equation with apparent exponent ta(psi) that exceeds t. The kinetics of electrical conductivity changes during the aggregation is discussed. In the range of concentration Pc(phi)<p<0.9527 the time of percolation cluster formation pi c decreases with increasing phi.

Evaluation of the fitting efficiencies of general effective medium equation and its modification

is always minimized by varying the fittable parameters in the predictive equation. Here eequ is the effective dielectric constant calculated by the predictive equation using appropriate variable parameters, n is the number of data points and p is the number of variable parameters in that equation [1]. Wu and McLachlan suggested that the fitting efficiencies of GEM and mGEM were nearly the same over the whole range of volume fractions [5]. We also observed the phenomenon that, when fitted to the same data set, GEM and mGEM had very close χ values, which means similar fitting efficiencies [4]. At each data point, however, are the fitting efficiencies of GEM and mGEM nearly the same as well? As χ can only reflect fitting efficiency in whole range of volume fraction, after the parameters of a predictive equation have been calculated, the quantity:

Evaluation of Mixing Rules for Dielectric Constants of Composite Dielectrics by MC-FEM Calculation on 3D Cubic Lattice

Effective dielectric constants of diphase composite dielectrics are simulated by Monte Carlo-finite element method on three-dimensional lattice. Effective dielectric constants with coefficients of variation less than 5.5% are obtained for different ratios of dielectric constants of the two phases, ranging from 10 to 700. Various mixing rules and equations are fitted to these data and the accuracy and relevance of the fits are thoroughly examined. Modified logarithmic rule loses its physical basis when fitted to three-dimensional data. As the ratio of dielectric constants of the two phases increases, the parameters in general Lichterecker mixing rule, general Bruggeman's symmetric equation, general effective media equation and its modified form all increase or decrease monotonously. General effective media equation and its modified form give the best fits to the effective dielectric constants simulated. The simulation results for the dielectric constants of some composite systems are in good agreement with experimental data.

Electrical conductivity of carbonaceous powders

The present paper deals with the electrical conductivity of a number of carbonaceous powders undergoing a low compaction. It is shown that the contributions to the conductivity of such samples are too numerous and too complex to be solved exactly. Hence, several criteria are proposed in order to quantify the behaviour of moderately compressed powders, and thus derive several useful parameters characterizing their grains. Thus, their morphological characterization is shown to be feasible from such electrical measurements. For that purpose, a known weight of material is poured into a thick glass cylinder and is moderately compressed between two stainless steel pistons. A number of different carbonaceous powders having various morphologies, from almost spherical to disk- or needle-shaped, have been investigated. During the compaction, both the electrical conductivity σ and the volume of the sample are simultaneously measured. The results are found to depend strongly on the initial height h of the powder column. The non-monotonic behaviour of σ versus h is studied for each available material. Arguments are given for retaining only the sample size which leads to the highest conductivity. The so-called general effective media (GEM) equation is then applied to the corresponding data of σ versus Φ, where Φ is the volume fraction of grains in the column. In this context, the grains are assumed to behave like either oblate or prolate spheroids, depending on the material. The fitting procedure requires the knowledge of both the apparent density of the powder in the non-compacted state, and that of the grains, and thus uses only two adjustable parameters. For each material, average aspect ratios are derived from the fits of the GEM equation and are shown to be strongly consistent with the expected morphologies.

Thermal volume expansion in polymeric PTC composites: a theoretical approach

Polymeric PTC (positive temperature coefficient) composites have been prepared by incorporating carbon black into polyethylene matrices. The dependence of conductivity upon filler loading level obeys a mixtures rule, which was determined by fitting the experimental data into the general effective medium (GEM) equation. A phenomenological model is proposed which is based on the GEM equation and the dilution effect of filler volume fraction due to thermal volume expansion. By using this model, the contribution of the thermal expansion of the matrix to the jump-like PTC switch transition of the composite is quantitatively estimated, and a mechanical explanation is given. The thermal-expansion-dependent percolation curves can also be predicted. Comparisons between predicted and measured PTC effects reveal that thermal expansion is one of the leading factors responsible for the thermal switch effect. Other factors influencing the composite resistivity are discussed.

Electrical properties and thermal expansion of LaCoO 3/La 2(Zr,Y) 2O 7 composites

Homogeneous fine grained La 2(Zr,Y) 2O 7/LaCoO 3− δ -based composite materials were obtained from simple mixture of the components and solid state reaction between LaCoO 3− δ and yttria-stabilized zirconia. Thermal expansion data can be fitted to simple mixture rules to predict the composite behavior from the properties of the individual components. The dependence of total and ionic conductivity on the volume fraction of the conductive phase can be described by the General Effective Media equation. The resistance to thermal cycling was evaluated and found satisfactory

Zirconia-tungsten composites: synthesis and characterisation for different metal volume fractions

Zirconia-tungsten composites have been obtained by the densification of ceramic-metal powders. The influence of the sintering conditions and the metal volume fraction on the morphology, the porosity, the linear shrinkage, the microhardness and on the linear expansion coefficient has been evidenced. Using a general effective medium equation to fit the electrical resistivity of the composites (1400°C-3h) a percolation threshold of 0.26 has been found. The distribution functions of relaxation times obtained from impedance spectroscopy measurements show that the higher is the thermal treatment the sharper are the peaks and the lower is the relaxation time τp. The intragrain zirconia ionic conductivity is improved for high tungsten volume fractions below the percolation threshold.

An analytical solution to general effective medium equation

An analytical solution to general effective medium equation

Electrical conductivity of two-phase composite material

It is a long standing problem to predict the effective electrical conductivity of binary mixtures or two-phase composites. The early history of this problem is often connected with the names of Clausius, Mossotti, Lorenz, and Lorentz. Currently, the general effective media (GEM) equation, proposed by McLachlan, seems to explain different results of experimental studies over an entire composition range. The GEM equation consists of electrical conductivity of constituent phases, their volume fractions and two free parameters. As has been pointed out by Boccaccini the free parameters of the GEM equation do not have yet a clear correlation with the real features of the microstructure. In this paper the author discusses such correlation using connections among the GEM equation, percolation theory and effective medium theory.

Evaluating the Microstructure of Conductor-Insulator Composites using Effective Media and Percolation Theories

ABSTRACTThe widely used Bruggeman symmetric and asymmetric media equations and the Clausius-Mossetti approximation are presented, and the idealized microstructures characterizing these media are examined.The percolation equations for the conductivity (σm) and complex dielectric constant (ɛm = ɛrm + iɛim) for ideal perfect conductor–conductor and conductor–perfect insulator continuum media (and lattices) are then introduced. The parameters characterizing the microstructure in the percolation equations are a critical volume fraction or percolation threshold (φc) and two exponents (t&amp;s). The equations characterizing the crossover region, which occurs whenever the ratio of the conductivities of the two components is finite, or dielectric measurements are made, are also introduced. A general effective media equation, which has the same parameters as the percolation equations is noted, and its success in obtaining φc, when the ratio of the conductivities is finite, examined.The microstructures that allow φc to have values between a few percent and over 60%, as well as the occurrence of non–universal values of t, will also be discussed.