Self-consistent Effective Medium Approximation Research Articles

Multiscale local porosity theory, weak limits, and dielectric response in composite and porous media

A mathematical scaling approach to macroscopic heterogeneity of composite and porous media is introduced. It is based on weak limits of uniformly bounded measurable functions. The limiting local porosity distributions that were introduced in the work [Adv. Chem. Phys. XCII, 299–424 (1996)] are found to be related to Young measures of a weakly convergent sequence of local volume fractions. The Young measures determine frequency dependent complex dielectric functions of multiscale media within a generalized self-consistent effective medium approximation. The approach separates scales by scale factor functions of regular variation. It renders upscaled results independent of the shape of averaging windows upon reaching the scaling limit.

On the effective medium model for particles with a complex structure

We consider a generalization of the effective medium approximation to the case of matrices containing macroscopically inhomogeneous particles with an arbitrary structure (cermet topology). The form of the result is determined to a considerable extent by the heuristic choice of effective cells used to estimate the field and induction values averaged over the volume. The simplest choice of a particle in an unperturbed field as a cell leads to the Maxwell–Garnett approximation, while the self-consistent effective medium approximation corresponds to the replacement of the unperturbed field by the mean field. As an example, we describe particles with a shell, as well as statistically anisotropic media with a single preferred direction.

Sound propagation in light-modulated carbon nanosponge suspensions

Single-walled carbon nanotube bundles dispersed in a highly polar fluid are found to agglomerate into a porous structure when exposed to low levels of laser radiation. The phototunable nanoscale porous structures provide an unusual way to control the acoustic properties of the suspension. Despite the high sound speed of the nanotubes, the measured speed of longitudinal-acoustic waves in the suspension decreases sharply with increasing bundle concentration. Two possible explanations for this reduction in sound speed are considered. One is simply that the sound speed decreases because of fluid heat induced by laser light absorption by the carbon nanotubes. The second is that this decrease results from the smaller sound velocity of fluid confined in a porous medium. Using a simplified description of convective heat transport, we estimate that the increase in temperature is too small to account for the observed decrease in sound velocity. To test the second possible explanation, we calculate the sound velocity in a porous medium, using a self-consistent effective-medium approximation. The results of this calculation agree qualitatively with experiment. In this case, the observed sound wave would be the analog of the slow compressional mode of porous solids at a structural length scale of order of 100 nm.

Extraordinary magnetoresistance in nonmagnetic semiconductors: The effective-medium approximation

The self-consistent effective-medium approximation is used to study the extraordinary magnetoresistance effect observed in nonmagnetic semiconductors. The inhomogeneous materials are treated as a three-dimensional resistor network of binary disorder, where the receptivity of each component is a tensor describing both the zero-field resistance and the Hall effect. The effective conductivity tensor of the total system is calculated with applied magnetic field. The resulting transversal magnetoresistance, longitudinal magnetoresistance and effective Hall coefficient are shown for different component concentrations and magnetic fields. When the components have two different types of charge carriers, and the mismatch between the zero-resistivity is enlarged, the macroscopic magnetoresisance exhibits complex behaviors which are related closely with the formation of the percolation structure in the inhomogeneous system.

The self-consistent effective medium approximation (SEMA): New tricks from an old dog

The fact that the self-consistent effective medium approximation (SEMA) leads to incorrect values for the percolation threshold, as well as for the critical exponents which characterize that threshold, has led to a decline in using that approximation. In this article I argue that SEMA has the unique capability, which is lacking in other approximation schemes for macroscopic response of composite media, of leading to the discovery or prediction of new critical points. This is due to the fact that SEMA can often lead to explicit equations for the macroscopic response of a composite medium, even when that medium has a rather complicated character. In such cases, the SEMA equations are usually coupled and nonlinear, often even transcendental in character. Thus there is no question of finding exact solutions. Nevertheless, a useful ansatz, leading to a closed form asymptotic solution, can often be made. In this way, singularities in the macroscopic response can be identified from a theoretical or mathematical treatment of the physical problem. This is demonstrated for two problems of magneto-transport in a composite medium, where the SEMA equations are solved using asymptotic analysis, leading to new types of critical points and critical behavior.

Effective-medium approach to the magnetic susceptibility of composites with ferromagnetic inclusions

A self-consistent effective-medium approximation is applied to the problem of ferromagnetic inclusions in the composite medium to determine the effective magnetic susceptibility. As the derivation utilizes the tensor form of ferromagnetic susceptibility, the obtained equations are valid in the vicinity of the ferromagnetic resonance, where the susceptibility of a ferromagnetic inclusion has significant off-diagonal components. In addition, the static magnetization of neighboring inclusions is taken into account through an effective magnetization of the effective medium. Three most interesting cases are examined, namely, a fully magnetized composite, a composite with single-domain particles, and a composite with demagnetized particles. The effective susceptibility of a magnetized composite is most interesting since the effective susceptibility has the tensor form, significantly varies with changing of the volume fraction of inclusions, which includes shifting of the resonance frequency, and differs from the noninteracting case even at very low volume fractions. In contrast, the effective susceptibility of a composite with isotropically oriented single-domain particles is scalar, without notable shift of resonance frequency, and varies only moderately with changing of the volume fraction. The case of composite with demagnetized inclusions is discussed only qualitatively since the derivation of effective susceptibility greatly depends on the particular physical state of the inclusions.

Self-consistent effective-medium approximation for strongly nonlinear media

A self-consistent effective-medium theory is proposed for random dielectric composites of arbitrary nonlinear constitutive law. It is based on a Gaussian approximation for the probability distributions of the electric field in each component, and on second-order Taylor expansions with an integral remainder of the local energies. The effective energy is exact to second order in contrast. With power-law media with constitutive relation $\mathbf{D}=\ensuremath{\chi}{E}^{\ensuremath{\gamma}\ensuremath{-}1}\mathbf{E},$ the critical exponents are $s=t=(\ensuremath{\gamma}+1)/2.$ The theory reduces to Bruggeman's in the linear case $\ensuremath{\gamma}=1,$ and its percolation threshold is independent of the nonlinearity.

Critical point in the strong-field magnetoresistance of a normal conductor/perfect insulator/perfect conductor composite with a random columnar microstructure

A recently developed self-consistent effective medium approximation, for composites with a columnar microstructure, is applied to such a three-constituent mixture of isotropic normal conductor, perfect insulator, and perfect conductor, where a strong magnetic field {\bf B} is present in the plane perpendicular to the columnar axis. When the insulating and perfectly conducting constituents do not percolate in that plane, the microstructure-induced in-plane magnetoresistance is found to saturate for large {\bf B}, if the volume fraction of the perfect conductor $p_S$ is greater than that of the perfect insulator $p_I$. By contrast, if $p_S<p_I$, that magnetoresistance keeps increasing as ${\bf B}^2$ without ever saturating. This abrupt change in the macroscopic response, which occurs when $p_S=p_I$, is a critical point, with the associated critical exponents and scaling behavior that are characteristic of such points. The physical reasons for the singular behavior of the macroscopic response are discussed. A new type of percolation process is apparently involved in this phenomenon.

High-field magnetotransport in composite conductors: Effective-medium approximation

The self consistent effective medium approximation (SEMA) is used to study three-dimensional random conducting composites under the influence of a strong magnetic field {\bf B}, in the case where all constituents exhibit isotropic response. Asymptotic analysis is used to obtain almost closed form results for the strong field magnetoresistance and Hall resistance in various types of two- and three-constituent isotropic mixtures for the entire range of compositions. Numerical solutions of the SEMA equations are also obtained, in some cases, and compared with those results. In two-constituent free-electron-metal/perfect-insulator mixtures, the magnetoresistance is asymptotically proportional to $|{\bf B}|$ at {\em all concentrations above the percolation threshold}. In three-constituent metal/insulator/superconductor mixtures a line of critical points is found, where the strong field magnetoresistance switches abruptly from saturating to non-saturating dependence on $|{\bf B}|$, at a certain value of the insulator-to-superconductor concentration ratio. This transition appears to be related to the phenomenon of anisotropic percolation.

Permittivity of lossy heterostructures: effect of shapeanisotropy

The complex values of the permittivity of two-component lossy(deterministic) heterostructures, composed of inclusions of permittivityε1 embedded in a host matrix of permittivity ε2, arerigorously evaluated with the use of the field calculation package PHI3D andthe resolution of boundary integral equations. Numerical results are providedconcerning spherical and rod-like inclusions with various radius-to-lengthratios, of finite conductivity, periodically arranged in a simple-cubiclattice configuration. For illustrative purposes, a single set ofpermittivities is investigated: ε1 = 80-i102 andε2 = 2-i0. The conduction threshold volume concentration isstrongly dependent on the shape of the inclusions. Increasing theradius-to-length ratio by one order of magnitude has the effect of shiftingthe conduction threshold upwards by two decades. The exponents which determinehow the real and imaginary parts of the effective permittivity scale with thedistance from the conduction threshold are determined and are compared withthe scaling predictions of the percolation theory for infinitethree-dimensional (random) lattices of insulator-normal metal compositesystems and the self-consistent effective-medium approximation. We also foundthat the data concerning the imaginary part of the effective permittivitycollapse on a single scaling plot over the range of aspect ratio investigated.The effect of the orientation of the rod-like inclusions is further studied.We observed that the conduction transition is shifted towards higherconcentrations as the angle between the rod axis and the direction of theapplied electric field increases.

Magnetotransport in conducting composite films with a disordered columnar microstructure and an in-plane magnetic field

Some exact relations are found between different elements of the bulk effective resistivity tensor of a two-component composite medium with a columnar microstructure. A self-consistent effective-medium approximation is constructed that incorporates those relations. This is then used to conduct a theoretical study of magnetotransport in such a medium, which can be implemented experimentally as a thin conducting film with perpendicular cylindrical inclusions. Detailed, explicit results are obtained for a conductor/insulator random mixture and a normal conductor/perfect conductor random mixture that are above the two-dimensional percolation threshold of the normal conducting constituent. At large magnetic fields B, the in-plane magnetoresistance components increase as ${\mathbf{B}}^{2}$ without any saturation in mixtures of the first kind, but tend to constant asymptotic values in mixtures of the second kind. The Clausius-Mossotti or Maxwell Garnett type of approximation, which predicts a different behavior, is shown to be unreliable at strong magnetic fields even for very dilute mixtures.

Effective Properties of Ferroelectric and/or Ferromagnetic Composites: A Unified Approach and Its Application

A recently developed multiple‐scattering approach for calculating the coupled electrical–mechanical response of composites is generalized to treat more‐complicated coupling between the constituent phases. This provides a unified theoretical framework for determining the effective properties of ferroelectric and/or ferromagnetic composites with any multifield coupled interaction between electrical, magnetic, mechanical, and thermal effects. Among the advantages of the theory is its simplicity of formulation, which is suited to computation. The common non‐self‐consistent approximation (NSCA) and self‐consistent effective medium approximation (SCEMA) of the theory are discussed through application to a number of piezoelectric, pyroelectricpyromagnetic, and magnetoelectric composites.

Effect of Interfacial Rheology on Foams Viscoelasticity, an Effective Medium Approach

We have computed in 3-dimensional foams the influence of interfacial rheology on the macroscopic foam viscoelasticity. We show that the two interfacial moduli play different roles. The dilational viscosity is not at all additive to the macroscopic modulus, while both the interfacial shear viscosity and the internal phase modulus are nearly additive. The dilational viscosity is shunted by a grid mode at high frequency. The calculations are done in a selfconsistent effective medium approximation.

The theory of dielectric functions and its application to amorphous Si 1- xN x

The similarity of shape between the dielectric function of a-Si and its amorphous alloys has led to the development of a scaling theory to describe these functions. The dielectric theory of Phillips is used to derive the scaling parameters for several models of the structure of amorphous silicon alloys, combining the contributions of the microscopic components using a self-consistent effective medium approximation. A homogeneous medium theory and two microscopic theories, one based on bonds and the other on tetrahedral units, are considered; the occurrence of voids is also considered. Comparison of predictions with experimental data for several thin film samples of amorphous Si 1- x N x alloys, taken over the photon energy range 0–12 eV, is reported. The homogeneous medium theory clearly fails to describe the data. The two microscopic theories both describe them fairly well, the tetrahedron model being somewhat better.

Quasi-one-dimensional percolation model for the conductivity of doped organic semiconductors

A quasi-one-dimensional percolation model for the conductivity of a doped organic semiconductor is proposed. A selfconsistent effective-medium approximation is used. The weak magnetic field effect in the conductivity is considered.

Effective medium approximation for flicker noise and weak nonlinearities in composite media

An unambiguous formulation of the self-consistent effective medium approximation (SEMA) is presented and then applied to the problem of flicker noise in a composite conductor where each component has a small nonlinear (power law) correction to the Ohmic conductivity. The results, while reasonable for dilute systems, lead to seemingly strange results (e.g., infinite flicker noise and infinite nonlinear conductivity) for a nondilute metal-insulator mixture. These are explained in terms of the special microstructure for which SEMA is exact and demonstrate the great sensitivity of flicker noise and nonlinear conductivity to microstructural details.

Hopping transport on lattices with random traps

The particle diffusion coefficient and the occupation distribution function for the ensemble of sites with definite energetic levels for random walks, starting from a fixed level (the case of selective initial excitation) are studied in the framework of a self-consistent effective medium approximation. In the long time limit the dispersions of the diffusion coefficient and of the occupation function are found, that reveal a slow power law decrease of these parameters fluctuations.

Effective medium approximation for elastic constants of porous solids with microscopic heterogeneity

Formulas for the scattering from an inhomogeneous sphere in a fluid-saturated porous medium are used to construct a self-consistent effective medium approximation for the coefficients in Biot’s equations of poroelasticity [J. Acoust. Soc. Am. 28, 168 (1956)] when the material constituting the porous solid frame is not homogeneous on the microscopic scale. The discussion is restricted to porous materials exhibiting both macroscopic and microscopic isotropy. Brown and Korringa [Geophysics 40, 608 (1975)] have previously found the general form of these coefficients. The present results give explicit estimates of all the coefficients in terms of the moduli of the solid constituents. The results are also shown to be completely consistent with the well-known results of Gassmann and of Biot and Willis, as well as those of Brown and Korringa.

Hopping transport on anisotropic random-bond lattices

Hopping transport on anisotropic perlolating lattices is considered in the framework of a self-consistent effective medium approximation. The particle diffusion coefficient is studied as a function of conducting bond concentrations and lattice anisotropy. The dependence of the total critical concentration on anisotropy is considered also for a Caylee tree.

Effective-Medium Approximation for Diffusion on a Random Lattice

A self-consistent effective-medium approximation is presented for the problem of diffusion and ac conductivity on a lattice characterized by random values of transfer rates between pairs of nearest-neighbor sites. The approximation is applied to a percolation model in which only a fraction of the bonds are assigned a finite transfer rate. The results reflect the existence of a percolation threshold in the system, and are consistent with the properties of clusters of bonds in the critical region.