Self-consistent Effective Medium Research Articles

Effective medium method in solution of the elastic and thermo-elastic homogenization problems for polycrystals with spheroidal transverse isotropic grains

Self-consistent effective medium method is used for determination of the effective elastic and thermo-elastic properties of polycrystalline materials with spheroidal transverse isotropic grains. The grains with arbitrary aspect ratios including very oblate (disks) and very prolate (fibers) spheroids are considered. The function of the Owens-March type is used for description of distribution of grains over orientations. The equations for the effective elastic stiffness tensor and the tensor of effective coefficients of thermo-expansion of polycrystals are derived using integral equations for elastic and thermo-elastic fields in heterogeneous media and the hypotheses of the effective medium method. The numerical algorithm of the method is described in detail. Numerical examples are presented for polycrystalline clay rock materials with strongly anisotropic grains and for polycrystalline zinc. Graphs of the dependences of the effective elastic constants and effective coefficients of thermo-expansion of the polycrystals on the parameters of the orientation distribution function and aspect ratios of grains are presented.

Understanding the electrical transports of p-type polycrystalline SnSe with effective medium theory

SnSe crystal is one of the most potential thermoelectric materials due to its excellent transport properties. The electrical conductivity of p-type SnSe crystal gradually decreases with increasing temperature, while that of the polycrystalline sample shows a completely different trend. We revealed that below 400 K, the existence of plentiful grain boundaries dominates the carrier scattering and determines the electrical transport of p-type polycrystalline SnSe, while at high temperatures, from 400 to 800 K, the electrical transport and conductivity curve still requires a clear understanding. In this study, by conducting the high-temperature synchrotron radiation x-ray diffraction (SR-XRD) measurements and refining the patterns, we obtained the phase fractions of Pnma and Cmcm phases in this temperature range. Using the derived single-phase theoretical electrical conductivity, combined with the self-consistent effective medium theory, the electrical conductivity of p-type polycrystalline SnSe between 400 and 800 K was simulated. This study provides a perspective and simulation method to understand the electrical transport of p-type polycrystalline SnSe.

Effective thermal conductivity of particulate composites for time-varying temperature fields

A homogeneous conductive medium containing random sets of identical spherical inclusions is considered. The self-consistent effective medium method is used for solution of the homogenization problem and calculation of the effective thermal conductivity parameters of the composite. For time-varying fields, the effective conductivity parameters are non-local operators with respect to time. The effective medium method allows reducing the homogenization problem to solution of a system of non-linear equations for the Laplace transforms of the kernels of these operators. An efficient numerical algorithm of solution of this system is proposed. Examples of construction of these operators for the composites with copper and glass phases are considered. The Green functions of the corresponding effective media are compared with the Green function of the media with static effective parameters of thermal conductivity.

Amount of gas hydrate estimated from rock physics analysis based on morphology and intrinsic anisotropy in area B, Krishna Godavari offshore basin, expedition NGHP-02

India's National Gas Hydrate Program (NGHP) Expedition-02 was conducted in 2015 to investigate the prospective locations of sand-rich gas hydrate reservoirs along the eastern offshore, India. We select Area-B among four areas A, B, C and E of expedition NGHP-02 as it was completely unexplored during NGHP-01. Logging while Drilling (LWD) data were collected at twelve sites in area B in Krishna Godavari (KG) offshore basin, where a huge amount of gas hydrate (up to 90% of the pore volume) is present as pore-fill, fracture-fill and both pore- and fracture-filled (mixed) morphology. The resistivity log generally overestimates gas hydrate saturation because of the increase of resistivity of fresh water near gas hydrate accumulation and also in the presence of fractured-filled gas hydrate. Though gas hydrate saturation obtained from pressure cores are assumed to be ground truth but they are discrete. Hence, the application of an advanced rock physics model is required to obtain continuous quantification of gas hydrate throughout the depth considering the gas hydrate morphology and microstructure of sediment. We apply a combination of self-consistent approximation (SCA) and differential effective medium (DEM) theories to model biconnected marine gas hydrate reservoir that can deal with inherent anisotropy due to the presence of clay platelets and pore-filling, fracture-filling or mixture of both morphologies of gas hydrate. First, we determine the distribution pattern of clay platelets from the observed background velocity (water-saturated sediment) and then estimate the gas hydrate saturation at four holes of area B, where pressure core measurements are available. We predict maximum gas hydrate saturation as 50–90% of the pore space above the BSR in this area. Comparison with other rock physics model shows that prediction from the combined SCA-DEM theory matches better with the pressure core measurements.

Elastic Anisotropy Modeling of Organic-Rich Lower Gondwana Shale in Eastern India

Analysis of cores from the Lower Gondwana basin in eastern India confirms the presence of abundant organic matter, indicating a high shale gas prospective zone. Shale is intrinsically (transversely) anisotropic due to the preferential orientation of anisotropic clay platelets depending on the geological processes of deposition. Such anisotropy needs to be incorporated in modeling for an accurate explanation of observed seismic anisotropy in shale. Combined anisotropic formulations of self-consistent approximation (SCA) and differential effective medium (DEM) theory is the most suitable among existing theories to deal with the complex microstructure of shale. We start by predicting the bulk and rigidity moduli of clay mineral aggregates as 21 GPa and 10 GPa from our model, which are difficult to measure in the laboratory due to the small crystal size of clay. A concept of host medium (HM) is presented here, which is constituted of clay mineral aggregates, kerogen, and unconnected pores. Other minerals are added in the biconnected composite constituted of HM and connected pores filled with water. An orientation distribution function (ODF) of clay particles is determined using the combined SCA-DEM theory from the observed ultrasonic velocity measurements. Our model shows strong intrinsic anisotropy at the shallow depth that decreases with depth because of the changes in the microstructure of the shale. The P-velocity predicted from our model, widely used Biot–Gassmann theory (BGT) and Biot–Gassmann theory modified by Lee (BGTL) match well with the measured data where P-wave anisotropy is insignificant. We also predict from our model the volume of kerogen and total organic carbon as 26–43% and 6–8%, respectively.

Effect of magnetic-field-induced restructuring on the elastic properties of magnetoactive elastomers

Composite materials where magnetic micrometer-sized particles are embedded into a compliant polymer matrix are known as magnetorheological (or magnetoactive) elastomers (MAEs). They are distinguished by huge variations in their physical properties, when in a magnetic field, which is commonly attributed to the restructuring of the filler. The process of the magnetic-field-induced restructuring in a magnetorheological elastomer is interpreted as progression towards percolation. Such a physical model was previously used to explain the dependence of the magnetic permeability and dielectric permittivity of MAEs on the magnetic field strength. Based on this hypothesis, the magnetorheological effect in MAEs is considered theoretically. The theoretical approach is built upon a self-consistent effective-medium theory for the elastic properties, extended to the variable (field dependent) percolation threshold. The proposed model allows one to describe the large variations (over several orders of magnitude) of the effective elastic moduli of these composite materials, known as the giant magnetorheological (MR) and field-stiffening effects. The existence of a giant magnetic Poisson effect is predicted. The relation of the proposed model to the existing theories of the MR effect in MAEs is discussed. The results can be useful for applications of MAEs in magnetic-field-controlled vibration dampers and isolators.

Effective dynamic properties of random complex media with spherical particles.

The effective dynamic bulk modulus and density are presented for random media consisting of particles in a viscous host fluid, using a core-shell, self-consistent effective medium model, under the large compressional wavelength assumption. These properties are relevant to acoustic or dynamic processes in nano- and micro-particle fluids including particle density determination, resonant acoustic mixing, and acoustic characterisation. Analytical expressions are obtained for the effective bulk modulus and mass density, incorporating the viscous nature of the fluid host into the core-shell model through wave mode conversion phenomena. The effective density is derived in terms of particle concentration, particle and host densities, particle size, and the acoustic and shear wavenumbers of the liquid host. The analytical expressions obtained agree with prior known results in the limit of both static and inviscid cases; the ratio of the effective bulk modulus to that of the fluid is found to be quasi-static. Numerical calculations demonstrate the dependence of the effective mass density on frequency, particle size (from nano- to micro-regime), and concentration. Herein it is demonstrated both theoretically and numerically that the viscosity, often neglected in the literature, indeed plays a significant role in the effective properties of nanofluids.

Designing BaTiO3/carbon nanotube core-shell hybrids to achieve polyvinylidene fluoride composites with high dielectric properties

The synthesized core–shell hybrids, BaTiO3 (shell) @ carbon nanotubes (core), (BT@CNT), can make the polyvinylidene fluoride (PVDF) matrix composite to exhibit high dielectric permittivity (ε′) but low dielectric loss (tan δ) when approaching the percolation threshold. High ε′ is attributed to BT coating which can strengthen the interfacial polarization in the composite. Low tan δ is attributed to the formation of the structure: CNT → BT → PVDF with the gradually decreasing dielectric properties from the centers of hybrid to the surrounding borders of PVDF. Additionally, AC conductivity of the composites can be deduced by the self-consistent effective medium (SCEM) approximation, which helps to understand the conductive behavior of CNT with a ceramic buffer layer in PVDF matrix more comprehensively.

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.

Modeling large permittivity of poly(vinylidenefluoride-co-trifluoroethylene) and nanospring single-walled carbon nanotube-polyvinylpyrrolidone nanocomposites

Highly dispersible nanospring single-walled carbon nanotubes (NS-CNTs) were incorporated in a P(VDF-TrFE) copolymer with up to 15 wt.% of nanofiller. The relative dielectric constant (K) of the polymer nanocomposite at 1 kHz was greatly enhanced from 12.7 to 62.5 at 11 wt.% of NS-CNTs, corresponding to a 492% increase over that of pristine P(VDF-TrFE) with only a small dielectric loss tangent (D) of 0.1. Based on two theoretical models, the Bruggeman equation and self-consistent effective medium theory (SC-EMT), experimental permittivity data for the P(VDF-TrFE) and NS-CNTs nanocomposites were simulated to estimate the dielectric constant of the NS-CNTs while changing both the shape of the nanofillers and the volume fraction of the interface when increasing the number of NS-CNTs in piled layers of P(VDF-TrFE). The number of NS-CNTs layers was counted from HR-TEM images to calculate the interfacial volume fraction, and used to infer the Eshelby tensor of the NS-CNTs in the SC-EMT model. The experimental dielectric constants of the composite films fit the Bruggeman equation and SC-EMT theory well for dielectric constants k=240–360, showing that the NS-CNTs nanofillers may be considered electrically semiconductive.

BC/rGO conductive nanocomposite aerogel as a strain sensor

In this work, bacterial cellulose (BC)/reduced graphene oxide (rGO) as a new class of electrical conductive nanocomposite aerogels were in-situ synthesized and dried by the supercritical CO2 (ScCO2) drying method. A modification has been performed on the self-consistent effective medium theory to predict overall electrical conductivity of nanocomposite aerogels. It was observed that the porosity had a distinct effect on the overall electrical conductivity of nanocomposite aerogel beneath percolation threshold, but had no effect beyond it. The latter observation was attributed to the formation of continuous pathway associated with electron tunneling effect, beyond percolation threshold. The results also indicated that in the presence of rGO, the density, specific surface area and Young's modulus of nanocomposite aerogel increased up to 0.025 g/cm3, 252 m2/g and 2020 MPa, respectively. Also, the gauge factor of 19 obtained for BC/rGO nanocomposite aerogel indicated the potency of fabricated nanocomposite aerogel as a strain sensor.

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.

Transport properties of multigrained nanocomposites with imperfect interfaces

Multigrained or polycrystalline composite materials have attracted a considerable attention due to their potential applications as advanced materials with outstanding thermal, mechanical, and electromagnetic properties. When the grains' morphology is displayed at the nanoscopic scale, the presence of imperfect interfaces plays a central role in determining the effective transport properties. Therefore, we develop here a self-consistent effective medium theory able to evaluate the influence of real contacts between the different phases of multigrained composite materials. This approach takes into account the classical interface schemes that have been introduced in literature, namely, the low and the high conducting interface models. The theoretical results have been compared with numerical and experimental data concerning the thermal conductivity of (1−x)Si:xGe mixtures and the electrical conductivity of (1−x)Li2O:xB2O3 composites.

Inclusion-Based Effective Medium Models for the Permeability of a 3D Fractured Rock Mass

Effective permeability is an essential parameter for describing fluid flow through fractured rock masses. This study investigates the ability of classical inclusion-based effective medium models (following the work of Sævik et al. in Transp Porous Media 100(1):115–142, 2013. doi: 10.1007/s11242-013-0208-0 ) to predict this permeability, which depends on several geometric properties of the fractures/networks. This is achieved by comparison of various effective medium models, such as the symmetric and asymmetric self-consistent schemes, the differential scheme, and Maxwell’s method, with the results of explicit numerical simulations of mono- and poly-disperse isotropic fracture networks embedded in a permeable rock matrix. Comparisons are also made with the Hashin–Shtrikman bounds, Snow’s model, and Mourzenko’s heuristic model (Mourzenko et al. in Phys Rev E 84:036–307, 2011. doi: 10.1103/PhysRevE.84.036307 ). This problem is characterised by two small parameters, the aspect ratio of the spheroidal fractures, $$\alpha $$ , and the ratio between matrix and fracture permeability, $$\kappa $$ . Two different regimes can be identified, corresponding to $$\alpha /\kappa <1$$ and $$\alpha /\kappa >1$$ . The lower the value of $$\alpha /\kappa $$ , the more significant is flow through the matrix. Due to differing flow patterns, the dependence of effective permeability on fracture density differs in the two regimes. When $$\alpha /\kappa \gg 1$$ , a distinct percolation threshold is observed, whereas for $$\alpha /\kappa \ll 1$$ , the matrix is sufficiently transmissive that such a transition is not observed. The self-consistent effective medium methods show good accuracy for both mono- and polydisperse isotropic fracture networks. Mourzenko’s equation is very accurate, particularly for monodisperse networks. Finally, it is shown that Snow’s model essentially coincides with the Hashin–Shtrikman upper bound.

Saturation effects on the joint elastic–dielectric properties of carbonates

We used a common microstructural model to investigate the cross-property relations between elastic wave velocities and dielectric permittivity in carbonate rocks. A unified model based on validated self-consistent effective medium theory was used to quantify the effects of porosity and water saturation on both elastic properties (compressional and shear wave velocities) and electromagnetic properties (dielectric permittivity). The results of the forward models are presented as a series of cross-plots covering a wide range of porosities and water saturations and for microstructures that correspond to different predominant aspect ratios. It was found that dielectric permittivity correlated approximately linearly with elastic wave velocity at each saturation stage, with slopes varying gradually from positive at low saturation conditions to negative at higher saturations. The differing sensitivities of the elastic and dielectric rock properties to changes in porosity, pore morphology and water saturation can be used to reduce uncertainty in subsurface fluid saturation estimation when co-located sonic and dielectric surveys are available. The joint approach is useful for cross-validation of rock physics models for analysing pore structure and saturation effects on elastic and dielectric responses.

Iterative approaches to extend dilute homogenization predictions of Willis materials to higher volume fractions

Dynamic homogenization of generally heterogeneous elastic materials leads to three different effective material properties: stiffness, density, and parameters that couple momentum to strain and stress to velocity. This formulation was first proposed by Willis [Wave Mot. 3, 1–11, (1981)] and is thus commonly referred to as Willis coupling. While there are some homogenization methods which account for Willis coupling, those models rely on either the assumption of periodic media or on dilute concentrations of coupled inhomogeneities in an otherwise homogeneous matrix. This work presents the use of dilute models with iterative micromechanical methods to predict general material properties in materials with no long-range order and elevated volume fractions of inhomogeneity. Two such methods, self-consistent and differential effective medium models, will be presented for the homogenization of composites of arbitrary anisotropy and Willis coupling resulting from the presence of Willis coupled inclusions in an el...

Anisotropic rock physics models for interpreting pore structures in carbonate reservoirs

We developed an anisotropic effective theoretical model for modeling the elastic behavior of anisotropic carbonate reservoirs by combining the anisotropic self-consistent approximation and differential effective medium models. By analyzing the measured data from carbonate samples in the TL area, a carbonate pore-structure model for estimating the elastic parameters of carbonate rocks is proposed, which is a prerequisite in the analysis of carbonate reservoirs. A workflow for determining elastic properties of carbonate reservoirs is established in terms of the anisotropic effective theoretical model and the pore-structure model. We performed numerical experiments and compared the theoretical prediction and measured data. The result of the comparison suggests that the proposed anisotropic effective theoretical model can account for the relation between velocity and porosity in carbonate reservoirs. The model forms the basis for developing new tools for predicting and evaluating the properties of carbonate reservoirs.

Effective medium modelling the effects of saturation on the joint elastic-dielectric properties of carbonates

SUMMARY The effects of saturation on the joint elastic-dielectric properties of porous medium is important for the understanding of elastic and electromagnetic wave propagation phenomena as well as quantifying hydrocarbon content in partially saturated reservoir rocks. We studied theoretically for the first time the crossproperty relations between elastic velocity and dielectric permittivity (the joint elastic-dielectric properties) of carbonates with a unified microstructure. The effects of porosity and water saturation on the joint elastic-dielectric properties were also studied using validated self-consistent effective medium models for elastic velocity and dielectric permittivity. The results offered an important new possibility for estimating in situ carbonate porosity and hydrocarbon saturation using joint velocity-permittivity crossplots from co-located sonic and dielectric surveys.

First-principles molecular spin dynamics study on the magnetic structure of Mn-based alloys with Cu3Au-type crystal structure

The magnetic and electronic structures of Mn3Pt and Mn3Rh, which are three-dimensional frustrated itinerant magnets with a Cu3Au-type crystal structure, have been investigated by means of the first-principles Molecular Spin Dynamics (MSD) method. The theory is based on the first-principles tight-binding linear muffin-tin orbital Hamiltonian combined with the functional integral method and the isothermal MSD technique, and allows us to determine automatically the magnetic structures of itinerant magnets at finite temperatures. The MSD calculations using a self-consistent site-dependent effective medium show that below the Néel temperature Mn3Pt with fixed crystal structure (Cu3Au structure) and volume exhibits a second-order transition from a triangular structure to another noncollinear phase with increasing temperature. Mn3Rh, on the other hand, shows no sign of a phase transition up to the Néel temperature. We found that the Mn-Eg DOS peak, which is responsible for the ferromagnetic couplings among the second nearest-neighbor Mn local moments, develops at the Fermi energy (EF) around 350K for Mn3Pt, while the peak development for Mn3Rh occurs with increasing temperature slightly above EF.

Dielectric behaviors of PHBHHx–BaTiO 3 multifunctional composite films

Multifunctional polymeric composites were investigated for potential use in high energy storage capacitors and tissue engineering. The polymeric composites were fabricated by employing biodegradable polyester, poly(3-hydroxybutyrate-co-3-hydroxyhexanoate) (PHBHHx), as the matrix. Ferroelectric BaTiO 3 ceramic powders were added to the composites as fillers. The dielectric spectroscopy of the composites was measured over a wide frequency range (10 0–10 7 Hz) from −100 to 60 °C. The composition dependent dielectric behavior was modeled by a self-consistent effective medium theory. A percolation threshold of 0.367 was observed in the composites. The glass transition relaxation of the composite was also discussed by comparing the popular Vögel–Fulcher–Tammann law with a new glass model. The composites show attractive ferroelectricity and piezoelectricity for biomedical applications.