Maxwell-Wagner model

In heterogeneous systems, an interfacial polarisation is Created due to the space charges. This polarisation corresponds to the electron motion inside conductive charges, dispersed in an insulated matrice (Maxwell-Wagner Model). In fact, this phenomenon will appear as soon as two materials I and 2 are mixed so that c7]/ei C2le.2 with a conductivity and e dielectric constant at zero frequency [ 123]. 

The value of e" polarisation is related to the molecular polarisation phenomena such as dipole rotation (Debye model), space charge relaxation (Maxwell-Wagner model), hopping of confined charges [42,126,127]. The physical origin of this polarisation term is often ambiguous [42],... 

For analysis of the dielectric properties of blood-cell suspensions, several classical models are usually used (11,14, 185-201). For small volume fractions of cells the Maxwell-Wagner model is used, while for larger ones (see Sec. II) the Hanai formula wouldbe preferable (14,186). It was shown (70, 72) that for dilute suspensions of human blood cells the dielectric spectra of a single cell can be successfully calculated from the Maxwell model of suspension, according to the mixture formula [Eq. (19)] ... 

This is the Maxwell—Wagner model of a capacitor with two dielectric layers. Even with only two layers, the equations are complicated with three layers, they become much... 

The first exact expression of this type was derived by Maxwell [1881] for the dc conductivity of a dispersion of spheres in a continnons medinm. Maxwell Garnett [1904] derived a similar expression for dielectric and optical properties. Wagner [1914] extended Maxwell s model to the complex domain and this model has thereafter been known as the Maxwell-Wagner model. It gives the following expression for complex conductivity ... 

Brailsford and Hohnke [1983] have applied the Maxwell-Wagner model to grain boundaries in two-phase systems. Their microstructural model, shown in Figure 4.1.8fc, consists of a spherical grain of radius r2 surrounded by a shell of outer radius a, which represents the grain boundary and has a volume fraction Xi = 1 - (jjrif. The authors observe that for Xi —> 0 and 1/ 2 V i the effective medium model becomes identical to case (i) of the brick layer model, namely Eq. (6). Further, we have found that for x —> 0 and y/i xffz, it reduces to case (ii) of the brick layer model, namely Eq. (7). ... 

Inspection of Figure 4.1.9 suggests that Eq. (20) generates spectra that are similar to those of simple RC circuits. This is indeed the case. In fact, Bonanos and Lilley [1981] showed that the Maxwell-Wagner model is formally identical to the two-element circuit of Figure 4.1.1, but with values of gi, g2, Ci, and C2 that can be expressed as rather complicated functions of Oi, [Pg.218]

Evaluation of this expression generates a spectrum rather similar to that for the Maxwell-Wagner model, but with a different weighting of the two phases. Figure 4.1.12 shows a modulus spectrum for the same input parameters as those that were used to produce the spectrum in Figure 4.1.9fc. 

Keywords impedance spectroscopy, meat aging, Fricke model, Maxwel Wagner, model parameters, circular electrode... 

Generally, the literature shows that, for magnetic composite systems, relative permittivity decreases as frequency increases, this behavior being attributed to the interfacial polarization predicted by Maxwell-Wagner model, derived for the permittivity of a homogeneous two-phase system. In addition, a decrease in electrical resistivity for these magnetic composites was recorded with an increase in frequency... 

The application of an electric field can be assessed by applying the Maxwell-Wagner model, in which the composite is approximated as a two-layer capacitor (Fig. 6.8). 

The Maxwell-Wagner model describes dispersions and composites as conducting spheres suspended in a continuous insulating medium [4, pp. 192-198]. Several different models for a two-phase microstructure are possible ... 

This approximation shows that for small O, absence of agglomeration, and Ep, the absolute permittivity of the colloidal medium e is independent of Ep. For these conditions every 1% increase in concentration of conducting particles 4> increases the total system high-frequency permittivity by 3%. A similar derivation by Schwarz [6] based on the Maxwell-Wagner model [7] resulted in an alternative expression ... 

Two other approaches have been taken to modelling the conductivity of composites, effective medium theories (Landauer, 1978) and computer simulation. In the effective medium approach the properties of the composite are determined by a combination of the properties of the two components. Treating a composite containing spherical inclusions as a series combination of slabs of the component materials leads to the Maxwell-Wagner relations, see Section 3.6.1. Treating the composite as a mixture of spherical particles with a broad size distribution in order to minimise voids leads to the equation ... 

For the very simplified situation that the sphere behaves electrically as a pure capacitor, and the solution as a pure resistance, the relaxation can be described by a Maxwell-Wagner mechanism, with T = e e/K, see (1.6.6.321. Although some success has been claimed by Watillon s group J to apply this mechanism for a model, consisting of shells with different values of e and K, generally a more detailed double layer picture is needed. In fact, this Implies stealing from the transport equations of secs. 4.6a and b. generedizing these to the case of a.c. fields. 

Another model of experimental interest concerns the case of a highly conductive shell around practically non-conductive material. It may be applied to macromolecules or colloidal particles in electrolyte solution which usually have counterion atmospheres so that the field may displace freely movable ionic charges on their surfaces. The resulting dielectric effect turns out to be equivalent to a simple Maxwell-Wagner dispersion of particles having an apparent bulk conductivity of... 

The three phase dielectric system backbone-waterlayer-air of a real RF aerogel is reduced to a two layer system. The third phase (air) is neglected because of its relative low influence (compared to the other two phases) on the compound dielectric permittivity according to its own material parameters e and k. In order to explain the measured spectra by Maxwell-Wagner polarization processes due to the absorbed water we propose the following model. ... 

Interfacial polarization in biphasic dielectrics was first described by Maxwell (same Maxwell as the Maxwell model) in his monograph Electricity and Magnetism of 1892.12 Somewhat later the effect was described by Wagner in terms of the polarization of a two-layer dielectric in a capacitor and showed that the polarization of isolated spheres was similar. Other more complex geometries (ellipsoids, rods) were considered by Sillars as a result, interfacial polarization is often called the Maxwell-Wagner-Sillars (MWS) effect. 

Maxwell model A mechanical model for simple linear viscoelastic behavior that consists of a spring of Young s modulus E) in series with a dashpot of coefficient of viscosity (ri). It is an isostress model (with stress 8), the strain (e) being the sum of the individual strains in the spring and dashpot. This leads to a differential representation of linear viscoelasticity as stress relaxation and creep with Newtonian flow analysis. Also called Maxwell fluid model. See stress relaxation viscoelasticity. Maxwell-Wagner efifect See dielectric, Maxwell-Wagner effect. 

The whole procedure of the human blood-cell suspension study is presented schematically in Fig. 50. The TDS measurements on the cell suspension, the volume-fraction measurement of this suspension, and measurements of cell radius are excecuted during each experiment on the sample. The electrode-polarization correction (see Sec. II) is performed at flie stage of data treatment (in the time domain) and then the suspension spectrum is obtained. The singlecell spectrum is calculated by the Maxwell-Wagner mixture formula [Eq. (88)], using the measured cell radius and volume fraction. This spectrum is then fitted to the single-shell model [Eq. (89)] in the case of erythrocytes or to the double-shell model [Eqs (94)-(98)] to obtain flie cell-phase parameters of lymphocytes. 

The erythrocyte and erythrocyte ghost suspensions are very similar systems. They differ in their inner solution (in the case of erythrocytes it is an ionic hemoglobin solution in the case of ghosts it is almost like the surrounding solution they were in while they were sealed). The cell sizes in a prepared suspension depend both on the ion concentration in the supernatant and in the cell interior (70). Thus, the dielectric spectra of erythrocytes and erythrocyte ghost suspensions have the same shape, which means that there are no additional (except Maxwell-Wagner) relaxation processes in the erythrocyte cytoplasm thus, the singleshell model (Eq. 89) can be applied. 

Figure 3.9 Equivalent circuits for the Maxwell—Wagner effect in a simple dielectric model, (a) The slabs in series, and the resistors cause the interface to be charged, (b) The slabs...

The models of Figures 4.30—4.32 will exhibit Maxwell—Wagner dispersion. The anisotropy of Figure 4.32 disappears at high frequencies because the capacitive membranes are short-circuited. For example, when anisotropy is caused by air in the lungs, the anisotropy may persist at virtually all frequencies. 

The nanocomposites showed an average grain size of 37 nm. SEM studies indicated the preponderance of spherical particles embedded in the matrix. The relaxation behavior can be explained by Maxwell-Wagner two-layered dielectric models. 

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