Maxwell-Wagner dielectric model

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. 

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]. 

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. 

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)] ... 

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...

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 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. 

Interfacial or Maxwell-Wagner polarization is a special mechanism of dielectric polarization caused by charge build-up at the interfaces of different phases, characterized by different permittivities and conductivities. The simplest model is the bilayer dielectric [1,2], (see Fig. 1.) where this mechanism can be described by a simple Debye response (exponential current decay). The effective dielectric parameters (unrelaxed and relaxed permittivities, relaxation time and static conductivity) of the bilayer dielectric are functions of the dielectric parameters and of the relative amount of the constituent phases ... 

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 ... 

Figure 23 The dielectric constant and loss vs. frequency calculated by using different models for spherical particle case at different particle volume fraction marked in the graphs, a) Maxwell-Wagner-Sillars equation b) Bdtlcher-Hsu equation c) Looyenga equation. Parameters are c , =2, p=8, Om=10 S/m, CTp=10 S/m. The particle volume fraction changes from 0.1 to 0.9 with the interval 0.2. Reproduced with permission from G. Banhegyi, Colloid Polym. Sci., 266(1988)11.

In addition to the Debye model for dielectric bulk materials, other dielectric relaxations expressed according to Maxwell-Wagner or Schwartz "interfacial" mechanisms exist. For example, the Maxwell-Wagner "interfacial" polarization concept deals with processes at the interfaces between different components of an experimental system. Maxwell-Wagner polarization occurs... 

The Maxwell-Wagner dispersion effect due to conductance in parallel with capacitance for two ideal dielectric materials in series Rj Cj - Rj Cj can also be represented by Debye dispersion without postulating anything about dipole relaxation in dielectric. In the ideal case of zero conductivity for both dielectrics (R, — , R —> ), there is no charging of the interfaces from free charge carriers, and the relaxation can be modeled by a single capacitive relaxation-time constant. 

Conducting particles held in a nonconducting medium form a system which has a frequency-dependent dielectric constant. The dielectric loss in such a system depends upon the build-up of charges at the interfaces, and has been modeled for a simple system by Wagner [8], As the concentration of the conducting phase is increased, a point is reached where individual conducting areas contribute and this has been developed by Maxwell and Wagner in a two-layer capacitor model. Some success is claimed for the relation... 

In addition, the Wagncr-Maxwcll equation is able to adequately describe the dielectric property of ER fluids. Weiss [44] and Filisko [45] measured the dielectric constant and dielectric loss of ER fluids, and found that the Wagner-Maxwell equation is a suitable model to describe the dielectric property. The Wagner-Maxwell equation can also explain the frequency dependence of yield stress of ER fluids. As given in Figure 9 in Chapter 5, the frequency dependence of the yield stress and the dielectric constant follow a similar trend, further indicating that it is the Wagner-Maxwell polarization that controls the dielectric property of the ER suspension and then the ER effect. 

The conduction model is thought to be only valid for ER. suspensions in reaction with dc or low frequency ac fields. For high frequency ac fields, the polarization model is dominant [55,56]. As shown in Eq. (25) and (26), once the Wagncr-Maxwcll polarization is taken into account, the parameter P is detennined by the conductivity mismatch in dc or low frequency ac fields, and by the dielectric mismatch in high frequency fields (the low or high frequency is relative to the relaxation time of the Wagner-Maxwell polarization). The parameter p in the conduction model is ... 

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