Wagner model

The rate of product formation can be predicted from a knowledge of the conductivity and transport numbers of cations in the product barrier phase and appropriate thermodynamic data. This interpretation is based on the Wagner model (Sect. 2.3). 

Oxidation of zinc to zinc oxide is another example whose kinetics have been interpreted in terms of the Wagner model (Wagner Grunewald, 1938). At 670 K, the reaction has been found to be independent of oxygen pressure between 0.02 and 1 atm. ZnO is a n-type semiconductor, having a stoichiometric excess of zinc accommodated as interstitials the defect equilibrium could be represented as... 

A study was made of the ability of viscoelastic models to describe the measured material functions of unplasticised PVC during extrusion and to determine whether it was possible to reproduce the elastic properties of the large entrance pressure drop and small extrudate swell during the extrusion of PVC using a capillary rheometer. Models used were the Phan-Thien and Tanner model and the K-BKZ-Wagner model with a single exponential damping... 

Considering these previous remarks, two network models, thought to be representative of each class of equation, have been investigated, namely the Wagner model and the Phan-Thien Tanner model,... 

A11 iat pral constitutive equation flie Wagner modeL 2 1.I1ie general K-BEZ modeL... 

XPEB1MENTAL validation of the WAGNER MODEL 4.1.Time>strain separabifity. 

Figure 6 shows the predictions of the Wagner model compared to experimental data for elongational viscosity, shear viscosity and first normal stress difference of LD. These have been calculated according to ... 

The previous set of comparisons shows that the Wagner model enables a good description of the experimental data in various experiments in simple shear and uniaxial elongation. However, it is worth pointing out that in most cases, there is a major difficulty concerning the determination of the parameters of the damping function. 

Two different constitutive equations, namely the Wagner model and the Phan Thien Tanner model, both based on network theories, have been investigated as far as their response to simple shear flow and uniaxial elongational flow is concerned. This work was primarily devoted to the determination of representative sets of parameters, that enable a correct description of the experimental data for three polyethylenes, to be used in numerical calculation in complex flows. Additionally, advantages and problems related to the use of these equations have been reviewed. 

R.G.Larson, K.Monroe, The BKZ as an alternative to the Wagner model for fitting shear and elongational behavior of a LDPE melt, Rheol. Acta 22 (1984), 10-13. 

Derived firom e Lodge s rubbeilike Hquid theory, the Wagner model is based on a concept of separabiHty, since it is assumed that the memory function is the product of a time-dependent hnear function by a strain-dependent nonlinear fimction. 

Another example with the Wagner model at a higher shear rate (y = 33s ) is presented in Fig. 27, for LDPE and LLDPE. For both materials, the final extrudate swell values are well-predicted. However, the computed radius close to the die exit increases faster than the experimental one and a constant value is reached in a shorter time than in the real case, especially with LDPE. 

Figure 27. Free surface computation for LDPE and LLDPE with stream-tube analysis and m Wagner model.

Figure 32. Comparison between theoretical and experimental extrudate swell.(o) GOB model ( ) PTT model (0,CD) Wagner model.

P. Hagenmuller, M. Pouchard and J.C. Grenier, Nonstroichiometry in the perovskite-type oxides An evolution from the classic Scottky-Wagner model to the recent high Tc superconductors. Solid State Ionics, 43 (1990) 7-18. 

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)] ... 

By means of the preliminary results it has been possible to thermodynamically model the presence of chlorine in nickel silicides. Chlorine-containing silicides may be taken as diluted solid solutions of chlorine in nickel silicides, which in the thermodynamic sense can be described by a modified Wagner model [5]. The molar free enthalpy of the solution of chlorine in silicide (Gm) is derived from Eq. 4. 

Eq. 4. The modified Wagner model [5] applied to chlorine-containing nickel silicides. 

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). ... 

---