Bruggeman factor

The assumption of local thermal equilibrium also means that an overall effective thermal conductivity is needed, because there is only a single energy equation. One way to calculate this thermal conductivity is to use Bruggeman factors. 

Physically, the tortuosity is defined as the ratio of the actual distance traversed by the species between two points to the shortest distance between those two points. By combining Eqs 9.7 and 9.9, the tortuosity can be evaluated. With the Bruggeman factor of 3.5, the pore path tortuosity representative of the resistance to oxygen diffusion is calculated around 4 and the proton path tortuosity indicative of the ion transport resistance through the electrolyte phase around 18. The high proton path tortuosity could be attributed to the significantly low volume fraction (e g. <20%) of the electrolyte phase typically considered in the standard PEFC catalyst layers. 

Table 2. These parameters refer to the Maxwell-Garnett (Eq.(6)) and Bruggeman calculations with a filling factor f 0.6. N denotes the geometrical factor. All other values are in eV.

A number of other surrogate procedures have been developed. These include the use of reverse-phase thin-layer chromatography (TLC) (Bruggeman et al. 1982 Renberg et al. 1985) to measure relative mobilities (Rm) or reverse-phase high-performance liquid chromatography (HPLC) to measure capacity factors (Veith et al. 1979b). These values are then correlated with experimentally established Pow values for standard compounds, and the correlation is then used to calculate values of Pow for the unknown compounds. 

The most popular effective medium theories are the Maxwell Garnett theory [18], which was derived from the classical scattering theory, and the Bruggeman theory [19]. With these theories, an effective dielectric function is calculated from the dielectric functions of both basic materials by using the volume filling factor. At some extensions of these theories, a unique particle shape for all particles is assumed. There is also an other concept based on borders for the effective dielectric functions. The borders are valid for a special nanostructure. Between these borders, the effective dielectric function varies depending on the nanostructure of the material. The Bergman theory includes a spectral density function g(x) that is used as fit function and correlates with the nanostructure of the material [20]. 

The HBG formulas, also known as the asymmetric Bruggeman formulas, are obtained assuming that the MG model is exact at low filling factors and then, following an iterative procedure, adding a small fraction of particles at each step [18, 19]. This model is recognized as valid at least for/< 0.8. 

The understanding of factors that lead to enhanced band intensities and dispersive band shapes is of central interest in studies with nanostructured electrodes. Effective medium theory has often been employed to identify mechanisms for enhanced infrared absorption [28, 128, 172, 174, 175]. Osawa and coworkers applied Maxwell-Garnett and Bruggeman effective medium models in early SEIRAS work [28, 128]. Recently, Ross and Aroca overviewed effective medium theory and discussed the advantages and disadvantages of different models for predicting characteristics of SEIRAS spectra [174]. When infrared measurements on nanostructured electrodes are performed by ATR sampling, as is typically the case in SEIRAS experiments, band intensity enhancements occur, but the band shapes are usually not obviously distorted. In contrast, external... 

Several factors may contribute to the substantial variability observed in measured BCF values, which may range over several orders of magnitude for the same compound for example, for pentachlorobenzene, BCF values between 900 and 250 000 have been reported (e.g. Bruggeman et a/., 1984 Gobas, Shiu and Mackay, 1987 Hawker, 1990). The evident variability in experimental BCF data may arise from ... 

Remarkably, the thermal conductivities of the composites filled with BP40 and BP80 are much higher than the prediction of Bruggeman equation. This demonstrates that the brush-like AlN particles enhance the thermal conductivity of the polymer matrix significantly. The intrinsic reason can be explored by Agari model [16], which considers the effect of dispersion state by introducing factors Cj and Cj ... 

Figure 7.8-1 shows plots of e/q versus z for the three models of Maxwell, Weisseberg and Bruggeman. These models are close to each other and they can be used to estimate the tortuosity factor when experimental value is not available. 

In the catalyst layer, the effective proton conductivity of the polymer electrolyte and the effective diffusion coefficient in the pores of the catalyst layer depend on the catalyst layer structure. These properties, and were given using the Bruggeman correction factor as follows ... 

It is well known that the Bruggeman EMA formula is derived by considering one of the constituents as a small sphere. A deviation from such an assumption required a modification the formula to include depolarization factor. Typically, a value of 0.333 is used as a default value in the EMA layer, which assumes a spherical shape of the inclusion. The other two extremes are 0, for a needle-like or columnar micro structure, and 1 for flat disks or a laminar microstructure. This type of transition was found for polyaniline/poly(methylmethacrylate) blend films presenting with a spherical-like microstructure at low sample concentration, whereas at relatively high concentrations the depolarization factor shifted to values closer to 1. This indicated the formation of flat microstructures due to aggregation of the polyaniline particles [8]. 

Fig. 8.61 Comparison of experimental data for permeability and ideal separation factor for COJ CH gas pair with the predicted values by the Maxwell and the Bruggeman model (Ultem 1000/ CMS MMMs). (From [23])...

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