Bruggeman formula

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. 

Several well-known dielectric equations are selected for this modeling. The two-phase mixtures are also represented by the Bottcher-Bruggeman formula (Bottcher, 1942) based on the spherical particle model where the filler is interacting with polymer. According to this formulae... 

Analysis of a few dozens of films obtained from a single or a double coating process was performed. The refi ction index of SN voids free material was determined to be n = 1.62, and for HP A compound n = 1.56 at wavelength 632.8 nm. These values and the Bruggeman formula [5] was used to calculate the values of refinction index and the corresponding values of voids given in Fig 1. 

The most often applied effective medium model is the Bruggeman formula [11]... 

The generalized Hanai-Bruggeman formula (11.7) for ellipsoidic inclusions and applied on a porous rock is ... 

The effective medium theory consists in considering the real medium, which is quite complex, as a fictitious model medium (the effective medium) of identical properties. Bruggeman [29] had proposed a relation linking the dielectric permittivity of the medium to the volumetric proportions of each component of the medium, including the air through the porosity of the powder mixture. This formula has been rearranged under a symmetrical form by Landauer (see Eq. (8), where e, is the permittivity of powder / at a dense state, em is the permittivity of the mixture and Pi the volumetric proportion of powder / ) and cited by Guillot [30] as one of the most powerful model. 

In the statistic route, an effechve dielectric function e(v) is calculated from the dielectric funchon of the metal Enie(v) and the of polymer material po(v) by using a formula, the effective medium theory. The most general effective medium theory is the Bergman theory in which the nanostructure of the composite material can be considered by a spectral density function. The Bergman theory includes the soluhons from the Bruggeman theory and the Maxwell Garnett theory for spherical, parallel-oriented, and random-oriented ellipsoidal parhcles. 

There are a number of formulas for the determination of the effechve dielectric constant Maxwell Garneh (MG) formula, Bruggeman (BG) equation, Hanai-Bruggeman (HBG) formulas, empirical Lichtenecker mixture equahons, and so on. 

Equation (2b) is a scaling law depicting the conductive behavior in the vicinity of the percolation threshold, the value of the critical exponent y being 1.6 to within 0.2. Equation (2c) expresses the composite conductivity dependence upon conductor concentration beyond the percolation threshold. Equation (2c) is a simplified form, valid in the case of conductor-insulator mixtures, of a more general equation derived in different ways by Bruggeman (70), Bottcher (71) and Landauer (72) and known as the Effective Medium Theory, (E.M.T.), formula ... 

Boned and Pcyrclassc [95] extended the asymmetric integration technique used by Bruggeman and Hanai to randomly orientated ellipsoidal mixture and obtained the following formula ... 

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]. 

Bruggeman s formula (v = l), Bottcher s formula and Polder-van Santen formula y = 2) and coherent potential formula (1/ = 3). ... 

Several authors have derived formula for dielectric permittivity of a two component mixture from differential analysis. Most of them considered the excess polarization due to a small sphere introduced to the effective medium where the volume of the sphere can become infinitesimal enabling establishment of a differential equation for the effective permittivity of the mixture [30], The particular solutions obtained for a 3D case by three of the most notable authors contain a fractional power of a third. These are Looyenga formula [31], Bruggeman [29] and Hanai formula [32] and formula by Sen et al. [33] given in equations 9.25, 9.26 and 9.27, respectively. 

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