The Looyenga equation

The symmetrical integral formula was first introdueed by Landau and LifshiU [96] on the basis of general electrodynamic arguments for a (wo-phase isotropic mixture. The same formula was obtained by Looyenga [87] using the model of concentric spheres and symmetrical integration. This equation is furtlier extended to orientated ellipsoidal systems by many researchers independently [97-99J  

It would be interesting to compare F.q.(189) with previous mixture equations that arc shown in Section 5. The special case of Hq. (189) for spherical 

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Numerical results from the above three type equations are compared by Banhcgyi [83]. The dielectric constant and loss of two-phase spherical particle mixture are calculated with the Maxwell-Wagner-Sillars equation, the Bottcher-Hsu equation, and the Looyenga equation using the parameters e i =2, p 8, S/m, CTp=10 S/m, and shown in Figure 23 against... 

The cyclohexane-adsorbent mixture gave a capacitance corresponding to an overall dielectric constant of 3.267. Based on the adsorbent volume fraction, the Looyenga (10) equation yielded a value of 16.4 for the dielectric constant of the adsorbent. 

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.

The capacitance and dielectric constant are extracted by using the relation Z" = l (oQ from the data measured in the high-frequency range of 10 -10 Hz [31], The bulk grain capacitance C of the sample is given by the slope of the straight line determined by the variation in Z" as a function of l/co. Then, the effective dielectric constant Seff of the porous structure is calculated based on the equation Seff = Cx/(8o. With the measured Setf, we can calculate the Snano-Si based on the Looyenga approximation [32] ... 

This is a simplification of the Lorentz-Lorenz equation. Looyenga showed that the expression (n2— 1 )/(n2 + 2) can, with high accuracy be approximated by the more simple expressions (n213 — 1) for the polymer refraction indices mentioned in Table 10.5, the differences vary from 2.9% (n = 1.35) to 8.8% (n = 1.654). 

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. 

It can be seen from equations 9.25, 9.28 and 9.30 that for a single dielectric mixture in air, the form of Lichtenecker s equation becomes same as that of Looyenga s formula and Birchak s formula for k equal to 1/3 and 1/2, respectively. It has been pointed out by various authors that the value of k may be related to the inclusion topology [39-41,43]. 

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