Fillers ellipsoidal

In passing from spherical particles to particles of an anisodiametrical shape (ellipsoids or fibers) the stress resisted by the filler is the higher the more pronounced the anisodiametricity of the particles [142]. 

Figure 10 Deformation of spherical filler particles into prolate (needle-shaped) ellipsoids see text for details. 

The value of geometrical percolation threshold pc. The volume fraction at random close packing, d>m, is identified with . The pc of a dispersion of randomly placed monodisperse ellipsoidal filler particles as a function of Af is approximated by Equation 13.36. Equation 13.37 can then be used for fibers with Af>10, and Equation 13.38 for platelets of aspect ratio 1/Af, with the results summarized in Figure 13.14. 

The dispersed CB particles had ellipsoidal shapes and they also calculated their approximate dimensions. The difference in size and shape of the ellipsoidal structures were due to the difference in attractive interactions and viscosity between the filler and polymer. They also found that the size of these aggregates increased after crosslinking and was unaffected by the nature of processing oils. 

The composite of nanofibriUar material formed with precipitated calcium carbonate (c-PCC), as described above in Sect. 5.2.2.1, seems to consist of precipitated nanocrystals aggregated into ellipsoidal shapes, as shown in Fig. 5.5. The precipitation occurs at random sites mostly on the end of fibrils and, hence, the fibrils are partially covered with calcium carbonate fillers. The particle size of the primary colloidal c-PCC particle is less than 100 nm. 

Dipole moment of each BNN particle is calctrlated using Claussis-Mossotti equations. Clausius-Mossotti approximation is one of the most commonly used equations for calculating the bulk dielectric properties of inhomogeneous materials (Ohad and David, 1997). It is useful when one of the components can be considered as a host in which, inclusions of the other components are embedded. It involves an exact calculation of the field induced in the tmiform host by a single spherical or ellipsoidal inclusion and an approximate treatment of its distortion by the electrostatic interaction between the different inclusiorts. The Clausius-Mossotti equation itself does not consider any interaction between filler and matrix. This approach has been extensively used for studying the properties of two-component mixtures in which both the host and the inclusions possess different dielectric properties. In recent years, this approximation has been extensively applied to composites involving ceramics and polymers. 

Available fillers are of various types and forms, namely, rigid, flexible, spherical, ellipsoidal, flakes, platelets, fibers or whiskers. They may be organic or inorganic in nature. All varieties of fillers have been tried in polymers in order to impart various advantageous benefits. The presence of the fillers in ttie polymer matrix alters the rheological properties of the polymer. 

The polymer-carbon black filler reinforcement depends widely on the polymer type, carbon black type and structure. Another factor affecting this reinforcement is the filler-filler interaction which leads to the formatimi of three dimensional aggregation structures within the bulk of the rubber matrix. Figure 12 shows the aggregation and agglomeration of carbon black in the rubber. These aggregations takes various shapes which may be spherical or ellipsoidal with different major and minor... 

The Mori-Tanaka model is derived based on the principles of Eshelby s inclusion model for predicting an elastic stress field in and around an ellipsoidal filler in an infinite matrix. The complete analytical solutions for longitudinal and transverse E elastic moduli of an isotropic matrix filled with aligned spherical inclusion are [45,46]. 

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