Effective filler volume fraction

The Kerner-Nielsen equation (119) is a useful alternative to Eqs. (11.4) and (11.5) in the case when the elastic modulus of an interphase is substantially greater than that of the matrix. Most often, interphase elastic modulus is assumed to be equal to that of the filler resulting in a slightly overestimated elastic modulus of the compound. The effect of an interphase is then introduced in the Kerner-Nielsen equation substituting an effective filler volume fraction, Veff = (Vf + v,), for the filler volume fraction, Vt, (71) ... 

The following expression for the j/ term in Eq. (11.6) indicates that there exists a maximum volume fraction, v at which rigid inclusions of a given shape and size distribution can occupy. Similarly, a maximum effective filler volume fraction has to be used in calculating / (119) ... 

Mechanical properties are determined by the effective filler aspect ratio and effective filler volume fraction when incomplete dispersion is accounted forSS 91 (rather than on the absolute filler loading and the aspect ratio of the individual fillers). 

Using silicon nitride powder in a polypropylene/microcrystalline wax/stearic acid binder formulation, the effect of filler volume fraction (V) (over the range 50 to 70%) on relative viscosity (rjj.) was predicted from Eq. 5 ... 

Langley, K.R., Martin, A. and Ogin, S.L. (1994). The effect of filler volume fraction on the fracture toughness of a model food composite. Composites Sci. Tech. 50, 259-264. 

It is shown below that p fulfills a scaling relation which involves the size and mass fractal dimension of the primary aggregates. Due to significant deviations of the solid fraction p from 1, the filler volume fraction of carbon black in rubber composites has to be treated as an effective one in most applications, i.e., 0eff=0/0p (compare [22]). 

The established concepts predict some features of the Payne effect, that are independent of the specific types of filler. These features are in good agreement with experimental studies. For example, the Kraus-exponent m of the G drop with increasing deformation is entirely determined by the structure of the cluster network [58, 59]. Another example is the scaling relation at Eq. (70) predicting a specific power law behavior of the elastic modulus as a function of the filler volume fraction. The exponent reflects the characteristic structure of the fractal heterogeneity of the CCA-cluster network. 

On surface it is very simple model but effective concentration of filler includes observation that some layer of polymer is bound to the surface of filler and the mechanisms of this bonding is mathematically expressed by effectiveness factor. The recent model assumes that filler particles are spheres which might be connected to form chain-like agglomerates. Each particle is surface coated with matrix polymer. The elastomeric layer is considered immobilized. The effective filler volume is higher than filler volume fraction by the amount of adsorbed polymer. The effectiveness factors is given by equation ... 

Figure 15.18. Effective heat conductivity of polyethylene vs. filler volume fraction. [Data from Privalko V P, Novikov V V, Adv. Polym. Sci., 119, 1995, 31-77.]...

Figure 13.15. Effects of volume fraction O and aspect ratio Af on the zero-shear viscosity rio(relative) for dispersions of infinitely rigid anisotropic filler particles. The curve labels denote Af. At any given d> and Af, T)o increases more with fibers than it does with platelets.

In the case of isolated spheres, the equations are quite straightforward, and bring out clearly the effects of field frequency co and filler volume fraction complex dielectric constant e are ... 

For lower filler volume fractions the first effect is stronger raising the sample modulus, while with the increase of filler content the second one becomes dominating and the modulus drops. Let us notice that the alternate explanation of the modulus increase for lower - and its decrease for higher filler concentrations seems to be invalid as it would require strong polymer filler adhesion in the initial stages of deformation decreasing with the increase of filler content. The shape of ef/ EP YF EP curves can be explained... 

Equation (6.60) was used to determine the effects on the bulk modulus of the relative polymer-filler interactions, r = the filler volume fraction, 4>2-... 

The large interest in the nanoscale range originates from outstanding properties. Enhanced properties can often be reached for low filler volume fraction, without a detrimental effect on other properties such as impact resistance or plastic deformation capability. Though industrial exploitation of nanocomposites is still in its infancy, the rate of technology implementation is increasing. 

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