Layered fillers elastic modulus

Equation (70) is a scaling invariant relation for the concentration-dependency of the elastic modulus of highly filled rubbers, i.e., the relation is independent of filler particle size. The invariant relation results from the special invariant form of the space-filling condition at Eq. (67) together with the scaling invariance of Eqs. (68) and (69), where the particle size d enters as a normalization factor for the cluster size only. This scaling invariance disappears if the action of the immobilized rubber layer is considered. The effect of a hard, glassy layer of immobilized polymer on the elastic modulus of CCA-clusters can be de-... 

Then, one obtains the following power law-dependency of the elastic modulus G on filler concentration cp, particle size d, and layer thickness A ... 

This equation predicts a strong impact of the layer thickness A on the elastic modulus G. Furthermore, the influence of particle size d becomes apparent. Obviously, the value of G increases significantly if d becomes smaller, i.e., if the specific surface of the filler increases. 

Unlike in the case of spherical inclusions where the rigid inclusions cause stiffening of the composite by excluding volume of a deformable mattix, the presence of an interphase layer affects the tme reinforcing efficiency of the inclusions. Hence, the effective filler modulus of the inclusions have to be calculated as a function of interphase thickness and elastic modulus. This can be done effectively using simple rule of mixture ... 

One of the reasons for deviation of theoretical equations, connecting elasticity modulus with filler amount from the experimental data, is the formation of surface layers at the poljmier-filler interface (interphase layers). The properties of these layers are different than in bulk. It is very important to estimate the contribution of the interphase to the viscoelastic properties of composites. 

The presence of filler in the rubber as well as the increase of the surface ability of the Aerosil surface causes an increase in the modulus. The temperature dependence of the modulus is often used to analyze the network density in cured elastomers. According to the simple statistical theory of rubber elasticity, the modulus should increase twice for the double increase of the absolute temperature [35]. This behavior is observed for a cured xmfilled sample as shown in Fig. 15. However, for rubber filled with hydrophilic and hydrophobic Aerosil, the modulus increases by a factor of 1.3 and 1.6, respectively, as a function of temperature in the range of 225-450 K. It appears that less mobile chain units in the adsorption layer do not contribute directly to the rubber modulus, since the fraction of this layer is only a few percent [7, 8, 12, 21]. Since the influence of the secondary structure of fillers and filler-filler interaction is of importance only at moderate strain [43, 47], it is assumed that the change of the modulus with temperature is mainly caused by the properties of the elastomer matrix and the adsorption layer which cause the filler particles to share deformation. Therefore, the moderate decrease of the rubber modulus with increasing temperature, as compared to the value expected from the statistical theory, can be explained by the following reasons a decrease of the density of adsorption junctions as well as their strength, and a decrease of the ability of filler particles to share deformation due to a decrease of elastomer-filler interactions. 

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