Scaling Behavior of the Small Strain Modulus

In the framework of the approximation given by the rigidity condition, a simple power law relation can be derived for the dependency of the small strain modulus G 0 of the composite on filler concentration . It is obtained, 

Accordingly, we expect a power law behavior G,0 (O/Op)3 5 of the small strain elastic modulus for 0 0. Thereby, the exponent (3+df [j)/(3—df)w3.5 reflects the characteristic structure of the fractal heterogeneity of the filler network, i.e., the CCA-clusters. The strong dependency of G 0 on the solid fraction Op of primary aggregates reflects the effect of structure on the storage modulus. 

The predicted scaling behavior Eq. (30) is also found to be well fulfilled for carbon black suspensions in ethylene-vinyl-acetate copolymers [51]. Furthermore, it is confirmed by viscoelastic data obtained for S-SBR composites with highly cross-linked BR-microgels of various size [57]. 

We finally note that the decomposition of G 0 into a local elastic constant and a geometrical factor implies that the same form as Eq. (30) must also hold for the loss modulus G o- This follows from the fact that in the linear 

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FIGURE 22.6 Payne effect of butyl composites with various amounts of N330, as indicated (left) [28]. Scaling behavior of the small-strain modulus of the same composites right). The obtained exponent 3.5 confirms the cluster-cluster aggregation model. (From Kliippel, M. and Heinrich, G., Kautschuk, Gummi, Kunststoffe, 58, 217, 2005. With permission.)... 

Since the stiffness of the bonds transfers to the stiffness of the whole filler network, the small strain elastic modulus of highly filled composites is expected to reflect the specific properties of the filler-filler bonds. In particular, the small strain modulus increases with decreasing gap size during heat treatment as observed in Fig. 32a. Furthermore, it exhibits the same temperature dependence as that of the bonds, i.e., the characteristic Arrhenius behavior typical for glassy polymers. Note however that the stiffness of the filler network is also strongly affected by its global structure on mesoscopic length scales. This will be considered in more detail in the next section. 

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