Mass fractal dimension volume fraction

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]). 

To further quantify the microstructures of AMF, MF-TAGs, and MF-DAGs, the concept of fractal dimensions was used. Fractal dimensions (D) have been used extensively by our group to characterize the spatial distribution of mass in fat crystal networks (Marangoni and Rousseau, 1996 Narine and Marangoni, 1999a). Equation (6) shows that the storage modulus (G ) of a fat scales with the volume fraction of solids ( = SFC/100) in a power law fashion. 

It is clear that the chain fragments with the molecular mass are considered in a plane (d = 2) and the formation of clusters is discussed in a volume (d = 3). The D value calculated from Equation (11.39) for a two-dimensional fractal can be used to find the value for a three-dimensional fractal by applying Equation (11.27) to the dimension df. This method is not highly accurate but simple. Equation (11.51) makes it possible to determine the fraction of volume P occupied by vortices of the scale. Assuming that the P and (p,i values are equal, the value can be calculated from the relationship ... 

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