Fractal Analysis of Primary Carbon Black Aggregates

Fractal Analysis of Primary Carbon Black Aggregates 

For a quantitative analysis of the structure of carbon blacks as shown in Fig. 17 it is useful to consider the solid volume Vp or the number of primary particles Np per aggregate in dependence of aggregate size d. In the case of fractal objects one expects the scaling behavior [1, 2] 

The exponent df is denoted mass fractal dimension or simply fractal dimension. It characterizes the mass distribution in three dimensional space and can vary between l df 3. This kind of fractal analysis of furnace blacks was performed, e.g., by Herd et al. [108] or Gerspacher et al. [109, 110]. The solid volume Vp of primary aggregates is normally determined (ASTM 3849) from the cross-section area A and the perimeter P of the single carbon black aggregates by referring to a simple Euclidean relation [108]  

However, it is not quite clear whether this relation can be applied for non-Euclidean, ramified structures. Simulation results of carbon black formation under ballistic conditions by Meakin et al. [14] indicate that a scaling equation is fulfilled, approximately, between the number of particles Np in a primary aggregate and the relative cross section area A/Ap  

Ap is the cross section area of a single primary particle. Dependent on the application of Eqs. (12) or (13), respectively, significantly different values for the mass fractal dimension are obtained. 

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