(4d) Characterization of the Structure of Fractal Aggregates

The aggregates formed in hierarchical cluster-cluster sequence (CCA) are similar geometrically to their constituents. So all they are characterized by the same fractal dimension [1]. When two clusters of equal sizes take part in each aggregation act, the fractal dimension of a growing aggregate is thus constant at each stage of the process including the first step producing dimer. Otherwise Mandelbrot [2] established that the structure of aggregates growing by particle-cluster aggregation (PCA) sequence, when the primary particles are added successively to a growing aggregate, is definitely not scale-invariant, becoming increasingly compact in the mass range investigated. For such aggregates the fractal dimension increases from the value characteristic for dimer, which is common for CCA and PCA sequences, changes to higher values and stabilizes for a very large mass. The majority of aggregation phenomena involve collisions of different clusters and primary particles with different sequences, producing aggregates which can be characterized by the fractal dimension changing with the aggregate mass. They are called aggregates with mixed statistics. The fractal dimension of aggregate usually differs from that of clusters which formed the aggregate in the last collision act. Mass-radius relation is applied to characterize the arrangement of primary particles in constituent clusters and the resulting aggregate, as well as to analyze the packing of clusters in aggregate. As a result a master equation is proposed, leading to the determination of the fractal dimension from the parameters of approaching clusters for a given degree of interpenetration. This equation, named aggregation act equation, is utilized to characterize the structure and hydrodynamic behavior of both self similar aggregates and that of mixed statistics.

[1] M. Kolb, R. Botet and R. Jullien, Phys. Rev. Lett., 51 (1983) 2026. [2] B. B. Mandelbrot, Physica A, 191 (1992) 95.