Generation and Geometrical Analysis of Dense Clusters with Variable Fractal Dimension

The generation and geometrical analysis of clusters composed of rigid monodisperse primary particles with variable fractal dimension, d(f), in the range from 2.2 to 3 are presented. For all generated aggregate populations, it was found that the dimensionless aggregate mass, i, and the aggregate size, characterized by the radius of gyration, R-g, normalized by the primary particle radius, R-p, follow a fractal scaling, i = k(f)(R-g/R-p)(df). Furthermore, the obtained prefactor of the fractal scaling, k(f), is related to d(f) according to k(f) = 4.46d(f)(-2.08), which is in agreement with literature data. For cases when d(f) cannot be directly determined from light scattering or confocal laser scanning microscopy, it can be estimated from its relation with a perimeter fractal dimension, d(pf), or a chord fractal dimension, d(cf), both obtained from 2D projection of aggregates. A relation between d(f), and d(pf) of the form d(f) = - 1.5d(pf) + 4.4 was developed by fitting data obtained in this work for 2.2 < d(f) < 3 together with data of Lee and Kramer [Adv. Colloid Interface Sci. 2004, 112(1-3), 49-57] for 1.8 < d(f) < 2.4. It was found that the method of determining d(f) via d(pf) is very robust with respect to an artificially introduced blur. In contrary, a relation between d(f) and d(cf) could only be established for the case of ideal optical analysis, while the introduction of blur results in a significant effect on the chord length distribution (and its moments), up to the point of impeding the evaluation of d(cf). Hence, for compact aggregates, it is recommended to determine d(f) from d(pf) by applying the proposed relation, which is valid in a broad range of d(f) relevant for industrial praxis, with little effect of blur on it. Apart from scaling relations with respect to aggregate mass and size, it was found that the 3D quantities, i and R-g, can be directly related to the area squared over perimeter, A(2)/P, and the 2D radius of gyration, R-g.2D, respectively, which are obtained from 2D projections. In particular, the following two relations are provided: i = 4.5(A(2)/P)(0.9) and R-g/R-p = 1.47(R-g.2D/R-p)(0.99).