Generation and Geometrical Analysis of Dense Clusters with Variable Fractal Dimension

Abstract

The generation and geometrical analysis of clusters composed of rigid monodisperse primary particles with variable fractal dimension, df, 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, Rg, normalized by the primary particle radius, Rp, follow a fractal scaling, i = kf(Rg/Rp)df. Furthermore, the obtained prefactor of the fractal scaling, kf, is related to df according to kf = 4.46df-2.08, which is in agreement with literature data. For cases when df cannot be directly determined from light scattering or confocal laser scanning microscopy, it can be estimated from its relation with a perimeter fractal dimension, dpf, or a chord fractal dimension, dcf, both obtained from 2D projection of aggregates. A relation between df and dpf of the form df = or-1.5dpf + 4.4 was developed by fitting data obtained in this work for 2.2 < df < 3 together with data of Lee and Kramer [Adv. Colloid Interface Sci. 2004, 112(1-3), 49-57] for 1.8 < df < 2.4. It was found that the method of determining df via dpf is very robust with respect to an artificially introduced blur. In contrary, a relation between df and dcf 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 dcf. Hence, for compact aggregates, it is recommended to determine df from dpf by applying the proposed relation, which is valid in a broad range of df 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 Rg, can be directly related to the area squared over perimeter, A2/P, and the 2D radius of gyration, Rg,2D, respectively, which are obtained from 2D projections. In particular, the following two relations are provided: i = 4.5(A2/P)0.9 and Rg/Rp = 1.47 (Rg,2D/Rp)0.99.

Effect of flow field heterogeneity in coagulators on aggregate size and structure

“…hR g i d f (1Àc) , with a value equal to 1.33 was obtained, which is in good agreement with the literature data. 77,84 By combining all this information, we can conclude that for the investigated colloidal system (surfactantfree polystyrene latex with relatively large surface charge density) aggregated under turbulent conditions, the structure and shape of aggregates are independent of the type of device, and further the CMD is indeed affected by the heterogeneity of the flow field, but identical scaling behavior of hR g i and I(0) n with the hydrodynamic stress for all devices has been obtained. This is in agreement with the conception of aggregate breakup occurring in the viscous subrange, where the characteristic length scale, g K , scales independently from device geometry with ffiffiffiffiffiffiffiffi ffi v 3 =e 4 p .…”

“…Subsequently, the perimeter fractal dimension can be used to evaluate the mass fractal dimension using correlations as presented by Lee and Kramer 83 or by Ehrl et al 84 As it was shown by Ehrl et al, 84 this is a very robust method to evaluate the structure of compact aggregates. Detailed description of the experimental methodology, the used measurement techniques, and measurement data analysis can be found in our previous work.…”

“…Hence, the aggregate structure was determined by analyzing images of aggregates obtained via off-line sampling. 84 Examples of confocal laser scanning microscopy images of aggregates are presented in Figure 10. Note that similar structure factors obtained by on-line and off-line scattering measurements (see Figures 6,7, and 9) demonstrate that the sampling procedure affects neither the size nor the structure of the aggregates.…”

Fractal-like structures in colloid science

“…The latter issue holds only for 2D pictures, as CM allows resolving a complete 3-D picture of the cluster [14]. However, even in the case of dense clusters, with a fractal dimension higher than 2, from the projection, it is possible to estimate a perimeter fractal dimension, which can be related to the real fractal dimension through known correlations [21,22].…”

Determination of maximum turbulent energy dissipation rate generated by a rushton impeller through large eddy simulation

“…In fact all aggregates can be characterized by the same perimeter fractal dimension d pf 18,64,65 which in this particular case was equal to 1:14 (see Figure 2). Having determined d pf the mass fractal dimension d f was subsequently estimated using a correlation developed by Ehrl et al, 66 which for this particular case was equal to 2.7, indicating very compact aggregates. Similarly to the d f also the aspect ratio of formed aggregates did not change (see Figure 3), which is indicating that populations of formed fragments are rather similar independent whether breakup occurs under turbulent or laminar conditions.…”