Hydrodynamic properties of rigid fractal aggregates of arbitrary morphology.

The hydrodynamic properties of rigid fractal aggregates are key ingredients in understanding the governing mechanism of their motion and the properties of their suspensions. In the present work we outline explicit equations for the estimation of the complete set of hydrodynamic properties of arbitrary shaped aggregates made of uniform sized spherical primary particles. The rigid body motion equations are coupled to Stokesian dynamics model to derive hydrodynamic rigid body properties. A wide library of clusters consisting of fractal aggregates of different morphologies (d(f)=1.8-3.0) and spheroidal clusters of different axes ratios with broad range of number of constitutive spheres (N(sphere)=10-1000) was used. Using the developed hydrodynamic rigid body properties' equations quantities such as the hydrodynamic radii of translational (R(H)) and rotational (R(ω)) motion, and all hydrodynamic information for each cluster, contained in its grand resistance matrix, are found. Furthermore, the relations between different hydrodynamic properties of average clusters and their morphology are investigated. In the effort to introduce a simplified model that accurately reproduces the complex hydrodynamic behavior of a fractal cluster, two approaches are discussed, namely, an equivalent sphere model and an equivalent ellipsoid model. The predicted hydrodynamic properties from both approaches, which can be computed exactly, closely match those of the clusters, for all cluster masses and morphologies, with the equivalent ellipsoid model being more effective whenever the cluster anisotropy is crucial. Therefore, this simplified approach provides an effective tool to predict the behavior of any cluster with complex structure.

Aggregate breakup in a contracting nozzle.

The breakup of dense aggregates in an extensional flow was investigated experimentally. The flow was realized by pumping the suspension containing the aggregates through a contracting nozzle. Variation of the cluster mass distribution during the breakage process was measured by small-angle light scattering. Because of the large size of primary particles and the dense aggregate structure image analysis was used to determine the shape and structure of the produced fragments. It was found, that neither aggregate structure, characterized by a fractal dimension d(f) = 2.7, nor shape, characterized by an average aspect ratio equal to 1.5, was affected by breakage. Several passes through the nozzle were required to reach the steady state. This is explained by the radial variation of the hydrodynamic stresses at the nozzle entrance, characterized through computational fluid dynamics, which implies that only the fraction of aggregates whose strength is smaller than the local hydrodynamic stress is broken during one pass through the nozzle. Scaling of the steady-state aggregate size as a function of the hydrodynamic stress was used to determine the aggregate strength.

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, 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.

Role of counterion association in colloidal stability.

A generalized model for colloidal stability has been validated against experimentally measured values of Fuchs stability ratio and critical coagulation concentration (ccc) for electrolytes with mono- or divalent cation, i.e., potassium chloride and magnesium chloride, respectively. Besides the classical DLVO theory, the generalized model accounts for the interplay between colloidal interactions and the association of cations with the particles surface charge groups. The model parameters are either obtained or estimated purely on the basis of independent information available in the literature. For the monovalent salt, the predictions agree well with literature experimental data, forecasting both the ccc values and stability ratios quantitatively. For the divalent salt the predictions for large values of the stability ratio tend to deviate from the experimental data produced in this work, but it is noted that the onset of stability, i.e., the ccc, and small stability ratios are correctly predicted. Moreover, a comparison of the above results with those neglecting the effect of counterion association with the particles surface charge groups indicates that the latter substantially overestimates stability ratios in the presence of high salt concentration in the case of the monovalent salt, and leads to unrealistic large values of the ccc for the divalent salt. Including the association of cations with the particles surface charge groups can explain the relatively low values of experimental ccc for divalent salts compared to the theoretical predictions by the classical DLVO theory neglecting ion association, which is apoint of interest in industrial coagulation processes.

Scattering properties of dense clusters of colloidal nanoparticles.

In this work, we present a new methodology to accurately calculate scattering properties of fractal clusters with arbitrary large fractal dimension, d(f) (up to 3), and arbitrary primary particle size and material optical properties. Our approach is based on a combination of Monte Carlo simulations to generate cluster structures and mean-field T-matrix theory for the calculation of scattering properties. We have used a conventional cluster-cluster aggregation algorithm to generate clusters with d(f) up to 2.1, a tunable cluster-cluster aggregation algorithm for clusters with d(f) up to 2.5 and a newly developed Voronoi tessellation-based densification algorithm for clusters with d(f) up to 3. The scattering properties of clusters have been computed by means of mean-field T-matrix code (proposed by Botet; et al. Appl. Opt. 1997, 36 , 8791 - 8797 ), which can account for intracluster multiple scattering at a very low computational cost, thus overcoming the major limitations of commonly used Rayleigh-Debye-Gans (RDG) theory. The results of the calculations show significant deviations of the scattering cross sections and zero-angle intensities as compared to RDG theory for large primary particle sizes and high d(f). Good accuracies of the method have been confirmed by comparisons with full T-matrix calculations. The proposed approach is an ideal compromise between accuracy and high computational efficiency, and is suitable for inversion of experimental scattering data.

Dependence of aggregate strength, structure, and light scattering properties on primary particle size under turbulent conditions in stirred tank.

The steady-state size and structure of aggregates produced under turbulent conditions in stirred tank, for primary particle diameter, d(p), equal to 420 nm and 120 nm, were studied experimentally for various values of the volume average shear rate, G, and solid volume fraction, phi, and compared with data for d(p) = 810 nm. To exclusively investigate the effect of dp, polystyrene latexes with same type and similar density of surface charge groups (sulfate) were used. The mass fractal dimension, d(f), obtained by image analysis, was found to be invariant of d(p) and G, with a value equal to 2.64 +/- 0.18. Small-angle static light scattering was used to characterize the cluster mass distributions by means of the root-mean-square radius of gyration, R(g), and the zero-angle intensity of scattered light, I(0), whose steady-state values proved to be fully reversible with respect to G. The absolute values of R(g) obtained for similar phi and G proved to be independent of d(p), and for all studied conditions, R(g) was proportional to G-1/2. At very low phi, a critical aggregate size for breakage was obtained and used to evaluate the aggregate cohesive force, as a characteristic for the aggregate strength. The aggregate cohesive force was found to be independent of aggregate size, with similar values for the investigated dp. Due to large d(p) and high d(f), the effect of multiple light scattering within the aggregates was found to be present, and by relating the scaling of R(g) with I(0) to d(f), the corresponding correction factors were evaluated. By combination of the independently measured aggregate size and structure, it is possible to experimentally determine the relation between the maximum stable aggregate mass and the hydrodynamic stresses independent of the multiple light scattering present for large d(p) and compact aggregates.

Effect of shear rate on aggregate size and morphology investigated under turbulent conditions in stirred tank.

Aggregation and breakage of aggregates produced from fully destabilized polystyrene latex particles in turbulent flow was studied experimentally in both batch and continuous stirred tank. Detailed investigation of the initial kinetics showed that the collision efficiency, alpha, depends on the shear rate according to alpha proportional to G(-b), with a power law exponent, b, equal to 0.18. After steady state was reached the dynamic response of the system on a change in stirring speed and solid volume fraction was investigated. It was found that the steady-state values of two measured moments of the cluster mass distribution (CMD) are fully reversible upon a change in stirring speed. This indicates that although the moments of CMD at steady-state depend on the applied shear rate, the aggregate structure is independent of the shear rate in the given range of stirring speeds. This was proved by independent measurement of the fractal dimension, d(f), using image analysis which provided a d(f) equal to 2.62 +/- 0.18 independent of applied stirring speed. The critical aggregate size, below which breakage is negligible, determined by dilution experiments was consequently used to evaluate the aggregate cohesive force holding the aggregate together, which was found to be independent of the aggregate size and equal to 6.2 +/- 1.0 nN.