William R. Heinson

Does Shape Anisotropy Control the Fractal Dimension in Diffusion-Limited Cluster-Cluster Aggregation?

Motivated by recent experiments of soot formation in premixed flames where a minority population of the “stringy” aggregates is found to have a fractal dimension as low as 1.2 instead of the classic diffusion-limited cluster-cluster aggregation (DLCA) value of 1.8, we address this same question in our simulation study: is there a distribution of fractal dimensions in a given ensemble of aggregates and does shape anisotropy of clusters control the fractal dimension? Our results, however, clearly show classic DLCA yields aggregates of a broad range of shapes all with Df≈1.8 independent of their shape, but with a shape dependent pre-factor k0 , in the relation between radius of gyration Rg , mass N, monomer radius a, and fractal dimension . Thus the pre-factor k0 gains in status as a descriptor of aggregate morphology so that aggregates should be described by the pair of parameters Df and k 0 , i.e., (Df, k0).

A three parameter description of the structure of diffusion limited cluster fractal aggregates

A three parameter description of fractal aggregates is derived from the pair correlation function of the monomeric units that compose the aggregate. The parameters describe the mass fractal scaling with linear size (the fractal dimension), the packing fraction density of the spherical monomers, and the overall shape of the aggregates. Values for these three parameters are determined for diffusion limited cluster aggregates (DLCAs) in three dimensions. The effects of these parameters are found in terms of measureable quantities in both real and reciprocal space.

Crossover from spherical particle Mie scattering to circular aperture diffraction.

This paper demonstrates the manner in which the Mie results for light scattering by a three-dimensional sphere of arbitrary size and refractive index crosses over to Fraunhofer diffraction by a two-dimensional circular aperture of the same radius in the limit of very large radius. Demonstration is feasible only because the graphical results are plotted in the manner of the Q-space analysis that plots scattered intensity versus the logarithm of the magnitude of the scattering wave vector rather than linear versus the scattering angle.

Light Scattering Shape Diagnostics for Nano-Agglomerates

Motivated by light scattering experiments showing enhanced intensity of electric field aligned nano-agglomerates versus randomly oriented nano-agglomerates, we address the theoretical basis for this effect by applying the theory of small angle Rayleigh-Gans-Debye light scattering to oriented nano-clusters generated by classical diffusion-limited cluster–cluster aggregation (DLCA). We show that the shape anisotropy of these clusters is related to the ratio of small angle light scattering for partially aligned and randomly oriented clusters. It is also shown that state-of-the-art small-angle aerosol scattering measurements would have the angular resolution required to measure the shape anisotropy of clusters with 30 to 1000 nano-monomers. For large q, it is shown from the simulations that S(q) for the partially aligned nano-clusters is not proportional to , where Df is the fractal dimension, as it is for randomly oriented clusters. Nano-clusters with a fixed orientation are shown to result in a structure factor with multiple peaks, which might be used to obtain more detailed information about particle structure than shape anisotropy. The measurements reported in the literature showing enhanced scattering for partially aligned soot agglomerates were for angle integrated measurements. Calculation of the integrated light scattering cross-section for the same range of angles and polarization direction as the experiments indicate a significant enhancement of 70% and 120% for two representative aspect ratios. The smaller value overlaps with measured values of the scattering enhancement for oriented soot agglomerates in an electric field. Copyright 2013 American Association for Aerosol Research

Shear history independence in colloidal aggregation.

Stimulated by experiments, we have carried out detailed simulations of aggregation in the presence of shear in a model colloidal system with a short-range attractive potential. For weak shear rates, we find that the shear enhanced the aggregation and that the long-time state of the system is independent of the shear history. For strong shear rates, precipitous fragmentation occurred after the shear was turned on and, after an induction period, the aggregation quickly rebounded in a stochastic manner similar to classical nucleation phenomena. However, the long-time state of the system is, once again, independent of the shear history. Thus, for both weak and strong shear cases, the shear rate acts as a state variable of the aggregating system. Shear rates employed in the simulations can be attained in laboratory experiments, as confirmed by computing the dimensionless Péclet numbers.

Light scattering Q-space analysis of irregularly shaped particles

We report Q-space analysis of light scattering phase function data for irregularly shaped dust particles and of theoretical model output to describe them. This analysis involves plotting the scattered intensity versus the magnitude of the scattering wave vector q = (4π/λ)sin(θ/2), where λ is the optical wavelength and θ is the scattering angle, on a double-logarithmic plot. In q-space all the particle shapes studied display a scattering pattern which includes a q-independent forward scattering regime; a crossover, Guinier regime when q is near the inverse size; a power law regime; and an enhanced backscattering regime. Power law exponents show a quasi-universal functionality with the internal coupling parameter ρ′. The absolute value of the exponents start from 4 when ρ′ < 1, the diffraction limit, and decreases as ρ′ increases until a constant 1.75 ± 0.25 when ρ′ ≳ 10. The diffraction limit exponent implies that despite their irregular structures, all the particles studied have mass and surface scaling dimensions of Dm = 3 and Ds = 2, respectively. This is different from fractal aggregates that have a power law equal to the fractal dimension Df because Df = Dm = Ds < 3. Spheres have Dm = 3 and Ds = 2 but do not show a single power law nor the same functionality with ρ′. The results presented here imply that Q-space analysis can differentiate between spheres and these two types of irregularly shaped particles. Furthermore, they are applicable to analysis of the contribution of aerosol radiative forcing to climate change and of aerosol remote sensing data.

Divine Proportion Shape Invariance of Diffusion Limited Cluster–Cluster Aggregates

Aggregation is a non-equilibrium process of fundamental importance for all dispersed particulate systems, and the aggregates so produced are common in nature and technology. The resulting aggregates show fractal morphology with a universal, quantifiable fractal dimension. Current quantitative explanations for the fractal aggregate structure are based on simulation models that successfully describe experimental findings and have the key feature that as aggregates form, they continue to move and eventually connect with other aggregates. If the motion between meetings is diffusive, which is very typical, the process is called diffusion limited cluster aggregation (DLCA). We have shown previously that a three-parameter description, including the fractal dimension, the scaling prefactor, and the shape of these aggregates, is both necessary and sufficient to completely describe their morphology. Here we define a new shape description and find geometric relations between the dimensions of the aggregates. Moreover, we show that with this new shape description the aggregate shape is described by the Divine Proportion and its generalization to any dimension. Finally, we recall a simple, analytical theory for DLCA aggregate structure, the Restricted Hierarchical Model, to find that it accurately predicts shape, fractal dimension, and scaling prefactor to yield a necessary and sufficient three-parameter description of aggregate structure. Copyright 2015 American Association for Aerosol Research

Crossover from ballistic to Epstein diffusion in the free-molecular regime

We investigate, through simulation, a system of aggregating particles in the free molecular regime that undergoes a crossover from ballistic to diffusive motion. As the aggregates grow, the aggregate mean free path becomes smaller and the motion between collisions becomes more diffusive. From growth kinetics, we find that when the ratio of the aggregate mean path to the mean aggregate nearest neighbor separation reaches of the order of unity, a crossover to diffusive motion occurs. This ratio, called the nearest neighbor Knudsen number, becomes an important parameter in understanding aerosol aggregation in the free molecular regime. Copyright 2014 American Association for Aerosol Research

Computer Simulation of Aggregation with Consecutive Coalescence and Non-Coalescence Stages in Aerosols

We report results from computer simulation of aggregation in a system where during the aggregation process particles at first coalesce to spherical particles and then subsequently the coalescence stops and the particles continue to aggregate to form ramified, fractal aggregates. In the model systems studied in our paper, the regime of coalescence and the regime of fractal aggregation are separated in time scales. Such a two-step aggregation process has recently been observed in the explosive generation of silica nanoparticles in a chamber. Our simple model captures the essential features of the experimental work and presents a non-material specific scaling description. We study growth kinetics, cluster morphology, and cluster size distributions as a consequence of both the coalescence and aggregation stages.