Fractal Dimension Of Clusters Research Articles

Dynamic Scaling of the Island-Size Distribution and Percolation in a model of Sub-Monolayer Molecular Beam Epitaxy

ABSTRACTThe results of a detailed study of the scaling and percolation behavior of the sub-Monolayer in a model of molecular beam epitaxy are presented. In our model adatoms are randomly deposited on a square lattice and then allowed to diffuse. Whenever an adatom encounters another adatom or an island, it is attached to them and becomes immobile. We have found and studied four distinct scaling regimes, corresponding to a low-coverage (nucleation) regime, intermediate-coverage regime, an aggregation regime, and coalescence and percolation regime. At low coverage the islands have a dendritic structure such as seen in Au/Ru (0001) while at higher coverages they become compact. The scaling of the cluster fractal dimension, island density, Monomer density, island size distribution, and structure factor and pair-correlation function are studied as a function of the coverage θ and the ratio R = D /F of the diffusion rate to the deposition rate.

Cooperative diffusion of animals on the square lattice

The collective diffusion of N-particle lattice animals without vacancies and mass up to N=86 is investigated on the square lattice. Using the transfer matrix technique, a cluster fractal dimension df=1.5618+or-0.0012 is found. The mean-square displacement of the centre of mass R and the pair correlation function g2 are determined by Monte Carlo simulations. This dynamical study gives the mass dependence of the diffusion coefficient D approximately N-1.12+or-0.03 and the correlation function g2 is shown to depend on many characteristic times tau k. Scaling assumptions for R2 and g2 lead to the distinguishment of two characteristic times: tau which is the time necessary for the cluster to be translated by its radius of gyration Rg, and tau 2, the time necessary for a particle of the cluster to be moved approximately by Rg inside the cluster.

Structural studies of vapor-phase aggregates via mercury porosimetry

The physical structure (i.e., primary particle size, cluster fractal dimension) of a series of vapor-phase aggregates (fumed silica, carbon black) has been studied using mercury porosimetry. By employing an isostatic pressing technique, intrusion at low pressure is shown to be the result of cluster compaction for a range of Cab-O-Sil fumed silica and two carbon black samples. Analysis of the intrusion/compaction curve indicates fractal behavior with a D value of 2.5–2.6. Primary particle sizes determined via porosimetry support the theory that the nitrogen surface areas of fumed silica samples differ as a result of both particle size and surface roughness variation. The ratio of the mercury porosimetry-derived primary particle size to that determined from nitrogen adsorption was found to correlate with the surface fractal dimension as determined from scattering for both carbon black and fumed silica.

Variable aggregation rates in colloidal gold: Kernel homogeneity dependence on aggregant concentration.

Dynamic light scattering is used to study the dependence of the aggregation kernel homogeneity \ensuremath{\lambda} on the aggregant concentration [HCl] for aqueous gold sols. We find the cluster growth kinetics are well described by a powerlaw, ${R}_{\mathrm{app}\mathrm{\ensuremath{\sim}}{t}^{z/D}}$, where ${R}_{\mathrm{app}}$ is the measured apparent radius, D the cluster fractal dimension, and z=1/(1-\ensuremath{\lambda}) for all aggregant concentrations. The values for the dynamic exponent z, and hence the homogeneity \ensuremath{\lambda}, are functions of HCl concentration. We find the larger HCl concentrations yield a fast-aggregation regime characterized by \ensuremath{\lambda}\ensuremath{\simeq}-0.6. Smaller HCl concentrations yield a continuum of aggregation regimes characterized by homogeneities evolving from \ensuremath{\lambda}\ensuremath{\simeq}-0.6 towards 1.0. Our results do not support the view that aggregation in gold colloids is based on two limiting regimes, diffusion-limited and reaction-limited aggregation.

Fractal geometry of marine snow and other biological aggregates

Fractal dimensions of aggregates can potentially be used to classify aggregate morphology as well as to identify coagulation mechanisms. Microbial aggregates of Zoogloea ramigera have a cluster fractal dimension of 1.8±0.3 (± SD), suggesting that these aggregates are formed through cluster-cluster coagulation. An analysis of size-porosity correlations for two types of marine snow aggregates yielded fractal dimensions of 1.39±0.06 and 1.52±0.19, which were lower than values describing inorganic colloidal aggregation.

Universality of fractal aggregates as probed by light scattering

Fractal colloid aggregates are studied with both static and dynamic light scattering. The dynamic light scattering data are scaled onto a single master curve, whose shape is sensitive to the structure of the aggregates and their mass distribution. By using the structure factor determined from computer-simulated aggregates, and including the effects of rotational diffusion, we predict the shape of the master curve for different cluster distributions. Excellent agreement is found between our predictions and the data for the two limiting régimes, diffusion-limited and reaction-limited colloid aggregation. Furthermore, using data from several completely different colloids, we find that the shapes of the master curves are identical for each régime. In addition, the cluster fractal dimensions and the aggregation kinetics are identical in each régime. This provides convincing experimental evidence of the universality of these two régimes of colloid aggregation.

Toxicity of a sodium silicate solution used as concrete hardener: An experimental study : Yokose T, Ozawa H, Ogata T, et al. Jpn J Acute Med 1986;10:1011–1013

Structural features and thus, optical transparency of silica hydro- and aerogels from sodium silicate solution strongly depend on the preparation parameters. This study is focused on the effect of pH, for values higher than the isoelectric point of silica acid (pH ∼ 2). Previous studies have presented SAXS data characterizing the primary particles and fractal properties in silica gels from a sodium silicate solution. This study presents an analysis focusing on the structural units >50 nm, which is the relevant structure for the process of gelation and for optical transparency. UV–vis–NIR spectra revealed that hydrogel transmission increases with rising pH and decreases with the fraction of SiO2. Scanning electron microscopy (SEM) analyses of the inner structure of the supercritical dried gels showed that the mean cluster size along with the deviation dropped substantially with rising pH. An equation which agrees with the experimental results and describes the correlation between cluster size, fractal dimension, and packing index in mass fractal cluster packages is provided.

Universal kinetics in reaction-limited aggregation.

Reaction-limited cluster aggregation is modeled with the kinetic rate (Smoluchowski) equations, using a kernel determined intrinsically by the clusters fractal geometry. The kernel scales with cluster mass as ${\mathrm{M}}_{1}$${\mathrm{M}}_{2}^{\ensuremath{\lambda}\mathrm{\ensuremath{-}}1}$ (${\mathrm{M}}_{1}$\ensuremath{\gg}${\mathrm{M}}_{2}$), and ${\mathrm{M}}_{1}^{\ensuremath{\lambda}}$ (${\mathrm{M}}_{1}$\ensuremath{\approxeq}${\mathrm{M}}_{2}$), with \ensuremath{\lambda}=1 in three dimensions, resulting in exponential kinetics and a cluster mass distribution ${\mathrm{C}}_{\mathrm{M}}$\ensuremath{\sim}${\mathrm{M}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\tau}}}$, with \ensuremath{\tau}=(3/2), in excellent accord with experiments. The singular nature of this solution forces the adjustment of the cluster fractal dimension, ${\mathrm{d}}_{\mathrm{f}}$, thereby determining its value.

Aggregation Kinetics of Immunoglobulins

The kinetics of aggregation as described by the Smoluchowski equation is discussed for monomer addition aggregation, cluster-cluster aggregation and for a new model where the irreversible aggregation process requires a preceeding reaction step that changes monomers into an active state. The consequences of these kinetic models on quasielastically scattered light is evaluated. For Smoluchowski aggregation we find that the cluster fractal dimension may be obtained from measurements of the effective hydrodynamic radius as determined from the intensity-intensity correlation function of the scattered light. Experiments on the scattering of light from aggregating pooled human gammaglobulin monomers (IgG), during heating, reveal that two fractions are present. The normal fraction forms aggregates by a Smoluchowski process, and the clusters formed have a fractal dimension D = 2.56 ± 0.3. The heat stable H-fraction follows a new type of kinetics where the monomers must first transform to the denatured state before they can participate in the irreversible aggregation process.

Cluster structure and conductivity of three-dimensional continuum systems.

While the cluster statistics exponents for three-dimensional continuum systems have been shown to be the same as those of lattices, the cluster structure and the conductivity exponents of such continuum systems have not been reported before. Here we present the first determination of both the fractal dimension of clusters, D, and the conductivity exponent, t, for these systems. We further describe the ``porcupine''-like morphology of the clusters and the conductivity behavior in continuum anisotropic systems.

Position space renormalisation group for aggregation fractals

A simple position space renormalisation group scheme is used to find the fractal dimension of Sutherland type aggregation clusters grown in low-dimensional space. Clusters which move along paths of fractal dimension dW=0, 1, 2 are considered and a systematic variation in cluster fractal dimension matching that of recent computer simulation is found.

Scaling Behavior and Cluster Fractal Dimension Determined by Light Scattering from Aggregating Proteins

The size distribution ${n}_{i}(t)$ for systems with Smoluchowski aggregation kinetics is shown to have the dynamic scaling form ${n}_{i}(t)\ensuremath{\sim}{t}^{\ensuremath{-}\ensuremath{\theta}}g(\frac{1}{{t}^{z}})$, with $\ensuremath{\theta}=2$ and $z=1$. This gives a light-scattering intensity $〈I〉\ensuremath{\sim}{t}^{z}$, and an effective hydrodynamic cluster radius $〈R〉\ensuremath{\sim}{t}^{\frac{z}{D}}$, where $D$ is the cluster fractal dimension. We find that human immunoglobulin (IgG) monomers exhibit Smoluchowski aggregation kinetics with $z=1$, and $D=2.56\ifmmode\pm\else\textpm\fi{}0.3$, upon heating to 47-62\ifmmode^\circ\else\textdegree\fi{}C.