Aggregates, random fractal

Figure 8. The aggregates composed of 50 (top) and 100 (bottom) CPs used in the model calculations. The first two are regular and have a tetrahedron lattice, the others are random fractals with different packing parameters (see text for details). The fractal dimensions, prefactors, and gyration radii (for 100 CPs) are shown in the footnote.

In our studies, we consider several types of aggregated structures such as bispheres, linear chains, plane arrays on a plane rectangular lattice, compact and porous body-centered clusters embedded on the cubic lattice (bcc clusters, the porosity was simulated by random elimination of monomers), and random fractal aggregates (RF clusters). To generate RE clusters, a three-dimensional lattice model with Brownian or linear trajectories of both single particles and intermediate clusters was employed for computer simulations of aggregation process. At the initial time moment, = 50,000 particles are generated at... 

Figure 3.9 Results from 2d random aggregation simulations in which the depth of the potential of interaction was either 7 kT (a) or 4 kT (h). At 4 kT the aggregates are roughly circular crystallites in equilibrium with a monomer phase, that is, a dissolved phase. At 7 kT the aggregates are fractal, with a fractal dimension of 1.4, the DLCA value, over large length scales, but they retain a dense crystal packing over small length scales. These are called fat fractals.

Jullien, R., Thouy, R. and Ehrburger-Dolle, R. (1994). Numerical investigation of two-dimensional projections of random fractal aggregates. Phys. Rev. E, 50, 3878-3885. 

The concept of diffusion-limited cluster-cluster aggregation (DLCA) is very useful and applied in many simulations. In this type of simulation process, particles are placed in a box and subjected to Browni m (random walk) movements. Aggregation (clustering) may occur when two or more particles/clusters come within the vicinity of each other and the combined cluster continues the random walk. The simulation is stopped at the gelation point (percolating system) or when all particles are combined in one final aggregate. The fractal dimension of the DLCA aggregates is approximately 1.8. 

Avnir et al. llbl have examined the classical definitions and terminology of chirality and subsequently determined that they are too restrictive to describe complex objects such as large random supermolecular structures and spiral diffusion-limited aggregates (DLAs). Architecturally, these structures resemble chiral (and fractal) dendrimers therefore, new insights into chiral concepts and nomenclature are introduced that have a direct bearing on the nature of dendritic macromolecular assemblies, for example, continuous chirality measure44 and virtual enantiomers. ... 

In this section the notion of an allometric relation is generalized to include measures of time series. In this view, y is interpreted to be the variance and x the average value of the quantity being measured. The fact that these two central measures of a time series satisfy an allometric relation implies that the underlying time series is a fractal random process and therefore scales. It was first determined empirically that certain statistical data satisfy a power-law relation of the form given by Taylor [17] in Eq. (1), and this is where we begin our discussion of the allometric aggregation method of data analysis. 

Figure 1. The logarithm of the variance is plotted versus the logarithm of the mean for the successive aggregation of 10s computer-generated random data points with Gaussian statistics. The slope of the curve is essentially one, determined by a linear regression using Eq. (9), so the fractal dimension of the time series is 1.5 using Eq. (8).

Figure 3. The logarithm of the standard deviation is plotted versus the logarithm of the average value for the heartbeat interval time series for a young adult male, using sequential values of the aggregation number. The solid line is the best fit to the aggregated data points and yields a fractal dimension of D = 1.24 midway between the curve for a regular process and that for an uncorrelated random process as indicated by the dashed curves.

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