Fractals

The concept of fractal geometry is used to describe serrated and rugged grain boundaries in the pure materials Zinc and Titanium after deformation and heat treatment. The fractal dimension of the grain boundaries are determined by application of optical and scanning electron microscopy over a wide range of magnifications. Measurements of the coarsening kinetics of the initially fractal-like grain boundaries during isochronous as well as isothermal annealing are presented. The results of the annealing experiments can be explained by an analytic fractal coarsening model yielding the observed dependency of the time law of grain boundary smoothing on the initially fractal dimension of the grain boundaries. The results are supported by a Monte Carlo simulation of the smoothing process of single initially fractal grains.

This paper presents a theoretical approach of scattering by fractal aggregates based on a model with four parameters. It assumes that the aggregates are fractal and polydisperse with a number distribution which has a power law dependence on the molar mass. In addition the extemal cut-offs for the number distribution and the fractal structure are supposed to be stretched exponential functions.

β-lactoglobulin aggregates formed after heat-induced denaturation were studied by light and neutron scattering techniques over a range of wave vectors covering more than 4 decades. The results demonstrate clearly that the aggregates have a fractal structure at larger length scales and their number distribution is self similar. Their local structures depend on the pH and the ionic strength of the solutions and can be either globular at pH 7 or rigid rodlike at pH2 with a persistent length which increases with decreasing ionic strength. The explicit analysis of the dynamic light scattering data allows one to characterise precisely the mass distribution. The calculated structure factors are in good agreement with the experimental data.

Free form techniques and fractals are complementary tools for modeling respectively man-made objects and complex irregular shapes. Fractal techniques, having the advantage of describing self-similar objects, suffer from the drawback of a lack of control of the fractal figures. In contrast, free form techniques provide a high flexibility with smooth figures. Our work focuses on the definition of an IFS-based model designed to inherit the advantages of fractals and free form techniques (control by a set of control points, convex hull) in order to manipulate fractal figures in the way as classical free form shapes (BÉZIER, spline). The work is essentially based on the study of the functional equation Φ(τ * t) = T Φ (t), where Φ is a continuous function, τ and T are both contractive affine operators. We prove that there is a strong relationship between this functional equation and IFS attractors. This relationship will be used for the construction of parametric fractal attractors.

Localization of optical excitations in fractal metal nanocomposites, such as colloidal aggregates, self-affine and semicontinuous films, results in strong enhancements of optical nonlinearities. The localized modes of fractals cover a broad spectral range, from the visible to the far-infrared. Results of both theoretical and experimental studies of enhanced nonlinear optical responses of fractal nanostructures are considered.

Natural organic matter in soils is usually found in intimate association with inorganic materials, primarily aluminosilicates, during the early stages of diagenesis to form an organo-mineral composite known as humin. Such composites typically represent more than 50% of the organic carbon present. Because of humin's insolubility and the surface reactivity of its organic components it is recognized as an important adsorbent of many anthropogenic organic compounds introduced into natural soil systems. Fractal analysis of small-angle X-ray scattering data was used to characterize the surface roughness of four humin samples following sequential removal of organic matter. The surface fractal dimensions were observed to decrease with removal of organic matter which also resulted in a decrease in surface pore size. These results suggest that the mineral components of humin have smooth surfaces over length scales of ~ 1 to ~ 15 nm, and that it is the organic matter coatings that are responsible for their surface roughness. Models clay-organic compounds exhibited results that are consistent with the observation made for natural samples. The surfaces of all the components of humin were found to be dominated by micro- and mesopores that could explain humin's high sorptive uptake of organic chemicals in the environment.

The influence of fractal geometry on superconductivity has been studied for layered superconductors. Superconducting multilayers consisting of alternating Nb and Cu layers with fractal stacking sequence and fractal dimension D_f=0.63 including the two limiting cases D_f= 0 (single superconducting film) and D_f=1 (periodic multilayers) were prepared by electron-beam evaporation in ultrahigh vacuum. The layers of Nb and Cu were put down alternately via computer control of the target shutter. The structure of the samples has been checked with in situ reflection high-energy electron diffraction (RHEED) and Auger depth profiling, confirmed the prescribed layering geometry. Superconductivity was investigated by measurements of the critical temperature of superconducting transition T_c, and of the temperature and of the angular dependence of the upper critical magnetic fields B_c2. The observed dependences of T_c on the parameters of fractal samples are in a good qualitative agreement with the proximity effect theory developed for layered superconductors with a self-similar fractal structure. The behavior of the upper critical magnetic field is directly related to the type of the layering. At low temperatures, all samples show the same two-dimensional behavior essentially governed by the topological dimension of the individual superconducting layers, independent of the fractal dimensionality D_f of the samples, whereas for temperatures near T_c the type of layering determines the dimensionality, resulting in a multicrossover behavior of fractal samples. The angular dependence of the upper critical magnetic field B_c2(θ) of fractals corresponds to the theory for a two-dimensional superconductor at all temperatures, reflecting the multicrossover behavior of the fractal multilayers, as long as the temperature-dependent coherence length is comparable with a certain scale of fractal.

Temporally irregular, large spatial scale, fluid advection of passive tracers occurs in a wide variety of situations. Our main point is that these situations can be conveniently conceptualized as resulting from successive application of a sequence of random maps. This viewpoint is numerically convenient and also provides a useful theoretical framework for dynamical-systems-based analyses of the resulting fractal patterns. Examples of three different situations are discussed: (i) the fractal distribution of passive scalar gradients, (ii) the fractal pattern formed by a scum floating on the surface of a moving fluid, and (iii) the pattern of particles entrained as they flow past an obstacle in an open flow.

Recent research on cloud structure and cloud-radiation interaction at NASA's Goddard Space Flight Center is presented as a show case of interdisciplinary work where fractals and multifractals play a central role. Focus has been primarily on stratocumulus because of their first-order effect on the Earth's energy balance (hence the global climate) due to their unusual horizontal extension and persistence. These cloud layers have quasi-flat upper/lower boundaries and appear to be quite uniform but are highly variable inside. The general strategy has been: utilization of spatial statistics of in situ and remotely sensed data pertaining to cloud structure to constrain stochastic cloud models used in turn for radiative transfer simulations where artificial radiation fields are generated; these fields are compared to actual measurements. and so on, until a degree of closure is achieved. The major trends have been: i) computation and understanding of cloudradiative properties from the large scales of interest to Global Climate Models (over 10² km) down to the smallest observable scales (less than 10 m); ii) from predicting the outcome of "ideal" measurements to those of "real" ones with limited accuracy, sampling and averaging; iii) from passive to active remote-sensing methods; and iv) shifting from standard to wavelet-based analysis/modeling techniques. In terms of potential for impact on geophysical research at large, the most important contributions are: a) criteria for and measures of nonstationarity and intermittency in scale-invariant data; b) so-called "bounded" multifractal cascade models having a continuously variable degree of nonstationarity: c) a parameterization of the bulk effect of fractal variability on large-scale planetary albedo; and d) the basic scaling theory of radiative "smoothing" that explains non-trivially related multiple scattering phenomena in both solar- and lidar-based remote sensing. The last item also suggests new methods of observing clouds and new ways of processing cloud radiance data to retrieve physical cloud properties.