Showing papers on "Fractal dimension published in 2008"

Fractal Dimension and Size Scaling of Domains in Thin Films of Multiferroic BiFeO 3

Abstract: Domains in ferroelectric films are usually smooth, stripelike, very thin compared with magnetic ones, and satisfy the Landau-Lifshitz-Kittel scaling law (width proportional to square root of film thickness). However, the ferroelectric domains in very thin films of multiferroic BiFeO3 have irregular domain walls characterized by a roughness exponent 0.5–0.6 and in-plane fractal Hausdorff dimension H||=1.4±0.1, and the domain size scales with an exponent 0.59±0.08 rather than (1/2). The domains are significantly larger than those of other ferroelectrics of the same thickness, and closer in size to those of magnetic materials, which is consistent with a strong magnetoelectric coupling at the walls. A general model is proposed for ferroelectrics, ferroelastics or ferromagnetic domains which relates the fractal dimension of the walls to domain size scaling.

Biologically Inspired Surfaces : Broadening the Scope of Roughness

Abstract: Superhydrophobicity can be used for many applications that require non-adhesive and water-repellent surfaces. A successful design of superhydrophobic surfaces requires a correct assessment of the surface roughness effect on wetting. Roughness is an important property in surface mechanics, physics, chemistry, and biology, and it is critical for many tribological applications. Roughness can be defined in different ways, and the definition should be adequate to the problem under investigation. Our recent studies of biological and biomimetic superhydrophobic surfaces show that traditional roughness parameters, such as the root-mean-square, correlation length, or fractal dimension, are not always appropriate for the analysis of wetting. This is, in particular, due to the hierarchical nature of wetting mechanisms and interfaces. We discuss the effect of roughness on wetting mechanisms and relevant roughness parameters and ways to broaden the concept and scope of surface roughness.

A study of radiative properties of fractal soot aggregates using the superposition T-matrix method

Abstract: We employ the numerically exact superposition T-matrix method to perform extensive computations of scattering and absorption properties of soot aggregates with varying state of compactness and size. The fractal dimension, Df, is used to quantify the geometrical mass dispersion of the clusters. The optical properties of soot aggregates for a given fractal dimension are complex functions of the refractive index of the material m, the number of monomers NS, and the monomer radius a. It is shown that for smaller values of a, the absorption cross section tends to be relatively constant when Dfo2 but increases rapidly when Df42. However, a systematic reduction in light absorption with Df is observed for clusters with sufficiently large NS, m, and a. The scattering cross section and single-scattering albedo increase monotonically as fractals evolve from chain-like to more densely packed morphologies, which is a strong manifestation of the increasing importance of scattering interaction among spherules. Overall, the results for soot fractals differ profoundly from those calculated for the respective volume-equivalent soot spheres as well as for the respective external mixtures of soot monomers under the assumption that there are no electromagnetic interactions between the monomers. The climate-research implications of our results are discussed.

From diffusion-weighted MRI to anomalous diffusion imaging.

Abstract: We present a novel interpretation of non-monoexponential diffusion-weighted signal decay with b-value in terms of the theory of anomalous diffusion. Anomalous diffusion is the theory of diffusing particles in environments that are not locally homogeneous, such as brain tissue. In such environments the model of restricted diffusion commonly employed in the analysis of diffusion MR data is not valid, leading to a nonlinear time dependence for the mean-squared displacement of spins, and to a prediction of a stretched exponential form for the signal decay. We show that this prediction leads directly to a new parameter, the anomalous exponent, which may be measured from scan data and from this we can estimate a fractal dimension, d(w), which categorizes the complexity of the excursions of diffusing spins. We construct images of the anomalous exponent and fractal dimension from in vivo human brain data. Distributions of exponents and dimensions are constructed in grey and white matter and cerebrospinal fluid. We observe that these distributions peak at biologically plausible values consistent with previous studies: grey matter dw = 2.366 +/- 0.31, white matter dw = 2.587 +/- 0.39, CSF dw = 1.970 (mode). Marked contrast is observed between grey and white matter when compared with lateral ventricle CSF. We then consider the anisotropy of the value of the anomalous exponent and define quantities analogous to the mean diffusivity and fractional anisotropy that are commonly generated from diffusion tensor images.

Foundations of residual-density analysis.

Abstract: New and concise descriptors of the residual density are presented, namely the gross residual electrons, the net residual electrons and the fractal dimension distribution. These descriptors indicate how much residual density is present and in what way it is distributed, i.e. the extent to which the distribution is featureless. The amount of residual density present accounts for noise in the experimental data as well as for modeling inadequacies. Therefore, the minimization of the gross residual electrons during refinement serves as a quality criterion. In the case where only Gaussian noise is present in the residual density, the fractal distribution is parabolic in shape. Deviations from this shape therefore serve as an indicator for systematic errors. The new measures have been applied to simulated and experimental data in order to study the effects of noise, model inadequacies and truncation in the experimental resolution. These measures, although designed and examined with particular regard to applications of space residual density, are very general and can in principle also be applied to space and momentum residual densities in a one-, two-, three- or higher-dimensional Euclidean space.

Solution properties of asphaltenes.

Abstract: Ultracentrifugation has been used to produce asphaltene fractions of reduced polydispersity. The structure of these asphaltene fraction solutions has been investigated using viscosity and X-ray scattering (SAXS) measurements as a function of concentration. The relative viscosities of the solutions were found to be fraction-dependent: intrinsic viscosities, radii of gyration, and second viriel coefficients followed a power law with molar mass Mw. A flat disc model succeeded in describing scattering data but failed to take viscosity data into account. By contrast, a fractal model has been found to be consistent with dependence of all measured parameters. Asphaltene-in-toluene solutions were found to form nanometric mass fractal aggregates of fractal dimension 2.1, which in consequence trapped solvent. When, instead of concentration, effective volume fractions are used, the relative viscosities of fractions merge on a master curve which can be fitted by a hard sphere model. In addition, the reduced osmotic moduli deduced from scattering measurements of the different solutions, when expressed as a function of a concentration adimensional parameter, merge again on a master curve which is in accordance with the hard sphere behavior. The viscosities of solutions can be fully predicted from structure considerations if the ratio of hydrodynamic to gyration radius is taken as 0.6. This ratio is found consistent with the fractal description of the aggregates.

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

Abstract: 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 G 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 df 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.

Interannual Consistency in Fractal Snow Depth Patterns at Two Colorado Mountain Sites

Abstract: Fractal dimensions derived from log–log variograms are useful for characterizing spatial structure and scaling behavior in snow depth distributions. This study examines the temporal consistency of snow depth scaling features at two sites using snow depth distributions derived from lidar datasets collected in 2003 and 2005. The temporal snow accumulation patterns in these two years were substantially different, but both years represent nearly average 1 April accumulation depths for these sites, with consistent statistical distributions. Two distinct fractal regions are observed in each log–log variogram, separated by a scale break, which indicates a length scale at which a substantial change in the driving processes exists. The lag distance of the scale break is 15 m at the Walton Creek site and 40 m at the Alpine site. The datasets show consistent fractal dimensions and scale break distances between the two years, suggesting that the scaling features observed in spatial snow depth distributions are largely determined by physiography and vegetation characteristics and are relatively insensitive to annual variations in snowfall. Directional variograms also show consistent patterns between years, with smaller fractal dimensions aligned with the dominant wind direction at each site.

Investigations of human EEG response to viewing fractal patterns.

Abstract: Owing to the prevalence of fractal patterns in natural scenery and their growing impact on cultures around the world, fractals constitute a common feature of our daily visual experiences, raising an important question: what responses do fractals induce in the observer? We monitored subjects' EEG while they were viewing fractals with different fractal dimensions, and the results show that significant effects could be found in the EEG even by employing relatively simple silhouette images. Patterns with a fractal dimension of 1.3 elicited the most interesting EEG, with the highest alpha in the frontal lobes but also the highest beta in the parietal area, pointing to a complicated interplay between different parts of the brain when experiencing this pattern.

Absence of self-averaging and of homogeneity in the large scale galaxy distribution

Abstract: The properties of the galaxy distribution at large scales are usually studied using statistics which are assumed to be self-averaging inside a given sample. We present a new analysis able to quantitatively map galaxy large scale structures while testing for the stability of average statistical quantities in different sample regions. We find that the newest samples of the Sloan Digital Sky Survey provide unambiguous evidence that galaxy structures correspond to large amplitude density fluctuations at all scales limited only by sample sizes. The two-point correlations properties are self-averaging up to approximately 30 Mpc/h and are characterized by a fractal dimension D=2.1 +- 0.1. Then at all larger scales probed density fluctuations are too large in amplitude and too extended in space to be self-averaging inside the considered volumes. These inhomogeneities are compatible with a continuation of fractal correlations but incompatible with: (i) a homogeneity scale smaller than 100 Mpc/h, (ii) predictions of standard theoretical models, (iii) mock galaxy catalogs generated from cosmological Nbody simuations.

Are fractal dimensions of the spatial distribution of mineral deposits meaningful

Abstract: It has been proposed that the spatial distribution of mineral deposits is bifractal. An implication of this property is that the number of deposits in a permissive area is a function of the shape of the area. This is because the fractal density functions of deposits are dependent on the distance from known deposits. A long thin permissive area with most of the deposits in one end, such as the Alaskan porphyry permissive area, has a major portion of the area far from known deposits and consequently a low density of deposits associated with most of the permissive area. On the other hand a more equi-dimensioned permissive area, such as the Arizona porphyry permissive area, has a more uniform density of deposits. Another implication of the fractal distribution is that the Poisson assumption typically used for estimating deposit numbers is invalid. Based on data sets of mineral deposits classified by type as inputs, the distributions of many different deposit types are found to have characteristically two fractal dimensions over separate non-overlapping spatial scales in the range of 5–1,000,km. In particular, one typically observes a local dimension at spatial scales less than 30–60 km, and a regional dimension at larger spatial scales. The deposit type, geologic setting, and sample size influence the fractal dimensions. The consequence of the geologic setting can be diminished by using deposits classified by type. The crossover point between the two fractal domains is proportional to the median size of the deposit type. A plot of the crossover points for porphyry copper deposits from different geologic domains against median deposit sizes defines linear relationships and identifies regions that are significantly under explored. Plots of the fractal dimension can also be used to define density functions from which the number of undiscovered deposits can be estimated. This density function is only dependent on the distribution of deposits and is independent of the definition of the permissive area. Density functions for porphyry copper deposits appear to be significantly different for regions in the Andes, Mexico, United States, and western Canada. Consequently, depending on which regional density function is used, quite different estimates of numbers of undiscovered deposits can be obtained. These fractal properties suggest that geologic studies based on mapping at scales of 1:24,000–1:100,000 may not recognize processes that are important in the formation of mineral deposits at scales larger than the crossover points at 30–60,km.

Flocculation model of cohesive sediment using variable fractal dimension

Abstract: A new flocculation model using variable fractal dimension is proposed and validated with several experimental data and an existing model. The proposed model consists of two processes: aggregation and breakup due to flow turbulence. For aggregation process, the aggregate structure is considered to have the characteristic of self-similarity, the main concept of fractal theory. Under this assumption, a variable fractal dimension instead of a fixed one adopted by previous studies is utilized here for general cohesive sediment transport. For breakup, similar concept is adopted in a more empirical manner because breakup is too abrupt to entirely apply the concept of variable fractal dimension. By a linear combination of the formulations for aggregation and breakup processes, a flocculation model which can describe the temporal evolution of floc size is obtained. Flocculation model using variable fractal dimension is capable of predicting equilibrium floc size when compared with several experimental data sets using different types of mud provided that empirical coefficients are calibrated. Through model-data comparison with Manning and Dyer (Marine Geology 160:147–170, 1999), it is also clear that some of the empirical coefficients may depend on sediment concentration. Model results for the temporal evolution of floc size are less satisfactory, despite model results shows a more smooth “S-curve” for the temporal evolution of floc size as compared with the previous model using fixed fractal dimension. The proposed model is limited to mono-size of primary particle and dilute flow condition. These other features shall be investigated as future work.

A mimetic porous carbon model by quench molecular dynamics simulation

Abstract: A mimetic porous carbon model is generated using quench molecular dynamics simulations that reproduces experimental radial distribution functions of activated carbon. The resulting structure is composed of curved and defected graphene sheets. The curvature is induced by nonhexagonal rings. The quench conditions are systematically varied and the final porous structure is scrutinized in terms of its pore size distribution, pore connectivity, and fractal dimension. It is found that the initial carbon density affects the fractal dimension but only causes a minor shift in the pore size distribution. On the other hand, the quench rate affects the pore size distribution but only causes a minor shift in the fractal dimension.

Proteins: Coexistence of Stability and Flexibility

Abstract: We introduce an equation for protein native topology based on recent analysis of data from the Protein Data Bank and on a generalization of the Landau-Peierls instability criterion for fractals. The equation relates the protein fractal dimension df, the spectral dimension ds, and the number of amino acids N. Deviations from the equation may render a protein unfolded. The fractal nature of proteins is shown to bridge their seemingly conflicting properties of stability and flexibility. Over 500 proteins have been analyzed (df, ds, and N) and found to obey this equation of state.

Fractal texture analysis of bread crumb digital images

Abstract: A fractal texture analysis technique was applied to bread crumb digital images. Fractal dimensions obtained from several methods (fractional Brownian motion, frequency domain, relative differential box-counting, morphological fractal, mass fractal and random walks methods) were investigated in order to determine their capability to accurately describe the surface roughness of bread crumb images or the visual appearance of bread crumb in meaningful terms. A total of 500 bread crumb images of different porosity and grain quality were analysed. It was found that bread crumb appearance could be effectively quantified by the fractal dimension of its digital image. Correlations of fractal dimensions with mean cell area, standard deviation of cell area and void fraction were variable for the fractal methods. While the mass fractal method measured better crumb heterogeneity, other methods quantified coarseness, cell–cell wall ruggedness and cell wall tortuosity. A vector comprising fractal dimensions would objectively depict crumb grain and would allow comparisons between different bread crumb images.

Radar Backscattering Computations for Fractal-Shaped Snowflakes

Abstract: We propose a model for large snowflakes based on the fractal nature of their particle shapes. Monte Carlo simulations were conducted to make particles with a fractal dimension of 1.8 to 2.4. The roundness parameter for the projected images of the modeled particles was derived, and an average roundness of approximately 0.4 for particles with fractal dimension 2.1 was indicated; this average roundness matched reported values measured for large ice aggregates. The finite difference time domain method was used to calculate the backscattering cross-sections of ice particles with a fractal dimension of 2.1 and a particle diameter of up to ∼20 mm at microwave frequencies of 95 GHz, 35 GHz, and 9.8 GHz. The results were compared with those of equivalent-volume spheres and randomly oriented equivalent-volume hexagonal columns. Our snowflake model had smaller values of radar backscattering cross-sections than did the equivalent-volume spheres in the size range out of the Rayleigh regime. Furthermore, large differences in backscattering cross-sections between the snowflake model and the equivalent-volume hexagonal column were confirmed, in particular at a frequency of 35GHz.

Lymphoma and Leukemia Cells Possess Fractal Dimensions That Correlate with Their Biological Features

Abstract: Background: Living cells can be viewed as complex adaptive systems that exhibit non-linear dynamics and fractal features. We investigated the fractal qualities of normal and maligna