Range Of Fractal Dimensions Research Articles

Condensation versus diffusion. A spatial-scale-independent theory of aggregate structures in edible oils: applications to model systems and commercial shortenings studied via rheology and USAXS

Understanding how solid fats structures come about in edible oils and quantifying their structures is of fundamental importance in developing edible oils with pre-selected characteristics. We considered the great range of fractal dimensions, from 1.91 to 2.90, reported from rheological measurements. We point out that, if the structures arise via DLA/RLA or DLCA/RLCA, as has been established using ultra small angle x-ray scattering (USAXS), we would expect fractal dimensions in the range ~1.7 to 2.1, and ~2.5 or ~3.0. We present new data for commercial fats and show that the fractal dimensions deduced lie outside these values. We have developed a model in which competition between two processes can lead to the range of fractal dimensions observed. The two processes are (i) the rate at which the solid fat particles are created as the temperature is decreased, and (ii) the rate at which these particles diffuse, thereby meeting and forming aggregates. We assumed that aggregation can take place essentially isotropically and we identified two characteristic times: a time characterizing the rate of creation of solid fats, , where is the rate of solid condensation (cm3 s−1), and the diffusion time of solid fats, , where is their diffusion coefficient and is the typical average distance that fats must move in order to aggregate. The intent of this model is to show that a simple process can lead to a wide range of fractal dimensions. We showed that in the limit of very fast solid creation, the fractal dimension is predicted to be that of DLCA, ~1.7, relaxing to that of RLCA, 2.0–2.1, and that in the limit of very slow solid creation, , the fractal dimension is predicted to be that obtained via DLA, ~2.5, relaxing to that of RLA, 3.0. We predict that, given a system which satisfies our model assumptions and which can either be cooled rapidly or cooled slowly to yield fractal dimensions and then . This is supported by both rheological [] and USAXS measurements [, ] even though the latter models do not conform to the assumptions of those presented here.

Fractal characteristics of the microstructure for porous graphite

The micro-pore structure of porous graphite presents remarkable heterogeneity and self-similarity, which is very difficult to be described by Euclidean geometry, but can be accurately characterised by fractal theory. The results show that the volume porosity range from 27·32 to 32·56%; the area fractal dimensions range from 1·683 to 1·829; the volume fractal dimensions range from 2·898 to 2·935; and the fractal dimensions of boundary projection range from 1·09 to 1·23, respectively. The fractal dimensions indicate the heterogeneity of the micro-pore structure. Moreover, the porosity of porous graphite is deduced by fractal theory, and the mercury injection test is developed to verify the fractal expressions. The results show that the experimental data agree well with theoretical values.

The mobilities of flame synthesized aggregates/agglomerates in the transition regime

Mobility analysis is frequently used to evaluate the size distribution of nanoparticle aggregates synthesized in high temperature, gas phase environments, such as flames. Theoretically, the scalar mobilities of such non-spherical entities, whose characteristic sizes are similar to the mean free path of their background gas (λ), are dependent on two size descriptors, the hydrodynamic radius (RH) and the orientationally averaged projected area (PA) (as well as λ). Unfortunately, the theoretical relationship linking the mobility to RH and PA for aggregates has not been experimentally tested, and has often been discarded in lieu of simpler yet theoretically invalid expressions. Here, we examine the mobilities of flame synthesized titanium dioxide (TiO2) aggregates to directly test the link between the mobility, RH, PA, and λ. Flame synthesized aggregates with mobility equivalent diameters in air in the 45–80nm range were classified with a differential mobility analyzer (DMA) and deposited on a transmission electron microscope (TEM) grid for subsequent imaging. Probable 3-dimensional structures for each imaged aggregate were constructed, based on the assumption that aggregates were quasifractal in nature and by comparing four 2-dimensional size descriptors calculated for images to those of computationally generated projections of quasifractal aggregates (of prescribed pre-exponential factor, fractal dimension, and number of primary particles). The calculated mobilities for the 3-dimensional structures inverted for each image are found to be in excellent agreement with their mobilities inferred from DMA classification, supporting the theoretically proposed relationship between the mobility and RH, PA, and λ in the Kn=πλRH/PA=1.27–4.11 range. Although a wide range of fractal dimensions and pre-exponential factors are inferred for the observed aggregates, the volumes of the reconstructed aggregates are still found to scale with the mobility diameters.

The application of multiscale quasi 4D CT to the study of SrCrO4 distributions and the development of porous networks in epoxy-based primer coatings

Transport paths for inhibitor release within a model strontium chromate (SrCrO4) inhibited/epoxy primer have been studied using a combination of tomography techniques. It has been found that the SrCrO4 particles form independent clusters within the model primer. The clusters have a range of fractal dimensions with the largest clusters (a few hundred microns in size) having a fractal dimension of 2.36. Leaching of the SrCrO4 from the primer appears to be initially through direct dissolution of particles in contact with the electrolyte but changes to diffusion through void pathways created by dissolution of the SrCrO4 phase. No evidence was found for the diffusion of chromate ions through the epoxy. Transport through such clusters does not follow Fickian diffusion, which has traditionally been employed to describe inhibitor release dynamics. Release kinetics typically follow a tm behaviour where t is time and m is an index which would be 0.5 for Fickian diffusion. Thus the overall release with time will evolve, being initially the result of direct dissolution, then at intermediate times, be dominated by transport through the fractal network and at the final stage go to zero since all the strontium chromate will be dissolved from the cluster connected to the surface. Clusters not connected to the surface remain undissolved and form additional reservoirs for further release in when local damage occurs in their vicinity. This new model of inhibitor transport creates new strategies for the development of self-healing properties for coatings.

Fraktálna analýza osteoporózy: prístup podielom vierohodnosti

Based on the traditional fractal theory and on the paper of Stehlík, (2009) the range of fractal dimension of osteoporosis vertebras is analysed. First we give an insight into the field of fractals and the usa­ge of fractals in medicine. After this we show how the analytical tool of Stehlík, (2009) may be applied to the osteoporosis vertebras. It turns out that the used method can be applied very well and that it could help with medical diagnosis. Real data example illustrates the methods discussed.

Fractally deforested landscape: Pattern and process in a tri-national Amazon frontier

Forest clearings in the Amazon are expanding along roads and are enhanced by the associated expansion of human settlements. The purpose of this research is to analyze the spatial patterns associated with this development process using fractal geometry and to partition this development process into different levels by a model-based classification scheme that can be applied to regions globally. A critical region of tropical forest cover in the tri-national frontier in the center of the southwestern Amazon was used as the study area. We utilized box-counting fractal dimensions to describe the spatial patterns of deforestation at a pixel level from 1986 to 2010 in the study region. The evolving pattern of development, as indicated by density-sliced fractal dimension, provides a unique and informative view of a deforesting landscape. The cleared areas have become increasingly compact from 1986 to 2010, where the low fractal dimensions typically represent little to no forest clearings and higher fractal dimensions are associated with more highly developed areas. Such differences are summarized by a classification scheme derived from a mathematical model that partitions the continuous range of fractal dimensions into five possible classes ranging from no or minimal development to highly developed. Such graphical representations of these stages of deforestation in the study region with such spatially explicit pixel-level information enables us to provide multi-level, local, adaptive, and flexible information to forest conservation groups, land managers and related programs.

Multifractality attributed to dual central limit-like convergence effects

Multifractals can be defined as fractal systems that express a range of fractal dimensions. The origins of multifractality in time series data have conventionally been attributed to fat-tailed probability distributions, and to long-range correlations. Multifractal sequences can be generated from the eigenvalue deviations of the Gaussian unitary and orthogonal ensembles of random matrix theory. These deviations can be resolved into component monofractal sequences governed by the Tweedie compound Poisson distribution, a statistical model that expresses a variance to mean power law related to long-range correlations. Fully multifractal descriptions of these deviations can be constructed, provided that the parameter of the compound Poisson model related to fractal dimension varies in accordance with an asymmetric Laplace distribution. Both the Tweedie compound Poisson distribution and the asymmetric Laplace distribution serve as foci of convergence in limit theorems on independent and identically distributed random variables. The hypothesis that multifractal sequences can be attributed to mathematical convergence effects that have as their focus these two statistical models is proposed.

Normal Revised Model of Joint Interface Distinguishing Elastic and Plastic Deformation

On the basis of Hertz elastic theory and fractal geometry theory, one normal contact modification fractal model of fretting joint interface, following the character of the harsh surface bulge's deflection, smoothness and continuity of normal contact load exerting surface asperity as well as differentiating severely elastic deformation and plastic deflection, is proposed. The power function equation between surface summit's normal elastic contact load and its normal deformation is presented. Furthermore, the solving rule of differential function not partial differential function to attain fretting joint interface two bulges' interacting normal contact stiffness is addressed. The digital simulation curves reveal that real contact area firstly increases and afterwards decreases with the augment of fractal dimension at the given normal contact load. Real contact area may outright add by increasing normal contact load or reducing surface roughness. Fretting joint interface normal contact stiffness enhances as the increase of real contact area, normal contact load, relating factor or material property parameter, yet an increase in fractal roughness causes a decrease in fretting joint interface normal contact stiffness. Fretting joint interface normal contact stiffness increases when fractal dimension increases from 1 in the smaller range of fractal dimension; nevertheless, fretting joint interface normal contact stiffness sometimes shrinks when fractal dimension increases close to 2 in the larger range of fractal dimension. Conceiving normal contact amendment fractal model of fretting joint interface investigates deeply the practicable contact states between fretting two coarse surfaces.

Animal Model of Neurodegeneration and Stress Cause by Aluminum Toxicity

Electrocortical group neuron activity describe changes in neurotransmision cause by different factors. Such changes could by qualitative described by spectral analysis of electrocortical activity as a variation of relative spectral power in different frequency ranges. We used fractal dimension to compare treated animals with control to quantitativily describe degree of pathophysiological state. The aim of this study was to qualitativily and quantitatively evaluate effect of the stress and the neurodegeneration in animal model of chronic intoxication by aluminum. This animal model is comparible with Alzheimer’s disease and Parkinson disease. By spectral and fractal analysis we described changes in electrocortical activity of aluminum intoxication compare to physiological control. We used adult animals, during 6 weeks intraperitonealy treated with aluminum. Stress was cause by experimental treatmant and immobilization. Neurodegeneration was observed histologicaly. Results shows that decrease in delta range of spectral power and fractal dimension might be used for evaluation of pathophysiological state of stress and neurodegeneration.

Performance assessment of methods for estimation of fractal dimension from scanning electron microscope images

Processing of gray scale images in order to determine the corresponding fractal dimension is very important due to widespread use of imaging technologies and application of fractal analysis in many areas of science, technology, and medicine. To this end, many methods for estimation of fractal dimension from gray scale images have been developed and routinely used. Unfortunately different methods (dimension estimators) often yield significantly different results in a manner that makes interpretation difficult. Here, we report results of comparative assessment of performance of several most frequently used algorithms/methods for estimation of fractal dimension. To that purpose, we have used scanning electron microscope images of aluminum oxide surfaces with different fractal dimensions. The performance of algorithms/methods was evaluated using the statistical Z-score approach. The differences between performances of six various methods are discussed and further compared with results obtained by electrochemical impedance spectroscopy on the same samples. The analysis of results shows that the performance of investigated algorithms varies considerably and that systematically erroneous fractal dimensions could be estimated using certain methods. The differential cube counting, triangulation, and box counting algorithms showed satisfactory performance in the whole investigated range of fractal dimensions. Difference statistic is proved to be less reliable generating 4% of unsatisfactory results. The performances of the Power spectrum, Partitioning and EIS were unsatisfactory in 29%, 38%, and 75% of estimations, respectively. The results of this study should be useful and provide guidelines to researchers using/attempting fractal analysis of images obtained by scanning microscopy or atomic force microscopy.

Fractal analysis of wear topography of brazed polycrystalline cBN abrasive grains during grinding nickel super alloy

In order to characterize quantificationally the wear topography of the brazed polycrystalline cBN (PcBN) grains during grinding, the reconstruction model of grain topography is established through the photographs grasped with three-dimensional (3D) optical video microscope. The relationship between 3D fractal dimension and the complicated topography change of the PcBN grains is investigated based on fractal theory. The results obtained show that it is reasonable to calculate 3D fractal dimension according to the protrusion height of the abrasive grain. The fractal dimension of the grain wear topography formed due to microfracture is higher than that formed due to large fracture and attritious wear. The fractal dimension range of the wear topography of the brazed PcBN grains is limited to 2.0325–2.0475 with a concentrated value of 2.04 under the present experimental condition.

Fractality and Self-Similarity in the Structure of Road Networks

Fractal geometry has been an important tool in multiple disciplines, but it is usually limited to the geometrical fractal: a shape made of parts similar to the whole in certain ways. Recently there has been increasing interest in structural fractality of complex networks (e.g., biological and Internet hyperlinks) for studying organizational principles and evolutionary rules. In this study, the structural fractality of road networks (a kind of complex geographical network) is examined for better understanding of the complexity and dynamics of the road system. Fifty road networks of the most populous counties in the United States have been used as cases. The Maximum Excluded Mass Burning (MEMB) algorithm rooted in physics has been employed to compute the fractal dimensions by covering the network structure with a series of box sizes (ℓ s ). It is found that road networks have structural fractality. The values of structural fractal dimension range from 2.94 to 4.90 spanning ℓ s = 5 to ℓ s = 15. To examine the self-similar property of the structure of a road network at several scales, small-world and scale-free analyses have been carried out. It was found that road networks are structural self-similarities. The fractal and self-similar structures in a road network help improve the efficiency of flow transmission in the system and keep a balance between chaos and order. These findings help us understand the complexity, organizational rules, and dynamic principles of an urban system and blur the borders across several disciplines. They also provide an empirical guide for urban design and transportation planning.

Research of Fractal Theory to AC-20 Asphalt Mixture Gradation

AC-20 is a widely used in Expressway Close-Graded asphalt mixtures. Based on fractal theory, the fractal dimension range is defined by Marshall Test result analysis of gradation fractal dimension and the Marshall Test parameters. According to the gradation curve of fractal dimension and the degree of fitting, selection of Target Proportioning Design of asphalt mixture, gradation design, optimization test method for reducing test volume has certain guiding role.

Scaling relations for watersheds

We study the morphology of watersheds in two and three dimensional systems subjected to different degrees of spatial correlations. The response of these objects to small, local perturbations is also investigated with extensive numerical simulations. We find the fractal dimension of the watersheds to generally decrease with the Hurst exponent, which quantifies the degree of spatial correlations. Moreover, in two dimensions, our results match the range of fractal dimensions 1.10≤d(f)≤1.15 observed for natural landscapes. We report that the watershed is strongly affected by local perturbations. For perturbed two and three dimensional systems, we observe a power-law scaling behavior for the distribution of areas (volumes) enclosed by the original and the displaced watershed and for the distribution of distances between outlets. Finite-size effects are analyzed and the resulting scaling exponents are shown to depend significantly on the Hurst exponent. The intrinsic relation between watershed and invasion percolation, as well as relations between exponents conjectured in previous studies with two dimensional systems, are now confirmed by our results in three dimensions.

Habitat complexity in aquatic systems: fractals and beyond

Despite the intensity with which ecological information involving habitat complexity has been amassed to date, much remains to be revealed for a comprehensive understanding of the mechanics and implications of the structural complexity of habitats and its influences on ecological communities. This review examines the multi-faceted characteristics of habitat complexity, focusing in particular on aquatic ecosystems. Habitat complexity in aquatic systems is characterised by at least five different traits of physical structure: (1) spatial scales, (2) diversity of complexity-generating physical (structural) elements, (3) spatial arrangement of elements, (4) sizes of elements, (5) abundance/density of elements. Of these five traits, the concept of fractal dimension fully encompasses only the last one; in this sense, habitat complexity is more complex than what fractal measures represent. It is therefore important to investigate exactly which traits of habitat structure are exerting influences on organisms/communities. We hypothesise that, where an entire range of possible fractal dimension D is considered, intermediate levels of D are most likely to be associated with the highest level of biodiversity, to which the body size spectra of assemblages would have a close bearing. In most aquatic ecosystems, broadly two-dimensional structures of bottom substrate at the scale of 1–10 m mean that the addition of vertical, three dimensional structures almost always implies an increase in both the ‘diversity’ and ‘abundance’ components of structural elements, resulting in more habitats being made available to organisms of different sizes and functional designs. The conservation and management of aquatic ecosystems would be facilitated by rigorous assessments of linkages between habitat complexity and aquatic communities, for which an integrative approach to habitat complexity seems to offer a useful and versatile framework.

Beauty and the beholder: A sensitive relationship

Despite a rich set of experimental findings suggesting that certain stimulus qualities, such as symmetry, proportion, or complexity, are related to aesthetic appeal, the exact functional relationship between aesthetic experience and any of these stimulus qualities remains unknown. Motivated by our previous findings that humans display a consistent preference for a certain range of fractal dimension across fractal images of various types we turn to scale-specific processing of visual information to understand this relationship. Our hypothesis is that the preference for certain spatial structures, including fractals, can be partially accounted for by the visual system's general sensitivity for spatial variations at the spatial scales that most closely approximate given spatial structures. Although we do not offer a comprehensive model of aesthetic experience, we demonstrate a strong relationship between visual preference for simple visual patterns and visual sensitivity. Our results support assertions that there is a close relationship between aesthetic experience and the coding of natural stimuli.

DIFFUSION-LIMITED AGGREGATION DRIVEN BY OPTIMAL TRANSPORTATION

In this article, we combine the DLA model of Witten and Sander with ideas from ramified optimal transportation. We propose a modification of the DLA model in which the probability of sticking is inversely proportional to the additional transport cost from the point to the root. We used a family of cost functions parameterized by a parameter α as studied in ramified optimal transportation. α < 0 promotes growth near the root whereas α > 0 promotes growth at the tips of the cluster. α = 0 is a phase transition point and corresponds to standard DLA. What makes this model interesting is that when α is negative enough (e.g. α < -2) the final cluster is an one-dimensional curve. On the other hand, when α is positive enough (e.g. α > 2) we get a nearly two dimensional disk. Thus our model encompasses the full range of fractal dimension from 1 to 2.

Frequency-dependent viscous flow in channels with fractal rough surfaces

The viscous dynamic permeability of some fractal-like channels is studied. For our particular class of geometries, the ratio of the pore surface area-to-volume tends to ∞ (but has a finite cutoff), and the universal scaling of the dynamic permeability, k(ω), needs modification. We performed accurate numerical computations of k(ω) for channels characterized by deterministic fractal wall surfaces, for a broad range of fractal dimensions. The pertinent scaling model for k(ω) introduces explicitly the fractal dimension of the wall surface for a range of frequencies across the transition between viscous and inertia dominated regimes. The new model provides excellent agreement with our numerical simulations.

Variations in fractal characteristics of active regions and flares

A multifractal analysis of the Hα images for an active region has been performed; the singularity spectra and segmented images for a narrow range of fractal dimensions have been computed for them. The segmented images show the presence of singular areas where the singularity index takes on maximum values. These areas mark the active sites of flares.

Textural variation in juvenile pyroclasts from an emergent, Surtseyan-type, volcanic eruption: The Capelas tuff cone, São Miguel (Azores)

Here I present textural data (i.e., vesicularity, vesicle size distributions (VSD), plagioclase crystallinity, crystal size distributions (CSD), combined with fractal analyses of particle outlines) from a natural succession of alternating fall and surge deposits in the emergent Capelas tuff cone (Azores). The textural variation in the Capelas succession is surprisingly small considering the wide variety of fragmentation processes, vent activity and emplacement mechanisms that are characteristic of emergent eruptions. The plagioclase crystal content varies between 24 and 33 vol.%. CSD analyses of plagioclase show near-linear trends with a slight increase in time for the smallest crystal sizes (with surge deposits having more groundmass plagioclase when compared with fall deposits). This is consistent with crystallization induced by degassing and decompression at lower eruption rates. The vesicularities of the Capelas pyroclasts are more variable (18 to 59 vol.%), with VSDs displaying kinked trends characteristic of coalescence. This is especially evident in the fall deposits, and consistent with being formed in continuous uprush (jetting) with an overall shallow fragmentation level within the conduit. Bubble coalescence can also be identified in the surge deposits, although to a much lesser extent. The amount of bubble coalescence is negatively correlated with the amount of groundmass crystallization (i.e., plagioclase) in the Capelas deposits. A relatively broad range of fractal dimensions (with average D box = 1.744 and σ = 0.032) for the outlines of pyroclastic fragments emplaced by fall or as surges indicate that there is little difference in the fragmentation process itself at Capelas. In addition to this, the fact that the fractal dimensions for both the fall and surge end-members completely overlap suggests that shape modification due to abrasion and chipping of grain edges was minor during emplacement of base surges. These results are consistent with emergent eruptions, building tuff cones, to be a relatively low-energy phreatomagmatic landform (e.g., at least when compared with more energetic phreatomagmatic eruptions producing tuff rings and maar volcanoes).