Larger Fractal Dimension Research Articles

Complexity gradients in the Yilgarn Craton: Fundamental controls on crustal-scale fluid flow and the formation of world-class orogenic-gold deposits

Fractal-dimension analysis is an effective means of quantifying complex map patterns of structures and lithological contacts, which are conduits for hydrothermal fluid flow during the formation of orogenic-gold deposits. In this study, fractal dimensions, calculated on a 10 km grid across a geologic map of the Yilgarn Craton of uniform data quality, highlight relationships between geologic complexity and the location and size of Archaean orogenic-gold deposits. In the Kalgoorlie Terrane and Laverton Tectonic Zone, the largest gold deposits occur along steep gradients defined by fractal-dimension values. These steep gradients in the greenstone belts occur between massive sedimentary rock sequences of low complexity, and volcanic and intrusive rock units with more complex map patterns. The formation of world-class orogenic-gold deposits requires that hydrothermal fluids become focused from a large volume of well-connected rocks at depth, towards narrow, high-permeability zones near the location of deposit formation. Connectivity is indirectly related to permeability, and the degree of connectivity is related to the density and orientation of fluid pathways, which are quantified in map patterns using fractal-dimension analysis. Thus, fractal dimensions are a measure of the potential for increased connectivity and the likelihood of increased permeability. Greater complexity, as measured by larger fractal dimensions, implies that a certain area has the potential to produce more interconnected pathways, or zones of high connectivity. Therefore, the steep complexity gradients defined in the Kalgoorlie Terrane and Laverton Tectonic Zone correspond to areas that focused large volumes of hydrothermal fluid and enhanced the potential for significant gold mineralisation. Fractal-dimension analysis thus provides a link between empirical map features and the processes that have enhanced hydrothermal fluid flow and resulted in the formation of larger orogenic-gold deposits.

Fragmentation phenomena in populations of magmatic crystals

Fragmentation of crystals is an important mechanism, and a component of particle dynamics in igneous and metamorphic rocks that has received surprisingly little attention. Recent advances in textural analysis, extraction techniques, digital imaging, and computer-assisted measurements enable rapid accumulation of 3D data on particle shapes and size distributions. This paper reviews fragment size distributions (FSD) that result from fragmentation: lognormal, fractal, loggamma, and Weibull; discusses their genesis mechanisms; and presents relevant examples of fragmentation from experimental physics. Next, the paper considers FSDs of feldspars on digitized images of thin sections and on published images, and quartz extracted from vesicular pumice by acid from eight well-known large-volume eruptive units. The acid solution of pumice enables examination of volume abundance, 3D shapes, proportions of fragmented crystals, and measurements of their CSDs and FSDs. FSDs were also measured in samples of welded tuff and a granite disaggregated by electric pulse. Products of syneruptive shock wave fragmentation, and fragmentation by an electric pulse are found to be fractal with large breakage probabilities, branching ratios, and fractal dimensions of 2 to 3. In contrast, most quartz fragments in pumice obey a lognormal distribution and fragmentation is driven by a melt inclusion decrepitation mechanism, which results in low breakage probability and small number (2–3) of fragments per breakage cycle. These results are consistent with one atmosphere heating experiments of quartz phenocrysts that led to melt inclusion decrepitation and caused quartz to break up into several smaller pieces collectively having lognormal FSD. Measured melt inclusion size distributions suggest decrepitation of outermost melt inclusions, and low survival rate for large inclusions, and inclusions with large radius/crystal size ratio. The modeling of periodic fragmentation of crystals with melt inclusions due to overheating and/or decompression, which may occur many times during the lifetime of a long-lived magma body, may explain concave-down, lognormal CSDs abundant in igneous rocks. The genesis of lognormality can be explained by the fragmentation algorithm of Kolmogorov (1941). Other algorithms may generate lognormal-like loggamma distributions. Fragmentation serves as an important size limiting factor, a nucleation aid, and it facilitates isotopic and trace elemental exchange.

An Assessment of the Greenwood-Williamson and Other Asperities Models, With Special Reference to Electrical Conductance

Although in principle simple and neat results are obtained with the classical Greenwood-Williamson (GW) model (linearity of real contact area and conductance with load), the definition of asperity as local maxima of the surface leads to uncertain results for multiscale surfaces, as suspected already by Greenwood in a recent self-assessment of his theory [Greenwood, J. A., and Wu, J.J., 2001, “Surface roughness and contact: an apology,” Meccanica 36(6), pp. 617–630]. Quoting the conclusions in the latter paper “The introduction by Greenwood and Williamson in 1966 of the definition of a ‘peak’ as a point higher than its neighbours on a profile sampled at a finite sampling interval was, in retrospect, a mistake, although it is possible that it was a necessary mistake”. Greenwood and Wu suggest that an alternative definition of asperity captures the mechanics of the contact more correctly, that of Aramaki-Majumbdar-Bhushan (AMB). Here, numerical experiments confirm that with a Weierstrass series fractal profile (taken as a 2D slice of a true fractal surface but then used to define a set of circular asperities), load and conductance for numerically measured asperities defined “à la Greenwood-Williamson” (3PP, 3-point peaks) differ significantly from the results obtained with the Aramaki-Majumbdar-Bhushan definition of asperity. The AMB definition, which is based on the bearing area intersection best parabola fitting, gives finite limits for all quantities and varies very little with small scale terms, and tends to coincide with the 3PP method only at unrealistically large fractal dimensions D, or at unrealistically large separations. However, it remains unclear how the AMB results compare with the proper treatment of the problem when interaction effects are fully taken into account.

Avalanche-driven fractal flux distributions in NbN superconducting films

Flux distributions in thin superconducting NbN films placed in a perpendicular magnetic field have been studied using magneto-optical imaging. Below 5.5K the flux penetrates in the form of abrupt avalanches resulting in dendritic structures. Magnetization curves in this regime exhibit extremely noisy behavior. Stability is restored both above a threshold temperature T* and applied field H*, where H* is smaller for increasing field than during descent. The dendrite size and morphology are strongly T dependent, and fractal analysis of the first dendrites entering into a virgin film shows that dendrites formed at higher T have larger fractal dimension.

Fractal dimension and photoluminescence of ZnO tetrapod nanowhiskers

ZnO tetrapod nanowhiskers were prepared and characterized using fractal dimension analysis. The fractal dimension of well-developed ZnO nanowhiskers was higher than 1.7, while that of ZnO nanoparticles was lower than 1.6. In addition to UV band-edge emission, nanowhiskers of large fractal dimension having well-developed ZnO crystalline feet and a Zn phase at the tetrapod center exhibited two visible emissions at wavelengths of 419 and 438 nm at room temperature.

Large-Scale Morphology of Dispersed Layered Silicates

ABSTRACTUltra small angle x-ray scattering is used to probe the morphology of highly dispersed montmorillonite (MMT) in water and polyamide-66. In water the scattered intensity, I(q) shows a q-2 dependence for q > 0.01 Å-1, where q is the magnitude of the scattering vector. This is as expected for a two dimensional sheet-like object. On larger scales (smaller q) mass-fractal character is evident up to the radius-of-gyration of the individual scattering entities. The scattering profile is interpreted using a semi-flexible sheet model in which flat, disk-like entities of radius = 80 Å (an areal persistence length) are fractally distributed on large scales with a mass fractal dimension of 2.65. These size scales correspond to a scattering entity comprised of one or a few crumpled sheets. No evidence of inter-particle correlations is found at concentrations below the gel point. In polyamide-66 loaded with organically modified MMT long-range fractal behavior is also observed but with larger fractal dimension.

On fractal mechanism of coastline

MANDELBROT enunciated the uncertainty of the length of a coastline in his paper “How long is the coastline of Britain?” published in “Science” in 1967. The fractal concept was presented for the first time in that paper and has been applied to many fields ever since. According to the fractal theory and conditions of fractal research of coastline, the controls of faults and biologic function on the fractal character of coastline are preliminarily discussed on the basis of GIS in this paper. Finally, some significant conclusions are drawn: 1) the faults control the basic trends of coastlines of two study areas; 2) the fractal dimension of coastline of Taiwan is smaller than that of Changle — Lufeng, because the faults of Taiwan more intensely control the trend and fractal dimension of the coastline; 3) the larger the fractal dimension of the faults or the major faults, the more the controlling effect of them on the trend and fractal dimension of coastline; 4) the larger fractal dimension of the coastline of Changle — Lufeng indicates that the biologic function intensely shapes the coastline. In a word, the controls of faults and biologic function on the fractal character of coastline are discussed with a case study of China in this paper, it can be seen that faults and biologic function both have influence over the trend and fractal dimension of coastline, the fractal mechanism of coastline of two study areas may be so.

Heterotic phase in ferromagnetism of a nerve-cell-like fractal-based complex

Ferromagnetism of a nerve-cell-like fractal-based complex, in which the dendritic portions characterized by a small fractal dimension are grown on a somatic nucleus characterized by a large fractal dimension, is investigated using Monte Carlo simulations of the octahedron-spin model. The difference in fractal dimensions in a nerve-cell-like fractal-based complex creates a heterotic phase in which the spin-ordered states of a somatic nucleus and the spin-disordered states of dendritic portions are orchestrated in the vicinity of the junction of the two regions.

Morphology of horizontal cells in the retina of the capuchin monkey, Cebus apella: How many horizontal cell classes are found in dichromatic primates?

The morphology of horizontal cells was studied in the retina of dichromatic capuchin monkeys, Cebus apella. The cells were labeled with the carbocyanine dye, 1,1′,dioctadecyl-3,3,3′,3′-tetramethylindocarbocyanine perchlorate (DiI), and the labeling was then photoconverted to a stable product by using a diaminobenzidine reaction. The sizes of cell body, dendritic field, and axon terminal, as well as the number of dendritic clusters and cone convergence, were measured at increasing distance from the fovea. Three distinct morphological classes of horizontal cells were identified. Their dendritic and axonal morphology resembles those of H1, H2, and H3 cells described in trichromatic primates. The size of the cell bodies, dendritic fields, and axon terminals of all cell classes increases towards retinal periphery. H3 cells have larger dendritic fields and more dendritic clusters than H1 cells. All labeled horizontal cells located in selected patches of retina were further analyzed to quantify the differences between H1 and H3 cells. H1 cells have smaller dendritic field area, smaller total length of primary dendrites, more dendritic branching points, and larger fractal dimension than H3 cells. We have distinguished H1 and H3 cells based solely in morphological criteria. Their physiology should be further analyzed with detail, but their presence in both dichromatic and trichromatic primates suggests that neither of them have a specialized role in the red-green color opponent channel of color vision. J. Comp. Neurol. 443:105–123, 2002. © 2002 Wiley-Liss, Inc.

Morphology of horizontal cells in the retina of the capuchin monkey, Cebus apella: How many horizontal cell classes are found in dichromatic primates?

The morphology of horizontal cells was studied in the retina of dichromatic capuchin monkeys, Cebus apella. The cells were labeled with the carbocyanine dye, 1,1',dioctadecyl-3,3,3',3'-tetramethylindocarbocyanine perchlorate (DiI), and the labeling was then photoconverted to a stable product by using a diaminobenzidine reaction. The sizes of cell body, dendritic field, and axon terminal, as well as the number of dendritic clusters and cone convergence, were measured at increasing distance from the fovea. Three distinct morphological classes of horizontal cells were identified. Their dendritic and axonal morphology resembles those of H1, H2, and H3 cells described in trichromatic primates. The size of the cell bodies, dendritic fields, and axon terminals of all cell classes increases towards retinal periphery. H3 cells have larger dendritic fields and more dendritic clusters than H1 cells. All labeled horizontal cells located in selected patches of retina were further analyzed to quantify the differences between H1 and H3 cells. H1 cells have smaller dendritic field area, smaller total length of primary dendrites, more dendritic branching points, and larger fractal dimension than H3 cells. We have distinguished H1 and H3 cells based solely in morphological criteria. Their physiology should be further analyzed with detail, but their presence in both dichromatic and trichromatic primates suggests that neither of them have a specialized role in the red-green color opponent channel of color vision.

Scale-dependent spatial heterogeneity of vegetation in Mu Us sandy land, a semi-arid area of China

To examine spatial scale-dependent variation in the vegetation in Mu Ussandy land, in the semi-arid area of NW China, two transects (5 kmin length, meeting at right angle) were set, and two vegetation attributes(quantitative descriptors), vegetation cover and number of plant species, wereinvestigated. The data were analyzed combining geostatistics and fractalgeometry. Nested spatial structures were revealed for both vegetationattributes, indicating the existence of multi-scale variations. The largefractal dimensions (> 1.70) commonly found in the vegetation suggest thedominance of small-scale variations. Shifting in fractal dimensions wasdetected, implying scale-dependent spatial variation of the vegetation, as wellas its hierarchical organization. Significant differences in spatial variationof the vegetation attributes between the different directions were also found,suggesting a zonal anisotropy of geomorphologic processes.

Transport and fate of nonaqueous phase liquid (NAPL) in variably saturated porous media with evolving scales of heterogeneity

A stochastic simulation is performed to study multiphase flow and contaminant transport in fractal porous media with evolving scales of heterogeneity. Numerical simulations of residual NAPL mass transfer and subsequent transport of dissolved and/or volatilized NAPL mass in variably saturated media are carried out in conjunction with Monte Carlo techniques. The impact of fractal dimension, plume scale and anisotropy (stratification) of fractal media on relative dispersivities is investigated and discussed. The results indicate the significance of evolving scale of porous media heterogeneity to the NAPL transport in the subsurface. In general, the fractal porous media enhance the dispersivities of NAPL mass plume transport in both the water phase and the gas phase while the influence on the water phase is more significant. The porous media with larger fractal dimension have larger relative dispersivities. The aqueous horizontal dispersivity exhibits a most significant increase against the plume scale.

Analysis of Fractal Dimension of O2A Glial Cells Differentiating in Vitro

Fractal dimension is a quantitative measure of morphological complexity. Glial cells of the oligodendrocyte-type 2 astrocyte (O2A) lineage exhibit increasing morphological complexity as they differentiate in vitro. Enriched populations of O2A progenitor cells isolated from neonatal rat cerebral hemispheres or optic nerves were allowed to differentiate in vitro, and their fractal dimensions were measured over time. The fractal dimensions of the maturing cells correlated with perceived complexity; cells with elaborate process branching had larger fractal dimensions than cells with a simpler morphology. An analysis of changes in fractal dimension revealed distinct rates of growth for both oligodendrocytes and type 2 astrocytes. The fractal dimension remained constant over a 10-fold range in optical magnification, demonstrating that cultured O2A glial cells exhibit self-similarity, a defining characteristic of fractal objects. These results illustrate that fractal dimension analysis of maturing cell populations is a useful method for quantitatively describing the process of cell differentiation.

Structure function and spectral density of fractal profiles

Spectral density and structure function for fractal profile are analyzed. It is found that the fractal dimension obtained from spectral density is not exactly the same as that obtained from structure function. The fractal dimension of structure function is larger than that of spectral density for small fractal dimension, and is smaller than that of spectral density for larger fractal dimension. The fractal dimension of structure function strongly depends on the spectral density at low and high wave numbers. The spectral density at low wave number affects the structure function at long distance, especially for small fractal dimension. The spectral density at high wave number affects the structure function at short distance, especially for large fractal dimension. This problem is more serious for bifractal profiles. Therefore, in order to obtain a correct fractal dimension, both spectral density and structure function should be checked.

Surface Fractal Dimension of Anodic Dissolved Nickel Films.

Quantitative characterization of surfaces of anodic dissolved nickel films was performed using the roughness exponent, the Hurst rescaled range analysis and the Fourier transform of their AFM images. Of three methods, the first and second methods give almost the same fractal dimension 2.06 and 2.02 respectively, which indicate the roughness exponent α interchanged with the Hurst exponent, but the third method determines a little larger fractal dimension 2.25. The value of the fractal dimension, 2.0, implies that the anodic dissolved nickel films surface have no fractal structure.

Viscoelastic Changes of Rice Starch Suspensions During Gelatinization

ABSTRACT:Dynamic rheometry was utilized to characterize viscoelastic changes during heat‐induced gelatinization of 2 domestic rice starch suspensions, 1 from waxy and the other from indica rice. Gelatinization included 4 stages: suspension into sol, sol transition to gel, network destruction, and network strengthening. Increase in storage modulus (G′) was observed as early as about 47°C. For indica rice, the maximum value (G′max) was higher, and the decrease afterwards was slower, owing to its higher amylose content. The effects of heating time and temperature were found additive. Sealing samples with oil affected the accuracy of measurement. Finally, the network of starch gels had a larger fractal dimension than that of soy protein isolates previously investigated, suggesting firmer food texture.

Application of Fractal Dimension to Optimum Deposition of NiCrAlY Coating by D-Gun Spray

NiCrAlY coatings were fabricated by detonation gun (D-gun) spray under several process conditions to examine process optimization. The spray processes were adjusted by altering the composition of the mixed C2H2/O2 fuel gas, and a higher fuel temperature was obtained by increasing the ratio of C2H2 to O2 To preduce complexity for process optimization, an attempt to apply fractal dimension (FD) to the quantitative description of the splat patterns for each process was performed. The results indicated a fully melted splat usually had a larger FD, which coincided with the process conditions of dense and adhesive coating deposition. The FDs were verified to be sensitive and efficient for D-gun spray process optimization, based on the analytical results of the coating microstructures and fractured morphologies.

Velocity Profiles of Slow Blood Flow in a Narrow Tube**The project supported by the National Basic Research Project “Nonlinear Science”, National Natural Science Foundation of China and the State Education Commission through the Foundation of Doctoral Training

A fractal model is introduced into the slow blood motion. When blood flows slowly in a narrow tube, red cell aggregation results in the formation of an approximately cylindrical core of red cells. By introducing the fractal model and using the power law relation between area fraction and distance from tube axis , rigorous velocity profiles of the fluid in and outside the aggregated core and of the core itself are obtained analytically for different fractal dimensions. It shows a blunted velocity distribution for a relatively large fractal dimension , which can be observed in normal blood; a pathological velocity profile for moderate dimension , which is similar to the Segre–Silberberg effect; and a parabolic profile for negligible red cell concentration , which likes in the Poiseuille flow.

Far-field intensity of electromagnetic waves scattered from random, self-affine fractal metal surfaces

By means of the rigorous Green theorem integral equation formulation, we study the far-field intensity of linearly polarized, monochromatic electromagnetic waves scattered from a one-dimensionally rough silver surface characterized by a self-affine fractal structure. These surface fractal properties are ensured for the entire range of relevant length scales, from the illuminated spot size down to a sufficiently small (in terms of the wavelength) lower cut-off length. A peak in the specular direction is found in the angular distribution of the diffuse component of the mean scattered intensity, which becomes broader and smaller with increasing fractal dimension. For large fractal dimensions, enhanced backscattering in the case of p-polarization is observed owing to the roughness-induced excitation of surface plasmon polaritons. The interplay of different length scales of the fractal surface in the scattering process is analysed for an intermediate fractal dimension.

Fractal Dimension of Thin-film Surfaces Obtained by Fourier Spectral Analysis

The applicability of models based on fractal geometry to characterize thin-film surfaces was investigated. The fractal geometry of sputtered chromium nitride and silicon nitride thin-film surfaces was described using Fourier spectral analysis of profiles from scanning tunnelling microscopy images and atomic force microscopy images. The CrN x and SiN x coatings were deposited on silicon wafers using reactive magnetron sputtering and varying the gas pressure. The columnar structure of the amorphous silicon nitride varied with deposition, similar to the structure of the polycrystalline chromium nitride films. The fractal dimension decreases with increasing pressure. Films with fine columnar structures are characterized by a larger fractal dimension than films with coarse columnar structures.