Evolution of the fractal structure during sintering of SnO2 compacted sol-gel powder

The structural evolution during sintering of compacted SnO2 sol-gel powder was investigated using nitrogen adsorption isotherm analysis. Results show that for sintering temperatures up to 400°C the samples have a fractal pore size distribution. As the sintering temperature increases, a structural rearragement occurs, allowing an increase of the efficiency of particle packing and the reduction of fractality. Above 400°C, the pore size growth associated with grain coalescence is the main structural change observed as the sintering temperature increases. © 1995.

A theoretical study on modelling the respiratory tract with ladder networks by means of intrinsic fractal geometry

Fractional order modeling of biological systems has received significant interest in the research community. Since the fractal geometry is characterized by a recurrent structure, the self-similar branching arrangement of the airways makes the respiratory system an ideal candidate for the application of fractional calculus theory. To demonstrate the link between the recurrence of the respiratory tree and the appearance of a fractional-order model, we develop an anatomically consistent representation of the respiratory system. This model is capable of simulating the mechanical properties of the lungs and we compare the model output with in vivo measurements of the respiratory input impedance collected in 20 healthy subjects. This paper provides further proof of the underlying fractal geometry of the human lungs, and the consequent appearance of constant-phase behavior in the total respiratory impedance.

Free inertial processes driven by Gaussian noise: Probability distributions, anomalous diffusion and fractal behavior

We study the motion of an unbound particle under the influence of a random force modeled as Gaussian colored noise with an arbitrary correlation function. We derive exact equations for the joint and marginal probability density functions and find the associated solutions. We analyze in detail anomalous diffusion behaviors along with the fractal structure of the trajectories of the particle and explore possible connections between dynamical exponents of the variance and the fractal dimension of the trajectories.

The population genetic structure of clonal organisms generated by exponentially bounded and fat-tailed dispersal

Long distance dispersal (LDD) plays an important role in many population processes like colonization, range expansion, and epidemics. LDD of small particles like fungal spores is often a result of turbulent wind dispersal and is best described by functions with power-law behavior in the tails ("fat tailed"). The influence of fat-tailed LDD on population genetic structure is reported in this article. In computer simulations, the population structure generated by power-law dispersal with exponents in the range of -2 to -1, in distinct contrast to that generated by exponential dispersal, has a fractal structure. As the power-law exponent becomes smaller, the distribution of individual genotypes becomes more self-similar at different scales. Common statistics like G(ST) are not well suited to summarizing differences between the population genetic structures. Instead, fractal and self-similarity statistics demonstrated differences in structure arising from fat-tailed and exponential dispersal. When dispersal is fat tailed, a log-log plot of the Simpson index against distance between subpopulations has an approximately constant gradient over a large range of spatial scales. The fractal dimension D-2 is linearly inversely related to the power-law exponent, with a slope of similar to -2. In a large simulation arena, fat-tailed LDD allows colonization of the entire space by all genotypes whereas exponentially bounded dispersal eventually confines all descendants of a single clonal lineage to a relatively small area.

Fractal-to-Euclidean crossover of the isotropy restoration feature in a family of fractal resistor networks

Fractal with microscopic anisotropy shows a unique type of macroscopic isotropy restoration phenomenon that is absent in Euclidean space [M. T. Barlow et al., Phys. Rev. Lett. 75, 3042]. In this paper the isotropy restoration feature is considered for a family of two-dimensional Sierpinski gasket type fractal resistor networks. A parameter xi is introduced to describe this phenomenon. Our numerical results show that xi satisfies the scaling law xi similar to l(-alpha), where l is the system size and alpha is an exponent independent of the degree of microscopic anisotropy, characterizing the isotropy restoration feature of the fractal systems. By changing the underlying fractal structure towards the Euclidean triangular lattice through increasing the side length b of the gasket generators, the fractal-to-Euclidean crossover behavior of the isotropy restoration feature is discussed.

Effects of poly(vinyl alcohol) additions on the structure of silica xerogels

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Mass fractal characteristics of silica sonogels as determined by small-angle x-ray scattering and nitrogen adsorption

A sample series of silica sonogels was prepared using different water-tetraethoxysilane molar ratio (r(w)) in the gelation step of the process in order to obtain aerogels with different bulk densities after the supercritical drying. The samples were analyzed by means of small-angle x-ray-scattering (SAXS) and nitrogen-adsorption techniques. Wet sonogels exhibit mass fractal structure with fractal dimension D increasing from similar to2.1 to similar to2.4 and mass-fractal correlation length xi diminishing from similar to13 nm to similar to2 nm, as r(w) is changed in the nominal range from 66 to 6. The process of obtaining aerogels from sonogels and heat treatment at 500degreesC, in general, increases the mass-fractal dimension D, diminishes the characteristic length xi of the fractal structure, and shortens the fractal range at the micropore side for the formation of a secondary structured particle, apparently evolved from the original wet structure at a high resolution level. The overall mass-fractal dimension D of aerogels was evaluated as similar to2.4 and similar to2.5, as determined from SAXS and from pore-size distribution by nitrogen adsorption, respectively. The fine structure of the secondary particle developed in the obtaining of aerogels could be described as a surface-mass fractal, with the correlated surface and mass-fractal dimensions decreasing from similar to2.4 to similar to2.0 and from similar to2.7 to similar to2.5, respectively, as the aerogel bulk density increases from 0.25 (r(w)=66) up to 0.91 g/cm(3) (r(w)=6).

Mass fractal characteristics of wet sonogels as determined by small-angle x-ray scattering and differential scanning calorimetry

Low density silica sonogels were prepared from acid sonohydrolysis of tetraethoxysilane. Wet gels were studied by small-angle x-ray scattering (SAXS) and differential scanning calorimetry (DSC). The DSC tests were carried out under a heating rate of 2 degrees C/min from -120 degrees C up to 30 degrees C. Aerogels were obtained by CO(2) supercritical extraction and characterized by nitrogen adsorption and SAXS. The DSC thermogram displays two distinct endothermic peaks. The first, a broad peak extending from about -80 degrees C up to practically 0 degrees C, was associated to the melting of ice nanocrystals with a crystal size distribution with pore diameter ranging from 1 or 2 nm up to about 60 nm, as estimated from Thomson's equation. The second, a sharp peak with onset temperature close to 0 degrees C, was attributed to the melting of macroscopic crystals. The DSC incremental nanopore volume distribution is in reasonable agreement with the incremental pore volume distribution of the aerogel as determined from nitrogen adsorption. No macroporosity was detected by nitrogen adsorption, probably because the adsorption method applies stress on the sample during measurement, leading to a underestimation of pore volume, or because often positive curvature of the solid surface is in aerogels, making the nitrogen condensation more difficult. According to the SAXS results, the solid network of the wet gels behaves as a mass fractal structure with mass fractal dimension D=2.20 +/- 0.01 in a characteristic length scale below xi=7.9 +/- 0.1 nm. The mass fractal characteristics of the wet gels have also been probed from DSC data by means of an earlier applied modeling for generation of a mass fractal from the incremental pore volume distribution curves. The results are shown to be in interesting agreement with the results from SAXS.

Structural characteristics of gels prepared from sonohydrolysis and conventional hydrolysis of TEOS: an emphasis on the mass fractal as determined from the pore size distribution

Silica gels were preparated from fixed proportion mixtures of tetraethoxysilane, water and hydrocloric acid, using either ultrasound stimulation (US) or conventional method (CO) in the hydrolysis step of the process. Wet gets were obtained with the same silica volume concentration and density. According to small-angle X-ray scattering, the structure of the wet gels can be described as mass fractal structures with mass fractal dimension D = 2.20 in a length scale xi = 7.9 nm, in the case of wet gels US, and D = 2.26 in a length scale 6.9 nm, in the case of wet gels CO. The mass fractal characteristics of the wet gels US and CO account for the different structures evolved in the drying of the gels US and CO in the obtaining of xerogels and aerogels. The pore structure of the dried gels was studied by nitrogen adsorption as a function of the temperature. Aerogels (US and CO) present high porosity with pore size distribution (PSD) curves in the mesopore region while xerogels (US and CO) present minor porosity with PSD curves mainly in the micropore region. The dried gels US (aerogels and xerogels) generally present pore volume and specific surface area greater than the dried gels CO. The mass fractal structure of the aerogels has been studied from an approach based on the PSD curves exclusively. (c) 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

Mass fractal characteristics of sonogels prepared from sonohydrolysis of tetraethoxysilane with additions of dimethylformamide

Wet silica gels with similar to 1.4 x 10(-3) mol SiO2/cm(3) and similar to 90 vol.% liquid phase were prepared from the sonohydrolysis of tetraethoxysilane (TEOS) with different additions of dimethylformamide (DMF). Aerogels were obtained by CO2 supercritical extraction. The samples were studied mainly by small-angle X-ray scattering (SAXS) and nitrogen adsorption. Wet gels exhibit a mass fractal structure with fractal dimension D increasing from 2.23 to 2.35 and characteristic length xi decreasing from similar to 9.4 nm to similar to 5.1 nm, as the DMF/TEOS molar ratio is increased from 0 to 4. The supercritical process apparently eliminates some porosity, shortening the fractality domain in the mesopore region and developing an apparent surface/mass fractal (with correlated mass fractal dimension D-m similar to 2.6 and surface fractal dimension D-s similar to 2.3) in the micropore region. The fundamental role of the DMF addition on the structure of the aerogels is to diminish the porosity and the pore mean size, without, however, modify substantially the specific surface area and the average size of the silica particle of the solid network. (c) 2006 Elsevier B.V. All rights reserved.

Structure of Hydrophobic Ambient-Pressure-Dried Aerogels Prepared by Sonohydrolysis of Tetraethoxysilane with Additions of N,N-Dimethylformamide

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Mass and surface fractal in supercritical dried silica aerogels prepared with additions of sodium dodecyl sulfate

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Characterization of rock joints by fractal analysis

From a physical perspective, a joint experiences fracturing processes that affect the rock at both microscopic and macroscopic levels. The result is a behaviour that follows a fractal structure. In the first place, for saw-tooth roughness profiles, the use of the triadic Koch curve appears to be adequate and by means of known correlations the JRC parameter is obtained from the angle measured on the basis of the height and length of the roughnesses. Therefore, JRC remains related to the geometric pattern that defines roughness by fractal analysis. In the second place, to characterise the geometry of irregularities with softened profiles, consequently, is proposed a characterisation of the fractal dimension of the joints with a circumference arc generator that is dependent on an average contact angle with regard to the mid-plane. The correlation between the JRC and the fractal dimension of the model is established with a defined statistical ratio.

Atomic-level structure and deformation in metallic glasses

Metallic glasses (MGs) are a relatively new class of materials discovered in 1960 and lauded for its high strengths and superior elastic properties. Three major obstacles prevent their widespread use as engineering materials for nanotechnology and industry: 1) their lack of plasticity mechanisms for deformation beyond the elastic limit, 2) their disordered atomic structure, which prevents effective study of their structure-to-property relationships, and 3) their poor glass forming ability, which limits bulk metallic glasses to sizes on the order of centimeters. We focused on understanding the first two major challenges by observing the mechanical properties of nanoscale metallic glasses in order to gain insight into its atomic-level structure and deformation mechanisms. We found that anomalous stable plastic flow emerges in room-temperature MGs at the nanoscale in wires as little as ~100 nanometers wide regardless of fabrication route (ion-irradiated or not). To circumvent experimental challenges in characterizing the atomic-level structure, extensive molecular dynamics simulations were conducted using approximated (embedded atom method) potentials to probe the underlying processes that give rise to plasticity in nanowires. Simulated results showed that mechanisms of relaxation via the sample free surfaces contribute to tensile ductility in these nanowires. Continuing with characterizing nanoscale properties, we studied the fracture properties of nano-notched MGnanowires and the compressive response of MG nanolattices at cryogenic (~130 K) temperatures. We learned from these experiments that nanowires are sensitive to flaws when the (amorphous) microstructure does not contribute stress concentrations, and that nano-architected structures with MG nanoribbons are brittle at low temperatures except when elastic shell buckling mechanisms dominate at low ribbon thicknesses (~20 nm), which instead gives rise to fully recoverable nanostructures regardless of temperature. Finally, motivated by understanding structure-to-property relationships in MGs, we studied the disordered atomic structure using a combination of in-situ X-ray tomography and X-ray diffraction in a diamond anvil cell and molecular dynamics simulations. Synchrotron X-ray experiments showed the progression of the atomic-level structure (in momentum space) and macroscale volume under increasing hydrostatic pressures. Corresponding simulations provided information on the real space structure, and we found that the samples displayed fractal scaling (r^d ∝ V, d < 3) at short length scales (< ~8 Å), and exhibited a crossover to a homogeneous scaling (d = 3) at long length scales. We examined this underlying fractal structure of MGs with parallels to percolation clusters and discuss the implications of this structural analogy to MG properties and the glass transition phenomenon.

Modeling of the lung impedance using a fractional order ladder network with constant phase elements

The self similar branching arrangement of the airways makes the respiratory system an ideal candidate for the application of fractional calculus theory. The fractal geometry is typically characterized by a recurrent structure. This study investigates the identification of a model for the respiratory tree by means of its electrical equivalent based on intrinsic morphology. Measurements were obtained from seven volunteers, in terms of their respiratory impedance by means of its complex representation for frequencies below 5 Hz. A parametric modeling is then applied to the complex valued data points. Since at low-frequency range the inertance is negligible, each airway branch is modeled by using gamma cell resistance and capacitance, the latter having a fractional-order constant phase element (CPE), which is identified from measurements. In addition, the complex impedance is also approximated by means of a model consisting of a lumped series resistance and a lumped fractional-order capacitance. The results reveal that both models characterize the data well, whereas the averaged CPE values are supraunitary and subunitary for the ladder network and the lumped model, respectively.