Self-affine Fractal Geometry Research Articles

Study on thermal contact resistance at liquid–solid interface based on fractal theory

The study of the thermal contact resistance at the liquid–solid interface is an important subject of the heat transfer of phase change materials. In this work, a fractal model for predicting the thermal contact resistance at the liquid–solid interface is established by considering the self-affine fractal geometry of the rough surface. Based on the fractal characteristic of roughness structures, topographical and mechanical analyses have been conducted to identify the position of the liquid–solid interface and determine the thermal contact resistance at the interface. The relationship between contact parameters and the thermal contact resistance are studied. Based on the analytical predictive model for thermal contact resistance at the liquid–solid interface, the three-dimensional melting process of a nanoparticle-enhanced phase change material with different thermal contact resistances was simulated by using the finite volume method, and the enthalpy-porosity model is employed. The effects of thermal contact resistance between the composite phase change material and the heat source are investigated. It is found that the augmentation of thermal contact resistance decreases the melting and heat transfer rates and the influence of thermal contact resistance becomes more pronounced with a higher volume fraction of nanoparticles.

A METHODOLOGY TO CHARACTERIZE FIBER PREFORM PERMEABILITY BY USING KARDAR–PARISI–ZHANG EQUATION

Permeability tensor describes the resistance to fluid flow through the fibrous porous media, which may not be spatially uniform. This nonuniformity in fiber architecture causes variation in the permeability value of the fibrous domain. The time evolution and geometry of the rough interfaces of the fluid flow in fibrous porous medium are analyzed using the concepts of dynamic scaling and self-affine fractal geometry, and they are shown to belong to the Kardar–Parisi– Zhang (KPZ) universality class. The resulting growth exponent, β is found to match the 1 + 1 KPZ values, and the roughness exponent, α, describes the standard deviation of the variation in fiber preform permeability. Additionally, this characterization is used to develop a tool to quantify the percentage and strength of defects within the fibrous porous media from flow front profile analysis.

Morphology Evolution Of ZnO Thin Films Deposited By Nitrogen Mediated Crystallization Method

We study the surface morphology of ZnO thin films deposited by nitrogen mediated crystallization method utilizing atomic force microscopy as a function of nitrogen flow rates. Initially, the surface morphology of ZnO thin film deposited without nitrogen exhibits a bumpy surface with spiky grains where the skewness and kurtosis values were found to be 0.48 and 4.80, respectively. By addition of small amount of nitrogen, the skewness and kurtosis values of the films significantly decrease associated with a flatter topography. Further increase in nitrogen flow rate to 16 sccm has roughened the surface shown mainly by the increase in kurtosis value to be 3.30. These results indicate that the addition of small amount of nitrogen during deposition process has enhanced the adatoms migration on the surface resulting in a superior film with a larger grain size. Two-dimensional power spectral density analysis reveals that all the films have self-affine fractal geometry with total fractal values in the range of 2.14 to above 3.00.

Three-dimensional geometry of thrust surfaces and the origin of sinuous thrust traces in orogenic belts: Insights from scaled sandbox experiments

Sinuous traces of emerging thrust tips, comprising multiple salients and recesses, are commonly observed in orogenic belts (e.g. Lesser Himalayas of India, Nepal and Bhutan) and in accretionary prisms (e.g. Nankai Trough off the coast of Japan). Lateral (along the strike of the deformation zone) variation in the depths of foreland basins (i.e. variable sediment thickness) or in the strength of the basal detachment, or presence of a curved indenter has been traditionally cited to explain the formation of salients in fold-and-thrust belts, although they are not applicable in all cases. In the present work, we have carried out four series of scaled analog model experiments using dry quartz sand, changing the dip of the basal decollément (β = 0° or 5°) and the basal friction (μb = 0.5 or 0.3) to investigate the 3D shape of thrust surfaces under varying overall boundary conditions, but without any lateral variation of these parameters, within the models. The experimental results show that under all boundary conditions, thrust surfaces are curved both in their dip and strike directions (i.e. spoon-shaped in 3D). Multiple concave-upward and convex-upward segments constitute a thrust surface, which produces a sinuous trace when the tip line intersects the Earth's surface. It is also shown that thrust surface curvatures occur at different scales, and the overall thrust surface roughness (corrugations) has a self-affine fractal geometry.

Surface Fractal Analysis for Estimating the Fracture Energy Absorption of Nanoparticle Reinforced Composites.

In this study, the fractal dimensions of failure surfaces of vinyl ester based nanocomposites are estimated using two classical methods, Vertical Section Method (VSM) and Slit Island Method (SIM), based on the processing of 3D digital microscopic images. Self-affine fractal geometry has been observed in the experimentally obtained failure surfaces of graphite platelet reinforced nanocomposites subjected to quasi-static uniaxial tensile and low velocity punch-shear loading. Fracture energy and fracture toughness are estimated analytically from the surface fractal dimensionality. Sensitivity studies show an exponential dependency of fracture energy and fracture toughness on the fractal dimensionality. Contribution of fracture energy to the total energy absorption of these nanoparticle reinforced composites is demonstrated. For the graphite platelet reinforced nanocomposites investigated, surface fractal analysis has depicted the probable ductile or brittle fracture propagation mechanism, depending upon the rate of loading.

Compact Multiband Planar Fractal Cantor Antenna for Wireless Applications: An Approach

A compact multiband fractal antenna which is a new criterion in communication is proposed. The optimized prototype measures 35 mm×31 mm×1.6 mm. The proposed antenna covers WLAN IEEE 802.11b, 802.15, PCS, GSM lower and higher bands, DCS, IMT, UMTS, Wi-Fi, and WLAN wireless applications. The proposed antenna exhibits multiband characteristics with anS11of−30.69 dB at design frequency and it is found that~70% of theS11graph below−10 dB reference is achieved. ExperimentalS11has been compared with the one which is obtained using method of moments. The aim of implementing self-affine fractal concept in antenna design makes it flexible in controlling the resonance and bandwidth. This paper investigates self-affine fractal geometry to miniaturize and to resonate multiband frequencies. The prototype model with a good agreement ofS11is reported.

On the Self-Affine Fractal Geometry of Plasma-Sprayed Surfaces

Surface roughness strongly controls essential properties of thermally sprayed wear- and corrosion-resistant coatings including their mechanical adhesion to the substrate, tribological performance, efficient retention of lubricating materials, and also the presence of sufficient carrying surface able to support the wear couple along the line of contact expressed by the Abbott-Firestone curve. The determination of the surface fractal geometry may yield useful information on the topography of plasma-sprayed coatings beyond that provided by a single roughness parameter such as R a or R z. The fractal geometry of atmospheric plasma-sprayed chromium oxide coatings, deposited according to two different statistical experimental design protocols, was assessed through determination of the Hurst exponent H of fractal Brownian motion (fBM), as well as the area-scaled fractal complexity (ASFC) obtained by triangular tessellation (“patchwork” method). Attempts were made to correlate fractality with coating adhesion strength.

Adhesive contact of rough surfaces: Comparison between numerical calculations and analytical theories

The authors have employed a numerical procedure to analyse the adhesive contact between a soft elastic layer and a rough rigid substrate. The solution to the problem, which belongs to the class of the free boundary problems, is obtained by calculating Green's function which links the pressure distribution to the normal displacements at the interface. The problem is then formulated in the form of a Fredholm integral equation of the first kind with a logarithmic kernel. The boundaries of the contact area are calculated by requiring the energy of the system to be stationary. This methodology has been employed to study the adhesive contact between an elastic semi-infinite solid and a randomly rough rigid profile with a self-affine fractal geometry. We show that, even in the presence of adhesion, the true contact area still linearly depends on the applied load. The numerical results are then critically compared with the predictions of an extended version of Persson's contact mechanics theory, which is able to handle anisotropic surfaces, as 1D interfaces. It is shown that, for any given load, Persson's theory underestimates the contact area by about 50% in comparison with our numerical calculations. We find that this discrepancy is larger than for 2D rough surfaces in the case of adhesionless contact. We argue that this increased difference might be explained, at least partially, by considering that Persson's theory is a mean-field theory in spirit, so it should work better for 2D rough surfaces rather than for 1D rough surfaces. We also observe that the predicted value of separation is in agreement with our numerical results as well as the exponents of the power spectral density of the contact pressure distribution and of the elastic displacement of the solid. Therefore, we conclude that Persson's theory captures almost exactly the main qualitative behaviour of the rough contact phenomena.

Application of a geocentrifuge and sterolithographically fabricated apertures to investigations of multiphase flow in complex fracture apertures

A geotechnical centrifuge was used to investigate unsaturated multiphase fluid flow in synthetic fracture apertures under a variety of flow conditions. The geocentrifuge subjected the fluids to centrifugal forces allowing the Bond number to be systematically changed without adjusting the fracture aperture or the fluids. The fracture models were based on the concept that surfaces generated by the fracture of brittle geomaterials have a self-affine fractal geometry. The synthetic fracture surfaces were fabricated from a transparent epoxy photopolymer using sterolithography, and fluid flow through the transparent fracture models was monitored by an optical image acquisition system. Aperture widths were chosen to be representative of the wide range of geological fractures in the vesicular basalt that lies beneath the Idaho National Laboratory (INL). Transitions between different flow regimes were observed as the acceleration was changed under constant flow conditions. The experiments showed the transition between straight and meandering rivulets in smooth walled apertures (aperture width= 0.508 mm), the dependence of the rivulet width on acceleration in rough walled fracture apertures (average aperture width = 0.25 mm), unstable meandering flow in rough walled apertures at high acceleration (20 g) and the narrowing of the wetted region with increasing acceleration during the penetration of water into an aperture filled with wetted particles (0.875 mm diameter glass spheres).

Self-affinity of fracture surfaces and implications on a possible size effect on fracture energy

As demonstrated within the last 15 years by numerous experimental studies, tensile fracture surfaces exhibit a self-affine fractal geometry in many different materials and loading conditions. In the last few years, some authors proposed to explain an observed size effect on fracture energy by this fractality. However, because they did not consider a lower bound to this scale invariance (which necessarily exists, at least at the atomic scale), they had to introduce a new definition of fracture energy with unconventional physical dimensions. Moreover, they were unable to reproduce the observed asymptotic behavior of the apparent fracture energy at large specimen sizes. Here, we show that this is because they considered self-similar fracture surfaces (not observed in nature) instead of self-affine. It is demonstrated that the ignorance of the self-affine roughness of fracture surfaces when estimating the fracture energy from the work spent to crack a specimen necessarily leads, if the work of fracture is proportional to the fracture area created, to a size effect on this fracture energy. Because of the self-affine (instead of self-similar) character of fracture surfaces, this size effect follows an asymptotic behavior towards large scales. It is therefore rather limited and not likely detectable for relatively large sample sizes ( 10 −1 m). Consequently, significant and rapid increases of the apparent fracture energy are more likely to be explained mainly by other sources of size effect.

The dynamics and geometry of solid–liquid reaction interface

The spreading of small droplets (50–200 μm of Hg on thin Ag films has been studied. The Hg front formation mechanism has been determined using optical microscope, Scanning Electron Microscope (SEM) and Atomic Force Microscope (AFM) analyses. The dynamics and geometry of the growing rough surfaces observed in our experiment, have been analyzed using the concepts of dynamic scaling and self-affine fractal geometry, in order to determine the corresponding roughness ( α) and growth ( β) exponents. We have studied two systems, one in which the silver thickness was 500–2000 Å and the other where the silver thickness was 0.1 mm. We have found that α=0.66±0.03 and β=0.46±0.02 in the first system, and α=0.77±0.04 and β=0.6±0.02 in the second one. Each of these pairs satisfies the scaling law α+ α/ β=2 within the experimental error. In addition, it was found, in both systems, that for final stages of the experiment, the roughness exponent α crosses over to α≈0.5 for relatively long length scales (order of a few microns). These findings indicate that different mechanisms govern the process in various time and length regimes: diffusion and wetting at early stages, and chemical reaction at final stages and short-length scales.

Self-affine fractal geometry of agate

Self-affine fractal geometry of agate

Statistical characteristics and origin of oscillatory zoning in crystals

Complex intracrystalline zoning patterns in hydrothermal garnet and vesuvianite and magmatic plagioclase were analyzed by statistical methods to test for fractal behavior. The zoning data were collected by electron and proton microprobe, and backscattered electron images and polarized micrographs were digitized. The analysis shows that self-affine fractal geometry can be used to characterize the zoning patterns of vesuvianite and some garnet patterns. The range of power-law scaling extended up to two decades. The results from the plagioclase samples were not sufficient to determine whether or not the zoning patterns were self-affine. The measured Hurst exponents are mostly in the range 0.25-0.45, indicating fractal scaling and anti-persistent behavior. This means that an increasing compositional trend in the past favors a decreasing trend in the future and vice versa. No distinct periodic components of the zoning patterns were found. The influence of environmental changes (external fluctuations) on a simple crystal growth model was investigated by numerical simulations. The concentration at the boundary of a diffusion layer was allowed to vary as a Brownian-motion curve, and the effect of the external fluctuation on diffusion and local growth kinetics was investigated. We conclude that factors operating on scales much larger than the local interface processes are most important in controlling the zonation.

Rock mechanics and the internet

Scale effects on the shear strength of rock joints have been recognised and investigated by many authors. The aim of this study is to explore the mechanisms underlying this phenomenon by taking into account the geometric and mechanical characteristics of the entire surface of the joint and by focusing in particular on the variation in the contact areas as a function of joint size. A special testing method has been developed to determine the contact areas on rock joint replicas subjected to direct shear tests. Test results were assessed through a mechanical model specially developed to simulate the contact between surfaces created through the application of self-affine fractal geometry. The comparisons performed on the initial results showed satisfactory agreement. The use of this mechanical model also made it possible to bring out the reduction in normal stiffness and in the ratio between contact areas and total area with increasing joint size.

Variation in contact areas of rock joint surfaces as a function of scale

Abstract Scale effects on the shear strength of rock joints have been recognised and investigated by many authors. The aim of this study is to explore the mechanisms underlying this phenomenon by taking into account the geometric and mechanical characteristics of the entire surface of the joint and by focusing in particular on the variation in the contact areas as a function of joint size. A special testing method has been developed to determine the contact areas on rock joint replicas subjected to direct shear tests. Test results were assessed through a mechanical model specially developed to simulate the contact between surfaces created through the application of self-affine fractal geometry. The comparisons performed on the initial results showed satisfactory agreement. The use of this mechanical model also made it possible to bring out the reduction in normal stiffness and in the ratio between contact areas and total area with increasing joint size.

Fractal characterization of two-dimensional aluminum corrosion fronts.

The corrosion of aluminum foils in a two-dimensional cell has been investigated experimentally. The corrosion was allowed to attack from only one side of an otherwise encapsulated metal foil. A 1M NaCl (pH=12) electrolyte was used and the experiments were controlled potentiostatically. Experiments were performed at two different potentials. Photographs from the experiments were digitized and the interfaces between the electrolyte and metal (the corrosion fronts) were identified. The corrosion fronts were analyzed using four different methods, which showed that the fronts can be described in terms of self-affine fractal geometry over a significant range of length scales. The width as a function of length showed self-affine scaling with a Hurst exponent of 0.65\ifmmode\pm\else\textpm\fi{}0.05 for both fronts. Results from corrosion experiments using zinc and copper are briefly mentioned.

Fractal dimensions of zero-noise diffusion-limited aggregation

We investigate the scaling of cluster size with mass for zero-noise needle-star DLA clusters on the square ( d = 2) and cubic ( d = 3) lattices. We find that the clusters are essentially planar ( D = 2). However, estimates of the isotropic self-similar fractal dimension via the radius of gyration yield D = 1.5 for d = 2 and D ⋍ 1.7 for d = 3. The same scaling exponents are derived from an algebraic model. We show that the planar result D = 2 is consistent with self-affine fractal geometry.

Fractal analysis of crystalline surfaces at atomic resolution

Abstract High-resolution images of ceramic oxide surfaces (e.g. MgO, NiO and BaTiO3) have been obtained in digitized form. Analytical techniques were designed to search for self-similar scaling (fractal) geometries. Three measures of surface roughness were investigated, namely coastline analysis, height-difference correlation function and variance scaling analysis. In some cases, changes in surface topology occur with time. In principle this should be analysed using self-affine fractal geometry and scaling concepts. The scaling behaviour was in all cases complicated by atomic surface faceting. Analytical results are reported for all crystalline surfaces imaged. The results are interpreted in terms of Brownian motion or self-diffusion and Brownian motion with geometrical restriction (lattice string animal models).

Dynamic scaling of the interface in two-phase viscous flows in porous media

The time evolution and geometry of rough interfaces observed in experiments on immiscible displacement of viscous fluids in porous media are analysed using the concepts of dynamic scaling and self-affine fractal geometry. The authors find that the development of the interface is governed by dynamic scaling and they have determined the corresponding surface exponents alpha and beta . The values of the exponents calculated for the experimental patterns are different from those obtained for a variety of two-dimensional models of marginally stable interfaces.

The growth of self-affine fractal surfaces

Rough surfaces generated by a wide range of different processes can be described quite well in terms of self-affine fractal geometry. Here ballistic deposition models are used to illustrate some of the fractal characteristics that are common to many rough surfaces. The models also show how computer simulations and theoretical approaches have been used in a complementary manner to obtain a better understanding of the structure and growth of random rough surfaces. Some results obtained for ballistic deposition at oblique incidence and for spatially correlated ballistic deposition represent some of the recent advances made in these directions.