Self-affine Fractal Structure Research Articles

Metallophosphazene Precursor Routes to the Solid-State Deposition of Metallic and Dielectric Microstructures and Nanostructures on Si and SiO2

We present a method for the preparation and deposition of metallic microstructures and nanostructures deposited on silicon and silica surfaces by pyrolysis in air at 800 degrees C of the corresponding metallophosphazene (cyclic or polymer). Atomic force microscopy studies reveal that the morphology is dependent on the polymeric or oligomeric nature of the phosphazene precursor, on the preparation method used, and on the silicon substrate surface (crystalline or amorphous) and its prior inductively couple plasma etching treatment. Microscale and nanoscale structures and high-surface-area thin films of gold, palladium, silver, and tin were successfully deposited from their respective newly synthesized precursors. The characteristic morphology of the deposited nanostructures resulted in varied roughness and increased surface area and was observed to be dependent on the precursor and the metal center. In contrast to island formation from noble metal precursors, we also report a coral of SnP(2)O(7) growth on Si and SiO(2) surfaces from the respective Sn polymer precursor, leaving a self-affine fractal structure with a well-defined roughness exponent that appears to be independent (within experimental error) of the average size of the islands. The nature of the precursor will be shown to influence the degree of surface features, and the mechanism of their formation is presented. The method reported here constitutes a new route to the deposition of single-crystal metallic, oxidic, and phosphate nanostructures and thin films on technologically relevant substrates.

Modeling the self-affine structure and optimization conditions of city systems using the idea from fractals

This paper demonstrates self-affine fractal structure of city systems by means of theoretical and empirical analyses. A Cobb–Douglas-type function (C–D function) of city systems is derived from a general urban response equation, and the partial scaling exponent of the C–D function proved to be the fractal dimension reflecting the self-affine features of city systems. As a case, the self-affine fractal model is applied to the city of Zhengzhou, China, and the result is satisfying. A fractal parameter equation indicative of structural optimization conditions is then obtained from the C–D function. The equation suggests that priority should be given to the development of the urban element with a lower fractal dimension, or a higher partial scaling exponent, for utility maximization. Moreover, the fractal dimensions of different urban elements tend to become equivalent to each other in the long term. Accordingly, it is self-similar fractals rather than self-affine fractals that represent the optimal structure of city systems under ideal conditions.

Strong water repellency effect on SiO2 films processed at a nanometric scale

The present study deals with the creation of nano-rough surfaces with stable and controlled high hydrophobicity. These surfaces were obtained by combining the ion track etching technique with a simple functionalization by grafting perfluoroctyltrichlorosilane (PFOTS) molecules. Surface morphology was investigated by AFM observations which evidenced a self-affine fractal structure with a fractal dimension Df ~ 2.6. The study of the wetting properties of these surfaces allowed to elucidate the conditions for observing a high hydrophobicity phenomenon and to predict the contact angle values for surfaces designed at a nanometric scale.

The fractal structure of a carbon deposit formed during graphite spraying in an electric arc

Experimental data obtained by atomic force microscopy showing that a carbon deposit formed during graphite spraying in an electric arc possesses a fractal structure. The deposit is composed of clusters with dimensions on the order of 1–5 nm. The surface and profile of the deposit exhibit a self-affine fractal structure. The fractal dimensions of the deposit surface and profile, as well as the Hurst index characterizing the self-affine fractal, are determined.

A study on the simulated diffusion-limited current transient of a self-affine fractal electrode based upon the scaling property

The power law relation I( t)∝ t − α , α=( D f−1)/2 between current I and time t has been widely used to analyse current transient (chronoamperometric) behaviour during atomic/ionic diffusion towards the electrode with a fractal dimension D f, irrespective of whether the electrode has a self-similar fractal structure or at best a self-affine fractal structure. We show that the self-affine fractal dimension D f,sa (or Hurst exponent H) is not always the sufficient condition required for describing the atomic/ionic diffusion behaviour to the self-affine fractal electrode: the current transient exhibits a more negative power dependence of current on time before temporal outer cut-off of fractality with increasing morphological amplitude (roughness factor) of the self-affine fractal electrode, rather than a unique power dependence corresponding to D f,sa. It is particularly noted that the current transients from the electrodes with comparatively large amplitudes are roughly characterised by a two-stage power dependence before temporal outer cut-off of fractality. In the present work, a practical method has been suggested to interpret the anomalous current transient from the self-affine fractal electrodes with various amplitudes. This method includes the determination of the apparent self-similar scaling properties of the self-affine fractal structure by the triangulation method.

Theory of rubber friction: Nonstationary sliding

When rubber slides on a hard, rough substrate, the surface asperities of the substrate exert oscillating forces on the rubber surface leading to energy ``dissipation'' via the internal friction of the rubber. In this paper we extend an earlier published theory [B.N.J. Persson, J. Chem. Phys. $115,$ 3840 (2001)] to nonstationary sliding, and present a discussion of how the area of real contact and the friction force depend on the nature of the substrate surface roughness and on the history of the sliding motion. We consider in detail the case when the substrate surface has a self-affine fractal structure.

Self-Affine Fractal Characterization of a TNT Fracture Surface

AbstractA trinitrotoluene (TNT) fracture surface image is characterized in terms of a self-affine fractal structure. The fracture surface was produced by high acceleration in an ultracentrifuge when the TNT strength was exceeded. An atomic force microscope (AFM) captured the topography of a 4 micron square region on the fracture surface. The present analysis supports a self-affine description of the TNT fracture surface (wavelengths of 0.016 micron to 4.0 micron) and provides a new prespective on fracture processes in TNT. An essential step in self-affine fractal characterization of surfaces is the determination of reference surfaces. A self-affine fracture surface can be described in terms of a single-valued height function. In the TNT fracture surface, single-valued height functions, which describe surface texture can only be defined with respect to curved reference surfaces. By employing curved reference surfaces, we have demonstrated that self-affine fractal scaling can be used to characterize the TNT fracture surface. This provides important information that is not evident in the analysis of individual surface scans.

Atomic Force Microscopy Study of Self-Affine Fractal Roughness of Porous Silicon Surfaces

Atomic force microscopy was used to measure the “roughness” exponents of the rough porous silicon surfaces formed by electrochemical anodization. It was found that the porous Si surfaces have a self-affine fractal structure characterized by roughness exponents α=0.53–0.64, fractal dimensions D=2.36–2.47 and a growth exponent β=0.24 ±0.02.

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.

Strong surface field enhancements in the scattering of p-polarized light from fractal metal surfaces

We study the light scattering from a one-dimensional rough surface characterized by a self-affine fractal structure. The rigorous Green theorem, integral equations formulation is employed to calculate the scattered electromagnetic field. An increase in the roughness-induced excitation of p-polarized surface plasmon polaritons is found with increasing fractal dimension, leading to very large surface field enhancements.

Atomic force microscopy study on the surface structure of oxidized porous silicon

We study the surface structure of porous silicon (PS) using atomic force microscopy (AFM), before and after oxidation in a HNO 3 solution. The AFM image shows the PS surface with a self-affine random fractal structure of wires, hillocks and voids in various scales. After oxidization the wires and hillocks of PS structures are glazed with oxide and the voids are filled. PS structure is altered to a simple self-affine fractal structure of hillock clusters. The fractal dimension D of PS is around 2.3, which decreases with increasing oxidization to about 2.0 of a smooth surface for the saturated oxidization. Our direct observation of the fractal structure of PS from AFM data reveals a good explanation for the recently found novel nonlinear dc-response in Ag thin films deposited on PS. We find the fractal surface structure of oxidized PS responds to the stepwise avalanche electric breakdown of the resistivity of Ag thin films deposited on oxidized PS.

Random Cantor set models for the elastic-perfectly plastic contact of rough surfaces

The objective of this study was to formulate discrete and continuous spatial models to describe the elastic-perfectly plastic deformation of two rough surfaces in contact. The two surfaces in contact are assumed to exhibit fractal behavior and are modeled as an effective fractal surface compressed into a smooth rigid substrate. The rough self-affine fractal structure of the effective surface is approximated by a random Cantor set representation embedded in two dimensions. Both of the proposed models admit analytical solutions whether the plastic deformation is volume conserving or not. Presented results illustrate the effects that volume conservation and initial surface structure have on the elastic-perfectly plastic deformation process. The results from the continuous model are compared with the results obtained from the discrete model, and existing experimental load displacement data for the deformation of a bead-blasted steel surface.

A Fractal Model for the Rigid-Perfectly Plastic Contact of Rough Surfaces

In this study a continuous asymptotic model is developed to describe the rigid-perfectly plastic deformation of a single rough surface in contact with an ideally smooth and rigid counter-surface. The geometry of the rough surface is assumed to be fractal, and is modeled by an effective fractal surface compressed into the ideally smooth and rigid counter-surface. The rough self-affine fractal structure of the effective surface is approximated using a deterministic Cantor set representation. The proposed model admits an analytic solution incorporating volume conservation. Presented results illustrate the effects of volume conservation and initial surface roughness on the rigid-perfectly plastic deformation that occurs during contact processes. The results from this model are compared with existing experimental load displacement results for the deformation of a ground steel surface.

Fractal models of elastic-perfectly plastic contact of rough surfaces based on the Cantor set

The objective of this study was to formulate discrete and continuous models to describe the elastic-perfectly plastic deformation of two rough surfaces in contact. The two surfaces in contact are assumed to exhibit fractal behavior and are modeled as an effective fractal surface compressed into a smooth rigid substrate. The rough self-affine fractal structure of the effective surface is approximated using a Cantor set representation. Both of the proposed models admit analytical solutionś for the cases when the plastic deformation is volume conserving or not. Results are presented that illustrate the effects that volume conservation and initial surface structure have on the elastic-perfectly plastic deformation process. The results from the continuous model are compared with the results obtained from the discrete model, and also with existing experimental load displacement results for the deformation of a ground steel surface.