Fractal Formation Research Articles

A design procedure for improving the effectiveness of fractal layouts formation

AbstractRecently, new types of layouts have been proposed in the literature in order to handle a large number of products. Among these are the fractal layout, aiming at minimization of routing distances. There are already researchers focusing on the design; however, we have noticed that the current approach usually executes several times the allocations of fractal cells on the shop floor up to find the best allocations, which may present a significant disadvantage when applied to a large number of fractal cells owing to combinatorial features. This paper aims to propose a criterion, based on similarity among fractal cells, developed and implemented in a Tabu search heuristics, in order to allocate it on the shop floor in a feasible computational time. Once our proposed procedure is modeled, operations of each workpiece are separated in n subsets and submitted to simulation. The results (traveling distance and makespan) are compared to distributed layout and to functional layout. The results show, in general, a trade-off behavior, that is, when the total routing distance decreases, the makespan increases. Based on our proposed method, depending on the value of segregated fractal cell similarity, it is possible to reduce both performance parameters. Finally, we conclude the proposed procedure shows to be quite promising because allocations of fractal cells demand reduced central processing unit time.

2D dendritic fractal patterns from an amphiphilic polysaccharide

Fractals are ubiquitous in nature and they are also useful in many modern applications because they are able to incorporate multiple length scales within the same material. However, the bottom-up construction of fractals has been difficult because the physico-chemical principles that underpin the fractal formation have remained elusive. Here, an experimental model setup was used to investigate the emergence of fractals in 2D: Langmuir-Schaefer deposition of an amphiphilic polysaccharide (namely, trimethylsilyl cellulose, TMSC) on substrates with systematically varying surface free energies. Continuous monolayers of TMSC on water appeared to turn into 2D fractal patterns when deposited horizontally on a solid substrate of low surface free energy such as hydrophobically modified (methylated) silica. The decreased surface energy results in higher entropy, which enables fractal formation, as fractals have the highest degree of uncertainty of all 2D patterns. Furthermore, morphological changes within the fractals during a chemical reaction and water immersion were investigated. The results provide a starting point for a better understanding of the bottom-up construction of 2D fractal patterns from soft materials.

The processes of self-organization and the formation of fractals in non-crystalline solids

The processes of self-organization and the formation of fractals in non-crystalline solids

Fractal Electrode Formation in Metal–Insulator Composites Near the Percolation Threshold

Over the past 20 years, there have been a variety of experiments that have revealed a large increase in the low-frequency capacitance of devices constructed from metal-insulator nanocomposite materials. These capacitive increases are typically reported as dramatic increases in the effective relative dielectric constants of the nanocomposite materials. A class of these materials that operate at room temperature and have metal particle concentrations near the percolation threshold have been shown to have dramatic increases in the effective dielectric constant on the order of 104-1010. The simulations in this paper reveal that electrical contact to large metal clusters inside the composite material near the percolation threshold form fractal-like electrodes that deeply penetrate into the host dielectric material, which result in an effective dielectric constant that is 104-105 times greater than the host dielectric. Furthermore, insulating the planar electrodes so that no electrical contact is made to the metallic clusters inside a near-percolation-threshold composite material reveals a much smaller increase (40×) in the effective dielectric constant. Finally, a new physical scaling model and a simple geometric model for capacitance estimation in metal-insulator composites are developed and used to enhance understanding of the physical effects behind the results of these numerical simulations.

Fractal Patterning of Nanoparticles on Polymer Films and Their SERS Capabilities

We demonstrate control, via electro-hydrodynamic (EHD) induced polymer instabilities and nanoparticle mobility, of hierarchical fractal arrangements of gold nanoparticles on patterned thin polymer films. The induced changes in the film curvature enhance fractal formation for high and not for low mobility nanoparticles. The high mobility nanoparticles cluster in circular fractal networks on the rims of a hexagonally ordered array of EHD-induced polymer peaks. These arrangements exhibit plasmonic properties for surface-enhanced Raman scattering (SERS) spectroscopy.

Detection of melamine on fractals of unmodified gold nanoparticles by surface-enhanced Raman scattering

A simple way of detecting melamine in raw milk is demonstrated via surface-enhanced Raman spectroscopy (SERS) using fractals of bare and nonfunctionalized ~30 nm gold nanoparticles (AuNP) distributed on a solid support. The technique demonstrates the formation of AuNP fractals, from a random distribution, upon exposure to melamine, that enhance the Raman scattering cross-section to enable detection by SERS. The agglomeration, which is pronounced at higher melamine concentrations, is demonstrated directly through imaging, and the red-shift of the plasmon absorption peak of the AuNP fractal away from 530 nm by finite difference time domain (FDTD) calculations. The agglomeration results in a strong plasmon field, shown by FDTD, over the interparticle sites that enhances the Raman scattering cross-section of melamine and ensures unambiguous detection. Limit of detection of 100 ppb could be achieved reproducibly.

Scaling theory of continuum dislocation dynamics in three dimensions: Self-organized fractal pattern formation

We focus on mesoscopic dislocation patterning via a continuum dislocation dynamics theory (CDD) in three dimensions (3D). We study three distinct physically motivated dynamics which consistently lead to fractal formation in 3D with rather similar morphologies, and therefore we suggest that this is a general feature of the 3D collective behavior of geometrically necessary dislocation (GND) ensembles. The striking self-similar features are measured in terms of correlation functions of physical observables, such as the GND density, the plastic distortion, and the crystalline orientation. Remarkably, all these correlation functions exhibit spatial power-law behaviors, sharing a single underlying universal critical exponent for each type of dynamics.

Copper Fractal Growth Pattern in Polymer Systems

This paper presents the effect of polymer (polyvinyl alcohol) on the growth of copper fractal formation. A thin layer electrochemical cell was designed for the study of copper electrodeposition process. The structure and morphology for the copper fractal patterns were investigated by scanning electron microscopy (SEM). The copper fractal dimension was measured by using box counting method. On the basis of diffusion limited aggregation (DLA) model, growth pattern of copper fractal is two dimensional in nature. The copper fractal dimension is affected by the composition of polyvinyl alcohol. The growth pattern of copper fractal formation was measured under constant potential condition.

RESEARCH ON FUTURES TREND TRADING STRATEGY BASED ON SHORT TERM CHART PATTERN

The main task of this paper is to examine a short term trend trading strategy in futures market based on chart pattern recognition, time series and computational analysis. Specifications of historical data for technical analysis and equations for futures profitability calculations together with position size measurement are also discussed in the paper. A contribution of this paper lies in a novel chart pattern related to fractal formation and chaos theory and its application to short term up-trend trading. Trading strategy was tested with historical data of the most active futures contracts. The results have given significantly better and stable returns compared to the change of market benchmark (CRB index). The results of experimental research related to the size of trading portfolio and trade execution slippage are also discussed in the paper. The proposed strategy can be attractive for futures market participants and be applied as a decision support tool in technical analysis.

Fractal forming species and hierarchical growth in polymer electrolyte composites: Raman mapping and role of seed particles

Diffusion limited aggregation (DLA) is being reported in a polymer electrolyte composites PEO:NH4I+Al2O3 (acidic, basic and neutral). The aggregating specie is found to be I3- by using XRD and Raman spectroscopy. The involvement of NH4+ ion in the fractal pattern formation has also been explained via ion pairing with I3- (i.e., M++I3-→M+I3-), by making ammonium tri-iodide specie. Raman mapping has been done to analyze the fractal forming species in detail. A qualitative model has been proposed to explain the observations. The aggregates that get frozen in the host polymer matrix by diffusive growth form different patterns and are found to have fractal dimension that varies from 1.6 to 1.8 which is consistent with diffusion limited aggregation. The steady state of ion diffusion in polymer electrolyte is analyzed in the framework of the Nernst–Planck equation. The ion transport mechanism in the polymer electrolyte composite (with and without fractal growth), has been studied with the help of ac conductivity (σ) and ionic/electronic transference number (ti/te).

Al-Induced Crystallization of Amorphous Ge and Formation of Fractal Ge Micro-/Nanoclusters

Results on Al-induced crystallization of amorphous Ge (a-Ge) deposited by vacuum thermal evaporation techniques under thermal annealing in N(2) atmosphere are presented in detail. The a-Ge crystallization and fractal Ge pattern formation on the free surface of annealed Al/Ge bilayer films deposited on single-crystal Si (100) substrates were investigated by using scanning electron microscopy (SEM), X-ray diffraction (XRD), atomic force microscopy (AFM), energy dispersive X-ray spectrometry (EDS), and Raman spectra. It is found that the temperature field effects play an extremely crucial role in a-Ge crystallization and fractal Ge formation process. The open branched structure of fractal Ge clusters in Al/Ge bilayer films was effectively prepared by Al-induced crystallization when they were annealed at 400 °C for 60 min. These films with fractal Ge clusters exhibit charming noninteger dimensional nanostructures, which differ from those of conventional integer dimensional materials such as one-dimensional nanowires/nanorods, nanotubes, nanobelts/nanoribbons, two-dimensional heterojunctions, thin films, and zero-dimensional nanoparticles. The SEM image shows that a big Al grain was found located near the center of a fractal Ge cluster after the films were annealed at 400 and 500 °C for 60 min. This suggests that the grain boundaries of polycrystalline Al films are the initial nucleation sites of a-Ge. It also validates the preferred nucleation theory of a-Ge at triple-point grain boundaries of polycrystalline Al at the interface. This discovery may be explained by the metal-induced nucleation (MIN) mechanism.

Fractals, coherent states and self-similarity induced noncommutative geometry

The self-similarity properties of fractals are studied in the framework of the theory of entire analytical functions and the q-deformed algebra of coherent states. Self-similar structures are related to dissipation and to noncommutative geometry in the plane. The examples of the Koch curve and logarithmic spiral are considered in detail. It is suggested that the dynamical formation of fractals originates from the coherent boson condensation induced by the generators of the squeezed coherent states, whose (fractal) geometrical properties thus become manifest. The macroscopic nature of fractals appears to emerge from microscopic coherent local deformation processes.

SPONTANEOUS SPATIAL FRACTAL PATTERN FORMATION IN ABSORPTIVE SYSTEMS

We predict, for the first time to our knowledge, that purely-absorptive nonlinearity can support spontaneous spatial fractal pattern formation. A passive optical ring cavity with a thin slice of saturable absorber is analyzed. Linear stability analysis yields threshold curves for Turing (static) instabilities with features proposed as characteristics of potential fractal pattern formation. Numerical simulations of the fully-nonlinear dynamics, with both one and two transverse dimensions, confirm theoretical predictions.

Ultrasonication Study for Suspending Single-Walled Carbon Nanotubes in Water

A systematic calorimetry-based technique was developed to standardize single-walled carbon nanotube (SWNT) dispersion protocol. Simple calorimetric experiments were performed to benchmark the performance of the ultra-dismembrator. Temperature profiles for the sonication period were utilized to estimate energy input to the system. Energy loss profile was generated for the ultradismembrator in use and a calibration relationship was formulated that could standardize the sonication process. The standardized protocol was used to prepare aqueous SWNT suspensions-sonicating SWNTs in a varied range of input energy (18-100 kJ) in water. SWNT mass fractions suspended for each energy input was accurately measured and the suspended SWNT samples were characterized for morphology, surface potential, cluster size and structure, and chemical functionality using high resolution transmission electron microscopy (HRTEM), electrophoresis, dynamic and static light scattering (DLS/SLS), and Raman spectroscopy. The study demonstrated that suspended mass of SWNTs increased up to 18 kJ of energy input with no further increase upon continued energy input. The physicochemical properties showed similar trend for energy input. The aggregate cluster size, surface potential behavior, as well as the Raman defect properties plateaued after the initial energy input. The significant changes observed were limited to morphological properties, i.e., shorter length, debundled, and sharp edged SWNTs and fractal cluster formation (lower D(f)) with increased input energy.

Fractal pattern formation in thermal grooving at grain boundaries in Ag films on Si(111) surfaces

Growth of Ag films on Br- and H-passivated Si(111) surfaces and the annealing behaviour have been investigated by Rutherford backscattering spectrometry, scanning electron microscopy and photoemission electron microscopy techniques. Upon annealing the phenomenon of thermal grooving was observed in the Ag films. Depending on the annealing temperature, at an intermediate annealing time Ag depletion (evaporation) from the grain boundaries produces fractal patterns of Ag-depleted regions. Continued annealing eventually produces a percolated network of Ag-depleted regions (thermal grooves) along the grain boundaries and isolated Ag grains appear as the depth of the grooves reaches the substrate. For the fractal structures produced by thermal grooving, the fractal dimension has been estimated to be 1.60±0.04. Observation of a fractal pattern in thermal grooving was not hitherto reported. A thorough analysis of the experimental results has been carried out in the context of current theories. These theories are inadequate to describe the experimental results.

Formation of fractals by the self-assembly of interpolymer adducts of polymethacrylic acid with complementary polymers in aqueous solution

Interpolymer adducts of poly(methacrylic acid), (PMAA), with poly(vinylpyrrolidone) in presence of sodium chloride or potassium chloride form highly ordered fractal patterns in films on glass surface on drying at ambient temperature. The structure, morphology and the conditions under which the formation of fractal patterns occurs were investigated by SEM, EDX and confocal microscopic techniques. Self-organization of PMAA with complementary polymers such as poly(vinylpyrrolidone) is well-known and in the presence of sodium chloride formation of the fractals in films of the adducts is a novel observation. Fractal formation occurs due to the aggregation of interpolymer adducts. The composition of the fractals in the film is studied by EDX and confocal microscopic images of the fluorophores covalently bound to PMAA. In presence of salts, sodium chloride or potassium chloride, micellar like entities of 80 nm size were formed which further aggregate to form fractal patterns. It is suggested that the fractals result from the interpolymer adduct by Diffusion Limited Aggregation mechanism.

ANNIHILATIVE FRACTALS FORMED IN RAYLEIGH-TAYLOR INSTABILITY

We report on a fractal pattern formed in Rayleight-Taylor instability. The fractal dimension is 1.88 ± 0.06, which is constant in time t > t1 = 70 s. The pattern area decreases according to exp (-t/τ), where t and τ are time and time constant 69.8 s, respectively. τ agrees with t1. This result leads that it is important for the fractal formation that sufficient annihilation of the heavier solution at the surface.

Is Dislocation Flow Turbulent in Deformed Crystals?

Intriguing analogies were found between a model of plastic deformation in crystals and turbulence in fluids. A study of this model provides remarkable explanations of known experiments and predicts fractal dislocation pattern formation. Further, the challenges encountered resemble those in turbulence, which is exemplified in a comparison with the Rayleigh-Taylor instability.

Investigation of the formation of fractal structures in SiO2 · SnO x · CuO y thin films prepared by the sol-gel method

The processes of fractal structure formation have been considered in SiO2 · SnOx · CuOy nanofilms prepared by the sol-gel method from a tetraethoxysilane alchohol solution modified by metal salts. The atomic force microscopy images of these films have been obtained. The surface morphology has been analyzed using the Takens embedding method and Grassberger-Procaccia algorithm. The correlation and fractal dimensions of the film samples and the coefficient of surface area increase have been calculated. The effects of the annealing temperature and concentration of the doping component on the formation of fractal structures in the materials under study have been estimated.

Formation of Fractal Structures from Silicon Dioxide Nanoparticles Synthesized by RF Atmospheric Pressure Plasma Enhanced Chemical Vapor Deposition

Fractal structures were formed on silicon substrates from SiO2 nanoparticles homogeneously synthesized in low temperature atmospheric pressure plasma from tetraethoxysilane (TEOS). RF discharge (power absorbed was about 10 W) sustained between two parallel mesh electrodes was used to generate plasma. The average size of nanoparticles was in the range of 8-20 nm and was determined by process parameters. The obtained products were analyzed by SEM (scanning electron microscopy) and XPS (X-ray photoelectron spectroscopy). Values of fractal dimension parameter of bidimensionals agglomerates formed on the substrate surface from nanoparticles were calculated with the use of Gwyddion and others. It was found that values of this parameter of the deposited structures varied in the range of 1.48-2 and were determined by combination of the process parameters. An empirical model explaining mechanism of the fractal structures formation and variation of the fractal dimension parameter with the process parameters was proposed.