Studies of fractal growth patterns in poly (ethylene oxide) and chitosan membranes

Polymer electrolyte membranes containing various weight percent of salt have been prepared in the authors’ laboratory for the study of their potential applications in solid state electrochemical devices; however, the polymer electrolyte membranes were found to become sort of media for fractal growth, months after they were prepared. Fractal growth patterns in polymer membranes of poly (ethylene oxide) and chitosan membranes have been analyzed, and their fractal dimensions were determined. A diffusion-limited aggregate model which is based on the Brownian motion theory is applied to simulate the experimentally obtained fractal patterns. The approach was extended by performing computer simulations with computing resources available. The fractal dimension values of the simulated and the experimentally obtained original fractal patterns were marginally close. This indicates that the simulation work has successfully produced fractal patterns that are in fairly acceptable conformity with the fractal patterns observed in the polymer membranes.     

一个关于分形的教学实验 --对大学物理实验课程创新的探索

设计了一个分形（粘性指进）的教学实验，让学生通过动手操作和计算机模拟得出各种粘性指进分形图形，并测量分形维数．从而使他们对目前活跃、有特色的科学概念——分形有所认识，有所学习．     

Fractal microstructures of light-emitting porous silicon films

By means of scanning electron microscopy and computer simulation, we have investigated the microstructures of a 23-μm-thick porous silicon (PS) film and a 6-μm-thick PS film. The two films give off strong visible emissions when excited by the 254 nm light. For the 23-μm-thick PS film, both of its cross-sectional and top-surface morphologies exhibit self-similarity whose small-scale and large-scale microstructures resemble one another. For the 6-μm-thick PS film, self-similar cracks are developed on its top surface. Our results have demonstrated that the microstructures of PS films exhibit the characteristics of fractals. With box counting method, the fractal dimensions of the PS films are calculated to be about 2.3-2.6. Based on the model of diffusion-limited aggregation, the fractal growth processes have been simulated for the PS films.     

Electric field driven fractal growth dynamics in polymeric medium

This paper reports the extension of earlier work (Dawar and Chandra, 2012) [27] by including the influence of low values of electric field on diffusion limited aggregation (DLA) patterns in polymer electrolyte composites. Subsequently, specified cut-off value of voltage has been determined. Below the cut-off voltage, the growth becomes direction independent (i.e., random) and gives rise to ramified DLA patterns while above the cut-off, growth is governed by diffusion, convection and migration. These three terms (i.e., diffusion, convection and migration) lead to structural transition that varies from dense branched morphology (DBM) to chain-like growth to dendritic growth, i.e., from high field region (A) to constant field region (B) to low field region (C), respectively. The paper further explores the growth under different kinds of electrode geometries (circular and square electrode geometry). A qualitative explanation for fractal growth phenomena at applied voltage based on Nernst–Planck equation has been proposed.     

The kinetics of adsorption on fractals: A numerical experiment

An approach similar to the “random rain” model was used to simulate the formation of a three-dimensional cluster from identical particles. The cluster had a fractal structure with the mean fractal number that increased as the number of constituting “atoms” grew. We simulated adsorption on this cluster and considered the relation between the structure of the cluster and the number of vacant centers on which adsorption could occur. By analogy with the coordination number of an atom, the notion of the coordination number of a fractal particle was introduced. The fractal and coordination numbers were shown to be interrelated. An equation for the kinetics of irreversible adsorption on fractals was suggested.     

一维随机成核生长模型

建立了一种一维随机成核生长模型，在三种不同的近邻条件下进行模拟，得到了一系列聚集生长图形，并计算了相应的分形维数．所得图形与多孔硅形成图样相似．对分形维数Ｄ随生长概率Ｘ变化的Ｄ-Ｘ曲线性质以及分形结构转变为均匀结构的临界阈值等作了初步的讨论 关键词：     

FRACTAL PATTERN GROWTH OF METAL ATOM CLUSTERS IN ION IMPLANTED POLYMERS

The fractal and multi-fractal patterns of metal atoms are observed in the surface layer and cross section of a metal ion implanted polymer using TEM and SEM for the first time. The surface structure in the metal ion implanted polyethylene terephthalane (PET) is the random fractal. Certain average quantities of the random geometric patterns contain self-similarity. Some growth origins appeared in the fractal pattern which has a dimension of 1.67. The network structure of the fractal patterns is formed in cross section, having a fractal dimension of 1.87. So it can be seen that the fractal pattern is three-dimensional space fractal. We also find the collision cascade fractal in the cross section of implanted nylon, which is similar to the collision cascade pattern in transverse view calculated by the TRIM computer program. Finally, the mechanism for the formation and growth of the fractal patterns during ion implantation is discussed.     

凝聚态物理中的分形

本文回顾了近年来与凝聚态物理相关的分形研究的一些主要工作,叙述了该领域的一些新进展。着重介绍了涉及分维和分形的一些基本概念、物理思想和研究方法,较详细地讨论了分形网络上的振动和弹性性质、磁序问题、生长和熔化、多分形以及周期分形等课题。     

液态In-80wt%Sn合金结构的分形分析

在X射线衍射实验的基础上，把分形理论用于分析In-80wt%Sn在300℃时的液态结构，发现液态结构是一种多度域分形结构，并提出液态结构多度域分形模型.通过分形分析，液态合金的整体结构被清晰地呈现出来.推测多度域两分形之间过渡区的大小和液态合金性质有关，过渡区的变化可能反映了液态合金性质的变化. 关键词： 液态合金结构 分形理论 多度域分形     

The study of fractal correlation method in the displacement measurement and its application

The classical digital speckle, or digital image, correlation method of deformation measurement is based on gray level correlation between unformed and deformed digital images. The pattern of artificial random speckles and the natural textures on some object's surfaces have fractal characteristics, and their fractal dimensions represent both gray and morph information. Furthermore, the fractal dimensions are stable feature parameters of the patterns. The digitized images of the patterns are confirmed to be also fractals. By this fact, a new method of displacement measurement is developed in the paper, based on the fractal dimensions correlation. The in-plane displacement fields of a body can be acquired. In order to verify the validity of the new method, an experiment has been designed and the results have been compared with those obtained from the classical digital image correlation method. The validity of the new method is not less than that of classical method. Further discussions about the traits and the developing vista of the method are given at the end.     

Koch fractals in physical optics and their Fraunhofer diffraction patterns

This paper gives an overview of the utilization of fractals in physical optics, especially of Koch fractals and their diffractals. The term fractal itself is defined and some basic characteristics of fractals are mentioned. Constructions of the most typical Koch curves are also depicted. Laser diffraction experiments using regular, random and modified Koch curves are described and the corresponding diffraction patterns (intensity distributions of diffractals) are shown. Some interesting properties of these diffraction patterns are discussed.     

Experimental observation and simulation of fractal patterns of Au/Ge bilayer films

In this paper, we report on the simulation of fractal clusters and the comparison with experimental fractal patterns. We found that multiple fractal clusters can be formed in Au/Ge bilayer films for different annealing times. The fractal crystallization area increases with the increase in the annealing time. The random successive nucleation model can simulate the actual growth processes of multiple growth sites. The simulating fractal clusters are in good agreement with our experimental fractal patterns. Received: 21 June 2001 / Accepted: 15 December 2001 / Published online: 3 June 2002 RID="*" ID="*"Corresponding author. Fax: +86-551/3602-803, E-mail: chenzw@ustc.edu.cn     

A tale of two fractals: The Hofstadter butterfly and the integral Apollonian gaskets

This paper unveils a mapping between a quantum fractal that describes a physical phenomena, and an abstract geometrical fractal. The quantum fractal is the Hofstadter butterfly discovered in 1976 in an iconic condensed matter problem of electrons moving in a two-dimensional lattice in a transverse magnetic field. The geometric fractal is the integer Apollonian gasket characterized in terms of a 300 BC problem of mutually tangent circles. Both of these fractals are made up of integers. In the Hofstadter butterfly, these integers encode the topological quantum numbers of quantum Hall conductivity. In the Apollonian gaskets an infinite number of mutually tangent circles are nested inside each other, where each circle has integer curvature. The mapping between these two fractals reveals a hidden D ₃ symmetry embedded in the kaleidoscopic images that describe the asymptotic scaling properties of the butterfly. This paper also serves as a mini review of these fractals, emphasizing their hierarchical aspects in terms of Farey fractions.     

Elastic properties of a disordered continuum of regular fractal structure: Self-consistent approach and finite-element method simulations

Macromolecular structures, as well as aggregation of filler in polymer-based composites, often may be described properly as fractals. Scaling behavior of the elastic moduli of a modeled fractal, the Sierpinski carpet, was the subject of this study. Sheng and Tao [1] and Patlazhan [2] found that, in the case of voids in on elastic host, axial and shear moduli exhibit distinct scaling dependencies on the size of the system. Nevertheless, it is widely accepted that moduli of random isotropic fractals (percolation clusters) scale with the same exponents. Explanation of the discrepancy is one of the main targets of the paper. The self-consistent approach and position space renormalization group technique (PSRG) have been applied for this goal. The mapping, corresponding to PSRG, was constructed numerically using the finite-element method (FEM) in the cases of voids and rigid inclusions. The self-consistent approach gives scaling behavior with exponents of values of about 0.11, independent of the modulus and type of inclusion, at developed stages of the fractal. It has been shown that mappings of PSRG on the plane, for two ratios of three independent moduli, have stable fixed points. This means that different elastic moduli exhibit scaling behavior with the same exponents (0.29 for voids and 0.17 for rigid squares) for developed fractal structure. The discrepancy in the exponent values obtained in the previous simulations is caused by the analysis of the initial stages of the structure. We believe that analogous results are valid for the wide class of self-similar fractals, and the dimension is the main parameter that governs the exponents and fixed point values.     

Subwavelength photonic band gaps from planar fractals

We show by both experiment and theory that a specific class of planar conducting fractals possesses a series of self-similar resonances, leading to multiple gaps and pass bands for electromagnetic waves over an ultrawide frequency range. A double stack of these fractal patterns exhibits polarization-independent absolute gaps over a wide range of incidence angles. These characteristics are retained even when the fractal patterns are significantly subwavelength in all dimensions. In addition, the transmittance can be modulated by an external current source.     

Diffraction Fields of Fractally Bounded Apertures

Experimental study is performed of optical diffractions by two-dimensional, self-similar mass and surface fractal apertures as well as mass fractal apertures bounded by surface fractals. Self-similar intensity distributions are observed in the Fraunhofer diffraction fields from surface and mass fractal apertures. Power law decays in average intensities of the diffraction patterns are also investigated for the mass, surface and combined fractal apertures in connection with the fractal dimensions of the objects.     

Impedance spectroscopy of PEO-lithium triflate confined in nanopores of alumina membranes

Polymeric electrolytes are very useful for their technological applications in different electrochemical devices such as batteries, electrochromic devices, smart windows, etc. One of the most studied solid electrolyte system is PEO (poly-ethylene oxide) complexed with various lithium salts. A limitation of this polymer electrolyte is low ionic conductivity. However, nanoscale manipulation of the solid polymer electrolyte has the potential to address this issue. This work discusses how it is possible to increase the PEO conductivity when this polymer is contained in nanostructures, specifically nanopores. The nanostructures used are alumina filtration membranes (thickness=6 μm, diameter=13 mm) with three different pore sizes 0.02 μm, 0.1 μm and 0.2 μm. Electrochemical characterization has been performed with an HP4194A Impedance/Gain phase analyser and Solartron 1260 Impedance/Gain phase analyser. The former instrument tests these films at a high frequency (from 100 Hz to 40 MHz) while the later at low frequency (from 1 Hz to 1 MHz). From these experiments, it has been determined that two regions of ion conduction exit. One is conduction through the bulk polymer electrolyte in the pores while the other is an interfacial conduction at the interface between the pore walls and the PEO electrolyte. The conductivity of the PEO is increased when confined in these nanostructures. Invited Scholar Research from: LiCryl — INFM (Liquid Crystal Regional Laboratory) c/o Department of Physics, University of Calabria, Via P. Bucci Cubo 31C, I-87036 Rende (CS) Italy Paper presented at the Patras Conference on Solid State Ionics - Transport Properties, Patras, Greece, Sept. 14 – 18, 2004.     

Fresnel diffraction by deterministic fractal gratings: An experimental study

Experimental results are presented for light diffraction under the Fresnel regime by two types of fractal gratings: triadic Cantor bars and Vicsek fractals. It is shown that the evolution of the diffraction patterns in the Fresnel regions reveals the hierarchical structure of deterministic fractal gratings.     

Geometric structure of ion-induced displacement cascades in solids

The geometric structure of collision cascades developed in solids by energetic ion bombardment is investigated from the view-point of fractal geometry using computer simulated trajectories and defect distributions. The fractal dimension of a collision cascade, as extracted from defect distributions, depends on the effective interaction potential governing the sequence of events in a subcascade, as shown in recent analytical predictions. Possible implications and further work along these lines are suggested.     

Adsorption of a flexible self-avoiding polymer chain: Exact results on fractal lattices

The large-scale behavior of surface-interacting self-avoiding polymer chains placed on finitely ramified fractal lattices is studied using exact recursion relations. It is shown how to obtain surface susceptibility critical indices and how to modify a scaling relation for these indices in the case of fractal lattices. We present the exact results for critical exponents at the point of adsorption transition for polymer chains situated on a class of Sierpinski gasket-type fractals. We provide numerical evidence for a critical behavior of the type found recently in the case of bulk self-avoiding random walks at the fractal to Euclidean crossover.