Construction of fractal nanostructures based on Kepler-Shubnikov nets

A system of information codes for deterministic fractal lattices and sets of multifractal curves is proposed. An iterative modular design was used to obtain a series of deterministic fractal lattices with generators in the form of fragments of 2D structures and a series of multifractal curves (based on some Kepler-Shubnikov nets) having Cantor set properties. The main characteristics of fractal structures and their lacunar spectra are determined. A hierarchical principle is formulated for modules of regular fractal structures.     

铌酸锂晶体中的分形几何观察

研究了铌酸锂晶体沿c轴的真实生长界面及腐蚀坑模式,在两种情况中均观察到了明显的塞尔宾斯基(Sierpinski)三角垫分形几何特征,两种情况的分形维度通过计算得出均为ln3/ln2≈1.58.     

分形凹角蜂窝结构声子晶体振动带隙特性

凹角结构具有优良的减振降噪特性,可以有效衰减结构振动响应,蜂窝结构具有优良的力学特性并已经较为普遍地被应用到工程之中,因此凹角蜂窝的复合结构引起了学者们的关注。本文通过对内凹蜂窝结构分形设计,构建了一种新型声子晶体模型。基于有限元方法对分形凹角蜂窝结构进行分析,计算能带结构以及振动传输特性,分析结构的负泊松比特性以及结构分形对振动带隙的影响。通过改变壁厚和内凹角度等参数,以及在分形结构顶点处填充钢材,可以更好地对某些频段振动产生抑制。结果表明:分形结构仍具有负泊松比特性,分形结构在二阶结构产生的带隙更宽,壁厚和凹角角度的增加会导致结构振动带隙向高频区域转移,填充钢材会使带隙变宽。     

USAXS studies of monosaccharide gels. II. The common features of structural changes

The results of SAXS studies are compared for two groups of samples: (i) obtained by in situ process of gelation of α-galactose-based gel with benzene as solvent, (ii) gels of various concentrations prepared from glucofuranose-based gelator with toluene. The experimental conditions of sample preparation and SAXS measurements were very similar for both groups. The results include fractal analysis of aggregates, and – for aggregate envelopes – the radii of gyration, pair functions, and simulated models. For both groups only two aggregate types were found to be formed: the first – of simple, loose fractal structure and disk-like shape, and the second – smaller aggregates, of well-developed fractal structure and cylinder-like shape. The conditions of the change from the first type to the second one are discussed. In addition, it was verified by the SAXS method on the basis of kind of fractal analysis for the loose structure that the fractal aggregate is built of gelator molecules joined by hydrogen interaction.     

SAXS investigation of porous nanostructures

Fractal and aggregate structures of aerogels were investigated by small angle X-ray scattering in order to analyze the various evaluation methods of the SAXS data for porous nanostructures. Scattering data (SAXS, USAXS) for aerogels measured with laboratory equipment as well as synchrotron technique were interpreted in the terms of Guinier, Porod, Freltoft, Teixeira, and Emmerling theories. We modified the Freltoft fit in order to get information about the structure of elementary units. The performances of the evaluation programs were studied for different aerogels structures such as fractal of wide range, fractal of limited size, and aggregate systems. The evaluation of the scattering measurements resulted in fractal dimensions, sizes of the elementary units, sizes of the fractal domains or aggregates. Quality of the fits to SAXS data was characterized by a mathematical parameter and proved by TEM photography. TEM images confirmed the sizes of the elementary building units and clusters.     

Surface nanoscale relief of mineral crystals and its relation to nonautonomous phase formation

The features of the nanoscale relief of crystals formed under natural and experimental conditions are considered by examples of pyrite, sphalerite and its solid solutions, and native gold. Nonautonomous phases (NAPs), which differ from the bulk mineral by their chemical composition, stoichiometry, and structure, are fixed within an approximately half-micron surface layer. For pyrite crystals, the NAP composition and morphology vary depending on the genetic conditions. In the case of sphalerite, hierarchical fractal structures arise on the crystal surface due to the formation of NAPs of sulfate or sulfoxide composition. An increased content of admixed elements (Cd, Hg) is observed on the surface at the largest NAP development. Gold sulfide NAPs are observed to be stable at a relatively high temperature under hydrothermal conditions.     

Observing the twinkling fractal nature of the glass transition

A fundamental understanding of the nature and structure of the glass transition in amorphous materials is currently seen as a major unsolved problem in solid-state physics. A new conceptual approach to understanding the glass transition temperature (T_g) of glass-forming liquids called the twinkling fractal theory (TFT) has been proposed in order to solve this problem. The main idea underlying the TFT is the development of dynamic rigid percolating solid fractal structures near T_g, which are said to be in dynamic equilibrium with the surrounding liquid. This idea is coupled with the concept of the Boltzmann population of excited vibrational states in the anharmonic intermolecular potential between atoms in the energy landscape. Solid and liquid clusters interchange or “twinkle” at a cluster size dependent frequency ω_TF, which is controlled by the population of intermolecular oscillators in excited energy levels. The solid-to-liquid cluster transitions are in accord with the Orbach vibrational density of states for a particular fractal cluster g(ω) ~ ω^df − 1, where the fracton dimension d_f = 4/3. To an observer, these clusters would appear to be “twinkling.” In this paper, experimental evidence supporting the TFT is presented. The twinkling fractal characteristics of amorphous, atactic polystyrene have been captured via atomic force microscopy (AFM). Successive two-dimensional height AFM images reveal that the percolated solid fractal clusters exist for longer time scales at lower temperatures and have lifetimes that are cluster size dependent. The computed fractal dimensions, ≈ 1.88, are shown to be in excellent agreement with the theory of the fractal nature of percolating clusters in accord with the TFT. The twinkling dynamics of polystyrene within its glass transition region are captured with time-lapse one-dimensional AFM phase images. The autocorrelation cluster relaxation function was found to behave as C(t) ~ t^− 4/3 and the cluster lifetimes τ versus width R were found to be in excellent agreement with the TFT via τ ~ R^1.42. This paper provides compelling new experimental evidence for the twinkling fractal nature of the glass transition.     

Structural characteristics of silica sonogels prepared with different proportions of TEOS and TMOS

Sonohydrolysis of mixtures of tetraethoxysilane (TEOS) and tetramethoxysilane (TMOS) with different TMOS/(TMOS + TEOS) molar ratio R was carried out to obtain ∼2.0 × 10⁻³ mol SiO₂/cm³ and ∼86%-volume liquid phase wet gels. Aerogels were obtained by supercritical CO₂ extraction in autoclave. The samples were analyzed by small-angle X-ray scattering (SAXS) and nitrogen adsorption. The structure of the wet gels can be described as a mass fractal structure with fractal dimension D ∼ 2.2 and characteristic length ξ increasing from ∼4.6 nm for pure TEOS to ∼6.4 nm for pure TMOS. A fraction of the porosity is eliminated with the supercritical process. The fundamental role of the TMOS/(TMOS + TEOS) molar ratio on the structure of the aerogels is to increase the porosity and the pore mean size as R changes from pure TEOS to pure TMOS. The supercritical process increases the mass fractal dimension and shortens the fractality domain in the mesopore region. A secondary structure appearing in the micropore region of the aerogels can be described as a mass/surface fractal structure with correlated mass fractal dimension D_m ∼ 2.6 and surface fractal dimension D_s ∼ 2.3.     

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

Wet silica gels with ∼1.4 × 10⁻³ mol SiO₂/cm³ and ∼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 ξ decreasing from ∼9.4 nm 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 ∼ 2.6 and surface fractal dimension D_s ∼ 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.     

Construction of 2D quasi-periodic Rauzy tiling by similarity transformation

A new approach to constructing self-similar fractal tilings is proposed based on the construction of semigroups generated by a finite set of similarity transformations. The Rauzy tiling—a 2D analog of 1D Fibonacci tiling generated by the golden mean—is used as an example to illustrate this approach. It is shown that the Rauzy torus development and the elementary fractal boundary of Rauzy tiling can be constructed in the form of a set of centers of similarity semigroups generated by two and three similarity transformations, respectively. A centrosymmetric tiling, locally dual to the Rauzy tiling, is constructed for the first time and its parameterization is developed.     

Ferromagnetism of Fractal Hierarchy Dissipative Structures Based on Metalcontaining Heteroaromatic Systems

Abstract

Fractal hierarchy dissipative structures (FHDS) can quite possibly manifest new non-phonon type of magnetic ordering. In this paper we formulate the principles of FHDS formation from microscopic to macroscopic levels, determine physico-chemical criteria for picking up a suitable heteroaromatic system as well as the type of metal to obtain fractal ferromagnetism. Conditions of fractal ferromagnetism existance can be deduced from the Stoner's criterion of ferromagnetism in Hubbard model applied to fractal geometry.     

Effect of synthesis conditions on the fractal structure of yttrium-stabilized zirconium dioxide

The effect of synthesis conditions (sequential precipitation, coprecipitation, and sol–gel method) on the fractal structure of yttrium-stabilized zirconium dioxide has been investigated. It has been shown that the difference between methods manifests itself clearly in xerogel nanostructures, viz. by the number of fractal aggregation levels (three levels in hydroxide precipitates and two in sol–gel) and the absence of mass-fractal aggregation in sol–gel. It is been determined that the mass-fractal aggregation of precursors contributes to good filtration of hydroxide precipitates and allows preparation of oxides with soft, readily destructible aggregates, and that surface-fractal aggregation makes for good pressability of oxides.     

Structural study of fractal silica humid gels

The structures of gels of silica obtained from solutions of TMOS, methanol and water have been studied by small angle X-ray scattering (SAXS) using synchrotron radiation. The SAXS results indicate that all the gels studied exhibit fractal structures. It is inferred that gels obtained under basic conditions are volume fractals built up by surface and volume fractal units for high and low water content, respectively. The gels obtained under acidic conditions are all volume fractals having structural units much smaller than those of basic gels. The influence of aging on the structure of basic gels has also been established.     

A SAXS study of the gelation process of silicon and titanium alkoxides

The gelation process of silicon ethoxide and titanium iso-propoxide solutions was studied as a function of water content and reaction time by small-angle X-ray scattering (SAXS). Approaching the gelation points, the SAXS intensities for titanium tetra-iso-propoxide solutions start to follow a power-law decay in the Porod region, except for a H₂O/Ti ratio greater than 4. For silicon ethoxide solutions, the fractal dimension, d_f, measured for aggregated clusters increases continually with the H₂O/Si ratio and can be related to the spinnability of the solutions. For solutions of both silicon and titanium alkoxides, a solution of fractal dimension d_f < 1.79 shows spinnability, whereas solutions having d_f > 1.79 and no fractal structures do not show spinnability.     

Nanoparticle Aggregation and Fractal Growth in Fluid Smectic Membranes

We observe directly the diffusion and aggregation of buckyball clusters dispersed in thin, freely-suspended films of smectic liquid crystal using reflected light microscopy. The buckyballs at early times are barely resolvable, nanoscale clusters, which diffuse stochastically in the film. Clusters eventually coalesce to form micron-scale, fractal aggregates whose effective radius increases approximately linearly with time, so that after several days, millimeter-size fractals extend across the entire film. The measured fractal dimension of these final networks suggests that the aggregation of the buckyball clusters in the film is a diffusion-limited process.     

Percolation of polar nanoregions: A dynamic approach to the ferroelectric phase transition

The percolative nature of the ferroelectric phase transition in a potassium tantalate niobate (KTN) crystal is studied using time domain dielectric spectroscopy. A relaxation process linked to the off-center niobium ions is observed. The dynamic nature of this relaxation shows well defined temperature regimes in which it progresses from independent (Arrhenius) to cooperative (Vogel–Fulcher–Tammann) behavior. A recursive fractal model was applied in order to interpret the data obtained from the dielectric measurements. The structural parameters, ν and μ, derived from the correlation functions, enable the investigation of the onset of the phase transition in terms of the fractal dimensions of the polarization excitation.     

数学在晶体形貌研究中的应用

本文介绍了在各个历史时期数学在晶体形貌研究中的应用,重点介绍了我们近年来对数学特别是分形数学在远离平衡形貌及集合体形貌的研究中的应用.     

Fractal slice model analysis for effective thermal conductivity and temperature distribution of porous crystal layer via layer crystallization

A fractal slice model was established based on the reported model to predict the effective thermal conductivity of a porous crystal layer via layer crystallization. The temperature distribution of the crystal layer was obtained by the fractal slice method. The simulation results agreed with the experimental data better than the other theoretical models. The results were helpful to enhance the thermal transport by mitigating the thermal resistance effectively.     

Three-dimensional reconstruction of aerogels from TEM images

The knowledge of the internal structure of porous materials is of main importance to compute their physical properties. This article focuses on base-catalyzed and colloidal silica aerogels, which are fractal materials and we use an original method for the reconstruction of these aerogels from TEM images. The method used is iterative and leads to the same fractal dimension as the real material, computed from the internal two-point correlation function. Unlike the reconstruction of porous materials found in literature, our method is based on the distribution of matter and not of the porous network, and has the additional advantage of being only half statistical, i.e. only the coordinates in the z direction are statistically obtained while the coordinates x and y of the elements of matter are accurately obtained from TEM images of our samples     

Kohlrausch and Arrhenius analysis for some solid and liquid materials

Time- and temperature-dependent kinetics are analyzed for four materials: a single crystal, two liquids and an amorphous structure. While all these materials have fractal time dependence, fractal temperature dependence is found in some liquids and amorphous materials. In the single crystal case, the temperature dependency follows the regular Arrhenius equation. Some new results are found on the time and temperature dependence from the analysis and will be discussed.