Synthesis, structure, and dielectric properties of KH2PO4 fractal aggregates

Macroscopic fractal aggregates of KH₂PH₄ (KDP) measuring up to 500 μm have been obtained. The fractal structure forms as a result of the precipitation of KDP particles from a supersaturated aqueous solution in the presence of a temperature gradient followed by a diffusioncontrolled mechanism of aggregation. The electron-microscopic analysis performed has shown that the fractals are formed predominantly from crystallites of the tetragonal modification measuring ∼1 μm. The dielectric constant (ɛ) of fractal KH₂PO₄ has been measured in the temperature range 80–300 K. A characteristic anomaly has been discovered on the ɛ(T) curve in the vicinity of 122 K, which attests to a ferroelectric phase transition. The absolute value of ɛ is significantly smaller than the components ɛ ₁₁ and ɛ ₃₃ for KH₂PO₄. Fiz. Tverd. Tela (St. Petersburg) 41, 2059–2061 (November 1999)     

Fractal structure of marine measuring networks

Summary This paper deals with the fractal character of several distribution points measuring the Earth's total magnetic field (ETMF) during some marine surveys that we carried out in the Gulf of Pozzuoli and around the island of Ischia (Naples, Italy). Previous studies evaluated the fractal dimension of networks constituted by fixed measuring stations. The examined distributions display a ?fractal scaling? regime on 3–4 decades of distances with fractal dimensions of 1.80 and 1.54. These values characterize the whole investigated area covered by the distribution of experimental data. Linked to fractal dimension is the possibility of reconstructing the field and the capacity to reveal sparse and intense phenomena (e.g. magnetic anomalies) with a low fractal dimension. In distributions with large areas without measuring points, Shannon's theorem is no longer applicable in 2D. The fractal evaluation enables a spatial limit to record the phenomena to be defined and therefore establishes the minimum wavelength recordable of the magnetic field.     

A shell-model approach to fractal-induced turbulence

We present a shell-model of fractal induced turbulence which predicts that structure function scaling exponents decrease in absolute value as the fractal dimension of the turbulence-inducing fractal object increases. This qualitative prediction is in agreement with laboratory measurements. Finer details of the fractal induced turbulence statistics and dynamics depend on the fractal force's phases, i.e. on the detailed construction of the fractal stirrer. In a case of deterministic forcing phases, a critical fractal dimension exists below which the average rate of inter-scale energy transfer <T _n> is a decreasing function of the wavenumber k_n and the structure function scaling exponents take close to Kolmogorov values. Above this critical fractal dimension, <T _n> is an increasing function of k_n and the structure function scaling exponents deviate significantly from Kolmogorov values. Received 25 June 2001 / Received in final form 5 April 2002 Published online 19 July 2002     

Comment on the equivalence between fracton/spectral dimensionality,and the dimensionality of recurrence

The fracton/spectral dimensionality ^D fracton is an important characteristic of fractal processes. Until now, it has not been interpreted as the fractal dimensionality of any well-defined fractal set, and it has been claimed that ^D fracton is more intrinsic than the fractal dimensionality. In fact, ^D fractol, is best understood as an originaldynamical reinterpretation of a well-defined previously knownkinetic dimensionality: ^D fracton 's twice the fractal codimensionality of the time instants when a fractal process returns to a point it had previously visited.Presented at theThird Conference on Fractals: Fractals in the Physical Sciences, held at the National Bureau of Standards, Gaithersburg, Maryland, on November 20–23, 1983.While this talk has been widely discussed, its text was not available for inclusion in these proceedings.     

On discontinuity and tail behaviours of the integrated density of states for nested pre-fractals

We consider a general finitely ramified fractal set called a nested fractal which is determined byN number of similitudes. Basic properties of the integrated density of statesN (x) for the discrete Laplacian on the associated nested prefractal are investigated. In particulard N is shown to be purely discontinuous ifM, whereM is the number of branches of the inverse of the rational function involved in the spectral decimation method due to Rammal-Toulouse. Sierpinski gaskets and the modified Koch curve are special examples.     

Bifurcation phenomena in a periodically driven current filament and a conjecture on the turbulent patterns by computer simulations

Bifurcation routes to chaos in a periodically driven current filament have been studied by computer simulations. By an impact ionization model, theS-shaped currentvoltage curve is perturbed by the dc+ac bias ofE ₀+E _acsin(27f ₀t). The bifurcation maps are described as a function ofE ₀. In the prebreakdown region, the fractal basin boundary, the crisis and the intermittency are discussed, based on the general considerations of the carrier dynamics on the catastrophe manifold. The intermittent burst of the current filament is explained by the destabilization of the weak turbulence generated in the lower branch. In the diffusion-reaction model, the spatio-temporal mode patterns of the transverse carrier profile have revealed the competitive evolution of the hyper-freezing and the firing.     

Enhancement of the backscattered intensity from fractal aggregates

Abstract

Comparative measurements were conducted for the backscattered intensities of light from uniform random and fractal aggregated media. Different features are found for the backscattered intensity peak shapes. A crossover between the θ^1?D and θ^?2 dependences of the backscattered intensity occurs in the case of fractal aggregated medium, where D indicates the fractal dimension.     

Stochastic Lagrangian Particle Approach to Fractal Navier-Stokes Equations

In this article we study the fractal Navier-Stokes equations by using the stochastic Lagrangian particle path approach in Constantin and Iyer (Comm Pure Appl Math LXI:330–345, 2008). More precisely, a stochastic representation for the fractal Navier-Stokes equations is given in terms of stochastic differential equations driven by Lévy processes. Based on this representation, a self-contained proof for the existence of a local unique solution for the fractal Navier-Stokes equation with initial data in \mathbb W^1,p{{\mathbb W}^{1,p}} is provided, and in the case of two dimensions or large viscosity, the existence of global solutions is also obtained. In order to obtain the global existence in any dimensions for large viscosity, the gradient estimates for Lévy processes with time dependent and discontinuous drifts are proved.     

An examination of the shrinking-core model of sub-micron aluminum combustion

We revisit the shrinking-core model of sub-micron aluminum combustion with particular attention to the mass flux balance at the reaction front which necessarily leads to a displacement velocity of the alumina shell surrounding the liquid aluminum. For the planar problem this displacement simply leads to an equal displacement of the entire alumina layer, and therefore a straightforward mathematical framework can be constructed. In this way we are able to construct a single curve which defines the burn time for arbitrary values of the diffusion coefficient of O atoms, the reaction rate, the characteristic length of the combustion field, and the O atom mass concentration within the alumina provided that it is much smaller than the aluminum density. This demonstrates a transition between a ‘d  ²–t’ law for fast chemistry and a ‘d–t’ law for slow chemistry. For the spherical geometry, the one of physical interest, the outward displacement velocity creates not a simple displacement, but a stress field which, when examined within the framework of linear elasticity, strongly suggests the creation of internal cracking. We note that if the molten aluminum is pushed into these cracks by the high internal pressure characteristic of the stress field, its surface, where reaction occurs, could be fractal in nature and affect the fundamental nature of the burning law. Indeed, if this ingredient is added to the planar model, a single curve for the burn time can again be derived, and this describes a transition from a ‘d  ²–t’ law to a ‘d  ^ν–t’ law, where 0<ν<1.     

Fractal modelling of turbulent mixing

The aim of this work is to propose a new model for turbulent flows, called the fractal model (FM), applicable both in a Reynolds averaged Navier–Stokes (RANS) and a large-eddy simulation (LES) formulation, with the ultimate goal of applying it to simulate turbulent combustion irrelevant of its mode (premixed or non-premixed). The model is able to turn itself off in the laminar zones of the flow, and in particular near walls. It is based on the fractal theory. It describes the physics of the smaller spatial scales and therefore represents a small-scales model.

FM describes the physics of the small scales of turbulence based on the phenomenological concept of vortex cascade and on the self-similar behaviour of turbulence in the inertial range. Such a model is used in each cell of a numerical calculation. A characteristic length Δ is associated to each cell, and the local energy u ³ _Δ/Δ is distributed over a certain number of eddies, which depends on the local Reynolds number Re _Δ. Each vortex of the cascade generates N _c vortices; the recursive process of vortex generation terminates at the dissipative scale level, i.e. when the eddy Reynolds number is equal to one. FM is also able to estimate the volume fraction occupied by the dissipative fine structures of turbulence; this quantity is critical in reactive turbulent flows.

The physics of small scales is summarized by a turbulent ‘viscosity’ μ_t, to be added to the molecular one. μ_t is zero where the flow is laminar and, in particular, goes to zero at solid walls. Assuming μ_t to be isotropic, FM is applicable in a RANS formulation (IFM, isotropic fractal model). The model can be extended to the anisotropic case (AFM, anisotropic fractal model) and therefore used to close the transport equations in an LES approach. In the present paper, the model (IFM) is used in a RANS approach and is validated through a test case studied experimentally by Johnson and Bennett, and numerically (with LES) by Akselvoll and Moin. The results obtained are in good agreement both with the experimental and the numerical ones. Other tests are being performed.     

测量单泡声致发光中气泡R(t)曲线的前向Mie散射技术

光干涉原理和Mie理论计算结果表明,单泡声致发光中气泡前向Mie散射的振荡信号,主要是由于气泡的透射光束和表面反射光束之间光干涉产生的. 这些干涉波峰形成了测量气泡半径的空间标尺,标尺的单位长度δR由散射角θ,检测光波波长和流体光折射率确定,而每个波峰就是标尺的刻线,它们与该时刻的气泡半径大小一一对应. 在30°—50°散射角范围内,利用前向Mie散射实验测定了单泡声致发光中气泡的最大半径,R(t)曲线及平衡半径,表明前向Mie散射是一种便捷的测定气泡运动特性的有效方法. 关键词： 单泡声致发光 前向Mie散射 光干涉     

Effect of the entrance channel on the synthesis of superheavy elements

Cross-sections for the synthesis of superheavy elements were analyzed using the concept of a dinuclear system. Experimental values for the production of elements Z = 104, 108, 110, 111 and 112 by cold fusion reactions with targets of ²⁰⁸Pb and ²⁰⁹Bi were reproduced. The model reveals the importance of entrance channel dynamics and competition between quasi-fission and complete fusion processes. Energy windows were observed which allow capture of the reacting nuclei and formation of the compound nucleus. The quantities were studied which are significant for the interaction dynamics of massive nuclei in the entrance channel.     

Dynamic behavior of the voter model on fractals: logarithmic-periodic oscillations as a signature of time discrete scale invariance

The understanding of the dynamic behavior of the voter model, in low-dimensional media, is a very interesting open topic. In fact, due to the absence of the interfacial tension, only the interfacial noise becomes relevant during the coarsening processes, bringing the possibility of studing a new physical process. In this way, it is known that below the upper critical dimension (d < 2) and starting from a disordered configuration, a critical coarsening process takes place, and the density of interfaces, ρ(t), decays as a power-law function of time. Recently published numerical studies performed on low-dimensional fractal substrates (d _F < 2) [Physica A 362, 338 (2006)] show the existence of logarithmic-periodic oscillations superimposed on the standard ρ(t) power-law behavior, but the origin of those oscillations remains unclear. In this work, we provide an explanation of these oscillations in terms of the interplay between the dynamics of the voter model and the discrete scale invariance of the underlying fractal substrate. Our arguments are verified by means of extensive numerical simulations carried out on different fractal substrates.     

Some fractal properties of the percolating backbone in two dimensions

A new algorithm is presented, based on elements of artificial intelligence theory, to determine the fractal properties of the backbone of the incipient infinite cluster. It is found that the fractal dimensionality of the backbone isd _f ^BB =1.61±0.01, the chemical dimensionality isd _t=1.40±0.01, and the fractal dimension of the minimum pathd _min=1.15 ± 0.02 for the two-dimensional triangular lattice.     

Compressibility and Fractal Dimension of Fine Coal Particles in Relation to Pore Structure Characterisation Using Mercury Porosimetry

Mercury porosimetry has been applied to characterize the pore structure of fine coals particles. Interparticle voids and compressibility effects on the mercury intrusion data were examined. It is found that coal compressibility has a significant effect on mercury porosimetry data when pressure P>20 MPa. The compressibility of the two coals used was determined to be 3.13×10⁻¹⁰ m² N⁻¹ and 2.50×10⁻¹⁰ m² N⁻¹ for CA and GO coals, respectively. Fractal dimension analysis provides a “fingerprint” to distinguish the effect of coal compression from the pore filling process during mercury intrusion. It is shown that fractal dimension can be evaluated from the compressibility corrected pore volume data. Results from the present study suggest that statistic self-similarity of the fractal dimension perspective is limited by certain artificial effects, such as crushing and grinding. Different surface irregularities exist over different pore size ranges, and a single fractal dimension value can only be used to describe the surface irregularity within a limited pore size range. The average fractal dimensions in the pore size range of 6–60 nm were found to be 2.71 and 2.43 for CA and GO coals, respectively.     

Linewidth-Resolved N HSQC, a Simple 3D Method to Measure N Relaxation Times from T1 and T2 Linewidths

A three-dimensional approach for measuring ¹⁵N relaxation times is described. Instead of selecting particular values for the relaxation period, in the proposed method the relaxation period is incremented periodically in order to create a 3D spectrum. This additional frequency domain of the transformed spectrum contains the relaxation time information in the T₁ and T₂ linewidths, and thus the longitudinal and transverse ¹⁵N relaxation times can be measured without determination of 2D cross peak volumes/intensities and subsequent curve fitting procedures.     

Renormalization group analysis of the global properties of a strange attractor

This paper considers the circle map at the special point: the one at which there is a trajectory with a golden mean winding number and at which the map just fails to be invertable at one point on the circle. The invariant density of this trajectory has fractal properties. Previous work has suggested that the global behavior of this fractal can be effectively analyzed using a kind of partition function formalism to generate anf versus curve. In this paper the partition function is obtained by using a renormalization group approach.     

Universal scaling behaviour for iterated maps in the complex plane

According to the theory of Schröder and Siegel, certain complex analytic maps possess a family of closed invariant curves in the complex plane. We have made a numerical study of these curves by iterating the map, and have found that the largest curve is a fractal. When the winding number of the map is the golden mean, the fractal curve has universal scaling properties, and the scaling parameter differs from those found for other types of maps. Also, for this winding number, there are universal scaling functions which describe the behaviour asn→∞ of theQ _n ^th iterates of the map, whereQ _n is then ^th Fibonacci number.     

Hierarchical description for a non-Markoff relaxation process

A phenomenal model is proposed in this paper to describe a non-Markoff relaxation process. The main feature of the model is the fractal time concept introduced to take account of the coupling of a relaxation mode with a heat bath. The theoretical investigation indicates that a weakly non-Markoff relaxation process can be related to a set of hierarchically distributed Markoff relaxation processes and the fractal exponential law exp [?(t/τ)^1?σ] is the character of a non-Markoff relaxation process. Calculations about the temperature and time scale dependence of the exponent σ have been made, and comparison with recent experiments is also given.     

Fractal modelling of turbulent combustion

In a previous paper we proposed a new model for turbulent flows, called the fractal model (FM), which is applicable both to RANS and LES formulations. Here, the model is extended to the reactive case with the goal of simulating turbulent flames, both premixed and non-premixed.

FM is a subgrid model that describes the physics of the small scales of turbulence building on the phenomenological concept of vortex cascade and on fractal theory. The physics of the small scales is summarized by a turbulent ‘viscosity’ μ_t, to be added to the molecular one. μ_t is zero where the flow is laminar and, in particular, goes to zero at solid walls.

The fundamental assumption in treating combustion in this work is that chemical reactions take place only at the dissipative scales of turbulence, i.e. near the so-called ‘fine structures’ (the eddy dissipation concept). FM predicts the growth of dissipative scales due to heat release; therefore, it enables a local DNS in the hot regions of the flow where the dissipative scale may grow up to the cell dimension. FM can also estimate the volume fraction γ* occupied by the ‘fine structures’; this quantity is critical for modelling the reaction rate, and therefore the source terms in the energy and species equations. FM can also estimate the local surface of the reactive ‘fine structures’, that is, the local flame front area. It also takes into account, although in approximate manner, the formation of islands of unburnt mixture. In this paper, the model (in the isotropic formulation (IFM)) is used in conjunction with a time-dependent LES (but with the limitations of an isotropic model) approach and is validated through a three-dimensional axisymmetric diffusion flame studied experimentally by Correa and Gulati and numerically by many researchers. The time-dependent results obtained are in good agreement with the experiments. Moreover, the IFM solution offers a possible explanation for the stabilization process of this flame, which undergoes local stretching of the order of 46 000 s^?1.