Deterministic Fractal Model Research Articles

Structural Properties of Vicsek-like Deterministic Multifractals

Deterministic nano-fractal structures have recently emerged, displaying huge potential for the fabrication of complex materials with predefined physical properties and functionalities. Exploiting the structural properties of fractals, such as symmetry and self-similarity, could greatly extend the applicability of such materials. Analyses of small-angle scattering (SAS) curves from deterministic fractal models with a single scaling factor have allowed the obtaining of valuable fractal properties but they are insufficient to describe non-uniform structures with rich scaling properties such as fractals with multiple scaling factors. To extract additional information about this class of fractal structures we performed an analysis of multifractal spectra and SAS intensity of a representative fractal model with two scaling factors—termed Vicsek-like fractal. We observed that the box-counting fractal dimension in multifractal spectra coincide with the scattering exponent of SAS curves in mass-fractal regions. Our analyses further revealed transitions from heterogeneous to homogeneous structures accompanied by changes from short to long-range mass-fractal regions. These transitions are explained in terms of the relative values of the scaling factors.

Competitive scission and cross linking in a solid polymer electrolyte exposed to gamma irradiation: Simulation by a fractal model

Gamma irradiation influences the microstructure of polymers that significantly affects the correlated properties of solid polymer electrolytes (SPEs). The present paper introduces a deterministic fractal model of polymer molecules and investigates the morphological changes due to gamma irradiation in SPE. The computer aided simulation is primarily based on radiation induced scission and/or cross linking of polymer molecules. The drift velocity of conducting ion which is proportional to conductivity, is calculated considering two competing effects, viz. trapping effect of ions diffusing inside a branched fractal molecule and the motion assisted by segmental motion of the polymer itself. Variations in ionic conductivity, molecular weight distribution, intrinsic viscosity etc., as a function of probability of scission and/or cross linking show significant correlation with microstructural change induced by gamma radiation of variable dose. The present theoretical study enables easy understanding for correlation of associated properties exhibited upon experimental system perturbation through gamma irradiation.

Bioinspired, mechanical, deterministic fractal model for hierarchical suture joints

Many biological systems possess hierarchical and fractal-like interfaces and joint structures that bear and transmit loads, absorb energy, and accommodate growth, respiration, and/or locomotion. In this paper, an elastic deterministic fractal composite mechanical model was formulated to quantitatively investigate the role of structural hierarchy on the stiffness, strength, and failure of suture joints. From this model, it was revealed that the number of hierarchies (N) can be used to tailor and to amplify mechanical properties nonlinearly and with high sensitivity over a wide range of values (orders of magnitude) for a given volume and weight. Additionally, increasing hierarchy was found to result in mechanical interlocking of higher-order teeth, which creates additional load resistance capability, thereby preventing catastrophic failure in major teeth and providing flaw tolerance. Hence, this paper shows that the diversity of hierarchical and fractal-like interfaces and joints found in nature have definitive functional consequences and is an effective geometric-structural strategy to achieve different properties with limited material options in nature when other structural geometries and parameters are biologically challenging or inaccessible. This paper also indicates the use of hierarchy as a design strategy to increase design space and provides predictive capabilities to guide the mechanical design of synthetic flaw-tolerant bioinspired interfaces and joints.

Optical properties of non-dilute metal–insulator composites

The description of the optical properties of metal–insulator composites in the non-dilute region is a long standing problem. In this letter we extract the spectral density function of cobalt–amorphous aluminum oxide composites from optical and near-infrared data. The spectral functions are accurately computed numerically with the help of a recently developed technique. It is observed that the spectral features of the prepared composites change with increasing cobalt content. For low concentrations of cobalt, only one depolarization peak is found that corresponds to the Maxwell Garnett approximation. For concentrations higher than 11% cobalt, three effective depolarization factors are resolved that move towards low spectral parameter values with increasing cobalt content. Such a multi-peak structure arises naturally in fractal equivalent circuit models for the optical properties. A comparison with a deterministic fractal model is presented to illustrate the strength of the spectral density representation and to better comprehend our results. We conclude that the observed behavior gives important information on the relation of the optical characteristics to the composite micro-structure.

Fractal Analysis of Teotihuacan, Mexico

The standardisation in architectural orientation and construction forms, as well as the symmetric and highly proportional spatial distribution of buildings, suggest that the major structures of Teotihuacan were laid out from its inception according to a master plan (or plans) intended to express a specific worldview in material form. Fractal analysis of the Teotihuacan radar image and aerial photographs has confirmed that a master plan does exist in this archaeological area. The Sierpinski carpet, one of the best known deterministic fractal models, is proposed as the model for Teotihuacan in two-dimensional Euclidean space. Its dimension (1·89) is also proposed as the basic unit which joins in space the Mesoamerican calendar and the buildings' geometrical measures. It is hypothesised that inside the master plan of Teotihuacan, the information pertaining to space is expressed across the two-dimensional planes, and that of time through the pyramid verticals. A value of 0·83m, which corroborates with the Teotihuacan measurement unit (TMU) proposed by different anthropologists, was derived from an analysis of the height of the Sun and Moon Pyramids. These results open new prospects for future comparisons between different settlements and monuments, that were supposedly constructed by the people from Teotihuacan.

Archie's law from a fractal model for porous rocks

Electrical conductivity of porous rocks is related to the porosity through the empirical relation known as Archie's law. The principal motivation behind the present paper is to show that the fractal nature of the pore space is important for the validity of this relation. A random walk, on two variations of a three-dimensional deterministic fractal model, is simulated and the result leads to a realistic exponent for Archie"s law.

Strongly nonlinear response of fractal clusters.

We have developed a differential effective medium approximation (DEMA) for the effective nonlinear response due to clustering of a strongly nonlinear conducting material of a current-field (J-E) response of the form J=\ensuremath{\chi}\ensuremath{\Vert}E${\mathrm{\ensuremath{\Vert}}}^{2\mathrm{\ensuremath{\beta}}}$E (\ensuremath{\beta}>0) in a host medium, where \ensuremath{\chi} is the nonlinear coefficient. The DEMA results are compared with numerical calculations in a deterministic fractal model. As a similar problem of a random medium, we further investigate the scaling behavior of the nonlinear response. It is shown that by choosing a relevant scaling variable properly, the nonlinear response function can be rescaled to collapse onto a universal curve.

Effective medium theory for strongly nonlinear composites: fractal clusters

We have developed an effective medium approximation (EMA) for the effective nonlinear response due to clustering of a strongly nonlinear conducting material with a current-field ( J-E ) response of the form J = χ … E… 2β E ( β > 0) in a host medium, where χ is the nonlinear coefficient. A cluster energy functional is constructed and a variational principle is invoked to obtain an expression for the effective nonlinear response of a fractal cluster. The EMA results are compared with numerical calculations in a deterministic fractal model and excellent agreement is found.

Hierarchical percolation model with anomalous multifractal measure

Hierarchical models are introduced to discuss the anomalous multifractal measure of the current fraction distribution on the percolating cluster analytically. The deterministic fractal model proposed by de Arcangelis, Redner and Coniglio (1986) is extended to take into account the faster decreasing minimum current fraction than power law. It is shown that, in the hierarchical models constructed from the generator of a ladder configuration with m plaquettes, the minimum current fraction has the dependence on L, imin approximately=exp(-cm(1n L)2), where L is the system size and c a constant. The other hierarchical model showing imin approximately=exp(-c(1n L)(In(In L))) is also found.

The electrical conductivity of binary disordered systems, percolation clusters, fractals and related models

Abstract We review theoretical and experimental studies of the AC dielectric response of inhomogeneous materials, modelled as bond percolation networks, with a binary (conductor-dielectric) distribution of bond conductances. We first summarize the key results of percolation theory, concerning mostly geometrical and static (DC) transport properties, with emphasis on the scaling properties of the critical region around the percolation threshold. The frequency-dependent (AC) response of a general binary model is then studied by means of various approaches, including the effective-medium approximation, a scaling theory of the critical region, numerical computations using the transfer-matrix algorithm, and several exactly solvable deterministic fractal models. Transient regimes, related to singularities in the complex-frequency plane, are also investigated. Theoretical predictions are made more explicit in two specific cases, namely R-C and RL-C networks, and compared with a broad variety of experimental resul...

Deterministic fractal models for transport properties, inspired by d=2 random walks.

Deterministic fractal models for transport properties, inspired by d=2 random walks.

A hierarchical model for scaling structure in generalised diffusion-limited aggregations

Deterministic fractal models are presented to be exactly solvable for the growth probability distribution on the surface of the cluster in a hierarchical lattice. The recursion relations of the electric field on the growth bond are obtained for diffusion-limited aggregation and the dielectric breakdown models. A hierarchy of generalised dimensions D(q) is calculated to describe the growth probability, by using the recursion relations. The partition of (q-1)D(q) into a density of singularities f(q) with singularity strength alpha (q) is made and the alpha -f spectra are studied for different dielectric breakdown models. The scaling of the highest growth probability pmax on the growth bond is analytically derived and the dependence of the fractal dimensions is found on the parameter eta describing the different dielectric breakdown models.

A regular model for cluster numbers and structure just below percolation threshold

A two-dimensional deterministic fractal model is proposed to imitate the geometric texture just below percolation threshold. The model is constructed for bond percolation via a rule of bond occupation on the square lattice. It show the typical percolation behaviour as a function of a parameter p (p is the bond concentration). The model describes the approach towards the threshold below criticality. The critical bond concentration pc and the correlation length exponent nu are found. The scaling property of the cluster size distribution is studied below the threshold. The critical concentration and critical exponents are shown to agree with those derived from the regular model proposed for the geometric texture above the threshold. The scaling relations for the critical exponents are shown to be satisfied.

Deterministic model for Eden trees

The author proposes and studies a deterministic fractal model on the square lattice for the Eden trees, recently proposed by Dhar and Ramaswamy (1985). The author finds the fractal and spectral dimensions of this fractal structure which are equal to 2 and 4 ln 2/ln 6 respectively.