scholarly journals Structural Properties of Vicsek-like Deterministic Multifractals

Symmetry ◽  
2019 ◽  
Vol 11 (6) ◽  
pp. 806 ◽  
Author(s):  
Eugen Mircea Anitas ◽  
Giorgia Marcelli ◽  
Zsolt Szakacs ◽  
Radu Todoran ◽  
Daniela Todoran
Keyword(s):  
Structural Properties ◽  
Fractal Model ◽  
Fractal Structures ◽  
Scaling Factors ◽  
Scaling Properties ◽  
Additional Information ◽  
Box Counting ◽  
Mass Fractal ◽  
Angle Scattering ◽  
Multifractal Spectra

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.

scholarly journals Small-Angle Scattering from Fractals: Differentiating between Various Types of Structures

Symmetry ◽  
2020 ◽  
Vol 12 (1) ◽  
pp. 65 ◽  
Author(s):  
Eugen Mircea Anitas
Keyword(s):  
Small Angle ◽  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Theoretical Models ◽  
Small Angle Scattering ◽  
X Rays ◽  
Fractal Structures ◽  
Practical Applications ◽  
Additional Information ◽  
Angle Scattering

Small-angle scattering (SAS; X-rays, neutrons, light) is being increasingly used to better understand the structure of fractal-based materials and to describe their interaction at nano- and micro-scales. To this aim, several minimalist yet specific theoretical models which exploit the fractal symmetry have been developed to extract additional information from SAS data. Although this problem can be solved exactly for many particular fractal structures, due to the intrinsic limitations of the SAS method, the inverse scattering problem, i.e., determination of the fractal structure from the intensity curve, is ill-posed. However, fractals can be divided into various classes, not necessarily disjointed, with the most common being random, deterministic, mass, surface, pore, fat and multifractals. Each class has its own imprint on the scattering intensity, and although one cannot uniquely identify the structure of a fractal based solely on SAS data, one can differentiate between various classes to which they belong. This has important practical applications in correlating their structural properties with physical ones. The article reviews SAS from several fractal models with an emphasis on describing which information can be extracted from each class, and how this can be performed experimentally. To illustrate this procedure and to validate the theoretical models, numerical simulations based on Monte Carlo methods are performed.

scholarly journals Fractal Modeling of Pore Structure and Ionic Diffusivity for Cement Paste

2016 ◽  
Vol 2016 ◽  
pp. 1-8 ◽  
Author(s):  
Yun Gao ◽  
Jin-yang Jiang ◽  
Kai Wu
Keyword(s):  
Pore Structure ◽  
Cement Paste ◽  
Mercury Intrusion Porosimetry ◽  
Fractal Model ◽  
Homogenization Method ◽  
Solid Mass ◽  
Fractal Modeling ◽  
Mass Fractal ◽  
Chloride Migration ◽  
Ionic Diffusivity

Pore structure in cement based composites is of paramount importance to ionic diffusivity. In this paper, pore structure in cement paste is modeled by means of the recently proposed solid mass fractal model. Moreover, an enhanced Maxwell homogenization method that incorporates the solid mass fractal model is proposed to determine the associated ionic diffusivity. Experiments are performed to validate the modeling, that is, mercury intrusion porosimetry and rapid chloride migration. Results indicate that modeling agrees well with those obtained from experiments.

Multi-Scale Regular-Fractal Topography Characterization and Modeling

2014 ◽  
Author(s):  
Jinya Liu ◽  
Vijaya Chalivendra ◽  
Charles L. Goldsmith ◽  
Wenzhen Huang
Keyword(s):  
Fractal Structure ◽  
Fractal Model ◽  
Rf Switch ◽  
Fractal Structures ◽  
Force Microscopy ◽  
Multi Scale ◽  
Atomic Force ◽  
Model Components ◽  
Sample Allocation ◽  
Random Irregularity

Regular-fractal topography on RF-switch MEMS surface is reported over different scale ranges. Surface topography is crucial in understanding underling physics associated with the surface contacts, switch working performance, and reliability. The complexity of these structures requires new techniques to characterize topography and then replicate the multi-scale regular-fractal structure for analysis. Topography on RF-switch contacting surfaces are scanned by atomic force microscopy (AFM) at different length scales (e.g. 1×1, 10×10 and 60×60 μm2). A sample allocation plan is designed to maximize the spatial representative of the AFM scanning patches with different resolutions and uniformly distributed sample patches. The scanning data are used for characterizing and model estimation. Hexagonal patterns are found on at coarser scales (e.g. 10×10 and 60×60 μm2). They were formed by the remnant (polymer) of etching process. Random irregularity is observed and the fractal structure at finer scales (e.g. 1×1 μm2) is identified. A regular-fractal model is proposed to decompose and characterize the regular and fractal structures with two model components: one for the regular geometric pattern and the other for fractal irregularity. The former uses a 2D cosine functions to characterize dominant modes in the regular (larger scale) patterns. The later summarizes random irregularity in finer scales with a statistical fractal model estimated from the data on the scattered sample patches. The model validation is made through the comparisons of topography and conventional roughness parameters between the results of simulation from the proposed model and that derived from AFM scanned data.

scholarly journals Small-Angle Scattering and Multifractal Analysis of DNA Sequences

2020 ◽  
Vol 21 (13) ◽  
pp. 4651
Author(s):  
Eugen Mircea Anitas
Keyword(s):  
Dna Sequences ◽  
Small Angle ◽  
Multifractal Analysis ◽  
Small Angle Scattering ◽  
Spatial Inhomogeneity ◽  
Chaos Game Representation ◽  
Model Based ◽  
Angle Scattering ◽  
Box Counting Dimension ◽  
Multifractal Spectra

The arrangement of A, C, G and T nucleotides in large DNA sequences of many prokaryotic and eukaryotic cells exhibit long-range correlations with fractal properties. Chaos game representation (CGR) of such DNA sequences, followed by a multifractal analysis, is a useful way to analyze the corresponding scaling properties. This approach provides a powerful visualization method to characterize their spatial inhomogeneity, and allows discrimination between mono- and multifractal distributions. However, in some cases, two different arbitrary point distributions, may generate indistinguishable multifractal spectra. By using a new model based on multiplicative deterministic cascades, here it is shown that small-angle scattering (SAS) formalism can be used to address such issue, and to extract additional structural information. It is shown that the box-counting dimension given by multifractal spectra can be recovered from the scattering exponent of SAS intensity in the fractal region. This approach is illustrated for point distributions of CGR data corresponding to Escherichia coli, Phospholamban and Mouse mitochondrial DNA, and it is shown that for the latter two cases, SAS allows extraction of the fractal iteration number and the scaling factor corresponding to “ACGT” square, or to recover the number of bases. The results are compared with a model based on multiplicative deterministic cascades, and respectively with one which takes into account the existence of forbidden sequences in DNA. This allows a classification of the DNA sequences in terms of random and deterministic fractals structures emerging in CGR.

Dynamical scaling in fractal structures in the aggregation of tetraethoxysilane-derived sonogels

2010 ◽  
Vol 43 (5) ◽  
pp. 949-954 ◽  
Author(s):  
Dimas R. Vollet ◽  
Dario A. Donatti ◽  
Alberto Ibañez Ruiz ◽  
Fabio S. de Vicente
Keyword(s):  
Factor Model ◽  
Fractal Structures ◽  
Saxs Data ◽  
Scaling Properties ◽  
Dynamical Scaling ◽  
X Ray Scattering ◽  
Linear Chains ◽  
Saxs Intensity ◽  
Scattering Factor ◽  
Kinetics Of

Dynamical scaling properties in fractal structures were investigated from small-angle X-ray scattering (SAXS) data of the kinetics of aggregation in silica-based gelling systems. For lack of a maximum in the SAXS intensity curves, a characteristic correlation distance ξ was evaluated by fitting a particle scattering factor model valid for polydisperse coils of linear chains andf-functional branched polycondensates in solution, so the intensity atq= ξ−1,I(ξ−1,t), was considered to probe dynamical scaling properties. The following properties have been found: (i) the SAXS intensities corresponding to different timest,I(q,t), are given by a time-independent functionF(qξ) =I(q,t)ξ−D/Q, where the scattering invariantQhas been found to be time-independent; (ii) ξ exhibited a power-law behavior with time as ξ ≃tα, the exponent α being close to 1 but diminishing with temperature; (iii)I(ξ−1,t) exhibited a time dependence given byI(ξ−1,t) ≃tβ, with the exponent β found to be around 2 but diminishing with temperature, following the same behavior as the exponent α. In all cases, β/α was quite close to the fractal dimensionDat the end of the studied process. This set of findings is in notable agreement with the dynamical scaling properties.

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