scholarly journals Scattering from surface fractals in terms of composing mass fractals

2017 ◽  
Vol 50 (3) ◽  
pp. 919-931 ◽  
Author(s):  
A. Yu. Cherny ◽  
E. M. Anitas ◽  
V. A. Osipov ◽  
A. I. Kuklin
Keyword(s):  
Fractal Dimension ◽  
Power Law ◽  
Scattering Amplitude ◽  
Three Dimensional ◽  
Power Law Distribution ◽  
Scattering Intensity ◽  
Three Dimensional Objects ◽  
Additional Information ◽  
Mass Fractal ◽  
Surface Fractal

It is argued that a finite iteration of any surface fractal can be composed of mass-fractal iterations of the same fractal dimension. Within this assertion, the scattering amplitude of a surface fractal is shown to be a sum of the amplitudes of the composing mass fractals. Various approximations for the scattering intensity of surface fractals are considered. It is shown that small-angle scattering (SAS) from a surface fractal can be explained in terms of a power-law distribution of sizes of objects composing the fractal (internal polydispersity), provided the distance between objects is much larger than their size for each composing mass fractal. The power-law decay of the scattering intensityI(q) ∝ q^{D_{\rm s}-6}, where 2 <Ds< 3 is the surface-fractal dimension of the system, is realized as a non-coherent sum of scattering amplitudes of three-dimensional objects composing the fractal and obeying a power-law distribution dN(r) ∝r−τdr, withDs= τ − 1. The distribution is continuous for random fractals and discrete for deterministic fractals. A model of the surface deterministic fractal is suggested, the surface Cantor-like fractal, which is a sum of three-dimensional Cantor dusts at various iterations, and its scattering properties are studied. The present analysis allows one to extract additional information from SAS intensity for dilute aggregates of single-scaled surface fractals, such as the fractal iteration number and the scaling factor.

Characterization of Pore Framework Structure in Monolithic Mesoporous Silica

2009 ◽  
Vol 1217 ◽  
Author(s):  
Ching-Mao Wu ◽  
Szu-Yin Lin ◽  
Chin-Cheng Weng ◽  
Kuo-Tung Huang
Keyword(s):  
Fractal Dimension ◽  
Mesoporous Silica ◽  
Power Law ◽  
Sol Gel ◽  
Spherical Model ◽  
Pore Network ◽  
Transmission Electron Microscopy Micrographs ◽  
Mass Fractal ◽  
Commercial Applications ◽  
The Individual

AbstractMesoporous silica materials have received much interest due to commercial applications in chemical separations and heterogeneous catalysis. Recent studies have reported via a sol-gel nanocasting technique, monolithic mesoporous silica with wormlike pore framework could be prepared by utilizing room-temperature ionic liquids (RTILs) as templates and solvents. Although previous reports have indicated that the wormlike pores would be formed in the silica, the detailed pore network structure still remained the crucial issues to be resolved. In the present study, we investigated the pore structure in the monolithic mesoporous silica, which was templated by RTIL (1-butyl-3-methyl-imidazolium-tetrafluoroborate). We revealed an open fractal pore network with a branched and self-similar appearance was formed by the aggregation of the individual spherical pores. Transmission electron microscopy micrographs displayed that the disordered wormlike pore framework was formed in the silica. Furthermore, the small angle X-ray scattering profile measured herein further exhibited three distinct regions of power-law scattering on the respective length scales. In the high-q region, the profile followed Power behavior and a power-law of -4 was observed for the surface fractal dimension of 2, manifesting the primary pore with a smooth surface and a spherical appearance. In the intermediate-q region, a power-law of -2.5 (mass fractal dimension of 2.5), indicating an open mass fractal network was formed by the aggregation of the individual primary pores. Moreover in the low-q region, the power-law of -4 was observed for mass-fractal agglomerates of aggregates. With the proceeding analysis of unified equation in terms of two structural levels, the radiuses of gyration of primary pore (Rg1)and its aggregates (Rg2) were fitted as ca. 0.9 nm and 5.5 nm, respectively. For a spherical-model pore, the radius of pore (R) was ca. 1.16 nm; thus, the averaged pore diameter (D) was 2.32 nm. The number of primary pores in a fractal aggregate (degree of aggregation, z) was calculated as ca. 67.

Three-Dimensional Flexure of Rectangular Plates Made of Functionally Graded Materials

2005 ◽  
Vol 72 (5) ◽  
pp. 788-791 ◽  
Author(s):  
Isaac Elishakoff ◽  
Cristina Gentilini
Keyword(s):  
Functionally Graded Materials ◽  
Power Law ◽  
Minimum Energy ◽  
Three Dimensional ◽  
Functionally Graded ◽  
Rectangular Plates ◽  
Power Law Distribution ◽  
Dimensional Solution ◽  
Energy Principle ◽  
Graded Materials

A three-dimensional solution for the problem of transversely loaded, all-round clamped rectangular plates of arbitrary thickness is presented within the linear, small deformation theory of elasticity. The Ritz minimum energy principle is employed to derive the governing equation of the plate made of functionally graded materials. In theory, if we employ an infinite number of terms in the displacement series, the exact solution can be determined. However, a practical limit always exists due to numerical implementation. The solution has a validity comparable to some higher order theories. A power-law distribution for the mechanical characteristics is adopted to model the continuous variation of properties from those of one component to those of the other. The displacements and stresses of the plate for different values of the power-law exponent are investigated.

scholarly journals Characterization of 2D and 3D Rough Fractal Surfaces from Backscattered Radar Data

2017 ◽  
Vol 23 (3) ◽  
pp. 53-58
Author(s):  
Apostolos Kotopoulis ◽  
Georgios Pouraimis ◽  
Evangelos Kallitsis ◽  
Panayiotis Frangos
Keyword(s):  
Fractal Dimension ◽  
Three Dimensional ◽  
Radar Data ◽  
Wave Number ◽  
Backscattering Coefficient ◽  
Surface Fractal Dimension ◽  
Fractal Surfaces ◽  
Surface Fractal ◽  
Number Frequency ◽  
2D And 3D

AbstractThe scattering of electromagnetic (EM) waves, emitted by a monostatic radar, from two - dimensional (2D) rough fractal surfaces is examined by using the Kirchhoff approximation. We examine the way that the level of roughness of the fractal surface affects the backscattered EM wave captured by a radar as a function of frequency (therefore, a ‘spectral method’) and whether the roughness of the surface can be estimated from these radar measurements. The backscattering coefficient is calculated for a number of radar frequencies and for different values of the surface fractal dimension. It is found that the values of the slopes between the main lobe and the first sidelobes of the backscattering coefficient as a function of the wave number (frequency) of the incident EM waves increase with the surface fractal dimension. Therefore, we conclude that the magnitude of the above slopes provides a reliable method for the classification of the rough fractal surfaces. Furthermore, concerning three - dimensional (3D) fractal surfaces, investigations similar to the above are already performed by the authors and will be presented during the Conference. The above are also investigated in the presence of electronic noise in the radar receiver (effect of SNR values in the above proposed technique).

Three-dimensional static analysis of thick functionally graded plates using graded finite element method

2013 ◽  
Vol 228 (8) ◽  
pp. 1275-1285 ◽  
Author(s):  
Hassan Zafarmand ◽  
Mehran Kadkhodayan
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Power Law ◽  
Three Dimensional ◽  
Functionally Graded ◽  
Published Data ◽  
Power Law Distribution ◽  
Various Boundary Conditions ◽  
Graded Finite Element Method ◽  
Element Method

In this paper, a thick functionally graded plate based on three-dimensional equations of elasticity and subjected to nonuniform transverse loading is considered. The Young’s modulus of the plate is assumed to be graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents and the Poisson’s ratio is assumed to be constant. Three-dimensional graded finite element method based on Rayleigh–Ritz energy formulation has been applied to study the static response of the plate. The plate deflection and in-plane stress for different values of the power law exponent, thickness-to-length ratio, and various boundary conditions have been investigated. To verify the presented method and data, the results are compared to published data.

FRACTAL CANTOR PATTERNS IN THE SEQUENCE STRUCTURE OF DNA

Fractals ◽  
2000 ◽  
Vol 08 (01) ◽  
pp. 15-27 ◽  
Author(s):  
A. PROVATA ◽  
Y. ALMIRANTIS
Keyword(s):  
Fractal Dimension ◽  
Power Law ◽  
Dna Sequences ◽  
Cantor Sets ◽  
Power Law Distribution ◽  
Sequence Structure ◽  
Random Cantor Sets

The coding parts of DNA sequences are regarded as clusters of connected sites of a random Cantor-like set, while the non-coding parts are regarded as the empty regions of the same set. Under this representation, we find that higher eucaryotes are mapped on random Cantor sets with fractal dimension around 0.85, while lower organisms are mapped on Cantor sets with fractal dimension 1. This result indicates that the coding/non-coding partition in the DNA sequences of lower organisms is homogeneous-like, while in the higher eucaryotes the partition is fractal. This result agrees with the power law distribution observed in the non-coding parts of higher eucaryotes.

scholarly journals Fast Overlap Detection between Hard-Core Colloidal Cuboids and Spheres. The OCSI Algorithm

Algorithms ◽  
2021 ◽  
Vol 14 (3) ◽  
pp. 72
Author(s):  
Luca Tonti ◽  
Alessandro Patti
Keyword(s):  
Colloidal Particles ◽  
Dynamic Properties ◽  
Wide Spectrum ◽  
Three Dimensional ◽  
Hard Core ◽  
Detection Algorithm ◽  
Three Dimensional Objects ◽  
Aggregation Behaviour ◽  
Sphere Radius ◽  
User Friendly

Collision between rigid three-dimensional objects is a very common modelling problem in a wide spectrum of scientific disciplines, including Computer Science and Physics. It spans from realistic animation of polyhedral shapes for computer vision to the description of thermodynamic and dynamic properties in simple and complex fluids. For instance, colloidal particles of especially exotic shapes are commonly modelled as hard-core objects, whose collision test is key to correctly determine their phase and aggregation behaviour. In this work, we propose the Oriented Cuboid Sphere Intersection (OCSI) algorithm to detect collisions between prolate or oblate cuboids and spheres. We investigate OCSI’s performance by bench-marking it against a number of algorithms commonly employed in computer graphics and colloidal science: Quick Rejection First (QRI), Quick Rejection Intertwined (QRF) and a vectorized version of the OBB-sphere collision detection algorithm that explicitly uses SIMD Streaming Extension (SSE) intrinsics, here referred to as SSE-intr. We observed that QRI and QRF significantly depend on the specific cuboid anisotropy and sphere radius, while SSE-intr and OCSI maintain their speed independently of the objects’ geometry. While OCSI and SSE-intr, both based on SIMD parallelization, show excellent and very similar performance, the former provides a more accessible coding and user-friendly implementation as it exploits OpenMP directives for automatic vectorization.

scholarly journals The structure of co-publications multilayer network

2021 ◽  
Vol 8 (1) ◽  
Author(s):  
Ghislain Romaric Meleu ◽  
Paulin Yonta Melatagia
Keyword(s):  
Power Law ◽  
Degree Distribution ◽  
Preferential Attachment ◽  
Small World ◽  
Generation Model ◽  
Small World Network ◽  
Power Law Distribution ◽  
Multilayer Networks ◽  
Multilayer Network ◽  
Scientific Papers

AbstractUsing the headers of scientific papers, we have built multilayer networks of entities involved in research namely: authors, laboratories, and institutions. We have analyzed some properties of such networks built from data extracted from the HAL archives and found that the network at each layer is a small-world network with power law distribution. In order to simulate such co-publication network, we propose a multilayer network generation model based on the formation of cliques at each layer and the affiliation of each new node to the higher layers. The clique is built from new and existing nodes selected using preferential attachment. We also show that, the degree distribution of generated layers follows a power law. From the simulations of our model, we show that the generated multilayer networks reproduce the studied properties of co-publication networks.

Simulation of small-angle scattering curves by numerical Fourier transformation

2007 ◽  
Vol 40 (1) ◽  
pp. 16-25 ◽  
Author(s):  
Klaus Schmidt-Rohr
Keyword(s):  
Form Factor ◽  
Small Angle ◽  
Fourier Transformation ◽  
Three Dimensional ◽  
Power Laws ◽  
Numerical Approach ◽  
Peak Position ◽  
Two Dimensions ◽  
Correlation Peak ◽  
Scattering Intensity

A simple numerical approach for calculating theq-dependence of the scattering intensity in small-angle X-ray or neutron scattering (SAXS/SANS) is discussed. For a user-defined scattering density on a lattice, the scattering intensityI(q) (qis the modulus of the scattering vector) is calculated by three-dimensional (or two-dimensional) numerical Fourier transformation and spherical summation inqspace, with a simple smoothing algorithm. An exact and simple correction for continuous rather than discrete (lattice-point) scattering density is described. Applications to relatively densely packed particles in solids (e.g.nanocomposites) are shown, where correlation effects make single-particle (pure form-factor) calculations invalid. The algorithm can be applied to particles of any shape that can be defined on the chosen cubic lattice and with any size distribution, while those features pose difficulties to a traditional treatment in terms of form and structure factors. For particles of identical but potentially complex shapes, numerical calculation of the form factor is described. Long parallel rods and platelets of various cross-section shapes are particularly convenient to treat, since the calculation is reduced to two dimensions. The method is used to demonstrate that the scattering intensity from `randomly' parallel-packed long cylinders is not described by simple 1/qand 1/q4power laws, but at cylinder volume fractions of more than ∼25% includes a correlation peak. The simulations highlight that the traditional evaluation of the peak position overestimates the cylinder thickness by a factor of ∼1.5. It is also shown that a mix of various relatively densely packed long boards can produceI(q) ≃ 1/q, usually observed for rod-shaped particles, without a correlation peak.

Sign in / Sign up

Export Citation Format

Share Document