Statistical Correlation Functions Research Articles

Mechanical and thermal behavior of nanoclay based polymer nanocomposites using statistical homogenization approach

In the present study, the effects of nanoclay additives on the effective mechanical and thermal properties of polymer/nanoclay composites have been investigated using experimental and simulation analyzes. In this research, we propose the use of strong contrast statistical continuum theory to predict the effective elastic and thermal properties. To validate our modeling approach, we conducted experimental measurements of these properties for polyamide/nanoclay nanocomposites with concentrations of 1, 3 and 5 wt.% of nanoclay particles. Three-dimensional isotropic nanocomposite samples with randomly oriented monolayer nanoclays were computer generated and used to calculate the statistical correlation functions of the realized model. These correlation functions have been exploited to calculate effective thermal and elastic properties of the nanocomposite. The simulation results have shown that effective stiffness can be increased significantly with small amounts of particle concentration for the exfoliated clay monolayers. The predicted effective conductivity and elastic modulus have been compared to our experimental results. Effective thermal conductivity shows satisfactory agreement with experimental data. However, the predicted results for the elastic modulus overestimate the experimental data, which might be due to the increasing intercalated structure for high concentration of nanofiller and to anisotropic properties of the nanoclay.

Three-dimensional reconstruction and homogenization of heterogeneous materials using statistical correlation functions and FEM

In this study, a previously developed reconstruction methodology is extended to three-dimensional reconstruction of a three-phase microstructure, based on two-point correlation functions and two-point cluster functions. The reconstruction process has been implemented based on hybrid stochastic methodology for simulating the virtual microstructure. While different phases of the heterogeneous medium are represented by different cells, growth of these cells is controlled by optimizing parameters such as rotation, shrinkage, translation, distribution and growth rates of the cells. Based on the reconstructed microstructure, finite element method (FEM) was used to compute the effective elastic modulus and effective thermal conductivity. A statistical approach, based on two-point correlation functions, was also used to directly estimate the effective properties of the developed microstructures. Good agreement between the predicted results from FEM analysis and statistical methods was found confirming the efficiency of the statistical methods for prediction of thermo-mechanical properties of three-phase composites.

Microstructure, property and processing relation in gradient porous cathode of solid oxide fuel cells using statistical continuum mechanics

This paper investigates the relation between microstructure, macroscopic transport properties, and fabrication processing for a gradient porous cathode of solid oxide fuel cells (SOFCs). Functionally graded porous cathode with smooth variations in pore size is composed of lanthanum strontium manganite (LSM) fabricated on yttria stabilized zirconia (YSZ) electrolyte substrate using a multi-step spray pyrolysis (SP) technique at various deposition conditions. Two-dimensional (2D) serial-sections of the gradient porous microstructure obtained by FIB–SEM are fully characterized using statistical correlation functions. Results of statistical analysis of the microstructures revealed that the SP processing technique is capable of generating statistically identical and homogeneous microstructures with smooth gradient in pore size resulting from changing the processing parameters. Strong contrast statistical approach is also used to predict the in-plane temperature dependent effective electrical conductivity of the gradient porous cathode and the results are compared to the experimental data.

Using SAXS approach to estimate thermal conductivity of polystyrene/zirconia nanocomposite by exploiting strong contrast technique

In this study, the effective thermal conductivity of a polymer/ZrO 2 nanocomposite material has been calculated based on small-angle X-ray scattering (SAXS) data. These SAXS results have been used to calculate statistical two-point correlation functions of the nanostructure. A statistical continuum approach based on higher-order correlation functions has been utilized to estimate the effective thermal conductivity of this zirconia-filled polymer. A strong-contrast formulation was used to predict the effective conductivity of the nanocomposite by means of two- and three-point statistical functions with an approximation of the three-point probability functions for a two-phase media. The thermal conductivity of ZrO 2 particles is estimated using a semi-inverse application of the strong contrast technique for a buckypaper material sample. Comparison of the simulated results with our experimental results for polystyrene/ZrO 2 nanocomposite samples with different volume fractions shows satisfactory agreement. This supports the capability of our proposed methodology to predict effective thermal conductivity of nanocomposites.

Development of YBCO Roebel cables for high current transport and low AC loss applications

We discuss production of lengths of up to 27 m of YBCO Roebel cable. Results for 5/2 (5 strands, 2 mm width), 9/2 and 15/5 cables produced from standard 12 mm commercial YBCO wire are presented. We discuss specifications for the input wire and suggest using a statistical correlation function, using data from magnetic field scanning, that is shown to produce high performance strands. We discuss advances in manufacturing techniques including cable insulation processes. Transport and magnetic AC loss data are presented for 5/2 cable which demonstrates the effectiveness of decreased strand width and the transposition of strands. Both losses are predominantly hysteretic in nature. Finally, the cable DC transport is presented and we discuss the possibilities for high current cables in high field applications.

Representation of correlation statistics functions in heterogeneous materials using layered fast spherical harmonics expansion

Statistical correlation function, including two-point function, is one of the popular methods to digitize microstructure quantitatively. This paper investigated how to represent statistical correlations using layered fast spherical harmonics expansion. A set of spherical harmonics coefficients may be used to represent the corresponding microstructures. It is applied to represent carbon nanotube composite microstructures to demonstrate how efficiently and precisely the harmonics coefficients will characterize the microstructure. This microstructure representation methodology will dramatically improve the computational efficiencies for future works in microstructure reconstruction and property prediction.

GENERALIZED UNCERTAINTY PRINCIPLE AND PARIKH–WILCZEK TUNNELING

We investigate the modifications of the Hawking radiation by the Generalized Uncertainty Principle (GUP) and the tunneling process. By using the GUP-corrected de Broglie wavelength, the squeezing of the fundamental momentum cell, and consequently a GUP-corrected energy, we find the nonthermal effects which lead to a nonzero statistical correlation function between probabilities of tunneling of two massive particles with different energies. Then the recovery of part of the information from the black hole radiation is feasible. From the other point of view, the inclusion of the effects of quantum gravity as the GUP expression can halt the evaporation process, so that a stable black hole remnant is left behind, including the other part of the black hole information content. Therefore, these features of the Planck-scale corrections may solve the information problem in black hole evaporation.

GALEX-SDSS CATALOGS FOR STATISTICAL STUDIES

We present a detailed study of the Galaxy Evolution Explorer’s (GALEX) photometric catalogs with special focus on the statistical properties of the All-sky and Medium Imaging Surveys. We introduce the concept of primaries to resolve the issue of multiple detections and follow a geometric approach to define clean catalogs with well understood selection functions. We cross-identify the GALEX sources (GR2+3) with Sloan Digital Sky Survey (SDSS; DR6) observations, which indirectly provides an invaluable insight into the astrometric model of the UV sources and allows us to revise the band merging strategy. We derive the formal description of the GALEX footprints as well as their intersections with the SDSS coverage along with analytic calculations of their areal coverage. The crossmatch catalogs are made available for the public. We conclude by illustrating the implementation of typical selection criteria in SQL for catalog subsets geared toward statistical analyses, e.g., correlation and luminosity function studies.

Benthic foraminiferal variability on a monthly scale in a subtropical bay moderately affected by urban sewage

Benthic foraminifera were sampled monthly during a one-year period in order to examine their biological response to the environmental factors in the Ubatuba Bay (northern coast of Sao Paulo State, Brazil). The area is a popular tourist destination with a population that varies during the year, as does the untreated sewage carried into the bay by the rivers. Four sites were analysed. Each station is near one of the rivers that discharge into the bay. Biological data were analysed with multivariate and univariate techniques. The influence of the abiotic parameters on the foraminiferal fauna was inferred through statistical methods and time correlation functions. Geochemical and populational parameter differences allowed the sites to be separated into two groups according to their stress conditions. One group was dominated by Ammonia tepida —a herbivore species—and showed higher densities indicating a more productive and less contaminated location. The other one was dominated by Buliminella elegantissima —a detritivore species. Species diversity did not seem to be a good indicator of environmental health in this area due to the low densities and the high dominance of few species. Nevertheless, density and richness were used as evidence of local productivity and environmental conditions. Quasi-azoic moments related to the high degree of contamination were observed. Anthropogenic effects were stronger in the austral summer period, when sewage input through the rivers increases due to mass tourism.

Characterization of the effects of friction stir processing on microstructural changes in DRA composites

The effect of friction stir processing (FSP) on the spatial homogeneity of SiC particles in discontinuously reinforced aluminum (DRA) metal matrix composites is investigated. DRA composites were produced with microstructures having controlled degrees of clustering, by carefully varying the particle size ratios (PSR) of the powders used in their fabrication. In this contribution, the spatial heterogeneity of these DRA microstructures is characterized by careful measurement of statistical two-point correlation functions and also via the multi-scalar analysis of area fractions (MSAAF) technique, at length scales up to 1 mm, by the use of high-resolution, large-area image montaging. These complimentary techniques yield similar results for the spatial heterogeneity of the DRA materials, both before and after FSP. The microstructural data suggest that significant microstructural modifications are imparted by FSP, including reductions in the measured spatial heterogeneity and microstructural anisotropy, as well as re-orientation of the reinforcement particles due to the stirring action. These microstructural modifications have a strong influence on the experimentally-measured tensile strengths and ductilities of the DRA composites. A general relationship exists between the measured tensile ductility and the microstructural parameters of homogeneous length scale ( L H) and cluster size ( d clus), independent of the DRA processing history.

Using microstructure reconstruction to model mechanical behavior in complex microstructures

A systematic approach to understanding the role of microstructure on the mechanical response of complex multiphase materials is presented based on the “microstructure reconstruction” method of Yeong and Torquato. Low-order statistical correlation functions from an experimental microstructure are used to generate, using a simulated annealing algorithm, a spectrum of statistically similar microstructures that are then tested numerically. To illustrate the method, a ductile iron microstructure is studied. All simulated structures have elastic behavior and average stress–strain behavior nearly identical to that predicted for the experimental microstructure. The structures differ in the strain at which nucleation of localization begins. To identify the characteristic critical “hot spot” driving localization in each specific microstructure, a new variant of the microstructure reconstruction method is proposed in which a potential “hot spot” region from a particularly weak microstructure is embedded into new statistical microstructures, which are then tested. If the selected “hot spot” region is large enough, the new microstructures fail in the same “hot spot” region and at comparable applied stress and strain levels. This method thus shows promise for capturing local failure initiation sites in complex microstructures.

Hamiltonian and Brownian systems with long-range interactions: I Statistical equilibrium states and correlation functions

We discuss the equilibrium statistical mechanics of systems with long-range interactions. We contrast the microcanonical description of an isolated Hamiltonian system from the canonical description of a stochastically forced Brownian system. We show that the mean-field approximation is exact in a proper thermodynamic limit N → + ∞ . The one-point equilibrium distribution function is solution of an integrodifferential equation obtained from a static BBGKY-like hierarchy. It also optimizes a thermodynamical potential (entropy or free energy) under appropriate constraints. In the case of attractive potentials of interaction, we show the existence of a critical temperature T c separating a homogeneous phase ( T ⩾ T c ) from a clustered phase ( T ⩽ T c ). The homogeneous phase becomes unstable for T < T c and this instability is a generalization of the Jeans gravitational instability in astrophysics. We derive an expression of the two-body correlation function in the homogeneous phase and show that it diverges close to the critical point. One interest of our study is to provide general expressions valid for a wide class of potentials of interaction in various dimensions of space. Explicit results are given for self-gravitating systems, 2D vortices and for the HMF model.

Computer Simulations of "Realistic" Microstructures Using Statistical Two-Point Correlation Functions

Computer Simulations of "Realistic" Microstructures Using Statistical Two-Point Correlation Functions

ON SOME APPROXIMATIONS FOR NONSTATIONARY PROBLEM OF THE ACOUSTICAL PULSE SCATTERING BY A RANDOM MEDIUM

Statistical boundary value problem of a scattering of sound pulses incident on the randomly fluctuating layered medium is considered on the basis of an exact wave formulation in the spatial-time domain. To solve this problem we developed an analytical-numerical approach in earlier papers, and results of a statistical simulation were presented both for various durations of the incident pulses and for various thickness of a random medium layer. Analysis for the statistical moments, correlation functions and power spectral densities of the backscattered wave field has been carried out. Comparison with the results of this problem approximate asymptotical analysis, previously carried out by the other authors, finds out the number of differences. In this paper we examine both the results of an exact statistical simulation and the approximate analytical ones and propose, as a generalization, some quite simple approximations to describe statistical moments of the backscattered field in the region of nonstationarity. Studying the problem considered is of a fundamental interest for theoretical acoustics as well as its outcomes can be applied to interpret the data on the ocean water and bottom sediments probing with the time pulse.

Non-linear systems of multiple degrees of freedom under both additive and multiplicative random excitations

A quasi-linearization procedure is developed for non-linear systems under both additive and multiplicative white-noise excitations to obtain response statistical properties, including moments, correlation functions, and spectral densities. In the proposed procedure, only the system properties are linearized, while all excitations are kept unchanged. In particular, the basic characteristics associated with the multiplicative excitations are preserved, which is the key factor for the accuracy of the approach. Numerical examples are given, and the accuracy of the procedure is substantiated by comparing the analytical results with those obtained from Monte Carlo simulations.

Statistical Description of Microstructures

▪ Abstract We review progress made in quantitatively ascertaining the various statistical correlation functions that are fundamental to determining the material properties of specific classes of disordered materials. Topics covered include the definitions of the correlation functions, a unified theoretical means of representing and computing the different statistical descriptors, structural characterization from two-dimensional and three-dimensional images of materials, scalar order metrics and particle packings, and reconstruction techniques.

Monte Carlo Simulations of Spin-Diffusion in a 2-D Heisenberg Paramagnet

We study spin diffusion and spin waves in paramagnetic quantum crystals (solid 3He, for example) by direct simulation of a square lattice of atoms interacting via a nearest-neighbor Heisenberg exchange Hamiltonian. Recently, Cowan and Mullin have used a moments method to study spin transport at arbitrary polarizations. We test their analytic results by calculating the statistical spin correlation function from molecular dynamics simulations using a Monte Carlo algorithm to average over initial spin configurations. Since it is not practical to diagonalize the S\({\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$2$}}\) exchange Hamiltonian for a lattice which is of sufficient size to study long-wavelength (hydrodynamic) fluctuations, we instead study the S→∞ limit and treat each spin as a vector with a classical equation of motion. We compare our simulations with the assumptions of the moments method regarding the short-time behavior and the long exponential tail of the correlation function. We also present our numerical results for the polarization dependence of the longitudinal spin diffusion coefficient and the complex transverse spin diffusion coefficient.

Generating microstructures with specified correlation functions

A stochastic optimization technique has recently been developed that can reconstruct or construct random heterogeneous materials with specified statistical correlation functions. We demonstrate how this technique can be used to reconstruct a digitized image of an interpenetrating, isotropic ceramic-metal composite. In this case, the two-point probability function displays no short-range order and the image is reconstructed by optimizing in two orthogonal directions only. However, this technique results in artificial anisotropy in the unoptimized directions when one (re)constructs an image in which the isotropic two-point probability function exhibits appreciable short-range order. We show that by optimizing in more than two directions, one can effectively eliminate the artificial anisotropic effects for a system possessing significant short-range order. This is done by optimizing in three directions on a hexagonal grid and by optimizing in four directions on a square grid. Finally, an aspect of the nonuniqueness of the resulting structures is quantitatively examined.

On the Geometry and Topology of 3D Stochastic Porous Media

The nature of geometric and topological information contained in statistical correlation functions was investigated systematically using simulated porous media, generated by the level-cut of Gaussian random fields. Pore space partitioning techniques based on multiorientation scanning were implemented to determine the pore and neck size distributions, coordination number distribution, and genus of a number of model porous media. These results were correlated with the statistical properties (porosity and correlation function) of the microstructure, revealing for the first time the extent of morphological diversity of a broad class of stochastically reconstructed porous media. It was found that the dominant factor explaining microstructural variability among the media studied is the dimensionless length of spatial correlation. Accordingly, the resolution at which the void space is discretized during simulation was shown to affect significantly the resulting pore and neck size distributions and specific genus. It was also found that the average coordination number of simulated porous media is independent of correlation length, but decreases slightly with decreasing porosity.

Model independent shape analysis of correlations in 1, 2 or 3 dimensions

A generic, model-independent method for the analysis of the two-particle short-range correlations is presented, that can be utilized to describe e.g. Bose–Einstein (HBT or GGLP), statistical, dynamical or other short-range correlation functions. The method is based on a data-motivated choice for the zeroth order approximation for the shape of the correlation function, and on a systematic determination of the correction terms with the help of complete orthonormal sets of functions. The Edgeworth expansion is obtained for approximately Gaussian, the Laguerre expansion for approximately exponential correlation functions. Multi-dimensional expansions are also introduced and discussed.