Intercept Length Distribution Research Articles

Analytical models for phonon mean free path in polycrystalline nanostructures based on mean square displacement

In this study, a numerical simulation method and analytical models for predicting the boundary scattering mean free path (MFP) of phonons in polycrystalline nanostructures are developed. The grain morphologies are assumed to be approximately equiaxed, i.e., forbidding needle-like or pancake-like morphologies. Adapting a technique from rarefied gas dynamics, the method evaluates the MFP from the mean square displacements of phonons that experience random motion and interface collisions in nanostructures. We confirm that the MFP in simple cubic polycrystalline nanostructures obtained by the simulations agrees with that reported in a previous study; this result supports the validity of the method. Two analytical models for high and low interfacial transmission probabilities at the crystal interfaces are also derived by considering the mean square displacements. We find that the grain-boundary intercept length distribution of polycrystalline structures is an essential parameter for determining this boundary scattering MFP. These analytical models reproduce the MFPs in simple cubic and Voronoi diagram polycrystalline nanostructures calculated by the numerical simulations. This result indicates that the boundary scattering MFP of phonons in polycrystalline nanostructures can be obtained once the intercept length distribution is evaluated, without any additional numerical simulations.

EFFECTS OF VOLUME OF INTEREST (VOI) SIZE ON TRABECULAR BONE QUANTIFICATION IN THE JUVENILE DISTAL FEMUR

Volumes of interest (VOI) are commonly assigned to image stacks generated from micro-computed tomography to specify areas for bone quantification. However, the size of the VOI can impact the values obtained for trabecular bone parameters. This study aims to investigate the effect of VOI size by applying VOIs of four different diameters (10%, 15%, 20% and 30% of the antero-posterior width). Ten juvenile right distal femora were included, aged from pre-natal to three years. Smaller VOIs were placed within the largest VOI, with multiple locations used for the same VOI size. The observed parameters included bone volume fraction (BV/TV), trabecular bone thickness (Tb.Th), separation (Tb.Sp), number (Tb.N), degree of anisotropy by mean intercept length (DA.MIL) and star volume distribution (DA.SVD). Statistically significant differences between VOI sizes were found for Tb.Sp, Tb.N, DA.MIL and DA.SVD. A possible effect of localized variation was found due to significantly different values for Tb.Th between VOI locations. The effect related to VOI geometry was reflected by DA.MIL and DA.SVD as no significant difference was found between locations. The minimum diametric strut quantity (DSQ, diameter multiplied by Tb.N) of 4 can be implied based on the zero DA.SVD value found in particular VOIs.

Description of induced anisotropy in microstructure of granular soils

The primary focus in this work is on developing a methodology for specification of an evolution law which correlates the changes in the fabric of granular soil with the continuing deformation. The fabric descriptors are based on lineal intercept measurements and include the areal pore size and the mean intercept length distribution. The approach involves performing a series of Discrete Element simulations for a granular assembly under evolving directions of the principal stress/strain and defining a correlation with the evolution of material axes. It is demonstrated that granular materials with spherical particles may become anisotropic as a result of a continuing deformation process. For dense assemblies characterized by a high macroscopic friction angle (e.g., dense sand), the evolution law governing this induced anisotropy may be defined by assuming coaxiality between the microstructure and the total strain tensors.

Effect of Cr addition on solid state sintering of WC–Co alloys

The effect of Cr addition and C content on densification and microstructural evolution of WC–Co alloys in the solid state was studied. Two alloys containing Cr and an excess of carbon or tungsten were investigated. They were compared with Cr free alloys. The intercept length distribution of the pore and binder phase was measured at several temperatures using image analysis. Cr addition delays the shrinkage whatever the C potential whilst better binder spreading and more effective rearrangement occur at lower temperature in the W rich alloy. When the temperature increases, the densification is accelerated in the C rich alloy. It is found that Cr addition has no significant effect on the solubility of W in the binder. The influence of Cr on the shrinkage behaviour is discussed putting emphasis on the grain boundary energies and on Cr oxides.

P.374 Dental implants in children with ectodermal dysplasia

The effect of Cr addition and C content on densification and microstructural evolution of WC–Co alloys in the solid state was studied. Two alloys containing Cr and an excess of carbon or tungsten were investigated. They were compared with Cr free alloys. The intercept length distribution of the pore and binder phase was measured at several temperatures using image analysis. Cr addition delays the shrinkage whatever the C potential whilst better binder spreading and more effective rearrangement occur at lower temperature in the W rich alloy. When the temperature increases, the densification is accelerated in the C rich alloy. It is found that Cr addition has no significant effect on the solubility of W in the binder. The influence of Cr on the shrinkage behaviour is discussed putting emphasis on the grain boundary energies and on Cr oxides.

Determination of the WC grain size in hardmetals using stereology – A critical analysis

Stereology is frequently applied to determine mean grain size and grain size distribution of the carbide phase in hardmetals. The spreading use of softwares based in stereological routines tends to intensify this practice. Nevertheless, the utilization of stereology must be done with criticism. This paper examines the use of both the intercept length and the cross section area to represent grain size. Moreover, it proposes the use of the section perimeter to do the same representation. The limitation of the mean intercept length to represent the mean grain size of a population of grains with different sizes is discussed. The intercept length distribution does not represent the true population of grains in the structure since part of the grains is excluded from the measurement.

Particle texture analysis using polarized light imaging and grey level intercepts

The usual characterization of particulate ore material with image analysis includes modal analysis and liberation analysis. Both methods include stereological corrections based on intercept length distributions within each phase of interest. The first principle of stereology relies on a simple assumption of uniform random sectioning, whereas the liberation models require all particles to have similar textural properties. However, for the sake of adequate prediction of ore behavior in industrial processes it is becoming more and more important to be able to describe particle populations in terms of texture classes. Among the texture analysis techniques, this paper focuses on mean intercept length (MIL). It brings two important improvements: the first one is to improve grain boundary imaging by using images of bireflectant minerals under plane polarized illumination, the second one is to consider intercept analysis on grey level images instead of binary ones. The method is illustrated on a selection of critical hematite textures found within banded iron formations.

Hemisphere Sizing from Linear Intercept Measurement

The stereological problem of unfolding the hemisphere radius distribution from the length distribution is analyzed. Let a stationary isotropic process of hemispheres be given. The hemispheres have random diameters and are isotropically uniformly randomly orientated in space. A straight line probe yields a process of intercepts. The inverse problem of re-obtaining the size distribution of the hemispheres in terms of an experimental intercept length distribution is solved. The length distribution of a single hemisphere, known analytically, is approximated by piecewise polynomials in two intervals. The solution of the inverse problem is traced back to a simple recurrence equation. Numerical checks with exact and simulated data are performed to demonstrate the applicability. Data of chord length sampling, resulting from image analysis procedures, from scattering methods or from other appropriate physical apparatuses, are applicable.

Angular orientation of trabecular bone in the femoral head and its relationship to hip joint loads in leaping primates

The elastic properties and mechanical behavior of trabecular bone are largely determined by its three-dimensional (3D) fabric structure. Recent work demonstrating a correlation between the primary mechanical and material axes in trabecular bone specimens suggests that fabric orientation may be used to infer directional components of the material strength and, by extension, the hypothetical loading regime. Here we quantify the principal orientation of trabecular bone in the femoral head and relate these principal fabric directions to loading patterns during various locomotor behaviors. The proximal femora of a diverse sample of prosimians were scanned using a high-resolution X-ray computed tomography scanner with resolution of better than 50 mum. Spherical volumes of interest were defined within the femoral heads and the 3D fabric anisotropy was calculated using the mean intercept length and star volume distribution methods. In addition to differences in bone volume and anisotropy, significant differences were found in the spatial orientation of the principal trabecular axes depending on locomotor behavior. The principal orientations for leapers (Galago, Tarsius, Avahi) are relatively tightly clustered (alpha(95) confidence limit: 8.2; angular variance s: 18.2 degrees ) and oriented in a superoanterior direction, while those of nonleapers are more variable across a range of directions (alpha(95): 16.8; s: 42.0 degrees ). The mean principal directions are significantly different for leaping vs. nonleaping taxa. These results further suggest a relationship between bone microstructure in the hip joint and locomotor behavior and indicate a similarity of loading across leapers despite differences in kinematics and phylogeny.

A Computer Simulation of the Hard-Metal Structure: the Interpretation of Structural Features

The intercept length measurement of WC grains in hardmetal alloys is simulated in computer. The distribution of intercept length is related to the distribution of WC grain size, but the latter can not be measured from a sintered alloy. It must be interpreted from the intercept length measurement. By using simulation, the complex grain size distributions of real alloys can be decomposed into more simple distributions, as the monomodal ones. The analysis of the monomodal structures is used to interpret the more complex structures. It is found that the lognormal intercept length distribution measured in hardmetal alloys in practice does not necessarily correspond to a lognormal grain size distribution, as commonly reported. The lognormal-like format of the intercept length distribution curve of real alloys is a natural trend of structures with multimodal grain size distributions.

Quantitative analysis of the effect of the binder phase composition on grain growth in WC-Co sintered materials

Microstructural evolution of WC-Co materials with different compositions and WC grain sizes is discussed from SEM observations and quantitative image analyses of intercept length distributions in the WC phase. Effect of the C/W ratio in the binder phase and of V- or Cr-dopants is particularly studied. Microstructural coarsening is shown to be more rapid and less abnormal when the C/W ratio is increased in the two-phase field {WC + liquid} and up to the three-phase field {WC + C + liquid}. The V-dopant does not completely suppress this effect. The composition dependence of the liquid stability—analysed from the calculated Gibbs energies of the phases—could explain the trend of the experimental results.

Analysis of the evolution of the grain size distribution in WC–Co sintered materials with random set models

Coarsening of WC grains in equilibrium with a Co-based liquid is characterised by a gradual evolution of the mean grain size, while a population of large grains can form under certain conditions. Coarsening phenomena in such cases cannot be analysed simply in terms of average grain size. In this paper, grain size distribution is followed using a Boolean model. WC grains are represented by triangular prisms, with a log-normal grain size distribution. Model parameters are estimated from image analysis measurements. Variation of size distribution with sintering time is analysed. Accuracy of the method is discussed and its efficiency in characterising the abnormal character of grain growth is compared with more conventional measurements of intercept length distributions.

Direct calculation of the surface-to-volume ratio for human cancellous bone

There are many diseases which cause detrimental changes in the trabecular structure of cancellous bone, leading to mechanical failure of the tissue. One approach to understanding the mechanisms of these diseases is to create idealized models that recreate the morphology of the tissue. This paper presents a partial development of such a model. Further histological methods must be developed before a complete definition of morphologically valid models is possible. In a histological section of cancellous bone, the orientation and length of the trabecular surfaces determine how a line drawn across the bone section will intersect the bone-marrow interface. The distribution of the average length between intersections for a set of parallel lines is defined as the mean intercept length distribution. In this paper, the average surface morphology and volume of the average structure of cancellous bone is determined from an examination of the mean intercept length. The average structure of cancellous bone contains a repeated structural element (SE). As a result, the basic bone structure is analogous to a brick wall made from many similar bricks. For a group of 107 specimens, a strong relationship between structural element volume (SE.V) and bone volume fraction ( BV TV ) is demonstrated, SE.V=0.017κ( BV TV ) −2.05 mm 3 , R 2=0.93, with κ a model-dependent constant. For the same specimens, the structural element surface (SE.S) showed the relationship, SE.S=0.144κ( BV TV ) −1.35 , R 2=0.92. As a result of the inverse square dependence of structural element volume on bone volume fraction, it is predicted that cancellous bone strength is inversely proportional to structural element volume.

A Significant Method for Estimation of the Grain Size of Polycrystalline Materials

A method has been proposed to estimate the grain size, which is defined as the grain size distribution in three dimensions, from the measured distribution of linear intercept length. The method is developed on the assumption that the grain size distribution is log normal in three dimensions. The three-dimensional grain size distribution is expressed as that of the “grain diameter” D defined as the equivalent volume diameter; the distribution of D is estimated from ln Dg and ln σg (where Dg is the geometric mean of D and σg is the geometric standard deviation for D). The estimation is based upon the theoretical distribution of linear intercept length calculated numerically from a tetrakaidecahedron grain structure model with log-normal distribution of grain diameter. The values of Dg and ln σg are determined by applying the ‘discrete’ least squares method to the measured linear intercept length distribution on the basis of the calculated results. The present method has made it possible to evaluate the fundamental parameters of Dg and ln σg defining three-dimensional distribution of grain size from measurements on the cross-sectional structure. Futher, the influence of grain size distribution on the estimation of mean grain sizes by the two representative methods, the intercept method and the planimetric method, has been discussed from the calculation in the above model and the analysis of the experimental results. As a result, it has been concluded that the grain size distribution has a significant effect on the estimation of the mean grain size.

One‐dimensional image analysis of soil structure. 11. Interpretation of parameters with respect to four forest soil profiles

SUMMARYThe one‐dimensional (1‐D) image analysis system described in the companion to this paper (Part I) was used to analyse horizontal sections of pore space (≥ 180 μm) in four forest soil profiles with impeded drainage. Parameters were interpreted with respect to structure in the images. Mean solid intercept length is inversely proportional to extent of pore space. It was longer for blocky structures than massive structures, which had extensive, fine pore space. It decreased with decreasing ped size, increasing development and increasing intrapedal porosity. The solid intercept length distribution separated the effects of intrapedal porosity. For blocky structures, the pore intercept length distribution indicated the sizes and quantities of fissures and intrapedal pores. All profiles had an horizon with blocky structure with lower porosity owing to greater distance between interpedal pore space and low extent of intrapedal pore space. The deeper, more massive horizons had extensive, fine pore space separating fine aggregates.

One‐dimensional image analysis of soil structure. I. Principles

SUMMARYA one‐dimensional (1‐D) image analysis system, ANOSOL, has been developed which uses test lines to measure the distances in a random direction across and between pore space in images of impregnated soil blocks. Suitable strategies are discussed for both random and systematic sampling using test lines. A comprehensive set of interrelated 1‐D parameters are derived for use as structural indices and their stereological relationship to 3‐D parameters is discussed. The intercept length distributions for both pore space and solid can be adequately described using a three‐parameter model consisting of the weighted sum of two negative exponential functions. The advantages of the parameters chosen are discussed with references to those used by other workers. The importance of choosing, as structural indices, parameters that are stereologically related to the desired 3‐D parameters is also discussed.

Intercept length distributions for several polygons

Intercept length distribution curves were calculated or determined graphically for a variety of polygons and shapes as an initial step in ascertaining the nature of intercept distributions for statistical grain sections, an intermediate goal being to provide a method of calculating section diameters from the more easily measured intercept distributions determined by lineal analysis, but with the ultimate aim of determining the spatial distribution. The following shapes were selected: an ellipse, an equilateral and an irregular triangle, a square, a 60° rhombus, a rectangle, a pentagon, a hexagon, an irregular seven sided polygon, and an irregular five sided polygon having convex sides. If the intercept length is designated as q and the distribution function as f(q), it is concluded that f(q) of a statistical grain section would have the following characteristics: f(q) finite at q=0, df(q)/dq≠0 at low values of q, f(q) a maximum at q < qmax and f(q)=0 at qmax.MST/1104

Intercept length distributions for several polygons

Abstract Intercept length distribution curves were calculated or determined graphically for a variety of polygons and shapes as an initial step in ascertaining the nature of intercept distributions for statistical grain sections, an intermediate goal being to provide a method of calculating section diameters from the more easily measured intercept distributions determined by lineal analysis, but with the ultimate aim of determining the spatial distribution. The following shapes were selected: an ellipse, an equilateral and an irregular triangle, a square, a 60° rhombus, a rectangle, a pentagon, a hexagon, an irregular seven sided polygon, and an irregular five sided polygon having convex sides. If the intercept length is designated as q and the distribution function as f(q), it is concluded that f(q) of a statistical grain section would have the following characteristics: f(q) finite at q=0, df(q)/dq≠0 at low values of q, f(q) a maximum at q < qmax and f(q)=0 at qmax.MST/1104

Does fabric tensor exist for a fabric?

It is shown that the mean intercept length distribution for planar fibre networks or for materials composed of a set of plates is not in general elliptic and cannot be expressed analytically in terms of a second-order tensor. However, our numerical computations indicate that the polar plot of the mean intercept length at the angle of measurement may become nearly ellipsoidal as the microstructure (fibres or plates) become less discretely organized, but yet remain orthotropic. The equations presented in this study may be used to obtain fibre (plate) orientation density functions from the experimental data on mean intercept length distribution.

Linear Intercept Length Distribution of Polycrystals

Linear Intercept Length Distribution of Polycrystals