Crystallite Distribution Function Research Articles

Real-space texture and pole-figure analysis using the 3D pair distribution function on a platinum thin film.

An approach is described for studying texture in nanostructured materials. The approach implements the real-space texture pair distribution function (PDF), txPDF, laid out by Gong & Billinge {(2018 ▸). arXiv:1805.10342 [cond-mat]}. It is demonstrated on a fiber-textured polycrystalline Pt thin film. The approach uses 3D PDF methods to reconstruct the orientation distribution function of the powder crystallites from a set of diffraction patterns, taken at different tilt angles of the substrate with respect to the incident beam, directly from the 3D PDF of the sample. A real-space equivalent of the reciprocal-space pole figure is defined in terms of interatomic vectors in the PDF and computed for various interatomic vectors in the Pt film. Furthermore, it is shown how a valid isotropic PDF may be obtained from a weighted average over the tilt series, including the measurement conditions for the best approximant to the isotropic PDF from a single exposure, which for the case of the fiber-textured film was in a nearly grazing incidence orientation of ∼10°. Finally, an open-source Python software package, FouriGUI, is described that may be used to help in studies of texture from 3D reciprocal-space data, and indeed for Fourier transforming and visualizing 3D PDF data in general.

Calculation of normal distributions on the sphere S2

Abstract In quantitative texture analysis normal distributions (NDs) on SO(3) group and on the sphere S2 are frequently used to calculate pole figures and orientation distribution function of crystallites in polycrystalline materials. Different motivations are used to define NDs on SO(3) and S2. This article considers a CLT-based motivation (CLT here is the central limit theorem on SO(m) rotation group, m≥2). This article defines ND on the sphere S2 as a “projection” of CLT-motivated ND on SO(3) group onto the sphere S2. The induction from the SO(3) group is performed by the way of transitioning from the NDs represented as expansions into series, where terms are matrix elements of group representations, to corresponding Fourier series for the class of functions invariant under rotation subgroup characterized by a fixed axis. This article studies NDs on the sphere S2 induced from SO(3) rotation group using corresponding NDs on the group. These NDs on S2 satisfy CLT on S2 as projections of NDs on SO(3) group that satisfy the corresponding CLT there. For central ND the corresponding induced ND on S2 matches the ND obtained during the study of Brownian motion. Different numerical algorithms are considered in order to calculate induced NDs on S2: Fourier series method, analytical approximation method and Monte Carlo method. These methods were used to approximate pole figures of NDs on S2. Also this article considers smoothing for modeling results. This task presents itself at the time of processing EBSD experimental data (obtained in the form of sample of orientations) when approximating NDs of orientations.

Biaxial Creep Behavior of Nb-Modified Zircaloys

Zirconium-niobium (Zr-Nb) alloys are used as cladding materials to encapsulate radioactive fuel in nuclear reactors. They possess excellent corrosion resistance at high temperatures making it possible to achieve high fuel burnup, directly increasing the thermal efficiency of the reactor. While they are commonly used in recrystallized (Rx) form in boiling water reactors, there is a need to understand the effect of cold work and stress relief (CWSR) on the biaxial creep characteristics of these materials due to their use in pressurized water reactors. In this study, the biaxial creep behaviors of as-received Zr-Nb alloys, HANA and Zirlo®, have been investigated at 500°C and 400°C, respectively, using internally pressurized tubing superimposed with axial load under varied hoop σθ to axial σz stress ratios of 0 to 2 while monitoring both the axial and hoop strains using a linear variable displacement transformer and a laser telemetric extensometer, respectively. The crystallographic textures and creep loci of these as-received Zr-Nb alloys have been evaluated to correlate with the previous studies on recrystallized HANA4 and CWSR Zircaloy-4. The creep locus of HANA4 was found to be unaffected by initial state (CWSR or Rx) and showed close correspondence to planar isotropy while the creep locus of CWSR Zirlo exhibited more resistance to axial deformation than diametrical as per CWSR Zircaloy-4 reported earlier. These differences are shown to arise from grain-shape anisotropy of the CWSR Zirlo and Zircaloy-4. The simulated creep loci using crystallite-orientation distribution functions in conjunction with prism slip models showed excellent agreement with experimental results.

Hashin-Shtrikman bounds with eigenfields in terms of texture coefficients for polycrystalline materials

The Hashin-Shtrikman bounds accounting for eigenfields are represented in terms of tensorial texture coefficients for arbitrarily anisotropic materials and arbitrarily textured polycrystals. This requires a short review of the Hashin-Shtrikman bounds with eigenfields, an investigation of the polarization field determined by the stationarity condition and, finally, the analysis of the resulting expressions of the Hashin-Shtrikman bound of the effective potential. The resulting expressions are given naturally in terms of symmetric second-order tensors and minor and major symmetric fourth-order tensors. These properties induce, based on the tensorial Fourier expansion of the crystallite orientation distribution function, a dependency of all Hashin-Shtrikman properties in terms of solely the second- and the fourth-order texture coefficients. This is a new result, which is not self-evident, since an alternative formulation of the polarization field would alter the implied algebraic properties of the Hashin-Shtrikman functional. The results obtained by the polarization field, determined through the stationarity condition of the Hashin-Shtrikman functional, are discussed and demonstrated with an example for linear thermoelasticity in which bounds for elastic and thermoelastic properties are illustrated.

Representation of Hashin–Shtrikman Bounds in Terms of Texture Coefficients for Arbitrarily Anisotropic Polycrystalline Materials

The present work generalizes the results of Bohlke and Lobos (Acta Mater. 67:324–334, 2014) by giving an explicit representation of the Hashin–Shtrikman (HS) bounds of linear elastic properties in terms of tensorial Fourier texture coefficients not only for cubic materials but for arbitrarily anisotropic linear elastic polycrystalline materials. Based on the HS bounds as given by Walpole (J. Mech. Phys. Solids 14(3):151–162, 1966) and tensor functions for the representation of the crystallite orientation distribution function, it is shown that the HS bounds are represented in terms of the exact same second- and fourth-order texture coefficients which appear in the consideration of the Voigt and Reuss bounds. The derived representations in terms of tensorial texture coefficients are valid for all symmetry classes in elasticity and present expressions highly attractive for the description of physical quantities in terms of tensorial variables. In order to make these results also accessible for the community of quantitative texture analysis, transformation relations between experimentally obtained Bunge’s or Roe’s coefficients and the tensorial texture coefficients are given. The representation of the present work offers a finite and low dimensional parametrization of the fully anisotropic Hashin–Shtrikman bounds, which can be used in inverse materials design problems in order to explore the set of possible materials properties or for the determination of optimal microstructural influence with respect to prescribed material properties. Examples for orthotropic polycrystals of cubic materials and transversely isotropic polycrystals of hexagonal materials (showing the connection and applicability of the results also to fiber orientation distributions) are discussed. Finally, an implementation in Mathematica® 11 of the HS bounds for arbitrarily anisotropic materials is offered, such that readers can reproduce all the results of this work and use them for their own purposes.

Fully Crystallized Silicon Nanostructured Film Prepared at Low Temperatures by Plasma-Enhanced Chemical Vapor Deposition

I studied the nanocrystalline silicon thin films by means of photoluminescent spectroscopy, Raman spectroscopy and Fourier-transformed infrared spectroscopy technique. The chemical bonding properties was studied by using the Fourier-transformed infrared spectroscopy in the range from500 cm-1 to 2300 cm-1. Spectral peak at wave number of 1100 cm-1 is related to the Si-O-Si bonding configuration. Hydrogenation of film can be estimated by using spectral lines at 2100 cm-1 and 2080 cm-1. The Si-O dipoles which are located into silicon film play great role because of electron affinity for oxygen. Photoluminescent (PL) properties are significant for the films which were made by using hydrogenation of silicon. Fourier-transformed infrared spectra of film’s absorption show the changes in chemicals in the film: from the oxygen incorporation into silicon to the elimination the Si-O bonding by adding the silicon tetra fluoride into electrochemical reactor, and increasing the Si-F density of bonds. The dissociation of SiF4 molecule causes the appearance of fluorine on the film surface, but the hydrogen atoms react with fluorine and excess fluorine is removed. By increase of SiF4 flow rate, the Si-O-Si bridges were etched by HF acid. The size effect in PL from nanocrystalline silicon film can be explained by means of statistical method. We suggested that the ƒ(x) is distribution function of crystallites in their sizes, θ(x) is a quantum efficiency of nanocrystals. Accordingly, the quantity of emitted hydrogenizednanocrystals by band-to-band radiative transitions is important to estimate the PL linear optical response. The processing of grown hydrogeneited silicon nanocrystalline film by annealing and etching the oxides results in preparation of fully crystallized silicon film for manufacturingthe photonic devices with significant quantum yield.

Low Temperature Fluorinated Silicon FILMS Synthesis

Second-harmonic generation (SHG), photoluminescence (PL) and Fourier-transformed infrared (FT IR) spectra from silicon films with different average size of nanocrystals were analized to develop new possible material for active channel in nonlinear optical devices. Silicon films were prepared with dominant concentrations of silicon bonding with oxygen or fluorine atoms. Also, the volume fraction of nanocrystals was varied from 30% to 70%. It is seen the spectral peak with energy 3.26 eV is related to defects appeared in interface area silicon-silicon dioxide. For films with small silicon crystals (less than 20 nm) the nonlinear optical response contains two spectral peaks. The second peak is caused by optical response from nanocrystal grain boundary that contains oxygen or fluorine atoms incorporated in silicon as dipoles inside film. The optical nonlinear switch device based on the nonlinear optical response of SiOxmedia inside film was proposed. Also, the silicon film with quartz micro-clusters was investigated as material for making the nonlinear optical transmitter device. The structural properties of silicon films can be easily illustrated by using the films morphology which was determined by using atomic-force microscopy. Figure 1 shows the atomic-force microscopic photos for Sample 1, Sample 2 and Sample3, respectively. It is seen, that the nanocrystal grains are in all types of investigated silicon films. The lateral distribution of nanocrystals with dominant orientation (111) are homogeneous for Sample 1 (see Fig. 1 a)), but for the Sample 2 is unhomogeneous due to appeared conglomerates which are created from the amorphous phase separately (see Fig. 1 b)). The Sample 3 has the triangular shape of nanocrystals (111) which are distributed in submicron crystalline (111) domens. It is clear, that there is great difference in morphological properties of three types of films. The HF etching the silicon surface is efficient because at Td=100oC the surface diffusion and desorption from the surface are low. By adding the SiF4 we change the H2 :SiF4 ratio and some amount of fluorine atoms are included into H2SiF6 assembly creation. Because, the etching process is low at [SiF4] =0.25 sccm and [SiF4] =0.375 sccm. After this values of silicon tetra-fluoride flow rate the H2:SiF4 ratio is changed to destruct intensive creation of molecular assemblies. This stage at [SiF4] =0.5 sccm the HF significant etches the silicon and causes the decreasing the grain sizes. It is assumed that there are two possibilities of fluorine removing: the first is hydrogen incorporation with fluorine, and, the second, the H2SiF6 assemble is creation by 300oC. The SHG is forbidden for centrosymmetric crystal such as bulk silicon because the sum dipole moment is zero, but it is possible due to the surface breaking symmetry and quadruple terms contributions. The opposite situation is for nanostructured oxidized (or fluorinated) silicon films for which the surface area of great quantity of nanocrystals is significant, and the breaking of symmetry is permanent and lateral isotropic. The spectral characteristic of optical nonlinear response from the thin silicon film strictly depends on the spectral properties of polarization and susceptibility tensors χ(2) distortion and χ(2) Si-O.F are the nonlinear susceptibilities produced by nonzero dipole moment caused by shape distortion from the spherical shape of nanocrystals and due to the existing of dangling bonds caused by fluorine or oxygen incorporation in grain boundary region, respectively. By these values of densities the dipole moments of such nanoscaled ellipsoidal dipoles covered by surface charges can be estimated as PSiO =0.04 Debye and PSiH =0.015 Debye. Therefore, the concentration of nanocrystals mainly plays a great role in PL and SHG signal intensity of radiation due to their great surface area covered by silicon bonds with contaminants (atoms of O, H, F) in grain boundary but not as a separate nanoscaled dipoles. By increasing the nanocrystal quantity inside the silicon film the ratio S/V grows significant. The size effect in PL from nanocrystalline silicon film can be explained by means of statistical method. We suggested that the ƒ(x) is distribution function of crystallites in their sizes, θ(x) is a quantum efficiency of nanocrystalls. It is supposed, that the large quantity of nanocrystals is Gauss distributed in their sizes. The Nnc-Si value is proportional to the square under the both functions ƒ(x) and θ(x) near the zero values. Accordingly, the quantity of emitted hydrogenized nanocrystals by band-to-band radiative transitions is important to esimate the PL linear optical response. For nanocrystalline silicon films for Pin àSout optical scheme χ(2) xxx component of susceptibility tensor is detected. By using optical scheme Pin àPout SHG spectra contains isotropic and anisotropic component of susceptibility tensor χ(2) zzz, χ(2) zxx,χ(2) xxz , and χ(2) xxx. Figure 1

Materials design of elastic properties of multiphase polycrystalline composites using model functions

AbstractFor a chosen combination of materials for a multiphase polycrystalline composite, a crystallite orientation distribution function for each phase is formulated as a superposition of analytic central model von Mises‐Fisher functions. The chosen central model functions allow the analytic integration of orientation averages of arbitrarily anisotropic fourth‐order tensors. Based on this result, the first‐order bounds of Voigt and Reuss and the geometric average of the stiffness can be expressed explicitly in closed forms for arbitrarily anisotropic polycrystalline materials and number of phases depending on the volume fractions and the influence of the crystallographic texture of each constituent. These expressions can then be used for the materials design aiming the determination of volume fractions and orientation distribution of selected materials for prescribed effective properties. (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Measurement and Evaluation of Crystallographic Texture in Ti-3Al-2.5V Tubing

Although the texture of Ti alloys is currently evaluated by the pole figure, it can only provide incomplete and qualitative information. In this investigation, the textures of Ti-3Al-2.5V tubing were studied to seek for an appropriate method for evaluating Ti tubing texture. The texture measurement was performed by thinning the tubing chemically into foils and flattening them, followed by x-ray diffraction analysis. A Bruker D8 Advance texture goniometer with Cu Kα radiation was employed for measuring the {0002}, {10-10}, {10-12}, {11-20} and {10-13} incomplete pole figures using the Schultz reflection technique. Based on the pole data, the crystallite orientation distribution function (ODF) was synthesized using TexEval V2.5 software, Bruker AXS GmbH. The texture strengths of the three specimens were compared using pole figures and ODF. It is found that it gives rise to a large error in the texture strength using the pole figure to evaluate texture. The orientation densities of the maxima in ODF and the tilt angle of their basal poles from the normal toward the tangential direction can describe the tubing texture accurately, which can evaluate the tubing mechanical properties. Subsequently, ODF can be used to predict the mechanical properties of batches of tubing.

Two-dimensional Rietveld analysis of celluloses from higher plants

The Rietveld method is a versatile tool to parameterize the fine structure of crystallites analyzed by diffraction. The method relies on a crystallographic model representing what is known a priori, and free coefficients determined from fits to experimental data. This article provides an introduction to Rietveld analysis of celluloses from higher plants that are adequately described by the cellulose Iβ crystal structure. Possibilities of Rietveld analysis have been recently enhanced by a tailored crystallographic model and computational algorithm, named Cellulose Rietveld Analysis for Fine Structure (CRAFS). From each two-dimensional diffraction pattern, CRAFS automated analysis outputs unit cell parameters, crystallite sizes, peak profile functions, integrated crystalline intensity (proportional to cellulose degree of crystallinity), and crystallite orientation distribution function. Two of the major hurdles for analysis of plant cellulose—overlapping of diffraction peaks and preferred crystallite orientation—are consistently treated by the two-dimensional Rietveld analysis. Hence, the method is a unique tool to explore cellulose fine structural variability, with differences arising from specimen conditioning, processing, and biological origins.

CRAFS: a model to analyze two-dimensional X-ray diffraction patterns of plant cellulose

Cellulose from higher plants is a vast renewable resource organized as crystals. Analysis of these crystals by X-ray diffraction poses very specific challenges, including ubiquitous crystallite texture and substantial overlapping of diffraction peaks. In this article, a tailor-made model named Cellulose Rietveld Analysis for Fine Structure (CRAFS) is developed to analyze two-dimensional X-ray diffraction patterns from raw and processed plant cellulose. One-dimensional powder diffractograms are analyzable as a particular case. The CRAFS model considers cellulose Iβ crystal structure, fibrillar crystal shape, paracrystalline peak broadening, pseudo-Voigt peak profiles, harmonic crystallite orientation distribution function and diffraction in fiber geometry. Formulated on the basis of the Rietveld method, CRAFS is presently written in the MATLAB computing language. A set of meaningful coefficients are output from each analyzed pattern. To exemplify model applicability, representative samples are analyzed, bringing some general insights and evidencing the model's potential for systematic parameterization of the fine structure of raw and processed plant celluloses.

ON THE SOLVABILITY OF MAXIMUM ENTROPY MOMENT PROBLEMS IN TEXTURE ANALYSIS

The estimation of the crystallite orientation distribution function based on the leading texture coefficients can be rephrased as a maximum entropy moment problem. In this paper, we prove the solvability of these moment problems under quite general assumptions on the moment functions which carries over to general locally compact and σ-compact Hausdorff topological groups.

Ultrasonic scattering for inverse characterization of complex microstructures

This paper overviews resent theoretical and experimental results for inverse characterization of duplex polycrystalline microstructures from measurements of ultrasonic attenuation and backscattering. The duplex microstructures are formed by elongated regions of clustering crystallites with preferred orientations and are typical for titanium alloys. New insights into the dependences of the backscattering and attenuation on frequency and averaged ellipsoidal grain radii are obtained. In particular the dominant effect of the averaged ellipsoidal radius in the direction of wave propagation, instead of the ellipsoidal cross-section, is shown. Also the opposite effect on attenuation and backscattering of grain size in the direction of wave propagation is demonstrated by theory and experiment. Both attenuation and backscattering depend on effective elastic properties of clusters, their size and orientation and the measurement system parameters. It is shown that for material property inversion it is advantageous to decouple elastic and geometrical properties by a special series of measurements and data inversion: first determine grain geometry and size from relative directional and/or attenuation-to-backscattering ratio measurements; this is followed by absolute attenuation and backscattering measurements to obtain the crystallite orientation distribution function in the clusters.

A software for diffraction stress factor calculations for textured materials

A software for the calculation of diffraction elastic constants (DEC) for materials both with and without preferred orientation was developed. All grain-interaction models that can use the crystallite orientation distribution function (ODF) are incorporated, including Kröner, Hill, inverse Kröner, and Reuss. The functions of the software include: reading the ODF in common textual formats, pole figure calculation, calculation of DEC for different (hkl,φ,ψ), calculation of anisotropic bulk constants from the ODF, calculation of macro-stress from lattice strain and vice versa, as well as mixture ratios of (hkl) of overlapped reflections in textured materials.

Residual Stress Analysis of Textured Materials by X-Ray Diffraction Method

The residual stress measurement by the conventional X-ray diffraction was formulated on the assumption that a specimen from polycrystalline materials was quasi-isotropic and homogeneous, and the stress was biaxial and almost constant within the X-ray penetration depth. Therefore, it was not available to analyze the stress state of the textured materials by the conventional measurement as a general rule. In resent years, advanced methods have been proposed for the X-ray stress measurement of textured materials. In some methods, it is assumed that the X-ray elastic constant is derived from the crystallite orientation distribution function of textured materials for solving the first anisotropic problem. However, there is a nonlinear problem in the stress analysis from the measured lattice strain. In present study, the X-ray elastic constants were averaged as the expected value around the normal direction of the X-ray diffraction in a similar way. A stress analysis was proposed by differential calculus of the X-ray elastic constant in order to the avoidance of nonlinear problem. The stress analysis was applied to residual stress measurements of a titanium carbide coating film with preferred orientation and a cold-rolled steel with texture. The calculated values of the X-ray elastic constants showed the linearity on some condition for the film. The X-ray stress determination was carried out by the fitting the gradients of the measured lattice strain.

Effective Flow Potentials for Anisotropic Polycrystals

AbstractIn this paper a micromechanically based flow potential for anisotropic fcc polycrystals is derived that takes into account the crystallite orientation distribution function (codf) in terms of tensorial texture coefficients. The effective flow potential is based on a representation theorem for anisotropic scalar functions depending on a 2nd‐order tensor. A priori unknown functions in the representation are determined by defining and solving explicitly a minimization problem over SO(3). Important analytical properties of the coefficient matrix of the minimization problem are discussed. (© 2009 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Representation of effective flow potentials for polycrystals based on texture data

In this paper, an effective micromechanically based flow potential for face-centered cubic crystals is pro- posed. In a first step, the single crystal flow potential is represented in terms of texture coefficients of the crystallite orientation distribution function. An optimization problem on SO(3) is formulated and solved in order to identify the best approximation of the single crystal flow potential. In a second step, assuming homogeneous stress or strain rate fields, the flow potential of the corresponding polycrystal is deduced explicitly from the tensorial representation of the single crystal potential. In contrast to phenomenological yield criteria, the advantage of such an approach lies in the fact that the macroscopic yield function can be estimated directly based on a single texture measurement and the critical resolved shear stress of the single crystal.

Estimation of Planar Anisotropy of the <I>r</I>-Value in Ferritic Stainless Steel Sheets

Planar anisotropies of r-values are calculated from crystallite orientation distribution functions (ODFs) by using the Taylor theory. These values are compared with experimentally obtained values for two kinds of ferritic stainless steel sheets with different planar anisotropies of the r-values and different texture gradients in the thickness direction. Moreover, the most suitable model for predicting the r-value of ferritic stainless steel sheets is examined. For a material with a remarkable texture gradient in the thickness, the ODF in a particular plane perpendicular to the normal direction was unsuitable for predicting the planar anisotropy of the r-value. On the other hand, r-value calculated using the average ODF through sheet thickness measured in a section perpendicular to the transverse direction shows good agreement with the experimentally obtained. The relaxed-constraints model, in which the shear strain components e13 and e23, are relaxed with a CRSS ratio of 1.1 in the {211}〈111〉 and {110}〈111〉 slip systems is the most suitable model for predicting the planar anisotropy of the r-value in ferritic stainless steel sheets.

Micromechanical Generalization of the Mises‐Hill Anisotropy Yield Criterion

AbstractThis paper proposes a new micromechanically motivated yield criterion expressed in terms of the leading tensorial texture coefficient of the crystallite orientation distribution function. The central finding is that although only a 4th–order anisotropy tensor occurs in the expression for the yield locus, the ansatz provides significantly better results in comparison to the quadratic Hill condition. The improvement is due to cubic and quartic tensorial contributions in terms of the stress direction. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Texture simulation based on tensorial Fourier coefficients

The main result of the paper is the derivation of the evolution equation of the tensorial texture coefficients of the crystallite orientation distribution function (codf). The evolution equation of each coefficient depends on the complete codf and the lattice spin, which is a constitutive quantity. For the solution of the differential equation based on a finite number of coefficients, the codf has to be estimated. This estimate is obtained here by the maximum entropy method. By this approach the texture evolution can be described by modeling some low-order Fourier coefficients. It will be shown that such a low-dimensional approach yields a reasonable description of the texture evolution and of mechanical properties like the Taylor factor.