Crystallite Distribution Function Research Articles

The use of fourier analysis of the x-ray diffraction line profile for calculating the distribution function of crystallites according to dimensions in oriented linear polyethylene

A study was made of the possibility and limitation of Fourier analysis of the X-ray diffraction line profile applied to calculate in oriented polymers the dimensional distribution function of crystallites. Calculations were made of distribution functions of crystallites according to dimensions in the longitudinal direction for model and experimental profiles of the diffraction line 002 of uniaxially oriented linear PE samples including the sample with large longitudinal dimensions of crystallites. It was shown that the average dimension which characterizes the distribution shows satisfactory agreement with the effective size of crystallites, calculated from integral line width. It was concluded from calculations that crystallites of large longitudinal dimensions are formed in oriented linear PE with an increase in the frequency of orientation elongation.

The crystallite orientation distribution functions of deformed polytetrafluoroethylene: Multiple‐pulse 19F NMR spectroscopy

AbstractA moment analysis of dipolar‐decoupled 19F NMR spectra of deformed polytetrafluoroethylene leads to an expansion which approximates the crystallite orientation distribution function. The resulting distributions are then interpreted in terms of this polymer's known mechanical properties, and an attempt is made to model these functions.

Rheo‐optical studies on the deformation mechanism of semicrystalline polymers. XIV. Alpha and beta mechanical dispersions of spherulitic high‐density polyethylene and the dynamic orientation distribution function of crystallites

AbstractThe dynamic tensile deformation mechanism of spherulitic high‐density polyethylene was investigated by dynamic x‐ray diffraction at various temperatures and frequencies in order to assign the α and β mechanical dispersions explicity. The uniaxial orientation distribution function q(ζj,0) of the jth crystal plane and its dynamic response Δq′j(ζj,0) in phase with dynamic strain were observed for the (110), (200), (210) and (020) crystal planes. Then the orientation distribution function w(ζ,0,η) of crystallites (crystal grains) and its dynamic response Δw′(ζ,0,η), also in phase with the dynamic strain, were determined by a mathematical transformation procedure proposed by Roe and Krigbaum on the basis of the Legendre addition theorem. The temperature and frequency dependences of w′(ζ,0,η) were analyzed in terms of the model parameters for dynamic spherulite deformation combining affine orientation of crystal lamellae with several types of preferential reorientation of the crystal grains within the orienting lamellae. The following assignments are made: (i) The α mechanical dispersion must be assigned to the dynamic orientation dispersion of crystal grains within the crystal lamellae, involving two types of preferential rotations of the grains about their own crystal b and a axes. The rotation about the b axis is associated with lamellar detwisting, mostly in the equatorial zone of uniaxially deformed spherulites; the rotation about the a axis is associated with intralamellar shearing, mostly in the polar zone of the spherulites. Thus both rotations are intralamellar grain‐boundary phenomena. (ii) The β mechanical dispersion must be assigned to the dynamic orientation dispersion of the crystal lamellae behaving as rigid bodies. It is not accompanied by reorientation of the crystal grains, but is associated with orientation dispersion of noncrystalline material between the lamellae. Thus it is an interlamellar grain‐boundary phenomena.

Static and dynamic deformation mechanisms of a high-density polyethylene having rownucleated crystalline texture of a-axis orientation as observed by orientation distribution function of crystallites

The rheo-optical properties of a high-density polyethylene having a row-nucleated crystalline texture of a-axis orientation were investigated in the machine direction (MD) and transverse direction (TD) of fabricating the specimen. The tensile complex dynamic compliance function J∗ displays two dispersions designated as the α 2 and α 2 mechanical retardation processes with activation energies of about 21 and 35 kcal mol −1, respectively, for both MD and TD specimens, and the complex dynamic stress-optical coefficient function M∗ exhibits a single discrete dispersion designated as the α optical retardation process with an activation energy of about 24 kcal mol −1 for both specimens over the same frequency and temperature ranges as those covered for the dynamic mechanical measurements. The real and imaginary components of M∗ converge from positive values to almost zero at the highest frequencies and lowest temperatures covered for both specimens, with the exception of the real component of the MD specimen which converges to a small but definitely negative value. The analysis of this anomalous behaviour of the MD specimen in terms of the optical retardation time spectrum has ascribed the negative convergence to a negative form birefringence that arises from a dynamic lamellar splaying or bending process responding in phase with the dynamic bulk stress. The dynamic X-ray diffraction studies have revealed the α optical dispersion and possibly the α mechanical dispersion to be mostly attributed to the crystal orientation dispersions involving intralamellar shearing processes, i.e. detwisting and tilting processes of the lamellar segments for the MD and TD specimens, respectively. The optical dispersion, if any, associated with the lamellar splaying or bending process may be expected at higher frequencies or lower temperatures than those covered here to assign the β mechanical dispersion to the interlamellar shearing process.

Rheo-Optical Studies on the Deformation Mechanism of Semicrystalline Polymers XVI. Alpha and Beta Mechanical Dispersions of Spherulitic Polypropylene and Polybutene-1 and the Dynamic Orientation Distribution Function of Crystallites

The uniaxial deformation mechanism of spherulitic polypropylene and polybutene-1 was investigated in regard to changes in the birefringence and orientation distribution function of crystal grains with the extension ratio of the bulk specimen. This mechanism can be represented by a spherulite deformation model combining the affine orientation of the crystal lamellae with the reorientation of the crystal grains within the orienting lamellae. The latter is caused by three kinds of intra-lamellar shearing mechanisms; i.e., lamellar detwisting mostly in the equatorial zone of the uniaxially deformed spherulites, and lamellar tilting and micronecking mostly in the polar zone of the spherulites. In contrast to spherulitic high-density polyethylene in which the lamellar detwisting and tilting mechanisms are dominant, the lamellar tilting and micronecking occur dominantly in spherulitic polypropylene and polybutene-1. The investigation was extended to dynamic measurements and good correspondence of the mechanical and optical dispersions designated as α and β dispersions was found, in contrast to the case of spherulitic high-density polyethylene for which an additional α2 mechanical dispersion having no correspondence to the optical dispersion was observed. The β dispersion became clearer the a dispersion did less so as the specimen changed from polyethylene to polypropylene and polybutene-1. The β dispersion was attributed to the dynamic orientation dispersion of the crystal lamellae constituting the spherulitic crystalline texture for any poly-alpha-olefin specimen, i.e., inter-lamellar crystal-grain-boundary phenomena possibly associated with orientational and/or distorsional dispersions of noncrystalline material between the lamellae. The α dispersion was ascribed to the dynamic reorientation dispersion of the crystal grains within the orienting lamellae, which occurs with a preferential orientation of the c-axis caused by dynamic lamellar detwisting and tilting in the case of polyethylene and only by dynamic lamellar tilting for polypropylene and polybutene-1, i.e., intra-lamellar crystal-grain-boundary phenomena associated with the inter-crystal friction at the boundaries.

Use of neutron diffraction texture measurements to establish a model for calculation of ultrasonic velocities in highly oriented austenitic weld material

Neutron diffraction has been used to measure the crystallite orientation distribution function for a cubic test sample of highly textured austenitic weld material. The longitudinal and transverse ultrasonic velocities in three orthogonal directions in the sample have been calculated from this function and compared with measured values for the sample. The Voigt approximation is found to give good agreement with experiment and appears to be the most useful approach for the calculation of ultrasonic velocities in highly textured materials.

Calculations from texture of earing in deep drawing for FCC metal sheets

The present study aims to calculate the height at each peripheral position of a cup drawn from polycrystalline sheet using texture data. In the analytical treatment the polycrystalline sheet is simplified to an aggregation of many single crystals with various orientations, and the texture is represented by a three-dimensional crystallite orientation distribution function. The ear is calculated crystallographically using the orientation distribution function as a volume fraction of a certain oriented crystal. In the experiment aluminium, Al-Mg alloy and copper sheets which are treated under various conditions of cold rolling and heat treatment are used. The average textures over the thicknesses of the sheets are measured by the Schulz reflection method. The calculated ears based on the measured textures are compared with the experimental ears for each material. The results show that all the principal features of the ears of drawn cups are predicted satisfactorily by the calculation.

Ultrasonic velocities in anisotropic polycrystalline aggregates

The ultrasonic velocities in a polycrystalline aggregate of cubic crystals with orthorhombic symmetry are derived. Use is made of the formalism of Roe (1965, 1966) for treating the texture of polycrystalline aggregates. It is shown how information about the crystallite orientation distribution function can be derived from ultrasonic velocity measurements. This enables the construction of ultrasonic pole figures, which may be compared with those obtained from neutron diffraction. Application is made to the effect of texture on ultrasonic propagation in austenitic welds.

Rheo-Optical Studies on the Deformation Mechanism of Semicrystalline Polymers. XIII. Uniaxial Deformation Mechanism of Polyethylene Spherulites as Observed by Orientation Distribution Function of Crystallites

Rheo-Optical Studies on the Deformation Mechanism of Semicrystalline Polymers. XIII. Uniaxial Deformation Mechanism of Polyethylene Spherulites as Observed by Orientation Distribution Function of Crystallites

Calcul de la FDOC à différents ordres de développement. Choix de l'ordre en fonction de l'acuité de la texture dans le cas de tôles d'aluminium

Quantitative analysis of textures of materials requires calculation of the crystallite orientation distribution function (CODF) by expanding pole figures as series of spherical harmonics; the accuracy of the function depends on the truncation order l of the developments. A criterion TRC has been used to obtain comparable truncation error estimations for different pole figures of a sample with a given order of development; in this way this criterion gives the value of the truncation order corresponding to a given accuracy. In the case of materials having cubic crystal symmetry, the expansion coefficients of the CODF have to satisfy relations of linear dependence: these relations have been determined for l up to 34, so that sharp textures may be considered. The calculation of the CODF at l = 34 is applied to textures having orthotropic symmetry in aluminium sheets. It is shown that an insufficient order of the series expansions may lead to important errors in the CODF values and in the nature of the preponderant preferred orientations.

The influence of texture on the mechanical response of commercial purity copper sheet in some simple forming processes

By controlling the combination of mechanical and thermal processing, different textures were obtained in commercial purity copper sheets. The degree of preferred orientation was determined from pole figure measurements and these data were analysed to provide a mathematical description of each type of texture by the crystallite orientation distribution function (CODF). The CODF, in conjunction with a chosen crystallographic slipping process, enables predictions to be made of upper- and lower-bound polycrystalline yield loci and other plastic properties of the material. Calculations of this type were performed for the copper sheets after the initial mechanical and thermal processing. Some simple deformation processes, e.g. uniaxial tension, “in-plane” balanced biaxial tension and hydrostatic bulging in circular dies, were performed on specimens prepared from each of the sheets. This enabled a comparison to be made of the influence of texture on the behaviour of the material. The properties investigated included the following: the yield strength, the stress-strain response and the plastic strain ratio (the r value) as determined from the tensile tests; the extent of uniform straining prior to instability (and/or fracture) in all the above-stated deformation modes and the development of texture ( i.e. how the texture changes) under a uniaxial and in-plane balanced biaxial straining mode. Some of the measured mechanical property data and the texture changes were compared with predictions based on the CODF of the initial sheets and a selected crystallographic slip mode. In some cases excellent agreement was obtained, particularly for texture development; the least successful comparison was the absolute values of the plastic strain ratio. However, a comparison of the trend in these values was reasonable.

An upper-bound evaluation of interactive creep and growth in textured Zircaloy

Uniform strain-rate, upper-bounding constitutive modeling techniques are applied to the study of irradiation creep in textured hexagonal metals with a specific application to Zircaloy. A crystallite orientation distribution function is used to describe the distribution of grains in a polycrystal for use in strain and stress averaging. Stress dependent creep compliances are constructed for a system exhibiting slip on a discrete set of crystallographic slip systems. Superposition of creep and stress independent sources of internal strain, such as those produced by the irradiation growth mechanism, is studied. A model for irradiation growth is proposed which accounts for grain shape and size effects as well as non-uniform dislocation densities arising from variable fabrication history. The internal strains are shown to interact with those dependent on applied stress in a manner which alters the constitutive behavior.

Textures in a superplastic Al-6Cu-0.3Zr alloy

Quantitative texture studies using the crystallite orientation distribution function have been made on a superplastic Al-6Cu-0.3Zr alloy over a range of strain rates at 450°C. It is shown that not only can such studies be used to determine the occurrence of grain boundary sliding and both single and multiple slip, but that they can also be used to identify and characterise the nature of the dynamic recrystallisation occurring on superplastic deformation. In this way multiple slip is shown to be a major deformation mode at the highest superplastic strain rates, whilst single slip is shown to contribute to the overall strain throughout Region II. The dynamic recrystallisation is shown to be continuous, in contrast to the discontinuous recrystallisation observed on long static anneals.

Microplasticity Predictions of r‐Values and Yield Loci of Low Carbon Sheet Steels Deforming by <111> Pencil Glide

The plastic deformation of low carbon sheet steels has been analyzed using microplasticity models and Hill′s theory of anisotropic continuum plasticity. The analytical methods used to determine the crystallite stress states which must be generated for the isostress (lower‐bound) and isostrain (upper‐bound) cases are briefly discussed and then applied to predict yield loci for three cold‐rolled, annealed low‐carbon steels (enameling iron, aluminum‐killed steel, and interstitial‐free steel) whose textures were characterized by crystallite orientation distribution functions. While measured r‐values correlate best with upper‐bound yield loci predictions for enameling iron, those for the aluminum‐killed steels show better agreement with lower‐bound yield loci predictions. In the case of the interstitial‐free steel, measured r‐values could not be accounted for by either upper‐ or lower‐bound predictions. The yield loci measurements and predictions are reviewed in the context of Hill′s continuum theory.

Studies on the Deformation Mechanism of Polyethylene Spherulites by the Orientation Distribution Function of Crystallites

Studies on the Deformation Mechanism of Polyethylene Spherulites by the Orientation Distribution Function of Crystallites

Quantitative texture studies of the superplastically deformed AlCu eutectic alloy

Quantitative texture measurements using crystallite orientation distribution functions have been made on both phases of the Al-33 wt% Cu eutectic alloy as a function of strain and strain-rate during superplastic deformation at 713 K. The details of the texture changes can be interpreted in terms of grain boundary sliding, dislocation motion and diffusion creep. Grain boundary sliding is shown to be a maximum at maximum m, but while multiple slip in the aluminium-rich phase is important at high strain-rates slip also makes a contribution to deformation at maximum m. In the CuAl 2 phase anisotropic diffusion creep is an important deformation mechanism.

Textures in superplastic Zn-40 wt.% Al

Quantitative texture measurements using the crystallite orientation distribution function have been made for both phases in a superplastic Zn-40 wt.% Al alloy as a function of strain rate. Although the overall level of the texture is reduced by superplastic deformation, certain components are retained. The results are interpreted in terms of grain boundary sliding combined with slip in both phases.

Earing in Deep-Drawing Steels

The variations of the fourth-order coefficients of the crystallite orientation distribution function, with rolling reduction have been determined after cold-rolling and annealing for a deep-drawing quality rimming steel and an aluminium-killed steel. These coefficients influence drawability and earing behaviour and by the manipulation of the coefficients in the distribution function of a 60% cold-rolled and annealed rimming steel, a hypothetical non-earing sheet texture has been derived. By comparison with the actual sheet texture those textural components which most affect earing behaviour are identified.

Crystal orientation in a semicrystalline polymer in relation to deformation of polymer spherulites. II. Orientation distribution function of crystallites within crystal lamella as a function of lamellar orientation

AbstractA model relating crystal orientation in a semicrystalline polymer to the deformation of polymer spherulites is proposed. The distribution function for orientation of crystallites within crystal lamellae is assumed to be a function of lamellar orientation. In addition to the orientation of crystal lamellae in affine fashion, several parameters are introduced to characterize the untwisting of the crystal lamellae and the four different types of orientations of the crystallites within the crystal lamellae in the undeformed and deformed states of the spherulite. The model was tested by experiments in uniaxial stretching of a low‐density polyethylene. The theoretical distributions of orientation of given reciprocal lattice vectors of the crystallites, such as the reciprocal lattice vectors of the (110) and (200) crystal planes, are compared with the results of x‐ray diffraction experiments. It was found that the most important factors in fitting the model to experimental results are: (a) the fraction of crystallites having random orientation within lamella and, in turn, representing the degree of imperfection of the lamella in the undeformed state; (b) the ease of transition of crystal orientation within lamella from b‐axis orientation parallel to the lamellar axis to two types of c‐axis orientations (type Ca and type Cr) parallel to the stretching direction; and (c) the fraction of crystallites having orientation in type Cr (unfolding mechanism) rather than type Ca (rotation mechanism).

Elastic and plastic anisotropy in sheets of cubic metals

Methods are described for the calculation of the elastic and plastic properties of textured polycrystalline metals. The calculations involve the crystallite orientation distribution function derived from X-ray texture data together with data which describe the single crystal behavior. Subsequently, it is shown how a knowledge of the elastic properties can be used to make an analytical prediction of the plastic properties. In making this prediction use is made of the fact that for cubic metals both the elastic and plastic properties are influenced predominantly through the zeroth and fourth order coefficients of the crystallite distribution function. The methods are illustrated by application to the analysis of the data of Stickels and Mould. The statistical correlation they observed is shown to have a justifiable analytical basis.