Diffraction-based modelling of microstructural size and strain effects in sintered ceramics
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Crystallite (or grain) size and strain within a polycrystalline material may have a profound influence on its physical properties, eg. the fracture toughness, wear and thermal shock resistance. A diffraction pattern for a material conveys information about the strain through the strain-induced changes in the shapes of the Bragg peaks and also through peak shifts. Crystallite size effects also influence the peak shape. Therefore, it is possible, in principle, to extract descriptions of crystallite size and strain from the peak broadening of a diffraction pattern. Various methods for size and strain evaluations have been proposed for extraction of the size and strain information in metals and ceramic powders. However, there appear to be no detailed amounts in the literature to be on the development of models appropriate for sintered ceramic materials. The objectives for this study were to critically examine the existing models for crystallite size and strain assessments and then to develop a new physically-based model which might be appropriate for sintered ceramics. The principal steps for the research, designed to fulfill the study objectives, were (1) acquiring high-quality diffraction data with synchrotron radiation, laboratory x-ray and neutron diffraction techniques for model evaluation; (2) performing preliminary evaluation using the existing models; (3) developing a new model and the non-linear least-squares calculation software; and (4) performing peak profile analyses using the existing and new models to evaluate the effectiveness of the new model. A convolution model for crystallite size and strain determination from diffraction line broadening has been developed with particular reference to the characterisation of sintered ceramics.The size profile component function for the convolution model involves the modal size and the size distribution appropriate for `normal' crystallite growth according to the mean-field theory, as proposed recently in a seminal publication by Dr. Brian York of IBM. A Gaussian strain profile component function was considered in the study on the basis that it has been widely used for specimens which exhibit small microstrain (ca. 10-3 or less). The overall profile describing the diffraction pattern involves convolution of the instrument, size and strain effects. A non-linear least-squares refinement program entitled MOZAIX has been developed for profile fitting with the model. Data simulations were performed with the model, and non-linear least-squares optimisations for fitting the simulated data showed that the calculations were reasonable for low-strain sintered ceramics. The convolution model for size and strain assessments from diffraction line broadening has been evaluated with synchrotron and laboratory x-ray radiation diffraction data (SRD and XRD, respectively). The study made use of MgO ceramics with three different purity levels which had been sintered at a range of temperatures in order to provide diffraction data with a range of microstructural strain and size effects. The cubic symmetry of MgO provided isotropic size and strain effects as had been anticipated. The Voigt function, a convolution of the Gaussian and Lorentzian functions, is widely used to extract crystallite size and strain information from powder diffraction data using (1) Fourier transforms, (2) the Rietveld method and (3) integral breadth methods. Size and strain model evaluation carried out using the Voigt-based Rietveld and integral breadth methods assumes that the size effect contributes only to the Lorentzian component and the strain contributes only to the Gaussian component.Size and strain assessment using the Voigt integral breadth single-line and Rietveld methods has been examined in this study with diffraction data for MgO ceramics. Two major outcomes from the evaluation confirmed impressions gained from the literature that: 1. the integral-breadth single-line method can be used as a reliable technique for size and strain analysis; 2. analysis using the Voigt function has no physical basis, is inappropriate for profiles with 'super-Lorentzian' character and is inadequate for size-strain analysis since the function does not take into account the size distribution parameter. There has been a strong trend recently towards whole-pattern size and strain evaluations which are progressively replacing single-line methods. However, due to time constraints, this study was confined to single-line analysis with the focus being on the development of the model, and with an expectation that the single-line model would readily be extended in the future to use with whole-powder pattern data. The size-strain analysis results using the convolution model showed that sintering (1) promotes crystallite growth and (2) relieves residual strains in low density sintered ceramics and introduces strains in dense ceramics, presumably due to grain-grain shear interactions. The effect of sintering on the size distribution clearly depends on the crystallite growth behaviour. Comparing the SRD convolution size results with those from scanning electron microscopy (SEM) showed that (1) the "grains" imaged using SEM contain clusters of crystallites and (2) the SEM-derived and convolution size distributions are in a satisfactory agreement.In general, despite the larger uncertainties due to instrument resolution, the XRD results are in agreement with those from SRD. The size and strain values obtained with the convolution model were compared with those calculated using the Voigt single-line integral-breadth method. The comparison showed that size and strain results for both methods were dependent upon the character of the diffraction peak shapes. The convolution model improves the Voigt model in terms of (1) reliability of models from a physical point of view, (2) the additional size distribution parameter and (3) its applicability to `super-Lorentzian' profiles. Subsequent research is suggested to further improve the model in dealing with large microstrains and developing a whole powder fitting procedure.
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