A. Borbély

Crystallite size distribution and dislocation structure determined by diffraction profile analysis: principles and practical application to cubic and hexagonal crystals

Two different methods of diffraction profile analysis are presented. In the first, the breadths and the first few Fourier coefficients of diffraction profiles are analysed by modified Williamson–Hall and Warren–Averbach procedures. A simple and pragmatic method is suggested to determine the crystallite size distribution in the presence of strain. In the second, the Fourier coefficients of the measured physical profiles are fitted by Fourier coefficients of well established ab initio functions of size and strain profiles. In both procedures, strain anisotropy is rationalized by the dislocation model of the mean square strain. The procedures are applied and tested on a nanocrystalline powder of silicon nitride and a severely plastically deformed bulk copper specimen. The X-ray crystallite size distributions are compared with size distributions obtained from transmission electron microscopy (TEM) micrographs. There is good agreement between X-ray and TEM data for nanocrystalline loose powders. In bulk materials, a deeper insight into the microstructure is needed to correlate the X-ray and TEM results.

The contrast factors of dislocations in cubic crystals: the dislocation model of strain anisotropy in practice

It has been shown recently that in many cases strain anisotropy in powder diffraction can be well accounted for by the dislocation model of the mean square strain. The practical application assumes knowledge of the individual contrast factors C of dislocations related to particular Burgers, line and diffraction vectors or to the average contrast factors C¯. A simple procedure for the experimental determination of C¯ has been worked out, enabling the determination of the character of the dislocations in terms of a simple parameter q. The values of the individual C factors were determined numerically for a wide range of elastic constants for cubic crystals. The C¯ factors and q parameters were parametrized by simple analytical functions, which can be used in a straightforward manner in numerical analyses, as e.g. in Rietveld structure refinement procedures.

Dislocations and grain size in ball-milled iron powder

The microstructure of ball-milled iron powder has been investigated by high resolution X-ray line profile analysis. Analysis of line breadths suggested that the contrast factors related to dislocations have to be taken into account in the Williamson-Hall procedure. This concluded to a modified Williamson-Hall plot which provided, in a straightforward manner, the grain size refinement of nanocrystals ball-milled for different periods of time. At the same time it has been shown that strain broadening, even in these nanoscale small-grain particles, is caused by the presence of dislocations. The line profiles were also studied by the method of Fourier analysis, which gave the absolute values of dislocation densities.

Computer program ANIZC for the calculation of diffraction contrast factors of dislocations in elastically anisotropic cubic, hexagonal and trigonal crystals

The computer program ANIZC has been developed using the Pascal programming language for the calculation of diffraction contrast factors of dislocations in elastically anisotropic cubic, hexagonal and trigonal crystals. The contrast factor is obtained numerically by integrating the angular part of the distortion tensor in the slip plane. The distortion tensor is calculated by solving the sextic equation provided by the mechanical equilibrium of a single dislocation in an infinite anisotropic medium. The contrast factors can be used for the interpretation of strain anisotropy as obtained from peak profile measurements made on either single crystals, textured polycrystals or powders.

An X-ray method for the determination of stored energies in texture components of deformed metals: Application to cold worked ultra high purity iron

Abstract An X-ray method has been developed to evaluate the stored energy of cold work in different texture components of plastically deformed metals. The dislocation density and the outer cut-off radius of dislocations are obtained from Bragg peaks recorded from single texture components. The stored energy is approximated by the energy of dislocations, which is calculated according to the anisotropic theory of elasticity. As an example the method is applied to the case of two major texture components developed in cold rolled ultra high purity (UHP) iron. The stored energy of the {111}〈112〉-γ fibre component of the 88% cold rolled UHP iron is about 3.6 times larger than that of the {001}〈110〉-α fibre component. The present results, of significantly higher accuracy than those of previous methods, are in good agreement with data obtained from microhardness and recent calorimetric measurements.

Particle-size, size distribution and dislocations in nanocrystalline tungsten-carbide

Abstract Ball-milled WC powder has been compacted and sintered. The particle size has been determined by transmission and scanning electron microscopy and high resolution X-ray diffraction. The X-ray diffraction profiles were evaluated for particle size and dislocation densities by the recently developed procedures of modified Williamson-Hall plot and modified Warren-Averbach method. Log-normal particle size distributions have been obtained by a newly developed numerical method from the size parameters provided by X-ray line profile analysis. The size distributions obtained by TEM and X-rays are compared and discussed. The dislocation density was found to be 1.9 × 1017 m−2 in the ball-milled powder which decreases to 2.0 × 1015 m−2 after annealing.

Dislocations and grain size in electrodeposited nanocrystalline Ni determined by the modified Williamson-Hall and Warren-Averbach procedures

Electrodeposited nanocrystalline Ni foils were studied by high-resolution X-ray diffractometry. The full width at half-maximum and Fourier coefficients were found to vary rather anisotropically as a function of diffraction order. The modified Williamson–Hall plot and the modified Warren–Averbach analysis, developed recently by taking into account the dislocation contrast in peak broadening, have been applied to interpret this anisotropic behaviour in terms of grain size, dislocation densities and twin boundaries. The average grain size has been found to range between 50 and 12 nm, in good agreement with transmission electron microscopy observations. The average dislocation density has been found to be 4.9 (5) × 1015 m−2 and the dislocations are of the screw character.

FE investigation of the effect of particle distribution on the uniaxial stress–strain behaviour of particulate reinforced metal-matrix composites

Abstract A multi-particle 2D finite element model of a 20% particulate reinforced metal-matrix composite was developed on a statistical basis taking into account the correlations between the position, size and orientation of the ceramic particles in the matrix. The stress–strain curves in tension and compression given by the clustered multi-particle model are compared with the curves obtained from one-particle unit cell simulations. It is shown that clustering of particles increases the plastic strain accumulated in the matrix leading to a higher strain hardening and thus to a higher flow stress. The size of the representative volume element (RVE) should be at least equal to the correlation length of the geometrically relevant correlation functions, which was ∼2.4 times larger than the average interparticle distance for the experimentally studied case. Reasonable agreement is obtained between computed residual strains and data available in the literature.

Long-range internal stresses in cell and subgrain structures of copper during deformation at constant stress

Long-range internal stresses in dislocation cell and subgrain structures were investigated experimentally. The transition of the dislocation structure from cells to subgrains was achieved by deforming copper polycrystals in compression creep tests at constant stress normalized by the shear modulus in the temperature range from 298 K to 633 K. The long-range internal stresses were investigated by two methods. The first one was the evaluation of characteristically asymmetric X-ray line profiles. The internal stresses are the result of the analysis of the X-ray line profiles. The second one was the measurement of local lattice parameters by convergent beam electron diffraction. The internal stresses can be determined from the changes in the local lattice parameters. The results obtained from both methods show that long-range internal stresses of the same type exist in the cell as well as in the subgrain structures.

Dislocation densities and internal stresses in large strain cold worked pure iron

Abstract Polycrystalline samples of Fe-0.005%-C were deformed by torsion at 300 K far into stage IV of deformation and investigated by X-ray peak profile analysis (XPA) for the microstructural evolution. The long-range internal stresses Δτ w −δτ c (τ w , τ c are the shear stresses in the cell wall and cell interior regions) pass through a maximum at the onset of stage IV, but reincrease within stage IV at higher deformation. A similar maximum is observed in the formal dislocation density derived directly from XPA which, however, is not seen in residual electrical resistivity. These results can be consistently explained by the assumption that in stages II and III the cell walls are formed as polarized dipole walls (PDW) which in stage IV change into polarized tilt walls (PTW), similarly to recent findings in cold rolled Cu. The significant constancy of total volume fraction of cell walls as well as of specific internal stresses in stage IV confirm the importance of the PTW s for stage IV strengthening.