Grain Size Distribution Function Research Articles

Finite elements in numerical micromagnetics: Part I: granular hard magnets

Micromagnetic finite element calculations give a quantitative treatment of the correlation between the microstructure and the magnetic properties. Variational principles and dynamic equations to calculate the hysteresis properties are discussed with respect to the finite element modeling of hard magnetic, granular materials. Simulations of domain formation in Co/Pt multilayer recording media strongly suggest that the boundaries between the columnar grains act as pinning sites and cause an irregular maze-like domain structure. Energy minimization provides upper bounds for the remanence and the coercive field of nanocomposite Nd 2Fe 14B/α-Fe/(Fe 3B, Fe 23B 6) magnets as a function of grain size and phase distribution. The dynamic equations reveal how reversed domains nucleate and expand, and thus provide a fundamental understanding of magnetization reversal in nanocomposite magnets. Adaptive refinement and coarsening of the mesh controls the discretization error and provides optimal girds for micromagnetic finite element calculations of domain wall motion in hard magnetic platelets.

Magnetization processes in small particles and nanocrystalline materials

Micromagnetic calculations using computational techniques are a pre-requisite to treat complex magnetization processes in ensembles of ferromagnetic grains. Applying the finite element method, the critical fields for the reversal of magnetization are determined as a function of grain size, isotropic and textured distribution of easy axes and different magnetic properties of grain boundaries. Adapting the mesh sizes of finite elements to the gradients of the direction cosines of the spontaneous magnetization, inhomogeneous spin distributions are determined within the grains as well as within grain boundaries. Coercive fields will be determined for nanocrystalline ensembles of grains taking into account exchange and dipolar coupling between grains and inhomogeneous magnetic material parameters within the grain boundaries. The results of these investigations will give information on the role of exchange coupling between the grains, dipolar long-range magnetic stray fields and the remanence enhancing by exchange coupling between the grains. From the model calculations general rules for the development of optimized nanocrystalline materials with large remanences and large coercivities will be derived.

Micromagnetism and the microstructure in nanocrystalline materials

Micromagnetic calculations using computational techniques are a pre-requisite to treat complex magnetization processes in ensembles of ferromagnetic grains. Applying the finite element method spin structures of small spherical prismatic and platelet-like particles and the critical fields for the reversal of magnetization have been determined as a function of grain size, isotropic and textured distribution of easy axes and different magnetic properties of grain boundaries. Adapting the mesh sizes of finite elements to the gradients of the direction cosines of the spontaneous magnetization, inhomogeneous spin distributions could be determined within the grains as well as within grain boundaries. Numerical results have been obtained for nanocrystalline single phase and composite materials taking into account exchange and dipolar coupling between grains and inhomogeneous magnetic material parameters within the grain boundaries. The results of these investigations show clearly that large coercivities require exchange decoupling between the grains. Dipolar long-range magnetic stray fields reduce the coercivity mainly for large grain sizes, whereas the remanence enhancing by exchange coupling becomes effective for grain sizes of the order of twice the domain wall widhh of the hard-magnetic phase. For larger grain sizes the coercivity breaks down. From the model calculations general rules for the development of optimized nanocrystalline materials with large remanences and large coercivities are derived.

A user-friendly PC program for evaluation of Mössbauer spectra

An integrated PC program for evaluation of Mössbauer spectra both from polycrystalline and from amorphous materials is proposed. The software structure and specifications, operating in a Microsoft Windows environment, are discussed. The program is applied to the Mössbauer analysis of a Fe - Cu alloy, using a version of the fitting procedure for disordered systems developed to include evaluation of the grain-size distribution function.

Hydraulic conductivity of homogenized tailings from hard rock mines

Tailings produced by milling for one extraction from hard rock mines, which range in size from clay to fine sand, have relatively low hydraulic conductivity, k. The value of k must be known with a certain degree of certainty to analyze consolidation and seepage conditions in and around tailings ponds. In this paper, the authors present the results of a laboratory investigation on hydraulic conductivity of homogenized tailings from hard rock mines. After describing some of the basic properties of four different materials, including mineralogy, grain size, Atterberg limits, compaction characteristics, and consolidation curves, permeability test results are given. The hydraulic conductivity value usually varies between 10–4 and 10–5 cm/s. The effect of various factors on this value, including void ratio and grain size, is then discussed in relation to predictive models. It is shown that a modified version of the Kozeny-Carman equation, in which a tortuosity factor and a grain-size distribution function are included explicitly, can represent the data very well. The equation is also checked against results taken from the literature on similar types of materials. Key words: tailings, hydraulic conductivity, laboratory tests, void ratio, grain size, tortuosity.

Void Sizes in Granular Soils

In an earlier paper, the writer presented a simple stochastic model of the grain and pore structure in cohesionless soils and similar materials. Equations that very adequately predict the void ratio as a function of grain-size distribution were derived. In this technical note, the stochastic model is used to find the distribution of void volume as a function of void size (void-chord length). An evaluation of laboratory tests shows that both the bubbling pressure of a soil and the size of the largest grain that can pass through the voids of a soil in a base-filter compatibility test can be correlated with the calculated mean void chord of the soil.

Grain-Size Distribution for Smallest Possible Void Ratio

In an earlier paper, the writer derived theoretical equations for void ratio as a function of grain-size distribution. Here, the equations are used to find the grain-size distributions that give the smallest possible void ratio for highly densified, cohesionless soils and similar materials that have different ratios, x100 /x0, for their largest and smallest grain sizes. Both numerical and analytical methods are used. Depending on the value of x100 /x0, materials that consist of two or three grain-size fractions with grading gaps between them have the smallest void ratios. Within the fine and coarse fractions, the grain sizes need to be as uniform as possible. The required grading of the medium fraction depends on the characteristics of the fine and coarse fractions.

Influence of microstructure on the resistivity of Al-Cu-Si thin-film interconnects

The rate of resistivity decay in Al-2Cu-1Si thin-film conductors was studied as a function of temperature and grain size distribution. The decay kinetics were assumed to be governed by the rate of precipitate reconfiguration to grain boundaries. This assumption was confirmed by transmission electron microscopy (TEM) observations of the microstructure during resistance decay, and by studies of lines of two different widths. The results can be explained qualitatively from the microstructure of the lines. In particular, increasing the mean grain size slows the rate of resistivity decay, and establishing a bimodal distribution with a significant population of relatively large grains has the same effect. A simple model was developed to treat these effects quantitatively. The model assumes a cylindrical grain geometry and a uniform initial distribution of Cu and ignores the effect of intragranular precipitation. The model yields reasonable values for the activation energy for Cu diffusion in thin films, and predicts the correct dependence of the decay rate on grain size and grain distribution; however, it appears to overestimate the value of the preexponential factor D0.

The Radioactivity of Sediments in Danube Reservoirs in Austria before and after the Chernobyl Accident

A major part of pollutants transported by rivers are normally adsorbed to the fine-grained particles in suspension. The coupling between pollutants and particles leads to an enrichment of radionuclides in sediments that settle in river reservoirs and lakes. The concentration of radionuclides is a function of grain-size distribution and of the mineralogical composition. In a survey, sediment samples from the lock area of each Danube reservoir were collected during the spring of 1985. Three grain-size fractions (<20 µm, 20–63 µm, 63–250 µm) were analyzed. On an average the estimated concentration of radionuclides is about 600 Bq/kg for K-40, 50 Bq/kg for Ra-226, and 40 Bq/kg for Th-232. A 17 m depth borehole has allowed us to trace the history of radionu-clide distribution for the past 25 years since the culmination and the end of atomic testing in the atmosphere, which coincides with the damming of the river Danube at Aschach. The content of Cs-137 in freshly deposited Danube sediments rose by two orders of magnitude following the accident at Chernobyl (i.e. up to 3000 Bq/kg).

Determination of grain-size distribution function using two-dimensional Fourier transforms of tone-pulse-encoded images: Generazio, E.R. Materials evaluation, vol. 46, no. 4, pp. 528–534 (Mar. 1988)

Microstructural images may be tone pulse encoded and subsequently Fourier transformed to determine the two-dimensional density of frequency components. A theory is developed relating the density of frequency components to the density of length components. The density of length components corresponds directly to the actual grain size distribution function from which the mean grain shape, size, and orientation can be obtained.

Scaling relations for grain autocorrelation functions during nucleation and growth.

The kinetics of grain growth associated with a first-order phase transformation is studied with use of the Kolmogorov model (constant nucleation rate within the metastable phase; constant growth rate of the nucleated grains) in one and two dimensions (d=1 and 2). The time-dependent grain-size distribution function and grain autocorrelation function, ${G}_{s}$(r,t), are calculated exactly for d=1. For d=2, ${G}_{s}$(r,t) is estimated by Monte Carlo simulation. Scaling relations, ${G}_{s}$(r,t)=F(r/R(t)), using forms for F(x) which are exact for t\ensuremath{\rightarrow}0 are investigated. It is found that while such scaling relations are no longer exact in late stages of growth, they continue to provide accurate simple approximations.

Effect of the initial texture on the secondary recrystallization in grain oriented silicon iron

Texture development has been determined by ODF analysis at two stages of the grain growth. The intensity of Goss texture increases during the process of grain growth before the secondary recrystallization. The secondary grain size is a function of both Goss texture intensity and Goss grain size distribution in the primary recrystallized matrix.

Electron-microscopic study of grain-size distribution function in splat-cooled aluminium

The one-dimensional grain-size distribution function in splat-cooled aluminium samples and its dependence on sample storage conditions (temperature and time) were determined by electron microscopy. In contrast to previously published results [1], the theoretical logarithmic normal distribution function gave a very good representation of the experimental data obtained in this work. In addition, some aspects of quenching efficiency and reproducibility of the results for the two-piston splat-cooling device are discussed.

Electron-microscopic study of grain-size distribution function in splat-cooled aluminium

Electron-microscopic study of grain-size distribution function in splat-cooled aluminium

Beziehungen zwischen der Korngrößenverteilung und der Beanspruchungsenergie bei der Einzelkornprallzerkleinerung

Investigations are reported concerning single impact crushing of irregularly shaped rock-salt particles. It is shown that with constant initial grain size not only the probability of fracture as function of the mass-related load energy follows a logarithmic normal function, but that the same relation also applies to the size-distribution function of the product being crushed for any undersize at any mesh aperture. The distributions resulting with different impact energies are thus functionally interconnected. By comparison with the results of other authors it can be shown that other substances also reveal the same behaviour with the most different initial grain sizes. Only glass spheres deviate from it. Further relations between initial size, load energy, and grain-size distribution function are indicated for rock-salt particles.