Misorientation Distribution Function Research Articles

Effect of a novel controlled thermomechanical treatment on the microstructure and mechanical properties of a high-carbon nanobainitic steel

The effect of the novel controlled thermomechanical treatment, including torsion components in the elastic strain range during the isothermal holding on the microstructure and mechanical properties of the high-carbon nanobainitic steel, was investigated. TEM observations of the thermo-mechanically treated steel revealed bainitic ferrite laths with an average size of 68 ± 40 nm and films of retained austenite with an average size of 34 ± 17 nm, along with the blocky morphology of retained austenite in sub-micron scale. The XRD synchrotron diffraction allows estimating the amount of retained austenite at 43.1 ± 1.2% volume fraction with a carbon concentration of 1.17 ± 0.09 wt.%. Furthermore, the deconvolution of (200) Fe-γ reflections corresponding to two different low-carbon and high-carbon retained austenite peaks and, simultaneously, the blocky and film-like retained austenite was performed. In addition, the Nishiyama–Wassermann (N–W) crystallographic orientation relationship between bainitic ferrite and retained austenite was described as dominant using the misorientation distribution function (MDF). The crystallographic texture results indicated that the main growth of bainitic ferrite plates occurred after removing external stress during isothermal holding. The tensile tests and hardness measurements showed a high tensile strength achieved mainly by nano-metric bainitic ferrite plates and a high dislocation density. The high level of elongation is most likely attained due to a high amount of retained austenite in steel and both TRIP and TWIP effects during tensile deformation.

Influence of grain boundary energy anisotropy on the evolution of grain boundary network structure during 3D anisotropic grain growth

In this paper, we present the results of grain growth simulations in three-dimensions using an existing level set method. Most of the previous grain growth studies have been either isotropic or, if they are not isotropic, consider simplified models of anisotropy. We perform these simulations using three different grain boundary (GB) energy functions (a realistic 5D GB energy function, a Read–Shockley energy model that depends only on disorientation angle, and an isotropic GB energy model). We compare the results in terms of several statistical microstructural descriptors (crystallographic texture, GB character distribution, triple junction distribution (TJD), etc.). In addition to considering different energy models, we also compare the evolution of microstructures that have different initial crystallographic textures (fiber texture and random texture). We find that the morphological evolution of individual grains can be completely different depending on the GB energy function employed. However, we also find that certain spatially independent microstructural statistics (e.g. orientation distribution function, misorientation distribution function, and TJD) are similar at steady state for all of the tested GB energy functions.

Interface stress in nanocrystalline materials

Abstract Based on a generalization of a capillary equation for solids, we develop a method for measuring the absolute value of grain-boundary stress in polycrystalline samples having a large interface-to-volume ratio. The grain-boundary stress in nanocrystalline Pd is calculated from X-ray diffraction measurements of the average grain size and the residual-strain-free lattice spacings, yielding a value of 1.2 ± 0.1 N /m. The random distribution of crystallite orientations in the sample in conjunction with calorimetric data for the area-averaged interfacial energy and knowledge of the grain-boundary misorientation distribution function suggest that this value is characteristic of random high-angle grain boundaries in Pd.

Interphase misorientation as a tool to study metamorphic reactions and crystallization in geological materials

AbstractInterphase boundaries are planar defects that separate two different minerals, which in general have different compositions and/or crystalline structures; they may play an important role as a pathway for fluids in rocks and affect their physical properties. To completely characterize interphase boundaries, one needs to define the misorientation between adjacent grains and the orientation of the grain boundary plane. The analysis performed here is limited to the misorientation characterization and the trace of the interphase boundary. Although the determination of possible orientation relationships between the two adjacent phases is routinely performed by selected-area electron diffraction in a transmission electron microscope, this method lacks statistical representativeness. With the advent of techniques like electron backscatter diffraction (EBSD), it is possible to calculate orientation relationships not only in single pairs of crystals of the same phase but in full thin sections and between different minerals. The interphase misorientation is calculated from two orientations of two adjacent crystals of different phases. A set of single misorientations is then used to calculate the misorientation distribution function (MDF), from where it is possible to identify a maximum and its crystallographic interpretation. If we then know the misorientation and the unit-cell parameters of the individual phases, the crystal-lographic relationships between the different phases can be described with the pairs of parallel crystal-lographic planes and the pairs of crystallographic directions. In this paper, I present examples of the use of interphase misorientation analysis on the transformation of calcite-aragonite, olivine-antigorite, magnetite-hematite, and on the study of orientation relationships between plagioclase-olivine-ilmenite in mid-ocean ridges gabbros (ODP Hole 735).

Computer simulation of microstructure evolution in strip-cast silicon steel with cube and fibre texture

ABSTRACTThin strip casting process which is already used for non-oriented grades of electrical steel, is relatively novel processing route for grain oriented electrical steel and to achieve the cube texture through suitable annealing. In the present investigation, a two-dimensional Monte Carlo model has been formulated to study texture evolution in strip-cast and annealed Fe–3.2%Si steel with a cube texture. Both random and non-random microstructures of Fe–3.2% silicon steel and corresponding Euler angles have been used as input parameters. Results obtained from the simulation characterised the evolving texture in terms of grain size distribution and misorientation distribution functions. It has been found that randomly textured microstructures follow power law growth kinetics with evolving grain size distribution similar to normal grain growth. Microstructures with random orientation texture evolve steady state misorientation distribution functions, while those with fibre texture narrows and shifts to low angles. The simulation results based on EBSD microstructure inputs have been compared and validated with microstructures obtained from computationally generated input microstructures as well as published literature for both random and non-random orientation texture and found to be in good agreement.

MISORIENTATION DISTRIBUTION FUNCTION FOR CUBIC CRYSTALS

A calculation method for obtaining the misorientation distribution function (MDF) for cubic crystals which can be used to estimate the presence or absence of special boundaries in the materials is presented. The calculation was carried out for two samples of Al-Mg-Si alloy subjected to various mechanical and thermal treatments: the first sample is subjected to rolling; the second sample is subjected to recrystallization annealing. MDF is calculated for each sample; the results are presented in the Euler space and in the angle-axis space. The novelty of the method consists in the possibility of gaining data on the grain boundaries from X-ray texture analysis without using electron microscopy. A calculation involving only mathematical operations on matrices was performed on the basis of the orientation distribution function restored from incomplete pole figures. It is shown that no special boundaries are observed in the deformed sample, whereas in the recrystallized alloy, special boundaries are detected at Ʃ = 23, 13, and 17. The shortcoming of the proposed method can be attributed to the lack of accurate data on grain boundaries, since all possible orientation in the polycrystal should be taken into account in MDF calculation.

Effect of grain boundary energy anisotropy in 2D and 3D grain growth process

Abstract Phase-field modeling of isotropic and anisotropic grain growth has been performed in 2D and 3D system. The multi order parameter grain growth model is adopted and OpenMP is implemented to enhance the computational efficiency. When the anisotropic grain growth phenomenon occurs, the misorientation distribution function of the low angle grain boundary is increased at the early stage and it converge to the certain value, 0.18 in 2D and 0.24 in 3D. The average numbers of faces of anisotropic grain growth is nearly same as the values of isotropic grain growth, which is 6.0 in 2D and 13.8 in 3D.

Misorientation Distribution Function of Crystals

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Analysis of grain misorientation distribution in polygonal ferrite of low-carbon steel

A method to deduce the origin and crystallographic features of the polycrystalline state basing on the analysis of misorientation distribution function is considered. According to this approach, the misorientation distribution is represented in the form of linear superposition of partial distributions, each distribution corresponding to the boundaries of a specific physical nature. Then the fractions of the boundaries of different types in the overall structure are determined from the best fit of the computed distribution to the experimental data. In the present work the method is applied to a low-carbon ferrite-pearlite steel. The emphasis is made on the boundaries, whose misorientations are predetermined by orientation relationship between parent austenite and ferrite, i.e. on the so-called crystallography-specified or inter-variant boundaries. It was shown that they constitute about one third of the total amount of boundaries in the polygonal ferrite microstructure.

A kinetic approach to modeling general-texture evolution in two-dimensional polycrystalline grain growth

Important statistical descriptors of grain boundary networks are misorientation distribution functions (often referred to as grain boundary character distributions), which show relative measures of interfaces with given misorientation parameters. This paper is aimed at extending the Boltzmann-type kinetic modeling framework developed in the previous work (Yegorov et al., 2016) to the two-dimensional case of general textures with no restrictions on possible grain orientations, but under the assumption that the grain boundary energy density depends only on the disorientation variable. In order to avoid an enormous computational complexity of treating the general-texture case, the kinetic model for polycrystalline grain growth is constructed from the very beginning in a specific discretized form with respect to the disorientation variable. A significant aspect of the developed approach is that its formulation requires a collection of a priori distributions which describe possible disorientations of grain boundaries connected in a triple junction. A separate numerical algorithm for approximating such distributions is proposed as applied to cubic crystal lattices. The numerical results obtained by the constructed model are in good agreement with the corresponding large-scale simulation results from the related literature.

Descriptive tools for the analysis of texture projects with large datasets using MTEX : strength, symmetry and components

Abstract This paper presents the background for the calculation of various numbers that can be used to characterize crystal-preferred orientation (CPO), also known as texture in materials science, for large datasets using the combined scripting possibilities of MTEX and MatLab®. The paper is focused on three aspects in particular: the strength of CPO represented by orientation and misorientation distribution functions (ODFs, MDFs) or pole figures (PFs); symmetry of PFs and components of ODFs; and elastic tensors. The traditional measurements of texture strength of ODFs, MDFs and PFs are integral measurements of the distribution squared. The M-index is a partial measure of the MDF as the difference between uniform and measured misorientation angles. In addition there other parameters based on eigen analysis, but there are restrictions on their use. Eigen analysis does provide some shape factors for the distributions. The maxima of an ODF provides information on the modes. MTEX provides an estimate of the lower bound uniform fraction of an ODF. Finally, we illustrate the decomposition of arbitrary elastic tensor into symmetry components as an example of components in anisotropic physical properties. Ten examples scripts and their output are provided in the appendix.

Grain Boundary Character Distribution of Nanocrystalline Cu Thin Films Using Stereological Analysis of Transmission Electron Microscope Orientation Maps

Stereological analysis has been coupled with transmission electron microscope (TEM) orientation mapping to investigate the grain boundary character distribution in nanocrystalline copper thin films. The use of the nanosized (<5 nm) beam in the TEM for collecting spot diffraction patterns renders an order of magnitude improvement in spatial resolution compared to the analysis of electron backscatter diffraction patterns in the scanning electron microscope. Electron beam precession is used to reduce dynamical effects and increase the reliability of orientation solutions. The misorientation distribution function shows a strong misorientation texture with a peak at 60°/[111], corresponding to the Σ3 misorientation. The grain boundary plane distribution shows {111} as the most frequently occurring plane, indicating a significant population of coherent twin boundaries. This study demonstrates the use of nanoscale orientation mapping in the TEM to quantify the five-parameter grain boundary distribution in nanocrystalline materials.

Simulations of anisotropic grain growth: Efficient algorithms and misorientation distributions

An accurate and efficient algorithm, closely related to the level set method, is presented for the simulation of Mullins’s model of grain growth with arbitrarily prescribed surface energies. The implicit representation of interfaces allows for seamless transitions through topological changes. Well-resolved large-scale simulations are presented, beginning with over 650,000 grains in two dimensions and 64,000 grains in three dimensions. The evolution of the misorientation distribution function (MDF) is computed, starting from random and fiber crystallographic textures with Read–Shockley surface energies. Prior work had established that with random texture the MDF shows little change as the grain network coarsened whereas with fiber texture the MDF concentrates near zero misorientation. The lack of concentration about zero of the MDF in the random texture case has not been satisfactorily explained previously since this concentration would decrease the energy of the system. In this study, very-large-scale simulations confirm these previous studies. However, computations with a larger cut-off for the Read–Shockley energies and an affine surface energy show a greater tendency for the MDF to concentrate near small misorientations. This suggests that the reason the previous studies had observed little change in the MDF is kinetic in nature. In addition, patterns of similarly oriented grains are observed to form as the MDF concentrates.

A Variant Selection Rule in Transformation in Steel and Prediction of Transformation Texture

Quantitative prediction of transformation textures in steel becomes possible if a variant selection rule is taken into account, in which ferrite nucleating on austenite grain boundaries prefers to have orientation relationship with two neighboring austenite grains at the same time. The mathematical model of the variant selection rule is described and some examples of simulation in transformation textures in hot-rolled steel sheets are presented using the textures of retrained austenite and the misorientation distribution function method. An excellent agreement is attained between the predicted and experimental ferrite textures.

Grain growth in thin films with a fibre texture studied by phase-field simulations and mean field modelling

The evolution of fibre textured structures is simulated in two dimensions using a generalised phase field model assuming two forms for the misorientation dependence of the grain boundary energy. In each case, a steady-state regime is reached after a finite amount of grain growth, where the number and length weighted misorientation distribution functions (MDF) are constant in time, and the mean grain area A as a function of time t follows a power growth law A − A 0 = kt n with n close to 1 and A 0 the initial mean grain area. The final shape of the MDF and value of the prefactor k in the power growth law clearly correlate with the misorientation dependence of the grain boundary energy. Furthermore, a mean field approach is worked out to predict the growth exponent for systems with non-uniform grain boundary energy. The conclusions from the mean field approach are consistent with the simulation results. In previous studies on grain growth in anisotropic fibre textured systems, this steady-state regime was often not reached, which resulted in wrong conclusions on the growth exponent n and evolution of the MDF.

The effect of deliberate aluminium additions on the microstructure of rolled steel plate characterized using EBSD

The use of aluminium as a deliberate alloying addition in steels has attracted increased attention recently as a possible replacement for Si in transformation-induced plasticity (TRIP) steels. In addition, some authors have suggested that it offers beneficial effects as a solid solution strengthener as well as galvanizability. In this work three low carbon (0.02 wt.%) manganese (1.4 wt.%) steels have been alloyed with very different aluminium contents (0.02, 0.48 and 0.94 wt.%) in order to study the effect of this alloying element on the final ferritic microstructure. Two different rolling schedules have been applied to these steels and the final microstructures have been characterized extensively by EBSD measurements. The results indicate that aluminium additions have a profound influence on ferrite grain size and the grain boundary misorientation distribution functions.

Misorientation texture development during grain growth. Part II: Theory

A critical event model for the evolution of number- and area-weighted misorientation distribution functions (MDFs) during grain growth is proposed. Predictions from the model are compared to number- and area-weighted MDFs measured in Monte Carlo simulations with anisotropic interfacial properties and several initial orientation distributions, as well as a dense polycrystalline magnesia sample. The steady-state equation of our model appears to be a good fit to all data. The relation between the grain boundary energy and the normalized average boundary area is discussed in the context of triple junction dynamics.

Misorientation texture development during grain growth. Part I: Simulation and experiment

Grain growth in two and three dimensions with anisotropic interfacial properties was simulated using the Monte Carlo method. The relative effects of grain boundary energy and mobility anisotropy on number- and area-weighted misorientation distribution functions (MDFs) were compared. Results indicate that energy anisotropy has a measurable effect on misorientation texture development, while mobility anisotropy does not. Qualitatively similar results are obtained in all simulations regardless of dimensionality or crystal symmetry. Microstructures with random orientation texture appear to evolve steady-state MDFs, while those with a preferred orientation do not. Experimentally measured number- and area-weighted MDFs in polycrystalline magnesia are shown to be comparable to those measured in our simulations.

Variable Elastic-Plastic Properties of the Grain Boundaries and Their Effect on the Macroscopic Flow Stress of Nano-Crystalline Metals

AbstractThe paper reports new experimental results describing properties and microstructure of nanocrystalline metals. Nano- and sub-micron aluminium has been produced by hydrostatic extrusion at ambient tempearture. The structures have been quantified in terms of size of grains and misorientation of the grain boundaries. Different average size of grains, variable normalized width of grain size distribution and changing grain boundary misorientation distribution functions have been revealed depending on processing parameters. The results of the tensile tests showed that the average grain size, grain size distribution and the distribution function of misorientation angles influence the flow stress of obtained nano-metals. In order to explain the observed difference in the properties of nano- and micro-sized aluminium alloys, a Finite Element Method models have been developed, which assumes that both grain boundaries and grain interiors may accommodated elastic and non-linear plastic deformation. These models assumed true geometry of grains (which differed in size and shape). Also, variable mechanical properties of grain boundaries have been taken into account (elastic modulus, yield strength and work hardening rate). The results of modelling explain in a semi-quantitative way macroscopic deformation of nano-crystalline aggregates. In particular, they illustrate the importance of the interplay between properties of grain boundaries and grain interiors in elastic and plastic regime.

Grain Refinement of C–Mn Steel to 1 μm by Rapid Cooling and Short Interval Multi-pass Hot Rolling in Stable Austenite Region

A noble grain refinement method to attain ferrite grain size of 1 μm in plain C-Mn steel by multi-pass hot rolling in a stable austenite temperature region has been explored. 0.1%C-1%Mn and 0.16%C-0.7%Mn steels were finish-rolled at about Ae 3 and then rapidly cooled. Special emphases have been placed on the interval, At, between the finish of rolling and the start of water spray cooling of more than 1000°C·s -1 in cooling rate and the interval between the last two rolling passes. By reducing At from 0.5 to 0.05 s alone, the grain sizes near the sheet surface and in the central region were markedly reduced to 1.3 and 2.2 μm respectively. By reducing At and the rolling pass interval together, a further reduction in grain size was then achieved. Consequently, the steel sheet in which the ultrafine grain structure of about 1 μm in grain size penetrated from the sheet surface to the depth of a quarter of the sheet thickness and the grain size in the central region was well below 2 μm was obtained. The transformation to the ultrafine ferrite is considered to be static and from deformed y. The detailed grain refinement mechanisms are discussed based on the microstructure observation of quenched C-Mn steels and a fcc Ni-30%Fe alloy as well as the calculation of prior y texture by the misorientation distribution function method.