On misorientation distribution evolution during anisotropic grain growth

In order to study the development of texture and boundary character during annealing, three-dimensional grain crystallography and crystallographically mediated grain boundary properties were incoporated into a finite temperature Monte Carlo model for grain growth. Randomly textured microstructures evolve normally, with growth exponent n=0.96. While texture remains random, the steady-state boundary misorientation distribution favors low-angle boundaries. To first order, low-angle boundaries increase by lengthening, not by proliferating. In contrast, microstructures with a strong single-component texture develop four-grain junctions and highly curved grain boundaries, which alter evolution. The boundary misorientation distribution narrows and shifts to low angles, and no steady state is observed. The accompanying decrease in mean boundary mobility causes growth to slow, resulting in a growth exponent n=0.62. The dependence of the growth exponent on average boundary mobility may explain experimental observations of exponents less than unity.     

Effect of grain refinement by severe plastic deformation on the next-neighbor misorientation distribution

Next-neighbor misorientation distributions (NNMD) in severely deformed polycrystalline materials are commonly measured by orientation imaging. A procedure is proposed which enables the separation of NNMD of ultrafine-grained materials into two parts: the distribution of misorientations between newly emerged grains within the original (“parent”) grain interior (“internal daughter grains”) and the distribution of misorientations between grains adjacent to an original grain boundary on its opposite sides (“grain boundary daughter grains”). The procedure is based on electron backscatter diffraction orientation map analyses carried out on different planes of deformed samples considering the evolution of the grain size and shape during severe plastic deformation. It was applied to copper processed by up to three passes of equal-channel angular pressing. A characteristic feature of the measured NNMD is the occurrence of a double peak, which is clearly due to the differences between the NNMD of the two distinct populations of new grains defined above. The peak at low angles represents mainly the continual grain subdivision process in the interior of a parent grain (and is associated with internal daughter grains), while the peak at large angles is due to the high angle misorientations of the grain boundary daughter grains.     

Spatial correlation of high-energy grain boundaries in two-dimensional simulated polycrystals

A polycrystal undergoes microstructural changes to reach a lower energy state. In particular, the system evolves so as to reduce the total grain boundary energy. A simple two-dimensional model of a polycrystal comprised of randomly oriented crystalline grains suggests that energy minimization reduces or eliminates any spatial correlation among high-energy grain boundaries. Thus grain boundary engineering not only reduces the density of high-energy boundaries, but also prevents their organization into a coarse, albeit discontinuous, network.     

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.     

Effect of deformation mode and grain orientation on misorientation development in a body-centered cubic steel

Strain-induced misorientation development was studied in an IF steel as a function of strain for two deformation modes, plane strain compression and simple shear. Using electron back-scattered diffraction, orientation maps of “large” areas were obtained, from which several individual grains associated with the principal texture components could be extracted so that only intragranular misorientations could be estimated for these orientations. It was observed that the increase of the misorientation angle was more prominent in simple shear than in plane strain compression and that the orientation influence was different for each mode. Considering texture evolution as a possible source of misorientation development, the lattice spin tensor was estimated with the Taylor model for the two deformation modes; both reorientation axis and angle were compared with misorientation angle and axis. The striking concordance of both quantities allows us to conclude that there is a direct contribution of texture evolution to misorientation accumulation with strain.     

Molecular dynamics simulations of grain boundary sliding: The effect of stress and boundary misorientation

Molecular dynamics simulations were used to study the effect of applied force and grain boundary misorientation on grain boundary sliding in aluminum at 750 K. Two grains were oriented with their 〈1 1 0〉 axes parallel to their boundary plane and one grain was rotated around its 〈1 1 0〉 axis to various misorientation angles. For any given misorientation, increasing the applied force leads to three sliding behaviors: no sliding, constant velocity sliding and a parabolic sliding over time. The last behavior is associated with disordering of atoms along the grain boundary. For the second sliding behavior, the constant sliding velocity varied linearly with the applied stress. A linear fit of this relationship did not intersect the stress axis at the origin, implying that a threshold stress for sliding exists. This threshold stress was found to decrease with increasing grain boundary energy. The ramifications of this finding for modeling grain boundary sliding in polycrystals are discussed.     

On the influence of the grain boundary misorientation on the plastic deformation of aluminum bicrystals

Aluminum bicrystals with symmetric 〈1 1 2〉 tilt boundaries and misorientations of 8.7° (small angle), 15.4° (transition), and 31.5° (large angle) were deformed in a channel die experiment in order to study the influence of misorientation on the deformation at grain boundaries. Samples were characterized by strain measurements and microtexture mappings. The experiments were compared to crystal plasticity finite element simulations. We studied strain heterogeneity at the macroscopic and at the microscopic level. Even macroscopically homogeneous areas showed microscopic heterogeneity in the form of bands of different sets of glide systems. We observed clear effects of the grain boundary misorientation on the deformation kinematics close to the boundaries. The 8.7° grain boundary did not show any orientation change which was interpreted in terms of free dislocation penetration. In contrast, the 15.4° and 31.5° bicrystals showed orientation changes which were attributed to dislocation pile-ups.     

Light Pre-deformation induced misorientation of grain boundary bcc-precipitates from the Kurdjumov-Sacks relationship in a Ni−Cr alloy

Crystallography of grain boundary (GB) bcc-precipitates formed in a lightly pre-deformed fcc-matrix phase was examined in a Ni-43Cr alloy by EBSD with emphasis on a misorientation from the Kurdjumov-Sachs orientation relationship. The misorientation of GB bcc-precipitates became somewhat larger when the matrix phase was lightly pre-deformed. The GB bcc-precipitates were preferentially formed at the intersection of geometrically necessary dislocation boundaries (GNB) with grain boundaries. In contrast, the pre-deformation did not influence the misorientation of intragranular bcc-precipitates. The preferential precipitation and the increased misorientation at the GNB-GB intersection were discussed.     

Evolution of misorientation spectrum of grain boundaries of submicrocrystalline molybdenum upon deformation under conditions of grain-boundary diffusion of nickel

Comparative investigations of the concurrent evolution of the structure and the misorientation spectrum of grain boundaries of submicrocrystalline molybdenum have been performed upon free annealing and under the creep conditions at a temperature of 1023 K. It has been established that changes in the misorientation spectrum of grain boundaries of submicrocrystalline molybdenum upon annealing and creep are observed concurrently with boundary migration. The grain-boundary diffusion of nickel upon annealing leads to an increase in the fraction of grain boundaries of the special type with Σ17a in the grain-boundary ensemble of submicrocrystalline molybdenum. Under the creep conditions grain-boundary diffusion fluxes of nickel atoms favor transformation of low-angle boundaries into high-angle boundaries and growth in the angle of misorientation of the high-angle boundaries of submicrocrystalline molybdenum to 45°–60°.     

The distribution of grain boundary planes in polycrystals

During the last ten years, techniques have been developed to measure the distribution of grain boundaries in polycrystals as a function of both lattice misorientation and grain boundary plane orientation. This paper presents a brief overview of the techniques used for these measurements and the principle findings of studies implementing these techniques. The most significant findings are that grain boundary plane distributions are anisotropic, that they are scale invariant during normal grain growth, that the most common grain boundary planes are those with low surface energies, that the grain boundary populations are inversely correlated with the grain boundary energy, and that the coincident site lattice number is a poor predictor of the grain boundary energy and population.     

Five-parameter grain boundary distribution of commercially grain boundary engineered nickel and copper

The five-parameter grain boundary distributions of grain boundary engineered nickel and copper specimens have been analyzed in detail. The relative areas of {1 1 1} planes in the entire population did not increase as a result of grain boundary engineering (GBE) and, in the Σ3-excluded population, decreased after GBE. This decrease occurred because the majority of the newly generated Σ3 grain boundaries were not coherent twins with {1 1 1} grain boundary plane orientations. GBE increased the proportion of Σ3 boundary length that was vicinal-to-{1 1 1} and the proportion of asymmetrical 〈1 1 0〉 tilt boundaries. There was a clear propensity for selection of particular planes or plane combinations which were associated with low energy. These plane types were analyzed in some detail, and it was shown that many of these boundaries were asymmetrical tilts comprising (or vicinal to) at least one low-index plane.     

On boundary misorientation distribution functions and how to incorporate them into three-dimensional models of microstructural evolution

The fundamental difficulties of incorporating experimentally obtained boundary misorientation distributions (BMDs) into three-dimensional microstructural models are discussed. An algorithm is described which overcomes these difficulties. The boundary misorientations are treated as a statistical ensemble which is evolved toward the desired BMD using a Monte Carlo method. The application of this algorithm to a number of complex arbitrary BMDs shows that the approach is effective for both conserved and non-conserved textures. The algorithm is successfully used to create the BMDs observed in deformation microstructures containing both incidental dislocation boundaries (IDBs) and geometrically necessary boundaries (GNBs). The application of an algorithm to grain boundary engineering is discussed.     

Self-similar grain size distribution in two dimensions: Analytical solution

Consideration of the physics and topology of two-dimensional grain growth suggests that a stochastic treatment is required to determine grain size distribution [Pande CS. Acta Metall 1987;35:2671]. In this paper, a size-based continuum stochastic formulation is presented based on topological considerations. As expected, this analysis leads to a Fokker–Planck equation for the size distribution, which should yield a unique self-similar asymptotic state that could be reached from any arbitrary initial state. The approximate solution of the Fokker–Planck equation presented here is limited to two dimensions and is based on the assumption of quasi-stationary distributions reached in the long time limit. The resulting grain size distribution is shown to be in agreement with that obtained from computer simulations, indicating the validity of the stochastic approach.     

The planar distribution of grain size in a polycrystalline ceramic

The planar distribution of grain size in polycrystalline magnesium oxide is accurately represented, over a wide range of average grain diameter, by a simple function of the square root of the grain diameter. An analysis of the experimental data shows that a similar function of the logarithm of the grain diameter does not fit them so well.     

Investigation of the correlation between growth restriction and grain size in Cu alloys

The growth restriction factor Q of the alloying elements of the Cu-system was determined using thermodynamic software tools to obtain accurate Q-values. A comprehensive list was given for a nominal solute content of 1 wt-%. Based on the calculations, melting experiments were carried out under defined casting conditions given by the TP-1 test to evaluate the correlation between Q and grain size in Cu alloys.     

Self-similar grain size distribution in three dimensions: A stochastic treatment

In this paper, a stochastic formulation of three-dimensional grain growth is presented. This formulation employs the recent extension of the von Neumann law to three dimensions, and leads to a Fokker–Planck equation for the size distribution. The self-similar solutions of the Fokker–Planck equation presented here are based on the assumption of quasi-stationary distributions reached in the long time limit. The resulting grain size distributions, obtained both numerically and analytically, are shown to be in good agreement with each other and also with those obtained from computer simulations, indicating the validity of the stochastic approach.     

Invariant distributions and stationary correlation functions of simulated grain growth processes

This paper first reviews and highlights a relatively large number of distributions and correlation functions that are characteristic for normal grain growth. Attention is then paid to the correlation between the average size of an n-sided grain and the average size of a grain next to an n-sided grain. Size correlations between neighbouring grains are examined using a nearest-neighbour Q=40 Potts model on a triangular 500×500 lattice. We show that few-sided grains tend to be surrounded by large grains, while the neighbours of six-sided grains are typically average-sized. The neighbours of many-sided grains tend to be somewhat smaller than an average-sized grain. This confirms the existence of spatial grain size correlations, which have previously been reproduced by the use of a fundamentally different simulation technique. Finally, the robustness of the stationary correlation functions is examined by incorporating artificial correlations into the starting pattern.     

Fracture roughness scaling and its correlation with grain boundary network structure

Roughness scaling laws for intergranular cracks deviate from self-affine (fractal-like) behavior at length scales related to the polycrystalline microstructure. We consider two versions of the same alloy material with many of the same microstructural length scales but differing in their processing history: one conventional and one grain boundary engineered. The engineered material, processed to contain a high fraction of “special” grain boundaries, fails more slowly and more isotropically. We present evidence that the difference is determined by processes related to clusters of twin-related grains, shown through analysis of scales of the fracture roughness measured with confocal microscopy and the special grain boundary network determined by electron backscatter diffraction. Above the cluster scale, the fracture roughness exponents in the two materials are nearly indistinguishable (confirming theoretical predictions); below this scale conventional cracks exhibit correlations indicating consistently weak paths for crack propagation, suggesting percolation of “random” boundaries.