Michael P. Anderson

Computer simulation of grain growth-II. grain size distribution, topology, and local dynamics

Abstract The microstructures produced by the grain growth simulation technique described in the previous paper are analyzed. The grain size distribution function is found to be time invariant when the grain size, R, is scaled by the mean grain size, R , and is shown to fit the experimental data better than either the log-normal function or the grain size distribution function suggested by Hillert. The grain size distribution peaks at approximately R and has a maximum at ~2.1 R . The topological class distribution (number of edges per grain) is monitored and found to reproduce the existing experimental data. Similarly, the experimentally observed linear relationship between edge class number and the means of the individual edge class distributions is reproduced. The mean curvature per grain is also measured. Finally, the temporal evolution of the sizes of individual grains is monitored to provide a link between the observed grain size distribution function and the macroscopic grain growth kinetics.

Computer simulation of normal grain growth in three dimensions

Abstract Computer modelling has been carried out to study normal grain growth in three dimensions. The approach consists of digitizing the microstructure by dividing the polycrystalline material into small volume elements and storing the spatial location and crystallographic orientation of each element. An energy is assigned between each element and its neighbours, such that neighbours having unlike orientations provide weaker bonding than neighbours of like orientations. The annealing treatment during which grain growth occurs is simulated using a Monte Carlo technique in which elements are selected at random and thermally activated transitions to other orientations are attempted. With time, the system evolves so as to reduce the total grain interface area. The microstructures produced are in good correspondence to observations of pure metals and ceramics which have undergone grain growth. Power-law kinetics ([Rbar] = ct n) are observed, with a growth exponent in three dimensions of n = 0·48 ±0·04 in the...

Simulation and theory of abnormal grain growth—anisotropic grain boundary energies and mobilities

Abnormal grain growth has been studied by means of a computer-based Monte Carlo model. This model has previously been shown to reproduce many of the essential features of normal grain growth. The simulations presented in this work are based on a modified model in which two distinct types of grains are present. These two grain types might correspond to two components of different crystallographic orientation, for example. This results in three classes of grain boundaries: (a) between unlike types, (b) between grains of the tirst type and (c) between grains of the second type, to which different grain boundary energies or different mobilities can be assigned. Most simulations started with a single grain of the first type embedded in a matrix of grains of the second type. Anisotropic grain boundary energies were modeled by assigning a higher energy to boundaries between like type than to boundaries between grains of unlike type. For this case, abnormal grain growth only occurred for an energy ratio greater than 2 and then wetting of the matrix by the abnormal grain occurred. Anisotropic grain boundary mobilities were modeled by assigning a lower mobility to boundaries between grains of like type than to boundaries between unlike type. For this case the extent of abnormal grain growth varied with the ratio of mobilities and it is tentatively concluded that there is a limiting ratio of size of the abnormal grain relative to the matrix, A simple treatment of anisotropic grain boundary mobility was developed by modifying Hillerts grain growth model (Acta meralf. 13,227 (1965)I. This theoretical treatment also produced a limiting ratio of relative size that is a simple function of the mobility ratio. RhsumP-La croissance anormale des grains a ete u (b) entre grains du premier type; (c) entre grains due second type, auxquels peuvent @tre assignees differentes energies intergranulaires ou differentes mobilitts. La plupart des simulations concernent un seul grain du premier type entoure par une matrice de grains du second type. On a tenu compte des energies intergranulaires anisotropes en assignment aux joints separant des grains du mm entsprechend konnen diese Korngrenzarten unterschiedliche Energien und Beweglichkeiten aufweisen. Die meisten der Simulationen begannen mit einem einzigen Kom des ersten Typs, welches in einer Matrix von Kornem des zweiten Typs eingebettet war. Anisotrope Komgrenzenergien wurden im Model1 beschrieben, indem den Komgrenzen zwischen gleichen Komem eine hohere Energie als zwischen ungleichen Kiimern zugeschrieben wurde. In diesem Fall ergab sich anormales Kornwachstum nur, wenn das Verhaltnis der Energien griil3er als 2 war und dann das anormale Kom die Matrix benetzte. Anisotrope Korngrenzbeweglichkeiten wurden im Model1 beschrieben, indem Komgrenzen zwischen Kdrnem des gleichen Typs mit einer geringeren Beweglichkeit als zwischen ungleichen Kiimern versehen wurden. In diesem Fall hing der Grad des anomalen Wachstums von dem Verhaltnis der Beweglichkeiten ab; daraus kann gefolgert werden, daB es eine Grenze

Computer simulation of grain growth—V. Abnormal grain growth

Abstract Monte Carlo computer simulation techniques have been utilized to investigate abnormal grain growth in a two dimensional matrix. The growth of abnormally large grains is modelled under two conditions: 1. (a) where the driving force is provided solely by curvature and 2. (b) where the driving force is provided by the difference in the gas-metal surface energy between grains of different crystallographic orientation. For curvature driven growth three cases are considered: 1. (a) the growth of abnormally large grains in microstructures without grain growth restraints, 2. (b) the growth of abnormally large grains in microstructures with particle dispersions, and 3. (c) grain growth in a particle pinned microstructure in which a sudden decrease in the number of particles occurs. In all these cases, the initiation of abnormal grain growth/secondary recrystallization is not found to occur. In systems free from grain growth restraints the normal grain size distribution is very robust and strongly resistant to perturbations. For systems which contain particle dispersions strong pinning of the grain boundaries is always observed. However, when a preferred surface energy orientation is introduced, abnormal grain growth/secondary recrystallization does take place. The microstructural evolution observed during secondary recrystallization is in good correspondence with experiment. The area fraction of secondary grains exhibits sigmoidal behavior as a function of time, and is characterized by an Avrami exponent of 1.8 ± 0.3 when fit to a modified Avrami equation.

Computer simulation of grain growth-III. Influence of a particle dispersion

Abstract A Monte Carlo computer simulation technique has been developed which models grain growth in the presence of a particle dispersion. The simulation allows for the monitoring of an evolving microstructure as a function of time. The model predicts normal grain growth, i.e. R = Ct n , where R is the average grain size and n is the grain growth exponent, followed by an abrupt transition to a pinned state. Both the exponent n and the grain size distribution are found to be in close agreement with that observed for grain growth in the absence of particles. The grain size distribution and kinetics are independent of particle concentration. The final average grain area and the time required for the microstructure to pin are both approximately proportional to the inverse of the particle concentration. The results are quantitatively accounted for in terms of a simple topological theory.

Computer simulation of grain growth—IV. Anisotropic grain boundary energies

Abstract A Monte Carlo computer simulation technique has been developed which models grain growth for the case in which the grain boundary energy is anisotropic. The grain growth kinetics, as represented by the growth exponent n ( R = Ct n ), is found to decrease continuously from 0.42 ± 0.02 to 0.25 ± 0.02 as the anisotropy is increased, where 0.42 is the growth exponent in the isotropic case. The grain size distribution functions become broader as the anisotropy is increased. For large anisotropy, the microstructure is described as consisting of large grains with extended regions of small grains. The small grains tend to have low angle grain boundaries. Anisotropic grain boundary energies can result in preferred crystallographic orientation, however the orientational correlations are limited to a few times the mean grain radius when potentials yielding reasonable microstructures are utilized.

Friction and wear of tough and brittle zirconia in nitrogen, air, water, hexadecane and hexadecane containing stearic acid

Abstract The friction and wear of zirconia sliding on zirconia at low speed (1 mm s−1) and moderate load (9.8 N) were studied with a pin-on-disk machine. Two materials were investigated, a brittle (2.5 MPa m 1 2 ), cubic phase doped with 5.5 mass% yttria and a tough (11.6 MPa m 1 2 ), tetragonal phase stabilized with 3 mass% yttria. Sliding occurred in dry nitrogen, where mechanical effects alone are expected, in laboratory air (relative humidity, 50% ± 10%), in water, in pure hexadecane and in hexadecane containing 0.5 wt.% stearic acid. Friction coefficients ƒ depend on the environment and somewhat on toughness ( ƒ = 0.1 in hexadecane containing stearic acid, 0.15 in pure hexadecane, 0.3–0.6 in air, 0.7 in water and 1.0 in dry N2). All wear rates decrease with increasing sliding distance. They are in the range 10−5-3 × 10−4mm3N−1m−1 for the tough material. Environmental effects are strong; they are compatible with stress corrosion cracking and not with the tribochemistry that governs environmental effects in non-oxide ceramics such as Si34. For the tough zirconia, the wear rate is lowest in dry nitrogen. For the other environments, it increases in the following order: hexadecane, hexadecane containing stearic acid, air, water. With the brittle material, wear is lowest in hexadecane and highest in air. The results are analyzed with the help of a model that relates the wear rate to the local contact stresses.

Grain growth in three dimensions: a lattice model

Grain growth is the term used to describe the increase in average grain size R which occurs upon annealing a polycrystalline aggregate after primary recrystallization is complete. In the long time limit, R increases with time t raised to a value n which is the growth exponent. Most existing grain growth theories implicitly assume that grains can be described as spherical and that growth occurs in an average environment. This, however, ignores the fact that adjacent grains share common boundaries, resulting in a microstructure that is topologically connected. These theories predict long term growth kinetics of the order n=1/2. Since experimental studies of grain growth yield an exponent less than 1/2, the discrepancy may be a consequence of neglecting topological constraints and the detailed environment. To examine this possibility a lattice model for 2-dimensional grain growth was developed in which topology and local environment are included. While this 2-dimensional model accurately simulates grain growth in thin encapsulated films, the validity of its predictions for 3-dimensional grain growth remains to be established. This paper reports the first results of our extension of the lattice grain growth model to 3 dimensions.

COMPUTER SIMULATION OF RECRYSTALLIZATION--III. INFLUENCE OF A DISPERSION OF FINE PARTICLES

Two-dimensional Monte Carlo simulations of recrystallization have been carried out in the presence of incoherent and immobile particles for a range of different particle fractions, a range of stored energies and a range of densities of potential nuclei (embryos). For stored energies greater than a critical value (H/J > 1) the recrystallization front can readily pass the particles leading to a random density of particles on the front and a negligible influence of particles on the recrystallization kinetics. At lower stored energies the particles pin the recrystallization front leading to incomplete recrystallization. However at very low particle fractions, when the new grain has grown much larger than the matrix grains, before meeting any particles, the new grains can complete the consumption of the deformed grains giving complete “recrystallization” by a process that appears to be similar to abnormal grain growth. Particles are, as reported previously, very effective at pinning grain boundaries, both of the deformed and recrystallized grains, when boundaries migrate under essentially the driving force of boundary energy alone. Such boundaries show a density of particles that rises rapidly from the random value found at the start of the simulation. As a consequence, particles very strongly inhibit normal grain growth after recrystallization. Such growth can only occur if the as-recrystallized grain size is less than the limiting grain size seen in the absence of recrystallization. Under these circumstances a small increment of grain growth occurs until the grain boundaries once again acquire a higher than random density of particles.

Abnormal grain growth in three dimensions

Normal grain growth is characterized by the self-similar coarsening of the microstructure. Therefore, knowledge of the microstructure at one time and the temporal evolution of the mean grain size, , provides all of the information required for a complete statistical description of the evolving microstructure. In contrast, abnormal grain grow:h is characterized by the growth of a small number of grains at a rate in excess to that of the mean grain size and, consequently, a lack of self-similarity. Abnormal grain growth is most frequently observed in samples containing a dispersed second phase and/or in sheet materials.