Coarsening Rate Constant Research Articles

The effects of elastic stress on coarsening in the Ni-Al system

The effects of elastic stresses on the kinetics of coarsening were investigated. The coarsening of γ′ particles in Ni-Al alloys was quantified by measuring the γ- γ′ interfacial area per volume, instead of a characteristic average particle size. This allowed the kinetics of coarsening to be measured meaningfully at very long times when most particles do not have an equiaxed shape. It was found that the interfacial area per volume follows a t − 1/3 rate law, with no change in the exponent or the rate constant even at long coarsening times when elastic energy is a significant contribution to the total energy of the system, and the microstructure is not self-similar. The coarsening rate constant was found to vary with volume fraction as predicted by theory in the absence of elastic stress, in contrast to previous experimental results which quantify coarsening kinetics using an average particle radius.

Volume fraction dependence on coarsening of Y 2BaCuO 5 particles in Ba–Cu–O melt

The effect of the volume fraction of the solid phase (Vf) on coarsening phenomena of Y 2BaCuO 5 (Y211) particles in Ba–Cu–O melt has been experimentally investigated. The rate constant of the coarsening cubic-law ( K) also depends strongly on Vf and is classified into two groups. The samples with Vf lower than 15% reveal higher coarsening rate constants than those with Vf higher than 20%. The coarsening properties of the lower Vf were in good agreement with the classical mean field (MF) model that considered the effect of Vf. It was clarified that K values of the higher Vf were suitable for those calculated numerically by the communicating neighbor (CN) model, in which the solute was assumed to diffuse directly among the particles. Thus, the coarsening phenomena of Y211 particles have been qualitatively explained by the difference in coarsening mechanisms as the MF and CN models.

Y-Ba-Cu-O系超電導体におけるY211相粒子の粗大化

It is important to understand the coarsening behavior of the Y2BaCuO5 phase particles in Ba-Cu-O melts, since the particle-size distribution of the Y2BaCuO5 particles entrapped in the YBa2Cu3Ox superconducting phase largely affect the superconducting properties. In this paper, we have investigated the dependence of the volume fractions of the high-temperature stable phase Y2BaCuO5 particles in the Ba-Cu-O melt on the coarsening behavior. Consequently, it was found that the coarsening rate constants between the volume fraction of Vf=10-15% and Vf=20-30% were significantly different. Furthermore, the particle-size-distributions for the Vf=20 and 30%, were investigated and discussed.

An improved small-angle X-ray scattering analysis of δ′ precipitation in an Al–Li alloy for growth kinetic studies

An improved method for small-angle X-ray scattering (SAXS) data analysis is developed to reconstruct the free-form particle size distribution of δ′ precipitation in an Al–Li alloy. This improved method consists of four iterative steps; the interparticle interference is also included. The indirect transform method (ITM) plus a hard-sphere (HS) model which considers the depleted zones are used in the analysis of δ′ precipitation in an Al–Li alloy. Two parameters, namely the hard-sphere volume fraction, ηHS, and the ratio of hard-sphere radius to the particle radius,RHS/R, which determine the structure factor of the interparticle effect, are iteratively calculated using the monodisperse assumption and Gaussian size distribution. These two parameters are finally used in reconstructing the particle size distribution by the ITM + HS method. This method is tested by analysing simulated SAXS data and shows a better agreement than found in similar studies. This improved method is applied to analyse a set of experimental SAXS intensities from δ′ (Al3Li particles) precipitation in an Al–9.7 at.% Li alloy. The monodisperse results are compared with the polydisperse ITM + HS results. The current ITM + HS method fits the SAXS data better than the other methods. The variations of average radii with aging time were found to follow the kinetic power law. The SAXS results are used to investigate the theoretical kinetic model of the volume-fraction effect on late-stage coarsening (Ostwald ripening). By comparing both experimentally obtained asymptotic size distributions of δ′ particles as well as coarsening rate constants with those predicted by the various kinetic models, the modified Lifshitz–Slyozov–Wagner (MLSW) theory is found to be in better agreement with the experimental results than the other theories.

Correspondence

The success of thixoforming mostly depends on the preparation of semisolid slurry, consisting of non-dendritic and spherical solid grains surrounded by liquid. Initial microstructure of alloys before remelting, especially the initial grain size and the grain boundaries, can influence microstructural coarsening of semisolid alloys. The investigation focused on the microstructural evolution during reheating of a fine-grained 7075Al alloy processed by friction stir process (FSP) into the semisolid state. The results showed that a semisolid slurry containing fine and spherical solid grains was prepared and the distribution of solid grains and liquid phase is relatively uniform. Owing to the penetration of liquid phase into high angle grain boundaries (HAGBs) and a large fraction of HAGBs in the FSPed alloy, a network structure of intragranular liquid film in the semisolid state can be obtained in a short soaking time. The coarsening rate constant of the alloy increased with increasing liquid fraction. The grain coarsening mechanism in the semisolid state was discussed, and the effect of grain boundary characteristics in the initial microstructure on coarsening kinetics of semisolid alloys was also described.

Ostwald ripening in non-spherical morphologies

A statistical theory of Ostwald ripening has been extended to describe the diffusion-limited coarsening of non-spherical convex interfaces. Distributional analysis is performed on a series of interfacial patches, where the patches and adjacent isopotential surfaces have a common fixed ratio of principal curvatures. Results of this analysis are expressed in terms of a dimensionless shape factor that ranges from zero for spherical interfaces to unity for cylindrical surfaces. Coarsening rate constants are calculated as a function of both the precipitate volume fraction and this geometric shape factor. Application of these results to ripening of ellipsoidal precipitates and other morphologies are discussed.

Element partitioning during coarsening of (γ − γ′) Ni-Al-Mo alloys

In this article, the coarsening rate constants for several of the model Ni-Al-Mo ternary alloys employed in an earlier investigation at 775 C are reported for three additional temperatures, 842 C, 872 C, and 903 C. Arrhenium plots of the coarsening rate constants provide strong experimental indication for a change in the rate-controlling mechanism for coarsening from the long-range diffusion of Al to the long-range diffusion of Mo. This is supported by an estimate of the relative contributions of the alloying elements, Al and Mo, to the overall coarsening resistance of the alloys.

Influence of coherency stress on microstructural evolution in model Ni-Al-Mo alloys

The influence of the sign and magnitude of the misfit strain on two-phase ( γ— γ′) microstructural evolution during coarsening is examined in a series of five low-volume-fraction Ni-Al-Mo alloys using TEM and SAXS. The evolution of the symmetry-conserving, sphere-to-cube transition and the symmetry-breaking cube-to-cuboid transition with increasing particle size were quantified and the general trends were found to be in excellent agreement with two-dimensional calculations of particle shapes. Both types of particle shape transitions and particle alignment occur at smaller particle sizes the larger the magnitude of the misfit strain. A decrease in the coarsening rate constant was measured with increasing Mo content and was attributed to Mo diffusion and partitioning rather than the increasingly negative misfit strain of the alloys with increasing Mo content.

Simulation studies of diffusion‐limited coarsening in two dimensions

AbstractThe kinetics of particle coarsening are studied by means of contour dynamics simulation in two dimensions. The current method is free from the usual restrictions, such as small area fraction, constant particle shape, and local particle motion. Growth or shrinkage of particles in a given realization is found to depend on the local environment of the particles. The concept of critical size central to the classical theories such as the LSW theory, therefore, appears to be inappropriate. Surprisingly, the ensemble‐averaged macroscopic behavior is consistent with mean‐field theory. The particle‐size distribution (PSD) tends toward an asymptotic shape, different from the mean field result. The normalized width of the PSD seems to be insensitive to volume fraction. Interfacial transport limitation can be significant for small particles, and it has the effect of altering the width of the PSD. The coarsening rate constant is found to be a function of the area fraction.

Design of M2C carbides for coarsening resistance

A model has been developed for coarsening kinetics of M 2C carbides and subsequently applied to optimize the composition of high Co-Ni steels with respect to the overaging resistance during secondary hardening. The model accounts for both effects of multicomponent diffusion and precipitate shape anisotropy. It is found that W additions to high Co-Ni steels significantly reduces the coarsening rate constant. Calculations for two W-free 14Co-10Ni steels at a typical tempering temperature of 783K reveals a beneficial effect of the Cr substitution with Mo, in a good agreement with available experimental data.

Ostwald ripening in a system with a high volume fraction of coarsening phase

Experiments on the coarsening behavior of two-phase mixtures in a model Pb-Sn system are reported. This system fulfills most of the assumptions of theory and has the particular advantage that all the materials parameters necessary for a comparison between the experimentally measured and theoretically predicted coarsening kinetics are known. We have examined the coarsening of Sn-rich and Pb-rich solid phases in contact with eutectic liquid in the volume fraction solid range above approximately 0.6 where the development of a solid skeletal structure inhibits sedimentation. Particle intercept distributions are measured and found to be time independent when scaled by the average intercept. This invariance is interpreted as evidence that scale factor coarsening is present. The intercept distributions are in good agreement with the predictions of theory. Measurements of average intercept diameter as a function of time establish unambiguously that the coarsening follows the theoretically predictedt 1/3 kinetics. The coarsening rate constants are measured as a function of volume fraction solid and are found to exceed the values calculated from theory using the known thermophysical properties of the Pb-Sn system by factors ranging from approximately 2 to 5.

Application of thermodynamic factors in analysis of particle coarsening kinetics in Fe–Cu and Co–Cu

Abstract The classical Lifshitz, Slyozov, and Wagner (LSW) theory has been modified by several authors to take account of a finite volume fraction of particles. Somewhat different dependencies of the coarsening rate constant on the volume fraction are predicted by these modified theories. To resolve these discrepancies, coarsening rates in the Fe–Cu and Co–Cu systems are measured and the interfacial energy was determined from the classical coarsening equation, volume fraction modified equations, and an equation containing additional thermodynamic factors. These values are compared with independently measured interfacial energy values and good agreement is found with volume fraction modified theories containing the thermodynamic factor. The dependency of the rate constant on volume fraction predicted by these theories is also found to be in good agreement with experiment. The volume fraction and thermodynamic factor modified models are applied to the analysis of coarsening data in the literature for the Ni...

Coarsening behavior of cementite particles in a ferrite matrix in 10B30 steel

The coarsening behavior of cementite particles in 10B30 boron steel was studied. The hardened steel specimens were tempered at 903, 923, 943, and 963 K for various lengths of time, and the cementite particle sizes were measured by electron microscopy. It was observed that at all the aging temperatures the log-log plots of cementite particle size versus tempering time were straight lines with a slope of approximately 1 3 . This suggests that the cementite particles in the steel grow by a diffusion-controlled coarsening process. The activation energy for the coarsening process was calculated to be 223 kJ mol −1, which is close to that for diffusion of Fe in α-ferrite. The interfacial energy γ of the cementite-ferrite interface was also calculated from the coarsening rate constants and was found to be in the range 0.248–0.417 J m −2.

Coarsening kinetics of silica in copper

Coarsening experiments have been conducted in the copper-silica system to study the aging behavior of compound precipitates in metallic systems under various conditions of temperature and atmosphere. Most of the experiments were conducted in a two-zone heating furnace, wherein a copper-cuprous oxide mixture was heated in one zone and the copper-silica sample was heated in the other zone, all enclosed in a quartz capsule. The temperature of the former zone determined the oxygen partial pressure in the system and the temperature of the latter zone determined the coarsening temperature. The silica particles from the coarsened samples were extraction-replicated and photographed in an electron microscope. The average radii obtained from these photographs were used to determine coarsening rate constants and activation energies. Isothermal coarsening kinetics followed the γ-3vs t law, indicating volume diffusion control. Activation energies were obtained for four different sets of experimental conditions. In each case the activation plot was linear. For coarsening under variable oxygen pressure the experimental activation energy of 530 kJ (127 Kcal) agrees fairly well with the predicted value of 514 kJ (123 Kcal) which takes into consideration the heats of solution and dissolution, as well as the activation energy for diffusion. For coarsening under constant oxygen pressures the experimental activation energies do not compare well with the predicted values, calculated on the basis of complete equilibration of oxygen between the gaseous phase and the sample. However, the experimental activation energies of 524 kJ (125 Kcal) and 447 kJ (107 Kcal) lie in the predicted range calculated on the basis of fixed amounts of oxygen in solution in copper. In no case is the activation energy for coarsening equal to that for diffusion of either species in the matrix as is sometimes assumed.

On the coarsening of grain boundary precipitates

The diffusion-controlled coarsening behavior of precipitates situated on both high-angle and low-angle grain boundaries has been investigated theoretically. The effect of volume fraction, φ, is included in the calculations. At high-angle grain boundaries the average particle size increases as t 1 4 after long aging times, irrespective of the value of φ. The coarsening rate constant increases, and the theoretical distribution of particle sizes broadens, as φ increases. These results are considered in the light of previous theoretical treatments of this problem. It is assumed that precipitate growth on low-angle grain boundaries is controlled by the diffusion of solute along the dislocation network in the boundary. In this case the average particle size increases as t 1 4 provided that the particle radii are much larger than the dislocation spacing. However, when the dislocation spacing is greater than the radii of all the particles, the average particle size increases as 1 5 . The effect of φ on the coarsening rate constants and particle size distributions is estimated semiquantitatively for the extreme cases of t 1 4 and t 1 5 kinetics, and discussed in a qualitative manner for intermediate ratios of particle size and dislocation spacing.