Large-scale simulations of Ostwald ripening in elastically stressed solids. II. Coarsening kinetics and particle size distribution

Ostwald ripening of misfitting second-phase particles in an elastically anisotropic solid is studied by large-scale simulations. The coarsening kinetics for the average particle size are described by a t^1/3 power law with a rate constant equal to its stress-free value when the particles are fourfold symmetric. However, the rate constant increases when the elastic stress is sufficient to induce a large number of twofold-symmetric particles. We find that interparticle elastic interactions at a 10% area fraction of particles do not affect the overall coarsening kinetics. A mean-field approach was used to develop a theory of Ostwald ripening in the presence of elastic stress. The simulation results on the coarsening kinetics agree well with the theoretical predictions. The particle size distribution scaled by the average particle size is not time invariant, but widens slightly with an increasing ratio of elastic to interfacial energies. No time-independent steady state under scaling is found, but a unique time-dependent state exists that is characterized by the ratio of elastic energy to interfacial energy.     

Phase-field simulation of coarsening of γ precipitates in an ordered γ′ matrix

The evolution of the microstructure and coarsening kinetics of disordered γ precipitates in an ordered monovariant γ′ matrix was studied using a diffuse-interface phase-field model in two dimensions. Specifically, we studied the morphological evolution, average precipitate size and size distribution as a function of time for a given temperature and for different concentrations. It was found that the coarsening kinetics for the average particle size are described by a t^1/3 power law with a rate constant that increases with the volume fraction of the γ phase. The particle size distribution scaled by the average particle size is time invariant. We find that the particle shapes are perturbed by elastic interaction and local distributions of γ particles. Comparison of the phase-field simulation results with experiments shows good qualitative agreements.     

Transient Ostwald ripening and the disagreement between steady-state coarsening theory and experiment

The coarsening of solid-Sn particles in a Pb–Sn liquid has been studied under microgravity conditions. These experiments permit an unambiguous comparison between theory and experiment to be made. In contrast to steady-state theories, such as those due to Lifshitz and Slyozov and Wagner, the scaled particle size distributions evolve in samples containing 0.1 and 0.2 volume fractions of solid. Steady state was not reached even though the average particle radius increased by a factor of three during the experiment. In addition, the scaled spatial correlation functions were also found to be time dependent in samples containing 0.1, 0.2, and 0.3 volume fractions of solid. The size distributions and correlation functions for all coarsening times at the fractions ≤0.3 agree with the predictions of a theory for transient coarsening. We show that the microstructures have not reached the steady-state regime for all volume fractions, are thus not self-similar, and that given our initial experimental conditions the time required to reach steady-state coarsening increases with increasing volume fraction. In these experiments, and we suspect in others as well, the transients are sufficiently long that steady-state theories cannot adequately describe the evolution of the microstructure.     

Coarsening of grain-refined semi-solid Al–Ge32 alloy: X-ray microtomography and in situ radiography

The Lifshitz, Slyozov and Wagner theory (LSW) describes the coarsening of low volume fraction dispersed particles in a supersaturated solution as governed by a t^1/3 power law, while stating that ripening occurs in a self-similar manner. Only a few experiments have reported three-dimensional (3D) coarsening in binary semi-solid alloys, which differs from the LSW theory. We report here on in situ coarsening of Al–Ge32 (wt.%), which is used as a model system for a large variety of technical alloys. Numerical analysis of 2D and 3D images of the microstructure measured by X-ray radiography and microtomography reveals the evolution of the solid particles during annealing. Ripening of a grain-refined particle network is found to be quite well described by LSW theory, although somewhat smaller exponents (t^1/4–t^1/5) are found. Changes in coarsening behavior are observed in samples which are thinner than 0.5 mm, as well as in non-equiaxed alloy microstructures, characterized by anisotropic dendrites.     

Large-scale three-dimensional simulation of Ostwald ripening

Large-scale three-dimensional simulations of Ostwald ripening were carried out using a multiphase-field model for accurate and efficient computation of multiparticle diffusional interaction in order to clarify coarsening kinetics in systems with finite particle fractions. The simulations were performed on 400³ grid systems in a particle fraction range of 0.25–0.86, where the initial microstructures with 20,000–100,000 solid particles in a liquid matrix evolve into structures with 1500–3000 particles. The simulation results were carefully analyzed and compared with those from the previous simulations and experiments, which have data range relevant to the present simulation range. The present results are in excellent agreement with the simulations by Akaiwa and Voorhees at low particle volume fractions, with the recent results from the 3-D reconstruction of serial sections at high fractions and with the grain size distribution of ideal grain growth at very high fractions.     

Phase-field simulation of 2-D Ostwald ripening in the high volume fraction regime

The microstructural evolution and kinetics of Ostwald ripening were studied in the high volume fraction regime by numerically solving the time-dependent Ginzburg–Landau (TDGL) and Cahn–Hilliard equations. It is shown that the growth exponent m is equal to 3, independent of the volume fraction, and the kinetic coefficient k increases as the volume fraction increases. The shape of size distributions changes significantly with increasing volume fraction of the coarsening phase; the skewness changes continuously from negative to positive while the kurtosis decreases in the low fraction regime and increases in the high volume fraction regime.     

Coarsening of γ (Ni–Al solid solution) precipitates in a γ′ (Ni3Al) matrix

The coarsening behavior of Ni–Al solid–solution precipitates in an Ni₃Al matrix was investigated in alloys containing 22.0–22.8 at.% Al aged at 650–800 °C for times exceeding 1800 h. The rate constant for coarsening increases with equilibrium volume fraction as predicted by the MLSW theory. The activation energy for coarsening, 314.1 ± 16.6 kJ mol⁻¹, agrees very well with results from conventional diffusion experiments. The particle size distributions are not in very good agreement with the predictions of any theory; possible reasons are discussed. The particles become more spherical with decreasing elastic self-energy. The results are consistent with the premise that a strong volume fraction effect is observed so long as diffusion in the matrix phase, and not through the precipitate–matrix interface, controls the kinetics.     

Liquid formation and microstructural evolution during re-heating and partial melting of an extruded A356 aluminium alloy

Thixoforming processes require feedstock having a non-dendritic, equiaxed microstructure, which in some cases can be obtained by a deformation–recrystallisation technique followed by partial melting. From the study of an extruded, re-heated and partially re-melted A356 alloy, a number of phenomena were observed and analysed: grain and particle growth, texture and low angle grain boundary formation, primary phase coarsening during isothermal holding in the semi-solid state and uneven distribution of Si particles and liquid phase. Both grain growth (solid state) and particle growth (semi-solid state) obey a R³ against time relationship. It was also found that microstructural coarsening follows an Ostwald ripening type of growth and that coalescence was also present, although exerting a minor role. Finally, phenomena such as the evolution with time of the proportion of low angle grain boundaries, and the relationship between recrystallisation, the onset of liquid formation and the coarse Si particle precipitation and dissolution are presented and discussed.     

Simulation of convection and ripening in a binary alloy mush using the phase-field method

Two-dimensional Ostwald ripening of an Al–4% Cu alloy solid/liquid mush in the presence of melt convection, and the influence of ripening on the flow, is studied numerically using a recent extension of the phase-field method that accounts for flow in the liquid phase. Through a parametric study, the ripening kinetics are investigated and compared for cases with and without melt convection. The cases without convection show good agreement with available coarsening theories for a finite fraction of solid. In the cases with flow the mean radius of the solid particles increases at a faster rate than without convection. The ripening exponent changes from 1/3 to 1/2, while the rate constant depends on the fraction of solid. Comparisons are made with the convective ripening theory of Ratke and Thieringer. Although the present analysis of coarsening is hampered by the limited number of particles in the domain, some qualitative results are presented for the effect of convection on the particle radius distribution. Finally, the present simulations allow for a determination of the permeability of the mush as a function of the fraction of solid, and the dependence of the permeability on the ripening kinetics is shown to be scalable using the specific surface area or the mean radius.     

Trans-interface-diffusion-controlled coarsening of γ′ precipitates in ternary Ni–Al–Cr alloys

Published data on the coarsening behavior of γ′ precipitates in three ternary Ni–Al–Cr alloys are examined in light of the theory of trans-interface-diffusion-controlled (TIDC) coarsening, in which the kinetics is controlled by diffusion through the coherent precipitate–matrix interface. The experimental data are independent of the equilibrium γ′ volume fraction, as expected for TIDC coarsening. Kinetics of the type 〈r〉ⁿ ∝ t for the growth of precipitates of average radius 〈r〉, and X_∞Al and X_∞Cr ∝ t^–1/n for the variations of the far-field matrix solute concentrations, X_∞Al,Cr, with aging time, t, are characteristic of TIDC coarsening. The temporal exponent n ≈ 2.4 was obtained from the fitting of published particle size distributions. Based on correlation coefficients, the dependencies of 〈r〉ⁿ on t and X_∞Al,Cr on t^?1/n were comparable for n = 2.4 and n = 3 (the temporal exponent for matrix-diffusion-controlled coarsening). The dependencies of volume fraction, f, and number density, N_v, on t are also compared with theoretical predictions. Using a thermodynamic model of the Ni–Al–Cr phase diagram, the interfacial free energy, σ, was estimated from analysis of the data; σ varies from ～14.5 to 19 mJ m^–2 in the three alloys. Interfacial diffusion coefficients, also obtained from analysis of the data, are greater than those in the γ′ phase but smaller than those in the γ phase, which is consistent with the demands of the TIDC theory. Comparisons with the results from previously published work are noted and all discrepancies are discussed.     

A theoretical model for digestive ripening

Recently, gram quantities of monodisperse gold or silver nanoparticles were reported to be produced through a digestive ripening process, in which colloidal particles of size from 2 to 40 nm are transformed to nearly monodisperse particles of 4–5 nm diameter. Digestive ripening, an example for inverse Ostwald ripening, is a puzzling phenomenon since it appears to avoid the usual capillary consequence, i.e., reduction of interfacial free energy. This paper presents a theoretical model which accounts for the self-assembled monodisperse state of such nanoparticles by considering the effect of charges, and therefore electrostatic energy, in the coarsening behavior of the particles. An appropriate Gibbs–Thomson equation is first derived, and then particle growth rates are calculated. The results show that success of a monodisperse state depends on, among other things, the initial particle distribution, and the size distribution at equilibrium follows some of the thermodynamic principles observed in binary phase diagrams of alloy systems.     

OSTWALD COARSENING UNDER CHANGING VOLUME FRACTION CONDITION

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The development of spatial correlations during Ostwald ripening: a test of theory

The coarsening of solid-Sn particles in a Pb–Sn liquid was studied under microgravity conditions. Spatial correlation functions were measured on plane sections in a low-volume fraction system undergoing Ostwald ripening. The correlation functions changed with time in a way that indicated that the microstructure initially consisted of clusters of particles and evolved into one which was more dispersed. The model by Akaiwa and Voorhees (AV) was used to study the effect of spatial correlations on the ripening process. We found that the initially highly correlated structure had no observable effect on the evolution of particle size distributions, but did have an effect on the coarsening rate of the system. Specifically, we determined that a structure consisting of clusters of particles coarsened faster than a system with a random, spatial arrangement of non-overlapping particles. We also found that the approach of the microstructure towards the steady-state regime could be monitored more sensitively using spatial correlations rather than using particle size distributions. The spacial correlations and the particle size distributions measured from the experiment agreed well with those calculated from the AV simulations using the initial experimental correlations and size distribution.     

Coarsening of ordered intermetallic precipitates with coherency stress

The morphological evolution and coarsening kinetics of ordered intermetallic precipitates with coherency stress were studied using a diffuse-interface phase-field model in two dimensions (2D). The emphasis is on the effects of precipitate volume fraction. The average aspect ratio of the precipitates in the microstructure is found to increase with time and decrease with volume fraction. Contrary to all the existing coarsening theories but consistent with a number of experimental measurements on the coarsening kinetics of ordered γ′ precipitates in Ni-base superalloys, we found that the coarsening rate constant from the cubic growth law decreases as a function of volume fraction for small volume fractions (≲20%) and is constant for intermediate volume fractions (20–50%). From the simulation results, we infer that the two length scales in a stress-dominated coherent two-phase microstructure, the average precipitate size and average spacing between arrays of aligned precipitates, follow different growth exponents. It is demonstrated that as the volume fraction increases, the precipitate size distributions become broader and their skewness become increasingly positive.     

The effect of rare earth elements on the kinetics of the isothermal coarsening of the globular solid phase in semisolid AZ91 alloy produced via SIMA process

In the present study, the effects of rare earth (RE) elements on the microstructure and coarsening kinetics of the solid globular particle in the semisolid slurry of AZ91 magnesium alloy have been studied at 570 °C and 580 °C. The results showed that the coarsening kinetics of the solid globular particles in semisolid slurry of AZ91 alloy satisfies the Ostwald ripening theory. It was shown that the coarsening rate of the solid particles decreases by adding RE elements into AZ91 alloy, specially at 580 °C, which results in the smaller particles size. It was attributed to the solid–liquid interfacial energy reduction due to the addition of RE elements.     

Phase-field modeling of isothermal dendritic coarsening in ternary alloys

The process of isothermal dendritic coarsening in a Ni–Al–Nb ternary system is simulated by using the phase-field method. The coarsening behaviors and coarsening mechanisms with different solid fractions f_S are investigated in detail. Simulated results show that, as f_S increases from 64% to 88%, the coarsened morphology of the liquid phase varies from connected plate to rod-like shape, which is similar to the lamellar to rod transition in eutectics due to the interface energy. Simulated results also indicate that the dendritic isothermal coarsening mechanisms are dominated by remelting of the third arms, coalescence and smoothing in the case of low solid fraction, while in the case of high solid fraction, coalescence, smoothing, Rayleigh instability, rounding and shrinking away of small liquid droplets are the main mechanisms of dendritic coarsening. The increase of liquid diffusion coefficient D^L will also trigger the Rayleigh instability to accelerate the morphology transition from plate to cylindrical for the liquid phase.     

Precipitation of Al3(Sc,Zr) particles in an Al–Zn–Mg–Cu–Sc–Zr alloy during conventional solution heat treatment and its effect on tensile properties

The effect of heat treatment on precipitation and growth of coherent nanometer-sized Al₃(Sc,Zr) particles and the effect of these particles on tensile properties of a direct chill (DC) cast Al–Zn–Mg–Cu–Sc–Zr alloy were studied. The size distribution, average size, number density and volume fraction of the Al₃(Sc,Zr) particles were determined as a function of the solution treatment temperature and time. An increase in the solution treatment temperature and time resulted in Al₃(Sc,Zr) particles with a larger mean diameter, higher volume fraction and lower number density. The particle size distributions were described well by normal (Gaussian) distributions. The kinetics of the phase transformation followed the Kolmogorov–Johnson–Mehl–Avrami law, with the Avrami exponent m = 0.404. Room temperature tensile properties were evaluated in the as-solution treated and artificially aged conditions. The coherent nanometer-sized Al₃(Sc,Zr) particles provided additional Orowan strengthening, which increased with increasing particle volume fraction and decreasing particle size, and varied from 75 to 118 MPa after different heat treatments.     

Three-dimensional simulation of isotropic coarsening in liquid phase sintering I: A model

A three-dimensional, Potts model of liquid phase sintering in a system with full solid wetting was introduced in order to investigate the coarsening kinetics and microstructures associated with this process. Kinetic Monte Carlo simulation was used to probe the coarsening dynamics and obtain the properties of solid particles, including the volume of critical nuclei and the distribution of particle sizes as a function of time. It was found that the average particle volume increased linearly with time and that the particle size distributions were consistent with those obtained experimentally, as in the W–Ni–Fe and Sn–Pb systems. In obtaining these results careful consideration was given to the role of the initial microstructural features in the subsequent evolution of the system.     

Coarsening of γ′ in Ni-Al alloys aged under uniaxial compression: III. Characterization of the morphology

The morphological evolution of γ′ precipitates during coarsening at 640 °C under applied compressive stress in the range 0–150 MPa was investigated in a monocrystalline Ni-Al alloy containing nominally 13.36 at. % Al. The strain was primarily elastic. The microstructures were examined by transmission electron microscopy, primarily in the section (100) perpendicular to the applied stress. The aspect ratio of a γ′ precipitate, as well as a shape parameter which provides a measure of how cuboidal it is, were used to characterize the γ′ morphology. As the stress increases precipitates of a given size generally tend to become more non-equiaxed and their interfaces are more planar, though this depends on the γ′ volume fraction. The applied compressive stress also promotes the coalescence of γ′ precipitates. This tendency is more pronounced the higher the γ′ volume fraction and appears to be its main influence on directional coarsening during elastic deformation.