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.     

Characterisation and modelling of precipitate evolution in an Al–Zn–Mg alloy during non-isothermal heat treatments

This paper describes the response of a precipitate microstructure to various types of non-isothermal temperature changes, namely reversion, ramp heating and thermal cycles, met in heat-affected zones (HAZs) of arc-welds, in an Al–Zn–Mg alloy in the T6 and T7 states. During these thermal histories, the precipitate size and volume fraction are quantitatively measured by small-angle X-ray scattering (SAXS). Reversion experiments are characterised by a fast dissolution stage at almost constant average precipitate size, followed by a coarsening stage that affects the surviving precipitates. Ramp heating experiments show a dissolution behaviour, which is temporarily interrupted for low heating rates, either due to phase transition when the initial precipitates are of a metastable nature (the case of T6 initial state), or due to a dynamic competition between the average and critical radii during the temperature increase (the case of the T7 material). The HAZ of arc-welds is characterised by a gradual increase of the dissolved precipitate fraction as one gets closer to the weld line. In the zone immediately before complete dissolution, precipitate coarsening is observed.

A simple model, based on the growth/dissolution of precipitates distributed in size classes, has been adapted to the present ternary alloy. This model, calibrated using the reversion experiments, has proven its predictive nature in all the other thermal cycles investigated (ramp heating and welding). The predictions of the model are used in order to improve the physical insight into the range of material behaviour observed in the experiments.     

Multiscale simulations on the coarsening of Cu-rich precipitates in α-Fe using kinetic Monte Carlo,molecular dynamics and phase-field simulations

The coarsening kinetics of Cu-rich precipitates in an α-Fe matrix for thermally aged Fe–Cu alloys at temperatures above 700 °C is studied using a kinetic Monte Carlo (KMC) simulation and a phase-field method (PFM). In this work, the KMC approach adequately captures the early stage of the system evolution which involves nucleation, growth and coarsening, while the PFM provides a suitable framework for studying late-stage coarsening at large precipitate volume fraction regimes. Hence, both models complement each other by transferring the results of KMC along with precipitate–matrix interface energies from a broken-bond model to a quantitative PFM based on a grand chemical potential formulation and the CALPHAD database. Furthermore, molecular dynamics simulations provide information on the structural coherency of the precipitates and hence justify the sequential parameter transfer. We show that our PFM can be validated quantitatively for the Gibbs–Thomson effect and that it also predicts the coarsening kinetics correctly. It is found that the kinetics closely follow the LSW (Lifshitz–Slyozov–Wagner) law, whereas the coarsening rate constant increases with an increase in volume fraction of precipitates.     

加热温度对含Nb中碳钢奥氏体晶粒长大的影响

利用微合金析出物与临界晶粒尺寸的定量关系,研究了Si含量较高的中碳Nb微合金钢在不同加热温度下的奥氏体晶粒长大规律。结果表明,随着加热温度的升高,试验钢中奥氏体晶粒逐渐长大,当温度高于1100℃时,晶粒开始粗化。由经验模型可得,随着温度的升高,析出相的体积分数逐渐减少,而颗粒半径逐渐增大,由于二者的共同作用导致了奥氏体晶粒在高于1100℃时迅速粗化;在实验的基础上,得到了适用于试验钢的晶粒长大模型。     

Analytical description of phase coarsening at high volume fractions

An analytical theory of volume fraction effects in diffusion-controlled phase coarsening is developed. It is based on the application of the Lifshitz–Slyozov–Wagner procedure of Ostwald ripening theory to a self-similar particle volume evolution equation which is approximated by a quadratic polynomial of the scaled particle size. A family of analytical particle size distributions with the scaled maximum particle size as a parameter is derived, which contains the size distribution of normal grain growth as a genuine limit distribution at ultra-high volume fraction without changing the t^1/3-Ostwald ripening kinetics. This reflects an important characteristic feature of particle coarsening at nearly space filling, which was first noted by Ardell and which is in principle in agreement with recent observations. The obtained analytical expressions for the maximum particle size, the particle size distribution and the coarsening rate constant are easy to use for evaluating coarsening data. The results compare well with a number of experimental and simulation results at intermediate and high volume fractions of the second phase.     

Coarsening kinetics of γ′ precipitates in the Ni–Al–Mo system

The effect of Mo on the microstructure evolution and coarsening kinetics of γ′ precipitates in the Ni–Al–Mo system is studied using phase-field simulations with inputs from thermodynamic, kinetic and lattice parameter databases. For alloys of different compositions, the precipitate morphology and the statistical information of precipitate sizes are predicted as a function of annealing time. It is observed that increasing Mo content leads to a change of the precipitate morphology from being cuboidal to spherical as well as a reduction in the coarsening rate. Comparison between simulated results and existing experimental microstructure morphologies and coarsening rates shows good agreements.     

Modelling the kinetics of strain induced precipitation in Nb microalloyed steels

Strain induced precipitation is a key phenomenon that controls the microstructure evolution during the finish rolling stages of microalloyed steels. Extensive research has shown that the precipitation of Nb(CN) delays the onset of recrystallisation. This paper presents a model to describe the precipitation kinetics during isothermal holding following high temperature deformation in Nb-containing steels. The model is based on the assumption that heterogeneous nucleation of precipitates on dislocations and enhanced coarsening due to pipe diffusion are responsible behind the accelerated kinetics observed in strain induced precipitation. Results show a very good agreement between reported experimental observations and predictions of the present model for precipitate size and volume fraction evolution.     

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.     

相场法模拟扩散场相互作用对沉淀相形貌及成分的影响

在把网格引入的各向异性的影响降低到合理范围的基础上,采用相场法研究了在无弹性场作用时,沉淀颗粒间的扩散场相互作用对沉淀相形貌及成分的影响.模拟结果表明:由于扩散场的相互作用,在沉淀相生长及粗化过程中,沉淀颗粒形貌可出现圆形→方形→圆形→菱形→圆形的多种变化,且沉淀内部呈现成分不均匀性.随着沉淀相体积分数的增大,沉淀颗粒间的扩散场相互作用加强,对沉淀相形貌及其成分的影响也更加显著.     

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.     

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.     

Determining the effect of microstructure and heat treatment on the mechanical strengthening behavior of an aluminum alloy containing lithium precipitation hardened with the δ′ Al3Li intermetallic phase

The effect of the thermal treatment and composition on microstructure and subsequent mechanical behavior of an Al-2.6 wt.% Li-0.09 wt.% Zr alloy that was solution heat treated (SHT) and artificially aged for a series of aging times and temperatures was studied. The underaged, peakaged, and overaged thermal heat treatments were studied to determine the effect of the microstructure and processing on the mechanical properties. The precipitates in the microstructure, which impede dislocation motion and control the precipitation strengthening response as a function of aging practice, were analyzed as the basis for controlling the strengthening depending on their size distribution, average size, and interparticle spacing. The average particle size, spacing, and size distribution were determined from the microstructure as a function of the thermal processing and composition. For the demonstration alloy, the primary strengthening was a direct consequence of ordered coherent Al₃Li (δ′) intermetallic precipitates, which are uniformly distributed throughout the microstructure and restrict the glide motion of dislocations during plastic deformation. The Al₃Li average particle size, distribution, spacing, and volume fraction are closely related to the overall mechanical behavior and are a result of the heat treating practice and composition. Consequently, a micromechanical model was developed for predicting the precipitation hardening response in terms of the variation in polycrystalline strength with aging time, aging temperature, and composition. The overall micromechanical model, which was determined from the particle coarsening kinetics, dislocation mechanics, thermodynamics, resolved shear stress, as well as the dislocation particle shearing and bypassing mechanisms, accurately predicted the mechanical strength in the underaged, peak-aged, and overaged tempers of the demonstration alloy.     

共格沉淀析出过程的模拟Ⅰ--微观结构演化

采用相场模型对共格沉淀析出过程微观结构演化进行模拟研究.模拟结果表明,应力场的存在将会对相变过程中析出相形态产生显著的影响,通过界面能与弹性应变能的相互竞争析出相在不同阶段呈现不同的形态如模量结构、网格结构、三明治多畴结构、沉淀宏观点阵结构以及板条状等;此外,点阵错配度中等(2%～4%)时,粒子的粗化过程将出现应力诱导反向粗化现象,这种粗化现象取决于粒子的点阵错配度与体积分数.     

含共格畸变的立方合金粗化机制的原子层面计算机模拟研究

基于微观相场动力学模型，编制了包含共格畸变能的二元立方合金沉淀过程微观组织演化的计算机模拟程序，开展了不同共格畸变能作用下，溶质浓度为20at％的二元镍基合金的粗化机制的计算机模拟。研究发现：共格畸变能为零时，粗化遵循LSW机制，仅仅由颗粒的尺寸大小决定：随着共格畸变能的增大，粗化过程遵循混合机制，由沉淀颗粒的大小和位向（颗粒间的相对位置）共同决定；当共格畸变增大到一定程度时，粗化过程纯粹由颗粒间的位向决定，处于弹性“软”方向上的颗粒优先长大，而处于弹性“软”方向外的颗粒将消失掉。     

Optimal precipitate shapes in nickel-base γ-γ′ alloys

Precipitate shapes in nickel-base superalloys vary substantially with alloy composition, partitioning of elements to the disordered gamma matrix and the ordered gamma prime precipitates and the degree of coarsening during elevated temperature exposures. These shapes, which vary from spherical to cuboidal to rod-like, have typically not been quantified in Ni-base alloys containing relatively high γ′ volume fractions, in spite of their importance to mechanical properties. Precipitate shapes in a series of new platinum group metal (PGM)-containing Ni-base alloys have been quantified by their two-dimensional moment invariants. Precipitate morphologies were characterized in a total of 17 PGM-containing alloys in the solution treated and aged condition. The average γ′ volume fraction was measured as 0.60, typical of highly creep resistant Ni-base alloys. PGM additions resulted in an unusually large range of precipitate shapes. Precipitate morphologies were quantitatively analyzed using a shape parameter derived from the absolute moment invariant. For the compositions examined, the shape parameter reaches a maximum value at a precipitate-matrix misfit magnitude of 0.4%. A large set of commercial single crystals exhibits the same range in shape parameter values around this misfit magnitude, suggesting that favorable high temperature properties are correlated with an optimum precipitate shape.     

Effect of Time-Dependent Pinning Pressure on Abnormal Grain Growth: Phase Field Simulation

The effect of the time-dependent pinning pressure of precipitates on abnormal grain growth has been investigated by multiphase field simulation with a simple precipitation model. The application of constant pinning pressure is problematic because it always induces abnormal grain growth or no grain growth, which is not reasonable considering the real situation. To produce time-dependent pinning pressure, both precipitation kinetics and precipitate coarsening kinetics have been considered with two rates: slow and fast. The results show that abnormal grain growth is suppressed at the slow precipitation rate. At the slow precipitation rate, the overall grain growth caused by the low pinning pressure in the early stage indeed plays a role in preventing abnormal grain growth by reducing the mobility advantage of abnormal grains. In addition, the fast precipitate coarsening rate tends to more quickly transform abnormal grain growth into normal grain growth by inducing the active growth of grains adjacent to the abnormal grains in the early stage. Therefore, the present study demonstrates that the time dependence of the pinning pressure of precipitates is a critical factor that determines the grain growth mode.     

Three-dimensional model of precipitation of ordered intermetallics

The development of the two-phase (f.c.c.+L1₂) coherent microstructure in the prototype Ni–Al superalloy is studied by using the three-dimensional computer simulation technique. The dynamics and morphology of the microstructure evolution are described by our three-dimensional version of the stochastic time-dependent kinetic equation which explicitly includes the coherency strain, elastic anisotropy and L1₂ ordering of the preciptate phase. The input parameters, the crystal lattice misfit, elastic moduli, interfacial energy and equilibrium compositions of the coexisting phases are taken from the published independent measurements. The simulation results demonstrate that the strain accommodation in the microstructure evolution results in the cuboidal-like precipitates faceted by the {100} planes. The size of the precipitates obtained in the simulation is of the order of 50 nm. The important conclusion is that the precipitates are always single-domain particles with no antiphase boundaries. This effect is associated with the ordered structure of precipitates. It causes the slowing down of the coarsening kinetics since it excludes the agglomeration of the out-of-phase precipitates in one particle. As has been shown previously, the latter is a very important coarsening mode in an absence of ordering.     

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.