Coarsening kinetics of γ' precipitates in a Re-containing Ni-based single crystal superalloy during long-term aging

The morphological evolution and coarsening kinetics ofγ'precipitates in a Re-containing Ni-based single crystal superalloy were investigated during isothermal aging at 900,950 and 1000℃.After heat treatment,well-defined cuboidalγ'precipitates with low misfit was obtained within the experimental alloy.Then coarsening rate constants and particle size distribution(PSD)ofγ'phases were calculated and specified based on the measured precipitate sizes for va rying periods of aging times from 100 to 2000 h.After aging for 2000 h,γ'precipitates maintained cubical shape at 900℃,while exhibited sphere at 950 and 1000℃.Coarsening models based on diffusion-controlled process with a functional relationship of r³ vs.t(classic Lifshitz-Slyozov-Wagner coarsening model)and interface-controlled model with a function of r² vs.t(trans-interface diffusion-controlled coarsening model)were investigated to fit between the experimental results and theoretical analysis.It was found that Re as the slowest diffusing solute in the alloy constituted the rate-limited step for coarsening based on LSW model,while the process limiting coarsening as governed by an interface diffusion process could possibly be related to the Al diffusion through theγ/γ'interface.The PSDs and coarsening exponent were discussed by comparing the experimental data with predictions of LSW and TIDC models.Finally,coarsening mechanism could be divided into four regimes:(i)coarsening by diffusion-controlled;(ii)coarsening by diffusion and interface cocontrolled;(iii)coarsening by interface-controlled;(iv)coarsening by interface-controlled accompanied withγ'coalescence.     

Coerced mechanical coarsening of nanoparticle assemblies

Coarsening is a ubiquitous phenomenon that underpins countless processes in nature, including epitaxial growth, the phase separation of alloys, polymers and binary fluids, the growth of bubbles in foams, and pattern formation in biomembranes. Here we show, in the first real-time experimental study of the evolution of an adsorbed colloidal nanoparticle array, that tapping-mode atomic force microscopy (TM-AFM) can drive the coarsening of Au nanoparticle assemblies on silicon surfaces. Although the growth exponent has a strong dependence on the initial sample morphology, our observations are largely consistent with modified Ostwald ripening processes. To date, ripening processes have been exclusively considered to be thermally activated, but we show that nanoparticle assemblies can be mechanically coerced towards equilibrium, representing a new approach to directed coarsening. This strategy enables precise control over the evolution of micro- and nanostructures.     

A cellular automaton model of steady-state columnar-dendritic growth in binary alloys

A two-dimensional cellular automaton model has been developed to examine the evolution and coarsening behaviour of solid-solution dendrites during steady-state columnar freezing. Using an empirical rule to account for interface growth, realistic dendrite geometries were obtained for different assumed compositions and process conditions. Coarsening occurred by a coalescence mechanism associated with bridging of adjacent dendrite arms.     

Simulation of microstructural evolution in materials with misfitting precipitates

To predict the behavior of directional coarsening and the temporal evolution of the shape of coherent precipitates in two-phase materials, a dislocation-free model is proposed, based on a combination of statistical mechanics and linear elasticity. This model takes elastic anisotropy and isotropic interfacial energy into account. Based on an example of isolated precipitates under plane strain condition, the influence of particle size, inhomogeneity, direction and sign of external loads on the equilibrium shape will be discussed in terms of a generalized thermodynamic force acting on the interface. To simulate the morphological diffusion process of typical microstructures with several random distributed misfitting inclusions, a computational technique in form of a finite element Monte Carlo simulation is presented. Within this numerical technique, no restrictions on the particle shape or the elastic anisotropy of both phases are made.     

Phase field formulations for modeling the Ostwald ripening in two-phase systems

Phase field formulations have been constructed for modeling Ostwald ripening in two-phase systems. The microstructural evolution and the kinetics of Ostwald ripening were studied by numerically solving the time-dependent Ginzburg-Landau (TDGL) equations. The simulated microstructures are in a striking resemblance with experimental observations. The shape accommodation of second phase particles occurs as the volume fraction increases. It was observed that these two-phase systems reach the steady state or scaling state after a short transient time and the scaling functions are independent of time for all volume fractions of the second phase. The kinetics of Ostwald ripening in a two-phase mixture have been studied over a range of volume fractions of the coarsening phase. It was found that the coarsening kinetics of second phase particles follows the power growth law R_t^m − R₀^m = kt with M = 3, which is independent of the volume fraction of the coarsening phase. The kinetic coefficient k increases significantly as the volume fraction of the coarsening phase increases.     

Coarsening Kinetics of γ' Precipitates in Dendritic Regions of a Ni_3 Al Base Alloy

Coarsening behavior of γ' precipitates in the dendritic regions of a Ni 3 Al base alloy containing chromium,molybdenum,zirconium and boron was investigated.Annealing treatment was performed up to 50 h at 900,1000 and 1100℃.The alloy was produced by vacuum-arc remelting technique.Results show that coarsening of the γ' precipitates in this complex alloy containing high volume fractions of γ' phase follows Lifshitz-Slyozov-Wagner(LSW) theory.Coarsening activation energy of the γ' precipitates was evaluated to be about 253.5 kJ.mol-1 which shows that the growth phenomenon is controlled by volume diffusion of aluminum.With an innovative approach,diffusion coefficient of the solute element(s) and the interfacial energy between γ' precipitates and γ'(matrix) were estimated at 900,1000 and 1100℃.Accordingly,the interfacial energies at 900,1000 and 1100℃ are 4.49±1.48,2.08±0.69 and 0.98±0.32 mJ.m-2,respectively.Also the diffusivities of solute element(s) at these temperatures are 3.41±1.08,30±9.5 and 145.15±45.85(10-15 m-2.s-1),respectively.     

Coarsening during solidification of aluminium-copper alloys

Coarsening of directionally solidified -phase dendrites and of particulate -phase/liquid mixtures was investigated in Al-4, 10 and 20 wt% Cu alloys, as a function of temperature, composition and presence or absence of forced convection. Isothermal dendritic coarsening in the absence of convection operated in two stages. In stage I the dendritic structure broke down through remelting into fragments which spheroidized quickly; in stage II the spherical particles coarsened slowly. The coarsening rate of the dendritic or particulate solid increased with temperature and copper dilution. Alloy inoculation with titanium slowed coarsening, yielding finer dendritic microstructures. The effect of turbulent flow on coarsening was manifested only for longer holding times. At higher impeller angular velocities the dendritic structure breaks down into fragments which spheroidize rapidly. At lower shear rates (below 650 rev min^–1) solid particles in solid-liquid mixtures coalesce into clusters, whereas at higher rates the clusters break up again into individual particles. A coarsening model was introduced which showed that coarsening is faster in the presence of forced convection, because of the resulting decrease in solute diffusion-boundary layer thickness.     

Three-dimensional phase-field simulation of microstructural evolution in three-phase materials with different diffusivities

The coarsening behavior of three-phase materials, such as eutectic material systems, is of high technological interest. Microstructure evolution simulations can help to understand the effect of different magnitudes of the diffusivities in the different phases. In this study, the evolution of a 3D three-phase morphology was modeled with equal interfacial energy and volume fraction and similar thermodynamic properties for the three phases, but the diffusion mobilities were taken different. It was observed that the phase with the lowest mobility has the highest growth rate and, on average, a larger number of grain faces, while the other two phases have a nearly equal growth rate and average number of grain faces. The simulation results are compared with results from experiments and simulation studies for single-phase and two-phase materials.     

A computational thermodynamics approach to the Gibbs–Thomson effect

In two-phase system, curvature of interface leads to increase of solute concentration in matrix. This effect plays a significant role in solidification, precipitation, nucleation and growth and coarsening. There are number of models and formulas for Gibbs–Thomson effect in binary alloys. In this paper with the help of CALPHAD calculations, new approach for describing this effect in binary and multicomponent systems is proposed. In this generalized method no traditional simplifying assumption are considered and this yield to more accurate result for Gibbs–Thomson phenomenon. This model is compared with previous formulas in some case alloying systems.     

Coarsening kinetics of coherent γ′ precipitates in ternary Ni-based alloys: the Ni–Al–Si system

Coarsening of coherent precipitates γ' in alloys such as Ni-based alloys has been studied extensively not only for its practical significance in the design of engineering alloys but also in an effort to understand the phenomenon of coarsening. However, a complete understanding of the role of the multiple factors that can affect the coarsening kinetics in such systems is still lacking. Although some advances have been made through computer simulations, studying experimentally the influence of the volume fraction of the second phase and that of coherency strains on the kinetics of coarsening has been particularly challenging.This paper will highlight some of the issues that are relevant to the study of coarsening in multi-component alloy systems. Recent results obtained for the kinetics of coarsening of γ' precipitates in different alloys within the Ni-Al-Si system will be presented. Compositions of these alloys have been chosen so as to vary both the magnitude and the sign of coherency strains between the precipitate and the matrix. Some anomalies related to the composition dependence of the kinetics of coarsening will be highlighted. This paper will conclude with a discussion on the role of the volume fraction of the second phase and coherency strains in contributing to these anomalies and identify directions for future work.     

Coarsening dynamics of phase-separating systems

When a system such as a binary liquid is cooled rapidly from a homogeneous phase into a two-phase region, domains of the two equilibrium phases form and grow ('coarsen') with time. In the absence of an external drive, such as gravity or an imposed shear flow, a dynamical-scaling regime emerges in which the domain morphology is statistically self-similar at different times, up to an overall length-scale (coarsening scale) that grows with time. In the first part of the paper, the scaling phenomenology will be reviewed and the time-dependence of the coarsening scale will be discussed in the context of a number of different physical systems and scaling regimes. In the second part, the influence of an external drive, in particular a shear flow, will be addressed and recent developments reviewed. Interesting open questions include the late-time behaviour under shear and whether the coarsening continues indefinitely or is ultimately arrested by the shear flow.     

Coarsening kinetics of coherent γ′ precipitates in ternary Ni–based alloys: the Ni–Al–Si system

Coarsening of coherent precipitates γ’ in alloys such as Ni-based alloys has been studied extensively not only for its practical significance in the design of engineering alloys but also in an effort to understand the phe omenon of coarsening. However, a complete understanding of the role of the multiple factors that can affect the coarsening kinetics in such systems i still lacking. Although some advances have been made through computer simulations, studying experimentally the influence o the volume fraction of the second phase and that of coherency strains on the kinetics of coarsening has been particularly challenging.

This paper will highlight some of the issues that are relevant to the study of coarsening in multi–component alloy systems. Recent results obtained for the kinetics of coarsening of γ’ precipitates in different alloys within the Ni-Al-Si system will be presented. Compositions of these alloys have been chosen so as to vary both the magnitude and the sign of coherency strains between the precipitate and the matrix. Some anomalies related to the composition dependence of the kinetics of coarsening will be highlighted. This paper will conclude with a discussion on the role of the volume fraction of the second phase and coherency strains in contributing to these anomalies and identify directions for future work.     

Model for growth and coarsening of two phase systems under diffusional control

Abstract

A model is described which is capable of simulating the two-dimensional evolution of microstructure in two phase systems undergoing diffusion controlled growth and surface tension driven coarsening. To solve the diffusion equation in the matrix phase, an integral equation method is employed. Thus, although it is necessary to describe the shapes of the second phase particles using a number of elements, it is not necessary to discretise the matrix phase as the particles evolve. This allows the computation times to be kept within reasonable limits. The boundary condition at the interface and the variation of the interfacial concentration with interface curvature are accounted for in a rigorous fashion. It is shown that the method can handle the ‘soft’ impingement of overlapping diffusion fields. A treatment of the ‘hard'’ physical impingement of particles is developed. To demonstrate the accuracy and stability of the method, the results from the model are compared with the exact solution for a spherical particle growing in a circular domain; it is shown that the agreement is reasonable. The results from a number of example computations are presented, which include (a) growth of a single particle in a finite domain, (b) soft and hard impingement of two particles in a finite domain, (c) coarsening of a significant number of particles at constant volume fraction, and (d) simultaneous nucleation, growth, and coarsening of second phase particles. Where appropriate, the results are compared with those from other models which have been published in the literature. The advantages and disadvantages of the present model are discussed.     

Vertically stratified flows in microchannels. Computational simulations and applications to solvent extraction and ion exchange

In this paper, we describe the conditions under which two immiscible fluids flow atop one another (viewed perpendicular to the plane on which the channel is inscribed) in a shallow microfluidic channel. First, we predict the behavior of a two-phase system using fluid dynamic simulations with water-butanol and water-chloroform as model systems. We numerically model the effect of various physical parameters, such as interfacial surface tension, density, viscosity, wall contact angle, and flow velocity on the type of flow observed and find that interfacial surface tension and viscosity are the parameters responsible for formation of vertically stratified, side-by-side, or segmented flows. As predicted by numerical simulations, a water-chloroform system never assumes a vertically stratified configuration, but a water-butanol system does when the two liquids flow at sufficiently high flow velocities. In actual experiments, we test conditions under which potentially useful two-phase systems form stable vertically stratified flows. We also demonstrate that compared to side-by-side flow schemes, shorter diffusion paths are achievable, and thus, the system can be used at higher flow rates to obtain the same performance. We then apply such findings to practical analytical problems, such as solvent extraction and ion exchange.     

Parallel refinement and coarsening of tetrahedral meshes

This paper presents a parallel adaptation procedure (coarsening and refinement) for tetrahedral meshes in a distributed environment. Coarsening relies upon an edge collapsing tool. Refinement uses edge‐based subdivision templates. Mesh optimization maintains the quality of the adapted meshes. Focus is given to the parallelization of the various components. Scalability requires repartitioning of the mesh before applying either coarsening or refinement. Relatively good speed‐ups have been obtained for all phases of the proposed adaptation scheme. Copyright © 1999 John Wiley & Sons, Ltd.     

时效组织演化的计算机模拟理论与模型

总结了时效组织的计算机模拟研究,基于Cahm-Hillard的非经典转变理论,提出一种新的动力学模型,不同于以往的连续介质模型,为离散格点形式,可租用于时效过程的所有阶段,包括形核、长大、粗化等,模型还可描述原子有序化和相分离,并考虑到弹性能对形貌演化和粗化过程的影响,且体积分数的影响、沉淀颗粒间的相互作用也在方程中自动体现。     

Effect of interfacial reaction rate on the morphogenesis of nanostructured coatings in a simulated electrodeposition process

Brownian dynamics simulations (BDSs) are performed to investigate the influence of interfacial electrochemical reaction rate on the evolution of coating morphology on circular fibres. The boundary condition for the fluid phase concentration, representing the balance between the rates of interfacial reaction and transport of ions by bulk diffusion, is incorporated into the BDS by using a reaction probability, P(s). Different modes of growth, ranging from diffusion limited ([Formula: see text]) to reaction controlled [Formula: see text], are studied. It is found that, consistent with experimental observations, two distinct morphological regimes exist, with a dense and uniform structure for [Formula: see text] (reaction limited deposition (RLD)) and an open and porous one as [Formula: see text] (diffusion limited deposition (DLD)). An analysis of the fractal dimension indicates that this morphological transition occurs at P(s)≈0.3. Long-time power-law scalings for the evolution of thickness [Formula: see text] and roughness (ξ) of the coating exist, i.e.?[Formula: see text] with 0.86≤α≤0.91 and 0.56≤β≤0.93 for 0.01≤P(s)≤1. These values are different from those reported for sequential, pseudo-time lattice simulations on planar surfaces, signifying the importance of multiparticle dynamics and surface curvature. The internal structure and porosity of the coating are characterized quantitatively by the radial density profile, pair correlation function, two-point probability function, void distribution function and pore area distribution. For RLD the radial density, ρ(n), remains nearly constant, while for DLD ρ(n) follows a power law, [Formula: see text]. The coating exhibits short ranged order in the RLD regime while a long range order is created by DLD. The void distribution function becomes broader with increasing P(s), indicating that in the RLD regime the coating consists of small and spherical pores, while in the DLD regime large and elongated pores are obtained. The pore area distribution shows narrower distributions in DLD for small pores, while the area of the largest pore increases by nearly three orders of magnitude as one moves from the RLD to the DLD regime. Such morphological diversity could be potentially exploited for applications such as percolation, catalysis and surface protection.     

Microstructural evolution in two-phase solids: morphological stability of near-periodic particle arrangements

Microstructural evolution in binary alloys is studied numerically using a sharp interface model and boundary integral techniques together with a time stepping scheme. Two-dimensional matrix-inclusion type microstructures are considered with an isotropic interfacial energy density and cubically orthotropic elasticity, being distinct in the two phases. An assumption of local equilibrium is applied which provides the coupling of the diffusion and the elasticity problems via an interfacial jump condition. Special attention is paid to the verification of a phenomenon observed in an earlier work: energy considerations suggested the possibility of sustained inverse coarsening leading to equilibrium microstructures with large numbers of small particles. It is found that, while such stationary states exist, their domain of attraction is rather small; reduction of surface area is the dominant driving force in the process.     

Computer simulation of grain growth by grain boundary migration during liquid phase sintering

From many experiments with mixtures of small and large particles, it can be concluded that during liquid phase sintering, smaller particles partially dissolve and a solid phase precipitates on the larger particles. Therefore, the number of smaller particles decreases due to coarsening. The growth rate can be controlled either by the solid-liquid phase boundary reaction or by diffusion through the liquid phase. This dissolution-reprecipitation process leads to further densification by rearrangement of smaller and larger particles. The microstructure may change either by larger particles growing during the Ostwald ripening process or by shape accommodation. In this study, two-dimensional simulation of grain growth by grain boundary migration based on such a physical and corresponding numerical modeling of liquid phase sintering was considered. The simulation method developed is based on the defined submodels for model system definition, for solution-precipitation, and for grain coarsening process.