A phase-field model coupled with a thermodynamic database

A phase-field model for the solidification of multi-component alloys, capable of being integrated with thermodynamic databases, has been developed. The solid–liquid interface is modelled as a mixture of solid and liquid phases. The solute concentration of the solid phase depends only on the local temperature while the solute concentration of the liquid phase is affected by solute rejection (or absorption) from the solid being formed together with the extent of any chemical diffusion in the liquid. The governing equations for the phase-field model have been derived in a thermodynamically consistent way such that the parameters in these equations are fully determined. By linking directly with the thermodynamic database MTDATA, the model is capable of simulating the microstructural evolution of real alloys. Numerical calculations for aluminium–silicon alloys show that the phase-field driving force changes in a similar manner to the thermodynamic driving force, although the phase-field driving force changes more quickly than the thermodynamic driving force when the system approaches equilibrium. The phase-field mobility is shown not to be a trivial function of constitutional undercooling. With the new phase-field model, the width of the liquid–solid interface region is allowed a much large value with the continuity of all parameters in the phase-field model and properties of interface maintained, and hence the model is suitable for simulating large-scale systems.     

Calculated phase diagrams of aluminum alloys from binary Al–Cu to multicomponent commercial alloys

More than three decades have passed since the publication of Alan Prince’s book on multicomponent phase equilibria. The most significant development in this time has been the use of a combined computational/experimental approach to calculate multicomponent phase diagrams. This has led to important advances in the design and processing of structural and functional materials for practical applications. In this paper, we present a few examples focusing on aluminum alloys from the classical Al–Cu binary to multicomponent alloys with a view toward practical applications.     

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.     

金属熔体凝固微观组织模拟的相场方法研究

全面阐述了现有的金属熔体凝固微观组织模拟的相场模型:纯物质、二元合金理想溶液或正规溶液和稀溶液相场模型,指出构造适合于具有复杂溶液模型的金属熔体凝固微观组织模拟相场模型的必要性.提出采用相图计算热力学模型构造相场模型中的体系自由能,并对相场参数做出相应修正,建立了能对具有复杂溶液模型的金属熔体凝固微观组织进行准确模拟的相场模型.利用该相场模型对Ti-Al金属熔体中晶体的自由生长和定向生长进行了模拟,得到许多与试验观察和理论分析相一致的结果.     

The phase-field approach and solute drag modeling of the transition to massive γ → α transformation in binary Fe-C alloys

The transition between diffusion controlled and massive transformation γ →α in Fe–C alloys is investigated by means of phase-field simulations and thermodynamic functions assessed by the Calphad technique as well as diffusional mobilities available in the literature. A gradual variation in properties over an incoherent interface, having a thickness around 1 nm, is assumed. The phase-field simulations are compared with a newly developed technique to model solute drag during phase transformations. Both approaches show qualitatively the same behavior and predict a transition to a massive transformation at a critical temperature below the T₀ line and close to the α/α + γ phase boundary. It is concluded that the quantitative difference between the two predictions stems from different assumptions on how the properties vary across the phase interface yielding a lower dissipation of Gibbs energy by diffusion in the phase-field simulations. The need for more detailed information about the actual variation in interfacial properties is emphasized.     

THEORETICAL PREDICTION OF PROEUTECTOID FERRITIC TRANSFORMATION IN HYPO-PROEUTECTOID STRUCTRAL STEELS

1.IntroductionInordertodevelopnewtypeofsteels,itisveryimportanttoderiveamuchgeneralruleoftheaustenitedecompositionfromexperimentaldataofknownsteels.Recently)someworkhasmadeencouragingprogressinthisfieldtl--6].Butdifficultiesandsuspicionsarestillobyiousbecausethethermodynamicaldataofmulticomponelltsteelarescatteredanddisordered.PuremathematicalsimulationmaybeusedtocalculateTTTandCCTc...es[4],butithasapremisethatsomespecialpointsofTTTcurvessuchasthenosepointmustbeknowninadvance,whichprevents…     

Effect of the Compositional Strain on the Diffusive Interface Thickness and on the Phase Transformation in a Phase-Field Model for Binary Alloys

A Cahn-Hilliard phase-field—elasticity model was used to study the effect of compositional strain on the diffusive interface thickness and on the solid state phase transformations in binary alloys. Compositional strain was introduced using the Vegard’s law. Mixed order finite element analyses and analytical solutions of an infinite diffusion couple with a flat interface were used to track the phase-field interface morphology. Both analytical and numerical calculations showed a substantial rate-increasing effect of compositional strain on the interface thickness, especially for low energy barrier values. Compositional strain was found to cause substantial patterning of single precipitates during their evolution in a parent matrix and significantly change the equilibrium size of the precipitates. Results show a considerable influence of compositional strain on the coarsening kinetics of coherent precipitates.     

U-Nb合金低温分解机制的相场法研究

低温时效处理导致U-Nb合金的力学性能发生显著改变,利用相场法研究了性能变化对应的微观组织演化。结果表明,在较高温度可能发生调幅分解,但必须满足其热力学条件,而且也仅出现在初期。发生普通型析出的范围可能更广,因为合金成分接近于发生结构转变的临界成分,时效对应的母相无论是类高温立方相还是类低温正交相,稳定核胚和母相之间的Nb含量差异都可以相当小。当相界面的迁移率较大而扩散率较低时,处于长大阶段的析出物的Nb含量明显不同于平衡态成分。因此,U-Nb合金的低温时效硬化效应可能是由相分解引起的Nb再分配所导致。     

用Wilson方程估算多元液态合金在给定温度下的组元活度

根据Ｗｉｌｓｏｎ参数与温度的关系，可将Ｗｉｌｓｏｎ方程扩展到只利用不同温度下测定的二元液太民合金各组元的活度，估算出给定温度下二元和多元液态合金各组元的活度，研究结果表明，多元液太民合金的活度算值与其实验数据和符合良好。     

Solidification microstructures: recent developments,future directions

The status of solidification science is critically evaluated and future directions of research in this technologically important area are proposed. The most important advances in solidification science and technology of the last decade are discussed: interface dynamics, phase selection, microstructure selection, peritectic growth, convection effects, multicomponent alloys, and numerical techniques. It is shown how the advent of new mathematical techniques (especially phase-field and cellular automata models) coupled with powerful computers now allows the following: modeling of complicated interface morphologies, taking into account not only steady state but also non-steady state phenomena; considering real alloys consisting of many elements through on-line use of large thermodynamic data banks; and taking into account natural and forced convection effects. A series of open questions and future prospects are also given. It is hoped that the reader is encouraged to explore this important and highly interesting field and to add her/his contributions to an ever better understanding and modeling of microstructure development.     

Phase-Field Modeling of Austenite-to-Ferrite Transformation in Fe-C-Mn Ternary AlloysEI北大核心CSCD

The effect of Mn on the austenite-to-ferrite transformation has been widely studied by both physical models and experiments due to its technological importance for alloy design in steel industries. In recent years, an increasing interest of this issue is moved onto the effect of alloying element on the migrating interface during the austenite-to-ferrite transformation. For ternary Fe-C-Mn alloys, the interfacial condition is more complicated than that of binary Fe-C alloys in view of the large difference in the diffusivity between the interstitial and substitutional alloying elements. Generally speaking, there are two main concepts, i.e. the paraequilibrium model and the local-equilibrium model, which have been proposed to describe the phase transformation kinetics in ternary Fe-C-Mn alloys based on different assumptions about the diffusion of the substitutional elements. And many modeling attempts have been made to study the effect of Mn on the migration kinetics by using these theories. In this work, a multi-phase-field (MPF) model coupling with a Gibbs-energy dissipation model was developed to simulate the isothermal austenite-to-ferrite transformation in ternary Fe-C-Mn alloys. This model has considered the Mn diffusion inside the migrating interface in a physical manner and takes its effect on the transformation kinetics into account. Comparison simulations were made to analyze the difference in the transformation kinetics and ferrite morphologies with and without considering the energy dissipation at the moving interface. It shows that the incomplete transformation phenomenon does occur due to the Mn diffusion inside interface. The modified MPF model was then used to study the effect of Mn contents on the microstructures and kinetics of the phase transformations. It is found that the ferrite growth along the austenite/austenite boundaries is faster than that in the perpendicular direction. This difference is intensified with increasing the Mn concentration, which hence leads to the ferrite morphology changed from elliptical to flat alike. It also produces a slower transformation kinetics and a larger degree of the incomplete transformation when increasing the Mn concentration.     

On dendritic solidification of multicomponent alloys with unequal liquid diffusion coefficients

A model has been developed previously by Rappaz and Thévoz (Acta metall., 1987, 35, 1478; 1987, 35, 2929) for the solidification of binary alloys having an equiaxed dendritic morphology. The extension of this model to multicomponent alloys is straightforward if the diffusion coefficients in the liquid of the various solute elements are equal. When they are different however, it becomes necessary to distinguish the concentrations at the dendrite tip position, governing the growth kinetics, from the average concentrations within the interdendritic liquid region. A model is presented which couples the dendrite growth kinetics with an overall solute balance performed at the scale of the grain. The diffusion profile outside the grain envelope is calculated using a Landau transformation and an implicit scheme. The calculated solidification paths during and after recalescence are compared with the Scheil approximation for various conditions. Additionally, a model predicting the secondary dendrite arm spacing in multicomponent alloys is briefly described.     

Diffusion processes in multicomponent nickel-base superalloy-nickel system

Optical and scanning electron microscopy, as well as electron microprobe analysis and electron backscatter diffraction, have been used to study diffusion processes that occur in a diffusion pair that consistsof a single-crystal CMSX-10 nickel-base superalloy and polycrystalline nickel, at temperatures of 1050–1250°C. It has been found that, in this system, the distributions of γ-stabilizing elements (Cr, Co, W, and Re) are described by the Boltzmann solution for diffusion between two semiinfinite plates of a binary alloy. The processing of these distributions has shown that the diffusion coefficients of Cr, Co, W, and Re in the multicomponent system are close to those in binary alloys of these elements with Ni. The diffusion redistribution of the elements leads to the dissolution of the γ′ phase in the nickel-base superalloy, growth of nickel grains toward the superalloy constituent of the diffusion pair, and the formation of porosity on both sides of the migrating interface, which is determined from a crystal misorientation of the alloy single crystal and nickel grains.     

A phase field model for isothermal solidification of multicomponent alloys

With the ability to model the kinetics and the pattern formation for solidification, a phase field model has been studied by many scientists. Currently available models, however, are restricted not only to binary alloys but also to those with substitutional solute elements. In this work, a new phase field model is developed to study solidification of a multicomponent alloy containing substitutional as well as interstitial solute elements. By employing the number of moles per unit volume as the concentration variable, the evolution equations of both the phase field and the concentration fields are derived from the free energy functional in the thermodynamically consistent way. In the model, the interfacial region is assumed to be a mixture of solid and liquid with the same composition, but with different chemical potentials. Based on this assumption, the phase field parameters are matched to the alloy properties and an interface thickness limitation is also deduced. Using the chemical rate theory, the phase field mobility is determined at a thin-interface limit condition under the assumption of negligible diffusivity in the solid phase. Another advantage of the model is that any thermodynamic database available in the literature can be directly ported to the model such that quantitative results for solidification of the real alloy systems could be made. As an example, a dendritic growth in an Fe-Mn-C ternary alloy is examined with the thermodynamic data from the commercial software Thermo-Calc code.     

相场法对比模拟二元合金等温/非等温凝固过程(英文)

基于熵函数建立二元合金的二维相场模型,采用基于均匀网格的有限差分法求解相场和溶质场控制方程;为了避免时间步长的限定,采用交替隐式差分法(ADI)求解温度场控制方程。对Ni-Cu合金非等温凝固过程的部分特征进行模拟研究,对比分析二元合金等温/非等温凝固过程。模拟结果表明:非等温模型更能有效地模拟二元合金的实际凝固过程,并且随着热扩散系数的减小,非等温相场模型逐渐向等温相场模型回归。     

Phase-field simulations of α → γ precipitations and transition to massive transformation in the Ti–Al alloy

A phase-field model for the solid–solid α → γ transition of Ti–Al binary alloys is presented based on analytical Gibbs free energies and couplings to the thermodynamical database ThermoCalc. The equilibrium values recover the α + γ phase boundaries. Morphological transitions from diffusive to massive (partitionless) growth are observed on increasing the initial mole fraction of aluminum. Temporal evolution of the interface shows a $\sqrt{t}$ behavior for diffusive and a linear behavior for massive growth, which is in accordance with theoretical predictions. An estimate of the interfacial mobility of Ti–Al based on the Burke–Turnbull equation is calculated. The expression of the mobility follows an Arrhenius law. Using the derived interfacial mobility, the calculated interfacial velocities of the massive transformation are in quantitative agreement with those observed in experiments.     

A New Kinetic Mode During the Austenite-to-Ferrite Transformation in Fe–Mn and Fe–Mn–Mo Alloys

The kinetics of isothermal austenite(γ)-to-ferrite(α) transformation at various temperatures in Fe-2 Mn and Fe-2 Mn-0.5 Mo(wt%) alloys is investigated via dilatometry experiments and phase-field modeling.It was interestingly found that Mo addition has a marginal effect on the transformation kinetics.Besides the well-known partitioning and partitionless transformation modes,a new kinetic mode,in which interface migration is controlled by interfacial diffusion,was identified.The phase-field model with considering interfacial segregation could well predict the transformation kinetics and the kinetic mode transition.     

定向凝固微观组织的相场法模拟研究进展

定向凝固过程中的界面形态对凝固组织有着决定性作用，相场法可以很好地展示凝固过程界面形态的演化。阐述了二元合金定向凝固相场方法数值模拟的基本原理及国内外研究现状，以Ni-Cu二元合金为例，采用相场法模拟了定向凝固过程中界面形态的演化，并指出该领域目前所面临的问题及进一步的研究方向。     

A general method for calculating deviation from local equilibrium at phase interfaces

A general method to calculate the deviation from local equilibrium at phase interfaces in multicomponent systems is suggested. The deviation is caused by solute drag and finite interfacial mobility. In the limit of low transformation rates the new method degenerates to the well-known local equilibrium condition. The phase interface is divided into three zones, each with a finite thickness. In each zone a variation in thermodynamic properties and diffusional mobilities is assumed.