Phase-Field Modelling in Extractive Metallurgy

The phase-field method has already proven its usefulness to simulate microstructural evolution for several applications, e.g., during solidification, solid-state phase transformations, fracture, etc. This wide variety of applications follows from its diffuse-interface approach. Moreover, it is straightforward to take different driving forces into account. The purpose of this paper is to give an introduction to the phase-field modelling technique with particular attention for models describing phenomena important in extractive metallurgy. The concept of diffuse interfaces, the phase-field variables, the thermodynamic driving force for microstructure evolution and the phase-field equations are discussed. Some of the possibilities to solve the equations describing microstructural evolution are also described, followed by possibilities to make the phase-field models quantitative and the phase-field modelling of the microstructural phenomena important in extractive metallurgy, i.e., multiphase field models. Finally, this paper illustrates how the phase-field method can be applied to simulate several processes taking place in extractive metallurgy and how the models can contribute to the further development or improvement of these processes.     

Investigation of Al-Cu-Ni alloy solidification: thermodynamics,experiments and phase-field modeling

The investigation of solidification in ternary Al-Cu-Ni alloys is carried out by means of experiments and phase-field modeling. For three alloys in the Al-rich corner of the phase diagram differential thermal analysis (DTA) is performed. Then the alloys were analyzed using scanning electron microscopy with energy dispersive X-ray microanalysis. For the understanding of the general features of the alloy solidification quantitative phase-field simulations are carried out additionally to the theoretical Scheil calculation. It is found that many experimental DTA signals and the microstructure parts cannot be explained by simple Scheil calculation. We apply the multi-phase-field model previously developed for the simulation of peritectic reaction and extended it to three components and four-phases reactions. The main advantage of this model is the application of the equilibrium parameters evaluated from the refined free energies of the phases. It is shown that the simulated microstructure is comparable to the experimental one for two investigated alloys. The final phase fractions in the modeling correspond to the theoretical predictions of Scheil calculation but the time evolution of fractions is more complicated. In particular the kinetics (relation between tangential and normal growth velocities) of the peritectic-like reaction in ternary Al-Cu-Ni alloys shows differences compared to the peritectic reaction in binary Al-Ni alloys.     

Bridging the phase-field and phase-field crystal approaches for anisotropic material systems

In this paper the amplitude representation of the anisotropic phase-field crystal (APFC) model recently proposed as a generalized model for isotropic as well as anisotropic crystal lattice systems is developed. The relationship between the amplitude equations and the standard phase-field model for solidification of pure substances with elasticity effects is derived which provide an explicit connection between the phase-field and APFC models. On the one hand we present a computationally more efficient model and highlight its potential as a bridge between the PFC and phase-field models with anisotropic interface energies and kinetics on the other hand.     

Phase-field simulation of dendritic growth in a binary alloy with thermodynamics data

This paper simulates the dendrite growth process during non-isothermal solidification in the Al-Cu binary alloy by using the phase-field model. The heat transfer equation is solved simultaneously. The thermodynamic and kinetic parameters are directly obtained from existing database by using the Calculation of Phase Diagram （CALPHAD） method. The effects of the latent heat and undercooling on the dendrite growth, solute and temperature profile during the solidification of binary alloy are investigated. The results indicate that the dendrite growing morphologies could be simulated realistically by linking the phase-field method to CALPHAD. The secondary arms of solidification dendritic are better developed with the increase of undercooling. Correspondingly, the tip speed and the solute segregation in solid-liquid interface increase, but the tip radius decreases.     

Coupling of the phase-field and CALPHAD methods for predicting multicomponent,solid-state phase transformations

The development of an effective microstructure design method for multicomponent alloys is of considerable importance for improving both the design of alloys and the design of processes for producing alloys with unique properties. The coupling of the phase-field method and the calculation of phase diagrams (CALPHAD) method can be used for predicting the evolution of microstructures in multicomponent alloys. Such predictions make use of CALPHAD thermodynamic information with the chemical free energy function in the phase-field method. This article reviews several of these coupling methods, focusing on solid-state phase transformations in multicomponent systems, such as phase separation and disordered or ordered phase precipitation from a matrix. When calculating disordered phase transformations, the Gibbs energy function derived from the CALPHAD database can be used directly in the phase-field method. On the other hand, when dealing with an order/disorder transition, the degrees of freedom of the element site fraction for an ordered phase in the CALPHAD method can be reduced using the Gibbs energy single formalism for constituent phases, by using a database that stores the Gibbs energy and chemical equilibrium conditions, or by obtaining the driving force calculated using the Thermo-Calc software. The current status and future directions for further development of these coupled methods are discussed.     

纳米尺度下Sm-Co合金体系中相组成与相稳定性的研究

建立了纳米晶合金相的热力学模型,可定量描述纳米尺度下合金体系中化合物相的热力学性质,并预测合金相的稳定性及其转变规律.利用该模型全面计算了纳米晶Sm-Co合金体系中各化合物相在不同晶粒尺寸下的摩尔吉布斯自由能随温度的变化关系,预测了纳米尺度下Sm-Co合金体系中各物相的相对稳定性及转变规律.模型预测结果示出,在室温附近,随着纳米晶粒尺寸的减小,某些纳米晶合金相的摩尔吉布斯自由能将由负值变为正值,预示着将向其他更稳定的纳米晶合金相转变,这是与传统粗晶材料中合金相的稳定性仅依赖于温度条件而完全不同的纳米晶合金 关键词： 纳米晶材料热力学 Sm-Co合金 相稳定性 相变     

An analytical study of the static state of multi-junctions in a multi-phase field model

We investigate the properties of the multi-order parameter phase field model of Steinbach and Pezzolla [I. Steinbach, F. Pezzolla, A generalized field method for multi-phase transformations using interface fields, Physica D 134 (1999) 385–393] with respect to the behavior in triple and higher order junctions. From the structure of this model, it was speculated that “dynamical” solutions may exist in the triple junction, which could lead to a violation of Young’s law. Here we confirm analytically recent numerical simulations showing that such dynamical states do not exist, and that an equilibrium solution therefore does indeed correspond to a minimum of the free energy; this implies that Young’s law must be satisfied in the framework of the model. We show that Young’s law is a consequence of the interface kinetic equilibrium and not due to a mechanical force balance, in agreement with earlier predictions [C. Caroli, C. Misbah, On static and dynamical Young’s condition at a trijunction, J. Phys. I France 7 (1997) 1259–1265].     

二元合金多晶粒的枝晶生长的等温相场模型

基于Ginzburg-Landau理论和单晶粒的枝晶生长模型,发展了一个单相场控制的多个晶粒的枝晶生长模型. 采用相场和溶质场耦合的方法,以Al-2%Cu合金为例模拟了二元合金等温凝固过程中多个晶粒的生长过程. 结果表明,这个模型的计算结果展现了多个晶粒枝晶的竞争生长,能较真实的再现凝固过程中的枝晶的生长过程. 关键词： 相场法 多晶粒 等温凝固 二元合金     

二元合金多晶粒的枝晶生长的等温相场模型

基于Ginzburg-Landau理论和单晶粒的枝晶生长模型，发展了一个单相场控制的多个晶粒的枝晶生长模型. 采用相场和溶质场耦合的方法，以Al-2%Cu合金为例模拟了二元合金等温凝固过程中多个晶粒的生长过程. 结果表明，这个模型的计算结果展现了多个晶粒枝晶的竞争生长，能较真实的再现凝固过程中的枝晶的生长过程.     

相场法模拟多元合金过冷熔体中的枝晶生长

在二元合金相场模型研究的基础上，进行扩展获得了多元合金相场模型.以Al-Si-Mg三元合金为例，采用该相场模型实现了逼真地模拟多元合金凝固过程的等轴枝晶生长，得到了二次或更高次晶臂生长等复杂的枝晶形貌.随着第三组元Mg含量的减少，枝晶的二次枝晶越发达，枝晶中溶质的偏析越严重，枝晶尖端的生长速率和半径越大，与丁二腈-丙酮体系中枝晶尖端生长速率、半径随溶质浓度变化关系的理论计算和实验结果相符合.另外，枝晶初生晶臂中心的溶质浓度最低，在被二次晶臂包围的界面区域的溶质浓度最高；固液界面区域具有较大的浓度梯度，其中枝晶尖端前沿的梯度最大. 关键词： 相场法 多元合金 凝固过程 枝晶生长     

Phase-field simulation of a Fe-Mn alloy under forced flow conditions

In this work a study through numerical simulation of dendritic growth for the system Fe-Mn under the influence of a forced flow field is presented. The investigations are based on an extension of the quantitative phase-field approach developed by Echebarria et. al. Phy. Rev. E 061604 (2004), to simulate the solidification of Fe-Mn under the influence of a forced flow field. The simulations are performed for isothermal conditions and the investigation concentrates on the effects of forced flow on the dendrite morphology during the growth dynamics. The effects of forced flow on microsegregation are also discussed.     

A thermal conductivity model for nanofluids including effect of the temperature-dependent interfacial layer

The interfacial layer of nanoparticles has been recently shown to have an effect on the thermal conductivity of nanofluids. There is, however, still no thermal conductivity model that includes the effects of temperature and nanoparticle size variations on the thickness and consequently on the thermal conductivity of the interfacial layer. In the present work, the stationary model developed by Leong et al. (J Nanopart Res 8:245–254, 2006) is initially modified to include the thermal dispersion effect due to the Brownian motion of nanoparticles. This model is called the ‘Leong et al.’s dynamic model’. However, the Leong et al.’s dynamic model over-predicts the thermal conductivity of nanofluids in the case of the flowing fluid. This suggests that the enhancement in the thermal conductivity of the flowing nanofluids due to the increase in temperature does not come from the thermal dispersion effect. It is more likely that the enhancement in heat transfer of the flowing nanofluids comes from the temperature-dependent interfacial layer effect. Therefore, the Leong et al.’s stationary model is again modified to include the effect of temperature variation on the thermal conductivity of the interfacial layer for different sizes of nanoparticles. This present model is then evaluated and compared with the other thermal conductivity models for the turbulent convective heat transfer in nanofluids along a uniformly heated tube. The results show that the present model is more general than the other models in the sense that it can predict both the temperature and the volume fraction dependence of the thermal conductivity of nanofluids for both non-flowing and flowing fluids. Also, it is found to be more accurate than the other models due to the inclusion of the effect of the temperature-dependent interfacial layer. In conclusion, the present model can accurately predict the changes in thermal conductivity of nanofluids due to the changes in volume fraction and temperature for various nanoparticle sizes.     

A Lagrangian description of transport associated with a front-eddy interaction: Application to data from the North-Western Mediterranean Sea

We discuss the Lagrangian transport in a time-dependent oceanic system involving a Lagrangian barrier associated with a salinity front which interacts intermittently with a set of Lagrangian eddies — ‘leaky’ coherent structures that entrain and detrain fluid as they move. A theoretical framework, rooted in the dynamical systems theory, is developed in order to describe and analyse this situation. We show that such an analysis can be successfully applied to a realistic ocean model. Here, we use the output of the numerical ocean model DieCAST from Dietrich et al. (2004) [17] and Fernández et al. (2005) [18] studied earlier in Mancho et al. (2008) [15] where a Lagrangian barrier associated with the North Balearic Front in the North-Western Mediterranean Sea was identified. The numerical model provides an Eulerian view of the flow and we employ the dynamical systems approach to identify relevant hyperbolic trajectories and their stable and unstable manifolds. These manifolds are used to understand the Lagrangian geometry of the evolving front-eddy system. Transport in this system is effected by the turnstile mechanism whose spatio-temporal geometry reveals intermittent pathways along which transport occurs. Particular attention is paid to the ‘Lagrangian’ interactions between the front and the eddies, and to transport implications associated with the transition between the one-eddy and two-eddy situation. The analysis of this ‘Lagrangian’ transition is aided by a local kinematic model that provides insight into the nature of the change in hyperbolic trajectories and their stable and unstable manifolds associated with the ‘birth’ and ‘death’ of leaky Lagrangian eddies.     

Green functions and Langevin equations for nonlinear diffusion equations: A comment on ‘Markov processes, Hurst exponents, and nonlinear diffusion equations’ by Bassler et al.

We discuss two central claims made in the study by Bassler et al. [K.E. Bassler, G.H. Gunaratne, J.L. McCauley, Physica A 369 (2006) 343]. Bassler et al. claimed that Green functions and Langevin equations cannot be defined for nonlinear diffusion equations. In addition, they claimed that nonlinear diffusion equations are linear partial differential equations disguised as nonlinear ones. We review bottom-up and top-down approaches that have been used in the literature to derive Green functions for nonlinear diffusion equations and, in doing so, show that the first claim needs to be revised. We show that the second claim as well needs to be revised. To this end, we point out similarities and differences between non-autonomous linear Fokker-Planck equations and autonomous nonlinear Fokker-Planck equations. In this context, we raise the question whether Bassler et al.’s approach to financial markets is physically plausible because it necessitates the introduction of external traders and causes. Such external entities can easily be eliminated when taking self-organization principles and concepts of nonextensive thermostatistics into account and modeling financial processes by means of nonlinear Fokker-Planck equations.     

Periodic entanglement of magnons in an antiferromagnet

The dynamics of entanglement for two-mode magnons in an antiferromagnet is investigated according to the entanglement criterion proposed by Duan et al. [L.M. Duan, G. Giedke, J.I. Cirac, P. Zoller, Phys. Rev. Lett. 84 (2000) 2722]. It is shown that entanglement between the two modes of magnons can be generated and occurs periodically with time.     

Effect of the experimental parameters on the number of free electrons in the drift region of a wire-cylinder electrostatic precipitator

The aim of this paper is to highlight the number evolution of free electrons in the drift region of a wire-cylinder electrostatic precipitator in negative voltage depends on the experimental parameters, more particularly of gas composition. A numerical model of the negative DC corona discharge developed by Chen et al. was used and modified to investigate the negative discharge corona for different gases. A parametric study was conducted to examine the effect on the electron distribution of operating conditions. The results showed the electron concentration increases with temperature, decreases when the pressure increases, and is closely related to gas composition.     

The Order of Curvature Operators on Loop Groups

For loop groups (free and based), we compute the exact order of the curvature operator of the Levi-Civita connection depending on a Sobolev space parameter. This extends results of Freed (J Differ Geom 28:223–276, 1988) and Maeda et al. (Riemannian geometry on loop spaces. arXiv:0705.1008v3, 2008).     

二元合金等温凝固过程的相场模型

基于Ginzberg-Landau理论，发展了一个与WBM模型和KKS模型一致的新相场模型.并利用该相场模型与溶质场耦合计算，以Al-65%Cu合金为例模拟了不同过冷度条件下，二元合金凝固过程的等轴枝晶生长过程.研究过冷度对二元合金等温凝固过程的等轴枝晶生长以及溶质场分布的影响.结果表明：随着过冷度的增大，枝晶的二次枝晶更加发达，浓度Peclet数和枝晶尖端的生长速率增大，而枝晶尖端的曲率半径减小，枝晶前沿的溶质富集现象也更严重；另外，计算结果与Ivantsov理论符合较好. 关键词： 相场法 枝晶生长 溶质场 Ivantsov理论     

On the equilibria of the MAPK cascade: Cooperativity, modularity and bistability

In this paper we present a discussion of a phenomenological model of the MAPK cascade which was originally proposed by Angeli et al. [D. Angeli, J.E. Ferrell, Jr., E.D. Sontag, PNAS 101 (2004), 1822]. The model and its solution are extended in several respects: (a) an analytical solution is given for the cascade equilibria, exploiting a parameter-based symmetry of the rate equations; (b) we discuss the cooperativity (Hill coefficients) of the cascade and show that a feedforward loop within the cascade increases its cooperativity. The relevance of this result for the notion of modularity is discussed; (c) the feedback model for cascade bistability by Angeli et al. is reconsidered. We argue that care must be taken in modeling the interactions and a biologically realistic phenomenological model cannot be too reductionist. The inclusion of a time-dependent degradation rate is needed to account for a switching of the cascade.     

Spectral variations in the optical transition matrix element and their impact on the optical properties associated with hydrogenated amorphous silicon

Using an empirical model for the density of states functions associated with hydrogenated amorphous silicon, in conjunction with an elementary model for the optical transition matrix elements, we aim to explore how variations in the matrix elements impact upon the spectral dependence of the optical properties associated with this material. We also wish to ascertain as to whether or not the hydrogenated amorphous silicon mobility gap result suggested by Jackson et al. [W.B. Jackson, S.M. Kelso, C.C. Tsai, J.W. Allen, S.-J. Oh, Phys. Rev. B 31 (1985) 5187] is consistent with the results of the experiment. We find that the mobility gap value suggested by Jackson et al. is too large. An upper bound on the mobility gap associated with hydrogenated amorphous silicon of 1.68 eV is suggested instead. Electrical measurements performed on undoped hydrogenated amorphous silicon yield a mobility gap value that is consistent with this bound.