Stochastic modeling of columnar dendritic grain growth in weld pool of Al‐Cu alloy

A multi‐scale model is used to simulate columnar dendritic growth in TIG (tungsten inert‐gas) weld molten pool of Al‐Cu alloy. The grain morphologies at the edge of the weld pool are studied. The simulated results indicate that the average primary dendrite spacing changes during the solidification process in the weld pool because of the complicated thermal field, solute diffusion field and competitive growth. And it is shown that the secondary dendrite arms grow insufficiently in the space between dendrite trunks if the primary dendrite spacing is small. And the phenomenon has been explained by analyzing the influence of the solute accumulation on the constitutional undercooling and undercooling gradient when there are two different opposite solute diffusion fields. (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Phase‐field simulation of dendrite growth under forced flow conditions in an Al–Cu welding molten pool

A quantitative phase‐field model for directional solidification is applied to study dendrite growth under forced flow conditions in an Al‐Cu Gas Tungsten Arc (GTA) welding molten pool. Evolution of the dendrite morphology and the solute field under forced flow conditions is simulated. Growth of columnar grains goes through three periods, including the initial instability period, the competitive growth period and the relatively stable period. The solute segregation, the solute redistribution and the solute concentration in the liquid side of the interface are investigated, respectively. For the given conditions, simulation results are in good agreement with experimental findings.     

Columnar grain growth pattern with fluid flowing in molten pool

A coupled model was used to simulate columnar grain growth in TIG (tungsten inert‐gas) molten pool of nickel base alloy. The cellular automaton algorithm for dendritic growth is incorporated with solute transport model to take fluid flow into consideration. The results indicate that shear flow changes the solute distribution at the S/L (Solid/Liquid) interface, leading to asymmetrical growth of columnar grains. The dendrite arms on the upstream side grow fast, while the growth of dendrite arms on the downstream side is much delayed. However, dendrite arms on both sides are not as well‐developed as the grain growth without flow. With inlet flow velocity increasing, the phenomenon becomes more obvious. In addition, shear flow also results in more severe coring segregation. (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical description for the recalescence of bulk-undercooled Cu70Ni30 alloy

Assuming thermal balance and solute conservation, a numerical model has been proposed to describe the recalescence behavior of bulk-undercooled Cu–Ni melts. Applying a finite-difference scheme, the transformed solid fraction upon recalescence is given as a function of the liquid temperature, while the average liquid concentration can be tracked by calculation of the liquid/solid (L/S) Gibbs energy difference, in combination with a dendrite growth model. Accordingly, a transition from non-equilibrium to equilibrium process has been described from the evolution of L/S Gibbs energy difference. Applying the present model, the experimentally observed maximum recalescence temperature can be well predicted.     

Growth of equiaxed dendritic crystals settling in an undercooled melt,Part 1: Tip kinetics

Experiments are conducted to measure the dendrite tip growth velocities of equiaxed crystals of the transparent model alloy succinonitrile–acetone that are settling in an undercooled melt. The tip velocities are measured as a function of the crystal settling speed and the Eulerian angle between the dendrite arms and the flow direction relative to the crystal. The ratio of the settling speed (or flow velocity) to the tip growth velocity ranges from 62 to 572. The ratio of the measured tip velocity to that predicted from a standard diffusion theory for free dendritic growth ranges from almost zero for dendrite tips growing in the wake of the crystal, about unity for dendrite tips with an orientation close to normal to the flow direction, and up to two for dendrite tips growing into the flow. Despite the relatively strong flow relative to the crystal, the average tip growth velocity of the six primary dendrite arms of an equiaxed crystal is found to be in excellent agreement with the standard diffusion theory result. The individual tip velocities are correlated using a boundary layer model of free dendritic growth in the presence of melt flow that is modified to account for the flow angle dependence. Using the same dendrite tip selection parameter, σ^*, as established previously under purely diffusive conditions (0.02), good agreement is achieved between the measured and predicted tip velocities. The model is also found to predict well the variations in the tip velocity that occur during settling due to crystal rotation and settling speed changes.     

Phase field investigation on sidebranching dynamics in transient growth

A computationally efficient quantitative phase field formulation is used to investigate the sidebranching dynamics under different transient conditions in directional solidification for realistic parameters of a dilute alloy. The sidebranch growths where the pulling speed or temperature gradient increases with constant increasing rates are simulated and discussed by using the noise amplification theory. The results show that in transient growth, the tip shape can rapidly adjust with the instantaneous conditions, and the tip velocity changes continuously due to the change of tip undercooling. Therefore, under our given transient conditions, the sidebranch spacing or sidebranching frequency depends strongly on the transient conditions and transient history, which is different with that in steady‐state conditions where the sidebranching frequency is found to be independent of the temperature gradient and primary spacing, but varies as a power law of pulling speed.     

Radiative Heat Transfer in a Resistance Heated Floating Zone Furnace: A Numerical Study with FIDAPTM

This paper presents a numerical study of radiative heat transfer in a floating zone (FZ) furnace which was performed by using the commercial finite element program FIDAP^TM. This resistance furnace should provide a temperature higher than the melting temperature of silicon (i.e. T_max ≈ 1500 °C) and a variable temperature gradient at the liquid/solid interface (≥ 25 K/cm). Due to the high working temperatures, heat radiation plays the dominant role for the heat transfer in the furnace. For this reason, the quality of view factors used in the wall‐to‐wall model was carefully inspected with energy‐balance checks. A numerical model with two control parameters is applied to study the influence of material and geometrical parameters on the temperature field. In addition, this model allows us to estimate the internal thermal conditions which were used as thermal boundary conditions for partial 3D simulations. The influences of an optical lens system on the radial symmetry of the temperature field were examined with these partial 3D simulations. Furthermore, we used the inverse modeling method to achieve maximum possible temperature gradients at the liquid/solid interface according to the limitation of maximum available power and the maximum stable height of a melt zone.     

Efficient adaptive phase field simulation of directional solidification of a binary alloy

Efficient adaptive phase field simulation based on a finite volume method is carried out to study the morphological development during directional solidification of a nickel/copper alloy. The adaptive nature of the method allows the calculation to cover different length scales for the interface, solute diffusion, and heat conduction. With the frozen temperature approximation, our calculated results are in reasonable agreement with previous ones (J. Crystal Growth 200 (1999) 583). However, the use of a much larger domain allows us to perform simulation at low speed near the onset of constitutional supercooling, where both solutal boundary layer and cell wavelength are large. For the same domain size, the calculated results without using the frozen temperature approximation remain about the same, even though the release of latent heat lowers the steady interface position and the thermal gradient in the melt side.     

Effect of a vertical magnetic field on the dendrite morphology during Bridgman crystal growth of Al–4.5 wt% Cu

The effect of a vertical high magnetic field (up to 10 T) on the dendrite morphology has been investigated during Bridgman growth of Al–4.5 wt%Cu alloys experimentally. It is found that the field causes disorder in dendrites and their tilt in orientation. Along with the increase of the magnetic field and decrease of the growth velocity, the dendrites became broken and orientated in 1 1 1 along the direction of solidification instead of 1 0 0. The field also enlarged the primary dendrite spacing and promoted the branching of the dendrites to form high-order arms. Above phenomena are attributed to the thermoelectromagnetic convection effect and orientation caused by the high magnetic field.     

金刚石衬底上GaN HEMT沟道温度建模:各向异性且非均匀热导率的影响

为了改善GaN HEMT的自热效应,集成高热导率的金刚石衬底有助于增强器件有源区的热量耗散。然而,化学气相淀积(CVD)生长的多晶金刚石(PCD)具有柱状晶粒结构,导致了各向异性的材料热导率,且其热导率值与生长厚度有关。为此,通过建模金刚石生长过程中晶粒尺寸的演变过程,计算了金刚石沿面内和截面方向的热导率。基于该PCD热导率模型,利用计入材料非线性热导率的GaN器件热阻解析模型,计算得到了GaN HEMT沟道温度的波动范围,并分析了其与器件结构(栅长、栅宽、栅间距、衬底厚度)和功耗的依赖关系。最后,通过与有限元(FEM)仿真结果对比,分区域提取了GaN HEMT器件中PCD衬底的有效热导率,分别为260~310 W/(m·K)和1 250~1 450 W/(m·K)。本文的计算为预测金刚石衬底上GaN HEMT器件的沟道温度提供了快速、有效的方法。     

An experimental investigation of the columnar-to-equiaxed grain transition in aluminum–copper hypoeutectic and eutectic alloys

Providing benchmark data of the thermal and metallographic parameters during the columnar-to-equiaxed transition (CET) in a wide range of alloy concentrations is of fundamental importance for understanding this phenomenon as well as for metallurgical and modeling purposes. The aim of this study was to investigate the columnar-to-equiaxed transition (CET) in aluminum–copper alloys of different compositions covering a wide range from 2 to 33.2 wt%Cu (eutectic composition), which were directionally solidified from a chill face. The thermal parameters studied included recalescence, cooling rates, temperature gradients and interphase velocities. We found that the temperature gradient and velocity of the liquidus interphase reached critical values at the CET; these critical values were between −0.44 and 0.09 K/mm and between 0.67 and 2.16 mm/s, respectively. The metallographic parameters analyzed were grain size, primary and secondary dendritic arm spacing and also eutectic spacing. The results obtained were compared with previous experimental studies, published predictions and models of the CET for similar alloys.     

Monte Carlo Simulation Algorithms of Grain Growth in Polycrystalline Materials

In the paper we present a variety of Monte Carlo algorithms, that can be employed to simulate the grain growth in polycrystalline materials during sintering. The simulation algorithms of monophase or two‐phase structure, on both the square and triangular distribution of lattice points, possibly with the second phase particles being either of static or dynamic nature, are described. The paper deals with oriented and anisotropic grain growth as well. A considerable number of input parameters in the simulation procedure makes it possible to set up a large amount of combinations of conditions under which we want to simulate the sintering process.     

Growth of equiaxed dendritic crystals settling in an undercooled melt,Part 2: Internal solid fraction

Experiments are conducted to measure the internal solid fraction evolution of equiaxed dendritic crystals that are freely growing and settling in an undercooled melt using the transparent model alloy succinonitrile–acetone. The internal solid fraction is determined from the measured settling speed and crystal envelope shape and size. Depending on the melt undercooling and acetone concentration, the internal solid fraction is found to vary between 0.55 and 0.1. In all experiments, the internal solid fraction decreases continually during settling. Based on heat and solute balances, a model is developed for predicting the internal solid fraction evolution under convective conditions. Nusselt and Sherwood number correlations are obtained that allow for the calculation of the thermal and solutal boundary layer thicknesses at the crystal envelope. The measured and predicted internal solid fraction evolutions are found to be in good agreement.     

Numerical simulation of the solidification processes of copper during vacuum continuous casting

A numerical simulation method is used to analyze the microstructure evolution of 8-mm-diameter copper rods during the vacuum continuous casting (VCC) process. The macro–microscopic coupling method is adopted to develop a temperature field model and a microstructure prediction model. The effects of casting parameters, including casting speed, pouring temperature, cooling rate, and casting dimension on the location and shape of the solid–liquid (S/L) interface and solidified microstructure are considered. Simulation results show that the casting speed has a large effect on the position and shape of the S/L interface and grain morphology. With an increase of casting speed, the shape of the S/L interface changes from a planar shape into an elliptical shape or a narrow, pear shape, and the grain morphology indicates a change from axial growth to axial–radial growth or completely radial growth. The simulation predictions agree well with the microstructure observations of cast specimens. Further analysis of the effects of other casting parameters on the position and shape of the S/L interface reveals that the casting dimension has more influence on the position and shape of the S/L interface and grain morphology than do pouring temperature and cooling rate. The simulation results can be summarized to obtain a discriminant of shape factor (η), which defines the shape of the S/L interface and grain morphology.     

Inversion of the Temperature Field in Optical Crystals

This article is devoted theoretical and experimental researches of temperature fields which are formed in area semi‐transparent, diffusely transmitting and scattering boundary of two optical environments. It is revealed, that thus can arise non‐monotonic or at certain conditions completely an inverse temperature field. The phenomenon of inversion of a stationary temperature field is revealed theoretically and subsequently is experimentally confirmed. The specific conditions of occurrence of the phenomenon of inversion are determined. During the crystal growth process behind the front of crystallization there can be a congestion of impurities or micro‐bubbles which are grasped by the moving front of crystallization. It results in occurrence diffusely transmitting and scattering boundary on which the radiating thermal flux going from the melt is dissipated in a growing crystal. In turn under the certain conditions it could result in non‐monotonic of temperature field in area of phase boundary and even in full inversion. The experimental equipment was developed and results of experimental measurements which completely confirm theoretical conclusions are given. The described phenomenon could be meet in growing of such optical crystals as sapphire, ruby, fluorides etc. It is specified, that the similar conditions can arise as well on boundary of solid‐gas and liquid‐gas.     

Mathematical modeling of the multi‐run process of crystal pulling from the melt by EFG (Stepanov) technique in dependence on the angle of inclination of the working edges of the dies

A mathematical model for the determination of melt hydrodynamics, impurity concentrations and thermal stresses in the multi‐run process of the growth of sapphire ribbons by EFG (Stepanov) technique with inclinated working surfaces of the dies is considered. The mathematical model deals with thermal conductivity equation, Navier‐Stokes equation, diffusion equation, capillary Laplace equation. This problem has been solved by the finite‐element method. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Three dimensional numerical study of the effect of large Grashof number on HEM crystal growth

Heat transfer and fluid flow in HEM crystal growth of silicon in cylindrical cavity is studied numerically. The walls of the crucible are heated to a fixed temperature. The exchanger that causes and induces natural convection is seated at the middle‐bottom of the crucible. The finite‐volume method is employed to solve the governing equations with proper boundary conditions. The effects of transport mechanism on the temperature distribution, melt flow, pressure and stream function are presented. We focus our work on the pressure field which has not yet been studied in HEM crucible. Also, we extend our work on a wide range Grashof number and for large numbers until 10¹² not yet studied in HEM furnace. It is found that the onset of flow fluctuations appears at Gr = 10¹⁰. Uniform temperature is observed in the entire melt at high Grashof number with development of a thermal boundary layer close to the exchanger. The thermal boundary layer thickness is calculated for strong buoyancy regime. Besides, for very high Gr number, buoyancy has less effect on temperature and then on melt‐crystal interface shape. During enlarging Gr, pressure evolution is related to temperature variation more than flow pattern.     

热屏间距对泡生法蓝宝石单晶生长影响研究

太空、军事和科研等高科技领域的持续发展极大促进了对蓝宝石晶体的需求,泡生法是蓝宝石晶体的主要制造方法之一;热场结构对所得蓝宝石晶体的质量具有重要影响.本文对采用泡生法工艺制造蓝宝石单晶过程中,具有内置7层氧化锆外置8层钼金属的新型热屏结构间距进行研究.通过数值模拟考察热屏间距对单晶炉功率、固-液界面形状和晶体热应力的影响确定了合理的热场结构;并与试验生产结果进行对比验证.结果表明:热屏间距增大使得单晶炉功率明显提升,并引起固-液界面凸度增大;而蓝宝石晶体热应力出现减小.综合考察三个影响因素的影响,最后确定热屏间距为5 mm时单晶炉能耗较低,可用于制造高质量的蓝宝石晶体.     

熔盐法合成SrBi2Nb2O9粉体

以分析纯的Bi2O3,Nb2O5 和 SrCO3为原料,以KCl和NaCl为熔盐,采用熔盐法在800～1000 ℃合成了片状SrBi2Nb2O9粉体.研究了熔盐含量及合成温度对晶体定向生长和粉体形貌的影响.结果表明:与固相法相比,熔盐法是一种有效的晶粒定向生长的方法.制备的粉体呈明显的片状和高度的各向异性,且无团聚现象产生.沿(00l)面择优生长适合的熔盐含量为60;质量分数,随着熔盐含量的增加,晶粒尺寸逐渐增大.合成温度在900 ℃为最优,可获得较大尺寸和高度各向异性的SrBi2Nb2O9粉体.