A critical examination of the dendrite growth models: Comparison of theory with experimental data

Dendrite tip temperature, dendrite tip radius and primary arm spacing data, and their variation with the growth speed and temperature gradient for directionally solidified succinonitrile-acetone, succinonitrile-salol, aluminum-copper, and lead-paladium alloys have been examined against their qualititative and quantitative fit with predictions from several dendrite growth models. The Burden and Hunt analysis while predicting the proper quantitative behavior, does not in general, yield a good quantitative agreement with experimental data. Models due to Trivedi, and more recently, Laxmanan (minimum dendrite tip undercooling approach as well as the tip stability approach) show a very good quantitative fit with the experimental data. Predictions of dendrite tip temperature and tip composition in the liquid have been shown to be inadequate to distinguish between the models within the experimentally feasible directional solidification conditions. Therefore, in order to determine which model is most appropriate, additional directional solidification experiments involving simultaneous measurements of dendrite tip radius, tip temperature, tip composition, and primary arm spacing in the low growth velocity regime are suggested.     

Dendrite characteristics in directionally solidified Pb-8 Pct Au and Pb-3 Pct Pd alloys

Pb-8 pct Au and Pb-3 pct Pd alloy specimens partially directionally solidified and then quenched have been examined in order to characterize their dendritic microstructural details and solute composition profiles. Dendrite tip radii have been measured by a controlled sectioning technique. Dendrite tip radius, solute content of quenched liquid at the dendrite tip, solute profile within the interdendritic region and ahead of the dendrite tip, cell length, and the primary arm spacing values obtained experimentally have been compared with the theoretical predictions. Two groups of models, one based on the minimum undercooled dendrite tip criterion and the other based on the marginal stability at the dendrite tip, have been examined. The Burden and Hunt model, based on the “minimum undercooling” approach, does not predict the observed behavior. However, a modification of the Burden and Hunt's model recently proposed by Laxmanan shows a good fit to the experimentally observed parameters. The models based on the marginal stability approach also predict most of the observed behavior well. It is concluded that quantitative comparison of the primary arm spacing measurements can not form the basis of distinguishing among the various dendrite growth models in a positive temperature gradient. There is a critical need to carry out carefully controlled directional solidification experiments in a well characterized metallic alloy system to help distinguish between the minium dendrite tip undercooling and the marginal stability approaches. New experiments based on simultaneous measurements of (a) dendrite tip radius, (b) dendrite tip temperature, and (c) the solute profile ahead of the dendrite tip—all in a convection free atmosphere—are required.     

Columnar dendritic solidification in a metal- matrix composite

Results are presented of a study on columnar dendritic solidification of the matrix of a fibrous metal-matrix composite, the fibers of which are aligned SiC fibers 140 m in diameter and the matrix of which is Al-4.5 wt pct Cu. Samples were produced by pressure infiltration of the metal-matrix into a preform of the fibers. The matrix was subsequently remelted and resolidified under controlled thermal gradient and growth rate. Dendrite growth begins in the center of the interstices left between the fibers. The dendrite tip temperature is not significantly influenced by the fibers, but the usual linear dependency of dendrite arm spacing ont ^1/3 (wheret is time during solidification) is altered significantly in the narrower interstices at long solidification times. The underlying mechanism is dendrite arm coalescence which takes place at a sufficiently rapid rate in the composite that the microstructure gradually becomes nondendritic. The solid/liquid interface then is parallel to the matrix/fiber interface. A model is presented for the kinetics of dendrite arm coalescence and compared with experimental results. The amount of microsegregation that was found in the matrix within interstices is significantly less than that found in the usual cast alloy, especially at long solidification times (low cooling rates). The mechanism responsible for the observed reduction in microsegregation is solid-state diffusion which is enhanced in the composite by the fact that the fibers place an upper limit on the dendrite arm spacing, and hence on the required diffusion distance.     

铝合金铸件定向凝固过程的数值模拟

为了研究铝合金定向凝固组织的变化规律,采用有限元软件ProCAST对Al Si Cu合金定向凝固过程进行模拟,分析了不同浇注温度和抽拉速率对铸件定向凝固过程中的温度梯度、固液界面前沿、糊状区宽度、枝晶生长速率和二次枝晶臂间距的影响。结果表明,当浇注温度越高时,温度梯度越大,而固液界面前沿下凹越小,糊状区宽度也越窄,从而越有利于顺序凝固的发生;随着抽拉速率的增大,枝晶生长速率先增大后减小,当抽拉速率为200 μm/s时,最大生长速度达到0.093 mm/s,铸件凝固组织最佳;当抽拉速率大于300或小于200 μm/s时,都会导致枝晶生长速率缓慢,枝晶生长不平稳,二次枝晶臂粗大。对模拟得到较优的工艺参数进行试验验证,可以制备出具有较好力学性能的铸件。     

定向凝固研究易切削钢中硫化物的生成行为

通过定向凝固实验研究了易切削钢中硫化物的生成过程,并建立了枝晶臂间距、硫化物的平均直径与凝固条件之间的关系.实验结果表明:随着温度梯度和拉速的增大,MnS的平均直径减小,数量增多;定向凝固过程中硫化物的完全析出温度大约在固相线温度下100℃左右,MnS形态的改变,主要受其组成元素的活度影响,即MnS形态的改变受其界面自由能的影响;对定向凝固实验中钢的枝晶臂间距、硫化物的平均直径与凝固条件的关系进行了拟合,拟合结果与前人研究结果相吻合.      

Interdendritic Spacing: Part I. Experimental Studies

Directional solidification experiments have been carried out in a succinonitrile-5.5 mol pct acetone system to characterize dendrite tip radius and interdendrite spacings as functions of growth rate and temperature gradient in the liquid. A maximum in primary dendrite spacing as a function of growth rate is observed, and this maximum is found to occur at the dendrite-cellular transition velocity. A scaling law is shown to exist between the secondary dendrite arm spacing, λ₂, near the tip and the dendrite tip radius, p, which is λ₂/ρ = 2.2 ± 0.3. Experimental results on ρ have been found to agree with the theoretical model based on the marginal stability criterion. This paper is based on a presentation made at the symposium “Establishment of Microstructural Spacing during Dendritic and Cooperative Growth” held at the annual meeting of the AIME in Atlanta, Georgia on March 7, 1983 under the joint sponsorship of the ASM-MSD Phase Transformations Committee and the TMS-AIME Solidification Committee.     

Cellular and dendritic growth: Part I. Experiment

An Al-4.1 mass pct Cu alloy was unidirectionally solidified under various growth rates. Features, such as tip radius of curvature, primary arm spacing, and tip concentration were measured as functions of growth rate. Dependence of tip radius on growth rate was different between cells and dendrites. Measured tip radius and primary arm spacing were maximum at the cellular-dendritic transition. Tip concentration, however, monotonously decreases with growth rate. Linear relationships between tip radius and characteristic dimensions of dendrites like core diameter, half length of tip arc, and the first secondary arm spacing are obtained to determine what affects growth rate, convection, and gravity segregation. Experimental results are compared with current theoretical models for dendrite growth under controlled solidification. It was determined that the measured tip radius is larger than that of theoretical predictions at fast growth rate, but the measured tip concentration is in good agreement with theoretical predictions.     

Numerical Determination of Secondary Dendrite Arm Spacing for IN738LC Investment Castings

A numerical model was developed to estimate the solidification conditions and the secondary dendrite arm spacing of equiaxed solidified IN738LC investment castings. The model, composed of geometric data, thermophysical properties, and boundary conditions, was verified by a comparison of calculated and measured process temperatures obtained from casting experiments. The computation of the secondary dendrite arm spacing was carried out from temperature gradient G, solidification rate v, and an alloy-specific parameter M, determined by means of an inverse approach. The calculated secondary dendrite arm spacing was found to be in very good agreement with metallographic measurements.     

强制对流影响下Fe?C合金定向凝固微观组织的相场法研究

定向凝固技术能够获得特定柱状晶结构,对于优化合金轴向力学性能具有非常显著的效果。本文采用耦合流场的相场模型模拟了定向凝固过程中枝晶的生长过程,研究了各向异性系数、界面能对定向凝固枝晶生长的影响以及强制对流作用下枝晶的生长行为。数值求解过程中,选用基于均匀网格的有限差分方法对控制方程进行离散,实现了格子中标记点算法（MAC）和相场离散计算方法的联合求解。处理微观速度场和压力场耦合时,采用MAC算法求解Navier-Stokes方程和压力Poisson方程,采用交错网格法处理复杂的自由界面。结果表明:随着各向异性系数的增大,枝晶尖端生长速度增大,曲率半径减小,枝晶根部溶质浓度逐渐降低;随着界面能的增大,枝晶尖端曲率半径增大,当界面能为最大（0.6 J·m?2）时,凝固呈现平界面的凝固方式向前推进;强迫对流对定向凝固枝晶生长方向影响较大,上游方向定向凝固枝晶粗大且生长速度更快,其现象随流速的增大而愈加明显。      

Simple model of microsegregation during solidification of steels

A simple analytical model of microsegregation for the solidification of multicomponent steel alloys is presented. This model is based on the Clyne-Kurz model and is extended to take into account the effects of multiple components, a columnar dendrite microstructure, coarsening, and the δ/γ transformation. A new empirical equation to predict secondary dendrite arm spacing as a function of cooling rate and carbon content is presented, based on experimental data measured by several different researchers. The simple microsegregation model is applied to predict phase fractions during solidification, microsegregation of solute elements, and the solidus temperature. The predictions agree well with a range of measured data and the results of a complete finite-difference model. The solidus temperature decreases with either increasing cooling rate or increasing secondary dendrite arm spacing. However, the secondary dendrite arm spacing during solidification decreases with increasing cooling rate. These two opposite effects partly cancel each other, so the solidus temperature does not change much during solidification of a real casting.     

<Emphasis Type="Italic">In-Situ</Emphasis> Analysis of Coarsening during Directional Solidification Experiments in High-Solute Aluminum Alloys

Coarsening within the mushy zone during continuous directional solidification experiments was studied on an Al-30 wt pct Cu alloy. High brilliance synchrotron X-radiation microscopy allowed images to be taken in-situ during solidification. Transient conditions were present during directional solidification. Under these conditions, solute-rich settling liquid flow affects the dendritic array and thus coarsening. Coarsening was studied by following the secondary dendrite arm spacing (SDAS) of a developing dendrite at different local solidification times according to the mush depth and instant interface velocity. Solute enrichment and liquid flow cause deceleration and acceleration of the solidification front, which in turn influences both the mush depth and local growth and coarsening due to variations in solutal gradients and thus local undercooling. In addition, spacing between neighboring dendrites (i.e., primary dendrite arm spacing), which determines permeability within the mushy zone, affects the development of high-order branches. This article is based on a presentation given at the International Symposium on Liquid Metal Processing and Casting (LMPC 2007), which occurred in September 2007 in Nancy, France.     

Microstructure Formation in AlSi7Mg Alloys Directionally Solidified in a Rotating Magnetic Field

To produce high stressed automotive components like engine frames and cylinder heads in foundry industry often AlSi7Mg alloys are used. During mould filling and casting melt flow affects the development of the microstructure, which defines the mechanical properties. In this paper the microstructure formation in AlSi7Mg0.3 and AlSi7Mg0.6 alloys during directional solidification is investigated. To induce a forced melt flow a rotating magnetic field is applied. For that purpose a Bridgman‐type gradient furnace is equipped with a rotary ring magnet. For detailed investigation of the shape of the solid‐liquid interface and the primary dendrite spacing a decanting device is used. As a result, the forced melt flow substantially changes the dendritic solidification microstructure. The rotating magnetic field generates a radial secondary flow in and ahead of the mushy zone, which causes an enrichment of eutectics in the centre of the samples. At lower solidification velocities this locally leads to the transition to mixed columnar‐equiaxed or even to equiaxed growth. In that case the solid‐liquid interfaces of the decanted samples show a significant depression in the centre part. In the out‐of‐centre region columnar growth still exists and the primary dendrite spacing decreases with increasing melt flow.     

Solidification modelling of microstructure in twin-roller thin strip casting of stainless steels

The influence of process parameters on the dendritic microstructure of thin strips cast by the twin-roll method is analyzed in the framework of a one-dimensional solidification model and compared with experimental results. As a relevant characteristic the secondary dendrite arm spacing Λ₂ as a function of the distance x from the roll surface is investigated. The difference between the local dendrite arm spacing near the strip surface and the strip centre, respectively, increases with the strip thickness and only depends on the casting temperature to a small extent. An increase in the strip/roller heat transfer coefficient due to a rising casting velocity or possibly enhanced roll-separating forces leads to a decrease in the dendrite arm spacing. The effect of a sudden decrease in heat transfer during the solidification process, on the Λ₂(x) characteristics, e.g. by a local separation of the solidified shell from the roller surface, is discussed.     

凝固组织对GCr15轴承钢220 mm×260 mm连铸坯中心偏析的影响

铸坯高中心等轴晶率及小的二次枝晶臂间距有利于降低高碳钢M+E-EMS连铸坯中心偏析。通过建立GCr15钢220 mm×260 mm连铸坯耦合有限元-元胞自动机模型(CAFE)及二次枝晶臂间距(SDAS)模型,研究结晶器电磁搅拌、过热度和拉速对中心等轴晶率及二次枝晶臂间距的影响。结果表明,相比于拉速,过热度和结晶器电磁搅拌对其影响明显。随着过热度降低及结晶器电磁搅拌强度增加,铸坯中心等轴晶率增加而二次枝晶臂间距减小,而拉速对凝固终点和中心固相率影响大。工业试验结果表明,采用结晶器与凝固末端电磁搅拌,相比于过热度35℃和拉速0.75 m/min,控制过热度小于25℃且拉速调整为0.8 m/min时,轴承钢GCr15铸坯中心等轴晶率由原27%增加至38%且二次枝晶臂间距细化,中心碳偏析指数由原1.06～1.39降至0.93～1.13。     

Refinement of dendrite arm spacings in aluminum ingots through heat flow control

A model for heat flow during solidification of alloys is presented which treats the heat of fusion released during solidification separately for three distinct regions of a casting: portions released isothermally at the liquidus temperature, between the liquidus and solidus in a specified manner and the remainder released at the solidus. The model is solved numerically by a finite difference technique for unidirectional and two-dimensional heat flow in end-chilled thin plates. Effects of heat transfer coefficient at the chill, superheat, heat input, liquid convection and amount of sidewise heat loss are considered. Results are presented in terms of position of liquidus and solidus isotherms as a function of time, width of the mushy zone and local solidification time and secondary dendrite arm spacingvs distance from the chill. Results from experimental castings made under controlled heat flow conditions are compared with computer calculations. The local solidification time and resultant dendrite arm spacing are shown to decrease at a given location as a) the chill heat transfer coefficient increases, b) superheat increases, c) the gradient of temperature at the solidification front increases, and d) the multidimensionality of the heat flow path increases. Formerly Graduate Students in the Department of Materials Science and Engineering, M.I.T., Cambridge, MA, Department of Metallurgy and Mining Engineering.     

Investigations of the solidification structure of continuously cast slabs

A study of solidification phenomena performed under industrial conditions is presented. The solidification structure of high carbon, medium carbon and microalloyed steel grades was observed and correlated to casting parameters such as superheat and secondary cooling. Experimental data of dendrite morphology are related to calculated solidification variables. The calculation is based on the solution of the enthalpy balance equation. An expression for the dependence of secondary dendrite arm spacing on solidification variables is proposed for columnar growth. The results are compared to the findings of other authors. The effect of chemical composition on secondary dendrite arm spacing was observed. Therefore the developed relationships are extended to carbon content on the basis of the presented experimental data.     

The dendrite arm spacings of aluminum-copper alloys solidified under steady-state conditions

Aluminum-copper alloys of different composition have been solidified under steady-state conditions with known growth velocities and temperature gradients. Specimens were quenched during solidification to reveal the dendrite growth morphology and to allow the solute content at dendrite tips to be analyzed. Primary and secondary arm spacings have been measured as a function of the distanced behind the growing dendrite tips, the tip velocityR the temperature gradient in the liquid G_l and the solute content. The primary arm spacing λ₁ is described for each alloy by the empirical relation: λ₁ = kG_L ^-aR^-b wherek is a constant, and both a and b are close to 1/2; λ₁ is also found to increase with solute content. The secondary arm spacing λ₂ for each alloy increases with timeθ spent in the liquid/solid region, being directly proportional to θⁿ where n is close to 1/3; increasing solute content however causes a reduction in λ₂. It is suggested that the observed “coarsening” of the secondary arms is primarily a coalescence phenomenon. Formerly at Sheffield University     

连铸钢坯一次枝晶间距及“鼓肚”的影响

 探讨了铸坯中一次枝晶间距的变化规律及其“鼓肚”量对枝晶间距的影响。通过确定凝固速度的表达式,定量计算了连铸过程中一次枝晶间距。结果表明,“鼓肚”的存在,使柱状晶生长初期一次枝晶间距出现增大、减小、再增大的周期性变化趋势,当固液界面向正弦波的波峰处运动的过程中,一次枝晶间距逐渐增大,当固液界面向正弦波的波谷处运动的过程中,一次枝晶间距逐渐减小。“鼓肚”量增加时,一次枝晶间距变化的幅度增大。在凝固的中后期,波峰和波谷处的一次枝晶间距基本相等,表明枝晶进入稳定生长的阶段,此时“鼓肚”对枝晶间距的影响基本可以忽略。计算结果与实验结果基本吻合。