Phase-field simulation of tip splitting in dendritic growth of Fe-C alloy

Two types of dendrite tip splitting including dendrite orientation transition and twinned-like dendrites in Fe-C alloys were investigated by phase-field method.In equiaxed growth, the possible dendrite growth directions and the effect of supersaturation on tip splitting were discussed;the dendrite orien-tation transition was observed, and it was found that the orientation regions of anisotropy parameters were reduced from three to two with increasing the supersaturation, which was due to the effect of interfacial anisotropy controlled by the solute in front of S/L interface changing with the increase of supersaturation.In directional solidification, it was found that the twinned-like dendrites were formed with the fixed anisotropy couples and no seaweed dendrites were observed;these were concluded from the results of competition between process anisotropy and inherent anisotropy.The formation process of twinned-like dendrite was investigated by tip splitting phenomenon, which was related to the chan-ges of dendrite tips growth velocity.Then, the critical speed of tips splitting and solute concentration of twinned-like dendrites were investigated, and a new type of microsegregation in Fe-C alloys was proposed to supplement the dendrite growth theories.     

Phase Field Simulation of Binary Alloy Dendrite Growth Under Thermal- and Forced-Flow Fields: An Implementation of the Parallel–Multigrid Approach

Dendrite growth and morphology evolution during solidification have been studied using a phase field model incorporating melt convection effects, which was solved using a robust and efficient parallel, multigrid computing approach. Single dendrite growth against the flow of the melt was studied under a wide range of growth parameters, including the Lewis number (Le) and the Prandtl number (Pr) that express the relative strengths of thermal diffusivity to solute diffusivity and kinematic viscosity to thermal diffusivity. Multidendrite growths for both columnar and equiaxed cases were investigated, and important physical aspects including solute recirculation, tip splitting, and dendrite tilting against convection have been captured and discussed. The robustness of the parallel–multigrid approach enabled the simulation of dendrite growth for metallic alloys with Le ~ 10⁴ and Pr ~ 10^?2, and the interplay between crystallographic anisotropy and local solid/liquid interfacial conditions due to convection on the tendency for tip splitting was revealed.     

Directional dendritic solidification of a composite slurry: Part I. Dendrite morphology

A simple analytical model to describe the morphology of a growing dendrite in the presence of an inert particle has been presented. The presence of a particle changes the solute concentration gradient at the tip of a growing dendrite and this, in turn, affects the dendrite tip radius. Results of the analysis show that the dendrite tip radius decreases at a high growth velocity due to the presence of a particle, while there is no influence on the tip radius at both low and intermediate growth velocities. Lower thermal conductivity of the particle decreases the tip radius, while a higher thermal conductivity increases the radius. However, the effect of thermal conductivity on the tip radius is only significant in the cellular growth regime. The analysis shows that the presence of SiC particles in Al-Cu alloys reduces the cell to dendrite transition velocity. Results of directional solidification experiments carried out on an Al-4.5Cu-SiC composite system agree with our model.     

A free dendritic growth model accommodating curved phase boundaries and high peclet number conditions

A steady-state free dendrite growth model accommodating nonlocal equilibrium tip conditions and curved liquidus and solidus has been developed. The developed model assumes a dendrite tip of a paraboloid of revolution and is applicable to dendrite growth in dilute binary alloys for all values of P _c , and reduces to the BCT model for linear liquidus and solidus. The marginal stability criterion of Trivedi and Kurz is shown to apply even in the presence of kinetic undercooling and curved phase boundaries when used with an appropriate concentration-dependent liquidus slope. The model is applied to Sn-Pb alloys to predict the tip velocity, tip radius, solute trapping, and four components of undercooling in the quasi-solutal, solutal-to-thermal transition and quasi-thermal regions.     

Primary spacing in directional solidification

A new analytical model is developed to explain the variation in primary spacing λ with growth velocity V. In this model, dendrite growth is resolved into two parts: the growth of the center core and that of the side arms, which are separately treated. In contrast to the assumption in the current models, it is only the dendrite core, not the entire dendrite, whose curvature radius at the tip is directly related to dendrite tip radius R. The primary spacing is considered to be the sum of core diameter and twice the sidearm length. As long as the growth of side arms is suppressed, it becomes cellular growth. As a result, this model gives a reasonable dependence of cell and dendrite spacing on the process parameters. The proposed model has been applied to several alloys to compare its predictions both with experimental data and with the analytical expression of the Hunt-Lu model.     

Effects of Thermoelectric Magnetic Convection on the Solidification Structure During Directional Solidification under Lower Transverse Magnetic Field

This work investigated the thermoelectric magnetic convection (TEMC) during directional solidification under a transverse magnetic field numerically and experimentally. Numerical results show that the TEMC will form in liquid near the liquid/solid interface and in the dendritic network. The value of the TEMC mainly depends on the crucible diameter, the temperature gradient, and the magnetic field intensity. The value of the TEMC increases as the crucible diameter and the temperature gradient are increased. The value of the TEMC on the sample scale increases to a maximum when the magnetic field is of the order of 0.1 T, and then decreases as the magnetic field still increases. However, the value of the TEMC on the cell/dendrite scale continues to increase with the increase of the magnetic field intensity when the applied magnetic field is less then 1 T. Two alloys are solidified directionally in the vertical configuration under a transverse magnetic field, and results show that the application of a lower transverse magnetic field (B < 1 T) modified the liquid/solid interface shape and the cellular/dendritic array significantly. Indeed, it was observed that, along with the refinement of the cell/dendrite, the magnetic field caused the deformation of the liquid/solid interfaces and the extensive segregations (i.e., channel and freckle) in the mushy zone. Comparison of the numerical and experimental results shows that the modification amplitude of the liquid/solid interface and the cellular/dendritic morphology is in good agreement with the value of the TEMC at the liquid/solid interface and in the dendritic network. This implies that changes of the interface shape and the cellular/dendritic morphology should be attributed, respectively, to the TEMC on the sample and the cell/dendrite scales.     

Primary spacing in directional solidification

A new analytical model is developed to explain the variation in primary spacing λ with growth velocity V. In this model, dendrite growth is resolved into two parts: the growth of the center core and that of the side arms, which are separately treated. In contrast to the assumption in the current models, it is only the dendrite core, not the entire dendrite, whose curvature radius at the tip is directly related to dendrite tip radius R. The primary spacing is considered to be the sum of core diameter and twice the sidearm length. As long as the growth of side arms is suppressed, it becomes cellular growth. As a result, this model gives a reasonable dependence of cell and dendrite spacing on the process parameters. The proposed model has been applied to several alloys to compare its predictions both with experimental data and with the analytical expression of the Hunt-Lu model.     

Cellular-to-dendritic transition during the directional solidification of binary alloys

The transition from a cellular to dendritic microstructure during the directional solidification of alloys is examined through experiments in a transparent system of succinonitrile (SCN)-salol. In a cellular array, a strong coupling of solute fields exists between the neighboring cells, which leads not only to multiple solutions of primary spacing, but also includes multiple solutions of amplitude, tip radius, and shape of the cell. It is found that these multiple solutions of different microstructural features in a cellular array, obtained under fixed growth conditions and compositions, play a key role in the cell-dendrite transition (CDT). The CDT is controlled not only by the input parameters of alloy composition (C ₀), growth rate (V), and thermal gradient (G), but also by microstructure parameters such as the local primary spacing. It is shown that the CDT is not sharp, but occurs over a range of growth conditions characterized by the minimum and maximum values of V/G. Within this transition range, a critical spacing is observed above which a cell transforms to a dendrite. This critical spacing is given by the geometric mean of the thermal, diffusion, and capillary lengths and is inversely proportional to composition in weight percent.     

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。      

Growth of solutal dendrites: A cellular automaton model and its quantitative capabilities

A model based on the cellular automaton (CA) technique for the simulation of dendritic growth controlled by solutal effects in the low Péclet number regime was developed. The model does not use an analytical solution to determine the velocity of the solid-liquid (SL) interface as is common in other models, but solves the solute conservation equation subjected to the boundary conditions at the interface. Using this approach, the model does not need to use the concept of marginal stability and stability parameter to uniquely define the steady-state velocity and radius of the dendrite tip. The model indeed contains an expression for the stability parameter, but the process determines its value. The model proposes a solution for the artificial anisotropy in growth kinetics valid at zero and 45° introduced in calculations by the square cells and trapping rules used in previous CA formulations. It also introduces a solution for the calculation of local curvature, which eliminates mesh dependency of calculations. The model is able to reproduce qualitatively most of the dendritic features observed experimentally, such as secondary and tertiary branching, parabolic tip, arms generation, selection and coarsening, etc. Computation results are validated in two ways. First, the simulated secondary dendrite arm spacing (SDAS) is compared with literature values. Then, the predictions of the classic Lipton-Glicksman-Kurz (LGK) theory for steady-state tip velocity are compared with simulated values as a function of melt undercooling. Both comparisons are found to be in very good agreement.     

Dendritic solidification—III. Some further refinements to the model for dendritic growth under an imposed thermal gradient

Some further refinements to a simple model for dendritic solidification (presented earlier by the author) in a binary alloy melt under an imposed positive thermal gradient are presented. Two new expressions for the dendrite tip undercooling have been obtained and shown to yield a limiting value of ΔT₀ and very small growth rates. Here ΔT₀ is the equilibrium solidification range of the alloy. At very large growth rates, all three tip undercooling expressions reach the same limiting value depending on the value of a dimensionless parameter λ which is related to the effective diffusion distance ahead of the dendrite tip. The predicted tip undercoolings are, however, somewhat lower at intermediate growth rates. An improved calculation for the solute buildup at the dendrite tip due to curvature effects is also included.     

Multistage Fatigue Modeling of Cast A356-T6 and A380-F Aluminum Alloys

This article presents a microstructure-based multistage fatigue (MSF) model extended from the model developed by McDowell et al.[1,2] to an A380-F aluminum alloy to consider microstructure-property relations of descending order, signifying deleterious effects of defects/discontinuities: (1) pores or oxides greater than 100 μm, (2) pores or oxides greater than 50 μm near the free surface, (3) a high porosity region with an area greater than 200 μm, and (4) oxide film of an area greater than 10,000 μm². These microconstituents, inclusions, or discontinuities represent different casting features that may dominate fatigue life at stages of fatigue damage evolutions. The incubation life is estimated using a modified Coffin–Mansion law at the microscale based on the microplasticity at the discontinuity. The microstructurally small crack (MSC) and physically small crack (PSC) growth was modeled using the crack tip displacement as the driving force, which is affected by the porosity and dendrite cell size (DCS). When the fatigue damage evolves to several DCSs, cracks behave as long cracks with growth subject to the effective stress intensity factor in linear elastic fracture mechanics. Based on an understanding of the microstructures of A380-F and A356-T6 aluminum alloys, an engineering treatment of the MSF model was introduced for A380-F aluminum alloys by tailoring a few model parameters based on the mechanical properties of the alloy. The MSF model is used to predict the upper and lower bounds of the experimental fatigue strain life and stress life of the two cast aluminum alloys. This article is based on a presentation made in the symposium entitled “Simulation of Aluminum Shape Casting Processing: From Design to Mechanical Properties,” which occurred March 12–16, 2006 during the TMS Spring Meeting in San Antonio, Texas, under the auspices of the Computational Materials Science and Engineering Committee, the Process Modeling, Analysis and Control Committee, the Solidification Committee, the Mechanical Behavior of Materials Committee, and the Light Metal Division/Aluminum Committee.

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Overview 12: Fundamentals of dendritic solidification—I. Steady-state tip growth

Systematic measurements of dendrite tip radius and growth velocity in succinonitrile reveal that consideration of dendrite tip stability should be incorporated into the heat transfer theory to determine the steady-state dendritic growth condition. The dendritic stability criterion measured is 2αd₀/VR² = 0.0195, where V is the dendritic growth velocity, R is the dendritic tip radius, a is the liquid thermal diffusivity, and d₀ is a capillary length defined in the text. Several dendritic stability models are reviewed and discussed in comparison to the present experimental results.     

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.     

A quantitative dendrite growth model and analysis of stability concepts

While a number of cellular automaton (CA) based models for dendrite growth have been proposed, none so far have been validated, casting doubt on their quantitative capabilities. All these models are mesh dependent and cannot correctly describe the influence of crystallographic orientation on growth morphology. In this work, we present an improved version of our previously developed CA based model for dendrite growth controlled by solutal effects in the low Péclet number regime. The model solves the solute and heat conservation equations subject to the boundary conditions at the interface, which is tracked with a new virtual front tracking method. It contains an expression equivalent to the stability constant required in analytical models, termed stability parameter, which is not a constant. The process determines its value, changing with time and angular position during dendrite formation. The article proposes solutions for the evaluation of local curvature, solid fraction, trapping rules, and anisotropy of the mesh, which eliminates the mesh dependency of calculations. Several tests were performed to demonstrate the mesh independence of the calculations using Fe-0.6 wt pct C and Al-4 wt pct Cu alloys. Computation results were validated in three ways. First, the simulated secondary dendrite arm spacing (SDAS) was compared with literature values for an Al-4.5 wt pct Cu alloy. Second, the predictions of the classic Lipton-Glicksman-Kurz (LGK) analytical model for steady-state tip variables, such as velocity, radius, and composition, were compared with simulated values as a function of melt undercooling for Al-4 wt pct Cu alloy. In this validation, it was found that the stability parameter approaches the experimentally and theoretically determined value of 0.02 of the stability constant. Finally, simulated results for succinonitrile-0.29 wt pct acetone (SCN-0.29 wt pct Ac) alloy are compared with experimental data. Model calculations were found to be in very good agreement with both the analytical model and the experimental data. The model is used to simulate equiaxed and columnar growth of Fe-0.6 wt pct C and Al-4 wt pct Cu alloys offering insight into microstructure formation under these conditions.     

The Evolution of As-cast Microstructure of Ternary Mg-Al-Zn Alloys: An Experimental and Modeling Study

A numerical formulation of solidification model which can predict the microsegregation and microstructural features for multicomponent alloys is presented. The model incorporates the kinetic features during solidification such as solute back diffusion, dendrite tip undercooling, and secondary arm coarsening. The model is dynamically linked to thermodynamic library for accurate input of thermodynamic data. The modeling results are tested against the directional solidification experiments for Mg-Al-Zn alloys. The experiments were conducted in the cooling rate range of 0.13 to 2.33 K/s and microstructural features such as secondary arm spacing, primary dendrite arm spacing, second phase fraction, and microsegregation were compared with the modeling results. Based on the model and the experimental data, a solidification map was built in order to provide guidelines for as-cast microstructural features of Mg-Al-Zn alloys in a wide range of solidification conditions.     

在强加来流作用下二元系中枝晶生长的稳态解

研究了远场来流作用下二元系中的枝晶生长.当Schmidt数很大时,应用渐近分析方法得到枝晶稳态生长的渐近解,其温度场和浓度场的首级解、一级解均为相似性解,枝晶形状为存在细微波动的旋转抛物面.远场来流的强弱影响着枝晶生长的Peclet数的大小,进而影响着枝晶的尖端半径与生长速度.当过冷度一定时,在枝晶尖端或在枝晶前沿处的温度随着流场的增大而减小,而溶质浓度随着流场的增大而增大.     

Relationship between contractile strain ratioR and texture in zirconium alloy tubing

The crystallographic textures of zirconium alloy tubing used as cladding in nuclear reactor fuel are commonly characterized by the quantitative texture numbersF (Källström) and f_r (Kearns) which are derived from the direct and inverse pole figures. The texture numbers of zircaloy 2 and 4 tubes have been correlated experimentally with the value of the contractile strain ratioR which is a measure of the plastic anisotropy of the tube. The correlations were based on the results of 20 different tubing lots. Thef _r-R correlation shows much less data scatter than theF-R correlation. By assuming a simple plastic deformation model for zirconium alloys the following relations between texture and anisotropy are obtained:F=R- 1/R+1 and f_r = R/R+1 The theoretically derived relations are in good agreement with the experimental data. The procedure of correlating texture with plastic anisotropy is not limited to zirconium alloy tubing, but should be equally applicable to textured sheet and plate materials and other alloys with a limited number of slip systems.     

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.