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.     

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.     

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.     

Effect of the primary phase on grain coarsening in undercooled Fe-Co alloys

Fe-Co alloy melts with Co contents of 10, 30, and 60 at. pct were undercooled to investigate the dependence of the primary phase on grain coarsening. A pronounced characteristic is that the metastable fcc phase in the Fe-10 at. pct Co alloy and the metastable bcc phase in the Fe-30 at. pct Co alloy will primarily nucleate when undercoolings of the melts are larger than the critical undercoolings for the formation of metastable phases in both alloys. No metastable bcc phase can be observed in the Fe-60 at. pct Co alloy, even when solidified at the maximum undercooling of ΔT = 312 K. Microstructural investigation shows that the grain size in Fe-10 and Fe-30 at. pct Co alloys increases with undercoolings when the undercoolings of the melts exceed the critical undercoolings. The grain size of the Fe-60 at. pct Co alloy solidified in the undercooling range of 30 to 312 K, in which no metastable phase can be produced, is much finer than those of the Fe-10 and Fe-30 at. pct Co alloys after the formation of metastable phases. The model for breakage of the primary metastable dendrite at the solid-liquid interface during recalescence and remelting of dendrite cores is suggested on the basis of microstructures observed in the Fe-10 and Fe-30 at. pct Co alloys. The grain coarsening after the formation of metastable phases is analyzed, indicating that the different crystal structures present after the crystallization of the primary phase may play a significant role in determining the final grain size in the undercooled Fe-Co melts.     

Fe-C合金凝固枝晶形貌及特性参数的相场模拟

为了实现对合金凝固过程中枝晶形态的定量表征、揭示凝固前沿溶质分布与过冷度对微观偏析的影响,进而实现对凝固枝晶间液相渗透率的量化研究,采用相场模型探讨了 Fe-0.5％C合金凝固过程中的显微组织和特征参数,并引入分形维数和无量纲周长定量分析了枝晶形貌、微观偏析和其糊状区的渗透性.结果表明,分形维数和无量纲周长可用于定量描...     

Dendritic growth of undercooled nickel-tin: Part I

Solidification of undercooled Ni-25 wt pct Sn alloy was observed by high-speed cinematography and results compared with optical temperature measurements. Samples studied were rectangular in cross-section, and were encased in glass. Cinematographic measurements were carried out on samples undercooled from 68 to 146 K. These undercoolings compare with a temperature range of 199 K from the equilibrium liquidus to the extrapolated equilibrium solidus. At all undercoolings studied, the high-speed photography revealed that solidification during the period of recalescence took place with a dendrite-like front moving across the sample surface. Spacings of the apparent “dendrite” were on the order of millimeters. The growth front moved at measured velocities ranging from 0.07 meters per second at 68 K undercooling to 0.74 meters per second at 146 K undercooling. These velocities agree well with results of calculations according to the model for dendrite growth of Lipton, Kurz, and Trivedi. It is concluded that the coarse structure observed comprises an array of very much finer, solute-controlled dendrites.     

Dendritic growth of undercooled nickel-tin: Part II

High-speed optical temperature measurements were made of the solidification behavior of levitated metal samples within a transparent glass medium. Two undercooled Ni-Sn alloys were examined, one a hypoeutectic alloy and the other of eutectic composition. Recalescence times for the 9 mm diameter samples studied decreased with increasing undercooling from the order of 1.0 second at 50 K under-cooling to less than 10⁻³ second for undercoolings greater than 200 K. Both alloys recalesced smoothly to a maximum recalescence temperature at which the solid was at or near its equilibrium composition and equilibrium weight fraction. For the samples of hypoeutectic alloy that recalesced above the eutectic temperature, a second nucleation event occurred on cooling to the eutectic temperature. For samples which recalesced only to the eutectic temperature, no subsequent nucleation event was observed on cooling. It is inferred in this latter case that both the α and β phases were present at the end of recalescence. The thermal data obtained suggest a solidification model involving (1) dendrites of very fine structure growing into the melt at temperatures near the bulk undercooling temperature, (2) thickening of dendrite arms with rapid recalescence, and (3) continued, much slower recalescence accompanying dendrite ripening.     

深过冷技术制备均质过偏晶合金及其形成机制的研究

采用熔融玻璃净化和循环过热相结合的方法使Ni 40 % (质量分数 )Pb合金获得 2 92K大过冷度 ,成功制备出大体积均质过偏晶合金。根据BCT模型和组织演化结果分析表明 :过冷粒状晶是在内应力的作用下 ,枝晶发生全面碎断 ,随后在枝晶段表面和应变能的驱动下使晶界移动发生再结晶的结果 ,即枝晶碎断 再结晶机制 ;试样基体上弥散分布的细密铅颗粒是由于快速凝固阶段溶质截留效应而形成的 ,少量较大尺寸铅颗粒的形成主要与慢速凝固阶段分布于枝晶骨架间残余富铅液相的聚合有关。     

Solidification of undercooled dilute tin-lead alloys

The rate of solidification of dilute tin-lead alloys has been measured as a function of the initial undercooling (up to 45°C) and the solute content (up to 2 wt pct lead). Solidified specimens were examined by metallography and X-ray diffraction to obtain information on the solidification process and the resulting grain structure. Over an intermediate range of undercoolings, it was found that dendrites grow in the tin-lead alloys as much as four times faster than in pure tin at the same undercooling. This result is inconsistent with any present theories for dendrite growth kinetics in binary alloys. At both lower and higher undercoolings there is no evidence for growth by simple extension of dendrites along the specimen, and solidification rate measurements made under these conditions are probably not indicative of normal dendrite growth kinetics. A. W. Urquhart and G. L. F. Powell were formerly at the Thayer School of Engineering.     

Effect of growth rate dependent partition coefficient on the dendritic growth in undercooled melts

The theory of alloy dendritic growth at large undercoolings is extended to include the effect of growth rate dependent partition coefficient on the growth rate, tip radius and composition of dendrites. Three distinct behaviors are observed depending on the value of the dimensionless rate v. For small values of v, k ≅ k₀. For the intermediate range, when v is between 0.01 and 100, a significant effect of the velocity dependent partition coefficient is found. When v > 100, k → 1 and the dendrite behavior in an alloy is found to approach that for the pure material. In undercooled alloy melts segregation free zones could then be obtained by dendrite growth.     

Solidification of undercooled molten Ni-based alloys

The effect of undercooling on grain structure is investigated in pure nickel, Ni₇₅Cu₂₅, and DD3 singlecrystal superalloy by employing the method of molten salt denucleating combined with thermal cycling. Meanwhile, a comparison of factors that may be related to structure formation is performed and the difference in the refined structure between Ni₇₅Cu₂₅ alloy and DD3 single-crystal superalloy is explained. Only one grain refinement occurs at the critical undercooling in pure nickel, whereas two take place at both low and high undercoolings in Ni₇₅Cu₂₅ and DD3 single-crystal superalloy melts. The first grain refinement at low undercoolings mainly originates from dendrite remelting driven by the chemical superheating produced in recalescence, and the second one at high undercoolings is due to the recrystallization process as a result of the high stress provided in the rapid solidification after high undercooling. Dislocation morphology evolution in as-solidified structure is also provided by the transmission electron microscopy (TEM) technique to further verify the recrystallization mechanism.     

An integrated model for dendritic and planar interface growth and morphological transition in rapid solidification

Rapid solidification can be achieved by quenching a thin layer of molten metal on a cold substrate, such as in melt spinning and thermal spray deposition. An integrated model is developed to predict microstructure formation in rapidly solidified materials through melt substrate quenching. The model solves heat and mass diffusion equations together with a moving interface that may either be a real solid/liquid interface or an artificial dendrite tip/melt interface. For the latter case, a dendrite growth theory is introduced at the interface. The model can also predict the transition of solidification morphology, e.g., from dendritic to planar growth. Microstructure development of Al-Cu alloy splats quenched on a copper substrate is investigated using the model. Oscillatory planar solidification is predicted under a critical range of interfacial heat-transfer coefficient between the splat and the substrate. Such oscillatory planar solidification leads to a banded solute structure, which agrees with the linear stability analysis. Finally, a microstructure selection map is proposed for the melt quenching process based on the melt undercooling and thermal contact conditions between the splat and the substrate.     

Phase Selection and Microstructure Evolution in Nonequilibrium Solidification of Fe40Ni40B20 Alloy

Phase selection and microstructure evolution in nonequilibrium solidification of ternary eutectic Fe₄₀Ni₄₀B₂₀ alloy have been studied. It is shown that γ-(Fe, Ni) and (Fe, Ni)₃B prevail in all the as-solidified samples. No metastable phase has been found in the deeply undercooled samples. This is explained as resulting from the size effect of undercooled solidification. At small and medium undercoolings, the dendrite γ-(Fe, Ni) appears as the leading phase. This is ascribed to the existence of the skewed coupled growth zone in FeNiB alloy. With increasing undercooling, the amount of dendrites first increases and then decreases, accompanied by a transition from regular eutectic to anomalous eutectic. The formation mechanisms of the anomalous eutectics are discussed. Two kinds of microstructure refinement are found with increasing undercooling in a natural or water cooling condition. However, for melts with the same undercooling, the as-solidified microstructure refines first, and then coarsens with an increasing cooling rate. The experimental results show that the nanostructure eutectic cell has been obtained in the case of Ga-In alloy bath cooling with an initial melt undercooling of approximately 50 K (50 °C).     

A solutal interaction mechanism for the columnar-to-equiaxed transition in alloy solidification

A multiphase/multiscale model is used to predict the columnar-to-equiaxed transition (CET) during solidification of binary alloys. The model consists of averaged energy and species conservation equations, coupled with nucleation and growth laws for dendritic structures. A new mechanism for the CET is proposed based on solutal interactions between the equiaxed grains and the advancing columnar front—as opposed to the commonly used mechanical blocking criterion. The resulting differences in the CET prediction are demonstrated for cases where a steady state can be assumed, and a revised isotherm velocity (V _T ) vs temperature gradient (G) map for the CET is presented. The model is validated by predicting the CET in previously performed unsteady, unidirectional solidification experiments involving Al-Si alloys of three different compositions. Good agreement is obtained between measured and predicted cooling curves. A parametric study is performed to investigate the dependence of the CET position on the nucleation undercooling and the density of nuclei in the equiaxed zone. Nucleation undercoolings are determined that provide the best agreement between measured and calculated CET positions. It is found that for all three alloy compositions, the nucleation undercoolings are very close to the maximum columnar dendrite tip undercoolings, indicating that the origin of the equiaxed grains may not be heterogeneous nucleation, but rather a breakdown or fragmentation of the columnar dendrites. An erratum to this article is available at .     

Containerless solidification of highly undercooled mullite melts: Crystal growth behavior and microstructure formation

A mullite (3Al₂O₃·2SiO₂) sample has been levitated and undercooled in an aero-acoustic levitator, so as to investigate the solidification behavior in a containerless condition. Crystal-growth velocities are measured as a function of melt undercoolings, which increase slowly with melt undercoolings up to 380 K and then increase quickly when undercoolings exceed 400 K. In order to elucidate the crystal growth and solidification behavior, the relationship of melt viscosities as a function of melt undercoolings is established on the basis of the fact that molten mullite melts are fragile, from which the atomic diffusivity is calculated via the Einstein-Stokes equation. The interface kinetics is analyzed when considering atomic diffusivities. The crystal-growth velocity vs melt undercooling is calculated based on the classical rate theory. Interestingly, two different microstructures are observed; one exhibits a straight, faceted rod without any branching with melt undercoolings up to 400 K, and the other is a feathery faceted dendrite when undercoolings exceed 400 K. The formation of these morphologies is discussed, taking into account the contributions of constitutional and kinetic undercoolings at different bulk undercoolings.     

Undercooling-Induced macrosegregation in directional solidification

The accepted primary mechanism for causing macrosegregation in directional solidification (DS) is thermal and solutal convection in the liquid. This article demonstrates the effects of under-cooling and nucleation on macrosegregation and shows that undercooling, in some cases, can be the cause of end-to-end macrosegregation. Alloy ingots of Pb-Sn were directionally solidified upward and downward, with and without undercooling. A thermal gradient of about 5.1 K/cm and a cooling rate of 7.7 K/h were used. Crucibles of borosilicate glass, stainless steel with Cu bottoms, and fused silica were used. High undercoolings were achieved in the glass crucibles, and very low undercoolings were achieved in the steel/Cu crucible. During under-cooling, large, coarse Pb dendrites were found to be present. Large amounts of macrosegregation developed in the undercooled eutectic and hypoeutectic alloys. This segre-gation was found to be due to the nucleation and growth of primary Pb-rich dendrites, continued coarsening of Pb dendrites during undercooling of the interdendritic liquid, Sn enrichment of the liquid, and dendritic fragmentation and settling during and after recalescence. Eutectic ingots that solidified with no undercooling had no macrosegregation, because both Pb and Sn phases were effectively nucleated at the start of solidification, thus initiating the growth of solid of eutectic composition. It is thus shown that undercooling and single-phase nucleation can cause significant macrosegregation by increasing the amount of solute rejected into the liquid and by the movement of unattached dendrites and dendrite fragments, and that macrosegregation in excess of what would be expected due to diffusion transport is not necessarily caused by convection in the liquid.     

A numerical analysis of time dependent isolated dendritic growth for conditions near the steady state

A numerical model has been developed to model time dependent array growth. As a test for the model the spacing was made very large and the temperature gradient was reduced to zero so that results could be compared to approximate analytic solutions for an isolated dendrite. It was found that there was good agreement both for shape and for the composition field at steady state. Instead of finding a family of solutions as was expected a unique solution was found for a given bath undercooling. The steady state solution corresponded precisely to the value of a parameter σ¹ previously used to describe marginal stability. The condition was obeyed over many orders of magnitude change in the growth conditions. When the dendrite was perturbed by momentarily changing the bath undercooling the interface gradually returned to the steady state shape. It is suggested that the unique solution is the result of correctly including the surface energy and that the stability parameter could instead be considered to be the steady state growth parameter.     

Investigation on dendritic solidification and microsegregation in presence of melt convection using phase field simulation

Abstract

A phase field model has been used to simulate dendritic solidification of a binary alloy in the presence of forced melt convection. The influence of melt flow on morphology and solute distribution was investigated for various conditions. The results showed that incorporation of fluid flow causes asymmetric dendritic growth which is amplified by increasing fluid velocity. Moreover, it has been found that the effects of melt flow on the growth of different arms depend on the preferred growth orientation of the dendrite with respect to flow direction. Solid microsegregation study of the growing dendrite arm perpendicular to the flow direction indicated that the position of the arm axis varied almost linearly with flow velocity. Introducing an adjusting term called an antitrapping current in the concentration equation prevents solute from being highly trapped in the solid phase and causes the phase field simulations to be more realistic, especially for high undercoolings.

On a utilisé un modèle de champ de phase pour simuler la solidification dendritique d’un alliage binaire en présence de convection forcée du bain. On a examiné l’influence de l’écoulement du bain sur la morphologie et la distribution de soluté sous des conditions variées. Les résultats ont montré que l’incorporation de l’écoulement du fluide produisait une croissance dendritique asymétrique qui est amplifiée par l’augmentation de la vitesse du fluide. De plus, on a trouvé que l’effet de l’écoulement du bain sur la croissance de différentes branches dépendait de l’orientation préférée de la croissance de la dendrite par rapport à la direction de l’écoulement. L’étude de la microségrégation solide de la branche de dendrite en croissance perpendiculaire à la direction de l’écoulement indiquait que la position de l’axe de la branche variait presque linéairement avec la vitesse de l’écoulement. L’introduction d’un terme d’ajustement, appelé courant de contre-piégeage, dans l’équation de concentration empêche le soluté d’être trop piégé dans la phase solide et rend les simulations de champ de phase plus réalistes, particulièrement dans les cas de surfusions élevées.     

Nonequilibrium Solidification,Grain Refinements,and Recrystallization of Deeply Undercooled Ni-20 At. Pct Cu Alloys: Effects of Remelting and Stress

Grain refinement phenomena during the microstructural evolution upon nonequilibrium solidification of deeply undercooled Ni-20 at. pct Cu melts were systematically investigated. The dendrite growth in the bulk undercooled melts was captured by a high-speed camera. The first kind of grain refinement occurring in the low undercooling regimes was explained by a current grain refinement model. Besides, for the dendrite melting mechanism, the stress originating from the solidification contraction and thermal strain in the FMZ during rapid solidification could be a main mechanism causing the second kind of grain refinement above the critical undercooling. This internal stress led to the distortion and breakup of the primary dendrites and was semiquantitatively described by a corrected stress accumulation model. It was found that the stress-induced recrystallization could make the primary microstructures refine substantially after recalescence. A new method, i.e., rapidly quenching the deeply undercooled alloy melts before recalescence, was developed in the present work to produce crystalline alloys, which were still in the cold-worked state and, thus, had the driven force for recrystallization.

     

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.