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.     

Heat flow parameters affecting dendrite spacings during unsteady-state solidification of Sn-Pb and Al-Cu alloys

Solidification thermal parameters and dendrite arm spacings have been measured in hypoeutectic Sn-Pb and Al-Cu alloys solidified under unsteady-state heat flow conditions. It was observed that both primary and secondary spacings decreased with increased solute content for Sn-Pb alloys. For Al-Cu alloys, the primary spacing was found to be independent of composition, and secondary spacings decrease as the solute content is increased. The predictive theoretical models for primary spacings existing in the literature did not generate the experimental observations concerning the Sn-Pb and Al-Cu alloys examined in the present study. The theoretical Bouchard-Kirkaldy’s (BK’s) equation relating secondary spacings with tip growth rate has generated adequately the experimental results for both metallic systems. The insertion of analytical expressions for tip growth rate and cooling rate into the predictive model, or into the resulting experimental equations in order to establish empirical formulas permitting primary and secondary dendritic spacings to be determined as functions of unsteady-state solidification parameters such as melt superheat, type of mold, and transient metal/mold heattransfer coefficient is proposed.     

An analytical model for solute redistribution during solidification of planar,columnar, or equiaxed morphology

Existing models for solute redistribution (microsegregation) during solidification were reviewed. There are no analytical models that take into account limited diffusion in both the liquid and the solid phases. A new analytical mathematical model for solute redistribution was developed. Diffusion in liquid and in solid was considered. This model does not require a prescribed movement of the interface. It can be used for one-dimensional (1-D) (plate), two-dimensional (cylinder), or three-dimensional (3-D) (sphere) calculations. Thus, it is possible to calculate microsegregation at the level of primary or secondary arm spacing for columnar dendrites or for equiaxed dendrites. The solution was compared with calculations based on existing models, as well as with some available experimental data for the segregation of base elements in as cast Al-4. 9 wt pct Cu, INCONEL 718, 625, and plain carbon (0. 13 wt pct C) steel.     

The squeeze casting of hypoeutectic binary Al-Cu

A combination of thermocouple measurements, optical microscopy, and image analysis has been used to investigate the effects of applied pressure, melt temperature, mould insulation, and addition of grain refiner on the cooling/solidification behavior and resulting macro-and microstructure in squeeze cast Al-4.5 wt pct Cu ingots. Channel macrosegregates are formed in Al-4.5 wt pct Cu squeeze castings because of an increased rate of heat removal due to the application of pressure. The increased density of solute-rich liquid in the Al-Cu system causes channel segregates to form with a characteristic V pattern. Pressure and melt superheat increase the temperature gradients and cooling rate during squeeze casting but have only a minor effect upon the formation of channel segregates. The addition of grain refiner disperses but does not eliminate channel segregates. The application of pressure during squeeze casting changes the solidification behavior from mushy to near-plane front, leading to normal rather than inverse solute concentration profiles.     

Correlation between unsteady-state solidification conditions, dendrite spacings, and mechanical properties of Al-Cu alloys

The wide range of operational conditions existing in foundry and casting processes generates as a direct consequence a diversity of solidification microstructures. Structural parameters such as grain size and interdendritic spacings are strongly influenced by the thermal behavior of the metal/mold system during solidification, imposing, as a consequence, a close correlation between this system and the resulting microstructure. Mechanical properties depend on the microstructural arrangement defined during solidification. Expressions correlating the mechanical behavior with microstructure parameters should be useful for future planning of solidification conditions in terms of a determined level of mechanical strength, which is intended to be attained. In the present work, analytical expressions have been developed describing thermal gradients and tip growth rate during one-dimensional unsteady-state solidification of alloys. Experimental results concerning the solidification of Al-4.5 wt pct Cu and Al-15 wt pct Cu alloys and dendritic growth models have permitted the establishment of general expressions correlating microstructure dendrite spacings with solidification processing variables. The correlation of these expressions with experimental equations relating mechanical properties and dendrite spacings provides an insight into the preprogramming of solidification in terms of casting mechanical properties.     

Infiltration of fiber preforms by a binary alloy: Part I. Theory

During infiltration of a fiber preform by a binary hypoeutectic alloy, solid metal can form in the composite because of cooling at the fibers or at the mold wall. Contrary to the case of an unalloyed matrix, temperature, composition, and fraction solid may vary in the composite. This results in macrosegregation and microstructural heterogeneity within the composite casting. It is shown that solid metal that forms because of cooling at the fibers grows gradually behind the infiltration front, while the local temperature increases. Metal superheat, when present, serves to progressively remelt solid metal in the composite during infiltration and increases compositional and microstructural heterogeneity within the composite. General expressions are derived to describe heat, mass, and fluid flow during the infiltration process. In the case of unidirectional adiabatic infiltration driven by a constant applied pressure, a similarity method can be used to reduce the mathematical complexity of the problem. Numerical solution of the resulting equations then allows us to predict temperature, fraction solid, and composition profiles within the composite. With the further assumption of negligible thermal conduction, the problem lends itself to an analytical solution. The analysis is performed for the case of unidirectional adiabatic infiltration under constant applied pressure of 24 vol pct δ-alumina preforms by Al-4.5 wt pct Cu. Results indicate that there is significant latitude for control of macro-segregation and microstructure within cast fiber-reinforced alloys.     

Predicting the Effect of Pouring Temperature on the Crystallite Density,Remelting, and Crystal Growth Kinetics in the Solidification of Aluminum Alloys

In the present work, we developed an analytical model to describe the effect of pouring temperature on the crystallite density, remelting, growth kinetics, and the resultant final grain size for aluminum (Al)-based alloys synthesized using gravity casting. The model predicts that there are three regimes of pouring temperature/grain size-related behavior: (i) at low superheats, grain size is small and relatively constant; (ii) at intermediate levels of superheat, there appears to be a transitional behavior where grain size increases in a rapid, non-linear fashion; and (iii) at high superheats, grain size increases linearly with increasing temperature. This general pattern is expected to be shifted upward as distance from the bottom of the casting increases, which is likely a result of the slower cooling rates and/or longer solidification times with increasing distance from the bottom of the casting. To validate the model, a set of experiments has been conducted using Al-Cu and Al-Si alloys (i.e., Al-3.0 wt pct Cu, Al-4.5 wt pct Cu, and Al-A356.2 alloys), and the experimental measurements showed consistent results with theoretical predictions.     

The critical amount of nitrogen on the formation of nitrogen gas pores during solidification of 25Cr-7Ni duplex stainless steels

The effects of casting thickness, nitrogen contents, cooling rate, and Mn contents on the formation of nitrogen gas pores during solidification of 25Cr-7Ni-1.5Mo-3W duplex stainless steels (DSS) were quantitatively investigated. In the case of a sand mold, the formation of nitrogen gas pore was not affected by the thickness of castings, which ranged from 13 to 52 mm, and the critical initial nitrogen content for the formation of gas pore was 0.30 wt pct. In the case of the molds made of a stainless steel (STS) and water-cooled Cu, the critical initial nitrogen content did not change much compared to the sand mold. The amount of nitrogen gas pores increased with initial nitrogen contents of castings. The segregation of nitrogen and alloying elements was calculated with Thermo-Calc. The calculated data and the experimental results were compared to estimate the critical nitrogen partial pressure in the residual melt for the nucleation of gas pores. The effect of Mn content on the formation of gas pores was also investigated. The increase of Mn content from 1 wt pct to 2.6 wt pct changed the critical initial nitrogen content 0.30 wt pct to 0.40 wt pct.     

Gaseous oxygen absorption by molten iron and some Fe-AI,Fe-Si,Fe-Ti,and Fe-V alloys

An experimental investigation of the initial rates of oxygen dissolution in molten iron and some Fe-(≤9 pct Al), Fe-(≤6 pct Si), Fe-(≤1 pct Ti) and Fe-(≤1 pct V) alloys was carried out in pure oxygen. Two experimental techniques were employed in this study: a modified constant-volume Sieverts method and a falling droplet technique. It was found that the oxidation behavior of liquid iron-based alloys in gaseous pure oxygen as a function of alloy composition was similar under conditions of the falling droplet and modified constant volume Sieverts methods. Marked declines in the oxygen absorption rates were observed for Fe-Al and Fe-Si alloys when the initial alloy compositions reached 6 wt pct Al and 3 wt pct Si in iron, respectively. This behavior indicated a change in mode of oxidation from a burning to a passive type. Fe-Ti and Fe-V alloys initially containing up to 1 wt pct solute in iron exhibited only a burning type behavior. The sudden decline in oxygen absorption rate in molten iron-aluminum and iron-silicon alloys is discussed in terms of changes in the nature of the surface oxide film with increasing amounts of alloying element in the metal.     

Solute segregation behavior in spray-atomized Pd-Rh-V(Co) powders

In the present study, two palladium-based ternary alloys, with nominal compositions of Pd-4.9 wt pct Rh-0.72 wt pct V and Pd-5 wt pct Rh-1 wt pct Co, were spray atomized into micron-sized powders using a high-energy gas atomization technique. Solute segregation in the spray-atomized powders was systematically investigated. It was found that, as the powder size decreases, the solute segregation level decreases, either in terms of standard deviation of solute content from the average value, or in terms of the percentage of segregation-free regions in the powder, or in terms of the maximum (Rh)/minimum (Co, V) solute content in the powder. Moreover, theoretical analyses were carried out to evaluate the percentage of segregation-free regions, as well as the maximum/minimum solute content, in each powder. The theoretical analysis indicated that, among the different mechanisms governing the solute segregation behavior, undercooling levels experienced by the droplets, both prior to and after nucleation, played the most important role in decreasing the solute segregation level as the powder size decreased.     

The columnar to equiaxed transition in tin-lead alloys

Tin-lead alloys were solidified directionally and the position of the columnar to equiaxed transition determined on vertical sections of the ingots. The columnar length was found to increase with decreasing lead concentration and increasing heat transfer coefficient. A mathematical model of the heat flow in the system was used to determine local temperatures, temperature gradients, and velocities in the solidifying alloy. Comparing the position of the columnar to equiaxed transition and local thermal conditions, it was found that the transition occurred when the temperature gradient in the melt at the liquidus temperature was 0.11°C/mm for Sn 10 wt pct Pb, 0.10°C/mm for Sn 5 wt pct Pb, and 0.13°C/mm for Sn 15 wt pct Pb. The position of the transformation was found to be independent of melt superheat for the conditions considered.     

Effects of Residual Elements Arsenic,Antimony, and Tin on Surface Hot Shortness

Scrap-based electric arc furnace (EAF) steelmaking is limited by a surface cracking problem in the recycled steel products, which is known as surface hot shortness. This problem originates from the excessive amount of copper (Cu) in the steel scrap, which enriches during the oxidation of iron (Fe) and consequently melts and penetrates into the austenite grain boundaries. In this article, the effects of arsenic (As), antimony (Sb), and tin (Sn) on surface hot shortness were investigated. A series of Fe-0.3 wt pct Cu-x wt pct (As, Sb, or Sn) alloys with x content ranging from 0.06 to 0.10 wt pct was oxidized in air at 1423 K (1150 °C) for 60, 300, and 600 seconds inside the chamber of a thermogravimety analyzer (TGA) where heat is supplied through infrared radiation. Scanning electron microscopy (SEM) investigations show that (1) the presence of Sb and Sn results in severe grain boundary cracking, whereas the presence of As does not, (2) open cracks with Fe oxides were found beneath the oxide/metal interface in the Sb and Sn alloys, and (3) the oxide/metal interfaces for all As, Sb, and Sn alloys are planar. Penetration experiments of pure Cu and Cu-30 wt pct Sn liquid were also conducted in the chamber of a hot-stage confocal laser scanning microscopy (CLSM) in nonoxidizing atmosphere: (1) on the Fe-35 wt pct manganese (Mn) alloys to study the correlation between cracking and grain boundary characters, and (2) on the pure Fe substrates to exclude the bulk segregation effects of Sn on grain boundary cracking. It was found that grain boundary cracking rarely took place on low-energy grain boundaries. The results also suggest that the bulk segregation of Sn in the substrate is not necessary to promote significant grain boundary cracking, and as long as the liquid phase contains Sn, it will be highly embrittling.     

Modeling superheat removal during continuous casting of steel slabs

To investigate superheat dissipation in a continuous slab casting machine, mathematical models have been developed to compute fluid flow velocities, temperature distribution within the liquid pool, heat transfer to the inside of the solidifying shell, and its effect on growth of the shell. Three-dimensional (3-D) velocity and heat-transfer predictions compare reasonably with pre-vious experimental measurements and two-dimensional (2-D) calculations. The results indicate that the maximum heat input to the shell occurs near the impingement point on the narrow face and confirm that most of the superheat is dissipated in or just below the mold. Superheat tem-perature and casting speed have the most important and direct influence on heat flux. The effects of other variables, including mold width, nozzle jet angle, and submergence depth, are also investigated. Calculated heat flux profiles are then input to a one-dimensional (1-D) solidifi-cation model to calculate growth of the shell. Shell thickness profiles down the wide and narrow faces are compared with the predictions of conventional heat conduction models and available measurements.     

Mathematical modeling of particle segregation during centrifugal casting of metal matrix composites

A one-dimensional transient heat-transfer model coupled with an equation for force balance on particles is developed to predict the particle segregation pattern in a centrifugally cast product, temperature distribution in the casting and the mold, and time for complete solidification. The force balance equation contains a repulsive force term for the particles that are in the vicinity of the solid/liquid interface. The solution of the model equations has been obtained by the pure implicit finite volume technique with modified variable time-step approach. It is seen that for a given set of operating conditions, the thickness of the particle-rich region in the composite decreases with an increase in rotational speed, particle size, relative density difference between particles and melt, initial pouring temperature, and initial mold temperature. With reduced heat-transfer coefficient at the casting/mold interface, the solidification time increases, which, in turn, results in more intense segregation of solid particulates. Again, with increased initial volume fraction of the solid particulates in the melt, both the solidification time and the final thickness of the particulate-rich region increase. It is noted that for Al-Al₂O₃ and Al-SiC systems, in castings produced using finer particles, lower rotational speeds, and an enhanced heat-transfer coefficient at the casting/mold interface, the volume fraction of particles in the outer layer of the casting remains more or less the same as in the initial melt. However, for castings produced with coarser particles at higher rotational speeds and reduced heat-transfer coefficients at the casting/mold interface, intense segregation is predicted even at the outer periphery of the casting. In the case of the Al-Gr system, however, intense segregation is predicted at the innermost layers.     

Quantification of microsegregation during rapid solidification of Al-Cu powders

A new technique is introduced to quantify microsegregation during rapid solidification. The quantification involves calculation of the average solute solubility in the primary phase during solidification of an Al-Cu binary alloy. The calculation is based on using volume percent eutectic and weight percent of second phase (in the eutectic), which were obtained experimentally. Neutron diffraction experiments and stereology calculation on scanning electron microscope images were done on impulse atomized Al-Cu alloys of three compositions (nominal), 5 wt pct Cu, 10 wt pct Cu, and 17 wt pct Cu, atomized under N₂ and He gas. Neutron diffraction experiments yielded weight percent CuAl₂ data and stereology yielded volume percent eutectic data. These two data were first used to determine the weight percent eutectic. Using the weight percent eutectic and weight percent CuAl₂ in mass and volume balance equations, the average solute solubility in the primary phase could be calculated. The experimental results of the amount of eutectic, tomography results from previous work, and results from the calculations suggest that the atomized droplets are in metastable state during the nucleation undercooling of the primary phase, and the effect of metastability propagates through to the eutectic formation stage. The metastable effect is more pronounced in alloys with higher solute composition.     

Modeling of macrosegregation caused by volumetric deformation in a coherent mushy zone

A two-phase volume-averaged continuum model is presented that quantifies macrosegregation formation during solidification of metallic alloys caused by deformation of the dendritic network and associated melt flow in the coherent part of the mushy zone. Also, the macrosegregation formation associated with the solidification shrinkage (inverse segregation) is taken into account. Based on experimental evidence established elsewhere, volumetric viscoplastic deformation (densification/dilatation) of the coherent dendritic network is included in the model. While the thermomechanical model previously outlined (M. M’Hamdi, A. Mo, and C.L. Martin: Metall. Mater. Trans. A, 2002, vol. 33A, pp. 2081–93) has been used to calculate the temperature and velocity fields associated with the thermally induced deformations and shrinkage driven melt flow, the solute conservation equation including both the liquid and a solid volume-averaged velocity is solved in the present study. In modeling examples, the macrosegregation formation caused by mechanically imposed as well as by thermally induced deformations has been calculated. The modeling results for an Al-4 wt pct Cu alloy indicate that even quite small volumetric strains (≈2 pct), which can be associated with thermally induced deformations, can lead to a macroscopic composition variation in the final casting comparable to that resulting from the solidification shrinkage induced melt flow. These results can be explained by the relatively large volumetric viscoplastic deformation in the coherent mush resulting from the applied constitutive model, as well as the relatively large difference in composition for the studied Al-Cu alloy in the solid and liquid phases at high solid fractions at which the deformation takes place.     

Microstructure of Cu-Co alloys solidified at various supercoolings

The effects of supercooling on the microstructure of Cu-Co alloys containing 10 to 65 wt pct Co were investigated. Supercooling of the alloys below a characteristic temperature,t _SEP, resulted in a metastable phase separation into two liquids: one Co rich (L1) and the other Cu rich (L2). The microstructure of the phase-separated alloys consisted of spherulites of one phase embedded in a matrix of the other. The spherulites in alloys containing less than 40 wt pct Co were solidified from the L1 melt and from L2 in alloys containing more than 55 wt pct Co. Supercooling of copper alloys containing around 50 wt pct Co resulted in a duplex structure of fine and coarse dendrites. Microstructural evidence was presented for the formation of aε-Cu metastable phase in alloys containing less than 30 wt pct Co.     

Interaction of porosity with a planar solid/liquid interface

In this article, an investigation of the interaction between gas porosity and a planar solid/liquid (SL) interface is reported. A two-dimensional numerical model able to accurately track sharp SL interfaces during solidification of pure metals and alloys is proposed. The finite-difference method and a rectangular undeformed grid are used for computation. The SL interface is described through the points of intersection with the grid lines. Its motion is determined by the thermal and solute gradients at each particular point. Changes of the interface temperature because of capillarity or solute redistribution as well as any perturbation of the thermal and solute field produced by the presence of non-metallic inclusions can be computed. To validate the model, the dynamics of the interaction between a gas pore and a solidification front in metal alloys was observed using a state of the art X-ray transmission microscope (XTM). The experiments included observation of the distortion of the SL interface near a pore, real-time measurements of the growth rate, and the change in shape of the porosity during interaction with the SL interface in pure Al and Al-0.25 wt pct Au alloy. In addition, porosity-induced solute segregation patterns surrounding a pore were also quantified.