Solute Redistribution Equations Research Articles

On the local solute redistribution equation of macrosegregation, remelting and the formation of channel segregates

In this paper, we reassess the local solute redistribution equation (LSRE) of macrosegregation which, since it first appeared in 1960s, has served as a cornerstone for understanding the composition variations that occur in the solidification of alloys. We highlight some anomalies in earlier literature, in particular as regards the prediction of remelting as a precursor to the formation of channel segregates (freckles, A-segregates and V-segregates) in casting processes. Also, we suggest extensions to the LSRE for situations where solute diffusion in the solid phase is not negligible, as well as when the mode of solidification is unconsolidated equiaxed dendritic, rather than columnar/consolidated equiaxed dendritic. In addition, the significance of the equation for latter-day numerical computations of macrosegregation is also discussed.

Nanoparticle promoted solution treatment by reducing segregation in AA7034

The zinc-rich aluminum alloy AA7034 offers the highest tensile strength among all commercial aluminum alloys. However, the commercial AA7034 alloy was produced by Rapid Solidification. At a slow cooling rate, the high-alloying of AA7034 produces coarse secondary phases, causing severe segregation at interdendritic regions, and difficulties in later solution treatment. Nano-treating, by introducing a small volume fraction of nanoparticles, was applied to AA7034 alloy to successfully modify the microstructure and promote solution treatment at a slow cooling rate of 0.25 K/s. Primary phases were refined and the volume fraction of eutectic phases was modified. A modified local solute redistribution equation (LSRE) was applied to show how nanoparticles change the volume fraction of the eutectic phase by altering the viscosity of the liquid and the permeability of the coherent solid network in the terminal stage of solidification. In particular, by adding 1.0 to 2.0 vol percent of nanoparticles, the eutectic phases were not only refined and much more homogeneously distributed, but also their volume fractions were reduced by 24.5% and 30.1% respectively after down-hill casting. The modified microstructures enabled by nanoparticles allowed an accelerated solution treatment as well. The dissolution of the refined and distributed secondary phases was more complete during solution treatment. The nano-treated AA7034 also exhibited high strength and reasonable ductility after extrusion and heat treatment. The nano-treating approach has great potential to make conventional casting applicable to the difficult-to-cast/heat-treatable alloys through refined microstructure and promoted solution treatment.

A model for coupling prediction of inverse segregation and porosity for up-vertical unidirectional solidification of Al–Cu alloys

A mathematical model for coupling prediction of inverse segregation and porosity for up-vertical unidirectional solidification of Al–Cu alloys is established, based on the analysis of the redistribution behaviors of both gas element and the alloying elements in the mushy zone. The model first investigates the mass variation during solidification, including the specific mass variation and the volume shrinkage during the solidification of both primary phase and eutectic, which give rise to the back flow towards the dendrite roots. Then, the porosity fraction is obtained by combing hydrogen segregation and pressure depression associated with the feeding flow through a porous media being solved with Darcy's equation. Consequently, both the feeding flow and the 'local solute redistribution equation' are modified with the presence of porosity. Finally, the solute distribution is obtained by coupling a finite difference solidification model with the modified segregation model. Numerical results show that the overall back-flow, which is needed to compensate the volume shrinkage, decreases with the increase of initially dissolved hydrogen concentration. As a result, the inverse segregation induced mainly by the back-flow of liquid enriched with solute, decreases with the increase in porosity fraction. The numerical predictions show rather good agreement compared with the experimental results reported in literature with the same condition. This model is capable of predicting inverse segregation with the presence of porosity in vertical unidirectional solidification of Al–Cu alloys.

Advanced Solute Conservation Equations for Dendritic Solidification Processes Part I: Experiments and Theory

AbstractThe macrosegregation formed in dendritic equiaxed structure during early stages of solidification of Al‐4.5%Cu alloy has been studied by experimental work and by metallurgical study of cast samples taken from the experimental work. An experimental work was conducted to study the coupled effect of natural convection streams, interdendritic strain and mushy permeability of Al‐4.5%Cu aluminum alloy solidified in horizontal rectangular parallelepiped cavity at different superheats. The metallurgical study includes macro‐microstructure evaluation, measurements of grain size of equiaxed crystals and macrosegregation analysis. This study shows that the level of surface segregation exhibiting as positive segregation varies with superheat whereas the rest of inner ingot areas show the light fluctuation in segregation values. In addition to experimental work, there is a mathematical study which contains a complete derivation of local solute redistribution equations based on Fleming's approach under different solute diffusion mechanisms in the dendritic solid. This derivation includes also the effects of interdendritic strain and mushy permeability on the local solute redistribution distribution. Owing to the length of the study, it is presented in two parts. The first part describes the experimental work and its results as well as a detail derivation of solute conservation equations. This part also involves comparison and discussion between existing and proposed solute conservation equations. The second part contains the mathematical analyses of a two dimensional mathematical model of fluid flow, heat flow, solidification, interdendritic strain and macrosegregation. Also, this part also contains the numerical simulations by using finite difference technique “FDT” to create convection patterns, heat transfer, interdendritic strain, and macrosegregation distributions. This part also includes comparisons between the available measurements and model predications as well as full discussion of different model simulations. The mechanism of interdendritic strain generation and macrosegregation formation during solidification of dendritic equiaxed structure under different diffusion mechanisms in dendritic solid has also been explained and discussed.

Growth mechanism of primary silicon in cast hypoeutectic Al-Si alloys

The microstructural features of hypoeutectic Al-10%Si alloy were observed using optical microscopy and electron backscatter diffraction. The results show that primary silicon particles are frequently found in hypoeutectic alloys. Hence, the nucleation and growth mechanisms of the precipitation of primary silicon of hypoeutectic Al-10%Si alloy melts were investigated. It was discovered that Si atoms are easy to segregate and form Si-Si clusters, which results in the formation of primary silicon even in eutectic or hypoeutectic Al-Si alloys. In addition, solute redistribution caused by chemical driving force and large pile-ups or micro-segregation of the solute play an important role in the formation of the primary silicon, and the solute redistribution equations were derived from Jackson-Chalmers equations. Once Si solute concentration exceeds eutectic composition, primary silicon precipitates are formed at the front of solid/liquid interface.

Solute redistribution during planar growth of intermetallic compound with nil solubility

Planar growth of a stoichiometric intermetallic compound, Al9Co2, is first experimentally confirmed in directionally solidified Al-11at%Co alloy. Based on the principle of solute conservation, the solute redistribution equations during planar growth of a stoichiometric intermetallic compound are obtained. When diffusion is the solute transport mode in the melt, a linear equation is obtained to evaluate the variation of the solute concentration in the liquid at the solid/liquid interface front with the freezing distance. For planar growth of intermetallic compound with nil solubility, there is no steady state boundary layer at the solid/liquid interface front, unless the initial composition of the melt is equal to the composition of the intermetallic compound. Experimental and theoretical work demonstrates that planar growth of the two constituent intermetallic compound phases can lead to a banded structure in a peritectic system.

Solute Redistribution Resulting in Growth of Primary Silicon in Cast Hypoeutectic Al-Si Alloys

The experimental data present that primary silicon is precipitated from eutectic and hypoeutectic Al–Si alloy melts. It attributes to solute redistribution on the chemical driving force. Si atoms are easy to segregate to form Si-Si clusters, resulting in primary silicon precipitated from hypereutectic Al–Si alloy. Another reason is that primary silicon is precipitated from solid-liquid interface front once solute concentration in the solidification front exceeds eutectic composition. Solute redistribution equations are derivate from Jackson-Chalmers equation. The third reason is that precious little impurity turn Heterogeneous Nucleation of Si atoms.

The Influence of Solid State Diffusion on Microstructural Development during Solidification

The primary factors that effect solid state diffusion during solidification are described and binary solute redistribution equations that permit estimation of the significance of solid state diffusion are discussed. Model calculations suggest that solid state diffusion of substitutional alloying elements in FCC alloys is insignificant under most processing conditions, while that of interstitial alloying elements is likely to be complete. Experimental data that supports these results are presented. Several cases that highlight the practical importance of microsegregation on performance of engineering alloys are described as well as methods for avoiding or minimizing microsegregation for improved properties. A solute redistribution model for handling the limiting cases of solute diffusion in ternary alloys is presented and model calculations are reviewed to reveal the strong influence diffusion can have on the solidification path and resultant microstructure.

Numerical Approaches for microscopic modelling of solute redistribution during solidification of binary alloys

In the present study, two numerical approaches for single-domain modelling of microsegregation during solidification of binary alloys are presented. In the first approach, the concentration jump at the moving solid/liquid interface is formulated using a volumetric term and a Boolean function. The governing solute redistribution equation, valid for the whole domain comprising the solid and liquid regions, is derived in terms of the liquid phase composition. The effects of microstructure coarsening on microsegregation has been described and included in the model. In the second approach, the continuum mixture theory is utilized to derive a single domain solute redistribution equation in terms of the mixture composition. The solidification front motion and dendrite arm coarsening effects are accommodated by considering the representative elementary volume to consist of solid, interdendritic, and extradendritic liquid phases. Numerical solutions have been obtained using a control-volume based finite-difference method with a fixed grid. Good agreement has been observed between the predictions of the present fixed-domain models and the exact analytical and experimental results.

Analytical, numerical, and experimental analysis of inverse macrosegregation during upward unidirectional solidification of Al-Cu alloys

The present work focuses on the influence of alloy solute content, melt superheat, and metal/mold heat transfer on inverse segregation during upward solidification of Al-Cu alloys. The experimental segregation profiles of Al 4.5 wt pct Cu, 6.2 wt pct Cu, and 8.1 wt pct Cu alloys are compared with theoretical predictions furnished by analytical and numerical models, with transient hi profiles being determined in each experiment. The analytical model is based on an analytical heat-transfer model coupled with the classical local solute redistribution equation proposed by Flemings and Nereo. The numerical model is that proposed by Voller, with some changes introduced to take into account different thermophysical properties for the liquid and solid phases, time variable metal/mold interface heat-transfer coefficient, and a variable space grid to assure the accuracy of results without raising the number of nodes. It was observed that the numerical predictions generally conform with the experimental segregation measurements and that the predicted analytical segregation, despite its simplicity, also compares favorably with the experimental scatter except for high melt superheat.

Theoretical and experimental analysis of inverse segregation during unidirectional solidification of an Al–6.2 wt.% Cu alloy

An analytical solidification model has been coupled with the local solute redistribution equation proposed by Flemings and Nereo, permitting a complete analytical description of the positional variation of the segregation as a function of solidification thermal parameters. The theoretical predictions were compared with the experimental inverse segregation profile of a vertical directionally solidified Al–6.2 wt.% Cu ingot.

A unified microscale parameter approach to solidification-transport process-based macrosegregation modeling for dendritic solidification: Part II. Numerical example computations

This article, as the second part of the present work, applied the unified Φ-parameter-based micro/macrosegregation modeling, which was proposed and described in the first part, to two groups of alloy solidification systems. The first group is on closed simultaneous solidification systems of an Al-4.5 pct Cu alloy and a Fe-0.5 pct C based plain carbon steel to investigate the microscale solute-redistribution behaviors with different influential factors including the solid phase morphologies. A new solute-redistribution equation was derived from the present microscale unified Φ-parameter-based modeling, which includes more microscale dendrite-solidification features while preserving the simple function form equivalent to the well-known lever rule and the Scheil, Flemings-Brody, Clyne-Kurz, and Ohnaka models. In the second example computation group on a directionally solidified, bladelike casting system, a previous continuum model and the related PC-based codes developed by the present authors were extended and modified by incorporating the present microscale Φ-parameter approach to any extent of solid back-diffusion effects. Numerical simulations on the directional-solidification transport phenomena and micro/macrosegregation formations in a bladelike Al-4.5 pct Cu casting show the feasibility and the efficiency of the present solidification-transport process (STP)-based two-scale segregation model and the numerical methods.

Modeling solute redistribution and microstructural development in fusion welds of Nb-bearing superalloys

Solute redistribution and microstructural evolution have been modeled for gas tungsten arc fusion welds in experimental Ni base and Fe base Nb-bearing superalloys. The multi-component alloys were modeled as a ternary system by grouping the matrix (Fe, Ni, Cr) elements together as the “solvent” to form the γ component of the γ-Nb–C “ternary system”. The variation in fraction liquid and liquid composition during the primary L→ γ and eutectic type L→( γ+NbC) stages of solidification were calculated for conditions of negligible Nb diffusion and infinitely fast C diffusion in the solid phase. The proposed model is based on modifications of solute redistribution equations originally developed by Mehrabian and Flemings. Results of the calculations were superimposed on the pseudo-ternary γ-Nb–C solidification surfaces to predict the solidification reaction sequences along with the total and individual amounts of γ/NbC and γ/Laves eutectic-type constituents which form during solidification. Comparison was made to experimentally measured values, and reasonable agreement was found. The model results permit quantitative interpretations of composition–microstructure relations in these Nb-bearing experimental alloys and should provide useful insight into comparable commercial alloy systems as well.

A combined solubility product/new PHACOMP approach for estimating temperatures of secondary solidification reactions in superalloy weld metals containing Nb and C

Solubility data are combined with the new PHACOMP calculation procedure and solute redistribution equations to propose a simple method for estimating temperatures of the secondary solidification reactions that occur in superalloy weld metals. The variation in the primary solid composition with temperature is calculated with the Scheil equation for substitutional elements, while the lever law is used for C. The calculated compositions are used to determine the temperature-dependent variation in the solubility parameter and the metal d-level parameter, which is used in the new PHACOMP routine. The solubility parameter and metal d-level profiles are compared to temperature-dependent critical values to predict temperatures of secondary solidification transformations involving carbides and topologically close-packed phases such as Laves. The general approach is applied to experimental Ni-base and Fe-base alloys containing systematic variations in Nb, Si, and C. The procedure correctly predicts formation of the NbC phase prior to the Laves phase and reveals the effect of nominal alloy content on reaction temperature. Calculated reaction temperatures are in reasonable agreement with experimentally measured values, particularly when considering the relatively large range of material constants that are needed to make the calculations.

Solute redistribution and growth velocity response in directional solidification process

Based on the principle of solute conservation during directional solidification of binary alloys, the solute redistribution equations are obtained. A differential equation to evaluate the actual growth velocity is derived and the equation is solved by the method of matched asymptotic expansions. The solution shows the effect of kinematic supercooling and the coupling of thermal and mass transportation on the real growth of a planar interface. Numerical results indicate that there exists not only a hysterical effect of the solid-liquid interface during solidification but also a transient zone of growth velocity in the initial transient process. The growth velocity response to the externally imposed velocity sharply increases at the very beginning of solidification, then develops successively and progressively during the transient zone.

Effect of stress acting upon the solid network in the two-phase zone during solidification

Having taken the solid network movement in the two-phase zone into consideration, this paper develops the original local solute redistribution equation. The additional term div V′s in the new equation is dependent on the history of an effective stress σ′ acting upon the solid network. The solidifying isotherm advances quickly and there is not enough time to make the solid network compressible at large cooling rates. With decreasing cooling rate, movement of the isotherm is gradually slowed and there is enough time to make the network deformed. It is suggested that the compressibility due to the normal stress may produce “channel space” filled by unstable flow under the conditions of small cooling rate during solidification. The calculated results have demonstrated that the back-flow toward the centerline increases when the stress is considered.

Further discussions on the solute redistribution during dendritic solidification of binary alloys

After a review over former works about the solute redistribution during dendritic solidification, a new“local solute redistribution equation ”is deduced based on Flemings's model, where lim-ited diffusion in solid during solidification is carefully treated. Because a form parameter is also included, the equation can be used for the solidification processes with different shapes of den-drites. By solving the equation at the condition of directional solidification, more completefs-C, functions for both needlelike and platelike dendritic solidifications with both linear and parabolic solidification rates are obtained. As examples, the volume fractions of nonequilibrium phase in Al-4.5 pct Cu alloy is evaluated with differentfs-Cl functions. On the thinking that the dendrites in actual solidification process is usually between needlelike and platelike ones, the volume fraction of the nonequilibrium phase is suggested to be in the region between the one calculated by the model for platelike dendrites and that for needlelike dendrites. The relationship between the region and local solidification time is also presented by figures, which are compared with the data of former researchers.

Modification of mathematical analyses and related physical descriptions used to describe channel segregation

AbstractThe aim in this paper is to develop further and extend the application of the mathematical descriptions based on the previously developed local solute redistribution equation’ for the alloy-solidification process. To this end, a modified local solute redistribution equation for unidirectional solidification is presented and, using channel perturbation analysis based on this equation, a condition is developed for unstable flow and possible channel segregation which relates the interdendritic fluid velocity to the isotherm velocity. On assuming this condition to hold, a physical description related to the modified local solute redistribution equation is proposed, and a possible mechanism is presented for local remelting and the formation of channel segregation.