Interdendritic Liquid Research Articles

Modeling the overall solidification kinetics for undercooled single-phase solid-solution alloys. I. Model derivation

Departing from the volume-averaging method, the equiaxed solidification model was extended to describe the overall solidification kinetics of undercooled single-phase solid-solution alloys. In this model, a single grain, whose size is given assuming site saturation, is divided into three phases, i.e. the solid dendrite, the inter-dendritic liquid and the extra-dendritic liquid. The non-equilibrium solute diffusion in the inter-dendritic liquid and the extra-dendritic liquid, as well as the heat diffusion in the extra-dendritic liquid, is considered. The growth kinetics of the solid/liquid interface is given by the solute or heat balance, where a maximal growth velocity criterion is applied to determine the transition from thermal-controlled growth to solutal-controlled growth. A dendrite growth model, in which the nonlinear liquidus and solidus, the non-equilibrium interface kinetics, and the non-equilibrium solute diffusion in liquid are considered, is applied to describe the growth kinetics of the grain envelope. On this basis, the solidification path is described.

Microstructure development during directional solidification of Ti–45Al–8Nb alloy

Microstructure development and phase transition were investigated by quenching during directional solidification (DS) of Ti–45Al–8Nb (at.%) alloy. The solidification route and process can be controlled by adjustment of the solidification condition. A coexisted region of L + β + α was found at a low growth rate. The high temperature α phase, produced by peritectic transformation or β → α solid-state transformation or direct solidification from interdendritic liquid, was determined by the solidification conditions and can be identified by checking the β-segregation pattern in dendritic region. Lamellar orientation control was achieved when the interdendritic α grew on the top of peritectically formed α. High temperature α can be aligned continuously although α partly grew from the region of primary β interdendrite. Some elongated B2 particles were generated due to β-stablizers were centralized in interdendritic liquid during DS process. The volume fraction of the B2 particle decreased with decreasing growth rate ( V).

On the Formulation of a Freckling Criterion for Ni-Based Superalloy Vacuum Arc Remelting Ingots

A criterion for freckling prediction that includes the effect of a tilted solidification front was proposed and evaluated with experimental data available in the literature. The criterion is based on the maximum local Rayleigh number in the mush layer and was developed using Flemings’ criterion and assuming that the interdendritic liquid flow is governed by the Darcy law. The proposed form preserves the anisotropic nature of the permeability tensor throughout the derivation and provides improved resolution on freckle prediction. A clear separation between the freckled and nonfreckled experiments was obtained for all compositions. The effect of the tilted solidification front over the freckling potential was corroborated, and the results suggested that the directionality of permeability affects the location within the mush layer of the potential nucleation sites for the channels leading to freckles. A threshold zone was determined from the enclosing experiments data, and the range contained one of the proposed critical values for superalloys, which previously was developed by a completely different method.

Effect of dendrite coarsening on the solidification of interdendritic liquid in iron alloys

The solidification of the interdendritic liquid in austenitic 110G13L steel and white cast iron is studied. In the absence of dendrite coarsening, the solidification mechanism of the interdendritic liquid in the manganese steel is shown to change and solidification occurs in the form of polycrystalline aggregates around dendrites from different centers. The relation between the standard solidification of the interdendritic liquid and the dendrite coarsening in iron alloys is grounded.

Permeability in the Mushy Zone

When modeled at macroscopic length scales, the complex dendritic network in the solid-plus-liquid region of a solidifying alloy (the “mushy zone”) has been modeled as a continuum based on the theory of porous media. The most important property of a porous medium is its permeability, which relates the macroscopic pressure gradient to the throughput of fluid flow. Knowledge of the permeability of the mushy zone as a function of the local volume-fraction of liquid and other morphological parameters is thus essential to successfully modeling the flow of interdendritic liquid during alloy solidification. In current continuum models, the permeability of the mushy zone is given as a deterministic function of (1) the local volume fraction of liquid and (2) a characteristic length scale such as the primary dendrite arm spacing or the reciprocal of the specific surface area of the solid-liquid interface. Here we first provide a broad overview of the experimental data, mesoscale numerical flow simulations, and resulting correlations for the deterministic permeability of both equiaxed and columnar mushy zones. A extended view of permeability in mushy zones which includes the stochastic nature of permeability is discussed. This viewpoint is the result of performing extensive numerical simulations of creeping flow through random microstructures. The permeabilities obtained from these simulations are random functions with spatial autocorrelation structures, and variations in the local permeability are shown to have dramatic effects on the flow patterns observed in such microstructures. Specifically, it is found that “lightning-like” patterns emerge in the fluid velocity and that the flows in such geometries are strongly sensitive to small variations in the solid structure. We conclude with a comparison of deterministic and stochastic permeabilities which suggests the importance of incorporating stochastic descriptions of the permeability of the mushy zone in solidification modeling.

Criterion for Dendrite Fragmentation of Carbon Steel under Imposition of Linear Travelling EMS

In this study the columnar-to-equiaxed transition (CET) of carbon steel with 0.22-0.98 mass%C is investigated, based on the dendrite fragmentation theory of T. Campanella et al. The following conclusions are obtained: The criterion for dendrite fragmentation under linear EMS is obtained and verified by the 0.22-0.98 mass%C steel experiments. Investigation is carried out on the relation between the superficial velocity of the liquid phase and the actual velocity of the interdendritic liquid and the volume fraction of solid at the time of dendrite fragmentation (CET occurrence). At the same superficial velocity, the dendrite fragmentation critical volume fraction of solid is smaller in the case of high carbon steel. As a result, the CET occurrence is more difficult in the case of high carbon steel. Dendrite fragmentation induced by EMS is confirmed by the fragments of dendrite arms observed in the macrostructure. The length of dendrite fragmentation observed is about 1mm.

On The Formation of Extruded Surface Segregation Layer in Aluminum Direct Chill Casting Process

The aim of this research work was to develop a mathematical model to study the formation of extruded surface segregation phenomenon during the early stages of solidification in an aluminum direct chill casting process. The mathematical analyses contain a one dimensional mathematical model of heat flow, solidification, interdendritic strain and macrosegregation. Some typical cases in conventional direct chill casting processes related to increase surface segregation and fluctuation in macrosegregation with distance from the ingot surface were simulated. Good agreement was found by previous measurements and simulated results. Model results were discussed to explain the effects of different thermal solidification factors and interdendritic strain on the extruded surface segregation layer phenomenon. The results show that this phenomenon is sensitive to alloy composition and forms in the gap between the ingot surface and mould wall. Also, the model results point out that this phenomenon appears as a direct effect of the extrusion mechanism of interdendritic solute liquid associated with the interdendritic strain distribution in the coherent region in the mushy zone.

Erratum to: Modeling of the Equiaxed Dendrite Coarsening Based on the Interdendritic Liquid Permeability during Alloy Solidification

In this investigation, a model has been developed for determination of liquid flow permeability through dendritic structures during equiaxed dendrite growth. Two separate computational models have been coupled for modeling the permeability and coarsening in mushy alloys. The dendrite growth has been simulated using a cellular automation finite difference analysis, while the fluid flow has been modeled employing computational fluid dynamic. A new approach has also been proposed in calculation of the coarsening phenomenon as well as modification of the micro fluid flow based on switching between permeability and pressure gradients at high solid fractions. The predictions show that in high solid fractions, an iterated algorithm on the permeability gradient at a given pressure is closer to the experimental results. Therefore, under these circumstances, the continuity equation would not be satisfied by prediction and correction of the pressure field. Instead, the continuity equation would be satisfied by iteration and correction of the local permeability at a constant pressure gradient.

Properties of stochastic permeability

When modeled at macroscopic length scales, the complex dendritic network in the solid-plus-liquid region of a solidifying alloy (the “mushy zone”) has been modeled as a continuum based on the theory of porous media. The most important property of a porous medium is its permeability, which relates the macroscopic pressure gradient to the throughput of fluid flow. Knowledge of the permeability of the mushy zone as a function of the local volume-fraction of liquid and other morphological parameters is thus essential to successfully modeling the flow of interdendritic liquid during alloy solidification. Permeability is usually treated as a deterministic function of parameters that can be calculated by the model (e.g., local solid fraction, dendrite arm spacing). However, recent results show that the length scales that must be resolved are too small for the assumption of deterministic behavior to be valid, and investigators must confront the stochastic behavior of the permeability field. We describe early work to investigate the spatial structure of the stochastic permeability at these small scales, with a view to develop a comprehensive treatment of stochastic permeability to enable improved modeling.

Phase-field simulation of micropores constrained by a solid network

A 2D phase-field model has been developed in order to describe the morphology of a pore forming within interdendritic liquid channels and the geometrical effect of mechanical contacts with neighboring solid. The distribution of the solid, liquid and gas phases is calculated with a multiphase-field approach which accounts for the pressure difference between the liquid and gas phases, as well as diffusion of dissolved gases in the liquid. The model incorporates the perfect gas and Sievert’s laws to describe the concentration and partitioning of gas molecules or atoms at the pore/liquid interface. The results show that the presence of solid can substantially influence the volume and pressure of the pore. A pore constrained to grow in narrow liquid channels exhibits a substantially higher mean curvature, a larger pressure and a smaller volume as compared with a pore grown under unconstrained conditions.

Influence of Al and Nb on castability of a Ni-base superalloy, IN713LC

The influence of Al and Nb on the castability of the Ni-base superalloy, IN713LC, has been experimentally studied. It has been observed that increased Al and Nb additions result in decreased inter-dendritic shrinkage porosity. Porosities were observed in the inter-dendritic channels between abutting secondary dendrite arms within the grain as well as at the grain boundary. For the alloys doped with higher Al and Nb concentration, MC carbides form corresponding to a greater weight fraction liquid. The analysis reveals that the increased fraction liquid for the doped composition improves mush permeability and reduces pressure drop in the inter-dendritic liquid and therefore reduces shrinkage porosity.

Mathematical Model for Prediction of Composition of Inclusions Formed during Solidification of Liquid Steel

Non-metallic inclusions originate mainly during secondary steelmaking due to deoxidation and other exogenous sources. Additional inclusions form during cooling and subsequent freezing of liquid steel. Rejection of solutes by the solidifying dendrites causes segregation of solutes in the interdendritic liquid with consequent build-up of their thermodynamic supersaturation. The work reported in the present paper was undertaken to develop a computation procedure for prediction of inclusion compositions formed during cooling and solidification of liquid steel. The model has been applied to an inclusion sensitive grade of steel. Segregation of various solutes with progress of freezing has been calculated using the Clyne–Kurz microsegregation equation. A sequential computation procedure involving segregation equation and thermodynamic equilibrium calculations by the Factsage thermodynamic software has been developed. Compositions of inclusions at various solid fractions have been determined. Model predictions have been compared with literature as well as with inclusion compositions determined in continuously cast billet samples using SEM-EDS. Reasonably good correspondence between model predictions and observed inclusions have been obtained.

Investigation of microporosity formation mechanisms in A356 aluminium alloy castings

The problem of microporosity formation during solidification of aluminum alloy castings has great commercial importance and engineering interest due to its detrimental effects on mechanical properties. The present study concentrates on the mechanisms of microporosity formation in aluminium alloy castings. The microstructural characterisation of the produced samples was carried out with image analysis and SEM. It was observed that formation of shrinkage induced interdendritic porosity depends on solidification mode of the sample. The precipitation of intermetallic s-Al5FeSi platelets in the interdendritic liquid were also found to promote the interdendritic porosity formation by blocking interdendritic feeding. The results reveal that the oxide inclusions have a significant effect on formation of hydrogen precipitation induced regular micropores.

Weld metal cracking in laser beam welded single crystal nickel base superalloys

The weldability of two single crystal nickel base superalloys CMSX-4 and CMSX-486 were investigated by autogenous bead-on-plate laser beam welding. The analysis of microsegregation that occurred during solidification in the fusion zone indicated that while W and Re segregated into the γ dendrites, γ′ forming elements Ti, Ta, and Al as well as Hf were rejected into the interdendritic liquid. Fusion zone cracking was observed in both the materials and was observed to have initiated and propagated along the high angle stray grain boundaries in the weld metal of the two alloys. Increased formation of stray grains and increased extent of fusion zone cracking were, however, observed in the CMSX-486 alloy. Extension of solidification temperature range mainly by microsegregation of grain boundary strengthening elements B, Hf, and Zr in CMSX-486 was suggested to be responsible for the increased susceptibility of the alloy to stray grain formation and weld metal cracking compared to CMSX-4 superalloy.

Investigation into microsegregation during solidification of a binary alloy by phase-field simulations

Microsegregation within the columnar dendritic array is analyzed by performing a two-dimensional phase-field simulation. The influence of microstructure morphology on microsegregation is studied for various back diffusion conditions. Under the condition of no back diffusion, it is found that for the region without second dendrite arms the simulation result agrees well with the Scheil equation, but for the region with well developed second dendrite arms there is a severe deviation. This deviation is attributed to the dendrtic coarsening and the inhomogeneity of interdendritic liquid concentration caused by various interface curvatures. Under the condition of moderate back diffusion, it is found that the effect of dendritic morphology on microsegregation can be accounted by enhancing the Fourier number which characterizes the solid-state diffusion. However, this effect decreases with enhancing the back diffusion of the system.

Modelling and simulation of equiaxed dendritic structures permeability for Pb–Sn alloys

In this investigation, the permeability of interdendritic liquid flow through the equiaxial mushy zone has been modelled for Pb–Sn alloys based on experimental measurements. In the present work by solving Navier–Stokes equation, the flow pattern around the equiaxed dendrite has been obtained and then permeability has been determined by applying Darcy's law. Numerical determined values of permeabilities have been analysed by the use of Statistical Package for the Social Sciences (SPSS) statistical software. Then an experimental method has been used to measure the permeability for flow through equiaxial mushy zone of Pb–Sn alloys. Results show that increasing the solid fraction and perturbation decreases the permeability, but unlike the solid fraction, perturbation has less effect on permeability. It seems that the empirical relationship presented in this article could be used to estimate permeability in the mushy Pb–Sn alloys.

Prediction of Macrosegregation in Steel Ingots: Influence of the Motion and the Morphology of Equiaxed Grains

Although a significant amount of work has already been devoted to the prediction of macrosegregation in steel ingots, most models considered the solid phase as fixed. As a result, it was not possible to correctly predict the macrosegregation in the center of the product. It is generally suspected that the motion of the equiaxed grains is responsible for this macrosegregation. A multiphase and multiscale model that describes the evolution of the morphology of the equiaxed crystals and their motion is presented. The model was used to simulate the solidification of a 3.3-ton steel ingot. Computations that take into account the motion of dendritic and globular grains and computations with a fixed solid phase were performed, and the solidification and macrosegregation formation due to the grain motion and flow of interdendritic liquid were analyzed. The predicted macrosegregation patterns are compared to the experimental results. Most important, it is demonstrated that it is essential to consider the grain morphology, in order to properly model the influence of grain motion on macrosegregation. Further, due to increased computing power, the presented computations could be performed using finer computational grids than was possible in previous studies; this made possible the prediction of mesosegregations, notably A segregates.

Modeling of the Equiaxed Dendrite Coarsening Based on the Interdendritic Liquid Permeability during Alloy Solidification

Modeling of the Equiaxed Dendrite Coarsening Based on the Interdendritic Liquid Permeability during Alloy Solidification

Microtomographic characterization of columnar Al–Cu dendrites for fluid flow and flow stress determination

During the twin roll casting of Al alloys, the interdendritic liquid may flow as the two solidification fronts are compressed together between the rolls. This can lead to defects such as centerline segregation. To understand the flow properties of the interdendritic liquid, samples of Al–12wt.% Cu were solidified directionally in a Bridgman furnace and quenched to capture the growing columnar dendritic structures. The quenched samples were scanned using a laboratory X-ray microtomography (XMT) unit to obtain the 3D structure with a voxel resolution of 7.2μm. Image analysis was used to separate the Al dendrite from the interdendritic Al–Al2Cu eutectic. Flow between the dendrites was simulated by solving the Stokes equation to calculate the permeability tensor as a function of the fraction solid. The results were compared to prior experimental measurements and calculations using synchrotron tomography observations of equiaxed structures. Elasto–plastic finite element (FE) simulations were performed on the dendritic structures to determine flow stress behavior as a function of fraction solid. It was found that the standard approximations for the reduction in flow stress in the semi-solid have a variation in excess of 100% from that calculated using the true structure. Therefore, it is critical to simulate the actual dendrite for effective flow stress determination.

An examination of effects of solidification parameters on permeability of a mushy zone in castings

A model describing the development of dendritic structure and the resulting gradient of flow resistance to interdendritic liquid is presented. The Hagen–Pousielle version of D’Arcy’s equation for flow through a porous structure is developed as a function of cooling rate and liquid volume fraction. Applied to finite elements in a unidirectionally cooled casting model, permeability gradient, feeding flow-rate required to prevent porosity, and mushy-zone liquid pressure drop at this flow rate are evaluated for the simple Fe–2Cr–0.5C and Al–5Cu castings exhibiting asymptotic and linear temperature profiles, respectively. The model shows permeability of the dendritic structure in the mushy zone dropping sharply, approaching the root of solidification front (solidus). Also shown is the effect of relative magnitude of primary and secondary arm spacing. If secondary dendrite arm spacing approaches primary arm spacing, the permeability for flow normal to primary dendrite arms approaches or even surpasses the permeability for flow parallel to primary dendrite arms.