Columnar Solidification Research Articles

A truncated-Scheil-type model for columnar solidification of binary alloys in the presence of melt convection

A two-phase truncated-Scheil-type model is developed for columnar solidification of binary alloys in the presence of melt convection and liquid undercooling ahead of the primary dendrite tips. The model is derived from a three-phase model, which takes into account liquid undercooling ahead and behind the primary tips. These models are used to simulate a numerical solidification benchmark problem, and their predictions are compared with those of a Scheil-type model that disregards liquid undercooling and solute diffusion in solid. Simulation results reveal that the predictions of the truncated-Scheil-type and three-phase models are nearly identical, indicating that undercooling behind the primary tips can be disregarded and the truncated-Scheil-type model can replace the significantly more complex three-phase model. The truncated-Scheil-type and three-phase models smoothly recover the Scheil-type model as the value of the dendrite tip selection parameter is increased. Taking into account liquid undercooling changes the melt convection pattern around the columnar front and the form of channel segregates but does not change the overall macrosegregation pattern.

Strongly Out-of-Equilibrium Columnar Solidification During Laser Powder-Bed Fusion in Additive Manufacturing

Laser-based additive manufacturing offers a promising route for 3D printing of metallic parts. We evidence experimentally a particular columnar solidification microstructure in a Laser Powder-Bed Fusion processed Inconel 718 nickel-based alloy, that we interpret using phase-field simulations and classical dendritic growth theories. Owing to the large temperature gradient and cooling rate, solidification takes places through dendritic arrays wherein the characteristic length scales, i.e tip radius, diffusion length and primary spacing, are of the same order. This leads to a weak mutual interaction between dendrite tips, and a drastic reduction of side-branching. The resulting irregular cellular-like solidification pattern then remains stable on time scales comparable to the complete melt pool solidification, as observed in the as-built material.

Permeability prediction for flow normal to columnar solidification structures by large–scale simulations of phase–field and lattice Boltzmann methods

Computer simulation is the most promising approach for the systematical permeability prediction of liquid flow in the mushy zone for various solidification conditions. In this study, we propose a permeability prediction method by using large–scale simulation of phase–field and lattice Boltzmann methods. Using the proposed method, permeability predictions are performed for the flow normal to the columnar solidification structures with multiple dendrites/cells. In addition, the prediction is also performed for columnar solidification structures with periodically arranged dendrites/cells to evaluate the permeability in the full range of solidification fraction and investigate the effect of primary arm array on the permeability. As a result, it is concluded that the dimensionless permeability using a specific interface area of columnar solidification structures can be well approximated by that of a regular hexagonal array of cylinders. Moreover, it is confirmed that the columnar dendrite/cell structure with a periodic regular hexagonal array provides realistic permeability predictions for columnar solidification structures.

Macrosegregation Formation in an Al\u2013Si Casting Sample with Cross-sectional Change During Directional Solidification

A unidirectional solidification experiment of hypoeutectic Al-7.0 wt% Si alloy against gravity direction in a cylindrical mold with cross-sectional change was made, and the macrosegregation in different parts of the as-solidified sample was investigated (Ghods et al. in J Cryst Growth 441:107–116, 2016; J Cryst Growth 449:134–147, 2016). The current study is to use a two-phase columnar solidification model to analyze the segregation mechanisms as used in this experiment. Following flow phenomena and their contributions to the formation of macrosegregation are simulated and compared: (1) solidification shrinkage-induced feeding flow; (2) thermo-solutal convection; and (3) combined thermo-solutal convection and shrinkage-induced feeding flow. The shrinkage-induced feeding flow leads to an inverse (positive) segregation in the bottom part, and a severe negative segregation in the part below cross-sectional change. Thermo-solutal buoyancy leads to a so-called steepling convection in the main part of the sample (away from the bottom and cross-sectional change), and this kind of flow leads to a positive macrosegregation near the sample surface. The calculations have successfully explained the experimental result of macrosegregation.

A finite element formulation for macrosegregation during alloy solidification using a fractional step method and equal-order elements

A finite element formulation of the “minimal” solidification model is presented for the prediction of macrosegregation during two-dimensional (2D) columnar solidification of binary alloys. A fractional step method is extended to solve the thermosolutal convection that has a damping in the mushy zone during solidification. Using this method, the velocity and pressure are decoupled and interpolated by linear equal-order triangular elements, resulting in decoupled systems that can be solved simply and efficiently. For convection-diffusion equations of energy, solute and momentum, the consistent streamline upwind Petrov-Galerkin (SUPG) method and the second-order Crank-Nicolson scheme are used for the discretization and integration over the spatial domain, respectively. A solution procedure is designed to couple the resolutions of conservations of energy, solute and momentum, as well as the microsegregation model at an overall computational efficiency and accuracy. The formulation is first validated and then applied to predict macrosegregation during solidification of Pb-18 wt%Sn and Sn-10 wt%Pb alloys in a rectangular mold. The formation of macrosegregation is investigated, and comparisons with another finite element method (FEM) based code are made.

Influence of natural and forced gravity conditions during directional columnar solidification

In a recent study, the present authors reported an analysis of a transient process of directional solidification of TiAl alloys in the absence of convection (Battaglioli et al., 2017). The adopted front tracking model, also coupled with indirect methods for predicting columnar to equiaxed transition (CET), showed how the development of different grain regions, namely axial columnar, radial columnar and equiaxed, depends on process parameters such as temperature distribution and applied cooling rates, as well as on properties such as the degree of inoculation of the melt and nucleation undercooling required for equiaxed growth. In this paper, the previous front tracking model is significantly developed, by including the solution of Navier-Stokes equations in order to predict thermal convection in the liquid region as well as in the columnar mush (treated as an isotropic porous medium). This improvement is introduced with the aim to investigate TiAl alloys solidification under different gravity conditions. Accordingly, the simulation setup employed in the study reproduces the one used in experimental campaigns carried out on the MAXUS 9 sounding rocket (microgravity) and on ESA’s Large Diameter Centrifuge (hypergravity), within the framework of ESA’s GRADECET (Gravity Dependence of CET in TiAl alloys) project. The ability of the model in predicting thermal convection is demonstrated by considering several case studies. Results show that the evolution of fluid flow patterns in the samples depends on the external forces considered, combined to the transient change of axial and radial temperature gradients that occur during the solidification process. A parametric study of the directional solidification process on a centrifuge is performed by changing the value of the centrifuge arm and rotation rate and results are compared to predictions derived from non-dimensional considerations.

Simulation of a macrosegregation benchmark with a meshless diffuse approximate method

PurposeThe purpose of this paper is to simulate a macrosegregation solidification benchmark by a meshless diffuse approximate method. The benchmark represents solidification of Al 4.5 wt per cent Cu alloy in a 2D rectangular cavity, cooled at vertical boundaries.Design/methodology/approachA coupled set of mass, momentum, energy and species equations for columnar solidification is considered. The phase fractions are determined from the lever solidification rule. The meshless diffuse approximate method is structured by weighted least squares method with the second-order monomials for trial functions and Gaussian weight functions. The spatial localization is made by overlapping 13-point subdomains. The time-stepping is performed in an explicit way. The pressure-velocity coupling is performed by the fractional step method. The convection stability is achieved by upstream displacement of the weight function and the evaluation point of the convective operators.FindingsThe results show a very good agreement with the classical finite volume method and the meshless local radial basis function collocation method. The simulations are performed on uniform and non-uniform node arrangements and it is shown that the effect of non-uniformity of the node distribution on the final segregation pattern is almost negligible.Originality/valueThe application of the meshless diffuse approximate method to simulation of macrosegregation is performed for the first time. An adaptive upwind scheme is successfully applied to the diffuse approximate method for the first time.

A Comprehensive Comparison of the Analytical and Numerical Prediction of the Thermal History and Solidification Microstructure of Inconel 718 Products Made by Laser Powder-Bed Fusion

Abstract The finite-element (FE) model and the Rosenthal equation are used to study the thermal and microstructural phenomena in the laser powder-bed fusion of Inconel 718. A primary aim is to comprehend the advantages and disadvantages of the Rosenthal equation (which provides an analytical alternative to FE analysis), and to investigate the influence of underlying assumptions on estimated results. Various physical characteristics are compared among the FE model, Rosenthal equation, and experiments. The predicted melt pool shapes compared with reported experimental results from the literature show that both the FE model and the analytical (Rosenthal) equation provide a reasonably accurate estimation. At high heat input, under conditions leading to keyholing, the reported melt width is narrower than predicted by the analytical equation. Moreover, a sensitivity analysis based on choices of the absorptivity is performed, which shows that the Rosenthal approach is more sensitive to absorptivity, compared with the FE approach. The primary reason could be the effect of radiative and convective losses, which are assumed to be negligible in the Rosenthal equation. In addition, both methods predict a columnar solidification microstructure, which agrees well with experimental reports, and the primary dendrite arm spacing (PDAS) predicted with the two approaches is comparable with measurements.

Investigation of laser metal deposited Alloy 718 onto an EN 1.4401 stainless steel substrate

This paper focuses on how process parameters affect the deposition of Alloy 718 onto an EN 1.4401 stainless steel substrate in terms of secondary phase formation, dilution and hardness. A columnar solidification structure with elongated grains growing in the direction normal to the substrate was observed for all parameters. In the interdendritic regions, phases with a high content of Niobium were identified. Scanning Electron Microscopy imaging and Energy Dispersive Spectroscopy measurements revealed these phases to most likely be Laves phase and Nb-carbides. Temperature measurements indicated no significant aging in the deposits. Considerable enrichment of iron was found in the initially deposited layers due to dilution from the substrate. The increased content of iron seemed to aid in forming constituents rich in niobium which, in turn, influenced the hardness. The highest mean hardness was noted in the sample with the lowest area fraction of Nb-rich constituents.

Uncertainty Quantification in Modeling Metal Alloy Solidification

Numerical simulations of metal alloy solidification are used to gain insight into physical phenomena that cannot be observed experimentally. These models produce results that are used to draw conclusions about a process or alloy and often compared to experimental results. However, uncertainty in model inputs cause uncertainty in model results, which have the potential to significantly affect conclusions drawn from their predictions. As a step toward understanding the effect of uncertain inputs on solidification modeling, uncertainty quantification (UQ) and sensitivity analysis are performed on a transient model of solidification of Al–4.5 wt % Cu in a rectangular cavity. The binary alloy considered has columnar solidification morphology, and this model solves equations for momentum, temperature, and species conservation. UQ and sensitivity analysis are performed for the degree of macrosegregation and solidification time. A Smolyak sparse grid algorithm is used to select input values to construct a polynomial response surface fit to model outputs. This polynomial is then used as a surrogate for the complete solidification model to determine the sensitivities and probability density functions (PDFs) of the model outputs. Uncertain model inputs of interest include the secondary dendrite arm spacing (SDAS), heat transfer coefficient, and material properties. The most influential input parameter for predicting the macrosegregation level is the dendrite arm spacing, which also strongly depends on the choice of permeability model. Additionally, the degree of uncertainty required to produce accurate predictions depends on the outputs of interest from the model.

Overview: Application of heterogeneous nucleation in grain-refining of metals.

In all of metallurgical processing, probably the most prominent example of nucleation control is the "inoculation" of melts to suppress columnar solidification and to obtain fine equiaxed grain structures in the as-cast solid. In inoculation, a master alloy is added to the melt to increase its solute content and to add stable particles that can act as nucleants for solid grains. This is important for alloys of many metals, and in other cases such as ice nucleation in living systems, but inoculation of aluminum alloys using Al-5Ti-1B (wt.%) master alloy is the exemplar. The key elements are (i) that the chemical interactions between nucleant TiB2 particles and the melt ensure that the solid phase (α-Al) exists on the surface of the particles even above the liquidus temperature of the melt, (ii) that these perfect nucleants can initiate grains only when the barrier for free growth of α-Al is surmounted, and (iii) that (depending on whether the melt is spatially isothermal or not) the release of latent heat, or the limited extent of constitutional supercooling, can act to limit the number of grains that is initiated and therefore the degree of grain refinement that can be achieved. We review recent studies that contribute to better understanding, and improvement, of grain refinement in general. We also identify priorities for future research. These include the study of the effects of nanophase dispersions in melts. Preliminary studies show that such dispersions may be especially effective in achieving grain refinement, and raise many questions about the underlying mechanisms. The stimulation of icosahedral short-range ordering in the liquid has been shown to lead to grain refinement, and is a further priority for study, especially as the refinement can be achieved with only minor additions of solute.

The spatial and temporal distribution of dendrite fragmentation in solidifying Al-Cu alloys under different conditions

The directional columnar solidification of Al-Cu alloy dendrites was studied in-situ by synchrotron X-ray radiography to investigate the spatial and temporal distribution of dendrite fragmentation, without and with an applied pulsed electromagnetic field (PEMF) and in different solidification directions. The differing arrangements had a strong and reproducible influence on fragmentation behaviour that was rationalised using a model of solute-driven re-melting of dendrite arms in microstructural regions of high vulnerability. Although absolute rates of fragmentation varied, all the fragmentation distributions had a characteristic peak in fragmentation rate at a distance of 500–800 μm into the mushy zone. Consistent with the need for solute-rich liquid flow for dendrite root re-melting, this peak was explained to occur under conditions where there was both comparatively steep liquid solute concentration gradients, measured directly from the radiographs, and relatively high permeability of the dendrite network to liquid flow.

Numerical analysis of macrosegregation in vertically solidified Pb-Sn test castings – Part I: Columnar solidification

Macrosegregation in a casting is caused by different flow and crystal movement phenomena during solidification. In order to get basic understanding of the macrosegregation mechanisms, 8 simple test cases of a 2D casting (50×60mm2) with unidirectional cooling from either top or bottom were configured and calculated by using a volume-average based three-phase solidification model (Wu and Ludwig, 2006). Two different alloys were considered: Sn-10wt.%Pb with solute element Pb heavier than the solvent Sn; Pb-18wt.%Sn with solute element Sn lighter than the solvent Pb. The work is presented in two parts. Part I (this paper) presents 4 cases of pure columnar solidification. Direction of the thermal-solutal buoyancy force of the interdendritic melt can be estimated according to the alloy (Sn-10wt.%Pb or Pb-18wt.%Sn); and the possible flow pattern can also be estimated when the solidification direction (upwards or downwards) is known. As consequence, the macrosegregation tendency should be estimated as well. However, the modelling results show that complicated details of macrosegregation distribution can develop due to the complexity of the transient flow. Other 4 cases of equiaxed solidification will be presented as Part II in next paper.

Numerical analysis of macrosegregation in vertically solidified Pb-Sn test castings – Part II: Equiaxed solidification

In order to get basic understandings of the macrosegregation mechanisms, 8 test cases of a 2D casting (50×60mm2) with unidirectional cooling from either top or bottom were simulated. Part I of this two-part study has presented 4 cases of pure columnar solidification. Part II will present the other 4 cases of equiaxed solidification. Two alloys were considered: Sn–10wt%Pb with solute element Pb heavier than the solvent Sn; Pb–18wt%Sn with solute element Sn lighter than the solvent Pb. The solidified equiaxed crystal is considered always heavier than the melt. With such simple configurations, it is believed that the macrosegregation tendency during equiaxed solidification can be estimated. However, the modeling results show that complicated macrosegregation patterns can develop due to the complexity of the event of equiaxed-to-columnar transition (ECT). Analysis of the segregation mechanisms for each case is made. The three-phase mixed columnar-equiaxed solidification model (Wu and Ludwig, MMTA, 2006, p. 1613) is modified and used for this study. One modification is to consider the heterogeneous nucleation of equiaxed crystals and transport of inoculants (embryos). The inoculants serve as heterogeneous nucleation sites for the equiaxed crystals, which can be transported as well. Another modification is to include the ECT. When all inoculants are consumed and the equiaxed crystals settle down in the bottom region, the remaining melt in the upper part can only solidify as columnar structure. Those columnar structures develop/grow from the packed equiaxed crystals by triggering the event of ECT.

Simulation of thermos-solutal convection induced macrosegregation in a Sn-10%Pb alloy benchmark during columnar solidification

In order to investigate the effect of thermo-solutal convection on the formation of macrosegregation during columnar solidification, simulations with a liquid-columnar two phase model were carried out on a 2D rectangular benchmark of Sn-10%Pb alloy. The solidification direction in the benchmark is unidirectional: (') downwards from top to bottom or (2) upwards from bottom to top. Thermal expansion coefficient, solutal expansion coefficient and liquid diffusion coefficient of the melt are found to be key factors influencing the final macrosegregation. The segregation range and distribution are also strongly influenced by the benchmark configurations, e.g. the solidifying direction (upwards or downwards) and boundary conditions, et al. The global macrosegregation range increases with the velocity magnitude of the melt during the process of solidification.

Modelling of Columnar-to-Equiaxed and Equiaxed-to- Columnar Transitions in Ingots Using a Multiphase Model

We present a new method to handle a representative elementary volume (REV) with a mixture of columnar and equiaxed grains in ingot castings in the framework of an Eulerian volume averaged model. The multiscale model is based on a previously established fully equiaxed model. It consists of a three-phase (extra-granular liquid, intra-granular liquid and solid) grain-growth stage coupled with a two-phase (solid and liquid) macroscopic transport stage accounting for grain and nuclei movement.In this context, we take into account the formation of a columnar structure and its development using a simplified front-tracking method. Columnar solidification is coupled with the growth of equiaxed grains ahead of the columnar front. The particularity of the model is the treatment of concurrent growth of mixed columnar and equiaxed structures only in the volumes that contain the columnar front. Everywhere else, the structure is considered either fully columnar or fully equiaxed. This feature allows for reasonable computational times even in industrial size castings, while describing the solutal and mechanical blocking phenomena responsible for the Columnar-to-Equiaxed Transition.After a validation of the model, we discuss the numerical results for a 6.2-ton industrial steel ingot by comparison with experimental measurements. Final maps for macrosegregation and grain structures size and morphology are analysed. Furthermore, we quantify the impact of nuclei formation through fragmentation along the columnar front on the result.An attempt at predicting the occurrence of the Equiaxed-to-Columnar Transition in the later phases of the process is also made.

Mesoscopic modeling of columnar solidification and comparisons with phase-field simulations

We use two complementary modeling approaches for the simulation of columnar growth in directional solidification of organic alloys: a phase field model and a mesoscopic envelope model of dendritic growth. While the phase-field method captures the details of the dendritic structure and of the growth dynamics, the mesoscopic model approximates the complex dendritic morphology by its envelope. The envelope growth is deduced from the velocities of the dendrite tips, calculated by an analytical LGK-type tip model that is matched to the temperature and concentration fields in the stagnant film around the envelope. The computational cost of the mesoscopic model is several orders of magnitude lower and can bridge the gap between phase-field and macroscopic models. We demonstrate the applicability of the mesoscopic model to columnar growth and we discuss in particular its capabilities to predict the primary dendrite arm spacing.

Characterization of microstructure and strain response in Ti–6Al–4V plasma welding deposited material by combined EBSD and in-situ tensile test

Additive layer manufacturing (ALM) of aerospace grade titanium components shows great promise in supplying a cost-effective alternative to the conventional production routes. Complex microstructures comprised of columnar remnants of directionally solidified β-grains, with interior inhabited by colonies of finer α-plate structures, were found in samples produced by layered plasma welding of Ti–6Al–4V alloy. The application of in-situ tensile tests combined with rapid offline electron backscatter diffraction (EBSD) analysis provides a powerful tool for understanding and drawing qualitative correlations between microstructural features and deformation characteristics. Non-uniform deformation occurs due to a strong variation in strain response between colonies and across columnar grain boundaries. Prismatic and basal slip systems are active, with the prismatic systems contributing to the most severe deformation through coarse and widely spaced slip lines. Certain colonies behave as microstructural units, with easy slip transmission across the entire colony. Other regions exhibit significant deformation mismatch, with local build-up of strain gradients and stress concentration. The segmentation occurs due to the growth morphology and variant constraints imposed by the columnar solidification structures through orientation relationships, interface alignment and preferred growth directions. Tensile tests perpendicular to columnar structures reveal deformation localization at columnar grain boundaries. In this work connections are made between the theoretical macro- and microstructural growth mechanisms and the observed microstructure of the Ti–6Al–4V alloy, which in turn is linked to observations during in-situ tensile tests.

Modelling the creation and destruction of columnar and equiaxed zones during solidification and melting in multi-pass welding of steel

The authors present a novel meso-scale (mm–cm) numerical model of multi-pass welding of stainless steel based on the front tracking formulation. During a typical multi-pass notched butt weld, multiple cycles of melting and solidification occur throughout the weld pool, where previously solidified equiaxed zones are destroyed by melting and subsequently resolidify in columnar form. These thermal cycles result from the thermal effects of the current deposition of liquid filler metal being superimposed onto those of the previous pass. The advancement of the liquidus isotherm is employed to characterize melting, while both columnar and equiaxed dendritic growth in the undercooled melt are considered in the model of solidification. Columnar solidification is simulated using a front tracking method where computational markers are employed to explicitly define the advancing columnar front. Competing equiaxed solidification is modelled using a volume averaging approach. During the first and second welding passes of a case study, the weld pool solidifies as purely columnar. This is due to the high thermal gradient present in the weld pool, which results in a very small region of undercooled liquid ahead of the columnar front and subsequently low levels of equiaxed grain nucleation/growth. Towards the end of the third and final pass the equiaxed grains have sufficient dwell time in the undercooled liquid to form a coherent network and present a physical barrier to the advancing columnar front. A Columnar to Equiaxed Transition (CET) occurs and the remainder of the weld pool solidifies in a fully equiaxed microstructure. The results are compared with Hunt’s analytical model of CET and the agreement is reasonably good. The occurrence of CET throughout the weld pool is directly predicted via this front tracking model without the need to rely on estimated thermal gradients and solidification rates. Further simulations indicate that CET can be promoted, even in early passes, by increasing the grain refiner level and/or by pre-heating the parent metal.

Observation of flow regimes and transitions during a columnar solidification experiment

Experimental data for the validation of numerical models coupling solidification and hydrodynamics are very rare. Many experiments made in the field of solidifications are performed with pure metals or alloys (Al-Cu, Pb-Sn, etc) which are opaque and do not allow direct observation of the hydrodynamic. Only the results related to solidification such as grain size and orientation, or macro-segregation are usually used for the validation. The present paper is dedicated to the description of well-controlled experiments where both solidification and fluid dynamic can be simultaneously observed. The important point is the almost purely columnar nature of the solidified mushy region. To our knowledge this is the very first reported macro-scale experiment with almost purely columnar solidification where the flow was measured with a PIV technique. The experiments consist in studying the hydrodynamics during the columnar solidification of a H2O-NH4Cl hypereutectic alloy in a die cast cell. Particle image velocimetry was employed to measure the flow velocity in the liquid bulk. Different flow regimes generated by complex thermo-solutal double diffusive convection were observed. In the beginning of the solidification the solutal buoyancy generates a turbulent flow, which is progressively replaced by the development of stratification from the top of the cell. Later, the stratification leads to the development of a long lasting meandering flow, which filled almost all the liquid region. The kinetic energy of the flow was calculated and it was found out that it decreased with time. The solidification front was smooth and no freckles appeared in the mushy zone. The evolution of the thickness of the mushy zone was measured. As this experiment showed a good reproducibility it represents an excellent benchmark for validation of the numerical models that target the simultaneous prediction of flow dynamics and solidification.