Planar Solid-liquid Interface Research Articles

Directional solidification of the Al0.8CrFeNi2.2 eutectic high-entropy alloy

The concepts of high-entropy alloys and eutectic in-situ composites have been applied as an interesting approach to achieve materials with an improved balance between strength and ductility. In this study, the Al0.8CrFeNi2.2 eutectic high-entropy alloy (EHEA), formed by the ordered body-centered cubic (B2) NiAl-rich and face-centered cubic (FCC) CrFeNi-rich phases, was fabricated through arc melting and directionally solidified in a Bridgman setup at translation rates of 1.4–14 μm/s. At the lowest rates, growth occurred with a planar solid–liquid interface, resulting in a highly regular lamellar array. As the translation rate increased, an interface transition from planar to cellular occurred, resulting in eutectic grains surrounded by layers of anomalous eutectic structure. The dependence of the lamellar spacing on the growth rate was established, along with the relationship between them and the Vickers hardness. Finally, the mechanical behavior was assessed by tensile tests, revealing that the translation rate increases the yield strength without sacrificing ductility. The directionally solidified Al0.8CrFeNi2.2 EHEA exhibited a maximum yield strength of approximately 558 MPa with reasonable ductility (total elongation of approximately 6.5 %) under the analyzed conditions.

The impact of convection on morphological instability of a planar crystallization front

A linear morphological stability analysis of a planar solid-liquid phase interface describing the solidification processes of a binary melt with convection is carried out. The developed theory includes conductive and convective heat and mass transfer mechanisms near the phase interface and generalizes previously known theories of morphological stability. The amplification rate as a function of wavenumber of perturbations and neutral stability curve that divides the stability/instability parametric domains are obtained. It is shown that these domains are highly dependent of convection intensity, which represents a stabilizing factor for solidification processes. A criterion of concentration supercooling in the steady-state solidification conditions with convection is found. The obtained dispersion relation and neutral stability curve define various crystallization scenarios such as (i) morphological instability and concentration supercooling appearing to the formation of a two-phase mushy layer, (ii) morphological stability and concentration supercooling leading to the existence of a slurry layer, (iii) morphological stability without concentration supercooling when the planar solidification front is stable, and (iv) morphological instability without concentration supercooling forming the mesoscopically rough phase interface.

Morphological stability of solid-liquid interfaces under additive manufacturing conditions

Understanding rapid solidification behavior at velocities relevant to additive manufacturing (AM) is critical to controlling microstructure selection. Although in-situ visualization of solidification dynamics is now possible, systematic studies under AM conditions with microstructural outcomes compared to solidification theory remain lacking. Here we measure solid-liquid interface velocities of Ni-Mo-Al alloy single crystals under AM conditions with synchrotron X-ray imaging, characterize the microstructures, and show discrepancies with classical theories regarding the onset velocity for absolute stability of a planar solid-liquid interface. Experimental observations reveal cellular/dendritic microstructures can persist at velocities larger than the expected absolute stability limit, where banded structure formation should theoretically appear. We show that theory and experimental observations can be reconciled by properly accounting for the effect of solute trapping and kinetic undercooling on the velocity-dependent solidus and liquidus temperatures of the alloy. Further theoretical developments and accurate assessments of key thermophysical parameters – like liquid diffusivities, solid-liquid interface excess free energies, and kinetic coefficients – remain needed to quantitatively investigate such discrepancies and pave the way for the prediction and control of microstructure selection under rapid solidification conditions.

Effects of Heating Power on Microstructure Evolution and Tensile Properties at Elevated Temperature by Directional Solidification for Ti2AlC/TiAl Composites

To improve the directionally solidified microstructure and tensile properties at elevated temperature, Ti46Al4Nb1C0.8Ta ingots are directional solidified by electromagnetic cold crucible with different heating powers. The microstructure is observed and characterized, tensile properties are tested, and related mechanisms are revealed. Results show that a relatively planar liquid–solid interface is formed under the low heating power (39 kW), which promoted formation of columnar crystals. When the heating power increases from 39 to 48 kW, degree of dent (H1) increases from 8.5 to 18.1 mm and influence length of the initial transition area (H2) increases from 4.3 to 28.6 mm. The formation of a concave liquid–solid interface is due to the high heating power increasing the temperature gradient. High heating power increases the electromagnetic stirring ability, which increases the time to reach stable growth. Tensile properties of the directionally solidified alloy are higher than those of the as‐cast alloy under the same testing temperature, e.g., the tensile strength is 342.5 MPa, an increase of 1.4 times, and the tensile strain is 4.4%, an increase of 1.3 times at 950 °C. This shows that the tensile properties and microstructure of TiAl‐based composites can be improved by the directional solidification technology.

High Accuracy Analysis of Nonconforming Mixed Finite Element Method for the Nonlinear Sivashinsky Equation

The fourth-order nonlinear Sivashinsky equation is often used to simulate a planar solid-liquid interface for a binary alloy. In this paper, we study the high accuracy analysis of the nonconforming mixed finite element method (MFEM for short) for this equation. Firstly, by use of the special property of the nonconforming EQ1rot element (see Lemma 1), the superclose estimates of order Oh2+Δt in the broken H1-norm for the original variable u and intermediate variable p are deduced for the back-Euler (B-E for short) fully-discrete scheme. Secondly, the global superconvergence results of order Oh2+Δt for the two variables are derived through interpolation postprocessing technique. Finally, a numerical example is provided to illustrate validity and efficiency of our theoretical analysis and method.

Matching time and spatial scales of rapid solidification: dynamic TEM experiments coupled to CALPHAD-informed phase-field simulations

A combination of dynamic transmission electron microscopy (DTEM) experiments and CALPHAD-informed phase-field simulations was used to study rapid solidification in Cu–Ni thin-film alloys. Experiments—conducted in the DTEM—consisted of in situ laser melting and determination of the solidification kinetics by monitoring the solid–liquid interface and the overall microstructure evolution (time-resolved measurements) during the solidification process. Modelling of the Cu–Ni alloy microstructure evolution was based on a phase-field model that included realistic Gibbs energies and diffusion coefficients from the CALPHAD framework (thermodynamic and mobility databases). DTEM and post mortem experiments highlighted the formation of microsegregation-free columnar grains with interface velocities varying from ∼0.1 to ∼0.6 m s−1. After an ‘incubation’ time, the velocity of the planar solid–liquid interface accelerated until solidification was complete. In addition, a decrease of the temperature gradient induced a decrease in the interface velocity. The modelling strategy permitted the simulation (in 1D and 2D) of the solidification process from the initially diffusion-controlled to the nearly partitionless regimes. Finally, results of DTEM experiments and phase-field simulations (grain morphology, solute distribution, and solid–liquid interface velocity) were consistent at similar time (μs) and spatial scales (μm).

Investigation on solidification path of Ni-based single crystal superalloys with different Ru contents

The solidification path of Ni-based single crystal superalloy with different Ru contents (0–4wt%) was systematically investigated under planar and dendrite solid-liquid interface solidification conditions, aiming to reveal the evolution rule of solidification microstructure and selection mechanism of phases of the fourth generation single crystal superalloys. The results indicated that under plane solidification interface, the fine γ/γ′ eutectic structure started to form after the solidification of γ phase and then further developed the coarse γ/γ′ structure. The solidification finally terminated with the formation of β-NiAl phase enveloped by coarse γ′ phase for Ru-free superalloy. However, when Ru was added in superalloy, the β-NiAl phase was firstly precipitated after the solidification of γ phase, and no fine γ/γ′ structure connecting the γ phase with β-NiAl phase was found. Under the dendrite solid-liquid interface condition, the solidification path was consistent with that in the plane one. The evolution mechanism of solidification path with different Ru contents was discussed, which was helpful to optimize the alloy composition design.

Influence of icosahedral short range order on diffusion in liquids: A study on Al-Zn-Cr alloys

It has been shown recently that minute additions of Cr in Al-Zn alloys refine the solidified grain structure and induce multiply-twinned grains which can only be explained if the icosahedral symmetry is considered (Kurtuldu et al., 2013). It has been suggested that Cr addition to Al-Zn alloys promotes the formation of icosahedral short-range order (ISRO) in the liquid, leading to the formation of icosahedral quasicrystals (iQC) acting then as a template for the nucleation of the fcc Al-phase. If ISRO exists in liquid Al-Zn-Cr alloys, this should influence diffusion of solute elements in the liquid. The present study focuses on the effect of Cr addition on Zn diffusion in liquid Al-Zn-Cr alloys. The solute diffusion coefficients of Zn and Cr in the liquid were deduced from the solute profiles in the quenched liquid ahead of a planar solid-liquid interface. By comparison with the same measurement in Al-Zn alloys, it is shown that Cr indeed slows down the diffusion (and mobility) of Zn atoms, an effect which reinforces the hypothesis of ISRO in Al-Zn liquid induced by Cr.

Effects of Solidification Rate and Settling Time of SiC Particles on the Macrosegregation of Carbon in Silicon Ingots

The macrosegregation of carbon in silicon was studied aiming for silicon refining for photovoltaic applications. Experiments of directional solidification upward and of settling of SiC particles in the melt were carried out in a Bridgman furnace using electronic grade silicon intentionally contaminated with carbon. Mold pulling velocities in the range between 5 and 80 μm/s and settling time periods of 1 and 6 hours were adopted in the experiments. In the resulting cylindrical ingots, the macrosegregation of carbon along the axial direction was measured and their macro and microstructures revealed and examined. In all directional solidification experiments, the carbon concentration increases to the ingot top as result of an accumulation of SiC particles in this region. For the lower velocities (<40 μm/s), a larger particle accumulation and columnar grains aligned axially and pointing to the ingot top are seen, which is consistent with the pushing of SiC particles by a planar solid–liquid interface. For the settling experiments, the carbon concentration increases from the ingot top to the bottom as a result of the accumulation of SiC particles at the bottom. This accumulation is probably caused by the settling of SiC particles, but an increase in the settling time from 1 to 6 hours have no significant effect in the macrosegregation profile. Finally, in the conditions of the present experiments, the directional solidification of silicon at mold pulling velocities lower than 40 μm/s is more effective to remove carbon from silicon than the settling of SiC particles during times up to 6 hours.

Morphological stability during solidification of silicon incorporating metallic impurities

We study the stability of a planar solidification front during pulsed laser melting-induced rapid solidification of silicon containing high concentrations of ion-implanted metallic impurities. We calculate the critical impurity concentration for destabilizing plane-front solidification, and introduce the “amplification coefficient,” which is an empirical parameter describing the degree of amplification that must occur between the time the planar liquid-solid interface first becomes unstable, and the time of formation of morphological features of interface breakdown that is later observed in the microstructure. By connecting our calculations to experimental observations from the literature, we determine this parameter for Au, Co, Cr, Fe, Ga, In, and Zn in (100) Si and Ti in (111) Si, and find that it increases with impurity diffusive speed vD approximately as vD0.56. We present an approximate but simple method of estimating the maximum impurity concentration that may be incorporated in a surface layer of a given thickness without the appearance of cellular breakdown.

Study of Soft Impingement of Diffusion Fields and Its Effect on Cast Microstructure

Rigorous Mullins and Sekerka perturbation theory has been applied to two solidification interfaces. The results indicate that as the distance between the two interfaces decreases, the rate of amplitude growth of perturbations on the planar solid–liquid interface becomes negative, which is a further verification of the stability of a planar interface due to soft impingement of diffusion fields during solidification. This work conclusively shows that spheroidal grains form due to grain refinement of Mg alloys by Zr addition because of the high density of nucleation.

Rapid colloidal solidifications under local nonequilibrium diffusion conditions

Partition coefficient for rapid solidification of colloidal suspensions has been calculated under local nonequilibrium diffusion conditions typically used when processing advanced materials. It has been demonstrated that the local nonequilibrium diffusion effects stabilize the planar solid liquid interface and lead to an abrupt transition to diffusionless solidification with complete particle trapping. The effective diffusion coefficient, which depends on interface velocity and particle size, has been introduced. It explains the strong dependences of the partition coefficient and the velocity leading to absolute stability of a planar solid–liquid interface on particle size.

Preparation, microstructure and dislocation of solar-grade multicrystalline silicon by directional solidification from metallurgical-grade silicon

A vacuum directional solidification with high temperature gradient was performed to prepare low cost solar-grade multicrystalline silicon (mc-Si) directly from metallurgical-grade mc-Si. The microstructure characteristic, grain size, boundary, solid-liquid growth interface, and dislocation structure under different growth conditions were studied. The results show that directionally solidified multicrystalline silicon rods with high density and orientation can be obtained when the solidification rate is below 60 μm/s. The grain size gradually decreases with increasing the solidification rate. The control of obtaining planar solid-liquid interface at high temperature gradient is effective to produce well-aligned columnar grains along the solidification direction. The growth step and twin boundaries are preferred to form in the microstructure due to the faceted growth characteristic of mc-Si. The dislocation distribution is inhomogeneous within crystals and the dislocation density increases with the increase of solidification rate. Furthermore, the crystal growth behavior and dislocation formation mechanism of mc-Si were discussed.

Numerical analysis of constitutional supercooling during directional solidification of alloys

Some limitations of Tiller’s morphological stability criterion are discussed in the present study. This criterion assumes a purely diffusive regime in the melt as well as a planar solid–liquid interface and a constant solidification rate. But experimental works in agreement with previous numerical modeling have shown a significant decrease of the growth rate and a variable interface curvature during the concentrated semiconductor alloys solidification. The mathematical expression of the morphological stability criterion was derived by using Tiller’s equation, predicting the solute distribution in the liquid. The numerical computations performed in this study show a significant disagreement between the numerical results and Tiller’s formula. Numerical modeling conducted in conditions when the supercooling should occur, show that the Tiller’s stability criterion cannot predict the moment of interface destabilization. The interface destabilization is numerically observed when some fluctuations appear in the liquid solutal profiles and cause the appearance of a supercooled zone inside the liquid at small distance from the interface. The present numerical results are not in contradiction with the basic elements of the classical constitutional supercooling theory, providing only that the stability criterion cannot predict the moment of the interface destabilization.

Macrosegregation of Impurities in Directionally Solidified Silicon

planar solid–liquid interface for mold velocities £10 lm seconds –1 . Concentration profiles of several impurities were measured along the ingots, showing that their bottom and middle are purer than the metallurgical Si from which they solidified. At the ingot top, however, impurities accumulated, indicating the typical normal macrosegregation. When the mold velocity decreases, the macrosegregation and ingot purity increase, changing abruptly for a velocity variation from 20 to 10 lm seconds –1 . A mathematical model of solute transport during solidification shows that, for mold velocities ‡20 lm seconds –1 , macrosegregation is caused mainly by diffusion in a stagnant liquid layer assumed at the solid–liquid interface, whereas for lower velocities, macrosegregation increases as a result of more intense convective solute transport.

Optimisation of the VGF growth process by inverse modelling

Numerical and experimental results on the thermal optimisation of vertical gradient freeze crystal growth are presented. An inverse modelling approach is described aimed at solidification with a constant growth rate and planar solid–liquid interface. As a result of modelling an optimised growth process characterised by a modified ampoule configuration and thermal regime was established. For experimental confirmation Ga-doped germanium single crystals were grown with the optimised process. In good agreement with the numerical results, solidification with an almost constant growth rate was achieved with the interface deflection being significantly lower than in conventionally grown crystals.

In-Situ and Real-Time Investigation of Columnar-to-Equiaxed Transition in Metallic Alloy

In this article, we present a review of observations during Al-3.5 wt pct Ni alloy solidification experiments performed at the European Synchrotron Radiation Facility (ESRF) in Grenoble. These experiments provide direct access to dynamical phenomena during columnar growth (initial transient and breakdown of a planar solid-liquid interface), and for the first time to the transition from columnar-to-equiaxed microstructure (nucleation ahead of a columnar front and blocking of a columnar front by an equiaxed microstructure) and fully equiaxed growth (propagation of an effective front). Based on these experimental observations, critical parameters such as columnar growth velocity variation during the transition or equiaxed-grain diameter are measured and discussed.

Phase field modeling of solidification under stress

We consider a phase field model that includes a stress field during nonisothermal phase transformation of a single-component system. The model is applied to the solidification and melting of confined spherical volumes, where sharp interface solutions can be obtained and compared with the results of the phase field simulations. Numerical solutions to the phase field model for a spherically symmetric geometry have been obtained, with particular emphasis on the computation of surface energy, surface stress, and surface strain. The analysis of the equilibrium states for the phase field model allows us to obtain the value of the surface energy in the presence of stress, which can then be compared to the analogous calculation of the energy of a planar liquid-solid interface. It is also demonstrated that modeling the liquid as a solid with zero shear modulus is realistic by comparing the long-range stress fields in phase field calculations with those calculated using sharp interface models of either a coherent or a relaxed liquid-solid interface.

Investigation of the initial transient in directional solidification of binary AlCu alloys

For the understanding of the formation of microstructures the initial transient, i.e. the very beginning of solidification, is of major relevance. At this stage a complex interaction between the velocity of the solidification front, the concentration pile-up and the temperature gradient takes place. In a Bridgman-Stockbarger furnace directional solidification experiments with binary AlCu 1.3 wt.% were performed allowing solidification with a planar solid–liquid interface. The velocity and temperature profile were measured during the experiments, concentration profiles afterwards. In this paper we present experimental results. Parallel to the experimental work numerical simulations based on the model of Coriell et al. were performed and first results are compared to the experiments.

Linear morphological stability analysis of the solid-liquid interface in rapid solidification of a binary system.

The interface stability against small perturbations of the planar solid-liquid interface is considered analytically in linear approximation. Following the analytical procedure of Trivedi and Kurz [Acta Metall. 34, 1663 (1986)]], which is advancing the original treatment of morphological stability by Mullins and Sekerka [J. Appl. Phys. 35, 444 (1964)]] to the case of rapid solidification, we extend the model by introducing the local nonequilibrium in the solute diffusion field around the interface. A solution to the heat- and mass-transport problem around the perturbed interface is given in the presence of the local nonequilibrium solute diffusion. Using the developing local nonequilibrium model of solidification, the self-consistent analysis of linear morphological stability is presented with the attribution to the marginal (neutral) and absolute morphological stability of a rapidly moving interface. Special consideration of the interface stability for the cases of solidification in negative and positive thermal gradients is given. A quantitative comparison of the model predictions for the absolute morphological stability is presented with regard to experimental results of Hoglund and Aziz [D. E. Hoglund and M. J. Aziz, in Kinetics of Phase Transformations, edited by M.O. Thompson, M. J. Aziz, and G. B. Stephenson, MRS Symposia Proceedings No. 205 (Materials Research Society, Pittsburgh, 1991), p. 325] on critical solute concentration for the interface breakdown during rapid solidification of Si-Sn alloys.