Solidification Of Binary Alloy Research Articles

Magnetohydrodynamic stationary and oscillatory convective stability in a mushy layer during binary alloy solidification

• Mushy layer growth during solidification of a bottom cooled binary alloy. • Magnetohydrodynamic stationary and oscillatory convective stability in the mushy layer • Results are presented for various values of mush Hartmann numbers in the range, 0 ≤ Ha m ≤ 50. • The critical Rayleigh number for stationary convection shows a linear relationship with increasing Ha m . • Increasing magnetic strength shows reduction in wavenumber and in the number of rolls in the mushy layer. For the case of solidification of a bottom cooled binary alloy, the magnetohydrodynamic stationary and oscillatory convective stability in the mushy layer is investigated analytically using normal mode linear stability analysis. In the limit of large Stefan number ( S t ), a near–eutectic approximation with large far field temperature is considered in the present research. To ascertain the instability in the mushy layer, the strength of the superimposed magnetic field is so chosen that it corresponds to a given mush Hartmann number ( Ha m ) of the problem. The results are presented for various values of mush Hartmann numbers in the range, 0 ≤ Ha m ≤ 50. The critical Rayleigh number for stationary convection shows a linear relationship with increasing Ha m . The magnetohydrodynamic effect imparts a stabilizing influence during stationary convection. In comparison to that of the stationary convective mode, the oscillatory mode appears to be critically susceptible at higher values of β (β = S t /℘ 2 ϒ 2 , ℘ is the compositional ratio, ϒ = 1 + S t /℘), and vice versa for lower β values. Analogous to the behavior for stationary convection, the magnetic field also offers a stabilizing effect in oscillatory convection and thus influences global stability of the mushy layer. Increasing magnetic strength shows reduction in the wavenumber and in the number of rolls formed in the mushy layer.

ThermoCalc Application for the Assessment of Binary Alloys Non-Equilibrium Solidification

Abstract The paper presents the possibility of application of the developed computer script which allows the assessment of non-equilibrium solidification of binary alloys in the ThermoCalc program. The script makes use of databases and calculation procedures of the POLY-3 module. A solidification model including diffusion in the solid state, developed by Wołczyński, is used to describe the non-equilibrium solidification. The model takes into account the influence of the degree of solute segregation on the solidification process by applying the so-called back-diffusion parameter. The core of the script is the iteration procedure with implemented model equation. The possibility of application of the presented calculation method is illustrated on the example of the Cr-30% Ni alloy. Computer simulations carried out with use of the developed script allow to determine the influence of the back-diffusion parameter on the course of solidification curves, solidus temperature, phase composition of the alloy and the fraction of each phase after the solidification completion, the profile of solute concentration in liquid during solidification process, the average solute concentration in solid phase at the eutectic temperature and many other quantities which are usually calculated in the ThermoCalc program.

Finite bandwidth, nonlinear convective flow in a mushy layer

Finite amplitude convection with a continuous finite bandwidth of modes in a horizontal mushy layer during the solidification of binary alloys is investigated. We analyze the nonlinear convection for values of the Rayleigh number close to its critical value by using multiple scales and perturbation techniques. Applying a combined temporal and spatial evolution approach, we determine a set of three coupled differential equations for the amplitude functions of the convective modes. A large number of new subcritical or supercritical stable solutions to these equations in the form of steady rolls and hexagons with different horizontal length scales are detected. We find, in particular, that depending on the parameter values and on the magnitude and direction of the wave number vectors for the amplitude functions, hexagons with down-flow or up-flow at the cells’ centers or rolls can be stable. Rolls or hexagons with longer horizontal wave length can be stable at higher amplitudes, and there are cases where hexagons are unstable for any value of the Rayleigh number, while rolls are stable only for the values of the Rayleigh number beyond some value. We also detected new stable and irregular flow patterns with two different horizontal scales in the form of superposition of either two sets of hexagons or two sets of inclined rolls.

Numerical testing of quantitative phase-field models with different polynomials for isothermal solidification in binary alloys

Quantitative phase-field models have been developed as feasible computational tools for solving the free-boundary problem in solidification processes. These models are constructed with some polynomials of the phase-field variable that describe variations of the physical quantities inside the diffuse interface. The accuracy of the simulation depends on the choice of the polynomials and such dependence is indispensable for high-performance computing and valuable for extending the range of applications of the model to several physical systems. However, little is known about the dependence of the accuracy on the choice of the polynomials. In this study, numerical testing is carried out for quantitative phase-field models with extensive sets of polynomials (24 different models) for isothermal solidification in binary alloys. It is demonstrated in two-dimensional simulations of dendritic growth that a specific set of polynomials must be employed to achieve high accuracy in the models with double-well and double-obstacle potentials. Both types of model with the best set of polynomials yield almost the same numerical accuracy.

Abstract A63: Collective and individual migration after the epithelial-mesenchymal transition in engineered microenvironments

Abstract Scattering and local dissemination of individual cells from a collective invasion front has been associated with the epithelial-mesenchymal transition (EMT). This phenotypic heterogeneity and plasticity has been challenging to reconstruct using classical assays which only measure population averages at endpoints. Here, we comprehensively track single cell invasion dynamics within engineered microenvironments such as microfabricated pillar arrays as well as 3D biomaterial scaffolds. After EMT induction through the master regulator Snail in mammary epithelial cells, we find distinct phenotypic subpopulations that display collective or individual migration behaviors. We show that these complex behaviors can be physically understood in terms of a phase transition during the solidification of binary alloys. Biologically, this conceptual framework addresses two essential mechanisms—phenotypic cell sorting and interconversion. We present new results to corroborate this model by systematically varying the composition of mosaic populations with both epithelial and EMT phenotypes. We also perturb these behaviors using migration-inhibiting compounds, revealing that different invasion phenotypes are correlated with differences in drug sensitivity. Altogether, this integration of engineered microenvironments with single cell resolution may permit new quantitative insights into the interplay of invasion, drug resistance and heterogeneity. Citation Format: Ian Y. Wong. Collective and individual migration after the epithelial-mesenchymal transition in engineered microenvironments. [abstract]. In: Proceedings of the AACR Special Conference on Engineering and Physical Sciences in Oncology; 2016 Jun 25-28; Boston, MA. Philadelphia (PA): AACR; Cancer Res 2017;77(2 Suppl):Abstract nr A63.

Accelerated onset of cellular breakdown at increased solidification speed after laser melting of indium in silicon

We report on the incidence of cellular breakdown in silicon wafers implanted with indium and pulsed laser melted at varying laser fluence. We predict from heat flow calculations that as laser fluence is decreased, (a) the melt depth decreases, (b) the solidification velocity increases, and (c) the solidification front should be more stable with respect to cellular breakdown (based on the binary alloy solidification theory), which should result in observed incidence of cellular breakdown at a relatively shallower depth below the surface. However, secondary ion mass spectrometry (SIMS) data show that cellular breakdown at lower fluence is observed deeper, indicating that the interface became unstable earlier in its solidification. The SIMS data show significant In evaporation, with greater evaporation occurring as fluence increases. Calculations show that surface evaporation can reduce the bulk liquid concentration during solidification, reducing the degree of instability. The longer melt durations after irradiation at higher fluence give more time for evaporation, which may result in a relatively larger delay in the onset of breakdown compared to lower fluence by suppressing the bulk liquid concentration and suppressing the growth of unstable perturbation. Under certain conditions, this effect could dominate over the role of reducing the interface speed, which would tend to accelerate amplification of a perturbation, which leads to breakdown. Calculations qualitatively support this interpretation.

Different Techniques of Determination of the Cracking Criterion for Solidification in Casting

Cracking which occurs during the solidification process is a serious defect in castings and welds. Alloys are susceptible to cracking during solidification. Al alloys, stainless steels and Ni-base alloys are notable examples. Cracking takes place in the semisolid region called the mushy zone. During solidification, shrinkage and thermal contraction of the mushy zone and its surrounding solid are hindered, tensile deformation can induce cracking (in the semisolid region) along grain boundaries that are not fed with sufficient liquid phase. The cracking criterion is focused on events occurring at the grain boundary, including separation of grains from each other, lateral growth of grains toward each other, and liquid feeding between grains. The susceptibility factor of a binary aluminum alloy to cracking during solidification was also described. The criterion has an effect upon: (a) the lateral growth rate of two neighboring grains toward each other to bond together to resist cracking, and (b) the length of the grain-boundary liquid channel through which feeding has to occur to resist cracking. In future work the criterion will be compared with experimental data and computer simulation during solidification of binary aluminum alloys in welding.

Directional solidification of aqueous TiO2 suspensions under reduced gravity

Porous materials exhibiting aligned, elongated pore structures can be created by directional solidification of aqueous suspensions—where particles are rejected from a propagating ice front and form interdendritic, particle-packed walls—followed by sublimation of the ice, and sintering of the particle walls. Theoretical models that predict dendritic lamellae spacing—and thus wall and pore width in the final materials—are currently limited due to an inability to account for gravity-driven convective effects during solidification. Here, aqueous suspensions of 10–30 nm TiO2 nanoparticles are solidified on parabolic flights under micro-, lunar (∼0.17 g; gl = 1.62 m/s2), and Martian (∼0.38 g; gm = 3.71 m/s2) gravity and compared to terrestrially-solidified samples. After ice sublimation and sintering, all resulting TiO2 materials exhibit elongated lamellar pores replicating the ice dendrites. Increasing the TiO2 fraction in the suspensions leads to decreased lamellar spacing in all samples, regardless of gravitational acceleration. Consistent with previous studies of microgravity solidification of binary metallic alloys, lamellar spacing decreases with increasing gravitational acceleration. Mean lamellar spacing for 20 wt% TiO2 nanoparticles suspensions under micro-, lunar, Martian, and terrestrial gravity are, respectively: 50 ± 8, 34 ± 11, 30 ± 6, and 23 ± 9 μm, indicating that gravity-driven convection strongly affects lamellae spacing under terrestrial gravity conditions. Gravitational effects on lamellar spacing are highest at low TiO2 fractions in the suspension; for 5 wt% TiO2 suspensions, the microgravity lamellar spacing is more than twice that under terrestrial gravity (182 ± 21 vs. 81 ± 23 μm). Results of this study are in good agreement with previous studies of binary metallic alloy solidification where primary dendrite spacing increases under microgravity. Literature data from ice-templating systems are used to discuss a dependence on lamellae spacing of the density ratio of particles and fluid.

Zone Partitioning Sequential Calculation Method for Three-Dimensional Phase-Field Material Simulation during Directional Solidification

Complex dendritic structures can be simulated directly by phase field method. However, phase field method needs a very fine mesh computing and memory requirements. A new calculation method named zone partitioning sequential calculation method is proposed to expand the simulation area of phase field method. The simulation area can be divided into several parts, and the parts can be calculated one by one in a certain order by the new method. This new method can reduce the computing and memory requirements of single calculation, because a part of the simulation area is less than the whole simulation area. Although this method could make error in the interface of different parts of the simulation area when the grains go through the interface, but the error has less effect on the grain growth. By using phase field method, coupled with zone partitioning sequential calculation method, the simulation of the directional solidification process of Al-Cu binary alloy is operated. The results show that the new method can be applied to phase field simulation of binary alloy solidification in a large area. The simulation results have certain accuracy and reliability.

Eutectic solidification patterns: Interest of microgravity environment

The solidification of binary eutectic alloys produces two-phase composite materials in which the microstructure, that is, the geometrical distribution of the two solid phases, results from complex pattern-formation processes at the moving solid–liquid interface. Since the volume fraction of the two solids depends on the local composition, solidification dynamics can be strongly influenced by thermosolutal convection in the liquid. In this contribution, we review our experimental and numerical work devoted to the understanding of eutectic solidification under purely diffusive conditions, which will soon be tested and extended during the microgravity experiment TRANSPARENT ALLOYS planned by the European Space Agency (ESA).

Phase-Field Modeling of Binary Eutectic Alloy Solidification with Convection

In the framework of a description of melt pool heat transfer under severe accident conditions, we introduce a computational fluid dynamics approach for the phase change based on the phase-field method. The approach is derived using the formalism of irreversible thermodynamics and depends on a phenomenological expression for the free energy of binary eutectic alloys. The free energy is constructed to describe sharp interfaces on sufficiently small length scales and is capable of representing the appearance of mushy layers in a volume-averaged large-scale perspective. In particular, a dynamic calculation procedure for the diffuse interface width is introduced based on free energy minimization. Numerical simulations using this approach are performed and compared with experimental and numerical results from the literature. These comparisons demonstrate that the new model improves numerical simulation results and is able to describe the dynamics of sharp and diffuse interfaces.

Identification of the heat transfer coefficient in the two-dimensional model of binary alloy solidification

The paper presents the procedure for solving the inverse problem for the binary alloy solidification in a two-dimensional space. This is a continuation of some previous works of the authors investigating a similar problem but in the one-dimensional domain. Goal of the problem consists in identification of the heat transfer coefficient on boundary of the region and in reconstruction of the temperature distribution inside the considered region in case when the temperature measurements in selected points of the alloy are known. Mathematical model of the problem is based on the heat conduction equation with the substitute thermal capacity and with the liquidus and solidus temperatures varying in dependance on the concentration of the alloy component. For describing this concentration the Scheil model is used. Investigated procedure involves also the parallelized Ant Colony Optimization algorithm applied for minimizing a functional expressing the error of approximate solution.

Numerical simulation of Bridgman solidification of binary alloys

A transient 2D axisymmetric numerical model for the Bridgman solidification process for a stationary furnace and moving sample is presented. The model is able to predict the evolution of temperature and solid fraction of binary alloys in cases where buoyancy induced convection is negligible, such as in microgravity conditions. A dimensionless form of the governing equations was derived in order to identify the dimensionless parameters that characterise the process, those being the Stefan, Péclet, and Biot numbers. The problem was solved using a finite volume method and an explicit time stepping scheme. To test the efficacy of the model, simulated results were compared with experimental data from the literature and acceptable agreement was obtained. Finally, a parametric analysis was performed for understanding the influence of the process parameters on solidification. One key feature of this study was the inclusion of a term describing the advection of latent heat due to the translation of the mushy zone with varying solid fraction. This thermal transport mechanism was shown to be significant, since its magnitude was comparable to the advection of sensible heat. It was also found that when small Biot numbers were due to low values of the heat transfer coefficients at the surface of the sample, rather than to small sample radii, advective mechanisms were enhanced resulting in more convex shapes of the liquidus isotherm. This highlighted the importance of considering both axial and radial heat fluxes when describing the process.

Numerical simulation of dendritic growth in directional solidification of binary alloys using a lattice Boltzmann scheme

A lattice Boltzmann (LB) based model is utilized to simulate three dimensional dendritic growth in directional solidification of alloys. The LB–D3Q19 lattice vectors are used to describe advancement of solid–liquid interface, coupling with a conservation equation for solute transport in solidification. After model validation, the dendritic growth under several conditions of directional solidification was investigated and a solidification entropy was proposed to quantitatively characterize solidification morphologies. The results show that the present model captures tip-splitting occurring at a relative fast solidification rate. Initial thermal undercooling and solute concentration are dominant factors influencing the final microstructure. The solidification entropy reflects complexity of dendritic morphologies and it is useful to characterize dendritic growth in solidification. A design map was proposed to predict dendritic growth with and without tip-splitting, offering potential for simulating dendritic growth and microstructure evolution in solidification of alloys.

Primary arm array during directional solidification of a single-crystal binary alloy: Large-scale phase-field study

Primary arm arrays formed during the directional solidification of a single-crystal binary alloy were investigated by performing large-scale phase-field simulations using the GPU supercomputer TSUBAME2.5 at Tokyo Institute of Technology. The primary arm array and spacing were investigated by Voronoi decomposition and Delaunay triangulation, respectively. It was concluded that a hexagonal array was dominant for both the dendrite and cell structures and that penta–hepta defects, which are typical defects in hexagonal patterns, were formed. The primary arms continuously moved such that the number of hexagons increased, and the distribution of primary arm spacing became uniform over time even after the number of primary arms was constant. The order of array was highest in the growth condition of the dendrite close to the cell-to-dendrite transition region. In addition, we proposed a realistic and accurate evaluation method of primary arm array by removing small sides from the Voronoi polygons.

Numerical Simulation of Solidification Microstructure based on Adaptive Octree Grids

Abstract The main work of this paper focuses on the simulation of binary alloy solidification using the phase field model and adaptive octree grids. Ni-Cu binary alloy is used as an example in this paper to do research on the numerical simulation of isothermal solidification of binary alloy. Firstly, the WBM model, numerical issues and adaptive octree grids have been explained. Secondary, the numerical simulation results of three dimensional morphology of the equiaxed grain and concentration variations are given, taking the efficiency advantage of the adaptive octree grids. The microsegregation of binary alloy has been analysed emphatically. Then, numerical simulation results of the influence of thermophysical parameters on the growth of the equiaxed grain are also given. At last, a simulation experiment of large scale and long-time has been carried out. It is found that increases of initial temperature and initial concentration will make grain grow along certain directions and adaptive octree grids can effectively be used in simulations of microstructure.

A Phase Field Model for Binary Alloy Solidification with Boundary Interface Intersection

A phase field model for binary alloy solidification with boundary interface intersection is developed. In the phase field model, the heat and solute conservation equations are appropriately modified to account for the presence of heat and solute rejection inside the diffuse interface, and a relaxation boundary condition for the phase field variable is introduced to balance the interface energy and boundary surface energy in the multiphase contact region. The thin interface asymptotic analysis is applied on the phase field model to yield the free interface problem with dynamic contact point condition.

Surface rippling during solidification of binary polycrystalline alloy: Insights from 3-D phase-field simulations

The mechanisms by which crystalline imperfections initiate breakdown of a planar front during directional solidification remain a topic of longstanding interest. Previous experimental findings show that the solid-liquid interface adjacent to a grain boundary provides a potential site where morphological instabilities initiate. However, interpretation of experimental data is difficult for complex 3-D diffusion fields that develop around grain multi-junctions and boundary ridges. We apply a phase-field approach to investigate factors that induce interfacial instabilities during directional solidification of a binary polycrystalline alloy. Using 2-D simulations, we establish the influence of solid-liquid interfacial energies on the spatial localization of initial interfacial perturbations. Based on parametric studies, we predict that grain misorientation and supersaturation in the melt provide major crystal growth factors determining solute segregation responsible for surface rippling. Subsequent breakdown of boundary ridges into periodic rows of hills, as simulated in 3-D, conform well with experiments. Finally, the significance of crystal misorientation relationships is elucidated in inducing spatial alignment of surface ripples.

A similarity solution for phase change of binary alloy with shrinkage or expansion

A similarity solution for solidification of an undercooled binary alloy is developed including the effect of shrinkage or expansion due to change in the density. The proposed similarity solution takes into account the change in density of the alloy while undergoing a phase change. Thermo-physical properties are assumed to be constant. An analytical solution (which is an outcome of the similarity solution) for temperature and concentration distribution has been established. The effect of the density ratio and Lewis number on both interface motion and conjugate heat and mass transfer is studied. It is found that the interface moves faster with the decrease in density ratio and initial temperature during solidification.

On nonlinear transient state of modal package convection in a mushy layer

Finite-amplitude transient convective flow with a continuous finite bandwidth of modes in a horizontal mushy layer during the solidification of binary alloys is investigated. Under a near-eutectic approximation and in the limit of large far-field temperature, we analyse the nonlinear transient convection for values of the Rayleigh number close to its critical value by using multiple scales and perturbation techniques. Applying the combined temporal and spatial evolution approach, we determine a set of coupled time-dependent partial differential equations for the amplitude functions of the convective modes. In the transient regime, we find a wide class of solutions to these equations in the form of transient rolls and transient hexagons with down-flow or up-flow at the cells’ centres that eventually either reach their steady-state form or decay to zero. We also apply a criterion of stability to examine the stability of these time-dependent solutions. Among the detected solutions, there are inclined large-scale transient rolls and large-scale transient hexagons that are realizable at higher values of their amplitude. Our main results agree with the available experimental observations.