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Cellular Spacing Selection During the Directional Solidification of Binary Alloys. A Numerical Approach

The evolution of cellular solid/liquid interfaces from an initially unstable planar front was studied by means of a two-dimensional computer simulation. The developed numerical model makes use of an interface tracking procedure and has the capability to describe the dynamics of the interface morphology based on local changes of the thermodynamic conditions. The fundamental physics of this formulation was validated against experimental microgravity results and the predictions of the analytical linear stability theory. The performed simulations revealed that in certain conditions, based on a competitive growth mechanism, an interface could become unstable to random perturbations of infinitesimal amplitude even at wavelengths smaller than the neutral wavelength, lambda(sub c), predicted by the linear stability theory. Furthermore, two main stages of spacing selection have been identified. In the first stage, at low perturbations amplitude, the selection mechanism is driven by the maximum growth rate of instabilities while in the second stage the selection is influenced by nonlinear phenomena caused by the interactions between the neighboring cells. Comparison of these predictions with other existing theories of pattern formation and experimental results will be discussed.

Catalina, Adrian V.↗

Numerical Analysis of Temperature Gradients and Interface Shape During Directional Solidification of Al and Al-Cu Alloy Under Microgravity Conditions

Numerical modeling was undertaken to analyze the influence of radial thermal gradient on solid/liquid (s/1) interface shape and convection patterns during solidification of pure Al and Al-4 wt% Cu alloy. The objective of the numerical task was to predict the influence of convective velocity on an insoluble particle near a s/l interface. These predictions would then be used to define the minimum gravity level (g) required to investigate the fundamental physics of interaction between a particle and a s/I interface. To satisfy this objective, steady state calculations were performed for different gravity levels and orientations with the gravity vector. ne furnace configuration used in this analysis is the proposed International Space Station Furnace, Quench Module Insert (QMI) 1. Results from a thermal model of the furnace core were used as initial boundary conditions for solidification modeling. General model of binary alloy solidification was based on the finite element code FIDAP. It was found that for the worst case orientation of 90 degrees with the gravity vector and a g level of 10(exp -4)g(sub o) (g(sub o) = 9.8 m/s(exp 2)) the dominant forces acting on the particle would be the fundamental drag and interfacial forces.

Bune, Andris V.↗

The modelling of heat, mass and solute transport in solidification systems

The aim of this paper is to explore the range of possible one-phase models of binary alloy solidification. Starting from a general two-phase description, based on the two-fluid model, three limiting cases are identified which result in one-phase models of binary systems. Each of these models can be readily implemented in standard single phase flow numerical codes. Differences between predictions from these models are examined. In particular, the effects of the models on the predicted macro-segregation patterns are evaluated.

Voller, V. R.↗

Directional Solidification of a Binary Alloy into a Cellular Convective Flow: Localized Morphologies

A steady, two dimensional cellular convection modifies the morphological instability of a binary alloy that undergoes directional solidification. When the convection wavelength is far longer than that of the morphological cells, the behavior of the moving front is described by a slow, spatial-temporal dynamics obtained through a multiple-scale analysis. The resulting system has a "parametric-excitation" structure in space, with complex parameters characterizing the interactions between flow, solute diffusion, and rejection. The convection stabilizes two dimensional disturbances oriented with the flow, but destabilizes three dimensional disturbances in general. When the flow is weak, the morphological instability behaves incommensurably to the flow wavelength, but becomes quantized and forced to fit into the flow-box as the flow gets stronger. At large flow magnitudes the instability is localized, confined in narrow envelopes with cells traveling with the flow. In this case the solutions are discrete eigenstates in an unbounded space. Their stability boundary and asymptotics are obtained by the WKB analysis.

Chen, Y.- J.↗

Phase-field model of freeze casting

Directional solidification of water-based solutions has emerged as a versatile technique for templating hierarchical porous materials. However, the underlying mechanisms of pattern formation remain incompletely understood. In this work, we present a detailed derivation and analysis of a quantitative phase-field model for simulating this nonequilibrium process. The phase-field model extends the thin-interface formulation of dilute binary alloy solidification with antitrapping to incorporate the highly anisotropic energetic and kinetic properties of the partially faceted ice-water interface. This interface is faceted in the basal plane normal to the ⟨0001⟩ directions and atomically rough in other directions within the basal plane. On the basal plane, the model reproduces a linear or nonlinear relationship between the interface growth rate and the kinetic undercooling that can be linked to experimental measurements. In both cases, spontaneous parity breaking of the solidification front is observed when the preferred growth direction is aligned with the temperature gradient. This phenomenon leads to the formation of partially faceted ice lamellae that drift laterally in one of the ⟨0001⟩ directions. Here, we demonstrate that the drifting velocity of the ice lamellae is controlled by the kinetics on the basal plane and converges as the thickness of the diffuse solid-liquid interface decreases. Furthermore, we examine the effect of the form of the kinetic anisotropy, which is chosen here such that the inverse of the kinetic coefficient varies linearly from a finite value in the ⟨0001⟩ directions to zero in all other directions within the basal plane, consistent with the assumption that the interface grows in local thermodynamic equilibrium in this plane. Our results indicate that the drifting velocity of ice lamellae is not affected by the slope of this linear relation, and the radius and undercooling at the tip of an ice lamella converge at relatively small slope values. Consequently, the phase-field simulations remain quantitative with computationally tractable choices of both the interface thickness and the slope assumed in the form of the kinetic anisotropy.

Materials science↗

Effect of Melt Convection at Various Gravity Levels and Orientations on the Forces Acting on a Large Spherical Particle in the Vicinity of a Solidification Interface

Numerical modeling was Undertaken to analyze the influence of both radial and axial thermal gradients on convection patterns and velocities claiming solidification of pure Al and an Al-4 wt% Cu alloy. The objective of the numerical task was to predict the influence of convective velocity on an insoluble particle near a solid/liquid (s/l) interface. These predictions were then be used to define the minimum gravity level (q) required to investigate the fundamental physics of interactions between a particle and a s/l interface. This is an ongoing NASA founded flight experiment entitled "particle engulfment and pushing by solidifying interfaces (PEP)". Steady-state calculations were performed for different gravity levels and orientations with respect to the gravity vector The furnace configuration used in this analysis is the quench module insert (QMI-1) proposed for the Material Science Research Facility (MSRF) on board the International Space Station (ISS). The general model of binary alloy solidification was based on the finite element code FIDAP. At a low g level of 10(exp -4) g(sub o) (g(sub o) = 9.8 m/square s) maximum melt convection was obtained for an orientation of 90 deg. Calculations showed that even for this worst case orientation the dominant forces acting on the particle are the fundamental drag and interfacial forces.

Bune, Andris V.↗

Thermosolutal convection during dendritic solidification of a binary alloy

The FEM model of Heinrich (1988) is presently extended to the case of dendritic vertical solidification under steady-state conditions. An analysis is conducted of the nonlinear governing equations for linear stability, and FEM calculations are performed in cases where instability has been predicted for Pb-Sn melt systems; the results obtained show that the model is more stable than the plane front liquid-solid model, and shows similar, finger-like covection when unstable. It is found that convection in the 'mush' zone is confined to the top of that region; convective motion is driven by convection in the bulk fluid, in the absence of natural convection.

Heinrich, Juan C.↗

Simulating the Effect of Space Vehicle Environments on Directional Solidification of a Binary Alloy

Space microgravity missions are designed to provide a microgravity environment for scientific experiments, but these missions cannot provide a perfect environment, due to vibrations caused by crew activity, on-board experiments, support systems (pumps, fans, etc.), periodic orbital maneuvers, and water dumps. Therefore, it is necessary to predict the impact of these vibrations on space experiments, prior to performing them. Simulations were conducted to study the effect of the vibrations on the directional solidification of a dendritic alloy. Finite element ca!cu!attie?ls were dme with a simd2titcr based on a continuum model of dendritic solidification, using the Fractional Step Method (FSM). The FSM splits the solution of the momentum equation into two steps: the viscous intermediate step, which does not enforce continuity; and the inviscid projection step, which calculates the pressure and enforces continuity. The FSM provides significant computational benefits for predicting flows in a directionally solidified alloy, compared to other methods presently employed, because of the efficiency gains in the uncoupled solution of velocity and pressure. finite differences, arises when the interdendritic liquid reaches the eutectic temperature and concentration. When a node reaches eutectic temperature, it is assumed that the solidification of the eutectic liquid continues at constant temperature until all the eutectic is solidified. With this approach, solidification is not achieved continuously across an element; rather, the element is not considered solidified until the eutectic isotherm overtakes the top nodes. For microgravity simulations, where the convection is driven by shrinkage, it introduces large variations in the fluid velocity. When the eutectic isotherm reaches a node, all the eutectic must be solidified in a short period, causing an abrupt increase in velocity. To overcome this difficulty, we employed a scheme to numerically predict a more accurate value for the rate of change of fraction of liquid as the liquid in an element solidifies. The new method enables us to contrast results of simulations in which the alloy is subjected to no gravity or a steady-state acceleration versus simulations when the alloy is subjected to vibration disturbances; therefore, the effect of vibration disturbances can be assessed more accurately. To assess the impact of these vibration-perturbations, transient accelerometer data from a space shuttle mission are used as inputs for the simulation model. These on-orbit acceleration data were obtained from the Microgravity Science Division at Glenn Research Center (GRC- MSD) and are applied to the buoyancy term of the momentum equation in a simulation of a Pb-5.8 wt. % Sb alloy that solidifies in a thermal gradient of 4000 K/m and a translation velocity of 3 p d s . Figure 2 shows the vertical velocity of a node that begins in the all-liquid region and subsequently solidifies; the vibrations are applied at 5000 seconds in this simulation. An important difficulty, common to all solidification models based on finite elements or 2 The magnitudes of the velocity oscillations that are vibration-induced are very small and acceptable. The biggest concern is whether the concentration of the liquid near the dendrite tips is distorted because of the vibration-induced perturbations. Results for this case show no concentration oscillations present in the all-liquid region.

Westra, D. G.↗

Dendritic solidification in a binary alloy melt: Steady-state versus morphological stability theories

A model for dendritic growth in a binary alloy melt, in the presence of a positive temperature gradient in the liquid, is presented. The model describes satisfactorily the transition from a dendritic interface to a planar interface at very low and very large growth rates. A dendrite tip stability parameter is derived, strictly from steady state considerations, without resorting to a perturbation analysis. The estimates of the parameter agree with those obtained by models based on perturbation analyses.

Laymanan, V.↗

Dendritic solidification in a binary alloy melt - Comparison of theory and experiment

A simple model for 'constrained' growth of an 'array' of cells or dendrites in a binary alloy melt, in the presence of a positive temperature gradient in the liquid ahead of the tips, is presented. The cell or dendrite tip radius is calculated by adopting both the 'ad hoc' assumption of minimum undercooling at the tips as well as the more recently proposed hypothesis of dendritic growth under conditions of 'marginal stability'. Theoretical predictions of the model have been compared with experimental data in binary succinonitrile-acetone alloys. It has been shown that, according to the present model, the 'marginally stable' state may be virtually indistinguishable from the 'minimum undercooled' state, under the usual conditions of dendritic growth.

Laxmanan, V.↗

Nonplanar interface morphologies during unidirectional solidification of a binary alloy. II - Three-dimensional computations

A finite difference method is used to obtain three-dimensional steady-state solutions for nonplanar interface morphologies in order to study the situation of equal thermal properties in the crystal and melt with negligible latent heat release. Stable steady-state solutions corresponding to two-dimensional bands and three-dimensional hexagonal nodes, as well as to rectangular interface planiforms, are found using a model of an aluminum-chromium alloy with a distribution coefficient of greater than one. Hexagonal nodes are predicted near the onset of instability, in agreement with weakly nonlinear theory.

Mcfadden, G. B.↗

Isothermal solidification in a binary alloy melt

A space shuttle experiment employing the General Purpose (Rocket) Furnace (GPF) in its isothermal mode of operation is manifested on MSL-3, circa 1989. The central aim of this experiment is to investigate the effect of reduced gravity levels on the segregation behavior in a slowly, and isothermally, cooled sample of a binary Pb-15 wt percent Sn alloy. This experiment would thus be able to simulate, in a small laboratory sample, about 20 mm dia 60 mm high and weighing about 150 grams, some aspects of the segregation phenomena occurring in large industrial ingots. Ground-based experiments conducted in the single-cavity simulator of the GPF, located at Marshall Space Flight Center (MSFC), in support of the microgravity experiment are described in detail. The results of the MSFC experiments are compared with other related experiments conducted at Case Western Reserve University (CWRU), wherein the isothermal constraints were relaxed.

Laxmanan, V.↗

Vertical solidification of dendritic binary alloys

Three numerical techniques are employed to analyze the influence of thermosolutal convection on defect formation in directionally solidified (DS) alloys. The finite-element models are based on the Boussinesq approximation and include the plane-front model and two plane-front models incorporating special dendritic regions. In the second model the dendritic region has a time-independent volume fraction of liquid, and in the last model the dendritic region evolves as local conditions dictate. The finite-element models permit the description of nonlinear thermosolutal convection by treating the dendritic regions as porous media with variable porosities. The models are applied to lead-tin alloys including DS alloys, and severe segregation phenomena such as freckles and channels are found to develop in the DS alloys. The present calculations and the permeability functions selected are shown to predict behavior in the dendritic regions that qualitatively matches that observed experimentally.

Heinrich, J. C.↗

Convective Influence on Radial Segregation During Unidirectional Solidification of the Binary Alloy HgCdTe

In order to simulate the space environment for basic research into the crystal growth mechanism, Hg(0.8)Cd(0.2)Te crystals were grown by the vertical Bridgman-Stockbarger method in the presence of an applied axial magnetic field. The influence of convection, by magneto hydrodynamic damping, on mass transfer in the melt and segregation at the solid-liquid interface was investigated by measuring the axial and radial compositional variations in the grown samples. The reduction of convective mixing in the melt through the application of the magnetic field is found to have a large effect on radial segregation and interface morphology in the grown crystals. Direct comparisons are made with a Hg(0.8)Cd(0.2)Te crystal grown without field and also in the microgravity environment of space during the second United States Microgravity Payload Mission (USMP-2).

Watring, D. A.↗

Effect of a Nonplanar Melt-Solid Interface On Lateral Compositional Distribution During Unidirectional Solidification of a Binary Alloy with a Constant Growth Velocity V: Theory - Pt. 1

Infrared detected materials, such as Hg(1-x)Cd(x)Te, Hg(1-x)Zn(x)Te have energy gaps almost linearly proportional to their composition. Due to the wide separation of liquidus and solidus curves of their phase diagram, there are compositional segregation in both of the axial and radial directions of these crystals grown in the Bridgman system unidirectionally with constant growth rate. It is important to understand the mechanisms, which affect lateral segregation such that large radially uniform composition crystal can be produced. Following Coriel, etc's treatment, we have developed a theory to study the effect of a curved melt-solid interface shape on lateral composition distribution. The model is considered to be a cylindrical system with azimuthal symmetry and a curved melt-solid interface shape which can be expressed as a linear combination of a series of Bessell's functions. The results show that melt-solid interface shape has a dominant effect on the lateral composition distribution of these systems. For small values of beta, the solute concentration at the melt-solid interface scales linearly with interface shape with a proportional constant of the produce of beta and (1 -k), where beta = VR/D, with V as growth velocity, R as the sample radius, D as the diffusion constant and k as the distribution constant. A detailed theory will be presented. A computer code has been developed and simulations have been performed and compared with experimental results. These will be published in another paper.

Wang, Jai-Ching↗

Effect of a Nonplanar Melt-Solid Interface on Lateral Compositional Distribution during Unidirectional Solidification of a Binary Alloy with a Constant Growth Velocity V: Theory - Part 1

Infrared detector materials, such as Hg(1-x)Cd(x)Te, Hg(1-x)Zn(x)Te have energy gaps almost linearly proportional to its composition. Due to the wide separation of liquidus and solidus curves of their phase diagram, there are compositional segregations in both of axial and radial directions of these crystals grown in the Bridgman system unidirectionally with constant growth rate. It is important to understand the mechanisms which affect lateral segregation such that large uniform radial composition crystal is possible. Following Coriell, etc's treatment, we have developed a theory to study the effect of a curved melt-solid interface shape on the lateral composition distribution. The system is considered to be cylindrical system with azimuthal symmetric with a curved melt-solid interface shape which can be expressed as a linear combination of a series of Bessell's functions. The results show that melt-solid interface shape has a dominate effect on lateral composition distribution of these systems. For small values of b, the solute concentration at the melt-solid interface scales linearly with interface shape with a proportional constant of the product of b and (1 - k), where b = VR/D, with V as growth velocity, R as sample radius, D as diffusion constant and k as distribution constant. A detailed theory will be presented. A computer code has been developed and simulations have been performed and compared with experimental results. These will be published in another paper.

Wang, Jai-Ching↗

A Phase-Field Model for In-Space Manufacturing of Binary Alloys

The integrity of the final printed components is mostly dictated by the adhesion between the particles and phases that form upon solidification, which is a major problem in printing metallic parts using available In-Space Manufacturing (ISM) technologies based on the Fused Deposition Modeling (FDM) methodology. Understanding the melting/solidification process helps increase particle adherence and allows to produce components with greater mechanical integrity. We developed a phase-field model of solidification for binary alloys. The phase-field approach is unique in capturing the microstructure with computationally tractable costs. The developed phase-field model of solidification of binary alloys satisfies the stability conditions at all temperatures. The suggested model is tuned for Ni-Cu alloy feedstocks. We derived the Ginzburg-Landau equations governing the phase transformation kinetics and solved them analytically for the dilute solution. We calculated the concentration profile as a function of interface velocity for a one-dimensional steady-state diffuse interface neglecting elasticity and obtained the partition coefficient, k, as a function of interface velocity. Numerical simulations for the diluted solution are used to study the interface velocity as a function of undercooling for the classic sharp interface model, partitionless solidification, and thin interface.

36 MATERIALS SCIENCE↗