Model Of Directional Solidification Research Articles

Gradient-Stable Algorithm for the Dynamics of the Phase Field in Modeling of Directional Solidification in Si–As

Gradient-Stable Algorithm for the Dynamics of the Phase Field in Modeling of Directional Solidification in Si–As

Convective effects on columnar dendritic solidification – A multiscale dendritic needle network study

Gravity-induced buoyancy, inevitable in most solidification processes, substantially alters the dynamics of crystal growth, such that incorporating fluid flow in solidification models is crucial to understand and predict key aspects of microstructure selection. Here, we present a multi-scale Dendritic Needle Network (DNN) model for directional solidification that includes buoyant flow in the liquid, and apply it to a range of alloys and growth conditions. After a brief presentation of the model, we study the selection of stable primary dendrite arm spacings in Al-4at.%Cu and in Ti-45at.%Al alloys under different gravity levels, comparing both applications to published phase-field results and experimental measurements. Then, we simulate the oscillatory growth behavior recently reported via X-ray in situ imaging of directional solidification of nickel-based superalloy CMSX-4. In this last application, the DNN simulations manage to reproduce the oscillatory growth behavior, and hence permit identifying the fundamental mechanisms behind the oscillatory growth regime. In particular, we show that sustained oscillations occur when the average liquid flow velocity is close to the crystal growth velocity, and that primary dendritic spacings also play a crucial role in the oscillatory behavior.

Gradient-Stable Algorithm for the Dynamics of the Phase Field at Modeling of Directional Solidification in Si-As

Gradient-Stable Algorithm for the Dynamics of the Phase Field at Modeling of Directional Solidification in Si-As

A Mathematical Analysis of Directional Solidification of Aqueous Solutions

Abstract Horizontal directional solidification techniques have been broadly utilized for the freezing of biological matter under conditions in which the freezing rate during solidification must be controlled and known. Directional solidification is used for diverse applications such as fundamental research on freezing of biological materials, cryopreservation of biological matter, and tissue engineering. This study is motivated by our intent to use directional solidification as a simplified model for the study of three-dimensional (3D) cryoprinting. In evaluating directional solidification in the context of 3D cryoprinting, we realized that current mathematical models of directional solidification are not adequately representative for this purpose, because they are simplified and one-dimensional (1D). Here, we introduce an experimentally verified and more representative two-dimensional (2D) mathematical model of directional solidification that can aid in the fundamental study of freezing of biological matter, in particular during 3D cryoprinting. The mathematical model was used to develop correlations between the freezing rates that a layer of an aqueous solution experiences during directional solidification and the various design parameters such as thickness of the sample and temperature gradients in the substrate. Results show that the freezing rates can be higher than those suggested by the previously used simplified 1D mathematical models. The results can be used for developing simplified models of 3D cryoprinting. In addition, the results suggest that many experimental studies on directional solidification of aqueous solutions and biological matter may require readjustment of analysis, in view of these findings.

Global heat transfer model and dynamic ray tracing algorithm for complex multiple turbine blades of Ni-based superalloys in directional solidification process

High-quality solidification microstructure during directional solidification relies on precise temperature gradient control, so accurate calculation of the temperature field is critical. In this study, a 3D transient global heat transfer model of directional solidification by the Bridgman method based on the finite difference method is developed. The radiation heat in this model is calculated by the discrete transfer method, and a modified method of external surface area for irregular geometric models is proposed to reduce the zigzag shape caused by finite difference grids. Considering the radiative heat transfer between any surface elements of all materials in the directional solidification furnace, a dynamic ray tracing algorithm is developed to simulate the entire process of directional solidification. Then, the simulated results are compared with the theoretical results and experimental results, respectively. Finally, based on the present model and method, the simulation program developed is applied to the directional solidification of actual castings. The simulated results are in good agreement with the experimental results, which indicate that the model and method developed in this study is effective and practical.

Phase‐field simulation of dendrite growth under forced flow conditions in an Al–Cu welding molten pool

A quantitative phase‐field model for directional solidification is applied to study dendrite growth under forced flow conditions in an Al‐Cu Gas Tungsten Arc (GTA) welding molten pool. Evolution of the dendrite morphology and the solute field under forced flow conditions is simulated. Growth of columnar grains goes through three periods, including the initial instability period, the competitive growth period and the relatively stable period. The solute segregation, the solute redistribution and the solute concentration in the liquid side of the interface are investigated, respectively. For the given conditions, simulation results are in good agreement with experimental findings.

Heat Transfer Model of Directional Solidification by LMC Process for Superalloy Casting Based on Finite Element Method

With the rapid development of the aviation industry, the turbine blade, a critical component of the aeronautical engine, has come to be widely produced by liquid-metal cooling (LMC) process. A temperature- and time-dependent heat transfer coefficient was used to represent the heat convection between the shell and the cooling liquid, and an improved Monte Carlo ray-tracing approach was adopted to handle the boundary of radiation heat transfer. Unstructured mesh was used to fit the irregular shell boundary, and the heat transfer model of directional solidification by LMC process based on finite element method (FEM) was established. The concept of local matrix was here proposed to guarantee computational efficiency. The pouring experiments of directional solidification by LMC process were carried out, then simulation and experimental results were compared here. The accuracy of the heat transfer model was validated by the cooling curves and grain morphology, and the maximum relative error between simulation and experimental cooling curve was 2 pct. The withdrawal rate showed an important influence on the shape of solidification interface, and stray grain is liable to be generated on the bottom of platform at an excessive withdrawal rate.

Advances in Modeling of Solidification Microstructures

In different manufacturing processes, such as casting, welding, and additive manufacturing, metallic materials go through solidification to produce parts with different sizes and shapes. Microstructures that develop during solidification control the property and performance of the manufactured parts, thereby predicting the solidification microstructures as functions of alloying composition and processing parameters is essential to control the quality of products. The multiscale nature of solidification microstructures, which depend on temperature distribution, solute concentration, capillary forces, and kinetic length, make the understanding and prediction of these microstructures exceptionally difficult. In recent years, and because of emergence of new and powerful supercomputers, it has become more feasible to computationally simulate the materials nanoand microstructures in large scales and with finest details. The computational models for solidification of metallic materials need to be developed based on the actual multiphysics of solidification, while considering the need for efficient numerical algorithms to solve the governing equations of the models. To quantitatively predict solidification microstructures, computational models at mesoscale need information from theory, experiments, lower scale models (e.g., density functional theory calculations, molecular dynamics simulations, etc.), and phase diagram calculations. The current computational models for simulating dendritic growth at the microscopic scale are based on different methods such as phase field (diffusive interface), level set, direct interface tracking, and cellular automaton methods. Each of these methods has its advantages and disadvantages: some can simulate the finest details of solidification microstructures with high accuracy, while others can simulate dendritic growth in large-scale domains with high computational efficiency. In this Journal of Metals topic, we present recent contributions in modeling of solidification microstructures. Damien Tourret et al. present a three-dimensional (3D) version of the dendritic needle network (DNN) model for directional solidification. They apply the DNN model to predict the stable range of primary dendritic spacings for an Al-9.8 wt%Si alloy over a range of growth velocities. They compare their predictions to spacings measured from in situ x-ray imaging of directional solidification experiments. Mohsen Eshraghi et al. present a parallel 3D lattice Boltzmann-cellular automaton model to simulate dendritic growth during solidification of metallic binary alloys. Their large-scale simulations show a great scale-up performance up to 40,000 computing cores and an excellent speed-up performance on up to 1000 computing cores. Yasushi Shibuta and co-workers performed simulations of solidification from atomic to microstructural levels using a graphics processing unit (GPU) architecture. They use million-atom molecular dynamics simulations to study nucleation of solid phase in undercooled melt and to capture evolution of anisotropy for solid seeds. Using a quantitative phasefield model, they simulate dendrite growth in directional solidification at millimeter scale in two– dimensional (2D) and 3D by multi-GPU computation on a supercomputer. These two articles present promising techniques to predict solidification microstructures in large macro-scale domains with good computational efficiency. Alexander Monas et al. present 3D phase-field simulations of solidification microstructures of MgAl alloy. They use the Calphad method to obtain the phase diagram and necessary parameters for their simulations. Michael Rappaz and Guven Kurtuldu discuss a new nucleation mechanism in liquid metallic alloys based on thermodynamics arguments. They explain that the two-step nucleation mechanism starts with Mohsen Asle Zaeem is the guest editor for the Solidification Committee of the TMS Materials Processing & Manufacturing Division, and coordinator of the topic Advances in Modeling of Solidification Microstructures in this issue. JOM, Vol. 67, No. 8, 2015

Anisotropy of flow and transition between mixing regimes in physical model of directional solidification

Anisotropy of flow and transition between mixing regimes in physical model of directional solidification

Coupled Cellular Automaton (CA) – Finite Element (FE) Modeling of Directional Solidification of Al-3.5 wt% Ni Alloy: A Comparison with X-ray Synchrotron Observations

Development of dendritic grain structure and mesa-segregation leading to localized intergranular eutectic is simulated during directional solidification of an Al - 3.5 wt% Ni alloy with two pulling velocities (4 mu m s(-1) and 8 mu m s(-1)) using a two-dimensional (2D) Cellular Automaton Finite Element (CAFE) model. 2D CAFE simulations are compared with in situ and real-time experiments characterized by means of X-ray radiography at the European Synchrotron Radiation Facility (Grenoble, France). Two nucleation models are used. The first model is established by listing all measured positions and orientations of experimentally-observed nucleation events. The undetermined value of the nucleation undercooling is then set to 0 degrees C for all nucleation sites. The other model considers a stochastic model with a Gaussian distribution of the nucleation site as a function of the undercooling. It is derived from series of experiments. The influences of the nucleation models and liquid convection on the grain structure characteristics are numerically investigated. A good agreement with experimental observations is achieved concerning the evolutions of both the dendritic and the eutectic growth fronts (i.e., the size of the mushy zone). The predicted grain size and elongation are compared with measurements for the two nucleation laws. Simulations using the list of nucleation events reach better agreement with the longitudinal profiles of the grain equivalent diameter and elongation factor compared to the stochastic nucleation model. Direct tracking of the eutectic growth front as well as three-dimensional analyses are found to be required for improvement of the predictions.

Modeling of Directional Solidification of Columnar Grain Structure in CMSX-4 Nickel-Based Superalloy Castings

The paper presents the analysis of numerical simulation of the Bridgman directional solidification process performed on CMSX-4 rods. The numerical simulation was studied applying the ProCAST software. The constitutive law parameters of the normal Gaussian distribution were used to describe the nucleation process. The coefficients of the equation were determined and used to calculate the growth rate of dendrite tip. The analysis of the as-cast microstructure was carried out with the use of Aphelion software in order to determine the average area of grains and their quantity. The experimental verification of both nucleation and grain growth coefficients used for the simulation of the directional solidification process confirmed that the model was correct and described well the investigated process of directional solidification using the Bridgman method.

Formation of microstructures in the spheroidal graphite cast iron

Pipeline systems for hydraulic networks are obtained via centrifugal casting of spheroidal graphite cast iron. The very high cooling rate that is achieved in the skin of the product can sometimes lead to carbide instead of graphite in cast iron. An experimental device has been built in the laboratory that allows reproducing the extreme thermal conditions encountered during formation of skin of centrifugally cast pipes. Liquid metal droplets fall on a cold substrate. Rapid directional solidification occurs. The temperature evolution of the lower surface of the droplet is recorded during the very first moment of the solidification (t < 200 ms) thanks to a photodiode, which is located below the substrate. The microstructures that are obtained in laboratory are characterised in both the as-cast state and the heat-treated state. They are compared to the centrifugally cast ones. A model of directional solidification of cast iron under a very large temperature gradient has been built. It allows explaining the transition from stable to metastable micro structure that was observed in the products and reproduced in the laboratory samples.

Numerical Simulation of TiAl-Nb Alloy Solidification Experiment in TEM 01-3M Facility Aboard MAXUS 8

The problem of numerical modeling of directional solidification of TiAl refractory intermetallics aboard the MAXUS 8 sounding rocket is considered. The research is of relevance to the FP6 Integrated project IMPRESS (Intermetallic Materials Processing in Relation to Earth and Space Solidification). Attention is paid to columnar-to-equiaxed microstructure transition (CET) phenomenon and mushy zone evolution in Ti-45.9Al-8Nb (at %) alloy being processed in TEM 01-3M high-temperature (up to 17000C) furnace. In this three-zone resistive furnace the “bent” temperature profile is applied with two strongly different axial thermal gradients, presumably allowing the achieving of CET conditions along the sample of 160 mm length. Temperature profile evolution is defined by power-down furnace operation. 2D-numerical study of heat transfer and realtime-scale solidification dynamics of TiAl-Nb under zero gravity approximation is performed. The approaches used for solution of Navier-Stokes equations and phase transition (Stefan) problem are briefly described. The solidification time is shown to be satisfying the 12-minute microgravity limit aboard a MAXUS. The position and the time at which CET may be triggered are predicted and confirmed in line with the Hunt diagram. The comparison is performed of model predictions with the real microstructure of TiAl-Nb reference sample solidified on-ground in TEM 01-3M facility.

Three-dimensional adaptive phase field modeling of directional solidification of a binary alloy: 2D–3D transitions

Morphological instability is an important subject in alloy directional solidification, and the development of cell allay is important to alloy properties. In nature, as a result of cell arrangement, this problem is often three-dimensional (3D), even for thin-film, and its simulation is rather difficult. We have developed a 3D adaptive phase field model to simulate efficiently the directional solidification of a binary alloy. The morphological transition from 2D to 3D approximations are illustrated, and compared with available analytical and simulated results. The effect of anisotropy on the cell arrangement is discussed as well.

Modeling of directional solidification in 2D and 3D cases

A method of numerical solution of the directional solidification problem for sharp interfaces is developed. The cellular-structure development is analyzed by calculating the concentration fields and consecutive profiles of the solid-liquid interface. The anisotropy of growth rate and surface tension is taken into account. The wavelength ranges in which distortions develop are determined and the dependences of the rate of increase in the distortion amplitude on the wavelength are found. The results obtained for the 2D and 3D cases are compared with the known experimental and calculated data.

Numerical study on the prediction of microstructure parameters by multi-scale modeling of directional solidification of binary aluminum–silicon alloys

In order to create a model to predict microstructural quantities like grain size, primary and secondary dendrite arm spacing a multi-phase and multi-scale model based on the work of Wang and Beckermann [C. Wang, C. Beckermann, Metallurgical and Materials Transactions A 27A (1996) 2754–2764] was combined with a front tracking technique [A. Wu, A. Ludwig, in: C.-A. Gandin, M. Bellet (Eds.), Modeling of Casting, Welding, and Advanced Solidification Processes – XI, TMS, 2006, pp. 291–298], micro-models for nucleation [M. Rappaz, P. Thevoz, Acta Metallurgica 35 (7) (1987) 1487–1497], primary [J. Hunt, S.-Z. Lu, Metallurgical and Materials Transactions A 27A (1996) 611–623], secondary [W. Kurz, D. Fisher, Fundamentals of Solidification, Trans Tech Publication, 1986, ISBN 0-87849-522-3] dendrite arm spacing and a control volume based finite element solver for axial-symmetric problems. As most of the micro-models are just valid for pure diffusive conditions, the model just takes into account macroscopic diffusion in the melt and thus neglects the influence of melt flow. The new software was used for a comprehensive comparison to several test cases. The validation includes investigation of the correlation of calculated and measured grain size distributions for inoculated alloys. Experimental and numerical data for the primary and secondary dendrite arm spacing for steady state and transient directional solidification were compared in a second step. A good correlation is found for all test cases.

Modeling of directional solidification on an atomic scale

Kinetic modeling of directional solidification on an atomic scale has been carried out by the Monte Carlo method for the model of lattice liquid. Presence of vacancies in the melt that realizes diffusion of the components is taken into account. Activation energies for the interexchange between the atoms and the vacancies were calculated after counting all the nearest neighbors of both. Rates of development of distortions with different wavelengths in initial interfaces have been determined. Chemical diffusivity and surface energy were determined for the model under consideration. Particularities of cellular structure of the interface at very high pulling velocities have been studied.

Rapid directional solidification in Sn-Cu lead-free solder

An experimental study on the microstructures of a rapid directionally solidified metallo-eutectic Sn-Cu alloy was carried out. This material is an important alloy that is used as a lead-free solder. The results showed that the kinetic undercooling due to the rapid solidification process led to the formation of a pseudoeutectic zone, whereas the hypereutectic reaction produced the regular lamellar structure in the hypereutectic Sn-1.0Cu alloy. The corresponding arm spacing in the obtained lamellar phases decreased gradually with the increase of the applied cooling rate, which corresponded well with the prediction of a rapid directional solidification model.

Asymptotic solutions for an axisymmetric, stagnant film model of directional solidification

A model is presented to describe the time-dependent transport of heat and solute in a directional solidification system for a binary alloy. The stagnant film concept is used to include the effects of melt flow and the transport of solute by convection. The geometry of the model is described by an axisymmetric coordinate system. The solution procedure involves a coupled asymptotic/numerical approach. The asymptotic expansions combine quasi-steady limits with a small ampoule aspect ratio. The boundary layer solutions around the solidifying front are obtained under additional assumptions that the heat exchange between the ampoule and sample is small, the latent heat is large, the solute diffusivity in the solid phase is small, and the thickness of the stagnant film is small. The core melt and solid temperature profiles and the mean position of the solidifying front are calculated numerically as functions of the latent heat, heater profile, and solute profile in the liquid phase. Due to the inclusion of curvature effects on the melting point, the non-planar shape of the interface is described by a superposition of an oscillatory profile onto a parabolic profile. The parabolic profile is set via the balance of axial and radial diffusive transports of heat. The oscillatory profile develops in the case of undercooling as in the classical morphological instability of a planar interface during directional solidification. The above results are obtained for varying stagnant film thickness to study melt regimes ranging from diffusion—to convection—controlled growth. The dependence on the film thickness of axial and radial segregation and the morphological stability of the interface are considered.

Pattern selection in a phase field model for directional solidification

A symmetric phase field model is used to study wavelength selection in two dimensions. We study the problem in a finite system using a two-pronged approach. First we construct an action and, minimizing this, we obtain the most probable configuration of the system, which we identify with the selected stationary state. The minimization is constrained by the stationary solutions of stochastic evolution equations and is done numerically. Secondly, additional support for this selected state is obtained from straightforward simulations of the dynamics from a variety of initial states.