Solidification Of Binary Alloy Research Articles

Exact phenomenological theory for thermotransport in a solid binary alloy

ABSTRACT In this paper, we study the theory of the diffusion behaviour of a binary alloy in a temperature gradient (thermotransport). Using the Onsager formalism of irreversible thermodynamics we develop exact relations for the heats of transport of each component in terms of various measurable diffusion quantities under steady state conditions and with the vacancies being at local equilibrium. Apparent inconsistencies with some earlier work are indicated. Further relations for heats of transport are then developed in terms of the random alloy model for the Darken and Manning approximations. Relations for the formal separation of the heats of transport are also developed.

Investigation of phase-field data-assimilation system using results of thin-film in situ observation during binary alloy solidification

Investigation of phase-field data-assimilation system using results of thin-film in situ observation during binary alloy solidification

Interface kinetics modeling of binary alloy solidification by considering correlation between thermodynamics and kinetics

Abstract By considering collision-limited growth mode and short-range diffusion-limited growth mode simultaneously, an extended kinetic model for solid−liquid interface with varied kinetic prefactor was developed for binary alloys. Four potential correlations arising from effective kinetics coupling the two growth modes were proposed and studied by application to planar interface migration and dendritic solidification, where the linear correlation between the effective thermodynamic driving force and the effective kinetic energy barrier seems physically realistic. A better agreement between the results of free dendritic growth model and the available experiment data for Ni−0.7at.%B alloy was obtained based on correlation between the thermodynamics and kinetics. As compared to previous models assuming constant kinetic prefactor, a common phenomenon occurring at relatively low undercoolings, i.e. the interface migration slowdown, can be ascribed to both the thermodynamic and the kinetic factors. By considering universality of the correlation between the thermodynamics and kinetics, it is concluded that the correlation should be considered to model the interface kinetics in alloy solidification.

Effect of magnetic field on stability in mushy layer during binary alloy solidification

During directional solidification of binary alloy mixtures, instability in the solid/liquid interface appears due to constitutional undercooling. As a result of this instability, a reactive porous medium, namely mushy layer, is formed, and it separates the liquid phase from the solid phase completely. The intrinsic structure of the mushy layer is of fine-scale dendritic crystal that shelters solute in the interstitial fluid. In a gravitational field, the rejection of lighter solute components from an advancing solidification front brings about unstable density gradient. Ensuing convective motions in the mush are driven by a density difference. The convection can change the solid matrix of the mushy layer. Hence, the dynamic response of the mushy layer is driven by interaction among heat transfer, solute transport and convection. As a contactless control tool, external magnetic field can change the heat and solute transport, which has a significant effect on the phase change process. Therefore, when magnetic field, thermal diffusion, solute transport and buoyancy convection are considered simultaneously in the phase transformation process, the mechanism of mushy region will become more complex and interesting. In this paper, the effect of external magnetic field on the stability of mushy layer during binary alloy solidification is studied. The coupling effects of magnetic field, temperature field, concentration field and convection are considered in the model. Including the direct mode and the oscillation mode, the resulting dispersion relation reveals the influence of magnetic field on the stability of mushy layer through linear stability analysis. It is found that the Lorentz force can reduce the instability effect which is caused by buoyancy convection. In the oscillation mode, an external magnetic field brings about a stabilizing effect on the mushy layer, but in the direct mode, the effect of external magnetic field on stability of the mushy layer is uncertain. In conclusion, the finding in this paper provides an important theoretical reference for improving products quality by applying an external magnetic field in the metals processing industry.

Revisiting dynamics and models of microsegregation during polycrystalline solidification of binary alloy

Microsegregation formed during solidification is of great importance to material properties. The conventional Lever rule and Scheil equation are widely used to predict solute segregation. However, these models always fail to predict the exact solute concentration at a high solid fraction because of theoretical assumptions. Here, the dynamics of microsegregation during polycrystalline solidification of refined Al-Cu alloy is studied via two- and three-dimensional quantitative phase-field simulations. Simulations with different grain refinement level, cooling rate, and solid diffusion coefficient demonstrate that solute segregation at the end of solidification (i.e. when the solid fraction is close to unit) is not strongly correlated to the grain morphology and back diffusion. These independences are in accordance with the Scheil equation which only relates to the solid fraction, but the model predicts a much higher liquid concentration than simulations. Accordingly, based on the quantitative phase-field simulations, a new analytical microsegregation model is derived. Unlike the Scheil equation or the Lever rule that respectively overestimates or underestimates the liquid concentration, the present model predicts the liquid concentration in a pretty good agreement with phase-field simulations, particularly at the late solidification stage.

A Monte Carlo approach to simulate dendritic microstructures during binary alloy solidification

A Monte Carlo (MC) approach to model the evolution of microstructures during the binary alloy solidification process has been presented. The evolution of these microstructures is comprehensively modelled by ensuring a local minimization of free energy taking into account the contributions of the free energy change during phase transformation, solid-liquid interfacial energy, interfacial energy anisotropy and grain boundary energy, which are some of the ignored aspects of the existing MC models. Further, the effect of imposed orientation is also represented in this model. The proposed method is able to simulate the preferential growth of cubic structures in ⟨100⟩ direction, growth of large number of grains and competition among them, and also incorporates the grain boundary energetics. The model is applied to simulate the growth of free and constrained dendrites, the transition from stable to cellular and dendritic growth under different growth conditions. Growth of multiple grains, the competition during columnar growth, columnar to equiaxed transition by seeding the undercooled melt with solid phase and the impingement of equiaxed grains has also been simulated using this model.

An enthalpy based model for microstructure evolution during binary alloy solidification

An enthalpy based solidification model is presented to simulate the evolution of binary alloy microstructure. The model is capable of simulating the nucleation and growth of multiple equiaxed and columnar grains subjected to a given undercooling or cooling rate. Thermal and species transport, forced convection, buoyancy driven convection, shrinkage driven flow and formation of secondary arms are incorporated in the model. The developed model uses a volume averaged enthalpy based solidification solver which is coupled with a pressure based flow solver and a probabilistic nucleation model. Simulations are performed to show the ability of the model in predicting microstructure evolution, flow pattern and microsegregation. Subsequently, the model is used to study the effect of cooling rate on grain size and solute segregation during microstructure evolution, formation of secondary arms and the effect of undercooling on primary arm spacing in columnar grain growth. The effect of natural convection on columnar dendrite growth is also studied. It is found that grain size and solute segregation decreases with increase in cooling rate. Also, it is observed that for columnar growth the primary arm spacing decreases with increase in initial undercooling but remains almost unaffected with change in specified number of nuclei. The developed model is extended to capture three-dimensional dendrite growth with secondary arms.

Numerical modeling of equiaxed crystal growth in solidification of binary alloys using a lattice Boltzmann-finite volume scheme

A lattice Boltzmann-finite volume scheme is proposed to numerically study the equiaxed crystal growth during binary alloy solidification. In present model, the lattice Boltzmann method is extended to numerically model the growth of equiaxed crystal coupled with solute transfer. An anisotropic lattice Boltzmann equation fitting into the lattice Bhatnagar-Gross-Krook (BGK) scheme is applied to describe the liquid-solid phase changes with interfacial anisotropic effect. Solute transfer is described by the diffusion equation including an anti-trapping current term, which is solved by the finite volume scheme and coupled straightly with the anisotropic lattice Boltzmann equation. After model validation, the hybrid model is applied to numerically simulate the equiaxed crystal growth with single and multiple seeds during Al-Cu solidification. The results demonstrate that the hybrid model is an alternative approach for numerical simulations of equiaxed crystal growth during binary alloy solidification with reliable numerical accuracy and excellent computational efficiency, which has important instructional significance for understanding the underlying mechanism of alloy solidification.

Interface kinetics of rapid solidification of binary alloys by atomistic simulations: Application to Ti-Ni alloys

Using molecular dynamics and Monte Carlo simulations we investigate the non-equilibrium interfacial kinetics during rapid solidification of Ti-Ni alloys. According to the existing theories, the kinetic coefficient is related, via an analytical expression, to the equilibrium and non-equilibrium solute concentration profiles across the crystal-melt interface, the interface temperature and velocity, and the drag coefficient. The kinetic coefficient was obtained by deriving these properties from specifically designed atomistic simulations and then fitting using the analytical expression. The results show that the kinetic coefficient is only weakly anisotropic and increases with increasing temperature. The velocity-dependent partition coefficient, as described by two solute trapping models, the continuous growth and the local non-equilibrium models, were fitted to the molecular dynamics simulation results. In addition, molecular dynamics and semi-grand canonical Monte Carlo simulations suggest that complete solute trapping might occur only under certain conditions. This can be explained by the fact that the maximum achievable effective free energy driving force for solidification, which defines the complete solute trapping, is limited by the chemical potential and free energy profiles for the Ti-Ni alloy. The investigations, using molecular dynamics simulations, of the dependence of crystal-melt interface width on solidification velocity show, at high velocities, similar trend to that predicted by the hyperbolic phase field model which is suitable for studies of rapid solidification of alloys.

Extended Stefan problem for the solidification of binary alloys in a sphere

We study the extended Stefan problem which includes constitutional supercooling for the solidification of a binary alloy in a finite spherical domain. We perform an asymptotic analysis in the limits of large Lewis number and small Stefan number which allows us to identify a number of spatio-temporal regimes signifying distinct behaviours in the solidification process, resulting in an intricate boundary layer structure. Our results generalise those present in the literature by considering all time regimes for the Stefan problem while also accounting for impurities and constitutional supercooling. These results also generalise recent work on the extended Stefan problem for finite planar domains to spherical domains, and we shall highlight key differences in the asymptotic solutions and the underlying boundary layer structure which result from this change in geometry. We compare our asymptotic solutions with both numerical simulations and real experimental data arising from the casting of molten metallurgical grade silicon through the water granulation process, with our analysis highlighting the role played by supercooling in the solidification of binary alloys appearing in such applications.

Phase field modelling of dendritic solidification by using an adaptive meshless solution procedure

A novel numerical procedure is developed for modelling two-dimensional dendritic solidification in dilute binary alloys. The evolution of the phases and the solute concentration is described by the partial differential equations, obtained from the phase field model. The meshless radial basis function-generated finite difference (RBF-FD) method is used for the spatial discretisation of the partial differential equations. The forward Euler scheme is used for the time-stepping. In order to reduce the computational cost, an adaptive procedure is developed, based on the quad-tree strategy, ensuring the highest density of the computational nodes at the solid-liquid interface. In the procedure, the computational domain is divided into overlapping sub-domains which can be dynamically refined or coarsened. The regular or scattered node distribution with constant node density is used for discretisation of each sub-domain. The h-adaptive procedure is ensured by the constant product between the area of a sub-domain and the computational node density. The accuracy and speedup in comparison to the solution on a uniform node distribution are assessed by solving the benchmark problem for dendritic solidification in dilute binary alloys. The main originality of the model represents the first use of RBF-FD method for the spatial discretisation of the PF equations in combination with adaptive solution procedure. The RBF-FD method can be used on unstructured node distributions, which is especially advantageous in the solution of PF model for dendritic growth, since the solution is very sensitive to the regularity of the node distribution. The developed spatial-temporal-adaptive numerical model represents an accurate and computationally efficient tool for the prediction of the dendrite morphology and micro-segregation during the solidification in binary alloys.

Numerical simulation of solidification of an iron-based binary alloy modified with nanoscale particles

A mathematical model of solidification of an iron-based binary alloy (Fe-C) modified with nanosize high-refractory particles is proposed. The processes of alloy heterogeneous nucleation on the surface of nanosize particles and those of alloy crystallization in a cylindrical mold are described. For liquidus temperature, a linear approximation of the concentration of dissolved carbon is adopted; the law of change of this temperature obeys the non-equilibrium lever rule. The volume of the solid phase formed around a nucleus during the crystallization determines the characteristic size of the grain structure in the solidified alloy. Numerical simulation of melt solidification in a cylindrical crucible was carried out, and the kinetics of heterogeneous nucleation and growth of the solid phase was analyzed. It was found that the conditions for nucleation and crystallization differ substantially within the casting. It is shown that, as the melt undergoes cooling, volume-sequential crystallization gets established.

Microsegregation model including Soret effect: an application for Sn-Bi alloy

We present an application of a microsegregation model that takes into account the Soret effect. This model, summarized here, lies in the context of the evaluation of thermodiffusion influence in alloys solidification. Initially, we deal with the thermodiffusion effect on the microsegregation scale. Subsequently, the Soret effect on macrosegregation shall be analyzed. The present model was conceived with similar considerations compared to the Gulliver-Scheil approach. A dimensionless parameter emerged from the model, which is potentially useful for assessing the Soret effect on the solidification of binary alloys. In the present work, this parameter was applied to an Sn-Bi alloy. The microsegregation model including the Soret effect can be widely used in the study of solute redistribution.

Comparison of two-phase and three-phase macroscopic models of equiaxed grain growth in solidification of binary alloy with electromagnetic stirring

Simulations of equiaxed solidification using two-phase and three-phase models are performed for the experimental benchmark AFRODITE with electromagnetic stirring. A three-phase model presented by authors elsewhere accounts for solid phase, inter- and extradendritic liquid phases. With respect to that model, the two-phase approach can be considered as reduced or simplified, yet, this implies also less number of assumptions regarding closure relations. In simulations, as expected, final segregation obtained with two–phase model is stronger, yet, it is qualitatively similar to the segregation pattern obtained with three-phase model.

Comparing mesoscopic models for dendritic growth

We present a quantitative benchmark of multiscale models for dendritic growth simulations. We focus on approaches based on phase-field, dendritic needle network, and grain envelope dynamics. As a first step, we focus on isothermal growth of an equiaxed grain in a supersaturated liquid in three dimensions. A quantitative phase-field formulation for solidification of a dilute binary alloy is used as the reference benchmark. We study the effect of numerical and modeling parameters in both needle-based and envelope-based approaches, in terms of their capacity to quantitatively reproduce phase-field reference results. In light of this benchmark, we discuss the capabilities and limitations of each approach in quantitatively and efficiently predicting transient and steady states of dendritic growth. We identify parameters that yield a good compromise between accuracy and computational efficiency in both needle-based and envelope-based models. We expect that these results will guide further developments and utilization of these models, and ultimately pave the way to a quantitative bridging of the dendrite tip scale with that of entire experiments and solidification processes.

Cellular automaton modelling to predict multi-phase solidification microstructures for Fe-C peritectic alloys

In this study, we developed a cellular automaton (CA) model to simulate the multiphase microstructural evolution of the solidification of Fe-C binary peritectic alloys. Three phases were considered in the model: δ, γ, and liquid (L). To simulate microstructures formed by the peritectic transformation, moving interfaces at δ-γ, δ-L, and γ-L transformations were introduced for diffusion-controlled growth. The coarsening growth of γ grains after solidification of peritectic alloys was modelled using the CA method. Numerical simulations of multi-phase microstructure evolution for hyper-peritectic and hypo-peritectic alloys were performed in two and three dimensions. In these simulations, the nuclei of γ phases were formed at the δ/L interface below the peritectic temperature. After nucleation of the γ phase in hyper-peritectic alloys, the δ phase in the calculation domain disappeared before γ solidification was completed. Conversely, in hypo-peritectic alloys, the L phase disappeared before the δ-γ transformation was complete. Moreover, the coarsening of γ grains occurred for both alloys after the end of the peritectic transformation. These outcomes agree with the fact that the phase transformation of alloys involves the solidification of peritectic alloys. Thus, we confirmed that the proposed model was a valid approach to simulating multi-phase microstructures formed by the solidification of peritectic alloys.

Evaluation of the Effect of Cooling Conditions and Heat Transfer Coefficients on Solidification of Al-Cu Binary Alloy in Static Casting Process

Source-based method for modelling solidification problems have been modified and presented in the current work. It coupled the effect of thermal radiation to macro-transport codes and was solved using finite volume method. The problem was formulated based on the classic continuum energy conservation equation for transient conduction controlled solidification system. Radiation heat transfer and latent heat evolution were added as source terms and solved with appropriate numerical treatments to obtain a system of linearized source terms. This circumvented the need for the application of any analytical solution to the intricate heat transfer regimes included in the model. The effect of cooling was carried out under various cooling conditions imposed on different surfaces of the mould for the solidifying metal. The resultant influence of cooling on the solid fraction evolution during static casting was then evaluated. The simulated cooling curves show that thermal radiation have no influence on the rate of heat extraction and the results show that the predicted cooling curves and solid fraction updates are similar to the results of previous models. The predicted curves at the top section of the open mould however show a little deviation due to effect of surface tension gradient forces. It was further revealed that heat transfer coefficients has more effect cooling curves and temperature contours at the lateral mould surfaces than the interior of the casting which is in agreement with theory of Newtonian cooling.

Reconstruction of the Boundary Condition in the Binary Alloy Solidification Problem with the Macrosegregation and the Material Shrinkage Phenomena Taken into Account

In this paper we solve the inverse problem of the binary alloy solidification including two phenomena: macrosegregation and creation of the air gap between the cast and the mold in result of the metal shrinkage during the solidification. Aim of inverse problem is to reconstruct the heat transfer coefficient on the external boundary of the mold separating the cast from the environment, by using the measurements of temperature taken from the sensor located in the middle of the mold. The solidification process is described by using the model of solidification in the temperature interval, the shrinkage of metal is modeled basing on the mass balance equation and the macrosegregation process is approximated with the aid of two models: the Scheil model and the lever arm model - the results received from those two approaches are then compared. For retrieving the sought coefficient, we minimize the proper functional by using the Artificial Bee Colony algorithm.

Modelling binary alloy solidification with adaptive mesh refinement

The solidification of a binary alloy results in the formation of a porous mushy layer, within which spontaneous localisation of fluid flow can lead to the emergence of features over a range of spatial scales. We describe a finite volume method for simulating binary alloy solidification in two dimensions with local mesh refinement in space and time. The coupled heat, solute, and mass transport is described using an enthalpy method with flow described by a Darcy-Brinkman equation for flow across porous and liquid regions. The resulting equations are solved on a hierarchy of block-structured adaptive grids. A projection method is used to compute the fluid velocity, whilst the viscous and nonlinear diffusive terms are calculated using a semi-implicit scheme. A series of synchronization steps ensure that the scheme is flux-conservative and correct for errors that arise at the boundaries between different levels of refinement. We also develop a corresponding method using Darcy's law for flow in a porous medium/narrow Hele-Shaw cell. We demonstrate the accuracy and efficiency of our method using established benchmarks for solidification without flow and convection in a fixed porous medium, along with convergence tests for the fully coupled code. Finally, we demonstrate the ability of our method to simulate transient mushy layer growth with narrow liquid channels which evolve over time.

Solidification of Binary Alloys and Nonequilibrium Phase Transitions

A model was constructed for reconstructing the initial stage of solidification of binary alloys treated as a nonequilibrium phase transition with a diffusion stratification mechanism. Numerical experiments concerning the self-excitation of a homogeneous state by applying a melt cooling boundary control condition were performed.