Solidification Of Multicomponent Alloys Research Articles

Accelerating phase-field simulation of multi-component alloy solidification by shallow artificial neural network

Low computing efficiency is a significant barrier in the phase-field modeling of multi-component alloys coupled with the CALPHAD (CALculation of PHAse Diagram) method. The fundamental issue is that the quasi-equilibrium thermodynamic data (QETD) required for calculating the chemical driving force must be acquired by repeatedly solving a large number of nonlinear equations. In this work, a novel method is developed to predict the QETD by a shallow neural network in a straightforward manner, so circumventing the repetitive numerical calculation of nonlinear quasi-equilibrium thermodynamic conditions. The numerical evaluation of a Ni-Cr-Al ternary alloy demonstrates that the proposed scheme can decrease the computing consuming to about 1/80 of that required by the conventional phase-field method when the computational domain size reaches the millimeter scale, while accurately reproducing the equiaxed dendritic growth and solute segregation kinetics during polycrystalline solidification. The method presented in this work will provide an effective tool for modeling the microstructure evolution of complex materials involved in practical engineering applications.

A novel treatment of solute drag and solute trapping at a solid–liquid interface during rapid solidification of multicomponent alloys

AbstractIn response to the renewed interest in solute drag and solute trapping models fueled by their applications to additive manufacturing, a novel treatment is proposed to describe the diffusional behaviors of solute at a migrating solid–liquid interface during rapid solidification of multicomponent alloys. While the treatment is still built on irreversible thermodynamics and linear kinetic law, its novelty lies in breaking up the classical trans‐interface diffusional flux into two separate fluxes one is the transferred‐back flux with its ending point at the interface and the other is the bumping‐back flux with its starting point at the interface. This novel treatment entails three significant improvements in reference to the existing models. Firstly, it reveals that the degree of solute drag is dependent on the ratio of liquid diffusive speed over interface diffusive speed. Secondly, a novel relation between the distribution coefficient and interface velocity is derived. It amends the confusing behavior seen in Aziz’s without‐drag continuous growth model. Thirdly, the proposed treatment eliminates the need of prescribing the degree of solute drag parameter for the kinetic phase diagram calculation. The numerical solution to the proposed model is presented, and it is ready to be used for the kinetic phase diagram calculation.

Kinetic effects during the plane-front and dendritic solidification of multicomponent alloys

A generalized model for the kinetics of a dendrite tip with a non-equilibrium interface is presented for multicomponent alloys. The model includes full coupling with thermodynamic databases to account for non-dilute non-ideal solutions, in addition to a full diffusion matrix in the liquid. The consequences of the computed non-equilibrium phase diagram boundaries on the dendrite tip kinetics are considered, and substantial deviations from existing theories are observed, especially in the case of zero solute drag. The model is applied to the rapid solidification of Inconel 718 (a nickel-based superalloy) and 316L (a stainless steel), which contain seven and five solute species, respectively. From the model, the phase diagram properties (i.e., partition coefficients and liquidus slopes) are able to be directly visualized as a function of velocity and observed to vary non-linearly and, in some cases, non-monotonically due to the combination of non-linear phase diagrams and kinetic effects. Finally, recent reports in the literature on the competition between the growth of ferrite and austenite from the melt in 316L are revisited with these new developments.

Interface Response Functions for multicomponent alloy solidification—An application to additive manufacturing

The near-rapid solidification conditions during additive manufacturing can lead to selection of non-equilibrium phases. Sharp interface models via interface response functions have been used earlier to explain the microstructure selection under such solidification conditions. However, most of the sharp interface models assume linear superposition of contributions of alloying elements without considering the non-linearity associated with the phase diagram. In this report, both planar and dendritic Calphad coupled sharp interface models have been implemented and used to explain the growth-controlled phase selection observed at high solidification velocities relevant to additive manufacturing. The implemented model predicted the growth-controlled phase selection in multicomponent alloys, which the other models with linear phase diagram could not. These models are calculated for different steels and the results are compared with experimental observations.

A numerical method for sharp-interface simulations of multicomponent alloy solidification

We present a computational method for the simulation of the solidification of multicomponent alloys in the sharp-interface limit. Contrary to the case of binary alloys where a fixed point iteration is adequate, we hereby propose a Newton-type approach to solve the non-linear system of coupled PDEs arising from the time discretization of the governing equations, allowing for the first time sharp-interface simulations of the multialloy solidification. A combination of spatially adaptive quadtree grids, Level-Set Method, and sharp-interface numerical methods for imposing boundary conditions is used to accurately and efficiently resolve the complex behavior of the solidification front. The convergence behavior of the Newton-type iteration is theoretically analyzed in a one-dimensional setting and further investigated numerically in multiple spatial dimensions. We validate the overall computational method on the case of axisymmetric radial solidification admitting an analytical solution and show that the overall method's accuracy is close to second order. Finally, we perform numerical experiments for the directional solidification of a Co-Al-W ternary alloy with a phase diagram obtained from the PANDAT™ database and analyze the solutal segregation dependence on the processing conditions and alloy properties.

Quantitative phase-field model for the isothermal solidification of a stoichiometric compound in a ternary liquid

The phase-field (PF) model is one of the most powerful mathematical models to simulate the solidification of multicomponent alloys. A major challenge faced by the PF model is the treatment of stoichiometric compounds (SCs). In the PF model, compositional derivatives of the free energies of phases need to be treated; however, those of SCs are thermodynamically defined only at a certain composition with respect to the stoichiometric components. Therefore, conventional studies used parabolic functions to provide a pseudo-compositional dependence to free energies. In this paper, we propose a methodology to treat SCs without using the parabolic free energy function. Time-evolution equations were rederived assuming that the free energy of the SC is independent of the composition of its stoichiometric component. We performed numerical calculations for a ternary system comprising two phases, olivine (stoichiometric solid phase) in a basaltic melt (liquid phase), whose free energies were gathered from a thermodynamic database, and we confirmed that the model represents the stoichiometry of the solid phase with high accuracy. This model provides a simple and straightforward methodology to handle SCs in the quantitative PF model.

A quantitative comparison between pseudo-binary and multi-component phase field models

We present a benchmark analysis to compare three different phase-field models for multi-component alloy solidification. This analysis is carried out between, two pseudo-binary approaches and one multi-component approach, with respect to solidification of a quaternary iron-containing Ti alloy. The first pseudo-binary PF model is a common approach based on the model of Karma et al. (Echebarria et al., 2004), while the second pseudo-binary PF model and the multi-component model are implemented based on the grand potential model of Provatas et al. (Shampur, 2017). A very good similarity in microstructure is achieved between the three phase field models during both isothermal and directional solidification. The two grand potential models also show an excellent agreement in solute segregation predictions, and they are able to predict both positively segregating and negatively segregating elements. Finally, the influence of solidification rate on the microstructure is also studied, with the results matching analytical predictions. The results demonstrate the usefulness of different PF modeling approaches, and highlight cases where a full multi-component model is needed.

Rapid Multicomponent Alloy Solidification with Allowance for the Local Nonequilibrium and Cross-Diffusion Effects.

Motivated by the fast development of various additive manufacturing technologies, we consider a mathematical model of re-solidification of multicomponent metal alloys, which takes place after ultrashort (femtosecond) pulse laser melting of a metal surface. The re-solidification occurs under highly nonequilibrium conditions when solutes diffusion in the bulk liquid cannot be described by the classical diffusion equation of parabolic type (Fick law) but is governed by diffusion equation of hyperbolic type. In addition, the model takes into account diffusive interaction between different solutes (nonzero off-diagonal terms of the diffusion matrix). Numerical simulations demonstrate that there are three main re-solidification regimes, namely, purely diffusion-controlled with solute partition at the interface, partly diffusion-controlled with weak partition, and purely diffusionless and partitionless. The type of the regime governs the final composition of the re-solidified material, and, hence, may serve as one of the main tools to design materials with desirable properties. This implies that the model is expected to be useful in evaluating the most effective re-solidification regime to guide the optimization of additive manufacturing processing parameters and alloys design.

Secondary dendritic arm spacing and cooling rate relationship for an ASTM F75 alloy

The solidification evolution of an ASTM F75 alloy was studied for samples quenched during directional cooling. These samples were obtained using five extraction rates (1.2–18 mm/min) and two temperature gradients (6 and 12 K/mm), leading to a 25-fold change in the cooling rate (0.13–3.31 K/s). The solidification range for this alloy is observed to be approximately 200 °C. The solid grows more rapidly in the first 100 °C of this interval, where a solid fraction of approximately 0.8–0.9 is reached. Coarsening of the Secondary Dendrite Arm Spacing, SDAS, occurs mainly during this growth period. This behavior defines an effective temperature interval for the SDAS coarsening instead of the whole solidification interval. Consideration of this effective temperature interval enables us to reconcile the measured SDAS-cooling rate relationship with that obtained from the data for the cast samples. A model of the SDAS coarsening during anisothermal solidification of multicomponent alloys is proposed. The predictions of this model compare well with the present experimental results.

Kinetic interface condition phase diagram for the rapid solidification of multi-component alloys with an application to additive manufacturing

Solid-liquid Interface response function or the analysis of kinetic interfacial conditions under rapid solidification setting occupies a central role in solidification research. With the concepts of solute drag and solute trapping and on the base of irreversible thermodynamics, the calculation of kinetic interface condition phase diagram for binary alloys has been well established owing to the continuous growth model by Aziz et al. Motivated by the fast development of various additive manufacturing (AM) technologies, clear needs to extend the kinetic phase diagram calculation towards multi-component alloy systems emerge to meet the challenges of processing parameter optimization encountered under those sub-rapid or rapid AM solidification conditions for new alloys. In this paper the irreversible thermodynamics analysis and the binary continuous growth model are reformulated into a form suitable for the coupling with the multi-component CALPHAD databases. Then the numerical solution to the CALPHAD-coupled multi-component model is described. The model and its numerical solution are verified by comparing its calculation results with those reported in literature for A-B hypothetical ideal solution phases, Ag–Cu and Al–Be alloys. The model predictive power is demonstrated by calculating the kinetic diagram of Al–Ti, Fe–Cr–Ni and Al–Cu–Mg–Si–Zn alloys. To illustrate one of the practical values of the proposed model, kinetic growth restriction factor is calculated from the predicted Al–Ti and Al–Cu–Mg–Si–Zn kinetic phase diagram. It is concluded that the proposed multi-component model and its numerical solution can calculate kinetic phase diagram of any multi-component alloys. Moreover, the proposed model can be used in evaluating solute effect for grain refinement under sub-rapid or rapid solidification conditions, which is great value to understand the solidification phenomenon in AM. The model is expected to be useful in many scenarios to guide the optimization of AM processing parameters and alloys design.

Identification of a multi-dimensional reaction coordinate for crystal nucleation in Ni3Al.

Nucleation during solidification in multi-component alloys is a complex process that comprises competition between different crystalline phases as well as chemical composition and ordering. Here, we combine transition interface sampling with an extensive committor analysis to investigate the atomistic mechanisms during the initial stages of nucleation in Ni3Al. The formation and growth of crystalline clusters from the melt are strongly influenced by the interplay between three descriptors: the size, crystallinity, and chemical short-range order of the emerging nuclei. We demonstrate that it is essential to include all three features in a multi-dimensional reaction coordinate to correctly describe the nucleation mechanism, where, in particular, the chemical short-range order plays a crucial role in the stability of small clusters. The necessity of identifying multi-dimensional reaction coordinates is expected to be of key importance for the atomistic characterization of nucleation processes in complex, multi-component systems.

On an expression for the growth of secondary dendrite arm spacing during non-equilibrium solidification of multicomponent alloys: Validation against ternary aluminum-based alloys

The technological importance of the microstructure length scale is directly related to the influence exerted on solute redistribution and microporosity formation and on mechanical properties, such as, toughness, ductility, ultimate and yield tensile strengths. There is a huge lack of literature concerning theoretical predictive dendritic growth models for unsteady-state solidification of multicomponent alloys. Most of the existing models have been proposed for steady-state solidification and for binary alloys. One of these models, initially restricted to binary alloys, has been extended for multicomponent alloys; however, it was shown to be valid only for low growth rates and small dendrite tip undercooling, that is, conditions that are very close to those of equilibrium cooling from the melt. In this paper, an extended approach is proposed, encompassing the back diffusion parameter β to allow back diffusion treatment to be included in the analysis. A technique based on Butler’s formulation and on thermodynamic databases is used to permit necessary thermophysical parameters, such as the surface energy and the Gibbs-Thomson coefficient to be calculated for Al-Cu-(Si;Mg) alloys. Directional solidification apparatuses are used to provide a wide range of experimental solidification cooling rates and growth rates along the length of the directionally solidified castings. The model predictions are validated against the experimental scatters of secondary dendrite arm spacings of Al-Cu-Si; Mg) alloys castings solidified under transient upward and horizontal heat flow conditions. It is shown that the predictions fit quite well the experimental results.

Phase field modeling of solidification in multi-component alloys with a case study on the Inconel 718 alloy

Abstract

Instabilities in rapid solidification of multi-component alloys

Rapid solidification of multi-component liquids occurs in many modern applications such as additive manufacturing. In the present work the interface departures from equilibrium consist of the segregation coefficient and liquidus slope depending on front speed, the one-sided, frozen-temperature approximation, and the alloy behaving as the superposition of individual components. Linear-stability theory is applied, showing that the cellular and oscillatory instabilities of the binary case are modified. The addition of components tends to destabilize the interface while the addition of a single large-diffusivity material can entirely suppress the oscillatory mode. Multiple minima in the neutral curve for the cellular mode occur.

Instabilities in solidification of multi-component alloys

Linear stability of the solid/liquid interface in directional solidification of a multicomponent alloy is examined. The front becomes more unstable as more components are present. The neutral stability boundary displays many minima compared to a single one for two-component alloys. Bounds are presented to guarantee stability in the multicomponent case.

Application of the thermodynamic extremal principle to phase-field modeling of non-equilibrium solidification in multi-component alloys

Modeling of non-equilibrium solidification in multi-component alloys is of singular importance in microstructure control, which however owing to the complex systems with complex additional constraints is still an open problem. In this work, the thermodynamic extremal principle was applied to solve the complex additional constraints self-consistently in thermodynamics. Consequently, short-range solute redistribution and long-range solute diffusion that share the same mobility are integrated naturally into the solute diffusion equations, thus avoiding the introduction of additional kinetic coefficients (e.g. interface permeability) to describe solute redistribution. Application to the non-equilibrium solidification of Al-Si-Cu alloys shows that anomalous solute trapping and anomalous solute profiles within the diffuse interface could occur, thus highlighting the important effect of the interaction among the component elements on the interface kinetics. The current phase-field model might be preferred for simulations not only because of its simplest form of evolution equations but also its feasibility to increase the simulation efficiency by the “thin interface limit” analysis.

Quantification of Epistemic Uncertainty in Grain Attachment Models for Equiaxed Solidification

Recent work has investigated various schemes for the attachment of free-floating grains in models of equiaxed solidification in multicomponent alloys. However, these models are deterministic in nature, and simply investigating their differences for a limited number of results would not constitute an adequate comparison of their predictions. Instead, the models are compared in the context of the uncertainty in the most important input parameters. This approach is especially important in light of the effort required to implement a new model. If the predictions are essentially the same, then either model will suffice, or one may be selected for ease of implementation, numerical robustness, or computational efficiency. If, however, the models are significantly different, then the most accurate should be selected. In order to investigate the effects of input uncertainty on the output of grain attachment models, the PRISM Uncertainty Quantification framework was employed. The three models investigated were a constant packing fraction (CPF) scheme, an average solid velocity method (AVM), and a continuum attachment approach. Comparisons were made between the CPF and AVM models to estimate the importance of the local velocity field and between the CPF and continuum models to determine the sensitivity of the macrosegregation to new parameters unique to the continuum model.

Application of Computational Thermodynamics to the Evolution of Surface Tension and Gibbs-Thomson Coefficient during Multicomponent Aluminum Alloy Solidification

Numerical simulation of multicomponent alloy solidification demands accuracy of thermophysical properties in order to obtain a numerical representation as close as possible to the physical reality. Some alloy properties are only seldom found in the literature. In this paper, a solution of Butler’s formulation for surface tension is presented for Al-Cu-Si ternary alloys, allowing the Gibbs-Thomson coefficient to be calculated as a function of Cu and Si contents. The importance of the Gibbs-Thomson coefficient is related to the reliability of predictions furnished by predictive microstructure growth models and of numerical computations of solidification thermal variables that will be strongly dependent on the values of the thermophysical properties adopted in the calculations. A numerical model based on Powell hybrid algorithm and a finite difference Jacobian approximation was coupled with a ThermoCalc TCAPI interface to assess the excess Gibbs energy of the liquid phase, permitting the surface tension and Gibbs-Thomson coefficient for Al-Cu-Si hypoeutectic alloys to be calculated. The computed results are presented as a function of the alloy composition.

The Solidification of Multicomponent Alloys.

Various topics taken from the author's research portfolio that involve multicomponent alloy solidification are reviewed. Topics include: ternary eutectic solidification and Scheil-Gulliver paths in ternary systems. A case study of the solidification of commercial 2219 aluminum alloy is described. Also described are modifications of the Scheil-Gulliver analysis to treat dendrite tip kinetics and solid diffusion for multicomponent alloys.

Rapid phase transformation under local non-equilibrium diffusion conditions

Phase transformation with a moving interface occurs under far from local equilibrium conditions when the interface moves with sufficiently high velocity. This deviation cannot be adequately described by the classical irreversible thermodynamics with diffusion equation of parabolic type because it assumes local equilibrium hypothesis. The local non-equilibrium diffusion model has been developed to take into account the deviation from local equilibrium during binary alloy solidification using the hyperbolic diffusion equation. The model introduces a finite propagation velocity of concentration disturbances in the bulk liquid VD as the characteristic diffusion parameter and predicts a sharp transition from diffusion controlled to diffusionless and partitionless solidification at V = VD due to the deviation from local equilibrium. This review does not intend to be extensive, but rather to illustrate the main advances of the local non-equilibrium diffusion approach in rapid alloy solidification. Applications of the local non-equilibrium model to other phase transformation processes such as solid–solid transformation, colloidal and multicomponent alloy solidification have been briefly discussed. The parallels and possible combinations of the model with molecular dynamics, phase field, phase field crystal and cellular automata methods have been considered.