Model Of Binary Alloy Solidification Research Articles

Hot cracking susceptibility prediction from quantitative multi-phase field simulations with grain boundary effects

Hot cracks are notorious defects in casting, welding, and additive manufacturing. Grain boundaries play a key role in hot cracking formation. Still, their effects are often neglected in hot cracking susceptibility predictions, and thus a quantitative picture of hot cracking susceptibility and liquid fracture is still lacking. In this work, we construct a multi-order parameter phase-field model for binary alloy solidification to simulate attractive, neutral, and repulsive grain boundaries in the thin-interface limit. Phase-field simulations of different combinations of grain boundary and bi-crystal growth types are used to obtain solidification path data, which is then used as the input into the Rappaz, Drezet and Gremaud (RDG) model (Rappaz et al., 1999). We study the so-called Λ-shape variation of hot cracking susceptibility as a function of solute concentration, showing that the magnitude of the Λ curve peak shifts to higher values and to lower concentrations as grain boundary energy increases. Meanwhile, the peak magnitude becomes higher and shifts to a higher value and higher concentrations when convergent grain growth is applied. Furthermore, in all cases examined, the corresponding pressure drops predicted for hot cracking exceed the theoretical liquid rupture stress, consistent with experimental observations. These results demonstrate that quantitative phase-field simulations, when coupled with the RDG model, are capable of relatively accurately predicting liquid rapture states relevant to hot cracking since the large disparity between the predicted pressure drop and critical stress of liquid film is eliminated through the consideration of grain boundary effects in the solidification path calculation.

Isotropic finite-difference approximations for phase-field simulations of polycrystalline alloy solidification

Phase-field models of microstructural pattern formation during alloy solidification are commonly solved numerically using the finite-difference method, which is ideally suited to carry out computationally efficient simulations on massively parallel computer architectures such as Graphic Processing Units. However, one known drawback of this method is that the discretization of differential terms involving spatial derivatives introduces a spurious lattice anisotropy that can influence the solid-liquid interface dynamics. We find that this influence is significant for the case of polycrystalline dendritic solidification, where the crystal axes of different grains do not generally coincide with the reference axes of the finite-difference lattice. In particular, we find that with the commonly used finite-difference implementation of the quantitative phase-field model of binary alloy solidification, both the operating state of the dendrite tip and the dendrite growth orientation are strongly affected by the lattice anisotropy. To circumvent this problem, we use known methods in both real and Fourier space to derive finite-difference approximations of leading differential terms in 2D and 3D that are isotropic at order h2 of the lattice spacing h. Importantly, those terms include the divergence of the anti-trapping current that is found to have a critical influence on pattern selection. The 2D and 3D discretizations use an approximated form of the anti-trapping current that facilitates the Fourier-space derivation of the associated isotropic differential operator at O(h2), but we also derive a 2D discretization of the standard form of this current. Finally, we present 2D and 3D phase-field simulations of alloy solidification, showing that the isotropic finite-difference implementations dramatically reduce spurious lattice anisotropy effects, yielding both the tip operating state and growth direction of the dendrite that are nearly independent of the angle between the crystal and finite-difference lattice axes.

Identification of the air gap thermal resistance in the model of binary alloy solidification including the macrosegregation and the material shrinkage phenomena

Goal of this elaboration is to investigate the mathematical model of the inverse problem of binary alloy solidification within the casting mould, with the material shrinkage and the macrosegregation phenomena included simultaneously. Major result of this paper is the solution of the inverse problem consisting in reconstruction of the following elements: the thermal resistance of the air gap created between the cast and the mould in the course of solidification process and the heat transfer coefficient on the boundary of heat exchange between the mould and environment. The additional information, necessary to solve the inverse task, is delivered by the temperature measurements read in the control point located in the centre of the mould. The theoretical discussion is supported by the numerical examples executed for various sets of input data.

Numerical study of crystal growth kinetics influence on prediction of different dendritic zones and macro-segregation in binary alloy solidification

PurposeProper selection of the stability parameter determines the accuracy of dendrite tip kinetics at a single crystal scale. Recently developed sophisticated phase field modelling of a single grain evolution provides evidence that this parameter is not constant during the process. Nevertheless, in the commonly used micro-macroscopic simulations of alloy solidification, it is a common practice to use a constant value of the stability parameter, resulting from the marginal stability theory. This paper aims to address the issue of how this inaccuracy in modelling crystal growth kinetics can influence numerically predicted zones of columnar and equiaxed dendrites and the macro-segregation formation.Design/methodology/approachUsing the original authors’ micro-macroscopic computer simulation model of binary alloy solidification, the calculations have been performed for the Kurz-Giovanola-Trivedi (KGT) crystal growth kinetics with two different values of the stability parameter, and for two different compositions of Al-Cu alloys. The computational model is based on single domain-based formulation of transport equations, which are discretized on control-volume mesh. To identify zones of different grain structures, developing within the two-phase liquid-solid region, an envelope of columnar dendrite tips is tracked on a fixed non-orthogonal, triangular control volume grid. The models of porous and slurry media are used, along with the concept of the switching function, to account for diverse flow resistances in the columnar and equiaxed crystal zones. The numerical predictions are carefully studied to address the question of how the chosen stability parameter influences macroscopic structures of a cast, the most important issue from the engineering point of view.FindingsThe carried-out comprehensive numerical analysis shows that the value of the stability parameter of the KGT-constrained dendrite growth model does not have a direct significant impact on the macrosegregation formation. It, however, visibly influences the undercooling along the front, separating different dendritic structures and the size of the undercooled melt region where the equiaxed grains can develop. It also affects the amount of eutectic phase created.Originality/valueTo the best of the authors’ knowledge, this is the first attempt at estimating the influence of some inaccuracies, caused by possible ambiguities in choosing the stability constant of the KGT law, on numerically predicted macroscopic fields of solute concentration, the developing zones of columnar and equiaxed crystals and the macrosegregation patterns.

Influence of crystal growth kinetics on prediction of macro segregation by micro-macroscopic simulation of binary alloy solidification

Using the micro-macroscopic computer simulation model of binary alloy solidification, based on a single domain control-volume calculations, and involving the tracking of columnar dendrite tips envelope for distinguishing different grain structures within the two-phase liquid-solid region, the paper reports the outcomes of the detailed analysis addressing the question of the influence of enhanced KGT crystal growth models on the predicted macro-segregation pattern in the Al-4wt.%Cu alloy cast. In particular, the analysis concerns the impact of a proper selection of the stability constant for the KGT model (based on a crystal-melt surface energy anisotropy strength and linear scaling law of the marginal stability theory) on the predicted dendrite tip under-cooling, volumetric solid fraction and actual solute concentration along the front of columnar dendrite tips. Differences in the resulting evolution of both: the under-cooled melt region within the mushy zone and the solute concentration fields calculated for the considered various kinetics of a grain growth are also reported and discussed.

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

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

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

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

Variational formulation and numerical accuracy of a quantitative phase-field model for binary alloy solidification with two-sided diffusion.

We present the variational formulation of a quantitative phase-field model for isothermal low-speed solidification in a binary dilute alloy with diffusion in the solid. In the present formulation, cross-coupling terms between the phase field and composition field, including the so-called antitrapping current, naturally arise in the time evolution equations. One of the essential ingredients in the present formulation is the utilization of tensor diffusivity instead of scalar diffusivity. In an asymptotic analysis, it is shown that the correct mapping between the present variational model and a free-boundary problem for alloy solidification with an arbitrary value of solid diffusivity is successfully achieved in the thin-interface limit due to the cross-coupling terms and tensor diffusivity. Furthermore, we investigate the numerical performance of the variational model and also its nonvariational versions by carrying out two-dimensional simulations of free dendritic growth. The nonvariational model with tensor diffusivity shows excellent convergence of results with respect to the interface thickness.

On the fully implicit solution of a phase-field model for binary alloy solidification in three dimensions

A fully implicit numerical method, based upon a combination of adaptively refined hierarchical meshes and geometric multigrid, is presented for the simulation of binary alloy solidification in three space dimensions. The computational techniques are presented for a particular mathematical model, based upon the phase-field approach, however their applicability is of greater generality than for the specific phase-field model used here. In particular, an implicit second order time discretization is combined with the use of second order spatial differences to yield a large nonlinear system of algebraic equations as each time step. It is demonstrated that these equations may be solved reliably and efficiently through the use of a nonlinear multigrid scheme for locally refined grids. In effect this paper presents an extension of earlier research in two space dimensions (J. Comput. Phys., 225 (2007), pp. 1271-1287) to fully three-dimensional problems. This extension is validated against earlier two-dimensional results and against some of the limited results available in three dimensions, obtained using an explicit scheme. The efficiency of the implicit approach and the multigrid solver are then demonstrated and some sample computational results for the simulation of the growth of dendrite structures are presented.

On the Fully Implicit Solution of a Phase-Field Model for Binary Alloy Solidification in Three Dimensions

AbstractA fully implicit numerical method, based upon a combination of adaptively refined hierarchical meshes and geometric multigrid, is presented for the simulation of binary alloy solidification in three space dimensions. The computational techniques are presented for a particular mathematical model, based upon the phase-field approach, however their applicability is of greater generality than for the specific phase-field model used here. In particular, an implicit second order time discretization is combined with the use of second order spatial differences to yield a large nonlinear system of algebraic equations as each time step. It is demonstrated that these equations may be solved reliably and efficiently through the use of a nonlinear multigrid scheme for locally refined grids. In effect this paper presents an extension of earlier research in two space dimensions (J. Comput. Phys., 225 (2007), pp. 1271-1287) to fully three-dimensional problems. This extension is validated against earlier two-dimensional results and against some of the limited results available in three dimensions, obtained using an explicit scheme. The efficiency of the implicit approach and the multigrid solver are then demonstrated and some sample computational results for the simulation of the growth of dendrite structures are presented.

Numerical computations for temperature, fraction of solid phase and composition couplings in ternary alloy solidification with three different thermodynamic data-acquisition methods

In this paper, a previously proposed numerical model for binary alloy solidification, and the corresponding algorithm for solving the strong coupling among the solidification temperature, solid fraction, and liquid composition (T–fS–CL), are extended to ternary alloys. The feasibility of the extended numerical method is demonstrated by sample simulations for the directional solidification of blade-like castings of Al–Cu–Si alloys using three different thermodynamic data-acquisition methods: direct coupling with the CALPHAD software Thermo-Calc via the TQ6-Interface, using the “mapping technique” and regression functions. The computational efficiency and the accuracy of the calculation results of the three different methods are compared. The present numerical simulations confirm that thermodynamic data-acquisition method of using regression functions and the “mapping technique” are much more efficient than that of direct coupling with Thermo-Calc.

Numerical solution and comparison to experiment of solute drag models for binary alloy solidification with a planar phase interface

A generalized solute drag model for binary alloy solidification with a planar phase interface was proposed as an extension of Hillert–Sundman model. Using a new thermodynamic parameter set of the Si–As system, the present model can, with the introduction of three new adjustable parameters, fit the available experimental data of Si–9 at.% As alloy. In addition, non-solute drag models with the new set of thermodynamic parameters cannot reproduce the experimental data.

Phase field modeling of shallow cells during directional solidification of a binary alloy

We report some computational results obtained by using the phase field model for binary alloy solidification. We study directional solidification and concentrate on the transition between a planar interface and steady shallow cells near the onset of morphological instability at low growth speeds. The model is formulated on the principle of local positive entropy production. This gives rise to a set of partial differential equations for the temperature, composition and the phase fields. We make a frozen temperature approximation and compute steady-state nearly periodic cellular crystal–melt interfaces in two dimensions. The cell shapes can be characterized by a small number of Fourier coefficients and become nearly sinusoidal as stability is approached. Wavelength selection by tip splitting or coarsening is also observed.

Influence Of Turbulent Natural Convection In Simulating CrystallizationOf II- VI Semiconductors

A theoretical model of binary alloy solidification is formulated using the mass, momentum and energy balances in the liquid and the mass and energy balances in the solid. The turbulent energy and momentum transfer are taken into account using two different turbulent k — e models for low Reynolds numbers. The interface between the solid and liquid phases is modeled as a discontinuity surface where the mass, momentum and energy balance conditions are expressed. The linear theory of mixtures is employed in modeling the two-component system. In a practical application of the computational model, the crystallization process of CdZnTe which can be considered as a II-VI pseudobinary mixture is simulated and the results obtained for the Bridgman method of crystal growth are discussed.

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 during 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 ( g) required to investigate the fundamental physics of interactions between a particle and a s/l interface. This is an ongoing NASA funded 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 −4g 0 ( g 0=9.8 m/s 2) maximum melt convection was obtained for an orientation of 90°. Calculations showed that even for this worst case orientation the dominant forces acting on the particle are the fundamental drag and interfacial forces.

Computational modeling of CdZnTe crystal growth from the melt

A theoretical model of binary alloy solidification is formulated using the mass, momentum and energy balances in the liquid, and the mass and energy balances in the solid. The interface between the solid and liquid phases is modeled as a discontinuity surface where the mass, momentum and energy balance conditions are expressed. The linear theory of mixtures is employed in modeling the two-component system. The numerical solution of the mathematical model is performed using the Galerkin finite element method in a 2-D approximation with the cylindrical symmetry and the moving boundary problem is solved by a front-fixing technique. Computer implementation of the numerical model is then performed. In a practical application of the computational model, the process of CdZnTe crystal growth using the vertical Bridgman method and the vertical gradient method is simulated. The influence of principal adjustable parameters of both crystal growth techniques is studied, and these parameters are optimized. Also, the effects of uncertainty of material data and of second-order material parameters are analyzed.

Experimental Validation of Continuum Mixture Model for Binary Alloy Solidification

Experiments were performed with binary metal alloys to validate a continuum mixture model for alloy solidification. Ingots of two compositions, Pb-20%Sn and Pb-40%Sn, were cast in a permanent mold, and the solidification process was simulated. Temperature histories were measured during casting, and composition profiles were found in the solidified ingot. Dendritic arm spacings were found from optical micrographs of the alloy microstructure and used to determine a constant in the Blake-Kozeny submodel for the mushy zone permeability in the liquid-solid interaction term of the momentum equations. Scaling analysis from a previous work and a large uncertainty in the permeability constant suggested that predictions of the composition are extremely sensitive to the choice of a permeability model. Three simulations of each casting were performed using the permeability constant as a parameter, and measured temperatures and compositions were compared with predictions based on different model constants. In the region of the liquidus interface, where all of the significant advection of solute takes place, the results suggest that the Blake-Kozeny model based on measured dendritic arm spacings significantly underpredicts the resistance of the dendritic array to fluid flow.

Existence for a stationary model of binary alloy solidification

A proof of existence is given for a stationary model of alloy solidification. The system is composed of heat equation, solute equation and Navier-Stokes equations. In rite latter Carman-Kozeny penalization of porous medium models the mushy zone. The problem is first regularized and a sequence of regularized solutions is built thanks to Leray-Schauder's fixed point Theorem. A solution is then extracted by compactness argument.

A model of binary alloy solidification with convection in the melt

The solidification of a binary alloy is studied on a three-phase model (solid, liquid, mushy zone). The balance equations of the model are formulated globally over the whole region occupied by the alloy and are treated in a distributional sense. Convective phenomena in the melt and the influence of gravitational field are taken into account. The significance of the model for studying solidification processes under different gravity conditions is discussed

A two-dimensional model of binary-alloy solidification with a Mushy-zone

The results of the experiments with solidification in space imply that the effect of gravity conditions should be included into the theoretical and computational models of solidification processes. The paper presents the results of computational simulation of binary-alloy solidification where the gravity acts as the driving force of chemical diffusion. A simplified version of a complex theoretical model constructed by the same authors was implemented on a computer and used to simulate the solidification of a two-dimensional sample of the pseudo-binary mercury-cadmiumtelluride alloy. The computational model was used to study the influence of gravity on the chemical diffusion in the sample. The results of our simulations have shown that the influence of gravity on the diffussion of the solute is a second-order effect only. Therefore, to describe the influence of the gravitational field on solidification processes adequately we would need a more complex model involving convective phenomena in the melt as well. In addition, a series of numerical experiments were performed to consider how the accuracy with which the material parameters had been determined affects the computed temperature and concentration fields. The results imply that the simulation of this kind could be seriously biased by insufficient accuracy in determining the primary material properties. Moreover, the results show that neglecting the thermal diffusion and the temperature and concentration dependence of the material parameters may lead to serious errors.