Solidification Of Binary Alloy Research Articles

Impact of chemical reaction on hybrid nanofluid (GO + MoS2) flow over an exponentially stretching sheet with Soret and Dufour effects

AbstractThe Soret and Dufour effects play a crucial role in various fields such as geosciences, groundwater pollutant migration, chemical reactor operations, binary alloy solidification, and isotope separation. This study focuses on examining the impact of mixed convective flow on hybrid nanofluid through an exponentially stretching sheet with Soret and Dufour effects. The flow is affected by factors like variable viscosity, radiation, viscous dissipation, and activation energy. Instead of the no‐slip condition at the boundary, velocity slip, thermal slip, and concentration slip are considered. The physical problem is modeled using boundary layer theory, and flow patterns are expressed using partial differential equations (PDEs). These governing fluid flow equations are transformed into non‐linearly coupled ordinary differential equations (ODEs) using exponential similarity transformations. These simplified ODEs are resolved using the MATLAB bvp4c package. The effects of physical parameters on velocity, temperature, and concentration are illustrated through figures. Additionally, the drag force coefficient and heat and mass transfer rates are calculated for various parameters and presented graphically and in tabular form. It is observed that compared to nanofluids, the drag force coefficient of hybrid nanofluids increases by up to 21.05% with various solute buoyancy parameters (δ). Also, the mass transfer rate of hybrid nanofluids can be increased by .96% by the chemical reaction rate (σm). A comparison of this work with previously published research has been reported.

An anisotropic lattice Boltzmann - phase field model for dendrite growth and movement in rapid solidification of binary alloys

A model coupling the lattice Boltzmann and the phase field methods with anisotropic effects is proposed, which is used to numerically describe the growth and movement of dendrites in rapid solidification of alloys. The model is quantitatively validated by the simulation of the continuous growth and the drafting-kissing-tumbling phenomenon of two falling particles, and then applied to investigate the effects of dendrite movement and interfacial non-equilibrium on evolution of dendritic patterns for Si-9.0at%As and the CET for Al-3.0wt%Cu alloys. Both the growth and remelt processes of isolated dendrites are studied, and the result reveals the remelting influences on dendrite growth and solute micro-segregation in the condition of directional solidification. This work demonstrates that the proposed model has a wide range of applicability and great potential to simulate the microstructure evolution with various solidification conditions.

Three dimensional phase-field simulation for non-isothermal binary alloy solidification: Comparison with LKT theory

Three dimensional phase-field simulation for non-isothermal binary alloy solidification: Comparison with LKT theory

Structure and diffuseness model of solid-liquid interface for binary alloys

Using the morphological instability at the solid–liquid interface as a basis by the maximum entropy production rate principle (MEPR), a model is presented on the morphological structure and diffuseness of the interface during directional solidification of binary alloys. It is shown that, the independent diffuseness theory of Cahn and the Jackson roughness criterion can be unified at a limiting condition under this new MEPR solidification model. The model under the principle of MEPR is applied to describe the evolution of atomistically smooth and rough interfaces through the evaluation of the size of the solid–liquid interface and the of number atomic layers. The model is tested with data for binary alloys of aluminium, lead, and tin at varying solute concentrations. The results showed strong agreement with available data from experimental measurements.

Phase-field lattice Boltzmann simulation of three-dimensional settling dendrite with natural convection during nonisothermal solidification of binary alloy

Understanding the motion and growth behaviors of equiaxed dendrites during solidification is important for predicting macrosegregation. In this study, we develop a phase-field lattice Boltzmann (PF-LB) simulation method for the settling and growth of an equiaxed dendrite during the nonisothermal solidification of a binary alloy. The PF-LB computations are accelerated by employing parallel computation using multiple graphic processing units (GPUs) and the octree block-structured adaptive mesh refinement method, which incorporates multiple mesh and time increment methods. By using the developed method, we can simulate the three-dimensional long-distance settling dendrite while considering the effects of latent heat release and natural convection. From the simulation results, we confirm that the natural convection due to the high solute concentration around a dendrite reduces the settling velocity. In addition, we observe that the temperature increase owing to latent heat release slows dendrite growth, which in turn slightly slows the settling velocity. From these results, we confirm that the effects of latent heat release and natural convection are not negligible in the quantitative evaluation of settling dendrites.

A model for nanoparticle transport and distribution with particle-interface interaction during binary alloy solidification

The interaction between nanoparticles and the solid-liquid interface during the solidification of binary alloys containing nanoparticles affects the nanoparticle transport and subsequently the final nanoparticle distribution and solid growth. Thus, a model to capture the engulfment or pushing of nanoparticles by the interface and the resulting change in nanoparticle transport and distribution is important. In this work, a numerical model to capture the nanoparticle transport and solute transport during the micro-scale solidification of binary alloy containing nanoparticles is presented. To develop the model, a sharp interface enthalpy method-based solidification model is used. The nanoparticle transport is governed by nanoparticle diffusion due to Brownian motion and by nanoparticle engulfment/pushing at the interface. The nanoparticle engulfment or pushing is modeled as a function of critical engulfment velocity and interface velocity. Thus, the effect of nanoparticle size and local interface movement velocity on nanoparticle engulfment can be predicted by the model. Simulation results show poor engulfment of smaller nanoparticles and relatively higher engulfment of larger nanoparticles. This results in slower dendrite growth rate in presence of smaller nanoparticles. It is also observed that as the dendrites grow, there is a transition from engulfment of nanoparticles to pushing of nanoparticles. This transition depends on the nanoparticle size and is delayed for larger nanoparticles. Thus, more uniform distribution of larger nanoparticles is observed in the solidified binary alloy nanocomposites.

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.

The roles of solidification cooling rate and (Mn,Cr) alloying elements in the modification of β-AlFeSi and hardness evolvements in near-eutectic Al-Si alloys

High recycling rates worldwide underline the economic and environmental value of aluminum scrap. However, several alloying elements in recycled Al negatively affect the final properties of castings, among which Fe has the most permissive effect on the mechanical properties. The extremely restricted solid solubility of Fe in Al leads to the formation of brittle Fe-containing intermetallic compounds (IMCs), and a possible strategy is to use modifiers of harmful IMCs by alloying. The present work aims to investigate the substitutional characteristic of Fe, Mn, Cr in the α-IMC taking the Al-12 %Si alloy as a reference, i.e., by analyzing the directional solidification (DS) of binary (Al-12 %Si), ternary (Al-12 %Si-1 %Fe) and quaternary (Al-12 %Si-1 %Fe-1 %Mn and Al-12 %Si-1 %Fe-0.6 %Cr) alloys. For all DS alloys castings, the macrostructure is shown to be characterized by columnar grains associated with solidification cooling rates (Ṫ) from 0.4 to 42.3 °C/s. The microstructure of the four alloys examined is shown to be typified by an α-Al dendritic matrix with interdendritic regions formed by α-Al, Si and different IMCs, which have been characterized and associated with the local Ṫ. The evolvement of iron-containing IMCs morphologies according to the addition of alloying elements in Al-Si alloys, is as follows: plate-like for Fe, plate-like and Chinese script for Mn, and fishbone and trefoil/blocky for Cr. Experimental power laws equations are determined relating the primary (λ1), secondary (λ2) and tertiary dendritic arm spacings (λ3) to Ṫ, for any experimentally examined DS alloy casting. The microhardness (HV) of the alloys is correlated with λ3 and Hall-Petch type equations are derived relating HV to both λ3 and Ṫ. The Al-12 %Si-1 %Fe-0.6 %Cr alloy achieved the highest HV among all the alloys examined.

In-situ experimental investigations of the morphological characteristics of plumes and chimneys formed during the directional solidification of a binary alloy

The defects such as freckles formed during the directional solidification of binary alloys severely affect the performance of cast products. The phenomenon of freckle formation is strongly influenced by the convective flows generated during solidification. The rejection of solute during the solidification at the solid/liquid interface induces the concentration gradients in the liquid. These concentration gradients in combination with existing temperature gradients, may induce buoyancy-driven convection in the mushy zone (plumes) and the parent liquid region (fingers). As the mushy zone evolves, small chimneys are formed, which leads to the formation of discrete plumes of interdendritic liquid. These plumes have a very complex flow structure which leads to the structural and compositional heterogeneities in the final cast product. The current study is an effort to investigate the morphological characteristics of plumes and chimneys formed during the directional alloy solidification. For this, controlled solidification experiments in a bottom-cooled rectangular cavity with an analogue transparent aqua ammonia system (NH4Cl + 74 wt% H2O) are performed. Particle Image Velocimetry (PIV), High Speed (HS) imaging and thermocouples temperature measurement techniques are used for real-time measurement of flow field, solidified and mushy zone dynamics and local temperature during the course of solidification. The dynamical nature of plumes is captured and the morphology of chimneys (shape, size, spacings, etc.) and plumes (velocity magnitude, shape, size, etc.) are quantified. The phase-locking with 180° phase difference due to the coupling between nearby plumes is observed. Also, the evolution of double-diffusive layers (DDLs) is captured and the critical value of thermal Rayleigh number for the onset of DDLs is estimated.

Data assimilation for dendritic solidification with melt convection: phase-field lattice Boltzmann study

Time-resolved in-situ X-ray tomography and high-performance phase-field simulations are state-of-the-art approaches to clarifying dendrite solidification. However, major issues persist, such as the insufficiency of spatiotemporal resolution in experiments and lack of material properties in simulations. To overcome these issues, in this study, we developed a data assimilation system using an ensemble Kalman filter based on the phase-field lattice Boltzmann method as a simulation model for the dendrite solidification of binary alloys with liquid flow. The validity of the developed system was confirmed through twin experiments to infer the kinematic viscosity in a two-dimensional dendrite growth problem with forced convection.

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

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

Modeling of dendritic solidification and numerical analysis of the phase-field approach to model complex morphologies in alloys

Dendrites are one of the most widely observed patterns in nature, and occur across a wide spectrum of physical phenomena—from snow flakes to river basins; from bacterial colonies to lungs and vascular systems; and in solidification and growth patterns in metals and crystals. The ubiquitous occurrence of these “tree-like” structures can be attributed to their excellent space-filling properties, and at times, dendritic structures also spatially manifest fractal-like distributions. As is the case with many fractal-like geometries, the complex multi-level branching structures in dendrites pose a modeling challenge, and a full resolution of dendritic structures is computationally very demanding. In the literature, extensive theoretical models of dendritic formation and evolution, essentially as extensions of the classical moving boundary Stefan problem exist. Much of this understanding is from the analysis of dendrites occurring during the solidification of metallic alloys, as this is critical for understanding microstructure evolution during metal manufacturing processes that involve solidification of a liquid melt. Motivated by the problem of modeling microstructure evolution from liquid melts of pure metals and metallic alloys during metal additive manufacturing, we developed a comprehensive numerical framework for modeling a large variety of dendritic structures that are relevant to metal solidification. In this work, we present a numerical framework encompassing the modeling of Stefan problem formulations relevant to dendritic evolution using a phase-field approach and a finite element method implementation. Using this framework, we model numerous complex dendritic morphologies that are physically relevant to the solidification of pure melts and binary alloys. The distinguishing aspects of this work are—a unified treatment of both pure metals and alloys; novel numerical error estimates of dendritic tip velocity; and the study of error convergence of the primal fields of temperature and the order parameter with respect to numerical discretization. To the best of our knowledge, this is a first of its kind study of numerical convergence of the phase-field equations of dendritic growth in a finite element method setting. Further, using this numerical framework, various types of physically relevant dendritic solidification patterns like single equiaxed, multi-equiaxed, single columnar and multi-columnar dendrites are modeled in two-dimensional and three-dimensional computational domains.

A study of dual phage lag model of heat transfer for solidification of binary eutectic alloy

In this paper, we developed a generalized non-Fourier model of energy transfer for solidification of a binary eutectic system (Al-Cu alloy). The mathematical model is divided into three stages: stage I (liquid region), II (liquid and mushy region), and III (liquid, mushy and solid region). The analysis of the model has been done in dimensionless form. This model is a boundary value problem of the system of second-order hyperbolic partial differential equations for Stage I, the moving boundary problem of the System of second-order hyperbolic partial differential equations for Stage II and III. For the solution, we applied Laplace transform technique in Stage I and Legendre wavelet spectral Galerkin method in Stage II and III. The effect of unit-less parameters: Relaxation time, Predvoditelev numbers, dimensionless thermal conductivity, Fourier numbers, Stefan number, solid fraction on temperature, and position of moving interface are discussed in detail.

The thermodynamics of non-equilibrium interfaces during phase transformations in concentrated multicomponent alloys

A unified thermodynamic description of moving non-equilibrium interfaces is developed for both solid/solid and solid/liquid transformations. The theory is applicable to concentrated multicomponent alloys where diffusion is possible in both phases or in just the parent phase, and where energy is dissipated due to solute drag. To be consistent with energy dissipation, we find that solute drag affects both the velocity of the interface and distribution coefficients for the compositions of the two phases at the interface. In the limit of binary alloy solidification, the theory predicts significant changes in the interfacial compositions from that given by the equilibrium phase diagram at velocities commonly found during additive manufacturing. Since the distribution coefficient is affected by solute drag, the solute-trapping behavior observed in molecular dynamics simulations implies that the interfacial diffusivity lies between the diffusivity of the bulk solid and liquid. A comparison to past work on non-equilibrium interfaces during solid/solid and solid/liquid phase transformations is also given.

Flow in oscillatory boundary layers over permeable beds

In fluid dynamics applications that involve flow adjacent to a porous medium, there exists some ambiguity in how to model the interface. Despite different developments, there is no agreed upon boundary condition that should be applied at the interface. We present a new analytical solution for laminar boundary layers over permeable beds driven by oscillatory free stream motion where flow in the permeable region follows Darcy's law. We study the fluid boundary layer for two different boundary conditions at the interface between the fluid and a permeable bed that was first introduced in the context of steady flows: a mixed boundary condition proposed by Beavers and Joseph [“Boundary conditions at a naturally permeable bed,” J. Fluid Mech. 30, 197–207 (1967)] and the velocity continuity condition proposed by Le Bars and Worster [“Interfacial conditions between a pure fluid and a porous medium: Implications for binary alloy solidification,” J. Fluid Mech. 550, 149–173 (2006)]. Our analytical solution based on the velocity continuity condition agrees very well with numerical results using the mixed boundary condition, suggesting that the simpler velocity boundary condition is able to accurately capture the flow physics near the interface. Furthermore, we compare our solution against experimental data in an oscillatory boundary layer generated by water waves propagating over a permeable bed and find good agreement. Our results show the existence of a transition zone below the interface, where the boundary layer flow still dominates. The depth of this transition zone scales with the grain diameter of the porous medium and is proportional to an empirical parameter that we fit to the available data.

Micro-macro modelling of thermal evolution during solidification of binary equiaxed alloys with experimental validation

A micro-macro model based on deterministic continuum mechanics and the modified source term method has been used to study the solidification problems involving binary Al-4.0 wt% Cu eutectic system. The model is further tested with other equiaxed eutectic binary aluminium alloys that include Al-3.0 wt% Si, Al-6.0 wt% Mg and Al-3.0 wt% Zn. All the heat transfer regimes including convection and radiation at the open top surface of the mould are considered. The nucleation steps are accounted for by considering continuous nucleation and growth kinetics. Information such as undercooling, nucleation rate, recalescence, growth rate, solid fraction, cooling curve, grain size and volumetric grain density which directly link to mechanical properties of cast components are accurately predicted. The model result is validated against the experimental data obtained from the current study and others published in the literature. From the results predicted in the current work, it is observed that the model gives the actual cooling history instead of a rough estimation of cooling curves as obtained from previous models. Among the aluminium-based alloys analysed, the Al–Cu alloy has the fastest nucleation and solidification rates. The predicted volumetric grain density suggests that magnesium has a more grain refinement effect on the aluminium matrix than copper, zinc and silicon. The model results are in agreement with the physical metallurgy of alloy theory and can be used by alloy manufacturers to improve the mechanical properties of alloy castings.

Solidification Simulation of Al-Si Alloys with Dendrite Tip Undercooling

A novel solution approach is proposed for the numerical simulation of the solidification process of binary Al-Si hypoeutectic alloys during upward and downward solidification modes. Undercooling is always observed during solidification, but the phenomenon could not be considered in the present-day numerical solution. In this approach, the temperature distribution in the mushy zone was used to define the fraction of solid, which enabled the evaluation of the effect of dendrite tip undercooling on the characteristics of the binary alloy solidification. The present numerical algorithm was found to significantly reduce the computation time. Transient temperature distribution and solidification time from the numerical analysis, with consideration of natural convection due to temperature and concentration gradients, have been successfully simulated and validated with experiment results. Numerical results with consideration of dendrite tip undercooling have better agreement with experimental results. The effect of dendrite tip undercooling on the fluid flow (velocity profile), G, R and λ1 for both upward and downward solidification modes of Al-Si alloys have been investigated and discussed. Consideration of undercooling was found to increase G and reduce R in both solidification modes. During downward solidification, considering undercooling significantly increased flow velocity and decreased λ1. The primary dendrite arm spacing could be validated with results from uni-directional solidification experiments only when dendrite tip undercooling was considered.

Fast prediction of phase equilibrium at varying temperatures for use in multi-component phase field models

A new method for temperature-dependent phase equilibrium prediction for use in multi-component phase field models of solidification is proposed. The method consists of two parts. First, the convex hull method is applied to predict the phase equilibrium at a single temperature. Second, a set of linear equations is developed to extend the equilibrium calculation over a range of temperatures. These linear equations are derived as an extension of the equation used for solidification of binary alloys in approximating the equilibrium state of multi-component systems. Phase field simulations of solidification of a Ti-Al-V-Fe alloy are performed to demonstrate the effectiveness of the present approach under isothermal and continuous cooling conditions. The results are compared against Thermo-Calc calculations, and indicate that a high accuracy of equilibrium prediction is achieved at a single and multiple temperatures, thus demonstrating that this approach can be successfully applied to the multi-component phase field models.

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.

Parallel-GPU-accelerated adaptive mesh refinement for three-dimensional phase-field simulation of dendritic growth during solidification of binary alloy

In the phase-field simulation of dendrite growth during the solidification of an alloy, the computational cost becomes extremely high when the diffusion length is significantly larger than the curvature radius of a dendrite tip. In such cases, the adaptive mesh refinement (AMR) method is effective for improving the computational performance. In this study, we perform a three-dimensional dendrite growth phase-field simulation in which AMR is implemented via parallel computing using multiple graphics processing units (GPUs), which provide high parallel computation performance. In the parallel GPU computation, we apply dynamic load balancing to parallel computing to equalize the computational cost per GPU. The accuracy of an AMR refinement condition is confirmed through the single-GPU computations of columnar dendrite growth during the directional solidification of a binary alloy. Next, we evaluate the efficiency of dynamic load balancing by performing multiple-GPU parallel computations for three different directional solidification simulations using a moving frame algorithm. Finally, weak scaling tests are performed to confirm the parallel efficiency of the developed code.