Simulation Of Dendritic Growth Research Articles

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

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

Numerical simulation of dendritic growth in directional solidification of binary alloys using a lattice Boltzmann scheme

A lattice Boltzmann (LB) based model is utilized to simulate three dimensional dendritic growth in directional solidification of alloys. The LB–D3Q19 lattice vectors are used to describe advancement of solid–liquid interface, coupling with a conservation equation for solute transport in solidification. After model validation, the dendritic growth under several conditions of directional solidification was investigated and a solidification entropy was proposed to quantitatively characterize solidification morphologies. The results show that the present model captures tip-splitting occurring at a relative fast solidification rate. Initial thermal undercooling and solute concentration are dominant factors influencing the final microstructure. The solidification entropy reflects complexity of dendritic morphologies and it is useful to characterize dendritic growth in solidification. A design map was proposed to predict dendritic growth with and without tip-splitting, offering potential for simulating dendritic growth and microstructure evolution in solidification of alloys.

Multi-scale simulation of directional dendrites growth in superalloys

The directional dendrite growth behavior and morphology were simulated based on the cellular automaton-finite difference (CA-FD) considering macro directional solidification (DS) parameters such as withdrawal rate and pouring temperature. Two types of directional dendrite growth models were proposed to realize the multi-scale simulation. A 3D dendrite growth model based on heat transfer and solute diffusion was built to simulate the dendrite growth during a real DS process, which is proved useful for the detail dendrite growth and morphology prediction at a micro 3D scale. A modified shaped function model was proposed to predict the evolution behavior of a large number of dendrites in the entire section of a DS cast sample at a macro scale, based on which the evolution behavior of thousands of dendrites was predicted in the 2D section during the DS process. In these models, the temperature interpolation algorithm and the dendrite orientation mathematical projection were used to transfer information between macro DS process calculation and micro dendrite growth simulation. The corresponding DS experiment was performed to verify the numerical models, and they agreed well. The DS dendritic evolution behavior and detail morphology, especially thousands of dendrites distribution at a large scale, were successfully simulated based on the two models proposed on the condition of a real DS process.

Erratum to: On the Origin of Grid Anisotropy in the Simulation of Dendrite Growth by a VFT Model

Erratum to: On the Origin of Grid Anisotropy in the Simulation of Dendrite Growth by a VFT Model

On the Origin of Grid Anisotropy in the Simulation of Dendrite Growth by a VFT Model

A virtual front tracking model, based on solute and heat diffusion in two dimensions, is chosen to capture the full microstructural behavior of dendritic solidification in a binary alloy. We use a simple method of calculation, easy to perform, with relatively high stable time step, to simulate the dendrite growth in an Al-8 wt pct Mg alloy for which no numerical simulation has been carried out in the past. Local equilibrium at the liquid solid interface and the buildup of solute ahead of the interface are solved, and the dendrite growth process is simulated in isothermal solidification conditions. We show that the artificial grid anisotropy originates from the four cell neighborhood method adopted for capturing the moving front. By a correct neighborhood configuration, a grid independent set of results and expected phenomena are reproduced for a free dendrite growing either aligned or inclined with the grid. The dendrite morphology and orientation, and the growth velocity are explored via physical simulation parameters such as undercooling and surface tension anisotropy.

Phase-Field Modeling of Metal Plating on Battery Electrodes

The growth of ramified metal deposits on battery electrodes during charging can lead to capacity loss as pieces of metal become electrically disconnected upon cycling or result in battery failure as metal dendrites penetrate the separator and short-circuit the cell. This undesirable metal plating can occur in Li-ion batteries under conditions of fast charging, overcharging, low temperature, or in the presence of battery separator defects. Nonuniform electrodeposition is also a challenge faced by next-generation battery technologies with pure metal anodes. Developing a better understanding of how different electrodeposit morphologies form inside batteries could lead to more effective strategies for preventing dendrite growth, making Li-ion batteries safer and metal anodes more practical. Phase-field modeling is a promising approach for modeling electrodeposition morphologies and has been used extensively to simulate a variety of complex microstructural evolution processes. We have developed a phase-field model of electrodeposition from a binary electrolyte that accounts for Butler-Volmer reaction kinetics, the metal-electrolyte surface energy, and ion transport via diffusion and migration. We validate our model with linear stability analyses and numerically simulate the formation of various morphologies under different conditions. In addition, we simulate dendrite growth through a porous battery separator and explore the effects of separator properties on the growth of the metal electrodeposits.

Phase field simulation of dendrite growth in binary Ni–Cu alloy under the applied temperature gradient

Phase field method was employed to investigate the effect of the applied temperature gradient on the microstructure evolution for the binary Ni–Cu alloy during the solidification process. As a comparison group, the isothermal solidification and the non-isothermal solidification taking into account the effect of latent heat were simulated at the same time. Simulation results showed that temperature distribution could greatly influence dendrite morphology, concentration distribution and tip velocity. Under the applied temperature gradient, a cone shape of dendrite, instead of the normal dendrite with four symmetric arms, could be found easily. The growth rate of dendrite and the segregation of concentration field increased with the increasing temperature gradient. This model was able to give a clear outlook for the influence of temperature gradient on the dendrite morphology and help a better design for the microstructure control in the field of direct chill casting.

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.

Numerical Simulation of Three-Dimensional Dendritic Growth of Alloy: Part I—Model Development and Test

To improve the computational efficiency of the three-dimensional (3D) cellular-automaton–finite-volume-method (CA-FVM) model for describing the dendritic growth of alloy, the block-correction technique (BCT) and the parallel computation approach are introduced. Accordingly, a serial of investigations on the efficiency of the optimized codes in dealing with the designed cases for the melt flow and the heat transfer problems is carried out. Moreover, the accuracy of the present codes is evaluated by the comparisons between the solution to the melt flow and the heat transfer problems and the results from analytical equations and the commercial software. Additionally, the capability of the present CA model is evaluated by comparing the steady growth parameters of the equiaxed dendritic tip and the morphology and the secondary dendrite arm spacing (SDAS) of columnar dendrites with the LGK analytical model and the experimental results of the unidirectional solidification of high-carbon steels. The results show that with the introduction of the 3D BCT, the iteration process of the serial tri-diagonal matrix algorithm (TDMA) code changes from the fluctuation type to the smooth one, and thus, the computational cost is reduced significantly. Moreover, the parallel Jacobi code with one two-dimensional (2D) iteration in 3D BCT is proved to be the most efficient one among the codes compiled in the present work, and therefore, accordingly it is employed to simulate the 3D dendritic growth of alloys. The calculated velocity distribution and temperature variation agree well with the results from the analytical equations and the commercial software. The predicted steady tip velocities agree with the LGK analytical model as the undercooling is 6 K to 7 K. Moreover, the predicted columnar dendritic morphology and SDAS of high-carbon Fe-C alloys during the unidirectional solidification agree with the experimental results.

Simulation of dendritic growth of Al-4 wt.% Cu alloy from an undercooled melt

Abstract Three-dimensional dendritic growth of an Al-4 wt.% Cu alloy during solidification in the presence of forced flow was simulated by using a three-dimensional cellular automaton model. The three-dimensional Lattice–Boltzmann method was used in computing the flow dynamics. The effects of forced flow on the growth Péclet number, dendrite tip selection parameter, and secondary arm spacing were investigated. The simulated relationships of growth Péclet number and dendrite tip selection parameter with the forced flow are in good agreement with the Oseen–Ivantsov solution. The results indicate that with the increase in forced flow velocity, the secondary arm spacing first increases and then decreases.

Simulations of three-dimensional dendritic growth using a coupled thermo-solutal phase-field model

Using a phase field model, which fully couples the thermal and solute concentration field, we present simulation results in three dimensions of the rapid dendritic solidification of a class of dilute alloys at the meso scale. The key results are the prediction of steady state tip velocity and radius at varying undercooling and thermal diffusivities. Less computationally demanding 2-dimensional results are directly compared with the corresponding 3-dimensional results, where significant quantitative differences emerge. The simulations provide quantitative predictions for the range of thermal and solutal diffusivities considered and show the effectiveness and potential of the computational techniques employed. These results thus provide benchmark 3-dimensional computations, allow direct comparison with underlying analytical theory, and pave the way for further quantitative results.

Quantitative phase-field modeling of dendritic electrodeposition.

A thin-interface phase-field model of electrochemical interfaces is developed based on Marcus kinetics for concentrated solutions, and used to simulate dendrite growth during electrodeposition of metals. The model is derived in the grand electrochemical potential to permit the interface to be widened to reach experimental length and time scales, and electroneutrality is formulated to eliminate the Debye length. Quantitative agreement is achieved with zinc Faradaic reaction kinetics, fractal growth dimension, tip velocity, and radius of curvature. Reducing the exchange current density is found to suppress the growth of dendrites, and screening electrolytes by their exchange currents is suggested as a strategy for controlling dendrite growth in batteries.

The Three-Dimensional Morphology of Growing Dendrites.

The processes controlling the morphology of dendrites have been of great interest to a wide range of communities, since they are examples of an out-of-equilibrium pattern forming system, there is a clear connection with battery failure processes, and their morphology sets the properties of many metallic alloys. We determine the three-dimensional morphology of free growing metallic dendrites using a novel X-ray tomographic technique that improves the temporal resolution by more than an order of magnitude compared to conventional techniques. These measurements show that the growth morphology of metallic dendrites is surprisingly different from that seen in model systems, the morphology is not self-similar with distance back from the tip, and that this morphology can have an unexpectedly strong influence on solute segregation in castings. These experiments also provide benchmark data that can be used to validate simulations of free dendritic growth.

Advances in Modeling of Solidification Microstructures

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

Simulation of Dendritic Growth with Melt Convection in Solidification of Ternary Alloys**Supported by the National Natural Science Foundation of China under Grant Nos 51306037 and 51371051.

A cellular automaton-lattice Boltzmann coupled model is extended to study the dendritic growth with melt convection in the solidification of ternary alloys. With a CALPHAD-based phase equilibrium engine, the effects of melt convection, solutal diffusion, interface curvature and preferred growth orientation are incorporated into the coupled model. After model validation, the multi dendritic growth of the Al-4.0 wt%Cu-1.0 wt%Mg alloy is simulated under the conditions of pure diffusion and melt convection. The result shows that the dendritic growth behavior, the final microstructure and microsegregation are significantly influenced by melt convection in the solidification.

Cellular automaton simulation of three-dimensional dendrite growth in Al–7Si–Mg ternary aluminum alloys

Due to the extensive applications in the automotive and aerospace industries of Al–7Si–Mg casting alloys, an understanding of their dendrite microstructural formation in three dimensions is of great importance in order to control the desirable microstructure, so as to modify the performance of castings. For this reason, a three-dimensional cellular automaton model (3-D CA) allowing for the prediction of dendrite growth of ternary alloys is presented. The growth kinetics of the solid/liquid (S/L) interface is calculated based on the solutal equilibrium approach. This proposed model introduces a modified decentered octahedron algorithm for neighborhood tracking, in order to eliminate the effects of mesh dependency on dendrite growth. The thermodynamic and kinetic data needed in the simulations are obtained through coupling with the Pandat software package in combination with thermodynamic/kinetic/equilibrium phase diagram calculation databases. The solute interactions between alloying elements are considered in the model. The model was first used to simulate the Al–7Si dendrite, followed by a validation using theoretical predictions. The influence of Mg content on the dendrite growth dynamics and dendrite morphologies was investigated. Also, the model was applied to Al–7Si–0.5Mg dendrite simulation both with and without a consideration of solute interactions between the Si and Mg alloying elements and the effects on dendrite growth process was analyzed using the simulation results. This model was finally used in order to simulate the dendrite growth in different crystallographic orientations in an Al–7Si–0.36Mg ternary alloy during polycrystalline solidification, resulting in a predicted secondary dendrite arm spacing (SDAS) and dendrite volume fraction data that show a reasonable agreement with experimental results. The single dendrite and polycrystalline growth simulations effectively demonstrate the capability of this model in predicting the three-dimensional dendrite microstructure of ternary alloys.

Cellular Automaton Modeling of Dendritic Growth Using a Multi-grid Method

A two-dimensional cellular automaton model with a multi-grid method was developed to simulate dendritic growth. In the present model, we used a triple-grid system for temperature, solute concentration and solid fraction fields as a new approach of the multi-grid method. In order to evaluate the validity of the present model, we carried out simulations of single dendritic growth, secondary dendrite arm growth, multi-columnar dendritic growth and multi-equiaxed dendritic growth. From the results of the grid dependency from the simulation of single dendritic growth, we confirmed that the larger grid can be used in the simulation and that the computational time can be reduced dramatically. In the simulation of secondary dendrite arm growth, the results from the present model were in good agreement with the experimental data and the simulated results from a phase-field model. Thus, the present model can quantitatively simulate dendritic growth. From the simulated results of multi-columnar and multi-equiaxed dendrites, we confirmed that the present model can perform simulations under practical solidification conditions.

GPU-accelerated 3D phase-field simulations of dendrite competitive growth during directional solidification of binary alloy

Phase-field method has emerged as the most powerful numerical scheme to simulate dendrite growth. However, most phase-field simulations of dendrite growth performed so far are limited to two-dimension or single dendrite in three-dimension because of the large computational cost involved. To express actual solidification microstructures, multiple dendrites with different preferred growth directions should be computed at the same time. In this study, in order to enable large-scale phase-field dendrite growth simulations, we developed a phase-field code using multiple graphics processing units in which a quantitative phase-field method for binary alloy solidification and moving frame algorithm for directional solidification were employed. First, we performed strong and weak scaling tests for the developed parallel code. Then, dendrite competitive growth simulations in three-dimensional binary alloy bicrystal were performed and the dendrite interactions in three-dimensional space were investigated.

A Three Dimensional Cellular Automata Model for Dendrite Growth in Non-Equilibrium Solidification of Binary Alloy

A three-dimensional (3D) cellular automata (CA) model has been developed for simulation of dendrite growth during non-equilibrium solidification. The heat and mass transports are calculated using 3D finite element (FE) method. The migration of solid/liquid (SL) interface is associated with effective free energy difference incorporating solute trapping and relaxation effects. The non-equilibrium solute partition coefficient involving solute trapping and relaxation effects is used to calculate solute redistribution at SL interface. The interface curvature and anisotropy of surface energy are introduced to represent the characteristics of dendrite tip. The CA model has been applied to simulate the microstructure evolution of Fe-1.5 wt.% C alloy. The grain morphology, solute and temperature distributions, and velocity of dendrite tip are characterized during both isothermal and non-isothermal conditions. The velocity of dendrite tip and secondary dendrite arm spacing during directional solidification are validated and are in good agreements with previous experimental and mathematical results.

Simulation of facet dendrite growth with strong interfacial energy anisotropy by phase field method

Numerical simulations based on a new regularized phase-field model were presented, to simulate the solidification of hexagonal close-packed materials with strong interfacial energy anisotropies. Results show that the crystal grows into facet dendrites, displaying six-fold symmetry. The size of initial crystals has an effect on the branching-off of the principal branch tip along the 〈100〉 direction, which is eliminated by setting the b/a (a and b are the semi-major and semi-minor sizes in the initial elliptical crystals, respectively) value to be less than or equal to 1. With an increase in the undercooling value, the equilibrium morphology of the crystal changes from a star-like shape to facet dendrites without side branches. The steady-state tip velocity increases exponentially when the dimensionless undercooling is below the critical value. With a further increase in the undercooling value, the equilibrium morphology of the crystal grows into a developed side-branch structure, and the steady-state tip velocity of the facet dendrites increases linearly. The facet dendrite growth has controlled diffusion and kinetics.