Dendrite Growth Model Research Articles

Mathematical modeling of dendrite growth in an Al–Ge alloy with convective flow

A theory of stable dendrite growth in an undercooled binary melt is developed for the case of intense convection. Conductive heat and mass transfer boundary conditions are replaced by convective conditions, where the flux of heat (or solute) is proportional to the temperature or concentration difference between the surface of the dendrite and far from it. The marginal mode of perturbation wavelengths is calculated using the linear morphological stability analysis. Combining this analysis with the solvability theory, we have derived a selection criterion that represents the first condition to define a combination of dendrite tip velocity and tip diameter. The second condition—the undercooling balance—is derived for intense convection. The theory under consideration determines the dendrite tip velocity and tip diameter for low undercooling. This convective theory is combined with the classical theory of dendritic growth (conductive boundary conditions), which is valid for moderate and high undercooling. Thus, the entire range of melt undercooling is covered. Our results are in good agreement with experiments on Al–Ge crystallization.

Phase-field modeling of dendritic growth of magnesium alloys with a parallel-adaptive mesh refinement algorithm

A two-dimensional phase field (PF) model was developed to simulate the dendritic solidification in magnesium alloy with hcp crystal structure. By applying a parallel-adaptive mesh refinement (Para-AMR) algorithm, the computational efficiency of the numerical model was greatly improved. Based on the PF model, a series of simulation cases were conducted and the results showed that the anisotropy coefficient and coupling coefficient had a great influence on the dendritic morphology of magnesium alloy. The dendritic growth kinetics was determined by the undercooling and equilibrium solute partition coefficient. A significant finding is acquired that with a large undercooling, the maximum solute concentration is located on both sides of the dendrite tip in the liquid, whereas the maximum solute concentration gradient is located right ahead of the dendrite tip in the liquid. The dendrite tip growth velocity decreases with the increase of the equilibrium solute partition coefficient, while the variation trend of the dendrite tip radius is the opposite. Quantitative analysis was carried out relating to the dendritic morphology and growth kinetics, and the simulated results are consistent with the theoretical models proposed in the previously published works.

Chemo-mechanical Model of Lithium Dendrite Growth Impacted By External Pressure and Plastic Deformation

High energy density and low coulombic efficiency of lithium metal anodes has made Li-ion batteries a standout area of research for improving commercial battery performance; however, there are operation and safety concerns that must be addressed before expanding their applications. During cell operation lithium dendrites form on surface inhomogeneities of the rough anode, which push through the separator, decrease cell efficiency, and eventually short-circuit the cell. Mechanical stress caused by an external pressure applied to the cell has been shown to mitigate lithium plating, though its addition leads to inhomogeneous ion transport between the separator and anode. As choice of electrolyte is also crucial for mitigating dendrite growth, the combined effects of stress and electrochemical transport properties on the propensity of plating are of interest. In this work we present a 3-D mesoscale model of a typical porous separator brought into contact with a lithium protrusion during cycling. The protrusion is characteristic of an inhomogeneity on a rough Li anode surface. The effects of lithium plastic deformation under high stress are modeled as the materials are brought into contact; previous work neglected this complexity. Partial contact between the materials creates an electrolyte-filled gap which drastically impacts the electrochemical response across the Li-separator interface. Propensity of plating is related to the combined effects of internal stress and various material properties; protrusion shape, separator porosity and modulus, separator/electrolyte ionic diffusivities, and applied current density are of specific interest in this study. The conditions for which dendrite formation is encouraged or suppressed are reported and may be used to inform choice of materials and cell manufacturing methods.Supported by the Laboratory Directed Research and Development program at Sandia National Laboratories, a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-NA-0003525.

Electrochemical-Mechanical Coupled Crack Propagation and Dendrite Growth in All-Solid-State Battery

Lithium metal all-solid-state batteries (ASSBs) are considered as one of the most promising candidates for future lithium batteries thanks to their high safety performance and energy density. However, dendrite growth and interfacial issues induced failure and poor cyclability are the two fatal problems on the road of commercialization of ASSBs. To further understand the underlying mechanisms, coupled electrochemical-mechanical models for crack propagation and lithium dendrite growth are proposed from both the cell level and electrolyte level. From the cell level, we discover that the overpotential induced stress may drive the crack to penetrate the solid electrolyte and the dendrite growth eventually causes the short circuit. The short circuit occurs earlier with increasing charging rate and decreasing electrolyte conductivity. The effect of Young’s modulus on crack is expressed by affecting the driving force and the fracture energy. Larger fracture toughness represents higher threshold energy for crack, thus the dendrite growth speed is reduced. From the electrolyte level, we observe that longer defect with sharp edge causes more severe crack propagation and leads to larger dendrite growth area due to the increased strain energy density. The initial defect within grain plays an irrelevant role in the dendrite growth within the grain boundary. Stacking pressure greater than 10 MPa significantly speeds up crack propagation as well as dendrite growth due to the nontrivial mechanical driving force. Mechanical stress-induced strain-energy would contribute to more than 15% of the total dendrite growth once the stacking pressure exceeds 20 MPa, while it is trivial if the stacking pressure is below 10 MPa. Results provide a fundamental tool for the design and evaluation of ASSB safety and cyclability from a more comprehensive perspective and clear the barrier for the development of next-generation ASSBs.

Understanding the formation process of shrinkage pores with a 3D dendrite growth model: from casting to additive manufacturing

Understanding the formation process of shrinkage pores with a 3D dendrite growth model: from casting to additive manufacturing

Multi-GPU implementation of a cellular automaton model for dendritic growth of binary alloy

A parallel three-dimensional (3D) cellular automaton (CA) model is developed for dendritic growth simulation during solidification of binary alloy. The CA model is implemented using the graphic processing unit (GPU). Compute Unified Device Architecture (CUDA) is combined with Message Passing Interface (MPI) to perform large-scale simulation on the multi-GPU architecture. A CUDA-Aware OpenMPI is used to reduce the communication cost between different GPU cards. The accuracy and the speedup ratio of the CA model are investigated. First, the simulated steady-state tip velocity under different undercooling is compared with the LGK analytical solution. Then, dendritic growth during directional solidification and equiaxed multi-dendrite growth are simulated. Dendritic growth simulation with a domain size up to 453 million grids is performed. Finally, the speedup ratio of the CA model using four GPUs is compared with that using four CPU processes via MPI, which leads to a maximal speedup ratio of 152.19. The developed CA model achieves high parallel performance compared with the MPI-based parallel model. The developed CA model is capable of predicting the columnar to equiaxed transition (CET) event during directional solidification. The results indicate that increasing cooling rate promotes the occurrence of the equiaxed dendrites in the melt ahead of the solidification front, and equiaxed dendrites are prone to nucleate in the columnar dendrite groove zone when a middle cooling rate is applied.

Field variable diffusion cellular automaton model for dendritic growth with multifold symmetry for the solidification of alloys

The substantial mesh dependency caused by the discrete properties of cells and limitations of the sharp interface model are the inherent disadvantages of the cellular automaton (CA) model, which leads to difficulties in simulating dendritic growth with multifold symmetry and random preferred orientations for the solidification of alloys. A novel field variable diffusion CA (FCA) model is proposed by referring to the concept of the gradient energy term for diffuse interfaces in the phase field method. The diffusion equation for the field variable based on the constructed gradient functional is derived using the gradient descent method and is introduced into the CA model to reduce the mesh dependency. The FCA model deals with the solidification/remelting growth kinetics of the solid–liquid interface according to the lever rule and by considering the constitutional supercooling and Gibbs–Thomson effect. Free dendritic growth from undercooled melts and competitive dendritic growth during the directional solidification of Al–4 wt% Cu and Mg–5.26 wt% Al alloys were calculated to validate the accuracy of the FCA model for four-fold, six-fold symmetric dendritic morphologies and growth kinetics. The FCA model can efficiently reproduce multi-equiaxed and columnar dendrites with multifold symmetry and random preferred orientations as well as graphically reveal essential dendritic growth information, such as coarsening of secondary arms and side branch fusion.

Transformation kinetics of the metastable bcc phase during rapid solidification of undercooled Fe-Co alloy melts

Solidification of metastable δ-phase (bcc) and subsequent transformation into the stable γ-phase (fcc) in undercooled Fe-Co alloy melts have been investigated on electromagnetically levitated samples. Rapid solidification is monitored by a high-speed video camera, which enables the direct observation of the solidification front. Metastable phase formation is revealed by a two-step recalescence event showing primary solidification of metastable bcc phase and subsequent formation of stable fcc phase within the mushy-zone of primary bcc phase. A decisive parameter describing the kinetics of phase transformation is the delay time between nucleation of the metastable and the stable phase. The delay time obtained from high-speed video recordings strongly decreases with rising undercooling prior to solidification. A model is proposed, which yields an explanation for the origin of the undercooling dependency. It is based on the classical nucleation theory combined with an estimation of the evolution of the number of heterogeneous nucleation sites within the expanding mushy-zone during dendritic growth of the primary phase. Thus, the delay time is not only determined by nucleation kinetics but it also depends on both the growth velocity v and the morphology of the dendritic network of the primary metastable phase, which both depend on undercooling ΔT. Using the relation v(ΔT) for the metastable bcc phase calculated by current models for dendritic growth and verified by measured growth velocities, the experimental data as a function of undercooling are well described by the calculated delay times in a wide range of compositions between 30 and 65 at.% Co.

Fully-discrete spectral-Galerkin scheme with decoupled structure and second-order time accuracy for the anisotropic phase-field dendritic crystal growth model

In this work, we consider numerical approximations of the anisotropic phase-field dendritic crystal growth model, which is a highly complex coupled nonlinear system consisting of the anisotropic Allen-Cahn equation and the heat equation. Through the combination of a novel explicit auxiliary variable IEQ approach for temporal discretization and the spectral-Galerkin approach for spatial discretization, we develop the first fully-discrete numerical scheme with linearity, decoupled structure, unconditional energy stability, and second-order time accuracy for the particular phase-field dendritic model. In the process of obtaining a full decoupling structure and maintaining energy stability, the introduction of two auxiliary variables and the design of two auxiliary ODEs play a vital role. The designed scheme is highly efficient because only a few elliptic equations with constant coefficients are needed to be solved at each time step. The unconditional energy stability of the scheme has been strictly proved, and the detailed implementation process is given. Through several numerical simulations of 2D and 3D dendritic crystal growth examples, we further verify the convergence rate, energy stability, and effectiveness of the developed algorithm.

Ion competition and limiting dendrite growth models of hybrid-ion symmetric cell

NaK symmetric cell test acts as an important detection method for NaK electrodes. It is vital and essential to confirm its suitable application conditions, which include identification of active-ion (Na+ or K+) and limitation of dendritic-free properties. Herein, ion priority is analyzed by electron distribution, electronic attraction and Gibbs free energy and then we draw a theoretical conclusion: K+ prefers to dissolve on anode while Na+ prefers to deposit on cathode, indicating the active-ion of NaK symmetric cell is K+. Furthermore, the upper limit for dendrite-free properties of NaK electrodes is detected: Na dendrite grows under Na+ based electrolyte due to the trans-metalation reaction; K dendrite forms when the K+ plating speed exceeds the alloying velocity. Overall, the discovery provides experimental basis and theoretical guidance for the application of NaK electrodes.

Multiphase and multiphysics modeling of dendrite growth and gas porosity evolution during solidification

Gas porosity is one of the most detrimental defects that can considerably deteriorate mechanical properties of casting parts. Simulating the interaction between gas porosity and solidifying microstructure is challenging due to multiphase and multiphysical characteristics. In this work, a solid-liquid-gas multiphase-field lattice-Boltzmann model is developed to describe the complex multiphase interaction during solidification. The model relaxes the assumption of pure metal system, simplified bubble shape and pure diffusion condition. It can consider solid growth, bubble motion, interface deformation, component transport, melt flow, and partition of both alloy solute and dissolved gas species. The model is validated in terms of mass conservation, mapping operation to the two-phase model, Laplace pressure condition, and bubble dynamics. The effectiveness of the model is further evaluated by comparing with other four models and experiments. The model is successfully used to describe the interaction between gas porosity and magnesium dendrite. The dependence of phase fractions on the characteristic parameters including melt undercooling, interface mobility coefficient and internal bubble pressure is quantified to explore the multiphase equilibrium. The proposed model can be regarded as complementary to the previous models and it is suitable for addressing the problems involving the solid-liquid-gas multiphase and multiphysical characteristics.

A Novel Crystal Growth Model with Nonlinear Interface Kinetics and Curvature Effects: Sensitivity Analysis and Optimization

With the ever-increasing engineering and biological applications of rapid droplet solidification on superhydrophobic surfaces, developing accurate dendritic growth models has become vital at lower ...

A numerical study of dendrite growth and microstructure transition in a non-equilibrium solidification

Controlling the thermal condition of a solidification process to obtain a desirable microstructure morphology and then prepare specific materials with excellent mechanical properties or special functions has been attracting many concerns, where the behavior of microstructure morphology transition, especially the columnar-to-equiaxed transition, is one of the most important fundamental issues. This paper introduced a kinetic model for dendrite growth with a large undercooling during a non-equilibrium solidification process using cellular automaton, in which the local thermodynamic equilibrium condition, non-equilibrium partitioned solute, curvature and interface energy anisotropy were considered. The growth rate of dendrite tip was numerically investigated by using the cellular automaton model, and the effects of temperature gradient and cooling rate on the growth rate and morphology features were found. By integrating the simulation results and analytical models, a new microstructure morphology transition criterion for the non-equilibrium solidification depending on the temperature gradient and cooling rate was proposed, in which both the planar-to-columnar and the columnar-to-equiaxed transition were presented.

Free dendritic growth model considering both interfacial nonisothermal nature and effect of convection for binary alloy

Considering both the effect of the nonisothermal nature of the interface as well as the effect of forced convection, an extended free dendritic growth model for binary alloys was proposed. Comparative analysis indicates that the effect of convection on solute diffusion is more remarkable compared with the ignorable effect of convection on thermal diffusion at low bath undercooling, due to the fact that solute diffusion coefficient is usually three orders of magnitude less than thermal diffusion coefficient. At high bath undercooling, the effect of convection on the dendritic growth is very slight. Furthermore, a satisfying agreement between the model predictions with the available experiment data for the Cu70Ni30 alloy was obtained, especially at low bath undercoolings, profiting from the higher values of interfacial migration velocity predicted by the present model with nonideal fluid case than that predicted by the one ignoring the effect of convection.

A CA-LBM model for simulating dendrite growth with forced convection

A two-dimensional coupled model of the cellular automaton (CA) and the lattice Boltzmann method (LBM) was developed to simulate the solute dendrite growth of Fe–C–Mn–S alloy in the presence of forced convection. The model describes the transport phenomenon by the evolution of moving pseudo-particles distribution functions and utilizes the LBM to solve fluid flow and solute transport under forced convection numerically. Based on the solute field calculated by the CA technique, the dynamics of dendrite growth were determined by the previously proposed local solute balance method. The accuracy of the forced convection dendrite growth model was verified by comparing the CA-LBM model with Lipton–Glicksman–Kurz analytical model. It is revealed that the dendrite symmetry structure is destroyed compared to free diffusion, and the upstream arm is more developed than the downstream arm of the dendrite. The enriched solute segregates more at the downstream side than at the upstream side of the dendrite. The length of the upstream dendrite arm increases firstly and then becomes stable with the increase in the flow velocity, the dendrite necking is restrained, and the vertical dendrite arm becomes longer.

On a novel full decoupling, linear, second‐order accurate, and unconditionally energy stable numerical scheme for the anisotropic phase‐field dendritic crystal growth model

AbstractThe anisotropic phase‐field dendritic crystal growth model is a highly nonlinear system that couples the anisotropic Allen–Cahn equation and the thermal equation together. Due to the high anisotropy and nonlinear couplings in the system, how to develop an accurate and efficient, especially a fully decoupled scheme, has always been a challenging problem. To solve the challenge, in this article, we construct a novel fully decoupled numerical scheme which is also linear, energy stable, and second‐order time accurate. The key idea to realize the full decoupling structure is to introduce an ordinary differential equation to deal with the nonlinear coupling terms satisfying the so‐called “zero‐energy‐contribution” property. This scheme is very effective and easy to implement since only a few fully decoupled elliptic equations with constant coefficients need to be solved at each time step. We rigorously prove the solvability of each step and the unconditional energy stability, and perform a large number of numerical simulations in 2D and 3D to demonstrate its stability and accuracy numerically.

Effect of High Pressure on the Solidification of Al–Ni Alloy

The effect of high pressure on the microstructure of hypo-peritectic Al–38wt.%Ni alloy was studied. The results show that Al3Ni and Al3Ni2 phases coexist at ambient pressure. However, it becomes a typical hyper-eutectic microstructure when synthesized at 2 GPa and 4 GPa. Meanwhile, the interface temperature of Al3Ni and Al3Ni2 phases was calculated with the combination of the BCT dendrite growth model, which is suitable for the Al3Ni2 phase. According to the highest interface temperature principle, the result shows that the Al3Ni phase dominates over 1–5 GPa. Finally, the Debye temperature and potential energy of the hypo-peritectic Al–38wt.%Ni alloy under different pressures were researched. Based on the low temperature specific heat-capacity curve. The Debye temperatures at ambient pressure, 2 GPa, and 4 GPa are 504.4 K, 508.71 K and 515.36 K, respectively, and the potential energy in the lowest point decreases with the increase of pressure.

Coupled crack propagation and dendrite growth in solid electrolyte of all-solid-state battery

Lithium metal all-solid-state batteries (ASSBs) are promising candidates for future lithium batteries thanks to their high safety performance and energy density. However, Li dendrite growth and interfacial failure are the two fatal issues deteriorating the cyclability that hinders the wide commercialization of ASSBs. To further understand the underlying mechanisms, a coupled electrochemical-mechanical phase-field model for grain crack propagation and lithium dendrite growth is proposed from the energy conservation perspective in this study. We discover that longer defect with sharp edge and angle ( θ ≥ 45 ° ) causes more severe crack propagation and leads to larger dendrite growth area due to the increased strain energy density. We observe that the initial defect within grain plays an irrelevant role in the dendrite growth within the grain boundary. Stacking pressure greater than 10 MPa significantly speeds up crack propagation as well as dendrite growth due to the nontrivial mechanical driving force. Mechanical stress-induced strain-energy would contribute to more than 15% of the total dendrite growth once the stacking pressure exceeds 20 MPa, while it is trivial if the stacking pressure is below 10 MPa. Large enough fracture threshold strain can prevent the crack propagation. Results provide a fundamental tool for the design and evaluation of ASSB safety and cyclability from a more comprehensive perspective and clear the barrier for the development of next-generation ASSBs. • A coupled model for crack propagation and Li dendrite growth is proposed. • Longer defect with sharp edge and angle promotes crack propagation and dendrite growth. • Stacking pressure above 10 MPa greatly assists crack propagation and dendrite growth. • Strain-energy contributes more than 17% of the dendrite growth if the stacking pressure is above 20 MPa.

Numerical Analysis of Stray Grain Formation during Laser Welding Nickel-based Single-crystal Superalloy Part I: Columnar/Equiaxed Morphology Transition

The thermo-metallurgical modeling of stray grain formation was further developed by couple of heat transfer model, dendrite selection model, multicomponent dendrite growth model, nonequilibrium solidification model and minimum undercooling model during three-dimensional nickel-based single-crystal superalloy weld pool nonequilibrium solidification over a wide range of welding conditions (laser power, welding speed and welding configuration). Welding configuration simultaneously influences distributions of stray grain formation and columnar/equiaxed transition (CET). The stray grain formation and dendrite morphology ahead of solid/liquid interface are symmetrically distributed about the weld pool centerline in the (001)/[100] welding configuration. The stray grain formation and dendrite morphology ahead of solid/liquid interface is asymmetrically distributed in the (001)/[110] welding configuration. Vulnerable [100] dendrite growth region is suppressed in favor of epitaxial [001] dendrite growth region to predominantly facilitate single-crystal dendrite growth with further reduction of heat input. Stray grain formation and solidification cracking are preferentially confined to [100] dendrite growth region. The smaller heat input is used, the less nucleation and growth of stray grain formation with decreasing constitutional undercooling ahead of solid/liquid interface is incurred with mitigation of metallurgical driving forces for solidification cracking and columnar dendrite morphology is increased and vice versa. Symmetrical crystallographic orientation of dendrite growth spontaneously ameliorates microstructure development, and improves resistance to solidification cracking. The mechanism of asymmetrical solidification cracking because of crystallography-dependent stray grain formation and morphology instability is therefore proposed. Optimum low heat input (low laser power and high welding speed) with (001)/[100] welding configuration essentially minimizes both stray grain formation and columnar/equiaxed morphology transition and is beneficial to weldability and weld integrity through morphology control, while undesirable high heat input (high laser power and slow welding speed) with (001)/[110] welding configuration leads to microstructure anomalies and worsens solidification cracking susceptibility. The stray grain formation and morphology transition in the [100] dendrite growth region on the right side of the weld pool are more severe than that in the [010] dendrite growth region on the left side, although the same heat input imposes on both sides of the weld pool in the (001)/[110] welding configuration. The theoretical predictions agree well with the experiment results. Moreover, the promising and reliable model is also applicable to other single-crystal superalloys with similar metallurgical properties for successful crack-free laser welding or laser cladding.

Numerical Simulation of Microstructure Evolution in Solidification Process of Ferritic Stainless Steel with Cellular Automaton

In order to study the basic principles of vibration-excited liquid metal nucleation technology, a coupled model to connect the temperature field calculated by ANSYS Fluent and the dendritic growth simulated by cellular automaton (CA) algorithm was proposed. A two-dimensional CA model for dendrite growth controlled by solute diffusion and local curvature effects with random zigzag capture rule was developed. The proposed model was applied to simulate the temporal evolution of solidification microstructures under different degrees of surface undercooling and vibration frequency of the crystal nucleus generator conditions. The simulation results showed that the predicted columnar dendrites regions were more developed, the ratio of interior equiaxed dendrite reduced and the size of dendrites increased with the increase of the surface undercooling degrees on the crystal nucleus generator. It was caused by a large temperature gradient formed in the melt. The columnar-to-equiaxed transition (CET) was promoted, and the refined grains and homogenized microstructure were also achieved at the high vibration frequency of the crystal nucleus generator. The influences of the different process parameters on the temperature gradient and cooling rates in the mushy zone were investigated in detail. A lower cooling intensity and a uniform temperature gradient distribution could promote nucleation and refine grains. The present research has guiding significance for the process parameter selection in the actual experimental.