Results Of Phase-field Simulations Research Articles

Phase-field numerical simulation of three-dimensional competitive growth of dendrites in a binary alloy

The normal vector of migration direction in the solid-liquid interface of dendrites was used to describe the phase-field governing equation. By using the three angles formed by the normal vector for the migration direction of the dendritic growth interface and the coordinate axes of the simulation region, the authors expressed the interfacial anisotropy equation, and built a phase-field model for the competitive growth of multiple grains. Taking a Al-2%mole-Cu binary alloy as an example, the competitive growth of multiple grains during isothermal solidification was simulated by applying parallel computing techniques. In addition, the phase field simulation results were verified by the experimental method. The simulation results show that the competitive growth of equiaxed dendrite is divided into two types: the first occurs during the process of competitive growth, the tips of primary dendrite on different grains taking part in the competition stop growing in their optimal growth direction; the second also occurs during competitive growth, the tips of primary dendrite which participate in the competition on different grains never stop growing in their optimal growth direction. The dendritic morphologies of the first competition growth type are divided into two types. Primary dendrites of grains taking part in the competition stop growing in their optimal growth direction and the competition plane enlarges when neither one wins the competition. However, when one wins the competition, the primary dendrites of grains with superiority go through the blocking grains and continue to grow in their optimal growth direction. The primary dendrites of inferior grains stop growing in their optimal growth direction and then instead grow in those areas without obstacles. The dendritic morphology of the second competition-growth type is shown to be the deformation of primary dendrites, which are mainly represented as the deflection and bending observed from different views. Compared with the metallographic picture, the simulation results can show the morphology of the competitive growth in all directions, so this simulation method can better characterize the competitive growth process.

Atomistic Determination of Anisotropic Surface Energy-Associated Growth Patterns of Magnesium Alloy Dendrites.

Because of the existence of anisotropic surface energy with respect to the hexagonal close-packed (hcp) lattice structure, magnesium alloy dendrite prefers to grow along certain crystallographic directions and exhibits a complex growth pattern. To disclose the underlying mechanism behind the three-dimensional (3-D) growth pattern of magnesium alloy dendrite, an anisotropy function was developed in light of the spherical harmonics and experimental findings. Relevant atomistic simulations based on density functional theory were then performed to determine the anisotropic surface energy along different crystallographic directions, and the corresponding anisotropic strength was quantified via the least-square regression. Results of phase field simulations showed that the proposed anisotropy function could satisfactorily describe the 3-D growth pattern of the α-Mg dendrite observed in the experiments. Our investigations shed great insight into understanding the pattern formation of the hcp magnesium alloy dendrite at an atomic level.

Phase field and analytical study of mushy zone solidification in a static thermal gradient: From dendrites to planar front

We present phase field simulation results of the directional solidification of an AlCu4wt% alloy followed by a holding stage during which the mushy zone solidifies in a static thermal gradient. During the holding stage, the initially dendritic morphology solidifies through the TGZM (Temperature Gradient Zone Melting) process, in accordance with recent in-situ X-ray measurements. This yields a planar solid/liquid interface that evolves towards a partial equilibrium where the liquid is homogeneous and the solid still shows a microsegregation pattern. A slow solid state diffusion then drives the subsequent evolution towards the final stationary state where chemical fluxes vanish. We study analytically the microsegregation in the solid, showing that TGZM has a major influence on its development during directional dendritic growth and on its evolution during the holding stage. Especially, we develop an expression for the concentration profile within a secondary dendrite arm during growth. We also develop theoretical arguments to determine the position in the thermal gradient of a transition region that separates the solid at partial equilibrium into two spatial domains. Close to the solid/liquid interface, the microsegregation exhibits a simple pattern resulting from a full remelting of the dendritic solid, while at lower temperatures the pattern is more complex and results from an only partial remelting.

Achieving High Energy Density in PVDF-Based Polymer Blends: Suppression of Early Polarization Saturation and Enhancement of Breakdown Strength.

Polymers with high dielectric strength and favorable flexibility have been considered promising materials for dielectrics and energy storage applications, while the achievable energy density (Ue) of polymer is rather limited by the intrinsic low dielectric constant and ferroelectric hysteresis. Polyvinylidene fluoride-trifluoroethylene-chlorofluoroethylene (P(VDF-TrFE-CFE)) with ultrahigh εr of >50 is considered promising in achieving high Ue of polymer dielectrics. However, P(VDF-TrFE-CFE) only exhibits moderate Ue due to the early saturation of electrical polarization at low electric field. In this contribution, we show that, by blending P(VDF-TrFE-CFE) with polyvinylidene fluoride (PVDF), the early saturation of P(VDF-TrFE-CFE) is substantially suppressed, giving rise to concomitant enhancement of dielectric permittivity and breakdown strength. An ultrahigh energy density of 19.6 J/cm3 is thus achieved at ∼640 kV/mm, which is 1600% greater than Ue of the benchmark biaxially oriented polypropylene (BOPP, 1.2 J/cm3 at 640 kV/mm). Results of phase field simulations reveal that the interfaces between PVDF and P(VDF-TrFE-CFE) play a critical role by not only suppressing early saturation of electrical polarization in P(VDF-TrFE-CFE) but also inducing additional interfacial polarization. Binary phase diagram of P(VDF-TrFE-CFE)/PVDF blends is also systematically explored with their dielectric and energy storage behavior studied.

Three-dimensional modeling of a thermal dendrite using the phase field method with automatic anisotropic and unstructured adaptive finite element meshing

Dendritic growth is computed with automatic adaptation of an anisotropic and unstructured finite element mesh. The energy conservation equation is formulated for solid and liquid phases considering an interface balance that includes the Gibbs-Thomson effect. An equation for a diffuse interface is also developed by considering a phase field function with constant negative value in the liquid and constant positive value in the solid. Unknowns are the phase field function and a dimensionless temperature, as proposed by [1]. Linear finite element interpolation is used for both variables, and discretization stabilization techniques ensure convergence towards a correct non-oscillating solution. In order to perform quantitative computations of dendritic growth on a large domain, two additional numerical ingredients are necessary: automatic anisotropic unstructured adaptive meshing [2,[3] and parallel implementations [4], both made available with the numerical platform used (CimLib) based on C++ developments. Mesh adaptation is found to greatly reduce the number of degrees of freedom. Results of phase field simulations for dendritic solidification of a pure material in two and three dimensions are shown and compared with reference work [1]. Discussion on algorithm details and the CPU time will be outlined.

On the Buoyancy-driven Detachment of a Wall-bound Pendant Drop: Results of Phase-field Simulations.

On the Buoyancy-driven Detachment of a Wall-bound Pendant Drop: Results of Phase-field Simulations.

Phase-field simulation of tilted growth of dendritic arrays during directional solidification

The tilted growth of dendritic arrays during directional solidification of non-axially oriented crystals has been simulated in two-dimensional system by using a quantitative phase-field model. A convergence study with respect to the diffuse interface width has been carried out, showing that the results are reasonably independent of the diffuse interface width for W0/d0⩽11.3. A parametric study is performed to investigate the effects of the primary spacing, the misorientation angle and the pulling velocity on the tip undercooling, the upper limit of primary spacing, and the tilted angle. Results show that the tip undercooling decreases when increasing the primary spacing or decreasing the misorientation angle. It has been found that the upper limit, resulting from tertiary branching, depends on both the solidification condition (the pulling velocity and the thermal gradient) and the preferred crystalline orientation (the misorientation angle) for the tilted growth of dendritic arrays. Previous rotation law, describing the relation between the tilted angle and the Péclet number, is evaluated according to phase-field simulation results. Moreover, the effect of the development of sidebranches on the growth direction selection was discussed. Further studies on the microstructure evolution from the onset of planar instability during directional solidification show that the selection of the steady-state primary spacing is sensitive to the misorientation angle.

Solidification in a Supercomputer: From Crystal Nuclei to Dendrite Assemblages

Thanks to the recent progress in high-performance computational environments, the range of applications of computational metallurgy is expanding rapidly. In this paper, cutting-edge simulations of solidification from atomic to microstructural levels performed on a graphics processing unit (GPU) architecture are introduced with a brief introduction to advances in computational studies on solidification. In particular, million-atom molecular dynamics simulations captured the spontaneous evolution of anisotropy in a solid nucleus in an undercooled melt and homogeneous nucleation without any inducing factor, which is followed by grain growth. At the microstructural level, the quantitative phase-field model has been gaining importance as a powerful tool for predicting solidification microstructures. In this paper, the convergence behavior of simulation results obtained with this model is discussed, in detail. Such convergence ensures the reliability of results of phase-field simulations. Using the quantitative phase-field model, the competitive growth of dendrite assemblages during the directional solidification of a binary alloy bicrystal at the millimeter scale is examined by performing two- and three-dimensional large-scale simulations by multi-GPU computation on the supercomputer, TSUBAME2.5. This cutting-edge approach using a GPU supercomputer is opening a new phase in computational metallurgy.

Charge compensation of head-to-head and tail-to-tail domain walls in barium titanate and its influence on conductivity

The effect of the polarization charge compensation by ionic and electronic space charges on domain properties in ferroelectrics with semiconducting features is considered, in particular, the conductivity of head-to-head and tail-to-tail domain walls is studied. It is shown that the domain wall conductivity that is enhanced by electrons or holes depends on the configuration and the types of domains as well as on the energy levels and concentrations of the defects involved. Phase field simulation results are used to explain recent equivocal experimental results on conductivity of charged domain walls in different ferroelectrics.

Anomalous overgrowth of converging dendrites during directional solidification

Competitive growth of converging dendrites in directional solidification was investigated by in situ observation in a succinonitrile–acetone alloy. The specific dendrite growth behavior was analyzed and compared with the results of phase field simulations. It is found that the growth of the well aligned grain boundary dendrite is retarded and the spacing between this dendrite and its immediate neighbors (λGB1) decreases continuously as the misaligned dendrites get to the grain boundary, one by one. The anomalous overgrowth only happens when the spacing λGB1 decreases to certain level. Under the same solidification condition, smaller initial primary spacing makes the anomalous overgrowth take place earlier and more frequently.

An elastic phase field model for thermal oxidation of metals: Application to zirconia

A multi-phase field model was developed for non-selective oxidation of metals which captures both the oxidation kinetics and stress generation. Phase field formulation involved a non-conserved phase field variable as the marker for the metallic substrate, oxide scale, and a fluid phase containing oxygen, and a conserved phase field variable representing the concentration of oxygen. The evolution equations of the phase field variables were coupled to the mechanical equilibrium equations to investigate the evolution of stress generation in both the oxide scale and the underlying metal. The governing equations were solved in a finite element framework. This phase field model predicts the oxygen composition depth and stress profiles in the oxide layer and at the metal–oxide interface. The model was proven successful in predicting the observed evolution of oxide thickness and growth stresses for Zircaloy-4 oxidized at 900°C. The results of phase field simulations showed that the generation of stresses upon oxidation tends to slow down the oxidation kinetics, and this substantially improved the model predictability of experimental data.

Phase-field modeling of the γ′-coarsening behavior in Ni-based superalloys

Microstructural changes during heat treatment of the Ni-based CMSX-4 and CMSX-6 superalloys have been investigated experimentally and simulated using a phase-field multi-component model incorporating elastic driving forces in the presence of a lattice misfit. Furthermore, a theoretical model of the coarsening of anisotropic particles is proposed for the prediction of the main kinetic parameters of the coarsening process. A comparison of the main characteristics of the microstructural evolution during non-directional γ′-coarsening, provided from both experiments and phase-field simulations, gives a good agreement of the coarsening kinetics of the CMSX-4 superalloy. However, for the CMSX-6 superalloy, phase-field simulation results and theoretical predictions are not entirely consistent with experimental results, and show that additional effects, for example, those caused by plastic deformation, might be a reason for the slow coarsening rate.

Phase field simulations of stress controlling the vortex domain structures in ferroelectric nanosheets

Effect of stress loads on the vortex domain structures (VDSs) in ferroelectric nanosheet has been investigated. Results of phase field simulations show that the different vortex domain structures can form in free-standing nanosheet from random perturbations. Applying stress loads on nanosheet, it is found that the domain morphology, especially the size and number of vortices, can be regularly controlled. These results indicate promising controlling the vortex domain structures in ferroelectric nanostructures by the mechanical loads.

Effect of crystallographic orientation on instability behavior of planar interface in directional solidification

The instability process of planar interface in directional solidification with respect to the crystallographic orientation is studied using a transparent model alloysuccinonitrile-acetone. Three typical crystal grains which have preferred dendrite, tilted dendrite and seaweed patterns at rapid pulling velocity respectively are chosen in our experiment. The experimental results show that the preferred dendrite grain has the shortest incubation time and the smallest initial perturbation wavelength of planar interface instability, the tilted dendrite grain has the largest ones and the seaweed grain has median ones. These results accord qualitatively with previous analytical results and phase-field simulation results. It is also found that the interfacial non-steady-state evolution behaviors of the preferred dendrite grain and the tilted dendrite grain are significantly different from that of the seaweed grain, suggesting that the non-steady-state evolution behavior of planar interface instability is closely related to the crystallographic orientation.

Quantitative investigation of cellular growth in directional solidification by phase-field simulation

Using a quantitative phase-field model, a systematic investigation of cellular growth in directional solidification is carried out with emphasis on the selection of cellular tip undercooling, tip radius, and cellular spacing. Previous analytical models of cellular growth are evaluated according to the phase-field simulation results. The results show that cellular tip undercooling and tip radius not only depend on the pulling velocity and thermal gradient, but also depend on the cellular interaction related to the cellular spacing. The cellular interaction results in a finite stable range of cellular spacing. The lower limit is determined by the submerging mechanism while the upper limit comes from the tip splitting instability corresponding to the absence of the cellular growth solution, both of which can be obtained from phase-field simulation. Further discussions on the phase-field results also present an analytical method to predict the lower limit. Phase-field simulations on cell elimination between cells with equal spacing validate the finite range of cellular spacing and give deep insight into the cellular doublon and oscillatory instability between cell elimination and tip splitting.

Precipitation Kinetics Study of Al – Zr – X(Sc or Ti) Alloys by Phase Field Simulations and Atom Probe Tomography

Phase field modeling of precipitation kinetics in Al – Zr – Sc and Al – Zr – Ti ternary alloys has been performed. The free energy was evaluated using the Thermo-calc data. Our simulations showed that L12 precipitates in Al – Zr – Sc alloy consists of Sc rich zone of in core and Zirconium rich zone at the precipitate / matrix interface. In Al – Zr – Ti system, Al3 (Zr-Ti) precipitates are homogeneous and no segregation is observed. Phase-field simulation results are compared with 3D APT data.

Phase-field modeling of two-dimensional solute precipitation/dissolution: Solid fingers and diffusion-limited precipitation

Two-dimensional dendritic growth due to solute precipitation was simulated using a phase-field model reported earlier [Z. Xu and P. Meakin, J. Chem. Phys. 129, 014705 (2008)]. It was shown that diffusion-limited precipitation due to the chemical reaction at the solid-liquid interface has similarities with diffusion-limited aggregation (DLA). The diffusion-limited precipitation is attained by setting the chemical reaction rate much larger compared to the solute diffusion to eliminate the effect of the interface growth kinetics. The phase-field simulation results were in reasonable agreement with the analytical solutions. The fractal solid fingers can be formed in the diffusion-limited precipitation and have a fractal dimension measured d(f)=1.68, close to 1.64, the fractal dimensionality of large square lattice DLA clusters.

Phase-field study for the influence of solute interactions on solidification process in multicomponent alloys

Abstract The industrial alloys, such as steel and aluminum alloy, always contain more than two main solutes. The phase equilibrium and solute diffusion behaviors in multicomponent alloys will be no longer the same as that in binary alloys because of the solute interactions. In order to study the influence of solute interactions on the solidification process, we develop a phase-field model for multicomponent alloy coupled with thermodynamic and diffusion mobility database, which can accurately predict the phase equilibria and solute diffusion transportations. The model is applied to Al–Cu–Mg and Al–Cu–Si systems respectively. There are totally different solute interactions in these two systems, that is, attractive interactions in Al–Cu–Mg alloy and repulsive interactions in Al–Cu–Si alloy. Their influences on the solidification process are quantitatively analyzed from the phase-field simulation results.

Phase-Field Modeling of Domain Structure of Confined Nanoferroelectrics

The results of phase-field simulation of domain structures (DSs) in ferroelectric nanorods of different shapes and sizes are presented. It is shown that equilibrium DSs consist of an electrostatically compatible circuit of 180 degrees and 90 degrees domains. A DS in a thin rod contains 90 degrees cubic elastic domains. The trend to minimize the residual stress and the stray field results in the formation of crater-shaped sets of closed circuits of 90 degrees domains, which can be mechanically incompatible but able to maintain electrostatic compatibility during the evolution under an applied electric field.

Influences of Marangoni effect on solidification of curved interfaces with constant curvature

The asymptotic method and phase field simulation are applied to study the influences of variation of the surface tension with temperature on the movements of solid-liquid interfaces in the solidification process of spherical and cylindrical nuclei, respectively. Results indicate that Marangoni effect will increase the critical nucleation radius, slow up the movement of interface. The tip speed of dendrite decreases linearly with Marangoni number for melt without convection. The results of phase field simulation are qualitatively in accord with that of asymptotic method.