Large-scale Phase-field Simulations Research Articles

Large-scale phase-field simulations for dendrite growth: A review on current status and future perspective

The current status of large-scale phase-field (PF) simulations for dendrite growth is reviewed by focusing on the study conducted by our group. The discussion includes the competitive growth of multiple columnar dendrites, dendrite growth with liquid flow and solid motion, permeability prediction, and cross-scale simulations using the PF method. All PF simulations introduced here were executed using a graphics processing unit (GPU) or a GPU supercomputer to significantly accelerate the PF simulations. Finally, the future perspectives of large-scale dendrite-growth PF simulations are discussed.

A stochastic framework for evolving grain statistics using a neural network model for grain topology transformations

We develop a stochastic framework to evolve grain statistics during 2D isotropic grain growth. Grain microstructures observed in simulations and experiments often demonstrate the existence of asymptotic distributions for descriptors such as grain size and topology (number of sides). Moreover, such grain descriptors/states are used to characterize a microstructure as they influence the mechanical properties of a polycrystal. This motivates us to discover laws that govern the distributions of grain microstructure descriptors. In particular, we focus on the evolution of distributions for grain sizes and the number of sides. Since the surface tension is assumed to be isotropic, the von Neumann–Mullins law provides an evolution equation for grain areas in terms of their topological states—number of edges. However, since grains change their topology as they evolve by interacting with neighbors, we construct a topology transformation model (TTM) that predicts the probability of topology transformation of a grain in terms of its current state and the states of its neighbors. The construction of the TTM relies on a data-driven approach using a fully connected deep neural network. Topology transformations recorded in phase field simulations are used as training data. The resulting neural network model is used in a Monte Carlo framework to evolve grain microstructures in a statistical sense. The stochastic method is validated using the asymptotic and transient grain statistics predicted by large-scale phase field simulations. In addition, our results suggest that the TTM incorporates, in a statistical sense, the geometric constraint that grains fill the space.

Validating a mean-field theory via large-scale phase-field simulations for abnormal grain growth induced by nonuniform grain boundary properties

Validating a mean-field theory via large-scale phase-field simulations for abnormal grain growth induced by nonuniform grain boundary properties

Testing an Abnormal Grain Growth Theory via Large-scale Phase-field simulations

The mean-field theory proposed by Humphreys is frequently used to predict abnormal grain growth caused by nonuniform grain boundary properties. This theory enables the abnormal growth conditions of a particular grain to be described using only three parameters: the size ratio, grain boundary energy ratio, and mobility ratio between the particular grain and its surrounding matrix grains. However, detailed validation of the theory has yet to be reported. In this study, we elucidate the validity of the mean-field theory by performing very large-scale phase-field simulations of 2D and 3D abnormal grain growth based on parallel GPU computing. Systematic simulations are carried out while varying the initial size ratio, boundary energy ratio, and mobility ratio of a particular grain and matrix grains. Comparisons of the simulated theoretical results demonstrate that the abnormal growth behaviors (i.e., whether the abnormal growth occurs and the limiting size that can be reached by an abnormally growing grain) is well described by the mean-field theory.

Phase-field study on an array of tilted columnar dendrites during the directional solidification of a binary alloy

An array of columnar dendrites is important to the microsegregation and permeability of interdendritic liquid flow. In this study, an array of tilted columnar dendrites growing during the directional solidification of a binary alloy was investigated via large-scale phase-field simulation. The main conclusion is that the hexagonal array is the dominant array for reasonably tilted dendrites as well as dendrites that grow along the temperature gradient. It is also concluded that the array ordering gradually improves with growth for all tilt angles, and the ordering rate is faster for a small tilt angle.

Equiaxed dendritic growth in nearly isothermal conditions: A study combining in situ and real-time experiment with large-scale phase-field simulation

The equiaxed dendritic growth of Al-Cu alloys in nearly isothermal temperature field under continuous cooling condition is studied using in situ and real-time observation of experiments by synchrotron X-ray radiography and large-scale quantitative two-dimensional (2D) phase-field (PF) simulations. It is revealed that the equiaxed dendritic morphology and the secondary dendritic arm spacing (SDAS) in the 2D PF simulations are in a reasonable agreement with the experimental data. Increasing the cooling rates results in a smaller SDAS, as predicted by the analytical Kattamis-Flemings model. The transformation kinetics of solid fraction can be described by the Johnson-Mehl-Avrami-Kologoromov (JMAK) theory, but quantitative differences between the experiments and 2D PF simulations are significant. The maximum solute concentration Cmax in liquid is approximately equal to the equilibrium concentration, which depends on the undercooling rather than the cooling rate. But the minimum solute concentration Cmin in solid decreases with the cooling rate, thus leading to a larger segregation ratio SR = Cmax/Cmin. Moreover, the liquid gravity-driven natural convection is considered in simulations. The liquid flow slightly increases the SDAS but has no apparent effect on solid fraction, and the segregation ratio is slightly reduced by the liquid convection, which could be attributed to the almost same Cmax and enlarged Cmin.

Phase-field study of eutectic colony formation in NiAl-34Cr

The formation of two-phase eutectic colonies is often observed in microstructures of directionally solidified ternary alloys. Their formation is driven by microscopic instabilities in a macroscopic planar solidification front, due to impurities of the minor component, diffusing from the two solidifying phases into the liquid. The growth conditions for eutectic colonies, their interactions and their responses to the microstructure during growth are the focus of the current work. Therefore, phase-field simulations based on a grand potential formalism are performed for the high-performance material NiAl-34Cr. To enable the evolution of eutectic colonies in two-dimensional simulations, a concentration-driven nucleation mechanism is introduced into a multiphase-field framework and is subsequently validated. With this mechanism, two-dimensional large-scale phase-field simulations are conducted to study the influence of the applied temperature gradient on the evolving colonies. The patterns are quantitatively analyzed by measuring their number, size and height. Furthermore, the adjustment processes between the eutectic colonies during the directional solidification are investigated. The results demonstrate the ability of the presented phase-field approach with integrated nucleation mechanism for the formation of eutectic colonies in two-dimensional simulations.

Large-scale phase-field simulation of three-dimensional isotropic grain growth in polycrystalline thin films

In this study, assuming an ideal system free from thermal grooving and anisotropy in grain boundary properties, we analyze thin-film grain growth via three-dimensional (3D) phase-field simulations with approximately one million initial grains. The large-scale simulations accelerated by multiple graphics processing units allow for the reliable statistical investigation of grain growth behaviors in films with various thickness. Over the transition from 3D to two-dimensional (2D) growth modes, variations in the averages and distributions of grain sizes are quantified and compared for different regions of the films. Furthermore, we propose a comprehensive scaling law of thin-film grain growth, by which the 3D–2D transition behaviors and grain growth kinetics can be described in a unified manner independent of film thickness.

Competitive growth during directional solidification of a binary alloy with natural convection: two-dimensional phase-field study

Due to the recent acceleration of phase-field simulations using a graphics processing unit (GPU), the mechanism of competitive growth among columnar grains has been better understood. In this study, the effects of natural convection, caused by the gravity driven buoyancy force, on the competitive growth of columnar grains during directional solidification of bi-crystal and polycrystal binary alloys are investigated by performing two-dimensional large-scale phase-field simulations using a GPU supercomputer. As a result, the downward flow accelerates and the upward flow decelerates the selection speed in competitive growth. It is also confirmed that these phenomena are caused by a characteristic liquid flow pattern at the converging and diverging grain boundaries.

3D Analysis of Observed and Simulated Microstructure Evolution in SOFC Anodes

An important degradation mechanism in solid oxide fuel cells (SOFCs) is the microstructural evolution of Ni-YSZ anodes caused by diffusional transport of nickel under operating conditions. In this work, we carry out 3D analysis on several data sets from FIB-SEM reconstructions of a full-ceramic co-sintered SOFC as well as from large-scale phase-field simulations, in order to obtain a quantitative understanding of the different microstructure evolution effects and their impact on anode degradation. The experimental data stem from FIB-SEM reconstructions of both newly fabricated anodes and anodes aged for ca. 1000h under operating conditions. The simulated data stem from large-scale phase field simulations that have been initialized with a reconstructed microstructure. The underlying phase-field model incorporates both volume diffusion of nickel and interface diffusion along the Ni surfaces, grain boundaries, and Ni-YSZ interfaces. The required material parameters (interface energies and diffusivities) are determined partly by ab-initio simulations and partly by thermal grooving experiments on bicrystalline and polycrystalline nickel. For the 3D analysis, we calculate standard microstructural features such as volume fractions, active and inactive three-phase-boundary (3PB) length, and tortuosity, and we use the transmission line model to calculate the expected anode performance. Comparison with the measured anode performance shows that the observed decrease of 3PB length explains the measured degradation only partly, while the main degradation factor is loss of connectivity of the nickel phase. To understand the connectivity loss in more detail, we carry out a bottleneck analysis to quantify the gradual tightening of the bottlenecks. In addition, we statistically assess the spatial homogeneity, and we calculate several shape descriptors such as interface areas and local structure size, which allows us to show that nickel relocates over considerable distances from small pores to larger pores within the YSZ skeleton.

First Evidence for Mechanism of Inverse Ripening from In-situ TEM and Phase-Field Study of \u03b4\u2032 Precipitation in an Al-Li Alloy

In-situ TEM investigation of aging response in an Al–7.8 at.% Li was performed at 200 °C up to 13 hours. Semi-spherical δ′ precipitates growing up to an average radius of 7.5 nm were observed. The size and number of individual precipitates were recorded over time and compared to large-scale phase-field simulations without and with a chemo-mechanical coupling effect, that is, concentration dependence of the elastic constants of the matrix solid solution phase. This type of coupling was recently reported in theoretical studies leading to an inverse ripening process where smaller precipitates grew at the expense of larger ones. Considering this chemo-mechanical coupling effect, the temporal evolution of number density, average radius, and size distribution of the precipitates observed in the in-situ experiment were explained. The results indicate that the mechanism of inverse ripening can be active in this case. Formation of dislocations and precipitate-free zones are discussed as possible disturbances to the chemo-mechanical coupling effect and consequent inverse ripening process.

Correlation between three-dimensional and cross-sectional characteristics of ideal grain growth: large-scale phase-field simulation study

Grain growth is one of the most fundamental phenomena affecting the microstructure of polycrystalline materials. In experimental studies, three-dimensional (3D) grain growth is usually investigated by examining two-dimensional (2D) cross sections. However, the extent to which the 3D microstructural characteristics can be obtained from cross-sectional observations remains unclear. Additionally, there is some disagreement as to whether a cross-sectional view of 3D grain growth can be fully approximated by 2D growth. In this study, by employing the multi-phase-field method and parallel graphics processing unit computing on a supercomputer, we perform large-scale simulations of 3D and 2D ideal grain growth with approximately three million initial grains. This computational scale supports the detailed comparison of 3D, cross-sectional, and 2D grain structures with good statistical reliability. Our simulations reveal that grain growth behavior in a cross section is very different from those in 3D and fully 2D spaces, in terms of the average and distribution of the grain sizes, as well as the growth kinetics of individual grains. On the other hand, we find that the average grain size in 3D can be estimated as being around 1.2 times that observed in a cross section, which is in good agreement with classical theory in the stereology. Furthermore, based on the Saltykov–Schwartz method, we propose a predictive model that can estimate the 3D grain size distribution from the cross-sectional size distribution.

Competitive grain growth during directional solidification of a polycrystalline binary alloy: Three-dimensional large-scale phase-field study

Competitive grain growth during the directional solidification of a polycrystalline binary alloy is investigated by performing systematic three-dimensional large-scale phase-field simulations with the GPU supercomputer TSUBAME2.5 at the Tokyo Institute of Technology. Contrary to two-dimensional investigations, in which an unusual growth of unfavorably oriented (UO) grains has been observed frequently in preference to favorably oriented (FO) ones, the grain selection in the present three-dimensional simulations follows essentially the Walton and Chalmers model, in which FO grains are predominant. The UO dendritic grains persist for longer times than the UO cellular grains, and the FO dendritic grains remain smaller than the FO cellular ones. The change in the number of grains follows a power law, with an exponent that is much lower than that of the Kolmogorov's model describing a purely geometrical growth selection.

Analysis of grain topology and volumetric growth rate relation in three-dimensional normal grain growth

The volumetric growth rate of individual grains as a function of their number of sides, as well as the average normalized volumetric growth rate for a given topological class are studied based on large-scale phase-field simulations of normal grain growth. For all cases, where different initial grain size distributions are considered, the same averaged normalized volumetric growth rate for a given topological class was obtained with the averaged normalized volumetric growth rate approximately zero for grains with Fo≈15 sides. The normalized average grain size as a function of topological class becomes constant within the transient regime and the relation in the transient and steady-state regime is similar and independent of initial grain structure. The relation does however not match Mullins' analytical model [W. Mullins, Acta Metall. 37 (1989) 2979–2984]. The effect of the average number of sides of the neighbouring grains on the average volumetric growth rate for a given topological class is analyzed, as well. Moreover, tracking of the evolution of four grains suggests that the normalized volumetric growth rate of a grain is strongly linked with the number of sides of the grain at the given time.

Computationally Efficient Phase-field Simulation Studies Using RVE Sampling and Statistical Analysis

For large-scale phase-field simulations, the trade-off between accuracy and computational cost as a function of the size and number of simulations was studied. For this purpose, a large reference representative volume element (RVE) was incrementally subdivided into smaller solitary samples. We have considered diffusion-controlled growth and early ripening of δ′ (Al3Li) precipitate in a model Al-Li system. The results of the simulations show that decomposition of reference RVE can be a valuable computational technique to accelerate simulations without a substantial loss of accuracy. In the current case study, the precipitate number density was found to be the key controlling parameter. For a pre-set accuracy, it turned out that large-scale simulations of the reference RVE can be replaced by simulating a combination of smaller solitary samples. This shortens the required simulation time and improves the memory usage of the simulation considerably, and thus substantially increases the efficiency of massive parallel computation for phase-field applications.

Comparison of coarsening behaviour in non-conserved and volume-conserved isotropic two-phase grain structures

Microstructure evolution in non-conserved and volume-conserved isotropic two-phase grain structures, with equal interfacial energy for all interfaces and equal volume fractions of the two-phases, is compared based on large-scale phase-field simulations. Two important observations are as follows: (1) the growth mechanism has a larger effect on the grain size distributions of the two alloys than on their grain topology distributions; (2) the growth rate and normalized grain size relation of the grains per topological class are much more scattered for the conserved system than the non-conserved system.

Accelerating large-scale phase-field simulations with GPU

A new package for accelerating large-scale phase-field simulations was developed by using GPU based on the semi-implicit Fourier method. The package can solve a variety of equilibrium equations with different inhomogeneity including long-range elastic, magnetostatic, and electrostatic interactions. Through using specific algorithm in Compute Unified Device Architecture (CUDA), Fourier spectral iterative perturbation method was integrated in GPU package. The Allen-Cahn equation, Cahn-Hilliard equation, and phase-field model with long-range interaction were solved based on the algorithm running on GPU respectively to test the performance of the package. From the comparison of the calculation results between the solver executed in single CPU and the one on GPU, it was found that the speed on GPU is enormously elevated to 50 times faster. The present study therefore contributes to the acceleration of large-scale phase-field simulations and provides guidance for experiments to design large-scale functional devices.

Investigation on the existence of a ‘Hillert regime’ in normal grain growth

The existence of a ‘Hillert regime’ in 3D normal grain growth, where the grain size distributions (GSDs) at different time-steps match the Hillert distribution during parabolic grain growth, is investigated for different initial GSDs using large-scale phase-field simulations. The short-lived ‘Hillert regime’ was present in the early-stage only in few cases. The GSDs obtained at a later-stage for all cases are self-similar over long period; independent of the initial GSDs; and wider than the Hillert distribution. Also, the topological properties in the ‘Hillert regime’ were different from that in the later-stage self-similar regime.

GPU-accelerated three-dimensional phase-field simulation of dendrite growth in a nickel-based superalloy

The microstructure formation of a nickel-based superalloy during solidification in three dimensions was investigated using the phase-field method. To accelerate the large-scale phase-field simulation, a parallel computing approach was developed using the graphic processing unit (GPU), and the limitation of insufficient GPU memory was circumvented by employing an asynchronous concurrent algorithm. The performance of the GPU-based parallel computing method was tested and the results demonstrate that a maximum performance of 774.292 GFLOPS (giga floating-point operations per second) can be obtained using a single NVIDIA GTX1080 GPU. In simulations of isothermal solidification, the microstructure evolution of a single and multiple dendrites under different undercooling levels was shown in detail. During the solidification, the dendrite tip growth velocity and fraction solid were recorded and then analyzed. In simulations of directional solidification, the formation of primary dendrite arms under different temperature gradients was investigated, and the simulated microstructure was in good agreement with experimental observations. Additionally, the distribution of primary dendrite arm spacing was quantitatively analyzed by Voronoi tessellation. Finally, simulation of polycrystalline growth in directional solidification was conducted to study the dendrite competitive growth. The unusual overgrowth phenomenon was observed in the initial growth stage, while as the solidification process proceeded, the dendrites with small inclination angles were more likely to overgrow the dendrites with large inclination angles.

Phase-field study of ripening and rearrangement of precipitates under chemomechanical coupling

We investigate the evolution of large number of ${\ensuremath{\delta}}^{\ensuremath{'}}$ coherent precipitates from a supersaturated Al-8 at.% Li alloy using large-scale phase-field simulations. A chemomechanical cross-coupling between mechanical relaxation and diffusion is taken into account by considering the dependence of elastic constants of the matrix phase onto the local concentration of solute atoms. The elastic constants as a function of solute concentration have been obtained using density functional theory calculations. As a result of the coupling, inverse ripening has been observed where the smaller precipitates grow at the expense of the larger ones. This is due to size-dependent concentration gradients existing around the precipitates. At the same time, precipitates rearrange themselves as a consequence of minimization of the total elastic energy of the system. It is found that the anisotropy of the chemomechanical coupling leads to the formation of new patterns of elasticity in the matrix thereby resulting in new alignments of the precipitates.