Grain Growth In Polycrystalline Materials Research Articles

Nonlinear inhomogeneous Fokker–Planck models: Energetic-variational structures and long-time behavior

Inspired by the modeling of grain growth in polycrystalline materials, we consider a nonlinear Fokker–Plank model, with inhomogeneous diffusion and with variable mobility parameters. We develop large time asymptotic analysis of such nonstandard models by reformulating and extending the classical entropy method, under the assumption of periodic boundary condition. In addition, illustrative numerical tests are presented to highlight the essential points of the current analytical results and to motivate future analysis.

Role of inclination dependence of grain boundary energy on the microstructure evolution during grain growth

The role of inclination dependence of grain boundary energy on the microstructure evolution and the orientation distribution of grain boundary planes during grain growth in polycrystalline materials is investigated by three-dimensional phase-field simulations. The anisotropic grain boundary energy model uses the description of the faceted surface structure of the individual crystals and constructs an anisotropic energy of solid-solid interface. The energy minimization occurs by the faceting of the grain boundary due to inclination dependence of the grain boundary energy. The simulation results for a single grain show the development of equilibrium shapes (faceted grain morphologies) with different families of facets which agrees well with the theoretical predictions. The results of grain growth simulations with isotropic and anisotropic grain boundary energy for cubic symmetry show that inclination dependence of grain boundary energy has a significant influence on the grain boundary migration, grain growth kinetics and the grain boundary plane distribution. It has been shown that the model essentially reproduces the experimental studies reported for NaCl and MgO polycrystalline systems where the anisotropic distribution of grain boundary planes has a peak for the low-index {100} type boundaries.

Preface

The Seventh International Conference on Recrystallization and Grain Growth is the continuation of a conference series that was initiated in 1990 with Recrystallization ’90, organized in Wollongong, followed in 1991 by ’Grain Growth in Polycrystalline Materials”, organized in Rome. Since 2001, both conference series were merged with the organization of the ‘1st International Conference on Recrystallization and Grain Growth’ in Aachen, which has ever since been a regularly recurring triannual event.During the three decades that have passed since its launch, this conference series has shaped an active and versatile research community with active participation from fundamental and applied research groups and industrial R&D organizations. Recrystallization and grain growth are defining factors in shaping the microstructure of both structural and functional polycrystalline materials, metals and nonmetals alike, and therefore of paramount importance for a wide variety of microstructurally sensitive material properties. Thus, an intensive and mutually fruitful collaboration has emerged between applied R&D groups, often associated with industrial organizations, and more fundamentally oriented scientists. Other scientific fields, such as geology, mineralogy and archeology have increasingly studied microstructures that ensue from thermally activated processes such as recrystallization and grain growth, which may provide clues as to how materials were created and what circumstances prevailed during their history.A more detailed understanding of the basic mechanisms of recrystallization and grain growth also contributes to the development of new materials with improved and more versatile functional properties. For newly emerging processing technologies, such as additive manufacturing, a better knowledge on recrystallization and grain growth, particularly in far out-of-equilibrium conditions, is of crucial importance for microstructural control of 3D printed metal objects and hence for the successful introduction of additive manufacturing to all its potential markets.

Synergy of elastic strain energy and electron wind force on thin film grain growth at room temperature

Grain growth in polycrystalline materials is commonly achieved with thermal annealing. Here, we explore an alternate athermal route through a synergy of electrical current and tensile strain at or near room temperature. Our experimental approach involves passing electrical current in 100 nm thick freestanding palladium films with about 5 nm initial grain size. The resulting Joule heating increases the temperature up to 470 K (homologous temperature of 0.25) at the middle section of the specimen. The massive heatsinks at the two ends of the specimen constrain them at the room temperature. At current density of 7 × 105 A/cm2, in-situ transmission electron microscopy shows more than two orders of magnitude grain growth in the high temperature regions and little growth at the room temperature regions. However, application of elastic strain (about 0.1%) dramatically increased the grain size to about 100 nm in a few seconds at the room temperature regions. Our finding suggests that the synergy of elastic strain energy and electrical current density may achieve grain growth in metallic materials even at the room temperature. Molecular dynamics simulation of this phenomenon reveals that the externally applied strain is localized at the grain boundaries in nanocrystalline metals, which promotes the effects of electron wind force on the grain boundary atoms. We conclude that a synergy of two or more stimuli can achieve grain growth at room or even lower temperatures.

Statistics of grain growth: Experiment versus the phase-field-crystal and Mullins models

We present a detailed comparison of multiple statistical grain metrics for previously reported experimental thin film samples of aluminum with 2D simulations obtained from the Phase-Field-Crystal (PFC) model and a Mullins grain boundary motion model. For all these results, “universality” is observed with respect to the dynamics and initial conditions. This comparison reveals that PFC reproduces geometric metrics such as area and perimeter, but does not capture grain shape and topology as accurately. Similarly, the Mullins model captures the number of sides distribution quite well but not other metrics. Our collective comparison of such measurements underscores the critical importance of the use of multiple metrics for comparison of experiments with all present and future models of grain growth in polycrystalline materials.

Simulation of grain growth under simultaneous grain boundary migration and grain rotation using multi-order parameter phase-field model

Study of microstructure evolution in the form of grain growth in polycrystalline materials has been an important goal for material scientists as it drastically affects physical and mechanical properties. Specifically, nanocrystalline materials, which are known for their superior mechanical properties, are highly susceptible to grain growth even at low temperatures and stresses. Various experiments and simulations carried out on nanocrystalline materials indicate that the microstructure evolution in these materials takes place due to grain boundary migration and grain rotation. Therefore, migration of grain boundaries and grain rotation-induced grain coalescence contribute in increasing the average grain size in the microstructure. In order to simulate microstructure evolution in polycrystalline materials, the multi-order parameter phase-field model is a popular approach and is widely used for studying evolution purely due to the grain boundary migration. In this work, we present a multi-order parameter phase-field model capable of capturing microstructure evolution due to grain boundary migration and grain rotation-induced grain coalescence. The model couples constitutive equation of viscous sliding-induced grain rotation and the classical phase-field model for curvature-driven grain boundary migration. This paper covers various topological and statistical aspects of the microstructure evolved in the presence of both these growth mechanisms in great detail.

Parallel multiphase field simulations with OpenPhase

The open-source software project OpenPhase allows the three-dimensional simulation of microstructural evolution using the multiphase field method. The core modules of OpenPhase and their implementation as well as their parallelization for a distributed-memory setting are presented. Especially communication and load-balancing strategies are discussed. Synchronization points are avoided by an increased halo-size, i.e. additional layers of ghost cells, which allow multiple stencil operations without data exchange. Load-balancing is considered via graph-partitioning and sub-domain decomposition. Results are presented for performance benchmarks as well as for a variety of applications, e.g. grain growth in polycrystalline materials, including a large number of phase fields as well as Mg–Al alloy solidification. Program summaryProgram Title: OpenPhaseProgram Files doi:http://dx.doi.org/10.17632/2mnv2fvkkk.1Licensing provisions: GPLv3Programming language: C++Nature of problem: OpenPhase[1] allows the simulation of microstructure evolution during materials processing using the multiphase field method. In order to allow an arbitrary number of phase fields active parameter tracking is used, which can cause load imbalances in parallel computations.Solution method: OpenPhase solves the phase field equations using an explicit finite difference scheme. The parallel version of OpenPhase provides load-balancing using over-decomposition of the computational domain and graph-partitioning. Adaptive sub-domain sizes are used to minimize the computational overhead of the over-decomposition, while allowing appropriate load-balance.Additional comments including Restrictions and Unusual features: The distributed-memory parallelism in OpenPhase uses MPI. Shared-memory parallelism is implemented using OpenMP. The library uses C++11 features and therefore requires GCC version 4.7 or higher. [1] www.openphase.de

Simulation of abnormal grain growth in polycrystalline materials

Certain events of low probability that occur in material systems have a considerable impact on system characterization. Though rare event simulation has been a well-researched problem in areas like financial risk assessment and communication systems, modeling and simulation of rare events in material systems remain under-explored.In this paper, we turn to large deviations theory and importance sampling to develop a solution to simulate an important rare event that arises in polycrystalline materials. More specifically, the microstructure of a polycrystalline material consists of grains that have different orientations associated with them. These grains evolve over time and this phenomenon is called grain growth. The growth of grains is a slow process, making direct observation expensive and impractical. To alleviate this problem, computational methods have been developed to simulate grain growth. However, one event of interest which occurs with low probability involves a single grain that grows abnormally large at the expense of other grains. Though Gibbs distribution based models exist for such abnormal grain growth, occurrence of this event under such models is still rare enough that we still need to draw many samples before an abnormal growth manifests.We propose an importance sampling distribution from which to draw samples to simulate abnormal grain growth, instead of the conventional Gibbs distribution used to model grain growth. Our proposed importance sampling distribution is based on an asymptotically efficient rare event probability estimator. With our method, we consistently generate abnormal grain growth, thus providing a reliable solution to this important materials science problem. Our solution can potentially be used in general to simulate rare events in any system that is modeled by a Gibbs distribution.

A role of interfacial energy balance in delta to gamma transformation kinetics in carbon steel with dispersed second phase particles analyzed by phase-field simulation

It was reported in the authors’ previous study that second phase particles in δ phase induce a retardation effect on migration of δ–γ interface in carbon steels as is similar to the pinning effect on the grain growth in polycrystalline materials and effects of volume fraction and size of the particle were clarified. However, several other factors are expected to affect the magnitude of the retardation effect. In this study, the effect of the different interfacial energies is systematically investigated by means of two-dimensional phase-field simulations, and it is found that the magnitude of the retardation effect can be described by the parameter, Δσ=σPγ−σPδ where σPγ (σPδ) is the interfacial energy between the particle and γ (δ) phases. The large positive value of Δσ yields the strong pinning effect. In addition, it is shown that the δ–γ interface migration is accelerated when Δσ is large negative. These results originate from the increase in the total interfacial energy when Δσ>0 and the decrease when Δσ<0 during the interface passing through the particles. A degree of the retardation, Δa was summarized in the relation, Δa=m1(r/l2)(1+m2Δσ), where m1 and m2 are constants and r is radius of the particle and l is particle spacing.

2D finite element modeling of misorientation dependent anisotropic grain growth in polycrystalline materials: Level set versus multi-phase-field method

Different full field models exist for the simulation of anisotropic grain growth in polycrystalline structures. In the present work, two full field methods i.e. level set (LS) and multi-phase-field (MPF) methods are implemented into finite element (FE) formulations. The model formulation and simulation results are compared on FE unstructured mesh for the two methods. Relationships between the parameters in both models representing same grain boundary properties are derived. Using equivalent model parameters, both methods can provide similar numerical accuracy in the case of different theoretical test of 2D anisotropic grain growth. In addition, two 2D large scale grain growth simulations are performed with respectively isotropic and anisotropic grain boundary energies by level set method to illustrate the performance of this full field approach.

Geometrical grounds of mean field solutions for normal grain growth

The classical mean field approach for normal grain growth in polycrystalline materials is revisited. We re-drive and study possible self-similar solutions and show that the grain size distribution can be determined only by the geometry of neighbouring grains for any given configuration. In three dimensions, it is shown that a single index 〈R〉2〈R2〉 can represent the geometrical characteristic of grains and has a one-to-one relationship with the mean field parameter γ. We reinvestigate the results of our recent phase-field study [Darvishi Kamachali R, Steinbach I. Acta Mater 2012;60:2719] in the light of new analytical results and found a value γ≈3.5–3.2 for the stable regime.

Unified Theoretical Framework for Polycrystalline Pattern Evolution

The rate of curvature-driven grain growth in polycrystalline materials is well known to be limited by interface dissipation. We show analytically and by simulations that, for systems forming modulated phases or nonequilibrium patterns with crystal ordering, growth is limited by bulk dissipation associated with lattice translation, which dramatically slows down grain coarsening. We also show that bulk dissipation is reduced by thermal noise and that this reduction leads to faster coarsening behavior dominated by interface dissipation for a high Peierls-Nabarro barrier to dislocation motion and high noise. Those results provide a unified theoretical framework for understanding and modeling polycrystalline pattern evolution in diverse systems over a broad range of length and time scales.

Grain growth and grain boundary dynamics in colloidal polycrystals

Grain boundary dynamics and grain growth play a pivotal role in the fabrication of functional polycrystalline materials. However, not much is known about the delicate interplay between various microscopic processes that drive grain boundary motion which eventually culminates in the desired grain morphology. Colloidal systems are ideally suited to bridge the gap between the microscopic and macroscopic processes underlying grain growth, since their dynamics can be followed in real space and real time with single-particle resolution. The present review aims at highlighting contributions from colloid experiments that have led to a holistic understanding of grain growth in polycrystalline materials.

Existence or nonexistence of thermal pinning effect in grain growth under temperature gradient

In order to clarify existence or nonexistence of the thermal pinning, viz., retardation of grain growth in polycrystalline materials due to steep temperature gradient, effects of the temperature gradient on the grain growth were systematically investigated by means of the phase-field simulations. It was found that the local grain growth is determined only by the local thermal history regardless of the temperature gradient, and hence, the temperature gradient does not directly influence the local grain growth. It was concluded that the thermal pinning does not exist when the grain boundary energy and mobility are isotropic, and the grain growth is driven only by the reduction of total grain boundary energy.

Effect of Matrix Size and Neighbour Sites on Microstructure Evolution in 3D Monte Carlo Grain Growth Simulation

Grain size is an important microstructural feature of materials; it contributes to its mechanical and physical properties. Quantitative information of three-dimensional topological evolution during the three-dimensional normal grain growth process is very rare. In this work, an isothermal Monte Carlo procedure is applied to study the kinetics of normal grain growth in three dimensions using the Monte Carlo technique. Monte Carlo simulation of the grain growth in polycrystalline materials employed maps the microstructure onto a discrete lattice. It consists in evaluation of potential energy of a lattice point. Therefore, the choice of the simulation parameters can affect the grain growth characteristics. We attempted to define specific simulation parameters for the three dimensional MC simulation of grain growth using theses lattices. The parameters include the choice of the lattice symmetry, the pixels number and the number of the nearest neighbor lattice sites in the calculation of the system energy. The goal of these computer simulations is to provide insights on the simulated structure and to analyze the grain growth kinetics for the 3D domain through several 2D cross-sections. Compact microstructures were developed and the first results show that the lattice symmetry and the neighbours number have a fundamental influence on the results of grain growth simulation. The saturation is too rapid with only the consideration of the 1st neighbours while acceptable similar evolutions are obtained for the 2nd or 3rd neighbours consideration. Grain size evolution obeys the power law kinetics for the 2D cross-section confirming the quasi-steady state of the 3D normal grain growth. The highest grain growth velocity and grain size were obtained at the cross-sections of the heart of microstructures. One other noted information is the existence of a minority of grains which continue to grow slowly leading up to the abnormal grain growth.

Homogeneous Abnormal Grain Growth in Polycrystalline Materials

The controversy about the start up of the abnormal growth as heterogeneously engendered in the microstructure or produced by peculiar continuous incubation process has still not been solved. In this work the statistical theory of grain growth has been applied to treat the case of grain growth in the presence of an homogeneously unstable Zener drag. By the simulations presented it will be shown as abnormal grain growth is a result out of a continuous and homogeneous process without requiring any heterogeneity in the microstructure presumed to give local advantages to some grains. The mechanism by which during an incubation period the preconditions for the unstable growth are built up in the microstructure is clarified and discussed. Moreover the peculiar shape of Grain Size Distribution (GSD) approaching the “structural instability” will be also analytically defined and compared with experimental results obtained in a grain oriented Silicon Iron just before the onset of abnormal grain growth.

Investigating the effects of grain boundary energy anisotropy and second-phase particles on grain growth using a phase-field model

A phase-field model was used to investigate the simultaneous effects of grain boundary energy anisotropy and the presence of second-phase particles on grain growth in polycrystalline materials. The system of grains with anisotropic grain boundary energies was constructed by considering models of low and high misorientation angles between adjacent grains. Systems without particles reached a steady state grain growth rate, and this rate decreased by including the grain boundary energy anisotropy. In addition, the presence of particles significantly altered the microstructures during grain growth. This study showed that for systems including particles, the critical average grain size to stop grain growth depends not only on the volume fraction and size of particles, but also on the grain boundary energy anisotropy.

Level set framework for the finite-element modelling of recrystallization and grain growth in polycrystalline materials

The paper describes a level set framework for the finite-element modelling of recrystallization in polycrystalline materials, including the nucleation, primary recrystallization and grain growth stages. A set of convection–reinitialization–diffusion equations is implemented within a finite-element formulation to describe boundary motion, according to a kinetic law relating the velocity of the boundary to a thermodynamic driving force. Different theoretical test cases illustrate the potential of the method for simulations of recrystallization and grain growth following large plastic deformation and/or considering complex microstructures.

Modeling grain growth and related phenomena with vertex dynamics

Grain growth is the simplest phenomena related to the evolution of a population of grains in crystalline materials. Some typical results obtained with vertex dynamics, a deterministic technique applied to the simulation of grain growth in polycrystalline materials, mostly in two dimensions (2D), are presented: (i) the dynamics of a population of grains interacting with various distributions of pinning obstacles; (ii) bulging as a possible recrystallization mechanism; and (iii) the influence of a confined geometry on grain growth as those found in electronic devices. Finally recent developments in 3D are presented.

Phase field modeling of grain growth: effect of boundary thickness, triple junctions, misorientation, and anisotropy

Phase-field models based on multiple order parameters are used extensively to study grain growth in polycrystalline materials. However, if simulations are to be carried out using experimentally obtained microstructures as the initial condition, and the resultant microstructures are to be carefully compared with those obtained from experiments, then the parameters used in the numerical simulations need to be benchmarked with analytical solutions. Furthermore, the models themselves need to be modified to incorporate the dependence of grain boundary energy on misorientation across the boundary as well as the anisotropy in the boundary energy for any given misorientation that stems from the planes of different grains that make up the boundary. In this article, we address both these issues and present some preliminary results from our 2D and 3D simulations.