Grain Growth In Polycrystalline Materials Research Articles

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.

Modeling the topological features during grain growth by cellular automaton

A two dimensional cellular automaton model, with advanced features for continuous tracking and precise estimation of curvature of grain boundary, has been developed to simulate and study the topological and kinetic features of grain growth in polycrystalline materials. The model simulations exhibited experimentally observed topological arrangements of grains as well as validated the two independently derived grain growth kinetics models, namely, Burke and Turnbull’s parabolic law and von Neumann–Mullins law for the growth rate of individual grains. Furthermore, the model was used to study the growth kinetics in graded and columnar microstructures. The growth kinetics was found to depend on the extent of gradation and the aspect ratio of columnar grains. It was observed that both the graded and columnar microstructures evolve towards the equiaxed microstructure during the grain growth.

Phase field study of the effect of grain boundary energy anisotropy on grain growth

Abstract We study the effect of grain boundary energy anisotropy on grain growth in polycrystalline materials using the multiphase field approach. In contrast to previous studies, the energy anisotropy here is not restricted to low misorientation angles between adjacent grains. Phase field simulations of grain growth in systems require a large number of order parameters to describe the individual grains. The concomitant computational expense restricted studies to small numbers of order parameters and, implicitly, the misorientation angles to a small finite number of discrete values. The treatment of large number of order parameters and continuous misorientation angles is made possible here using the active parameter tracking algorithm [S. Vedantam, B.S.V. Patnaik, Phys. Rev. E 73 (2006) 016703]. The high angle grain boundary energy is taken to be given by an extension of the Read–Shockley energy for low angle grain boundaries [D. Wolf, Scripta Metall. 23 (1989) 1713]. It is found that while the Read–Shockley anisotropic grain boundary energy decreases the growth rate over the isotropic case, the extended Read–Shockley energy system has significantly lower the growth rates.

Three‐dimensional phase field simulations of grain growth in materials containing finely dispersed second‐phase particles

AbstractPhase field modelling has proven to be a versatile tool for simulating microstructural evolution phenomena, such as grain growth in polycrystalline materials. However, the computational requirements of a phase field model impose strong limitations on the number of phase field variables employed in a practical implementation. In this paper, a bounding box algorithm is proposed allowing the use of a large number of phase field variables without excessive computational requirements. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Bounding box algorithm for three-dimensional phase-field simulations of microstructural evolution in polycrystalline materials

Phase-field modeling has proven to be a versatile tool for simulating microstructural evolution phenomena, such as grain growth in polycrystalline materials. However, the computing time and computing memory requirements of a phase-field model pose severe limitations on the number of phase-field variables that can be taken into account in a practical implementation. In this paper, a sparse bounding box algorithm is proposed that allows the use of a large number of phase-field variables without excessive memory usage or computational requirements. The algorithm is applied to a three-dimensional model for grain growth in the presence of second-phase particles.

Rate of Change of Boundary Area in Coarsening Three Dimensional Cellular Structures

Coarsening cellular structures serve as models of grain growth in polycrystalline materials. Based on the assumption that growth is driven by the grain boundary curvature, we obtain a general formula for the rate of change of boundary area. The formula allows for deriving a new relationship for the rate of area change in individual cells with spherical-cap faces. In the case of 'regular' cells, the rate is a function of the number of faces, i.e., it is given by a von Neumann-type relationship. We also apply the general formula to whole structures with arbitrary cells. In this general instance, the rate of total area reduction is expressed via structure topological parameters and one geometric entity (the average difference between principal curvatures of faces). Moreover, assuming self-similarity of the structure, the dependence of some structure parameters (total area, number of cells per unit volume, average radius) on time is obtained. The formula for the rate of area change leads to parabolic law of growth of self-similar structures.

A 3-D Monte-Carlo (Potts) model for recrystallization and grain growth in polycrystalline materials

The evolution of microstructure during annealing of deformed polycrystalline metallic materials was modeled using a 3-D Monte-Carlo (Potts) approach in which the initial microstructure, texture (ODF), spatial distribution of the stored energy of deformation, and nucleation mechanism were carefully quantified . The modeling technique was tested using several special cases of 3-D stored-energy distributions and different textures of the as-deformed state.

Monte Carlo simulation of grain growth in polycrystalline materials

Abstract Understanding the kinetics of grain growth, under the influence of second phase (such as impurities, voids and bubbles) is fundamental to advances in the control of microstructural evolution. As a precursor to this objective, we have investigated the grain growth kinetics in a polycrystalline material using a standard Q -state Potts’ model under Monte Carlo settings. Based on physical reasoning, new modifications are suggested to circumvent some of the disadvantages in the basic Potts model. The efficacy of these modifications vis-a-vis the basic model is verified. The influence of second phase particles on the impurity loaded grain boundaries is investigated for the study of grain growth kinetics.

A variational formulation and a double-grid method for meso-scale modeling of stressed grain growth in polycrystalline materials

A deterministic approach for meso-scale modeling of grain growth in stressed polycrystalline materials based on the principle of virtual power is presented. The variational equation is formulated based on the power balance of the system associated with grain boundary surface tension and curvature, rate of strain energy stored in each grain, strain energy density jump across the grain boundaries, and external work rate. The numerical solution of stressed grain growth variational equation requires discretization of grain interiors and grain boundaries. This cannot be effectively modeled by Lagrangian, Eulerian, or Arbitrary Lagrangian Eulerian finite element method if grain boundary migration (moving interfaces) and topological changes of grain boundary geometry are considered. This paper presents a double-grid method to resolve the above mentioned difficulty. In this approach, the material grid points carry material kinematic variables, whereas the grain boundary grid points carry grain boundary kinematic variables. The material domain is discretized by a moving least squares reproducing kernel approximation with strain discontinuity enrichment across the grain boundaries. The grain boundaries, on the other hand, are discretized by the standard finite elements. An interface enrichment function to accurately capture strain jump conditions across the grain boundaries is introduced. A reproducing kernel approximation that includes the periodicity of the unit cell in the construction of reproducing kernel shape function for material velocity is also presented. This proposed double-grid method allows modeling of arbitrary evolution of grain boundaries without remeshing.

Annihilation mechanisms in thin film grain growth

Grain growth in succinonitrile thin films is dominated by topological effects [ Scripta Metall. Mater. 30 (1994) 633; Metall. Mater. Trans. A 26 (1995) 1061; Philos. Mag. 79(4) (1999) 763]. The topological transformations are effected by the shortened length of the grain boundaries [ R.T. De Hoff, in: H. Weiland, B.L. Adams, A.D. Rollet, Grain Growth in Polycrystalline Materials III, TMS, 1998, p. 225; Acta Mater. 48 (1998) 5175]. To explain how these events occur in three dimensions each face of a grain is assumed to behave as predicted by Smith [ Metal Interf. 3 (1952) 65; Contemporary Phys. 25 (1984) 59], and placed in the context of work by Fortes and Ferro [ Acta Metall. 33 (1985) 1697; 1683].

結晶粒成長のコンピュータ・シミュレーションにおけるモビリティーの影響

Grain-boundary mobility, which plays a significant role during grain growth in polycrystalline materials, is related to activation energy for grain-boundary migration. From this aspect, the authors simulated grain growth considering mobility into an activated state accompanying reorientation. The mobility is affected by impurities in the activation process and is represented as a function of interaction factor between grain boundary and impurities. The authors confirmed that grain boundary migration is slower for a larger interaction factor. Assuming a constant driving pressure of the grain boundary, the mobility suggests Arrhenius type dependence of temperature. The direction of grain-boundary migration at a triple junction depends on not only the grain boundary energy but also the interaction factor. Therefore, equilibrium at the triple junction of grain boundaries should be established based on the grain-boundary energy compensated by the mobility.

Monte Carlo Simulation Algorithms of Grain Growth in Polycrystalline Materials

In the paper we present a variety of Monte Carlo algorithms, that can be employed to simulate the grain growth in polycrystalline materials during sintering. The simulation algorithms of monophase or two-phase structure, on both the square and triangular distribution of lattice points, possibly with the second phase particles being either of static or dynamic nature, are described. The paper deals with oriented and anisotropic grain growth as well. A considerable number of input parameters in the simulation procedure makes it possible to set up a large amount of combinations of conditions under which we want to simulate the sintering process.

Computer simulation of grain growth

A variety of computer simulation techniques have been developed to described grain growth in polycrystalline materials and the evolution of cellular patterns in other systems. Most techniques have now been developed to the degree that they can be used to analyze idealized microstructural evolution processes, and, in some cases, answer questions which arise from experimental observations or engineering goals. The development of crystallographic texture, the effect of dispersions of second phase particles, and the nature of grain structure evolution in thermally inhomogeneous environments have also attracted considerable interest.

Grain geometry evolution during grain growth in polycrystalline materials: Variation in the degree of grain size uniformity

This paper reports the results of grain growth studies in polycrystalline metals: Al, Fe-[alpha], Fe-[gamma] (316L type steel), and NiMn2 alloy, characterized by significantly different values of a stacking fault energy. Detailed descriptions of changes in the geometry of grains have been obtained for the specimens recrystallized and annealed in a wide range of temperatures. The results are used to suggest some general tendencies observed during grain growth in polycrystalline materials.

The number of contiguous neighbors of grains in polycrystalline materials

The stochastic field approach to grain growth in polycrystalline materials permits to combine the stochastic element of the process with physical mechanisms causing grain growth without violating volume conservation, connectivity, and continuity. However, a stochastic field analysis requires knowledge of the number of nearest neighbors of the grains. Based on the assumption that the number of neighbors is determined by their relative size, the present article derives a linear relationship in two dimensions and a parabolic relationship in three dimensions between the number of contiguous neighbors and the grain radius. The models are found to describe simulated 2D and 3D Voronoi polygons as well as the observed number of neighbors–grain size relationships in polycrystalline aluminum.

Grain growth as a stochastic process

The stochastic field approach describes grain growth in polycrystalline materials as a stochastic process in volume time space. A stochastic process represented by a Fokker-Planck equation in volume time space automatically conserves the volume of the specimen. The stochastic component of grain growth becomes proportional to the variance of the volume growth rate and to unit time. The present paper examines various stochastic models of grain growth in the light of the stochastic field approach.