Dana Zöllner

Grain Size Distributions in Normal Grain Growth

Abstract An analytical distribution function characterising the grain sizes of polycrystalline microstructures is presented. Contrary to standard mathematical probability functions that are still often used for description of experimentally obtained size distributions, this one is based on a statistical mean-field theory of grain growth and is fully consistent with the fundamental physical conditions of total-volume conservation and the existence of a finite average grain volume. It is found that this distribution function describes the grain size distribution obtained by Monte Carlo Potts model simulations better than standard mathematical distributions. Additionally, two-dimensional plane sections from the simulated three-dimensional grain structures are considered and compared with experimental data, and the analytic size distribution function is also compared with an experimental grain size distribution for pure iron obtained by serial sectioning.

Computer Simulations and Statistical Theory of Normal Grain Growth in Two and Three Dimensions

A modified Monte Carlo algorithm for single-phase normal grain growth is presented, which allows one to simulate the time development of the microstructure of very large grain ensembles in two and three dimensions. The emphasis of the present work lies on the investigation of the interrelation between the local geometric properties of the grain network and the grain size distribution in the quasi-stationary self-similar growth regime. It is found that the topological size correlations between neighbouring grains and the resulting average statistical growth law both in two and three dimensions deviate strongly from the assumptions underlying the classical Lifshitz- Sloyzov-Hillert theory. The average local geometric properties of the simulated grain structures are used in a statistical mean-field theory to calculate the grain size distribution functions analytically. By comparison of the theoretical results with the simulated grain size distributions it is shown how far normal grain growth in two and three dimensions can successfully be described by a mean-field theory and how stochastic fluctuations in the average growth law must be taken into account.

Normal Grain Growth: Monte Carlo Potts Model Simulation and Mean-Field Theory

Grain growth in polycrystals is modelled using an improved Monte Carlo Potts model algorithm. By extensive simulation of three-dimensional normal grain growth it is shown that the simulated microstructure reaches a quasi-stationary self-similar coarsening state, where especially the growth of grains can be described by an average self-similar growth law, which depends only on the number of faces described by a square-root law. Together with topological considerations a non-linear effective growth law results. A generalized analytic mean-field theory based on the growth law yields a scaled grain size distribution function that is in excellent agreement with the simulation results. Additionally, a comparison of simulation and theory with experimental results is performed.

The Kinetics of Individual Grains in Polycrystalline Materials

Abstract Based on a mean-field approach a 3D grain growth model is presented which predicts the growth history of individual grains of a grain ensemble in terms of linear grain size and time. The analytical results are compared with the results of numerical simulations using the Monte-Carlo-Potts model. In addition, based on a stochastic model of grain growth, the diffusivity of the very discontinuous movements of individual quadruple and triple junctions is set in relation to the rate of growth. This allows the calculation of the average growth law of an ensemble of grains solely from the measurement of the stochastic growth kinetics of individual grains. The results from simulation and theory are compared with experimental results of in-situ scanning electron microscopy observations of grain growth in polycrystalline metals.

Monte Carlo Simulation of Normal Grain Growth in Three Dimensions

A Monte Carlo algorithm for single-phase normal grain growth has been implemented, which allows one to simulate and observe the temporal development of large grain microstructures in three dimensions. The relaxation process to the self-similar coarsening regime has been studied by following the temporal development of quantities like the average grain size, the standard deviation of the grain sizes and topological correlations.

Topology of grain microstructures in two dimensions: a comparison of grain boundary and triple junction controlled grain growth

The topology of polycrystalline grain microstructures is compared for the cases of normal, boundary controlled grain growth and triple junction controlled grain growth as it may occur in nanocrystalline materials. To this end, Monte Carlo Potts model simulations have been performed in two dimensions. Independent of the microstructural feature (respectively, its mobility controlling the evolution) growth regimes with topological self-similarity can be observed. In particular, it is found that each growth regime is characterized by its own distinct topology, enabling a deduction of the associated growth kinetics. The results are in very good agreement with theoretical predictions and computer simulations found in the literature.

Self-Similarity as a Feature of Nanocrystalline Grain Growth

In the present work it is revealed by modified Potts model simulations and theoretical considerations that self-similarity is a feature of junction controlled grain growth as it can be found in nanocrystalline materials. To this aim the influence of the grain junctions – boundary faces, triple lines and quadruple points – on grain growth is analyzed by attributing each type of junction an own specific energy and mobility yielding nine types of growth kinetics, each characterized by a self-similar scaling form of the growth law and a corresponding self-similar grain size distribution.

Grain Size Distributions and Evolution Equations in Nanocrystalline Grain Growth

Size effects observed in nanocrystalline grain growth are modeled by attributing each type of grain boundary junction an own specific energy and finite mobility. By considering grain growth as a dissipative process that is driven by the reduction of the Gibbs free interface and junction energy a general grain evolution equation is derived that separates into nine types of possible growth kinetics. The corresponding self-similar grain size distributions are derived and compared with results from modified Monte Carlo Potts model simulations taking into account size effects in triple and quadruple junction limited grain growth.

Studying the influence of triple junction energy and mobility on annealing processes

The kinetics of grain growth in nanocrystalline two-dimensional polycrystals and thin films can be controlled by the triple junctions of the microstructure. In the present work the influence of energy and mobility of grain boundaries and their junctions on annealing is investigated by analytic theories and modified Monte Carlo Potts model simulations. To that aim each structural feature of a polygonal grain is assigned its own specific energy and finite mobility, which results in four different limiting cases of self-similar growth kinetics each characterized by its own metrical and topological properties.

Monte Carlo Potts Model Simulation and Statistical Theory of 3D Grain Growth

An improved Monte Carlo (MC) Potts model algorithm has been implemented allowing an extensive simulation of three-dimensional (3D) normal grain growth. It is shown that the simulated microstructure reaches a quasi-stationary state, where the growth of grains can be described by an average self-similar volumetric rate of change, which depends only on the relative grain size. Based on a quadratic approximation of the volumetric rate of change a generalized analytic mean-field theory yields a scaled grain size distribution function that is in excellent agreement with the simulation results.