Theory Of Normal Grain Growth Research Articles

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Simple mean-field deterministic theory of normal grain growth in both 3-dimension and 2-dimension cases is presented, predicting the Rayleigh size distribution. The main idea is an application of standard thermodynamics to normalized size space with unit length proportional to average size. In the normalized space, the change of normalized free energy caused by the size change of an arbitrary grain is independent of the reservoir, which consists of all the other grains. Such “decoupling” enables us to use the original Burke-Turnbull approach to relate driving force to velocity of grain growth in the normalized space.

A mean field theory of grain growth: I

The initial part of a theory of normal grain growth previously published by Saetre (2002 Acta Mater. 50 1539) contained a minimum grain size of uncertain physical interpretation. In this work, the previous derivation is modified in order to eliminate the minimum grain size. The present theory thus yields a quasi-stationary state size distribution function similar to the classical Hillert distribution in the range u ϵ [0, umax), where u is the relative grain size. This theory which, under certain conditions, is valid both for two and three dimensions, is compared to the Hillert theory and to experimental data. It is concluded that the present theory yields a size distribution, which is similar, although not identical, to the Hillert distribution. Hence, the present version of the theory fails to represent the experimental distribution in the same manner as the Hillert theory. However, the strength of the present theory is that it can be further modified, as will be shown in the accompanying paper, which is the third part of the theory. The accompanying paper offers a modification of the theory, which produces a much-improved agreement with experimental data.

A mean field theory of grain growth: II

The initial part of a theory of normal grain growth previously published by Saetre (2002 Acta Mater. 50 1539) contained a minimum grain size of uncertain physical interpretation. In the accompanying paper to the present one, the theory has been improved by eliminating the constraint of a minimum size class and thus enabling the range u ϵ [0, umax), where u is the relative grain size. It has been shown that the resulting size distribution skews to the right in a manner similar to that of the Hillert distribution. The theory is further developed in this paper. Instead of assuming that all the grains in the ensemble have contiguous neighbouring grains of the same size, the present theory allows the contiguous grains to have sizes that vary as a function of the grain size. In principle, the smaller grains can be surrounded by larger grains and the larger grains can be surrounded by smaller grains, or vice versa. The distances between the triple junctions determine the neighbourhood of a grain. It is shown that a number of distribution functions exist that meet the Lifshitz and Slyozov quasi-stationary state criteria. Theoretical grain size distribution functions are compared with an experimental distribution found in the literature, and the theoretical distribution with the best agreement with the experimental data has been determined. The best theoretical distribution is skewed to the left and can account for major parts of the experimental distribution, but the tail towards the larger grain sizes approaches zero faster than does the experimental distribution.

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.

Curvilinear polygons, finite circle packings, and normal grain growth

A key element of analytical models for curvature-driven grain growth is the formulation of a growth law for individual grains. Grain size distributions can be derived through the use of such growth laws. The present paper reviews some recent progress in developing a mean-field statistical theory of normal grain growth. A paradox related to curvilinear polygons is shown to support the expectation that the grain size distribution has a finite cutoff. The circumference of the circumscribing circle and the perimeter of the curvilinear polygon become exactly equal when the number of sides is 12. The boundary of an average 12-sided grain has an integral curvature that is equal in magnitude to the integral boundary of a completely circular grain. For this reason, grains with more than 12 sides are suggested to be energetically unfavourable, a result which is in excellent agreement with simulation data. In the concluding part, finite circle packings are employed to interpret a topological relationship found for Potts-model-type simulations of 2D grain growth. It is briefly indicated how spherical packings may help to develop some intuition about topological relationships found for normal grain growth in three dimensions.

Grain growth studies of thin Cu(In, Ga)Se 2 films

In order to analyze grain growth on the basis of a kinetic growth model, this study reports on annealing experiments with thin Cu(In, Ga)Se 2 films. The substrate heater in an ultra-high vacuum chamber serves to anneal Cu(In, Ga)Se 2 films of 1 μm thickness for varying times t and temperatures T. An ultra-high resolution scanning electron microscope visualizes the cross-section of the film in order to determine the average grain radius r. The theory of normal grain growth explains the increase of the average grain radius r in dependence on annealing time t and temperature T. For an annealing time t=3 h at T=610°C the average grain size increases by a factor of 2.4. Grains with a (1 1 2)-surface orientation are slightly energetically preferred and grow at the expense of other orientations. Grain growth is thermally activated. The Cu-content of the film determines the activation energy Q for grain boundary motion. Increasing the Cu-content of the film from 17.9% to 25.7% lowers Q from 3.5 to 3.0 eV.For high-efficiency Cu(In, Ga)Se 2 solar cell absorbers with a Cu-content below 22% the temperature T has to exceed at least 550°C in order to obtain sufficient atomic diffusion and therefore significant grain growth.

Theory of normal grain growth in normalized size space

Simple mean-field deterministic theory of normal grain growth in both 3-dimension and 2-dimension cases is presented, predicting the Rayleigh size distribution. The main idea is an application of standard thermodynamics to normalized size space with unit length proportional to average size. In the normalized space, the change of normalized free energy caused by the size change of an arbitrary grain is independent of the reservoir, which consists of all the other grains. Such “decoupling” enables us to use the original Burke-Turnbull approach to relate driving force to velocity of grain growth in the normalized space.

On the theory of normal grain growth in two dimensions

A two-dimensional theory of grain growth has been derived based on the concept of curvature-driven migration of the grain boundaries. Grains with six neighbouring grains are of the critical grain size and will neither shrink nor grow. The rate equation is thus in accordance with the von Neumann–Mullins law. An analytical expression for the steady state distribution has been derived based on the same principles as Hillert's theory. The present theory contains both a minimum and a maximum grain size.

On the theory of normal grain growth

Abstract The theory of normal grain growth that invokes random walk is further extended by allowing for biases in step direction. It is concluded that the predictions of this theory are in overall quantitative accord with experiment. However, the degree of agreement between theory and experiment as to the exact form of the grain size distribution function remains in doubt because of a lack of precision in the definition and measurement of grain size. Possibly, this difficulty may be surmounted by replacing measurements of grain size by those of the lengths of grain boundary edges as found in planar sections of polycrystals.

A note on the stochastic theory of normal grain growth

The treatment of grain growth as a diffusion-drift process in time size space has recently received renewed attention. In the present paper we emphasize that the grains move in only one dimension in time-size space, while in the diffusion-drift model introduced by Pande and Dantsker [ Acta metall. mater. 38, 945 (1990)], the grains are allowed to move in 1, 2 and 3 dimensions. We also demonstrate that their theory predicts grain size distribution functions that do not satisfy all the requirements normally attributed to such functions.

The scaling theory of normal grain growth: Beyond the mean-field approximation

Computer simulations of domain coarsening have given excellent agreement with patterns observed experimentally in grain-boundary networks. In contrast, the statistical distribution functions predic...

A statistical theory of grain growth

By extending the theory of Kurtz and Carpay, a detailed statistical theory of normal grain growth has been developed. The theory utilizes their concept of multiple distributions, i.e. the division of grains into topological classes (14 planar, 34 spatial) but unlike their treatment it does not require the individual distributions of each class to be lognormal. An alternate multiple distribution of grain sizes, where the grains are classified according to a given value of size, rather than a given value of sides, or faces is also proposed. The overall distributions (both in grain shapes and sizes) are then obtained by summing the individual distributions, and are shown to be approximately lognormal. Alternate origins of lognormality are also discussed.

A stochastic theory of grain growth

A stochastic theory of normal grain growth is proposed. The model is based on the concept that the migration of kinks and ledges should cause a Brownian motion of the grain boundary. This motion results in a drift of the grain size distribution to larger sizes. The kinetics of grain growth is thus related to the kinetics of kinks and ledges; specifically, via the rates of nucleation, recombination and sink annihilation. A variety of growth exponents are obtained from a scaling analysis, but only one universal grain size distribution is applicable in all cases. The specific predictions of this model are in total agreement with the recent computer simulations of domain growth, and are consistent with experimental observations of normal grain growth.

Electron microscopic examination of multifilamentary bronze-processed Nb3Sn

A detailed examination of the grain structure of commercial multifilamentary N3Sn-wire has been carried out. This examination as well as the various electron microscopic preparation techniques used are discussed. Ion-milling was found to be a suitable method especially when combined with electrolytic pre-thinning. This yielded almost 100% acceptable samples. The grain size of the samples examined ranged from 30 to 200 nm. Based on the theory of normal grain growth an analytical description of grain growth versus reaction time,t R, and reaction temperature,T R, was determined and thus important parameters such as apparent activation energy, nucleation grain size and ultimate grain size of grain growth of Nb3Sn were obtained. The grain structure itself was found to be free of dislocations or other distortions whereas outstanding grain growth textures were evident, resulting in a certain grain size distribution over the Nb3Sn area. Strong evidence of abnormal grain growth of Nb3Sn grains was found depending on the heat treatment, for which some possible mechanisms are discussed. Based on the results of the analytical description of grain growth versus reaction conditions and an examination of the corresponding superconducting properties a “two-stage” annealing process is proposed which hypothetically should result in improved superconducting properties in the high-field range. Initial orientation examinations seem to confirm this hypothesis.

Microstructure and normal grain growth in metals and ceramics. Part I. Theory

By extending the theory of Feltham and combining it with the work of Rhines and Craig, a detailed statistical theory of normal grain growth has been constructed. The theory exhibits all four attributes of normal grain growth: uniformity, scaling, stability, and lognormality. A prime new feature of the theory is the division of the grains into topological classes (14 planar, 34 spatial), each with a lognormal distribution of grain sizes. Growth is found to be controlled by the rate of loss of grains from the lowest topological class. Complete solutions are found for the grain growth kinetics of each class, as well as the transfer rates between classes. The latter result is used to explain how the median diameter of those classes in which grains are shrinking still manages to increase in the manner required to keep their number a constant fraction of the total population. A parabolic growth law is found for the median grain size of the whole population as well as the median grain size in each topological class. The growth constant for each class is found to increase approximately as the cube of the planar topological parameter or the square of the spatial topological parameter. The Rhines-Craig structural gradient is shown to be independent of time and hence a basic constant of normal grain growth. Stability is due to a maximum in the grain boundary velocity with increasing grain size. The ratio of the maximum to median grain diameter is found to be e(=2.718). A comparison of the present theory is made with that of Hillert. Possible origins of the lognormality are discussed.

Microstructure and normal grain growth in metals and ceramics. Part II. Experiment

An extensive set of measurements on the grain growth and microstructure evolution of a series of high-density Ni-Zn ferrites annealed over periods ranging from 1/4 to 48 h is reported. A comparison of the data with predictions of the statistical multiple-lognormal theory of normal grain growth presented in Part I is made. The time invariance of the standard deviations of both the grain size and population of individual topological classes is confirmed, demonstrating that the scaling and stability of normal grain growth are critically related to the dispersion of the grain sizes and that at least two parameters are needed to characterize a given microstructure. A parabolic growth law for the median diameter in both two and three dimensions is confirmed as well as the predicted power law dependence of the growth constants for each class on the topological index. A series of sectioning experiments verifies the near equivalence of the planar and spatial size parameters.

Antiphase domain growth in Cu 3Au: Quantitative comparison between theory and experiment

The Hillert theory of normal grain growth has been modified for antiphase domain (APD) growth in Cu 3Au. A semi-empirical correction factor, β, is introduced into the theory in order to incorporate the effect of the extreme breadth of the experimentally observed APD size distributions on the rate of APD growth. The mobility. M, of APD boundaries in an ordered alloy, which is proportional to the chemical diffusion coefficient, \\ ̄ tD, is shown to be inversely proportional to the long-range order parameter, S. The theory is compared with literature data on APD growth in Cu 3Au, as well as our own measurements made at 325, 350 and 375°C for times up to 10 h on pure Cu 3Au and Cu 3Au-Co alloys containing 1.5 and 2.5 at.%Co. Using estimated values of \\ ̄ tD, measured values of S, values of β determined by fitting the theory to existing APD distributions and another parameter 〈 u〉 which is a measure of the relative values of the average and ‘critical’ APD sizes, the experimentally measured rate constants for APD growth were used to calculate values of the APD boundary interfacial energy, σ, for comparison with recently derived theoretical values. The quantitative agreement between theory and experiment is satisfactory, considering the uncertainty in \\ ̄ tD. The theory also predicts that APD growth should increase in the Co-containing alloys. Such an effect is observed experimentally, but is greater in magnitude than predicted by the theory, for reasons that are not understood.

On the theory of normal grain growth

Grain growth is analysed on the basis that the necessary boundary motion is random in its essentials. This analysis leads to a growth rate proportional to the reciprocal of the square root of time. It also leads to a common distribution function for grain diameters and grain edges. This distribution accords well with experiment.