Theory Of Grain Growth Research Articles

Topological Approach to Abnormal Grain Growth in three Dimensions

Theories of abnormal grain growth (AGG) in three dimensions usually approximate an abnormal grain by a sphere. The abnormal grain is then represented by its spherical equivalent grain radius. This study, by contrast, treats AGG in terms of concepts that include both the boundary curvature and the number of faces of the abnormal grain. We treat AGG for the case of pinned matrices, including the phenomena of initiation and growth kinetics. The influence of interfacial energy and mobility of the abnormal grain boundary are also discussed.

Abnormal Grain Growth and New Core–Shell Structure in (K,Na)NbO3‐Based Lead‐Free Piezoelectric Ceramics

A unique core–shell structure was observed in coarse grains in (K,Na)NbO3 (KNN)‐based lead‐free piezoelectric ceramics. It is morphologically different from the chemical inhomogeneity‐induced core–shell grain structure reported previously in BaTiO3‐based ceramics. The core region is composed of highly parallel nanosized subgrains, whereas the shell region consists of larger‐sized but similar self‐assembled subgrains. The electron‐backscattered diffraction analysis and selected area electron diffraction pattern confirmed that coarse grains with a core–shell structure were single‐crystalline‐like grains. The formation process of such coarse grains was then discussed based on mesocrystal growth along with the classical theory of grain growth. The two studied KNN‐based systems showed a similar grain growth transformation: from self‐assembled aggregation clusters with nanosized subgrains to a typical core–shell grain structure when the sintering temperature was increased only by a range of 10°–20°C. The volatilized alkali metal oxides and liquid phase were supposed to accelerate such grain growth transformation. When abnormal grown grains with a core–shell structure occurred, both systems showed the highest densities and dielectric constants along with the lowest dielectric losses, while their piezoelectric properties tended to decline.

Abnormal Grain Growth Topology and Kinetics

One common point amongst extant theories of abnormal grain growth (AGG) is that they treat this phenomenon in terms of the relative grain size, or grain radius, of the abnormal grains. Topological and metrical quantities of abnormal grains, such as the number of their faces, or their grain boundary curvature, are taken into account only indirectly through the grain size itself. This paper, by contrast, treats AGG in terms of concepts, that include both the boundary curvature and the number of faces of the abnormal grain. Two cases are examined: 1) AGG, in which the matrix grains are fully pinned, so normal grain growth cannot occur; 2) AGG in which the matrix grains are free to evolve, so that normal grain growth ensues simultaneously in the matrix.

Topological theory of abnormal grain growth

Theories of abnormal grain growth (AGG) treat this interesting phenomenon in terms of the relative grain size, or grain radius, of the abnormal grains. This study, by contrast, treats AGG in terms of concepts that include both the boundary curvature and the number of faces of the abnormal grain. First, we formulate a topological AGG criterion: if a “candidate” grain is undergoing abnormal growth, then its number of faces is increasing with time. AGG initiation in a pinned matrix is analyzed. An “AGG map” summarizes the results of this analysis. Finally, we derive analytical expressions for the number of faces as a function of time for AGG in a pinned matrix, and for AGG in a matrix that is free to undergo normal grain growth. Consideration of the topological features of polyhedral grains introduces new aspects important to our understanding of the kinetics of AGG.

The theory of anomalous grain growth in submicrocrystalline materials produced by severe plastic deformation

A new approach to describing anomalous grain growth based on the idea of transition of migrating grain boundaries into a highly nonequilibrium state is proposed. The effect of the grain-size scatter and the initial dislocation density on the evolution of this process has been considered. Conditions that are required for the realization of an anomalously rapid grain growth have been analyzed.

Effects of addition participation in the interfacial reaction on the growth patterns of Cu 6Sn 5-based IMCs during reflow process

The growth pattern of intermetallic compounds generated between Cu substrate and the eutectic Sn–3.5Ag doped with small amount of additives (0.1 mass%), say, Ni or Co, was explored in this paper. During reflow experiments, the dwell time at peak temperature 250 °C was set up for 1, 2, 5, 10, 20, 30 and 60 s, respectively, to investigate the intermetallic morphology growth behavior. The intermetallics formed between Cu substrate and three kinds of solders, namely, Sn–3.5Ag, Sn–3.5Ag–0.1Co and Sn–3.5Ag–0.1Ni, were all identified as Cu 6Sn 5-based, although the additives participated in the interfacial reaction. However, the rounded shape of Cu 6Sn 5 was observed, while the (Cu, Ni) 6Sn 5 or (Cu, Co) 6Sn 5 intermetallics were polyhedral shaped (or faceted). The presence of additives, namely Ni or Co, at the outer region of (Cu, Ni) 6Sn 5 or (Cu, Co) 6Sn 5 IMCs was responsible for the morphology transition, which might lead to the increase of enthalpy change and thus result in the Jackson's parameter larger than 2. Another attractive result was the evolution of grain size distribution. For the grains of typical Cu 6Sn 5 intermetallic, the size distribution tended to be more narrow following with the extended reflow time, although the number of intermetallic was reduced. However, for the (Cu, Ni) 6Sn 5 or (Cu, Co) 6Sn 5 intermetallics, the grain size distribution became wider and spread out. Based on the basic theory of grain growth, the AGG (abnormal grain growth) pattern was suggested for the (Cu, Ni) 6Sn 5 or (Cu, Co) 6Sn 5 intermetallic growth, while Cu 6Sn 5 IMC exhibited the normal grain growth.

Grain microstructure evolution and grain boundary junction engineering

The grain microstructure evolution in the course of two dimensional (2D) grain growth is considered in greater detail, taking into account the influence of grain boundary triple junctions. It is shown that there are two limiting regimes of grain growth in polycrystals: the first one is associated with the situation when the kinetics of grain growth are controlled by the motion of grain boundaries, while the second one is defined by the motion of grain boundary triple junctions, i.e. when the mobility of triple junctions determines the kinetics of grain growth. A generalised theory of 2D grain growth including a limited triple junction mobility is presented. The theoretical predictions are compared with results of computer simulations by a virtual vertex model. We introduce a new branch of grain boundary engineering, namely, grain boundary junction engineering that utilises junction properties for microstructure control.

Abnormal Grain Growth and Texture Development

A theory for abnormal grain growth (AGG) in polycrystalline materials is revisited and extended in order to predict AGG in textured polycrystals. The motivation for the work is to improve our understanding of the origins of the Goss texture component, {110}<001>, during annealing of Fe-Si sheet. Since the AGG phenomenon in grain-oriented electrical steels is known to be dependent on the presence of a dispersion of fine second phase particles, the grain boundary properties are treated as representative of the homogenized behavior of the material, and not necessarily the properties that would be measured directly. The predictions of AGG are presented in the form of maps in Euler space, showing which texture components are most likely to grow abnormally. For different models of grain boundary properties applied to a theoretically derived texture, different sets of texture components are predicted to grow; neither model, however, predicts growth of the Goss component.

Triple junction motion and grain microstructure evolution

The classical concepts of grain growth in polycrystals are based on the dominant role of grain boundaries. This is reflected by the well known von Neumann–Mullins relation. According to this approach triple junctions do not affect grain boundary motion, and their role in grain growth is reduced to maintaining the thermodynamically prescribed equilibrium angles at the lines where boundaries meet. In the current study the experimental data of triple junction mobility are considered with respect to the process of grain growth in 2D systems, in particular with regard to the controlling kinetics. When boundary kinetics prevails grain growth in a polycrystal complies with the von Neumann–Mullins relation. When grain growth is governed by the mobility of triple junctions the kinetics change, and the von Neumann–Mullins relation does not hold anymore. This is the more pronounced the smaller the triple junction mobility. We present a generalized theory of 2D grain growth including a limited triple junction mobility. In this concept the criterion Λ plays a central role. It reflects the ratio of boundary to triple junction mobility but is proportional to the grain size as well. The generalized von Neumann–Mullins relation can be expressed in terms of Λ. For small values of Λ, conspicuous changes of microstructure evolution during grain growth and of microstructural stability are predicted. The theoretical predictions are compared to results of computer simulations by a virtual vertex model.

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.

The Correlation Theory of 2D Normal Grain Growth

Computer simulations of 2D normal grain growth have shown that size correlations between adjacent grains exist in 2D grain structures. These correlations prevail during the coarsening process and influence on the kinetics of the process and on the grain size distribution. Hillert’s analysis starts with the assumption that all grains in the structure have the same environment. Since computer simulations contradict this assumption, the mean-field theory for normal grain growth needs to be modified. A first attempt was made by Hunderi and Ryum, who modified Hillert’s growth law to include the effect of spatial grain size correlations. In the 1D case the distributions derived by means of the modified growth law agreed well with simulation data. However, the distribution derived for 2D grain growth retained unwanted properties of the Hillert distribution. We review some recent progress in developing a mean-field statistical theory. A paradox related to curvilinear polygons is shown to support the expectation that the grain size distribution has a finite cutoff.

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.

Effect of Sintering Cooling Rate on V Segregation Amount at WC/Co Interface in VC-doped WC-Co Fine-Grained Hardmetal

In other researchers' studies by HRTEM observation and XMA analysis on VC-doped WC-Co fine-grained hardmetal, the phenomenon that V segregated at total WC(0001)/Co interface was reported and such segregation itself was suggested to be the direct cause for V inhibition effect on WC grain growth. This suggestion, however, seemed to be inadequate, because it is not clear whether this segregation was generated by an equilibrium adsorption mechanism during the temperature-raising-stage and/or temperature keeping-stage of sintering, i.e., during the socalled grain growth-stages, or by a heterogeneous precipitation (hetero-epitaxial nucleation and growth) mechanism during the cooling-stage after the temperature keeping-stage.In this study, the effect of the cooling rate on the amount of V segregation was investigated to solve the above problem. It was found that the amount depended on each WC grain even at a certain cooling rate and the arithmetic average amount became smaller with increasing cooling rate. These findings, a consideration on the basis of the general theory of grain growth in multi-phase system, etc., suggested the following. All of the V segregation at total WC(0001)/Co interface is not necessarily generated by an equilibrium adsorption mechanism, but mainly by the heterogeneous precipitation mechanism. Therefore, such V segregation itself should not be directly correlated to the VC inhibition effect on WC grain growth. The segregation, etc., however, suggested that V atoms adsorb and desorb intermittently at the step and/or kinks, i.e., WC growth site, of WC(0001) crystal surface and thus WC grain growth which occurs according to Ostwald-ripening mechanism is inhibited during the V adsorption and desorption.

Dust in the Early Universe: Dust Formation in the Ejecta of Population III Supernovae

Dust grains play a crucial role in the formation and evolution history of stars and galaxies in the early universe. We investigate the formation of dust grains in the ejecta of Population III supernovae, including pair-instability supernovae, which are expected to occur in the early universe, applying a theory of non-steady state nucleation and grain growth. Dust formation calculations are performed for core-collapse supernovae with progenitor mass Mpr ranging from 13 to 30 M☉ and for pair-instability supernovae with Mpr = 170 and 200 M☉. In the calculations, the time evolution of gas temperature in the ejecta, which strongly affects the number density and size of newly formed grains, is calculated by solving the radiative transfer equation, taking account of the energy deposition of radioactive elements. Two extreme cases are considered for the elemental composition in the ejecta, unmixed and uniformly mixed cases within the He core, and formation of CO and SiO molecules is assumed to be complete. The results of calculations for core-collapse supernovae and pair-instability supernovae are summarized as follows: in the unmixed ejecta, a variety of grain species condense, reflecting the difference of the elemental composition at the formation site in the ejecta; otherwise only oxide grains condense in the uniformly mixed ejecta. The average size of newly formed grains spans a range of 3 orders of magnitude, depending on the grain species and the formation condition, and the maximum radius is limited to less than 1 μm, which does not depend on the progenitor mass. The size distribution function of each grain species is approximately lognormal, except for Mg silicates, MgO, Si, and FeS in the unmixed case and Al2O3 in both cases. The size distribution function summed up over all grain species is approximated by a power-law formula whose index is -3.5 for the larger radius and -2.5 for the smaller one; the radius at the crossover point ranges from 0.004 to 0.1 μm, depending on the model of supernovae. The fraction of mass locked into dust grains increases with increasing the progenitor mass: 2%-5% of the progenitor mass for core-collapse supernovae and 15%-30% for pair-instability supernovae whose progenitor mass ranges from 140 to 260 M☉. Thus, if very massive stars populated the first generation of stars (Population III stars), a large amount of dust grains would be produced in the early universe. We also discuss the dependence of the explosion energy and the amount of 56Ni in the ejecta, as well as the efficiency of formation of CO and SiO molecules, on the formation of dust grains in the ejecta of supernovae.

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.

Theory of abnormal grain growth in thin films and analysis of energy anisotropy

A theoretical model is proposed for the abnormal grain growth in thin films of columnar grains. In addition to the grain boundary energy usually considered for the grain growth in bulk materials, the influences of surface energy, interface energy and strain energy are also considered in the model. A reviewed analysis is given out for the anisotropy of energy. For fcc and bcc metal films, surface energy minimization generally favours (111) and (110) textures, while strain energy minimization generally favours (110) and (100) textures.

Ice crystals grown from vapor onto an orientated substrate: application to snow depth-hoar development and gas inclusions in lake ice

AbstractA laboratory experiment was conducted in which new ice crystals were nucleated from the vapor phase onto large existing ice crystals obtained from Antarctic lake ice. Flat, smooth ice-crystal surfaces were prepared, with c axes oriented either vertically or horizontally. When these were subjected to a supersaturated vapor environment, multiple individual crystals nucleated onto the substrates adopting the same crystallographic orientation as the parent. A dominant grain-growth scenario for kinetic-growth metamorphism in snow, which in some ways is analogous to the oriented morphologies in lake ice, is hypothesized. In the lake-ice-growth scenario, optimally oriented crystals will grow at the expense of those less preferentially positioned.The proposed dominant grain-growth theory for snow is in agreement with the observed decrease in the number of grains and the proximal similarity of crystal habit in kinetic-growth metamorphism in snow. Similarly, kinetic crystal growth on the interior of gas inclusions in Antarctic lake ice will also acquire the crystallographic orientation of the substrate ice. These small-faceted interior crystals significantly influence light scattering and penetration in the lake-ice cover.