Grain Growth Law Research Articles

On the basic mechanism of grain growth: the role of texture on grain boundary migration rate

In this paper we will review some of the many achievements of Statistical Theory of Grain Growth (which is 40 years old, this year!), based on the classic law of grain growth. For instance, we will show that grain growth in presence of texture (which is the usual feature in a real microstructure) is not exclusively depending on the curvature of the grain boundary, as in this case the effect of curvature at the vertexes (in 2-D case) contributes as well (which, depending on the texture pattern, sometimes may be more relevant than curvature along the boundary). Such phenomenon requires a reformulation of Von Neuman’s equation in 2-D by involving, beside the different boundary mobilities, the different energies of the “third” grain boundary, which is pulling the vertex and inducing an extra curvature which is not balanced along the grain perimeter (for simplicity’s sake we are referring to a 2-D case, but the analysis can also be done in 3-D). We can then observe that boundary movement is not only depending on a couple of grains in contact (with the boundary curvature acting like in a bicrystal) but also includes the network of “third” grains operating at the vertexes. The overall consequence is that the growth kinetics and the grain size distribution shape are unique and variable for each texture component which contradicts classic growth laws as well as the expected results for the whole grain size distribution.

Grain growth of natural and synthetic ice at 0 °C

Abstract. Grain growth can modify the microstructure of natural ice, including the grain size and crystallographic preferred orientation (CPO). To better understand grain-growth processes and kinetics, we compared microstructural data from synthetic and natural ice samples of similar starting grain sizes that were annealed at the solidus temperature (0 ∘C) for durations of a few hours to 33 d. The synthetic ice has a homogeneous initial microstructure characterized by polygonal grains, little intragranular distortion, few bubbles, and a near-random CPO. The natural ice samples were subsampled from ice cores acquired from the Priestley Glacier, Antarctica. This natural ice has a heterogeneous microstructure characterized by a considerable number of air bubbles, widespread intragranular distortion, and a CPO. During annealing, the average grain size of the natural ice barely changes, whereas the average grain size of the synthetic ice gradually increases. These observations demonstrate that grain growth in natural ice can be much slower than in synthetic ice and therefore that the grain-growth law derived from synthetic ice cannot be directly applied to estimate the grain-size evolution in natural ice with a different microstructure. The microstructure of natural ice is characterized by many bubbles that pin grain boundaries. Previous studies suggest that bubble pinning provides a resisting force that reduces the effective driving force of grain-boundary migration and is therefore linked to the inhibition of grain growth observed in natural ice. As annealing progresses, the number density (number per unit area) of bubbles on grain boundaries in the natural ice decreases, whilst the number density of bubbles in the grain interiors increases. This observation indicates that some grain boundaries sweep through bubbles, which should weaken the pinning effect and thus reduce the resisting force for grain-boundary migration. Some of the Priestley ice grains become abnormally large during annealing. We speculate that the contrast of dislocation density amongst neighbouring grains, which favours the selected growth of grains with low dislocation densities, and bubble pinning, which inhibits grain growth, are tightly associated with abnormal grain growth. The upper 10 m of the Priestley ice core has a weaker CPO and better-developed second maximum than deeper samples. The similarity of this difference to the changes observed in annealing experiments suggests that abnormal grain growth may have occurred in the upper 10 m of the Priestley Glacier during summer warming.

Estimation of the Critical Value of the Second-Phase Particles in the Microstructure of AZ31 Mg Alloy by Phase-Field Methods

In this study, phase-field models were employed to simulate the effects of second-phase particles (SPPs) on grain growth of the AZ31 Mg alloy, under realistic spatial and temporal scales, at 350 °C, during annealing. The particle sizes ranged from 0 to 7 μm, and the particles with large volume fractions were used in the paper. The results reveal that the volume fractions and sizes of the SPP affect grain growth and that the volume fractions and sizes of the SPP on pinning exhibited critical values. When the SPP volume fraction is f = 5%, the SPP is at the maximum critical size, rμmmax; when the SPP size is r=1 μm, the SPP minimum critical volume fraction is fmin = 0.25% and the maximum critical volume fraction is fmax = 20%. The critical values increase with the increase of the sizes or volume fractions of the second-phase particles. Finally, the average grain size, particle size, and particle volume fraction obtained from the simulation were fitted according to the Zener relationship, and the obtained results showed that the fitting indices were in the range of 0.33–0.50. The results were compared with the experimental results. The simulation results obtained in this study will provide an important academic reference for understanding the mechanism and law of grain growth, an important reference for accurate control of grain size and properties of the material, a reference for the development of the annealing treatment process of Mg alloy, and a theoretical guide for the use of recrystallization process to control the microstructure of Mg alloy and improve the plastic-forming properties.

A comprehensive analysis of three-dimensional normal grain growth of pure iron via multi-phase field simulation

A three-dimensional (3D) multi-phase field model was established to simulate normal grain growth in pure iron. The advanced visualization technology was used to extract the related data for individual grains, which can clearly display the grain morphology with distinct grain boundary surface as well as the space distribution of neighboring grains?Based on the simulation results, the grain growth kinetics model was described, which is in conformity with Burke and Turnbull?s parabolic law. The phenomenon of a ?Hillert regime? in 3D grain growth and the topological transformation mechanism were investigated. The grain size distributions under different time evolution showed a good agreement with the Hillert distribution. The details of grain growth, especially grain size distribution and volume growth rate, were analyzed. The models of von Neumann-Mullins and Hilgensfeldt for predicting the volumetric growth rate were compared. The volumetric growth rate was approximately zero when the number of grain sides was close to 13.7. The multi-phase field simulation can be used to analyze the dynamic evolution of the topological relationship of grains and reveal the general law of normal grain growth quantitatively.

Tracing the recrystallization of warm temper-rolled Fe–6.5 wt% Si non-oriented electrical steel using a quasi in situ EBSD technique

Fe–6.5 wt% Si non-oriented electrical steel is an excellent soft magnetic material due to the low core losses at high frequencies and near-zero magnetostriction. In this study, an Fe–6.5 wt% Si non-oriented electrical steel was processed by hot rolling, warm cross rolling, intermediate annealing, warm temper rolling and final annealing. The evolution of microstructure and microtexture during final annealing was investigated using a quasi in situ EBSD (electron backscatter diffraction) technique. After warm temper rolling, an area on the ND–RD (normal direction–rolling direction) cross section was marked by micro-hardness indents, and the recrystallization of individual grains in this area was traced under EBSD when the annealing time was increased. Due to the differences in the microstructure, texture and stored energy after warm temper rolling, the surface and center regions showed different recrystallization behaviors. The recrystallization of the surface region was essentially initiated by the growth of existing crystallites having lower stored energy than their neighboring areas (essentially no nucleation), while the center region showed both nucleation and grain growth during annealing. The growth rates of individual grains were evaluated with respect to the number of neighboring grains, and they approximately followed the von Neumann–Mullins law of grain growth. The surface region showed a much faster growth rate than the center region. The final texture was dominated by //ND and //ND in both the surface and center regions, due to the preferred growth of these grains.

The effect of Y2O3 on the grain growth and densification of W matrix during low temperature sintering: Experiments and modelling

In this work, the effect of Y2O3 addition on the grain growth and densification of W matrix was explored from a kinetic perspective by comparing the low temperature sintering behavior of W-Y2O3 composite nanopowder system with that of pure W nanopowder system. By combining nonisothermal sintering experiment with the power law of grain growth, it was found that grain boundary diffusion is responsible for the mass transport mechanism in both systems. The inhibition of W grain growth after Y2O3 addition is closely related to the lower surface diffusivity of W skeleton. The densification rate of W skeleton in these two systems were modelled using a periodic 3D arrangement of identical particles and a much smaller grain boundary diffusivity of W skeleton was obtained for W-Y2O3 sample. Firstly, it is the decrease in dihedral angle of W matrix after Y2O3 addition that causes a decrease in thermodynamic driving force for densification. Secondly, from the view of atomic mobility, the inhibited W grain growth caused by limited surface diffusivity is one of the reasons impeding densification. Besides, Y2O3 grains distributed at W grain boundary coarsen through wetting W grain boundary, which also has an inhibition effect on the W skeleton densification.

Large-scale phase-field simulation of three-dimensional isotropic grain growth in polycrystalline thin films

In this study, assuming an ideal system free from thermal grooving and anisotropy in grain boundary properties, we analyze thin-film grain growth via three-dimensional (3D) phase-field simulations with approximately one million initial grains. The large-scale simulations accelerated by multiple graphics processing units allow for the reliable statistical investigation of grain growth behaviors in films with various thickness. Over the transition from 3D to two-dimensional (2D) growth modes, variations in the averages and distributions of grain sizes are quantified and compared for different regions of the films. Furthermore, we propose a comprehensive scaling law of thin-film grain growth, by which the 3D–2D transition behaviors and grain growth kinetics can be described in a unified manner independent of film thickness.

B2 Grain Growth Behavior of a Ti-22Al-25Nb Alloy Fabricated by Hot Pressing Sintering

Grain growth behavior of a powder metallurgy (P/M) Ti-22Al-25Nb alloy was investigated by applying a series of isothermal treatment tests over a wide range of temperatures and holding times. An isothermal treatment scheme was conducted in the B2 phase region (1070-1110 °C) and α2 + B2 phase region (1010-1050 °C) at holding times of 10, 30 min, 1, 2, and 3 h, respectively. The effects of temperature and holding time on the microstructure evolution and microhardness of the P/M Ti-22Al-25Nb alloy at elevated temperatures were evaluated using optical microscope, scanning electron microscope, x-ray diffraction, and Vickers hardness test techniques. The results revealed that the alloy’s treated microstructure was closely linked to temperature and holding time, respectively. The change law of B2 grain growth with holding time and temperature can be well interpreted by the Beck equation and Hillert equation, respectively. The B2 grain growth exponent n and activation energy Q were acquired based on experimental data in the α2 + B2 and B2 phase regions. In addition, the grain growth contour map for the P/M Ti-22Al-25Nb alloy was constructed to depict variations in B2 grain size based on holding time and temperature.

Temperature dependent grain growth of forsterite–nickel mixtures: Implications for grain growth in two-phase systems and applications to the H-chondrite parent body

Abstract Grain growth experiments in the system forsterite (Fo) + nickel (Ni) have been performed on two analogue mixtures of ordinary chondrites, with volume % of Fo:Ni (95:5) and (80:20). These two mixtures have been studied at temperatures of 1390 °C and 1340 °C, at an oxygen fugacity ( f O 2 ) three orders of magnitude below the Ni–NiO buffer, for durations between 2 h and 10 days. Microstructures and grain size distributions show that grain growth is normal and that for durations >10 h the Zener relation is verified (i.e., the ratio of Fo and Ni grain size is independent of time). Comparison with results previously obtained at 1440 °C shows a similar grain growth exponent ( n ∼ 5 ) for both phases, consistent with growth of forsterite by grain boundary migration, limited by the growth-rate of nickel. The details of size distribution frequencies and the value of grain-growth exponent indicate that the nickel grains, which pin forsterite grain boundaries, grow by diffusion along one-dimensional paths (i.e., along forsterite triple junctions). The derived activation energies for nickel and forsterite are 235 ± 33 kJ / mol and 400 ± 48 kJ / mol respectively. Within the framework of the Zener relation, this unexpected difference of activation energy is shown to be related to temperature-dependent variations in the ratio of Ni and Fo grain-size that are consistent with observed variations in Fo–Ni–Fo dihedral angle. These data thus indicate that the presence of all phases should be taken into account when considering the activation energy of growth rate of individual phases. As an application, the experimentally derived growth law for metal has been used in conjunction with temperature–time paths taken from models of the thermal history of the H-chondrite parent body to estimate the grain size evolution of metal in H-chondrites. A remarkably self-consistent picture emerges from experimentally derived grain-growth laws, textural data of metal grains in well characterised H-chondrite samples, and geochemically constrained temperature–time paths, providing the potential to use textural data of metal as a window into the thermal history of chondritic samples.

Effect of grain size on the phase structure and electrical properties of PZT–PNZN quaternary systems

The quaternary systems 0.74Pb(Zr0.47Ti0.53)O3–0.26Pb[(Ni0.6Zn0.4)1/3Nb2/3]O3(PZT–PNZN) have been prepared using the conventional oxide mixing method in a wide temperature region between 1050 and 1250°C. All specimens present high relative density above 95%, indicating a wide sintering window for this system. Investigation of the microstructure reveals a continuous increase in grain size from 1.07 to 2.77μm when the sintering temperature is increased from 1050 to 1250°C, which obeys the common grain-growth law well. However, unlike compositional component PZN–PZT and PNN–PZT, the phase structure of quaternary PZN–PNZN is not sensitive to the grain size. The specimens with different grain size retain the morphotropic phase boundary (MPB) structure and the size induced phase transition does not appear. Excluding the influence of the phase structure, the size effect plays a dominant role in improving the piezoelectric properties, which facilitates the motion of domain walls due to the decreasing number of grain boundary. The optimum piezoelectric parameters are obtained at 1150°C: d33=307pC/N, kp=0.56.

Constitutive modeling of stress-driven grain growth in nanocrystalline metals

In this work, we present a variational multiscale model for grain growth in face-centered cubic nanocrystalline (nc) metals. In particular, grain-growth-induced stress softening and the resulting relaxation phenomena are addressed. The behavior of the polycrystal is described by a conventional Taylor-type averaging scheme in which the grains are treated as two-phase composites consisting of a grain interior phase and a grain boundary-affected zone. Furthermore, a grain-growth law that captures the experimentally observed characteristics of the grain coarsening phenomena is proposed. To this end, the grain size is not taken as constant and varies according to the proposed stress-driven growth law. Several parametric studies are conducted to emphasize the influence of the grain-growth rule on the overall macroscopic response. Finally, the model is shown to provide a good description of the experimentally observed grain-growth-induced relaxation in nc-copper.

Research on Key Technology of Normal Grain Growth for Second Phase Particle Materials by Cellular Automata Simulation

Based on cellular automata, a model of simulating grain growth was established and the key technologies of simulation was studied which including second phase particle of single size, multi-size distribution and different shapes generation technologies. The simulation result can accurately reflect the influence law of the second phase particle grain growth and its pinning mechanism. Grain boundaries can therefore more easily break free from the particles than in purely two-dimensional systems, resulting in fewer grain boundaryparticle intersections and a larger final grain size. For a given volume fraction f and size of the particles r, the final grain size increases with film thickness. Moreover, it was found that particles located in the middle of the film are most efficient in pinning grain boundaries. The simulation results are compared with Zener type relations and previous simulation results.

Kinetics of Initial Coarsening During Sintering of Nanosized Powders

Grain growth during sintering is a critical issue for the manufacture of nanocrystalline bulk materials from nanosized powders. The grain growth process during sintering can be viewed as consisting of two parts: initial coarsening during early and intermediate stages of sintering and latter stage grain growth during the final stage of sintering. The latter stage grain growth is the normal grain growth that has been well studied and reported in the literature. The initial coarsening, which often inevitably causes a material to lose nanoscaled grain size characteristics, however, is not well studied at all. In this investigation, the initial coarsening during sintering of nanosized powders was studied by both nonisothermal and isothermal experimental techniques using tungsten as an example material. The results show that the initial coarsening during the heat-up process of a sintering cycle is sufficient to increase the grain size beyond the nanoscale. The kinetics of initial coarsening is found to be linear rather than polynomial, as predicted by the conventional power law of grain growth. The analysis of activation energies showed that surface diffusion is the primary mechanism for interparticle mass transport during the initial coarsening. The linear kinetic behavior could be attributed to the pinning of grain boundaries by surface grooves and high concentration of defects as the result of the synthesis of nanosized powders.

Isothermal crystallization kinetic of ZnO thin films

Zinc oxide (ZnO) thin films deposited by DC magnetron sputtering were annealed in nitrogen atmosphere at different temperatures ranging from 100 to 500 °C with a step of 100 °C; the annealing time was 6 h. In order to study the film’s crystallization kinetic, their structures were monitored by means of X-ray diffraction (XRD) analysis each hour. Variation in grain size, calculated from the XRD patterns, with annealing time and temperature, obeys the classical parabolic law of grain growth. Exponent n was found to be dependent on the annealing temperature; it ranged from 5.13 to 3.8 with increase in annealing temperature. From the obtained exponent n values we inferred that the grain growth mechanism is mainly governed by the atom jumping across the grain boundary. We have found that the grain growth is characterized by a low activation energy ranging from 22 to 24 kJ/mol.

Topology Based Growth Law and New Analytical Grain Size Distribution Function of 3D Grain Growth

Based on topological considerations and results of Monte Carlo Potts model simulations of three-dimensional normal grain growth it is shown that, contrary to Hillert’s assumption, the average self-similar volume change rate is a non-linear function of the relative grain size, which in the range of observed grain sizes can be approximated by a quadratic polynomial. In particular, based on an adequate modification of the effective growth law, a new analytical grain size distribution function is derived, which yields an excellent representation of the simulated grain size distribution.

Effect of sintering temperature on the phase transition and dielectrical response in the relaxor-ferroelectric-system 0.5PZN–0.5PZT

Relaxor ferroelectrics of 0.5Pb(Zn1∕3Nb2∕3)O3–0.5Pb(Zr0.47Ti0.53)O3 (0.5PZN–0.5PZT) near the morphotropic phase boundary (MPB) were prepared using the conventional oxide mixing method. The influence of sintering temperature on the phase transition and dielectrical response in the relaxor-ferroelectric-system 0.5PZN–0.5PZT has been investigated. X-ray diffraction analysis indicated that the phase of the material changed from an initially rhombohedral structure to a MPB structure and then a tetragonal structure when the sintering temperature is increased from 900to1250°C. The corresponding grain size is enlarged from 1.1μm for predominantly rhombohedral to 2.5μm for more than 60% tetragonal, which obeys the common grain-growth law. The dielectric studies revealed that the indicator of the degree of diffuseness γ decreased with the increase of the sintering temperature from 900to1150°C, indicating that the dielectric relaxor behavior was weakened, while at high sintering temperature above 1250°C, γ increased subsequently, which was attributed to the Pb vacancy due to the formation of a pyrochlore phase. Raman analysis on the B-site cation order correlates well with the dielectric measurement results.

Modeling microstructure evolution in three dimensions with Grain3D and LaGriT

This paper will describe modeling microstructure evolution using a combination of our gradient-weighted moving finite elements code, Grain3D and our 3-D unstructured grid generation and optimization code, LaGriT. Grain boundaries evolve by mean curvature motion, and Grain3D allows for the incorporation of grain boundary orientation dependence modeled as anisotropic mobility and energy. We also describe the process of generating an initial computational grid from images obtained from electron backscatter diffraction. We present the grid optimization operations developed to respond to changes in the physical topology such as the collapse of grains and to maintain uniform computational grid quality. For 3-D columnar microstructures, validation of the method is demonstrated by comparison with experiments. For large systems of fully 3-D microstructures, simulations compare favorably to the parabolic law of normal grain growth.

Grain-growth law during Stage 1 sintering of materials

This work investigates the grain-growth behaviourof powder compact during Stage 1 sintering (<90{%} theoreticaldensity). It is widely accepted that grain size is an important statevariable in the constitutive modelling in material sintering. However, itis noted that all the existing grain-growth laws proposed in theliterature do not incorporate the effect of externally applied stressindependently. In this work, a grain-growth law with externally appliedstress as a variable was proposed. Alumina powders were forge-sinteredat different applied stresses to examine the proposed grain-growthrelationship. The proposed grain-growth law was then applied to modelthe grain-growth process on the sinter forging of tool steel. It isshown that the present proposed grain-growth law provides a gooddescription on the experimental results.

Excess Gibbs free energy and kinetics of grain growth of nanostructured silver

Samples of nanostructured silver were prepared by a mechanochemical technique and by electrodeposition. In the former, the microstructure was stabilized by intergranular cosegregation of Mg and O atoms. The excess Gibbs free energy of these samples was investigated by e.m.f. measurements at 453 K in a solid state electrochemical cell with AgI as a solid electrolyte. Electrodeposited pure silver samples had a labile microstructure even at 298 K. Their structural relaxation process was monitored in situ by chronopotentiometric measurements in an aqueous iodoargentate electrolyte. The time dependence of the e.m.f. could be quantitatively described by a parabolic law of grain growth without any adjustable parameter.

Microestrutural evolution in a CuZnAl shape memory alloy: kinetics and morphological aspects

The microstructural evolution of the CuZnAl shape memory alloys was studied by indirect techniques relating to the atomic migration rate of grain boundaries. Addition elements were used in a Cu-15,5Zn-8,0Al alloy to provide a comparison with the same alloy without microelement additions. The alloys were melted in an induction furnace of 24 kVA. After casting, the bulk samples of the alloys were homogenized. Then they were solution treated and hot-rolled followed by water-quenching to initiate the recrystallization. Finally, annealing produced at different temperature ranges was made in different samples in order to establish a law for the grain growth. Following the heat treatments, all annealed samples were examined by statistical metallography and the grain sizes were measured. After measurements, the same empirical law of grain growth was found for the different alloys and the ln [D-Do] x 1/T diagrams were plotted in order to establish the kinetic behavior. Based on the estimated values of the activation energy, important conclusions were obtained concerning the addition elements.