# Grain boundary energy anisotropy: a review

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## Abstract

This paper reviews findings on the anisotropy of the grain boundary energies. After introducing the basic concepts, there is a discussion of fundamental models used to understand and predict grain boundary energy anisotropy. Experimental methods for measuring the grain boundary energy anisotropy, all of which involve application of the Herring equation, are then briefly described. The next section reviews and compares the results of measurements and model calculations with the goal of identifying generally applicable characteristics. This is followed by a brief discussion of the role of grain boundary energies in nucleating discontinuous transitions in grain boundary structure and chemistry, known as complexion transitions. The review ends with some questions to be addressed by future research and a summary of what is known about grain boundary energy anisotropy.

## Keywords

Triple Junction Misorientation Angle Boundary Energy Energy Anisotropy Coincident Site Lattice## Introduction

The vast majority of the solid materials used in engineered systems are polycrystalline. In other words, they are comprised of many single crystals joined together by a three-dimensional (3D) network of internal interfaces called grain boundaries. Because the performance and integrity of a material are often determined by the structure of this network, grain boundaries have been of interest to materials scientists for many decades. There have been several recent review articles surveying grain boundary phenomena [1, 2, 3, 4, 5]. The current review is more narrowly focused on the topic of grain boundary energy anisotropy and provides an account of advancements since the publication of Sutton and Balluffi’s [6] critical review of grain boundary energy data in 1987. The paper contains a brief survey of historical concepts and grain boundary energy measurements. Next, findings from experiments and simulations are reviewed. This is followed by an introduction to recent findings about complexion transitions at grain boundaries. The review concludes with a prospectus for future studies of grain boundary energy anisotropy and a summary.

_{s}. However, the grain boundary energy will be less than this because of the binding energy (

*B*) gained when the two surfaces are brought together and new bonds are formed. The grain boundary energy is then:

_{s}/2 ≤

*B*≤ 3γ

_{s}/4 and γ

_{s}/2 ≤ γ

_{gb}< γ

_{s}. Using some simple approximations, Mullins [7] estimated γ

_{s}as the elastic work done to create a free surface and found it to be (

*E*/8) × 10

^{−10}m, where

*E*is the elastic modulus. Assuming an elastic modulus of 80 GPa, the resulting surface energy is approximately 1 J/m

^{2}and the average grain boundary energy (γ

_{gb}) should then be in the range of 0.5–1.0 J/m

^{2}. According to this estimate, the grain boundary energy should scale with the stiffness of the material.

The excess energy of the grain boundary provides the driving force for grain growth [8]. As grains shrink and disappear, the average grain size increases and the total grain boundary area per volume decreases. The capillary driving force, 2γ_{gb}/〈*r*〉, where 〈*r*〉 is a characteristic grain radius, decreases as 〈*r*〉 increases. Thus, as the average grain size increases, the driving force diminishes and it is more and more difficult to eliminate additional grain boundaries. This is why grain boundaries are nearly always found in solid materials, even though they are non-equilibrium defects.

The paragraphs above refer to an average grain boundary energy, but this review focuses on the anisotropy of the grain boundary energy. The anisotropic characteristics of the energy have been recognized since at least the time of Smith [9] and it has recently been shown that the probability that a grain boundary is annihilated during grain growth is related to its energy, and this leads to an anisotropic distribution of grain boundary types [10]. The energy anisotropy arises because different grain boundaries have different microscopic structures; following the line of reasoning that leads to Eq. 1, anisotropy in the grain boundary energy can arise from either γ_{s} or *B*. Macroscopically observable crystallographic parameters are used to classify boundaries with different microscopic structures. To classify the boundaries, five independent parameters must be specified. Three describe the misorientation of the crystal lattice and two describe the orientation of the grain boundary plane. The implication of having five independent parameters is that the number of different grain boundary types is large [4]. If the five dimensional domain of grain boundary types is discretized in 10° intervals, then there are roughly 6 × 10^{3} different grain boundaries for a material with cubic symmetry. The number of distinct boundaries increases rapidly for finer discretizations and for crystals with reduced symmetry.

Throughout this review, the so-called “axis-angle” description will be used to specify the three parameters of grain boundary lattice misorientation, Δ*g*. In other words, a misorientation will be specified by a crystallographic axis common to both crystals, [*uvw*], and a rotation about that axis, θ. The grain boundary plane orientation is specified by the unit vector, **n**. While **n** can assume any orientation within a hemisphere, certain special grain boundary plane orientations are sometimes referred to with the terms “tilt” and “twist”. For any lattice misorientation, the twist boundary is the one for which [*uvw*] and **n** are parallel. The tilt grain boundaries are those for which **n** is perpendicular to [*uvw*]. The term “symmetric” tilt means that the crystallographic planes bounding the grains on each side of the boundary are identical. All other tilts are asymmetric. The grain boundary character distribution (GBCD) is defined as the relative areas of grain boundaries as a function lattice misorientation and grain boundary plane orientation, λ(Δ*g*, **n**). Analogously, the grain boundary energy distribution (GBED) is defined as the relative energies of grain boundaries as a function lattice misorientation and grain boundary plane orientation, γ_{gb} (Δ*g*, **n**).

## Historical concepts for grain boundary structure and energy

Many years later, having knowledge of the atomic structure of crystals, Hargreaves and Hill [12] proposed that every atom in the boundary region could be associated with the crystal on one side or the other. Atoms in a transition zone (five to six plane on either side of the boundary, according to the illustration in their paper) near the grain boundary would be slightly displaced from their ideal positions (see Fig. 2c). More significantly, Hargreaves and Hill [12] also recognized the existence of coincidence boundaries, where some of the atomic sites in each crystal overlap within the boundary plane. They wrote an equation to find the rotational angles for different coincidence boundaries and illustrated a grain boundary where every fifth site was in coincidence (see Fig. 2b).

If we generalize these ideas using contemporary terminology, we can say that the Rosenhain and Ewen [11] model suggests that the atoms in the grain boundary region have a disordered structure that accommodates the orientation transition between the two crystals and the positions of these boundary atoms are not extensions of lattices of either of the adjoining crystals. The Hargreaves and Hill [12] model suggests more perfect order, with some relaxation in atomic positions, until an atomically abrupt transition. The two models could be characterized as a disordered boundary model and an ordered boundary model. In the past, ordered and disordered grain boundary models have been pitted against one another as if it must be one or the other. However, the current state of knowledge suggests that neither model is always a good description of a boundary. The increasingly powerful microscopic probes and computational models that have been applied to study grain boundaries have provided strong evidence for the existence of both ordered and disordered grain boundaries in polycrystals.

*E*

_{0}and

*A*have a dependence on the boundary plane orientation, but the principal variable is the misorientation angle, θ [13]. Gjostein and Rhines [14] measured the energies of tilt grain boundaries in Cu and found that for boundaries with θ < 6°, agreement with Eq. 2 was reasonable, but at higher misorientation angles the theory underestimated the actual energy. At misorientations greater than 6°, the dislocation cores are separated by less than 10 atomic planes and the elastic theory applied to derive Eq. 2 is not expected to accurately depict the energy.

The Read–Shockley [13] model is widely accepted as providing a good explanation for the energies of low misorientation angle grain boundaries. Beyond this, the theory actually makes it possible to represent any possible grain boundary as a collection of dislocations. Given the lattice misorientation (Δ*g*), grain boundary plane orientation (**n**), and three non-coplanar Burgers vectors, Frank’s formula [15, 16] can be used to determine the density of these dislocations needed to create the boundary. In the simplest approximation, the boundary energy can be assumed to be proportional to the minimum geometrically necessary dislocation density.

The CSL concept does not explicitly address grain boundary energy and the authors of the earliest papers do not suggest that the special boundaries have low energy. However, a reduced grain boundary energy was suggested by Brandon [21, 22], who extended to high coincidence boundaries Read and Shockley’s [13] concept of using dislocations to compensate small orientation differences. Using this concept, Brandon [21] was able to define an angular width for the assumed region of reduced grain boundary energy. While some coincident site boundaries clearly have lower energies than general boundaries (the Σ3 twin is an example), the phenomenon is not general. For example, Goodhew et al. [23] found that the CSL concept could not explain the configuration of tilt grain boundaries in a gold foil and Chaudhari and Matthews [24] concluded that coincidence site density is not a good guide to the energies of coincidence boundaries in MgO. In Sutton and Bulluffi’s [6] seminal review of models for interfacial boundary energy, they concluded that there was “no support for the general usefulness of criteria” based on coincident site density.

## Measuring grain boundary energies

_{i}is the energy of the

*i*th interface. This is a vector balance of forces tangential (\( \vec{t}_{i} \)) and normal (\( \vec{n}_{i} \)) to the interfaces, as labeled in Fig. 7a. The tangential forces, shown by solid lines, amount to a three-way tug-of-war at the triple junction. If one interface has a higher energy, it will pull the triple line along its tangent, annihilating the relatively higher energy interface and replacing it by the lower energy interfaces. The normal forces, shown by dashed lines and often referred to as torques, result from anisotropy. If the differential of the energy with respect to rotation angle β is large, there is a normal force to rotate the boundary in the direction that lowers the energy. For the junction to be in equilibrium, the six forces must balance, and this is reflecting in Eq. 3.

*x*and

*y*directions. Solving the equations yields a simplified form of Eq. 3 usually referred to as Young’s equation:

While the simplified approaches have facilitated numerous measurements, it should be emphasized that it is not possible to evaluate the full anisotropy of the grain boundary energy without using the complete form of the Herring [32] equation. To do this, one needs to examine triple junctions involving all different types of grain boundaries. Because the necessary number of junctions is in the range of 10^{3}–10^{4}, it was not feasible to conduct such measurements using manual techniques. However, the development of automated electron backscatter diffraction orientation mapping in the scanning electron microscope has made it possible to characterize the crystallography of 10^{4}–10^{5} triple junctions in a reasonable amount of time [39, 40, 41]. When coupled with serial sectioning, it is possible to determine the complete geometry for triple lines involving all boundary types and apply the Herring equation [42, 43, 44].

A method for doing this was developed by Moraweic [45]. The domain of grain boundary types is discretized so that there is a finite number of unknown boundary energies. The equilibrium condition in Eq. 3 can be written for each observed triple junction. As long as the number of equilibrium equations (observed triple junctions) exceeds the number of unknown grain boundary energies, it is possible to determine a set of energies that best satisfy the equations. The implementation of the method has been described in detail in Ref. [45] and has resulted in the determination of several complete grain boundary energy distributions from metals and ceramics [44, 46, 47, 48]. The method has also been applied to determine surface energy anisotropy [49, 50]. The method is sensitive to the amount of data available, so the energies of the most rarely observed boundaries are the most uncertain. Furthermore, in places where the energy varies rapidly with angle, the depth of the minimum or height of the maximum will be underestimated.

## Measurements of grain boundary energy anisotropy

Compared to the [100] tilt series, there is more variation in the energies for the [110] symmetric tilt grain boundaries. In Fig. 8b, data from Al [34], again very near the melting point, is compared to data for Cu at a range of temperatures [51]. All of the data agrees that for a 70° rotation about [110] there is a deep minimum in the energy; this is the coherent twin. For Al and the Cu at the highest temperature, there is also agreement that there is a minimum at 130° that corresponds to the Σ11 (113) boundary. The low energy of this boundary also agrees with observations reported by McLean [53]. However, for Cu at lower temperatures, this minimum moves closer to the Σ9 boundary at 140° [51]. One significant difference between Al and Cu is that the ratio of the energies of the symmetric Σ9 (at 40°) and the coherent twin (at 70°) is much larger in Cu than Al. Despite the differences, the similar overall appearance of the data indicates that the energy anisotropy might have a strong link to the ideal crystal structure. Finally, it should be noted that recent calculations of the energies of the symmetric tilt boundaries in Cu and Al agree well with the high temperature data [54].

_{100}) were determined for every observed boundary. The energy was then discretized into equal partitions, and the mean and standard deviation of ln(λ + 1) and θ

_{100}were determined for all of the boundaries within each partition. The results in Fig. 11 show that as γ increases, θ

_{100}increases and λ decreases; this illustrates that the trends observed in Fig. 10 persist throughout the data set.

_{2}O

_{3}, Ni, yttria stabilized ZrO

_{2}(YSZ), and SrTiO

_{3}. The energy distributions are also compared to grain boundary plane distributions averaged in the same way. Note that the averaging compresses the true anisotropy. However, the basic result is that certain low index planes have lower energies and that boundaries comprised of these planes appear more frequently in the microstructure. In fact, the most commonly occurring grain boundary plane orientations are correlated with low index, low energy surface planes. This is illustrated in Fig. 13, which compares free surface energies with grain boundary populations for MgO (cubic) [42, 43, 44], TiO

_{2}(rutile, tetragonal) [60], and Al

_{2}O

_{3}(corundum, trigonal) [61]. On average, rather than seeking high symmetry configurations, grain boundaries tend to favor configurations in which at least one side of the interface can be terminated by a low index plane [62]. Recall that the energy cost for making a grain boundary can be thought of as the energy to create the two surfaces on either side of the interface, minus the binding energy that is recovered by bringing the two surfaces together. The observation that the total grain boundary energy is correlated to the surface energies suggests that surface energy anisotropies makes a significant contribution to the total anisotropy that, on average, is greater than the binding energy anisotropy. This is consistent with the suggestions made by Wolf and Philpot [31] based on the results of atomistic calculations.

_{3}[64]. In conclusion, the current findings suggest that low energy, low index grain boundary planes are good predictors of relatively low energy boundaries, while interface coincidence is not.

Atomistic simulations have recently been used to calculate grain boundary energies in Cu, Al, Ni, and Au [65, 66]. These new calculations cover the 388 highest symmetry boundaries and, therefore, represent a more complete sampling of the entire space of grain boundary types than was available in the past. This work as led to a number of important conclusions that are consistent with the experimental observations. First, grain boundaries with the same misorientation, but different grain boundary plane orientations may have very different energies. This leads to the observation that disorientation angle is not a good predictor of grain boundary energy. Similarly, boundary coincidence is also not a good predictor of grain boundary energy. When the calculated boundary energies are plotted as a function of Σ, there is no correlation and at any fixed value of Σ, the range of energies is nearly as wide as the total anisotropy [66].

The calculations also reveal that when all of the grain boundary energies of crystallographically identical grain boundaries in different metals are compared, there is a strong linear correlation. For a pair of metals, the ratio of the energies of the vast majority of boundaries is very nearly equal to the ratio of *a*_{0}*G*, where *a*_{0} is the lattice constant and *G* is the shear modulus [65]. Recall that the rough estimate for the grain boundary energy discussed in “Introduction” section also scaled with the elastic properties of the material. For boundaries with stacking fault character, the boundary energy ratio is closer to the stacking fault energy ratio. One possible implication of this observation is that there is a single, scalable, grain boundary energy distribution for any given structure type. Based on the fact that the energies scale with the shear modulus, the second implication is that a dislocation model for the grain boundary might be valuable in predicting the energy.

Even with some outliers, the correspondence between calculated and observed grain boundary energies is gratifying. One of the disadvantages of the experiment is that if a grain boundary does not appear frequently in the polycrystal and is not sufficiently sampled in the data set, the energy cannot be reliably determined. The simulations are obviously not subject to this constraint. On the other hand, if a boundary appears frequently enough in the microstructure, its energy can be determined by the experiment, regardless of its symmetry or the size of its repeat unit. However, the calculation has an upper limit in the size of the simulation cell and is not able to determine the energies of low symmetry boundaries with large repeat units.

## Grain boundary complexions

Grain boundary composition is known to affect grain boundary energy and it has been shown that the grain boundary population is influenced by segregating impurities [60, 67]. Nevertheless, it had generally been assumed that the properties of grain boundaries changed continuously with temperature and composition. Over the last several years, a body of evidence has been published demonstrating that grain boundaries can undergo discontinuous changes in structure and composition and that these transitions can be associated with transitions in mobility and energy. These grain boundary states have been referred to by some authors as “complexions” [68, 69, 70, 71, 72, 73]. The breakthrough experimental studies by Dillon and Harmer [69, 74, 75] focused on alumina with controlled impurity concentrations. High resolution microscopy was used to demonstrate that boundaries could be pure, could have a single adsorbed monolayer of solute, or an adsorbed bilayer of solute. Other boundaries had multilayer adsorption, thin intergranular wetting films of constant thickness, or films of arbitrary thickness. While some of these boundary structures had been observed previously, the breakthrough was associating them with very different grain boundary mobilities and observing that they could co-exist in the same microstructure. Based on these findings, Harmer and Dillon [76, 77] have described a plausible mechanism for abnormal grain growth.

One outstanding question has been, why do some boundaries enriched in solute and undergo complexion transitions while others simply deposit the solute in the form of precipitate phases? A recent experiment has explored this issue and found that the relative energy of the interface between a precipitate and the host lattice is an indicator of whether or not a complexion transition will occur [80]. Chemistries that produce low-energy interphase boundaries tend to suppress complexion transitions, while those nucleating precipitates with high interfacial energies promote them. This may be explained in the context of a phase selection competition in which the activation barrier to the complexion transition and precipitation compete with one another. The interphase boundary energies tend to be intermediate to the energies of the grain boundaries in the component systems. These facts lead to a proposed selection criterion for additives based on knowledge of the interfacial energies. Namely, complexion transitions should be sought in systems where the solute strongly segregates to the boundary and where precipitates with coherent, low energy interfaces do not form.

## Future prospects for grain boundary energy studies

There has been significant progress in understanding the anisotropy of grain boundary energies since the initial experimental studies by Dunn and Leonetti [33]. One significant opportunity for further study is to explore the relationship between grain boundary energy and population. The measurements of the grain boundary population are more accurate than measurements of the energy. If there is a fixed relationship between the two, it would be possible to calculate the energy directly from the population measurement, rather than fitting the Herring equation to the observed geometries of the triple junctions. This would allow the generation of a greater quantity of higher quality grain boundary energy data. This has been suggested by Gruber et al. [81, 82] and demonstrated in one-dimension, but the method has not yet been extended to all five crystallographic dimensions.

A related question is whether or not grain boundary plane distributions represent grain boundary Wulff shapes. In other words, if the energy and population are inversely correlated, it is possible that the statistical grain boundary distribution is a measure of the grain boundary energy anisotropy just as the shape a small crystal in equilibrium is a measure of the surface energy anisotropy. Although grain boundaries are moving at the high temperatures where the grain boundary distribution is determined, local equilibrium is obtained at the triple junctions and this may be the feature that creates statistically equivalent grain boundary population distributions at different grain sizes [83]. If so, this would also represent a path toward measuring grain boundary energies.

As mentioned previously, there is evidence that the grain boundary energy anisotropies of isostructural materials are roughly the same. If we can understand this scaling and the bounds on the correlation, it will be of great value to generalize the past findings. This will require the exploration of materials beyond fcc metals. Such a finding could be applied to grain boundary engineering [84], where changes in the grain boundary character distribution with sequential thermomechanical processing are thought to be closely related to the grain boundary energy distribution [85, 86, 87].

Finally, chemical effects on the grain boundary energy anisotropy remain largely unexplored. This is particularly important with respect to understanding complexion transitions. Because of the effect that complexion transitions have on grain boundary mobility, they offer the possibility of controlling and designing new multicomponent materials. However, our currently incomplete knowledge of the mechanisms governing complexion transitions makes it impossible to predict their occurrence. Grain boundary energy measurements are likely to play an important role in this emerging research area.

## Summary

- 1.
The energy anisotropy that results from variations in the grain boundary plane orientation are greater than the anisotropy that results from variations in the lattice misorientation.

- 2.
The Read–Shockley model for the energies of small misorientation angle grain boundaries provides reliable predictions for relative grain boundary energies.

- 3.
Models based lattice or boundary coincidence are not good predictors of grain boundary energies.

- 4.
There is an inverse correlation between the grain boundary energy and the grain boundary population.

- 5.
Grain boundaries comprised of low energy surfaces have relatively low energies.

- 6.
Isostructural materials have similar grain boundary energy anisotropies.

## Notes

### Acknowledgements

The work was supported by the MRSEC program of the National Science Foundation under Award Number DMR-0520425. The author acknowledges collaboration with S.J. Dillon, M.P. Harmer, J. Gruber, C.S. Kim, S.B. Lee, J. Li, H.M. Miller, T. Sano, D.M. Saylor, A.D. Rollett, and P. Wynblatt in studies of grain boundary energies at Carnegie Mellon University.

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