The relative free energies of grain boundaries in magnesia as a...

Acta Materialia 51 (2003) 3675–3686www.actamat-journals.com

The relative free energies of grain boundaries in magnesiaas a function of five macroscopic parameters

David M. Saylora, 1, Adam Morawiecb, Gregory S. Rohrera,∗

a Materials Science and Engineering Department, Carnegie Mellon University, Pittsburgh, PA 15213, USAb Institute of Metallurgy and Materials Science, Polish Academy of Sciences, Cracow, Poland

Received 2 December 2002; accepted 21 March 2003

Abstract

Using measurements of the geometric and crystallographic characteristics of approximately 104 triple junctions in aMgO polycrystal, the relative grain boundary energy has been determined as a function of five macroscopic parameters.The relative energy of a particular grain boundary is inversely correlated with its frequency of occurrence. At allmisorientations, grain boundaries with {1 0 0} interface planes have relatively low energies. For low misorientationangle grain boundaries, the results are consistent with the predictions of dislocation models. At high misorientationangles, the population and energy are correlated to the sum of the energies of the free surfaces that comprise the bound-ary. 2003 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved.

Keywords: Grain boundary energy; Ceramics; Microstructure; Dislocation boundaries; Coincident lattices

1. Introduction

Grain boundaries in materials with centrosym-metric crystal structures can be distinguished onthe basis of five macroscopically observable para-meters[1]. Three parameters are needed to specifythe transformation that brings two adjacent misori-ented crystals into coincidence; the remainingparameters specify the orientation of the boundaryplane separating the crystals. Based on measure-

∗ Corresponding author. Tel.:+1-412-268-2696; fax:+1-412-268-3113.

E-mail address: [email protected] (G.S. Rohrer).1 Present address: National Institute of Standards and Tech-

nology, Gaithersburg, MD, USA.

1359-6454/03/$30.00 2003 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved.doi:10.1016/S1359-6454(03)00182-4

ments of dihedral angles at triple junctions wherethree grain boundaries meet[2–6] or at thermalgrooves where a grain boundary meets a free sur-face [7–14], it has been shown that the grainboundary free energy per unit area varies with themacroscopically observable parameters. However,the relative grain boundary energy has not yet beenmeasured as a function of all five degrees of free-dom. The basic problem is that there are so manyphysically distinct boundaries in the five-dimen-sional domain that if one applies traditional micro-scopies under human control, it was not feasibleto make enough dihedral angle measurements tocharacterize all of the distinguishable interfaces.

In a companion paper referred to as Part I, anexperimental method was described for mapping

3676 D.M. Saylor et al. / Acta Materialia 51 (2003) 3675–3686

the geometry of the grain boundary network andmeasuring the frequency with which differenttypes of grain boundaries occur [15]. This semi-automated technique has been applied to MgO andwas used to measure the grain boundary dihedralangles of 1.9 × 104 triple junctions assumed to bein local thermodynamic equilibrium; thesemeasurements are used here to determine the grainboundary energies. Thus, the purpose of thepresent paper is to report on the relative grainboundary energy of MgO as a function of the fivemacroscopic grain boundary parameters. Afterdescribing the methods for determining the relativeenergies in the next section, the characteristics ofthe five-dimensional energy distribution arepresented in Section 3. In Section 4, the observedenergies are compared to existing models for theanisotropy of the grain boundary energy and ourfindings are summarized in Section 5.

2. Methods

2.1. Sample

The polycrystalline magnesia sample has beendescribed in Part I [15]. As part of the earlieranalysis, we measured the function l(�g, n), whichis the relative frequency of occurrence of a grainboundary with misorientation (�g) and boundaryplane normal (n), in units of multiples of a randomdistribution (MRD). These data serve as a basis forthe present analysis, whose goal is to determineg(�g, n), the relative grain boundary energy perunit area as a function of misorientation andboundary plane normal. It is important to recognizethat the grain boundary dihedral angles were estab-lished during a 48 h high temperature anneal in airat 1600 °C and that before analysis, the surfaceregion was removed by grinding. Thus, the ener-gies that we extract from these data are representa-tive of the local thermodynamic equilibrium at thistemperature, in the bulk region of the sample.Although grain boundary positions were identifiedby observations of grain boundary grooves formedon a polished plane by a shorter anneal at 1400 °C,there is no evidence for grain boundary migrationduring this lower temperature treatment.

2.2. Triple junction geometry data

The triple junction geometries were extractedfrom the previously described grain boundary net-work data assembled from five adjacent parallelsections through the microstructure [15]. Pointswhere the same triple line intersects adjacent layerswere identified by comparing the lateral positionsof the points and the orientations of the three grainsbounding the triple junctions on each layer. Basedon the lateral offset of the junction positions andthe spacing between the two sections (�h), the tri-ple line direction (l) was determined (see Fig. 1).Taking the grain boundary tangents (ti) directlyfrom the planar sections, it was possible to deter-mine the grain boundary plane normal (ni = l×ti).Because of boundary curvature and experimentaluncertainties, the apparent grain boundary tangentsare not necessarily identical on adjacent layers. Toaverage out the effect of these differences, thegeometry of each triple junction was determinedtwice; once using the boundary tangents in the toplayer and a second time using the tangentsobserved in the bottom layer. Thus, each valid tri-ple junction was considered twice in our analysisand this produces the set of 19,094 observed tri-ple junctions.

2.3. Energy reconstruction

The key assumption underpinning this and pre-vious measurements of the relative grain boundary

Fig. 1. Schematic representation of the geometric parametersrequired to characterize a grain boundary triple junction.

3677D.M. Saylor et al. / Acta Materialia 51 (2003) 3675–3686

energy is that the interfacial junctions are in localthermodynamic equilibrium and, therefore, obeythe Herring [16] condition. Under these circum-stances, the relative energies can be determinedusing the capillarity vector reconstruction method,which is described in detail elsewhere [17]. Here,we provide a brief account of the method’s under-pinnings. Hoffman and Cahn [18,19] expressed theHerring condition in the following way:

(x1 � x2 � x3) � l � 0, (1)

where x1, x2, and x3 are the capillarity vectors asso-ciated with the three grain boundaries. In general,the component of x perpendicular to a boundaryis x� = g(n)n, where g(n) is inclination dependentenergy of a boundary with a given misorientation.The component of x parallel to the boundary, x��

= (∂g /∂b)maxt0, points in the direction of maximumincrease in g(n), where b is the right-handed angleof rotation about l. For every observed triple junc-tion, there is a separate equilibrium expression, asgiven in Eq. (1). For each such equation, we havemeasured t and n for each boundary and l for thejunction. Therefore, the only unknowns are the sca-lar components of the three x vectors. If we par-tition the range of distinguishable grain boundaries,then there is a discrete x vector for each boundarytype. Using an iterative procedure, it is then poss-ible to find the set of capillarity vectors that mostnearly satisfies this system of linear equations [17].

A specific type of grain boundary is dis-tinguished by its misorientation (�g) and itsboundary plane (n). The method used to partitionthe domain of grain boundary types was describedin Part I [15] and the iterative procedure used tofind the set of x that satisfy the equilibrium equa-tions has already been described by Morawiec [17].The relaxation factor, which controls the magni-tude of changes applied to each capillarity vectorduring subsequent iterations, was chosen as tentimes the inverse of the maximum number of equa-tions that any one interfacial character wasinvolved in (here, 0.0585). The iteration processwas stopped when the change in the sum of themagnitude of all of the deviation vectors is lessthan 1% of the change during the first iteration(here, 279 steps). Because the observed grainboundary types are not distributed uniformly, some

cells are poorly populated and this creates non-physical discontinuities in the capillarity vectorfield. We eliminate these discontinuities after thefinal step by smoothing the result as follows. Thevalue of x in each cell is replaced by the averageof that vector and the vectors in the adjacent cells.Finally, for each grain boundary type, the energy(gx·n) was taken as the average of all 36 symmetri-cally equivalent cells that correspond to the bound-ary type.

The capillarity vector reconstruction method wasoriginally tested on simulated data containing 2× 105 triple junctions [17]. Since our experimentaldata set contains about an order of magnitudefewer triple junctions, we have again tested theperformance of the method on a smaller simulateddata set. Following Ref. [17], simulated data isgenerated using a model function for the grainboundary energy over all five parameters that isbased on a uniform energy distribution with cuspsat certain misorientations, shaped by analogy to theRead–Shockley (RS) [20] expression for theenergy of low angle boundaries. In the presentcase, the function contained one cusp at a mis-orientation of 60° about [1 1 1] with a boundaryplane normal to [1 1 1] and another at a misorien-tation of 45° about [1 1 0] with a boundary planenormal to [1 1 0]. The model function also had aspecial cusp at zero misorientation that was inde-pendent of the boundary plane. Simulated obser-vations from 2 × 104 triple junctions were gener-ated using this function. To simulate experimentalerrors, all of the model observations were per-turbed by a random angle generated from a Gaus-sian distribution with a standard deviation of 5°.The energy reconstruction was conducted using thesame set of discrete grain boundary types, relax-ation parameter, and convergence criterion as theactual measurements.

The reconstructed and model energies are com-pared in Fig. 2. Note that while the trends in themodel function are reproduced, the magnitudes ofthe energy differences are not. In particular, thereconstructed energy has cusps that are more shal-low than those in the model function. Based on thissimulation, we conclude that the energy recon-structed from our experimental data will exhibit the


Fig. 2. (a) Model and (b) reconstructed energies for boundaries with a 45° rotation about [1 1 0]. (c) Model and (d) reconstructedenergies for 60° misorientation about [1 1 1].

same qualitative features as the true function, butit will be less anisotropic.

3. Results

Results for the low angle boundaries formed byrotations about �1 1 0� are illustrated in Fig. 3.When the relative energies (a) are compared to thepopulation (b), it is clear that there is an approxi-mate inverse correlation between the energies andthe populations. The RS [20] model predicts thatthe energies of grain boundaries with low anglemisorientations will vary with the density andenergy of the dislocations that compensate for themisfit. While the famous RS dependence of theenergy on the misorientation will not be visible

because of the relatively coarse discretization, theinfluence of the grain boundary plane on the energyis expected to be observed at a fixed misorien-tation. Therefore, we can determine if the RSmodel provides a plausible explanation for theobservations in the following way.

Given a specific type of grain boundary (�g,n),and three non-coplanar Burgers vectors (bi),Frank’s [21] formula can be used to determine adensity of these dislocations needed to make theboundary. In the analysis that follows, we willassume that edge and screw dislocations have thesame energy per length and that the energy per unitarea of the boundary is proportional to the dislo-cation density (r). Following RS [22], we calculatethe vectors Ni = b∗

i × u�n(n·b∗i × u) which are

simultaneously perpendicular to the ith dislocation


Fig. 3. (a) The reconstructed energies, (b) the observed population, and (c) geometrically necessary dislocation densities for a 5°misorientation about [1 0 1]. The quantities are plotted in stereographic projection, with the [1 0 0], [1 1 0], and [1 1 1] directionsmarked with a solid black, gray, and white circle, respectively.

line and the boundary plane normal (n). In theexpression for Ni, u is the misorientation axis andb∗

i are reciprocal vectors. For example, b∗1 = (b2

× b3) / (b1·(b2 × b3)) and b∗2 and b∗

3 are obtained bypermuting subscripts. For small misorientationangles (q), the magnitude of Ni is 1/(qDi), whereDi is the distance between adjacent dislocationlines of the ith type. Therefore |Ni|q is the numberof dislocation lines cut per length of Ni. To get adislocation density in the interface (r), we considerhow many dislocations lines are cut by the three Ni

in a circular area of interface with a unit radius, r

r �

�i

2rqNi

pr2 �2qpr�i

Ni. (2)

In practice, we have more than one possible set ofthree non-coplanar dislocations, so we perform thecalculations for all sets, and select the smallestdensity. To complete the calculation, we have tospecify the types of dislocations. Based on obser-vations of deformed magnesia [23] and other rocksalt compounds [24], we assume that the bound-aries are made up of dislocations of the type b =a /2 � 1 1 0 �. We also explored the possibilitythat dislocations with b = a /2 � 1 0 0 � couldcontribute to boundary formation. However, thenon-coplanar sets of dislocations with the mini-mum density for each boundary never containedthis second type of dislocation.

The result of this calculation for a 5° misorien-tation around the [1 1 0] axis is shown in Fig. 3c.The minimum dislocation density occurs for tilt

boundaries with (1 1 0) and (1 1 0) planes.Within the resolution of our partitioning scheme,these are indistinguishable from symmetric tiltboundaries (the ideal symmetric case has ninclined from (1 1 0) and (1 1 0) by 2.5°). Theseorientations are also positions of relatively lowenergy (Fig. 3a) and high population (Fig. 3b) inthe experimental data. The highest dislocation den-sities are found for the pure twist boundaries on(1 1 0) and (1 1 0) planes, positions of maximumenergy and minimum population.

Fig. 4 shows the relative energies of grainboundaries with misorientations of 10°, 20°, and30° around the [1 1 0] axis. In each case, the ener-gies are relatively low for boundaries with planesnear {1 0 0} orientations. If one compares theseresults to the grain boundary populations for thesame boundaries (for example, compare Fig. 8a inpart I with Fig. 4b in the present paper), there isan obvious inverse correlation. Boundary planeorientations with relatively low grain boundaryenergy correspond to orientations with relativelyhigh populations near {1 0 0} and the higherenergy orientations have relatively low popu-lations. To illustrate that these relationships persistthroughout the data set, we compare all boundariesin a fixed range of energy to the population of eachtype of boundary (l) and to the minimum incli-nation of the boundary normal from the �1 0 0�direction (q100). For this comparison, we considerthe boundary types corresponding to the center ofthe cells in the five parameter space. After discret-izing the range of energies into equal partitions, we


Fig. 4. The reconstructed energies for boundaries with misorientations of (a) 10°, (b) 20°, and (c) 30° around the [1 1 0] axis. Theenergies, which are in arbitrary units (a.u.), are plotted in stereographic projection.

determined the mean and standard deviation ofln(l + 1) and q100 for all boundaries in each par-tition. This procedure allows us to examine the cor-relation over the entire space. The results illus-trated in Fig. 5a show that as g increases, q100

increases and l decreases. A second way to lookat these data is to average the energy over all mis-orientations, and plot the dependence of this mis-orientation averaged energy as a function ofboundary plane, as in Fig. 5b. While the averagingcompresses the dynamic range of the data, the{1 0 0} orientations are clear minima.

To quantify the extent to which the recon-structed energy function correlates to the distri-bution, we use the Spearman rank-order correlationcoefficient, rs [25]. The values of rs range from �1to 1, which indicate perfect negative and positivecorrelation, respectively, and for rs = 0, no corre-lation exists. The comparison is based on values ofthe distribution (l(�g, n)) that correspond to thecenter of each cell in the parameter space. Compar-ing the reconstructed energies to the frequency ofoccurrence of each type, we find an rs equal to–0.77; this indicates a high degree of (negative)correlation and a marked relationship between thetwo distributions [26]. It should be noted that thisstrong negative correlation arises principally fromthe three misorientation parameters. In otherwords, if we average over the boundary planes andcompare the population and the misorientationdependence of the reconstructed energies, the cor-relation rises to a remarkably high value of rs =�0.98. On the other hand, if we examine the corre-lation over only the two inclination parameters by

Fig. 5. (a) The grain boundary population (squares) and theminimum inclination of the boundary normal from a �1 0 0�direction (circles) plotted as a function of the reconstructedgrain boundary energy. For each quantity, the average of allvalues within a range of 0.022 a.u. is represented by the point;the bars indicate one standard deviation above and the mean.(b) The misorientation averaged reconstructed grain boundaryenergies plotted in stereographic projection.


comparing the reconstructed energies to nor-malized values of the distribution (λ(�g, n) /l(�g)), we find rs to be –0.40, which indicates alow to moderate inverse correlation.

Our ability to determine the misorientationdependence of the energy more accurately than theinclination dependence is related both to the differ-ence in the experimental uncertainties in measuringthese parameters and to the difference in the totalvariance of the energy over these parameters. Theestimated experimental errors in the misorientationcalculation (� 2°) are significantly less than thesize of partitions in the five-dimensional domainof distinct boundary types (~10°). On the otherhand, the uncertainty in the determination of theboundary plane normal can be larger than the sizeof the partition (the uncertainty varies with incli-nation from the sample normal and has a maximumvalue of about 15° for planes in the zone of thesurface normal). Furthermore, we expect theenergy to vary over a wider range of values as afunction of misorientation than as a function ofboundary plane orientation. Nevertheless, there isa high degree of (negative) correlation between thereconstructed relative grain boundary energies andthe frequency of occurrence of specific types ofboundaries over the five-parameter domain.

The data show that grain boundaries with{1 0 0} planes have lower energies and occur morecommonly in this microstructure. This behavior isreminiscent of the orientation dependence of thesurface energy of MgO. Earlier measurements ofthe surface energy anisotropy of magnesia, con-ducted in our laboratory on the same sample, indi-cated that the (1 0 0) plane has the minimumenergy. The energies of the (1 1 0) and (1 1 1) sur-faces are 7% and 17% higher than (1 0 0), respect-ively [27]. Since these relative surface energies areconsistent with the increase in the density ofbroken bonds, we take the surface energy ani-sotropy of MgO to be a quantitative indicator ofthe extent to which the ideal bonding is disruptedby forming a particular surface.

If the grain boundary energy also reflects thelocal disruption in bonding, then we can create ahypothetical energy by summing of the energies ofthe two surfaces planes comprising the interface,and subtracting a binding energy that reflects the

gain in bonds that occurs when the two surfacesare brought together to create the grain boundary[28]. For general boundaries, where the repeatunits of the two free surfaces do not form a com-mensurate structure (i.e., there is no special coinci-dence relation) and the misorientation cannot becreated by a set of non-overlapping dislocations,the density of reformed bonds should be more orless constant. In these cases, the binding energyshould also be constant and this means that theenergy anisotropy of general boundaries should bereproduced by summing the measured energies ofthe two free surfaces that make up the boundary.We therefore compare hypothetical boundary ener-gies computed based only on measured surfaceenergies to the experimental observations to testthe idea that the density of broken bonds in theinterface determines the grain boundary energyanisotropy.

The results in Fig. 6 compare the hypotheticalenergy, the measured energy, and the observedpopulation for boundaries with 10°, 20°, and 30°of misorientation around [1 0 0]. These plots showthat the anisotropy of the hypothetical grain bound-ary energy is correlated with the reconstructedgrain boundary energy and has an inverse corre-lation with the grain boundary population. For{1 0 0} boundary planes, the sum of the surfaceenergies is low, as are the reconstructed energies.The distribution, on the other hand, exhibits max-ima at these positions.

To quantify the extent to which the hypotheticalenergy function correlates to the distribution, weagain use the Spearman rank-order correlationcoefficient, rs [26]. Here it must be recognized thatthe hypothetical energies are expected to reproduceonly the dependence of the energy on the grainboundary plane (g(n)); they will not decrease sys-tematically with misorientation angle. Therefore,we compare the hypothetical energies to the nor-malized values of the distribution (l(�g, n) /l(�g)) and find rs equal to �0.76. This value alsoindicates a high degree of (negative) correlation.

It is noteworthy that the hypothetical energiesare more strongly correlated to the inclinationdependence of the grain boundary population thanthe reconstructed energy. As illustrated by theresults in Fig. 2, the energy reconstruction pro-


Fig. 6. The hypothetical grain boundary energies based on the surface energies at fixed misorientations corresponding to rotationsabout [1 0 0] of (a) 10°, (b) 20°, and (c) 30°. For comparison, the reconstructed energies (d–f) and the observed population (g–i) atthe same fixed misorientations are also shown. The quantities are plotted in stereographic projection.

cedure reproduces the general trends in the modelfunction, but underestimates the relative aniso-tropies. Furthermore, while the simulated data wasevenly distributed in the five-dimensional space,the real data was concentrated such that for axesdifferent from �1 1 1�, the data is sparse forrotations greater then 35°. In these poorly popu-lated regions, the reconstructed energy is notexpected to be accurate and this contributes to thelower degree of negative correlation with the incli-nation distribution.

4. Discussion

For the case of low misorientation angle grainboundaries, the conventional RS [20] model pro-vides a reasonable explanation for the relativelylow energy of the symmetric tilt boundaries on(1 1 0) and (1 1 0) planes. This is because incrystals with the rock salt crystal structure in gen-eral, and in MgO in particular, dislocations withBurgers vectors of a /2 � 1 1 0 � are most com-mon. While any boundary can be formed by three


dislocations with non-coplanar Burgers vectors,boundaries with the minimum dislocation densityoccur when the misorientation can be created bya single set of parallel dislocations. In networkscontaining two or more types of dislocations, anti-parallel contributions to the displacement canceleach other so that more dislocations are requiredto create the same angular rotation of the lattice.The effect of this is seen most clearly in Fig. 3c,where the minimum dislocation density occurs forpure, symmetric tilt boundaries on (1 1 0) and(1 1 0) planes. In each case, the boundary can becreated by a single set of parallel edge dislocations.As the plane is changed along the zone of pure tiltboundaries to orientations where the boundary isasymmetric, more and more dislocations with asecond Burgers vector must be added and this sys-tematically increases the density to a maximumwhen n is ~90° from the symmetric plane. Finally,we recall that the dislocation model applied hereis based on the assumption that boundaries are for-med by the minimum number of geometricallynecessary dislocations. While the results in Fig. 3indicate that this model provides a consistentexplanation for the present observations, it doesnot rule out the possible existence of more complexmicroscopic defect structures.

The low energy of {1 1 0} tilt boundaries is alsoconsistent with previous studies of crystals with therock salt structure that have been annealed afterdeformation. In these cases, annealing usuallyleads to polygonization, where the resulting lowangle grain boundaries have tilt character and mostfrequently lie on {1 1 0} type planes [23]. In thepresent case, these boundaries also dominated thepopulation, but they appeared as a result of graingrowth rather than polygonization. This is consist-ent with the idea that grain growth involves notonly the elimination of total grain boundary area,but the elimination of the highest energy bound-aries while the lower energy boundaries are pre-served.

Our observations of high angle grain boundariesshould be compared to the theory of the coincidentsite lattice (CSL), which has had a profound influ-ence on grain boundary studies for the past threedecades [29]. There is some evidence that bound-aries with these special geometries have lower

energies [3,4,10,12–14] and even that materialswith high fractions of these boundaries haveproperties that are superior to materials with morerandom grain boundary networks [30]. However,there are also observations showing that some lowΣ CSL boundaries (boundaries with highcoincidence) do not have reduced energies[3,7,8,11]. While it is common practice to considera boundary to be “special” on the basis of mis-orientation alone, it must be recognized that a CSLboundary only has high interface coincidence oncertain coherent planes, and these are relativelyrare. For example, using the discretization schemeemployed here, there are 6561 distinct boundarytypes and fewer than 1% correspond to coherentlow sigma CSL boundaries (Σ�19).

The findings illustrate that in MgO, the CSLboundaries are nearly insignificant in comparisonto the boundaries terminated on one side by a{1 0 0} plane. The latter boundaries comprise a fargreater part of the five-dimensional space (theyoccur at every misorientation), they are on averagepopulated with an equal or greater frequency, andthey have lower energies. The present results seemto contradict previous observations of boundariesin MgO smoke, which have extraordinarily highfractions of low Σ CSL twist boundaries [31].However, one must remember that grain bound-aries in the present case are formed as the resultof extensive grain growth. The twist boundariesfound in aggregates of smoke particles are thoughtto arise from a nucleation process in the hot plumeof burning Mg [31]. It should be noted, however,that one characteristic of the smoke aggregates isremarkably similar to the present observations: thegrain boundaries in MgO smoke are always termin-ated on a {1 0 0} plane.

In comparing the relative importance of the CSLboundaries and the boundaries terminated by{1 0 0} planes, it is instructive to examine theenergy and distribution of boundary planes at theΣ5 misorientation. Fig. 7 shows the population andenergy of pure tilt Σ5 boundaries. The boundariesthat couple the highest population and the lowestenergy are asymmetric tilt boundaries of the type{1 0 0}/{4 3 0}. At the positions of the coherent,high coincidence symmetric tilts, {1 2 0}/{1 2 0},the population reaches a minimum and the energy


Fig. 7. Comparison of the reconstructed energies (squares)and the observed distribution (circles) for Σ5 tilt boundaries.The quantities are plotted in 5° intervals as a function of theangle between the boundary plane normal and(0 1 0)/ (0 4 3). For reference, the location of the symmetrictilts, both the {3 1 0} and {2 1 0} types, as well as the asymmet-ric boundaries terminated by {1 0 0} planes are indicated. Notethat the function has mirror symmetries at the symmetric tiltorientations, {2 1 0} and {3 1 0}. The deviations from this aredue to the relatively large intervals of the plotted points.

reaches a maximum. This indicates that asymmet-ric boundaries with {1 0 0} planes are favored overthe high coincidence boundaries. The other sym-metric boundary in this tilt system, which has onlypartial coincidence, is the {3 1 0}/{3 1 0}. Thisboundary has an energy similar to the asymmetric{1 0 0} /{4 3 0}, but occurs at a minimum in thepopulation. It should also be noted that the{1 0 0} /{1 0 0} twist boundary (not shown),which also has a high coincidence, does occur ata maximum in the population. However, {1 0 0}/{1 0 0} twist boundaries exhibit high populationsand low energies at all �1 0 0� misorientations(see Fig. 6) and we therefore argue that the specialnature of this particular boundary derives from itsdual {1 0 0} interface plane rather than its parti-cular misorientation.

The observation that grain boundaries with{1 0 0} planes have relatively low energies at allmisorientations can be quite simply explained byassuming that the grain boundary energy is pro-portional to the surface energies of the planes thatmake up the boundary, which is also proportionalto the density of broken bonds on these planes. Thedominant influence of broken bond density or free

surface characteristics on the grain boundaryenergy may be surprising in comparison to theoriesfor grain boundary structures based on repeat unitsthat are distinct from arrangements occurring in thebulk or on a free surface [32,33]. However, earliermicroscopic studies of Σ5 grain boundaries inbicrystals of the isostructural compound, NiO, areentirely consistent with our conclusions. Merkleand Smith [34] noted that while repeat structurescan be located in Σ5 tilt boundaries, they occuronly in boundary segments where the symmetric{2 1 0}/{2 1 0} configuration is realized and thatasymmetric segments along the same boundarieswith the {4 3 0}/{1 0 0} configuration were morecommon. It is worth noting that the success ofstudies seeking the properties and structure of lowΣ CSLs or repeat units in highly symmetric bound-aries does not imply that these interfaces areimportant, special, or even particularly commoninterfaces in real polycrystalline microstructures.

The dominance of boundaries with {1 0 0} sur-face planes and the ability of the surface energyanisotropy to predict the relative energies of grainboundaries is reasonable, since the excess energyof the interface derives from incomplete bonding.The {1 0 0} plane has the highest coordinationnumber (fewest broken bonds) and for the mostcommon case of a general boundary where the sur-face repeat units in parallel planes on either sideof the interface do not form a periodic structure, itis easy to imagine that a relatively low energywould result by fixing one of the planes in the{1 0 0} orientation. The fact that the surfaceenergy is a good predictor of the grain boundaryenergy must derive from the fact that the amountof energy gained by bringing two surface planestogether to form a boundary is more or less con-stant. Boundary atoms in incommensurate inter-faces experience a wide range of possible atomicconfigurations and this range is not expected todepend strongly on the type of boundary. As longas the details of the atomic relaxations in a specificboundary are as widely varied along the aperiodicstructure as they are from boundary to boundary,they average to about the same value. Thus, thesurface energies provide a good measure of thedensity of unsatisfied bonds in the interface and,therefore, the grain boundary energy.


The findings suggest that for MgO, it is possibleto create a model for the grain boundary energybased on the knowledge of the dislocation structureand the surface energy anisotropy. Knowing themost prevalent types of dislocations in the system,the five parameter energy distribution can be esti-mated for low angle boundaries using RS [23]theory, and the surface energy anisotropy can beused to compute the energies of all higher angleboundaries. The generality of such a model is cur-rently not known. It is possible that because ofmore directional bonding in covalent systems andshorter range forces in metallic materials, the sur-face energy will not be a reliable predictor of thegrain boundary energy in other types of materials.We plan to test this in the future. However, themodel has several appealing features. First, the sur-face energy is only a two parameter function andit is therefore much easier to measure than the fiveparameter grain boundary function. Second, if thegrain boundary energy anisotropy can really beestimated from the energies of the two surfaces thatcomprise the boundary, then the grain boundaryenergy is reduced to a function of only four para-meters. In other words, the energy for high anglemisorientations is independent of rotations aboutn. We note that the approximate independence ofthe grain boundary energy on the twist angle (whenthe misorientation axis and n are parallel) for highmisorientation angle grain boundaries has beenobserved in Cu [7].

The extended annealing of the sample prior toanalysis was intended to ensure local thermodyn-amic equilibrium at the boundaries. While thispaper has focused on the geometry and energies ofthe boundaries, we assume that chemical equilib-rium was also achieved. In other words, we assumethat boundaries with the same five macroscopicallyobservable parameters have the same composition.As part of a separate experiment, the compositionof this sample has already been studied and theprincipal impurity is Ca, which is present at a bulkconcentration of 0.2% [35]. We are presently sur-veying grain boundary compositions and prelimi-nary results, based on the analysis of hundreds ofboundaries, indicate that Ca segregation is aniso-tropic and that the most frequently observedboundaries accumulate the least Ca [36].

Perhaps the most exciting outcome of this studyis the inverse correlation between the grain bound-ary energy and the population. If this correlationis a general feature of late stage microstructures,then the direct observation of the grain boundarypopulation will be a good way to estimate the rela-tive grain boundary energy. The correlation impliesthat as grain growth proceeds, higher energyboundaries are eliminated preferentially so thatlower energy boundaries persist and dominate thepopulation. However, the idea that the grainboundary population is refining toward a lowerenergy can be misleading, since no grain bound-aries actually persist throughout the entire graingrowth process; whenever a grain disappears, newboundaries appear. For example, as the grain sizeincreases from 1 to 100 µ, only 1 in 106 grainssurvive and the probability that any boundarybetween two crystals in the initial configurationsurvives this process is extraordinarily small. Theinverse correlation of the population and energy isprobably better explained as a kinetic phenomenon.The time that any given grain boundary exists inmicrostructure is, on average, equal to the time ittakes to move through a neighboring grain of aver-age size. High mobility boundaries move throughneighbors in a shorter time interval and are, there-fore, annihilated more rapidly than low mobilityboundaries. Assuming energy and mobility to becorrelated, then, at any instant in time, there shouldbe more low energy (mobility) boundaries thanhigh energy (mobility) boundaries.

5. Summary

In the five-dimensional domain of grain bound-ary types, the boundary energy and populationexhibit an inverse correlation. This implies thatduring grain growth, higher energy boundaries arepreferentially eliminated in favor of low energyboundaries. At low misorientation angles, the dis-tribution and energy of the observed grain bound-aries can be explained by the conventional Read–Shockley dislocation model. One overarchingcharacteristic of the distribution in MgO is that atall misorientations, the boundaries with {1 0 0}interface planes have relatively low energies and


have higher populations, even in comparison tomore symmetric, high coincidence boundaries. Bysumming the energies of the two surfaces thatcomprise the boundary, a hypothetical energy func-tion is created that reproduces the inclinationdependence of the distribution and energy.

Acknowledgements

This work was supported primarily by theMRSEC program of the National Science Foun-dation under Award Number DMR-0079996.

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