Direct observation of grain rotations during coarseningof a semisolid Al–Cu alloyJules M. Dakea, Jette Oddershedeb, Henning O. Sørensenc, Thomas Werza, J. Cole Shattoa, Kentaro Uesugid,Søren Schmidtb,1, and Carl E. Krill IIIa,1

aInstitute of Micro and Nanomaterials, Ulm University, 89081 Ulm, Germany; bDepartment of Physics, Technical University of Denmark, 2800 KongensLyngby, Denmark; cNano-Science Center, Department of Chemistry, University of Copenhagen, 2100 Copenhagen, Denmark; and dResearch and UtilizationDivision, Japan Synchrotron Radiation Research Institute, Sayo, Hyogo 679-5198, Japan

Edited by Frans Spaepen, Harvard University, Cambridge, MA, and accepted by Editorial Board Member Tobin J. Marks August 3, 2016 (received for reviewFebruary 19, 2016)

Sintering is a key technology for processing ceramic and metallicpowders into solid objects of complex geometry, particularly in theburgeoning field of energy storage materials. The modeling ofsintering processes, however, has not kept pace with applications.Conventional models, which assume ideal arrangements of constit-uent powders while ignoring their underlying crystallinity, achieveat best a qualitative description of the rearrangement, densification,and coarsening of powder compacts during thermal processing.Treating a semisolid Al–Cu alloy as a model system for late-stagesintering—during which densification plays a subordinate role tocoarsening—we have used 3D X-ray diffraction microscopy to trackthe changes in sample microstructure induced by annealing. Theresults establish the occurrence of significant particle rotations, drivenin part by the dependence of boundary energy on crystallographicmisorientation. Evidently, a comprehensive model for sintering mustincorporate crystallographic parameters into the thermodynamic driv-ing forces governing microstructural evolution.

3D microstructural evolution | sintering | Ostwald ripening | grain rotation |x-ray imaging

Whether we realize it or not, our daily lives are marked byencounters with sintering, ranging from natural rock for-

mations and glaciers to artificial products like the porcelain fromwhich we eat or the amalgam fillings in many teeth. Sintering isan efficient method for combining loose granular precursors intosolid objects of predetermined shape at relatively low tempera-ture, circumventing the processing route of melting, casting, andmachining. The applications of sintering are widespread. For ex-ample, it is the predominant method for producing bulk technicalceramics (1), and, thanks to advances in powder metallurgy, sinteringis of growing importance in the fabrication of metals, as well (2). Thefocus in recent years on nanostructured materials—specifically, onmaterials for energy storage and conversion—has intensified thescientific investigation and modeling of sintering processes. Thelatter find application in the production of fuel cells (3) and batteryelectrodes (4). In other applications, such as catalysis (5), sinteringcan be highly undesirable, as it reduces the connectivity of pores andthe free surface of nanoparticles. In all of these cases, we need a firmunderstanding of sintering fundamentals to achieve and retain de-sired materials properties during thermal processing.The reduction in free energy that accompanies the removal of a

powder compact’s surface area is the driving force behind sinter-ing (2, 6). When two particles come into contact, two free surfacesare replaced by a single solid/solid boundary. If the particles arecrystalline, the new interface is a grain boundary, the (excess)energy of which is generally on the order of one-third of that of a(single) free surface of equal area (6); consequently, the sinteringprocess results in a net release of energy. As free surface areadecreases, the powder compact becomes denser, which usuallyleads to geometric shrinkage (6). Sintering is generally dividedinto stages characterized by various observable phenomena: for-mation and widening of a neck at the contact between particles;

particle rearrangement (i.e., rigid-body motion); shrinkage of theinterconnected network of pores, followed by pore closure and theshrinkage/coarsening of isolated pores; and grain growth. Sometype of diffusion (surface, grain boundary, lattice) is typically as-sumed to govern the kinetics of each these component processes,several of which may occur simultaneously. Because the compu-tational simulation of any of these complex processes is a dauntingtask in its own right, it should come as no surprise that currentalgorithms fall well short of predictive accuracy for the sintering ofrealistic powder compacts (2, 7).Our current inadequate understanding of sintering kinetics

can in part be traced to the experimental difficulty of trackingmicrostructural changes in three dimensions (8). With recentadvances in 3D imaging, however, researchers can now performtime-resolved measurements of the true size, shape, and inter-connectivity of particles and pores (9–12), from which it may bepossible to pin down the atomic-level mechanism(s) underlyingeach stage of sintering. Of particular interest in this regard is the3D rearrangement of particles that occurs during early-stagesintering (8, 13), which can significantly affect subsequent den-sification but is hardly accessible from 2D cross-sections. In arecent experiment, Grupp et al. (12) observed individual particlerotations as large as 8° during the sintering of loosely packed

Significance

Computational modeling of materials phenomena promises toreduce the time and cost of developing new materials andprocessing techniques—a goal made feasible by rapid advancesin computer speed and capacity. Validation of such simulations,however, has been hindered by a lack of 3D experimental dataof simultaneously high temporal and spatial resolution. In thisstudy, we exploit 3D X-ray diffraction microscopy to capturethe evolution of crystallographic orientations during particlecoarsening in a semisolid Al–Cu alloy. The data confirm a long-standing hypothesis that particle rotation is driven (in part) bythe dependence of grain boundary energy on misorientation.In addition, the results constitute an experimental foundationfor testing the predictive power of next-generation computa-tional models for sintering.

Author contributions: J.M.D., S.S., and C.E.K. designed research; J.M.D., J.O., H.O.S., J.C.S.,K.U., S.S., and C.E.K. performed research; J.M.D., J.O., H.O.S., T.W., S.S., and C.E.K. ana-lyzed data; and J.M.D., J.O., and C.E.K. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission. F.S. is a Guest Editor invited by the EditorialBoard.

Data deposition: The 3D reconstructed datasets reported in this paper can be downloaded incompressed text format from the Materials Data Facility (https://www.materialsdatafacility.org) using the link http://dx.doi.org/doi:10.18126/M25P46.1To whom correspondence should be addressed. Email: carl.krill@uni-ulm.de or ssch@fysik.dtu.dk.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1602293113/-/DCSupplemental.

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monocrystalline Cu spheres. The changes in 3D orientation ofthese spherical grains were extracted from marker holes drilledby a focused ion beam. In the literature, such particle rotations aremost frequently attributed to the formation of asymmetric necks(13); however, the dependence of grain boundary energy oncrystallographic misorientation (the discontinuous change in lat-tice orientation at the boundary separating adjoining particles) canalso induce particle rotations (Fig. 1), as captured under labora-tory conditions by so-called ball-on-plate experiments (14–17).Indeed, Grupp et al. (12) proposed that the particle rotations intheir experiment resulted primarily from a misorientation-baseddriving force, but experimental evidence for the validity of thisassertion was lacking, as the measurement technique they used—absorption-contrast X-ray computed tomography—is insensitiveto the crystal orientation of sintering particles.With the advent of diffraction-based imaging techniques, such as

3D X-ray diffraction (3DXRD) microscopy (18, 19), high-energydiffraction microscopy (20), and diffraction-contrast tomogra-phy (21), it is now feasible to perform nondestructive, 3D mappingof the shapes and lattice orientations of thousands of crystalliteswithin a polycrystalline specimen. From a sequence of such mea-surements applied to the same sample, we can track its micro-structural evolution and determine whether a particular processingprotocol [such as a heat treatment (22–24) or a mechanical de-formation (25–27)] induced grain rotation. Furthermore, we cancorrelate changes in grain orientation to the morphologies andmisorientations of a rotating grain’s immediate neighbors.In this article, we report the application of 3DXRD micros-

copy to the detection and statistical analysis of grain rotationstaking place during the coarsening of a semisolid Al–Cu alloy(crystalline Al particles embedded in a liquid phase). Second-phase materials that melt below the sintering temperature arefrequently added to powder compacts to accelerate densification(28, 29). Owing to their 100% density, our Al–Cu specimens areexpected to mimic the late-stage behavior of a liquid-phase sin-tering system, which is marked by a negligible amount of residualporosity and by coarsening of the solid particles via Ostwald

ripening (30, 31). In such a sample, we expect to encounter few,if any, particle rearrangements driven by the aforementionedformation of asymmetric necks, as each particle is surrounded bya liquid layer. Effectively, such a microstructure isolates crys-tallographic misorientation as a potential driving force for par-ticle rotation. Over the course of stepwise heat treatments atvarious temperatures, we have mapped the internal volumes ofAl–Cu specimens using 3DXRD as well as X-ray computed to-mography (CT). The resulting datasets yield time-dependenttrajectories for the size and orientation of hundreds of particlesat various volume fractions of the liquid phase. Not only do thesetrajectories represent a time-resolved observation of lattice ori-entation variations in a semisolid bulk polycrystal, but they alsoreveal—through a statistical analysis—the simultaneous occur-rence of two distinct mechanisms for particle rotation.

ResultsTwo fully dense, cylindrical samples of Al–5 wt% Cu were annealedin the semisolid state, one at 630 °C and the other at 619 °C. Ap-plying the lever rule to the Al–Cu phase diagram (32), we estimatethe volume fraction VV of the solid, Al-rich phase to be 70% at630 °C and 82% at 619 °C; the remaining volume in each sample isoccupied by a liquid having a higher concentration of Cu than in thesolid phase. From here on, we refer to the two samples as S-70 andS-82, respectively. Sample S-70 was annealed in three steps for anoverall duration of 60 min, and sample S-82 in five steps for a totalof 75 min.

Coarsening. Exemplifying the particle growth that occurredduring annealing, Fig. 2 A and B show the initial and final mi-crostructures of S-70 after reconstruction of 3DXRD data.Because each solid particle in these specimens is mono-crystalline, we refer to solid particles alternatively as grains. Thedistribution of grain misorientations in each sample was foundto agree quite closely with the Mackenzie distribution for ran-domly orientated cubic crystals (33), implying an absence ofsignificant crystallographic texture (Fig. 2C). At the temperaturesof annealing, most grains are separated from neighboring particlesby a relatively thick layer of liquid, but upon cooling these liquidregions solidify into thinner layers of elevated Cu concentration,which can be detected by high-resolution CT but not by 3DXRD(Fig. S1). For this reason, each grain in a 3DXRD reconstructionappears to be in direct contact with many grain neighbors (13.5on average), all of which are included in the calculation of mis-orientation distributions in Fig. 2C. In the semisolid state, theaverage number of direct particle/particle contacts lies well below13.5, which can be deduced from tomographic reconstructions ofsimilar Al–5 wt% Cu samples measured at elevated temperatureby CT (Fig. S2).In Fig. 2 as well as in subsequent 3D images, we index

the color of each grain to its crystallographic orientation bymapping the components of the Rodrigues vector representationof orientation (34) to RGB color coordinates. Consequently, themisorientation of any two grains having similar colors is small.The irregularity visible at the top and bottom surfaces of thecylindrical sample results from the removal of grains that werenot completely contained within the volume irradiated by X-rays.As a result, the total volume under consideration varies slightlyover the course of annealing. The number of grains withinsample S-70 was initially 682, which dropped to 418 by the endof the final annealing step; likewise, sample S-82 initially con-tained 637 grains, which eventually fell to 431. Thus, the numberof grains in each sample was large enough to warrant a statisticalanalysis of the coarsening behavior.In the semisolid state of a material, solid particles coarsen

primarily by Ostwald ripening, which is characterized by a netdiffusive flux of atoms (through the liquid) from smaller to largerparticles. Growth trajectories of 100 grains, chosen at random, are

misorientation

boun

dary

ene

rgy

2~1 J/m

Fig. 1. Schematic illustration of a grain rotation being driven by the mis-orientation dependence of the grain boundary energy γ. At misorientationsϑ above ∼15°, γ takes on an approximately constant value, except nearspecial coincident site lattice (CSL) boundaries. For two crystallites with initialmisorientation ϑ1 < 15°, a relative rotation to ϑ2 < ϑ1 leads to a significantreduction in γ, owing to the gradient dγ=dϑ.

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plotted in Fig. 3 A and B for S-70 and S-82, respectively. Here, wequantify the grain size using the equivalent radius R of a sphere ofequal volume. Superimposed in black is the evolution of the av-erage grain size, which increases monotonically in both samples.The growth rates of individual particles are plotted against

their normalized size in Fig. 3C. For both S-70 and S-82, thecloud of data points takes on a nonlinear shape with a slightlyupward curvature. This behavior contradicts the conventionalLifshitz, Slyozov, and Wagner (LSW) model (35, 36) for Ostwaldripening, according to which the time derivative of a particle’svolume, dV=dt∼R2dR=dt, varies linearly with the normalizedgrain size, intersecting the horizontal axis at a normalized radiusof unity. However, the LSW prediction was derived for a van-ishingly small volume fraction of the coarsening phase—that is,in the limit VV → 0. Experiments (37) and simulations (38, 39)performed at higher volume fractions agree qualitatively with thenonlinearity evident in Fig. 3C.

Grain Rotations. Just as with the grain size, we can track the ori-entation of each grain throughout the course of microstructuralevolution. After each ex situ annealing step, we corrected the setof grain orientations determined by 3DXRD microscopy forglobal sample rotation and translation (Materials and Methods);consequently, any difference in an individual grain’s orientationwith respect to a prior orientation must reflect a rotation Δθ ofthe grain’s lattice planes.* In Fig. 4A, we plot trajectories of grainrotations for samples S-70 and S-82. In this figure, we also in-clude the rotation trajectories (green) from a third, single-phase

Al–1 wt% Mg sample. Like the two Al–Cu specimens, the Al–Mgsample was heated ex situ and characterized by 3DXRD betweenannealing steps. Because the Al–Mg sample is completelyrecrystallized, with a relatively large grain size (∼200 μm) and noliquid phase (VV = 1), its constituent crystallites must be highlyconstrained; at the moderate annealing temperature of 400 °C, itis safe to assume that any grain rotations that occur in this sampleduring coarsening are well below the detection limit of the3DXRD technique (∼ 0.05°) (40). Therefore, based on repeatedmeasurements of this sample, we can estimate the uncertaintyinherent in the Δθ values yielded by our experimental procedure.From measurement to measurement, the median Δθ obtained forgrains in the Al–Mg sample was only 0.06°, with 95% of the (ap-parent) rotations lying below 0.18°. In contrast, for samples S-70and S-82, we find median changes in grain orientation of 0.37° and0.17°, respectively. From this comparison, we conclude that thevast majority of orientation changes measured in the semisolidsamples represent true grain rotations.To compare grain rotations from annealing intervals of dif-

ferent duration, we construct probability plots for the rate ofrotation Δθ=Δt (Fig. 4B). Evidently, the rotation rate increasessignificantly with the volume fraction of the liquid phase: themedian rotation rate in sample S-82 (18% liquid phase) is0.014°=min, whereas in sample S-70 (30% liquid phase) one-halfof the grains rotate faster than 0.023°=min.Furthermore, we observe a roughly inverse correlation between

rotation rate and grain size (Fig. 4C). Comparison of prematurelyterminating trajectories plotted in Fig. 4A to their correspondingdata points in Fig. 4C reveals that large rotations often occur justbefore a shrinking grain vanishes. For example, the three highest-rotation trajectories of S-70 at the 30-min annealing mark corre-spond to the top three red points plotted in Fig. 4C. The associatedgrains disappeared by the time the next 3DXRD measurement wasperformed (at 60 min).

DiscussionThe results plotted in Fig. 4 indicate that grain rotations—significantboth in number and magnitude—occur during the coarsening ofsemisolid Al–Cu at liquid volume fractions of 18% and 30%. Wepostulate that such rotations are driven by gradients in the grainboundary energy γ with respect to misorientation ϑ. As illustratedschematically in Fig. 1, grain rotations can lower the free energybecause γ is a function of the change in lattice orientation across

0 20 40 600

0.01

0.02

0.03

0.04

freq

uenc

y

(deg)

S70 (initial)S70 (final)Mackenzie

60 minat 630°C

500 m

A B C

Fig. 2. Evolution of microstructure and texture in sample S-70. The (A) initial and (B) final 3D reconstructions illustrate the microstructural evolution thatoccurred during a 60-min heat treatment at 630 °C. Grain colors are indexed to the components of the Rodrigues vector representation of grain orientation;consequently, similarly colored grains have similar crystal orientations. (C) A comparison of the initial and final (nearest-neighbor) misorientation distribu-tions to the Mackenzie distribution for randomly oriented cubic crystals reveals little evidence for crystallographic texture.

*A note on our terminology and notation for grain orientations and misorientations: Theorientation of any single-crystalline grain can be specified by the transformation re-quired to rotate the axes of a reference coordinate system into the unit cell axes ofthe grain. In the axis-angle representation, we denote the rotation angle of such atransformation by the symbol θ, which is inherently greater than or equal to zero.Changes in grain orientation from an earlier time t to a later time t +Δt can also bedescribed in axis-angle notation, but in this case we use Δθ to denote the (usually small)rotation angle. (Again, the latter quantity is nonnegative.) In a similar manner, themisorientation between two grains can be described as a rotation of the unit cell axesof one grain into those of the other; to avoid confusion with the previous quantities, wedenote a given misorientation relationship by the corresponding rotation angle ϑ andchanges in misorientation by δϑ= ϑðt+ΔtÞ− ϑðtÞ. Unlike Δθ, the quantity δϑ can take onpositive or negative values. When calculating the orientation as well as the misorienta-tion, we account for the cubic symmetry of the fcc Al lattice, thus ensuring that theminimum equivalent rotation angle is reported.

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the boundary (41, 42). When the difference in orientation betweengrains is larger than about 15°, γ takes on a roughly constant value;however, when the misorientation lies within the low-angle rangeðϑ< 15°Þ, the boundary energy decreases ever faster as ϑ approaches

zero (43). In the latter case, the steep energy gradient would beexpected to generate equal and oppositely directed torquesacting on the grains in contact, which could potentially inducea relative rotation.To assess the relevance of this mechanism to our data, we begin

by examining pairs of neighboring grains for which the postulateddriving force might be expected to generate a significant torqueat the common boundary. Although strong gradients in γ havebeen measured near misorientation values of special coincidentsite lattice (CSL) boundaries and twin boundaries, the shape(depth and width) of these so-called “energy cusps” is not wellunderstood and may disappear at elevated temperatures (42, 44).In addition, aluminum is known to have a high stacking faultenergy, which limits twinning (45). In light of these considerations,we restrict our attention in the following discussion to rotatinggrains having nearest neighbors of low misorientation ðϑ< 15°Þ,for which the deep minimum that appears in γ as ϑ→ 0 is ame-nable to description by the Read–Shockley dislocation model (41).An example of a rotating grain with a low-misorientation

neighbor ðϑ= 10.1°Þ is shown in Fig. 5. The two grains pictured inthis image are indexed to similar colors because their latticeorientations are nearly the same. During a 30-min anneal at630 °C, the lower grain in Fig. 5A rotates by Δθ= 5.4° to theposition shown in Fig. 5B, thereby changing its misorientationwith respect to the upper grain by δϑ=−2.7° [see footnote (*) foran explanation of the notation used here]. Of course, our pos-tulated mechanism of misorientation energy-driven grain rota-tion presupposes that the boundary between the two grains is wetby no more than a few atomic layers of liquid, as a thicker liquidlayer would essentially decouple the grains from each other. Ingeneral, a “dry” boundary can form between two neighboringgrains whenever the solid/solid boundary energy is lower than theoverall energy of the alternative solid/liquid boundaries: that is,γsol=sol < 2γsol=liq. This criterion is especially likely to be fulfillednear the deep minimum in γsol=sol at low misorientation angles.Inspection of the corresponding high-resolution CT scan (Fig.5C) reveals no sign of an enrichment of Cu (light-gray shading)at the shared boundary, as would be expected had it been wet bythe liquid phase. Moreover, a well-defined (formerly) liquid layeris seen to coat the lower grain’s other boundaries, despite thesample having been measured at room temperature. At the el-evated temperature of annealing, the liquid layers are signifi-cantly thicker than shown in Fig. 5C (Fig. S2), which apparentlyfacilitates the accommodation of a rigid-body rotation of 5.4°.Consequently, we conclude that the example shown in Fig. 5 isconsistent with a grain rotation having been driven by a gradientin the grain boundary energy landscape.We now examine the set of all grains that have a single im-

mediate neighbor with initial misorientation less than 15°. Ifgradients in grain boundary energy drive grain rotations, then wewould expect the misorientations of these low-ϑ grain pairs totend toward smaller values over time, at least on average. Tominimize effects of melting and solidification and better isolatethe influence of boundary energy on grain rotations, we consideronly the misorientation changes that occurred during the finalannealing step, which was the longest interval and equal in dura-tion (30 min) for both specimens. In sample S-70, 72 grain pairsfulfill the low-ϑ criterion, and a histogram of the measured changesin misorientation, δϑ=ϑfinal −ϑinitial (red bars), is plotted in Fig. 6.A positive value of δϑ indicates that the misorientation betweenneighboring grains increased, whereas a negative value signifiesbehavior consistent with the operation of a grain boundary energy-induced torque. There is a clear skew toward negative values, withthe mean change in misorientation lying at −0.25°.If, instead of restricting our selection of grain pairs to those with

a low misorientation neighbor, we examine changes in mis-orientation relative to neighboring grains chosen at random, thenwe obtain a significantly more symmetric histogram for δϑ (gray

A

B

C

Fig. 3. Kinetics of coarsening: growth trajectories for 100 grains selected atrandom from samples (A) S-70 and (B) S-82. Most grains larger than themean grain size hRi (black diamonds) are observed to grow, whereas theopposite is true of smaller-than-average-sized grains. (C) In both samples, aslight upward curvature is evident in plots of the volumetric growth rateagainst the normalized grain size.

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bars in Fig. 6) with a mean of −0.014°. (Because the overallsampling includes not only general grain pairs but also low-ϑneighbors, we might expect the mean value to lie slightly belowzero.) Considering this distribution to be the parent population forδϑ, we apply Student’s t test (46) to calculate the probability ofobtaining a mean misorientation change equal to −0.25° for72 grain pairs drawn at random from the parent population. Thet test value of only 0.22% indicates that we can reject the nullhypothesis—that the low-ϑ population and the parent populationhave the same mean value for δϑ—with over 99% certainty.Applying the same analysis to the low-ϑ grain pairs of sampleS-82, we obtain qualitatively similar results with a certainty of97% for rejecting the null hypothesis.This statistical analysis points to gradients in the grain boundary

energy landscape as having been instrumental in driving the rotationof grains in semisolid Al–Cu. Further evidence in support of thisconclusion may be gained from an examination of the rate at whichgrains sharing a low-angle boundary reduce their misorientation.Analogous to the dependence of a curved boundary’s migration rateon capillary pressure, a grain’s rotation rate can be related to thetorque that acts on its periphery (47). Because each grain in thepairs considered above has only one low-angle neighbor, we assumeits remaining boundaries to be wet by the liquid phase duringannealing, leaving only a single dry boundary of area A and gradientdγ=dϑ to generate a torque:

τ=Adγdϑ

. [1]

The speed of rotation is then given by

dϑdt

=Mrτ=MrAdγdϑ

, [2]

where the proportionality constant Mr denotes a rotational mobil-ity. Shewmon (48, 49) derived an expression for Mr for a sphericalparticle having a single boundary in contact with a plate, under theassumption that the material transport needed to accommodateparticle rotation occurs by lattice diffusion. Inserting Shewmon’sexpression into Eq. 2, we obtain

dϑdt

=8DLΩkBTa3

dγdϑ

, [3]

with lattice diffusion coefficient DL, atomic volume Ω, Boltzmannconstant kB, absolute temperature T, and radius a of the contactarea between sphere and plate. More recently, this equation hasbeen extended to the case of columnar grain structures in single-phase polycrystals (47, 50); however, owing to the decoupling pro-vided by a wetting liquid phase, we expect Shewmon’s originalexpression to be more applicable to our specimens. In addition,the results of recent molecular dynamics simulations (51) areconsistent with the dependency dϑ=dt∝ a−3. For these reasons,we use Eq. 3 to obtain an order-of-magnitude estimate for thegrain rotation rate.Assuming that the grain boundary energy increases from zero

to 460 mJ/m2 (52) over the angular range 0–15°, we obtaindγ=dϑ≈ 30.7 mJ/(m2·deg). Inserting a= 30 μm into Eq. 3 along withtypical values (53) for the diffusion coefficient and atomic volume ofAl at T = 630 °C, we estimate a rotation rate dϑ=dt≈ 0.003 °/min forsample S-70. Thus, over the course of a 30-min anneal, one couldexpect low-angle grain pairs in this specimen to reduce their mis-orientation by about 0.09° on average. Although this is less than themean value observed experimentally (−0.25°), the estimate is cer-tainly plausible, considering the sensitivity of the right-hand side ofEq. 3 to the contact radius a and the difficulty in determiningan accurate value for a from ex situ measurements. Adjusting T andDL in Eq. 3 to reflect the conditions of the 30-min anneal of sample

B

A

C

Fig. 4. Grain rotations during annealing. (A) Rotation trajectories recordedfor grains in samples S-70 and S-82, manifesting individual rotations upto 12°. At each annealing time, the angle of rotation Δθ is calculated relativeto the given grain’s previous orientation. For comparison, we also plot ro-tation trajectories recorded for grains in a single-phase Al–1 wt% Mg alloy(S-100). Assuming that no grains actually rotate in the latter sample, wetreat the Δθ values as a measure for the uncertainty inherent to the 3DXRDtechnique. (B) Log-normal probability plots for the rotations shown in A,demonstrating that both the median rotation rate Δθ=Δt and the maximumobserved rotation rate increase with volume fraction of the liquid phase(0–30%). (C) Grains of smaller normalized radius R=hRi have a tendency torotate more rapidly.

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S-82, we obtain an expected reduction in misorientation of 0.07° forthis specimen’s low-angle grain pairs, which is nearly identical to themeasured mean value of −0.067°. The overall good agreementbetween experimental values and those predicted by Eq. 3 lendsfurther credence to our interpretation of grain rotations as beingdriven by gradients in boundary energy.This mechanism cannot, however, explain all of the observed

rotations. Many grains rotate without having low-ϑ neighbors,and some grains even rotate away from their low-ϑ neighbors(i.e., δϑ> 0). The symmetry about δϑ= 0 that is evident in thehistogram for randomly chosen pairs of contiguous grains (graybars in Fig. 6) indicates that most rotating grains are equally likelyto undergo a misorientation increase or decrease with respect to aneighboring grain. This observation suggests that there is a ran-dom component to the observed grain rotations, as well.Over the course of interrupted annealing experiments, sev-

eral mechanisms may plausibly lead to undirected grain rota-tion. For example, the melting and solidification processes thattake place during each sample’s transitional heating and cool-ing stages could conceivably induce rotations. When our Al–Cuspecimens are heated above the eutectic temperature (548 °C),a liquid phase begins to form in small pockets, which grow andcoalesce upon further heating, eventually spreading throughoutthe sample. The expansion and flow of this liquid will likelycause slight rearrangements of the ensemble of solid grains (54).Similarly, upon cooling, nonuniform solidification of the liquidphase could induce small rotations, reminiscent of those thatresult from asymmetric neck formation during solid-state sinter-ing (13). Because melting and solidification are but transitoryphenomena in this experiment, the magnitude of the resultingrotations should not depend on the duration of annealing.For this reason, we were surprised to observe that trajectories

of nearest-neighbor grain misorientation tend to spread out withannealing time, indicating that at least one additional source ofrandom grain rotation must be active under isothermal condi-tions. As with the gray histogram of Fig. 6, we can calculate thechange in misorientation δϑ for all nearest-neighbor grain pairsbefore and after each annealing step. By construction, such aquantity is insensitive to the collective grain motion associatedwith slumping or flotation of grain clusters in the semisolid state.As mentioned above, a random rotation of a given particle isequally likely to result in a positive or negative change in mis-orientation with respect to a grain neighbor; indeed, whenever

misorientation histograms are compiled for a statistical samplingof grain pairs taken from our specimens, the histograms are in-variably centered about zero. There is, however, a clear increasein the width of such distributions with annealing time, which wequantify by plotting the standard deviation (SD) of δϑ against Δt(Fig. 7). Because a thicker average liquid layer allows for greaterrotational freedom, it seems reasonable to observe larger mis-orientation changes in the sample containing the greater volumefraction of liquid (S-70).With increasing annealing time, coarsening induces significant

changes in the local neighborhood of any given particle, as nearbygrains disappear or neighbor switching events occur. Such phe-nomena are likely accompanied by redistribution of the liquid phaseand rigid-body grain movement, which, although smaller in magni-tude, is similar in nature to the grain rearrangement that occursduring the early stages of liquid-phase sintering (2, 28). Contributing

Fig. 6. Changes in misorientation of low-angle grain pairs in sample S-70compared with the same quantity evaluated for randomly chosen grainneighbors. (Red) Histogram of the change in misorientation, δϑ= ϑfinal − ϑinitial,of 72 grain boundaries with ϑinitial < 15°, showing a distinct bias toward negativevalues. (Gray) Histogram of the same distribution for 418 randomly chosenboundaries in the same sample. The overall distribution of misorientationchanges is much more symmetric than that of the subset of low-ϑ boundaries.

100 m

30 min

A B C

Fig. 5. Example grain rotation at a low-angle boundary in sample S-70. (A) Two grains are shown that share a low-angle grain boundary of initial mis-orientation ϑ=10.1°. Over the course of a 30-min anneal at 630 °C, the lower grain rotates by 5.4° about the indicated axis. (B) In the final configuration ofthis grain pair, the misorientation between the two grains has decreased to 7.4°. (C) A CT slice through the two grains shows no enrichment of Cu at theircommon boundary, indicating that the respective crystal lattices are in direct contact rather than separated by a liquid layer. The formation of a solid/solidboundary is a prerequisite for boundary energy-driven grain rotation.

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to such grain shifts could be strains coupled to the migration of dryboundaries between particles (55, 56). Buoyancy forces, on the otherhand, are expected to be of negligible importance, because thedensities of the solid and liquid phases differ by only a small amount(<1% for S-70 and <3% for S-82) (57).The above findings support the hypothesis that gradients in

boundary energy can drive grain rotations during sintering, but—atleast in semisolid specimens—such rotations are superimposed on abackground of random orientation fluctuations. Our analysis iden-tifies two primary modes of particle rotation. First, grains meeting atlow-angle boundaries have a strong tendency to undergo a decreasein misorientation over time, because of torques generated by thesteep gradient in grain boundary energy at ϑ< 15°. Second, solidgrains suspended in a liquid matrix manifest undirected, randomfluctuations in orientation, likely as a result of thermal processing(sequential melting and solidification) as well as the changes in localgrain environment that accompany coarsening. To the extent thatrandom rotations enable a grain to sample a certain region in mis-orientation space, the second mechanism may actually contribute tothe formation of the low-angle grain boundaries that are conduciveto the first mechanism.As a result, even though the semisolid Al–Cu samples exam-

ined in this study were fully compact (negligible pore density)and devoid of significant crystallographic texture, we observed anotable number of grain rotations driven at least in part by theconcomitant reduction in interfacial energy. During the earlyagglomeration and rearrangement stages of sintering, when thedensity is well below 100% [compare with rotations measured byGrupp et al. (12) and McDonald et al. (23)], we would expectboundary energy-driven grain rotations to be even more prevalent.The same may hold true for samples manifesting a higher-than-average population of low-energy boundaries—for example, incases of significant texture or twinning. In addition, the depth ofthe energy minimum could play a role: aluminum is characterizedby a rather moderate variation in boundary energy with misori-entation, compared with that of other metals (45, 52, 58). In ma-terials with larger high-angle boundary energies and, hence, deeper

cusps in the energy landscape—for example, Cu or Ni (58)—theinfluence of boundary energy on grain rotations should be evenstronger than in Al. For these reasons, we anticipate that grainrotation-assisted phenomena will influence densification andcoarsening in a large number of systems, with correspondingimplications for the accurate modeling of sintering and semisolidmaterials processing.

Materials and MethodsSample Preparation. Cylindrical samples 1.4mm in diameter were cut via sparkerosion from a high-purity plate of Al–5 wt% Cu, which had been homog-enized for 24 h at 500 °C and cold-rolled to a thickness reduction of 50%.The longitudinal axis of the cylindrical samples was chosen to lie in therolling direction. After spark erosion, the cylinders were annealed for 20 minat 640 °C, at which temperature the sample consists of solid, Al-rich particlessurrounded by a liquid matrix of higher Cu concentration. In our experience,this final processing step in the semisolid state removes most of the crys-tallographic texture that arises from cold rolling.

By adjusting the annealing temperature within the solid–liquid co-existence region, we can set the volume fraction of solid-phase particles,VV , to any desired value: at 619 °C, VV = 0.82 (sample S-82), whereas for anannealing temperature of 630 °C, VV =0.70 (sample S-70). These volumefractions were calculated from the Al–Cu phase diagram (32) using a cor-rection described by Werz et al. (37) for small portions of liquid trappedinside the solid grains. Isothermal heat treatments—performed in air—ranged from 10 to 30 min in duration. Sample S-70 was annealed for threeincreasingly long intervals: first 10 min, then 20 min, and finally 30 min.Sample S-82 was initially annealed three times for 10 min, followed bysingle anneals for 15 min and then 30 min.

Data Acquisition. Before the first anneal and after each subsequent heattreatment, we carried out room temperature 3DXRD and CT measurementsof each sample at beamline BL20XU of the synchrotron radiation facilitySPring-8 in Japan. A monochromatic (32-keV) box beam, 300 μm in heightand 1.6 mm in width, was used to illuminate the samples during 3DXRD map-ping. Diffraction data were collected in 10 adjacent layers with 50 μm of verticaloverlap between layers, resulting in a total measured height of 2.55 mm. Foreach layer, we collected both far-field (Hamamatsu C7942CA-22 detector with50-μm pixel size) and near-field (Photron SA2 detector with 4.56-μm pixel size)data, rotating the specimen by 360° in 0.48° steps. The scan time for one layerof far-field data collection was 750 s and 75 s per layer for near-field data. Theflux density for both scans was approximately 1013 photons/(s ·mm2).

High-resolution (1-μm) CT scans of the same sample volume were per-formed in three layers, each of dimensions 1.0 mm × 1.6 mm (height bywidth), again with 50-μm overlap in the vertical direction. For the collectionof CT data, the flux density was reduced to 1012 photons/(s ·mm2), and 1,800projections were recorded while rotating the sample 180°. Under theseconditions, we were able to scan one layer in 340 s. Contrast in the tomo-graphic reconstructions arises from the fact that, upon cooling, the liquidphase surrounding the solid, aluminum-rich particles solidifies to a mixtureof the θ-phase (Al2Cu) and Al. Although the presence of Cu atoms in the θ-phasegenerates strong absorption contrast, the thickness of the θ-phase regions isonly a fraction of that of the liquid phase, which tends to retreat upon coolinginto triple junctions. Particle boundaries that are observed at room temperatureto be decorated by the θ-phase were certainly wet by the liquid phase duringannealing; on the other hand, particle boundaries unmarked by the θ-phasemay or may not have been wet by the liquid matrix at elevated temperature.

Data Reconstruction and Grain Tracking. The reconstruction of individual layersof 3DXRD data began with the filtering and segmentation (59, 60) of far-fielddiffraction spots, which were then indexed to obtain the orientation and centerof mass of each grain in the irradiated volume. To this end, we iterativelyapplied GrainSpotter (61) from the FABLE suite of software (62) until no furthergrains were found. The 3D morphology of the grains is next determined usingthe near-field diffraction images and a 3D generalization of GrainSweeper (63,64). This second algorithm carries out voxel-by-voxel growth of the seed grainssupplied by GrainSpotter. For voxels with more than one plausible grain as-signment, the orientation is chosen that results in the most complete diffrac-tion pattern. Here, completeness is defined as the number of diffraction spotsobserved at spatially correct locations on the near-field detector divided by theexpected number of diffraction spots when the voxel in question is assigned toa particular lattice orientation (19). Near the center of a grain, we typicallyobtain completeness values close to 100%, but this value falls off sharply in thevicinity of grain boundaries.

10 15 20 25 30t (min)

0

0.2

0.4

0.6

0.8S

D o

f (

deg)

S-70

S-82

S-100

Fig. 7. Width of the histogram of misorientation changes versus annealingtime. In samples S-70 and S-82, the SD of the distribution of δϑ values isfound to increase with the duration Δt of the annealing interval. Evidently, agreater volume fraction of liquid phase is conducive to larger rotations. Thefinal point for S-70 at 30 min is equal to the SD of the gray histogram shownin Fig. 6. Error bars denote the uncertainty in determination of the SD(calculated from the variance of the variance). For the Al–1 wt% Mg alloywith no liquid phase (S-100), we expect the misorientation between any twograins to remain fixed over time; therefore, the nonzero value of this sam-ple’s SD must reflect experimental uncertainties that are characteristic of the3DXRD technique.

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The 10 individually reconstructed 3DXRD layers were then stitched to-gether, yielding a 3D map of the entire illuminated volume. For each timestep, each grain in the specimen was assigned a unique label. Grains weretracked backward in time using an automated algorithm, comparing thecurrent time step to the previous one. If more than 70% of the volume of agrain in the current time stepwas found to overlapwith a grain in the previoustime step, and if the misorientation between the two grains was smallerthan 1.5 °, then the two grains were assumed to be identical. Grains nottracked automatically were viewed individually in three orthogonalcross-sections to determine the correct tracking label.

Registration of Datasets. Care was taken to remount samples in the synchrotronbeam in the same position following each ex situ annealing step, but we wereunable to avoid slight translation of the specimen’s center of mass (20–50 μm)and rotation of the sample (1–5°), primarily about its longitudinal axis. Tocorrect for global translations, we used MATLAB (65) to register thereconstructed 3D volumes. By averaging the measured rotation of each trackedgrain between annealing time steps, we were also able to determine the globalrotation of the sample. The inverse of this rotation was then applied to both the3D spatial reconstruction and its constituent grains to register their individu-al orientations. These corrected values were then used to determine the true

rotation of each grain from its change in orientation over the course of eachannealing step. To estimate the uncertainty in our measurement of grain ro-tations, we tested a single-phase Al–1 wt%Mg sample, for which we expect anygrain rotations to be far smaller than the detection limit. Applying the sameexperimental and reconstruction procedures to this specimen, we obtained amedian variation in grain orientation between measurements of 0.06°. Underthe assumption that no grains in this sample actually rotate, we conclude thatthe distribution of measured changes in orientation reflects the reproducibilitywith which we can determine a crystallite’s true orientation. The value of 0.06°is close to other values reported in the literature for the 3DXRD technique (40).

ACKNOWLEDGMENTS. We thank the Japan Synchrotron Radiation ResearchInstitute for the allotment of beam time on beamline BL20XU of SPring-8(Proposals 2012A1427 and 2013A1506). We are most grateful to D. Molodovof the Institute of Physical Metallurgy and Metal Physics, Rheinisch-Westfälische Technische Hochschule Aachen, for providing the Al–5 wt% Cuand Al–1 wt% Mg specimens and to the Deutsche Forschungsgemeinschaftfor granting financial support through the National Science Foundation/Deutsche Forschungsgemeinschaft Materials World Network Program(Project KR 1658/4-1). J.O., H.O.S., and S.S. acknowledge the Danish Agencyfor Science, Technology and Innovation via Danscatt for travel support.

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