Direct observation of grain rotations during coarsening of a semisolid Al–Cu alloy Jules M. Dakea, Jette Oddershedeb, Henning O. Sørensenc, Thomas Werza, J. Cole Shattoa, Kentaro Uesugid, Søren Schmidtb,1, and Carl E. Krill IIIa,1 a Institute of Micro and Nanomaterials, Ulm University, 89081 Ulm, Germany; bDepartment of Physics, Technical University of Denmark, 2800 Kongens Lyngby, Denmark; cNano-Science Center, Department of Chemistry, University of Copenhagen, 2100 Copenhagen, Denmark; and dResearch and Utilization Division, 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 review February 19, 2016)
Sintering is a key technology for processing ceramic and metallic powders into solid objects of complex geometry, particularly in the burgeoning field of energy storage materials. The modeling of sintering processes, however, has not kept pace with applications. Conventional models, which assume ideal arrangements of constituent powders while ignoring their underlying crystallinity, achieve at 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-stage sintering—during which densification plays a subordinate role to coarsening—we have used 3D X-ray diffraction microscopy to track the changes in sample microstructure induced by annealing. The results establish the occurrence of significant particle rotations, driven in part by the dependence of boundary energy on crystallographic misorientation. Evidently, a comprehensive model for sintering must incorporate crystallographic parameters into the thermodynamic driving forces governing microstructural evolution.
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hether we realize it or not, our daily lives are marked by encounters with sintering, ranging from natural rock formations and glaciers to artificial products like the porcelain from which we eat or the amalgam fillings in many teeth. Sintering is an efficient method for combining loose granular precursors into solid objects of predetermined shape at relatively low temperature, circumventing the processing route of melting, casting, and machining. The applications of sintering are widespread. For example, it is the predominant method for producing bulk technical ceramics (1), and, thanks to advances in powder metallurgy, sintering is of growing importance in the fabrication of metals, as well (2). The focus in recent years on nanostructured materials—specifically, on materials for energy storage and conversion—has intensified the scientific investigation and modeling of sintering processes. The latter find application in the production of fuel cells (3) and battery electrodes (4). In other applications, such as catalysis (5), sintering can be highly undesirable, as it reduces the connectivity of pores and the free surface of nanoparticles. In all of these cases, we need a firm understanding of sintering fundamentals to achieve and retain desired 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 sintering (2, 6). When two particles come into contact, two free surfaces are replaced by a single solid/solid boundary. If the particles are crystalline, 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 sintering process results in a net release of energy. As free surface area decreases, the powder compact becomes denser, which usually leads to geometric shrinkage (6). Sintering is generally divided into stages characterized by various observable phenomena: formation and widening of a neck at the contact between particles;
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particle rearrangement (i.e., rigid-body motion); shrinkage of the interconnected network of pores, followed by pore closure and the shrinkage/coarsening of isolated pores; and grain growth. Some type of diffusion (surface, grain boundary, lattice) is typically assumed to govern the kinetics of each these component processes, several of which may occur simultaneously. Because the computational simulation of any of these complex processes is a daunting task in its own right, it should come as no surprise that current algorithms fall well short of predictive accuracy for the sintering of realistic powder compacts (2, 7). Our current inadequate understanding of sintering kinetics can in part be traced to the experimental difficulty of tracking microstructural changes in three dimensions (8). With recent advances in 3D imaging, however, researchers can now perform time-resolved measurements of the true size, shape, and interconnectivity of particles and pores (9–12), from which it may be possible to pin down the atomic-level mechanism(s) underlying each stage of sintering. Of particular interest in this regard is the 3D rearrangement of particles that occurs during early-stage sintering (8, 13), which can significantly affect subsequent densification but is hardly accessible from 2D cross-sections. In a recent experiment, Grupp et al. (12) observed individual particle rotations as large as 8° during the sintering of loosely packed Significance Computational modeling of materials phenomena promises to reduce the time and cost of developing new materials and processing techniques—a goal made feasible by rapid advances in computer speed and capacity. Validation of such simulations, however, has been hindered by a lack of 3D experimental data of simultaneously high temporal and spatial resolution. In this study, we exploit 3D X-ray diffraction microscopy to capture the evolution of crystallographic orientations during particle coarsening in a semisolid Al–Cu alloy. The data confirm a longstanding hypothesis that particle rotation is driven (in part) by the dependence of grain boundary energy on misorientation. In addition, the results constitute an experimental foundation for testing the predictive power of next-generation computational 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. analyzed 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 Editorial Board. Data deposition: The 3D reconstructed datasets reported in this paper can be downloaded in compressed text format from the Materials Data Facility (https://www.materialsdatafacility. org) using the link http://dx.doi.org/doi:10.18126/M25P46. 1
To whom correspondence should be addressed. Email: [email protected] 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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misorientation Fig. 1. Schematic illustration of a grain rotation being driven by the misorientation dependence of the grain boundary energy γ. At misorientations ϑ above ∼15°, γ takes on an approximately constant value, except near special coincident site lattice (CSL) boundaries. For two crystallites with initial misorientation ϑ1 < 15°, a relative rotation to ϑ2 < ϑ1 leads to a significant reduction in γ, owing to the gradient dγ=dϑ.
Dake et al.
Results Two fully dense, cylindrical samples of Al–5 wt% Cu were annealed in the semisolid state, one at 630 °C and the other at 619 °C. Applying the lever rule to the Al–Cu phase diagram (32), we estimate the volume fraction VV of the solid, Al-rich phase to be 70% at 630 °C and 82% at 619 °C; the remaining volume in each sample is occupied by a liquid having a higher concentration of Cu than in the solid phase. From here on, we refer to the two samples as S-70 and S-82, respectively. Sample S-70 was annealed in three steps for an overall duration of 60 min, and sample S-82 in five steps for a total of 75 min. Coarsening. Exemplifying the particle growth that occurred during annealing, Fig. 2 A and B show the initial and final microstructures of S-70 after reconstruction of 3DXRD data. Because each solid particle in these specimens is monocrystalline, we refer to solid particles alternatively as grains. The distribution of grain misorientations in each sample was found to agree quite closely with the Mackenzie distribution for randomly orientated cubic crystals (33), implying an absence of significant crystallographic texture (Fig. 2C). At the temperatures of annealing, most grains are separated from neighboring particles by a relatively thick layer of liquid, but upon cooling these liquid regions 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 reconstruction appears to be in direct contact with many grain neighbors (13.5 on average), all of which are included in the calculation of misorientation distributions in Fig. 2C. In the semisolid state, the average number of direct particle/particle contacts lies well below 13.5, which can be deduced from tomographic reconstructions of similar Al–5 wt% Cu samples measured at elevated temperature by CT (Fig. S2). In Fig. 2 as well as in subsequent 3D images, we index the color of each grain to its crystallographic orientation by mapping the components of the Rodrigues vector representation of orientation (34) to RGB color coordinates. Consequently, the misorientation of any two grains having similar colors is small. The irregularity visible at the top and bottom surfaces of the cylindrical sample results from the removal of grains that were not completely contained within the volume irradiated by X-rays. As a result, the total volume under consideration varies slightly over the course of annealing. The number of grains within sample S-70 was initially 682, which dropped to 418 by the end of the final annealing step; likewise, sample S-82 initially contained 637 grains, which eventually fell to 431. Thus, the number of grains in each sample was large enough to warrant a statistical analysis of the coarsening behavior. In the semisolid state of a material, solid particles coarsen primarily by Ostwald ripening, which is characterized by a net diffusive flux of atoms (through the liquid) from smaller to larger particles. Growth trajectories of 100 grains, chosen at random, are PNAS | Published online September 26, 2016 | E5999
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ripening (30, 31). In such a sample, we expect to encounter few, if any, particle rearrangements driven by the aforementioned formation of asymmetric necks, as each particle is surrounded by a liquid layer. Effectively, such a microstructure isolates crystallographic misorientation as a potential driving force for particle rotation. Over the course of stepwise heat treatments at various temperatures, we have mapped the internal volumes of Al–Cu specimens using 3DXRD as well as X-ray computed tomography (CT). The resulting datasets yield time-dependent trajectories for the size and orientation of hundreds of particles at various volume fractions of the liquid phase. Not only do these trajectories represent a time-resolved observation of lattice orientation variations in a semisolid bulk polycrystal, but they also reveal—through a statistical analysis—the simultaneous occurrence of two distinct mechanisms for particle rotation.
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monocrystalline Cu spheres. The changes in 3D orientation of these spherical grains were extracted from marker holes drilled by a focused ion beam. In the literature, such particle rotations are most frequently attributed to the formation of asymmetric necks (13); however, the dependence of grain boundary energy on crystallographic misorientation (the discontinuous change in lattice orientation at the boundary separating adjoining particles) can also induce particle rotations (Fig. 1), as captured under laboratory conditions by so-called ball-on-plate experiments (14–17). Indeed, Grupp et al. (12) proposed that the particle rotations in their experiment resulted primarily from a misorientation-based driving force, but experimental evidence for the validity of this assertion was lacking, as the measurement technique they used— absorption-contrast X-ray computed tomography—is insensitive to 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-energy diffraction microscopy (20), and diffraction-contrast tomography (21), it is now feasible to perform nondestructive, 3D mapping of the shapes and lattice orientations of thousands of crystallites within a polycrystalline specimen. From a sequence of such measurements applied to the same sample, we can track its microstructural evolution and determine whether a particular processing protocol [such as a heat treatment (22–24) or a mechanical deformation (25–27)] induced grain rotation. Furthermore, we can correlate changes in grain orientation to the morphologies and misorientations of a rotating grain’s immediate neighbors. In this article, we report the application of 3DXRD microscopy to the detection and statistical analysis of grain rotations taking place during the coarsening of a semisolid Al–Cu alloy (crystalline Al particles embedded in a liquid phase). Secondphase materials that melt below the sintering temperature are frequently added to powder compacts to accelerate densification (28, 29). Owing to their 100% density, our Al–Cu specimens are expected to mimic the late-stage behavior of a liquid-phase sintering system, which is marked by a negligible amount of residual porosity and by coarsening of the solid particles via Ostwald
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Fig. 2. Evolution of microstructure and texture in sample S-70. The (A) initial and (B) final 3D reconstructions illustrate the microstructural evolution that occurred 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 distributions to the Mackenzie distribution for randomly oriented cubic crystals reveals little evidence for crystallographic texture.
plotted in Fig. 3 A and B for S-70 and S-82, respectively. Here, we quantify the grain size using the equivalent radius R of a sphere of equal volume. Superimposed in black is the evolution of the average 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, the cloud of data points takes on a nonlinear shape with a slightly upward curvature. This behavior contradicts the conventional Lifshitz, Slyozov, and Wagner (LSW) model (35, 36) for Ostwald ripening, according to which the time derivative of a particle’s volume, dV =dt ∼ R2 dR=dt, varies linearly with the normalized grain size, intersecting the horizontal axis at a normalized radius of unity. However, the LSW prediction was derived for a vanishingly 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 the nonlinearity evident in Fig. 3C. Grain Rotations. Just as with the grain size, we can track the orientation of each grain throughout the course of microstructural evolution. After each ex situ annealing step, we corrected the set of grain orientations determined by 3DXRD microscopy for global sample rotation and translation (Materials and Methods); consequently, any difference in an individual grain’s orientation with respect to a prior orientation must reflect a rotation Δθ of the grain’s lattice planes.* In Fig. 4A, we plot trajectories of grain rotations for samples S-70 and S-82. In this figure, we also include the rotation trajectories (green) from a third, single-phase
*A note on our terminology and notation for grain orientations and misorientations: The orientation of any single-crystalline grain can be specified by the transformation required to rotate the axes of a reference coordinate system into the unit cell axes of the grain. In the axis-angle representation, we denote the rotation angle of such a transformation 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 be described 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, the misorientation between two grains can be described as a rotation of the unit cell axes of one grain into those of the other; to avoid confusion with the previous quantities, we denote a given misorientation relationship by the corresponding rotation angle ϑ and changes in misorientation by δϑ = ϑðt + ΔtÞ − ϑðtÞ. Unlike Δθ, the quantity δϑ can take on positive or negative values. When calculating the orientation as well as the misorientation, we account for the cubic symmetry of the fcc Al lattice, thus ensuring that the minimum equivalent rotation angle is reported.
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Al–1 wt% Mg sample. Like the two Al–Cu specimens, the Al–Mg sample was heated ex situ and characterized by 3DXRD between annealing steps. Because the Al–Mg sample is completely recrystallized, with a relatively large grain size (∼200 μm) and no liquid phase (VV = 1), its constituent crystallites must be highly constrained; at the moderate annealing temperature of 400 °C, it is safe to assume that any grain rotations that occur in this sample during coarsening are well below the detection limit of the 3DXRD technique (∼ 0.05°) (40). Therefore, based on repeated measurements of this sample, we can estimate the uncertainty inherent in the Δθ values yielded by our experimental procedure. From measurement to measurement, the median Δθ obtained for grains in the Al–Mg sample was only 0.06°, with 95% of the (apparent) rotations lying below 0.18°. In contrast, for samples S-70 and S-82, we find median changes in grain orientation of 0.37° and 0.17°, respectively. From this comparison, we conclude that the vast majority of orientation changes measured in the semisolid samples represent true grain rotations. To compare grain rotations from annealing intervals of different duration, we construct probability plots for the rate of rotation Δθ=Δt (Fig. 4B). Evidently, the rotation rate increases significantly with the volume fraction of the liquid phase: the median rotation rate in sample S-82 (18% liquid phase) is 0.014°=min, whereas in sample S-70 (30% liquid phase) one-half of 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 prematurely terminating trajectories plotted in Fig. 4A to their corresponding data points in Fig. 4C reveals that large rotations often occur just before a shrinking grain vanishes. For example, the three highestrotation trajectories of S-70 at the 30-min annealing mark correspond to the top three red points plotted in Fig. 4C. The associated grains disappeared by the time the next 3DXRD measurement was performed (at 60 min). Discussion The results plotted in Fig. 4 indicate that grain rotations—significant both in number and magnitude—occur during the coarsening of semisolid Al–Cu at liquid volume fractions of 18% and 30%. We postulate that such rotations are driven by gradients in the grain boundary energy γ with respect to misorientation ϑ. As illustrated schematically in Fig. 1, grain rotations can lower the free energy because γ is a function of the change in lattice orientation across Dake et al.
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Fig. 3. Kinetics of coarsening: growth trajectories for 100 grains selected at random from samples (A) S-70 and (B) S-82. Most grains larger than the mean grain size hRi (black diamonds) are observed to grow, whereas the opposite is true of smaller-than-average-sized grains. (C) In both samples, a slight upward curvature is evident in plots of the volumetric growth rate against the normalized grain size.
the boundary (41, 42). When the difference in orientation between grains is larger than about 15°, γ takes on a roughly constant value; however, when the misorientation lies within the low-angle range ðϑ < 15°Þ, the boundary energy decreases ever faster as ϑ approaches Dake et al.
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zero (43). In the latter case, the steep energy gradient would be expected to generate equal and oppositely directed torques acting on the grains in contact, which could potentially induce a relative rotation. To assess the relevance of this mechanism to our data, we begin by examining pairs of neighboring grains for which the postulated driving force might be expected to generate a significant torque at the common boundary. Although strong gradients in γ have been measured near misorientation values of special coincident site lattice (CSL) boundaries and twin boundaries, the shape (depth and width) of these so-called “energy cusps” is not well understood and may disappear at elevated temperatures (42, 44). In addition, aluminum is known to have a high stacking fault energy, which limits twinning (45). In light of these considerations, we restrict our attention in the following discussion to rotating grains having nearest neighbors of low misorientation ðϑ < 15°Þ, for which the deep minimum that appears in γ as ϑ → 0 is amenable 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 in this image are indexed to similar colors because their lattice orientations are nearly the same. During a 30-min anneal at 630 °C, the lower grain in Fig. 5A rotates by Δθ = 5.4° to the position shown in Fig. 5B, thereby changing its misorientation with respect to the upper grain by δϑ = −2.7° [see footnote (*) for an explanation of the notation used here]. Of course, our postulated mechanism of misorientation energy-driven grain rotation presupposes that the boundary between the two grains is wet by no more than a few atomic layers of liquid, as a thicker liquid layer would essentially decouple the grains from each other. In general, a “dry” boundary can form between two neighboring grains whenever the solid/solid boundary energy is lower than the overall energy of the alternative solid/liquid boundaries: that is, γ sol=sol < 2γ sol=liq. This criterion is especially likely to be fulfilled near the deep minimum in γ sol=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 by the liquid phase. Moreover, a well-defined (formerly) liquid layer is seen to coat the lower grain’s other boundaries, despite the sample having been measured at room temperature. At the elevated temperature of annealing, the liquid layers are significantly thicker than shown in Fig. 5C (Fig. S2), which apparently facilitates the accommodation of a rigid-body rotation of 5.4°. Consequently, we conclude that the example shown in Fig. 5 is consistent with a grain rotation having been driven by a gradient in the grain boundary energy landscape. We now examine the set of all grains that have a single immediate neighbor with initial misorientation less than 15°. If gradients in grain boundary energy drive grain rotations, then we would expect the misorientations of these low-ϑ grain pairs to tend toward smaller values over time, at least on average. To minimize effects of melting and solidification and better isolate the influence of boundary energy on grain rotations, we consider only the misorientation changes that occurred during the final annealing step, which was the longest interval and equal in duration (30 min) for both specimens. In sample S-70, 72 grain pairs fulfill the low-ϑ criterion, and a histogram of the measured changes in misorientation, δϑ = ϑfinal − ϑinitial (red bars), is plotted in Fig. 6. A positive value of δϑ indicates that the misorientation between neighboring grains increased, whereas a negative value signifies behavior consistent with the operation of a grain boundary energyinduced torque. There is a clear skew toward negative values, with the 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 misorientation relative to neighboring grains chosen at random, then we obtain a significantly more symmetric histogram for δϑ (gray
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bars in Fig. 6) with a mean of −0.014°. (Because the overall sampling includes not only general grain pairs but also low-ϑ neighbors, we might expect the mean value to lie slightly below zero.) Considering this distribution to be the parent population for δϑ, we apply Student’s t test (46) to calculate the probability of obtaining a mean misorientation change equal to −0.25° for 72 grain pairs drawn at random from the parent population. The t test value of only 0.22% indicates that we can reject the null hypothesis—that the low-ϑ population and the parent population have the same mean value for δϑ—with over 99% certainty. Applying the same analysis to the low-ϑ grain pairs of sample S-82, we obtain qualitatively similar results with a certainty of 97% for rejecting the null hypothesis. This statistical analysis points to gradients in the grain boundary energy landscape as having been instrumental in driving the rotation of grains in semisolid Al–Cu. Further evidence in support of this conclusion may be gained from an examination of the rate at which grains sharing a low-angle boundary reduce their misorientation. Analogous to the dependence of a curved boundary’s migration rate on capillary pressure, a grain’s rotation rate can be related to the torque that acts on its periphery (47). Because each grain in the pairs considered above has only one low-angle neighbor, we assume its remaining boundaries to be wet by the liquid phase during annealing, leaving only a single dry boundary of area A and gradient dγ=dϑ to generate a torque: τ=A
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The speed of rotation is then given by dϑ dγ = Mr τ = Mr A , dt dϑ
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where the proportionality constant Mr denotes a rotational mobility. Shewmon (48, 49) derived an expression for Mr for a spherical particle having a single boundary in contact with a plate, under the assumption that the material transport needed to accommodate particle rotation occurs by lattice diffusion. Inserting Shewmon’s expression into Eq. 2, we obtain dϑ 8DL Ω dγ = , dt kB Ta3 dϑ
Fig. 4. Grain rotations during annealing. (A) Rotation trajectories recorded for grains in samples S-70 and S-82, manifesting individual rotations up to 12 °. At each annealing time, the angle of rotation Δθ is calculated relative to the given grain’s previous orientation. For comparison, we also plot rotation 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, we treat the Δθ values as a measure for the uncertainty inherent to the 3DXRD technique. (B) Log-normal probability plots for the rotations shown in A, demonstrating that both the median rotation rate Δθ=Δt and the maximum observed rotation rate increase with volume fraction of the liquid phase (0–30%). (C) Grains of smaller normalized radius R=hRi have a tendency to rotate more rapidly.
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with lattice diffusion coefficient DL, atomic volume Ω, Boltzmann constant kB, absolute temperature T, and radius a of the contact area between sphere and plate. More recently, this equation has been extended to the case of columnar grain structures in singlephase polycrystals (47, 50); however, owing to the decoupling provided by a wetting liquid phase, we expect Shewmon’s original expression to be more applicable to our specimens. In addition, the results of recent molecular dynamics simulations (51) are consistent with the dependency dϑ=dt ∝ a−3. For these reasons, we use Eq. 3 to obtain an order-of-magnitude estimate for the grain rotation rate. Assuming that the grain boundary energy increases from zero to 460 mJ/m2 (52) over the angular range 0–15°, we obtain dγ=dϑ ≈ 30.7 mJ/(m2·deg). Inserting a = 30 μm into Eq. 3 along with typical values (53) for the diffusion coefficient and atomic volume of Al at T = 630 °C, we estimate a rotation rate dϑ=dt ≈ 0.003 °/min for sample S-70. Thus, over the course of a 30-min anneal, one could expect low-angle grain pairs in this specimen to reduce their misorientation by about 0.09° on average. Although this is less than the mean value observed experimentally (−0.25°), the estimate is certainly plausible, considering the sensitivity of the right-hand side of Eq. 3 to the contact radius a and the difficulty in determining an accurate value for a from ex situ measurements. Adjusting T and DL in Eq. 3 to reflect the conditions of the 30-min anneal of sample Dake et al.
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S-82, we obtain an expected reduction in misorientation of 0.07° for this specimen’s low-angle grain pairs, which is nearly identical to the measured mean value of −0.067°. The overall good agreement between experimental values and those predicted by Eq. 3 lends further credence to our interpretation of grain rotations as being driven 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 the histogram for randomly chosen pairs of contiguous grains (gray bars in Fig. 6) indicates that most rotating grains are equally likely to undergo a misorientation increase or decrease with respect to a neighboring grain. This observation suggests that there is a random component to the observed grain rotations, as well. Over the course of interrupted annealing experiments, several mechanisms may plausibly lead to undirected grain rotation. For example, the melting and solidification processes that take place during each sample’s transitional heating and cooling stages could conceivably induce rotations. When our Al–Cu specimens are heated above the eutectic temperature (548 °C), a liquid phase begins to form in small pockets, which grow and coalesce upon further heating, eventually spreading throughout the sample. The expansion and flow of this liquid will likely cause slight rearrangements of the ensemble of solid grains (54). Similarly, upon cooling, nonuniform solidification of the liquid phase could induce small rotations, reminiscent of those that result from asymmetric neck formation during solid-state sintering (13). Because melting and solidification are but transitory phenomena in this experiment, the magnitude of the resulting rotations 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 with annealing time, indicating that at least one additional source of random grain rotation must be active under isothermal conditions. As with the gray histogram of Fig. 6, we can calculate the change in misorientation δϑ for all nearest-neighbor grain pairs before and after each annealing step. By construction, such a quantity is insensitive to the collective grain motion associated with slumping or flotation of grain clusters in the semisolid state. As mentioned above, a random rotation of a given particle is equally likely to result in a positive or negative change in misorientation with respect to a grain neighbor; indeed, whenever Dake et al.
misorientation histograms are compiled for a statistical sampling of grain pairs taken from our specimens, the histograms are invariably centered about zero. There is, however, a clear increase in the width of such distributions with annealing time, which we quantify by plotting the standard deviation (SD) of δϑ against Δt (Fig. 7). Because a thicker average liquid layer allows for greater rotational freedom, it seems reasonable to observe larger misorientation changes in the sample containing the greater volume fraction of liquid (S-70). With increasing annealing time, coarsening induces significant changes in the local neighborhood of any given particle, as nearby grains disappear or neighbor switching events occur. Such phenomena are likely accompanied by redistribution of the liquid phase and rigid-body grain movement, which, although smaller in magnitude, is similar in nature to the grain rearrangement that occurs during the early stages of liquid-phase sintering (2, 28). Contributing
Fig. 6. Changes in misorientation of low-angle grain pairs in sample S-70 compared with the same quantity evaluated for randomly chosen grain neighbors. (Red) Histogram of the change in misorientation, δϑ = ϑfinal − ϑinitial, of 72 grain boundaries with ϑinitial < 15°, showing a distinct bias toward negative values. (Gray) Histogram of the same distribution for 418 randomly chosen boundaries in the same sample. The overall distribution of misorientation changes is much more symmetric than that of the subset of low-ϑ boundaries.
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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 misorientation 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 of this 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 their common boundary, indicating that the respective crystal lattices are in direct contact rather than separated by a liquid layer. The formation of a solid/solid boundary is a prerequisite for boundary energy-driven grain rotation.
cusps in the energy landscape—for example, Cu or Ni (58)—the influence of boundary energy on grain rotations should be even stronger than in Al. For these reasons, we anticipate that grain rotation-assisted phenomena will influence densification and coarsening in a large number of systems, with corresponding implications for the accurate modeling of sintering and semisolid materials processing.
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Fig. 7. Width of the histogram of misorientation changes versus annealing time. In samples S-70 and S-82, the SD of the distribution of δϑ values is found to increase with the duration Δt of the annealing interval. Evidently, a greater volume fraction of liquid phase is conducive to larger rotations. The final point for S-70 at 30 min is equal to the SD of the gray histogram shown in 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 alloy with no liquid phase (S-100), we expect the misorientation between any two grains to remain fixed over time; therefore, the nonzero value of this sample’s SD must reflect experimental uncertainties that are characteristic of the 3DXRD technique.
to such grain shifts could be strains coupled to the migration of dry boundaries between particles (55, 56). Buoyancy forces, on the other hand, are expected to be of negligible importance, because the densities of the solid and liquid phases differ by only a small amount (
Edited by Frans Spaepen, Harvard University, Cambridge, MA, and accepted by Editorial Board Member Tobin J. Marks August 3, 2016 (received for review February 19, 2016)
Sintering is a key technology for processing ceramic and metallic powders into solid objects of complex geometry, particularly in the burgeoning field of energy storage materials. The modeling of sintering processes, however, has not kept pace with applications. Conventional models, which assume ideal arrangements of constituent powders while ignoring their underlying crystallinity, achieve at 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-stage sintering—during which densification plays a subordinate role to coarsening—we have used 3D X-ray diffraction microscopy to track the changes in sample microstructure induced by annealing. The results establish the occurrence of significant particle rotations, driven in part by the dependence of boundary energy on crystallographic misorientation. Evidently, a comprehensive model for sintering must incorporate crystallographic parameters into the thermodynamic driving forces governing microstructural evolution.
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hether we realize it or not, our daily lives are marked by encounters with sintering, ranging from natural rock formations and glaciers to artificial products like the porcelain from which we eat or the amalgam fillings in many teeth. Sintering is an efficient method for combining loose granular precursors into solid objects of predetermined shape at relatively low temperature, circumventing the processing route of melting, casting, and machining. The applications of sintering are widespread. For example, it is the predominant method for producing bulk technical ceramics (1), and, thanks to advances in powder metallurgy, sintering is of growing importance in the fabrication of metals, as well (2). The focus in recent years on nanostructured materials—specifically, on materials for energy storage and conversion—has intensified the scientific investigation and modeling of sintering processes. The latter find application in the production of fuel cells (3) and battery electrodes (4). In other applications, such as catalysis (5), sintering can be highly undesirable, as it reduces the connectivity of pores and the free surface of nanoparticles. In all of these cases, we need a firm understanding of sintering fundamentals to achieve and retain desired 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 sintering (2, 6). When two particles come into contact, two free surfaces are replaced by a single solid/solid boundary. If the particles are crystalline, 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 sintering process results in a net release of energy. As free surface area decreases, the powder compact becomes denser, which usually leads to geometric shrinkage (6). Sintering is generally divided into stages characterized by various observable phenomena: formation and widening of a neck at the contact between particles;
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particle rearrangement (i.e., rigid-body motion); shrinkage of the interconnected network of pores, followed by pore closure and the shrinkage/coarsening of isolated pores; and grain growth. Some type of diffusion (surface, grain boundary, lattice) is typically assumed to govern the kinetics of each these component processes, several of which may occur simultaneously. Because the computational simulation of any of these complex processes is a daunting task in its own right, it should come as no surprise that current algorithms fall well short of predictive accuracy for the sintering of realistic powder compacts (2, 7). Our current inadequate understanding of sintering kinetics can in part be traced to the experimental difficulty of tracking microstructural changes in three dimensions (8). With recent advances in 3D imaging, however, researchers can now perform time-resolved measurements of the true size, shape, and interconnectivity of particles and pores (9–12), from which it may be possible to pin down the atomic-level mechanism(s) underlying each stage of sintering. Of particular interest in this regard is the 3D rearrangement of particles that occurs during early-stage sintering (8, 13), which can significantly affect subsequent densification but is hardly accessible from 2D cross-sections. In a recent experiment, Grupp et al. (12) observed individual particle rotations as large as 8° during the sintering of loosely packed Significance Computational modeling of materials phenomena promises to reduce the time and cost of developing new materials and processing techniques—a goal made feasible by rapid advances in computer speed and capacity. Validation of such simulations, however, has been hindered by a lack of 3D experimental data of simultaneously high temporal and spatial resolution. In this study, we exploit 3D X-ray diffraction microscopy to capture the evolution of crystallographic orientations during particle coarsening in a semisolid Al–Cu alloy. The data confirm a longstanding hypothesis that particle rotation is driven (in part) by the dependence of grain boundary energy on misorientation. In addition, the results constitute an experimental foundation for testing the predictive power of next-generation computational 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. analyzed 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 Editorial Board. Data deposition: The 3D reconstructed datasets reported in this paper can be downloaded in compressed text format from the Materials Data Facility (https://www.materialsdatafacility. org) using the link http://dx.doi.org/doi:10.18126/M25P46. 1
To whom correspondence should be addressed. Email: [email protected] or ssch@fysik. dtu.dk.
This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. 1073/pnas.1602293113/-/DCSupplemental.
www.pnas.org/cgi/doi/10.1073/pnas.1602293113
boundary energy
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misorientation Fig. 1. Schematic illustration of a grain rotation being driven by the misorientation dependence of the grain boundary energy γ. At misorientations ϑ above ∼15°, γ takes on an approximately constant value, except near special coincident site lattice (CSL) boundaries. For two crystallites with initial misorientation ϑ1 < 15°, a relative rotation to ϑ2 < ϑ1 leads to a significant reduction in γ, owing to the gradient dγ=dϑ.
Dake et al.
Results Two fully dense, cylindrical samples of Al–5 wt% Cu were annealed in the semisolid state, one at 630 °C and the other at 619 °C. Applying the lever rule to the Al–Cu phase diagram (32), we estimate the volume fraction VV of the solid, Al-rich phase to be 70% at 630 °C and 82% at 619 °C; the remaining volume in each sample is occupied by a liquid having a higher concentration of Cu than in the solid phase. From here on, we refer to the two samples as S-70 and S-82, respectively. Sample S-70 was annealed in three steps for an overall duration of 60 min, and sample S-82 in five steps for a total of 75 min. Coarsening. Exemplifying the particle growth that occurred during annealing, Fig. 2 A and B show the initial and final microstructures of S-70 after reconstruction of 3DXRD data. Because each solid particle in these specimens is monocrystalline, we refer to solid particles alternatively as grains. The distribution of grain misorientations in each sample was found to agree quite closely with the Mackenzie distribution for randomly orientated cubic crystals (33), implying an absence of significant crystallographic texture (Fig. 2C). At the temperatures of annealing, most grains are separated from neighboring particles by a relatively thick layer of liquid, but upon cooling these liquid regions 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 reconstruction appears to be in direct contact with many grain neighbors (13.5 on average), all of which are included in the calculation of misorientation distributions in Fig. 2C. In the semisolid state, the average number of direct particle/particle contacts lies well below 13.5, which can be deduced from tomographic reconstructions of similar Al–5 wt% Cu samples measured at elevated temperature by CT (Fig. S2). In Fig. 2 as well as in subsequent 3D images, we index the color of each grain to its crystallographic orientation by mapping the components of the Rodrigues vector representation of orientation (34) to RGB color coordinates. Consequently, the misorientation of any two grains having similar colors is small. The irregularity visible at the top and bottom surfaces of the cylindrical sample results from the removal of grains that were not completely contained within the volume irradiated by X-rays. As a result, the total volume under consideration varies slightly over the course of annealing. The number of grains within sample S-70 was initially 682, which dropped to 418 by the end of the final annealing step; likewise, sample S-82 initially contained 637 grains, which eventually fell to 431. Thus, the number of grains in each sample was large enough to warrant a statistical analysis of the coarsening behavior. In the semisolid state of a material, solid particles coarsen primarily by Ostwald ripening, which is characterized by a net diffusive flux of atoms (through the liquid) from smaller to larger particles. Growth trajectories of 100 grains, chosen at random, are PNAS | Published online September 26, 2016 | E5999
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ripening (30, 31). In such a sample, we expect to encounter few, if any, particle rearrangements driven by the aforementioned formation of asymmetric necks, as each particle is surrounded by a liquid layer. Effectively, such a microstructure isolates crystallographic misorientation as a potential driving force for particle rotation. Over the course of stepwise heat treatments at various temperatures, we have mapped the internal volumes of Al–Cu specimens using 3DXRD as well as X-ray computed tomography (CT). The resulting datasets yield time-dependent trajectories for the size and orientation of hundreds of particles at various volume fractions of the liquid phase. Not only do these trajectories represent a time-resolved observation of lattice orientation variations in a semisolid bulk polycrystal, but they also reveal—through a statistical analysis—the simultaneous occurrence of two distinct mechanisms for particle rotation.
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monocrystalline Cu spheres. The changes in 3D orientation of these spherical grains were extracted from marker holes drilled by a focused ion beam. In the literature, such particle rotations are most frequently attributed to the formation of asymmetric necks (13); however, the dependence of grain boundary energy on crystallographic misorientation (the discontinuous change in lattice orientation at the boundary separating adjoining particles) can also induce particle rotations (Fig. 1), as captured under laboratory conditions by so-called ball-on-plate experiments (14–17). Indeed, Grupp et al. (12) proposed that the particle rotations in their experiment resulted primarily from a misorientation-based driving force, but experimental evidence for the validity of this assertion was lacking, as the measurement technique they used— absorption-contrast X-ray computed tomography—is insensitive to 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-energy diffraction microscopy (20), and diffraction-contrast tomography (21), it is now feasible to perform nondestructive, 3D mapping of the shapes and lattice orientations of thousands of crystallites within a polycrystalline specimen. From a sequence of such measurements applied to the same sample, we can track its microstructural evolution and determine whether a particular processing protocol [such as a heat treatment (22–24) or a mechanical deformation (25–27)] induced grain rotation. Furthermore, we can correlate changes in grain orientation to the morphologies and misorientations of a rotating grain’s immediate neighbors. In this article, we report the application of 3DXRD microscopy to the detection and statistical analysis of grain rotations taking place during the coarsening of a semisolid Al–Cu alloy (crystalline Al particles embedded in a liquid phase). Secondphase materials that melt below the sintering temperature are frequently added to powder compacts to accelerate densification (28, 29). Owing to their 100% density, our Al–Cu specimens are expected to mimic the late-stage behavior of a liquid-phase sintering system, which is marked by a negligible amount of residual porosity and by coarsening of the solid particles via Ostwald
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Fig. 2. Evolution of microstructure and texture in sample S-70. The (A) initial and (B) final 3D reconstructions illustrate the microstructural evolution that occurred 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 distributions to the Mackenzie distribution for randomly oriented cubic crystals reveals little evidence for crystallographic texture.
plotted in Fig. 3 A and B for S-70 and S-82, respectively. Here, we quantify the grain size using the equivalent radius R of a sphere of equal volume. Superimposed in black is the evolution of the average 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, the cloud of data points takes on a nonlinear shape with a slightly upward curvature. This behavior contradicts the conventional Lifshitz, Slyozov, and Wagner (LSW) model (35, 36) for Ostwald ripening, according to which the time derivative of a particle’s volume, dV =dt ∼ R2 dR=dt, varies linearly with the normalized grain size, intersecting the horizontal axis at a normalized radius of unity. However, the LSW prediction was derived for a vanishingly 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 the nonlinearity evident in Fig. 3C. Grain Rotations. Just as with the grain size, we can track the orientation of each grain throughout the course of microstructural evolution. After each ex situ annealing step, we corrected the set of grain orientations determined by 3DXRD microscopy for global sample rotation and translation (Materials and Methods); consequently, any difference in an individual grain’s orientation with respect to a prior orientation must reflect a rotation Δθ of the grain’s lattice planes.* In Fig. 4A, we plot trajectories of grain rotations for samples S-70 and S-82. In this figure, we also include the rotation trajectories (green) from a third, single-phase
*A note on our terminology and notation for grain orientations and misorientations: The orientation of any single-crystalline grain can be specified by the transformation required to rotate the axes of a reference coordinate system into the unit cell axes of the grain. In the axis-angle representation, we denote the rotation angle of such a transformation 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 be described 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, the misorientation between two grains can be described as a rotation of the unit cell axes of one grain into those of the other; to avoid confusion with the previous quantities, we denote a given misorientation relationship by the corresponding rotation angle ϑ and changes in misorientation by δϑ = ϑðt + ΔtÞ − ϑðtÞ. Unlike Δθ, the quantity δϑ can take on positive or negative values. When calculating the orientation as well as the misorientation, we account for the cubic symmetry of the fcc Al lattice, thus ensuring that the minimum equivalent rotation angle is reported.
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Al–1 wt% Mg sample. Like the two Al–Cu specimens, the Al–Mg sample was heated ex situ and characterized by 3DXRD between annealing steps. Because the Al–Mg sample is completely recrystallized, with a relatively large grain size (∼200 μm) and no liquid phase (VV = 1), its constituent crystallites must be highly constrained; at the moderate annealing temperature of 400 °C, it is safe to assume that any grain rotations that occur in this sample during coarsening are well below the detection limit of the 3DXRD technique (∼ 0.05°) (40). Therefore, based on repeated measurements of this sample, we can estimate the uncertainty inherent in the Δθ values yielded by our experimental procedure. From measurement to measurement, the median Δθ obtained for grains in the Al–Mg sample was only 0.06°, with 95% of the (apparent) rotations lying below 0.18°. In contrast, for samples S-70 and S-82, we find median changes in grain orientation of 0.37° and 0.17°, respectively. From this comparison, we conclude that the vast majority of orientation changes measured in the semisolid samples represent true grain rotations. To compare grain rotations from annealing intervals of different duration, we construct probability plots for the rate of rotation Δθ=Δt (Fig. 4B). Evidently, the rotation rate increases significantly with the volume fraction of the liquid phase: the median rotation rate in sample S-82 (18% liquid phase) is 0.014°=min, whereas in sample S-70 (30% liquid phase) one-half of 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 prematurely terminating trajectories plotted in Fig. 4A to their corresponding data points in Fig. 4C reveals that large rotations often occur just before a shrinking grain vanishes. For example, the three highestrotation trajectories of S-70 at the 30-min annealing mark correspond to the top three red points plotted in Fig. 4C. The associated grains disappeared by the time the next 3DXRD measurement was performed (at 60 min). Discussion The results plotted in Fig. 4 indicate that grain rotations—significant both in number and magnitude—occur during the coarsening of semisolid Al–Cu at liquid volume fractions of 18% and 30%. We postulate that such rotations are driven by gradients in the grain boundary energy γ with respect to misorientation ϑ. As illustrated schematically in Fig. 1, grain rotations can lower the free energy because γ is a function of the change in lattice orientation across Dake et al.
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Fig. 3. Kinetics of coarsening: growth trajectories for 100 grains selected at random from samples (A) S-70 and (B) S-82. Most grains larger than the mean grain size hRi (black diamonds) are observed to grow, whereas the opposite is true of smaller-than-average-sized grains. (C) In both samples, a slight upward curvature is evident in plots of the volumetric growth rate against the normalized grain size.
the boundary (41, 42). When the difference in orientation between grains is larger than about 15°, γ takes on a roughly constant value; however, when the misorientation lies within the low-angle range ðϑ < 15°Þ, the boundary energy decreases ever faster as ϑ approaches Dake et al.
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zero (43). In the latter case, the steep energy gradient would be expected to generate equal and oppositely directed torques acting on the grains in contact, which could potentially induce a relative rotation. To assess the relevance of this mechanism to our data, we begin by examining pairs of neighboring grains for which the postulated driving force might be expected to generate a significant torque at the common boundary. Although strong gradients in γ have been measured near misorientation values of special coincident site lattice (CSL) boundaries and twin boundaries, the shape (depth and width) of these so-called “energy cusps” is not well understood and may disappear at elevated temperatures (42, 44). In addition, aluminum is known to have a high stacking fault energy, which limits twinning (45). In light of these considerations, we restrict our attention in the following discussion to rotating grains having nearest neighbors of low misorientation ðϑ < 15°Þ, for which the deep minimum that appears in γ as ϑ → 0 is amenable 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 in this image are indexed to similar colors because their lattice orientations are nearly the same. During a 30-min anneal at 630 °C, the lower grain in Fig. 5A rotates by Δθ = 5.4° to the position shown in Fig. 5B, thereby changing its misorientation with respect to the upper grain by δϑ = −2.7° [see footnote (*) for an explanation of the notation used here]. Of course, our postulated mechanism of misorientation energy-driven grain rotation presupposes that the boundary between the two grains is wet by no more than a few atomic layers of liquid, as a thicker liquid layer would essentially decouple the grains from each other. In general, a “dry” boundary can form between two neighboring grains whenever the solid/solid boundary energy is lower than the overall energy of the alternative solid/liquid boundaries: that is, γ sol=sol < 2γ sol=liq. This criterion is especially likely to be fulfilled near the deep minimum in γ sol=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 by the liquid phase. Moreover, a well-defined (formerly) liquid layer is seen to coat the lower grain’s other boundaries, despite the sample having been measured at room temperature. At the elevated temperature of annealing, the liquid layers are significantly thicker than shown in Fig. 5C (Fig. S2), which apparently facilitates the accommodation of a rigid-body rotation of 5.4°. Consequently, we conclude that the example shown in Fig. 5 is consistent with a grain rotation having been driven by a gradient in the grain boundary energy landscape. We now examine the set of all grains that have a single immediate neighbor with initial misorientation less than 15°. If gradients in grain boundary energy drive grain rotations, then we would expect the misorientations of these low-ϑ grain pairs to tend toward smaller values over time, at least on average. To minimize effects of melting and solidification and better isolate the influence of boundary energy on grain rotations, we consider only the misorientation changes that occurred during the final annealing step, which was the longest interval and equal in duration (30 min) for both specimens. In sample S-70, 72 grain pairs fulfill the low-ϑ criterion, and a histogram of the measured changes in misorientation, δϑ = ϑfinal − ϑinitial (red bars), is plotted in Fig. 6. A positive value of δϑ indicates that the misorientation between neighboring grains increased, whereas a negative value signifies behavior consistent with the operation of a grain boundary energyinduced torque. There is a clear skew toward negative values, with the 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 misorientation relative to neighboring grains chosen at random, then we obtain a significantly more symmetric histogram for δϑ (gray
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bars in Fig. 6) with a mean of −0.014°. (Because the overall sampling includes not only general grain pairs but also low-ϑ neighbors, we might expect the mean value to lie slightly below zero.) Considering this distribution to be the parent population for δϑ, we apply Student’s t test (46) to calculate the probability of obtaining a mean misorientation change equal to −0.25° for 72 grain pairs drawn at random from the parent population. The t test value of only 0.22% indicates that we can reject the null hypothesis—that the low-ϑ population and the parent population have the same mean value for δϑ—with over 99% certainty. Applying the same analysis to the low-ϑ grain pairs of sample S-82, we obtain qualitatively similar results with a certainty of 97% for rejecting the null hypothesis. This statistical analysis points to gradients in the grain boundary energy landscape as having been instrumental in driving the rotation of grains in semisolid Al–Cu. Further evidence in support of this conclusion may be gained from an examination of the rate at which grains sharing a low-angle boundary reduce their misorientation. Analogous to the dependence of a curved boundary’s migration rate on capillary pressure, a grain’s rotation rate can be related to the torque that acts on its periphery (47). Because each grain in the pairs considered above has only one low-angle neighbor, we assume its remaining boundaries to be wet by the liquid phase during annealing, leaving only a single dry boundary of area A and gradient dγ=dϑ to generate a torque: τ=A
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where the proportionality constant Mr denotes a rotational mobility. Shewmon (48, 49) derived an expression for Mr for a spherical particle having a single boundary in contact with a plate, under the assumption that the material transport needed to accommodate particle rotation occurs by lattice diffusion. Inserting Shewmon’s expression into Eq. 2, we obtain dϑ 8DL Ω dγ = , dt kB Ta3 dϑ
Fig. 4. Grain rotations during annealing. (A) Rotation trajectories recorded for grains in samples S-70 and S-82, manifesting individual rotations up to 12 °. At each annealing time, the angle of rotation Δθ is calculated relative to the given grain’s previous orientation. For comparison, we also plot rotation 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, we treat the Δθ values as a measure for the uncertainty inherent to the 3DXRD technique. (B) Log-normal probability plots for the rotations shown in A, demonstrating that both the median rotation rate Δθ=Δt and the maximum observed rotation rate increase with volume fraction of the liquid phase (0–30%). (C) Grains of smaller normalized radius R=hRi have a tendency to rotate more rapidly.
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[2]
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with lattice diffusion coefficient DL, atomic volume Ω, Boltzmann constant kB, absolute temperature T, and radius a of the contact area between sphere and plate. More recently, this equation has been extended to the case of columnar grain structures in singlephase polycrystals (47, 50); however, owing to the decoupling provided by a wetting liquid phase, we expect Shewmon’s original expression to be more applicable to our specimens. In addition, the results of recent molecular dynamics simulations (51) are consistent with the dependency dϑ=dt ∝ a−3. For these reasons, we use Eq. 3 to obtain an order-of-magnitude estimate for the grain rotation rate. Assuming that the grain boundary energy increases from zero to 460 mJ/m2 (52) over the angular range 0–15°, we obtain dγ=dϑ ≈ 30.7 mJ/(m2·deg). Inserting a = 30 μm into Eq. 3 along with typical values (53) for the diffusion coefficient and atomic volume of Al at T = 630 °C, we estimate a rotation rate dϑ=dt ≈ 0.003 °/min for sample S-70. Thus, over the course of a 30-min anneal, one could expect low-angle grain pairs in this specimen to reduce their misorientation by about 0.09° on average. Although this is less than the mean value observed experimentally (−0.25°), the estimate is certainly plausible, considering the sensitivity of the right-hand side of Eq. 3 to the contact radius a and the difficulty in determining an accurate value for a from ex situ measurements. Adjusting T and DL in Eq. 3 to reflect the conditions of the 30-min anneal of sample Dake et al.
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S-82, we obtain an expected reduction in misorientation of 0.07° for this specimen’s low-angle grain pairs, which is nearly identical to the measured mean value of −0.067°. The overall good agreement between experimental values and those predicted by Eq. 3 lends further credence to our interpretation of grain rotations as being driven 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 the histogram for randomly chosen pairs of contiguous grains (gray bars in Fig. 6) indicates that most rotating grains are equally likely to undergo a misorientation increase or decrease with respect to a neighboring grain. This observation suggests that there is a random component to the observed grain rotations, as well. Over the course of interrupted annealing experiments, several mechanisms may plausibly lead to undirected grain rotation. For example, the melting and solidification processes that take place during each sample’s transitional heating and cooling stages could conceivably induce rotations. When our Al–Cu specimens are heated above the eutectic temperature (548 °C), a liquid phase begins to form in small pockets, which grow and coalesce upon further heating, eventually spreading throughout the sample. The expansion and flow of this liquid will likely cause slight rearrangements of the ensemble of solid grains (54). Similarly, upon cooling, nonuniform solidification of the liquid phase could induce small rotations, reminiscent of those that result from asymmetric neck formation during solid-state sintering (13). Because melting and solidification are but transitory phenomena in this experiment, the magnitude of the resulting rotations 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 with annealing time, indicating that at least one additional source of random grain rotation must be active under isothermal conditions. As with the gray histogram of Fig. 6, we can calculate the change in misorientation δϑ for all nearest-neighbor grain pairs before and after each annealing step. By construction, such a quantity is insensitive to the collective grain motion associated with slumping or flotation of grain clusters in the semisolid state. As mentioned above, a random rotation of a given particle is equally likely to result in a positive or negative change in misorientation with respect to a grain neighbor; indeed, whenever Dake et al.
misorientation histograms are compiled for a statistical sampling of grain pairs taken from our specimens, the histograms are invariably centered about zero. There is, however, a clear increase in the width of such distributions with annealing time, which we quantify by plotting the standard deviation (SD) of δϑ against Δt (Fig. 7). Because a thicker average liquid layer allows for greater rotational freedom, it seems reasonable to observe larger misorientation changes in the sample containing the greater volume fraction of liquid (S-70). With increasing annealing time, coarsening induces significant changes in the local neighborhood of any given particle, as nearby grains disappear or neighbor switching events occur. Such phenomena are likely accompanied by redistribution of the liquid phase and rigid-body grain movement, which, although smaller in magnitude, is similar in nature to the grain rearrangement that occurs during the early stages of liquid-phase sintering (2, 28). Contributing
Fig. 6. Changes in misorientation of low-angle grain pairs in sample S-70 compared with the same quantity evaluated for randomly chosen grain neighbors. (Red) Histogram of the change in misorientation, δϑ = ϑfinal − ϑinitial, of 72 grain boundaries with ϑinitial < 15°, showing a distinct bias toward negative values. (Gray) Histogram of the same distribution for 418 randomly chosen boundaries in the same sample. The overall distribution of misorientation changes is much more symmetric than that of the subset of low-ϑ boundaries.
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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 misorientation 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 of this 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 their common boundary, indicating that the respective crystal lattices are in direct contact rather than separated by a liquid layer. The formation of a solid/solid boundary is a prerequisite for boundary energy-driven grain rotation.
cusps in the energy landscape—for example, Cu or Ni (58)—the influence of boundary energy on grain rotations should be even stronger than in Al. For these reasons, we anticipate that grain rotation-assisted phenomena will influence densification and coarsening in a large number of systems, with corresponding implications for the accurate modeling of sintering and semisolid materials processing.
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Fig. 7. Width of the histogram of misorientation changes versus annealing time. In samples S-70 and S-82, the SD of the distribution of δϑ values is found to increase with the duration Δt of the annealing interval. Evidently, a greater volume fraction of liquid phase is conducive to larger rotations. The final point for S-70 at 30 min is equal to the SD of the gray histogram shown in 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 alloy with no liquid phase (S-100), we expect the misorientation between any two grains to remain fixed over time; therefore, the nonzero value of this sample’s SD must reflect experimental uncertainties that are characteristic of the 3DXRD technique.
to such grain shifts could be strains coupled to the migration of dry boundaries between particles (55, 56). Buoyancy forces, on the other hand, are expected to be of negligible importance, because the densities of the solid and liquid phases differ by only a small amount (
