Atomic structure characterization of an incommensurate grain boundary

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Acta Materialia 61 (2013) 5078–5086 www.elsevier.com/locate/actamat

Atomic structure characterization of an incommensurate grain boundary A. Gautam a,⇑, C. Ophus a, F. Lancßon b, V. Radmilovic c, U. Dahmen a,⇑ a National Center for Electron Microscopy, LBNL, Berkeley, CA 94720, USA Laboratoire de Simulation Atomistique (L_Sim), SP2M, INAC, CEA-UJF, 38054 Grenoble, France c University of Belgrade, Faculty of Technology and Metallurgy, Nanotechnology and Functional Materials Center, Belgrade, Serbia b

Received 25 January 2013; received in revised form 8 April 2013; accepted 11 April 2013 Available online 22 May 2013

Abstract The structure of an incommensurate 90°h1 1 0i tilt grain boundary in gold was characterized by atomic resolution aberration-corrected electron microscopy and compared with atomistic simulations. Based on a periodic hyperspace description, the non-periodic structure can be described by the Aubry hull function, which plots atomic relaxations at the core of the boundary relative to an unrelaxed structure, folded into a single repeat unit of the neighboring grain. By measuring the hull functions from atomic resolution images, we were able to make quantitative comparisons of experimental observations with molecular statics simulations of this boundary. The results show good agreement in the pattern of atomic relaxations, replicate features of the hull functions that are characteristic of a boundary with superglide behavior and demonstrate the experimental feasibility of this approach for analysis of interfaces. Ó 2013 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Interface; Grain boundaries; HRTEM; Atomistic simulations; Hypofriction

1. Introduction The microscopic structure of grain boundaries holds the key to many macroscopic properties such as strength, ductility, corrosion resistance or thermal stability. This is the reason for the large research effort aimed at understanding the local atomic structure of grain boundaries employing both experimental and computational approaches [1–3]. Major recent advances in electron microscopy have improved characterization techniques to the point that atomic resolution imaging of grain boundaries in metals [4], semiconductors (e.g. Ref. [5]) and even complex oxides [6–12] has become almost routine. Parallel to these experimental developments, considerable progress has been made on the computational side, leading to a convergence of ⇑ Corresponding authors. Tel.: +1 510 495 2901; fax: +1 510 486 5888 (A. Gautam), tel.: + 1 510 486 4627; fax: +1 510 486 5888 (U. Dahmen). E-mail addresses: [email protected] (A. Gautam), [email protected] (U. Dahmen).

capabilities at the nanometer scale where structures of similar size are accessible to both experiment and theory. This is making the quantitative comparison between simulated and observed structures increasingly important. Originally, such comparisons were made visually by identifying characteristic features such as structural units and their spatial arrangement [13]. To account for the non-straightforward relationship between observed image intensity and underlying atomic structure, image simulations are necessary, even for high-angle annular dark field scanning transmission electron microscopy (HAADFSTEM) Z-contrast images, which are simpler to interpret directly than high resolution transmission electron microscopy (HRTEM) phase contrast images. Techniques for quantitative image analysis have been applied to many grain boundaries and compared to atomistic simulations. Most of these comparisons have been made for symmetrical tilt grain boundaries with relatively short repeat distances [14,15]. The periodicity of such boundaries makes them particularly suitable for simulation and experimental

1359-6454/$36.00 Ó 2013 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.actamat.2013.04.028

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analysis alike. Short repeat periods make it possible to record high resolution images containing several periods of the same grain boundary, offering better statistics and smaller error bars on atomic positions than a single noisy image of an isolated structural unit. However, this method fails for boundaries that have large repeat distances or are incommensurate. While such interfaces still exhibit characteristic structural features [16], the irregular spatial arrangement and variable configuration of these features make standard techniques inapplicable. In this work, we present a quantitative analysis of an incommensurate grain boundary. The 90°h1 1 0i tilt grain boundary in gold tends to form preferred facets along (1 1 0)/(0 0 1) planes, where the lattice periodicities of the two grains p along the incommensurate direction are in the ratio of 1: 2. This boundary offers a model system to study the phenomenon of hypofriction/superlubricity [17], which is of interest to many technological and scientific problems such as grain boundary engineering, fracture, contact formation and atomic-scale friction [18,19]. Previous investigations of this interface by high resolution electron microscopy [20] and in situ nanocompression [21] have confirmed its incommensurate structure and the prediction of superglide [20,22] along the incommensurate direction. By analogy with the Frenkel–Kontorova model of an elastic chain on a periodic potential [23], it was shown that the Aubry transition, which marks the transition between friction and hypofriction, could be seen as a change between discontinuity and continuity of lines in the hyperspace description of this interface. Based on this description, the present work uses Aubry hull functions as a way of measuring the degree of atomic relaxation quantitatively. The hull function and hyperspace descriptions of the grain boundary [22] represent the same data in different axis systems. The hyperspace description shows both grain modulations and their interplay at once, and is directly related to the diffraction pattern [20]. But, the hull functions are more sensitive to subtle changes in the atomic relaxations because they emphasize the deviation from perfect lattice positions. This framework, which is useful for the quantitative description of an incommensurate boundary, can also be applied to describe a periodic boundary and therefore offers a general approach for quantitative characterization of the structure of a grain boundary. 2. Experimental procedure The samples were prepared by physical vapor deposition of high purity Au onto heated h1 0 0i Ge substrates. The Ge substrate was later dissolved in a 1:1 solution of hydrofluoric acid and hydrogen peroxide, resulting in freestanding Au films [24,25]. This method produced h1 1 0i oriented bicrystals with a crystal misorientation of 90°. The structure of these films was that of a mazed bicrystal, characterized by grain boundaries with fixed misorientation but variable inclination [25,26]. The typical thickness of the Au films used for imaging the grain boundary structure

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was 4 nm. Selected grain boundaries were imaged using the aberration-corrected TEAM microscopes [27,28] at the National Center for Electron Microscopy in both the HAADF-STEM as well as the HRTEM mode. Imaging conditions in either mode were tuned to fully correct for aberrations up to third order, while a small objective defocus and negative spherical aberration (Cs = 10 lm) were used in the HRTEM mode to obtain the best contrast [29]. These images were then processed using the MATLABÒ and the MacTempasÒ software to extract atomic column positions with sub-pixel accuracy and subsequently calculate the grain boundary relaxations. Atomic columns were identified by a template matching procedure, using an average motif from the image itself as a template [30]. A two-dimensional (2-D) Gaussian distribution fitted to these positions defined atomic column positions with subpixel accuracy. Based on these atomic positions, two lattices on either side of the grain boundary were defined through a least squares fit in an undistorted region at a distance from the grain boundary. These lattices were then extrapolated to the interface and the measured distance of atomic columns from the nearest lattice point was registered as the atomic displacement. 3. Modeling Models of the grain boundary were constructed by joining together two suitably oriented slabs of face-centered cubic (fcc) gold and minimizing the total potential energy. The grain with (0 0 1) planes parallel to the interface has a periodicity equal to the nearest neighbor distance, ro, along the incommensurate direction x, while the grain with p (1 1 0) planes parallel to the interface has a periodicity 2ro. The nearest neighbor distance in fcc gold at room temperature is 0.288 nm. Instead of truncating an infinite incommensurate interface, we use periodic approximants with the number of periods in the two grains denoted nS and nL, where p the ratio nS/nL is a Diophantine approximation of 2. Any rational approximant implies a residual strain along x, but by using the 239:169 approximant, the strains in the grains are only ± 4  106. To test possible size effects, we also considered a smaller 17:12 approximant. The grain boundary is periodic in the common h1 1 0i direction, z, and therefore we used periodic condition for z too. In the direction y perpendicular to the interface, a free boundary condition was used to allow relative positioning of the two grains during energy minimization. The total number of atoms included in the 239:169 approximant is 11,482, which implies the use of an interatomic potential to model the energy. We used an embedded-atom method (EAM) potential for gold, which agrees well with the experimental lattice parameters and elastic constants but predicts lower vacancy formation energy and melting temperature [31]. To induce an Aubry transition in the grain boundary structure, we also applied compressive strains perpendicular to it (see Section 4.3) [22]. From the equilibrium positions, a given uniaxial strain is

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first applied, followed by a new energy minimization, but now the atoms at the free surfaces in the planes parallel to the interface have their y-coordinates fixed in order to maintain the global strain. 4. Results and discussion 4.1. Characteristics of the grain boundary An HRTEM image of a typical grain boundary in a mazed bicrystal film is shown in Fig. 1a. Owing to the stability of the aberration-corrected microscope, the image has a very high signal/noise (S/N) ratio and allows measurement of atomic column position with a precision of 4 pm. The image displays several features that are characteristic of this boundary, including fivefold structural units (outlined), a step (arrowed) and a chevron defect [32] where the boundary meets the edge of the foil (zigzag line). Phasecontrast HRTEM images such as that shown in Fig. 1a depend strongly on the imaging condition, and hence require image simulations for proper interpretation. By comparison, Z-contrast HAADF-STEM images are less sensitive to small variations in imaging conditions and are often directly interpretable. Fig. 1b shows a Z-contrast image of a grain boundary segment without any additional defects at the boundary. Comparing the two images clearly shows that it is easier to identify the incommensurate boundary and its defects in the Z-contrast STEM images. However, an important limitation of using STEM images for this study results from distortions introduced by various scan artifacts. Distortions caused by instabilities faster than the frame time and with zero mean can be removed by averaging over multiple frames [33] while improving the signal/noise ratio of the images such as that shown in Fig. 1b, obtained by averaging over ten frames, acquired at a rate of 1 frame s1 using a dwell time of 1 ls per pixel. To find the optimum imaging condition for Z-contrast, a series of images from a defect-free region of a thin film was recorded under systematically varying imaging parameters such as probe current, scan rate and scan direction. Post processing of these images involved alignment and

averaging of multiple images, extraction of atomic column positions by quantitative image analysis, lattice fitting and measurement of the error distribution. Acquisition conditions that minimized the error in locating the atomic columns were then used to image the grain boundary. From these measurements the error parallel and perpendicular to the scan direction was found to be similar, with a zero mean. The full width at half maximum (FWHM) of the error distribution for optimum STEM imaging conditions was found to be on the order of 10 pm, significantly higher than the 4 pm measurement error found in HRTEM images. In addition, it was observed that the microscopic structure of the boundary changed continuously under the influence of the 300 kV electron beam. Such atomic displacements were observed in both TEM and STEM imaging modes, but were found to be more pronounced in STEM. The convolution of displacements due to static grain boundary relaxation with those due to dynamic atomic motion increases the measurement error. To minimize this error, HRTEM imaging conditions were chosen for further investigation of the grain boundary structure, since this mode reduces both the atomic motion and the measurement error. Although electron-beam-induced atomic motion was reduced in the HRTEM mode, fluctuations in atomic column positions were still found to be significant, both at 300 kV and at 80 kV. Instabilities were more pronounced near steps and edges, and therefore grain boundary segments containing such additional defects were excluded from this study. The electron-beam-induced dynamics were mostly limited to local atomic column displacements, but sometimes resulted in coordinated motion such as small but noticeable shift and rotation of the two grains. To minimize the effect of coordinated motion on the measurement of the boundary structure, individual events were identified in a time sequence, and images acquired during that event were excluded from the data set. For instance, specific atomic columns were occasionally observed to appear and disappear at the grain boundary. Detailed analysis by atomistic and image simulations revealed this effect to be due to nucleation of steps in the boundary along the

Fig. 1. (a) Atomic resolution HRTEM image of incommensurate (1 1 0)/(0 0 1) grain boundary in a 90°h1 1 0i mazed bicrystal of Au. (b) HAADF-STEM image of the boundary obtained by averaging 10 fast scan (1 ls dwell time) images. Both images correspond to a total exposure of 1 s.

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direction of projection. Based on the characteristic contrast from these defects, such transitional events could therefore be excluded from the dataset. The dynamic atomic structure of the grain boundary under the influence of the electron beam is being analyzed in detail and will be published separately. Here we focus on the static structure and its relationship to atomistic simulations. 4.2. Simulated structure The incommensurate boundary was modeled using commensurate rational interfaces that approximate the structure with increasing accuracy. Results from molecular statics simulations of two rational approximants, 17:12 and 239:169, were used in this study. Fig. 2 shows the 17:12 simulation of the boundary. In this structure, calculated at 0 K, high-energy sites are shown in red. The pentagonal units outlined in the figure are characteristic of this boundary. The excess volume associated with these sites is reflected in their relatively open structure and high energy. In a typical analysis, this structure would be used for an image simulation to compare directly with the experimental observation (e.g. Refs. [20,34]). The simulation in Fig. 2b was performed for the imaging conditions used in the experiment (Cs = 0.03 mm, Df = 14 nm, convergence angle 0.5 mrad, spread of defocus 2 nm). The best match was obtained for a foil thickness of 4 nm. A visual comparison of the experimental and simulated structures shows good agreement in local areas such as the pair of fivefold units highlighted in the image. However the non-periodic nature of the interface prevents any exact match between model and experiment. To quantify the identification of structural units, a valuable approach has been the use of template matching [30] to obtain multiple examples of a particular structural unit such as those outlined in the image. In this method, a cross-correlation map is calculated by comparing a small template image of a selected unit pixel by pixel with a larger image of the

Fig. 2. (a) Model structure calculated by energy minimization of a 17:12 rational approximant, showing local energy using the same color scale shown in Fig. 4; (b) image simulation based on this model for a foil thickness of 4 nm and a defocus of 14 nm. Fivefold structural units corresponding to those in the experimental image (Fig. 1a) are outlined. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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boundary. The highest peaks in this map mark the positions of the most similar structural units in the image. By averaging over multiple similar units, it is possible to obtain an average image of the unit and its standard deviation image, even if the units are arranged in a non-periodic pattern. If all units are identical and differ only by random noise, this procedure improves the S/N, and the standard deviation image is featureless. If, however, there is a systematic variation in the structure of the units, then the standard deviation image will display structural features. In this case, non-periodic averaging does not improve the accuracy of the comparison between model and experiment. This is the case in the present boundary. Close inspection of Fig. 2a shows that even in the absence of noise, the fivefold units vary in structure along the boundary. This variation can easily be seen by superimposing similar structural units, as shown in Fig. 3a. The units were overlaid with their center of mass and allowed some rotation to achieve the best possible alignment. Atomic positions are marked in red. The lines connecting atomic positions are seen to be fanned out, indicating considerable variation between structural units. By comparison, a similar overlay for a short-period boundary would have sharp lines, since there are fewer variants of such structural units. Fig. 3b shows that a compressive strain

Fig. 3. Overlay of fivefold structural units from the incommensurate 239:169 approximant grain boundary simulated (a) without any applied strain and (b) with 3.5% compressive strain normal to the boundary. (c) Smoothed histogram plotting the relative frequency of the geometrical shear that transforms a regular pentagon to the pentagonal units found in the relaxed boundary. For an nS/nL (239:169) approximant, there are nS  nL = 70 pentagonal units.

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of 3.5% normal to the boundary gives rise to a subtle but significant change in the structure. In addition to an increased aspect ratio of the structural unit, the distribution of shapes develops gaps, as evidenced by discontinuities (arrowed) in the fanned distribution of connecting lines. This can be seen more easily in a plot of the geometrical shear that transforms a regular pentagon into the range of fivefold structural units in the incommensurate boundary as shown in Fig. 3c. The minima in the curve for the 3.5% strained boundary correspond to the two discontinuities marked in Fig. 3b. Similar plots for an unstrained and a 1.2% strained grain boundary show no such discontinuity. The analysis illustrated in Fig. 3 shows why for an incommensurate boundary, a quantitative comparison between model and experiment cannot be performed directly or by template matching. Although the structural unit model of an interface can still be applied to an incommensurate boundary [1,35,36], the accuracy with which it describes structural relaxations can be drastically limited. To overcome this limitation, we propose a characterization based on the Aubry hull function. 4.3. Measurement of the hull function The Aubry hull function was introduced to describe the incommensurate ground state of the Frenkel–Kontorova model [37] and to demonstrate the existence of a transition by “breaking of analyticity” [37,38] now also called “Aubry transition”. In this model, a linear chain of interacting atoms with a mean lattice spacing ro is embedded in an external potential field of periodicity a. When ro/a is an irrational number, the ground state structures are incommensurate and indexed by an arbitrary phase a 2 R, and the atomic coordinate xi of atom i is given by xi = i ro + a + g(i ro + a). The integer i is the equivalent of a lattice index and i ro + a are the ideal undistorted atomic positions. The phase a can be seen as a global shift with respect to the potential-minimum locations j a (j 2 Z), one of which is arbitrarily taken as the origin. The modulation hull function g(x) represents the distortion from the ideal positions and it must have the periodicity a of the external field since each atom will keep the same environment if the total chain is shifted by a. The complete Aubry hull function f(x) = x + g(x) includes the linear increase of the atom locations with the index i plus the modulation g(x). Once a given ground state {xi} is known, the modulation hull function g(x) can be sampled by plotting the relaxation displacement Di = xi  i ro  a vs. x = (i ro + a) modulo a, i.e. x is the undistorted position folded back into [0, a]. The global shift a is determined once by choosing the origin. In the following discussion the modulation hull function will be referred as the hull function for the sake of simplicity. For our grain boundary analysis, each atomic row parallel to the incommensurate direction x of the interface plays the role of the linear chain, while the adjacent grain

plays the role of the periodic potential. We measure the displacements of the atoms in the chain from their ideal lattice positions, due to the interactions with the adjacent grain. These displacements correspond to the hull function. Since the displacements Di are three-dimensional (3-D) vectors, the hull function itself is vectorial. Simulations have shown that no displacements occur along the periodic direction z of the grain boundary, which is the common h1 1 0i direction of both grains. Thus we only have to determine the components of g in the incommensurate direction x and perpendicular to the interface y. We can consider all the atomic rows parallel to the x-direction but it is only near the core of the boundary that the hull functions will deviate significantly from a flat curve. Unlike the Frenkel–Kontorova model, we have two systems that are mutually interacting. So we need to consider the periodic reference lattices of both undistorted grains. We measure the coordinate x along the grain boundary of the ideal position of the atoms of one grain, say B, with respect to a reference lattice site of the opposite grain A. The displacements D for grain B are then plotted vs. x modulo the periodicity a of grain A so that g(x + a) = g(x). The choice of the reference lattice point in A is arbitrary because of the folding modulo a. Different choices for the global shift a correspond to different but undistinguishable incommensurate structures described by the same function g(x) [39]. However, it is important to set the origin to a lattice site because a is then coherently determined between different experimental images. Thus hull plot data from different images of a grain boundary can be simply grouped together, which allows a direct quantitative comparison between experiment and simulation. As long as the hull functions are continuous, one grain can slide against the other without static friction, like an elastic chain can slide on a periodic substrate. The appearance of discontinuities in the hull functions, due to atomic rearrangement influenced either by a change in interaction strength or by any external influence such as stress normal to the interface, corresponds to the Aubry transition and the onset of pinning [22,40,41]. Fig. 4 shows the hull function of the simulated incommensurate boundary, determined from two rational approximants. The graph plots gx(x), the atomic column displacements Dx (ordinate) vs. ideal lattice positions x (abscissa), for the (0 0 1) atomic plane parallel and closest to the interface. This plot is for the grain with periodicity ro in the incommensurate direction x, normalized to the p periodicity 2ro of the opposite grain, ro being the interatomic distance in gold. The large open circles correspond to the 17:12 rational approximant while the continuous line is for the 239:169 approximant. The excellent match between the two models validates the fact that these rational approximants are already large enough to represent the incommensurate grain boundary. Fig. 4 also illustrates the Aubry transition when a compressive strain perpendicular to the boundary is applied to simulate an increased atomic interaction across

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Fig. 4. Modulation hull function Dx = g(x), for the nearest atomic plane to the interface in the grain having (0 0 1) parallel to the grain boundary. The plots are for three models: 17:12 and 239:169 approximants (open circles and continuous curve respectively), as well as 239:169 approximant with a 3.5% strain (small solid circles) applied perpendicular to the boundary (large discontinuities are indicated by vertical line segments). The continuous and discontinuous functions g(x) are representatives of the two possible states of an incommensurate interface, which are separated by an Aubry transition. The points are shaded by the energy E of the corresponding atoms like in Fig. 2a.

the interface. With increasing strain, the hull plot develops discontinuities and an associated transition from unpinned to pinned behavior. It has been shown that for the present boundary, the onset of the Aubry transition is predicted to occur at 2% strain, with the largest magnitude of these discontinuities reaching a maximum at a strain of 3.5% [22]. The discontinuities imply the p pinning behavior. For example, at the origin (x = 0 or 2ro), atoms that are directly aligned across the boundary in the unrelaxed structure are displaced by ±18 pm, requiring a 36 pm jump at the slightest motion of the grains relativep to each other. A similar effect occurs at positions x/( 2ro) = 0.3 and 0.7. These discontinuities correspond to the gaps in the distribution of the structural units, as illustrated in Fig. 3. Note that the hull function of a periodic grain boundary has a discrete distribution along the x-axis. This is why the 17:12 approximant shows only 17 data points in Fig. 4. For short-period boundaries, the hull plot would have even fewer data points, indicating discrete, well-defined structural units. Fig. 4 shows only one of several characteristic hull plots. Indeed, four separate plots are needed to characterize the Dx- and Dy-components of the atomic displacements that describe the lattice relaxations in the two atomic planes immediately adjacent to the interface. 4.4. Analysis from HRTEM images HRTEM images such as that shown in Fig. 1a display the structure of the boundary well, but due to the effect of the electron beam, the structure changes dynamically. In order to quantify this dynamic effect on the measurement of the hull function, images of the same boundary segment were recorded in a time sequence during beam

(a)

(b)

Fig. 5. (a) Hull plot from atomic columns at the boundary in the grain with the (0 0 1) plane parallel to the interface. The data were measured from a 32-image time sequence of a single boundary segment under the influence of the electron beam. (b) Root mean square error obtained from these measurements and comparison to the hull function for a simulated boundary using the 239:169 approximant under a uniaxial strain of 1.2% (solid line).

exposure to sample many configurations. The measured spread in atomic column positions was then used to quantify the error in measuring the displacement due to the grain boundary relaxation. This scatter is illustrated in Fig. 5a, which shows a hull plot measured from 32 images of the same boundary segment recorded over a period of 32 s. The sources of error in measuring the relative lattice positions include statistical errors in the lattice fitting procedure, relative motion of the two lattices and beam-induced

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displacement of atomic columns. Relative motion of the two lattices is easy to identify and exclude from the data. To minimize statistical errors and reduce sensitivity to error from beam induced displacements of a few atomic columns, lattice fitting was done on a relatively large number of atomic positions, away from the grain boundary. This procedure reduces the error to 4 pm. On the other hand, the error in measuring the displacement of an atomic column at the grain boundary from a corresponding lattice point was found to be 10 pm. The measured atomic column displacement includes a component due to the grain boundary relaxation (ideally constant during a time sequence) and a component due to the atomic motion. Therefore a large fluctuation at the grain boundary significantly increases the error. Based on this observation it was found that the accuracy of the current measurement is mostly limited by these atomic motions and can be further improved only by minimizing atomic fluctuations. However, attempts to reduce atomic motion by decreasing the electron beam intensity within a practical range or reducing the accelerating voltage to 80 kV had no visible effect on the dynamics. Fig. 5b shows the same data as in Fig. 5a, binned into discrete data points representing individual atomic columns with horizontal and vertical error bars. The sparse distribution of data points does not indicate that the boundary is periodic, but simply reflects the limited statistics due to the small number of atomic columns sampled in this boundary segment. For comparison with the model, Fig. 5b also shows the hull plot for the best matching grain boundary, simulated with a 1.2% normal strain (solid line). It is apparent that the experimental observations follow a similar trend to the simulated structure. The limited number of measurements clearly shows the need for better sta-

tistics to obtain a denser distribution of data points. This is why similar time sequences were obtained from several boundaries, as described below. Fig. 6 shows images from six different grain boundary segments between 4 and 8 nm long, used to measure the hull functions for this boundary. Each of the six segments was recorded in a dynamical time sequence, as described above, so that the total data set represents measurements from 240 individual images. To avoid systematic errors, these images were scrutinized for coordinated atomic motion during image acquisition. Such motion typically results in a very small relative displacement and rotation of the two adjoining grains, or nucleation and migration of steps at the grain boundary. Images around these events were discarded from the data set and the remaining images were used to determine the position, displacement and error bars for every atomic column adjacent to the grain boundary. Fig. 7 shows the hull plots obtained from these measurements. The four columns present four separate hull plots, one each for the x- and y-components of the atomic displacements in the two planes immediately adjacent to the interface. The three rows compare these data with three different models (shown as solid black lines). The solid red curves are trend lines from a data fit using a weighted kernel smoothing method. The horizontal and vertical error bars associated with each point were calculated by tracking individual atomic columns over multiple images as described in Fig. 5. Thus, every data point in these plots corresponds to an individual atomic column in one of the six HRTEM images. The measurements from experimental images were compared with the hull functions for the simulated boundary

Fig. 6. HRTEM images of six different incommensurate grain boundary segments. Atomic relaxation measurements from these six images were combined to calculate the hull plots.

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a

b

c

d

e

f

g

h

i

j

k

l

Fig. 7. Hull plots of atomic displacements along the x-direction parallel to the boundary (columns I, II) and y-direction perpendicular to the boundary (columns III, IV) for the two planes immediately adjacent to the boundary. Displacements for (0 0 1) plane shown in columns I and III, and for (1 1 0) plane in columns II and IV. The red solid curves are trend lines calculated using a weighted kernel smoothing method. The three rows show the same experimental data and trend lines, compared to three different models (black solid lines) simulated with strains of 0%, 1.2% and 3.5%, in rows 1, 2 and 3 respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

structure before and after the Aubry transition. In the simulations, the Aubry transition was modeled by applying a compressive strain perpendicular to the interface. These simulations indicated the onset of the Aubry transition at 2% strain, attaining a maximum discontinuity in the hull function at 3.5% strain [22]. In Fig. 7, the hull functions for 0%, 1.2% and 3.5% compressive strain are shown as black solid lines in rows 1, 2 and 3, respectively. As can be seen from the graphs in row 1, for 0% strain the experimental measurements and the simulated boundary exhibit a good match, especially for y-displacements perpendicular to the boundary (c, d). The x-displacements parallel to the boundary follow a similar trend, but with a significant difference in amplitude (a, b). For a 1.2% strain (row 2 of Fig. 7) this mismatch is minimized, as indicated by a lower root mean square error. This improvement is particularly apparent in (e) and to a smaller extent in (g). For a 3.5% strain (row 3 of Fig. 7), the hull functions of the model show discrete gaps, which are characteristic of the pinning effect associated with the Aubry transition. These discontinuities are especially prominent in (i) and (k), which correspond to the row with the short average p interatomic distance ro reacting to the longer period 2ro of the other grain. The magnitude of these gaps is more than 30 pm, much larger than the experimental measurement error of the current approach. Therefore any discontinuity due to the Aubry transition should be apparent in the hull plots obtained from the experimental images. The comparison of simulated and measured data in row

3 of Fig. 7 shows a poorer match than the other two models, supporting the conclusion that the structure of this boundary is unpinned and below the Aubry transition. The detailed comparison of experimental and simulated structures outlined in Fig. 7 is not unambiguous. This is not surprising for several reasons. First, the structure was simulated for 0 K, which may have a bearing on the magnitude of atomic displacements. Second, beam-induced atomic fluctuations at the grain boundary led to a sampling of multiple structural configurations. Third, the property of superglide for this boundary [21] makes it easy for boundary segments to slide in response to nearby defects or stresses, leading to experimental scatter or uncertainty. Fourth, long segments of defect-free boundary were rarely found. It has been shown that in similar grain boundaries in Au, defects such as disconnections lead to local lattice rotations and stress concentrations [42]. Local stresses from such defects could explain how a globally unstrained bicrystal could develop locally strained structures. The fact that the best fit between experiment and simulation was obtained for a strain of magnitude 1.2% is not considered significant, however, because it is likely that over the course of a dynamic time sequence the boundary samples many different configurations, representing many different states of strain. This diversity of states is increased by adding data from several different boundary segments (although no systematic difference between data from different segments was detected). Thus, given the incommensurate and dynamic nature of this boundary, the match between

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experimentally measured and theoretically predicted structures is striking. We conclude p that the experimentally 2 incommensurate grain measured structure of the boundary in gold is well described by the model that predicts the interface to be on the unpinned side of the Aubry transition. 5. Summary and conclusions We have introduced an approach based on the Aubry hull function to analyze an incommensurate grain boundary. This approach uses the two lattices of the grains far from the grain boundary as a reference for the unrelaxed ideal positions at the interface. Atomic relaxations are measured relative to these reference positions. Atomic locations that are spread along the interface are folded back into one cell of the periodic reference lattice, thus accumulating quantitative information that can be compared to models. The possibility to join together data from different images is particularly well suited to non-periodic interfaces. Similar surroundings of two atoms along the interface will result in folded positions that are close in the hull function domain. With sufficient statistics, discontinuities can be detected and discriminate between pinned and unpinned types of incommensurate structures. Following this approach, we have analyzed atomic resolution images of an incommensurate 90°h1 1 0i tilt grain boundary in gold. By combining data from many aberration-corrected HRTEM images of different grain boundary segments, we were able to achieve sufficient accuracy to make a quantitative comparison with theoretical predictions. The agreement between experiment and simulation was found to be close, indicating that the atomic relaxations at the grain boundary were continuous and the structure corresponded to the unpinned state where an incommensurate boundary is capable of frictionless glide. The ability to link characteristic features of the hull plot with the behavior of an interface provides a convenient tool to study structure–property relationships in grain boundaries. Acknowledgements The National Center for Electron Microscopy, Lawrence Berkeley National Lab, is supported by the US Department of Energy under Contract # DE-AC0205CH11231. VRR also acknowledges support of Nanotechnology and Functional Materials Center, funded by the European FP7 project No. 245916 and of the Ministry of Education and Science of Republic of Serbia, under contract No 172054. FL acknowledges support by the Structure Fe´de´rative de Recherche CEA-UJF (Inac; # FED 4177) of the French Commissariat a` l’E´nergie Atomique et aux E´nergies Alternatives (CEA).

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