On the atomic structure of an asymmetrical near Sigma = 27 grain boundary in copper

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On the atomic structure of an asymmetrical near Σ = 27 grain boundary incopperJean-Philippe Couzinié a; Olivier Hardouin Duparc b; Sylvie Lartigue-Korinek a; Jany Thibault-Pénissonc; Brigitte Décamps d; Louisette Priester a

a ICMPE, CNRS, UMR 7182, Université Paris 12 Val de Marne, Thiais, France b LSI, CEA-X-CNRS,UMR 7642, École Polytechnique, Palaiseau Cedex, France c TECSEN, CNRS, UMR 6122, Faculté desSciences et Techniques de St. Jérôme, Marseille Cedex, France d CSNSM, CNRS/IN2P3, UMR 8609,Université Paris-Sud 11, Orsay, France

To cite this Article Couzinié, Jean-Philippe, Hardouin Duparc, Olivier, Lartigue-Korinek, Sylvie, Thibault-Pénisson, Jany,Décamps, Brigitte and Priester, Louisette'On the atomic structure of an asymmetrical near Σ = 27 grain boundary incopper', Philosophical Magazine Letters, 89: 12, 757 — 767To link to this Article: DOI: 10.1080/09500830903282925URL: http://dx.doi.org/10.1080/09500830903282925

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Philosophical Magazine LettersVol. 89, No. 12, December 2009, 757–767

On the atomic structure of an asymmetrical

near '^ 27 grain boundary in copper

Jean-Philippe Couziniea, Olivier Hardouin Duparcb*, Sylvie Lartigue-Korineka, Jany Thibault-Penissonc, Brigitte Decampsd and Louisette Priestera

aICMPE, CNRS, UMR 7182, Universite Paris 12 Val de Marne, 2-8 rue Henri Dunant,94320 Thiais, France; bLSI, CEA-X-CNRS, UMR 7642, Ecole Polytechnique, 91128Palaiseau Cedex, France; cTECSEN, CNRS, UMR 6122, Faculte des Sciences et

Techniques de St. Jerome, 13397 Marseille Cedex, France; dCSNSM, CNRS/IN2P3,UMR 8609, Universite Paris-Sud 11, 91405 Orsay, France

(Received 12 March 2009; final version received 19 August 2009)

The atomic structure of an asymmetrical near � ¼ 27 {525} tilt grainboundary (GB) in copper is determined by coupling high-resolutiontransmission electron microscopy and molecular dynamics simulation. Theaverage GB plane is parallel to {414} in crystal (1) and {343} in crystal (2).The detailed GB structure shows that it is composed of facets alwaysparallel to {101} and {111} in crystals (1) and (2), respectively. The atomicstructure of one facet is described using the structural units model. Eachfacet is displaced with respect to its neighbours by a pure step, giving rise tothe asymmetry of the GB plane orientation. The energy of this asymmet-rical GB is significantly lower than that of both the {525} symmetrical andthe {11,1,11}/{111} asymmetrical �¼ 27 GBs. One GB region displaysanother atomic structure with a dislocation that accounts for the misfitbetween interatomic distances in the {414} and {343} GB planes.

Keywords: atomic structure; atomistic stimulation; copper; grain boundarystructure; HRTEM

1. Introduction

In the face-centred cubic (fcc) materials with a low stacking-fault energy, prone tothermal twinning, the proportion of �¼ 3n grain boundaries (GBs) in polycrystalsmust be relatively high [1]. The coherent twins �¼ 3 {111} develop for energeticreason. The �¼ 9, �¼ 27 and �¼ 81 GBs appear for geometrical reason since theynecessarily connect the �¼ 3 GBs at triple junctions in agreement with thecoincidence index combination rule [2]. In the case of copper, histograms of the GBdistributions in annealed polycrystals reveal the following percentages: �¼ 3 (55%),�¼ 9 (10%), �¼ 27 (5%), yielding 70% of �¼ 3n GBs [3]. These proportions maybe managed by sequential thermomechanical treatments [4].

The predominance of �¼ 3n GBs most likely affects the global response of thematerial to any stimulus. The �¼ 3–�¼ 3–�¼ 9 or �¼ 3–�¼ 9–�¼ 27 triple

*Corresponding author. Email: [email protected]

ISSN 0950–0839 print/ISSN 1362–3036 online

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junctions, which are numerous in the microstructure, may also have a specific rolein the plastic behaviour of polycrystals. For example, at the triple junctionbetween near �¼ 3, near �¼ 9 and near �¼ 27 coincidence GBs, the three intrinsicdislocation networks perfectly connect to each other [5]. This must favourcontinuous GB sliding associated with small GB displacement, resulting from themotion of secondary dislocations within the GBs [6].

The response to stress differs from one GB to another. In particular,experimental studies performed on nickel and copper bicrystals have clearly shownthat the interactions between dislocations and GBs strongly depend on theGB structure [7–9]. They allow us to distinguish between hard GBs such as �¼ 3{111} for which the entrance and accommodation of dislocations are not easy,relatively soft GBs such as near �¼ 11 {323} and near �¼ 9 {212} for which theaccommodation processes occur more easily but incompletely, and really soft generalGBs where the dislocation stresses relax rapidly. The detailed processes of theseinteractions have been established [7–9] and are supported by simulations [10,11].

Even though it is not straightforward to relate the behaviours of individualGBs to the behaviour of the material as a whole, advances in the knowledge ofmaterial properties imply a better understanding of the structures, the defects and theproperties of individual GBs.

Two complementary approaches may be attempted to understand the role of theGB ensemble in the polycrystal properties. At the mesoscopic level, it implies to betterestablish the distributions of the GB characteristics (including GB planes)in polycrystals. At the atomistic level, it is necessary to improve our knowledge ofthe GB structures, as a base for describing the intergranular defects and theinteractions between GBs and other crystal defects. For this latter point of view, thestructures and the defects of symmetrical tilt (STGB) and asymmetrical (ATGB)�¼ 3 GBs in copper have been extensively studied [12–14]. Concerning �¼ 9 GBs, itis noteworthy that the calculated energies of the ATGBs near the {212} STGB arealmost the same until an inclination angle of the boundary plane of about 16�

including the {11,1,11}//{111} ATGB [15–17]. For higher GB plane inclinationangles with respect to {212}, the intergranular energies are generally lower than theprevious range. Some numerical discrepancies appear between the simulation datafor only a few GBs [15,16]. Similar energy variations have been found experimentally[18]. It is remarkable that, despite the large gap between the simulation and theexperiment scales (atomistic description in one case and average GB plane estimationin the other), both types of results agree for the low energy of {151}//{111} ATGB[15,16,18]. However, this GB has been found unstable against dissociation reactions[15,19,20]. A recent combination of simulations and High-Resolution TransmissionElectron Microscopy (HRTEM) allowed the descriptions of the atomic structuresof several �¼ 9 h101i tilt GBs in copper [17]. ATGB facets with {111} plane in onegrain are mainly observed. The symmetrical facets which appear between theasymmetrical ones have a translational state imposed by the ATGBs and displaya structure with an energy which is not the lowest possible one. Locally asymmetricalfacets may also display a 3D structure [21].

On the basis of atomistic simulations, structures of �¼ 27 STGBs have beenproposed for metals with low and large SFE [22,23]. To our knowledge, there areno experimental results to be compared with the predicted structures in the case

758 J.-P. Couzinie et al.


of metals. �¼ 27 ATGBs are totally unknown at the atomic level. By couplingmolecular dynamics simulations and HRTEM observations, the present workattempts to fill the lack of information about the structures and energies of some�¼ 27 GBs that constitute, together with �¼ 3 and �¼ 9 GBs, the major elementsof the GB network in copper.

2. Experimental and computational procedure

2.1. Experimental details

The thin foils containing the GB are obtained from disks of about 130 mm thick usingtwin-jet electropolishing and ion-milling. HRTEM observations of the �¼ 27GB are performed on a JEOL3010 at 300 kV. The electron microscope characteristicsare as follows: spherical aberration Cs¼ 0.6mm; chromatical aberrationCc¼ 1.3mm; focus spread of 10 nm; semi-divergence angle of 1mrad; point-to-point resolution rP¼ 0.17 nm. All the micrographs are taken close to the Scherzerdefocus and along the common [101] axis zone. HRTEM image simulations ofatomistic models are extracted from defocus-thickness maps using MacTempassoftware [24]. The sample thickness has been estimated from the comparison betweenthe experimental image and images simulated for different values of defocus andthickness. In order to have a quantitative estimation of the matching between theobserved and the simulated images, positions of the intensity maxima are superposedafter alignment of the two images by cross correlation [24].

2.2. Numerical methodology

The basic methodology has already been described in previous works [17,25]. A semi-empirical n-body potential well fitted for copper is used [26,27] to simulateconfigurations containing a few thousands of atoms. Free surfaces are created in thedirection perpendicular to the GB plane and periodic boundary conditions areconsidered in the two other directions. Simulated annealing and quenching are usedto minimise the energies of various atomic configurations.

3. Results

The observed �¼ 27 GB comes from an intended �¼ 9 GB, the latter beingobtained by solidification of a �¼ 3–�¼ 3–�¼ 9 tricrystal grown by verticalBridgman technique [28]. This �¼ 27 GB probably comes from a dissociationprocess.

Figure 1 gives a general view of the interface. In this region, the misorientationangle has been determined via a series of measurements with DigitalMicrograph� tobe 32.7� 1.0�. It is close to the angle corresponding to the exact �¼ 27 coincidence(�¼ 31.59�). The local atomic arrangement of this structure is presently analysedin terms of structural units (SUs) by coupling observations and simulations. SUs areindicated by full white lines. At the top and at the bottom of the micrograph, the GBdisplays an average plane parallel to ð�4�14Þ in crystal C1 and to ð�3�43Þ in crystal C2.

Philosophical Magazine Letters 759


Figure 1. HRTEM micrograph of the near �¼ 27 GB. Atomic columns are imaged in white.Thin white lines indicate the SUs, they reveal the occurrence of periodic structures at the topand at the bottom of the GB; the central region is not periodic but still composed of the sameSUs. The average orientation of the GB plane ð�4�14Þ in C1 and ð�3�43Þ in C2 is indicated by twowhite coarse lines at the bottom. Black crosses superimposed on the image are atomicpositions along the tilt axis corresponding to the black circles of Figure 2. The sub-regionlimited by dashed lines corresponds to the analysis carried out in Figure 3.



In fact, the interface is composed of asymmetrical facets shifted by a step of one{111} interplanar distance with respect to its neighbours.

The incommensurate ð�4�14Þ1==ð�3�43Þ2 GB has been simulated using its simplest

approximant (1:1). The translation period vector on the ð�4�14Þ side isffiffiffiffiffi

33p

a=ffiffiffi

2p

,whereas its counterpart on the ð�3�43Þ side is

ffiffiffiffiffi

34p

a=ffiffiffi

2p

, where a is the cubic latticeparameter [25]. The ratio between these two periods is an irrational numberffiffiffiffiffiffiffiffiffiffiffiffi

34=33p

� 1:015. It is close to 1, with a misfit factor �� 1.5%. The h110i projectedview of the energy-minimised structure shows that each facet corresponds to oneperiod as follows: |EEEA| with E the �¼ 9 SU and A akin to the �¼ 1 SU(Figure 2a). This period is appreciably shorter than the periods of the two exact�¼ 27 GBs given in Table 1: the {525} STGB [23] and the {11,1,1}//{111} ATGB(Figure 2b). The comparison between the experimental micrograph and the HRTEMsimulated image corresponding to the model of Figure 2a provides a good qualitativefit as shown in Figure 3. The period reproduced in this figure corresponds to theSU sequence: jEAEEj in order to explicitly include the step (compare Figure 3 toFigure 1). Whereas the matching is especially good in the GB region, there are somediscrepancies in crystal C2 that probably result from a small twist component of theGB. The energy of the ð�4�14Þ1==ð

�3�43Þ2GB is lower than the energies of the two exact�¼ 27 GBs given in Table 1. The �¼ 27 {11,1,11}//{111} given in Figure 2b clearlywould not fit the experimental micrograph. The SU sequence is |EEEAE0�EEEAE0|where E0 has been identified as a Lomer SU in [17].

For the sake of simplicity, here we use only the E symbol even for distortedE SUs. Taking into account the topological versatility of the E units [17,29], whichbecome E1 by deformation, the period of the three GBs may be alternatelyconsidered as follows: |EE1AEE1A| for the STGB and |EE1E1AE0�EE1E1AE0| and|EE1E1A| for the two asymmetrical GBs. The HRTEM image of the GB supportsthis topological analysis (Figure 1). Yet, the E and E0 SUs correspond to anadditional {010} half plane on the C1 side whereas the E1 SU does not, and can besimply considered as made of two A units.

In the central part of the micrograph, the GB is still composed of E and A SUsbut their arrangement is more complicated. A circuit drawn around this region,starting and finishing at the top of an E unit (Figure 4) yields a closure failure wellmapped in the dichromatic pattern of the �¼ 27 GB. The associated Burgers vector~b ¼ a

54 ½7�8�7�1 is nearly contained in the GB plane and its length is 85 pm. The defect

can be visualised as an additional (010) plane in C1.

4. Discussion

The structure observed at the top and at the bottom of the near �¼ 27 GB (Figure 1)corresponds to several units of the periodic structure of {414}//{343} GB and fits wellwith the calculated structure. This goes well with the three following remarks:

(i) the misorientation of the asymmetrical {414}//{343} (33.29�) is very close tothat of the observed near �¼ 27 GB (32.7�) – within the measurementaccuracy;

(ii) its energy is significantly lower than for the energies of two other modelsproposed in Table 1; and



Figure 2. Two-dimensional projected images of the energy-minimised structures (full linesindicate the structural units): (a) the (1:1) approximant of the incommensurate ATGBð�4�14Þ1==ð�3�43Þ2. One period is described by the following simplified sequence: |EEEA|and forms a facet parallel to {101} (namely ð�101Þ1) in C1 and {111} (namely ð11�1Þ2) in C2.Each facet is shifted with respect to its neighbours by a step of one {111} (namely ð11�1Þ2)interplanar distance. Three periods are given for the sake of clarity, in order to explicitly showthe step and to fit the lower part of Figure 1. In the lowest period, eight ð0�10Þ1 atomic planesin C1 are overlined versus seven ð1�1�1Þ2 atomic planes in C2. The additional ð0�10Þ1 planecorresponds to the true E SU. The eight versus seven matching on the ð�4�14Þ1==ð�3�43Þ2mathematical plane actually implies an extra 1.5% mismatch; (b) the commensurateð�1�1, �1, 11Þ1==ð�1�11Þ2 �¼ 27, described by the following sequence: |EEEAE0� EEEAE0|.



Figure 3. (a) Simulated HRTEM image according to the model of the near �¼ 27

ð�4�14Þ1==ð�3�43Þ2GB in Figure 2 (a) (defocus �50 nm and thickness 11 nm). Thin white linesindicate the SUs; (b) comparison between the experimental image and the simulated one.Superimposed crosses correspond to the intensity maxima of the simulated image (determinedusing peak finding analysis in MacTempas software [24]); (c) plot of both experimentalpeak positions (circles) and simulated ones (crosses). The error bars on the intensity maximafor both experimental and simulated positions are within the size of the crosses/circles(about 25pm for the experimental positions and less than 15 pm for the simulated ones).

Table 1. Interfacial energies and atomic structures of symmetrical and asymmetrical near andexact �¼ 27 GBs in copper.

GB plane � (�)Structural unitdescription

Periodlength (nm)

Energy(mJ m�2)

f525g1==f525g2 31.59 (�¼27) |EEAEEA| 1.88 690f11, 1, 11g1==f111g2 31.59 (�¼27) |EEEAE0� EEEAE0| 3.98 704f414g1==f343g2 33.29 |EEEA| 1.48 640

Notes: The f414g1==f343g2 interface corresponds to the (1:1) approximant with �� 1.5%.The cubic lattice parameter a of copper is equal to 361.5 pm.



(iii) the real GB locally adopts an incommensurate structure with a shorterperiod than the symmetrical {525} and asymmetrical {11,1,11}//{111}�¼ 27 GBs, provided the occurrence of secondary dislocations. Similarbehaviour also occurs in a near �¼ 11 GB of nickel, although the observedincommensurate GB had a higher energy than the other two [25].

The following discussion is divided into two parts: first the incommensuratestructure is analysed from an atomic point of view, then, the presence of intrinsicdefects inherent to this structure is considered.

The plane of the {414}//{343} GB decomposes into facets that are strictlyparallel to the {101} planes in C1 and {111} C2 dense planes providing the formation

Figure 4. Enlargement of the central part of Figure 1, with a circuit drawn around thedefectuous region starting and finishing at the top of an E unit (points A and B). A defectappears that may be associated to the extra (010) planes noted X in the crystal C1.Its associated Burgers vector is given in the text.



of steps. This tendency for the GB planes in fcc metals to form facets parallel to{101} in one crystal and {111} in the other has already been described even forSTGBs [23]. For instance, the �¼ 27 {525} and the �¼ 19 {313} symmetrical GBsare composed of {101}//{111} facets more or less extended and alternately inclinedon one direction and on the other with respect to the GB plane. Locally the mirrorsymmetry is broken, but globally the imposed symmetry is maintained. The observed{414}//{343} GB is composed of only one type of facet, which are parallel to eachother and separated by steps to account for the GB asymmetry. It is worth notingthat these steps actually allow for a continuous prolongation of the {101} (namelyð�101Þ1) and {111} (namely ð11�1Þ2) planes along the {414}//{343} interface, bycontrast to what happens in the case of the {11,1,11}/{111} interface (compareFigures 2a and b). The formation of these pure steps appears as a relaxationphenomenon which avoids the introduction of the E0 SUs. The E0 SU, present in the{11,1,11}//{111} and in the {101}//{111} ATGBs (see Figure 5 of [17]), is not to befound in the energy minimised STGBs of comparable misorientations. Theseconsiderations are in agreement with the small energy of this {414}//{343} GBcompared to the other GBs given in Table 1, and also to the energy of the {101}//{111} GB which is slightly higher, about 676mJm�2 (for the approximant (9:11) [25]with a mismatch as small as �� 0.2%, see [17]). It is noteworthy that a realisation oflimited sections of {101}//{111} facets with pure steps appears to be energeticallypreferable to the extended {101}//{111} interface.

This structure requires the presence of intrinsic defects. The {414}//{343} ATGBmay be analysed as an interphase interface with the almost matching of eight {010}1planes in C1 versus seven inclined {111}2 planes in C2, with a 1.5% � mismatch.Thus, with respect to the approximant (1:1), the real {414}//{343} GB structure mustcontain some misfit dislocations. The defect whose Burgers vertical length is 85 pm,analysed in the previous section, accounts for a 5.7 nm approximant length. As thewhole GB length in Figure 1 is 11 nm, only one defect located in the middle of theimage can be detected, as observed.

Finally, it must be noted that the bicrystalline thin foil that has been observedwas intended to contain a �¼ 9 GB as one component of a �¼ 3–�¼ 3–�¼ 9 triplejunction [24]. The observation of a near �¼ 27 GB may result from the dissociationof the near �¼ 9 component to form a triangular prism bounded by one near �¼ 27GB and two �¼ 3 GBs, one coherent {111} twin and one incoherent {112} twin.This type of GB dissociation was not mentioned in previous works in pure Cu or Cu–Si alloys that described and/or observed the formation of two coherent and oneincoherent �¼ 3 GBs coming from several �¼ 9 STGBs [19,20,30,31]. In this article,this dissociation could be explained by the relative low energy of the observed {414}//{343} near �¼ 27 GB (640mJm�2) compared to the energies of most �¼ 9 regionspreviously observed in the same sample (from 669 to 716mJm�2 [17]). The energy ofthe coherent twin in copper being estimated to be 20–25mJm�2 [17], its formationtogether with the {414}//{343} GB is thus likely to occur. Alternatively, a �¼ 27 GBmay be formed by coalescence if a coherent twin, formed in crystal (1) or crystal (2),encounters the �¼ 9 GB between the two crystals. This type of coalescence hasalready been mentioned for two �¼ 3 GBs giving rise to a �¼ 9 GB in stainlesssteels [32]. Whatever the process that has really been involved during the tricrystalgrowth, the existence of a �¼ 27 GB implies the occurrence along the �¼ 9 GB of a



triple junction �¼ 3–�¼ 9–�¼ 27, which is not visible in the observable region ofthe thin foil.

5. Concluding remarks

Thanks to the coupling between HRTEM observations and simulations, interestingfeatures concerning the atomic structures and the energies of exact and near �¼ 27ATGBs are summarised in the following; some comparisons with the resultsobtained on exact and near �¼ 9 ATGBs [17] are highlighted.

. Compared to two exact �¼ 27 GBs, the slightly deviated from �¼ 27misorientation {414}//{343} {1:1} approximant GB displays the lowestenergy.

. The realisation of limited sections of {101}//{111} facets with pure stepsin between, such as in the {414}//{343} {1:1} approximant GB, appears tobe energetically preferable to the extended {101}//{111} interface despitethe mismatch of 1.5%. The occurrence of these short period sections thusrequires the presence of ‘misfit like’ dislocations. One such defect has beenfound in the central part of the experimental micrograph.

. All the �¼ 27 ATGBs are composed of the same SUs arranged in differentsequences. These units are mainly the A units of the single crystal and theE unit of �¼ 9 {212} STGB, deformed to different degrees. Furthermore,an E0 unit appears in the �¼ 27 ð11, �1, 11Þ==ð11�1Þ atomic structure that wasalready present in the �¼ 9 ð11, �1, 11Þ==ð1�1�1Þ core structure and in ametastable variant of the �¼ 9 {212} STGB.

. The exact and near �¼ 9 GBs previously studied display the same SUs withvarious local distribution leading to similar energies [17]. These propertieshave to be related to their observed easy extrinsic dislocation accommoda-tion [9]. Here one GB clearly seems preferable, namely the {414}//{343}.It would be interesting to study in detail the deformation behaviour ofthis ATGB.

Finally, the atomistic results on �¼ 27 GBs, together with those on �¼ 9 GBs,are a prerequisite to further understanding the atomic arrangement at the �¼ 3–�¼9–�¼ 27 triple junctions that constitute a major feature in polycrystalline metalswith a low stacking-fault energy.

Acknowledgement

The authors would like to thank Professor S. Hashimoto (Department of Intelligent MaterialsEngineering, Osaka City University, Japan) for providing them with the �¼ 3–�¼ 3–�¼ 9tricrystal of copper.

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On the atomic structure of an asymmetrical near Sigma = 27 grain boundary in copper

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