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Acta Materialia 63 (2014) 1–15

The roles of grain boundary dislocations and disclinationsin the nucleation of {10�12} twinning

Christopher D. Barrett ⇑, Haitham El Kadiri

Center for Advanced Vehicular Systems, Mississippi State University, Mississippi State, MS 39762, USA

Department of Mechanical Engineering, Mississippi State University, Mississippi State, MS 39762, USA

Received 24 June 2013; received in revised form 6 September 2013; accepted 6 September 2013Available online 9 October 2013

Abstract

Consider a magnesium bicrystal created from two single crystal lattices referred to as black and white. The bicrystal abuts a basalplane of the black crystal with a first-order prismatic plane of the white crystal. At zero stress, the boundary relaxes by developing coher-ent terraces, while two types of disconnection dipoles and a misfit dislocation above each dipole nucleate, in combination, to remove theattendant long-range elastic strains and maximize local coherency. When the bicrystal is stretched normal to its semicoherent interface,stress motivates the misfit dislocation at the far side of the boundary to glide and annihilate at the triple point. The removal of thisbounding misfit dislocation allows the positive step of the disconnection dipole to conservatively move away. The negative step of thisdipole, however, remains sessile at the grain boundary as it cannot glide conservatively. This sessile step facilitates nucleation of a newdisconnection dipole. The first process of disconnection dipole widening then repeats, and the new residual negative step coalesces withthe first one to form a novel disconnection of double step height. The whole process cycles once more, and two of these double-heightdisconnections pile-up at the boundary. The pile-up quickly relaxes into a coherent boundary along the {10�12} plane; a {10�12} twinfacet thus nucleates, and a disclination dipole arises to bound the twin facet at the two junctions. Soon thereafter, the lower disclination,at the opposite side from where the misfit was removed, begins emitting twinning disconnections toward the upper disclination. Thesetwinning disconnections caused their source junction to recede in favor of an overwhelming lengthening of the twin facet. All twinningdisconnections end up moving through the upper disclination, thereby undergoing a dislocation transformation event. The new trans-formed disconnection glides freely along the remaining basal-prismatic boundary segment, and enables this boundary to be dragged withthe propagating twin facet. This steady stage proceeds until the entire bicrystal transforms in to a single twinned crystal. This paperexamines the fundamentals of these twin nucleation mechanisms at the grain boundary on the basis of the interfacial defect theoryadvanced by Pond and co-workers [1,2]. It introduces an important twin nucleation mechanism with dynamics of twin faceting in relationto an important asymmetric low-energy boundary in hexagonal close-packed materials.� 2013 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Magnesium; Twin nucleation; Interfacial dislocations; Grain boundary; Faceting

1. Introduction

Recent considerations of Mg and Ti as candidate mate-rials for weight reduction in the automotive and aerospaceindustries have reinvigorated the study of deformationtwinning and slip in hexagonal close-packed (hcp) materi-als [3,4]. Central to this work is the {10�12} twin mode,

1359-6454/$36.00 � 2013 Acta Materialia Inc. Published by Elsevier Ltd. All

http://dx.doi.org/10.1016/j.actamat.2013.09.012

⇑ Corresponding author. Address: Center for Advanced VehicularSystems, 200 Research Blvd., Mississippi State University, MississippiState, MS 39762, USA.

E-mail address: cdb333@cavs.msstate.edu (C.D. Barrett).

which is mediated by the most glissile disconnections ofall twin modes in hcp metals [5]. The reason for the rapidgrowth of this twin mode is not yet entirely clear as the dis-connection core, though wide, is still relatively narrowcompared to other less readily propagating twinningmodes, such as the {11�21} twinning observed in Ti andZr. The relatively small shear magnitude of {10�12} par-tially explains its predominance; however, {10�11} pos-sesses an almost equally small characteristic shear butstill shows a very high critical resolved shear stress (CRSS).{10�12} requires four small shuffles to produce the twinned

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2 C.D. Barrett, H. El Kadiri / Acta Materialia 63 (2014) 1–15

lattice, while {10�11} needs eight. Therefore, it is perhapsthe combined simple shuffles and the small characteristicshear that may make the best argument for the universalityand fast growth of {10�1 2} twinning. However, many pecu-liarities of this twin mode, such as detwinning and trans-mission through grain boundaries (GBs), remainunexplained.

Recently, Zhang et al. [6] and Li and Ma [7] have pro-posed that twinning disconnections do not accomplish twingrowth in the traditional manner, i.e. by propagation, butinstead disconnections separately nucleate at each localpoint on the twin boundary (TB) in what they call a shuf-fle-dominated twin growth process. Zhang et al. [6] haveobserved {10�12} twin boundaries deviating significantlyfrom the twin plane. According to Zhang et al. [6], thegrowth of {10�12} twinning on these boundaries cannotbe explained by the classical theory of twinning disconnec-tions. However, Li and Ma [7]’s theory has been called intoquestion by Serra et al. [8], who state that Li and Ma [7]have not understood interfacial defect theory and havefailed to account for experimental and simulated observa-tions of the traditional twinning mechanism.

An alternate perspective of twin boundaries deviatingfrom the twin plane is given by Li et al. [9]. They observedf10�11g twins faceting onto f10�13gkf10�10g boundariesand identified disconnections which can potentially accom-plish the associated twin growth on this faceted plane.However, these authors did not address the mechanismsof facet nucleation, and it is unclear how disconnectionactivities on two facets can coordinate with each other.Interactions between facets and twins are importantbecause faceting of grain boundaries into twin segmentscan be considered as an important aspect of twin nucle-ation in engineering polycrystalline alloys. Currently,despite the profound effect of grain boundaries and corre-sponding texture on twinning activity, the fundamentalsof twin nucleation at grain boundaries remain largely unex-plored. Twin nucleation is still limited in the literature toan effect of dislocation pile-ups, which is relatable to theeffect of grain size on the twinning stress [10–12]. Thismechanism provides an explanation of the negative sensi-tivity of twinning stress to the grain size, as larger grainsallow longer pile-ups to build up, and thus sufficient stresslevels for twin nucleation. However, the profound effect ofthe atomic structure of the various types of tilt, twist andmixed boundaries is yet to be studied and understood.

Studies performed in 2013, published after the accom-plishment of the analysis in this paper, have noted theimportance of faceting in {10�12} twin formation [13–16],and stressed the prominent role of the low-energy basal–prismatic (BP) boundary (164 mJ m�2). {10�12} twins werefound to grow on both the twin plane facet and the BPfacet. However, these authors did not identify the discon-nection-based mechanisms through which the two facetsinteract, and how either facet may nucleate from an ini-tially flat interface. Moreover, the misorientation differenceacross the BP facet and the twin boundary has not been

thoroughly accounted for in these works. A change in mis-orientation requires either an interfacial disclination or anarray of grain boundary dislocations (GBDs) in at leastone facet to accommodate the rotation that upsets the idealmisorientation. Any studies concerned with faceting shouldclearly identify the activities and transformation mecha-nisms of disconnections through facet transitions.

Motivated by these important fundamental gaps, thispaper examines on the basis of interfacial defect theoryadvanced by Pond and co-workers [1,2] the effect of theatomic structure inherent to the BP boundary on {10�1 2}twin nucleation and subsequent propagation. To this end,we used molecular dynamics (MD) which simulated uniax-ial tension applied to a bicrystal normal to its BP bound-ary. MD is a valuable simulation tool that allows us todecipher the topological features of complex dislocationmechanisms such as those correlated to moving interfaces.While transmission electron microscopy techniques remainindispensable, they are still limited compared to MD whencapturing the three-dimensional dynamic behavior of dislo-cation motion and twin growth.

2. Atomistic simulations

LAMMPS [17] was used to perform atomistic simula-tions of Mg deformation with an embedded-atomicmethod (EAM) potential by Liu et al. [18]. To capture dif-ferent aspects of the {10�12} twin growth behavior, weused bicrystals ranging in size from 1.6 to 11 million atoms,with free surfaces in all dimensions. While free-surface sim-ulations introduce debate regarding the effects of long-range elastic distortions and image stresses on simulationresults, simulations using periodic boundary conditionsinstead have artifacts caused by the wrapping of stressfields across the periodic boundaries, making it impossibleto determine the properties of an isolated disconnectionloop. We have concluded that the best method currentlyavailable for our study is simply to use cell sizes as largeas are computationally possible to reduce the free surfaceeffects.

Simulations with twin-plane and f000 1gkf10�10g (BP)initial interfaces were analyzed with loading on the hci-axisin uniaxial tension. The simulation with a twin-plane inter-face was also loaded on the twin plane in simple shear.Prior to loading, the simulation cells underwent conjugategradient energy minimization and at least 10 ps of equili-bration. The system stresses and local stress fields were cal-culated using the virial definition by Thompson et al. [19].Visualization and analysis were done using Ovito [20].

3. Synopsis of the deformation

For practicality, this paper focuses analysis on thebicrystal initially having a BP boundary. The conclusionsarising from the analysis of this simulation are applicableto all of the various simulations we performed. For thebicrystal with an existing BP boundary, the crystal that

Fig. 1. Molecular dynamic simulations of uniaxial tension applied to aMg bicrystal normal to the BP boundary showing the atomic positions in a{11�20} projection (a) at zero stress and after relaxation of the BP whichthen developed a series of interfacial dislocation dipoles, and (b)immediately after yield at 2.7% strain characterized by a sudden formationof a {10�12} twin facet near the upper free surface, splitting the BPboundary into upper (UBP) and lower (LBP) segments by virtue of adisclination dipole. The initial position of the BP is marked by the orangedashed line. (For interpretation of the references to color in this figurelegend, the reader is referred to the web version of this article.)

C.D. Barrett, H. El Kadiri / Acta Materialia 63 (2014) 1–15 3

has the basal plane aligned with the tensile axis is termedwhite, k, while the other having the basal plane normalto the tensile axis is termed black, l. In the 2-D projectionsof Figs. 1, 2, 5, 6, 7, 8, the white and black crystals are sit-uated on the left- and right-hand sides, respectively. InFig. 3, the white crystal is on top of the black one.

MD simulations of uniaxial tension normal to the BPled to the formation of {10�1 2} twinning at the yield stressrecognized by a sudden drop in stress. It is important tonote that the twinning stresses varied based on the typeand size of the cell, but was significantly lower with apre-existing TB (�300 MPa) than with a pre-existing BPboundary (�1.2 GPa). The yield stress was, however, thehighest of all when no GBs or TBs were preliminarily intro-duced [21]. This suggests that {10�12} twin nucleation ispreceded by an incubation regime which depends on thenature and state of pre-existing inhomogeneities. This isconsistent with the idea of the necessity of stress concentra-tions at local regions induced by dislocation patterns suchas pile-ups [5,22,23].

{10�12} twin nucleation in bicrystals occurred as a verysmall facet which suddenly seized a segment of the BPboundary (Fig. 1). This facet did not exceed a few interpla-nar spacings and was located at approximately 15 prismaticlayers below the triple point of the bicrystal free surface. Anatural outcome of the facet nucleation in the BP boundarywas the formation of two junctions which separated the ini-tial BP interface into two segments with the new facetbetween them. These segments are referred to as the upperBP boundary segment (UBP) and the lower and longer BPboundary segment (LBP). As the deformation proceeded,the twin facet grew in area via two simultaneous processes:

1. The first process corresponded to the migration of theUBP boundary, together with the upper junction, wherethe black crystal recedes in favor to the white crystal (theBP boundary moves to the right in Figs. 1 and 2). Thisadvance of the upper junction enabled the twin facet toextend to the right.

2. The second process corresponded to the recession of theLBP in favor of lengthening the twin facet. By the timethe LBP boundary vanished, the twin facet had reachedthe lower free surface of the crystal, at which point thenew triple junction began to migrate to the right. As aresult of these processes, the shear offsets developeddifferently at the upper and lower free surfaces, with asingle atomic plane offset at the upper surface, and allof the rest of the offset, which constituted five atomicplanes, migrating to the bottom surface. The steadystage of twin growth following the elimination ofthe LBP boundary corresponded to the simultaneousmigration of the twin boundary and UBP boundaryaccommodating hci-axis tension required by the uniaxialtension test.

The deformation scenario in this simulation clearly com-prises a {10�12} twin nucleation mechanism at the BP

boundary. The growth was subsequently dictated by thedeformation constraints. In this paper, we analyze thetopological details of the creation and dynamics of theinterfacial defects involved in the nucleation and growthregimes of these facets. To prepare for these analyses, wefirst describe the atomic structure of the unstressed BPboundary and its inherent defect structure, which dictatesthe twin nucleation and growth mechanisms. We thenexamine the interfacial dislocation patterning when a stressis applied. This patterning may be viewed as an incubationregime which is precursor to the twin facet nucleation pro-cess. Finally, in order to understand the growth process ofthe twin boundary, the dynamics related to the disclinationdipole movements and interfacial dislocation–disclination

Fig. 2. Snapshots of the same molecular dynamic simulations as in Fig. 1showing the atomic positions in a {11�20} projection (a) at 3.0% strainwhich were marked first by the propagation of both the twin fact and theupper BP boundary segment (UBP), and second the lengthening of the{10�12} twin facet at the expense of a receding lower disclination, and (b)at 3.5% strain where the steady stages of the twin–BP facet migration werereached after the complete recession of the lower junction and disappear-ance of the lower BP segment (LBP). The lower free surface is offset as aresult of slip on the LBP facet. In both images, the initial position of theBP where the twin facet first nucleated is highlighted by the orange dashedline. (For interpretation of the references to color in this figure legend, thereader is referred to the web version of this article.)

4 C.D. Barrett, H. El Kadiri / Acta Materialia 63 (2014) 1–15

interactions which follow twin facet nucleation aredescribed in detail.

To proceed we begin by recalling some basic aspects ofinterfacial defect theory which will be used throughout thispaper.

4. Interfacial defect operations

The interfaces and deformation of the bicrystals wereanalyzed using interfacial defect theory introduced byPond [24,25], Pond and Vlachavas [1], Pond and Bastawe-esy [26], and Pond and Hirth [2] from the groups of gener-alized symmetry. Analytical expressions for the

dislocations and disclinations in interfaces may be com-puted in terms of Seitz symbols extended by Pond and Hir-th [2]. The nonsymmorphic group operation required tobring the two surfaces into coincidence is an affine transfor-mation denoted, in general, by [Fjs], where F is a lineartransformation matrix describing a point group operation,and s is a translation which may comprise the congruencemoduli of the first system. It is a function of the affinetransformations which describe the operation in the whitelattice (Fk), and the black lattice (Fl). The transformationbelonging to the nonsymmorphic Federov group, [Fjs], iscomputed by the following equation using a bicrystal refer-ence coordinate system:

½Fjs� ¼ F lF k�1

¼ ½Tljtl�½Fljsl�½Tljtl��1½Tkjtk�½Fkjsk��1½Tkjtk��1 ð1Þ

Here [Tljtl] and [Tkjtk] are the operations which convert theblack and white lattices to coordinates of the bicrystal. In thefollowing analysis, Fl will be expressed in coordinates of thebicrystal, allowing [Tljtl] to be the identity, and Tk will be arotation about the x-axis which relates the two crystals. Allallowable defects in an interface may be computed from bro-ken symmetry using Eq. (1) by requiring Fl = Wk and Fk = -Wl where Wl and Wk are symmetry operations in the twocrystals. However, in analysis of multiple facets, this ap-proach will not always be the most convenient, and so we re-tain the notation of all affine transformations.

5. The unstressed BP boundary

The bicrystal with an initial BP boundary has a 90� mis-orientation between the two grains, which differs from thatof the twin plane boundary by only �4�. The excess poten-tial energy of the BP boundary is unusually low (164 mJ/ m2

with EAM) compared to other asymmetric tilt boundaries,and it is relatively close to the perfect symmetric tilt {10�1 2}twin boundary energy (124 mJ/ m2 with EAM) [27].

Serra et al. [28] have previously described the BP bound-ary as incommensurate, i.e. they did not observe any signif-icant displacements of atoms along the boundary fromtheir crystal positions except for the presence of screwdipoles. One may speculate that the incommensuratedescription of the BP boundary by Serra et al. [28] mayhave been due to the interatomic pair potential they used.However, utilizing a different well-established potentialfor Mg, our MD simulations and analyses prompt a differ-ent conclusion, but one which will be of important consid-eration in the twin nucleation process. The BP boundary isactually semicoherent, i.e. it contains coherent terraceswhere the long-range strains are removed by a complexarray of interfacial dislocations with localized cores [29].

In fact, as the relaxed boundary indicates (Fig. 3), the semi-coherent terraces penetrate alternately in the black and whitelattices which gives the boundary a crenelated appearance.After each two penetrations, approximately 15–16 interpla-nar distances along the boundary, there is a misfit dislocation

Fig. 3. The atomic structure of the relaxed basal–prismatic (BP) boundary and schematic illustrations of the intrinsic defect structure developed therein.The continuous array of blue dislocations correspond to infinitesimal dislocations which represent the coherency strains, ginf(y)dy in [29], in the coherentterraces. To remove all coherency stains at long range, an alternating sequence of an edge and a mixed disconnection dipoles (in black), and a misfitdislocation (in red) above each of these dipoles arise. Here the view of the interface is rotated 90� clockwise from Figs. 1 and 2. (For interpretation of thereferences to color in this figure legend, the reader is referred to the web version of this article.)

Table 1The observed disconnections in the BP boundary.

Relaxed boundary disconnections

Mixed disconnections

bð1Þ1=1a ¼ a

12ffiffi3p�c

21ffiffi3p

0B@

1CA, h ¼ c

2 bð1Þ1=1b ¼ a

12ffiffi3p�c

2� 1ffiffi

3p

0B@

1CA, h ¼ c

2

Edge disconnections

bð2Þ1=1a¼ a

0ffiffi3p�c

216c�15

ffiffi3p

2 þ 1ffiffi3p

0B@

1CA, h ¼ c

2 bð2Þ1=1b¼ a

0ffiffi3p�c

216c�15

ffiffi3p

2 � 1ffiffi3p

0B@

1CA, h ¼ c

2

Misfit dislocation

bmisfit ¼ a

120�ffiffi3p

2

0@

1A

Stressed boundary disconnections

b2=2 ¼ a0ffiffiffi

3p� c

0

0@

1A, h = c b0=0 ¼ a

00

c�ffiffiffi3p

0@

1A, h = 0

C.D. Barrett, H. El Kadiri / Acta Materialia 63 (2014) 1–15 5

(in red). This period of the penetrations, however, does notcorrespond to the pseudo-periodic distance (PPD) along thez-axis. In fact, each terrace penetrating the black lattice corre-sponds to a disconnection dipole, but, as we will demonstrate,one dipole is pure edge while the neighboring ones are ofmixed character, so only every other step exhibits a mixedcharacter or an edge character. Thus, the PPD correspondsto two square indentations (four terraces) every 30 dl

f10�10g or32 dk

f0001g interplanar spacings of the planes specified by thesubscript in the crystal color specified by the superscript.

Both mixed and edge disconnections have a step heightof a single interplanar spacing, i.e. dl

f0001g ¼ c=2. Multipledisconnections, however, may combine to form defectswith a step height equal to two interplanar spacings. Insum, the misfit and the two types of disconnection dipoles,in combination, meet the constraint to remove the long-range elastic strains that would be associated with thecoherent terraces. All these disconnections are identifiableusing the interfacial defect theory [1], and the correspond-ing analyses are expounded in the next section.

The complete intrinsic defect structure of the BP boundarydescribed above is schematically illustrated over the atomicstructure of the relaxed configuration obtained by MD simu-lations and reported in Fig. 3. The Burgers vectors for eachinterfacial dislocation are summarized in Table 1. The discon-nection with the misfit superimposed on it (which is the rightdisconnection in Fig. 3, and the upper disconnection in allother figures) is referred to as positive sense, and the otheras negative sense. Most of the Burgers vectors are understoodand illustrated by means of the dichromatic complex reportedin Fig. 4. Here we prefer the dichromatic complex instead ofthe dichromatic pattern, because the former enables bettervisualization of the b

ð1;2Þ1=1a defects based on mirror symmetries

as well as all the translation symmetry defects. The latticedepicted in Fig. 4 is rotated about an atomic point, rather thanthe point of inversion, leaving a line of coincidence betweenthe two lattices normal to the view plane ([2�1�1 0] direction).

The disconnections in the BP boundary have a nomen-clature given as b

ðiÞm=nl. Here i is the identifier for b1/1 to

indicate mixed or edge nature, m is the interplanar stepin the l lattice, n is the interplanar step in the k lattice,and l indicates the stacking sequence position of {00 01}l

planes on which the disconnection acts.

5.1. Disconnection dipole with screw component

The Burgers vector for the disconnection containing ascrew component is identified using a mirror-glide (orglide-reflection which is equivalent to screw-axis for onesided bands) symmetry operation on the f10�10gl planeof the black lattice, and a combination of a {0001}k mirroroperation and a translation operation in the white lattice.

W lW k�1 ¼ ðMl;wlÞðR90; 0ÞðMk; tkÞ�1ðR90; 0Þ�1 ð2Þ

Fig. 4. The dichromatic complex of the basal–prismatic (BP) boundaryshowing the b

ð1Þ1=1a; b

ð2Þ1=1a; b2=2; b0=1 and b0/0 disconnections. The dichro-

matic complex was used here as it is better than the dichromatic pattern invisualizing the b

ð1Þ1=1a disconnections (or b

ð1Þ1=1b, but not both).

6 C.D. Barrett, H. El Kadiri / Acta Materialia 63 (2014) 1–15

This defect can be visualized by joining the incompati-ble, demistep on the (0001) surface of the black crystaland step on a (10�10) surface of the white crystal. Eq. 2yields:

bð1Þ1=1a ¼ a

12ffiffi3p�c

21ffiffi3p

0BB@

1CCA ð3Þ

The different between this defect and the bð1Þ1=0 defect by

Serra et al. [28] is noteworthy. These authors did not use astep on the white lattice so proper combination of the properand improper operations in the black and white lattice led toa defect of the following type, which they labeled as bð3Þz :

bð1Þ1=0 ¼

0c2

0

0B@

1CA ð4Þ

However, since these authors defined rotational operationsusing the inversion point1 as the bicrystal center of rota-

1 Situated at c/4 below the tetrahedral site of three mutually touchingatoms in the hard sphere model.

tion, and not an atomic coincident point, the defect formedfrom the same operations would be expressed in our bicrys-tal as:

bð2Þ1=0 ¼

0c2

� affiffi3p

0B@

1CA ð5Þ

The difference between the two defects of Eqs. (4) and(5) is only due to the relation of the mirror symmetry oper-ations involved in the defect, which depend on the rotationcenter of the two lattices. Our choice of the rotation centerwas stimulated by the fact that our interface is semicoher-ent, whereas that of Serra et al. [28] was identified asincommensurate. We chose an atomic coincident pointbecause it suffered no shift during relaxation on the basisof the local excess potential energy field in our relaxed con-figuration. To alleviate any confusion, the reader may useoperations described by Eq. (2) to derive Eq. (4). Thiscan be done by setting the white lattice translation to zero,and defining the center of rotation as the point of inver-sion, but one should be careful in properly defining the mir-ror glide symmetry operations for this origin. The observeddefect in our bicrystal is derived by combining Eq. (5) witha white lattice translation, b0/1:

bð1Þ1=1 ¼ b0=1 � b

ð2Þ1=0; which yields b

ð1Þ1=1a ¼ a

12ffiffi3p�c

21ffiffi3p

0BB@

1CCA ð6Þ

Here c is the ca ratio, and bi/j represents the Burgers vector

of a disconnection with step heights in each of the whiteand black lattices equal to idl and jdk, respectively. The i

and j are two integers while dl and dk are the interplanarspacings of the black and white lattices, respectively. Inour case, dl ¼ df10�10g and dk = d{0001}.

As we will see, the bð1Þ1=1a is glissile in the BP boundary. Thus,

in case this disconnection glides away, the nucleation of a dis-connection of the b

ð1Þ1=1 type in the slipped bicrystal shall be

revisited. If the latter, however, glides away in turn, there willbe no need to expound the same exercise as all other equiva-lent disconnections of type b

ð1Þ1=1 can be derived from these

two and additional translation symmetry operations.The second b

ð1Þ1=1 type disconnection, designated by b

ð1Þ1=1b,

arises because of the mirroring used in defining the defects.After the passage of a b

ð1Þ1=1a disconnection, the black and

white lattices have both undergone mirror symmetry oper-ations. This results in a valid disconnection since mirroroperations on the same axis are their own inverse, but italters the BP interface, resulting in an interface that is mir-rored across the z-axis. Thus the z axis components of thebð1Þ1=1b disconnections are reversed:

bð1Þ1=1b ¼ a

12ffiffi3p�c

2

� 1ffiffi3p

0BB@

1CCA ð7Þ

C.D. Barrett, H. El Kadiri / Acta Materialia 63 (2014) 1–15 7

These disconnection dipoles nucleate and grow to astable spacing in the relaxed boundary by a conserva-tive glide process. Despite the difference in interplanarspacing between the two systems, they convert basalplanes to prismatic planes and vice versa. In orderto expound upon this process, it is practical todescribe the {0001}k{10�10} boundary in terms ofcoherency strains and misfit dislocations as describedby Hirth et al. [29]. The misfit dislocations and dis-connections display wide cores, indicating that theycan glide easily, but also that the boundary is stillnear incoherency. The effect of the coherency strainsand misfit dislocations is no longer negligible whenplastic deformation begins because the disconnectiondipoles interact with these dislocations.

In order to analyze the formation of these steps duringrelaxation, it is necessary to analyze the glide of b

ð1Þ1=1 discon-

nections. These disconnections possess a wide core and canglide conservatively in the boundary, but require theboundary surrounding the core of the disconnection to becoherent, imposing a coherency strain to maintain this con-dition. Using the disconnection glide/diffusion formulasdeveloped by Hirth and Pond [30], we may calculate thatconservative glide for this disconnection eliminates thecoherency strain on the initial boundary plane, moving itup with the step to the final plane of the interface. Thiscalculation is expounded in Appendix A and shows thatthe material flux, I, required for the glide of this disconnec-tion is zero. Thus, the disconnection glides conservatively.bð1Þ1=1 disconnection dipoles form at points in the boundary

which occur approximately every 30–32 interplanar spac-ing planes, when the atoms are already nearly coherent.The dipole core spreads from these points, imposing acoherency strain on the step. An equal and opposite misfitstrain is created between the steps to maintain dislocationcontent conservation on the boundary and eliminate far-field stresses.

5.2. Disconnection dipole without screw component

The second disconnection dipole type, bð2Þ1=1a, is nearly

identical to the bð1Þ1=1 but does not include a screw com-

ponent. In fact, after nearly 15 interplanar distancesalong ½10�10�l or 16 interplanar distances along[0001]k from the coincident atomic point, the whitebasal planes and black prismatic planes almost, butnot exactly, coincide. Calculation of b

ð2Þ1=1a is achieved

by adding symmetry translation vectors in both latticescorresponding to this distance from the coincident pointto b

ð1Þ1=1:

bð2Þ1=1a¼ ððI;slÞðMl;tlÞÞðR90;0ÞððI;skÞðMk; tkÞÞ�1ðR90;0Þ�1

bð2Þ1=1a¼ b

ð1Þ1=1aþ sl�R90s

k; where sl¼0

0

8c

0B@

1CA; sk¼ a

12

15ffiffi3p

2

0

0B@

1CA ð8Þ

This yields:

bð2Þ1=1a ¼ a

0ffiffi3p�c

2

16c�15ffiffi3p

2þ 1ffiffi

3p

0BB@

1CCA ð9Þ

bð2Þ1=1b is likewise defined in relation to b

ð1Þ1=1b. Because the y-

component of this disconnection, which is the componentnormal to the interface, is identical to the y-componentof b

ð1Þ1=1, the material flux calculation is identical to the cal-

culation given above for bð1Þ1=1. Therefore, the formation of a

stable dipole with this disconnection enforces a local coher-ency as well.

The b1/1 disconnections create coherency which equal-izes the interplanar distances of the prismatic planes of land the basal planes of k, as seen in Fig. 3.

5.3. The coherency and misfit dislocations

Both types of planes form a line of intersection with theBP interface plane which lies on the x-axis which is ½2�1�10�in both lattices. The basal and prismatic planes have ABABand AaBbAaBb stacking sequences, respectively. If the b

ð1Þ1=1

disconnection is defined as bringing the A-planes of theblack lattice into coincidence with the B-planes of the whitelattice (as in Fig. 3), then the b

ð2Þ1=1 disconnection brings A-

planes of the black lattice into coincidence with A-planesof the white lattice. Therefore, the b

ð2Þ1=1 disconnections

nucleate at points where the A prismatic planes of the blacklattice are nearly in coincidence with the A basal planes ofthe white lattice. These points are spaced 30–32 planesapart, as are the b

ð1Þ1=1 disconnection dipoles, but are offset

from them by 15–16 planes. Therefore, the combinationof these two sets of disconnection dipoles creates the rug-ged appearance of the boundary which has steps every15–16 planes. Nevertheless, only every other step exhibitsa screw dipole component.

Between these steps, there is a coherency strain betweenBa- or Ab-corrugated planes of the black lattice with B-planes of the white lattice. The prismatic plane type (a orb) depends upon whether the point being brought intocoincidence has a b

ð1Þ1=1 dipole above it and a b

ð2Þ1=1 dipole

below it, or vice versa.Importantly, in order to eliminate the overall dislocation

content of the BP boundary induced by the coherencystrains, a misfit dislocation arises along with each discon-nection dipole. This misfit is given by:

bmisfit ¼ a

12

0

�ffiffi3p

2

0B@

1CA ð10Þ

This misfit dislocation superimposes on the positivesense b1/1a disconnections (or the negative sense b1/1b dis-connections) because the z-axis components of the discon-nection and the misfit dislocation partially cancel eachother. The z-axis component of the negative sense discon-nection locally eliminates the coherency strain. These

8 C.D. Barrett, H. El Kadiri / Acta Materialia 63 (2014) 1–15

reactions between the disconnections, misfit dislocationand coherency strain critically make disconnection nucle-ation energetically favorable in equilibrium conditions.

6. Deformation and twin incubation

The low energy and high stability of the BP boundary isprobably largely due to the significant portion of atomswhich are brought into semicoherency. The disconnectionshave a wide core and a small Burgers vector, so that theentire dipole array can easily glide along the boundary, caus-ing only a very small elastic distortion along the interface asit moves. However, the presence of the misfit dislocationsimpedes this ability to glide, as the disconnections cannotconservatively pass these dislocations in the boundary.

The earliest stages of the deformation proceeded as fol-lows, and are depicted schematically in Fig. 5.

Image forces and the triple point exert attractive forceson the misfit and the glissile positive sense disconnectionof the uppermost dipole. As soon as stress was applied,they migrated upward to the free surface. The wideningof this dipole promoted stress relaxation due to the portionof the Burgers vector that is normal to the BP boundary,thereby causing an expansion of the lattice in the directionof the applied tensile stress. However, significant wideningonly occurred near the top and bottom surfaces. For thedipole nearest the top surface, the negative sense step

Fig. 5. Schematic illustration of the interfacial dislocation process governing tcycles, each comprising five major consecutive mechanisms: (a) glide of the mis(b) glide of the positive dislocation of type b

ð2Þ1=1a, which accommodates tensi

boundary is tracked by the upper arrows); (c) core extension of the negative-snucleation of a new dipole of type b

ð2Þ1=1b on the stepped boundary; and (e) glide

of the negative sense with the former residual negative sense disconnection (of b

(f) The (b)–(e) process is repeated again until two b2/2 disconnections pile-up

barely glided downward because it was restricted by a mis-fit dislocation, and likewise the positive step nearest thebottom surface could not glide upward to expand thedipole. The dipoles inside the bicrystal were pinned by mis-fit dislocations on both sides, restricting their expansion.

The expansion of this dipole resulted in a section of the BPboundary about 15 planes long immediately below the upperfree surface which was fully coherent, and stepped relative tothe initial boundary. Following the migration of the misfitdislocation and the positive sense disconnection, b

ð2Þ1=1a, out

to the upper free surface, a new dipole of the form bð2Þ1=1b nucle-

ated on the stepped boundary, but without a misfit disloca-tion. This nucleation event promoted additional expansionof the lattice. It also relaxed the dislocation content of thenegative sense disconnection b

ð2Þ1=1a because the two discon-

nections, bð2Þ1=1a and b

ð2Þ1=1b combined to form a new disconnec-

tion with a step height of two planes, b2/2:

b2=2 ¼ bð1Þ1=1a þ b

ð1Þ1=1b ¼ a

0ffiffiffi3p� c

0

0B@

1CA ð11Þ

This defect is shown in the dichromatic complex of Fig. 4.However, it possible to define it in the dichromatic patternas identified by Serra et al. [28] using defect operation be-tween parallel tk = c and tl ¼ a

ffiffiffi3p

. In the semicoherentboundary the b2/2 disconnection is glissile as may be veri-fied in the same manner as for b

ð1Þ1=1a using Appendix A. This

win facet incubation close to the upper free surface which consists of twofit dislocation toward the triple junction under the action of external stress;on of the bicrystal and movement of UBP segment (the position of theense disconnection which is unable to glide by a conservative process; (d)of the positive sense of the new dipole to the free surface, and combinationð2Þ1=1a type). This last stage leads to the formation of a new defect of type b2/2.on the BP boundary, setting the stage for twin facet nucleation.

Fig. 6. Detailed deformation snapshots of MD simulations evidencingnucleation of a {10�12} facet by a relaxation process which converted thedisconnection pile-up and coherency strain into an equivalent disclinationdipole. The difference in the interface is initially very small.

C.D. Barrett, H. El Kadiri / Acta Materialia 63 (2014) 1–15 9

is evident by noting that both the Burgers vector compo-nent normal to the interface, and the step height are dou-bled relative to b

ð1Þ1=1a.

After the nucleation of the bð2Þ1=1b disconnection dipole,

and the combination of the negative sense disconnectionwith a negative sense b

ð2Þ1=1a to form a negative sense b2/2, this

dipole continued to widen, accommodating further expan-sion of the lattice. Thus, the positive sense disconnectionglided out to the free surface which acted as an effective dis-location sink. The negative sense disconnection b2/2 waspinned by the misfit dislocation, restricting it from gliding.

This process was then repeated again by the nucleation ofanother b

ð2Þ1=1a disconnection dipole on the stepped BP bound-

ary immediately below the upper free surface. Again, the posi-tive sense b

ð2Þ1=1a disconnection glided out to the free surface,

and the negative sense disconnection was pinned by boththe misfit dislocation and the b2/2 disconnection. Hence, themisfit dislocation began to cause disconnection pile-up.

A fourth disconnection dipole formed following thisevent. The negative sense b

ð2Þ1=1b disconnection again com-

bined with the bð2Þ1=1a disconnection below it, resulting in

the pile-up of two b2/2 disconnections on the misfit disloca-tion. This pile-up immediately relaxed on its nearly medianplane, which was the {10�12} plane.

7. Twin nucleation

The {10�12} facet nucleation may be described as the dis-sociation of the interfacial step with dislocation characterinto an equivalent disclination dipole with disclination coresat the upper and lower facet junctions. Two effects distin-guish the transition from step character to facet character:

1. Constructing symmetry operation circuits through thestep before facet nucleation reveals two distinct interfa-cial dislocations, whereas, after facet nucleation, con-structing circuits through the step do not reveal anyinterfacial dislocations unless the circuit encompassesboth junctions and the entire facet, therefore includingthe entire disclination dipole.

2. Constructing circuits through the step before facetnucleation does not reveal any disclination content,whereas, after facet nucleation, constructing a circuitencompassing one facet junction reveals one disclina-tion, and encompassing the other junction reveals thecorresponding disclination of the opposite sign.

Because the disclination dipole is equivalent to a disloca-tion, a circuit constructed around the entire step beforefacet nucleation, or around the facet afterward reveal thesame total dislocation content. The interface before andafter facet nucleation is shown in Fig. 6.

Before the facet nucleated, the surfaces of the black andwhite lattices across the facet are described by the latticetranslations:

sl ¼ ½20�22�; sk ¼ ½20�2�2� ð12Þ

These vectors represent the total step in both crystalsacross the piled-up disconnections and including infinitesi-mal coherency strains. The corresponding linear transfor-mations are merely identity matrices, as these basal-to-basal translations can be defined in the dichromatic pat-tern. Therefore, we have the following equations:

W l ¼ ½Ijsl�; ½Fkjsk� ¼ ½Ijsk� ð13ÞApplying this to Eq. 1, we obtain:

½Fjs� ¼ ½Ij � Tksk þ sl� ð14ÞSince the linear transformation is the identity matrix, thedefect must be a dislocation with Burgers vector s. It isnow computed to be:

s ¼ 2a

0ffiffiffi3p� c

c�ffiffiffi3p

0B@

1CA ð15Þ

This dislocation content is, naturally, equal to the Burgersvector of 2b2/2 plus the coherency strain dislocation contentover a projected distance of 2

ffiffiffi3p

a from the BP boundary.After nucleation of the facet, the surfaces of the black

and white crystals had a different relationship, and thusthe interfacial defects must be reanalyzed. The analysismust now be split into three parts: (i) the junction of thefacet and the UBP boundary; (ii) the junction of the facetand the LBP boundary; and (iii) the intermediate facetlength.

The surface at the upper junction transitioned from thebasal plane to the twin plane in the parent (black) lattice,and from the prismatic plane to the twin plane in the twin(white) lattice. This transition can be described by the fol-lowing equations:

Fig. 7. Schematic illustration of the twin propagation process showing (a)nucleation of twinning disconnection dipoles near the lower partialdisclination; (b) transformation of the negative-sense disconnection acrossthe LBP into a b0/0-type dislocation which glided toward the lower freesurface; (c) movement of the positive-sense disconnection to the upperjunction which resulted in an expansion of the twin; and (d) transforma-tion of the positive-sense disconnection across the upper disclination into ab2/2-type disconnection which continued gliding along the UPB toward thefree surface.

10 C.D. Barrett, H. El Kadiri / Acta Materialia 63 (2014) 1–15

F l¼ ½Rlj0�; F k¼ ½Tkj0�½Rkj0�½Tkj0��1 ð16Þ

Rl¼

1 0 0

0ffiffi3pffiffiffiffiffiffiffic2þ3p � cffiffiffiffiffiffiffi

c2þ3p

0 cffiffiffiffiffiffiffic2þ3p

ffiffi3pffiffiffiffiffiffiffic2þ3p

0BBB@

1CCCA; Rk¼

1 0 0

0 cffiffiffiffiffiffiffic2þ3p �

ffiffi3pffiffiffiffiffiffiffic2þ3p

0ffiffi3pffiffiffiffiffiffiffic2þ3p cffiffiffiffiffiffiffi

c2þ3p

0BBB@

1CCCAð17Þ

Since both Fl, and Fk are of the form [Rj0], the interfacialdefect is necessarily a disclination. Using Eq. 1, we have:

F lF k�1¼½RlTkRk�1

Tk�1 j0�¼Rj0�; R¼

1 0 0

0 2ffiffi3p

cc2þ3

� c2�3ð Þc2þ3ð Þ

0c2�3ð Þc2þ3ð Þ

2ffiffi3p

cc2þ3

0BBB@

1CCCAð18Þ

This corresponds to a counterclockwise rotation:

h ¼ p2� 2 arctan

cffiffiffi3p� �

� 4� ð19Þ

The same analysis when applied to the lower junctionshows that the two lattices rotate back to the BP orien-tation and so may be described by Rl�1 and Rk�1. Putt-ing these into Eq. 1 reveals that the disclination at thelower junction is described by R�1, which fulfills therequirement that the long-range character of the disclina-tion dipole has dislocation character with no rotationaldeviation.

The intermediate facet length is described by a {10�12}twin boundary and thus has no defect content. This maybe verified using Eq. 1, leading to:

W lW k�1 ¼ ½Ij0� ð20ÞWe emphasize that the total dislocation content of the facetmay be calculated using the same method as previouslyused to find the dislocation content of the interfacial step,and that the dislocation content increases as the facetlength increases. This facet nucleation process is dominatedby energy reduction, due to the energy of the twin planeboundary being significantly lower than that of a BPboundary with disconnection pile-up. The disconnectionpile-up on the BP boundary is eliminated by the nucleation,but replaced with a disclination dipole. The dipole inducedonly a small distortion in the surrounding lattice, makingthe stepped boundary energetically unfavorable. After for-mation of the twin facet, the LBP endured a stressconcentration.

8. Twin growth

8.1. Twin propagation and the LBP

Immediately following nucleation of the {10�12} facet,the lower BP–{10�12} junction quickly acted as a streamingsource of disconnections gliding upward toward the uppersegment of the BP boundary (UBP) which moved with theTB at high velocity. This process is illustrated schemati-cally in Fig. 7.

The upward glide of each twinning disconnectioninduced the lower BP–{10�12} junction to shift downward.The nucleation of each of these disconnections, whichmoved the junction downward, was accompanied by dislo-cation nucleation in the LBP boundary. These dislocationsglided in the LBP boundary to the lower free surface. Theshift of the lower junction proceeded until the LBP com-pletely vanished, leaving a free surface offset resulting fromthe glide of these dislocations. The {10�1 2}–UBP facet con-tinued to quickly advance toward the right by a movementof disconnection generation and glide on both boundariesuntil the entire invasion of the bicrystal by the twin latticewas complete (Fig. 2).

The disconnections emitted by the lower junction had astep of two interplanar spacings. The Burgers circuit (BC)technique illustrated in Fig. 8 yielded the Burgers vector bt

to be:

bt ¼3� c2

3þ c2h�1011i ð21Þ

This expression corresponds to the Burgers vector of the{10�12} twinning dislocation [31].

The dichromatic complex here given has a misorienta-tion of �86� corresponding to the {10�12} twin rather thanthe 90� misorientation of the initial bicrystal. The represen-tation of this disconnection in a reference frame local to the{10�12} facet is advantageous for several reasons. The firstreason resides in the ability to straightforwardly illustratethe identity of the disconnection as the twinning disloca-tion. The second reason relates to the redundancy of choos-ing which facet is the reference interface and which one isthe derived interface, while local defects, such as disclina-tions, remain unaffected by this choice. Therefore, a localmisorientation is used to define interfacial defects on eachfacet. This approach has the disadvantage that the long-range strains are not clear, and cannot be found without

Fig. 8. Identification of the Burgers vector of the twinning disconnectionby FS/RH atom-to-atom steps Burgers circuits drawn similarly (a) aroundthe disconnection, and (b) around the twin boundary in the dichromaticcomplex where the closure failure FS demonstrates that the Burgers vectorof the disconnection corresponds to the b2 = bt {10�12} twinningdislocation.

C.D. Barrett, H. El Kadiri / Acta Materialia 63 (2014) 1–15 11

first defining the global misorientation. If the 90� misorien-tation reference is used, the defect found will be the sum ofthe twinning dislocation bt and the dislocation contentequivalent to the disclination dipole running along the twinplane through the circuit.

Finally, the twin misorientation is used because it is themost appropriate reference structure for this type of situa-tion. In the language of Hirth et al. [29], an array of edgedislocations with Burgers vectors normal to a tilt boundarycreates a distortion field, rotating both crystals and

changing the misorientation of the interface. When suchan array is present, additional defects in the interfaceshould be identified in a rotated coherent dichromatic pat-tern (RCDP) rather than the original dichromatic pattern,or a coherent dichromatic pattern (CDP).

When an array of disconnections rather than dislocationsis present, both the misorientation of the bicrystal and theinterface angle are changed by the array [29]. In this casealso, the RCDP is the relevant DP which is needed to defineadditional defects which are superimposed on the new inter-face. For our situation, the twin facet was formed conserva-tively from an array of disconnections without adding anydefects. Moreover, at long ranges, the disclination dipoleis still elastically equivalent to a dislocation array [32]. Thusthe RCDP is the appropriate DP for this situation. In Fig. 8,the RCDP, which has the misorientation of the {10�12}twin, is depicted with the twin plane interface.

At the lower junction, the emission of each disconnec-tion into the {10�12} facet results in the junction migratingdownward. The movement of the disclination pole requiresa change in dislocation content equal to the Frank vector.This situation may be visualized by representing the dipoleas a dislocation array. In order to move the endpoint of thearray by a certain given vector, r, additional dislocationcontent normal to that vector with a half-plane lying alongthe vector must be added:

f ¼ r� Rr ð22ÞHere R is the disclination rotation, and f is the Frank vec-tor which is equal to the dislocation content added to thedisclination by its motion along r. This equation was suc-cinctly derived by Hirth et al. [32] in their Appendix, butwe expound it in our Appendix B to the general case ofthe dispiration with conserved disclination character. Atthe lower junction, if the disclination pole moves down-ward along the z-axis by a value equal to:

r1 ¼ a

0

0

c

0B@

1CA ð23Þ

The vector has a magnitude of c rather thanffiffiffi3p

a because c

is the magnitude of the final configuration as the disconnec-tion motion converts l material to k. Recalling that thelower disclination is described by RT, the resulting Frankvector in the coordinates of the bicrystal is:

f1 ¼ a

0

c c2�3c2þ3

�cc�ffiffi3pð Þ2

3þc2

0BB@

1CCA ð24Þ

When the disconnection, bt, is added to this value, theresulting dislocation content is:

�b0=0 ¼ f1 þ bt ¼ a

0

0ffiffiffi3p� c

0B@

1CA ð25Þ

12 C.D. Barrett, H. El Kadiri / Acta Materialia 63 (2014) 1–15

This dislocation is the coherency strain in the LBP bound-ary, which is eliminated by the movement of the junction.It is also a permissible defect based on broken translationsymmetry [28]. The glide of this dislocation induces glideof the misfit dislocation array in the LBP downward to-ward the lower free surface, where these glides are eventu-ally eliminated. The entire process of disconnectionnucleation at the LBP junction therefore satisfies the con-servation criterion of the dislocation content.

Fig. 9. Disconnections on the twin-plane and BP interfaces. Interfaceplanes are colored red through orange, green and light-blue, to dark blue.Colors illustrate the interplanar distance of atoms on the BP or twin planefrom a reference plane. Four dislocation lines separate these five planes onthe twin-plane and BP interfaces. As may be seen, two b1/1 dislocationlines sometimes merge to form a stable core of b2/2, i.e. the planetransitions directly from dark blue to green or green to red, skipping thelight blue and yellow, respectively. (For interpretation of the references tocolor in this figure legend, the reader is referred to the web version of thisarticle.)

8.2. Twin propagation and the UBP

At the upper facet junction, the disconnection on the{10�12} facet moves upward from the bottom, causingthe disconnection to migrate over toward the black crystalalong the right the y-axis. This process creates a disconnec-tion in the UBP given by b2/2. Using the previously exer-cised method, the change in dislocation content from thedisclination motion may be calculated as for the LBPjunction:

r2 ¼ a

0ffiffiffi3p

0

0B@

1CA; f2 ¼ a

0

�cc�ffiffi3pð Þ2

c2þ3

�ffiffiffi3p

c2�3c2þ3

0BB@

1CCA ð26Þ

Adding this to the emerging twinning disconnection, �bt,gives the resulting disconnection on the UBP interface:

�b2=2 ¼ f2 � bt ¼ a

0

c�ffiffiffi3p

0

0B@

1CA ð27Þ

We emphasize that the two types of interaction between thedisconnection on the {10�12} facet and each of the twojunctions are different. At the lower junction the interactionmoves the junction down, while at the upper junction, itmoves the junction normal to the UBP boundary towardthe black crystal. Geometrically, moving downward or tothe right into the black crystal is equally feasible. However,the twin plane boundary has a lower energy than the BPboundary and also migrates easier. Therefore, the interac-tion between disconnections and the junctions never de-creases the length of the {10�12} facet, but may increase it.

The propagation of the UBP facet shows two main note-worthy phenomena:

1. bt disconnection loops nucleated at the bottom of the{10�12} facet, propagated along the perfect twin inter-face, and finally cross-slipped across the {10�12}-UBPjunction to the vertical UBP boundary to undergo dis-connection transformation.

2. The UBP boundary underwent nucleation of disconnec-tion loops at the center, and these propagated to bothends of the UBP. The negative-sense disconnectionswhich propagated toward the {10�12}-UBP junctionwere annihilated by bt disconnections, leaving positive-sense b1/1 and b2/2 disconnections.

Here we introduce the term disconnection transforma-tion to refer to a nonconservative slip process wherebythe disconnection glides from one lattice plane to another.Disconnection transformation also differs from dislocationtransmutation [3,33,34] in that the defect is moving fromone interface plane into another, rather than transformingby crossing an interface. In this process, unlike cross-slip,the two glide planes’ line of intersection is not necessarilyparallel to the Burgers vector, but the disconnection maystill move conservatively if the material flux is null (Appen-dix A). The governing equations for disconnection trans-formation are derived in Appendix C, although theyconstitute an important development in this paper.

In their reply to Serra et al. [8] and Li and Ma [35]claimed that dislocation loops were not observed for{10�12} twinning. However, under similar simulation con-ditions, Fig. 9 clearly shows four bt disconnection loops,each of which advanced the interface by two layers, as wellas the faceted BP boundaries with four b1/1 disconnection

C.D. Barrett, H. El Kadiri / Acta Materialia 63 (2014) 1–15 13

lines, which combine along part of their loops to form twob2/2 disconnection lines.

Faceted boundary segments are limited by the increasingelastic distortion proportional to their length, due to themagnitude of the disclination dipole. However, many smal-ler segments can join to form a long, stable, faceted bound-ary in bulk specimens. In our simulations, the facetedboundary interaction with the free surfaces was significant,as the free surfaces acted as sinks for the misfit dislocationsin the initial boundary. However, real materials often havesinks, and the mechanisms we observed may be activatedeven without sinks under sufficient local stresses. Longerfaceted boundaries with BP and PB character may be seenin bulk materials, such as some of the TEM figures ofZhang et al. [6], who referred to them as twin boundariesdeviating from the twin plane. They also occur in twinnucleation studies using MD, as seen in Wang et al. [36].The occurrence of these particular boundaries is not ran-dom, but energetically and dynamically favorable for dislo-cation-controlled twin growth. Growth of the twins alongthese faceted boundaries may proceed in many circum-stances when growth along the twin plane is not favorable.Therefore, the migration and growth of these facets isvitally important for a complete description of {10�12}twinning in hcp metals.

9. Conclusions

This paper proposes a mechanism of twin nucleationand propagation in Mg from the {00 01}k{10�10} (BP)asymmetric tilt boundary forming a bicrystal stretched nor-mal to this boundary. In order to decipher the intricateinterfacial dislocation and disclination activities involvedin this mechanism, a detailed analysis of the atomic struc-ture of the BP boundary in the relaxed condition was nec-essary. The following conclusions can be retained from thiswork:

1. In the relaxed stress-state, the BP boundary develops asemicoherent atomic structure containing terracesbounded by disconnection dipoles with a characteristicspacing. The disconnection dipoles, however, cannotalone remove the effect of coherency strains at longrange, and thus a misfit dislocation nucleates above eachdipole to relieve the coherency strains. The disconnec-tion dipoles alternate in screw character: one is pureedge and the next has mixed character. The BP bound-ary was found to be one of the lowest–energy bound-aries in Mg, having an excess potential energy ofapproximately 164 mJ m�2 according to the EAM, vs.75 mJ m�2 for the extremely low-energy {10�11} twinboundary [27].

2. When tension was applied normal to the BP boundary,the misfit dislocation closest to the free-surface sinkglided away, and thus unlocked several activities of var-ious interfacial dislocations with profound consequencesfor the plasticity and texture of the bicrystal. In

particular, the disconnection dipoles widened by conser-vative glide, but remained pinned by the misfit disloca-tions. When one of these disconnection dipoles wasable to grow to a sufficient width, a new disconnectiondipole nucleated on the stepped boundary. A repetitionof this process led to a pile-up of a new type of discon-nections, which built up on the misfit dislocations. In thevicinity of sinks, the pile-up may drive the misfit disloca-tions to glide toward sinks such as free surfaces, causing,meanwhile, migration of the BP boundary upward. Thisprocess has been regarded in this paper as twin-facetincubation.

3. Once the disconnection pile-up has grown to a stepheight of four planes, it relaxed into a {10�1 2} twinfacet, thereby triggering a bounding disclination dipole.Having a lower energy than the BP boundary, the twinfacet favorably replaced the disconnection pile-up with asmall disclination dipole, which kept the same disloca-tion content at a lower energy level. The disclinationdipole compensated for the difference in the �86� mis-orientation of the {10�12} twin and the 90� misorienta-tion of the BP bicrystal.

4. Further deformation induced one of the two disclina-tions to emit disconnections gliding toward the otherdisclination. These twinning disconnections caused thetwin boundary to advance forward, forcing the sourcedisclination to recede in favor of an increase in the twinboundary length.

5. As they crossed the disclination, twinning disconnec-tions on the {10�12} plane transformed by a conserva-tive process into a b2/2 disconnection, which happenedto be glissile on the BP boundary. This disconnectionpromoted migration of the remaining BP segment withthe twin boundary. The process continued until theentire bicrystal eventually transformed into a twinnedsingle crystal.

The interaction between the {0001}k{10�10} asymmet-ric tilt boundary and the {10�1 2} twinning should be con-sidered as an important aspect of plasticity in polycrystalswhen hci-axis tension of one of the crystals is required bythe macroscopic stress or locally by grain–grain interac-tions. The conservative transformation of twinning discon-nections from the twin facet to the BP facet through thedisclination greatly enhances the importance of the BPmobility, and thus, accounts for some recent transmissionelectron microscopy observations of faceting in hcp metals.Most importantly, nucleation of the {10�12} twin by meansof pile-up of interfacial disconnection pinned along asym-metric tilt grain boundaries is of substantial significanceto the fundamental science related to the nucleation oftwinning in hcp polycrystals.

Acknowledgments

The authors thank Professor R.C. Pond for his guidancewhich was critical to the preparation of this manuscript. The

14 C.D. Barrett, H. El Kadiri / Acta Materialia 63 (2014) 1–15

authors also thank Carlos Tome for providing resources andinsight which led to the development of this article.

Appendix A

The material flux equation for glide of a disconnection isgiven by [30]:

I ¼ Lv hDX þ bnX k� �

ð28Þ

Here, Xk is the number of atoms per unit volume of theright lattice unit cell, and DX is the change in atoms perunit volume accomplished by the disconnection glide alongone unit cell, h is the step height of the Burgers vector andbn is the component of the Burgers vector normal to theinterface. For the case of b1/1 glide, because l is locallycoherent, the prismatic interplanar distance is compressedto a distance of c. Xk and DX are given by:

X k ¼ N

acffiffi3p

2a; DX ¼ N

acffiffi3p

2a� N

ac c2

¼ 2Nac

1ffiffiffi3p

a� 1

c

� �ð29Þ

Putting these values into Eq. 28 gives:

I ¼ LvNa

1ffiffiffi3p

a� 1

c

� �þ N

ffiffiffi3p

a� c

acffiffiffi3p

a

!¼ 0 ð30Þ

The disconnection glides conservatively because the mate-rial flux required to move it is null.

Appendix B

The flux of dislocation content in a boundary containinga dispiration dipole as either core moves can be calculated.Consider the initial and final step character in each of thewhite and black lattices:

F l0 ¼ ½Rljsl

0 �; F l1 ¼ ½Rljsl

1 �; F k0 ¼ ½R

kjsk0�; F k

1 ¼ ½Rkjsk

1�:ð31Þ

Here, we obviously had to assume that the disclinationcharacter of the dispirations did not change along theboundary. Now, we may define the change in step charac-ter as the difference between the two dispirations:

F l1F k�1

1 � F l0F k�1

0 ¼ ½RlTkRk�1Tk�1j � RlTkRk�1sk1 þ sl

1 �� ½RlTkRk�1Tk�1j � RlTkRk�1sk

0 þ sl0 � ð32Þ

That is, if R = RlTkRk�1Tk�1, then:

F l1F k�1

1 �F l0F k�1

0 ¼ ½R�Rj�RlTkRk�1sk1þ sl

1þRlTkRk�1sk0�sl

0 � ð33ÞF l

1F k�11 �F l

0F k�10 ¼ ½0jRTkðsk

0�sk1Þþsl

1�sl0 � ð34Þ

Now we define:

r ¼ sl1 � sl

0 ¼ Tkðsk1 � sk

0Þ ð35ÞNote that this definition asserts that the junction moves bythe same distance in both lattices.

F l1F k�1

1 � F l0F k�1

0 ¼ ½0j � Rrþ r� ð36Þ

Thus we have that the change in interfacial defect contentfrom moving a disclination core by a vector r is equal to theFrank vector:

f ¼ r� Rr ð37Þ

Appendix C

The requirements for a disconnection to transformacross a facet junction conservatively may be derivedas follows. Define the disconnection in the originalfacet:

½Wljwl�½Tkj0�½Wkjwk��1½Tk�1j0� ð38ÞWe will define the new facet by rotating the white lattice, sothat the black lattice remains in the bicrystal coordinates.

½Wljwl�½Dkj0�½Tkj0�½Wkjwk��1½Tk�1j0�½Dk�1j0� ð39ÞSimplifying this, we obtain:

½WlDkTkWkTk�1Dk�1jwl �WlDkTkWk�1wk� ð40ÞSince we are relating disconnections on each facet, the finaldefect must be translational; this translates to the followingrequirement:

WlDkTkWkTk�1Dk�1 ¼ I ð41ÞIt is evident at this stage that the disconnections of typeb1/1 cannot undergo disconnection transformation be-cause Eq. (41) is not satisfied. However, b2/2 may be de-rived from translation symmetry operations in eachlattice, and all such defects have a corresponding defectin the second facet, as Eq. (41) is automatically satisfiedwhen Wl = I and Wk = I. Let us focus here on DSC de-fects (defects formed from displacements which are sym-metry-conserving):

½Ijwl �WlDkTkWk�1wk� ¼ ½Ijwl �DkTkwk� ð42ÞNow we derive the change in the Burgers vector of the dis-connection by subtracting Eq. 38 from Eq. 42.

D ¼ wl �DkTkwk � wl þ Tkwk ¼ �DkTkwk þ Tkwk ð43ÞNow, we recall that the action of the disconnection movesthe disclination core by a vector r. We postulate the valueof this vector as:

r ¼ Tkwk; f ¼ r�Dkr ð44ÞThis expression for the change in dislocation content frommoving the disclination core is identical to Eq. 43, andtherefore the process is conservative.

It should be noted that the process of disconnectiontransformation is a function of wl and Tkwk, unlikethe individual disconnections which are functions of wl -� Tkwk. For this reason, there is not always a one-to-one correspondence between translation symmetry-baseddisconnections on different facets. For example, thetwinning disconnection bt may be derived in multipleways:

C.D. Barrett, H. El Kadiri / Acta Materialia 63 (2014) 1–15 15

bt ¼0

c

0

0B@

1CA� T86

0

0

�ffiffiffi3p

a

0B@

1CA ¼

0

0ffiffiffi3p

a

0B@

1CA� T86

0

c

0

0B@

1CAð45Þ

These translation vector sets both produce the sametwinning disconnection, but when the disconnection istransformed onto the BP facet, they produce differentdefects: b2/2 and b0/0. Either of these defects may transforminto the twinning disconnection, and the twinning discon-nection may transform into either of these disconnections.

References

[1] Pond R, Vlachavas D. Proc R Soc Lond A: Math Phys Sci1983;386(1790):95–143.

[2] Pond R, Hirth J. Solid State Phys 1994;47:287–365.[3] El Kadiri H, Oppedal A. J Mech Phys Solids 2010;58(4):613–24.[4] Oppedal AL, El Kadiri H, Tome CN, Kaschner GC, Vogel SC, Baird

JC, et al. Int J Plasticity 2012;30–31:41–61.[5] Christian J, Mahajan S. Prog Mater Sci 1995;39(1–2):1–157.[6] Zhang X, Li B, Wu X, Zhu Y, Ma Q, Liu Q, et al. Scripta Mater

2012.[7] Li B, Ma E. Phys Rev Lett 2009;103(3):35503.[8] Serra A, Bacon D, Pond R. Phys Rev Lett 2010;104(2):29603.[9] Li Y, Chen Y, Walmsley J, Mathinsen R, Dumoulin S, Roven H.

Scripta Mater 2010;62(7):443–6. http://dx.doi.org/10.1016/j.scripta-mat.2009.11.039. <http://www.sciencedirect.com/science/article/pii/S1359646209007519>.

[10] Vohringer O. Z Metall 1976;67(8):518–24.[11] Armstrong RW. Can Metall Quart 1974;13(1):187–202.[12] Meyers MA, Vohringer O, Lubarda VA. Acta Mater

2001;49(19):4025–39.

[13] Xu B, Capolungo L, Rodney D. Scripta Mater 2013;68(11):901–4.[14] Wang J, Liu L, Tome C, Mao S, Gong S. Mater Res Lett

2013;1(2):81–8.[15] Wang J, Yadav S, Hirth J, Tome C, Beyerlein I. Mater Res Lett

2013:1–7.[16] Tu J, Zhang X, Wang J, Sun Q, Liu Q, Tome CN. Appl Phys Lett

2013;103(5). 051903-051903.[17] Plimpton S. J Comput Phys 1995;117(1):1–19.[18] Liu X, Adams J, Ercolessi F, Moriarty J. Modell Simul Mater Sci Eng

1996;4:293.[19] Thompson A, Plimpton S, Mattson W. J Chem Phys

2009;131:154107.1–6.[20] Stukowski A. Modell Simul Mater Sci Eng 2010;18. <http://ovito.org>.[21] Barrett C, El Kadiri H, Tschopp M. J Mech Phys Solids

2012;60(12):2084–99.[22] Capolungo L, Beyerlein I. Phys Rev B 2008;78(2):024117.[23] Wang J, Beyerlein I, Tome C. Scripta Mater 2010;63(7):741–6.[24] Pond R. Philos Mag A 1983;47(6):49–55.[25] Pond R. Dislocat Prop Real Mater 1984:71–93.[26] Pond R, Bastaweesy A. Le J Phys Colloques 1985;46(C4). 4-4.[27] Wang J, Beyerlein I. Modell Simul Mater Sci Eng 2012;20(2). 024002.[28] Serra A, Bacon D, Pond R. Acta Mater 1999;47(5):1425–39.[29] Hirth J, Pond R, Hoagland R, Liu XY, Wang J. Prog Mater Sci

2013;58(5):749–823.[30] Hirth J, Pond R. Acta Mater 1996;44(12):4749–63.[31] Serra A, Bacon D, Pond R. Acta Metall 1988;36(12):3183–203.[32] Hirth J, Pond R, Lothe J. Acta Mater 2006;54(16):4237–45.[33] El Kadiri H, Kapil J, Oppedal A, Hector Jr L, Agnew SR, Cherkaoui

M, et al. Acta Mater 2013;61(10):3549–63.[34] El Kadiri H, Baird J, Kapil J, Oppedal A, Hector Jr L, Cherkaoui M,

et al. Int J Plasticity 2013;44:111–20.[35] Li B, Ma E. Phys Rev Lett 2010;104:029604. http://dx.doi.org/

10.1103/PhysRevLett.104.029604. <http://link.aps.org/doi/10.1103/PhysRevLett.104.029604>.

[36] Wang J, Hirth J, Tome C. Acta Mater 2009;57(18):5521–30.