Available online at www.sciencedirect.com

www.elsevier.com/locate/actamat

ScienceDirect

Acta Materialia 77 (2014) 258–268

Atomic simulation of bcc niobium R5h001i 310f g grain boundaryunder shear deformation

Bo-Wen Huang a, Jia-Xiang Shang a,⇑, Zeng-Hui Liu a, Yue Chen b

a School of Materials Science and Engineering, Beihang University, Beijing 100191, People’s Republic of Chinab Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY 10027, USA

Received 8 March 2014; received in revised form 19 May 2014; accepted 19 May 2014Available online 1 July 2014

Abstract

The shear behaviors of grain boundaries are investigated using molecular dynamics simulations. The R5h001i 310f g symmetric tiltgrain boundary (GB) of body-centered cubic (bcc) Nb is investigated and the simulations are conducted under a series of shear directionsat a wide range of temperatures. The results show that the GB shearing along ½1�30�, which is perpendicular to the tilt axis, has a coupledmotion behavior. The coupling factor is predicted using Cahn’s model. The critical stress of the coupling motion is found to decreaseexponentially with increasing temperature. The GB under shear deformation along the ½00�1� direction, which is parallel to the tilt axis,has a pure sliding behavior at most of the temperatures investigated. The critical stress of sliding is found to be much larger than that ofthe coupled motion at the same temperature. At very low temperatures, pure sliding is not observed, and dislocation nucleating andextending is found on GBs. We observed mixed behaviors when the shear direction is between ½1�30� and ½00�1�. The transition regionbetween GB coupled motion and pure sliding is determined. If the shear angles between the shear direction and the tilt axis are largerthan a certain value, the GB has a coupled motion behavior similar to the ½1�30� direction. A GB with a shear angle smaller than thecritical angle exhibits mixed mechanisms at low temperatures, such as dislocation, atomic shuffle and GB distortion, whereas for the½00�1�-like GB pure sliding is the dominating mechanism at high temperatures. The stresses to activate the coupling and gliding motionsare analyzed for shear deformations along different directions at various temperatures.� 2014 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Grain boundary; Shear deformation; Molecular dynamics; Bcc

1. Introduction

Grain boundaries (GBs) play a significant role indetermining the mechanical properties of polycrystallinematerials[1,2], e.g. nanocrack healing by migrating GBs[3]. Numerous investigations have been conducted in orderto understand GB structures, energies and thermodynamics.Molecular dynamics (MD) simulation is a powerfulmethod, providing a comprehensive understanding of themicroscopic mechanisms of GB movements [4–8], plastic

http://dx.doi.org/10.1016/j.actamat.2014.05.047

1359-6454/� 2014 Acta Materialia Inc. Published by Elsevier Ltd. All rights r

⇑ Corresponding author. Tel./fax: +86 10 8231 6500.E-mail address: shangjx@buaa.edu.cn (J.-X. Shang).

deformations [9,10] and other atomic behaviors [11–13].One of a GB’s most important features is its mobility,which depends on the GB crystallography and externalconditions. It has been found that the normal GB motionis often coupled to the tangential translation of grains(referred to as coupled GB motion).

Stress-induced GB motions have been studied in greatdetail, both in simulations and experimentally. GB motionscan be classified into different groups based on analysis ofthe atomic structure evolution [14–16], which includes GBcoupled motion, GB sliding, grain rotation and dislocationemission. GB coupled motion was first observed in experi-ments by Li et al. [17] in small-angle Zn GBs. Molteni and

eserved.

B.-W. Huang et al. / Acta Materialia 77 (2014) 258–268 259

Shiga et al. investigated the tilt and twist GB motions inGe, Al and Ni using density functional theory and foundthat the sliding and migration motions exhibit a stick–slipbehavior [18,19].

The Frank–Bilby equation defines the dislocation of aninterface between two crystals. The solutions of this equa-tion are meaningful only for low-angle tilt GBs [20], thoughrecently the equation has been extended to high-angle tiltGBs [21,22]. Cahn et al. [23] proposed a geometric modelto predict the GB coupled motion. The model considersthe upper grain as a parallelogram. In particular, one faceof the parallelogram is along the GB plane, while anotherone is normal to the tilt axis, and the third one is parallelto the slip plane. To calculate the coupling factor b, whichis defined as the ratio of the normal to the lateral motion,the parallelogram is first sheared along the direction paral-lel to the slip plane by a magnitude of B (the sum of theBurgers vectors). Then the parallelogram is rotated clock-wise by h (tilt angle) to align the upper grain with the lowergrain. The model gives two opposite signs for the couplingfactor, which refer to the different slip mechanisms. For theslip plane of GB dislocation along the f100g plane:

b ¼ 2tanðh=2Þ ð1ÞFor the slip plane of GB dislocation along the f110g plane:

b ¼ �2tanðp=4� h=2Þ ð2ÞIt can be seen that the coupling factor only depends on themisorientations. Caillard et al. [7] have further proposed amodel of shear-migration coupling for general GBs.

A series of MD simulations of the h100i tilt GBs in Cuhave been performed by Cahn et al. [5,6,21], confirming theabove models. The coupling factors predicted by the geo-metric theory show excellent agreements with MD simula-tions and experimental data [24–27]. In addition, threetypical tilt GBs (h100i; h110i and h111i) in Ni have beeninvestigated [28,29] using a synthetic driving force method.It was found that the GBs exhibit excellent agreements withthe theory for most of the h100i boundaries, whereas theh110i and h111i boundaries did not obey the previousmodels.

Wan and Wang [30] studied the R9h001i 221f g GB inCu, Al and Ni with different shear directions at roomtemperature. They observed various GB motions includ-ing coupling and sliding, while the motion types thatare activated depended on the shear directions and mate-rials. In another report, occasional sliding has beenobserved during coupled motions as the strain rateincreases [6]. In other words, coupling and sliding takeplace once the stress has accumulated to a critical level.If the critical stress for coupling is larger than that neededto activate sliding, sliding motion will be dominant;otherwise, coupled motion will take place. In certain con-ditions, e.g. low temperatures or along some particularshear directions, if the critical stresses for coupling orsliding exceed the yield stress, dislocations accompaniedby atomic shuffles will take place.

Most of the investigations of GB motions focus onmaterials with face-centered cubic (fcc) structure such asNi, Al and Cu. The GB motions of body-centered cubic(bcc) metals have rarely been studied except for the bcc-Fe R5h001i 310f g symmetric tilt GB, which has been inves-tigated using MD under shear deformation from 1 to600 K. The nucleation and gliding of partial GB disloca-tions were found in GB migration [31]. Do the GBs inbcc materials exhibit a behavior similar to that seen infcc materials? Does the geometric theory hold valid in met-als with bcc structure? What is the relation between GBmotions, temperatures and shear directions?

To answer these questions, we choose the R5h001i 310f gsymmetric tilt GB of niobium (Nb) with bcc structure as amodel in this paper. Nb is a refractory metal, and one ofthe most important elements in superalloys with promisingapplications. The GB is constructed using the coincidencesite lattice (CSL) model. By applying shear loads parallelto the GB plane at a wide range of temperatures and differ-ent directions, we observe that there are various kinds ofGB motion which depend on the shear conditions. We haveshown that the shear deformation along the ½1�30� axis isconsistent with the prediction of geometric theory. Sheardeformation along the ½00�1� axis displays a GB pure slidingbehavior. Other simulations whose directions are betweenthe ½1�30� and ½00�1� axes show a mixed behavior. A transi-tion region between the coupling and sliding motions,which depends on the temperatures and directions, isdescribed. The paper is organized as follows. In Section 2,the general atomistic simulation method is described. InSection 3, the results and discussions are presented. Finally,a summary of the present work and conclusions are givenin Section 4.

2. Simulation methodology

The shear deformation of the Nb R5h001i 310f g GB isinvestigated in this paper. Simulation models are con-structed using the CSL method. We create the GB modelby concatenating two separate grains with specific crystallo-graphic orientations. The orientation of the lower grain isshown in Fig. 1. The upper grain is built by rotating thelower grain around the [310] axis for 180�. The size ofthe simulation model is approximately 10 nm � 10 nm �20 nm, which contains about 120,000 atoms. The periodicboundary condition (PBC) is applied along three dimen-sions to mimic the bulk material conditions. In order toavoid the interference of the second GB which is causedby the PBC, we fix the atoms on the top and bottom layersof the model. The fixed atoms are located in their perfectlattice positions. We define the thickness of each fixedregion as twice of the cut-off distance.

Two grains undergo a rigid-body translation within theGB plane to find the position with the lowest energy. A con-jugate gradient algorithm is applied for energy minimizationin this work. Once the optimized structure is obtained, wethen run MD simulations for sufficiently long times to ensure

Fig. 1. Schematic illustration of the GB simulation model that used in this work. The crystallographic directions are defined with reference to the lowergrain. The ½00�1� axis is the tilt axis.

260 B.-W. Huang et al. / Acta Materialia 77 (2014) 258–268

the stability of the model. The initial equilibrium GB struc-ture is shown in Fig. 2, which is viewed along the tilt axis.According to their CSL notation and the GB normaldirection, this GB is termed R5h001i 310f g; its structurewas observed experimentally in Ref. [32]. In Fig. 2, thestructural unit, which contains six atoms located on theGB plane, is outlined and the orientation vectors for boththe GB period and normal directions are given. The blackand white atoms define two adjacent atomic planes alongthe tilt axis. It should be noted that each kite-like structureunit is composed of atoms belonging to two neighboringatomic planes.

MD simulations are then performed to deform the GBmodel at a constant shear strain rate of 1�108 s�1. The

Fig. 2. The R5h001i 310f g symmetric tilt GB structure in Nb at 0 K. Filled andstructural units are outlined in solid blue lines. (For interpretation of the referenthis article.)

shear direction is parallel to the GB plane. Table 1 liststhe shear directions and the included angles between the½00�1� tilt axis and the shear directions. The shear processdeforms the simulation box as a whole. Stresses on alldirections except the shear direction are allowed to relaxduring the simulations. A canonical ensemble (NPT) withthe Nose–Hoover thermostat is applied [33]. Stress is calcu-lated using the standard viral expression. We adopt the Nbembedded atom method potential [34] which has been usedin other studies [35,36]. The accuracy of this potential isconfirmed by testing the elastic constants, melting point,lattice constant and thermal expansion. Common neighboranalysis (CNA) is used to display the atomic structure. MDsimulations are realized using the LAMMPS code [37].

empty circles represent the Nb atoms in two adjacent atomic planes. Theces to color in this figure legend, the reader is referred to the web version of

Table 1The shear directions, critical stresses and GB motion types at 100 K.

Direction ½00�1� ½1�3 �30� ½1�3 �20� ½1�3 �15� ½1�3 �10� ½1�3�6� ½�39�5� ½1�3�1� ½1�30�Angle h 0� 6.02� 8.99� 11.90� 17.55� 27.79� 62.21� 72.45� 90�Critical stress (GPa) 3.50 5.00 4.80 4.10 2.80 2.20 1.18 1.10 1.22Motion type Slide Mix Mix Mix Mix Couple Couple Couple Couple

B.-W. Huang et al. / Acta Materialia 77 (2014) 258–268 261

3. Result and discussion

3.1. Shear along the ½1�30� direction

Shear deformation is applied along the ½1�30� directionwith a constant strain rate. The ½1�30� direction is locatedon the GB plane and is perpendicular to the ½00�1� tilt axis.We perform MD simulations at different temperatures toanalyze the mechanisms of GB motion. Fig. 3 shows thestress–time relations at temperatures between 1 and2400 K (the bulk melting point 2750 K). It is obvious thatthe curves display a strong temperature dependence. Theyield stress decreases exponentially with increasing temper-ature as shown in the inset of Fig. 3.

At temperatures between 1 and 2100 K, the stress–timecurves display a sawtooth behavior. The stress drops whenthe critical value is reached. The magnitude of the droppingand the period of each curve are temperature independentbetween 300 and 2100 K. Ivanov [6] reported that the grainsize in the normal direction affects the stress behavior dur-ing shear deformation. At temperatures below 100 K, thestress–time curves display larger stress drops and longerperiods. The atoms at temperatures below 100 K are diffi-cult to move; thus the GBs can accumulate and releasemore elastic energy in one period. When the temperatureis above 2400 K, the stress is close to zero. After checkingthe GB structure at 2400 K using the radial distributionfunction (RDF), we find that the GB has already pre-melted, and this explains the abnormal behavior of thestress–time relation at 2400 K.

In order to understand the relation between GB motionand stress, we plotted the time dependency of the shear

Fig. 3. Stress–time curves for shear along the ½1�30� direction at temperatures berate of 1 � 108 s�1. The inset of the plot shows the exponential fitting of the c

stress and GB displacement at 300 K (see Fig. 4). TheGB position is tracked by the CNA computation, whichgives values of 3 and 5 for atoms in the bcc lattice andthe GB region, respectively [38]. It is obvious that the GBmigration curve displays a regular serrated profile, whichis the so-called “stick–slip” behavior. While the “stick”stages correspond to elastic straining, the “slip” stagesrelate to certain structural transformations. During theelastic deformation stage, the shear stress increases almostlinearly. After the critical stress is reached, the GB rapidlymoves to a new position. As the deformation continues, thestress drops after GB migration; this is then followed by anew increase in the stress until the next peak. Each peak ofthe stress correlates exactly with an increment of the GBmotion. One possible explanation is that the GB becomesmechanically unstable as the deformation continues andrequires some mechanism to release the excess energy.GB migration is one of the mechanisms that is likely tooccur in this condition. The accompanying grain transla-tion produces a permanent shear deformation. This typeof GB motion is trapped in one energy minimum until itloses stability and jumps to a new minimum.

Fig. 5 shows the GB migrations at different tempera-tures. It can be seen that the GB migration is similar formost of the temperatures considered here. An obviousstick–slip behavior is seen in our simulations at low tem-peratures. The displacement steps firstly decrease and thenmaintain at a certain value as the temperature increases.Thermal fluctuation tends to smooth out the discontinuousstick–slip behavior. The smallest displacement step is con-firmed to be the distance between two neighboring atomicplanes in the GB region.

tween 1 and 2400 K. The simulations are performed with a constant strainritical stress.

Fig. 4. The shear stress and the GB migration as functions of time forshearing along ½1�30� at 300 K. The process of migration exhibits a stick–slip behavior. The blue line is the stress–time curve and the red linerepresents the GB displacement which normal to the GB plane. (Forinterpretation of the references to color in this figure legend, the reader isreferred to the web version of this article.)

Fig. 5. Migrations of the GB as functions of time at different temperaturesfor shearing along ½1�30�. Migrations exhibit stick–slip behavior at mosttemperatures. The GB exhibits a triple jumps and double jumps at 1 and100 K, respectively. At 2400 K, the GB shows random motions.

Fig. 6. Sliding of the GB as functions of time at different temperatures forshearing along ½1�30�. Sliding exhibits stick–slip behavior at most temper-atures. The stick–slip behavior is weakened as temperature increases.

262 B.-W. Huang et al. / Acta Materialia 77 (2014) 258–268

At 100 K, the GB jumps by twice the smallest height andthe drop in stress is also doubled. The atoms cannot moveeasily at this low temperature; therefore, the accumulatedelastic strain energy is so large that one single jump cannotrelease it completely. It is expected that at temperaturesbelow 100 K, higher multiple jumps may be observed. Sim-ulation at 1 K shows a triple-distance migration which con-firms this expectation. At 2400 K, the migration isinterrupted because of the GB pre-melting. These phenom-ena are consistent with a recent publication that reports thetransition from stick–slip to driven Brownian dynamics forfcc materials [5].

We measure the GB displacement on the GB plane tostudy whether sliding is also occurring.The magnitude ofsliding is calculated using relative positions between two

predefined atoms located on different sides of the GB.The relative distance between this two marked atoms isequal to zero at time ðtÞ ¼ 0.

In all of our simulations, GB sliding along the ½1�30�direction is found to accompany GB migration. The tem-perature dependence of sliding (see Fig. 6) has a behaviorsimilar to that of migration, i.e. stick–slip behavior. TheGB migration and sliding velocities can be obtained fromFigs. 5 and 6 (displacement–time curves), respectively.Our results indicate that the GB velocity is independentof temperature, and is determined by the shear rate andGB structure. The shear rate is also referred to as the slid-ing velocity by other authors [5,6,21].

Considering that the velocities of migration and slidingremain constant, we assume that a coupled relation existsin these two motions. Fig. 7(a) shows the GB motions inboth of the ½1�30� and ½310� directions at 100 K. It can beseen that the migration and the sliding motions take placesimultaneously, which means they couple perfectly. Thecoupled motion is similar to other stick–slip motions, e.g.the tip movements in atomic force microscopy. The cou-pling factor b for GB introduced by Cahn et al. [21] isdefined as the ratio between the GB sliding velocity (v//)and the GB migration velocity (vn). From the geometricmodel of coupling, the perfect coupling factor of our GBis b ¼ �2tanðp=4� h=2Þ ¼ �0:99. Our simulations give aconsistent result that b ¼ v===vn � �1. In the range of1–2100 K, b is found to be practically independent of tem-perature and in good agreement with its geometrical value(see Fig. 7(b)). This indicates that the coupling is nearlyperfect for this temperature range. When the temperatureis above 2400 K, the coupling behavior is destroyed byGB pre-melting.

It was mentioned in a previous publication for fcc Cu[21] that the GB coupling behavior is determined solelyby the GB geometric structure. In other words, the cou-pling factor is fixed once the GB structure is determined.

Fig. 7. The GB coupled motion for shearing along ½1�30�. (a) The GBmigration and sliding as functions of time at 100 K. The blue line describesmigration and the red line represents the sliding process. (b) The couplingfactor b as a function of temperature. The dashed line is the b predictedfrom the geometric model. (For interpretation of the references to color inthis figure legend, the reader is referred to the web version of this article.)

B.-W. Huang et al. / Acta Materialia 77 (2014) 258–268 263

As our simulations reveal similar behaviors in Nb, the samerule may also apply to other bcc materials. However, moreGBs in other bcc materials need to be further tested inorder to confirm this.

To study the structure evolution of the GB, snapshots ofthe atomic positions are recorded at fixed time-step inter-vals. Fig. 8 shows a plot of the vector field of theR5h001if310g GB, which is viewed along the ½00�1� direc-tion. It records the position variation before and afterone migration.

The green unit (abcdef) belongs to the lower grain and isinterlocked with the kite-like blue unit (ABCDEF). Thegreen unit is a slightly distorted version of the perfect lat-tice. Importantly, the green and blue units are topologicallyidentical and can transform to each other by relativelysmall atomic displacements. At each step of boundarymotion, the green unit changes its shape and transformsto a kite-like blue unit, whereas the blue unit simulta-neously transforms to an orange unit. Note that the latteris a mirror reflection of the green unit.

As a result, the GB position shifts one step down whilethe upper grain translates to the right to accommodate the

deformation of the units. This process can be viewed as theglide of the parallel arrays of the GB dislocations along½00�1�. The coupling factor can be obtained, which is theratio of the unit displacement length on plane to that inthe vertical direction.

3.2. Shear along the ½00�1� direction

The ½00�1� direction is parallel to the tilt axis. We applyshear deformation along this direction as was done forthe ½1�30� direction. Several phenomena that are differentfrom those we have discussed in the ½1�30� direction areobserved. It is determined that the GB motion is pure slid-ing along the ½00�1� direction.

Stress–time curves at different temperatures are shownin Fig. 9. The critical stress is much larger than that ofthe ½1�30� direction, which means pure sliding along the½00�1� direction is harder to activate than coupled motion.The critical stress does not strongly depend on temperaturein the range 100–300 K. A periodic sliding behavior similarto the GB coupled motions is observed at most tempera-tures except 1 K. Based on the analysis on the varianceof the GB position, it is found that there is no displacementperpendicular to the GB plane. The GB can only slidealong the ½00�1� direction within the GB plane. Similar tothe ½1�30� direction, the GB sliding velocities display temper-ature independence and remain constant.

The shear stress and the GB in-plane displacement at900 K are shown as functions of time in Fig. 10. The slidingalong ½00�1� can also be regarded as a stick–slip process. TheGB initially stays in a stick stage, and it accumulates strainthroughout the crystal. The slip process takes place whenstrain reaches the critical level.

Simulation at T = 1 K shows an abnormal behavior. Inorder to understand the mechanisms in detail, we monitorthe crystal structure evolution at this temperature. Fig. 11shows the deformation along the ½00�1� direction at 1 K. Itcan be seen that a partial dislocation nucleates and is emit-ted from the GB, due to its lower activation energy com-pared to pure GB sliding at this temperature. Once thepartial dislocation moves toward the fixed boundary, theregion occupied by the stacking fault will extend. Mean-while, detailed analysis shows that limited GB sliding andlocal atomic shuffle also take place simultaneously. Thenucleation and gliding of partial GB dislocations has previ-ously been observed in bcc Fe at low temperature [31].

3.3. Shear between the ½1�30� and ½00�1� directions

In this section, we conduct a series of simulations withshear directions between the ½1�30� and ½00�1�. The specificinformation about the shear directions are summarized inTable 1. We have divided the simulations into two groupsbased on the results at T = 100 K. The first group repre-sents shearing along the ½1�3�6�; ½�39�5� and ½1�3�1� directions.These simulations behave similarly to the ½1�30� direction(coupled motion). The second group consists of simulations

Fig. 8. Vector field of atomic displacements for shearing along ½1�30� at 300 K. The black dots represent the original positions of the atoms, while the redarrows show the atomic trajectories. The blue and green lines represent the GB structural units before and after migration takes place, respectively. (Forinterpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 9. The stress–time relations for shearing along the ½00�1� direction at temperatures between 1 and 2100 K. The simulations are performed with aconstant strain rate of 1 � 108 s�1. The inset shows the exponential fitting of the critical stress.

Fig. 10. The shear stress and the GB sliding in the ½00�1� direction asfunctions of time for shearing along ½00�1� direction at 900 K.

264 B.-W. Huang et al. / Acta Materialia 77 (2014) 258–268

along the ½1�3 �30�; ½1�3 �20�; ½1�3 �15� and ½1�3 �10� directions; theseshow complex mechanisms at low temperatures and ½00�1�-like pure sliding at high temperatures.

In the first group, we have studied the ½1�3�6� and ½�39�5�directions in detail. The ½1�3�6� direction is about 27:79�

clockwise to the ½00�1� direction and the ½�39�5� direction is62:21� anticlockwise to the ½00�1� direction. Two kinds ofsimulations behave similarity. The GB migrates and slideswith stick–slip behavior which is similar to the deformationalong the ½1�30� direction. Furthermore, the GB in-planemovements do not only take place in the shear direction,but also in the vertical direction. The GB can move alongthe ½1�3�6� and ½�39�5� directions simultaneously. As we havediscussed previously, the GB exhibits a coupled motionand a pure sliding when it is sheared along the ½1�30� and½00�1� directions, respectively. Thus, we have decomposedthe velocities of sliding into the ½00�1� and ½1�30� directions.

Fig. 12 shows how we have decomposed the velocity andstress. For shear along ½1�3�6�, the resolved velocity in ½00�1� isclose to zero and the resolved velocity in ½1�30� is 3.81 m s�1.The GB migration velocity is �3.85 m s�1. Therefore, thecoupling factor is �0.99, which is consistent with that ofthe ½1�30� direction. A similar decomposition is applied to

Fig. 11. Selected snapshots of dislocation nucleation, motion and atomic shuffle on the GB at 1 K. Atoms are colored according to the centrosymmetryparameters.

Fig. 12. Decomposition of the ½1�3�6� displacement into the ½00�1� and ½1�30�directions.

B.-W. Huang et al. / Acta Materialia 77 (2014) 258–268 265

the critical stress. The resolved critical stress in ½00�1� is1.94 GPa, which is much smaller than the stress neededto activate the ½00�1� pure sliding (about 3.50 GPa), at100 K. On the other hand, the resolved critical stress in½1�30� is 1.02 GPa, which is comparable to the stress needed

Table 2The critical stress Fc, stress F and resolved stress f (in GPa) at different tempe

Temperature Fc½00�1� Fc½1�30� F ½1�3�6�

100 K 3.50 1.22 2.19300 K 4.20 0.58 1.07600 K 3.20 0.31 0.61900 K 2.50 0.18 0.43

to activate the ½1�30� coupled motion (about 1.22 GPa). Thesmall difference could be compensated by atomic shuffling.

For further investigations, we have performed MD sim-ulations at 100, 300, 600 and 900 K. Table 2 lists theresolved critical stresses and the critical stresses of defor-mation in the ½00�1�; ½1�30� and ½1�3�6� directions. The MDresults are consistent with experiment in that the resolvedstress in the ½00�1� is much smaller than the critical stressof pure sliding. The resolved critical stress in the ½1�30�direction is comparable to the critical stress of coupledmotion. Moreover, Fig. 13 shows that the GB is not dis-placed in the ½00�1� direction. Combining the coupling fac-tor and the atomic structure evolution, we conclude thatshearing along the ½1�3�6� direction shares the same mecha-nism as that along the ½1�30� direction.

It can also be seen from Table 2 that the shear along the½�39�5� direction shows behavior similar to the deformationin the ½1�3�6� direction. The resolved critical stress in the½00�1� direction is always much smaller than the stressneeded to activate pure sliding, while the resolved criticalstress in the ½1�30� direction and the critical stress alongthe ½1�30� direction are comparable. Thus we conclude thatthe deformation along the ½�39�5� direction shares the samemechanism as the coupled motion in the ½1�30� direction.

ratures.

f½00�1� f½1�30� F ½�39�5� f½00�1� f½1�30�

1.94 1.02 1.18 0.55 1.040.95 0.50 0.58 0.27 0.520.54 0.29 0.32 0.15 0.280.38 0.20 0.22 0.10 0.19

Fig. 13. The stress–time relations as well as the GB displacements in different directions for shearing along ½1�3�6� at various temperatures.

Fig. 14. The stress–time relation for shearing along the ½1�3 �10�; ½1�3 �15�; ½1�3 �20� and ½1�3 �30� directions at 100 K.

266 B.-W. Huang et al. / Acta Materialia 77 (2014) 258–268

Fig. 15. The stress and the GB displacement as functions of time forshearing along the ½1�3 �30� direction at 900 K. Arrow 1 is the ½1�30�-likecoupled motion.

B.-W. Huang et al. / Acta Materialia 77 (2014) 258–268 267

By investigating the deformation mechanisms in the firstgroup at 100 K, we find that the simulations exhibit a ½1�30�-like GB coupled motion once the angle between the sheardirection and the ½00�1� tilt axis is larger than the criticalangle.

In the second group, the angles between the shear direc-tions and the tilt axis are smaller than the critical value, andthe stress–time curves of the four shear directions at 100 Kare shown in Fig. 14. The strain–stress curves display anon-periodic behavior, which indicates that the sheardeformations belong neither to the ½1�30�-like GB coupledmotion, nor to the ½00�1�-like GB pure sliding.

For the ½1�3 �30� shear direction (Fig. 14(a)), whoseincluded angle h is 6:02�, there are three different stagesof deformation. After the critical stress is reached, atomicshuffle appears first to partially release the stress. A largenumber of dislocations then nucleate on the GB. As the

Fig. 16. Different types of GB motions with re

deformation proceeds, finally, there is a GB distortionwhich makes the stress drop rapidly.

For the shear along the ½1�3 �20� direction (Fig. 14(b)),which has an included angle of h = 8:99�, there are twostages of deformation, which correspond to stages oneand three of the ½1�3 �30� direction. Dislocation nucleationsare not observed in our simulations.

For the shears along the ½1�3 �15� and ½1�3 �10� directions(Fig. 14(c) and (d)), the included angles h are 11:90� and17:54�, respectively. The GB coupled motions are observedto release the stress. However, the GB coupled motionshave limited effects on the deformation, and atomic shuffleand GB distortion still dominate the shearing process.

Based on the above analysis, we conclude that a transi-tion region exists between the GB pure sliding and the GBcoupled motion, which has different shear deformationmechanisms. With a larger included angle, the proportionof GB coupled motion increases. Nevertheless, we havenot observed pure GB sliding, even for an extremely smallangle. This is similar to the simulation along the ½00�1�direction at 1 K. A possible reason is that a highertemperature is needed to activate the GB pure sliding inthose small-angle simulations, as shown in the followingdiscussion.

The stress and the GB position are shown as functionsof time in Fig. 15 for shearing along the ½1�3 �30� directionat 900 K. Sliding along ½1�3 �30� and ½�39�1� (perpendicular to½1�3 �30�) as well as GB migration are observed in the simula-tion. The displacements along the ½1�3 �30� and ½�39�1� direc-tions are decomposed to the ½00�1� and ½1�30� directions.Based on the analysis of the GB position, we find thatthe deformation contains ½1�30�-like GB coupled motionand ½00�1�-like GB pure sliding. The stress–time curve dis-plays a sawtooth behavior. The first small peak corre-sponds to GB sliding on the ½1�30� direction andmigration in the vertical direction. For the second and

spect to temperatures and shear directions.

268 B.-W. Huang et al. / Acta Materialia 77 (2014) 258–268

fourth peaks, the GB slides along the ½1�30� and ½00�1� direc-tions and simultaneously migrates; the ½00�1�-like GB puresliding dominates and releases most of the stress. The thirdand the fifth peaks indicate that the GB displays a puresliding along the ½00�1� direction. Since the barrier of the½1�30� direction coupled motion is much smaller than the½00�1� direction GB pure sliding, it is reasonable that ahandful of ½1�30� direction coupled motions take place atsuch high temperature. Thus, we consider that this simula-tion is similar to the ½00�1� direction GB pure sliding.

We have summarized the types of the GB motions fordifferent temperatures and shear directions as shown inFig. 16. The MD simulations with ½1�3�6�; ½�39�5�; ½1�3�1� sheardirections exhibit ½1�30�-like GB coupled motion. The defor-mations with shear directions along the ½1�3 �30�; ½1�3 �20�; ½1�3 �15�and ½1�3 �10� are similar to that along ½1�3 �30� at T = 900 K. Inaddition, shear along the ½1�3 �30� direction is a ½00�1�-like puresliding at 300 K. It is found that the ½00�1� pure sliding isdominating in small-angle simulations at high tempera-tures, while GB dislocations, distortion and atomic shuffleare the main deformation mechanisms at low temperatures.

4. Conclusion

We have performed MD simulations to study the shearresponses of the R5h001i 310f g symmetric tilt GB in bcc Nbover a wide range of temperatures. Nine shear directionsparallel to the GB plane have been studied, namely½00�1�; ½1�3 �30�; ½1�3 �20�; ½1�3 �15�; ½1�3 �10�; ½1�3�6�; ½�39�5�; ½1�3�1� and½1�30�.

For shear deformation along the ½1�30� direction, the GBalways shows a coupled motion regardless of the tempera-ture.The critical stress decreases exponentially with increas-ing temperature. The coupled motion displays a stick–slipbehavior, i.e. the system is trapped in an energy minimumuntil the GB becomes unstable and jumps to a new mini-mum. At very low temperatures, we observe multiplejumps, which are different from the single jump at hightemperatures. The GB coupling factor is found to be inde-pendent of temperature and can be predicted from geomet-ric calculations. Therefore, we propose that shearing alongthe direction perpendicular to the tilt axis shares a mecha-nism with the geometric theory proposed by Cahn.

For the shear deformation along the ½00�1� direction, theGB movement is pure sliding at most of the temperaturesstudied. The sliding is more difficult to activate than thecoupled motion because its critical stress is much larger.Pure sliding is not observed at T = 1 K, and the GB exhib-its an unusual behavior caused by the dislocation nucleat-ing and extending.

The shear deformations between the ½1�30� and ½00�1�directions are more complicated, and we have divided theminto two groups according to their deformation mecha-nisms. The first group has large angle between the sheardirections and the tilt axis, which exhibits a GB coupledmotion and a geometrically predictable coupling factor.

We find that this type of deformation shares the sameatomic structural evolution with the ½1�30� shear direction.The second group has smaller angles between the sheardirections and tilt axis. Dislocation, atomic shuffle andGB distortions are dominant at low temperatures, whilethe ½00�1�-like GB pure sliding is the principal mechanismat high temperatures.

Acknowledgment

This work was financially supported by the NationalScience Foundation of China (Grant Nos. 51071011 and51371017).

References

[1] Sutton AP, Vitek V. Philos Trans R Soc A 1983;309:1.[2] Mishin Y, Asta M, Li J. Acta Mater 2010;58:1117.[3] Xu GQ, Demkowicz MJ. Phys Rev Lett 2013;111:145501.[4] Bishop GH, Harrison RJ, Kwok T, Yip S. J Appl Phys 1982;53:5596.[5] Mishin Y, Suzuki A, Uberuaga BP, Voter AF. Phys Rev B

2007;75:224101.[6] Ivanov VA, Mishin Y. Phys Rev B 2008;78:064106.[7] Caillard D, Mompiou F, Legros M. Acta Mater 2009;57:2390.[8] Trautt ZT, Adland A, Karma A, Mishin Y. Acta Mater 2012;60:6528.[9] Spearot DE, Jacob KI, Mcdowell DL. Acta Mater 2005;53:3579.

[10] Spearot DE, Tschopp MA, Jacob KI, Mcdowell DL. Acta Mater2007;55:705.

[11] Pan Z, Li Y, Wei Q. Acta Mater 2008;56:3470.[12] Fr£seth A, Swygenhoven HV, Derlet PM. Acta Mater 2004;52:2259.[13] Farkas D, Fr£seth A, Swygenhoven HV. Scr Mater 2006;55:695.[14] Cahn JW, Taylor Jean E. Acta Mater 2004;52:4887.[15] Rajabzadeh A, Legros M, Combe N, Mompiou F, Molodov DA.

Philos Mag 2013;93:1299.[16] Sansoz F, Molinari JF. Acta Mater 2005;53:1931.[17] Li Choh Hsien, Edwards EH, Washburn J, Parker ER. Acta Metall

1953;1:223.[18] Molteni C, Marzari N, Payne MC, Heine V. Phys Rev Lett

1997;79:869.[19] Shiga M, Shinoda W. Phys Rev B 2004;70:054102.[20] Sutton AP, Balluffi RW. Interfaces in crystalline materi-

als. Oxford: Clarendon Press; 1995.[21] Cahn JW, Mishin Y, Suzuki A. Acta Mater 2006;54:4953.[22] Zhang H, Du D, Srolovitz DJ. Philos Mag 2008;2:243.[23] Cahn JW, Mishin Y, Suzuki A. Philos Mag 2006;86:3965.[24] Molodov DA, Ivanov VA, Gottstein G. Acta Mater 2007;55:1843.[25] Molodov DA, Gorkaya T, Gottstein G. J Mater Sci 2011;46:4318.[26] Winning M, Gottstein G, Shvindlerman LS. Acta Mater 2001;49:211.[27] Mompiou F, Legros M, Caillard D. J Mater Sci 2011;46:4308.[28] Homer ER, Foiles SM, Holm EA, Olmsted DL. Acta Mater

2013;61:1048.[29] Olmsted DL, Holm EA, Foiles SM. Acta Mater 2009;57:3704.[30] Wan L, Wang S. Phys Rev B 2010;82:214112.[31] Hyde B, Farkas D, Caturla MJ. Philos Mag 2005;85:3795.[32] Campbell GH, Foiles SM, Gumbsch P, Ruhle M. Phys Rev Lett

1993;70:449.[33] Melchionna S. Phys Rev E 2000;61:6165.[34] Fellinger MR, Hyoungki P, Wilkins JW. Phys Rev B 2010;81:144119.[35] Wang LG, van de Walle A. Phys Chem Chem Phys 2012;14:1529.[36] Zhang RF, Wang J, Beyerlein IJ, Germann TC. Philos Mag Lett

2011;91:731.[37] LAMMPS code homepage: <http://lammps.sandia.gov/>.[38] Faken D, Jonsson H. Comput Mater Sci 1994;2:279.