Dynamics of grain boundary motion coupled to shear deformation: An analytical model and its verification by molecular dynamics

Transcription

1 PHYSICAL REVIEW B 78, Dynmis of grin boundry motion oupled to sher deformtion: An nlytil model nd its verifition by moleulr dynmis V. A. Ivnov* nd Y. Mishin Deprtment of Physis nd Astronomy, George Mson University, MSN 3F3, 44 University Drive, Firfx, Virgini 223, USA Reeived 4 Februry 28; revised mnusript reeived 25 June 28; published 11 August 28 Mny tomilly ordered grin boundries GBs ouple to pplied mehnil stresses nd re moved by them, produing sher deformtion of the lttie they trverse. This proess does not require tomi diffusion nd n be implemented t low tempertures by deformtion nd rottion of struturl units. This so-lled oupled GB motion ours by inrements nd n exhibit dynmis similr to the stik-slip behvior known in tomi frition. We explore possible dynmi regimes of oupled GB motion by two methods. First, we nlyze simple one-dimensionl model in whih the GB is mimiked by prtile tthed to n elsti rod nd drgged through periodi potentil. Seond, we pply moleulr dynmis MD with n embedded-tom potentil for Al to simulte oupled motion of prtiulr tilt GB t different tempertures nd veloities. The stress-veloity-temperture reltionships estblished by both methods re qulittively similr nd indite highly nonliner dynmis t low tempertures nd/or lrge veloities. At high tempertures nd/or slow veloities, the hrter of the GB motion hnges from stik slip to driven rndom wlk nd the stressveloity reltion beomes pproximtely liner. The MD simultions lso revel multiple GB jumps due to dynmi orreltions t high veloities, nd trnsition from oupling to sliding t high tempertures. DOI: 1.113/PhysRevB PACS number s : Bb, Mm, Af I. INTRODUCTION Mny grin boundries GBs hve the property tht their norml motion requires simultneous reltive trnsltion of the djent grins prllel to the GB plne. 1 4 Conversely, ny reltive trnsltion of the grins uses norml displement of the boundry. Suh GBs re sid to be oupled nd their response to pplied driving fores n be different from tht of unoupled GBs. Speifilly, if sher stress is pplied prllel to oupled GB, it retes driving fore for its norml motion. 5 This motion, in turn, produes sher deformtion of the mteril swept by the boundry. Conversely, if oupled GB is driven by pillry fores or volume driving fore due to elsti or mgneti nisotropy, the boundry shers the lttie region it trverses nd produes rigid trnsltion of the grins. It n be shown tht oupled motion of urved GB indues grin rottion nd vie vers. 1,6 The oupling effet is hrterized by ftor equl to the rtio of the tngentil grin trnsltion veloity v to the ompnying norml GB displement veloity v n. The oupling is lled perfet if is geometri onstnt tht depends only on the GB rystllogrphy but not on the GB veloity, driving fore or ny other physil prmeters. The oupled GB motion is rther ommon effet. Atomisti omputer simultions hve reveled dozens of oupled GBs. 2 4,6 Stress-indued GB motion hs been observed in experiments on birystls in both metls 7 12 nd ermi mterils. 13 The experimentl oupling ftors were found 7,8 to mth the perfet vlues predited by the geometri theory. 2 4 It is believed tht the oupling effet might be responsible for the stress-driven GB motion nd stressindued grin growth in nnorystlline mterils Among the unsolved problems nd future work diretions outlined in Ref. 4, it ws pointed out tht oupled GB motion n exhibit rih vriety of dynmis tht need to be identified nd understood. In prtiulr, it ws suggested tht the stress-veloity reltion n hnge signifintly s the stopnd-go stik-slip hrter of motion observed t low tempertures 3,4 trnsforms to driven rndom wlk t high tempertures. The stress-veloity reltion in the stik-slip regime ws reently studied by elerted moleulr dynmis MD over wide veloity rnge 17 t one fixed temperture. The temperture dependene of GB dynmis ws not exmined in tht work. In this pper we ontinue to investigte the dynmis of oupled GB motion, now fousing on its temperture dependene. We pply two different pprohes to the problem: First, we nlyze one-dimensionl nlytil model of oupling whih, despite its simplisti hrter, permits derivtion of useful stress-veloity reltions tht n be tested ginst tomisti simultions. Seond, we ondut MD simultions of stress-driven nd spontneous oupled motion of prtiulr GB over wide temperture rnge nd revel the trnsition from the stik-slip regime to Brownin dynmis. This ombintion of the nlytil nd MD pprohes is the entrl point of this pper. While the previous MD simultions 2 4,17 employed opper s model mteril, in this work we swith to luminum GBs in order to be more omptible with reent 7,8 nd ongoing 18 experiments on Al birystls. This lso gurntees semless onnetion to our urrent work fousing on rystllogrphi spets of oupling in Al boundries. We emphsize, however, tht most of our results re generi nd should not be dependent on the mteril. In this pper we tret oupled GB motion s motion through periodi energy lndspe. To explin the origin of this lndspe, onsider n exmple of plnr tilt boundry in pure metl t zero temperture. Assume for simpliity tht the tilt ngle is suh tht oinident site lttie CSL 19 rises nd the GB lies in CSL plne. Suh boundries typilly hve periodi struture onsisting of polyhedrl struturl units. 19 Due to trnsltionl symmetry of /28/78 6 / The Amerin Physil Soiety

2 V. A. IVANOV AND Y. MISHIN PHYSICAL REVIEW B 78, L n GB v ( ) ( b) t Stress τ τ Grins trnsltion the birystl, there is n infinite number of GBs tht hve extly the sme energy nd identil tomi strutures obtinble from one nother by rigid trnsltions. All these GBs re prllel to eh other nd only differ in the trnsltionl stte of the grins. Energetilly, these equivlent GBs orrespond to potentil energy minim in the 3N-dimensionl onfigurtion spe of the birystl, with N being the number of toms in the system. The GB motion n be oupled if the boundry struture permits trnsltions between neighboring equivlent positions by reltively smll tomi displements without tomi diffusion. These displements n usully be desribed s deformtions nd rottions of the struturl units nd require overoming ertin energy brrier E. If driving fore is pplied nd the GB is set to motion, it moves by jumping between the equilibrium positions. The ritil stresses required for this proess n depend on temperture nd the GB veloity. Investigting these reltions is the gol of this pper. II. DYNAMICS OF COUPLED BOUNDARY MOTION In this setion we outline some problems relted to dynmis of oupled GB motion. We will onsider prtiulr oupling mode, in whih neighboring GB positions re distne H prt while the oupled grin trnsltions re by S in diretion t norml to the tilt xis Fig. 1. The perfet oupling ftor in this mode is =S/H. A. Zero-temperture dynmis GB displement FIG. 1. A gednken experiment in whih GB shown s dshed line is moved by sher stress. The tilt xis of the GB is norml to the viewer, n is unit vetor norml to the GB plne, nd t is unit vetor norml to both n nd the tilt xis. The dshed res designte two slbs used s lmps. The stress is pplied by moving the upper slb with fixed veloity v prllel to t while the lower slb remins fixed. L is the norml size of the dynmi region, whih plys the role similr to the grin size. b If the oupling is perfet, the GB moves by inrements of H ompnied by reltive grin trnsltions S. The stress exhibits sw-tooth behvior with pek vlue nd n verge. At K nd in the bsene of pplied stresses, the GB is initilly in prtiulr equilibrium position. Suppose sher stress is pplied to the birystl by holding the lower grin nd moving the upper grin with onstnt veloity v prllel to t Fig. 1. The fore ting on the upper grin is trnsmitted through the lttie regions nd retes lol stress ˆ t the GB. This stress elstilly deforms the GB S H struturl units nd redues the energy brrier for the GB displement to new position, simultneously rising the brrier in the opposite diretion. We re prtiulrly interested in the sher stress = ˆ n t resolved in the diretion of the grin trnsltion. When rehes ritil vlue, 2 the stress-redued brrier E turns to zero. The GB beomes mehnilly unstble nd jumps to new position. The ompnying grin trnsltion produes permnent sher deformtion of the birystl nd the stress drops. As the upper grin ontinues to move, the stress builds up gin until it rehes nd the GB mkes nother step. As this proess ontinues, the GB moves forwrd by inrements of H while the stress displys sw-tooth behvior s indited in Fig. 1 b. 21 This inrementl motion, in whih the GB is trpped in one energy minimum until it loses stbility nd jumps to new minimum, n be lssified s stik-slip proess by nlogy with other similr phenomen Between the inrements of the GB motion, the stress inreses s liner funtion of time ssuming liner elstiity. The mgnitude of the stress drop t eh step is then proportionl the inrement of the sher deformtion, S/L, nd thus inversely proportionl to the system size L in the norml diretion. It follows tht the stress verged over yle of this proess is = KS/2L, where K is the pproprite elsti modulus. If L inreses, the stress drop dereses nd in the limit of L the GB moves under nerly onstnt stress. In the other limit, when the grins re smll, the stress drops to smll vlue nd n even beome negtive. 4 In the ltter se, the stress rising immeditely fter jump is driving the GB bk to the previous position. Importntly, the vlue of the pek stress should be insensitive to L sine this vlue depends only on the GB struture nd the oupling mode. These preditions regrding the grin-size dependene of the stress behvior need to be tested by tomisti simultions. B. Finite temperture dynmis At finite tempertures, therml flututions ssist the GB in overoming the brrier nd it n jump to new position when E is still positive, i.e., before the stress rehes. This effetively redues the ritil stress required for the GB motion, mking it. By ontrst to, the redued ritil stress is stohsti quntity tht should be hrterized by its verge nd sttistil distribution round it. This stress is ntiipted to derese with temperture s therml flututions intensify. At fixed temperture, should inrese with v sine higher veloities give the GB less time to overome the brrier before it vnishes. Understnding suh reltionships mong the veloity, stress, nd temperture is n importnt tsk of GB dynmis. Some of these reltions will be investigted lter in this pper. At high tempertures, there is finite probbility for the GB to mke spontneous jump bk to the previous position. We expet tht suh bkwrd jumps, whih should beome more frequent s temperture inreses, should

3 DYNAMICS OF GRAIN BOUNDARY MOTION COUPLED eventully destroy the sw-tooth behvior of the stress nd reple it by rndom noise with some verge. In this regime, the ritil stress loses its physil signifine while beomes the most meningful mesure of the stress driving the boundry motion. Reduing the GB veloity t fixed temperture should produe similr effet by giving the GB more time to smple both forwrd nd bkwrd jumps ording to their probbilities. In ft, if v tends to zero t high temperture, the forwrd nd bkwrd jumps beome eqully probble, resulting in rndom wlk of the boundry between its equilibrium positions. 2,4 In this limit must turn to zero by symmetry. If oupled GB motion t smll veloities n be treted s driven Brownin proess, one should expet liner reltion between v nd. This predition 4 seems to be onsistent with reent experiments 7,12 but ws never verified by omputer simultions. III. ONE-DIMENSIONAL MODEL OF BOUNDARY DYNAMICS Before we explore some of the regimes of oupled GB motion by tomisti simultions, it is useful to exmine simple one-dimensionl model tht predits some importnt reltions between the key dynmi prmeters. In reent pper, 17 simple mehnil nlog of oupled GB motion hs been proposed in whih the GB is represented by prtile tthed to n elsti rod. The prtile is drgged through sptilly periodi potentil U x by pulling the other end of the rod in diretion x. The rod models the elstilly deformed grins while U x mimis the potentil energy lndspe of the GB in the bsene of pplied stresses. The mss m of the prtile represents the effetive mss of the moving grins while the prtile frition ginst the potentil surfe nd the energy dissiption in the rod pture the dmping proesses t the GB nd inside the grins, respetively. This simple one-dimensionl model is rih enough to pture number of dynmi regimes whose detiled nlysis is deferred to seprte publition. Here, we will limit the nlysis to few prtiulr ses nd derive nlytil expressions tht will lter be ompred with tomisti simultions. Following Refs. 26 nd 27, we desribe U x by osine with period Fig. 2 : U x = E 2 1 os 2 x. The minim of U x define equilibrium lotions of the prtile, whih re seprted by the energy brrier E. Suppose the prtile is initilly t x= nd the elsti rod begins to exert fore on it, pulling the prtile to the right. We pproximte the modified potentil energy round the prtile by the tilted osine U x x. This pproximtion is vlid if the rod is soft enough tht 2 /k, where k is the spring onstnt of the rod otherwise there is nonnegligible hnge of between the right nd left mxim round x=. This tilt of the potentil energy redues the brrier E + for the forwrd jump to the right nd rises the brrier E for the bkwrd jump to the left. It lso shifts the equilibrium point of the prtile. 2 b - -/2 /2 no stress -/2 stress τ -/2 PHYSICAL REVIEW B 78, x 2 -/4 τ -/4 τ The new mxim nd minim of the potentil energy stisfy the ondition U x =, i.e., sin 2 x =, where we denote E / Fig. 2. From this eqution, the new equilibrium position is x = 2 rsin, nd the mxim re t x 1 =/2 x nd x 2 = /2 x. The energy brriers re found s work done when moving the prtile from x to x 1 nd x 2, respetively: x 1 E + U x dx = x = E rsin, x 2 E U x dx = x = E rsin. Consider limiting ses of these expressions. When rehes, the brrier E + turns to zero while x 1 x /4. The equilibrium beomes unstble nd the prtile is bound to mke jump forwrd. This revels the mening of s U E U' /4 U' - τ x x 1 /2 /4 /2 FIG. 2. The potentil energy U x, b its derivtive U x nd the totl fore U x in the one-dimensionl model of oupling. x x x

4 V. A. IVANOV AND Y. MISHIN PHYSICAL REVIEW B 78, the ritil fore for the prtile motion through the potentil lndspe t zero temperture. When =, the brrier of the bkwrd jump is E = E. Thus, the therml motion of the prtile tht strts t = ours by forwrd jumps only. To determine extly how E + tends zero nd E to E when, we expnd Eqs. 5 nd 6 in powers of the smll prmeter 1 / nd keep only leding terms, obtining Veloity T > T 1 2 T T 1 2 Nonliner dynmis E + 2E 1 3/2, 7 E E. Thus, the brrier E + vnishes s 3/2, not s s one ould expet. 28 In the other limit, when, Eqs. 5 nd 6 give E E 1 2, inditing tht smll fore shifts the forwrd nd bkwrd jump brriers in opposite diretions by the sme mount proportionl to. At finite temperture T, the prtile n mke jumps over the brriers by therml flututions. Applying the trnsition-stte theory, the rtes of the forwrd nd bkwrd jumps number of jumps per unit time re W + = exp E + /k B T nd W = exp E /k B T, respetively. Here, is jump ttempt frequeny nd k B the Boltzmnn ftor. The verge veloity of the prtile is v = W + W. In the smll-fore limit, we n pply Eq. 9 to obtin v =2 exp E k B T sinh E 2k B T Expnding the hyperboli sine to the first order in leds to the liner fore-veloity reltion v M T, 11 where the M T needs not to be detiled for this disussion. This liner reltion is vlid if E 1, 2k B T 12 i.e., when either the driving fore is smll or the temperture is high. This regime is best desribed s driven rndom wlk, or driven Brownin motion. In the limit of, we hve W + W nd bkwrd jumps n be negleted. Then, v W + v exp 2E 1 3/2 k B T, 13 Liner dynmis Stress FIG. 3. Veloity-stress reltions t high T 1 nd low T 2 tempertures predited by the one-dimensionl model of oupled GB motion. The regions of the liner dynmis mobility regime nd nonliner exponentil dynmis re indited. where v =. We emphsize tht the fore-veloity reltions obtined re only vlid s long s the ssumptions of the trnsitionstte theory re stisfied. In prtiulr, fter eh jump the prtile must hve enough time to ome to therml equilibrium with the environment before the next jump my our. This ssumption n be violted if the verge veloity is too high nd/or the energy dissiption proesses re too slow. Trnslting this nlysis from n bstrt prtile to moving GB, the fore should be referred to unit GB re nd beomes the pplied sher stress. The ttempt frequeny, whih for prtile is proportionl to U x /m, for GB hs more omplex mening tht involves tomi vibrtions both in the GB nd in the grins. Sine it enters the veloity expressions s pre-exponentil ftor, it is less importnt thn the jump brriers nd for this disussion is ssumed to be onstnt. Also, insted of fixing nd omputing the verge veloity, we re interested in the verge stress or the verge pek stress under onstnt imposed veloity v. Note tht in the limit of lrge grins L these stresses beome lmost identil. Figure 3 summrizes the stress-veloity reltions expeted from this model. In the region of smll stresses nd smll veloities, we expet the liner dynmis ording to Eq. 11, where M T is often referred to s GB mobility. 32 The veloity rnge dominted by this mobility regime expnds with temperture. The GB dvnes in jittery mnner in whih mny forwrd jumps re retrted. At lrge veloities nd/or reltively low tempertures, the GB moves by predominntly or exlusively forwrd jumps. The stressveloity reltion is then strongly nonliner nd n be desribed by n eqution similr to Eq. 13 with either the pek stress or the verge stress. The pre-exponentil ftor in Eq. 13 estblishes the upper bounds of the veloity in the limit of, but in relity this eqution loses its signifine in this limit sine the trnsition-stte theory does not pply when the energy brrier is smller thn k B T. Thus, this eqution is vlid s long s E + is signifintly lrger thn k B T but t the sme time muh smller thn E. Numeril estimtes bsed on Eq. 5 indite tht even if E + is not muh smller thn E, v is still n exponentil funtion of stress but the power of 1 / is between 3/2 nd

5 DYNAMICS OF GRAIN BOUNDARY MOTION COUPLED IV. METHODOLOGY OF MD SIMULATIONS B B PHYSICAL REVIEW B 78, [11] In the rest of the pper, the dynmi reltions bsed on the one-dimensionl model will be tested ginst tomisti simultions. The MD simultions were performed with n embedded-tom method EAM potentil 33 tht urtely reprodues vriety of properties of Al, inluding the elsti onstnts, phonon frequenies, the intrinsi stking fult energy, nd others. Sine the melting temperture of Al with this EAM potentil ws not lulted in the originl pper, 33 this ws done in this work using method similr to Ref. 34. The melting temperture ws found to be T m =142 K, whih is pproximtely 1% higher thn the experimentl melting temperture of pure Al 933 K. 35 This implies tht omprison of our simultions with experiment should be bsed on homologous tempertures T/T m. To study GB motion, simultion blok with two grins seprted by flt GB ws onstruted. The blok hd n orthorhombi shpe with periodi boundry onditions imposed in the diretions prllel to the GB plne. To stisfy these onditions, the GB hd to be CSL boundry. Depending on the gol of prtiulr simultion, two types of boundry ondition in the diretion norml to the GB plne were pplied. We refer to them s the fixed nd free boundry onditions. For the fixed boundry ondition, the grins re sndwihed between two slbs prllel to the GB plne, in whih the toms re fixed in their perfet-lttie positions reltive to eh other. Eh slb n be either fixed or llowed to move s rigid body. All other toms of the blok re dynmi. The thikness of eh slb is twie the utoff rdius of tomi intertions, whih equls.126 nm for this potentil. The thikness L of the dynmi region ws hosen to be 12.2 nm, exept in few runs exmining the size effet on the stress behvior s will be explined lter. The fixed boundry ondition is onvenient for pplying sher stress prllel to the boundry plne. To this end, the upper onfining slp is moved with onstnt veloity v in hosen diretion prllel to the GB plne, wheres the lower remins fixed. This boundry ondition prohibits spontneous rigid trnsltion of the grins, lthough lttie regions djent to the GB n still trnslte reltive to eh other. Most of the simultions reported here were performed with v=1 m/s norml to the tilt xis. Some runs were lso mde with smller veloities down to.1 m/s nd up to 3 m/s in order to explore the veloity dependene of the sher stress. In the free boundry ondition, the previously fixed toms of the upper slb re mde dynmi, so tht the upper grin now termintes t free surfe. The lower grin still remins tthed to its fixed slb. Thus, the stress in the simultion blok is lwys zero. This sheme llows free trnsltions of the upper grin reltive to the lower one. This boundry ondition ws pplied to study spontneous GB displements in oupled mode. The MD simultions were implemented in the nonil onstnt temperture, volume nd number of toms ensemble using the ITAP Moleulr Dynmis Progrm IMD. 36 The MD integrtion time step ws.2 fs nd the totl simultion time ws.5 2 ns. Therml expnsion oeffiients t different tempertures were determined by seprte zero-pressure Monte Crlo simultions. Prior to the MD simultions, the blok ws uniformly expnded to inlude the effet of therml expnsion t the simulted temperture. Tht this proedure prtilly elimintes therml stresses in the blok ws dditionlly onfirmed by omputing the verge stress tensor without ny pplied lods nd heking tht ll omponents were muh smller thn typil stresses ompnying GB motion. The symmetril tilt GB =44.42 ws hosen s model. 37 Eh grin hd n pproximtely ubi shpe nd the entire blok ontined 24,19 toms. The ground-stte struture of the GB ws determined by stti minimiztion of the totl potentil energy with respet to lol tomi displements nd reltive trnsltions of the grins. In ddition, we heked tht the ground stte did not produe ny long-rnge stresses in the grins. The stress tensor ˆ verged over ll dynmi toms ws omputed using the stndrd viril expression nd ws onstntly monitored during the simultions. The quntity of prime interest ws the resolved sher stress prllel to the sher diretion. During the simultions, the GB position ws trked using the entrosymmetry prmeter P proposed in Ref. 38. The tomi lyer prllel to the GB plne whose toms hd the lrgest vlue of P ws identified with the boundry position. V. SIMULATION RESULTS [111] [112] [241] A. Grin boundry struture nd migrtion mehnism [312] FIG. 4. Atomi struture of the GB t K. The six different symbols represent tomi rows with different depth long the tilt xis 112 norml to the viewer. The order of the lternting 112 plnes is s follows: solid tringle solid squre solid irle open tringle open squre open irle. The openirle lyer is the deepest from the viewer. The struturl units in the GB kites nd in the grins lbeled B re outlined. Note tht their orners lie in different 112 plnes. The tomi struture of the GB t K is shown in Fig. 4. The boundry onsists of identil struturl units with the shpe of kites. Eh row of suh units running prllel to the tilt xis nd thus norml to the viewer n be thought of s n edge dislotion with the Burgers vetor b= 1 1. This Burgers vetor ws determined by Burgers iruit onstrution using the upper grin s the referene lttie. This interprettion of the Burgers vetor is onsistent with the Chn- Tylor work. 1 The GB energy lulted t K is.443 J/m

6 V. A. IVANOV AND Y. MISHIN PHYSICAL REVIEW B 78, b/4 ( ) A B Coupling ftor β=.8165 T m ( b ) ( ) A FIG. 5. Atomi mehnism for the oupled GB motion. Initil stte, b trnsition stte, finl stte. B is the lttie struturl unit onverting to the boundry unit A. A nd B indite the sme struturl units fter the trnsformtion. When sher ws pplied by trnslting the upper fixed slb to the right, the GB ws found to move up with n verge veloity v n proportionl to the grin trnsltion veloity v. Multiple snpshots stored during the MD simultions were nlyzed to determine the tomi mehnism of the boundry motion, whih is shown in Fig. 5. Consider the tringulr struturl unit B whih belongs to the upper grin nd is interloked with the kite-shped unit A. Note tht B is slightly distorted version of the perfet-lttie unit f. Fig. 4. Importntly, units A nd B re topologilly identil nd n be trnsformed to eh other by reltively smll in-plne tomi displements. At eh step of boundry motion, eh unit B hnges its shpe nd trnsforms to kite A, wheres eh unit A simultneously trnsforms to bulk unit B in the lower grin. Note tht the ltter is mirror refletion of the B unit of the upper grin. As result, the GB position shifts one step up while the upper grin trnsltes to the right to ommodte the deformtions of the units. This proess n lso be viewed s glide of the prllel rry of GB dislotions long 111 plnes of the upper grin. This mehnism of GB motion ws found to operte t tempertures 1 K nd bove. At K, the pplied sher stress produed rigid sliding of the upper grin without ny GB displement. This ft indites tht the ritil stress of sliding t K is smller thn the ritil stress of oupled GB motion. Both stresses derese with temperture nd pprently rossover ours t some point below 1 K. From the geometri theory of oupling, 4 it n be shown tht the perfet oupling ftor for this GB is =2 tn /2 = 2/3 1/2 = The tul oupling ftor ws determined from the MD simultions s the rtio of the imposed v nd the verge GB migrtion veloity v n. At tempertures up to 9 K.86T m, ws found to be prtilly independent of temperture nd in good greement B T [K] FIG. 6. Coupling ftor s funtion of temperture obtined by MD simultions with v=1 m/s. The dshed line shows the idel geometri vlue of. T m is the bulk melting point of Al with this EAM potentil. with its geometril vlue Fig. 6, inditing tht the oupling t these tempertures is nerly perfet. At 1 K.96T m, dul behvior 3 ws observed in whih the GB moved by spordilly swithing bk nd forth between two oupling modes: the fmilir one with.817 nd new mode tht only ppered t high tempertures. The multipliity of oupling modes is seprte topi tht will be disussed elsewhere. In this pper we fous on oupled motion in one prtiulr mode with =.8165 nd thus the temperture intervl 1 9 K. The verge step H of the GB motion obtined from the MD simultions is bout.133 nm. To find the geometri vlue of H for this oupling mode, note in Fig. 5 tht the GB dislotions glide long the 111 plnes by inrements of b/4= 1 1 /4. These plnes mke the ngle /2 with the GB norml. Thus, H= b/4 os /2 = 3/28 1/2, whih using the Al lttie prmeter =.45 nm gives H=.133 nm in exellent greement with our MD results. The verge inrement S of grin trnsltions lso urtely mthes its geometri vlue S= H=.18 nm. B. Boundry dynmis: temperture dependene Turning to the dynmis of the oupled GB motion, we will first disuss the effet of temperture under fixed veloity v=1 nm/s. Figure 7 indites tht the sher stress required for moving the GB dereses with temperture. Simultneously, the hrter of the GB motion hnges from inrementl stop-nd-go t reltively low tempertures to more stohsti t high tempertures. At low tempertures, stik-slip dynmis re lerly seen, with sw-tooth behvior of the stress nd stepwise behvior of the GB position. Eh pek of the stress orreltes extly with n inrement of the GB motion: When ritil stress is rehed, the boundry quikly moves distne H up nd stops, while the stress drops to minimum vlue. It ws interesting to exmine the grin-size effet on this stress behvior. To this end, some of the low-temperture simultions were repeted with L=6.4, 9.6, 12.8, 16, nd

7 DYNAMICS OF GRAIN BOUNDARY MOTION COUPLED PHYSICAL REVIEW B 78, GB displement [nm] ( ) stress GB displement L=9.6 nm L=16 nm GB displement [nm] ( b ) GB displement [nm] ( ) GB displement GB displement stress stress Grin trnsltion [nm] 19.2 nm while keeping ll other simultion onditions identil. The simultions onfirm ll trends predited in Se. II A. We find tht the mgnitude of the stress drop t eh inrement of the GB motion dereses with inresing L. At the sme time, the pek stress hrdly hnges with L. As n exmple, Fig. 8 ompres the stress behvior for two seleted vlues of L. All subsequent simultions will be reported for just one grin size L=12.2 nm. Returning to Fig. 7, we see tht t 9 K.86T m the stepwise hrter of the GB motion is lmost destroyed by therml flututions nd the sw-tooth behvior of the stress is muh less pronouned thn t low tempertures. Although the GB is still perfetly oupled mthes its geometri vlue, the boundry dynmis hve obviously undergone trnsition FIG. 7. GB displement nd sher stress t 1 K, b 5 K, 9 K nd the imposed grin trnsltion veloity v=1 m/s. The rrows indite the orreltion between the peks of stress nd the inrements of the GB motion Time [ns] FIG. 8. Sher stress s funtion of time in MD simultions of oupled GB motion with two different simultion blok sizes L. In both ses, the temperture is 3 K nd the imposed grin trnsltion veloity is v=1 m/s. Note tht the mount of the stress drop dereses with L wheres the pek stress remins the sme. To understnd the nture of this dynmi trnsition, note tht ording to the stik-slip model disussed in Se. III, log v is expeted to be pproximtely liner in 3/2 /k B T s long s is lose to its zero-kelvin vlue. At fixed v, the pek stress is therefore expeted to be liner in T 2/3, = BT 2/3, 14 where the onstnt B depends on the ttempt frequeny, the energy dissiption rte nd other ftors. To test this reltion, the pek stress t eh temperture ws verged over 16 2 stik-slip events nd plotted in Fig. 9 s funtion of T 2/3. We lso plot the stress verged over the entire time intervl ontining those stik-slip events. Although the bsolute vlue of depends on the grin size, this verge stress should lso be liner in T 2/3 in the stik-slip regime. We observe tht both plots re indeed liner up to t lest 5 K.48T m. Together with the very len sw-tooth behvior of the stress f. Figs. 7 nd 7 b, this linerity onfirms tht this temperture intervl is indeed dominted by stik-slip dynmis. Extrpoltion of the liner reltions to T gives the therml vlues =.99 GP nd =.87 GP. Using this, we find tht the rtio / vries between.74 t 1 K nd.21 t 5 K. Rell tht Eq. 14 ws derived ssuming tht ws lose to Se. III ; it my not be urte for suh smll vlues of /. Nevertheless, this eqution does tully desribe the MD results firly well. At tempertures 6 K.57T m nd higher, both nd devite from the stright lines nd tend to level out. Combined with the noisy behvior of the stress nd the GB position Fig. 7, these devitions indite trnsition to dynmi regime different from stik-slip. We suggest tht this regime is strongly driven Brownin dynmis

8 V. A. IVANOV AND Y. MISHIN PHYSICAL REVIEW B 78, Sher stress [GP] T/T m Indeed, while ll GB jumps in the stik-slip mode our in one diretion in our simultions, only upwrd, the driven Brownin regime is hrterized by rndom jumps up nd down, lthough the jumps up dominte. If the stress is ompletely removed, the GB must ontinue to jump up nd down, implementing rndom wlk indued by ontinul therml flututions. This predition ws tested by stress-free MD simultions t 9 K.86T m with the free boundry ondition. The GB ws indeed found to implement rndom wlk Fig. 1 ; in ft, given enough time it ould wnder quite fr wy from its initil position. These spontneous GB movements were ompnied by simultneous trnsltions of the upper grin due to the oupling effet. By ontrst, t 4 K.38T m temperture inside the stik-slip rnge the boundry did not mke ny spontneous movements. Figure 1 demonstrtes tht there is lose orreltion between the boundry displements nd the upper grin trnsltions during the rndom wlk t 9 K. To quntify this orreltion, we plot in Fig. 11 the GB displement reltive to the initil position versus the trnsltion of the enter of mss of the upper grin perpendiulr to the tilt xis. An exellent liner orreltion is pprent, with the slope of.843 whih is lose to the geometri oupling ftor These observtions re onsistent with the ft 4 tht the oupling ftor does not depend on whether the GB motion is indued by therml flututions or driven by n pplied stress. When stress is pplied t 9 K, it bises the existing rndom jumps of the boundry nd drives it on verge upwrd. Attempts were mde to diretly observe bkwrd GB jumps during the stress-driven simultions t high tempertures. It should be noted tht in suh simultions, the fixed pek verge /3 2/3 T [K ] FIG. 9. Pek stress solid irles nd verge stress open irles s funtions of temperture to the power 2/3 obtined by MD simultions with onstnt veloity v=1 m/ s. The numbers indite the tempertures. The error brs represent the stndrd devition of verging over 16 2 stik-slip events. The lines show the liner orreltion t tempertures 1 5 K. b Displement [nm] K 9 K Time [ns] Time [ns] 4 K 4 K FIG. 1. Spontneous oupled motion of the GB: upper grin trnsltion nd b GB displement nd t 4 K nd 9 K. The simultions were performed with the free boundry ondition. boundry ondition onstrins spontneous rigid trnsltions of the grins, lthough lol trnsltions ner the GB re still possible. Sine suh lol trnsltions ome t the expense of elsti deformtion of the surrounding lttie regions, the spontneous GB movements nnot be s extensive s they re with the free boundry ondition. Nevertheless, they re expeted to our t high enough tempertures. Unfortuntely, we were not ble to see bkwrd GB jumps in the stress-driven simultions. Presumbly, the bis imposed by the stress ws too strong nd mde the bkwrd jumps rre events. Given lso the signifint distortions of the GB struture produed by the stress, the few bkwrd jumps tht still hppened were not deteted by our visuliztion method. One wy to revel them would be to drstilly redue the GB veloity using elerted MD methods, 17 but GB displement [nm] β= Grin trnsltion [nm] FIG. 11. Grin boundry displement s funtion of GB trnsltion during spontneous oupled GB motion t 9 K. The slope of the orreltion line gives n estimte of the orreltion ftor

9 DYNAMICS OF GRAIN BOUNDARY MOTION COUPLED this ws not pursued in this work. We emphsize tht under rel onditions, the GB veloities enountered in experiments on birystls or during rerystlliztion nd growth in polyrystlline mterils re orders of mgnitude smller thn in our simultions. 32 At suh smll veloities, spontneous GB displements oupled to lttie trnsltions n our muh more redily. Thus, in rel mterils t high tempertures, the oupled GB motion is likely to our by driven Brownin motion. C. Boundry dynmis: veloity dependene The low- nd high-temperture regimes of the GB motion were lso studied by fixing temperture nd vrying the imposed grin trnsltion veloity. Two different tempertures were tested, 4 K.38T m nd 9 K.86T m, eh with veloities rnging from.1 up to 3 m/s. The results were nlyzed in terms of veloity-stress reltions t these tempertures. The verge stress ws used sine it ould be omputed more urtely thn. 4 At T=4 K, the GB remined oupled with.817 nd exhibited sw-tooth behvior of the stress over the entire veloity rnge. The veloity-stress reltion obtined is highly nonliner s shown in Fig. 12. At low stresses, v inreses with extremely slowly until.18 GP. Although we expet v to beome liner funtion of t smll enough veloities, this liner regime ws not tully reveled t this temperture. Implementing this regime would require reduing v signifintly below.1 m/s. To reh suh smll veloities, the MD time hd to be longer thn 5 1 ns the time to move the GB over t lest few nnometers. Implementing this regime ws beyond our omputtionl resoures. Above.18 GP, the growth of v with rpidly elertes nd by.2 GP whih orresponds to v 5 m/s the plot beomes lmost vertil. The log v versus plot shown in Fig. 12 b indites tht t v 5 m/s the rpid growth of v n be desribed s pproximtely exponentil in stress. Anlysis shows tht it n lso be desribed s exponentil with respet to 3/2 within the stter of the points. Although the limited sttistis do not permit us to distinguish between the powers of 1 nd 3/2, the importnt point is tht the growth of v is exponentilly fst in greement with the nlysis in Se. III. The deprture from the exponentil growth t higher veloities v 1 m/s is explined by the effet of dynmi orreltions between the GB jumps. At suh high veloities, the energy dissiption rte nnot th up with the elsti strin energy relese t eh step of the boundry motion. The undmped energy, existing in the form of sound wves bouning bk nd forth between the two fixed regions, ssists the GB in overoming the next tivtion brrier. This produes derese in the stress required for moving the GB in omprison with the low-veloity regime in whih the GB ompletely thermlizes fter eh jump. In the strongly underdmped regime observed t high veloities, the trnsitionstte theory does not pply nd the GB motion does not hve to follow the exponentil reltions derived in Se. III bsed on this theory. At even higher veloities, the urve ould turn Veloity [m/s] Veloity [m/s] b K PHYSICAL REVIEW B 78, K K 4 K FIG. 12. The GB veloity nd its logrithm b s funtions of the verge sher stress t tempertures 4 nd 9 K. The open squres indite veloities t whih multiple jumps re observed. The dshed line in indites the zero veloity; the solid lines re liner fits. over nd produe regime in whih the veloity dereses with inresing stress. This regime ws indeed found in the reent MD study of opper GB. 17 We emphsize tht, lthough this effet is generi, the veloity rnge in whih it ours depends on the dissiption mehnisms, temperture, grin size nd mny other ftors. A onvining proof of the existene of the dynmi orreltions is the observtion of double jumps of the GB t v 1 m/s. In suh events, illustrted in Fig. 13, the boundry mkes jump by the double mount 2H nd the stress drops to lower level thn it does fter single jump. This hppens beuse the elsti strin energy relesed fter the first jump is lrge enough to immeditely produe nother jump. In ft, t veloities higher thn 15 m/s we sw triple nd even higher multiple jumps. There is n interesting nlogy between suh multiple jumps in oupled GB motion nd the multiple slip events found reently in tomi-sle frition experiments. 22,41,42 At the temperture of 9 K, the GB motion remins perfetly oupled t veloities up to 1 m/s. Above 1 m/s, the oupled motion begins to be interrupted by o

10 V. A. IVANOV AND Y. MISHIN PHYSICAL REVIEW B 78, GB Displement [nm] Time [ns] Time [ns] FIG. 13. Stik-slip GB motion t 4 K nd v=7.5 m/s. The double jumps re indited by rrows. sionl sliding events s indited in Fig. 14 the sliding events re mnifested by nerly horizontl prts of the urve. Between the sliding events, the GB ontinues to move in the oupling mode with As v inreses further, the frequeny of the sliding events lso inreses until t v 3 m/s sliding beomes the dominnt response of the boundry to the pplied sher stress. Although the veloity dependene of sliding ws not studied in this work, these results suggest tht the stress required for sliding is less sensitive to v thn is, produing rossover of the two stresses t bout.15 GP orresponding to v 1 m/s t this temperture. In the veloity rnge of perfet oupling t 9 K, the veloity-stress reltion is overll nonliner but it does exhibit nerly liner prt below v 3 m/s. This liner regime is well-onsistent with the driven Brownin hrter of the GB motion t this temperture Se. V B. The nonliner behvior exhibited t higher veloities mrks trnsition to the stik-slip dynmis, hrterized by rpid roughly, exponentil inrese in veloity with stress. Overll, the stress-veloity reltions found by the MD simultions Fig. 12 ompre well with preditions of the one-dimensionl model disussed in Se. III Fig. 3. GB displement [nm] sliding sliding oupling Time [ns] VI. CONCLUSIONS oupling FIG. 14. Coupled GB motion interrupted by sliding events t 9 K. The imposed grin trnsltion veloity is 12 m/s. We hve nlyzed possible dynmi regimes of oupled GB motion using two different pprohes. The onedimensionl model disussed in Se. III is rude nlog of the omplex multidimensionl proess tking ple during the GB motion, but it n predit simple nlytil reltions between the GB veloity, stress, nd temperture. The MD simultions bring us loser to relity, but the results re only numeril nd the simulted onditions re subjet to the time nd length-sle limittions of the method. As result, only limited re of the prmeter spe n be explored by MD. Despite these differenes, the two methods give qulittively onsistent results tht n be summrized s follows: At low tempertures nd/or high migrtion veloities, oupled GB exhibits stik-slip behvior hrterized by inrementl stop-nd-go motion nd sw-tooth time dependene of the stress. The verge veloity inreses with the verge stress in highly nonliner mnner, lose to exponentil. The GB mkes jumps only forwrd nd stops s soon s the pplied stress is removed. As temperture rises nd/or the veloity slows down, the GB begins to mke osionl reverse jumps nd eventully swithes from the stik-slip regime to driven Brownin motion. The stress-veloity reltion pprohes liner, with oeffiient whih is often lled mobility. 32 When the stress is ompletely removed, the GB ontinues to implement rndom wlk due to therml flututions. We emphsize tht throughout ll these dynmi hnges, the boundry motion still remins perfetly oupled, with the rtio of the norml GB veloity to the grin trnsltion veloity being geometri onstnt. The perfet oupling remins even for the stress-free rndom wlk t high tempertures. Most of the GB migrtion experiments, s well s tomisti simultions, reported in the literture e.g., Refs. 7, 8, nd 43 45, hve been onduted t reltively high tempertures. They typilly disply liner stress-veloity reltion inditive of Brownin dynmis. There re ses, however, when nonliner dynmis were lso observed, but their physil origin nd tomi mehnisms were not investigted

11 DYNAMICS OF GRAIN BOUNDARY MOTION COUPLED In the future, the one-dimensionl model of Se. III n be exmined in greter detil by inluding the effets of inerti nd underdmping, perhps in mnner similr to Refs. 31 nd 48. This might help better understnd the double jumps nd other interesting dynmi effets observed t high veloities. Another model tht we re exploring is the driven Frenkel-Kontorov 49 model. We will lso exmine the grinsize effet on GB dynmis, prtiulrly the effet of the GB re. The ltter ould be importnt sine the GB dimensions in omprison with the ritil nuletion size of disonnetion loops 4 responsible for the oupled motion n ffet the dynmis regimes. PHYSICAL REVIEW B 78, ACKNOWLEDGMENTS We would like to thnk J. W. Chn nd D. A. Molodov for helpful disussions. We re grteful to the Center for Theoretil nd Computtionl Mterils Siene of the Ntionl Institute of Stndrds nd Tehnology for mking their omputtionl luster vilble for this work. This work ws supported by the U.S. Deprtment of Energy Offie of Bsi Energy Sienes, Division of Mterils Sienes. We knowledge useful disussions filitted through oordintion meetings sponsored by the DOE-BES Computtionl Mterils Siene Network CMSN progrm. *vivnov@gmu.edu ymishin@gmu.edu 1 J. W. Chn nd J. E. Tylor, At Mter. 52, A. Suzuki nd Y. Mishin, Mter. Si. Forum 52, J. W. Chn, Y. Mishin, nd A. Suzuki, Philos. Mg. 86, J. W. Chn, Y. Mishin, nd A. Suzuki, At Mter. 54, If the rystl is elstilly nisotropi nd the boundry symmetri, the sher stress n rete volume driving fore due to the differene between the elsti strin energy densities in the grins. This driving fore is qudrti in the stress nd is not onsidered in this pper. The intertion of oupled GBs with stresses is first-order effet, in whih the driving fore is liner in stress. 6 S. G. Srinivsn nd J. W. Chn, in Siene nd Tehnology of Interfes, edited by S. Ankem, C. S. Pnde, I. Ovidko, nd R. Rngnthn TMS, Settle, 22, pp D. Molodov, A. Ivnov, nd G. Gottstein, At Mter. 55, D. Molodov, T. Gorky, nd G. Gottstein, Mter. Si. Forum , M. Winning, G. Gottstein, nd L. S. Shvindlermn, At Mter. 49, M. Winning, G. Gottstein, nd L. S. Shvindlermn, At Mter. 5, M. Winning nd A. D. Rollett, At Mter. 53, M. Winning, Philos. Mg. 87, H. Yoshid, K. Yokoym, N. Shibt, nd Y. I. A. T. Skum, At Mter. 52, J. Monk, B. Hyde, nd D. Frks, J. Mter. Si. 41, K. J. Hemker nd W. N. Shrpe, Annu. Rev. Mter. Res. 37, T. Zhu, J. Li, A. Smnt, H. G. Kim, nd S. Suresh, Pro. Ntl. Ad. Si. U.S.A. 14, Y. Mishin, A. Suzuki, B. P. Uberug, nd A. F. Voter, Phys. Rev. B 75, D. A. Molodov privte ommunition. 19 A. P. Sutton nd R. W. Blluffi, Interfes in Crystlline Mterils Clrendon, Oxford, The supersript in is reminder tht this ritil stress refers to zero temperture. 21 It is implied tht the elsti strin energy relesed t eh inrement of the GB motion is immeditely dissipted nd the temperture omes bk to K. 22 C. M. Mte, G. M. MClellnd, R. Erlndsson, nd S. Ching, Phys. Rev. Lett. 59, E. Gneo, R. Bennewitz, T. Gylog, C. Loppher, M. Bmmerlin, E. Meyer, nd H.-J. Güntherodt, Phys. Rev. Lett. 84, E. Gneo, R. Bennewitz, T. Gylog, nd E. Meyer, J. Phys.: Condens. Mtter 13, R A. Sooliu, E. Gneo, S. Mier, O. Pfeiffer, A. Brtoff, R. Bennewitz, nd E. Meyer, Siene 313, D. Tomnek, W. Zhong, nd H. Thoms, Europhys. Lett. 15, K. Johnson nd J. Woodhouse, Tribol. Lett. 5, One ould ttempt to pproximte the fore-redued brrier by E + E /2 sine in the bsene of fores the energy mximum ours t x=/2. This brrier would go through zero s liner funtion of, whih is inorret. The liner pproximtion of E + is vlid only when is smll but fils when is lrge enough to eliminte the brrier. The reson is tht lso shifts the positions of the equilibrium point x nd the mximum x 1, bringing them loser to eh other until they merge s the brrier tends to zero. This merger of x nd x 1 mkes the derese of E + with slower thn liner, whih is refleted by the power of 3/2. 29 A. Grg, Phys. Rev. B 51, Y. Sng, M. Dube, nd M. Grnt, Phys. Rev. Lett. 87, P. Reimnn nd M. Evstigneev, New J. Phys. 7, G. Gottstein nd L. S. Shvindlermn, Grin Boundry Migrtion in Metls CRC, Bot Rton, Y. Mishin, D. Frks, M. J. Mehl, nd D. A. Pponstntopoulos, Phys. Rev. B 59, J. Morris nd X. Song, J. Chem. Phys. 116, Smithells Metls Referene Book, 8th ed., edited by W. Gle nd T. Totemeier Elsevier, New York/Butterworth, Wshington, DC/ Heinemnn, London, J. Stdler, R. Mikull, nd H. R. Trebin, Int. J. Mod. Phys. C 8, In this pper, ll rystllogrphi indies re given reltive to the lttie of the upper grin. 38 C. L. Kelhner, S. J. Plimpton, nd J. C. Hmilton, Phys. Rev. B

12 V. A. IVANOV AND Y. MISHIN PHYSICAL REVIEW B 78, , It n be shown tht for this GB sin /2 = 1/7 1/2, os /2 = 6/7 1/2 nd thus 2 tn /2 = 2/3 1/2. 4 At high tempertures nd/or lrge veloities, the pek stress beomes noisy nd in some ses nnot be even resolved. 41 Y. Hoshi, T. Kwgishi, nd H. Kwktsu, Jpn. J. Appl. Phys., Prt 1 39, S. N. Medynik, W. K. Liu, I. H. Sung, nd R. W. Crpik, Phys. Rev. Lett. 97, D. A. Molodov, G. Gottstein, F. Heringhus, nd L. S. Shvindlermn, At Mter. 46, V. Ivnov, D. Molodov, L. Shvindlermn, nd G. Gottstein, At Mter. 52, B. Shönfelder, D. Wolf, S. R. Phillpot, nd M. Furtkmp, Interfe Si. 5, H. Zhng, M. I. Mendelev, nd D. J. Srolovitz, At Mter. 52, B. Shönfelder, Ph.D. thesis, IMM, RWTH-University, M. Evstigneev nd P. Reimnn, Phys. Rev. B 73, O. M. Brun nd Y. S. Kivshr, The Frenkel-Kontorov Model: Conepts, Methods, nd Applitions, Theoretil nd Mthemtil Physis Springer-Verlg, Berlin,