Dislocation transmission across the Cu/Ni interface: a hybrid atomistic continuum study

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1 Philoophical Magazine, Vol. 87, No. 10, 1 April 2007, Dilocation tranmiion acro the Cu/Ni interface: a hybrid atomitic continuum tudy M. A. SHEHADEH*y, G. LUy, S. BANERJEEz, N. KIOUSSISy and N. GHONIEMx ydepartment of Phyic and Atronomy, California State Univerity, Northridge, CA , USA zengineering Technology Department, Saint Loui Univerity, Saint Loui, MO 63103, USA xdepartment of Mechanical and Aeropace Engineering, Univerity of California, Lo Angele, CA 90095, USA (Received 12 July 2006; in final form 6 October 2006) The trengthening mechanim in bimetallic Cu/Ni thin layer are invetigated uing a hybrid approach that link the parametric dilocation dynamic method with ab initio calculation. The hybrid approach i an extenion of the Peierl Nabarro (PN) model to bimaterial, where the dilocation preading over the interface i explicitly accounted for. The model take into account all three component of atomic diplacement of the dilocation and utilize the entire generalized tacking fault energy urface (GSFS) to capture the eential feature of dilocation core tructure. Both coherent and incoherent interface are conidered and the lattice reitance of dilocation motion i etimated through the ab initio-determined GSFS. The effect of the mimatch in the elatic propertie, GSFS and lattice parameter on the preading of the dilocation onto the interface and the tranmiion acro the interface are tudied in detail. The hybrid model how that the dilocation diociate into partial in both Cu and Ni, and the dilocation core i queezed near the interface facilitating the preading proce, and leaving an interfacial ledge. The competition of dilocation preading and tranmiion depend on the characteritic of the GSFS of the interface. The trength of the bimaterial can be greatly enhanced by the preading of the glide dilocation, and alo increaed by the pre-exitence of mifit dilocation. In contrat to other available PN model, dilocation core preading in the two diimilar material and on their common interface mut be imultaneouly conidered becaue of the ignificant effect on the tranmiion tre. 1. Introduction The influence of interface on the mechanical propertie of multiphae and polycrytalline material i ubiquitou. The mechanical propertie of an interface are determined, in large part, by the nature of the chemical bonding at the interface, which may be ignificantly different from that within either of the material meeting *Correponding author. mutaem@cun.edu. Philoophical Magazine ISSN print/issn online ß 2007 Taylor & Franci DOI: /

2 1514 M. A. Shehadeh et al. at the interface. The variation of the generalized tacking fault energy urface (GSFS) of the interface, the exitence of mifit dilocation and the lattice mimatch can act a barrier to dilocation motion and tranmiion acro the interface. In recent year, there ha been coniderable interet in the mechanical and tructural propertie and the deformation mechanim of metallic multilayer ytem, which diplay remarkably high mechanical trength and hardne comparable to their theoretical trength [1]. It ha been found experimentally that the hardne and ultimate tenile trength of nano-layered tructure increae with decreaing bilayer thickne, in a relation analogou to the Hall Petch behaviour to ome critical layer thickne. At maller wavelength, the hardne i een to increae more rapidly, with Hall Petch exponent of the order of unity or greater, to ome peak tre value that i much greater than that attainable by traditional microtructure. Thu, multilayer compoed of alternating layer made of metal uch a Cu, Ni, Cr and Nb exhibit peak trength on the order of few GPa at layer thickne of few nanometre [2, 3], compared to the yield trength value of few ten of MPa in bulk Cu, Ni, Cr and Nb. The dramatic enhancement of multilayer trength ha been generally attributed to the following factor: the mimatch in the elatic propertie which reult in image force on the dilocation; the mimatch in the GSFS between incoming and outgoing plane which play a major role in determining the core propertie of the dilocation; the mimatch in the lattice parameter that lead to the generation of coherency tre acro the interface; and the GSFS of the interface which may uppre or enhance the preading of the dilocation core from the glide plane to the interface. Additionally, the exitence of mifit dilocation affect the overall trength of multilayer a a reult of their mutual interaction with glide dilocation. A pronounced ize effect ha been oberved in thin multilayer ytem a different deformation mechanim operate at different length cale. Wherea the behaviour of multilayer can be decribed by a caling law in the ubmicron length cale, deviation from thi caling law occur at the nanocale and the effect of dicrete or even ingle dilocation trengthening mechanim applie [4]. At the nanocale, the trengthening mechanim fall into everal broad categorie. The firt mechanim i the claic Hall Petch model of dilocation pileup [5 7]. The econd mechanim wa introduced by Koehler [8], where the image force impoed by the layer of alternating material retrict the motion of dilocation [9]. The dilocation are attracted toward (repelled from) the interface by the decreae (increae) in line energy a a dilocation move toward the material with lower (higher) elatic contant. Another deformation mechanim involve the formation and propagation of o-called Orowan bow within the layer [10]. From the point of view of theory, there have been approache baed on the continuum elaticity approach for both welded and lipping interface [11]. Recently, Han and Ghoniem [12] have developed a Green function approach for the elatic field of three-dimenional dilocation loop in aniotropic multilayer material. A expected, the image force, which i accurately decribed by linear elaticity [13, 14], diverge in the vicinity of the dilocation core. In order to overcome thi difficulty, an arbitrary cut-off radiu, r 0, i generally introduced but it actual value i highly uncertain. Conequently, important quantitie, uch a the

3 Dilocation tranmiion acro the Cu/Ni interface 1515 critical tre required to make the dilocation cro the interface, are not accurately determined unle r 0 i calibrated with atomitic calculation. On the other hand, molecular dynamic (MD) imulation have been ued extenively to tudy different deformation mechanim in bimaterial. The MD imulation of Hoagland et al. [15] have hown that coherency tree and interface dilocation play a critical role in determining how the layered microtructure behave under an applied load. The MD imulation of Rao and Hazzledine [16] demontrated that a crew dilocation in Cu form interfacial dilocation in the Cu/Ni twinned interface and hence prefer to pread on the interface rather than to tranmit to Ni. Other MD imulation have alo indicated that the intrinic reitance to lip tranmiion for Cu/Nb ytem can originate from the low hear trength of the interface [17]. Although MD imulation are very ueful in revealing the atomitic mechanim for the trengthening effect in multilayer, they uffer from the lack of reliable empirical potential for treating interatomic interaction acro the interface [18], epecially when one conider new material for which empirical interatomic potential are not available. Over the pat few year, the Peierl Nabarro (PN) model combined with ab initio-determined GSFS ha been ued to tudy the dilocation core propertie in bulk material [18 21] and the effect of chemitry on the dilocation core propertie in aluminium [22]. The GSFS repreent the two-dimenional energy profile when the two crytal halve above and below the glide plane are hifted rigidly againt each other by a contant diregitry vector, u. The GSFS-baed approach i eentially a local formulation of the PN model and it aume lowly varying lip ditribution. For a dilocation with a wide core a in the cae of Cu and Ni, the train gradient i relatively mall, and therefore, the local formulation hould give a reaonable decription of the dilocation core tructure. Although a non-local formulation of the PN model ha been propoed [23], it ha not been widely ued due to it complexity. Anderon et al. [24 26] have extended the PN model to invetigate the tranmiion of a crew dilocation acro a coherent, lipping bimaterial interface. Anderon and Li [24] and Shen and Anderon [26] developed a 2D PN model for the tranmiion of a crew dilocation acro a lipping and welded interface. Their model were able to predict everal trend that give more inight to undertanding the trengthening mechanim in bimetallic material. A general obervation made i that lipping interface delocalize and trap the core of an incoming dilocation. Anderon model i limited in two repect: (i) rather than uing an ab initio decription of the atomitic hear on both the glide and interface plane, it employ a imple inuoidal form of the urface which in turn doe not allow diociation, and (ii) the poibility of a pure crew dilocation to diociate with a Burger vector containing an edge component i neglected. The preent model i imilar to that of Anderon and Li, and make ue of ab initio and parametric dilocation dynamic (PDD) [27] computation to reolve thee two iue. We have developed an extenion of the PN model which integrate the atomitic nature from ab initio electronic tructure calculation into the PDD method [28]. The GSFS provide a two-dimenional repreentation of the tacking fault energy at zero temperature. Both coherent and incoherent interface are conidered and the

4 1516 M. A. Shehadeh et al. lattice reitance of dilocation motion i captured through the ab initio-determined GSFS. In thi tudy, the core propertie of a pure crew dilocation a it move from Cu to Ni are invetigated. The effect of the GSFS of the interface on dilocation core preading and on the tranmiion tre i alo determined. Additionally, the effect of pre-exiting mifit dilocation on interfacial trength and dilocation core preading i invetigated. 2. PN model for the Cu/Ni interface The PN model ued in thi work i an extenion to the parametric dilocation dynamic model for bulk material developed by Banerjee et al. [28]. In thi approach, the full dilocation i repreented by a et of N fractional Volterra dilocation with fractional Burger vector db ¼ b/n. A right handed coordinate ytem a hown in figure 1a i ued. The dilocation line i choen to be along the z direction. In the cae of a pure crew dilocation a ufficient amount of edge component i added by introducing N/2 poitive and N/2 negative fractional edge (ee figure 1b). In contrat to the treatment of Anderon, the atomic diplacement of the dilocation have component in all three direction rather than only along the direction of the Burger vector. The diplacement component of the lip u x and u z are determined from the poition of the fractional dilocation of edge and crew type with fractional Burger vector. The equilibrium tructure of the dilocation core i obtained by eeking the equilibrium configuration of thee fractional dilocation. Phyically, thi correpond to balancing the elatic force and the lattice retoring force for each infiniteimal dilocation acro the glide plane. For a mixed dilocation with component (b e, b ) the diplacement component can be approximated a u e ðxþ ¼u x ¼ XN i¼1 u ðxþ ¼u z ¼ XN i 0 ¼1 b i e tan 1 ðx x i Þþ b e 2 b i0 tan 1 ðx x i 0Þþ b 2 where b i e ¼ (b e/n), b i0 ¼ (b /N) are the Burger vector of the fractional dilocation, N i the total number of dilocation of edge and crew type, and x i, x i 0 are the correponding poition. The net elatic force reulting from the interaction between thee fractional dilocation i balanced againt the lattice retoring force derived from the GSFS acro the glide plane. In thi formulation, the GSFS of Cu, Ni, and the Cu/Ni interface are calculated uing firt-principle ab initio method. The PN model ha been modified to invetigate the trengthening mechanim in lipping and rigid Cu/Ni bimaterial. The effect of the mimatch in the elatic propertie, the mimatch in the GSFS, the mimatch in the lattice parameter and the exitence of mifit dilocation are explicitly taken into account. ð1þ ð2þ

5 Dilocation tranmiion acro the Cu/Ni interface 1517 Figure 1. (a) The Carteian coordinate ytem ued in the imulation; (b) chematic howing the idea of adding extra poitive and negative edge component for pure crew dilocation; (c) chematic of the dicretized Peierl crew dilocation during the tranmiion proce from Cu to Ni. The Burger vector and the line ene of the dilocation are along the z direction. The fractional Burger vector in Cu i b Cu /N, inniib Ni /N and on the interface b Ni /2N þ (b Cu b Ni )/2N. For a lipping interface, part of the dilocation content can be accommodated by the interface through dilocation core preading. Any dilocation that move from the glide plane to the interface i divided into two identical fractional dilocation, ymmetrically placed with repect to the lip plane. In particular, the mimatch in lattice contant between Cu and Ni i accommodated by the fractional reidual lip on the interface. The continuity of the Burger vector require that: b Cu ¼ n Cu db Cu þ n Ni db Ni þ 2n Int db Int, where N ¼ n Cu þ n Int þ n Ni, n Cu, n Ni, n Int are the number of fractional dilocation in Cu, Ni, and interface, repectively, and db Cu ¼ b Cu /N, db Ni ¼ b Ni /N, and db Inti ¼ b Cu /2N. For the Cu/Ni bimaterial problem, the equilibrium condition of a fractional dilocation i depend on whether the fractional dilocation i in Cu, Ni or ð3þ

6 1518 M. A. Shehadeh et al. on the interface. The total force on a fractional crew dilocation i can be expreed a: " # F T i Cu ¼ ext þ Xn Cu Cu Cu ðx i,x j Þþ X2n int Cu Int ðx i,y j Þþ Xn Ni Cu Ni ðx i,x j Þ b icu ð4þ þ f i Cu þ f Coh i Cu " # Int Cu ðy i,x j Þþ X2n int Int Int ðy i,y j Þþ Xn Ni Int Ni ðy i,x j Þ F T i Int ¼ ext Int þ Xn Cu þ f i Int þ f Coh i Int " # F T i Ni ¼ ext þ Xn Cu þ f i Ni þ f Coh i Ni Ni Cu ðx i,x j Þþ X2n int Ni Int ðx i,y j Þþ Xn Ni Ni Ni ðx i,x j Þ where x i and y i are the poition of the fractional dilocation on the glide and the interface plane, repectively. The firt term in each equation ( ext ) i the externally applied tre along the dilocation line; the econd, third and fourth term are the tre exerted on the fractional dilocation from other fractional dilocation located in Cu, on the interface, and in Ni, repectively; the fifth term i the lattice retoring force derived from the GSFS, and the lat term i the coherency tre. The expreion for the tre in the above equation are normalized with repect to ¼ ( Cu þ Ni )/2. Note that the fourth term in Þ and the econd term in equation (6) ð Ni Cui Þ repreent the image tree (Koehler tree) reulting from the mimatch in the elatic propertie between Cu and Ni. The explicit expreion of the image tre for the crew and edge component are given by Anderon and Li [24] and Head [14], repectively. In our formulation we aume that there i no dilocation climb and therefore only crew dilocation can pread out onto the interface. Additionally, the external tre and coherency tre are applied only to the crew component on the glide the average hear modulu equation (4) ð Cu Ni ¼ 0. The expreion for the net force on a fractional edge dilocation i i imilar to thoe for a fractional crew dilocation in equation (4) and (6), but with the following difference: no external tre i applied on the edge component and there i no tre contribution from the interface becaue the fractional edge dilocation i confined to the glide plane. The edge and crew component of the lattice retoring force, obtained from the GSFS are given by plane; i.e. ext Int ¼ 0 and f Coh i Int f e i ¼ b e x¼xi f i ¼ x¼xi b iint b ini ð5þ ð6þ ð7þ ð8þ where i the GSFS.

7 Dilocation tranmiion acro the Cu/Ni interface 1519 Table 1. Material propertie of Cu and Ni ued in the imulation: i the hear modulu; b i the Burger vector; int and un are the intrinic and untable tacking fault energie, repectively; and B i the drag coefficient. Material (GPa) b (A ) int (mj m 2 ) un (mj m 2 ) B (N m 1 ) Ni Cu The force due to the coherency tre reult from the mimatch in the Cu/Ni lattice parameter. The calculated uniaxial coherency tre in Cu/Ni bimaterial i around 2.5 GPa [29], which i equivalent to 1.0 GPa when reolved on the glide plane. In thi work the coherency tre i aumed to be of a tep function centered at the interface with the value i Coh ¼1.0 GPa. The plu (minu) ign indicate compreive (tenile) tre in Cu (Ni). It i worth mentioning that the geometry ued in the current work aume that the interfacial (100) plane i perpendicular to the glide plane (111) to avoid additional computational complexitie. However, in reality the glide plane i inclined with repect to the interfacial plane by an angle ¼ 54.5, which in turn lead to aymmetric dilocation core preading due to the modification of the image force. In thi quaitatic framework, the dilocation glide i controlled by drag, and hence the dilocation velocity i proportional to the reolved hear tre. Having determined all the force component, the equilibrium poition of each fractional dilocation i computed according to the drag dilocation dynamic relation, F T i ¼ B dx i dt, ð9þ where B i a drag coefficient to update the poition x i of the dilocation at each time tep until the ytem reache equilibrium (ee table 1). The interfacial trength i determined by computing the critical value of the external tre required to tranmit the dilocation from Cu to Ni. The external tre i applied incrementally to puh all fractional dilocation acro the interface. When the leading dilocation reache a critical ditance of 0.04b Cu from the interface, two pathway can be taken by the dilocation: it can either pread onto the interface or continue gliding on the original lip plane into Ni. The pathway with a lower energy will be taken by the dilocation. The proce i repeated for all fractional dilocation while the applied tre keep increaing until it reache a critical value that allow dilocation tranmiion. Thi critical value i defined a the tranmiion tre critical. 3. Reult and dicuion 3.1. Generalized tacking fault energy urface The Cu/Ni bimaterial ytem i modelled a two emi-infinite homogenou and iotropic region connected at the interface, a hown in figure 1c. The glide plane of

8 1520 M. A. Shehadeh et al. Figure 2. Ab initio tacking fault energie for Ni and Cu along the [112] direction and for the interface along [100] direction. Cu and Ni are aumed to be coplanar and normal to the interface. A pure crew dilocation of Burger vector 1/2 [110] i placed in Cu (oft material) gliding in (111) plane. The electronic tructure calculation of the GSFS were done uing the projector augmented-wave (PAW) method [30] a implemented in the VASP code [31, 32]. The ab initio-determined GSFS projected along the [121] direction for the pure Cu and Ni and along the [001] direction for the Cu/Ni interface are hown in figure 2. The firt energy maximum encountered along the [121] direction for the Cu and Ni i the untable tacking fault energy ( un ) which repreent the lowet energy barrier to nucleate a dilocation from a crack tip. The firt energy maximum encountered along the [121] direction for the Cu and Ni i the untable tacking fault energy which repreent the lowet energy barrier to nucleate a dilocation from a crack tip at 0 K. At finite temperature, the effective energy barrier for dilocation nucleation will be reduced by both thermal excitation and the fact that dilocation nucleu may take a three-dimenional hape. The local minimum on the other hand, correpond to the intrinic tacking fault energie ( intrin ). The calculated value of un are 225 mj m 2 and 350 mj m 2 for Cu and Ni, repectively, wherea the value of intrin for Cu and Ni are 53 mj m 2 and 163 mj m 2, repectively. Thee value are in good agreement with other calculation [33]. A expected, the GSFS of the interface i ymmetric along the [110] direction and it ha untable tacking fault energy of 730 mj m 2, which i much higher than the correponding value of Ni and Cu. The abence of a addle point in the GSFS of the interface ugget that the full dilocation on the interface doe not diociate. The hear modulu, Burger vector and the energy value for the variou tacking fault energie for Ni and Cu are lited in table 1.

9 Dilocation tranmiion acro the Cu/Ni interface Diplacement and denity profile The large increae in the mechanical trength of nano-layered material i widely attributed to the preence of interface. Several factor can affect the mechanical and phyical propertie of the interface uch a: the untable tacking fault energy of the interface, int, which i a meaure of the propenity of interfacial liding and which i directly related to the electron charge bonding acro the interface, the preence of mifit dilocation, and the preence of impuritie. The maller int i the eaier it i for the interface to lide, thu allowing the dilocation to pread onto the interface. Figure 3 how the equilibrium edge and crew diplacement and the correponding Burger vector denity (x) of the dilocation for three value of applied tre. The edge and crew Burger vector denity are defined by e (x) ¼ du e /dx, and (x) ¼ du /dx. The crew dilocation, originally placed in the oft material (x40) i puhed toward the Cu/Ni interface. For relatively low value of applied tre (around 2.0 GPa), the dilocation core in Cu diociate into two partial bounding a tacking fault with a eparation ditance of about 7b (figure 3a). A the external tre increae, the dilocation approache the interface but remain diociated. However; the dilocation core tructure ha changed ignificantly. Firt, the dilocation Burger vector denity accumulate on the leading partial at the expene of the trailing partial (figure 3b and 3c). Second, the dilocation core contrict teadily and the two partial become ignificantly overlapped (figure 3c). Note, that the maximum value of the crew component of the diplacement in Cu i 2.35 A, wherea the Burger vector of Cu i 2.6 A. Thi reduction of Burger vector i a reult of the energetically favourable preading of the core onto the interface. Our reult ugget that the dilocation preading proce proceed via the following mechanim: when the leading fractional dilocation reache the vicinity of the interface it pread on it, if it i energetically favourable. A the external tre i increaed, the trailing fractional dilocation follow and pread onto the interface. The preading proce continue until the interface can no longer accommodate additional lip. At the critical value of the applied tre, once the leading dilocation on the glide plane overcome the interfacial barrier and i tranmitted to the Ni crytal, the remaining fractional dilocation follow. In figure 4 we how the time evolution of the diplacement u(x) and the Burger vector denity profile (x) and e (x) of the dilocation when the applied tre ha reached it critical value of 3.35 GPa. At the initial tage of the dilocation tranmiion proce, mot of the fractional dilocation are localized in the vicinity of the interface in the Cu hot (figure 4a). A the fractional dilocation relax, they get tranmitted through the interface toward the Ni hot till all of them pa. Note that after the dilocation get tranmitted, the denity profile how the formation of two partial with a eparation ditance of about 6b (figure 4c). The peak in the denity profile at x ¼ 0 indicate the formation of a ledge on the interface, in agreement with MD imulation for edge or mixed dilocation for the Cu/Ni interface [16] Effect of untable tacking fault of the interface Interface can be coherent, emi-coherent or fully non-coherent. In the cae of fully non-coherent tructure, dilocation motion i retricted to individual layer [9],

10 1522 M. A. Shehadeh et al. Figure 3. The diplacement u(x) and denity (x) profile for the dilocation a it move from Cu toward Ni. The GSFS of the interface i equal to the ab initio value ð int ¼ int abþ. The profile how the equilibrium poition of the dilocation at (a) 2.0 GPa, (b) 2.8 GPa and (c) 3.30 GPa. The continuou line repreent the crew or the edge diplacement and the dotted line repreent the correponding denitie.

11 Dilocation tranmiion acro the Cu/Ni interface 1523 Figure 4. The dynamic evolution of the dilocation during the tranmiion proce. The critical tre for penetration i 3.35 GPa and a the imulation time pae, the dilocation penetrate to Ni. i.e. the interface act a a dilocation ink. For a emi-coherent interface, a in the cae of Cu/Ni, dilocation tranmiion acro the interface i poible depending on the tacking fault energy of the interface. In order to explore the effect of interfacial liding on the critical tre required for dilocation tranmiion, we have varied the value of int with repect to it ab initio value int ab ¼ 730 mj m 2. Figure 5 diplay the dilocation diplacement profile for the crew component on the glide plane, u (x), and on the interface plane, u (y), for different value of the ratio, int =int ab, of 1.2 (hard interface), 1.0 and 0.8 (oft interface). In each panel we how alo the reult for the diplacement profile for variou applied tre value. A expected, a the interfacial energy barrier for liding i reduced and the interface become le bonded, the percentage of the dilocation preading on the interface increae from about 13% for int ¼ 1:2int ab to 30% for int ¼ 0:8int ab. The preading of fractional dilocation on the interface impoe an extra energy barrier on the tranmiion of the glide dilocation. The extra barrier i due to the repulive elatic interaction between the glide and interfacial fractional dilocation and the formation of the interfacial ledge which hinder the tranmiion proce from Cu to Ni. Conequently, the critical value of applied tre for dilocation tranmiion increae (decreae) to the value of 4.2 GPa (3.2 GPa) a the interfacial GSFS decreae (increae) compared to it correponding ab initio value. The ledge formation i partially due to the accommodation of the mimatch in Burger vector between Cu and Ni which i about 0.10 A. A cloer examination of figure 5b, however, how that the crew diplacement profile a the dilocation tranmit from Cu to Ni ha a tep at the interface whoe height i 0.25 Å, which i larger than the lattice contant mimatch of 0.1 A.

12 1524 M. A. Shehadeh et al. Figure 5. Snaphot of the diplacement and the denity profile along the glide plane, u (x) and (x), and along the interface plane, u (y), for different value of the GSFS of the interface (a) int ¼ 0:8int ab, (b) int ¼ int ab and (c) int ¼ 1:2int ab.

13 Dilocation tranmiion acro the Cu/Ni interface 1525 Figure 6. The diplacement and the denity profile along the glide plane, u (x) and (x), and along the interface plane, u (y), for a pure crew dilocation, with and without the aociated edge component Effect of dilocation plitting Previou PN-baed imulation [24 26] did not take into conideration the development of the edge component of a pure crew dilocation upon diociation. Thi poibility reult in dilocation plitting into a partial with mixed Burger vector, with an overall reduction of the energy, a compared to diociation along crew component alone, which render the dilocation with higher mifit energy. In order to explore the effect of the dilocation plitting on the preading proce, we have carried out everal imulation in which we diallow the dilocation to plit by conidering the crew diplacement only (i.e. removing the edge component aociated with each crew dilocation). The reult are then compared with our previou reult in figure 3 5, where the edge component wa taken into account explicitly. Figure 6 how the crew diplacement and denity profile on the glide plane, u (x) and y x), and on the interface plane, u (y), with and without the edge component. The removal of the edge component change the dilocation core tructure coniderably. Firt, the dilocation core doe not diociate. Second, the dilocation core become much narrower a reflected on the dramatic increae in the magnitude of the Burger vector denity. Finally, more dilocation core preading onto the interface take place which in turn lead to a ignificant increae in the tranmiion tre. By removing the edge component, the critical tre i doubled compared to the cae where the edge component i taken into account explicitly. The reult ugget that a contricted dilocation core can pread onto the interface much eaier, which i analogou to the cro lip mechanim [21].

14 1526 M. A. Shehadeh et al. Figure 7. Effect of pre-exiting mifit dilocation on the diplacement and the denity profile for a crew dilocation. The diplacement and denity profile after dilocation tranmiion are hown on the glide plane, u (x) and (x), in the top panel, and on the interfacial plane, u (y), in the lower panel, in the preence and abence of pre-exiting mifit dilocation, repectively Effect of pre-exiting mifit dilocation Interfacial mifit dilocation, interacting with the applied tre, and incoming glide dilocation can be a potent barrier to lip tranmiion [15]. In thi ection we explore the effect of pre-exiting mifit dilocation on the dilocation core propertie and on the tranmiion tre. Two imulation are carried out in the abence and the preence of pre-exiting mifit dilocation a follow:. Initially, the interface ha zero diplacement content and there i only a ingle glide dilocation on the lip plane having a maximum diplacement of 2.6 A. After the dilocation tranmit from Cu to Ni a hown in figure 6, it leave part of it diplacement on the interface. Therefore, the original diplacement on the glide plane i reduced from 2.6 A to 2.35 A, wherea that on the interface i increaed from 0 to 0.25 Å.. Initially, we conider both a glide and pre-exiting mifit dilocation on the lip and the interfacial plane, repectively. The maximum diplacement content on both the interfacial and glide plane are equal (2.6 A ). The fractional dilocation on the interface are allowed to interact with the glide fractional dilocation, where the glide fractional dilocation are allowed to pread on the interfacial plane and vice vera. After the dilocation tranmiion a hown in figure 6, the diplacement content on the glide and interfacial plane

15 Dilocation tranmiion acro the Cu/Ni interface 1527 Figure 8. The variation of the critical hear tre with the GSFS of the interface. Rigid interface doe not accommodate core preading and therefore it i not affected by the change in interfacial GSFS. The removal of the edge component lead to a ignificant increae in the tranmiion tre for the lipping interface cae and minor increae for the rigid interface cae. do not change from their initial value, indicating that the preence of a pre-exiting dilocation prevent any preading from or to the glide plane. The crew diplacement and denity profile on the glide plane, u (x) and (x), and on the interfacial plane, u (y) after the glide dilocation ha been tranmitted are hown in figure 7, in the abence (dotted curve) and the preence (olid curve) of the mifit interfacial dilocation. The reult indicate that the tranmiion tre increae dramatically compared to it correponding value in the abence of the mifit dilocation. Note, that there i no ledge formation on the interface when the mifit dilocation i preent. Figure 8 how the critical tranmiion tre veru the ratio of int with repect to it ab initio value for the cae of a non-lipping (rigid) and a lipping interface, with and without the edge component taken into account. The number in parenthei indicate the percentage dilocation content on the interface for the cae of lipping interface with the edge component taken into account. A expected, the critical tranmiion tre i independent of int for the cae of a rigid interface, and it value i higher when the edge component i neglected in the imulation. For the cae of a lipping interface, i.e. when int <int ab, the interface allow more dilocation content to pread from the glide plane to the interface. Thi in turn lead to a dramatic increae in the critical tre for tranmiion. For example, the critical tre i increaed by a factor of three for int ¼ 0:60int ab, compared to the correponding value at int ¼ int ab. On the other hand, when int >int ab, the GSFS of the interface increae, the interfacial lipping become more difficult and the critical tre

16 1528 M. A. Shehadeh et al. decreae more lowly aturating to a value of about 2.8 GPa. Thu, thee reult clearly demontrate that the increae of tranmiion tre i directly related to the preading proce on the interface. Note that the percentage of dilocation content on the interface increae from about 10% at int ¼ 1:50int ab to 98% at int ¼ 0:60int ab. A imilar trend of the critical tranmiion tre a a function of the interfacial energy barrier i alo found for the cae of lipping interface without taking into account the edge component (plain circle), but with higher value of critical tre. Our calculation how for the firt time that the removal of the edge component reult in dramatically different value for the tranmiion tre compared to thoe if the diociation i included. 4. Concluion We have preented a hybrid approach to tudy the dilocation tranmiion/ preading in both coherent and emi-coherent Cu/Ni bimaterial. Thi approach combine the parametric dilocation dynamic baed of the PN framework with ab initio-determined GSFS. The model take into account all three component of atomic diplacement of the dilocation and utilize the entire GSFS to reveal outtanding feature of dilocation diociation. The effect of the mimatch in the elatic propertie, gamma urface, and mifit dilocation on the preading of the dilocation at the interface and on the tranmiion acro the interface are accounted for. We are able to reproduce everal MD imulation trend and make further prediction about the trength of Cu/Ni laminate, without the reliance on empirical potential. Our calculation how that the dilocation diociate into partial in both Cu and Ni. The dilocation core i queezed near the interface facilitating the preading proce, and leaving an interfacial ledge during the tranmiion proce. The dependence of the critical tranmiion tre on the dilocation preading/tranmiion i examined. The competition of dilocation preading and tranmiion depend on the GSFS of the interface. It i found that the decreae of the interfacial GSFS enhance core preading which in turn increae the tranmiion tre. Moreover, it i found that the trength the bimaterial can be ignificantly enhanced by the preence of pre-exiting mifit dilocation. In contrat to other available PN model, it i hown that dilocation core preading in the two diimilar material and on their common interface mut be imultaneouly conidered becaue of the ignificant effect on the tranmiion tre. Acknowledgement The author are very grateful to R. Hoagland for hi helpful comment. Thi reearch wa performed under the aupice of the United State Air force Office of Scientific Reearch (AFOSR) grant number F and the US Army Reearch Office (ARO) under grant No. W911NF G. Lu acknowledge the upport from the ACS Petroleum Reearch Fund.

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