Interface dislocation patterns and dislocation nucleation in face-centered-cubic and body-centered-cubic bicrystal interfaces

International Journal of Plasticity xxx (2013) xxx–xxx

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International Journal of Plasticity

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Interface dislocation patterns and dislocation nucleationin face-centered-cubic and body-centered-cubic bicrystalinterfaces

0749-6419/$ - see front matter Published by Elsevier Ltd.http://dx.doi.org/10.1016/j.ijplas.2013.07.002

⇑ Corresponding author. Tel.: +1 505 667 1238.E-mail address: [email protected] (J. Wang).

Please cite this article in press as: Wang, J., et al. Interface dislocation patterns and dislocation nucleation in face-centered-cubic ancentered-cubic bicrystal interfaces. Int. J. Plasticity (2013), http://dx.doi.org/10.1016/j.ijplas.2013.07.002

J. Wang a,⇑, R.F. Zhang b, C.Z. Zhou d, I.J. Beyerlein b, A. Misra c

a MST Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USAb Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USAc MPA-CINT, Los Alamos National Laboratory, Los Alamos, NM 87545, USAd Department of Materials Science and Engineering, Missouri University of Science and Technology, Rolla, MO 65409, USA

a r t i c l e i n f o a b s t r a c t

Article history:Received 29 April 2013Received in final revised form 30 June 2013Available online xxxx

Keywords:InterfaceDislocationFrank–BilbyAtomistic simulation

Nanolayered metallic composites exhibit unusual high strength at the layer thickness innanometers. Plastic deformation including nucleation, glide, and transmission of disloca-tions is strongly related to interface structure and properties. Combining atomistic simula-tions with the classical Frank–Bilby theory, we studied dislocation structures of semi-coherent interfaces between face-centered-cubic (fcc) and body-centered-cubic (bcc) crys-tals. An atomically informed Frank–Bilby theory is proposed for quantitative analysis ofinterface dislocations. The results showed that (1) seven sets of interface dislocations arepresent in the Nishiyama–Wasserman (NW) interface and two sets of interface dislocationin the Kurdjumov–Sachs (KS) interface although they are misoriented by only �5.6�; (2)Burgers vectors of interface dislocations can be well defined in a commensurate/coherentdichromatic pattern (CDP) lattice corresponding to the NW interface and the Rotation CDP(RCDP) lattice corresponding to the KS interface; (3) the CDP and RCDP lattices are not sim-ply a geometric average of the two natural lattices; finally we demonstrated that (4) thenucleation of dislocations, including interface dislocation loops corresponding to interfacesliding and lattice dislocation loops corresponding to plastic deformation in crystals, arestrongly correlated with interface dislocation patterns.

Published by Elsevier Ltd.

1. Introduction

Grain and interphase boundaries in metals can act as sources, sinks, barriers, and storage sites for point and line defects(Sutton and Balluffi, 1995; Howe, 1997; Beyerlein et al., 2012a; Demkowicz et al., 2008b; Wang and Misra, 2011; Wang et al.,2011b). Nanolayered metallic composites take the advantage of the role of interfaces, and exhibit unusual high strength atthe layer thickness in nanometers. Experimental, theoretical, and modeling works have demonstrated that plastic deforma-tion including nucleation, glide, and transmission of dislocations, referred to as interface-mediated plasticity, is strongly re-lated to interface structure and properties (Wang and Misra, 2011; Zbib et al., 2011; Shen and Anderson, 2006, 2007;Anderson and Li, 2001; Chu et al., 2013). These processes can result in the instability of interface structures, the migrationand sliding of interfaces, deformation twinning, shear bands, and fracture (Zheng et al., 2012; Han et al., 2012; Vo et al.,2012; Wang et al., 2010a,b, 2011a, 2012a,b; Beyerlein et al., 2012b; Li et al., 2012; Bieler et al., 2009). Their activities areheavily dependent on the orientation relation and dislocation structures of interfaces. Molecular dynamics (MD) simulations

d body-

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Fig. 1. High-resolution transmission electronic microscopic image of Copper–Niobium multilayers shows the co-existence of two thermodynamicallypreferred interfaces that adopt Kurdjumov–Sachs (KS) or Nishiyama–Wasserman (NW) orientation relationships.

2 J. Wang et al. / International Journal of Plasticity xxx (2013) xxx–xxx

using reliable empirical interatomic potentials are capable of studying the kinetic processes of each deformation event ormultiple deformation modes, as well as the influences of interface structure, stress, temperature on deformation processes;and have shown that interface-mediated plasticity is strongly connected to characteristics of interface structures, i.e., theinterfacial dislocation network (Zhang et al., 2011, 2012, 2013; Wang et al., 2011d; Demkowicz et al., 2008a,b; Li et al., 2013).

For semi-coherent interfaces, interface dislocations (IDs) are required in the interface to remove incompatibilities and/orthe dissimilarities between the two lattices. They act as stress annihilators to free the stresses far from the interfaces (Hirthet al., 2013). Frank (1953) and Bilby et al. (1955) provided a theory in which the net Burgers vector content BðpÞ of the stressgenerators crossing a probe vector p lying in the interface could be determined. The O-lattice (Bollmann, 1970), which maybe regarded as a quantized form of the Frank–Bilby (FB) theory, was developed to deal with the Burgers vectors content BðpÞin a discrete fashion rather than continuous distribution (Bollmann, 1962, 1967a,b; Hirth and Lothe, 1982; Hirth et al., 1996).The principle is based on the concept of ‘‘well matching’’, i.e., optimal structures are produced in interfaces in regions cen-tered on points where the two lattices adjoining the interface are ‘‘well matched’’ (Sutton and Balluffi, 1995). However, boththeories (FB and O-Lattice) were developed with respect to geometry considerations and have one major weak point corre-sponding to the physical assumption that determines the choice of the reference lattice. Moreover, even if the reference lat-tice can be reasonably chosen, several sets of interface dislocations can be predicted by using the FB theory and O-latticetheory (Hirth et al., 2013). The predicted dislocation patterns and their characters may or may not agree with experimentalobservation although there have been many successful examples (Fujita, 1977; Wang et al., 2009, 2010a,b; Gleiter, 1982;Goodhew et al., 1976; Kluge-Weiss and Gleiter, 1978; Forwood and Clarebrough, 1991; Goodhew and Balluffi, 1980). Wehave developed an approach that combines atomistic simulations with the FB theory to characterize interface dislocations.For a relaxed interface, the disregistry analysis and/or the relative displacement analysis of the relaxed structure with re-spect to the un-relaxed structure can provide insights in understanding the relaxation processes (Kang et al., 2012a,b; Wanget al., 2013). Consequently, the coherent regions presented within the interface can be identified from the relaxed interface.Thus, the regions separating the coherent regions correspond to interface defects, showing that the line sense and the spac-ing can be determined from atomistic simulations (Wang et al., 2013). The Burgers vectors and reference lattice can be ob-tained by solving FB theory while satisfying MD analysis (Hirth et al., 2013).

The most commonly occurring face-centered-cubic/body-centered-cubic (fcc/bcc) interfaces adopt Kurdjumov–Sachs(KS) or Nishiyama–Wasserman (NW) orientation relationships (Hall et al., 1972; Misra and Thilly, 2010). These are thermo-dynamically preferred interfaces formed by joining the crystallographic planes, fcc {111} and bcc {110}, as observed in mul-tilayers fabricated by physical vapor deposition (Fig. 1). Many atomistic simulations with empirical interatomic potentials(Johnson and Oh, 1989; Liu et al., 2010) have been performed to study the structures, properties and dislocation–interfaceinteractions for KS interfaces (Wang et al., 2008a,b,c, 2011c; Mastorakos et al., 2009; Akasheh et al., 2007; Overman et al.,2009). However, dislocation networks of KS interface reported earlier (Demkowicz et al., 2008a), satisfying the FB theory,seem not to represent the relaxed atomic structures, because one of the two sets of interface dislocations characterized crossthe coherent region. In this paper, we adopted an atomistically informed Frank–Bilby (AIFB) theory to study dislocation net-works for KS and NW interfaces and the correlation of the networks with interface-mediated plasticity.

2. Atomically informed Frank–Bilby theory

Atomistic simulations can provide the details of relaxed atomic structures of an interface, which can serve as the evidencein determining the physical solution obtained from the FB theory. Atomistic simulations include two stages, (a) interfacerelaxation and (b) defect characterization. The bicrystal simulation model is employed with the periodic in-plane x and zdimensions, and then relaxed at zero temperature by quenching molecular dynamics with the embedded atom method(EAM) potentials. Details can be found in Wang et al. (2008a,b,c, 2011c). To characterize the interface dislocations, three

Please cite this article in press as: Wang, J., et al. Interface dislocation patterns and dislocation nucleation in face-centered-cubic and body-centered-cubic bicrystal interfaces. Int. J. Plasticity (2013), http://dx.doi.org/10.1016/j.ijplas.2013.07.002


J. Wang et al. / International Journal of Plasticity xxx (2013) xxx–xxx 3

structures are required in atomically informed Frank–Bilby theory, including the un-relaxed structure (or a nature dichro-matic pattern (NDP)), the relaxed structure (or real bicrystal interface), and the CDP structure (a commensurate/coherentdichromatic pattern). A CDP is constructed by applying the biaxial displacements to two crystals with opposite signs. Fromenergy viewpoint, the minimum displacements are desired in choosing the CDP. For the classical FB theory, NDP and CDPare required. However, the discrete dislocations cannot be well defined by the two structures. To characterize interfacedislocations with the AIFB theory, we first construct a trial reference lattice. With a trial reference lattice, the disregistryanalysis is performed to characterize interface defects. In the disregistry analysis, pairs of atoms that straddle the desiredinterface plane are first identified in the un-relaxed interface. Three sets of vectors with respect to three structures (un-relaxed structure (or NDP), relaxed structure (or real bicrystal interface), and CDP structure) are calculated with the fol-lowing identified pairs: rCDP

ij describes the relative displacements of the i–j pair in the reference lattice, rUnRij describes the

relative displacements of the i–j pair in the un-relaxed interface (referred to be the natural dichromatic pattern, NDP), andrRlx

ij describes the relative displacements of the i–j pair in the relaxed, equilibrium interface. The vector rU�Cij ¼ rUnR

ij � rCDPij

describes the relative displacements corresponding to the distortion from a NDP to a CDP. Along a given probe vector,it will define the uniformly continuous distribution of interface Burgers vectors. The vector rR�C

ij ¼ rRlxij � rCDP

ij describesthe relative displacements corresponding to the distortion from the relaxed interface to a CDP. Along a given probe vector,it will define the non-uniformly continuous distribution of interface Burgers vectors along the probe vector. The net vectorrU�C

ij is equal to the interface Burgers vectors. The vector rR�Uij ¼ rRlx

ij � rUnRij describes the relative displacements correspond-

ing to the distortion from the relaxed interface to a NDP. It defines the non-uniform relaxation displacement due to theformation of discrete interface dislocations. Within a periodic length of the BUC, the net vector rR�U

ij along the probe vectoris zero. The displacements in the left and the right of the dislocation line have opposite sign. Thus, the two vector fieldsrR�C

ij and rR�Uij along the probe vector will provide the necessary information for determining the position of discrete inter-

face dislocations in a real bicrystal interface. In principle, the probe vector can be arbitrary. However, it is better to choosea probe vector that is not aligned with a discrete interface dislocation, because the net vectors rR�U

ij will be affected by thediscrete interface dislocation.

FB theory determines the Burgers vector content BðpÞ of an equilibrium interface in a bicrystal (or a real bicrystal inter-face). The lattices A and B of an interface can be related to a reference lattice by homogeneous distortion transformationmatrices SA and SB. For an atomically flat interface, the transformation is represented by 2 � 2 matrices and transforms vec-tors in lattices A and B to the reference lattice. Let p be a probe vector in the interface where the vector must be large com-pared with any substructure within the interface. The BðpÞ crossing p can be calculated as

Fig. 2.NishiyaorientaNDPs. Tdescribweb ve

Pleasecenter

BðpÞ ¼ ðS�1B � S�1

A Þp ð1Þ

(a) A commensurate/coherent dichromatic pattern (CDP) and natural dichromatic patterns (NDPs) between Cu and Nb interface that adopts thema–Wasserman (NW) orientation relationship. (b) The Rotation CDP (RCDP) and the corresponding NDPs with respect to the Kurdjumov–Sachs (KS)tion relationship. The black solid circles outline a unit cell of Nb crystal, and the blue empty circles outline a unit cell of Cu crystal. They form thehe mixed circles outline the CDP and RCDP. Possible Burgers vectors bI and bI

R are denoted in the CDP/RCDP lattices. Three variables are employed toe the CDP and RCDP lattices with respect to the NDPs. (For interpretation of the references to color in this figure legend, the reader is referred to thersion of this article.)

cite this article in press as: Wang, J., et al. Interface dislocation patterns and dislocation nucleation in face-centered-cubic and body-ed-cubic bicrystal interfaces. Int. J. Plasticity (2013), http://dx.doi.org/10.1016/j.ijplas.2013.07.002



where S�1A and S�1

B are the inverse matrices of SA and SB. Following Knowles’ suggestion (Knowles, 1982) that the discreteBurgers vectors of interface dislocations are the lattice vectors in the reference lattice, BðpÞ can be expressed in the referencelattice as

Fig. 3.relation½1�10�Cu

Pleasecenter

BðpÞ ¼XN

i¼1

n� ni

di� p

� �bi ð2Þ

where N sets of discrete interface dislocations are assumed to exist in the interface, bi is the Burgers vector of interface dis-location, and n is the interface normal, ni and di are the line sense and spacing of interface dislocation with Burgers vector bi,respectively.

The procedure for the application of AIFB theory is described as follows. (1) Construct a trial reference lattice. For an inter-face in which their boundary unit cells are similar and can form a CDP lattice without a twist, two unknown variables vx andvz correspond to the geometric partition of the mismatch strain between the two crystals in the two commensurate direc-tions (Fig. 2(a)). If a twist is involved in creating an interface, the third variable vh is introduced to describe the partition ofthe twist angle between the two crystals (Fig. 2(b)) (Hirth et al., 2013; Wang et al., 2013). According to the trial referencelattice, we calculate the two distortion matrices SA, SB, and the inverse matrices S�1

A and S�1B . (2) Corresponding to the FB the-

ory, there are only two independent probe vectors (orthogonal) for a flat interface. Thus, there are only four governing equa-tions with respect to the two independent probe vectors (four terms in the matrix B). However, there are more than fourunknown variables: vx, vz, vh, ni, bi, di and N.

If there are only two sets of lines presented within the simulated interface (N = 2), the FB equations can be directly solvedand often give several solutions. The real solution must be the one that matches the prediction from atomistic simulations. Ifthe solutions cannot satisfy the disregistry analysis of the simulated interfaces, we then adjust the three variables, vx,vz andvh until the solutions are consistent with the disregistry analysis of the simulated interfaces. If there are more than two dis-location lines present within the atomic interface (N > 2), the FB equations cannot be directly solved. Firstly, we assign theline sense and Burgers vector to one of all lines that are characterized from atomistic simulations. Then using the FB equa-tion, we search for the possible line senses and Burgers vectors for the other two lines. If there does not exist any solution inagreement with atomistic simulations, we will re-assign the line sense and Burgers vector for the line and repeat the aboveprocedure.

Atomic structures of (a) the un-relaxed and (b) relaxed interfaces between Cu and Nb that adopts the Nishiyama–Wasserman (NW) orientationship. Cu atoms are on the top. Atoms are colored according to the excess energy. The x-axis is along ½112�Cu and ½110�Nb , and the z-axis is alongand ½001�Nb .




3. Dislocations patterns within fcc(111)||bcc(110) interfaces

3.1. Interface geometry

For the NW interface in Fig. 2(a), ~eCux ¼

aCu2 ½112�, ~eNb

x ¼aNb

2 ½110�, ~eCuz ¼

aCu2 ½1�10�, and ~eNb

z ¼ aNb½001�, the final dimensions ofthe bicrystal model are 7.592 and 7.923 nm in the x and z directions, corresponding to a super boundary unit cell at equi-librium. It contains 21 periodic units for Cu and 23 for Nb in the x-direction, and 31 periodic units for Cu and 24 for Nbin the z-direction. For the KS interface in Fig. 2(b),~eCu

x ¼aCu2 ½11�2�,~eNb

x ¼aNb

2 ½1�12�, aCu2 ½�110�, and~eNb

z ¼ aNb½1�11�, the final dimen-sions are 18.596 and 4.858 nm in the x and z directions, containing 42 periodic units for Cu and 23 for Nb in the x-direction,and 19 periodic units for Cu and 17 for Nb in the z-direction.

Since the NW interface holds the parallelism of the commensurate directions, this interface can be used to calibrate thetwo parameters vx and vz related to the partition of the mismatch strain. The KS interface can thus be treated as a twist inter-face (twist angle, Dh = 5.6�) with respect to the NW interface, with which the third parameter vh can be determined. Adopt-ing the same reference lattice (CDP), the RCDP in the KS interface can be obtained by rotating the CDP abouthCDP ¼ hNb þ vhðhCu � hNbÞ.

For the NW interface, the six sites in the CDP lattice can be written in the x–z coordinate system as

Fig. 4.magnitinterfacdifferenposition

Pleasecenter

Disregistry analysis and characterization of interface dislocations. (a) The disregistry plot of the vector fields rR�Uij . The length of arrows indicates the

ude of the displacements and the color is assigned with respect to the sign of the x-component of the displacement. The solid lines indicate thee dislocations. (b) The disregistry vectors rR�C

ij and rU�Cij as the probe vector is along the x-direction. (c) The disregistry rR�U

ij corresponding to thece between the two disregistry vectors rR�C

ij and rU�Cij in (b). rij;x and rij;z present the x-component and the z-component of the disregistry vector. The

s of interface dislocations are marked as a dislocation symbol.



Fig. 5. Disregistry analysis and characterization of interface dislocations. (a) The disregistry plot of the vector fields rR�Uij . The length of arrows indicates the

magnitude of the displacements and the color is assigned with respect to the sign of the z-component of the displacement. The solid lines indicate theinterface dislocations. (b) The disregistry vectors rR�C

ij and rU�Cij as the probe vector is along the z-direction. (c) The disregistry rR�U

ij corresponding to thedifference between the two disregistry vectors rR�C

ij and rU�Cij in (b). rij;x and rij;z present the x-component and the z-component of the disregistry vector. The

positions of interface dislocations are marked as a dislocation symbol.


Pleasecenter

Cð1Þ ¼ffiffiffi6p

2aNb � vx

ffiffiffi6p

2aNb �

ffiffiffi6p

4aCu

!;

ffiffiffi3p

4aNb � vz

ffiffiffi3p

4aNb �

ffiffiffi2p

4aCu

! !

Cð2Þ ¼ 0;

ffiffiffi3p

2aNb � vz

ffiffiffi3p

2aNb �

ffiffiffi2p

2aCu

! !

Cð3Þ ¼ �ffiffiffi6p

2aNb þ vxð

ffiffiffi6p

2aNb �

ffiffiffi6p

4aCuÞ;

ffiffiffi3p

4aNbvzð

ffiffiffi3p

4aNb �

ffiffiffi2p

4aCuÞ

!

Cð4Þ ¼ �ffiffiffi6p

2aNb þ vx

ffiffiffi6p

2aNb �

ffiffiffi6p

4aCu

!;�

ffiffiffi3p

4aNb þ vz

ffiffiffi3p

4aNb �

ffiffiffi2p

4aCu

! !

Cð5Þ ¼ 0;�ffiffiffi3p

2aNb þ vz

ffiffiffi3p

2aNb �

ffiffiffi2p

2aCu

! !

Cð6Þ ¼ffiffiffi6p

2aNb � vx

ffiffiffi6p

2aNb �

ffiffiffi6p

4aCu

!;�

ffiffiffi3p

4aNb þ vz

ffiffiffi3p

4aNb �

ffiffiffi2p

4aCu

! !

ð3Þ

The distortion matrix is constructed by mapping the lattice vectors from the real lattices to the reference lattice using.



Fig. 6.Burgers


Pleasecenter

SNb ¼

ffiffi6p

2 aNb�vx

ffiffi6p

2 aNb�ffiffi6p

4 aCu

� �ffiffi6p

4 aCu0

0ffiffi3p

2 aNb�vz

ffiffi3p

2 aNb�ffiffi2p

2 aCu

� �ffiffi2p

2 aCu

2664

3775 ð4Þ

SNb ¼

ffiffi6p

2 aNb�vx

ffiffi6p

2 aNb�ffiffi6p

4 aCu

� �ffiffi6p

2 aNb0

0ffiffi3p

2 aNb�vz

ffiffi3p

2 aNb�ffiffi2p

2 aCu

� �ffiffi3p

2 aNb

2664

3775 ð5Þ

The possible Burgers vectors defined in the CDP lattice (Fig. 2c) are

bI ¼ �ffiffiffi6p

2aNb þ vx

ffiffiffi6p

2aNb �

ffiffiffi6p

4aCu

!;

ffiffiffi3p

4aNb � vz

ffiffiffi3p

4aNb �

ffiffiffi2p

4aCu

! !

bII ¼ �ffiffiffi6p

2aNb þ vx

ffiffiffi6p

2aNb �

ffiffiffi6p

4aCu

!;

ffiffiffi3p

4aNb � vz

ffiffiffi3p

4aNb �

ffiffiffi2p

4aCu

! !

bIII ¼ 0;

ffiffiffi3p

2aNb � vz

ffiffiffi3p

2aNb �

ffiffiffi2p

2aCu

! !ð6Þ

For the KS interface, the four lattice sites in the RCDP lattice are

CRðiÞ ¼cosðhNb þ vhðhCu � hNbÞÞ sinðhNb þ vhðhCu � hNbÞÞ� sinðhNb þ vhðhCu � hNbÞÞ cosðhNb þ vhðhCu � hNbÞÞ

� �CðiÞ ð7Þ

The distortion matrixes SR;Cu and SR;Nb are constructed by mapping the lattice vectors from the real lattices to the RCDP, con-taining one more unknown variable, vh. The four Burgers vectors defined in the RCDP lattice (Fig. 2d) are

biR ¼

cosðhNb þ vhðhCu � hNbÞÞ sinðhNb þ vhðhCu � hNbÞÞ� sinðhNb þ vhðhCu � hNbÞÞ cosðhNb þ vhðhCu � hNbÞÞ

� �bi; i ¼ I; II; and III ð8Þ

With these possible Burgers vectors and the distortion matrices, the characters of discrete interface dislocations includingthe Burgers vectors, the line senses, and the spacing can be obtained by solving the FB equations when varying the threevariables. In AIFB, the iteration regarding the reference lattice stops while the solutions obtained by solving the FB equationsare identical with the prediction from atomistic simulations.

Four cases of interface dislocation patterns solved by the FB theory corresponding to the Nishiyama–Wasserman (NW) orientation relationship. Allvectors are characterized according to the CDP in Fig. 2(a).



Fig. 7. Interface dislocation networks (a) constructed according to the interface disregistry analysis, (b) constructed according to the formation of junctionsin case 3 predicted by the FB theory, and (c) relaxed by using the PDD model. The black arrows indicate the line sense of interface dislocations, and the redarrows indicate the Burgers vectors. The dislocations segments, bIV

intf ; bVintf ; b

VIintf ; b

VIIintf , further act as sources for the nucleation of lattice dislocations, as

demonstrated in Fig. 12. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)


3.2. The CDP lattice and interface dislocations in the NW interface

Atomic structures of the un-relaxed and relaxed interface structures are shown in Fig. 3. First, a quasi-repeating pattern isobserved, indicated by blue1 atoms (Fig. 3b). Second, the highly distorted regions correspond to these blue atoms, as exploredin Fig. 4(a). The disregistry plot of the vector fields rR�U

ij is shown in Fig. 4(a), which is independent of the choice of the CDPlattice. Fig. 4(b) shows the disregistry vectors rR�C

ij and rU�Cij when the probe vector is along the x-direction. Fig. 4(c) shows

the disregistry rR�Uij corresponding to the difference between the two disregistry vectors rR�C

ij and rU�Cij in Fig. 4(b). The non-uni-

form relaxation in Fig. 4(c) is associated with the formation of discrete interface dislocations, as indicated by the solid lines inFig. 4(a) and dislocation symbols in Fig. 4(b) and (c). The magnitudes of these displacements cannot be used to predict the Bur-gers vector of these interface dislocations, because these displacements are a summation of all dislocations at a position. Fromthe above analysis we find that there are four straight dislocations, which can be categorized into two sets corresponding to thesign of z-component of the relative displacement rR�C

ij;z . The average spacing of the two sets of interface dislocations is 8.634 nm.In Fig. 5(a) we re-colored the disresgistry vectors according to the sign of the z-component. Fig. 5(b) shows rR�C

ij and rU�Cij when

1 For interpretation of color in Figs. 3 and 5, the reader is referred to the web version of this article.



Table 1Four cases of interface dislocation patterns solved by the FB theory corresponding to the Nishiyama–Wasserman (NW) orientation relationship, and the realsolution identified by AIFB. All Burgers vectors are characterized according to the CDP in Fig. 2(a) with the geometric factors vx = 0.555, vz = 0.467, andvh = 0.530.

Dislocation cases Interface dislocations

biintf

Line sense, n (angle with respect to the x-axis)

Spacing, d(nm)

Burgers vector(Fig. 2a)

Case 1 I 90 8.634 bI

II 90 8.634 bII

III 0 1.132 bIII

Case 2 I 90 8.634 bI

II 90 8.634 bII

III / 2.264cos/ bIII

IV �/ 2.264cos/ bIII

Case 3 I 90 8.634 bI

II 90 8.634 bII

III 180 1.132 bI

IV 0 1.132 bII

Case 4 I 90 8.634 bI

II 90 8.634 bII

III �172.65 2.245 bI

IV 7.35 2.245 bII

V �7.35 2.245 bI

VI 172.65 2.245 bII

AIFB (A result of case 3 accompanying the formation ofjunctions.)

I 90 8.634 bI

II 90 8.634 bII

III 0 1.132 bIII

IV 150 0.980 bII

V 30 0.980 bI

VI �30 0.980 bI

VII �150 0.980 bII

/ can be any angle between 0 and 90 deg.


the probe vector is located at x = 0 along the z-direction. Fig. 5(c) shows the rR�Uij along the z-direction corresponding to the dif-

ference between the two disregistry vectors rR�Cij and rU�C

ij in Fig. 5(b). By varying the position of the probe vector, we reveal thatthe dislocation lines, indicated by the black lines, and their average spacing 1.132 nm. These black lines cannot be closed, whichsuggests the other dislocation segments indicated by the green lines.

To determine all these dislocations in terms of the line senses and Burgers vectors, we have to rely on the FB theory.Substituting Eqs. (4), (7) and (8) to (1), we get

Pleasecenter

B ¼ S�1Nb � SCu ¼

aNb�12aCu

aNb�vx aNb�12aCuð Þ 0

0ffiffi2p

2 aNb�ffiffi3p

3 aCu

aNb�vx aNb�12aCuð Þ

264

375 ð9Þ

When the probe vector has the magnitude of Lx and is along the x-direction,

B ¼ Bx;x Bx;z½ � ¼ aNb�12aCu

aNb�vxðaNb�12aCuÞ

Lx 0h i

ð10Þ

From Fig. 4, we can choose the Burgers vectors, bIIntf = bI and bII

Intf = bII (Fig. 2a). Bx;z = 0 is automatically satisfied. Substitutingthe two Burgers vectors and the average spacings determined in Fig. 4 into Eq. (2), the line sense of the two dislocations, b1

and b2, is along the negative z-direction, and the variable vx is equal to 0.555.When the probe vector has the magnitude Lz and is along the z-direction,

B ¼ ½Bz;x Bz;z � ¼ 0ffiffi2p

2 aNb�ffiffi3p

3 aCu

aNb�vx aNb�12aCuð Þ Lz

� �ð11Þ

Corresponding to the features identified in Fig. 5, four sets of solutions are obtained to satisfy Eq. (11) by solving Eq. (2). Thecase 1 solution is schematically plotted in Fig. 6(a) and includes three types of interface dislocations. Since the Burgers vectorbIII in Fig. 2a is along the z-direction, Bz;x can be automatically satisfied if the line sense is parallel to the x-axis. SubstitutingbIII

Intf = bIII into Eq. (2), the Burgers vector bIII = (000.290) and vz = 0.467, when the average spacing from atomistic simulation1.132 nm is used. For case 2, there are four types of interface dislocations. If the dislocation bIII

Intf in case 1 is alternatively ro-tated upwards and downwards with the same rotation angle /, Bz;x is automatically equal to zero, which corresponds to two



Fig. 8. Atomic structures of (a) the un-relaxed and (b) relaxed interfaces between Cu and Nb that adopts the Kurdjumov–Sachs (KS) orientationrelationship. Cu atoms are on the top. Atoms are colored according to the excess energy. The x-axis is along ½11�2�Cu and ½1�12�Nb, and the z-axis is along½�110�Cu and ½1�11�Nb. (c) The disregistry plot of the vector fields rR�U

ij . The length of arrows indicates the magnitude of the displacements and the color isassigned with respect to the sign of the x-component of the displacement. The solid lines indicate the interface dislocations. (d) The disregistry vectors rR�C

ij

and rU�Cij as the probe vector is along the z-direction. rij;x and rij;z present the x-component and the z-component of the disregistry vector. The positions of

interface dislocations are marked as a dislocation symbol.




Table 2Three cases of interface dislocation patterns solved by the FB theory corresponding to the Kurdjumov–Sachs (KS) orientation relationship, and the real solutionidentified by AIFB. All Burgers vectors are characterized according to the RCDP in Fig. 2(b) with the geometric factors vx = 0.555, vz = 0.467, and vh = 0.530.

Dislocation cases Interface dislocations

biintf

Line sense, n (angle with respect to the x-axis) Spacing, d (nm) Burgers vector (Fig. 2b)

Case 1 I 90 1.245 bIIIR

II �152 2.131 bIR

Case 2 I 90 1.245 bIIR

II �112 0.905 bIR

Case 3 I 68 0.905 bIIIR

II �152 2.130 bIIR

AIFB (Case 1) I 90 1.245 bIIIR

II �152 2.131 bIR

Fig. 9. (a)–(c) Three cases of interface dislocation networks predicted by the FB theory corresponding to the Kurdjumov–Sachs (KS) orientation relationship.All Burgers vectors are characterized according to the RCDP in Fig. 2(b). (d) The interface dislocation networks relaxed by using the PDD model. The blackarrows indicate the line sense and the Burgers vectors of interface dislocations.


different dislocations bIIIIntf and bIV

Intf with line senses at the angles �/ and /, respectively. Using Eq. (2), we found that there areinfinite solutions when the relation, d = 2.264cos/, is held. Fig. 6(b) shows one solution (/ = 14.7�), in which the two sets ofdislocations cross through the intersections within the interface plane. For case 3, we take case 1 but treat the dislocation bIII

Intf

as a combination of dislocations bI and bII . Correspondingly, there are four types of interface dislocations. Two types of dis-locations bIII

Intf and bIVIntf in Fig. 6(c) are created with the line sense parallel to the x-axis. To satisfy Bz;x = 0 and the net Burgers

vector content along the z-direction, the average spacing between two adjacent dislocations bI and bII is equal to 0.066 nm,which equals half the spacing determined in atomistic simulation. For case 4, we take a similar idea as in case 2. The twotypes of dislocations bIII

Intf and bIVIntf identified in case 3 are alternatively rotated upwards and downwards with the same rota-

tion angle, which corresponds to four different dislocations, bIIIIntf , bIV

Intf , bVIntf and bVI

Intf . Bz;x is automatically equal to zero. UsingEq. (2), we found that there are infinite sets of solutions when the relation, d = 1.132cos/, is held. Fig. 6(d) shows one set ofsolution (/ = 7.35�) in which the six types of interface dislocations cross through the intersections within the interface plane.

To test whether these dislocation networks can be relaxed into the network identified in MD simulations, we performedelastic relaxation by using the Parameterized Dislocation Dynamics (PDD) (Ghoniem et al., 2000; Zhou and LeSar, 2012).Although the PDD code was developed for isotropic elastic solid, for the first order approximation it could capture the reac-tion details between interface dislocations and the anisotropic dislocation line energies. The elastic relaxation leads to an



Fig. 10. (a) The in-plane shear strength of the Cu/Nb NW interface with respect to the shear direction. (b) and (c) Show the variation of shear stress with theshear strain with respect to the shear directions along the x- and z-axis, respectively.


unexpected dislocation pattern. Although all solutions obtained by solving FB equations do not satisfy the feature character-ized in atomistic simulations, these solutions enable us to determine the interface dislocations. First, two interface disloca-tions bI

Inft and bIIIntf in all cases can be part of the solution, which matches the two dislocations identified in Fig. 4(a). Second, if

bIIIIntf and bIV

Intf determined in case 3 can form junctions, as indicated in Fig. 7(b), the junction has the same Burgers vector asthe bIII

Intf defined in case 1 and the rest of the two dislocations will have the zigzag segments as the green lines identified inFig. 5(a). We schematically plot the dislocation network in Fig. 7(a). Using the dislocation network in Fig. 7(b) as an initialstructure, the relaxed, equilibrium dislocation pattern from PDD in Fig. 7(c) is consistent with that observed in the atomisticsimulations (Figs. 4 and 5). Thus the NW interface can be considered to consist of seven sets of interface dislocations (in Ta-ble 1) that result from four sets of interface dislocations (case 3) accompanying the formation of junctions.

3.3. The RCDP lattice and interface dislocations in the KS interface

Atomic structures of the un-relaxed and relaxed structures of the KS interface, and the vector field rR�Uij are shown in

Fig. 8. It is clearly observed that the regions surrounded by the distorted regions takes the coherent structure. Thus the inter-face probably contains two sets of dislocations, as denoted in Fig. 8(b) and (c). One set of dislocation is identified in the dis-registry vectors rR�C

ij and rU�Cij (Fig. 8(d)) as the probe vector along the z-direction. Following the same method in Section 3.2,

we first performed FB analysis and obtained three sets of solutions, as summarized in Table 2 and shown in Fig. 9. Except forset 1, the other three sets of dislocations will cross through the coherent regions within the interface. Starting with set 1



Fig. 11. The relative displacements of interface under the applied shear stress along the x-direction. Irreversible sliding begins within the intersectionregions of two interface dislocations (in Fig. 8) with respect to the shear strain at (a) 0.20, (b) 0.025 and (c) 0.035. The solid lines approximate the locationsof interface dislocation loops, separating the slipped and non-slipped regions in the interface.


solution, we performed AIFB analysis to determine the variable vh. The disregistry analysis (Fig. 8) shows that the two sets ofdislocations bI

Intf and bIIIntf have the lines at the angles 90� and 151.89�, and their spacings 1.245 and 2.131 nm, respectively.

Substituting these parameters into Eq. (2), vh is calculated to be 0.530. Using PDD, we relaxed the dislocation structure(Fig. 9(a)) that is only driven by the dislocation interaction stresses and find that in Fig. 9(d) it is consistent with that ob-served in the atomistic simulations.

4. The effects of interface dislocations on interface-mediated plasticity

The nucleation of dislocations, including interface dislocation loops, corresponding to interface sliding, and lattice dislo-cations, corresponding to plastic deformation in crystals, are strongly correlated with the interface dislocations. Previouslywe have performed a series of studies on KS interfaces: interface structures, interface shear, dislocation nucleation and thereactions of defects with interfaces (Wang et al., 2008a,b, 2011c; Demkowicz et al., 2008a,b). The results show that KS inter-face has the low shear resistance at 0.6 GPa and strong anisotropic with respect to the shear direction (Wang et al., 2008a).The low shear resistance has been ascribed to the easy nucleation and glide of interface dislocations (Wang et al., 2008a;Demkowicz et al., 2008a). In addition, the nucleation and emission of lattice dislocations have been characterized fromthe intrinsic interface dislocations (Zhang et al., 2011, 2012). In this section, we study the interface shear strength and nucle-ation of lattice dislocations from the NW interface.



Fig. 12. Atomistic simulations show (a) the nucleation of Shockley partial dislocations in Cu from interface, and (b) nucleation sites that are associated withthe interface dislocations, zigzag segments shown in Fig. 7.


The shear strength of an interface is the critical shear stress at which irreversible sliding of one crystal with respect to theother commences along the interface. To calculate it, the relaxed interface is subjected to the uniaxial stress state, in whichthe non-loaded surfaces were constrained to have zero stress and the resolved shear stress on the slip systems lying in theinterface plane is zero. MD simulations are performed at a certain strain at temperature 1 K for 50 ps, and then followed bydynamic quenching MD until the maximum residual force on each atom becomes less than 5 pN (Wang et al., 2008a). The in-plane shear strength for the NW interface is plotted in Fig. 10(a) with respect to the shear direction. Fig. 10(b) and (c) showthe variation of shear stress with the shear strain with respect to the shear directions along the x- and z-axis, respectively.The results revealed that the shear strength of Cu/Nb interface is significantly lower than the theoretical shear strength inperfect crystals of Cu and Nb (>2.0 GPa). When the applied shear stress reaches a critical value, irreversible sliding occursnonuniformly, beginning within the intersection regions of interface dislocations as shown in Fig. 11. The spatially nonuni-form sliding in the interface suggests an interface sliding mechanism in which dislocation loops nucleate in interfaces andsubsequently expand by gliding in the interface plane. The lines in Fig. 11 approximate the locations of interface dislocationloops, separating the slipped and non-slipped regions in the interface. As sliding commences, the dislocation loops nucleateat the intersection region of two interface dislocations, then expand, coalesce, and glide in the interface.

We also use MD simulation to study the nucleation of lattice dislocations from the NW interface. The relaxed interface issubjected to a tensile stress along the x-axis. Dynamic boundary conditions were adopted to achieve the uniaxial tensilestress state, in which the non-loaded surfaces were constrained to have zero stress and the resolved shear stress on the slipsystems lying in the interface plane is zero (Zhang et al., 2011). Under the applied stress state, atomistic simulations showthe nucleation of Shockley partial dislocations in Cu from the interface (Fig. 12). They nucleate at the zigzag segments shownin Fig. 7, which are parallel to the traces of glide planes in Cu with the interface.

5. Conclusions

Dislocation patterns of semi-coherent fcc/bcc interfaces that adopt Kurdjumov–Sachs (KS) or Nishiyama–Wasserman(NW) orientation relationships have been studied by combining classical Frank–Bilby theory and atomistic simulations.We found that




(1) Both interfaces share the same CDP lattice defined in the NW interface. For the KS interface, RCDP is a rigid rotation ofthe CDP about the interface normal. The CDP and RCDP are a non-equal partition of the lattice misfit and twistbetween the two lattices.

(2) The Burgers vectors of the interface dislocations can be well defined in the reference lattices (CDP or RCDP).(3) Unlike the KS interface, which has two sets of interface dislocations, for the NW interface, we found seven sets of inter-

face dislocations, forming the complicated dislocation network. The four sets of interface dislocations are parallel tothe trace of the glide planes in Cu with the interface, which are preferred nucleation sites for lattice dislocations.The intersections between the other three sets of interface dislocations are relative weak to the rest of interface, actingas the preferred sites for nucleating interface dislocation loops responsible for in-plane sliding.

Finally, it is worth pointing out that interface-mediated plasticity in nanolayered metallic composites includes nucleationand glide of interface/lattice dislocations, transmission of lattice dislocations across interface, and their reaction within inter-faces. By using atomistic simulations and AIFB theory, people can characterize interface structures, interface properties, andinterface dislocations. However, atomistic scale models are not able to predict macroscopic properties of nanolayered com-posites. The high length scale models are required. Dislocation Dynamics models have the unique advantage of exploring thedislocation activity, but currently they do not incorporate deformation processes related to interfaces. Thus a predictivematerials modeling tool by advancing the DD method and coupling with atomistic-level deformation mechanisms within/at/across boundaries would be promising to bridge the length-scale gap from atomic-scale to meso/macro-scale.

Acknowledgments

This work was supported by the Center for Materials at Irradiation and Mechanical Extremes, an Energy Frontier ResearchCenter funded by the US Department of Energy, Office of Science, Office of Basic Energy Sciences under Award Number2008LANL1026. JW was also supported by LDRD projects ER20140450 and DR20110029.

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Documents

Interface dislocation patterns and dislocation nucleation in face-centered-cubic and body-centered-cubic bicrystal interfaces