JOM, Vol. 66, No. 1, 2014

DOI: 10.1007/s11837-013-0808-8 Ó 2013 The Minerals, Metals & Materials Society

Modeling Interface-Dominated Mechanical Behavior of Nanolayered Crystalline Composites JIAN WANG,1,4 CAIZHI ZHOU,3 IRENE J. BEYERLEIN,2 and SHUAI SHAO1 1.—MST Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA. 2.—Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA. 3.—Department of Materials Science and Engineering, Missouri University of Science and Technology, Rolla, MO 65409, USA. 4.—e-mail: [email protected]

Interface-dominated nanolayered crystalline composites exhibit extraordinary strength and hardness, far beyond those of their constituent materials. Modeling the deformation of such materials would aid in understanding and designing them for future applications. This task is a multiscale effort. Up to now, most modeling efforts lie at either the atomic scale or the mesoscale. Models that link the two scales are missing. In this work, we develop some tools that aim to help in making this important connection.

INTRODUCTION Extensive investigations over the past decade indicate that nanolayered metallic composites have unprecedented levels of strength, ductility, and damage tolerance in extreme environments.1–10 As the length scale is reduced from microscale to nanoscale, interfaces become crucial in determining mechanical properties of nanoscale materials due to the change of deformation mechanisms from phase dominated to interface dominated.10 Atomic-scale modeling can reveal unit process (involving single or a few defects) occurring at interfaces during deformation with respect to kinetics and energetics, but it is limited to time scale (ns) and length scale (nm).11–21 Current state-of-the-art microscale, mesoscale, and continuum-scale modeling works well for microscale materials because of the dominated deformation processes by dislocation activities in phases. Due to the less dependence of deformation on interfaces, interfaces often are phenomenologically treated as boundaries without any structural characteristics.22–28 For materials containing the high density of interfaces, it is still a challenge to incorporate interface physics, such as nucleation, motion, reactions, etc., in current models.29 The bottleneck is ascribed to two reasons: (I) A generic interface model, representing the interface structures and the resultant properties of interfaces, does not exist and (II) knowledge of interface-dominated deformation mechanisms is lacking because of the complexity of dislocation–interface interactions.

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To correlate the characteristics of interface with mechanical properties and behavior related to interfaces, it is essential to understand interfacial structure–property relationships at different scales and develop new materials modeling tools that incorporate interface physics and can address interface roles in terms of dislocation activity during mechanical deformation. Dislocation dynamics (DD) simulations offer a way to simulate dislocation activity and extend length and time scales beyond those of atomistic simulations.30–40 Recently, several groups started considering dislocation–interface interactions in DD simulations.40 However, most physical properties of interfaces are still missing. In this article, we summarized the effort in developing a new materials modeling tool at the microscale by combining atomistic studies with the DD simulation—in accounting for the roles of interfaces on storage, recovery, nucleation, emission of dislocations within/at/across interfaces, and their correlation with interface structures and properties. We highlighted three aspects of our efforts: (I) interface physics at atomic scale, (II) the framework of interface–dislocation dynamics model, and (III) numerical examples related to interface physics. Finally, we discussed the future work. INTERFACE PHYSICS AT ATOMIC SCALE In the past two decades, experimental and modeling studies have endeavored to discover that

(Published online November 26, 2013)

Modeling Interface-Dominated Mechanical Behavior of Nanolayered Crystalline Composites

details of atomic structure matter when it comes to the response of the interface to mechanical loading and, in particular, its influence on plastic deformation.12–21 However, it is still a challenge to correlate structure–property relationships because achieving this knowledge by exploring the entire parameter space via calculation alone would be exhaustive. We thus devised a purely geometrical scheme to predict some general attributes of interface structure including characteristics of interface plane and interfacial dislocations. With this scheme, we demonstrated that interfaces are classified into certain types without having to carry out atomistic simulations. More importantly, the same type interface shares the unique and common features. This classification scheme, described next, is intended to help efforts in multiscale modeling. Classification Scheme of Interfaces Atomic environments—structural units defined by the relative configuration of an atom and its neighbors—inside and in the vicinity of an interface may be distinctly different from those in the neighboring perfect crystals. Several theories and atomic-scale models have been put forth to describe the defect structure of an interface with a given orientation relation (OR) and interface plane (IP), in terms of the number of sets of interfacial dislocations, their spacing, Burgers vector, and orientation.41–45 It is realized that geometric characters of the habit planes of the two joined crystals determine the principal characteristics of interface structures. Although describing interfaces in this general way would sacrifice atomic-scale detail, it would permit broad classification of these interfaces into a few structural types. From the viewpoint of thermodynamics, interfaces composed of low-energy surfaces and/or low-energy ledges are energetically favorable, close to thermodynamic equilibrium. Two geometric factors are thus chosen to classify interfaces: compact plane and compact direction (Fig. 1a, b). In addition, the third factor is the similarity of unit cell because it is likely to form a coherent structure for the two habit planes with similar atomic structures (Fig. 1c, d). With respect to the three geometric factors, we classified interfaces into four types. Type I, the interface consists of the compact planes of both crystals. Type II, the interface is not Type I. That is, at least one of the two planes selected from each crystal for the interface is noncompact for the respective crystal. The interface plane contains, however, the compact directions of the two crystals and they are aligned. For Type III, the interface is neither Type I nor Type II. The interface does not match the compact planes or the compact directions. The unit cells, however, are similar in that they have the same basic shape and the same number of atoms. Type IV, the interface does not belong to Type I, Type II, or Type III. To access this scheme,

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we use the experimentally observed Cu/Nb interfaces as a prototypic material system because of the available experiment data. Two ORs: Kurdjumov– Sachs (K–S) and Nishiyama–Wassermann (N–W)15 and different IPs are studied. Four types of interfaces are listed in Table I. Employing molecular dynamics (MD) simulations and the developed atomistically informed FrankBilby theory (AIFB),42–45 we demonstrated that different crystal pair combinations within the same class lead to atomic interfacial structures sharing the same characteristics and interfaces within the same class share the similar mechanical responses. Characteristics of Interfaces Type I interface is atomically flat due to the compactness of the two habit planes. Second, interface contains a quasi-repeating pattern corresponding to the intrinsic interfacial dislocations (or misfit dislocation pattern). Third, atoms around the intersection of misfit dislocations undergo nonuniform or complicated displacements that are different from other regions. An explanation was eventually found based on the relaxation of interface dislocations intersection.20,44–46 The Burgers vectors of the intrinsic interfacial dislocations only have in-plane components corresponding to the flat atomic interface plane. The places where the misfit dislocations from different sets intersect are regions of high energy, where many nearest-neighbor relationships are broken. The line orientation, spacing, and Burgers vectors of these misfit dislocations and the density of intersection points are obtained by the AIFB method and determined by the crystal misorientation and the lattice structure of the two crystals. Figure 2 shows the misfit dislocation networks for the Type I K–S and N–W interfaces.43 Type II interfaces tend to be faceted or terraced because the compact directions of the two crystals are aligned and the compact planes contain the compact direction. In this case, either one or both of the planes constituting the interface are noncompact. To accommodate the lattice mismatch and the crystal rotation, Type II interfaces contain at least two sets of interfacial dislocations with the line senses parallel to the aligned compact direction. One set of interfacial dislocations has Burgers vectors that lie within the terrace planes and the other set of dislocations has Burgers vectors out of the terrace planes corresponding to the crystal rotation. The latter set also has a step character and, hence, is an array of disconnections. As shown in Fig. 3a, both unrelaxed planes to be joined are not flat but have a saw tooth profile. For instance, the {112} plane of Cu consists of alternating {100} and {111} planes, and likewise, the {112} plane of Nb alternates two different variants of the {110} planes. When the bcc and fcc {112} planes are joined and the interface relaxed, an array of disconnections is generated (Fig. 3b, c). The regions between the

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Fig. 1. Geometry factors used in the classification scheme of interface types: compact plane, compact direction, and the similarity of atomic structures of unit cells in habit planes. (a) Wulff construction of face-centered cubic structure. Compact planes are {111} and {100}, and compact direction along h110i. (b) Wulff construction of body-centered cubic structure. Compact planes are {110} and {100}, and compact directions along h111i and h100i. (c) and (d) Similarity of unit cells.

Table I. Types of face-centered cubic (FCC)/body-centered cubic (BCC) interfaces with respect to geometrical classification scheme Orientation Relation

Interface Plane

Compact Plane

Compact Direction

Similarity of Unit Cell

Interface Type

Kurdjumov– Sachs (K–S)

{110}fcc||{111}bcc {112}fcc||{112}bcc {111}fcc||{110}bcc {110}fcc||{001}bcc {112}fcc||{110}bcc {111}fcc||{110}bcc

No No Yes No No Yes

No Yes Yes No Yes Yes

No No Yes Yes No Yes

Type IV Type II Type I Type III Type II Type I

Nishiyama– Wassermann (N–W)

disconnections are facets that join the compact planes of the two crystals, e.g., {111} of Cu and {110} of Nb. The interface also lowers its total energy by forming regions of relatively low energy terraces, albeit at the expense of creating the disconnections.44,45,47 Type III interfaces, unlike the other interfaces studied, have the distinct characters corresponding to their unit cells of crystal planes, which are similar in that they have the same basic shape and the same number of atoms. Corresponding to this, such interfaces tend to be semicoherent and flat, and they

contain more than one set of interfacial dislocations with Burgers vectors lying within the interface plane. Atomistic simulations are performed on two Cu/Nb interfaces and shown to demonstrate these characteristics (Fig. 4). It is noticed that interface dislocations may have a nonplanar core because glide planes in the crystals are often weaker than the coherent interface in terms of shear resistance.47 This is different from interfacial dislocations in the Type I interface, where interfacial dislocations have a planar core due to the easy shear interface.

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Fig. 2. The relaxed Sachs (KS) and (b) Nishiyama-Wasserman (N–W) interface. In both figures, the   atomic structures of the (a) Kurdjumov    x-direction is 112 Cu and the y-direction is 110 Cu. The solid lines mark interface dislocations where segments of the same color have the same Burgers vector. The three edges of the triangle in the center indicate the traces where the three {111} planes of Cu intersect the interface (Reprinted from Ref. 43).

Mechanical Response of Interfaces Under mechanical loading, interfaces act as sources for defects that result in interface shear sliding and/or crystal plasticity in association with nucleation and emission of lattice dislocations. We developed the correlation of such mechanical responses with interface types using atomistic simulations and dislocation theory. The correlation of mechanical response with interface type is summarized in Fig. 7. The shear strength of an interface is the critical shear stress at which irreversible sliding of one crystal with respect to the other commences along the interface. This is determined by gradually increasing a homogeneous shear strain applied to bilayer models of Cu/Nb.14 For Type I interfaces, the results in Fig. 5b revealed that the shear strength is significantly lower than the theoretical estimates of shear strengths on glide planes in perfect crystals of Cu and Nb and (ii) is anisotropic (i.e., dependent on the applied in-plane shear direction). Details of the atomic-scale sliding in interfaces (Fig. 5c) indicate

that the low shear resistance is due to the easy nucleation and glide of interface dislocations at the intersection of misfit dislocations or associated with the glide of intrinsic interfacial dislocations because the interface plane is bonding both compact habit planes (or glide planes in the two crystals). For Type II interface, the in-plane shear resistance is plotted in Fig. 5e. The results show that (I) the interface likely slides along the compact direction (Fig. 5d) and (II) lattice dislocations can be emitted from interfaces due to the dissociation of interface disconnections that are shown in Fig. 4b. For the Type III interface, when the applied shear stress reaches a critical value, the interface does not slide, instead, lattice dislocations nucleate and emit from interfacial intrinsic dislocations. For Type IV interfaces, the shear response of the interface is complicated due to the disordered nature of their atomic structure, and a high shear resistance (>2.0 GPa) corresponds to either atomic reconstruction or the nucleation and emission of lattice dislocations. The magnitude of shear resistance and the corresponding

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Fig. 3. Atomic structures of Kurdjumov–Sachs (K–S) {112}fcc||{112}bcc interface: (a) the unrelaxed interface, (b) the relaxed interface, and (c) plan view of interface. The dashed lines mark three sets of interface dislocations.

Fig. 4. Atomic structures of (a) the relaxed Nishiyama–Wasserman (N–W) {110}fcc||{100}bcc interface within the interface plane [100]Cu||[010]Nb     and ½110 Cu jj½100Nb , (b) a twisted interface with respect to (a) within interface plane ½111Cu jj½110Nb and ½112Cu jj½110Nb . The solid lines mark interface dislocations where segments of the same color have the same Burgers vector (Reprinted from Ref. 42).

Modeling Interface-Dominated Mechanical Behavior of Nanolayered Crystalline Composites

shear mechanisms for the three types of interfaces are summarized in Fig. 7. When an interface is strained, nucleation and emission of lattice dislocations can occur. We studied nucleation events for the three types of interfaces by using atomistic simulations47–50 and developed a theoretical model51 that reveals novel connections between the nucleation site, preferred dislocation types, and intrinsic interface structure. In every axial loading condition studied, dislocations only nucleated on slip systems associated with the maximum Schmid factors. So called ‘non-Schmid’ effect, where lattice dislocations with lowranked or zero Schmid factors are not observed. In other words, the model and MD simulations show that interface structure can prevent some systems with maximum, positive Schmid factors from nucleating but cannot promote systems with zero or negative Schmid factors to nucleate. Significantly, we find that the interface nucleation behavior can be classified by type. For Type I interfaces,48,49 dislocation nucleation is along the intrinsic dislocation lines because they are commonly aligned with the traces of glide planes with the interface and driven by local intensive shear because of the low shear resistance of interface. For Type II and III interfaces,42,47 interface dislocations have no-planar core and/or have out-of-plane Burgers vectors. When interfacial dislocation lines are parallel or nearly parallel to the traces of glide planes with the interface, nucleation is relatively

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easy via dissociation mechanisms. When the trace is not parallel to any intrinsic dislocation line, nucleation is preferred at the intersection of interfacial dislocations. Figure 6 gives some examples of MD simulations and the corresponding mechanisms discussed above. The correlation of dislocation nucleation with interface type is summarized in Fig. 7. Dislocation–Interface Interactions For the case of dislocation trapping in the interface plane via core spreading, slip transmission is achieved through three unit processes: (I) a glide dislocation is attracted into the interface in association with the core spreading within the interface, (II) the extended core has to shrink in order to nucleate a glide dislocation in the adjacent crystal, and (III) the nucleated dislocation loop bows out from the interface under stress, overcoming the attraction force due to the interface shear and the residual dislocation at interface.19 For certain slip systems, the slip transmission process is studied using the chain-of-state method. First, a straight dislocation is introduced into one crystal in the bicrystal model close to the interface. After fully relaxing the dislocated configuration, the equilibrium configuration containing a lattice dislocation is obtained, and this acts as the initial configuration. Second, the final configuration containing a dislocation loop in the adjacent crystal is

Fig. 5. (a) Atomic structures of Type I N–W interface, (b) the in-plane shear resistance, and (c) interface sliding mechanisms: nucleation and propagation of interface dislocation loops at the intersection of misfit dislocations. (d) Atomic structures of Type II K–S {112} interface and (e) the in-plane shear resistance and shear response (sliding and emission of lattice dislocation).

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Fig. 6. Nucleation mechanisms of lattice dislocations with respect to the relation between interfacial dislocation lines and the traces of glide planes with the interface. (a) The trace is parallel to the interfacial dislocation line, (b) the trace is nearly parallel to the interfacial dislocation line, and (c) the trace is non-parallel to the interfacial dislocation lines. (Reprinted from Ref. 51).

Fig. 7. Correlation of interface structures and properties with interface types. ID = interfacial dislocation, LD = lattice dislocation.

created and relaxed at a certain applied stress. Finally, molecular dynamics is performed for the final configuration while reducing the applied stress to obtain a series of configurations between the

initial and the final configurations. The change of potential energy is then calculated as a function of the size of the dislocation loop.19 Corresponding to the different states, the change in potential energy

Modeling Interface-Dominated Mechanical Behavior of Nanolayered Crystalline Composites

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of the bicrystal is calculated. The critical stress corresponding to the stabilized dislocation loop is determined from the first derivative of the potential energy with respect to the dislocation loop and found to be on the order of 0.8 GPa in terms of the resolved shear stress. Regarding different interfaces, this critical stress will increase as the interface shear resistance decreases due to the expansion of the dislocation core within the weaker interface.19 For the case of dislocation blocking by the interface plane via the reaction with interface disconnections, slip transmission processes are complicated, involving different mechanisms such as dislocation climb along interface, dissociation and/or recombination of interface defects, and nucleation and emission of dislocation. Correspondingly, temperature and local stresses become more important because most of these mechanisms mentioned above are thermally activated processes. INTERFACE–DISLOCATION DYNAMICS (DD) MODEL Deformation processes related to interfaces could be captured based on the activities of dislocations within interfaces. Differing from the conventional DD method, the most important improvement in the interface–DD model (Fig. 8a) is that we implemented interface physics including structure and properties of interfaces and the reaction and nucleation rules for dislocations within/at/across interfaces. According to the classification scheme described in the previous sections, several important aspects, such as dislocation characters, dislocation motion, and slip within interfaces and in the bulk related to interfaces (slip transmission across interfaces, slip reflection at interfaces, interface shear/sliding, and interface migration), can be correlated to interface types. In interface–DD model, interface plane is treated as a special plane where dislocations can move through glide and climb. Two types of dislocations, lattice dislocation and interfacial dislocation, are considered. Lattice dislocations are well defined in crystals. Interfacial dislocation refers to a dislocation lying within an interface, including a nucleated dislocation whose Burgers vector lies entirely within interface due to interface shear, a lattice dislocation after entering interfaces, or intrinsic dislocations in association with the formation of interfaces. These dislocations can react with each other through their glide and climb, resulting in the recovery of interfacial dislocations and the nucleation and emission of lattice glide dislocation from interfaces into the bulk. Nucleation and Motion of Interfacial Dislocations Interfacial dislocations associated with the given OR and IP could be a glide-type, climb-type, or mixed-type interfacial dislocations.41 Their characters, such as Burgers vector, line sense, and spacing,

Fig. 8. (a) Schematic of interface–DD model. Interface plane is treated as a special plane where dislocations can move through glide and climb. Two types of dislocations, lattice dislocation and interfacial dislocation, are considered. (b)–(d) Nucleation of lattice dislocations from interfacial dislocations (See details in text).

can be determined with respect to interface types. In addition, climb-type and mixed-type interfacial dislocations can also be created as a result of the dislocations reactions within interfaces. Glide-type interfacial dislocations are associated with the interface shear or interface sliding. They are created via the absorption and subsequent dissociation of lattice dislocations within interface, or the cross-slip of screw-type lattice dislocation onto interface, or the nucleation within interface as the shear stresses within interface plane exceed the interface shear strength. The first two mechanisms are related to the dislocation reactions that take place within the interface. The last one corresponds to interface shear. Interfacial dislocations move on the interface via either glide or climb mechanisms. The mobility depends on interface structure and properties. Glide is controlled by drag, whereas climb is controlled by vacancy diffusivity. Both the drag coefficient and the formation and migration energy of a vacancy within an interface for a particular interface type have to be calculated using atomistic modeling. Atomistic simulations were used to provide these parameters for the DD calculations presented next.

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We also consider that within interfaces, interfacial dislocations react, resulting in annihilation (recovery), zipping (forming junctions), and the redissociation corresponding to the unzipping of a junction and the nucleation of a lattice dislocation. These reactions could be energetically favored or unfavored. For energetically unfavored reactions, if the resultant net Burgers vector is bigger than a lattice dislocation, then the reaction will lead to the nucleation and emission of a lattice dislocation. As a result, the net Burgers vector at interface will decrease due to the nucleation and emission of a lattice dislocation. In addition, we use a greater critical distance regarding the athermal reaction for the climb-type interfacial dislocations than lattice dislocations. As explored in atomistic simulations and confirmed by in situ transmission electron microscopy (TEM) observations,18,52 the two climb-type dislocations can athermally react at the separating distance of 1.7 nm within interface, which is greater than the critical distance  0.4 nm in the bulk. Nucleation of Lattice Dislocations from Interfaces As discussed in the Mechanical Response of Interfaces section, lattice dislocations in nanolayered composites mainly nucleate from interfacial dislocations. In the interface–DD model, we check the parallelism of the interfacial dislocation line to the possible traces of glide planes with an interface at each step during DD simulation, and then modify the dislocation segments that are closely parallel to the trace. For those segments that are parallel to the traces, we assume that the nucleation event takes place from the segment parallel to the traces via the Frank-Read source mechanism. The choice of slip systems is determined by comparing the residual dislocations and the Peach-Koehler (PK) forces. For the example of lattice dislocation nucleation from the interface found in Fig. 3, dashed lines indicate the traces of glide planes with the interface, and the brown solid line indicates the lines of interfacial dislocations. We model nucleation following the mechanisms described in the Mechanical Response of Interfaces section (Fig. 8b–d). 1. We check the parallelism of the interfacial dislocation line to the possible traces at each step during DD simulation, and then modify the dislocation segments that are closely parallel to the trace. 2. For the segment longer than a critical segment length, we determine the numbers n of slip systems that contain the segment. Among these i slip systems, we compute the PK force FPK corresponding to the slip system that could be activated and the residual Burgers vector, bires , accompanying with the emission of the lattice

Wang, Zhou, Beyerlein, and Shao i points toward the dislocation. If the PK force FPK interface, then the slip system will not be considered. For the other slip systems, if one slip system has b2res + b2lattice < b2inter, then the slip system will be preferred. If all slip systems exhibit b2res + b2lattice > b2inter, then we use a Monte Carlo method to determine which slip system will be activated. (In carrying out the Monte Carlo method, system hP each slip i. j has an j1 1 Pj 1 where assigned span: i¼1 bires ; i¼1 bires S, Pn 1 S = i¼1 bi and n is the number of all possible res slip systems. The slip system will be activated, if the chosen random number falls into its span. This process will guarantee slip systems with smaller residual Burges vectors have higher probability to be activated.) 3. Once the activated slip system is determined, we then introduce a dislocation loop corresponding to the selected slip system. This operation can be described as the dissociation of the original interfacial dislocation segment and the emission of the lattice dislocation, corresponding to the nucleation of lattice dislocation at an interface.

NUMERICAL EXAMPLES OF INTERFACE– DD MODEL This work focuses on the interaction of dislocations with Cu/Nb interfaces. The typical {111} K–S interface has been chosen as the reference system in our simulations. A rectangular bilayer model composed of Cu and Nb crystals is adopted with periodic boundary condition along the XYZ axes. The material properties of Cu used are as follows: shear modulus l = 48 GPa, Poisson’s ratio m = 0.34, and lattice constant a = 0.36 nm. The material properties of Nb used are as follows: shear modulus l = 38 GPa, Poisson’s ratio m = 0.4, and lattice constant a = 0.33 nm. Nucleation of Interfacial Dislocation and Lattice Dislocation Figure 9 shows the nucleation of the glide-type interfacial dislocation when a lattice dislocation approaches the interface. In the interface–DD model, the interface was meshed into grids to examine the shear stress on the interface and determine where a dislocation loop would be nucleated. Once the shear stress exceeds the interface strength in the regions on the interface, the region will be sheared in association with the nucleation of the glide-type interfacial dislocations. The Burgers vector of the nucleated loop can be decided by computing the glide forces with respect to all interface dislocations, fi = seffbinter. The Burgers vector corresponding to the max (fi) will be chosen as the ‘‘trial Burgers vector’’ of the nucleated loop. In Fig. 9a, the two symmetric zones under shear have opposite shear directions; thus, two opposite-signed Burgers

Modeling Interface-Dominated Mechanical Behavior of Nanolayered Crystalline Composites

vectors are assigned on the two zones as different colors in this figure. Due to the interaction with interfacial dislocations l2 and l3, the lattice dislocation l1 moves toward the interface, enters the interface, reacts with l2, and forms a new interface segment l4 as marked in Fig. 9b.53 This reaction process is energetically favorable due to the reduction in line energies of dislocations according to Frank’s rule. l4 grows in association with the reaction of l1 and l2, and it reacts with l3 to form l5 in Fig. 9d. With increasing applied loading to 2 GPa, according to the nucleation rules, only the middle segment l5 on the interface can perform the nucleation process because the line direction is parallel to one trace between the interface plane and gliding planes. The Burgers vector of the dislocation seg   ð11 ment l5 is a2 110  1Þ and the line direction of the  segment is in the 111 direction based on the bcc structure. The lattice dislocation in bcc should be activated, as the direction of PK force on this segment is pointing toward the bcc side. According to the rules for nucleation of lattice dislocations described above,  a lattice dislocation l6 with Burgers   is generated in bcc phase vector as 12 111 ð011Þ shown in Fig. 5f. This nucleation process left a residual dislocation l7 on the interface with a  This calculation result Burgers vector12 ½001ð011Þ.

111

matches well with the results from atomistic simulations.19 Stress–Strain Response With the dislocation–interface interaction rules described in the previous sections, we studied the mechanical response of a Cu/Nb bilayer system under load by using our interface–DD model. In this case, the dimensions of the simulation box are 500 nm 9 500 nm 9 100 nm. Both Cu and Nb phases have the same thickness of 50 nm. Periodic boundary conditions are applied in all three directions. So there are two interfaces located at 25 nm (lower) and 75 nm (upper), respectively, with the normal parallel to z direction. The Cu phase is set in the middle of the composite, whereas the Nb phase fills up the rest of the simulation box (top and bottom parts). Initially, dislocation loops were generated at both interfaces with dislocation density at 1.8 9 1015 m2, and the Burgers vectors for these loops were randomly chosen from Burgers vectors in both Cu and Nb phases. The external load was applied on both x and y directions (in-plane stretch) following the strain rate control method, i.e., e_ xx ¼ e_ yy ¼ e_ app and rzz = 0. To save computation time, the applied strain rate was set to 106 s1,

Fig. 9. (a)–(d) The processes of absorption of lattice dislocation into interface in association with interface shear with new loop nucleation, and (e) and (f) slip transmission across interface in association with the nucleation of lattice dislocation (see details in text).

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Fig. 10. (a) Stress–strain curves for Cu, Nb phases and the Cu/Nb nanolayered composite; (b) the evolution of dislocation densities in Cu, Nb phases, two interfaces and the whole composite; (c) a snapshot showing nucleation and glide of lattice dislocations nucleating from interfaces.

which is still much smaller than the strain rate used in MD simulations. Figure 10a shows the stress versus total strain curves for Cu, Nb phases and the composite. For inplane stretch, the total stress on the composite is calculated based on the conventional rule of mixtures: rcomposite = VCurCu + VNbrNb, where VCu and VNb are the volume fraction; rCu and rNb are the stresses for Cu and Nb phases, respectively; and rcomposite is the stress on the overall composite. From Fig. 10a, we can see that Cu phase yielded earlier than Nb phase during deformation because the lattice dislocations nucleate initially from the Cu phase. After nucleation, the lattice dislocation continues propagating in the Cu phase following the operation behavior of Frank-Read sources. On approaching another interface, the front part of the half dislocation loop was absorbed by the interface and two threading dislocations were formed after the front part of the dislocation loop was truncated by the interface. Under external load, these two threading dislocations propagated in the Cu phase and generated plastic strain to minimize the internal strain energy. Figure 10b shows the evolution of dislocation densities in Cu, Nb phases, two interfaces, and the composite corresponding to the stress–strain curves in Fig. 10a. An increase in the

dislocation density is observed for both the Cu and Nb phases caused by the nucleation of lattice dislocations from interfaces. Once the nucleated half loop touches interfaces, it will form two threading dislocations. In layered composites, the motion of threading dislocations will not contribute to the increase of the dislocation density in phases because the total dislocations length in phases almost stays constant during motion. However, when threading dislocation move in phases, their tails are left on the interface. This action will form interfacial dislocations. This is the main reason for the increase of dislocation densities within interfaces. Figure 10c shows the dislocation structure of a deformed Cu/Nb sample at 2% strain. CONCLUSION In this work, several models are introduced to demonstrate the link between atomic-scale and dislocation-scale interface physics that operate in nanolayered crystalline composites. We first develop a model that classifies interfaces into four structural groups. We then show that these structural groups have characteristic behaviors with regard to interface shear strength, dislocation nucleation, and dislocation–interface interactions. Based on this

Modeling Interface-Dominated Mechanical Behavior of Nanolayered Crystalline Composites

classification scheme, we build an interface–DD model. We demonstrate the ability of this DD model to capture the nucleation of lattice dislocations from interfaces, the glide of threading dislocations within these composites, and the associated stress–strain behavior. Further work regarding interface–DD model development is ongoing.54 ACKNOWLEDGEMENTS The authors acknowledge the support provided by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences. J. W. also acknowledge the support provided by Los Alamos National Laboratory Directed Research and Development projects ER20140450. The authors sincerely appreciate the discussions with Dr. Amit Misra, Profs. J. P. Hirth, and R. G. Hoagland at Los Alamos National Laboratory. REFERENCES 1. A. Misra and A. Gibala, Metall. Mater. Trans. A 30A, 991 (1999). 2. R.G. Hoagland, T.E. Mitchell, J.P. Hirth, and H. Kung, Philos. Mag. 82, 643 (2002). 3. S.J. Zheng, I.J. Beyerlein, J.S. Carpenter, K. Kang, J. Wang, W.Z. Han, and N.A. Mara, Nat. Commun. 4, 1696 (2013). 4. I.J. Beyerlein, N.A. Mara, J.S. Carpenter, T. Nizolek, W.M. Mook, T.A. Wynn, R.J. McCabe, J.R. Mayeur, K. Kang, S.J. Zheng, J. Wang, and M.P. Tresa, J. Mater. Res. 28, 1799 (2013). 5. J. Wang and A. Misra, Curr. Opin. Solid State Mater. Sci. 15, 20 (2011). 6. W.D. Sproul, Science 273, 889 (1996). 7. B.M. Clemens, H. Kung, and S.A. Barnett, MRS Bull. 24, 20 (1999). 8. A. Misra and H. Kung, Adv. Eng. Mater. 3, 217 (2001). 9. M.A. Phillips, B.M. Clemens, and W.D. Nix, Acta Mater. 51, 3171 (2003). 10. A. Misra, J.P. Hirth, and R.G. Hoagland, Acta Mater. 53, 4817 (2005). 11. J. Wang, R.G. Hoagland, J.P. Hirth, and A. Misra, Acta Mater. 56, 5685 (2008). 12. R.G. Hoagland, R.J. Kurtz, and C.H. Henager, Scripta Mater. 50, 775 (2004). 13. J. Wang, R.G. Hoagland, and A. Misra, Scripta Mater. 60, 1067 (2009). 14. J. Wang, R.G. Hoagland, J.P. Hirth, and A. Misra, Acta Mater. 56, 3109 (2008). 15. M.J. Demkowicz, J. Wang, and R.G. Hoagland, Interfaces between Dissimilar Crystalline Solids, in Dislocations in Solids, ed. J.P. Hirth (Los Alamos, NM: Materials Science and Technology Division, Los Alamos National Laboratory, 2008), pp. 141–207. 16. P.M. Derlet, P. Gumbsch, R.G. Hoagland, J. Li, D.L. McDowel, H. Van Swygenhoven, and J. Wang, MRS Bull. 34, 184 (2009). 17. X.Y. Liu, R.G. Hoagland, J. Wang, T.C. Germann, and A. Misra, Acta Mater. 58, 4549 (2010). 18. J. Wang, R.G. Hoagland, and A. Misra, Appl. Phys. Lett. 94, 131910 (2009). 19. J. Wang, A. Misra, R.G. Hoagland, and J.P. Hirth, Acta Mater. 60, 1503 (2012).

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