Deformation mechanisms, length scales and optimizing the mechanical properties of nanotwinned metals

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Acta Materialia 59 (2011) 6890–6900 www.elsevier.com/locate/actamat

Deformation mechanisms, length scales and optimizing the mechanical properties of nanotwinned metals Z.X. Wu a,b, Y.W. Zhang b,⇑, D.J. Srolovitz b a

NUS Graduate School for Integrative Sciences and Engineering, National University of Singapore, 117456 Singapore, Singapore b Institute of High Performance Computing, 138632 Singapore, Singapore Received 17 December 2010; received in revised form 20 May 2011; accepted 16 July 2011 Available online 26 August 2011

Abstract Refinement of microstructural length scales and modification of interface character offer opportunities for optimizing material properties. While strength and ductility are commonly inversely related, nanotwinned polycrystalline copper has been shown to possess simultaneous ultrahigh strength and ductility. Interestingly, a maximum strength is found at a small, finite twin spacing. We study the plastic deformation of nanotwinned polycrystalline copper through large-scale molecular dynamics simulations. The simulations show that plastic deformation is initiated by partial dislocation nucleation at grain boundary triple junctions. Both pure screw and 60° dislocations cutting across twin boundaries and dislocation-induced twin boundary migration are observed in the simulation. Following twin boundary cutting, 60° dislocations frequently cross-slip onto {0 0 1} planes in twin grains and form Lomer dislocations. We further examine the effect of twin spacing on this Lomer dislocation mechanism through a series of specifically designed nanotwinned copper samples over a wide range of twin spacings. The simulations show that a transition in the deformation mechanism occurs at a small, critical twin spacing. While at large twin spacings, cross-slip and dissociation of the Lomer dislocations create dislocation locks that restrict and block dislocation motion and thus enhance strength, at twin spacings below the critical size, cross-slip does not occur, steps on the twin boundaries form and deformation is much more planar. These twin steps can migrate and serve as dislocation nucleation sites, thus softening the material. Based on these mechanistic observations, a simple, analytical model for the critical twin spacing is proposed and the predicted critical twin spacing is shown to be in excellent agreement both with respect to the atomistic simulations and experimental observations. In addition, atomistic reaction pathway calculations show that the activation volume of this dislocation crossing twin boundary process is consistent with experimental values. This suggests that the dislocation mechanism transition reported here for the first time can be a source of the observed transition in nanotwinned copper strength. Ó 2011 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Twins; Dislocations; Plastic deformation; Ultrahigh strength; Molecular dynamics simulations

1. Introduction Optimizing the strength of metallic materials is often achieved through the manipulation of microstructural length scales. Unfortunately, increases in alloy strength are usually accompanied by concomitant decreases in ductility. Most commonly, the microstructural length scales ⇑ Corresponding author. Permanent address: Institute of High Performance Computing, 1 Fusionopolis Way, #16-16 Connexis, 138632 Singapore, Singapore. Tel.: Tel.: +65 64191478; fax: +65 64674350. E-mail address: [email protected] (Y.W. Zhang).

that affect mechanical properties of metals are manipulated through changes in interparticle spacing (precipitates, second phase particles, etc.) or interface (grain boundaries, twins, etc.) separation. Twins are especially good for controlling strength because of their extraordinary stability (relative to other microstructural features) [1]. The small microstructural length scales inherent in nanomaterials open the door to the development of ultrahigh-strength metals [1–7,9–13]. Interestingly, some nanotwinned metals paradoxically exhibit both high strength and high ductility, e.g. in Cu [2–5] and Co [6]. The increase in strength with decreasing grain size/twin spacing is based upon the

1359-6454/$36.00 Ó 2011 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2011.07.038

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interfaces serving as barriers to dislocation migration, resulting (in some cases) in dislocation pile-ups at the interfaces; this is the so-called Hall–Petch effect [14,15] in which the yieldpstress ry scales with the grain/twin size d as ry ¼ ffiffiffi r0y þ A= d , where r0y and A are constants. Surprisingly, Lu et al. [16] demonstrated that in pure, nanotwinned copper, the yield strength exhibits a maximum at a small, finite twin spacing. In addition, they found that while the strength goes through a maximum at a critical twin spacing kc, the strain hardening and ductility increase monotonically with decreasing twin spacing. Earlier simulations [17,18] were unable to reproduce this transition in strengthening behavior at twin spacings below the experimentally observed critical twin spacing kc  15 nm. A recent molecular dynamics (MD) study by Li et al. [37] demonstrated a softening mechanism arising from dislocation nucleation at the intersection of twin and grain boundaries and subsequent dislocation glide along the twin boundaries at small twin spacings (k  0.63–1.25 nm). They suggested that plastic deformation in this unique microstructure switched from strengthening based on Hall–Petch dislocation pile-up to softening through twin migration (i.e. the glide of dislocations/steps along twin boundaries). However, high-resolution transmission electron microscopy (TEM) observations [16] of post-deformation nanotwinned copper samples with small twin spacing (k  4 nm) showed a high density of dislocations blocked at twin boundaries. The presence of these dislocations is not consistent with the MD simulation results and model [37]. Apart from the reported mechanism switch [37,38], other plastic deformation mechanisms were observed to be operative in the simulations with increasing twin spacing. The existence of the experimentally observed maximum in the strength of nanotwinned copper (without extrema in either the hardening coefficient or strain to failure) with respect to twin spacing, the discrepancy between previous simulation results and the experiments, and the experimental observations of many dislocations within the twins all suggest the presence of another dislocation mechanism operating in nanotwinned metals. It is this new dislocation mechanism and its importance in explaining the experimentally observed plastic deformation behavior of nanotwinned metals that is the subject of this paper. In this work, we perform large-scale MD simulations of nanotwinned polycrystalline copper. These simulations show that nanotwinned copper deforms not only through the previously reported twin migration mechanism [37], but also through various types of dislocations cutting across twin boundaries. The present polycrystalline simulations demonstrate the frequent, widespread formation of Lomer dislocations that dissociate and propagate across multiple grain boundaries, as we discussed earlier based upon bicrystal simulations [13]. To understand this mechanism more clearly, we perform an additional series of simulations of multiply twinned grains with several different twin spacings. A major change in the deformation mechanism is observed when the twin spacing is reduced below a critical value.

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Analysis of this mechanism change demonstrates its good correlation with the experimentally observed maximum in the yield strength vs. twin spacing in nanotwinned copper. Using atomistic reaction pathway calculations [11,39–43], we also show that the activation volume of this important process is consistent with experimental values [8]. Our simulation results, together with previous simulations [11,13,37] and experimental observations [16], paint a detailed picture of the plastic deformation mechanisms that operate in nanotwinned copper and distinguish which changes in mechanism are responsible for the experimentally observed change in the strength of this new and interesting class of material as a function of twin spacing. 2. Methods In nanotwinned crystals, the two grains, labelled matrix (M) and twin (T), have a mirror symmetry about the (1 1 1) plane as shown in Fig. 1a. We use the classical Thompson tetrahedron [13,20,21] construction to visualize the facecentered cubic (fcc) dislocation slip systems for the above matrix and twin grains (see Fig. 1b). The surfaces of the tetrahedra in the figure represent {1 1 1} slip planes. Plane a is opposite corner A and its center is labeled a; similarly for the other slip planes in the matrix grain. The notation for Thompson tetrahedron features in the twin grain is represented by the superscript T; hence, aT, AT and aT. The twin (mirror) plane is alternatively described as ABC (i.e. d) or BTATCT (i.e. dT) plane in the matrix and twin grains, respectively. Dislocation Burgers vectors are indicated by ordered pairs of points, e.g. Ac or AD. Dislocations with Burgers vector AB form pure screw dislocations when they encounter the twinning boundary ABC while those with Burgers vector DA or DB form 60° dislocations. We refer to Fig. 1b whenever dislocation Burgers vectors and notations are used in this paper. MD simulations were performed using the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) [29] in which the atomic trajectories are determined by integrating the equations of motion for each atom in the system. The interactions between copper atoms are described using the embedded atom method (EAM) [30] potential parametrized by Mishin et al. [19]. This potential was fit to experimental and ab initio data for copper and has been shown to accurately reproduce many material properties, including the stacking fault energy cSF—an important property in deformation studies. In the MD simulations in this paper, periodic boundary conditions were imposed in all directions. The temperature and the stress tensor were controlled through a Nose´–Hoover temperature thermostat and pressure barostat [31–34], respectively. Uniaxial tensile loading was simulated at a constant true strain rate of 0.1 ns1 by stretching the simulation box in one direction while the other two dimensions were adjusted through the Nose´–Hoover pressure barostat such that zero normal stress was maintained. Atomic configurations were visualized in AtomEye [35]. In the following, we describe

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(a)

(b)

Fig. 1. Schematic illustrations of the (a) crystallographic orientation between nanotwins and (b) slip systems for nanotwinned grains. (a) The crystal consists of a matrix (M) and a twin (T) grain. (b) The slip systems in the nanotwinned crystal represented by the fcc twin hexahedron, formed from the Thompson tetrahedra for the matrix and twin grains meeting on the {1 1 1} mirror plane, ABC. The upper tetrahedron ABCD represents the slip systems in the matrix grains. The four {1 1 1} slip planes opposite vertices A, B, C and D in the matrix grains are labeled a, b, c and d and their midpoints are denoted as a, b, c and d, respectively. Burgers vectors for perfect dislocation are formed by pairs of Roman letters (e.g. DB), while partial dislocation Burgers vectors are formed by Greek–Roman pairs (e.g. cB). We use symbols with superscript T to describe the slip systems for twin grains in the lower Thompson tetrahedron.

the plastic deformation process of the polycrystalline simulation first, followed by simulations focusing on the twin spacing variation. 3. Polycrystalline molecular dynamics simulations We constructed a polycrystalline simulation cell through a Voronoi procedure as illustrated in Fig. 2. We first created a group of Voronoi sites arranged in a body-centered cubic (bcc) lattice (note: this is not the copper crystal lattice). The Voronoi polyhedron associated with each Voronoi site is a truncated octahedron as shown in Fig. 2a— these polyhedra represent the grains in the polycrystal. The cubic simulation cell encloses 16 such Voronoi sites and its dimensions were scaled to be 90  90  90 nm as shown in Fig. 2b. The 16 truncated octahedral grains were populated with nanotwinned fcc crystals of random

(a)

(b)

orientations (note: this twinned fcc lattice represents the actual positions of the atoms) such that the total number of atoms in the simulation unit cell is 61,386,312 (the grain size is 45 nm and the twin spacing is 11.27 nm). The atomic configuration was relaxed at 0 K via a conjugate gradient algorithm. The system was then heated to a temperature of 900 K and the hydrostatic pressure was increased to 1 GPa over 100 ps followed by cooling to room temperature (300 K) and the reduction of the hydrostatic pressure to zero within 50 ps. The system was then subject to uniaxial tension along the z-axis (see Fig. 2a). Fig. 2c and d show two views of the atomistic configuration of the simulation cell in which only atoms that are not in perfect fcc local environments are shown. Fig. 3 shows the evolution of the defect structure in the nanotwinned polycrystalline copper sample during tensile loading. The onset of plastic deformation is seen in

(c)

(d)

Fig. 2. The polycrystalline molecular dynamics simulation cell. (a and b) The polycrystal was constructed from 16 Voronoi cells (red frames) that were each seeded from one of the dark blue points in (a); these points were arranged on a body-centered cubic lattice. (c and d) The Voronoi cell grains were populated with copper atoms on fcc lattices (each grain has a unique, random orientation). The crystals that make up each grain are twinned (the twin boundaries are on (1 1 1) planes and are uniformly spaced). The atoms are shown only if their central symmetry parameters [22] differ from that of the perfect fcc crystal; the colors indicate the local symmetry: twin boundaries, dislocations, intrinsic and extrinsic stacking faults are shown in dark brown, green or yellow (depending on dislocation type), magenta and dark brown, respectively. The same coloring scheme is applied throughout this paper. (c and d) only differ by a uniform rotation of the simulation cell (that in (d) is viewed along the h1 1 1i direction). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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Fig. 3. Dislocation evolution in the nanotwinned polycrystalline copper during tensile loading. (a) Dislocation nucleations at grain boundary triple junctions. (b) Dislocations passing through some twin boundaries and forming Lomer dislocations on the {0 0 1}T plane. Twin migration through a twinning dislocation is also observed in some of the grains. (c and d) show the dislocation/twin/grain boundary microstructure at true strain of 7.5% and 8.5%, respectively. In (b–d), the blue ellipses highlight sites where Lomer dislocations form and dissociate. For details of the evolution of the microstructure during plastic deformation and more of such mechanisms, see the movie available as online Supplementary material.

Fig. 3a, where leading partial dislocations first nucleate at grain boundary triple junctions. These partial dislocations glide until they impinge twin boundaries and leave behind wide stacking faults (10 nm). Fig. 3b shows the sample at a later stage of deformation, where dislocations (both pure screw and 60°) start to cross twin boundaries. Those pure screw dislocations (e.g. AB in Fig. 1b) pass from matrix {1 1 1} to twin {1 1 1}T planes each time they cross a twin boundary. This process leaves the twin boundary intact. In addition, our polycrystalline simulation also shows that pure screw dislocations that cross-slip onto twin boundaries (forming Ad and dB) created pairs of steps on these twins. The above two mechanisms are known as the transmission and absorption of pure screw dislocation at twin boundary as previously studied by Jin et al. [10] and Zhu et al. [11] in bicrystal set-ups. The interactions of 60° dislo-

cations (e.g. DA, DB) with twin boundaries are different from those for pure screws [12]. In our polycrystalline simulation, Lomer dislocations frequently form on {0 0 1}T planes after 60° dislocations cross twin boundaries. Locations where such a dislocation mechanism can be seen are highlighted by blue ellipses in Fig. 3. The Lomer dislocation can be seen to glide on the unusual {0 0 1}T plane in the twinned material (see the blue ellipses in Fig. 3b) because of the very high Schmid factor at these locations. Lomer dislocation generation from twin boundaries was also reported previously in Cu/Ni composite [24], Cu nanowires [25], Au nanopillars [17] and by Ezaz et al. [26]. Wu et al. [13] demonstrated that such Lomer dislocations were able to cross-slip and dissociate into Shockley and stair-rod dislocations, provided (of course) that the dislocation line segments were pure screw.

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In addition to the dislocation–twin boundary interactions, twin boundary migration was also observed in some grains via a step (partial dislocation) mechanism (see Fig. 3b) in the simulation. Li et al. [37] attributed the softening that occurs in nanotwinned copper at small twin spacings to this twin boundary migration mechanism. However, close examination of Fig. 3b only reveals two twin steps despite the fact that the total twin area in the sample is quite large (the initial total twin boundary area is 7 lm2). On the other hand, Fig. 3 shows many dislocations crossing between twins. Analysis of the deformation at 3% strain suggests that the ratio of the plastic strain associated with twin boundary migration to that from dislocation migration between twin boundaries is approximately 0.15. The simulation here and those of Li et al. [37] show that dislocation nucleation at the twin boundary/grain boundary junction and subsequent twin boundary migration is not the dominant deformation mechanism for twin spacings above 3 nm. (Note: dislocations that cross the twin boundaries can also leave steps on the twin boundaries and these steps can also contribute to twin boundary migration. This is a consequence of the dislocations cutting twin boundaries and is a different mechanism to that suggested by Li et al. [37].) Fig. 3c and d show the system at a true strain of 7.5% and 8.5%, respectively. In these figures, many leading partial dislocations were blocked by twin boundaries while the trailing partial dislocations had not yet been nucleated. In some cases, the trailing partial does form, combines with the leading partial at a twin boundary (under the action of the applied stress), cuts through the twin boundaries and forms a Lomer dislocation (on a {0 0 1}T plane). The number of dislocations that cut through twin boundaries is small relative to the total number of dislocations formed. Throughout the deformation process, the total plastic strain in the polycrystal was created mainly by the migration of three types of dislocations, namely twinning dislocation, pure screw dislocation and 60° dislocation. Our simulation demonstrates that the migration of various types of dislocations within each grain is necessary to satisfy the general plastic strain. (An animation showing the evolution of the microstructure during plastic deformation is available as online Supplementary material.). The fact that the yield stresses in the experimental [16] nanotwinned polycrystalline copper samples (k P 15 nm) follow a Hall–Petch relation with respect to twin spacing (i.e. the yield strength increases as the inverse square root of the twin spacing) suggests that the twin boundaries are finite barriers to dislocations and it is dislocations crossing twin boundaries that are critical in the yielding process. Since the formation and dissociation of Lomer dislocations through 60° full dislocations passing through twin boundaries occur readily in our simulation (and its dissociation mechanism depends on its radius of curvature and the twin spacing) and can be the event that determines the flow stress, we designed a series of simulations to explicitly examine the effect of twin spacing on this mechanism. We do this by tracing the positions and type of dislocations

(and their reactions) through the plasticity evolution process in each of the simulations. Interestingly, a dislocation mechanism transition occurs when the twin spacing reaches a lower critical value in our simulation. We describe these simulations in detail in the following section. 4. Dislocation deformation mechanism as a function of twin spacing 4.1. Simulation model We designed a set of simulations with simulation cells constructed as shown schematically in Fig. 4. The simulation cells consist of two types of twin-related grains M (matrix) and T (twin). A pair of dislocations with Burgers vectors DB and BD were introduced in the upper and lower grains in Fig. 4, respectively. This is achieved by displacing atoms from their original perfect fcc lattice positions to new positions according to their isotropic, linear elastic displacement fields [20]. If two atoms were too close together, one was removed and, if voids were detected, additional atoms were inserted in the structure to fill the voids (as dictated by crystal symmetry) prior to the beginning of the simulations. The unstrained simulation cell has dimensions 70.8  170  20.4 nm in the x, y and z directions, respectively, corresponding to 20,850,080 copper atoms, and the twin boundary spacing k is varied between 1.88 and 31.3 nm (i.e. the number of twins varies with the choice of k). These atomic configurations were relaxed at 0 K via a conjugate gradient algorithm. These systems were then heated to room temperature (300 K) within 100 ps and held at room temperature for another 50 ps. Uniaxial

Fig. 4. Schematic illustrations of the molecular dynamics unit cell. The simulation cell consists of alternating matrix (M) and twin (T) grains, each of thickness k. The crystallographic orientations of the matrix and twin grains are also shown. h1 1 0i dislocations DB and BD are introduced in the top and bottom grains at the beginning of the simulations, which then dissociate into Shockley partial dislocations (Dc + cB). (Refer to Fig. 1b for Burgers vectors and notations.)

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tensile loading was then carried out along the y direction in Fig. 4 (see Section 2 for details on the MD ensemble). The Burgers vector of the dislocations in the grains were determined using the classical Burgers circuit construction, while those on the twin boundaries were deduced by applying the conservation of Burgers vector between slip transfers. 4.2. Simulation Results Fig. 5 shows an atomic view (only atoms in non-fcc local environments are shown) of the evolution of the dislocations as they pass through twin boundaries for (a) large (k = 18.8 nm) and (b) small (k = 1.88 nm) twin boundary spacing simulations. Fig. 6 is an idealized illustration of the dislocation processes exhibited in Fig. 5. In both figures, time and strain evolve from left to right. 4.2.1. Deformation at large twin spacings We first summarize the evolution of the dislocation microstructure at large twin spacings (k = 18.8 nm) in atomistic detail and in schematic form in Fig. 5a and Fig. 6a. A pair of partial dislocations from the dissociated 60° dislocation (Dc + cB on slip plane c—see Fig. 6a1) constrict into a full dislocation (DB) at the intersection of slip plane c and twin boundary TB1 (Fig. 6a2). This full dislocation cross-slips onto a {0 0 1}T plane in the twin grain, thus forming a {0 0 1}Th1 1 0iT Lomer dislocation (CTDT in Fig. 6a3). While the remaining DB segments on the twin boundary continue to cross-slip, the cross-slipped Lomer dislocation glides further and adopts a semicircular shape under the resolved shear stress (see Fig. 6a4). As the Lomer dislocation line evolves, part of the dislocation line (near the

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twin boundary) rotates such that its line direction becomes parallel to its Burgers vector, thus forming a pure screw Lomer dislocation segment. Since {0 0 1}T, aT and bT planes intersect at a single line, this screw segment cross-slips onto one of the {1 1 1}T slip planes in the twin grain (bT in Fig. 6a5) and dissociates into the usual Shockley partial pair (CTbT and bTDT). These cross-slip and dissociation events are energetically favorable (based on Frank’s dislocation reaction criterion [20]). The newly formed Shockley partial pair glide on their usual {1 1 1}T slip plane and form a 60° dislocation (equivalent to the initial DB) when they impinge on the next twin boundary, allowing the Lomer dislocation generation–dissociation process to repeat. Cross-slips of pure screw segments onto the aT plane with Shockley partials CTaT and aTDT are equally possible under the loading condition in our simulation. In addition, we observe the pure screw Lomer dislocations cross-slipping onto both aT and bT to form a stair-rod segment at the intersection of the above two slip planes. This is illustrated in Fig. 6a7– a10, where a different view with all the Burgers vectors labeled is also shown (see the inset of Fig. 5a5 and Fig. 6a7). The Shockley dislocations, cross-slipped from the Lomer, glide on their respective {1 1 1}T slip planes while dragging the sessile stair-rod dislocation aTbT through a junction zipping–unzipping process [23]. Those Shockley partial dislocations and remaining Lomer dislocation segments glide on different slip planes, restricted by the sessile junctions that are formed where they meet. The restriction on the motion of the Lomer dislocation further curves the dislocation line, making cross-slip and dissociation favorable [13]. Through the cross-slip and dissociation of Lomer dislocations, the system develops a complicated 3D dislocation structure containing both glissile and sessile

Fig. 5. Atomistic view of dislocations passing twin boundaries for case (a) large (k = 18.8 nm) and (b) small (k = 1.88 nm) twin boundary spacing (the image shows the same size section of the simulation cell in each case). In both figures, time evolves from left to right (1–6). For k = 18.8 nm, (a1) a segment of the dissociated 60° dislocation constricts to pass through a twin boundary. (a2) The dislocation segment cross-slipped onto a {0 0 1}T plane forming a semicircular {0 0 1}Th1 1 0iT Lomer dislocation. (a3) A pure screw segment of the Lomer dislocation starts to cross-slip onto bT plane. (a4) Additional pure screw segments of the Lomer dislocation cross-slip onto aT and bT. (a5) Pure screw segments of the Lomer dislocation cross-slip onto aT and bT and form a stair-rod dislocation at the junction (see inset of Fig. 6a7 for their dislocation Burgers vectors). (a6) A late stage of the deformation. For k = 1.88 nm, (b1) shows the dissociated 60° dislocation constricting to pass through a twin boundary, as in (a1). (b2 and b3) show a dislocation passing subsequent twin boundaries, leaving behind intrinsic stacking faults in the matrix grains, but not the twin grains. (b4) shows a segment of the Lomer dislocation gliding in the twin grain without cross-slip or dissociation (see Fig. 6b for details). (b4 and b5) show that the intrinsic stacking faults in the matrix grain are converted to extrinsic stacking faults by the passage of an additional Shockley partial. (b6) A late stage of the deformation showing a structure with extrinsic stacking faults bridging matrix grains and Shockley partial dislocations on each twin boundary.

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Fig. 6. Schematic illustration of dislocations passing twin boundaries with different twin spacings, corresponding to the atomistic images in Fig. 5. The keys identifying the slip planes and stacking faults are shown above the dislocation schematics. The inset of (a7) shows a Lomer dislocation segment crossslipped onto the aT and bT planes, forming a stair-rod dislocation segment at the intersection. The inset of (a10) illustrates the above dislocation structure migrating from “p1” to “p2” through a junction zipping–unzipping process.

dislocations on various slip planes. Fig. 5a6 shows such a dislocation structure at a later stage of the deformation. 4.2.2. Deformation at small twin spacing Fig. 5b shows the case where the twin spacing k = 1.88 nm. We use Fig. 6b as a schematic illustration to describe the dislocation activity. A Lomer dislocation CT DT is formed as shown in Fig. 6b1–b3 through the same mechanism as described earlier in Fig. 6a1–a3. However, instead of dissociation or cross-slip, CTDT glides on the {0 0 1}T plane in the twin grain until it impinges the next twin boundary TB2 (see Fig. 6b4). The Lomer dislocation cross-slips to the usual {1 1 1} slip plane with a Shockley partial dislocation Dc in the matrix grain after passing through TB2. This leaves a dislocation with combined Burgers vector cB + dA on that twin boundary. An intrinsic stacking fault is generated as Dc glides in the matrix grain (Fig. 6b5). Dc is blocked as it meets the next twin boundary, i.e. TB3. At a later stage, the blocked leading Shockley partial dislocation Dc cross-slips onto a {0 0 1}T plane in the next twin grain, forming a new {0 0 1}T h1 1 0iT Lomer dislocation CTDT as shown in Fig. 6b7. This leaves a dislocation of combined Burgers vector Bc + Ad on TB3. The newly formed Lomer CTDT glides in the twin grain until it meets the next twin boundary TB4 (see Fig. 6b8). As the Lomer glides, a Shockley partial dislocation cB glides from TB2 in the matrix grain and annihilates with dislocation Bc on TB3. However, cB glides on the {1 1 1} plane adjacent to the one on which Dc previously glided, thus forming an extrinsic stacking fault. This process is illustrated in Fig. 6b9. The resulting configuration is shown

in Fig. 6b10. The Lomer dislocation CTDT on TB4 now has the same configuration as the previous one on TB2 in Fig. 6b4. This means that the whole process repeats at every second twin boundary as observed in the MD simulations and shown in Fig. 5b1–b5. In summary, the dislocation glides on {1 1 1} and {0 0 1}T planes, alternately in the matrix and twin grains, leaving behind a Shockley partial dislocation on each twin boundary and an extrinsic stacking fault in each of the matrix grains. Examining the deformation process in simulations performed over a wide range of twin spacing k suggests that this process of cross-slip and dissociation does not occur at small twin spacings. We suggest that this change in dislocation mechanism is a source of the experimentally observed maximum in the strength of nanotwinned copper as a function of twin spacing, as discussed below. Comparison of Fig. 5a6 and b6 suggests a different dislocation structure results depending on the operative dislocation mechanism. The former mechanism involves the cross-slip and dissociation of a Lomer dislocation that forms a complex 3-D dislocation network, while in the later case, slip is constrained to the original slip system such that a 2-D dislocation microstructure is maintained. The feature that is common between these two cases is that both processes repeat. 5. Analytical model and atomistic reaction pathway calculations The simulations in Section 4 show that the transition in deformation mechanism resulted from the cross-slip and

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dissociation of the Lomer dislocation. But what governs this change? As the applied stress increases and eventually reaches a critical level rc, the constricted 60° dislocation segment passes through the twin boundary and cross-slips to form a Lomer dislocation segment. The newly formed Lomer segment adopts an arc shape under the action of this stress rc, the two ends of which are on the twin boundary (see Fig. 7a). The equilibrium radius of curvature rc of the arc can be approximated through the balance between the applied shear stress rc and its line tension T: rc ¼

T ; brc

ð1Þ

where b is the Burgers vector of the dislocation. As time evolves, more of the original 60° dislocation passes through the twin boundary and cross-slips onto the {0 0 1}T plane; the Lomer dislocation arc grows larger and adopts a new configuration (this is the evolution from the dashed arc “s1” to “s2” in Fig. 7a). However, Eq. (1), governing the radius of curvature of those arcs, remains valid and the evolving dislocation arc in Fig. 7a retains the same radius of curvature. As this process proceeds, the Lomer dislocation eventually attains a semicircular shape (denoted as the solid halfcircle in Fig. 7a) and the sections of the dislocation arcs that are near the twin boundary become parallel to the Burgers vector and hence become pure screw dislocation segments. It is these pure screw segments that can cross-slip and dissociate easily, as observed in the MD simulations. The Lomer dislocation is able to form this semicircular configuration (and form pure screw segments) if the expanding loop can continue to grow into the semicircular geometry before it hits the next twin boundary. In other words, the cross-slip process can only occur if the twin spacing is larger than the semicircular arc radius of curvature rc. In cases where the twin spacing k is smaller than rc, such as those shown in Fig. 7b and c, the leading Lomer dislocation arc is blocked by the next twin boundary. In

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this case, the minimum angle between the dislocation Burgers vector and its line direction /m does not go to zero (i.e. a pure screw does not form). Hence, the Lomer dislocation will not form a semicircular shape, no screw segments are formed, and no cross-slip and dissociation are possible. Therefore, it is the relative size of the critical radius of curvature rc and the spacing between twin boundaries k that determines which deformation mechanism will operate. We now validate this conjecture through quantitative comparisons between this prediction and our MD simulations. The simulations show that for all cases where k P 3.13 nm, Lomer dislocation cross-slip and dissociation occur. For the smallest twin boundary spacing for which MD simulations were performed k = 1.88 nm, no cross-slip or dissociation occurs. This implies that the critical twin spacing for the Lomer dislocation to form pure screw segments (a necessary condition for cross-slip) lies between 1.88 and 3.13 nm. In order to calculate the critical twin boundary spacing/critical radius of curvature, we must calculate the dislocation line tension T. In the isotropic elastic limit (and neglecting the dependence of line tension on line direction), we can write T = alb2 [20], where a  0.5 [27,28]. For copper, we use this definition of the line tension and the experimental values of the parameters: l = 48 GPa and b = 0.256 nm. For rc in Eq. (1), we take rc = 1.5 GPa, which is the minimum applied stress resolved onto the slip system needed to form the Lomer dislocation into the twin grain. This stress, corresponding to a 5.5 GPa uniaxial tensile stress, is the minimum stress required to pass dislocations through the entire twinned structure and hence can be thought of as the yield strength of the materials for the present simulation conditions (the yield stress for the polycrystalline simulation described in Section 3 is 2.2 GPa). Using this approach, the predicted transition in deformation mechanism should occur at k = rc  2.4 nm. This theoretical value is consistent with our MD simulation observation for where the dislocation mechanism changes, i.e. 1.88 nm < k < 3.13 nm.

(b)

(c)

Fig. 7. Schematic illustrations of the Lomer dislocation gliding in the twin grain at different twin spacings. /m is the minimum angle between the dislocation Burgers vector b and its line direction. (a) The Lomer dislocation evolves from the dashed arc “s1” of size L1 to “s2” of size L2. The expanding Lomer dislocation arch becomes a semicircular half-loop (shown as a solid arc) at size L3. The line segments of the half-loop in contact with twin make an angle /m = 0 with respect to the Burgers vector and hence become locally pure screws (denoted as segments with magenta color). The radius of the semicircular loop rc is equal to the twin spacing k (measured in the slipping plane) in this case. (b) The Lomer dislocation gliding in a twin grain with a twin spacing that is too small for the arc to become semicircular and no pure screw segments are formed. In this case, the leading Lomer dislocation arc is block by twin boundary TB2. (c) The Lomer dislocation gliding in a grain with even smaller twin spacing. As in (b), no screw segments are formed. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

It is not reasonable to expect that the simulation prediction will carry directly over to experiments, given the fact that MD simulations are performed on a much shorter time scale, necessitating much higher loading rates than those used experimentally. Since flow stress typically increases with strain rate, this necessitates re-examining how rc is determined for the sake of comparison with experiment. If we replace our estimate of rc with the experimentally measured tensile yielding stress 1 GPa of nanotwinned copper [3] in Eq. (1) and use the line tension above, we find a critical twin spacing kc approximately 5 times that obtained directly from the MD simulations. This suggests that the critical twin spacing for the deformation mechanism transition occur is 13 nm. This is in excellent agreement with the experimentally observed maximum strength occurring at k = 15 nm. The above analysis demonstrates that the prediction for the twin spacing corresponding to the maximum strength obtained from the MD simulations is consistent with the experimental observations. However, this does not prove that the dislocation mechanism discussed above is indeed the dislocation mechanism that controls the yielding behavior of nanotwinned Cu. Zhu et al. [11] used atomistic reaction pathway calculations to study pure screw dislocation interactions with a twin boundary in a bicrystal and concluded that it is this interaction that controls the plasticity of nanotwinned Cu. Since our large-scale MD simulation suggests that both 60° and pure screw dislocations contribute to the plasticity of the polycrystalline nanotwinned material, it is imperative to inquire as to which mechanisms are more important. To address this question, we perform nudged elastic band method calculations for a 60° dislocation passing through a twin boundary. The resultant activation path data is then used to relate the results from high strain-rate MD simulations to the much lower strain-rate experiments. The activation path simulations were performed using the same procedure as employed by Zhu et al. [11]. Fig. 8 shows the activation energy and activation volume obtained from these nudged elastic band calculations. The zero-temperature activation energy Q0 is accurately fitted by the standard functional form [43,44]  a r ; ð2Þ Q0 ðrÞ ¼ A 1  rath where r is the applied shear stress, a, A and rath are fitting parameters. A and rath represent the zero stress activation energy and athermal shear stress. This procedure yields an activation energy of A = 14.8 eV, an athermal shear stress of rath = 2.0 GPa and a = 1.9. The activation volume, X = oQ/or, is a (nearly linear) function of the applied stress. We now compare the activation volumes for the 60° dislocation transmission and that for the screw dislocation transmission at the same activation energy, 0.67 eV (corresponding to the laboratory strain rate [8,11]). For the screw dislocation mechanism, this yields an activation volume of 44b3. For the 60° dislo-

1.5

46

1.4

44

1.3

42

1.2

40

1.1

38

1

36

0.9

34

0.8 0.7

32

0.6

30

0.5 1.4

1.45

1.5

1.55

1.6

Activation volume Ω (b3)

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Activation energy Q0 (eV)

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28 1.65

Shear stress σ (GPa) Fig. 8. Nudged elastic band calculation of the activation energy and volume for a 60° dislocation passing through a twin interface as a function of the applied stress. The red circles represent the calculated activation energy as a function of shear stress, the blue line is the best fit to the functional form of Eq. (2), and the purple line is the activation volume, X = o Q/or. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

cation mechanism considered here, the activation volume is 31b3. While the activation volume for the 60° dislocation mechanism is high compared with experiment [8], it is much closer to the experimental value than that for the screw dislocation mechanism of Zhu et al. [11]. Nonetheless, given that the required stress for the screw mechanism is lower than that for the 60° dislocation mechanism, and the uncertainty in extrapolation to the experimental conditions, we can only conclude that the 60° and screw dislocation mechanisms are competitive at the experimental conditions. This is consistent with our observation that both mechanisms operate in the large-scale, polycrystal MD simulations of nanotwinned Cu. 6. Discussion The large-scale MD simulations of polycrystalline, nanotwinned Cu suggest that there are three important deformation mechanisms operating: twin migration, a pure screw dislocation transmission mechanism, and the transmission of 60° dislocations. The first two mechanisms were considered previously by Li et al. [37] and Zhu et al. [11]. In this paper, we describe the details of the 60° dislocation transmission mechanism. Using activation pathway calculations and extrapolation to experimental conditions, we demonstrated that this new mechanism is (at least) competitive with the screw dislocation mechanism. Based on the critical stress criterion, plastic deformation via the glide of pure screw dislocations is much easier and hence more favored in the nanotwinned structure. However, the polycrystal simulation in Section 3 suggests that both types of dislocation mechanisms, in addition to twinning dislocations, operate in order to satisfy the required general plastic strain condition. TEM observations of experimental sam-

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ples (with grain sizes an order of magnitude larger than those simulated here) show that a large density of dislocations exist in the grain interior between twin boundaries. This demonstrates that transmission of dislocations through twin boundaries is widespread during the largestrain plastic deformation of nanotwinned metals. While the twin boundary migration mechanism has been suggested to explain the observed peak in the yield stress vs. twin spacing data [37], the screw dislocation mechanism has not been able to explain this strength transition. On the other hand, in the present study, we have demonstrated that the existence of a critical twin size can be explained in terms of the dislocation half-loop argument in 60° dislocation transmission mechanism and that this spacing is in good agreement with the experimental observations. We now examine the detailed mechanisms by which the 60° dislocation transmission mechanism gives rise to the observed transition in the strength. The fact that the three slip planes, {0 0 1}T, aT and bT, intersect at a single line and the dislocation energy is reduced by dissociation leads to the spontaneous cross-slip and dissociation of the pure screw Lomer dislocation. The cross-slip and dissociation of the Lomer dislocation simultaneously activates several slip systems and leads to the evolution of a complex 3-D dislocation structure. This structure favors the formation of many Lomer–Cottrell dislocation locks [20] and hardening. Sessile dislocations resulting from the Lomer dislocation cross-slip and dissociation can restrict the motion of the Lomer dislocation itself and also serve as barriers for other dislocations in the system. In the small k case, where the Lomer dislocation does not form a pure screw segment and no cross-slip and dissociation occur, Shockley partial dislocations with alternating Burgers vectors are produced on each twin boundary during Lomer dislocation glide. The number (and density) of these Shockley partials at twin boundaries increases with decreasing twin spacing. These Shockley partial dislocations correspond to steps on the twin (their Burgers vectors lie in the twinning plane). While these twin steps may lead to softening of the material via slip along the twin boundary [16], they can also serve as dislocation emission sites, as found in MD simulations [18]. Our simulations thus indicate the operation of both a strengthening mechanism (Lomer cross-slip and dissociation) and a mechanism for increased ductility (through twin step formation), both of which operate above and below a critical twin spacing. The above analysis on the critical radius for reaching a pure screw configuration is demonstrated and verified for this special case of Lomer dislocation half-loop. However, we would expect the above theory to be generally applicable to other types of dislocation half-loops. The proposed critical radius of curvature controls the formation of pure screw dislocation segments, which in turn determines the cross-slip/non-cross-slip behavior of dislocations in the nanotwinned microstructure. Earlier studies [11,16,36] suggest that the plastic deformation process in this unique microstructure is related to

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dislocation–twin interactions and those interactions become dominant in samples with small twin spacings. Lu et al. [16] showed that while dislocation tangles and networks are formed in samples with large twin spacings, dislocations form stacking faults along parallel planes and do not cross-slip in situations where the twin spacing is small. The two different dislocation mechanisms operating at large and small twin spacings first reported here result in dislocation patterns which are consistent with the above experimental facts. Hence the mechanism suggested in this paper matches closely, not only in the transition length scale, but also in the resultant dislocation pattern. The experimental data on the deformation of pure, nanotwinned copper [16] suggest that it is possible to obtain simultaneously ultrahigh strength and ductility. The peak in the yield stress occurs at a twin spacing k  15 nm. While no such peak is observed in either the hardening coefficient nor the strain to failure, both of these quantities show a monotonic increase with decreasing k. Nonetheless, close examination of the stress–strain curves from the nanotwinned copper experiments reported in Ref. [16] indicates a rapid rise in the ductility (strain to failure) when k is decreased below 20 nm (there is only one experimental data point reported for 10 nm < k < 35 nm). This observation is consistent with a change in deformation mechanism at a critical twin spacing that is very close to the measured (and predicted) kc. The pronounced increase in the ductility at k < kc may be attributed to the formation of steps on the twin boundary in this twin spacing regime, as discussed above. 7. Conclusion We present large-scale MD simulations of plastic deformation of polycrystalline nanotwinned copper. It is found that dislocation nucleation from grain boundary triple junctions initiates plastic deformation in the simulations. Twin boundary migration via twin steps formed either by Shockley partial dislocation nucleation at twin–grain boundary junctions or by absorption of pure screw dislocations are observed. Pure screw and 60° dislocations crossing twin boundaries are more dominant in our simulations and the latter generates Lomer dislocations which further dissociate into Shockley and stair-rod partial dislocations. Simulations of copper samples over a wide range of twin spacing were also carried out and these simulations reveal a transition in the deformation mechanism at a small, critical twin spacing. Based upon these observations, we proposed an analytical model for the magnitude of the critical twin spacing. The critical twin spacing and activation volume derived from the atomistic simulation results are in good agreement with the experimental data. This supports our proposal that the experimentally observed peak in the strength of nanotwinned copper with twin spacing is associated with a change in the dominant dislocation mechanism.

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