Nanostructure and surface effects on yield in Cu nanowires

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Acta Materialia 61 (2013) 1831–1842 www.elsevier.com/locate/actamat

Nanostructure and surface effects on yield in Cu nanowires Z.X. Wu a, Y.W. Zhang a, M.H. Jhon a, J.R. Greer b, D.J. Srolovitz c,d,⇑ a Institute of High Performance Computing, A*STAR, Singapore 138632, Singapore Division of Engineering and Applied Science, California Institute of Technology, Pasadena, CA 91125, USA c Department of Materials Science and Engineering, University of Pennsylvania, Philadelphia, PA 19104, USA d Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia, PA 19104, USA b

Received 1 November 2012; received in revised form 16 November 2012; accepted 17 November 2012 Available online 8 January 2013

Abstract The yield strengths of nanomaterials are highly sensitive to their internal and surface structures. However, it is difficult to identify a priori which structural feature will govern plastic yield. We employ very large scale molecular dynamics simulations to explicitly identify the relevant yield mechanisms for Cu nanowires with four distinct, experimentally realizable nanostructures: single crystal (SC), nanotwinned single crystal (NTSC), nanocrystal (NC) and nanotwinned nanocrystal (NTNC). By characterizing the deformation at the yield point on the atomic scale, our simulations elucidate the effects of surface defects, nanotwins and grain boundaries on the commencement of yield and reveal several critically important features of the yielding process. First, the initial yields in all nanowires occur via dislocation nucleation at different characteristic nanostructural features. SC and NTSC nanowires yield via dislocation nucleation from surfaces or surface defects, while NC and NTNC nanowires yield via dislocation nucleation from grain boundary triple junctions. Second, our simulations highlight the relative potency of stress concentrators arising from different imperfections in modulating the yield strength of nanowires. Grain boundary triple junctions are as effective as surface defects at acting as stress concentrators. However, the higher density of triple junctions in NC and NTNC nanowires renders these structures considerably weaker than their SC and NTSC counterparts. Third, the presence of nanotwins only marginally enhances the yield strength of nanocrystalline Cu nanowires, which is in line with experimental observation in NTNC Cu nanowires but contrary to that in bulk ultrafine-grain nanotwinned Cu. The reason for this divergent behavior is that in nanowires yield strength is governed by dislocation nucleation from triple junctions in contrast to dislocation propagation in the bulk. Finally, excellent agreement is obtained between the relative yield strengths, stress–strain behavior and dislocation nucleation conditions of nanowires in our simulations and existing experimental data. This suggests that our predicted atomistic processes controlling yield in our simulations may also control yield in experiments. Ó 2012 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Nanostructure effect; Yield; Copper nanowire; Molecular dynamics simulations

1. Introduction Understanding the yield behavior of metallic materials is central for their structural applications. In coarse-grained metals, yield is usually dictated by dislocation migration. Hence, yield behavior is manipulated by addition of barriers to migration; such barriers include interfaces, precipi⇑ Corresponding author at: Department of Materials Science and Engineering, University of Pennsylvania, Philadelphia, PA 19104, USA. Tel.: +1 215 850 6551. E-mail address: [email protected] (D.J. Srolovitz).

tates, secondary phase particles or other dislocations. In general, decreasing the barrier spacing leads to increased yield strength; this is captured in the classic Hall–Petch [1,2] and Orowan strengthening [3] laws for the cases of grain boundaries and second-phase particles, respectively. Yield processes in nanomaterials are, however, less well understood [4–9]. One illustration of this complexity is that both single-crystal nanopillars [10,11] and pristine (dislocation-free) microparticles [12] show a power-law increase of yield strength with decreasing size. The nanostructural origin of these size effects is dominated by dislocation nucleation rather than dislocation migration. In the case of

1359-6454/$36.00 Ó 2012 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.actamat.2012.11.053

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nanopillars, nucleation occurs either at the free surface or from a limited number of internal dislocation sources [13–16], while in the case of microparticles, dislocation nucleation is governed by stress gradients associated with particle shapes and loading mechanisms. In nanostructured materials containing twin and grain boundaries, the yielding process is even more obscure. For example, Lu et al. [17–20] demonstrated that the yield strength of pure copper can be increased by 1000% by introducing a very high density of coherent twin boundaries in each grain. Most surprisingly, their samples also exhibited substantial work-hardening during tensile deformation. While the effect of twin boundaries on strength can be simply understood in terms of the ability of twin boundaries to block dislocations, the mechanism for work-hardening remains an active research topic [21–23]. Yield processes in nanomaterials are further complicated when there are multiple length scales involved, e.g. nanowire/pillar diameter, grain size, and twin spacing [24–27]. In particular, Jang et al. [27] reported that twin boundaries only increased the yield strength of polycrystalline Cu nanopillars by 10% and the yield strength of nanotwinned single-crystal nanopillars by 300% compared to their untwinned counterparts, while Lu et al. [17–20] showed that nanotwinned ultrafine-grained samples exhibit strengths 1000% larger than similar samples without nanotwins. It is not clear how nanotwin strengthening interacts with grain size and sample size strengthening. One reason why bulk and nanoscale metals exhibit very different yield behavior [6,7] is that at large grain sizes there is an abundance of dislocations and dislocation sources within each grain, but as the grain size is reduced, the total dislocation length and the number of sources within a grain become too small to accommodate the imposed deformation. If the source density is low, mechanical deformation drives pre-existing dislocations out of the system [28,29]. This state of dislocation starvation means that the mechanism of plastic deformation has to change from the standard dislocation migration-limited plasticity to some other mechanism such as dislocation nucleation-limited plasticity [30–32]. As the grain size decreases further, it is possible to achieve a shift from dislocation plasticity to other forms of plastic deformation [33–35] (e.g. associated with grain boundary sliding, diffusion or migration). Shifts in yield mechanism result in corresponding shifts in deformation behavior. It is therefore no surprise that nanomaterials can exhibit extraordinary mechanical properties. In fact, extensive experimental study suggests that the yielding process in nanomaterials is strongly dependent on size [36], strain rate [37–41,32] and temperature [42,43]. One signature of the shift in deformation mechanism is the decrease in activation volumes to as small as 1–10b3 in nanocrystalline materials as opposed to 1000b3 in their large-grained counterparts [44,32]. In the cases of nanowires and nanopillars with small diameters, additional complexity arises from the interplay between internal defects (e.g. dislocations, grain bound-

aries and twin boundaries) and the free surface (e.g. grain boundary/surface triple junctions and surface steps). For example, grain boundaries intersecting the free surface undergo sliding more easily than those within the nanostructure because of the relaxation of the mechanical constraint against sliding normally imposed by continuity between grains (especially at grain boundary triple junctions). This implies the possibility of a shift in the relative importance of different deformation mechanisms simply as a result of the proximity of the free surface to other defects in nanostructures. Unfortunately, it is difficult to experimentally resolve individual yield mechanisms (e.g. dislocation nucleation, propagation and grain boundary sliding and migration) due to their inherent small length and time scales. Molecular dynamics (MD) simulations provide an excellent approach to directly identify operative yield mechanisms in nanomaterials. This approach has been widely applied to study yield in single-crystal metallic nanopillars, nanowires and thin films [45–53,32,13,14]. While providing excellent insights, most MD simulations have focused on defect-free (pristine) samples, whereas most experimental samples contain various defects which can strongly affect dislocation nucleation conditions. Simulations of polycrystalline nanostructures are typically limited to small simulated samples because of limited computational resources [54–60]. Unfortunately, unrealistically small grain sizes can shift the relative importance of one yield mechanism with respect to others (e.g. grain boundary sliding and grain rotation are increasingly important with decreasing grain size), which may obscure the real deformation mechanism. Despite the great attention devoted to understanding the underlying yield mechanisms of nanomaterials and the nanostructural features that control their behavior, the situation is far from clear. Nonetheless, the strong system-size dependence, temperature sensitivity and low activation volumes associated with yield suggest that their yielding is likely controlled by thermally activated nucleation processes [44]. MD simulations of the deformation of nanomaterials provide an excellent approach to studying yield mechanisms and their dependence on nanostructural features, provided that they are done at the appropriate scale (and the material is not strongly rate sensitive). We demonstrate the applicability of such MD simulations below by comparing them with a wide range of experimental measures. It is the spatial and temporal resolutions of MD simulations that make them an ideal complement to experimental observations in such systems. In this work, we apply very large scale MD simulations to identify the most important nanostructural features and yield mechanisms in Cu nanowires with several distinct nanostructures (all of which have previously been studied experimentally). In particular, we focus on four distinct Cu nanostructures: (i) single crystals (SC), (ii) nanotwinned single crystals (NTSC), (iii) polycrystalline nanocrystals (NC) and (iv) nanotwinned polycrystalline nanocrystal

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(NTNC) nanowires. We also examine the effect of surface defects in otherwise perfect nanowires. These structures reflect experimentally realizable Cu nanowire nanostructures [24,25,27]. To control possible effects associated with the sample dimension [61], all the simulations were performed on nanowires of the same length and diameter. 2. Nanowire structures and simulation methods In the simulation results, presented below, we consider the tensile deformation of cylindrical Cu nanowires with the nanostructures illustrated in Fig. 1a–d. SC nanowires were constructed with their long axis oriented in the [1 1 1] crystallographic direction, as shown in Fig. 1a. NTSC nanowires containing an array of parallel twin boundaries on {1 1 1} planes were constructed as shown in Fig. 1b. The NC and NTNC nanowires, as shown in Fig. 1c and d, were created from a polycrystalline nanostructure

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constructed using a Voronoi procedure [21]. We first created a set of randomly distributed points in a periodic cell, as shown in Fig. 2a. The Voronoi construction was used to associate a cell or grain with each of these points. Fig. 2d shows the distribution of grain diameters within the simulation cell, where the grain diameter is determined as the diameter of the sphere with the same volume as the grain. The simulation cell contains 648 grains and its size was scaled such that the average grain diameter is 13 nm. Each polyhedron was then populated with Cu face-centered cubic (fcc) or nanotwinned fcc crystals of random orientation. Nanowires were obtained by cutting cylinders from this periodic cell as shown in Fig. 2c. In all of the simulations presented here, the nanowires share the same initial length of 190 nm and same diameter of 40 nm. The NTSC and NTNC nanowires also share the same twin spacing of 2.5 nm. Each nanowire contains 20 million Cu atoms.

Fig. 1. Schematic illustrations of (a) single crystal, (b) nanotwinned single crystal, (c) (polycrystalline) nanocrystal and (d) (polycrystalline) nanotwinned nanocrystal nanowires of length L = 190 nm and diameter D = 40 nm. Nanowires were simulated with and without circumferential surface defects in the ˚ t-shaped rings. form of 5 A

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Fig. 2. Schematic illustration of the construction of polycrystalline nanowires: (a) 648 points are randomly distributed in a periodic unit cell, (b) the corresponding Voronoi polyhedra constructed from (a) showing only the edges, and (c) a cylindrical nanowire of diameter D = 40 nm cut from the simulation cell. (d) The grain diameter distribution of the 648 grains/polyhedra in (b).

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Since experimental studies of even the most uniform nanowires must contain at least some surface steps, we ˚ deep t-shaped defect along the cirinscribed a single 5 A cumference of each of the nanowires. In addition to representing defects present along the surfaces of experimentally produced samples, this surface defect also serves the role of localizing the position of surface dislocation nucleation in the single crystal samples. In total, we examined eight types of simulation samples corresponding to those structures shown in Fig. 1a–d, with and without surface defects. The MD simulations were performed using the Largescale Atomic/Molecular Massively Parallel Simulator (LAMMPS) [64], in which interactions between Cu atoms are described using the embedded atom method (EAM) [65] potential parameterized by Mishin et al. [66]. This Cu potential was fit to experimental and ab initio data, including the lattice parameter a0, cohesive energy E0 and the stacking fault energy cSF, which are important material properties controlling their plastic deformation. Periodic boundary conditions were imposed along the nanowire axis, while the surface of the nanowire was free. All of the nanowires were equilibrated at 300 K before loading [61]. A uniaxial tensile load was applied by stretching the nanowires in the axial direction at a constant true strain rate of 0.1 ns1. During tensile loading, the temperature was held at 300 K using a Nose´–Hoover thermostat [67– 70]. 3. Stress–strain behavior Fig. 3a–d shows the engineering stress–strain curves for nanowires corresponding to those shown schematically in Fig. 1a–d. In each subpanel, two curves are shown, corresponding to nanowires with and without surface defects, respectively. We first compare the SC and NTSC nanowires, as shown in Fig. 3a and b. The stress–strain curves for the SC and NTSC nanowires without surface defects are very similar; nearly linear elastic up to the yield point which is the same as the ultimate tensile strength. These ultimate tensile strengths differ by only 1%. This suggests that the high density of twin boundaries in the NTSC sample has little effect on the yield behavior. The stress–strain curves for the SC and NTSC nanowires with surface defects are elastic up to the first, rapid drop in stresses. The samples with the surface defect show a significant drop in the peak stress as compared with their surface-defectfree counterparts. Additionally, the yield strengths of the SC and NTSC samples with surface defects are also notably different from each other; the SC sample shows a drop in yield strength of 36% relative to its surface-defect-free counterpart, while the NTSC shows a 24% drop. The presence of surface defects drastically reduces the SC and NTSC nanowires yield/ultimate stress and the effect is more significant in the SC nanowire. Furthermore, the differences in peak stress between the SC and NTSC nano-

Fig. 3. Engineering stress–engineering strain curves for the (a) SC, (b) NTSC, (c) NC and (d) NTNC nanowires. The blue curves are for ˚ t-shaped surface defects and the yellow curves are nanowires with the 5 A their counterparts without the surface defects. The stress–strain curves for the SC and NTSC nanowires are strongly affected by the presence of the surface defect, while those for the NC and NTNC nanowires show almost no surface defect effect. The circles indicate the strains at which the defect structures are shown in Figs. 4 and 6. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

wires with a surface defect indicate a possible change in atomistic yielding processes between these two nanowires. Fig. 3c and d show the stress–strain curves for the NC and NTNC nanowires with and without surface defects. The curves for the cases with and without surface defects are nearly identical in every respect. This suggests that surface defects play a negligible role in the yield behavior of the NC and NTNC nanowires, contrary to the previous cases of SC and NTSC nanowires. The stress–strain curves of the NC and NTNC nanowires are also similar to each other, showing a smooth transition from elastic to plastic deformation without any sharp drops. The ultimate strength of the NTSC nanowires is 10% higher than that of the NC nanowires. Comparison of the NC and NTNC ultimate strengths (Fig. 3c and d) with those of the SC and NTSC nanowires (Fig. 3a and b) shows that the polycrystalline wires have ultimate strengths less than one-third of that of the SC nanowires. In addition, the sharp drops in the stress–strain curve seen in the SC and NTSC nanowires are completely absent in the NC and NTNC nanowires. These differences in the stress–strain curves suggest different yielding

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processes among these nanowires with different nanostructures. In the next section, we examine the mechanistic origins of these differences. 4. Atomic-level observations of yielding processes The stress–strain curves in Fig. 3 show that nanowires with different nanostructures have very different yield behaviors. Surface defects also have very different effects on the yield behavior of the single-crystal and nano (poly) crystal nanowires. In order to understand the mechanisms responsible for these differences, we examine the yielding processes for each nanowire on the atomic scale. Figs. 4 and 6 show the evolution of the defect structures within the nanowires during yielding. Figs. 5 and 7 show the corresponding dislocation activity viewed along the axial direction. In each of these figures, subpanels in the top row are for nanowires without surface defects, while those in the bottom row are for nanowires with surface defects. In each subpanel, four images are shown, corresponding to the four (circular) symbols on their respective stress– strain curves in Fig. 3. 4.1. SC and NTSC nanowires Fig. 4a and b show atomistic views of the SC and NTSC nanowires without surface defects. Fig. 4a1 and b1 (and Fig. 5a1 and b1) show the point where the first partial dislocation nucleates from the surface. In both nanowires, the

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first partial dislocation is nucleated at similar stresses (see the first circular symbols on the orange curves of Fig. 3a and b), suggesting that the twin boundaries play little role in the nucleation of dislocations at the surface. Fig. 4a2 and b2 show that dislocation propagation is accompanied by rapid drops in the stress (see Fig. 3a and b). This is also seen in Fig. 5a2 and b2. We note that in the SC nanowire, the first partial dislocation propagates throughout the entire nanowire diameter. However, in the NTSC nanowire, the nucleation of the leading partial is immediately followed by the formation of the trailing partial dislocation and that together they are able to cross twin boundaries (see Ref. [21] for a detailed description of this dislocation activity). The stress required to nucleate both the leading and trailing partials at the surface is higher than that required to drive the full dislocation through the twin boundary. Hence, the yield stress for this nanowire is governed by the stress required to nucleate the first full dislocation, i.e. the nucleation of the trailing partial. Based on this observation, we can define the first partial and trailing partial dislocation nucleations as the yield points for the SC and NTSC nanowires (without surface defects), respectively. Subpanels a3, a4 and b3, b4 of Figs. 4 and 5 show the nanowires at later stages of yielding, where more dislocations are present in the nanowires. Both nanowires exhibit only small amounts of necking with a 3% reduction in minimum cross-sectional area. The NTSC nanowire exhibits more localized plastic deformation than does its SC counterpart, as seen in Fig. 4b4.

Fig. 4. Atomistic illustration of the onset of yield in SC and NTSC nanowires: (a and b) SC and NTSC nanowires without surface defects; (c and d) SC and NTSC nanowires with surface defects. In each subpanel, the brown images show the nanowire profile and surface relief, while the image on their right shows an axial cross-section of the same configuration taken along the mid-section of the nanowire. Atoms are shown only if their central symmetry parameters [71] differ from that of the perfect face-centered cubic crystal; the colors indicate the local symmetry. Atoms on twin boundaries, dislocations, intrinsic and extrinsic stacking faults are shown in light blue, dark blue or green (depending on dislocation type), orange and light blue, respectively. The numbers below each pair of images give the per cent strain. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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Fig. 5. Atomistic views of the dislocation activity inside nanowires viewed along the axial direction (i.e. not cross-sections). The four subpanels correspond to the images shown in Fig. 4. In these figures, only atoms on dislocations and in intrinsic stacking faults are shown.

Fig. 4c and d show SC and NTSC nanowires with surface defects. In both nanowires, partial dislocations are first nucleated from the surface defects at similar stress levels as seen in Fig. 4c1 and d1, and the corresponding first circular symbols in their respective stress–strain curves in Fig. 3a and b. This suggests that the twin boundaries have only a very weak effect on the first partial dislocation nucleation; this is consistent with the observation in nanowires without surface defects. However, the presence of surface defects reduces the dislocation nucleation stress by more than one-third as compared with that of nanowires without surface defects. We note that the images in Fig. 4c1 for the SC nanowire with a surface defect correspond to the peak of its stress–strain curve. Partial dislocations propagate immediately following their nucleation (see Fig. 4c1, c2 and 5 c1, c2), resulting in a rapid drop in stress. This is similar to the case of the SC nanowire without surface defect. Hence, we can define the first partial dislocation nucleation as the yield point for this SC nanowire with surface defects as well, the same as noted for the previous case of a SC nanowire without surface defects. The NTSC nanowire in Fig. 4d does not actually yield at the strain at which the first partial dislocation is nucleated. The image in Fig. 4d1 does not correspond to the peak point in its stress–strain curve. In this nanowire, a leading partial dislocation is nucleated and is able to propagate until it meets a twin boundary, which is 2.5 nm away from their nucleation point. No appreciable stress drop corresponding to the initial partial dislocation nucleation is observed in the stress–strain curve (blue curve in Fig. 3b). Instead, the peak point of its stress–strain curve corresponds to Fig. 4d2 and 5d2, where a trailing partial is nucleated at a higher stress. The trailing partial combines with the early leading partial and they form a full dislocation segment. The resulting full dislocation is able to cross twin boundaries immediately following their formation (see Supplementary Material for animations of nanostructure evolution of this and other nanowires, and Ref. [21] for a detailed description of this dislocation activity). This suggests that the tensile stress required to drive the full dislocation through twin boundaries is lower than that required to nucleate its trailing partial from the current constructed surface defect. Based on this, we can define the yield point for this NTSC nanowire

as the stress required to nucleate the trailing partial dislocation from surface defects (not the leading partial and not the stress required to push the dislocation through the twin boundary). Apart from having different yield stresses, these SC and NTSC nanowires also exhibit very different deformations following yielding, as shown in Fig. 4c2–c4, d3 and d4. The plastic deformation in the NTSC nanowire is localized to 10 twin spacings (i.e. 25 nm) while the deformation in the SC nanowire extends throughout the total nanowire length (i.e. 190 nm). The reduction in minimum cross-sectional area in the NTSC nanowire during the first yield drop is 40%, an order of magnitude larger than that of the SC nanowire. The closely spaced twin boundaries effectively constrained dislocation propagation, resulting in rapid, localized necking accompanied by some twin coarsening immediately following yield. We note that in this NTSC nanowire with the surface defect, the deformation features of highly localized necking and twin coarsening match very well with recent electron microscopy studies of the tensile deformation of NTSC nanowires [26] (cf. Fig. 4d4, and 3b of Ref. [26]). 4.2. NC and NTNC nanowires The stress–strain curves for the SC and NTSC nanowires show clear yield points followed by rapid, large stress drops. Each of these yield points can be related to a distinct dislocation event. However, the yield behavior of NC and NTNC nanowires is quite different. This suggests that the yielding processes in NC and NTNC nanowires are very different from those of SC and NTSC nanowires. Next, we examine the nanostructure evolution during yielding for the NC and NTNC nanowires, as shown in Fig. 6. Fig. 7 shows the dislocation activity corresponding to the nanowires shown in Fig. 6. Since there is no distinct yield point in the stress–strain curves of these nanowires, we choose to define their yield point as the strain at which the first partial dislocation crosses an entire grain diameter anywhere in the nanowire. This is shown in the first image in each of the subpanels in Figs. 6 and 7. These first subpanels also correspond to the first points shown in the stress–strain curves in Fig. 3.

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Fig. 6. Atomistic views of nanostructure evolution of NC and NTNC nanowires: (a and b) NC and NTNC nanowires without surface defects; (c and d) NC and NTNC nanowires with the surface defects. In both (b) and (d), many dislocations and stacking faults are not visible due to blocking by the high density of twin boundaries. See Fig. 7 for details on dislocations and stacking faults.

Fig. 7. Atomistic views of dislocation activities inside the nanowires viewed in the axial direction (i.e. not cross-sections). The four subpanels correspond to the four nanowires shown in Fig. 6. In these figures, only atoms on dislocations and intrinsic stacking faults are shown.

A comparison of Fig. 6a with 6c and 6b with 6d for nanowires without and with surface defects shows some detailed differences in dislocation dynamics. However, these differences are very minor, and no appreciable difference in their overall nanostructure evolution is observed. This suggests the effect of surface defects on the yield behavior of the NC and NTNC nanowires is very weak. This is consistent with the observation (made above) that the stress–strain curves of the NC and NTNC nanowires are not influenced by the presence of surface defects. Close examination of the dislocation activity in Fig. 6 shows that most partial dislocations are nucleated in the interior grains, instead of at the surface, as in the SC and NTSC nanowires. This is also evident in Fig. 7, where only atoms on dislocations and intrinsic stacking faults are shown. A comparison of Fig. 6a with 6b and c with 6d for NC and NTNC nanowires shows the effect of twin boundaries on the evolution of the dislocation nanostructure. In the NC nanowires, partial dislocations are nucleated inside

many grains and are able to propagate towards grain boundaries on the opposite sides of grains, leaving behind intrinsic stacking faults that span entire grains (see Fig. 6a2–a4 and c2–c4). However, in the NTNC nanowires, as shown in Fig. 6b2–b4 and d2–d4, only dislocations moving on slip planes parallel to the twin boundaries are able to propagate through the entire grain. Those with slip planes intersecting twin boundaries are blocked by the twin boundaries, resulting in a much lower density of stacking faults in the early stage of yielding. In Fig. 6, the density of stacking faults crossing entire grains in the NTNC nanowires appears to be much less than that in the NC nanowires (also see Fig. 7). Taking the nanostructure configuration at a tensile strain of 4% as an example, we find that the intrinsic stacking fault density in NTNC and NC is 1:4. In order to reveal the atomistic details associated with dislocation nucleation and propagation in the NC and NTNC nanowires, a cross-sectional view of the nanowires

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Fig. 8. Detailed illustration of dislocation nucleation in (a) NC and (b) NTNC nanowires. (a1) and (b1) illustrate the tensile (ryy) stress (scale on the right), where the largest stresses are focused at triple junctions. (a2) and (b2) show an atomistic view of dislocation activity close to the onset of plasticity, while (a3) and (b3) show detailed views near some grain boundaries; (a4) and (b4) schematically illustrate the grain boundaries (blue lines), intrinsic stacking faults (orange lines) and twin boundaries (cyan lines) in the detailed images. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

with surface defects is shown in Fig. 8. Fig. 8a1 and b1 show the spatial distribution of atomic-level (virial) stresses in the loading direction, ryy, at a strain prior to the onset of yield (2.5%). Overall, the stress distributions or inhomogeneities in the NC and NTNC nanowires are very similar. In both nanowires, there are appreciable stress concentrations at many grain boundary triple junctions (as high as twice the average tensile stress). The corresponding dislocation activities at later stages are shown in Fig. 8a2 and b2. Fig. 8a3 and b3 show detailed, atomistic-level images of the dislocation activity. Their schematics are shown in Fig. 8a4 and b4. Clearly, most of the dislocations are nucleated at grain boundary triple junctions where there are significant stress concentrations (consistent with earlier studies [72,73,44]). We also note that in these nanowires, the stress concentration associated with surface defects is much smaller than that at many grain boundary triple junctions.

nanostructures. Further, since nanowires in experiments have imperfect surfaces, we only consider the yield strengths of the nanowires with surface defects from the MD simulations. Fig. 9 shows a comparison of the yield strengths normalized by the yield strength of the NTSC nanowire, both from our MD simulations and from existing experimental data [27]. The MD yield stresses are defined as in Section 4. The relative yield stresses of NC and NTNC nanowires calculated in our simulations correspond very well with experimental data. Both methods report that the yield strengths of these nanowires are about one-third of the yield strengths of the NTSC sample.

5. Yield strength After understanding the atomistic phenomena that determine the nature of yield in the present set of nanowires, we compare the yield stresses between them. We note that it is difficult to make a direct comparison between the nanowire yield stresses found in the MD simulations and measured in experiments under laboratory conditions. This is because of the many differences in sample geometry, testing methods and conditions, and also because of the vast differences between the MD and experimental strain rates. To compare against experiment, it is more relevant to measure the relative strengths of the nanowires with different

Fig. 9. Comparison of yield stresses determined from the simulations with experimental data [27,74–76]. The yield stress values from the MD and experiment data sets are scaled by the yield strength of the NTSC nanowire in MD and in experiment [27], respectively. In both MD and experiment, the nanocrystalline nanowires are much weaker than the NTSC nanowires.

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However, although both our simulations and experiment find that the SC should be weaker than the NTSC, the SC experimental value in Ref. [27] predicts a much lower yield stress than that in our simulation. We attribute this discrepancy to the strong sensitivity of SC nanowires to pre-existing defects. Since our simulation cell contains a single, well-controlled defect, we expect it to overestimate the strength of the experimentally fabricated samples with a rough surface and pre-existing dislocations in Ref. [27]. It is no surprise to note that other experimental measurements of the yield strength of Cu nanowires [74–76] are spread over a wide range, with some pristine ones having yield stress as high as 6 GPa. 6. Roles of nanostructure and surface defects The roles of both surface defects and twin boundaries are very different in the single-crystal (SC and NTSC) and the nanocrystal (NC and NTNC) cases. For instance, we find that yield strength is highly sensitive to surface defects in the SC and NTSC nanowires, while the NC and NTNC nanowires are insensitive to surface defects. In nanowires with surface defects, twin boundaries enhance yield stress more effectively in single-crystal than in nanocrystal nanowires. The difference between the yield behavior in the single-crystal and nanocrystal nanowires is related to the origin of the initial yield event. In the case of SC and NTSC nanowires, the surface defect is a stress concentrator that reduces the macroscopic load required to nucleate dislocations. In the case of NC and NTNC nanowires, the surface defects are rarely the dominant and/or earliest source of dislocations; rather, dislocations tend to nucleate first at grain boundary triple junctions. This is clearly shown by reference to the resolved shear stresses close to these defects.

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Fig. 10a–d show sectional views of the shear stress resolved along the slip system of the first dislocation emitted, i.e. the (1 1 1) slip plane along the ½1 1 2 slip direction. The four images show the regions in the vicinity where the first dislocation will nucleate at an applied nominal strain of 2.5% (i.e. before the nucleation event) for all four nanostructures. Note that in all cases, a significant stress concentration exists at the defects (i.e. the surface defects or the triple junctions). The stress concentration in SC and NTSC nanowires originates from the constructed surface steps, while the stress concentration at triple junctions in NC and NTNC nanowires originates from both constrained grain boundary sliding and the high elastic anisotropy of Cu [62,63]. Fig. 10e–h show the initial stages of dislocation nucleation corresponding to the subpanels above. The positions of the initial dislocation nucleation correlate well with areas under local high resolved shear stress. These micrographs illustrate that dislocations nucleate at the surface defects in SC and NTSC nanowires and at grain boundary triple junctions in NC and NTNC nanowires. The magnitude of shear stress enhancement along the slip plane in the direction of the Burgers vector (i.e. ½1 1 2) of the first nucleated dislocation can be determined quantitatively, as shown in Fig. 11. This illustrates that the surface defects in SC and NTSC nanowires and the triple junctions in NC and NTNC nanowires concentrate stress equally effectively. Both the magnitude and the gradient of the stress decay near the two types of defects are very similar. The resolved shear stresses decay according to a power law, with an exponent of 0.3. A simple analysis [12] shows that the critical external stress required to nucleate a partial dislocation is 2.8 GPa, consistent with our MD simulations. However, the critical dislocation loop radius is estimated to be less than 0.5 nm (corresponding to

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Fig. 10. Illustration of the shear stress near a surface defect for SC (a) and NTSC (b) nanowires and near a grain boundary triple junction in NC (c) and NTNC (d) nanowires at a nominal strain of 2.5%, before the first plastic event. These shear stresses are resolved onto the slip plane of and parallel to the Burgers vector of the first dislocation to be nucleated. Note that the magnitudes of the stress concentration for all cases are of the same order. Panels (e–h) illustrate the first plastic events that occur at later stages of deformation. The location of the first dislocation correlates well with the location of the highest stress.

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Fig. 11. The resolved shear stress s projected on the slip plane and in the direction of the Burgers vector of the first dislocation emitted as a function of distance from the peak stress. The four data sets correspond to the four cases shown in Fig. 10. a0 is the fcc Cu lattice parameter. The solid line corresponds to a power-law decay with exponent 0.3.

an activation volume of 17b3), a length scale much smaller than any nanostructural feature in the simulated nanowires. Our estimate of the critical dislocation loop radius is also consistent with previous estimates of nanomaterial yielding process activation volumes, on the order of 10b3 [43,50,32]. This strongly suggests that dislocation nucleation from these stress concentrators is probably the mechanism controlling nanomaterial yield. Because the critical loop size is so small, the twin boundaries in the NTSC nanowire are expected to have a negligible effect on the nucleation of the initial partial dislocation. The critical loop size is much smaller than the twin spacing. This density of twin boundaries is expected to enhance the strength of copper by blocking the motion of the leading partial dislocation. This analysis is consistent with the observation that yield is governed by the nucleation of the trailing partial in the NTSC nanowire. In the NTNC nanowires, leading partial dislocations are effectively blocked when twin planes intersect their slip planes. We support this hypothesis by measuring the amount (area) of stacking fault in the deformed crystal. At a fixed applied strain (4%), the total area of intrinsic stacking faults in the NTNC nanowire is only 1/4 of that in the NC nanowire. This is consistent with our estimate that 1/4 of the dislocations nucleated in the NTSC nanowires have slip planes parallel to twin planes. One would therefore expect the yield strength of the NTNC nanowire to be strongly enhanced compared to the NC nanowire, since the majority of dislocations are blocked by twin boundaries in the NTNC case. However, our simulations and the experimental data [27] show contradictory results. Two factors may explain why the high density of twin boundaries only marginally enhances the yield strength of the nanocrystalline nanowires. First, and most importantly, the yielding process in the nanocrystalline systems is governed by dislocation nucleation from grain boundary triple junctions, rather than the propagation of existing dislocations. This is because the initial nanowires (and grains) are dislocation free. This is as expected for grains of the

small diameter studied here. The initial state of our MD samples could bias the yield mechanism to favor the dislocation nucleation mechanism. Second, the small grain size and small nanowire diameter may allow other plastic relaxation mechanisms to operate. These include grain boundary sliding and grain rotation. If grain boundary sliding is active, grains can rotate to an orientation that can accommodate the applied deformation. This can effectively increase the number of active slip systems, allowing for general deformation of the nanostructure. In fact, our simulations suggest that only one slip system is activated in most grains throughout the yielding process. The influence of grain boundaries on yield is intriguing because they play multiple roles. On the one hand, triple junctions concentrate stress and serve as dislocation sources, reducing the dislocation nucleation stress and the yield stress of the NC and NTNC nanowires. On the other hand, grain boundaries themselves can also act as barriers to block dislocations from propagation through the entire nanowire, thereby increasing the strength. These two competing roles of grain boundaries in NC and NTNC nanowires cause a gradual yield, rather than a sharp drop in the stress–strain curve corresponding to a large, dramatic plastic event in the SC and NTSC nanowires. This results in a nearly perfect plastic deformation in the NC and NTNC nanowires (i.e. nearly constant flow stress over a large range of strain), consistent with the stress–strain curves in Fig. 3c and d and experimental observations [77]. 7. Conclusion In this work, we studied the effects of surface defects, nanotwins and grain boundaries on the plastic yield of Cu nanowires using very large scale MD simulations. We focused on nanowires of identical dimensions and four nanostructures: SC, NTSC, NC and NTNC. Our simulations show that although NC and NTNC nanowires exhibit much lower yield strengths than the SC and NTSC nanowires, their yield behavior is also much less sensitive to surface defects. Likewise, the nanocrystalline nanowires are much less sensitive to the presence of nanotwins within the grains. By atomic-level observation of the initial stages of plastic yield, we found that dislocation nucleation controls the yield behavior of all nanowires studied. The differences in mechanical properties come from the nanostructural features from which dislocations nucleate. In the SC and NTSC nanowires, yield is initiated by dislocation nucleation from surfaces (or defects on the surfaces), whereas in the NC and NTNC nanowires, yield is initiated by dislocation nucleation from grain boundary triple junctions. Both surface defects and grain boundary triple junctions serve as effective stress concentrators. Since there are a greater number of triple junctions than surface defects within the nanowires, the NC and NTNC nanowires are weaker than the SC and NTSC nanowires. In addition, our simulation results showed quantitative agreement with recent experimental study in several aspects, including

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