Letter pubs.acs.org/NanoLett
Nanowire Failure: Long = Brittle and Short = Ductile Zhaoxuan Wu,† Yong-Wei Zhang,† Mark H. Jhon,† Huajian Gao,*,‡ and David J. Srolovitz*,† †
Institute of High Performance Computing, 1 Fusionopolis Way, #16-16 Connexis, Singapore 138632 School of Engineering, Brown University, Providence, Rhode Island 02912, United States
‡
S Supporting Information *
ABSTRACT: Experimental studies of the tensile behavior of metallic nanowires show a wide range of failure modes, ranging from ductile necking to brittle/localized shear failureoften in the same diameter wires. We performed large-scale molecular dynamics simulations of copper nanowires with a range of nanowire lengths and provide unequivocal evidence for a transition in nanowire failure mode with change in nanowire length. Short nanowires fail via a ductile mode with serrated stress−strain curves, while long wires exhibit extreme shear localization and abrupt failure. We developed a simple model for predicting the critical nanowire length for this failure mode transition and showed that it is in excellent agreement with both the simulation results and the extant experimental data. The present results provide a new paradigm for the design of nanoscale mechanical systems that demarcates graceful and catastrophic failure. KEYWORDS: Copper nanowire, length effect, ductile necking, brittle shear failure, molecular dynamics simulations
W
taken from the literature. Our study provides a simple, mechanistic understanding of the different modes of nanowire tensile failure and reveals why previous simulations were only able to observe ductile failure. We simulated cylindrical, single crystal copper nanowires with their long axis oriented in the [111] direction, as illustrated in Figure 1. Their initial diameter d is chosen to
hen the characteristic size of a structure is reduced to the submicrometer and nanometer range, its mechanical properties can be very different from that of bulk materials.1−6 For example, as the diameter of a face-centered cubic (FCC) metal nanowire decreases, its failure strength increases (Au,7−12 Ag,13,14 Cu,15−18 Ni19,20). However, although such nanowires fail at higher stresses, they may exhibit a range of ductilities and failure modes. Some experiments showed that even though bulk FCC metals tend to be ductile, their single-crystal whisker21 and nanowire22−24 counterparts can fail through either brittle fracture or localized shear failure, without noticeable ductile necking or extensive plasticity. In contrast, other experiments found that FCC metal nanowires can fail through a completely different failure mode involving extensive plasticity15 and ductile necking.12,18 Unfortunately, atomistic simulations8,13,25 have not been able to resolve why FCC metal nanowires can fail through these two distinct modes. In fact, all existing atomistic simulations have predicted that single crystal nanowires should fail through ductile necking. Why were previous atomistic simulations unable to observe nanowire failure through brittle or localized shear failure? Further, what factors control how nanowires fail under tension? In the present work, we address these questions by employing large-scale molecular dynamics (MD) simulations on nanowires of unprecedentedly large lengths. We find that nanowires undergo a ductile-to-brittle transition with increasing nanowire length: a short nanowire fails via a ductile necking mechanism with extensive plasticity, while a long nanowire fails via a localized shear failure, without noticeable necking. To explain this observation, we develop a simple model that predicts the critical length at which the failure mode changes. The predicted critical length is in excellent agreement with both the simulation results as well as a survey of experimental data © 2012 American Chemical Society
Figure 1. Schematic illustration of the simulation cell. Nanowires have a cylindrical cross section with their long axis oriented in the [111] direction. A 5 Å deep U-shaped notch was created along the circumference to represent surface defects in the wire. Periodic boundary conditions were used in the y direction.
be 20 nm, and their initial lengths L0 are chosen from a range between 188 and 1503 nm. In order to control the position of the initial plastic deformation, we placed a surface defect (a 5 Å deep U-shaped notch along the circumference) in each wire. The MD simulations were performed using the large-scale atomic/molecular massively parallel simulator (LAMMPS),26 where interactions between copper atoms were described using Received: November 11, 2011 Revised: December 9, 2011 Published: January 3, 2012 910
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Figure 2. Panels a−d illustrate the engineering stress−strain curves for the four nanowires at different initial lengths L0. The stress−strain curves for all four nanowires are similar up to the first stress peak. The stress−strain curves exhibit different levels of serrations depending on the nanowire length. These serrations are associated with discrete bursts of dislocation activities. Longer nanowires fail at progressively smaller strains. The inset to (a) shows the true stress−true strain curve, where the true stress is based upon the instantaneous minimum cross-sectional area of the nanowire. Panel e shows the size of the first stress drop as a function of nanowire length. Analysis of the stress−strain curves reveals a nearly linear relationship between the size of the drop and the nanowire length.
the embedded atom method (EAM)27 potential parametrized by Mishin et al.28 Periodic boundary conditions were imposed along the y direction. To minimize spurious vibrations that arise from the rapid thermal expansion during heating, the nanowires were carefully equilibrated at 300 K: we first increased the temperature from 0 to 300 K over 100 ps using Langevin dynamics and then annealed at 300 K for 50 ps using molecular dynamics under the NPT ensemble, where constant temperature and zero normal stress in the y direction were maintained using a Nosé−Hoover temperature thermostat and pressure barostat.29−32 After the nanowires were equilibrated, we applied uniaxial tensile loading by stretching the nanowires in the y direction at a constant true strain rate of 0.1 ns−1. Figure 2a−d shows the engineering stress−strain curves of four samples at length L0 = 188, 376, 751, and 1503 nm. Although all four samples show a similar yield stress of ∼6 GPa at ∼3% strain, other features in the stress−strain curves are substantially different. The nanowire with length 188 nm fails after many serrations in the stress−strain curve, indicating the presence of extensive plasticity; in contrast, the nanowire with length 1503 nm fails at the first stress drop, suggesting that it fails in an abrupt (brittle) manner. The ductility, or strain at failure, clearly increases with decreasing nanowire length. In addition, the size of the first stress drop appears to be a nearly linear function of nanowire length L0, as shown in Figure 2e. The true stress−true strain curve for the L0 = 188 nm nanowire is shown as an inset to Figure 2a (true stress is calculated based upon the instantaneous minimum cross-sectional area). Note that apart from the serrations the stress−strain behavior can be viewed as elastic−perfectly plastic and that the maximum and minimum stresses in all of the serrations are very similar. In order to support our inferences about the failure modes from the stress−strain curves, we describe the atomic level processes that occur during deformation and the final fracture surfaces of the different nanowires, as illustrated in Figure 3. By
Figure 3. Fracture surfaces of nanowires with diameter d = 20 nm and different lengths. The L0 = 188, 376, and 751 nm nanowires exhibit facets at many positions and directions. This corresponds to dislocation activities on many slip systems, and consequently the failure of the nanowire exhibits the characteristics of ductile necking. In contrast, the L0 = 1503 nm nanowire shows a flat fracture surface, indicating that plastic activity is confined along a single shear plane. Atomic configurations are rendered using POV-Ray.33
doing so, we show that nanowires with different lengths exhibit different levels of plastic deformation under the same loading conditions. All of the nanowires show failure via slip mechanisms over a region of order 100 nm. The behavior of short wires is consistent with previous atomistic simulations.8,13,25 Stress concentration causes the nucleation of a Shockley partial dislocation at the notch root. After the first nucleation event, the three shorter wires (188, 376, and 751 nm) exhibit multiple bursts of dislocation activities on different slip systems at different locations, creating many shear facets over a large region of the wire. These observations can be used to interpret the stress−strain behavior shown in Figure 2. The maxima of the true-stress serrations in Figure 2a correspond to the stress required to nucleate dislocation slip processes, and the minima of this curve indicate where the stress drops below the level required to continue these slip processes within a single band. Each serration in the stress−strain curve 911
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{111} planes.35 One more dislocation of either category can transform the extrinsic stacking fault plane into a twin with three {111} atomic layers. The newly formed nanotwin is further expanded through the same dislocation mechanisms as discussed above (see Figure 4c). Such dislocation−stacking fault plane interactions widen the stacking fault region (as shown in Figure 4d) and create many defects consisting of stacking faults and dislocations. Shear deformation along the stacking fault region becomes dominant as the region grows to 8−10 atomic planes in width. Eventually, a localized shear failure occurs, causing the nanowire to rupture without noticeable necking near its fracture surface. Our simulations unambiguously demonstrate that the length of a copper nanowire helps to control both its ductility and its mode of tensile failure. By increasing the nanowire length, we observe a transition in failure mode from ductile necking to shear failure along a single plane. Our simulations indicate that for copper nanowires with diameter of 20 nm this transition occurs at a length less than or equal to 1503 nm. Note that the present study is the first atomistic simulation of failure of nanowires longer than this transition length. The origins of this length-induced ductile−brittle transition can be understood with the following simple, mechanistic model. This length effect may be explained by noting that a longer nanowire can store more elastic energy for a given strain. Consider a nanowire with initial length L0 and Young’s modulus E loaded to a total strain ε. For simplicity, we take the cross section to be square (calculations with a cylindrical cross section are more complex but lead to equivalent results) with side length d and assume the slip direction has an angle α with the nanowire axis. We further assume that all plastic deformation occurs on the same slip plane. This simple model captures the essential physics of the problem. In the analysis, we choose to ignore all dynamic effects as well as these nonessential plastic deformation mechanisms. We denote the plastic deformation in terms of the total slip length, s (see Figure 4d), such that the plastic strain along the axial direction is s cos α/L0 and the effective cross-sectional area decreases by sd sin α. During the slip process, the tensile stress in the slipped region σm changes because the cross-sectional area and the force in the nanowire decrease simultaneously.
corresponds to slip along a single band; multiple serrations correspond to multiple, distinct bands that conspire to create the observed (ductile) necked failure mode. In contrast, plastic deformation in the longest (1503 nm) nanowire has a flat fracture surface, signifying a localized shear failure via dislocation activity along a single plane, as shown in Figure 4. This is consistent with the nonserrated stress−strain
Figure 4. Snapshots of the plastic deformation of a 1503 nm nanowire during tensile loading. Each panel corresponds to a labeled point in Figure 2d. In order to observe plastic mechanisms inside the nanowire, we made a planar cut along the midsection of the wire along its axial direction and draw those atoms only on one side of this plane. Panel a shows the nanowire after the first yield event. Panels b−f show subsequent yielding events, mostly localized on a single slip plane. Panel d illustrates the angle of the slip plane α and slip length s. Atoms are shown only if their central symmetry parameters34 differ from that of the perfect FCC crystal; the colors indicate the local symmetry. Atoms on surfaces, twin boundaries, dislocations, intrinsic stacking faults and extrinsic stacking faults are shown in dark blue, light blue, green or yellow (depending on dislocation type), orange, and light blue, respectively.
curve seen for long nanowires (see Figure 2d; note in experiments the single stress drop seen here would be perfectly sharp). Animations showing the evolution of the nanowire during plastic deformation are available as Supporting Information. Since the localized shear deformation in the 1503 nm wire is unique and not observed in previous MD simulations, we further provide a detailed illustration of the atomic-level processes that occur during the deformation. After the first Shockley partial dislocation is emitted from the notch root, the dislocation quickly propagates through the nanowire, leaving behind an intrinsic stacking fault. Subsequent plastic deformation is primarily carried by two categories of Shockley partial dislocations. The first category includes Shockley partial dislocations with the same Burgers vector as the first dislocation on {111} planes parallel to and adjacent to the intrinsic stacking fault. The other category includes dislocations on {111} planes intersecting the intrinsic stacking fault. Dislocations of the first category can transform the intrinsic stacking fault plane into an extrinsic stacking fault plane. Such transformations can also be achieved through a cross slip of pure screw dislocations or 60° dislocations gliding on different
σm =
Ed L0 ε − s cos α L0 d − s sin α
(1)
The stress in the slipped region always increases with increasing slip length provided that the aspect ratio satisfies the condition L0/d > cot α/ε. In other words, if held at the yield strain εy, the slip length will continue to grow provided the nanowire is suffciently long (i.e., Lc > d cot α/εy). For the copper nanowires considered here (d = 20 nm, εy = 0.03, sin α = 2/3√3), Lc = 1600 nm. If this critical condition is not met, the slip is arrested and the nanowire undergoes a finite stress drop. Roughly, the nanowire will continue to plastically deform as long as σm exceeds the stress required to activate the dislocation source σc. This leads to an equilibrium slip length
seq =
E ε − σc L0 E cos α − (L0 / d)σc sin α
(2)
The corresponding tensile stress drop due to this slip is then simply 912
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E ε − σc Δσ = 1 − (L0 /d)(σc /E) tan α
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machine stiffness37,38 and nanowire length are both important. The deformation of a long nanowire on a stiff machine is analogous to testing a short nanowire on a compliant machine. The long nanowire is able to provide sufficient stored energy to drive localized dislocation activities once the instability condition is reached. Therefore, the length of the nanowire controls how much energy is available to drive plastic deformation to failure. In summary, we studied the uniaxial tensile failure of single crystal copper nanowires using very large scale MD simulations. Our simulations show that the length plays an essential role in determining nanowire tensile failure modes and fracture surface morphologies. A failure mode transition occurs with increasing nanowire length. Short nanowires fail via dislocation plasticity along multiple slip planes and ductile necking, while long nanowires fail by an unstable localized shear along a single slip plane without appreciable necking. A simple mechanical model was developed to predict the critical length for the failure mode transition and the magnitude of the stress drop in the stress− strain curve. The model predictions are in excellent agreement with both the simulation results and a wide body of experimental results in the literature. The present simulations are the first to span a sufficient range of nanowire length to observe both ductile and brittle failure modes in the same system. The present work suggests a basic engineering principle for the design of nanoscale mechanical systems. That is, there exists a critical length scale below which nanoscale systems fail gracefully and above which, catastrophically.
(3)
This expression shows how the size of the serrations in the stress strain curve depend on the aspect ratio of the nanowire. As the nominal nanowire length increases, the stress drop also increases. Likewise, as the nominal nanowire width increases, the size of the load drop is expected to decrease. In cases where the aspect ratio L0/d is small, Δσ ≈ a + (L0/d)b, where a and b are constants. This is consistent with our MD simulations as shown in Figure 2e. It is noted that the mechanism of dislocation plasticity is essentially the same for all four nanowires. The main difference is whether these dislocation activities are dispersed on different slip systems and at different locations or predominantly on the same slip system and concentrated along a localized shear region. Our MD simulations show that the nanowire with length 1503 nm fails due to a single shear event, while the shorter nanowires fail via dispersed ductile necking failure. The failure mode transition occurs at about 1500 nm, which is in excellent agreement with our model prediction for the critical length, 1600 nm. Since all the atomistic simulations focused on nanowires shorter than the critical length, this explains why all the previous MD studies8,13,25 fail to observe this transition. Now, we employ the above results to interpret the extant experimental literature on nanowire failure modes.21−24 The experimental data are summarized in Figure 5, together with
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ASSOCIATED CONTENT
S Supporting Information *
Additional movies. This material is available free of charge via the Internet at http://pubs.acs.org.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected] (H.G.);
[email protected]. edu.sg (D.J.S.).
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ACKNOWLEDGMENTS The authors gratefully acknowledge the financial support from the Agency for Science, Technology and Research (A*STAR), Singapore, and the use of computing resources at the A*STAR Computational Resource Centre, Singapore. H.G. acknowledges support by the A*STAR Visiting Investigator Program, “Size Effects in Small Scale Materials”, hosted at the Institute of High Performance Computing in Singapore.
Figure 5. Summary of literature data of FCC nanowire tensile failure modes as a function of their length and diameter. Filled points represent ductile/necking failure modes; empty points represent brittle failure modes; triangular points represent observations in Cu; and circular points represent observations in Au. The results of the present study are the upward-facing triangular points. The solid line is the critical length predicted by the present study. It is assumed that these nanowires are of arbitrary orientations.
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