Letter pubs.acs.org/NanoLett
Cyclic Deformation in Metallic Glasses Z. D. Sha,† S. X. Qu,‡ Z. S. Liu,† T. J. Wang,*,§ and H. Gao*,∥ †
International Center for Applied Mechanics, State Key Laboratory for Strength and Vibration of Mechanical Structures, Xi’an Jiaotong University, Xi’an 710049, China ‡ Department of Engineering Mechanics, Zhejiang University, Hangzhou 310027, China § State Key Laboratory for Strength and Vibration of Mechanical Structures, School of Aerospace Engineering, Xi’an Jiaotong University, Xi’an 710049, China ∥ School of Engineering, Brown University, Providence, Rhode Island 02912, United States ABSTRACT: Despite the utmost importance and decades of experimental studies on fatigue in metallic glasses (MGs), there has been so far little or no atomic-level understanding of the mechanisms involved. Here we perform molecular dynamics simulations of tension−compression fatigue in Cu50Zr50 MGs under strain-controlled cyclic loading. It is shown that the shear band (SB) initiation under cyclic loading is distinctly different from that under monotonic loading. Under cyclic loading, SB initiation takes place when aggregates of shear transformation zones (STZs) accumulating at the MG surface reach a critical size comparable to the SB width, and the accumulation of STZs follows a power law with rate depending on the applied strain. It is further shown that almost the entire fatigue life of nanoscale MGs under low cycle fatigue is spent in the SB-initiation stage, similar to that of crystalline materials. Furthermore, a qualitative investigation of the effect of cycling frequency on the fatigue behavior of MGs suggests that higher cycling frequency leads to more cycles to failure. The present study sheds light on the fundamental fatigue mechanisms of MGs that could be useful in developing strategies for their engineering applications. KEYWORDS: Metallic glass, fatigue mechanism, shear band formation, molecular dynamics simulation
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or mixed-mode microcracks emanating from stress concentrators, such as pores, surface flaws, and sample edges.7,12 However, previous experiments and simulations on MGs have mostly focused on the investigations of SB formation under monotonic loading.21−23 There has been comparatively little work on SB formation under cyclic deformation, because of the limited experimental capability as well as computing power.7 To our knowledge, there is so far little or no atomic-level understanding of the mechanism of SB formation in MGs under cyclic deformation. The atomic-scale processes underlying shear localization under cyclic deformation are difficult to study experimentally, due to the small length and time scales involved. For example, the SBs can be as thin as ∼10 nm, with internal structures on the atomic scale. In contrast, this scale is perfectly amenable to molecular dynamics (MD) simulations. It has been shown that MD simulations can provide critical atomistic insights into the underlying mechanisms of fatigue failure in MGs,6,9,24−27 even though they should not be directly compared to experimental measurements which usually occur on very different time scales.28
etallic glasses (MGs) have shown great promise for applications that require materials with concurrent high strength and a high elastic limit.1−5 In practical applications, however, service conditions that introduce cyclic variations in stresses are inevitably involved,6 and fatigue is believed to account for more than 90% of all mechanical failures in structural materials.7−9 Consequently, the resistance of MGs to the onset and progression of localized deformation under cyclic loading is of considerable scientific and technological interest.6,10 Despite the utmost importance and decades of experimental studies on fatigue in MGs,7,8,10−15 the exact nature of fatigue characteristics and mechanisms of MGs still remains elusive. Given the high strength of MGs under monotonic loading, their reported low fatigue limits ranging from 5%16,17 to 50%18−20 of the ultimate tensile strength (UTS) are surprising. The fatigue−failure process usually can be categorized into three stages: damage initiation, stable damage growth, and fast fracture.7,13,20 The lack of a damage-initiation stage in macroscopic MG samples was shown to account for their low fatigue limits, in striking contrast to the crystalline materials which possess a significant damage-initiation stage and hence exhibit much higher fatigue limits.11,13 Obviously, the damageinitiation mechanism is a key to the fatigue performance of MGs.7 Damage in MGs typically initiates as shear bands (SBs) © XXXX American Chemical Society
Received: August 1, 2015 Revised: September 17, 2015
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DOI: 10.1021/acs.nanolett.5b03045 Nano Lett. XXXX, XXX, XXX−XXX
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Nano Letters In the present study, we aim to address the following questions via MD simulations: (1) How do SBs initiate and propagate under cyclic loading? (2) What are the differences in SB formation under cyclic loading versus under monotonic loading? (3) What are the basic deformation characteristics and underlying mechanisms of MGs under cyclic loading? (4) What is the effect of cycling frequency on the fatigue behavior of MGs? To this goal, MD simulations of tension−compression fatigue tests on Cu50Zr50 MG up to 175 cycles under straincontrolled loading are performed. Simulations for monotonic loading are also conducted under the same conditions for comparison. It is found that the MG surface undergoes large localized shear strain and hosts a high density of shear transformation zones (STZs) under cyclic loading. The SBs initiate when the size of STZ aggregates accumulating at the MG surface reaches a critical value comparable to the SB width. Our work shows that the growth of SBs under cyclic loading is distinctly different from that under monotonic loading, in which SBs form by the coalescence of localized regions with concentrated STZs. It is further shown that nearly the entire fatigue life of MGs is spent in the SB-initiation stage, which has important implications for improving the fatigue life of MGs. Furthermore, we examine the basic deformation characteristics of MGs under cyclic loading. It will be shown that the accumulation of STZs follows a power law, regardless of the applied strains within elastic or plastic regime, and the accumulation of STZs is strongly related to the applied strain. Lastly, the effect of cycling frequency on the fatigue behavior of MGs is discussed, with results suggesting more cycles to failure at higher cycling frequencies. MD simulations are performed using the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS).29 The simulated system is a nanoscale Cu50Zr50 MG sample containing ∼0.55 million atoms in a slab of dimensions of 56.2 (x) × 28.0 (y) × 6.23 (z) nm3. A constant integration time step of 2 fs is used in all simulations. The atomic interactions within the Cu−Zr MGs are modeled using the embedded atom method (EAM) potential.23,30,31 In constructing the sample, a small cube (∼13 000 atoms) with periodic boundary conditions (PBCs) along all three dimensions is first equilibrated at 2000 K for 2 ns and then cooled at a quenching rate of 109 K/s to 50 K, at zero external pressure. The simulation samples are constructed by replications of the small cube, then annealed for 0.5 ns at 800 K, and finally brought back to 50 K.30,32 Such treatment is supposed to eliminate potential artifacts due to the replication process. Cheng et al. have compared MD simulations of a small sample and those of a large sample constructed by replication of the small sample under pure shear and uniaxial tension/compression and have shown that both simulations resulted in similar stress−strain curves.31 For cyclic as well as monotonic loadings, PBCs are imposed along the xand z-directions while the free surface condition is kept along the y-direction. A strain rate of ∼109 s−1 along the x-direction is imposed at a temperature of 50 K. The deformation is carried out under displacement control. Prior to the fatigue tests, monotonic uniaxial loading is performed and the tensile stress−strain curve is depicted in Figure 1a. The red dotted line indicates the initial linear elastic region. The open symbols mark the strains corresponding to those used in the strain-controlled cyclic tests, ranging over elastic and plastic regimes from 1.71% to 6.40%. The typical strain−time plots for tension−compression fatigue tests with applied strains of 1.71%, 4.27%, and 6.40% are shown in Figure
Figure 1. (a) Monotonic uniaxial tensile stress−strain curve for Cu50Zr50 MG at a strain rate of ∼109 s−1. The red dotted line indicates the initial linear elastic region. The open symbols mark the strains corresponding to those used in the strain-controlled cyclic loading. (b) Typical strain−time plots for tension−compression fatigue tests with applied strains of 1.71%, 4.27%, and 6.40%.
1b. Note that a positive strain value indicates tensile loading while a negative strain value indicates compressive loading. Figure 2a shows a sequence of snapshots that demonstrate deformation in Cu50Zr50 MG under monotonic uniaxial tensile loading. The deformation process is monitored by using the socalled local atomic shear strain ηMises .30,31,33,34 The presence of i regions with relatively large local atomic shear strain indicates a high density of STZs,30 which represents the collective and inelastic shearing of atoms in response to an applied shear stress.35 Previous studies have shown that the deformation process of MGs is strongly dependent on the cooling rate in MG sample preparation and the strain rate in tensile loading.23,30,31 For comparison purposes, monotonic uniaxial loading is performed under the same conditions used in the cyclic loading in the present work. In the elastic regime, only a few STZs occur initially. With further loading, more STZs pop up at various places in the sample including both surface and interior, similar to that previously reported by Cao et al.30 At an overall ε of ∼10%, regions with concentrated STZs are linked up to form localized shear regions known as SBs. In order to describe the growth rate of STZs, Figure 2b plots the fraction of atoms with relatively large local atomic shear strain, ηMises > 0.2,32 during deformation up to the onset of the i SB formation. It is found that a power law with an exponent of 5.678 can fit the behavior, as shown in Figure 2b. The STZs grow slowly in the elastic regime but develop rapidly in the plastic regime until the SB formation, as shown in the corresponding panels in Figure 2a. We are particularly interested in SB formation in low cycle fatigue tests under high applied strains. Figure 3 shows fatigue tests under an applied strain of 6.40%, where the maximum stress is 96% of the UTS under monotonic loading (see Figure 1a). A snapshot at the strain of 6.40% under monotonic loading colored by the atomic local shear strain is shown in Figure 2a, where STZs are seen to pop up at various places. The stress as a B
DOI: 10.1021/acs.nanolett.5b03045 Nano Lett. XXXX, XXX, XXX−XXX
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function of cycle number is plotted in Figure 3a. After dozens of loading cycles, a stress drop indicated by the arrow is observed. Usually, a drop in stress corresponds to a rapid localization of plastic strain into a single dominant SB.32,36,37 Figure 3b shows a sequence of snapshots that clearly capture the SB growth under cyclic loading. Regions with large ηMises i (i.e., STZs) predominately occur at the MG surface and some also inside the sample. With increasing loading cycles, several embryonic STZ regions at the MG surface merge and keep growing. Then, as the aggregate size of STZs accumulating at the MG surface reaches a critical value, SBs initiate and propagate across the sample. To quantify this critical state, we measure the size of STZs at the onset of SB initiation by equating their area to that of a circle πd2/4, where d is an effective diameter defined as the STZ size. In Figure 3c, the red line outlines the area of the critical STZ aggregates at the onset of SB initiation. Our calculation shows that d is ∼6.2 nm. After an SB is fully formed as shown in Figure 3d, its width is measured in the range of 5.6−6.3 nm. Hence, we conclude that the SB initiation criterion is dictated by the size of critical STZ aggregates. More specifically, when the size of the STZ aggregates accumulating at the MG surface during cyclic loading reaches a critical value comparable to the SB width, the SB initiation can take place. Interestingly, as the STZs propagate across the sample, they are simultaneously driven toward the boundaries of the SB under the repeated loading and unloading fatigue process, as shown in the bottom panels of Figure 3b. These observations in Figures 2−3 highlight that the SB initiation under cyclic loading is distinct from that under monotonic loading.
Figure 2. Deformation of Cu50Zr50 MG under monotonic uniaxial loading. (a) A sequence of snapshots capturing the occurrence of shear localization. The color indicates local atomic shear strain. (b) The fraction of atoms with relatively large atomic shear strain during deformation, i.e., the growth rate of STZs. The red dotted line indicates a fitting curve. The symbol of X indicates SB formation.
Figure 3. Nucleation and growth of an SB under cyclic loading. (a) Stress as a function of cycle number for a fatigue test with applied strain of 6.40%. The stress drop indicated by the arrow corresponds to the SB formation. (b) A sequence of snapshots capturing the SB formation under cyclic loading. Color indicates the atomic local shear strain. (c) The area of an STZ aggregate marked by the red line at the onset of SB formation. (d) Boundaries of the SB marked in red lines. C
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investigated. Figure 5a displays the fraction of atoms with relatively large atomic shear strain as a function of cycle number
Fatigue damage in MGs was observed to initiate as mixedmode cracks or SBs.7,9,12 In the present work, no apparent crack formation has been observed, and shear banding is found to be the fatigue mechanism for nanoscale MGs under lowcycle fatigue shown in Figure 3. This is consistent with recent experiments on fatigue behaviors of microsized MGs,12 as well as MD simulations on low-cycle fatigue fracture of MG nanowires.9 The process of shear banding is comprised of SB initiation and SB propagation. In the present work, the SB initiation occurs as the size of STZ aggregates accumulating at the MG surface reaches a critical value comparable to the SB width, and SB propagation takes place when the STZ aggregates subsequently propagate across the sample, corresponding to what is traditionally referred to as the front propagation mechanism of SB initiation.28,35 In spite of similar stress drops in the stress−strain curve,28,38 the mechanism for SB propagation studied here is different from that of SB propagation in experiments, the latter typically involving two parts of a sample sliding relative to each other.28,35 The time of SB initiation and that of SB propagation as a function of the applied strain amplitude are plotted in Figure 4. In order to
Figure 5. Deformation characteristics of MGs under cyclic loading. (a) The fraction of atoms with relatively large atomic shear strain, corresponding to the accumulation rate of STZs, as a function of cycle number for fatigue tests with applied strains from 1.71% to 6.40%. The red dotted lines are fitting curves. The symbol of X indicates SB formation. (b) The exponent c of the power law in (a) as a function of applied strain. The red dotted line is a fitting curve.
for all fatigue tests with the applied strain amplitude ranging from 1.71% to 6.40%. Regardless of the applied strains within the elastic or plastic regime, all of the curves can be fitted using the same type of power law as in Figure 5a. Such power-law form can be rationalized from the point of view of damage accumulation, which depends on the applied strain. In the present work, no apparent crack is introduced or formed. Therefore, fatigue damage evolution in terms of STZ development, instead of crack propagation, should be considered. Atoms with a higher than 20% local shear strain are considered to be damaged.9 Thus, the fatigue damage can be quantified as the ratio of the number of atoms in the damaged region relative to the total number of atoms in the sample, as shown in Figure 5a. Our simulation results reveal that the damage accumulation is controlled by the applied strain via a power law. However, the accumulation rate of STZs is quite different. For fatigue tests with applied strain amplitudes within the elastic regime, the STZs accumulate quickly in the early stage but the rate slows down in the later stage. On the other hand, for fatigue tests with applied strain amplitudes within the plastic regime, the STZs accumulate quickly until SB formation. The accumulation rate of STZs can be measured by the exponent c of the power law (note that the parameters a and b in the power law have very small values). Figure 5b plots the exponent c as a function of the applied strain amplitude. It can be seen that the accumulation rate of STZs is strongly related to the applied strain. This is perhaps not surprising because the applied strain governs the growth rate of STZs, as shown in Figure 2b. The accumulation rate of STZs as a function of the applied strain amplitude also follows a
Figure 4. Time of SB initiation and that of SB propagation as functions of applied strain amplitude in fatigue tests. The SB initiation occurs as the size of STZ aggregates accumulating at the MG surface reaches a critical value comparable to the SB width, and SB propagation takes place when the STZ aggregates subsequently propagate across the sample. Note that almost the entire fatigue life of the nanoscale MGs is spent in the SB-initiation stage.
observe shear banding, the applied strains in the fatigue tests are chosen to range in plastic regime from 5.76% to 6.62%. It is found that the time of SB initiation is very sensitive to the applied strain. With increasing applied strain, the time of SB propagation decrease slowly, while the time of SB initiation decrease sharply. An important finding is that most of the fatigue life of MGs is spent in the SB initiation. This observation is similar to the fatigue life of crystalline materials which is also dominated by the damage-initiation stage.39 In contrast to macroscopic MGs for which the damagepropagation process takes up almost the entire fatigue life,11 our results suggest that the fatigue life of nanoscale MGs is dominated by the loading cycles needed for SB initiation at a critical size (d) of STZ aggregates, which may have important implications for improving the fatigue life of MGs by suppressing SB initiation. The statistics of accumulating deformations can help us understand the growth behavior of defects during cyclic loading.26 As MGs typically fail by the formation of highly localized shear regions, the fraction of atoms with relatively large atomic shear strain (i.e., STZs) during cyclic loading is D
DOI: 10.1021/acs.nanolett.5b03045 Nano Lett. XXXX, XXX, XXX−XXX
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Figure 6. Effect of cycling frequency on the fatigue deformation of MGs. (a) Typical strain−time plots for various cycling frequencies at an applied strain of 6.40%. (b) Stress versus cycle number curves at various cycling frequencies. Stress drops indicated by the arrows correspond to the SB formation. (c) Fatigue life as a function of the cycling frequency. The results demonstrate that a higher cycling frequency leads to more cycles to failure. (d) The fraction of atoms with relatively large atomic shear strain as a function of cycle number for various cycling frequencies at an applied strain of 6.40%.
quality, cycling frequency, and surface condition.7,12,20,27 As the deformation and failure mechanisms of MGs also depend on the sample size,40,41 it will be interesting to consider different sample sizes in future work. In summary, we have performed a series of MD simulations of low-cycle fatigue of Cu50Zr50 MGs with the applied strain amplitude ranging across the elastic and plastic regimes from 1.71% to 6.40%. It is found that almost the entire fatigue life of MGs is spent in the SB-initiation stage, and the SB initiation under cyclic loading is distinctly different from that under monotonic loading. It is shown that the SB initiation criterion under cyclic loading is dictated by the size of STZ aggregates. In this process, the MG surface undergoes large localized shear strain and hosts a high density of STZs. When the size of critical STZ aggregates accumulating at the MG surface during cyclic loading reaches a critical value comparable to the SB width, SB initiation takes place. Our work further reveals that the STZ accumulation as a function of the loading cycle follows a power law, regardless of the applied strain amplitude within the elastic or plastic regime. Lastly, we have demonstrated the effect of cycling frequency on the fatigue behavior of MGs, with results suggesting that a higher cycling frequency leads to more cycles to failure. Our work not only provides an important platform to probe the atomic-level understanding of the fundamental mechanisms of fatigue in MGs but also may help develop strategies for engineering applications of MGs.
power law with an exponent of 5.05, similar to that for the growth rate of STZs shown in Figure 2b. To better understand or predict the fatigue behavior of MGs, it is important to make a qualitative investigation of the effect of cycling frequency. Typical strain−time plots for various cycling frequencies with the applied strain of 6.40% are shown in Figure 6a. The stresses as a function of cycle number for various cycling frequencies are plotted in Figure 6b. The stress drop indicated by the arrow corresponds to the SB formation shown in Figure 3. Our results clearly demonstrate the significant influence of cycling frequency on the fatigue life of MGs. Increasing cycling frequency leads to more cycles to failure, as shown in Figure 6c. This finding is qualitatively consistent with Jang et al.’s experimental studies demonstrating that MGs exhibit more cycles to failure at higher cycling frequencies, by suppressing SB propagation.12 In order to explore the origin of this phenomenon (i.e., more cycles to failure at higher cycling frequencies), Figure 6d displays the fraction of atoms with relatively large atomic shear strain as a function of cycle number for various cycling frequencies at an applied strain of 6.40%. It can be seen that the accumulation rate of STZs is very sensitive to the cycling frequency. With decreasing cycling frequency, the STZs accumulate quickly because of the adequate duration of each pulse shown in Figure 6a; SB formation thus occurs early. The effect of cycling frequency on the fatigue behavior of MGs shown in Figure 6 suggests that a higher cycling frequency leads to more cycles to failure. The power law reported in the present work has also been recently observed in Luo et al.’s MD simulation on Ti−Zr MG nanowires.9 In addition, the trend of the effect of cycling frequency on the fatigue behavior is also consistent with Jang et al.’s experimental studies on microsized Zr−Cu−Al MGs.12 Despite these favorable comparisons, the fatigue behavior of MGs is likely to be influenced by a number of factors such as chemical composition, specimen geometry and size, material
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AUTHOR INFORMATION
Corresponding Authors
*E-mail:
[email protected]. *E-mail:
[email protected]. Notes
The authors declare no competing financial interest. E
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(35) Homer, E. R. Acta Mater. 2014, 63, 44−53. (36) Sha, Z. D.; He, L. C.; Xu, S.; Pei, Q. X.; Liu, Z. S.; Zhang, Y. W.; Wang, T. J. Scr. Mater. 2014, 93, 36−39. (37) Zhou, H. F.; Zhong, C.; Cao, Q. P.; Qu, S. X.; Wang, X. D.; Yang, W.; Jiang, J. Z. Acta Mater. 2014, 68, 32−41. (38) Maass, R.; Klaumunzer, D.; Loffler, J. F. Acta Mater. 2011, 59, 3205−3213. (39) Polak, J.; Vasek, A. Int. J. Fatigue 1994, 16, 403−408. (40) Jang, D. C.; Gross, C. T.; Greer, J. R. Int. J. Plast. 2011, 27, 858− 867. (41) Tonnies, D.; Maass, R.; Volkert, C. A. Adv. Mater. 2014, 26, 5715−5721.
ACKNOWLEDGMENTS Z.D.S., S.X.Q., Z.S.L., and T.J.W. are grateful for financial support from the National Natural Science Foundation of China through Grant Nos. 11321062, 11402189, 11321202, and 11372236.
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