English

• 0.   Introduction

  Metal nanomaterials, as a new class of materials, have been studied extensively in recent years to explore their specific properties for application^[1-4]. Among them, Cu nanowires (NWs) are widely used in composites^[5], wearable devices^[6], sensors^[7], and solar cells^[8] due to their low production cost, excellent flexibility, high light transmission, and electrical conductivity^{[4, 9]}. To increase the service time of the nano-devices, the basic research on the correlation between initial microstructural defects and breaking failure of Cu NWs is particularly significant.

  The externally applied loads to the nanodevices would promote a series of microstructural defects to generate within the structure of nanomaterials, which would have a significant impact on the mechanical properties of the material^[10] and have been studied by a series of experimental techniques. For example, Li et al. ^[11] demonstrated a technique for observing the nucleation of individual dislocations during in-situ transmission electron microscopy (TEM) tensile testing. Wang et al. ^[12] observed the sliding of the grain boundaries in in-situ atomic - resolution TEM experiments, contributing to understanding interface-mediated deformation and failure mechanisms in polycrystalline materials. However, the capability of experimental methods for large samples and detailed evolution of the microstructure is limited due to complicated surface morphology and the small size of NWs^[13-14]. Moreover, the experimental works require advanced techniques, and it usually takes more time and funds but results in fewer samples. In contrast, molecular dynamics (MD) simulation could be a promising method for solving the above problems, which have been widely used in materials science^[15-18]. In particular, Cheng et al. ^[19] simulated the tensile of a series of NWs with different initial structures using MD simulations, finding that the local geometry of nanostructures plays a dominant role in determining the nature and location of mechanical failure. Moreover, Pang et al. ^[20] found that the plastic deformation modes transform with the variation of NW orientations by comparing simulated atomic snapshots for different oriented NWs.

  On the other hand, with continuous loads applied to the nanomaterials, structural defects will spontaneously occur, leading to a final breaking of the nanodevice. In experiments, fracture behaviors are usually observed in in-situ TEM nanomechanical tests^[21-22] and scanning electron microscope (SEM) tensile tests^[23]. However, the limitation comes from the fact that enormous nanoscale samples would not provide by conventional experiment techniques. In addition, the conventional experimental methods are hard to perform precisely temperature-dependent^[24]. Therefore, in our previous works, the MD simulations were used to simulate the tensile of a large number of samples to study the statistical results of fracture positions, indicating the dependence of fracture mechanism on the temperature^[25], cross-section^[26], and strain rate^[27].

  Based on these works, a study on the effect of microstructural defects generated at the early stage of plastic deformation on the breaking failure is required, which is beneficial for structural strengthening and repair as well as optimizing design principles. As Yi et al. ^[28] reported, the emission of the unclosed shear loop plays a crucial role in not only the tensile deformation but also the fracture process of NWs. In our previous work^[29], for the tensile of single crystal Cu NWs under an extreme temperature of 10 K, an apparent correlation between the initial microstructural defects and breaking failure has been shown based on the statistical distribution. By considering the investigation for mechanical properties of metal NWs under a low-temperature range conducive to application, such as the low-temperature wearable strain sensors^[30], two following aspects need to be discussed in detail. Firstly, one would answer the question of how the temperature affects the distribution features of initial microstructural defects and breaking failure. Secondly, it is essential to know the relationship between the initial microstructural and breaking failure at a temperature range based on the statistical analysis.

  The direct observation of the deformation mechanism and statistic feature would help us to answer the above two questions. Therefore, in this work, as shown in Scheme 1, the temperature is set as a variable, and the tensile MD simulations of a series of single-crystal Cu NWs with different initial thermal states have been performed to carry out the statistical analysis of initial microstructural defects and breaking failure. Besides, a series of properties and microstructural deformation behaviors also have been studied in detail to give a comprehensive understanding, which is contributing to promoting the fundamental research for the prediction of breaking failure.

  Scheme 1

  

  Scheme 1.  Schematic explanation of this research

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  1.   Methodology

  1.1   Molecular dynamics simulation

  The single - crystal Cu NWs with face - centered cubic (fcc) lattice along [111] orientation have been modeled for this study, in which with a dimension of 6.25 nm×5.23 nm×19.17 nm. To study the relationship between initial microstructural defect and breaking failure at different temperatures, the temperatures of six ensembles have been set from 20 to 300 K, and all the tensile simulations were performed with a constant strain rate of 0.1%·ps^-1, corresponding to an absolute speed of 19.1 m·s^-1 along the axial direction. Before applying the tensile loads, the free relaxation for an average relaxation time of ca. 10 000 MD steps was carried out by each sample to reach an equilibrium state^[27]. Following thermal equilibration, the relaxation time for each sample increased by 50 MD steps to ensure the randomness and independence in samples. The ensemble corresponding to each temperature included 300 samples with the same initial atomic arrangement but different initial states, which were characterized by the different thermal movements of atoms. The tensile stress loading along the axial direction was applied by the fixed layers which are set by every four atoms layer at the ends of the NW. In this work, the embedded - atom method (EAM) potential^[31] has been used to describe the interatomic interactions which are expressed as

  $ E = \frac{1}{2}\sum\limits_{ij} {V({r_{ij}})} + \sum\limits_i {F({{\begin{array}{*{20}{l}} \rho \end{array}}_i})} $

  (1)

  $ {\rho _i} = \sum\limits_{j \ne i} {\begin{array}{*{20}{l}} {\phi ({r_{ij}})} \end{array}} $

  (2)

  where E is the total potential energy of the system, V(r_ij) is the pair potential energy between atoms i and j, and r_ij is the distance between them, ϕ(r_ij) is the electron density at atom i due to atom j as a function of distance r_ij, ρ_i is the electron density at atom i due to all other atoms, F(ρ_i) is the energy to embed atom i in an electron density ρ_i. The parameters of EAM potential for Cu used in this paper were referred to as that developed by Johnson^[32-33], which have been widely used in published works^[34-35].

  The tensile loading was simulated using the self- developed software NanoMD^[36], in which the free boundary condition was used to simulate the working condition of the NW structure. The Verlet and Cell - linked list^[37] was performed to build the adjacent list. The Leap-Frog^[38] algorithm with an MD step of 2.5 fs was adopted to solve Newton′s equations of motion. During the stretching of the NW, the types and coordinates of atoms, stress, strain, and energy in the NW system have been recorded every 500 MD steps for further analysis.

  1.2   Data analysis

  The average stress within the NW in the z-direction has been calculated using the virial scheme^[39], in which the atomic stress is expressed by the EAM potential function as

  $ \sigma _i^{zz} = \frac{1}{{{\Omega _i}}}\{ - {m_i}{(v_i^z)^2} + \frac{1}{2}\sum\limits_{j \ne i} {[\frac{{\partial \varphi }}{{\partial {r_{ij}}}}} + (\frac{{\partial F}}{{\partial {\rho _j}}} + \frac{{\partial F}}{{\partial {\rho _i}}})\frac{{\partial f}}{{\partial {r_{ij}}}}]\frac{{{{(r_{ij}^z)}^2}}}{{{r_{ij}}}}\} $

  (3)

  where $\sigma _i^{zz}$ is the atomic stress tensor of the atom i in the zz direction (tensile direction), $r_{ij}^z$ is the distance between the atoms i and j in the zz direction, Ω_i and m_i are the volume and mass of the atom i, and $v_i^z$ is the velocity component in the z-direction of the atom i. The parameters of φ, F, ρ, and f in the above equation are from the EAM potential^[40]. The strain ε and strain rate ${\boldsymbol{\mathop \varepsilon \limits^.}} $ (the strain per unit time) of the tensile are defined as

  $ \varepsilon = \frac{{l - {l_0}}}{{{l_0}}} $

  (4)

  $ \mathop \varepsilon \limits^. {\rm{ }} = {\rm{ }}\frac{{{\rm{d}}\varepsilon }}{{{\rm{d}}t}} $

  (5)

  where l is the actual length of the NW in the z-direction during stretching, l₀ is the length after free relaxation but before stretching, and t is the time.

  For the analysis of the crystal structure, the radial distribution function (RDF) has been used to identify the crystal order, in which the g(r) function is defined as

  $ g(r) = \frac{{2U}}{{{N_{\rm{m}}}}}\langle \sum\limits_{i < j} {(r - {r_{ij}})} \rangle $

  (6)

  where U is the system volume and N_m is the number of atoms. The detailed calculation procedures have referred to in the reference report^[41].

  The study of the relationship between the breaking failure and initial microstructural defects is depended on positions among them. The breaking position of the NW has been calculated using the atom number ratio (ANR)^[26], which is defined as follows

  $ ANR{\rm{ }} = \frac{{{N_{{\rm{bottom}}}}}}{{{N_{{\rm{total}}}}}} $

  (7)

  Where N_bottom is the atom number in the bottom part after breaking and N_total is the total atoms number of the NW.

  The initial microstructural defects are referred to as the slips generated at the stress yield point (YP), called initial slips. To get the position of the initial slip, the centrosymmetric parameter (CSP) ^[42] has been used to obtain all hcp atoms of the NW at the stress YP. The CSP value of each atom is defined as follows

  $ p = \frac{1}{{D_0^2}}{\sum\limits_{i = 1, 6} {\left| {{{\boldsymbol{R}}_i} + {{\boldsymbol{R}}_{i + 6}}} \right|} ^2} $

  (8)

  Where R_i and R_i+6 are the vectors corresponding to the six pairs of opposite nearest neighbors in the fcc lattice, and D₀ is the distance of the nearest neighbors. The CSP value of an atom i is defined as the p_i. For reference, p_i < 0.4 represents the fcc atoms, 0.7 < p_i < 1.1 stands for stacking faults, corresponding to the hcp atoms; p_i > 2.0 corresponds to the surface atoms of the NWs^[29].

  Following all hcp atoms of the NW at stress YP are obtained, based on the coordinates of these atoms, then cluster all the hcp atoms into every slip to get the positions. Considering the initial slips are various in shape and size, and the number of initial slips for each sample is random, the density-based spatial clustering of applications with noise (DBSCAN) algorithm^[43] has been used to cluster closely-linked hcp atoms into one slip plane. There is no need for giving the clustering numbers beforehand. However, the parameters (e, M) are requisite, which describe the compactness within a group of neighborhoods, where e is the neighborhood distance threshold for a particular hcp atom and M is the minimum number of atoms within a neighborhood of this particular hcp atom^[29]. In this paper, the e was set as 0.08, and the M was set as 8. After the clustering, the coordinate z of every initial slip plane intersected with the lateral edge of the NW is defined as the initial slip position. This method was applied to the analysis of initial slips at the temperature range from 20 to 300 K, however, it provides an element of uncertainty for the analysis of initial slips under a higher temperature than 300 K because of the thermal movement. The positions of initial slips and breaking failure were based on the normalized length of the NW at the stress YP and when it breaks, respectively.

  2.   Results and discussion

  2.1   Temperature dependence of the distributions of initial slip positions

  While the tensile stress applied to the NW system is exceeded the ability to resist plastic deformation, a series of microstructural defects such as slip planes are formed in the NW. Generally, the dislocation first occurs at 4/5 of yield stress, which refers to the true YP. With continuous tensile loading, the movement of dislocations promotes the forming of slip planes. When the stress reaches the maximum (YP), the deformed structure shows several {111} slip planes which refer to the initial slips discussed in this work. Fig. 1 shows the initial slips of three representative samples at the stress YP. It could be found that these initial slips are usually generated from the sides of the NWs and developed upwards or downwards at an angle to the surface^[10]. The published works^[44-45] have demonstrated that the physical and mechanical properties of NWs are strongly affected by surface effects and nanosize effects. Therefore, in this paper, the free boundary condition was used to simulate the NW structures. In Fig. 1a, two initial slips are generated at the ends of the NW. Moreover, three initial slips in Fig. 1b are generated at both the two ends and the middle part of the NW. In Fig. 1c, it is seen that several initial slips are generated near the middle part of the NW, indicating that the fixed - end effect on initial microstructure defects is reduced as much as possible even though the tensile load is applied by the two fixed ends. Moreover, the results demonstrate that the number of initial slips generated is random and the positions of initial slips generated could be any part of the NW.

  Figure 1

  

  Figure 1.  Initial slips generated at the stress YP at (a) 20, (b) 50, and (c) 150 K

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  By considering the above representative characteristics of initial slips, the DBSCAN algorithm was used to calculate the normalized position of initial slips. Based on the MD simulations, the normalized positions of initial slips at different temperatures from 20 to 300 K have been analyzed to further investigate the temperature dependence of initial microstructural defects, as shown in Fig. 2. The ratio between the statistical number of initial slips (N_slips) and the total number of samples (N_samples) for each temperature ensemble is defined as

  Figure 2

  

  Figure 2.  Distributions of initial slip positions for 300 independent samples at different temperatures

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  $ \varphi = \frac{{{N_{{\rm{slips}}}}}}{{{N_{{\rm{samples}}}}}} $

  (9)

  The φ under these six temperatures are 2.77, 2.80, 3.36, 3.87, 3.85, and 3.92, respectively. It means that the statistical number of initial slips for each distribution is much more than 300 samples, which is usually 2-4 times the number of samples during this temperature range, and the number of initial slips increases in the temperature range from 20 to 150 K. From 150 to 300 K, the φ fluctuates around 3.88 with its existing statistical errors. In Fig. 2a and 2b, it can be observed that high statistical peaks are formed at the ends of distributions, appearing skewed distributions at two ends under the temperatures of 20 and 50 K. Comparatively, the initial slips generated in the interval of 0.2 to 0.8 are not forming the statistical peaks, but following a uniform distribution. It infers the fact that the initial slips of the NWs simulated in this paper tend to generate at the two ends while the temperature is less than 50 K. Under the temperature of 100 K (Fig. 2c), there is a tendency of equalization toward the peaks at the ends to the middle part. While the temperature increases to 150 K (Fig. 2d), the peak at the left end is almost disappeared, and the peak at the right end significantly decreases with which the height approaching the middle part. At the higher temperatures of 200 and 300 K (Fig. 2e, 2f), the positions of initial slips are concentrated on the middle part of the NW rather than the ends. The ratio (θ) of the average peak values at the ends (C_ends) and corresponding average counts in the interval of 0.2 to 0.8 (C_center) is defined as

  $ \begin{array}{*{20}{l}} {\theta = } \end{array}\frac{{{C_{{\rm{ends}}}}}}{{{C_{{\rm{center}}}}}} $

  (10)

  The θ at the temperatures of 20, 50, 100, 150, 200, and 300 K are 8.14, 7.56, 3.15, 1.34, 0.74, and 0.94, respectively. At temperatures of 200 and 300 K, the θ is less than 1, demonstrating that the generation of initial slips at the ends is restrained at high temperatures. The temperature dependence of peak values is much larger than the fixed - end effect, in which the fixed-end effect is reduced as much as possible in MD simulations. As is well - recognized, thermal motion is related to the change of temperature, leading to the micro atomic fluctuation ways and movements being different^[25]. As the above analysis, these microscopic structural changes are also shown statistically in initial slip distributions. Furthermore, a detailed analysis of mechanical properties is necessary to understand how temperature affects the generation and propagation of initial microstructural defects.

  2.2   Mechanical properties

  The effect of temperature on the deformation mechanism is attributed to the effect on the mechanical properties^{[10, 46]}. To further investigate the temperature dependence of the initial slip distribution and deformation mechanism, the correlative properties of the system have been studied.

  Fig. 3a illustrates how the temperature affects the stress-strain curves of six representative samples. While the stress increases to the maximum value, it is referred to as the stress YP. Generally, the deformation before the YP is referred to as elastic deformation, which is labeled in Fig. 3a as region E. At the YP, as above mentioned, the initial slips have been developed sufficiently, promoting the stress to release during the plastic deformation. The plastic deformation corresponds to different deformation states: stress release region, high - stress fluctuation region, and low - stress fluctuation region, which are called regions R, HS, and LS, respectively.

  Figure 3

  

  Figure 3.  Stress-strain curves of the Cu NW at different temperatures with a constant strain rate of 0.1% ps^-1: (a) stress-strain curves of representative samples; (b) average yield stress with error bars for every 300 samples

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  Comparing all stress - strain curves, due to the induced atomic vibrations at elevated temperatures, the fluctuations in these curves increase with the increasing temperature^[47]. In region E, the slopes of the stress- strain curves are reflected Young′s modulus of the NWs. The statistical average Young′s modulus (Fig.S1, Supporting information) for each temperature was calculated, and it is found that Young′s modulus decreases with the increase in temperature^{[10, 25, 47]}, which infers that the stiffness of the NW structure is weakened with the increasing of the temperature. Fig. 3b marks the average yield stress and average yield strain, showing a deviation from each sample. As shown in Table 1, with temperature increases, the average yield stress decreases. It is further observed that the mechanical strength of the metallic material reduces with an increase in temperature. From 20 to 100 K, an abrupt decrease is identified and the average yield strains are 0.054 (20 K) and 0.046 (100 K) respectively. From 100 to 300 K, it is found that the average yield strain keeps almost constant with an increase in the temperature. A similar temperature dependence of the yield strain has also been observed in the study of breaking behaviors of the Cu NWs with the dependence of temperature, as Wang et al.^[25] reported, the first yield strain almost keeps constant at the temperature range from 100 to 600 K, whereas, the higher temperature (700 K) leads to significant changes in yield strain. At low temperatures, the bigger yield strain and bigger yield stress demonstrate that the stage of elastic deformation is longer. Whereas, the smaller yield stress and smaller Young′s modulus infers that the mechanical strength of the NW reduces at high temperatures, resulting that the yield strain being smaller and keeping almost constant in the temperature range from 100 to 300 K. Such change in the YP demonstrates that the structural change of the NW at different temperatures, especially for the fluctuation of the position and the number of the initial microstructural defect. After the YP, further plastic deformation forces the stress of each system to decrease abruptly in region R and then follows a continuous process of increasing and then decreasing in HS. In the region HS, the stress curves both fluctuated around 2 GPa. However, what is significantly different is that the increase in stress is greater at lower temperatures and the stress tends to decrease as the temperature increases, reflecting the positive effect of higher temperatures on viscoelastic properties. According to that illustrated above, the changes in distributions of initial slips are likely caused by the influence of temperature on the variation in properties of the NW structure. In our previous work^[48], it is reported that the generation and development of initial slips are closely related to yielding behaviors. To understand the variation in initial slips more clearly than is possible, more properties of these samples have been analyzed.

  Table 1

  

  Table 1.  Average yield strain and average yield stress for 300 samples at different temperatures

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  Parameter	20 K	50 K	100 K	150 K	200 K	300 K
  Average yield strain	0.054±0.001	0.051±0.001	0.046±0.002	0.044±0.002	0.045+0.003	0.045±0.005
  Average yield stress	3.84±0.07	3.58±0.06	3.22±0.09	3.03±0.09	3.02±0.1	2.87±0.11

  Fig. 4a shows the energy-strain curves under different temperatures. The average potential energy of the NW increases with the increase in temperature, which is caused by the atomic movement overcoming the interatomic cohesive energy in the tensile deformation process of the NW. At a higher temperature, the average kinetic energy of the atoms increases due to the thermal motion^[14], making the atoms more likely to move relative to each other, leading to a significant reduction in the interaction forces and bond strengths between the atoms^[46]. Therefore, the average yield stress (Fig. 3b) decreases with the increase in temperature. Another important point, it is well-known that the relationship between the average potential energy and temperature is also reflected by the influence of temperature on the degree of lattice order. To clarify the influence of temperature on the potential energy more clearly to further investigate the structural changes at the YP, Fig. 4b shows the RDF at the stress YP for these six samples. At a low temperature of 20 K, the peak value of the RDF curve is very high and sharp, indicating a well-ordered crystalline structure of the Cu NW. With increasing temperature, the peak values of RDF curves gradually decrease and the width increase, which demonstrates that the number of ordered crystal structures decreases with temperature and that of amorphous structures increases^[49]. Under a higher temperature, there is a larger magnitude of atomic oscillation, contributing to the atoms overcoming the interatomic cohesive energy to form a disordered amorphous structure, which further induces the increase of the average potential energy^[25]. In general, with the temperature increasing, the increases in the average kinetic energy of the atoms and the generation of disordered crystalline structures have collectively resulted in the reduction in the ability to resist plastic deformation of the NW. It means that the initial slips are easier to generate within the NW. Therefore, the average yield strain (Fig. 3b) decreases as well as the total number of initial slips increases with the increase of the temperature.

  Figure 4

  

  Figure 4.  Energy-strain curves during stretching and the curves of RDF: (a) variation of the energy of representative samples with the strain at different temperatures; (b) RDF curves of representative samples at different temperatures

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  The above analysis has demonstrated the temperature dependence of mechanical properties and yield behaviors, resulting in the changing of the initial slip distribution. At a lower temperature, the yield stress and Young′s modulus in the NW system are higher, showing the brittleness of the NW. As the increasing of the temperature, the decreasing yield stress and Young′s modulus, lead to a reduction in the bonding force within the system. Therefore, the brittleness of the NW is diminished. Accordingly, it shows a positive effect on the ductility of the NW, which leads to more initial slips tending to generate at a lower yield strain. Moreover, it has suppressed the generation of initial slip at both ends. Thus, in the distributions of initial slips under the temperatures from 20 to 300 K (Fig. 2a- 2f), the peak values at two ends gradually decrease and average out towards the middle part.

  To further study the generation and development of initial slips, the variation of the numbers of hcp atoms and fcc atoms in the tension process has been analyzed, as shown in Fig. 5. As the average yield strain of samples decreases with the increase in temperature (Fig. 3b), it could be seen that the strains of samples at which the number of the hcp atoms begin to increase are different in Fig. 5. After the yield stress, the number of the hcp atoms for each sample both linearly increases, correspondingly, the number of fcc atoms decreases, which demonstrates the sign of generation of initial slips. Furthermore, the first peak values of the number of hcp atoms for these samples are significantly different. Compared with the sample at the temperature of 150 K, the numbers of hcp atoms for others samples continuously increase. It further indicates that the total area of initial slips or the number of initial slips within the system is different from each other (Fig. 1)^[48].

  Figure 5

  

  Figure 5.  Variation of the number of hcp and fcc atoms with the strain

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  The above analysis has shown the effect of temperature on the mechanical properties of the Cu NWs, which was also reported by a series of published studies^{[20, 25, 46]}. Particularly, it needs to be pointed out the temperature dependence of the initial slips. With further increase in temperature, on the one hand, the positive effect on the ductility of the NW promotes the initial slips generating at a smaller strain to release the stress in the system. On the other hand, it contributes to the changing of initial slip positions from two ends to the middle part of the NW.

  2.3   Temperature effect on breaking failure behaviors

  With continuous plastic deformation, the NW materials would finally fracture and become useless. If the breaking position is predictable, the NW could be strengthened timely to avoid fracture failure^[50]. Likewise, it significantly contributes to optimizing design principles in manufacturing. Given that the breaking behavior of a single NW sample is not enough to demonstrate the most probable breaking position, the breaking cases of plenty of samples are essential for showing a result of statistical physical chemistry.

  In this work, the breaking positions of 300 samples for each simulated temperature ensemble from 20 to 300 K have been counted, as shown in Fig. 6. At each temperature, the characteristics of breaking distribution are similar, which follows two high but asymmetric peaks at two ends. Similarly, Cheng et al. ^[19] reported that the Cu NWs at the temperature of 10 K tend to break at two ends of the NW, in which the simulated strain rate is similar to this work. In addition, as Liu et al. ^[50] investigated the effect of the length on the breaking behavior of the Cu NW, the NWs underwent a high-rate stretching with a constant temperature of 300 K, and it was found that as the length of the NW increases, the breaking position gradually shifts to the ends. As above mentioned, it is inferred that the breaking positions of the Cu NWs in present simulated conditions tend to form at two ends of the NW. To give insight into the breaking behavior, it is necessary to understand the properties of the Cu NWs.

  Figure 6

  

  Figure 6.  Distributions of breaking positions for 300 samples at different temperatures

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  In general, the simulation conditions, such as temperature and strain rate, are essential for investigating the deformation behaviors and mechanical properties of the metal NW^{[25, 47]}. It is known that the melting temperature of copper is about 1 350 K^[51]. However, the maximum temperature simulated is 300 K which is below room temperature and less than a quarter of the melting temperature. Therefore, in this simulated condition, it still shows the brittleness of the Cu NW. Moreover, at this lower range of temperature, there is strong symmetric mechanical impact strength within the ends of the NW, so the stretching shock waves had still not propagated repeatedly through the whole NW, while the NW was already showing the sign of fracture failure at the ends. Thus, the strengthening of the ductility is negligible, the NW simulated in this temperature ranging is still tended to break at the ends.

  On the other hand, although the breaking positions of quite a few samples follow distributions at two ends, noted that the number of the discrete samples which break near the middle part of the NW increases with the increase in temperature. Compared with NWs that break at two ends, these discrete samples have better ductility, in which the breaking failure is forming in a larger strain range. Fig. 7 illustrates the average breaking strains of 300 samples under different temperatures, which follow the skewed distributions. At the lower temperatures of 20 and 50 K (Fig. 7a, 7b), for the average breaking strains, only a few samples are greater than 1.0. While the temperature increases to 100 K (Fig. 7c), the number of samples whose breaking strain is greater than 1.0. The results demonstrate that the increase in temperature promotes the ductility of a few samples, however, the structure of the Cu NW is brittle, leading to the breaking positions at the temperature range from 20 to 300 K both following two statistical peaks at the ends of distributions.

  Figure 7

  

  Figure 7.  Distributions of average breaking strain for 300 samples at different temperatures

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  2.4   Relationship between initial microstructural defects and breaking failure

  While initial slips are generated in the NW system, it means the beginning of the plastic deformation, and then the further plastic deformation would make the NW break. If the breaking position could be predictable from the initial microstructural defects, it is beneficial for increasing the service life of the materials. However, few works reported whether there is a relationship between the breaking failure and initial microstructural defects. To get a result of statistical analysis to further investigate the relationship between them, in this work, the breaking position distributions and initial slip position distributions under the temperature range from 20 to 300 K have been analyzed as above mentioned. It is found that there are both two obvious statistical peaks in the initial slip distributions and breaking distributions at temperatures of 20, 50, and 100 K, respectively, inferring that there may exist a statistical correlation between them. It means that the initial microstructural defects are tended to generate near two ends of the NW, further leading to most of the samples being broken at two ends. Whereas, with further increase in temperature, the consistency of distribution characteristics among them gradually disappeared due to the different temperature dependence of initial microstructural defects and breaking behaviors. On the one hand, the increasing temperature has restrained the generation of initial slips at two ends. On the other hand, the single-crystal Cu NW simulated in this paper shows the property of stiffness during the temperature range of 20 to 300 K, so most of the samples still prefer to break at two ends even the temperature increases to 300 K. It indicates the statistical correlation between the initial microstructural defects and breaking behaviors is not well understood. To understand this correlation comprehensively, the microstructural deformation behaviors are studied, which depends on the observation of atomistic configuration with different temperatures during tension, as shown in Fig. 8.It shows the microstructures at the YP, region HS, and region LS.

  Figure 8

  

  Figure 8.  Snapshots of the deformed microstructure with the temperatures of 20, 100, and 200 K at different applied strains: (a) strains in the order of 0.058, 0.1, 0.15, 0.20, 0.25, 0.30, 0.31, 0.40; (b) strains in the order of 0.046, 0.10, 0.15, 0.20, 0.25, 0.30, 0.40, 0.48; (c) strains in the order of 0.042, 0.10, 0.15, 0.20, 0.25, 0.30, 0.40, 0.478

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  Fig. 8a shows the deformation process of the sample at the temperature of 20 K. At the stress YP (0.058), a group of initial slip planes is generated that is parallel and at an angle to the side edges. Generally, the stress within the NW system is released due to the generation of initial slips, so the stress in Fig. 3 decreases abruptly after the YP. While the stress is released during region R, the initial slips have been developed adequately, accompanied by the generation of a series of new slip planes. Thus, there is an increase in the number of hcp atoms at the strain from 0.058 to 0.1 in Fig. 5. Therefore, it is seen that many slips are intersected in the region HS. At the early stage of region HS (0.1 strain), noted that one of the initial slips has propagated to the top of the NW and reflected, following the intersecting with other slips. Further increasing strain, although several slips are intersecting at the bottom of the NW, however, the molten cluster is forming near the top of the NW at the strain of 0.2 which shows up as a stress concentration in the macroscopic view. At the last of region HS, it shows the local melting in the NW, resulting in the necking region and final fracture failure at the top of the NW in region LS. It infers that the breaking failure is related to the propagation of initial slips.

  The deformation behavior of the sample in Fig. 8b under the temperature of 100 K is similar to that in Fig. 8a. At the YP (0.046 strain), there are two slip planes generated on the left and right sides of the NW, respectively. At the 0.1 strain (region HS), the two initial slips on the left side quickly propagate to the top of the NW and intersect. With further stretching, it is observed that the initial slips on the left and right sides intersect with each other near the top of the NW at 0.2 strain. Finally, for the deformation in region LS, most atoms are irregularly arranged and local atomic dislocations result in atomic stacking and the final necking region. Similarly, as shown in Fig. 8c, the initial slips propagate to the bottom of the NW in region HS. In addition, a series of slips intersect near the top of the NW. However, the necking is finally forming at the bottom of the NW in the region LS. As a result, the initial slips would be generated anywhere within the NW and propagate with further stretching. According to that illustrated above, it seems that the final necking region is related to the propagating and intersecting of the initial slips, and it always shows in HS. The microstructural deformation of the samples at the temperatures of 50, 150, and 300 K gives a similar result (Fig.S2). It is illustrated that the deformation mechanism in region HS plays a crucial role in breaking failure. To give insight into the deformation mechanism in region HS, the effect of the boundary could be served as a reference to understand the propagation mechanism of slip planes.

  Scheme 2

  

  Scheme 2.  Schematic explanation of the blocking effect of the ends

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  In general, nanomaterial structures are with boundaries, which include side boundaries and the boundaries of the ends, and that is why the free boundary condition is used to simulate the single-crystal Cu NW in this paper rather than the periodic boundary. While the slips propagate to the side of the NW, the side effect promotes slips to reflect and further develop. In this paper, the initial slips are always developed upwards or downwards, so the slips are easier to develop to the ends of the NW, as shown in Scheme 2. While several slips extended to the ends, a series of atoms is disordered due to the blocking effect from the boundaries of the end, resulting in the local melting in the NW structure and further enhancement of plasticity^[52]. Finally, the local necking is easier to form here even though the whole NW system is under the same stress. This indicates that the primary cause that most samples are broken at two ends is the blocking effect from the ends. Furthermore, this also infers that the breaking failure is affected by the propagation of initial slips and new slips generated in region HS rather than the positions where they first appeared. As published works reported, the metal NWs fracture through extensive plasticity and ductile necking^[52-54]. Furthermore, the breaking failure of the NW is significantly affected by the middle and late stages of plastic deformation^[55]. Therefore, there is no direct causal relationship between the fracture and the initial microstructural defects. It just shows the consistency between the initial slip distributions and breaking distributions while the temperature is less than 100 K.

  3.   Conclusions

  The study on the effect of microstructural defects generated at the early stage of plastic deformation on the breaking failure is beneficial for structural strengthening and optimizing design principles, which has significant implications for the application of metallic nanomaterials. In this work, based on the statistical analysis, the temperature dependence of the initial microstructural defects (also called initial slips) generated at the stress YP and the temperature dependence of the final breaking positions have been investigated using the MD simulations. At the temperature range of 20 to 300 K, each temperature ensemble includes 300 independent single-crystal Cu NWs. We have calculated the position of the initial slip using the DBSCAN algorithm. Considering the profound significance of predicting the breaking failure of nanostructural materials, we have performed a comprehensive analysis to investigate the relationship between the initial slips and breaking behaviors of the NWs. The results indicate that, with the increase in temperature, the average yield stress decreases due to the decrease in Young′s modulus, the increase in average kinetic energy of atoms, and the increase in amorphous structures, further promoting the average yield strain decreased and more initial slips generated. It is found that the positive effect on the ductility of the NW during the early stage of plastic deformation has restrained the generation of initial slips at two ends. The positions of initial slips are averaged out from the ends toward the middle part of the NW. For the final breaking positions, the structure of the Cu NW is brittle due to the properties of the copper, resulting that the breaking positions under different temperatures are both following two statistical peaks at the ends of distributions. As a result of statistical distributions, it shows a consistency between the initial slip distributions and breaking distributions while the temperature is less than 100 K. However, the consistency between them gradually disappeared with the increase in temperatures because of their different temperature dependence. The microstructural deformation behaviors further demonstrated that there is no direct causal relationship between the fracture and the initial microstructural defects, and the breaking failure is affected by the surface effect and blocking effect of the ends.

  Supporting information is available at http://www.wjhxxb.cn

  
  Acknowledgments: This work is supported by the National Natural Science Foundation of China (Grant No.51861145202).
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• 

  Scheme 1  Schematic explanation of this research

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  Figure 1  Initial slips generated at the stress YP at (a) 20, (b) 50, and (c) 150 K

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  Figure 2  Distributions of initial slip positions for 300 independent samples at different temperatures

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  Figure 3  Stress-strain curves of the Cu NW at different temperatures with a constant strain rate of 0.1% ps^-1: (a) stress-strain curves of representative samples; (b) average yield stress with error bars for every 300 samples

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  Figure 4  Energy-strain curves during stretching and the curves of RDF: (a) variation of the energy of representative samples with the strain at different temperatures; (b) RDF curves of representative samples at different temperatures

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  Figure 5  Variation of the number of hcp and fcc atoms with the strain

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  Figure 6  Distributions of breaking positions for 300 samples at different temperatures

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  Figure 7  Distributions of average breaking strain for 300 samples at different temperatures

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  Figure 8  Snapshots of the deformed microstructure with the temperatures of 20, 100, and 200 K at different applied strains: (a) strains in the order of 0.058, 0.1, 0.15, 0.20, 0.25, 0.30, 0.31, 0.40; (b) strains in the order of 0.046, 0.10, 0.15, 0.20, 0.25, 0.30, 0.40, 0.48; (c) strains in the order of 0.042, 0.10, 0.15, 0.20, 0.25, 0.30, 0.40, 0.478

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  Scheme 2  Schematic explanation of the blocking effect of the ends

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  Table 1.  Average yield strain and average yield stress for 300 samples at different temperatures

  Parameter	20 K	50 K	100 K	150 K	200 K	300 K
  Average yield strain	0.054±0.001	0.051±0.001	0.046±0.002	0.044±0.002	0.045+0.003	0.045±0.005
  Average yield stress	3.84±0.07	3.58±0.06	3.22±0.09	3.03±0.09	3.02±0.1	2.87±0.11

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