Internal stress and its effect on mechanical strength of metallic glass nanowires

Available online at www.sciencedirect.com

ScienceDirect Acta Materialia 91 (2015) 174–182 www.elsevier.com/locate/actamat

Internal stress and its effect on mechanical strength of metallic glass nanowires Qi Zhang,a,b Qi-Kai Lib and Mo Lia,

⇑

a

School of Materials Science and Engineering, Tsinghua University, Beijing 100084, China School of Materials Science and Engineering, Georgia Institute of Technology, Atlanta, GA 30332, United States

b

Received 24 November 2014; revised 30 January 2015; accepted 11 March 2015 Available online 31 March 2015

Abstract—Size induced strength increase in crystalline nanowires is related to crystal defects at small length scales. But how this phenomenon manifests in metallic glasses, an amorphous solid without obvious intrinsic structural defects, still remains an open issue. Here we show that in the amorphous nanowires free of obvious structural defects or cracks, reduction in wire diameter generally leads to weakening of the strength compared to the bulk sample. Our analysis shows that the surface stress as well as surface induced internal stress that scales inversely with the wire diameter become significant. It is the complex interplay between the applied stress and the internal residual stresses induced by the nanoscale surface that affects the mechanical behavior in the amorphous nanowires at the small length scale regime. The role of the localized shear banding played in relation to the size effect is also discussed in the context of the deformation mechanisms in amorphous wires with different sizes. Ó 2015 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Metallic glass; Nanowire; Molecular dynamics simulation; Mechanical properties

1. Introduction When the characteristic dimension of a crystalline material is reduced to nanometer scale, such as a wire with the diameter of hundreds of nanometers or below, the mechanical strength could increase dramatically [1–5]. The size dependence of the mechanical property is thought to be related to crystal dislocations in the small sized sample as they are responsible for yielding. Therefore, the less the number of the defects or less active they are, the stronger the material is [1–11]. But whether this conclusion can be extended to non-crystalline materials such as metallic glass (MG) remains unanswered. There are several fundamental differences between the two types of materials that prevent us from making a generalization. One is that MG has no long-range translational order, which deprives amorphous solids of the possibility to have extended structural defects such as dislocations and grain boundaries that are the underlying structural units for yielding and plastic deformation. From this point of view, sample size should not be a sensible parameter in influencing the strength of MGs. Nevertheless, a number of experiments have reported increasing strength in small dimensions, including micron sized pillars and nanowires (NWs) [12–15]. However, little or even opposite effect of size reduction in MGs has also

⇑ Corresponding author. Tel.: +1 404 385 2472; e-mail: mo.li@gatech. edu

been observed [16–19]. To date the subject still remains contentious and therefore motivates us to explore the underlying mechanisms. One of the difficulties to resolve the issue in experiment is ironically related to the sample size itself. Since samples with small dimensions are very difficult to fabricate, processing defects and imperfections often show up and play a significant role in the changes of size related mechanical behaviors than in bulk samples, such as the geometric shape in micro pillars and surface imperfections [12–21]. The situation is exacerbated by extrinsic factors in sample structure and morphology characterization, such as the high energy electron beam used in observing in-situ mechanical testing of NWs. Electron irradiation may change the atomic structure in small samples and subsequently the mechanical properties [22]. Therefore, before the quality of the samples and the characterization process can be improved, this challenge will likely stay for some time. On the other hand, these factors can be easily controlled and even eliminated in theoretical treatment. Then it brings up the possibility to answer the following questions: Should the strength of MG nanowire increase if the above issues are not present? Or simply, must the MG nanowire be strong intrinsically? And what are the underlying deformation mechanisms in amorphous solids when the characteristic dimensions are reduced? The answers to these questions are fundamental to our understanding of deformation in amorphous nanowires, especially amid the

http://dx.doi.org/10.1016/j.actamat.2015.03.029 1359-6462/Ó 2015 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Q. Zhang et al. / Acta Materialia 91 (2015) 174–182

formidable difficulties encountered presently in real experiment. Here we use a model system and atomistic modeling to show that in the MG nanowires without processing defects and imperfections as well as external perturbation, size effect on the strength indeed manifests. But instead of showing strengthening as expected in crystalline nanowires, the amorphous wires exhibit universally weakening compared to that of the similarly prepared bulk samples. Our analysis suggests that this unique behavior at the small size is a result of the intrinsic response of MG nanowires with less than 30 nm in diameter. The size effect is found to originate primarily from the surface stress and internal residual stress induced by the surface and their complicated interplay with the externally applied stress in deformation process of the amorphous material. In a broader sense, this work gives another interesting case of how a material adapts to external stimuli with regard to its unique atomic structure when confined in size. When the sample size is larger than 30 nm, different mechanisms may operate, as the surface induced internal stress becomes less significant. This transition is dictated by the surface stress via the Young–Laplace-like relation, as we show below. As known, surface plays a key role in nanoscale materials at small sizes. The broken bonds of the atoms on and near the surface contribute to the increase of the total free energy via surface energy. The surface atoms with fewer neighbors are also subject to imbalanced forces and thus generate stress on the surface. The surface stress is counterbalanced by the stress developed inside the wire. As a result, compressive internal stress is established inside the wire which is otherwise stress-free in the bulk, whereas the surface develops tensile stress. The internal compressive stress is called residual stress when no external stress is applied to the wires. Since the self-balancing of the surface and internal stress occurs in the entire sample, the magnitudes of these two types of stresses must be related to the characteristic dimension of the wire, the diameter of a wire, which results in a relation between the stress and the sample size, which resembles the Young–Laplace relation for capillary effect in liquid droplets. The overall mechanical behavior of the wire is thus determined by the effective stress composed of the internal and the applied stress. The size effect, therefore, must manifest through the internal or residual stresses. The relatively large surface-to-bulk volume ratio in nanoscale materials is expected to amplify this effect dramatically. In the following, we shall examine this process and how it affects the deformation mechanisms in model metallic glass nanowires.

2. Methods To validate the above conceptual argument, we carried out extensive molecular dynamics (MD) simulation on MG nanowires made of Cu64Zr36. We chose the binary alloy primarily for its high thermal stability [23–25] and easiness to handle in atomistic modeling [26,27]. Bulk samples were first prepared by cooling the equilibrated liquid from 2000 K to 300 K with a cooling rate of 1 K/ps. A MD time step is 103 ps. The nanowires were then cut off from the bulk sample at 300 K. Periodic boundary conditions (PBCs) are used in the bulk sample but only along the axial direction of the wires. The freshly cut wires are

175

then relaxed for long enough time, up to 100,000 steps, until reaching mechanical equilibrium. This is done by allowing the wires to change both the length and diameter gradually until the total stress in the wire approaches zero. A series of nanowires with diameter from 4 to 50 nm were prepared this way and the aspect ratio of length to diameter for freshly cut nanowire is kept at 3 for all samples. Mendelev interatomic potential [28] is used here to compute interatomic forces; the same potential has been used extensively for the Cu–Zr MGs that supports its applications [29–35]. NPT ensemble MD is used throughout the simulation to keep the pressure at zero and temperature at 300 K. Both tensile (ten) and compressive (com) deformation at a strain rate about 2.5  108 s1 is applied subsequently to the equilibrated wires with PBC in axial direction, while the other two perpendicular directions are set free to mimic free surfaces. Note that different strain rates result in quantitatively different mechanical responses such as yield stress and maximum strength, but no qualitative difference is found in the mechanisms reported below. When a fully relaxed wire is subject to uniaxial deformation, the mechanical equilibrium also needs to be reached. It is done by using the same relaxation procedure mentioned above to get rid of the stresses perpendicular to the wire at each given deformation strain (or stress). It is also worth to mention that the nanowires used in this work are obtained from cutting them from bulk samples. They can also be produced using casting/rapid cooling liquid or hot pressing into a mold in undercooled temperature. The atomic structure and property are strongly dependent on processing. As a result, the mechanical behaviors are naturally affected. Their dependence on detailed processing parameter in those wires will be examined and reported elsewhere. Here to avoid these complications in our search for intrinsic deformation mechanisms we shall focus on the cut-and-relaxed wire.

3. Results 3.1. Internal stress induced by surface in nanowires Fig. 1 shows the profile of the axial stress rzz for the nanowires with different diameters before and after relaxation in sample preparation. The stress in the wires at position r is taken as those from all atoms in the concentric cylindrical ring between r and r + Dr along the radial ˚ . Since the atomic diamedirection of a wire with Dr of 1 A ˚ , the bin size of 1 A ˚ would ters of Cu and Zr are around 3 A be sufficient in obtaining converging statistics for the stress profiles for disordered atomic positions in the amorphous sample. When a nanowire is cut from the bulk, tensile stress is generated instantly in nanowires’ surface; but the rest of the wire still remains stress free. This can be seen in Fig. 1 by the dashed lines. The compressive stress inside the wires has a negative value, while the tensile stress has a positive one. After relaxation, surface tension drops down. In order to maintain the mechanical equilibrium, compressive stress inside the wire starts to develop to balance the surface tensile stress. This can be seen in the axial components of the stress shown as the solid lines in Fig. 1. For illustration we only plotted the initial and final

176

Q. Zhang et al. / Acta Materialia 91 (2015) 174–182

stress profile during first 1000 steps in the relaxation process; very little changes are found beyond this relaxation time. All three normal stresses and the pressure in the cylindrical coordinate are shown. One can see that during relaxation, as the tensile stress is reduced on the surface, compressive stress develops gradually inside which was absent in the freshly cut wire. This self-equilibrium process can be described by the following expression for the total stress tensor rab tot of a wire, " # N N tot tot X X a b ab 1 a b @U rkl rkl rtot ¼ V tot m k vk vk  @rkl rkl k¼1 N surf

¼

V surf 1 V tot V surf

X

l>k

"

mk vak vbk



stress profiles in relaxation; more detailed time evolution during deformation will be shown below. As the diameter of a nanowire becomes smaller, the larger the internal compressive stress develops. This can be seen in the more negative values in the smaller wires close to the center of the wires. Fig. 1 also reveals that for all nanowires with different diameters, the stresses in surface region remain tensile. They maintain almost the same value and distribution before and after relaxation. This feature ˚ in allows us to define the outermost layer of about 5 A thickness as “surface layer”. The nanowire prepared this way resembles that made by focused ion beam (FIB) but without the known processing defects and geometric complication [12–21]. The internal stress in the nanowires is calculated from the atomic stress. In the volume formed by each concentric shell between r and r + Dr from the central axis of the wire with length h, we sum over all atomic stresses of the atoms P k k in the volume, rab ðrÞ ¼ V 1ðrÞ NðrÞ k¼1 rab , where rab is the atomic stress tensor of atom k, N ðrÞ is the number of atoms inside the volume, and V ðrÞ ¼ p½ðr þ DrÞ2  r2 h. It is known that atomic stress exhibits large variations if the averaging volume is small, that may lead to concern of convergence. But in the nanowire, this method is adequate because the concentric rings contain a sufficient number of atoms. The number of the atoms can reach over hundreds or more in the rings close to the surface for wires with diameter larger than 8 nm. So rather good convergence at and close to the surface can be reached (see Fig. 1). In the region close to the center of the wires where the number of atoms is the smallest, as the volume is the smallest, we see larger fluctuations in the internal stress. This systematic trend, however, does not affect our analysis as surface regions are our main focus. 3.2. Scaling relation between internal stress and surface stress To have a better understanding of relaxation process and internal stress, we plot in Fig. 2 the evolution of the

þ VV int tot

1 V int

" N int X

a b @U rkl rkl @rkl rkl

l>k

k¼1

Fig. 1. The axial stress profile for the nanowires with different diameters before (dashed-line) and after relaxation (solid-line). The different colors represent the nanowires with different diameters. The position of each peak corresponds approximately to the surface position (or radius) of the wire. The thickness of each concentric ˚. cylindrical ring, or the bin size, is 1 A

N tot X

mk vak vbk 

k¼1

N tot X

a b @U rkl rkl @rkl rkl

#

#

ð1Þ

l>k

ab ¼ xsurf rab surf þ xint rint ab where rab surf , and rint are the surface and internal stress, V respectively, while xsurf ¼ Vsurf and xint ¼ VV int are the volume tot tot fraction of the surface and the core or interior regions of the nanowire. V tot , V surf , and V int are the volume of these regions and N tot , N surf , and N int the corresponding number of atoms inside these volumes. mk is the mass of the k-th atom, vak , rakl are the Cartesian components of its velocity and the separation distance rkl between the k- and l-th atom. And U is the interatomic interaction potential. ab Both the surface stress rab surf and the internal stress rint can be obtained using Eq. (1) when the atoms in the surface region and interior of the wire are properly allocated. Eq. (1) shows that the total stress in the wire is the sum of those from the surface and interior weighted by their volume fractions. As shown in Fig. 2, when the wire is just cut, ab the internal stress rab int ¼ 0 but rsurf –0, thus the surface ab stress is induced and the total wire stress rab tot ¼ rsurf –0. As the wire relaxes, surface stress rab surf decreases and interab ab –0 and r nal stress develops, rab surf –0, and thus rtot ¼ int ab ab xsurf rsurf þ xint rint –0. And finally when the wire is fully relaxed, i.e., the total stress in the wire reaches zero, that ab ab is, rab tot ¼ xsurf rsurf þ xint rint ¼ 0, and both surface stress ab ab rsurf and internal stress rint are not zero. The self-equilibration condition allows us to get a scaling relation,

~ab r int ¼ 

xsurf ab V surf ab ~ ~ ; ¼ r r xint surf V int surf

ð2aÞ

from which we could obtain the relation between the internal residual stress and the surface stress and their scaling relation with respect to the wire diameter. It is done as follows. The fully relaxed nanowire exhibits two regions along ~surf along the radial direction, one with a tensile stress r the axis at the surface and the other in the interior of the ~int . In general, wire with the residual compressive stress r for a wire with radius r and the thickness of the surface V surf ab ~ab ~surf region, Dr, from Eq. (2a) we have r int ¼  V int r 2

2

pðrDrÞ  ab ~surf for all components of the stress tensors r ¼  ½pr pðrDrÞ 2 labeled by the Cartesian indices a and b. For r significantly larger than Dr, we can obtain the scaling relation or size

Q. Zhang et al. / Acta Materialia 91 (2015) 174–182

177

Fig. 2. The surface and internal (a) radial, (b) tangential, (c) axial stress and (d) hydrostatic pressure profiles along the radial direction of the nanowire of d = 12 nm at different relaxation times (in MD steps). Step = 0 when the wire is just cut. The thickness of each concentric cylindrical ˚. ring, or the bin size, is 1 A

effect of the residual stress in a fully relaxed wire with diameter d, 2Dr ab 4Dr ab ~ ~ : ¼ ð2bÞ r r r surf d surf This relation resembles the well-known Young–Laplace relation for internal pressure and surface tension in liquid droplets. In solids, such relation is expected to hold also, although the mechanical anisotropy needs to be taken into consideration as shown above in the tensorial relations. Like in liquid droplet, the key quantity is the magnitude ~surf , which determines the effective of the surface stress r range of the surface stress at certain wire diameter d. We check this relation in MG wires using MD simulation. The result in Fig. 1 indicates that the surface region

~ab r int  

Dr remains nearly constant, about a half nanometer for wires with different diameters. This allows us to set Dr = 0.5 nm and identify atoms on the surface and inside the wire. We validate the relation in two ways. One is by calculating the internal stress directly in wires with different ~int can be diameter and then fit Eq. (1). The internal stress r calculated directly by summing the atomic stresses in the core region by excluding the surface atoms using Eq. (1). Fig. 3 plots the internal stress along the axial direction, ~zz r int , versus d obtained using this approach. It appears that n ~zz ~zz r int scales with d as r int ¼ A=d . By fitting the data in Fig. 3 with A and n as free parameters to be determined, we obtain the constants n ¼ 1 and A  5:22  0:07. From 4Dr zz ~zz ~surf , we obtain the surface stress Eq. (2b), r int ¼  d r zz ~surf ¼ 2:61  0:035 GPa. The result indicates that the surr face stress can indeed play an important role in mechanical behavior of metallic glass nanowires via the surface induced internal stress. For example, when the wire diameter is reduced to about 20 nm, the internal stress can reach to 20% of the yield stress (see Fig. 4 below). As the wire size becomes smaller, this effect is expected to become larger. ~surf The second way is to calculate the surface stress r directly from the MD simulation by summing over the atomic stresses of the atoms in the surface region. The ~zz surface stress obtained this way is r surf 2.30 GPa. This method has larger variance, especially when the wire diameter d gets small (<10 nm). 3.3. Mechanical behavior of metallic glass nanowires

~zz Fig. 3. The axial residual internal compressive stress, r int , versus d. The calculated data are fitted to an inverse power law relation (dotted line). The setting for the calculation is shown schematically.

Having established the surface-internal stress relation, we now can proceed to examine how the surface induced internal residual stress and surface stress itself affect the overall mechanical properties of the nanowires. In this case,

178

Q. Zhang et al. / Acta Materialia 91 (2015) 174–182

Fig. 4. (a) The tensile and compressive stress–strain relations for the nanowires with different diameters and the bulk sample. The solid line is the overall stress–strain for the nanowires, dotted-line is for surface stress and dashed-line for internal stress. (b) An enlarged version of the (a) for tension and (c) compression. Note that in (c) we use positive sign of the compressive stress and strain; and the tensile surface stress has negative sign.

under applied external stress the mechanical behavior of the nanowire must be determined by the effective stress, i.e., the applied stress plus the residual surface and internal stresses. As the reference, we obtained the stress–strain relation for the bulk sample (Fig. 4(a)). The maximum tensile and compressive stresses of the bulk sample are 3.06 and 3.36 GPa, respectively, indicating obvious asymmetry between tension and compression. Young’s modulus is 80.05 GPa. As a comparison, the experimental compressive yield stress is about 2 GPa at 2.2% strain and Young’s modulus 92.3 GPa [23]. There is no experimental bulk tensile yield stress available yet due to the apparent brittleness. Our calculated

bulk compressive (yield) stress at about 3% strain limit is 2.37 GPa and the tensile yield stress is about 1.63 GPa at about 2% strain limit. The differences between the simulation and experiment are mainly the result of the interatomic potential and the high strain rate used in MD simulation. But the results are consistent with other simulation works [34,35]. Fig. 4 shows the total axial tensile and compressive stress–strain relations for the nanowires with different diameter d. In the same figures, we also plot the surface zz stress rzz surf and the internal stress rint versus d. We focus primarily on the axial stresses for their importance in uniaxial deformation of the nanowires. In Fig. 4(a), we plot the tensile stress and strain with positive signs and the compressive stress and strain with negative signs. In Fig. 4(b) and (c), we plot both the tensile and compressive stress and strain with positive sign by following the convention that is used in experiment. Under uniaxial loading, the nanowires exhibit similar trends in the stress–strain relations as in bulk samples. However, one can see clearly that both the compressive and tensile yield stress and maximum stress are smaller than those of the bulk samples (Fig. 4(a)). We have examined the wires with diameter from 4 to 32 nm (only three nanowires are shown in Fig. 4). Both the yield and maximum stress show consistent decreasing trend with the decreasing wire diameter, which indicates weakening of the MG nanowires. In addition, there is one surprising result, that is, the tensile strength of the nanowires is much resilient to the decreasing size than the compressive one as shown in Fig. 4: The tensile stress decreases more gradually than the compressive stress after passing the maximum stress. We shall analyze these findings in the next section. In Fig. 5 we plot the yield stress, the maximum stress and Young’s modulus versus the wire diameter. The yield stress is obtained by using 0.2% off-set strain method from the overall stress–strain curves of the wires; the maximum stress is the peak stress in the stress–strain curve; and Young’s modulus is obtained from the slope of the stress–strain curve at the small strain limit. The trend is clearly captured in the change for the tensile and compressive strengths mentioned above. When d > 15 nm the maximum stresses show a slow decreasing magnitude with decreasing d. Below 15 nm, the compressive peak stress decreases rapidly with d, much faster than the tensile peak stress, indicating that the wire under compression is losing stability more quickly than under tension. And most importantly, both peak stresses are clearly smaller than these of the bulk sample. The same trend is also found for yield stresses (Fig. 5a). Young’s modulus also decreases with d (Fig. 5b). The maximum (compressive) stress and Young’s modulus data are fitted with inverse power law relation with respect to wire diameter d via a power law relation, A  B=d n , where A, B, and n are free parameters to be determined. The best fit is found to follow the relations, rcom max ¼ 3:18  5:28=d and E ¼ 79:5  76:6=d, respectively. As can be seen, the value of A coincides well with the values of the bulk samples, that is, as d ! 1, the relations give the correct values of the maximum stress and the modulus of the bulk sample. The decreasing trend in Young’s modulus for the wires can also be understood with a “core–shell” model. As shown in Figs. 1 and 2, as the wire diameter decreases,

Q. Zhang et al. / Acta Materialia 91 (2015) 174–182

the surface shows little change in its thickness and surface stress during deformation. We can thus consider the surface as a separate entity in addition to the core region. This division allows us to treat Young’s modulus as that from the “composite” made of the core and the surface shell region, that is, E ¼ xcore Ecore þ xsurf Esurf , where xcore and xsurf are the volume fraction and Ecore and Esurf are Young’s modulus of the core and the surface region. Since the surface region ˚, remains almost constant with a thickness d = 5 A 2 2 2 2 xcore ¼ ðd  2dÞ =d and xsurf ¼ ð4dd  d Þ=d depend on wire diameter d only. Young’s modulus Ecore and Esurf can be obtained directly from numerically fitting the slopes of the stress–strain relations of the core and surface shown in Fig. 4. The result is shown in Fig. 5c. The modulus of the core is close to that of the bulk till d goes below 10 nm while the surface modulus is small, about 40% lower than that of the core (see the inset of Fig. 5c). The effective Young’s modulus calculated from the composite model shows the same trend as that from the direct calculation (Fig. 5b) with only a slightly higher value due to the neglect of the contribution from the interaction between atoms in

Fig. 5. (a) The maximum or peak stresses, yield stresses and (b) Young’s modulus under tension (ten) and compression (com) obtained from the results shown in Fig. 4, versus the wire diameter. The star represents the corresponding value in the bulk metallic glass (BMG) sample. Fitted curve: rcom max ¼ 3:18  5:28=d for the maximum stress and E ¼ 79:5  76:6=d for Young’s modulus, respectively. (c) Young’s moduli calculated from the “composite” made of the core region and the surface shell region and from the direct calculation on the entire nanowires.

179

the core and surface regions. The softening in the wire’s modulus can be seen originating from the surface which not only has a lower modulus Esurf but also an increasingly larger relative volume fraction xsurf and decreasing fraction xcore from the core as the wire diameter becomes smaller. Note that although larger data scatter occurs when the wire diameter becomes less than 10 nm and also for surface due to the small number of atoms included in the region [36], the effective modulus obtained behaves well. 3.4. Effect of internal stress and surface stress Next we try to understand the mechanisms of the general weakening behavior of the amorphous wires. In addition, we also focus on other two salient features reported above: one is the extra stability in wires under tension and the second is the possible surface “defects” that causes yielding in tension and internal deformation in compression. Our analysis will follows the effects of the residual stresses in the interior of the wire and surface separately. We have learned that (1) the core of the wires is generally under residual compressive stress r ~int , (2) the surface ~surf , (3) the internal stress is inversely is under tensile stress r scales with the wire diameter; and (4) the internal stress becomes significant at the wire diameter below 15 nm. In addition, we also know that the (nearly constant) surface stress also contributes to the overall mechanical behavior. Both the internal and surface stress contributions are weighted by the relative volume fraction of these two regions via. Eq. (1). Based on these results, we shall try to find plausible mechanisms for mechanical properties of metallic glass nanowires. 3.4.1. Role of residual internal stress When the wires are under tension, the compressive residual stress inside the wires counterbalances the applied stress, thus stabilizing the wire (see Figs. 4(b) and 6). In other words, part of the external applied tensile stress is needed in order to compensate the compressive residual stress. This phenomenon is well-known in strengthening of brittle materials such as concrete and rocks. By pre-stressing these materials, they could exhibit certain resistance to tensional loading which otherwise would be impossible. The extra stability enables the wires to have relatively high resistance to degradation of its strength. Indeed, Figs. 4(b) and 5 show the slow decrease of the peak and yield strength with decreasing d when the wires are under tension. This finding may explain recent experimental findings where MG wires with decreasing diameter become more ductile [14,15]. In light of the results, we could attribute the apparent ductility to the enhanced stability of the wires under tension, which enables the plastic displacement to continue rather than come to an abrupt stop caused by brittle failure. When under compression, the wires would reach yielding and maximum stress or strain early since the effective stress inside the wire is larger than the applied stress. This is because the wire interior has already been stressed by the pre-existing residual compressive stress (see Figs. 4 and 6). Therefore, the net amount of the externally applied stress needed to make the wire yield is smaller than that needed to cause yielding. As shown in Fig. 3, the preexisting compressive stress increases through an inverse power law and becomes significant at d < 15 nm. So the

180

Q. Zhang et al. / Acta Materialia 91 (2015) 174–182

with the wires with larger diameters. As the wire becomes smaller (d < 8 nm), surface contribution becomes significant: surface stress rzz surf remains nearly constant but its relative volume becomes larger, especially after yielding (see Fig. 4(c)). However, the power law relation d 1 still holds in the wires’ maximum strength and yield stress, as inherited from or dominated by the internal residual stress, which is indeed the case as shown by the results in Fig. 5. Secondly, under tension, the nanowire surface is stretched the same as the interior but with much higher stress value as it is already subject to the pre-existing residual surface tension. In this case, the surface stress does not provide the stability to the wires anymore as in the case of compression. In this case, it plays the role of a “defect”. That is, the surface yields before the core does since the effective stress already reaches the yield stress in the surface region. This can be seen clearly in the earlier drop of the surface stress in tension deformation in Fig. 4(b) after 3% strain. As a contrast, the surface stress in compression remains nearly constant after 3% overall strain, signaling no effect provided by the surface to the stability of the wires. This unexpected discovery reveals very different mechanisms of surface and surface stress in yielding of MG nanowires under tension and compression.

4. Discussions

Fig. 6. The profiles of the surface and internal stress along the axial direction at different deformation strains in (a) compression and (b) tension for the nanowire with d = 12 nm.

residual internal stress contributes a significant part to yielding. Thus we saw a large and rapid decrease of the compressive peak stress with decreasing wire diameter (Fig. 5). 3.4.2. Roles of the surface as defect and surface stress In MG nanowires, we found that surface stress is present and the thickness of the surface region remains nearly invariant for all different wire sizes (Figs. 1, 4 and 6). These two features tell us that the role of the surface stress played out in NW mechanical behavior is very different from that of the residual compressive stress discussed above. First, surface stress contributes to the overall stress of the wires via the weighted volume contribution or the relative surface-to-volume ratio as explained in Eq. (1). Therefore, the smaller the wire becomes, the larger the relative contribution of the surface stress, since its relative volume fraction is larger. This is indeed the case for the nanowires under compression as shown by Figs. 4(c) and 6: while the internal stress contribution still follows d 1 scaling relation, the compressive peak stress at d < 10 nm decreases rapidly due to the increasing contribution from the surface tensile stress to the overall stress in the wire – the tensile surface stress balances some of the compressive interior compressive stress. The counterbalancing surface stress contributes more to the overall stress of the wire at the small wire size, via Eq. (1) (see Fig. 3). An example can be seen clearly in the case of the wire with d ¼ 8 nm in Fig. 4(c) where rab tot decreases significantly as compared

The results obtained above are from metallic glass wires without any imperfections or defects except the surface itself. It is deliberately done to probe the intrinsic deformation mechanism in nanowires. Obviously this is an idealized scenario. Nevertheless, the mechanisms revealed here would still serve its purpose in guiding us in searching for the fundamental understanding of deformation in amorphous nanoscale materials. In real samples, imperfections, defects, structural or chemical change induced by processing occur more often than not, which demand further clarification and understanding. One example is the MG nanowires made via high temperature impressing. We have found both atomic structure change and the chemical segregation in these samples that give rise to discernible changes in mechanical properties [37]. Another complication is shear band, which is a dominant structural entity underlying mechanical deformation in bulk samples. Shear banding in small dimensions below 20 nm is less frequently observed in the MG wires, which may be related to the size incompatibility between the wire and the band [38]. We have observed shear banding in wires with d c > 10 nm. This is the main reason for the drop of the surface stress in large sized wires under tension (Fig. 4(c)). Below this size, no well-defined shear zone is seen in the NWs. So the wires deform nearly homogeneously, which is why the surface stress for small NWs drops later (see Fig. 4). It is generally believed that strengthening should be found in the small length scale due to absence of shear localization in MG, that is, the smaller the stronger [1– 15]. Our result so far, however, rebuts this thinking. The weakening of the strength occurs more dramatically at the smaller size regime where well-defined shear band is not found – in this regime, the wires are expected to be strong for the lack of shear banding [38]. We show that the weakening is caused by the internal stress and surface

Q. Zhang et al. / Acta Materialia 91 (2015) 174–182

defect which one cannot avoid in nanoscale even when other defects are absent. Our results, therefore, suggest two different governing mechanisms for MG NW deformation: Below certain critical size d c , say about 10 or 15 nm the surface and internal stress may play a more important role. Above d c both shear banding and internal stresses contribute to overall mechanical response of MG wires, although the latter may have a diminishing role as the size becomes larger. d c is governed by the surface stress of a material via the Young–Laplace relation.

5. Conclusions We have validated the idea using atomistic modeling that internal stress and surface stress should play key but different roles in deformation of amorphous nanowires while any obvious extended structural defects are absent. Previous works on mechanical deformation of MG nanowires have focused on either high temperature behaviors [39] or internal defect process [40]. No attention is paid to internal stress and its systematic effects as the function of wire sizes. As shown here, surface and internal stress play a vital role in deformation of MG nanowires. The main function of the internal residual stress is to affect the effective stress and thus the overall mechanical behavior of MG nanowires. This effect becomes increasingly important as the diameter of the wires becomes smaller. The transition from internal-stress-dominated to shear banding mechanism is governed by the surface stress and the related Young– Laplace relation (Eq. 2). The existence of compressive internal residual stress enhances the strength and also stability of the wires under tension, therefore, making the wires appear stronger when they have more ductility. In addition, we discovered that besides being the source of internal and surface stresses, the surface acts as an effective defect in tension. As a result, in both tension and compression the amorphous material is generally weakened instead of strengthened as in crystalline materials.

[7] [8] [9] [10]

[11]

[12] [13]

[14] [15] [16] [17]

[18]

[19]

[20]

Acknowledgment The work is supported by the National Thousand Talents Program of China. We would also like to thank the anonymous referee for the suggestion which motivated us to add Fig. 5c.

References [1] M.D. Uchic, D.M. Dimiduk, J.N. Florando, W.D. Nix, Sample dimensions influence strength and crystal plasticity, Science 305 (2004) 986–989. [2] J. Greer, W. Nix, Nanoscale gold pillars strengthened through dislocation starvation, Phys. Rev. B 73 (2006) 245410. [3] S.M. Han, T. Bozorg-Grayeli, J.R. Groves, W.D. Nix, Size effects on strength and plasticity of vanadium nanopillars, Scr. Mater. 63 (2010) 1153–1156. [4] B. Wu, A. Heidelberg, J.J. Boland, Mechanical properties of ultrahigh-strength gold nanowires, Nat. Mater. 4 (2005) 525– 529. [5] R. Dou, B. Derby, A universal scaling law for the strength of metal micropillars and nanowires, Scr. Mater. 61 (2009) 524– 527. [6] D.M. Norfleet, D.M. Dimiduk, S.J. Polasik, M.D. Uchic, M.J. Mills, Dislocation structures and their relationship to

[21]

[22] [23]

[24] [25] [26] [27]

181

strength in deformed nickel microcrystals, Acta Mater. 56 (2008) 2988–3001. A.H.W. Ngan, K.S. Ng, Transition from deterministic to stochastic deformation, Philos. Mag. 90 (2010) 1937–1954. A.A. Benzerga, An analysis of exhaustion hardening in micron-scale plasticity, Int. J. Plast. 24 (2008) 1128–1157. H. Tang, K. Schwarz, H. Espinosa, Dislocation-source shutdown and the plastic behavior of single-crystal micropillars, Phys. Rev. Lett. 100 (2008) 185503. Z.W. Shan, R.K. Mishra, S.A. Syed, S.A. Syed Asif, O.L. Warren, A.M. Minor, Mechanical annealing and sourcelimited deformation in submicrometre-diameter Ni crystals, Nat. Mater. 7 (2008) 115–119. C. Chisholm, H. Bei, M.B. Lowry, J. Oh, S.A.S. Asif, O.L. Warren, Z.W. Shan, E.P. George, A.M. Minor, Dislocation starvation and exhaustion hardening in Mo alloy nanofibers, Acta Mater. 60 (2012) 2258–2264. C.J. Lee, J.C. Huang, T.G. Nieh, Sample size effect and microcompression of Mg65Cu25Gd10 metallic glass, Appl. Phys. Lett. 91 (2007) 161913. Y.H. Lai, C.J. Lee, Y.T. Cheng, H.S. Chou, H.M. Chen, X.H. Du, C.I. Chang, J.C. Huang, S.R. Jian, J.S.C. Jang, T.G. Nieh, Bulk and microscale compressive behavior of a Zrbased metallic glass, Scr. Mater. 58 (2008) 890–893. D. Jang, J.R. Greer, Transition from a strong-yet-brittle to a stronger-and-ductile state by size reduction of metallic glasses, Nat. Mater. 9 (2010) 215–219. D. Jang, C.T. Gross, J.R. Greer, Effects of size on the strength and deformation mechanism in Zr-based metallic glasses, Int. J. Plast. 27 (2011) 858–867. C.A. Volkert, A. Donohue, F. Spaepen, Effect of sample size on deformation in amorphous metals, J. Appl. Phys. 103 (2008) 083539. K.S. Nakayama, Y. Yokoyama, T. Ono, M.W. Chen, K. Akiyama, T. Sakurai, A. Inoue, Controlled formation and mechanical characterization of metallic glassy nanowires, Adv. Mater. 22 (2010) 872–875. B.E. Schuster, Q. Wei, T.C. Hufnagel, K.T. Ramesh, Sizeindependent strength and deformation mode in compression of a Pd-based metallic glass, Acta Mater. 56 (2008) 5091– 5100. C.Q. Chen, Y.T. Pei, J.T.M. De Hosson, Effects of size on the mechanical response of metallic glasses investigated through in situ TEM bending and compression experiments, Acta Mater. 58 (2010) 189–200. Y. Yang, J.C. Ye, J. Lu, F.X. Liu, P.K. Liaw, Effects of specimen geometry and base material on the mechanical behavior of focused-ion-beam-fabricated metallic-glass micropillars, Acta Mater. 57 (2009) 1613–1623. D.Z. Chen, D. Jang, K.M. Guan, Q. An, W.A. Goddard, J.R. Greer, Nanometallic glasses: size reduction brings ductility, surface state drives its extent, Nano Lett. 13 (2013) 4462– 4468. D.J. Magagnosc, R. Ehrbar, G. Kumar, M.R. He, J. Schroers, D.S. Gianola, Tunable tensile ductility in metallic glasses, Sci. Rep. 3 (2013) 16–18. D. Xu, B. Lohwongwatana, G. Duan, W.L. Johnson, C. Garland, Bulk metallic glass formation in binary Cu-rich alloy series – Cu100xZrx (x = 34, 36, 38.2, 40 at.%) and mechanical properties of bulk Cu64Zr36 glass, Acta Mater. 52 (2004) 2621–2624. O.J. Kwon, Y.C. Kim, K.B. Kim, Y.K. Lee, E. Fleury, Formation of amorphous phase in the binary CuZr alloy system, Met. Mater. Int. 12 (2006) 207–212. Y. Li, Q. Guo, J.A. Kalb, C.V. Thompson, Matching glassforming ability with the density of the amorphous phase, Science 322 (2008) 1816–1819. Q.-K. Li, M. Li, Atomic scale characterization of shear bands in an amorphous metal, Appl. Phys. Lett. 88 (2006) 241903. Q.-K. Li, M. Li, Free volume evolution in metallic glasses subjected to mechanical deformation, Mater. Trans. 48 (2007) 1816–1821.

182

Q. Zhang et al. / Acta Materialia 91 (2015) 174–182

[28] M.I. Mendelev, D.J. Sordelet, M.J. Kramer, Using atomistic computer simulations to analyze X-ray diffraction data from metallic glasses, J. Appl. Phys. 102 (2007) 043501. [29] M.I. Mendelev, D.K. Rehbein, R.T. Ott, M.J. Kramer, D.J. Sordelet, Computer simulation and experimental study of elastic properties of amorphous Cu–Zr alloys, J. Appl. Phys. 102 (2007) 093518. [30] M.I. Mendelev, R.T. Ott, M. Heggen, M. Feuerebacher, M.J. Kramer, D.J. Sordelet, Deformation behavior of an amorphous Cu64.5Zr35.5 alloy: a combined computer simulation and experimental study, J. Appl. Phys. 104 (2008) 123532. [31] D. S ß opu, K. Albe, Y. Ritter, H. Gleiter, From nanoglasses to bulk massive glasses, Appl. Phys. Lett. 94 (2009) 191911. [32] M.I. Mendelev, M.J. Kramer, Reliability of methods of computer simulation of structure of amorphous alloys, J. Appl. Phys. 107 (2010) 073505. [33] H.L. Peng, M.Z. Li, W.H. Wang, Structural signature of plastic deformation in metallic glasses, Phys. Rev. Lett. 106 (2011) 135503. [34] D. S ß opu, Y. Ritter, H. Gleiter, K. Albe, Deformation behavior of bulk and nanostructured metallic glasses studied

[35]

[36] [37] [38] [39]

[40]

via molecular dynamics simulations, Phys. Rev. B 83 (2011) 100202. K. Albe, Y. Ritter, D. S ß opu, Enhancing the plasticity of metallic glasses: shear band formation, nanocomposites and nanoglasses investigated by molecular dynamics simulations, Mech. Mater. 67 (2013) 94–103. K.S. Cheung, S. Yip, Atomic-level stress in an inhomogeneous system, J. Appl. Phys. 70 (1991) 5688. Q. Zhang, Q.-K. Li, M. Li, Chemical segregation in metallic glass nanowires, J. Chem. Phys. 141 (2014) 194701. Q.-K. Li, M. Li, Assessing the critical sizes for shear band formation in metallic glasses from molecular dynamics simulation, Appl. Phys. Lett. 91 (2007) 231905. Y. Wei, A.F. Bower, H. Gao, Analytical model and molecular dynamics simulations of the size dependence of flow stress in amorphous intermetallic nanowires at temperatures near the glass transition, Phys. Rev. B 81 (2010) 125402. F. Delogu, Molecular dynamics study of size effects in the compression of metallic glass nanowires, Phys. Rev. B 79 (2009) 184109.