Mechanical Properties of Glassy Nanopillars: A Comparative, Computational Study of Size Effects in Nanoglasses and Homogeneous Bulk Glasses

We study the mechanical properties of nanoglass (NG) nanopillars with diameters ranging from 4.5 to 54 nm by means of molecular dynamic simulations and compare the results with those obtained for nanopillars prepared from homogeneous glasses. NG nanopillars of two different types of glasses, namely, Cu64Zr36 and Pd80Si20, were cut from samples prepared by nanoparticle consolidation. The influence of nanopillar diameter on the deformation behavior and strain localization is investigated. Moreover, cyclic loading is used to explore the origin of stress overshoots in the stress–strain curves of NGs. Finally, from the calculated properties, a deformation map for NG and homogeneous glass nanopillars is derived.


INTRODUCTION
Nanoglasses (NGs) are glassy materials with microstructural features on the nanoscale, which consist of glassy grains connected by glass-glass interfaces (Ivanisenko et al., 2018). These interfaces are characterized by an excess volume (Jing et al., 1989;Şopu et al., 2009), a defective short-range order (SRO) , and a different composition as compared with the glassy grains (Adjaoud and Albe, 2016;Wang et al., 2016a;Adjaoud and Albe, 2018). NGs can be synthesized by cold compaction of nanometer-sized metallic glassy particles obtained via inert gas condensation (Jing et al., 1989;Weissmüller et al., 1992;Gleiter, 2008;Fang et al., 2012;Gleiter, 2013;Gleiter et al., 2014;Gleiter, 2016;Nandam et al., 2017;Ivanisenko et al., 2018). Microcompression test revealed considerable plasticity in an Sc 75 Fe 25 NG, while the homogeneous glass (HG) with identical composition fails in a brittle manner (Fang et al., 2012). When reducing the sample size to the nanoscale, 15% plastic strain was observed in a 400 nm Sc 75 Fe 25 NG nanopillar using in situ tensile tests in a transmission electron microscope . Moreover, a systematic study of nanosized and microsized Sc 75 Fe 25 HG and NG pillars by means of compression tests showed that both, yield strength and deformation mode, are size-dependent in HG pillars. These properties, however, are size-independent in the NG pillars (Wang et al., 2016b), which was attributed to the microstructural features present in the NG. Indeed, molecular dynamic (MD) simulations revealed that glass-glass interfaces in NGs act as nucleation sites for shear transformation zones (STZs) and prevent strain localization, which leads to the more homogeneous deformation of NGs as compared with HGs Adibi et al., 2013;Albe et al., 2013;Adibi et al., 2014;Şopu and Albe, 2015;Adjaoud and Albe, 2019;Cheng and Trelewicz, 2019b).
MD simulations were also used in the past to investigate the effect of grain size on the deformation behavior of a 50 nm Cu 64 Zr 36 NG nanopillar (Adibi et al., 2015a;Adibi et al., 2016). In these studies, the nanopillar was prepared by cutting a cylinder from an NG produced from an HG using the Poisson-Voronoi tessellation method (Brostow et al., 1978;Finney, 1979;Tanemura et al., 1983). However, this model of an NG underestimates the volume fraction of the interfaces as compared with the NGs prepared by consolidation of nanoparticles (Adjaoud and Albe, 2018;Cheng and Trelewicz, 2019a;Cheng and Trelewicz, 2019b;Adjaoud and Albe, 2019) which is more consistent with the experiment (Fang et al., 2012;Wang et al., 2015;Wang et al., 2016b).
In this work, we report the results of a systematic study on the influence of nanopillar diameter on the mechanical properties of NG and HG nanopillars with various diameters by means of MD simulations. The NG nanopillars were cut from NGs prepared by nanoparticle consolidation (Adjaoud and Albe, 2018). We first investigate the influence of nanopillar diameter on the deformation behavior and strain localization. Then, we examine the effects of nanopillar diameter on Young's modulus and yield stress of NG and HG nanopillars. We finally use the obtained mechanical properties to derive a deformation map for NG and HG nanopillars. In order to see whether the mechanical properties of NG and HG nanopillars depend on the chemical composition, all simulations were done for two different types of NGs and HGs, namely, Cu 64 Zr 36 and Pd 80 Si 20 .

METHODOLOGY
MD simulations were carried out using the LAMMPS simulation package (Plimpton, 1995).The interatomic interactions are described by the Finnis-Sinclair-modified-type potential for Cu-Zr (Mendelev et al., 2009) and the embedded atom model potential for Pd-Si (Ding et al., 2012). These potentials have been optimized to predict accurately the structure of liquid and amorphous Cu-Zr and Pd-Si alloys and were successfully applied to Cu-Zr and Pd-Si glasses (Cheng et al., 2013;An et al., 2016;Tang et al., 2018;Adjaoud and Albe, 2019).
First, Cu 64 Zr 36 and Pd 80 Si 20 HGs were obtained by quenching a melt, which was already equilibrated at 2,000 K, to a temperature of about 50 K at a quenching rate of 0.01 K ps −1 . Next, we prepared Cu 64 Zr 36 and Pd 80 Si 20 NGs by cold compaction of several glassy spheres with diameters ranging from 6 to 8 nm as described by Adjaoud and Albe, 2018. The resulting NGs have dimensions of about 18 nm × 18 nm × 18 nm, and their microstructure consists of glassy regions connected by glass-glass interfaces. These interfaces are characterized by a defective SRO in a zone with a width of at least 2 nm (Adjaoud and Albe, 2018;Cheng and Trelewicz, 2019a The NGs and HGs were replicated and relaxed in order to construct larger samples. After that, the nanopillars were prepared by cutting a cylinder followed by structural relaxation. The diameters of the nanopillars are ranging from D 4.5-54 nm, which corresponds to 17 million atoms at maximum. The HG nanopillars have a homogeneous microstructure, while the NG nanopillars have an inhomogeneous microstructure (see Supplementary Figure S1). All nanopillars have an aspect ratio (length-to-diameter) of 2, except the nanopillars with D 4.5 nm which have an aspect ratio of 4. The later aspect ratio is chosen in order to still keep the periodicity in the direction of the axis of the NG nanopillar.
In order to characterize the mechanical properties of NG and HG nanopillars, a series of tensile deformation simulations were performed. The uniaxial tensile load was applied along the axis of the nanopillars, which was chosen in the z-direction, with a constant strain rate of 4.10 7 s −1 at 50 K. The atomic scale deformation mechanisms were analyzed in terms of the local atomic von Mizes shear strain calculated with the OVITO analysis and visualization software (Stukowski, 2010).

Deformation Behavior and Strain Localization
In NG nanopillars, the glass-glass interfaces as well as the free surface can act as nucleation sites for shear events (Albe et al., 2013). Figures 1A,B present the stress-strain curves for Cu 64 Zr 36 and Pd 80 Si 20 NG nanopillars with an average grain size of d 7 nm and diameters ranging from D 4.5-54 nm. It can be seen that the maximum stress, σ max , is higher for Pd 80 Si 20 NG nanopillars than for Cu 64 Zr 36 NG nanopillars, which is essentially a modulus effect. Apart from σ max , stress-strain curves of Cu 64 Zr 36 are similar to those of Pd 80 Si 20 with the same nanopillar diameter; both types of NG nanopillars do not show a stress drop upon yielding.
Nanopillars with a diameter smaller than the average grain size (d 7 nm) exhibit the highest σ max . When D ≥ 9 nm, the stress-strain curves exhibit pronounced strain softening. The onset of strain softening occurs earlier in the large nanopillars: the stress of 36 and 54 nm NG nanopillars starts decreasing at an engineering strain of about 13-15%, in agreement with experimental results on a 400 nm Sc 75 Fe 25 NG nanopillar showing about 15% plastic strain .
Figures 1C,D show the engineering stress-strain curves for the Cu 64 Zr 36 and Pd 80 Si 20 HG nanopillars with diameters ranging from 4.5 up to 54 nm. The Cu 64 Zr 36 and Pd 80 Si 20 HGs are shown for comparison. It can be seen that the Cu 64 Zr 36 HG exhibits a stress drop after the maximum stress, indicating the formation of a critical shear band (Adjaoud and Albe, 2019). The Pd 80 Si 20 HG, in contrast, does not show a stress drop, suggesting that the Pd 80 Si 20 HG is more ductile than the Cu 64 Zr 36 HG. This is in agreement with the bending experiment and compression test on Pd-Si glasses (Yao et al., 2006;An et al., 2016). The nanopillars with smaller diameters, 4.5 nm for Cu 64 Zr 36 and up to 9 nm for Pd 80 Si 20 , show strain softening upon yielding. Nanopillars with larger diameters, D ≥ 9 nm for Cu 64 Zr 36 and D ≥ 18 nm for Pd 80 Si 20 , exhibit a pronounced stress drop and strain softening, while the magnitude of the stress drop is increasing with increasing nanopillar diameter. This size dependence can be understood in terms of the elastic energy release right after the stress drop, which is proportional to D 2 (Liu et al., 2011). In the case of Cu 64 Zr 36 HG nanopillars, strain hardening occurs right after the stress drop at about 9% engineering strain, which is most pronounced in the 30 and 54 nm HG nanopillars. This effect can be explained by the structural relaxation which occurs after the elastic energy release (Ritter and Albe, 2012).
Next, we inspect the strain localization in NG nanopillars. By analyzing von Mises strains, three deformation modes can be defined: (1) pure necking, when the deformation plane is perpendicular to the deformation axis; (2) pure shear banding, when the angle between the deformation plane and the deformation axis is about 40°-60°; and (3) if a nanopillar is not long enough to allow strain localization to occur in a form of necking or shear banding, then von Mises strain distributes over the whole nanopillar which exhibits homogeneous deformation.
Figures 1E,F show the atomic von Mises shear strain at 27% engineering strain for Cu 64 Zr 36 and Pd 80 Si 20 NG nanopillars. It can be seen that the 4.5 nm nanopillars deform by necking. This necking deformation mode can be explained by the large surfaceto-volume fraction which leads to the nucleation of a high number of STZs at the surface of the nanopillar (Albe et al., 2013). The 9 and 18 nm NG nanopillars, where the diameter is about or twice the average grain size, exhibit nearly homogeneous deformation, confirming the weak strain softening observed in the stress-strain curves (see Figures 1A,B). When the nanopillar diameter is significantly larger than the average grain size, as in the case of the 36 and 54 nm NG nanopillars, the nanopillars deform again by necking. This is consistent with the strain softening occurring in the stress-train curves. By comparing the Cu 64 Zr 36 and Pd 80 Si 20 NG nanopillars, it can be seen that the deformation behavior is similar and thus does not depend on the type of the NG nanopillars. Moreover, even after deformation, the interfaces are still characterized by compositional variation (see Supplementary Figure S2).
Recent MD studies (Adjaoud and Albe, 2019;Cheng and Trelewicz, 2019b) revealed that the mechanical properties of bulk NG samples are rather independent of the grain size which was attributed to the constant volume fraction of interfaces for NGs with different grain sizes because of the varying interfaces' width. Based on those results, one could expect that the deformation behavior of the NG nanopillars in the present study is grain-size independent as their bulk NGs counterparts. It is interesting to note that a different result was obtained for bulk NGs and NG nanopillars produced by the Poisson-Voronoi tessellation method, which exhibits a grainsize-dependent deformation behavior because the glass-glass interfaces are assumed to have a constant width of 1 nm independent of the grain size (Adibi et al., 2013;Adibi et al., 2015a).
If we compare the NG nanopillars and bulk NG samples of both glasses, we see that bulk NG samples show no strain softening and a homogeneous deformation behavior (Adjaoud and Albe, 2019;Cheng and Trelewicz, 2019b) similar to the 9 nm NG nanopillars (see Figures 1A,B).
In order to see whether the necking deformation mode, which is observed in the Cu 64 Zr 36 and Pd 80 Si 20 NG nanopillars, is related to the presence of glass-glass interfaces or free surface, we inspected the atomic von Mises shear strain in the HG nanopillars which is shown in Figures 1G,H at 16% engineering strain. It can be seen that, with smaller diameters, the nanopillars deform by necking, while strain localization in the nanopillars with larger diameters occurs in the form of a shear band. This is in line with previous MD simulation results (Adibi et al., 2015b). Again, because of the large surface-to-volume fraction, the 4.5 nm HG nanopillars deform by necking independent of the chemical composition, similar to the 4.5 nm NG nanopillars. Previous atomistic simulations on Cu 64 Zr 36 HGs have shown that a sample thickness of at least 5 nm is needed for shear band formation (Cheng and Ma, 2011).
Comparing the Cu 64 Zr 36 HG nanopillars with the Pd 80 Si 20 HG nanopillars, we find that the stress drop is steeper and strain localization is more pronounced in the Cu 64 Zr 36 HG nanopillars. Moreover, σ max is higher for the Pd 80 Si 20 HG nanopillars than for the Cu 64 Zr 36 HG nanopillars, similar to NG nanopillars. This difference in the mechanical properties of Cu 64 Zr 36 and Pd 80 Si 20 HG nanopillars might be attributed to the difference in the chemical and topological SRO. In fact, recent bending experiments and MD simulations have shown that Pd 82 Si 18 HG is significantly tougher than Cu 46 Zr 54 HG; the higher toughness of the Pd 82 Si 18 HG is related to its topological SRO . Moreover, our results reveal that the shear-band thickness in the Pd 80 Si 20 HG nanopillars is larger than in the Cu 64 Zr 36 HG nanopillars (see Figures 1H,G). This is consistent with previous experimental and simulation studies which have shown that the shear-band thickness is ranging from 5 to 20 nm or even 200 nm depending on the chemical composition of the metallic glass (Zhang and Greer, 2006;Cao et al., 2009;Murali et al., 2012;Shao et al., 2013;Liu et al., 2017).
Our results show that the main differences between the NG nanopillars and HG nanopillars are maximum stress σ max and the deformation mode: strain localization in the HG nanopillars occurs in the form of a shear band, whereas the NG nanopillars deform by necking or homogeneously, depending on the nanopillar diameter. These results are in agreement with a recent experimental study on the mechanical properties of Sc 75 Fe 25 NG and HG nanopillars with diameter of 400 nm using in situ tensile tests in a transmission electron microscope .

Stress Overshoot during Cyclic Loading
It is known that cyclic loading may significantly affect the deformation behavior of materials. Cyclic loading is usually performed in the elastic regime (Sha et al., 2015;Sha et al., 2017) or in the plastic regime (Zhu et al., 2017;Tang et al., 2018). The cyclic loading of metallic glasses in the plastic regime leads usually to the phenomenon of stress overshoot (Maaß et al., 2012).
In order to investigate the stress overshoot and the remaining plastic deformation in the NG nanopillars in more detail, we performed cyclic loading on the 18 nm Cu 64 Zr 36 and Pd 80 Si 20 NG nanopillars by unloading them from 15% engineering strain and then reloading them. Figures 2A,B present the corresponding stress-strain curves during the cyclic loading. In both glasses, the cyclic loading leads to a slight increase of flow stress after reloading caused by structural relaxation: the fractions of the Cu-centered icosahedra in the Cu 64 Zr 36 NG nanopillar, including full icosahedra (index [0,12,0,0]) and disordered icosahedra (indexes [0,2,8,2] and [0,3,6,3]) and the most prominent Voronoipolyhedra Si[0,3,6,0] in the Pd 80 Si 20 NG nanopillar, significantly increase after the first loading stage (see Figures 2C,D), which indicates that defective interfacial areas are relaxed during the initial stage of plastic deformation. The correlation between the increase of flow stress and structural relaxation is also observed in the HG nanopillars (see Figures 2E-H). However, in the HG nanopillars, the stress increases only in a small region of engineering strains as compared with the NG nanopillars: after the initial stress increases, the stresses of the monotonic tensile loading and the cyclic loading merge together in the Cu 64 Zr 36 HG nanopillar, while in the Pd 80 Si 20 HG nanopillar, the stress of the cyclic loading is slightly lower than that of the monotonic tensile loading. Moreover, the fractions of the Cu-centered icosahedra in the Cu 64 Zr 36 HG nanopillar and the Voronoipolyhedra Si [0,3,6,0] in the Pd 80 Si 20 HG nanopillar for the cyclic loading remain higher than those for the monotonic tensile loading after the stress overshoot, indicating the recovery of the local order upon unloading.
These results are in agreement with recent MD simulation results which showed that a cyclic loading after the formation of a shear band in a Cu 50 Zr 50 HG leads to structural relaxation, and consequently, a stress increase occurs in the stress-strain curve (Tang et al., 2018). Experiments and theoretical models have shown that this phenomenon depends on temperature, strain rate, and structural relaxation (Kawamura et al., 1997;Lu et al., 2003;Maaß et al., 2012;Jiang et al., 2015;Yang et al., 2019).

Size Effects on Young's Modulus and Yield Strength
We, next, explore the effect of nanopillar diameter on Young's modulus and yield strength. We obtained Young's modulus from the slope of stress-strain curves at the small strain where the stress is proportional to the strain. We used 0.2% offset strain to calculate the yield strength. Figures 3A,B (Mendelev et al., 2008) and 94 Gpa for Pd 80 Si 20 (Mizubayashi et al., 1998). It can be seen that Young's modulus depends slightly on the nanopillar diameter when D ≤ 9 nm. However, in the HG, surface stresses are the dominant contribution to this size effect, and the elastic softening of the NG is due to the different elastic contributions of the glass-glass interfaces and the action of surface stresses, which are counteracting. However, the nanopillars with D ≥ 18 nm show This can be explained by the existence of the glass-glass interfaces in NGs, which act as a source for STZs (Albe et al., 2013). Recently, Wang et al. (2015) investigated the mechanical properties of Sc 75 Fe 25 NG and HG nanopillars by using in situ tensile tests in a transmission electron microscope and found a similar trend in yield strength: the yield strength of the NG nanopillar is lower than that of the HG nanopillar with identical chemical composition. Strength softening was also observed in Ni 78 P 22 NG pillars prepared by the electrodeposition technique (Li et al., 2018).

Critical Stress
In order to further investigate the difference in deformation modes between the NG and HG nanopillars, we calculated the critical stress required for SB formation (Volkert et al., 2008), where E is Young's modulus, Γ is the SB energy per unit area, A is the aspect ratio, and D is the diameter of the nanopillar. The values for E are reported in Figure 3. For Γ, we used a value of 1.30 J/m 2 for Cu-Zr determined in a previous study (Adjaoud and Albe, 2018) and a value of 2.34 J/m 2 for Pd-Si. We calculated Γ for Pd-Si from the stress-strain curve of the 18 nm Pd 80 Si 20 HG nanopillar, using the same method as that for Cu-Zr (Adjaoud and Albe, 2018). Our values for Γ are close to 0.98 ± 0.04 J/m 2 reported in tensile simulation for a Cu 50 Zr 50 metallic glass film (Zhong et al., 2015) and 0.56 J/m 2 reported in uniaxial compression experiments for Cu-based and Zr-based metallic glasses (Chen et al., 2010). On the other hand, our calculated Γ values are much smaller than 10 J/m 2 for a Pd-based metallic glass (Volkert et al., 2008) and 9 J/m 2 for a Zr-based metallic glass (Dubach et al., 2009) reported in uniaxial compression experiments. This suggests that the energy Γ depends on the composition of metallic glasses and their processing route.
The predicted critical stress values for the homogeneous glasses (HGs) match very well with the observed deformation modes as shown in Figure4. The only exception is the Cu-Zr nanopillar with D 9 nm. In case of the NG nanopillars, the data fall in the homogeneous regime. The only clear exceptions are the data for the nanopillars with D ≥ 54 nm. Interestingly, the Cu-Zr and Pd-Si NG nanopillars have the same critical stress values. This is consistent with the results in Figures 1E,F which show that the deformation modes for the Cu-Zr and Pd-Si NG nanopillars are similar. The relative good agreement with the model prediction is implying that the more homogeneous deformation of NGs can also be understood as a mere modulus effect.

DISCUSSION
The results presented above are for nanopillars with diameters up to 54 nm. This size is much smaller than that used in experiment which is about 400 nm . Sample preparation and mechanical testing in experiments make it difficult to prepare nanopillars with diameters below 100 nm (Shi, 2019). On the other hand, the increase of the nanopillar size in MD simulation leads to the increase of the number of atoms, and this becomes computationally expensive. Although the sample size mismatched between MD simulation and experiment, MD simulation is useful to understand experimental observations (Liontas et al., 2016). From the present MD results, one can see that, for both systems, Cu-Zr and Pd-Si, the 36 and 54 nm NG nanopillars exhibit a similar mechanical behavior. This may be attributed to the small surface-to-volume fraction which leads to the deformation to be dominated by the interfaces rather than by the surface. Indeed, a recent MD simulation study on HG nanopillars has shown that the surface stress becomes less significant when the nanopillar diameter is larger than 30 nm . Thus, one could expect that NG nanopillars with diameters larger than 54 nm should show a similar mechanical behavior as the 36 and 54 nm nanopillars.
Present MD simulation results on HG nanopillars show that the transition from necking to shear banding occurs at a diameter between 10 and 20 nm. These results are in agreement with previous MD simulation results which reported that sizeindependent shear banding until their dimensions approach the shear-band thickness which is about 5-20 nm (Adibi et al., 2016;Liontas et al., 2016;Shi, 2019). This transition appears in experiments at a lager sample diameter, between 100 and 400 nm, as compared with MD simulations (Tian et al., 2016). This difference in the sample size on the change of the deformation mode between experiments and simulations can be related to the different processing and testing conditions (Shi, 2019).
Simulations of NG nanopillars reveal no shear banding: NG nanopillars exhibit necking or homogeneous deformation. Homogeneous deformation occurs in the 9 and 18 nm NG nanopillars, where their length is not long enough to allow strain localization to form necking. Necking deformation mode was also reported in previous MD simulations on the Cu 64 Zr 36 NG nanopillars with a diameter of 50 nm (Adibi et al., 2015a;Adibi et al., 2016). The NG nanopillars studied in the present study are cut from an as-prepared NG. Structural relaxation due to the high compaction rate used in simulation during the preparation of the NG may change the deformation mode. Indeed, a recent study on the compression test of Pd-Si NG nanopillar with a diameter of 36 nm showed a change in the deformation mode from homogeneous deformation to shear banding when the nanopillar is cut from an annealed NG (Nandam et al., 2020). The effect of annealing on the change of the deformation mode was also reported in a recent experiment and simulation study on Zr-Ni-Al nanosized metallic glasses (Liontas et al., 2016).

CONCLUSIONS
We have investigated the influence of structure size on the mechanical properties of NG and HG nanopillars with 7 nm grain size and diameters ranging from 4.5 up to 54 nm by means of MD simulations. Simulations were done for two different glasses, namely, Cu 64 Zr 36 and Pd 80 Si 20 , as representatives of metal-metal and metal-metalloid systems, respectively. Different from previous studies, the NGs were produced by consolidation rather than Voronoi tessellation and thus have a more realistic microstructure.
Our results show a clear difference in the deformation mode between NG and HG for the 36 and 54 nm nanopillars, independent of the glass type. While HG nanopillars exhibit a stress drop and strain localization developing in a shear band, NG nanopillars show ductile deformation behavior with softening at larger engineering strains and deformation by necking. The tensile ductility of about 13-15% found in our simulations is in agreement with 15% plastic strain observed for a 400 nm Sc 75 Fe 25 NG nanopillar using in situ tensile tests in a transmission electron microscope . HG and NG nanopillars with D 4.5 nm, where the pillar diameter is smaller than the average grain size d 7 nm of the NG, deform by necking since the nucleation of STZs on the surface is dominating. In the HG nanopillars with D 9 and 18 nm, shear banding is more obvious in Cu 64 Zr 36 than in Pd 80 Si 20 . When reducing the NG nanopillar diameter to near or double the average grain size, strain softening appears at the larger engineering strain (>20%). Moreover, structural relaxation after a cyclic loading leads to local recovery, and the stress increases upon reloading.
We determined Young's modulus and yield strengths from stress-strain curves of tensile deformations. We find that both properties are smaller in the NG nanopillars as compared with their homogeneous counterparts in both glasses. From Young's modulus values and the shear band energy, the critical stress for shear band formation is estimated. We find that the predicted critical stress values are quite consistent with the observed deformation modes.

DATA AVAILABILITY STATEMENT
All datasets generated for this study are included in the article and Supplementary Material.

AUTHOR CONTRIBUTIONS
OA and KA contributed equally to the conception and design of this study; OA set up and ran the simulations, performed the analysis, and wrote the first draft of the manuscript; KA supervised the work and reviewed and edited the manuscript.

FUNDING
The authors acknowledge the financial support of the Deutsche Forschungsgemeinschaft (DFG) through project no. AL 578/ 15-2.