Reformation Capability of Short-Range Order and Their Medium-Range Connections Regulates Deformability of Bulk Metallic Glasses

Metallic glasses (MGs) typically have high yield strength while low ductility, and the latter is commonly considered as the Achilles’ heel of MGs. Elucidate the mechanism for such low ductility becomes the research focus of this field. With molecular level simulations, we show the degree of short-range order (SRO) of atomic structure for brittle Fe-based glass decreases dramatically during the stretch, while mild change occurs in ductile Zr-based glass. The reformation capability for SRO and their medium-range connections is found to be the primary characteristics to differentiate the deformability between the two metallic glasses. We suspect that, in addition to the strength of networks formed by SRO structure, the reformation capability to reform SRO networks also plays the key role in regulating the ductility in metallic glasses. Our study provides important insights into the understanding about the mechanisms accounting for ductility or brittleness of bulk metallic glasses.

Bulk metallic glasses (BMGs) have captured significant interests due to their unique properties such as high stiffness, high toughness and superior corrosion resistance [1][2][3][4][5][6][7] . However the low ductility of BMGs below their glass transition temperature T g restricts their wide applications. Searching for BMGs with both high strength and high ductility thus becomes the current focus of research. So far, the properties associated with deformability in BMGs have been explored from both macroscopic and atomistic scales. For the former, it is found that the plasticity of BMGs is closely linked to the Poisson's ratio of the material: BMGs show significant toughness when the Poisson's ratio exceeds a critical value of 0.31-0.32 8,9 , with greater Poisson's ratio leading to more diffusive shear bands 10 . Further studies suggest specimen aspect ratio 11 , free volume [12][13][14] , loading rate 11,14 and energy dissipative mechanisms 12 maybe have influence on the plasticity of MGs. On the atomic level, Zhang et al. 15 explained the plasticity observed in Zr-Cu(Ni)-Al MGs based on internal atomistic structures from simulations. A strategy was then proposed to design BMGs compositions with the desired local order which is considered as the structural basis for BMGs 16,17 . How the local order structures improve the ductility, however, remains elusive. Cheng et al. 18 then resolved the atomic structure of Cu 46 Zr 47 Al 7 through molecular dynamics (MD) simulation and found the small percentage of Al leads to dramatically increased population of Cu/Al centered full icosahedra (FI) as well as their spatial connectivity and medium-range order (MRO). Kumar et al. 19 found that substituting a few percent of Al for Zr in Cu 46 Zr 54 MGs leads to remarkable increase in strength and ductility. Follow this line, Lee et al. 20 investigated the evolution of MRO during applied loading and found the interpenetrating connection of icosahedra (ICOI) network is resistant to stress-induced shear transformation. The disruption of ICOI network would result in obvious strain localization and lower

Results
Ductility of MG nanowires. Two MG nanowire samples, Ni 30 Zr 70 and P 30 Fe 70 , are studied respectively with MD simulations using LAMMPS 22 . Figure 1a shows the stress-strain profiles during the elongation for Ni 30 Zr 70 and P 30 Fe 70 MGs. For comparison the stresses are normalized by their respective yield strength σ c . It is seen that after yielding, the stress of P 30 Fe 70 nanowires drops abruptly to one-tenth of σ c as the strain increases to 0.4 (Fig. 1a, the red line), indicating brittle nature of the material. For Ni 30 Zr 70 , however, the stress drops mildly to about one half of σ c as strain reaches to 0.4, indicating the material is relatively ductile. Fig. 1b,c show the close-up view of the two samples at ε = 0.144. The color of individual atoms represents the local von Mises equivalent strain 23 . The results show that during elongation, P 30 Fe 70 glass develops one localized shear zone across the sample (Fig. 1b) while Ni 30 Zr 70 glass has multiple small shear bands distributing along the sample. The size and the distribution of shear bands reveal distinct strain softening mechanism: for P 30 Fe 70 the softening is mainly due to the localized deformation; for Ni 30 Zr 70 however, softening is due to the initiation of multiple small bands which gives rise to rather homogeneous deformation in the sample.
We are interested in the mechanisms that regulate the initiation and propagation of tiny shear bands. One key question is why there emerge diffusive shear bands in ductile Ni 30 Zr 70 rather than localized bands seen in P 30 Fe 70 sample. One possible way to reveal the mechanism is to examine the internal structure of glasses during deformation at different stages. It has been shown the SRO and their medium-range connections, i.e., the atomic full icosahedra (FI) formed superclusters, would directly affect the mechanical behavior of MGs 17,18,20,[24][25][26] . Here we also adopt the scheme and use the degree of SRO as the indicator to probe the properties of MGs. Figure 1d,e show the examples of superclusters for NiZr and PeFe samples, where FIs interpenetrate each other. The capability of forming those FI superclusters serves as an indicator for plastic deformability, which will be illustrated in detail next. Figure 2a,b show some snapshot configurations of both glasses during stretching. For clarification only the atoms in FI superclusters within a slab of 1 nm thick are shown. The states I-IV are the deformation snapshots when ε = 0, 0.08, 0.144, 0.29, respectively. It is seen that at state I, i.e. ε = 0, Ni 30 Zr 70 sample has relatively dense network of FI clusters compared to that in P 30 Fe 70 sample. As the sample is stretched, the network in P 30 Fe 70 becomes even sparse (state II, III, and IV), which is in distinct contrast to the situation in Ni 30 Zr 70 where the network remains dense. It seems there exists a strong correlation between FI network and ductility: when the network becomes sparse due to the breakage of FI supercluster linkage, the glass shows brittle behavior in stretching.
To quantitatively analyze the stretch induced change for FI networks, we count the number of FI clusters during stretching. As seen in Fig. 2c in Ni 30 Zr 70 the population of Zr-centered FI clusters decreases mildly (< 7% when ε = 0.4). For P 30 Fe 70 , however, the population of Fe-based FI clusters first drops quickly then recovers during plastic yielding and eventually reaches to about 90% as ε = 0.4. We attribute the transition of FI cluster population to the propagation of STZ, which releases the stored strain energy and enables the reformation of FI clusters. Since the propagation of STZ is highly localized, we summarize the distribution of FI clusters along the axis of the nanowires in Fig. 2d to further capture the homogeneous/heterogeneous deformation behavior of FI network. It is seen that for Ni 30 Zr 70 , the density of FI clusters along the axis of nanowire remains almost constant at each state of I-IV. For P 30 Fe 70 , at states III and IV, the distribution of FI is highly differentiated, indicating the breaking of FI networks is localized and deformation becomes heterogeneous.
Fracture behavior of MGs. Since the tensile ductility is closely related to the fracture toughness of materials, we repeat the simulations described in the work by Murali et al. 21 to check if the evolution of SRO structure would affect the fracture behavior of MGs. In the new simulation systems, P 30 Fe 70 and Ni 30 Zr 70 samples with dimensions of 200 nm × 100 nm × 2 nm are constructed by multiplying the quenched glass unit as obtained in the nanowire samples. The samples are loaded in tension along y-direction after introducing a sharp crack with initial length 34 nm. All the other details are the same as those described in Ref 21. From the results we also observed the distinct fracture behavior of Ni 30 Zr 70 and P 30 Fe 70 BMGs: the former is ductile while the latter is brittle, as described in Ref 18. With the MRO structure presented at specific strain (Fig. 3), it is seen at the crack tip that the Ni 30 Zr 70 sample has dense network of FI clusters, which effectively prohibits the propagation of crack. For P 30 Fe 70 sample, however, the network of FI clusters at the crack tip is sparser than those far from the tip, indicating the network of FI clusters at the tip is destroyed while crack propagating. Figure 3c shows the density of FI along the x-direction where one apparent drop occurs at the crack tip (pink arrow). For Ni 30 Zr 70 sample, however, the FI density has no obvious change at the crack tip. We note that the drop ahead of the crack tip is due to the increased volume of the crack. So for bulk metallic glasses, the fracture behavior also has strong correlation with FI networks, which is similar to the case of nanowires under stretching.

Reformation capability of SRO and their medium-range connections. One question remaining
elusive is what induces the distinct behavior of FI networks of the two BMGs. We have shown that the variation in population of FI clusters during stretching Ni 30 Zr 70 nanowire is small (Fig. 2c). One may postulate that the network of Ni 30 Zr 70 FI clusters could resist breakage due to their strong bonding. However, it seems to be contradictory to our observation that P 30 Fe 70 has even higher resistance than that of Ni 30 Zr 70 (Fig. 1a, the inset). Furthermore this postulation is also inconsistent with the observation by Lee et al. 20 that the disruption of network would result in strain localization and lower plasticity. Here we propose that the reformation capability of FI networks has major role in regulating the ductility of glasses. We note that in the work of Shimizu et al. 27 , they have proposed there is a rejuvenation zone ahead of the crack tip, where the glass undergoes transition from well-aged state to rejuvenated state. To illustrate this mechanism, we examine the evolution of typical interpenetrating connections of FI networks within STZs for the two glasses, as shown in Fig. 4. For Ni 30 Zr 70 glass, the FI network does break in the backbone during stretching (Fig. 4b). Yet the broken backbones form a long backbone again ( Fig. 4c,d). Such reformation behavior, however, is not seen in P 30 Fe 70 FI networks: as the strain increases, the broken backbone of the network is continuously falling into segments (Fig. 4f,g).
To further distinguish the reformation capability of FI networks between Zr-and Fe-based glasses, we count the bonding event between two interpenetrating Fe-/Zr-centered FIs during loading of nanowires and bulk MGs (see Methods for details). Figure 5b shows the bonding event φ occurring at different time interval of 1-10 fs, 11-20 fs, 21-30 fs, etc in the stretched nanowires as ε = 0.144. It is seen that in Zr-based glass, the bonding events are much higher than those in Fe-based sample, indicating the Zr-based sample has much higher reformation capability. For a bulk MG with a crack, the bonding event has the same trend as that in nanowires (Fig. 5b). We further count the total bonding event occurred within 1 ps at various stretched state for nanowires (Fig. 5c). It is seen that the bonding event for Zr-based glasses keeps higher than that of Fe-based ones during the stretch. Such reformation capability is thought to promote reconnection of FI clusters and reformation of MRO networks, subsequently enhances the ductility of glasses. In the calculations, we have selected Fe-/Zr-centered full icosahedrons as the connection units since the population of such FIs is dominant: the fraction of Zr-centered FIs reaches as high as 89% and that of Fe-centered FIs reaches almost 100%. In Ni 30 Zr 70 glasses, we also calculate the bonding propensity of Ni-centered FIs and the results show the difference for bonding propensity between Ni-centered and Zr-centered FIs is within 4%. Here we have considered the connection as the interpenetrating connection of icosahedra (ICOI) 20 . To further see how the other connection types, i.e. vertex sharing (VS), edge sharing (ES) and face sharing (FS) 26 , are affected in the calculating of bonding event, we add more calculations (see Methods). The results show the bonding event for non-ICOI has mild change: for P 30 Fe 70 glass, the bonding event of ICOI is about 3.92 while that of non-ICOI is about 5.04; for Ni 30 Zr 70 glass, the bonding event of ICOI is about 12.59 while that of non-ICOI is about 10.8. We also change the time interval from 1 fs to 10 fs and keep the total time as 1 ps (see Methods), the results show that the bonding event decreases accordingly: it changes from 12.59 to 9.2 for Ni 30 Zr 70 glass, and from 3.92 to 3.6 for P 30 Fe 70 glass. It is seen that the P 30 Fe 70 glass still has much lower bonding event.
Cooling rate/free volume effect on the rejuvenation capability. To further investigate if such high reformation capability is possessed solely by ductile glasses, we change the simulation protocol with additional simulations for P 30 Fe 70 nanowires. The new sample prepared at cooling rate of 1800 K/ns which is two orders of magnitude higher than the previous ones. The results show that the ductility for the fast-cooling-rate sample is higher than that of the slow-cooling-rate one (Fig. 6a). Since fast cooling rate would lead to more free volume or higher concentration of vacancies, our results are consistent with previous observation that pre-existing shear bands/defects can enhance the ductility of BMGs 10,13 . Through comparing the evolution of FI clusters, we find the FI bonding propensities are higher in fast-cooling-rate Fe-based MGs (Fig. 6b). We also repeat this protocol on Ni 30 Zr 70 glass and find the same trend that fast-cooling rate enhances the bonding propensity (Fig. 6c,d). Thus the reformation capability for the fast-cooling samples is higher than that in the slow-cooling ones. It seems that the reformation capability depends not only on the components, but also on free volume of glasses. The former determines if the glasses would form dense FI clusters (like Zr-based glasses), which enhances the possibility for isolated FIs to connect each other. The latter, however, determines if the FIs could overcome steric hindrance to establish connection.
Temperature effect on the reformation capability. Apart from the cooling rate/free volume effect, the temperature during loading is another factor influencing the deformability of MGs. It is well known that the increase of testing temperature results in the brittle-to-ductile transition of MGs 14 . To see if such temperature-induced transition has correlation with the bonding propensity of FIs, we repeat the simulations of stretching Ni 30 Zr 70 nanowires at various temperatures. From the stress-strain profiles (Fig. 7a), it is seen that the ductility of MGs enhances as temperature increases. Meanwhile, the corresponding bonding event φ of FIs also increases with the temperature (Fig. 7b), indicating the temperature could enhance the bonding propensity efficiently. Subsequently, the re-bonding of FIs greatly enhances the ductility of MGs. Such observed temperature dependent bonding propensity can be explained by Arrhenius theory, , where k is the bonding reaction rate, E a is the activation energy, k B is the Boltzmann constant, and T the temperature. It proves again that the deformability has strong correlation with the reformation capability of SRO structures.

Discussion
In the comparative study we have used FI, i.e. a cluster of atoms with coordination number 12 (CN = 12), as the index to monitor the propensity of reformation of glasses. Although for Zr-based glasses the population of FIs is demonstrated to be dominant 17,18 , there is still a lack of such information for Fe-based glasses. To elucidate it we calculate the number of clusters with different CN for P 30 Fe 70 and find the fraction of FIs (CN = 12) in all clusters is about 52%, much higher than that of the second highest  cluster with CN = 11 (19.6%). So it is appropriate to use FI as the index in calculating the propensity of reformations of glasses.
To extend the finding that ductility of Fe-and Zr-based MGs is induced by the reformation capability of MRO structures, we also conduct some additional calculations based on published result. It has been shown that the strength and ductility of Cu x Zr 100−x MGs vary with the Cu composition: the strength increases while ductility decreases with x 28 . We calculate the bonding event φ for different x (x = 20, 30, 46, 64) during the deformation. The results agree with our conclusion that high bonding event φ leads to high ductility (see Supplementary Material). So for some typical studied MG samples, we find there is strong correlation between ductility and reformation capability of SRO structures. With this hint, we may explore further to see if such correlation is general to all the MG glasses.
We note that in the work by Murali et al. 21 , the fluctuation in the stress field near a crack tip due to atomistic level structural inhomogeneity was believed to govern the brittle fracture behavior. Our observations could potentially explain why there is so much fluctuation in strength of cavitations in brittle BMG. With MRO becoming more sensitive to stretch, the strength of cavitations of material would presumably become more sensitive to the local atomic environment: with confinement of dense FI clusters, the fluctuation in strength of cavitations is effectively alleviated.
In summary, we have studied the ductile versus brittle mechanisms of Zr-and Fe-based metallic glasses through molecular dynamics simulations. With the degree of SROs and their medium-range connections as an indicator, we have shown that the brittle behavior of Fe-based glass is associated with the dramatically decrease of FI clusters, which induces the initiation and propagation of STZs. For Zr-based glass, however, our results show that the degree of FI clusters has mild change during the stretch. We propose that, instead of the strength of networks of FIs, the reformation capability to form networks of FIs plays the major role in regulating the ductility of glasses. For glasses with high reformation capability, the FIs can re-bonding instantaneously after the breakage of FI networks, which effectively prevents the propagation of STZs and the heterogeneous deformation of glasses. Our study provides important insights into the understanding about the mechanisms accounting for ductility or brittleness of BMGs.

MD simulations.
Atomic interactions are modeled by embedded atom method (EAM) potentials with parameters given by Mendelev et al. 29,30 . Glass samples consisting of 16000 atoms with randomly substituted solid solution in a face-centered cubic (FCC) lattice are used in a melting-and-quenching simulation, whose temperature is raised gradually from 0 to 2100 K and is then cooled down to 300 K. The cooling rate is set at 18 K/ns and the periodic boundary condition (PBC) is applied in all directions. The samples are then replicated three times along x-direction to form nanowires following 1 ns equilibrium at 300 K with the PBC applied along x-direction only. The time step for integration is chosen to be 1 fs. In all simulations the lateral pressure is controlled at 0 bar. The MGs nanowire samples with final dimensions of 20 nm × 6.7 nm × 6.7 nm are then loaded under uniaxial tension at the strain rate of 10 8 s −1 .

Bonding event calculations.
A criterion to judge the occurrence of bonding is defined that when two isolated FIs approach each other within a critical distance, r cr . To determine r cr , we calculate the pair distribution function (PDF) of Fe-and Zr-based MGs as shown in Fig. 5a. For the case of interpenetrating connection of icosahedra (ICOI), i.e., a Fe-/Zr-centered FI overlapping with another Fe-/Zr-centered FI, r cr is the distance to the first valley in the PDF profile (Fig. 5a). Via this approach, we obtain r cr = 3.2 Å and r cr = 3.7 Å for Fe-and Zr-based MGs, respectively. For other non-ICOI like vertex sharing (VS), edge sharing (ES) and face sharing (FS), the r cr is doubled. To analyze bonding events, in each of the MD simulations we track all atoms' trajectory at each femtosecond (fs) for one picosecond (ps). At time t i + Δ t (i = 1,…1000), the structure deforms from its reference configuration (at time t i ) and the bonding events are counted with the criterion. Here we define a parameter, φ i (t i , Δ t), to indicate the propensity of bonding, which is the bonding event occurred in ∆t normalized by the mean number of FIs in the glasses. Then φ φ = ∑ = i n i 1 counts the total bonding events occurred within time interval Δ t.