Dynamic Evolution of Local Atomic Environments in a Cu₆₆Zr₃₄ Bulk Metallic Glass

Abstract

This study presents a molecular dynamics (MD) investigation of the evolution of local atomic environments (LAEs) in a Cu₆₆Zr₃₄ bulk metallic glass (BMG), both at rest and under constant shear deformation. LAEs were characterized using Voronoi polyhedra analysis. Even in the absence of external load, LAEs frequently transformed into one another due to short-ranged atomic position fluctuations. However, as expected, each transition from one polyhedra to another was balanced by the reverse transition, thereby preserving the proportions of the different polyhedra. Cu-centered icosahedral LAEs were observed to preferentially transform into and from <1,0,9,3,0>, <0,1,10,2,0>, and <0,2,8,2,0> LAEs. Upon applying pure shear, the simulation box was first deformed in one direction up to a strain of 25% and then in the opposite direction to the same strain level. Shear deformation induced large nonaffine atomic displacements in the directions parallel to the shear, which were concentrated in specific regions of the BMG, forming band-like regions. From the onset, shear deformation led to the destabilization of Cu-centered icosahedral LAEs, as indicated by more frequent transitions to and from other polyhedra. Unlike other Cu-centered LAEs, icosahedra were also found to be more sensitive to yielding. The destruction of Cu-centered icosahedra was primarily a result of net transformations into <1,0,9,3,0> and <0,2,8,2,0> LAEs in the BMG subjected to pure shear, with a minor contribution of transformations involving the <0,1,10,2,0> polyhedra.

1. Introduction

Bulk metallic glasses (BMGs) are amorphous alloys with unique properties, making them promising candidates for various applications. They exhibit superior corrosion resistance compared to crystalline materials and possess remarkable mechanical properties, such as a low elastic modulus and a high elastic limit [1,2]. The disordered structure of BMGs determines their primary deformation mechanism, which differs from those in crystalline materials, as it relies not on slip or twinning but rather on strain concentration within shear bands. Shear banding originates from local heterogeneities called shear transformation zones (STZs), which comprise softer regions in the amorphous alloy. However, plasticity in metallic glasses is limited, as these materials typically fail shortly after shear bands form, particularly under tension and at low temperatures, which prevents their use as structural materials [3,4,5]. Extensive research has been conducted on STZs, shear banding, and other aspects involved in the plasticity of BMGs [6,7,8,9,10,11,12,13].

The microscopic structural disorder exhibited by metallic glasses makes it significantly more challenging to establish the structure-property relationship for these materials. Lacking long-range order and the symmetries that facilitate the analysis of deformation processes in crystalline materials, atomic-scale analyses of BMGs require both a local and a global (statistical) perspective on how neighboring groups of atoms are structured. Without excluding other approaches, the local atomic environment (LAE) of each atom in a material can be characterized by topological constructions known as Voronoi polyhedra [14,15,16]. A Voronoi polyhedron can be visualized as a cage with an atom at its center, where each nearest neighbor of this atom is associated with a face of the polyhedron. In a crystalline phase of a metallic alloy, the LAE of each atom, provided it is not located in the vicinity of a lattice defect, is unique and clearly defined. On the other hand, amorphous materials possess a diversity of LAEs [17,18]. The different types of Voronoi polyhedra associated with these LAEs are identified by indices [19], and their distribution can be highly uneven, depending on factors such as the composition of the BMG.

Among the polyhedra, the icosahedra (Voronoi index: <0,0,12,0,0>) are known to be a critical structural unit for the properties of both liquid metals and metallic glasses [20,21,22,23]. In supercooled liquid metals, icosahedral clusters act as obstacles to crystal nucleation [24,25,26] and affect the formation of amorphous alloys during rapid solidification [3,27,28]. Moreover, in recent years, several works employing molecular dynamics (MD) simulations found that icosahedra play a crucial role in the development of shear banding in metallic glasses. For instance, simulations of various amorphous binary and ternary alloys have provided evidence that the tendency to undergo plastic flow is closely linked to the fraction of icosahedra [29]. Generally speaking, an increase in the concentration of icosahedra is associated with greater resistance to STZs. On the other hand, the softening of BMGs has been observed to result in the destruction of these polyhedra [30]. Fan and Li investigated the topology of icosahedral networks in CuZr metallic glasses, unveiling their role on yielding [31]. Wang and Wong, in turn, found that icosahedra are created and destroyed during deformation and can participate in STZs when they are not part of interconnected icosahedral networks [32].

MD simulations of a Cu₄₅Zr₄₅Al₁₀ BMG under compression have identified some mechanisms of shear band formation associated with variations in the proportion of predominant Voronoi polyhedra [9]. The sensitivity to deformation of these LAEs, although not very pronounced, provided an indication of their importance in the plasticity of the BMG. For example, consistent with previous studies, a decrease in Cu-centered icosahedra was observed. In another study, Tercini and colleagues used MD simulations to investigate the kinetics of icosahedra destruction in BMG-based nanocomposites undergoing plastic deformation, where the matrix was the Cu₄₅Zr₄₅Al₁₀ metallic glass and CuZr with a B2 structure served as the filler [33].

In this article, we present a computational study of the evolution of LAEs using MD simulations, with a focus on icosahedral polyhedra. We monitored the transitions between icosahedra and other Voronoi polyhedra throughout the simulations, both when the BMG was in a resting state (i.e., not subjected to an external load) and when it was under pure shear. This allowed us to statistically characterize the main topological transformations at the atomic level in the BMG when a load is applied and to correlate these transformations with the yielding behavior of the material.

2. Computational Method

Classical molecular dynamics (MD) simulations were performed using the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) open-source code [34], which is known for its efficiency in handling systems with hundreds of thousands or even millions of atoms. The Newtonian equations of motion were integrated using the Velocity Verlet algorithm [35] with a timestep of 1 fs.

Interatomic Cu-Cu, Zr-Zr, and Cu-Zr forces were obtained from the Embedded Atom Method (EAM) potential for this binary system, as described in Ref. [36]. Initially proposed by Daw and Baskes [37], the total energy E_tot of a collection of atoms in the context of EAM is given by the following formula:

$$E_{tot}={\displaystyle \frac{1}{2}}{\sum }_{i,j}\phi \left(r_{ij}\right)+{\sum }_i{F}_i\left[{\sum }_{i\ne j}\rho \left(r_{ij}\right)\right]$$

(1)

where φ (r_ij) represents the pairwise energy between atoms i and j, F_i denotes the embedding energy of atom i when inserted into the electronic density of the surrounding atoms, and ρ (r_ij) signifies the electron density contribution of atom j at the position of atom i.

The simulation box used in the MD simulations contained a representative volume of amorphous Cu₆₆Zr₃₄. The entire system comprised a cube with a volume of approximately 16 nm³ enclosing 250,000 atoms. To obtain the amorphous structure, the quenching method seen in Ref. [9] was employed. In this method, an initially crystalline, body centered cubic (BCC) Cu₆₆Zr₃₄ alloy with Zr randomly distributed within the Cu matrix was used as the starting material. The process of producing the BMG model consisted of a thermal cycle simulated using MD in the isothermal-isobaric (NPT) ensemble at zero pressure. Periodic boundary conditions (PBCs) were applied to all directions. The cycle started by rapidly increasing the system’s temperature from 300 K to 2000 K in 1 ns. The molten Cu₆₆Zr₃₄ system was then maintained at this high temperature of 2000 K for an additional 1 ns. The next step involved rapidly cooling the melt to 300 K over 1 ns. Finally, the system was thermally and mechanically equilibrated for 2 ns. At the end of this simulated thermal cycle, it was confirmed that the final state of the Cu₆₆Zr₃₄ system was indeed a completely disordered metallic glass.

Subsequent MD simulations were carried out at 300 K. The simulations consisted of three sequential stages, as depicted in Figure 1, considering Lees–Edwards boundary conditions [38] in a triclinic simulation box, with initial tilts set to zero. In the first stage, pure shear was applied to the simulation box in the positive x-direction, relative to the z-direction, deforming it at a rate of ~10⁸/s until a shear strain ε_xz = 25% was reached. Then, with the same deformation rate, the strain was reversed, and the simulation box was brought back to its original shape. Finally, in the last stage, also at a deformation rate of ~10⁸/s, the system continued to be sheared in the negative x-direction, ending with a strain ε_xz = −25%.

It should be mentioned that this unrealistic deformation rate (~10⁸/s) is a consequence of the short timescales accessible in MD simulations. It implies that the deformation of the BMG occurs primarily in an athermal regime where the thermal activation of plasticity events is mostly suppressed, and the deformation is almost entirely stress-driven. As a result of the very short simulation times, the transitions between LAEs captured in the MD simulations, both at rest and under shear application, represent only a low-energy barrier subset of the possible transitions. Further investigation, using methods capable of capturing transitions involving high energy barriers (e.g., NEB [39] or ART [40]), will be necessary to effectively map the transitions that may occur over longer timescales and lower strain rates.

The primary tools for analyzing the atomic trajectories generated by the MD simulations were the computations of nonaffine atomic displacements and Voronoi polyhedra. The latter is particularly important since it is the typical method for characterizing LAEs in MD simulations of amorphous systems. In the MD simulations, each Cu and Zr atom was designated as the center of a Voronoi polyhedron, with its nearest neighbor atoms associated with the faces of the polyhedron. Throughout the simulations, not only the coordinates but also the Voronoi polyhedra were recorded at intervals Δt = 20 ps. This enabled us to monitor the changes in the LAEs as the simulations progressed, thus cataloging the transitions between the polyhedra under dynamic conditions. Both analyses, along with other visualization methods of the simulated structures, were conducted using the corresponding features implemented in the Ovito software, version 3.10 [41].

3. Results and Discussion

3.1. Cu₆₆Zr₃₄ Metallic Glass at Rest

An initial analysis of the distribution of the LAEs was carried out over 200 ps of an MD simulation of the Cu₆₆Zr₃₄ BMG in the NVT ensemble at T = 300 K. Under this condition of thermal equilibrium with the surroundings, with the metallic glass maintained at rest, the average proportions of Voronoi polyhedra remained roughly constant throughout the simulation, as one would expect. The four most common polyhedra for each type of atom are listed in

Table 1. Average percentage of the predominant Cu- and Zr-centered Voronoi polyhedra in the Cu₆₆Zr₃₄ BMG at rest.

Cu-Centered		Zr-Centered
Polyhedra	%	Polyhedra	%
<0,0,12,0,0>	17.1	<0,1,10,5,0>	9.2
<0,1,10,2,0>	11.3	<0,2,8,6,0>	7.3
<0,2,8,2,0>	10.0	<0,1,10,4,0>	7.1
<0,3,6,4,0>	9.1	<0,2,8,5,0>	5.7

, along with their corresponding average percentages at the end of the simulation. It can be observed that the four predominant LAEs constituted almost half of the Cu-centered LAEs, with the icosahedra being the most numerous, making up about 17%. On the other hand, the four most common Zr-centered LAEs collectively account for less than 30%. It is worthwhile to mention that the amount of Zr-centered icosahedral LAEs was negligible (~0.02%).

As previously mentioned, icosahedra are expected to play a significant role in the mechanical properties of metallic glasses, with regions rich in these polyhedra being more resistant to shear banding. A local analysis of the distribution of Cu-centered icosahedral LAEs was conducted for the BMG at rest, considering subvolumes of 5 nm³ (containing between 4800 and 5200 copper atoms) at different moments during the simulation. Overall, it was observed that within each of these subvolumes, the number of icosahedra remained highly stable throughout the MD simulation, with a variation of less than 3% around the mean. When comparing different subvolumes, the average percentage of icosahedra ranged between 16% and 19%, which is quite close to the global average of 17% reported above. This indicates a reasonably homogeneous distribution of icosahedral LAEs within the Cu₆₆Zr₃₄ metallic glass at rest.

To assess the stability of the LAEs, the population of polyhedra was monitored by comparing the atomic environments of all Cu and Zr atoms between every two consecutive simulation snapshots, taken 20 ps apart. On average, approximately 64% of the copper atoms and no less than 78% of the zirconium atoms changed their LAEs during this short period of time. These intense transitions between LAEs within the Cu₆₆Zr₃₄ amorphous alloy is a striking contrast to the long-standing LAEs found in crystalline materials. Given the low temperature and short simulation duration, diffusion was not expected to play a significant role in these transformations. Indeed, our analysis confirmed that about 99.8% of the nonaffine atomic displacements between consecutive snapshots were less than 0.1 nm. This implies that the structure of the BMG should be characterized by a statistical distribution of polyhedra that are not static but continuously evolve due to short-range atomic position fluctuations.

Next, we examine in greater detail the changes that the predominant Cu- and Zr-centered LAEs underwent between two consecutive MD simulation snapshots. These findings are summarized in

Table 2. Average percentage of the predominant Cu- and Zr-centered LAEs that remained unaltered in the Cu₆₆Zr₃₄ BMG at rest between consecutive MD simulation snapshots.

Cu-Centered		Zr-Centered
Polyhedra	%	Polyhedra	%
<0,0,12,0,0>	77.5	<0,1,10,5,0>	38.9
<0,1,10,2,0>	54.6	<0,1,10,4,0>	35.8
<0,2,8,2,0>	40.5	<0,2,8,5,0>	29.7
<0,3,6,4,0>	39.7	<0,2,8,6,0>	29.1

. One can observe that, on average, less than 1/4 of Cu atoms with an icosahedral LAE transitioned to a different polyhedron every 20 ps. Over the same time interval, approximately 60% of Cu atoms with <0,3,6,4,0> and <0,2,8,2,0> LAEs were found to have shifted to a different local environment, on average. In the case of Zr atoms, when considering only those associated with the four most common polyhedra, their stability was even lower. On average, between 60 and 70% of these Zr atoms did not retain their LAE between consecutive simulation snapshots. These results indicate that icosahedra were by far the most stable LAEs in the metallic glass, exhibiting less sensitivity to atomic position fluctuations compared to other Cu- and Zr-centered polyhedra.

In Figure 2, we present a statistical picture of the most frequent transformations involving the Cu-centered icosahedra during the MD simulation of the metallic glass at rest. Considering that Cu atoms made up 2/3 of the alloy’s composition, one should expect the transformations undergone by the predominant Cu-centered polyhedra, particularly the icosahedra, to have a greater impact on the properties of the Cu₆₆Zr₃₄ amorphous alloy. Figure 2 should be interpreted as follows. In the center, the icosahedral LAE represents either the starting or ending point, depending on the direction of the arrows. If an arrow points from the icosahedral LAE to another LAE, it indicates that at time t, the copper atom had an icosahedral environment, and at time t + Δt, where Δt = 20 ps, it transitioned to a different LAE. Conversely, if an arrow points from another LAE to the icosahedral LAE, it means that the copper atom initially was enclosed within that atomic environment at time t, and at time t + Δt, it underwent a transformation into an icosahedral atomic environment. The numbers accompanying the arrows indicate, on average, how many times these transformations between the icosahedral and other LAEs occurred within the 20 ps interval between consecutive snapshots in the MD simulation.

One can see that, on average, 5.8% of the Cu atoms designated as having an icosahedral LAE at time t ended with a <0,1,10,2,0> LAE after 20 ps. For the opposite transition, 5.8% of the Cu atoms that ended with an icosahedral LAE after the same 20 ps were also atoms with a <0,1,10,2,0> LAE at time t. The equilibrium between forward and reverse transformations during the simulation was observed for all other pairs of polyhedra depicted as well, with very small variations. This is not surprising since the BMG was at rest and the number of forward LAE transformations must be balanced by the number of reverse transformations to prevent any net flux. Considering the predominant Cu-centered LAEs presented on Table 1 besides the icosaedral polyhedron itself, the transformations of icosahedral environments into/from <0,1,10,2,0> and <0,2,8,2,0> environments (with an average frequency of 5.8% and 5.4%, respectively) were much more common than transformations into/from the <0,3,6,4,0> environment (which occurred with a frequency of only 0.6%). Notably, the most common transformation occurred between the icosahedral LAEs and the <1,0,9,3,0> polyhedra. On average, every 20 ps, just under 7% of the icosahedral LAEs transformed into <1,0,9,3,0> LAEs or originated from these LAEs in the preceding time step. As a percentage of the total Cu-centered polyhedra, the <1,0,9,3,0> LAEs accounted for approximately 3%. It has been observed to be one of the less stable Cu-centered LAEs in the Cu₆₆Zr₃₄ amorphous alloy, as around 80% of the <1,0,9,3,0> LAEs, on average, underwent transformations into other polyhedra every 20 ps, primarily into <0,1,10,2,0> (approximately 20%) and <0,0,12,0,0> (approximately 40%). Along with the <0,1,10,2,0> and <0,2,8,2,0> polyhedra, the <1,0,9,3,0> polyhedra have fewer five-edged faces than the complete fivefold <0,0,12,0,0> polyhedral, but they still are structurally closer to these than the <0,3,6,4,0> polyhedra. Although direct transformations between <0,3,6,4,0> LAEs and icosahedral LAEs were rare in the MD simulation of the metallic glass at rest, it is worth noting that the former exhibited significantly more transitions with the <0,1,10,2,0> and <0,2,8,2,0> LAEs, averaging 6.7% and 6.6%, respectively. This observation suggests that these LAEs may have served as intermediate atomic environments between the <0,3,6,4,0> and icosahedral LAEs at various points during the simulation, highlighting the complexity of tracking these structural local transformations over extended periods. However, considering that approximately 22.5% of the Cu-centered icosahedra transformed to or originated from other polyhedra every 20 ps of the MD simulation, about 80% of these transitions directly involved the <1,0,9,3,0>, <0,1,10,2,0>, and <0,2,8,2,0> LAEs. Thus, it is possible to characterize the icosahedral environments in terms of greater stability and a strong preference for the specific polyhedra they transform into or originate from when the BMG is not under external load, clearly distinguishing them from the higher randomness observed in other LAEs.

3.2. Cu₆₆Zr₃₄ Metallic Glass under Pure Shear

MD simulations were performed in which the BMG was subjected to pure ε_xz shear in three steps, as illustrated in Figure 1. During the first step, the simulation box was sheared in the positive x-direction, relative to the z-direction, starting from its initial state at rest, up to a strain of 25%. Because pure shear was applied, the volume remained constant, and only the shape of the simulation box was altered. Figure 3a presents the corresponding stress–strain curve. It can be seen that the metallic glass deformed elastically up to a strain of approximately 6%. Even before yielding, a significant number of atoms (up to 1.3%) exhibited nonaffine displacements greater than 0.1 nm, in contrast to the behavior of the BMG in its undeformed state. Previous studies have shown that large nonaffine displacements are correlated with indicators of plasticity in amorphous alloys, such as the degree of local fivefold symmetry (LFFS) [42]. However, it should be mentioned that localized plasticity in what is nominally the elastic regime has been already observed in metallic glasses [43,44], highlighting the peculiarities of the structural–mechanical property relation of these materials. Hereafter, we use the x-component of the nonaffine atomic displacements, which is parallel to the applied shear, to characterize the effects of deformation in the metallic glass. For the remainder of this section, atoms that displaced more than 0.1 nm in either the positive or negative x-direction will be referred to as having large positive displacements (LPDs) or large negative displacements (LNDs), respectively.

It is worth noting that, during deformation in the nominally elastic regime, among the relatively few copper atoms exhibiting LPD or LND (less than 0.5% of the total just before yielding), less than 3%, on average, had an icosahedral LAE, compared to ~17% of the total copper atoms in the BMG. This demonstrates the elevated resistance of icosahedral LAEs to localized strains associated with large nonaffine atomic displacements. In this same set of copper atoms, the <0,1,10,2,0> and <0,2,8,2,0> polyhedra were also under-represented when their proportions were compared to those of the entire system, whereas the <0,3,6,4,0> polyhedra were present in slightly larger proportions. Before yielding, these LPD and LND atoms exhibited a much higher proportion of less common and more unstable LAEs, such as <0,2,8,1,0>, <0,4,4,4,0>, and <0,3,6,3,0>. The onset of plastic flow, however, was accompanied by a gradual increase in the percentage of atoms with the most common Cu-centered LAEs among those with LPD or LND. For instance, by the end of the first shearing stage, the percentage of LPD and LND Cu atoms with icosahedral LAEs (13.2%) was much closer to that observed in the entire BMG (14.6%).

The onset of yielding originated from the formation of an increasing number of LPD and LND atom clusters, the majority of which (>80%) were Cu atoms. The maximum stress of |σ_xz| = 1.2 GPa occurred at a strain of ~12%, after which these clusters gradually coalesced, leading to the emergence of three distinct regions within the simulation box for strains larger than ~15%. The first region was a band-like region where most atoms exhibited LPD, i.e., they were displaced in the direction of the applied shear. There was a second band-like region where the majority of nonaffine atomic displacements occurred in the opposite direction and were larger than 0.1 nm. Both regions can be clearly visualized in Figure 3a, when the strain reached 25%, as two parallel blocks. The third region was found separating the previous two, and it was characterized by random nonaffine atomic displacements in either direction. Band formation was associated with softening, and, by the end of the first step, at 25% strain, the absolute stress had decreased to approximately 0.9 GPa. Additionally, around 34% of the atoms had nonaffine displacements greater than 0.1 nm in either the positive or negative x-direction compared to their positions in the initial BMG state at rest.

In the second and third steps, illustrated in the stress–strain curve shown in Figure 3b, the direction of the applied shear was reversed to the negative x-direction. A reduction in strain from 25% to ~20% brought the shear stress in the simulation box to zero. Beyond this point, as the simulation box continued to be sheared, the stress began to increase, and, just before the simulation box returned to its initial shape (zero strain), the metallic glass entered a regime of near-constant flow stress, which saturated around 0.9 GPa. Back to zero strain, the second step was completed, and it was observed that the band-like region, where most atoms presented LPD (therefore, displaced in the direction opposite to the direction of applied shear), was still present in the simulation box but had visibly thinned. The LND band-like region, on the other hand, vanished, although large clusters of LND atoms were seen to form in another part of the simulation box. Using the initial resting state of the metallic glass as a reference, approximately 34% of the atoms exhibited LPD or LND. The continued shearing of the simulation box in the negative x-direction ultimately led to the reconstruction of the two band-like regions characterized by large nonaffine atomic displacements in opposite directions. These bands continued to thicken until strain reached ε_xz = −25% (end of third step), at which point they became well-defined blocks within the simulation box. With the resting state of the metallic glass still as a reference, the percentage of LPD and LND atoms increased to 64%.

An effect of shear manifested in the destabilization of the Cu-centered icosahedral atomic environments. In Figure 4a, it can be seen that even before yielding, the percentage of icosahedra that remained unchanged between consecutive snapshots of the MD simulation showed a consistent, albeit slight, decline, from nearly 78% to approximately 75.5%. Once the metallic glass began to yield, this percentage of unchanged icosahedra decreased to just above 70% when the strain reached ~15%. At this level of strain, where banding started to form, the decline in the percentage of unchanged icosahedra between consecutive MD snapshots started to slow down. As can be seen in Figure 4b, this trend can be directly associated with the transformations of the <0,0,12,0,0> polyhedra into the preferred LAEs, which remained the same as the BMG at rest. Shear deformation promoted an increase in the frequency of transformations from icosahedral polyhedra to <0,1,10,2,0> and <0,2,8,2,0> polyhedra, rising from 5.8% to approximately 7% and from 5.6% to around 6.5%, respectively. Transformations to <1,0,9,3,0> polyhedra, which were the most common in the BMG at rest, showed a slight increase, from just below to slightly above 7%. Finally, the frequency of transitions from icosahedra to <0,3,6,4,0> LAEs, which was about 0.6% in the metallic glass’s resting state, reached more than 1% by the end of the first stage of shear deformation. The effect of shear was more pronounced in the initial phase of deformation, still within the elastic regime and shortly after the onset of plastic flow, before band formation. Banding can be linked to the deceleration in the increase in transition frequencies for the <0,1,10,2,0>, <0,2,8,2,0>, and <0,3,6,4,0> polyhedra, which displayed a tendency to saturate in the second half of the first shear step.

Next, we examine how the quantity of icosahedral LAEs and the other predominant Cu-centered polyhedra was affected by shear deformation. Figure 5 illustrates the overall variations in the percentages of the four predominant Cu-centered polyhedra throughout the three stages of shear application. Points related to significant phenomena that occurred during shear deformation, such as the onset of yielding and the initial formation of bands, were also marked on the curves. Of the four most common polyhedra, <0,2,8,2,0> and <0,3,6,4,0> were relatively insensitive to deformation, showing only minor variations and generally retaining nearly the same proportion in the Cu₆₆Zr₃₄ amorphous alloy. In contrast, the most abundant type of LAE, the Cu-centered icosahedral polyhedra, showed a much more pronounced decrease during the first stage of shearing, dropping from 17.1% to 14.6%. This decline in the number of icosahedra was steeper in the strain range corresponding to higher stresses, between the onset of yielding and before the formation of bands. When the direction of shear was reversed, the amount of icosahedral LAEs experienced a brief recovery, associated with stress relief, up to 15.7% as the strain went from 25% to about 17%, before declining again to around 14.9% when the strain in the simulation box reached zero. During the third stage, a slight but steady decrease in the number of Cu-centered icosahedra was observed, ultimately reaching 14.2% at ε_xz= −25%. Regarding the second most abundant Cu-centered LAEs, <0,1,10,2,0>, we observe similar trends to those of the icosahedra, but with much less variation in their proportion within the BMG, decreasing from around 11.3% (undeformed) to approximately 10.5% at ε_xz= −25%. For the sake of completeness, regarding the Zr-centered LAEs, all of the other most common polyhedra listed in Table 1, except for <0,2,8,5,0>, experienced some depletion due to deformation, mainly during the first step of shearing. However, only the <0,1,10,5,0> LAE showed a significant variation (reduction from 9.2% to 7.6%) compared to the BMG’s resting state.

The destruction of Cu-centered icosahedra depicted in Figure 5 implies, of course, that a certain amount of these LAEs transformed into other LAEs. The effect of shear deformation on the frequency of this kind of transition was shown above, in Figure 4b. Conversely, icosahedra creation also took place in the MD simulations, meaning that other LAEs have transformed into them. We have previously shown that Cu-centered icosahedra frequently transformed into and from other LAEs in the course of MD simulations, even when the metallic glass is not subjected to any external load. The ratio between the number of icosahedra destroyed and created between consecutive MD snapshots (Δt = 20 ps) of the BMG under shear allows us to characterize the dominant types of local structural transformations involving these polyhedra during the deformation process. If we track the total forward and reverse transitions between icosahedra and other LAEs from one MD snapshot to the next, a ratio greater than 1 indicates the net destruction of icosahedra, while a ratio less than 1 indicates their creation at the expense of other LAEs. If it is exactly 1, there is no net destruction or creation of icosahedra with respect to another polyhedra. The destruction-to-creation ratio of icosahedra between consecutive MD snapshots with respect to the most relevant polyhedra—<1,0,9,3,0>, <0,1,10,2,0>, <0,2,8,2,0>, and <0,3,6,4,0>—showed significant fluctuations, making it difficult to identify clear trends. To address this, we performed a statistical analysis, grouping and averaging the data on the ratio of destroyed and created icosahedra into relevant strain ranges that correspond to key stages in the deformation history of the amorphous alloy. The results are shown in Figure 6.

Consistent with the findings illustrated in Figure 4 and Figure 5, and also with the stress–strain relationship shown in Figure 3 and Figure 6, our study reveals that the destabilization of Cu-centered icosahedra is also related to an unfavorable (i.e., greater than 1) average destruction-to-creation ratio of icosahedra for all predominant transitions, with the noticeable exception of the transitions to/from <0,1,10,2,0> LAEs. The strain range where the BMG began to yield, just before banding occurred, which corresponds to the highest stresses (see Figure 3a), was the one with the most pronounced asymmetry in the destruction and creation of icosahedra. This was also the only strain range in which the average ratio of <0,0,12,0,0> to/from <0,1,10,2,0> transformations was noticeably larger than 1. For other strain ranges, the number of forward and reverse transitions involving these LAEs were, on average, the same and, in one case (during icosahedra recovery), more <0,1,10,2,0> polyhedra transformed into icosahedra than the inverse. This suggests that these transformations made only a minor contribution to the observed net destruction of icosahedra. Although the destruction of icosahedra in favor of <0,3,6,4,0> is notable, as it had the highest ratios, this was the least frequent transition involving icosahedra (barely exceeding 1%). Thus, its impact on the overall net destruction of icosahedra was less significant compared to other transitions. As depicted in Figure 5, icosahedra recovery occurred when the shear strain was reversed, reaching a peak when the strain dropped from 25% to ~17%. Figure 6 shows that this recovery can be attributed to the reversal of the average destruction-to-creation ratios, which fell below 1 for all transitions. It is clear that the overall asymmetry between icosahedra destruction and creation shown in Figure 6 was quite small, with an average ratio that barely reached 1.01 for the most important transitions (those involving the <1,0,9,3,0>, <0,1,10,2,0>, and <0,2,8,2,0> LAEs) per MD step. Moreover, these effects impacted only a small fraction of the icosahedra population, as at least 70% of these LAEs remained unchanged between consecutive MD snapshots. Nevertheless, the cumulative effect of the asymmetry in the forward and reverse transitions to and from the <1,0,9,3,0>, <0,2,8,2,0>, and, to a lesser extent, <0,1,10,2,0> polyhedra throughout the MD simulations was the primary cause of the depletion of icosahedra observed in the Cu₆₆Zr₃₄ BMG under pure shear since other transitions involving the icosahedra were much less frequent.

4. Conclusions

In summary, this article presented a molecular dynamics study on the evolution of LAEs, characterized as Voronoi polyhedra, in the Cu₆₆Zr₃₄ metallic glass at rest and under pure shear, with a focus on the Cu-centered icosahedral LAEs due to their well-known role on the mechanical properties of these materials. In the amorphous alloy at rest, the four most prevalent Cu-centered LAEs account for nearly half of all LAEs, while the four most common Zr-centered LAEs account for less than 30%. Even in the resting state, the polyhedra underwent frequent transitions from one to another. However, their proportions remained constant over time because the number of forward transitions was roughly equal to the number of reverse transitions. Among the LAEs, the Cu-centered icosahedral ones were the most stable, showing a higher persistence rate over time and predominantly undergoing transformations to and from a small set of polyhedra.

The BMG was sheared first in the positive x-direction and then in the negative x-direction, with the strain reaching up to 25% in both cases. Yielding was associated with large nonaffine atomic displacements in these directions, taking the BMG at rest as the reference state. Regions with high concentrations of atoms exhibiting large displacements formed bands within the metallic glass. Shear deformation destabilized the icosahedral polyhedra, increasing the frequency of transformations of these LAEs into or from other types of LAEs. The dominant polyhedra in the metallic glass in the absence of load remained dominant when the glass was subjected to shear deformation, with only slight variations in their proportions. An exception was the Cu-centered icosahedral polyhedra, which experienced a greater decline (from 17.1% up to 14.2%) compared to other LAEs. This reduction in the number of icosahedra resulted primarily from the net transformations of icosahedra into <1,0,9,3,0> and <0,2,8,2,0> polyhedra, with a minor contribution of transformations involving the <0,1,10,2,0> polyhedra.