It has been well known that the discovery of dislocation and the development of dislocation theory established the relationship between structure and mechanical properties of crystalline materials, such as the structure origin of strength and plasticity. However, for amorphous solids, such as metallic glasses, due to the disordered nature and lack of a structural indicator such as dislocation in crystalline counterparts, it is difficult to define a local geometrical parameter as energy dissipative units for understanding the microscopic mechanism of mechanical behavior ^{[1], [2], [3] and [4]}.

The model of shear transformation zone (STZ) proposed by Argon provides a dynamical picture for the understanding of deformation mechanism in metallic glasses ^{[5], [6] and [7]}. Falk and Langer introduced mean-field equations of motion for the number density of STZs and their two-state transition ^[6]. All these efforts have been successful in explaining many phenomena, such as shear localization and emergence of yield in amorphous solids ^{[2] and [3]}. However, STZ theory did not provide any information on the specific structural feature accounting for plastic domains in amorphous solids. Based on the potential energy landscape theory and Frenkel's analysis for the shear strength of crystal solids, Johnson and Samwer introduced a cooperative shearing model of STZs for predicting the temperature dependence of yield strength of metallic glasses ^[8], correlating STZs with potential energy barrier and thus providing a simple explanation of plasticity of metallic glasses at low temperatures.

So far, a couple of structure indicators have been proposed for understanding the mechanical behavior in metallic glasses, such as free volume and icosahedral short-range order ^{[9], [10] and [11]}. Based on the concept of free volume envisioned by Turnbull and Cohen ^[12], the free volume is defined as excess atomic volume surrounding each atom in amorphous states. Based on a competition between creation and annihilation of free volumes in metallic glasses driven by shear stress, Spaepen developed a steady-state inhomogeneous flow model to investigate the mechanical behavior of metallic glasses ^[8]. Because of simple and clear physical picture, free volume model has been widely adopted to understand various mechanical properties in metallic glasses ^{[2], [4] and [13]}. However, as a structural indicator, free volume is a statistical or average variable, unable to provide specific structure information of the atomic symmetry or chemical ordering of local environments ^[11]. On the other hand, free volume model did not provide a clear picture for the atomic rearrangements during mechanical deformation ^{[4] and [11]}.

Although amorphous alloys lack of long-range order, it has been demonstrated that metallic glasses contain significant short-range order ^{[10], [14] and [15]}. Icosahedral short-range order (ISRO) with five-fold symmetry has been suggested as the most favorable local structural unit in metallic liquids and glasses, because of its specific close-packed atomic arrangement ^[16]. So far, ISRO and its spatial distribution have been found to closely correlate with structural heterogeneity, glass-forming ability, dynamical and mechanical properties in metallic liquids and glasses ^{[11], [14], [17], [18], [19], [20], [21] and [22]}. Therefore, ISRO is also adopted as a structural indicator for the fundamental process underlying structural relaxation and atomic rearrangements ^{[11] and [23]}. The effect of ISRO characterized by icosahedral clusters on the mechanical behavior of Cu–Zr-base metallic glasses has been widely investigated with computer simulations and found that ISRO is responsible for the local stiffness, yield strength and mechanical stability ^{[13], [24] and [25]}. However, the so-called ISRO is mainly represented by the icosahedral clusters characterized in terms of Voronoi tessellation method ^[14]. Most icosahedra in metallic glasses are not perfect, but distorted ^[21]. Perfect icosahedron is five-fold symmetric. However, the distorted icosahedra even contain partial face-centered cubic symmetry ^[21]. The local atomic packing of perfect and distorted icosahedra may not be the same, either. Thus, they may exhibit significantly different mechanical behavior. On the other hand, ISRO may not be the representative of main local structures or may be even absent in some metallic glasses ^{[14], [26] and [27]}.

In fact, local atomic symmetry may be a more general concept in metallic liquids and glasses. Recently, direct experimental evidence for the existence of five-fold symmetry has been obtained in liquid Pb ^[28]. Nuclear magnetic resonance (NMR) experiments have also shown that local cluster symmetry plays an important role in glass-forming ability (GFA) and anelastic deformation in metallic glasses ^{[29] and [30]}. Recent statistical analysis of pair distribution functions of many metallic glasses shows the local translational symmetry feature in atomic packing in metallic glasses ^[31]. Even for the icosahedral clusters in metallic glasses, they may contain both local five-fold symmetry and face-centered cubic symmetry ^[21]. In addition, the degree of local five-fold symmetry in local structures has been demonstrated to correlate with the initiation and propagation of plastic deformation in metallic glasses ^[32]. Therefore, local atomic structures in metallic glasses may contain both local five-fold and translational symmetries.

For an atomic cluster, the structural configuration can be characterized by the Voronoi tessellation which divides the space between central atom and its nearest neighbors by construction of bisecting planes along the lines joining the central atoms and all its neighbors. The constructed polyhedron can be identified by the Voronoi index < n₃, n₄, n₅, n₆, …>, where n _i ( i = 3, 4, 5, 6, …) denotes the number of i-edged faces of the polyhedron. Apparently, an i-edged face in a Voronoi polyhedron reflects the atomic packing nature and local atomic symmetry of the central atom with some nearest neighbor atoms in the direction normal to the face. Therefore, the triangle, tetragon, and hexagon faces should reflect the local translational symmetry feature, while pentagon faces reflect the local five-fold symmetry. Computer simulations for some realistic metallic glasses have demonstrated that the evolution of the fraction of pentagons with temperature or shear stress is totally different from that of other type faces ^{[24] and [33]}, implying the competition and transformation between local five-fold symmetry and local translational symmetry in local structures during glass formation or deformation process. Therefore, local atomic symmetry could be a more general structural indicator in metallic liquids and glasses for better understanding the structure-property relationship in amorphous alloys.

Currently it is difficult to measure the local atomic symmetry in metallic glasses in experiments. It is not easy for numerical analysis, either. Several structural parameters have been proposed to characterize the atomic symmetry feature for local atomic clusters, such as bond-orientational order ^[34], common-neighbor analysis ^[35], and Honeycutt–Andersen (HA) index ^[36]. Common-neighbor analysis and HA index are quite similar. Steinhardt et al. proposed bond-orientational order based on spherical harmonics to quantify the bond-orientational symmetry around the central atom ^[34]. The normalized parameter Wˆl( l = 2, 4, 6, 10) was derived, which is able to differentiate various local atomic clusters. Different atomic clusters may correspond to different values of Wˆl. For example, Wˆ6 values of fcc, bcc, hcp, and icosahedron clusters are -0.01316, 0.01316, -0.01244, and -0.16975, respectively. For local structures in metallic glasses, however, Wˆ6 exhibits a broad distribution between -0.2 and 0.1 ^[20]. Thus, it is not so clear how to quantify the local atomic symmetry for an atomic cluster with irregular configuration in the frame of bond-orientational order.

HA index was proposed by Honeycutt and Andersen by using four integers ijkl to characterize local atomic arrangements ^[36]. The first index denotes the root pair of two atoms. i = 1 if two atoms are the nearest neighbors, otherwise i = 2. The second one represents the number of near neighbors shared by the root pair. The third index denotes the number of the nearest neighbor bonds formed among the shared neighbors. The fourth index is used to distinguish configurations with the same first three indices but with different topologies. Typically the first minimum of the pair correlation function is taken as the distance cutoff to determine whether a bond is formed between root pairs or two common nearest neighbor atoms. This method is able to distinguish between various local structures like fcc, hcp, bcc, and icosahedral environments. In most analyses, only the case of i = 1 is often considered. Thus, the 1551 index represents a five-fold ring of common neighbors with all nearest neighbors bonded. 1541 or 1531 corresponds to the five-fold ring with one or two bonds broken because the distance between them is beyond the cutoff. The types of 1661, 1651, 1441, 1421, and 1422 are present in the bulk bcc, fcc, and hcp crystals, respectively. For an atomic cluster, HA index characterizes the local atomic packing and symmetry feature between the central atom and a part of nearest neighbor atoms, does not provide full atomic symmetry information for an atomic cluster.

To quantify the local atomic symmetry of an atomic cluster, we simply define the degree of local five-fold symmetry (LFFS) in a cluster as the ratio of the number of pentagons to the number of the nearest neighbor atoms ^[29],

d5=n5∑i=36ni (1)

As mentioned above, n _i ( i = 3, 4, 5, 6) denotes the number of i-edged face of a polyhedron. In this work, we used the degree of LFFS, d₅, as structural indicator to investigate structural signature of plastic deformation in metallic glasses and the relationship between normal modes and local atomic symmetry.

Molecular dynamics (MD) simulation was performed for the model system of Cu₅₀Zr₅₀ alloy to generate metallic liquid and glass samples and conduct deformation for metallic glass sample ^[37]. For Cu–Zr alloy system, a realistic embedded-atom method (EAM) potential was chosen for the interatomic interactions, which has been proved to be reliable and accurate to describe the structural and mechanical properties ^[38]. The sample is cubic cell and contains 40,000 atoms with periodic boundary conditions. In the process of sample preparation, it was first melted and equilibrated at T = 2000 K, then cooled down to 300 K with various cooling rates. During cooling the cell size was adjusted to give zero pressure in Isothermal–Isobaric (NPT) ensemble. In the MD study of deformation, compressive deformation was applied to the generated metallic glass samples along Y axis with a strain rate of 0.25/ns at T = 300 K and the period boundary conditions on three dimensions. After each compression of 0.0111 nm, the structure was relaxed for 10 ps (5000 MD steps).

In a periodic lattice the deformation is affine under a uniform strain. In disordered materials, however, the displacements are not affine even under uniform strain. The non-affine displacements of atoms in metallic glasses under deformation are associated with the plastic deformation. To evaluate the irreversible atomic rearrangement of an atomic cluster under deformation, the non-affine displacement of the central atom relative to its neighbor atoms is calculated ^[6]:

Di2(t,Δt)=1N∑j[rj(t)-ri(t)-ε⇀(rj(t+Δt)-ri(t+Δt))]2 (2)

where idenotes the central atom, and jruns over the nearest neighbor atoms of central atom i( Ndenotes the number of the nearest neighbors of each atom) defined by Voronoi tessellation analysis, r _i( t) is the position of atom iat time t, Δ tis the time interval for the plastic rearrangement, and ε→is the maximum local elastic strain tensor ^[6]. Here Δ t= 10 ps was used in the calculations.

Fig. 1 clearly demonstrates that the response of the metallic glass structure to the compressive deformation is inhomogeneous, and the inhomogeneous deformation is highly localized. This also implies that the local structural environments of a metallic glass and their irreversible rearrangements are quite different. Furthermore, the regions with bigger values of D² are expanding with strain increasing. This indicates that the metallic glass structures are correlated with the strain localization in deformation.

Fig. 1.

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	Fig. 1. Atomistic configurations of the non-affine displacement D2 at the strain of 5% with time interval of Δ t = 10 ps (a) and 40 ps (b). The color indicates the values of non-affine displacement each atom experiences. Reproduced from Ref. [32].

On the other hand, non-affine displacement D² is closely correlated with the degree of LFFS d₅ of atomic clusters as shown in Fig. 2. It is clearly seen that D² decreases as d₅ increases in local structures. In contrast, as the fraction of triangle, tetragon, or hexagon faces increases in atomic clusters, the non-affine displacements increase. Therefore, the non-affine displacement often takes place in the regions with more degree of the local translational symmetry, or with less degree of LFFS. The higher the degree of LFFS, the lower the non-affine displacement is in these regions. Such a correlation between irreversible atomic rearrangements and the degree of LFFS is also illustrated clearly in Fig. 3 where the degree of LFFS and the spatial distribution of plastic events are compared. It is clearly seen that the irreversible atomic rearrangements during deformation are trying to avoid the local structures having higher degree of LFFS, preferring to occur in regions with less degree of LFFS. This is consistent with the above analysis. Therefore, the structural indicator of LFFS in metallic glasses can simply capture the local structural features responding to the plastic deformation.

Fig. 2.

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	Fig. 2. Dependence of non-affine displacements on the fraction of i-edged faces in Voronoi clusters. Reproduced from Ref. [32].

Fig. 3.

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Fig. 3. Correlation between local structures with more degree of LFFS and the irreversible atomic rearrangement during deformation. A slice in the middle of the samples along Z direction with a thickness of the first minimum of the pair correlation function was taken for the illustration. Black points represent the atoms having more degree of LFFS ( d5 ≥ 0.5) at the strain of 5% (a) and 10% (b). The red and blue areas are the mostly deformed regions emerged in the time interval of 40 ps and the less deformed regions, respectively. Reproduced from Ref. [32].

The plastic deformation is initiated in the regions with less degree of LFFS. As strain increases, the plastic deformation is propagating in metallic glasses. Fig. 4 shows the distribution of the average d₅ with the non-affine displacement. For smaller and moderate D², the averaged d₅ is relatively higher. d₅ decreases exponentially as D² increases. For very large D², however, averaged d₅ is almost not changed with it. Therefore, in the early stage of deformation, the local structures response to it accordingly, lowering the degree of LFFS in them. As they are deformed to some extent, the local structures reach a saturated situation in which the degree of LFFS cannot be reduced anymore. Further deformation will be transformed into the less deformed regions where the degree of LFFS should be relatively higher. Therefore, the behavior of plastic deformation propagation is also correlated with the local atomic symmetry. Fig. 5 illustrates the propagation trend of plastic events. Here Dc2 is defined as critical value of non-affine displacements to justify whether a local structure experiences a plastic deformation (D2>Dc2) or not (D2<Dc2). With this threshold, one may examine how many plastically deformed local structures are emerging in the time intervals of ( t, t + iΔ t, i = 1, 2, 3, …), and then examine the structural properties of the newly deformed local structures in the time intervals and analyze the propagation trends of the plastic events during deformation. As shown in Fig. 5, with strain increasing, the plastic events take place gradually from the regions with less degree of LFFS toward those with relatively more degree of LFFS. In short time interval, the plastic events tend to take place in the regions with less degree of LFFS. However, as strain increases, the plastic events will finally become saturated in those regions, and have to propagate to the regions with higher degree of LFFS.

Fig. 4.

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	Fig. 4. Change of the averaged LFFS d5 with non-affine displacement D2. Reproduced from Ref. [32].

Fig. 5.

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Fig. 5. Variation of the averaged LFFS d5 with strain increasing for different values of $D_c^2$. Reproduced from Ref. [32]. For different choice of $D_c^2$, the propagation trend of the plastic events is consistent as shown in Fig. 3(a). However, as $D_c^2$ increases, the average degree of LFFS d5 decreases. This also implies that the larger plastic events tend to take place and propagate in the regions with less degree of LFFS.

The local atomic symmetry also closely correlates with the yield strength of metallic glasses. Cheng et al. ^[25] have demonstrated by MD simulation study that both the yield strength and the fraction of icosahedral clusters in metallic glasses increase with cooling rate getting slower. Therefore, it is expected that the higher the LFFS in metallic glasses, the larger the yield strength becomes. Fig. 6 shows the strain–stress curves for samples obtained through different cooling rates. The yield strength increases as cooling rate decreases, consistent with previous results. The analysis of LFFS for these samples shows that the degree of LFFS is increasing as cooling rate is decreasing. Fig. 7 shows that the more the degree of LFFS a metallic glass contains, the higher the yield strength of the metallic glass is. The yield strength increases almost linearly with the degree of LFFS in metallic glasses. Therefore, the degree of LFFS directly correlates with the yield strength of metallic glasses.

Fig. 6.

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	Fig. 6. Strain–stress curves of samples obtained through different cooling rates under deformation.

Fig. 7.

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	Fig. 7. Correlation between the yield strength and the average degree of LFFS measured with different cooling rates (CR). Reproduced from Ref. [32].

In supercooled liquids and glasses an anomalously enhanced soft low-frequency modes are present in vibrational density of states (VDOS) compared to the Debye squared-frequency law in crystals, which is also called boson peak. Plenty of experimental and theoretical work have been done to explore the nature of the soft low-frequency modes and the physical origin of the boson peak ^[39]. It has been demonstrated that the soft low-frequency modes are localized or quasi-localized. Laird and Schober found that these soft modes are localized around atoms whose neighborhood structure differs significantly from the average glass environment and might be associated with interstitial defects in crystals ^[40]. It was also found that the atoms with larger free volume contribute to boson peak over the Debye value ^[41]. Li et al. observed the effect of annealing and quenching processes on boson peak and assumed that the boson peak mainly originated from the random dense cluster-packing structure ^[42]. It is also found that the boson peak in glasses is equivalent to the transverse acoustic van Hove singularity in crystals ^[43]. Although it is still unknown what specific structures are related to the excess of the soft modes in disordered materials, the previous studies have indicated that they might be associated more with loosely packed local structures with less degree of LFFS. On the other hand, experiments on colloidal glasses and computer simulation studies have demonstrated that the low-frequency modes are spatially correlated with particle rearrangements, and thus the mechanical instability of amorphous materials ^{[44], [45], [46] and [47]}. To examine the correlation between local atomic symmetry and soft modes in metallic glasses may get deep insight into the deformation mechanism.

To obtain soft modes in metallic glasses, the vibrational density of state (VDOS) g( ω) should be analyzed. Here the direct diagonalization method was adopted, in which the steepest-descent method was carried out for the final configuration. The dynamical matrix corresponding to the potential energy minimum reached by LAMMPS line search algorithms minimization ^[37] is given by

Dij=1MiMj∂2U(x1,y1,z1,…,xN,yN,zN)∂Ri∂Rj (3)

where M _iis the mass of atom iand R _iis the coordinate x, y, or zof atom i. g( ω) can be calculated by directly diagonalizing the dynamical matrix as

g(ω)=13N∑λδ(ω-ωλ) (4)

where ω _λis the eigenfrequency of eigenmode λ.

From the diagonalization of the dynamical matrix of Eq.(3), we can obtain a set of eigenvalues and corresponding eigenvectors e _i, from which we can get the normal modes. These give the structure of the vibration, provide deep understanding of the dynamical excitations of a system, and can be also used to analyze the properties of the vibrations. The localization of vibrational modes can be evaluated by the participation ratio in the form of ^[40]

pλ=1N{∑i=1N[ei(λ)·ei(λ)]2}-1 (5)

It is a measure of the number of atoms participating in a given vibrational mode. p _λ ≈ 1/ N for localized modes, p _λ ≈ 1 for extended modes, and p _λ ≈ 2/3 for an ideal standing plane wave ^[44].

Fig. 8(a) and (b) shows a typical VDOS and PR of all frequency modes in metallic glasses obtained from the average of 10 independent structural configurations. In the very low and high-frequency regimes, PRs are quite small, indicating the normal modes in these frequency regimes are localized. It has been well known that high-frequency modes are truly localized ^[48]. However, some studies have demonstrated that the low-frequency modes are not localized, but quasi-localized. Although PRs of the low-frequency modes are small, they are spatially extended to some extent ^{[48] and [49]}.

Fig. 8.

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	Fig. 8. (a) Vibrational density of state in Cu–Zr metallic glass, (b) participation ratio of vibration modes.

To elucidate the relationship between local atomic symmetry and soft modes, we introduced a partial PR, p _m, to analyze the participation of atoms with different local atomic symmetry in soft modes ^[48],

pm=13∑i=1N∑λ=1n∑α=13eiα(λ)·eiα(λ)·δ(d5i-m) (6)

Here N is the number of atoms, eiα is the component of the eigenvector in direction α on atom i. n is the number of selected soft modes. d5i represents the LFFS value of atom i. m is a given value of d₅. Fig. 9 shows the dependence of partial PRs on LFFS d₅ for structures with different system size. As d₅ increases, the partial PRs decrease. As d₅ = 1, partial PRs reduce to the minimum. This indicates that the higher the LFFS of atoms, the less the atoms participate in soft modes, vice versus. Therefore, the local structures participated more in soft modes are mechanically unstable, which may be due to low LFFS of such local structures. On the other hand, even for the local structures with the largest LFFS, that is d₅ = 1, their partial PRs are not zero. This indicates that almost all local structures participate in soft modes. The local structures with smaller LFFS make much more contributions than those with larger LFFS. This feature is totally different from the high-frequency modes in which only small fraction of atoms participate. This also implies that almost all local structures response to external deformation, but with different behavior.

Fig. 9.

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	Fig. 9. Change of the partial PR of the selected soft modes with LFFS d5 for samples with number of atoms N = 5000 (a) 10,000 (b) and 100,000 (c) n = 300, 600, and 4404 are the selected soft modes in three samples, respectively.

To analyze the vibration properties of local structures with different degree of LFFS, local spectrum of these atoms were calculated ^[48]:

g(m,ω)=13∑i=1N∑λ=13N∑α=13eiα(λ)eiα(λ)δ(ω-ωλ)δ(d5i-m) (7)

Fig. 10 shows VDOS and reduced VDOS ( g( ω)/ ω²) of local structures with different values of d₅, respectively. As d₅ increases, the VDOS at low-frequency regime is getting lower, while VDOS at moderate- and high-frequency regime is getting higher. This indicates that the local structures with larger LFFS participate more in moderate and high-frequency modes. The reduced VDOS of different local structures shown in Fig. 10(b) indicates that boson peak is more related to the local structures with smaller LFFS, and therefore might be also correlated with mechanical properties of metallic glasses.

Fig. 10.

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	Fig. 10. VDOS (a) and reduced VDOS (b) of local structures with different LFFS d5.

We also investigated the relationship between the soft modes and irreversible atomic rearrangements in metallic glasses during deformation. To do this, the athermal and quasi-static shear deformation algorithm was applied. First the simulation cell is deformed by a small amount of strain Δ γ₀. Next the potential energy of the system is minimized with the shape of the simulation cell held. The minimization is terminated when no component of force on any atoms exceeds 10^-7 eV/nm. Here Δ γ₀ = 5 × 10^-5. Fig. 11 illustrates the comparison of the spatial distribution of non-affine displacements and soft modes. Here 600 lowest frequency modes were selected (2% of the lowest modes) and non-affine displacements were calculated in the time interval of Δ γ₀. The red areas represent the most deformed regions emerged in the time interval of Δ γ₀, while the blue areas are the less deformed regions. The black points represent the atoms with the largest values of PRs in the selected 600 soft modes. As shown in Fig. 11, it is clearly seen that the black points are highly consistent with the most deformed red regions, indicating that soft modes are closely correlated with the irreversible atomic rearrangements. Note that soft modes reflect information of static structures, while non-affine displacements reflect the response of local structures to external deformation. Fig. 11 indicates that the most deformed regions and the atoms participating in soft modes share similar structure feature. This is consistent with previous numerical analysis ^{[44], [45] and [47]}. Comparing Fig. 3 and Fig. 11, it is naturally expected that the atoms which participate significantly in the soft modes might exhibit low degree of LFFS, that is, high degree of local translational symmetry. These results are consistent with some experimental observations ^[43].

Fig. 11.

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Fig. 11. Correlation between the spatial distribution of non-affine displacements and the selected 600 (the lowest 2%) soft modes. The red area represents the regions with larger non-affine displacements, while the black spheres represent the atoms with large PR in the selected soft modes. (a–f) are the slices in the deformed sample at strain 10% extracted from different locations along z direction.

Local atomic symmetry represented by the parameter LFFS d₅ reflects specific atomic packing nature and symmetry information in local structural environments. In this sense, local atomic symmetry in local clusters should have included the information of potential energy and free volume. A perfect icosahedron has the lower potential energy compared to a 13-atom fcc or hcp cluster ^[16]. On the other hand, an atomic cluster containing more degree of LFFS exhibits longer lifetime ^{[19] and [50]}. All these results show that LFFS has lower potential energy, resulting in the local structural stability ^{[14] and [21]}. As for free volume, it has been demonstrated that the free volume decreases as the fraction of pentagons increases ^[21], which indicates that the more pentagons a structural configuration contains, the more densely the atoms are packed. Therefore, the atomic clusters having more degree of LFFS pack more densely and contain less free volume.

It is often perceived that in metallic glasses the regions with more free volumes are softer, while the regions with less free volume are harder, reflecting structural heterogeneity in metallic glasses. The corresponding hard and soft zones have been observed in plastic metallic glasses ^[51], and various experimental observations have indicated mechanical heterogeneity and structural heterogeneity in metallic glasses ^{[51], [52], [53], [54], [55], [56], [57], [58] and [59]}. Such mechanical heterogeneity and structural heterogeneity are also correlated to the local atomic symmetry in metallic glasses. Previous studies also show that the structures containing more degree of LFFS tend to form string-like order, leading to the inhomogeneous structures of metallic glasses ^[18]. Fig. 1 shows the inhomogeneous localized deformation in metallic glasses, indicating the inhomogeneous distribution of local atomic symmetry. The heterogeneity of the local atomic symmetry leads to the heterogeneity of the mechanical response.

It has been demonstrated that as temperature decreases, the population of pentagons increases, while the population of triangles, hexagons, and tetragons decreases ^{[21] and [32]}, indicating a competition between local five-fold symmetry and local translational symmetry in both local and global structures. Therefore, the understanding of the evolution of local atomic symmetry with temperature may provide deep insight into the nature of glass transition.

Note that according to Eq.(1), the degree of LFFS in an atomic cluster can be explicitly quantified. However, it cannot distinguish the degree of LFFS in atomic clusters with the same Voronoi index but different atomic packing. For example, for a perfect icosahedron and a distorted one, although the Voronoi indices are the same, they may have quite different configurations. As mentioned above, a perfect icosahedron is five-fold symmetric, while a distorted one even contains partial fcc symmetry ^[21]. As shown in Fig. 4, most atomic irreversible rearrangements in deformation try to avoid the regions with higher degree of LFFS, preferring to occur in regions with less degree of LFFS. However, some irreversible rearrangements still occur in the regions with more degree of LFFS. This indicates the local structures with the same degree of LFFS could exhibit different vibrational properties and mechanical response to deformation. To characterize the local atomic symmetry of local structures in more details, a new method is desired.

We have shown that local atomic symmetry in metallic glasses may be a good structural indicator to characterize mechanical and vibrational properties. The local structures with less degree of LFFS are packed loosely, while those with more degree of LFFS are packed more densely, which could correspond to the soft and hard regions, respectively, in metallic glasses. The regions with less degree of LFFS accommodate larger plastic deformation, and the irreversible rearrangement is more difficult to be induced in the regions with more degree of LFFS. On the other hand, the soft modes can predict mechanical behavior in metallic glasses well. The local structures with less degree of LFFS participate more in soft low-frequency modes and make more contributions to boson peak. The local structures with more degree of LFFS participate more in moderate- and high-frequency modes. The correlation between local atomic symmetry and free volume, structural heterogeneity, mechanical heterogeneity, and glass transition is also discussed. All these indicate that local atomic symmetry as a structural indicator may be more general for understanding structure-property relationship in metallic glasses.

References