**2. Structure of metallic glasses**

The essential feature of MG is considered as disordered, but there is still no exactly model to describe the total features of amorphous structure. The little research on the atomic structure of amorphous system, as well as the correlation of amorphous structure and mechanical behaviors, has obstructed the development on design and implication of amorphous materials seriously. Thus, the characterization and modeling of amorphous structure has been one of the most important and challenge research works in condense physics and material science area, and it is also the basis of research work on the mechanical behavior of MGs.

The first model of atomic structure in MG was developed by Bernal, and it was described as a dense random packed structure of equal-sized spheres with the absence of medium-range and long-range order [5]. Bernal used the active hard balls to describe a liquid-like amorphous system, and analyzed the radial distribution function of the model, and found to have a good consistence with that of liquid system. This model provided a possibility of modeling and characterization based on the computer science. The further development of amorphous structure modeling has been based on Bernal's work, and more developed characterization methods have been raised from the aspects of statistic and topology with the aids of MD simulations. The most used characterization methods of MGs with MD simulation are presented as the following in this section, including pair distribution function, Honeycutt-Andersen analysis, and Voronoi tessellation method.

#### **2.1. Pair distribution function**

both experimentally and theoretically. Meanwhile, the corresponding simulation methods with aids of computer science have also been developed to provide assistance for the experimental research from the theoretical aspect. The most used simulation methods for the MGs are finite element method (FEM) and molecular dynamics (MD) method. FEM constructs a constitutive model of MG at the macroscopic scale on time and space, which is coincident with experimental works. For this advantage, it is usually employed to investigate the mechanical properties and shear bands (SBs) formation in the bulk MG systems [1–4]. Although a MG is usually regarded as isotropic and homogeneous from the macroscopic scale, a homogeneous constitutive model is not sufficient for the generation and localization of SBs in a typical brittle MG system. For the improvement, two different SB formation theories, free volume theory [1, 2] and shear transition zone (STZ) theory [3, 4] were employed in constitutive model to provide nucleation possibilities in the MG model, and they are proved quiet effective when depicting the formation process of SBs. However, these FEM simulations are basing on a fuzzy description of flow events according to the two immature theories, but neglect intrinsic dynamics of the flow events from atomic aspect. Furthermore, the theoretical frames and constitutive model for the research of MGs have to build on accurate description of their atomic structure. Since, the lack of effective experimental tools on characterizing the atomic structure of MGs, resolving the intrinsic structure only from experimental methods is unreasonable. Recently, MD methods are developed to offer possibility for the investigation on MGs from atomic level. By simulating the atoms movements and tracking atoms trajectories, MD simulation can calculate the basic thermal information of a MG ensemble, including temperature, potential energy, as well as some derived information such as atom shear stress, atom shear strain, and structure evolution and so on. Thus, MD simulation method is an effective tool when investigating the intrinsic structure of MG, and its connection with mechanical proper-

In this chapter, we mainly take a review on the former research work on the connections between structure and properties in MGs. We began with the current understanding in the structure of MGs and then discuss their proper connection with some mechanical behaviors. Afterwards, more specific discussion on the SBs formation and development will be given based on various mechanics of MD simulations. Finally, based on the limitation of MGs in their applications, we discussed several multi-component materials derived from cast MGs,

The essential feature of MG is considered as disordered, but there is still no exactly model to describe the total features of amorphous structure. The little research on the atomic structure of amorphous system, as well as the correlation of amorphous structure and mechanical behaviors, has obstructed the development on design and implication of amorphous materials seriously. Thus, the characterization and modeling of amorphous structure has been one of the most important and challenge research works in condense physics and material science

and have an outlook on the development trend of MG preparation and application.

area, and it is also the basis of research work on the mechanical behavior of MGs.

ties, as well as the elastic to plastic deformation transition.

**2. Structure of metallic glasses**

28 Metallic Glasses - Properties and Processing

Pair distribution function (PDF) is the most classical and important statistical analysis method in amorphous systems; it is widely used in the characterization of liquids and amorphous materials. It is a pair correlation representing the probability of finding atoms, and described as a function of distance *r* from an average center atom. In a monatomic system, PDF is defined as:

$$\mathcal{g}(r) = \frac{1}{4\pi \, r^2 \rho N} \sum\_{l=1}^{N} \sum\_{\rho l}^{N} \delta \left( r - \left| \overline{r}\_{\left| l \right|} \right| \right), \tag{1}$$

where |*<sup>r</sup>* ¯*ij*| is the interatomic distance between atom *i* and atom *j*, and *ρ* is the number density of atoms in the system with *N* atoms. According to the PDF curve of a MG, the short-to-medium range order information can be manifested by the peak position, peak width, and relative intensity, etc. Conventionally, the nearest-neighbor shell atoms contribute to the first peak, which represents short range order (SRO) in the MG. A further distance up to 1–2 nm contributes the medium-range order (MRO). With distance *r* going larger, PDF gradually converges to unity, which means the atoms are randomly distributed, represents the long-range disorder. The structure evolution from liquid to glass can be shown from the PDF curves during the cooling procedure. **Figure 1** displays the PDF curves of a binary Cu64Zr36 MG in different temperature regions during its preparation process. It can be clearly seen that with decreasing temperature during quenching process, the second peak of total PDF gradually splits and becomes more pronounced, which indicates the formation of the glass phase and enhancement of SRO. Furthermore, the appearing temperature of the split is usually relevant to the glass transition temperature (Tg ), indicating the generally formation and stability of the glass structure.

PDF is concise and clear for the characterization of amorphous materials, especially when detecting the amorphous phase transferred from a crystalline system; thus, PDF method is widely used to verify the effective MG models in MD simulations [6]. However, this method is only established on a statistic basis, without considering the exactly certain structure of MGs. It cannot provide the specific topology description of a certain amorphous system, thus has a limitation when further revealing the atomic structure geometry of MGs.

**2.3. Voronoi tessellation**

and center atom [10]. Usually, *i*

**2.4. MRO structures in MG**

*k*

a Voronoi polyhedral, and a four-number vector < *i*

H-A analysis method can reveal the topology SRO structure effectively, but fail to obtain the specific geometry features of SRO structure. Voronoi tessellation is a polyhedral analysis method developed by introducing the concept of Voronoi polyhedral, which is defined as a closed convex polyhedral enclosed by the vertical bisectors of the nearest-neighbor atoms

to describe the arrangement and symmetry of the nearest-neighbor atoms around the center atom. For example, an icosahedral with Voronoi index < 0,0,12,0 > has 12 pentagons enclosed, and a polyhedral with index < 0,3,6,0 > has 3 quadrilaterals and 6 pentagons enclosed. The polyhedral with same Voronoi index can be concluded as one type cluster. After analyzing all the local clusters compositions in MG, the motifs are summarized as a series of typical Voronoi polyhedral types and can be regard as mainly contributions to SRO structure.

Among these mentioned analysis methods, the Voronoi tessellation method is the most accurate and effective tool for typical structure identification in MGs, and also be used by many researchers [10–13]. Cheng et al. used this method to describe the structure evolution of Cu-Zr MG during quenching process [11]. Five most popular Voronoi polyhedral types were elected, which accounting for over 75% of all Cu-centered polyhedral. These clusters had common characteristics of high symmetry and a CN of 12, while the other fragmented clusters have low symmetry and usually irregular polyhedral with unfavorable CNs or highly distorted shapes. The polyhedral with Voronoi index < 0,0,12,0 > have a symmetrical fivefold geometry and icosahedral structure, and defined as "full icosahedra" (FI). FI clusters were found increase sharply during the quenching process, especially during the supercooled liquid regime. While the cluster motifs of MG with lower symmetry had a gradually change with the sample cooling. But the fragmental ones, which might the motifs of equilibrium liquid at melting temperature, decreased sharply accompanying with formation of glass state. These observations suggested that FI cluster mostly contributed to SRO and dominated the solid-like nature of MGs, and with lower cooling rate, the fraction of FI increased and made the generated MG a more stable one, which was consistent with glass formation cooling rate dependence. We will have a more detailed discuss on the cooling rate effect in Section 3.

By combining experimental characterization tools such as XRD, EXAFS, and RMC together with *ab initio* simulation method, Sheng et al. resolved the MRO structure of various MGs [6]. With application of Voronoi tessellation, the formation of local solute-centered coordination polyhedral with strong chemical favoring unlike bonds was found and defined with "quasi-equivalent cluster". On the basis of the quasi-equivalent clusters, the MRO was further proposed to be dense packing of these clusters by sharing shell atoms, and regard as general feature of MG. Three basic icosahedra linkages were concluded as face-sharing (FS), edge-sharing (ES) and vertex-sharing (VS). **Figure 2** shows the typical connections of SRO and MRO structure in three MGs. Liu analyzed various MGs structure by PDF data and MD simulation. They indicated that the MRO structure can be interpreted as spherical-periodic

3 *,i*4 *,i*5

is used to represent the number of polygon with *k* edges in

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**Figure 1.** Partial distribution function of Cu-Zr MG during quenching process.

### **2.2. Honeycutt-Andersen analysis**

A widely used and effective analysis method for the topology structure of crystalline systems is common neighbor analysis (CNA), which is proposed by Clarke and Jónsson [7]. CNA is mainly to describe a bonding correlation between neighboring atoms in highly regular ordered systems with integers *jkl*: (a) *j* for the number of near neighbors they have in common; (b) *k* for the number of bonds among the shared neighbors; (c) and *l* represents the number of bonds in the longest bonding chain in *k* type bonds. It can effectively detect general structure and abnormal structure in crystalline materials, but when used for various disordered structures, it cannot describe an accurate amorphous local structure and might have a misleading to structure analysis of MG.

Similar with CNA analysis method, Honeycutt-Andersen (H-A) analysis method uses a multiple integers *ijkl* to describe more unregularly ordered and complicated bond types [8]. The additional number *i* in the H-A indices *ijkl* indicates whether or not the near neighbors are bonded, *i* = 1 for bonded pairs and *i* = 2 for nonbonding atoms, and *l* is used to distinguish different bond geometries in case the first three numbers are same. The cutoff distance is derived from PDF curves for the particular pair of atoms. Different structures have different H-A indices to represent distinguished bond types. In the MGs, 1551 and 2331 are usually the characteristics of icosahedral ordering, which mainly contribute to SRO in MGs.

H-A indices can be employed to analyze SRO structures of MG. Duan et al. took the H-A analysis on Cu50Zr50 MG during the preparation process [9]. At the liquid state, the bond types with H-A indices 1441 and 1661 dominated the initial B2 phase crystalline model, and had the proportions 42 and 56%, respectively. When the temperature went to the totally melting liquid phase, the proportions dropped to 7 and 6%. The amorphous bond type 1431, 1541, and 1551 increased to 16, 12, and 14%, respectively. During a quenching process, the amorphous H-A indices were monitored, they found out that 1431 and 1541 pairs do not change much, while the icosahedral 1551 and 2331 pairs increased uniformly as the system being supercooled until a local maxima at 700 K, indicating the final state came to be stable amorphous state. Therefore, as the MG system is cooled from its liquid state, the icosahedral symmetry keeps increasing and this SRO structure has been proved a correlation with stability and generation of a MG system.

## **2.3. Voronoi tessellation**

**2.2. Honeycutt-Andersen analysis**

30 Metallic Glasses - Properties and Processing

**Figure 1.** Partial distribution function of Cu-Zr MG during quenching process.

eration of a MG system.

A widely used and effective analysis method for the topology structure of crystalline systems is common neighbor analysis (CNA), which is proposed by Clarke and Jónsson [7]. CNA is mainly to describe a bonding correlation between neighboring atoms in highly regular ordered systems with integers *jkl*: (a) *j* for the number of near neighbors they have in common; (b) *k* for the number of bonds among the shared neighbors; (c) and *l* represents the number of bonds in the longest bonding chain in *k* type bonds. It can effectively detect general structure and abnormal structure in crystalline materials, but when used for various disordered structures, it cannot describe an accurate amorphous local structure and might have a misleading to structure analysis of MG. Similar with CNA analysis method, Honeycutt-Andersen (H-A) analysis method uses a multiple integers *ijkl* to describe more unregularly ordered and complicated bond types [8]. The additional number *i* in the H-A indices *ijkl* indicates whether or not the near neighbors are bonded, *i* = 1 for bonded pairs and *i* = 2 for nonbonding atoms, and *l* is used to distinguish different bond geometries in case the first three numbers are same. The cutoff distance is derived from PDF curves for the particular pair of atoms. Different structures have different H-A indices to represent distinguished bond types. In the MGs, 1551 and 2331 are usually the

characteristics of icosahedral ordering, which mainly contribute to SRO in MGs.

H-A indices can be employed to analyze SRO structures of MG. Duan et al. took the H-A analysis on Cu50Zr50 MG during the preparation process [9]. At the liquid state, the bond types with H-A indices 1441 and 1661 dominated the initial B2 phase crystalline model, and had the proportions 42 and 56%, respectively. When the temperature went to the totally melting liquid phase, the proportions dropped to 7 and 6%. The amorphous bond type 1431, 1541, and 1551 increased to 16, 12, and 14%, respectively. During a quenching process, the amorphous H-A indices were monitored, they found out that 1431 and 1541 pairs do not change much, while the icosahedral 1551 and 2331 pairs increased uniformly as the system being supercooled until a local maxima at 700 K, indicating the final state came to be stable amorphous state. Therefore, as the MG system is cooled from its liquid state, the icosahedral symmetry keeps increasing and this SRO structure has been proved a correlation with stability and genH-A analysis method can reveal the topology SRO structure effectively, but fail to obtain the specific geometry features of SRO structure. Voronoi tessellation is a polyhedral analysis method developed by introducing the concept of Voronoi polyhedral, which is defined as a closed convex polyhedral enclosed by the vertical bisectors of the nearest-neighbor atoms and center atom [10]. Usually, *i k* is used to represent the number of polygon with *k* edges in a Voronoi polyhedral, and a four-number vector < *i* 3 *,i*4 *,i*5 *,i*<sup>6</sup> > is chosen as the Voronoi index, to describe the arrangement and symmetry of the nearest-neighbor atoms around the center atom. For example, an icosahedral with Voronoi index < 0,0,12,0 > has 12 pentagons enclosed, and a polyhedral with index < 0,3,6,0 > has 3 quadrilaterals and 6 pentagons enclosed. The polyhedral with same Voronoi index can be concluded as one type cluster. After analyzing all the local clusters compositions in MG, the motifs are summarized as a series of typical Voronoi polyhedral types and can be regard as mainly contributions to SRO structure.

Among these mentioned analysis methods, the Voronoi tessellation method is the most accurate and effective tool for typical structure identification in MGs, and also be used by many researchers [10–13]. Cheng et al. used this method to describe the structure evolution of Cu-Zr MG during quenching process [11]. Five most popular Voronoi polyhedral types were elected, which accounting for over 75% of all Cu-centered polyhedral. These clusters had common characteristics of high symmetry and a CN of 12, while the other fragmented clusters have low symmetry and usually irregular polyhedral with unfavorable CNs or highly distorted shapes. The polyhedral with Voronoi index < 0,0,12,0 > have a symmetrical fivefold geometry and icosahedral structure, and defined as "full icosahedra" (FI). FI clusters were found increase sharply during the quenching process, especially during the supercooled liquid regime. While the cluster motifs of MG with lower symmetry had a gradually change with the sample cooling. But the fragmental ones, which might the motifs of equilibrium liquid at melting temperature, decreased sharply accompanying with formation of glass state. These observations suggested that FI cluster mostly contributed to SRO and dominated the solid-like nature of MGs, and with lower cooling rate, the fraction of FI increased and made the generated MG a more stable one, which was consistent with glass formation cooling rate dependence. We will have a more detailed discuss on the cooling rate effect in Section 3.

#### **2.4. MRO structures in MG**

By combining experimental characterization tools such as XRD, EXAFS, and RMC together with *ab initio* simulation method, Sheng et al. resolved the MRO structure of various MGs [6]. With application of Voronoi tessellation, the formation of local solute-centered coordination polyhedral with strong chemical favoring unlike bonds was found and defined with "quasi-equivalent cluster". On the basis of the quasi-equivalent clusters, the MRO was further proposed to be dense packing of these clusters by sharing shell atoms, and regard as general feature of MG. Three basic icosahedra linkages were concluded as face-sharing (FS), edge-sharing (ES) and vertex-sharing (VS). **Figure 2** shows the typical connections of SRO and MRO structure in three MGs. Liu analyzed various MGs structure by PDF data and MD simulation. They indicated that the MRO structure can be interpreted as spherical-periodic

Zr atoms, the fraction of Cu-centered FI decreased severely, and the formed networks tended to be diffusive with decrease in size and increase in number. At 50% fraction for Zr atoms, the number of networks reached the maximum and had a downward tendency with the further increase of Zr atoms fraction. It was noted that the compositions of Cu70Zr30 and Cu50Zr50 both had high glass-forming ability, differences in network forming proved that the presence of networks is neither a sufficient nor necessary condition for the formation of glass phase, but networks formation does have dependence on the preparing procedures, and a strong effect on mechanical behavior of MG as well. The detailed explanation will be given in Section 3.

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Another found by Cheng is that an addition of a small percentage of Al in the original Cu-Zr binary MG can lead to dramatically increased population of FI clusters and their spatial connectivity (shown in **Figure 3**) [12]. It has been known that the ideal icosahedral dense packing requires several factors: an atomic size ratio of 0.902 for a hard-packing model [16], and negative mixing heat of atoms. For binary Cu-Zr MG, the atomic size ratio is 0.804, Al atom has

**Figure 3.** (a) Histogram showing differences in SRO components and fractions in Cu-Zr-Al ternary MG and Cu-Zr binary MG; (b) combinations of Cu-Zr-Al atoms in Cu-centered FI SRO in Cu-Zr-Al ternary MG and (c) Cu-Zr binary MG [13].

**Figure 2.** Typical connections of SRO clusters and MRO structures in MGs [6].

order and local translational symmetry, and the increasing of MRO structures proceeded throughout the glass transition process [14].

The combining of PDF and topological analysis method can also reveal the formation of MRO. Liang identified icosahedral clusters from an Mg-Zn MG using H-A analysis method, and found some neighbor clusters have linkage with structure packed and atoms shared [15]. The icosahedra linkages of intercross-sharing (IS), FS, ES, and VS were also detected. By evaluating the average distances between center atoms in each linking cluster, Liang found the distribution of the distances had a coincidence with the second peak splitting of PDF curve. With more clusters interacted into a network, MRO came to formation in the glass.

With the increase in simulation dimension, it was also found that the clusters tend to form large and interconnecting networks. Ward further extended the range of correlated packing of the icosahedral clusters, and found the diversities of the network diffusivity in compositions about the binary Cu-Zr MGs [12]. With 30% fraction of Zr atoms, the MG had higher fraction of Cu-centered FI clusters, which tended to form a single large network. With the promotion of Zr atoms, the fraction of Cu-centered FI decreased severely, and the formed networks tended to be diffusive with decrease in size and increase in number. At 50% fraction for Zr atoms, the number of networks reached the maximum and had a downward tendency with the further increase of Zr atoms fraction. It was noted that the compositions of Cu70Zr30 and Cu50Zr50 both had high glass-forming ability, differences in network forming proved that the presence of networks is neither a sufficient nor necessary condition for the formation of glass phase, but networks formation does have dependence on the preparing procedures, and a strong effect on mechanical behavior of MG as well. The detailed explanation will be given in Section 3.

Another found by Cheng is that an addition of a small percentage of Al in the original Cu-Zr binary MG can lead to dramatically increased population of FI clusters and their spatial connectivity (shown in **Figure 3**) [12]. It has been known that the ideal icosahedral dense packing requires several factors: an atomic size ratio of 0.902 for a hard-packing model [16], and negative mixing heat of atoms. For binary Cu-Zr MG, the atomic size ratio is 0.804, Al atom has

order and local translational symmetry, and the increasing of MRO structures proceeded

The combining of PDF and topological analysis method can also reveal the formation of MRO. Liang identified icosahedral clusters from an Mg-Zn MG using H-A analysis method, and found some neighbor clusters have linkage with structure packed and atoms shared [15]. The icosahedra linkages of intercross-sharing (IS), FS, ES, and VS were also detected. By evaluating the average distances between center atoms in each linking cluster, Liang found the distribution of the distances had a coincidence with the second peak splitting of PDF curve.

With the increase in simulation dimension, it was also found that the clusters tend to form large and interconnecting networks. Ward further extended the range of correlated packing of the icosahedral clusters, and found the diversities of the network diffusivity in compositions about the binary Cu-Zr MGs [12]. With 30% fraction of Zr atoms, the MG had higher fraction of Cu-centered FI clusters, which tended to form a single large network. With the promotion of

With more clusters interacted into a network, MRO came to formation in the glass.

throughout the glass transition process [14].

32 Metallic Glasses - Properties and Processing

**Figure 2.** Typical connections of SRO clusters and MRO structures in MGs [6].

**Figure 3.** (a) Histogram showing differences in SRO components and fractions in Cu-Zr-Al ternary MG and Cu-Zr binary MG; (b) combinations of Cu-Zr-Al atoms in Cu-centered FI SRO in Cu-Zr-Al ternary MG and (c) Cu-Zr binary MG [13].

a radius in between of Zr and Cu atoms, the three atomic sizes can adjust the coordination polyhedron around the Cu atom, and increases the possibility of comfortable arrangements to reach FI. Furthermore, the negative mixing heat of Al with Zr and Cu drives Al to scatter in the Cu-Zr matrix. A more ideal ratio of 0.905 of Zr-Al atoms make Al surrounded by Zr become the topologically optimal way for FI packing. Similar with the binary Cu-Zr MG mentioned above, the FIs in this ternary MG overlapped and connected with the bonding forms of tetrahedral sharing (TS, same as IS), FS, ES, and VS, and then interconnected with each other resulting in the formation of networks (shown in **Figure 4**). The degree of connectivity of FI might serve as the backbone of the MG structures, and induced the more stable structure and improvement of mechanical properties in ternary glass compared with the binary glass. A further study taken by Tang suggested that those SRO clusters with low degree five-fold symmetry structure could also tend to form interconnected networks [17]. But different with the solid-like networks formed by FI clusters, the low five-fold symmetry clusters built up liquid-like networks. Compared with the solid-like networks, the liquid-like ones were less resistant to shear events and usually fertile sites for plastic deformation.

**3. Correlation of structure and mechanical properties**

heterogeneity and decide the mechanical behavior of the MG [10, 19].

the degree of strain localization parameter was defined as:

*<sup>ψ</sup>* <sup>=</sup> <sup>√</sup>

According to Wakeda's investigations on the shear deformation of Cu*<sup>x</sup>*

The correlation of structure and mechanical properties is a central theme of materials research, no matter crystalline materials or amorphous ones. The properties of MGs change pronouncedly with the internal structures change. Several works have been done for the influence of changes in composition proportions and variation in the processing conditions for MGs of a fixed composition, but what really governs the properties of MG from the sight of atomic-level structures has to be concerned for its quantitative and predictive theory establishment. The icosahedral cluster structures usually have a more dense packing efficiency, and the distributions are not even in space with connection of icosahedral existing. Thus, the MG structure has intrinsic fluctuations, and the local atomic-level area has heterogeneity in structure and dynamics, and the nanoscale mechanical heterogeneity had been proved with the using of dynamics force microscopy [18]. The mechanical response of MG sample can be significantly influenced by various local structure motifs, relative fractions, and their distributions in space. In other words, the structural heterogeneity necessarily leads to mechanical

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MGs, the local geometrical structures had large variation with the change of proportions, and affected the yield and fracture behavior of the MGs [20]. With application of Voronoi polyhedral analysis method, the local structures were characterized by pentagons and free volume, and these two factors were found related to each other. The pentagonal regions corresponded to the densely packed structure and nonpentagonal regions corresponded to the free volume structure. With shear simulation taken on the various MGs, it was found the pentagon-rich region tends to undergo elastic deformation, while the pentagon-poor was easily deformed plastically. Pentagon-rich local area had more formation of fivefold symmetry structure, and more relevant to the FI cluster, thus the aggregation of pentagons could be regard as formation of SRO. The results indicated the pentagonal SRO contributed the structural stability as well as elastic strength, while the opposite ones closely related to yield and fracture behavior in MGs. The preparation processing of MG also has a significant influence on the formation of SRO cluster types and contents, and induces local SRO structure heterogeneity in MG. Cheng found that the mechanical behavior and dynamics responses varied with different cooling rates of MG [11]. When applied the same tensile loading procedure to the MG samples prepared under different cooling rates, the sample with lower cooling rate showed a strong tendency to form a single and highly localized SB, while the one with higher cooling rate had a more homogeneous deformation. To obtain a quantitative evaluation of strain localization,

\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \_\_1

where, a larger *ψ* means larger fluctuations in the atomic strain and a more localized deformation mode. The strain localization degree *ψ* was found negatively correlated with cooling rates and positively correlated with the fraction of Cu-centered FI clusters. A shear deformation

*Mises* − *ηave*

*Mises*)2, (2)

*<sup>N</sup>* ∑*<sup>i</sup>*=1 *<sup>N</sup>* (*η<sup>i</sup>* Zr1*-x* (*x =* 0.30–0.85)

**Figure 4.** (a) A supercluster consisting four types of connections VS, ES, FS and TS; (b) a large supercluster consisting of over 700 atoms; (c) higher degree of connectivity of FI shown in Cu-Zr-Al MG compared with (d) Cu-Zr MG [13].
