Critically-Percolated, Cluster-Packed Structure in Cu–Zr Binary Bulk

Materials Transactions, Vol. 53, No. 6 (2012) pp. 1113 to 1118 © 2012 The Japan Institute of Metals

Critically-Percolated, Cluster-Packed Structure in Cu­Zr Binary Bulk Metallic Glass Demonstrated by Molecular Dynamics Simulations Based on Plastic Crystal Model Akira Takeuchi and Akihisa Inoue WPI-Advanced Institute for Materials Research (WPI-AIMR), Tohoku University, Sendai 980-8577, Japan Molecular dynamics (MD) simulations based on a plastic crystal model (PCM) were performed for a Cu0.618Zr0.382 binary bulk metallic glass (BMG). The local atomic arrangements of the Cu0.618Zr0.382 BMG were demonstrated with MD-PCM under an assumption that the BMG is composed of randomly-rotated as well as distorted hypothetical clusters. The Cu-rich Cu0.618Zr0.382 alloy was computationally created from a tentatively-created Zr-rich Zr0.73Cu0.27 alloy through two steps. The first step includes the Zr0.73Cu0.27 alloy quenched from a liquid through conventional MD simulation, whereas the second step has a replacement of the Zr and Cu atoms in the Zr0.73Cu0.27 alloy with randomly-rotated icosahedral and tetrahedral clusters, respectively, and subsequent structural relaxation. The Cu0.618Zr0.382 alloy, thus created with MD-PCM, was formed in a noncrystalline state as a critically-percolated cluster-packed structure. The analyses with total pair-distribution and interference functions revealed that the calculation results tend to reproduce the experimental data in an as-quenched state. The results explain that the glassforming ability of the Cu0.618Zr0.382 BMG is due to (1) a critically-percolated and distorted tetrahedral clusters forming a network and (2) atomistic-level inhomogeneity for the local atomic arrangements with keeping macroscopic homogeneity in terms of the chemical composition and dense random atomic arrangements. [doi:10.2320/matertrans.M2012025] (Received January 12, 2012; Accepted March 1, 2012; Published April 18, 2012) Keywords: copper­zirconium alloy, metallic glass, cluster, percolation, molecular dynamics simulation

1.

Introduction

In the last decade, several bulk metallic glasses (BMGs) have sequentially been fabricated in binary systems, such as Cu­Zr,1­4) Cu­Hf,3,5,6) Ni­Nb7) and Ca­Al.8) These findings of binary BMGs have greatly contributed to performing an unprecedented easy and deep analysis of the features of BMGs due to the less number of constituent elements than multicomponent BMGs9) with three or more elements. Among these binary BMGs, an especial attention has been paid to Cu­Zr BMGs because of the close relationships to Zr-based multicomponent BMGs as a representative BMG. As a result, a great progress has recently been achieved for clarifying the features of the local atomic arrangements of the Cu­Zr binary BMGs10) as well as Zr-based multicomponent BMG11) by molecular dynamics (MD) simulations. These researches using MD methods for liquids, supercooled liquids and glasses for monatomic systems are represented by early studies by Steinhardt et al.12) and Tomida and Egami,13) and are succeeded to recent literature10,11,14­17) for metallic glasses. These represented and recent results greatly contributed to the progress in understanding the nature of the local atomic arrangements of metallic glasses that contain icosahedral clusters. Here, it should be noted that recent literature10,11,14­17) was also greatly influenced by a concept of medium-range order (MRO)18­20) proposed in the mid2000s. In a framework to clarify the compositional features of BMGs, the authors reported21) that a Cu0.62Zr0.38 BMG2,4) exhibits a peculiar characteristic in terms of its fraction of the constituent elements. Specifically, the composition of the Cu0.62Zr0.38 is approximated as Cuº¹1Zrº¹2 with a value of Golden Mean (º ³ 1.618).21) A further research of the Cu0.62Zr0.38 alloy for its composition by Golden Mean analysis predicts a possibility of the BMG to possess

cluster-packed structure. In other words, Golden Mean analysis revealed that the BMG can be formed as an ensemble of icosahedral and tetrahedral clusters, and the ideal number-density ratio of the icosahedra to tetrahedra is º2 : 1, respectively.21) Here, it should be noted that the latter result regarding number-density yields the fraction of the minor components (tetrahedral clusters) of 0.276, which is considerably near the critical percolation concentration for site (pcsite) ³ 0.27 for the dense random packing (DRP) structure.22,23) On the basis of these early works,21) the authors have recently confirmed the above prediction by demonstrating a molecular dynamics (MD) simulations for a Zr­Al­Ni BMG24) by adopting an especial technique named as plastic crystal model (PCM) proposed by Takeuchi et al.25) The present study aims to apply the MD-PCM techniques to the Cu0.618Zr0.382 binary BMG. The purpose of the present study is to clarify the characteristics of the local atomic arrangements of the Cu0.618Zr0.382 BMG with MD-PCM by utilizing the features of the alloy composition and to analyze the mechanism of the high glass-forming ability of the Cu-rich Cu­Zr binary BMG. 2.

Methods

The present study dealt with two kinds of binary alloys: a Zr0.73Cu0.27 alloy and a Cu0.618Zr0.382 alloy. It should be noted that the former is a tentatively-considered Zr-rich alloy whereas the latter is a Cu-rich alloy. The reason for considering the Zr-rich alloy is due to the aspects that Zr has a larger atomic radius (rZr = 0.160 nm) than Cu (rCu = 0.128 nm),26) which would be appropriate for the subsequent replacements of the icosahedral and tetrahedral (greater and smaller) clusters with Zr and Cu atoms, respectively, in a process of MD-PCM. The explanations of the aspects are given in Fig. 1 and its legends. The density (µ) of the

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Cu

Zr Fig. 1 Ball and stick views for an icohsahedral cluster comprising 8 Cu atoms and 5 Zr atoms (left) and a tetrahedral cluster comprising 3 Cu and 2 Zr atoms (right). The edge-length (a) of the clusters are determined to be 0.320 nm (= 2rZr) to avoid collisions of atoms. The icosahedral and pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi tetrahedral polyhedra have circumradus (rcircum) of ð5 þ 5Þ=8a and pffiffiffi 31) ( 6/4)a, respectively. Thus, the longest cirmumradius (rcircumlongest) of the icosahedral and tetrahedral clusters are determined to be 0.464 and 0.356 nm, respectively, as the ones including the atoms at the vertexes from the center site (rcircum for polyhedra comprising vertexes plus rZr). These evaluated values for rcircumlongest give the ratio of (rcircumicosah/ rcircumtetrah)longest to be 1.30, which is close to the ratio of rZr/rCu of 1.25.26)

Zr0.73Cu0.27 and Cu0.618Zr0.382 alloys were determined to be 7.156 and 8.002 Mg/m3, respectively, by averaging the µ values of pure Zr and Cu (6.5 and 8.93 Mg/m3)26) with their mole fractions. The MD simulations were performed for the Cu0.618Zr0.382 alloy with commercial software (Fujitsu, Materials Explorer Ver. 5). The computation techniques were the same to those carried out by the authors’ previous work24) for Zr­Al­Ni BMG. The Cu0.618Zr0.382 alloy was created by the following two steps (Steps 1 and 2) with rapid quenching and subsequent structural relaxations, respectively. It is possible that these two steps correspond to ¡ and ¢ relaxations27,28) of glassy materials, respectively, which deal with the motions of groups of atoms for the former and atoms for the latter relaxations. The details of Steps 1 and 2 are as follows. Step 1 deals with the Zr0.73Cu0.27 alloy, and includes conventional atomistic MD simulation method which causes the Zr0.73Cu0.27 alloy with critically-percolated state for Cu atoms in a DRP. More specifically, the Zr0.73Cu0.27 glassy alloy comprising 125 atoms with µ = 7.156 Mg/m3 was created by a conventional MD simulation at a quenching rate of 1015 K/s from 2500 to 500 K under Universal Force Field (UFF) potential,29) the Number of atoms, Temperature, and Volume (constant-NTV) ensemble and periodic boundary conditions with a super-cell size of 1.344 nm. Here, the UFF potential is a Lennard­Jones (L­J) potential, which was used in the authors’ previous work for C6Cr23 alloy30) for the hypothetical clusters. On the other hand, Step 2 deals with the Cu0.618Zr0.382 alloy comprising 1353 atoms with µ = 8.002 Mg/m3. The Cu0.618Zr0.382 alloy was created from the Zr0.73Cu0.27 glassy alloy through MD-PCM under Generalized Embedded Atom Method (GEAM) potential after the Zr and Cu atoms in the Zr0.73Cu0.27 glassy alloy had been replaced with the icosahedral and tetrahedral clusters shown in Fig. 1, respectively. After the replacements, structural relaxations were performed under GEAM potential at 100 K for 0.2 ps under NTV ensemble and periodic boundary conditions with a super-cell size of 2.752 nm. The replace-

ments of the atoms with the clusters are regarded as renormalization group in the field of statistical mechanics, which is discussed in the authors’ work.21) As shown in Fig. 1, the icosahedral and tetrahedral clusters, which were used for the replacements with Zr and Cu atoms, respectively, are supposed to be near a shape of a sphere due to their high symmetry with the longest circumradius from the center atom of 0.464 and 0.356 nm, respectively, from geometric features of the polyhedra.31) We selected Cu atom as the element at the center site in the clusters. This selection is due to the result in literature10) for an icosahedral cluster, whereas the other elements at the vertexes are placed randomly. In addition to the investigations with MD-PCM, the conventional atomistic MD simulations were also performed for the Cu0.618Zr0.382 alloy for comparisons. The conditions of the conventional MD simulation include the total number of atoms of 1353, the density of 8.002 Mg/m3, quenching process (2500 to 500 K, 1015 K/s), NTV ensemble and subsequent structural relaxation at 100 K for 1 ps under GEAM potential. The structures of the alloys were examined by total pair-distribution and interference functions in the software. 3.

Results and Discussion

Figure 2(a) shows the atomic arrangements of the Zr0.73Cu0.27 glassy alloy created with atomistic MD simulations from a liquid through Step 1. It is possible that the Zr0.73Cu0.27 alloy loses long-range periodicity inherent to a crystalline phase. The structure was investigated by the total pair-distribution function, gtotal(r), as shown in Fig. 2(b). Figure 2(b) indicates that the alloy was formed in a glassy structure due to the tendencies that gtotal(r) profile exhibits rather sharp first peak accompanied by the second peaks around r = 0.27 and 0.5 nm, respectively, and that the gtotal(r) profile approaches the value ³ 1. Then, the Zr0.73Cu0.27 glassy alloy was investigated whether the Cu atoms form percolation. Figure 2(c) shows atomic arrangements of the Cu atoms in the Zr0.73Cu0.27 glassy alloy, in which the lines between the Cu atoms are drawn for Cu atoms at a distance ranging 0.2 ¯ r/nm ¯ 0.4 in Fig. 2(b). This range approximately corresponds to the first peak of gtotal(r), indicating that the Cu atoms connected with lines in Fig. 2(c) are placed in the nearest neighbor distance. Ideally, these lines to connect Cu atoms in Fig. 2(c) should spread with keeping spatial homogeneity and closely approach at least a set of two boundaries of the super-cell to form a network structure where the two boundaries can be the following cases: two x­y, y­z or z­x planes facing each other, or two planes from x­y, y­z and y­z with edge shared. The network structure, thus created in the super-cell, will exhibit high electrical conductivity as pure Cu between the boundaries in the Zr0.73Cu0.27 metallic glass with DRP structure and pcsite = 0.27. This is because the network structure formed in the Zr0.73Cu0.27 metallic glass due to percolation is in the same state to a historic demonstration for exhibiting percolation of DRP.21,23) Specifically, the schematic illustration of the demonstration drawn in literature23) as well as its reproduced one in our previous work21) indicates that a simple test was carried out by using Al and oxide-glass

Critically-Percolated, Cluster-Packed Structure in Cu­Zr Binary Bulk Metallic Glass

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(a) (b)

Zr

Cu

isolated bond

(c)

dangling bond isolated Cu atom isolated bond

x

z y

Fig. 2 Calculation results of the Zr0.73Cu0.27 alloy. Figures 2(a) and 2(c) show ball and stick views for atomic arrangements of the Zr0.73Cu0.27 alloy and those of the Cu components, respectively, in the super-cell with a size of 1.344 nm. Figure 2(b) shows the total pair-distribution function, gtotal(r). The lines connecting Cu atoms in Fig. 2(c) shows the correlations of the Cu atoms, which were placed in a distance of nearest neighbors ranging 0.2­0.4 nm in Fig. 2(b).

(a)

(b)

Random Rotations of Clusters

Structural Relaxation

Fig. 3 Ball and stick views for arrangements of components (atoms and clusters) for the Cu0.618Zr0.382 alloy, in which (a) and (b) exhibits the initial arrangements of components (a) before and (b) after the structural relaxations. The size of the super-cell is 2.752 nm.

particles in a contacting material of insulator, such as a beaker, covered by Al-foil at its top and bottom where electric conduction took place due to the metallic Al particle when Al content reaches to 0.27 in DRP structure. In reality, Fig. 2(c) shows the presence of some isolated- and danglingbonds and isolated Cu atoms. However, the authors think that Cu atoms are in a percolated state to some extent, since the most of the lines excepting for isolated- and dangling-bonds in Fig. 2(c) are widely spread in the super-cell. Accordingly, it is possible that the Zr0.73Cu0.27 alloy was formed mostly in a state of critically-percolated Cu atoms in DRP structure. Then, the Cu0.618Zr0.382 alloy was created with MD-PCM through Step 2. Figures 3(a) and 3(b) show atomic arrangements of the Cu0.618Zr0.382 alloys created by MD-PCM. Figure 3(a) corresponds to Fig. 2(a) and exhibits the replacements of the icosahedral and tetrahedral clusters, respectively, with the Zr and Cu atoms in Fig. 2(a). After MD-PCM was

performed, the local atomic structure shown in Fig. 3(b) demonstrates the randomly distributed atomic arrangements, which is different from regular atomic arrangements inherent to a crystalline structure. In order to examine the local atomic arrangements of the alloys carefully, gtotal(r) was calculated and is demonstrated in Fig. 4 for the following alloys and their statuses. They are, (a): the Cu0.618Zr0.382 alloys in a state of after relaxation. (b): experimental data for a Cu0.6Zr0.4 alloy.32) (c): calculation results for a Cu0.588Zr0.412 alloy created from the Cu10Zr7 structure with MD-PCM.33) In addition, the Cu0.618Zr0.382 alloys created in the present study by conventional atomistic MD method are also shown for comparisons for processes of (d) quenching and (e) subsequent structural relaxation. Furthermore, profile (f ) exhibits the data for the Zr0.73Cu0.27 alloy while profile (g) shows those for the Cu0.618Zr0.382 alloys in a state of before relaxation. The results in Fig. 4

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Fig. 4 Total pair-distribution functions of the Cu0.618Zr0.382 alloy created with MD-PCM and other related alloys.

Fig. 5 Total interference function, Qi(Q), of the Cu0.618Zr0.382 alloy created with MD-PCM and other related alloys.

show the following tendencies. The profiles except for (g) tend to show that gtotal(r) profile sharply increases near r = 0.2 nm and reaches the maximums of gtotal(r) of at r = 0.26 nm, followed by converging variation in gtotal(r) with sub-peaks in the range of r ² 0.35 nm. Furthermore, the variation of the gtotal(r) at r = 1.4 nm has a tendency to reach unity, indicating that a liquid or amorphous-like structure was formed for (a)­(f ). On the other hand, the profile (g) exhibits sharp peaks at r ³ 0.3 and 0.5, indicating that short range order of the constituent atoms still remains between the atoms at the center and the vertex in polyhedra, as is illustrated in Fig. 1. The explanation for this sharp peak is that the random rotations of clusters only applied as operations in MD-PCM. The profiles except for (g) show considerably similar tendencies in terms of the variations of gtotal(r) values as a function of r. However, the shapes of the second peaks at r ranging 0.4 to 0.6 nm characterize each profile. Specifically, the profile (b) exhibits the clear splitting of the second peak whereas the profile (a) tends to show the symptom of the splitting of the second peak by the presence of a small shoulder at r ³ 0.47 nm. Furthermore, the profiles (a) and (b) show asymmetric shapes of gtotal(r) profiles across r ³ 0.48 nm. This tendency underlying profiles (a) and (b) for splitting and asymmetry is in contrast to those of the profiles (c)­(e). Thus, it is possible that the profile (a) obtained in the present study are formed in an amorphouslike structure that exhibits the best fit with the experimental data in profile (b). Further analyses with interference function, Qi(Q), were performed for the Cu0.618Zr0.382 alloy created with MD-PCM and other related alloys to check them for the formation of noncrystalline phases. As shown in Fig. 5, the Cu0.618Zr0.382 alloy created with MD-PCM, profile (a), tends to reproduce the experimental data of the alloy in an as-quenched state, profile (b), for Q ranging 0 to 50 nm¹1. In particular, profiles (a) and (b) exhibit good agreement for the first peak at Q ³ 30 nm¹1 than the authors’ previous result for Cu0.588Zr0.412 alloy created from Zr10Cu7 intermetallic compound.33) The calculation data of Qi(Q) also support the Cu0.618Zr0.382 alloy created with MD-PCM reproduces the features of an experimental as-quenched alloy.

The reasons for the Cu0.618Zr0.382 alloy to have a high glass-forming ability are discussed as follows. The calculation results shown in Figs. 4 and 5 imply that the criticallypercolated tetrahedral clusters can be included in the local atomic arrangements in the Cu0.618Zr0.382 noncrystalline alloy. This is supported by p a ffiffiffiffiffiffiffiffiffiffiffi fact that square root of the mean square displacements LMSD of the constituent elements during MD-PCM is approximately half of the interatomic pffiffiffiffiffiffiffiffiffiffiffi distances.34) The actual LMSD for Cu and Zr atoms during structural relaxation (100 K, 0.2 ps) for the Cu0.618Zr0.382 alloy was evaluated as 0.07 and 0.06 nm, respectively, which are approximately a half of the rCu = 0.128 nm. Hence, it is possible that shapes of the clusters distorts due to the displacement of the constituent atoms of the clusters and that the network of the distorted tetrahedral cluster would remain and play a key to enhancing the glass-forming ability of the Cu0.618Zr0.382 noncrystalline alloy. Furthermore, the glass-forming ability of the Cu0.618Zr0.382 noncrystalline alloy can be affected by the atomistic-level inhomogeneity,35) which was reported for Pd-based metallic glass, due to the presence of mixture of the icosahedral and tetrahedral clusters and by the macroscopic homogeneity for chemical composition and atomic arrangements as dense random packing. Finally, significance of MD-PCM used in the present study for the Cu–Zr BMG is discussed by comparing with early MD simulation studies in terms of the following four items: icosahedral clusters, network structure, MRO and atomisticlevel inhomogeneity. To the best of authors’ knowledge, early studies regarding the four items would date back to 1980s when some of the foresighted research results were obtained about the presence of clusters with icosahedral topology in liquid, supercooled and glassy phases. For instance, Steinhardt et al.12) performed MD simulations for bondorientational order in supercooled liquids, and reported the presence of essentially perfect icosahedral bond correlations. On the other hand, MD simulations by Tomida and Egami13) clarified the number of icosahedral clusters to increases progressively with decreasing temperature, and its abrupt arrest at a glass transition temperature. As for the MD results for metallic glasses, Shimono and Onodera14) have recently clarified the changes of the number of icosahedral clusters as

Critically-Percolated, Cluster-Packed Structure in Cu­Zr Binary Bulk Metallic Glass

functions of annealing time, atomic size difference and negative heat of mixing, the latter two of which correspond to the factors for evaluating the glass-forming ability. Besides, the MD simulations for the Cu­Zr binary system have aggressively been performed after the experimental findings of the Cu­Zr BMGs.1­4) For instance, Cheng et al.15) demonstrated the presence of icosahedral ordering, and discussed the correlation between the structural and dynamical heterogeneities. Wakeda and Shibutani16) reported the relationships between the icosahedral clustering with MRO in amorphous metals and local elastic properties. Lee et al.17) demonstrated the presence of network structure in metallic glasses by focusing on interpenetrating connection of icosahedra. Furthermore, Tian et al.10) performed ab initio MD simulation for a binary Cu64Zr36 BMG based on the cluster-plus-glue-atom model, and clarified a gradual dominance of the Cu-centered icosahedral Cu8Zr5 clusters at the glass state. These studies10­17) with a research topic of clusterpacked structure or network structure comprising icosahedral clusters are deeply related to the concepts of MRO18­20) and atomistic level inhomogeneity that has recently reported by Ichitsubo et al.35) from the experimental results. Thus, it is possible that the conventional MD simulation results mentioned above greatly contributed to the progress in understanding the local atomic arrangements of BMGs. However, there is every reason to perform MD-PCM due to the concept of MD-PCM. The concept of creating a glassy phase by MD-PCM with a process of random rotations of clusters and subsequent structural relaxation is absolutely different from that by the conventional MD simulations10­17) with a process of quenching from a liquid. Specifically, the unique aspect of MD-PCM in the present study is that MD-PCM deals with the distorted or deformed icosahedral clusters from the regular icosahedral clusters given as an initial structure. In other words, MD-PCM deals with the decay process of the regular icosahedral clusters. In strong contrast, all the conventional MD simulations for amorphous and glassy alloys deal with the formation of the icosahedral clusters from a random phase, such as a liquid phase. Thus, the approaches to analyze the stability of the icosahedral clusters are different between the conventional MD and MD-PCM. Besides, the states of the icosahedral clusters in a glassy phase are also considerably different between the conventional MD and MD-PCM. Specifically, one can use the conventional MD and MD-PCM, respectively, for investigating an initial and late stage in an incubation time region for crystallization of a glassy phase in the time-temperaturetransformation diagram.36) As discussed in our previous work,36) an advantageous aspect of MD-PCM against the conventional MD is that MD-PCM can treat experimentally observed intermetallic crystalline structure or compounds, such as metastable Zr2Ni for Zr-based BMGs, as the initial state. In other words, MD-PCM can create a glassy structure that is just before the crystallization. Thus, the authors strongly believe that it is of great importance to study the local atomic arrangements with MD-PCM as well as the conventional MD simulations in order to have deep understanding the nature of BMGs. It is possible that the mutual studies using both conventional MD and MD-PCM will

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clarify the nature of the local atomic arrangements of BMGs by compensating for shortcoming inherent to each method. 4.

Conclusions

The Cu0.618Zr0.382 glassy alloy describable as Cuº¹1Zrº¹2 with the value of Golden Mean (º ³ 1.618) was computationally created with molecular dynamics simulations based on a plastic crystal model (MD-PCM). The analysis revealed that the Cu0.618Zr0.382 alloy can be formed in an amorphouslike structure through MD-PCM, which gives the best fit with the experimental data in an as-quenched state. The results provide the following reasons of the Cu-rich Cu­Zr binary BMG to have a high glass-forming ability: (1) a criticallypercolated and distorted tetrahedral clusters forming a network structure and (2) atomistic-level inhomogeneity for the local atomic arrangements due to the presence of clusters with keeping macroscopic homogeneity in terms of chemical composition and random atomic arrangements. Acknowledgements This work was supported by Japan Society for the Promotion of Science (JSPS) under grant numbers 20226013 and 21560715. REFERENCES 1) D. Wang, Y. Li, B. B. Sun, M. L. Sui, K. Lu and E. Ma: Appl. Phys. Lett. 84 (2004) 4029­4031. 2) D. Xu, B. Lohwongwatana, G. Duan, W. L. Johnson and C. Garland: Acta Mater. 52 (2004) 2621­2624. 3) A. Inoue and W. Zhang: Mater. Trans. 45 (2004) 584­587. 4) L. Yang, J. H. Xia, Q. Wang, C. Dong, L. Y. Chen, X. Ou, J. F. Liu, J. Z. Jiang, K. Klementiev, K. Saksl, H. Franz, J. R. Schneider and L. Gerward: Appl. Phys. Lett. 88 (2006) 241913. 5) G. Duan, D. Xu and W. L. Johnson: Metall. Mater. Trans. 36A (2005) 455­458. 6) L. Xia, D. Ding, S. T. Shan and Y. D. Dong: J. Phys. Condens. Matter 18 (2006) 3543­3548. 7) L. Xia, W. H. Li, S. S. Fang, B. C. Wei and Y. D. Dong: J. Appl. Phys. 99 (2006) 026103. 8) F. Q. Guo, S. J. Poon and G. J. Shiflet: Appl. Phys. Lett. 84 (2004) 37­ 39. 9) A. Inoue: Bulk Amorphous Alloys: Preparation and Fundamental Characteristics, Materials Science Foundations, eds. M. Magini and F. H. Wöhlbier, (Trans Tech Publications, Switzerland, 1998) pp. 1­ 116. 10) H. Tian, C. Zhang, L. Wang, J. Zhao, C. Dong, B. Wen and Q. Wang: J. Appl. Phys. 109 (2011) 123520. 11) Y. Q. Cheng, E. Ma and H. W. Sheng: Phys. Rev. Lett. 102 (2009) 245501. 12) P. J. Steinhardt, D. R. Nelson and M. Ronchetti: Phys. Rev. B 28 (1983) 784­805. 13) T. Tomida and T. Egami: Phys. Rev. B 52 (1995) 3290­3308. 14) M. Shimono and H. Onodera: Mater. Sci. Eng. A 448­451 (2007) 717­ 721. 15) Y. Q. Cheng, H. W. Sheng and E. Ma: Phys. Rev. B ­ Condens. Matter Mater. Phys. 78 (2008) 014207. 16) M. Wakeda and Y. Shibutani: Acta Mater. 58 (2010) 3963­3969. 17) M. Lee, C.-M. Lee, K.-R. Lee, E. Ma and J.-C. Lee: Acta Mater. 59 (2011) 159­170. 18) D. B. Miracle: Nature Mater. 3 (2004) 697­702. 19) D. B. Miracle: Acta Mater. 54 (2006) 4317­4336. 20) H. W. Sheng, W. K. Luo, F. M. Alamgir, J. M. Bai and E. Ma: Nature 439 (2006) 419­425.

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21) A. Takeuchi, A. R. Yavari and A. Inoue: Intermetallics 17 (2009) 696­ 703. 22) J. P. Fitzpatrick, R. B. Malt and F. Spaepen: Phys. Lett. A 47 (1974) 207­208. 23) R. Zallen: The Physics of Amorphous Solids, (Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim 2004) pp. 1­304. 24) A. Takeuchi, K. Yubuta, M. Ogata and A. Inoue: J. Mater. Sci. 45 (2010) 4898­4905. 25) A. Takeuchi, Y. Yokoyama, H. Kato, K. Yubuta and A. Inoue: Intermetallics 16 (2008) 819­826. 26) W. F. Gale and T. C. Totemeier (eds.): Smithels Metals Reference Book, 8th edition, (ASM International, Elsevier, Oxford, 2004) pp. 4­44. 27) P. G. Debenedetti: Metastable Liquids, Concepts and Principles, (Princeton University Press, New Jergey, U.S.A., 1996). 28) J. Rault: J. Non-Cryst. Solids 271 (2000) 177­217. 29) A. K. Rappe, C. J. Casewit, K. S. Colwell, W. A. Goddard, III and

W. M. Skiff: J. Am. Chem. Soc. 114 (1992) 10024­10035. 30) A. Takeuchi, K. Yubuta, Y. Yokoyama, A. Makino and A. Inoue: Intermetallics 16 (2008) 283­292. 31) D. Sutton: Platonic & Archimedean Solids ® The Geometory of Space ®, (Walker & Company, New York, U.S.A., 2002) pp. 1­64. 32) J. F. Sadoc and A. Lienard: Proc. 3 Int. Conf. on Rapidly Quenched Materials, ed. by B. Cantor, (The Metals Society Vol. 2, 1978) pp. 405­409. 33) A. Takeuchi, K. Yubuta, A. Makino and A. Inoue: J. Alloy. Compd. 483 (2009) 102­106. 34) A. Takeuchi, K. Yubuta, Y. Yokoyama, A. R. Yavari and A. Inoue: Intermetallics 16 (2008) 774­778. 35) T. Ichitsubo, E. Matsubara, T. Yamamoto, H. S. Chen, N. Nishiyama, J. Saida and K. Anazawa: Phys. Rev. Lett. 95 (2005) 245501. 36) A. Takeuchi, M. Ogata and A. Inoue: Prog. Nat. Sci. 20 (2010) 87­96.