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Growth of icosahedral medium-range order in liquid TiAl alloy during rapid solidification
Journal of Non-Crystalline Solids 394–395 (2014) 16–21
Contents lists available at ScienceDirect
Journal of Non-Crystalline Solids journal homepage: www.elsevier.com/ locate/ jnoncrysol
Growth of icosahedral medium-range order in liquid TiAl alloy during rapid solidification Zhuo-Cheng Xie a, Ting-Hong Gao b, Xiao-Tian Guo b, Xin-Mao Qin a, Quan Xie b,⁎ a b
College of Materials and Metallurgy, Guizhou University, Guiyang 550025, China Institute of New Type Optoelectronic Materials and Technology, College of Electronic Information, Guizhou University, Guiyang 550025, China
a r t i c l e
i n f o
Article history: Received 30 January 2014 Received in revised form 13 March 2014 Available online xxxx Keywords: TiAl alloy; Rapid solidification; Icosahedral medium-range order; Icosahedral connectivity; Hexagonal connected formation
a b s t r a c t The formation and evolution of icosahedral medium-range order structures of TiAl alloy during the rapid solidification are investigated based on molecular dynamics simulations. The icosahedral medium-range order structural evolutions are described in detail by coordination number and icosahedral connectivity parameters. Simultaneously, the evolution of the medium-range order icosahedral clusters is analyzed by nano-clusters tracing method. The results reveal that the full growths of icosahedral medium-range order structures improve the glass forming ability and stability of metallic glass. Meanwhile, the hexagonal connected icosahedral clusters which link via volume-sharing have excellent structural stability, and the dendritic icosahedral clusters based on the hexagonal formation keep continuous growth below the Tg. The junctions which consist of the hexagonal formations enhance the compactness of icosahedral medium-range order structure. Moreover, the icosahedral medium-range order is an appropriate structural characterization which accurately defines the characteristics of amorphous TiAl alloy during the rapid solidification. © 2014 Elsevier B.V. All rights reserved.
1. Introduction Metallic glass (MG) has many excellent properties that are unusual for metallic crystal, the glass transition will improve the material performances of metal [1–5]. Since microstructure is an accessible signature of stability in MGs, completed structural information is necessary to understand both the atomic arrangement and the formation of clusters. In 1960, J. D Bernal [6] proposed a widely cited structural model for MGs — Bernal's dense random packing of hard spheres. After that, short-range order (SRO) was proposed by P. H Gaskell [7] as a conceptual structural unit, and considered as the fundamental structural unit in MGs. With the development of structural study, researchers realized that the SRO structural unit did not presuppose spatial distributions and linking patterns to be some other important structural parameters affecting the structural stability in MGs, so it could not describe the structures beyond the nearest-neighbor atomic shell [8]. The defects naturally lead to the introduction of medium-range order (MRO), the extended structural organization in a hierarchical sense beyond the SRO. The concept of atomic MRO in amorphous structure was proposed on the basis of structural and thermodynamic considerations [9–11]. In recent years, several MRO model ideas have been investigated by experimental and computational studies [12–14]. As a major MRO structural unit in metallic liquids and glasses, the growth of connected icosahedral ⁎ Corresponding author. Tel./fax: +86 851 3623 248. E-mail address: [email protected] (Q. Xie).
http://dx.doi.org/10.1016/j.jnoncrysol.2014.03.027 0022-3093/© 2014 Elsevier B.V. All rights reserved.
clusters was demonstrated to play an essential role in the glass forming ability (GFA) of amorphous alloys [15,16]. Since the structure of interpenetrating icosahedra clusters is more stable than other types of connected icosahedral clusters, the icosahedral MRO (IMRO) formation is considered to be the most important type of MRO affecting the mechanical properties of MGs [8,16–17]. Over the past few years, TiAl alloy has received much attention due to the attractive properties, which is considered as the next-generation high-temperature structural material in the aerospace and automotive fields [18–20]. However, the main factors limiting the commercial manufacture of TiAl-based components are the intrinsic characteristics of TiAl alloy [21–23]. The glass transition is an efficient method to improve the material properties of TiAl alloy as an excellent structural material. Until recently, few have reported on the MRO structures of TiAl alloy during the rapid solidification. In this study, computer simulation based on molecular dynamics (MD) is applied as an effective tool to detect the detailed IMRO evolutions of liquid and amorphous TiAl alloy at atomic level. The radial distribution function (RDF) is applied to indicate the formation of MG. The evolutions of IMRO structures are investigated by coordination number (CN), and icosahedral connectivity parameter (C). We further extend the C to indicate the triangular connected center atoms (Ctri) and paired connected center atoms (Cpair) of connected icosahedral clusters. Moreover, nano-cluster tracing method is applied to trace the dendritic growth of IMRO clusters based on hexagonal connected icosahedral formation. The rest of the paper is arranged as follows: the simulation details are presented in Section 1. The results
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Fig. 1. (a) RDF and partial RDF curves of TiAl alloy at 700 K. (b) Relation of the R with decreasing temperature during the rapid solidification. The inset is a snapshot of the simulated box at 2000 K. The purple and light blue balls represent Ti and Al atoms, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 3. Relative numbers of CNs of TiAl alloy during the rapid solidification. The inset shows the evolution of average CN of TiAl alloy during the solidification.
obtained from our simulations are presented and discussed in Section 2. Finally, concluding remarks are given in Section 3.
3.1. Formation of MG
2. Simulation details The MD technique has been widely used in studying glass formation processes of TiAl alloys during rapid solidification [24,25]. The system, with initial cell arranging as a L10 structure (a = 4.00 Å, c = 4.06 Å) [26], contained 32,000 atoms (16,000 Al atoms and 16,000 Ti atoms) in a tetragonal box subject to the periodic boundary condition. The embedded-atom model (EAM) was developed for the TiAl alloys to describe the interatomic interactions in a many-body framework [27]. Simulation was performed under constant pressure and constant temperature (NPT ensemble) with zero pressure and the time step was 1.0 fs. Since the melting point of TiAl alloy is around 1753 K [19], the simulation was started at 2000 K. Firstly, the system was run for 200 ps at 2000 K to guarantee an equilibrium liquid state, and the microstructure was proved to be in a liquid state after this operation. From the snapshot of the simulated box at 2000 K in the inset of Fig. 1(b), the liquid state structure with random atomic arrangement at the initial temperature can be clearly seen. Subsequently, in order to form the MG, the system was cooled down from 2000 K to 200 K under the cooling rate of 1011 K/s. The atomic positions and other relevant data in the system were recorded with an interval of 100 K during the cooling process. Finally, several structural characterization methods were applied to indicate the IMRO structural evolutions of TiAl alloy.
3. Results and discussions
RDF describes the probability of finding a neighboring atom in the spherical shell of a central atom. In Fig. 1(a), a notable split in the second peak of gtotal(r) curve can be observed at 700 K. It is well known that the split of the second peak in gtotal(r) curve is the evidence of glass formation and corresponds to the glass transition temperature (Tg) for many metals. The Wendt–Abraham ratio (R), which is defined as R = g(r)min/g(r)max, leads to a better estimate of Tg [28,29]. Fig. 1(b) shows the trend of R during the rapid solidification, an inflection of the fitting curve is clearly seen at 700 K, which thus determines the Tg ≈ 700 K. As shown in Fig. 1(a), the first peak of gAl–Ti(r) is higher compared with gTi–Ti(r) and gAl–Al(r), the g(r) curves of partial RDFs further reveal that a strong interaction between Al and Ti atoms and some chemical SRO structures indeed existed in the rapidly solidified TiAl alloy. The result emerging from the present analysis indicates that the system has already become MG at 700 K. 3.2. MRO icosahedral structures MRO has been defined as the next highest level of structural organization beyond SRO, and a comprehensive understanding of characteristics of MRO is one of the most important purposes in MG research [13]. In previous studies [8,14,17], interpenetrating connection of icosahedral clusters has the lowest average potential energy and smaller average atomic volume compared with other types of connected icosahedral clusters, which make the interpenetrating clusters of icosahedra not only energetically more favorable, but also structurally more stable.
Fig. 2. Sectional drawings of center atoms of icosahedral clusters during the rapid solidification.
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The center atoms of icosahedral clusters interconnect with each other to form the string-like and ring-like networks during the rapid solidification. However, the recent studies in structural evolution of IMRO during the rapid solidification were confined to configurations of the connected center atoms. In order to gain deeper insight into the characteristics of network structures, some quantitative methods and visualization technology were applied in this study to indicate the structural evolutions of IMRO.
3.2.1. Coordination number (CN) CN is defined as the number of neighboring atoms surrounding any atom in microstructure. It was originally applied to characterize the SRO structures in liquid and amorphous alloys as one of the most essential parameters. In this study, we extended the CN to describe the detailed evolution of center atoms of icosahedral clusters during the rapid solidification. The CN provides a new method, which only considers the center atoms of icosahedral clusters in the system, for investigating the integral distributions of center atoms in MRO icosahedral clusters. Fig. 2 shows the growths of kinds of center atoms of icosahedral clusters in the rapid quenching process. The CN = 1 center atoms had notable development above the Tg, and the CN = 2 and CN = 3 center atoms had marked increases below the Tg. As shown in Fig. 3, the percentage of the CN = 0 center atoms kept reducing with decreasing temperature, especially above the Tg, and had a moderate decrease below the Tg. The proportion of the CN = 0 center atoms decreased from 79.70% at 1400 K to 18.06% at 200 K. The percentage of the CN = 1 center atoms had a significant increase above the Tg, but it slightly decreased below the Tg, and it accounted for 34.38% at 200 K. Simultaneously, the CN = 2, CN = 3, and CN = 4 center atoms increased during the rapid cooling process, especially around the Tg, the proportions of the CN = 2, CN = 3, and CN = 4 center atoms came up to 29.89%, 13.61%, and 4.06% at the end of the solidification. The inset of Fig. 3 shows the evolution of average CN of center atoms of icosahedral clusters during the rapid solidification. The average CN had a remarkable increase with decreasing temperature, especially above the Tg, and it increased from 0.23 at 1400 K to 1.53 at 200 K.
Fig. 5. Evolution of the Cico during the rapid solidification. The inset shows the snapshot of connected icosahedral cluster at 200 K. The purple and light blue balls represent Ti and Al atoms, respectively.
The results indicate that the rapid solidification made the IMRO clusters of TiAl alloy a more ordered structure with a higher atomic packing density. Since the efficient atomic packing reduces the energy of the amorphous structure and produces a more viscous liquid which can significantly decrease the kinetics of nucleation and growth of the competing crystalline state, the GFA of MG was enhanced during the rapid solidification [30–33]. 3.2.2. Icosahedral connectivity parameter (C) To investigate the detailed evolution of IMRO, the characteristics of the icosahedral network structures were quantitatively assessed through the connectivity parameters. As shown in Fig. 4, the networks connected by center atoms of icosahedral clusters increased rapidly from 1200 K to 700 K, and this increase kept slow growth below the
Fig. 4. Evolution of the connected center atoms of IMRO clusters during the rapid solidification. The purple and light blue balls represent Ti and Al atoms, respectively, and the gray balls represent the unconnected center atoms of icosahedral clusters. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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Fig. 6. Evolutions of the triangular and “lone pair” connected center atoms of IMRO clusters during the rapid solidification. The red and light blue balls represent atoms in triangular formation and “lone pair” formation, respectively, and the gray balls represent the center atoms of other IMRO configurations. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Tg. At the end of the solidification, the vast majority of center atoms of icosahedral clusters were the connected center atoms. M. Lee [17] proposed the icosahedral connectivity parameter (Cico) to indicate the connectivity of interconnected icosahedral clusters, which is defined as Cico ¼
n1 N1
ð1:1Þ
Here N1 is the total number of icosahedral clusters, and n1 is the number of the connected icosahedral clusters via connection of center atoms. The Cico increased remarkably at temperature above the Tg, and kept moderate growth below the Tg, the Cico came up to 82% at 200 K as
shown in Fig. 5. H. Tanaka [34] proposed that the stable icosahedral order in the supercooled liquid prevents the formation of a long-range crystalline order. The result reveals that the fully grown IMRO structures stunted the nucleation and growth of the crystalline nucleus, thus enhancing the GFA [30,35]. As shown in Fig. 4, most ring-like networks consist of three center atoms in a triangular formation (inset of Fig. 7(a)) at the end of the rapid solidification, and the majority of string-like networks were formed by two center atoms as “lone pair” formation (inset of Fig. 7(b)) at the beginning of the evolution. Fig. 6 illustrates the variations of triangular and “lone pair” connected center atoms of icosahedral clusters, respectively. The triangular connected center atoms kept increasing during the rapid solidification, as well as the “lone pair” connected center atoms.
Fig. 7. (a) Evolution of the Ctri during the rapid solidification. The inset shows the configuration of triangular connected icosahedral clusters. (b) Evolution of the Cpair during the rapid solidification. The inset shows the configuration of “lone pair” connected icosahedral clusters. The purple and light blue balls represent Ti and Al atoms, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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Fig. 8. Schematic of junctions formed by hexagonal connected icosahedral clusters. The inset shows the schematic of hexagonal connected icosahedral clusters. The red and orange balls represent Ti and Al atoms in hexagonal connected icosahedral clusters, respectively. The blue and green balls represent Ti and Al atoms in dendritic icosahedral clusters, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
In order to further represent the detailed structural evolution of IMRO during the rapidly cooling process of TiAl alloy, we extended the C to indicate the Ctri and Cpair of MRO icosahedral clusters, which are defined as Ctri ¼
ntri n1
Cpair ¼
npair n1
ð1:2Þ
ð1:3Þ
Here n1 is the number of the connected icosahedral clusters via connection of center atoms, ntri is the number of three connected icosahedral clusters via triangular connection of center atoms, and npair is the number of two connected icosahedral clusters via volume-sharing. As shown in Fig. 7(a), the Ctri increased significantly during the solidification, especially above the Tg, and kept moderate growth below the Tg, the Ctri elevated to 36% at 200 K. Simultaneously, the Cpair had a remarkable decrease at temperature above the Tg, and kept slight reduction below the Tg, the Cpair dropped from 89% at 1400 K to 17% at 200 K as shown in Fig. 7(b). As the triangular connected structure is more stable than the “lone pair” connected structure and the single icosahedral cluster in the system [36–38], at the same time, it also made up a high proportion in icosahedral connected structures, we can conclude that the growth of triangular icosahedral formation enhanced the stability of MG during the rapid solidification. Some other connected icosahedral structures also exist in MG of TiAl alloy, such as hexagonal connected icosahedral structure in which center atoms of icosahedral clusters form a hexagonal ring as shown in the inset of Fig. 8. Since the hexagonal connected icosahedral structure which consists of 38 atoms has lowest energy compared with other connected icosahedral structures in this system, the stability of the hexagonal formation is better than other formations in IMRO [37–39]. During the rapid solidification, although a small amount of hexagonal connected icosahedral structure existed in the system, the hexagonal formations formed bulky nano-clusters which built the strongest junctions in the IMRO structure as shown in Fig. 8. The junctions closely connected the nearby extended icosahedral clusters, thus making the IMRO structure more compact after the rapid cooling process. 3.2.3. Nano-clusters tracing method A complete understanding of the formation and growth in hexagonal connected icosahedral formation is the foundation to explore the microstructure of IMRO clusters. The evolution of one of the hexagonal connected icosahedral clusters during the rapid solidification is shown in Fig. 9. The initial connected icosahedral clusters were formed by 10 center atoms and 53 peripheral atoms at 700 K, and the hexagonal connected icosahedral structure had not already been formed. With
Fig. 9. Evolution of the IMRO clusters based on hexagonal connected icosahedral formation during the rapid solidification. The red and orange balls represent Ti and Al atoms in hexagonal connected icosahedral clusters, respectively. The blue and green balls represent Ti and Al atoms in dendritic icosahedral clusters, respectively. The schematic configurations of center atoms of IMRO clusters are at bottom right corner of each figure. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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decreasing temperature, the formation of hexagonal structure had been completed, the dendritic icosahedral clusters began to grow, and the initial clusters aggregated into extended icosahedral clusters which contained 18 icosahedral clusters with 74 peripheral atoms at 600 K. After that, the extended icosahedral clusters, based on the hexagonal connected icosahedral formation, kept stable and continual growth, and the hexagonal connected icosahedral formation maintained stability during the rapid solidification. At the end of the solidification, the extended icosahedral configuration was combined by 30 icosahedral clusters consisting of 127 peripheral atoms, and dendritic icosahedral clusters had fully grown after the rapid quenching. The evolution of the IMRO clusters indicates the excellent structural stability of hexagonal connected icosahedral formation. Meanwhile, the dendritic icosahedral clusters based on hexagonal icosahedral formation had good growing ability, and promoted the continuous growth of IMRO clusters below the Tg. 4. Conclusion The present study investigated the evolutions of IMRO structures in TiAl alloy during the rapid solidification, the following conclusions are obtained. (1) With decreasing temperature, the atoms in IMRO structures packed more efficiently, and the full growths of IMRO structures in TiAl alloy stunted the crystal nucleation and improved the GFA. Meanwhile, the stability of MG strengthened markedly with the increase of triangular icosahedral connectivity during the rapid quenching process. (2) The hexagonal icosahedral formation as the foundation structure of IMRO clusters had good structural stability during the rapid solidification. Meanwhile, the dendritic icosahedral clusters based on hexagonal icosahedral formation as the potential growth of IMRO clusters had good configural continuity. Furthermore, the hexagonal icosahedral clusters formed junctions of IMRO structure which improved the compactness of IMRO clusters. In conclusion, the full growths of IMRO structures improved the GFA and stability of MG. Furthermore, the IMRO is a more appropriate definition of the characteristics of amorphous TiAl alloy compared with SRO during the rapid solidification. Acknowledgments This work is supported by the National Natural Science Foundation of China (grant no. 61264004), the Special Fund for Construction of Sci-Tech Innovative Talents Team of Guizhou Province of China (grant no. [2011]4002), and the Fund for International Sci-Tech Cooperation Program of Guizhou Province of China (grant no. [2012]7004). References [1] A.L. Greer, Metallic glasses, Science 267 (1995) 1947–1953. [2] A. Inoue, Stabilization of metallic supercooled liquid and bulk amorphous alloys, Acta Mater. 48 (2000) 279–306. [3] C. Qin, W. Zhang, K. Asami, N. Ohtsu, A. Inoue, Glass formation, corrosion behavior and mechanical properties of bulk glassy Cu–Hf–Ti–Nb alloys, Acta Mater. 53 (2005) 3903–3911. [4] P. Jia, H. Guo, Y. Li, J. Xu, E. Ma, A new Cu–Hf–Al ternary bulk metallic glass with high glass forming ability and ductility, Scr. Mater. 54 (2006) 2165–2168. [5] D.V. Louzguine-Luzgin, D.B. Miracle, L. Louzguina-Luzgina, A. Inoue, Comparative analysis of glass-formation in binary, ternary, and multicomponent alloys, J. Appl. Phys. 108 (2010) 103511.
21
[6] J. Bernal, Geometry of the structure of monatomic liquids, Nature 185 (1960) 68–70. [7] P. Gaskell, A new structural model for transition metal–metalloid glasses, Nature 276 (1978) 484–485. [8] M. Lee, H.-K. Kim, J.-C. Lee, Icosahedral medium-range orders and backbone formation in an amorphous alloy, Met. Mater. Int. 16 (2010) 877–881. [9] P. Gaskell, Local and medium range structures in amorphous alloys, J. Non-Cryst. Solids 75 (1985) 329–346. [10] J. Phillips, Topology of covalent non-crystalline solids II: medium-range order in chalcogenide alloys and A Si (Ge), J. Non-Cryst. Solids 43 (1981) 37–77. [11] T. Hamada, F.E. Fujita, Interference function of crystalline embryo model of amorphous metals.–I, Japan, J. Appl. Phys. 20 (1982) 981–986. [12] D.B. Miracle, A structural model for metallic glasses, Nat. Mater. 3 (2004) 697–702. [13] H. Sheng, W. Luo, F. Alamgir, J. Bai, E. Ma, Atomic packing and short-to-mediumrange order in metallic glasses, Nature 439 (2006) 419–425. [14] M. Wakeda, Y. Shibutani, Icosahedral clustering with medium-range order and local elastic properties of amorphous metals, Acta Mater. 58 (2010) 3963–3969. [15] Y. Cheng, E. Ma, Atomic-level structure and structure–property relationship in metallic glasses, Prog. Mater. Sci. 56 (2011) 379–473. [16] M. Shimono, H. Onodera, Icosahedral order in supercooled liquids and glassy alloys, Materials Science Forum, Trans Tech Publ, 2007, pp. 2031–2035. [17] M. Lee, C.-M. Lee, K.-R. Lee, E. Ma, J.-C. Lee, Networked interpenetrating connections of icosahedra: effects on shear transformations in metallic glass, Acta Mater. 59 (2011) 159–170. [18] G. Chen, W. Zhang, Z. Liu, S. Li, Y. Kim, Y. Kim, D. Dimiduk, M. Loretto, Gamma Titanium Aluminides, TMS, Warrendale, PA, 1999. 371. [19] F. Appel, U. Brossmann, U. Christoph, S. Eggert, P. Janschek, U. Lorenz, J. Müllauer, M. Oehring, J.D. Paul, Recent progress in the development of gamma titanium aluminide alloys, Adv. Eng. Mater. 2 (2000) 699–720. [20] J. Aguilar, A. Schievenbusch, O. Kättlitz, Investment casting technology for production of TiAl low pressure turbine blades—Process engineering and parameter analysis, Intermetallics 19 (2011) 757–761. [21] P.C. Priarone, S. Rizzuti, L. Settineri, G. Vergnano, Effects of cutting angle, edge preparation, and nano-structured coating on milling performance of a gamma titanium aluminide, J. Mater. Process. Technol. (2012) 2619–2628. [22] T. Tetsui, T. Kobayashi, T. Mori, T. Kishimoto, H. Harada, Evaluation of yttria applicability as a crucible for induction melting of TiAl alloy, Mater. Trans. 51 (2010) 1656. [23] J. Lapin, Z. Gabalcová, Solidification behaviour of TiAl-based alloys studied by directional solidification technique, Intermetallics 19 (2011) 797–804. [24] M. Shimono, H. Onodera, Molecular dynamics study on liquid-to-amorphous transition in Ti-Al alloys, Mater. Trans. JIM 39 (1998) 147–153. [25] Q. Pei, C. Lu, M. Fu, The rapid solidification of Ti3Al: a molecular dynamics study, J. Phys. Condens. Matter 16 (2004) 4203. [26] M. Grujicic, S. Batchu, A crystal plasticity materials constitutive model for polysynthetically-twinned γ-TiAl + α2-Ti3Al single crystals, J. Mater. Sci. 36 (2001) 2851–2863. [27] R.R. Zope, Y. Mishin, Interatomic potentials for atomistic simulations of the Ti-Al system, Phys. Rev. B 68 (2003) 024102. [28] B. Shen, C. Liu, Y. Jia, G. Yue, F. Ke, H. Zhao, L. Chen, S. Wang, C. Wang, K. Ho, Molecular dynamics simulation studies of structural and dynamical properties of rapidly quenched Al, J. Non-Cryst. Solids (2013) 13–20. [29] H. Wendt, F.F. Abraham, Empirical criterion for the glass transition region based on Monte Carlo simulations, Phys. Rev. Lett. 41 (1978) 1244. [30] T. Scopigno, G. Ruocco, F. Sette, G. Monaco, Is the fragility of a liquid embedded in the properties of its glass? Science 302 (2003) 849–852. [31] S. Mukherjee, J. Schroers, Z. Zhou, W. Johnson, W.-K. Rhim, Viscosity and specific volume of bulk metallic glass-forming alloys and their correlation with glass forming ability, Acta Mater. 52 (2004) 3689–3695. [32] D. Miracle, W. Sanders, O. Senkov, The influence of efficient atomic packing on the constitution of metallic glasses, Philos. Mag. 83 (2003) 2409–2428. [33] D. Turnbull, Under what conditions can a glass be formed? Contemp. Phys. 10 (1969) 473–488. [34] H. Tanaka, Roles of local icosahedral chemical ordering in glass and quasicrystal formation in metallic glass formers, J. Phys. Condens. Matter 15 (2003) L491. [35] V. Novikov, A. Sokolov, Poisson's ratio and the fragility of glass-forming liquids, Nature 431 (2004) 961–963. [36] Z.-Y. Hou, L.-X. Liu, R.-S. Liu, Z.-A. Tian, J.-G. Wang, Short-range and medium-range order in rapidly quenched AlbsubN 50b/subN MgbsubN 50b/subN alloy, J. NonCryst. Solids 357 (2011) 1430–1436. [37] I.A. Solov'yov, A.V. Solov'yov, W. Greiner, A. Koshelev, A. Shutovich, Cluster growing process and a sequence of magic numbers, Phys. Rev. Lett. 90 (2003) 053401. [38] J.P. Doye, A model metal potential exhibiting polytetrahedral clusters, J. Chem. Phys. 119 (2003) 1136–1147. [39] J.P. Doye, M.A. Miller, D.J. Wales, The double-funnel energy landscape of the 38-atom Lennard-Jones cluster, J. Chem. Phys. 110 (1999) 6896–6906.
Contents lists available at ScienceDirect
Journal of Non-Crystalline Solids journal homepage: www.elsevier.com/ locate/ jnoncrysol
Growth of icosahedral medium-range order in liquid TiAl alloy during rapid solidification Zhuo-Cheng Xie a, Ting-Hong Gao b, Xiao-Tian Guo b, Xin-Mao Qin a, Quan Xie b,⁎ a b
College of Materials and Metallurgy, Guizhou University, Guiyang 550025, China Institute of New Type Optoelectronic Materials and Technology, College of Electronic Information, Guizhou University, Guiyang 550025, China
a r t i c l e
i n f o
Article history: Received 30 January 2014 Received in revised form 13 March 2014 Available online xxxx Keywords: TiAl alloy; Rapid solidification; Icosahedral medium-range order; Icosahedral connectivity; Hexagonal connected formation
a b s t r a c t The formation and evolution of icosahedral medium-range order structures of TiAl alloy during the rapid solidification are investigated based on molecular dynamics simulations. The icosahedral medium-range order structural evolutions are described in detail by coordination number and icosahedral connectivity parameters. Simultaneously, the evolution of the medium-range order icosahedral clusters is analyzed by nano-clusters tracing method. The results reveal that the full growths of icosahedral medium-range order structures improve the glass forming ability and stability of metallic glass. Meanwhile, the hexagonal connected icosahedral clusters which link via volume-sharing have excellent structural stability, and the dendritic icosahedral clusters based on the hexagonal formation keep continuous growth below the Tg. The junctions which consist of the hexagonal formations enhance the compactness of icosahedral medium-range order structure. Moreover, the icosahedral medium-range order is an appropriate structural characterization which accurately defines the characteristics of amorphous TiAl alloy during the rapid solidification. © 2014 Elsevier B.V. All rights reserved.
1. Introduction Metallic glass (MG) has many excellent properties that are unusual for metallic crystal, the glass transition will improve the material performances of metal [1–5]. Since microstructure is an accessible signature of stability in MGs, completed structural information is necessary to understand both the atomic arrangement and the formation of clusters. In 1960, J. D Bernal [6] proposed a widely cited structural model for MGs — Bernal's dense random packing of hard spheres. After that, short-range order (SRO) was proposed by P. H Gaskell [7] as a conceptual structural unit, and considered as the fundamental structural unit in MGs. With the development of structural study, researchers realized that the SRO structural unit did not presuppose spatial distributions and linking patterns to be some other important structural parameters affecting the structural stability in MGs, so it could not describe the structures beyond the nearest-neighbor atomic shell [8]. The defects naturally lead to the introduction of medium-range order (MRO), the extended structural organization in a hierarchical sense beyond the SRO. The concept of atomic MRO in amorphous structure was proposed on the basis of structural and thermodynamic considerations [9–11]. In recent years, several MRO model ideas have been investigated by experimental and computational studies [12–14]. As a major MRO structural unit in metallic liquids and glasses, the growth of connected icosahedral ⁎ Corresponding author. Tel./fax: +86 851 3623 248. E-mail address: [email protected] (Q. Xie).
http://dx.doi.org/10.1016/j.jnoncrysol.2014.03.027 0022-3093/© 2014 Elsevier B.V. All rights reserved.
clusters was demonstrated to play an essential role in the glass forming ability (GFA) of amorphous alloys [15,16]. Since the structure of interpenetrating icosahedra clusters is more stable than other types of connected icosahedral clusters, the icosahedral MRO (IMRO) formation is considered to be the most important type of MRO affecting the mechanical properties of MGs [8,16–17]. Over the past few years, TiAl alloy has received much attention due to the attractive properties, which is considered as the next-generation high-temperature structural material in the aerospace and automotive fields [18–20]. However, the main factors limiting the commercial manufacture of TiAl-based components are the intrinsic characteristics of TiAl alloy [21–23]. The glass transition is an efficient method to improve the material properties of TiAl alloy as an excellent structural material. Until recently, few have reported on the MRO structures of TiAl alloy during the rapid solidification. In this study, computer simulation based on molecular dynamics (MD) is applied as an effective tool to detect the detailed IMRO evolutions of liquid and amorphous TiAl alloy at atomic level. The radial distribution function (RDF) is applied to indicate the formation of MG. The evolutions of IMRO structures are investigated by coordination number (CN), and icosahedral connectivity parameter (C). We further extend the C to indicate the triangular connected center atoms (Ctri) and paired connected center atoms (Cpair) of connected icosahedral clusters. Moreover, nano-cluster tracing method is applied to trace the dendritic growth of IMRO clusters based on hexagonal connected icosahedral formation. The rest of the paper is arranged as follows: the simulation details are presented in Section 1. The results
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Fig. 1. (a) RDF and partial RDF curves of TiAl alloy at 700 K. (b) Relation of the R with decreasing temperature during the rapid solidification. The inset is a snapshot of the simulated box at 2000 K. The purple and light blue balls represent Ti and Al atoms, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 3. Relative numbers of CNs of TiAl alloy during the rapid solidification. The inset shows the evolution of average CN of TiAl alloy during the solidification.
obtained from our simulations are presented and discussed in Section 2. Finally, concluding remarks are given in Section 3.
3.1. Formation of MG
2. Simulation details The MD technique has been widely used in studying glass formation processes of TiAl alloys during rapid solidification [24,25]. The system, with initial cell arranging as a L10 structure (a = 4.00 Å, c = 4.06 Å) [26], contained 32,000 atoms (16,000 Al atoms and 16,000 Ti atoms) in a tetragonal box subject to the periodic boundary condition. The embedded-atom model (EAM) was developed for the TiAl alloys to describe the interatomic interactions in a many-body framework [27]. Simulation was performed under constant pressure and constant temperature (NPT ensemble) with zero pressure and the time step was 1.0 fs. Since the melting point of TiAl alloy is around 1753 K [19], the simulation was started at 2000 K. Firstly, the system was run for 200 ps at 2000 K to guarantee an equilibrium liquid state, and the microstructure was proved to be in a liquid state after this operation. From the snapshot of the simulated box at 2000 K in the inset of Fig. 1(b), the liquid state structure with random atomic arrangement at the initial temperature can be clearly seen. Subsequently, in order to form the MG, the system was cooled down from 2000 K to 200 K under the cooling rate of 1011 K/s. The atomic positions and other relevant data in the system were recorded with an interval of 100 K during the cooling process. Finally, several structural characterization methods were applied to indicate the IMRO structural evolutions of TiAl alloy.
3. Results and discussions
RDF describes the probability of finding a neighboring atom in the spherical shell of a central atom. In Fig. 1(a), a notable split in the second peak of gtotal(r) curve can be observed at 700 K. It is well known that the split of the second peak in gtotal(r) curve is the evidence of glass formation and corresponds to the glass transition temperature (Tg) for many metals. The Wendt–Abraham ratio (R), which is defined as R = g(r)min/g(r)max, leads to a better estimate of Tg [28,29]. Fig. 1(b) shows the trend of R during the rapid solidification, an inflection of the fitting curve is clearly seen at 700 K, which thus determines the Tg ≈ 700 K. As shown in Fig. 1(a), the first peak of gAl–Ti(r) is higher compared with gTi–Ti(r) and gAl–Al(r), the g(r) curves of partial RDFs further reveal that a strong interaction between Al and Ti atoms and some chemical SRO structures indeed existed in the rapidly solidified TiAl alloy. The result emerging from the present analysis indicates that the system has already become MG at 700 K. 3.2. MRO icosahedral structures MRO has been defined as the next highest level of structural organization beyond SRO, and a comprehensive understanding of characteristics of MRO is one of the most important purposes in MG research [13]. In previous studies [8,14,17], interpenetrating connection of icosahedral clusters has the lowest average potential energy and smaller average atomic volume compared with other types of connected icosahedral clusters, which make the interpenetrating clusters of icosahedra not only energetically more favorable, but also structurally more stable.
Fig. 2. Sectional drawings of center atoms of icosahedral clusters during the rapid solidification.
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The center atoms of icosahedral clusters interconnect with each other to form the string-like and ring-like networks during the rapid solidification. However, the recent studies in structural evolution of IMRO during the rapid solidification were confined to configurations of the connected center atoms. In order to gain deeper insight into the characteristics of network structures, some quantitative methods and visualization technology were applied in this study to indicate the structural evolutions of IMRO.
3.2.1. Coordination number (CN) CN is defined as the number of neighboring atoms surrounding any atom in microstructure. It was originally applied to characterize the SRO structures in liquid and amorphous alloys as one of the most essential parameters. In this study, we extended the CN to describe the detailed evolution of center atoms of icosahedral clusters during the rapid solidification. The CN provides a new method, which only considers the center atoms of icosahedral clusters in the system, for investigating the integral distributions of center atoms in MRO icosahedral clusters. Fig. 2 shows the growths of kinds of center atoms of icosahedral clusters in the rapid quenching process. The CN = 1 center atoms had notable development above the Tg, and the CN = 2 and CN = 3 center atoms had marked increases below the Tg. As shown in Fig. 3, the percentage of the CN = 0 center atoms kept reducing with decreasing temperature, especially above the Tg, and had a moderate decrease below the Tg. The proportion of the CN = 0 center atoms decreased from 79.70% at 1400 K to 18.06% at 200 K. The percentage of the CN = 1 center atoms had a significant increase above the Tg, but it slightly decreased below the Tg, and it accounted for 34.38% at 200 K. Simultaneously, the CN = 2, CN = 3, and CN = 4 center atoms increased during the rapid cooling process, especially around the Tg, the proportions of the CN = 2, CN = 3, and CN = 4 center atoms came up to 29.89%, 13.61%, and 4.06% at the end of the solidification. The inset of Fig. 3 shows the evolution of average CN of center atoms of icosahedral clusters during the rapid solidification. The average CN had a remarkable increase with decreasing temperature, especially above the Tg, and it increased from 0.23 at 1400 K to 1.53 at 200 K.
Fig. 5. Evolution of the Cico during the rapid solidification. The inset shows the snapshot of connected icosahedral cluster at 200 K. The purple and light blue balls represent Ti and Al atoms, respectively.
The results indicate that the rapid solidification made the IMRO clusters of TiAl alloy a more ordered structure with a higher atomic packing density. Since the efficient atomic packing reduces the energy of the amorphous structure and produces a more viscous liquid which can significantly decrease the kinetics of nucleation and growth of the competing crystalline state, the GFA of MG was enhanced during the rapid solidification [30–33]. 3.2.2. Icosahedral connectivity parameter (C) To investigate the detailed evolution of IMRO, the characteristics of the icosahedral network structures were quantitatively assessed through the connectivity parameters. As shown in Fig. 4, the networks connected by center atoms of icosahedral clusters increased rapidly from 1200 K to 700 K, and this increase kept slow growth below the
Fig. 4. Evolution of the connected center atoms of IMRO clusters during the rapid solidification. The purple and light blue balls represent Ti and Al atoms, respectively, and the gray balls represent the unconnected center atoms of icosahedral clusters. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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Fig. 6. Evolutions of the triangular and “lone pair” connected center atoms of IMRO clusters during the rapid solidification. The red and light blue balls represent atoms in triangular formation and “lone pair” formation, respectively, and the gray balls represent the center atoms of other IMRO configurations. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Tg. At the end of the solidification, the vast majority of center atoms of icosahedral clusters were the connected center atoms. M. Lee [17] proposed the icosahedral connectivity parameter (Cico) to indicate the connectivity of interconnected icosahedral clusters, which is defined as Cico ¼
n1 N1
ð1:1Þ
Here N1 is the total number of icosahedral clusters, and n1 is the number of the connected icosahedral clusters via connection of center atoms. The Cico increased remarkably at temperature above the Tg, and kept moderate growth below the Tg, the Cico came up to 82% at 200 K as
shown in Fig. 5. H. Tanaka [34] proposed that the stable icosahedral order in the supercooled liquid prevents the formation of a long-range crystalline order. The result reveals that the fully grown IMRO structures stunted the nucleation and growth of the crystalline nucleus, thus enhancing the GFA [30,35]. As shown in Fig. 4, most ring-like networks consist of three center atoms in a triangular formation (inset of Fig. 7(a)) at the end of the rapid solidification, and the majority of string-like networks were formed by two center atoms as “lone pair” formation (inset of Fig. 7(b)) at the beginning of the evolution. Fig. 6 illustrates the variations of triangular and “lone pair” connected center atoms of icosahedral clusters, respectively. The triangular connected center atoms kept increasing during the rapid solidification, as well as the “lone pair” connected center atoms.
Fig. 7. (a) Evolution of the Ctri during the rapid solidification. The inset shows the configuration of triangular connected icosahedral clusters. (b) Evolution of the Cpair during the rapid solidification. The inset shows the configuration of “lone pair” connected icosahedral clusters. The purple and light blue balls represent Ti and Al atoms, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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Fig. 8. Schematic of junctions formed by hexagonal connected icosahedral clusters. The inset shows the schematic of hexagonal connected icosahedral clusters. The red and orange balls represent Ti and Al atoms in hexagonal connected icosahedral clusters, respectively. The blue and green balls represent Ti and Al atoms in dendritic icosahedral clusters, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
In order to further represent the detailed structural evolution of IMRO during the rapidly cooling process of TiAl alloy, we extended the C to indicate the Ctri and Cpair of MRO icosahedral clusters, which are defined as Ctri ¼
ntri n1
Cpair ¼
npair n1
ð1:2Þ
ð1:3Þ
Here n1 is the number of the connected icosahedral clusters via connection of center atoms, ntri is the number of three connected icosahedral clusters via triangular connection of center atoms, and npair is the number of two connected icosahedral clusters via volume-sharing. As shown in Fig. 7(a), the Ctri increased significantly during the solidification, especially above the Tg, and kept moderate growth below the Tg, the Ctri elevated to 36% at 200 K. Simultaneously, the Cpair had a remarkable decrease at temperature above the Tg, and kept slight reduction below the Tg, the Cpair dropped from 89% at 1400 K to 17% at 200 K as shown in Fig. 7(b). As the triangular connected structure is more stable than the “lone pair” connected structure and the single icosahedral cluster in the system [36–38], at the same time, it also made up a high proportion in icosahedral connected structures, we can conclude that the growth of triangular icosahedral formation enhanced the stability of MG during the rapid solidification. Some other connected icosahedral structures also exist in MG of TiAl alloy, such as hexagonal connected icosahedral structure in which center atoms of icosahedral clusters form a hexagonal ring as shown in the inset of Fig. 8. Since the hexagonal connected icosahedral structure which consists of 38 atoms has lowest energy compared with other connected icosahedral structures in this system, the stability of the hexagonal formation is better than other formations in IMRO [37–39]. During the rapid solidification, although a small amount of hexagonal connected icosahedral structure existed in the system, the hexagonal formations formed bulky nano-clusters which built the strongest junctions in the IMRO structure as shown in Fig. 8. The junctions closely connected the nearby extended icosahedral clusters, thus making the IMRO structure more compact after the rapid cooling process. 3.2.3. Nano-clusters tracing method A complete understanding of the formation and growth in hexagonal connected icosahedral formation is the foundation to explore the microstructure of IMRO clusters. The evolution of one of the hexagonal connected icosahedral clusters during the rapid solidification is shown in Fig. 9. The initial connected icosahedral clusters were formed by 10 center atoms and 53 peripheral atoms at 700 K, and the hexagonal connected icosahedral structure had not already been formed. With
Fig. 9. Evolution of the IMRO clusters based on hexagonal connected icosahedral formation during the rapid solidification. The red and orange balls represent Ti and Al atoms in hexagonal connected icosahedral clusters, respectively. The blue and green balls represent Ti and Al atoms in dendritic icosahedral clusters, respectively. The schematic configurations of center atoms of IMRO clusters are at bottom right corner of each figure. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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decreasing temperature, the formation of hexagonal structure had been completed, the dendritic icosahedral clusters began to grow, and the initial clusters aggregated into extended icosahedral clusters which contained 18 icosahedral clusters with 74 peripheral atoms at 600 K. After that, the extended icosahedral clusters, based on the hexagonal connected icosahedral formation, kept stable and continual growth, and the hexagonal connected icosahedral formation maintained stability during the rapid solidification. At the end of the solidification, the extended icosahedral configuration was combined by 30 icosahedral clusters consisting of 127 peripheral atoms, and dendritic icosahedral clusters had fully grown after the rapid quenching. The evolution of the IMRO clusters indicates the excellent structural stability of hexagonal connected icosahedral formation. Meanwhile, the dendritic icosahedral clusters based on hexagonal icosahedral formation had good growing ability, and promoted the continuous growth of IMRO clusters below the Tg. 4. Conclusion The present study investigated the evolutions of IMRO structures in TiAl alloy during the rapid solidification, the following conclusions are obtained. (1) With decreasing temperature, the atoms in IMRO structures packed more efficiently, and the full growths of IMRO structures in TiAl alloy stunted the crystal nucleation and improved the GFA. Meanwhile, the stability of MG strengthened markedly with the increase of triangular icosahedral connectivity during the rapid quenching process. (2) The hexagonal icosahedral formation as the foundation structure of IMRO clusters had good structural stability during the rapid solidification. Meanwhile, the dendritic icosahedral clusters based on hexagonal icosahedral formation as the potential growth of IMRO clusters had good configural continuity. Furthermore, the hexagonal icosahedral clusters formed junctions of IMRO structure which improved the compactness of IMRO clusters. In conclusion, the full growths of IMRO structures improved the GFA and stability of MG. Furthermore, the IMRO is a more appropriate definition of the characteristics of amorphous TiAl alloy compared with SRO during the rapid solidification. Acknowledgments This work is supported by the National Natural Science Foundation of China (grant no. 61264004), the Special Fund for Construction of Sci-Tech Innovative Talents Team of Guizhou Province of China (grant no. [2011]4002), and the Fund for International Sci-Tech Cooperation Program of Guizhou Province of China (grant no. [2012]7004). References [1] A.L. Greer, Metallic glasses, Science 267 (1995) 1947–1953. [2] A. Inoue, Stabilization of metallic supercooled liquid and bulk amorphous alloys, Acta Mater. 48 (2000) 279–306. [3] C. Qin, W. Zhang, K. Asami, N. Ohtsu, A. Inoue, Glass formation, corrosion behavior and mechanical properties of bulk glassy Cu–Hf–Ti–Nb alloys, Acta Mater. 53 (2005) 3903–3911. [4] P. Jia, H. Guo, Y. Li, J. Xu, E. Ma, A new Cu–Hf–Al ternary bulk metallic glass with high glass forming ability and ductility, Scr. Mater. 54 (2006) 2165–2168. [5] D.V. Louzguine-Luzgin, D.B. Miracle, L. Louzguina-Luzgina, A. Inoue, Comparative analysis of glass-formation in binary, ternary, and multicomponent alloys, J. Appl. Phys. 108 (2010) 103511.
21
[6] J. Bernal, Geometry of the structure of monatomic liquids, Nature 185 (1960) 68–70. [7] P. Gaskell, A new structural model for transition metal–metalloid glasses, Nature 276 (1978) 484–485. [8] M. Lee, H.-K. Kim, J.-C. Lee, Icosahedral medium-range orders and backbone formation in an amorphous alloy, Met. Mater. Int. 16 (2010) 877–881. [9] P. Gaskell, Local and medium range structures in amorphous alloys, J. Non-Cryst. Solids 75 (1985) 329–346. [10] J. Phillips, Topology of covalent non-crystalline solids II: medium-range order in chalcogenide alloys and A Si (Ge), J. Non-Cryst. Solids 43 (1981) 37–77. [11] T. Hamada, F.E. Fujita, Interference function of crystalline embryo model of amorphous metals.–I, Japan, J. Appl. Phys. 20 (1982) 981–986. [12] D.B. Miracle, A structural model for metallic glasses, Nat. Mater. 3 (2004) 697–702. [13] H. Sheng, W. Luo, F. Alamgir, J. Bai, E. Ma, Atomic packing and short-to-mediumrange order in metallic glasses, Nature 439 (2006) 419–425. [14] M. Wakeda, Y. Shibutani, Icosahedral clustering with medium-range order and local elastic properties of amorphous metals, Acta Mater. 58 (2010) 3963–3969. [15] Y. Cheng, E. Ma, Atomic-level structure and structure–property relationship in metallic glasses, Prog. Mater. Sci. 56 (2011) 379–473. [16] M. Shimono, H. Onodera, Icosahedral order in supercooled liquids and glassy alloys, Materials Science Forum, Trans Tech Publ, 2007, pp. 2031–2035. [17] M. Lee, C.-M. Lee, K.-R. Lee, E. Ma, J.-C. Lee, Networked interpenetrating connections of icosahedra: effects on shear transformations in metallic glass, Acta Mater. 59 (2011) 159–170. [18] G. Chen, W. Zhang, Z. Liu, S. Li, Y. Kim, Y. Kim, D. Dimiduk, M. Loretto, Gamma Titanium Aluminides, TMS, Warrendale, PA, 1999. 371. [19] F. Appel, U. Brossmann, U. Christoph, S. Eggert, P. Janschek, U. Lorenz, J. Müllauer, M. Oehring, J.D. Paul, Recent progress in the development of gamma titanium aluminide alloys, Adv. Eng. Mater. 2 (2000) 699–720. [20] J. Aguilar, A. Schievenbusch, O. Kättlitz, Investment casting technology for production of TiAl low pressure turbine blades—Process engineering and parameter analysis, Intermetallics 19 (2011) 757–761. [21] P.C. Priarone, S. Rizzuti, L. Settineri, G. Vergnano, Effects of cutting angle, edge preparation, and nano-structured coating on milling performance of a gamma titanium aluminide, J. Mater. Process. Technol. (2012) 2619–2628. [22] T. Tetsui, T. Kobayashi, T. Mori, T. Kishimoto, H. Harada, Evaluation of yttria applicability as a crucible for induction melting of TiAl alloy, Mater. Trans. 51 (2010) 1656. [23] J. Lapin, Z. Gabalcová, Solidification behaviour of TiAl-based alloys studied by directional solidification technique, Intermetallics 19 (2011) 797–804. [24] M. Shimono, H. Onodera, Molecular dynamics study on liquid-to-amorphous transition in Ti-Al alloys, Mater. Trans. JIM 39 (1998) 147–153. [25] Q. Pei, C. Lu, M. Fu, The rapid solidification of Ti3Al: a molecular dynamics study, J. Phys. Condens. Matter 16 (2004) 4203. [26] M. Grujicic, S. Batchu, A crystal plasticity materials constitutive model for polysynthetically-twinned γ-TiAl + α2-Ti3Al single crystals, J. Mater. Sci. 36 (2001) 2851–2863. [27] R.R. Zope, Y. Mishin, Interatomic potentials for atomistic simulations of the Ti-Al system, Phys. Rev. B 68 (2003) 024102. [28] B. Shen, C. Liu, Y. Jia, G. Yue, F. Ke, H. Zhao, L. Chen, S. Wang, C. Wang, K. Ho, Molecular dynamics simulation studies of structural and dynamical properties of rapidly quenched Al, J. Non-Cryst. Solids (2013) 13–20. [29] H. Wendt, F.F. Abraham, Empirical criterion for the glass transition region based on Monte Carlo simulations, Phys. Rev. Lett. 41 (1978) 1244. [30] T. Scopigno, G. Ruocco, F. Sette, G. Monaco, Is the fragility of a liquid embedded in the properties of its glass? Science 302 (2003) 849–852. [31] S. Mukherjee, J. Schroers, Z. Zhou, W. Johnson, W.-K. Rhim, Viscosity and specific volume of bulk metallic glass-forming alloys and their correlation with glass forming ability, Acta Mater. 52 (2004) 3689–3695. [32] D. Miracle, W. Sanders, O. Senkov, The influence of efficient atomic packing on the constitution of metallic glasses, Philos. Mag. 83 (2003) 2409–2428. [33] D. Turnbull, Under what conditions can a glass be formed? Contemp. Phys. 10 (1969) 473–488. [34] H. Tanaka, Roles of local icosahedral chemical ordering in glass and quasicrystal formation in metallic glass formers, J. Phys. Condens. Matter 15 (2003) L491. [35] V. Novikov, A. Sokolov, Poisson's ratio and the fragility of glass-forming liquids, Nature 431 (2004) 961–963. [36] Z.-Y. Hou, L.-X. Liu, R.-S. Liu, Z.-A. Tian, J.-G. Wang, Short-range and medium-range order in rapidly quenched AlbsubN 50b/subN MgbsubN 50b/subN alloy, J. NonCryst. Solids 357 (2011) 1430–1436. [37] I.A. Solov'yov, A.V. Solov'yov, W. Greiner, A. Koshelev, A. Shutovich, Cluster growing process and a sequence of magic numbers, Phys. Rev. Lett. 90 (2003) 053401. [38] J.P. Doye, A model metal potential exhibiting polytetrahedral clusters, J. Chem. Phys. 119 (2003) 1136–1147. [39] J.P. Doye, M.A. Miller, D.J. Wales, The double-funnel energy landscape of the 38-atom Lennard-Jones cluster, J. Chem. Phys. 110 (1999) 6896–6906.