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Cluster distribution entropy in Ti50Cu50 and Ti50Cu45Ni5 metallic glasses
Journal of Non-Crystalline Solids 398–399 (2014) 16–18
Contents lists available at ScienceDirect
Journal of Non-Crystalline Solids journal homepage: www.elsevier.com/ locate/ jnoncrysol
Cluster distribution entropy in Ti50Cu50 and Ti50Cu45Ni5 metallic glasses Yong Li Sun a,⁎, Ya Juan Sun b, Ping Yang a, Hua Ming Li a a b
College of Physics and Optoelectronics, Taiyuan University of Technology, Taiyuan, 030024, China School of Science, Tianjin Polytechnic University, Tianjin, 300160, China
a r t i c l e
i n f o
Article history: Received 12 February 2014 Received in revised form 18 April 2014 Available online xxxx Keywords: Amorphous metals; Metallic glasses; Molecular dynamics simulations
a b s t r a c t It remains unclear how a cluster packing pattern affects the glass forming ability of bulk metallic glasses. The topological structures of Ti50Cu50 and Ti50Cu45Ni5 metallic glasses obtained by molecular dynamics simulations are compared to investigate the relation between the atomic structure and glass forming ability. The structural analysis shows that the addition of Ni in Ti50Cu50 alloy can significantly change the topological structure. Cluster distribution entropy is proposed to quantitatively descript the disordered degree of cluster distribution. It is shown that the cluster distribution entropy provides a statistical approach for assessing the microstructures of metallic glasses. © 2014 Elsevier B.V. All rights reserved.
1. Introduction Recently, Ti-based bulk metallic glasses (BMGs) have attracted significant attention because of their advantageous properties, such as high specific strength and low density [1,2]. The potential applications promote the exploration of new BMGs [3]. It is proved that the glass forming ability (GFA) of BMGs is quite sensitive to composition. Element addition could significantly improve the GFA and enhance the plasticity and strength of BMGs [4]. By this method, a number of Ti-based BMGs with the critical diameters range from 1 mm to 32 mm have been prepared by copper mold casting [5–9]. However, the reasons why alloy composition affects the GFA are still obscure. Atomic structure is one of the significant reasons affecting GFA. Generally, element additions may effectively improve the GFA due to the formation of a denser atomic packing or an increased complexity in the glass-forming liquid. Icosahedral clusters in BMGs have been confirmed by both experiments and simulations [10,11]. Therefore, more efforts have been made to construct the cluster packing models. Under these conditions, efficient cluster packing model and fivefold symmetry packing scheme were proposed to describe the cluster packing pattern for metallic glasses [12,13]. These models depict the cluster packing geometrically, however, how to statistically express the cluster packing pattern remains ambiguous. The effect of element additions on GFA in the view of microstructure would be worth investigating further. Simple alloys are convenient for analyzing the structural properties. Although no binary Ti-based BMG has been formed so far, several ternary alloys have been prepared in Ti–Cu–Ni alloy system [14,15]. The
⁎ Corresponding author. E-mail address: [email protected] (Y.L. Sun).
http://dx.doi.org/10.1016/j.jnoncrysol.2014.04.020 0022-3093/© 2014 Elsevier B.V. All rights reserved.
addition of Ni can effectively improve the GFA of Ti-based alloys. In this paper, we study the atomic substitution effect in Ti50Cu50 and Ti50Cu45Ni5 metallic glasses (MGs). Atomic level structure can be investigated utilizing classical molecular dynamics (MD) simulations, in which three-dimensional atomic configurations could be constructed. The structural distinction between the two alloys is obtained via statistical analysis of cluster distributions. We intend to provide some structural understanding of the GFA of BMGs. 2. Simulations and methods The accuracy of MD simulation depends on the selected atomic interaction potential. In this work, generalized embedding atomic method potential was adopted [16]. The total energy of this potential takes the ! form: Ei ¼ F α ∑ ραβ r ij þ 12 ∑ ϕα β r ij , where α and β denote the j≠i
j≠i
elemental types of atom i and j, rij is the interatomic distance. ϕαβ(rij) is the two-body potential, F is the embedding energy and ραβ(rij) is electron density function. The details and parameters about this potential can refer to Ref. [16]. Molecular dynamics simulations were performed in a cubic cell of 500 atoms with periodic boundary condition. The simulations were carried out in the NPT (constant pressure and temperature) ensemble using Nose thermostat to control the pressure and temperature. The Gear predictor-corrector algorithm was used to integrate the equations of motion. The pressure was kept at 0 Pa. The system was first melted and equilibrated at 2000 K for 20 ps with a time step of 1 fs and then cooled down to 300 K with the cooling rate 1.7 × 1014 Ks−1. At 300 K, the system was relaxed for 50 ps to collect the relative thermal and structural information.
Y.L. Sun et al. / Journal of Non-Crystalline Solids 398–399 (2014) 16–18
17
3. Results and discussion 5
Ti-Cu Cu-Cu
4
PPCF
Generally, high packing fraction implies large GFA of metallic materials. The density (mass of the system divided by the system volume) of Ti50Cu45Ni5 MG derived from the simulation is ρ = 6.122 g/cm3, which is slightly smaller than that of Ti50Cu50 MG, ρ = 6.137 g/cm3. But the packing fractions of the two alloys are identical, both of them are φ = 0.54. Thus, the two alloys have similar degree of atomic packing. The structural reasons for distinct GFA may lie in the topological structures of the two alloys. Fig. 1 shows the total pair correlation functions g(r) of Ti50Cu50 and Ti50Cu45Ni5 MGs and the difference between them, i.e. Δg ðr Þ ¼ g Ti50 Cu45 Ni5 ðrÞ−g Ti50 Cu50 ðr Þ. The spitting second peaks in the g(r) curves indicate the formation of glassy state in both model systems. It is observed that the two curves overlapped especially well, indicating that the local structure of Ti50Cu45Ni5 MG is nearly same as that of Ti50Cu50 MG in spite of the substitution of Ni for Cu. The slight difference is illustrated in the Δg(r) curve. There is a peak at r = 2.4 Å and a valley at r = 2.65 Å. It indicates that Ti50Cu45Ni5 MG has more short bonds than Ti50Cu50 MG, involving the Ti–Ni, Cu–Ni and Ni–Ni bonds. The addition of short bonds is in favor of large GFA of Ti50Cu45Ni5 MG. In order to achieve the addition effect of Ni, partial pair correlation functions (PPCF) of the Cu–Cu, Ti–Ti and Ti–Cu pairs of the two alloys are depicted in Fig. 2. Slight changes are detected in the peak heights of the two alloys. The peak heights of first nearest neighbor for Cu–Cu and Ti–Ti pairs increase, but that for Ti–Cu pair decreases with the addition of Ni. The result implies that the substitution of Ni for Cu atoms leads to a change in the local short order. The Ni addition makes the Cu and Ti more ordered. The reduction of the peak height of Ti–Cu pair is due to the decrease of the Ti–Cu pair which is substituted by Ti–Ni pair. The Voronoi method could be used to characterize the nearestneighbor coordination number (CN) and local atomic environment [17,18]. The result of voronoi analysis is sensitive to the selected cutoff distance. When a cutoff distance of 3.6 Å is used, the dominant CN is 13 in both BMGs. To obtain a cluster distribution dominated by icosahedron-like clusters, the cutoff distance was selected to be rc = 3.4 Å. After relaxation for 20 ps, 30 configurations with the interval of 1 ps were extracted and averaged. Fig. 3 shows the distributions of clusters with different CNs in the Ti50Cu50 and Ti50Cu45Ni5 BMGs. In both metallic glasses, the dominant clusters are icosahedron-like clusters with CN = 12, which involve the ideal icosahedra with the index of b0,0,12,0N and distorted icosahedra with the index of b0,2,8,2N and b0,3,6,3N. The distribution of CNs changes when substituting Ni
3
Ti-Ti
2
1
0 2
3
5
6
Fig. 2. Partial pair correlation functions of Ti50Cu50 and Ti50Cu45Ni5 MGs. The solid lines correspond to Ti50Cu50 alloy and the dashed lines to Ti50Cu45Ni5 alloy.
atoms for a portion of Cu atoms. The fractions increase for clusters with CNs of 10, 11 and 15, but decrease for clusters with CNs of 12, 13 and 14. This occurs because atomic size ratio affects the local orders in metallic alloys [19]. The size ratio of Cu atom to Ti atom is 0.886, while the size ratio of Ni atom to Ti atom is 0.871. The atomic radius of Ni is less than that of Cu. For Ti50Cu50 alloy, some of the clusters with 13, 14 or 15 atoms are centered by Cu atoms. By substituting smaller Ni atoms for Cu atoms in Ti50Cu45Ni5 MG, the size of such clusters and their CNs decrease, resulting in a fraction of increase of the smaller clusters with 11 and 12 atoms. The clusters centered by Ti atoms have comparatively the larger atomic size and CN. When substituting smaller Ni atoms for Cu atoms, more atoms have chance to occupy the first nearest-neighbor sites, thus the CNs of the larger Ti atoms increase. Correspondingly, the number of 16-atom clusters increases. A two-dimensional cluster transformation is illustrated in Fig. 4. The clusters are taken in such a way that they contain the first nearest neighbor atoms. Each cluster has a different local topological condition. The numerical density within the first nearest-neighbor distance is inhomogeneous. The cluster distributions are diverse for alloys with distinct components. Obviously, the clusters can be classified by the CNs. To quantitatively represent the disordered degree of cluster distribution, a cluster distribution entropy (CDE) is evaluated in terms of the expression of
Ti50Cu45Ni5 Ti50Cu50
3
4
r(Å)
Ti50Cu45Ni5
35
Ti50Cu50
Δ g(r) 30 25
Fraction
PCF
2
1
20 15 10 5
0 2
4
6
8
(Å)
0 9
10
11
12
13
14
15
Coordination number Fig. 1. Pair correlation functions g(r) of Ti50Cu50 and Ti50Cu45Ni5 MGs. The green line shows the difference △g between the two functions.
Fig. 3. Coordination number distributions of the Ti50Cu50 and Ti50Cu45Ni5 MGs.
16
18
(a)
Y.L. Sun et al. / Journal of Non-Crystalline Solids 398–399 (2014) 16–18
(b)
4. Conclusions In summary, the microstructures of Ti50Cu50 and Ti50Cu45Ni5 metallic glasses are investigated by numerical simulations. Cluster distribution entropy is proposed to descript the disorder degree of BMGs. It is proved that the parameter can effectively describe the microstructures of metallic glasses from the perspective of cluster distribution. It is hoped that the investigations will give some inspiration to the composition design of new BMGs.
Fig. 4. Two-dimensional schematics of clusters in (a) Ti50Cu50 and (b) Ti50Cu45Ni5 alloys. The dashed circles represent the cutoff distance that defines the clusters. The centered Ti, Cu, and Ni atoms are plotted in red, green, and blue respectively.
the Shannon entropy H(P) = − ∑j Pj ln Pj, where P = (P1,…,PN) is the probability distribution of clusters with ∑jPj = 1 and j = 1,…,N [20]. Using above expression, the CDE of Ti50Cu50 and Ti50Cu45Ni5 MGs are obtained to be H = 1.48 and H = 1.56 respectively. The addition of Ni leads to an increase of the CDE. Element addition may increase the structural complexity by offering more atomic size discrepancy. According to the atomic-level stress theory [21,22], the topological fluctuation of local structure is well described by the atomic level strains. The local atomic strains increase with the size differences between the component elements. When adding small atom Ni in Ti50Cu50 MGs, the size difference between Ni and Ti increases, but the size difference between Ni and Cu decreases. The atomic environment of each atom is severely distorted, enlarging the local atomic strain fluctuation. Thus the probabilities of the clusters with larger or smaller CNs, for instance, CN = 10, 11 and 15, tend to increase, correspondingly, the CDE of the metallic glass increases. Here, we propose a statistical approach to describe the structure of BMGs. This approach is based on the fluctuation of cluster distribution. Under certain simulation conditions, CDE represents the inherent structural property of BMG. This parameter makes it possible to compare the atomic structures of different alloys quantitatively.
Acknowledgment This work was financially supported by National Natural Science Foundation of China (NSFC) under grant No. 11204200. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
[17] [18] [19] [20] [21] [22]
Z.Y. Zhang, Y. Wu, J. Zhou, H. Wang, X.J. Liu, Z.P. Lu, Scr. Mater. 69 (2013) 73. P. Gong, K.F. Yao, X. Wang, Y. Shao, Prog. Nat. Sci. 22 (2012) 401. G. He, M. Hagiwara, Mater. Sci. Eng. C 26 (2006) 14. W.H. Wang, Prog. Mater. Sci. 52 (2007) 540. Y.L. Wang, E. Ma, J. Xu, Philos. Mag. Lett. 88 (2008) 319. H. Men, S. Pang, A. Inoue, T. Zhang, Mater. Trans. 46 (2005) 2218. E. Yin, S. Pang, X. Hui, M. Zhang, L. Zhuo, C. Chen, T. Zhang, J. Alloys Compd. 546 (2013) 7. P. Gong, X. Wang, Y. Shao, N. Chen, X. Liu, K.F. Yao, Intermetallics 43 (2013) 177. P. Li, G. Wang, D. Ding, J. Shen, Mater. Des. 53 (2014) 145. J. Li, X. Gu, T.C. Hufnagel, Microsc. Microanal. 9 (2003) 509. P.J. Steinhardt, D.R. Nelson, M. Ronchetti, Phys. Rev. Lett. 47 (1981) 1297. D.B. Miracle, Nat. Mater. 3 (2004) 697. H.W. Sheng, W.K. Luo, F.M. Alamgir, J.M. Bai, E. Ma, Nature 439 (2006) 419. P. Li, G. Wang, D. Ding, J. Shen, J. Non-Cryst. Solids 358 (2012) 3200. X.F. Wu, Z.Y. Suo, Y. Si, L.K. Meng, K.Q. Qiu, J. Alloys Compd. 452 (2008) 268. X.W. Zhou, H.N.G. Wadley, R.A. Johnson, D.J. Larson, N. Tabat, A. Cerezo, A.K. Petford-long, G.D.W. Smith, P.H. Clifton, R.L. Martens, T.F. Kelly, Acta Mater. 49 (2001) 4005. J.L. Finney, Nature 266 (1977) 309. V.A. Borodin, Philos. Mag. A 79 (1999) 1887. H.J. Lee, T. Cagin, W.L. Johnson, W.A. Goddard, J. Chem. Phys. 119 (2003) 9858. H. Maassen, J.B.M. Uffink, Phys. Rev. Lett. 60 (1988) 1103. T. Egami, J. Non-Cryst. Solids 317 (2003) 30. T. Egami, JOM 62 (2010) 70.
Contents lists available at ScienceDirect
Journal of Non-Crystalline Solids journal homepage: www.elsevier.com/ locate/ jnoncrysol
Cluster distribution entropy in Ti50Cu50 and Ti50Cu45Ni5 metallic glasses Yong Li Sun a,⁎, Ya Juan Sun b, Ping Yang a, Hua Ming Li a a b
College of Physics and Optoelectronics, Taiyuan University of Technology, Taiyuan, 030024, China School of Science, Tianjin Polytechnic University, Tianjin, 300160, China
a r t i c l e
i n f o
Article history: Received 12 February 2014 Received in revised form 18 April 2014 Available online xxxx Keywords: Amorphous metals; Metallic glasses; Molecular dynamics simulations
a b s t r a c t It remains unclear how a cluster packing pattern affects the glass forming ability of bulk metallic glasses. The topological structures of Ti50Cu50 and Ti50Cu45Ni5 metallic glasses obtained by molecular dynamics simulations are compared to investigate the relation between the atomic structure and glass forming ability. The structural analysis shows that the addition of Ni in Ti50Cu50 alloy can significantly change the topological structure. Cluster distribution entropy is proposed to quantitatively descript the disordered degree of cluster distribution. It is shown that the cluster distribution entropy provides a statistical approach for assessing the microstructures of metallic glasses. © 2014 Elsevier B.V. All rights reserved.
1. Introduction Recently, Ti-based bulk metallic glasses (BMGs) have attracted significant attention because of their advantageous properties, such as high specific strength and low density [1,2]. The potential applications promote the exploration of new BMGs [3]. It is proved that the glass forming ability (GFA) of BMGs is quite sensitive to composition. Element addition could significantly improve the GFA and enhance the plasticity and strength of BMGs [4]. By this method, a number of Ti-based BMGs with the critical diameters range from 1 mm to 32 mm have been prepared by copper mold casting [5–9]. However, the reasons why alloy composition affects the GFA are still obscure. Atomic structure is one of the significant reasons affecting GFA. Generally, element additions may effectively improve the GFA due to the formation of a denser atomic packing or an increased complexity in the glass-forming liquid. Icosahedral clusters in BMGs have been confirmed by both experiments and simulations [10,11]. Therefore, more efforts have been made to construct the cluster packing models. Under these conditions, efficient cluster packing model and fivefold symmetry packing scheme were proposed to describe the cluster packing pattern for metallic glasses [12,13]. These models depict the cluster packing geometrically, however, how to statistically express the cluster packing pattern remains ambiguous. The effect of element additions on GFA in the view of microstructure would be worth investigating further. Simple alloys are convenient for analyzing the structural properties. Although no binary Ti-based BMG has been formed so far, several ternary alloys have been prepared in Ti–Cu–Ni alloy system [14,15]. The
⁎ Corresponding author. E-mail address: [email protected] (Y.L. Sun).
http://dx.doi.org/10.1016/j.jnoncrysol.2014.04.020 0022-3093/© 2014 Elsevier B.V. All rights reserved.
addition of Ni can effectively improve the GFA of Ti-based alloys. In this paper, we study the atomic substitution effect in Ti50Cu50 and Ti50Cu45Ni5 metallic glasses (MGs). Atomic level structure can be investigated utilizing classical molecular dynamics (MD) simulations, in which three-dimensional atomic configurations could be constructed. The structural distinction between the two alloys is obtained via statistical analysis of cluster distributions. We intend to provide some structural understanding of the GFA of BMGs. 2. Simulations and methods The accuracy of MD simulation depends on the selected atomic interaction potential. In this work, generalized embedding atomic method potential was adopted [16]. The total energy of this potential takes the ! form: Ei ¼ F α ∑ ραβ r ij þ 12 ∑ ϕα β r ij , where α and β denote the j≠i
j≠i
elemental types of atom i and j, rij is the interatomic distance. ϕαβ(rij) is the two-body potential, F is the embedding energy and ραβ(rij) is electron density function. The details and parameters about this potential can refer to Ref. [16]. Molecular dynamics simulations were performed in a cubic cell of 500 atoms with periodic boundary condition. The simulations were carried out in the NPT (constant pressure and temperature) ensemble using Nose thermostat to control the pressure and temperature. The Gear predictor-corrector algorithm was used to integrate the equations of motion. The pressure was kept at 0 Pa. The system was first melted and equilibrated at 2000 K for 20 ps with a time step of 1 fs and then cooled down to 300 K with the cooling rate 1.7 × 1014 Ks−1. At 300 K, the system was relaxed for 50 ps to collect the relative thermal and structural information.
Y.L. Sun et al. / Journal of Non-Crystalline Solids 398–399 (2014) 16–18
17
3. Results and discussion 5
Ti-Cu Cu-Cu
4
PPCF
Generally, high packing fraction implies large GFA of metallic materials. The density (mass of the system divided by the system volume) of Ti50Cu45Ni5 MG derived from the simulation is ρ = 6.122 g/cm3, which is slightly smaller than that of Ti50Cu50 MG, ρ = 6.137 g/cm3. But the packing fractions of the two alloys are identical, both of them are φ = 0.54. Thus, the two alloys have similar degree of atomic packing. The structural reasons for distinct GFA may lie in the topological structures of the two alloys. Fig. 1 shows the total pair correlation functions g(r) of Ti50Cu50 and Ti50Cu45Ni5 MGs and the difference between them, i.e. Δg ðr Þ ¼ g Ti50 Cu45 Ni5 ðrÞ−g Ti50 Cu50 ðr Þ. The spitting second peaks in the g(r) curves indicate the formation of glassy state in both model systems. It is observed that the two curves overlapped especially well, indicating that the local structure of Ti50Cu45Ni5 MG is nearly same as that of Ti50Cu50 MG in spite of the substitution of Ni for Cu. The slight difference is illustrated in the Δg(r) curve. There is a peak at r = 2.4 Å and a valley at r = 2.65 Å. It indicates that Ti50Cu45Ni5 MG has more short bonds than Ti50Cu50 MG, involving the Ti–Ni, Cu–Ni and Ni–Ni bonds. The addition of short bonds is in favor of large GFA of Ti50Cu45Ni5 MG. In order to achieve the addition effect of Ni, partial pair correlation functions (PPCF) of the Cu–Cu, Ti–Ti and Ti–Cu pairs of the two alloys are depicted in Fig. 2. Slight changes are detected in the peak heights of the two alloys. The peak heights of first nearest neighbor for Cu–Cu and Ti–Ti pairs increase, but that for Ti–Cu pair decreases with the addition of Ni. The result implies that the substitution of Ni for Cu atoms leads to a change in the local short order. The Ni addition makes the Cu and Ti more ordered. The reduction of the peak height of Ti–Cu pair is due to the decrease of the Ti–Cu pair which is substituted by Ti–Ni pair. The Voronoi method could be used to characterize the nearestneighbor coordination number (CN) and local atomic environment [17,18]. The result of voronoi analysis is sensitive to the selected cutoff distance. When a cutoff distance of 3.6 Å is used, the dominant CN is 13 in both BMGs. To obtain a cluster distribution dominated by icosahedron-like clusters, the cutoff distance was selected to be rc = 3.4 Å. After relaxation for 20 ps, 30 configurations with the interval of 1 ps were extracted and averaged. Fig. 3 shows the distributions of clusters with different CNs in the Ti50Cu50 and Ti50Cu45Ni5 BMGs. In both metallic glasses, the dominant clusters are icosahedron-like clusters with CN = 12, which involve the ideal icosahedra with the index of b0,0,12,0N and distorted icosahedra with the index of b0,2,8,2N and b0,3,6,3N. The distribution of CNs changes when substituting Ni
3
Ti-Ti
2
1
0 2
3
5
6
Fig. 2. Partial pair correlation functions of Ti50Cu50 and Ti50Cu45Ni5 MGs. The solid lines correspond to Ti50Cu50 alloy and the dashed lines to Ti50Cu45Ni5 alloy.
atoms for a portion of Cu atoms. The fractions increase for clusters with CNs of 10, 11 and 15, but decrease for clusters with CNs of 12, 13 and 14. This occurs because atomic size ratio affects the local orders in metallic alloys [19]. The size ratio of Cu atom to Ti atom is 0.886, while the size ratio of Ni atom to Ti atom is 0.871. The atomic radius of Ni is less than that of Cu. For Ti50Cu50 alloy, some of the clusters with 13, 14 or 15 atoms are centered by Cu atoms. By substituting smaller Ni atoms for Cu atoms in Ti50Cu45Ni5 MG, the size of such clusters and their CNs decrease, resulting in a fraction of increase of the smaller clusters with 11 and 12 atoms. The clusters centered by Ti atoms have comparatively the larger atomic size and CN. When substituting smaller Ni atoms for Cu atoms, more atoms have chance to occupy the first nearest-neighbor sites, thus the CNs of the larger Ti atoms increase. Correspondingly, the number of 16-atom clusters increases. A two-dimensional cluster transformation is illustrated in Fig. 4. The clusters are taken in such a way that they contain the first nearest neighbor atoms. Each cluster has a different local topological condition. The numerical density within the first nearest-neighbor distance is inhomogeneous. The cluster distributions are diverse for alloys with distinct components. Obviously, the clusters can be classified by the CNs. To quantitatively represent the disordered degree of cluster distribution, a cluster distribution entropy (CDE) is evaluated in terms of the expression of
Ti50Cu45Ni5 Ti50Cu50
3
4
r(Å)
Ti50Cu45Ni5
35
Ti50Cu50
Δ g(r) 30 25
Fraction
PCF
2
1
20 15 10 5
0 2
4
6
8
(Å)
0 9
10
11
12
13
14
15
Coordination number Fig. 1. Pair correlation functions g(r) of Ti50Cu50 and Ti50Cu45Ni5 MGs. The green line shows the difference △g between the two functions.
Fig. 3. Coordination number distributions of the Ti50Cu50 and Ti50Cu45Ni5 MGs.
16
18
(a)
Y.L. Sun et al. / Journal of Non-Crystalline Solids 398–399 (2014) 16–18
(b)
4. Conclusions In summary, the microstructures of Ti50Cu50 and Ti50Cu45Ni5 metallic glasses are investigated by numerical simulations. Cluster distribution entropy is proposed to descript the disorder degree of BMGs. It is proved that the parameter can effectively describe the microstructures of metallic glasses from the perspective of cluster distribution. It is hoped that the investigations will give some inspiration to the composition design of new BMGs.
Fig. 4. Two-dimensional schematics of clusters in (a) Ti50Cu50 and (b) Ti50Cu45Ni5 alloys. The dashed circles represent the cutoff distance that defines the clusters. The centered Ti, Cu, and Ni atoms are plotted in red, green, and blue respectively.
the Shannon entropy H(P) = − ∑j Pj ln Pj, where P = (P1,…,PN) is the probability distribution of clusters with ∑jPj = 1 and j = 1,…,N [20]. Using above expression, the CDE of Ti50Cu50 and Ti50Cu45Ni5 MGs are obtained to be H = 1.48 and H = 1.56 respectively. The addition of Ni leads to an increase of the CDE. Element addition may increase the structural complexity by offering more atomic size discrepancy. According to the atomic-level stress theory [21,22], the topological fluctuation of local structure is well described by the atomic level strains. The local atomic strains increase with the size differences between the component elements. When adding small atom Ni in Ti50Cu50 MGs, the size difference between Ni and Ti increases, but the size difference between Ni and Cu decreases. The atomic environment of each atom is severely distorted, enlarging the local atomic strain fluctuation. Thus the probabilities of the clusters with larger or smaller CNs, for instance, CN = 10, 11 and 15, tend to increase, correspondingly, the CDE of the metallic glass increases. Here, we propose a statistical approach to describe the structure of BMGs. This approach is based on the fluctuation of cluster distribution. Under certain simulation conditions, CDE represents the inherent structural property of BMG. This parameter makes it possible to compare the atomic structures of different alloys quantitatively.
Acknowledgment This work was financially supported by National Natural Science Foundation of China (NSFC) under grant No. 11204200. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
[17] [18] [19] [20] [21] [22]
Z.Y. Zhang, Y. Wu, J. Zhou, H. Wang, X.J. Liu, Z.P. Lu, Scr. Mater. 69 (2013) 73. P. Gong, K.F. Yao, X. Wang, Y. Shao, Prog. Nat. Sci. 22 (2012) 401. G. He, M. Hagiwara, Mater. Sci. Eng. C 26 (2006) 14. W.H. Wang, Prog. Mater. Sci. 52 (2007) 540. Y.L. Wang, E. Ma, J. Xu, Philos. Mag. Lett. 88 (2008) 319. H. Men, S. Pang, A. Inoue, T. Zhang, Mater. Trans. 46 (2005) 2218. E. Yin, S. Pang, X. Hui, M. Zhang, L. Zhuo, C. Chen, T. Zhang, J. Alloys Compd. 546 (2013) 7. P. Gong, X. Wang, Y. Shao, N. Chen, X. Liu, K.F. Yao, Intermetallics 43 (2013) 177. P. Li, G. Wang, D. Ding, J. Shen, Mater. Des. 53 (2014) 145. J. Li, X. Gu, T.C. Hufnagel, Microsc. Microanal. 9 (2003) 509. P.J. Steinhardt, D.R. Nelson, M. Ronchetti, Phys. Rev. Lett. 47 (1981) 1297. D.B. Miracle, Nat. Mater. 3 (2004) 697. H.W. Sheng, W.K. Luo, F.M. Alamgir, J.M. Bai, E. Ma, Nature 439 (2006) 419. P. Li, G. Wang, D. Ding, J. Shen, J. Non-Cryst. Solids 358 (2012) 3200. X.F. Wu, Z.Y. Suo, Y. Si, L.K. Meng, K.Q. Qiu, J. Alloys Compd. 452 (2008) 268. X.W. Zhou, H.N.G. Wadley, R.A. Johnson, D.J. Larson, N. Tabat, A. Cerezo, A.K. Petford-long, G.D.W. Smith, P.H. Clifton, R.L. Martens, T.F. Kelly, Acta Mater. 49 (2001) 4005. J.L. Finney, Nature 266 (1977) 309. V.A. Borodin, Philos. Mag. A 79 (1999) 1887. H.J. Lee, T. Cagin, W.L. Johnson, W.A. Goddard, J. Chem. Phys. 119 (2003) 9858. H. Maassen, J.B.M. Uffink, Phys. Rev. Lett. 60 (1988) 1103. T. Egami, J. Non-Cryst. Solids 317 (2003) 30. T. Egami, JOM 62 (2010) 70.