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A criterion for the glass-forming ability of binary bulk metallic glasses
Journal of Non-Crystalline Solids xxx (xxxx) xxx–xxx
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A criterion for the glass-forming ability of binary bulk metallic glasses Jae Bok Ria,⁎, Zi Wenb, Qing Jiangb a
Faculty of Materials Engineering, Kim Chaek University of Technology, Kyogu 60, Yonggwang Street, Pyongyang, DPR of Korea Key Laboratory of Automobile material of Education and Department of Materials Science and Engineering, Jilin University, Changchun 130022, People's Republic of China b
A R T I C L E I N F O
A B S T R A C T
Keywords: Bulk metallic glasses (BMGs) Glass-forming ability (GFA) Binary phase diagram Eutectic composition Critical cooling rate Crystallization
We proposed a parameter ϕ = (ΔHmc − RΔTemix)/ΔHmc as a criterion for glass-forming ability (GFA) of binary alloys near eutectic composition using the thermodynamic data of the equilibrium phase diagram by considering the relative thermodynamic stability of the liquid phase. The larger φ is, the larger GFA is in different binary alloy systems. A linear fit of Rc = 7.73 × 1023 exp(−65.26φ) is obtained, where Rc denotes the critical cooling rate to form glass from liquid. Since φ is determined directly from thermodynamic data of elements and well known equilibrium phase diagram, GFA of binary alloys can be easily estimated without any thermal measurement after fabrication of BMGs.
1. Introduction Bulk metallic glasses (BMGs) have been extensively developed due to their excellent physical and mechanical properties, which leads to the foundation of many new BMGs experimentally [1,2]. Correspondingly, theoretical progress for finding new BMGs has also been made, which is related to the evaluation on the glass forming ability (GFA) of alloys. To do that, many criteria have been established. The earliest criterion is the reduced glass transition temperature Trg = Tg / Tm or improved Trg = Tg / Tl, where Tg, Tm and Tl are the glass transition temperature, melting temperature and liquidus temperature, respectively [3,4]. This criterion directly relates GFA to the relative value of Tg from Tm or Tl. Note that the liquidus temperature denotes a temperature in an alloy phase diagram, above which the phase is fully liquid. Thus, the improved criterion has clarified the actual definition of Tm. Let Tx denote the crystallization temperature for a heated glass with a certain heating rate, K = (Tx − Tg) / (Tl − Tx) [5] characterizes the relative stability of the liquid in terms of the corresponding Tl. γ = Tx / (Tg + Tl) is another more complicated consideration [6], where the importance of both Tg and Tl characterizing the stability of liquid is emphasized. These models are widely used as indicators of GFA for BMGs although there are some differences in their sense of GFA when the alloy system changes from one to another, or alloy composition changes for a given alloy system [7]. Since these temperatures need to be determined after a thermal analysis for BMGs, these parameters can hardly supply a composition-design principle to find a new BMG or a suitable composition for alloying design with the best GFA. The above models only consider some related characterized ⁎
temperatures. Other models suggest the close relationship between GFA and the thermodynamic data of alloys, γ∗ = Hamor/(Hinter − Hamor) is defined where Hamor and Hinter are the glass formation enthalpy and intermetallic compound formation enthalpy, respectively [8]. This parameter reveals that GFA is achieved against crystallization. Another criterion is ε = − ΔSmix/ΔHmix with Smix and Hmix being the mixing entropy and the mixing enthalpy, respectively [9]. This criterion reflects the atomic rearrangement-inhibiting ability (ARIA) of the alloys. Both parameters describe GFA of metallic glasses as a competition of disordering (amorphization) against ordering (crystallization) from two aspects. The problem of the above two criteria is that these thermodynamic data need to be calculated with many assumptions. In light of the above models, one can see there is a need to find a criterion for GFA of alloys, which is only based on well-known elemental thermodynamic data and phase diagram. It is clear that all elements and binary phase diagrams are well studied. Thus, once a model is based on the two sources, a best GFA of an alloy with a certain composition can be determined. In this contribution, a parameter φ as criterion for GFA of binary alloys is developed in light of the stability of liquid phase. This criterion can be determined by well-known thermodynamic data of elements and phase diagram without any data from an obtained BMG. Usefulness of φ is confirmed from result that φ is related to logRc linearly. 2. Model It is well known that the stability of a liquid phase decides GFA of alloys. This could be reflected by drop of Tl induced by alloying
Corresponding author. E-mail address: [email protected] (J.B. Ri).
http://dx.doi.org/10.1016/j.jnoncrysol.2017.06.004 Received 26 March 2017; Received in revised form 4 June 2017; Accepted 5 June 2017 0022-3093/ © 2017 Elsevier B.V. All rights reserved.
Please cite this article as: Ri, J.B., Journal of Non-Crystalline Solids (2017), http://dx.doi.org/10.1016/j.jnoncrysol.2017.06.004
Journal of Non-Crystalline Solids xxx (xxxx) xxx–xxx
J.B. Ri et al.
Table 1 Related parameters in Eq. (4). Hmc is the melting enthalpy of the alloy, Tmmix the ideal melting temperature of the alloy. Dmax is the typical thickness (or diameter), from references reported previously, and logRc is the critical cooling rate, calculated by using approximate formula Rc = 10/D2 (D in unit of cm) from Ref. 14.
Fig. 1. Schematic definition of ΔTemix in binary eutectic phase diagram.
compared with the corresponding i-th element melting temperature Tm,i. Since the minimum of Tl is the eutectic temperature of a given alloy Te, we define ΔTemix = Tmmix − Te as a scale reflecting the biggest stability ability of alloying for a liquid phase, where Tmmix = ∑inxiTm , i is the ideal melting temperature for a mechanically mixed alloy with xi being molar fraction of i-th component in an n-component alloy. Fig. 1 shows such a schematic binary eutectic-phase diagram for the above definition. The ΔTemixrepresents the stable degree of liquid phase and how deep the eutectic is at the eutectic composition in given alloy system. On the other hand, as well-known, the Gibbs free energy difference, ΔGl − s, between liquid(denote, l) and corresponding crystal(denote, s) has also played an important role in predicting the GFA of a glass former. At melting temperature, the Gibbs free energy difference, ΔGml − s = ΔHm − TmΔSm, where ΔHm is the molar melting enthalpy and ΔSm is the molar melting entropy. At any temperature T below Tm, ΔG can be expressed as follows [10]:
ΔG = ΔHm ⎛ ⎝ ⎜
Tm − T ⎞ Tm ⎠
φ
Dmax (mm)
logRc, (K/s)
Ref.
Cu46Zr54 Cu50Zr50 Cu60Zr40 Cu64Zr36 Cu64.5Zr35.5 Cu55Hf45 Cu65Hf35 Cu66Hf34 Ni60Nb40 Ni61.5Nb38.5 Ni62Nb38 Ni62.4Nb37.6 Ni80P20 Pd81Si19 Pd82Si18 Zr62Ni38 Zr65Be35 Zr70Fe30 Zr70Cu30 Zr70Pd30 Zr80Ni20
17.49 17.18 16.39 16.08 16.04 19.53 18.14 18.00 22.48 22.30 22.24 22.18 14.12 23.10 22.76 19.72 16.52 18.91 18.75 19.80 20.37
1773.61 1742.81 1665.80 1635.00 1631.15 1874.47 1759.65 1748.17 2137.00 2121.65 2116.55 2112.46 1446.10 1801.21 1802.62 1976.02 1929.10 2032.80 1896.83 2037.90 2047.94
1201 1196a 1158 1158 1158 1253 1243 1243 1451 1451 1451 1451 1143 1137 1093 1283 1238 1201 1268 1303 1233
4.76 4.54 4.22 4.00 3.93 5.16 4.29 4.20 5.70 5.57 5.53 5.50 5.47 5.52 6.02 5.76 6.91 6.91 5.23 6.11 6.77
0.73 0.74 0.74 0.75 0.76 0.73 0.76 0.77 0.75 0.75 0.75 0.75 0.61 0.76 0.73 0.71 0.58 0.63 0.72 0.69 0.67
2.0 2.0 1.5 2.0 2.0 1.5 2.0 2.0 1.0 1.5 2.0 2.0 – 6.0 – – – – – – –
2.4 2.4 2.6 2.4 2.4 2.6 2.4 2.4 3 2.6 2.4 2.4 6 1.4b 3.4 4 7 5.7 4.9 6 5.9
[14] [15] [16] [17] [18] [16] [20] [21] [22] [23] [8] [24] [25] [26] [24] [8] [27] [25] [28] [29] [25]
(4)
3. Results and discussion
(1)
As seen in Eq. (4), the values of φ will be changed from 0 to 1.0. As parameter φ reflects the relative decreasing degree of the drive force for crystallization near eutectic composition, the larger φ is, the better GFA will be. In terms of Eq.(4), values of φ of some binary alloys and BMGs are calculated and shown in Table 1 where the calculated Tmmixare also given, where Te is cited value [12] and the related elemental data are shown in Table 2. In the calculations, for Cu50Zr50 BMG, the Table 2 For some individual elements, its number Z, structure, melting temperature Tm,i, molar melting entropy ΔSm,i, molar melting enthalpy ΔHm,i, calculated by Eq. (4). The data for Tm,i and ΔSm,i cited from Ref. [12].
(2)
n i=1
RΔTemix (KJ/ mol)
where R is the gas constant to convert parameter φ to dimensionless measure.
ΔTemix
∑ (xi ΔHm,i)
Te (K)
Tmmix
In the light of Ziman liquid theory, supposing that the melt is an ideal solution, ΔHm can be calculated by following expression [11]:
ΔHmc =
Tmmix (K)
Now we can define the parameter φ for the estimate of GFA as follows:
⎟
ΔHm
Hmc (KJ/ mol)
a For Cu50Zr50 at the line compound composition, because of the absence of eutectic line, the melting point of the compound CuZr is introduced by using its binary phase diagram instead of Te. b For Pd81Si19, in order to unify, using its Dmax value, Rc is estimated though its Rc is reported to 8 K/s.
As shown in the Eq. (1), ΔHm ∝ ΔG and thus is related to the driving force for crystallization and reversely to GFA. In addition, one can simply consider that the ΔHm at Tm is the driving force for crystallization, and then, the larger ΔHm is, the more facile crystallization is, consequently, the more difficult the glass formation is. As mentioned above, metallic glass formation is a competing process of amorphous against crystallization and a destroying process of the stability of liquid phase. Therefore, we can here introduce a measure, ΔTemix/ΔHm, reflecting the decreasing degree of the drive force for the crystallization against the stability of liquid phase, and then;
GFA ∝
Alloy
(3)
where xi is the mole percent of i-th component, n is component number, ΔHm , i is molar melting enthalpy of i-th component and subscript c denotes a calculated value, but not an experimental one. On the other hand, Hm,i = Tm,iSm,i where Tm,i and Sm,i are melting temperature and the molar melting entropy of i-th component of the system [12]. 2
Z
Element
Structure
Tm,i, (K)
Sm,i (J/mol·K)
Hm,i (KJ/mol)
4 14 15 26 28 29 40 41 46 60 72
Be Si P Fe Ni Cu Zr Nb Pd Nd Hf
hcp fcc monocl. bcc fcc fcc hcp bcc fcc hcp hcp
1560.00 1687.00 317.30 1811.00 1728.30 1357.77 2127.85 2750.00 1828.00 1289.00 2506.00
5.061 29.762 2.077 7.623 10.114 9.768 9.868 10.909 9.155 5.541 10.852
8.00 50.21 0.66 13.81 17.48 13.26 21.10 30.00 16.73 7.14 27.20
Journal of Non-Crystalline Solids xxx (xxxx) xxx–xxx
J.B. Ri et al.
metastable eutectic composition [19]. In their results, great interest is that for bulk glass former Cu50Zr50 alloy, the main competitor crystalline phase is in the primary compound phase Cu51Zr14 of copper-rich side in metastable eutectic system. This is in good accordance with the fact that primary crystalline phase competing glass-forming phase in simple binary eutectic system such as AueSi, in which the first metallic glass was found. Nevertheless, in our current study, primary phase was considered as the compound CuZr, corresponding to the composition of amorphous phase, and its melting point and GFA of the alloy were evaluated using phase diagram and ideal mixing enthalpy, respectively. In other words, on the basis of the earlier results that the glass-forming process of binary alloys is the competitive process between crystallization and amorphization process, the primary phase disturbing the glass-forming at the melting point was considered to be a crystalline phase with the same composition as the corresponding glass alloy. This is based on the fact that Gibbs free energy difference of liquid and solid phase is related to the GFA, no matter whether the crystalline phase is solid solution or compound as seen in Eq. (2) and Eq. (3). The parameter φ defined in the current study cannot identify the primary phase of solidification as seen in Ref. [19]. If a crystalline phase competing with a glass former is known and the thermodynamic data were added to the parameter φ, the reliability and accuracy will be higher. But the glass forming process is very complicated and the study on the physical property of the melt and its metallurgy of solidification is still under development, so our assumptions suggested in the definition of parameter φ do not seem to be too unreasonable. Though parameter φ is yet theoretically derived and not experimentally proved, it is confirmed that φ involves practical and potential utilities in terms of its effectiveness in the reliability and availability as seen in Fig.2. The use of this criterion is simple relatively to compare with the previous criteria because it can be determined by well-known elemental thermodynamic data and the equilibrium phase diagram without any additional measurement. Since all of the above consideration has not limited the component numbers of alloy systems although the present results concerned with only binary systems, it could be extended to ternary or multi-component systems when the corresponding phase diagrams are available.
Fig. 2. log Rc(φ) function for binary BMGs where the data cited from Table 1.
composition is just a line compound. Its Tm here is used instead of Te. Table 1 also includes experimental data of the typical thickness (or diameter) Dmax of BMGs under the critical cooling rates Rc where Rc = 10/Dmax2 is assumed [13]. For Pd81Si19 BMG, although Rc = 8 K/ s [27], Rc is still estimated by its Dmax value in order to unify the method. Based on the data shown in Table 1, logRc(φ) is plotted, which is shown in Fig. 2 where logRc in terms of Fig. 2 has been fitted as a linear function.
logRc = (23.88 ± 2.12)–(28.34 ± 2.94) φ
(5-a)
or;
R c = 7.73 × 1023 exp( −65.26φ
(5-b)
4. Conclusions
As shown in Fig. 2, φ ∝ 1/log Rc is found. The result suggests that φ ≥ 0.73 is the condition to form BMGs. The parameter φ is sensitive to the alloy composition. For example, the composition difference of Pd82Si18 and Pd81Si19 alloys is only 1 at. %, the corresponding φ values however are 0.73 and 0.76, respectively. In our knowledge, no any criterion for GFA could show a so large change with this small composition difference. On the other hand, there are a few eutectics and intermetallic compounds in binary phase diagrams with better glass formers; for example, CueZr, typical glass-forming binary alloy system, has five eutectics and six intermetallic compounds are found. The influences of such eutectics and intermetallic compounds on the GFA of binary alloys should be taken into consideration over the wide range of composition. No glass formers have been found at eutectic composition of various binary systems with unsolved puzzle so far, better glass former (i.e., bulk glass formers) can be found within the narrow range of off-eutectic composition in the binary alloy systems with metallic elements [18]. In lights of these results, the present work suggests the way to research composition with the better GFA within the near eutectic. Wang et al. suggested that, for CueZr alloy system, Cu50Zr50(at.%) alloy, which is corresponding to composition of line-type compound CuZr, can be cast fully amorphous into a copper model in rod up to 2 mm in diameter and its formation can be explained by using deep metastable eutectic phase diagram, consisting of compound Cu51Zr14, the highest melting point among compounds in the system and terminal solid solution α-Zr, and the glass can be formed in the wide range of the
In summary, φ as a criterion for GFA of BMGs has been proposed in the present study, which reflects the decreasing degree of the drive force for the crystallization in light of the stability of liquid phase. It is found that GFA characterized by logRc could be a positive linear function of 1/φ. Evident advantages of the criterion are that φ reflects well the change of GFA when alloy composition varies a little in the same alloy system and the criterion only needs well-known elemental data and phase diagram. Acknowledgments The authors gratefully acknowledge the financial support from National Key Basic Research and Development Program (Grant No. 2004CB619301) and ″985 project″ of Jilin University. References [1] [2] [3] [4] [5] [6] [7] [8] [9]
3
A. Inoue, Acta Mater. 48 (2000) 279. W.H. Wang, C. Dong, C.H. Chek, Mat. Sci. Eng. A 44 (2004) 45. D. Turnbull, Contemp. Phys. 10 (1969) 473. Z.P. Lu, H. Tan, Y. Li, S.C. Ng, Scripta Mater. 42 (2000) 667. A. Hurby, Czech. J. Phys., Sect. B 22 (1972) 1187. Z.P. Lu, C.T. Liu, Acta Mater. 50 (2002) 3501. W. Chen, Y. Wang, J. Qiang, C. Dong, Acta Mater. 51 (2003) 1889. L. Xia, W.H. Li, S.S. Fang, B.C. Wei, Y.D. Dong, J. Appl. Phys. 99 (2006) 026103. Mingxu Xia, Shuguang Zhang, Chaoli Ma, Jianguo Li, Appl. Phys. Lett. 89 (2006) 091917.
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[21] G. Duan, D.H. Xu, W.L. Johnson, Metal. Mater. Trans. 36A (2005) 455. [22] H.J. Chang, E.S. Park, Y.S. Jung, M.K. Kim, D.H. Kim, J. Alloys Compd. 434-435 (2007) 156. [23] Z.W. Zhu, H.F. Zhang, D.G. Pan, W.S. Sun, Z.Q. Hu, Adv. Eng. Mater. 8 (2006) 953. [24] A. Takeuchi, A. Inoue, Mater. Sci. Engi. A 304-306 (2001) 446. [25] K. Lu, Acta Metall. Sin. 28 (1992) B17. [26] Yao Ke-Fu, Ruan Fang, Chin. Phys. Lett. 22 (2005) 1481. [27] K. Saksl, H. Franz, P. Jóvári, K. Klementiev, E. Welter, A. Ehnes, J. Saida, A. Inoue, J.Z. Jiang, Appl. Phys. Lett. 83 (2003) 3924. [28] B.Q. Chi, Q. Jiang, Adv. Eng. Mater. 7 (2005) 512. [29] J. Saida, M. Matsushita, A. Inoue, Appl. Phys. Lett. 79 (2001) 412.
[10] [11] [12] [13] [14] [15] [16] [17]
D. Turnbull, J. Appl. Phys. 21 (1950) 1022. A.-H. Cai, H. Chen, W.-K. An, J.-Y. Tan, Y. Zhou, Mater. Sci. Engi. A 457 (2007) 6. H.-J. Hoffmann, Mat.-wiss. u. Werkstofftech 34 (2003) 517. X.H. Lin, W.L. Johnson, J. Appl. Phys. 78 (1995) 6514. D.H. Xu, G. Duan, W.L. Johnson, Phys. Rev. Lett. 92 (2004) 245504. M.B. Tang, D.Q. Zhao, M.X. Pan, W.H. Wang, Chin. Phys. Lett. 21 (2004) 901. A. Inoue, W. Zhang, Mater. Trans. JIM 45 (2004) 584. D.H. Xu, B. Lohwongwatana, G. Duan, W.L. Johnson, C. Garland, Acta Mater. 52 (2004) 2621. [18] D. Wang, Y. Li, B.B. Sun, M.L. Sui, K. Lu, E. Ma, Appl. Phys. Lett. 84 (2004) 4029. [19] W.H. Wang, J.J. Lewandowski, A.L. Greer, J. Mater. Res. 20 (2005) 2307. [20] L. Xia, D. Ding, S.T. Shan, Y.D. Dong, J. Phys.:Condens. Matter. 18 (2006) 3543.
4
Contents lists available at ScienceDirect
Journal of Non-Crystalline Solids journal homepage: www.elsevier.com/locate/jnoncrysol
A criterion for the glass-forming ability of binary bulk metallic glasses Jae Bok Ria,⁎, Zi Wenb, Qing Jiangb a
Faculty of Materials Engineering, Kim Chaek University of Technology, Kyogu 60, Yonggwang Street, Pyongyang, DPR of Korea Key Laboratory of Automobile material of Education and Department of Materials Science and Engineering, Jilin University, Changchun 130022, People's Republic of China b
A R T I C L E I N F O
A B S T R A C T
Keywords: Bulk metallic glasses (BMGs) Glass-forming ability (GFA) Binary phase diagram Eutectic composition Critical cooling rate Crystallization
We proposed a parameter ϕ = (ΔHmc − RΔTemix)/ΔHmc as a criterion for glass-forming ability (GFA) of binary alloys near eutectic composition using the thermodynamic data of the equilibrium phase diagram by considering the relative thermodynamic stability of the liquid phase. The larger φ is, the larger GFA is in different binary alloy systems. A linear fit of Rc = 7.73 × 1023 exp(−65.26φ) is obtained, where Rc denotes the critical cooling rate to form glass from liquid. Since φ is determined directly from thermodynamic data of elements and well known equilibrium phase diagram, GFA of binary alloys can be easily estimated without any thermal measurement after fabrication of BMGs.
1. Introduction Bulk metallic glasses (BMGs) have been extensively developed due to their excellent physical and mechanical properties, which leads to the foundation of many new BMGs experimentally [1,2]. Correspondingly, theoretical progress for finding new BMGs has also been made, which is related to the evaluation on the glass forming ability (GFA) of alloys. To do that, many criteria have been established. The earliest criterion is the reduced glass transition temperature Trg = Tg / Tm or improved Trg = Tg / Tl, where Tg, Tm and Tl are the glass transition temperature, melting temperature and liquidus temperature, respectively [3,4]. This criterion directly relates GFA to the relative value of Tg from Tm or Tl. Note that the liquidus temperature denotes a temperature in an alloy phase diagram, above which the phase is fully liquid. Thus, the improved criterion has clarified the actual definition of Tm. Let Tx denote the crystallization temperature for a heated glass with a certain heating rate, K = (Tx − Tg) / (Tl − Tx) [5] characterizes the relative stability of the liquid in terms of the corresponding Tl. γ = Tx / (Tg + Tl) is another more complicated consideration [6], where the importance of both Tg and Tl characterizing the stability of liquid is emphasized. These models are widely used as indicators of GFA for BMGs although there are some differences in their sense of GFA when the alloy system changes from one to another, or alloy composition changes for a given alloy system [7]. Since these temperatures need to be determined after a thermal analysis for BMGs, these parameters can hardly supply a composition-design principle to find a new BMG or a suitable composition for alloying design with the best GFA. The above models only consider some related characterized ⁎
temperatures. Other models suggest the close relationship between GFA and the thermodynamic data of alloys, γ∗ = Hamor/(Hinter − Hamor) is defined where Hamor and Hinter are the glass formation enthalpy and intermetallic compound formation enthalpy, respectively [8]. This parameter reveals that GFA is achieved against crystallization. Another criterion is ε = − ΔSmix/ΔHmix with Smix and Hmix being the mixing entropy and the mixing enthalpy, respectively [9]. This criterion reflects the atomic rearrangement-inhibiting ability (ARIA) of the alloys. Both parameters describe GFA of metallic glasses as a competition of disordering (amorphization) against ordering (crystallization) from two aspects. The problem of the above two criteria is that these thermodynamic data need to be calculated with many assumptions. In light of the above models, one can see there is a need to find a criterion for GFA of alloys, which is only based on well-known elemental thermodynamic data and phase diagram. It is clear that all elements and binary phase diagrams are well studied. Thus, once a model is based on the two sources, a best GFA of an alloy with a certain composition can be determined. In this contribution, a parameter φ as criterion for GFA of binary alloys is developed in light of the stability of liquid phase. This criterion can be determined by well-known thermodynamic data of elements and phase diagram without any data from an obtained BMG. Usefulness of φ is confirmed from result that φ is related to logRc linearly. 2. Model It is well known that the stability of a liquid phase decides GFA of alloys. This could be reflected by drop of Tl induced by alloying
Corresponding author. E-mail address: [email protected] (J.B. Ri).
http://dx.doi.org/10.1016/j.jnoncrysol.2017.06.004 Received 26 March 2017; Received in revised form 4 June 2017; Accepted 5 June 2017 0022-3093/ © 2017 Elsevier B.V. All rights reserved.
Please cite this article as: Ri, J.B., Journal of Non-Crystalline Solids (2017), http://dx.doi.org/10.1016/j.jnoncrysol.2017.06.004
Journal of Non-Crystalline Solids xxx (xxxx) xxx–xxx
J.B. Ri et al.
Table 1 Related parameters in Eq. (4). Hmc is the melting enthalpy of the alloy, Tmmix the ideal melting temperature of the alloy. Dmax is the typical thickness (or diameter), from references reported previously, and logRc is the critical cooling rate, calculated by using approximate formula Rc = 10/D2 (D in unit of cm) from Ref. 14.
Fig. 1. Schematic definition of ΔTemix in binary eutectic phase diagram.
compared with the corresponding i-th element melting temperature Tm,i. Since the minimum of Tl is the eutectic temperature of a given alloy Te, we define ΔTemix = Tmmix − Te as a scale reflecting the biggest stability ability of alloying for a liquid phase, where Tmmix = ∑inxiTm , i is the ideal melting temperature for a mechanically mixed alloy with xi being molar fraction of i-th component in an n-component alloy. Fig. 1 shows such a schematic binary eutectic-phase diagram for the above definition. The ΔTemixrepresents the stable degree of liquid phase and how deep the eutectic is at the eutectic composition in given alloy system. On the other hand, as well-known, the Gibbs free energy difference, ΔGl − s, between liquid(denote, l) and corresponding crystal(denote, s) has also played an important role in predicting the GFA of a glass former. At melting temperature, the Gibbs free energy difference, ΔGml − s = ΔHm − TmΔSm, where ΔHm is the molar melting enthalpy and ΔSm is the molar melting entropy. At any temperature T below Tm, ΔG can be expressed as follows [10]:
ΔG = ΔHm ⎛ ⎝ ⎜
Tm − T ⎞ Tm ⎠
φ
Dmax (mm)
logRc, (K/s)
Ref.
Cu46Zr54 Cu50Zr50 Cu60Zr40 Cu64Zr36 Cu64.5Zr35.5 Cu55Hf45 Cu65Hf35 Cu66Hf34 Ni60Nb40 Ni61.5Nb38.5 Ni62Nb38 Ni62.4Nb37.6 Ni80P20 Pd81Si19 Pd82Si18 Zr62Ni38 Zr65Be35 Zr70Fe30 Zr70Cu30 Zr70Pd30 Zr80Ni20
17.49 17.18 16.39 16.08 16.04 19.53 18.14 18.00 22.48 22.30 22.24 22.18 14.12 23.10 22.76 19.72 16.52 18.91 18.75 19.80 20.37
1773.61 1742.81 1665.80 1635.00 1631.15 1874.47 1759.65 1748.17 2137.00 2121.65 2116.55 2112.46 1446.10 1801.21 1802.62 1976.02 1929.10 2032.80 1896.83 2037.90 2047.94
1201 1196a 1158 1158 1158 1253 1243 1243 1451 1451 1451 1451 1143 1137 1093 1283 1238 1201 1268 1303 1233
4.76 4.54 4.22 4.00 3.93 5.16 4.29 4.20 5.70 5.57 5.53 5.50 5.47 5.52 6.02 5.76 6.91 6.91 5.23 6.11 6.77
0.73 0.74 0.74 0.75 0.76 0.73 0.76 0.77 0.75 0.75 0.75 0.75 0.61 0.76 0.73 0.71 0.58 0.63 0.72 0.69 0.67
2.0 2.0 1.5 2.0 2.0 1.5 2.0 2.0 1.0 1.5 2.0 2.0 – 6.0 – – – – – – –
2.4 2.4 2.6 2.4 2.4 2.6 2.4 2.4 3 2.6 2.4 2.4 6 1.4b 3.4 4 7 5.7 4.9 6 5.9
[14] [15] [16] [17] [18] [16] [20] [21] [22] [23] [8] [24] [25] [26] [24] [8] [27] [25] [28] [29] [25]
(4)
3. Results and discussion
(1)
As seen in Eq. (4), the values of φ will be changed from 0 to 1.0. As parameter φ reflects the relative decreasing degree of the drive force for crystallization near eutectic composition, the larger φ is, the better GFA will be. In terms of Eq.(4), values of φ of some binary alloys and BMGs are calculated and shown in Table 1 where the calculated Tmmixare also given, where Te is cited value [12] and the related elemental data are shown in Table 2. In the calculations, for Cu50Zr50 BMG, the Table 2 For some individual elements, its number Z, structure, melting temperature Tm,i, molar melting entropy ΔSm,i, molar melting enthalpy ΔHm,i, calculated by Eq. (4). The data for Tm,i and ΔSm,i cited from Ref. [12].
(2)
n i=1
RΔTemix (KJ/ mol)
where R is the gas constant to convert parameter φ to dimensionless measure.
ΔTemix
∑ (xi ΔHm,i)
Te (K)
Tmmix
In the light of Ziman liquid theory, supposing that the melt is an ideal solution, ΔHm can be calculated by following expression [11]:
ΔHmc =
Tmmix (K)
Now we can define the parameter φ for the estimate of GFA as follows:
⎟
ΔHm
Hmc (KJ/ mol)
a For Cu50Zr50 at the line compound composition, because of the absence of eutectic line, the melting point of the compound CuZr is introduced by using its binary phase diagram instead of Te. b For Pd81Si19, in order to unify, using its Dmax value, Rc is estimated though its Rc is reported to 8 K/s.
As shown in the Eq. (1), ΔHm ∝ ΔG and thus is related to the driving force for crystallization and reversely to GFA. In addition, one can simply consider that the ΔHm at Tm is the driving force for crystallization, and then, the larger ΔHm is, the more facile crystallization is, consequently, the more difficult the glass formation is. As mentioned above, metallic glass formation is a competing process of amorphous against crystallization and a destroying process of the stability of liquid phase. Therefore, we can here introduce a measure, ΔTemix/ΔHm, reflecting the decreasing degree of the drive force for the crystallization against the stability of liquid phase, and then;
GFA ∝
Alloy
(3)
where xi is the mole percent of i-th component, n is component number, ΔHm , i is molar melting enthalpy of i-th component and subscript c denotes a calculated value, but not an experimental one. On the other hand, Hm,i = Tm,iSm,i where Tm,i and Sm,i are melting temperature and the molar melting entropy of i-th component of the system [12]. 2
Z
Element
Structure
Tm,i, (K)
Sm,i (J/mol·K)
Hm,i (KJ/mol)
4 14 15 26 28 29 40 41 46 60 72
Be Si P Fe Ni Cu Zr Nb Pd Nd Hf
hcp fcc monocl. bcc fcc fcc hcp bcc fcc hcp hcp
1560.00 1687.00 317.30 1811.00 1728.30 1357.77 2127.85 2750.00 1828.00 1289.00 2506.00
5.061 29.762 2.077 7.623 10.114 9.768 9.868 10.909 9.155 5.541 10.852
8.00 50.21 0.66 13.81 17.48 13.26 21.10 30.00 16.73 7.14 27.20
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metastable eutectic composition [19]. In their results, great interest is that for bulk glass former Cu50Zr50 alloy, the main competitor crystalline phase is in the primary compound phase Cu51Zr14 of copper-rich side in metastable eutectic system. This is in good accordance with the fact that primary crystalline phase competing glass-forming phase in simple binary eutectic system such as AueSi, in which the first metallic glass was found. Nevertheless, in our current study, primary phase was considered as the compound CuZr, corresponding to the composition of amorphous phase, and its melting point and GFA of the alloy were evaluated using phase diagram and ideal mixing enthalpy, respectively. In other words, on the basis of the earlier results that the glass-forming process of binary alloys is the competitive process between crystallization and amorphization process, the primary phase disturbing the glass-forming at the melting point was considered to be a crystalline phase with the same composition as the corresponding glass alloy. This is based on the fact that Gibbs free energy difference of liquid and solid phase is related to the GFA, no matter whether the crystalline phase is solid solution or compound as seen in Eq. (2) and Eq. (3). The parameter φ defined in the current study cannot identify the primary phase of solidification as seen in Ref. [19]. If a crystalline phase competing with a glass former is known and the thermodynamic data were added to the parameter φ, the reliability and accuracy will be higher. But the glass forming process is very complicated and the study on the physical property of the melt and its metallurgy of solidification is still under development, so our assumptions suggested in the definition of parameter φ do not seem to be too unreasonable. Though parameter φ is yet theoretically derived and not experimentally proved, it is confirmed that φ involves practical and potential utilities in terms of its effectiveness in the reliability and availability as seen in Fig.2. The use of this criterion is simple relatively to compare with the previous criteria because it can be determined by well-known elemental thermodynamic data and the equilibrium phase diagram without any additional measurement. Since all of the above consideration has not limited the component numbers of alloy systems although the present results concerned with only binary systems, it could be extended to ternary or multi-component systems when the corresponding phase diagrams are available.
Fig. 2. log Rc(φ) function for binary BMGs where the data cited from Table 1.
composition is just a line compound. Its Tm here is used instead of Te. Table 1 also includes experimental data of the typical thickness (or diameter) Dmax of BMGs under the critical cooling rates Rc where Rc = 10/Dmax2 is assumed [13]. For Pd81Si19 BMG, although Rc = 8 K/ s [27], Rc is still estimated by its Dmax value in order to unify the method. Based on the data shown in Table 1, logRc(φ) is plotted, which is shown in Fig. 2 where logRc in terms of Fig. 2 has been fitted as a linear function.
logRc = (23.88 ± 2.12)–(28.34 ± 2.94) φ
(5-a)
or;
R c = 7.73 × 1023 exp( −65.26φ
(5-b)
4. Conclusions
As shown in Fig. 2, φ ∝ 1/log Rc is found. The result suggests that φ ≥ 0.73 is the condition to form BMGs. The parameter φ is sensitive to the alloy composition. For example, the composition difference of Pd82Si18 and Pd81Si19 alloys is only 1 at. %, the corresponding φ values however are 0.73 and 0.76, respectively. In our knowledge, no any criterion for GFA could show a so large change with this small composition difference. On the other hand, there are a few eutectics and intermetallic compounds in binary phase diagrams with better glass formers; for example, CueZr, typical glass-forming binary alloy system, has five eutectics and six intermetallic compounds are found. The influences of such eutectics and intermetallic compounds on the GFA of binary alloys should be taken into consideration over the wide range of composition. No glass formers have been found at eutectic composition of various binary systems with unsolved puzzle so far, better glass former (i.e., bulk glass formers) can be found within the narrow range of off-eutectic composition in the binary alloy systems with metallic elements [18]. In lights of these results, the present work suggests the way to research composition with the better GFA within the near eutectic. Wang et al. suggested that, for CueZr alloy system, Cu50Zr50(at.%) alloy, which is corresponding to composition of line-type compound CuZr, can be cast fully amorphous into a copper model in rod up to 2 mm in diameter and its formation can be explained by using deep metastable eutectic phase diagram, consisting of compound Cu51Zr14, the highest melting point among compounds in the system and terminal solid solution α-Zr, and the glass can be formed in the wide range of the
In summary, φ as a criterion for GFA of BMGs has been proposed in the present study, which reflects the decreasing degree of the drive force for the crystallization in light of the stability of liquid phase. It is found that GFA characterized by logRc could be a positive linear function of 1/φ. Evident advantages of the criterion are that φ reflects well the change of GFA when alloy composition varies a little in the same alloy system and the criterion only needs well-known elemental data and phase diagram. Acknowledgments The authors gratefully acknowledge the financial support from National Key Basic Research and Development Program (Grant No. 2004CB619301) and ″985 project″ of Jilin University. References [1] [2] [3] [4] [5] [6] [7] [8] [9]
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