Simplified thermodynamic method to predict the glass formation of the ternary transition metal systems

Abstract

By considering the different effects of enthalpies on the glass formation of ternary transition metal systems, a thermodynamic method is proposed to predict the glass-forming regions as well as the glass-forming abilities. Cu–Zr–Ti and ten more ternary systems, as well as the corresponding binary subsystems, were studied, and the predictions agree well with the experimental observations.

---

Recently, bulk metallic glasses (BMGs) have attracted considerable attention due to their novel properties in physical and chemical aspects, such as ultra-high strength, excellent soft magnetic properties and good corrosion resistance.^1,2 In order to produce BMGs, one basic issue is to search the systems and compositions with which metallic glasses could be obtained, i.e., to predict the glass-forming range/region (GFR) of a metal system. Initially, BMGs were developed by experimental methods, e.g. liquid melt quenching (LMQ) and mechanical alloying (MA).^3,4 Obviously, it is time-consuming and inconvenient to find whether metallic glass could be formed in a specific system or not by the try and error process. Thus, in order to effectively predict the GFR, several empirical criteria and models have recently been proposed from both theoretical and experimental aspects for ternary systems, as most of the BMGs are obtained in systems with more than three constituent elements.^5–7 Among all the criteria and models developed, the extended Miedema's macroscopic atom models for ternary systems were widely acknowledged.⁸

According to Miedema's model, the GFR of a system can be determined by comparing the enthalpies of the solid solutions and glass phases, since from the thermodynamic point of view, the GFR of a system can be determined by comparing the Gibbs free energies of the possible competing phases (usually the substitutional solid solutions and the glass phases) and the entropy terms for the concentrated solid solutions and glasses are both regarded as the same as ideal solutions.^8,9 However, different results were obtained for ternary systems by different extended models.^10–14 Among these models, Gallego's method is widely adopted.¹⁵ Further considering the prediction of glass forming abilities (GFA), arguments, such as γ* and P_HS, were raised based on the extended models.^13,14,16 However, these arguments are either incompatible with binary subsystems or complicated to produce. Inoue and Lu argued that the GFA of the alloys depends not only on the liquidus and glass-transition temperature but also on the elastic enthalpy arising in a solid solution and the stability of the competing crystalline phases.^17,18 Thus, in order to simplify the calculation and develop the compatibility with binary subsystems, a simple method is proposed based on Gallego's method and Inoue and Lu's argument in this paper.

In Gallego's model, the mixing enthalpies for a ternary system A–B–C are obtained using Miedema's model for binary system and the enthalpy of solid solution (ΔH^SS_ABC) is written in the following form:

ΔH^SS_ABC = ΔH^SS(chem)_ABC + ΔH^SS(elastic)_ABC + ΔH^SS(struct)_ABC, (1)

where ΔH^SS(chem)_ABC, ΔH^SS(elastic)_ABC and ΔH^SS(struct)_ABC stand for chemical, elastic and structural contributions to enthalpy. These three enthalpy terms are obtained in the following equations:where ΔH^SS(struct)_ij originates from the valence and the crystal structure of the solvent and solute atoms and are assumed to be zero,¹² ΔH^SS(chem)_ij and ΔH^SS(elastic)_ij stand for chemical and elastic contributions to the enthalpies of binary systems calculated by following equations:

ΔH^SS(chem)_ij = c_ic_j(c^s_jΔH^inter(i in j) + c^s_iΔH^inter(j in i)), (5)

ΔH^SS(elastic)_ij = c_ic_j(c_iΔH^elastic(i in j) + c_jΔH^elastic(j in i)), (6)

where

Here, c_i and c_j are the compositions of the constituent elements, c^s_i and c^s_j are the surface fractions, V_i and V_j are the molar volumes, P and Q are empirical constants. Δϕ stands for the difference of work function of the electrons (φ_i − φ_j), n_ws stands for the electron density at the boundary of the Wigner–Seitz cell. K is the compressibility, G is the shear modulus. W_{i in j} and W_{j in i} are the corrected molar volumes due to the electron transfer and can be given by

where α depends on the values of V and n_ws.

For the metallic glass phases, both the elastic and the structural terms are absent, but the topological term should be taken into account. The enthalpy of formation (ΔH^Amor_ABC) can thus be written as:

ΔH^Amor_ABC = ΔH^Amor(chem)_ABC + ΔH^topo_ABC, (12)

where Tⁱ_m is the melting temperature of constituent element. According to the Inoue's discovery, short-range order effect could be neglected in ternary transition metal systems, except for Pd-based systems.¹⁸ Thus, for ternary transition metal systems besides Pd-based systems, ΔH^Amor(chem)_ABC is equal to ΔH^SS(chem)_ABC. Accordingly, the GFR of a ternary metal system could be determined quantitatively by

ΔH^SS_ABC − ΔH^Amor_ABC = ΔH^SS(elastic)_ABC − ΔH^topo_ABC > 0. (14)

It means that when the stabilizing ability of amorphous phases contributed by the mismatch enthalpy is bigger than the stabilizing ability of crystalline phases contributed by the topological enthalpy, metallic glasses are prone to be obtained.

Concerning the prediction of the GFA of each alloy, a parameter κ is proposed by comparing the positive effect of chemical enthalpy and the negative effect of topological enthalpy on glass formation.

where ΔH^Amor(chem)_ABC is equal to ΔH^SS(chem)_ABC which can be calculated by eqn (2). The larger the parameter κ, the higher the GFA.

The proposed method is first applied to the Cu–Zr–Ti system. The GFR of the system is first identified by eqn (14) and then the parameters κ is calculated by eqn (15). Fig. 1 shows the calculated κ in the GFR of the Cu–Zr–Ti system. The colourful areas with dots are the determined GFRs and different colours indicate the distribution of the values of κ. The yellow circles represent the compositions with which metallic glasses were obtained by LMQ or MA, and the blue star is the composition Cu₆₀Zr₃₀Ti₁₀ which was considered to have the best GFA by Inoue.¹⁹ One can see that all the experimental compositions are located in the GFR predicted. Meanwhile, it is found that the alloy with a composition of Cu₅₄Zr₃₆Ti₁₀ has the largest κ, as show in the Fig. 1 and Table 1. As mentioned above, large parameter κ stand for good GFA. Thus, the alloy with composition of Cu₅₄Zr₃₆Ti₁₀ is assumed to own the optimized GFA. The composition as well as vicinity compositions are marked by black dots in the Fig. 1. Apparently, the prediction by the proposed method and the experimental result are well accordant with each other.

Fig. 1 Calculated κ in the GFR of the Cu–Zr–Ti system. The black spots are the composition which has the largest κ, i.e., the best GFA, as well as vicinity compositions; the yellow circles are the compositions with which metallic glasses were obtained by LMQ or MA; the blue star is the composition which has the best GFA determined by experimental results.

Table 1 The compositions predicted with the largest κ, i.e., the superior GFAs, for some systems, as well as the corresponding experimental results^16,19,20

Systems	Largest κ	Predicted compositions	Experimental compositions
Cu–Zr–Ti	2.96	Cu54Zr36Ti10	Cu60Zr30Ti10
Cu–Hf–Ti	1.89	Cu52Hf30Ti18	Cu60Hf25Ti15
Cu–Hf–Ni	5.02	Cu12Hf36Ni52	Cu10Hf30Ni60
Cu–Zr–Fe	2.88	Cu54Zr38Fe8	Cu50Zr40Fe10
Fe–Hf–Ta	1.95	Fe52Hf36Ta12	Fe50Hf40Ta10
Ni–Nb–Ta	3.94	Ni52Nb28Ta20	Ni60Nb30Ta10

---

In order to further testify the validity of the proposed method, the glass formation of other ternary transition metal systems are predicted similarly. The predicted compositions with the largest κ, i.e., the superior GFAs and the experimental results for these systems are included for comparison in the Table 1.^10,19,20 From Table 1, it can be concluded that agreements were obtained in these ternary transition metal systems. It also can be seen that there are some little discrepancies between the predicted compositions and experimental observations. These discrepancies are thought to be based on the following three reasons: (1) limitation of the number of experimental results compared to the entire compositions of the systems, (2) simplification inherent in the present model, (3) kinetic factors were not taken in to account for the formation of glass phases. The first reason indicates that not all the compositions have been studied because of scientific and application interests. The second and the third reason would not affect the application of the proposed method, as the distribution of the values of κ is a continuous curved surface, i.e., the alloys with compositions nearby the one predicted with superior GFA may also have superior GFAs. Based on the discussion above, a conclusion can be drawn that the prediction is in well accordance with the experimental result.

To further check its compatibility with the binary subsystems, the GFRs and GFAs of Cu–Zr and Cu–Ti calculated by the proposed method are studied. For the Cu–Zr system, it predicted that the GFR is 6–86 at.% Zr and the Cu₆₀Zr₄₀ alloy own the largest κ, i.e., the superior GFA, which is nearly the same as the LMQ experiment result of 10–80 at.% Zr and Cu₆₀Zr₄₀.^21–23 For the Cu–Ti system, it turns out that the GFR is 24–60 at.% Ti and the Cu₅₆Ti₄₄ alloy own the largest κ, which is accordant with observed in experiment in the previous study.²³ Further, the comparison between the predicted GFRs of binary subsystems and those obtained by experiments, such as LMQ, ion beam mixing (IBM) and MA, is shown in the Table 2. Although some little deviations were observed for some binary subsystems, the predictions are accordant with the experiment results. These discrepancies are assumed to be caused by the limitation of experimental compositions and the non-ignorable short-range order effect for binary systems. Generally speaking, the proposed method for the ternary transition metal systems is compatible with the corresponding binary subsystems.

Table 2 The predicted GFRs of binary subsystems and the corresponding experimental results for some ternary transition metal systems^21–23

Systems	GFRs of binary subsystems
	Predicted			Experimental
A–B–C	A–B (at.% B)	A–C (at.% C)	B–C (at.% C)	A–B (at.% B)	A–C (at.% C)	B–C (at.% C)	Method
Cu–Zr–Ti	6–86	24–60	—	10–80	25–65	—	LMQ
Cu–Hf–Ti	6–86	24–60	—	30–70	25–65	—	LMQ
Cu–Hf–Ni	6–86	—	10–92	30–70	—	30–70	LMQ
Cu–Zr–Fe	6–86	—	12–90	10–80	—	18–81	LMQ
Cu–Fe–Ta	—	16–62	16–70	—	30–50	25–80	IBM
Fe–Hf–Ta	8–88	18–64	—	24–86	25–80	—	IBM
Cu–Zr–Ta	6–86	16–62	—	10–80	20–50	—	MA
Ni–Nb–Ti	12–76	20–78	—	38–60	28–72	—	MA
Ni–Nb–Ta	12–76	10–78	—	38–60	20–90	—	MA

---

Compared the results calculated for the ternary systems in the ref. 16, one can see clearly that the compositions with superior GFA obtained by the simplified method were accordant with the prediction by Wang's method. However, the GFR of Cu–Zr–Ti ternary system calculated by Wang's method is much small than the reasonable one. Further, the GFR of Cu–Ti binary subsystem and the composition with best GFA calculated by Wang's method are 58–62 at.% Ti and Cu₄₂Ti₅₈, respectively, indicating the incompatibility with the binary subsystems. Thus, the simplified method to predict the glass formation of ternary transition metal systems is much more effective.

Conclusions

In summary, considering the different effects of enthalpy terms, i.e., mismatch enthalpy, topological enthalpy and chemical enthalpy, on glass formation, and the negligible short-range order effect on ternary transition metal systems, a thermodynamic method was proposed to predict the GFRs as well as the GFAs of ternary transition metal systems. Applying the proposed method, the glass-formation in ten more ternary metal systems were predicted. With very few exceptions e.g. Pd-based systems, not only the GFR but also the alloys with superior GFA predicted in the present study are in good accordance with those reported from experimental observations.

Acknowledgements

The authors are grateful for the financial support from the National Natural Science Foundation of China (51131003), the Ministry of Science and Technology of China (973 Program 2011CB606301, 2012CB825700), and the Administration of Tsinghua University.

Notes and references

1. S. Pauly, J. Das, N. Mattern, D. H. Kim and J. Eckert, Intermetallics, 2009, 17, 453–462.
2. W. H. Wang, C. Dong and C. H. Shek, Mater. Sci. Eng., R, 2004, 44, 45–89.
3. H. W. Kui, A. L. Greer and D. Turnbull, Appl. Phys. Lett., 1984, 45, 615–616.
4. A. Inoue, T. Zhang and T. Masumoto, Mater. Trans. JIM, 1989, 30, 965–972.
5. W. C. Wang, J. H. Li, H. F. Yan and B. X. Liu, Scr. Mater., 2007, 56, 975–978.
6. Z. P. Lu and C. T. Liu, Phys. Rev. Lett., 2003, 91, 1–4.
7. A. Takeuchi and A. Inoue, Mater. Trans. JIM, 2000, 41, 1372–1378.
8. F. R. De Boer, R. Boom and W. C. M. Mattens, Cohesion in metals: Transition metal alloy, Elsevier, North Holland, Amsterdam, 1989, ch. 2, pp. 79–82.
9. B. X. Liu, W. S. Lai and Q. Zhang, Mater. Sci. Eng., R., 2000, 29, 1–48.
10. Z. Sniadecki, J. W. Narojczyk and B. Idzikowski, Intermetallics, 2012, 26, 72–77.
11. L. M. Wang, Y. J. Tian, R. P. Liu and W. H. Wang, Appl. Phys. Lett., 2012, 100, 261913.
12. N. Das, J. Mittra, B. S. Murty, S. K. Pabi, U. D. Kulkarni and G. K. Dey, J. Alloys Compd., 2013, 550, 483–495.
13. M. F. de Oliveira, J. Appl. Phys., 2012, 111, 23509.
14. S. Vincent, D. R. Peshwe, B. S. Murty and J. Bhatt, J. Non-Cryst. Solids, 2011, 357, 3495–3499.
15. L. J. Gallego, J. A. Somoza and J. A. Alonso, J. Phys.: Condens. Matter, 1990, 2, 6245–6250.
16. T. L. Wang, J. H. Li and B. X. Liu, Phys. Chem. Chem. Phys., 2009, 11, 2371–2373.
17. Z. P. Lu and C. T. Liu, Phys. Rev. Lett., 2003, 91, 115505.
18. A. Takeuchi and A. Inoue, Mater. Trans., 2001, 42, 1435–1444.
19. A. Inoue, W. Zhang, T. Zhang and K. Kurosaka, Acta Mater., 2001, 49, 2645–2652.
20. M. H. Lee, J. Y. Lee, D. H. Bae, W. T. Kim, D. J. Sordelet and D. H. Kim, Intermetallics, 2004, 12, 1133–1137.
21. J. H. Li, Y. Dai, Y. Y. Cui and B. X. Liu, Mat. Sci. Eng., R., 2011, 72, 1–28.
22. K. Brunelli, M. Dabalà, R. Frattini, G. Sandonà and I. Calliari, J. Alloys Compd., 2001, 317–318, 595–602.
23. C. Politis and W. L. Johnson, J. Appl. Phys., 1986, 60, 1147–1151.

---