­­A thermodynamic approach to assess glass-forming ability of bulk metallic glasses

JI Xiu-lin (纪秀林)¹, PAN Ye(潘 冶)²

1. Engineering Research Center of Dredging Technology of Ministry of Education, Hohai University,

Changzhou 213022, China;

2. School of Materials Science and Engineering, Southeast University, Nanjing 211189, China

Received 28 September 2008; accepted 6 March 2009

Abstract: According to the Gibbs free energy difference between liquid and crystal, a thermodynamic glass-forming ability(GFA) parameter related to characteristic temperatures, onset crystallization temperature(T_x) and liquidus temperature(T_l), was proposed for evaluating the GFA of bulk metallic glasses(BMGs). The new parameter defined as ω=T_l(T_l+T_x)/(T_x(T_l-T_x)) has good correlation with the critical section thickness(Z_c) of Ca-Mg-Cu BMGs. Being verified by the glasses data, including oxide glasses, which were used to validate the former GFA parameters, ω is one of the most reliable and applicable GFA parameters among T_rg (=T_g/T_l), γ(=T_x/(T_l+T_g)), α (=T_x/T_l), δ (=T_x/(T_l-T_g), and so on. Finally, predicting GFA of Cu-Ag-Zr-Ti and Cu-Zr-Ti-Al BMGs using ω was compared with the experimental results.

Key words: glass-forming ability; bulk metallic glass; Gibbs free energy

1 Introduction

Whether the vitrification of an alloy is easy or difficult, glass-forming ability(GFA) is vital to develop new bulk metallic glasses(BMGs). Scientific efforts for quantification of GFA have been started when the first Au-Si metallic glass was reported[1]. TURNBULL[2] identified reduced glass transition temperature(T_rg), which is the ratio of glass transition temperature(T_g) to liquidus temperature(T_l), as a GFA gauge. INOUE et al [3] proposed the supercooled liquid region ?T_x (=T_x-T_g, where T_x is the onset crystallization temperature of the glass) to measure GFA. Since 2002, LU and LIU[4-6] have published a serial of papers to show the strongest correlation of the GFA parameter γ (=T_x/(T_l+T_g)) with the GFA of various glass formers among the parameters suggested so far. Therefore, γ has been frequently used to predict the GFA of alloys today.

However, the predominant situation of γ is challenged in recent two years by other GFA indicators, α(=T_x/T_l)[7], δ(=T_x/(T_l-T_g)[8], (=T_rg(?T_x/T_g)^0.143)[9], T_rx(=T_x/T_s, where T_s is onset temperature of solidification)[10], and so on. Origins of those GFA parameters mentioned above obtained from time—temperature—transformation(TTT) diagram analysis[5, 7-10] or kinetics analysis, such as viscosity[11], fragility[9] and homogenous nucleation and growth[4, 8]. Lately, LU et al[1] restated that γ was still the best GFA indicator in terms of reliability and applicability.

In this work, based on thermodynamic analysis, mainly Gibbs free energy difference between liquid and crystal, a new GFA parameter was proposed and the new glass criterion was validated with lots of experimental data to assess the GFA of BMGs.

2 Origins of new parameter

Gibbs free energy difference(?G_l-s) between liquid and crystal means the driving force of crystallization. In a supercooled alloying liquid, the less ?G_l-s means the more stable and the better GFA of the liquid. Thus, ?G_l-s plays an important role in appraising the GFA of BMGs and the GFA parameter maybe has a solid interrelationship with ?G_l-s. In this way, a new GFA indicator was developed by this approach.

Scientific efforts for ?G_l-s estimation of the super- cooled alloying liquid have started for a long time and many approximate expressions of ?G_l-s have been derived [12-14]. The expression developed by THOMPSON and SPAEPEN[12] is one of the most important and reliable:

                      (1)

where T_m is the melting temperature; ?H_m is the enthalpy of fusion; and T is the temperature of the supercooled liquid.

As discussed above, GFA of metallic glasses is associated with ?G_l-s. So, their relationship can be expressed as follows:

GFA∝∝                (2)

At slower cooling rates, the glass freezes at lower temperature. But the glass produced at higher cooling rates undergoes structural relaxation readily at lower temperature[15]. So, T_g detected from DSC (differential scanning calorimetry) is different from the actual freezing temperature of an metallic glass. But the expression of ?G_l-s is derived from continuous cooling process. Therefore, it is not surprising that the GFA parameter directly using Eq.(2) has no good correlation with GFA.

For glass transition of an alloying system, two equilibrium states, amorphous and liquid states, should be emphasized. From the viewpoint of heating process, the glass is still in amorphous state before system temperature exceeds T_x and the alloying system is not in a real liquid state before it reaches T_l. That is to say, T_x and T_l are thermal stability gauges of the glass and the liquid, respectively[7]. Thus, T_l but T_m characterizes the temperature of a liquid state and T_x but T_g characterizes the temperature of an amorphous state. Therefore, T and T_m in Eq.(2) are substituted by T_x and T_l, respectively, meaning that GFA is in inverse proportion to the Gibbs free energy difference of the alloying system from T_l to T_x. So, the following expression can be obtained from Eq.(2):

GFA∝?                      (3)

If there is no inner relationship between ?H_m and T_l (or T_x), the above expression can be simplified as

GFA∝                            (4)

Hence, the new parameter ω for inferring the relative GFA among BMGs is defined as

                              (5)

To reveal the validity of ω as a GFA parameter, Eq.(5) is deduced as:

 (6)

Compared with T_l, the difference between T_x and T_g is very little, usually less than 5% T_l for the characteristic temperatures of BMGs. Therefore, the GFA parameters, T_rg=T_g/T_l[2], γ=T_x/(T_g+T_l)[4-6] and α=T_x/T_l[7] are close, respectively, to T_x/T_l, T_x/(T_x+T_l) and T_x/T_l. All the GFA parameters, T_rg, γ and α can be treated as functions of T_x/T_l. On the other hand, Eq.(6) indicates that ω is also the function of T_x/T_l. And the positive correlation between ω and α is illustrated in Fig.1. So, ω not only has the accordant trends of GFA description with α, but also expands the difference of values between descriptive parameters. And it may be more effective than other GFA parameters.

Fig.1 Function relationship between GFA parameters ω and α

3 Results and discussion

3.1 Correlation between new parameters and GFA

The most direct GFA indicator is the critical cooling rate(R_c) and the critical section thickness(Z_c). Unfortunately, it is very difficult to measure R_c experimentally and Z_c is rough. Table 1 shows the glass transition temperature(T_g), onset crystallization temperature(T_x), liquidus temperature(T_l) and Z_c of Ca-Mg-Cu BMGs. The data in Table 1 were obtained from Ref.[16]. And the calculated GFA indicators (γ and ω) are also listed in Table 1. The relationships between ω, γ and Z_c are shown in Fig.2. In order to reveal how closely the GFA indicators ( γ and ω) correspond to the actual experimental data of Z_c, the statistical correlation parameter, R², was also computed. The higher the R², the more reliable the GFA parameter. As shown in Fig.2, R² of ω is 0.70, but the corresponding R² of γ is 0.64, indicating the new GFA parameter ω is more reliable than γ for Ca-Mg-Cu BMGs.

Table 1 T_g, T_x, T_l, T_m, Z_c, ?H_m and calculated γ and ω for Ca-Mg-Cu BMGs

Fig.2 Correlation between Z_c and ω (a), γ (b) for Ca-Mg-Cu BMGs

3.2 Comparison of ω and other GFA parameters

To verify the reliability of the new GFA parameter and to compare it with others, more BMGs data with their characteristic temperatures (T_g, T_x and T_l) and Z_c, together with more calculated GFA parameters are listed in Table 2. The data in Table 2 were obtained from Table 1 in Ref.[4] and Table 1 in Ref.[8], which are used to interpret γ and α. The relationships between GFA indicators (T_rg, γ, α, δ, and ω) and the actual experimental data of Z_c, are shown in Fig.3. By comparing R² of every GFA indicator shown in Figs.3(a)-(f), R² of ω is obviously higher than others, and the order of GFA indicators from the best to the worst is ω, δ, α, , γ and T_rg. This result suggests the ω is the best GFA parameter. The relationship between the GFA and ω for these BMGs can be expressed as

Z_c=-77.98+11.93ω                            (7)

Table 2 T_g, T_x, T_l, Z_c and calculated T_rg, γ, α, δ,  and ω for reported BMGs

Fig.3 Correlations between Z_c and T_rg (a), γ (b), α (c), δ (d), (e) and ω (f) for BMGs

Besides T_rg, γ, α, δ and , other GFA parameters, such as ?T_rg (=(T_x-T_g)/(T_l-T_g))[17], ?T_x, T_rgx (=(T_gT_x)/ (T_lT_m))[18] and K_gl (=(T_x-T_g)/(T_m-T_x))[19] are also compared with ω. R² of ?T_rg and ?T_x calculated with data in Table 2 are 0.27 and 0.13, respectively, which are clearly lower than 0.53 of ω. From the viewpoint of emphasis on the two states, the relationships between GFA and those GFA parameters which import other characteristic temperatures beyond the amorphous and the liquid state will be injured. For example, T_g/T_l is a better representation of GFA than T_g/T_m[20]. R² of T_rgx and K_gl calculated with data in Table 1 are 0.53 and 0.42, respectively, which are obviously lower than 0.64 of γ and 0.70 of ω. On the other hand, except α and ω, all other GFA parameters mentioned above need the knowledge of T_g or T_s. However, T_g is not usually available for some glasses[7] and T_s needs to be measured during cooling of liquid, which increases the operational complexity. Based on above comparison, it can be concluded that ω is one of the most reliable and applicable GFA parameters.

3.3 Comparison of ω in glasses including oxides

To verify the applicability of the new GFA parameter, T_g, T_x, T_l and R_c of some BMGs and oxide glasses are listed in Table 3, together with the calculated GFA parameters. The data of BMGs and oxide glasses in Table 3 were obtained from Table 3 in Ref.[4] and  Table 2 in Ref.[6], respectively, which are applied to interpreting γ. According to the data in Table 3, six plots of the critical cooling rate as the functions of the parameters, T_rg, γ, α, δ, and ω, respectively, are shown in Fig.4. The statistical factor R² for these regressions are high among which R² of ω is the highest. This suggests a reliable relationship between the GFA and ω for these BMGs and oxide glasses. And this relationship is expressed in an approximation formula:

lg R_c=6.72-0.80ω                             (8)

Table 3 T_g, T_x, T_l, R_c and calculated T_rg, γ, α, δ, and ω for reported BMGs and oxide glasses

Fig.4 Correlations between R_c and T_rg (a), γ (b), α (c), δ (d), (e) and ω (f) for glasses including BMGs and oxides

3.4 Applications of ω in bulk metallic glasses

As an applicable GFA parameter, ω should indicate the actual GFA of BMGs. A centimeter-diameter Cu-based BMG had been developed[21]. Their experimental data and the calculated values are listed in Table 4. Values of ω and the calculated values of Z_c are computed with Eq.(5) and Eq.(7), respectively. The relationship between the experimental values of Z_c (Z_c^Exp) and the calculated values of Z_c (Z_c^Cal) is shown in Fig.5. Although the values of Z_c^Exp are not usually equal to the values of Z_c^Cal for every alloying composition, they present a solid linear relationship with 0.92 of R². Therefore, with the knowledge of T_x and T_l, the critical section thickness and the critical cooling rate of the alloy can be estimated approximately using Eqs.(7) and (8), respectively. Of course, these two expressions would be refined by more data of glasses to predict GFA of BMGs.

Table 4 T_g, T_x, T_l, actual critical section thickness Z_c^Exp and calculated ω and Z_c^Cal of reported Cu-Ag-Zr-Ti alloys

Fig.5 Actual critical section thickness Z_c^Exp vs calculated critical section thickness Z_c^Cal for Cu-based BMGs

Moreover, GFA of Cu-Zr-Ti-Al alloys [22] is described by ω parameter. The characteristic temperatures of the alloys are listed together with GFA parameters in Table 5. The results of scanning electron microscopy(SEM) and X-ray diffractometry(XRD) indicated the GFA of Cu-Zr-Ti-Al alloys are enhanced firstly and then depressed with minor additions of Al. Compared with the GFA parameters in Table 5, ΔT_x, γ and ω are accordant to these experimental results, but except T_rg, showing that ω is one of the best applicable and reliable GFA parameters.

Table 5 Characteristic temperatures and GFA parameters of prepared Cu-Zr-Ti-Al alloys

4 Conclusions

1) Based on the analysis of Gibbs free energy between liquid and crystal, a new GFA parameter, ω=T_l(T_l+T_x)/[T_x(T_l-T_x)], is suggested for bulk metallic glasses.

2) Together with other GFA parameters proposed formerly, including T_rg, γ, α, δ and , ω was verified with alloying and oxide glasses. Results indicate the new GFA parameter ω is the most reliable and applicable approach to assess the GFA of various glasses.

3) ω is applied to estimating the GFA of Cu-Ag-Zr-Ti and Cu-Zr-Ti-Al BMGs and the predicted data are consistent with the experimental results.

Acknowledgments

The authors would like to thank Dr. CAI An-hui (Institute of Powder Metallurgy, Central South University, China) for helpful discussions.

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Corresponding author: JI Xiu-lin; Tel: +86-519-85191969; E-mail: xiulinji@gmail.com

DOI: 10.1016/S1003-6326(08)60438-0

(Edited by YANG Hua)