Elsevier

Journal of Alloys and Compounds

Volume 550, 15 February 2013, Pages 221-225
Journal of Alloys and Compounds

Characterizing thermodynamic properties of Ti–Cu–Ni–Zr bulk metallic glasses by hyperbolic expression

https://doi.org/10.1016/j.jallcom.2012.09.105Get rights and content

Abstract

The hyperbolic expression of specific heat difference, ΔCp, in the supercooled liquid and corresponding crystalline phase of bulk metallic glasses can be deduced based on the expression of ΔCp in the hole theory of the liquid state. According to the hyperbolic expression of ΔCp, novel expressions of the Gibbs free energy difference, ΔG, enthalpy difference, ΔH, and entropy difference, ΔS, between the supercooled liquid and corresponding crystalline phase may be estimated. The experimentally measured thermodynamic parameters of Ti35.37Cu45.11Ni8.88Zr10.64 and Ti37.65Cu43.25Ni9.6Zr9.5 bulk metallic glasses exhibit a perfect fit with these novel expressions for ΔG, ΔH and ΔS. This finding clearly suggests that the hyperbolic expression of ΔCp provides an optimum mathematical model for the elucidation of glass forming ability, with increased accuracy and ease of modeling compared to previously reported expressions.

Highlights

► A hyperbolic expression of ΔCp is deduced based on the hole theory of liquid. ► Novel expressions of the ΔG, ΔH, and ΔS are deduced. ► The hyperbolic expression of ΔCp provides a mathematical model for the elucidation of GFA.

Introduction

In the field of metallic materials, bulk metallic glasses (BMGs) have garnered increased research attention due to their superior physical and chemical properties compared to oxide glasses and crystalline metals [1], [2]. BMGs generally have a high glass forming ability (GFA) [3], [4], [5], [6], allowing nucleation from the supercooled liquid to be suppressed at low cooling rates. This effect favors the formation of the glassy phase. The GFA of alloys may be characterized suing the Gibbs free energy difference, ΔG; entropy difference, ΔS; and enthalpy difference, ΔH, that exist between the supercooled liquid and corresponding crystalline phase. Together, these parameters can be used to comprehensively reflect both the nucleation behavior in the supercooled liquid and the crystal growth phenomena [7], [8]. Notably, nucleation rate is known to be associated with the GFA of alloys, exhibiting an exponential dependence on ΔG [8]. Decreasing ΔG acts as a driving force for nucleation, causing an increase in critical nucleation work and a reduction in nucleation rate [7]. The result is an improved GFA in alloy materials. The values of ΔG, ΔS, and ΔH can be calculated by measurement of the specific heat difference, ΔCp, between the supercooled liquid and the corresponding crystalline phase as functions of temperature.

Determination of the specific heat difference is critical to further investigation of the GFA of BMGs in the framework of thermodynamics. The metastable nature of the supercooled liquid, however, makes it difficult to attain experimentally accurate values for ΔCp in the supercooled liquid [9]. As a result, the experimental ΔCp value of the supercooled liquid is often roughly estimated by fitting several experimental data points. This lack of detailed specific heat data in the supercooled liquid region necessitates the theoretical estimation of the functional dependences of ΔG, ΔS and ΔH values, resulting in a variety of expressions based on different ΔCp estimations [10], [11], [12], [13], [14], [15], [16], [17], [18]. Thompson et al. and Hoffman [16], [17] assumed that the value of ΔCp was a constant; Mondal et al. [14] proposed that the value of ΔCp was a linear function of temperature; Patel et al. [15] states that the value of ΔCp was a linear and hyperbolic function of temperature. Each of these expressions, however, was based on a ΔCp value taken from experimental results only, lacking a firm theoretical explanation. Recently, Dubey et al. [11], [12] proposed a theoretical expression of ΔCp based on the hole theory of the liquid state, resulting in calculated values of ΔG, ΔH, and ΔS that closely approximated experimental results throughout the entire temperature range (TmTg) for the BMG Zr57Cu15.4Ni12.6Al10Nb5.

Application of the hole theory of the liquid state provides a method for the theoretical elucidation of ΔCp. The current study details the development of a novel hyperbolic expression for ΔCp as a function of temperature using a Taylor’s series expansion on the ΔCp expression proposed by Dubey et al. Based on this hyperbolic expression of ΔCp, novel expressions for ΔG, ΔH, and ΔS were also deduced. The model BMG materials Ti35.37Cu45.11Ni8.88Zr10.64 and Ti37.65Cu43.25Ni9.6Zr9.5 were selected to experimentally evaluate the accuracy of this novel expression. Comparative evaluation of deviations observed in thermodynamic behaviors between experimental results and the theoretical calculations of current and previous studies [10], [13] was conducted.

Section snippets

Expressions for the thermodynamic parameters ΔG, ΔS, and ΔH

The free energy difference, ΔG, plays an important role in the characterization of glass forming ability (GFA). As a necessary parameter for the calculation of the ΔG value, ΔCp is expected to be obtained experimentally. Recently, Dubey et al. proposed an expression for ΔCp based on the hole theory of the liquid state [11], [12], shown in Eq. (1),ΔCp=ΔCpmTmT2exp-σΔTT,where Tm is the melting temperature of alloy; ΔT = TmT is the degree of supercooled liquid; ΔCpm is the specific heat difference

Results and discussion

In the present study, values of ΔG, ΔH, and ΔS in the BMGs Ti35.37Cu45.11Ni8.88Zr10.64 and Ti37.65Cu43.25Ni9.6Zr9.5 BMGs were calculated based on experimentally measured values of ΔCp, expressed as [5],ΔCP=AT+BT-2+CT2,where A, B and C are constants. These constants and parameters have been reported in Ref. [5], summarized in Table 1.

Fig. 1(a) and (b) plot the values of ΔCp for two BMGs as functions of temperature calculated based on Dubey’s expressions [Eq. (1)], hyperbolic expressions [Eq. (3)

Conclusion

Based on the hole theory of the liquid state, a hyperbolic expression of ΔCp was deduced. Using the recently determined hyperbolic relationship, a novel expression of ΔG, ΔH, and ΔS was obtained. According to the experimentally determined thermodynamic parameters of the BMGs Ti35.37Cu45.11Ni8.88Zr10.64 and Ti37.65Cu43.25Ni9.6Zr9.5, more accurate calculations of ΔCp, ΔG, ΔH, and ΔS were able to be obtained using the hyperbolic expression model, evidencing a closer fit between calculated and

Acknowledgements

The work described in this paper was supported by the NSF of China (Nr. 51171098). G. Wang also thanks the financial support by Shanghai Pujiang Program (Nr. 11PJ1403900), the Innovation Program of Shanghai Municipal Education Commission (Nr. 12ZZ090), and the Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning.

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