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

Journal of Alloys and Compounds 550 (2013) 221–225

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Characterizing thermodynamic properties of Ti–Cu–Ni–Zr bulk metallic glasses by hyperbolic expression Peiyou Li a, Gang Wang b,⇑, Ding Ding b, Jun Shen a a b

School of Materials Science and Engineering, Harbin Institute of Technology, 150001 Harbin, People’s Republic of China Laboratory for Microstructures, Shanghai University, 200444 Shanghai, People’s Republic of China

a r t i c l e

i n f o

Article history: Received 24 August 2012 Accepted 21 September 2012 Available online 29 September 2012 Keywords: Bulk metallic glass Thermodynamics Specific heat Hyperbolic expression

a b s t r a c t The hyperbolic expression of specific heat difference, DCp, in the supercooled liquid and corresponding crystalline phase of bulk metallic glasses can be deduced based on the expression of DCp in the hole theory of the liquid state. According to the hyperbolic expression of DCp, novel expressions of the Gibbs free energy difference, DG, enthalpy difference, DH, and entropy difference, DS, 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 DG, DH and DS. This finding clearly suggests that the hyperbolic expression of DCp provides an optimum mathematical model for the elucidation of glass forming ability, with increased accuracy and ease of modeling compared to previously reported expressions. Ó 2012 Elsevier B.V. All rights reserved.

1. 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) [3–6], 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, DG; entropy difference, DS; and enthalpy difference, DH, 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 DG [8]. Decreasing DG 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 DG, DS, and DH can be calculated by measurement of the specific heat difference, DCp, 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, ⇑ Corresponding author. Tel./fax: +86 21 66135269. E-mail address: [email protected] (G. Wang). 0925-8388/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jallcom.2012.09.105

however, makes it difficult to attain experimentally accurate values for DCp in the supercooled liquid [9]. As a result, the experimental DCp 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 DG, DS and DH values, resulting in a variety of expressions based on different DCp estimations [10–18]. Thompson et al. and Hoffman [16,17] assumed that the value of DCp was a constant; Mondal et al. [14] proposed that the value of DCp was a linear function of temperature; Patel et al. [15] states that the value of DCp was a linear and hyperbolic function of temperature. Each of these expressions, however, was based on a DCp value taken from experimental results only, lacking a firm theoretical explanation. Recently, Dubey et al. [11,12] proposed a theoretical expression of DCp based on the hole theory of the liquid state, resulting in calculated values of DG, DH, and DS that closely approximated experimental results throughout the entire temperature range (Tm–Tg) for the BMG Zr57Cu15.4Ni12.6Al10Nb5. Application of the hole theory of the liquid state provides a method for the theoretical elucidation of DCp. The current study details the development of a novel hyperbolic expression for DCp as a function of temperature using a Taylor’s series expansion on the DCp expression proposed by Dubey et al. Based on this hyperbolic expression of DCp, novel expressions for DG, DH, and DS 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

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P. Li et al. / Journal of Alloys and Compounds 550 (2013) 221–225

experimental results and the theoretical calculations of current and previous studies [10,13] was conducted.

DS ¼ DSm  DC m p

#  " DT DT r DT 1 þ 23TDT 1þ 1 ; 2 T 1 þ D2TT T 2T

ð13Þ

where the r value is given by [11,12], 2. Expressions for the thermodynamic parameters DG, DS, and DH

r¼ The free energy difference, DG, plays an important role in the characterization of glass forming ability (GFA). As a necessary parameter for the calculation of the DG value, DCp is expected to be obtained experimentally. Recently, Dubey et al. proposed an expression for DCp based on the hole theory of the liquid state [11,12], shown in Eq. (1),

DC p ¼ DC m p



Tm T

2

  DT exp r ; T

DC p ¼ DC m p

r  1 þ ð2  rÞ

ð2Þ



Tm : T

DG ¼ DH m

DS ¼

T m  T g DC gp =DC m p ; Tm  Tg

ð4Þ

DC gp

where is the specific heat difference at Tg. Then, DG is the free energy difference between the supercooled liquid and the corresponding crystalline phase, which can be expressed as,

DG ¼ DH  T DS; DH ¼ DH m 

Z

DC p dT;

Tm T

DC p dT; T

DC m p



 Tm ð3  rÞDT þ ðð3  rÞT m  ðr  1ÞDTÞ ln ; T

  Tm ; DH ¼ DH m  DC m p ðr  1ÞðT m  TÞ þ ð2  rÞT m ln T

ð7Þ

ð8Þ

ð9Þ

and

  Tm  T Tm : DS ¼ DSm  DC m þ ðr  1Þ ln p ð2  rÞ T T

ð10Þ

In addition to the above expressions, another three models for characterization of the thermodynamic properties of BMGs exist. These expressions will be used to comparatively predicate the GFA of BMGs. Dubey et al. [11,12] provided simplified expressions for DG, DS and DH,

DG ¼ DS m DT  DC m p

  DT 2 DT 1r ; 2T 3T

ð11Þ

  DT rDT 1 ; T 2T

ð12Þ

DH ¼ DH m  DC m p Tm and

2

ð15Þ

m 2 2 DHm DC p ðT þ 2TT m  3T m Þ þ ; 2 Tm ðT m þ TÞ

DC m p T m DTðT m þ 3TÞ ðT m þ TÞ2

:

ð16Þ

ð17Þ

Singh et al. assumed that DCp was a linear function of temperature [10]. Thus, the expressions of DG, DH and DS were approximated as [10],

DT 7T ; T m T m þ 6T

ð18Þ

DS ¼

7DHm 6T 2 þ 2TT m  T 2m : Tm ðT m þ 6TÞ2

ð19Þ

DH ¼

7DHm T 2 7T m þ 5T : Tm ðT m þ 6TÞ2

ð20Þ

DG ¼ DH m

To clearly exhibit the deviation between the model-based calculated values and the experimental values, the deviation percentage values, D, between the calculated and experimental values of DCp, DG, DH, and DS were calculated using the following equation,

  VðmodÞ  VðexpÞ ; D ¼   VðexpÞ

ð21Þ

where V (mol) and V (exp) are the model-calculated and experimental values of DCp, DG, DH, and DS, respectively.

ð6Þ

where DHm is the enthalpy of fusion, DSm = DHm/Tm is the entropy of fusion. Substituting Eq. (3) into Eqs. (5)–(7) brings out the novel expressions for DG, DS and DH,

DG ¼ DS m DT 

ð14Þ

3. Results and discussion

Tm T

Z

# DC m p g : DC p

ð5Þ

and

DS ¼ DSm 

2

DT DC p ðDTÞ ;  Tm þ T Tm

DH ¼ DH m 

ð3Þ

The present expression of DCp shown in Eq. (3) is clearly a hypothesis form of DCp = A + B/T, originally proposed by Patel et al. [15], where A and B are the coefficients of the hyperbolic expression. Similarly, the evaluation of the r parameter is similar to that used in the method proposed by Dubey et al. [11,12]. Since the experimental value of DCp is usually measured in the vicinity of Tg, the value of DCp can be employed in conjunction with Eq. (3) to yield,

r¼1þ

Tm Tg

Jones et al. assumed that the specific heat difference is a constant between the supercooled liquid and corresponding crystal phase, yielding the following relationships [13],

producing the expression,



"

m

ð1Þ

where Tm is the melting temperature of alloy; DT = TmT is the degree of supercooled liquid; DC m p is the specific heat difference at Tm; and r is the coefficient related to hole formation energy. In BMG systems, the glass transition temperature, Tg, is usually larger than 0.5 Tm. Thus, when the temperature is decreased from Tm to Tg, the value of DT/T is less than one. In this case, it is reasonable to speculate that the exponential term in Eq. (1) can be expanded using a Taylor series. As the exponential term for temperature is a non-periodic function, this can be accomplished by neglecting part of the high-order terms (n > 2),

      DT DT r 2 DT 2 r n DT n exp r þ ::: þ ; 1r þ 2 2 T T T T

Tg ln ðT m  T g Þ

In the present study, values of DG, DH, and DS in the BMGs Ti35.37Cu45.11Ni8.88Zr10.64 and Ti37.65Cu43.25Ni9.6Zr9.5 BMGs were calculated based on experimentally measured values of DCp, expressed as [5],

DC P ¼ AT þ BT 2 þ CT 2 ;

ð22Þ

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 DCp for two BMGs as functions of temperature calculated based on Dubey’s expressions [Eq. (1)], hyperbolic expressions [Eq. (3)], and experimental fitting [Eq. (22)], respectively. Subsequently, deviation values of DCp deduced from different expressions (DCp) as functions of temperature were plotted in Fig. 1(c) and (d). In Fig. 1(a) and (b), the calculated values of the hyperbolic expression of DCp were shown to be closer to experimental values throughout the entire temperature range (Tg–Tm) than values produced using Dubey’s expression of DCp. Furthermore, Fig. 1(c) and (d) show maximum deviation values of DCp in temperature ranges between Tg and Tm. The maximum deviation values based on the hyperbolic expression of DCp were only 0.2% and 0.3% for Ti35.37Cu45.11Ni8.88Zr10.64 and Ti37.65Cu43.25Ni9.6Zr9.5 alloys, respectively, across the entire temperature range from Tg to Tm. Maximum deviation values based on the Dubey’s expression of DCp, however, approached approximately 10% and 11% for Ti35.37Cu45.11Ni8.88Zr10.64 and Ti37.65Cu43.25Ni9.6Zr9.5 alloys, respectively. Therefore, hyperbolic expression of DCp with

223

P. Li et al. / Journal of Alloys and Compounds 550 (2013) 221–225 Table 1 Thermodynamic parameters for evaluation of DG, DH, and DS in Ti35.37Cu45.11Ni8.88Zr10.64 and Ti37.65Cu43.25Ni9.6Zr9.5 bulk metallic glasses (BMGs). Alloy BMGs

A (J mol1 K2)

B (J mol1 K2)

C (J mol1 K3)

Tg (K)

Tm (K)

DC m p (J mol1 K1)

DC gp (J mol1 K1)

D Hm (kJ mol1)

DSm (J mol1 K1)

Ti35.37Cu45.11Ni8.88Zr10.64 Ti37.65Cu43.25Ni9.6Zr9.5

0.0122 0.0087

1.237  107 1.170  107

1.1862  105 0.925  105

673 673

1092 1097

9.62 8.15

30.46 27.51

11.033 11.040

10.103 10.064

45

35

experiment Dubey et al. Hyperbolic

30 25

(c) Ti35.37Cu45.11Ni8.88Zr10.64

Tg

20

10

D-Cp (%)

Δ Cp (J/mol K)

40

12 (a) Ti35.37Cu45.11Ni8.88Zr10.64

8 6

Tm

15

2

10 5

experiment Dubey et al. Hyperbolic

Tg

20

Tm

(d) Ti37.65Cu43.25Ni9.6Zr9.5

8 6

Dubey et al. Hyperbolic

4

15

Tm 2

10 5 600

Tg

10

30 25

0 12

(b) Ti37.65Cu43.25Ni9.6Zr9.5

D-Cp (%)

Δ Cp (J/mol K)

35

Dubey et al. Hyperbolic

4

700

800

900

1000

1100

Temperature (K)

0 600

Tg

Tm

700

800

900

1000

1100

Temperature (K)

Fig. 1. The specific heat difference, DCp, and the deviation values of D–Cp as functions of temperature were derived for Ti35.37Cu45.11Ni8.88Zr10.64 and Ti37.65Cu43.25Ni9.6Zr9.5 BMGs using reported experimental results and a theoretical model based on the hole theory of the liquid state. This model pertains to the relationship between specific heat and temperature.

temperature clearly produced results more consistent with experimental findings compared with those produced by the Dubey expression of DCp. The DG between the supercooled liquid and corresponding crystalline phase as a function of temperature for the two BMGs is shown in Fig. 2(a) and (b). The values of DG calculated by the hyperbolic expression of DCp exhibited the closest relationship, in many cases near identical, with the experimental values of DG across the entire temperature range from Tg to Tm compared with values produced by Dubey’s, Jones’, and Singh’s expressions for DG. Fig. 2(c) and (d) show that the deviation values of DG (D–G) decrease with increasing temperature from Tg to Tm. For these two BMGs, the deviation values for DG attained using the hyperbolic expression of DCp were less than 1% at Tg see Fig. 2(c) and (d). Notably, the deviation values for DG in Ti35.37Cu45.11Ni8.88Zr10.64 were much larger when the value was derived from Dubey’s, Jones’, and Singh’s expressions for DCp, with values of 3.9%, 12.4%, and 17.6% at Tg, respectively. Similarly, Dubey’s, Jones’, and Singh’s expressions for DCp produced values for Ti37.65Cu43.25Ni9.6Zr9.5 of 3.8%, 15.5%, and 31.5%, respectively. Smaller deviation values for DG suggest that the hyperbolic expression for DCp is a better fit for the experimental values than previous models. Furthermore, the DH values for the two BMGs derived from the hyperbolic expression of DCp as functions of temperature

exhibited better fits to the experimental results than those produced by Dubey’s, Jones’, and Singh’s models, as shown in Fig. 3(a) and (b). The deviation values of DH (D–H) at the Tg temperature derived from the hyperbolic expression of DCp for the two BMGs were less than 3%, a similar low deviation value to that observed in the DG parameter [see Fig. 3(c) and (d)]. Conversely, the deviation values at Tg based on all other expressions of DH were larger than 10%, as shown in Fig. 3(c) and (d). This finding further indicates that the hyperbolic expression of DCp is more accurate than other models for fitting values of DH. Results and analysis of DS and D–S derived from these models between the supercooled liquid and corresponding crystalline state showed similar results to those observed for DH and D–H, as shown in Fig. 4. Thus, when DS and D–S at Tg are considered, the results show that the values of DS calculated using the hyperbolic expression model for DCp are very close to the observed experimental values when compared to the other models. 4. Conclusion Based on the hole theory of the liquid state, a hyperbolic expression of DCp was deduced. Using the recently determined

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P. Li et al. / Journal of Alloys and Compounds 550 (2013) 221–225

40

(a)

Experiment Dubey et al. Jones et al. Singh et al. hyperbolic

Δ G (J/mol)

3000 Tg

2000

(c)

Ti35.37Cu45.11Ni8.88Zr10.64

Ti35.37Cu45.11Ni8.88Zr10.64 Dubey et al. Jones et al. Singh et al. Hyperbolic

30

D-G (%)

4000

20

10

1000

Tg

Tm

Tm

0 (b) Ti37.65Cu43.25Ni9.6Zr9.5 Experiment Dubey et al. Jones et al. Singh et al. hyperbolic

3000 Tg

2000

(d) Ti37.65Cu43.25Ni9.6Zr9.5

30

D-G (%)

Δ G (J/mol)

0 4000

Dubey et al. Jones et al. Singh et al. Hyperbolic

20

10 Tg

1000 Tm

0 600

700

800

900

1000

0 600

1100

Tm

700

800

900

1000

1100

Temperature (K)

Temperature (K)

Fig. 2. Gibbs free energy difference, DG, and the deviation values of D–G as functions of temperature were derived for Ti35.37Cu45.11Ni8.88Zr10.64 and Ti37.65Cu43.25Ni9.6Zr9.5 BMGs using reported experimental results and three theoretical models. Each of these three models employed different assumptions about the relationships between specific heat and temperature.

12

60 (a) Ti35.37Cu45.11Ni8.88Zr10.64

(c) Ti35.37Cu45.11Ni8.88Zr10.64 50

10

6 Experiment Dubey et al. Jones et al. Singh et al. Hyperbolic

4 2

Tg

Dubey et al. Jones et al. Singh et al. Hyperbolic

40

D-H (%)

Δ H (kJ/mol)

Tm

8

30 20 10 0

0

Tm Tg

(b) Ti37.65Cu43.25Ni9.6Zr9.5 Tm

8 Experiment Dubey et al. Jones et al. Singh et al. Hyperbolic

6 4 Tg

2 600

700

Ti37.65Cu43.25Ni9.6Zr9.5

40

D-H (%)

Δ H (kJ/mol)

10

(d)

Dubey et al. Jones et al. Singh et al. Hyperbolic

30 20 10

Tm

Tg

0 800

900

1000

Temperature (K)

1100

700

800

900

1000

1100

Temperature (K)

Fig. 3. Enthalpy difference, DH, and the deviation values of D–H as functions of temperature were derived for Ti35.37Cu45.11Ni8.88Zr10.64 and Ti37.65Cu43.25Ni9.6Zr9.5 BMGs using reported experimental results and three theoretical models. Each of these three models employed different assumptions about the relationships between specific heat and temperature.

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P. Li et al. / Journal of Alloys and Compounds 550 (2013) 221–225

100

12 (a) Ti35.37Cu45.11Ni8.88Zr10.64

(c)

10 Δ S (kJ/mol)

6 4

Experiment Dubey et al. Jones et al. Singh et al. Hyperbolic

2 0 Tg

D-S (%)

80 Tm

8

60 40

0

Tm Tg (d) Ti37.65Cu43.25Ni9.6Zr9.5

(b) Ti37.65Cu43.25Ni9.6Zr9.5 Tm

6 4

Experiment Dubey et al. Jones et al. Singh et al. Hyperbolic

2 0

Tg

-2 600

700

D-S (%)

80

8 Δ S (kJ/mol)

Dubey et al. Jones et al. Singh et al. Hyperbolic

20

-2 10

Ti35.37Cu45.11Ni8.88Zr10.64

Dubey et al. Jones et al. Singh et al. Hyperbolic

60 40 20

Tm

Tg 0

800

900

1000

1100

700

Temperature (K)

800

900

1000

1100

Temperature (K)

Fig. 4. Entropy difference, DS, and the deviation values of D–S as functions of temperature were derived for Ti35.37Cu45.11Ni8.88Zr10.64 and Ti37.65Cu43.25Ni9.6Zr9.5 BMGs using reported experimental results and three theoretical models. Each of these three models employed different assumptions about the relationships between specific heat and temperature.

hyperbolic relationship, a novel expression of DG, DH, and DS 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 DCp, DG, DH, and DS were able to be obtained using the hyperbolic expression model, evidencing a closer fit between calculated and experimental data than that observed in expression models previously proposed by Dubey, Jones, and Singh. 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.

References [1] W.H. Wang, Prog. Mater Sci. 57 (2012) 487. [2] W.H. Wang, C. Dong, C.H. Shek, Mater. Sci. Eng. 44 (2004) 45. [3] Q.K. Jiang, X.D. Wang, X.P. Nie, G.Q. Zhang, H. Ma, H.J. Fecht, J. Bendnarcik, H. Franz, Y.G. Liu, Q.P. Cao, J.Z. Jiang, Acta Mater. 56 (2008) 1785. [4] Y.J. Sun, D.D. Qu, Y.J. Huang, K.D. Liss, X.S. Wei, D.W. Xing, J. Shen, Acta Mater. 57 (2009) 1290–1299. [5] P.Y. Li, G. Wang, D. Ding, J. Shen, J. Non-Cryst. Solids 358 (2012) 3200. [6] W.H. Wang, Prog. Mater Sci. 52 (2007) 540. [7] G.J. Fan, J.F. Löffler, R.K. Wunderlich, H.J. Fecht, Acta Mater. 52 (2004) 667. [8] F.H. Stillinger, J. Chem. Phys. 88 (1988) 7818. [9] R. Busch, Y.J. Kim, W.L. Johnson, J. Appl. Phys. 77 (1995) 4039. [10] H.B. Singh, A. Holz, Solid State Commun. 45 (1983) 985. [11] K.S. Dubey, AIP Conf. Proc. 1249 (2010) 211. [12] P.K. Singh, K.S. Dubey, Thermochim. Acta 530 (2012) 120. [13] D. Jones, G. Chadwick, Philos. Mag. 24 (1971) 995. [14] K. Mondal, U.K. Chatterjee, B.S. Murty, Appl. Phys. Lett. 83 (2003) 671. [15] T.A. Patel, A. Pratap, AIP Conf. Proc. 1249 (2010) 161. [16] C.V. Thompson, F. Spaepen, Acta Metall. 27 (1979) 1855. [17] J.D. Hoffman, J. Chem. Phys. 21 (1956) 1022. [18] X.L. Ji, Y. Pan, J. Non-Cryst. Solids 353 (2007) 2443–2446.