1 Introduction

Due to the presence of a large supercooled liquid region, bulk metallic glasses (BMGs) usually exhibit high thermal stability against crystallization. As a result, a large range of experimental time and temperatures for nucleation and crystalline growth processes exist in metallic glass forming melts. Characterization of the three thermodynamic parameters including Gibbs free energy difference ΔG, entropy difference ΔS; and enthalpy difference ΔH; are important in evaluation of nucleation and crystal growth processes in BMGs between the supercooled liquid and corresponding crystalline phases ^{[1, 2]}. Nucleation rates have been shown to have an exponential dependence on ΔG ^[3], acting as a driving force of nucleation. When ΔG is small, the critical nucleation work is improved, and nucleation rates are reduced ^[4]. As a result, the glass forming ability (GFA) of these materials is improved. The values of ΔG; ΔS; and ΔH are routinely calculated by measuring changes in the specific heat difference, ΔC_p, between the supercooled liquid and corresponding crystalline phases across a range of temperatures.

The metastable nature of supercooled liquids, however, makes accurate experimental values for ΔC_p difficult to determine ^[5]. Thus, most ΔC_p values for the supercooled liquid regions of various BMGs are only rough approximations generated by fitting limited experimental data to the melting temperature, T_m, in the vicinity of the glass transition temperature T_g. Because the accurate specific heat data in the supercooled region are notably absent, the functional dependences of ΔG; ΔS; and ΔH on temperature are generally estimated theoretically ^[5]. Several models for calculating ΔG; ΔS; and ΔH values have been previously proposed based on different expressions for ΔC_p ^{[6, 7, 8, 9, 10, 11, 12, 13, 14]}. In these expressions, Thompson et al. ^[12] and Hoffman et al. ^[13] assumed that ΔC_p was constant with temperature. Whereas Mondal et al. ^[10] and Patel et al. ^[11] suggested that ΔC_p value depended lineaσ_ly or hyperbolically on temperature, respectively. Each of these expressions, however, is deduced strictly from experimental data ^{[6, 10, 11, 12, 13, 14]} without theoretical support.

Recently, Dubey et al. ^{[7, 8]} proposed a theoretical expression for ΔC_p based on the hole theory of the liquid state, thus calculating more accurate values for ΔG, ΔH, and ΔS from experimental results collected among the temperature range from T_g to T_m in the Zr₅₇Cu_15.4Ni_12.6Al₁₀Nb₅ BMG ^[8]. Furthermore, a hyperbolic expression for ΔC_p was deduced that provided an optimal mathematical model for elucidating GFA based on the theoretical expression for ΔC_p proposed by Dubey et al. ^{[7, 8]}. According to the hyperbolic expression for ΔC_p based on the hole theory of the liquid state ^[15], the current study further deduced a linear expression for ΔC_p. The values of ΔG; ΔH; ΔS and for BMGs in the temperature range from Tg to T_m were calculated based on the hyperbolic, linear, and Dubey's expression for ΔC_p. Sixteen BMG materials ^{[16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26]} were selected as models for using in experimental evaluation of the accuracy of these 3 expressions for ΔC_p (Dubey's, hyperbolic, linear). The deviations observed in thermodynamic parameters between experimental results and these three models ^{[7, 8, 15]} were comparatively evaluated. 2 Expressions for the thermodynamic parameters ΔC_p, ΔG, ΔS, and ΔH

Since ΔG is vital to the study of GFA in BMGs, expressions for ΔC_p used in the calculation of ΔG values are important. The authors ^[15] previously proposed a hyperbolic expression for ΔC_p based on the hole theory of the liquid state ^{[7, 8]}, shown as follows:

where ΔC_p^m is the specific heat difference between the supercooled liquid and the corresponding crystalline phase at T_m, σ_h is a coefficient related to the hole formation energy in the hyperbolic express; ΔT is the degree of supercooling (ΔT = T_m - T; where T is the temperature). In BMG systems, the T_m - T values are smaller than T_m when T is decreased from T_m to T_g, suggesting that the value of ΔT/T_m is less than one. In this case, it is reasonable to approximate that the hyperbolic term in Eq. (1) can be expanded using a Taylor series, as follows:

thus producing the expression by neglecting a portion of the higher-order terms (n > 1 ),

where σ_l is a coefficient related to the hole formation energy in the linear expression. Since the portion of the higher-order terms (n > 1 ) in Taylor's series, i.e., Eq. (2), is neglected, which can modify the coefficient of (2 - σ_h) in Eq. (1), σ_l is used to replace σ_h. The linear expression of ΔC_p shown in Eq.(3) is similar to the linear form of ΔC_p = A + BT; where and B are the coefficients for linear expression, proposed by Patel et al. ^[11]. Evaluation of the parameter σ_l also results in a method similar to that proposed by Dubey et al. ^{[7, 8]}. Since experimental values of ΔC_p^g are usually measured in the vicinity of T_g, the ΔC_p^m value can be employed in conjunction with Eq. (3) to yield

where T_rg = T_g/T_m is the reduced glass transition temperature; and ζ = ΔC_p^g=ΔC_p^m ; where ΔC_p^g is the specific heat difference between the supercooled liquid and the corresponding crystalline phase at T_g.

ΔG; ΔH and ΔS are the differential values between the supercooled liquid and the corresponding crystalline phase, which can be expressed as

where ΔH_m is the enthalpy of fusion, and ΔS_m = ΔH_m=T_m is the entropy of fusion. Substituting Eq. (3) into Eqs. (5)- (7), the novel expressions for ΔH, ΔS and ΔG can be obtained as

Comparative studies were conducted on the expressions for ΔG, ΔS, and ΔH produced by the hyperbolic expression and Dubey's expression for ΔC_p using the framework of the hole theory of the liquid state as a basis. This technique allowed for further characterization of the thermodynamic behaviors of BMGs. Based on the hyperbolic expression for ΔC_p, the expressions for ΔG, ΔS, and ΔH, respectively, are ^[15]

and

Based on the hole theory of the liquid state, Dubey et al. ^{[7, 8]} provided an expression for the ΔC_p as

where σ_D is a coefficient related to the hole formation energy in the Dubey's expression. Substituting Eq. (15) into Eqs. (5)-(7), the simplified Dubey's expressions for ΔG; ΔH and ΔS are ^{[7, 8]}

The value σ_D is given by ^{[7, 8]}

The deviation percentage D, between the theoretical calculated value and the experimental value can be expressed as

where VemodT and VeexpT represent the calculated values and experimental values of ΔC_p, ΔG; ΔH and ΔS respectively. As shown in Eqs. (4), (14), and (19), most of the BMG materials in this study exhibit T_rg values that can be considered to be a constant equal to 0.65 (discussed in detail in later sections). Thus the deviation values, D, for ΔC_p, ΔG; ΔH; and ΔS occurring between temperatures from T_g to T_m correlate with the σ_l, σ_h and σ_D values as well as the ζ value. 3 Deviation between theoretical model-based calculated

and experimental values of ΔC_p in BMGs The thermodynamic behavior of BMGs is studied using expressions for thermodynamic parameters ΔC_p, ΔG; ΔS; ΔH; based on experimental results from 16 different BMGs. The values of ΔG, ΔS, and ΔH are calculated using experimentally measured ΔC_p values, which can be expressed as ^{[16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26]}

where A, B, and C are constant. These constants and parameters can be found in Refs. ^{[16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26]} and are summarized in Table 1. The calculated ΔC_p value, experimental ΔC_p value, and deviation value between calculated and experimental values from T_g to T_m can be generated by Eqs. (1), (3), (15), (20), and (21). It is prolix that all figures of ΔC_p, ΔG; ΔS; ΔH are described for the 16 alloys. In consideration of the different ζ values and alloy compositions in the present study, the 4 BMGs Zr₄₆Cu₄₆Al₈ ^[19], La₅₅Al₂₅Cu₂₅Ni₅Co₅ ^[17], Ti_36.89Cu_43.87Ni_9.36Zr_9.88 ^[22], and Pd₄₃Ni₁₀Cu₂₇P₂₀ ^[26] were representatively plotted in Figs. 1 and 2, respectively (2 materials per figure). These images are representative of results from all 16 BMGs (see Table 1).

Figures 1 and 2 show experimentally fitted ΔC_p values and calculated values deduced from the hyperbolic, Dubey's, and linear expressions. Initially, the deviations in ΔC_p for these 4 BMGs increased, followed by an immediate reduction with further temperature increased from Tg to T_m. The deviations in ΔC_p achieved a maximum deviation value, D_max.; at an uncertain temperature in the range of T_g and T_m. Notably, this value was achieved approximately at the midpoint between T_g and T_m in each sample. For Zr₄₆Cu₄₆Al₈ and La₅₅Al₂₅Cu₂₅Ni₅Co₅, the D_max. values of ΔC_p, D_max - ΔC_p, from hyperbolic, linear, and Dubey's expressions were each smaller than 4 %, indicating that calculated values of ΔC_p closely approximated experimental values, as compared with the D_max - ΔC_p values for Ti_36.89Cu_43.87Ni_9.36Zr_9.88 and Pd₄₃Ni₁₀Cu₂₇P₂₀ BMGs (see Fig. 2).

A comparison of the calculated and experimental values of ΔC_p in BMGs is provided by the D_max - ΔC_p values for 16 BMGs (see Table 2). The relationship between D_max - ΔC_p values and ζ values for the 16 alloys is summarized in Fig. 3. In addition, D_max - ΔC_p values derived from the linear expression demonstrated a maximum value of 16 % of the initial value, and the values derived from Dubey's expression increased with increasing ζ value for ζ > 2. Notably, the maximum value of Dubey's expression approached 38 % of the initial value for a ζ value of 6. For the hyperbolic expression, D_max - ΔC_p values were generally less than 11 % of the initial values. Cumulatively, these findings indicated that the hyperbolic expression for ΔC_p fitted well with the experimental values compared with both linear and Dubey's expressions. When ζ < 2, the D_max - ΔC_p derived from all 3 expressions was smaller than 11 % of the initial value, suggesting that all 3 theoretical models were applicable.

4 Deviations, calculated values, and experimental

values of ΔG, ΔS, and ΔH in BMGs The deviation values D of parameters ΔG; ΔS; ΔH; D - ΔG; D - ΔS; and D - ΔH; respectively, between calculated and experimental values for 4 BMGs were determined. As shown in Figs. 4-7, the values of D - ΔG; D - ΔH; and D - ΔS exhibited maximum deviations at T_g in the temperature range from T_g to T_m. A notable exception to this trend was the deviation value of ΔG from Dubey's expression for ΔC_p for Zr₄₆Cu₄₆Al₈ (see Fig. 4d) and Mg₆₅Cu₂₅Y₁₀ (not shown). Thus, deviation values for ΔG; ΔS; and ΔH at T_g could reasonably denote the degree of fit between ΔG; ΔS; and ΔH values in both calculated expressions and experimental results. D - ΔG; D - ΔH; and D - ΔS values at T_g for all 16 metallic glasses are listed in Table 2.

The D - ΔG; D - ΔH; and D - ΔS values at T_g and n values for 16 BMGs are listed in Table 2. The maximum D - ΔG values were achieved in the hyperbolic expression, whereas Dubey's and linear expression values for ΔC_p were smaller than 8 % of initial values, indicating the accuracy of calculated values using these 3 expressions for ΔG relative to experimental values. The maximum D - ΔH and D - ΔS values achieved by the hyperbolic expression for ΔC_p in 16 BMGs were less than 10 % and 15 % of initial values (see Table 2), respectively. The maximum D - ΔH and D - ΔS values derived from linear and Dubey's expressions for ΔC_p (see Table 2) presented bigger values of 16.8 % and 45.7 %, respectively, compared with those derived from the hyperbolic expression. Thus, calculations for ΔG; ΔH; and ΔS values also suggested that the hyperbolic expression for ΔC_p was the most accurate predictor of experimental values.

Results and analysis of D - ΔH and D - ΔS derived from Dubey's expression for ΔC_p produced similar findings to those of D - ΔC_p derived from Dubey's expression. The majority of D - ΔH and D - ΔS values derived from Dubey's expression for ζ < 2 were very close to the D - ΔH and D - ΔS values derived from the linear and hyperbolic expressions, suggesting that the difference between D - ΔH and D - ΔS values in each of the three expressions was very small for ζ < 2 (see Table 2). The majority of D - ΔH and D - ΔS values derived from Dubey's expression for ζ > 2 were much larger than the D - ΔH and D - ΔS values derived from the linear and hyperbolic expressions, suggesting that most ΔH; and ΔS values derived from Dubey's expression did not accurately predict experimental values for ζ > 2.

Thus, the hyperbolic expression for ΔC_p represents are relatively universal expression, compared to the linear expression and Dubey's expression for ΔC_p which are only accurate under certain conditions. Notably, experimental values more closely fit values produced by the linear expression for ΔC_p than values produced by Dubey's expression for ΔC_p. Dubey's expression for ΔC_p was, however, a good approximation of experimental values for ζ < 2, though not for ζ > 2. 5 Discussion

D_max. between the calculated and experimental values for ΔC_p, ΔG; ΔH; and ΔS from T_g to T_m was found to be associated with the ζ parameter according to Fig. 3 and Table 2. Equations (4), (14), and (19) showed that in addition to the effects of the ζ parameter, D values were also influenced by the T_rg values, which ranged from 0.56 to 0.73 and could be expressed as a function of the n value (see Fig. 8). Thus, equidistant low, medium, and high T_rg values of 0.56, 0.65, and 0.73 were selected to further characterize the effect of T_rg on the value of D (or D_max.). Figure 9a shows the relationship between the σ_h, σ_D, and σ_l values and the ζ values for T_rg values of 0.56, 0.65, and 0.73.

For ζ < 2, the change in the 3 σ_h, σ_D, and σ_l values with T_rg was negligible, suggesting that ΔC_p, ΔG; ΔH; and ΔS values calculated using different models were virtually identical (see Fig. 3 and Table 2). For ζ > 2, however, the values of σ_h, σ_D, and σ_l at the 3 different T_rg values revealed a decreasing trend. As ζ values increased from 0.5 to 7, σ_D values exhibited only small decreases, while larger decreases were exhibited by σ_h values. Moderate decreases in σ_l values were observed in between those of σ_D and σ_h compared with σ_h, σ_D, and σ_l values at T_rg = 0.65. Notably, this change can be neglected to simplify analysis.

Thus the majority of BMGs T_rg can be considered constant (T_rg = 0.65). Figure 9b demonstrates the relationship between σ_h, σ_D, and σ_l values and the ζ parameter at T_rg = 0.65. When T_rg = 0.65, the changes in σ_h, σ_D, and σ_l values are revealed to be very large, leading to variation in the deviation values for ΔC_p, ΔG; ΔH; and ΔS between calculated and experimental values. As shown in Fig. 3 and Table 2, the D_max - ΔC_p values can be used to reveal the fit of ΔG; ΔH; and ΔS between the calculated and experimental values (see Fig. 3 and Table 2). Due to this observation, only D_max - ΔC_p values are discussed.

Graphs of C_p as a function of temperature when ζ values equal to 2 and 3 (n = 2; ζ = 3) are shown in Fig. 10a, in which C_p^l and C_p^s represent the heat capacity of the supercooled liquid and crystal, respectively. C_p^l and C_p^s values were derived by fitting the experimental data. Figure 10b shows the ΔC_p value evolution and temperature increases for n values of 2 and 3, where the ζ parameter is the change rate of DC_p^s and ΔC_p^m, In order to characterize the difference in D_max. values produced by Dubey's expression and the hyperbolic expressions for ΔC_p from T_g to T_m, corresponding temperatures for D_max - ΔC_p values can be inferred.

Temperatures corresponding to the D_max - ΔC_p values for Zr₄₆Cu₄₆Al₈, La₅₅Al₂₅Cu₂₅Ni₅Co₅, Ti_36.89Cu_43.87Ni_9.36Zr_9.88, and Pd₄₃Ni₁₀Cu₂₇P₂₀ BMGs are generally between T_g and T_m (see Figs. 1 and 2). Thus T_max = 0.5 T_g + T_m), where T_max is the hypothetical maximum temperature of D_max - ΔC_p. All 16 studied BMGs also exhibit trends similar to those of the 4 representative materials shown (data not shown). ΔC_p values calculated using the linear expression closely approximate experimental values for ΔC_p (see Fig. 3). Thus, for temperature of 0.5 T_g + T_m), a calculated value closely fits to the experimental ΔC_p^{0.5 T_g + T_m)} (exp.), can be expressed as (see Fig. 10b)

Substitution of T_max into Eqs. (1) and (15), the hyperbolic expression and Dubey's expression for ΔC_p^{0.5 T_g + T_m)} (hyper.) and ΔC_p^{0.5 T_g + T_m)} (Dubey) can be deduced as

Thus, expressions of D_max. (hyper.) and D_max. (Dubey) generated by the hyperbolic expression and Dubey's expression for ΔC_p between the calculated and experimental values, respectively, can be written as

where T_rg is treated as a constant with a value of 0.65. Figure 11 shows the relationship between D_max. from the hyperbolic expression and Dubey's expression for ΔC_p as well as the ζ parameter based on Eqs. (25) and (26).When ζ < 2, the majority of D_max. (Dubey) and D_max. (hyper.) values are less than 10 %, indicating that these values closely approximate the D_max - ΔC_p values in Fig. 3. When 2<ζ<7, the D_max. (Dubey) values dramatically increase from 10 % to 43.7 %, and the D_max. (hyper.) values gradually increase from 8 % to 16 %. These findings suggest that the accuracy of values calculated using the hyperbolic expression for ΔC_p are higher than those calculated using Dubey's expression for ΔC_p. Thus, calculated D_max. values (see Fig. 11) from the hyperbolic expression and Dubey's expression vary according to trends very similar to those of experimental D_max. values (see Fig. 3). Based on the error scale determined by these findings, the expressions of D_max. (hyper.) and D_max. (Dubey) can be used to indicate changes in D_max - ΔC_p according to the ζ parameter values shown in Fig. 3 and Table 2 for all 16 examined BMGs in the current study. 6 Conclusions

A linear expression for ΔC_p derived from the hyperbolic expression for ΔC_p was deduced and used to obtain a novel expression for ΔG; ΔH; and ΔS: According to the experimentally determined thermodynamic parameters of the 16 examined BMGs in the current study, more accurate calculations of ΔC_p, ΔG; ΔH; and ΔS were obtained using the linear, hyperbolic, and Dubey's expression for ΔC_p. These results suggest that the hyperbolic expression for ΔC_p can be applied as a universal expression for ΔC_p, while linear and Dubey's expressions for ΔC_p are condition-dependent. Notably, Dubey's expression for ΔC_p also closely approximated experimental values when ζ < 2, though values were shown to deviate from experimental values for ζ > 2.

Acknowledgments The work described in this paper was supported by the grant from the National Natural Science Foundation of China (Grant No. 51025415).