Adv. Manuf. (2013) 1:293–304 DOI 10.1007/s40436-013-0040-3
Determination of accurate theoretical values for thermodynamic properties in bulk metallic glasses Pei-You Li • Gang Wang • Ding Ding Jun Shen
•
Received: 15 July 2013 / Accepted: 23 September 2013 / Published online: 31 October 2013 Ó Shanghai University and Springer-Verlag Berlin Heidelberg 2013
Abstract Deviation values of specific heat difference DCp ; the Gibbs free energy difference DG; enthalpy difference DH; and entropy difference DS between the supercooled liquid and corresponding crystalline phase produced by the linear, hyperbolic, and Dubey’s expressions of DCp and the corresponding experimental values are determined for sixteen bulk metallic glasses (BMGs) from the glass transition temperature Tg to the melting temperature Tm : The calculated values produced by the hyperbolic expression for DCp most closely approximate experimental values, indicating that the hyperbolic DCp expression can be considered universally applicable, compared to linear and Dubey’s expressions for DCp ; which are accurate only within a limited range of conditions. For instance, Dubey’s DCp expression provides a good approximation of actual experimental values within certain conditions (i.e., n ¼ DCpg =DCpm \2; where DCpg and DCpm represent the specific heat difference at temperatures Tg and Tm ; respectively). Keywords Bulk metallic glass (BMG) Specific heat Linear expression Hyperbolic expression
P.-Y. Li J. Shen School of Materials Science and Engineering, Harbin Institute of Technology, Harbin 150001, People’s Republic of China e-mail:
[email protected] G. Wang (&) D. Ding Laboratory for Microstructures, Shanghai University, Shanghai 200444, People’s Republic of China e-mail:
[email protected]
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 DG, entropy difference DS; and enthalpy difference DH; 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 DG [3], acting as a driving force of nucleation. When DG 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 DG; DS; and DH are routinely calculated by measuring changes in the specific heat difference, DCp ; 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 DCp difficult to determine [5]. Thus, most DCp values for the supercooled liquid regions of various BMGs are only rough approximations generated by fitting limited experimental data to the melting temperature, Tm ; in the vicinity of the glass transition temperature Tg : Because the accurate specific heat data in the supercooled region are notably absent, the functional dependences of DG; DS; and DH on temperature are generally estimated theoretically [5]. Several models for calculating DG; DS; and DH values have been previously proposed based on different expressions for DCp [6–14].
123
294
P.-Y. Li et al.
In these expressions, Thompson et al. [12] and Hoffman et al. [13] assumed that DCp was constant with temperature. Whereas Mondal et al. [10] and Patel et al. [11] suggested that DCp value depended linearly or hyperbolically on temperature, respectively. Each of these expressions, however, is deduced strictly from experimental data [6, 10–14] without theoretical support. Recently, Dubey et al. [7, 8] proposed a theoretical expression for DCp based on the hole theory of the liquid state, thus calculating more accurate values for DG, DH, and DS from experimental results collected among the temperature range from Tg to Tm in the Zr57Cu15.4Ni12.6Al10Nb5 BMG [8]. Furthermore, a hyperbolic expression for DCp was deduced that provided an optimal mathematical model for elucidating GFA based on the theoretical expression for DCp proposed by Dubey et al. [7, 8]. According to the hyperbolic expression for DCp based on the hole theory of the liquid state [15], the current study further deduced a linear expression for DCp : The values of DG; DH; DS and for BMGs in the temperature range from Tg to Tm were calculated based on the hyperbolic, linear, and Dubey’s expression for DCp : Sixteen BMG materials [16–26] were selected as models for using in experimental evaluation of the accuracy of these 3 expressions for DCp (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 DCp, DG, DS, and DH Since DG is vital to the study of GFA in BMGs, expressions for DCp used in the calculation of DG values are important. The authors [15] previously proposed a hyperbolic expression for DCp based on the hole theory of the liquid state [7, 8], shown as follows: Tm DCp ¼ DCpm rh 1 þ ð2 rh Þ T ! 1 m ¼ DCp rh 1 þ ð2 rh Þ ; ð1Þ 1 DT Tm DCpm
where is the specific heat difference between the supercooled liquid and the corresponding crystalline phase at Tm; rh is a coefficient related to the hole formation energy in the hyperbolic express; DT is the degree of supercooling (DT ¼ Tm T; where T is the temperature). In BMG systems, the Tm T values are smaller than Tm
123
when T is decreased from Tm to Tg ; suggesting that the value of DT=Tm 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: 2 n Tm 1 DT DT DT ¼ ¼1þ þ þ þ ; ð2Þ DT Tm Tm Tm T 1T m
thus producing the expression by neglecting a portion of the higher-order terms (n [ 1), T DCp ¼ DCpm 3 rl ð2 rl Þ ; ð3Þ Tm where rl 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 ð2 rh Þ in Eq. (1), rl is used to replace rh : The linear expression of DCp shown in Eq.(3) is similar to the linear form of DCp ¼ Tg DCpm Tm DCpg Tg Tm DCpm ðrl 2Þ ) are the Tm
A þ BT; where A (A ¼ (B ¼
DCpg DCpm
¼
Tg Tm
¼ DCpm ð3 rl Þ) and B coefficients for linear
expression, proposed by Patel et al. [11]. Evaluation of the parameter rl also results in a method similar to that proposed by Dubey et al. [7, 8]. Since experimental values of DCpg are usually measured in the vicinity of Tg ; the DCpm value can be employed in conjunction with Eq. (3) to yield rl ¼ 2 þ
1n ; 1 Trg
ð4Þ
where Trg ¼ Tg =Tm is the reduced glass transition temperature; and n ¼ DCpg =DCpm ; where DCpg is the specific heat difference between the supercooled liquid and the corresponding crystalline phase at Tg. DG; DH and DS are the differential values between the supercooled liquid and the corresponding crystalline phase, which can be expressed as DH ¼ DHm
Z
Tm
DCp dT;
ð5Þ
T
and DS ¼ DSm
Z
Tm T
DG ¼ DH TDS;
DCp dT; T
ð6Þ ð7Þ
where DHm is the enthalpy of fusion, and DSm ¼ DHm =Tm is the entropy of fusion. Substituting Eq. (3) into Eqs. (5)– (7), the novel expressions for DH, DS and DG can be obtained as
Thermodynamic properties in bulk metallic glasses
295
Table 1 Thermodynamic parameters for evaluation of DG, DH, and DS in the 16 BMGs Alloys
A/(J mol-1 K-2)
B/(J mol-1 K-3)
C/(J mol-1 K-2)
Tg /K
Tm /K
DCpm /(J mol-1 K-1)
DCpg /(J mol-1 K-1)
La62Al14Cu24 [16]
0.03071
4.16 9 105
-1.49 9 10-5
401
673
0.60
La55Al25Ni20 [17]
0.02190
1.24 9 10
6
-1.01 9 10-5
491
712
0.69
6.835
14.840
12.510
7.477
12.940
Cu47Ti34Zr11Ni8 [18]
0.01650
2.83 9 106
-6.82 9 10-6
673
1114
13.460
0.60
11.300
12.187
14.257
Zr46Cu46Al8 [19]
0.12850
4.61 9 106
-5.59 9 10-6
715
La55Al25Cu10Ni5Co5 [17]
0.02520
1.69 9 10
6
-1.18 9 10-5
466
979
0.73
8.035
13.490
16.410
661
0.70
6.095
12.610
La62Al14(Cu5/6Ag1/6)24 [16]
0.03340
7.54 9 105
-3.04 9 10-5
15.580
404
656
0.62
6.118
10.600
Mg65Cu25Y10 [20]
0.01750
1.8 9 106
13.160
-1.02 9 10-5
410
730
0.56
8.650
10.730
Zr57Cu15.4Ni12.6Al10Nb5 [18]
0.01630
15.700
6.32 9 106
-8.37 9 10-6
682
1091
0.63
9.400
13.130
Zr46(Cu4.5/5.5Ag1/5.5)46 Al8 [19]
20.810
0.01620
8.17 9 106
-2.08 9 10-5
620
937
0.66
8.200
11.060
17.870
Pt57.3Cu14.6Ni5.3P22.8 [21]
0.01010
5.77 9 106
-1.20 9 10-5
488
776
0.63
11.400
10.190
26.280
Zr52.5Cu17.9Ni14.6Al10 Ti5 [18]
0.00260
6.43 9 10
6
-16.8 9 10-6
675
1072
0.63
8.200
7.520
19.830
Ti36.89Cu43.87Ni9.36Zr9.88 [22]
0.02140
1.02 9 107
-0.97 9 10-5
678
1093
0.62
10.949
8.400
27.920
Ti37.65Cu43.25Ni9.6Zr9.5 [22]
0.00870
1.17 9 107
-9.25 9 10-6
673
1097
0.61
11.040
8.150
27.510
Zr41.2Ti13.8Ni10Cu12.5Be22.5 [23, 24]
0.01620
8.17 9 10
6
-2.08 9 10-5
620
937
0.66
8.200
6.220
23.300
Zr58.5Cu15.6Ni12.8Al10.3Nb2.8 [25]
0.01280
5.70 9 106
-1.37 9 10-5
660
1083
0.61
8.700
2.655
15.550
Pd43Ni10Cu27P20 [26]
0.03320
4.89 9 106
-5.00 9 10-5
576
790
0.73
7.200
2.860
17.280
DH ¼ DHm
DCpm
ð3 rl ÞðTm TÞ rl 1 2 2 ðT T Þ ; 1 2 Tm m Tm m rl 2 DS ¼ DSm DCp DT þ ð3 rl Þ ln ; Tm T DT 2 m DG ¼ DSm DT DCp ðrl 2Þ 2Tm Tm þð3 rl Þ DT T ln : T
DS ¼ DSm
DHm /(kJ mol-1)
Trg
DCpm
ð8Þ
Tm T Tm þ ðrh 1Þ ln ð2 rh Þ ; T T ð13Þ
and ð9Þ
ð10Þ
Comparative studies were conducted on the expressions for DG, DS, and DH produced by the hyperbolic expression and Dubey’s expression for DCp 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 DCp ; the expressions for DG, DS, and DH, respectively, are [15] m DG ¼ DSm DT DCp ð3 rh ÞDT þ ðð3 rh ÞTm ð11Þ Tm ðrh 1ÞDTÞ ln ; T Tm m DH ¼ DHm DCp ðrh 1ÞðTm TÞ þ ð2 rh ÞTm ln ; T ð12Þ
rh ¼ 1 þ
1 Trg n : 1 Trg
ð14Þ
Based on the hole theory of the liquid state, Dubey et al. [7, 8] provided an expression for the DCp as 2 Tm DT DCp ¼ DCpm exp rD ; ð15Þ T T where rD 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 DG; DH and DS are [7, 8] DT 2 DT 1 rD ; 3T 2T DT rD DT m 1 ; DH ¼ DHm DCp Tm T 2T DG ¼ DSm DT DCpm
ð16Þ ð17Þ
and DS ¼ DSm
DCpm
! DT DT rD DT 1 þ 2DT 3T 1þ 1 : T 2T 2 T 1 þ DT 2T ð18Þ
123
296
P.-Y. Li et al.
Fig. 1 Specific heat difference DCp a, b, and the deviation values of D DCp c, d as functions of temperature derived for 2 of 16 representative BMGs, Zr46Cu46Al8 and La55Al25Cu10Ni5Co5 alloys
The value rD is given by [7, 8] ! Trg 1 rD ¼ ln 2 : Trg n 1 Trg
3 Deviation between theoretical model-based calculated and experimental values of DCp in BMGs ð19Þ
The deviation percentage D, between the theoretical calculated value and the experimental value can be expressed as VðmodÞ VðexpÞ ; ð20Þ D¼ VðexpÞ where VðmodÞ and VðexpÞ represent the calculated values and experimental values of DCp ; DG; DH and DS respectively. As shown in Eqs. (4), (14), and (19), most of the BMG materials in this study exhibit Trg 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 DCp ; DG; DH; and DS occurring between temperatures from Tg to Tm correlate with the rl ; rh and rD values as well as the n value.
123
The thermodynamic behavior of BMGs is studied using expressions for thermodynamic parameters DCp ; DG; DS; DH; based on experimental results from 16 different BMGs. The values of DG, DS, and DH are calculated using experimentally measured DCp values, which can be expressed as [16–26] DCp ¼ AT þ BT 2 þ CT 2 ;
ð21Þ
where A, B, and C are constant. These constants and parameters can be found in Refs. [16–26] and are summarized in Table 1. The calculated DCp value, experimental DCp value, and deviation value between calculated and experimental values from Tg to Tm can be generated by Eqs. (1), (3), (15), (20), and (21). It is prolix that all figures of DCp ; DG; DS; DH are described for the 16 alloys. In consideration of the different n values and alloy
Thermodynamic properties in bulk metallic glasses
297
Fig. 2 Specific heat difference DCp a, b, and the deviation values of D DCp c, d as functions of temperature derived for 2 of 16 representative BMGs, Ti36.89Cu43.87Ni9.36Zr9.88 and Pd43Ni10Cu27P20 alloys
compositions in the present study, the 4 BMGs Zr46Cu46Al8 [19], La55Al25Cu10Ni5Co5 [17], Ti36.89Cu43.87Ni9.36Zr9.88 [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 DCp values and calculated values deduced from the hyperbolic, Dubey’s, and linear expressions. Initially, the deviations in DCp for these 4 BMGs increased, followed by an immediate reduction with further temperature increased from Tg to Tm : The deviations in DCp achieved a maximum deviation value, Dmax ; at an uncertain temperature in the range of Tg and Tm : Notably, this value was achieved approximately at the midpoint between Tg and Tm in each sample. For Zr46Cu46Al8 and La55Al25Cu10Ni5Co5, the Dmax values of DCp ; Dmax DCp ; from hyperbolic, linear, and Dubey’s expressions were each smaller than 4 %, indicating that calculated values of DCp closely approximated
experimental values, as compared with the Dmax DCp values for Ti36.89Cu43.87Ni9.36Zr9.88 and Pd43Ni10Cu27P20 BMGs (see Fig. 2). A comparison of the calculated and experimental values of DCp in BMGs is provided by the Dmax DCp values for 16 BMGs (see Table 2). The relationship between Dmax DCp values and n values for the 16 alloys is summarized in Fig. 3. In addition, Dmax DCp 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 n value for n [ 2. Notably, the maximum value of Dubey’s expression approached 38 % of the initial value for a n value of 6. For the hyperbolic expression, Dmax DCp values were generally less than 11 % of the initial values. Cumulatively, these findings indicated that the hyperbolic expression for DCp fitted well with the experimental values compared with both linear and Dubey’s expressions. When n \ 2, the Dmax DCp derived from all 3 expressions was smaller
123
123 2.7 1.6 1.3 1.1 0.6 2.5 4.2 1.3 0.7 2.4 5.4 5.3 1.1 4.1
10.5
10.6
Cu47Ti34Zr11Ni8
Zr46Cu46Al8
La55Al25Cu10Ni5Co5
La62Al14(Cu5/6Ag1/6)24
Mg65Cu25Y10
Zr57Cu15.4Ni12.6Al10Nb5
Zr46(Cu4.5/5.5Ag1/5.5)46Al8
Pt57.3Cu14.6Ni5.3P22.8
Zr52.5Cu17.9Ni14.6Al10Ti5
Ti36.89Cu43.87Ni9.36Zr9.88
Ti37.65Cu43.25Ni9.6Zr9.5
Zr41.2Ti13.8Ni10Cu12.5Be22.5
Zr58.5Cu15.6Ni12.8Al10.3Nb2.8
Pd43Ni10Cu27P20 2.4
10.4
9.6
15.9
9.1
9.0
13.8
5.5
6.7
10.3
0.7
2.4
2.7
3.2
0.6 1.9
37.5
34.0
18.8
11.1
15.4
11.0
3.5
2.0
4.2
10.8
3.3
3.1
3.1
7.5
9.3 5.1
1.1
1.7
1.0
0.2
1.7
1.8
0.4
0.2
0.4
0.9
0.7
0.1
0.3
0.3
0.7 0.3
2.5
3.9
2.5
0.9
5.2
5.5
1.2
0.8
1.9
2.3
1.5
0.3
0.7
0.8
2.0 0.5
3.2
6.4
4.4
2.7
14.6
14.0
2.0
2.5
7.4
4.5
2.5
0.4
2.0
1.4
3.8 0.9
D DS
0.3
1.9
2.4
5.1
3.1
2.0
2.7
2.2
2.8
2.4
0.0
0.5
0.6
0.8
0.1 0.3
0.7
4.9
7.2
16.8
10.9
7.0
6.6
8.7
10.7
5.9
0.2
1.2
1.7
2.1
0.2 0.8
D DH
D DG
D DH
D DG
Dubey
Hyper.
Line
Line-D (T = Tg)
Hyper.-D (T = Tg)
Dmax DCp
La62Al14Cu24 La55Al25Ni20
Alloys
0.9
8.4
13.0
45.7
31.9
18.0
10.7
25.0
41.0
11.3
0.3
1.8
2.3
3.9
0.5 1.1
D DS
5.4
7.1
5.4
3.8
5.5
3.8
0.6
0.1
0.3
0.5
1.4
0.1
0.2
1.0
4.4 0.4
D DG
15.6
20.4
18.1
13.2
19.5
12.5
1.5
2.2
4.1
4.1
9.1
2.0
1.3
10.8
32.8 4.2
D DH
Dubey-D (T = Tg)
20.7
36.0
33.4
36.7
57.1
32
2.4
8.0
21.1
11.2
18.4
3.3
2.3
23.8
74.7 6.7
D DS
6.04
5.85
3.74
3.38
3.28
2.64
2.58
1.62
1.58
1.46
1.24
1.24
1.22
1.17
0.84 1.04
n
Table 2 The maximum deviation values Dmax DCp of DCp ; deviation values D DG; D DH; and D DS; of DG; DH; and DS at Tg ; and n values for 16 alloys (Dmax DCp ; D DG; D DH; and D DS denote the deviation values of DCp ; DG; DH; and DS; respectively, between the calculated and experimental values)
298 P.-Y. Li et al.
Thermodynamic properties in bulk metallic glasses
Fig. 3 Relationships between the maximal deviation values Dmax DCp (the maximum deviation of DCp between the calculated and experimental values), and the n values for the 16 alloys in Table 2
than 11 % of the initial value, suggesting that all 3 theoretical models were applicable.
4 Deviations, calculated values, and experimental values of DG, DS, and DH in BMGs The deviation values D of parameters DG; DS; DH; D DG; D DS; and D DH; respectively, between calculated and experimental values for 4 BMGs were determined. As shown in Figs. 4–7, the values of D DG; D DH; and D DS exhibited maximum deviations at Tg in the temperature range from Tg to Tm : A notable exception to this trend was the deviation value of DG from
299
Dubey’s expression for DCp for Zr46Cu46Al8 (see Fig. 4d) and Mg65Cu25Y10 (not shown). Thus, deviation values for DG; DS; and DH at Tg could reasonably denote the degree of fit between DG; DS; and DH values in both calculated expressions and experimental results. D DG; D DH; and D DS values at Tg for all 16 metallic glasses are listed in Table 2. The D DG; D DH; and D DS values at Tg and n values for 16 BMGs are listed in Table 2. The maximum D DG values were achieved in the hyperbolic expression, whereas Dubey’s and linear expression values for DCp were smaller than 8 % of initial values, indicating the accuracy of calculated values using these 3 expressions for DG relative to experimental values. The maximum D DH and D DS values achieved by the hyperbolic expression for DCp in 16 BMGs were less than 10 % and 15 % of initial values (see Table 2), respectively. The maximum D DH and D DS values derived from linear and Dubey’s expressions for DCp (see Table 2) presented bigger values of 16.8 % and 45.7 %, respectively, compared with those derived from the hyperbolic expression. Thus, calculations for DG; DH; and DS values also suggested that the hyperbolic expression for DCp was the most accurate predictor of experimental values. Results and analysis of D DH and D DS derived from Dubey’s expression for DCp produced similar findings to those of D DCp derived from Dubey’s expression. The majority of D DH and D DS values derived from Dubey’s expression for n \ 2 were very close to the D DH and D DS values derived from the linear and
Fig. 4 Parameters DG a, DH b, and DS c as well as deviation values D DG d, D DH e, and D DS f as functions of temperature derived for the Zr46Cu46Al8 alloy using reported experimental results and three different expressions
123
300
P.-Y. Li et al.
Fig. 5 Parameters DG a, DH b, and DS c and deviation values D DG d, D DH e, and D DS f as functions of temperature derived for the La55Al25Cu10Ni5Co5 alloy using reported experimental results and 3 different expressions
Fig. 6 Parameters DG a, DH b, and DS c and deviation values D DG d, D DH e, and D DS f as functions of temperature derived for the Ti36.89Cu43.87Ni9.36Zr9.88 alloy using reported experimental results and 3 different expressions
hyperbolic expressions, suggesting that the difference between D DH and D DS values in each of the three expressions was very small for n \ 2 (see Table 2). The majority of D DH and D DS values derived from Dubey’s expression for n [ 2 were much larger than the D DH and D DS values derived from the linear and
123
hyperbolic expressions, suggesting that most DH; and DS values derived from Dubey’s expression did not accurately predict experimental values for n [ 2. Thus, the hyperbolic expression for DCp represents are relatively universal expression, compared to the linear expression and Dubey’s expression for DCp which are only
Thermodynamic properties in bulk metallic glasses
301
Fig. 7 Parameters DG a, DH b, and DS c and deviation values D DG d, D DH e, and D DS f as functions of temperature derived for the Pd43Ni10Cu27P20 alloy using reported experimental results and 3 different expressions
Fig. 8 Relationship between Trg and n values
accurate under certain conditions. Notably, experimental values more closely fit values produced by the linear expression for DCp than values produced by Dubey’s expression for DCp : Dubey’s expression for DCp was, however, a good approximation of experimental values for n \ 2, though not for n [ 2.
5 Discussion Dmax between the calculated and experimental values for DCp ; DG; DH; and DS from Tg to Tm was found to be associated with the n parameter according to Fig. 3 and Table 2. Equations (4), (14), and (19) showed that in addition to the effects of the n parameter, D values were
also influenced by the Trg 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 Trg values of 0.56, 0.65, and 0.73 were selected to further characterize the effect of Trg on the value of D (or Dmax ). Figure 9a shows the relationship between the rh ; rD ; and rl values and the n values for Trg values of 0.56, 0.65, and 0.73. For n \ 2, the change in the 3 rh ; rD ; and rl values with Trg was negligible, suggesting that DCp ; DG; DH; and DS values calculated using different models were virtually identical (see Fig. 3 and Table 2). For n [ 2, however, the values of rh ; rD ; and rl at the 3 different Trg values revealed a decreasing trend. As n values increased from 0.5 to 7, rD values exhibited only small decreases, while larger decreases were exhibited by rh values. Moderate decreases in rl values were observed in between those of rD and rh compared with rh ; rD ; and rl values at Trg = 0.65. Notably, this change can be neglected to simplify analysis. Thus the majority of BMGs Trg can be considered constant (Trg ¼ 0:65). Figure 9b demonstrates the relationship between rh ; rD ; and rl values and the n parameter at Trg = 0.65. When Trg = 0.65, the changes in rh ; rD ; and rl values are revealed to be very large, leading to variation in the deviation values for DCp ; DG; DH; and DS between calculated and experimental values. As shown in Fig. 3 and Table 2, the Dmax DCp values can be used to reveal the fit of DG; DH; and DS between the calculated and experimental values (see Fig. 3 and Table 2). Due to this observation, only Dmax DCp values are discussed.
123
302
Fig. 9 Relationship between rh ; rD ; and rl ; and n values for different Trg a and Trg ¼ 0:65 b
Graphs of Cp as a function of temperature when n values equal to 2 and 3 (n ¼ 2; n ¼ 3) are shown in Fig. 10a, in which Cpl and Cps represent the heat capacity of the supercooled liquid and crystal, respectively. Cpl and Cps values were derived by fitting the experimental data. Figure 10b shows the DCp value evolution and temperature increases for n values of 2 and 3, where the n parameter is the change rate of DCps and DCpm : In order to characterize the difference in Dmax values produced by Dubey’s expression and the hyperbolic expressions for DCp from Tg to Tm ; corresponding temperatures for Dmax DCp values can be inferred. Temperatures corresponding to the Dmax DCp values for Zr46Cu46Al8, La55Al25Cu10Ni5Co5, Ti36.89Cu43.87Ni9.36Zr9.88, and Pd43Ni10Cu27P20 BMGs are generally between Tg and Tm (see Figs. 1 and 2). Thus Tmax ¼ 0:5 Tg þ Tm ; where Tmax is the hypothetical maximum temperature of Dmax DCp : All 16 studied BMGs also exhibit trends similar to those of the 4 representative materials shown (data not shown). DCp values calculated using the linear
123
P.-Y. Li et al.
Fig. 10 Graphs of Cp a and DCp b for n = 2 and n = 3, respectively (The inset shows the fit of the linear expression for DCp at different n values in the temperature range from Tg to Tm )
expression closely approximate experimental values for DCp (see Fig. 3). Thus, for temperature of 0:5 Tg þ Tm ; a calculated value closely fits to the experimental DCp ; 0:5ðTg þTm Þ
DCp
ðexp :Þ; can be expressed as (see Fig. 10b)
DCp0:5ðTg þTm Þ ðexp :Þ ¼
DCpg þ DCpm 1 þ n DCpm : ¼ 2 2
ð22Þ
Substitution of Tmax into Eqs. (1) and (15), the hyperbolic expression and Dubey’s expression for DCp ; 0:5ðT þT Þ
0:5ðT þT Þ
DCp g m ðhyper:Þ and DCp g m ðDubeyÞ can be deduced as 2 rh 0:5ðTg þTm Þ DCp ðhyper:Þ ¼ rh 1 þ DCpm ; 0:5ð1 þ Trg Þ
DCp0:5ðTg þTm Þ
ð23Þ DCpm 1 Trg ðDubeyÞ ¼ exp r : 2 D 1 þ Trg 0:5ð1 þ Trg Þ ð24Þ
Thermodynamic properties in bulk metallic glasses
303
6 Conclusions
Fig. 11 Relationship between Dmax from the hyperbolic expression and Dubey’s expression for DCp and the n parameter for Trg = 0.65 using Eqs. (25) and (26)
Thus, expressions of Dmax ðhyper:Þ and Dmax ðDubeyÞ generated by the hyperbolic expression and Dubey’s expression for DCp between the calculated and experimental values, respectively, can be written as rh 1 þ 2rh 0:5ð1þTrg Þ ð25Þ 1 ; Dmax ðhyper:Þ ¼ 0:5ð1 þ nÞ 1Trg 1 exp r D 1þTrg 0:5ð1þT Þ 2 rg Þ ð Dmax ðDubeyÞ ¼ ð26Þ 1 ; 0:5ð1 þ nÞ where Trg is treated as a constant with a value of 0.65. Figure 11 shows the relationship between Dmax from the hyperbolic expression and Dubey’s expression for DCp as well as the n parameter based on Eqs. (25) and (26).When n \ 2, the majority of Dmax ðDubeyÞ and Dmax ðhyper:Þ values are less than 10 %, indicating that these values closely approximate the Dmax DCp values in Fig. 3. When 2 \ n \ 7, the Dmax ðDubeyÞ values dramatically increase from 10 % to 43.7 %, and the Dmax ðhyper:Þ values gradually increase from 8 % to 16 %. These findings suggest that the accuracy of values calculated using the hyperbolic expression for DCp are higher than those calculated using Dubey’s expression for DCp : Thus, calculated Dmax values (see Fig. 11) from the hyperbolic expression and Dubey’s expression vary according to trends very similar to those of experimental Dmax values (see Fig. 3). Based on the error scale determined by these findings, the expressions of Dmax ðhyper:Þ and Dmax ðDubeyÞ can be used to indicate changes in Dmax DCp according to the n parameter values shown in Fig. 3 and Table 2 for all 16 examined BMGs in the current study.
A linear expression for DCp derived from the hyperbolic expression for DCp was deduced and used to obtain a novel expression for DG; DH; and DS: According to the experimentally determined thermodynamic parameters of the 16 examined BMGs in the current study, more accurate calculations of DCp ; DG; DH; and DS were obtained using the linear, hyperbolic, and Dubey’s expression for DCp : These results suggest that the hyperbolic expression for DCp can be applied as a universal expression for DCp ; while linear and Dubey’s expressions for DCp are condition-dependent. Notably, Dubey’s expression for DCp also closely approximated experimental values when n \ 2, though values were shown to deviate from experimental values for n [ 2. Acknowledgments The work described in this paper was supported by the grant from the National Natural Science Foundation of China (Grant No. 51025415).
References 1. Zallen R (1973) The physics of amorphous solids. Wiley, New York 2. Machlin E (2007) An introduction to aspects of thermodynamics kinetics relevant to materials science. Elsevier, Science or Technology Books, Amsterdam 3. Stillinger FH (1988) Supercooled liquids, glass transitions and the Kauzmann paradox. J Chem Phys 88:7818–7825 4. Turnbull D (1950) Formation of crystal nuclei in liquid metals. J Appl Phys 21:1022–1028 5. Paul A (1982) Chemistry of glasses. Chapman and Hall, London 6. Singh HB, Holz A (1983) Stability limit of supercooled liquids. Solid State Commun 45:985–988 7. Dubey KS (2010) Thermodynamic and viscous behaviour of glass forming melts and glass forming ability. AIP Conf Proc 1249:211–232 8. Singh PK, Dubey KS (2012) Thermodynamic behaviour of bulk metallic glasses. Thermochim Acta 530:120–127 9. Jones D, Chadwick G (1971) An expression for the free energy of fusion in the homogeneous nucleation of solid from pure melts. Philos Mag 24:995–998 10. Mondal K, Chatterjee UK, Murty BS (2003) Gibb’s free energy for the crystallization of glass forming liquids. Appl Phys Lett 83:671–673 11. Patel TA, Pratap A (2010) Study of thermodynamic properties of Pt57.3Cu14.6Ni5.3P22.8 bulk metallic glass. AIP Conf Proc 1249:161–165 12. Thompson CV, Spaepen F (1979) On the approximation of the free energy change on crystallization. Acta Metall 27:1855–1859 13. Hoffman JD (1958) Thermodynamic driving force in nucleation and growth processes. J Chem Phys 29:1192–1193 14. Ji X, Pan Y (2007) Gibbs free energy difference in metallic glass forming liquids. J Non-Cryst Solids 353:2443–2446 15. Li PY, Wang G, Ding D et al (2013) Characterizing thermodynamic properties of Ti-Cu-Ni-Zr bulk metallic glasses by hyperbolic expression. J Alloys Compd 550:221–225
123
304 16. Jiang QK, Zhang GQ, Yang L et al (2007) La-based bulk metallic glasses with critical diameter up to 30 mm. Acta Mater 55:4409–4418 17. Lu ZP, Hu X, Li Y (2000) Thermodynamics of La based La-AlCu-Ni-Co alloys studied by temperature modulated DSC. Intermetallics 8:477–480 18. Glade SC, Busch R, Lee DS et al (2000) Thermodynamics of Cu47Ti34Zr11Ni8, Zr52.5Cu17.9Ni14.6Al10Ti5 and Zr57Cu15.4Ni12.6Al10Nb5 bulk metallic glass forming alloys. J Appl Phys 87:7242–7248 19. Jiang QK, Wang XD, Nie XP et al (2008) Zr-(Cu, Ag)-Al bulk metallic glasses. Acta Mater 56:1785–1796 20. Busch R, Liu W, Johnson WL (1998) Thermodynamics and kinetics of the Mg65Cu25Y10 bulk metallic glass forming liquid. J Appl Phys 83:4134–4141 21. Legg BA, Schroers J, Busch R (2007) Thermodynamics, kinetics, and crystallization of Pt57.3Cu14.6Ni5.3P22.8 bulk metallic glass. Acta Mater 55:1109–1116
123
P.-Y. Li et al. 22. Li PY, Wang G, Ding D et al (2012) Glass forming ability and thermodynamics in the new Ti-Cu-Ni-Zr bulk metallic glasses. J Non-Cryst Solids 358:3200–3204 23. Busch R, Kim YJ, Johnson WL (1995) Thermodynamics and kinetics of the undercooled liquid and the glass transition of the Zr41.2Ti13.8Cu12.5Ni10.0Be22.5 alloy. J Appl Phys 77:4039–4043 24. Cai AH, Chen H, Li X et al (2007) An expression for the calculation of Gibbs free energy difference of multi-component bulk metallic glasses. J Alloys Compd 430:232–236 25. Gallino I, Shah MB, Busch R (2007) Enthalpy relaxation and its relation to the thermodynamics and crystallization of the Zr58.5Cu15.6Ni12.8Al10.3Nb2.8 bulk metallic glass-forming alloy. Acta Mater 55:1367–1376 26. Fan GJ, Loffler JF, Wunderlich RK et al (2004) Thermodynamics, enthalpy relaxation and fragility of the bulk metallic glassforming liquid Pd43Ni10Cu27P20. Acta Mater 52:667–674
