New criteria of glass forming ability, thermal stability and characteristic temperatures for various bulk metallic glass systems

Materials Science and Engineering A 459 (2007) 196–203

New criteria of glass forming ability, thermal stability and characteristic temperatures for various bulk metallic glass systems W.Y. Liu a,b , H.F. Zhang a,∗ , A.M. Wang a , H. Li a , Z.Q. Hu a a

Shenyang National Laboratory for Materials Science, Chinese Academy of Sciences, Institute of Metal Research, Shenyang 110016, PR China b Graduate School of the Chinese Academy of Sciences, Beijing 100039, PR China Received 4 December 2006; received in revised form 20 December 2006; accepted 11 January 2007

Abstract Six mathematical criteria used for quantitatively measuring their glass forming ability, thermal stability and characteristic temperatures (temperature of glass transition Tg , crystallization onset Tx and liquidus Tl ) of various bulk metallic glasses (BMGs) have been proposed in this article. It is found that the criteria are suitable for evaluating these quantities for BMGs. The estimated results are better than those calculated by Fang et al.’s models and Lu et al.’s γ criterion, but our analyses indicate that these criteria, defined by the data (Pauling electronegativity xi , atomic radius ri , electron concentration ni and melting temperature Tmi ) easily found in literatures, still need to be further refined in future research for shortcomings of the parameters. © 2007 Elsevier B.V. All rights reserved. Keywords: Bulk metallic glasses; Glass forming ability; Thermal stability

1. Introduction Glass forming ability (GFA) is very crucial for understanding the origins of glass formation and developing new bulk metallic glasses (BMGs). The GFA of a melt can be efficiently evaluated by the critical thickness for glass formation (Zc ), which is the maximum size to keep the melt amorphous without precipitation of any crystals during solidification and is dependent on its fabrication technique. The larger the Zc of a glassy system, the higher the GFA. However, Zc is a parameter that cannot be obtained before freezing the glass former. So a great deal of effort has therefore been devoted to searching for a simple and reliable gauge for quantifying GFA of metallic glasses. As a result, many criteria [1–3], such as Txg (Tx − Tg ), Trg (Tg /Tl ) and γ (Tx /(Tg + Tl )), have been proposed to reflect relative GFA among BMGs on the basis of the characteristic temperatures (glass transition temperature Tg , crystallization onset temperature Tx and liquidus temperature Tl ) measured by differential scanning calorimetry (DSC) or/and differential thermal analysis (DTA). These parameters have been successfully used as GFA indicators for different metallic glasses, but they cannot be com-

∗

Corresponding author. Tel.: +86 24 23971783; fax: +86 24 23971783. E-mail address: [email protected] (H.F. Zhang).

0921-5093/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2007.01.033

puted before DSC or DTA analysis. Hence, further investigation is necessary to obtain a better and more applicable criterion to reflect the GFA of BMGs, which can be computed quantitatively based on existing data to guide the design of BMG composition. Based on their experimental work, Inoue et al. [4] have firstly proposed two empirical rules for exploring multicomponent BMGs with high GFA and large supercooled liquid region (SLR), i.e. (1) a large mismatch in atomic size, and (2) a large negative heat of mixing between the main constituents. Recently, Fang et al. [5] have suggested two simple parameters to account for these two factors in multi-component Mg based systems, i.e. the electronegativity difference x and the atomic size parameter δ. According to their views, the electronegativity difference x and atomic size parameter δ are defined as   n n   x =  Ci (xi − x¯ )2 , x¯ = Ci xi , (1) i=1

  n   ri  2 δ =  Ci 1 − , r¯ i=1

i=1

r¯ =

n 

C i ri ,

(2)

i=1

where n is the number of component in the alloy system; Ci , xi and ri are the atomic percentage, Pauling electronegativity and covalent atomic radius of element i, respectively. The quantities

W.Y. Liu et al. / Materials Science and Engineering A 459 (2007) 196–203

xi and ri can be obtained in Table 1 of Ref. [5]. In following work [6], Fang et al. have introduced another new parameter on valence electron difference that is defined as n1/3 =

n  i=1

1/3

Ci (ni

− n¯ 1/3 ),

n¯ =

n 

Ci ni ,

(3)

i=1

where ni , the element electron concentration, is the number of (s + d) electrons and (s + p) electrons for transition metals and elements with p-electron structure, respectively. Interestingly, these authors have demonstrated that there is a linear equation between these parameters and BMG thermal stability (Txg ). But Fang et al. have not used these parameters to evaluate Zc value of any BMG system. In this paper, the authors aim at determining that, to what extent, these parameters can be utilized to measure the GFA of BMGs. The Ca-, Mg-, Zr-, Ti-, Cu-, Au-, Pt-, Pd-, Fe-, Ni- and Re (rare earth)based alloy systems were chosen for the study because of the variety and availability of experimental data. The authors found that Zc and Txg have poor relationship with these parameters in glassy systems listed in Table 1. So these parameters were refined by introducing two additional variables and rewriting Fang et al.’s models. Through the refined parameters, better results were obtained which had an advantage of γ parameter. The authors also used the models to predict the characteristic temperatures of various BMG systems and obtained rather good results. 2. Criterion analysis As mentioned previously, Zc is an effective GFA gauge in metallic glasses but it can not be acquired before X-ray diffraction (XRD) or transmission electron microscope (TEM) analysis. Moreover, Zc can only be determined once the composition and fabrication technique for glass formation are known, just like Rc (the critical cooling rate for glass formation), which is inverse to Zc value: a higher GFA, a lower Rc value. Rc is another effective parameter to indicate GFA, but it is difficult to be measured precisely. It is thus necessary to establish a simple and reliable parameter that correlates well with GFA and can be calculated using more accessible existing data, such as xi , ri , ni , or Tmi (all symbols stand for the same quantities in the full paper). In return, such a criterion can be utilized as a guideline for exploring new bulk glassy compositions. In the following, a new criterion for representing the Zc or Rc value will be proposed. Other criteria are also established, which are used to evaluate the GFA, thermal stability and characteristic temperatures of typical BMG systems. Glass formation is always a competing process between supercooled melt and crystalline phases [1,7,8]. The GFA of BMGs is associated with two competing aspects: (1) the stability of liquid phase and (2) the stability of competing crystalline phases. Factors either increasing the liquid phase stability or destabilizing the competing crystalline phases can increase Zc or decrease Rc value (enhance GFA). Based on the Hume–Rothery rules [8,9], the electronegativity difference actually reflects the bonding nature of atomic pairs in alloys. Elements with similar electronegativity tend to form

197

solid solutions, and elements with sufficiently different electronegativity tend to form compounds. That is to say, the GFA of both alloys will be extremely low due to the strong tendency for crystalline phase formation. So the proper mismatch in electronegativity can promote the formation of certain atomic pairs (i.e. clusters), which is favored to restrain the solubility of these elements in the competing crystalline phases. Thus, long-range interdiffusion is required during crystallization upon cooling, leading to an enhancement in GFA (a higher Zc or lower Rc value). On the other hand, large atomic size mismatch also confines the solubility of the constituent elements in the competing crystalline phases [8,9]. During the crystallization process upon cooling, these elements have to be redistributed. It retards the nucleation process and, in turn, promotes glass formation, hence leading to an increase in GFA. In addition, large atomic size difference in an alloy system can support a new supercooled liquid structure [8,10]: high dense random packing, new local configuration and long-range homogeneity. The new atomic configurations tighten the packed structure of undercooled liquids and increase their packing densities, which lower the groundstate energy of the undercooled liquids and thus stabilize the supercooled liquids, leading to enhance GFA. As we all know, the packing density in an alloy does not always keep up with the value of atomic size mismatch in full range. In other words, there exists an optimum atomic size ratio/distribution in a given alloy system as far as GFA is concerned, which agrees with Miracle’s topological  model [11] and λn criteria [12,13]. The λn criteria (λn = ni=2 Ci |1 − (ri /r1 )3 |) is equal to 0.1 [12] for binary system and 0.18 [13] for multicomponent system. As mentioned above, the GFA is not always increased with the increase of the difference in electronegativity and the atomic size ratios among constituent elements. Excessively large electronegativity difference triggers compound formation while excessive difference in atomic size may relax the high randomly packed structure of the undercooled liquids. So the authors have combined two electronegativity difference parameters (L and L ) and three atomic size ratio parameters (W, W and λn ), which can support a proper balance between L and L , as well as W, W and λn values. These parameters are defined as L=

n 

Ci |xi − x¯ |,

(4)

xi Ci 1 − , x¯

(5)

i=1 

L =

n  i=1

W=

n 

Ci |ri − r¯ |,

(6)

ri Ci 1 − , r¯

(7)

i=1

W =

n  i=1

λn =

n  i=2



3 ri Ci 1 − , r1

(8)

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W.Y. Liu et al. / Materials Science and Engineering A 459 (2007) 196–203

Table 1 The glass transition temperature (Tg ), crystallization temperature (Tx ), liquidus temperature (Tl ) and the critical glass formation thickness (Zc ) for various BMGs System

Tg (K)

Tx (K)

Tl (K)

Ca67 Mg19 Cu14 Ca57 Mg19 Cu24 Mg80 Ni10 Nd10 Mg75 Ni15 Nd10 Mg70 Ni15 Nd15 Mg65 Ni20 Nd15 Mg65 Cu25 Y10 Mg65Cu25Tb10 Mg65Cu25Tb9Y1 Mg65Cu25Tb8Y2 Mg65Cu25Tb7Y3 Mg65 Cu15 Ag5 Pd5 Gd10 Mg65 Cu25 Gd5 Y5 Mg65 Cu15 Ag5 Pd5 Y10 Mg65 Cu20 Zn5 Y10 Mg65 Cu20 Ni5 Gd10 Mg80 Cu10 Y10 Zr66 Al8 Ni26 Zr66 Al8 Ni19 Cu7 Zr66 Al8 Ni14 Cu12 Zr66 Al9 Ni9 Cu16 Zr65.5 Al5.6 Ni6.5 Cu22.4 Zr65 Al7.5 Ni10 Cu17.5 Zr65 Al7.5 Ni10 Cu17.5 Zr57 Ti5 Al10 Ni8 Cu20 Zr41.2 Ti13.8 Cu12.5 Ni10 Be22.5 Zr38.5 Ti16.5 Cu15.25 Ni9.75 Be20 Zr42.63 Ti12.37 Cu11.25 Ni10 Be23.75 Zr44 Ti11 Cu10 Ni10 Be25 Zr45.38 Ti9.62 Cu8.75 Ni10 Be26.25 Zr46.75 Ti8.25 Cu7.5 Ni10 Be27.5 Ti55 Zr10 Cu9 Ni8 Be18 Ti50 Zr15 Cu9 Ni8 Be18 Ti40 Zr25 Cu9 Ni8 Be18 Ti34 Zr11 Cu47 Ni8 Ti50 Ni24 Cu20 B1 Si2 Sn3 Cu60 Zr30 Ti10 Cu60 Hf25 Ti15 Cu47 Ti34 Zr11 Ni8 Cu47 Ti33 Zr11 Si1 Ni8 Cu47 Ti33 Zr11 Si1 Ni6 Sn2 Cu46 Al7 Zr45 Y2 Cu46 Al7 Zr42 Y5 Cu54 Ni6 Zr22 Ti18 Pd40 Ni40 P20 Pd40 Cu30 Ni10 P20 Pd81.5 Cu2 Si16.5 Pd79.5 Cu4 Si16.5 Pd77.5 Cu6 Si16.5 Pd73.5 Cu10 Si16.5 Pd71.5 Cu12 Si16.5 Pd77 Cu6 Si17 La70 Al14 (Cu,Ni)16 La68 Al14 (Cu,Ni)18 La66 Al14 (Cu,Ni)20 La62 Al14 (Cu,Ni)24 La55 Al25 Ni10 Cu10 La55 Al25 Cu20 La55 Al25 Ni5 Cu10 Co5 La66 Al14 Cu20 La55 Al25 Ni15 Cu5 La55 Al25 Ni5 Cu15 Nd60 Fe20 Al10 Y10 Nd55 Fe20 Al10 Y15

387 404 454.2 450 467.1 459.3 424.5 414 416 417 416 430 413 437 404 420 427 672 662.3 655.1 657.2 630 650 656.5 676.7 623 630 623 625 623 622 629 622 621 698.4 726 720 730 671 720 720 693 672 712 590 576.9 633 635 637 645 652 642.4 404 405 405 417 467.4 455.9 465.2 395 473.6 459.1

407 440 470.5 470.4 489.4 501.4 479.4 487 489 492 488 472 486 471 456 481 448 707.6 720.7 732.5 736.7 733 750 735.6 720 672 678 712 739 740 727 667 662 668 727.2 800 757 795 717 757 765 770 772 769 671 655.8 670 675 678 685 680 686.4 429 431 431 446 547.2 494.8 541.8 449 541.2 520

1251 1200.8 1172.1 1170.6 1211 1153 1167.6 1145.2 996 1003 1057 1206 1239 1185 1013 1009 1009 1169.2 1310 1160 1160 1160 1282 1140 1143 1113 1287 991 836 1097.3 1086 1058.1 1135.9 1153.6 1128.4 763 724 674 738 835 896.1 822.5 731 899.6 878.1

776

834

1004

878 789.8 844.3 804.9 770.9 733 732 734 736 748 755

786

Zc (K)

References

2 4 0.6 2.8 1.5 3.5 7 5 5 5 5 10 5 7 6 5 1.5

[14] [14] [2,7] [2,7] [2,7] [2,7] [2,7] [15] [15] [15] [15] [16] [17] [18] [19] [20] [21] [2,7] [2,7] [2,7] [2,7] [22] [22] [2,7] [2,7] [2,7] [2,7] [2,7] [2,7] [2,7] [2,7] [23] [23] [24] [2,7] [2,7] [25] [25] [26] [27] [27] [28] [28] [29] [2,7] [2,7] [2,7] [2,7] [2,7] [2,7] [2,7] [2,7] [30] [30] [30] [30] [2,7] [2,7] [2,7] [2,7] [2,7] [2,7] [31] [31]

3 3 16 10 50

6 6 8 4.5 1 4 4 3 4 6 8 10 6 25 72 2 0.75 1.5 2 2 2 0.5 1 1.5 10 5 3 9 2

3 2

W.Y. Liu et al. / Materials Science and Engineering A 459 (2007) 196–203

199

Table 1 (Continued ) System

Tg (K)

Tx (K)

Tl (K)

Zc (K)

References

Nd60 Al15 Ni10 Cu10 Fe5 Nd61 Al11 Ni8 Co5 Cu15 Nd60 Fe30 Al10 Pr68 Cu25 Al7 Pr68 (Cu,Ni)25 Al7 Pr72 (Cu,Ni)25 Al3 Pr72 (Cu,Ni)21 Al7 Fe61 B15 Mo7 Zr8 Co7 Y2 Fe61 B15 Mo7 Zr8 Co6 Y2 Al1 Fe61 B15 Mo7 Zr8 Co5 Y2 Cr2 Pt57.5 Cu14.7 Ni5.3 P22.5 Pt42.5 Cu27 Ni59.5 P21 Pt60 Cu16 Co2 P22 Au49 Ag5.5 Pd2.3 Cu26.9 Si16.3 Au55 Cu25 Si20 Cu50 Zr50 Ni62 Nb38 Ce70 Al10 Cu20 Ce68 Al10 Cu20 Fe2 Sc36 Al24 Co20 Y20 Er50 Al24 Co20 Y6 Sm40 Al25 Co20 Y15 Ho35 Al24 Co20 Y21 Dy46 Al24 Co18 Fe2 Y10 Tb36 Al24 Co20 Y20 Gd36 Al24 Co20 Y20

430 445 591 382 399 367 395 904.6 899.5 901.1 508 515 506 401 348 670 892 341 352 662 651 590 644 627 619 603

475 469 722 402 416 402 410 916.4 955.6 958.9 606 589 569 459 383 717 932 401 423 760 702 657 696 677 868 658

779 744 958 705 703 743 760 1490.2 1495.7 1490.1 795 873 881 644 654

5 6 15 1.5 1.5 1.5 1.5 5 5 5 16 20 16 5 0.5 2 2 2 5 3 8 3 5 5 5 3

[2,7] [2,7] [32] [33] [33] [33] [33] [34] [34] [34] [35] [35] [35] [36] [36] [37] [38] [39] [40] [40] [40] [40] [40] [40] [40] [40]

1483 708 1048 1079 950 1074 1023 1021 1048

Most of temperature data were obtained by DSC or/and DTA at heating rate of 20 K/min; most of glass formation was obtained by the copper mould casting methods.

where both x¯ and r¯ parameters are still according with Fang et al.’ definitions [5,6]. It is important to note that an unfavorable relative size or relative electronegativity factor alone is sufficient to limit GFA to a low value. If both factors above are favorable, other factors should be considered in deciding on the probable degree of GFA. So the authors introduce another two parameters, the valence electron difference Y and reduced melting temperature Trm . Y is defined by Eq. (9). The Hume–Rothery rules also tell that [9]: a metal of lower valence tends to dissolve a metal of higher valence and vice versa. So the relative valence factor should be considered too. The electron number of an element is counted by Fang et al.’s method. Trm (defined by Eq. (10)) represents the fractional departure of Tmi from the simple rule of mixtures melting temperature T¯ m . Ci and Tmi are the mole fraction and melting point, respectively, of the ith component of an n-component alloy system. Y=

n  i=1

Trm =

1/3 Ci |ni

n  i=1

− n¯

1/3

|,

Tmi , Ci 1 − T¯ m

(9)

T¯ m =

n 

Ci Tmi .

(10)

i=1

Considering the melt as a regular solution, its Gibbs free energy G can be expressed as G = H − TS,

(11)

As mentioned above, there is a relationship between enthalpy H, temperature T, entropy S and electronegativity difference,

atomic size mismatch, valence electron difference, respectively. So the authors have combined these seven parameters according to the mode of Eq. (11) as ln f (u) = A0 + A1 L + A2 L + Trm (A3 W + A4 W  + A5 λn ) + A6 Y,

(12)

where f(u) stands for the as-calculated value: ln Zc , ln Rc , ln Txg , ln Tg , ln Tx or ln Tl ; Ai (i = 0–6) represents the ith value among a series of constant, which can be obtained by computed data list in Table 1 [2,7,14–36]. So Eq. (12) is a potential criterion for developing and designing new BMG composition.

3. Empirical criteria calculation Table 1 shows the glass transition temperature Tg , crystallization onset temperature Tx , liquidus temperature Tl and critical glass formation thickness Zc for Ca-, Mg-, Zr-, Ti-, Cu-, Au-, Pt-, Pd-, Fe-, Ni- and Re (rare earth)-based BMGs. Most of their characteristic temperatures were measured by DSC or/and DTA at a heating rate of 20 K/min. Most of their Zc values were gotten by copper mould casting method. In some Pd-based system, flux melting technique was applied. All Rc values used in the paper were extracted from Table 3 of Ref. [7], which are not listed in Table 1. Using Eq. (12), the authors evaluated its effect on GFA (Zc and Rc ), thermal stability (Txg value) and the characteristic temperatures (Tg , Tx and Tl ) of various BMGs listed in Table 1.

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W.Y. Liu et al. / Materials Science and Engineering A 459 (2007) 196–203

Table 2 Summary of R2 values for various mathematical criteria

Zc Rc Txg Tg Tx Tl

The authors’

Lu et al.’s γ

0.61 0.63 0.40 0.70 0.70 0.73

0.49 0.64

Fang et al.’s

0.05

Their empirical formulas are listed as follows:

This prediction interval, which was also computed by the same regression program, describes the range where the data values will fall a percentage of the time for repeated measurements. A narrower band at a fixed confidence level (normally 95%) implies a less scatter of the experimental data and a stronger correlation between independent variables [7]. exp Fig. 2 is a plot of the Rc experimental value (Rc ) as a function of the Rc calculation value (Rcal c derived by Eq. (14)) for typical bulk metallic glasses. A linear relationship between calculation values and experimental values is obtained with considerable scatter. The relationship can be expressed as following:

ln Zccal = −1.50 + 27.84L − 36.43L + Trm (−1995.35W + 475.55W  − 25.15λn ) + 11.10Y,

(13)

  ln Rcal c = 10.81 − 136.15L + 177.69L + Trm (−3337.61W + 288.21W − 48.64λn ) − 17.38Y,

(14)

cal ln Txg = 2.79 + 18.79L − 27.02L + Trm (539.92W + 29.97W  − 23.47λn ) + 6.51Y,

(15)

ln Tgcal = 6.26 + 8.11L − 11.32L + Trm (0.78W + 16.05W  − 5.29λn ) − 4.31Y,

(16)

ln Txcal = 6.27 + 8.86L − 12.43L + Trm (13.28W + 21.56W  − 6.90λn ) − 3.40Y,

(17)

ln Tlcal = 6.85 + 6.95L − 9.31L + Trm (−244.75W + 48.45W  − 7.63λn ) − 4.31Y.

(18)

In order to reveal how closely the estimated values for the regression calculation correspond to the actual experimental data, the statistical correlation parameter, R2 , was also computed using the regression method. These values are listed in Table 2. In order to compare with other criteria, their R2 values are also summarized in Table 2. The R2 value, also known as the coefficient of determination, is an indicator ranging from 0 to 1. The higher the R2 value, the more reliable the regression calculation [7]. As can be observed from Eqs. (13)–(18), their R2 values are not less than 0.60, which all are practicable criteria for developing new BMG composition except for Txg criterion (it is 0.40). 4. Discussion

cal 2 Rexp c = (0.59 ± 0.06)Rc + (8.62 ± 20.50), R = 0.78 (20)

The dash line in Fig. 2 also shows the 95% prediction limits for this correlation, as expressed in Eq. (20). The data in Fig. 2 are scattered too. The widely spread data result in a lower R2 value of 0.78 and a larger prediction band. This can presumably be attributed to that Rc values were difficult to measure precisely. Most of them were estimated by empirical formula. On the other hand, Rc values used here were not complete (33 data in total), which only reflected partial BMG systems. In order to compare with γ parameter, the authors also computed γ regression formula for Zc and Rc value, which are listed as follows: ln Zc = −10.78 + 30.35γ,

4.1. The correlation between new criteria (Zc and Rc ) and GFA

R2 = 0.49,

(21)

exp

The relationship between the Zc experimental value (Zc ) and Zc calculation value (Zccal derived by Eq. (13)) in representative metallic glasses is shown in Fig. 1. A linear relationship is observed between them, as demonstrated by the solid line. This relationship is expressed in an approximation formula, where Zc is in millimeter: Zcexp = (1.02 ± 0.04)Zccal − (2.24 ± 0.49),

R2 = 0.85. (19)

In order to reveal how closely the estimated values for the regression line correspond to the actual experimental data, the statistical correlation parameter, R2 , was also computed using the regression method. As is clear in the graph, the R2 value is as high as 0.85 for this fitting, suggesting that there is a solid correlation between the Zc calculation value and the novel indicator for Zc value. The predicted error band in Eq. (19) obtained at 95% confidence interval is shown in Fig. 1 as two dashed lines.

Fig. 1. The relationship between the Zc experimental value and Zc calculation value in representative BMGs.

W.Y. Liu et al. / Materials Science and Engineering A 459 (2007) 196–203

Fig. 2. The plot of the Rc experimental value as a function of Rc calculation value for typical BMGs.

201

exp

Fig. 4. The relationship between Tg

and Tgcal in various BMGs.

following: ln Rc = 35.57 − 81.54γ,

R = 0.64, 2

(22)

As shown in Table 2, the new criteria can be successfully used in Zc and Rc estimation than γ criterion. So γ criterion can be replaced by Zc and Rc criteria to predict Zc and Rc values. 4.2. The correlation between new criterion and thermal stability (Txg value) Unlike GFA, the thermal stability of metallic glasses is determined by crystallization processes upon reheating, and thus is mainly controlled by the difficulties in atomic rearrangement during heating [8]. Fig. 3 is a plot of the Txg experimental value exp cal derived by (Txg ) as a function of the Txg calculation value (Txg Eq. (15)) for typical bulk metallic glasses. A linear relationship between calculation values and experimental values is obtained with considerable scatter. The relationship can be expressed as

exp cal Txg = (1.01 ± 0.12)Txg + (2.96 ± 6.08), R2 = 0.44.

(23)

The dash line in Fig. 3 also shows the 95% prediction limits for this correlation, as expressed in Eq. (23). Compared with Figs. 1 and 2, the data in Fig. 3 are the most scattered. The widely spread data result in a lowest R2 value of 0.44 and a largest prediction band. This can presumably be attributed to that both Tg and Tx are dependent on but have different functions of heating rates during DSC measurements. Tg and Tx values listed in Table 1 were not obtained by the same heating rate, which affected their evaluating precision. On the other hand, different fabrication methods also vary the values of Tg and Tx . In order to compare with Fang et al.’s parameters, the authors also computed Fang et al.’s regression formula for Txg , which is expressed as following: ln Txg = 3.80 − 3.27x2 + 12.03δ2 + 3.40n2/3 , R2 = 0.05,

(24)

As shown by Eqs. (15), (23) and (24), the R2 value of new criterion is higher than that of Fang et al.’s estimated Txg value. So the new Txg criterion is much better than Fang et al.’s criterion to evaluate thermal stability for BMGs. 4.3. The correlation between new criteria and characteristic temperatures The relationships between the experimental value (Texp ) and calculation value (Tcal derived by Eqs. (16)–(18)) of their characteristic temperatures in representative metallic glasses are shown in Figs. 4–6. A linear inter-relationship is observed between Texp and Tcal for each temperature, as demonstrated by the solid line in every figure. These relationships are expressed in the approximation formulas as follows: Fig. 3. The plot of the Txg experimental value as a function of Txg calculation value for typical BMGs.

Tgexp = (1.01 ± 0.07)Tgcal − (0.70 ± 37.66),

R2 = 0.70, (25)

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W.Y. Liu et al. / Materials Science and Engineering A 459 (2007) 196–203

As is clear in the three graphs, the R2 values are not lower than 0.70 for their fittings, and the predicted error band in Eqs. (25)–(27) obtained at 95% confidence intervals are shown in the three figures. These prediction intervals are narrow, although wider than that in Fig. 1 because of lower R2 value, which also can imply less scatter of the experimental data and a good correlation between the calculation values. So the new criteria are useful to evaluate their characteristic temperature for various BMGs. It is a guideline for setting experimental parameters for DSC or DTA analysis. The authors thus concluded that the novel criterion can evaluate the characteristic temperatures for various BMG systems, which give another strong confirmation that the criterion can be used widely in the field of designing and developing of new BMGs with proper characteristic temperatures. exp

Fig. 5. The relationship between Tx

exp

Fig. 6. The relationship between Tl

and Txcal in various BMGs.

4.4. Application of new criteria in bulk glass formation of Mg–Cu–Ag–(Gd, Y) and Mg–(Cu, Ag)–Er system

and Tlcal in various BMGs.

Txexp = (0.99 ± 0.07)Txcal + (10.90 ± 40.40),

R2 = 0.70, (26)

exp

Tl

= (1.00 ± 0.07)Tlcal + (2.34 ± 67.79),

R2 = 0.72. (27)

As mentioned above, the novel criteria can be used as indicators for designing and developing new BMG systems. The authors applied the criteria to Mg–Cu–Ag–(Gd, Y) and Mg–(Cu, Ag)–Er bulk glass formers. Table 3 presents the summary of Tg , Tx , Tl , Txg and Zc calculated values (based on Eqs. (13), (15)–(18)) and experimental values obtained from Refs. [41,42]. Table 4 summarizes their error gaps between the experimental value and calculation values for both alloy systems. From the two tables, it is proved that there is a good application in these systems. Among the evaluated values, Zc for Mg65 Cu25 Er10 alloy has a large departure of 4.9 mm between the calculation value and experimental value. Txg calculation values for all alloys are lower (about 13–25 K) or higher (0–10 K) than their experimental values because of Txg estimation with a low R2 value of 0.44. The Txg error gaps are narrower than those derived by Txcal − Tgcal , although both calculation values have higher R2 values than Txg estimation. Their error gaps for the characteristic temperatures are rather scattered for their medium R2 value of 0.70. This shows that the new criterion needs to be refined further. It is concluded that it only partially reflects the effects of the electronegativity difference, atomic or covalent radius distribution and reduced melting temperature on the GFA and thermal stabilities of the BMGs.

Table 3 1 derived by Eq. (15); T 2 derived Summary on calculation values (left column) and experimental values (right column in parentheses) of Tg , Tx , Tl , Zc and Txg (Txg xg by Tx − Tg ) for Mg–Cu–Ag–(Gd, Y) and Mg–(Cu, Ag)–Er alloy systems System

Tg (K)

Tx (K)

Tl (K)

Zc (mm)

1 (K) Txg

2 (K) Txg

Mg65 Cu15 Ag10 Y10 Mg65 Cu15 Ag10 Y8 Gd2 Mg65 Cu15 Ag10 Y6 Gd4 Mg65 Cu15 Ag10 Y4 Gd6 Mg65 Cu15 Ag10 Y2 Gd8 Mg65 Cu15 Ag10 Gd10 Mg65 Cu25 Gd10 Mg65 Cu25 Er10 Mg65 Cu15 Ag10 Er10

432 (428) 432 (428) 432 (427) 431 (424) 431 (420) 431 (416) 418 (408) 426 (422) 419 (427)

481 (469) 481 (472) 480 (472) 480 (467) 479 (464) 478 (459) 467 (478) 476 (480) 470 (465)

736 (694) 739 (690) 741 (685) 742 (682) 743 (683) 746 (686) 739 (755) 753 (766) 722 (733)

9.7 (7) 9.4 (7) 9.0 (7.5) 8.6 (8) 8.2 (9) 8.1 (7.5) 7.6 (8) 7.9 (3) 9.8 (6)

46 (41) 45 (44) 45 (45) 44 (43) 44 (44) 43 (43) 45 (70) 45 (58) 48 (38)

50 (41) 49 (44) 49 (45) 48 (43) 48 (44) 48 (43) 49 (70) 50 (58) 51 (38)

The temperature data were obtained by DSC at heating rate of 20 and 40 K/min for Mg–Cu–Ag–(Gd, Y) [37] and Mg–(Cu, Ag)–Er [38] alloy systems, respectively; glass formation was obtained by the copper mould casting methods.

W.Y. Liu et al. / Materials Science and Engineering A 459 (2007) 196–203

203

Table 4 exp Summary of error gaps between calculation values and experimental values for Mg–Cu–Ag–(Gd, Y) and Mg–(Cu, Ag)–Er alloy systems, where Tg = Tgcal − Tg , 



 (T 1 derived by Eq. (15); T 2 derived by T  − T  ) as well as Tx , Tl , Zc and Txg xg xg x g

System

Tg (K)

Tx (K)

Tl (K)

Zc (mm)

Mg65 Cu15 Ag10 Y10 Mg65 Cu15 Ag10 Y8 Gd2 Mg65 Cu15 Ag10 Y6 Gd4 Mg65 Cu15 Ag10 Y4 Gd6 Mg65 Cu15 Ag10 Y2 Gd8 Mg65 Cu15 Ag10 Gd10 Mg65 Cu25 Gd10 Mg65 Cu25 Er10 Mg65 Cu15 Ag10 Er10

4 4 5 7 11 15 10 4 −8

12 9 8 13 15 19 −11 −4 5

42 49 56 60 60 60 −16 −13 −11

2.7 2.4 1.5 0.6 −0.8 0.6 −0.4 4.9 3.8

5. Conclusions It has proposed six criteria on designing and developing of new BMG systems in this article. The criteria combine the effects of electronegativity mismatch, atomic or covalent ratios, valence electron difference and reduced melting temperatures in the glassy former systems. These criteria were utilized to evaluate the glass forming ability (Zc or Rc ), thermal stability (Txg value) and characteristic temperatures (Tg , Tx and Tl ) for BMGs. A reasonable application was undertaken in the alloy systems of Mg–Cu–Ag–(Gd, Y) and Mg–(Cu, Ag)–Er. This is the first time to calculate the critical thickness for glass formation of the glass former candidates through ri , xi , ni and Tmi , which is easily available in existing data. The new criteria, especially ln Zc criterion, provide several useful guidelines for locating composition of new bulk metallic glasses. Acknowledgements

[10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27]

The authors gratefully acknowledge the financial support from the Ministry of Science and Technology of China (Grants Nos. 2006CB605201, 2005DFA50806) and the National Natural Science Foundation of China (Grant No. 50471077).

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1 (K) Txg

5 1 0 1 0 0 −25 −13 10



2 (K) Txg

9 5 4 5 4 5 −21 −8 13

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