Evaluation of glass-forming ability for bulk metallic glasses based on characteristic temperatures

Journal of Non-Crystalline Solids 355 (2009) 2183–2189

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Evaluation of glass-forming ability for bulk metallic glasses based on characteristic temperatures Ping Zhang a, Hongqing Wei a,c, Xiaolin Wei d, Zhilin Long a,b,*, Xuping Su b a

Institute of Fundamental Mechanics and Material Engineering, Xiangtan University, Xiangtan, Hunan 411105, China Key Laboratory of Materials Design and Preparation Technology of Hunan Province, Xiangton University, Hunan 411105, China c Department of Mechanical and Energy Engineering, Shaoyang University, Shaoyang 422000, China d Faculty of Materials, Optoelectronics and Physics, Xiangton University, Xiangton, Hunan 411105, China b

a r t i c l e

i n f o

Article history: Received 9 September 2008 Available online 8 August 2009 PACS: 64.70.pe 68.60.Dv 64.70.P67.25.de Keywords: Amorphous metals Metallic glasses Glass formation Glass transition

a b s t r a c t A new criterion x2, defined as Tg/(2TxTg)Tg/Tl (wherein Tg is the glass transition temperature, Tx the onset crystallization temperature, and Tl the liquidus temperature), has been proposed to assess the glass-forming ability (GFA) of bulk metallic glasses (BMGs) based on the classical crystallization theory and the crystallization resistance. The analysis indicates that the factors Tg/(2TxTg) and Tg/Tl could reflect the crystallization resistance and liquid phase stability of metallic glasses, respectively. From the available experimental data in literatures, the new criterion x2 has a better correlation with the GFA of metallic glasses than all other existing criteria such as Trg(=Tg/Tl), DTx(=TxTg), c(=Tx/(Tg+Tl)), DTrg(=(Tx  Tg)/ (Tl  Tg)), a(=Tx/Tl), b(=Tx/Tg + Tg/Tl), d(=Tx/(Tl  Tg)), u(=Trg(DTx/Tg)0.143), cm(=(2Tx  Tg)/Tl), b(=Tx  Tg/ (Tl  Tx)2) and n(=DTx/Tx+Tg/Tl). It has also been demonstrated that this x2 parameter is a simple and efficient guideline for exploring new BMG formers. Ó 2009 Elsevier B.V. All rights reserved.

1. Introduction Over the past two decades, a great deal of efforts has been devoted to exploring and synthesizing bulk metallic glasses (BMGs) because of the scientific and engineering significance. To date, many BMGs have been successfully developed and some of them have applications in engineering utilizing their exceptional properties [1–5]. However, one of the biggest obstacles for the use of these BMGs is still the low glass-forming ability (GFA) of many systems [5]. Understanding the nature of glass formation and GFA is the key for designing and developing new BMGs with improved properties. It is well known that GFA can be directly evaluated by the critical cooling rate (Rc) or the critical section thickness (Dmax) for glass formation. The smaller Rc or the larger Dmax is, the higher GFA of the alloys should be. However, Rc is difficult to measure experimentally, and Dmax strongly depends on the fabrication condition. Hence, a number of parameters which can easily be determined from differential thermal analysis (DTA) or differential scanning calorimetry (DSC) have been proposed to reflect the GFA of alloys. Their representatives include Trg(=Tg/Tl) [6,7],

DTx(=Tx  Tg) [8], c(=Tx/(Tg+Tl)) [9,10], DTrg(=(Tx  Tg)/(Tl  Tg)) [11], a(=Tx/Tl) [12], b(=Tx/Tg+Tg/Tl) [12], d(=Tx/(Tl  Tg)) [13], u(=Trg(DTx/Tg)0.143) [14], cm(=(2Tx  Tg)/Tl) [15], b(=Tx  Tg/ (Tl  Tx)2, hereafter referred to as b2) [16] and n(=DTx/Tx+Tg/Tl) [17]. Here, Tg, Tx and Tl are the glass transition, onset crystallization and liquidus temperatures, respectively. Among these GFA parameters/criteria, the reduced glass transition temperature Trg, the supercooled liquid region DTx, and the parameter c are the most widely used. Although each of those parameters can correlate quite well with GFA of some alloys, the correlation can be poor for other alloy systems. Therefore, further investigation is necessary to obtain a better and more precise criterion to infer the GFA of BMGs. The purpose of this paper is to investigate the interrelationship of characteristic temperature Tg, Tx, Tl with Rc and to propose a better criterion than the current GFA parameters for metallic glasses based on the classical crystallization theory and the crystallization resistance. Additionally, the validity of the criteria mentioned above will be compared and vindicated by the available experimental data of metallic glasses. 2. Theoretical analysis

* Corresponding author. Address: Institute of Fundamental Mechanics and Material Engineering, Xiangtan University, Xiangtan, Hunan 411105, China. Tel.: +86 732 8298287; fax: +86 732 8293240. 0022-3093/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2009.06.001

As discussed by Lu and Liu [10], glass formation is always a competing process between liquid phase and the resulting crystal-

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line phases. Therefore, GFA involve the liquid phase stability and the resistance to crystallization, and the former include two aspects: the stability of the liquid at the equilibrium state (i.e., stable state), and the stability of the liquid during undercooling (i.e., metastable state). Both the liquid phase stability and the crystallization resistance are related to characteristic temperatures Tg, Tx and/or Tl. In this sense, GFA could be well characterized by a proper combination of the above-mentioned three temperatures. According to the classical crystallization theory, when a liquid is cooled from the liquidus temperature Tl to the glass transition temperature Tg, the time t which required for the formation of a crystalline phase of volume fraction X is given by following equation [18,19]:

( )1=4 2 9:3g a90 X exp½ð16pr3m Þ=ð3NA kTGm Þ t ; kT f 3 NV ½1  expðGm =RTÞ3

ð1Þ

where g is the viscosity of the melt, k Boltzmann’s constant, T the transformation temperature, a0 the atomic diameter, NV the number of atoms per unit volume, R the universal gas constant, f the fraction of sites on the interface where atoms may preferentially be added and moved, Gm the molar free energy driving force for liquid to crystal growth, NA Avogadro’s number and rm the liquid/crystal interfacial energy per molar surface area. The time– temperature-transformation (TTT) curves can be calculated by Eq. (1). As shown in Fig. 1, in order to avoid crystallization, the liquid must be cooled fast enough from above Tl through Tg without intersecting the TTT curve, and Rc can be calculated using the following equation:

Rc ¼ ðT l  T n Þ=t n ;

ð2Þ

where Tn and tn are the temperature and the time at the nose of the TTT curve, respectively. Tn can be expressed in the light of Tl and Tg as Tn = a(Tg + Tl), where parameter a is between 0.45 and 0.55, and it is generally close to 0.5, i.e., Tn = 0.5(Tg + Tl) [20]. Thus, Eq. (2) can be rewritten as following:

Rc  0:5ðT l  T g Þ=t n ;

ð3Þ

Generally, a liquid with a high viscosity between the Tg and Tl favor glass formation. Since the viscosity at Tg is constant (1012 Pa s), a narrow of undercooled liquid region (Tl  Tg) or a high value of the reduced glass transition temperature Trg(=Tg/Tl) implies that the viscosity of undercooled melt can reach about 1012 Pa s by a lower degree of undercooled, or a rapid increase of viscosity with falling temperature below the Tl, which would result

in a high viscosity in the undercooled liquid state [8,18,21], and, consequently, lead to a high liquid phase stability and a low Rc. Thus, Trg can be used to indicate the liquid phase stability, and there was a rough linear correlation between Trg and log10Rc for metallic glasses [7,8,22]:

Rc / log10 Rc / T rg ¼ T g =T l ;

It is to be noticed that in any circumstance, the GFA of the glassforming liquids cannot be attributed to the liquid phase stability alone. The crystallization resistance must be considered as far as the GFA is concerned. It is well known that the supercooled liquid region DTx (=Tx  Tg) is an indication of the tendency of a glass upon heating above Tg [23]. A large DTx value may indicate that the supercooled liquid can remain stable in a wide temperature range without crystallization and has a high resistance to the nucleation and growth of crystalline phases. Since crystallization is actually a competitive process with respect to glass formation, a large DTx would lead to a high GFA. This speculation has been well confirmed in several glass-forming alloy systems in which the supercooled liquid region correlate reasonably well with the GFA of alloys [24,25]. Therefore, the correlation between the GFA and DTx can be expressed as:

GFA / DT x ;

Tl

Temperature /K

cooling TTT curve

Undercooled Liquid Tn

Crystall

Tx Rc Tg heating

Glass

tn

log10(time /s) Fig. 1. Schematic time–temperature-transformation (TTT) diagram for the onset of crystallization of a glass-forming liquid. Crystallization occurs between Tl and Tg, and it can be avoided when the liquid is chilled with the cooling rate RPRc.

ð5Þ

Moreover, based on Louzguine and Inoue’s investigation [26], the alloys with a higher crystallization activation energy usually have a higher Tx at the certain heating rates (e.g., 0.33 and 0.67 K/s). Thus, the onset crystallization temperature Tx on reheating could also be used to indicate the crystallization resistance during glass formation for metallic liquids [10,15]. This leads to the following correlation between GFA and Tx, i.e.

GFA / T x ;

ð6Þ

Base on the above analysis, it is reasonably to assume that the summation of (DTx + Tx) is a measure of the crystallization resistance of metallic glass. In order to make possible comparisons between various glasses showing different Tg, (DTx + Tx) should be normalized with Tg leading to the dimensionless factor (DTx+Tx)/ Tg. Considering that GFA is inversely proportional to Rc, the relationship between Rc and the factor (DTx + Tx)/Tg can be expressed as follows:

Rc / T g =ðDT x þ T x Þ ¼ T g =ð2T x  T g Þ;

ð7Þ

Combining Eq. (4) with (7), a new dimensionless criterion, which links the Rc of an alloy with its characteristic thermal temperatures Tg, Tx, and Tl, can be defined as:

x2 ¼ T g =ð2T x  T g Þ  T g =T i ; Liquid

ð4Þ

ð8Þ

According to Eqs. (4)–(8), Rc increases with an increase in x2, implying that x2 is inversely proportional to the GFA. In the following section, we shall show the strong correlation between x2 and GFA by analyzing the readily available experimental data. 3. The efficiency of the new criterion In order to compare the efficiency of the currently proposed x2 criterion with other GFA criteria reported so far, such as Trg, DTx, c, DTrg, a, b, d, u, cm, b2 and n, statistical analysis were made on the basis of the available experimental data of metallic glasses. The number of data points used for Rc and Dmax are 66 and 278, respectively. Table 1 tabulates characteristic temperatures Tg, Tx, Tl and the critical cooling rate Rc for the already reported 66 metallic glasses based on Au, Ca, Ce, Cu, Fe, La, Mg, Ni, Pd, Pr, Ti and Zr. The data in Table 1 were obtained from Refs. [27–43] and most of characteristic temperatures were measured using differential scanning calorimetry (DSC) and/or differential thermal analysis

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Table 1 The characteristic temperatures Tg, Tx, Tl, calculated x2, critical cooling rates determined experimentally Rc and calculated Rcx2 values using Eq. (10). Most of data were obtained by DSC or/and DTA at a heating rate of 20 K/min. Based type AuCaCeCuFe-

La-

Mg-

Ni-

Pd-

PrTiZr-

Alloy (at.%)

Tg (K)

Au77.8Ge13.8Si8.4 Ca65Mg15Zn20 Ce70Al10Ni10Cu10 Cu47Ti34Zr11Ni8 Cu50Zr50 Fe48Cr15Mo14Er2C15B6 Fe83B17 Fe91B9 La55Al25Cu20 La55Al25Ni10Cu10 La55Al25Ni15Cu5 La55Al25Ni20 La55Al25Ni5Cu10Co5 La55Al25Ni5Cu15 La66Al14Cu20 Mg65Cu25Gd10 Mg65Cu25Y10 Mg65Cu7.5Ni7.5Zn5Ag5Y10 Mg65Ni20Nd15 Mg70Ni15Nd15 Mg75Ni15Nd10 Mg77Ni18Nd5 Mg80Ni10Nd10 Mg90Ni5Nd5 Ni Ni59Zr16Ti13Si3Sn2Nb7 Ni60Nb40 Pd30Pt17.5Cu32.5P20 Pd37.5Cu30Ni12.5P20 Pd40Cu25Ni15P20 Pd40Cu30Ni10P20 Pd40Cu32.5Ni7.5P20 Pd40Ni40P20 Pd42.5Cu27.5Ni10P20 Pd42.5Cu30Ni7.5P20 Pd43.2Ni8.8Cu28P20 Pd43Ni10Cu27P20 Pd44Ni10Cu26P20 Pd45Cu25Ni10P20 Pd45Cu30Ni5P20 Pd75Si25 Pd77.5Cu6Si16.5 Pd77Cu6Si17 Pd79.5Cu4Si16.5 Pd82Si18 Pd95Si5 Pr60Cu20Ni10Al10 Ti63Be37 Zr38.5Ti16.5Ni9.75Cu15.25Be20 Zr39.88Ti15.12Ni9.98Cu13.77Be21.25 Zr41.2Ti13.8Cu12.5Ni10Be22.5 Zr42.63Ti12.37Cu11.25Ni10Be23.75 Zr44Ti11Cu10Ni10Be25 Zr45.38Ti9.62Cu8.75Ni10Be26.25 Zr46.75Ti8.25Cu7.5Ni10Be27.5 Zr52.5Cu17.9Ni14.6Al10Ti5 Zr55Al22.5Co22.5 Zr57Cu15.4Ni12.6Al10Nb5 Zr57Ti5Al10Cu20Ni8 Zr65Al7.5Cu17.5Ni10 Zr65Al10Cu15Ni10 Zr65Be35 Zr66Al8Cu12Ni14 Zr66Al8Cu7Ni19 Zr66Al8Ni26 Zr66Al9Cu16Ni9

Tx (K)

293 375 359 698.4 670 844 760 600 456 467.4 473.6 491 465.2 459.1 395 423 413 426 459 467 450 429 454 426 425 845 891 540 572 596 586 568 578 584 574 579 576 587 595 577 656 637 642 635 648 647 409 673 630 629 623 623 625 623 622 675 753 682 676.7 656.5 652 623 655.1 662.3 672 657.2

x2

Tl (K)

293 410 377 727.2 717 880 760 600 495 547.2 541.2 555 541.8 520 449 484 473 464 501 489 470 437 471 449 425 885 924 614 647 668 660 654 651 665 660 693 660 667 675 659 656 686 686 678 658 647 452 673 678 686 672 712 739 740 727 727 808 742 720 735.6 757 623 732.5 720.7 707.6 736.7

629 630 714 1169.2 1219 1446 1448 1628 896 835 899.6 941 822.5 878.1 731 740 760 717 805 844 790 887 878 919 1725 1301 1478 807 929 910 856 932 973 871 834 859 845 874 884 861 1343 1058 1128 1086 1071 1688 719 1353 1003 1006 996 1057 1206 1239 1185 1091 1323 1115 1145.2 1167.6 1121 1238 1172.1 1200.8 1251 1170.6

Rc (K/s) 6

0.5342 0.2475 0.4061 0.3265 0.3273 0.3377 0.4751 0.6314 0.3450 0.1857 0.2515 0.2714 0.1867 0.2675 0.2449 0.2045 0.2314 0.2545 0.2751 0.3606 0.3487 0.4804 0.4132 0.4390 0.7536 0.2640 0.3282 0.1157 0.1765 0.1505 0.1138 0.1581 0.2043 0.1123 0.0812 0.0434 0.0925 0.1142 0.1150 0.1085 0.5115 0.2646 0.3103 0.2960 0.3650 0.6167 0.2574 0.5026 0.2397 0.2213 0.2386 0.1884 0.2145 0.2241 0.2227 0.2478 0.3034 0.2387 0.2956 0.2436 0.1748 0.4968 0.2500 0.2985 0.3670 0.2438

3.000  10 2.000  101 4.450  102 1.000  102 2.500  102 2.780  101 8.300  105 2.600  107 7.230  101 2.250  101 3.450  101 6.750  101 1.880  101 3.590  101 3.750  101 1.000 5.000  101 5.000  101 3.000  101 1.782  102 4.610  101 4.900  104 1.251  103 5.300  104 3.000  1010 4.000  101 1.400  103 6.700  102 1.330  101 1.500  101 1.000  101 1.330  101 1.600 8.300  102 6.700  102 9.000  103 1.000  102 1.000101 1.000  101 8.300  102 1.000  106 1.000  102 1.250102 5.000102 1.800103 5.000  107 1.600  102 6.300  106 1.400 1.400 1.400 5.000 1.250  101 1.750  101 2.800  101 1.000  101 1.700  101 1.000  101 1.000  101 1.500 4.100 1.000  107 9.800 2.270  101 6.660  101 4.100

Rcx2 (K/s)

Ref.

1.204  106 1.364  101 7.422  103 3.146  102 3.255  102 4.916  102 1.154  105 5.735  107 6.566  102 1.173  100 1.599  101 3.533  101 1.219 3.022  101 1.233  101 2.478 7.216 1.801  101 4.091  101 1.219  103 7.619  102 1.422  105 9.873  103 2.746  104 7.345  109 2.632  101 3.368  102 7.290  102 8.149  101 2.894  101 6.740  102 3.923  101 2.456 6.370  102 1.850  102 4.100  103 2.900  102 6.850  102 7.080  102 5.470  102 4.899  105 2.692  101 1.655  102 9.378  101 1.454  103 3.193  107 2.025  101 3.433  105 1.000  101 4.828 9.581 1.305 3.677 5.398 5.100 1.382  101 1.257  102 9.635 9.244  101 1.168  101 7.596  101 2.724  105 1.506  101 1.037  102 1.575  103 1.178  101

[27] [27] [28,29] [27] [30] [29,31] [22] [27] [27] [27] [27] [27] [27] [27] [27] [27] [27] [27] [27] [27] [27] [27] [27] [27] [27] [27] [27,32] [33] [27] [27] [27] [27] [29,34] [27] [27] [35] [36,37] [35] [27] [27] [27] [27] [27] [27] [27,38] [27] [29,39] [27] [27] [27] [27] [27] [27] [27] [27] [40] [41] [40] [27] [27] [42,43] [27] [27] [27] [27] [27]

Table 2 The values of the statistical correlation parameter R2 corresponding to plots of Rc against Trg, DTx, c, DTrg, a, b, d, u, cm, b2, n and x2, respectively for 66 metallic glasses listed in Table 1. Parameter

Trg

DTx

c

DTrg

a

b

d

u

cm

b2

n

x2

R2

0.753

0.642

0.909

0.713

0.884

0.918

0.742

0.888

0.906

0.504

0.920

0.926

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Table 3 The values of the statistical correlation parameter R2 corresponding to plots of Rc against Trg, DTx, c, DTrg, a, b, d, u, cm, b2, n and x2, respectively for 53 metallic glasses from Ref. [27]. Parameter

Trg

DTx

c

DTrg

a

b

d

u

cm

b2

n

x2

R2

0.547

0.786

0.879

0.797

0.854

0.886

0.667

0.863

0.868

0.417

0.885

0.890

12

Critical cooling rate Rc, K/s

10

2

R =0.753

(a)

8

10

10

4

10

0

-4

10

0.2

0.3

0.4

0.5

0.6

0.7

T rg

Critical cooling rate Rc, K/s

10

12

10

2

R =0.642

8

10

4

10

0

10

(b)

-4

0

20

40

60 ΔTx

80

100

120

2 cooling rate values (Rx c ) based on regressive results are also summarized in Table 1. From the regression analysis of plots between the above-mentioned GFA criteria and Rc, the statistical correlation parameter R2 has been determined. R2 is a parameter in the range of 0–1 and can be obtained by a regression program. The R2 value can give an idea of the effectiveness of different GFA parameters. The higher the R2 value, the better is the correlation between the GFA parameter with Rc. Table 2 compares the R2 values for Rc with various GFA parameters based on the data listed in Table 1. It is obvious from Table 2 that the newly proposed x2 parameter gives an R2 value of 0.926 with Rc, which is the highest among all the GFA criteria, indicating that there is the strongest correlation between Rc and x2 among all the GFA criteria. The parameter n follows closely with an R2 value of 0.920 suggesting that it is the next best. It is important to note that the parameters b, c and cm are very close to x2 and n with R2 values of more than 0.9 with Rc. All the above five parameters are much better than DTx or Trg (see Table 2). Even if the data used by Lu and Liu [27,44] is considered (53 data sets for Rc), x2 still exhibits the best relation with Rc when compared with those of Trg, DTx, c, DTrg, a, b, d, u, cm, b2 and n (see Table 3). Fig. 2(a–c) show the representative plots of Rc against Trg, DTx and x2, respectively, for 66 metallic glasses listed in Table 1. The predicted error band, a narrower interval implying lower scatter of data and thus a more reliable correlation between the parameters, is shown as two dashed lines in each figure at a fixed confidence level of 95%. A linear interrelationship is observed between x2 values and log10Rc, as demonstrated by the solid line, that is,

log10 Rc ¼ ð3:134  0:193Þ þ ð17:250  0:609Þx2 ;

ð9Þ

Rc can be derived from Eq. (9)

Rc ¼ 7:347  104 exp ð39:720x2 Þ;

Critical cooling rate Rc, K/s

10

12

10

ð10Þ

2

R =0.926

(c)

8

x2

where Rc is in K/s. As shown in Table 1, Rc and Rc have the same order of magnitude for most of alloys, indicating that x2 can be utilized to estimate Rc. Similarly, the relation between Rc with GFA parameters c, a, b, /, cm and n can also be expressed as,

Rc ¼ A expðB GFA parameterÞ; 10

4

10

0

10

-4

0.0

0.2

0.4 ω2

0.6

0.8

Fig. 2. Relationships between the critical cooling rates Rc and GFA parameters: Trg (a), DTx (b) and x2 (c), respectively for 66 metallic glasses listed in Table 1.

(DTA) at a heating rate of 20 K/min. The same heating rate for measurements is emphasized due to the strong dependence of these characteristic parameters on it. Moreover, the predicted critical

ð11Þ

where A and B are fitting coefficients for various parameters given in Table 4, and GFA parameter is dimensionless and stands for c, a, b, /, cm and n, respectively. Fig. 3 shows relationships between the parameter Trg, DTx, c, DTrg, a, b, d, u, cm, b2, f, x2 and Dmax for Cu-, Ca-, Mg-, Ti-, Pd-, La-, Gd-, Pr-, Y-, Co-, Zr-, Fe- and Nibased 278 BMG data from Ref. [16]. As can be see, all GFA indicators exhibit a direct proportional or inverse relationship with respect to the Dmax value, indicating that all these parameters can, to a certain extent, reflect the GFA of the alloys. Table 5 presents the values of correlation coefficient, R2, corresponding to plots of Dmax against Trg, DTx, c, DTrg, a, b, d, u, cm, b2, n and 1/x2, respectively, as shown in Fig. 3. It is evident from Table 5 that the reciprocal value of the newly proposed x2, 1/x2, gives an R2 value of 0.599 or 0.373 with Dmax, which is also the highest

Table 4 The values of coefficients A and B used in the expression of Rc with GFA parameters. Parameter

c

a

b

/

cm

n

x2

A B

9.245  1020 112.050

1.453  1016 53.297

1.440  1036 47.710

4.822  107 36.974

1.573  1014 42.311

9.339  1015 50.343

7.347  104 39.720

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80

(a)

60

Dmax, mm

Dmax, mm

80

40 20

(b )

60 40 20 0

0 0.50

0.55

0.60

0.65

0

0.70

20

40

60

Dmax, mm

Dmax, mm

40 20

0.36

0.38

0.40

0.42

0.44

0.46

40 20

0.00

γ

0.05

0.10

0.15

80

(e)

0.25

0.30

(f)

60

Dmax, mm

60 40 20 0

40 20 0

0.55

0.60

0.65

0.70

0.75

1.55

0.80

1.60

1.65

1.70

α

1.75

1.80

1.85

β

80

80

(g) 60

(h)

60

Dmax, mm

Dmax, mm

0.20

Δ T rg

80

Dmax, mm

60

0

0 0.34

40 20 0

40 20 0

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

0.30

0.35

0.40

δ

0.45

0.50

φ

80

80

(i) 60

Dmax, mm

Dmax, mm

120

(d)

(c)

60

40 20 0

(j)

60 40 20 0

0.55

0.60

0.65

0.70

γm

0.75

0.80

0.85

0.90

0

2

4

6

8

10

12

ß2 80

(k)

60

(l)

60

Dmax, mm

Dmax, mm

100

80

80

80

80

ΔTx

T rg

40 20

40 20 0

0 0.55

0.60

0.65

0.70

0.75

0.80

ξ

2

4

6

1/ ω 2

8

10

Fig. 3. The correlations between the critical section thickness (Dmax) and the parameters Trg (a), DTx (b), c (c),DTrg (d), a (e), b (f), d (g), u (h), cm (i), b2 (j), f (k) and x2 (l) for 278 BMGs from Ref. [16].

among all the GFA criteria. Moreover, as shown in Tables 2, 3 and 5, it is interesting to note that the R2 values for the plots of different GFA criteria with Dmax are not as high as those with Rc for any of the criteria. This may be due to the fact that Rc does not depend on the conductivity of material and hence has a better cor-

relation to the GFA criteria than the Dmax [12]. From the above results, it is clear that our proposed x2 criterion is a better indicator in reflecting GFA than Trg, DTx, c, DTrg, a, b, d, u, cm, b2 and n. This will be further confirmed by the following application of the x criterion in Pd-based BMGs.

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Table 5 The values of the linear correlation coefficient R2, corresponding to plots of Dmax against Trg, DTx, c, DTrg, a, b, d, u, cm, b2, n and 1/x2, respectively for Cu-, Ca-, Mg-, Ti-, Pd-, La-, Gd, Pr-, Y-, Co-, Zr-, Fe- and Ni-based BMGs from Ref. [16]. Alloys

N

Dmax  Trg

DmaxDTx

Dmax  c

Dmax  DTrg

Dmax  a

Dmax  b

Dmax  d

Dmax  u

Dmax  cm

Dmax  b2

Dmaxn

Dmax  1/x2

A group B group

214 278

0.144 0.085

0.171 0.100

0.396 0.257

0.377 0.243

0.365 0.229

0.397 0.257

0.300 0.173

0.391 0.251

0.416 0.268

0.573 0.329

0.392 0.252

0.599 0.373

N is the number of samples. There are 214 alloys based Cu, Ca, Mg, Ti, Pd, La, Pr, Y and Co in A group. There are all 278 alloys based Cu, Ca, Mg, Ti, Pd, La, Pr, Y, Co, Zr, Fe and Ni in B group.

(a)

0.2

2

R =0.800

0.15

0.1

Critical cooling rate Rc, K/s

Critical cooling rate Rc, K/s

0.2

0.05

(b)

2

R =0.811

0.15

0.1

0.05 0.43

0.44

0.45

0.46

0.47

1.74

1.76

1.78

γ

(c)

0.2

2

R =0.767

0.15

0.1

0.05 0.72

0.74

0.76

0.78

0.80

0.82

Critical cooling rate Rc, K/s

Critical cooling rate Rc, K/s

0.2

1.80

1.82

1.84

0.16

0.18

β

(d) R2=0.849

0.15

0.1

0.05

0.08

0.10

0.12

ξ

0.14

ω2

Fig. 4. Relationships between the critical cooling rates Rc and GFA parameters: c (a), b (b), n (c) and x2 (d), respectively for Pd–Cu–Ni–P alloys system [44].

As is known, the best glass-forming alloys are found in Pd-based system. In order to explain their high GFA, the Rc has been measured for several Pd-based glasses, though tedious experiments are still unavoidable to determine Rc for each alloy composition [27,29,34–38]. Some investigations indicated that the formation of the centimeter-sized Pd-based glasses is mainly due to their extremely low Rc. Fig. 4(a–d) shows the relationships between Rc with c, b, n and x2, respectively for Pd–Cu–Ni–P alloys system. Apparently, x2 has the strongest ability in reflecting the GFA of these alloys. As demonstrated previously, the term GFA should not only mean the easiness with which a glass can be formed from the liquid, but also a high resistance to crystallization of the glass on its subsequent thermal exposure [9]. Thus, GFA is related to the liquid phase stability and the crystallization resistance. Similar to c, a, b, /, cm and n, the x2 parameter which include both the aspects of glass formation, namely, the easy glass formation and its stability, has better correlation with Rc or Dmax in comparison to those, which deal with only one of these two aspects of glass formation (DTx or Trg). On the other hand, as shown in Tables 2 and 3 and Table 5, the x2 parameter exhibit the best relation with Rc or Dmax among all the GFA criteria investigated in the present study. This may be because of the fact that x2 possesses a more proper combination of three characteristic temperatures (i.e., Tg, Tx and Tl) than the other parameters (e.g., c, a, b, /, cm and n). Based the above results and discussion, it can be concluded that the x2 criterion can be used as a guideline for exploring new BMG formers.

4. Conclusions A new criterion x2, defined as Tg/(2Tx  Tg)  Tg/Tl, which is proportional to log10Rc, has been proposed to evaluate the GFA for metallic glasses. The present results show that x2 exhibits the best correlation with GFA among all other existing criteria. The GFA of BMGs could be in proportion to the Tg/Tl value and (Tx + DTx)/Tg value, which can be used to characterize the liquid phase stability and the crystallization resistance respectively. Since the parameter x2 can be calculated simply by data on Tg, Tx and Tl, the current parameter is a simple and efficient guideline for exploring new BMG formers. Acknowledgement This work was supported by a Project Supported by Scientific Research Fund of Hunan Provincial Education Department (No.09A088), Project supported by Hunan Provincial Natural Science Foundation of China (No.09JJ6069), the Planned Science and Technology Project of Hunan Province (No. 2008FJ3095), and National Natural Science Foundation of China (Nos. 50671088, 50771089). References [1] Z.L. Long, B.L. Shen, Y. Shao, C.T. Chang, Y.Q. Zeng, A. Inoue, Mater. Trans. 47 (2006) 2566.

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