A criterion for evaluating glass-forming ability of alloys
Guo-Hua Zhang and Kuo-Chih Chou
Citation: Journal of Applied Physics 106, 094902 (2009); doi: 10.1063/1.3255952
View online: http://dx.doi.org/10.1063/1.3255952
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JOURNAL OF APPLIED PHYSICS 106, 094902 共2009兲
A criterion for evaluating glass-forming ability of alloys
Guo-Hua Zhang1 and Kuo-Chih Chou1,2,a兲
1
University of Science and Technology Beijing, Beijing 100083, China
Shanghai University, Shanghai 200072, China
2
共Received 4 August 2009; accepted 29 September 2009; published online 9 November 2009兲
A criterion defined as ␪ = 共Tx + Tg兲 / Tl关共Tx − Tg兲 / Tl兴␣ 共where Tx is the onset crystallization
temperature, Tg is the glass transition temperature, Tl is liquidus temperature, and ␣ is an exponent兲
is proposed to evaluate the glass-forming ability of alloy systems. Regression analysis between the
logarithm of the critical cooling rate ln Rc and ␪ yields value of ␣ = 0.0728. There exists a fine linear
relationship between criterion ␪ and ln Rc with a coefficient of determination R2 = 0.942. The
obtained relationship can be expressed as: ln Rc = 共30.10⫾ 0.91兲 − 共27.93⫾ 0.97兲 ␪. The comparisons
of ␪ with all other relevant parameters: ␸兵=Trg关共Tx − Tg兲 / Tg兴0.143其, ␥关=Tx / 共Tl + Tg兲兴, Trg共=Tg / Tl兲,
␤关=TxTg / 共Tl − Tx兲2兴, ␦关=Tx / 共Tl − Tg兲兴 show that this criterion has the best ability for evaluating the
GFA of alloys. © 2009 American Institute of Physics. 关doi:10.1063/1.3255952兴
I. INTRODUCTION
II. THEORETICAL ANALYSIS
With the rapid development of bulk metallic glasses, it is
urgent to find a valid criterion to evaluate the glass-forming
ability 共GFA兲 of alloys. The critical cooling rate Rc is the
most direct and accurate parameter for evaluating the GFA,
however, it is always very difficult to obtain this value for
glass-forming alloys. The critical thickness Zc can also reflect GFA to some extent. In general, the higher the Zc, the
lower the Rc, and then the higher the GFA will be. However,
it is also difficult to obtain an accurate value of Zc in practice. Therefore, a valid and convenient criterion is required,
based on which one can directly distinguish the GFA for
various alloys. It can save lots of time in finding glassforming alloys.
Inoue et al.1 have proposed a GFA parameter, supercooled liquid width ⌬Tx 共Tx − Tg, where Tx is the onset crystallization temperature of the glass, and Tg is the glass transition temperature兲, and concluded that a good glass-forming
alloy system should have a large ⌬Tx. The critical cooling
rate Rc is the characteristic of good glass forming alloys, and
a low Rc means a high GFA. However, a high degree of
scatter has been found in a number of cases when ⌬Tx is
plotted against Rc.2 Recently, many criteria have been proposed in terms of some combination of the characteristic
temperatures, such as, glass transition temperature Tg, onset
crystallization temperature Tx, peak crystallization temperature T p, onset melting point Tm, liquidus temperature Tl, and
so forth: ␸兵=Trg关共Tx − Tg兲 / Tg兴0.143其,3 ␥ = Tx / 共Tl + Tg兲,2 reduced
glass transition temperature Trg共=Tg / Tl兲,4 ␤关=TxTg / 共Tl
− Tx兲2兴,5 ␦关=Tx / 共Tl − Tg兲兴.6 Those criteria have been widely
used and obtained good results in exploring the metal glass
materials.7–9 In this paper, a criterion ␪, defined as 共Tg
+ Tx兲 / Tl关共Tx − Tg兲 / Tl兴␣ has been suggested, and its validity
has been checked. The comparisons of this criterion with
other ones are also given in this article.
It has been well recognized that the glass formation is a
competing process between supercooled liquid and crystalline phase that is thermodynamically stable. Based on the
classical theories of nucleation and crystal growth, the homogeneous nucleation rate, I, and the growth rate, U, of the
crystalline phase forming at the supercooled liquid can be
expressed as follows:10,11
a兲
Author to whom correspondence should be addressed. Tel.: 86-10-62332646. Electronic mail: kcc126@126.com.
0021-8979/2009/106共9兲/094902/4/$25.00
I=
1035
U=
␩
冋
冋 冉
册
3 2
T
− 16␲ ⌬S f ␣m
,
3 NAk共Tl − T兲2
exp
共Tl − T兲⌬S f
kT
1 − exp −
2
R gT
3 ␲ a 0␩
共1兲
冊册
共2兲
,
where ␩, k, ␣m, a0, NA, ⌬S f , and Rg are viscosity, the Boltzmann constant, a constant of 0.86, mean atomic diameter,
Avogadro’s number, the molar fusion entropy, and the gas
constant, respectively.
The critical cooling rate required for glass formation is
determined as12
Rc =
再
4␲
3f c
冕
Tg
Tl
I共T⬘兲dT⬘
冋冕
Tg
T⬘
册冎
3
U共T⬙兲dT⬙
1/4
,
共3兲
where f c is the crystallized volume fraction 共commonly,
10−6兲.
From Eqs. 共1兲–共3兲, it can be seen that the viscosity ␩ has
an inverse proportion to the nucleation rate I and the crystalline growth rate U. An increase of the viscosity will lead to
rapid decreases of I and U, which decreases the critical cooling rate Rc, and therefore, enhances the GFA. By convention,
the viscosity at glass transition temperature Tg is a constant
共=1012 Pa s兲.13 Therefore, a high value of Tg / Tl 共Trg, for
short兲 means a rapid increase of viscosity of supercooled
liquid when it is cooling from liquidus temperature, which
dramatically lowers the diffusion ability of atoms. Besides,
the high value of Trg also implies a lower degree of super-
106, 094902-1
© 2009 American Institute of Physics
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094902-2
J. Appl. Phys. 106, 094902 共2009兲
G.-H. Zhang and K.-C. Chou
cooling. The nucleation rate I is small when the degree of
supercooling is low. All above analyses indicate that the following relationship should be kept:
GFA ⬀ Tg/Tl .
共4兲
The reason why using liquidus temperature Tl rather than the
onset temperature of melting 共Tm兲 is that GFA is more
closely related to Tg / Tl rather than Tg / Tm.14
According to the analysis of Wakasugi et al.,15 the Tx / Tl
ratio increases with increasing viscosity of the supercooled
liquid, fusion entropy, and activation energy of viscous flow.
It can be seen from Eqs. 共1兲–共3兲 that increasing the viscosity
of supercooled liquid, the fusion entropy, as well as the activation energy of viscous flow will dramatically decrease the
nucleation frequency and growth rate of crystal, thus decrease Rc, which means an enhancement of GFA. Therefore,
Tx / Tl can reflect the GFA either. It can be expressed as
GFA ⬀ Tx/Tl .
共5兲
Since both Tg / Tl and Tx / Tl can reflect the GFA of alloys,
adding them together, a dimensionless parameter 共Tg
+ Tx兲 / Tl will be generated to characterize the GFA
GFA ⬀ 共Tx + Tg兲/Tl .
共6兲
The stability of a glass can be qualified by the width of the
supercooling liquid region when it is reheated. The large Tx
− Tg reflects that the undercooled liquid can be kept in a wide
temperature range without crystallization and has strong resistance against the nucleation and growth of crystalline
phase.1 Thus, Tx − Tg may be related to GFA of alloys, a large
Tx − Tg means high GFA. In order to make it comparable for
different alloys, normalizing Tx − Tg by Tl is necessary.
Therefore the GFA should be proportional to the factor of
共Tx − Tg兲 / Tl, expressed as follows:
GFA ⬀ 共Tx − Tg兲/Tl .
␪ = 共Tx + Tg兲/Tl · 关共Tx − Tg兲/Tl兴0.0728 .
共9兲
From Fig. 1, it can be seen that there exists a fine linear
relationship between the logarithm of the critical cooling rate
ln Rc and parameter ␪, with a coefficient of determination
R2 = 0.942. In order to compare the parameter ␪ with any
other parameters mentioned above, all the parameters are
plotted against ln Rc 共figures are not given here兲, and coefficients of determination R2 are listed in Table II. It can be
seen that the criterion ␪ possesses the highest R2. It is known
to us, the value of the R2 that varies from 0 to 1 can statistically reflect how strong the linear correlation is. The higher
the value of R2, the better the linear relationship is. Therefore, it can be concluded that the parameter ␪ is valid for
evaluating the GFA of alloys. The obtained relationship can
be expressed as follows:
LnRc = 共30.10 ⫾ 0.91兲 − 共27.93 ⫾ 0.97兲␪ .
共10兲
共7兲
As discussing both cooling and reheating processes for an
undercooled liquid, the GFA of an alloy could be proportional to both factors: 共Tx + Tg兲 / Tl and 共Tx − Tg兲 / Tl. Therefore,
a product of them, as shown in Eq. 共8兲, is adopted for evaluating the GFA of alloys
␪ = 共Tx + Tg兲/Tl · 关共Tx − Tg兲/Tl兴␣ ,
shows the glass transition temperature Tg, onset crystallization temperature Tx and liquid temperature Tl of Mg-, Zr-,
La-, Ti-, Ni-, Fe-, Pd-, and Ca-based metallic alloys. The
used data were measured by differential thermal analysis
共DTA兲 at a heating rate of 20 K/min.2,16
In Table I, parameters ␪, ␸, ␥, Trg, ␤, and ␦ are also
calculated. From the definition of ␪ and ␸, it can be seen that
both of them are proportional to the width of supercooled
liquid region, ⌬Tx. Many metallic alloys do not show a supercooled liquid region calorimetrically as ⌬Tx → 0. In order
to have a meaningful ␪ value for the statistical treatment of
all related data, ⌬Tx is set to be unity in their calculation,
which is exactly the same as what Fan et al. did.5 Through
the linear regression analysis between ln Rc and ␪, the value
of ␣ is generated to be 0.0728. Therefore the parameter ␪ can
be expressed as
共8兲
where ␣ is a positive parameter that can be obtained by
regressing the experimental data. The validity of the parameter ␪ will be checked in the next section.
III. VALIDITY OF THE CRITERION
The critical cooling rate Rc is much better than the critical thickness Zc in reflecting the GFA of alloys since Zc has
a weaker relation with the GFA. It is noted that, except for
the inherent glass-forming ability, many other factors can
affect the value of Zc either, for example, the large casting
variations for glass formation and different approaches to the
maximum casting sizes.2 Besides, most of values of Zc are
integer. Therefore it is more reasonable to evaluate GFA with
Rc rather than Zc. In the following paragraph, the data of Rc
will be used to check the validity of the criterion ␪. Table I
IV. COMPARISON AMONG ⌰, ⌽, ⌫, TRG, ⟩, ⌬
From Table II, it can be seen that parameters Trg, ␤, and
␦ have weaker linear relationships with ln Rc, R2 of which
are 0.724, 0.49, 0.722, respectively. The parameter of ␤ is
not a good criterion with a coefficient of determination R2
= 0.49, and the authors have also pointed out that the ␤ criterion cannot evaluate the GFA for La-, Fe-, and Ni-based
alloys very well.5
Parameter ␥ proposed by Lu and Liu is very good17 and
has been used widely now in the literatures, the coefficient of
determination of which can reach high to 0.914. Followed Lu
and Liu’s idea, the criterion suggested in this paper, ␪, can
also reach a very high value with R2 = 0.942. It is expected
that it may be useful in evaluating the GFA for alloys in
future.
V. CONCLUSION
A criterion ␪ is proposed for evaluating the GFA of alloys, the application of which to 53 groups of metallic alloy
systems shows that an excellent linear relationship between
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094902-3
J. Appl. Phys. 106, 094902 共2009兲
G.-H. Zhang and K.-C. Chou
TABLE I. Characteristic temperatures and critical rate Rc as well as calculated parameters.
Composition
Tg共K兲
Tx共K兲
Tl共K兲
Rc共K / s兲
␪
␸
␥
Trg
␤
␦
Pd82Si18
Pd75Si25
Pd95Si5
Pd79.5Cu4Si16.5
Pd77.5Cu6Si16.5
Pd77Cu6Si17
Pd40Ni40P20
Pd42.5Cu30Ni7.5P20
Pd40Cu30Ni10P20
Pd42.5Cu27.5Ni10P20
Pd40Cu32.5Ni7.5P20
Pd40Cu25Ni15P20
Pd45Cu25Ni10P20
Pd45Cu30Ni5P20
Pd37.5Cu30Ni12.5P20
Mg80Ni10Nd10
Mg75Ni15Nd10
Mg65Ni20Nd15
Mg65Cu25Y10
Mg77Ni18Nd5
Mg70Ni15Nd15
Mg90Ni5Nd5
Mg65Cu25Gd10
Mg65Cu7.5Ni7.5Zn5Ag5Y10
Zr65Be35
Zr66Al8Ni26
Zr66Al8Cu7Ni19
Zr66Al8Cu12Ni14
Zr66Al9Cu16Ni9
Zr65Al7.5Cu17.5Ni10
Zr57Ti5Al10Cu20Ni8
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
Zr46.25Ti8.25Cu7.5Ni10Be27.5
La55Al25Ni20
La55Al25Ni15Cu5
La55Al25Ni10Cu10
La55Al25Ni5Cu15
La55Al25Cu20
La55Al25Ni5Cu10Co5
La66Al14Cu20
Ti34Zr11Cu47Ni8
Ti63Be37
Ni
Ni59Zr16Ti13Si3Sn2Nb7
Fe91B9
Au77.8Si8.4Ge13.8
Ni60Nd40
Ca65Mg15Zn20
648
656
647
635
637
642.4
590
576
576.9
584
568
596
595
577
572
454.2
450
459.3
424.5
429.4
467
426
423
426
623
672
662.3
655.1
657.2
656.5
676.7
630
629
623
623
625
623
622
490.8
473.6
467.4
459.1
455.9
465.2
395
698.4
673
425
845
600
293
933
375
648
656
647
675
678
686.4
671
658
655.8
665
654
668
675
659
647
470.5
470.4
501.4
479.4
437.2
489
449
484
464
623
707.6
720.7
732.5
736.7
735.6
720
678
686
672
712
739
740
727
555.1
541.2
547.2
520
494.8
541.8
449
727.2
673
425
885
600
293
933
410
1071
1343
1688
1086
1058.1
1128.4
991
884
836
871
932
910
884
861
929
878
789.8
804.9
770.9
886.9
844
919
740
717
1238
1251
1200.8
1172.1
1170.6
1167.6
1145.2
1003
1006
996
1057
1206
1239
1185
941.3
899.6
835
878.1
896.1
822.5
731
1169.2
1353
1725
1301
1628
629
1484
630
1800
1 ⫻ 106
5 ⫻ 107
500
100
125
0.167
0.067
0.1
0.083
0.133
0.15
0.1
0.083
0.133
1251.4
46.1
30
50
4.9⫻ 104
178.2
53 000
1
50
1 ⫻ 107
66.6
22.7
9.8
4.1
1.5
10
1.4
1.4
1.4
5
12.5
17.5
28
67.5
34.5
22.5
35.9
72.3
18.8
37.5
100
6.3⫻ 106
3 ⫻ 1010
40
2.6⫻ 107
3 ⫻ 106
1400
20
0.759
0.604
0.467
0.968
1.000
0.948
1.076
1.191
1.259
1.206
1.102
1.154
1.206
1.210
1.092
0.807
0.913
0.980
0.983
0.712
0.888
0.744
1.037
1.002
0.625
0.869
0.941
0.987
0.995
0.996
0.980
1.064
1.079
1.063
1.070
0.966
0.940
0.968
0.929
0.949
1.039
0.939
0.860
1.045
0.970
0.952
0.614
0.300
1.032
0.450
0.606
0.772
1.027
0.240
0.193
0.152
0.394
0.407
0.388
0.448
0.493
0.519
0.505
0.465
0.484
0.505
0.507
0.460
0.321
0.366
0.405
0.411
0.273
0.357
0.305
0.433
0.420
0.200
0.353
0.390
0.412
0.415
0.415
0.399
0.435
0.444
0.435
0.446
0.406
0.396
0.407
0.390
0.398
0.435
0.392
0.358
0.437
0.406
0.379
0.196
0.104
0.420
0.148
0.207
0.236
0.424
0.377
0.328
0.277
0.392
0.400
0.388
0.424
0.451
0.464
0.457
0.436
0.444
0.456
0.458
0.431
0.353
0.379
0.397
0.401
0.332
0.373
0.334
0.416
0.406
0.335
0.368
0.387
0.401
0.403
0.403
0.395
0.415
0.420
0.415
0.424
0.404
0.397
0.402
0.388
0.394
0.420
0.389
0.366
0.421
0.399
0.389
0.332
0.198
0.412
0.269
0.318
0.386
0.408
0.605
0.488
0.383
0.585
0.602
0.569
0.595
0.652
0.690
0.670
0.609
0.655
0.673
0.670
0.616
0.517
0.570
0.571
0.551
0.484
0.553
0.464
0.572
0.594
0.503
0.537
0.552
0.559
0.561
0.562
0.591
0.628
0.625
0.626
0.589
0.518
0.503
0.525
0.521
0.526
0.560
0.523
0.509
0.566
0.540
0.597
0.497
0.246
0.650
0.369
0.466
0.629
0.595
2.347
0.912
0.386
2.537
2.989
2.257
3.866
7.420
11.650
9.152
4.806
6.798
9.194
9.319
4.654
1.287
2.075
2.500
2.395
0.928
1.812
0.866
3.124
3.088
1.026
1.610
2.071
2.483
2.572
2.588
2.695
4.044
4.214
3.9884
3.727
2.118
1.851
2.156
1.827
1.995
3.088
1.862
1.401
3.199
2.230
2.600
0.980
0.107
4.321
0.341
0.760
2.867
3.177
1.532
0.955
0.622
1.497
1.610
1.412
1.673
2.136
2.531
2.317
1.797
2.127
2.336
2.320
1.812
1.110
1.384
1.451
1.384
0.956
1.297
0.911
1.527
1.594
1.013
1.222
1.338
1.417
1.435
1.439
1.537
1.818
1.820
1.802
1.640
1.272
1.201
1.291
1.232
1.270
1.488
1.241
1.124
1.516
1.336
1.545
0.990
0.327
1.941
0.584
0.872
1.693
1.608
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J. Appl. Phys. 106, 094902 共2009兲
G.-H. Zhang and K.-C. Chou
25
ACKNOWLEDGMENTS
LnRc=30.10-27.93 θ
2
R =0.942
The authors wish to thank the Chinese Natural Science
Foundation for their kind support under the Contract No.
50774004, and the “Changjiang Scholars and Innovative Research Team in University,” “PCSIRT” support under the
Grant No: IRT0708. Thanks are also given to the Key Laboratory of Ecologic & Recycle Metallurgy, Ministry of Education.
LnRc/(K/s)
20
15
10
5
0
-5
0.2
1
0.4
0.6
0.8
1.0
1.2
1.4
0.0728
θ, (Tx+Tg)/Tl ((Tx-Tg)/Tl)
FIG. 1. The logarithm of the critical cooling Rc as a function of
parameters ␪.
ln Rc and ␪ 共value of ␣ is 0.0728兲 can be reached. Comparisons among parameters ␪, ␸, ␥, Trg, ␤, and ␦ are also given,
and the results show that this criterion is the best one among
them. It is anticipated that the criterion ␪ will play a useful
role in exploring BMG materials.
TABLE II. Comparison among different criterion.
Criteria
R2
␪
␸
␥
Trg
␤
␦
0.942
0.924
0.914
0.724
0.49
0.722
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