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Intermetallics 18 (2010) 933–937

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Intermetallics

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Evaluation of liquid fragility for glass-forming alloys based on mixing enthalpyand mismatch entropy

Jing Guo a, Xiufang Bian b,*, Xuelian Li b, Chunzhi Zhang b

a School of Mechanical and Electronic Engineering, Shandong Agricultural University, Taian, Shandong 271018, Chinab Key Laboratory of Liquid Structure and Heredity of Materials, Ministry of Education, Shandong University, 73 Jingshi Road, Jinan, Shandong 250061, China

a r t i c l e i n f o

Article history:Received 5 February 2009Received in revised form28 December 2009Accepted 8 January 2010Available online 19 February 2010

Keywords:B. Metallic glassesB. Thermodynamic and thermochemicalproperties

* Corresponding author. Tel.: þ86 531 88392748; faE-mail address: xfbian@sdu.edu.cn (X. Bian).

0966-9795/$ – see front matter � 2010 Elsevier Ltd.doi:10.1016/j.intermet.2010.01.004

a b s t r a c t

A calculable parameter d, defined as the absolute multiplication of the chemical mixing enthalpy andmismatch entropy, has been proposed to evaluate the liquid dynamic fragility of glass-forming alloys. Thealloy with a larger d exhibits a stronger liquid characteristic in an alloy system with the same baseelement. The proposal of the parameter d gives a much easier and more available method in evaluatingliquid fragility without any experiments.

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1. Introduction

During the last few decades the concept of liquid fragility hasbeen attracting considerable attention from both materials scien-tists and physicists [1–5]. The concept of dynamic fragility of glass-forming liquids, proposed by Angell [6], refers to deviations fromArrhenius temperature dependence of relaxation time or viscosity.Strong liquids exhibit nearly Arrhenius temperature dependence ofthose dynamic properties. Fragile liquids generally display Vogel–Fulcher–Tamman behavior [7–9]. The dynamic fragility of a liquidcan be expressed by the fragility parameter [10],

m ¼ dlog10sðTÞd�Tg=T

� jT¼Tg

¼dlog10h

d�Tg=T

� jT¼Tg

(1)

which characterizes the rapidity with which a liquid’s dynamicproperties change as the glass transition temperature isapproached. The liquid fragility is of significance for the study ofmetallic glasses, and so far the investigation on the liquid fragilityhave enhanced the understanding of the nature of the glass tran-sition and the dynamics of glasses [4]. Interestingly, recent reportsdemonstrated that m has close correlations with the glass proper-ties such as vibration [11], Poisson’s ratio, etc. [12], and there even isa negative correlation between the glass-forming ability and m for

x: þ86 531 88395011.

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some metallic glass formers [13–15]. It seems that the fragility isbecoming one of the crucial subjects in the field of metallicmaterials.

Several kinds of method have been used to get the liquiddynamic fragility. The parallel plate rheometry and three-pointbeam bending methods [14] based on the viscosity measurementare the main methods in getting the liquid fragility and they requirethe amorphous solids should be prepared beforehand and thesample size should be several millimeters at least. For the alloyswith low glass-forming ability, it is difficult to obtain the fragility bythe two methods. Also, the thermodynamics method can obtain thedynamic fragility, by which the values of liquid fragility parameterare similar to those obtained by viscosity method [5], but itpresupposes that the alloy can be prepared into metallic glass andthe glass should display obvious glass transition during the heatingprocess. All in all, the methods in obtaining liquid fragility alldisplay some deficiencies, which confine the application of liquidfragility in the field of metallic glass. So further investigation isnecessary to search a more effective and available parameter toevaluate the liquid fragility of glass-forming alloys.

Rao [16] has demonstrated that electronegativities and bondstrength between elements are two important parameters that canbe used to determine liquid fragility. Unfortunately, how to deter-mine the two parameters for metallic glasses is not given. Inspiredby the study, we select the mismatch entropy normalized byBoltzmann constant (Ss/kB) and the chemical mixing enthalpy(DHmix) as the investigated objects in this work. A parameter (d) forevaluating the fragility of glass-forming alloys will be proposed by

Table 1The chemical mixing enthalpy (DHmix), mismatch entropy normalized by Boltzmannconstant (Ss/kB), the absolute multiplication of chemical mixing enthalpy andmismatch entropy (d), the parameter of supercooled liquid fragility (m) and theslope of the fitting line about d and m (K).

Alloy DHmix(KJ/mol)

Ss/kB d m K

Pd-basedalloys

Pd82Si18 �32.47 0.007 0.227 106a

Pd40Ni40P20 �22.72 0.077 1.749 46b

Pd48Ni32P20 �22.85 0.078 1.782 44b �0.0255Pd39Ni10Cu30P21 �25.34 0.062 1.571 55b

Pd40Ni10Cu30P20 �24.88 0.062 1.543 57b

Ni-basedalloys

Ni62.4Nb37.6 �27.93 0.202 5.642 121a

Ni75Si8B17 �22.60 0.301 6.803 116a �0.3219Ni60Zr30Al10 �45.84 0.421 19.30 78e

Zr-basedalloys

Zr55Al22.5Co22.5 �45.92 0.264 12.12 73f

Zr41.2Ti13.8Cu12.5Ni10Be22.5 �34.91 0.549 19.17 50b �0.3854Zr46.75Ti8.25Cu7.5Ni10Be27.5 �39.06 0.627 24.47 43b

La-basedalloys

La55Al25Ni20 �37.18 0.739 27.48 33c

La55Al25Ni15Cu5 �35.35 0.728 25.73 36d �0.5535La55Al25Ni5Cu15 �31.93 0.704 22.48 42d

Fe-basedalloys

Fe83B17 �14.67 0.317 4.650 103a

Fe41.5Ni41.5B17 �15.48 0.304 4.690 99a �0.0085Fe79Si10B11 �20.71 0.219 4.536 117a

Gd-basedalloys

Gd55Al20Ni25 �38.61 0.647 24.98 36g

Gd55Al25Ni10Co10 �37.21 0.588 21.86 58h �0.1227Gd55Al25Co20 �34.93 0.583 20.37 74h

a Reference [22].b Reference [23].c Reference [13].d Reference [15].e Reference [24].f Reference [25].g Reference [26]h Reference [27].

J. Guo et al. / Intermetallics 18 (2010) 933–937934

combining the chemical mixing enthalpy and the difference inatomic sizes in the alloys.

2. Calculation and results

Inoue [17] proposed three factors in favor of the stabilization ofsupercooled liquid and the formation of metallic glass: (1) multi-component systems consisting of more than three elements; (2)significant difference in atomic sizes with the size ratios aboveabout 12% among the three main constituent elements; and (3)negative heats of mixing among the three main constituentelements. The parameters, mismatch entropy normalized byBoltzmann constant (Ss/kB) and chemical mixing enthalpy (DHmix),correspond to the second and the third term of the empirical rules,respectively. The parameter, Ss/kB, relates to the elastic energy ina system and has an effect on the stable structure. In other words,the increase of Ss/kB leads to a more random structure resultingfrom large difference in atomic size. DHmix is a parameter relatedwith bond strength in the system. The large Ss/kB and negativeDHmix are believed to be helpful for stabilizing the supercooledliquid structure [18]. Consequently, there should be a correlationbetween Ss/kB, DHmix and liquid fragility parameter m since liquidfragility reflects the rapidity of change in the supercooled liquidstructure near the Tg.

The chemical mixing enthalpy is calculated according to Eqs. (2)and (3) on the basis of the extended regular solution model [19]:

DHmix ¼ DHc ¼Xn

i¼1;isj

Uijcicj (2)

where Uij is the regular solution interaction parameter between iand j elements, which can be expressed by the mixing enthalpy ofbinary alloys DHmix

AB(Uij ¼ 4 DHmixAB) referred in the reference

[20], and ci is the composition of the i element. Mismatch entropycan be calculated,

Ss=kB ¼32

�z2 � 1

�y1 þ

32ðz� 1Þ2y2 �

�12ðz� 1Þðz� 3Þ

þ lnz

�ð1� y3Þ (3)

where kB is the Boltzmann constant; z is a parameter being definedas z ¼ 1/(1�x) with a packing fraction x. Dimensionless parametersy1, y2 and y3 being defined as Eqs. (4)–(6) have a relation:y1 þ y2 þ y3 ¼ 1.

y1 ¼1s3

X3

j_i¼1

�di þ dj

��di � dj

�2cicj (4)

y2 ¼s2

�s3�2

X3

j_i¼1

didj�di � dj

�2cicj (5)

y3 ¼�s2�3

�s3�2 (6)

sk ¼X3

i¼1

cidki ; k ¼ 2;3 (7)

Here, di is an atomic diamerer of i element and is quoted fromthe reference [21]. According to Eqs. (2) and (3), chemical mixingenthalpy and mismatch entropy of Pd-, La-, Fe-, Zr-, Ni- and Gd-based glass-forming alloys are calculated, as shown in Table 1.

Fig. 1 shows the relationships between the liquid fragilityparameter m and the absolute of chemical mixing enthalpy jDHmixj,the mismatch entropy normalized by Boltzmann constant Ss/kB ofPd-, La-, Ni- and Fe-based glass-forming alloys, respectively. For thePd and La alloy systems, there are both linear correlations betweenjDHmixj and m, Ss/kB and m. It is noticed that for Pd-based alloys,with m increasing, jDHmixj increases and Ss/kB decreases, but for La-based alloys, with m increasing, both jDHmixj and Ss/kB decrease.Also, for Ni-based and Fe-based glass-forming alloys, there aredifferent change tendency of jDHmixj and Ss/kB with m increasing.Therefore, it is impossible to build a direct correlation between onlyjDHmixj and m. In Fig. 1(a) and (b), there is a good linear relationshipbetween Ss/kB and m, and with increasing m, Ss/kB alwaysdecreases. However, in Fig. 1(c) and (d) no good linear correlationcan be observed between Ss/kB and m. The Gd- and Zr-based alloysexhibit the same behaviors as the Pd and Ni-based alloys, respec-tively. From the results mentioned above, it can be concluded thatonly jDHmixj or Ss/kB cannot indicate the change tendency of m.Then, since the large Ss/kB and negative jDHmixj are both in favor ofstabilizing the supercooled liquid structure, it is possible thata correlation exists between the combination of chemical mixingenthalpy and mismatch entropy and m. Accordingly, We define theabsolute multiplication of chemical mixing enthalpy and mismatchentropy as a parameter d,

d ¼ jDHmix � ðSs=kBÞj (8)

Based on the data listed in Table 1, we plot the liquid fragilityparameter, m, against the absolute multiplication of chemical mix-ing enthalpy and mismatch entropy, d, for Pd-, La-, Fe-, Zr-, Ni- andGd-based glass-forming alloys in Fig. 2. Strikingly, a good negativelinear correlation between m and d as demonstrated by the dash line

14

15

16

17

18

19

20

21

S/k

B

Liquid Fragility parameter m (d)112 114 11898 100 102 104 106 108 110 116

0.22

0.24

0.26

0.28

0.30

0.32

15

20

25

30

35

40

45

50

55

60

65

70

75

80

Hm

ix

Liquid Fragility parameter m (c)70 80 90 100 110 120

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

S/k

B

30

31

32

33

34

35

36

37

38

39

40

Liquid Fragility parameter m (b)

S/k

BHm

ix

32 34 36 38 40 420.68

0.69

0.70

0.71

0.72

0.73

0.74

22

24

26

28

30

32

34

S/k

BHm

ixH

mix

Hm

ix

Liquid Fragility parameter m(a)40 50 60 70 80 90 100 110

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Fig. 1. The correlations between m and DHmix, Ss/kB of Pd-(a), La-(b), Ni-(c) and Fe-(d) based glass-forming alloys.

J. Guo et al. / Intermetallics 18 (2010) 933–937 935

is observed for each alloy system, indicating that the alloy witha larger absolute multiplication of chemical mixing enthalpy andmismatch entropy exhibits a stronger liquid characteristic. It showsus that, for a single system with different contents of compositions,the liquid fragility denoted by m can be formulated as

mfKd (9)

120 130 14020 30 40 50 60 70 80 90 100 110 120 130 140

0

5

10

15

20

25

30

m

Pd-based alloys

Ni-based alloysZr-based alloys

Gd-based alloys

La-based alloys

Fe-based alloys

Fig. 2. The relationship between the liquid fragility parameter m and d values of 6kinds of glass-forming alloys.

K is the slope of the fitting line on Fig. 2. The equation implies thatthe parameter, d, can be used to predict the liquid fragility in analloy system with the same base element. From Eq. (9), it isconcluded that the chemical mixing enthalpy and mismatchentropy govern the liquid fragility, since K is a constant parameter.The result agrees with Rao’s idea that the fragilities of liquids can bedetermined by the nature of the chemical bonding in them [14].The two quantities, the significant difference in atomic size ratiosand the negative heats of mixing among the constituent elements,are both related with the bond strength.

3. Discussion

Angell has found that dynamic fragilities of different liquids arewell correlated with their thermodynamic fragilities, whichsuggests the dynamic fragility has a close relationship with theexcess entropy [28]. Some experimental results also demonstratethat the lowering of the viscosity of the alloy liquids may beattributed to the excess entropy [29]. It can be concluded that thebehaviors of glass-forming liquids are determined by the entropy ofthe system [30]. Therefore, the excess entropy is rather importantphysics quantity in the glass-forming alloys. Recently, the jDHmixjhas been demonstrated to correlate with the excess entropy at Tg

[31]. The increasing of mixing enthalpy can reflect the excitedincreasing excess entropy at the glass transition temperature.However, DHmix is not the unique parameter to govern the excessentropy. The random mixing of hard-spheres differing in radii may

Table 2The mechanism of the strong liquid behavior resulting from the combined effect oflarge negative heat of mixing and large atomic size ratios.

A combined effect of large negative heat of mixing and large atomic size ratios

Increase in the degree of dense Formation of liquid with

random packed structure from the strong interactions on

topological and chemical points of view a short-range scale

Difficulty of atomic rearrangement

High stability of supercooled liquid

Strong liquid behavior1.45 1.50 1.55 1.60 1.65 1.70 1.75 1.80 1.85

3.2

3.4

3.6

3.8

4.0

4.2

4.4

Pd40Ni10Cu30P20

Pd40Ni40P20

Pd48Ni32P20

Cp,

s-l J

/mol

.K

Fig. 3. The correlation between d and DCp,s�l of Pd-based glass-forming alloys, thevalues of DCp,s�l are cited from Ref. [33]. Here only lists three kinds of Pd-based glass-forming alloys due to the absence of data of the other alloys.

J. Guo et al. / Intermetallics 18 (2010) 933–937936

also contribute to the excess entropy of disorder [28], whichsuggests that Ss/kB is also an important parameter influencing theexcess entropy. Consequently, the parameter d, the absolutemultiplication of chemical mixing enthalpy and mismatch entropy,should closely correlate with the excess entropy. Then, a relation-ship between the parameter d and the dynamic fragility shouldexist due to the correlation between the excess entropy and theliquid fragility.

From the dynamics point of view, a large negative DHmix valueresponds to the formation of multicomponent chemical interac-tion on a short range scale, which is supposed to increase thedifficulty of atomic rearrangement of the structure regions ofa metallic glass against glass transition [13,21]. Therefore, thelarge negative DHmix is in favor of the stability of supercooledliquid structure approaching glass transition temperature. Thelarge Ss/kB in an alloy system can be in favor of tightening thepacked structure of undercooled liquids and increasing theirpacking densities and coordination numbers, which lowers theground-state energy of the undercooled liquids and thus stabilizesthe undercooled liquids. Consequently, it can be understood thatthe strong liquid behavior is attributed to the strong interactionand the highly dense packing resulting from a combined effect oflarge negative heat of mixing and large atomic size ratios, asshown in Table 2.

On the other hand, from the view of the heat capacity changescaused by the glass transition (DCp,s�l), we also can understand thecorrelation between m and d. DCp,s�l is a dominant and positivefactor to determine liquid fragility strength [32], directly relatedwith the excess entropy, which is determined by the plethora ofpossible distinct packing states accessible at the temperature T. Agenerally positive relationship is found between m and DCp,s�l [32],suggesting that large heat capacity difference between the glassand the supercooled liquid near Tg corresponds to large liquidfragility. It means that an alloy with a large heat capacity differenceshows relative instability in the supercooled liquid. TakingPd-based glass-forming alloys as an example, we discuss therelationship between d and DCp,s�l. As shown in Fig. 3, there isa negative linear correlation between them and the alloy witha larger d exhibits a smaller DCp,s�l in some Pd-based glass-formingalloys. From the view of liquid stability, an alloy with a largerd means more stable supercooled liquid characteristics in one alloysystem, moreover, an alloy with a small heat capacity differenceshows a relative stability liquid characteristics as mentioned above.Accordingly, the negative correlation between d and DCp,s�l iscomprehensible. Since a positive relationship between m andDCp,s�l had been found, it can be deduced that the alloy with

a larger d exhibits a smaller liquid fragility parameter in one alloysystem.

In addition, it is noticed that for the alloy systems with thedifferent base element the slope, K, in Eq. (8) is different, as shownin Table 1. It may be related to the different interaction strengthbetween the based element and the other major elements in thealloy systems since the parameter d is the multiplication ofchemical mixing enthalpy and mismatch entropy, which are thefunctions of composition in the alloy systems [19] and bothcorrelated with the strength of the interaction between the rele-vant elements.

As a parameter of evaluating liquid fragility, d overcomes thedeficiencies of other methods in obtaining liquid dynamic fragility.Particularly, it can be obtained by calculation without any experi-ments, which makes it easy to apply the liquid dynamic fragility tothe metallic glasses field.

4. Conclusions

A parameter d, based on the combination of the chemical mixingenthalpy and the difference in atomic sizes in the alloys, has beenproposed to evaluate the liquid dynamic fragility of glass-formingalloys. The alloy with a larger d exhibits a stronger liquid charac-teristic in an alloy system with the same base element. Significantly,the parameter d is easy to be obtained, and it can evaluate thefragile or strong characteristics of all kinds of liquids even for thosedifficultly m-described alloys by experimental methods.

Acknowledgements

This work was supported by the National Natural ScienceFoundation of China (Grant Nos. 50831003 and 50871062), NationalBasic Research Program of China (2007CB613901) and YouthScience and Technology Innovation Fund of Shandong AgriculturalUniversity.

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