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Materials Science and Engineering A 449–451 (2007) 211–214

Identification of compositions with highest glass forming abilityin multicomponent systems by thermodynamic

and topological approaches

B. Srinivasa Rao, Jatin Bhatt, B.S. Murty ∗Department of Metallurgical and Materials Engineering, Indian Institute of Technology Madras, Chennai 600036, India

Received 21 August 2005; received in revised form 28 October 2005; accepted 22 December 2005

bstract

Compositions with easy bulk metallic glass formation have been identified in quinary systems with the Gibbs-energy change between themorphous and solid solution phases as the thermodynamic parameter. The Gibbs-energy change has been calculated with the help of Miedema,iracle, mismatch entropy, and configurational entropy models. The best glass forming composition has been identified by drawing iso-Gibbs-

nergy change contours by representing quinary systems as pseudo-ternary ones. Attempts have been made to correlate the Gibbs-energy change

ith different existing glass forming criteria and it is found that the present thermodynamic parameter has good correlation with the reduced glass

ransition temperature. Further, encouraging correlations have been obtained between the energy required for amorphization during mechanicallloying to the Gibbs-energy change between the amorphous and solid solutions.

2006 Elsevier B.V. All rights reserved.

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eywords: Bulk metallic glass; Glass forming criteria; Thermodynamics; Topo

. Introduction

Ever since the pioneering works of Inoue [1] and Johnson2] on the formation of bulk metallic glasses, there has been aot of interest to identify parameters to assess the glass form-ng ability (GFA) of various alloy systems and compositions.

number of parameters such as Trg (Tg/Tl) [3], T mixl [4], �Tx

Tx − Tg) [5], γ (Tx/(Tg + Tl)) [6], α (Tx/Tl) [7] and Trx (Tx/Ts)8] have been used by various investigators to predict good glassorming alloy compositions. However, all the above parametersxcepting T mix

l need the alloy to be first prepared in glassy formo be able to measure the glass transition temperature Tg and/orhe crystallization temperature Tx. Hence, the above parame-ers are not predictive in nature, as they cannot predict a goodlass forming composition without actually making that alloynd rapidly solidifying it into the glassy state. However, theyre useful in identifying the compositions with good GFA, so as

o be able to make them in bulk glassy form. Recently, Park etl. has suggested a σ parameter, which is based on the meltingemperature and atomic size data [9].

∗ Corresponding author. Tel.: +91 44 2257 4754.E-mail address: murty@iitm.ac.in (B.S. Murty).

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921-5093/$ – see front matter © 2006 Elsevier B.V. All rights reserved.oi:10.1016/j.msea.2005.12.092

l approach; Mechanical alloying

The present paper attempts to identify the best glass formingomposition by identifying the composition with the largest freenergy change between amorphous and solid solution phasesith the help of Miedema [10], Miracle [11], mismatch entropy

12], and configurational entropy models. This can be a predic-ive model as it does not need any experimental data such asg or Tx. Based on the Meidema model, Murty et al. [13] havealculated the GFR for Ti–Ni–Cu ternary system and could suc-essfully correlate it to that obtained by mechanical alloying.akeuchi and Inoue [14] have applied this model to predict theFR for a number of ternary systems and showed that the exper-

mentally obtained GFR falls within the calculated region.

. Experimental details

The elemental powder blends of various compositions witharticle sizes of −325 mesh (<45 �m) were mechanicallylloyed (MA) in a high-energy ball mill (Fritsch Pulverisette-

5). The milling was carried out in WC vials with WC ballsf 10 mm diameter in toluene medium. Milled powder samplesollected at regular intervals were subjected to X-ray diffractionXRD) with Cu K� (0.1542 nm) radiation in order to confirmhe formation of the amorphous phase.

212 B.S. Rao et al. / Materials Science and Eng

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Fig. 1. Correlation of α (Tx/Tl) parameter with critical cooling rate, Rc.

. Results and discussion

The present authors have recently developed an expression toalculate the free energy of an undercooled liquid and it has beenhown in Ref. [15] that the �G calculated using this expression isery close to the experimental value in a variety of glass forminglloys ranging from binary to quinary systems, in comparisono the existing models. The present authors have also recentlyhown that a simple new parameter, α (Tx/Tl), can have strongorrelation with the critical cooling rate for glass formation,c, as shown in Fig. 1 [7]. This is because of that fact thatparameter combines both easy glass formation from liquid

nd the thermal stability of the glass. A high Tx relates to theigher stability of glass and low Tl denotes higher stability ofupercooled liquid and hence easy glass formation. Thus, highx and low Tl leads to higher value of α or higher GFA. From

he results of this work it is clear that the GFA parameters whichnclude both the aspects of glass formation, namely, the easylass formation and its stability, have better correlation with Rcn comparison to those, which deal with only one of these twospects of glass formation (�Tx or Trg). This parameter is veryseful, particularly in cases where a distinct Tg is not observed.he work also demonstrated that the heating rate in DSC doesot significantly affect the GFA criteria.

However, the above parameter also does not have the predic-ive capabilities, without one making a glass out of a compositionnd obtained the Tx of it. The free energy difference betweenhe amorphous phase and solid solution phases can be expresseds, �G = �H − T�S. The enthalpy of mixing of solid solutions suggested by Gallego et al. [16] and Murty et al. [13] can bexpressed as

HSS = �Hc + �He + �H s (1)

here the superscripts c, e and s corresponds to the chemical,

lastic and structural contributions, respectively. The chemicalontribution to the enthalpy of mixing of solid solution andmorphous phase has been obtained following the approach ofiedema model [10]. The elastic contribution was first given

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ineering A 449–451 (2007) 211–214

y Eshelby [17] with the help of sphere and hole model basedn continuum elastic theory. Recently, Senkov and Miracle [11]odified the Eshelby’s model to take into account the interstitial

nd substitutional occupancy of atoms. For the present calcula-ion, Senkov and Miracle’s topological model, which dependsn the atomic size differences between the constituent elements,as used to evaluate the elastic contribution to enthalpy of mix-

ng. The elastic contribution is given as follows:

�He =n∑

i = 1

j �= i

xixj(�heiinj + �he

jini) and

�hekinj = Xkinj

s Ekinjs + X

kinji E

kinji (2)

here Xs and Xi are the fraction of solute atoms occupied atubstitutional and interstitial sites and Es and Ei are the energytates of the substitutional and interstitial sites, respectively. Thetructural contribution to the enthalpy change has been assumedo be negligible. In the case of the amorphous phase, the elasticnd structural contributions to its enthalpy of mixing are absent.he total entropy change is taken as the sum of configurationalnd mismatch entropies. Mismatch entropy is calculated basedn the hard sphere model of Mansoori et al. [12], while the con-gurational entropy is calculated by the conventional statisticalpproach.

In Fig. 2(a–d), an attempt has been made to correlate the var-ous thermodynamic parameters such as T�S, �He, �Hc and

G, respectively, with the reduced glass transition temperature,rg, for a number of glass forming Zr based alloys. The data forbout 30 compositions is used in these figures. The figures indi-ate that the entropy term and chemical enthalpy terms do nothow strong correlation with Trg, while the elastic enthalpy termhows a reasonable correlation. The fact that �He shows a betterorrelation than �Hc suggests that topological factors (atomicize mismatch) have probably a better control on the glass forma-ion than the chemical factors. The �G, which is combinationf all the above parameters shows a strong correlation with aorrelation factor (R2) of 0.88. In addition to the linear correla-ion between the Trg and �G, it is also interesting to note thatmong all other compositions considered, the composition withhe highest GFA, Zr41.2Ti13.8Cu12.5Ni10Be22.5 (number indicatet.%) has not only the highest Trg but also the highest �G. Thisuggests that we can use �G, which is a combination of thermo-ynamic (�Hc, T�S) and the topological (�He) factors safelys the parameter to assess the GFA. Most importantly, this GFAarameter has predictive capabilities, as it does not need anyxperiment to be performed to demonstrate glass formation inny composition and obtain values of Tg, Tx and/or Tl as in thease of the previous models.

Thus, one can identify the best glass forming compositionn any system by simply calculating �G and identifying the

omposition with the highest negative �G. This approach haslso been used to calculate the glass forming composition rangeGFR) in a number of binary, ternary, quaternary and quinary Zr,i, Fe based systems and good correlations have been observed

B.S. Rao et al. / Materials Science and Engineering A 449–451 (2007) 211–214 213

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low has been observed in the XRD patterns in the range of 4–10 hof milling. The total milling energy required for amorphizationhas been calculated using the model developed earlier [18], andit has been plotted as a function of �G in Fig. 4(b). The fig-

Fig. 2. Correlation of (a) �Hc, (b

n most of the systems with the experimentally reported GFR.s systems higher than ternary are difficult to be represented

n two dimensions, quasiternary diagrams are drawn by com-ining elements that have similar nature at one corner of theriangle. For example, in case of Zr–Ti–Cu–Ni–Al system, Zrnd Ti are located at one corner, Cu and Ni at the second andl at the third corner of the triangle. Compositions with the

ame �G values are joined together by lines and thus iso-freenergy contours are plotted on such quasiternary diagrams. Fromhese contours, the composition with the highest negative �Gas been identified. Fig. 3 shows such an iso-free energy con-our map for Zr–Ti–Cu–Ni–Al system, in which the compositionith the highest negative �G has been found to be Zr19.6Ti17.4i25.2Cu16.8Al21. Similar contour maps have been obtained in aumber of ternary, quaternary and quinary systems and the bestlass forming compositions have been identified. It has beenn general observed that the �G becomes more negative withncrease in the number of elements, which suggests an increasen GFA and the �G has strong correlation with it.

The binary, ternary, quaternary and quinary compositionsith the highest negative have been mechanically alloyed to pre-

are amorphous alloys in the Zr–Ti–Cu–Ni–Al quinary system.he evolution of amorphous phase in a representative com-osition (Ti60Cu20Ni20Al10) in the above system is shown inig. 4(a). In all the compositions studied, amorphous broad hal-

Ft

S, (c) �He and (d) �G with Trg.

ig. 3. Iso-free energy contour map for the Zr–Ti–Ni–Cu–Al system showinghe composition with the highest negative �G.

214 B.S. Rao et al. / Materials Science and Engineering A 449–451 (2007) 211–214

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[16] L.J. Gallego, J.A. Somoga, J.A. Alonso, J. Phys. Condens. Matter. 2 (1990)

ig. 4. (a) XRD patterns of quaternary Ti60Ni20Cu20Al10 as a function of duratietween �G and total energy required for amorphization by milling for a numb

re shows that there is a general trend that the energy requiredmorphization during MA decreases with increasingly negativeG. The scatter in the plot could be due to two reasons. A care-

ul observation of Fig. 4(b) also shows that the compositionshat needed higher energy for amorphization are the ones withower amounts of Zr and Ti (31 and 32 at.%, respectively). Asr and Ti have greater affinity to oxygen, it is possible that ther- and Ti-rich compositions might pick up some oxygen dur-

ng milling, which could help in early amorphization of Zr- andi-rich compositions. Secondly, as Ti and Zr are brittle compo-ents in comparison to Ni, Cu, Fe, Co and Al during milling, thei- and Zr-rich compositions can become nanocrystalline much

aster than the Ti- and Zr-leaner compositions. As amorphiza-ion always follows nanocrystallization, this could explain theeason for the higher energy required for amorphization in com-ositions with lower Ti and Zr. However, further investigationsan throw light on the reasons for the above observation.

. Conclusions

(i) The free energy change between amorphous and solidsolution, �G, calculated using Meidema (�Hc), Mira-cle (�He) and Mansoori (entropy mismatch) approachesshows strong correlation with reduced glass transition tem-perature (Tg/Tl) in Zr-base metallic glasses. Thus, �G can

be used as a predictive GFA parameter to identify compo-sitions with the highest GFA.

(ii) The compositions with the highest GFA have been iden-tified in a number of quinary systems by iso-free energy

[[

MA showing the evolution of amorphous phase and (b) showing the correlationZr and Ti based alloys.

contour maps by representing quandary systems as qua-siternary systems.

iii) A general correlation has been observed between the totalmilling energy required for amorphization and the �G.

eferences

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