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Materials Science and Engineering A 454–455 (2007) 654–661

On the prediction of solid–liquid interfacial energy of glass formingliquids from homogeneous nucleation theory

K. Mondal a, B.S. Murty b,∗a Department of Metallurgical and Materials Engineering, Indian Institute of Technology, Kharagpur 721 302, India

b Department of Metallurgical and Materials Engineering, Indian Institute of Technology Madras, Chennai 600 036, India

Received 5 January 2006; received in revised form 15 November 2006; accepted 16 November 2006

bstract

A new model has been proposed to estimate the solid–liquid interfacial energy on the basis of classical nucleation theory. The present model

oes not need any experimental data on nucleation kinetics, as is the case with the earlier models. The solid–liquid interfacial energy for both pureetals and glass forming liquids has been estimated using the present model. The estimated interfacial energy values have been used to calculate

he critical cooling rates required for the formation of glass. 2006 Elsevier B.V. All rights reserved.

g

t(pa

σ

wfcBlnt

f

eywords: Metallic glass; Liquid; Crystallization; Surface energy; Undercoolin

. Introduction

The solid–liquid interfacial free energy, defined as σ, is theeversible work done for the creation of a unit area of interfaceetween a crystal and its coexisting liquid and it plays an impor-ant role in determining the kinetics of crystal nucleation androwth [1]. Unfortunately, direct experimental determinationf σ is very difficult and the literature available in this direc-ion is very scanty. For most materials, knowledge of the σ isbtained indirectly from measurement of crystal nucleation ratesrom undercooled liquids, within the approximation of classicalucleation theory. Turnbull [2] tried to calculate the σ for a num-er of materials (most of which are pure elements) indirectlyrom homogeneous nucleation experiments. Turnbull expressedolar interfacial tension (σm) of an interface (one atom thick)

s σm = KσVm2/3N0

1/3, where K is a structure factor of the orderf unity, Vm the molar volume and N0 is Avogadro’s number. σman also be expressed as [2], σm = α�Hm, where α is the reducednterfacial tension, defined as the number of monolayers per unitrea of crystal, which would be melted by an enthalpy equivalent

o σm and �Hm is the enthalpy of fusion. Turnbull [2] found thathe value of α is 0.45 for metals (especially close packed metals)nd 0.32 for many non-metals. However, he calculated σ from

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

nbmGbml

921-5093/$ – see front matter © 2006 Elsevier B.V. All rights reserved.oi:10.1016/j.msea.2006.11.123

he homogeneous nucleation theory with basic assumptions of:a) spherical nucleus, (b) exp(−�FA/kT) = 10−2 and (c) inde-endency of entropy of fusion with temperature and arrived atn equation for σ as [2].

=[(

3 × 2.303�Hm(�T )2kT

16πTm2

)

× log

(nkT exp(−(�FA/kT ))

Ih

)]1/3

(1)

here �T is the extent of supercooling and I was measuredrom droplet experiment with 50 �m particle at maximum super-ooling. Tm, k, �FA, �Hm, n and h are melting temperature,oltzmann’s constant, Gibb’s free energy difference between

iquid and crystal for the undercooled liquid, melting enthalpy,umber of atoms in a mass of liquid and Plank’s constant, respec-ively.

Vinet et al. [3] also determined σ of a number of pure elementsrom the drop tube experiment with help of the homogeneousucleation theory using a similar expression as that of Turn-ull [2] (Eq. (1)). Table 1 shows the values of σ of some pureetals calculated by Turnbull [2] as well as Vinet et al. [3].

licksman and Vold [4] had calculated the absolute σ for pureismuth (Bi) to be 61.3 × 10−3 J/m2 based on the dihedral angleeasurements at the root of tilt boundaries. This value is 15%

arger than the value calculated by Turnbull [2] and 38% lower

K. Mondal, B.S. Murty / Materials Science and Engineering A 454–455 (2007) 654–661 655

Table 1Comparison of the σ (10−3 J/m2) of pure metals considering different approximations for �G

Element Turnbull [2] Vinet et al. [3] Present work

Hoffmann [11] Thompson and Spaepen [10] Ramachandra Rao [12]

η = 102 P η = 103 P η = 102 P η = 103 P η = 102 P η = 103 P

Al 93 106 111 108 106 103 132 128Ag 126 131 123 120 119 115 156 152Au 132 128 137 132 132 128 138 134Bi 54 99 58 56 55 54 – –Co 234 289 237 230 228 222 – –CFM

th[

gbt

σ

wcaofti

fhTinatt

2

[

I

wt1moe�

c

emh

I

w1

ac

D

wtS

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�

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w

waus

u 177 186 180 174e 204 269 205 199n 206 162 172 167

han the value calculated by Vinet et al. [3]. In addition, thereave also been hard sphere models for the calculation of σ

5–7].Till now, there are limited attempts to calculate the σ in case of

lass forming liquids. The σ of Au77Si9Ge14 has been calculatedy Thompson et al. [8] based on Spaepen’s model [5]. Accordingo this model, σ can be expressed as,

= �Smαm

(N0V 2m)1/3 T (2)

here �Sm is the molar entropy of fusion and αm = 0.86 for fccrystals. In evaluating σ for the above alloy, �Sm has been takens that for pure gold and Vm was calculated from the densityf liquid Au77Si9Ge14 [8]. Based on these calculations, σ wasound to be 63.6 × 10−3 Jm−2 for the above alloy. In contrasto this work, most of the other investigations consider σ to bendependent of temperature [2–4,15,16,23,24].

In the present paper, σ of a number of pure metals and glassorming liquids has been predicted from a simple model, whichas been derived from classical homogeneous nucleation theory.he advantage of this model over the existing models is that

t is very simple and does not need any experimental data onucleation kinetics, as is the case with many existing models. Inddition, from the measured σ values, critical cooling rates forhe formation of glass has been calculated and compared withhe values predicted by previous investigators.

. The model

The nucleation rate I in condensed systems can be expressed9] as

= nvv

(n∗

s

n∗

)(�G∗

3πkT

)1/2

exp

(−�Gd + �G∗

kT

)(3)

here nv = the number of molecules or formula units of nuclea-ing phase per unit volume of parent phase (typically 1028 −029 m−3); v = vibration frequency ∼1013 s−1; n∗

s = number ofolecules on the surface of a critical nucleus; n* = number

f molecules in the critical nucleus; �Gd = activationnergy for the transport across the nucleus/matrix interface;G* = thermodynamic barrier for nucleation; k = Boltzmann’s

onstant.�

173 167 – –197 192 204 197166 161 238 231

The quantity (n∗s /n*)(�G*/3πkT)1/2 is within one or two pow-

rs of ten for all nucleation problems of interest [9], which isuch smaller than the other components of the equation and

ence can be ignored. So, Eq. (3) can be written as

= nvv exp

(−�Gd + �G∗

kT

)(4)

here the pre-exponential factor (nvv) is typically041 − 1042 m−3 s−1 [9].

Assuming that the molecular rearrangement for the nucle-tion process can be described by an effective diffusionoefficient, D, one can have [9]

= vλ2 exp

(−�Gd

kT

)(5)

here λ is the jump distance, of the order of atomic dis-ance. D can also be related to the viscosity (η) by means oftokes–Einstein equation [9]:

= kT

3πλη(6)

Combining Eqs. (5) and (6) one can get

Gd = kT ln

(3πvλ3η

kT

)(7)

For spherical nuclei, the activation barrier for nucleation isiven by

G∗ = 16πσ3

3(�Gv)2 (8)

here �Gv = �G

Vm

(9)

here �G is the Gibb’s free energy difference between liquidnd crystal for the undercooled liquid and Vm is the molar vol-me. Various expressions for �G are available in the literature,ome of which are given below.

Thompson and Spaepen model [10]

G = 2�HmT (Tm − T )

Tm(Tm + T )(10a)

6 ence

�

�

�

webrsg

tnnncifTosvna

I

a

I

(

ab[ci

Ai

η

w

oPal

hfttTvpRtucCRtmtuot

fcpafTosmdt[pcoiti

56 K. Mondal, B.S. Murty / Materials Sci

Hoffman model [11]

G = �HmT (Tm − T )

T 2m

(10b)

Ramachandra Rao’s model (Taylor series) [12]

G = �Hm(Tm − T )

Tm− �Cm

p(Tm − T )2

Tm + T

+ ∂�Cp

∂T

∣∣∣∣Tm

(Tm − T )3

2(Tm + T )(10c)

Mondal et al. model [13]

G = �Hm(Tm − T )⌊4T (Tm − Tα)2 − Tα(Tm − T )2

⌋2Tm(Tm − Tα)2(Tm + T )

(10d)

here Tα is the isenthalpy temperature, �Hm is the meltingnthalpy and �Cm

p is the difference in molar heat capacityetween liquid and solid at melting point. Mondal et al. [13] haveecently shown that the �G values calculated from their expres-ion are much closer to experimental values for a wide range oflass forming liquids, in comparison to the earlier models.

Buckle [14] has shown that the threshold nucleation tempera-ure (Tr) for homogeneous nucleation is about 0.8Tm. Thresholducleation temperature is the temperature at which homoge-eous nucleation starts, wherein detectable quantity of criticalucleus is available in the liquid. This approximation of under-ooling has also been considered in the recent times by severalnvestigators [15,16] while drawing TTT diagrams for glassorming alloys even for bulk glass forming alloys. Hence, when

tends to Tr, nucleation frequency (I) is very small. On thether hand, when a supercooled liquid approaches glass tran-ition temperature (Tg) I becomes very small because of theery low atomic diffusion at Tg in spite of very low thermody-amic barrier for nucleation. Therefore, one can, to a reasonablepproximation, equate nucleation frequencies at Tr and Tg.

At Tr, Eq. (4) becomes

Tr = nvv exp

(−�GTr

d + �GTr

kTr

)(11)

nd at Tg, Eq. (4) becomes

Tg = nvv exp

(−�GTg

d + �GTg

kTg

)(12)

Equating ITr and ITg and combining Eqs. (7), (8), (11) and12), one can get

16πσ3

3

[Tg

(�GTrv )2 − Tr

(�GTgv )2

]= TrTgk ln

(Trη

Tg

TgηTr

)(13)

In order to solve Eq. (13), one has to have the viscosity levelst Tr and Tg. Viscosity at the glass transition temperature can

e assumed to be 1013 poise (P) [17]. Ramachandra Rao et al.17] have shown that the temperature dependence of viscosityould be represented by a Doolittle-type [18] expression involv-ng the relative free volume. According to this approach, for

at[I

and Engineering A 454–455 (2007) 654–661

u77Ge13.6Si9.4 (Tm = 625), temperature dependence of viscos-ty could be written as

= 1.268 × 10−2 exp

(0.9025

fT

)(14)

here fT = 3.046 exp

(−2707.4

RT

)(15)

According to Eqs. (14) and (15), at Tr, the viscosity turnsut to be ∼1.076 P for the above alloy. For Pd77.5Cu6Si16.5 andd82Si18 alloys, the viscosity level is within 102–103 (η ∼ 740 P)t Tr = 0.8Tm [17]. So, one can reasonably assume the viscosityevel at Tr to be in the range of 102 and 103 P.

Finally, one can calculate the σ of any pure metals with theelp of Eqs. (10) and (13). One can also extend this to glassorming liquids. The only data required for calculating the σ ishe �G values at Tr and Tg and the value of Tg. For pure metals,he Kauzmann temperature (TK) can be taken as the measure ofg and the expressions developed by Murty et al. [19] for TK forarious �G models have been used in the present study. In theresent work, Thompson and Spaepen [10], Hoffman [11] andamachandra Rao [12] models have been used for the calcula-

ion of �G and the σ of some pure elements has been calculatedsing the present model (Eq. (13)) and compared with that cal-ulated by Turnbull’s model in Table 1. However, for Cu, Bi ando, the value of σ turns out to be imaginary with Ramchandraao’s model [12] because the TK value is far below the melting

emperature and hence they are not shown in Table 1. In case ofodel by Mondal et al. [13], one needs to solve a cube equa-

ion for getting the TK values. Hence, this model has not beensed for finding the σ for pure metals. However, later in casef multicomponent glasses where experimental Tg is available,his model has been used.

The thermodynamic data for the calculation of σ was takenrom Barin et al. [21]. The molar volumes of the elements arealculated from the elemental density [22]. Table 1 also com-ares the values of σ at two levels of viscosity (η = 102 and 103 P)t Tr = 0.8Tm. It has been observed that the σ values calculatedrom Taylor series model [12] show larger deviation from theurnbull values [2] in a number of cases, in comparison to thether models. Murty et al. [19] have shown earlier that Tayloreries model loses its validity far away from Tm. As the presentodel uses the �G values at Tg, which is far away from Tm large

eviation in the σ is expected from the Taylor series model. Forhe other two cases (Hoffman [11] and Thompson and Spaepen10]), the calculated σ using the present model are within closeroximity with those obtained by Turnbull [2]. In most of theases, the percentage deviation falls within 8% and only in casef Mn, the deviation is about 20%. Turnbull [2], while calculat-ng �Tr for Fe and Mn, pointed out the difficulty in predictinghe σ in these cases due to the allotropic transitions that existsn these cases.

Table 2 shows the σ of 38 pure elements (for which Vinet et

l. [3] have already calculated σ using homogeneous nucleationheory), calculated from the present model using Hoffman model11] for the calculation of �G with Tr = 0.8Tm and ηTr = 102 P.t is observed that the most of the values of σ calculated in

K. Mondal, B.S. Murty / Materials Science and Engineering A 454–455 (2007) 654–661 657

Table 2Comparison of σ of pure metals

Element Turnbull [2] Vinet et al. [3] Present worka % Deviation fromVinet et al. [3]

% Deviation fromTurnbull [2]

Experimental results ofother investigators

Al 93 106 111 5 16Bi 54 99 58 42 7 61 [4]Cu 177 186 180 3 2Co 234 289 237 18 1Au 132 128 137 7 4Fe 204 269 205 24 1Mn 206 162 172 6 20Ni 255 306 248 18 3Ag 126 131 124 5 2Hg 24 31 20 35 20Ga 56 82 44 46 27Sn 54 77 49 36 10 61 [20], 49 [20]Pb 33 47 37 21 11 34 [20], 42 [20]Sb 101 137 111 19 9Ge 181 314 224 29 19Pd 209 199 214 12 2Pt 240 213 229 8 5Ti 168 173 3V 242 259 7Cr 230 245 7Zn 87 82 6Zr 158 120 24Nb 262 274 5Mo 315 349 11Tc 330 289 12Ru 329 302 8Rh 279 267 4Cd 58 55 5Hf 229 216 6Ta 335 328 2W 436 402 8Re 463 394 15Os 402 390 3Ir 322 318 1Si 477 334 30Se 76 47 38In 37 29 21Te 127 87 31

P.

tedeo1p

pi(oJsi

v

odbavpcd[pcfb

a Hoffmann model is considered for �G calculation, Tr = 0.8Tm and ηTr = 102

he present work are quite close to those calculated by Vinett al. [3]. Out of 38 elements, only 8 elements have shown aeviation more than 25% (Table 2). The σ of all the remaininglements is quite close to that calculated by Vinet et al. [3]. Outf 17 elements for which σ was calculated by Turnbull [2], onlyelement (Ga) has shown a large deviation of 27% from the

resent result, and all others show reasonable closeness.It is important to note that the σ values calculated using the

resent model match quite closely with those measured exper-mentally, wherever data is available (Bi, Pb and Sn) [4,20]Table 2). Glicksman and Vold [4] found out σ for Bi basedn dihedral angle measurements at the root of tilt boundaries.ordan and Hunt [20] found out σ for Pb and Sn from the lamellar

pacing measurement during growth of eutectic structure. Thisndicates the usefulness of the present model.

Both Turnbull [2] as well as Vinet et al. [3] have calculated σ

alues from the undercooling experiments, which are not devoid

laat

f inaccuracies. Careful observation suggests a few inherentrawbacks of the method used by Vinet et al. [3] and Turn-ull [2]. Firstly, they considered measured nucleation frequencyt maximum undercooling and they have assigned a certainalues of pre-exponential factor in the Eq. (1). For example,re-exponential factor considered by Vinet et al. [3] is typi-ally 3 × 1031 for bulk droplets of container-less processing (e.g.rop-tube experiments, electromagnetic levitation) and Turnbull2] considered 2 × 1027 for tiny particles prepared by the dis-ersion technique. Both Vinet et al. [3] and Turnbull [2] haveonsidered homogeneous nucleation theory. However, it is clearrom the different values of pre-exponential factor consideredy both of them that differences in experimental conditions can

ead to differences in σ values. The present model is devoid ofll the above difficulties of critical undercooling experimentss carried out by either Turnbull [2] or Vinet et al. [3]. Evenhe difference in σ arising due to the difference in the values of

658 K. Mondal, B.S. Murty / Materials Science

Table 3Undercooling level used by Vinet et al. [3] for eight elements which have largedeviation from the values of σ calculated using the present model

Elements % Deviation of σ valuesfrom Vinet et al. [3]

Undercooling level (�T/Tm)considered by Vinet et al. [3]

Bi 42 0.42Hg 35 0.38Ga 46 0.50Sn 49 0.37Ge 29 0.27Si 30 0.27ST

pba

i[nseipG(�

ta[l�

[mHiwmcTe2

ibc

T[lums

gleawfcσ

lbhtNhae[

I

andc�

o

TC

A

Z

T

NA

e 38 0.20e 31 0.33

re-exponential factor is also not present in the present modelecause the pre-exponential factor gets cancelled while arrivingt the final expression.

Another important aspect, viscosity of liquid, is consideredn the present model, which is not considered by both Turnbull2] and Vinet et al. [3]. Both Turnbull [2] and Vinet et al. [3] haveot used the other term (�Gd) present in the expression of clas-ical homogeneous nucleation theory (Eq. (1)). This may lead torror in the calculated values of σ of pure metals. Another mostmportant factor is the use of over-simplified model by both therevious investigators [2,3] for the calculation of solid liquidibb’s free energy difference, which is the Turnbull’s expression

�Hm�T/Tm) [2]. At higher undercooling Turnbull’s model ofG deviates significantly from the experimental values, while

he other models [10–13] are closer to experimental results event higher undercooling. In the present case, Hoffmann model11] is used to calculate �G. Even at a lower undercoolingevel of Tr = 0.8Tm, there is a great difference in the values of

G as calculated by Turnbull model [2] and Hoffmann model11]. For example, if �G for Si is ∼40 kJ/mol by Turnbullodel [2], which is 5 times higher than that calculated usingoffmann model [11]. This would lead to higher values of σ

n the work of Vinet et al. [3]. All the nine elements in theork of Vinet et al. [3], which give deviation of about 25% orore in σ values from the values as calculated in the present

ase, have undercooling level higher than that corresponding tor = 0.8Tm except for Se as shown in Table 3, which could easilyxplain the differences. For Se, the value of �G at Tr = 0.2Tm is680 J/mol when Turnbull model [2] is used whereas the value

Ntt

able 4omparison of σ (10−3 J/m2), cooling rates, Rc (K s−1), present and past results

lloys Assumed value of σ taking�G* = 50 kT at �Tr = 0.2 (×10−3 J/m2)

Presen(×10−

r65Be35 124 [24] 137

i63Be37 157 [24] 170

i74.5Si8.5B17 153 [23] 160u65.45Cu11.55Si9Ge14 63 [7] 58

a Using Hoffman model for �G and η = 103 P.b Experimentally measured values.

and Engineering A 454–455 (2007) 654–661

s 2144 J/mol when Hoffmann [11] model is used. It shoulde mentioned that for lower values of σ, the influence of thehoice of the model for the calculation of �G is higher.

The present model considers a generalized assumption ofr = 0.8Tm, which is also being considered by many investigators15,16], while carrying out nucleation modeling of undercoolediquid, including glassy alloys. However, in some cases the actualndercooling deviates from this value. In spite of this, the presentodel gives reasonable values of σ of pure elements in a much

impler way.The present model can also be extended to calculate the σ of

lass forming alloys successfully by using Eq (13). This calcu-ation is carried out in the same way as was done for pure metals,xcepting that the Tg was not calculated from thermodynamics,s was done for pure metals, but the actual experimental valuesere used. Table 4 shows the σ of these three alloys calculated

rom the present model using Hoffmann [11] model for cal-ulating the �G. There have been earlier efforts to calculate

for glass forming alloys by using Eq. (8). In these calcu-ations, the most crucial parameter is the critical free energyarrier to crystal nucleation (�G*) and many investigatorsave assumed different values for it. Morris [23] has assumedhe value of �G* in the range of 30–60 kT at �Tr = 0.2 fori74.5Si8.5B17, where �Tr = (Tm − Tr)/Tm. Tanner and Ray [24]ave proposed the values of �G* of about 35–50 kT for Ti63Be37nd 50 to 65 kT for Zr65Be35 alloys at �Tr = 0.2. In gen-ral, steady state homogeneous nucleation can be approximated25] as

≈ 1030

ηexp

(− B

T 3r �T 2

r

)(16)

Uhlmann [25] used this equation and the constant B was takens 1.024, corresponding to a free energy of forming the criticalucleus (�G*) of 50 kT at �Tr = 0.2, based on the results ofroplet nucleation experiments on a variety of simple liquids. Healculated the activation energy barrier for nucleation as 55 kT atTr = 0.2 from the nucleation experiments (direct measurement

f nucleation frequency).

The σ values calculated for Zr65Be35, Ti63Be37 and

i74.5Si8.5B17 alloys, by taking 50 kT for �G* at �Tr = 0.2, withhe help of Eq. (8) using Hoffmann model [11] for calculatinghe �G, are shown in Table 3. Both Morris [23] and Tanner

t modela3 J/m2)

Rc calculated by previous investigatorusing �G* = 50 kT (K s−1)

Rc by presentmodel (K s−1)

1.0 × 107b (original DU treatment) [24]1.3 × 108 (modified DU treatment) [24]

4.4 × 105

6.3 × 106b (original DU treatment) [24]7.7 × 107 (modified DU treatment) [24]

5.7 × 105

3 × 105b [23] 2.7 × 105

– –

and

at

cbripatAioltt

ittafbdTft

f[t

X

wqta

I

waNjb

U

wDg

aiv

(g

t

By using Eq. (20), TTT diagram for Zr65Be35, Ti63Be37 andNi74.5Si8.5B17 alloys have been plotted when Xc is 10−6 andshown in Fig. 1(a–c) respectively. Critical cooling rate (Rc) can

K. Mondal, B.S. Murty / Materials Science

nd Ray [24] have used the Hoffman model for calculatinghe �G.

If one tries to calculate the �G* by taking the values of σ

alculated with the present model, the �G* values turn out toe 67, 63 and 57 kT for Zr65Be35, Ti63Be37 and Ni74.5Si8.5B17espectively, which are close to the range considered by othernvestigators. The difference in the σ calculated from theresent model and that of earlier investigators [23,24] for thebove alloys turn out to be around 4–7% which speaks abouthe efficiency of the present model. In addition to this, foru–Cu–Si–Ge alloy [7], the σ has also been calculated by apply-

ng the present model using Hoffmann model for the calculationf �G and it shows around 8% deviation from the value calcu-ated by Thompson et al. [8]. Thompson et al. [8] have takenhe �Sm value of pure gold for that of the alloy and otherhermodynamic parameters are extrapolated.

Another way to validate the σ calculated from this model is anndirect one. One can calculate the critical cooling rate by usinghe σ values obtained from the present model and compare withhe experimentally obtained critical cooling rates for the samelloys. In the present work, the critical cooling rate for glassormation in Zr65Be35, Ti63Be37 and Ni74.5Si8.5B17 alloys haseen estimated from the time temperature transformation (TTT)iagrams calculated for these three glass forming compositions.he TTT diagram gives the time required to form Xc volume

raction of the crystals during an isothermal treatment at variousemperatures.

The fraction of transformed material, Xc, when it is a smallraction, may be described by the Johnson-Mehl [26] and Avrami27] relationships for the case where no composition changeakes place during transformation, is given by

c = π

3IU3t4 (17)

here I is the thermally activated homogeneous nucleation fre-uency per unit volume, U the crystal growth rate and t is theime required for the formation of Xc quantity. I can be expresseds (combining Eqs. (4) and (5))

= DnNv

λ2 exp

(−�G∗

kT

)(18)

here Dn is the atomic diffusion coefficient characteristic fortomic motion necessary for crystallization at temperature T,v is the average atomic concentration per unit volume, λ is

ump distance or mean atomic distance and �G* and k haveeen defined earlier.

Crystal growth rate (U) can also be described by [27,28]

= fDg

λ

[1 − exp

(−�Tr�Hm

RT

)](19)

here f is the fraction of sites available at the growth interface,g a diffusion coefficient for atomic motion required for crystalrowth. �Tr and �Hm have been defined earlier.

It is generally assumed that the diffusion coefficient for nucle-tion (Dn) and growth (Dg) processes are the same [28] anddentical to the bulk diffusion coefficient D, which is related toiscosity (η) by the Stokes–Einstein equation as defined by Eq.

Engineering A 454–455 (2007) 654–661 659

6). Incorporation of Eqs. (6), (18) and (19) into (17), one canet the expression for time scale at the nose region (t).

≈ 9.3η

kT

{λ9Xc/f 3Nv[exp(�G∗/kT )]

[1 − exp(−�Hm�Tr/RT )]3

}1/4

(20)

Fig. 1. TTT diagram of: (a) Zr65Be35 and (b) Ti63Be37 and Ni74.5Si8.5B17.

6 ence

b

R

wnofettη

[Zontnfawf5oagvqTt

ibrt[woaD

mTo

lvauAcBMd[um

talcahvttiTevoafr[difoctTib

TC

A

ALMMPZZ

60 K. Mondal, B.S. Murty / Materials Sci

e expressed as

c = Tm − Tn

tn(21)

here Tm is the melting temperature, Tn the temperature at theose of the TTT diagram and tn is the time scale at the nosef the TTT diagram. With Eq. (21), the critical cooling ratesor the three alloys have been calculated by using the surfacenergy term found out in the present case and compared withhe critical cooling rates found out by the previous investiga-ors in Table 4. The datasets (Tg, Tm, λ, Nv, f, Xc, �Hm and) required for the calculation have been obtained from Morris23] for Ni74.5Si8.5B17 alloy and from Tanner and Ray [24] forr65Be35 and Ti63Be37 alloys. Tanner and Ray [24] have foundut the Rc for Zr65Be35 and Ti63Be37 alloys by applying origi-al Davies [28] and Uhlmann [29] treatment and modified DUreatment [23] and arrived at values of 1.0 × 107 K s−1 (origi-al DU treatment) and 1.3 × 108 K s−1 (modified DU treatment)or Zr65Be35 alloy and 6.3 × 106 K s−1 (original DU treatment)nd 7.7 × 107 K s−1 (modified DU treatment) for Ti63Be37 alloyhen �G* is 50 kT at �Tr = 0.2. In the present calculation the Rc

or Zr65Be35 and Ti63Be37 alloys turns out to be 4.4 × 105 and.7 × 105 K s−1, respectively, which are close to the lower limitf Rc previously calculated. In case of ternary Ni74.5Si8.5B17lloy, Morris [23] has found out experimentally that the Rc forlass formation is 3 × 105 K s−1. The present model gives aalue of critical cooling rate is 2.7 × 105 K s−1, which matchesuite well with the experimentally measured critical cooling rate.his result also is a strong proof that the σ calculated according

o present model is well suited for glass forming liquids also.In the present work, one basic assumption is that the σ is

ndependent of temperature, which is also considered by a num-er of investigators such as Davies [28] and Uhlmann [29]. Theecent experiments on the large undercooling of droplets of cer-ain low-melting point metals and alloys by Perepezko et al.30] do not support the concept of Spaepen and Mayer [31],ho have considered the σ as entropy term and also dependentn the temperature. In Table 4, the data for Zr65Be35, Ti63Be37nd Ni74.5Si8.5B17 alloys are calculated according to originalU model considering independency of σ on the temperature.

In addition, σ of a few more glass forming systems (binary to

ulticomponent glasses) has also been calculated and shown inable 5. Here also different models are used for the calculationf the free energy difference liquid and crystal for undercooled

tσ

ao

able 5omparison of the σ (10−3 J/m2) of different metallic glasses considering different a

lloys Hoffmann TS

η = 102 P η = 103 P η = 102 P

u81.4Si18.6 73 71 64a50Al30Ni20 108 105 109g65Cu25Y10 72 69 69g50Ni30La20 105 101 100

d40Ni40P20 57 55 57r65Cu17.5Ni10Al7.5 95 92 94r41.2Ti13.8Cu12.5Ni10Be22.5 90 87 90

and Engineering A 454–455 (2007) 654–661

iquid. It has been observed that there is little difference in thealues of σ at a fixed viscosity level even if different modelsre used for the calculation of the free energy difference liq-id and crystal for undercooled liquid. Thermodynamic data foru81.4Si18.6, La50Al30Ni20 and Zr41.2Ti13.8Cu12.5Ni10Be22.5 are

ollected from Thompson and Spaepen [10], Sommer [32] andusch et al. [33], respectively. For the alloys Mg65Cu25Y10,g50Ni30La20, Pd40Ni40P20 and Zr65Cu17.5Ni10Al7.5, thermo-

ynamic data were collected from Busch et al. [34], Sommer32], Wilde [35] and Sommer [32], respectively. The molar vol-me of the glasses has been calculated according to rule ofixture of pure elements.The viscosity levels at Tr are considered to be 102–103 P and

he change in the σ with the change in the viscosity levels at Tr arelso shown in Tables 1 and 3. The difference at the two viscosityevels is within 3%. So, one can take any of the two levels of vis-osity at Tr with minimum of error. Apart from the uncertainty inssigning values for �G*, the second difficulty in the kinetics ofomogeneous nucleation arises from the limited availability ofiscosity data, hence the diffusion data. Usually, the viscosity inhe liquid state above the melting point is known. It is assumedhat the viscosity at Tg is 1013 P and some interpolation processs used find out the viscosity in the temperature range of interest.he present model overcomes both these two problems (knowl-dge of �G* and η) to a great extent. It needs some assumedalues of viscosity, but, it does not consider any assigned valuesf �G*. The present model uses the thermodynamic and materi-ls parameters, which in turn lead to the calculation of the �G*

or the glass forming liquids and finally the critical cooling rateequired for the formation of glass. Turnbull [2] and Vinet et al.3] have found out σ from the nucleation frequencies of smallroplets on the basis of homogeneous nucleation theory, whichnvolves very cumbersome process of measuring I (nucleationrequency) at Tr. The current method overcomes the tedious taskf measuring nucleation frequency. Glickman et al. [4] have cal-ulated the absolute σ of bismuth by establishing equilibrium athe intersection of a low-angle grain boundary with the interface.his process involves a lot of electron microscopy at the melt-

ng point to measure the contact angle between the low angleoundary and solid–liquid interface and the difficulty in high

emperature electron microscopy itself puts a barrier to measurefor high meting point metals. The present model can overcome

ll these difficulties to the fullest extent. This model follows mostf the basic assumptions taken by Turnbull [2] while calculating

pproximations for �G

Taylor series Mondal et al.

η = 103 P η = 102 P η = 103 P η = 102 P η = 103 P

62 61 59 63 61106 108 105 109 10667 67 65 69 6797 103 100 103 10055 56 54 57 5591 94 91 95 9287 89 86 89 87

and

tmawk

3

1

2

3

R

[

[[

[

[[[[

[[[[

[[[[[[

[[[

[[32] W. Sommer, Mater. Sci. Eng. A 226–228 (1997) 757–762.

K. Mondal, B.S. Murty / Materials Science

he σ for interface such as spherical nucleus, complete isotropicaterial and independency of fusion entropy on the temperature

nd comes out with a simple expression for the calculation of σ,ithout the need of any experimental data related to nucleationinetics.

. Conclusions

. A simple model based on the homogeneous nucleation theoryis proposed for the estimation of solid–liquid σ without thenecessity of any data on nucleation.

. The present model has been shown to be successful for theprediction of σ of both pure metals and glass forming alloys.

. The present model has also been extended to predict thecritical cooling rate for the glass forming alloys.

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