Prediction of glass-forming ability in terms of a critical contact angle

JOURNAL

ELSEVIER

OF

Journal of Non-Crystalline Solids 215 (1997) 283-292

Prediction of glass-forming ability in terms of a critical contact angle Z. Rivlin a, H.G. Jiang b,l, M.A. Gibson c, N. Froumin a, j. Baram a,* Materials Engineering Department, Ben-Gurion Universi~ of the Negev, P.O. Box 653, 84105 Beer-Sheva, Israel Nuclear and Chemwal Engmeermg Department, Materials Section, University of California lrvine, USA c Division of Materials Science and Technology, CSIRO, Melbourne, Australia b

.

•

.

Received 14 December 1995; revised 26 February 1997

Abstract

A numerical simulation treating rapid solidification as controlled by heterogeneous nucleation, has been applied to predict the compositional dependence of the glass-forming ability in melt-spun binary and ternary alloys, in terms of a critical contact angle related to the wetting properties of the solid nuclei and quenching substrate in contact with the solidifying liquid. The smaller the critical contact angle, the higher the glass-forming ability. A criterion linking glass-forming ability to the computed critical contact angle has been inferred (for melt-spinning), as follows: easy glass-forming alloys (or compositions) are those for which 0crit is _< 60 °. There is good agreement between experimentally determined and the computation deduced glass-forming abilities in a series of binary and ternary alloy systems. © 1997 Elsevier Science B.V.

I. Introduction

Metallic glasses are commonly prepared by rapid quenching from the liquid state. For the glass transition to occur during the quenching process, the nucleation of a crystalline phase must be suppressed until the glass transition temperature is reached. Glass-forming ability (GFA) is a term used as a measure of the ease with which an alloy melt can be undercooled below the glass transition temperature during solidification. The estimation of the GFA in metallic alloy systems is of special importance since it can provide guidelines for the production of metal-

* Corresponding author. Tel.: +972-7 461 474; fax: +972-7 461 474: e-mail: [email protected]. L On leave from the Institute of Metal Research, Chinese Academy of Sciences, Shenyang. People's Republic of China.

lic glasses, in the bulk form in particular. A number of empirical and theoretical criteria regarding the evaluation of GFA have been suggested in the literature and these include: crystallization should be suppressed [1], as a result of high undercooling; the solute concentration of constituent B in the A matrix should be more than 10 at.% [2,3]; the atomic radii mismatch of the glass-forming alloy components should exceed 15% [2,4]; the constituents A and B are each glass-forming elements and should consist in a combination of metal and metalloid elements [4]; the reduced glass temperature Trg = T g / T m (Tg and Tm are the glass transition and the melting temperature, respectively) is subjected to the following constraints: for pure metals: Tr_ = 0 . 2 5 ; for glass-forming alloys: 0.5 < Trg < 0.8 il,4-11]; the AB composition is (close to) the eutectic one [4]; for non-eutectic compositions, the lower the T~g, the

0022-3093/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved. PII S 0 0 2 2 - 3 0 9 3 ( 9 7 ) 0 0 1 0 8 - 7

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Rivlin et aL / Journal of Non-Crystalline Solids 215 (1997) 283-292

worse the glass-forming ability [6]; the ratio K = (Tx - Tg)/(Tm - Tx) where Tx is the crystallization temperature, should be as large as possible [12]. In addition to the empirical ones, several theoretical criteria have also been put forward in the past. Time-temperature-transformation (TTT) curves have been constructed by Uhlmann [13], corresponding to some barely detectable degree of crystallinity, about 10 -6 vol.%, from which an estimate of the lowest cooling rate to just avoid crystallization have been deduced. Saunders et al. [14] extended Uhlmann's method to multicomponent systems, by introducing a thermodynamic phase diagram calculation which enables prediction of glass-forming range (GFR) and critical cooling rate for glass-formation. Morris [15] modified Uhlmann's kinetic treatment by introducing the Thompson-Spaepen [16] temperature dependence for the volume free energy difference between the liquid and the solid phases, AG v. Morris has calculated the volume fraction crystallized during cooling a melt at constant cooling rate. The calculations were fitted to the experimental data for Ni74.sSis.sB17 by adjusting values of the free energy barrier for homogeneous nucleation, AG* [15]. It is worthwhile to note that such an adjustment is equivalent to assuming a certain value for the solid-liquid interracial energy o5_ s as inferred from the relation: AG* = (16~/3)(o'13s/AG2v). Nash et al. [17] have numerically integrated the transformation kinetic equation and established continuous-cooling-transformation curves as a function of composition in Ni-Ti melts. The glass-forming range of rapidly quenched binary alloys can then be predicted from a 'T' criterion'. T' curves have been defined as the temperature-composition locii for a given volume fraction of crystallized material formed during cooling at a constant rate and can be constructed as a function of the binary composition x. The T' criterion states that the glass-forming range is expressed by: T'(x)< Tg(x), where Tg(x) is the glass transition temperature for the composition x. This criterion has been shown to predict 'quite well' the glass-forming range of Ni I _xTix [17]. All the afore-mentioned theoretical criteria have been derived from Uhlmann's kinetic treatment, assuming that the rapid solidification process is dominated by steady-state homogeneous nucleation. However, in numerous rapid solidification processes het-

erogeneous nucleation is the rate limiting factor for glass formation. In that case, undercooling is limited to temperatures much higher than those where homogeneous nucleation can be thermodynamically active [18]. Saunders [14] mentions that the deviations of the calculated critical cooling rates presented in Ref. [14], based on homogeneous nucleation, from the actual cooling rates for glass formation in the studied material systems are due to an underestimatation of the 'resistance to homogeneous nucleation'. He suggests that the discrepancy for the cooling rates stems from the assumptions made on the numerical values of the interfacial energy as input parameter in the calculation. In the present work, heterogeneous nucleation, induced by a catalyst such as a quenching substrate or a solid impurity within the liquid, is considered to be the dominant process during rapid solidification. In that case, surface tensions involved in the thermodynamics of nucleation include the liquid and the evolving nucleus solid phases, the evolving nucleus solid and the heterogeneous catalyst solid phases and finally the liquid and the catalyst solid phases. The glass-forming ability of binary and ternary metallic alloys solidifying heterogeneously during rapid quenching by melt spinning has been investigated, in terms of the involved interfacial energies. For comparison with experimental results, special emphasis has been given to the easy-glass-forming magnesium-based alloys for which good correlation between the computational and experimental GFR have been found.

2. Theoretical background When a solid spherical cap-shaped nucleus is formed on a catalytic substrate surface (in contact with the liquid melt), the contact angle 0 is expressed by Young's equation, an energy balance involving surface tensions which depend upon the melt, the catalytic substrate and the solid nucleus interfaces: cos 0 =

~c-I

-- ~c-s

(1)

Orl - s

~_1, crc_s and o-1_s are the interfacial energies

Z Riulin et al./Journal of Non-Crystalline Solids 215 (1997) 283-292

between the catalytic substrate and the liquid, the catalytic substrate and the evolving solid nucleus and between the liquid and the evolving solid nucleus, respectively. This contact angle plays a crucial role in the heterogeneous nucleation process. Following the classical theory for heterogeneous nucleation, the number of atoms in a spherical capshaped nucleus is

ncrit

32 xro-13_~ 3VmAG3 F(O).

(2)

The term F(O)=(2+cosO)(l-cosO)2/4 is a measure of a 'catalytic efficiency', Vm is the atomic volume, 0 is the contact, or wetting, angle between the spherical cap-shaped nucleus (within the liquid melt) and the catalytic substrate, AG is the free energy difference between the bulk phases. AG serves as a driving force for homogeneous nucleation, when F( O) = 1. The nucleation rate as a function of temperature J(T) is proportional to the probability of forming solid nuclei with excess energy AG * :

[-ac*l].

J(T)---Joexp[~BT

(3)

k B is the Boltzmann constant and T the temperature. The pre-exponential factor J0 depends upon the density of available heterogeneous nucleation sites on the catalyst-liquid interface. Heterogeneous nucleation can be basically classified into two categories, namely surface heterogeneous nucleation and bulk heterogeneous nucleation. In both cases, the physical factors which determine J0 in Eq. (3) are experimentally inaccessible in the case of deeply undercooled liquid melts. This could be one of the reasons why rapid solidification modeling has been often treated as being controlled by homogeneous nucleation. Moreover, the role played by heterogeneous nucleation during solidification process depends on the time scale. At increasing cooling rates, the dwell time of the liquid metal at higher temperatures where heterogeneous nucleation is operative becomes shorter. Heterogeneous nucleation may then somehow be suppressed, allowing the solidification process to be dominated by homogeneous nucleation. This issue has been examined [19].

285

In the rapidly quenched material produced by melt-spinning (or other rapid solidification techniques), there is a competition between two different mechanisms, namely: (1) the diffusion mechanism that is responsible for the growth of embryo clusters [21-24] and (2) the steep temperature drop at the liquid-substrate interface. The nucleation is considered to be heterogeneous, the nuclei having a spherical cap shape geometry. Kelton et al. [24] assumed that the total number of atoms in the sample available for the embryos to grow to nuclei is the Avogadro number. In the model adapted to melt-spinning [25], the total number of atoms in the sample available for the embryos to grow into nuclei is limited. This number depends on and is calculated according to the geometrical and hydraulic conditions prevailing in melt-spinning. These conditions define the surface area of the extracted ribbon as a function of the time (temperature) interval [21]. The clusters in an isothermal control volume, which is computed in the numerical simulation and in which the size of the embryos, i.e., the number of atoms included within each embryo, can be calculated. At any temperature, the largest existing embryo cluster has a number of atoms labeled nmax . The solid-liquid interfacial energy tr~_~ may be computed for an undercooling regime by means of the Spaepen equation [20]:

t-/e Orl-s --

,

(4)

,~xl/3 "

(NAVm)

NA is the Avogadro number and a a constant, the value of which is generally approximately to 0.71 for a number of glass-forming systems [7]. A Hf equates to ASfTm, where ASf is the entropy of fusion and is taken as m s f ~ - x ( m n A / / r m A) + (1 - - x X A H B / / T B) for binary alloy AxB ~_x [16] and can be extended to maf x(AHA//TmA) -~- y(Anfa/Tma) + (1 - x yXA HfC/TC m) for ternary alloy A xB yC l - x- ~.. The temperature dependence of viscosity in the undercooled liquid melts is given by the Doolittle equation: =

[ "q(T) =3.34×

10-4exp

1

cexp[_W//eT]

]

.

(5)

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Rivlin et al. / Journal of Non-Crystalline Solids 215 (1997) 283-292

c and W are constants which can be calculated using the isofree volume model [7]:

RT~Tg [ Vf(Zm) ] In W = Tin- Tg [ Vf(Tg)l'

(6)

c = Vf(T~,)exp

(7)

.

The free volumes are Vf(Tm) = 0.35, Vf(Tg) = 0.03 [7]. In Eqs. (6) and (7), the only unknowns are Tm and ~ . Tm is a physical property of the material system, Tg is either known from experiments or from the empirical relationship: Tg = 0.56Tm [26].

3. Numerical simulation The purpose of the numerical simulation is to evaluate a theoretical critical contact angle 0crit for a particular material system. If the actual contact angle in a melt-spinning experiment equals, or exceeds the critical contact angle, the resulting ribbon is predicted to be amorphous. If the actual contact angle is less than the critical contact angle, the resulting ribbon is predicted to be crystalline. The algorithms used in this study have been described elsewhere [21-23], with some minor modifications. During the rapid solidification melt-spinning process, the competition between crystalline and amorphous phases depends on the thermal gradient along the ribbons thickness and the resulting number of atoms available within an isothermal control layer. The surface of such a control layer in contact with the substrate has been shown to depend on the melt-spinning process parameters [21-23]. The layer thickness is determined by the time step in the numerical simulation procedures, chosen in a way that the temperature gradient in the layer thickness will not exceed 0.1°C. If the quenching wheel rotating speed is increased, the thermal gradient becomes steeper and solidifying isothermal liquid layers become thinner. The amount of atoms in any isothermal layer is therefore limited and may not be sufficient to supply the atoms necessary for embryo clusters to grow into critical nuclei. Nucleation is then suppressed. With the continuing decrease of

temperature, there is a continuous increase in the liquid metal viscosity and glass formation may occur. The evolution of any solid phase embryo cluster to form a nucleus of critical size has been treated by a series of absorption/desorption reactions. The number of atoms in clusters was calculated by Eq. (8), that describes the atomic balance in the embryo cluster:

dt

+ ---NE"-lkn-I "-~ NE n-+-Ikn+l,

+ NE

)

(8)

where the index E n relates to the embryo that contains n atoms, t is the time, NE, the number of embryos of size E,, k~+ the rate at which one single atom is absorbed into an embryo cluster of size n, k~ the rate at which one single atom desorbs from an embryo of size n. The rates k~ can be obtained from the reaction rate theory [24]. The number ncrit of atoms in any critical nucleus decreases as rapid cooling proceeds, whereas the maximum available number nmax of atoms in the largest embryo that is found in the liquid at temperature T increases. A temperature is reached, the nucleation temperature TN, at which ncrit equals nmax, i.e., an embryo has reached the state where it has grown up to the critical size, has become a nucleus and crystallization occurs. However, during on-going undercooling, as temperature drops, there is a continuous increase in the liquid metal viscosity, becoming drastic as the glass transition temperature Tg is approached. The glass transition takes place when the viscosity of the undercooled liquid reaches the value of 10 jl Pa s and the temperature at which this value is attained is Tg. Heterogeneous nucleation can also take place by the 'site saturation' mechanism, i.e., with zero nucleation rate, when nucleation sites that already exist within the liquid phase start to grow, as the appropriate temperature for growth is reached. In that specific case, growth of the existing nuclei may be suppressed in the same way that nucleation has been suppressed in the previous case, due now to the temperature dependent competition between crystal growth (not nucleation) and viscosity increase. The

287

z. Rivlin et al. / Journal of Non-Crystalline Solids 215 (1997) 283-292

final microstructure of the melt-spun ribbons results from the afore-mentioned competition between either nucleation or crystal growth and amorphisation. The kinetic aspects of that competition have been treated in Ref. [19]. The critical contact angle 0crit is the contact angle that satisfies, as an input value, the condition ncrit = nmax, at a temperature, which is Tg (or at the temperature where the viscosity has attained the critical value of l 0 II Pa s). From Eq. (2) it can be seen that the number of atoms in a critical nucleus decreases with decreasing contact angle. There is therefore an enhanced (or reduced) probability for amorphisation (or crystallization) when the contact angle reaches high values, depending on the nature and compositions of the material systems involved (liquid, solid and substrate). The processing variables used in the calculations were kept constant for all alloy systems investigated and are as follows: copper quenching substrate rotating speed: 20 m s - I ; ejection velocity of the melt: 10 m s - l ; ribbon width and thickness: 2 m m and 20 Ixm, respectively; heat transfer coefficient: 10 6 W K I s - l , corresponding to a typical cooling rate in melt spinning of 10 6 K s -1. A data bank containing various physical quantities for elemental constituents, such as melting temperatures, thermal conductivities, densities, elemental masses etc., has been built into the program. The melting temperatures for the selected alloy systems have been extracted either from existing phase diagrams in the literature or from C A L P H A D computations performed specially for this study. Glass transition temperatures were either chosen from reported experimental values or from the J i a n g - B a r a m empirical relationship [26].

(a) Gla~ transition mmpemmre lOl~

Tg

"~ 107 .0 it

~ --

The procedure for assessing glass-forming ability of binary or ternary metallic systems, in terms of a critical contact angle 0crir is presented by taking the easy glass-forming Fe80 Silo B l0 alloy as an example. A similar procedure has been presented [27] for a MgCu alloy. Fig. l a - c show the evolution of the viscosity, of the number of atoms Refit necessary to form critical nuclei and of the maximum available number of atoms nmax in the isothermal layers, as a

assumed eoumet angl©0 = 40°

i0~

t

to~

~-~,lo5

t

~

k

n cm

)"

~O' 1o4

I°~ 1o' r-"

10. 3

102 102

r

i

i

9oo

1ooo

L , = i

1~oo

L t I = ,

I =

l~

Te=~mmlmr~

I ,

13oo

1{2~

l~

1~o

[K]

assumed contact angle 0 = 5fi°

107

ZO~ n

102

"~

to" .*

o 7-

o tO3 z 102

102

10-I

lOt

10.3

.::J.:::,.:::l 800

900

1000

(c)

1100

:::!.

1200

~,,,~,-,~

1300

::2

1400

lOo

1500

t,,l

Glass transition tcmperatu~ 10ti 109

['-T

lO~ assumed c°°tact a~ie 0 = 45°

i~t

t

~

106

net"

107

~0~

i

t° 3

.~

10~

102

lO-X 10-3

4. Results

10s

•. 800

!

)

900

I000

: , , 1100

: •. . ! :

1200

Temptt'wam~

1300

: . ! 1400

:

._J 10o 1500

[KI

Fig. 1. Evolution with temperature of the viscosity, the number of atoms necessary to form a critical nucleus ncrit and the maximum number of atoms, nma` in existing clusters, for different assumed contact angles: (a) 40°, (b) 50°, (c) 45° in FesoSiloBto. function of temperature, for the FesoSiloB10 alloy. The varying input parameter in the computations that resulted in these figures was only the value of the contact angle, chosen as 40 °, 50 ° and 45 ° respec-

288

Z Rivlin et al. / Journal of Non-Crystalline Solids 215 (1997) 283-292

tively. The figures depict the concurrent occurrence of competing processes, as temperature lowers, namely: the n u m b e r of atoms ncrit necessary for a nucleus to b e c o m e of critical size approaches the m a x i m u m n u m b e r of available atoms nmax in embryos; the viscosity of the undercooled liquid approaches the critical value of 1011 Pa s, same graph for all three Fig. 1. The final structure of the rapid solidification product will d e p e n d on which process is finally prevailing. Fig. l a represents the case of a contact angle input o f 40 °. As rapid cooling proceeds to a certain degree, a temperature is reached, still above the glass transition temperature, where nmax = ncd t. The n u m ber of atoms necessary to form a critical nucleus has already accumulated in a n y o n e o f the largest existing e m b r y o cluster, which itself becomes therefore a nucleus. Nucleation has taken place and crystallization follows, before occurrence of the glass transition, i.e., before viscosity reaching its critical value of 101! Pa s for glass formation. Fig. l b represents the case of a contact angle input of 50 °. Here the glass transition temperature is reached at a stage where the n u m b e r o f atoms necessary to form a critical n u c l e u s is still higher than the n u m b e r of atoms present in the largest existing e m b r y o cluster, nmax < ncrit. Glass transition takes place, prior to crystallization. This leads to the obvious conclusion that the actual contact angle for the Fe80Sil0Bt0 system rapidly quenched, with processing parameters as described above, on a copper substrate lies between 40 ° and 50 °. Fig. l c has been computed for a critical contact angle 0crit = 45 °, as an intermediate value. In that case, the c o m p u t i n g accuracy does not allow a clear definition of the prevailing process, crystallization or amorphisation. It can therefore be inferred that the critical contact angle for Fe80 Si10 B l0 ribbons m e l t - s p u n on a copper substrate is 45 ° _+ 3 °. The computational procedure was applied to a series of binary and ternary alloys, including n u m e r ous m a g n e s i u m - b a s e d systems, for the prediction of the compositional dependence on the glass-forming ability. Table 1 lists the melting temperatures, glass transition temperatures and the computed critical contact angles for the selected binary and ternary alloys. The temperatures marked by an asteric were

Table 1 Melting temperatures Tin, glass transition temperatures Tg and critical contact angles 0crit for selected binary and ternary metallic alloys Tm (K)

Tg (K)

1300 853 1618 1448 1321 1683 873 1413 1658 1533 1713 1090 720 793 790 1393 733 613 743

727 477 906 740 650 820 488 790 928 730 959 626 403 444 442 780 410 343 416

* * * * * * *

67 51 75 48 44 60 42 55 70 46 55 37 67 72 47 72 53 44 55

726 696 724 744 738 1083 708 695 702

406 397 442 432 428 606 * 425 424 428

39 43 37 41 42 54 39 40 40

Mg85Ca4 AIIl

995 900 953

557 * 504 * 533 *

68 58 55

Mg75CalsSi7 Mg85CaaSijl Mg9oCa2Si8

880 1120 1090

492 * 627 * 610 *

55 86 92

Mg73Alr iSi6 Mg 83A112Si5 Mg85AIi e6 Si24

1250 1030 990

700 * 576 * 554 "

170 160 115

Mg65Ni2oNd15 MgzTNilsNd5

738 717

467 456

35 36

1340 1420 916 1309 1170

782 818 602 707 655

48 45 25 42 37

Ca4oAl6o C475AI25 Cr58Ni42 FesoB2o Fes3P17 FessZr12

Laso Au2o

Mg34Ni66 Ni 76A124 NisoZrso

Ni78ZF22 Pd84Si16 Mgs0 A150 Mg80AI20 Mg3sCa~5 Mg 66Si 34 MgroZn4o

Mg72Zn28 MgsoZn2o Mg65Cu3oY5 MgsoCulsY5 Mg55Cu35Ylo

MgroCu3oYIo Mg65Cu25Yw Mg65Cu35Ylo MgzoCu2oYio Mg75Culj 1Iio MgsoCuIoYIo Mg 70Ca10A120 Mg7oCa25AI5

Ni74Si8Bts FesoSiw Bw

ed40 Ni40P20 TisoNi25Cu25 Zr6oNi2oCu2o

* * *

* * * *

Ocrit (o)

* Calculated from the Jiang-Baram relationship [26].

289

Z. Rivlin et al. / Journal o f Non-Crystalline Solids 215 (1997) 2 8 3 - 2 9 2

80 &

70

iI . O

O

30

800 Expmmetal Glass Forn~ogRange

2.00

!

i

20

i

i

40

Mg

i

i

i

60

I

Expenmeatal

700

400

i

80

Ca [at% ]

i 5

0

100

I l0

MgToCs~o

Ca

15

20

25

AI [at%] (AI + Ca = 30%)

3O MsToAI3o

Fig. 2. Critical contact angle 0crit versus composition, superposed to the binary MgCa phase diagram.

Fig. 4. Critical contact angle 0crit versus composition, superposed to the ternary MgToCaxAl30_ x phase diagram.

calculated from the Jiang-Baram relationship [26]. Materials systems with 0 < 60 ° are shown in italics. Figs. 2 and 3 show the computed dependence of the critical contact angle with composition for the MgCa and CaA1 binary systems, superposed to the respective phase diagrams. The composition range

for which the critical contact angle values are < 60 ° is labeled the 'theoretical glass-forming range', which is to be compared with the composition range for which the particular system has been reported to be glassy, and this is labeled the 'experimental glass-

;7o 8O

11

JI

Theore~cal

8O

Thcote~cal

60

Y

~o

50 0

O

40

30 1200

8O0

Experimental 1000

I r, 400

t

0

AI

20

,40

60 Ca [at%l

i

i

~)

i

o lO0 Ca

Fig. 3. Critical contact angle 0crit versus composition, superposed to the binary A1Ca phase diagram.

M~sCazs

5

10

15

Ai [at%l (A! + Ca = 25%)

20

25 MDysAi2s

Fig. 5. Critical contact angle 0crit versus composition, superposed to the ternary Mg75CaxA125_x phase diagram.

Z Rivlin et aL / Journal of Non-Crystalline Solids 215 (1997) 283-292

290

I

9O Theoretical 70

50 800 m 700

Ir0o J

g 5oo 400

i

5 Mg,t sSiz$ (Si

10 15 Ca [at%] + C a = 2;%)

i

20

25 Mg7 5Caz$

Fig. 6. Critical contact angle 0crit versus composition, superposed to the ternary Mg75SixCa25_ x phase diagram.

forming range'. The MgCa and CaAI systems are glass-forming alloys in a relative wide compositional range. Conversely, the computation show that for the binary AIMg the critical contact angle is above 60 ° for all compositions. There is indeed no report on any A1Mg glass. The computed dependence of the critical contact angle with composition for ternary MgAICa and MgSiCa systems, also superposed to specific pseudo-binary phase diagrams is shown in Figs. 4-6. Figs. 4 and 5 are for the MgToCaxAl30_~ and Mg75Ca~A125_ x compositional ranges, respectively. Fig. 6 is for the Mg75SixCa25_ x compositional ranges.

5. Discussion

For glass formation to take place during rapid quenching, it is necessary that homogeneous and heterogeneous nucleation processes of any phase in competition with glass formation be avoided before the melt is cooled below Tg. The widely adopted kinetic treatment model, as originally proposed by Uhlmann [13], applies only to homogeneous nucleation, neglecting the possibility of eventual occur-

rence of heterogeneous nucleation. However, rapid cooling can be free from heterogeneous nucleation in only some specific cases, such as emulsified samples or containeriess solidification. Even when the processing conditions are arranged to produce maximum undercooling, most experiments suggest that solidification is still initiated by heterogeneous sites associated with the sample surface [27]. The present model treats the rapid cooling process in light of the both afore-mentioned conditions, by considering heterogeneous nucleation to be the dominant process during rapid cooling. In that case, the 'catalytic efficiency' F(O), a process parameter that depends on the wetting properties of the solidifying liquid and the quenching substrate plays now a decisive role. The 'thermo-kinematic' competition between nucleation and glass transition is primarily affected by the value of F(O), i.e., by the value of the contact angle 0. The formerly introduced concept of a critical contact angle 0crit enables the evaluation of the relative ease of glass-forming in binary and ternary alloys. The evaluated critical contact angle is a lower limit for the actual contact angle in order to induce glass formation (suppression of crystallization) in the considered alloy during rapid quenching by melt-spinning. As seen from Table 1 and Figs. 2-6 glass-forming alloys have lower values of 0crit. Accordingly, a criterion linking the glass-forming ability to the computed critical contact angle 0crit c a n be inferred (for melt-spinning under the imposed solidification conditions), as follows: easy glass-forming alloys (or compositions) are those for which Ocrit is < 60 °. Table 1 shows that the well known very easy glass-forming alloys, such as PdsaSi16 [28], MgssCu35Y10 [29,30], Mg65Ni20Nd15 [31], Mg77NilsNd5 [31], Pd40Ni40P2o [32] and Zr60Nie0Cu20 [33], have 0crit values lower than 40 °. Melt-spun Mg65Cu30Y5 ribbons have been produced lately, and shown to be glassy too [34]. A series of alloys listed (in italic) in Table 1 have critical contact angles ranging from 40 ° to 60°. All these alloys, either binaries or ternaries, have been reported to exist in the glassy structure. Well documented cases are the PdSi [29], the NiZr [35] and the MgCuY systems [29,30]. Conversely, the binaries NiA1, MgA1 and MgSi and to some extent even CaAI, have critical contact angles exceeding 60 °. The ternary

z. Rit,lin et aL / Journal of Non-Cr)..stalline Solids 215 (1997) 283-292

Mg-based systems that contain both A1 and Si have the highest computed critical angles, making that system a non-glass-forming system, together with MgCaSi. The physical reasons for such an alloying behavior is still a matter for research. The comparison between the expected and the reported compositional ranges for glass-forming in the binary MgCa [6], AICa [36] and MgA1 systems and in the ternary Mg70CaxAl30_x, Mg75CaxA125_, and Mg75Si~Ca25_ x systems, as depicted in Figs. 2-6, show fair correlations. The results shown in Fig. 2 seem to imply that the experimental glass-forming range for the MgCa system corresponds to a critical contact angle value less than 60 °. The discrepancy is however well within the error range of the numerical computations. Fig. 3 shows an almost perfect correspondence between the predicted and the experimentally observed compositional range for glass formation in the A1Ca system. Figs. 4 - 6 predict glass-forming ranges in the vicinity of the pseudo-binary eutectic compositions in the ternary Mg75CaxA125_ x, the Mg70CaxAI30_x and the Mg75SixCa25_x systems. Fig. 4 shows a very narrow theoretical glass-forming range in Mg70CaxA130_x, though the evaluation of the critical contact angle is well within the computational inaccuracy. There is however a report [37] on a glassy MgToCal0A120 composition, apparently in discordance with the present study. So far, we have no explanation for the observation that there is no glass-forming ability in the AIMg system over the entire composition range. It is interesting to note that a decreasing in Mg content from 75 to 70 at.% in the ternary MgCaAI system causes a narrowing of the theoretical glassforming range in the vicinity of the pseudo-binary eutectic composition (see Figs. 4 and 5). To the authors knowledge, there is no report 2 on glassy compositions in the Mg70CaxA13o x and the Mg75SixCa25_~ systems. An investigation of the relation between glass formation and alloy composition in these systems is currently in progress [39]. Considering the results presented so far, it seems that the present model indeed enables a fairly good estimate of the glass-forming ability in binary and

2 A melt-spunMg94AIsSiI alloy,quenchedat a coolingrate of 106°C s- i has been reportedto be crystalline[38].

291

ternary metallic systems, in terms of the critical contact angle 0crit , and can easily be extended to other compositions not treated in this paper. The model has however some shortcomings. The critical contact angle model assesses for the compositional dependence of the glass-forming ability, but cannot provide an estimate for a critical cooling rate, as the kinetic model did [13]. Moreover, the crystalline to amorphous transition which has been observed in melt spun ribbons [40,41], is also a kinetic problem that the present model does not account for. The variance in the computed critical contact angle is affected by the accuracy of the constituents thermodynamic and physical parameters and the correctness (and availability) of the relevant phase diagrams. Finally, accurate knowledge of the glass transition temperature and of the temperature dependence of viscosity are also necessary. The uncertainties related to these issues are discussed elsewhere [27] 3

6. Conclusions A numerical simulation can be used to evaluate the glass-forming ability of melt-spun binary or ternary metallic alloys, in terms of a critical contact angle 0rot derived from the liquid-solid wetting theory. The glass-forming compositions are those for which the critical contact angle 0crit is _< 60 °. When the actual contact angle is equal to or higher than the theoretical 0crit value, melt-spun ribbons have an amorphous structure. When it is lower than 0crit, the melt-spun ribbons have an crystalline structure. The smaller 0~rit, the higher is the glass-forming ability. The model can be used to predict the compositional range for the glass-forming ability of binary and ternary alloy systems.

Acknowledgements H.G.J. is grateful for a Post Doctorate fellowship at Ben-Gurion University. The highly valuable assis-

3 Surface roughness of the quenching substrate has been reported to affect wetting [42-44]. This issue has not been considered in the present paper.

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tance of Professor N. Fraga, Materials Engineering Department, Ben-Gurion University, in producing several missing literature magnesium-based ternary phase diagrams by using the ThermoCalc software, has been a decisive contribution to this research.

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