Journal of Non-Crystalline Solids 143 (1992) 140-146 North-Holland

]OURNA L O F

NON-CRYSTALLINE SOLIDS

Estimation of viscosity in undercooled liquid metal alloys

Fabio Miani and Paolo Matteazzi Istituto di Chimica Universita' di Udine, Ha Cotonificio 108, 1-33100 Udine, Italy

Received 30 August 1991 Revised manuscript received 29 November 1991

A simple procedure for calculating the viscosity of undercooled liquid metals exploiting available experimental data is presented. In the case of alloys at eutectic composition an empirical correlation may give approximate values in case viscosity data are not available. The values of the Vogel-Fulcher equation presently used in literature are discussed and some simple estimates proposed.

1. Introduction

The problem of having reliable data for the determination of the viscosity (or the diffusivity related by the Stokes-Einstein equation) of highly undercooled liquids has been a serious limitation to the kinetic approach for evaluating the glass- forming ability in metallic liquids since the very beginning [1]. Steps have been done but in a non-systematic way; we hope that a straightfor- ward procedure for an order of magnitude esti- mate of this critical parameter will be useful for researchers working in the field of ultra-rapid quench from a liquid, particularly in laser glazing of metallic materials [2].

Present calculations are founded on the 'rather shadowy' [3] free volume concept; it is not the aim of this paper to discuss such theories [4,5]. It might be useful to accept the fundamental as- sumption of the theory but to avoid the indirect steps of the original procedure: materials scien- tists cannot directly check the physical quantities involved in such an approach.

The following lists the symbols that will be used hereafter: r/: viscosity Tin: melting or liquids temperature Tr: reduced temperature T / T m

Na: Avogadro number Vmol: molar volume

R: universal constant of gases To: ideal glass temperature A: atomic or molecular weight Units used: S.I. system.

Presently we know that a Vogel-Fulcher be- haviour [6],

r /= F exp( T_~OT0 ), (1)

is appropriate for undercooled liquids. These em- pirical parameters are usually obtained from re- laxation analysis of metallic glasses and high-tem- perature behaviour of liquid and extrapolated for undercooled liquids. On the other hand available data on viscosity above the melting temperature fit an Arrhenius-type equation. The Arrhenius fit is usually expressed as

r /= r/0 exp ~-~ . (2)

Viscosity data for elements are available in the literature. The Andrade model is the most suc- cessful model able to fit them with errors compa- rable to experimental uncertainties [7].

The Andrade semi-empirical model relates the viscosity at T m with the characteristic frequency of the liquid; this frequency is taken to be of the same order of magnitude as that of the solid. In such a way a close correlation with the Debye

0022-3093/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved

F. Miani, P. Matteazzi / Viscosity in undercooled liquid metal alloys 141

Table 1 Empirical correlations for the dependence of the activation energy for viscous flow in eq. (2) and the temperature T m of melting (dev. ind. = % / Ymean)

Ref. Fit Expression Comment

[10] log10 E (kcal) 1.348 log10 T m -2 .366 [27] log10 E (kcal) 1.36 log10 T m -3 .418 [7] E (cal) 1.21T 1"2

E (cal) 0.75Tin 12 [6] E (J) (2 .4+ 1.0)RT m

E (J) (2.4 + 0.8)RT m E (J) (2.4 + 1.0)RT m E (J) 3.5RT m

This work E (kJ) 23.97 × 10-3Tm This work E (kJ) - 11.0+35.2X 10-3Tm

Lantanides, actinides Sb Bi excl. 'Normal ' metals Semimetals Pure metals Intermetallic compounds 'Normal eutectics' 'Deep eutectics' Pure metals, dev. ind. = 270 × 10 +3 Eutectics, dev. ind. = 3 4 0 x 10 -3 Si, NiPd alloys excluded

characteristic temperature is obtained; further, this may be expressed using Lindemann's approx- imation. At last one can obtain a dependence on the atomic weight and the liquid molar volume at Tm:

C A ~ n(rm) v2/3 (3)

v tool

The Andrade coefficient C A is quite constant for pure metals melts and has a mean value [6] of

1.8 × 10 -7 ( J / K mo11/3). It is surprising that in the review by Wilson [8] in 1965 this correlation was not quoted even though it had been present in papers such as the one [9] of Bondi, dealing with a different field of materials science. For the same class of materials, several empirical rela- tionships have been proposed for the tempera- ture dependence of the activation energy for vis- cous flow in eq. (2) starting from Grosse [10] in 1961 up to Shimoji [11] and Battezzati [6]. We can see some quantitative data in table 1. While for

LJJ

60

40-

20-

• mmm m

• • mm mm •

40,

20-

• mm m

,oo 660 860 lo'oo 12'oo 14oo doo 8oo B6o Io'oo 12'oo 14oo Tm K Tm K

Fig. 1. (a) Linear regression for the activation energy for viscous flow; E ( k J ) = 2 3 . 9 7 × 1 0 - 3 T m for pure metals. (b) Linear regression for the activation energy for viscous flow for eutectic alloys; data from ref. [6] excluding Si- and NiPd-based alloys;

E(kJ) = - 11 .0+35 .2× 10-3Tin.

142 F. Miani, P. Matteazzi / Viscosity in undercooled liquid metal alloys

melts with compound-forming composition the model still holds well [6], at eutectic composition there is a wider range of variation, still good enough to give order-of-magnitude estimates.

For eutectic compositions we propose to fol- low a least-square procedure for a correlation between the melting temperature and the activa- tion energy for viscous flow, as has been done for pure metallic liquids (fig. l(a)). This procedure, applied to classes of liquid metallic alloys, may give acceptable results as we can see in fig. l(b). Here we exclude Si- and PdNi-based alloys from eutectics, taking data from ref. [6]. In table 1 we consider also the scatter from the experimental data; we express the index of the deviation from the linear fit as the standard deviation of the regression normalized by the mean of the data considered: dev. ind. = O'y/Ymean.

In the case of pure metals, excluding liquid semiconductors, we found such deviation index to be of the same order as that for eutectic alloys.

2. A simple procedure for an estimation of the viscosity of undercooled metallic mel t s

Battezzati and Greer [6] proposed to 'link' the Arrhenius and Vogel-Fulcher interpretations in a way that such data are required only for equi- librium conditions above T m. It is possible there- fore to calculate the viscosity at T m and its derivative from the Arrhenius model; moreover at Tg, the glass transition temperature, r/ is gen- erally assumed as 1012 Pa s. In the same paper [6] in which rates interesting for rapid quenching applications were given, it was proposed to use lower viscosities (from 109 to 1011 Pas).

To obtain the parameters for the Vogel- Fulcher equation we solved a system of three non-linear equations [2]: (o) ~(Tm) = F exp T m T T ° , (4)

o

(rm -- r0)

-r/(Tg) = 1010 Pas.

(5)

(6)

This leads to T o , i.e. the ideal glass temperature,

770= [Z 2 - z g r m - Tg ln(71(Tm)/ ' r l (rg))

X TI( Tm) / ( O~q /OT)Tm]

/ [ T m - Tg - r l ( T m ) / ( O r l / O T ) r m

× (rg))], (7) from which also 0 and F are calculated in such a way:

(~)~/3T) rm (Zm - T0) 2 ( 8 )

(rm)

and

F = n(rm)

e x p [ O / ( T m _ To)] . (9)

This analytical relationship is explainable in terms of slopes of the two representations of viscosity. If plotted in a log(3?) vs. 1 0 3 / T graph, the plot may bring, as we will see, useful information. In the same previous work [2], we exploited this approach to calculate the fraction of volume crys- tallized in a laser glazing treatment of a Ni60Nb40 alloy. Equation (7) requires an estimation for Tg or conversely, of T 0. In section 5 we will see that it is possible to have an estimate of it exploiting heat capacity measurements above T~.

3. Calculation versus experimental data

We compared calculated properties of the glass-forming alloy, Au78G%4Si s, one of the few alloys on which experimental work has been per- formed, measuring viscosities of the liquid [12] and the corresponding metallic glass [13] (fig. 2). As we can see it is not possible to accept the Arrhenius behaviour in the undercooled r e g i m e (T m = 625 K). To do so would lead to a viscosity at Tg (295 K) 10 orders of magnitude less than the actual one. In the same figure the extrapo- lated viscosity from the one above T m and eqs. (7)-(9) are plotted. This alloy, being one of the easiest glass formers, it is not possible to use the correlation cited in table 1 for eutectic alloys. We note that excluding Si- and NiPd-based alloys

F. Miani, P. Matteazzi / Viscosity in undercooled liquid metal alloys 143

12

B 10

8

0. 6

~ 2

0

-2 A

- 4 J ~ i

295 395 495 595 695 Temperature K

Fig. 2. Alloy Au78Ge14Si 8. Logarithmic plot of viscosity vs. temperature: (A) Arrhenius behaviour, (B) according to Polk [12], and (C) calculated Vogel-Fulcher extrapolation from

viscosity above T m [12].

leads to a dispersion of the data taken f rom ref. [6] that is significantly less than for the one ex- cluding me ta l -me ta l lo id alloys.

Consider ing the extreme simplicity of the hy- pothesis, the fit is useful when no o ther experi- menta l data are available for a glass-forming alloy

Table 2 Values of To, O and log F for alloy AuysGex4Si s from literature data and as calculated from eqs. (7)-(9). (a) using experimental [12] Tg=295 K, ~7(Tm) , EExp=31000 J; (b) using experimental Tg, ~7[Tm]; simulated E = 3RT m = 15 600 J; (c) using experimental Tg, ~7(Tm); simulated E = 9RTm= 46800 J; (d) T o value calculated from eq. (20) and heat-capac- ity data in ref. [28]; (e), (f) experimental values for the glass form; (g), (h) values Calculated with the free volume theory

T o (K) 0 (K) log F (Pas)

(a) 253 1278 - 7.46 (b) 274 590 - 5.70 (c) 231 2310 - 9.79 (d) 265 (e) [12] 266 1186 - 7.31 (f) [28] 241 1360 - (g) [3] 193 - - (h) [25] 198 - -

and for which an estimate of viscosity in the undercoo led regime, for example in building T T T curves, is needed. In table 2 we can see a compar- ison be tween calculated and experimental values of To, F and O. Quo ted values f rom ref. [13] refer to the glass form. In our opinion the only pa ramete r that should not be expected to vary in the l iquid-glass transit ion is To, which has a direct physical meaning, being the ideal glass transit ion tempera ture . Moreover we might say that [6] for marginal glass formers, with a lower value of Trg = Tg/Tm, the model should work more satisfactorily. In fact for those alloys that show less association in the liquid state the corre- lation for E = 3 R T m proves to work better.

4. D i scus s ion on the value o f T O

This simple calculation seems to fit best To, the ideal glass transit ion tempera ture . This tem- pera ture is also defined as the K a u z m a n n tem- pera ture or the iso-entropic t empera tu re for a material. It is the limiting t empera tu re at which an undercoo led liquid can exist. Below this tem- perature , under any circumstance, the under- cooled liquid collapses to a f rozen glass.

Let us consider a certain value, K, for the order o f magni tude of r l (Tg) /V(Tm) . This value is impor tant because Tg is defined in an isoviscous way [6,13]. However the glass transit ion is a ki- netic process and the viscosity during quenching f rom the liquid could be some order of magni- tude less at Tg. K will vary f rom the highest values, 12, of pure metallic liquids to 9 for such alloys as Pdv8Cu68i16. In particular, assuming the correlation, E = 2.4RTrn, holds for melts with pure and intermetall ic composi t ion [6] and ap- proximating this proport ional i ty with ln(10), then f rom eq. (7)

( K + 1 ) T ~ g ) - 1

T~r°) = ( K - 1) + T(rg ) (10)

This relation does not only mat te r theoretically, since recently some pure metals have been ob- tained f rom a liquid quench [15].

144 F. Miani, P. Matteazzi / Viscosity in undercooled liquid metal alloys

The series expansion

T(ro) 1 K 2 - 2

( K + 1~ + ( K - 1) 2Trg

( K 2 - 2) ( X - 1) 3 Tr2"'" (11)

truncated to the first term is very close to the value (K = 12 for pure metals): Tr0 = Trg - 0.0793 quoted in ref. [15].

For the reduced value of O it seems that the usual values assumed for O r = 3.3 [15,17] might be upgraded as

O r = 3 (K2(1 -- r(rg,)2) (12)

( ( K - 1) + T(rg)) 2"

5. Non- l inear express ion of the free v o l u m e

According to several authors (for example refs. [12,18]) it is not possible to fit the Vogel-Fulcher equation over the entire interval T m to Tg. Such a fit would correspond to a more complicated rela- tionship between the free volume, ft, and the temperature. We recall that the empirical Doolit- tle equation [4] for viscosity should account for the temperature dependence over the entire in- terval,

~1 =A exp( B/ft). (13)

We are not interested in developping a discussion about this value but in reviewing some approxi- mations that have been carried out to date:

ft 1 + (14) ~mm = T-m~T0

(where fm is the relative free volume at T m) comes from a reorganization of the entropy dif- ference, AS = Smelt- Ssolid, according, for exam- ple, to in ref. [19].

Whereas, Ramachandrarao et al. [4,5] pro- posed

ft _ 1 + sin (15) -/

Instead of calculating the free energy of hole formation and then using the series of hypothesis outlined in refs. [4,5] to evaluate the parameters A, B, and T O in eqs. (13) and (15) and then link them to viscosity data, we propose a more direct approach which may give directly the results one needs.

The Doolittle equation leads to a set of equa- tions very close to eqs. (5)-(7):

"q(Tm) = A exp , (16)

(a~/) (Of~aT)fro = - B "q ( rm ) , (17)

V fT2m and

~7(Tg) = 1012 Pas. (18)

If we then accept, eq. (15), we can calculate T o solving the non-linear equation

T~-r0 "rr( r/( rm ) ]

- in 2 ,(Tg) J

[w ( T g - T r n ) ) 1 + s i n / . . . .

'2 ( Tm - To ) (rl(Tm) )

s'nt (19)

which may be solved numerically. Now if one can give a value for T o with a

different procedure, it is then possible to give directly an estimate for Tg. This is done solving the equation as in ref. [14]:

Y J T m r O q - ( A C p mk-'-~JTm ) -T-'m~)

+ AS(Tm) - Tm( 0ACp 1 ~-~T-~] T m = O, (20)

where A stands for the difference of the physical values between the solid and the liquid. In fig. 3 we can see a comparison of ft/fm values, under different hypotheses. In the case of alloy Auv8Gex4Si8, by means of eqs. (8), (9), (10) and

F. Miani, P. Matteazzi / Viscosity in undercooled liquid metal alloys 145

(27), the data of Chen and Turnbull reported in ref. [13] and those of Chen and Turnbull [12], we estimate a value for Tg of 273 K to be compared with the value 295 K quoted in the literature [12].

Again in table 2 for the same alloy we note that T o values obtained by viscosity or heat ca- pacity extrapolation are more consistent with ex- perimental determinations than are free-volume calculations, even in the same that the activation energy for viscous flow is set fictitiously to the extreme values of its range.

6. Viscosity data in the kinetic approach of glass formation

A first major step in the critical assessment of the possibilities of the kinetic approach was es- tablished by Uhlmann" [20] twenty years ago. Davies [1,21] exploited such an approach to draw some estimates for glass formation in metal alloy systems. Even if some estimates were perhaps too optimistic, the discussion still holds. There [21] the critical cooling rate was estimated to be pro-

1

o.9.

E o.8

0.7

~ 0.6

~ 0.5

~ 0.4

_: 0.3

"~ o.2 ~ - - - ~ c 0.1-

% oso Temperature K

Fig. 3. Alloy Au7sGe14Si s. Relative free-volume calculated under (A) linear hypothesis according to, eq. (8), and (B) and (C) non-linear hypothesis, according to eq. (15) and ref. [4],

respectively.

-2

~'-3

o .4 .

-5-

-6 350 460 450 s60

Temperature K

Fig. 4. Alloy Au78Ge~4Si s. Behaviour of the TTT curves (for a crystallized fraction X~ = 10 - 6 ) for simulated variations of the activation energy for viscous flow; (A) E = 3RTm,

(B) experimental value E = 6RTm, (C) E = 9RT m.

portional to the reciprocal of the value viscosity; uncertainties were estimated to _+ 1 order of mag- nitude. We may say that closer limits in the estimates for viscosity should reduce the uncer- tainties. In fig. 4 we compare the shift in the T I T curves (in the case of homogenous nucleation) for AuysGe14Si 8 alloy for different values of the acti- vation energy. It is very important to note that the values are extreme variations of the activation energy for viscous flow. Estimated values may easily be included in a smaller range giving a reliable tool to model nucleation and therefore critical cooling rates for producing the amor- phous state in metallic alloys.

For eutectic alloys, we suggest that when it is not possible to have data in the high temperature range either, then the Andrade model should be used in conjunction with an estimate of the acti- vation energy for viscous flow using, if possible, extrapolations from data for structurally similar alloys. Once a proper order of magnitude for viscosity and its temperature dependence have been set, the error in evaluating physical pro- cesses depending on this parameter should be acceptable.

146 F. Miani, P. Matteazzi / ~scosity in undercooled liquid metal alloys

7. Conclusions

The estimation of the value of viscosities in undercooled metallic liquids on a theoretical or semi-empirical basis is still not settled. As pointed out in refs. [7,11], much more experimental work is needed. We propose, in a direct way, a tool to model important processes such as nucleation from a liquid [17]. In this case, the model is directly usable in contrast to free-volume theories [4,5,25,26] notwithstanding their more thorough interpretation of metastability of metallic liquids.

References

[1] H.A. Davies, J. Aucote and J.B. Hull, Scr. Metall. 8 (1974) 1179.

[2] P. Matteazzi, F. Miani, A.M. Lashin and I.Yu. Smurov, in: Proc. 3rd Eur. Conf. on Laser Treatment of Materials (Sprechsaal Publishing Group, Coburg, 1990) pp. 927.

[3] R.W. Cahn, in: Physical Metallurgy, 3rd Ed., ed. R.W. Cahn and P. Haasen (Elsevier, Amsterdam, 1983) ch. 28, p. 1815.

[4] P. Ramachandrarao, B. Cantor and R.W. Cahn, J. Non- Cryst. Solids 24 (1977) 109.

[5] P. Ramachandrarao, B. Cantor and R.W. Cahn, Mater. Sci. 12 (1977) 2488.

[6] L. Battezzati and A.L. Greer, Acta Metall. 37 7 (1989) 1791.

[7] T. Iida and R.I.L. Guthrie, The Physical Properties of Liquid Metals (Clarendon, Oxford, 1988) ch. 6.

[8] J.R. Wilson, Metall. Rev. 10 (40) (1965) 381. [9] A. Bondi, Theories of viscosity, in: Rheology: Theory and

Applications, Vol. 1 (Academic Press, New York, 1952) 321.

[10] A.V. Grosse, J. Inorg. Nucl. Chem. 23 (1961) 333. [11] M. Shimoji and T. Itami, Atomic Transport in Liquid

Metals (Trans. Techn., Aedermannsdorf, 1986) pp. 189- 206.

[12] D.E. Polk and D. Turnbull, Acta Metall. 20, (1972) 493. [13] H.S. Chen and K.A. Jackson, in: Treatise on Materials

Science and Technology, 7, Vol. 20 (Academic Press, New York, 1981) p. 215.

[14] P. Ramachandrarao, K.S. Dubey and S.L. Lele, Acta Metall. 37 (1989) 2795.

[15] Y.W. Kim, Hong-Ming Lin and T.F. Kelly, Acta Metall. 37 (1989) 247.

[16] A. Van Den Beukel and J. Sietsma, Acta Metall. 38 (1990) 383.

[17] D.R. Uhlmann and H. Yinnon, in: Glass: Science and Technology, Vol. 1 (Academic Press, 1983) ch. 1, pp. 1-47.

[18] P. Chaudhari, F. Spaepen and P.J. Steihardt, in: Glassy Metals II, ed. H. Beck and H.J. Guntherodt (Springer, Berlin, 1983) ch. 5, pp. 127-168.

[19] I. Gutzow, D. Kashchiev and I. Avramov, J. Non-Cryst. Solids 73 (1985) 477.

[20] D.R. Uhlmann, J. Non-Cryst. Solids 7 (1972) 337. [21] H.A. Davies and B.G. Lewis, Scr. Metall. 9 (1975) 1107. [22] W. Kurz and D.J. Fisher, Fundamentals of Solidification,

3rd Ed. (Trans. Techn., Aedermannsdorf, 1989). [23] B. Feuerbacher, Mater. Sci. Rep. 4 (1989) 1. [24] A.L. Greet, J. Less-Common Met. 145 (1988) 131. [25] K.S. Dubey and P. Ramachandrarao, Acta Metall. 32

(1984) 91. [26] J. Wachter and F. Sommer, J. Non-Cryst. Solids 117&118

(1990) 890. [27] E.T. Turkdogan, Physical Chemistry of High Tempera-

ture Technology (Academic Press, New York, London, 1980).

[28] H.S. Chen, Rheology and Relaxation in Metal Glasses, in: Glass: Science and Technology, Vol. 3 (Academic Press, New York, London, 1986).