On the relation between surface tension and viscosity for liquid metals

Scripta METALLURGICA et MATERIALIA

Vol.

28, pp. 1273-1276, 1993 Printed in the U.S.A.

Pergamon Press Ltd. All rights reserved

ON THE RELATION BETWEEN SURFACE TENSION AND VISCOSITY FOR LIQUID METALS

Ivan Egry Institute for Space Simulation, German Aerospace Research Establishment 5000 K61n 90, Germany (Received January 28, 1993) (Revised March 8, 1993) Introduction With the advent of powerful mathematical modelling techniques for metallurgical phenomena r l ] , there is renewed interest in reliable data on the physical properties of liquid metals. Presently, the knowledge and accuracy of these data are the limiting factor in the models. Surface tension and viscosity are two such important quantities. Unfortunately, both their measurement, and their calculation, are rather difficult. Therefore empirical relations are a useful tool to assess the available data. In a recent paper [2], we derived a relation between surface tension and viscosity, based on two well known expressions due to Fowler I-3,] and Born I-4-] respectively. This relation makes use of the fact that both, surface tension and viscosity, can be expressed as integrals over the product of interatomic forces and the pair distribution function. We have checked its validity using available data at the melting point and found agreement to within 10-20%. In this paper we extend this analysis to a finite temperature range of approximately 300 K by using data available in the literature. Results Fowler derived a useful formula [3] for the surface tension ~' of a fluid, approximating the smooth density profile n(z) by a step function and replacing the gas by vacuum. It reads:

Y -

n 2 j'0~°

8

dR

R4 d~(R)

d--"-'R~ g(R)

(1)

Here, n is the particle number density, q~the pair potential and g(R) the pair correlation function. All these quantities depend on the temperature which gives rise to the temperature dependence of ~'. Very similarly, Born and Green 1-4,] have derived an expression for the viscosity r/of a fluid using kinetic theory. They found

,1 =

27: n 2

15

I?

dR

R4 dq~(R)

d~

g(R)

(2)

where m is the atomic mass, k is Boltzmann's constant and T is the temperature. Both of these equations essentially use a hard sphere model: Fowler's formula neglects the realistic density profile of finite thickness at the surface, and the Born-Green formula does not contain contributions from the soft, long-range part of the potential I-5,]. This seems to restrict their applicability to simple systems with shorf-range, isotropic binding, such as liquid metals.

1273 0956-716X/93 $6.00 + .00 Copyright (c) 1993 Pergamon Press Ltd.

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SURFACE TENSION AND VISCOSITY

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It should also be noted that both the Fowler formula and the Born-Green formula are derived for pure elements and are not valid for alloys. Although one may think of an effective medium approximation, such an approximation can clearly not account for surface segregation effects. In addition, three particle correlations are neglected in equations (1) and (2). Therefore, we expect that they hold only at moderate undercoolings and break down near the glass transition, where these correlations become essential. A striking feature of these two expressions is the fact that the two integrals are equal. Therefore, by forming y/t/, the integral cancels and we obtain the following simple relation: Y r/

-

15 16

~/kT m

(3)

This relation has been found by several authors [6, 7, 8, 2]. In Turkdogan's derivation [8], the factor 15116 (=0.938) is replaced by (3/5). 1.485 ( = 0.891). As pointed out in the introduction, we have checked this relation in an earlier publication [2] for the transition metals Fe, Co, Ni and the noble metals Cu, Ag, Au at the melting point. Surface tension data were taken from Nogi et al [9] except for gold, where the value measured by Keene [10] was used. The data are given as the surface tension value at the melting point ~'(Tm), and the slope - dy/dT at T,,. These authors have measured the surface tension by the levitated drop technique, which seems the most reliable technique at these high temperatures, and is the only one working in the undercooled regime. Viscosity data were taken from the recent compilation by Chhabra and Sheth [11]. In this contribution, we use the same input sources; however, we have taken the viscosity data points from the original publications [12, 13, 14] rather than from Chhabra and Sheth's list, which displays viscosities in a parametrized form assuming an Arrhenius-type temperature dependence. For convenience, the input data are listed in tables 1 and 2. TABLE 1 Surface tension and viscosity of noble metals T [K]

1273

Cu, T,, = 1356,

Tm =

I 1.37 3.9 1234,

I

r/ [mPas]

I 3.66

7 IN/m] r/ [mPas]

I

11573

I 1.34 J 1.31 3.19

2.86

Mr = 107.87

y IN/m]

0.917

Au, T,, = 1336,

11473

Mr = 63.54

7 [N/m] r/ [mPas] Ag,

1373

0.898 3.19

I 0.879 2.83

I 0.86 2.59

Mr = 196.97 1.12

1.10

1.09

5.13

4.64

4.24

We present the results in figures 1 and 2. According to (3), the points should lie on a straight line indicated by "prediction". In order to display all metals on one graph, we have actually plotted (7/~/)~/(Mr) , where Mr is the relative atomic mass. Therefore, the theoretically predicted slope of this expression as function of ~ - - is given by

~ ~/

_ 15 ~ 16

j~-

(4)

where M, is the absolute atomic mass. Figure 1 shows the measured 7/t/ ratio for the noble metals Cu, Ag, Au as a function of ~ - , while Figure 2 shows the results for the transition metals Fe, Co, Ni.

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SURFACE TENSION AND VISCOSITY

IN LIQUID METALS

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TABLE 2 Surface tension and viscosity of selected transition metals T [K]

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Fe, Tm = 1809,

1723

ri [mPas] Co, Tm = 1768,

M r

=

11873

I 1923

11973

12023

2.02

2.00

1.97

5.7

5.3

4.9

4.5

4.3

4.1

3.9

58.93

I

2.02

ri [mPas] I Ni, Tm = 1726,

I 1823

M, = 55.85

~' IN/m]

[N/m]

1773

4.5 M, = 58.71

2.00

1.97

1.95

4.1

3.9

3.6

1.92 3.4

I 1.90 3.2

I 1.88 3.0

1.78 4.2

1.76 4.0

I 1.74 3.8

I

[N/m~

1.87

1.85

1.83

1.80

ri [mPas]

5.4

5.1

4.8

4.5 Discussion

The striking feature of the results is that the experimental data seem to obey the linear relationship between -•/r/ and q/T- very well. However, the slope differs from the one predicted by theory, and the agreement gets worse at lower temperatures, as expected. This linear relationship is particularly surprising, because it is not compatible with the generally accepted formulae for the temperature dependence of surface tension and viscosity, namely: d~

=

"~(Tm) + - - ~

=

rio e AIRT

(T -

(5)

Tin)

and

ri(T)

(6)

Note, however, that equation (5) contains only the first term in a Taylor expansion and does not necessarily imply a strict linear temperature dependence over a finite temperature range. On the other hand, our findings could also be taken as an argument to support recent mode-coupling theories which predict a power law behaviour of the viscosity at moderate undercoolings [15]. As far as the absolute accuracy is concerned, the agreement is satisfactory, i.e. it is within 20 % over the entire temperature range. A better agreement would be purely fortuitous, since this is about the accuracy of the input data. It should be also mentioned that a major problem in using published data for checking a given relation, is the unbiased choice of the input data. As is well known, r5], data on surface tension and, in particular, viscosity, are widely scattered and differ sometimes by more than 100%. We have tried to avoid a biased choice by using just three sets of well accepted measured data, one for the surface tension (Nogi), one for the viscosities of transition metals (Cavalier) and one for the viscosities of noble metals (Gebhardt). In summary, relation (3) should provide a useful estimate in cases where either surface tension or viscosity data are not available. Acknowledgement The author is grateful to K. Mills and T. lida for pointing out additional references regarding the derivation of equation (3). Part of this work was carried out while the author was with K. Mills at NPL, UK. He enjoyed his kind hospitality and wants to thank him and his group.

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SURFACE TENSION AND VISCOSITY IN LIQUID METALS

I

I

I

Vol. 28, No. i0

"Y E N / m ] i / M r

I

~ [mPoS]

3.5

2.5 2 35

I 36

I 37

I 38

I 39

40

~/T[KI Figure 1.

Surface tension over viscosity for the noble metals Cu, Ag, Au as a function of 5

,

,

,

,

YrN/ml

~ [m PoS] 4--

Fe

2

!

I

I

I

41

42

43

44

45

/TEK] Figure 2.

Surface tension over viscosity for the transition metals Fe, Co, NI as a function of

References

[1] [2l [3l [4,1 [5,1 [6] [7]

[8,1 [9] [10,1 [11] [12,1 [13] [14] [15]

M.I. Baskes Modelling and Simulation in Materials Science and Engineering first issue, October 1992 I. Egry Scripta metallurgica et materlalia 26, (1992), 1349 R.H. Fowler Proc.Roy.Soc. A159, (1937), 229 M. Born, H.S. Green Proc.Roy.Soc. A190, (1947), 455 T. lida, and R.I.L Guthrie, The Physica/Properties of Liquid Metals, Clarendon Press, Oxford, 1988 M. Shimoji, Adv. Phys., 16, (1967), 705 Y. Waseda, M. Ohtani, Tetsu-To-Hagan~, 61, (1975), 46 (in Japanese) E.T. Turkdogan, Physical Chemistry of High Temperature Technology, Academic Press, New York, 1980 K. Nogi, K. Ogino, A. McLean, W.A. Miller Metallurgical Transactions B 17, (1986), 163 B.J. Keene NPL Internal Report DMA(A) 56, (1982) R.P. Chhabra, D.K. Sheth Z. Metallkunde 81, (1990), 264 G. Cavalier Comptes Rendu 256, (1963), 1308 E. Gebhardt, M. Becker Z. Metallkde. 42, (1951), 111 E. Gebhardt, G. WOrwag Z. Metallkde. 42, (1951), 358 E. ROssler Phys. Rev. Letts. 65, (1990), 1595