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The Viscosity of Liquid-Metals and Alloys
Article in Acta Metallurgica · July 1989
DOI: 10.1016/0001-6160(89)90064-3
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Arm metall. Vol. 37, No. 7, pp. 1791-1802,
Printed in Great Britain. All rights reserved
1989
THE VISCOSITY
Copyright
OF LIQUID
L. BAT’TEZZATI’
c
OOOl-6160/89 $3.00 f0.00
1989 Pergamon Press plc
METALS AND ALLOYS
and A. L. GREER’
di Torino, Dipartimento di Chimica Inorganica, Chimica Fisica e Chimica dei Materiali, Via
Pietro Giuria 9, 10125 Torino, Italy and %niversity of Cambridge, Department of Materials Science and
Metallurgy, Pembroke Street, Cambridge CB2 3QZ, England
‘UniversitB
(Received 15 December 1988)
Abstract-A
large body of published data is presented on the viscosity of alloy liquids at compositions
corresponding to elements, intermetallic compounds and eutectics. In all these cases, with the exception
of liquids showing strong association and in particular of good glass-formers, it is found: that the melting
point viscosity is well matched by the Andrade formula; and that the temperature-dependence of the
viscosity above the melting point is Arrhenius, with approximately constant pre-exponential viscosity, and
with activation energy proportional to the melting point. For glass-forming systems the temperaturedependence of the viscosity is best fitted by the Vogel-Fulcher-Tammann
equation, and a procedure is
outlined for estimating the parameters in the equation. Some qualitative correlations are demonstrated
for the composition-dependence of isothermal viscosity in binary alloys.
prtsentons un vaste ensemble de rCsultats publies sur la visconsitk d’alliages liquides dont
les compositions correspondent i des Clkments, $ des composts interm&talliques ou i des eutectiques. Dans
tous ces cas (i I’exception des liquides qui prCsentent une forte association, en particulier ceux qui forment
de bons verres), on montre que la viscositk au point de fusion correspond bien B la formule d’Andrade,
et que l’influence de la temptrature sur la viscositt, au dessus du point de fusion, suit une loi d’Arrh&nius,
avec une viscositk pri-exponentielle a peu prts constante, et une knergie d’activation proportionnelle au
point de fusion. Pour les systemes qui forment des verres, l’influence de la tempirature sur la viscositt
est reprtsentCe dans les meilleures conditions par l’equation de Vogel, Fulcher et Tammann, et nous
indiquons comment estimer les parametres de cette Cquation. Nous dtmontrons quelques corr6Jations
qualitatives pour l’influence de la composition sur la viscositk isothenne dans des alliages binaires.
Rhm~Nous
Fiille von vertiffentlichten Ergebnissen zur Viskositlt von Metallschmelzen mit
Zusammensetzungen entsprechend den Elementen, intermetallischen Verbindungen und Eutektika wird
zusammengestellt. In allen Flllen, ausgenommen Schmelzen mit starker Assoziation und insbesondere die
von guten Glasbildnern, ergibt sich, dal3 die Viskositlt am Schmelzpunkt gut mit der Andrade-Formel
wiedergegeben werden kann, AuBerdem entspricht die Temperaturabhgngigkeit der ViskositIt oberhalb
des Schmelzpunktes einer Arrheniusbeziehung,
mit einer Aktivierungsenergie proportional zum
Schmelzpunkt. Bei den glasbildenden Systemen wird die Temperaturabh&igkeit
am besten durch die
Vogel-Fulcher-Tammann-Gleichung
beschrieben; hierzu wird ein Verfahren zur Abschltzung der
Parameters in dieser Gleichung angegeben. Einige qualitative Korrelationen werden fiir die Abhlngigkeit
der isothermen Viskositlt bingrer Legierungen von der Zusammensetzung aufgezeigt.
Zusammenfassung-Eine
1. INTRODUCTION
The viscosity of liquid metals has been investigated
both experimentally
and theoretically
for many
decades, and there is an extensive literature on the
subject [l, 21. Recently there has been renewed interest in viscosity in connection with glass-formation
in
alloy systems. Glass-formation
is promoted
by a
rapid increase of viscosity on undercooling [3], and
such an effect in particular composition
ranges may
be necessary, in addition to thermodynamic
effects, to
explain the variation of glass-formability
in alloys.
(For example, in Cu-Ti good glass formability
is
found at the composition of the CuTi compound [4].)
It seems appropriate,
therefore, to survey the available data on the viscosity of alloy liquids and to
re-examine earlier analyses of it [5]. It is useful that,
in addition to the very dispersed earlier data, a large
collection of references on atomic transport in liquid
metals has recently appeared [6]. In this paper we
check some empirical correlations for the viscosity of
pure liquid elements, and extend them to the cases of
intermetallic
compounds
and eutectics. We discuss,
further, the available formulae attempting to describe
the viscosity as a function of composition
and temperature, with particular attention to glass-forming
alloys.
2. MELTING POINT VISCOSITY
2.1. The Andrade formula
The viscosity of pure liquids does show correlations with other quantities. For example, the melting
point viscosity of the elements varies periodically with
atomic number, in the same way as the melting
temperature and the surface energy, and oppositely to
the molar volume [5,7,8]. Such correlations
are not
useful for a quantitative prediction of viscosity, however. By far the most successful quantitative correla1791
1792
BATTEZZATI and GREER:
VISCOSITY OF LIQUID METALS
tion, also for the viscosity at the melting point, is that
of Andrade [9]. He pointed out that the near-equality
of the specific heats of solid and liquid at the melting
point suggests similar atomic vibration in each. Andrade assumed that the characteristic vibration frequency in the liquid vL would, at the melting point
T,,,, be equal to that in the solid vs. He derived an
expression for the shear viscosity q by considering
that momentum would be transferred from one layer
of fluid to another with different velocity, not by
transfer of atoms as in gases, but by contact between
atoms in neighbouring layers as a result of vibrational
displacement from their mean positions
(1)
(1.80 5 0.4) x lo-’ (J/K moli~3)ii2. (The ranges are
&-1 SD.) The values of C, for chromium, manganese
and the refractory metals are highly uncertain, but
are nevertheless included in the calculation of the
average C,. The small value of the standard deviation of C, shows that for pure metals equation (3) is
an effective correlation. It should be noted that for
pure metals q (T,,,) itself varies by more than an order
of magnitude.
If it should happen that a value is available for the
atomic diffusivity in the liquid at T,, then this can be
used to estimate q(T,) as an alternative to the use of
the Andrade formula. The Stokes-Einstein relation
kT
’ = 3xaD
In this expression m is the molecular mass and e the
average inte~ol~ular
distance. Andrade stated that
vs could be obtained in a number of ways, but he
proposed that it be estimated from the Lindemann
formula
where C, is an empirical constant, A the atomic
weight, and V the molar volume at T,,,. Combining
equations (1) and (2), setting v,(T,) = v,(T,,,), and
taking m = A/N, and Q = (VjN,)“3 (where NA is
Avogadro’s number), Andrade obtained his formula
He showed that the coefficient C, was roughly constant for pure metal melts and for liquified diatomic
he estimated
its value to be
gases, and
1.655 x lo-’ (J/K mo11~3)1~2.
The assumptions made in Andrade’s derivation of
equation (3) are questionable. In particular, given the
inverse relationship between r/ and atomic diffusivity
in liquids, it is disturbing (as pointed out by
Mott [lo]) that in equation (1) q is proportional to vL
and not inversely proportional to it. Nonetheless,
there does seem to be a good empirical basis for the
formula, equation (3). It is worth noting that a
formula of this form can be derived by other means,
for example from more recent treatments of dense,
hard-sphere fluids [2]. In the following sections we
attempt the most comprehensive assessment to-date
of the Andrade formula in its application not only to
pure metals, but aiso to intermetallic compounds and
eutectics.
2.2. Elements
Relevant data for pure metals, for germanium and
silicon, and for some semi-metals (Sb, Se, Te) are
cohected in Table 1. The values of the Andrade
coefficient C, can be seen to be roughly constant,
particularly if only the metals are considered. The
metals
is
for
the
value
of
C,
is often used to relate the atomic diffusivity D and the
viscosity in liquid metals [ill. It is successful if the
characteristic distance a is taken to be an ionic
diameter. Table 2 shows the reasonable correlation
between measured values of q(?;n) and values calculated from equation (4) using available data for
D(T,), showing that this may be a useful way of
estimating ~(7’~) if D(T,,,) data are available. There
is also the possibility of estimating q from D at
temperatures other than T,, but reliable D(T) data
are sparse [12].
The semiconductors silicon and germanium have
abnormally low values of C, in Table 1. For these
elements the solid is a covalent tetrahedral network,
whereas the liquid is metallic. It is unreasonable to
suppose that vL(Tm) = v,(T,,,). The different bonding
in solid and liquid states is also reflected in anomalously high entropies of fusion. Grimvall[l3]
has
suggested that for such materials the entropy of
fusion per mole, AS,,,, should be given by
AS, = R[l + 3 ln(@s/O,)]
(5)
where R is the gas constant and 0, and & the Debye
temperatures of the solid and liquid phases. Since the
ratio B,/@, is Q/V,, the measured values of AS,,, for
silicon and geranium
can be used to estimate vsjvt.
Taking vs from equation (2), a value of vL can
be computed for insertion in equation (I). When
this correction is applied, the values of C, for
silicon and germanium
become 1.3 x lox7 and
0.9 x 10m7(J/K mo11i3)1izrespectively, in somewhat
better agreement with the pure metals.
Notwithstanding
the different bonding in liquid
and solid silicon or germanium, the relevant particles
(i.e. atoms) are the same in each case. This is not true,
for example, for selenium, where the high degree of
association in the liquid leads to anomalously high
values of q and C,. In terms of the Andrade derivation of equation (3), this can be interpreted as being
due to different effective molecular masses in equations (1) and (2). For melting of the solid the vibration of atoms is relevant [equation (2)], but for
viscous flow of the liquid the effective mass in equa-
BATTEZZATI
and
GREER:
Table
VISCOSITY
OF
LIQUID
METALS
1793
I. Data for liquid elements
V
IO
Element
Group
Li
Na
K
Rb
Cs
Group
Bc
Mg
Ca
Group
La
Ce
Pr
Yb
Group
Ti
ZI
Hf
u
PU
Group
V
Group
Cr
Group
Mn
Group
Fe
co
Ni
Pd
Group
CU
.4g
AU
Group
Zn
Cd
Hg
Group
Al
Ga
In
TI
Group
Si
Ge
So
Pb
Group
Sb
Bi
Group
Se
Te
(mPa s)
I .47
I .34
I .80
I .99
I .92
0.146
0.153
0.134
0.094
0.102
References”
454
471
336
312
302
13.2
24.8
48.3
59.5
71.7
0.57
0.68
0.51
0.67
0.68
1.01
I .97
1.83
I .98
I .x4
I560
922
III2
5.33
15.3
29.4
I .25
I .63
1.73
1193
1071
1205
1097
23.2
21 SJ
21.3
1958
2125
2500
1405
913
II.6
II.7
15.7
16.1
I3 3
I4 7
2.2
5.2
3.5
5.0
I.18
2.77
1.58
1.51
6.5
6.0
2.0
2.41
2175
x.9
2.4
0.98
2133
8.28
5.7
9.58
5
5.56
5.24
5.02
5.15
4.81
Ila
I .22
-51
30.5
27.2
-3.93
3.98
2.94
25.2
14.0
2.54
I .57
II.2
23.8
I.12
2.61
[391
-0.1
0.025
0.065
IIIa
2.45
2.88
1.55
1.79
I .65
2.80
2.68
0.209
-0.675
0.936
0.197
[401
[401
[401
[401
-4.18
-0.034
[411
-4.98
- 5.34
2.60
1.70
- 0.024
- 0.024
0.485
I.089
[4ll
[411
-13
- 4.04
- 0.042
[411
-185
- 10.43
1.7x IO 4
(421
1.63.7
0.12-1.02
[43.44]
2.75
IVa
-68
-88
-III
30.4
12.9
[401
Va
Via
-2.2
Vlla
1517
2.5
20-46.5
2.18
I .58
1.85
- I.4
41.4
I .72
VIII
I x09
I765
1728
1825
IO.1
5.5
4.18
4.90
-4.2
1356
1234
I336
7 94
11.5
II.3
4.0
3.88
5.0
h93
594
234
9.94
14.0
14.6
933
303
429
577
7.96
7.59
7 43
0.370
0.255
0.166
-0.156
44.4
50.2
-50
3.03
3.49
- 3.29
I .72
2.1 I
30.5
22.2
15.9
2.71
2.16
I .43
0.301
0.453
I.132
3.85
2.28
2.10
2.64
12.7
1.54
1.62
10.9
2.51
2.21
1.29
I.413
0.300
0.557
II 3
Il.4
16.3
I8 I
I .30
2.04
I .x9
2.64
1.30
2.25
1.74
I .68
16.5
4.0
2.13
1.59
0.149
0.436
6.65
10.5
I .86
2.19
0.302
0.298
1685
1210
505
600
II.2
13.0
17.0
194
0.8
0.73
I .85
0.6
0.4
1.58
2.65
1.71
27.-36
II.5
5.4
X.61
I .9%2.6
I.14
I .29
I .73
904
544
18.8
20.8
I .22
I .80
0.82
1.27
22.0
6.45
2.93
I .43
0.08I
0.446
494
723
I98
22.3
16.5
4.02
-0.447
0.669
[451
Ib
lib
1.83
lllb
IVb
o.ou.
12
-0.23
0.538
0.464
[42,461
[461
Vb
Vlb
Table 2. Comparison.
viscosities with those
Ag
cu
Ga
Hg
In
K
Li
Na
Pb
Rb
Sn
Zn
‘lo
B = E/Iv,
Ia
“Unless otherwise
Element
6x
(mZmolm’)
24.X
29
1.8G2.1
I .6-2.4
5.3
0.88
[461
[471
specified the data are from Ref. [38]
at the melting point, of measured liquid
calculated from liquid diffusivity
via the
Stokes-Einstein
relation
D(T,)“‘ed
IO-’ x (m*s-‘)
2.56
3.98
I .6X
I .02
I .6?
3.64
6.3
3.95
2.20
2.47
2.31
2.04
q(Tm)‘““
tion
(1) may be considerably greater than the atomic
mass.
q( T,,,jmeas 2.3. Intermetallic compounds
(mPa sl
(mPa s)
3.12
2.60
2.13
1.50
2.04
0.51
0.92
0.89
2.38
0.62
2.16
3.00
3.88
4.0
2.04
2.10
I .89
0.5 I
0.57
0.68
2.65
0.67
1.85
3.85
Congruently
opportunity
melting compounds
provide a further
to test the Lindemann
melting criterion
and the Andrade
interest in studies
correlation,
and are of particular
of concentrated
alloys for potential
glass formation.
The values of Debye temperature
needed to test the Lindemann
criterion directly are
not often available for compounds.
Where they are
available, as for a series of Ni-Zr compounds [14],
the criterion appears to work well, with the constant
8.854 x lo8 m SC]
having
the
same
value,
c,
1794
BATTEZZATI and GREER:
VISCOSITY OF LIQUID METALS
molar volume are available, but mostly the values
were estimated by linear interpolation of the elemental values. This approximation is expected to lead to
a variation in C, of not more than 5%.
The C, values for the compounds are, with few
exceptions,
closely
grouped.
The
value
is
(1.88 f 0.5) x lo-’ (J/K mol”3)1/2, very close to that
for pure metals. This shows that the Andrade correlation can be applied not only to pure metals, but also
(kg/K)“* n101-~‘~, as for pure metals. Relevant data
and computed values of the Andrade coefficient C,
for intermetallic compounds are collected in Table 3.
This compilation includes incongruently
melting
compounds. For these the Lindemann criterion is
expected to concern the metastable congruent melting
point. Since this is usually not far below the liquidus
temperature, the latter has been used in equation (3)
in computing C,,. In some cases measured values of
Table 3. Data for liouid allovs at
V
10-f
Compound
CA
x
(m3mol~‘)
AI,Mg,
AlNi
AuSn
&Cd&u,
c-Cd&u,
CdSb
968/1118’
997/l 103
7531958
1901
863
732
1913
692
835
670/8 IO
739
12.5
12.3
12.8
9.2
9.2
12.6b
8.gb
13.9
12.0
12.6
16.2
Cd,Sb,
688C
17.7
/?-Agln
fl-AgSn
y-AgSn
AlCo
AI&u
CrxC,
C1,Ge
Cr,Ge,
Cr,,Ge,
CrGe
Cr,,Ge,,
y-CuSn
6-CuSn
L-CuSn
rj -CuSn
Fe,B
-Fe,P
_ Fe, ,P
FeSi
GaSb
Hg,K
HgIn
Hg,TI,
In,Bi
InSb
MgK+e
Mg,Pb
Mg,Si
Mg,Sn
Mn,Si,
Ni,Mg
Ni,Si
p-PbTI
Pd,Sn
Pd,Sn
Se,Ge
Sn,Au
Sn,Au
Te,Ga,
Te,In,
Te,TI,
Zn,Sb,
-1800
152311673
1433/1573
129811443
1228/1283
1028/1063
870/1015
950/1000
688/903
1662
1323/ I423
1323/1423
1683
979
538
254
287
363
798
1387
823
1393
1051
1558
-1413
1592
653
9.4
IO.1
10.1
10.4
11.0
8.92
9.19
9.55
12.7
-7.6
7.4
7.4b
8.1
16.9b
20.0b
15.2
15.6
18.4
20.5b
13.8
15.6b
13.5
15.7b
9.9
8.1
7.9
18.9b
1013
5251543
5821618
1063
940
703
839
19.7
16.8
15.0
18.4b
21.3b
21.4b
13.6
Zn,Sb,
8361839
13.9
ZnSb
819/830
14.5
VU-,)
@Pa s)
10-7x
(JijiKv)
E
(kJ mol-‘)
‘lo
(mPa s)
17.7
0.404
14.5
T-range
(K)
1118-1258
1103-1573
958-l 573
4.16
4.50
3.41
-4.2
2.1
1.90
-8.5
4.2
-9.6
-4.7
-2.2
2.67
-3.3
2.6
-5.3
3.0
3.3
2.3
2.2
1.9
5.0
5.1
5.95
3.15
- 10
-18
16.5
-4
2.0
3.4
2.0
3.3
6.7
2.0
I .03
2.28
0.63
-0.8
I.81
3.2
-6
-7.5
2.5
2.12
2.18
I .80
-2.04
1.6
2.37
-4.0
2.32
* 5.7
* 2.88
-1.5
1.84
2.33
1.82
0.516
21.8
0.541
1.3
1.5
1.1
1.1
1.0
2.5
2.5
3.04
I .83
4.6
8.1
0.591
1.62
0.452
0.422
0.365
0.386
0.655
0.414
0.756
24.9
9.8
21.1
20.0
17.5
22.8
17.6
22.2
10.7
2073
1800-1960
168l&l990
1573-2020
150&2000
132t%1950
1173%1573
1033-1573
100~1573
903-l 573
1712
1.324
29.8
1425-1540
0.140
14.2
553623
304
293
0.138
18.8
873-l 173
0.066
0.68
28.6
39
1073-I 173
1573-1773
0.437
5.5
2.60
3.92
2.3
-1.2
7.4
2.17
2.51
3.08
3.00
1.48
2.59
4.5
2.03
2.6
I.6
-0.9
5.2
1.44
1.67
2.06
2.01
1.0
1.75
0.422
0.476
9.3
50
42
241
8.1
10.4
0.267
0.034
19.4
28.4
75&l 150
84&924
0.240
17.7
X4&955
0.043
24. I
831-910
1.96
1.4
2.8
1.9
2.7
6.4
1.6
0.8
I .70
0.6
-0.8
1.48
1.8
-3
-3.4
1.5
0.114
17.0
773-973
0.533
11.8
773-I 173
References
[481
$
I501
1251
[511
[521
653-773
573-l I73
673-I 173
dWhen two temperatures are indicated the compound melts incongruently.
The lower temperature is the decomposition
higher the liquidus. The liquidus has been taken to approximate the (me&table)
congruent melting point.
%alculated assuming ideal mixing of the elements.
‘Metastable congruent melting temperature.
[531
[541
[541
t::;
[541
[551
[561
t::;
[571
[571
1571
I581
l581
[581
[581
[591
[591
WI
(611
WI
b31
b541
1651
WI
WI
b71
WI
b71
b91
[701
[711
Lb91
k591
~721
[451
[451
[731
1531
[531
WI
WI
[471
[741
[551
[741
[551
[741
[551
temperature,
the
BATTEZZATI
and
GREER:
VISCOSITY
to intermetallic compounds. The exceptions, all having anomalously high values of C,, are some semiconducting compounds such as Se,Ge, and some
compounds close to relatively easy glass-forming
compositions such as Fe,B and Fe,P. It is important
to note that other metal-metalloid systems, such as
Fe&i and Cr-Ge, do not show deviations from the
Andrade formula. This could be suggestive of a
difference in behaviour of interstitial-like and substitutional-like liquid solutions, but many more data are
needed to justify such a conclusion. More data are
needed also to demonstrate whether there is a correlation between a high value of C, and a glass-forming
system. The anomalously high values of C, in Table
3 may have the same origin as for selenium in Table
1, i.e. they may be due to association in the liquid.
The regular associated solution model, based on an
assumption of the existence of particular clusters or
associates in the liquid, has been very successful in
modelling the thermodynamic properties of liquid
alloys [I S-181. There is further evidence for such
association from the variation of many physical
properties [16]. It is interesting to note that a tendency to association, arising from a negative enthalpy
of mixing, is correlated with glass-formation [16].
OF
LIQUID
METALS
1795
2.4. Eutectics
In attempting a comprehensive analysis of the
viscosity of alloy liquids, the behaviour at eutectic
compositions is of particular importance. It is additionally of interest because deep eutectics are associated with glass-formation [19]. Data for eutectics,
including several metal-metalloid glass-forming systems, are collected in Table 4. It is clear that the
values of C, fall into two groups. For “normal”
C, has a value of (1.85 k 0.4) x
eutectics
lo-’ (J/K mol’13)“2, very similar to the behaviour for
pure metals. For “deep” eutectics the values of C,
are much higher, (6.52 k 1.0) x lo-‘(J/K mol’ ‘)’ ‘,
giving the same percentage variation as for normal
eutectics. The experimental difficulties associated
with viscosity measurements on deep eutectics seem
to be particularly severe. Large discrepancies are
found between data obtained by different authors, for
example, between the data for Pd,,Si,, and Pd,,Si,,
in Table 4. Yet other data [20] on the Pd-Si system
suggest viscosity values in the composition range
15-30 at.% Si intermediate between those in the
Table. Because of the uncertainty in the data, the
three highest C, values have been excluded from the
Table 4. Data for liaidd eutectic allovs
CA
V
'x
10-6x
Eutectic
(m’mol
‘)
(~F?~)
Ag,@>,
4, I Mg,, P
A4,Ak.u~
4dil
I i
AW%, 3, L
Au,, A, o
‘&Cd,,
Fed,,
b2,G,3
Fe,Ni,B,,
Fe,,NI,,P,,B,
Fe,&,
Fe,,P&,
Ge&r,,
In,, ,Cdx 7
MgwPb,,
Mgw,Sn,,,
Ni,,P,,
Ni,,Pd,,Px,
PbxCd,,
Pb&‘g,,
Pb,, ,Sb,o o
Pd,,Cu,Si,,
Pd,&
Pd&%
Pug0 sFe, 5
SbvCd,,
924
723
123
850
625
553
810
1447
1426
1413
II80
(kJ:l
‘)
90
(mPa s)
T-range
(W
5.49
2.55
19.4
0.595
1073-1573
11.2
4.01
2.10
16.8
0.456
9241384
12.4
1.44
1.78
13.9
0.136
713-913
12.7
1.25
1.58
14.0
0.121
773-973
10.9"
1.31
1.34
12.2
0.233
923-1173
II.9
..33
0.774
8.93
Ag,,Cuw,
(,,/dh,
11.0
11.0
10.7
6.3
11.5
0.90
-29
5.6
5.28
-1.7
11.5
4.1
14.6
6.0
0.157
1460-1560
37.6
0.352
1173.-1273
8.0
35.4
0.706
1325Sl540
7.7
24.0
0.066
1218-1430
22.6
0.141
1290
12.6
5.3
7.23
I6
1.3
7.23
16.1
7.8
15.1
7.3
II.0
5.0
17.6
16.0
7.3"
1218
1193
401
741
834
II53
1050
522
523
520
7.24"
1497
52.1
6.46"
1323
11.8
1.38
I.1
15.7"
2.12
1.96
4.x
0.499
15.2
2.04
1.87
14.1
0.211
15.1
1.54
1.77
9.3
0.369
8.25
I5
8.44
43
573-1173
823.923
1.9
21
1930
173ml173
873-1173
32.0
0.54
130&1450
63.1
0.031
120&1450
18.0
2.83
1.99
7.34
0.482
623.~823
17.5
2.60
1.84
7.4
0.489
573-1173
19.0
2.78
1.99
6.R3
0.572
523-873
0.059
115~1500
0.062
1153~1500
1015
10.1
169
1071
10.2
14.2
1071
10.2
66
31
43.1
684
14.0
25
11.5
23.6
0.400
729
16.5
82
6.7
2.2
I.5
2.0
1.4
67.2
48.8
52
1073.1273
705-1081
Sn,, &u, 1
Sn,, sCdv 5
Sn,, Mg,
Sn,,Pb,,
490
16.5
2.31
1.93
7.0
0.433
573~1173
449
15.5
2.01
1.13
5.11
0.458
623-823
479
16.7
1.82
1.64
5.2
0.478
523-1173
456
17.6
2.67
2.25
7.15
0.532
3.16
2.66
7.53
0.433
S%,Z%2
Zn,, 4,
472
16.2
1.86
1.65
4.93
0.517
653
10.0'
3.02
2.22
9.39
0.472
“Calculated
/3
assuming
ideal mixing of the elements
473-973
673-973
References
[751
[761
[511
[511
1511
[771
[531
[541
(591
(781
1601
[791
[781
[791
[591
[601
[601
[571
WOI
1681
1701
[XII
1811
[821
16x1
B31
[X41
[841
[791
[X51
[541
[551
[531
P21
[701
F61
1831
[821
[871
BATTEZZATI
1796
and GREER:
VISCOSITY OF LIQUID METALS
calculation of the average C, value for deep eutectics.
These high C, values, however, are for the good
glass-forming compositions, Pds2Sir,, Pd,,, CusSi,,
and Ni,, Pd,, P,, , and may be related to their high
glass-formability,
It is not clear that the Andrade correlation should
work for eutectics. The Lindemann criterion, used in
the derivation of equation (3), is applicable to onecomponent systems and considers melting to be due
to molecular vibration in the solid. For eutectics,
however, there is an additional factor; the chemical
mixing in the liquid will act to lower T,. The degree
of non-ideality of the mixing in the liquid can be
characterised quantitatively by the depression of the
actual eutectic temperature, T,, below an ideal eutectic temperature T. The temperature T would apply
for a system with ideal mixing in the liquid and no
solid solubility, and is given by
-AH,,,
“(R
lnx,-AS,)
where AH, and AS, are the enthalpy and entropy of
fusion of the main component, and x, its mole
fraction at the eutectic. The values of C, for eutectics
are plotted in Fig. I as a function of the deviation
from ideality K - T,. The category of “normal”
eutectics, for which the Andrade correlation is obeyed
with the usual C, value, is seen to be characterised by
near-ideal mixing. Deep eutectics, however, show
non-ideal mixing, with a tendency to association in
the liquid. This leads to a T, abno~ally
below that
predicted by the Lindemann criterion and consequently to a high value of C,. The unusually high
viscosity in the vicinity of deep eutectics may also be
expected
since high-melting
compounds
and
metfzstable eutectics may often occur at similar compositions. The already high viscosity at the melting
point of the compound should become even higher on
cooling to the temperature of the metastable (and
deep) eutectic which would be reached if the compound did not exist.
Thus for eutectics, as for intermetallic compounds,
the Andrade correlation appears to work well for
i
200
0
Deviation
400
600
from Ideality (EC)
Fig. 1. The Andrade coefficient C, [lo-’ x (J/K mo1”3)“2]
for liquid eutectic alloys as a function of the deviation of the
eutectic temperature from its ideal value (T, - T,).
systems with approximately ideal mixing. For these
systems
the
Andrade
coefficient
C,
is
(1.85 f 0.4) x lo-’ (J/K mol”3)1i2. When there is association in the liquid, corresponding to a negative
enthalpy of mixing in alloys or to molecular bonding
in elemental liquids (e.g. Se), unusually high values of
C, are obtained.
3. COMPOSITION-DEPENDENCE
VISCOSITY
OF
Ciebhardt and Kiistlin first related the viscosity of
alloys to phase diagram features [S]. Their conclusions, which are based on much experimental data,
may be summarised as follows. In systems showing
complete miscibility in solid and liquid states, the
viscosity approximates to the weighted average viscosity of the two elements [Fig. 2(a)]. Simple eutectics
show a negative deviation from this behaviour [Fig.
2(b)]. Systems with intermetallic compounds show
more complex behaviour, with maxima of viscosity in
the liquid state at compositions corresponding to
those of crystalline compounds [Fig. 2(c)].
The three types of behaviour shown in Fig. 2 may
be related to the correlations described in Section 2.
It is known from Section 2 that for many alloys the
Andrade formula is obeyed with the normal value of
C, at all the compositions at which it has been tested,
whether for pure metal, eutectic or compound. For
such a binary alloy the imposition-de~ndence
of
the melting point viscosity can be moderately well
inferred from the known points. The composition-dependence of q( T,,,) reflects the smooth variation of A
and V, and the more complex variation of T,. Since
the composition-dependence
of the melting point
viscosity is less strong than the temperature-dependence of the viscosity at any composition, and since
the temperature-de~ndence
of the viscosity is not
greatly different from composition to composition
(see Section 4), the qualitative dependence on composition of the isothermal viscosity in the liquid phase
field may be estimated. The isothermal viscosity
should be greater at compositions with higher melting
points (since the melting point is closer to the measurement temperature), and vice versa, in agreement
with the behaviour shown in Fig. 2.
It is possible that systems with complete miscibility
in the solid and liquid states may, unlike eutectic and
~ompo~d-foxing
systems, permit some interpolation of the melting point viscosity. Application of
equation (3) with average values of A and of V may
yield estimates for the viscosity of the (slightly undercooled) liquid at the To temperature (between the
liquidus and the solidus).
The composition-de~ndence
of the viscosity in
more complex systems is very difficult to describe
quantitatively. An attempt was made by MoelwynHughes [21] by calculating the viscosity from the
interdiffusion coefficient using the Stokes-Einstein
relation. Making the doubtful assumption of a com-
BATTEZZATI and GREER:
VISCOSITY OF LIQUID METALS
1797
a
A
u
AQ
AU
b
PD
Cu Mg
Ag
Fig. 2. The composition dependence of isothermal liquid viscosity in three types of binary alloy (after
Ref. [5]).
position-independent
tained the formula
interdiffusion coefficient, he ob-
(7)
where qA and qs are the viscosities of the elements, xA
and xs the mole fractions and R the regular solution
interaction parameter. The Moelwyn-Hughes formula has been widely used in the literature, but we
can now attempt a further assessment of it. It predicts
a negative deviation of viscosity from the linear
interpolation for systems having a positive heat of
mixing AH,,, in the liquid, and vice versa. The
negative deviation is indeed found for some simple
systems with a positive AH,,,i,, but this effect is also
found (for example in Au-Cu [5]) for some systems
having a negative AH,,,,,. For systems having a
strongly negative AH,,,,, equation (7) predicts an
increased viscosity for concentrated alloys, which
may seem to be in accord with the observed high
viscosity at compound-forming compositions. However, equation (7) cannot predict the lower viscosities
expected for eutectic compositions between compounds. It gives only a smooth parabolic variation of
q with composition, and furthermore predicts that the
maximum viscosity will be at the equiatomic composition. On the other hand, high-melting compounds
or associates in the liquid, believed to correspond to
high viscosity, may occur at compositions other than
equiatomic. More recent modifications to the Moelwyn-Hughes formula, designed to account for hardcore interactions in addition to the chemical effect, do
not remove the objections to it raised here [22].
In systems with a strongly positive heat of mixing,
leading to a liquid miscibility gap, it may be more
reasonable to expect a smooth variation of q with
composition, as solid compound or liquid association
formation is less likely. With the strong interatomic
interactions the atomic motion will show some correlation. For diffusion in crystals this is taken into
account by introducing a correlation factor which
reduces the effective atomic mobility. A similar effect
may be responsible for the viscosity maxima observed
just above the liquid miscibility gaps in the systems
Cd-Ga, Bi-Ga and Ga-Mg[23].
In glass-forming liquids a strongly negative heat of
mixing is expected. The thermodynamic mixing properties of such liquids can be well represented by the
regular associated solution model. We expect that an
extension of the model to atomic transport properties
could describe the correlated atomic motion responsible for the anomalously high melting point viscosities
found for such systems (as described in Section 2). It
should be emphasised, however, that in systems of
this type for which isothermal viscosity data are
available in a wide composition range, i.e. Fe-B,
Fe-P and Fe-C [24,25], the general behaviour of Fig.
2 is still observed. Viscosities at eutectic compositions
1798
BATTEZZATI and GREER:
VISCOSITY OF LIQUID METALS
are close to, or slightly lower than that of the major
component (the pure metal), while at compound
compositions the viscosity is somewhat higher. Thus
it seems that at the compositions of deep eutectics
the viscosity is not intrinsically high. Rather the
liquid phase is stable at abnormally low temperature,
and the melting point viscosity is enhanced by the
temperature-dependence.
At compound compositions, if it were possible to undercool the liquid to
temperatures comparable to the deep eutectic points,
is likely that the viscosities would be even higher.
Metallic glasses at the same composition as a
compound can in a few cases be formed by meltquenching, but more often nucleation of the compound can be avoided only by solid state
amorphisation. In this process a homogeneous amorphous phase is obtained by reaction of multilayers of
the pure polycrystalline elements on low temperature
annealing. The formation of the amorphous phase in
this way is made possible by the strongly negative
heat of mixing in the liquid, and is promoted by the
presence of a fast diffusing species. In addition to
these effects, glass formation may be favoured at
compound compositions by the fact that the undercooled liquid should show a relatively high glass
transition temperature; its viscosity is already unusually high at the melting point, and should increase
further on undercooling. The high glass transition
temperature implies that the glass formed will be
relatively resistant to crystallisation. If there is an
initial amorphous phase at the multilayer interfaces it
will be very difficult for the compound to nucleate at
the low annealing temperature and to compete
against glassy phase formation as the reaction of the
elements proceeds.
4. TEMPERATURE-DEPENDENCE OF
v1sc0s1TV
4.1. Non -glass -forming systems
Quantitative discussion so far has been limited to
the consideration of melting point viscosity. This
limitation to a single temperature is a direct consequence of Andrade’s derivation of equation (3) by
equating the vibrational frequencies in solid and
liquid at T,,,. Extension of the Andrade treatment to
other temperatures is not successful. Application of
equation (1) would suggest a very weak dependence
of viscosity on temperature, in marked disagreement
with experiment. Andrade himself recogised this
problem, and in a paper on the temperature-dependence of viscosity [26] he did not use his earlier
derivation of equation (3). He supposed that the
viscosity of liquids would fall as the temperature were
raised because molecules would lose local orientational order. This does not seem tenable for metallic
liquids, though it is of interest to note that the
temperature-dependence
of the viscosity of liquid
alloys is much less than that of molecular liquids in
which orientational effects may arise [26].
One method for calculating the temperature-dependent viscosity of a liquid is through a corresponding states correlation. Such a correlation for pure
metal melts has been developed with some success by
Chapman [27]. Although this demonstrates that the
viscosity-temperature
relationship (with the quantities suitably reduced using distance and energy
parameters characterising the interatomic pair potential) for pure metals falls on a universal curve, the
approach does not readily yield a shape for the curve.
Andrade [26] proposed an equation for the temperature-dependent viscosity of the form
where C, and C, are constants and v the specific
volume. This differs little from an Arrhenius behaviour of fluidity
rl(T) = fhexp
(gT)
where E is the activation energy for viscous flow, ‘lo
the pre-exponential viscosity and R the gas constant.
Many other multiparameter formulae have been proposed [ 11,but equation (9) seems adequate to fit most
of the data for liquid alloys above the equilibrium
melting point. For non-glass-forming alloys the temperature range below T,,, is not normally accessible.
The viscosity of undercooled liquid alloys is considered in the Section 4.2 for glass-forming systems.
In Table 1 values of E and qO,calculated from data
taken in the vicinity of T,,, , are given for pure metals.
For some refractory elements (e.g. Cr, Be) the values
are rather uncertain, and in other cases (e.g. Mn)
large discrepancies exist in the literature. It is of
importance to determine whether a reliable correlation exists which can be used to predict E in cases
where it is not known with confidence. Such a
correlation has been attempted for the activation
energy for atomic diffusion in crystalline materials [28]. It is suggested [28] that the activation energy
for diffusion should be a linear function of the
enthalpy of melting AH,,, (reflecting the difficulty of
local atomic rearrangement) and of the enthalpy of
sublimation AH, (reflecting the cohesion of the material). Since both the entropies of fusion (=AH,/T,)
and of sublimation (=AHJT,,,)
at T,,, may be expected to be approximately constant for a given class
of material, a simple correlation is suggested
E = B(RT,,,)
(10)
where B is a constant. A plot of E against T,,, is shown
in Fig. 3. There clearly is a correlation, and the values
of E/RT,,, (given in Table 1) have a value of 2.4 + 1.O.
The correlation is noticeably better than with AH, or
with AH,,, . The deviation from the rough correlation
in the latter case is particularly severe for silicon and
germanium which have anomalously high values of
AS,,,. There is no evidence that E is a linear function
of AH, and AH,,,, and deviations from the average
BATTEZZATI and GREER:
VISCOSITY OF LIQUID METALS
1799
120
.
100 -
.
50 ,
.
80 -
40 -
.
.
.
60 -
30 .
401.
.
1’.
: . ,
20 .
0
.
.,+m
.
. .
%
20-
.
.
-
.
.
.s
.
10 ;
.
.
.
.
.
.
.
.
.
. . .
=:
.
8
n
,
0
.
.
I
1000
2000
Melting
0
500
3000
“~*‘~~~~“~“_
1000
Point (K)
1500
Melting
2000
Point (K)
Fig. 3. The activation energy E for viscosity as a function
of the melting temperature 7’,,,for liquid elements.
Fig. 5. The activation energy E for viscosity as a function
of the melting temperature r, for liquids of compoundforming compositions.
value of B do not appear to show any correlation
with AH, or AH,. As can be seen from Fig. 3,
however, there is a tendency for higher values of B
at higher r,,,. Indeed, there is an acceptably linear
correlation
between E and Ti, shown in Fig. 4.
This correlation seems to offer the best prospect
for estimation of E in pure metal liquids, with the
constant
of proportionality
being (2.3 + 1.4) x
10M5kJ mall’ Km2. Nonetheless, it may be expected
that better correlations would be found if the elements were considered in groups as in Table 1. For
the parameter B (=E/RT,,,) there is a discernable
variation, as follows: group Ia, B = 1.3-2.0; group
IIa, 2.940; group IVa (Ti, Zr, Hf), 4.2-5.3; group
VIII, 2.7-3.5; groups Ib, IIb and IIIb, 1.3-2.7. The
pre-exponential viscosity ‘lo varies by less than two
orders of magnitude for all the elements for which it
is known. Its value is 0.37 &-0.29 mPa s. The variation from group to group may be seen in Table 1. The
pre-exponential factor ‘lo can also be expressed as a
kinematical viscosity. It is found that the percentage
variation of its value is very similar to that of
viscosity. Thus density is not a significant normalising
factor, as is consistent with the low dependence on
volume shown in equation (8).
For compounds the correlation between E and T,,,
is shown in Fig. 5. The value of B is 2.4 f 0.8,
matching the behaviour of the elements. The values
of B do not tend to be higher for higher values of T,,
and a correlation
of E with T,!,, is not sought. The
pre-exponential
viscosity
value
of
has
a
0.46 + 0.36 mPa s, again close to the behaviour found
for the elements.
For eutectics the correlation between E and T,,, is
shown in Fig. 6. Excluding good glass-forming alloys
from the calculation, as in Section 2.4, the value of
B is 2.4 f 1.O. Although this overall average matches
those for the elements and compounds, it is possible
to distinguish the “normal” and “deep” eutectic
groups identified on the basis of their C, values in
Section 2.4. Normal eutectics have an average B of
2.0, whereas deep eutectics have an average of 3.45.
Glass-forming eutectics have still larger values of B,
the average for Pds2Si,,, Pd,lSi,,, Pd,,Cu,Si,, and
Ni5,Pd2,Pzo being 6.4. For eutectics as a whole,
however, there is no discernible correlation of B with
T,,,. The pre-exponential viscosity for eutectics (excluding good glass-formers)
has a value of
0.41 + 0.18 mPa s. The average value is again close to
those for elements and compounds.
4.2. Glass-forming systems
For some compound and eutectic compositions the
Arrhenius equation, equation (9), does not fit the
data well over the measured temperature range; the
,
.
.
.
.
.
0
2
Square
6
4
of Melting
.
.
.
x
Temperature
400
Fig. 4. The activation energy E for viscosity as a function
of the square of the melting temperature (lo6 x K2) for
liquid elements. The linear fit is E (kJ/g-atom) =
1.808 + 1.55 x 1O-5K2.
600
800
Melting
1000
1200
1400
1600
Point(K)
Fig. 6. The activation energy E for viscosity as a function
of the melting temperature IF,,,for eutectic liquids.
1800
BATTEZZATI and GREER:
VISCOSITY OF LIQUID METALS
behaviour is such that the viscosity increases more on
decreasing the temperature than equation (9) would
predict [S]. This increase in viscosity increases as the
liquid is undercooled below T,, and is of particular
importance in glass-forming systems. In such systems
the viscosity below T,,, may also be directly measurable. If nucleation and growth of the crystal phase, or
phases, are avoided, a glass is formed as the liquid
viscosity becomes so high that the configuration is
frozen. This transition from liquid to glass has been
analysed in terms of the excess configurational entropy of the liquid [29], and in terms of its free volume
(i.e. the excess of its volume over that of the ideal
close-packed liquid) [30-321. In either case, the predicted temperature-dependence of the liquid viscosity
has the Vogel-Fulcher-Tammann
form
(11)
where r and 0 are constants, and T’ is the temperature at which the excess configurational entropy or
the free volume is zero. Equation (11) does seem to
fit measured data on glass-forming systems [33-371,
and modifications to it, for example by incorporating
an Arrhenius term, do not significantly improve the
fit [36]. It should be noted that in these very viscous
systems relaxation effects (i.e. time-dependence of the
viscosity) are significant [36,371. The Vogel-FulcherTammann equation applies strictly only to the temperature-dependence of the equilibrium viscosity [36].
When equation (11) is fitted to the data, the temperature T’ is found to lie somewhat below the measured
glass transition temperature Tg; it may be thought of
as an ideal T,.
For glass-forming systems it is of interest to know
q(T) in the range T, to T,,,, as this may, for example,
be used in calculating the critical cooling rate for glass
formation [3]. When, as is usually the case, q(T) has
not been measured, it is useful to be able to estimate
the parameters in equation (11). To calculate r, 0
and T’ knowledge of q, or drJ/dT, at three points
is required. These three points can conveniently be
taken to be q(T,),
(dq/dT)rm, and q(Tg). The
Andrade formula, equation (3), can be used to estimate v(T,,,), with C, = 1.85 x 10m7 (J/K molri3)“‘.
The correlation, E = 2.4 RT,,,, can be used to estimate (dq /dT),.
The conventional value for q at the
glass transition is lOI Pas. This would apply for
conditions of very slow cooling. At the measured Tg,
typically determined from the specific heat capacity
on heating, the viscosity has been found to be somewhat lower, typically IO9 to 10” Pas [33-361. A
suitable value in the general case is probably 10”’Pa s.
For good glass-formers the estimates of q(T,,,) and
(dq/dT),
based on equations (3) and (10) will be
erroneously low. This is because the deviation from
Arrhenius behaviour is already apparent at T,,,. However, for such systems u(T) may comparatively easily
be measurable in at least part of the range Tg to T,,,.
The procedure outlined here will be of most use for
marginal glass-formers, for which measurement of
the viscosity of the undercooled liquid will not be
feasible. For marginal glass-formers the estimates of
q(T,) and (dq /dT)r” should be reasonably good.
However, as a procedure for predicting T, does not
exist, the calculation of q(T) can be performed only
when Tg is known, and cannot be used a priori to
assess the glass-forming ability of a system.
5. CONCLUSIONS
The Andrade formula for the melting point viscosity, q(T,,,) of liquids has been reassessed using a
substantial body of data on metallic elements, intermetallic compounds and eutectics (for alloys taking
the liquidus as the melting point). The formula
matches the data well for all these systems, with the
coefficient C, having the value (1.85 + 0.4) x low7
(J/K mol”3)112. Deviations from this correlation, in
each case giving higher values of C, , arise when there
is association in the liquid. This occurs for elements
such as selenium, for intermetallic compounds close
to glass-forming compositions, for deep eutectics,
and for glass-forming liquids in general.
For elements, if the atomic diffusivity in the liquid
is known, this can be used with confidence to calculate the viscosity using the Stokes-Einstein relation.
In a binary alloy the composition-dependence
of
the isothermal viscosity is not strong. There is, for
example, no indication of abnormally high viscosity
at glass-forming compositions, which have high values of the melting point viscosity only because the
melting points are very low and the temperaturedependence of the viscosity becomes strong at low
temperature. The isothermal viscosity tends to reflect
the liquidus temperature, being high for high-melting
compositions, and vice versa. Thus good glassforming compositions, being correlated with deep
eutectics, may be the positions of minima in the
isothermal viscosity vs composition curve.
For most liquid metals and alloys above the
melting point the temperature-dependence
of the
viscosity is satisfactorily described by an Arrhenius
equation. The activation energy for viscous flow,
E, is correlated with the melting temperature T,.
For elements, compounds and eutectics B( = E/RT,,,)
is found to be 2.4 f 1.O. There is a deviation to larger
values of B for deep eutectics and glass-forming
liquids. For elements, B tends to be higher for
higher T,, and there is a fair correlation with
E(kJ mol-I) = 2.3 x 10m5[T,,,(K)12.
The pre-exponential
viscosity is approximately
constant for elements, compounds and eutectics, with
the value 0.4 mPa s.
Good glass-forming
systems lie outside the
correlations demonstrated here for q(T,,,) and E.
Their values of t](T,) and E(T,,,) are high, and
their q(T) is non-Arrhenius
and best fitted by
the Vogel-Fulcher-Tammann
equation. By rapid
quenching good glass-forming ability may be found
BATTEZZATI and GREER:
VISCOSITY
at deep eutectics, and by solid state amorphisation at
compound compositions near metastable deep eutectics. In either case the deviation from the normal
correlations is related to the depression of the eutectic
temperature below its ideal value.
For marginal
glass-formers
the correlations
demonstrated for ~(7’~) and E can be used to make
a reasonable estimate of p(T) in the range T, to T,
if the glass transition temperature rB is known.
Acknowledgements-L.B.
is grateful to the Consiglio
Nazionale delle Ricerche, Rome for support and to the
British Council for research funding.
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