See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/223071408 The Viscosity of Liquid-Metals and Alloys Article in Acta Metallurgica · July 1989 DOI: 10.1016/0001-6160(89)90064-3 CITATIONS READS 195 3,430 2 authors, including: A. Lindsay Greer University of Cambridge 501 PUBLICATIONS 20,246 CITATIONS SEE PROFILE Some of the authors of this publication are also working on these related projects: Giant Magnetoresistance View project All content following this page was uploaded by A. Lindsay Greer on 09 October 2017. The user has requested enhancement of the downloaded file. 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 , 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 .) It seems appropriate, therefore, to survey the available data on the viscosity of alloy liquids and to re-examine earlier analyses of it . 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 . 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 . 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 . 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 . 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.  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 , 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 . It is interesting to note that a tendency to association, arising from a negative enthalpy of mixing, is correlated with glass-formation . 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 . 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  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  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. ). 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 ) 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 . 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. 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  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 . 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 . 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  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 . It is suggested  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 , 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 . 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 . 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 . 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. REFERENCES I. R. T. Beyer and E. M. Ring, in Liquid Metals, Chemistry and Physics (edited by S. Z. Beer), pp. 431-460. Dekker, New York (1972). 2. T. E. Faber, An Introduction to the Theory of Liquid Met& Cambridge Univ. Press, Cambridge (1972). 3. H. A. Davies, Phys. Chem. Glasses 17, 159 (1976). 4. J. Reeve, G. P. Gregan and H. A. Davies, in Rapidly Quenched Metals (edited bv S.. Steeb and H. WarEmont), pp. 2033206. North Holland, Amsterdam 5. 6. 7. 8. 9. 10. 11. (1985). E. Gebhardt and K. Kostlin, Z. Metaffk. 49,605 (1958). M. Shimoji and T. Itami. in Diffusion and Defecf Data (edited by F. H. Wohlbier). Trans Tech, Aedermannsdorf (1986). J. R. Wilson, Metal1 Ret]. 10, 381 (1966) 1. J. van Loef. J. Phvs. F4. Ll90 (1974). E. N. da C. Andrade, Phil.‘Msg. 17, 497 (1934). N. F. Mott, Proc. R. Sot. 215A, 1 (1952). N. H. Nachtrieb, Ber. Bunsenges. Phys. Chim. 80, 678 Z. Metaflk. 75, 446 (1984). 18. S. L&e, G. V. S. Sastry and P. Ramachandrarao, Z. Metaffk. 75, 897 (1984). 19. D. Polk and B. C. Giessen, in Metallic Glasses (edited 20. 21. 22. 23. 24. 25. 26. 27. 28. by J. J. Gilman and H. J. Leamy), p. 1. Am. Sot. Metals, Metals Park, Ohio (1978). E. L. Dubinin, 0. A. E&n, N. A. Vatolin and A. A. Kuranov, Zh. Neorg. Khim. 12, 3156 (1967). E. A, Moelwyn-Hughes, Physical Chemistry, p. 793. Pergamon, Oxford (1964). Z. Morita, T. Iida and M. Ueda, Inst. Phys. Conf. Ser. 30, Ghan. 1. Part I. 0. 600 (1977). W. Me& and F. Sauerwald; Z. Ph.ys. Chem. 232, 134 (1966). 0. I. Ovtrovskii, V. M. V’yunov and V. A. Grigoryan, Russ. J. Phvs. Chem. 56. 13 (1982). F. D. Richardson, Physical Chemistry of Melts in Metallurgy, Vol. 1. Academic Press, New York (1974). E. N. de C. Andrade. Phil. Mag. 17, 698 (1934). T. W. Chapman, A.I.Ch.E.Ji. 12, 395 (1966). A. M, Brown and M. F. Ashby, Acta metall. 28, 1085 METALS 1801 32. D. Turnbull and M. H. Cohen, J. them. Phys. 52, 3038 (1970). 33. H. S. Chen and D. Turnbull, J. them. Phys. 48, 2560 (1968). 34. D. E. Polk and D. Turnbull, Acta metalf. 20,493 (1972). 35. H. S. Chen, J. non-tryst. Solids 29, 223 (1978). 36. S. S. Tsao and F. Spaepen, Acta metall. 33, 881 (1985). 37. S. S. Tsao and F. Spaepen, Acta metafl. 33, 891 (1985). 38. E. A. Brandes (editor), Smitheffs Metals Reference Book, 6th edn. Butterworths, London (1983). 39. 1. V. Milov, Yu. F. Klimov and D. M. Skorov, Russ. Metaff. 54, no. 4 (1963). 40. L. J. Wittenberg and R. Dewitt, in The Properties of Liquid Metafs (edited by S. Tacheuchi), p, 555. Taylor & Francis, London (1973). 41. A. D. Agaev, V. I. Kostikov and V. N. Bobkovski, Izv. Akad. Nouk. SSSR Metall. 43, No. 3 (1980). 42. B. A. Baum. P. V. Geld and P. V. Kocherov. Russ. Metali. 27, No. 1 (1967). 43. E. S. Levin, V. N. Zamarayev and P. V. Gel’d, Russ. Metaii. 86, No. 2 (1976). 44. K. &bagi, H. Schlarb and M. Frohberg, Archs Eisenhurt. 55, 109 (1984). 45. 0. V. Demidovich, A. A. Zhuchenko, E. L. Dubinin, N. A. Vatolin and A. I. Timofeev, Izv. Akad. Nauk. SSSR MeiaN. 73, No. 1 (1979). 46. V. M. Glazov, S. N. Chizhenskaya and N. N. Giagoleva, Liquid Semicandu~tors. Plenum, New York (1969). 47. T. N. Adrianova, 48. 49. 50. (1976). 12. A. D. Pasternak, Phys. Chem. Liquids 3, 41 (1972). 13. G. Grimvall, Inst. Phyr. Con.6 Ser. 30, Chap. 1, Part 1, p. 90 (1977). 14. R. Kuentzler. J. Phys. F 14, L79 (1984). 15. F. Sommer, Z. Metailk. 73, 72 (1982). 16. F. Sommer, Z. Metallk. 73, 77 (1982). 17. S. Lele. G. V. S. Sastrv and P. Ramachandrarao. OF LIQUID 51. A. A. Aleksandrov, B. S. Okhotin, L. A. Razumeichenko and Z. 1. Panina, High Temp. 9,850 (1971). H. Nakajima, Trans. Japan Inst. Merals 17, 403 (f976f. ‘E. Gebhardt, M. Becker and E. Trlgner, Z. Metaffk. 44, 379 (1953). E. S. Levin, M. S. Petrushevskii, P. V. Gel’d and G. D. Ayushina, Russ. J. Phys. Chem. 46, 807 (1972). E. Gebhardt and K. Dietering, Z. Metallk. 50, 379 (1959). 52. M. S. Petrushevskii, E. S. Levin and P. V. Gel’d, Russ. J. Phys. Chem. 45, 1719 (1971). 53. E. Gebhardt, M. Becker and K. Kostlin, Z. Metaflk. 47, 684 (1956). 54. D. K. Bel~hchenko, Z. Fiz. Khim. 31, 2269 (1957). 55. S. N. Chizhenskava. V. A. Danil’chenko and V. P. Kurbatov, IN. Akad. Nauk. SSSR Neorg. Mater. IO, 1397 (1974). 56 G. V. Tegunov, B. A. Baum and P. V. Gel’d, IX. Vys. Uch. Zau. Fiz. 159, no. 8 (1971). 57. E. S. Levin, V. N. Zamaraev and P. V. Gel’d. Russ. J. Phys. Chem. 51, 817 (1977). 58. E. Gebhardt, M. Becker and S. Schafer, Z. Metallk. 43, 292 (1952). 59. 0. I. Ostrovskii, V. M. V’yunov and V. A. Grigoryan, Russ. J. Phys. Chem. 56, 13 (1982). 60. Y. Nishi, H. Watanabe, K. Suzuki and T. Masumoto, J. Phys. COB. c8, 359 (1980). 61. B. A. Baum, P. V. Gel’d, P. V. Kocherov and M. A. Ryss, Russ. Metall. 43, no. 5 (1968). 62. V. M. Glazov and S. N. Chizhenskaya, Izta.Akad. Nauk. SSSR. Met. i Tom. 3. 154 (1961). 63. M. Katajima, T. Itami and ‘M. Shimoji, Phil. Msg. 30, 285 (1974). 64. V. I. Kononenko, S. P. Yatsenko, V. A. Sudakov, L. F. Kozin and V. V. Khromtsov, Russ. J. Phys. Chem. 46, 188 (1972). (1980). 29. G. Adams and J. H. Gibbs, J. them. Phys. 43, 139 65. B. Predel and I. Arpshofen, Z. Nuturforsch. 29a, 1206 (1965). 30. M. H. Cohen and D. Turnbull, J. &em. Pfzys. 31, 1I64 66. Ya, I. Dutchak, N. M. Klym and A. F. Mykolaychuk, ( 1959). 31. D. Turnbull and M. H. Cohen, J. them. Phys. 34, 120 (1961). (1974). Physica Meralf. 14(5), 132 (1962). 67. V. M. Glazov, N. N. Glagoleva and L. A. Romantseva, Izv. Akad. Nauk. SSSR, Neorg. Mater. 2, 1953 (1966). 1802 BATTEZZATI and GREER: VISCOSITY OF LIQUID METALS 68. E. Gebhardt, M. Becker and E. Tragner, Z. Metalik. 46, 90 (1955). 69. S. 0. Safolov, V. F. Ukhov and N. A. Vatolin, Trudy. Inst. Met. Sverdl. 98, no. 27 (1972). 70. E. Gebhardt, M. Becker and H. Sebastian, Z. MetaNk. 46, 669 (1955). 71. Yu. N. Akshenaev, B. A. Baum and P. V. Gel’d, Zh. Priklad. Khim. 43, 68 (1969). 72. F. A. Kanda and J. A. Domingue, .I. less-common Metals 64, 135 (1979). 73. A. Laugier, G. Chaussemy and J. Fornazero, J. nontryst. Solids 23, 419 (1977). 74. A. Sinha and E. Miller, Metall. Trans. 1, 1365 (1970). 75. E. Gebhardt and G. W&wag, Z. Met&k. 42, 358 (1951). 76. L. Martin-Garin, R. Martin-Garin and P. Desre, J. less-common Metals 59, 91 (1978). 77. D. E. Polk and D. Tumbull, Acta metall. 20,493 (1972). 78. P. P. Arsent’ev, Yu. A. Anikin, N. N. Drozdov, V. P. View publication stats 79. 80. 81. 82. 83. Ovcharov and B. N. Kulagin, Phys. Metals 5, 597 (1985). J. Steinberg, S. Tyagi and A. E. Lord, Acta metall. 29, 1309 (1981). B. Diemili, L. Martin-Garin, R. Martin-Garin and P. Des& J. less-common Metals 19, 29 (1981). Y. Nishi and A. Yoshihiro. Scrinra metall. 19. 1023 (1985). A. F. Crawley, Metall. Trans. 3, 971 (1972). E. Gebhardt and K. KBstlin, Z. Metallk. 48, 636 . (1957). 84. Y. Nishi, N. Kayama, S. Kiuchi, K. Suzuki and T. Masumoto, J. Japan Inst. Metals 44, 1336 (1980). 85. L. J. Wittenberg, D. Ofte and C. F. Curtiss, J. them. Phys. 48, 3253 (1968). 86. H. R. Thresh and A. F. Crawley, Metall. Trans. 1, 1531 (1970). 87. E. Gebhardt, M. Becker and S. Dormer, Z. Met&k. 45, 83 (1954).