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Melting point trends in intermetallic alloys
.I. Phys. Printed
Chem. Solids Vol. in Great Britain.
48,
No.
2, pp.
197-205,
1987 0
MELTING
POINT TRENDS
IN INTERMETALLIC
0022-3697/87 1987 Pergamon
$3.00 + 0.00 Journals Ltd.
ALLOYS
JAMESR. CHELIKOWSKY~and KAREN E. ANDERSON Science Laboratories, Exxon Research and Engineering Company, Annandale, NJ 08801, U.S.A.
Corporate
Research
(Receiued 12 May 1986; accepted 10 July 1986)
Abstract-We have examined the melting points of approximately 500 intermetallic binary alloys. We attempted to correlate the melting point behavior of the binary (1: 1) alloys with a number of elemental variables including electron number, atomic size, orbital radii, electronegativity, etc. We find that a “Vegards’s Law” of melting points works very well for predicting the melting points of binary transition metal alloys, i.e. the melting point of the alloy correlates with the linear average of the elementary melting points. However, this “law” works only moderately well for alloys involving simple metals. In addition, we find that transition metal alloys tend to have melting points below the averaged elemental melting points. This finding is in sharp contrast to simple metal alloys where the opposite trend is observed and it is indicative of fundamental differences between transition metal and simple metal binding. Finally, we have attempted to correlate deviations from a Vegard’s law of melting with elemental variables. We found no strong correlation with elemental variables (or the heats of formation of the alloy in question) with the possible exception being a correlation with elemental volume changes upon alloying. The consequences of this correlation upon alloy design and metallic alloy formation are briefly discussed. Keywork
Intermetallic
alloys,
melting
points,
chemical
Although it is not possible, at present, to predict the phase diagrams of alloys from a knowledge of the elemental constituents, considerable progress toward this goal has been made in the past few years. Most of the progress has centered on the creation of “chemical scales” with which one may examine trends and predict properties based on existing data. For example, Miedema [l] has recently developed a scheme based on two scales: an electronegativity scale and a “surface tension” scale. One can make accurate predictions using this scheme for the heats of formation of intermetallic alloys. Specifically, Miedema’s theory can be used to predict the existence of an ordered phase given the constituent species. Therefore, given a metal A and a metal B one can predict whether an ordered phase of the type A, B, will exist. However, within the context of Miedema’s theory, it is not possible to predict the stoichiometry of the phase, the structural properties of the phase, or the thermodynamic properties of the phase, e.g. melting points. Some of these issues have been addressed within the last few years [2]. Namely, the creation of chemical coordinates such as orbital radii has made possible the prediction of crystal structures for a given composition. For example, the work of Villars [3] has identified chemical indices based on electronegativety scales and orbital radii which appear superior to the currently available ones for structural properties.
to whom
correspondence
should
thermodynamic
trends.
Villars has made a number of predictions as to the structural properties of phases which are known to be ordered, but have not yet been experimentally verified to exist as a specific type of structure. However, little success has been achieved with respect to the other issues raised above. We are not in a position to predict whether specific phases are stable or to predict solubility trends [4], nor are we in a good position to predict melting or boiling points as a function of alloy composition. It is this latter issue which we will concentrate on in this manuscript. We have sought to predict the melting points of alloys from a knowledge of elemental properties. There has been little work in this area; a notable exception is the work of Van Vechten [5]. He has invented a scaling theory of melting in covalent crystals, e.g. Si, GaAs, ZnS, etc. While his theory is very accurate for these crystals, it is not directly applicable to metallic alloys as his theory is based on a dielectric theory appropriate for semiconducting or insulating materials. In this work, we shall examme the melting points of some 500 alloy systems. We shall concentrate on binary (1: 1) alloy systems in order to establish a significant and uniform data base. We have used the metallurgical handbooks of Hansen, Elliot, Shunk and Moffat [6] to extract the melting points for our study. For the most part determining the melting points is straight-forward. We simply read off the melting point at the (1: 1) composition. However, for some cases the melting point determination is not trivial. For example, in the case where the liquidus and solidus do not coincide, there is no well defined melting point. As a first approximation, we have
1. INTRODUCTION
t Author
trends,
be addressed. 197
JAMES R. CHELIKOWSKY
198
and KAREN E. ANDEIWN
taken the liquidus curve at the (1: 1) composition for the “melting point”. This choice allows us to include a number of systems which otherwise would be omitted. One could have chosen instead to use the solidus curve as our “melting point” with little change in the essential findings of our study. Another issue, related to the constructing of a melting point data base is the accuracy of the experimental data. It can be a nontrivial exercise to determine whether the system of interest has reached thermodynamic equilibrium. We have assumed that equilibrium has been achieved in the cases at hand and that the number of nonequilibrium determinations are not sufficient to alter our essential findings.
2. CHEMICAL COORDINATES MELTING POINTS
FOR
In choosing chemical coordinates for correlating melting points for examining melting point trends, we were guided by the recent work of Villars [3]. In his study of crystal structures, he examined a number of indices and attempted to categorize such coordinates into general classes. He grouped the coordinates into five broad classes which can be associated with variables such as atomic radii, atomic number, cohesive energy, electronegativity and the number of valence electrons. We attempted to use combinations of these classes to look for correlations of melting points with these elemental variables. In Table 1 we list the coordinates used in our study. In all, we examined ten sets of coordinates and combinations thereof. Not surprisingly, we found very poor correlations with the melting point for the vast majority of combinations. Like, crystal structures, variations in melting points involve subtle differences in total energies and wide variations between seemingly similar materials exist. As an example of how poor the correlations can be, in Fig. 1 we display melting point data for in the order of 500 alloys involving simple and transition metals [7], but no rare earths. The correlation is so poor that we have not attempted to quantify the results. However, we note that the result is not atypical. If we had used atomic radii, the
Table 1. Chemical coordinates used to examine melting points. For a discussion of chemical coordinates see Chelikowsky J. R. and Phillips J. C., Phys. Rev. B17, 2453 (1978) Atomic radii Orbital radii Electronegativity Electron number Electron density Principal quantum Cohesive energy Melting points Crystal structure Bulk moduli
Fig. 1. Binary alloy melting points vs differences in the elemental electronegativity as taken from the scale by Pauling L., The Nature of the Chemical Bond (3rd edn), Cornell, Ithaca (1960).
results would be equally bad. Of all the combinations we used, only those involving cohesive energies or elemental melting points yielded quantitative results. In determining our correlations, we found it useful to categorise the results in terms of the types of metals involved in the alloy. We will discuss the metal systems in terms of transition-transition metal alloys, simple-transition metal alloys and simple-simple metal alloys. The elemental melting points we used in establishing the correlations are given in Table 2.
3. USING ELEMENTAL MELTING POINTS AS CHEMICAL COORDINATES FOR PREDICTING INTERMETALLIC ALLOY MELTING POINTS
3.1. Transition metal-transition
metal alloys
Given 24 transition metals, there exist 276 different binary alloy systems. We do not include the noble metals: Cu, Ag and Au within this group or the divalent metals: Zn, Cd and Hg. We were able to find data for 129 systems; the alloys are listed in Table 3 along with the melting points We have not distinguished in this group between those alloys which are ordered and those which are not. We will address this question in a later section, but we note here that the trends we find are not a function of whether ordering occurs or not. In Fig. 2 we present our correlation with melting points. We have presented in this figure the observed melting points of the alloy and the elemental average of the individual constituents. For reference, we indicate the best least squares fit of the linear average to the observed melting points. We have used the following expression for the melting points:
number
T’(M) where T/(AB) melting point
= a [ T’(‘4 ) + T’(B)]/2 + b,
(1)
is the alloy melting point, T’(A) is the for the elemental metal A and T’(B)
Melting
point
trends
in intermetallic
199
alloys
Table 2. Elemental melting points from Gschneidner K., Jr., Solid St. Phys. 16, 275 (1964). Also given are elemental crystal structures from Wyckoff R. W. G., Crystal Sfructurer (2nd edn), Interscience, New York (1963); the nomenclature for the crystal structure is from Kittel C., So/id St. Phys. (5th edn), Wiley, New York (1976)
Element Li Be Na Mg Al Si K Ca SC
Ti V Cr Mn Fe 2 CU Zn Ga Ge Rb Sr Y Zr Nb MO Tc Ru Rh Pd Ag Cd In Sn cs Ba La Hf Ta W Re OS Ir Pt Au Hg Tl Pb
Melting point (“C) 181 1284 98 650 660 1412 63 839 1539 1668 1905 1875 1244 1535 1492 1453 1083 420 30 936 39 772 1502 1850 2468 2615 2170 2280 1960 1552 961 321 156 232 29 725 920 2495 3271 3653 3433 3300 2716 2042 1336 -39 576 601
Lattice parameters Structure bee
W bee hcp fee diam bee fee hcp hcp bee bee complex bee hcp fee fee hcp complex diam bee fee hcp hcp bCC bcC hCp hCp
fee fee fee hcp tetr diam bee bee hex hcp bee bee hcp hcp fee fee fee rhom hcp fee
(A) 3.49 2.27,3.59 4.23 3.21,5.21 4.05 5.43 5.23 5.58 3.31, 5.27 2.95,4.68 3.03 2.88 2.87 2.51,4.07 3.52 3.61 2.66,4.95 5.66 5.59 6.08 3.65, 5.73 3.23,5.15 3.30 3.15 2.74,4.40 2.71,4.28 3.80 3.89 4.09 2.98,5.62 3.25,4.95 6.49 6.05 5.02 3.77 3.19,5.05 3.30 3.16 2.76,4.46 2.74,4.32 3.84 3.92 4.08
Fig. 2. Transition metal-transition metal alloy melting points as observed experimentally plotted vs the average of the elemental melting points. The dashed line is a least squares fit to the data.
3.2. Simple metal-transition
metal alloys
We found that the same chemical coordinates for the transition-transition metal alloys which gave a good correlation for the melting points were the best coordinates for the simple metal-transition metal alloys. However, the correlation between the average of the elemental melting points and the alloy melting points is not as accurate. In Fig. 3, we present the average melting points vs the observed melting points for 172 alloy combinations. The rms deviation of the observed melting point from the elemental average is about 412°C. The range is from 620°C for BiPd to 3928°C for HfC or about a 12% error. This error estimate is somewhat misleading in that many alloy combinations occur in the lOOO_1500°C range and the distribution of melting points over the range spanned by BiPd and HfC is not uniform. In Table 4 we present the alloys involved in Fig. 3. It is interesting to observe that the largest discrepancies between the average elemental melting points and the
3.46,5.52 4.95
is the melting
point for the elemental metal B. (a&) are constants determined by a least square fit. We find a = 0.96 and b = - 133°C; the rms error in the fit is cu 215°C. The range of alloy melting points runs from LaNi (723°C) to TaW (3240°C): a span of ca 2500°C. Thus, the fit is good to slightly better than 9%. If one takes the average elemental temperatures the error is slightly larger, i.e. in the order of a 12% error. In either case, the correlation is quite good considering the nature of the problem especially when contrasted to the results presented in Fig. 1.
Fig. 3. Transition vs the average
metal-simple metal alloy melting points of the elemental melting points.
JAMES R. CHELIKOWSKY and KAREN E. ANDERSON
200
Table 3. Transition metal-transition metal alloy melting points. The melting points were defined as in the text. The first column gives the experimental values from Ref. [6]. The second column gives the average of the elemental melting points.
The temperatures are in “C CoCr CoHf Colr CoLa CoMn CoMo
1420 1640 1400 730 1190 1580
1683 1857 1967 1206 1368 2053
CoNb CoNi
1825 1470
1980 1472
CoPd CoPt CoTi cov COY CoZr CrHf CrIr CrMo CrNb CrNi CrOs CrPd CrPt CrRe CrRh CrRu Cr.% CrTi CrTa CrV
1217 1500 1300 1310 1020 1380 1730 2050 2140 1740 1375 1960 1320 1750 2400 1650 1780 1360 1980 1390 1900
1522 1630 1580 1698 1497 1671 2048 2159 2245 2171 1664 2451 1713 1822 2517 1917 2077 1707 2436 1771 1890
CrZr FeHf FeLa
1550 1700 1240
1862 1878 1227
FeMo FeNb FePd FePt Fe& FeTa FeTi FeV FeZr HfIr HfMn HfMo HfNb HfNi HfPd HfRe HfTi HfV HfZr
1850 1580 1310 1590 1460 1520 1320 1500 1500 2400 1640 2040 2100 1530 1720 2900 1840 1500 2025
IrMo IrNb IrPt IrRh IrTa IrTi IrV IrW LaMn LaNi LaRh LaSc MnPd
2240 1950 2100 2260 2060 2140 1920 2480 1020 715 1440 960 1515
2075 2001 1543 1652 1537 2266 1601 1720 1692 2332 1733 2418 2345 1837 1887 2691 1945 2063 2036 2529 2455 2106 2201 2720 2055 2174 2911 1082 1186 1440 1229 1398
observed melting points involve the transition metals La and Y. For example, LaPb is observed to melt at ca 1300°C while the average melting point of La and Pb is ca 600°C. Of the 15 alloy combinations with the largest discrepancies between the observed melting point and the linear average of the elemental melting points, 10 involve the elements La and Y.
MnPt MnTi MnY MoNb MoPd MoPt MoRe MoRh MoRu MoTa MoTi MOW MoZr NbNi NbPd NbPt NbRe NbRh NbRu NbSc NbTa NbV NbW NbZr NiPd NiPt NiTa NIV
1500 1270 1040 2560 1820 2175 2440 2050 2020 2800 2190 2950 1880 1260 1580 1750 2500 1560 1942 1970 2700 1900 3000 2000 1237 1520 1675 1220
NiY NiZr OsRu OsTi
1065 1270 2750 2160
1506 1456 1373 2541 2083 2192 2887 2287 2447 2806 2141 2997 2232 1960 2010 2118 2814 2214 2374 2003 2733 2186 2924 2159 1502 1611 2225 1679 1477 1651 2653 2347
osv osw OsZr PdSc PdTi PdY PdZr PtRh PtRu PtTi Ptw PtZr ReZr RhTa RhW RuSc RuTi RuV RuW RuZr ScTi scv SCY TaTi TaW TaZr TcV TiV TiY TiZr VZr WZr YZr
1960 2800 2200 1600 1400 1440 1600 1940 2075 1830 2350 2100 2450 1800 2240 2200 2150 1860 2220 2100 1300 1640 1365 2250 3240 2040 1900 1670 1560 1575 1250 2640 1370
2466 3203 2438 1545 1610 1527 1701 1864 2024 1718 2574 1809 2505 2479 2670 1909 1974 2092 2830 2065 1603 1722 1520 2333 3189 2424 2037 1786 1585 1759 1877 2615 1676
4. CHEMICAL TRENDS IN THE MELTING POINTS OF INTERMETALLIC ALLOYS
In examining Figs 2-4, we can make several generalizations about alloying and the resulting stability as indicated by the melting point trends. For example, the transition metal-transition metal alloys melt at temperatures below what one would expect from a simple average of the melting points. If we examine
3.3. Simple metal-simple metal alloys We have considered ca 240 simple metal alloys in Fig. 4 where we display the average melting point vs the observed melting point. The rms deviation of the observed melting point of the alloy from the average of the melting points is 305°C. The range is from CsK which melts at ca - 38°C to the BSi alloy which melts at about 1750°C. Thus, the results for the simple metal-simple metal alloys is the poorest of the three combinations with an average error of ca 17%. Again, there exist some well defined trends for those alloys which exhibit the largest deviations from the average of the melting points. Of the top 12 alloys with respect to the deviations, all but one (BiCa) involve either Ga, or a chalcogen. For example, the lead salts such as PbS or PbSe have melting points of about 1100°C; the average of the elemental melting points for these species is about 250°C.
Fig. 4. Simple metal-simple metal alloy melting points the average of the elemental melting points.
vs
201
Melting point trends in intermetallic alloys
Table 4. Transition metal-simple metal alloy melting points. The melting points were defined as in the text. The first column gives the experimental values from Ref. [a]. The second column gives the average of the elemental melting points. The temperatures are in “C AgLa AgPt AgSc A8Y ABZr AlCo AlCr AlFe AlHf AlLa AlMn AlMo AlNb AlNi AlPd AlPt AlRu AlSc AlTi AlY AlZr AsCo AsFe AsMn AsMo AsNi AsPd AsPt AuCo AuFe AuLa AuMn AuNi AuPd AuTi AuV AuZr BCo BCr BHf BMn BMo BNb
880 1550 1230 1160 1135 1645 1600 1340 1300 1100 1205 1760 1700 1638 1645 1554 2040 1300 1470 1290 1500 1180 1030 935 1010 962 730 1375 1160 1140 1325 1260 975 1450 1490 1360 1360 1460 2100 3040 1890 2575 2917
940 1365 1250 1231 1405 1076 1267 1097 1441 790 952 1637 1564 1056 1106 1214 1470 1099 1164 1081 1255 1154 1176 1030 1716 1135 1184 1293 1277 1299 991 1153 1258 1307 1365 1484 1456 1858 2050 2223 1743 2420 2346
BNi BRe BRu BTa BTi BV BW BZr BeCo BeCr BeFe BeNi BePd BiIr BiLa BiNi BiPd BiPt BiRh BiY BiZr CHf CLa CMo CNb CTa CTi cv CY CaLa CaNi CdLa cocu CoGe cos CoSb CoSe CoSi CoSn CoTe CrGa CrGe
1100 2080 1600 3090 2790 2580 2640 2700 1505 1650 1290 1472 1465 1420 1650 1050 1200 620 765 1150 2020 1380 3928 1410 2620 3530 3880 3020 2625 1825 845 955 946 1380 1160 1200 1055 1450 1140 1000 1380 1200
1839 2692 2252 2611 1946 2065 2802 2037 1388 1579 1409 1368 1418 1357 595 757 862 911 1020 1115 886 1060 3024 2373 3221 3147 3412 2747 2866 2664 879 1146 620 1287 1214 805 1061 854 1452 862 971 952 1405
the simple-transition metal alloys in Fig. 4, we find a very interesting result: the observed melting points are higher than the average melting for the elemental constituents. Although the detailed comparison between the average and the observed melting points is only fair, the number of melting points above the average far exceeds that below the average melting point. We also note that the melting points of the simple-simple metal alloys reside below the melting point of the transition melting points. The overlap between the two alloy groups exists mostly because we have included some elements in our data base such as boron, silicon and carbon which can hardly be described as metallic, but are included in many of the binary alloy phase diagram data bases. The simpletransition metal alloys span a range which strongly overlaps both groups of alloys and contain the highest melting points. We can consider an empirical approach for some
CrSb CrSi CuFe CuHf CuLa CuMn CuNi CuPd cusc CuTi CUY CuZr FeGa FeGe FeS FeSb FeSi GaMn GaNi GaPt GaTi GaV GaZr GeHf GeLa GeMn GeMo GeNi GePd GePt GeY HlSi HfSn HgLa InLa InMn InNi InPd InPt LaMg LaPb LaSb LaSe
1110 1450 1440 1130 700 935 1300 1240 1125 985 935 935 1010 1030 1170 1010 1410 845 1180 1104 1230 1470 1510 2100 1370 760 1720 910 830 1075 1900 2200 1725 1078 1125 860 940 1260 1020 745 1370 1600 1980
1253 1643 1309 1652 1001 1163 1268 1317 1311 1375 1292 1466 782 1235 827 1083 1473 637 741 899 849 967 940 1579 928 1090 1775 1194 1244 1352 1219 1817 1227 440 538 700 805 854 963 785 623 775 568
LaSn LaTl LaZn MnP MnSb MnSi MnSn MnTe MnZn NbGa NbGe NbSb NbSn NiS NiSb NiSe Nisi NiSn NiSr NiZn PbPd PbPt PbY PdSb PdSi PdSn PdTe PdTi PtSb PtSi PtSn PtTe PtTl RhSb RuSi SbY SiTi SiV SiY SiZr SnTi SnY YZn
1260 1180 815 1147 830 1275 820 1165 960 1680 1940 1675 1930 975 1153 980 992 1200 1250 1060 620 860 1590 805 1090 850 720 875 1185 1229 1305 1030 900 1310 1800 2310 1880 1860 1835 2220 1480 1720 1095
576 611 670 644 937 1328 738 847 832 1249 1702 1549 1350 786 1042 835 1432 842 1112 936 939 1048 914 1091 1482 892 1001 927 1200 1590 1000 1109 1036 1295 1846 1066 1540 1658 1457 1631 950 867 961
of the observed changes upon melting. Consider the case of an elemental metal. We know from previous studies [8] that the melting point T’ follows the cohesive energy E_,,,, i.e. Tf x aE,,,. This is a physically understandable result as it suggests that stronger bonds, which are correlated with the cohesive energy, should result in higher melting points. In general, this can be seen by an examination of some representative elements. For a binary alloy, following the discussion above, we hypothesize: T’(AB) = a OEco&)
+ &h(B)112 + dE(AB)),
(2)
where AE(AB) is defined as the difference in the cohesive energy of the solid from the sum of the two elemental cohesive energies. This formulation is clearly unsatisfactory for the transition-transition metal alloys. Given that the heat of formation in this sign convention would be positive for an increase in
JAMESR. CHEL~KOWSKY and
202
Table 5. Simple metal-simple metal alloy melting points. gives the experimental values from Ref. 161. The second
KAREN
E.
ANDER~~N
The melting points were defined as in the text. The first column column gives the average of the elemental melting points. The
temperatures are in “C AgAl AgAs AgBa AgBe AgCa AgCd Agcu Aga AgGe AgHg AgIn AgLi AgMg AgNa AgSb AgSi AgSn AgSr AgTe AgTl AgZn AlAu AlB AlBa AlBe AlCu AlGa AlGe AlHg AiLi AtMg AlSb AISe AlSr AlTe AlZn AsBi AsCd AsCu AsGa AsGe AsIn ASK AsPb AsSb AsSe AsSi AsSn AsTe AsZn AuBe AuBi Au&a AuCd AuCs AuCu AuGa AuGe AuHg AuIn
660 660 120 1030 666 710 823 450 800 600 300 480 820 743 1:: 480 680 460 630 690 820 1600 773 1140 810 410 635 363 718 4fio IO65 960 820 830 510 630 670 640 1238 737 942 62.3 300 683 295 1083 393 860 390 730 330 990 627 390 890 461 610 400 510
810 889 343 1122 900 641 1022 493 948 461 539 571 805 529 796 1186 596 866 703 632 690 861 1442 692 972 871 345 798 310 420 633 643 438 716 553 540 544 369 950 423 876 487 440 372 724 317 II14 324 633 618 1173 667 931 692 346 1073 546 999 512 610
AUK AuLi AuMg
AuNa AuPb AuRb AuSb AuSe AuSi AuSn AuSr AuTe AuTI AuZn BSi BaCa BaCd BaCu BaGa BaGe BaHg BaIn BaMg BaNa BaPb BaSi BeCu BiCa BiCs BiCu BiGa BiHg BiIn BiK BiLi BiMg BiNa BiPb BiRb BiS BiSe BiSr BiTe BiTl BiZn CaCu CaGa CaGe CaHg Cain CaLi CaMg CaNa CaPb CaSb CaSi CaSn CaTl CaZn CdCu
630 643 1150
870 400 510 430 780 940 418 960 500 460 723 1730 603 581 570 675 1145 822 683 320 313 900 925 980 1240 390 830 230 160 110 340 620 600 520 140 338 710 607 830 565 193 490 373 845 1320 961 895 670 640 1050 980 983 1243 987 970 580 550
363 622 836 580 695 531 847 640 1237 647 917 7.56 683 741 1818 782 323 904 317 830 343 441 687 411 326 1068 1 la3 535 150 677 130 116 214 167 226 460 184 299 135 195 244 321 360 287 343 961 434 887 400 498 310 744 468 383 733
1123 333 371 629 702
binding energy, we would expect transition alloys to exhibit a higher melting point than the average of the elemental melting points. Since this is not observed, we must conclude that other effects are important. One might conclude on the basis of our study that entropic considerations must play a significant role in
CdIn CdLi CdMg CdNa CdP CdPb CdSb Cd.% CdSr CdTe CdZn cswg CsK CsNa CsRb CsSb CsTl CllGa CuGe CuIn CuLi CuMg CuPb CuSb CuSe cusn cusr CUR? CuZn GaK Gati GaMg GaNa GaS GaSb GaSe GaSr GaTe GeLi GeS GeSe GeSi GeSr GeTe GeTl CeZn HgIn HgK. HgLi HgMkt HgNa HgPb HgRb HgSn RgSr HgTl HgZn InK InLi InMg
310 160 -38 42 9 386 375 660 750 640 880 670 975 570 670 615 605 620 880 640 740 380 540 1015 706 930 910 824 328 665 750 1270 1163 724 830 790 -19 180 595 627 220 160 160 135 830 85 225 440 633 675
239 231 485 209 182 324 476 269 546 385 370 -5 46 63 34 330 166 556 1009 620 632 866 705 837 650 657 921 766 751 46 105 340 64 74 330 123 401 240 558 527 376 1174 854 693 6t9 678 59 I2 71 305 29 144 0 96 366 132 190 110 169 403
InNa InPb InSb InSe InSn InSr InTe InTt InZn KNa KPb KSb KTl LiMg LiPb LiSl3 LiTl LiZn MgPb MgSb MgSn MgSr MgTl MgZn NaFb NaRb NaSb NaSe NaSn NaTe NaTl PbS PbSe PbSr PbTe PbTl RbSb RbTf SSe ssn STe ST1 SbSe SbSi SbSn SbSr SbTe SbTl SbZn SeSn SeSr SeTl sisr SiTe SnTe SnTi TeTl TeZn TlZn
400 240 530 640 130 760 696 190 835 5 370 610 290 540 4x2 485 510 485 450 870 630 620 358 323 370 42 465 330 580 430 303 1110 1083 940 917 370 610 360 130 883 383 260 531 1280 430 833 603 390 550 860 1600 330 1140 1150 806 215 370 1290 790
127 242 394 187 194 464 303 230 288 80 195 347 183 415 254 206 242 300 488 640 441 71I 476 535 212 68 364 157 165 274 200 223 272 549 388 31.5 335 171 168 175 284 211 424 1021 431 701 540 467 525 224 494 260 1092 931 341 267 376 435 361
determining melting points of transition metal alloys. We can estimate the size of these contributions by writing the following. AS%4B) = fA&(A) + AS,,(B)]/2 + AS(M),
(3)
where AS~{~B) is the entropy change on melting for
203
Melting point trends in intermetallic alloys the alloy, AS,,(A) is the change in entropy upon melting for the elemental metal A; likewise AS,,(B) is the entropy change for metal B, and AS(AB) is the deviation of the entropy of fusion of an alloy from the average of the changes in elemental entropies. If AE(AB) and AS(AB) are small compared to the corresponding elemental values, we can make an expansion and obtain: T’(AB)
x T’d,, (AB) [1 - AS(AB)/AS,,(AB) + AS(AB)
(4)
5. EFFECTS OF ORDER MELTING POINTS
ON
Given that we have not distinguished between ordered and disordered alloys in our correlations, one might raise some rather strong objections to our correlations. However, the trends presented in our figures are valid for at least one case of ordered alloys, We examined the melting points of approximately 50 CsCl structures. The structures examined are listed in Table 6 and the melting points are illustrated in Fig. 5. We find that the general trends in Figs 24 are preserved for our CsCl structures. For example, we obtain a good correlation between the
where E,,, (AB) = L% (A) +
L,W1/2
and AS,,, (AB) = [AS, (A) + AS,
(BW.
This expression contains the qualitative physics of melting point trends in that the melting points are depressed from T<,,, (AB) by the increased entropy over the average entropy change and are elevated from T{,, (AB) by an increase in the binding energy term. The interesting term for our discussion is the issue of the entropy term. In semiconductors, we can estimate this term by following the discussion of Van Vechten [5]. We take AS(AB) to be the entropy of mixing. Then the ratio AS,,,(AB)/AS,,, is about 0.1; however, in metals the ratio is much larger than in semiconducting solids. AS,,., = 4-6 Typically, cal/mole.deg for transition metal alloys [9] with AS,,,,, (AR) = 2R In (2) z 2 cal/mole.deg. Thus, in transition metal alloys, we find the ratio to be a factor of 3 or more larger than in the case of semiconductors. In both cases we have made a rather strong assumption, i.e. that the melt does not phase separate or have short range order. In the case of tetravalent semiconductors where As is the anion, Van Vechten found his scaling law to break down completely. If the ratio AE(AB)/E,,,(AB) is the same for both the semiconductor and transition metal case, we would expect to see a rather different behavior between the observed melting points and the average melting points. We would expect to see a considerable reduction in the observed melting points relative to the average melting points in the metal case. In fact, this is what we have observed in Figs 2 and 4. It is not trivial to estimate AE(AB)/E,,(AB) from our definition. However, we would expect intuitively that this ratio would be large for semiconductors and simple metal compounds involving metalloids than for transition metal alloys. For example, if we consider the heat of formation as an estimate for AE(AB), then for many transition metal alloys we would expect AE(AB) z 0. In this case, the melting points clearly would be depressed from the average of the melting points.
Table 6. Binary alloys which occur in the cesium chloride structure. The melting points are from Ref. [6]. The lattice parameters are from Wyckoff R. W. G., Crystal Structures (2nd edn), Interscience, New York (1963) Compound AgLa AgSc
A8Y AlCo AlLa AlNi AlRu AuCs AuLa AuMg AuRb AuZn BaCd BaHg BeCo BeNi BePd CaHg Caln CaTl CdLa CdSr CoHf cusc CUY FeTi GaNi HgLa HgLi HgMg HgSr HfRu InNi InPd LaMg LaTl LaZn LiTl MgTl OsTl osv OsZr PdSc RuSc RuTi TcV YZn
Melting point (“C) 800 1230 1160 1645 1100 1638 2040 590 1325 1150 510 725 581 822 1505 1472 1465 961 895 970 946 600 1640 1125 935 1320 1180 1078 595 627 850 2400 940 1260 145 1180 815 510 358 2150 1960 2200 1600 2200 2150 1870 1095
Lattice
constant (A) 3.80 3.41
3.62 2.86 3.19 2.89 2.99 4.26 3.14 3.21 4.10 3.13 4.22 4.13 2.61 2.62 2.82 3.15 3.86 3.86 3.91 4.01 3.17 3.25 3.48 2.98 2.88 3.85 3.29 3.45 3.93 3.23 3.09 3.25 3.96 3.93 3.76 3.43 3.64 3.07 3.01 3.26 3.28 3.20 3.07 3.03 3.58
204
JAMESR. CHELIKOWSKY and KARENE. ANDERSON
Fig. 5. Cesium chloride structure alloy melting points vs the average melting point of the elemental metal constituents. The squares are transition metal-transition metal alloys, the triangles are transition metal-simple metal alloys and the circles are simple metal-simple metal alloys.
average melting points and the observed melting points. We find that for simple-simple metal alloys, the majority of the observed melting points reside above the average of the elemental melting points; for transition-transition metal alloys the reverse is observed. Perhaps more significant is the existence of three general regions of melting points which arise in such a plot. The simple-simple metal alloys melt at lower temperatures than the simple-transition metal alloys which melt lower than the transition-transition metal alloys. This is some what surprising in that it suggests that some of the characteristics of the elemental bonding are carried over into the alloys. We do not feel that this would be the case for binary compounds which are nonmetallic. For example, this would not be the case for oxides.
melting point deviations would not correlate with the heat of formation. The only elemental variable with which we found a “reasonable” correlation with experiment was with volume changes upon compound formation. To examine such trends we had to restrict our study to ordered alloys where we could easily determine volume change upon alloy formation. Consider two elemental metals A and B, with volumes V(A) and V(B). From Vegards Law [IO], we would expect the (1: 1) alloy to have a volume V,,(AB) = [V(A) + V(B)]/2. We felt on the basis of studies of metallic glass [ 111, which have attempted to correlate glass forming ability and melting points with volume changes, that the deviations of the observed melting points from the average of the elemental melting points might correlate with the deviations from Vegards Law. Thus, we examined a plot of AT = T’(AB) - T&.(AB) vs AV = V(AB) - V,,(AB). Unlike the case of our correlations of alloys where we did not distinguish between ordered and disordered alloys, we do not have a large data base. In our study, we have included some compounds which disorder before melting. We argue that the volume changes upon disordering should be small compared to the volume changes formed upon achieving the binary alloy itself. In Fig. 6, we present the results of a correlation involving only the transition metaltransition metal alloys. We also examined the simple-simple metal alloys and the simple-transition metal alloys. The results were far from satisfactory. This conclusion is not too surprising given the rather poor correlations with the average melting points for the simple-simple and simple-transition metals. The general trend in Fig. 6 is consistent with what one might expect on intuitive grounds. Namely, that
6. CORRELATION OF VEGARD’S LAW FOR THE VOLUME OF INTERMETALLIC ALLOYS AND MELTING POINTS Given deviations in the averages of the melting points with the observed melting points, one can ask whether it is possible to correlate these deviations with the elemental variables listed in Table 1. We proceeded in a similar fashion to the general case of melting points. We found no elemental variable or combination of variables which led to a reasonable correlation. In particular, we attempted to correlate the deviations with the heat of formation of the alloy; we used both theoretical values for heats of formation and the experimentally known values. On the basis of our eqn (2) we felt that a large heat of formation might lead to a greater variation from the average of the elemental temperatures. However, given our discussion concerning the role of entropic contributions, it should not be surprising that the
Fig. 6. The difference in the observed melting points for transition metal alloy cesium chloride structures from the average of the elemental melting points vs the difference of the alloy atomic volume from the elemental average. The differences are normalized to the alloy melting point temperature and volume, respectively.
Melting point trends in intermetallic alloys the melting point is elevated in alloys which show a considerable contraction of volume upon alloying. As an example, RuSc which undergoes a significant contraction relative to the elemental volumes, is one of the few transition-transition metal alloys to exhibit an exceptionally high melting point, much above the average melting point of the constituents.
7. CONCLUSIONS In summary, we would like to note some consequences of our study with respect to contemporary chemical coordinates within the literature. First, with respect to Miedema’s scheme for predicting heats of formation, we would note that the transitiontransition metal alloys are easier to understand, i.e. obtain simple correlations with elemental properties, than are the alloys which involve simple metals. This conclusion that alloys with simple metals are not easily understood is not new. Often the bonding in simple metals is partially covalent and not amenable to a straight-forward description. Our second point is that the scheme of Villars, while promising for structural trends, is not applicable to melting points. This suggests that one might consider alloying energies to contain two elements: an isotropic contribution which does not depend on the details of the structural properties and an anisotropic contribution which does. Miedema’s theory of alloying is probably appropriate for the isotropic contributions and the gross features of the bonding process in metals, but is well known not to accurately reflect the structural properties of alloys. In this case, Villars’ theory would be appropriate only for structural energies. As pointed out by Villars, the combination of the two theories has resulted in the prediction of a number of yet to be discovered ordered alloys. Our present study should aid in providing a first estimate of the melting point of such phases.
205 REFERENCES
1. Miedema A. R., Boom R. and De Boer F. R., J. lesscommon Metals 4, 283 (1975); 45, 231 (1976); 46, 271 (1977) and Refs therein. 2. For example, see Burdett J. K., Price G. D. and Price S. L., Phys. Rev. B24, 2903 (1981); Chelikowsky J. R. and Phillips J. C., Phys. Rev. B17,2453 (1977); St. John J. and Bloch A. N., Phys. Rev. Lett. 33, 1095 (1974); Bloch A. and Schattemann G. C. in Structure and Bonding in Crystals (Edited by M. O’Keeffe and A. Navrotsky), Academic Press, New York (1981); Zunger A., Phys. Rev. Lett. 44, 582 (1980) and Phys. Rev. B22, 5839 (i980); Pettifor D., Phys. Rev. Lett. 42,846 (1979); Pettifor D. G. and Podlouckv R., Phvs. Rev. Lett. 53, 1080 (1984); and Varma C. My, Solid &ate Commun. 31, 295 (1979). 3. Villars P., J. less-common Metals 92, 215 (1983); 99. 33 (1984) and to published. 4. Chelikowskv J. R.. Phvs. Rev. B19. 686 (1979): Alonso J. A. and Simozar’ S.,.Phys. Rev. B22, i583 (i980). 5. Van Vechten J. A., Phys. Rev. Lett. 29, 769(1972); the classic paper is from Lindemann F. A., Z. Phys. 11,609 (1910). Recent work on elemental materials may be found in Ubbelohde A. R., The Molton State of Matter: Melting and Crystal Structures. Wiley, New York (1978). 6. Hansen M., Constitution of Binary Alloys, McGraw Hill, New York (1958); Elliot R. P., Constitution of Binary Alloys-First Supplement, McGraw Hill, New York (1965); Shunk F. A., Constitution of Binary Alloys~Sec&d Supplement, McGraw Hill, Gew York (1965); Moffatt W. G., Binary Phase Diagrams Handbook, General Electric, Schenectady, New York (1976). in the phase 7. We have included all the binary-alloys diagram books of Ref. [6]. Therefore, we include some elements which would not normally be considered metallic, e.g. C, Si, B, etc. We define simple metals to be those which do not have d valence electrons. 8. Guinea F., Rose J. H., Smith J. R. and Ferrante J., Appl. Phys. Lett. 44, 53 (1984). 9. Hultgren R., Orr R. L., Anderson P. D. and Kelly K. K., Selected Values of Thermodynamic Properties of Metals and Alloys, Wiley, New York (1963). 10. A discussion of Vegard’s law may be found in W. B. Pearson’s book: The Crystal Chemistry and Physics of Metals and Alloys, Wiley, New York (1972). 11. Yavari A. R., Hitter P. and Desre P., i de Chim. Phys. 79. 579 11982): Donald L. W. and Davies H. A.. J. of non-crysi. Solids 30,77 (1978) and Ramachandraiao P, Z. Metallkde. 71, 172 (1980).
Chem. Solids Vol. in Great Britain.
48,
No.
2, pp.
197-205,
1987 0
MELTING
POINT TRENDS
IN INTERMETALLIC
0022-3697/87 1987 Pergamon
$3.00 + 0.00 Journals Ltd.
ALLOYS
JAMESR. CHELIKOWSKY~and KAREN E. ANDERSON Science Laboratories, Exxon Research and Engineering Company, Annandale, NJ 08801, U.S.A.
Corporate
Research
(Receiued 12 May 1986; accepted 10 July 1986)
Abstract-We have examined the melting points of approximately 500 intermetallic binary alloys. We attempted to correlate the melting point behavior of the binary (1: 1) alloys with a number of elemental variables including electron number, atomic size, orbital radii, electronegativity, etc. We find that a “Vegards’s Law” of melting points works very well for predicting the melting points of binary transition metal alloys, i.e. the melting point of the alloy correlates with the linear average of the elementary melting points. However, this “law” works only moderately well for alloys involving simple metals. In addition, we find that transition metal alloys tend to have melting points below the averaged elemental melting points. This finding is in sharp contrast to simple metal alloys where the opposite trend is observed and it is indicative of fundamental differences between transition metal and simple metal binding. Finally, we have attempted to correlate deviations from a Vegard’s law of melting with elemental variables. We found no strong correlation with elemental variables (or the heats of formation of the alloy in question) with the possible exception being a correlation with elemental volume changes upon alloying. The consequences of this correlation upon alloy design and metallic alloy formation are briefly discussed. Keywork
Intermetallic
alloys,
melting
points,
chemical
Although it is not possible, at present, to predict the phase diagrams of alloys from a knowledge of the elemental constituents, considerable progress toward this goal has been made in the past few years. Most of the progress has centered on the creation of “chemical scales” with which one may examine trends and predict properties based on existing data. For example, Miedema [l] has recently developed a scheme based on two scales: an electronegativity scale and a “surface tension” scale. One can make accurate predictions using this scheme for the heats of formation of intermetallic alloys. Specifically, Miedema’s theory can be used to predict the existence of an ordered phase given the constituent species. Therefore, given a metal A and a metal B one can predict whether an ordered phase of the type A, B, will exist. However, within the context of Miedema’s theory, it is not possible to predict the stoichiometry of the phase, the structural properties of the phase, or the thermodynamic properties of the phase, e.g. melting points. Some of these issues have been addressed within the last few years [2]. Namely, the creation of chemical coordinates such as orbital radii has made possible the prediction of crystal structures for a given composition. For example, the work of Villars [3] has identified chemical indices based on electronegativety scales and orbital radii which appear superior to the currently available ones for structural properties.
to whom
correspondence
should
thermodynamic
trends.
Villars has made a number of predictions as to the structural properties of phases which are known to be ordered, but have not yet been experimentally verified to exist as a specific type of structure. However, little success has been achieved with respect to the other issues raised above. We are not in a position to predict whether specific phases are stable or to predict solubility trends [4], nor are we in a good position to predict melting or boiling points as a function of alloy composition. It is this latter issue which we will concentrate on in this manuscript. We have sought to predict the melting points of alloys from a knowledge of elemental properties. There has been little work in this area; a notable exception is the work of Van Vechten [5]. He has invented a scaling theory of melting in covalent crystals, e.g. Si, GaAs, ZnS, etc. While his theory is very accurate for these crystals, it is not directly applicable to metallic alloys as his theory is based on a dielectric theory appropriate for semiconducting or insulating materials. In this work, we shall examme the melting points of some 500 alloy systems. We shall concentrate on binary (1: 1) alloy systems in order to establish a significant and uniform data base. We have used the metallurgical handbooks of Hansen, Elliot, Shunk and Moffat [6] to extract the melting points for our study. For the most part determining the melting points is straight-forward. We simply read off the melting point at the (1: 1) composition. However, for some cases the melting point determination is not trivial. For example, in the case where the liquidus and solidus do not coincide, there is no well defined melting point. As a first approximation, we have
1. INTRODUCTION
t Author
trends,
be addressed. 197
JAMES R. CHELIKOWSKY
198
and KAREN E. ANDEIWN
taken the liquidus curve at the (1: 1) composition for the “melting point”. This choice allows us to include a number of systems which otherwise would be omitted. One could have chosen instead to use the solidus curve as our “melting point” with little change in the essential findings of our study. Another issue, related to the constructing of a melting point data base is the accuracy of the experimental data. It can be a nontrivial exercise to determine whether the system of interest has reached thermodynamic equilibrium. We have assumed that equilibrium has been achieved in the cases at hand and that the number of nonequilibrium determinations are not sufficient to alter our essential findings.
2. CHEMICAL COORDINATES MELTING POINTS
FOR
In choosing chemical coordinates for correlating melting points for examining melting point trends, we were guided by the recent work of Villars [3]. In his study of crystal structures, he examined a number of indices and attempted to categorize such coordinates into general classes. He grouped the coordinates into five broad classes which can be associated with variables such as atomic radii, atomic number, cohesive energy, electronegativity and the number of valence electrons. We attempted to use combinations of these classes to look for correlations of melting points with these elemental variables. In Table 1 we list the coordinates used in our study. In all, we examined ten sets of coordinates and combinations thereof. Not surprisingly, we found very poor correlations with the melting point for the vast majority of combinations. Like, crystal structures, variations in melting points involve subtle differences in total energies and wide variations between seemingly similar materials exist. As an example of how poor the correlations can be, in Fig. 1 we display melting point data for in the order of 500 alloys involving simple and transition metals [7], but no rare earths. The correlation is so poor that we have not attempted to quantify the results. However, we note that the result is not atypical. If we had used atomic radii, the
Table 1. Chemical coordinates used to examine melting points. For a discussion of chemical coordinates see Chelikowsky J. R. and Phillips J. C., Phys. Rev. B17, 2453 (1978) Atomic radii Orbital radii Electronegativity Electron number Electron density Principal quantum Cohesive energy Melting points Crystal structure Bulk moduli
Fig. 1. Binary alloy melting points vs differences in the elemental electronegativity as taken from the scale by Pauling L., The Nature of the Chemical Bond (3rd edn), Cornell, Ithaca (1960).
results would be equally bad. Of all the combinations we used, only those involving cohesive energies or elemental melting points yielded quantitative results. In determining our correlations, we found it useful to categorise the results in terms of the types of metals involved in the alloy. We will discuss the metal systems in terms of transition-transition metal alloys, simple-transition metal alloys and simple-simple metal alloys. The elemental melting points we used in establishing the correlations are given in Table 2.
3. USING ELEMENTAL MELTING POINTS AS CHEMICAL COORDINATES FOR PREDICTING INTERMETALLIC ALLOY MELTING POINTS
3.1. Transition metal-transition
metal alloys
Given 24 transition metals, there exist 276 different binary alloy systems. We do not include the noble metals: Cu, Ag and Au within this group or the divalent metals: Zn, Cd and Hg. We were able to find data for 129 systems; the alloys are listed in Table 3 along with the melting points We have not distinguished in this group between those alloys which are ordered and those which are not. We will address this question in a later section, but we note here that the trends we find are not a function of whether ordering occurs or not. In Fig. 2 we present our correlation with melting points. We have presented in this figure the observed melting points of the alloy and the elemental average of the individual constituents. For reference, we indicate the best least squares fit of the linear average to the observed melting points. We have used the following expression for the melting points:
number
T’(M) where T/(AB) melting point
= a [ T’(‘4 ) + T’(B)]/2 + b,
(1)
is the alloy melting point, T’(A) is the for the elemental metal A and T’(B)
Melting
point
trends
in intermetallic
199
alloys
Table 2. Elemental melting points from Gschneidner K., Jr., Solid St. Phys. 16, 275 (1964). Also given are elemental crystal structures from Wyckoff R. W. G., Crystal Sfructurer (2nd edn), Interscience, New York (1963); the nomenclature for the crystal structure is from Kittel C., So/id St. Phys. (5th edn), Wiley, New York (1976)
Element Li Be Na Mg Al Si K Ca SC
Ti V Cr Mn Fe 2 CU Zn Ga Ge Rb Sr Y Zr Nb MO Tc Ru Rh Pd Ag Cd In Sn cs Ba La Hf Ta W Re OS Ir Pt Au Hg Tl Pb
Melting point (“C) 181 1284 98 650 660 1412 63 839 1539 1668 1905 1875 1244 1535 1492 1453 1083 420 30 936 39 772 1502 1850 2468 2615 2170 2280 1960 1552 961 321 156 232 29 725 920 2495 3271 3653 3433 3300 2716 2042 1336 -39 576 601
Lattice parameters Structure bee
W bee hcp fee diam bee fee hcp hcp bee bee complex bee hcp fee fee hcp complex diam bee fee hcp hcp bCC bcC hCp hCp
fee fee fee hcp tetr diam bee bee hex hcp bee bee hcp hcp fee fee fee rhom hcp fee
(A) 3.49 2.27,3.59 4.23 3.21,5.21 4.05 5.43 5.23 5.58 3.31, 5.27 2.95,4.68 3.03 2.88 2.87 2.51,4.07 3.52 3.61 2.66,4.95 5.66 5.59 6.08 3.65, 5.73 3.23,5.15 3.30 3.15 2.74,4.40 2.71,4.28 3.80 3.89 4.09 2.98,5.62 3.25,4.95 6.49 6.05 5.02 3.77 3.19,5.05 3.30 3.16 2.76,4.46 2.74,4.32 3.84 3.92 4.08
Fig. 2. Transition metal-transition metal alloy melting points as observed experimentally plotted vs the average of the elemental melting points. The dashed line is a least squares fit to the data.
3.2. Simple metal-transition
metal alloys
We found that the same chemical coordinates for the transition-transition metal alloys which gave a good correlation for the melting points were the best coordinates for the simple metal-transition metal alloys. However, the correlation between the average of the elemental melting points and the alloy melting points is not as accurate. In Fig. 3, we present the average melting points vs the observed melting points for 172 alloy combinations. The rms deviation of the observed melting point from the elemental average is about 412°C. The range is from 620°C for BiPd to 3928°C for HfC or about a 12% error. This error estimate is somewhat misleading in that many alloy combinations occur in the lOOO_1500°C range and the distribution of melting points over the range spanned by BiPd and HfC is not uniform. In Table 4 we present the alloys involved in Fig. 3. It is interesting to observe that the largest discrepancies between the average elemental melting points and the
3.46,5.52 4.95
is the melting
point for the elemental metal B. (a&) are constants determined by a least square fit. We find a = 0.96 and b = - 133°C; the rms error in the fit is cu 215°C. The range of alloy melting points runs from LaNi (723°C) to TaW (3240°C): a span of ca 2500°C. Thus, the fit is good to slightly better than 9%. If one takes the average elemental temperatures the error is slightly larger, i.e. in the order of a 12% error. In either case, the correlation is quite good considering the nature of the problem especially when contrasted to the results presented in Fig. 1.
Fig. 3. Transition vs the average
metal-simple metal alloy melting points of the elemental melting points.
JAMES R. CHELIKOWSKY and KAREN E. ANDERSON
200
Table 3. Transition metal-transition metal alloy melting points. The melting points were defined as in the text. The first column gives the experimental values from Ref. [6]. The second column gives the average of the elemental melting points.
The temperatures are in “C CoCr CoHf Colr CoLa CoMn CoMo
1420 1640 1400 730 1190 1580
1683 1857 1967 1206 1368 2053
CoNb CoNi
1825 1470
1980 1472
CoPd CoPt CoTi cov COY CoZr CrHf CrIr CrMo CrNb CrNi CrOs CrPd CrPt CrRe CrRh CrRu Cr.% CrTi CrTa CrV
1217 1500 1300 1310 1020 1380 1730 2050 2140 1740 1375 1960 1320 1750 2400 1650 1780 1360 1980 1390 1900
1522 1630 1580 1698 1497 1671 2048 2159 2245 2171 1664 2451 1713 1822 2517 1917 2077 1707 2436 1771 1890
CrZr FeHf FeLa
1550 1700 1240
1862 1878 1227
FeMo FeNb FePd FePt Fe& FeTa FeTi FeV FeZr HfIr HfMn HfMo HfNb HfNi HfPd HfRe HfTi HfV HfZr
1850 1580 1310 1590 1460 1520 1320 1500 1500 2400 1640 2040 2100 1530 1720 2900 1840 1500 2025
IrMo IrNb IrPt IrRh IrTa IrTi IrV IrW LaMn LaNi LaRh LaSc MnPd
2240 1950 2100 2260 2060 2140 1920 2480 1020 715 1440 960 1515
2075 2001 1543 1652 1537 2266 1601 1720 1692 2332 1733 2418 2345 1837 1887 2691 1945 2063 2036 2529 2455 2106 2201 2720 2055 2174 2911 1082 1186 1440 1229 1398
observed melting points involve the transition metals La and Y. For example, LaPb is observed to melt at ca 1300°C while the average melting point of La and Pb is ca 600°C. Of the 15 alloy combinations with the largest discrepancies between the observed melting point and the linear average of the elemental melting points, 10 involve the elements La and Y.
MnPt MnTi MnY MoNb MoPd MoPt MoRe MoRh MoRu MoTa MoTi MOW MoZr NbNi NbPd NbPt NbRe NbRh NbRu NbSc NbTa NbV NbW NbZr NiPd NiPt NiTa NIV
1500 1270 1040 2560 1820 2175 2440 2050 2020 2800 2190 2950 1880 1260 1580 1750 2500 1560 1942 1970 2700 1900 3000 2000 1237 1520 1675 1220
NiY NiZr OsRu OsTi
1065 1270 2750 2160
1506 1456 1373 2541 2083 2192 2887 2287 2447 2806 2141 2997 2232 1960 2010 2118 2814 2214 2374 2003 2733 2186 2924 2159 1502 1611 2225 1679 1477 1651 2653 2347
osv osw OsZr PdSc PdTi PdY PdZr PtRh PtRu PtTi Ptw PtZr ReZr RhTa RhW RuSc RuTi RuV RuW RuZr ScTi scv SCY TaTi TaW TaZr TcV TiV TiY TiZr VZr WZr YZr
1960 2800 2200 1600 1400 1440 1600 1940 2075 1830 2350 2100 2450 1800 2240 2200 2150 1860 2220 2100 1300 1640 1365 2250 3240 2040 1900 1670 1560 1575 1250 2640 1370
2466 3203 2438 1545 1610 1527 1701 1864 2024 1718 2574 1809 2505 2479 2670 1909 1974 2092 2830 2065 1603 1722 1520 2333 3189 2424 2037 1786 1585 1759 1877 2615 1676
4. CHEMICAL TRENDS IN THE MELTING POINTS OF INTERMETALLIC ALLOYS
In examining Figs 2-4, we can make several generalizations about alloying and the resulting stability as indicated by the melting point trends. For example, the transition metal-transition metal alloys melt at temperatures below what one would expect from a simple average of the melting points. If we examine
3.3. Simple metal-simple metal alloys We have considered ca 240 simple metal alloys in Fig. 4 where we display the average melting point vs the observed melting point. The rms deviation of the observed melting point of the alloy from the average of the melting points is 305°C. The range is from CsK which melts at ca - 38°C to the BSi alloy which melts at about 1750°C. Thus, the results for the simple metal-simple metal alloys is the poorest of the three combinations with an average error of ca 17%. Again, there exist some well defined trends for those alloys which exhibit the largest deviations from the average of the melting points. Of the top 12 alloys with respect to the deviations, all but one (BiCa) involve either Ga, or a chalcogen. For example, the lead salts such as PbS or PbSe have melting points of about 1100°C; the average of the elemental melting points for these species is about 250°C.
Fig. 4. Simple metal-simple metal alloy melting points the average of the elemental melting points.
vs
201
Melting point trends in intermetallic alloys
Table 4. Transition metal-simple metal alloy melting points. The melting points were defined as in the text. The first column gives the experimental values from Ref. [a]. The second column gives the average of the elemental melting points. The temperatures are in “C AgLa AgPt AgSc A8Y ABZr AlCo AlCr AlFe AlHf AlLa AlMn AlMo AlNb AlNi AlPd AlPt AlRu AlSc AlTi AlY AlZr AsCo AsFe AsMn AsMo AsNi AsPd AsPt AuCo AuFe AuLa AuMn AuNi AuPd AuTi AuV AuZr BCo BCr BHf BMn BMo BNb
880 1550 1230 1160 1135 1645 1600 1340 1300 1100 1205 1760 1700 1638 1645 1554 2040 1300 1470 1290 1500 1180 1030 935 1010 962 730 1375 1160 1140 1325 1260 975 1450 1490 1360 1360 1460 2100 3040 1890 2575 2917
940 1365 1250 1231 1405 1076 1267 1097 1441 790 952 1637 1564 1056 1106 1214 1470 1099 1164 1081 1255 1154 1176 1030 1716 1135 1184 1293 1277 1299 991 1153 1258 1307 1365 1484 1456 1858 2050 2223 1743 2420 2346
BNi BRe BRu BTa BTi BV BW BZr BeCo BeCr BeFe BeNi BePd BiIr BiLa BiNi BiPd BiPt BiRh BiY BiZr CHf CLa CMo CNb CTa CTi cv CY CaLa CaNi CdLa cocu CoGe cos CoSb CoSe CoSi CoSn CoTe CrGa CrGe
1100 2080 1600 3090 2790 2580 2640 2700 1505 1650 1290 1472 1465 1420 1650 1050 1200 620 765 1150 2020 1380 3928 1410 2620 3530 3880 3020 2625 1825 845 955 946 1380 1160 1200 1055 1450 1140 1000 1380 1200
1839 2692 2252 2611 1946 2065 2802 2037 1388 1579 1409 1368 1418 1357 595 757 862 911 1020 1115 886 1060 3024 2373 3221 3147 3412 2747 2866 2664 879 1146 620 1287 1214 805 1061 854 1452 862 971 952 1405
the simple-transition metal alloys in Fig. 4, we find a very interesting result: the observed melting points are higher than the average melting for the elemental constituents. Although the detailed comparison between the average and the observed melting points is only fair, the number of melting points above the average far exceeds that below the average melting point. We also note that the melting points of the simple-simple metal alloys reside below the melting point of the transition melting points. The overlap between the two alloy groups exists mostly because we have included some elements in our data base such as boron, silicon and carbon which can hardly be described as metallic, but are included in many of the binary alloy phase diagram data bases. The simpletransition metal alloys span a range which strongly overlaps both groups of alloys and contain the highest melting points. We can consider an empirical approach for some
CrSb CrSi CuFe CuHf CuLa CuMn CuNi CuPd cusc CuTi CUY CuZr FeGa FeGe FeS FeSb FeSi GaMn GaNi GaPt GaTi GaV GaZr GeHf GeLa GeMn GeMo GeNi GePd GePt GeY HlSi HfSn HgLa InLa InMn InNi InPd InPt LaMg LaPb LaSb LaSe
1110 1450 1440 1130 700 935 1300 1240 1125 985 935 935 1010 1030 1170 1010 1410 845 1180 1104 1230 1470 1510 2100 1370 760 1720 910 830 1075 1900 2200 1725 1078 1125 860 940 1260 1020 745 1370 1600 1980
1253 1643 1309 1652 1001 1163 1268 1317 1311 1375 1292 1466 782 1235 827 1083 1473 637 741 899 849 967 940 1579 928 1090 1775 1194 1244 1352 1219 1817 1227 440 538 700 805 854 963 785 623 775 568
LaSn LaTl LaZn MnP MnSb MnSi MnSn MnTe MnZn NbGa NbGe NbSb NbSn NiS NiSb NiSe Nisi NiSn NiSr NiZn PbPd PbPt PbY PdSb PdSi PdSn PdTe PdTi PtSb PtSi PtSn PtTe PtTl RhSb RuSi SbY SiTi SiV SiY SiZr SnTi SnY YZn
1260 1180 815 1147 830 1275 820 1165 960 1680 1940 1675 1930 975 1153 980 992 1200 1250 1060 620 860 1590 805 1090 850 720 875 1185 1229 1305 1030 900 1310 1800 2310 1880 1860 1835 2220 1480 1720 1095
576 611 670 644 937 1328 738 847 832 1249 1702 1549 1350 786 1042 835 1432 842 1112 936 939 1048 914 1091 1482 892 1001 927 1200 1590 1000 1109 1036 1295 1846 1066 1540 1658 1457 1631 950 867 961
of the observed changes upon melting. Consider the case of an elemental metal. We know from previous studies [8] that the melting point T’ follows the cohesive energy E_,,,, i.e. Tf x aE,,,. This is a physically understandable result as it suggests that stronger bonds, which are correlated with the cohesive energy, should result in higher melting points. In general, this can be seen by an examination of some representative elements. For a binary alloy, following the discussion above, we hypothesize: T’(AB) = a OEco&)
+ &h(B)112 + dE(AB)),
(2)
where AE(AB) is defined as the difference in the cohesive energy of the solid from the sum of the two elemental cohesive energies. This formulation is clearly unsatisfactory for the transition-transition metal alloys. Given that the heat of formation in this sign convention would be positive for an increase in
JAMESR. CHEL~KOWSKY and
202
Table 5. Simple metal-simple metal alloy melting points. gives the experimental values from Ref. 161. The second
KAREN
E.
ANDER~~N
The melting points were defined as in the text. The first column column gives the average of the elemental melting points. The
temperatures are in “C AgAl AgAs AgBa AgBe AgCa AgCd Agcu Aga AgGe AgHg AgIn AgLi AgMg AgNa AgSb AgSi AgSn AgSr AgTe AgTl AgZn AlAu AlB AlBa AlBe AlCu AlGa AlGe AlHg AiLi AtMg AlSb AISe AlSr AlTe AlZn AsBi AsCd AsCu AsGa AsGe AsIn ASK AsPb AsSb AsSe AsSi AsSn AsTe AsZn AuBe AuBi Au&a AuCd AuCs AuCu AuGa AuGe AuHg AuIn
660 660 120 1030 666 710 823 450 800 600 300 480 820 743 1:: 480 680 460 630 690 820 1600 773 1140 810 410 635 363 718 4fio IO65 960 820 830 510 630 670 640 1238 737 942 62.3 300 683 295 1083 393 860 390 730 330 990 627 390 890 461 610 400 510
810 889 343 1122 900 641 1022 493 948 461 539 571 805 529 796 1186 596 866 703 632 690 861 1442 692 972 871 345 798 310 420 633 643 438 716 553 540 544 369 950 423 876 487 440 372 724 317 II14 324 633 618 1173 667 931 692 346 1073 546 999 512 610
AUK AuLi AuMg
AuNa AuPb AuRb AuSb AuSe AuSi AuSn AuSr AuTe AuTI AuZn BSi BaCa BaCd BaCu BaGa BaGe BaHg BaIn BaMg BaNa BaPb BaSi BeCu BiCa BiCs BiCu BiGa BiHg BiIn BiK BiLi BiMg BiNa BiPb BiRb BiS BiSe BiSr BiTe BiTl BiZn CaCu CaGa CaGe CaHg Cain CaLi CaMg CaNa CaPb CaSb CaSi CaSn CaTl CaZn CdCu
630 643 1150
870 400 510 430 780 940 418 960 500 460 723 1730 603 581 570 675 1145 822 683 320 313 900 925 980 1240 390 830 230 160 110 340 620 600 520 140 338 710 607 830 565 193 490 373 845 1320 961 895 670 640 1050 980 983 1243 987 970 580 550
363 622 836 580 695 531 847 640 1237 647 917 7.56 683 741 1818 782 323 904 317 830 343 441 687 411 326 1068 1 la3 535 150 677 130 116 214 167 226 460 184 299 135 195 244 321 360 287 343 961 434 887 400 498 310 744 468 383 733
1123 333 371 629 702
binding energy, we would expect transition alloys to exhibit a higher melting point than the average of the elemental melting points. Since this is not observed, we must conclude that other effects are important. One might conclude on the basis of our study that entropic considerations must play a significant role in
CdIn CdLi CdMg CdNa CdP CdPb CdSb Cd.% CdSr CdTe CdZn cswg CsK CsNa CsRb CsSb CsTl CllGa CuGe CuIn CuLi CuMg CuPb CuSb CuSe cusn cusr CUR? CuZn GaK Gati GaMg GaNa GaS GaSb GaSe GaSr GaTe GeLi GeS GeSe GeSi GeSr GeTe GeTl CeZn HgIn HgK. HgLi HgMkt HgNa HgPb HgRb HgSn RgSr HgTl HgZn InK InLi InMg
310 160 -38 42 9 386 375 660 750 640 880 670 975 570 670 615 605 620 880 640 740 380 540 1015 706 930 910 824 328 665 750 1270 1163 724 830 790 -19 180 595 627 220 160 160 135 830 85 225 440 633 675
239 231 485 209 182 324 476 269 546 385 370 -5 46 63 34 330 166 556 1009 620 632 866 705 837 650 657 921 766 751 46 105 340 64 74 330 123 401 240 558 527 376 1174 854 693 6t9 678 59 I2 71 305 29 144 0 96 366 132 190 110 169 403
InNa InPb InSb InSe InSn InSr InTe InTt InZn KNa KPb KSb KTl LiMg LiPb LiSl3 LiTl LiZn MgPb MgSb MgSn MgSr MgTl MgZn NaFb NaRb NaSb NaSe NaSn NaTe NaTl PbS PbSe PbSr PbTe PbTl RbSb RbTf SSe ssn STe ST1 SbSe SbSi SbSn SbSr SbTe SbTl SbZn SeSn SeSr SeTl sisr SiTe SnTe SnTi TeTl TeZn TlZn
400 240 530 640 130 760 696 190 835 5 370 610 290 540 4x2 485 510 485 450 870 630 620 358 323 370 42 465 330 580 430 303 1110 1083 940 917 370 610 360 130 883 383 260 531 1280 430 833 603 390 550 860 1600 330 1140 1150 806 215 370 1290 790
127 242 394 187 194 464 303 230 288 80 195 347 183 415 254 206 242 300 488 640 441 71I 476 535 212 68 364 157 165 274 200 223 272 549 388 31.5 335 171 168 175 284 211 424 1021 431 701 540 467 525 224 494 260 1092 931 341 267 376 435 361
determining melting points of transition metal alloys. We can estimate the size of these contributions by writing the following. AS%4B) = fA&(A) + AS,,(B)]/2 + AS(M),
(3)
where AS~{~B) is the entropy change on melting for
203
Melting point trends in intermetallic alloys the alloy, AS,,(A) is the change in entropy upon melting for the elemental metal A; likewise AS,,(B) is the entropy change for metal B, and AS(AB) is the deviation of the entropy of fusion of an alloy from the average of the changes in elemental entropies. If AE(AB) and AS(AB) are small compared to the corresponding elemental values, we can make an expansion and obtain: T’(AB)
x T’d,, (AB) [1 - AS(AB)/AS,,(AB) + AS(AB)
(4)
5. EFFECTS OF ORDER MELTING POINTS
ON
Given that we have not distinguished between ordered and disordered alloys in our correlations, one might raise some rather strong objections to our correlations. However, the trends presented in our figures are valid for at least one case of ordered alloys, We examined the melting points of approximately 50 CsCl structures. The structures examined are listed in Table 6 and the melting points are illustrated in Fig. 5. We find that the general trends in Figs 24 are preserved for our CsCl structures. For example, we obtain a good correlation between the
where E,,, (AB) = L% (A) +
L,W1/2
and AS,,, (AB) = [AS, (A) + AS,
(BW.
This expression contains the qualitative physics of melting point trends in that the melting points are depressed from T<,,, (AB) by the increased entropy over the average entropy change and are elevated from T{,, (AB) by an increase in the binding energy term. The interesting term for our discussion is the issue of the entropy term. In semiconductors, we can estimate this term by following the discussion of Van Vechten [5]. We take AS(AB) to be the entropy of mixing. Then the ratio AS,,,(AB)/AS,,, is about 0.1; however, in metals the ratio is much larger than in semiconducting solids. AS,,., = 4-6 Typically, cal/mole.deg for transition metal alloys [9] with AS,,,,, (AR) = 2R In (2) z 2 cal/mole.deg. Thus, in transition metal alloys, we find the ratio to be a factor of 3 or more larger than in the case of semiconductors. In both cases we have made a rather strong assumption, i.e. that the melt does not phase separate or have short range order. In the case of tetravalent semiconductors where As is the anion, Van Vechten found his scaling law to break down completely. If the ratio AE(AB)/E,,,(AB) is the same for both the semiconductor and transition metal case, we would expect to see a rather different behavior between the observed melting points and the average melting points. We would expect to see a considerable reduction in the observed melting points relative to the average melting points in the metal case. In fact, this is what we have observed in Figs 2 and 4. It is not trivial to estimate AE(AB)/E,,(AB) from our definition. However, we would expect intuitively that this ratio would be large for semiconductors and simple metal compounds involving metalloids than for transition metal alloys. For example, if we consider the heat of formation as an estimate for AE(AB), then for many transition metal alloys we would expect AE(AB) z 0. In this case, the melting points clearly would be depressed from the average of the melting points.
Table 6. Binary alloys which occur in the cesium chloride structure. The melting points are from Ref. [6]. The lattice parameters are from Wyckoff R. W. G., Crystal Structures (2nd edn), Interscience, New York (1963) Compound AgLa AgSc
A8Y AlCo AlLa AlNi AlRu AuCs AuLa AuMg AuRb AuZn BaCd BaHg BeCo BeNi BePd CaHg Caln CaTl CdLa CdSr CoHf cusc CUY FeTi GaNi HgLa HgLi HgMg HgSr HfRu InNi InPd LaMg LaTl LaZn LiTl MgTl OsTl osv OsZr PdSc RuSc RuTi TcV YZn
Melting point (“C) 800 1230 1160 1645 1100 1638 2040 590 1325 1150 510 725 581 822 1505 1472 1465 961 895 970 946 600 1640 1125 935 1320 1180 1078 595 627 850 2400 940 1260 145 1180 815 510 358 2150 1960 2200 1600 2200 2150 1870 1095
Lattice
constant (A) 3.80 3.41
3.62 2.86 3.19 2.89 2.99 4.26 3.14 3.21 4.10 3.13 4.22 4.13 2.61 2.62 2.82 3.15 3.86 3.86 3.91 4.01 3.17 3.25 3.48 2.98 2.88 3.85 3.29 3.45 3.93 3.23 3.09 3.25 3.96 3.93 3.76 3.43 3.64 3.07 3.01 3.26 3.28 3.20 3.07 3.03 3.58
204
JAMESR. CHELIKOWSKY and KARENE. ANDERSON
Fig. 5. Cesium chloride structure alloy melting points vs the average melting point of the elemental metal constituents. The squares are transition metal-transition metal alloys, the triangles are transition metal-simple metal alloys and the circles are simple metal-simple metal alloys.
average melting points and the observed melting points. We find that for simple-simple metal alloys, the majority of the observed melting points reside above the average of the elemental melting points; for transition-transition metal alloys the reverse is observed. Perhaps more significant is the existence of three general regions of melting points which arise in such a plot. The simple-simple metal alloys melt at lower temperatures than the simple-transition metal alloys which melt lower than the transition-transition metal alloys. This is some what surprising in that it suggests that some of the characteristics of the elemental bonding are carried over into the alloys. We do not feel that this would be the case for binary compounds which are nonmetallic. For example, this would not be the case for oxides.
melting point deviations would not correlate with the heat of formation. The only elemental variable with which we found a “reasonable” correlation with experiment was with volume changes upon compound formation. To examine such trends we had to restrict our study to ordered alloys where we could easily determine volume change upon alloy formation. Consider two elemental metals A and B, with volumes V(A) and V(B). From Vegards Law [IO], we would expect the (1: 1) alloy to have a volume V,,(AB) = [V(A) + V(B)]/2. We felt on the basis of studies of metallic glass [ 111, which have attempted to correlate glass forming ability and melting points with volume changes, that the deviations of the observed melting points from the average of the elemental melting points might correlate with the deviations from Vegards Law. Thus, we examined a plot of AT = T’(AB) - T&.(AB) vs AV = V(AB) - V,,(AB). Unlike the case of our correlations of alloys where we did not distinguish between ordered and disordered alloys, we do not have a large data base. In our study, we have included some compounds which disorder before melting. We argue that the volume changes upon disordering should be small compared to the volume changes formed upon achieving the binary alloy itself. In Fig. 6, we present the results of a correlation involving only the transition metaltransition metal alloys. We also examined the simple-simple metal alloys and the simple-transition metal alloys. The results were far from satisfactory. This conclusion is not too surprising given the rather poor correlations with the average melting points for the simple-simple and simple-transition metals. The general trend in Fig. 6 is consistent with what one might expect on intuitive grounds. Namely, that
6. CORRELATION OF VEGARD’S LAW FOR THE VOLUME OF INTERMETALLIC ALLOYS AND MELTING POINTS Given deviations in the averages of the melting points with the observed melting points, one can ask whether it is possible to correlate these deviations with the elemental variables listed in Table 1. We proceeded in a similar fashion to the general case of melting points. We found no elemental variable or combination of variables which led to a reasonable correlation. In particular, we attempted to correlate the deviations with the heat of formation of the alloy; we used both theoretical values for heats of formation and the experimentally known values. On the basis of our eqn (2) we felt that a large heat of formation might lead to a greater variation from the average of the elemental temperatures. However, given our discussion concerning the role of entropic contributions, it should not be surprising that the
Fig. 6. The difference in the observed melting points for transition metal alloy cesium chloride structures from the average of the elemental melting points vs the difference of the alloy atomic volume from the elemental average. The differences are normalized to the alloy melting point temperature and volume, respectively.
Melting point trends in intermetallic alloys the melting point is elevated in alloys which show a considerable contraction of volume upon alloying. As an example, RuSc which undergoes a significant contraction relative to the elemental volumes, is one of the few transition-transition metal alloys to exhibit an exceptionally high melting point, much above the average melting point of the constituents.
7. CONCLUSIONS In summary, we would like to note some consequences of our study with respect to contemporary chemical coordinates within the literature. First, with respect to Miedema’s scheme for predicting heats of formation, we would note that the transitiontransition metal alloys are easier to understand, i.e. obtain simple correlations with elemental properties, than are the alloys which involve simple metals. This conclusion that alloys with simple metals are not easily understood is not new. Often the bonding in simple metals is partially covalent and not amenable to a straight-forward description. Our second point is that the scheme of Villars, while promising for structural trends, is not applicable to melting points. This suggests that one might consider alloying energies to contain two elements: an isotropic contribution which does not depend on the details of the structural properties and an anisotropic contribution which does. Miedema’s theory of alloying is probably appropriate for the isotropic contributions and the gross features of the bonding process in metals, but is well known not to accurately reflect the structural properties of alloys. In this case, Villars’ theory would be appropriate only for structural energies. As pointed out by Villars, the combination of the two theories has resulted in the prediction of a number of yet to be discovered ordered alloys. Our present study should aid in providing a first estimate of the melting point of such phases.
205 REFERENCES
1. Miedema A. R., Boom R. and De Boer F. R., J. lesscommon Metals 4, 283 (1975); 45, 231 (1976); 46, 271 (1977) and Refs therein. 2. For example, see Burdett J. K., Price G. D. and Price S. L., Phys. Rev. B24, 2903 (1981); Chelikowsky J. R. and Phillips J. C., Phys. Rev. B17,2453 (1977); St. John J. and Bloch A. N., Phys. Rev. Lett. 33, 1095 (1974); Bloch A. and Schattemann G. C. in Structure and Bonding in Crystals (Edited by M. O’Keeffe and A. Navrotsky), Academic Press, New York (1981); Zunger A., Phys. Rev. Lett. 44, 582 (1980) and Phys. Rev. B22, 5839 (i980); Pettifor D., Phys. Rev. Lett. 42,846 (1979); Pettifor D. G. and Podlouckv R., Phvs. Rev. Lett. 53, 1080 (1984); and Varma C. My, Solid &ate Commun. 31, 295 (1979). 3. Villars P., J. less-common Metals 92, 215 (1983); 99. 33 (1984) and to published. 4. Chelikowskv J. R.. Phvs. Rev. B19. 686 (1979): Alonso J. A. and Simozar’ S.,.Phys. Rev. B22, i583 (i980). 5. Van Vechten J. A., Phys. Rev. Lett. 29, 769(1972); the classic paper is from Lindemann F. A., Z. Phys. 11,609 (1910). Recent work on elemental materials may be found in Ubbelohde A. R., The Molton State of Matter: Melting and Crystal Structures. Wiley, New York (1978). 6. Hansen M., Constitution of Binary Alloys, McGraw Hill, New York (1958); Elliot R. P., Constitution of Binary Alloys-First Supplement, McGraw Hill, New York (1965); Shunk F. A., Constitution of Binary Alloys~Sec&d Supplement, McGraw Hill, Gew York (1965); Moffatt W. G., Binary Phase Diagrams Handbook, General Electric, Schenectady, New York (1976). in the phase 7. We have included all the binary-alloys diagram books of Ref. [6]. Therefore, we include some elements which would not normally be considered metallic, e.g. C, Si, B, etc. We define simple metals to be those which do not have d valence electrons. 8. Guinea F., Rose J. H., Smith J. R. and Ferrante J., Appl. Phys. Lett. 44, 53 (1984). 9. Hultgren R., Orr R. L., Anderson P. D. and Kelly K. K., Selected Values of Thermodynamic Properties of Metals and Alloys, Wiley, New York (1963). 10. A discussion of Vegard’s law may be found in W. B. Pearson’s book: The Crystal Chemistry and Physics of Metals and Alloys, Wiley, New York (1972). 11. Yavari A. R., Hitter P. and Desre P., i de Chim. Phys. 79. 579 11982): Donald L. W. and Davies H. A.. J. of non-crysi. Solids 30,77 (1978) and Ramachandraiao P, Z. Metallkde. 71, 172 (1980).