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Pair potential model of substitutional solid solutions
PAIR POTENTIAL MODEL OF SUBSTITUTIONAL SOLID SOLUTIONS E. S. MACHLIN H. Krumb School of Mines, Columbia Universtty. New York. NY 10027, U.S.A. (Received 7 July 1975; in reused form ?I January 1976) Abstract-The pair potential model previously developed [i-3] is applied to concentrated random solid solutions. it is found that the predicted lattice parameters are in agreement with observed values within an r.m.s. deviation of 0.3Sq:. A correction is developed to allow the model to be applied to heteroelectronic phases (i.e. components have mixed electron character I It is found that except for combinations between (Ni, Pd, Pt: and (IB, IIB: groups and for solutions where the simple metal component is the solvent that the corrected model correctly predicts the lattice parameter. The ordering energies of the Al - Ll, transitions are correctly predicted. Those for the AZ-+ 82 transitions deviate numerically from the observed values in such a way as to suggest the existence of another unique contribution to the ordering energy of the latter transition. R&urn&On applique ie modkle de potentiel de paires dCveloppC antirieurement [l-4] aux solutions solides aleatoires concentrees. Les valeurs privues pour le paramitre cristallin sont en accord avec les valeurs expirimentales, B 0.3892 pr6s. On prtsente une correction qui permet d’appliquer le modele aux phases hettroelectroniques (c’est P dire aux phases dont les composants ont un caracthre tlectromque mixte). Lc modele corrigk prbvoit blen le paramttre cristallin. g l’exception des combinaisons entre les groupc3 : Nl, Pd, Pt: et {iB, IIB: et des solutions dans lesquelles le composant m&allique simple est le soltant. On privoit correctement les inergies des transitions Al -ct Liz. Les &arts entre les Cnergies prCvues pour les transitions A’_-* B2 et les valeurs exp&mentales suggtrent l’existence d’une autre contribution a l’energie de mise en ordre dans ce cas. Zusammenfassung-Das friiher entwickelte Paarpotentialmodell wird auf konzentrierte feste Zufallslijsungen angewendet. Es ergibt sich, da13 die vorausgesagten Gitterparameter mit den beobachteten Werten innerhalb einer mittferen quadratischen Abweichung von 0.38”: ~bereinstimmen. Urn die Anwendung des Modelles auf heteroelektronische Phasen (d.h., die Komponenten haben gemischten Elektronencharakter) zu ermiiglichen, wird eine Korrektur entwickelt. Das korrigierte Model1 sagt die Gitterparameter richtig voraus, auDer bei Kombinationen zwischen {Ni, Pd, Pt) und den Gruppen {IB, IIB} und bei LGsungen. bei dene! die einfache Metallkomponente das LGsungsmittel ist. Die Ordnungsenergien Nr die Al -+ LI, Ubergange werden korrekt vorausgesagt. Diejenigen ftir die A2 -+ B2 Uberglnge weichen von den beobachteten Werten ab..Die Art der Abweichung deutet auf einen weiteren einheidichen Beitrag zur Ordnungsenergie dieses Uberganges an.
I>TRODUCTIOU In previous papers (I. 3,4] in this sequence [l-4], it was shown that the iattice parameters and relative stability of h~moeIectronic~ inrerffzetai~ic co~po~l~s were accurately predicted by a pair potential model [l]. Thus, the model might be expected to apply to homoelectronic solid solutions. One object of this paper is to evaluate this application of the pair potential mode1 El]. It has also been shown that although the pair potential model does not predict the lattice parameters of heteroelectronic? inturmetoallic cotnpounrfs. it does predict their relative stability [Z-4]. However. there is reason to believe that the model will not sucteed in the latter prediction for heteroelectronic so/id ~~~rio~~. Thus, there may be a need to modify the *Combinations of components which have the same outer electron character (e.g. all s.p. etc.) t Combinations of components which have mixed outer electron character (e.g. s.p and d.)
pair potential model for the latter type of metallic solid solutions. Another objective of this paper is to develop and evaluate such a modified pair potential model applicable to heteroel~tronic solid solutions. Finally. if an adequate pair potential model of a solid solution is developed, then it should be possible to predict ordering energies for the Al - Liz and AZ-9 B2 ordering transformations. Thus, another objective of the present paper is to evaluate this possibifity.
HOMOELECTRONIC
SOLID
SOLtTTIONS
The model will be applied to concentrated (atom per cent solute concentration greater than reciprocal coordination number) solid solutions, which are random. The first limitation is for the sake of clarity. In concentrated
solid solutions
there is no ambig&y
about the extent of electron transfer due to differences in electronegativity between components. atoms take part in this exchange. However, 543
All the
in dilute
544
MACHLIN:
PAIR POTENTIAL
solutions, there is an ambiguity arising from the uncertainty as to whether only those solvent atoms that are nearest neighbors to a solute atom exchange electrons with it, or whether further neighbor solvent type atoms also can contribute to this electron transfer. The application of the model to random solid solutions is dictated by the lack of short-range order data for the wide variety of solutions that will be investigated. We will find that because of the relative insensitivity of the lattice parameter to short-range order, this assumption will be acceptable for most of the alloy systems explored. Application of the pair potential model to homoelectronic solid solutions is straightforward. Use is made of equations (15) and (16) of Ref. 1, together with the data listed in Appendix 1 of Ref. 1. Also, the applicable structure sums are
S*A.i = SP’C,
(1)
SBB,i= SP’CB (i = 4,8), where for the Al structure S”, = 25.338305, Si = 12.801937 and for the A2 structure S”, = 22.63872, Si = 10.3552. Computer program “CUBIC’ yielding values of the lattice parameter and cohesive energy of cubic solid solutions has been written.* Lattice parameters obtained from application of this program are compiled in Table 1. In evaluating these results reference should be made to an analysis of previously proposed methods of predicting the lattice parameters of solid solutions [S]. According to this analysis, for hosts of Cu, Ag, Au, Al and Mg: (a) where the solute atom is smaller than the solvent, the sign of the predicted
MODEL
ation relative to Vegard’s Law is incorrectly
Table 1. Predicted
*1
where the atomic volume of the solute is greater than that of the solvent, the predicted and observed values of this deviation agreed to within 5% in only hay of
*Computer
program denoted
by CUBIC obtained at mailing cost from author.
can
be
Cohesir*
Energiar,
chl/r&o.
L12
EUl
l
-79,406
-81,116
+
-831326
-84,2’13
+
-51,554
-53,140
1786
E&12)
)
9
EWl)
- E&12)
1710 947
+
-II,41
-74,026
2607
+
49,579
-51,536
19s7
+
-131,698
-137,095
5437
-90,328
-102,659
433t
l
this random selection of alloy systems.
cohesive energies of homoelectronic phases ?r*dicted
Stable structu.
change in atomic volume relative to that given by VegarJs Law was always found to be in error; (b)
For the same and additional alloy systems, the results obtained using the pair potential method may be summarized as follows: (a) where the atomic diameter of the solute is smaller than the host, the correct sign of the volume change relative to that given by Vega& Law is predicted for all systems (i.e. a total of 11 systems have the atomic diameter of the solute smaller than that of the host); (b) where the atomic diameter of the solute exceeds the host, the deviation between predicted and observed lattice parameters is a r.m.s. value of 0.38x, the maximum deviation is less than 1.660/b(equivalent to a 5% volume deviation) and the sign of the devi-
predicted
for 4 systems (Ag-Sn. Ag-In, Au-Sn. Ni-V), is correctly predicted for 16 systems and cannot be predicted for an additional 16 systems because the deviation from Vegard’s Law is less than the uncertainty in the prediction. Thus, we may conclude that the pair potential model of homoelectronic solid solutions is an appreciable improvement over previous models for predicting the lattice parameters of solid solutions. Another test to which we may subject the pair potential model of homoelectronic solid solutions is that concerning the relative stability of competing structures. We show in Table 1 the cohesive energy predicted by the model for the Al and L12 structures at the AB3 composition and also compare the predicted stability with the observed stability. If we assume that equilibrium cannot be achieved below some fraction,J of the melting point, T,, then in order to observe the Lll structure, the latter structure must be more stable than the Al structure by f T; 1.118 Cal/g atom”C. If we set this quantity equal to 615ca1,g atom then we find only one system in disagreement with the observed stability for a total of 31 systems. This result, which is in agreement with prior experience concerning the ability of the model to predict relative stability of competing intermetallic compounds. suggests that it may well be worthwhile to reinvestigate experimentally the single system at odds with the prediction (LiAg,).
+
-132,i%
-137,424
4666
+
-134,bTG
-1JS,123
3447
+
-135,213
-138,290
3077
+
-131
-113,056
146:
+
-140,148
-146,504
6356
+
-146,725
-150,870
4145
l
-li5,123
-1 Si ,867
6144
c
-130,636
-154,991
4157 3808
,591
+
-154,619
-158,427
+
-151,031
-152,110
1679
+
-160,704
-166,715
6051
+
-166,804
-170,858
3974
l
-165,740
-172,159
6433
l
-63,089
-64,658
1569
+
-71,696
-73,001
1305
+
-82,402
-82,886
484
l
-72,945
-72,934
-1
l
-56.911
-36,773
-13.8
+
-57,757
-57,949
162
+
-w,s42
-53,964
-578
+
-65,601
-65,274
407
l
-68.339
-70,867
252s
l
-58,439
-59,051
611
-68,353
-68,573
220
t
MACHLIN:
PAIR
We may thus conclude that the pair potential model of homoelectronic solid solutions correctly predicts the Al-Lt2 relative stability. A concomitant of this conclusion is that the influence of factors not taken into account in the model (e.g. Fermi surfaceBrillouin zone interaction. etc.) on this relativ*e stability must be very small indeed. HETEROELECTRONIC
SOLID SOLUTIONS
Unfortunately, there is no first principle theory available for predicting the discrepancy between the lattice parameters predicted by the pair potential model and those observed for heteroelectronic intermetallic compounds. Thus. we are forced to rely on an empirical method. In devising such a method, it is useful to note that these lattice parameter deviations for Class V type phases (late transition metals combined with simple metals) are sensitive to the identity of the simple metal component and are insensitive to the identity of the transition element component [2]. This result suggests that the change in Iattice parameter is related to a change in the atomic diameter of the simple metal atom when it is in contact with transition element atoms. We shall adopt this hypothesis and explore its consequences. Table 2. Derived simple metal modified atomic diameters Phase HTn
AINij
Derived Individual
2.696
AlCo
2.4?5
AlRh
2.695
AlPd
2.71
AlOs
2.69
AlIr
2.66
Amu
2.78
AlRe
2.50
MSPd
3.065
Mem
3,024
CuPd
2.522
cup
2.520
AIF,
2.69
Ae3Pt
2.75
BeNi
2.173
B&O
2.153
ZnXi
2.676
ikPt3
2.67
=% PbPd3
SnPd3 S@? cm,
Angstroms Averaes
2.72
AlNi
AUp
IIn(
2.79
2.682
3.045
2.521
2.72
2.163
2.6-n
2.19
3.10 3.10
3.10
2.97 2.97 2.95
2.97
2.95
POTEXTIAL
MODEL
Table 3. Class V simple metal parameters Al
A2
EM** 2.616
-99,454
-85,695
2.459
-84,574
2.720
-87,050
2.653
-85,909
2.790
-99,673
2.721
-98,366
Al.
2.682
-100,776
ou
2.521
43
AU Zn
2.670
-32,626
2.604
-32,198
Pb
3.100
-76,045
3.024
-75,048
2.897
-110,949
Sn
2.970
-112,424
Be
2.163
-87,990
2.11
-86,836
MS
3.045
-43,980
2.97
43,403
Cd
2.95
-30,167
2.878
-29,771
In
2.999
-67,082
2.925
-66,202
Ha
3.045
-18,223
2.97
-17,984
* Angstroms ** cs1/gatom Another hypothesis we shall make is that the attractive pair potential parameter is invariant. This hypothesis requires that E. D’ = constant,
(2)
Thus, there is one parameter that requires fitting. We choose this parameter to be the modified atomic diameter of the simple metal component. Values of this parameter have been calculated using observed lattice parameters of Class V type intermetallic compounds. The results of these calculations are compiled in Table 2. It is apparent from a study of these results that the modified atomic diameter in Class V type intermetallic compounds for a given simple metal component is constant to within about l?$ independent of the identity of the transition element and the crystal structure. (There are two esceptions to this rule (AIRu and AIRe). Perhaps a recheck of the lattice parameters of these phases is in order to ascertain whether these are in fact exceptions to the rule.) The generality of this result justifies an application of such modified atomic diameters to the prediction of lattice parameters of Class V type solid solutions. A list of the modified atomic diameters and modified E values for the simple metal elements evaluated from the data in Table 2 and equation (2) is given in Table 3. (Application of these average atomic diameters yield deviations between observed and calculated lattice parameters such that the r.m.s. deviation is about 0.4x, which is consistent with prior experience with the pair potential model [l]. Hence, use of more refined (varying) values of Dw is not justified.) Application of these simple metal atomic diameters to the evaluation of lattice parameters for Class V
546
MACHLIN:
PAIR POTENTIAL
Table 4. Observed and predicted lattice parameters of At solid solutions class V (Xi, Pd. PtHCu, Ag, Auf combinatxons i.3) System
AsPt, AgPt &3Pt AgPd AiW3 Cu3Pd A% i-=0 3 Auo'7Pdo'3 AuC',CO~'~ Au0',Ni0'9 Auoq3Ni0'7 Au~'~P~~'~ 'Pt0.25 * A"o,?5 c"o. aCoo. 8
Unmodified a*
Observed *v
3.963 3.981 4.042 3.990 3.940 3.704 3.952 4.027 3.618 3.602 3.739 4.027 4.033 3.559
3.947 3.9Sl 4.027 3.9772 3.9328 3.102 3.943 4.020 3.6015 3.594 3.729 4.020 4.035 3.564
Modified * a 3.922 3.921 3.906 3.894 3.882 3.671 3.849 3.929 3.598 3.582 3.685 3.929 3.936 3.550
solid solutions revealed two groups of behavior. First, it was noted that combinations formed with one component from the (Ni, Pd. Pt) group and the other component from either Group IB (noble metals) or Group IIB of the periodic table consistently yielded a different result from the other possible ClassV component combinations. Accordingly, the results for these two groups are separately presented in Tables 4 and 5. Table 4 compares the calculated and observed lattice parameters for the (Ni, Pd. Pt)-(Group IB, Group IIB) types of component combinations and Table 5 for the other possible Class V component combinations. From the rest&s in TabIe 4, it is apparent that the Class V contraction is absent in the solid solutions formed from the (Ni, Pd. Pt)-(Group IB, Group IIB) component combinations. This result contrasts with the Class V contraction found in ordered intermetallic compounds formed by these component combinations. The results listed in Table 5 demonstrate that for the other possible Class V solid solutions, the Class V contraction does occur in solid solutions and is accounted for quantitatively using the parameters in Table 5. Observed and predicted lattice parameters of Al solid solutions class V combinations not containing groups IB and IIB elements
CoAl
10
3.559
3.564
3.595
3.568
Pal
10.5
2.871
2.8855
2.903
2.8923
1r.5
2.873
2.8942
Tfesa
10
2.942
2.936
2.953
3%
NiSn
9.75
3.636
1.605
3.638
3.726
NiAl
10
1.541
3.541
3.577
3.575
PdSn
13
3.950
3.936
16.5
3.967
3.942
17
3.969
3.963
4.070
4.079
14
3.957
3.960
4.039
4.094
PdPb
2.9047
4.074
MODEL
Table 6. Observed and predicted relative stabiiitv of the Al and Ll;! structures-lass Phase
Stable strscturs Al
ALvi3
Ll*
Predicted E(Al)
V component corn&nations Cohesive E(Ll2)
Energies,
cal/gatom
B(M)
- E(Ll2)
+
-104,958
-107,349
2390
AlPt3
+
-128,281
-130,869
2588
&IF%3
+
-128,370
-128,704
334
ma3
+
-95,504
-96,361
a57
PbPt3
+
-117,391
-118,059
568
FWd,
*
-85,285
-86,055
770
PWu3
+
-95,243
-97,867
2624
pa3
c
-98,355
-99,125
770
me3 cdPt3
+
-108,975
-109,728
753
+
-107,604
-107,945
341
ZnPf3
+
-107,477
-108,763
1286
PdCu,
+
-85,334
-66,571
1237
COP+)
+
-125,060
-125,658
590
-88,306
-88,646
338
CuPd,
+
AsPd3
+
-69,872
-90,033
161
p-3
l
-87,814
-88,079
2G5
-
Table 3 and the pair potential model. Thus, the constant modified atomic diameters listed in Table 3 for simple metal components are applicabie both to these Class V solid solutions as well as to Class V intermetallic compounds. The above results suggest that the Class V contractions for these different groups of Class V intermetallit compounds have different origins. However, the comparison of the energies of formation using the modified parameters do not indicate any difference in behavior of these two types of component combinations. (See Table 6). The results indicate that the model yields correct prediction of the Al-L12 ordering energy for 14 out of 16 phases, with one error in each group. It thus seems reasonable to apply the model in order to evaluate the relative s~biljty of Class V type phases. Let us now consider Class VI type phases (combinations between early transition elements and simple metals.) According to Ref. 2, we may expect that an expansion should occur reiative to the values predicted on the basis of the Class V modified atomic diameters. Further, this expansion does not seem to be very sensitive to the identity of the transition element component. Consequently, the simpIest assumption to make is that this expansion represents a change in the atomic diameter of the simple metal component. Accordingly, new values of atomic diameters have been calculated for the simple metal components of Class VI intermetallic compounds by fitting to the observed lattice parameters. As shown in Table 7, the expansions reIative to the Class V values of the modified atomic diameters range from about 0 to about 0.5 A, increasing with increase in the principal quantum number and with decrease in the group number of the transition element. The effect of the transition principal quantum number was previously noted in Ref. 2 where it was called the “Class VI-principal quantum number effect”. Data available allow the prediction of lattice parameters for only one Class VI type solid solution.
MACHLIN:
PAIR POTESTI,\L
MODEL
Table 7. Expansion of simple metal atomic dtameter m class VI phases relattve to value m class V phases (.A) Elemnt
Trsnsiticn
Simple Netal compancnt
SC
Y
Al
0.084
0.131
-Q
0.073
0.376
Ti
La
0.611
AA
0.20
dc
0.132
CU
0.1‘83
8: In
lif
, O.?,,
xb
T8
0
0
0
O.li
-
.? 0.13:0.1m ---
O.C57-%5R0.001
0.006
---
0.233
Pb
-0.13
-.-
5n Zx
V
0.208 ---0.201 0.163
::a
Cmponent Zr
0.20.
0.43 6
0.033
The lattice parameter predicted using the expanded simple metal atomic diameter of Au in a Nb matrix IS given m Table S. (The expanded sample metal atomic diameter is obtained by adding the appropriate value of the expansion from Table 7 to the corresponding atomic diameter given in Table 3. The expanded atomic diameter is then substituted into equation (2) to obtain the corresponding cohesive energy of the expanded simple metal component.) According to the data in Table 8 the Class VI solid solution of Au as a solute in Nb obeys the sa?ne Tuzlation as does the Class VI intermetallic compound (NbsAu). That is, the same value of expanded simple metal atomic diameter applies to both the Class VI solid solution and intermetallic compound, at least for the transition metal rich solid solution. There are a few systems which can be used to evaluate the concentration dependence of the expanded atomic diameter for Class VI solid solutions where the simple metal is the solvent. Table 9 compares the calculated lattice parameters based on the relation D,,, = ‘*cr. [D,(Table 7) - D,&(unmodified)], (3) with the observed lattice parameters. The term cr is the atomic fraction of transition element component in the binary solid solution. It appears from the data in Table 9 that this relation describes the concentration dependence of the expansion in the atomic diameter in Class VI solid solutions, at least for the few cases available to test it. The Al-L12 relative stability predictions for Class VI phases are compared to the observed stabilities m Table 10. Except for the case of HgTi3 the predictions are in agreement with observation. It is possible that the observation is wrong because HgTi, is
---
O.lT
_o.fyj-y 0. O-FxTl-
observed to be stable in the A15 structure at [arc temperature and the model does predict that the A15 structure (predicted cohesive energy equal to -95.6OOcal:g atom) is stable relative to both the Al and L12 structures. Perhaps, the single observation of a stable L12 structure at high temperature in this alloy is wrong. It deserves an independent check.
DIFFERESCE IN PREDICTIOS OF RELATIVE STABILITY LSISG MODIFIED VS UNMODIFIED SDIPLE METAL ATOMIC DLtiIETERS In the comparisons of relative stability for Classes V and VI (Tables 6 and 10) there was no indication that the predictions of the pair potential model using the modified simple metal atomic diameters differed from those using the pure component atomic diameters. This question has been explored further. One clear example of a difference in predictions of stability for these cases has been discovered. This example involves the relative stability of pseudo-binary solid solutions with respect to a mixture of terminal pseudo-binary solid solutions, i.e. the problem of development of a miscibility gap in pseudo-binary solid solutions. The pseudo-binary explored was that involving Al5 phases of the type Nb3(X, Y) and V,(X, Y). where X and Yare simple metals. The results of the calculations are given in Table 11. According to these results, the use of modified simple metal atomic diameters yields a prediction in agreement with observation, whereas the use of unmodified simple metal atomic diameters produces a prediction in disagreement with observation.
Table 8. Observed and predicted lattice parameters of Al solld solutions class VI component combmations
So2~:%lute
Lattice
s4
s&e
Umodified a*
Nb
AU
23
3.279
-
Modified D* 3.304
Panmeters, Observed k 3.297
bgstroms Vagsrd’s 5-L 3.291
LOW
MACHLIX:
PAIR POTENTIAL
MODEL
Tabie 9. Observed and predicted lattice parameters of Al solid solutions class VI component combinations
AU
Ti
12
4.047
4.254
4.096
4.065
Au
v
10
4.045
4.130
4.069
4.067
24.5
3.998
4.070
4.039
4.039
40
3.948
4.007
4.002
4.003
Table 10. Observed and predicted relative stability of the Al and Ltz structures. class VI component combinations Phas*
Stable structure Al Ll2
Cohsafvr E(U)
Enereios,
csI/~atom
E(L12)
E(M)
- E(Ll2)
TiZlA,
+
-50,653
-52,753
2100
N-bzs,
+
-64,542
-69,163
4641
WI3
+
-51,648
-53,217
1569
BeTi,
*
-91,948
-58,779
-3169
An,
+
-130,668
-131,644
976
Al-L12
ORDERIKG
ENERGY
The success in predicting the relative stability of the Al-Ll, competition raised the hope that the model might just be good enough to p&ict the ordering energy E(AI)-E(Ll& quanti~tively, as well. Although some values of this ordering energy have been measured, a more extensive measure of it can be obtained by using either the observed Ll,-Al transition temperature (7;) or the Liz-liquid transition temperature (Tw). In the latter case, T, must exceed 7&. We know that at the transition temperature, z;, that the free energies of the Al and Liz phases are equal. Hence, E(A1) - ?;.S(Al) = E(L12) - T,[S(L12)] Of
E(Ai) - E(L1,) = T,[S(Al) - S(Ll,)].
4.083
3.963
potential model parameters to predict cohesive energy values, implies either that the scheme we have used to correct for the heteroelectronic interaction is too rough an approximation, or that it neglects some important effect, or that the entropy difference between the phases varies markedly. According to the previous relation the slope of the line in Fig. 1 should equal SfAl)-S(L12). Now, the configurational contribution to this difference in entropy is 1.118 Cal/g atom/Y. The best value of this quantity given by the data in Fig. t is 1.9cal/g atom/‘C. The difference between these two numbers (0.782cal/g atom/‘C) corresponds to an excess of other than configurational entropy of the Al structure over that of the Lll structure, on the areruge. This result is reasonable. The data points in Fig. 1 deviate from the best straight line through them. Where the sign of the deviation is constant for a given class of component combination it appears that two interpretations may be offered for the deviation. One is that the predicted AE is in error by the average deviation for the class or that the AS for that class differs from the average AS value. Study of the deviations imply that the latter is the more likely explanation. Thus, for Class I it appears that the value of AS z 2.2 Cal/g atom/“C; for Class II combinations the apparent value is about 1.67 Cal/g atom,K.* If the previous explanation for the deviations by class is correct, then the remaining random deviations represent the uncertainty in the prediction of AE, i.e.
Thus. a plot of predicted values of E(A1jE(L12) versus obserced values of 7; should yield a straight line through the origin of slope S(A1jS(Llt). Table 11. Predicted values of relative energy of formation We show in Fig. 1, for homoelectronic phases, a of some pseudobinary Al5 phases. cal’g atom plot of predicted values of &Al)-E(L1,) vs 1; or xw:,, Iredieted Encrw of gz:pon Belatite to Binrrr whichever of the latter has been measured. It is Pswdcbirury obvious that the data, on the average, obey the f( E(T,M’) + E(T,X*) ) E(T,#$l;) expected relation. This is an extraordinary result! The Modiiied Umcdiiied pair potential model certainly represents a breakthrough in the prediction of the relative stability of 1099’ 311 b-b,(AQB.i) alloy phases and the significance of its unexpected 1926 205 h%3(AliPbi) success deserves first-principle exploration. 147 3 Nb,(Pb+Sn+) 264 Figure 2 shows a similar plot for heteroelectronic 2829 P,(A’fPb$ phases. The much greater scatter, which incidentally 390 1814 P3(Alpyf is unchanged even upon using the unmodified pair 3 146 V3(Pb+Sni) * Class I type phases contain simple metal components only. Class II type phases involve combinations between early and late transition elements.
?vfACHLIN:
P;\IR POTENTIAL
MODEL
519
6 000
E 0 5000 z \ ;; u G- 4000
p
3ooc
t
Ir
0
0” 5
2ooc
0
,” si!
0
0
0 0
w” K ?COC P
0
0
500
1000 OBSERVED
1500
2000
TRANSITION
2500
TEMPERATURE,
3000
OK
Fig. 1. Correlatton between predicted ordermg ene_rgyand observed transItion temperature. Homoelectromc type phases. 0 Class I (spsp); 0 Class 11(tf-4 component combinations. Al-Ll; tfansformatton.
about 5OOcal/g atom. This also must be considered to be an extraordinary result. To understand why it is extraordinary and totally unexpected, dwell for a moment on the fact that the cohesrve energy values are on the order of IM.Ocal g atom. A2 + B2 ORDERING
ENERGY
The ability to calculate the cohesive energy of random solid solutions and the success noted in the previous section using this ability to predict the Al-Liz relative stability suggests an evaluation of the AI-B2 relative stabihty to provide inFormation concerning
+
the ordering energy driving this transformation. A comparison of the predicted cohesive energies for these structures is given in Table 12. The data in Table 12 reveal that, for homoelectronic phases, the model predicts the correct sense of the A2 --t B2 transformation for 7 out of 9 Class I type combinations and 13 out of 17 Class II type combinations. The magnitudes of the incorrectiy predicted negative ordering energies are all small (less than lOOOcal/g atom.) Thus, the model yields only a fair prediction for the sense of this transfo~ation. The scatter in ordering energy values in a piot of predicted ordering energies YStransition temperature
+
-t
a
+
+ ++ ++
iz
+ @ t
+
OL 0
I
I
I
500
1000
1500
OBSERVED
TRANSITION
I
2000 TEMPERATURE,
I
2500 *K
I
3000
Fig. 2. Correlation between predicted ordering energy and observed transition temperature. Heteroekctronic type phases. c Class V: D Class VI component combinations. Al-LIZ transformation.
550
MACHLIN:
PAIR POTENTIAL iMODEL
Table 11. Predicted relative stabilities of Al and 52 structures for ~omoel~cronic phases that are observed to be stable in the B2 structure Phase
Predictad
Cohesive
E(A2)
Energies,
E(B2)
cal/gstom
EbZ) -
LiAg
-66, t 70
-67,703
1533
LiPb
-46,934
-52,387
5453
LiTl
41,:GZ
-51,941
4679
W.&J NgAu
-56,378
-56,182
-196
-69,597
-70,551
HgTl
-42,164
44,011
1847
BeCu
-76,419
-80,591
4172
CsTl
-55,561
-55,358
-203
SrTl.
-55,034
-55,672
638
scco
-108,231
-111,685
3454
E(B2)
954
SCM
-105,013
-109,219
4206
SoRu
-134,462
-133,974
-488
SC&h
-128,131
-127,s21
-310
ScPd
-104,676
-105,076
ScIr
-142,449
-141 ,478
-971
TiRU
-141,067
-140,908
-159
TiOs
-157,971
-158,372
395
TiCo
-116,923
-138,054
1131
400
TiNi
-114,591
-116,338
t 747
HfCO
-136,562
-14G,991
10,429 3603
HfRu
-163,834
-167,437
HfOS
-182,035
-183,813
i 7-78
zrC0
-126,906
-136,455
9549
zrnu
-154,146
-f 57,296
3148
zros
-172,206
-173,580
1374
VOS
-154,135
-155,008
873
(see Fig. 3) suggests that another factor contributing to the ordering energy must be present in the A2 --+ B2 tmnsformation. This factor must be volume independent, because the model predicts the lattice parameters of these phases correctly. DISCUSSION
OF RESULTS
The results for homoelectronic solid sofutions reveal why previous attempts to predict lattice parameters of solid solutions have failed. Namely, these attempts did not incorporate the effect of a difference in electronegativity between the components upon the tattice parameter of the solid solution. The pair potential model does incorporate this effect [I] and, as shown in the present paper, is in essential agreement with observations of both lattice parameters and relative stabilities of solid solutions and intermetallic compounds for homoelectronic component combinations. It should be noted that no adjustable parameters are involved in these predictions! The fact that the pair potential model predicts the relative stability and the correct value of the ordering energy for the Al -, Lt2 transformation has an important implication concerning the driving force for this transformation. As shown in Table i3. the predicted change in lattice parameter for this transformation often exceeds the r.m.s. uncertainty,‘in the lattice
parameter (OX,) for those phases where the L12 structure is stable and is less than this value for the cases where the Al structure is stable (i.e. with the exception of LiAg,.) The implication of these results is that the Al--t L l2 order-disorder transformation is driven by the tendency to reduce strain energy upon ordering. Although the same driving force exists in the A2 -c B2 order-disorder transformation, it is obviously not the only contribution to the ordering energy of this transformation as shown by the many cases where the predicted difference in lattice parameters is negligible but for which the BZ structure is stable-see Table 14. We shall call this additional contribution to the ordering energy of the AZ-* B2 transformation the A2 - B2 volume independent ordering energy factor. A possible candidate for this factor is the electrostatic type interaction considered by Blandin and Deplante [6] or that considered by Inglesfield [7]. The fact that no evidence of this factor was found in the relative stability comparisons previously made (AlILl*, Cl5C32, Ll,/Alj, A2/AiS [l--l. S] suggests that it may be unique to the B2 structure. i.e. another possible candidate for this factor may be a Fermi surface-Brilloin zone stabilization. The nature of the stabilization associated with intermetallic compounds formed from Class V combinations between the (Ni, Pd, Pt) group and Group IB or IIB elements is unknown. It seems unlikely that this stabilization is similar to that applicable to other Class V type phases. Fortunately, the rule seems to be for “regular” behavior, with the exception corresponding to “irregular” behavior. Study of the results suggests that the Class V contraction in this special group of irregular behaving phases cannot be described in terms of modified simple metal atomic diameters and the pair potential model. Were this to be the case then we would expect that the predicted &Al)-E(L1,) would correspond to EtAI) (unmodifiedtE(L1,) (modified}. The latter vatues are much larger than the observed values of this difference in cohesive energies. This interpretation of the results is consistent with the change in the trend of Class V contractions with group number of the simple metal component, as exhibited in Fig. 7 of Ref. 2. CONCLUSIONS 1. The predicted lattice parameters of the pair potential model are in agreement with the observed vaiues for solid solutions within an r.m.s. deviation of 0.38%. As such, the pair potential mode1 represents a significant improvement over previous methods [5) of predicting the lattice parameters of solid solutions. This improvement is due to incorporation of the effect of eiectronegativity difference in the pair potential model and its neglect in prior models. 2. The lattice parameters of Class V phases, solid solutions as well as intermetallic compounds, are con-
MACHLIN:
PAIR POTENTIAL
410DEL
7 000
i IO;423
6 000 A
E ; 5000 P
A
.
2 5 - 4000 :: 5 ‘:
3000
E ” :: o d
2000
A
: u iti
i
A I
IO00
A
ii
0
A
-1000
A
A
I
I
I
I
I
500
1000
1500
2000
2500
OBSERVED TRANSITION
TEMPERATURE,
Fig. 3. Plot of predicted ordering energies for A2-BZ transformation perature. A Class I (spsp); i Class II (i-3 component combinations.
Table
System AUCU, CUAU, CaPb, CaSn, CaTI, ScRhJ ScPd, ScPt, TiRh, TtPt, Vir, ZrIr> NbIr, HfIr, hIgAg, ZnAu, AgAu, AuAg, AgCd,
CdAg, HgAg, ZnAl, LiAg, ZnAg, ZnCu,
13. Predicted
contraction transformation
[@(Al)-a*(LlJ]: o*(Al), “/, 0.57 0.31 0.87 0.85 1.01 0.99 1.14 0,83 0.63 0.56 0.28 0.96 0.61 1.00 0.56 0.51 0.15 0.00 -0.15 0.03 -0.37 -0.11 0.88 0.31 0.04
due
to
OK
%s observed transition temHomoelectronic type phases.
AI-LIZ Table l-1. Predicted contraction due to ordering for phases stable in the BZ structure
Structure
System
LIZ
LiAg Lifb LiTl MgAg MgAu MgTl BeCu CaTl SrTl scco SCNI ScRu ScRh ScPd Stir TiRu TiOs TiCo TtNi HfCo HfRU HfOs ZrCo ZrRu ZrOs vos
Al
[a*(A2h~*(BZ)]ja*(A21. go 0.50 2.38 2.44 - 0.20 0.17 1.12 1.41 - 0.03 0.41 0.94 I 14 -0.20 -0.11 OlS - 0.30 - 0.06 000 0.35 0.49 3.35 0.66 0.37 2.19 0.59 0.20 015
551
MACHLIN:
PAIR POTENTIAL MODEL
sistent with the use of constant simple metal atomic diameters (modi~ed values) that differ from the values applicable to homoelectronic type phases and pure components, except for solid solutions formed by components one of which belongs to the (Ni, Pd, Pt) group and the other to Group 1B or 1IB of the periodic table. For this exception. the unmodified pure component atomic diameters are applicabie. 3. The modified simple metal atomic diameters deduced for Class VI type phases depend upon the identity of the transition element component, but for a given combination of components is independent of structure. 4. The pair potential model, using either unmodified or modified simple metal atomic diameters correctly predicts the AI/Lll relative stability competition for 49 phases out of a total of 52 phases. 5. The predicted value of the ordering energy for the .41+ LIZ transformation is found to be proportional to the experimental value of the Ll 2- Al or L12 --+ liquid transition temperature. The value of the average difference in entropy deduced from this proportionality is S(Al)-S(L12) = 1.9 Cal/g atom/C 6. The use of modified simple metal atomic diameters yields agreement with the observed -relative energy of formation of Class VI pseudobinary solid solutions having the Al5 structure, whereas the use of unmodified diameters produces prediction~in disagreement with observations. 7. The pair potential model does not consistently predict the AZ/B2 relative stability. The cases that violate the predictions are characterized by having
nearly equal v-alues of lattice parameters for both ordered and disordered phases. Comparison to those phases exhibiting the Al,‘LI? transition and that also obey the condition of nearly equal lattice parameters in the ordered and disordered structures leads to the conclusion that there exists a constant voiume ordering energy component that contributes to the stability of the B2 structure, but plays RO role in the Al - L12 order transfo~ation. 8. The source of the stabilization effect that is unique to intermetallic compounds formed between components one of which belongs to the (Ni, Pd, Pt) group and the other to Group IB or IIB is unknown. Because this eFfect appears to be absent in disordered solid solutions formed by these component combinations, it is likely that the effect requires an ordered arrangement of the components.
REFERENCES I. Machlin E. S.. Acta Met. 22, 95 (19711. 2. klachlin E. S.. Acta Met. 22. 109 1197-U. 3. lLfach&n E. S.. Acfa Met. 22, 367 i197-r). 4. Machlin E. S.. Acta Met. 22. f433 (19-4). 5. Kin8 I-I. W.. in A/ME Symposium on .-lilo.ving Beharior and Eficts 111Concentrated Solid Solutions. Gordon &
Breach, NY (1967). 6. Bland~n A. and Deplante J. L., J. Pfi_rs.Radium 23, 609 (1962). 7. Inglesfield J. E., Acta Met. 17, 1395 (1969). 8. ;Machlin E. S., J. Phys. Chem. Solids. To be publIshed. 9. Bachner A., Goodenough J. and Gatos H.. J. Ph.w. Chem. Solids 28, 889 (1967).
I>TRODUCTIOU In previous papers (I. 3,4] in this sequence [l-4], it was shown that the iattice parameters and relative stability of h~moeIectronic~ inrerffzetai~ic co~po~l~s were accurately predicted by a pair potential model [l]. Thus, the model might be expected to apply to homoelectronic solid solutions. One object of this paper is to evaluate this application of the pair potential mode1 El]. It has also been shown that although the pair potential model does not predict the lattice parameters of heteroelectronic? inturmetoallic cotnpounrfs. it does predict their relative stability [Z-4]. However. there is reason to believe that the model will not sucteed in the latter prediction for heteroelectronic so/id ~~~rio~~. Thus, there may be a need to modify the *Combinations of components which have the same outer electron character (e.g. all s.p. etc.) t Combinations of components which have mixed outer electron character (e.g. s.p and d.)
pair potential model for the latter type of metallic solid solutions. Another objective of this paper is to develop and evaluate such a modified pair potential model applicable to heteroel~tronic solid solutions. Finally. if an adequate pair potential model of a solid solution is developed, then it should be possible to predict ordering energies for the Al - Liz and AZ-9 B2 ordering transformations. Thus, another objective of the present paper is to evaluate this possibifity.
HOMOELECTRONIC
SOLID
SOLtTTIONS
The model will be applied to concentrated (atom per cent solute concentration greater than reciprocal coordination number) solid solutions, which are random. The first limitation is for the sake of clarity. In concentrated
solid solutions
there is no ambig&y
about the extent of electron transfer due to differences in electronegativity between components. atoms take part in this exchange. However, 543
All the
in dilute
544
MACHLIN:
PAIR POTENTIAL
solutions, there is an ambiguity arising from the uncertainty as to whether only those solvent atoms that are nearest neighbors to a solute atom exchange electrons with it, or whether further neighbor solvent type atoms also can contribute to this electron transfer. The application of the model to random solid solutions is dictated by the lack of short-range order data for the wide variety of solutions that will be investigated. We will find that because of the relative insensitivity of the lattice parameter to short-range order, this assumption will be acceptable for most of the alloy systems explored. Application of the pair potential model to homoelectronic solid solutions is straightforward. Use is made of equations (15) and (16) of Ref. 1, together with the data listed in Appendix 1 of Ref. 1. Also, the applicable structure sums are
S*A.i = SP’C,
(1)
SBB,i= SP’CB (i = 4,8), where for the Al structure S”, = 25.338305, Si = 12.801937 and for the A2 structure S”, = 22.63872, Si = 10.3552. Computer program “CUBIC’ yielding values of the lattice parameter and cohesive energy of cubic solid solutions has been written.* Lattice parameters obtained from application of this program are compiled in Table 1. In evaluating these results reference should be made to an analysis of previously proposed methods of predicting the lattice parameters of solid solutions [S]. According to this analysis, for hosts of Cu, Ag, Au, Al and Mg: (a) where the solute atom is smaller than the solvent, the sign of the predicted
MODEL
ation relative to Vegard’s Law is incorrectly
Table 1. Predicted
*1
where the atomic volume of the solute is greater than that of the solvent, the predicted and observed values of this deviation agreed to within 5% in only hay of
*Computer
program denoted
by CUBIC obtained at mailing cost from author.
can
be
Cohesir*
Energiar,
chl/r&o.
L12
EUl
l
-79,406
-81,116
+
-831326
-84,2’13
+
-51,554
-53,140
1786
E&12)
)
9
EWl)
- E&12)
1710 947
+
-II,41
-74,026
2607
+
49,579
-51,536
19s7
+
-131,698
-137,095
5437
-90,328
-102,659
433t
l
this random selection of alloy systems.
cohesive energies of homoelectronic phases ?r*dicted
Stable structu.
change in atomic volume relative to that given by VegarJs Law was always found to be in error; (b)
For the same and additional alloy systems, the results obtained using the pair potential method may be summarized as follows: (a) where the atomic diameter of the solute is smaller than the host, the correct sign of the volume change relative to that given by Vega& Law is predicted for all systems (i.e. a total of 11 systems have the atomic diameter of the solute smaller than that of the host); (b) where the atomic diameter of the solute exceeds the host, the deviation between predicted and observed lattice parameters is a r.m.s. value of 0.38x, the maximum deviation is less than 1.660/b(equivalent to a 5% volume deviation) and the sign of the devi-
predicted
for 4 systems (Ag-Sn. Ag-In, Au-Sn. Ni-V), is correctly predicted for 16 systems and cannot be predicted for an additional 16 systems because the deviation from Vegard’s Law is less than the uncertainty in the prediction. Thus, we may conclude that the pair potential model of homoelectronic solid solutions is an appreciable improvement over previous models for predicting the lattice parameters of solid solutions. Another test to which we may subject the pair potential model of homoelectronic solid solutions is that concerning the relative stability of competing structures. We show in Table 1 the cohesive energy predicted by the model for the Al and L12 structures at the AB3 composition and also compare the predicted stability with the observed stability. If we assume that equilibrium cannot be achieved below some fraction,J of the melting point, T,, then in order to observe the Lll structure, the latter structure must be more stable than the Al structure by f T; 1.118 Cal/g atom”C. If we set this quantity equal to 615ca1,g atom then we find only one system in disagreement with the observed stability for a total of 31 systems. This result, which is in agreement with prior experience concerning the ability of the model to predict relative stability of competing intermetallic compounds. suggests that it may well be worthwhile to reinvestigate experimentally the single system at odds with the prediction (LiAg,).
+
-132,i%
-137,424
4666
+
-134,bTG
-1JS,123
3447
+
-135,213
-138,290
3077
+
-131
-113,056
146:
+
-140,148
-146,504
6356
+
-146,725
-150,870
4145
l
-li5,123
-1 Si ,867
6144
c
-130,636
-154,991
4157 3808
,591
+
-154,619
-158,427
+
-151,031
-152,110
1679
+
-160,704
-166,715
6051
+
-166,804
-170,858
3974
l
-165,740
-172,159
6433
l
-63,089
-64,658
1569
+
-71,696
-73,001
1305
+
-82,402
-82,886
484
l
-72,945
-72,934
-1
l
-56.911
-36,773
-13.8
+
-57,757
-57,949
162
+
-w,s42
-53,964
-578
+
-65,601
-65,274
407
l
-68.339
-70,867
252s
l
-58,439
-59,051
611
-68,353
-68,573
220
t
MACHLIN:
PAIR
We may thus conclude that the pair potential model of homoelectronic solid solutions correctly predicts the Al-Lt2 relative stability. A concomitant of this conclusion is that the influence of factors not taken into account in the model (e.g. Fermi surfaceBrillouin zone interaction. etc.) on this relativ*e stability must be very small indeed. HETEROELECTRONIC
SOLID SOLUTIONS
Unfortunately, there is no first principle theory available for predicting the discrepancy between the lattice parameters predicted by the pair potential model and those observed for heteroelectronic intermetallic compounds. Thus. we are forced to rely on an empirical method. In devising such a method, it is useful to note that these lattice parameter deviations for Class V type phases (late transition metals combined with simple metals) are sensitive to the identity of the simple metal component and are insensitive to the identity of the transition element component [2]. This result suggests that the change in Iattice parameter is related to a change in the atomic diameter of the simple metal atom when it is in contact with transition element atoms. We shall adopt this hypothesis and explore its consequences. Table 2. Derived simple metal modified atomic diameters Phase HTn
AINij
Derived Individual
2.696
AlCo
2.4?5
AlRh
2.695
AlPd
2.71
AlOs
2.69
AlIr
2.66
Amu
2.78
AlRe
2.50
MSPd
3.065
Mem
3,024
CuPd
2.522
cup
2.520
AIF,
2.69
Ae3Pt
2.75
BeNi
2.173
B&O
2.153
ZnXi
2.676
ikPt3
2.67
=% PbPd3
SnPd3 S@? cm,
Angstroms Averaes
2.72
AlNi
AUp
IIn(
2.79
2.682
3.045
2.521
2.72
2.163
2.6-n
2.19
3.10 3.10
3.10
2.97 2.97 2.95
2.97
2.95
POTEXTIAL
MODEL
Table 3. Class V simple metal parameters Al
A2
EM** 2.616
-99,454
-85,695
2.459
-84,574
2.720
-87,050
2.653
-85,909
2.790
-99,673
2.721
-98,366
Al.
2.682
-100,776
ou
2.521
43
AU Zn
2.670
-32,626
2.604
-32,198
Pb
3.100
-76,045
3.024
-75,048
2.897
-110,949
Sn
2.970
-112,424
Be
2.163
-87,990
2.11
-86,836
MS
3.045
-43,980
2.97
43,403
Cd
2.95
-30,167
2.878
-29,771
In
2.999
-67,082
2.925
-66,202
Ha
3.045
-18,223
2.97
-17,984
* Angstroms ** cs1/gatom Another hypothesis we shall make is that the attractive pair potential parameter is invariant. This hypothesis requires that E. D’ = constant,
(2)
Thus, there is one parameter that requires fitting. We choose this parameter to be the modified atomic diameter of the simple metal component. Values of this parameter have been calculated using observed lattice parameters of Class V type intermetallic compounds. The results of these calculations are compiled in Table 2. It is apparent from a study of these results that the modified atomic diameter in Class V type intermetallic compounds for a given simple metal component is constant to within about l?$ independent of the identity of the transition element and the crystal structure. (There are two esceptions to this rule (AIRu and AIRe). Perhaps a recheck of the lattice parameters of these phases is in order to ascertain whether these are in fact exceptions to the rule.) The generality of this result justifies an application of such modified atomic diameters to the prediction of lattice parameters of Class V type solid solutions. A list of the modified atomic diameters and modified E values for the simple metal elements evaluated from the data in Table 2 and equation (2) is given in Table 3. (Application of these average atomic diameters yield deviations between observed and calculated lattice parameters such that the r.m.s. deviation is about 0.4x, which is consistent with prior experience with the pair potential model [l]. Hence, use of more refined (varying) values of Dw is not justified.) Application of these simple metal atomic diameters to the evaluation of lattice parameters for Class V
546
MACHLIN:
PAIR POTENTIAL
Table 4. Observed and predicted lattice parameters of At solid solutions class V (Xi, Pd. PtHCu, Ag, Auf combinatxons i.3) System
AsPt, AgPt &3Pt AgPd AiW3 Cu3Pd A% i-=0 3 Auo'7Pdo'3 AuC',CO~'~ Au0',Ni0'9 Auoq3Ni0'7 Au~'~P~~'~ 'Pt0.25 * A"o,?5 c"o. aCoo. 8
Unmodified a*
Observed *v
3.963 3.981 4.042 3.990 3.940 3.704 3.952 4.027 3.618 3.602 3.739 4.027 4.033 3.559
3.947 3.9Sl 4.027 3.9772 3.9328 3.102 3.943 4.020 3.6015 3.594 3.729 4.020 4.035 3.564
Modified * a 3.922 3.921 3.906 3.894 3.882 3.671 3.849 3.929 3.598 3.582 3.685 3.929 3.936 3.550
solid solutions revealed two groups of behavior. First, it was noted that combinations formed with one component from the (Ni, Pd. Pt) group and the other component from either Group IB (noble metals) or Group IIB of the periodic table consistently yielded a different result from the other possible ClassV component combinations. Accordingly, the results for these two groups are separately presented in Tables 4 and 5. Table 4 compares the calculated and observed lattice parameters for the (Ni, Pd. Pt)-(Group IB, Group IIB) types of component combinations and Table 5 for the other possible Class V component combinations. From the rest&s in TabIe 4, it is apparent that the Class V contraction is absent in the solid solutions formed from the (Ni, Pd. Pt)-(Group IB, Group IIB) component combinations. This result contrasts with the Class V contraction found in ordered intermetallic compounds formed by these component combinations. The results listed in Table 5 demonstrate that for the other possible Class V solid solutions, the Class V contraction does occur in solid solutions and is accounted for quantitatively using the parameters in Table 5. Observed and predicted lattice parameters of Al solid solutions class V combinations not containing groups IB and IIB elements
CoAl
10
3.559
3.564
3.595
3.568
Pal
10.5
2.871
2.8855
2.903
2.8923
1r.5
2.873
2.8942
Tfesa
10
2.942
2.936
2.953
3%
NiSn
9.75
3.636
1.605
3.638
3.726
NiAl
10
1.541
3.541
3.577
3.575
PdSn
13
3.950
3.936
16.5
3.967
3.942
17
3.969
3.963
4.070
4.079
14
3.957
3.960
4.039
4.094
PdPb
2.9047
4.074
MODEL
Table 6. Observed and predicted relative stabiiitv of the Al and Ll;! structures-lass Phase
Stable strscturs Al
ALvi3
Ll*
Predicted E(Al)
V component corn&nations Cohesive E(Ll2)
Energies,
cal/gatom
B(M)
- E(Ll2)
+
-104,958
-107,349
2390
AlPt3
+
-128,281
-130,869
2588
&IF%3
+
-128,370
-128,704
334
ma3
+
-95,504
-96,361
a57
PbPt3
+
-117,391
-118,059
568
FWd,
*
-85,285
-86,055
770
PWu3
+
-95,243
-97,867
2624
pa3
c
-98,355
-99,125
770
me3 cdPt3
+
-108,975
-109,728
753
+
-107,604
-107,945
341
ZnPf3
+
-107,477
-108,763
1286
PdCu,
+
-85,334
-66,571
1237
COP+)
+
-125,060
-125,658
590
-88,306
-88,646
338
CuPd,
+
AsPd3
+
-69,872
-90,033
161
p-3
l
-87,814
-88,079
2G5
-
Table 3 and the pair potential model. Thus, the constant modified atomic diameters listed in Table 3 for simple metal components are applicabie both to these Class V solid solutions as well as to Class V intermetallic compounds. The above results suggest that the Class V contractions for these different groups of Class V intermetallit compounds have different origins. However, the comparison of the energies of formation using the modified parameters do not indicate any difference in behavior of these two types of component combinations. (See Table 6). The results indicate that the model yields correct prediction of the Al-L12 ordering energy for 14 out of 16 phases, with one error in each group. It thus seems reasonable to apply the model in order to evaluate the relative s~biljty of Class V type phases. Let us now consider Class VI type phases (combinations between early transition elements and simple metals.) According to Ref. 2, we may expect that an expansion should occur reiative to the values predicted on the basis of the Class V modified atomic diameters. Further, this expansion does not seem to be very sensitive to the identity of the transition element component. Consequently, the simpIest assumption to make is that this expansion represents a change in the atomic diameter of the simple metal component. Accordingly, new values of atomic diameters have been calculated for the simple metal components of Class VI intermetallic compounds by fitting to the observed lattice parameters. As shown in Table 7, the expansions reIative to the Class V values of the modified atomic diameters range from about 0 to about 0.5 A, increasing with increase in the principal quantum number and with decrease in the group number of the transition element. The effect of the transition principal quantum number was previously noted in Ref. 2 where it was called the “Class VI-principal quantum number effect”. Data available allow the prediction of lattice parameters for only one Class VI type solid solution.
MACHLIN:
PAIR POTESTI,\L
MODEL
Table 7. Expansion of simple metal atomic dtameter m class VI phases relattve to value m class V phases (.A) Elemnt
Trsnsiticn
Simple Netal compancnt
SC
Y
Al
0.084
0.131
-Q
0.073
0.376
Ti
La
0.611
AA
0.20
dc
0.132
CU
0.1‘83
8: In
lif
, O.?,,
xb
T8
0
0
0
O.li
-
.? 0.13:0.1m ---
O.C57-%5R0.001
0.006
---
0.233
Pb
-0.13
-.-
5n Zx
V
0.208 ---0.201 0.163
::a
Cmponent Zr
0.20.
0.43 6
0.033
The lattice parameter predicted using the expanded simple metal atomic diameter of Au in a Nb matrix IS given m Table S. (The expanded sample metal atomic diameter is obtained by adding the appropriate value of the expansion from Table 7 to the corresponding atomic diameter given in Table 3. The expanded atomic diameter is then substituted into equation (2) to obtain the corresponding cohesive energy of the expanded simple metal component.) According to the data in Table 8 the Class VI solid solution of Au as a solute in Nb obeys the sa?ne Tuzlation as does the Class VI intermetallic compound (NbsAu). That is, the same value of expanded simple metal atomic diameter applies to both the Class VI solid solution and intermetallic compound, at least for the transition metal rich solid solution. There are a few systems which can be used to evaluate the concentration dependence of the expanded atomic diameter for Class VI solid solutions where the simple metal is the solvent. Table 9 compares the calculated lattice parameters based on the relation D,,, = ‘*cr. [D,(Table 7) - D,&(unmodified)], (3) with the observed lattice parameters. The term cr is the atomic fraction of transition element component in the binary solid solution. It appears from the data in Table 9 that this relation describes the concentration dependence of the expansion in the atomic diameter in Class VI solid solutions, at least for the few cases available to test it. The Al-L12 relative stability predictions for Class VI phases are compared to the observed stabilities m Table 10. Except for the case of HgTi3 the predictions are in agreement with observation. It is possible that the observation is wrong because HgTi, is
---
O.lT
_o.fyj-y 0. O-FxTl-
observed to be stable in the A15 structure at [arc temperature and the model does predict that the A15 structure (predicted cohesive energy equal to -95.6OOcal:g atom) is stable relative to both the Al and L12 structures. Perhaps, the single observation of a stable L12 structure at high temperature in this alloy is wrong. It deserves an independent check.
DIFFERESCE IN PREDICTIOS OF RELATIVE STABILITY LSISG MODIFIED VS UNMODIFIED SDIPLE METAL ATOMIC DLtiIETERS In the comparisons of relative stability for Classes V and VI (Tables 6 and 10) there was no indication that the predictions of the pair potential model using the modified simple metal atomic diameters differed from those using the pure component atomic diameters. This question has been explored further. One clear example of a difference in predictions of stability for these cases has been discovered. This example involves the relative stability of pseudo-binary solid solutions with respect to a mixture of terminal pseudo-binary solid solutions, i.e. the problem of development of a miscibility gap in pseudo-binary solid solutions. The pseudo-binary explored was that involving Al5 phases of the type Nb3(X, Y) and V,(X, Y). where X and Yare simple metals. The results of the calculations are given in Table 11. According to these results, the use of modified simple metal atomic diameters yields a prediction in agreement with observation, whereas the use of unmodified simple metal atomic diameters produces a prediction in disagreement with observation.
Table 8. Observed and predicted lattice parameters of Al solld solutions class VI component combmations
So2~:%lute
Lattice
s4
s&e
Umodified a*
Nb
AU
23
3.279
-
Modified D* 3.304
Panmeters, Observed k 3.297
bgstroms Vagsrd’s 5-L 3.291
LOW
MACHLIX:
PAIR POTENTIAL
MODEL
Tabie 9. Observed and predicted lattice parameters of Al solid solutions class VI component combinations
AU
Ti
12
4.047
4.254
4.096
4.065
Au
v
10
4.045
4.130
4.069
4.067
24.5
3.998
4.070
4.039
4.039
40
3.948
4.007
4.002
4.003
Table 10. Observed and predicted relative stability of the Al and Ltz structures. class VI component combinations Phas*
Stable structure Al Ll2
Cohsafvr E(U)
Enereios,
csI/~atom
E(L12)
E(M)
- E(Ll2)
TiZlA,
+
-50,653
-52,753
2100
N-bzs,
+
-64,542
-69,163
4641
WI3
+
-51,648
-53,217
1569
BeTi,
*
-91,948
-58,779
-3169
An,
+
-130,668
-131,644
976
Al-L12
ORDERIKG
ENERGY
The success in predicting the relative stability of the Al-Ll, competition raised the hope that the model might just be good enough to p&ict the ordering energy E(AI)-E(Ll& quanti~tively, as well. Although some values of this ordering energy have been measured, a more extensive measure of it can be obtained by using either the observed Ll,-Al transition temperature (7;) or the Liz-liquid transition temperature (Tw). In the latter case, T, must exceed 7&. We know that at the transition temperature, z;, that the free energies of the Al and Liz phases are equal. Hence, E(A1) - ?;.S(Al) = E(L12) - T,[S(L12)] Of
E(Ai) - E(L1,) = T,[S(Al) - S(Ll,)].
4.083
3.963
potential model parameters to predict cohesive energy values, implies either that the scheme we have used to correct for the heteroelectronic interaction is too rough an approximation, or that it neglects some important effect, or that the entropy difference between the phases varies markedly. According to the previous relation the slope of the line in Fig. 1 should equal SfAl)-S(L12). Now, the configurational contribution to this difference in entropy is 1.118 Cal/g atom/Y. The best value of this quantity given by the data in Fig. t is 1.9cal/g atom/‘C. The difference between these two numbers (0.782cal/g atom/‘C) corresponds to an excess of other than configurational entropy of the Al structure over that of the Lll structure, on the areruge. This result is reasonable. The data points in Fig. 1 deviate from the best straight line through them. Where the sign of the deviation is constant for a given class of component combination it appears that two interpretations may be offered for the deviation. One is that the predicted AE is in error by the average deviation for the class or that the AS for that class differs from the average AS value. Study of the deviations imply that the latter is the more likely explanation. Thus, for Class I it appears that the value of AS z 2.2 Cal/g atom/“C; for Class II combinations the apparent value is about 1.67 Cal/g atom,K.* If the previous explanation for the deviations by class is correct, then the remaining random deviations represent the uncertainty in the prediction of AE, i.e.
Thus. a plot of predicted values of E(A1jE(L12) versus obserced values of 7; should yield a straight line through the origin of slope S(A1jS(Llt). Table 11. Predicted values of relative energy of formation We show in Fig. 1, for homoelectronic phases, a of some pseudobinary Al5 phases. cal’g atom plot of predicted values of &Al)-E(L1,) vs 1; or xw:,, Iredieted Encrw of gz:pon Belatite to Binrrr whichever of the latter has been measured. It is Pswdcbirury obvious that the data, on the average, obey the f( E(T,M’) + E(T,X*) ) E(T,#$l;) expected relation. This is an extraordinary result! The Modiiied Umcdiiied pair potential model certainly represents a breakthrough in the prediction of the relative stability of 1099’ 311 b-b,(AQB.i) alloy phases and the significance of its unexpected 1926 205 h%3(AliPbi) success deserves first-principle exploration. 147 3 Nb,(Pb+Sn+) 264 Figure 2 shows a similar plot for heteroelectronic 2829 P,(A’fPb$ phases. The much greater scatter, which incidentally 390 1814 P3(Alpyf is unchanged even upon using the unmodified pair 3 146 V3(Pb+Sni) * Class I type phases contain simple metal components only. Class II type phases involve combinations between early and late transition elements.
?vfACHLIN:
P;\IR POTENTIAL
MODEL
519
6 000
E 0 5000 z \ ;; u G- 4000
p
3ooc
t
Ir
0
0” 5
2ooc
0
,” si!
0
0
0 0
w” K ?COC P
0
0
500
1000 OBSERVED
1500
2000
TRANSITION
2500
TEMPERATURE,
3000
OK
Fig. 1. Correlatton between predicted ordermg ene_rgyand observed transItion temperature. Homoelectromc type phases. 0 Class I (spsp); 0 Class 11(tf-4 component combinations. Al-Ll; tfansformatton.
about 5OOcal/g atom. This also must be considered to be an extraordinary result. To understand why it is extraordinary and totally unexpected, dwell for a moment on the fact that the cohesrve energy values are on the order of IM.Ocal g atom. A2 + B2 ORDERING
ENERGY
The ability to calculate the cohesive energy of random solid solutions and the success noted in the previous section using this ability to predict the Al-Liz relative stability suggests an evaluation of the AI-B2 relative stabihty to provide inFormation concerning
+
the ordering energy driving this transformation. A comparison of the predicted cohesive energies for these structures is given in Table 12. The data in Table 12 reveal that, for homoelectronic phases, the model predicts the correct sense of the A2 --t B2 transformation for 7 out of 9 Class I type combinations and 13 out of 17 Class II type combinations. The magnitudes of the incorrectiy predicted negative ordering energies are all small (less than lOOOcal/g atom.) Thus, the model yields only a fair prediction for the sense of this transfo~ation. The scatter in ordering energy values in a piot of predicted ordering energies YStransition temperature
+
-t
a
+
+ ++ ++
iz
+ @ t
+
OL 0
I
I
I
500
1000
1500
OBSERVED
TRANSITION
I
2000 TEMPERATURE,
I
2500 *K
I
3000
Fig. 2. Correlation between predicted ordering energy and observed transition temperature. Heteroekctronic type phases. c Class V: D Class VI component combinations. Al-LIZ transformation.
550
MACHLIN:
PAIR POTENTIAL iMODEL
Table 11. Predicted relative stabilities of Al and 52 structures for ~omoel~cronic phases that are observed to be stable in the B2 structure Phase
Predictad
Cohesive
E(A2)
Energies,
E(B2)
cal/gstom
EbZ) -
LiAg
-66, t 70
-67,703
1533
LiPb
-46,934
-52,387
5453
LiTl
41,:GZ
-51,941
4679
W.&J NgAu
-56,378
-56,182
-196
-69,597
-70,551
HgTl
-42,164
44,011
1847
BeCu
-76,419
-80,591
4172
CsTl
-55,561
-55,358
-203
SrTl.
-55,034
-55,672
638
scco
-108,231
-111,685
3454
E(B2)
954
SCM
-105,013
-109,219
4206
SoRu
-134,462
-133,974
-488
SC&h
-128,131
-127,s21
-310
ScPd
-104,676
-105,076
ScIr
-142,449
-141 ,478
-971
TiRU
-141,067
-140,908
-159
TiOs
-157,971
-158,372
395
TiCo
-116,923
-138,054
1131
400
TiNi
-114,591
-116,338
t 747
HfCO
-136,562
-14G,991
10,429 3603
HfRu
-163,834
-167,437
HfOS
-182,035
-183,813
i 7-78
zrC0
-126,906
-136,455
9549
zrnu
-154,146
-f 57,296
3148
zros
-172,206
-173,580
1374
VOS
-154,135
-155,008
873
(see Fig. 3) suggests that another factor contributing to the ordering energy must be present in the A2 --+ B2 tmnsformation. This factor must be volume independent, because the model predicts the lattice parameters of these phases correctly. DISCUSSION
OF RESULTS
The results for homoelectronic solid sofutions reveal why previous attempts to predict lattice parameters of solid solutions have failed. Namely, these attempts did not incorporate the effect of a difference in electronegativity between the components upon the tattice parameter of the solid solution. The pair potential model does incorporate this effect [I] and, as shown in the present paper, is in essential agreement with observations of both lattice parameters and relative stabilities of solid solutions and intermetallic compounds for homoelectronic component combinations. It should be noted that no adjustable parameters are involved in these predictions! The fact that the pair potential model predicts the relative stability and the correct value of the ordering energy for the Al -, Lt2 transformation has an important implication concerning the driving force for this transformation. As shown in Table i3. the predicted change in lattice parameter for this transformation often exceeds the r.m.s. uncertainty,‘in the lattice
parameter (OX,) for those phases where the L12 structure is stable and is less than this value for the cases where the Al structure is stable (i.e. with the exception of LiAg,.) The implication of these results is that the Al--t L l2 order-disorder transformation is driven by the tendency to reduce strain energy upon ordering. Although the same driving force exists in the A2 -c B2 order-disorder transformation, it is obviously not the only contribution to the ordering energy of this transformation as shown by the many cases where the predicted difference in lattice parameters is negligible but for which the BZ structure is stable-see Table 14. We shall call this additional contribution to the ordering energy of the AZ-* B2 transformation the A2 - B2 volume independent ordering energy factor. A possible candidate for this factor is the electrostatic type interaction considered by Blandin and Deplante [6] or that considered by Inglesfield [7]. The fact that no evidence of this factor was found in the relative stability comparisons previously made (AlILl*, Cl5C32, Ll,/Alj, A2/AiS [l--l. S] suggests that it may be unique to the B2 structure. i.e. another possible candidate for this factor may be a Fermi surface-Brilloin zone stabilization. The nature of the stabilization associated with intermetallic compounds formed from Class V combinations between the (Ni, Pd, Pt) group and Group IB or IIB elements is unknown. It seems unlikely that this stabilization is similar to that applicable to other Class V type phases. Fortunately, the rule seems to be for “regular” behavior, with the exception corresponding to “irregular” behavior. Study of the results suggests that the Class V contraction in this special group of irregular behaving phases cannot be described in terms of modified simple metal atomic diameters and the pair potential model. Were this to be the case then we would expect that the predicted &Al)-E(L1,) would correspond to EtAI) (unmodifiedtE(L1,) (modified}. The latter vatues are much larger than the observed values of this difference in cohesive energies. This interpretation of the results is consistent with the change in the trend of Class V contractions with group number of the simple metal component, as exhibited in Fig. 7 of Ref. 2. CONCLUSIONS 1. The predicted lattice parameters of the pair potential model are in agreement with the observed vaiues for solid solutions within an r.m.s. deviation of 0.38%. As such, the pair potential mode1 represents a significant improvement over previous methods [5) of predicting the lattice parameters of solid solutions. This improvement is due to incorporation of the effect of eiectronegativity difference in the pair potential model and its neglect in prior models. 2. The lattice parameters of Class V phases, solid solutions as well as intermetallic compounds, are con-
MACHLIN:
PAIR POTENTIAL
410DEL
7 000
i IO;423
6 000 A
E ; 5000 P
A
.
2 5 - 4000 :: 5 ‘:
3000
E ” :: o d
2000
A
: u iti
i
A I
IO00
A
ii
0
A
-1000
A
A
I
I
I
I
I
500
1000
1500
2000
2500
OBSERVED TRANSITION
TEMPERATURE,
Fig. 3. Plot of predicted ordering energies for A2-BZ transformation perature. A Class I (spsp); i Class II (i-3 component combinations.
Table
System AUCU, CUAU, CaPb, CaSn, CaTI, ScRhJ ScPd, ScPt, TiRh, TtPt, Vir, ZrIr> NbIr, HfIr, hIgAg, ZnAu, AgAu, AuAg, AgCd,
CdAg, HgAg, ZnAl, LiAg, ZnAg, ZnCu,
13. Predicted
contraction transformation
[@(Al)-a*(LlJ]: o*(Al), “/, 0.57 0.31 0.87 0.85 1.01 0.99 1.14 0,83 0.63 0.56 0.28 0.96 0.61 1.00 0.56 0.51 0.15 0.00 -0.15 0.03 -0.37 -0.11 0.88 0.31 0.04
due
to
OK
%s observed transition temHomoelectronic type phases.
AI-LIZ Table l-1. Predicted contraction due to ordering for phases stable in the BZ structure
Structure
System
LIZ
LiAg Lifb LiTl MgAg MgAu MgTl BeCu CaTl SrTl scco SCNI ScRu ScRh ScPd Stir TiRu TiOs TiCo TtNi HfCo HfRU HfOs ZrCo ZrRu ZrOs vos
Al
[a*(A2h~*(BZ)]ja*(A21. go 0.50 2.38 2.44 - 0.20 0.17 1.12 1.41 - 0.03 0.41 0.94 I 14 -0.20 -0.11 OlS - 0.30 - 0.06 000 0.35 0.49 3.35 0.66 0.37 2.19 0.59 0.20 015
551
MACHLIN:
PAIR POTENTIAL MODEL
sistent with the use of constant simple metal atomic diameters (modi~ed values) that differ from the values applicable to homoelectronic type phases and pure components, except for solid solutions formed by components one of which belongs to the (Ni, Pd, Pt) group and the other to Group 1B or 1IB of the periodic table. For this exception. the unmodified pure component atomic diameters are applicabie. 3. The modified simple metal atomic diameters deduced for Class VI type phases depend upon the identity of the transition element component, but for a given combination of components is independent of structure. 4. The pair potential model, using either unmodified or modified simple metal atomic diameters correctly predicts the AI/Lll relative stability competition for 49 phases out of a total of 52 phases. 5. The predicted value of the ordering energy for the .41+ LIZ transformation is found to be proportional to the experimental value of the Ll 2- Al or L12 --+ liquid transition temperature. The value of the average difference in entropy deduced from this proportionality is S(Al)-S(L12) = 1.9 Cal/g atom/C 6. The use of modified simple metal atomic diameters yields agreement with the observed -relative energy of formation of Class VI pseudobinary solid solutions having the Al5 structure, whereas the use of unmodified diameters produces prediction~in disagreement with observations. 7. The pair potential model does not consistently predict the AZ/B2 relative stability. The cases that violate the predictions are characterized by having
nearly equal v-alues of lattice parameters for both ordered and disordered phases. Comparison to those phases exhibiting the Al,‘LI? transition and that also obey the condition of nearly equal lattice parameters in the ordered and disordered structures leads to the conclusion that there exists a constant voiume ordering energy component that contributes to the stability of the B2 structure, but plays RO role in the Al - L12 order transfo~ation. 8. The source of the stabilization effect that is unique to intermetallic compounds formed between components one of which belongs to the (Ni, Pd, Pt) group and the other to Group IB or IIB is unknown. Because this eFfect appears to be absent in disordered solid solutions formed by these component combinations, it is likely that the effect requires an ordered arrangement of the components.
REFERENCES I. Machlin E. S.. Acta Met. 22, 95 (19711. 2. klachlin E. S.. Acta Met. 22. 109 1197-U. 3. lLfach&n E. S.. Acfa Met. 22, 367 i197-r). 4. Machlin E. S.. Acta Met. 22. f433 (19-4). 5. Kin8 I-I. W.. in A/ME Symposium on .-lilo.ving Beharior and Eficts 111Concentrated Solid Solutions. Gordon &
Breach, NY (1967). 6. Bland~n A. and Deplante J. L., J. Pfi_rs.Radium 23, 609 (1962). 7. Inglesfield J. E., Acta Met. 17, 1395 (1969). 8. ;Machlin E. S., J. Phys. Chem. Solids. To be publIshed. 9. Bachner A., Goodenough J. and Gatos H.. J. Ph.w. Chem. Solids 28, 889 (1967).