HEATS OF FORMATION OF RARE-EARTH METALS ALLOYS:
SEMI-EMPIRICAL CALCULATIONS
AND EXPERIMENTAL DATA
A.B. Shubin, K.Yu.Shunyaev
Institute of Metallurgy, Ural Division of Russian Academy of Sciences
Ekaterinburg, Russia
Abstract. The semi-empirical calculation of the heats of formation of solid and liquid
alloys is a way to obtain good estimations of unknown values if one already have
representative set of the known experimental data. Considering the rare-earth metals
alloys we have used semi-empirical models – mainly, the Miedema’s approach.
The standard heats of formation for more than 200 rare-earth metal-p-metal alloys
were calculated. Further, our results were used to assess the experimental data of the
different research groups for some binary intermetallides.
Earlier we considered the ability of “classic” Miedema’s model [1-3] to predict
the enthalpies of formation of binary alloys of rare-earth metals with some p-metals
(Me = Al, Ga, In, Tl, Sn, Pb, Sb,Bi). We found that original Miedema’s model
parameters are not useful to obtain good estimations of the experimental data for RMe systems. To calculate the optimal parameters ( electronegativity, electron density
etc.) we analyzed the data set consisted of 87 experimental points (the values of
standard integral enthalpy of formation fH) [4]. The aim of the present work is to
apply these “better” model parameters to the calculation of fH for the binary systems
which have not been used for model optimization.
The Miedema’s basic formula is the following:
fH = f (CAS, CBS) g(CA, CB)  F  P ( - (*)2 + (Q0/P)  ( nws1/3)2 – R/P).
Here f (CAS, CBS) g(CA, CB) is the composition function; F – the Faraday constant
96484.56 C/mol. The values P, Q0/P and R/P are the empirical parameters for
different groups of binary alloys.
The main contribution to the heat effect of alloying is caused by the negative term
- (*)2 where * is the electronegativity parameter for given metal. Using the
positive term (Q0/P)  ( nws1/3)2 one takes into account the electron density difference
on the boundaries of the Wigner-Zeits cells of pure metals. The additional part – R/P
is related to the p-d hybridization of the valent electrons and depends on the valency
of the p-metal. To calculate the composition function one must know also the values
of the pure metals molar volumes VM (and use them in the parameters VM2/3).
The optimal values of the model constants and parameters obtained in [4] by
special procedures are listed in Table 1 (for p-metals) and Table 2 (for rare-earth
metals).
The other constants (optimal values) were the following: P=0.25 V-1cm-2 (density
unit)-1/3 and Q0/P =1.00 ( V2/(density unit)2/3).
The initial experimental data set of 87 points can be well described by Miedema’s
formula using the optimal parameters mentioned. The standard deviation was 6.7
kJ/molat. The maximum deviation between calculated and experimental values were
about 20 kJ/molat.
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The attempts to estimate standard enthalpies of formation for binary
intermetallides using Miedema model were performed later. For example, Colinet [5]
considered about 218 intermetallic compounds (IMC) from the R-Me group. The
experimental and calculated data presented in [5] show significant disagreements.
Table 1. Model parameters nws1/3 , VM2/3 , * and R/P for p-metals.
Metal
Al
Ga
In
Tl
Sn
Pb
Sb
Bi
nws1/3,(density
1/3
1.39
1.31
1.17
1.12
1.24
1.15
1.26
1.16
4.6
5.2
6.3
6.6
6.4
6.9
6.6
7.2
*, V
4.20
4.10
4.0
3.90
4.15
4.10
4.25
4.15
R/P, V2
0.0
0.45
0.25
0.25
0.10
0.0
0.30
0.30
units)
VM2/3, cm2
Table 2. Model parameters nws1/3 , VM2/3 , * and R/P for rare earth metals.
Metal
nws1/3,(density
units)1/3
VM2/3,
cm2
*, V
Metal
nws1/3,(density
units)1/3
VM2/3,
cm2
*, V
Sc
1.32
6.10
3.30
Gd
1.19
7.35
3.18
Y
1.22
7.36
3.20
Tb
1.20
7.20
3.15
La
1.09
7.98
3.05
Dy
1.23
7.13
3.23
Ce
1.07
7.54
3.02
Ho
1.24
7.07
3.20
Pr
1.08
7.57
3.03
Er
1.26
6.99
3.21
Nd
1.11
7.51
3.04
Tm
1.27
6.90
3.24
Sm
1.10
7.36
3.10
Yb
0.95
8.50
3.20
Eu
0.90
9.42
3.16
Lu
1.30
6.81
3.40
These disagreements, probably, are caused by the internal model limitations. The
situation, most likely, can not be essentially improved. We must also take into account
the low precision of the experimental data for the number of R-Me systems. Often the
data obtained by e.m.f. and calorimetric measurements also show differences for the
same alloys.
The attempt to describe the wide data set presented in [5] using our model
parameters allow to verify this conclusion and, undoubtedly, is of considerable
interest.
The results of calculations can be presented as the figures shown below. Figure 1
demonstrates the deviations between experimental and calculated standard enthalpies
for R-Pb alloys. One can see that the array of points lies slightly higher than the
bisector of coordinate angle. At the same time, the agreement between experimental
and calculated values is satisfactory for this and some other systems (e.g. R-Sb (Fig.
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2)). The experimental and calculated data for all the 218 points available are shown in
Fig. 3.
Fig. 1. Comparison between experimental values fH (exp) and predicted values fH
(calc) of the enthalpies of formation of R-Pb compounds.
Fig. 2. Comparison between experimental values fH (exp) and predicted values fH
(calc) of the enthalpies of formation of R-Sb compounds.
So, we can see that our calculations based on the limited initial set of data (87
values of the standard enthalpy of formation) allow to predict “unknown” (by
convention) enthalpies (about 218 values in total). The accuracy of such a prediction
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is not so well, but, in our opinion, it is enough for some estimating procedures.
Besides that, the enthalpies calculated using our optimized model parameters allow to
assess the experimental data of different scientific groups if these data are in
contradiction.
Fig. 3. Comparison between experimental values fH (exp) and predicted values fH
(calc) of the enthalpies of formation of all the compounds considered (218 points).
For example, consider the comparison and assessment of our experimental data for
Sc-Al system[6] obtained by e.m.f. method and the enthalpies of formation of Sc-Al
intermetallides measured by two different calorimetric methods [7,8]. The analyze of
experimental values was performed using Miedema model with optimal parameters
presented in this work.
Four intermetallic compounds were found in Sc-Al system: ScAl3, ScAl2, ScAl,
Sc2Al [7]. Measured integral heats of formation of the IMC in Sc-Al system taken
from different experimental works are shown in Table 3.
One can see that the data of the study [6] (e.m.f.) are in good agreement with the
data of the work [7] (calorimetry) for IMC ScAl3 and ScAl2. At the same time, the
data differences for the other intermetallics are significant. The data [8] (HCl solution
calorimetry) are considerably different also for ScAl3 and ScAl2 compound (see
Fig.4). Nevertheless, these data were taken into account. The accepted values of
enthalpies of formation of intermetallides have been calculated by the following
procedure.
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Fig. 4. Standard enthalpies of formation in Sc-Al system (fH) vs the atomic fraction
of Sc (XSc). Experimental data [6] (triangles), [7] (squares) and [8] (circles). Semiempirical calculation (our data) – dashed line; accepted values – solid line.
Initially we found the squared deviations of the experimental points from the
theoretical curve calculated using optimized Miedema model parameters. Further we
assumed the statistical weight for each experimental value. This weight was inversely
proportional to the squared deviations mentioned. Of course, the weights for each
intermetallide were normalized and their sum was reducted to 1. The accepted value
of the enthalpy for each IMC (see Table 3) has been calculated by averaging of the
experimental values with the corresponding weights.
Table 3. Assessment results for the intermetallic compounds in scandium-aluminium
system.
-fH (exp), kJ/molat
-fH
(accepted),
kJ/molat
[6]
[7]
[8]
-fH (calc),
kJ/molat
ScAl3
41.3
43.5
59.8
36.3
42.6
ScAl2
47.6
48.0
94.1
45.7
47.8
ScAl
67.1
46.0
62.0
51.5
51.1
Sc2Al
42.6
37.0
28.2
40.0
41.0
IMC
It can be easily seen that accepted values of the heats of formation are very close
to those calculated by the Miedema model adapted in this work for R-Me alloys
group.
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Considering the improved original semi-empirical calculations [9, 10] based on
Miedema model we can conclude that if one attempts to describe very wide alloy
groups using the same set of model parameters, only crude approximation of
experimental data can be obtained. So, the predictions in this case are not useful. If the
investigator takes into account some specific alloy group he can be more successful
and this approach can give satisfactory results both for describing of the known data
and for predicting the unknown values. Apparently, this conclusion is valid also for
some another models similar to Miedema’s approach but more simple and developed
specially for rare-earth – p-metal IMC group [11].
Here, the Miedema approach to calculation of standard formation enthalpies was
applied to R-Me alloys. Model parameters found on the basis of the known
experimental data allowed to describe these data with acceptable confidence interval.
The prediction accuracy is considerably higher than the imprecision of the
experimental fH determination. Nevertheless, taking into account great systematic
differences between the results of different scientific groups, the model prediction
may be very useful to estimate unknown values or to analyze strongly different
experimental data.
This research is supported by the Russian Foundation for Basic Research(RFBR)
grant 08-03-00941-a.
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