ARBITRARY STOICHIOMETRY ASSOCIATES AND MIXTURE
THERMODYNAMICS IN LIQUID ALLOYS
. Shunyaev K.Yu1,. Lisin V.L1, Zinigrad M.I2.
Institute of metallurgy, Ural’s Division of Russian Academy of Science, 101Amundsen Str.,
Ekaterinburg 620016, Russia, shun@ural.ru
2
College of Judea and Samaria, Science Park, Ariel, 44837 Israel, zinigrad@research.yosh.ac.il
1
Role of arbitrary stoichiometry associates solutions in formation of the
mixture thermodynamic characteristics of melts is analyzed in terms of ideal
associated solution model. Calculation of the properties depending on size
and sign of single energy parameter are carried out as an example of model.
It’s shown, that the model allows to describe not only positive or negative
deviations from ideality, but also simultaneous positive deviation of one of
properties and negative another. The calculation results of thermodynamic
properties of mixture and liquidus curve positions for eutectic systems Na-K
and Ag-Bi and system with unlimited solubility Cs-K are presented.
One of models successfully used for accounts of the thermodynamic characteristics of
multicomponent liquid systems, is the model of the ideal associated solution. Traditionally
model is applied to solutions with stable compounds in solid phase. Thus it’s usually assumed,
that melt consists of single atoms of initial components and one or several associates with
fixed stoichiometry and minimal size. The equilibrium constants for formation reaction of
associates from initial components and often and stoichiometry associates are fitting
parameters [1-4]. The existence possibility of single atoms associates was considered by the
general theory of the ideal associated solution [1,5,6], however only recently began to occur
works, in which self-associates were taken into account at calculation of specified systems [78]. In works [9-14] was shown, that the account of self-association, even limited only by
configuration contributions in entropy, is sufficient for qualitative explanation of
thermodynamic properties behavior characteristics of metals melting both thermodynamic
melting characteristics and liquid eutectic alloys mixture.
Successful use of self-associates for calculation of pure metals and simple eutectic
properties allows to assume, that in multicomponent melts can exist also any stoichiometry
associates. If such associates existence is supposed, than, it is obvious, that their influence first
of all should have an effect in systems having unlimited solubility in solid and liquid states.
However, for simple eutectic, as well as for systems with stable compounds in solid phase, the
account of any stoichiometry associates, generally speaking, can have an effect on size of the
calculated properties, as well as on qualitative picture of their behavior. The present work goal
is the general scheme development of the account of arbitrary stoichiometry associates in the
ideal associated solution model and analysis of their influence to behavior of the
thermodynamic characteristics of mixture and position of liquidus curve.
Let's consider binary system AcB1-c, which components in a liquid phase form a
solution with complete mutual dissolution. Let's present it as ideal solution of associates An (i),
Bn (j) and AnBm (i, j, q), (n, m - number of the appropriate atoms in a complex, i, j and q number of nearest neighbours pairs such as AA, BB and AB in a complex, accordingly).
1 - 12
Accepting the energy of a complex as determined by the pairs energy sum of nearest
neighbours and limiting entropy only by configuration contributions, we obtain mole fractions
of complexes related by following equations [10]:
 i 
x An ,i  K An ,i x An1  exp  A  x An1  t Ai x An1
 kT 
 j 
x Bn , j  K Bn , j x Bn1  exp  B  x Bn1  t Bj x Bn1
(1)
 kT 
  i   B j   AB q  n m i j q n m
x An Bm (i, j , q )  K An Bm x An1 x Bm1  exp  A
 x A1 x B1  t A t B t AB x A1 x B1
kT


Where x An,ш ; x Bn, j ; x An Bm ; x A1 ; x B1
- mole fractions of complexes An, i, B
n, j,
AnBm and of
single atoms A1 and B1, accordingly, A, B and AB – bond energy of nearest neighbours
pairs AA, BB and AB taken with opposite sign, K An,i ; K Bn,i ; K An Bm - constants of
appropriate equilibrium.
Then the balance equations system for definition of single atoms concentration in
solution has the following form:
 x
n
c
i
An ,i
  x B n , j    x An Bm  1
n
j
n ,m i , j ,q
 nx
 x
n
i
n
An ,i
i
An ,i
   nx An Bm
(2)
n ,m i , j ,q
  x B n , j    (n  m) x An Bm
n
j
n ,m i , j ,q
The equations system solution (2) with the account of (1) allows to find mole fraction
of single atoms in solution. Then it’s easy to calculate the thermodynamic characteristics of
system (see, for example, [1-4, 10-13]).
For realization of calculations it’s necessary to know energy parameters A, B and
AB, and also nearest neighbours pairs number of a various type in associates. The energy
parameters A, and B can be estimated from pure components melting temperatures [9].
Then, as varied parameter remains only AB. It is possible to calculate total number of pairs in
associate in assumption, that the appropriate crystal local structure remains the in a liquid [9,
12]. For the preliminary analysis of opportunities of model it is possible to choose linear chain
approximation for associate structure, as it was done in [10] for simple eutectic. Such
simplification allows to carry out summation in (2) easily. Thus, as shows the analysis, which
has been carried out in [10], the accuracy loss for the calculated properties does not exceed 10
%.
Some properties calculation results for model alloy which components have melting
temperatures 700 and 1000 K are shown at figures 1-4 and in the table 1.
Parameters A, and B were determined from melting temperatures of components, and
parameter W=[AB – 0.5(A + B)] was varied. As well as it was expected, the model allows
to describe both positive, and negative deviations from ideality. Thus the negative deviations
can be arbitrary large, and at the large positive W values behaviour of the mixture
thermodynamic characteristics becomes same, as for systems with strong interaction of
components. Other situation is observed at negative W values, i.e. when the formation of pairs
such as AB is energetically unprofitable. The mixture enthalpy positive value growth take
1 - 13
place at small W values of is observed as long as the energy loss can be compensated by the
configuration entropy. Mixture enthalpy begins to decrease at the further W values increase.
Mixture entropy also passes through maximum, but at others, much smaller W values, while
activity of components and mixture free energy change monotonously (table 1). Moreover, it
has appeared, that the model allows to describe a situation, when mixture enthalpy is negative,
and components activities deviations from the Raul law are the positive. It is an additional
illustration of that circumstance, that the offered model, in spite of the fact that it contains only
one varied parameter, such as interchange energy, is capable to explain much more various
behaviour of properties, than other similar models. In difference, for example, from model of a
regular solution, where the sign of all properties deviation from ideality is determined by
energy parameter sign, in the model, considered in present work, such unequivocal dependence
is not present. The value of property is defined as result from the several contributions, say, of
direct pair contribution to energy and indirect configuration contribution. Thus the result
depends on energy parameter value, melt temperatures and component melting temperature.
The calculations of mixture thermodynamic properties and liquidus curve position for
binary Na-K, Ag-Bi and Cs-K alloys were carried out. The Na-K is eutectic system [15]. Its
components are completely soluble in a liquid phase and practically are insoluble in solid state.
The thermodynamic characteristics of mixture are practically symmetric relatively equiatomic
composition. The calculation results, which have been carried out by offered model, are given
in figures 5-7. It follows from comparison with experiment, that the model in general
reproduces behaviour of properties concentration dependencies and liquidus curve position. It
is obvious, that for the good quantitative agreement between calculated and experimental
results, the account of mixed associates with various stoichiometry appears to be necessary.
The offered model clearly demonstrates importance of the autoassociation effects account at
calculation of eutectic systems properties and allows to carry out the calculation of mixture
thermodynamic characteristics, as well as melting diagrams basing on the same concepts.
The values of mixture thermodynamic characteristics and liquidus curve position for
binary system Ag-Bi calculated with parameter W = -3500 joule/mole are displayed in the
table 2. The results of calculations by model with account only autoassociation of components
are given for comparison. It’s can be seen, that the calculated results become much closer to
the experimental vales for all set of the calculated properties.
The Cs-K system has unlimited solubility in solid and liquid state close to liquidus
curve. The calculated results of mixture thermodynamic characteristics compared with
experimental data for this system (W parameter was accepted as -150 cal/mole) are given in
the table 3. It can be seen from the table 3, that the calculated results are in satisfactory
agreement with experimental data in this case too.
The results of work show, that the offered variant of the ideal associated solution
model can be applied for absolutely new objects for this model: eutectic systems and systems
with unlimited solubility. Thus the qualitative interpretation of the thermodynamic properties,
as well as quantitative calculations are possible.
The work is realized within the framework grant " Integration”
(project № 00-15-92420)
1 - 14
REFERANCES
1. Prigogine I., Defay R. Chemical thermodynamics, Longmans Green and Co., London, 1954.
2. Wasai K., Mukai K. // J.Jap. Inst. Metals, 1982, v.46, № 3, p.266-274.
3. Sommer F., Z.Metallkunde // 1982, v.73, № 2, p.72-86.
4. Schmid R., Chang Y.A. // CALPHAD, 1985, v.9, № 4, p.363-382.
5. H.Kehiaian // Bull. Acad. Polon. Sci., Ser. sci. chim., 1964, v.12, №7, p.497-501.
6. Morachevsky A.G., Mokrievich A.G., Majorova E.A. // JOCH, 1989, v.59, № 9, p. 19271934. (in Russ.)
7. Ivanov M. // Z.Metallkunde, 1991, v.82, № 1, p.53-58.
8. Singh, R.N., Sommer F. // Z.Metallkunde, 1992, v.83, № 7, p.533-540.
9. N.K. Tkachev, K.Yu. Shunyaev, A.N. Men, N.A. Vatolin. //Melts, 1988, v.2, №.1, p. 3-11.
10. N.K. Tkachev, K.Yu. Shunyaev, A.N. Men, N.A. Vatolin. // DAN USSR, 1988, v.302,
№1, p. 153-157.
11. K.Yu. Shunyaev, N.A. Vatolin // DAN RAS, 1993, v 332, №2, p. 167-169.
12. K.Yu. Shunyaev, N.A. Vatolin //, Melts, 1993, №5, p. 28-31.
13. K.Yu. Shunyaev, N.A. Vatolin // Metals, 1995, №5, p. 96-103. (in Russ.)
14. Shunyaev K.Yu., Tkachev N.C., Vatolin N.A. // Thermochimica Acta, 1998, v.314, p.299306.
15. Hultgren R., Desai P., Hawkins D., Gleiser M., Kelley K. Selected values of the
thermodynamic properties of binary alloys. Metal Park; Ohio: ASM, 1973.
16. Alblas B.P., van der Lugt W. Small-angle x-ray scattering from sodium-potassium alloys //
J. Phys. F, 1980, V.10, P.531.
17. Kagan D.N. High temperatures thermal physics, 1988, v.26, №3, p.478-491. (in Russ.)
1 - 15
Table 1. Dependence of properties for equiatomic composition melt on energy parameter value
W at T=1100 K.
-60000 -30000 -10000
-6000
-3000 -2000
0
3000
10000
30000
W
Дж/моль
289
515
641
534
370
295
104
-300
-1987
-12460
HM
Дж/моль
-3535
-3670
-4159
-4382
-4600 -4684 -4872 -5210
-6329
-12590
GM
Дж/моль
3.476
3.805
4.364
4.469
4.518 4.526 4.524 4.464
3.947
0.122
SM
Дж/мольK
0.68
0.669
0.635
0.619
0.605
0.54
0.587 0.566
0.501
0.253
аА
Table 2. The calculated and experimental properties of liquid Ag-Bi alloys at T=1000K.
1 – calculation with self-associates account only; 2 - calculation with the arbitrary
stoichiometry associates account. The experimental results are taken from [15].
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
cBi
398
474
514
525
508
462
380
246
0
H1
Дж/моль
276
352
396
410
395
350
273
160
0
H2
473
699
946
1540
1711
1582
1046
0
1222210
Hэксп
1728
2017
2175
2226
2177
2022
1735
1243
0
-G1
Дж/моль
3047
3655
3992
4102
3995
3660
3053
2055
0
-G2
4305
4489
5100
4515
3920
3096
1954
0
49451050
-Gэксп
0.568
0.653
0.715
0.765
0.808
0.847
0.887
0.932
1.0
aBi-1
0.330
0.437
0.528
0.610
0.687
0.76
0.834
0.911
1.0
aBi-2
0.231
0.419
0.579
0.743
0.802
0.869
0.930
1.0
0.6790.08
aBi-эксп
0.888
0.849
0.809
0.766
0.716
0.655
0.569
0.424
0
aAg-1
0.834
0.761
0.687
0.611
0.529
0.437
0.330
0.195
0
aAg-2
0.734
0.604
0.509
0.402
0.348
0.272
0.184
0
0.4480.05
aAg-эксп
Table 3. The calculated and experimental properties of liquid Cs-K alloys at T=384K. H, G
–cal/mole. Experiment H [15], activities [17].
K
H
Hэкс
-G
aK
aK-экс
aCs
aCs-экс
Scc(0)
0
0
0.1
9.4
0.2
16.0
0.3
20.5
0.4
23.1
0.5
24.0
0.6
23.1
0.7
20.5
0.8
16.1
0.9
9.5
1
0
0
8
14
19
23
28
27
23
17
0
0
194
290
350
381
265
392
381
349
290
194
0
0
0
1
1
0
0.18
0.11
0.91
0.90
0.12
0.31
0.22
0.83
0.80
0.22
0.42
0.32
0.76
0.70
0.29
0.51
0.42
0.68
0.61
0.34
0.60
0.50
0.60
0.52
0.35
0.68
0.58
0.51
0.43
0.34
0.76
0.69
0.42
0.32
0.29
0.83
0.79
0.31
0.20
0.22
0.91
0.90
0.18
0.10
0.12
1
1
0
0
0
1 - 16
1
Scc(0)
aA, aB 0,9
0,4
2
0,35
0,8
0,7
0,6
1
3
0,3
2
0,25
0,5
5
0,2
0,1
6
0,1
5
0,15
4
0,3
4
0,2
3
0,4
6
0,05
0
0
0
0,5
0
1
0,2
0,4 c 0,6
A
0,8
1
cA
Fig. 1. Components activity of model alloy at
T=1100 K at various energy parameter
values: W =-3000 (2); 3000 (3); 10000 (4);
13000 (5); 30000 (6).
T, K 1200
H 1000
Дж/моль
500
2
0.2
3
0.4
-1000
-1500
1000
1
0
-500 0
Fig. 2. Long wave limit of the partial
structure factor concentration - concentration
of model alloy. Curves numbers the same, as
in a fig. 1, dash - ideal solution.
cA
0.6
0.8
1
800
600
4
-2000
400
-2500
-3000
5
200
-3500
0
0
0.2
0.4 cA 0.6
0.8
1
Fig. 3. Mixing enthalpy dependence of model Fig. 4. Liquidus curve position for simple
alloy at T=1100 and various values of energy eutectic. Downwards curves for W = +3000;
parameter: W=0 (curve 1), 3000 - (3), 10000 0; -3000; -30000
- (4), 13000 - (5), -3000 - (2).
1 - 17
aK, aNa
1
Scc(0)
1
1
0.8
0,8
0.6
0,6
2
0.4
0,4
0.2
3
0,2
0
0
0
0
0,5
cNa
1
Fig. 5. Activity of components for liquid NaK alloys at T=384K. A continuous curve is
calculation in model with self-associates only,
dashed curve is model with mixed associates,
●, + - experimental activities of K and Na,
accordingly [15].
0.5
cNa
1
Fig. 6. Long wave limit of the partial
structural factor concentration - concentration
of liquid Na-K alloys at T=384K, (1) - model
with self-associates only, (2) – model with
mixed associates, (3) - ideal solution, o, x experiment [16].
Fig. 7. The phase diagram of a binary Na-K alloy [12] (X - liquidus curve position calculation
in model which self-associates only, O - calculation with the arbitrary stoichiometry associates
account).
1 - 18