THE IDEAL ASSOCIATED SOLUTION MODEL TAKING
INTO ACCOUNT OF COMPLEXES OF DIFFERENT
SHAPE AND SIZE
K. Yu. Shunyaev, N.A. Vatolin, L.I. Leontiev
Institute of Metallurgy, Ural’s Division of Russian Academy of Sciences,
101, Amundsen Str., 620016, Ekaterinburg, Russia, shun@ural.ru
Abstract
The review on authors’ works on the development of a
model variant of an ideal associated solution dealing with complexes of various compound, sizes and shape is presented. The
analysis has shown the possibility of using the model for estimation of configurational contribution into the melting characteristics
of monatomic systems. Later the model was used for calculation
for thermodynamic characteristics of binary and ternary eutectic
mixings. It turned out that it shows qualitatively right the peculiarities of thermodynamic characteristics for these systems mixing:
positive enthalpy of mixing, positive deviations of activities from
ideality, the increase of entalphy of mixing and the decrease of
activities with the increase of temperature. Besides, it turned possible to make the calculations for the liquidus line location both
for eutectic systems. In all cases the model parameters were not
adjusted but were estimated from melting temperatures of individual substances.
Introduction
There are many papers dedicated to the calculation of
thermodynamic characteristics of binary alloys mixing in the
frames of associated models [1-5]. Typically the model is applied
to the systems having stable compound in solid state. Common
theory of ideal associated solutions was developed for the associates with different compositions, sizes and shapes [1]. As the rule,
the practical calculations are taking into account only those associates with minimum size and are ignoring the possibility of selfassociation. Though the theoretical analysis of the influence of
self-association on thermodynamic functions of mixing behavior
202
was carried out earlier [1, 6, 7], it was taken practically into consideration only in numerical calculations of the last years [8, 9].
We had suggested earlier [10, 11] the variant of the model
of associated solutions allowing to take into account the presence
of associates of different sizes and shapes in the liquid phase. The
calculation of the associate's energy was reduced to pair interaction of the nearest neighbors. This consideration was limited also
by taking into account only configurational contributions into the
entropy. It was found out that for such associate of infinite size as
crystal it is possible to derive the energy parameter of the model
from melting temperature of stable compound. Moreover, we succeeded to extend the number of properties, which are typically
calculated in associated models on the base of melting characteristics [10-12]. It was shown also that the taking of self-association
into consideration allows applying the given model to any system
including the eutectic one.
Monoatomic systems
With the growth of the crystal's temperature some part of
the bonds between the nearest neighbors becomes longer due to
oscillation effects and, therefore, strongly weakens. The number of
such bonds at the given moment of time is as much higher as the
temperature is. Lets suppose that the melting takes place when the
number of weak bond becomes critically high and the system can’t
stay as the crystal any long. We shall simulate such system by the
potential with the shape of rectangular well. Then all the bonds
will be separated into two groups - one group containing the same
bonds as the crystal and the other group containing very weak
bonds. Hence the liquid phase represents the set of noninteracting
(or weakly interacting) crystal fragments of different sizes and
shapes. It is quite reasonable to describe this liquid phase using the
apparatus of the model of ideal associated solutions modified in
such way as to allow to carry out the calculations taking into account the possibility of coexistence of complexes of different sizes
and shapes.
Lets consider the monatomic system according to [13].
The crystal A in the liquid phase represents the set of noninteracting complexes {An,i}, where n is the number of atoms in the com203
plex, i is the number of the bonds between the nearest neighbors
and can vary from n-1 (the number of the bonds between the nearest neighbors in the linear chain) to some value mn determined by
the type of crystal lattice. Let NA to be the number of atoms in the
system, N A to be the number of complexes with the size n and
n,i
with i bonds with the nearest neighbors and NC to be the total
number of complexes

mn
NC    NA
n 1 i  n 1

n,i
(1)
mn
N A    nN A
n 1 i  n 1
n,i
Let internal energy of liquid phase counted from noninteracting system of separate atoms to be
 mn
EA   A 
 iN An,i ,
(2)
n 1 i n 1
where A is the energy of the bond for the couple of the nearest
neighbors AA taken with the opposite sign. The configurational
entropy of the system written using the point approximation is
SA  k ln
N C!
 N An,i !
(3)
n,i
Therefore the free energy of the system is

mn
FA  E A  TSA    A 
 iN An,i  kT ln
n 1 i  n 1
N C!

 N An,i !
n,i

 A 
mn


N A n,i ln N A n,i 
n 1 i  n 1


mn
 iN An,i  kT N C ln N C   
n 1 i  n 1

204
(4)
Minimization of (4) by
N An,i
taking into account expres-
sions (1) gives the correlation between molar fractions of the complexes with different sizes and shapes in the form of acting masses law
  i
x An,i  exp - A  x nA0  t iA x nA0 ,
 kT 
(5)
where x A  N A NC is the molar fraction of the complexes
n,i
n,i
with the size n and with i bonds with the nearest neighbors,
is the molar fraction of separate atoms,
x A0  x A
i,0
tA =exp(A/kT). Therefore molar fractions of all complexes are
expressed by molar fraction of separate atoms. It is possible to
calculate the equilibrium value of the latter by solving the balance
equation
 mn

n 1 i n 1
 mn
x An,i  
 x nA0t iA 1
(6)
n 1 i n 1
The solution of eq.(6) requires to know the value of energy
parameter A and the sequence of the figures mn. The values of mn
for some types of crystal lattice are tabulated in [13]. As far as the
rows in eqs.(2), (4), (6) are rapidly coinciding, the usage of the
first 10-15 terms is enough to get good accuracy in practical calculations [13].
As regards energy parameter A, it is possible to determine
it from melting temperature of the element A [13]. Indeed, for
such complex of infinite size as solid body one may determine A
by comparing free energies of solid and liquid phases in the melting point
-AzA/2 = kTAln(xAT),
(7)
where TA is the melting temperature of component A, zA is the
coordination number of component A in solid state, xAT is the molar fraction of separate atoms A in liquid at the temperature of
melting. One may write the following equation for xAT by inserting
A from eq.(8) into balance eq.(6)
205


n 1
x nAT
mn
 2i z A 
1
 x -AT
(8)
i  n 1
The solving of the last equation allows to find out the fractions of separate atoms in the liquid at melting temperature and to
determine using eq.(7) pair's bond energy A. The results for some
lattices are given in the [13].
As far as the difference between the entropy of liquid state
and of crystal in the described model is supposed to be only configuration's contribution, the entropy of liquid phase from eq.(3) in
the melting point is the entropy of melting. One can derive the
following entropy of liquid phase per atom from eq.(3) using Stirling's formula and taking into account eqs.(1) and (5) Hence, taking into consideration eq.(7), the configurational entropy of melting per atom in the melting point is

SM
A   klnx AT
mn

2i  n-2i zA 
 x AT
A 
   n - z
n 1 i n 1 
 mn
(9)
n -2i z A 
  nx AT
n 1 i n 1
The calculated values of configurational entropy of melting for the lattices under consideration are also shown in the [13].
For instance, for elements with bcc, fcc and hcp the values of
melting entropy are 0.83-0.85R.
Binary systems
Lets consider now some kind of binary system according
to [14-17]. Let it be the binary alloy AcB1-c containing noninteracting complexes {An,i} and {Bn,i} in the liquid phase. Here c is the
atomic concentration of the component A in solution. The molar
fractions of the complexes may be expressed by molar fractions of
separate atoms as follows
206
  i
x A n,i  exp - A  x nA1  t iA x nA1
 kT 
,
  i
x Bn,i  exp - B  x nB1  t iB x nB1
 kT 
(10)
where xA1 and xB1 are the molar fractions of the separate atoms A
and B in the solution correspondingly.
Let’s introduce some symbols for the following characteristic sums

mn

mn
S1L  

x Ln,i  

mn
nx Ln,i  
n 1 i n 1
S2L  

n 1 i n 1

S3L  
 x nL1t iA
n 1 i n 1

mn
 nx nL1t iL
,
(11)
n 1 i n 1

mn
mn
 ix Ln,i    ix nL1t iL
n 1 i  n 1
n 1 i n 1
where L = A,B indicates what sum belongs to one or another component of solution. The system of balance equations is
S1A + S1B = 1
(1-c)S2A – cS2B
(12)
The first equation represents the rationing (normalization)
of molar fractions of the complexes and the second one is the
equation of material balance.
Free energy of alloy's mixing is described by the following
expression
G  RT[clna A  (1 - c)lna B ]  RT[cln(x A1/x A0 )  (1 - c)lnx B1/x B0 ] (13)
Here aA, aB are the activities of the components A and B
correspondingly, xA0 and xB0 are the molar fractions of separate
atoms A and B in pure component's melts at the considered temperature.
One can express the enthalpy of mixing as follows
207
H  -
 AS3A   BS3B
S2A  S2B
 c A
S3A0
S2A0
 (1  c) B
S3B0
(14)
S2B0
The symbol 0 shows that corresponding sums are calculated here for pure components at the considered temperature. The
entropy of mixing may be calculated as usually
Lets assume now that solid phase's components are absolutely non-soluble in each other. Then, by setting the chemical
potentials to be equal each other and by neglecting the temperature
dependence of energy parameter, one can find out for the part of
liquidus line (Tliq) adjacent to the component L that
LzL
2
 kTliq ln x L1
(15)
Now it is possible to calculate thermodynamic characteristics of mixing from eqs.(13-14) and to determine the concentration
dependence of liquidus line by combined solving of the system of
balance equations (12) and the equation (15), written at first for
one component and later for another component.
Ternary systems
The model is simply applying to multicompounds alloys,
which have a phase diagram of eutectic type. The structure of material balance solutions and formulas for thermodynamics properties is the same, as for binary system. As an example the systems
of Ag-Bi-Cu and Bi-Cd-Sn was examined [18].
The calculations for the location of the liquidus line of
these eutectic systems were made. For example, the location of a
eutectic point of Ag-Bi-Cu is
cAg = 0.036,
cAg = 0.05,
cBi = 0.939,
cBi = 0.945,
cCu = 0.025 (calculation)
cCu = 0.005 (experiment)
Complexes of arbitrary stoichiometry in binary systems
Let now the binary alloy AcB1-c containing except for
complexes {An,i} and {Bn,i} in the liquid phase else complexes
AnBm(i,j,q).(n, m – number of atoms A and B in the complex, i, j
and q – number of pairs of the nearest neighbors types AA, BB
208
and AB in the complex, accordingly). Now set of equations (10)
will look like
 i 
x An ,i  exp  A  x An1  t Ai x An1
 kT 
 j 
x Bn , j  exp  B  x Bn1  t Bj x Bn1
 kT 
(16)
  i   B j   AB q  n m i j q n m
x An Bm (i, j , q)  exp  A
 x A1 x B1  t A t B t AB x A1 x B1
kT


where x An Bm (i, j , q ) - is the molar fraction of the complex
AnBm(i,j,q), AB – is the energy of the bond for the couple of the
nearest neighbors AB taken with the opposite sign.
Then the system of the balance equations for definition of
concentrations of single atoms in a solution looks like the following:
  x    x   x
n
A n,i
i
c
n
B n, j
j
n,m i, j ,q
A n Bm
  nx    nx
n
A n,i
i
n,m i, j ,q
1
(17)
A n Bm
  x    x    ( n  m) x
n
i
A n,i
n
j
B n, j
n,m i, j ,q
A n Bm
Such modernization of the model does not revise significantly the structure of accounting expressions, but only gives a
rise to additional terms, corresponding to mixed associates. Beside
this, there appears an additional energy parameter characterizing
bonding energy of АВ pair. This parameter happens to be used as
a parameter of fitting.
Results and discussion
The analysis of experimental data on entropies of melting
indicates some interesting regularities. It is well known that the
entropies of melting for metal systems with the same lattices in the
solid phase are typically close to each other [1]. Numerical values
of entropy of melting for metals are in common about R. The same
is true for inert gases. The following entropies of melting are rec209
ommended in [20] for metals with different lattices: 1.21  0.08
(HCP), 1.41  0.11 (FCC) and 0.84  0.08 (BCC) (in Rs).
As regards the configurational entropy of melting determined in the present model, it is seen from eqs.(8), (9) that the
numeric value of entropy of melting doesn't depend neither on
energy parameter nor on melting temperature. The magnitudes of
entropy of melting are determined only by coordination number
and by the set of values mn, i.e. they depend only on the type of
crystal lattice. The calculated values of configurational entropy of
melting are about 1R. Thus the developed model adequately follows in qualitative sense the observed features of metal systems
entropy of melting behavior. As for quantitative conformity, the
experimental values of entropy of melting are some higher than
the calculated ones. This is probably the result of nonconfigurational contributions. It should be noted that the observed
entropies of melting for inert gases Ar and Kr having FCC lattice
in solid phase (0.84 and 0.83 R correspondingly) agree well with
the calculated magnitudes. As for semiconducting elements Si and
Ge (experimental value of S is 3.5R) and for -Sn (S is 2R), the
calculated entropies of melting are significantly inconsistent with
the experimental results. One can explain such discrepancy by the
necessity of taking into account electron contributions while calculating the entropy of melting for those elements whose melting
is accompanied by significant reconstruction of electron system
(e.g. semiconductor-metal transition).
It is quite reasonable to apply the developed version of
liquid phase model to eutectic systems being composed by microgroups of the same atoms in liquid phase and being non-soluble
or hardly soluble in each other in solid phase. The developed model was found to be able to describe qualitatively the main features
of such alloys equilibrium characteristics behavior, i.e. positive
values of enthalpy of mixing, positive deviations of activities from
Raul's law, the increase of enthalpy of mixing and the decrease of
activities with the growth of temperature, the decrease of melting
temperature as the result of small additions of the second component, degeneration of eutectic point when component's melting
temperatures differs each other insignificantly. The model demonstrates the arising of thermodynamic characteristics asymmetry
210
due to the difference of alloy-forming components lattices. This
circumstance is illustrated by calculation of eutectic point's position as the function of the type of alloy component's lattice (Table
1) and example of calculation of liqudus line position (Fig.1).
The quantitative results have the following peculiarities:
calculated values of entropy of mixing and of heat of mixing are
systematically underestimated, the values of activities and the
utmost values of activity coefficients at infinite dilution are typically overestimated; though the quantitative agreement with experiment is observed in some cases. One may consider these results to
be satisfactory as far as the model takes into account only configurational contributions into entropy and the positive heat of mixing
provides only indirect configurational repulsion because the positive interaction between different kinds of the atoms in this model
is neglected. As for liquidus line and eutectic point positions, the
obtained results appears often to be quite satisfactory in the quantitative sense. The same resume is related to ternary eutectic systems.
As it was expected, the results of calculations become better quantitative for model, which is taking into account complexes
arbitrary stoichiometry. It is well visible from calculations for
systems Ag-Bi and Na-K (Table 2, Fig.2-4). Such model allows to
calculate satisfactorily all collection of thermodynamic properties,
using only by one varied parameter.
We shall note especially, that apart the proposed model
clearly demonstrates the importance of consideration of selfassociation effects when calculating eutectic systems and alloys
characteristics.
Acknowledgements
The work is under financial support of the Russian Foundation for Basic Research (grant № 01-03-32621)
211
Fig.1. Phase diagram for Ag-Tl system [19], x – calculations.
Table 1
Location of eutectic points of simple eutectics
System
Ag-Bi
Ag-Cu
Ag-Pb
Ag-Si
Ag-Tl
Al-Be
Al-Ga
Al-Si
Al-Sn
Au-Tl
Bi-Cd
Bi-Cu
Bi-Hg
Bi-Sn
Cd-Tl
Cd-Zn
Cs-Na
Ga-Zn
Hg-Pb
In-Zn
Experiment [19]
Te (K)
ce
535.6
0.951
1052.0
0.399
577.0
0.953
1113.0
0.106
564.0
0.960
919.0
0.025
299.8
0.973
850.4
0.123
501.5
0.978
404.0
0.720
417.8
0.550
543.0
0
233.0
0.999
415.0
0.570
476.6
0.710
539.0
0.265
241.3
0.209
298.4
0.037
234
0
416.6
0.038
212
Calculation
Te (K)
ce
523.3
0.964
1001.6
0.405
572.8
0.943
1070.4
0.269
553.0
0.951
861.1
0.115
298.7
0.987
866.9
0.112
471.6
0.926
558.0
0.963
434.4
0.384
528.3
0.026
226.2
0.969
393.9
0.573
454.6
0.530
494.2
0.350
255.6
0.302
291.4
0.035
227.8
0.023
393.6
0.130
Table 2
The calculated and experimental [19] functions of mixing
liquid alloys Ag-Bi at T=1000K.
1 – The results in the model taking into account only selfassociates; 2 - ones in the model taking into account complexes of
arbitrary stoichiometry (G and H – in 4.18-1 kJ/mol)
cBi
H1
H2
Hexp
-G1
-G2
-Gexp
aBi-1
aBi-2
aBi-exp
aAg-1
aAg-2
aAg-exp
0.2
398
276
473
0.3
474
352
699
0.4
514
396
946
1728
2017
2175
3047
4305
0.568
0.330
0.231
0.888
0.834
0.734
3655
4489
0.653
0.437
0.419
0.849
0.761
0.604
3992
5100
0.715
0.528
0.579
0.809
0.687
0.509
0.5
525
410
1222210
2226
4102
49451050
0.765
0.610
0.6790.08
0.766
0.611
0.4480.05
0.6
508
395
1540
0.7
462
350
1711
0.8
380
273
1582
0.9
246
160
1046
1.0
0
0
0
2177
2022
1735
1243
0
3995
4515
0.808
0.687
0.743
0.716
0.529
0.402
3660
3920
0.847
0.76
0.802
0.655
0.437
0.348
3053
3096
0.887
0.834
0.869
0.569
0.330
0.272
2055
1954
0.932
0.911
0.930
0.424
0.195
0.184
0
0
1.0
1.0
1.0
0
0
0
Fig. 2. Liquidus curve for Na-K system [19] (x - theoretical calculations
in the model, taking into account only self-association, o – calculations
taking into account arbitrary stoichiometry).
cNa
GM
(J/mol)
0
0.5
1
0
-500
-1000
-1500
-2000
Fig. 3. Gibbs free energy of mixing of Na-K liquid alloys at T=384
K. Solid line - model calculation taking into account only self213
associates, dash line – calculation with provision for the mixed
associates, rhombic dots - experimental data [19].
aK, aNa
1
0,8
0,6
0,4
0,2
0
0
0,5
cNa
1
Fig. 4. Activities of components of Na-K liquid alloys at T=384 K.
Light circles and dark squares - K and Na experimental activities
respectively [19].
214
References
1. Prigogine I., Defay R. Chemical thermodynamics, Longmans
Green and Co., London, 1954.
2. Wasai K., Mukai K. Consideration of thermodynamic properties of binary liquid alloys with negative deviation of activities from Raoul’t law based on associated solution model,
J.Jap.Inst.Metals, 1982, 46, (3), 266-274.
3. Sommer F. Association model for the description of the thermodynamic functions of liquid alloys, Z.Metallkunde, 1982,
73, (2), 72-86.
4. C.Bergman, R.Castanet, H.Said, M.Gilbert, J.-C.Mathieu.
Configurational entropy and the regular associated model for
compound-forming binary systems in the liquid state, J. LessCommon Metals, 1982, 85, 121-135.
5. Schmid R., Chang Y.A. A thermodynamic study on a associated solution model for liquid alloys, CALPHAD, 1985, 9, (4),
363-382.
6. H.Kehiaian. Thermodynamics of chemically reacting mixtures. XII. Chemical equilibrium in ideal associated mixtures
of in type A+A2+…+Al+B, Bull. Acad. Polon. Sci., Ser. sci.
chim., 1964, 12, (7), 497-501.
7. Morachevsky A.G., Mokrievich A.G., Mayorova E.A. Analysis
of behavior of thermodynamic functions based of ideal associated solution model. Systems A1+B1+AI and B1+Ai+AlBm,
Journal of common chemistry, 1989, 59, (9), 1927-1934 (in
Russ.).
8. Ivanov M., Thermodynamics of self-associated liquid alloys,
Z.Metallkunde, 1991, 82, (1), 53-58.
9. Singh, R.N., Sommer F. A simple model for demixing binary
liquid alloys, Z.Metallkunde, 1992, 83, (7), 533-540.
10. Tkachev N.C., Shunyaev K.Yu., Men A.N., Thermodynamic
behavior of liquid alloys with stable compound of AB-type
and size distribution of complexes // Phys. Chem. Liquids,
1986, 15,(4), 271- 282.
11. .Shunyaev K.Yu., Tkachev, N.C., Men A.N., Thermodynamics
of a ideal associated solution containing complexes of differ215
12.
13.
14.
15.
16.
17.
18.
19.
20.
ent sharp and size, Rasplavy, 1988, v.2, №5, p.11-20, (in
Russ.).
Tkachev N.C., Shunyaev K.Yu., Katznelson A.M., Krylov A.S.,
Men A.N., Kashin V.I., Estimation of configurational contribution in the melting entropy compounds with the fluoritetype structure, Russian Journal of Physical Chemistry, 1989,
63, (5), 1372-1374.
Tkachev N.C., Shunyaev K.Yu., Men A.N., Vatolin N.A., Configurational entropy of melting of body centred cubic and
hexagonal metals, Melts, 1989, 2, (1), 1-8.
Tkachev N.C., Shunyaev K.Yu., Men A.N., Vatolin N.A., Mixing thermodynamics of liquid eutectic systems, Reports of
Academy of Science of USSR, 1988, 302, (1), 153-157.
Tkachev N.C., Shunyaev K.Yu., Men A.N., Theoretical aspects
of the associated liquid model, High Temperatures-High
Pressures, 1990, 22, .207-210.
Shunyaev K.Yu., Vatolin N.A., Mixing thermodynamics and
melting of liquid eutectic systems, Reports of Academy of Science of USSR, 1993, 332, (2), 167-169
. Shunyaev K.Yu., Vatolin N.A., Calculation of thermodynamic characteristics of mixing and liqudus line location of simple eutectic, Melts, 1993, (5), 28-34.
Shunyaev K.Yu., Tkachev N.C., Vatolin N.A., Liquidus surface
and association in eutectic ternary alloys, Thermochimica
Acta,1998, 314, 299-306
Hultgren R, Desai P, Hawkins D, Gleiser M, Kelley K, Selected values of the thermodynamic properties of binary alloys. Metal Park; Ohio: ASM, 1973
.Gschneidner K.A., Entropies of transformations and fusion
of metallic elements, J. Less-Common Metals, 1975, 43, (1-2),
179-189.
216