CALCULATIONS OF THERMODYNAMIC PROPERTIES AND
LIQUIDUS LINES OF BINARY AND TERNARY MELTS TAKING
INTO ACCOUNT OF COMPLEXES OF DIFFERENT SHAPE AND
SIZE
K. Yu. Shunyaev
Institute of Metallurgy, Ural’s Division of Russian Academy of Sciences,
101, Amundsen Str., 620016, Ekaterinburg, Russia
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 the 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 the
thermodynamic functions of mixing behavior 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 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
737
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 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. Let us suppose that the melting takes place when the number of weak bonds
becomes critically high and the system can not 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 complex, i is the number of the bonds between the nearest neighbors
which 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 An,i to be the number of complexes with the size n and
with i bonds with the nearest neighbors and NC is the total number of complexes

NC  
mn

n 1 i  n 1
NA 

N A n,i
(1)
mn
 
n 1 i  n 1
nN A n,i
Let the internal energy of liquid phase counted from noninteracting system of
separate atoms to be

EA   A 
mn

n 1 i  n 1
iN A n,i
(2)
where A is the energy of the bond for the pair of the nearest neighbors AA taken with
the opposite sign. The configurational entropy of the system is written using the point
approximation
738
S A  k ln
NC!
 N An,i !
(3)
n, i
Therefore the free energy of the system is equal to

FA  E A  TS A    A 
mn

n 1 i  n 1
iN A n,i  kT ln
NC!

 N An,i !
n, i

iN A n,i  kT  N C ln N C 
n 1 i  n 1


  A 
mn


N A n,i ln N A n,i 
n 1 i  n 1


(4)
mn
 
Minimization of (4) by N An,i with account of expressions (1) gives the
correlation between molar fractions of the complexes with different sizes and shapes in
the form of acting masses law
  i
x A n,i  exp - A  x nA0  t iA x nA0
 kT 
(5)
where x An,i  N An,i N C is the molar fraction of the complexes with the size n and
with i bonds with the nearest neighbors, x A0  x Ai,0 is the molar fraction of separate
atoms, 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

x A n,i  
mn
 x nA0 t 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 for the 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
739
temperature of melting. One may write the following equation for xAT by inserting A
from eq.(8) into balance eq.(6)

 x nAT
n 1
mn
 x -AT2i z A   1
(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 pair bond energy A using eq.(7).
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 the configurational 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 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

2i  n - 2i z A 
 n x
z A  AT
n 1 i  n 1 
mn
 

mn
n - 2i z A 
(9)
  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
Let us 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
  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 us introduce some symbols for the following characteristic sums
740

S1L  
mn

n 1 i  n 1

S 2L  
mn

mn
ix Ln,i  
n 1 i  n 1
 x nL1t iA

nx Ln,i  

mn
n 1 i  n 1

n 1 i  n 1
S3L  

x Ln,i  
mn
 nx nL1t iL
(11)
n 1 i  n 1

mn
 ix nL1t iL
n 1 i  n 1
where L = A,B indicates what a sum belongs to one or another component of solution.
The system of balance equations is
S1A  S1B  1
(12)
(1 - c)S2A - cS2B  0.
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 the alloy 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 melts at the considered
temperature.
One can express the enthalpy of mixing as follows
H  -
 A S3A   BS3B
S2A  S2B
 c A
S3A0
S2A0
 (1  c) B
S3B0
S2B0
(14)
The symbol 0 shows that corresponding sums are calculated here for the pure
components at the considered temperature. The entropy of mixing may be calculated as
usually.
Let us assume now, that solid phase components are absolutely nonsoluble in
each other. Then, by setting the chemical potentials to be equal to each other and
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.
741
TERNARY SYSTEMS
The proposed model can be simply applied to multicomponent alloys, which
have a phase diagram of eutectic type. The structure of material balance solutions and
formulas for thermodynamics properties are the same, as for the binary system. As an
example the systems of Ag-Bi-Cu and Bi-Cd-Sn were examined in (18).
The calculations for the location of the liquidus line of these eutectic systems
were performed. For example, the location of a eutectic point of Ag-Bi-Cu is equal to
cAg = 0.036,
cBi = 0.939,
cCu = 0.025 (calculation)
cAg = 0.05,
cBi = 0.945,
cCu = 0.005 (experiment)
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 of the order R. The same suggestion is true for
the inert gases. The following entropies of melting are recommended 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 to the configurational entropy of melting being determined in the
present model, it is seen from eqs.(8), (9) that the numeric value of entropy of melting
does not depend neither on the energy parameter nor the on melting temperature. The
magnitudes of entropy of melting are determined only by the coordination number and the
set of values mn, i.e. these values are depended only on the type of crystal lattice. The
calculated values of configurational entropy of melting are about R. 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. Probably it is 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 b-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 the
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 of microgroups of the same atoms in liquid phase and
being non-soluble or hardly soluble in each other in the 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 law, the increase of enthalpy of mixing and the decrease of activities
742
with the increase of temperature, the decrease of melting temperature as the result of
small additions of the second component, degeneration of eutectic point when component
melting temperatures differ each other insignificantly. The model demonstrates the arising
of thermodynamic characteristics asymmetry due to the difference of alloy component
formation lattices. This circumstance is illustrated by calculation of eutectic point
position as the function of the type of alloy component lattice (Table 1), calculation of
functions of mixing (Table 2) 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 the configurational contributions into the 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 far as the 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.
ACKNOWLEDGEMENTS
The work is under financial support of the Russian Fund of Fundamental
Research (grant № 99-03-32707)
Fig.1. Phase diagram for Ag-Tl system (19),
x – calculations.
743
TABLE 1. LOCATION OF EUTECTIC POINTS OF SIMPLE EUTECTICS.
Experiment (19)
Calculation
System
Te (K)
ce
Te (K)
ce
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
535.6
1052.0
577.0
1113.0
564.0
919.0
299.8
850.4
501.5
404.0
417.8
543.0
233.0
415.0
476.6
539.0
241.3
298.4
234
416.6
0.951
0.399
0.953
0.106
0.960
0.025
0.973
0.123
0.978
0.720
0.550
0
0.999
0.570
0.710
0.265
0.209
0.037
0
0.038
523.3
1001.6
572.8
1070.4
553.0
861.1
298.7
866.9
471.6
558.0
434.4
528.3
226.2
393.9
454.6
494.2
255.6
291.4
227.8
393.6
0.964
0.405
0.943
0.269
0.951
0.115
0.987
0.112
0.926
0.963
0.384
0.026
0.969
0.573
0.530
0.350
0.302
0.035
0.023
0.130
744
TABLE 2. THE CALCULATED AND EXPERIMENTAL (19) FUNCTIONS OF
MIXING. (G and H – in 4.18-1 kJ/mol)
A
P
c - concentration
Alloy Proper 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
ty
Al1-cGac Hcal
Hexp
45
71
70
119
85
148
94
160
97
15750
94
143
85
120
70
89
45
49
Al1-cSnc -Gcal
-Gexp
314
320
441
452
513
543
550
611
560
657100
544
678
502
661
427
590
303
430
Bi1-cHgc Hcal
Hexp
48
37
73
62
87
81
94
99
95
11850
90
131
80
133
64
122
39
89
Sn1-cHgc -Gcal
-Gexp
225
343
243
355
250
345100
246
317
229
274
197
216
141
139
aHg-cal
aHg-exp
0.64
0.56
0.70
0.67
0.75
0.760.08
0.80
0.82
0.84
0.87
0.88
0.91
0.93
0.95
Zn1-cInc -Gcal
-Gexp
211
181
297
260
346
320
373
363
382
39250
374
409
347
406
297
370
212
275
aZn-cal
aZn-exp
0.93
0.94
0.89
0.92
0.85
0.89
0.80
0.85
0.76
0.820.03
0.71
0.77
0.65
0.69
0.56
0.55
0.41
0.33
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