ASSOSIATION AND EQUILIBRIUM DIAGRAMS
OF BINARY ALLOYS
K.Yu. Shunyaev
Institute of metallurgy, Ural’s Division of Russian Academy of Science,
101Amundsen Str., Ekaterinburg 620016, Russia, shun@ural.ru
The model of ideal associated solution taking into account an
existence of complexes of a various composition, size and shape was
applied for the analysis of possible types of the equilibrium diagrams. It
was supposed that the solid phase solution is regular and the energies of
pair interaction do not vary as a result of a melting. It was shown that
depending on the sign and value of energy of interchange the occurrence
of different types of fusibility curves - eutectic, “cigar”- type, with the
upper or lower azeotropic points - is possible.
The model of ideal associated solution is widely used in modern
practice for calculations of the thermodynamic properties of liquid multicomponent systems [1-3]. The version of model of ideal associated
solution taking into consideration existence of associates of a various
composition, size and shape is developing by the authors within many
years [4-7]. It was shown that one can evaluate parameters of the model
from melting points of individual substances considering an associate of
the infinite size as a crystal and restricting oneself to a pair approximation of the nearest neighbours when calculating an energy of an associate. Besides, such approach allows calculating not only thermodynamic
characteristics of mixing, but also fusibility curves of solutions. It turned
out that one can get a quite realistic picture of hierarchy of the phase
diagrams if he considers the systems having mutual solubility only in
liquid phase but not in solid one. This is true for eutectic systems and
systems having stable compound in solid phase. The determining role of
associates of the same atoms (self-associates) in formation of properties
of simple eutectics was established. The further development of the
model has allowed associates of an arbitrary stoichiometry to be included into consideration [8]. In this case there appears the only one fitting
parameter like interchange energy of regular solution [9]. The introduc236
tion of associates of an arbitrary stoichiometry into consideration allows
not only to describe miscellaneous behaviour of concentration dependencies of the thermodynamic characteristics of mixing of liquid alloys,
but also to expand a circle of systems, to which one the model is applicable. First of all it refers to the alloys having unlimited solubility in solid and liquid states close to melting.
Let's consider a binary mixture AcB1-c, whose components in
liquid and solid phases will form solution with complete mutual solubility. Let solution in liquid phase to be a regular solution of associates
An(i), Bn(j) and AnBm(i,j,q). (n, m – numbers of the appropriate atoms in
a complex, i, j and q – number of pairs of the nearest neighbours of AA,
BB and AB type in a complex respectively). If one assume that the energy of a complex is determined by the sum of energies of pairs of the
nearest neighbours and restrict oneself only to configuration contributions to an entropy, then the molar fractions of complexes will be connected one another by following ratio [4, 5, 8]:
 i 
x An ,i  K An ,i x nA  exp A  x nA  t iA x nA
1
1
 kT  1
 j 
xBn , j  K Bn , j xBn  exp B  xBn  t Bj xBn
1
1
 kT  1
(1)
x An Bm ( i , j , q )  K An Bm x nA xBm 
1
1
  i   B j   ABq  n m i j q n m
 exp A
 x A xB  t A t B t AB x A xB
1
1
kT

 1 1
where x An ,i ; x Bn , j ; x An Bm ; x A1 ; x B1 - molar fractions of An,i, Bn,j,
AnBm complexes and single atoms A1 and B1 respectively,
A, B и AB – bond energies of pairs of the nearest
neighbours AA, BB and the AB taken with a reversed
sign,
K An ,i ; K Bn ,i ; K An Bm - constants of the appropriate equilibrium.
237
The set of equations of the material balance for determination of
single atoms concentrations in solution is:
  x A n ,i    x B n , j    x A n B m  1
n i
n j
cA 
n ,mi , j ,q
  nx A n ,i    nx A n B m
n i
n ,mi , j ,q
(2)
  x A n , i    x B n , j    ( n  m )x A n B m
n i
n j
n ,mi , j ,q
where cA – molar fraction of a component A.
The solution of the set of equations (2) taking into account the
equation (1) allows to find molar fractions of single atoms in solution,
through which ones it is possible to express the thermodynamic characteristics of a system [10].
For making calculations it is necessary to know energy parameters A, B and AB, and also a number of pairs of the nearest neighbours
of a different type in associates. The energy parameters A and B can
be evaluated from melting points of pure components [4, 5]. Then AB
will be the only one varied parameter. It is possible to count up a total
number of pairs in an associate under the assumption that the local structure of the conforming crystal is saved in a liquid phase [4, 5]. However
it is not sufficient for summation in (2), as it is still necessary to know,
how these pairs are divided into AA, BB and Ab types. This difficulty
can be overcome assuming that the associate belongs to a kind of a linear chain. Then the summation in (2) may be performed analytically.
Let's mark at once that such assumption does not reflect the authors’ position concerning possible structure of associates and it is merely approach to simplify calculations. Such simplification allows easily performing summation in (2). Thus, the loss of accuracy for the calculated
properties is no more than 10 %, as it was demonstrated, for example, by
the analysis conducted in [11]. The behaviour of thermodynamic characteristics of mixing for binary liquid solutions within the framework of
the proposed model has been analysed in [8]. This work is dedicated to
the study of capabilities of calculation of the phase diagrams.
There was shown in [5] that equilibrium condition for melting
points of components
 A z
 B z
 RT A ln x A1 0 ;
 RT B ln x B1 0
2
2
238
(3)
allow to calculate energy parameters A and B (TA and TB - melting
points of A and B components, x A and x B
- fractions of single
10
10
atoms in melts of these components).
Let's restrict ourselves to consideration of a case of regular solid
solution and suppose that the interchange energy does not change under
melting. In this case it appears to be the only parameter of model. Let's
introduce the value W:
W = 2AB - A - B
(4)
Chemical potentials of mixing of regular solution components
will be [9]:
zW
 A   A0  RT ln c A  ( 1  c A ) 2
;
2
,
(5)
zW
 B   B0  RT ln( 1  c A )  c A 2
2
where z – coordination number.
The requirement of equality of chemical potentials of components in solid and liquid phases results in occurrence of a following set
of equations:
 Az
2 zW
RT ln x A1 ( c A )  
 RT ln C A  ( 1  C A )
;
2
2
 z
zW
RT ln x B1 ( c A )   B  RT ln( 1  C A )  C A 2
2
2
,
(6)
where CA – molar fraction of component A on a solidus line.
Solution of this set one together with balance equations (2) allows to calculate a position of liquidus and solidus lines at given parameter W.
The possible kinds of the equilibrium diagrams depending on
value and sign of interchange energy are shown in figure 1. It was supposed that the components have melting points 700 and 1000 K and
form solution with f.c.c lattice in a solid state. It is seen that even the use
of only one parameter of model allows getting fusibility curves of various kinds: eutectic, “cigar”-type, with upper and with lower azeotropic
point. Moreover, all four mentioned kinds of the diagrams can take place
in a case, when the sign of interchange energy provides positive heat of
239
T (K) 2500
T (K) 1200
2000
1000
1500
800
1000
600
500
400
200
0
0
0,2
0,4
0,6
0,8
1
cA
0
0
0,2
a
0,4
cA
0,6
0,8
1
b
T (K) 1200
T (K) 1200
1000
1000
800
800
600
600
400
400
200
200
0
0
0
0,2
0,4
cA
0,6
0,8
1
0
0,2
0,4
cA
0,6
0,8
1
0,6
0,8
1
d
c
T (K) 1200
T (K) 1200
1000
1000
800
800
600
600
400
400
200
200
0
0
0
0,2
0,4
cA
0,6
0,8
1
e
0
0,2
0,4
cA
f
Fig. 1. Change of the equilibrium diagram of regular solution depending on
value of interchange energy.
W = 3000 J/mol (а); W = 0 (b); W = -800 J/mol (c); W = -1000 J/mol (d);
W = -1500 J/mol (e); W = -1700 J/mol (f).
240
mixing both in solid and in liquid phases. In a case of negative deviations of a heat of mixing from ideality the only diagrams with the upper
azeotropic point will be realised. Such situation rather often happens in
practice. The eutectic systems practically always have positive deviations of the thermodynamic characteristics of mixing from ideality [12].
The systems having the lower azeotropic point (for example, K-Rb, CsK) as well as “cigar”-type diagram (Bi-Sb) can also demonstrate positive
deviations from ideality [12]. Of course, in practice the connection between the kind of the equilibrium diagram and the nature of concentration dependencies of thermodynamic properties of mixing is much more
diversified. Let's remark, that the analysis of fusibility curves of solutions being regular both in solid and in liquid phases with different interchange energies done by D.S.Kamenetzkaya (see report on result in
[13]), demonstrates a much poorer picture. In this case there can be only
diagrams of a “cigar”-type or diagrams with the upper azeotropic point
when using four variable parameters.
For demonstrating capabilities of the model the calculated phase
diagram of K-Rb system is shown in a fig. 2. It has appeared rather similar to experimental one [12] in the region of a melting. The calculated
values of an azeotropic point are caz=0.62 и Taz=310
K (experimental values are caz=0.7 и Taz=307 K). At low temperatures the division into two solid solutions is observed as obviously
should take place in a system with positive heat of mixing. Parameter of
the model W=-140 cal/mol was estimated from experimental data on an
enthalpy of mixing (table 1).
Table 1
Comparison of calculated and experimental values of an enthalpy of mixing of
liquid binary alloy K-Rb at T = 384 K
СRb
Hcalc
(cal/mol)
Hexp
(cal/mol)
[12]
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0
12.1
20.3
25.8
29
30
29
25.8
20.3
12.1
0
0
11
20
26
29
305
28
25
20
12
0
241
400
T, K
350
300
250
200
150
100
50
0
0
0,2
0,4
cRb
0,6
0,8
1
Fig. 2. The calculated phase diagram of K-Rb system.
Acknowledgements
The author is thankful to the Russian Foundation for Basic Research (grant № 01-03-32621) and Foundation “Leading Scientific
Schools” (НШ-2022.2003.3) for financial support.
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