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,
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 self-association. Though the theoretical analysis of the influence of self-association on 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'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 complex, 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, to be the number of complexes with the size n and with i bonds with the nearest neighbors and NC to be the total number of complexes
(1)
Let internal energy of liquid phase counted from noninteracting system of separate atoms to be
, (2)
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
(3)
Therefore the free energy of the system is
(4)
Minimization of (4) by taking into account expressions (1) gives the correlation between molar fractions of the complexes with different sizes and shapes in the form of acting masses law
, (5)
where is the molar fraction of the complexes with the size n and with i bonds with the nearest neighbors, 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
(6)
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)
(8)
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
(9)
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
, (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
, (11)
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
(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
(14)
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
(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,cBi = 0.939,cCu = 0.025 (calculation)
cAg = 0.05,cBi = 0.945,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 and AB in the complex, accordingly). Now set of equations (10) will look like
where- 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:
(17)
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 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 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 non-configurational 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 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 self-association 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)
Fig.1. Phase diagram for Ag-Tl system [19], x – calculations.
Table 1Location of eutectic points of simple eutectics
Experiment [19] / Calculation
System / Te (K) / ce / Te (K) / ce
Ag-Bi / 535.6 / 0.951 / 523.3 / 0.964
Ag-Cu / 1052.0 / 0.399 / 1001.6 / 0.405
Ag-Pb / 577.0 / 0.953 / 572.8 / 0.943
Ag-Si / 1113.0 / 0.106 / 1070.4 / 0.269
Ag-Tl / 564.0 / 0.960 / 553.0 / 0.951
Al-Be / 919.0 / 0.025 / 861.1 / 0.115
Al-Ga / 299.8 / 0.973 / 298.7 / 0.987
Al-Si / 850.4 / 0.123 / 866.9 / 0.112
Al-Sn / 501.5 / 0.978 / 471.6 / 0.926
Au-Tl / 404.0 / 0.720 / 558.0 / 0.963
Bi-Cd / 417.8 / 0.550 / 434.4 / 0.384
Bi-Cu / 543.0 / 0 / 528.3 / 0.026
Bi-Hg / 233.0 / 0.999 / 226.2 / 0.969
Bi-Sn / 415.0 / 0.570 / 393.9 / 0.573
Cd-Tl / 476.6 / 0.710 / 454.6 / 0.530
Cd-Zn / 539.0 / 0.265 / 494.2 / 0.350
Cs-Na / 241.3 / 0.209 / 255.6 / 0.302
Ga-Zn / 298.4 / 0.037 / 291.4 / 0.035
Hg-Pb / 234 / 0 / 227.8 / 0.023
In-Zn / 416.6 / 0.038 / 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 self-associates; 2 - ones in the model taking into account complexes of arbitrary stoichiometry (G and H – in 4.18-1 kJ/mol)
cBi / 0.2 / 0.3 / 0.4 / 0.5 / 0.6 / 0.7 / 0.8 / 0.9 / 1.0H1 / 398 / 474 / 514 / 525 / 508 / 462 / 380 / 246 / 0
H2 / 276 / 352 / 396 / 410 / 395 / 350 / 273 / 160 / 0
Hexp / 473 / 699 / 946 / 1222210 / 1540 / 1711 / 1582 / 1046 / 0
-G1 / 1728 / 2017 / 2175 / 2226 / 2177 / 2022 / 1735 / 1243 / 0
-G2 / 3047 / 3655 / 3992 / 4102 / 3995 / 3660 / 3053 / 2055 / 0
-Gexp / 4305 / 4489 / 5100 / 49451050 / 4515 / 3920 / 3096 / 1954 / 0
aBi-1 / 0.568 / 0.653 / 0.715 / 0.765 / 0.808 / 0.847 / 0.887 / 0.932 / 1.0
aBi-2 / 0.330 / 0.437 / 0.528 / 0.610 / 0.687 / 0.76 / 0.834 / 0.911 / 1.0
aBi-exp / 0.231 / 0.419 / 0.579 / 0.6790.08 / 0.743 / 0.802 / 0.869 / 0.930 / 1.0
aAg-1 / 0.888 / 0.849 / 0.809 / 0.766 / 0.716 / 0.655 / 0.569 / 0.424 / 0
aAg-2 / 0.834 / 0.761 / 0.687 / 0.611 / 0.529 / 0.437 / 0.330 / 0.195 / 0
aAg-exp / 0.734 / 0.604 / 0.509 / 0.4480.05 / 0.402 / 0.348 / 0.272 / 0.184 / 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).
Fig. 3. Gibbs free energy of mixing of Na-K liquid alloys at T=384 K. Solid line - model calculation taking into account only self-associates, dash line – calculation with provision for the mixed associates, rhombic dots - experimental data [19].
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].
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