Estimation of Activity Interaction Parameters in Fe-S-j Systems

It is important to know the activity interaction parameters between components in melts in the process of metallurgy. However, it’s considerably difficult to measure them experimentally, relying still to a large extent on theoretical calculations. In this paper, the first-order activity interaction parameter (esj) of j on sulphur in Fe-based melts at 1873 K is investigated by a calculation model established by combining the Miedema model and Toop-Hillert geometric model as well as considering excess entropy and mixing enthalpy. We consider two strategies, with or without using excess entropy in the calculations. Our results show that: (1) the predicted values are in good agreement with those recommended by Japan Society for Promotion of Science (JSPS); and (2) the agreement is even better when excess entropy is considered in the calculations. In addition, the deviations of our theoretical results from experimental values |eS(exp)j−eS(cal)j| depend on the element j’s locations in the periodic table.


Introduction
Sulphur is one of the most detrimental impurity elements in metallurgy that typically causes the deterioration of hot ductility [1] and the degradation of the corrosion resistance [2] of steels. The content of sulphur in steels is normally required to be extremely low. "Inclusion engineering" [3] could be one of the ways to reduce the harmful effects of sulphur [4] with a relatively low cost. However, implementation of this technique needs to well understand the basic thermodynamics behavior of sulphur in iron-based melts.
The activity interaction parameter, which is first introduced by Wagner [5] in dilute solution to account for the effects of an added alloying element on the activity coefficient of a solute, provides more useful information in the process of metallurgy computation. Previously, only first-order activity interaction parameters had been considered in Wagner's formalism, resulting in inadequacy to describe the behavior of solutions that are "not very diluted". This phenomenon was observed by Lupis and Elliott [6], who then proposed an introduction of higher order interaction coefficients to the mathematical apparatus. Darken [7] also observed that the Wagner's formalism was not The models combined with the Toop/Toop-Kholer geometric model, such as Ding's model [12] and Wang's model [16], have mathematical difficulties in the deduction process when the solvent is chosen as an asymmetric component and one has to resort to other geometric models. For this reason, in our present work, we adopted the Toop-Hillert geometric model [22] in our model establishment.

Basic Relations
In a ternary system, i-j-k, k is a solvent, the activity interaction parameter ε j i can be expressed as: and: where R and T are the gas constant and absolute temperature, respectively; is the activity interaction parameter of j on i that the composition coordinate is in a molar fraction; g E and S E are the excess Gibbs free energy and excess entropy, respectively; ∆H is the mixing enthalpy of solution. Generally, the thermodynamics properties of a multi-component system are obtained from all the sub-binary systems with an assigned probability weights, which is called geometric model method, as follows: Therefore, when the excess Gibbs free energy of the binaries is available, the excess Gibbs free energy of the i-j-k system, g E , can be obtained, and then the activity interaction parameter ε j i can be calculated.
In liquid binary alloys, a satisfactory equation relating S E and ∆H has been deduced by Tanaka et al. [23], based on the free volume theory and excess volumes of the alloys, as follows (supposing and i-j binary alloy): where T mi and T mj are the melting points of pure elements A and B under the standard pressure respectively. Therefore, if: then:

Miedema Model
In a binary system i-j, the mixing enthalpy ∆H y can be obtained from the Miedema model [24][25][26], which was proposed by Miedema and his colleagues for estimating the heat of formation of solid or liquid metal alloys. For simplicity, the equation is deduced as follows: where: where, x is the atom fraction; V, n ws , and φ are the basic physical parameters of elements, representing mole volume, electron density, and electronegativity, respectively; p, q, and β ij are the empirical parameters defined by Miedema, and their data are correlate to the constituents. The values of molar volume, electron density, and electronegativity of elements, except for O, S, Se, Te, as well as values of all the constants, are obtainable in reference [26].

Hillert-Toop Geometric Model
The Toop-Hillert geometric model [22] is used in this work to represent the excess Gibbs free energy of a ternary system, i-j-k, from the three sub-binaries i-j, i-k, j-k. The Toop-Hillert geometric model is an asymmetric model. Hence, the exact expression depends on the selected asymmetric component. If the component i is an asymmetric component, the excess Gibbs free energy g E can be expressed as:

Calculation Model
Inserting Equations (5) and (7) into Equation (6), then expanding the formalism as Equation (8) to calculate the excess Gibbs free energy in a ternary system, finally, according to the Equation (1), the formalism for activity interaction parameter calculation is obtained:

1.
If the asymmetric component is solute, the activity interaction parameter can be calculated by: 2.
If the solvent is an asymmetric component, the activity interaction parameter is: where where the α ij is identical to Equation (5).
If the mass fraction (wt.%) is used, the activity interaction parameter is often denoted as e j i , and it can be obtained by applying the below transformation from ε j i : where M j and M k are the molecular weight of solute j and solvent k, respectively. In this paper, e j i is used.

Results and Discussion
The Miedema parameters [19] of O, S, Se, Te, and Po required in the calculation of e j i are listed in Table 1. The rule for selecting the asymmetric component is according to the criterion described in [27]. The temperature for the calculations of e j i is 1873 K, and calculations are performed with (S E = 0, case 1) or without considering (S E = 0, case 2) the excess entropy. When S E = 0, the α ij is equal to 1, else it is identical to Equation (5). The calculated results and experimental values recommended by JSPS [20] are listed in Table 2. Plotting calculated values (for S E = 0 and S E = 0) and experimental values according to incremental order of elements in each period of the periodic table, one can see the same trends among them (Figures 1-5). Our results, however, differ from those by Silva [10], who showed the experimental e i O (ε j O ) increases linearly with increasing atomic number. In addition, it's obvious (Figures 1-5) that a better agreement is achieved when S E = 0 (case 1) between theoretical values and experimental ones. Thus, theoretically, considering the excess entropy is more favorable.            To establish our calculation method, we noticed that the precision of the calculated results heavily relies on the Miedema model and the geometric model. The Miedema model is one of the most successful models to predict the formation enthalpy of alloys. It may be owing to that more physical quantities such as electronegativity, electron density, and molar volume than other models like the Pauling electronegativity model (only electronegativity considered) have been considered [24]. However, it's not sufficient when the constituents' physical properties are of large differences, some minor contribution terms which are usually neglected now can't be ignored. In the Fe-S-j system, is not only dependent on the physical property differences between sulphur and elements j, but also on the differences between elements j and iron as well as between iron and sulphur. Consequence, the elements j with large deviations between calculated and experimental values ( ) − ( ) are mainly located in the periodic table far from the Fe group, especially in the left side ( Figure 6), for example, the elements Ca, Ce, La, Y, Zr, etc. Moreover, the elements H, B, C, N, P, etc. are dealt with in Miedema's model in a complicated way, and in current model, this may be also need some appropriate corrections to be made. Aiming at the deviation of the Miedema's model, it can be improved by adding some terms such as a volume correction term [30] and an improved atomic size term [31], as well as by modifying the Miedema parameters of a specified element [32].
In addition, the energy of triplet interactions is neglected in our calculation model due to the contribution of this term to the excess Gibbs free energy g E of ternary elements is usually very small [33]. To see the influence of the geometric model, the results from the Ding's model [12], which also includes a geometric model, are listed in Table 2 for comparison. One can see large deviations from experiments in both the present and Ding's models. This means the deviations come mainly from the Miedema model basis instead of the geometric model. We are attempting to modify the Miedema model by adding some terms such as a volume correction term [30] and/or improving the atomic size factor [31] to optimize the calculated values. This work is now ongoing. To establish our calculation method, we noticed that the precision of the calculated results heavily relies on the Miedema model and the geometric model. The Miedema model is one of the most successful models to predict the formation enthalpy of alloys. It may be owing to that more physical quantities such as electronegativity, electron density, and molar volume than other models like the Pauling electronegativity model (only electronegativity considered) have been considered [24]. However, it's not sufficient when the constituents' physical properties are of large differences, some minor contribution terms which are usually neglected now can't be ignored. In the Fe-S-j system, e j S is not only dependent on the physical property differences between sulphur and elements j, but also on the differences between elements j and iron as well as between iron and sulphur. Consequence, the elements j with large deviations between calculated and experimental values e j S(exp) − e j S(cal) are mainly located in the periodic table far from the Fe group, especially in the left side ( Figure 6), for example, the elements Ca, Ce, La, Y, Zr, etc. Moreover, the elements H, B, C, N, P, etc. are dealt with in Miedema's model in a complicated way, and in current model, this may be also need some appropriate corrections to be made. Aiming at the deviation of the Miedema's model, it can be improved by adding some terms such as a volume correction term [30] and an improved atomic size term [31], as well as by modifying the Miedema parameters of a specified element [32].
In addition, the energy of triplet interactions is neglected in our calculation model due to the contribution of this term to the excess Gibbs free energy g E of ternary elements is usually very small [33]. To see the influence of the geometric model, the results from the Ding's model [12], which also includes a geometric model, are listed in Table 2 for comparison. One can see large deviations from experiments in both the present and Ding's models. This means the deviations come mainly from the Miedema model basis instead of the geometric model. We are attempting to modify the Miedema model by adding some terms such as a volume correction term [30] and/or improving the atomic size factor [31] to optimize the calculated values. This work is now ongoing.

Conclusions
Because the activity interaction parameter (1) A model for calculating the activity interaction parameter in a ternary system was established based on the Ding's method, wherein the Toop-Hillert model was used.
(2) The calculated results for j S e in Fe-based melts (Fe-S-j) by current model at 1873 K, with or without considering the excess entropy, show that better results would be obtained with considering the excess entropy. And better results would be obtained for the elements j located in the middle of periodic table nearby the Fe group.

Conclusions
Because the activity interaction parameter e j S is very important to understand the thermodynamic properties of Sulphur-contained iron-based melts, a great deal of work has been done on the experimental measurements. However, important data such as e Ru S , e Re S , e Os S , etc. are still lacking. Considering the complexity of measurements and the experimental data depend strongly on the experimental techniques, in this work we employed a theoretical method and systematically calculated the activity interaction parameter e j S in the Fe-S-j systems. Based on our study, we conclude: (1) A model for calculating the activity interaction parameter in a ternary system was established based on the Ding's method, wherein the Toop-Hillert model was used.
(2) The calculated results for e j S in Fe-based melts (Fe-S-j) by current model at 1873 K, with or without considering the excess entropy, show that better results would be obtained with considering the excess entropy. And better results would be obtained for the elements j located in the middle of periodic table nearby the Fe group.
(3) The reason for the large deviations between calculated and experimental values is because of the inaccuracy of Miedema's model when the constituents' physical properties are of large differences.