Thermodynamic and structural properties of Bi-based liquid alloys
Introduction
It is well known that last twenty to twenty-five years a lot of world projects were dedicated to investigate lead free solders. Arising from the transition to Pb-free and the EU Reduction of Hazardous substances (RoHs) Directive, safety and environmental compliances issues exercise a dominant role and it was addressed aiming to develop new lead free alloys having equal characteristics or better than Sn–Pb solders widely used in the electronics industry. In Europe two COST projects (COST 531 and COST MP0602) were dedicated to study lead free alloys and for that purpose, a lot of work have been done. The alloys of bismuth (Bi) have low melting points. This feature makes the alloys of bismuth suitable for uses as solders and toxic solder containing lead can be replaced by safer bismuth [1], [2], [3], [4], [5]. Additionally, Pb is being replaced from its alloys by Bi in numerous applications, such as pigments for paints, fishing sinkers, bullets and shots, brass for plumbing and as ingredients in grease for lubrications for safety purposes. The Indium Bismuth (In–Bi) alloy is a crystalline solid which is used as a semiconductor and in photo-optic applications. It also possesses the superconducting properties [6]. These various uses have drawn the consent of most modern day researches to study and predict the thermodynamic and structural behaviours of the Bi-based liquid alloys. Several theoretical models so far have been proposed to understand the complexities in thermodynamic and structural properties of the liquid alloys [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18]. In this paper, the regular associated solution model [19], [20], [21], [22], [23], [24] has been used to study and predict the thermodynamic and structural properties of two Bi-based liquid alloys, In–Bi at 900 K and Tl–Bi at 759 K.
According to the regular associated solution model, when atoms of types A and B are mixed in liquid state there is probability of associations among A, B; A, A and B, B. The regular solution can thus be regarded as the ternary mixtures of the complex (A–B) and the free monomers (A and B) and the three species are in equilibrium with each other. These associations are named as, 'complexes', 'pseudomolecules', 'clusters' or 'associations' [12]. It is also assumed that there exist unequal interactions among the associated and unassociated atoms. The expressions for different thermodynamic and structural functions are derived on this basis. Among the microscopic structural functions, the concentration fluctuation in long wavelength limit () has evolved as an important tool to predict the structure of the liquid alloys. Bhatia et al. [18] has shown that the concentration fluctuation in the long wavelength limit can be obtained from the model parameters even if the data from the low angle X-ray and neutron diffraction experiments are not available. The expressions for the and the thermodynamic functions of compound forming liquid binary alloys were derived by Bhatia and Hargrove [18]. However, the procedure of estimating the complex concentration and interaction energy parameters involves selection of initial values for the unknown parameters and spontaneous iteration to get the best fit values to explain the experimental data which is quite tedious. This procedure was significantly simplified by Lele and Ramchandrarao [21] which involves the estimation of the pairwise interaction energy parameters and the equilibrium constant from the activity coefficients of the components at infinite dilution and the activity data at one another intermediate compositions. The dependency of concentration fluctuation in long wave length limit () with the activity [14] and their relationship with the Gibbs free energy of mixing () can be understood in terms of the presence of the complexes. The phase diagram of In–Bi alloys [25] indicates the existence of different phases, such as "" (with tetragonal structure isotypic with ; , "" (with the hexagonal structure isotypic with ) and , "" (with the complex tetragonal structure isotypic with ). Whereas, the phase diagram of Tl–Bi alloys [25] indicates the existence of phases, such as '''' with a hexagonal structure and '''' with the fcc (Al) structure isotypic with Cu. Among these phases, this model well explains the thermodynamic and microscopic structural behaviours of the alloys considering the existence of complex in In–Bi and complex in Tl–Bi in the initial melts. We have thus estimated the stated behaviours assuming the existence of complexes in the In–Bi system and in the Tl–Bi system in the initial melts.
The expressions for the various thermodynamic and structural functions are presented in Section 2, the results and discussion is presented in Section 3 and the conclusion of this work is outlined in Section 4.
Section snippets
Theory
Let one mole of binary liquid solution contain atoms of monomer A (=In, Tl) and atoms of monomer B (=Bi) of the form A–B. According to Lele and Ramchandrarao [21], it is assumed that the chemical complexes () exists in the melt, where is a small integer whose value is determined from the compound forming concentration (=/(+1)) in the solid state. The liquid mixture thus can be considered to be the ternary mixtures of free atoms A and B and the complex . Also let ,
Thermodynamic properties
The model parameters for the In–Bi liquid alloy are determined from Eqs. (15a) and (15b) by the use of the experimental values of activity coefficients at infinite dilutions [25]. The true mole fractions of the complex for this system are obtained from Eq. (16) and the experimental data of activities by iterative procedure. Whereas, the model parameters and the true mole fractions of the complex for Tl–Bi liquid alloy are obtained by simultaneously solving the Eqs. (18) and (19) with the
Conclusions
The theoretical analysis of the thermodynamic and structural properties of Bi–In and Bi–Tl alloy melts, performed by the regular associated solution model, for temperatures of 900K and 750 K respectively suggests that there is comparatively greater tendency of complex formation in liquid Tl–Bi alloys. It further reveals that both of the liquid systems are found to be complete ordering in natures. Moreover, all the interaction energy parameters are found to be temperature dependent and both the
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