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Page 12 of 25 Park et al. J Mater Inf 2023;3:5 https://dx.doi.org/10.20517/jmi.2022.37

THERMODYNAMIC MODELS
Liquid phase
In previous assessments of the Fe-Sn system [18-22] , the BW random mixing model was used for describing the
positive deviation of the enthalpy of mixing (Δh > 0) of the liquid phase. Kang and Pelton showed that
[24]
unless many empirical temperature-dependent parameters are used in the BW model, the shape of the
miscibility gap is often too high and rounded. The limited reproducibility of flat-shaped miscibility gaps
[24]
results from neglecting short-range ordering (SRO) as well as clustering. In this case, it was demonstrated
that the accuracy of the liquidus phase boundary could be significantly improved using the Modified
[61]
Quasichemcal Model (MQM). Detailed information on SRO in liquid solutions can be found elsewhere .
[62]
In optimizing Fe-based phase diagrams, Shubhank and Kang and Tafwidli and Kang successfully
[7]
applied the MQM modeling of the miscibility gap in the Fe-Cu and the Fe-C-S system, respectively. The
miscibility gap in the present Fe-Sn system may not be seen as a significantly “flattened”-shape, compared
to those discussed by Kang and Pelton . However, Kang and Pelton also showed that the MQM predicts
[24]
the miscibility gap (positive deviation) in ternary systems better than the BW model does [24,61] . Since the
present study is a part of developing a thermodynamic database for a larger Fe-alloy system with tramp
elements, the MQM was employed in the present study.

[25]
A detailed description of the MQM can be found elsewhere . Hence, the model will be explained only
briefly in this section. In the pair approximation for a binary solution consisting of Fe and Sn atoms, the
following pair exchange reaction on the sites of a quasi-lattice is considered:

where (i - j) represent the first nearest neighbor pair and Δg is the non-configurational Gibbs free energy
FeSn
change forming two moles of (Fe - Sn) pairs. If n and n are the numbers of moles Fe and Sn, n is the
Fe
Sn
ij
number of (i - j) pairs and Z and Z are the coordination numbers of Fe and Sn, then the following mass
Fe
Sn
balances are considered:

The pair fractions, mole fractions and coordination-equivalent fractions are defined as:

The Gibbs energy of the solution is given by:

where and are the molar Gibbs energies of the pure components and ΔS config is the configurational
entropy of mixing given by randomly distributing the (Fe-Fe), (Sn-Sn) and (Fe-Sn) pairs in the one-
dimensional Ising approximation :
[63]