Assessment of the Ti-rich corner of the TiSi phase diagram using two sublattices to describe the Ti 5 Si 3 phase

The thermodynamic optimization of Ti-X-Si systems requires that their respective binary systems be constantly updated. The most recent assessments of the Ti-Si phase diagrams used three sublattices to describe the Ti5Si3 phase. The stable version of this phase diagram indicated the presence of Ti(β)+Ti5Si3→Ti3Si and Ti(β)→Ti(α)+Ti3Si reactions in the Ti-rich corner, while the metastable version featured the presence of a Ti(β)→Ti(α)+Ti5Si3 reaction. The present investigation assessed these phase diagrams using two sublattices to describe the Ti5Si3 phase in order to simplify the optimization of Ti-X-Si systems.


Introduction
There is a technological interest in the Ti-Si system promoted by the beneficial effect of Si addition for the oxidation and creep resistance of Ti-X-Si alloys (Azevedo, 1996).The earliest Ti-Si experimental phase diagram was obtained in 1952 (Hansen et al., 1952), indicating in the Ti-rich corner the presence of a eutectoid reaction at 1133K, Ti(β) → Ti(α) + Ti 5 Si 3 .In 1954, another work confirmed the presence of this eutectoid reaction at 1129K (Sutcliffe, 1954).In 1970, a new experimental version of this phase diagram was proposed (Svechnikov et al., 1970), indicating in the Ti-rich corner the presence of two new reactions (a peritectoid reaction at 1444K, Ti(β) + Ti 5 Si 3 → Ti 3 Si and a eutectoid reaction at 1133K, Ti(β) → Ti(α) + Ti 3 Si), instead of the eutectoid reaction previously observed.
In late 70´s, however, careful investigations of the eutectoid reaction of the Ti-Si system were performed without showing any evidence on the presence of the Ti 3 Si phase (Plitcha et al. 1977;Plitcha and Aaronson, 1978).They confirmed instead the presence of Ti 5 Si 3 phase at 1148K, Ti(β) → Ti(α) + Ti 5 Si 3 .The first thermodynamic assessment of the Ti-Si phase diagram was performed in 1976 (Kaufmann, 1976) considering the Ti 5 Si 3 phase as a stoichiometric intermetallic.Murray (Murray, 1987) assessed the Ti-Si system assuming the Ti 5 Si 3 phase as a non-stoichiometric phase and the calculated phase diagram was in agreement with one of the previous results (Svechnikov et al., 1970).In 1996, Seifert et al. (Seifert et al., 1996) employed an optimization method for the determination of the variables used for the thermodynamic description of the phases in order to assess the Ti-Si phase diagram from selected experimental data.They described, for instance, the Ti 5 Si 3 phase as a nonstoichiometric compound containing three sublattices, (Ti) 3 (Ti,Si) 2 (Si,Ti) 3 , to represent its D8 8 crystal structure.Their calculated phase diagram was in good agreement with previous calculated (Murray, 1987) and experimental (Svechnikov et al., 1971) phase diagrams, presenting Ti 3 Si as the stable phase of the eutectoid reaction.The dispute over the stability of the Ti 3 Si phase in Ti-Si and Ti-X-Si systems was, however, far from over.Azevedo (Azevedo, 1996;Azevedo and Flower, 1999;Azevedo and Flower, 2000;Azevedo and Flower, 2002) and Bulanova (Bulanova et al., 1997) identified the presence of Ti 5 Si phase (instead of Ti 3 Si) after long isothermal heat treatments below the eutectoid temperature.By contrast, the presence of Ti 3 Si phase was observed by other investigations (Kozlov and Pavlyuk, 2004;Ramos et al., 2006;Costa et al.;2010;Li et al., 2014).In 2010, the stability of intermetallic phases in the Ti-Si system was studied by ab-initio calculations, indicating that the stability of Ti 3 Si phase was controversial (Colinet and Tedenac, 2010).Recent ab-initio calculation showed that Ti 5 Si 3 phase was actually more stable than Ti 3 Si phase at 0 K (Poletaev et al., 2014).
The present work will calculate and compare the Ti-rich corner of the stable and metastable Ti-Si phase diagrams, using two sublattices, (Ti,Si) 5 (Si,Ti) 3 , to describe the Ti 5 Si 3 phase, assuming that Ti 3 Si is the stable phase in the eutectoid decomposition of Ti(β) phase.These results will be compared to previous calculated phase diagrams using three sublattices to describe the Ti 5 Si 3 phase (Cost, 1998;Fiori et al., 2016).

Methodology
The liquid, Ti(α) and Ti(β) phases are described using Equations 1 to 5. The Gibbs free energy of reference (G ref ) is described by Equation 2, while the Gibbs free energy of the ideal solution (G id ) is described by Equation 3 and the excess Gibbs free energy (G ex ) of the regular solution is described using the Redlich-Kister polynomial (see Equations 4 and 5) [23].Additionally, the Gibbs energy for formation of the stoichiometric Ti 3 Si phase is described using the Kopp-Neumann rule (see Equation 6) and the non-stoichiometric Ti 5 Si 3 phase is described by the Compound Energy Formalism (Lukas, 2007), using a two-sublattices containing Ti and Si, see Equations 7 to 10.

Si Ti
Where: G i ref = G i SER and x Si and x Ti are the molar fraction of the elements.
Where: L phase is the Ti-Si interaction parameter in the phase.
The parameters and variables used for the thermodynamic description of the Ti 5 Si 3 and Ti 3 Si phases are listed in Table 1.These variables were calculated from selected experimental data (see Tables 2 and 3) using the Parrot module of the Thermo-Calc software.The variables related to the Ti 5 Si 3 phase were initially calculated during the assessment of the metastable phase diagram (suspending the presence of the Ti 3 Si phase).These variables were then fixed during the assessment of the stable phase diagram for the calculation of the variables related to the Ti 3 Si phase.These diagrams were compared to the stable and metastable Ti-Si phase diagrams obtained by Thermocalc software using COST 507 database (Cost, 1998), whose Ti-Si system was based on the assessed version by Seifert et al. (Seifert et al., 1996).(Meschel and Kleppa, 1998;Coelho et al., 2006) Table 2 Enthalpy for the formation of intermetallic phases, Ti-Si system (kJ/mol of phase).

Results and discussion
The calculated values of the variables are shown in Table 4.According to Thermo-Calc User Guide (Thermo, 2015), the order of magnitude of Vi1type variables should not be higher than 10 5 and the Vi2-type variables should not be higher than 10 1 .In the present assessments V11 presented an order of magnitude above 10 5 ; and V52 above 10 1 .This Vi2-type variable, however, was used to describe the excess term of the enthalpy rather than the entropy for the formation of intermetallic phases.The values of the reduced sum of squares (~ 5 for both optimization procedures) exceeded the advisable maximum value of one (Thermo, 2015).These results indicate that the optimization procedures of the Ti-Si system using two sublattices to describe the Ti 5 Si 3 phase were successful but they can be further improved.Table 5 Main experimental and calculated values of the Ti-Si system.
Table 5 compares the values of the experimental and the calculated equilibria and the enthalpies for the formation of Ti 3 Si and Ti 5 Si 3 phases.Six out of the 38 calculated values presented relative deviation above 5% in relation to the experimental data.Two of these deviations were origi-nated in the equilibria involving the liquid phase and they could be decreased by the use of a more complex model for the thermodynamic description of the liquid phase (Lukas, 2007;Seifert et al., 1996;Fiori et al., 2016).The other values were found for the β +Ti 5 Si 3 →Ti 3 Si, β→α+Ti 3 Si and β→α+Ti 5 Si 3 reactions, indicating that further experiments in these critical regions of the Ti-rich corner of the Ti-Si phase diagram are needed to improve the results of the present optimization procedures; and to define which one of the eutectoid reactions is actually the stable one (β→α+Ti 3 Si or β→α+Ti 5 Si 3 ).
Figure 1-a shows a general view of the calculated stable Ti-Si phase diagram, indicating that the position of the phase boundaries are in fair agreement with previous results (Svechnikov et al. 1970;Fiore et al. 2016), except for the narrower solubility range of the Ti 5 Si 3 phase field.Figure 1-b shows a detail of the Ti-rich corner near the eutectoid reaction, indicating that there are no experimental data to validate the position of the calculated Ti(α) and Ti(β) solvus lines.The present assessment showed lower Si-solubility in the Ti(α) and Ti(β) phases when compared to the calculated phase diagram using COST 507 database (Cost, 1998), without any change in the eutectoid temperature.
Figure 2-a shows the calculated metastable Ti-Si phase diagram, indicating that the position of the phase boundaries are in good agreement with previous experimental (Hansen et al, 1952;Sutcliffe, 1954) and calculated (Fiore et al. 2016) phase diagrams, except for the narrower sol-ubility range of the Ti 5 Si 3 phase field.The shape of this phase field resembles a previous result, which described the Ti 5 Si 3 phase as Ti 3 Ti 2 (Ti,Si) 3 (Beneduce et al., 2016).Figure 2-b shows a detail of the Ti-rich corner near the eutectoid reaction, comparing the present assessment with previous experimental (Plitcha et al. 1977;Plitcha and Aaronson, 1978) and calculated (Cost, 1998;Fiore et al. 2016) phase diagrams.The present assessment showed smaller Si-solubility in the Ti(α) and Ti(β) phases when compared to the calculated phase diagram using COST 507 database (Cost, 1998) and a slightly higher value for the eutectoid temperature.The slope of the Ti(α) solvus line showed a typical inclination, unlike the one obtained by COST 507 database (Cost, 1998), indicating that the Si solubility of the Ti(α) phase decreased with decreasing temperature.This result is agreement with the most recent assessment of the metastable Ti-Si phase diagram (Fiore et al. 2016).
The position of the Ti 5 Si 3 phase field in both assessments was slightly shifted towards smaller Si contents.Additionally, its Si-solubility range was comparatively narrower and presented a maximum of 37.5at%.This maximum Si-solubility value suggests that the present thermodynamic description of the excess terms of the (Ti,Si) 5 (Si,Ti) 3 phase was not able to induce the presence of Si atoms on the Ti sublattice.In this sense, the hypothesis that the interaction between Si and Ti on each sublattice is independent of the occupation of the other sublattice , see Table 1) should be further analyzed.
For instance, another hypothesis, assuming that the interaction parameters on the two sublattices are symmetrical , can be investigated.Finally, the description of the Ti 5 Si 3 phase using only two sublattices presented promising results for the assessment of Ti-X-Si phase diagrams.

Conclusions
• The assessed versions of the stable and metastable Ti-Si phase diagrams, using only two sublattices to describe the Ti 5 Si 3 phase, were in fair agreement with previous experimental and calculated phase diagrams.
• The slope of the Ti(α) solvus line of the assessed metastable Ti-Si phase diagram showed a typical inclination, indicating that the Si-solubility of the Ti(α) phase decreased with decreasing temperature.
• The position of the Ti 5 Si 3 phase field in both assessments was slightly shifted towards smaller Si contents.Additionally, its Si-solubility range was comparativelly much narrower than expected and presented a maximum value of 37.5at%.
• The assessment of the Ti-Si phase diagram using two sublattices to describe the Ti 5 Si 3 phase might be further improved by the inclusion of new experimental data near the eutectoid reaction of the Ti-rich corner of the Ti-Si phase diagram.In this sense, further experimental work is needed to define which eutectoid reaction (β→α+Ti 3 Si or β→α+Ti 5 Si 3 ) is stable.
• Finally, the use of a more complex description for the liquid phase and another thermodynamic description for the excess terms of the Ti 5 Si 3 phase might be useful to improve the quality of the assessed phase diagrams.

Table 3 Experimental
values of the Ti-Si invariant reactions (X Si phase : atomic fraction of Si).