1. Introduction
There is a technological interest in the TiSi system promoted by the beneficial effect of Si addition for the oxidation and creep resistance of TiXSi alloys (^{Azevedo, 1996}). The earliest TiSi experimental phase diagram was obtained in 1952 (^{Hansen et al., 1952}), indicating in the Tirich 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 Tirich 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 TiSi 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 TiSi phase diagram was performed in 1976 (Kaufmann, 1976) considering the Ti_{5}Si_{3} phase as a stoichiometric intermetallic. Murray (^{Murray, 1987}) assessed the TiSi system assuming the Ti_{5}Si_{3} phase as a nonstoichiometric 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 TiSi 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 TiSi and TiXSi 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 TiSi system was studied by abinitio calculations, indicating that the stability of Ti_{3}Si phase was controversial (^{Colinet and Tedenac, 2010}). Recent abinitio 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 Tirich corner of the stable and metastable TiSi 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}).
2. 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 RedlichKister polynomial (see Equations 4 and 5) [23]. Additionally, the Gibbs energy for formation of the stoichiometric Ti_{3}Si phase is described using the KoppNeumann rule (see Equation 6) and the nonstoichiometric Ti_{5}Si_{3} phase is described by the Compound Energy Formalism (^{Lukas, 2007}), using a twosublattices containing Ti and Si, see Equations 7 to 10.
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 TiSi interaction parameter in the phase.
Where: L^{v}_{phase} = a+b.T+...
Where: y_{j}^{n} is the site fraction of the element (j) in the sublattice (n).
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 ThermoCalc 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 TiSi phase diagrams obtained by Thermocalc software using COST 507 database (^{Cost, 1998}), whose TiSi system was based on the assessed version by Seifert et al. (^{Seifert et al., 1996}).
Gibbs energy for the formation of Ti_{5}Si_{3} 

VI1 +V12.T 
Gibbs energy for the formation of hypothetic Ti_{8} 

(40,000 + 20.T)* 
Gibbs energy for the formation of hypothetic Si_{8} 

V21 +V22.T 
Gibbs energy for the formation of hypothetic Ti_{3}Si_{5} 

V31 + V32.T 
Excess Gibbs energy for the (Si,Ti:Ti) interaction 

V41 + V42.T** 
Excess Gibbs energy for the (Si,Ti:Si ) interaction 

V41 +V42.T** 
Excess Gibbs energy for the (Ti: Si,Ti) interaction 

V51 + V52.T** 
Excess Gibbs energy for the (Si:Si,Ti) interaction 

V51 +V52.T** 
Excess Gibbs energy for the (Si,Ti:Si,Ti ) interaction 

V61 +V62.T** 
Gibbs energy for the formation of Ti_{3}Si phase 

V71 + V72.T 
^{*}(^{Cost, 1998});
^{**}
50.0   72.67  79.0  77.76  57.036  Optimization  (Svechnikov et al. 1970) 
 47.11   72.53  74.63   72.23  49.87  Abinitio  (Colinet and Tedenac, 2014) 
   72.52       56.97  Experimental  (Robins and Jenkins, 1955; Topor and Kleppa, 1996; Maslov et al., 1978) 
   78.1   75.9  71.5   53.5  Experimental  (Kematick and Myers, 1996) 
 49   73.8   78.5   72.6  55  Experimental  (Meschel and Kleppa, 1998; Coelho et al., 2006) 
L → Ti_{5}Si_{3}  2401 


  (Hansen et al., 1952; Sutcliffe, 1954; Svechnikov et al. 1970; Plitcha et al. 1977; Plitcha and Aaronson, 1978; Seifert et al. 1996) 
L → TiSi_{2}  1773 


  
L → β+Ti_{5}Si_{3}  1613 




L → TiSi_{2}+Si  1603 




L → TiSi_{2}+TiSi  1747 




L+Ti_{5}Si_{3} → TI_{5}Si_{4}  2193 




L+Ti_{5}Si_{3} → TiSi  1843 




β+Ti_{5}Si_{3} → Ti_{3}Si  1443 




β → α+Ti_{3}Si  1149 




β → α+Ti_{5}Si_{3} (metastable) 
1133 



3. Results and discussion
The calculated values of the variables are shown in Table 4. According to ThermoCalc User Guide (^{Thermo, 2015}), the order of magnitude of Vi1type variables should not be higher than 10^{5} and the Vi2type 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 Vi2type 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 TiSi system using two sublattices to describe the Ti_{5}Si_{3} phase were successful but they can be further improved.
Gibbs energy for the formation of Ti_{5}Si_{3}  V11  592,126.51 
V12  6.055  
Gibbs energy for the formation of hypothetic Si_{8}  V21  36,405.66 
V22  12.595  
Gibbs energy for the formation of hypothetic Ti_{3}Si_{5}  V31  15,172.13 
V32  5.055  
Excess terms, (Si,Ti:Ti) and (Si,Ti:Ti) interactions, Ti_{5}Si_{3}  V41  49,843.69 
V42  16.271  
Excess terms, (Si:Si,Ti) and (Ti:Si,Ti) interactions, Ti_{5}Si_{3}  V51  624.781 
V52  335.78  
Excess term, (Si,Ti:Si,Ti) interaction, Ti_{5}Si_{3}  V61  226.340 
V62  14.165  
Gibbs energy for the formation of Ti_{3}Si  V71  200,788.05 
V72  2.737 
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 originated 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 Tirich corner of the TiSi 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}).
L → Ti_{5}Si_{3}  T(K)  2403  2394  0.4 

0.375  0.375  0  
L → TiSi_{2}  T(K)  1773  1757  1 

0.667  0.667  0  
L → β+ Ti_{5}Si_{3}  T(K)  1613  1626  0.6 

0.137  0.127  7  

0.047  0.044  4  

0.36  0.35  5  
L → TiSi_{2}+Si  T(K)  1603  1604  0.1 

0.86  0.815  5  

0.667  0.667  0  

1  1  0  
L → TiSi_{2}+TiSi  T(K)  1747  1747  0 

0.641  0.637  0.7  

0.5  0.50  0  

0.667  0.667  0  
L+Ti_{5}Si_{3} → Ti_{5}Si_{4}  T(K)  2193  2210  0.8 

0.477  0.476  0.2  

0.405  0.375  6  

0.444  0.444  0  
L+Ti_{5}Si_{4} → TiSi  T(K)  1843  1843  0 

0.60  0.604  0.7  

0.444  0.444  0  

0.5  0.5  0  
β +Ti_{5}Si_{3} → Ti_{3}Si  T(K)  1443  1457  0.1 

0.04  0.033  18  

0.36  0.353  2  

0.25  0.25  0  
β → α+ Ti_{3}Si  T(K)  1149  1142  0.8 

0.009  0.010  6  

0.004  0.0042  5  

0.25  0.25  0  

47,850  50,197  5  
β → α+ Ti_{5}Si_{3}(metastable)  T(K)  1133  1138  0.4 

0.011  0.013  18  

0.005  0.0054  8  

0.365  0.36  1.4  

73,874  74,016  0.2 
Figure 1a shows a general view of the calculated stable TiSi 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 1b shows a detail of the Tirich 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 Sisolubility 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 2a shows the calculated metastable TiSi 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 solubility 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 2b shows a detail of the Tirich 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 Sisolubility 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 TiSi 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 Sisolubility range was comparatively narrower and presented a maximum of 37.5at%. This maximum Sisolubility 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 (
4. Conclusions
The assessed versions of the stable and metastable TiSi 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 TiSi phase diagram showed a typical inclination, indicating that the Sisolubility 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 Sisolubility range was comparativelly much narrower than expected and presented a maximum value of 37.5at%.
The assessment of the TiSi 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 Tirich corner of the TiSi 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.