SIMPLIFICATION OF THE THERMODYNAMIC DESCRIPTION OF THE TiSi SYSTEM

The thermodynamic optimization of ternary or higher order systems containing Ti requires that the Ti-X binary systems are continuously updated. The Ti-Si system has been studied and thermodynamically optimized since de 1950s. The Ti5Si3 phase was initially considered stoichiometric to facilitate the thermodynamic calculations, although experimental results showed that this phase was a non-stoichiometric intermetallic. The most recent optimization of the Ti-Si system described this phase as a non-stoichiometric intermetallic with three sublattices, using the sub-lattice model. The problem of this approach is that it increases considerably the number of variables to be optimized during the calculation of the Ti-Si-X phase diagrams, hindering the convergence of the computational processing. The present work simplifies the optimization of the Ti-Si system, assuming that Ti5Si3 phase is mostly hyper-stoichiometric in relation to Ti. The results showed that this simplification did not significantly affect the phase diagram and the thermodynamic properties calculated for the system.


INTRODUCTION
The interest on the Ti-Si system is fundamentally related to the beneficial effects of Si on high temperature properties of Ti-X-Al alloys, such as creep and oxidation resistance [1][2][3].Hansen [4] experimentally investigated the Ti-Si phase diagram in the 50s and they showed the presence of non-stoichiometric Ti 5 Si 3 phase and indicated the presence of an eutectoid reaction, represented by Ti(β)->Ti(a)+Ti 5 Si 3 .Almost 20 years later, Svechnikov et al. [5] presented a new experimental version of the Ti-Si system, showing the presence of a peritectoid reaction, represented by Ti(β)+ Ti 5 Si 3 -> Ti 3 Si; and an eutectoid reaction, represented by Ti(β)->Ti(a)+Ti 3 Si.The first thermodynamic optimization of the system was performed by Kaufman [6], which used the experimental results of Hansen [4] (without the presence of Ti 3 Si and Ti 5 Si 4 phases), but considering the Ti 5 Si 3 phase as stoichiometric.Vahlas et al. [7] conducted a thermodynamic optimization Tecnol.Metal.Mater.Miner., São Paulo, v. 13, n. 1, p. 91-97, jan./mar.2016 based on data from Svechnikov et al. [5], but considering the Ti 5 Si 3 phase as stoichiometric.Only in the mid-90s that Seifert et al. [8] performed a thermodynamic optimization of the Ti-Si system, which were more consistent with previous experimental results [4][5].The variables calculated in this optimization were incorporated into the thermodynamic databases of computational thermodynamic software.The major distinction of the thermodynamic modeling made by Seifert et al. [8] was to consider the non-stoichiometric intermetallic Ti 5 Si 3 phase with three sublattices, with a configuration represented by (Ti,Si) 2 (Si,Ti) 3 (Ti) 3 .Most intermediate phases of the Si-X systems are, however, hyper-stoichiometric with respect to the element X [9] such as Y 5 Si 3 and V 3 Si phases.
The ternary system Ti-Al-Si was investigated experimentally by Crossley et al. [10] Crossley and Turner [11], Schob et al. [12], Wu et al. [13], Azevedo [1], Azevedo and Flower [2,14], Bulanova et al. [15] and Perrot [16].All these investigations showed that Al was soluble in non-stoichiometric Ti 5 Si 3 phase.This ternary system was thermodynamically optimized by Azevedo [1] and Azevedo and Flower [2,3], assuming, however, that the Ti 5 Si 3 phase was stoichiometric and without Al solubility.For the future thermodynamic optimization of the ternary Ti-Al-Si system with the presence of non-stoichiometric Ti 5 Si 3 phase with Al solubility, a better description of this phase should be used.The description of Ti 5 Si 3 phase with 3 sublattices as proposed by Seifert et al. [8] would lead to the use of eight hypothetical phases and at least 40 missing interaction parameters, which could hinder the convergence of the optimization procedure during the calculation of these parameters.
The objective of this paper is to simplify the optimization of Ti-Si system using 3 sublattices but assuming that Ti occupies the Si sublattice in the following description: (Ti) 2 (Si,Ti) 3 (Ti) 3 .The number of variables to be optimized for the calculation of the Ti-Si and Ti-Al-Si systems would decrease considerably, simplifying the thermodynamic optimization of the ternary Ti-Al-Si phase diagram.

MATERIALS E METHODS
The invariant equilibria considered in present thermodynamic optimization are presented in Table 1 and the enthalpies of formation of intermediate phases are shown in Table 2.
The liquid and solid solutions have been described thermodynamically using the model of substitutional solution [21].Thus (Equations 1-4): Where: The interaction parameter was described by a Redlich-Kister polynomial (see Equations 5 and 6).

. .ln( ) .ln( )
Ti Si Exc ' '' '' ''' Ti Ti Si Ti Ti:Si,Ti:Ti Where: y is the site fraction in the sublattice and the superscripts ', " and "' refer to the 1st, 2nd and 3rd sublattices of the compound, respectively.As in the 1st and 3rd sublattices there is only Ti, the sites fraction of Ti in them are equal to one.The free energies for the formation of the end-members, Ti Si Ti Ti Ti G 5 3   : : : :

ref
, were considered to be the same as the ones calculated by Seifert et al. [8].For the description of the stoichiometric intermediate phases, the Kopp-Neumann rule was used resulting in the following equation for a Ti m Si n phase (Equation 9) [25]: The thermodynamic parameters used in the present study are listed in Table 3.The parameters of GHSER were obtained from SGTE data [26].Data were optimized using the CALPHAD technique with the help of the Parrot module of Thermo-calc.

RESULTS E DISCUSSION
The optimization for the calculation of the Ti Si Ti Ti Si Ti L 5 3   0 : , : interaction parameters resulted in the coeficients a = +1.74102703E+05(J/mol) and b = -9.77215453E+01(J/mol.K).The results of invariant equilibria are shown in Table 4, indicating that the calculated values are very    close to the experimental data [5].From the nine invariant equilibria of the Ti-Si system, only one of the 32 calculated values presented a considerable relative deviation (over 10%) in relation to the experimental values: the value of the limit of solubility of Ti(a) phase in the eutectoid reaction Ti(β) = Ti(a)+Ti 3 Si, due to the lack of reliable experimental data concerning this equilibrium.Additionally, there is an essential difference of ~ 6% in the eutectic composition of the liquid (L = TiSi 2 + Si) given in Table 4.This difference is probably due to the high tendency of the liquid of the Ti-Si system to local ordering in high concentrations of Si.This same deviation between experimental and calculated values using the regular solution model to describe the liquid phase was observed Seifert et al. [8].They solved this discrepancy by using a partially ionic liquid model to describe the liquid phase, which is not a common procedure for metallic systems.If ordering of the liquid is a feasible hypothesis, which seems to be in agreement by the large amount of intermediate phases in the Ti-Si system, the use of other thermodynamic models to describe the liquid phase, such as the associated liquid or quasi-chemical, would result in a better fit between calculated and experimental data, especially in equilibria involving the liquid phase.
The enthalpies of formation of the compounds are presented in Table 5 and these values were compared to previous results [8].The calculated values for the enthalpies of formation of intermediate phases are nearly equal to those obtained by Seifert et al. [8] and very close to the experimental values shown in Table 2.The calculated Ti-Si phase diagram is shown in Figure 1, showing, additionally, the experimental data used in the present optimization.Figure 2 presents the calculated integral enthalpy of mixing

CONCLUSIONS
• The simplification in the description of the Ti 5 Si 3 phase ((Ti) 2 (Si,Ti) 3 (Ti) 3 ) produced a good assessment of the Ti-Si phase diagram, allowing its use for the optimization of the ternary Ti-Si-Al.
• The use of other thermodynamic models to describe the liquid phase, such as the associated liquid or quasi-chemical, might result in a better fit between calculated and experimental data, especially in equilibria involving the liquid phase.The present simplification of the description of the Ti 5 Si 3 phase ((Ti) 2 (Si,Ti) 3 (Ti) 3 ) produced good results when compared to previous assessment [8], allowing its use for the optimization of the Ti-Si-Al phase diagram.This simpler phase description will facilitate the thermodynamic optimization of Ti-Al-Si system, which is the next step of our work.

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(experimental -calculated)/experimental. Simplification of the thermodynamic description of the Ti-Si system 95 Tecnol.Metal.Mater.Miner., São Paulo, v. 13, n. 1, p. 91-97, jan./mar.2016 for the liquid at 2000K along with the experimental data of Esin et al. shown by Vahlas et al. [7].Both figures indicate that the experimental data are consistent with the calculated values of the present optimization procedure.

Figure 2 .
Figure 2. Calculated variation of integral enthalpy of mixture of the liquid of the Ti-Si system at 2000K (J/mol).Experimental data of Esin et al. shown by Vahlas et al. [7].

Table 2 .
Enthalpies of formation at 298.15K of intermediate phases of Ti-Si system (kJ/mol of atoms)

Table 5 .
Calculated enthalpies of formation at 298.15K of intermediate phases of Ti-Si system (kJ/mol of atoms)