Thermodynamic description of the Ga-Yb binary system

Pure gallium metal and many gallium based alloys and intermetallic compounds have extensive technological applications and fundamental interest. Gallium is used as a doping component in electronic devices (as in transistors or photovoltaic cells) [1]. The development of multi-alloy material depends now for most of them on preliminary numerical simulations. Such simulations are only useful when accurate thermodynamic databases are available. Such databases are developed by the CALPHAD (CALculation of PHAse Diagram) method on a basis of experimental thermodynamic and phase diagram measurements. The excess term of the Gibbs energy of the liquid phase was assessed with the recent exponential temperature dependence of the interaction energies by Kaptay [2-4] and compared with the Redlich-Kister [5] polynomial equation results. The calculations based on the thermodynamic modelling and optimisation are in good agreement with the phase diagram data available in the literature.


Introduction
This study is a part of a thermodynamic investigation of the Ga-R systems (R= La [6], Y [7] and Ce [8]) which is intended to give a better understanding of the constitutional properties and potential technological applications of these alloys.Thermodynamic description is also needed to obtain the thermodynamic parameters via the CALPHAD approach [9].No thermodynamic description of the Ga-Yb system is available in the literature.In the present work, a thermodynamic optimization for the Ga-Yb binary system is performed based essentially on the experimental phase diagram data.

Experimental
A recent assessment of the Ga-Yb system by Palenzona and Cirafici [10] is based on the original references, essentially the works of [11][12][13][14][15]. [11,12,14] investigated the Ga-Yb phase diagram by DTA and X-ray analysis.They reported the existence and structure of Ga 4 Yb and Ga 2 Yb.Ga 4 Yb melts peritectically at 785 °C and Ga 2 Yb melts congruently at 1120 °C.They also reported the peritectic formation of two other intermediate phases of tentative stoichiometry, GaYb and Ga 3 Yb 5 , with respective peritectic temperatures of 905 and 680°C.Palenzona and Cirafici [11] investigated the Ga-Yb system by DTA, microscopy, and X-ray diffraction.Pelleg et al. [13], in a careful investigation of the Ga-rich side of the RE-Ga systems, found an additional Ga-rich phase Ga 6 RE, common to almost all RE metals, and having the Ga 6 Pu-type structure.In the Ga-Yb system, Ga 6 Yb is formed by a peritectic reaction at 282 °C.Cirafici and Fornasini [16] investigated the 20 to 32 at.%Yb region of the Ga-Yb system by X-ray and metallographic methods, see Table 1.As Ga 6 Y, Ga 6 Yb is a Pauli paramagnet.Two new phases were identified: Ga 8 Yb 3 _HT=Ga 0.727 Yb 0.273 , which exists in a narrow range of temperatures estimated between 830 and 870 °C, and Ga (3-x) Yb_LT=Ga 0.737 Yb 0.263 , which melts peritectically at a temperature of about 740 °C.No thermal analysis data were reported; the characteristic temperatures for these two new phases were determined approximately by examining a high number of alloys in the as-cast and annealed conditions.Ga (3-x) Yb presents a narrow homogeneity range (~0.025 at.) around 0.263 at.Yb and Ga 8 Y 3 _HT is not the high temperature polytype of Ga (3-x) Yb_LT.[16] also confirmed the existence of Ga 4 Yb and resolved its crystal structure.[13] confirmed the existence of Ga 2 Yb and studied structures of GaYb and GaYb 2 .These two phases were found to be isomorphous with the AuCu-I and Co 2 Si type, respectively.Besides the above intermediate phases, the liquid, and two terminal solid solutions (Ga) and ( Yb then ( Yb) with negligible solubility exist.

Unary phases
The Gibbs energy function ) for the element i (i=Ga,Yb) in the Ф phase (Ф = Liquid, BCC, FCC and ORTHORHOMBIC is described by an equation of the following form: SER i H (298.15K) is the molar enthalpy of the element i at 298.15K in its standard element reference (SER) state, FCC for Ytterbium and ORTHORHOMBIC for Gallium.In this paper, the Gibbs energy functions are taken from the SGTE compilation of Dinsdale [17].

Liquid phase
The liquid phase was assessed with the recent exponential temperature dependence of the interaction energies by Kaptay [2][3][4] and compared with the linear model using the Redlich-Kister [5] The excess terms of all the phases were modelled by the Redlich-Kister model [5].
) )( ( , 0 where j i L , is the th interaction parameter between the elements Ga and Yb which is evaluated in the presented work according to: ** the linear model of Redlich-Kister of temperature dependence h λ enthalpy part and s λ entropy part to be optimized.In order to avoid the formation of an artificial inverted miscibility gap above the liquidus line as suggested by [18][19][20], in the case of the linear model, the stability constraint was enforced by requiring that the Gibbs energy of the liquid phase had a positive curvature, i.e.,

Results
The experimental phase diagram is shown in Fig. 1 and the calculated one in Fig. 2. A very good agreement is noted.The optimized enthalpies of mixing of the liquid phase (at 1750 K) are shown in Fig. 3 as well as the infinite dilution enthalpies   excess parameters (linear + positive curvature of the liquidus and exponential) are shown in Table 2.A satisfactory agreement is observed between the two models.
The enthalpies of formation of the intermediate phases are shown in Fig. 4.These latter are consistent with the thermodynamic parameters of the terminal solid solutions because when the liquid phase is suspended, the calculated phase diagram shows that they are no more stable at very high temperature [17][18][19].For this system, no predictive value for the enthalpies of mixing of the liquid phase and for the enthalpies of formation was calculated by the Miedema model [21].
polynomial equation results.The Gibbs energy of one mole of formula unit of phase is expressed as the sum of the reference part ref G, the ideal part id G and the excess part xs G a i and b i are the coefficients to be optimized, or.** the Kaptay model of exponential temperature dependence energy of the pure elements Ga and Yb respectively; a and b are parameters to be determined.The intermediate phases Ga 6 Yb, Ga 4 Yb, Ga 8 Yb 3 (high temperature phase), Ga 2 Yb, GaYb and GaYb 2 have been treated as stoichiometric compounds while a solution model has been used for the description of the liquid phase and the (Ga) and (Yb) solid solutions.The intermetallic compound Ga (3-x) Yb which has a homogeneity range, was treated as the formula (Ga,Yb) 0.263 :(Yb,Ga) 0.737 by a two-sublattice model with a mutual substitution of Ga and Yb on both sublattices.

Figure 4 .
Figure 4. Calculated enthalpies of formation of the intermediate phases.The lower value (-67 kJ/mol.at.) around x(Yb)~0.27 at.corresponds to the Ga8Yb3_HT phase, the temperature range of existence of which is not well known.

Table 1 .
Symbols and crystal structures of the solid phases in the Ga-Yb alloys.Enthalpy of mixing of the liquid phase at 1750 K. Enthalpy at infinite dilution at 1750 K.

Table 2 .
Comparison of the excess parameters optimized with the two models of temperature dependence.