Thermodynamic Description of the Gd-Sb and Gd-Bi-Sb Systems

Jinming Liu1,2*, Xiaoyang Chen2, Yinghui Zhang1, Ming Zhang1, Lan Zhang1, Tao Zhong1, Juan Lin1 and Caifang Cao3 1School of Material Science and Engineering, Jiangxi University of Science and Technology, Ganzhou, 341000, PR China 2Xingle Group Co. Ltd., Yueqing, Zhejiang, 325604, PR China 3School of Metallurgy and Chemical Engineering, Jiangxi University of Science and Technology, Ganzhou, 341000, PR China


Introduction
In past years, the advanced magnetic materials and its potential as an energy savings technology greatly stimulated the interest of researchers and promoted the rapid development of magnetic refrigeration. Magnetic refrigeration could be realized by utilizing the heat release or absorption caused by the magnetic entropy change DSM of a magnetic material due to a magnetic field change DH [1]. The recent discovery of the giant magnetocaloric effect in Gd 5 (Si 2 Ge 2 ) [1], gave further impulse towards the development of new materials [2]. Gd 4 Sb 3 was one of the candidates for magnetic refrigerant near room temperature (266 K). Substitution of Bi for Sb in Gd 4 Sb 3 increased its Curie temperature T c up to 330 K [3]. The calculation of phase diagrams (CALPHAD) method, which was a powerful approach to save cost and short time during development of materials, effectively provided a clear guideline for material design. So in order to better understand the interactions of Gd and Sb with Bi and design high performance thermoelectric materials, it was important to study the thermochemical properties and the phase equilibria concerning the Gd-Bi-Sb system and to obtain the thermodynamic parameters of the system.

Binary Systems
To obtain a thermodynamic description of a ternary system, the thermodynamic description of each involved binary system was necessary.

Gd-Sb system
The Gd-Sb system was firstly determined in the concentration range 0-60 at% Sb by Gambino [4]. Later, Gerasimov [12] updated all the invariant equilibria temperatures. The whole composition and temperature ranges in the Gd-Sb phase diagram were established by Abdusalyamova et al. [13]. The four intermediate compounds, Gd 5 Sb 3 , Gd 4 Sb 3 , GdSb 2 , and GdSb were reported in this system. Recently, the previously called "GdSb 2 '' phase was determined to be Gd 16 Sb 39 [14], which was confirmed in the Gd-Bi-Sb ternary system [15]. So the Gd-Sb phase diagram was revised by Borzone et al. [14]. All of the information of the Gd-Sb system was compiled by Li et al. [16]. The updated phase diagram was assessed by Li et al. [16]. The thermodynamic parameters of the Gd-Sb system were well reproduced [16], but the parameters of Bcc(Gd) were difficult to reproduce the temperature of the Bcc(Gd). In the present work, to assess the Gd-Bi-Sb ternary system, the model of the five intermediate compounds, Gd 5 Sb 3 , Gd 4 Sb 3 , Gd 16 Sb 39 , α-GdSb (GdSb-low temperature as GdSb-LT) and β-GdSb(GdSbhigh temperature as GdSb-HT), was re-built to cosistent with other thermodynamic parameters of the relative compounds in the Gd-Bi system [6]. So the thermodynamic parameters of the five intermediate compounds were re-assessed in the present work. Figure 3 presented the calculated phase diagram of Gd-Sb system using the thermodynamic parameters of the five intermediate compounds calculated in this work and the thermodynamic parameters of the liquid assessed by Li et al. [16].

Experimental Information on the Gd-Bi-Sb System
The information about the Gd-Bi-Sb system was very scarce. Recently, the phase diagram of the Gd-Bi-Sb ternary system had been determined at room temperature using X-ray powder diffraction and differential scanning calorimetry analysis by Jian et al. [15]. There were composed of six single-phase regions, six two-phase regions and one three-phase region. No ternary compound was reported in the Gd-Bi-Sb ternary system. The compounds, Gd 5 Sb 3 , Gd 4 Sb 3 and GdSb (α-GdSb and β-GdSb) in the Gd-Sb system and their relative compounds, Gd 5 Bi 3 , Gd 4 Bi 3 and GdBi in the Gd-Bi system were formed the continuous solid solutions, respectively. So the six single-phase regions are (Gd, Bi, Sb) solid solution (α phase), Gd 5 (Bi, Sb) 3 solid solution (β phase), Gd 4 (Bi, Sb) 3 solid solution (γ phase), Gd(Bi, Sb) solid solution (δ and   δ′ phases), Gd 16 Ga 39 (ε phase) phase and Gd (Hcp phase) phase in the Gd-Bi-Sb system. The maximum solid solubility of Gd in (Gd, Bi, Sb) at room temperature was determined to be 7.5 at %. The solid solubility of Gd in the other phases could not be detected. The above information was described in detail in the Ref. [15].

Unary phases
The Gibbs energy function for the element i (i=Gd, Bi, Sb) in the phase φ (φ = liquid, hcp, rhomb) was described as follows, In the present work, the Gibbs energy functions were taken from the SGTE(Scientific Group of Thermodata Europe) pure elements database compiled by Dinsdale [17] and listed in Table 1.

Solution phases
In the Gd-Bi-Sb system, there are three solution phases, liquid, hcp, and rhomb. Their molar Gibbs energies are described by the following expression: where R is the gas constant; x Gd , x Bi and x Sb were the mole fractions of the pure elements Gd, Bi and Sb, respectively; E m G φ was the excess Gibbs energy, expressed by the Redlich-Kister polynomial [18].
and Bi,Sb j L φ were the binary interaction parameters between elements Gd and Bi, Gd and Sb, and Bi and Sb, respectively. Its general form was L φ =a+bT+cTlnT+dT 2 +eT 3 +fT -1 (5) but in most case only the first one or two terms were used according to the temperature dependence of the experimental data. was the ternary interaction parameter expressed as: where Gd,Bi,Sb . a j and b j were the parameters to be optimized in this work.

Intermetallic compounds
The compounds, Gd 5 Sb 3 , Gd 4 Sb 3 and GdSb (GdSb-high temperature named as GdSb-HT and GdSb-low temperature named as GdSb-LT) in the Gd-Sb system and their relative compounds, Gd 5 Bi 3 , Gd 4 Bi 3 and GdBi in the Gd-Bi system were formed the continuous solid solutions, respectively. So the three continuous solid solutions single-phase (named as Gd m (Bi, Sb) n ), β-Gd 5 (Bi, Sb) 3 , γ-Gd 4 (Bi, Sb) 3 , δ-Gd(Bi, Sb) and δ′-Gd(Bi, Sb) in the Gd-Bi-Sb system were treated as two-sublattice model (Gd) m/m+n (Gd, Bi, Sb) n/m+n . In the Gd-Bi-Sb system, δ-Gd(Bi, Sb) and δ′-Gd(Bi, Sb) were the relative compounds, GdSb-high temperature (as GdSb-HT) and GdSb-low temperature (as GdSb-LT) in the Gd-Sb system. The Gibbs energy per mole of formula unit Gd m (Bi, Sb) n was given by the following expression: represented the jth interaction parameters (j=0) between the elements Bi or Sb on the second sublattice. The other binary intermetallic compounds, GdBi 2 and Gd 16 Sb 39 in the Gd-Bi and Gd-Sb system were treated as stoichiometric compounds, and were used as the two-sublattice model, which was consistented with the relative binary system [6,16].

Assessment procedure
A general rule for selection of the adjustable parameters was that only those coefficients determined by the experimental values should be adjusted [19]. The assessment was carried out by means of the optimization module PARROT of the thermodynamic software Thermo-Calc [10], which could deal with various kinds of experimental information. A careful examination of thermodynamic descriptions of the Gd-Bi [6] and the Bi-Sb [7] systems were made. The thermodynamic optimization of the Gd-Sb and Gd-Bi-Sb system were carefully performed in this work. The thermodynamic parameters for the Gd-Sb system were optimized on the basis of the experimental information available in the experimental data. The thermodynamic parameters of liquid were taken from the assessed data [16]. The compounds, Gd 5 Sb 3 , Gd 4 Sb 3 , Gd 16 Sb 39 , α-GdSb and β-GdSb in the Gd-Sb system were assessed in the present work. The experimental results of Jian et al. [15] were given more weight during the process of optimization. The thermodynamic parameters for the Gd-Bi-Sb system were optimized on the basis of the experimental information available in the experimental data [15]. The experimental results of Jian et al. [15] were given more weight during the process of optimization. The thermodynamic parameters of liquid, bcc, rhomb and hcp in the Gd-Bi-Sb system, are obtained by a combination of the corresponding Gibbs energy functions from the assessments of the binary systems using Muggianu interpolation of binary excess terms [20]. The interaction parameters of the Gd-Bi and Bi-Sb system and the liquid of the Gd-Sb system were taken from the calculated data assessed by Wang et al. [6], Dinsdale et al. [7] and Li et al. [16]. The binary parameters of the compounds of the Gd-Sb system and the ternary parameters in the Gd-Bi-Sb system were optimized according to the experimental data [15].  [17] 298.14−3000 G(hcp_A3, Bi)= +9900-11.8T+GHSER Bi [17] 298.14−2000 G(hcp_A3, Sb)= +19874-13T+GHSER Sb [17]

Results and Calculations
The Gd-Bi and the Bi-Sb systems were assessed by Wang et al. [6], and Dinsdale et al. [7], respectively. Their thermodynamic descriptions were accepted in the present work and listed in Table 1. Figures 1 and 2 presented the Gd-Bi and Bi-Sb phase diagrams using the thermodynamic description of Wang et al. [6], and Dinsdale et al. [7], respectively. There was some different at low temperature using the Thermo-Calc Software [10] and the Pandat software [11] to reproduce the Bi-Sb phase diagram, as shown in Figures 2a and 2b. Figure 3 presented the Gd-Sb phase diagram using the thermodynamic description of liquid assessed by Li et al. [16] and the calculated result of the compounds in the present work, and comparison with the experimental data [13]. Figure 4 presented the calculated standard enthalpies of formation in The Gd-Sb system at 300 K and comparison with the experimental data [14]. The reference states are hcp for Gd and rhomb for Sb. Reasonable agreement was obtained between the calculated results and the experimental data [14]. Figure 5 was the calculated enthalpies of mixing of liquid in the Gd-Sb system at 6000 K.
The calculated standard enthalpies of formation in the Gd-Sb system and comparison with the experimental data [14,[21][22][23][24][25] were shown in Figure 4 and Table 2. The calculated invariant equilibria in the Gd-Sb system were listed in Table 3. As shown in the tables, a very good agreement was obtained between the calculated results and the experimental data [14]. The Gd-Bi-Sb system were optimized on the basis of the available experimental data [15]. The thermodynamic description of the Gd-Bi-Sb system obtained in the present work was shown in Table 1. Figure 6 were the calculated isothermal section at 300 K using the present thermodynamic description in comparison with experimental data [15] in the Gd-Bi-Sb system. Satisfactory agreements were obtained between the calculated results and the experimental data [15] in Figure 6. For the Bi-Sb phase diagram at low temperature (under 300 K), the rhomb was broken down into two component parts rhomb1 and rhomb2, as shown in Figure 2b. So there was some different at the rhomb phase side from the experiment data [15].

Conclusions
The thermodynamic parameters in the Gd-Sb and Gd-Bi-Sb ternary system were critically evaluated from the experimental information available in the literature. A set of self-consistent thermodynamic    [16], and comparison with the experimental data [13].

Figure 4:
Calculated enthalpies of formation at 300 K in the Gd-Sb system and comparison with the experimental data [14] and the calculated data [16]. The reference states for the elements were hcp for Gd and rhomb for Sb. parameters describing the Gibbs energy of each individual phase as a function of composition and temperature was derived.