TherModynaMIc ModeLIng of The In–Sc and In–y SySTeMS SupporTed By fIrST-prIncIpLeS caLcuLaTIonS

Based on an assessment of the phase equilibria and thermodynamic data in the literature, the thermodynamic modeling of the In–Sc and In–Y systems was carried out by means of the calculation of phase diagram (CALPHAD) method supported by first-principles calculations. The solution phases, i.e., liquid, (In), (αSc), (βSc), (αY) and (βY), were modeled with the substitutional regular solution model. Ten intermetallic compounds, including InSc 3 , InSc 2 , In 4 Sc 5 , InSc, In 2 Sc, In 3 Sc, InY 2 , InY, In 5 Y 3 , and In 3 Y were described as stoichiometric phases, while In 3 Y 5 was modeled with a sublattice model with respect to its homogeneity range. The enthalpies of formation of the intermetallic compounds at 0 K were computed using first-principle calculations and were used as input for the thermodynamic optimization. A set of self-consistent thermodynamic parameters for both the In–Sc and In–Y systems were obtained and the calculated phase diagrams are in good agreement with the experimental data.


Introduction
Light alloys, like aluminum and magnesium alloys, have witnessed a huge number of applications in aviation, aerospace and automotive industry in recent years [1][2][3]. It is found that indium, yttrium, and scandium can be used as trace addition elements to improve the alloys' properties [4][5][6][7][8][9]. The addition of In in Mg-based alloys induces grain size refinement and precipitation hardening, thereby enhancing their mechanical properties [4,5]. The introduction of Sc cannot only increase the yield and tensile strength but also results in a reduction in weld and solidification cracking for both Al and Mg alloys [6,7]. Y addition refines the grain size of the alloys, leading to higher strength and better ductility [8,9]. An alloy's property can be understood by its microstructures and it is critical to know the effect of addition elements on its phase formation and phase stability. For establishing the thermodynamic database of the Al and Mg alloys, the thermodynamic description of the In-Sc-Y system is of great interest. Because the thermodynamic descriptions of In-Sc and In-Y systems are not available in the literature, it is our goal to develop a set of sound thermodynamic parameters for these systems by means of the CALPHAD method.   No other experimental work is available in the literature and the phase equilibria data in Ref. [10] were adopted in the present work. The crystal structure information of the stable phases in the In-Y system is summarized in Table 1.
In this work, the first-principles calculations were conducted within the generalized gradient approximation (GGA) [35] suggested by Perdew et al. [36] for the exchange and correlation contributions and with projector augmented plane-wave (PAW) potentials [37] for valence electrons, as implemented in the Vienna ab initio simulation package (VASP) [38]. The atoms were relaxed toward equilibrium until the Hellman-Feynman forces were less than 10 -2 eV/Å. Brillouin-zone integrations were performed using a Monkhorst-Pack mesh [39]. The energy cut off at 400 eV was sufficient to ensure the total energy difference less than 1meV/atom. The k-points meshes for Brillouin zone sampling were constructed using Monkhorst-Pack scheme and at least 10000 per reciprocal atom were employed. The structures were fully relaxed using the Methfessel-Paxton method [40] and a final self-consistent static calculation via the tetrahedron method with Blochl corrections [41] was performed. The enthalpies of formation of the compound In m X n (X=Sc or Y) at 0 K can be defined as the following equation: (1) where E(In m X n ) is the total energy of the compound In m X n . E(In) and E(X) represent the total energy of In and X in their stable structures, respectively. The calculated results are summarized in Table 2.

Thermodynamic modeling 4.1 Unary phases
The Gibbs energy description of the pure element i (i=In, Sc or Y) in the phase φ has been given in the SGTE database [42] and the function is expressed by the following equation: (2) where is the mole enthalpy of the element i at 25 ℃ and 1 bar in its standard element reference (SER) state, and T is the absolute temperature.

Solution phases
The liquid, (αSc), (βSc), (αY) and (βY) phases are modeled as completely disordered solutions. The Gibbs energy is described by Redlich-Kister polynomial [43]: where , x In and x i are the mole fractions of In and component i (i=Sc, Y), respectively.
suggests the mechanical mixing of the pure elements to the Gibbs energy. The middle term RT(x In Inx In + x i Inx i ) denotes the ideal entropy of mixing to the Gibbs energy and R is the gas constant. The last term indicates the excess Gibbs energy.
is the jth interaction parameter between In and component i of the phase φ, and it is equal to .The parameters, a j and b j , are to be optimized in the present work.

Compounds
InSc 3 , InSc 2 , In 4 Sc 5 , InSc, In 2 Sc, In 3 Sc, InY 2 , InY, In 5 Y 3 , and In 3 Y are modeled as stoichiometric compounds. The Gibbs energy of the compound In m X n (X=Sc or Y) is given by the following equation: (4) where a and b are the parameters to be optimized in the present work.
Considering the reported homogeneity ranges [10], In 3 Y 5 are modeled with two-sublattice models, (In) m (In,Y) n , m, n represent the number of sites in each sublattice. The Gibbs energy of this compound can be described by the following equation: where the parameter is the site fraction of the species i (i = In or Y) on the sublattice j. The superscript '' denote the second sublattices of the presented model. and is the Gibbs energy of the ideal compound In 3 Y 5 and In 3 In 5 , respectively. and can be described as the equation (4).

results and discussion
The first-principles calculated lattice parameters and formation enthalpies of the phases in the In-Sc and In-Y systems are shown in Table 1 and 2, respectively. Compared with the experimental lattice parameters, the first-principles results exhibit the deviation of 0.01%-4.47%, which is a good agreement.
Evaluation of the thermodynamic parameters was performed by the optimization program PARROT [44] of the Thermo-Calc software, which is based on minimizing the squared sum of differences between experimental values and computed ones. The step-bystep optimization procedure [45] was utilized in the present work. The reliable experimental data and the enthalpies of formation for the compounds via firstprinciples calculations were employed as input data. In the process of optimization, each piece of reliable input data was given a certain weight which was varied in different steps during the assessment.
Taking the In-Y system as an example, the optimization began with introducing InY 2 and In 3 Y in the modeling one by one. In the first steps, the parameter a in Eq. (4) was set to be the ΔH f calculated by the first-principles method because of its wellknown good accuracy. Then, parameter b in Eq. (4) for InY 2 and In 3 Y was optimized using the invariant reactions and melting points. Subsequently, the other compounds, In 3 Y 5 , In 3 Y 5 , InY, and In 5 Y 3 were introduced in the modeling one by one. The parameters a and b in Eq. (4) for these compounds were obtained as the same procedure. Meanwhile, the parameters for liquid, a 0 , a 1 , a 2 , b 0 , b 1 , and b 2 in Eq. (3) were evaluated to fit for the experimental liquidus data. For (In) phase, the interaction parameters were fixed to be zero due to its negligible solubility ranges. For (αY) and (βY), which contain certain solubility ranges, its parameters were optimized using invariant reactions and experimental data. Finally, all the paraments were optimized simultaneously by taking into consideration all the phase equilibria and thermodynamic data. The procedure of optimization for the In-Sc system was analogous with that of the In-Y system. The optimized thermodynamic parameters are summarized in Table 4.
The calculated enthalpies of formation for the In-Sc and In-Y binary compounds are shown in the  Fig. 2, respectively, which are comparable with the first-principles results. The calculated In-Sc and In-Y phase diagram compared with the experimental data [10,18] are shown in the Fig. 3 and Fig. 4, respectively. The calculated invariant equilibria along with experimental data are listed in Table 3. Most of the experimental data can be well reproduced by the present calculation. In the In-Sc system, it is shown that the melting temperature of the InSc 2 is calculated to be 1397 ℃, the eutectic reactions Liquid ↔ (βSc) + InSc 3 and Liquid ↔ In 3 Sc + (In) are located at 1327 ℃ and 158 ℃, respectively, the eutectoid reaction (βSc) ↔ (αSc) + InSc 3 is located at 1107 ℃, the peritectic reactions Liquid + InSc 2 ↔ InSc 3 , Liquid + InSc 2 ↔ In 4 Sc 5 , Liquid + In 4 Sc 5 ↔ InSc, Liquid + InSc ↔ In 2 Sc and Liquid + In 2 Sc ↔ In 3 Sc are located at 1347 ℃, 1277 ℃, 1127 ℃, 967 ℃ and 937 ℃, respectively. The deviation between the calculated and experimental invariant reaction points is less than 15 ℃ and 0.1 at.% In. In the In-Y systems, the melting temperature of the In 3 Y 5 , In 5 Y 3 , and In 3 Y compounds are calculated to be 1337 ℃，1147 ℃ and 1107 ℃, respectively. The eutectic points of four eutectic reactions Liquid ↔ (βY) + InY 2 , Liquid ↔ InY + In 5 Y 3 , Liquid ↔ In 5 Y 3 + In 3 Y and Liquid ↔ In 3 Y + (In) lie at 1057 ℃, 1127 ℃, 1087 ℃ and 158 ℃, respectively. Two eutectoid reactions Liquid + In 3 Y 5 ↔ InY 2 , Liquid + In 3 Y 5 ↔ InY are calculated to be 1317 ℃ and 1217 ℃. The difference between the calculated and experimental data of the other invariant reaction is less than 15 ℃ and 0.1 at.% In, which is a good agreement. There is a large difference (39 ℃) between the calculated (1127 ℃) and reported (1088 ℃) reaction point of the Liquid ↔ InY + In 5 Y 3 . The present calculation can be considered reasonable because the reported temperature is an assessed value of 9 DTA data points within the range from 1079 to 1136 ℃.

conclusions
(1) The experimental phase equilibria and thermodynamic data available for the In-Sc and In-Y systems were critically evaluated.
(2) The enthalpies of formation for InSc 3 , InSc 2 , In 4 Sc 5 , InSc, In 2 Sc, In 3 Sc, InY 2 , In 3 Y 5 , InY, In 5 Y 3 , and In 3 Y were obtained by means of first-principles calculations, which were also set as reliable input values used in the present thermodynamic modeling.
(3) A set of self-consistent thermodynamic parameters have been obtained to represent the phase      . .