Thermodynamic assessment of the Fe-Nb-Si system

Nb-Si based alloys have drawn continuously increasing attention due to their
 excellent high-temperature mechanical properties. The addition of element Fe
 could improve their poor high-temperature oxidation resistance which largely
 restricts their applications. With the aim to study the effect of Fe
 addition on the Nb-Si system and to design appropriate alloy composition,
 the Fe-Nb-Si ternary system has been thermodynamically investigated using
 the CALPHAD (CALculation of PHAse Diagrams) approach aided with the
 formation enthalpies for ternary compounds at 0 K computed via ab initio
 calculations. A self-consistent thermodynamic description of the Fe-Nb-Si
 system was obtained in this work. Key isothermal sections and liquidus
 projection were presented, and the calculation results show a good agreement
 with available experimental data.


Introduction
Nb-Si based alloys attract great attention for their high melting point, low density and excellent high-temperature mechanical properties, which makes them candidates for the next generation of turbine airfoil materials [1][2][3][4][5][6]. However, room temperature brittleness and poor high-temperature oxidation resistance restrict their applications. It is reported that the addition of Fe element could improve the high-temperature oxidation resistance of the Nb-Si alloys [7]. On the other side, Fe-Si based alloys (silicon steels) are excellent soft magnetic materials and have been widely applied in motors and transformers [8]. With the increase of market demand for silicon steels, further improvement to achieve a higher permeability and lower power losses is required [9]. Alloying with carbonitride formers, such as Nb and Ti [10][11][12][13][14], has been recognized as a crucial mechanism to obtain enhanced properties for silicon steels. In other words, the Fe-Nb-Si system is also an important subsystem for the silicon steels.
Thermodynamic calculation in the frameworks of the CALPHAD (CALculation of PHAse Diagrams) approach allows to spare time and efforts required to design new materials [15][16][17]. The Fe-Nb-Si system has been experimentally investigated for several times [18][19][20][21][22][23][24][25][26][27][28][29][30]. The obtained experimental data are used as the basis of thermodynamic calculations. Ab initio calculations are used to obtain the formation enthalpies of the ternary compounds in this system in order to provide reliable endmembers during optimization. The purpose of this study is to obtain a set of self-consistent parameters to describe the Fe-Nb-Si system and provide theoretical basis of thermodynamics for alloy design.

Literature review 2.1 The Fe-Nb system
There are two intermetallic phases εNbFe2, μFeNb and four solid solution phases (δFe), (γFe), (αFe) and (Nb) in the Fe-Nb system, as listed in Table 1 [31]. Huang [32] first assessed the Fe-Nb system where the ε phase was treated as stoichiometric phase and the homogeneity range of the μ phase was limited with a congruent melting, which was inconsistent with the later experimental data. In later studies, many researchers reassessed this system by using different sublattice models for the ε or μ phase to fit the available experimental data [33][34][35][36].
Liu et al. [37] optimized the Fe-Nb system where ab initio calculations were used to obtain the formation enthalpies of the ε and μ phases at 0 K. Subsequently, Khvan et al. [38,39] re-assessed this system twice in order to describe the solubility of Nb in the Fcc phase reasonably. In the present work, the parameters from Khvan et al. [39] were adopted and the calculated phase diagram is shown in Fig. 1a.

The Fe-Si system
There are five intermetallic phases Fe2Si, Fe5Si3, FeSi, FeSi2, Fe3Si7 and three solid solution phases (αFe), (γFe) and (Si) in the Fe-Si system, as listed in Table 1 [40]. The Fe-Si phase diagram is one of the most complicated binary phase diagrams because the Fe rich part of this system involves the low temperature magnetic transition, A2/B2 transition and B2/D03 transition. In the early report, Lacaze and Sundman [41] assessed the Fe-Si system including the A2/B2 ordering reaction of the Bcc phase, but the B2/D03 transition was neglected. Subsequently, Miettinen [42] proposed new values for the interaction parameter of the liquid phase for a better modeling of the Fe-Si-C system based on the work by Lacaze and Sundman [41]. Ohnuma et al. [43] experimentally investigated the Fe-Si system and carried out a new assessment.
Recently, Cui and Jung [44] re-optimized the Fe-Si system and obtained two sets of model parameters with MQM and BW, where both the A2/B2 transition and B2/D03 transition were considered. Since the evaluation of Lacaze and Sundman [41] shows good agreements with experimental data and has been widely accepted, the thermodynamic assessment of the system by Lacaze and Sundman [41] was adopted, and the calculated Fe-Si phase diagram is shown in Fig. 1b.

The Nb-Si system
There are four intermetallic phases Nb3Si, αNb5Si3, βNb5Si3, NbSi2 and two solid solution phases (Nb) and (Si) in the Nb-Si system, as listed in Table 1 [45]. The thermodynamic description of the Nb-Si system has been established by many researchers [46][47][48][49][50][51][52][53][54][55][56]. In the early reports, all the intermetallic compounds of the Nb-Si system were regarded as stoichiometric phases due to the negligible solubility ranges [45,46,57]. Fernandes et al. [47] first calculated this system considering the solubility of βNb5Si3. Then, David et al. [51] re-modeled αNb5Si3 and NbSi2 as non-stoichiometric phases. The related transformations βNb5Si3+Nb3Si↔αNb5Si3 and βNb5Si3↔αNb5Si3+NbSi2 were re-described which were widely accepted by later researchers. Based on the work of David et al. [51], Geng et al. [52] modified the Nb-Si system in order to match the experiments and thermodynamic data better. In the present work, the parameters reported by Geng et al. [52] are adopted, and the calculated Nb-Si phase diagram is shown in Fig. 1c.

The Fe-Nb-Si system
The Fe-Nb-Si ternary system has not been thermodynamically assessed yet.
Denham [26] found that the extent of the Laves field in the Fe-Nb-Si system was similar at 1273K and 1573K but smaller than previously reported. Singh and Gupta [27] reported a partial isothermal section at 1373K based on the experimental results.
However, the homogeneity ranges of Laves phase and μ phase were fairly large, and both extended along constant niobium lines. Raghavan and Ghosh [28] gave a tentative isothermal section at 1423K, where the six ternary compounds were considered and the extent of the ternary Laves phase region was consistent with that given by Denham and Singh [26]. Combining the results of Goldschmidt [18] and Haour [30], and the binary data [31,40,45], a tentative liquidus projection was also constructed by Raghavan and Ghosh [28].

Solution phases
The molar Gibbs free energies for (αFe), (γFe), (δFe), (Nb) and (Si), modeled as substitutional solutions, are described by the following equation: Where 0 i G  is the molar Gibbs free energy of pure element i in the phase φ, xi is the mole fraction of the components i, the term ex G  is the excess free energy, which is expressed by the Redlich-Kister polynomial [60] as: ,, 0 () ,, is the ternary interaction parameter and its coefficients a and b are to be evaluated in the present work.

Ternary compounds
The NbFeSi2, Nb2FeSi2, Nb4Fe3Si5, Nb4FeSi, Nb4Fe4Si7 and NbFeSi phases are described as stoichiometric compounds. The molar Gibbs free energies of these phases can be expressed as:

Results and discussion
Since no experimental data for the ternary compounds were available in the Fe-Nb-Si system, ab initio calculations were employed. Formation enthalpies computed via ab initio calculations (Table 2) in this work were treated as end-members during thermodynamic optimization. On the basis of experimental isothermal sections at 1473, 1373 and 1273K, the Fe-Nb-Si system was thermodynamically assessed in the present work and the thermodynamic models and optimized parameters are listed in Table 3.
The optimization was conducted using the PARROT module of Thermo-Calc [61]. With these parameters, the isothermal sections at different temperatures and the liquidus projection are calculated and presented as follows.
In the present work, all the thermodynamic parameters were optimized using the PARROT module of Thermo-Calc [61]. The step-by-step optimization method was used [62,63]. Firstly, as for the binary Fe-Zr, Nb-Zr, and Si-Zr systems, the thermodynamic model of εNbFe2, μFeNb, and αNb5Si3 phases was adjusted to describe their ternary solubilities. Secondly, all the ternary stoichiometric phases, NbFeSi2, Nb2FeSi2, Nb4Fe3Si5, Nb4FeSi, Nb4Fe4Si7, and NbFeSi phases, were added into the isothermal sections at 1423 K and 1473 K. Their thermodynamic parameters were optimized according to the experimental determined phase relationships. Thirdly, the experimental data on phase boundaries are used to optimize the thermodynamic parameters of the εNbFe2, μFeNb, and αNb5Si3 phases. A set of thermodynamic parameters for the Fe-Nb-Si system was finally obtained, as listed in Table 3.

Formation enthalpies computed via the ab initio calculations
Ab initio calculations based on the density functional theory (DFT) can provide insight into the characteristics of the thermodynamic and structural properties of phases, using only the atomic numbers and crystal structure information as input. The generalized gradient approximation (GGA) based on the Perdew-Burke-Ernzerhof (PBE) approach [64] for the exchange-correlation potential was employed. The valance electrons were explicitly treated by projector augmented plane-wave (PAW) potentials [65]. As implemented in the Vienna ab initio simulation package (VASP) [66]. A planewave cutoff energy of 550 eV and an energy convergence criterion of 10 -5 eV for electronic structure self-consistency were used in the calculation. The integration in the Brillouin zone was done on the special k-points determined from the Monkhorst-Pack scheme [67]. The system was fully relaxed, including the unit cell sizes and the ionic coordinates, to find the stable state.
The formation enthalpy ΔHf at 0 K for the compounds in the ternary system can be calculated by the following equation:   [63] are also listed in Table 2. Fig.2 shows the formation enthalpy of ternary compounds in the Fe-Nb-Si system calculated from ab inito and CALPHAD methods. The good agreement shows that the ab initio calculations generated data are reliable and can supplement the lack of experimental data. compared with the experimental data reported by Wang [58] (Fig. 3b is a tentative isothermal section constructed by Raghavan [28]). Three binary phases (εNbFe2, μFeNb, αNb5Si3) are calculated to have obvious ternary solid solubilities. Six ternary compounds (τ1, τ2, τ3, τ4, τ5 and τ6) are obtained at these temperatures.

Isothermal sections
The main difference between the calculated phase diagrams and experimental measurements is the single-phase region of μFeNb. According to [58], the solubility of Si in the μFeNb phase decreases from 14.7 at.% to 10.0 at.% with temperature decreasing from 1473K to 1273K. But in the present work, the morphology of μFeNb phase region mainly stays the same as temperature range from 1473K to 1273K, which conflicts with the experimental data. It may be attributed to the thermodynamic parameter of μFeNb phase. Generally, the calculated results have good agreements with the experimental data.

Liquidus projection and invariant reaction scheme
The calculated liquidus projection with nineteen primary phases regions is shown in Fig. 7. There are twenty-one eutectic grooves, eighteen peritectic ridges and eleven saddle points. Sixteen transitional invariant reactions and seven eutectic reactions are involved in the Fe-Nb-Si ternary system. The reaction scheme for the liquidus projection of the Fe-Nb-Si system is given in Fig.8. Therefore, the liquidus projection together with the reaction scheme of the whole ternary system has been constructed for the first time. Although the data are tentative and further experiments are required to validate the prediction, they provide key information for alloy processing, e.g., composition design and selection of smelting temperature.

Conclusion
Based on reliable experimental data, the Fe-Nb-Si system has been thermodynamically investigated via the CALPHAD approach for the first time. The lattice constants and formation enthalpies of ternary compounds in this system are calculated by ab initio calculations, which serve as the supplement to the lack of experimental value. A set of self-consistent thermodynamic parameters was obtained.
The calculated isothermal sections show reasonable agreements with the experimental data at 1473, 1373 and 1273K, respectively. The calculated liquidus projection and invariant reaction scheme can provide useful guidance for alloy processing.