Optimization method 83 

The optimization of the phase diagram was done using the PARROT module implemented in the Thermo-Calc software [144]. PARROT is based on the least-squares method and allows fitting parameters in the mathematical model chosen for the description of the phases so that they represent the available experimental data.

Each phase of the Fe-W system is modeled separately. For the pure elements Fe and W, we used the unary database from SGTE according to Dinsdale [81] in order to stay in consistency with other work and existing databases. The two intermetallic compounds (i.e. C14 Laves and - phases) are modeled using the sublattice models as previously described.

For the modeling of Fe-W, two different optimizations of the phase diagram are proposed; one including the DFT calculations for the description of the energy of the end-members, the second including the use of the phonon results for a description of the temperature dependence of the end- members. For both phase diagrams, the same sublattice models were used.

4.1 BCC, Liquid, FCC

For these phases, we used the values of the excess energy given by Gustafson [48] as starting values for our optimization. We started the assessment procedure with the liquid. In addition to the phase boundary data [44], there are thermodynamic data given on the Fe-rich [148–150] side of the system.

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We used these data to obtain the excess energy of the liquid phase using the L0 parameter with

temperature dependence and the L1 and L2 parameters without temperature dependence. The optimized

liquid phase was the basis to optimize the BCC phase which with the limited experimental knowledge cannot easily be optimized. Hence, the parameters for BCC, were optimized together with those for the liquid. The different data points from Sinha and Hume-Rothery [44] on the Fe-rich side describing the liquid + BCC equilibrium were used. The local minimum given at 1802 K at 4.4 at. % W [44] forming the congruent melting point was taken with a higher weight in order to correctly fix this part of the phase diagram. Additionally, we also used a higher weight on the peritectic reactions P1 and P2 (see Table 20). While data are well known on the Fe-rich side on the phase diagram, there are less data on the W-side due the experimental difficulties caused by the high temperature and slow diffusivity. In order to model this side of the phase diagram, we used the data from Sinha and Hume-Rothery [44] for the temperature and the composition of the liquid (i.e. 1910 K and 20.6 at. % W). However, the composition of W-BCC was taken from Ichise et al. [143] (i.e. 97.4 at. %), as these data are in agreement with the extrapolation of the data from Antoni-Zdziobek et al. [47] to higher temperature. The thermodynamic data were used with lower weight than the phase boundaries. The data of Sudavstova [148] were only taken with a weight of 20 % of the data from the phase boundary data. A higher weight on these data resulted in a too stable liquid phase which would be inconsistent with the experimental phase diagram data. As previously mentioned in the literature part, the data from [148] are not well documented and can only be used with care. Lowering the weight on the thermodynamic data allowed to converge the optimization of the excess Gibbs energy for the BCC and Liquid and to provide a consistent description in good agreement with experimental data points from the literature [44,47].

At the last step of the optimization of the liquid and BCC phases, we introduced phase boundary data at lower temperature from Takayama et al. [151] and Antoni-Zdziobek et al. [47], in order to check their reliability with the actual description of the BCC phases. As these results immediately were in good agreement, full convergence and agreement between the experiments and the calculated values were quickly reached.

4.2 FCC

The excess Gibbs energy of the FCC-phase was optimized separately from the BCC and Liquid. This phase forms a closed -loop within the BCC-phase and thus highly depends on its correct description. Various experimental data of the phase boundaries [45,152,153] were used for the determination of the excess Gibbs energy. For the optimization of the FCC-phase, only the L0 parameter with linear

temperature dependence was used. The use of the L1 parameter was tested but resulted in a

stabilization of the phase around 50 at. % W so that it was finally rejected.

For the Liquid, BCC and FCC phases, the Gibbs energy curves were checked for several temperatures in order to verify the stability of the phases and their correct behavior.

4.3 The intermetallic phases: Laves- and µ-phases

As already mentioned in the previous section, two different approaches were taken for the description of these two phases. They will be described in the following section.

a) Using 0K enthalpies of formation from DFT calculations only

The calculated values of the enthalpy of formation from DFT simulations were used in the description of the different G-parameters of the end-members of the intermetallic phases. The data describing “pure” end-members, i.e. those containing one element only, were taken from the literature [38,137]. For -Fe2Fe (i.e. C14 Laves phase only containing Fe), the value from Liu et al. [38] was used as it

had already been used in several other assessments containing an Fe-based Laves phase, whereas the data from Sluiter [137] were taken for the pure λ-W2W and µ-W13. As shown in Table 24 and Table 25, these values are in good agreement with the calculated end-member energies from the present work.

In the Calphad formalism, the enthalpy of formation in the Gibbs energy descriptions of the end- members typically refers to the pure elements (i.e. Fe, W) as given by the GHSERXX functions from Dinsdale [81]. These functions do not include magnetic contributions, which are added separately (for Fe in this case). In the DFT calculations magnetism is taken into account. If the enthalpies of

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formation for the end-members from the DFT calculations are used directly as enthalpy parameters (with GHSERFE as reference) the resulting discrepancy would cause the non-stability of the Laves and µ-phases at low temperature in the calculated phase diagram, while DFT calculations (present work) and experiments [154] suggest the opposite. The enthalpies of formation of the end-members calculated by DFT in the present work were therefore treated as experimental values and the corresponding parameters were optimized referring to the pure element GHSERXX functions as described by Dinsdale [81].

b) Finite temperature DFT

The results from the phonon calculations (harmonic and quasi-harmonic approximations) were used to obtain a temperature dependent description of the end member Gibbs energies. They were treated as mathematical functions following the polynomial conventionally used in the Calphad approach as shown in Eq. 57:

Eq. 57

This polynomial can in principle be extended to a higher power series as a function of temperature, but becomes more and more complex without contributing to the improvement of the fit to experimental or calculated values. It has to be noted that in any case this polynomial cannot be extended to low temperatures (~100K). The heat capacity is derived from this function and is given in Eq. 58:

2 6 2 Eq. 58

Close to the melting temperature of the compound, the experimental heat capacity curves often show a nonlinear tendency. The D3 parameter allows to take into account this effect. In the present phonon

results, this non-linear behavior of the heat capacity was not found so that the D3 parameter was not

Our phonon heat capacity data were fitted according to Eq. 58 from 100 K to their high temperature limits using a nonlinear least squares fit. The B parameter of the Gibbs energy function (Eq. 57) was then obtained by fitting the previously obtained parameters (C and Dn) to the entropy curve of the

phonon calculations. The resulting parameters (Table 26 and Table 27) were used as starting values for the thermodynamic assessment of the Fe-W system. For the hypothetical end-members, the parameterized Gibbs energy functions were used directly without modification in the assessment. The parameters of the λ-Fe2W and µ-Fe7W6 end-members were optimized as they were found to highly

influence the stability of the phases. Nevertheless, no big deviations resulted from the optimization. Additionally, we included a D3 parameter for the description of the Gibbs energy function of µ-Fe7W6

as the minimization over the sublattices showed a nonlinear tendency at high temperature (Figure 40). The obtained parameters are given in Table 27.