Experimental and computed phase diagrams of the Fe–Re system

The calculations reported in this work are based on the DFT. They have been performed both for ordered intermetallics and solid solution phases. The total energies were calculated using the Projector Augmented Wave method [24], implemented in the Vienna Ab initio Simulation Package (VASP) [25, 26]. The exchange-correlation energy of electrons was described in the generalized gradient approximation (GGA) using the functional parametrization of Perdew–Burke–Ernzerhof [27]. After necessary tests to control the stability of the energy differences between phases, the energy cut-off for the PAWs was set to 400 meV. These convergence tests for the plane-wave cut-off and the number of the k-points were essential to assure reliable total energy differences [28]. We have intentionally kept the space group fixed whilst relaxing each particular system. As a consequence, a spontaneous lowering of the symmetry was not possible. The total energies of every compound were also minimized in order to equilibrate their volume, assuming a zero pressure environment. Calculations with and without spin polarization (SP) have been considered in the collinear magnetic description.

For the intermetallics, the four phases D8_b, A12, A13 and P were considered. These phases were selected as they have been observed experimentally in this system. In particular, the calculation done on the P phase was continued from the experimental study of the present work. This phase would never have been calculated if it had not been evidenced experimentally, particularly because it has never been reported as a binary phase. This is an example of the limitations of systematic high-throughput theoretical approaches such as that adopted by Curtarolo [29] which are always performed within a given set of crystal structures that may never be exhaustive. The crystallographic structures studied are summarized in table 1. The distribution of the two elements (Re,Fe) on the s different sites generates N^φ = 2^s ordered configurations which have to be considered for each phase φ. For the complex P phase, because of the huge number of configurations, a simplification has been adopted. All sites of similar coordination number (CN): CN12, CN14 and CN15 + 16 have been merged resulting in only three sublattices of multiplicity 24, 20 and 12 respectively (see table 2) and thus 2³ = 8 compounds to compute according to the model (A,F,H,I,J)₂₄(B,C,D,G)₂₀(E,K,L)₁₂.

Table 1. Details of the structures considered in the present study: Strukturbericht, Prototype, Space group, usual name and N^φ, the number of configurations considered in the DFT calculations.

Strukturbericht	Prototype	Usual name	Space group	Nφ
A1	Cu	γ-Fe, fcc	$Fm\bar{3}m$	7
A2	W	α-Fe, δ-Fe, bcc	$Im\bar{3}m$	7
A3	Mg	-Re, hcp	P63/mmc	7
D8b	CrFe	σ	P42/mnm	25 = 32
A12	α-Mn	χ	$I\bar{4}3m$	24 = 16
A13	β-Mn		P4132	22 = 4
—	Cr18Mo42Ni40	P	Pnma	23 = 8

A mesh of 80, 68, 220 and 120 special k-points for σ, χ, β-Mn and P, respectively, were taken in the irreducible wedge of the Brillouin zone for the total energy calculation [30, 31]. Hence, the cell-shape relaxation was considered wherever possible whereas the internal coordinates were optimized using the conjugate gradient algorithm, a highly recommended scheme to relax the ions to their instantaneous groundstate, especially where ionic relaxation is problematic. The stable element reference (SER) states of Fe and Re are their respective ground states: Fe ferromagnetic bcc (α-Fe) and Re nonmagnetic hcp. The enthalpy of formation $\Delta H_{f}^{\varphi}$ for each configuration in φ phase has been calculated from the total energies, as:

$$\begin{eqnarray} &&\fl \Delta H^{\varphi}_{f}({\rm Fe}_{1-x}{\rm Re}_x) = E^{\varphi}_{0}({\rm Fe}_{1-x} {\rm Re}_x) \nonumber\\ &&- (1-x)E^{{\rm ferro}-bcc}_0({\rm Fe}) - xE^{hcp}_0({\rm Re}) \end{eqnarray} \tag{ 1 }$$

where $E^{\varphi}_0$ is the total energy of the DFT calculation at 0 K and x, the Re mole fraction.

In this work, the SQS approach was adopted to simulate the fcc, bcc and hcp solution phases. It consists of making a choice on how to match random atomic configurations with a periodic structure. Normally, the optimizing technique, in which the volume, shape and atomic positions are driven by energy and forces/stress minimization, is restricted to the condition where the structure keeps its original symmetry. Hence, we have relaxed the relevant phases, keeping the original symmetries allowing only volume and ionic relaxation for the cubic cell. An additional step has been performed allowing cell shape relaxation for the hexagonal structure. The SQS structures of fcc, bcc and hcp were taken from [32–34], without any further adaptation. A mesh of 172, 144 and 250 special k-points for fcc, bcc and hcp, respectively, were taken in the irreducible wedge of the Brillouin zone for the total energy calculation. In addition to compounds calculated by the SQS methodology for 25, 50 and 75% compositions, we have also simulated dilute solutions using a 16 atom supercell containing 15 similar and 1 dissimilar atom, corresponding to 6.25% and 93.75% of Re. Discrepancies between the first-principles and the CALPHAD values [35] for the lattice stabilities have often been noticed and have been discussed elsewhere [36, 37]. In order to address this issue, mixing parameters are used rather than the SQS-formation energies. The mixing enthalpy $\Delta H_{\rm mix}^{\varphi}$ of each composition has been calculated from the relative total energies as:

$$\begin{eqnarray} &&\fl\Delta H_{\rm mix}^{\varphi}({\rm Fe}_{1-x}{\rm Re}_x) = E^{\varphi}_{0}({\rm Fe}_{1-x}{\rm Re}_x) \nonumber\\ &&- (1-x)E^{\varphi}_0({\rm Fe}) - xE^{\varphi}_0({\rm Re}) \end{eqnarray} \tag{ 2 }$$

where $E^{\varphi}_0({\rm Fe})$ and $E^{\varphi}_0({\rm Re})$ are the energies of the pure elements in the φ phase under consideration.

To automatically generate first-principles input files of all the ordered configurations of the given crystal structures generated, by assigning each element to each crystal site, the script-tool 'ZenGen' was used [38].

In order to use the DFT results to calculate a phase diagram, continuous thermodynamic models have to be introduced. The Gibbs energy of the solution phases (fcc, bcc, hcp and liquid) has been expressed through a regular solution model as follows

$$\begin{eqnarray} &&\fl {G^{\varphi}(x_i)-\sum_i H_i^{\rm SER}} = \sum_{i} x_i \left( G^\varphi_{i} -H^{\rm SER}_{i} \right) \nonumber\\&&+ RT \sum_i x_i\ln x_i + G^{\rm mix} + G^{\rm mag}. \end{eqnarray} \tag{ 3 }$$

In order to describe the evolution of the relative stability of the different phases, the lattice stabilities of the scientific group thermodata Europe (SGTE) [35] have been used for $\left( G^\varphi_{i} -H^{\rm SER}_{i} \right)$. The mixing energy of the solution has been expressed by a Redlich–Kister expression $G^{\rm mix}=\sum_i\sum_{j>i}x_ix_j L^{\varphi,\nu}_{ij}$, where the SQS results have been used to estimate $L^{\varphi,\nu}_{ij}=\sum_\nu (x_i-x_j)^\nu\,{^\nu}L^\varphi_{ij}$ parameters. The magnetic contribution, G^mag, was taken from the pure SGTE description without binary excess contribution. The separation of the magnetic contribution implies that the expression of the lattice stabilities, $\left( G^\varphi_{i} -H^{\rm SER}_{i} \right)$, describes the paramagnetic state of the elements yielding some difficulties when trying to extract FP information to use in this formalism as discussed by Körmann et al [39].

The Gibbs energy of the intermetallic phases for a mole of formula unit has been expressed in the modified-CEF [40] i.e. as follows:

$$\begin{eqnarray} &&\fl {G^{\varphi} - \sum_{i={{\rm Fe},Re}} x_i H^{\rm SER}_{i} = \sum_s a^s {G^{CI\varphi}(x_i)} + {G^{CD\varphi}(y^{s}_i)}} \end{eqnarray} \tag{ 4 }$$

with

$$\begin{eqnarray} &&\fl {G^{CI\varphi}(x_i)} = \sum_{i} x_i \left(G^{CI\varphi}_i-H^{\rm SER}_{i}\right) +\sum_{ij}x_ix_jL_{i,j}^{\varphi} + G^{\rm mag}\nonumber\\ \end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray} &&\fl {G^{CD\varphi}(y^{s}_i)} = \sum_{C} \prod_s y_i^s \Delta G^\varphi_{C} + RT \sum_s a^{s} \sum_i y_i^{s}\ln y_i^{s}. \end{eqnarray} \tag{ 6 }$$

This approach splits the Gibbs energy of the phase under consideration into two contributions: G^CIφ(x_i) and $G^{CD\varphi}(y^{s}_i)$. G^CIφ(x_i) is the molar configuration independent contribution; it is a function of the overall composition of the phase expressed by x_i the molar fraction of the element i. It is multiplied by the number of moles of the atom in a formula unit of the phase, ∑_sa^s, a^s being the multiplicity of the site s. $G^{CI\varphi}_i$ corresponds to the Gibbs energy of the pure element i in the φ structure. In the present work, they have been expressed using the DFT results as follows:

$$\begin{eqnarray} &&\fl G^{CI\varphi}_i-H^{\rm SER}_{i} = \left( E_0^\varphi(i) - E_0^{\rm SER}(i)\right) - T \Delta S_i^\varphi\nonumber\\ &&+ \left(G^{\rm SER}_i-H^{\rm SER}_{i}\right) +\Delta G^{\rm SER\, mag}_i \end{eqnarray} \tag{ 7 }$$

$\left( E_0^\varphi(i) - E_0^{\rm SER}(i)\right)$ comes directly from the DFT results while $\left(G^{\rm SER}_i-H^{\rm SER}_{i}\right)$ is the SGTE lattice stability for the pure element i in its SER state. $\Delta S_i^\varphi$ is the difference of vibrational entropy of i between the two structures φ and SER. The influence of this term will be discussed in the last part of this work. The last term, $\Delta G^{\rm SER\, mag}_i$, is introduced to take into account that the temperature dependence $\left(G^{\rm SER}_i-H^{\rm SER}_{i}\right)$ describes the paramagnetism of the SER state while, $E_0^{\rm SER}(i)$ is obtained for the ferromagnetic state. This term will only be used for the magnetic elements, i.e. only for Fe in the present case. In the present work, a constant value of −9.5 kJ.mol⁻¹ was used, allowing us to reproduce the FP results with the stable configuration enthalpy curves calculated at low temperature.

The first term of G^CIφ(x_i) thus corresponds to the mechanical mixture of pure elements in the φ structure. The second term $\sum_{ij}x_ix_jL_{i,j}^{\varphi}$ corresponds to an excess term from the interaction of the element i and j independent of the configuration. This term has not been used in the present work. G^mag is the magnetic contribution. It is expressed with the same relation as for the solid solutions in equation (3). Its influence will be discussed in the results section for the case of the σ phase.

$G^{CD\varphi}(y^{s}_i)$ is the configuration dependent contribution function of the site fractions. $y_i^{s}$ stands for the fraction of the element i in the site s. The first term of this contribution expresses the sum of the Gibbs energy of formation of all the compounds, C, defined by the model weighted by a site fraction product. When calculating the site fraction product for the compound C, the element i occupying the site s in C has to be considered for each site. $\Delta G^\varphi_{C}$ is the Gibbs energy of formation of the stoichiometric compound C referring to the pure elements in the phase φ. The second term corresponds to the ideal configurationnal mixing.

During this work, the Gibbs energy of formation of the stoichiometric compound C has been expressed using the results of DFT calculation for this configuration making the assumption that the difference of total energies at 0 K corresponds to the difference of the enthalpies.

$$\begin{equation} \Delta G^\varphi_C = \Delta H^\varphi_C - T\Delta S^\varphi_C \end{equation} \tag{ 8 }$$

$$\begin{equation} \Delta H^\varphi_C = E^\varphi_0(C) - \sum_s a^{s} E^\varphi_0(i) \end{equation} \tag{ 9 }$$

where the $\sum_s a^{s} E^\varphi_0(i)$ takes the element i present on the site s in the compound C under consideration. In the present work, $\Delta S^\varphi_C$ were not considered. The vibrational entropy of the stoichiometric compounds is thus the weighted mean of the value for the pure elements in φ structure.

As a first approximation, The Gibbs energy is calculated at a finite temperature using the Bragg–Williams Approximation (BWA). In this approximation, BWA considers that both interactions within sublattices and non-configurational entropies are negligible; i.e. only the mixing entropy is considered.

The proposed phase diagram in figure 3 is limited to the temperature range studied in the present work. The experimental study of this system is particularly difficult. This is due to the fact that some phases that cannot be quenched (fcc and hcp) transform completely or partly by a martensitic transformation into the bcc phase. Other phases (σ, P) have quite complex crystal structures making their identification from XRD difficult when present simultaneously. Moreover, their identification by EPMA is also difficult as their compositions are similar; P and β may both have the composition of σ. Also, rhenium is very much a refractory element. The determination of the melting equilibria is difficult. The reactions at low temperature, in particular in the range 900–1100 °C where the invariants involving σ, P and β are located, are sluggish and the equilibrium is difficult to establish. This means that we are still far from having completely established the phase relations in the whole composition and temperature ranges. However, we believe we have clearly established new facts that are important for understanding this system. These are summarized below:

• the existence of the P phase which has never been reported in Fe–Re and, as far as we know, it is the first time that this structure type is evident in a binary system;
• non existence of the χ phase as previously reported;
• the martensitic nature of the transformation from hcp to bcc;
• the homogeneity range of the σ phase at high temperature (1100–1500 °C) larger than previously reported and the nature of the equilibria with other phases, including the absence of χ and β-Mn in this temperature range as reported before;
• reliable site occupancies have been obtained in the complete homogeneity range of the σ phase which makes Fe–Re one of the systems in which the σ phase has been best characterized (see below).

The large x-ray scattering contrast between Re and Fe allows us to precisely determine the Re occupancy on the five different sites in the σ phase of samples annealed from 1000 to 1500 °C (figure 4) by using the Rietveld refinement. The results are presented in a table in the supplementary material. All the sites show a mixed occupation in the studied composition range; this is characteristic of a rather low degree of order. The different site occupancies are clearly separated except for those of 2a and 8i₂. Sites 4f and 8i₁ are preferentially occupied by Re and site 4f of the highest coordination number (CN15) has the highest Re content. Site 8j is close to being randomly occupied. Sites 2a and 8i₂ are preferentially occupied by Fe. This is basically attributed to an atomic size effect: the largest atom shows a preference for sites with a large coordination number [43].

Zoom In Zoom Out Reset image size

Our results are in excellent agreement with the recent paper of Cieslak et al [44], as shown in figure 4, with a very comparable scattering of occupancies in the two datasets; however, we strongly extended the studied composition range.

The SQS derived enthalpies of the fcc, bcc and hcp phases, calculated without (blue symbols) and with spin polarization (red symbols) are displayed in figure 5. For the fcc and bcc phases, the states considering the spin polarization are more stable than those without SP while there is no difference for the hcp phase. The fcc and bcc phases are thus stabilized by magnetic ordering while the hcp phase is not. Otherwise, the fcc and hcp phases present similar behaviours. The mixing energies in these phases are essentially positive indicating repulsive interactions; they become slightly negative only for composition close to pure Fe.

Zoom In Zoom Out Reset image size

Equation (3) was considered to model the mixing enthalpy in the solution phases: fcc, bcc and hcp, based on the SQS results obtained with spin polarization, after full relaxation. However, because of the dynamical instability of Re in bcc [45, 46], only volume relaxation results have been considered for the bcc phase. A single regular ⁰L interaction term has been chosen for the bcc and hcp phases equal to ⁰L^bcc = −7 and ⁰L^hcp = +7 kJ.mol⁻¹. Because of the stronger asymmetrical shape for the fcc structure, an additional term has been considered yielding ⁰L^fcc = +3 and ¹L^fcc = −10 kJ.mol⁻¹. The dashed lines in figure 5 represent the resulting enthalpy curves for the different solution phases. The choice made may be considered rather subjective as none of the SQS sets follow a very regular behaviour. However, it is expected that they indicate an order of magnitude allowing us to build a reasonable thermodynamic database.

The formation enthalpies of all the configurations calculated for the four phases, σ, χ, β-Mn and P phases are shown in figure 6. For each phase, a solid line is drawn between the configuration of lowest energy; it corresponds to the so-called convex hull. Two sets of results corresponding to different approximations are shown: with and without spin polarization. The spin polarized calculations are expected to describe the 0 K and low temperature energetics more precisely. The results obtained without spin polarization help us to understand more clearly what happens at higher temperatures where the magnetic ordering vanishes. All the phases have significantly lower enthalpies in the spin polarized calculation. This stabilization is weaker for configurations with higher Re content.

Zoom In Zoom Out Reset image size

Both approximations predict the stability of only one intermetallic phase at 0 K: the β-Mn phase with a negative enthalpy of formation of −1.8 kJ.mol⁻¹ in the spin polarized and −1.72 kJ.mol⁻¹ in the nonmagnetic calculations. The stability of this phase, at low temperature, is in agreement with the experimental observations at low temperatures by Iyutina [19] and in the present work. Moreover, the minimum of the β-Mn phase convex hull corresponds to the configuration with Fe in CN12 and Re in CN14, at x_Re = 0.6, very close to the composition reported experimentally for this phase. The stability of the P phase at low temperature can be understood from the fact that its convex hull is the lowest in energy after the β-Mn phase. However, according to the present DFT results, this stability should not extend down to 0 K. Moreover, the closeness of the different convex hulls explains that the structures not stable at 0 K may appear at a high temperature.

In the following, we analyze the energetic dependence of the σ phase configurations as a function of the (Fe,Re) occupations of its five sublattices. The configuration FeReReFeRe⁴, lying on the convex hull has the lowest formation enthalpy, as predicted by spin polarized calculations. It is however very close to the ReReReFeRe configuration that is the most stable in the approximation without SP. The most stable σ phase groundstate configurations correspond to those with the high CN sites occupied by Re, while icosahedral sites are occupied by Fe. Among these, the most stable configuration corresponds to the composition Re₂Fe (i.e. FeReReFeRe). Even for this configuration the calculated enthalpy value is positive indicating that the σ phase is not stable at 0 K relative to Fe and Re in their respective ground states, in agreement with the experimental observation.

Within the BWA at 1500 K, the σ site occupancies have been computed using the two sets of heat of formation, with and without spin polarization. Figure 4 shows a good agreement between the experimental and calculated data, confirming the reliability of both the calculation and experimental measurements. The agreement is slightly better with the non-spin polarized calculations. This is understood by the fact that at the experimental temperature, the σ phase is not magnetic. This satisfactory explanation raises the question of the best DFT dataset to use within the continuous thermodynamic models. The results using spin polarization (SP) approximation would describe the low temperature region better and fails to reproduce the experimental results at finite and high temperatures. Linking both regimes in a continuous way constitutes one of the challenges of the use of DFT results in the CALPHAD approach.

During the calculations, the relaxed cell parameters of the different phases are also obtained and can be found in the tables in the supplementary material. The composition dependence of the cell parameters manifests a quasi-linear behaviour. They increase upon replacing Fe with Re. For the σ phase, the c/a ratio fluctuates around 0.52 with a deviation at x_Re = 0.4. In fact, site 8j of CN14 is characterized by a very short inter-atomic distance along the z-axis, which renders it different from its analogue, 8i₁ of CN14. This explains the different occupancy observed experimentally between these two sites. Another interesting feature is the negative deviation of the atomic volume from the Vegard's law. This behaviour is attributed to binding and short bond lengths which occur when Re atoms preferentially occupy high CN sites in the σ phase structure.

Within the GGA and for non-spin polarized (nonmagnetic) calculations, Korzhavyi et al [47] and Havránková et al [48] predicted the lattice stability of σ Fe to be 38.2 and 40.6 kJ.mol⁻¹, respectively, which is at variance with our prediction, 25.65 kJ.mol⁻¹. We attribute this striking disagreement to the internal coordinates and the tetragonal ratio which were held fixed at the experimental values (taken from σ-CrFe stable composition [49]), throughout the calculations of [47, 48]. In the same study Korzhavyi et al [47] demonstrated the importance of a spin polarized treatment of the paramagnetic state of iron-based σ phases. They modelled the paramagnetic state employing the Disordered Local Moment (DLM) [50] approach based on the Coherent Potential Approximation (CPA) [51]. The electronic structure calculations were based on DFT and the Korringa–Kohn–Rostoker (KKR) method [52, 53], in conjunction with the Multipole-corrected Atomic Sphere Approximation (ASA + M) [54]. After applying this procedure, the σ Fe lattice stability reduced to 21 kJ.mol⁻¹. This points out the importance of calculating the enthalpies of iron-based σ phases in a spin polarized paramagnetic scheme.

The average magnetic moment of every configuration obtained by DFT is shown in figure 7(a). The dotted red line links the most stable ones and the full red line links the configuration on the convex hull.

Zoom In Zoom Out Reset image size

To estimate the magnetic entropy contribution to the Gibbs energy at high temperatures where σ Fe–Re is experimentally observed, we assumed the crude hypothesis that Fe and Re (induced)-magnetic moments, calculated at 0 K in a spin polarized ferrimagnetic solution, $M_i^{(s)}$, conserve their values at high temperatures where σ is stable in the paramagnetic state. The magnetic entropy is approximated, at high temperatures [39], as:

$$\begin{equation} S_{\rm mag}=k_{\rm B}\!\sum_s\sum_ia^{(s)}y_i^{(s)}\ln[M_i^{(s)}+1]. \end{equation} \tag{ 10 }$$

The DFT magnetic entropy for each configuration, calculated using equation (10), is shown in figure 7(b) as red rhombuses. The red dotted line connects the magnetic entropies that belong to the most stable configurations and the solid red line connects those belonging to the convex hull, while the blue dashed line corresponds to the magnetic entropy determined by the CALPHAD approach. It has the most pronounced values in the Fe-rich side and decreases in a quasilinear behaviour upon replacing Fe by Re atoms in the different sublattices.

In the rest of this section, we show how we modelled the enthalpy in the finite temperature limit, based on CALPHAD methodology and starting from the DFT-inputs, namely the magnetic moments. This modelling is necessary to simulate the finite-temperature section of the phase diagram. The blue dashed line represented in figure 7(a) corresponds to the magnetic moment calculated using the magnetic contribution introduced in equation (5). It is expressed by the classical magnetic contribution to the Gibbs energy used in the CALPHAD approach introduced by Hillert and Jarl's expression [55] based on Inden work [56]. In this contribution, the averaged magnetic moment is a function of the composition of the phase.

$$\begin{equation} \mu^\varphi = \sum_i x_i\, \mu^\varphi_i +\sum_{i,j} x_i \,x_j\ \mu^\varphi_{i,j}. \end{equation} \tag{ 11 }$$

The dashed blue curve in figure 7(a) is obtained using $\mu^\varphi_{\rm Fe}=2.5\mu_{\rm B}$, $\mu^\varphi_{\rm Re}=0\mu_{\rm B}$ and $\mu^\varphi_{\rm Fe,Re}=-2.5\mu_{\rm B}$. The magnetic energy contribution is determined by the fit to the magnetic moment. Using this contribution together with the 0 K non-spin polarized DFT enthalpies allows us to model the formation enthalpy in the finite temperature limit. This is illustrated in the blue dashed curve of figure 7(c), where the enthalpies are less positive than the DFT non-spin polarized ones. The red curve is calculated using the DFT results in the spin polarization approximation. The full blue curve corresponds to the case using the DFT results without spin polarization, without CALPHAD magnetic contribution. The dashed blue curve is calculated adding to this one the magnetic contribution. The shift from the full blue line is determined by the fit to the magnetic moment. It comes quite close to the red curve calculated with SP, in particular in the range where the phase is the most stable. A good agreement between the magnetic entropies determined by the DFT (solid red line) from equation (10) and the one determined by CALPHAD (blue dashed line), is shown in figure 7(b).