Computing solubility products using ab initio methods

It should be mentioned that the magnetic disordering that occurs in ferrite in the temperature range of interest here strongly affects the free energies [1, 5, 13]. When considering the effect of magnetic disordering on the excess mixing properties, such as \(\Updelta G^{\alpha}_{\rm emb}[{\rm Nb}]\) and \(\Updelta G^{\alpha}_{\rm emb}[{\rm C}]\), the effect can be very significant also. This is illustrated by using the “proportional conversion method” of Nishizawa et al. [34] which has been applied to Fe–C and other systems [9, 37]. The “proportional conversion method” describes the effect of alloying as,where c _Q is the atomic fraction of the atomic species “Q” (Ni and Co species are not counted in the sum), T _C ^* is the Curie temperature of the Fe-rich bcc alloy, T _C ⁽⁰⁾ is the same for pure bcc Fe (1043 K), and T ^* = TT _C ⁽⁰⁾ / T _C ^* (this is Eq. 49 in Ref. [34]). For many sufficiently dilute alloys the Curie temperature T _C ^* is well described by a linear function of the compositionwhere \(\Updelta T_{\rm C,Q}\) is the negative slope of T _C ^* associated with species “Q”. Assuming that the alloying elements in their reference states are not magnetic, the magnetic free energy then iswhere \(r_{\rm Q} = \Updelta T_{\rm C,Q}/{T^{(0)}_{\rm C}}\). To relate G _mag ^* (T) to the mixing free energy we need to remove the linear part (1 − ∑_Q c _Q) G _mag ⁽⁰⁾ (T) which leaves,As r _Q is the only solute-specific parameter, the magnetic effect on the embedding free energy of species Q,In Fig. 1 G _mag ⁽⁰⁾ (T) is visualized using the pure bcc Fe parameters from Ref. [5]. For elements that strongly reduce T_C, so that r _Q is of order unity, it implies that the magnetic contribution to the embedding free energy is of the same order as the magnetic free energy of pure bcc Fe.

At finite temperature the Fermi–Dirac distribution broadens around the Fermi-level which signifies that some electronic states above the Fermi-level are populated and some states below it are vacated. Obviously this has consequences for the energy and entropy of the system. For insulators and semiconductors the presence of a band gap at the Fermi-level makes electronic excitation effects negligible, but for metallic materials the Sommerfeld approximation [2] gives a good representation of the free energy contributionwhere \(n(\epsilon_{\rm F})\) is the density of electronic states at the Fermi-level, see Table 1. The contribution to the embedding free energy is computed by approximating the composition derivative with a finite differencewhere Q represents either Nb or C, and m, n, and p are chosen sufficiently large. In this case we selected for Nb; m = 53, n = 54, p = 54, and for C; m = 54, n 54, p = 64. The vibrational contribution to the formation free energy of NbC is given by a similar equation where for NbC a Nb₃₂C₃₂ supercell was used.

Vibrational effects are generally considered to be the origin of the close-packed to bcc transitions with increasing temperature in several metals because the relatively open bcc structure tends to have a greater vibrational entropy than the close-packed structures. Computationally, vibrational excitation effects are usually included in the harmonic or the quasi-harmonic approximation. In the latter approximation, the crystal structure is allowed to expand and relax as the temperature is increased but otherwise it corresponds to the harmonic approximation. In the earliest implementations the Debye model was often used [32, 66]. Here, we will use the so-called direct method [39, 53] where interatomic force parameters are determined in a supercell by moving each individual atom away from its equilibrium position, one atom at a time, and monitoring the resulting forces on the remaining atoms. The force parameters are then Fourier transformed to obtain a dynamical matrix which, upon diagonalization, yields the vibrational density of states g(ω) as function of the vibrational frequency ω. The vibrational density of states provides the free energy due to vibrations[53]where \( \nu = \exp( -\hbar \omega / k_{\rm B} T )\). \(\Updelta G_{\rm vib,emb} \) is evaluated through finite differences just as for the electronic free-energy contribution, see Eq. 29.

The embedding free energy of Nb in ferrite at T = 0 K can be computed ab initio with a variety of methods, such as via Fe bcc supercell calculations in which a single-Nb atom is placed among a great number of Fe atoms [51, 55], by determining the mixing enthalpy from special quasi random structures [55], and by determining the mixing enthalpy from a cluster expansion (CE) for bcc Fe–Nb substitutional alloys [55]. As the second and third approaches are both based on the CE formalism, we will consider the first and second methods here.

Supercells can be used to model dilute solutions, particularly when we consider a series of supercells of increasing size. Such a sequence of larger and larger cells then models increasingly dilute solid solutions, and by taking a limit the infinite solution can be evaluated [30, 54, 55] using arguments derived from linear elastic theory. Furthermore, in the case of dilute alloys one can consider both fixed-volume (energy) and -pressure (enthalpy) cells. Fixed volume cells, when viewed as being embedded in the matrix material represent the case of the embedding material being infinitely stiff, whereas fixed-pressure cells (typically with P = 0 GPa) represent embedding in an infinite-compliant matrix material. As can be expected, the most realistic representation is some intermediate in between these two extremes, although a little tilted toward the fixed pressure side [30]. Supercells can be created in a variety of ways. In Fig. 2 we show supercells with sc, bcc, and fcc, arrangements of lattice points. Supercells as a representation of dilute solid solutions has as main shortcoming artificial periodicity, and consequently highly specific directions in which the dilute species “feel” each other. Figure 2 illustrates that the dilute species interact primarily along 〈100〉,  〈111〉, and 〈110〉 directions in the sc, bcc, and fcc type supercells, respectively. In highly anisotropic materials this means that sc, bcc, and fcc type supercells will each have their own convergence toward the infinite-dilution limit. This is evident from Fig. 3 where sc, bcc, and fcc type cells clearly follow distinct trends. This is not surprising in the case of ferrite, it is elastically far from isotropic (A = 2.41 [49]) and the magnetization is strongly coupled to elastic deformation. Hence, it is to be expected that the interaction between Nb atoms with their strong elastic distortions is highly sensitive upon the direction of interaction. It appears that for constant volume cells fcc and bcc type cells are best, whereas for constant pressure cells sc type supercells are optimal. When a series of supercells is considered of various types without regard for the various supercell types a rather jumbled sequence of formation energies and enthalpies is found, as shown in a recent study of Fe–Mo alloys by Lejaeghere et al. [26]. Ozolins and Asta [38] observed oscillatory interactions in dilute solutions of Sc in Al which they attributed to a combination of geometric volume expansion and electronic Friedel oscillations. While electronic Friedel oscillations do not provide an explanation here, one could suspect oscillations in the spin density. However, this did not show up in our analysis of the spin density in the various cells considered: Nb atoms aligned antiferromagnetically with the Fe matrix and the local moment centered on the Nb atom, computed as the spin density integrated over the Nb-centered Voronoi volume, was about −0.7 μ_B per Nb atom. Aside from the local moment on the Nb atom the spin density in the Fe matrix appeared rather unperturbed from the case without Nb, with the spin density strongly peaking near the Fe atoms and the spin density taking moderately negative values in the interstitial regions, without showing any sign of significant anisotropy. Therefore, lattice relaxations in elastically strongly anisotropic bcc Fe are expected to be main reason for the jumbled sequence of formation energies and enthalpies. A much more troubling aspect of Fig. 3 is that while constant volume and constant pressure results do seem to converge to the same infinite-dilution limit for each type of supercell, as in the study of Mishin et al. [30], the sc, bcc, and fcc type supercells each converge to their own distinct limit. It is evident that bcc, fcc, and sc type cells when extrapolated to n ⁻¹ = 0, which corresponds to \(\Updelta H^\alpha_{\rm emb}[{\rm Nb}]\), give values of about −2.4, −5.2, and −10.9 kJ/mol, respectively. This is the case not only for the embedding energy, but also for other properties such as the magnetization change per Fe replaced by Nb. This is in contrast to earlier defect energy supercell calculations on non-magnetic materials, where generally accurate and consistent results were obtained [30, 51, 54, 55]. In keeping with the previously mentioned role of elastic relaxations, it should be noted that the bcc, fcc, sc sequence is seen also in the elastically stiffest directions, which are in decreasing order 〈111〉,  〈110〉, and 〈100〉. When considering all constant volume results we extract a \(\Updelta H^\alpha_{\rm emb}[{\rm Nb}]\) value of −7.01 kJ/mol with a squared correlation coefficient of 0.60, while the zero pressure results give a \(\Updelta H^\alpha_{\rm emb}\) value of −10.06 kJ/mol with a squared correlation coefficient of 0.17. The energies can be extrapolated more reliably than the enthalpy results—presumably because changes of volume affect the systematic error in the enthalpy calculations. As the supercell results are not conclusive, another method is used also: evaluation of the mixing enthalpy in the dilute limit using special quasi-random structures (SQS) [69]. Details of how these SQS were designed will not be reported here, the description of these structures is given in Table 2. The ab initio computed formation enthalpies are shown in Fig. 4. As described in Ref. [55], the mixing enthalpy can be obtained by fitting a low order polynomial in the composition to the SQS enthalpies. Both third- and fourth-order polynomials were considered giving the following expressions for the mixing enthalpy,with h _0,Nb ^α  = 22.5 kJ/mol, h _1,Nb ^α  = 4.7 kJ/mol, andwith h _0,Nb ^α = 18.1 kJ/mol, h _1,Nb ^α = 57.1 kJ/mol, and h _2,Nb ^α = −17.5 kJ/mol. While both of these equations have about the same squared correlation coefficients, 0.958 and 0.960, respectively, they give \(\Updelta H^\alpha_{\rm emb}[{\rm Nb}]\) values that differ somewhat, 22.5 and 18.1 kJ/mol, respectively. More important than this difference between the two fits is the fact that it differs in sign from the results obtained from the supercell calculations. In some alloys, such as Fe–Cr [29, 68], it is well-known that mixing and unmixing tendencies may change with composition so that dilute and more concentrated alloys behave differently. Such tendencies have never been reported for Fe–Nb however, and the isoelectronic Fe-V system has been reported to not feature such a change with composition [28, 29]. Although Fe–Nb is a compound forming system, and the bcc-based FeNb B2 and Fe₃Nb DO₃ compounds form weakly exothermically from the pure elements, the actually occurring Fe–Nb compounds are not bcc superstructures but structures stabilized by atomic size differences. The size mismatch between the small Fe atoms and the much larger Nb atoms makes unmixing tendencies in the disordered solid solution more plausible. It should be remarked that the SQSs derived mixing enthalpy also agrees in sign with that obtained from thermodynamic modeling of the experimental phase equilibria [40] over the entire composition range. A reason for the apparent discrepancy between supercell and SQS derived mixing and embedding enthalpies may stem from the magnetic order. The magnetic order relevant for experimental observations is invariably in the neighborhood of the Curie temperature and thus includes significant magnetic disorder. Such disorder is known to strongly affect mixing and unmixing tendencies, as is apparent from studies on Fe–Cr (see Fig. 3c) [19]. One could speculate that in the SQSs the magnetic disorder which follows from the quasi-random distribution of Nb atoms, to some extent mimics the solid solution near the Curie temperature. The supercell results might then strictly speaking be correct at T = 0 K, but actually be irrelevant to the actual finite temperature Fe–Nb solution. Therefore, we prefer the SQS-based results over the supercell results. Hence, we proceed with an embedding enthalpy at T = 0 K for Nb of 20 ± 2 kJ/mol (average of the third- and fourth-order polynomials).

The effect of magnetic disordering to the embedding free energy can be evaluated with Eq. 27. The important parameter here is the rate at which the Curie temperature changes with Nb concentration. This property was measured by Stoelinga et al. [57] to be an increase of 0.8 K per atomic percent Nb. If we assume that the Curie temperature exhibits the same compositional trends as the magnetization per atom and make an estimate of the latter on the basis of the SQS results a contrasting picture emerges, see Fig. 5. The magnetization per atom of the Fe–Nb SQS structures in the limit of vanishing Nb concentration decays with 0.033 μ_B per atomic percent Nb. In amorphous Fe–Nb alloys also a significant decrease in the Curie temperature is measured [61]. Here, we will use the Stoelinga results [57]. We thus arrive at an estimate of r _Nb = 0.8 × 100/1043 = 0.077 which indicates that near the Curie temperature the embedding free energy is decreased by \(2 \times (0.077) \times (1.6) \approx\) 0.25 kJ/mol. Clearly, magnetic disordering contributes a rather minor term to the embedding free energy.

The vibrational effects have been evaluated also. Figure 6a shows the vibrational free energy per atom computed for Fe₅₄, Fe₅₃Nb, Nb₅₄, C₆₄, (NbC)₃₂, and Fe₅₄C using Eq. 30. The very stiff NbC and especially diamond C lattices are evident from the much smaller vibrational free energies of these structures. The curves for Fe₅₄,Fe₅₃Nb, and Fe₅₄C largely overlap. The vibrational contribution to the Nb embedding energy is displayed in Fig. 6b. Nb expands the Fe lattice and thereby introduces lower frequency modes so that the vibrational contribution to the embedding free energy, particularly at higher temperatures is negative (exothermic). In the temperature range from 500 to 1100 K, the vibrational contribution is well described by \(\Updelta G^\alpha_{\rm vib,emb}[{\rm Nb}] = - 0.1011-0.0042 \, {\rm T}\) kJ/mol. At T = 1000 K, the vibrational contribution is seen to be about −4 kJ/mol. However, bear in mind that this result is based on a supercell calculation, which in the previous subsection was shown to be unreliable. The vibrational contribution has not yet been evaluated via the SQS structures.

Electronic contributions to the embedding free energy can be obtained from Eq. 28. Fe _n Nb supercells give a \(n(\epsilon_{\rm F})\) typically about 4 states/eV less than in the sum of \(n(\epsilon_{\rm F})\) of Fe _n and Nb₁. Analysis of \(n(\epsilon_{\rm F})\) of the SQS structures shows some scatter, but a least squares fit to the second-order polynomial in the composition gives a decrease of about 1.3 states/eV per Nb atom at infinite dilution. Again, a significant difference between supercell and SQS results, with the latter appearing more likely as pure Fe and pure Nb have \(n(\epsilon_{\rm F})\) values of only about 1 and 1.5 states/(eV atom). Taking the SQS results, we obtain thus \(\Updelta G_{\rm elec} = 1.5 \times 10^{-6} \,{\rm T}^2\) kJ/mol Nb dissolved in ferrite. At T = 1000 K this amounts to an endothermic contribution of about 1.5 kJ/mol. The magnetic disordering, vibrational, and electronic contributions are not small relative to the ab initio computed embedding enthalpy at T = 0 K. Figure 7 shows the total embedding free energy obtained from summing the T = 0 K enthalpy, the magnetic, vibrational, and electronic free energies.

The embedding free energy of C in ferrite at T = 0 K has been computed by two ab initio methods: the afore mentioned supercell method, and the CE method. At least two supercell results are available in the literature, h _0,Nb ^α = 0.74 eV = 71 kJ/mol [14] (Table III) and h _0,C ^α = 0.81 eV = 78 kJ/mol [44] (Table IV). Our own calculations are essentially identical to the latter. Both references [14, 44] find their values confirmed by thermodynamic calculations based on experimental data [9, 48], but some caution is required. When the solubility limit of C in ferrite c _lmt is measured, as reported in Table I of Hasebe et al. [9], and the embedding free energy is extracted according to \(\Updelta G^\alpha_{\rm emb}[{\rm C}] = k_{\rm B} T \ln( c_{\rm lmt}/3 ), \) one obtains \(\Updelta H^\alpha_{\rm emb}[{\rm C}]\) = 91 kJ/mol and \(\Updelta S^\alpha_{\rm emb}[{\rm C}]\) = 2.9 k _B. Other solubility limit measurements give comparable values: \(\Updelta H^\alpha_{\rm emb}[{\rm C}]\) = 101 kJ/mol and \(\Updelta S^\alpha_{\rm emb}[{\rm C}]\) = 4.0 k _B [45] and \(\Updelta H^\alpha_{\rm emb}[{\rm C}]\) = 99 kJ/mol and \(\Updelta S^\alpha_{\rm emb}[{\rm C}]\) = 3.8 k _B [58]. As the embedding free energy computed with supercells for Nb was deemed unreliable for for T ≠ 0 K, here also, a comparison with another method is called for. A CE can be carried out over the octahedral interstitial sublattice of bcc Fe. Of course, the interstitial sublattice can be vacant, or occupied by C. The highest possible C concentration occurs for FeC₃ when all interstices are occupied by C. We select a CE from a pool of clusters which satisfy two criteria: (a) up to four sites in a cluster, (b) no two sites within a cluster are farther apart than the fifth nearest neighbor. These criteria yield a pool of 26 clusters. Furthermore we compute the energies of all Fe _n C _m structures that can be generated by imposing that the primitive translations are 1.5 a_bcc or shorter, this yields eight distinct types of periodic cells, and 389 distinct structures. After fully structurally relaxing these structures, we eliminate structures with Fe lattices that can no longer be classified as being bcc-based. By imposing that the CEs are complete [50], it is possible to examine all possible CEs that can be generated with our pool of clusters, we select the CE that has the smallest “predictive error” [52] or better known as leave-one-out cross-validation score [65]. We use this CE to compute the enthalpy of mixing, see Fig. 8, as described elsewhere [55]. Taking the slope of the mixing enthalpy at zero concentration C yields h _0,Nb ^α = 90 kJ/mol.

The magnetic contribution to the C embedding free energy can be estimated with Eq. 27 if the change in the Curie temperature per fraction C is known. Here we resort to the estimate by Hasebe et al. [9], \(\Updelta T_{\rm C}\) = −500 K, so that r _C = −500/1043 = −0.48. It should be noted that the magnetization per atom decays more rapidly, ∂(M/M ₀)/ ∂c _C = −1.37 in the limit of vanishing carbon concentration c _C. Proceeding with r _C = −0.48, we find that at T = 1000 K the embedding free energy is increased by \(2 \times (-0.48) \times (-1.6) \approx\) 1.5 kJ/mol.

The vibrational contribution to the C embedding energy, as derived from the data shown in Fig. 6a, is displayed in Fig. 6b. C atoms leave the stiff diamond lattice when they are inserted in the Fe lattice and therefore clearly lower energy vibrational modes are created so that the vibrational contribution to the embedding free energy is strongly negative. In the temperature range of interest, from 500 to 1100 K, the vibrational free energy is approximately linear with T: \(\Updelta G^\alpha_{\rm emb}[{\rm C}]\) = −3.068–0.0285 T kJ/mol. At T = 1000 K, the vibrational contribution is seen to be almost −32 kJ/mol.

Electronic contributions to the embedding free energy are obtained from Eq. 28. In Fe₁₂₈C supercells \(n(\epsilon_{\rm F})\) is about 10 states/eV greater than in the appropriate sum of Fe₁₂₈ and C diamond cells. An analysis of \(n(\epsilon_{\rm F})\) of the almost 400 structures for the CE gave an increase also in \(n(\epsilon_{\rm F})\) but just about 0.5 states/eV per dissolved C atom. If we accept the CE results we obtain thus \(\Updelta G_{\rm elec} = -6 \times 10^{-7} \, {\rm T}^2\) kJ/mol C dissolved in ferrite. The magnetic disordering and electronic contributions are small relative to the ab initio computed embedding enthalpy at T = 0 K, but the vibrational contribution is somewhat more significant. Figure 10 shows the total embedding free energy obtained from summing the T = 0 K enthalpy, the magnetic, vibrational, and electronic free energies (Fig. 9).

The formation of NbC from the reference states gives a formation enthalpy of −104.3 kJ/mol NbC at T = 0 K, see Table 1. There is no magnetic free-energy contribution, so we focus on the vibrational and electronic terms. The vibrational free energy is favorable for NbC formation. The reason is that although NbC is hard and has high energy vibrational modes associated with the C species, it is still is not nearly as stiff as diamond so that the average of Nb-bcc and C-diamond has higher energy vibrational modes than NbC. In the temperature range of interest, from 500 to 1100 K, we may well represent the vibrational free energy by a linear function \(\Updelta G_{\rm vib}[{\rm NbC}]\) = −1.706–0.0024 T kJ/mol NbC.

The electronic contribution is easily evaluated from the data in Table 1, where it is evident that NbC formation leads to the reduction of 0.86 states/(eV formula unit) at the Fermi level. Using Eq. 28 we find \(\Updelta G_{\rm elec}[{\rm NbC}] = 1 \times 10^{-6} \, {\rm T}^2\) kJ/mol, which is a rather minor effect. The ab initio formation free energy can thus be given as \(\Updelta G_{\rm f}{[\rm NbC}]\) = −106.0–0.0024 T + \(1 \times 10^{-6} \, {\rm T}^2\) kJ/mol.