Understanding element solution energies in nickelbase alloys using machine learning

The design of nickelbase superalloys requires to tune the content of different phases with a composition of Ni3A, either to strengthen the alloy ( γ′ -phase, Ni3Al, γ″-phase, Ni3Nb) or to influence its grain size and avoid embrittlement (δ-phase, Ni3Nb, η-phase, Ni3Ti). Here, we use a machine-learning-inspired approach to understand the influence of elemental properties on the energy of an alloying element in the respective phases. It is shown that the energy in γ″, δ, and η can be fitted well using the Bader charge and the volume of the element in a nickel matrix. In the case of γ′ , the small lattice mismatch requires a fit not involving the volume, but the bond order instead. We also show that the frequently used M d -parameter can be predicted from the properties of an element in a pure nickel matrix. Finally, the physical basis of the results is discussed in detail.


Introduction
Nickelbase superalloys are a versatile class of alloys due to their ability to form a large number of intermetallic phases that allow to tune their properties. Phases with a composition of Ni 3 A (where A denotes the phaseforming element) are especially important [1]: By definition, nickelbase superalloys contain the g¢-phase (Ni 3 Al) for strengthening. In some wrought alloys, the γ″-phase (Ni 3 Nb) may be used instead. Furthermore, the δ-phase (Ni 3 Nb) and the η-phase (Ni 3 Ti) have to be considered in alloy design because they can be employed to ensure a fine-grained structure during forging in the case of wrought alloys, but may also be detrimental because they can embrittle the alloy [2]. In addition, the formation of topologically closed-packed phases is an important concern in nickelbase superalloy design [3].
Due to the large number of alloying elements used in nickelbase superalloys, predicting their properties from the alloy composition can be difficult. Two physically-based quantities that are frequently used, especially in the prediction of topologically closed-packed (TCP) phases, are the electron vacancy number [3] or the M d -parameter [4]. Alternatively, machine learning can be used in alloy design, usually by using a large experimental input dataset. (Un-)desirable quantities can then be related to the composition of the alloy and a predictive tool relating composition to properties can be derived [5]. In [6], a large database of superalloys is used to predict the formation of phases based on physical quantities that are calculated for the given alloy composition (melting temperature, the atomic radius difference, the structural entropy).
Understanding the influence of physical quantities on the stability of different phases is thus an important step in designing new alloys. Density functional theory (DFT) calculations can be employed for this, see for example [7,8] for the g¢-, [9] for the γ″-, and [10] for the δand η-phase.
Here, we use machine-learning tools to predict the influence of alloying elements on the energy of different phases in nickelbase superalloys. We consider the four phases of composition Ni 3 A that are technically important: g¢, γ″, δ, and η. Data for the energy of the alloying elements in the γ″-, δ-, and η-phase are taken from [9,10], data for the g¢-phase were produced for this study. The aim is to find physical quantities that allow a prediction of the energy of the element in the Ni 3 A-phase. In addition to standard atomic quantities (like number of delectrons or electronegativity), we use DFT to calculate several properties of the alloying elements in a pure nickel matrix (atomic volume, atomic charge, etc.). We study correlations between the properties considered and show that the influence of the element on the solution energy can be predicted well using two element properties for elements of the 5th and 6th period; for elements of the 4th period, a separate fit is needed due to the different properties of the 3d orbitals. In most cases, the Bader charge [11] and the atomic volume, both calculated in a nickel matrix, provide a reasonable fit. We also show that the frequently used M d -parameter can be predicted using the properties considered.

Calculation method
All density functional theory reported in this paper were performed using VASP (Vienna Ab initio Simulation Package) [12][13][14]. A projected augmented wave (PAW) pseudopotential plane wave basis set [15,16] was used with Perdew-Burke-Ernzerhof [17] exchange correlation functionals. For all atoms, the maximum valency available was used (i.e. lower-shell s or p electrons were included if possible). The energy cutoff (ENCUT) was chosen as 520 eV. Calculations were spin-polarized with magnetic moments usually being initialized to 1 μ B . Elements for which larger magnetic moments can be expected (Fe, Co) were initialized with larger moments, all cells containing Cr, which has a tendency to have an antiferromagnetic orientation, were initialized both with ferro-and antiferromagnetic orientation and the state with the lower final energy was selected. The k-spacing of final calculations was always chosen as 0.1 Å −1 , real space operators were calculated to a precision of 10 −4 (ROPT=1e-4). The electronic precision was set to 10 −4 meV (EDIFF = 1.E-7), the ionic loop during relaxation was stopped when the energy change was below 10 −2 meV (EDIFFG = 1.E-5).
To avoid errors due to Pulay stresses, five calculations at different scale factors were performed for each configuration, usually using a larger k-spacing. Using a Birch-Murnaghan state equation [18], the optimum scale factor was determined and a final calculation was performed at this scale factor with high precision. Calculations were performed with Gaussian smearing and a smearing parameter of 0.07 eV.
Data for the energy difference between the element in the nickel matrix and in the Ni 3 A-phase (called transfer energy in the following) have already been shown in [9,10] for the case of the δ-, η − and γ″-phase. These energies were calculated using a 32-atom supercell. For some of these calculations, an older version of the pseudopotentials (from VASP 4.6) had been used. These calculations were repeated with the pseudopotential from VASP 5.4.
Data for the g¢-phase have been published by other groups in the past [7,8]. Since these publications did not list the transfer energy considered in this paper, calculations have been performed using 108-atom supercells (Ni 107 X, Ni 81 Al 26 X, Ni 80 Al 27 X). The transfer energy for substituting Al and Ni in the g¢-phase was calculated as using the same method as in [9,10]. Here and in the following, E( · ) denotes the energy of the respective structure. Throughout the paper, A denotes the phase-forming element (Al, Ti, Nb) and X denotes the alloying element. Figure 1 shows the unit cells of all phases considered.

Elemental properties
Basic elemental properties like period, number of d electrons, or electron affinity, which were taken from the literature. However, since the elements are used as alloying elements, these quantities may not be significant to describe the behaviour of the element in a nickel matrix. We therefore measured several quantities for each element in a Ni 107 X-cell and in the Ni 106 AX-cell where the element and the phase-forming element were positioned in a nearest-or next-nearest-neighbour position.
It is well-known that elastic distortions play an important role in determining the energy of an alloy. Therefore, we chose the Voronoi volume (and the nearest neighbour distance, which correlates with the volume) since it is directly related to the elastic straining of bonds. An alternative measure, the volume calculated by a Bader charge analysis, was also looked at, but this quantity did not give as consistent results as the Voronoi volume. The reason for this is that the Bader volume is not representative of the bond length; for example, the Bader volume of an Al atom in a nickel matrix is rather small (5.54 Å 3 ).
It is well-known that elastic distortions play an important role inAnother quantity that is known to determine bond energies is the charge transfer [19]. This is related to the electronegativity and the electron affinity of an element. However, a better measure is the charge transfer of the alloying element in a nickel matrix. The charge of an element was calculated using two different methods: The Bader method (calculated with [11]) and the chargemol charge (calculated with chargemol [20]). Both quantities are used since it is not a priori clear which of these measures correlates best with the energy of an alloying element.
It is well-known that elastic distortions play an important role inAnother quantity that may be important for the energy, especially in the formation of bonds between d-orbitals, is the bond order, which was also calculated using chargemol [20].
It is well-known that elastic distortions play an important role inFinally, pure nickel is strongly ferromagnetic, whereas the phases considered are only weakly ferromagnetic or paramagnetic. The energy of transferring an element from the nickel matrix to an intermetallic phase may therefore be influenced by the change in the magnetic moments. For this reason, the magnetic moment of each atom (as calculated by VASP from the difference of the two spin channels, projected onto the atomic site, see [21]) have been used. Alternatively, atomic magnetic moments can be calculated using chargemol [20], suppl. inf., but since the results of the two calculations usually agree quite well, only one method was used.
In summary, the following quantities were calculated for each element in a Ni 107 X-cell and in the Ni 106 AX-cell where the element and the phase-forming element were positioned in a nearest-or next-nearestneighbour position: Voronoi volume of the atom; distance to next atom; charge transfer calculated by Bader charge (calculated with [11]), in the following briefly denoted as 'Bader charge transfer'; chargemol charge and bond order (calculated with chargemol [20]); magnetic moment (taken from the VASP output).
A further quantity of interest is the interaction energy E int between the phase-forming element A and the alloying element X. This was calculated from a Ni 106 AX cell with A and X in nearest-neighbour position using the following formula:

Transfer energies
In a nickelbase alloy with a nickel matrix and an intermetallic Ni 3 A-phase, the quantity that decides the favoured position of the alloying element is the difference between the energy of the element in the nickel matrix and in the intermetallic phase, called transfer energy in the following. In Ni 3 A, the alloying element X can take either the position of a nickel atom or of the phase-forming element A. 1 It should be noted that these two possibilities of substitution are strongly different physically: If we consider only nearest-neighbour-bonds, an element that moves from the nickel matrix to the A position in Ni 3 A does not change its bonds as long as the Ni 3 A-phase is close-packed since the element will have 12 nickel neighbours in both cases. The transfer energy in this case can be expected to be determined by a change in the bond state due to next-nearest neighbour interactions and by the different lattice constant in the intermetallic phase.
On the other hand, transferring an element from the nickel matrix to a Ni position in Ni 3 A removes 4 Ni-A and 4 Ni-X bonds in total and adds 4 Ni-Ni bonds and 4 A-X bonds. This substitution is thus dominated by nearest-neighbour bonding effects. From this argument alone, it should be clear that most elements will not favour a Ni position in Ni 3 A because the large number of d electrons in Ni makes both Ni-A-bonds and Ni-X- bonds more favourable than Ni-Ni bonds in most cases. This implies that transferring an element to the Ni position in Ni 3 A will be energetically very costly for elements with a small number of d electrons. Figure 2 shows the calculated transfer energies. In the g¢-phase, all Al-and all Ni-positions are equivalent. In the γ″and δ-phase, there is only one distinct Nb atom position, but there are two configurationally different nickel positions [9,10]. In the η-phase, there are two distinct positions for Ti and Ni each.
From the plot, it can be seen that the curves are generally similar for the Zr-and Hf-period; for the Ti period the picture is less clear, probably mainly due to the influence of magnetic interactions between the elements, with Fe replacing Nb in the δ-phase having a particularly high energy.
When replacing the phase-forming element, the energy is low for elements with a small number of d electrons in all phases considered. The energy for Nb or Ti in Ni 3 Al is actually lower than the formation energy of the δor η-phase. This does not imply that these phases cannot form in the presence of Al since these energies are calculated in the dilute limit. Furthermore, finite-temperature effects may be involved as well. Experimentally, it is well-known that the δor η-phase do not form when a certain amount of Al is exceeded. This is the reason why wrought alloys such as Inconel 718 and VDM alloy 780 that utilize these phases for grain refinement contain limited amounts of Al [22].
The shape of the energy curves is rather similar for the γ″-, δand η-phase as expected since these phases involve elements with a small number of d electrons. In the case of g¢, the shape is different with a plateau for the first elements of the Zr-and Hf-period.
For Ni substitution, the shape of the curves for the different phases is again rather similar. As expected from the argument above, the transfer energy can take large values, especially for elements with a small number of d electrons. Substituting Ni is slightly favourable only for elements with a number of d electrons similar to Ni as expected because this substitution creates additional Ni-Ni-bonds which are usually less favourable than bonds to other transition elements.  Table 1 shows the quantities calculated for the alloying elements in a Ni 107 X cell and the interaction energy in the nearest-neighbour configuration Ni 106 AX. (All quantities were also calculated in Ni 106 AX-cells with the atoms A and X in the nearest or next-nearest neighbour position. These are not shown in the table since the subsequent calculations showed that using them does not significantly improve the fits).

Alloying element properties
To understand the variation of the data, we first performed a principal component analysis (PCA) on all quantities from the table except for the interaction energy. For the analysis, the quantities were normalized using standard minmax-preprocessing so that all variables were scaled to the interval [0; 1].
The first three principal components have an explained variance of 41.7%, 26.1%, and 17.0%. The loading scores of these components are shown in table 2. It can be seen that the first principal component is dominated by properties related to the charge transfer of the element, the second by the size of the atom, and the third by the bond order and number of d electrons.
Of course, some of the properties looked at are correlated, the most obvious example being the Ni-X bond length and the volume of the X atom in Ni 107 X. To study these correlations, the Spearman's correlation coefficients between all variables in table 1 have been calculated.
In the following, we look at all property combinations with a Spearman correlation coefficient greater than 0.8, see figure 3. The correlation between bond length and volume is perfect as expected. A correlation coefficient slightly above 0.8 is found between the Bader and the chargemol charge, two properties which both measure the charge transfer to or from the alloying atom. It is known that the correlation between these variables is not too strong in many cases [25], so the less than perfect value of the correlation coefficient is not surprising. A similar correlation coefficient is observed between the period of the element and the bond order. The data show that the bond order indeed tends to be larger for elements in the 5th and 6th period. This can be explained by the increased size of the d orbitals relative to the s orbitals.
Furthermore, a high correlation coefficient between 0.875 and 0.914 is observed between the Bader charge transfer and the interaction energy as calculated from equation (3), see figure 4 for the case of Nb as phaseforming element. Since the interaction energy can be expected to predict the transfer energy between the alloying atom in the nickel matrix and a nickel position in the Ni 3 A-phase (as explained in the previous section), this suggests that the Bader charge transfer should allow a reasonable prediction of the transfer energy.
Because we will try to fit the transfer energy of the alloying elements using linear fits in the following sections, it is also interesting to see whether the correlated properties can be related by a linear fit. For this, a linear fit was performed for all property pairs, and the r 2 -values were calculated. With the obvious exception of bond length and volume, the r 2 -values of the correlated properties are below 0.75. This, however, is not unexpected because the properties of 3d-orbitals strongly differ from those of 4d-or 5d-elements. It should thus not be expected that a linear fit can be performed using the same parameters for elements of the Ti period and of the Zr-and Hf-period. Therefore, we calculated the linear fit for elements of the latter two periods (Zr-Hg) only. In this case, the r 2 -value between the Bader charge transfer and the interaction energy with the phase-forming elements is between 0.866 and 0.940, showing that the Bader charge transfer is a good proxy for the interaction energy for these elements. A separate fit for the elements of the Ti period yields r 2 -values between 0.752 and 0.850, showing that the correlation is not as clear for these elements. This can be expected because for some of the 3d-elements, magnetic interactions play an important role for the energy.

Fitting solution energies with a single parameter
In this section, we fit the transfer energies from figure 2 using a single property from table 1.
For the case of nickel substitution, it was discussed in section 3.1 that the interaction energy between the alloying and the phase-forming element should determine the calculated transfer energy. A linear fit of the interaction energy to the transfer energy for Ni substitution for all elements considered results in r 2 -values between 0.897 for g¢ and 0.975 for one of the Ni positions in the γ″ phase. If the transfer energy were completely determined by the nearest-neighbour interaction, it should be proportional to four times the interaction energy. The actual slope of the fits lies between 3.955 for g¢ and 6.345 for one of the substitutions in the η phase, a reasonable agreement with the simple picture of nearest-neighbour interactions. If only the elements from the 5th and 6th period are used, the quality of the fit using the interaction energy improves only slightly to 0.905 for g¢, for all other phases, the r 2 -value lies between 0.963 and 0.990.
Since the Bader charge transfer in turn is correlated with the interaction energy as shown in the previous section, a linear fit with this property can also be expected to yield reasonable results. The r 2 -values for a linear fit of the Bader charge transfer lie between 0.767 and 0.828. However, if we restrict the fit to elements of the 5th and 6th period only (Zr-Hg), the quality become much better with r 2 -values between 0.846 and 0.922. In the case of substituting the phase-forming element, there is no single variable that can be expected to work similarly well. When all elements are used for the fit, the only variable with an r 2 -value above 0.7 is the chargemol charge in the case of Ti substitution in the η phase with values of 0.700 and 0.733.
If only the elements from the 5th and 6th period are used, the chargemol charge has r 2 -values of 0.704 for δ, 0.853 for γ″, and 0.795 and 0.916, respectively, for the two titanium substitutions in the η-phase. For the g¢-phase, the chargemol charge does not provide a good fit with an r 2 -value of only 0.268. When using only the elements from the 5th and 6th period, the Bader charge provides a reasonable linear fit in most cases, with r 2 -values between 0.702 for g¢ and 0.894 for δ. The exception is one of the titanium substitutions in the η-phase where the value is only 0.468. From this, it can be expected that the Bader charge   transfer together with another element property may be able to give a good two-parameter fit. This will be explored further in the following sections.

Two-parameter fits
In this section, we fit the transfer energy from figure 2 using several properties from table 1. One possibility to perform a fit using several variables is the Lasso-method [26] which is able to determine an optimal number of fit parameters. However, the additional constraints used to reduce the influence of the less important variables also affect the overall quality of the fit. In the current study, it was found that the fit quality detoriated noticeably when using the Lasso-method. Since the number of element properties is not very large, an alternative method is to simply fix the number of properties used in the fit and to try all possible combinations. Doing this, it was found that a two-variable fit, using two properties from table 1, provides reasonable fits in most cases, with no large gains from adding another property (and thus another free fit parameter). We therefore look at two-variable linear fits of the transfer energy, using three fit parameters. (Later on, we will reduce the number of fit parameters to two.) As already explained, a single fit cannot be expected to work well both for the elements Zr-Hg and for the 4th period (and Al). Therefore, we first performed all fits on the 5th and 6th period only. In the plots shown in the following, the same two properties were then used in a separate fit for the 4th period (Ti-Zn). Al was never considered in these fits due to its lack of d electrons.

Substituting Ni
Again, we consider the case of Ni substitution first. From the results shown in the previous section, it should be expected that the Bader charge transfer is one of the properties involved in the best fits. Figure 5 shows the best two-variable fits that can be achieved for Ni substitution in the different phases (the smaller open symbols denote the fit technique descibed in the next section). For each substitution, fits were performed separately for Zr-Hg and Ti-Zn as explained above.
Since the Bader charge transfer alone already provides a good fit for Ni substitution, it can be expected that adding another variable yields an excellent fit. With the exception of the g¢-phase, the best fit is achieved with the Bader charge transfer and the magnetic moment, with consistently high r 2 -values between 0.957 and 0.970, see figure 5.
A physical interpretation of the high fit quality when using the magnetic moments is difficult. Elements with a low magnetic moment in the nickel matrix reduce the magnetic moments of the surrounding Nickel atoms and can thus be expected to have an increased energy in the nickel matrix. However, the coefficient of the magnetic moment term in the fit is negative, showing that the transfer energy is reduced for elements with large magnetic moments in nickel, not increased. Since the Bader charge transfer alone provides a reasonable fit for the transfer energy, it may be that the good fit value achieved when adding the magnetic moment is somewhat accidental and mainly due to the deviation between the Bader charge transfer fit and the transfer energy being largest at elements with about five d electrons.
For the g¢-phase, the bond order together with the Bader charge transfer provides the best fit with an r 2 -value of 0.960.
Using the volume together with the Bader charge transfer also leads to good fits in all cases, with r 2 -values between 0.914 and 0.935. Since this is the property combination that also yields good results for the case of substituting the phase-forming element, the resulting fits are shown in figure 6. Another parameter combination that works well in all cases is the chargemol charge together with the bond order.
The results show that the charge transfer between the alloying element and the surrounding atoms (measured either by the Bader or the chargemol charge) is the most important property determining the transfer energy for the Ni position in Ni 3 A. This will be discussed further in section 4.

Substituting A
Next we consider the case of substituting the phase-forming element A in Ni 3 A. For the γ″, δ, and η, a good fit can be achieved using the volume and the Bader charge transfer, with r 2 -values between 0.917 for δ (where fits involving the number of d electrons and the volume or the electronegativity and the Bader charge transfer perform marginally better) and 0.992 for one of the titanium substitutions in η. For the g¢-phase, the fit is considerably worse with r 2 equal to 0.839. The best fit in this phase is again achieved using the Bader charge transfer and the bond order. This can be interpreted as showing that for all phases except for g¢, the volume of the element plays an important role when substituting the phase-forming element, whereas this is not the case in g¢. This is plausible because the lattice mismatch of the g¢-phase is rather small compared to that of the other phases. Figure 7 shows the resulting fits using the Bader charge and the volume. For the g¢-phase, the better fit using the Bader charge transfer and bond order is also shown.

Reduced two-parameter fits
The two-parameter fits from the previous section use three fit parameters (two slopes and one intersect). It is possible to reduce this number to two by using the known properties of the pure Ni 3 A-phase.
Consider first the case of A substitution. From equation (1), this energy is the energy of adding a Ni 3 X-cell to Ni 3 A. If we consider the phase-forming element itself as an 'alloying element', this is the formation energy E form of the respective phase. In the case of a fit using the volume and the Bader charge transfer, this suggests that the transfer energy E be fitted using the following equation: where V X and V A are the volume of the elements in the nickel phase, Q X and Q A their Bader charge transfers and a and b are two fit parameters. If the element X is equal to A, both sides of the equation are zero. This thus allows to remove the intersect from the fit parameters.
In the case of nickel substitution, we can look at the transfer of Nickel from the matrix to the Ni 3 A-phase. The energy required for this is obviously zero. Therefore, the fit can be performed using the properties of a nickel atom in a pure nickel matrix: Both equations can of course be used for using other quantities like the magnetic moments or the bond order. Figures 5, 6, and 7 also show the results of these reduced fits using smaller open symbols. In most cases, the fit quality is approximately the same with only a small reduction in the r 2 -value. There are some exceptions for the fits using the Bader charge transfer and volume: One of the Ti substitutions in the η-phase, where the r 2 -value drops from 0.992 to 0.809, and one of the two Ni substitutions in the γ″and δ-phase each, with the r 2 -value dropping to 0.86 in both cases. For the g¢-phase, the reduced fit using Bader charge and volume is bad both for Al and for Ni substitution. This is probably because in g¢, the volume of the atom is not a good physical property to determine the energy because of the small lattice mismatch, as already discussed above.

Comparison with Morinaga approach
In a series of papers [4,27,28], Morinaga et al have established the so-called M d -parameter that can be used to predict the formation of topologically closed-packed phases and can also be used to understand the strength of nickelbase alloys. The M d -parameter is calculated from a Ni 3 Al-cluster with composition Ni 12 Al 6 X with L1 2 structure in which the central Al atom is replaced by the alloying element. Physically, the parameter is a measure of the electronegativity of the element and of its volume [29].
Since the calculation of the M d -parameter involves a finite cluster, not a bulk material, and is performed using a Ni 3 Al-phase, it is interesting to see whether the parameter can be approximated using the properties of the alloying atom in a pure nickel matrix. To do so, we performed a linear fit of all pairs of properties from table 1 to the M d -parameter. When fitting all transition elements, the best fit is achieved with the number of d electrons and the atomic volume (or the bond distance), with r 2 = 0.944; the second-best fit with the volume and the Bader charge (r 2 = 0.906). If only the 5th and 6th period are used, the fit using the number of d electrons and the volume is almost perfect (r 2 = 0.991); the fit using Bader charge transfer and volume improves to an r 2 -value of 0.937. The results from these fits are shown in figure 8.
This result shows that the M d parameter, which is calculated in a cluster containing Al, can be predicted from properties calculated in a pure nickel matrix. The result also confirms that the combination of Bader charge transfer and volume of an element in a nickel matrix is indeed a quantity with high predictive power.  For the fit data points, + -symbols denote a fit performed for all elements, ×-symbols a fit performed for the elements of the 5th and 6th period (Zr-Hg) only. Different from other plots, no separate fit for the elements Ti-Zn was performed, the datapoints marked with × denote the prediction of the fit for these elements.

Discussion
The results from the previous section show that the transfer energy of an element between the nickel matrix and one of the Ni 3 A-phases can be understood from physical quantities of the element calculated in a nickel matrix.
In general, good fit parameters can be found to describe the transfer energy of the elements of the 5th and 6th period simultaneously, but elements of the 4th period usually need a different parameter set, although the same properties can be used for the fit. Physically, this is easily explained by the smaller size of the 3d orbitals compared to 4d and 5d orbitals which reduces the number of d-bonds and gives rise to magnetic phenomena. (Analoguously, the frequently used electron vacancy number [3] also takes different values for the 3d-elements than for 4d-and 5d-elements.) Interpreted in another way, the machine learning approach used in this paper thus indicates the differing physical properties of the elements in the different periods.
The results show that the Bader charge transfer of an alloying element in a Ni 107 X-cell is a useful quantity to understand the transfer energy of the element. The importance of the local charge (measured by the Bader charge transfer) of the alloying element can be understood from the rectangular-band model of the bonding of AB-type alloys of transition elements [19,30]: Alloying two elements with a different number of d electrons widens the d band and skews the local density of states of the element with fewer d electrons to higher energies and the band of the element with more d electrons to lower energies, causing a net charge transfer. Because of this, the binding energy is larger when the difference in the number of d electrons of the elements is large. An alternative way of looking at this, yielding qualitatively the same result, is to use the moment theorem and Hückel theory [31,32].
In the case of nickel substitution in Ni 3 A, the transfer energy of equation (2) is dominated by the exchange of Ni-A and Ni-X bonds with X-A and Ni-Ni bonds and is thus a direct measure of the bond strengths. A large difference between the number of d electrons of X and Ni causes a large charge transfer, making Ni-X bonds highly favourable, resulting in a high interaction energy. As already discussed, this correlation between the Bader charge transfer and the interaction energy provides the rationale for the excellent fits achieved with the Bader charge transfer when the alloying atom replaces Ni in Ni 3 A.
To understand the importance of the Bader charge transfer when substituting the phase-forming element A in Ni 3 A, instead of equation (1) we calculate the energy to exchange an X element in the nickel matrix with the phase-forming element in Ni 3 A, here written in general form when using 108-atom supercells: Quantitatively, this amounts to simply subtracting the formation energy of Ni 3 A from the transfer energy, thus shifting the curves in figures 2 and 7 so that the zero point of the energy lies at the data point of the phase-forming element. This idea was already used in equations (4) and (5) to achieve the reduced fits with only two fitting parameters. Therefore, if the Bader charge transfer of the element X is larger than that of the phase-forming element A, its tendency to move to the Ni 3 A-phase is larger than that of A and the exchange energy is negative. A positive exchange energy means that the tendency of the element to move to Ni 3 A is smaller, but the transfer energy from equation (1) can still be negative so that the element staibilizes Ni 3 A. In addition to the Bader charge transfer, the volume of the alloying element (measured in a nickel matrix) was found to be a useful fit parameter with the exception of the g¢-phase. This is of course to be expected [19]. In [6], the atomic radius was found to be a crucial parameter in determining the tendency of an element to form TCP phases. In our case of a Ni 3 A-phase, the size difference between nickel and the phase-forming element causes a chemical pressure [33], with Ni-Ni bonds being strained (for example, the distance between two nickel atoms in γ″ in our calculations is 2,57Å and 2.61 Å, compared to a bond length of 2.489 Å in pure Ni) and the Ni-A bonds being compressed. Replacing the phase-forming element with a smaller atom or Ni with a larger atom thus relieves this chemical pressure and is energetically favourable. This picture is corroborated by the fact that the coefficient of the volume term in the fit is positive for Ni substitution and negative for A substitution.
The exception to this is the g¢-phase. Here, a fit using the volume term has a rather low r 2 -value. This is not surprising because the g¢-phase has a bond length of 2.523 Å in our calculations, resulting in a low mismatch of 1.4%. The fact that the reduced fit from equations (4) and (5) does not work at all for this phase further indicates that the volume is not a good physical quantity in this case.
On first sight, it might seem suprising that we use a linear fit for the volume term, although the elastic energy is quadratic in the strain. The reason for this is that due to the strained bonds, we are not close to the minimum of the strain-versus-energy curve so that a linear fit can work reasonably well. Adding a quadratic term to the fit does not significantly improve the r 2 -values, so this was not done to reduce the number of fit parameters.
We can also look at the results in light of the principal component analysis from section 3.2. PC 1 was dominated by the Bader charge, PC 2 by the volume, so these properties can be expected to cover the variations of elemental properties best. In the case of g¢, where the volume term is less important due to the small lattice mismatch, the third principal component, which is dominated by the bond order, thus provides a good second fit variable. This argument is of course only heuristic because a linear fit to a physical quantity may be dominated by any single property regardless of its contribution to the principal components (the most obvious example being a fit of one of the properties themselves, the fit of M d in section 3.6 being another). Still, it is interesting that the three variables that dominate the principal components are involved in the fits.

Summary
In this paper, we have studied the transfer energy of an alloying element between a nickel matrix and different phases of composition Ni 3 A (g¢, γ″, δ, η) and tried to gain a physical understanding of the transfer energy by fitting it using properties of the element.
The main results from the study are: • The curves for the transfer energy have a similar shape for all phases considered. For most elements, replacing the phase-forming element is energetically more favourable than replacing nickel.
• A PCA analysis of the elemental properties shows that there are three main components related to the charge transfer, the volume, and the bond order.
• The Bader charge transfer is correlated with the interaction energy.
• When replacing a nickel atom in Ni 3 A, the transfer energy can be described well by the interaction energy between the alloying and the phase-forming element, equation (3). The Bader charge transfer alone still provides a reasonable fit, especially if the fit is restricted to elements of the 5th and 6th period.
• In general, it is considerably more difficult to find a fit for all elements considered than a fit that excludes elements from the 4th period. This is due to 3d-orbitals strongly differing from 4d-or 5d-orbitals because they are smaller than the highest occupied s-orbital.
• For substituting nickel, a 2-parameter fit using the Bader charge and the magnetic moment is usually excellent, although there seems to be no clear physical reason for the magnetic moment term. A fit using the Bader charge transfer and the volume is usually very good as well.
• When substituting the phase-forming element, the best fits are achieved with the Bader charge transfer and the volume.
• The physical motivation behind the predictive power of Bader charge and volume is the charge transfer and the lastic strain as elaborated in the discussion.
• The g¢-phase differs from the others due to its small lattice mismatch. For this phase, the volume term is not so important and the best fit is achieved with Bader charge transfer and bond order.
• A reduced fit that is based on the formation energy of the phase and uses no intercept fit parameter is almost as good as a full linear fit in most cases.
• The frequently used M d -parameter can be fitted well by the elemental properties considered, using the number of d electrons and the volume or the volume and Bader charge transfer.
This paper has thus shown that using a machine-learning inspired approach to understand the transfer energy of alloying elements allows to gain (or re-discover) a physical understanding of the influence of the alloying element on phase formation. The results imply that elemental properties that are calculated in an environment that is directly relevant to the materials studied (in our case, calculated in a Ni 107 X-cell) may be more useful in machine learning than general quantities like electronegativity. Future work might apply this appoach to the formation of TCP phases in nickelbase superalloys or to other alloys.