The Impact of Vibrational Entropy on the Segregation of Cu to Antiphase Boundaries in Fe3Al

We performed a quantum mechanical study of segregation of Cu atoms toward antiphase boundaries (APBs) in Fe3Al. The computed concentration of Cu atoms was 3.125 at %. The APBs have been characterized by a shift of the lattice along the ⟨001⟩ crystallographic direction. The APB energy turns out to be lower for Cu atoms located directly at the APB interfaces and we found that it is equal to 84 mJ/m2. Both Cu atoms (as point defects) and APBs (as extended defects) have their specific impact on local magnetic moments of Fe atoms (mostly reduction of the magnitude). Their combined impact was found to be not just a simple sum of the effects of each of the defect types. The Cu atoms are predicted to segregate toward the studied APBs, but the related energy gain is very small and amounts to only 4 meV per Cu atom. We have also performed phonon calculations and found all studied states with different atomic configurations mechanically stable without any soft phonon modes. The band gap in phonon frequencies of Fe3Al is barely affected by Cu substituents but reduced by APBs. The phonon contributions to segregation-related energy changes are significant, ranging from a decrease by 16% at T = 0 K to an increase by 17% at T = 400 K (changes with respect to the segregation-related energy difference between static lattices). Importantly, we have also examined the differences in the phonon entropy and phonon energy induced by the Cu segregation and showed their strongly nonlinear trends.

Regarding experimental studies of APBs, there were two types found in the D0 3 superlattice of Fe 3 Al using transmission electron microscopy (TEM) techniques [65,66]. The first one is characterized by a shift of the interfacing grains in the 100 direction by a half of the lattice parameter defined for a 16-atom cube-shaped D0 3 cell (see red arrows in Figure 1b,c) and is associated solely with the D0 3 superlattice (D0 3 -type of APBs). We have recently computed properties of D0 3 -APBs in (i) Fe 3 Al with and without Cr additions [67] as well as (ii) in Fe-Al-Ti compounds [68]. The second type can appear both in the D0 3 superlattice and in the B2 lattice (a B2-type of APBs) when two parts of the crystal are shifted in the 111 direction by a half of the lattice parameter of the B2 cell containing two atoms (we recently examined them in Fe 70 Al 30 alloy [69]). Both types were found separating partials of superdislocations in Fe-Al alloys [70][71][72]. Other findings related to APBs may be found, for example, in papers [73][74][75][76][77].
The properties of iron aluminides can be fine-tuned by alloying, see, e.g., [78][79][80][81][82][83][84], but the complexity of interactions of the solute atoms with extended defects are quite rarely studied (see an example in [85]). Therefore, our study is aimed at filling in this gap in the common knowledge. We have chosen to focus on Fe 3 Al because (i) it belongs to industrially important iron aluminides [1][2][3], (ii) it is known to contain extended defects including APBs, and (iii) properties of Fe 3 Al are routinely modified by additions, but (iv) a thorough theoretical analysis of interactions of copper atoms with APBs (possibly segregation tendencies to them) in Fe 3 Al has not been performed yet. Therefore, we have applied quantum mechanical calculations to compute properties of Fe 3 Al compound with Cu atoms (see Figure 1a) with and without APBs. Regarding the coexistence of both Cu atoms and APBs, we have examined both the case with Cu atoms at the APB interfaces (see Figure 1b) and far away from them (see Figure 1c). Importantly, we have performed our calculations not only for static lattices, but we have also evaluated contributions related to thermal vibrations. Our analysis has allowed us to thoroughly assess segregation tendencies of Cu atoms toward APBs in Fe 3 Al even at elevated temperatures.

Methods
Our first-principles calculations were performed with the help of the Vienna Ab initio Simulation Package (VASP) [86,87] implementing the density functional theory [88,89]. Projector augmented wave (PAW) pseudopotentials [90,91] were used together with the generalized gradient approximation (GGA) in the parametrization according to Perdew and Wang [92] (PW91) in combination with the Vosko-Wilk-Nusair correction [93] for the exchange and correlation energy. This specific setup was chosen as it correctly predicts the ground state of Fe 3 Al to be the D0 3 structure (its energy is lower than that of Fe 3 Al with the L1 2 structure by about 5.5 meV/atom [94]). We have employed 64-atom supercells as a stacking of 16-atom cube-shaped cell of Fe 3 Al (see Figure 1). They are 1 × 1 × 4 multiples of the 16-atom cell ( Figure 1). For phonon calculations, 256-atom supercells were generated by the PHONOPY software [95] as 2 × 2 × 1 multiples of the above-mentioned 64-atom supercells. Having computed the phonon frequencies ω(qν) for a dense set of reciprocal-space vectors q and all modes ν, we evaluated also the Helmholtz free energy F, the entropy S, harmonic phonon energy E, and constant-volume heat capacity C v [95]: Our calculations were performed with the plane-wave energy cutoff of 400 eV and the product of the number of Monkhorst-Pack k-points and the number of atoms was equal to 27,648 (e.g., 12 × 12 × 3 k-point mesh in the case of 64-atom supercells shown in Figure 1). We have fully relaxed all studied supercells, i.e., the energy was minimized with respect to internal atomic positions, cell shape, and the volume (forces acting upon atoms were reduced under 0.001 eV/Å in the case of 64-atom supercells). All local magnetic moments were initially set up as parallel (a ferromagnetic state).

Results
Starting with calculations of static lattices, i.e., without phonons, we first examined the impact of Cu solutes on surrounding atoms. As mentioned in the introduction above, there are two sublattices of iron atoms in the defect-free Fe 3 Al. The Fe I atoms have eight iron atoms in the first nearest neighbors (1NN) shell and their magnetic moment is equal to 2.37 µ B . Twice more abundant Fe II atoms have four iron atoms and four aluminum atoms in their 1NN shell and their magnetic moment is lower, equal to 1.87 µ B . We have computed the local magnetic moments of all Fe atoms in all three supercells shown in Figure 1 and the resulting values are compared with the above-mentioned Fe 3 Al-bulk values in Figure 2. The Cu atoms substitute the Fe II atoms because it was shown that it is energetically preferred over the Fe I site (see, e.g., [83]). As seen in Figure 1a, there are three Fe II atoms per one Cu atom in the (001) atomic planes. The supercell Figure 1a has a simple tetragonal symmetry with only eight symmetry operations (point group D2 d ). Two of these three Fe II atoms have the Cu atoms as second nearest neighbors and their local magnetic moments are slightly reduced (see Figure 2a). The third of these three Fe II atoms has the Cu atoms as the third nearest neighbor and its magnetic moment is slightly increased (Figure 2a). Magnetic moments of other Fe atoms are influenced to a lesser extent.
The same trends, i.e., either increased or decreased local magnetic moments of Fe atoms, were found also in the case when the APBs are introduced (see Figure 2b,c). However, the APBs have their own rather complex type of impact on iron atoms close to the APB interfaces. The effect of APBs is better separated from that related to the Cu atoms when the Cu substituents are located away from the APBs (see Figure 2c). The local magnetic moments of Fe atoms in the same {001} plane as the Cu atoms are similar for both Cu substituents in the supercell shown in Figure 1c. A qualitatively similar statement is true also in the case of the two APB interfaces; they lead to similar changes of the local magnetic moments as shown in Figure 2c. Interestingly, when the impact of Cu substituents combines with that of APBs, their overall influence is nontrivial and rather complicated trends emerge (see Figure 2b). For example, the local magnetic moments of Fe I atoms in the upper parts of Figure 2a,c are either equal to or slightly higher than the value in the defect-free Fe 3 Al bulk. In contrast to that, the combined impact of both Cu atoms and APB interfaces leads to local magnetic moments that are lower for some of the Fe I atoms, see the upper part of Figure 2b. Interactions of point defects and extended defects are, therefore, rather complex in Fe 3 Al and their influence on magnetic moments of Fe is not additive in any straightforward manner. This lack of a simple additivity in the case of the two types of studied defects is visible also in the case of other local magnetic moments of atoms, particularly those of the Fe II atoms in the lower part of Figure 2b.
When using the energies of static lattices U static , we have also evaluated the static lattice antiphase boundary energy. Comparing the two APB-containing supercells in Figure 1b,c, we found that the energy of the APB-containing supercell with the Cu atoms located directly at the APB interfaces (see Figure 1b) is lower than that when the Cu atoms are farther apart from the APB interfaces (see Figure 1c). Therefore, we have used the energy difference ∆U static between (i) the APB-containing supercell shown in Figure 1b and (ii) that one without any APB in Figure 1a for our evaluation of the APB energy. As there are two APBs per supercell, we have divided the above-mentioned energy difference by twice the area of the supercell within the {001} planes, i.e., the crystallographic orientation of the studied APB interfaces. The static lattice APB interface energy turns out to be equal to 84 mJ/m 2 . This rather low value obtained for Fe 3 Al with 3.125 at % of Cu atoms (i.e., 2 per 64 atoms) is slightly higher than the APB energy of 80 mJ/m 2 that we obtained for Fe 3 Al with the APBs characterized by the same shift in the 100 crystallographic direction and the same 64-atom supercell size in [67]. But the difference in the APB energy induced by the Cu atoms is very small and within an expected error bar of our calculations. This statement stems from the results of our comparative analysis of a series of supercells with different sizes and different separation of APBs in our previous study [67]. The APB energy in Fe 3 Al in our previous research exhibited values from the range of 80 ± 25 mJ/m 2 . Therefore, we conclude that the above discussed increase of the static lattice APB energy due to the presence of Cu atoms is rather insignificant. Now, focusing on the main topic of our work, that is, the segregation of Cu atoms toward the antiphase boundaries in Fe 3 Al, we evaluated the segregation-related energy difference ∆U static determined from the static lattice calculations. We found this quantity to be small, only −4.2 meV per Cu atom (the negative value means an energy gain). This difference can be formulated as −0.81 kJ/mol as an energy difference between the supercells in Figure 1b,c with one mole defined as that of 64-atom supercell with two segregating Cu atoms. In what follows, we do not express the segregation energy in a traditional manner per one segregating Cu atom. We rather analyze energy differences between 64-atoms supercells ∆U static in order to be able to smoothly combine static lattice energy differences discussed here with thermodynamic phonon-related terms as obtained directly from the PHONOPY software (see the discussion below).
Next, we have determined phonon modes for the supercells visualized in Figure 1. Starting with the APB-free Fe 3 Al with Cu substituents, the computed phonon spectra and the overall density of phonon states are shown in Figure 3. The band gap in vibrational frequencies above 8 THz is matching that in perfect Fe 3 Al bulk that we reported in our previous study [59]. Interestingly, the point defects represented here by Cu atoms do not close this gap in phonon frequencies in contrast to some APBs that, as we have shown, can close it nearly completely (see [59]). Importantly, the obtained phonon spectrum does not contain any soft phonon modes with imaginary frequencies. Therefore, we assume that the studied system of APB-free Fe 3 Al with Cu atoms is mechanically stable. Next, we evaluated Helmholtz free energy F, the entropy S, harmonic phonon energy E, and constant-volume heat capacity C v (see them visualized in Figure 4).  As our phonon calculations are performed only for the static lattice minimum-energy volume (the harmonic approximation), we present these quantities only up to T = 400 K (for which, we hope, thermal expansion does not alter the results too much). The dependences are qualitatively similar to those that we previously obtained for Fe 3 Al [59].
Next, we have determined phonon modes for both supercells containing APBs visualized in Figure 1b,c. The calculated spectra along selected higher-symmetry directions in the reciprocal space and the overall phonon densities of states are shown in Figure 5. Again, there are no soft phonon modes and, therefore, both studied systems with APBs (with the Cu atoms located directly at the APB interfaces or positioned away from them) are mechanically stable. The APBs make the band gap of phonon frequencies narrower, in particular when the Cu substituents are located directly at the APB interfaces, see Figure 5a. These findings are in line with the results of our previous study in which APBs (of a different type) nearly closed the gap of phonon frequencies [59]. In the present study, the gap is narrower but not completely closed.
Having detailed information related to phonon modes of both APB-containing systems available, it is possible to focus on the main topic of our research, segregation trends of Cu atoms toward the studied APBs. These are expected to occur as the formation energy of point defects represented by Cu atoms in Fe 3 Al is positive and the point-defect formation energy is equal to 0.192 eV/atom as shown by Park and coworkers in [83]. The ab initio calculations in [83] are relevant despite of the fact that a higher concentration of Cu atoms (6.25 at %) was studied (there is a decent solubility of a few atomic percent of Cu in Fe 3 Al when 5 at % was reported in [82]). The segregation-related difference of the static lattice energy added to the phonon free energy ∆(U static + F) per 64 atoms (including 2 segregating Cu atoms) is shown in Figure 6. The comparison with the energy difference of static lattice ∆U static (see the horizontal dashed line in Figure 6) indicates that the phonon contributions are reducing the segregation-related energy gain (when compared with the static lattice value ∆U static ) to −0.69 kJ/mol at T = 0 K. As the temperature increases from T = 0 K, the vibronic contributions into the segregation-related energy difference enhance the segregation tendency (the energy gain due to segregation is increased). The static lattice energy difference ∆U static (the above-discussed value of −0.81 kJ/mol) is reached at approximately T = 180 K and the free-energy gain due to the segregation then steadily increases for yet higher temperatures. Due to the fact that the static lattice segregation-related energy difference ∆U static is so small, vibrational degrees of freedom that are rather small themselves, represent an important part of the combined ∆(U static + F) energy difference. In particular, they can represent a decrease by 16% for T = 0 K or an increase by 17% for T = 400 K (expressed relatively with respect to the segregation-related ∆U static value). Figure 6. Calculated segregation-related energy difference (per 64-atom supercell) that includes both the static lattice energy U static and the phonon free energy F. The difference is between the states with (i) two Cu atoms at the APB interfaces and (ii) far away from them as a function of temperature. It is compared with the segregation-related energy difference of static lattices ∆U static (see horizontal dashed line). As one mol, we mean one mol of 64-atom supercells.
The above-discussed segregation-related energy differences ∆(U static + F) can be linked to the changes in the phonon entropy S and phonon energy E. Their differences between the two supercells containing APBs with differently located Cu atoms are shown in Figure 7. Their trends are nonlinear and nonmonotonous. They increase from their value at T = 0 K up to about T = 95 K and then decrease for higher temperatures. It is worth noting that the studied type of APBs alternates the distribution of atoms at the APB interfaces only slightly when compared to the APB-free lattice. Therefore, interatomic bonds are modified only weakly and the changes in thermal vibrations (and related terms in the free energy) are small. The phonon-related contributions into the segregation-related ∆(U static + F) are nevertheless significant (up to 16-17% discussed above). In contrast, their importance is much smaller (up to 1.5%) in the case the APB-related energy differences (again expressed as the energy difference between 64-atom supercells ∆(U static + F)), as see Figure 8.  The APB-related static lattice energy difference ∆U static is about 40 times bigger than the segregation-related energy difference discussed above, whereas the phonon-related con-tributions into the APB-related ∆(U static + F) difference are similar in their magnitude. Consequently, the vibronic contributions into the APB-related ∆(U static + F) are relatively much smaller than the segregation-related ones as seen in the comparison of Figures 6 and 8. The phonon contributions into the APB-related energy difference ∆(U static + F), particularly the entropy difference ∆S and phonon energy difference ∆E, are shown in Figure 9. They exhibit qualitatively similar trends as the segregation-related ones in Figure 7, but their combined impact is positive for the whole studied temperature range, i.e., they only increase the APB-related energy difference (see Figure 8). Finally, while we are not aware of any quantitative experimental data related to segregation of Cu atoms toward APBs in Fe 3 Al, we hope that our theoretical predictions will initiate such a research in future.

Conclusions
We performed an ab initio study of segregation of Cu atoms toward antiphase boundaries in Fe 3 Al. The studied concentration of Cu atoms was 3.125 at % (two Cu atoms per 64-atom supercells). When simulating APBs, one Cu atoms was approaching each of the two identical APBs inside the supercells. The Cu atoms substitute Fe II atoms, i.e., sublattice with the Fe atoms where the first nearest neighbor shell contains an equal amount of both Fe and Al atoms. The APBs have been characterized by a shift of the lattice along the 001 crystallographic direction and have the {001} orientation of their sharp interfaces. The APB energy turns out to be lower for Cu atoms located directly at the APB interfaces and we found it equal to 84 mJ/m 2 . Both Cu atoms (as point defects) and APBs (as extended defects) have their specific impact on local magnetic moments of Fe atoms (mostly reduction of the magnitude) close to these defects. Their combined impact was found not to be a simple linear addition of the impact of each of the defect types. The Cu atoms are predicted to segregate toward the studied APBs, but the related energy gain is small and amounts to only 4.2 meV per Cu atom.
As an extension of our study of static lattices, we have also performed phonon calculations for lattices with and without APBs (and the former with and without segregated Cu atoms) at their equilibrium volume (the harmonic approximation). All studied systems are mechanically stable without any soft phonon modes. The band gap in phonon frequencies of Fe 3 Al is barely affected by Cu substituents but slightly reduced by APBs. The phononrelated contributions into segregation-related energy difference were found significant, ranging from a decrease by 16% at T = 0 K to an increase by 17% at T = 400 K when compared with the segregation-related energy difference determined for static lattices. We have examined the differences in the phonon entropy and phonon energy between the states with (i) Cu atoms segregated to APB interfaces and (ii) the Cu atoms being away from them and found strongly nonlinear trends in both quantities.

Data Availability Statement:
The data presented in this study are available on request from the corresponding author.