Crystal Chemistry and Physical Properties of A Quaternary Intermetallic Compound, θ -(Al 0.8718 Cu 0.0256 Si 0.1026 ) 13 Fe 4

: θ -Al 13 Fe 4 particles form as a primary Fe intermetallic compound (Fe-IMC) during the casting of commercial Al metals and alloys that inevitably contain Fe and Si as impurities. Moreover, the excellent mechanical performances of the Al-Cu alloys demand knowledge about the structural chemistry of the Fe-IMCs, including the θ -phase in the quaternary Al-Cu-Fe-Si system. Here, we investigate the stability, crystal structure, and electronic and mechanical properties of the Cu and Si co-doped θ -phase using a ﬁrst-principles density-functional theory approach. The calculations reveal high stability of a quaternary θ -phase with chemical composition (Al 0.8718 Cu 0.0256 Si 0.1026 ) 13 Fe 4 at ambient conditions. Thermodynamics and statistical analysis show a broad range of Si content in the structure at the casting temperature. The Cu and Si (co-)doping enhances the bulk modulus of the compounds. The calculated bulk modulus of the quaternary θ -phase is 129 GPa. The ﬁndings help characterize the θ -phase in the quaternary Al-Si-Fe-Cu system and understand the formation of the θ -phase and related phase transformations in the various Al alloys during casting.

θ-Al 13 Fe 4 is also the prototype of the family of θ-M 13 Fe 4 (M = Co, Fe, Ni, Pt) [22]. Moreover, the rich volatility of the metallic species indicates the possibility of dissolving

Methods
The previous first-principles study revealed the high stability of θ-Al 76 Cu 2 Fe 24 in the ternary Al-Fe-Cu system [36]. This compound is used here as the starting point. The solid-solid reactions with Si solution at the Al sites can be defined as, Here ∆H is the enthalpy of the reaction (1). E(θ-(Al 76-n Si n )Cu 2 Fe 24 ), E(θ-Al 76 Cu 2 Fe 24 ), E(Si), and E(Al) represent the calculated total valence-electron energies of θ-(Al 76-n Si n )Cu 2 Fe 24 , θ-Al 76 Cu 2 Fe 24 , and the elemental solids Si and α-Al, respectively. The unit in Equation (2) is eV/cell. A negative value of ∆E means that the reaction (1) is exothermal and favored. At ambient conditions (T = 0 K, P = 0 Pa), the formation energy is equal to the negative value of the reaction enthalpy in Equation (1), ∆E = −∆H, when the zero-point vibration energy term is ignored.
For the calculations, we employ a plane-wave method implemented in the firstprinciples code VASP (Vienna Ab initio Simulation Package) [37,38]. We also use the spinpolarized generalized gradient approximation [39] within the projector-augmented wave method [40] for the exchange and correlation energy terms. This is because the generalized gradient approximation describes the 3d transition metals well [41,42]. Considering the localized nature of the Cu 3d electrons, we employed the Hubbard U correction [43,44] with U = 4 eV, according to the previous study, to avoid the unphysical interaction of the localized Cu 3d electrons with the neighboring atoms [36,45]. The cut-off energy of the wave functions was set at 550.0 eV and the cut-off energy of the augmentation functions was 700.0 eV. These energies are significantly higher than the corresponding default values for the elements. The electronic wave functions were sampled on a 4 × 8 × 6 grid with 70-110 k-points in the irreducible Brillouin zone of the conventional cell of the θ-phase, depending on the symmetry using the Monkhorst-Pack scheme [46]. Both the lattice parameters and the coordinates of the atoms were fully relaxed. Different k-meshes and cut-off energies were used for the waves and augmentation waves, respectively. The tests showed good convergence (<1 meV/atom).

Results and Discussions
Using the above mentioned code and settings, the calculations produced lattice parameters for the elemental solids [47]: a = 3.039 Å for α-Al with a face-centered cubic cell (experimental value, a = 3.0325 Å at 0 K [48], the same for the rest); a = 5.468 Å for Si of the diamond-type structure (a = 5.42982 Å [48]); a = 2.831 Å for the ferromagnetic α-Fe with a body-centered cubic cell (a = 2.8607 Å [48]); and a = 3.622 Å for Cu (a = 3.6032 Å [48]). The calculated magnetic moment for α-Fe is 2.18 µ B /Fe, the same as the previous work [27]. The calculated lattice parameters agree with the experimental values with deviations within 1%. The calculations also produced total valence-electron energies for these elemental solids, which will be used to study the formation energies of the related θ-phases.
Calculations for the Si solutions in the Fe sites in θ-Al 76 Cu 2 Fe 24 , with respect to this parent ternary compound and the elemental solids, α-Al and α-Fe, showed that the Si solution at the Fe sites is costly, with formation energies high than 0.5 eV/Si. This indicates that it is unlikely for the Si solution to form in the Fe sites in the θ-phase.

Si Solution in θ-Al 76 Cu 2 Fe 24
The Al atoms at each Al site are fully replaced by Si. In this way, the symmetry of the systems is maintained. The calculations revealed non-spin-polarized solutions for all the θ-phases. The obtained results (lattice parameters, cell volume, and formation energies, according to Equation (2)) are listed in Table 1. The coordination numbers of the nearest neighbors for the Si atoms in the optimized structures are included in Table 1. Table 1. The calculated results (lattice parameters, atomic coordinates, cell volumes, coordination numbers (CNN), Bader's charges at the Si sites (q) and formation energies according to Equation (2) for Si solutions at the Al sites. The Wyckoff sites are fully replaced by Si and thus, the symmetry of the systems is the same as that of the θ-Al 13 Fe 4 (space group, C2/m) [19].  Table 1 shows the overall trends of the calculated results with the Si solutions in the Al sites in θ-Al 76 Cu 2 Fe 24 . There are only three configurations with Si at the Al8 (denoted as Si8, same for the rest), whereas Al9 and Al6 sites have negative formation energies (Table 1). We also employ the (n Si m) to represent the condition of the calculation; 'n' means the number of Si atoms, and 'm' the Al sites. The 4Si4 has a small positive formation energy (0.174 eV/cell). The remaining configurations have high values of positive formation energies ranging between 0.48 eV/cell and 1.50 eV/cell. The order of the stability of the stable configurations (from high to low) is 4Si8 > 4Si9 > 4Si6. This order of stability is different from that of Si solutions in pristine θ-Al 78 Fe 24 , where the configuration with 4Si9 is the most stable one [35].

Si-Sites
The coordination of the Si atoms in the configuration of 4Si8 is similar to that of 4Si9. Each Si in the configurations has four Fe neighbors and seven Al neighbors (Table 1). In the 4Si4, each Si also has four Fe neighbors. In 4Si6, each Si atom has three Fe neighbors. The analysis indicates that each Si has three or four Fe neighbors in the stable configurations.
As shown in Table 1, the volume of the Si solutioned configurations decreases with increasing Si contents in general, with the exception of the highly unstable configuration of 4Si2. This corresponds to the smaller atomic volume of Si with respect to Al [48]. Table 1 also shows that the most stable configuration (4Si8) has the smallest volume among the configurations with the four Si solutions at the Al1 sites.
Next, we focused on the stable configurations with Si solutions at Al8, Al9 and Al6. We performed structural optimizations and total energy calculations for the Si solutions with various contents at the three Al sites. Moreover, we also investigated the configurations of the Si solutions at the mixed Al8, Al9, and Al6 sites. The obtained configurations and related formation energies for the more stable structures are shown in Figure 1. 50 eV/cell. The order of the stability of the stable configurations (from high to low) is 4Si8 > 4Si9 > 4Si6. This order of stability is different from that of Si solutions in pristine θ-Al78Fe24, where the configuration with 4Si9 is the most stable one [35]. The coordination of the Si atoms in the configuration of 4Si8 is similar to that of 4Si9. Each Si in the configurations has four Fe neighbors and seven Al neighbors (Table 1). In the 4Si4, each Si also has four Fe neighbors. In 4Si6, each Si atom has three Fe neighbors. The analysis indicates that each Si has three or four Fe neighbors in the stable configurations.
As shown in Table 1, the volume of the Si solutioned configurations decreases with increasing Si contents in general, with the exception of the highly unstable configuration of 4Si2. This corresponds to the smaller atomic volume of Si with respect to Al [48]. Table  1 also shows that the most stable configuration (4Si8) has the smallest volume among the configurations with the four Si solutions at the Al1 sites.
Next, we focused on the stable configurations with Si solutions at Al8, Al9 and Al6. We performed structural optimizations and total energy calculations for the Si solutions with various contents at the three Al sites. Moreover, we also investigated the configurations of the Si solutions at the mixed Al8, Al9, and Al6 sites. The obtained configurations and related formation energies for the more stable structures are shown in Figure 1. In details, the marks represent the formation energies for the configurations with (the order of) Si atoms at the corresponding Al sites: The black filled spheres represent the formation energies relating to that Si atoms first occupy the Al8 sites and gradually the Al9 sites secondly, and the Al6 at last, the red spheres that Si atoms occupy the Al9 sites first and gradually the Al8 sites secondly, the blue spheres that Si first occupies the Al6 sites and gradually the Al8 sites secondly, and the Al9 at last. The yellow spheres represent the formation energies with the configurations with Si atoms at the Al8 and Al9 sites in a mixed way. The green sphere represents the formation energy of the 4Si8, 3Si9 and 1Si6 configuration. Figure 1 shows that the formation energy decreases with the increased Si content for Si solutions at the individual Al sites and with the same order of stability, as shown in Table 1, (from high to low): Si8 > Si9 >> Si6, being consistent with the results in Table 1. In details, the marks represent the formation energies for the configurations with (the order of) Si atoms at the corresponding Al sites: The black filled spheres represent the formation energies relating to that Si atoms first occupy the Al8 sites and gradually the Al9 sites secondly, and the Al6 at last, the red spheres that Si atoms occupy the Al9 sites first and gradually the Al8 sites secondly, the blue spheres that Si first occupies the Al6 sites and gradually the Al8 sites secondly, and the Al9 at last. The yellow spheres represent the formation energies with the configurations with Si atoms at the Al8 and Al9 sites in a mixed way. The green sphere represents the formation energy of the 4Si8, 3Si9 and 1Si6 configuration. Figure 1 shows that the formation energy decreases with the increased Si content for Si solutions at the individual Al sites and with the same order of stability, as shown in Table 1, (from high to low): Si8 > Si9 >> Si6, being consistent with the results in Table 1. The Si solutions at mixing Al8 and Al9 have the formation energies in-between those at the corresponding Al8 and Al9 sites with the same Si content.
The investigation revealed that the most stable configurations have 4Si8 and 4Si9 with the composition θ-Al 68 Cu 2 Si 8 Fe 24 . This has been justified for more arrangements with the same Si content. The next stable configuration with this composition has 4Si8 + 3Si9 + 1Si, the formation energy of which is approximately 0.16 eV/cell higher than the stable one, as shown in Figure 1.
The solutions of Si at the Al6 sites in θ-Al 68 Cu 2 Si 8 Fe 24 produced the most stable configurations with the same Si content. The calculations showed that the formation energy increases with the Si content. The maximum Si content with negative formation energy is 12Si at the Al9, Al8, and Al6 sites. The calculations also showed that further Si solutions at the other Al sites, e.g., Al4 in θ-Al 68 Cu 2 Si 8 Fe 24 , increase the formation energy notably, indicating such solutions as being unfavored.

Crystal Chemistry of θ-Al 68 Cu 2 Si 8 Fe 24
The optimized structure for the most stable θ-Al 68 Cu 2 Si 8 Fe 24 , schematically along its [010] orientation, is shown in Figure 2. The atomic coordinates are shown in Table 2. The Si solutions at mixing Al8 and Al9 have the formation energies in-between those at the corresponding Al8 and Al9 sites with the same Si content. The investigation revealed that the most stable configurations have 4Si8 and 4Si9 with the composition θ-Al68Cu2Si8Fe24. This has been justified for more arrangements with the same Si content. The next stable configuration with this composition has 4Si8 + 3Si9 + 1Si, the formation energy of which is approximately 0.16 eV/cell higher than the stable one, as shown in Figure 1.
The solutions of Si at the Al6 sites in θ-Al68Cu2Si8Fe24 produced the most stable configurations with the same Si content. The calculations showed that the formation energy increases with the Si content. The maximum Si content with negative formation energy is 12Si at the Al9, Al8, and Al6 sites. The calculations also showed that further Si solutions at the other Al sites, e.g., Al4 in θ-Al68Cu2Si8Fe24, increase the formation energy notably, indicating such solutions as being unfavored.

Crystal Chemistry of θ-Al68Cu2Si8Fe24
The optimized structure for the most stable θ-Al68Cu2Si8Fe24, schematically along its [010] orientation, is shown in Figure 2. The atomic coordinates are shown in Table 2.
The frame of the structure of θ-Al68Cu2Si8Fe24 is the same as that of the pristine θ-Al78Fe24 [19,33]. Here, we focus on the special Al sites, Al7 (2Cu7), Al8 (4Si8), and Al9 (4Si9), in the Cu-Si co-solutioned θ-phase. The chemical bonds of those selected atoms in the different configurations are shown in Table 3.

−2.82
The frame of the structure of θ-Al 68 Cu 2 Si 8 Fe 24 is the same as that of the pristine θ-Al 78 Fe 24 [19,33]. Here, we focus on the special Al sites, Al7 (2Cu7), Al8 (4Si8), and Al9 (4Si9), in the Cu-Si co-solutioned θ-phase. The chemical bonds of those selected atoms in the different configurations are shown in Table 3.  Tables 2 and 3 showed that the coordination numbers of the atoms at the same site in the different compositions are the same. However, there are some subtle differences between θ-Al 68 Cu 2 Si 8 Fe 24 and the other configurations.
The Cu7-Fe bond length is 2.61 Å in θ-Al 76 Cu 2 Fe 24 , which is close to that of (2.64 Å) in θ-Al 68 Cu 2 Si 8 Fe 24 . However, these Cu7-Fe bond lengths are larger than the corresponding Al7-Fe bonds (~2.46 Å) in θ-Al 78 Fe 24 and θ-Al 74 Si 4 Fe 24 . Meanwhile, the Cu7-Al bonds, on average (~2.63 Å), in the Cu-containing configurations are shorter than the Al7-Al bonds in the binary Al-Fe and the ternary Al-Fe-Si compounds (~2.74 Å), as shown in Table 3. This phenomenon corresponds well to the smaller Cu volume than Al [44].

Electronic Properties of θ-Al 68 Cu 2 Si 8 Fe 24
Using the settings in Section 2, we performed electronic band structure calculations for the stable θ-Al 68 Cu 2 Si 8 Fe 24 . The obtained electron density distributions and partial density of states (pDOS) for selected atoms and the total density of states (tDOS) are shown in Figures 3 and 4, respectively.
To better understand the interatomic interactions in θ-Al 68 Cu 2 Si 8 Fe 24 , we also performed Bader charge analysis, which divides an atom into solid via the zero-flux surfaces between the atom and its surrounding atoms (δρ(r)/δr = 0) [49]. Based on the calculated electron density distribution (Figure 3), the obtained charges at the atomic sites are included in Table 2.
As shown in Figure 4, the Si 3s and 3p states are between −11.6 eV and −0.5 eV and the conduction bands. There is a valley between −0.5 eV and 0 eV, such as a pseudo-gap. The Si 3s states are positioned at the lower valence band (−11.6 eV to −6.0 eV). There are also some Si 3s states at the lower part of the conduction band. Meanwhile, the Si 3p state dominates the upper part of the valence bands and the conduction bands from 2.5 eV. There is admixing between the Si 3s and 3p states, corresponding to the sp3 hybrid.
The Cu 3d states dominate the curve with a peak at −4.5 eV and with a bandwidth of approximately 1.0 eV. This value peak is notably lower than that in pure Cu (−2.3 eV) from the first-principles calculations with Hubbard U correction and the photoemission spectrum measurements [45]. This indicates that the Cu in the structure gains electrons from the environments, which is confirmed by Bader's charge analysis (Table 2). Moreover, the narrow bandwidth indicates the localized nature of the Cu 3d electrons, which contribute little to the chemical bonding between the Cu and the neighboring atoms.
The Fe 3d states dominate the upper part of the valence band (from −4.0 eV to −0.5 eV). A considerable density of Fe 3d states is located at the Fermi level. Such broad peaks indicate the rather delocalized nature of the Fe 3d states in the compound. The Fe 3d states are almost fully occupied, consistent with the significant gains of electrons from their neighboring atoms. This also means that Fe atoms obtain electrons from the surrounding      Table 2, we obtained the following results: (1) Al atoms have few electrons around them, corresponding to their metallic nature, as such metals are composed of free electrons and ions. Bader's charge analysis showed the Al atoms are positively charged with a loss of 1.1 e/Al to 1.5 e/Al (Table 2). (2) Cu atoms have almost spherically shaped clouds of dense electrons around them. This is due to the itinerant 3d electrons (Figure 4). Each Cu gains 1.0 e/Cu (Table 2) from neighboring Al atoms. (3) Fe atoms also exhibit irregularly shaped clouds of high densities, indicating interactions between Fe 3d states with neighboring atoms, including Si, Cu and Al. Bader charge analysis showed more significant electron gains for the Fe atoms with charges ranging between −2.82 e/Fe and −3.54 e/Fe, corresponding to its electronegativity value being larger than Al. Correspondingly, the Fe atoms are non-spin-polarized in the compounds.

Stability and Mechanical Properties of the θ-Phases
The dependences of the total valence-electron energies on the volumes are investigated for the binary, ternary and quaternary θ-phases. Using the Murnaghan equation of states [50,51], we obtained the parameters (ground electronic energies, ground state volume, bulk modulus) of the related compounds. The obtained relation between the total valence-electron energies and volumes is plotted in Figure 5 and Table 4.    As shown in Figure 5, the energy-volume values fit well using the Murnaghan equation of states [51]. As summarized in Table 4, the solutions of the Si-Cu atoms at the Al sites decrease the volume of the pristine θ-Al 13 Fe 4 unit cell. This corresponds to the shorter M-Al and M-Fe (M = Si or Cu) than the corresponding Al-Al and Al-Fe bonds (Table 3). Table 4 also shows that solutions of Si or Cu enhance the bulk modulus of the θ-phases. The quaternary θ-Al 68 Cu 2 Si 8 Fe 24 has the maximum bulk modulus (129 GPa) which is higher than that (122 GPa) of the pristine θ-Al 13 Fe 4 and those of the ternary compounds (123 to 127 GPa).
To provide a comparison of the stability of the binary, ternary and quaternary θstructures, we investigated their formation energies with respect to the elemental solids in a systematic way. The formation energy of a compound Al 78-n-m Si n Cu m Fe 24 (0 ≤ n, 0 ≤ m ≤ 2) with respect to the elemental solids, α-Al, α-Fe, Si and Cu is here defined as here, E(Al 76-n-m Si n Cu m Fe 24 ), E(Fe), E(Al), E(Si) and E(Cu) are, respectively, the calculated total energy for θ-Al 76-n-m Si n Cu m Fe 24 , and the elemental solids, α-Al, α-Fe, Si and Cu. The calculated formation energies are shown in Table 4. As shown in Table 4, the Si and Cu solutions in θ-Al 13 Fe 4 form, respectively, the ternary θ-Al 74 Si 4 Fe 24 and θ-Al 76 Cu 2 Fe 24 compounds, are more stable than the parent binary phase. Table 4 also shows that the Si and Cu co-solution in θ-Al 13 Fe 4 form, the quaternary θ-Al 72 Cu 2 Si 4 Fe 24 and θ-Al 68 Cu 2 Si 8 Fe 24 , are more stable than the binary and the ternary phases, with the latter being the most stable under ambient conditions.
The calculated bulk modulus of the θ-phases provided us information about their mechanical properties. Meanwhile, for better knowledge regarding the elastic properties of these complex compounds, the related elastic constant coefficients and related phonon spectra require further investigation.

Formation Range and Stability of θ-(Al 76-n Si n Cu 2 )Fe 24 at Casting Temperature
The first-principles calculations showed that Si atoms prefer the Al8 and Al9 sites in the θ-phase. Meanwhile, the energy difference between the Si at the Al8 and Al at the Al9 sites is insignificant. For n = 1 in θ-(Al 76-n Si n Cu 2 )Fe 24 , the energy difference value between the 1Si8 and 1Si9 is small (0.05 eV/cell) and increases with the increasing the Si content. The energy difference reaches its maximum value with n = 4 (0.184 eV/cell). Meanwhile, the formation energies for configurations Si at the mixed Al8 and Al9 sites with -n = 2 to 4 are in-between those at Al8 or Al9. This indicates extra freedom of Si solutions in θ-Al 76 Cu 2 Fe 24 at elevated temperatures. With higher the Si concentration (n = 5 to 8), the energy differences between the 4Si8 + mSi9 configurations and the mSi8 + 4Si9 (m = 1 to 3) decrease with the Si content. For m = 1, the energy difference is 0.133 eV/cell and for m = 4 it becomes 0 eV. This analysis also means extra freedom for Si solutions in θ-Al 76 Cu 2 Fe 24 with higher Si contents. Next, we discuss the stability of the Cu-Si co-solutioned θ-phase at the casting temperature.
The experiments showed that the θ-phases are formed during casting, typically at around 1000 K, at which temperature the extra freedom of the Si solutions in the Fe-IMC contributes to the free energy of the system: where ∆S = R ln w, and w is the number of configurations of the same Si content and R (= 8.617 × 10 −5 eV/K) is the Boltzmann constant.
To simplify the analysis, we employed the random model to obtain the number of configurations for the θ-(Al 76-n Si n )Cu 2 Fe 24 . The obtained dependence of the estimated free formation energies on the Si content at 1000 K is shown in Figure 6. the 1Si8 and 1Si9 is small (0.05 eV/cell) and increases with the increasing the Si content. The energy difference reaches its maximum value with n = 4 (0.184 eV/cell). Meanwhile, the formation energies for configurations Si at the mixed Al8 and Al9 sites with -n = 2 to 4 are in-between those at Al8 or Al9. This indicates extra freedom of Si solutions in θ-Al76Cu2Fe24 at elevated temperatures. With higher the Si concentration (n = 5 to 8), the energy differences between the 4Si8 + mSi9 configurations and the mSi8 + 4Si9 (m = 1 to 3) decrease with the Si content. For m = 1, the energy difference is 0.133 eV/cell and for m = 4 it becomes 0 eV. This analysis also means extra freedom for Si solutions in θ-Al76Cu2Fe24 with higher Si contents. Next, we discuss the stability of the Cu-Si co-solutioned θ-phase at the casting temperature.
The experiments showed that the θ-phases are formed during casting, typically at around 1000 K, at which temperature the extra freedom of the Si solutions in the Fe-IMC contributes to the free energy of the system: where ΔS = R ln w, and w is the number of configurations of the same Si content and R (= 8.617 × 10 −5 eV/K) is the Boltzmann constant.
To simplify the analysis, we employed the random model to obtain the number of configurations for the θ-(Al76-nSin)Cu2Fe24. The obtained dependence of the estimated free formation energies on the Si content at 1000 K is shown in Figure 6. Thermodynamics and statistical analysis revealed that, at the casting temperature, in a Si-poor alloy, the Si content ranges between 3.9 at.% and 6.0 at.% or n = 4 to 6 in the θ-(Al 76-n Si n )Cu 2 Fe 24 . The exact composition depends on the chemical environment and the formation temperature.
The present investigations revealed that the most stable configuration at ambient conditions is θ-(Al 68 Si 8 )Cu 2 Fe 24 , with Cu at the Al7 and Si at the Al8 and Al9 sites. The calculations also produced a bulk modulus to be 129 GPa for this stable configuration. The study also provided detailed information about the structural and electronic properties of this stable configuration. This obtained information is useful to characterize this phase in the cast alloys and to obtain insight into the contribution of this phase to the mechanical properties of the products.
Thermodynamics and statistical analysis revealed that the extra freedom of Si substitutions at the Al sites strongly impacts the stability and chemical composition of the formed θ-phase at the casting temperature. Using the random model for the Si at the Al8 and Al9 sites, we found that at 1000 K, the Si partial occupations at the Al8 and Al9 sites, particularly the former, are preferred. The estimated chemical compositions are θ-(Al 76-n Si n )Cu 2 Fe 24 with 4 < n < 6 at 1000 K. Moreover, the configurational entropy contributions further stabi-lize the θ-phase. This information is helpful in investigating the formation of the θ-phase at the phase transformations of the θ-phase to other Fe-IMCs during the casting and related annealing processes of various Al-based alloys [7,52,53].

Conclusions
The present first-principles study revealed a quaternary compound of high stability with the chemical composition θ-Al 68 Si 8 Cu 2 Fe 24 or θ-Al 11.3333 Si 1.3333 Cu 0.3333 Fe 4 at ambient conditions. The calculated lattice parameters are a = 15.390 Å, b = 7.910 Å, c = 12.309 Å, β = 108.06 • and volume = 1424.53 Å 3 /cell. The lengths of the axis and the cell volume are smaller than the corresponding axis of the binary θ-Al 13 Fe 4 . This compound exhibits a chemically ionic, metallic, and covalent triple nature. The calculated bulk modulus for this quaternary compound is 129 GPa, higher than the related binary θ-Al 13 Fe 4 (118GPa), ternary θ-compounds θ-Al 12.6667 Cu 0.3333 Fe 4 (122GPa) and θ-Al 12.3333 Si 0.6667 Fe 4 (124 GPa). The obtained information helps characterize the θ-phase in the quaternary Al-Fe-Si-Cu system and the role of this phase in the mechanical performance of the related cast Albased alloys. The analysis also showed that configurational entropy enhances the stability of the partially Si solutions in this phase, forming θ-(Al 76-n Si n )Cu 2 Fe 24 with 4 < n < 6 at 1000 K. This information is helpful in investigating the formation of the θ-phase at the phase transformations of the θ-phase to other Fe-IMCs during casting and related annealing processes.

Conflicts of Interest:
The authors declare no conflict of interest.