Forming mechanism of equilibrium and non-equilibrium metallurgical phases in dissimilar aluminum/steel (Al–Fe) joints

Forming metallurgical phases has a critical impact on the performance of dissimilar materials joints. Here, we shed light on the forming mechanism of equilibrium and non-equilibrium intermetallic compounds (IMCs) in dissimilar aluminum/steel joints with respect to processing history (e.g., the pressure and temperature profiles) and chemical composition, where the knowledge of free energy and atomic diffusion in the Al–Fe system was taken from first-principles phonon calculations and data available in the literature. We found that the metastable and ductile (judged by the presently predicted elastic constants) Al6Fe is a pressure (P) favored IMC observed in processes involving high pressures. The MoSi2-type Al2Fe is brittle and a strong P-favored IMC observed at high pressures. The stable, brittle η-Al5Fe2 is the most observed IMC (followed by θ-Al13Fe4) in almost all processes, such as fusion/solid-state welding and additive manufacturing (AM), since η-Al5Fe2 is temperature-favored, possessing high thermodynamic driving force of formation and the fastest atomic diffusivity among all Al–Fe IMCs. Notably, the ductile AlFe3, the less ductile AlFe, and most of the other IMCs can be formed during AM, making AM a superior process to achieve desired IMCs in dissimilar materials. In addition, the unknown configurations of Al2Fe and Al5Fe2 were also examined by machine learning based datamining together with first-principles verifications and structure predictions. All the IMCs that are not P-favored can be identified using the conventional equilibrium phase diagram and the Scheil-Gulliver non-equilibrium simulations.

www.nature.com/scientificreports/ according to Pugh's criterion 11,12 , i.e., the ratio of bulk modulus versus shear modulus (B/G) based on the present first-principles calculations (cf., "Details of first-principles calculations" section). It indicates the ductile Al 6 Fe, Al 5 Fe 8 , and AlFe 3 ; the less ductile Al 13 Fe 4 and AlFe; and the brittle Al 5 Fe 2 and Al 2 Fe. Table 1 summarizes the Al-Fe IMCs formed in different processes reported in the literature. The metastable, ductile Al 6 Fe was observed in the processes of direct chill casting (example #1 in Table 1) 10 , high-pressure die casting (#2) 13 , equal channel angular extrusion (#3) 14 , tungsten inert gas (TIG) welding-brazing (#4) 15 , and additive manufacturing (AM) via laser powder bed fusion (#5) 16 . These observations suggest that Al 6 Fe is an IMC existing at high pressures. Table 1 further depicts that most of the stable and even metastable Al-Fe IMCs were observed in AM processes. For example, Al 6 Fe, Al 13 Fe 4 , Al 2 Fe, Al 5 Fe 2 , AlFe, and/or AlFe 3 were formed during the processes of laser powder bed fusion 16 , laser cladding 17 , direct energy deposition 18 , laser metal deposition 19 , and/or wire-arc AM 20,21 (see examples #5 to #10 in Table 1). In particular, the ductile (or less brittle) Al 13 Fe 4 , AlFe, and AlFe 3 20-22 (examples #9 to #11) were observed in Al-Fe based functional graded materials fabricated by additive manufacturing. These experiments indicate that AM is an exceptional process to tailor compositions and in turn the desired IMCs. Note that the AM induced residual stress is usually less than 1 GPa, for example, 290-416 MPa in 304L stainless steel 23 , and up to ~ 660 and 200 MPa for tensile and compressive, respectively, in 316L stainless steel 24 . These stresses are usually negligibly small to induce solid state phase transition. In the fusion and/or solid-state welding joints, Al 5 Fe 2 is the most observed IMC (usually adjacent to iron/steel) followed by Al 13 Fe 4 (usually adjacent to Al) processed by, for example, laser welding [25][26][27] (see examples #13 to #15 in Table 1), friction-type    Table S3. Note that Pugh's criterion 11 of 1.75 is a rough value to separate the ductile and brittle materials, as discussed in the authors' responses to Reviewers in Ref. 12 www.nature.com/scientificreports/ solid state welding [28][29][30][31][32][33][34][35][36] (#16 to #24), cold metal transfer fusion welding 37 (#25), and double electrode gas metal arc welding 38 (#26). The other IMCs such as Al 2 Fe and AlFe 3 were also observed in welding processes, depending on welding conditions (e.g., energy inputs) 7 ; see examples #14, #21, and #22 in Table 1. The same as those in welding processes, Al 5 Fe 2 (majorly) and Al 13 Fe 4 were also observed in immersion testing with solid Fe and liquid Al [39][40][41] (see examples #27 to #29 in Table 1), Al-Fe diffusion couples [42][43][44] (#30 to #32), high-temperature reactive sintering 45 (#12), and aluminized steel 46 (#33). Despite considerable observations as shown in Table 1, the underlying mechanism regarding the formation of Al-Fe IMCs is still lacking, albeit phase stability is known to be regulated by processing history involving T and P profiles for a given chemistry 47 . The phase diagram, as a foundational guide for any work in materials  48 , is the most used tool to analyze equilibrium IMCs under a given temperature and composition (usually under external pressure P = 0 GPa). Additionally, non-equilibrium simulations in terms of the Scheil-Gulliver model 49,50 can be used to analyze the forming IMCs in fast cooling processes by assuming that no diffusion takes place in the solid and that solute redistribution in the liquid is infinitely fast [51][52][53] . The Scheil simulations have been used, for example, to predict the formed IMCs in additively manufactured functionally graded metals 51,52 and to predict the temperatures of liquidus and solidus in steel 53 . In addition to the phase diagram, non-equilibrium IMCs can be predicted by calculating thermodynamic driving forces for the phases of interest with respect to supercooled liquid and associated solid phases; see the predicted interface phases at the Cu/solder joints by Lee et al. 54 . Also based on thermodynamics, non-equilibrium IMCs can be tailored by partitionless solidification or by chemical partition solidification with limited atomic diffusions; for example, the non-equilibrium solidification predicted in the Al-Sm system by Zhou and Napolitano 55 . It should be remarked that thermodynamic knowledge in the literature is usually at the ambient pressure or external pressure P = 0 GPa, thus hindering the analysis of P-favored phases such as Al 6 Fe in the present work. In addition to thermodynamics, kinetics (diffusion) is another factor to regulate nucleation, growth, and coarsening of IMCs 56,57 . For example, Al 5 Fe 2 and Al 13 Fe 4 were formed due mainly to Al and/or Fe interdiffusion in some processes; see the examples #13, #17, #18, #30, #31, and #32 in Table 1.
The present work aims to unveil the forming mechanism of equilibrium and non-equilibrium IMCs in dissimilar aluminum to steel joints based on thermodynamic knowledge in the Al-Fe system from (1) the present first-principles and phonon calculations based on density functional theory (DFT) and (2) the CALPHAD modeling by Sundman et al. 9 and also based on kinetic (diffusion) knowledge reported in the literature 42,58,59 . Special attention in the present DFT calculations is paid to the P-included Gibbs energy in addition to the variable of temperature. The challenge for the present DFT calculations is the unknown atomic configurations of (i) Al 5 Fe 2 caused by the partially occupied Wyckoff sites 4b and 8f. of space group Cmcm 60 and (ii) Al 2 Fe caused by the disordered Al and Fe in one of the Wyckoff sites 2i of space group P1 61 . To address this challenge, we adopt the following three approaches: (1) DFT-based USPEX (Universal Structure Predictor: Evolutionary Xtallography) predictions 62 , (2) DFT-based examinations of all possible configurations for a given supercell, and (3) datamining by examining all possible configurations in the literature with their formation energies predicted by machine learning. In addition to the conventional equilibrium phase diagram, non-equilibrium Scheil simulations were also used to analyze the formation of Al-Fe IMCs. The present work indicates that the forming mechanism of IMCs in dissimilar Al-Fe joints (see examples in Table 1) can be explained well using phase diagram, Scheil simulations, thermodynamic driving forces, P-and T-included Gibbs energies, and atomic diffusion coefficients in the Al-Fe system.  Table S1. However, Al 5 Fe 2 is an IMC with vacancies (Va) in its Wyckoff sites for Al atoms 60 . The structure of Al 5 Fe 2 can be described by the following sublattice model according to its Wyckoff sites 4c, 8 g, 4b (occupation of 0.32 by Al), and 8f. (occupation of 0.24 by Al) of space group Cmcm 60 , respectively, For another IMC of Al 2 Fe, Chumak et al. 61 indicated that it belongs to space group P1 with one of its Wyckoff sites 2i mixed with Fe (occupation of 0.705) and Al (occupation of 0.295), Atomic configurations of Al 5 Fe 2 were determined as follows in the present work. First, all the independent Al 5 Fe 2 configurations were generated by the ATAT code 65 using a 24-atom supercell, see Eq. (1). Second, we performed DFT calculations for the 14-to 16-atom configurations with one or two Al atoms in the Wyckoff sites 4b and 8f., respectively. For the composition of Al 5 Fe 2 , we also used the universal structure predictor-USPEX 62,66 -to predict the lowest energy configuration in terms of a 14-atom supercell; where the computational engine of USPEX is DFT-based calculations ( "Details of first-principles calculations" section). In addition, we also examined the low energy configurations of Al 5 Fe 2 suggested by Vinokur et al. 67 .

Methodology
Atomic configurations of Al 2 Fe were also examined by the ATAT code 65 based on the mixing of Al and Fe in Wyckoff site 2i (see Eq. 2) by using both the 38-and 57-atom supercells of Al 2 Fe. In addition, the MoSi 2 -type configuration suggested by Tobita et al. 68 was included in the present work. Aiming to search for the possible configurations of Al 2 Fe, we also adopted a datamining approach by considering all the AB 2 -type configurations (~ 1.3 million) in the Materials Project (MP) database 63 , the Open Quantum Materials Database (OQMD) 64 , the Crystallography Open Database (COD) 69,70 , and the Joint Automated Repository for Various Integrated Simulations (JARVIS) database 71 . The enthalpies of formation (ΔH 0 ) of these AB 2 -type configurations were predicted by machine learning (ML) in terms of the tool of SIPFENN (structure-informed prediction of formation energy using neural networks) 72 . Here, SIPFENN requires only atomic configurations and atomic species, which allows efficient integration into datamining study within minutes. On a random 5% subset in the OQMD structures, SIPFENN could achieve a mean absolute error of 28 meV/atom (2.7 kJ/mol-atom) to predict ΔH 0 72 . For the SIPFENN suggested A 2 B-type configurations with lower ΔH 0 values (more than 500 configurations were selected by considering the SIPFENN error bar up to 28 meV/atom), we performed DFT-based verifications. Notably, the present datamining approach found that the lowest energy configuration of Al 2 Fe is also the MoSi 2 -type. www.nature.com/scientificreports/ First-principles thermodynamics. Thermodynamic properties at finite temperatures can be predicted by the DFT-based quasiharmonic approach, i.e., Helmholtz energy F for a given phase as a function of volume V and temperature T is determined by 73,74 , Correspondingly, the Gibbs energy can be evaluated by G(P, T) = F(V , T) | P=fix + PV at the given pressure of interest. Here, E vib (V , T) and S vib (V , T) are vibrational contributions (internal energy and entropy, respectively) determined by phonon densities of states (DOS's, about 6 volumes were calculated for each phase) 73,75 . E el (V , T) and S el (V , T) are thermal electronic contributions (internal energy and entropy, respectively) determined by electronic DOS's 73,75 . S conf is ideal configurational entropy introduced to account for the IMCs with partially occupied Wyckoff sites, i.e., Al 5 Fe 2 (described by Eq. 1) and Al 2 Fe (Eq. 2), where R is the gas constant and y the site fraction with the superscript being Wyckoff site (i.e., also the sublattice). Based on experimental measurements for Al 5  where k 1 , k 2 , k 3 , and k 4 are fitting parameters. Equilibrium properties for each phase (configuration) from this EOS include the equilibrium energy E 0 , volume V 0 , bulk modulus B 0 , and the pressure derivative of bulk modulus B′. Usually, eight reliable data points were used for each EOS fitting in the present work.
It is worth mentioning that we ignored the contribution of anharmonicity to first-principles thermodynamics in Eq. (3), which can be accounted for by using such as molecular dynamics simulations 76,77 . In the present work, the relative Gibbs energy with respect to reference states (e.g., Al and Fe) was adopted to study phase stability, making the contribution of anharmonicity cancelled to some extent. In addition, we were trying to search for the possible "low energy atomic configurations" of Al 5 Fe 2 and Al 2 Fe (cf., "Atomic configurations of Al-Fe IMCs" section), and we used the ideal configurational entropy in first-principles thermodynamics for the sake of simplicity (cf., Eq. 3) for both Al 5 Fe 2 and Al 2 Fe, although the actual configurational entropy should be considered in terms of statistical mechanics (i.e., the partition function) by including all ergodic microstates (configurations) for a system (phase) of interest 78,79 . Note that even using the lowest energy atomic configurations of Al 5 Fe 2 and Al 2 Fe, we still need to consider configurational entropy due to the partially occupied Wyckoff sites. In summary, the sources of error in the present first-principles thermodynamics (Eq. 3) include the ignorance of anharmonicity, the adoption of ideal configurational entropy, the unknown atomic configurations of Al 5 Fe 2 and Al 2 Fe, and the approximations used in density functional theory such as the exchange-correlation (X-C) functional 80 . Nevertheless, the DFT-based quasiharmonic approach is still a predictive tool with great success to study thermodynamics in solid phases, see the examples in our review article 47 .

Details of first-principles calculations.
All DFT-based first-principles calculations in the present work were performed by the Vienna Ab initio Simulation Package (VASP) 81 with the ion-electron interaction described by the projector augmented wave method 82 and the X-C functional described by the generalized gradient approximation (GGA) developed by Perdew, Burke, and Ernzerhof (PBE) 83 . The same as those in the Materials Project 63 , three electrons (3s 2 3p 1 ) were treated as valence electrons for Al and fourteen (3p 6 3d 7 4s 1 ) for Fe. In the VASP calculations, a plane wave cutoff energy of 293.2 eV was employed for structural relaxations and phonon calculations in terms of the Methfessel-Paxton method 84 . Final calculations of total energies and electronic DOS's were performed by the tetrahedron method with a Blöchl correction 85 using a wave cutoff energy of 520 eV. The employed k-points meshes for each structure are listed in the Supplementary Table S1. The self-consistency of total energy was converged to at least 10 −6 eV/atom. Due to the magnetic nature of Fe, all Fe-containing materials were performed by the spin polarization calculations.
Phonon calculations were performed for each structure using the supercell approach 86 in terms of the YPHON code 87 . Here, the VASP code was again the computational engine in calculating force constants using the finite differences method. The employed supercell for each structure and the corresponding k-point meshes are given in the Supplementary Table S1. In addition, the single crystal elastic constants C ij 's in the Al-Fe system were determined by applying the stress-strain method with the non-zero strains being ± 0.01; see details in 88,89 . The aggregate polycrystal properties were determined by using the Hill (H) approach 90,91 based on the predicted C ij values, including bulk modulus (B H ), shear modulus (G H ), B H /G H ratio, Poisson's ratio (ν H ), and the anisotropy index A U92 . Note that the suggested DFT settings by USPEX 62,66 were used in the present work, aiming to search for the low energy configurations of Al 5 Fe 2 by USPEX. The IMC with the highest thermodynamic driving force of formation can be selected as the IMC that would form first, making the driving force D a reasonable criterion to predict the first-forming IMC 54 . Similarly to the analysis of interface phases formed at the Cu/solder joints by Lee et al. 54 , for example, Fig. 1 shows that at 1000 K of the Al-Fe system, the supercooled liquid has a composition x Fe = 0.163 (mole fraction of Fe in the metastable liquidus), which is in equilibrium with the supersaturated BCC phase (i.e., the metastable solidus) with x Fe = 0.281. At this composition (x Fe = 0.281), we can calculate thermodynamic driving forces of the IMCs (such as Al 13 Fe 4 , Al 5 Fe 2 , Al 2 Fe, and Al 8 Fe 5 ) formed from the supersaturated BCC phase -the higher the driving force, the larger the possibility to form this IMC. In the present work, thermodynamic driving forces to form IMCs from the supersaturated BCC phase were calculated as a function of temperature using the modeled Al-Fe system by Sundman et al. 9 and the Thermo-Calc software 57 .
In addition to thermodynamic driving forces, we can also use the non-equilibrium phase diagram, predicted by the Scheil-Gulliver simulations 49,50 (see its definition in the Introduction section), to predict the formation of IMCs in fast cooling processes, such as the AM process 51,52 . Here, we used the PyCalphad software 52,94 to calculate the Scheil non-equilibrium phase diagram with the Al-Fe thermodynamic description modelled by Sundman et al. 9 .

Results and discussion
DFT-based phase stability of Al-Fe IMCs. In this section, we show first the phase stability of Al-Fe IMCs at temperature T = 0 K and pressure P = 0 GPa ("DFT-based phase stability of IMCs at T = 0 K and P = 0 GPa") aiming to demonstrate the capable of DFT-based calculations; and then, we show the phase stability of Al-Fe IMCs at finite temperatures and finite pressures ("DFT-based phase stability of IMCs at finite temperatures and finite pressures" section) through both the case studies of three reactions and the predicted P-T phase diagram. Figure 3 shows the predicted values of enthalpy of formation (ΔH 0 ) for Al-Fe IMCs based on the present DFT calculations at T = 0 K and P = 0 GPa. Detailed atomic configuration and ΔH 0 value of each IMC are given in the Supplementary Table S1; in particular, the predicted 14-atom configuration of Al 5 Fe 2 by USPEX 62,66 is listed in the Supplementary Table S2. Figure 3 shows also the convex hull by DFT calculations to display the stable IMCs, the experimental ΔH 0 values collected by Sundman et al. 9 to measure the quality of the present DFT calculations, and the unstable configurations judged by imaginary phonon modes (not shown). It can be seen that (i) the DFT-predicted ΔH 0 values agree well with the experimental data which are scattered; (ii) Al 6 Fe is close to but above the convex hull, indicating that it is metastable at T = 0 K and P = 0 GPa, and more attentions need to be paid to its phase stability at high temperatures and high pressures; (iii) Al 9 Fe 2 is an unstable structure due to the existence of imaginary phonon modes and hence ignored in the present work; (iv) Al 5 Fe 2 is a metastable phase at T = 0 K and P = 0 GPa, although various configurations have been examined in the present work (see the green open squares as well as the details in Table S1); (v) the MoSi 2 -type Al 2 Fe possesses the lowest energy and on the convex hull at T = 0 K and P = 0 GPa, but this configuration doesn't belong to space group P1 as suggested by Chumak et al. 61 ; and (vi) the IMCs of Al 13 Fe 4 , AlFe, and AlFe 3 are stable based on the convex hull. Figure 3 implies that, at the conditions of T = 0 K and P = 0 GPa, the DFT predicted ΔH 0 values for Al 5 Fe 2 and non-MoSi 2 -type Al 2 Fe (i.e., the triclinic Al 2 Fe 61 ) are close to but above the convex hull, indicating that (a) the supercells used in the present work may be too small to    Table S1). Note that the convex hull was plotted using the DFT results, the unstable IMCs were judged by imaginary phonon modes, and the experimental data (Expt.) were collected by Sundman et al. 9 .

DFT-based phase stability of IMCs at T = 0 K and P = 0 GPa.
Scientific Reports | (2021) 11:24251 | https://doi.org/10.1038/s41598-021-03578-0 www.nature.com/scientificreports/ search for the lowest energy atomic configurations, and (b) additional effects on phase stability such as temperature, pressure, and new approaches need to be considered. To this end as well as the suggestions by Fig. 3, phase stabilities of Al 6 Fe, Al 5 Fe 2 , and Al 2 Fe are further examined at finite temperatures and finite pressures (see Fig. 4).

DFT-based phase stability of IMCs at finite temperatures and finite pressures.
Phase diagram at a given temperature and pressure can be constructed using the convex hull approach, i.e., by examining all reaction Gibbs energies, G reac , for a system of interest. Note that in general one reaction cannot determine phase stability in the whole temperature and pressure ranges. As test cases, Fig. 4 shows only the changes of G reac as a function of temperature and pressure (P = 0 and 6 GPa as two examples) for the following three reactions, aiming to understand phase stability of Al 6 Fe as well as Al 5 Fe 2 and Al 2 Fe with respect to the given reference phases (instead of building the convex hull), Here we choose the stable phases of Al, Al 13 Fe 4 , and AlFe (the B2 structure) as the reference states to examine phase stability of Al 6 Fe, Al 5 Fe 2 (using the configuration predicted by USPEX), and Al 2 Fe (using the MoSi 2 -type configuration predicted by SIPFENN). As mentioned at the end of "First-principles thermodynamics" section, the ideal configurational entropies together with the possible "low energy configurations" were used for Al 5 Fe 2 and Al 2 Fe, resulting in a large contribution of configurational entropy than the actual case. However, the predicted configurations of Al 5 Fe 2 and Al 2 Fe are still not the lowest energy ones based on the present approach, making the error by using the larger ideal configurational entropy cancelled to some extent. Also the G reac values with and without the contributions of ideal configurational entropy form an uncertainty range to analyze phase stability of Al-Fe IMCs. Figure 4 shows that Al 6 Fe is a T-unfavored (see R1) but a P-favored phase, which can be understood through phonon density of states as detailed in Supplementary Materials. Figure 4b. It shows that with increasing pressure (even less than 1 GPa) instead of increasing temperature, Al 6 Fe becomes stable with respect to Al and Al 13 Fe 4 (cf., the reaction R1). Based on experimental observations such as the examples #1 to #5 in Table 1, Al 6 Fe was formed in the processes associated with pressures (such as die casting and equal channel angular extrusion) and in high Al-containing samples (e.g., x Al > 0.9). The reaction R2 (see Eq. 8) in Fig. 4a and b shows that Al 5 Fe 2 is a    These results indicate that factors including atomic configuration, temperature, pressure, and S conf make Al 5 Fe 2 more stable. Figure 4 also shows that the MoSi 2 -type Al 2 Fe is T-unfavored, but it is a strong P-favored phase.
In addition, the S conf has less contribution to G reac in comparison with that for Al 5 Fe 2 , due to the less partially occupied Wyckoff site of Al 2 Fe; see Eqs. (4) and (5). The T-unfavored behavior is caused by the lower phonon DOS of Al 2 Fe than those of AlFe and Al 13 Fe 4 ; see details in Supplementary Material. With increasing pressure, Fig. 4 shows that the G reac value of reaction R3 decreases greatly; for example, dropping more than 2 kJ/molatom at T = 0 K as well as at other temperatures. Experimentally, the MoSi 2 -type Al 2 Fe was synthesized through the laser-heated diamond-anvil cell at 10 GPa and 1873 K 95 , and it was suggested that it is a high pressure phase existing at P > 5 GPa 68 ; these experiments agree with the present conclusion that Al 2 Fe is a T-unfavored but a strong P-favored phase, albeit it is stable at T = 0 K and P = 0 GPa (Fig. 3). Figure 5 shows a schematic P-T phase diagram (demonstrated with P = 0 and 6 GPa) for the Al-Fe system based on the present DFT calculations using Eq. (3) based on the convex hull approach by considering all G reac values. As an example, the G reac values at P = 0 GPa for six reactions are shown in the Supplementary Figure S2, where the reaction R4 can be used to determine the critical temperatures of Al 5 Fe 2 in some temperatures and pressures. Figure 5 indicates that Al 13 Fe 4 , AlFe, and AlFe 3 are always the stable IMCs marked by the shaded regions. However, at low pressures and low temperatures (e.g., P = 0 GPa and T < 165 K), the L1 2 -type AlFe 3 is more stable than the D0 3 -type AlFe 3 . It is worth mentioning that AlFe 3 from DFT-based predictions is either a L1 2 structure or a D0 3 structure depended on the selected X-C functional 96,97 . The commonly used X-C functional of GGA predicts that the L1 2 -AlFe 3 is more stable at 0 K with respect to the D0 3 -AlFe 3 (see Table S1 as well as the results in the literature 63,96,97 ). However, the energy difference between the L1 2 and D0 3 structures of AlFe 3 is very small (< 0.1 kJ/mol-atom, see Table S1), which is within the uncertainty of DFT predictions. Regardless of the stable structure at 0 K for AlFe 3 (L1 2 vs. D0 3 ), the present work shows that vibrational entropy makes the D0 3 structure more stable at high temperatures (> 165 K and P = 0 GPa) and/or at high pressures (> ~ 1 GPa); agreeing with the experimentally observed AlFe 3 with the D0 3 structure 9 . Over the entire temperature range in Fig. 5, Al 6 Fe is not stable at P = 0 GPa, but is stable at higher pressures. Al 5 Fe 2 (configuration predicted by USPEX) is stable at high temperatures (e.g., T > 345 K with P = 0 GPa), while higher pressures decrease its stability slightly. The MoSi 2 -type Al 2 Fe is a T-unfavored but a strong P-favored phase.
In comparison with the IMCs reported in the CALPHAD modelled Al-Fe phase diagram at P = 0 K and low temperatures (e.g., < 1000 K in Fig. 1), the DFT-based predictions in Fig. 5 agree reasonably well with those by the CALPHAD modeling, including the existed Al 13 Fe 4 , AlFe, and AlFe 3 , as well as the absent Al 6 Fe. The deviations are only for Al 5 Fe 2 and Al 2 Fe, which are stable at low temperatures (e.g., < 1000 K) by CALPHAD modeling but are not always stable by DFT-based predictions, indicating that the present configurations of Al 5 Fe 2 and Al 2 Fe are still not the lowest energy ones. It should be mentioned that the presently predicted Al 5 Fe 2 configuration (see Table S2) has the lowest energy than the configurations reported in the literature (see Table S1) 67 , while the presently predicted MoSi 2 -type Al 2 Fe is the same as the one suggested by Tobita et al. 68 To the best of our knowledge, the present configurations are the lowest energy ones which can be currently predicted for Al 5 Fe 2 and Al 2 Fe, but future efforts are still needed to predict new lower energy configurations by using larger supercells or new approaches. www.nature.com/scientificreports/ Table 2 summarizes phase stability of Al-Fe IMCs as a function of pressure and temperature as shown in Figs. 1, 3, and 5; together with their ductility/brittleness judged by Pugh's criterion 11,12 as shown in Fig. 2, which were determined by the presently predicted elastic constants in Table S3.

Non-equilibrium Al-Fe IMCs by thermodynamic and kinetic analyses.
In this section, we show first the formation of non-equilibrium IMCs by thermodynamic driving forces and kinetic analyses ("Non-equilibrium IMCs by thermodynamic driving forces and kinetic analyses" section); and then, we show the formation of non-equilibrium IMCs by Scheil simulations ("Non-equilibrium IMCs by Scheil simulations" section). Figure 6 shows the predicted thermodynamic driving forces of the Al-Fe IMCs as a function of temperature (T = 920-1320 K) as well as the associated mole fraction of Fe (x Fe = 0.28-0.40) along the metastable solidus line as shown in Fig. 1. Note that the eutectic reaction temperature is 927 K and the chosen thermodynamic description was that modelled by Sundman et al. 9 . It is seen that both Al 13 Fe 4 and Al 5 Fe 2 have the higher thermodynamic driving forces of formation at lower temperatures (< 1280 K) than those of Al 2 Fe and Al 8 Fe 5 . By examining atomic diffusivity in Al-Fe IMCs, the interdiffusion coefficients in Al 5 Fe 2 are at least two orders of magnitude faster than those in the other IMCs (AlFe, Al 2 Fe, and Al 13 Fe 4 ) at T = 823 -913 K 42 and are comparable with the diffusion coefficients of dilute Fe in FCC Al; see Fig. 7 the diffusion coefficients reported in the literature 42,58,59 . In addition, Al atoms have higher diffusivity in Al 5 Fe 2 than Fe atoms 38 . The fastest atomic diffusivity, especially Al atoms, in Al 5 Fe 2 is due mainly to the rich Al vacancies in Al 5 Fe 2 60 ; see Eq. (1). However, considerable vacancies have not been reported in the other Al-Fe IMCs. By considering both the high thermodynamic driving force of formation (Fig. 6) and the fastest interdiffusion coefficients (Fig. 7), the brittle Al 5 Fe 2 is the IMC with the largest possibility to be formed; see the Al-rich examples in Table 1, excepting those with extremely high Al contents, formed below the eutectic reaction temperature of 927 K, or processed by AM (examples #1 to #5, and #7 to #11).

Non-equilibrium IMCs by Scheil simulations.
As two examples of fast cooling solidification, Fig. 8 shows the calculated mole fractions of solid phases by Scheil simulations using the thermodynamic description modelled Table 2. Summary of phase stability of key Al-Fe IMCs with respect to pressure (P) and temperature (T) shown in Figs. 1, 3, and 5 (or not shown); together with their ductility/brittleness according to Pugh's criterion 11,12 as shown in Fig. 2. a Names used in the present work together with the names in the parentheses used in the literature. b These IMCs are always stable and on the convex hull in the present P and T of studied.   Table 1. For the case of x Fe = 0.6, the first solid phase formed with decreasing temperature is BCC (or the B2 phase), which reaches a maximum mole fraction of 0.95, and then Al 8 Fe 5 forms in a small temperature range of 1505-1493 K. Similar to the case of x Fe = 0.3, the predicted Al 8 Fe 5 was also not observed in the processes in Table 1 due probably to the small temperature range of phase formation. Figure 9 shows the complete non-equilibrium phase diagram by Scheil simulations using the modelled Al-Fe system by Sundman et al. 9 . This non-equilibrium phase diagram shows the temperatures of the forming phases, though the lever rule cannot be used to determine phase fractions. Both the equilibrium phase diagram (Fig. 1) and the Scheil non-equilibrium phase diagram (Fig. 9) can be used to determine the forming phases in the slow/equilibrium and the fast cooling processes, respectively.
As an example to examine equilibrium and Scheil simulations, Fig. 10 show the forming phases as a function of temperature with x Fe = 0.4. The forming phases are BCC and Al 8 Fe 5 (majorly) based on Scheil simulations (see also Fig. 9), while the forming phases are Al 8 Fe 5 (when T > 1360 K), BCC, and Al 2 Fe based on equilibrium calculations (see also Fig. 1). Therefore, the forming phases could be BCC, Al 2 Fe, and/or Al 8 Fe 5 depended on different processes. For instance, Stein et al. 99 observed the eutectoid reaction of Al 8 Fe 5 ↔ Al 2 Fe + BCC (B2) at 1368 K by the differential thermal analyses for the Al-40 at.% Fe alloy (x Fe = 0.4) at the heating rates of 5 and 10 K/min.
It should be remarked that the forming phases depend mainly on compositions (especially the local compositions) in addition to temperature, pressure, and atomic diffusivity for the system of interest. Table 1 shows that AM is superior to the other processes in achieving desired phases such as AlFe and AlFe 3 through varying the compositions. To predict the forming IMCs under a given composition and a given processing history, the combined thermodynamic and kinetic simulations are needed. For example, Lindwall et al. 100 simulated the

Summary
The present work investigated the forming mechanism of equilibrium and non-equilibrium intermetallic compounds (IMCs) in dissimilar aluminum/steel (Al-Fe) joints by means of Gibbs energy as a function of temperature (T) and pressure (P) from (i) first-principles phonon calculations, (ii) equilibrium Al-Fe phase diagram in the literature and the presently predicted non-equilibrium phase diagram by Scheil simulations, (iii) atomic diffusivity in Al-Fe, and (iv) experimentally observed IMCs in various processes (cf., Table 1). In particular, the unknown atomic configurations of Al 2 Fe and Al 5 Fe 2 were examined in the present work by machine learning based datamining together with first-principles verifications and structure predictor (using USPEX). To the best of our knowledge, the presently predicted configurations of Al 2 Fe and Al 5 Fe 2 possess lower energies in comparison with the configurations reported in the literature. However, the present configurations are still not the lowest energy ones, hence appealing for future efforts. In addition, the predicted MoSi 2 -type Al 2 Fe is a pressure-favored IMC, instead of the phase with space group P1 shown in the experimental phase diagram. Note that the present DFT-based thermodynamics is based on the quasiharmonic approach with the possible sources of error from such as the ignorance of anharmonicity, the adoption of ideal configurational entropy, the unknown atomic configurations of Al 5 Fe 2 and Al 2 Fe, and the approximations adopted in density functional theory. Al-Fe IMCs formed in various experimental processes are summarized in Table 1 ("Introduction" section). The present work concludes that the formation of IMCs can be explained well by phase diagrams, thermodynamic driving forces, P-and T-included Gibbs energy, and atomic diffusion coefficients. Specifically, the metastable and ductile Al 6 Fe is a P-favored IMC, which was observed in Al-dominant samples and the processes involving   Figure 9. Predicted Al-Fe non-equilibrium phase diagram by Scheil simulations uisng the modelled data by Sundman et al. 9 , showing the forming temperatures for the phases indicated by the lines. Note that the lever rule cannot be used to determine phase fractions (see Fig. 8 for two examples). www.nature.com/scientificreports/ pressures such as direct-chill casting, die casting, equal channel angular extrusion. Here the ductility and brittleness of IMCs were judged by Pugh's criterion 11,12 using the presently predicted elastic constants. The MoSi 2 -type Al 2 Fe is a brittle and strong P-favored IMC observed at high pressures. The stable but brittle η-Al 5 Fe 2 is the most observed IMC usually adjacent to steel (Fe) in almost all the processes as detailed in Table 1, such as fusion or solid-state welding, immersion testing, diffusion couples, and additive manufacturing (AM), since Al 5 Fe 2 is a T-favored phase with a high thermodynamic driving force of formation and the fastest atomic diffusivity among all Al-Fe IMCs. The slightly brittle θ-Al 13 Fe 4 is the second most observed IMC usually adjacent to Al shown in most of the processes, possessing the highest thermodynamic driving force of formation in Al-rich side. Notably, the ductile AlFe 3 , the less ductile AlFe, and almost all the other IMCs were observed in the AM processes, making AM an exceptional way to tailor composition and in turn achieve the desired IMCs in dissimilar materials. All the IMCs (without the P-favored phases) formed in the Al-Fe joints can be identified using the equilibrium and the Scheil non-equilibrium phase diagrams, together with kinetic considerations. License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/.