Forming Mechanism of Equilibrium and Non-equilibrium Metallurgical Phases in Dissimilar Materials: Illustrated With 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 the dissimilar aluminum/steel joints with respect to processing history (e.g., the pressure- and temperature-profiles) and chemical composition, where the used 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 while ductile (judged by the presently predicted elastic constants) Al6Fe is a pressure (P) favored IMC observed in the processes involving high pressures. The MoSi2-type Al2Fe is a brittle and a strong P-favored IMC observed at high pressures. The stable, brittle h-Al5Fe2 is the most commonly observed IMC (followed by q-Al13Fe4) in almost all processes, such as fusion/solid-state welding and additive manufacturing (AM), since h-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 predictor. All the IMCs, which are not P-favored, can be identified using the conventional equilibrium phase diagram and the Scheil-Gulliver non-equilibrium simulations.


Introduction
Joining of dissimilar materials has become increasingly important to create lightweight, highperformance, and economic structures applied in various industries, for example, automotive 1 , aerospace 2,3 , marine 4 , and information technology 5 . Specially, joining of aluminum (Al) to steel/iron (Fe) is of eminent technical interest due to the use of two essential engineering materials in the same design 1, 6 . It is known that mechanical properties of dissimilar materials are strongly affected by the type, amount/thickness, and morphology of the metallurgical phases formed at the bonding interfaces. For example, the formation of brittle intermetallic compounds (IMCs), such as -Al5Fe2 7,8 , is usually detrimental to the performance of dissimilar materials joints owing to the reduction of materials' strength, ductility, and fracture toughness. A great deal of efforts in chemistry and process design is hence required to avoid or reduce their formation in dissimilar materials, demanding fundamental understanding of phase stability of IMCs during various processes, for example, different pressure (P) and temperature (T) profiles under a given chemical composition.
Relevant to the present focus of Al-Fe joints, there are six IMCs shown in the equilibrium Al-Fe phase diagram under external pressure P = 0 GPa; see Figure 1 which was modelled by the CALPHAD (calculations of phase diagram) approach by Sundman et al. 9 . It includes the stable IMCs of -Al13Fe4, -Al5Fe2, Al2Fe, AlFe (in B2 structure), AlFe3 (D03), and the metastable ε-Al8Fe5 (D82). In addition, the other metastable IMCs include Al6Fe and AlmFe (4  m  4.4) 10 , which are absent in Figure 1. It is believed that the Al-rich IMCs (Al13Fe4, Al5Fe2, and Al2Fe) are brittle and favor crack nucleation in the joints, while the Fe-rich IMCs (i.e., the BCC based AlFe and AlFe3) show higher ductility and strength 7,8 . The ductility and brittleness of these IMCs are shown in Figure 2 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., Sec. 2.3). It indicates the ductile Al6Fe, Al5Fe8, and AlFe3; the less ductile Al13Fe4 and AlFe; and the brittle Al5Fe2 and Al2Fe. Table 1 summarizes the Al-Fe IMCs formed in different processes reported in the literature.
The metastable, ductile Al6Fe was observed in the processes of direct chill casting (example #1 shown 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 the laser powder bed fusion (#5) 16 . These observations suggest that Al6Fe is an IMC existed at high pressures. Table 1 further depicts that most of the stable and even metastable Al-Fe IMCs were observed in the AM processes. For example, the Al6Fe, Al13Fe4, Al2Fe, Al5Fe2, AlFe, and/or AlFe3 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) Al13Fe4, AlFe, and AlFe3 [20][21][22] (examples #9 to #11) were observed in the Al-Fe based functional gradient materials fabricated by additive manufacturing. These experiments indicate that AM is an exceptional process to tailor compositions and in turn the desired IMCs. In the fusion and/or solid-state welding joints, Al5Fe2 is the most observed IMC (usually adjected to iron/steel) followed by Al13Fe4 (usually adjacent to Al) processed by, for example, laser welding [23][24][25] (see examples #13 to #15 in 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 45 . Phase diagram, as the beginning of wisdom to guide any work in materials science and engineering 46 , 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 47,48 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 [49][50][51] . The Scheil simulations have been used to, for example, design optimal composition for additively manufactured functionally graded metals 49,50 and predict liquidus and solidus temperatures of steel 51 . In addition to 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. 52 . 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 53 . 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 Al6Fe in the present work. In addition to thermodynamics, kinetics (diffusion) is another factor to regulate nucleation, growth, and coarsening of IMCs 54,55 . For example, Al5Fe2 and Al13Fe4 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) 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.

Atomic configurations of Al-Fe IMCs
Most of the  69 . The enthalpies of formation (H0) of these AB2-type configurations were predicted by machine learning (ML) in terms of the tool of SIPFENN (structure-informed prediction of formation energy using neural networks) 70 . 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 H0 70 . For the SIPFENN suggested A2B-type configurations with lower H0 values (more than 500 configurations were selected herein), we performed DFT-based verifications. Notably, the present datamining approach found that the lowest energy configuration of Al2Fe is also the MoSi2-type.

First-principles thermodynamics
Thermodynamic properties at finite temperatures can be predicted by 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 71,72 , Eq. 4 Al 2 Fe = − 4 ( Al log( Al ) + Fe log( Fe )) 38 Eq. 5 where R is gas constant and y the site fraction with the superscript being Wyckoff site (sublattice).
Based on experimental measurements for Al5Fe2 58  ( ) in Eq. 3 is the static energy at 0 K without the zero-point vibrational energy, which was determined by fitting the DFT calculated energy-volume (E-V) data points using a four-parameter Birch-Murnaghan equation of state (EOS) 71 , where k1, k2, k3, and k4 are fitting parameters. Equilibrium properties for each phase from this EOS include the equilibrium energy E0, volume V0, bulk modulus B0, and the pressure derivative of bulk modulus B. Usually, eight reliable data points were used for each EOS fitting in the present work.

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) 74 with the ion-electron interaction described by the projector augmented wave method 75  Phonon calculations were performed for each structure using the supercell approach 79 in terms of the YPHON code 80 . 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-points meshes are given in the Supplementary  60,64 were used in the present work, aiming at searching for the low energy configurations of Al5Fe2 by USPEX.

Formation of non-equilibrium IMCs through thermodynamic analysis
The decrease in Gibbs energy, −∆ , for the precipitation of a new phase  (e.g., IMC) from a supersaturated solution (e.g., the supercooled liquid), is the thermodynamic driving force of formation, D, of the new  phase, i.e., = −∆ 86 . 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 52 . Similarly to the analysis of interface phases formed at the Cu/solder joints by Lee et al. 52 , for example, Figure 1 shows that at 1000 K of the Al-Fe system, the supercooled liquid has a composition xFe = 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 xFe = 0.281. At this composition (xFe = 0.281), we can calculate thermodynamic driving forces of the IMCs (such as Al13Fe4, Al5Fe2, Al2Fe, and Al8Fe5) 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 of the formation of 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 55 .
In addition to thermodynamic driving force, we can also use the non-equilibrium phase diagram, predicted by the Scheil-Gulliver simulations 47,48 (see its definition in the Introduction section), to predict the formation of IMCs in fast cooling processes, such as the AM process 49,50 . Here, we used the PyCalphad software 50,87 to calculate this non-equilibrium phase diagram with the thermodynamic description modelled by Sundman et al. 9 .

Results and Discussion
3.1 DFT-based phase stability of IMCs with respect to temperature and pressure   Figure 3 shows also the convex hull by DFT calculations to display the stable IMCs, the experimental H0 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 is seen (in Figure 3) that (i) the DFT-predicted H0 values agree well with the experimental data which are scattered; (ii) Al6Fe 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 for its phase stability at high temperatures and high pressures; (iii) Al9Fe2 is an unstable structure and hence ignored in the present work; (iv) Al5Fe2 is a metastable phase at T = 0 K and P = 0 GPa, albeit various  Figure 3 implies that, at the conditions of T = 0 K and P = 0 GPa, the presently predicted H0 values for Al5Fe2 and non-MoSi2-type Al2Fe (i.e., the triclinic Al2Fe 59 ) are close to but above the convex hull, indicating that (a) the supercells used herein may be too small to search for the stable atomic configurations, and (b) more effects on phase stability such as temperature and pressure need to be considered. To this end as well as the suggestions by Figure 3, phase stabilities of Al6Fe, Al5Fe2, and Al2Fe are further examined at finite temperatures and finite pressures (see Figure 4). where is phonon frequency. Eq. 10 indicates that the higher the phonon DOS in the low region, the higher the contribution to Gibbs energy will be (see Eq. 3) 54,88,89 . Figure 5 shows the phonon DOS's of FCC Al and the selected Al-Fe IMCs at P = 0 GPa. In general, it shows that the ( ) of Al has the highest density than Al-Fe IMCs at the low frequency region (e.g., < 6 THz), since FCC Al is the softest material with the largest equilibrium volume V0 and the smallest bulk modulus B0 in the Al-Fe system; see the Supplementary Table S 1. Relevant to the reaction R1 (Eq. 7) and at the low frequency region ( < 6 THz), Figure 5 shows that the ( ) of Al is much higher than that of Al6Fe with 3.5 < < 6 THz, but the ( ) of Al6Fe has higher values in a small region with < 2 THz. In addition, the phonon DOS's of Al and Al13Fe4 do not have significant differences at the low frequency region. These features imply that the contributions to both Al and Al13Fe4 should be slightly higher than that of Al6Fe, resulting a slight increase of ∆ reac for reaction R1 with increasing temperature (at P = 0 GPa). At higher pressures such as P = 6 GPa, ∆ reac for reaction R1 keeps roughly constant since the contributions to both Al and Al13Fe4 are nearly identical to that of Al6Fe. Figure 4b shows that with increasing pressure (even less than 1 GPa) instead of increasing temperature, Al6Fe becomes stable with respect to Al and Al13Fe4 (cf., the reaction R1). Based on experimental observations such as the examples #1 to #5 in Table 1, Al6Fe 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., xAl > 0.9).
The reaction R2 (see Eq. 8) in Figure 4a and b shows that Al5Fe2 is a T-unfavored but P-favored phase by ignoring the contribution of configurational entropy Sconf; see Eq. 3 and Eq. 4. The Tunfavored Al5Fe2 is due mainly to the lower phonon DOS of Al5Fe2 compared to that of AlFe with frequency around 4 THz (Figure 5). With Sconf contribution to ∆ reac for reaction R2, Al5Fe2 becomes both the T-and P-favored phase (see the blue dash lines of R2). These results indicate that the factors, including atomic configuration, temperature, pressure, and Sconf, make Al5Fe2 more stable.

Figure 4
shows that the MoSi2-type Al2Fe is T-unfavored, but it is a strong P-favored phase. In addition, the Sconf has less contribution to ∆ reac in comparison with that for Al5Fe2, due to the less partially occupied Wyckoff site of Al2Fe; see Eq. 4 and Eq. 5. The T-unfavored behavior is caused by the lower phonon DOS of Al2Fe than those of AlFe and Al13Fe4; see the range from 2 to 7 THz (Figure 5). With increasing pressure, Figure 4 shows that the ∆ reac value of reaction R3 decreases greatly; for example, dropping more than 2 kJ/mol-atom at T = 0 K as well as at other temperatures. Experimentally, the MoSi2-type Al2Fe was synthesized through the laser-heated diamond-anvil cell at 10 GPa and 1873 K 90 , and it was suggested that it is a high pressure phase existed with P > 5 GPa 66 ; agreeing with the present conclusion that Al2Fe is a T-unfavored but a strong P-favored phase, albeit it is stable at T = 0 K and P = 0 GPa (Figure 3). AlFe3 are already 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 L12-type AlFe3 is more stable than the D03type AlFe3. In all temperature range in Figure 6, Al6Fe is not stable when P = 0 GPa but stable at higher pressures. Al5Fe2 (configuration predicted by USPEX) is stable at high temperatures (e.g., T > 345 K with P = 0 GPa), while the pressure decreases its stability slightly. The MoSi2-type Al2Fe is a T-unfavored but a strong P-favored phase. Table 2 summarizes phase stability of Al-Fe IMCs as a function of pressure and temperature as shown in Figure 1, Figure 3, and Figure 6; together with their ductility/brittleness judged by Pugh's criterion 11,12 as shown in Figure 2, which were determined by the presently predicted elastic constants in Table S 3. Figure 7 shows the predicted thermodynamic driving forces of the Al-Fe IMCs in the temperature range, 920 K < T < 1320 K, together with the associated mole fraction of Fe (xFe) along the metastable solidus line in Figure 1. Note that the eutectic reaction temperature is 927 K and the used thermodynamic description was modelled by Sundman et al. 9 . It is seen that both Al13Fe4  Table 1, except for the samples with extremely high Al contents, or formed below the eutectic reaction temperature of 927 K, or processed by AM (examples #1 to #5, and #7 to #11).

Phase stability of Al-Fe IMCs by thermodynamic and kinetic analyses
As two examples, Figure 9 shows the calculated mole fractions of solid phases by Scheil simulations using the thermodynamic description modelled by Sundman et al. 9 . With decreasing tempeature at the fixed composition of xFe = 0.3, the solid phase of Al5Fe2 forms first and reaches a maximum mole fracition about 0.5 at T = 1427.5 K, and then the second solid phase of Al8Fe5 forms at almost the fixed temperature of 1427.5 K. Due to the exteremely small temperture range (<< 1 K) for phase transition, Al8Fe5 was not observed in all the processes in Table 1. For the case of xFe = 0.6, the first formed solid phase is BCC (or B2 phase) with decreasing temperature and reaches a maximum mole fraction of 0.95, and then Al8Fe5 forms in a small temperature range of 1505 ~ 1493 K. Similar to the case of xFe = 0.3, the predicted Al8Fe5 was also not observed in the processes in Table 1 due probably to the small temperature range of phase formation. Figure 10 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, while the lever rule cannot be used to determine phase fractions. Both the equilibrium phase diagram (Figure 1) and the Schiel non-equilibrium phase diagram (Figure 10) can be used to determine the forming phases in the slow/equilibrium and the fast cooling processes, respectively.
As an example, Figure 11 show the forming phases as a function of temperature with xFe = 0.4.
The forming phases are BCC (minor) and Al8Fe5 (major) based on Schiel simulations (see also  It should be mentioned 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 a superior process than the other processes to achieve desired phases such as AlFe and AlFe3 through varying the compositions. Aiming 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. 92 simulated the time-temperaturetransformation (TTT) diagram and the forming IMCs in the additively manufactured Ni-based Inconel 625. However, these simulations are beyond the scope of the present work.

Summary
The present work investigated the forming mechanism of equilibrium and non-equilibrium We conclude that the formation of IMCs can be explained well by using phase diagrams, thermodynamic driving forces, P-and T-included Gibbs energy, and atomic diffusion coefficients.
Specifically, the metastable while ductile Al6Fe is a P-favored IMC, which was observed in Aldominant samples and the processes involving 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 MoSi2-type Al2Fe is a brittle and a strong P-favored IMC observed at high pressures. The stable but brittle -Al5Fe2 is the most observed IMC usually adjacent to steel (Fe) in almost all the processes (see Table 1), such as the fusion or solid-state welding, immersion test, diffusion couple, and additive manufacturing (AM), since Al5Fe2 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 -Al13Fe4 is the second most observed IMC usually adjacent to Al shown in most of the processes, which possesses the highest thermodynamic driving force of formation in Al-rich side. Notably the ductile AlFe3, 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.

Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.    Table S 3. Note that the 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 12 .  Table S 1). 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 .        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 Figure 9 for two examples).  Al5Fe2 and Al2Fe; and AlFe (long-term annealing) 13 23 Al-steel overlap joints Laser welding (up to 1200C) Al5Fe2 (assuming diffusion from Fe to Al only) 14 24 Al alloy 6061-T6 and galvanized steel DP590 Laser welding without filler Al13Fe4 and Al5Fe2 with linear energy density of 162 J/mm; Al13Fe4, Al5Fe2, and AlFe with 309 J/mm 15 25 Al alloy 5083 and low alloy steel (XF350) plates Fiber laser welding with 8 kW of max power Al5Fe2 near steel (main) and Al13Fe4 near Al 16 26 Pure Al (1100) and low carbon steel Friction stir welding Al5Fe2 and Al13Fe4 17 27 Al alloy (5186) and low carbon steel Friction stir welding Al5Fe2 (adjacent to Fe) and Al13Fe4 (adjacent to Al, facilitated by Fe diffusion) 18 28 Al sheet (6061) and galvannealed steel sheet Friction stir welding Al13Fe4 (large size, diffusion induced) and AlFe3 (small size) 19 29 Al alloy 5754 with coated DP600 or 22MnB5 steel Diffusion bonding by friction stir welding Al5Fe2 in low welding speeds (16 mm/min) and AlFe in 45 mm/min 20 30 Al alloy 5083 and steel (< 0.1 wt.% C) sheets Annealing of friction stir lap welds Al5Fe2 (major) and Al13Fe4 annealed at 673 K for 180 min 21 31 Al alloy 6061-T6 and AISI 1018 steel Friction welding Al5Fe2 and AlFe (suggested based on compositions) 22 32 Al sheet (6016) and galvanized IF-steel sheet Friction stir spot welding Al13Fe4, Al5Fe2, and Al2Fe 23 33 Al alloy (surfalex 6 s) and ultrahigh strength steel Friction stir scribe welding Al5Fe2 (in the middle) or Al13Fe4 with Fe/Al solid solution depending on the weld regions 24 34 Al alloy (1050) sheets and Fe particles Friction stir processing Al5Fe2 close to Fe particle; and Al13Fe4 close to Al matrix 25 35 Al sheet (6061 T4) and coated steel sheet Hot-dip Al-coated steel Aluminized steel at 800C for 60 s, then 680C for 60 s Al13Fe4 just beneath Al cover layer and Al5Fe2 just underneath steel a Addition of Mn promotes the formation of Al6(Fe,Mn). Table 2. Summary of phase stability of key Al-Fe IMCs with respect to pressure (P) and temperature (T) shown in Figure 1, Figure 3, and Figure 6 (or not shown); together with their ductility/brittleness according to Pugh's criterion 11,12 as shown in Figure 2.