First-principles study of tensile and shear strength of an Fe2Al5//Fe interface

The interfacial strengths of a low misfit Fe2Al5//Fe interface structure found at aluminum-steel joints has been studied using density functional theory. An interface between Fe and Fe2Al5 was selected based on a criteria of low lattice misfit and number of atoms. Through virtual tensile testing of bulk Fe2Al5 and the interface structures we show that the energy-displacement curve can be well described by including extra polynomial terms in the Universal Binding Energy Relation (UBER). It is shown that the Fe2Al5//Fe interface has a higher tensile strength than the bulk Fe2Al5 phase. We also find that the shear deformation process potentially can be initiated from an Fe-terminated interface.


Introduction
Owing to the increased interest in light-weight and environmentalfriendly technology, Fe-Al compounds have been gaining increased industrial interest due to their light-weight, corrosion resistance and hightemperature resistance behavior [1][2][3]. However, the joining of aluminum and steel by traditional fusion welding techniques has been considered a main challenge due to the significant differences between their physical and chemical properties [4,5]. The intermetallic compounds (IMCs) which develop at the interface are normally not wanted, but unavoidable when welding aluminum and steel.
Various methods have been proposed and studied to join aluminum and steel [3,6]. For any method which requires high temperatures, a brittle layer of different types of Fe-Al IMCs is developed at the joint, making it difficult to obtain the desired joint strength. Although solidstate welding techniques can suppress the formation of Fe-Al IMCs at joints due to the low temperature, these methods can still not completely limit the formation of IMCs and can thus only produce Fe-Al joints with limited strength.
The thickness of the IMC layers also plays an important role in the strengthening of Fe-Al joints. It has been reported that the thickness of Fe-Al IMC layers formed in a brazed interface can be limited to less than 10 μm, which is considered as the critical thickness of a Fe-Al IMC layer for Fe-Al joints with good mechanical strength [7]. Analyses of Fe-Al joints suggest that the micro-structures and distribution of Fe-Al IMCs at the interface are dependent on heat input, and play an important role in determining the mechanical and/or corrosion behavior of the joints [8,9]. In general, most of the experimental and theoretical studies on Fe-Al IMC layers focus on, (i) heat input and thickness of the IMC layers [10] (ii) welding methodology [11,12] (iii) tensile and shear strength of IMC layers at the joint [3] and (iv) extended isothermal treatment [13][14][15].
Despite all these studies, the interfacial strength of intermetallics such as Fe 2 Al 5 //Fe has not been studied much in literature. Since it is thermodynamically possible to produce a range of Fe-Al compounds at the interface [16,17], it is necessary to understand the basic mechanical and interfacial strength of all these compounds to clarify their roles for the joint strength. The lack of convincing results for the interfacial strength is not due to a lack of academic and industrial interests on this important subject. However, due to the small thickness (2.3 ± 0.6 μm) of the IMC layers [18], it is very difficult to experimentally predict the interface strength of these compounds.
The above brief review indicates that the understanding of the behavior and strength of the individual interfaces of these compounds is far from complete, and it is evident that an atomistic study of these interfaces could provide useful new insight. The lack of atomistic studies is due to the complex atomic structure of the intermetallic compounds. It is, therefore, challenging to develop an interface model which is periodic, simple and has a low lattice misfit. There have been many studies in literature for the determination of the interface structures between two bulk phases, e.g. [19][20][21][22][23][24]. Many different approaches such as Olattice theory [25,26], the edge to edge model [27], the Conincidence of Reciprocal lattice point (CRLP) model [28] based on the Zur algorithm [22] have been developed to find and characterize the OR and conincidence of lattice between phases and grains. A major disadvantage of O-lattice theory is the lack of predictive capabilities, however the other approaches can successfully predict the OR, but they do not match the full structures. The edge to edge model considers the high density or nearly closed pack planes and direction and the CRLP and Zur algorithm match the underlying lattices of the structure ignoring the atoms inside them.
In this work, we have used a face-to-face matching technique to predict a possible Orientation Relationship (OR) between Fe 2 Al 5 and Fe suitable for atomistic calculations.
This work is a follow-up of a project working on the role of IMC layers on the joining strength of aluminum and steel. Many distinct IMC layers have been observed at aluminium and steel joints, and computational calculations on several other Fe-Al IMC layers have already been published [29,30]. In order to make a consistent comparison between different Fe-Al IMC layers, the same assumption, methodology, and computational techniques were applied in this study and as in the other studies related to the Fe-Al IMCs [29,30].
The scope of this paper is limited to establish and test the modeling methodology for finding a good atomistic interface structure and to study the mechanical and interfacial properties of the relevant Fe 2 Al 5 // Fe interface. The structure of the paper is as follows. First, we present the procedure for finding a low misfit interface structure between Fe and Fe 2 Al 5 . In Section 3 we present the calculation methodology and procedure for performing virtual tensile calculations. In Section 4, we present results of the strength of the bulk Fe 2 Al 5 as well as the Fe// Fe 2 Al 5 interface structure. In the last section, we discuss the results before presenting a summary and conclusions.

First-principles calculations
The first-principle calculations based on DFT were performed using the Vienna ab initio Simulation Package (VASP) [31]. The exchange-correlation energy was evaluated using the Generalized Gradient Approximation (GGA) by Perdew, Burke and Ernzerhof (PBE) [32] and with the Projector Augmented Wave (PAW) [33] method. By using the method proposed by Monkhorst-Pack to characterize energy integration as the first irreducible Brillouin zone [34] mesh size of 9 × 5 × 2 for bulk Fe 2 Al 5 and 9 × 5 × 1 for Fe 2 Al 5 //Fe interface structures.Maximum energy cutoff value of 450 eV was used for the plane wave expansion in reciprocal space. During the optimization process, the change in total energy were converged to 10 − 5 eV. Furthermore, the average force per atom was reduced to 0.009 eV/Å using a smearing factor of 0.2 and firstorder Meth-Paxton for the smearing of the partial occupation. Due to the magnetic behavior of Fe atoms, spin-polarized calculations were performed for the interface structures and bulk Fe by specifying the initial local magnetic moment of Fe.

Determination of bulk Fe 2 Al 5
Fe 2 Al 5 has an orthorhombic unit cell which contains single crystallographic Fe sites (four per cell) and three Al sites [35]. The Al1 site, which contains eight atoms per cell, is fully occupied, while Al2 and Al3 are too close to be occupied simultaneously, resulting in a partial occupancy factor of 1/6 for each of them [36]. We performed ground state energy calculations to find a stable crystal structure by calculating the formation enthalphy (ΔH R ) and cohesive energy (E c ) and used this structure further for bulk and interfacial calculations. The results of the bulk strength calculations of Fe 2 Al 5 have been reported in a previous work [37] and can also be seen in Table 1.

Prediction of orientation relationships
In order to create a good representative periodic interface structure for the DFT calculations, a common supercell of two crystal surfaces forming an interface is required. However, in the general case the two crystals have different lattice constants, it is necessary to rotate the two crystals relative to each other in order to obtain an interface with as little strain as possible in order to get an as realistic as possible interface.it is challenging to find low strain interface structure without an excessive number of atoms in the supercell due to the huge number of possible orientation relationships and orientations of the habit plane. The benefit with our algorithm is that it actually finds, in a very efficient way, the optimal OR and habit plane orientation with minimum strain and a manageable number size of the supercell. Stradi et al. [21] developed an algorithm for the efficient and systematic search for common supercells between two crystalline surfaces. The method presented in this work is based on the same principles as presented by Stradi et al. First the equivalent directions of the two crystal surfaces are determined and rotated to match at the interface. Then, both crystals are equally strained to match at the interface. This results in different interface structures and ORs based on the number of atoms and low lattice misfit as presented in Table 6. This method provides an advantage of predicting a number of ORs between two crystals without an excessive number of atoms and low lattice misfit interface.
The first step in creating the interface structure is to establish an OR between the two phases in question. We modeled the interface as an atomically sharp defect-free interface between two crystals 1 (Fe) and 2 (Fe 2 Al 5 ). To find possible ORs, a large number of possible sets of crystallographic directions were explored. The possible interface planes in Fe are defined by all pairs of lattice vectors, u 1 and v 1 , in Fe. Similarly, u 2 and v 2 define all possible interface planes of Fe 2 Al 5 . To obtain a periodic interface, the following relations must be fulfilled: where ∠ γ n = ∠(u n , v n ), with n = 1, 2 for crystal 1 and 2, respectively, and it is defined as the angle between vector directions u and v. We have added a vacuum layer along the normal direction to avoid periodic interaction. For this reason, angles ∠ α n = ∠(v n , w n ) and ∠ β n = ∠(u n , w n ) are not relevant, as the interface structures do not need to be periodic along the normal direction to the interface.
In the general case, it is not possible to find an OR satisfying these conditions exactly. The resulting interface structure depends on how well these conditions are fulfilled using the strains along direction u and  [38] b [39] c [40] direction v: and the difference in angles γ between the lattice directions; These angles between two crystals are illustrated in Fig. 1. The two structures (red and brown) are strained to match the angles to form a coherent interface structure (γ 1,2 ∕ = 90 • ).
In general an interface structure has 9 degrees of freedom (3 degrees related to the possible OR, 2 degrees for the possible interface plane, 2 for lateral translation along the interface plane and 2 degrees for position of where the interface cuts each phase). Ideally, all Δγ = 0, but when these conditions are not fulfilled, the minimum difference between angles (min Δγ) can be considered.
To construct good interface models, ORs are obtained by looping through all possible combinations of orientations up to a given crystal lattice vector length and testing them against the criteria listed above. We can thereby choose an interface structure with a low misfit and a corresponding supercell structure with low enough number of atoms so that DFT calculations are feasible.
By using the methodology presented above, we have predicted the possible interface structures between Fe 2 Al 5 and Fe. The DFT-relaxed bulk structure of Fe 2 Al 5 was used as input for finding the interface structures. To reduce computational cost, we only considered interface structures where the number of atoms and misfit are relatively small (see Appendix Table 6). We considered different orientation relationships based on their compact planes and directions and calculated the work of separation. Since some of the interfaces are quite large, optimizing all structures with atomic relaxations is a computationally expensive and difficult task. However, to make reasonable comparisons and to find the best low energy interface OR, we built all interfaces using the same criteria, these criteria where that the interplanar spacing between two bulk phases at the interface were set to be 2.86 Å and the transverse layers were assured to have a thickness of 10 Å on each side of the phase. In addition a vacuum layer of 10 Å was added to avoid periodic interaction along the normal direction. In order to motivate the selection of a good representative interface, we calculated the work of separation for an assortment of low misfit ORs. The calculated values are shown in Table 2. We finally selected the interface structure that has a low misfit, the least number of atoms and and the largest value of the work of separation (3.58 J/m 2 ) for further investigations.

Fe 2 Al 5 (020)//Fe(121) interface
The atomic structure of Fe 2 Al 5 //Fe was constructed using the procedure described above (Sec. 2.3). To ensure the bulk-like interior of atomic interfaces, six layers of Fe and Fe 2 Al 5 were tested. It is worth mentioning that Fe 2 Al 5 can be terminated either by Al or Fe at the interface. Both terminations were used for the interfaces shown in Fig. 2. To avoid periodic interactions, a vacuum layer of >10 Å was added along the normal direction to remove the effect of the two artificial interfaces. For the strength calculations, relaxed interface structures were used as an input for virtual tensile and shear test calculations.

Virtual tensile test calculations
Ab-initio virtual tensile calculations of the Fe 2 Al 5 //Fe interface were carried out in the framework of the Rigid Grain Shift (RGS) and RGS + relaxation methodology [41,42]. In this approach, the equilibrium structure was separated along the [020] direction. For each displacement, two kinds of calculations were performed: (1) RGS, without any atomic relaxations, and (2) RGS followed by atomic relaxations with a fixed supercell. We did not consider Poisson's effect in this study [43]. The top two layers are fixed, while the remaining middle layers are allowed to relax during the RGS + relaxation procedure, and a vacuum is added at the interface to imitate the tensile tests, as illustrated in Fig. 3. The same procedure was applied for both bulk and interface structures.
In the RGS approach, the interface structure was modeled by rigidly separating the Fe slab along the normal c direction at the interface and performing static calculations without any electronic and atomic relaxations, while in the RGS + relaxation method, atoms were allowed to relax. The slabs were initially separated by gradually adding vacuum at the interface in steps of 0.2 Å. The tensile displacement step size was selected based on the following criteria: (i) try to sample fairly dense near zero displacement to get a good estimate for the second derivative, (ii) try to sample fairly dense near where we expect the inflection point to be (iii) have at least one point at high displacements for good determination of the binding energy. Due to the computational cost of the RGS + relaxation methodology, a non-uniform step size was selected at the higher separation distances to find the fracture zone of the interface structures. The fitting of calculated values with the analytical expression provides a reasonable approach for reducing computational cost by reducing the considered step sizes and it can also further be useful for providing qualitative comparative analyses with other Fe-Al IMC interfaces [29,30].
Rose et al. [44] observed that the separation energy of metals has a universal form; where E e b is the separation energy of the equilibrium structure, d is the displacement defined with respect to the equilibrium structure and a is the re-scaled displacement, given by a = d/l, where l is a characteristic  length which can be approximated by the curvature of the energydisplacement curve at its minimum. Eq. (6) is used as a starting point for the fitting procedure, If the functional form g(a) is known, we can determine the theoretical strength and critical displacement of any material from the parameters E e b , and E ′ ′ b . This virtual tensile testing provides separation energy versus tensile displacement. The results obtained from these calculations can then be fitted to the UBER curve using Eqs. (5) and (7) (below). As Rose et al. observed, the metallic bonding-energy curve can be approximately scaled into the universal binding energy relation for the following cases: (i) metallic or bimetallic adhesion (ii) chemisorption on a metal surface, and (iii) cohesion of bulk metals [44]. Although UBER describes well separation energy versus displacement for unrelaxed metal surfaces, it is unable to describe the behavior of tightly bound intermetallics [45]. To find a good fit which captures the behavior of the separation energy versus displacement curve, we used a generalized form which includes two polynomials [46,30]: For the hydro-static compression/expansion, g(a) was determined to be [46]: where a is the rescaled displacement and P and Q are polynomials of order two or larger with positive (leading) coefficients. This expression for g(a) ensures that g(0) = − 1, g(a → ∞) = 0 and g ′ (0) = 0. The firstorder terms are excluded from P and Q since they are related to each other as well as to the characteristic length.
To ensure that the fitting behaves well, one should only include oddorder terms in the polynomials P and Q and make sure that all coefficients are zero or positive.
By differentiating the fitted energy-displacement curve, the theoretical tensile strength of the atomic structures can be evaluated [47]; The theoretical strength σ th at its maximum value is defined as the Ultimate Tensile Strength (σ UTS ). The value of d at σ UTS is defined as the critical length d c .

Bulk strength
To compare the bulk and interface structures, we also calculated the tensile properties of Fe 2 Al 5 (020) using the rigid shift (RGS) and RGS + relaxation methodology as explained in Sec. 3.1. We studied the virtual tensile strength of both the Al-and the Fe-terminated Fe 2 Al 5 structures as shown in Fig. 4. Fig. 5 (a) and (b) show the separation energy versus tensile displacement curve for Al and Fe-terminated fractures using the RGS and RGS + relaxation methodologies, respectively. In the stable configuration of the Fe 2 Al 5 phase, the Fe-Al bond distance is 2.50 Å and the Fe-Fe bond distance is 2.96 Å. During the virtual tensile testing, this bond distance at the cutting plane is stretched further until the bulk structure fractures and separates into two free surfaces. Fig. 4 (b) and (c) show the procedure for introducing a crack with Al and Fe-terminations. Table 3 lists the work of separation (W sep ) and the work of adhesion (W ad ). The former is defined as the work needed to separate a bulk phase without atomic relaxations, and the latter is the energy needed to separate a bulk interface into two relaxed surfaces [48].
The binding energy increases with tensile displacement. RGS without atomic relaxation produces a steeper curve which was fitted using Eqs. (5) and (7). During tensile displacement, the separation energy increases sharply until it stabilizes at larger displacements (>̃5 Å). Table 3 lists the calculated values of σ UTS . Fig. 6 shows the stress-strain curves for Fe 2 Al 5 along with bulk strengths for the RGS and RGS + relaxation methodologies. With increasing tensile strain, the tensile stress increases until its maximum value (σ UTS ). One can note that σ UTS calculated with the RGS + relaxation methodology is lower than that for the RGS methodology. For comparisons, we also present the strength of the bulk Fe (111) plane. The Al-terminated Fe 2 Al 5 bulk phase shows higher strength (20.09 GPa for RGS and 15.48 GPa for RGS + relaxation) as compared to the Fe-terminated structure (17.72 GPa for RGS and 13.28 GPa for RGS + relaxation). Moreover, the bulk Fe structure shows higher values of W sep and σ UTS , which signify the higher strength of bulk Fe than that of the Fe 2 Al 5 phase. A lower strength of the Fe-terminated bulk Fe 2 Al 5 structure indicates a weaker bonding between Fe-Fe atoms which will be discussed in sub-Section 4.3. Besides, the long bonding distance between Fe-Fe also contributes to the weakening of the bond.

Energy-displacement curves
Energy-displacement curves are shown in Fig. 7 for Al-terminated and Fe-terminated interface structures using the above-mentioned fitting technique. Fig. 7 (a) and (c) show the energy-displacement curve for RGS and Fig. 7 (b) and (d) show the same curves for the RGS + relaxation methodology. A steep and continuous curve is obtained for the RGS methodology without any atomic relaxations, which can be fitted well by using Eq. (5). As can be seen from Fig. 7(b) and (d), with increasing in tensile displacement, the energy required to fracture the interface structure decreases until the structure separates into two surfaces at larger displacements (>3 Å). The separation length at this point is defined as the final fracture length (d f ). Even though there is no unique way of determining d f , we here define it to be at the point where the binding energy curve reaches − 0.003 eV/Å 2 [30].
The minimum value of the binding energy gives -E b (0) = W sep for RGS and -E b (0) = W ad for the RGS + relaxation methodology. Table 4 lists the W sep and W ad values for the Fe 2 Al 5 //Fe interface. As given in Table 4, the Al-terminated interface shows higher W sep (4.45 J/m 2 ) as compared to the Fe-terminated interface (3.82 J/m 2 ). Lazar [41] postulated the rough approximation that W sep = 1.06 W ad by linear fitting of DFT results of RGS and RGS + relaxation methodologies for different compounds and materials. This fits perfectly for the Alterminated interface but less so for the Fe-terminated interface.
An optimal fit for the relaxed surfaces is shown in Fig. 7 (b) and (d). For the relaxed-type virtual tensile tests, crack opening is initiated by separating two blocks by introducing vacuum and subsequently allowing atoms to relax while keeping the plane area fixed. The initial crack introduced during RGS can potentially be healed by atomic relaxations if the separation between the two blocks is smaller than the critical length (d c ) [47]. In Fig. 7 (b) and (d), d c is located at the border of Region I (d < d c ). Table 4 lists the critical (d c ) and fracture lengths (d f ) for the two relevant interface structures.
Region II is defined for separations d c < d < d f . In this region, the structure is neither separated nor being able to heal by elastic relaxations, which is why it is defined as the instability region. The range of this instability region is determined by taking the difference between d f and d c . The width of Region II is related to the brittleness/ductility of the interface structure [41]. For the Al-terminated interface structure, the length of the instability region is approximated to be 0.84 Å, while for the Fe-terminated interface, it is 0.79 Å. The shorter range of the instability region for the Fe-terminated interface indicates a more brittle fracture than that of the Al-terminated interface.
At longer separation distances (d > d f ), the interface structures are completely separated into two relaxed bulk surfaces. This region is defined as Region III in light grey color ( Fig. 7 (b) and (d)). In this region, there is no interaction at the interface, and relaxation of the atomic positions relaxes the bulk surfaces into stable configurations. For this reason, the binding energy versus separation curve stabilizes, and no further increase in binding energy can be seen. In order to elucidate the bonding characteristics of the interfacial and bulk atoms, total charge density isosurfaces and charge density difference plots for all surfaces were constructed as shown in Fig. 8. A high charge density cloud (labeled as B in Fig. 8(c)) can be seen for the Feterminated interface as compared to the Al-terminated interface (labeled as A in Fig. 8(a)). Moreover, there is a higher charge transfer for the Fe-terminated interface, while there is a weak charge transfer zone for the Al-terminated interface as shown in yellow color in Fig. 8(b). This high charge density and transfer rate at B indicates stronger bonding between interfacial Fe-Fe atoms at the Fe-terminated interface, which explains the higher σ UTS for this interface as compared to the Alterminated interface.

Tensile strength
For the bulk Fe 2 Al 5 structure as shown in Fig. 8(e-f), Fe-Fe bonding (labeled as C) was found to be weaker than the Al-Fe bonding (labeled as D). This observation is consistent with the lower σ UTS for the Feterminated bulk structure. Generally, Fe-Al atoms are found to have higher charge density and charge transfer regions at the interfaces and in the bulk structures. However, in the Al-terminated interface, the Al atoms move towards the Fe atoms and develops a bond at the interface by compromising the bonding strength at the first layer of the Fe 2 Al 5 side, labeled as I in Fig. 8(a). This fracture plane can be a weak link of the overall Al-terminated interface structure.

Ideal shear strength
To calculate the ideal shear strength a series of incremental shear where E s (d) and E 0 are the energy of the displaced and unsheared structure, respectively, d is the shear displacement, and k n = 2πn λ , where λ is the periodicity along the shear direction. Appendix Table 5 and 6 gives the Fourier series coefficient values and the value of λ for both interface structures.
The shear stress is given by (10) where A is the interface area. The maximum value in the resulting sheardisplacement curve corresponds to the ideal shear strength, which is defined as the interface resistance to the shear displacement after which it starts to deform. Fig. 9 shows the stress-displacement curve for the shear stress as a function of shear displacement for both Al-and Fe-terminations. Initially, stress increases with the increase in the shear displacement until it reaches a maximum value for both cases, which is taken as the ideal shear strength of the interface structure. Table 5 summarizes the ideal shear strength of the Fe 2 Al 5 //Fe interface structure for the different cases discussed in this work. Results are quite different for both interface terminations. The Fe-terminated interface shows low shear strength (0.97 GPa) along < 001 > and larger shear strength along < 100 > (4.74 GPa), while the Al-terminated interface shows high shear strength (2.51 GPa) along the < 100 > direction and a slightly lower shear strength along < 001 > (3.97 GPa). In general the Al-terminated structure shows higher shear strength than the Fe-terminated interface structure. These calculations, therefore, indicate that the Fe-terminated < 001 > interface is more prone to shear failure than the Al-terminated interface.
Comparing shear strength with tensile strength indicates that the Alterminated interface shows higher tensile and shear strength than the Fe-terminated interface. From Table 4 and 5, it can be seen that shear  instability can occur earlier than normal decohesion. This is consistent with the experimental observations of an Al-Fe welded system [52]. The shear strength calculated in this study for loading parallel to the interface is lower than the perpendicular loading direction (σ UTS ). The same trend has been observed experimentally and theoretically in the literature [16,52,53].

Discussion
Before discussing the implications of these results, some limitations are worth to be mentioned. These simulations have been performed without considering dislocations, micro-voids, and other effects occurring at larger length scales, that will obviously influence the strength of real joints [54]. Hence, the calculated strengths are thus generally overestimated. Still, these calculations provide important insights about the crack formation mechanism of the interface structure at the atomic   scale. The role of crystal defects on the mechanical properties is proposed to be a subject of future studies.
In this work we have studied the effect the Fe 2 Al 5 intermetallic phases has on the strength of an aluminium-steel joints. It is a very difficult task to identify the fractured layer of aluminum and steel joints by experiments. For this reason, to predict the weak zone of the Fe 2 Al 5 // Fe interface structure, two zones were studied: (i) bulk Fe 2 Al 5 and (ii) Fe 2 Al 5 //Fe interface. Based on bulk and interface calculations, the interface between Fe 2 Al 5 // Fe showed higher strength as compared to bulk Fe 2 Al 5 and smaller than bulk Fe [49]. Virtual tensile testing, therefore, indicates that bulk Fe 2 Al 5 is more prone to induce fracture than the interface and bulk Fe side. Mechanical strength inferred from the virtual tensile calculations indicates that fracture is most likely to be initiated from the Fe-terminated side of the bulk Fe 2 Al 5 due to weak bonding between Fe-Fe atoms. Shear strength is seen to be lower than the tensile strength, which is also consistent with the experimental observation of Fe 2 Al 5 [55].
We have performed more calculations with the strained interface structures to study the effects of elastic strain on work of separation. All calculations were performed considering the optimized equilibrium interface structures. As discussed in Section 2, both bulk slabs were strained to match at the interface. In order to determine the elastic contribution to the work of separation, calculations were performed to determine the work of separation for the strained interface structures and the results are compared with those of the equilibrium interface structure. In the case of the Fe-terminated interface, the work of separation was reduced from 5.54 J/m 2 to 3.58 J/m 2 . However, for the Alterminated interface the work of separation was drastically reduced: from 6.16 J/m 2 to 2.91 J/m 2 . This indicates that the failure mechanism for the interface structure might be more complex than what can be described using this method and should be investigated in more detail in further work.
Moreover, the theoretical tensile strengths of an interface structure depends on the number of crystallographic layers of the model. A recent study [56] has indicated the decrease in fracture stress with increasing supercell size with localized strain models. Effect of supercell size is beyond the scope of this study. Moreover, as the main objective is to make a comparative analysis of Fe-Al IMCs interfaces, we adopted a consistent methodology and approach for all interface structures to make appropriate qualitative comparisons. However, further investigations are needed to find the influence of the number of layers on the strength values for the RGS + relaxation methodology. Since this study is limited to the DFT methodology, it presents an extra challenge of computational cost. This is why the values obtained from the RGS + relaxation method only provides qualitative comparative strength of the Fe//IMC interface as compared to the Al/IMC [29], IMC//Fe and pure IMC//IMC interfaces [30]. Still we believe this provides useful insights into the role of Fe-Al IMCs on the joining of aluminium and steel.
In general, these results have a particular significance for the welding of aluminum and steel joints for different welding methodologies, where the presence of an Fe 2 Al 5 intermetallic layer has been reported along the steel side. However, the defects at the IMC layers also play a significant role in deteriorating the joint strength and have to be included in the calculations to give more reliable predictions for real systems in the future.

Summary and conclusions
To summarize, we have performed DFT calculations of tensile and shear strength of the Fe 2 Al 5 //Fe interface. The interface structure with the lowest lattice misfit and number of atoms was selected for the DFT calculations of this work. Virtual tensile tests were performed with the rigid grain shift (RGS) methodology without atomic relaxations and RGS + relaxation methodology with atomic relaxations. Polynomial terms were introduced into the UBER to find a reasonable fit for the tensile stresses. Based on RGS calculations, the Al-terminated interface showed higher strength than bulk Fe 2 Al 5 and the Fe-terminated interface structure. During the relaxation of atomic positions in the RGS + relaxation methodology, the tensile strength decreased for all structures except for the Fe-terminated interface. Moreover, the charge density maps indicated a weaker bonding between Fe-Fe atoms in the bulk Fe 2 Al 5 structure, which contributed to a lower tensile strength. We also analyzed the shear strength for the interface along < 001 > and < 100 > directions. We found that < 001 > has lower shear strength for the Fe-terminated interface while it showed higher strength for the Alterminated interface.
Overall the Fe bulk side was found to be the strongest zone of the Fe 2 Al 5 //Fe interface structure followed by the interface and bulk Fe 2 Al 5 . Based on these calculations, it can be anticipated that during a mechanical failure, fracture is most likely to be initiated at the bulk Fe 2 Al 5 side. This study can potentially be the starting point for further investigations of the effects of crystal defects and temperature on the joint strength of aluminum-steel joints.

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 6 Some of the predicted ORs between Fe 2 Al 5 and Fe atoms. m 1 , m 2 and m 3 are the direction vectors for Fe 2 Al 5 phase and n 1 , n 2 and n 3 for Fe atoms. length and strain (%) are the length of supercell and misfit percentage (as defined in Eq. (2) and (3)) of the interface structures respectively. Normally m 1 , m 2 , m 3 and n 1 , n 2 , n 3 are integers, but due to sub-lattice translations in the conventional cell, fractions are also possible. Tables 7 and 8 lists the Fourier series coefficient values for the shear strength calculations for both interface structures.