Mechanical properties and fracture behavior of Al(111)/MgAlB4(0001) interface in Al matrix composites: a first-principle calculation study

Hexagonal crystal MgAlB4 is a strengthening phase in Al matrix composites, which can significantly improve ultimate tensile strength. In this paper, the surface perform, interfacial bonding characteristic, fracture mechanism, and electronic properties of the Al(111)/MgAlB4(0001) interface were thoroughly investigated by the first principles method. The results reveal that the top-site and bridge-site configurations were more unstable than the hollow-site. Besides, from the calculated results of interfacial energy and work of adhesion, the hollow-stacked Al(111)/B(Al)-terminated/MgAlB4(0001) interface expresses stronger stability than other interfacial models, which is attributable to the higher work of adhesion and lower interfacial energy of the hollow-stacked Al(111)/B(Al)-terminated/MgAlB4(0001) interface. Analysis of electronic structure reveals that the Al-termination and Mg-termination Al(111)/MgAlB4(0001) interface presents Al-Al and Al-Mg metallic bonds at the interface, respectively, but the B(Al)-termination Al(111)/MgAlB4(0001) interface expresses strong Al-B covalent bonds characteristic, which leads to the highest interface stability. The results of tensile fracture revealed that the HCP stacked B(Al)-termination interface transferred the external stress to Al bulk, due to the Al-B covalent bond formed near the interface. Therefore, ceramic phase MgAlB4 can effectively promote the particle reinforcement of Al matrix composites.


Introduction
With the continuous progress of science and technology, the requirements for aluminum alloys and mechanical properties of aluminum alloys are increasing ceaselessly [1,2]. Aluminum alloys are widely used in transportation [3], aerospace [4], and other fields [5] due to their superior high strength, plasticity, weldability, and machinability, especially in the automotive industry [6,7], which is of great significance to aluminum alloys to realize lightweighting. However, the problem of coarse grain structure continuously existed in the traditional aluminum alloy after casting and solidification, and the general casting defects, such as component shrinkage cavity, crack, and deformation, are often caused by coarse grain, which will decrease the mechanical property of Al-based composite [8][9][10]. In modern industry, adding grain refiners into aluminum melt has become the most simple and effective method to refine aluminum alloy grains and improve the mechanical properties of aluminum alloy [11,12].
Currently, particle reinforcement, as a promising method, has been used to improve the strength and plasticity of metal based hetero-structure composites [13][14][15]. Moreover, the MgAlB 4 as a ceramic phase has been used to strengthen the performance of Al alloys through reinforce Al matrix composites. Wang et al [16]

Computational methodology
In this paper, the Cambridge Sequential Total Energy Package (CASTEP) of materials studio [27,28] is used to calculate the interfacial properties of Al(111)/MgAlB 4 (0001), such as interfacial bonding and work of adhesion. Besides, the generalized gradient approximation (GGA) with Perdew-Burke-Ernzerhof (PBE) functional was implied for the exchange correlation between electrons in this investigation [29]. Moreover, Ultra-soft Pseudo Potential, which describes the wave function of electrons with a lower cutoff frequency, was used [30,31]. Besides, the self-consistent field (SCF) was used to solve Kohn-Sham equation, and the SCF tolerance of energy was set to 5.0 × 10 −6 eV/atom, and the Broyden-Fletcher-Goldfarb-Shannon (BFGS) algorithm was adopted to reach ground state. After convergence, the value of plane-wave cutoff energy, the calculated tolerances for the total energy change, and the force per atom are set to 550 eV, 5.0 × 10 −7 eV/atom and 0.03 eV/atom respectively. Besides, the k point for MgAlB 4 bulk, surface and Al/MgAlB 4 interface are set to 16 × 16 × 8, 16 × 16 × 1 and 18 × 18 × 1 respectively. In the tensile test process, the increment of the strain is 0.005, the vertical interface structure is implied to relax the position of the atom, and the lattice parameters of the transverse interface structure are fixed.

Bulk properties of Al bulk and MgAlB 4 bulk
In this section, the crystal parameters for face-centered cubic Al and hexagonal structure MgAlB 4 were firstly calculated to obtain the accurate calculate results, and the crystal structure of Al and MgAlB 4 are shown in figure 1. From the figure 1, one can see that Al is a face-centered cubic, and the space group symmetry of MgAlB 4 is a P6/mm hexagonal structure, which is similar to AlB 2 [32]. For MgAlB 4 bulk, our calculated results of the crystal parameters are a = 3.046 Å, c = 6.759 Å, B = 176.8 GPa and formation enthalpy is ΔH f = −0.357 in accordance with other calculated value (a = 3.039 Å, c = 6.764 Å, B = 170 GPa and ΔH f = −0.35) by DFT calculation [33], and experimental value (a = 3.052 Å, c = 6.722 Å) [16]. Moreover, for the bulk of Al, the lattice constant is a = 4.037 Å, which is practically consistent with other DFT result (4.04 8 Å) [34] and the experimental result (4.05 Å) [35]. To sum up, the calculate results are both consistent with the simulated and experimental results, and confirm that the selected calculated methodology is reliable and accurate.
3.2. Surface properties of Al(1 1 1) slab and MgAlB 4 (0 0 0 1) slab During the relaxation optimization of the surface model, the obvious differences for the atomic force at different positions exist in the surface model, which results in the movement and position change of the surface atoms relative to the atoms at other positions. After sufficient relaxation optimization, the steady-state structure is achieved. Therefore, the surface protons relaxation optimization is an indispensable property of the surface model. The convergence of the surface model with different layers can be tested by the relaxation optimization of Al and MgAlB 4 surface atomic layers with different layers, so the optimal surface layers can be determined. The relaxation calculation process of the surface atomic layer can be expressed as [36]: here, d ij is the atomic spacing between the i-th layer and j-th layers after relaxation optimization of the surface model, and d ij slab , is the atomic spacing between the i-th layer and j-th layers of the initial surface model. From the above formula, we can deduce that d ij D will be positive when the surface layer spacing expansion after the relaxation of the surface model, whereas the value of d ij D will be negative when the surface layer spacing contraction.
According to a previous investigation [37], the surface energy of Al (111) slabs tend to converge gradually when the thickness is seven layers. Table 1 shows the variation of the layer spacing of MgAlB 4 (0001) surface model after the optimization. From table 1, there are four different termination (Mg-termination, Altermination, B(Al)-terminated and B(Mg)-termination) for MgAlB 4 (0001) slab, and the distance between atomic layer has changed obviously. Moreover, for Al-termination and Mg-termination, as the number of layers of MgAlB 4 (0001) surface model is greater than 13, the distance of interlayer of MgAlB 4 (0001) slabs begins to converge. But that of B(Al)-terminated and B(Mg)-termination of MgAlB 4 (0001) slabs are both 17. Therefore, we constructed 13 layers Al-terminated MgAlB 4 (0001) slab, 13 layers Mg-terminated MgAlB 4 (0001) slab, 17 layers B(Al)-terminated MgAlB 4 (0001) slabs, and 17 layers B(Mg)-terminated MgAlB 4 (0001) slabs, which is shown in figure 2.
Generally, the crystal structures of the MgAlB 4 surface plays a critical function in the stability of the interface. Thus, the stability of Mg-terminated, Al-terminated, B(Al)-terminated, and B(Al)-terminated MgAlB 4 (0001) slabs were considered to characterize interfacial stability. According to the reports [38], the energy of MgAlB 4 (0001) surfaces can be calculated as: Where, N Al , N Mg , and N B are the numbers of Al, Mg, and B atoms of the MgAlB 4 (0001) surface respectively; μ Al , μ Mg , and μ B are the chemical potential of Al, Mg, and B respectively; E slab is the total energy of MgAlB 4 (0001) surface after relax; and A is the area of the MgAlB 4 (0001) surface. In the same system, the chemical potential of a single atom is in equilibrium with that of the bulk structure, the chemical potential and the surface energy of bulk MgAlB 4 can be written by:     Therefore, the equation (2) can be finally written by: In general, the chemical potentials of the bulk MgAlB 4 have to be larger than that of atoms, otherwise the slab is unstable; thus, Al m , According to the calculation results, the formation enthalpy ΔH f of MgAlB 4 bukl is −0.366 eV. Therefore, as shown in figure 3, the relationship between Δμ Al and surface energies of the Mg-terminated, Al-terminated, B(Al)-terminated, and B(Mg)-terminated MgAlB 4 (0001) surface models are calculated to estimate surface stability. Besides, the surface energy and the number of each kind of atoms in four terminated surface models are listed in table 2. From figure 3, the surface energy of MgAlB 4 (0001) surface changes within a range with the increase of chemical potential Δμ Al . Moreover, the surface energy of B(Mg)-terminated, B(Al)-terminated, and Al-terminated MgAlB 4 (0001) surfaces decrease with the increase of Δμ Al , but that of Mg-terminated MgAlB 4 (0001) surface increase with the increase of Δμ Al . From table 2, the surface energies of different termination at the Al-rich condition can be ranked as follows: B(Al)-termination > B(Mg)-termination > Mgtermination > Al-termination. At the Al-poor condition, the order of surface energy is: B(Al)-  Table 2. The calculated results of four different surface models, the surface energy of γ 1 at Δμ Al = −0.366 eV, and γ 2 at Δμ Al = 0 eV.  The work of adhesion (W ad ) is defined reversible work required to separate the interface into two independent surfaces, which is commonly considered to evaluate the interfacial stability, and it can be obtained from the following formula [39] :  Universal Binding Energy Relation (UBER) curve [37]. Figure 5 shows the relationship between adhesion work and interface spacing of the unrelaxed Al(111)/MgAlB 4 (0001) interface through UBER method. The unrelaxed and relaxed W ad and d 0 values determined for Al(111)/MgAlB 4 (0001) interface are listed in table 3.
From figure 5, one can see that there exist obvious difference on the interfaces separation distance and work of adhesion for four different Al(111)/MgAlB 4 (0001) interfacial models. Moreover, the interfaces separation distance of hollow-site is the smallest among three different stacking sites, but the W ad of the hollow-site is the largest among them. All these results prefer to the conclusion that the interfacial stability can be ranked as: HCP > MT > OT. The HCP stacked interfacial model has the highest bond strength, while the OT stacked interfacial structure has the minimum bonding strength. Compared four terminated interface models, we can  found that the B(Al)-terminated Al(111)/MgAlB 4 (0001) interface with HCP stacking has the minimum separation distance and the maximum work of adhesion, indicating that this interface model is the most consistent interfacial structure and has the highest binding strength. Additionally, Al-terminated Al(111)/MgAlB 4 (0001) interface with HCP stacking has the smallest work of adhesion among four HCP stacked interface models, which indicates that it is the most unstable interfacial structure.
To deeply understand the interfacial stability of Al(111)/MgAlB 4 (0001) interface, we calculated the interfacial energies (γ int ) of all the Al(111)/MgAlB 4 (0001) interfacial models, and the interfacial energy can be defined as the free enthalpy of interface per unit area. Generally, interfaces with a positive and smaller γ int value will cause more stable interfacial structure, and the more negative the γ int value, the more unstable the interfacial structure. Hence, the interfacial energy of Al(111)/MgAlB 4 (0001) interface can be expressed as:       atoms at the interface of Al(111) slab have obvious polarization characteristic, which presents a violently vibrating waveform and higher PDOS value at Fermi level. The above characteristics show that Al atoms at the interface express more metallic characteristics. Moreover, from figures 9(c) and (d), one can see that the PDOS value at Fermi level of Al atoms at the interface of Al(111) slab decreased compared with other locations, but the B atoms at the interface have the opposite trend. By comparing with other positions, the PDOS curve of B atoms at the interface is flatter, and the PDOS value at Fermi level is higher, indicating that the B atoms at the interface obtain more charges, and lead to more metallicity. Significantly, for Al(111)/B(Al)-terminated/MgAlB4(0001) interface, a hybrid peak appears at the interface between Al and B near 3.6 eV, especially at Al-s and B-p orbits. In summary, it can be concluded that the covalent bond characteristics of B(Al)-terminated interface in HCP stacking mode are stronger than those of other terminations. Therefore, interface binding is strong, which is consistent with the results obtained from the analysis of binding energy, interface energy and interface spacing θ.

Tensile strength
To investigate the mechanical properties of the Al(111)/MgAlB 4 (0001) interface at the atomic level, the HCP stacked Al(111)/B(Al)-terminated/MgAlB 4 (0001) interface and Al(111)/B(Al)-terminated/MgAlB 4 (0001) interface as the most stable interfacial models, were calculated for the uniaxial tension by using first principle method. In this investigation, the mechanical properties and the failure mechanism of Al(111)/MgAlB 4 (0001) interfacial models under the tensile load were considered, and the strength of tensile failure can be obtained by calculating the corresponding stress values under different strain loads. Besides, the stress of the interface can be given by the following formula: Where E is the total energy of the entire interface structure, and Vε is the corresponding system volume when the tensile strain is ε. Figures 10(a) and (b) show the tensile fracture curves of the HCP stacked Al(111)/B(Al)-terminated/MgAlB 4 (0001) interface and Al(111)/B(Al)-terminated/MgAlB 4 (0001) interface respectively. All the tensile structures of Al(111)/MgAlB 4 (0001) interface were taken from the full relaxed structures, and the strain was created in a quasistatic way [40] with a step of 0.005. To ensure the continuity of the whole stretch process, the stretch of the interfacial model was based on the previous optimized structure. From figure 10, one can see that the fracture strain of the HCP stacked Al(111)/B(Al)-terminated/MgAlB 4 (0001) interface and Al(111)/B(Mg)-terminated/MgAlB 4 (0001) interface are 0.08 and 0.073 respectively, and the corresponding critical tensile stresses are 11.6 GPa and 10.7 GPa respectively. From figure 10(a), it can be seen that the critical stress of HCP stacked Al(111)/B(Al)-terminated/MgAlB 4 (0001) interface is 11.6 GPa. In addition, the stress increases with strain when the strain is below 8%, and a sudden change occurs when the strain reaches 0.08, and then the strain increases linearly with the stress until fracture, thereby indicating that the yield strength of the interface is 11.6GPa and the elastic limit is 0.08. Figure 11 shows the charge density difference of HCP stacked Al(111)/B(Al)-terminated/MgAlB 4 (0001) interface in the tensile test when the strain are 0%, 2.5%, 5%, and 8% respectively. From figure 11, one can see that the charge density difference changes unobvious when the strain is less than 8%, which implies that the phenomenon of tensile fracture is still not formed at HCP stacked Al(111)/B(Al)-terminated/MgAlB 4 (0001) interface. However, compared with figure 11(a), the charge density between two Al atoms near the interface decreases significantly in the Al bulk of figure 11(d), indicating that the final mechanical fracture may be caused by the fracture of metallic bond between Al and Al atoms. To sum up, Al(111)/B(Al)-terminated/MgAlB 4 (0001) interface has successfully transferred the external stress to Al bulk, which results to the increasing of the strength and plasticity of the composite.
Therefore, the particle reinforcement may occurred at Al composite when the ceramic phase MgAlB 4 is added into Al-based composite, due to that Al/MgAlB4 interface can effectively perform load transfer to improve the strength and plasticity of the Al-based composite significantly. The results is consistent with the MgAlB 4 whiskers formed along the [0001] direction and then strengthening the Al matrix in the literature, the new whiskers nucleated on the surface of the existing MgAlB 4 whisker and the growth steps were shown in figure 12.

Conclusions
In this work, the surface properties, interfacial bonding and electronic properties, and the Interfacial tensile fracture mechanism are investigated by using first principle method, and the main conclusions show that: (1) The Al(111)/B(Al)-terminated/MgAlB 4 (0001) interfaces expresses stronger interface stability than Mgtermination, Al-termination, and B(Mg)-termination, adhesion work can be ranked as HCP > MT > OT. Thus, the HCP stacked Al(111)/B(Al)-terminated/MgAlB 4 (0001) interface has the strongest interface stability.
(3) The Al(111)/B(Al)-terminated/MgAlB 4 (0001) interface with HCP structure has the highest tensile performance among all the Al(111)/MgAlB 4 (0001) interfaces. When the strain is 8%, the maximum value of the tensile stress is about 11.6 GPa, and the final mechanical fracture may be caused by the fracture between Al-Al metallic bond in Al bulk.
(4) The ceramic phase MgAlB 4 plays a role in particle reinforcement of Al-based composite due to the Al(111)/B(Al)-terminated/MgAlB 4 (0001) interface transferred the external stress to Al bulk.