Understanding the Interface Characteristics Between TiB₂(0001) and L1₂-Al₃Zr(001): A First-Principles Investigation

Abstract

This study employs first-principles calculation methods to explore the characteristics of the TiB₂(0001)/L1₂-Al₃Zr(001) interface, including the atomic structure, adhesion work, interfacial energy, and electronic structure of various interface models. Considering four different terminations and three different stacking positions, twelve potential interface models were investigated. Surface tests revealed that a stable interface could be formed when a 9-layer TiB₂(0001) surface is combined with a 7-layer ZrAl-terminated and a 9-layer Al-terminated Al₃Zr(001) surface. Among these interfaces, the bridge-site stacking at the T/Al termination (TAB), hollow-site stacking at the Ti/ZrAl termination (TZH), top-site stacking at the B/Al termination (BAT), and hollow-site stacking at the B/ZrAl termination (BZH) were identified as the optimal structures. Particularly, the TAB interface exhibits the strongest adhesion strength and the lowest surface energy, indicating the highest stability. A Detailed analysis of the electronic structure further reveals that most interfaces predominantly exhibit covalent bonding, with the TAB, TZH, and BZH interfaces primarily featuring covalent bonds, while the BAT interface displays a combination of ionic and covalent bonds. The study ultimately ranks the stability of the interfaces from highest to lowest as TAB, BZH, TZH, and BAT.

1. Introduction

Aluminum–silicon alloys, characterized by low density, high strength, high wear resistance, and good machinability, are extensively used in mechanical manufacturing, automotive, maritime, and aerospace industries [1,2]. The performance of the alloy largely depends on the microstructure of its grains. However, in aluminum–silicon alloys, the coarse eutectic silicon phase can lead to grain coarsening, thereby diminishing the material’s performance [3]. Studies have shown that the addition of Zr to aluminum alloys can significantly enhance their performance. In these Zr-doped aluminum alloys, nanoscale Al₃Zr precipitates have been observed and are considered to contribute to grain refinement [4].

Al₃Zr in aluminum alloys exists in two structures, D0₂₃ and L1₂ [5,6]. D0₂₃-Al₃Zr is a stable structure formed by tetragonal lattice distortion and intra-cell atomic migration, and it exhibits higher brittleness [7]. L1₂-Al₃Zr is a metastable cubic structure, primarily precipitated from supersaturated aluminum solid solutions during rapid cooling processes [8]. L1₂-Al₃Zr is completely coherent with the α-Al matrix, and it significantly contributes to inhibiting grain recrystallization and enhancing alloy strength through its shearing suppression effect on dislocation movement [9,10]. Studies have indicated that L1₂-Al₃Zr serves as an effective heterogeneous nucleation substrate for α-Al at the atomic level [11,12], and first-principles calculations have confirmed that liquid Al atoms tend to nucleate on the L1₂-Al₃Zr(001) surface at the ZrAl terminated [13].

Adding Al-Ti-B grain refiners to aluminum melt is also a widely applied method for improving the microstructure of grains, which contain strong nucleating particles like TiB₂ [14,15]. TiB₂ serves as an effective nucleating particle and plays a crucial role in the refinement of α-Al grains [16]. It has been reported that the refining mechanism of Al-Ti-B grain refiners is the formation of a two-dimensional Al₃Ti compound on the (0001) TiB₂ surface, which effectively promotes heterogeneous nucleation of α-Al [17]. Reports indicate that the distribution of TiB₂ particles follows a log-normal distribution, with most nanoparticles being smaller than 100 nanometers [18]. Nanoscale TiB₂ particles can pin dislocations at grain boundaries and inhibit the growth of recrystallized grains. Additionally, TiB₂ is a boride with a hexagonal structure, known for its excellent stiffness, hardness, and wear resistance, making it an outstanding reinforcing material [19,20].

When both Al-Ti-B grain refiners and excess Zr and Si elements are present in the aluminum melt, the grain refining capability is significantly reduced, leading to the so-called “poisoning” phenomenon of the grain refiner [21,22]. The prevailing view on the mechanism of the “poisoning” phenomenon is that Zr atoms in the Al melt cause the dissolution of Al₃Ti on the TiB₂(0001) surface, thereby reducing the nucleation capability of the refiner [15,23,24,25]. Some studies also speculate that TiB₂ loses its ability to refine Al grains due to the formation of interfacial nucleation with Al₃Zr [26]. Therefore, to further elucidate the mechanism of the “poisoning” phenomenon, it is necessary to study the relationship between the two nucleating particles, TiB₂ and Al₃Zr. Current research on the interface relationship between TiB2 and D0₂₃-Al₃Zr is already quite extensive. Researchers have observed the precipitation of D0₂₃-Al₃Zr at the TiB₂ interface using scanning electron microscopy, with the orientation relationships identified as (0001)_TiB2//($1(\_)$0$3(\_)$)_D023-Al3Zr, (10$1(\_)$0)_TiB2//(10$5(\_)$)_D023-Al3Zr, and [12$1(\_)$0]_TiB2//[010]_D023-Al3Zr [27,28]. However, research on the interface relationship between TiB₂ and L1₂-Al₃Zr remains quite scarce. During the solidification process of the alloy melt, the precipitation of the D0₂₃-Al₃Zr phase at the TiB₂ interface nearly depletes Zr from the matrix, leading to scant precipitation of L1₂-Al₃Zr around the TiB₂ clusters. This situation makes it difficult to observe the interface relationship between TiB₂ and L1₂-Al₃Zr through experimental means [29]. Therefore, simulations based on Density Functional Theory can provide a convenient means to study the potential interface relationship between these two phases.

This article employs first-principles calculation methods to systematically investigate the structural and electronic properties of the TiB₂(0001)/L1₂-Al₃Zr(001) interface at the atomic level. To gain a more comprehensive understanding of the bonding behavior at the TiB₂(0001)/L1₂-Al₃Zr(001) interface, we also predicted the ideal adhesive work and interfacial energy. Finally, the electronic structure of TiB₂(0001)/L1₂-Al₃Zr(001) was analyzed through the differential charge density and a density of states analysis.

2. Calculation Method

First-principles calculations were performed using the Cambridge Serial Total Energy Package (CASTEP) code [30]. The interactions between ionic cores and valence electrons were described using ultrasoft pseudopotentials [31]. Within the framework of the Generalized Gradient Approximation (GGA), the Perdew–Burke–Ernzerhof (PBE) function was employed as the exchange-correlation potential [32]. Theoretical calculations were performed with a plane-wave cutoff energy of 380 eV. For the calculations of bulk and surfaces, Brillouin zone sampling was carried out using the Monkhorst–Pack method, with K-points set to 10 × 10 × 10 for bulk and 6 × 6 × 1 for surfaces [33]. During the relaxation of all atomic models, the Broyden–Fletcher–Goldfarb–Shanno (BFGS) minimization scheme was used for ensemble optimization steps, with all atomic structures and volumes being fully relaxed [34]. The structural optimizations of these systems were relaxed until the total energy was less than 1 × 10⁻⁵ eV/atom. The convergence tolerances for the forces between the atoms, maximum stress, and maximum ionic displacement were set to 0.01 eV/Å, 0.02 GPa, and 5 × 10⁻⁴ Å, respectively. Additionally, for the interfaces, a vacuum layer of 14 Å was chosen to isolate the free surfaces, ensuring sufficient space to prevent their interactions.

3. Results and Discussion

3.1. Crystal Properties

The schematic diagram of the crystal structure model used in the calculations is shown in Figure 1. From Figure 1a, it can be seen that the crystal structure model of the bulk TiB₂ was of the AlB₂ type, with the space group P6/mmm. As shown in Figure 1b, the crystal structure model of the bulk Al₃Zr was of the Cu₃Au type, with the space group Pm-3m.

To ensure the accuracy of the calculations in this paper, especially the accuracy of the pseudopotentials used, a series of calculations were performed on the lattice constants and formation enthalpies of TiB₂ and Al₃Zr using the Perdew–Burke–Ernzerhof (PBE) within the Generalized Gradient Approximation. The lattice constants were obtained directly from the calculations, and the formation enthalpies of TiB₂ and Al₃Zr were derived using Equations (1) and (2) [35]:

$$\Delta \gamma H_{Ti{B}_2}={\displaystyle \frac{E_{bulk}^{Ti{B}_2}-N_{bulk}^{Ti}\left(Ti{B}_2\right)\times \frac{E_{bulk}^{Ti}}{N_{bulk}^{Ti}\left(Ti\right)}-N_{bulk}^B\left(Ti{B}_2\right)\times \frac{E_{bulk}^B}{N_{bulk}^B\left(B\right)}}{N_{bulk}^{Ti}\left(Ti{B}_2\right)+N_{bulk}^B\left(Ti{B}_2\right)}}$$

(1)

$$\Delta \gamma H_{{Al}_{3}Zr}={\displaystyle \frac{E_{bulk}^{{Al}_{3}Zr}-N_{bulk}^{Al}\left({Al}_{3}Zr\right)\times \frac{E_{bulk}^{Al}}{N_{bulk}^{Al}\left(Al\right)}-N_{bulk}^{Zr}\left({Al}_{3}Zr\right)\times \frac{E_{bulk}^{Zr}}{N_{bulk}^{Zr}\left(Zr\right)}}{N_{bulk}^{Al}\left({Al}_{3}Zr\right)+N_{bulk}^{Zr}\left({Al}_{3}Zr\right)}}$$

(2)

where $\Delta \gamma H_{Ti{B}_2}$ and $E_{bulk}^{Ti{B}_2}$ represent the formation enthalpy and the total energy of bulk TiB₂, respectively; $E_{bulk}^{Ti}$ denotes the energy of a Ti atom in the bulk, and $E_{bulk}^B$ is the energy of boron in its conventional cell; $N_{bulk}^{Ti}\left(Ti{B}_2\right)$ and $N_{bulk}^B\left(Ti{B}_2\right)$ are the numbers of Ti and B atoms in the bulk model of TiB₂, respectively; and $N_{bulk}^{Ti}\left(Ti\right)$ and $N_{bulk}^B\left(B\right)$ represent the numbers of atoms in the conventional cells of titanium and boron, respectively. Equation (2) follows the same rationale and will not be reiterated here.

Table 1. Calculated and experimental lattice constants and formation energies of bulk TiB₂ and bulk Al₃Zr.

Phases	Method	Lattice Constant (Å)			${\Delta }_{\mathit{r}}\mathit{H}(\mathbf{e}\mathbf{V}/\mathbf{a}\mathbf{t}\mathbf{o}\mathbf{m})$
		a	b	c
TiB2	Present	3.026	3.026	3.229	−1.072
	Calculation a	3.034	3.034	3.224	−1.079
	Calculation b	3.030	3.030	3.221	−1.085
	Experiment c	3.032	3.032	3.229	−1.134
Al3Zr	Present	4.082	4.082	4.082	−0.487
	Calculation d	4.117	4.117	4.117	−0.471
	Calculation e	4.096	4.096	4.096	−0.466
	Experiment f	4.090	4.090	4.090	-

^a Ref. [36] GGA-PBE. ^b Ref. [37] GGA-PBE. ^c Ref. [38]. ^d Ref. [11] GGA-PBE. ^e Ref. [39] GGA-PW91. ^f Ref. [40].

presents the lattice constants and formation enthalpies of TiB₂ and L1₂-Al₃Zr, comparing them with other computational and experimental results. In this work, the lattice constants of TiB₂ differ by less than 0.004 Å from the theoretical values reported in the literature [36,37], and the deviation from the experimental values reported in the literature does not exceed 0.006 Å [38]. The calculated formation enthalpy of TiB₂ had a maximum deviation of 0.013 eV/atom from other theoretical values obtained from calculations and a difference of 0.062 eV/atom compared with the experimental formation enthalpies. For the L1₂-Al₃Zr unit cell structure, the maximum discrepancy in lattice constants compared to theoretical values from the other literature was 0.035 Å, with a formation enthalpy error within 0.021 eV/atom [11,39], and the discrepancy from the experimental lattice constant reference values was 0.008 Å [40]. These results indicate that our current computational methods and settings are reliable, and the calculation accuracy was sufficient.

3.2. Surface Convergence and Surface Energy

Due to the significant difference in lattice constants between the primitive cells of TiB₂(0001) and the Al₃Zr(001) surfaces, direct construction of the interface model was not feasible. To satisfy the periodicity conditions and lattice relationships, it was necessary to find matching lattice constants for the TiB₂(0001) and Al₃Zr(001) surfaces to build the interface model. After testing, it was found that the lattice constants of the TiB₂(0001)-($\sqrt{3}\times 2$)R30° surface unit and the Al₃Zr(001)-($\sqrt{2}\times \sqrt{2}$)R45° surface unit are numerically similar, and the mismatch rate at the interface can be calculated as follows:

$$\xi ={\displaystyle \frac{2A}{A_{A{l}_{3}Zr}+A_{Ti{B}_2}}}-1$$

(3)

where in the above equation, $A$ represents the interfacial area after the TiB₂(0001) and Al₃Zr(001) surfaces form an interface, while $A_{A{l}_{3}Zr}$ and $A_{Ti{B}_2}$, respectively, refer to the areas of the Al₃Zr(001) and TiB₂(0001) surfaces before forming the interface. Using Equation (1), the mismatch rate of this interface was calculated to be less than 1%. Therefore, the TiB₂(0001)/Al₃Zr(001) interface model was considered reasonable.

Before establishing the interface model, it was necessary to perform convergence tests on the atomic thickness of the phases on both sides of the interface to ensure that the atoms deep within the surface exhibited characteristics similar to those of bulk phase atoms. Typically, the convergence of the surface energy for both the TiB₂(0001) and Al₃Zr(001) surfaces was used to determine this. Since both the TiB₂(0001) and Al₃Zr(001) surfaces are non-polar, the convergence could be assessed by measuring the percentage change in relaxation between the atomic layers (${\Delta }_{ij}$, where $i$, $j$ represent the atomic layers in the surface model).

The change in interlayer distance ($\Delta ij$) is defined as follows:

$${\Delta }_{ij}={\displaystyle \frac{d_{ij}-d_{ij}^0}{d_{ij}^0}}\times 100\%$$

(4)

where ${\Delta }_{ij}$ is the initial distance between layers $i$ and j in the ideal crystal structure and is the vertical distance between the adjacent $i$ and j layers after relaxation.

After the geometric optimization of the atoms, the changes in the interlayer distances of the TiB₂(0001) and Al₃Zr(001) surfaces are shown in

Table 2. Interlayer relaxation percentages for TiB₂(0001) and Al₃Zr(001) surfaces at different slab thicknesses.

Surface	Termination	Interlayer	Slab Thickness, n
			3	5	7	9	11
TiB2(0001)	Ti	Δ12	−6.96	−7.69	−7.54	−7.73	−7.63
		Δ23		−1.61	−2.09	−2.00	−1.86
		Δ34			−1.98	−2.51	−2.29
		Δ45				−2.72	−2.14
		Δ56					−2.73
	B	Δ12	−8.15	−10.20	−9.53	−9.61	−9.71
		Δ23		0.47	0.11	−0.11	0.02
		Δ34			−2.31	−2.24	−2.40
		Δ45				−3.16	−2.69
		Δ56					−2.71
Al3Zr(001)	Al	Δ12	−1.45	3.02	1.33	0.63	0.91
		Δ23		−1.52	−1.20	1.68	1.05
		Δ34			0.65	2.89	2.62
		Δ45				−0.81	−0.46
		Δ56					0.67
	Zr-Al	Δ12	−15.33	−18.17	−16.38	−16.68	−16.92
		Δ23		−4.95	−5.10	−4.43	−4.73
		Δ34			4.28	2.51	3.21
		Δ45				0.09	−0.39
		Δ56					0.89

. For TiB₂, when the number of atomic layers nn in the Ti-terminated and B-terminated TiB₂(0001) surfaces was ≥9, the changes in the interlayer distances in TiB₂(0001) were essentially converged. A comparison of the percentage changes in the interlayer distances of the same number of atomic layers shows that the relaxation of the outermost interlayer distance on the B-terminated surface was much greater than that on the Ti-terminated surface. The results indicate that the outermost interlayer distances on the Ti-terminated and B-terminated surfaces with nine atomic layers decreased by 7.7% and 9.6% of their bulk volumes, respectively. From Table 2, it can be seen that for Al₃Zr, the interlayer distances on the Al-terminated side essentially converged when nn was ≥9, while on the ZrAl-terminated side, a good convergence effect was achieved when nn was ≥7.

For simulations related to surfaces, the more atomic layers the surface model contains, the more precise the results become. However, as the number of atoms increases, the computational resources required also increase. Therefore, to further determine the minimum thickness of the TiB₂(0001) and Al₃Zr(001) surfaces that satisfy the properties of the bulk phase, and to reduce the error in determining the number of layers, we further discussed the impact of different layer thicknesses on the convergence of surface energy. This approach makes the selection of convergence layer numbers more reliable [41].

Surface energy is an important parameter for measuring surface stability. For non-stoichiometric TiB₂ surface models, the chemical potentials of both Ti and B should be considered. Therefore, the surface energy of the TiB₂(0001) surface was calculated using a generalized definition of surface energy [42,43]:

$${\gamma }_{Ti{B}_2\left(0001\right)}={\displaystyle \frac{E_{surf}^{Ti{B}_2\left(0001\right)}-N_{Ti{B}_2\left(0001\right)}^{Ti}\times {\mu }_{bulk}^{Ti{B}_2}+\left({2N}_{Ti{B}_2\left(0001\right)}^{Ti}-N_{Ti{B}_2\left(0001\right)}^B\right)\times {\mu }_{slab}^B}{2\times A_{surf}^{Ti{B}_2\left(0001\right)}}}$$

(5)

In the formula, ${\gamma }_{Ti{B}_2\left(0001\right)}$ represents the surface energy of the TiB₂(0001) surface. $E_{surf}^{Ti{B}_2\left(0001\right)}$ refers to the total energy of the TiB₂(0001) surface used in the calculation. $N_{Ti{B}_2\left(0001\right)}^{Ti}$ is the number of Ti atoms on the TiB₂(0001) surface. $N_{Ti{B}_2\left(0001\right)}^B$ represents the number of B atoms on the TiB₂(0001) surface. ${\mu }_{bulk}^{Ti{B}_2}$ is the chemical potential of bulk TiB₂. ${\mu }_{slab}^B$ represents the chemical potential of B in the surface. $A_{surf}^{Ti{B}_2\left(0001\right)}$ is the area of the TiB₂(0001) surface model.

It is well known that the chemical potentials of various elements in the surface composition are always lower than the corresponding bulk chemical potentials because the surface optimizes energy through a specific ordered arrangement [44]. This implies that ${\mu }_{slab}^B\le {\mu }_{bulk}^B$ and ${\mu }_{slab}^{Ti}\le {\mu }_{bulk}^{Ti}$. Therefore, the range of chemical potential for B is as follows:

$${\mu }_{bulk}^{{TiB}_2}={\mu }_{bulk}^{Ti}+2{\mu }_{bulk}^B+\Delta \gamma H_{Ti{B}_2},{\displaystyle \frac{1}{2}}\Delta \gamma H_{Ti{B}_2}\le {\mu }_{slab}^B-{\mu }_{bulk}^B\le 0$$

(6)

In the formula mentioned above, $\Delta \gamma H_{Ti{B}_2}$ represents the formation enthalpy of TiB₂ previously discussed. ${\mu }_{bulk}^{Ti}$ is the chemical potential of Ti in the bulk phase. ${\mu }_{bulk}^B$ is the chemical potential of B in the α-B₁₂ bulk phase. ${\mu }_{bulk}^{{TiB}_2}$ is the chemical potential of the TiB₂ bulk phase.

The Al₃Zr(001) surface model used in the calculations was also non-stoichiometric, hence the surface energy of the Al₃Zr(001) surface could be determined using Equation (7):

$${\gamma }_{Zr{Al}_3\left(001\right)}={\displaystyle \frac{E_{surf}^{Zr{Al}_3\left(001\right)}-N_{Zr{Al}_3\left(001\right)}^{Zr}\times {\mu }_{bulk}^{Zr{Al}_3}+\left({3N}_{Zr{Al}_3\left(001\right)}^{Zr}-N_{Zr{Al}_3\left(001\right)}^{Al}\right)\times {\mu }_{slab}^{Al}}{2\times A_{surf}^{Zr{Al}_3\left(001\right)}}}$$

(7)

Similarly, the range for the chemical potential of aluminum is as follows:

$${\mu }_{bulk}^{{Al}_{3}Zr}={\mu }_{bulk}^{Zr}+3{\mu }_{bulk}^{Al}+\Delta \gamma H_{{Al}_{3}Zr},{\displaystyle \frac{1}{3}}\Delta \gamma H_{{Al}_{3}Zr}\le {\mu }_{slab}^{Al}-{\mu }_{bulk}^{Al}\le 0$$

(8)

Table 3. The surface energy (J/m²) calculation results of TiB₂(0001) and Al₃Zr(001) surfaces.

Layer	TiB2(0001)		Al3Zr(001)
	Ti-Terminated	B-Terminated	ZrAl-Terminated	Al-Terminated
3	5.442	3.014	1.768	1.193
5	5.447	2.987	1.721	1.113
7	5.392	2.914	1.707	1.073
9	5.354	2.876	1.705	1.053
11	5.305	2.836	1.637	1.017
13	5.264	2.779	1.686	1.062

lists the surface energy data for different structures of the TiB₂(0001) surface and the Al₃Zr(001) surface under conditions rich in B and Al, respectively. The surface energies of the Ti-terminated and B-terminated ends of TiB₂(0001) converged as the number of atomic layers increased, reaching convergence when the atomic layers were ≥9. Similarly, the Al-terminated end of Al₃Zr(001) converged at the atomic layers ≥7; the ZrAl-terminated end converged when the atomic layers were ≥9. The smaller the surface energy value, the better the stability of the surface, thus the stability of the surface was consistent with the assessment of the interlayer distances. This implies that choosing a 9-layer TiB₂(0001) in conjunction with a 7-layer ZrAl-terminated end and a 9-layer Al-terminated end of Al₃Zr(001) to construct the TiB₂/Al₃Zr interface is feasible for subsequent research.

The surface energy of non-stoichiometric surfaces is influenced by the chemical potential difference between the surface atoms and the bulk atoms. Therefore, we calculated the surface energy for a 9-layer TiB₂(0001) with different B chemical potential differences (${\mu }_{slab}^B-{\mu }_{bulk}^B)$, and also for a 7-layer ZrAl-terminated and a 9-layer Al-terminated Al₃Zr(001) with different Al chemical potential differences ${(\mu }_{slab}^{Al}-{\mu }_{bulk}^{Al})$. The relationship between the calculated surface energies and the different chemical potential differences is shown in Figure 2. From Figure 2a, it can be seen that the surface energy range for the B-terminated TiB2(0001) was from 6.124 to 2.876 J/m²; for the Ti-terminated TiB₂(0001), the range was from 2.106 to 5.354 J/m². Under Ti-rich conditions, the surface energy of the Ti-terminated side was lower than that of the B-terminated side, indicating that the Ti-terminated TiB₂(0001) surface was more stable at this time. However, as the B chemical potential difference increased, the surface energy of the Ti-terminated side gradually increased, while that of the B-terminated side gradually decreased. When the B chemical potential difference was greater than −0.613 eV, the B-terminated TiB₂(0001) surface became more stable. This is consistent with the results reported in the literature [45]. From Figure 2b, it is evident that the surface energy of L1₂-Al₃Zr(001) also showed a linear relationship with the chemical potential difference in Al. The surface energy range for the Al-terminated Al₃Zr(001) was from 1.545 to 1.053 J/m²; for the ZrAl-terminated Al₃Zr(001), the range was from 1.213 to 1.705 J/m². The results indicate that as the chemical potential difference shifted from Zr-rich conditions to Al-rich conditions, the surface energy of the Al-terminated end showed a decreasing trend, while the surface energy of the ZrAl end showed an increasing trend. Within the range of −0.487 eV to −0.322 eV, the surface energy of the Al end was approximately lower than that of the ZrAl end, and in the range from −0.322 eV to 0 eV, the opposite was true. This suggests that the Al-terminated end of Al₃Zr(001) is more stable in an aluminum-rich environment, while the ZrAl-terminated end is more stable in a zirconium-rich environment.

3.3. Interface Characteristics of TiB₂(0001)/Al₃Zr(001)

3.3.1. Atomic Structure of the Interface

Based on the results of the convergence tests, the TiB₂/Al₃Zr interface was constructed as follows: nine layers of TiB₂(0001) were stacked on top of the Al₃Zr(001) surface, with nine layers chosen for the Al-terminated surface of Al₃Zr(001) and seven layers for the ZrAl-terminated surface. The mismatch degree of this interface was less than 1%, indicating that it was a typical coherent interface. Considering that the TiB₂(0001) and Al₃Zr(001) surfaces each have two different terminations (Ti-terminated, B-terminated and Al-terminated, ZrAl-terminated) and three different stacking sequences—top site (TS), bridge site (BS), and hollow site (HS)—a total of 12 interface models were established. These models are illustrated in Figure 3. The “interface name” was designated based on the first letter of the terminations and stacking order of TiB₂(0001) and Al₃Zr(001).

3.3.2. Interface Adhesion Work

The strength of the bonding at the interface can be described by the adhesion work. The adhesion work between the TiB₂(0001) surface and the Al₃Zr(001) surface can be determined using the following definition [46]:

$$W_{ad}={\displaystyle \frac{E_{A{l}_{3}Zr}^{slab}+E_{Ti{B}_2}^{slab}-E_{\frac{A{l}_{3}Zr}{Ti{B}_2}}^{total}}{A}}$$

(9)

where $E_{A{l}_{3}Zr/Ti{B}_2}^{total}$ and $A$ represent the total energy and the interface area of the TiB₂(0001)/Al₃Zr(001) interface, respectively. $E_{Ti{B}_2}^{slab}$ and $E_{A{l}_{3}Zr}^{slab}$ are the energies of the relaxed TiB₂(0001) and Al₃Zr(001) slabs, respectively.

Table 4. The comparison of interfacial distance (d₀) and adhesion work (W_ad) for 12 different interfaces.

Terminations of TiB2(0001)	Terminations of Al3Zr(001)	Stacking Sequence	Interface Name	d0 (Å)	Wad (J/m2)
Ti	Al	TS	TAT	2.32	2.37
		BS	TAB	2.34	2.41
		HS	TAH	2.18	2.40
Ti	Zr-Al	TS	TZT	2.60	1.26
		BS	TZB	2.87	1.26
		HS	TZH	2.41	1.97
B	Al	TS	BAT	2.07	1.92
		BS	BAB	1.98	1.87
		HS	BAH	1.92	1.88
B	Zr-Al	TS	BZT	2.27	2.22
		BS	BZB	2.16	2.22
		HS	BZH	2.18	2.29

displays the interfacial distances and corresponding adhesion work for 12 different interfaces. From Table 4, it is evident that the stacking position significantly impacted both the equilibrium distance and the adhesion work of the interfaces. Within the same stacking order, the adhesion work of the interfaces between the Ti-terminated and Al-terminated sides of the TiB₂/Al₃Zr was higher than that of other terminated interfaces, indicating better stability for interfaces involving the Ti termination compared with those with the Al termination.

For interfaces between the B-terminated and ZrAl-terminated sides, the adhesion work for the BZT, BZB, and BZH interfaces were very close, at 2.22, 2.22, and 2.29 J/m², respectively, suggesting that the stacking order had a minimal impact on the bonding strength of these interfaces. However, for interfaces between the Ti-terminated and ZrAl-terminated sides, the effect of the stacking order was relatively more significant. The TZH interface, for example, had an adhesion work of 1.97 J/m², which was substantially higher than that of the TZT and TZB interfaces, yet their interfacial distances were the opposite. This is because, in the TZH interface, the Ti atoms interacted with the surrounding two Zr atoms and two Al atoms, affecting the interfacial distance. Generally, the higher the $W_{ad}$, the stronger the bonding force between the interface atoms, hence the TAB interface was the most stable among the 12 interface structures, possessing the greatest bonding strength.

3.3.3. Interfacial Energy

Interfacial energy is a key thermodynamic quantity used to measure the stability of an interface. Generally, the lower the interfacial energy, the more stable the interface structure. The formula for calculating the interfacial energy of the TiB₂(0001)/Al₃Zr interface is as follows [47,48]:

$${\gamma }_{int}={\gamma }_{Ti{B}_2\left(0001\right)}+{\gamma }_{Zr{Al}_3\left(001\right)}-W_{ad}$$

(10)

In the formula, ${\gamma }_{Ti{B}_2\left(0001\right)}$ represents the interfacial energy; ${\gamma }_{Ti{B}_2\left(0001\right)}$ and ${\gamma }_{Zr{Al}_3\left(001\right)}$, respectively, denote the surface energies of the TiB₂(0001) and Al₃Zr(001) surfaces; and $W_{ad}$ represents the adhesion work of the corresponding interface.

Based on the previous analysis, we have identified interface structures with the lowest interfacial energies for further study: the TAB, TZH, BAT, and BZH interfaces. Figure 4 illustrates the variation in interfacial energy for these four interfaces as a function of the chemical potential difference between boron in the slab and bulk $({\mu }_{slab}^B-{\mu }_{bulk}^B)$. Throughout the range of $({\mu }_{slab}^B-{\mu }_{bulk}^B)$, the interfacial energies for TAB, TZH, BAT, and BZH were, respectively, 1.730 to 4.430 J/m², 1.848 to 5.509 J/m², 2.455 to 6.118 J/m², and 2.895 to 5.595 J/m². Evidently, for interfaces on the titanium side, TAB exhibits the lowest interfacial energy and the highest interface stability, consistent with the results obtained from the adhesion work analysis. For interfaces on the boron side, under titanium-rich conditions, BZH displayed relatively lower interfacial energy, indicating a more stable interface structure. In contrast, under boron-rich conditions, BAT showed better thermodynamic stability.

3.4. Electron Structure

The bonding at interfaces fundamentally involves interactions among the electrons located at the interface. To further analyze the interactions within the TiB₂(0001)/Al₃Zr(001) interface, we selected representative models from four different terminated interfaces and calculated their differential charge density maps. These interfaces were the TAB, BAT, TZH, and BZH interfaces. Each of these interface models exhibited the highest adhesion work and the lowest interfacial energy within their respective terminated interface models. The differential charge density for the TiB₂(0001)/Al₃Zr(001) interface can be expressed as follows [47]:

$$\Delta \rho ={\rho }_{toal}-{\rho }_{TiB2\left(0001\right)}-{\rho }_{Al3Zr\left(001\right)}$$

(11)

In this formula, ${\rho }_{toal}$ is the total charge density of the TiB₂(0001)/Al₃Zr(001) interface; and ${\rho }_{TiB2\left(0001\right)}$ and ${\rho }_{Al3Zr\left(001\right)}$ represent the independent charge densities of the TiB₂(0001) surface and the Al₃Zr(001) surface that make up the interface structure, respectively.

Figure 5 shows the differential charge density results for the four interface models. Red and blue represent the accumulation and loss of charge, respectively. From Figure 5a, it can be seen that there was a significant charge accumulation between the Ti and Al atoms at the TAB interface, which was due to the loss of charge from the Al and Ti atoms. This indicates the formation of strong Ti−Al covalent bonds at the L1₂-Al₃Ti(111) surface. Additionally, some charge accumulated between the Al and Zr atoms, suggesting the formation of Al−Zr covalent bonds. As shown in Figure 5b, electron transfer occurred between the Al and B atoms, indicating the presence of ionic bonds; furthermore, there was a small accumulation of charge between the Al and B atoms, and the presence of B-Al covalent bonds. Figure 5c describes the differential charge density at the TZH interface, clearly showing Ti-Zr covalent bonding. On the BZH interface, strong covalent bonds were formed between the Zr and B atoms. It is noteworthy that Zr formed covalent bonds with both Al atoms in the same layer and those in the second layer, but the covalent bond between Zr and Al in the same layer was stronger than that between Zr and the second-layer Al, which was related to the distances between the atoms.

To further investigate the electronic structure and analyze the bonding nature of the atoms at the TiB₂(0001)/Al₃Zr(001) interface, we further analyzed the density of states (DOSs) of the stable interfaces, with results shown in Figure 6. Notable distinctions were observed in the sharpness of the peaks associated with the same elements in the TiB₂(0001) slab near the interface junction, and similarly in the Al₃Zr(001) slab. This indicates that the bonding process in these four interface models was primarily contributed by the outer electrons of the first layer of atoms at the interface end.

Continuing the analysis with a focus on the TAB interface model, we aimed to understand the bonding characteristics of the interface. From Figure 6a, it is evident that the first layer of Ti at the TiB₂(0001) end and the first layer of Al at the Al₃Zr(001) end exhibited a peak between 0 and 5 eV, primarily contributed by the Ti 3d and Al 3p orbitals. Moreover, hybridization between Ti 3d and Al 3p also occurred around 10 eV. Notably, there was intense hybridization near 8 eV between the first layer of Ti at the TiB₂(0001) end and the second layer of Zr at the Al₃Zr(001) end, mainly contributed by the Ti 3d and Zr 4d orbitals. The peak intensities indicated strong hybridization at the TAB interface.

In Figure 6b, the hybridization between the first layer of B on the TiB₂(0001) side and the first layer of Al on the Al₃Zr(001) side was also evident. A distinct peak appeared near the Fermi level, mainly due to the contributions of the B 2p and Al 3s orbitals. Additionally, the 2p orbital of the first-layer B atoms showed a prominent peak around 5 eV, and there were indications of increased orbital density around 5 eV in the 3p orbital of the first-layer Al atoms. This suggests a possible overlap between the B 2p and Al 3p orbitals, potentially pointing to the presence of hybridization. Such hybridization could lead to the formation of strong B-Al covalent bonds at the interface.

Figure 6c shows significant hybridization between Ti 3d and Zr 4d in the 5 to 10 eV range, forming Ti-Zr covalent bonds. The interface composition in Figure 6d, which consisted of the first layer of B at the TiB₂(0001) end and the first layer at the ZrAl end of Al₃Zr(001), however, showed no significant common peaks, with only slight hybridization near the Fermi level.

The density of states analysis in Figure 6 revealed variations in the electronic states of Al atoms when adjacent to Zr compared with their presence in layers without Zr. This observation suggests differences in orbital interactions, likely due to hybridization, between Al and neighboring atoms under different compositional contexts. Notably, the analysis showed that the hybridization patterns were more pronounced in the TAB interface, where peak intensities suggested a stronger covalent character in the bonding interactions, particularly involving Al 3p and Zr 4d orbitals. This finding underscores the impact of atomic positioning and local chemical environment on the electronic properties at the interface. Therefore, it can be concluded that the hybridization at the TAB interface exhibited a higher intensity and complexity; based on the analysis of the density of states and differential charge density, the order of interface stability is as follows: TAB interface > BZH interface > TZH interface > BAT interface. This is consistent with the previous analyses of adhesion work and interfacial energy.

4. Conclusions

To understand the interfacial bonding characteristics of the TiB₂(0001)/Al₃Zr(001) interface, first-principles studies were conducted focusing on the adhesion work, interfacial energy, and electronic structure. This study considered 12 different TiB₂/Al₃Zr interfaces featuring four different terminations and three stacking sequences, with a detailed investigation on four of the most stable structures. The main findings of the study are summarized as follows:

(1)

Simulation tests have demonstrated that a stable interface structure can be formed when a 9-layer TiB₂(0001) surface is combined with a 7-layer ZrAl-terminated and a 9-layer Al-terminated Al₃Zr(001) surface.

(2)

For interfaces with the same termination in TiB₂/Al₃Zr, the study identifies the bridge-site stacking (TAB) at the T/Al termination, hollow-site stacking (TZH) at the Ti/ZrAl termination, top-site stacking (BAT) at the B/A termination, and hollow-site stacking at the B/ZrAl termination as the optimal structures. Among these interfaces, the TAB interface has the strongest adhesion strength due to the highest adhesion work of 2.41 J/m².

(3)

The interfacial energies of TAB, TZH, BAT, and BZH across the entire range of ${(\mu }_{slab}^B-{\mu }_{bulk}^B)$ are, respectively, 1.730 to 4.430 J/m², 1.848 to 5.509 J/m², 6.118 to 2.455 J/m², and 5.595 to 2.895 J/m². Evidently, TAB exhibits the lowest surface energy and the highest interface stability.

(4)

Electronic structure analysis indicates that the TAB, TZH, and BZH interfaces primarily feature covalent bonding, while the BAT interface shows both ionic and covalent bonds. According to the analysis of the electronic structure, the stability order of the TiB₂(0001) and Al₃Zr(001) interfaces is as follows: TAB interface > BZH interface > TZH interface > BAT interface.