First-Principles Investigation of the Shear Properties and Sliding Characteristics of c-ZrO₂(001)/α-Al₂O₃($\mathbf{1}\overline{\mathbf{1}}\mathbf{02}$) Interfaces

Abstract

The ideal mechanical shear properties and sliding characteristics of c-ZrO₂(001)/α-Al₂O₃($1\overline{1}02$) interfaces are examined through simulated shear deformations using first-principles calculations. We investigate three types of interface models, abbreviated as O-, 2O-, and Zr- models, when shear displacements are applied along the <$1\overline{1}01$> and <$11\overline{2}0$> directions of their Al₂O₃ lattice. The theoretical shear strength and unstable stacking energy of the ZrO₂/Al₂O₃ interfaces are discussed. In the process of the ZrO₂/Al₂O₃ interfacial shear deformation, we find that the sliding of the ZrO₂ atomic layers, accompanied by the shifting of Zr atoms and Al atoms near the interface, plays a dominant role; in addition, the ZrO₂/Al₂O₃ interfaces fail within the ZrO₂ atomic layer. Among the three models, the O- model exhibits the strongest shear resistance; whereas the Zr- model is the most prone to slip. Furthermore, their tensile and shear strengths are compared; moreover, their potential applications are provided.

1. Introduction

Thermal barrier coatings (TBCs) are widely used in the protection of hot components in gas turbine engines; they can reduce the corrosive degradation of turbine blades and improve the efficiency of the engine [1,2,3,4,5,6,7]. Typically, TBCs consist of four material layers: (i) superalloy substrate; (ii) metallic bond coat (BC); (iii) thermally grown oxide (TGO); and (iv) ceramic top coat (TC) [8].

Yttria-stabilized zirconia (YSZ) is a frequently used TC material owing to its low thermal conductivity and high thermal expansion coefficient (TEC). When TBCs operate at high temperatures, there are many factors affecting their lifetime; such as the growth of TGO [9,10], phase transformation [11,12], the influence of calcium-magnesium-alumina-silicate (CMAS) [13,14], sintering at elevated temperatures [15,16], and residual stress [17,18]. The lifetime of TBCs at high temperatures is reduced for the above factors, so that the TBCs cannot efficiently protect turbine engines. Guan et al. proposed a master–slave model with a fluid-thermo-structure.

(FTS) interaction for the thermal fatigue life prediction of TBCs [19]. As the thickness of the TGO increased, the stresses at the TC/TGO and TGO/BC interfaces increased; resulting in the fracture of the TC/TGO or TGO/BC interfaces [20]. It is widely recognized that the lifetime of TBCs at high temperatures is mainly affected by the mechanical properties of the TC/TGO and TGO/BC interfaces. Investigating the mechanical properties of interfaces directly from experiments remains a challenge because of the lack of experimental conditions and difficulty of sample preparation. Atomistic modeling and simulation, which provide the intrinsic atomic properties of interface structures, are often used to investigate the mechanical properties of interfaces and identify their failure mechanisms [21]. The shear strength and sliding behavior of the Ni(111)/α-Al₂O₃(0001) interface were investigated by Guo et al. [22] using first-principles calculations. The ideal mechanical strength of the ZrO₂(111)/Ni(111) interface was calculated through simulated tensile and shear deformations using first-principles calculations [23]. However, despite these previous efforts, the mechanical strength and shear behavior of ZrO₂/Al₂O₃ interfaces under first-principles shear simulation are still not well examined. The difficulty of this type of research is mainly due to the complexity of the ceramic/ceramic interface structures and the large number of computations involved in the shear simulation process.

In this study, shear simulations of c-ZrO₂(001)/α-Al₂O₃($1\overline{1}02$) interfaces were carried out using first-principles calculations. The shear directions of the three models were applied along the <$1\overline{1}01$> and <$11\overline{2}0$> directions of the Al₂O₃ lattice. For all three models, we examined the ideal shear strength, unstable stacking energy, and evolution of the atomic structures to clarify the sliding mechanisms of the c-ZrO₂(001)/α-Al₂O₃($1\overline{1}02$) interfaces. With these results, we further compared their shear and tensile properties to clarify the overall features of the c-ZrO₂(001)/α-Al₂O₃($1\overline{1}02$) system.

2. Methodology

In this study, we examine c-ZrO₂(001)/α-Al₂O₃($1\overline{1}02$) interfaces using the SIESTA [24] code based on density functional theory (DFT). We adopted the all-electron projector augmented wave (PAW) method within the generalized gradient approximation (GGA) for the electron exchange and correlation. For our calculations, a 5 × 5 × 1 Monkhost-Pack scheme was used for the k-point sampling of the Brillouin zone. The cutoff energy for the plane-wave basis was 500 eV, and the self-consistent energy relaxation criterion for electron was 10⁻⁵ eV. Relaxation is performed using the conjugate gradient (CG) algorithm, and the convergence criterion is a Hellman–Feynman force of less than 0.01 eV/Å.

Ideal coherent c-ZrO₂/α-Al₂O₃ interface models are discussed in ref. [25]. The c-ZrO₂ surface was cut from its crystal on the (001) plane, the α-Al₂O₃ surface was cut from its crystal on the ($1\overline{1}02$) plane, and the lattice misfit of the c-ZrO₂(001)/α-Al₂O₃($1\overline{1}02$) interface was acceptable [26]. The layering sequence in the α-Al₂O₃($1\overline{1}02$) surface termination is |O—Al—O—Al—O|O—Al—O—Al—O|... This α-Al₂O₃($1\overline{1}02$) surface termination is a compact surface with the lowest energy. Compared to the α-Al₂O₃($1\overline{1}02$) surface termination, the c-ZrO₂(001) surface termination is more complex. The layering sequence in the c-ZrO₂(001) surface termination was Zr-O-Zr-..., O-Zr-O-..., and 2O-Zr-... [27]. In this study, three types of c-ZrO₂(001)/α-Al₂O₃($1\overline{1}02$) interface models were considered. In the following, these three models are named Zr-, O-, and 2O-, respectively. Figure 1 shows a schematic of the c-ZrO₂(001)/α-Al₂O₃($1\overline{1}02$) interfacial shear simulation. The supercell consists of a slab of α-Al₂O₃($1\overline{1}02$) sandwiched between two slabs of c-ZrO₂(001); furthermore, this unit cell is orthogonal, 5.11 Å × 4.86 Å in dimension. According to preliminary calculations, the atomic relaxation layers of the c-ZrO₂ slab and the α-Al₂O₃ slab were determined to be three and five layers, respectively. The atom layers outside the relaxation layers are considered to be rigid regions, represented by the two shaded rectangles in Figure 1; namely, the c-ZrO₂ and α-Al₂O₃ blocks. The c-ZrO₂ block moves a distance of δ relative to the α-Al₂O₃ block along the <$1\overline{1}01$> or <$11\overline{2}0$> direction in the α-Al₂O₃ lattice. The atoms outside the rectangles were then relaxed until the convergence criterion was satisfied.

3. Results and Discussion

3.1. Shear Strength Parameters

Figure 2 shows the shear stress–shear displacement curves for the c-ZrO₂(001)/α-Al₂O₃($1\overline{1}02$) interface. The peak shear stresses are 5.04 and 5.69 GPa at the critical shear displacements; and δ_c, 1.43 Å, and 1.85 Å along the <$1\overline{1}01$> and <$11\overline{2}0$> slip directions of the O- model (see Figure 2a,b), respectively. In addition, for the 2O- model, the peak shear stresses along the <$1\overline{1}01$> and <$11\overline{2}0$> directions are 3.66 and 4.52 GPa at 1.43 and 2.24 Å (see Figure 2c,d), respectively. Furthermore, for the Zr- model shown in Figure 2e,f, the peak shear stresses for the Zr- model are 2.94 and 3.95 GPa at 1.23 and 2.24 Å in the <$1\overline{1}01$> and <$11\overline{2}0$> directions, respectively. Compared with the 2O- and Zr- shear deformation models, the peak shear stress in the O- model is greater in the same slip direction. This implies that the termination of c-ZrO₂(001) has a significant influence on the adhesive strength of the interface. In addition, the peak shear stress in the <$11\overline{2}0$> direction is higher than that in the <$1\overline{1}01$> direction for all three models. This indicates that the <$11\overline{2}0$> direction has a stronger shear resistance than the <$1\overline{1}01$> direction in the interface slip process. In Figure 2a–f, the maximum shear stress values in region II are 3.13, 5.11, 2.65, 2.98, 2.51, and 2.59 GPa, respectively. The maximum shear stress value in region I is larger than that in region II in the same model.

The shear moduli of the α-Al₂O₃ single crystals obtained from the calculations and experiments are 168 GPa and 163 GPa, respectively [28,29]. The shear moduli of the c-ZrO₂ single crystals obtained by calculation and experiment are 106 and 81 GPa, respectively [30,31]. The maximum theoretical shear modulus of the supercell is estimated to be 38.7 GPa. Compared with the shear moduli of the α-Al₂O₃ and c-ZrO₂ single crystals, this value is much smaller. These results indicate that the c-ZrO₂(001)/α-Al₂O₃($1\overline{1}02$) interfaces are weaker than those of bulk α-Al₂O₃ and c-ZrO₂. Furthermore, this implies that the c-ZrO₂(001)/α-Al₂O₃($1\overline{1}02$) interfaces are not simply complements of bulk α-Al₂O₃ and c-ZrO₂.

The energy per unit area of the slip plane Φ is defined as $\Phi ={\displaystyle \int \tau d\delta }$; τ and δ are the shear stress and displacement, respectively. The maximum value of Φ is termed the unstable stacking energy γ_us [32], which is the area between δ = 0 and the first δ = δ_t at which τ = 0 again. It is an important parameter for characterizing the interfacial shear strength. The mechanical properties of ZrO₂/Al₂O₃ interfaces can be more comprehensively understood using the ideal shear strength combined with the unstable stacking energy. As shown in Figure 3, the unstable stacking energy γ_us of the O- model is larger than that of the other two models in the same slip direction; whereas the unstable stacking energy γ_us in the <$11\overline{2}0$> direction is higher than that in the <$1\overline{1}01$> direction for all three models. For the O- model, the calculated unstable stacking energies γ_us along the <$1\overline{1}01$> and <$11\overline{2}0$> slip directions are 399 and 612 mJ/m², respectively. For the 2O- model, the calculated unstable stacking energies γ_us along the <$1\overline{1}01$> and <$11\overline{2}0$> slip directions are 349 and 494 mJ/m², respectively. Furthermore, the calculated unstable stacking energies γ_us for the Zr- model along the <$1\overline{1}01$> and <$11\overline{2}0$> slip directions are 245 and 437 mJ/m², respectively.

The mechanical properties of the three interface models can be analyzed by combining the calculated ideal shear strength with the unstable stacking energy. The present results indicate that the O- model has the strongest shear resistance, and the Zr- model is the most prone to slip among the three models. It can also be concluded from the results that the shear resistance in the <$11\overline{2}0$> slip direction is higher than that in the <$1\overline{1}01$> direction in the same model. The O-<$11\overline{2}0$> model c-ZrO₂(001)/α-Al₂O₃($1\overline{1}02$) interface may be useful for optimizing TBC structures to meet harsh requirements.

3.2. Failure Mechanism

The evolution of the atomic structures at the c-ZrO₂(001)/α-Al₂O₃($1\overline{1}02$) interface during the shearing process is shown in Figure 4. The atom indices Zr1 and Zr2 denote the first Zr layer atom and second Zr layer in the c-ZrO₂ block from the interface, respectively. The same procedure is applied to Al1 and Al2. The capital letters of the markers in Figure 4 correspond to those in Figure 2. The failure process of the interfaces is investigated by combining Figure 2 and Figure 4.

From Figure 2, it can be seen that the shear stresses exhibit an obvious rebound phenomenon in region I. The shear stress increases gradually until it reaches the first peak value (point B), drops slightly to point C, and then continues to increase to the maximum value (point D). This rebound is mainly due to the shifting of Zr atoms, which represents the process of breaking and rebonding of the specific Zr-O and Zr-Al bonds near the interface. For the O-<$1\overline{1}01$> model, Figure 2a shows the breaking of the Zr1-O2 and Zr2-Al2 bonds; it also shows the rebonding of the Zr1-O1 and Zr1-Al2 bonds. For the O-<$11\overline{2}0$> model shown in Figure 2b, the Zr1-O1 and Zr1-O3 bonds break; in addition, the Zr1-Al2, Zr1-O2, and Zr2-O4 bonds rebond. For the 2O-model, the Zr1-O3 and Al1-O2 bonds break; the Zr1-O1, Zr1-O4, Zr1-Al1, and Al2-O2 bonds rebond along the <$1\overline{1}01$> direction, as shown in Figure 2c; the Zr1-O3, Zr1-O4, and Zr1-O5 bonds break; and the Zr-O1 and Zr1-Al1 bonds rebond along the <$11\overline{2}0$> direction (Figure 2d). For the Zr-model, the Zr2-O3 bond breaks, the Zr2-O2 bond rebonds, and the Zr2-Al2 bond rebonds along the <$1\overline{1}01$> direction (Figure 2e); the Zr2-O1 bond breaks and the Zr2-Al2 bond rebonds along the <$11\overline{2}0$> direction, as shown in Figure 2f.

The ZrO₂ atomic layers slide along the shear direction, accompanied by the shifting of Zr atoms, until the shear stresses reach the maximum value; this is shown in Figure 4 from A to D. The ZrO₂ atom layers undergo reconstruction from D to E, as shown in Figure 4. It is apparent that the models failed within the ZrO₂ atomic layer. The shear stresses decrease rapidly after point D in Figure 2, mainly because the ZrO₂ atom layers reconstruct intensively in the direction opposite to that of the shear accompanying the shifting of the Al atoms. Furthermore, the interfacial atomic configuration is transformed near both stress peaks; this indicates that the reconstruction of the interface structure has a significant influence on the mechanical strength of the interface region to resist sliding deformation. This phenomenon is consistent with the conclusions of Farkas [33]. Thus, it is concluded that in the process of the ZrO₂/Al₂O₃ interfacial shear deformation, the sliding of the ZrO₂ atomic layers, accompanied by the shifting of Zr and Al atoms near the interface, will play a dominant role. According to the comparison results of the maximum shear stress in regions I and II, it can also be concluded that the mechanical strength of the reconstructed interfaces is lower than that of the initial interfaces. Therefore, the reconstructed interfaces are more unstable and prone to slip than the initial interface.

3.3. Discussion

In this study, shear simulations of the c-ZrO₂(001)/α-Al₂O₃($1\overline{1}02$) interface models along the <$1\overline{1}01$> and <$11\overline{2}0$> directions of the α-Al₂O₃ lattice were performed using first-principles calculations. The O-<$11\overline{2}0$> model has the strongest shear resistance and the Zr-<$1\overline{1}01$> model is the most prone to slip. It indicates that the mechanical properties of c-ZrO₂(001)/α-Al₂O₃ ($1\overline{1}02$) interfaces are affected by the interfacial atomic configuration and shear direction. In the process of the ZrO₂/Al₂O₃ interfacial shear deformation, the ZrO₂ atomic layers slide along the shear direction, accompanied by the shifting of Zr and Al atoms near the interface; moreover, shear failure generally occurs in the ZrO₂ atomic layers. Furthermore, the reconstruction of the interface structure has a significant influence on the mechanical strength of the interface region that resists sliding deformation. In the Ni(111)/α-Al₂O₃ (0001) interface, Guo et al. [22] also observed the similar phenomenon. However, in the Al-terminated O-site interface, they observed that the maximum shear strength was along the Ni <110> direction; while the maximum unstable stacking energy was along the Ni <112> direction. This an intriguing observation; the reason for this phenomenon is the shifting of Al atoms near the interface during the sliding process. For our models, the shifting of Zr and Al atoms near the interface makes the sliding process more stable. In addition, the shear strength of the Ni/Al₂O₃ interfaces and the ZrO₂/Al₂O₃ interfaces are weak; indicating that shear failure occurs at any interface in TBC.

We examined the ZrO₂/Al₂O₃ interfaces using first-principle calculations to understand the tensile strength and failure properties [25]. In that work, the ideal tensile strengths and work of separation of the c-ZrO₂(001)/α-Al₂O₃ ($1\overline{1}02$) interfaces were obtained.

Table 1. The basic mechanical properties of the c-ZrO₂(001)/α-Al₂O₃ ($1\overline{1}02$) interfaces, including the tensile strength (σ_max), the work of separation (W_sep), the shear strength (τ_max), and the unstable stacking energy (γ_us).

Models	σmax (GPa)	Wsep (mJ/m2)	Shear Direction	τmax (GPa)	γus (mJ/m2)
O-	9.012	1208	<$1\overline{1}01$>	5.04	399
			<$11\overline{2}0$>	5.69	612
2O-	7.452	829	<$1\overline{1}01$>	3.66	349
			<$11\overline{2}0$>	4.52	494
Zr-	6.893	715	<$1\overline{1}01$>	2.94	245
			<$11\overline{2}0$>	3.95	437

summarizes the ideal tensile strength, the work of separation, the ideal shear strength, and the unstable stacking energy. For all the models in Table 1, the O-<$11\overline{2}0$> model is the most stable model; having the maximum tensile strength and shear strength. The Zr-<$1\overline{1}01$> model is the most unstable model; it has the lowest tensile strength and shear strength. These properties are closely related to the strength of the bonds at the interface. Compared with the tensile strengths, the shear strengths of the same model are much smaller. It indicates that ZrO₂/Al₂O₃ interfaces are more prone to shear failure than tensile failure. This is because of the different failure mechanisms between tensile failure and shear failure. Owing to the tensile failure of the interface, all the atomic bonds across the interface need to be broken; whereas shear failure only needs to be partially broken. Therefore, shear failure requires less stress and energy than tensile failure does.

The cohesive zone model (CZM) is a widely accepted model in the elastic–plastic fracture mechanics community. The core task of developing a material-dependent CZM is to extract material parameters, such as the cohesive energy and maximum cohesive strength in the cohesive laws. In the continuum mechanics framework, there are many ways to develop cohesive zone models; in addition, more accurate cohesive zone models can be constructed based on first-principles results [34,35,36]. Figure 5 shows the shear stress-displacement curves fitted by the Fourier series; the fitting curve is basically consistent with the exponential cohesive zone model [37] in the inset. We chose the O- and Zr- models for the highest and lowest mechanical strengths, respectively. The cohesive shear strength τ’_max and work of shear separation ϕ_t in the exponential CZM can be identified by the ideal shear strength τ_max and unstable stacking energy γ_us, respectively. Hence, it is necessary to determine the mechanical properties of the interface, such as the theoretical tensile strength, work of separation, theoretical shear strength, and unstable stacking energy.

In this study, shear simulations of c-ZrO₂(001)/α-Al₂O₃($1\overline{1}02$) interface models along the <$1\overline{1}01$> and <$11\overline{2}0$> directions of the α-Al₂O₃ lattice were performed using first-principles calculations. In practice, c-ZrO₂(001)/α-Al₂O₃($1\overline{1}02$) interfaces may slip along other directions and undergo different structural transformations. Therefore, further studies on the shear behavior of c-ZrO₂(001)/α-Al₂O₃($1\overline{1}02$) interfaces along other directions can provide more information for optimizing the TBC structure to meet the harsh requirements. In addition, the shear simulation method implemented in this study is the constrained shear method; which produces normal stress in the simulation process. Umeno [38] observed that the mechanical properties of silicon are affected by the internal displacement of atoms and normal stresses inside the cell. Based on this, we can investigate the effect of normal stresses on the ideal shear strength of the c-ZrO₂(001)/α-Al₂O₃($1\overline{1}02$) interface using the unconstrained shear method.

4. Conclusions

Shear simulations of the c-ZrO₂(001)/α-Al₂O₃($1\overline{1}02$) interfaces were carried out using first-principles calculations. The O- model sliding along the Al₂O₃ <$11\overline{2}0$> direction is the most stable, with a maximum shear strength of 5.69 GPa and an unstable stacking energy of 612 mJ/m². The Zr- model sliding along the Al₂O₃ <$1\overline{1}01$> direction is the most unstable; it has the minimum shear strength (2.94 GPa) and minimum unstable stacking energy (245 mJ/m²). In the process of c-ZrO₂(001)/α-Al₂O₃($1\overline{1}02$) interfacial shear deformation, the sliding of the Zr atomic layers, accompanied by the shifting of Zr atoms and Al atoms near the interface, plays a dominant role; in addition, shear failure generally occurs in the ZrO₂ atomic layers. The shear strength values of the c-ZrO₂(001)/α-Al₂O₃($1\overline{1}02$) interface are much smaller than the theoretical tensile strength in the same model; this indicates that the ZrO₂(001)/Al₂O₃ interfaces are more prone to shear failure than tensile failure.