## 2 Methodology and Model
### Simulation Model  
Considering that the structure and composition of real TC/TGO interfaces could be very complex, we focused on the relevant ZrO\({}_{2}\)/Al\({}_{2}\)O\({}_{3}\) interfaces in this paper. For Al\({}_{2}\)O\({}_{3}\), only \(\alpha\)-Al\({}_{2}\)O\({}_{3}\) was considered, since \(\alpha\)-Al\({}_{2}\)O\({}_{3}\) is the most stable phase, and other Al\({}_{2}\)O\({}_{3}\) phases have a more complicated, mostly ill-characterized bulk structure. ZrO\({}_{2}\) typically has three phases at low pressure conditions, i.e., cubic, tetragonal, and monoclinic. On the \(\alpha\)-Al\({}_{2}\)O\({}_{3}\) substrate, the epitaxial growth of the deposited YSZ(001) has been observed as \(\alpha\)-Al\({}_{2}\)O\({}_{3}\)(\(1\bar{1}02\))/YSZ(001) [33]. Moreover, the lattice constant of YSZ(001) is approximately the same as that of c-ZrO\({}_{2}\)(001). In addition, the mismatch ratio between c-ZrO\({}_{2}\)(001) and \(\alpha\)-Al\({}_{2}\)O\({}_{3}\)(\(\bar{1}\bar{1}\)02) is about 3.7%. Taking the above aspects into consideration, we focused on the c-ZrO\({}_{2}\)(001)/\(\alpha\)-Al\({}_{2}\)O\({}_{3}\)(\(\bar{1}\bar{1}\)02) interfaces. According to preliminary calculations, the calculated lattice constants of c-ZrO\({}_{2}\) were \(a_{0}=5.07\) A, while those of \(\alpha\)-Al\({}_{2}\)O\({}_{3}\) were \(a_{0}=4.809\) A and \(c_{0}=13.115\) A. In addition, the suitable atomic relaxation layers of the c-ZrO\({}_{2}\) slab and the \(\alpha\)-Al\({}_{2}\)O\({}_{3}\) slab were nine and fifteen layers, respectively. Our calculated values agree well with available experimental values and other first-principle results [30; 34; 35; 36]. The thickness of fifteen layers of \(\alpha\)-Al\({}_{2}\)O\({}_{3}\) is 10.5 A, corresponding to three \(\alpha\)-Al\({}_{2}\)O\({}_{3}\)(\(\bar{1}\bar{1}\)02) units. This thickness is usually sufficient to simulate a bulk ceramic surface in terms of electronic structure and ionic relaxation effects [37]. Figure 1 displays the schematic diagrams of the c-ZrO\({}_{2}\)(001)/\(\alpha\)-Al\({}_{2}\)O\({}_{3}\)(\(\bar{1}\bar{1}\)02) interfaces. The supercell consists of a slab of \(\alpha\)-Al\({}_{2}\)O\({}_{3}\)(\(\bar{1}\bar{1}\)02) sandwiched between two slabs of \(\alpha\)-Al\({}_{2}\)O\({}_{3}\)(\(\bar{1}\bar{1}\)02). The layering sequence in the \(\alpha\)-Al\({}_{2}\)O\({}_{3}\)(\(\bar{  
9.012 GPa when the strain was 9%, while for the Zr-terminated and 2O-terminated interfaces, the stresses reached a maximum value of 6.893 GPa and 7.452 GPa at the strains of 5% and 6%, respectively. Among the three models, the O-terminated interface has the greatest theoretical tensile strength and critical strain. This implies that the adhesion of the O-model is stronger compared to the other two models, and the tensile strength of the superlattice is strongly influenced by the termination of c-ZrO\({}_{2}\)(001). Moreover, the curve shapes of the three terminated interfaces are rather similar to each other, which was also mentioned by [PERSON] [22]. In addition, the curve also indicates some information about the failure behavior. Different from the O-terminated and Zr-terminated interfaces, the stress of the 2O-terminated interface has a sudden drop after the critical strain. This implies that the fracture type of the 2O-terminated interface is different from that of the other two terminated interfaces during the tension loading. The work of separation, \(W_{sep}\), is an important parameter for analyzing the interface strength, which is defined as the energy required to separate the interface into two free surfaces. Furthermore, the area, \(A\), under the stress-strain curve can approximate the value of the work of separation [20]. In this work, the calculated work of separation for the O-terminated interface is 1208 mJ/m\({}^{2}\), while that for the Zr-terminated and 2O-terminated interfaces are 715 mJ/m\({}^{2}\) and 829 mJ/m\({}^{2}\), respectively.