Computational Details and Structural Models

DFT calculations were carried out using the Vienna ab initio Simulation Package (VASP). [266] The projector augmented wave (PAW)[39] pseudopotentials were employed for electron-ion interactions. The generalized gradient approximation (GGA) parametrized by Perdew-Burke- Ernzerhof (PBE) [40] was used to describe the electron exchange-correlation potentials. A 400 eV cut-off energy for the plane wave basis set was used during structural relaxation, and a 2 x 2 x 1 Γ-centered k-point mesh is well-converged for self-consistent calculations. Subsequent single-point total energy calculations were carried out using the Gaussian method with a 4 x 4 x 1 Γ-centered k-point mesh, with a 480 eV cut-off energy for the plane wave basis set. All the atomic positions were relaxed until all components of the residual forces were smaller than 0.03 eV/Å, and the convergence threshold for self-consistent-field iteration was set to 10−5 eV. The Σ 5 (310)/[001] GB was selected for this study due to its relatively low energy and high symmetry, enabling the construction of large GB models suitable for impurity segregation analysis via DFT. Several prior MD studies have considered the Σ 5 (310)/[001] for these reasons. [262, 267] The lattice constants of our 240 atom ZrO2GB model are 5.14 Å, 8.13 Å, and 65.02 Åfor a, b, and c respectively.

We generated Σ 5 (310)/[001] symmetric tilt GBs of cubic ZrO2from an optimized unit cell.

he optimized lattice parameter of the ZrO2unit cell is 5.14 Å, consistent with the bulk experimental

value of 5.10 Å.[268] We build c–YSZ Gs from this pristine ZrO2 GB and introduce a suitable

number of VOand YZr, making the composition comparable with the experimental case. Considering

the intrinsic symmetry of the periodic ZrO2GB model, we introduced two Y atoms and one oxygen

vacancy in one half region of one ZrO2grain, and then applied mirror symmetry operation to get

the same distribution of Y atoms and oxygen vacancy in the other grain. That is, substituting two Zr atoms with two Y atoms from 20 cation sites and removing one O atom from 40 anion sites leads to a total number of all the possible structural configurations of C₂₀2 x C₄₀1 =7600, which corresponds to a 5 mol% Y2O3, or 5YSZ.

Figure 4.1: Energy distribution of 7600 structural configurations of 5YSZ GB model. All these structures are relaxed by empirical potential calculations.

empirical-potential calculations to relax all the 7600 structures using General Utility Lattice Program (GULP).[269] The related force field parameters were adopted from Ref.[270] The calculated energies are shown in Figure 4.1, which follows a Gaussian distribution with a spanning of about 13 eV. This result is in good agreement with prior empirical-potential calculations on the supercell model of 8YSZ bulk material.[271]

Next, we analyzed the contributions of vibrational and configurational entropies of the 7600 configurations. The vibrational entropy at 1000K was estimated from the phonon spectra calculated in GULP. As discussed below, the vibrational entropy will be considered by adding their contributions in the total energy of each configuration for estimating the configurational entropy, following the same approach in Dong’s et al. simulation work.[271] The configurational entropy term, ST, can be calculated with the Boltzmann entropy formula,

S= −kB W

∑

i=1

P_iln Pi, (4.1)

the system existing in microstate i. We consider our system to be a canonical ensemble, where the system is in thermal equilibrium with a heat bath at a constant temperature. In a canonical ensemble, Pican be calculated from

Pi=

g_ie−∆E/kBT ∑Wi=1gie−∆E/kBT

, (4.2)

where giis the degeneracy of microstate i according to structural symmetry, ∆E is the microstate

energy difference relative to the lowest energy configuration, and T is the temperature.[272] ∆E is obtained directly from optimized structures, and T is set to 1000K (about 1/3 of the melting point of ZrO2).[271]

4.3

Results and Discussion

From 4.1 and 4.2, we calculated the configurational entropy term of our 7600 YSZ grain- boundary structures to be 0.0165 eV. This value is less than the results of Dong et al. who calculated a configurational entropy term of 0.08 eV because our 5YSZ system has far fewer possible con- figurations compared to their 100,000 randomly sampled 8YSZ configurations. By adding the contributions of vibration entropy to the total energy, we re-calculated the configurational entropy to be 0.02 eV. This indicates that only states within about 0.02 eV above the lowest energy are present at 1000K. In other words, only few structures with total energy lying at the very left end of the distribution in Figure 4.1 can be formed, and interestingly, the Y atoms are near the GB core in all these structures. Therefore, the YSZ GB structure with the lowest energy was selected for studying the impurity segregation behavior using DFT calculations, see its local structures from different view angles in Figure 4.2

Figure 4.2: Schematic illustrations of local structures of YSZ GB from different view angles.

4.4

Conclusion

In summary, we have performed a statistical analysis of 7600 YSZ GB configurations from empirical-potential calculations. From vibrational and configurational antropy calculations of these microstates at 1000K, we determined that only states within about 0.02 eV above the lowest energy are present at 1000K. These states correspond to only a few structures with total energy lying at the lowest energy point of the total energy distribution. We found that these states contain Y atoms near the GB core. Knowing only these states are probably to exist, we were able to study the defect segregation tendencies in Σ 5 (310)/[001] YSZ GB structures.

Chapter 4, in part, is a reprint of the material as it appears in the Journal of the European Ceramic Society, 2019. Behtash, Mazier; Wong, Joseph; Jiang, Sicong; Luo, Jian; Yang, Kesong. Dr. Mazier Behtash was the primary investigator of this material.