First-Principles Calculations of the Structural , Mechanical and Thermodynamics Properties of Cubic Zirconia

The structural, mechanical and thermodynamics properties of cubic zirconium oxide (cZrO2) were investigated in this study using ab initio or first-principles calculations. Density functional theory was used to optimize the crystal structure of cZrO2 and thereafter, simulations were conducted to predict the lattice parameters and elastic constants. The Zr-O bond distance was calculated as 2.1763 Å with unit cell density of 6.4179 g/cm3. The data obtained were used to determine Young’s modulus, bulk modulus, Poisson’s ratio and hardness of cZrO2 as 545.12 GPa, 136.464 GPa, 0.1898 and 12.663(Hv) respectively. The result indicates that cZrO2 is mechanically stable with thermodynamics properties of a refractory material having potential for structural and catalytic applications in various forms as a nanomaterial.


Introduction
Recently, zirconium oxide in cubic polymorph has attracted interest due to its mechanical and thermal properties.These interests have resulted in the use of cZrO 2 to various applications such as catalysts, fuel cells, oxygen sensors and others [1]- [4].Thus cZrO 2 has been synthesized at nanoscale in various forms.The sol-gel method was used to produce ZrO 2 nanoparticles with fairly uniform dimension ranging from 50 nm to 90 nm [5].ZrO 2 nanosheets with thickness in the range of 3.2 -4.2 nm were produced through bottom-up synthesis by impregnation of graphene oxide in cyclohexane containing Zr-based alkoxides [6].Also, ZrO 2 nanotubes have been prepared via several methods such as direct anodization, template-assisted depositions and hydrothermal treatments [2]; the nanotubes obtained have different inner and outer diameters, thicknesses and lengths.
For efficient and cost-effective applications of cZrO 2 , its structural and mechanical properties are required at molecular and atomic scale.But due to complexity and high cost of equipment, only limited experiments have been conducted to characterize required properties of cZrO 2 at nanoscale and/or atomic level [2] [4].This led to computational modeling and simulation of various properties of cZrO 2 based on atomistic and continuum approaches [1] [2].However, most of the simulations already conducted are limited to properties such as electronic structures, phonon dispersion, surface adsorption and diffusion [7]- [13] with limited emphasis on structural and mechanical properties.
Therefore, in this study the first-principles calculations are utilized to numerically predict the structural characteristics, mechanical properties and thermodynamic properties of cZrO 2 .The values obtained are used to analyze the structural stability of cZrO 2 and compared available data for further simulation(s).

Computational Methods
The 3D structure of cZrO 2 was modeled from available data [2] using the surface builder tools of the Material Studio software [14].The crystal structure ofcZrO 2 was optimized geometrically in order to obtain initial lattice parameters and density based on density function theory (DFT) as implemented in Cambridge Sequential Total Energy Package (CASTEP) code [15].In this calculation, the Generalized Gradient Approximation (GGA) having Burke-ErnZerhof Potential (PBE) for solids was used [13].To obtain accurate structures, the calculations were conducted in the irreducible Brillouin zone with 8 × 8 × 8 k point mesh Monk horst-Pack scheme [16].In order to obtain plane wave expansions, a kinetic energy cut-off value of 380 eV was used.Thereafter the Broyden-Fletcher-Goldfarb-Shannon (BFGS) optimization method was used with fixed basis quality to obtain the symmetric crystal structure of cZrO 2 .During the optimization process, the total energy was designed to converge to 5 × 10 −6 eV and the force per atom diminished to 0.002 eV/Å.The optimized structure of cZrO 2 was then simulated to obtain required mechanical and thermodynamics properties.

Crystal Structure
The bulk crystal structure of cZrO 2 belongs to the Fm-3m space [4] [9] and is modeled in various forms as seen in Figure 1.The geometric structure has influence on the properties of the material even at nanoscale [2].After optimization, all the lattice parameters were recorded and compared with values obtained from other simulations and experiments as summarized in Table 1.It was observed that the deviation of the lattice parameters is less than 3% from the experimental values, which may be due to the approximation method used during optimization.

Mechanical Properties
The mechanical properties of cZrO 2 are based on the elastic constants of the crystal which indicates response to external forces.There are six components of stress and a corresponding six components of strain for the general 3-D case.Thus, Hooke's law may be expressed as: where C = stiffness or elastic constant, S = compliance, σ = stress and ε = strain.In matrix format, the stressstrain relation showing the 36 (6 × 6) independent components of stiffness can be represented as [3]: The number of independent elastic constants in cZrO 2 is three (3), i.e.C 11 , C 12 and C 44 which are computed as 596.33, 137.04 and 74.34 GPa respectively.Also the bulk modulus (B) was calculated as 290.134GPa using the Voigt model, which is comparable to experimental value [17].
With reference to Born-Huang's lattice dynamic theory [19], the mechanical stability of a cubic system is based on the conditions expressed in Equations ( 4) and (5).0, 0, 0, Based on conditions outlined in Equations ( 4) and ( 5), the structure is mechanically stable due to lattice parameters and orientation of the crystal.The stability will make polycrystalline cZrO 2 less vulnerable to generating micro cracks [19].
For cZrO 2 , computed Young's modulus (E) is 545.12GPa, shear modulus (G) based on Voigh model is 136.464GPa and the Lame modulus based on Reus model is 222.195GPa.The G/B ratio indicates the ductility or brittleness of a material [19] and was found out to be 0.469, which indicates that cZrO 2 ionic and brittle.This is becausefor covalent and ionic materials, the typical relations between bulk modulus and shear modulus are as G ≈ 1.1B and G ≈ 0.6B, respectively [20].The Poisson's ratio (υ) of cZrO 2 vary from 0.14 to 0.189, which indicates less metallic and ionic character in the Zr-O bond.
In crystals, hardness quantifies the resistance to deformation and may be predicted using microscopic models [17].The hardness (Hv) of cZrO 2 is determined using a semi-empirical equation defined as follows [21] [22]: The obtained hardness (Vickers number) for cZrO 2 is 12.663, which is consistent with the chemical bonding and elastic modulus as analyzed above and in similar other studies [19]- [22].

Thermodynamic Properties
The thermodynamic properties of a solid have direct relevance to its phonon characteristics as it indicates the quantum of elastic strain energy [19].Hence, the phonon dispersion curve of a material elucidates the thermal properties based on the concept of lattice vibrations and interpretation of lattice dynamics [15].Using the method of linear response with 0.05 1/Å q-vector grid spacing, the phonon dispersion curves (PDC) for cZrO 2 was generated as shown in Figure 2. The lower section of G-X direction indicates higher longitudinal energy while most of the branches in the X-K direction converge as non-degenerate, thus leading to the expected nine branches for a typical cubic structure [19].
With the aid of data from the phonon dispersion curves (PDC), quasi-harmonic approximation was used to evaluate some temperature dependent properties of cZrO 2 such as entropy, enthalpy, free energy and Debye temperature as illustrated in Figure 3 and Figure 4. From Figure 3, the zero-point energy was computed to be 0.20385 eV and the heat capacity per unit cell had a maximum value of 16.8957 cal/K (0.07069 kJ/K) at 992.25 K. Debye temperature gives an approximation for the low temperature heat capacity of insulating crystalline solids.The maximum Debye temperature for cZrO 2 was found to be 946 K which is similar to 963 K for Y 2 O 3 stabilized cZrO 2 determined experimentally using neutron powder diffraction [23].All the computed thermodynamics properties of cZrO 2 indicates it has less thermal conductivity despite having high ionic conductivity thus confirming its major characteristic as a refractory material.

Conclusion
Using the first principles calculations, the properties of cZrO 2 were investigated.The calculated lattice parameters showed conformity with available experimental data.The computed properties based on elastic constants   indicate that cZrO 2 has satisfied the mechanical stability requirements; however, the young's modulus and hardness are high when compared to similar material such as cerium oxide.As expected, the thermodynamic properties obtained confirmed the ceramic characteristics of cZrO 2 .The data obtained may further be used to predict properties in relation to defects and/or dopants and setting parameters for processing nanocomposites related to cZrO 2 .

Figure 1 .
Figure 1.Crystal structure of cZrO 2 in various forms: (a) Unit Cell; (b) Primitive Cell; (c) Zr-O Bond (O 2 and Zr atoms are represented by red and light blue balls respectively).

Figure 3 .
Figure 3. Enthalpy, free energy and entropy (a) and heat capacity of cZrO 2 .