Characterizing Property of States: Effect of Defects on the Coefficient of Thermal Expansion and the Specific Heat Capacity of ZrB 2

Thermal storage potential and thermal expansion are characteristic properties for extreme applications. ZrB 2 is a candidate for advanced applications in aircraft and fusion reactors. This article presents density functional theory calculations of its states, microstructure and quasi-harmonic levels calculations of thermophysical properties. Band structure highlighted dynamical instability with metallic impurities in ZrB 2 structure based on frequency modes. The observed projected density of states (PDOS) appropriate 4d orbital of Zr dominated at low frequency both in perfect crystal in the presence or absence of covalent impurities while B 2s and 2p orbitals dominate higher frequency states. Temperature dependency and anisotropy of coefficient of thermal expansion (CTE) were evaluated with various impurities. Various thermodynamic properties like entropy and free energy were explored for degrees of freedom resulting from internal energy changes in the material. Computed results for heat capacity and CTE were compared to available numerical and experimental data.


Introduction
Zirconium diboride (ZrB 2 ) attracts much attention as it is used in various high temperature applications including nuclear reactors, turbine engines and leading-edge aircraft due to its unique thermal properties at extreme conditions. This ultra-high temperature ceramics (UHTC) reportedly melts at the range of 3000˚C -3245˚C [1] [2]. The characteristic properties of expansion and heat capacity are therefore critical to inform life design methods for thermal stress optimization. This article is concerned with the impact of defects on its thermal properties and the hexagonal lattice structure of ZrB 2 . With relatively low density, ZrB 2 is a list of few candidates for hypersonic flight which generates high temperatures at its leading edges.
Its atoms do not deviate from their lattice sites hence the thermal and mechanical stability which is ascribed to its well layered in-plane and out-of-plane bonding in the hexagonal honeycomb ( Figure 1). The behavior of the material under thermal stress is quantified using heat capacity and thermal expansion. However, the perfect crystal of ZrB 2 rarely exists due to its partially covalent nature, and induced defects from synthesis and sintering processes [3] [4] [5]. Defects such as surface oxides, metallic impurities and carbides are unavoidably created during processing stages or exist in nature. These impurities, when present, result in microscopic structural changes, and hence redistribution in localized electron density. This induced variation in the mechanical reliability of the material creates phonon or electron (carrier) scattering differences evident in its thermal properties or processes. First principle calculations are critical at such microscopic levels to contrast the perfect ZrB 2 crystals to one with impurities and correlate electrical and thermodynamic properties. In this article, band structures and density of states (DOS) were investigated and the defect commonalities to heat capacity and thermal expansion explored.

Coefficient of Thermal Expansion of ZrB2
In considering ZrB 2 as the leading favorite in the internal design of nuclear reactors, one should critically evaluate its lifecycle potential based on expansivity [6] [7]. Studies have shown that the thermal expansion of ZrB 2 is anisotropic along the various lattice directions [8]. This directionally dependent property is attributed to the varying bond strengths along the different axes which determine stiffness and therefore the expansion [9] [10]. Researchers have attributed the characteristic stiffness of ZrB 2 crystal structure to the strong in-plane boron-boron sigma covalent bonds ( Figure 1). [9] [11]. Generality of the expression for coefficient of expansion holds true on the assumptions: (i) that the expansion is insensitive to the individual axial-direction (ii) the individual molecules do not expand but merely vibrates due to kinetic energy. Therefore, for ZrB 2 , the essence of linear expansion is also considered to quantify the anisotropic thermal expansion especially in the z-direction. The coefficient of linear thermal expansion is given as where L i represents specific axial-length (x, y, z) at temperatures T and initial.
For a typical solid, the coefficient of thermal expansion (CTE) (or volumetric expansivity) is generally expressed as: where α is the volumetric expansion coefficient, where l dimension of the crystal lattice in any direction x, y, z and ( ) l α is respective linear thermal expansion. In leading edge applications, the sharp-noses ( Figure 2) of supersonic aircraft moving through space with speed many times the speed of sound are subjected to extremely high thermal stress. Uneven expansion and contraction across the temperature gradient poses a threat to structural reliability of the material. Thus, this work investigated temperature dependency of CTE for ZrB 2 and the impact of impurities.

Specific Heat Capacity of ZrB2
Several studies have focused on thermal conductivity and diffusivity of ZrB2-based ceramics but only a few have given attention specific heat capacity. Figure 2. Depicting temperature gradient and regional component in ramping temperatures [12]. New Journal of Glass and Ceramics This property is essential in characterizing the state of the microstructures of materials for improved tailoring [13] [14]. It can be defined at constant volume or pressure and correlated respectively to the internal energy or enthalpy of the crystal structure. Phonon-based (lattice) constant volume heat capacity was formulated by Einstein and later Debye as shown in Equations (3) and (4) respectively: where D θ and N represent the Debye temperature and the Avogadro and . This relationship is simplified at low and can be written as: Both models show that the transfer of phonons in lattices contributes to measurable change in the energy. And Debye's model further attributes the interaction between harmonic oscillating atoms to the exponential increase in heat capacity with temperature. Also, analyzed in this article was analyzed the free energy, entropy and enthalpy of the structures in a bid to explain the heat capacity and internal energy.

Computational Details
The density function theory (DFT) calculations for this study were performed using Vienna Ab-initio Simulation Package (VASP) [  to ensure that the crystal lattice generated (see Figure 3(b)) does not deviate was applied and VESTA was used to load and generate XRD patterns. Figure 3 compares calculated to published experimental XRD patterns for perfect ZrB 2 [8].

Thermal Expansion Coefficient of ZrB2
At near room temperatures (300 K), the CTE for all configurations coincide at 2.5~3.0 × 10 −6 K −1 . Thermal expansion shows early stability between temperature ranges from 300 to 1200 K and suffers contraction at temperature exceeding 2000 K, approaching melting values. CTE for Hf impurity stands out in its rate of expansion and with notable breakdown at temperatures beyond 3000 K for all structures. NPT allows for dimension stabilization at given temperature which helps to predetermine the timesteps duration for thermodynamic equilibrium.
Correlation showed early dependency on temperature which is consistent with reported analysis by Okamoto et al. [19]. However, with temperatures reaching ~1500 K, almost twice the Debye value for ZrB 2 (750 K), the expansivity peaked for all configurations (see Figure 4(a)). Hafnium (Hf) impurities in ZrB 2 was highest 7.9 × 10 −6 K −1 at ~1500 K while carbon (C), boron vacancies, boron interstitials defects were 6.1 × 10 −5 , 6.25 × 10 −6 , 7.7 × 10 −6 K −1 respectively. There is a slight correlation of CTE values to atomic radius (Hf > B > C) of the impurities added. The control simulation of perfect ZrB2 reported values within the ranges previously reported in experiments (6.66 -6.93 × 10 −6 K −1 ) compared to simulations (5.9~6.68 × 10 −6 K −1 ) [8] [10] [12] [19]. Figure 4(a) also shows at temperatures with melting range (2900˚C -3000˚C), there is a gradual breakdown, potentially structurally with uncontrollable expansivity. Also, the independent axial contribution to the expansion coefficient (see Figure 4(b)) was investigated using the normalized values of the lattice parameters. Figure 4(b) shows the normalized lattice parameter with very uniform slow contraction along x and overlapping dimensions for all crystal lattice configurations. Along z, one can attribute expansion and dimension offsets to structural changes. It aligns with the results obtained from X-ray diffraction study on ZrB 2 anisotropy with matching reflections [8].

Specific Heat Capacity and Structure of ZrB2
To understand the effect of the impurities on the structural characteristics, the calculated total density of states (DOS) are plotted in Figure 5. The number of states for the perfect ZrB 2 reduced with increasing energy level. DOS for the covalently bonding impurities are closely overlapped with the pure crystal but with more dominating states likely from the projected orbitals of silicon and carbon at higher energy. The metallic impurities show states dominating at lower energy and slow increased. Potential bonding interaction between the covalent nature of where ( ) F ω represents the phonon DOS, lattice heat capacity ( v C ),  represents Planck's constant and B k denotes Boltzmann's constant.
The phonon dispersion for ZrB 2 structure was calculated along the high symmetry directions Γ-K-M-Γ as presented in Figure 6(a). The result for perfect crystal, shows dynamic stability as there are no imaginary modes, thus consistent with M. S. Daw et al. data [20]. Such stability is expected for structures with high atomic defect energy, and low tendency for atomic distortion. In the presence of impurities, there are observable differences. C and Si impurities show dynamical Figure 6. Phonon dispersion (a) normal mode frequencies of perfect ZrB 2 (b) imaginary (dynamic) and normal modes for impurities. New Journal of Glass and Ceramics stability similar to perfect structure but in addition to the dominant acoustic modes, there exist more low frequency optic modes. The added optic modes for Si are at relatively higher frequency than C. The case is different for the metallic impurities (Hf and W). As shown in Figure 6(b), negative phonons (imaginary branches) demonstrating dynamic instability for both configurations. It is noteworthy that some optic modes are also created by both Hf and W. The evolution of near zero frequency phonons transitioning, both normal and imaginary modes, points to changes in charge density area indicating that both of these metallic impurities have significantly distorted the original charge density distribution of ZrB 2 . No phonon softening is exhibited in band structure due to increased weight from added impurities.
The computed specific heat is plotted for the respective impurities as show in Figure 7(a). The specific heat reaches a hard limit near the debye temperature (~750 K) following Dulong-Petit prediction. When lattice vibration reaches its maximum, phonon contribution will not increase even with temperature based on Debye model. Phonopy applies quasi-harmonic approximations (QHA) which is not applicable at very high temperatures of complete anharmonicity. Hence in this study, specific heat capacity in the temperature range of 0 -1000 K was investigated. The presence of 5% vol of Hafnium (Hf) and Tungsten (W) impurities shows decreased specific heat capacity (158 J/mol/K) at reference 750 K temperature. This value is significantly lower than calculated for Carbon (C) and Silicon (Si) impurities. This can be described as stronger phonon scattering effects and reduced group velocity by Hf and W impurities in the ZrB 2 crystal.
The subplot in Figure 7(a) also shows the offset between Si and C in C v . Figure 7(b) shows the changes in the thermodynamic properties with respect to lattice temperature. The dependences of free energy and entropy on lattice vibration and temperature are expressed in Equations 8 and 9. The internal energy changes in the material are clearly observed in entropy dispersed. It is non-trivial but shows differences in the microstates (degrees of freedom) of the various material. Translating it by equipartition theorem, the small offsets in heat capacity values (Figure 7(a)) is equivalent to changes in its degrees of freedom. The overlaps, further align with the differences in the phonon bands between the covalent (C and Si) and metallic impurities (Hf and W).

Conclusion
Having presented the approaches used for heat capacity and thermal expansion calculations with microstructural explanation using phonon properties, it has New Journal of Glass and Ceramics been shown that volumetric coefficient of thermal expansion is driven by anisotropic z-direction expansion with temperature. Impurity effects add to the knowledge gathering for better forecasting of thermal properties of ZrB 2 -based materials. Exponential elevation of CTE near the melting temperatures shows structural breakdown and needs further investigation for phase transition. The challenge surrounding specific heat calculation, particularly using Phonopy is the quasi-harmonic level considerations, approximating anharmonic dependencies. Where available, computed thermal expansion and heat capacity data are compared to numerical and experimental data and are found to be within the range of experimentally reported results. Noteworthy is the redistribution of available states around the fermi level by metallic impurities, different from the covalent impurities in the electronic density of states. Further interest in structural differentiation in ZrB 2 and other transition metal diborides by analytical development is justified as the production process presents wide variations in density and purity level of the material.