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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.

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 [_{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 (_{2} rarely exists due to its partially covalent nature, and induced defects from synthesis and sintering 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.

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 [_{2} is anisotropic along the various lattice directions [

the characteristic stiffness of ZrB_{2} crystal structure to the strong in-plane boron-boron sigma covalent bonds (_{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 α x , y , z = ( 1 / L i ) ( d L / d T ) , 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:

α = V f − V i V i ( T f − T i ) (1)

where α is the volumetric expansion coefficient, V f , i , T f , i are the respective initial and final volumes and temperatures. However, in potentially anisotropic materials, the linear (directional) expansion along the different directions can be calculated using:

α ( l ) = 1 l ( δ l ( T ) δ T ) (2)

where l dimension of the crystal lattice in any direction x, y, z and α ( l ) is respective linear thermal expansion.

In leading edge applications, the sharp-noses (_{2} and the impact of impurities.

Several studies have focused on thermal conductivity and diffusivity of ZrB2-based ceramics but only a few have given attention specific heat capacity.

This property is essential in characterizing the state of the microstructures of materials for improved tailoring [

Phonon-based (lattice) constant volume heat capacity was formulated by Einstein and later Debye as shown in Equations (3) and (4) respectively:

C E v = 3 N K B x 2 e x ( e x − 1 ) 2 (3)

C D v = 9 N K B T ( T θ D ) 3 ∫ 0 θ D / T ( x 4 e x ( e x − 1 ) 2 ) d x (4)

where θ D and N represent the Debye temperature and the Avogadro and x = ℏ ω / k B T . This relationship is simplified at low and can be written as:

C v = β T 3 (5)

The coefficient, β (specific heat coefficient) is correlated to the Debye temperature θ D ( 0 ) via:

θ D ( 0 ) = ( 1944 β ) 1 3 (6)

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.

The density function theory (DFT) calculations for this study were performed using Vienna Ab-initio Simulation Package (VASP) [^{−4} eV between steps. For a more accurate band structure and density of states, higher k-point grid 12 × 12 × 12 was further used. A classical molecular dynamic simulation data was also employed for comparison. Phonopy (open source package) was used for phonon calculations. Package uses statistical thermodynamic expressions to compute free energy (F), heat capacity (C), entropy (S), and enthalpy (H). Smearing method was used via Phonopy to calculate the phonon density of states (DOS) on a sampling mesh.

The classical MD simulations were performed using Large-Scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) to calculate the coefficient of thermal expansion (CTE) of perfect ZrB_{2} and contributions from impurities [_{2} and SiC [_{2} crystal structure to enable interaction across boundary. Langevin thermostat was used to control the temperature within NPT ensemble. A timestep of 5 fs was employed with 0.6 ns equilibration time for within a simulation duration of 15 ns. Literature valures for ZrB_{2} lattice constant were adopted, a o , b o = 3.035 Å and c o = 3.223 Å. All atoms with same z-coordinates are termed same plane, with alternating B and Zr layer. Added validation procedure to ensure that the crystal lattice generated (see _{2} [

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. [_{2} (750 K), the expansivity peaked for all configurations (see _{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}) [_{2} anisotropy with matching reflections [

To understand the effect of the impurities on the structural characteristics, the calculated total density of states (DOS) are plotted in _{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 ZrB_{2} with both Si and C is noticeable in the overlap which is not the same for metallic impurities. Si extended the conduction band to 34.1 eV, while C impurities made negligible changes to the conduction band. Both Hf and W impurities extended both the conduction and the valence band. The conduction band in all cases are dominated by B orbitals. Increased width of valence band denotes increase in electron delocalization and therefore reduced band gap. On the other

hand, decreasing width of conduction band indicates electron delocalization weakening. Occurrence of these impurities generated additional localized states and potential changes to conductive properties. Lattice contribution to heat capacity is computed from it’s the phonon DOS as shown below:

C v ( t ) = K B ∫ ( ℏ ω / K B T ) 2 e ℏ ω / K B T [ e ℏ ω / K B T − 1 ] 2 F ( ω ) d ω (7)

where F ( ω ) represents the phonon DOS, lattice heat capacity ( C v ), ℏ represents Planck’s constant and k B 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

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 _{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 _{2} crystal. The subplot in _{v}.

F ( T ) = E t o t + E z p + K B T ∫ F ( ω ) ln [ 1 − e − ℏ ω K B T ] d ω (8)

S ( T ) = K B { ∫ ℏ ω K B T e ℏ ω K B T − 1 F ( ω ) d ω − ∫ F ( ω ) [ 1 − e − ℏ ω K B T ] d ω } (9)

Having presented the approaches used for heat capacity and thermal expansion calculations with microstructural explanation using phonon properties, it has

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.

This work used Oregon State University computing resources and also resources of the Bioinformatics Computational Research Group in University of California Riverside. Intel Corporation Education Assistance Program made this research work possible.

The findings in this study are backed by computed data that are available from the corresponding author, [J.O. Ighere], upon reasonable request.

The authors declare no conflicts of interest regarding the publication of this paper.

Ighere, J.O. and Greaney, P.A. (2020) Characterizing Property of States: Effect of Defects on the Coefficient of Thermal Expansion and the Specific Heat Capacity of ZrB_{2}. New Journal of Glass and Ceramics, 10, 15-27. https://doi.org/10.4236/njgc.2020.102002