In considering ZrB₂ 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 ZrB₂ 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 ZrB₂ 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 ZrB₂, 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 (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 ZrB₂ 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 [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:

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.

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 ZrB₂ (750 K), the expansivity peaked for all configurations (see Figure 4(a)). Hafnium (Hf) impurities in ZrB₂ 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 ZrB₂ anisotropy with matching reflections [8].

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 ZrB₂ 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 ZrB₂ 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 ZrB₂ 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

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 ZrB₂. 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 ZrB₂ 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).

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)