Structural Stability, Thermodynamic and Elastic Properties of Cubic Zr_0.5Nb_0.5 Alloy under High Pressure and High Temperature

Abstract

Structural stability, sound velocities, elasticity, and thermodynamic properties of cubic Zr_0.5Nb_0.5 alloy have been investigated at high pressure and high temperature by first-principles density functional calculations combined with the quasi-harmonic Debye model. A pronounced pressure-induced shear wave velocity stiffening in Zr_0.5Nb_0.5 alloy is observed at pressures above ~11 GPa, owing to its structural instability under high pressure, whose anomalous behavior is also observed in the end members of Zr-Nb alloys for Zr at ~13 GPa and for Nb at ~6 GPa upon compression, respectively. In addition, high-pressure elasticity and elastic-correlated properties of cubic Zr_0.5Nb_0.5 are reported, as compared with previous studies on Zr-Nb alloys with different compositions. A comprehensive study of the thermodynamic properties of cubic Zr_0.5Nb_0.5, such as heat capacity (C_v), thermal expansion coefficients (α), and Debye temperature (Θ_D), are also predicted at pressures and temperatures up to 30 GPa and 1500 K using the quasi-harmonic Debye model.

1. Introduction

Zr-Nb alloys, as important transition-metal alloys, have been widely used in the aerospace industry, nuclear reaction reactor, and biomedical applications because of their excellent properties such as good strength at high temperatures, corrosion resistance in high-temperature pure water, high melting point, and good mechanical properties under high-internal pressure, especially small thermal neutron absorption cross-section [1,2,3,4,5]. Characterizing the mechanical properties of metals and alloys under high-pressure and high-temperature conditions is fundamental for multiple technological applications [6,7,8]. It is known that structural stability/evolution, elastic moduli and their pressure dependences, and thermoelastic properties are important parameters in understanding the structural behavior and physical properties of materials under high pressure.

Regarding the end-member of Zr-Nb alloys, niobium (Nb) possesses a body-centered-cubic (bcc) structure in ambient conditions with rather low neutron absorption cross-sections, but zirconium (Zr) is preferred to adopt a hexagonal close-packed (hcp) structure (also called α phase) [9,10]. Since zirconium oxidizes easily in high-temperature water, zirconium alloys used in light-water reactors (LWR) today are usually adding a limited extent niobium (near 1 wt% or less than 5 wt%) to improve oxidation resistance and hydrogen absorption [11,12]. Based on different performance requirements, different Zr-Nb alloy compositions are designed, exploring the correlations between the microstructure and physical/mechanical properties of Zr_xNb_(1−x) alloys against pressure, temperature, and compositions [3,4,13,14,15,16,17].

Recent first-principles calculations showed that various Nb contents concentrated in Zr-Nb alloys would eventually result in significant effects on the elastic and thermodynamic properties [18]. It was found that the increase of Nb concentration in Zr-Nb alloys may lead to an enhancement of the bulk modulus to ~172 GPa for Nb metal, in contrast to an almost constant value of the shear rigidity in the Zr-Nb system with increasing Nb content from 5 at% to 100 at% [18]. High-pressure torsion experiments on Zr-2.5%Nb alloy showed that the grain was refined to nanograin size through the distorted deformation and the α-to-ω phase transition of Zr metal. It was clearly seen that the amount of ω-Zr content decreased with increasing temperature, and the thermal stability and microhardness of Zr-2.5%Nb alloy were significantly enhanced through the deformation [3].

For an Nb end-member in the Zr-Nb system, Struzhkin et al. [19] revealed two anomalous superconducting-transition temperature (T_c) in Nb at 5 and 60 GPa, respectively. This anomaly was proposed to be attributed to the electronic topological changes of the Fermi level upon compression [19]. Recently, sound velocities and thermoelasticity of Nb metal have been studied by Zou et al. [20,21] at simultaneous high-pressure and high-temperature conditions using ultrasonic interferometry in conjunction with synchrotron X-ray techniques, where new thermoelasticity data were reported and the pressure-induced anomaly in the shear velocity at ~4.8 GPa was explored. For Zr metal, phase transitions and elasticity have been measured at high pressure using ultrasonic measurements and in situ X-ray diffraction, where an anomalous velocity jump was observed at pressures between 3–6.8 GPa owing to the pressure-induced α-ω phase transition in Zr [22], in addition to the new elasticity data of ω-Zr [23]. To date, very few studies have been focused on the structural stability and thermoealsticity of Zr-Nb alloys in simultaneous high P-T conditions. The only one study on the phase stability/diagram of Nb₉₀Zr₁₀ alloy at high pressures and temperatures has been carried out by Landa et al. [15] using a self-consistent ab-initio lattice dynamics (SCAILD) approach combined with density-functional theory, where pressure-induced structural instability in cubic Nb₉₀Zr₁₀ was observed at ~50 GPa, and a rhombohedral phase was formed upon compression.

Despite of the importance of Zr-Nb alloys, the previous studies on Zr-Nb are mainly focused on the phase changes, microstructure, and mechanical properties at ambient/high pressure and high/room temperature [15,24,25], or the structural stability and thermoelasticity of the end-member of Zr-Nb alloys at high pressure [21,22,23,26]. To date, there is still a lack of a comprehensive study of sound velocities, thermoelasticity, and thermodynamic properties of cubic Zr_0.5Nb_0.5 alloy at high pressure and high temperature. Here, we have performed comprehensive studies on the structural stability, sound velocities, elastic properties, and thermodynamic properties of cubic Zr_0.5Nb_0.5 at high pressure and high temperature by first-principles calculations combined with the quasi-harmonic Debye model, in comparison with those for bcc-Nb and hcp-Zr end member in the Zr-Nb system.

2. Theoretical Calculations’ Details

Our theoretical calculations were performed using CASTEP code [27], which is based on the density functional theory (DFT) with the electronic density described by a plane-wave-basis. Vanderbilt-type ultrasoft pseudopotentials and a plane-wave expansion of wave functions were used [28]. Exchange and correlation potentials for electron interactions were employed by the generalized gradient approximation (GGA) of Perdew–Burke–Ernzerhof (PBE) [29]. In constructing the pseudopotentials, pseudo-atomic calculations were chosen for Zr: $4d^{2}5s^2$ and Nb: $4d^{4}5s^1$, respectively. Electronic wave functions were expanded in a plane-wave basis set with an energy cut-off of 520 eV. The crystal reciprocal-lattice and integration over the first Brillouin zone were performed using 24 × 24 × 24 Monkhorst–Pack [30]. In the current calculations, the self-consistent total energies were converged to 10⁻⁶ eV/atom, the maximum ionic Hellmann–Feynman force was converged to less than 0.01 eV/Å, and the total stress tensor was ~0.02 GPa by using the finite basis-set corrections.

The structure of metal element Nb and Zr is based on the original structure model in CASTEP, the space group and lattice constant are IM-3M, a = b = c = 3.301 Å, and P63/MMC, a = b = 3.231 Å, c = 5.148 Å, respectively. However, the Nb-Zr alloy is based on the virtual crystal approximation—building crystal structure models and then performing convergence tests and geometry optimizations. Based on their lattice parameters, total energies (E) and the corresponding unit-cell volumes (V) are derived. Fitting the energy-volume (E-V) to Birch–Murnaghan equation-of-state (EOS), we can obtain their structural parameters at equilibrium conditions.

Elastic constants (C_ij) are derived by calculating stress tensors against strains.

For single crystals, there are 21 independent elastic coefficients/constants, but they can be reduced according to the symmetry of the crystal. For cubic Zr_0.5Nb_0.5 and Nb crystals, due to the cubic symmetry, only three independent constants of C₁₁, C₁₂, and C₄₄ exist. By contrast, the hcp-Zr metal has five independent elastic constants of C₁₁, C₁₂, C₁₃, C₃₃, and C₄₄.

Elastic constants (C_ij) are calculated by the CASTEP program, and B, G, E, and σ are derived from Voigt–Reuss–Hill (VRH) approximations, as shown below [31,32]:

$$B_{}=\frac{1}{2}(B_V+B_R)$$

(1)

$$G_{}=\frac{1}{2}(G_V+G_R)$$

(2)

where G_V = (C₁₁ − C₁₂ + 3C₄₄)/5, G_R =15 [(12/(C₁₁ − C₁₂) + 9 C₄₄)]⁻¹ for cubic structure and B_V = [2(C₁₁ + C₁₂) + 4C₁₃ + C₃₃]/9, B_R = [(C₁₁ + C₁₂) C₃₃ − 2C₁₃²]/(C₁₁ + C₁₂ + 2C₃₃ − 4C₁₃), G_R = 5[((C₁₁ + C₁₂)·C₃₃ − 2C₁₃²)·C₄₄·(C₁₁ − C₁₂)/2]/6B_V·C₄₄·(C₁₁ − C₁₂)/2 + ((C₁₁ + C₁₂)·C₃₃ − 2C₁₃²)·(C₄₄ + (C₁₁ − C₁₂)/2) and G_V = (C₁₁ + C₁₂ + 2C₃₃ − 4C₁₃ + 12C₄₄ + 6(C₁₁ − C₁₂))/30 for hexagonal structure are the Reuss and Voigt [33] averages, which can be described by elastic constants of C_ij.

Based on the derived B and G, we calculate Young’s modulus (E) and Poisson’s ratio (σ) by the following formulas [16,31,32]:

$$E=\frac{9B{G}_{}}{3B_{}+G_{}}$$

(3)

$$\sigma =\frac{3B_{}-2G_{}}{2(3B_{}+G_{})}$$

(4)

In this study, the quasi-harmonic Debye model [34] is used to calculate the thermodynamic properties of Zr_0.5Nb_0.5 alloy and Nb and Zr metals; and their heat capacity (C_v), thermal expansion coefficient (α), and Debye temperature (Θ_D) are expressed as

$$\alpha =\frac{\gamma C_v}{B_{T}V}$$

(5)

$$C_v=3n{k}_B\left[4D\left(\Theta /T\right)-\frac{3\Theta /T}{e^{\Theta /T}-1}\right]$$

(6)

where the Grüneisen parameter (γ) is defined as

$$\gamma =-\frac{d\mathit{ln}\Theta \left(V\right)}{d\mathit{ln}V}$$

(7)

3. Results and Discussion

To investigate equilibrium structures of Zr-Nb alloys at zero pressure and zero temperature, we use their initial lattice parameters (a) to calculate the total energy (E) and the corresponding unit-cell volumes (V). The equilibrium volume (V₀), lattice parameters (a₀), bulk modulus (B₀), and its first-order pressure derivative (B₀′) can be derived by fitting the calculated total energy–volume E(V) data to the Birch–Mürnaghan equation-of-state (EOS) [35], which are summarized in

Table 1. Summary of lattice parameters (a, c), unit-cell volume (V₀), bulk modulus (B₀), and its pressure derivative (B₀′) at ambient pressure.

Material	a (Å)	c (Å)	V0 (Å3)	B0 (GPa)	B0′	Ref.
Nb0.5Zr0.5 (cubic)	3.419		39.957	134.2	3.3	This work
	3.432		40.424			Zhao et al. a (Theor.)
	3.447		40.957			Ikehata et al. b (Theor.)
Nb (cubic)	3.308		36.306	169.8	3.7	This work
	3.307		36.184	174.9(3.2)	3.97	Zou et al. c,d (Exp.)
	3.308		36.199	170.5	3.85	Zhao a; Daniel e et al. (Theor.)
hcp-Zr (α phase)	3.231	5.168	46.953	93.6	1.6	This work
	3.233	5.146	46.57	95.3	3.0(10)	Zhao et al. f (Exp.); Liu et al. g (Exp.)

^a Ref. [18]; ^b Ref. [38]; ^c Ref. [20]; ^d Ref. [21]; ^e Ref. [39]; ^f Ref. [10]; ^g Ref. [22].

, in comparison with previously experimental and theoretical results [10,18,20,21,36,37,38,39]. As shown in Table 1, our calculated lattice parameter of a₀ = 3.419 Å and unit-cell volume of V₀ = 39.957 ų for cubic Zr_0.5Nb_0.5 are well consistent with the previously predicted results of a₀ =3.432 Å, a₀ =3.447 Å, but ~2.5% smaller than the values of V₀ = 40.424~40.957 ų [18,38]. As clearly shown in Table 1, our theoretically calculated bulk modulus (B₀ = 169.8 GPa) and its pressure derivative (B₀′ = 3.7) for bcc-Nb are in good agreement with our recent synchrotron-based ultrasonic results [20,21], the previously experimental value, and theoretical simulations [39] (in Table 1). By contrast, the theoretical bulk modulus of 93.6 GPa for hcp-Zr is ~1.8% lower than the experimental data at ambient conditions, and the associated pressure derivative (B₀′ = 1.6) is generally consistent with the experimental synchrotron and ultrasonic result of B₀′ = 3.0 ± 1.0 within the relatively large uncertainty [10,22]. For cubic Nb_0.5Zr_0.5, our predicted bulk modulus (B₀) is equivalent to 169.8 GPa, and its pressure derivative is ~3.3, which is slightly smaller than that for an Nb end member in the Nb-Zr system.

To further understand the compressibility of the Nb_0.5Zr_0.5 alloy, the normalized lattice parameters (a/a₀) and volumes (V/V₀) versus pressures are plotted in Figure 1, as compared with those for bcc-Nb and hcp-Zr, respectively. Clearly, both the lattice constants (a/a₀) and volume (V/V₀) of Nb_0.5Zr_0.5 alloy, Nb, and hcp-Zr metals decrease with the increase of pressures up to 30 GPa without any pressure-induced volume collapse/jump. As shown in Figure 1, the value of a/a₀ in Nb_0.5Zr_0.5 decreases more steeply than the c/c₀ for hcp-Zr, indicating that hcp-Zr is more incompressible than the a-axis and exhibits an anisotropic compression behavior.

Based on the calculated elastic constants of C_ij, in the special case of hydrostatic pressure (P) applied to a cubic crystal, the elastic stiffness coefficients in the Voigt notation B_ij are: B₁₁ = C₁₁ − P, B₁₂ = C₁₂ + P and B₄₄ = C₄₄ − P, where P is the hydrostatic pressure. The conditions for elastic stability at a given pressure P can be evaluated by the following criteria for a cubic crystal [40,41]:

$$B_{11}-B_{12}>0,\left(B_{11}+2B_{12}\right)>0,B_{44}=C_{44}-P>0.$$

(8)

To understand the high-pressure elastic behavior in Zr_0.5Nb_0.5 alloy, elastic constants of C_ij are displayed in Figure 2 as a function of pressure. Clearly, the bcc structured Nb and Zr_0.5Nb_0.5 alloy are mechanically stable at pressures up to ~30 GPa. For Zr_0.5Nb_0.5 alloy, both C₁₁ and C₁₂ increase with the increase of pressure, whereas C₄₄ exhibits an apparent anomaly at pressures above 10–11 GPa in Figure 2a, which is proposed to be attributed to the structural instability of cubic Zr_0.5Nb_0.5 upon compression, or the precursor of the formation of the high-pressure new phase. This anomaly behavior is also observed in compressed cubic Nb and hcp-Zr under high pressure, as shown in Figure 2b,c.

To further know about elastic behavior in compressed Zr-Nb alloys, compressional and shear wave velocities at various pressures are calculated by Equations (9) and (10), as summarized in the Supplementary Materials of Table S1. For clear understanding, acoustic velocities and elastic moduli are plotted as a function of pressure, as shown in Figure 3 and Figure 4:

$$V_S=\sqrt{\frac{G}{\rho }}$$

(9)

$$V_P=\sqrt{\frac{B+\frac{4}{3}G}{\rho }}$$

(10)

As shown in Figure 3a, a pronounced pressure-induced shear wave velocity stiffening is observed in Zr_0.5Nb_0.5 at 10–11 GPa, which is proposed to be attributed to its structural instability upon compression. This pressure-induced anomaly is also found in Nb at ~6 GPa and Zr at ~13 GPa by our first-principles calculations (in Figure 3b,c) and was further supported by the previous synchrotron X-ray diffraction experiments where the phase-transition-induced velocities changes occurred during α-Zr to ω-Zr transition at high pressure [22,23]. For Nb metal, the above-mentioned pressure-induced anomaly in the phonon velocities at ~6 GPa was not derived from structural transition but attributed to the subtle change in the topology of the Fermi surface of Nb with pressures [21]. Based on our obtained acoustic velocities and densities data, elasticity of bulk modulus (B), shear modulus (G) and Young’s modulus (E) for cubic Zr_0.5Nb_0.5 alloy, Nb and hcp-Zr at various pressures are derived, as shown in Figure 4a–c, respectively. Similar to the behavior in the acoustic velocities, the pressure-induced anomalies in shear-related elastic properties for Zr_0.5Nb_0.5 and Nb/Zr metal are also observed (in Figure 4).

It is accepted that elastic modulus and its pressure dependence of materials are important parameters for their uses at extreme high-pressure conditions. Our predicted elastic properties of cubic Zr_0.5Nb_0.5 alloy at various pressures are summarized in

Table 2. Summary of elasticity of bulk and shear moduli, and their pressure derivatives for cubic Nb_0.5Zr_0.5, in comparison with those for Nb and hcp-Zr metals.

Material	B0 GPa)	G0 (GPa)	B0′	G0′	Ref.
Nb0.5Zr0.5 (cubic)	134.2	27.6	3.3	0.48	This work
Nb (cubic)	169.8	30.5	3.7	0.63	This work
	174.9	37.1	3.97	0.83	Zou et al. a (Exp.)
hcp-Zr (α phase)	93.6	34.3	1.6	−0.05	This work
	95.3	36.3	3.0 (10)	−0.1 (2)	Liu et al. b (Exp.)

^a Ref. [21]; ^b Ref. [22].

, as compared with the current theoretical results for cubic Nb and hcp-Zr (also called α phase), as well as the previous studies [10,21,22,36] (see details in Tables S2–S4 of Supplementary Materials). We note that the bulk moduli for Nb (B₀ = 169.8 GPa) and hcp-Zr (B₀ = 93.6 GPa) by our first-principles calculations are in good agreement with the previous experimental results of B₀ =174.9 (3.2) GPa for Nb and B₀ =95.3 GPa for hcp-Zr, respectively, from the synchrotron-based ultrasonic measurements [21]. Clearly, the calculated bulk moduli for Nb and Zr metals are around 1.8–3% lower than the previously experimental results at ambient conditions, respectively [10,21,22,36]. Moreover, the calculated shear moduli for both Nb and hcp-Zr both are obviously underestimated, as compared with the experimental counterparts [10,21,22,36]. These discrepancies might be originated from the under-binding of GGA, which yields larger volumes and/or lower elastic moduli than experimental values at high pressures. As clearly shown in, we find that the increase of Zr content in the Nb-Zr system results in an apparent reduction of the bulk modulus by about 21% when compared with that (B₀ = 169.8 GPa) for Nb metal, equal to ~134.2 GPa for Zr_0.5Nb_0.5 alloy. On the other hand, the addition of Zr in the Nb-Zr system is eventually lowering the shear rigidity of Zr_0.5Nb_0.5 alloy to ~27.6 GPa, which is significantly lower than those for the Zr (~34.3 GPa) and Nb (30.5 GPa) end members.

For linear fits of the current high-pressure bulk and shear moduli data of cubic Zr_0.5Nb_0.5, we obtain its pressure derivative, yielding B₀′ = 3.3 and G₀′ = 0.48 (in Table 2). As seen in Table 2, our calculated pressure derivative of bulk (B₀′ = 3.7) and shear (G₀′ = 0.48) moduli for Nb are generally consistent with the experimental results [21]. For hcp-Zr, however, its pressure derivative for bulk modulus yields as B₀′ = 1.6, which is also generally consistent with the experimental ultrasonic result of B₀′ =3.0 ± 1.0 within the uncertainty [22]. Its associated pressure dependence of the shear modulus exhibits a negative value of G₀′ = −0.05, which agrees with the result of G₀′ = −0.1 (2) by ultrasonic interferometry measurements within the relatively large uncertainty [22]. Clearly, this anomalous behavior is quite different from that of the Nb metal and/or other metals, which might be attributed to the progressive-d-electron-transfer-induced topological changes in the Fermi surface upon compression [42,43].

It is known that the ratio of shear modulus to bulk modulus (G/B) is usually used to evaluate the ductility or brittleness of materials. According to the Pugh’s criterion [44], when the G/B ratio of materials is less than 0.5, it is defined as a ductile behavior, otherwise as a brittle one. As shown in Tables S2–S4, the G/B values for Zr_0.5Nb_0.5, Nb, and Zr materials are all less than ~0.5 at various pressures up to 25–30 GPa, indicating a ductile nature even at the peak pressures. With increasing pressure, we find that the G/B values decrease gradually, suggesting that the Zr-Nb alloys become more ductile upon compression.

Using the quasi-Debye model, bulk moduli of cubic Zr_0.5Nb_0.5 at various pressures and temperatures are calculated, as shown in Figure 5a, which are compared with those for the Nb-Zr alloys end member of Nb (Figure 5b) and hcp-Zr (Figure 5c) metals. Similar to Nb and Zr metals, the bulk modulus (B) of Zr_0.5Nb_0.5 monotonically increases with increasing pressures, and gradually decreases with temperatures, without any observable bulk modulus collapse/jump at P-T conditions up to 25 GPa and 1200 K. As shown in Figure 5c, at the peak P-T conditions of 25 GPa and 1500 K, a pronounced bulk modulus collapse occurred in hcp-Zr, which might be due to the fact that the structural instability in hcp-Zr and/or a structural phase transformation is happening at such extreme pressures and temperatures.

It is accepted that the elastic anisotropy of crystals has important applications in technology and engineering, which may play an important role for the formation of microcracks in materials. For a single-crystal with cubic symmetry, an elastic anisotropy factor defined as A = 2C₄₄/(C₁₁−C₁₂) is shown in Tables S2–S4. For a locally isotropic crystal, A is equivalent to 1. However, for Zr_0.5Nb_0.5 alloy, its elastic anisotropy factor (A) is ~2.5 at 0 GPa, and is almost ~8 times as large as that (~0.31) for cubic Nb metal (in Tables S2–S4). It is clearly found that this factor becomes smaller with increasing pressure, indicating that this alloy becomes more isotropic upon compression, similar to the behavior in Nb at pressures less than ~16 GPa. As shown in Tables S2–S4, the Possion’s ratio (σ) of Zr_0.5Nb_0.5 is lower than 0.5 and greater than 0.25, indicating that the interatomic force in this alloy is central, which is in accordance with the above-mentioned anisotropy factor discussion. Moreover, we find that the Poisson’s ratio (σ) of Zr_0.5Nb_0.5 increases with increasing pressure up to ~10 GPa, further supporting the structural instability of cubic Zr_0.5Nb_0.5 at pressures above 10–12 GPa, as shown in Figure 3 and Figure 4.

For hcp-structured Zr, the anisotropy of compressional wave is obtained from [45]

$$\Delta p=\frac{C_{33}}{C_{11}}$$

(11)

The anisotropies of the wave polarized perpendicular to the basal plane (s₁) and the polarized basal plane (s₂) are described as

$$\Delta s_1=\frac{C_{11}+C_{33}-2C_{13}}{4C_{44}},\Delta s_2=\frac{2C_{44}}{C_{11}-C_{12}}$$

(12)

High-pressure anisotropies of the compressional wave ($\Delta p$) and shear wave (∆s₁ and ∆s₂) are shown in Table S5. It is found that the compressional wave $\Delta p$ increases with the increase of pressures, which is contrary to the high-pressure behavior of shear wave ∆s₂ that is reduced with pressures. This result indicates that the hcp-Zr has a highly elastic anisotropy at pressures ranging from 0 to 25 GPa.

Thermal expansivity is an especially important parameter for interpreting the thermodynamic and thermoelastic properties of materials at high temperature. Using the quasi-harmonic Debye model, we have calculated the thermal expansion (α) of Zr_0.5Nb_0.5 at high-pressure and high-temperature, which is plotted as a function of pressure and temperature in Figure 6a,b, as compared with those for Nb and hcp-Zr metals (seen in Figures S1 and S2 of the Supplementary Materials). As is clearly seen in Figure 6, Zr_0.5Nb_0.5 possesses a high ambient-condition thermal expansion of ~6.2 × 10⁻⁵ K⁻¹, which is about five times as large as those for Nb and hcp-Zr metals (in Figures S1 and S2). Similar to Nb and hcp-Zr metals in Figures S1 and S2, the thermal expansion of Zr_0.5Nb_0.5 exhibits an exponential increase with increasing temperature up to 200–300 K and then shows an almost linear behavior at a higher temperature range of up to ~1500 K, for which the behavior is significantly different from an almost exponential pressure dependence (in Figure 6 and Figures S1 and S2).

Debye temperature (Θ_D) is one of the important thermodynamic properties of solid materials, which is described as follows [46]:

$$V_m={\left[\frac{1}{3}\left(\frac{2}{V_S{}^3}+\frac{1}{V_P{}^3}\right)\right]}^{-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$$

(13)

$${\Theta }_D=-\frac{h}{k}{\left[\frac{3n}{4\pi }\left(\frac{N_A}{M}\right)\right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}{V}_m$$

(14)

where V_S and V_P represent the shear and compressional elastic wave velocity, which are obtained from Navier’s equations [47], V_m is average wave velocity, h is Planck’s constant, k is Boltzmann’s constant, N_A is Avogadro’s number, ρ is the density, M is the molecular weight, and n is the number of atoms in the molecule.

Based on the quasi-harmonic Debye model, the heat capacity (C_v) of Zr_0.5Nb_0.5 alloy at simultaneous high-pressure and high-temperature are calculated and plotted in Figure 7, in comparison with those for Nb and hcp-Zr metals. As shown in Figure 7, it is clearly seen that the heat capacity (C_v) of Nb-Zr alloys increase gradually with increasing temperature and decreasing pressure. However, the effect of temperature on C_v is much more pronounced than that of pressure, which is attributed to the anharmonic approximations of the Debye model. At a higher temperature range, the anharmonic effect on C_v is suppressed, and the heat capacity C_v is very close to the Dulong–Petit limit. As clearly seen in Figure 7, the Zr_0.5Nb_0.5 alloy reaches the Dulong–Petit limit faster than Nb/Zr metal with increasing temperature at certain pressures.

To further gain the thermodynamic properties at high pressure, our calculated heat capacity (C_v) and Debye temperature (Θ_D) of Zr_0.5Nb_0.5 are normalized by (X-X₀)/X₀ (where X and X₀ are heat capacity or Debye temperature at high pressure–temperature (P-T) and zero P–T conditions), and are plotted as a function of pressure at certain temperatures up to 1500 K, in comparison with those for Nb and hcp-Zr metals (in Figure 8). Clearly, the heat capacity (C_v) of Nb-Zr alloy/metal shows an almost linear reduction with increasing pressure at certain temperatures of 100 K and 1500 K, which is in contrast to a nearly linear increasing trend with pressures in Debye temperature. Moreover, with the addition of Zr into Nb metal and formation of the Nb-Zr solid solutions, the Zr_0.5Nb_0.5 alloy eventually exhibits a stronger pressure dependence of Debye temperature, as compared with the Nb and Zr end members of Zr-Nb alloys.

4. Conclusions

Structural instability, acoustic velocities, elasticity, and thermodynamic properties of Zr-Nb alloys have been studied at high pressure and high temperature by first-principles density functional calculations combined with the quasi-harmonic Debye model. In this work, we have observed a pronounced pressure-induced shear behavior stiffening in Zr_0.5Nb_0.5 at pressures above ~11 GPa, which is attributed to its structural instability upon compression. This anomalous behavior is also observed in the end members of Zr-Nb alloys for Nb at ~6 GPa and for Zr at ~13 GPa, respectively. Moreover, high-pressure elasticity and elastic-related properties of cubic Zr_0.5Nb_0.5 are reported, as compared with previously reported results on Zr-Nb alloys with different compositions. Using the quasi-harmonic Debye model, the pressure and temperature dependences of heat capacity (C_v), thermal expansion coefficients (α), and Debye temperature (Θ_D) for Zr_0.5Nb_0.5 are comprehensively studied, in comparison with those for Nb and hcp-Zr end members in the Nb-Zr system, which are of great importance for the understanding of thermoelastic properties of Nb-Zr alloys for their uses at extreme high-pressure and high-temperature conditions.