First-Principles Study of Different Polymorphs of Crystalline Zirconium

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First-Principles Study of Different Polymorphs of Crystalline Zirconium Hydride Weihua Zhu,*,† Rongshan Wang,‡ Guogang Shu,§ Ping Wu,⊥ and Heming Xiao† Institute for Computation in Molecular and Materials Science and Department of Chemistry, Nanjing UniVersity of Science and Technology, Nanjing 210094, China, Suzhou Nuclear Power Research Institute, Suzhou 215004, China, China Nuclear Power Engineering Company LTD, Shenzhen 518031, China, Institute of High Performance Computing, 1 Fusionopolis Way, #16-16 Connexis, Singapore 138632, Singapore ReceiVed: September 26, 2010; ReVised Manuscript ReceiVed: NoVember 6, 2010

We present a detailed study on the electronic structure, mechanical properties, phase stability, and thermodynamic properties of four polymorphs of crystalline zirconium hydride by using density functional theory within the generalized gradient approximation. An analysis of electronic structure shows that the zirconium hydrides retain metallic bonding over the whole hydrogen composition range. The calculated mechanical properties indicate that ζ-Zr2H and γ-ZrH are ductile, while δ-ZrH1.5 and ε-ZrH2 are brittle compared with R-Zr. The hydrides change from ductile to brittle in the order of ζ-Zr2H, γ-ZrH, ε-ZrH2, and δ-ZrH1.5. The formation enthalpies are negative for the four hydrides at the ambient pressure indicating that they are thermodynamically stable. Only the δ phase is thermodynamically stable in the whole pressure range. As the temperature increases, the decomposition reactions of the four hydrides are more and more favorable thermodynamically. The δ phase is thermodynamically easier to decompose than the others in the whole temperature range. 1. Introduction The absorption of hydrogen by zirconium alloys, used as the cladding devices of light water reactors, may lead to significant embrittlement and severely affect their useful life in storage and service conditions.1 This is related to the formation of zirconium hydrides in the zirconium alloy. Zirconium hydride is also widely used as a neutron moderator in nuclear reactors.2 Recently, zirconium hydride is included in the hydride fuels because of its excellent stability under irradiation conditions.3 Therefore, there are a large number of studies4-26 devoted to the characteristics of zirconium hydrides. Zirconium hydride is known to have a phase transition as a function of hydrogen concentration.27 A hydrogen-rich ε-ZrH2 phase has a tetragonal I4/mmm space group.28 As the hydrogen concentration decreases, the ε phase transits to nonstoichiometric δ-ZrH2 phase, which is the dominant stable hydride phase in zircaloy-2 and zircaloy-4.29 It is a face-centered cubic phase with Fm3jm space group and has four H atoms randomly occupying the eight available tetragonal (0.25, 0.25, 0.25) sites. In addition, a metastable γ-ZrH phase has a tetragonal P42/n space group.30 More recently, a new ordered compound (called ζ) fully coherent with R-Zr was detected with a composition close to Zr2H.31 Unfortunately, many fundamental problems of these compounds are still not well understood. For different polymorphs of crystalline zirconium hydride, there is a variation in the structure. Thus, understanding the differences between the structure and the fundamental properties of the hydrides is important for the development of new zirconium alloys and zirconium hydride fuels. * To whom correspondence should be addressed. E-mail: zhuwh@ mail.njust.edu.cn. † Nanjing University of Science and Technology. ‡ Suzhou Nuclear Power Research Institute. § China Nuclear Power Engineering Company LTD. ⊥ Institute of High Performance Computing.

In comparison with the experimental works, theoretical calculations can also play an important role in investigating the physical and chemical properties of zirconium hydrides at the atomic level and the establishment of the relationships between their structure and properties. Darby et al.32 calculated the band structures of the R and δ phases of zirconium hydride using the augmented plane wave (APW) method. Ackland33 studied the stability of the ε phase of zirconium hydride using the pseudopotential plane wave method. Kul’kova et al.34 employed the self-consistent linear muffin-tin orbital method in the atomic sphere approximation (LMTO-ASA) to calculate the electronic structure and lattice stability in the dihydrides of titanium, zirconium, and hafnium. Afterward, they also studied the electronic structure of zirconium dihydride in the cubic and the tetragonal phases using the same method.35 Wolf and Herzig36 investigated the energetics and chemical bonding of transitionmetal dihydrides ScH2, TiH2, VH2, YH2, ZrH2, and NbH2. Yamanaka et al.37 analyzed the electronic structure of zirconium hydride using cluster model and a discrete-variational-XR molecular orbital method. Domain et al.38 studied the electronic structure and formation energies of γ-ZrH, δ-ZrH1.5, and ε-ZrH2. Konashi et al.39 used the nonequilibrium molecular dynamics (NEMD) method to investigate the thermal conductivity of ZrH1.6. Recently, Holliger et al.40 presented the existence of several metastable hydride superstructures in the two-phase R-δ equilibrium region of H-Zr by a combination of cluster expansion and electronic structure methods. As the basic solidstate properties of the series of ζ-Zr2H, γ-ZrH, δ-ZrH1.5, and ε-ZrH2 are not systematically investigated and compared, there is a clear need to gain an understanding of those at the atomic level. In this study, we reported a systematic study of the electronic structure, mechanical properties, phase stability, and thermodynamic properties of the series of ζ-Zr2H, γ-ZrH, δ-ZrH1.5, and ε-ZrH2 from density functional theory (DFT). Our main purpose here is to examine the differences in the microscopic

10.1021/jp109185n  2010 American Chemical Society Published on Web 11/30/2010

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Figure 1. Unit cells for (a) ζ-Zr2H, (b) γ-ZrH, (c) δ-ZrH1.5, and (d) ε-ZrH2. Gray and dark cyan spheres stand for H and Zr atoms, respectively.

properties of the hydrides and to understand their structureproperty relationships. The remainder of this paper is organized as follows. A brief description of our computational method is given in section 2. The results and discussion are presented in section 3 followed by a summary of our conclusions in section 4. 2. Computational Method The calculations were performed in this study by using the DFT method with Vanderbilt-type ultrasoft pseudopotentials41 and a plane-wave expansion of the wave functions as implemented in the CASTEP code.42 The self-consistent ground state of the system was determined by using a band-by-band conjugate gradient technique to minimize the total energy of the system with respect to the plane-wave coefficients. The electronic wave functions were obtained by a density-mixing scheme,43 and the structures were relaxed by using the Broyden, Fletcher, Goldfarb, and Shannon (BFGS) method.44 The GGA functional proposed by Perdew and Wang,45,46 named PW91, was employed. The cutoff energy of plane waves was set to 800.0 eV. Brillouin zone sampling was performed by using the Monkhost-Pack scheme with a k-point grid of 6 × 6 × 6. The values of the kinetic energy cutoff and the k-point grid were determined to ensure the convergence of total energies. The δ phase is a nonstoichiometric phase. Here, the hydrogen contents of the δ phases were selected as 1.50 H/Zr written as δ-ZrH1.5. Figure 1 displays the unit cell of the ζ-Zr2H, γ-ZrH, δ-ZrH1.5, and ε-ZrH2 phases. Starting from the above-mentioned experimental structures, the geometry relaxation was performed to allow the ionic configurations, cell shape, and volume to change. In the geometry relaxation, the total energy of the system was converged less than 2.0 × 10-5 eV, the residual force less than 0.05 eV/Å, the displacement of atoms less than 0.002 Å, and the residual bulk stress less than 0.1 GPa. The elastic stiffness constants were calculated by stress-strain method. Both stress and strain have three tensile and three shear components giving six components in total. The linear elastic constants form a 6 × 6 symmetric matrix, having 27 different components, such that σi ) Cijεj for small stresses σ and strains ε.47 Properties such as the bulk modulus (response to an isotropic compression), shear modulus, Young’s modulus, and Poisson’s ratio may be computed from the values of Cij. The phonon frequencies at the gamma point were calculated from the response to small atomic displacements.48 The results of the phonon spectra can be used to compute enthalpy, entropy, free energy, and heat capacity at constant pressure as functions of temperature. The single-point energy calculation yields the total electronic energy at 0 K. The vibrational contributions to the thermodynamic properties are evaluated to compute enthalpy, entropy, free energy, and heat capacity at finite temperatures.49

TABLE 1: Experimental and Relaxed Lattice Constants (Å) for r-Zr, ζ-Zr2H, γ-ZrH, δ-ZrH1.5, and ε-ZrH2 Crystals this work

expt

previous work

a

c

a

c

a

c

R-Zr

3.223

5.175

3.231a

5.147a

3.16b 3.23c

5.10b 5.18c

ζ-Zr2H γ-ZrH δ-ZrH1.5 ε-ZrH2

3.254 4.586 4.670 4.724

10.872 4.980

3.30d 4.586e 4.768f 4.968g

10.29d 4.948e

4.207

4.449g

a The experimental values are from ref 50. b The calculated values (local density approximation, LDA) are from ref 38. c The calculated values (generalized gradient approximation, GGA) are from ref 38. d The experimental values are from ref 31. e The experimental values are from ref 30. f The experimental value is from ref 29. g The experimental values are from ref 28.

3. Results and Discussion 3.1. Bulk Properties. R-Zr is a hexagonal close-packed structure with P63/mmc space group.50 The calculated lattice parameters are given in Table 1 together with experimental and previous theoretical results. It is found that our calculated results are in good agreement with experimental values28-31,50 and other theoretical calculations.38 Table 1 also presents the calculated lattice constants of crystalline ζ-Zr2H, γ-ZrH, δ-ZrH1.5, and ε-ZrH2 along with corresponding experimental data. It is seen that the lattice constants are consistent with experimental values. The comparisons confirm that our computational parameters are reasonably satisfactory. 3.2. Electronic Structure. The calculated total density of states (DOS) and partial DOS (PDOS) for ζ-Zr2H, γ-ZrH, δ-ZrH1.5, and ε-ZrH2 are displayed in Figure 2. The DOS and PDSO of R-Zr crystal are also presented in Figure 2 for comparison. The DOS of the four hydrides are finite at the Fermi energy level as occurs in the DOS of R-Zr. This indicates that the hydrides retain metallic bonding over the whole H composition range. Also, the total DOS displays the most important H s and transition-metal d components. In the bottom valence band, the DOS of the four hydrides are superimposed by the s and d states. The top of the DOS valence band shows some main peaks, which are predominately from the d states. After that, several main peaks in the bottom conduction band are dominated by the d states. This shows that the d states for the four hydrides make a very important contribution to the DOS. Compared to the DOS of R-Zr, the DOS of ζ-Zr2H presents a peak at -5.83 eV from the hydrogen states. Similarly, the DOS of γ-ZrH shows one peak at -7.10 eV and another peak at -6.23 eV from the hydrogen states, and the DOS of δ-ZrH1.5 does one peak at -7.14 eV and another peak at -6.30 eV from the hydrogen states. However, the DOS of ε-ZrH2 presents a very small peak at -8.15 eV from the hydrogen states. Since these

Polymorphs of Crystalline Zirconium Hydride

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Figure 2. Total and partial density of states (DOS) of (a) R-Zr, (b) ζ-Zr2H, (c) γ-ZrH, (d) δ-ZrH1.5, (e) ε-ZrH2, and their Zr states and H states. The Fermi energy is shown as a dashed vertical line.

peaks are far from the upper valence band, the hydrogen states indirectly affect the DOS of R-Zr. This will be further discussed in the atom-resolved DOS and PDOS of the four hydrides below. The Fermi level is at the position of the DOS peaks for ζ-Zr2H and δ-ZrH1.5. This suggests that the two phases are not stable

at low temperatures. For γ-ZrH and ε-ZrH2, the Fermi level falls into a minimum of the DOS, which leads to a lowering of the total energy. The atom-resolved DOS and PDOS of the four hydrides are also shown in Figure 2. The main features can be summarized

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TABLE 2: Elastic Constants Cij (GPa), Bulk Modulus B (GPa), Shear Modulus G (GPa), Young’s Modulus E (GPa), and Poisson’s Ratio W for the r-Zr, ζ-Zr2H, γ-ZrH, δ-ZrH1.5, and ε-ZrH2 Crystals C11 R-Zr

ζ-Zr2H γ-ZrH δ-ZrH1.5 ε-ZrH2 a

d

159.38 155a 155b 145c 142d 97.02 127.86 62.97 101.53

C22

C33

C44

102.81

180.93 173a 182b 177c 164d 179.93 186.81 65.10 107.58

17.52 36a 25b 22c 29d 2.93 54.55 93.47 35.59

C55

C66

C12

C13

125.57 117.89 27.84 20.36

66.07 65a 76b 83c 64d 70.36 93.53 44.02 11.44

50.71 44a 45b 36c 39d

35.58

64.34 100.69 23.67

C23

B

G

E

V

B/G

38.99

103.04

0.32

2.47

11.94

96.16 97a 103b 105c 92d 100.73 116.94 46.97 44.38

5.49 43.86 62.53 36.84

16.17 116.96 129.93 86.58

0.47 0.33 0.04 0.17

18.34 2.67 0.75 1.20

b

The experimental values are from ref 51. The calculated values (LDA) are from ref 52. c The calculated values (LDA) are from ref 38. The calculated values (GGA) are from ref 38.

as follows. (1) At the low-energy side of the DOS, only the H states contribute to the DOS for the four hydrides. (2) Some strong peaks occur at the same energy in the PDOS of a particular H atom and a particular Zr atom. This shows that the H and Zr atoms are bonded. (3) There are some differences in the PDOS of the Zr atoms for the four hydrides. This is due to the differences in their local molecular packing. Therefore, it may be inferred that the hydrogen states affect the electronic structure of the hydrides by modifying the DOS of the Zr atoms. (4) In the conduction band region, the DOS of the four hydrides are overlapped by the H and Zr states. (5) Overall, the DOS shape of ζ-Zr2H is identical with that of its Zr states (Figure 2b). The rigidity of the DOS of Zr states thus implies that the modifications in the DOS of the hydride are mainly due to the changes in the DOS of Zr states. The same is true of γ-ZrH, δ-ZrH1.5, and ε-ZrH2. (6) In the upper valence band from -10.0 to 0 eV, the DOS of the H states becomes broader and gradually shifts toward the Fermi level with the increment of H composition. It may be inferred that the interactions between the Zr and H states become stronger as the H composition in the hydrides increases. 3.3. Mechanical Properties. Table 2 presents the calculated elastic constants of crystalline R-Zr, ζ-Zr2H, γ-ZrH, δ-ZrH1.5, and ε-ZrH2 along with available experimental data. It is seen that on the whole, reasonable agreement is obtained both with available experimental51 and calculated values by others.38,52 The errors between theory and experiment seem to be a common feature of DFT calculations within the plane-wave pseudopotentials framework. As a rule, the diagonal elements Cii(i ) 1-6) describe the stiffness to uniaxial compression and shear, while the offdiagonal elements Cij(i * j) describe the biaxial compression and distortion of the crystal. It is seen in Table 2 that the diagonal elements of elastic constants for R-Zr are anisotropic (C11 ≈ 0.88C33; C44 ≈ 0.35C66). This may result from the crystal packing. Similar situations are also found in the four hydrides. Therefore, it may be concluded that the zirconium hydrides are anisotropic to compression. In addition, it is found that the four hydrides have larger off-diagonal elements C12 and C13. This shows that the four hydrides have a greater stiffness to compression in the a and b directions than in the c direction. When the H composition changes, the hydrides present different rigidity but are still anisotropic. The mechanical properties including bulk modulus, shear modulus, Young’s modulus, and Poisson’s ratio for the four hydrides are also listed in Table 2. The bulk moduli B for the hydrides ζ-Zr2H, γ-ZrH, δ-ZrH1.5, and ε-ZrH2 are 100.73, 116.94, 46.97, and 44.38 GPa, respectively. The fracture strength for a material is proportional to the bulk modulus. The larger

the bulk modulus, the stiffer the material, and the smaller the deformation of the material is. Therefore, the fracture strength of the zirconium hydrides becomes weaker with the increment of the H composition. The bulk moduli of ζ-Zr2H and γ-ZrH are close to that of R-Zr (B ) 96.16 GPa), whereas δ-ZrH1.5 and ε-ZrH2 have smaller bulk moduli than R-Zr. This suggests that higher hydrogen composition can impair the strength of zirconium. The resistance to plastic deformation is proportional to the shear modulus. ζ-Zr2H has smaller shear modulus G than R-Zr, while γ-ZrH, δ-ZrH1.5, and ε-ZrH2 have higher shear moduli than R-Zr or have shear moduli close to it. It is thus deduced that higher hydrogen composition does not destroy the plastic strength of pure zirconium. The Young’s modulus of ζ-Zr2H is lower than that of R-Zr, whereas the hydrides γ-ZrH and δ-ZrH1.5 with high H composition have higher Young’s moduli than R-Zr. When the H composition reaches maximum, the hydride ε-ZrH2 is slightly smaller than pure zirconium. The Poisson’s ratio for the four hydrides decreases in the following order: ζ-Zr2H > γ-ZrH > ε-ZrH2 > δ-ZrH1.5. B/G ratio is related to brittleness (ductility) for a material. A high B/G value indicates ductility, while a low value shows brittleness. The critical value is about 1.75.53 It is seen in Table 2 that the B/G ratio for the four hydrides decreases in the following order: ζ-Zr2H > γ-ZrH > ε-ZrH2 > δ-ZrH1.5. R-Zr, ζ-Zr2H, and γ-ZrH have higher B/G values than the critical one. This suggests that they are ductile. δ-ZrH1.5 and ε-ZrH2 have smaller B/G values than the critical value showing that they are brittle. ζ-Zr2H and γ-ZrH have higher B/G values than R-Zr, while δ-ZrH1.5 and ε-ZrH2 have smaller B/G values than R-Zr. The hydrides change from ductile to brittle in the order of ζ-Zr2H, γ-ZrH, ε-ZrH2, and δ-ZrH1.5. This may be because higher hydrogen composition induces more covalent bonding. Also, δ-ZrH1.5 has the smallest B/G value among the hydrides. This shows that the nonstoichiometric hydride δ-ZrH1.5 is the most brittle phase among the four hydrides. It was experimentally observed54 that the dominant stable hydride phase in zircaloy-2 and zircaloy-4 at typical hydrogen concentrations of 300-600 ppm and room temperature is the nonstoichiometric δ-ZrH2-x phase. The existence of the nonstoichiometric hydrides may be the cause of significant embrittlement of zirconium alloys. These suggestions support our conclusion here. 3.4. Phase Stability. As is known, the application of a compound requires accurate knowledge of the thermodynamic stability of all relevant phases particularly the phase stability.55-57 In this section, we investigate the relative stability of the zirconium hydrides both at zero and at high pressures. When one hydride was studied, the others were chosen as the competing phases because they were synthesized experimentally



Figure 3. The formation enthalpy-pressure diagram of (a) ζ-Zr2H, (b) γ-ZrH, (c) δ-ZrH1.5, (d) ε-ZrH2, and their respective competing phases with respect to elemental Zr and H.

and were thermodynamically stable. The formation enthalpy H ) E + PV under pressure is shown in Figure 3. It is seen in Figure 3a that the formation enthalpies for ζ-Zr2H are negative in the pressure range from 0 to 31.84 GPa suggesting that the elements (Zr + H) are able to form the ζ-Zr2H phase, whereas above 31.84 GPa, they become positive indicating that the elements (Zr + H) are unable to form the ζ-Zr2H phase. The formation enthalpies for the γ-ZrH + Zr, δ-ZrH1.5 + Zr, and ε-ZrH2 + Zr phases are negative in the pressure range from 0 to 100 GPa. This shows that the phases γ-ZrH + Zr, δ-ZrH1.5 + Zr, and ε-ZrH2 + Zr are unable to form the ζ-Zr2H phase. Among them, the phase δ-ZrH1.5 + Zr is the most difficult to produce the ζ-Zr2H phase. Therefore, it can be concluded that the mixture of δ-ZrH1.5 + Zr is the most stable phase in the whole pressure range. ζ-Zr2H is the second stable phase below 11.28 GPa. At 11.28 GPa, the ζ-Zr2H phase and the mixture of ε-ZrH2 + Zr are isoenergetic. At 13.55 GPa, the ζ-Zr2H phase and the mixture of γ-ZrH + Zr are isoenergetic. Above 11.28 GPa, the competing phase ε-ZrH2 + Zr becomes the second stable one. From Figure 3b, it is found that the formation enthalpies for γ-ZrH are negative in the pressure range from 0 to 50.50 GPa showing that the elements (Zr + H) are able to form the γ-ZrH phase, whereas above 50.50 GPa, they become positive indicating that the elements (Zr + H) are unable to form the γ-ZrH phase. The formation enthalpies for the ζ-Zr2H + H2 phase are positive in the pressure range from 0 to 72.92 GPa, while they become negative above 72.92 GPa. This shows that the ζ-Zr2H

+ H2 phase is stable above 72.92 GPa. A similar situation is found in the mixture of ε-ZrH2 + Zr. The ε-ZrH2 + Zr phase is stable above 8.81 GPa. The formation enthalpies for the δ-ZrH1.5 + Zr phase are negative in the pressure range from 0 to 100 GPa. This indicates that the phase δ-ZrH1.5 + Zr is difficult to form the γ-ZrH phase. Thus, among them, the mixture of δ-ZrH1.5 + Zr is the most stable phase in the whole pressure range. γ-ZrH is the second stable phase below 41.15 GPa. At 41.15 GPa, the γ-ZrH phase and the mixture of ε-ZrH2 + Zr are isoenergetic. Above 41.15 GPa, the competing phase ε-ZrH2 + Zr becomes the second stable one. It is observed from Figure 3c that the formation enthalpies for the ζ-Zr2H + H2, γ-ZrH + H2, and ε-ZrH2 + Zr phases are positive as the pressure increases. This suggests that the ζ-Zr2H + H2, γ-ZrH + H2, and ε-ZrH2 + Zr phases are able to form the δ-ZrH1.5 phase in the pressure range from 0 to 100 GPa. However, the formation enthalpies of the δ-ZrH1.5 phase are negative with the increment of the pressure. This shows that the elements (Zr + H) are able to form the δ-ZrH1.5 phase. Accordingly, among them, the δ-ZrH1.5 is the most stable phase in the whole pressure range. From Figure 3d, it is seen that the formation enthalpies for ε-ZrH2 are negative in the pressure range from 0 to 65.66 GPa showing that the elements (Zr + H) are able to form the ε-ZrH2 phase, whereas above 65.66 GPa, they become positive indicating that the elements (Zr + H) are unable to form the ε-ZrH2 phase. The formation enthalpies for the ζ-Zr2H + H2 phase are positive in the pressure range from 0 to 84.12 GPa, while they

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Figure 4. Formation enthalpy of Zr-H systems at the pressure of (a) 0 GPa and (b) 50 GPa.

become negative above 84.12 GPa. This shows that the ζ-Zr2H + H2 phase is stable above 84.12 GPa. A similar situation is found in the mixture γ-ZrH + H2. The γ-ZrH + H2 phase is stable above 93.96 GPa. The formation enthalpies for the δ-ZrH1.5 + H2 phase are negative as the pressure increases. This indicates that the phase δ-ZrH1.5 + H2 is difficult to form the ε-ZrH2 phase. Thus, among them, the mixture of δ-ZrH1.5 + H2 is the most stable phase in the whole pressure range. ε-ZrH2 is the second stable phase below 65.66 GPa. Above 84.12 GPa, the competing phase ζ-Zr2H + H2 becomes the second stable one. From the above suggestions, it is concluded that the δ-ZrH1.5 phase, the mixture of δ-ZrH1.5 + Zr, and the mixture of δ-ZrH1.5 + H2 are stable in the whole pressure range. This shows that the δ-ZrH1.5 phase is the most easy to form in zirconium alloys. Figure 4 presents the formation enthalpies for the four hydrides at 0 and 50 GPa using H and Zr as the end components. As expected, Figure 3a shows a convex hull defined by the four structures ζ-Zr2H, γ-ZrH, δ-ZrH1.5, and ε-ZrH2, which were observed experimentally.28-31 This is also in good agreement with the results of enthalpy-pressure shown in Figure 3. The formation enthalpies present an asymmetrical shape with respect to the hydrogen composition of 0.5. The minimum of formation enthalpy reaches the hydrogen composition of 0.6 (δ-ZrH1.5). The formation enthalpy is lower in the hydrogen-rich side. This shows that the zirconium hydrides with higher hydrogen composition are more stable at zero pressure. However, the situation is different at high pressure. The formation enthalpy of ζ-Zr2H becomes positive, whereas the formation enthalpies of γ-ZrH and ε-ZrH2 are still negative. The formation enthalpy of δ-ZrH1.5 is more negative at 50 GPa compared with zero pressure. This indicates that ζ-Zr2H is unable to form at high pressure, but the γ-ZrH, ε-ZrH2, and δ-ZrH1.5 phases are able to form. 3.5. Thermodynamic Properties. In this section, the thermodynamic functions including enthalpy, entropy, heat capacity, and free energy for the four hydrides are evaluated and presented in Figure 5. With the increase of temperature, the calculated enthalpies of the four hydrides monotonically increase. This is because the main contributions to the enthalpy are from the translations and rotations at lower temperature, whereas the vibrational motion is intensified and makes more contributions to the enthalpy at higher temperature. At a given temperature, the enthalpy for the four hydrides increases in the following order: ζ-Zr2H < ε-ZrH2 < γ-ZrH < δ-ZrH1.5. This is because δ-ZrH1.5 contains the most zirconium and hydrogen atoms among the hydrides, and these atoms make more contributions

to the enthalpy. The difference of the enthalpy between ζ-Zr2H and ε-ZrH2 is small. As the temperature increases, the entropies of the four hydrides gradually increase. The γ-ZrH phase has the same entropy as δ-ZrH1.5 at about 425 K, while the ζ-Zr2H phase has the same entropy as δ-ZrH1.5 at about 684 K and as γ-ZrH at about 853 K. Similarly, the heat capacities of the four hydrides also gradually increase with the increment of temperature. The ζ-Zr2H phase has the same heat capacity as γ-ZrH at about 84 K, as δ-ZrH1.5 at about 95 K, and as ε-ZrH2 at about 219 K. The δ-ZrH1.5 has the same heat capacity as γ-ZrH at about 199 K and as ε-ZrH2 at about 31 K. When the temperature is over about 219 K, the heat capacity for the four hydrides increases in the following order: ζ-Zr2H < ε-ZrH2 < γ-ZrH < δ-ZrH1.5 at a given temperature. This is because the main contributions to the heat capacity are from the acoustic mode of lattice vibration at lower temperature, whereas the optical mode of lattice vibration, which is hydrogen vibrational contribution, makes more contributions to the heat capacity at higher temperature. For the free energy, the case is quite the contrary. As the temperature increases, the free-energy values of the four hydrides gradually decrease. The decreasing order of the free energy is δ-ZrH1.5 > γ-ZrH > ε-ZrH2 > ζ-Zr2H at a given temperature within the range of 0-718 K, whereas above about 718 K, the decreasing order changes as δ-ZrH1.5 > ε-ZrH2 > γ-ZrH > ζ-Zr2H. The free energy of a material strongly depends on the geometry of the atomic configurations. Since the four hydrides have similar crystal packing, their free energies are very similar. On the basis of the calculated thermodynamic functions, we evaluated the enthalpy of formation and the free energy of formation for the decomposition eqs 1-4 of the four hydrides.

ζ - Zr2H f 2Zr + 1/2H2

(1)

γ - ZrH f Zr + 1/2H2

(2)

δ - ZrH1.5 f Zr + 3/4H2

(3)

ε - ZrH2 f Zr + H2

(4)

The calculated enthalpy and free energy of formation for the four hydrides are shown in Figures 6 and 7, respectively. It can



Figure 5. Entropy (S), heat capacity (Cp), enthalpy (H), and free energy (G) of (a) ζ-Zr2H, (b) γ-ZrH, (c) δ-ZrH1.5, and (d) ε-ZrH2 as a function of temperature.

Figure 6. Enthalpy of formation for ζ-Zr2H, γ-ZrH, δ-ZrH1.5, and ε-ZrH2 as a function of temperature at zero pressure.

be seen that the enthalpies of formation for the ε-ZrH2 phase become more and more positive with the increasing temperature. This shows that the decomposition eq 4 is exothermic. The enthalpies of formation for ζ-Zr2H are negative in the range of 0-240 K, whereas they become positive above about 240 K. For γ-ZrH, its enthalpies of formation are negative in the range of 0-487 K, while they become positive above about 487 K. Similarly, the enthalpies of formation for δ-ZrH1.5 are negative

Figure 7. Free energy of formation for ζ-Zr2H, γ-ZrH, δ-ZrH1.5, and ε-ZrH2 as a function of temperature at zero pressure.

in the range of 0-412 K, whereas they become positive above about 412 K. This indicates that the decomposition eqs 1-3 are endothermic at low temperature but are exothermic at high temperature. It is found from Figure 7 that the free energies of formation for the four hydrides become more and more negative with the temperature increasing. This shows that the decomposition eqs 1-4 are thermodynamically favorable in the whole temperature range. Additionally, the free energy of formation versus tem-

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perature curve of γ-ZrH crosses that of ζ-Zr2H at about 184 K and of ε-ZrH2 at about 328 K. Above about 675 K, the free energies of formation for ζ-Zr2H and ε-ZrH2 are nearly equal. δ-ZrH1.5 has the smallest free energies of formation among the four hydrides at a given temperature. This shows that δ-ZrH1.5 is thermodynamically easier to decompose than the others in the whole temperature range. Therefore, it may be concluded that the nonstoichiometric δ-ZrH2 phase is thermodynamically unstable at high temperature. 4. Conclusions In this study, we have performed a detailed first-principles study of electronic structure, mechanical properties, phase stability, and thermodynamic properties for four polymorphs of crystalline zirconium hydrides in the generalized gradient approximation. The results show that the zirconium hydrides retain metallic bonding over the whole hydrogen composition range. The hydrogen states affect the electronic structure of the zirconium hydrides by modifying the DOS of the Zr atoms. As the H composition in the hydrides increases, the interactions between the Zr and H states become stronger. An analysis of elastic constants indicates that the zirconium hydrides are anisotropic to compression. Higher hydrogen composition makes pure zirconium become brittle. The nonstoichiometric hydride δ-ZrH1.5 is the most brittle phase among the four hydrides. The calculated formation enthalpies suggest that the four hydrides are stable at zero pressure. Only the δ-ZrH1.5 phase, the mixture of δ-ZrH1.5 + Zr, and the mixture of δ-ZrH1.5 + H2 are stable in the whole pressure range. This shows that the δ-ZrH1.5 phase is the most easy to form in zirconium alloys. As the temperature increases, the calculated enthalpies, entropies, and heat capacities of the four hydrides gradually increase, but the free energies monotonously decrease. The decomposition reactions of the four hydrides are more and more favorable thermodynamically with the temperature increasing. δ-ZrH1.5 is thermodynamically easier to decompose than the others in the whole temperature range. Acknowledgment. This work was supported by Suzhou Nuclear Power Research Institute. References and Notes (1) Aladjem, A. Solid State Phenom. 1996, 49-50, 281. (2) Simnad, M. T. Nucl. Eng. Des. 1981, 64, 403. (3) Konashi, K. Ikeshoji, T.; Kawazoe, Y.; Matsui, H. DeVelopment of Actinide-Hydride Target for Transmutation of Nuclear Waste; International Conference on Back-End of the Fuel Cycle from Research to Solutions, GLOBAL2001, Paris, France, September 9-13, 2001. (4) Puls, M. P.; Shi, S. Q.; Rabier, J. J. Nucl. Mater. 2005, 336, 73. (5) Domain, C.; Besson, R.; Legris, A. Acta Mater. 2004, 52, 1495. (6) Yamanaka, S.; Yamada, K.; Kurosaki, K.; Uno, M.; Takeda, K.; Anada, H.; Matsuda, T.; Kobayashi, S. J. Alloys Compd. 2002, 330-332, 99. (7) Yamanaka, S.; Setoyama, D.; Muta, H.; Uno, M.; Kuroda, M.; Takeda, K.; Matsuda, T. J. Alloys Compd. 2004, 372, 129. (8) Go´mez, M. P.; Domizzi, G.; Lo´pez Pumarega, M. I.; Ruzzante, J. E. J. Nucl. Mater. 2006, 353, 167. (9) Gulbransen, E. A.; Andrew, K. F. J. Electrochem. Soc. 1954, 101, 474. (10) Woo, O. T.; Carpenter, G. J. C. Microsc. Microanal. Microstruct. 1992, 3, 35. (11) Kim, J. H.; Lee, M. H.; Choi, B. K.; Jeong, Y. H. J. Alloys Compd. 2007, 431, 155. (12) Bowman, R. C., Jr.; Craft, B. D.; Cantrell, J. S.; Venturini, E. L. Phys. ReV. B 1985, 31, 5604.

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JP109185N

First-Principles Study of Different Polymorphs of Crystalline Zirconium

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