A DFT study of structural, elastic and lattice dynamical properties of Fe2Zr and FeZr2 intermetallics

Journal of Alloys and Compounds 723 (2017) 611e619

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Journal of Alloys and Compounds journal homepage: http://www.elsevier.com/locate/jalcom

A DFT study of structural, elastic and lattice dynamical properties of Fe2Zr and FeZr2 intermetallics Kawsar Ali a, b, P.S. Ghosh a, b, *, A. Arya a, b a b

Materials Science Division, Bhabha Atomic Research Centre, Mumbai, 400085, India Homi Bhabha National Institute, Mumbai, 400094, India

a r t i c l e i n f o

a b s t r a c t

Article history: Received 6 January 2017 Received in revised form 12 June 2017 Accepted 13 June 2017 Available online 15 June 2017

We report plane-wave based density functional theory (DFT) investigation of structural, elastic and lattice dynamical properties of Fe-rich C15-Fe2Zr and Zr-rich FeZr2 intermetallic compounds. These intermetallics dominantly appear in the Fe-Zr alloys which are potential candidates for immobilization of nuclear metallic waste. The calculated elastic constants and phonon dispersions show that both phases are mechanically and dynamically stable. The calculated bulk moduli as a function of temperature show that the Fe-rich C15 phase is less compressible and also less sensitive to temperature than the Zr-rich C16 phase. Furthermore, the C15-Fe2Zr phase has a lower thermal expansion co-efficient but higher thermal conductivity as compared to the C16-FeZr2 phase; while the heat capacities (CP and CV) of both the phases are almost identical above 450 K. © 2017 Elsevier B.V. All rights reserved.

Keywords: Thermodynamic properties Fe-Zr alloys Thermal conductivity Phonons

1. Introduction Iron-Zirconium alloys play an important role in the nuclear industry. Zr-rich Fe-Zr alloys are used as structural materials for light and heavy water based thermal reactors due to their low neutron absorption cross section and good mechanical and corrosion properties [1e3]. Zr-rich alloys, e.g., Zr-8 Stainless steel (SS) and Ferich alloys, e.g., SS-15 Zr alloys are promising candidates for immobilization of metallic solid waste originating after the reprocessing of spent nuclear fuel [4,5]. The domain of applications of Fe-Zr alloys and their ternary derivatives is not limited to the nuclear industry, as in Fe based aluminides, the presence of intermetallic phases provides an additional strength which makes these alloys a suitable structural material for high temperature applications [6e8]. Fe-Zr alloys are also excellent glass formers over a broad range of compositions [9,10] and the presence of intermetallic phases acts as a storage material for hydrogen and its isotopes [11,12]. Furthermore, Zircaloy-2 and Zircaloy-4 alloys, which are used as structural and cladding materials in light and heavy water based thermal reactors, contain some amount of Fe and due to its fast diffusion and very low solubility in a-Zr [13,14], several

* Corresponding author. Materials Science Division, Bhabha Atomic Research Centre, Mumbai, 400085, India. E-mail address: [email protected] (P.S. Ghosh). http://dx.doi.org/10.1016/j.jallcom.2017.06.154 0925-8388/© 2017 Elsevier B.V. All rights reserved.

intermetallic compounds, viz., hexagonal and cubic (Fe,Cr)2Zr Laves phases, tetragonal (Fe,Ni)Zr2 phase and orthorhombic FeZr3 are formed during different heat treatments [15e18]. The presence of Fe2Zr and FeZr2 phases in SS-15 Zr and Zr-8 SS is prominent which in turn can incorporate the long living readionuclides present in the metallic waste [19e21]. In Fe-Zr binary phase diagram, three polymorphs of Fe2Zr, viz., cubic (C15), dihexagonal (C36) and hexagonal (C14) phases and two polymorphs of FeZr2, viz., a tetragonal (C16) phase and a cubicstructured phase, which only crystallizes in the presence of nonmetals like O or N, have been observed [4,22,23]. Among the three polymorphs of Fe2Zr, the C15-phase has been shown experimentally to be the most stable phase under ambient conditions, while the other two phase are stable at high temperatures [22,23]. Similarly, the C16-FeZr2 phase has been experimentally shown to be the most stable phase [22,24,25]. The evaluation of structural, mechanical and thermodynamic properties of C15-Fe2Zr and C16-FeZr2 phases is crucial for establishing their thermodynamic and elastic stability from the point of view of their potential use as metallic wasteforms in nuclear industry. In our earlier publication, we calculated the ground state structural, mechanical and elastic properties of all the stable and metastable intermetallics of Fe-Zr system [26]. However, thermodynamic properties of Fe-Zr intermetallics have been insufficiently reported in the literature. Lück et al. [27] measured the constant pressure specific heat (CP) of C14-Fe2Zr phase using differential

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scanning calorimetry in the temperature range of 310e660 K. They showed that the CP exhibited a l-shaped variation as a function of temperature and at 315 K, the measured value of CP was 27.214 J/ mole/K. Mittal et al. [28] studied the superconducting properties of C16-FeZr2 phase by measuring phonon density of states using inelastic neutron scattering as well as density functional theory (DFT) based tight binding linear muffin-tin orbital (TB-LMTO). They employed lattice dynamic calculations using an empirical pair potential model to calculate the low temperature (0e20 K) specific heat at constant volume (CV) of C16-FeZr2 phase. Tao et al. [29] studied the phase stability and elastic properties of FeZr2, AlZr2 and Al2FeZr6 using DFT. They have found that all the phases are thermodynamically and mechanically stable. Chattaraj et al. [30] studied the structural, electronic, elastic and thermodynamic properties of FeZr2 and FeZr2H5 using planewave PAW potential based DFT calculations. They observed that both phases are thermodynamically and mechanically stable and exhibit anisotropy in their elastic properties. Motta et al. [31] calculated the temperature dependence of electric and magnetic hyperfine interactions of Ta181 atoms substituted at Zr site in Fe2Zr Laves phase using perturbed angular correlation spectroscopy. They also reported anisotropic behaviour of cubic Fe2Zr by observing a weak strongly damped electric-quadrupole interaction. As mentioned earlier, it has been observed that these phases are the potential hosts for radionuclides present in the metallic waste [19e21]. These phases are expected to experience the heat generated by the radioactive decay of different radionuclides. Therefore, evaluation of their lattice dynamical stability and thermal properties is of great significance. The thermal expansion coefficient (TEC), e.g., gives an indication about the extent of thermal stresses generated in these phases on incorporation of radionuclides. Similarly, the thermal conductivity, which is intimately related to the specific heat, gives an estimate of heat distribution in the system to rule out any possible local melting. The material to be used for metallic waste immobilization should have low TEC and high thermal conductivity and specific heat. In this work, we have studied the lattice dynamical stability, elastic and thermodynamic properties, viz., specific heat capacities (CV , CP ), thermal expansion coefficients (a), variation of bulk moduli (B) with temperature (T), free energy (F), entropy (S) and internal energy (E) variation with temperature of C15-Fe2Zr and C16-FeZr2 phases using ab-initio plane wave based methodology. The paper is organized as follows: In the next section, we will briefly present the computational approach employed to calculate the thermodynamic and elastic properties of the studied phases followed by results and discussion of these properties. Finally, the main conclusions of the work will be presented.

2. Computational method The Helmholtz free energy (F(V,T)) of a given phase as a function of temperature and volume can be approximated as

FðV; TÞ ¼ Fel ðV; TÞ þ Fvib ðV; TÞ

(1)

where Fel ð¼ Eel  TSel Þ denotes the electronic contribution to the free energy. Generally, the entropy contribution due to electrons to the electronic free energy is negligible and Eel contribution can be calculated accurately from density functional theory (DFT). Fvib ð¼ Evib  TSvib Þ is the phononic contribution to the free energy due to the vibrations of atoms. The Fvib , under quasi-harmonic approximation, can be represented as

Fvib ¼

X    1X Zuq;n ðVÞ þ kB T ln 1  exp  Zuq;n ðVÞ=kB T 2 q;n q;n (2)

where q and n are the wave vector and band index, respectively. uq;n is the phonon frequency with a wave vector q and band index n. T, kB and Z denote the temperature, the Boltzmann constant and the reduced Planck's constant, respectively. The vibrational specific heat at constant volume CV and entropy S can be calculated as a function of temperature using the following relations

CV ¼ T

v2 Fvib vT 2

! ¼ V

X q;n

 kB

Zuq;n kB T

2

   exp Zuq;n kB T  2    exp Zuq;n kB T  1 (3)

  vFvib S¼ vT V  

  X X Zuq;n Zuq;n 1  kB ¼ Zuq;n coth ln 2 sinh 2T q;n 2kB T 2kB T q;n

(4)

The constant pressure specific heat CP and constant volume specific heat CV are related as

CP ¼ CV þ a2 BVT

(5)

where a, B, V and T represent the linear thermal expansion coefficient, bulk modulus, volume and temperature of the system, respectively. The linear thermal expansion co-efficient ðaÞ and volume expansion co-efficient ðaV Þ have been calculated using the following relations

1 3

a ¼ aV ¼

1 dV 3V dT

(6)

where, L and V represent the length and the volume of the cell. The bulk modulus has been calculated using

BT ¼ V

v2 F vV 2

! (7) T

We have employed plane wave DFT based calculations using the Vienna Ab-initio Simulation Package (VASP) [32] to perform all static and dynamic calculations. The ion-electron interactions using projector Augmented Wave (PAW) [33] potentials, as implemented in VASP, have been used for all the calculations. The ½3d7 4s1  state of Fe and ½4s2 4p6 5s2 4d2  state of Zr have been treated as valence states. For exchange and correlation potential, the generalized gradient approximation (GGA) using the parametrization scheme of Perdew, Burke and Enzerhof (PBE) [34] has been employed. We have performed spin-polarized calculations for both C15-Fe2Zr and C16-FeZr2 phases. A converged plane-wave cut-off energy (Ecutoff) of 450 eV has been used for both the phases. The structural and elastic constant calculations have been performed using 12  12  12 (No. of k-points in the irreducible Brillouin Zone (NIBZ) ¼ 84) and 12  12  14 (NIBZ ¼ 170) k-point meshes for Fe2Zr and FeZr2 phases, respectively. The phonon calculations have been performed using the supercell approach where the Hellmann Feynman forces on the atoms have been calculated using the Parlinski-Li-Kawazoe method [35] with a finite displacement (FD) of 0.01 Å. Density functional perturbation theory (DFPT) [36] as implemented in VASP package has also been employed for phonon calculations. The phonon calculations with 2  2  2 supercells have been performed using 8 

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8  8 (NIBZ ¼ 150) and 7  7  7 (NIBZ ¼ 172) k-point meshes for C15-Fe2Zr and C16-FeZr2 phases, respectively. The phonon calculations with 3  3  3 supercells have been performed using 6  6  6 (NIBZ ¼ 68) and 5  5  5 (NIBZ ¼ 63) k-point meshes for C15-Fe2Zr and C16-FeZr2 phases, respectively. The PHONOPY [37,38] software package has been used to derive the phonon dispersion and phonon density of states and hence thermodynamic properties of both the intermetallic phases. The phonon contribution to the thermal conductivity has been calculated by using Slack expression [39]. 3

M q ðTÞ dðTÞ k¼A 2 D g ðTÞ T n2=3

(8)

where A is a constant having a value of 3.04  107 W mol kg1 m2 K3, qD ðTÞ, M, dðTÞ and gðTÞ are the Debye temperature, mean atomic mass, cube root of average atomic volume and Grüneisen parameter, respectively. The Debye temperature under the isotropic approximation has been determined using

sffiffiffiffiffiffiffiffiffiffi  1=3 BðTÞ 1=2 2 qD ðTÞ ¼ Z 6p VðTÞ n f ðnÞ k2B M

(9)

where, Z, B(T), M and V(T) are reduced Planck's constant, bulk modulus, molar mass and molar volume, respectively. In the above equation f(n) is a scaling function which can be determined by

"    #1=3  2ð1 þ nÞ 3=2 ð1 þ nÞ 3=2 f ðnÞ ¼ 31=3 2 þ 3ð1  2nÞ 3ð1  nÞ

(10)

where n is the Poisson's ratio of the material. The first-principles calculations of single crystal elastic constants have been performed using the stress-strain approach, where the crystal has been given a set of strains ε ¼ ðε1 ε2 ε3 ε4 ε5 ε6 Þ and the elastic constants have been calculated within the Hooke's law (s ¼ ε C). The detailed procedure has been discussed elsewhere [40]. The elastic constants have been calculated for the different values of lattice strains, viz., 0.015, 0.020, 0.025 and 0.030. The convergence of single crystal elastic constants with respect to k-point mesh has been ascertained. The bulk and shear moduli have been calculated using Voigt, Reuss and Hull method [41e43]. The Young modulus and poisson's ratio have been calculated using the following relations [44]:

Y¼

9BG ; ð3B þ GÞ

n¼

ð3B  2GÞ ð6B þ 2GÞ

(11)

3. Results and discussions

613

Table 1 Equilibrium ground state properties of C-15-Fe2Zr and C16-FeZr2 phases calculated using VASP. The experimental and DFT calculated results are given in the parentheses and braces, respectively. Bulk modulus (GPa)

Wyckoff Positions

Fe2Zr a ¼ 7.046 Space group: a ¼ (7.074) [46] Fd3 m

149.24

FeZr2 a ¼ 6.276 Space group: a ¼ (6.385) [46] I4/mcm c ¼ 5.738 c ¼ (5.596) [46]

121.98 {126.2} [29]

Fe (16d): 0.6250 0.6250 0.6250 (0.6250 0.6250 0.6250) Zr (8a): 0.0000 0.0000 0.0000 (0.0000 0.0000 0.0000) Fe (4a): 0.0000 0.0000 0.2500 (0.0000 0.0000 0.2500) Zr (8h): 0.1794 0.6794 0.0000 (0.1728 0.6728 0.0000)

Phase

Cell parameters (Å)

The calculated single crystal elastic constants (Cij's) are listed in Table 1 along with other DFT results obtained using the energystrain approach. It may be recalled that our results have been derived using the stress-strain approach. As can be seen from Table 2, the calculated single crystal elastic constants are in good agreement ð < 5%Þ with other DFT results [26,29] obtained using the energy-strain approach. The calculated bulk, shear and Young modulus of both intermetallics are also given in Table 2. The calculated values are in good agreement ð < 6%Þ with other DFT results [26,29]. The G/B ratio is often used to indicate the ductility of a material and empirical G/B ratio of less than 0.57 indicates a material to be ductile in nature [47]. Further, it can be seen that the C16-FeZr2 phase is more ductile than C15-Fe2Zr phase as the G/B ratio of C16FeZr2 is smaller than for C15-Fe2Zr. It can also be seen from Table 2 that C15-Fe2Zr phase is less compressible than C16-FeZr2 phase as the bulk modulus of C16-FeZr2 phase is less than C15-Fe2Zr phase. A material with high bulk modulus is essential for its use as a metallic wasteform, hence the C15-Fe2Zr phase is more preferable than the C16-FeZr2 phase. Debye temperature (qD ) of a material is closely related to the elastic properties and thermal conductivity of the material. A material with higher values of qD signifies a stiffer material with a higher related thermal conductivity. The qD of both the intermetallic phases has been calculated using equation (9). The calculated qD for both intermetallic phases are listed in Table 2. It can be seen that the calculated qD 's are in good agreement with our previous published results [26]. A small difference between the present results and our earlier published results is due to the difference in the way the bulk modulus has been calculated here using the second derivative of total energy with respect to the volume. The qD of C15Fe2Zr phase is higher than that for the C16-FeZr2 phase which indicates that the C15-Fe2Zr phase is stiffer than C16-FeZr2 phase as also indicated by the values of their relative bulk moduli.

3.1. Ground state structural and mechanical properties 3.2. Phonon dispersions The equilibrium lattice parameter and bulk modulus have been obtained by fitting the calculated total energy as a function of volume to the Birch-Murnaghan equation of states [45]. Calculated lattice parameters for C15-Fe2Zr and C16-FeZr2 phases are given in Table 1. Our calculated lattice parameters and bulk modulus are in good agreement ð < 2%Þ with the available experimental results. Furthermore, the atomic positions of C15-Fe2Zr and C16-FeZr2 phases are listed in Table 1. The x and y coordinates of Zr atoms in C16-FeZr2 phase show a slight deviation from the experimental values. The unit cells of C15-Fe2Zr and C16-FeZr2 phases are shown in Fig. 1.

The phonon dispersions and corresponding DOS of C15-Fe2Zr and C16-FeZr2 phases along the high symmetry directions, viz., L-GX-W-L-K-G for the C15-Fe2Zr phase and M-G-X-P-N for the C16FeZr2 phase in the Brillouin zone are depicted in Fig. 2. In order to check convergence of our DFT calculated phonon frequencies as related to the supercell size, we have calculated phonon dispersion curve of both C15-Fe2Zr and C16-FeZr2 phases using 2  2  2 (48 atoms) and 3  3  3 (162 atoms) supercells. A comparison of DFT calculated phonon frequencies is presented in the supplementary material file (Figure SM-Fig. 1) which shows that the calculated

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Fig. 1. Unit cells of (a) C15-Fe2Zr phase and (b) C16-FeZr2 phase and primitive cells of (c) C15-Fe2Zr and (d) C16-FeZr2 phase. The smaller red spheres correspond to Fe atoms and the bigger green spheres correspond to Zr atoms. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Table 2 Calculated single crystal elastic constants, elastic properties and Debye temperatures (qD ) of C15-Fe2Zr and C16-FeZr2 phases using the stress-strain approach along with previous literature results. The other DFT results [26,29] correspond to those obtained using the energy-strain approach. Properties

C11 C33 C12 C13 C44 C66 Bulk modulus (B) Shear modulus (G) Young modulus (Y)

n G/B

qD

C15-Fe2Zr

C16-FeZr2

DFT this study

DFT [26]

DFT this study

DFT [26]

DFT [29]

220.90 e 111.70 e 81.30 e 148.10 70.62 182.80 0.294 0.476 409.00

219.99 e 116.59 e 82.23 e 150.86 68.32 178.08 0.300 0.450 405.76

178.20 195.70 135.10 75.80 24.80 36.80 124.58 32.25 89.06 0.381 0.259 278.00

174.98 195.04 132.07 90.56 25.66 51.21 129.86 33.23 91.85 0.380 0.260 275.42

171.50 197.70 143.10 80.90 21.50 34.40 127.50 27.60 77.20 0.400 0.220 e

phonon frequencies using 2  2  2 supercells agree within 3% with those of the 3  3  3 supercells. Therefore, all further phonon calculations were performed using smaller in size 2  2  2 supercells. The primitive cells of both C15-Fe2Zr phase and C16-FeZr2 phase (see SM-Table 1 in supplementary material and Fig. 1) contain 6 atoms, hence, both intermetallics have 18 independent modes of vibrations. The phonon dispersion curves (Fig. 2) show that these two intermetallics are dynamically stable as no imaginary frequencies have been found. The calculated DOS of C15-Fe2Zr and C16-FeZr2 phases are distributed over 0e9.9 THz and 0e7.4 THz, respectively. Our PAW-PBE calculated phonon DOS of C16-FeZr2 phase is compared with that obtained using inelastic neutron scattering and calculated using lattice dynamics methods [28] in Fig. 2 (b). Our calculated phonon DOS of C16-FeZr2 phase is in good

agreement with that obtained by Mittal et al. [28]. Our DFT calculated and experimental DOS of C16-FeZr2 phase are distributed over 0e7.4 THz and 0e7.6 THz, respectively. Our DFT calculated DOS shows a maximum around 5 THz, whereas the maximum of the experimental DOS is around 5.6 THz. The underestimation of our DFT calculated peak position as compared to experimental DOS can be attributed to the underestimated lattice parameter at 0 K calculated using GGA compared to the experimental lattice parameter which was measured at 300 K. Moreover, the calculated DOS shows a second peak at around 4 THz which is consistent with the experimental peak at around 3.8 THz. Therefore, the underestimation of lattice parameter has a more significant effect on high frequency optical modes compared to low frequency acoustic modes. The partial phonon DOS of both the intermetallics is shown in Fig. 3. The partial DOS plot of C15-Fe2Zr phase shows that the total DOS mainly comes from Zr states at lower frequencies, whereas at higher frequencies the total DOS is dominated by the Fe states. This behaviour is expected as Fe atoms are lighter than Zr atoms. Similar behaviour has been found for the C16-FeZr2 intermetallic phase. The partial DOS of Zr atoms in C15-Fe2Zr intermetallic phase beyond 8 THz is negligible compared to DOS of Fe atoms, whereas in C16-FeZr2 intermetallic phase the partial DOS of Zr atoms becomes negligible beyond 6 THz.

3.3. Thermodynamic properties Our finite displacement (FD) calculated Helmholtz free energy of both phases is converged within 2% of that of DFPT calculated free energy at equilibrium volumes (Table 3). Therefore, further calculations of free energy at different volumes (either compressional or dilational) under the quasi-harmonic approximation (QHA) was performed using 2  2  2 supercells and FD phonon calculation method. The free energy of the intermetallics are calculated using

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615

Fig. 2. Phonon dispersion curve along with DOS of (a) C15-Fe2Zr phase and (b) C16-FeZr2 phase. The red and green curves in (b) are the experimental (red curve) and calculated values using lattice dynamics (green curve), respectively, taken from Ref. [28]. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

equation (1). The Helmholtz free energies of C15-Fe2Zr and C16FeZr2 phases have been calculated for 11 different values of volumes. In Fig. 4, the free energies of both the intermetallics as a function of volume at temperatures ranging between 0 and 1000 K in steps of 100 K have been plotted. The solid lines are the fitted curves at respective temperature and the filled round symbol in each curve corresponds to the minimum energy and simultaneously the equilibrium volume. The red lines are just guide to the eye. The vibrational free energies of C15-Fe2Zr and C16-FeZr2 phases under the QHA have been calculated using relation (2) and plotted in Fig. 5(a). The internal energies of both the phases as a function of temperature are plotted in Fig. 5(b). The free energy at 0 K is simply the internal energy of the system which in turn is the zero point energy of the system. The free energy of both the intermetallics decreases as the temperature increases whereas the internal energy increases with temperature. As the internal energy of both intermetallics is equal beyond 200 K, the difference in their free energies is due to difference in their vibrational entropies. The zero point energy of C15-Fe2Zr phase is 3.38 kJ/mol and that of the C16FeZr2 phase is 2.69 kJ/mol. The free energy becomes negative at ~270 K for the C15-Fe2Zr phase and at ~215 K for the C16-FeZr2

Table 3 Calculated Helmholtz free energies of C15-Fe2Zr and C16-FeZr2 phases using finite displacement method (FD) and density functional perturbation theory (DFPT) at different temperatures with different supercells sizes. Supercell size

222

System

Fe2Zr FeZr2

333

Fe2Zr FeZr2

Method

FD DFPT FD DFPT FD FD

Free energy (kJ/mole) 0K

1000 K

3.3762 3.3698 2.6861 2.7005 3.3767 2.6891

33.7899 33.8499 39.7365 39.5677 33.7908 39.7186

phase. The C16-FeZr2 phase becomes dynamically more stable than the C15-Fe2Zr phase as the free energy of the former becomes more negative. The vibrational entropy of the intermetallics as calculated using relation (3), are plotted in Fig. 6(a). The vibrational entropy of both the intermetallics increases with temperature. The entropy of C16-FeZr2 phase is higher than that of C15-Fe2Zr phase and the difference increases with temperature. The bulk modulus as a function of temperature for both C15Fe2Zr and C16-FeZr2 phases as calculated using equation (7), is

Fig. 3. Partial Phonon DOS of (a) C15-Fe2Zr phase and (b) C16-FeZr2 phase.

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Fig. 4. Helmholtz free energy of (a) C15-Fe2Zr phase and (b) C16-FeZr2 phase as a function of primitive cell volume at different temperatures ranging between 0 and 1000 K. The filled round red circles represent minima of respective curves. The volume of the primitive cells is calculated using the relation ða0 :ðb0  c 0 ÞÞ, where a0 , b0 and c0 are the lattice vectors of the primitive cell as defined in the supplementary file. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 5. (a) Helmholtz free energy and (b) Internal energy of C15-Fe2Zr and C16-FeZr2 phases as a function of temperature.

plotted in Fig. 6(b). The bulk modulus of both the intermetallics decreases with temperature. The bulk moduli of C15-Fe2Zr and C16-FeZr2 phases at 0 K is 145.45 GPa and 120.88 GPa, respectively, which are in good agreement with our earlier results [26] 149.24 GPa and 121.98 GPa. As there are no experimental values available to compare our QHA calculated B0(T), we compare these values with experimentally determined B0 versus temperature data for the bcc Fe [48], hcp Zr [51], C16 structured NiZr2 [50] and DFT calculated values under quasi-harmonic Debye model of C16 structured CuZr2 [49]. It can be seen from Fig. 6(b) that the bulk modulus of C15-Fe2Zr phase is less than that for bcc Fe and higher than that for hcp Zr. The C15-Fe2Zr phase contains 75% Fe and 25% Zr atoms and as shown in Ref. [26], the bonding between Fe and Zr atoms is predominantly metallic in nature, hence the bulk modulus of C15-Fe2Zr is expected to be a weighted sum of the bulk moduli of Fe and Zr. The bulk modulus of C16-FeZr2 phase is 8e9% higher compared to the bulk modulus of C16 structured NiZr2 and CuZr2 phases though the rate of decrease of bulk modulus of C16-FeZr2

phase is same as that of NiZr2 and CuZr2 phases. It has been noted that the bulk moduli of C15-Fe2Zr phase and C16-FeZr2 phase are weighted sums of the bulk moduli of Fe and Zr at their respective compositions. Further, the bulk modulus of C15-Fe2Zr phase is less sensitive to the temperature than that of C16-FeZr2 phase. For both phases, the bulk moduli exhibit a small variation as a function of temperature in the temperature range between 0 and 1000 K, which is a favorable feature for their use as wasteforms. Also, as the bulk modulus of C15-Fe2Zr phase is less sensitive to the temperature than that of the C16-FeZr2 phase, the formation of C15-Fe2Zr is more desirable. The linear thermal expansion coefficient (a) of both intermetallics as calculated from equation (6), is plotted in Fig. 7 (a). Initially the linear thermal expansion coefficient of both the intermetallics increases sharply and beyond 300 K, then the rate of increase slows down tending towards a constant value. The plot also shows that beyond 600 K, the increase in a for the C16-FeZr2 phase is more significant than that for C15-Fe2Zr phase. At 300 K,

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Fig. 6. (a)Entropy and (b)Bulk modulus of C15-Fe2Zr and C16-FeZr2 phases as a function of temperature. The bulk modulus as a function of temperature of bcc Fe, C16 structured CuZr2, C16 structured NiZr2 and hcp Zr has been also plotted in 5(b) and taken from Refs. [48e51], respectively.

Fig. 7. (a)Linear thermal expansion coefficients and (b) Specific heat at constant volume of Fe2Zr and FeZr2 phases as a function of temperature. The specific heats at constant volume of C15 Ni2Zr [52] and C16 CuZr2 [49] are also plotted in 6(b).

the linear thermal expansion coefficient is 9.77  106 K1 for the C15-Fe2Zr phase whereas for the C16-FeZr2 phase it is 10.74  106 K1. Therefore, the C15-Fe2Zr structure will generate less thermal stress than the C16-FeZr2 structure. A material with low thermal expansion coefficient is desirable as a metallic wasteform, hence the formation of C15-Fe2Zr is preferable as compared to the C16-FeZr2 phase. The specific heat capacity at constant volume of the intermetallics as calculated using relation (4), is plotted in Fig. 7 (b). The curves show that at low temperatures, the constant volume specific heat (CV ) of both the intermetallics increases rapidly and beyond 400 K, it becomes almost constant (~25 J/mole/K) which value is equal to limit of the specific heat of these phases. Since there are no experimental or theoretical results of CV available in the literature for C15-Fe2Zr and C16-FeZr2 phases, we have compared our results with CV of C15 structured Ni2Zr and C16 structured CuZr2. It can be seen from Fig. 7 (b) that the CV of C15Fe2Zr phase is the same as that of the C15 structured Ni2Zr. The CV of C16-FeZr2 phase is slightly smaller than that of C16 structured CuZr2 below 500 K, but at higher temperature both have the same value of CV . The plot also shows that at high temperature (T > 400 K), The CV of C15-Fe2Zr and C16-FeZr2 phases are almost

equal. The specific heat at constant pressure (CP ) of C15-Fe2Zr and C16FeZr2 phases are plotted in Fig. 8(a) and (b), respectively. As can be seen that the behaviour of CP with temperature is same as the behaviour of CV . In Fig. 8(a) we have also compared our calculated CP of C15-Fe2Zr phase with DFT calculated CP of C15 structured Ni2Zr and experimentally measured C14 structured Fe2Zr. In Fig. 8(b) we have compared our calculated CP of C16-FeZr2 phase with DFT calculated as well as experimentally measured CP of hcp Zr. Our calculated values of CP of C15-Fe2Zr phase is in good agreement with C15 structured Ni2Zr. The CP of C14-Fe2Zr phase is higher than that of C15-Fe2Zr phase. The hump in the CP curve of C14-Fe2Zr phase is due to the magnetic transformation that occurs at 584.2 K. It is emphasized here that our calculated CP does not include the electronic contribution. Our calculated value of CP of C16-FeZr2 phase is in good agreement with DFT calculated CP of hcp Zr. The difference in CP values from the experimentally measured CP is due to the contributions from electronic heat capacity which we have not considered in our calculations. The CP initially increases rapidly with temperature and at high temperature becomes constant to ~25 J/mole/K.

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Fig. 8. Specific heat at constant pressure of (a) Fe2Zr and (b) FeZr2 phases as a function of temperature. The specific heat at constant pressure (CP ) of C14 Fe2Zr [27], C15 Ni2Zr [52] are plotted in 7(a) and CP of hcp Zr [53] has been plotted in 7(b).

3.4. Thermal conductivity The thermal conductivity of a given material has contributions from both electrons and phonons. In this paper, we have calculated only the phononic contribution to the thermal conductivity for both the intermetallic phases using relation (8). Hence the calculated thermal conductivity in this study are the lower limit to the thermal conductivity of these intermetallic phases. The calculated thermal conductivity due to phonons in the temperature range 0e1000 K has been plotted in Fig. 9. The variation of thermal conductivity of both the intermetallic phases show similar trends. Fig. 9 shows that at low temperature (T < 30 K), the thermal conductivity increases with temperature, while at T > 30 K, the thermal conductivity decreases with temperature. At high T (T > Debye Temperature (qD )), the thermal conductivity of both intermetallic phases show a variation of 1/T with temperature. The thermal conductivity of C15-Fe2Zr and C16-FeZr2 phases above the qD can be fitted to 0:706 þ 5213:86 and 0:779 þ 2115:75 , respectively. It can T T be seen in Fig. 9 that the thermal conductivity of C15-Fe2Zr phase is higher than that of C16-FeZr2 phase throughout the temperature range. In Table 2, we have already shown that the Debye

Fig. 9. Thermal conductivity due to phonons for C15-Fe2Zr and C16-FeZr2 phases.

temperature of C15-Fe2Zr phase is higher than that of C16-FeZr2 phase, which implies that the thermal conductivity of C15-Fe2Zr is expected to be higher compared to that of C16-FeZr2 phase. Therefore, our DFT calculated lattice thermal conductivity values of these phases are consistent with those predicted from our Debye temperature values. The high specific heat of both the phases makes them suitable to be a material for metallic waste immobilization but from the point of view of their thermal conductivities, the formation of C15-Fe2Zr phase is more preferable in the wasteform as its thermal conductivity is higher than that of the C16-FeZr2 phase. 4. Conclusions We calculated the lattice dynamical properties of C15-Fe2Zr and C16-FeZr2 phases which are invariably found in Fe-Zr alloys; the latter being a promising material for immobilization of metallic waste originating from the nuclear reactors. The fact that the radionuclides in the metallic waste are always found to be embedded in these phases makes evaluation of their thermal and elastic properties, viz., thermodynamic and lattice dynamical stability, specific heat and thermal expansion and conductivity, etc., highly important. It was shown that both the intermetallics are thermodynamically and lattice dynamically stable up to 1000 K. The Helmholtz free energy of C15-Fe2Zr and C16-FeZr2 becomes negative at ~270 K and ~215 K, respectively, and beyond this temperature the free energy of both the intermetallics decreases further with the increase of the temperature. The internal energy and entropy of both the intermetallics increase with temperature and beyond 200 K their internal energies are exactly equal. Our calculated phonon DOS of FeZr2 is in good agreement with the experimental values obtained by Mittal et al. [28]. A comparison of their bulk modulus indicates that C15-Fe2Zr phase is less compressible than C16-FeZr2 phase and the bulk moduli decrease with temperature. Furthermore, the calculated G/B ratios indicated that the C16-FeZr2 phase is more ductile than C15-Fe2Zr phase. A comparison of linear thermal expansion coefficients of both the phases showed that the thermal stresses generated in C16-FeZr2 phase will be slightly higher that those generated in C15-Fe2Zr phase. The calculated specific heats, CP and CV , of both intermetallics are almost equal (~25 J/mole/K) above 450 K; while at lower temperatures, the CP and CV of FeZr2 phase is higher than that of Fe2Zr phase. Moreover, our calculated results showed higher thermal

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