Thermodynamic and physical properties of Zr3Fe and - DiVA

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Thermodynamic and physical properties of Zr3Fe and - DiVA

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Thermodynamic and physical properties of Zr3Fe and ZrFe2 intermetallic compounds
B. O. Mukhamedov, I Saenko, A. V Ponomareva, M. J. Kriegel, A. Chugreev, A. Udovsky, O. Fabrichnaya and Igor Abrikosov
The self-archived postprint version of this journal article is available at Linköping University Institutional Repository (DiVA):
N.B.: When citing this work, cite the original publication.
Mukhamedov, B. O., Saenko, I, Ponomareva, A. V, Kriegel, M. J., Chugreev, A., Udovsky, A., Fabrichnaya, O., Abrikosov, I., (2019), Thermodynamic and physical properties of Zr3Fe and ZrFe2 intermetallic compounds, Intermetallics (Barking), 109, 189-196.
Original publication available at: Copyright: Elsevier

Thermodynamic and physical properties of Zr3Fe and ZrFe2 intermetallic compounds
B.O. Mukhamedov1,*, I. Saenko2,3, A.V. Ponomareva1, M.J. Kriegel2, A. Chugreev4, A. Udovsky3,5, O. Fabrichnaya2, I.A. Abrikosov6
1 National University of Science and Technology ‘MISIS’, Moscow, 119049, Russia 2 Freiberg University of Mining and Technology, Freiberg, Sachsen, 09599, Germany 3 Baikov Institute of Metallurgy and Material Science of the Russian Academy of Sciences, Moscow, 119991, Russia 4 IFUM - Institute of Forming Technology and Machines, Leibniz Universitaet Hannover, An der Universitaet 2, Garbsen 30823, Germany 5 National Research Nuclear University MEPhI, Moscow, 115409, Russia 6 Linköping University, Linköping, 581 83, Sweden
*Corresponding author: B.O. Mukhamedov, Materials Modeling and Development Laboratory, National University of Science and Technology ‘MISIS’, Moscow, Russia, 119049
E-mail address: [email protected]
Experimental differential scanning calorimetry measurements and ab-initio simulations were carried out to define the heat capacities of Zr3Fe and C15-ZrFe2 compounds from 0 K up to their maximum stability temperatures. Experimental measurements of heat capacity of each compound were performed for the first time in wide range of temperatures. Density functional theory and quasi-harmonic approximation (QHA) were employed to calculate the free energy of the studied systems as a function of volume and temperature. A good agreement was observed between theoretical and experimental heat capacities within validity range of the QHA. This makes it possible to combine theoretical and experimental data to determine the standard entropies of intermetallic compounds.
1 Introduction
Zr-based alloys, so-called zircaloys, containing small amount of Fe and other alloying elements are widely used in nuclear industry due to their low absorption cross-section of thermal neutrons, high corrosion resistance and hardness [1–3]. Moreover, Zr-Fe intermetallic compounds in zircaloys are known to improve specific mechanical properties. Because of this, the Zr-Fe system plays an important role in modern steel design. For example, in the manufacturing of composite materials consisting of TRIP (transformation-induced plasticity) matrix and reinforcing ZrO2 based ceramic, the phase relations in the Zr-Fe-O system are essential for the stability of ceramic particles [4]. In fact, both TRIP matrix and ceramic particles may undergo martensitic transformation during compression, which thereby improves the mechanical properties of composite material, such as strength and energy absorption during compression [5].
Therefore, Zr-Fe system is becoming the subject of intense research, and the detailed knowledge on its thermodynamic properties and phase relations is essential for the modern industry of composite and structural steels. A systematic experimental study on binary Zr-Fe system was performed in Ref. [2] for the whole range of compositions. The authors reported that there were four stable intermetallic compounds in the phase diagram: two polymorphic modifications of ZrFe2 phase, Zr3Fe and high-temperature Zr2Fe. Cubic ZrFe2 C15-Laves phase, which is stable

up to 1503 K, transforms into hexagonal C36 polymorph at 1563 K [2]. Similar result was observed in Ref. [6], where the authors found the hexagonal Laves phase in Zr22.5Fe77.5 alloy after annealing at 1523 K. Intermetallic Zr3Fe phase was found to be orthorhombic with Re3Btype symmetry [2]. The Zr3Fe phase is formed from peritectoid reaction Zr2Fe + β-Zr ⇄ Zr3Fe; and the reported temperature of this reaction varied in several measurements: values of 1124 K [2], 1158 K [7], and 1213 K [8] were reported. Another intermetallic compound Zr2Fe is a constituent of above mentioned peritectoid reaction; it is a high-temperature phase with Ti2Nitype crystalline structure.
According to the several thermodynamic assessments [9,10], there is limited available information on thermodynamic properties of intermetallic compounds in binary Zr-Fe system. In Ref. [11] the enthalpy of formation Hf of ZrFe2 at 298 K was determined using direct reaction calorimetry. Enthalpy of formation of ZrFe2 was also derived from indirect measurements by solution calorimetry in Refs. [12], [13] and levitation alloying calorimetry in Ref. [14]. The values of Hf from Refs. [11] and [14] are quite consistent with each other, while the values of Hf in Refs. [12] and [13] are more positive. Several ab-initio calculations predicted fairly consistent results on formation enthalpy of Zr-Fe intermetallics: for C15-ZrFe2 phase the Density Functional Theory (DFT) calculations gave -0.280 eV/atom (-27.02 kJ/mol) [15] and -0.290 eV/atom (-27.98 kJ/mol) [16]. These results are in a good agreement with experimental data from Refs. [11] and [14]. For Zr3Fe phase the theoretical enthalpy of formation ∆Hf was found to be equal to -0.140 eV/atom (-13.51 kJ/mol) [15] and -0.150 eV/atom (-14.47 kJ/mol) [16], however, there were no available experimental results. Regarding the heat capacity, in Ref. [17] the experimental measurements were performed for C15-ZrFe2 phase in the range of temperatures between 313 K and 653 K. At the same time, we are not aware of any available experimental data on the heat capacity of Zr3Fe intermetallic phase. Moreover, it is worth noting that there is no available information on standard entropies for these intermetallic phases.
Thermodynamic modeling of Zr-Fe system without information on the thermodynamic properties of the phases becomes unreliable. At least, thermodynamics of the intermetallic compounds, that are stable in the wide temperature rage, like Zr3Fe and ZrFe2, should be defined for further improvement of the thermodynamic description of the Zr–Fe system. In this respect, the topic of the next generation of databases of materials properties created based on CALPHAD approach in combination with reasonable description of thermodynamic properties starting from zero temperature attracts increasing attention. However, the experimental measurements of thermodynamic properties of materials at such low temperatures are rather difficult and timeconsuming; and most of all they are very expensive. On the other hand, the predictive power of state-of-the-art first-principles simulations has greatly improved, and they can be used efficiently to describe the thermodynamic properties of materials [18]. For example, the DFT approach in combination with quasi-harmonic approximation allows one to simulate the materials thermodynamic and thermal properties, such as heat capacity, entropy, etc. However, the QHA has its limitation, since it does not account for the anharmonic effects, which might be significant at high temperature [19]. Thus, to define the heat capacities of studied Zr3Fe and C15-ZrFe2 compounds we apply DFT+QHA method for the low-temperature regions and experimental measurements for the high temperatures. The reliability of DFT+QHA results and the boarders between low- and high-temperature regions are defined from the comparison between theoretical

and experimental data. In addition to the heat capacity, we have determined the formation enthalpies and standard entropies of both compounds.

2 Methods

2.1 Calculation of thermodynamic properties

The total free energy Gtot(V,T) of a system as a sum of electronic, vibrational and magnetic terms:
Gtot (V ,T ) = Eel (V ) − TSel (V ,T ) + Gvib (V ,T ) + Gmagn (V ,T ) (1)
considering each term as a function of volume V and temperature T. In Eq. 1, the terms [Eel(V) – TSel], Gvib(V,T) and Gmagn(V,T) denote electronic, vibrational, and magnetic contributions to the total free energy, respectively. Zr3Fe phase is non-magnetic; therefore, the last term in the righthand side of Eq. 1 can be omitted in this case.

The electronic entropy can be estimated as:

Sel = −k B  n( )[ f ln f + (1 − f ) ln(1 − f )]d (2)
− 

Here, n(ε) is the electronic density of states, and f =f(ε, εF, T) is the Fermi distribution function.

Vibrational (or phonon) contribution Gvib(V, T) to the free energy was simulated within quasi-

harmonic approximation (QHA) where the phonon frequencies ωq were derived from interatomic

Hellmann-Feynman forces induced by finite atomic displacements. In QHA, the phonon

frequencies are volume-dependent parameters, hence vibrational energy Gvib becomes volume-

dependent, which makes it possible to account for the thermal expansion effects in the lattice.

The vibrational energy as function of volume and temperature can be defined as:

 q (V )

 q (V ) 

Gvib (V ,T ) =  q 


+ kBT ln1 − exp −


 


where ωq is the phonon frequency with wave vector q, and kB is the Boltzmann constant.

At low temperatures, which are of primary interest for theoretical simulations in this study, the

dominant magnetic excitations are spin waves. A contribution from the spin-waves excitations to

the total free energy of ZrFe2 phase was taken into consideration within Heisenberg model with

nearest neighbors interactions. According to the model, the magnon-dispersion relations can be

calculated as:

k = 4JS[1 − cos(ka)] (4)

where J denotes an effective (pair) exchange interaction between the nearest neighbors; S is the
on-site spin value of atom; k is a magnon frequency with wave vector k; and a is a lattice
parameter. Computation of the J parameter has been performed using the Exact Muffin-Tin Orbitals (EMTO) method [20], where the exchange parameter J can be found analytically from the change in energy of embedding spins induced by rotation of them an infinitely small angle [21–23].


At the low temperatures, i.e. in the long wavelength limit k → 0 , the magnon dispersion has a parabolic form: k = Dstiff  k 2 , where Dstiff = 2SJa 2 denotes the spin-wave stiffness at T = 0 K.

According to Kittel [24], at the low temperatures the internal energy of magnons Emag can be

expressed as: where x =  k



 ( ) E

(V ,T ) =


 


2 

X max

x 2 dx (5)


2 2

 


 


ex −1

. Since the effective exchange interaction J is volume-dependent parameter, the

spin-wave frequency also becomes volume-dependent k = k (V ) , which allows one to

implicitly calculate the magnon energy Emag as a function of volume V.

The evaluation of the spin-wave free energy Gmag(V,T) can be done numerically by means of the internal energy Emag(V,T) via:
T 1  Emag (V , T ) 
Gmag (V , T ) = Emag (V ,0) − T 0 T  T P dT (6)

By knowing the total free energy Gtot (see Eq. 1) as a function of volume V and temperature T it is possible to find the minimum free energy and corresponding volume Vmin at each temperature. Thus, having the information on the ground state volume Vmin as a function of the temperature we can determine the volumetric thermal expansion coefficient as:
V = V1 ddVTmin (7) 0
where V0 denotes the ground state volume at T = 0 K.

Bulk modulus as a function of the temperature can be expressed as:

B(T ) = V




 


min  V 2 T

Specific heat capacity at constant volume, i.e. isochoric heat capacity, can be defined by formula

given below:

 2G  C = −T  


V  T 2 V 0

Further, using the parameters obtained from Eqs. (7) – (10) one can determine the heat capacity

at constant pressure, i.e. isobaric specific heat capacity, as:

CP = CV + V2 BTVmT (10)

where Vm denotes the molar volume of the unit cell.

An electronic contribution to the heat capacity can be expressed from electronic entropy as:




T 


 


el  T P

The spin-waves contribution to the heat capacity can be defined as the temperature derivative of

Emag (V ,T ) :

 Emag 

Cmag =  


 



2.2 Calculations details

Based on the previous experimental data [2], the Zr3Fe and ZrFe2 phases were simulated as orthorhombic and cubic C15 crystals, respectively, with 16 and 6 atoms per unit cell. In Fig.1, we depict the unit cells of these intermetallic phases. Thermodynamic properties of Zr3Fe and ZrFe2 phases were calculated using projector augmented wave (PAW) [25] potentials implemented in Vienna Ab-initio Simulation Package (VASP) [26]. The exchange-correlation effects were treated using the generalized gradient approximation [27]. Magnetic properties were accounted for within the collinear picture. The magnitude and orientations of the collinear local moments were calculated self-consistently. The cutoff energies for plane waves were set to 700 and 500 eV for Zr3Fe and ZrFe2 compounds, correspondingly. The integration over the irreducible part of Brillouin zone has been carried out using Monkhorst-Pack method [28] on the grids of 18×6×6 k-points for Zr3Fe and 8×8×8 for ZrFe2. The convergence criterion for electronic subsystem was chosen to be equal to 10-3 eV/atom for subsequent iterations. The relaxation of atomic positions was realized by calculation of Hellman-Feynman forces [29,30] and stress tensor with using the conjugated gradient method. Relaxation was stopped when the forces became on the order of 10-3 eV/Ǻ.

Figure 1 – Unit cells of orthorhombic Zr3Fe (a) and cubic C15-ZrFe2 (b) intermetallic compounds.
Regarding to the phonon calculations, we applied supercells approach where Zr3Fe and ZrFe2 crystals were simulated as 3×2×2 (192 atoms) and 2×2×2 (48 atoms) supercells, respectively. To calculate the interatomic forces in supercells we set the values of atomic displacements to be equal to 0.01 Ǻ. Postprocessing calculations of phonons was realized within Phonopy code [31]. To check the accuracy of phonon calculations we considered the larger in size supercells, 4×2×2 for Zr3Fe and 3×3×3 for ZrFe2, and performed convergence test of heat capacities at the ground state volume. The test revealed that difference in heat capacities CV between larger and smaller supercells is about ~ 1.5 % at low-temperatures between 0 and 5 K; and this difference gradually

vanishes for the higher temperatures. Therefore, further phonon calculations were realized only for smaller in size supercells, e.i. 3×2×2 for Zr3Fe and 2×2×2 for ZrFe2 phase. Total free energy Gtot from Eq. (1) was calculated for the 7 volume points in the range between -6 and +6 % with respect to the ground state volume at zero temperature. Computation of effective exchange parameter J in cubic C15 phase was performed using the EMTO method [20]. Simulations were carried out for a basis set including the valence s, p, d and f orbitals and by means of the frozen-core approximation, i.e. the core states were kept fixed. The integration over the irreducible part of the Brillouin zone has been performed using a grid of 6×6×6 k-points. The energy integration has been carried out in the complex plane using a semielliptic contour comprising 12 energy points. The convergence of the energy with respect to the calculation parameters has been set to 10-4 eV.
2.3. Experimental methods
2.3.1. Sample preparation
Two binary Fe-Zr alloys with the nominal compositions of ZrFe2 and Zr3Fe were prepared by arc-melting method in the Ar-atmosphere. Pieces of Fe (99.99%, Alfa Aesar) and slugs of Zr (99.5%, Alfa Aesar) were weighed in accordance to the nominal compositions. In order to achieve a good homogeneity of the chemical composition in the bulks, the samples were turnedover and re-melted three times. The samples were sealed in the quartz tubes with the reduced Ar atmosphere. The pressure of the Ar in the quartz tubes was chosen in order to reach 1 atm. at the homogenization temperature. Afterwards, the samples were homogenized during 3 hours at 873 K for Zr3Fe and 1123 K for ZrFe2.
2.3.2. Sample treatment and characterization
The samples were analysed by X-ray diffraction (XRD) and scanning electron microscopy combined with an energy dispersive X-ray spectrometry (SEM/EDX). Phase assemblages of specimens after homogenization annealing have been identified by X-ray powder diffraction (XRD) using the URD63 diffractometer (Seifert, FPM, Freiberg, Germany) equipped with the graphite monochromator and the CuK radiation ( = 1.5418 Å). The goniometer of the diffractometer had the Bragg-Brentano geometry. The Rietveld refinement was applied for the characterisation of all measured diffraction patterns using Maud software [28]. We have investigated the sample microstructures after homogenization annealing using scanning electron microscopy combined with dispersive X-ray spectrometry (SEM/EDX; Leo1530, Carl Zeiss/ Bruker AXS Mikroanalysis GmbH). Chemical compositions of samples and present phases have been determined using a signal from EDX detector with an accuracy of  2 at. %. Based on the XRD and SEM/EDX results the lattice parameters of Zr3Fe were found to be a = 3.322 Å, b = 10.973 Å, c = 8.822 Å; and lattice parameter of C15-ZrFe2 – a = 7.075 Å. XRD patterns are presented in the Fig.2. Measured lattice parameters exhibit a good agreement with the literature data [33–36]; and correspond to the stoichiometric intermetallic compounds. Microstructural analysis confirms the XRD results. Chemical composition of the samples measured by EDX method matches to the initial chemical composition within the error of the

measurement. Chemical composition analysis shows that Zr3Fe samples have 77 at.% of Zr; and ZrFe2 samples – 34 at.% of Zr. In both cases, we observed the deviations toward Zr side. However, the deviation was in the range of the reliable accuracy.
Figure 2 – XRD results of Zr3Fe (a) and C15-ZrFe2 (b) intermetallic compounds.
2.3.3. Thermal expansion measurements
In order to verify thermal expansion coefficients calculated in the frame of the QHA, additional experimental measurements of the thermal expansion of the investigated compounds have been performed. The thermal expansion measurements were carried out in the temperature ranges: 353–1023 K for Zr3Fe and 373–1523 K for ZrFe2. In total, two different dilatometric devices were involved, namely DIL 802 (Netzsch GmbH, thermocouple type S, Ar flow, 10 K/min) and DIL 805A/D/T (TA Instruments Inc. formerly Bähr-Thermoanalyse GmbH, thermocouple type K, vacuum atmosphere, heating rate 300 K/min). Temperature correction of DIL 802 was performed using temperatures of magnetic and solid transformations in Fe (99.95% Alfa Aesar) and Co (99.995% Alfa Aesar). Certified reference material of sapphire (Bähr-Thermoanalyse GmbH) was used for calibration of linear temperature expansion. In the case of DIL 805A/D/T system, no additional calibration is required due to the specific design of the experimental procedure, since the thermocouple is placed directly onto the sample via spot welding. The measurements of two different samples were repeated two times with maximal uncertainty 6%. In order to avoid possible temperature gradients in the sample, we used quenching mode of DIL805A/D/T with hollow quartz push rods. Due to the principle of induction heating, only the sample is heated during the measurement.
2.3.4. Calorimetric measurements
For heat capacity measurements we used the classical three-step method (continuous method) with a constant heating rate [37]. The measurements were performed in the wide range of temperatures using two different devices. In the temperature range from 235 K to 675 K we have used DSC 8000 device (Perkin Elmer, Pt/Rh crucible, He/Ar flow, heating rate 10 K/min). The measurements in the temperature range from 235 K to 675 K were divided into smaller intervals 100–150 K. The heat capacity measurements in the temperature range from 623 K to 1220 K were performed in the one step using the device DSC Pegasus 404C (NETZSCH, Pt/Rh crucible,

Ar flow, heating rate 10 K/min). Calibration was performed using the certified standard materials depending on the reliable temperature range of their heat capacity: copper standard was used in the temperature range from 100 to 320 K; molybdenum – 300 to 673 K and platinum – 573 to 1473 K. The measurements of two different samples were repeated two times with maximal uncertainty value of 3 %. It should be mentioned that at high temperatures the CP measurement with DSC equipment becomes less reliable due to increase of heat radiation which decreases a registered signal. This effect was considered during interpretation of the experimental data.
3 Results
3.1. Zero temperature electronic structure calculations
Let us start with calculations of thermodynamic properties of Zr3Fe and ZrFe2 phases at zero temperature. Theoretical data on lattice parameters, magnetic moments and formation energies for both phases are given in Table 1. Here, for comparison we also show the available experimental results at room temperature [34,35]. For orthorhombic Zr3Fe phase, the calculated lattice parameters a and b are slightly underestimated, while parameter c is slightly overestimated compared to experimental data [34]. Simultaneously, the theoretical volume of Zr3Fe is in excellent agreement with experiment. Zr3Fe is found to be non-magnetic at zero temperature. For cubic C15-ZrFe2 phase, the calculated lattice parameter a is 7.065 Å, while experiment gives a = 7.061 Å [35]. This compound is found to be ferrimagnetic with average magnetic moment 1.12 μB/atom, which is in good agreement with experimental result 1.04 μB/atom measured at the liquid helium temperature [38]. The calculated magnetic moments on Fe atoms with the average value of 1.924 μB are parallel to each other, but antiparallel to magnetic moments induced on Zr atoms with the values of -0.501 μB.
Formation enthalpies of the compounds are also listed in Tab. 1. They were calculated with respect to pure elements: non-magnetic hcp Zr and ferromagnetic bcc Fe. Our theoretical data on formation enthalpy -0.121 eV/at (-11.67 kJ/mol) and -0.288 eV/at (-27.79 kJ/mol) for Zr3Fe and ZrFe2 phases, respectively, are in good agreement with previous DTF calculations [15,16] and experimental results for ZrFe2 [11–14].

Table1 – Ground state properties of Zr3Fe and ZrFe2 compounds at T = 0 K and available experimental data.




Exp. [34]


Exp. [35]

Lattice parameter, Å

a = 3.300 b = 10.900 c = 8.945

a = 3.321 b = 10.966 c = 8.825

a = 7.065

a = 7.061

Volume, Å3/atom





Atomic basis coordinates

Fe1 [0.000 0.730 0.250] Zr1 [0.000 0.138 0.060] Zr2 [0.000 0.421 0.250]

Fe1 [0.000 0.744 0.250] Zr1 [0.000 0.135 0.062] Zr2 [0.000 0.426 0.250]

Fe1 [0.625 0.625 0.625] Zr1 [0.000 0.000 0.000]

Fe1 [0.625 0.625 0.625] Zr1 [0.000 0.000 0.000]

Magnetic moments, μB/atom



Total: 1.12 Fe: 1.924 Zr: -0.501

Total: 1.04 [38] -



energy, eV/at

-0.121 (-11.62)



-0.288 (-27.79)

-0.308 (-29.72) [11] -0.256 (-24.70) [12] -0.228 (-22.00) [13] -0.331 (-31.93) [14]

3.2. Finite-temperature effects

Because of the limitations of the experimental techniques to obtain data on the thermodynamic properties of the studied materials at low temperature, we calculate their thermodynamic properties in the temperature range from 0 K up to room temperature. We have accounted for the finite-temperature effects by determining the contributions of electronic entropy Sel, magnetic excitations Gmag and vibrational energy Gvib to the total free energy Gtot of the system. The electronic entropy Sel was estimated using Fermi distribution which depends on Fermi energy and temperature (see Eq. 2).

3.2.1 Magnetic excitations in ZrFe2 phase

According to zero-temperature calculations, the ZrFe2 phase is ferrimagnetic with opposite orientation of local magnetic moments on Zr and Fe atoms; and there are two types of magnetic moments in this phase. However, since the pure Zr atom on its own is non-magnetic, we assume that in ZrFe2 phase the magnetic moment on Zr is induced by iron atoms in the local environment. Thus, for calculations of the magnon-dispersion relations we considered only the Fe-Fe pair exchange interactions. The plot of magnon-dispersion relations as a function of wave vector k is shown in Fig. 3. Here, the value of effective exchange parameter J between Fe-Fe pair in the nearest neighborhood is


NN Fe−


= 3.747 meV. For the next nearest Fe-Fe pairs the exchange interaction becomes

comparable too small


NNN Fe − Fe

















Therefore, in the long wavelength limit k → 0 , the theoretical value of spin-waves stiffness for ZrFe2 phase was found to be Dstiff ≈ 750 meV·Å2.

A simple estimation of the magnetic transition temperature TC can be performed using pair

 exchange interactions within the mean-field approximation, T MFA = 2 1  J  m m ,


3 k B j Fe−Fe Fe Fe

where summation runs over the first coordination shell and on-site magnetic moment is denoted

as mFe. The obtained value for the Curie temperature TC is 772 K. This result is overestimated as compared to the experimental value of 585 K [17], however, such a deviation is typical for the first-principles calculations of TC.

PropertiesTemperatureMeasurementsHeat CapacityTemperatures