Elsevier

Calphad

Volume 44, March 2014, Pages 62-69
Calphad

Laves phases in the V–Zr system below room temperature: Stability analysis using ab initio results and phase diagram

https://doi.org/10.1016/j.calphad.2013.08.003Get rights and content

Abstract

V–Zr is a well known system where a phase transformation from rhombohedral V2Zr structure to cubic C15 Laves phase occurs during heating at about 115 K. Here we provide a thermodynamic description of this phenomenon, supported by ab initio calculations. We utilize our new method of extension of the Scientific Group Thermodata Europe (SGTE) unary thermodynamic database to zero Kelvin and demonstrate that it may be applied also to complicated intermetallic phases. To keep our analysis on equal footing with previous results for higher temperatures, data regarding a recent thermodynamic assessment of the V–Zr system (valid for temperatures above 298.15 K) were reassessed. With the help of ab initio approach, we demonstrate that the ZrV2 rhombohedral phase is not stable at 0 K and transforms to C15 Laves phase. The stability of C15 Laves phase is confirmed by analysis of elastic stability criteria and phonon spectra calculations.

Introduction

Topologically close-packed intermetallic compounds such as Laves phases are promising candidates for high-temperature materials due to their interesting mechanical and corrosion resistance features [1] which can be influenced by their possible polymorphism and peculiar magnetic and electrical properties [2], [3]. The solubility of hydrogen in Laves phases [4] and superconductivity [5] are also important properties of these compounds. Electronic structure, elastic properties and total energies of C15 Laves phases of V2M type (M=Zr, Hf, or Ta) were studied with the help of ab initio calculations in [6], [7], [8] and the relations between electronic structure, elastic moduli and stability were analyzed. It was found that the V2Zr C15 Laves phase compound, which is cubic at room temperature, undergoes a structural transformation to a non-Laves rhombohedral phase at low temperatures [9], [10], [11]. The transformation temperature was determined to be 116.7 K by Moncton [11] and 110 K by Rapp [10], who also reported the corresponding latent heat of transformation as 31 J/mol. In general, it is very difficult to explore low-temperature phase diagrams describing such transformations both experimentally and theoretically and it is great challenge to perform thermodynamic modeling in this region. It is the reason for doing this work.

The first step towards the low-temperature predictions of thermodynamic functions was done in works [12], [13], where equations for Gibbs energy valid at low temperatures and necessary values of Debye temperatures for many elements are given. In our recent work [14], the expressions of Gibbs energies of 52 elements were extended to zero Kelvin on this basis and they may form the base for modeling of phase equilibria by the CALPHAD method at low temperatures.

Furthermore, one of the options how to model the Gibbs energy of any intermetallic phase at low temperatures is based on the extension of its Gibbs energy from high-temperature region towards zero Kelvin which also takes into account the value of its Debye temperature (TD). A fundamental prerequisite for the success of this method is the existence of precise expression for the Gibbs energy of studied phase at high temperature i.e. above 298.15 K.

In the V–Zr system the data of phase equilibria above room temperature were determined in [15], [16], [17] and thermodynamic assessments were published for equilibria above room temperature by Servant [18] and Zhao et al. [19], where the description of Gibbs energy of V2Zr C15 Laves phase was evaluated by the ATAT software [20], [21].

In this work, we have taken the results of Zhao [19] as reliable and improved the description of C15 Laves phase and a hexagonal close-packed (HCP_A3) phase above room temperature. However, Rapp [10], Moncton [11], Keiber [22] and Geibel [23] have found rhombohedral phase as stable phase below 113 K, which cannot be drawn in high-temperature phase diagram. Therefore, we have performed the extension of calculation of phase diagram down to zero Kelvin, using the description of unary data below room temperature [14] and the new extension of expression of Gibbs energy of C15 Laves phase and rhombohedral phase to zero Kelvin compatible with Gibbs energy expressions above 298.15 K [19] and based on respective values of Debye temperatures [22], [23], [24].

In addition to that, our ab initio calculations of elastic constants and phonon spectra bring new findings concerning the stability and behavior of C15 and rhombohedral V2Zr phase.

Section snippets

Ab initio calculations of stability of V2Zr phases

The energy of formation of chosen phase at 0 K can be obtained with the help of ab initio electronic-structure calculations, performed within the Density Functional Theory (DFT). We used the pseudopotential method [25] incorporated into the Vienna Ab initio Simulation Package (VASP) code [26], [27] combined with the Projector Augmented Wave–Perdew–Burke–Ernzerhof (PAW-PBE) pseudopotential [28], [29], [30] (i.e. we employed the Generalized Gradient Approximation (GGA) for the exchange-correlation

Elastic constants and phonon spectra of V2Zr C15 Laves phase

The phonon spectra of V2Zr C15 Laves phase were calculated using the Phonon software [36]. The behavior of phonon density of states (DOS) is displayed in Fig. 1a together with the dispersion relations of phonons (Fig. 1b).

It may be seen that the V2Zr in C15 Laves phase structure is dynamically stable at zero Kelvin which is in agreement with the findings published in [24] where the V2Zr C15 structure is presented as mechanically stable according to the elastic stability criteria: C11>0; C44>0; C

Thermodynamic modeling and phase diagram above 298.15 K

In this temperature region, we adopted a recent assessment of thermodynamic parameters of V–Zr system [19] except for the overestimated number of parameters of C15 Laves phase which was reduced in our work. Subsequently, the corresponding parameters for HCP_A3 phase had to be also reoptimized. The obtained data are presented in Table 5 together with data published in [19].

It is obvious that lower values of 0LV,Zr and 1LV,Zr parameters of HCP_A3 phase are sufficient in our modeling. In addition

Thermodynamic modeling of V2Zr phases below 298.15 K

The Compound Energy Formalism (CEF) [41], [42] was also employed for thermodynamic modeling of the V–Zr system down to zero Kelvin. In this temperature region, the Gibbs energy of elemental constituents may be expressed with respect to SER state by equation [14]G0(T)=E0+32RTE+3RTln(1eTE/T)a2T2b20T5c6T3.where TE is the Einstein temperature and a, b and c are constants.

It is assumed that Gibbs energy of stoichiometric phases can be expressed also in the form of Eq. (1) and it is evaluated to

Calculation of phase diagram

The thermodynamic basis of the CALPHAD method relies explicitly on the assumption that the equilibrium phase composition arises as a result of a minimization of Gibbs energy in a closed system at constant external conditions (temperature and pressure) [42].

For the modeling of C15 Laves phase in the V–Zr system below Tlim, we employed the model of stoichiometric phase, where a continual extension of Gibbs energy from temperature region above Tlim=298.15 K (Section 5) was included. The V2Zr

Conclusions

A thorough ab initio analysis of phases found in V–Zr system was performed at 0 K. Ab initio calculated values of lattice parameters, bulk moduli and energies of formation of Laves phases with respect to the SER states, i.e. BCC_A2 for V and HCP_A3 for Zr, correspond reasonably well to both experimental data wherever available and previous theoretical results. It is shown from ab initio calculations that the ZrV2 rhombohedral phase is not stable at 0 K and its structure transforms to C15 Laves

Acknowledgments

Financial support of the Grant Agency of the Czech Republic (Project no. P108/10/1908) and of the Project “CEITEC – Central European Institute of Technology” (CZ.1.05/1.1.00/02.0068) from the European Regional Development Fund is gratefully acknowledged. This research was also supported by the Academy of Sciences of the Czech Republic (Project no. RVO: 68081723). The access to computing and storage facilities owned by parties and projects contributing to the National Grid Infrastructure

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