Theoretical Approach to Investigate Temperature Dependent Ultrasonic and Thermophysical Properties of Ti-Zr-Hf Ternary Alloy

Abstract

In this study, we have computed temperature dependent ultrasonic and thermophysical properties of hcp medium entropy alloy Ti-Zr-Hf in temperature range of 0K-900K. Second order and third order elastic constants (SOECs and TOECs) have been calculated using lattice parameters using the Lennard-Jones potential model. With the help of SOECs and TOECs, the elastic parameters such as bulk modulus, shear modulus, Young’s modulus, Poisson’s ratio have been computed. SOECs were also utilized to determine the ultrasonic velocities at different angle along unique axis. Further, thermophysical properties such as Debye temperature, Debye heat capacity, energy density in temperature range of 0K-900K and thermal conductivity in temperature range of 300K-900K have also been theoretically estimated. Additionally, the ultrasonic attenuation due to phonon-phonon interaction in both longitudinal and shear mode and thermoelastic mechanism have been computed for chosen alloy in the temperature range 300K-900K and attenuation due to the phonon-phonon interaction was found to be dominating over that due to thermoelastic mechanism.

Introduction

I. INTRODUCTION

In recent decades, refractory high entropy alloys (RHEAs) have gained a lot of attention from researchers due to their excellent thermophysical properties and widespread applications[1]–[3]. High entropy alloys (HEAs) are defined as solid solution of five or more metal elements in single phase[4]. These alloys are considered to be stabilized by high configurational entropy of mixing[5]. Alloys consisting of three or four elements are known as medium entropy alloys (MEAs). HEAs demonstrate many remarkable properties such as high tensile strength[6], ductility[7], excellent resistance to wear, corrosion, fatigue and oxidation and these properties make them promising class of materials for applications in extreme environments such as cryogenics[8], [9]. Refractory elements such as Ti, Zr, Hf, Nb etc. are known for their excellent resistant to heat, wear and to high temperature oxidation[10] and these properties make them good elements for HEAs. In literature, most of the studies of RHEAs have been based on face-centred-cubic (FCC) or body-centred-cubic (BCC)[6], [7], [11]–[13] with a very few on hexagonal closed-packed (HCP) structures[5], [14]–[16].

Elastic properties provide fundamental insight into crystal structure and nature of bonding in materials, and help in predicting the mechanical behaviour of materials. Thermophysical properties such as Debye temperature, Debye heat capacity, thermal conductivity help estimate how they behave at different temperature.

Considering these remarkable properties of RHEAs, HCP Ti-Zr-Hf becomes a promising alloy for high pressure and temperature industrial applications. Duan et al.[5] have studied the second order elastic constants (SOECs), elastic moduli at room temperature of HCP ternary alloy Ti-Zr-Hf using first principles method. Huang et al.[17] have investigated HCP ternary and quaternary alloys of Sc-Ti-Zr-Hf for their thermoelastic properties including SOECs, elastic moduli and coefficients of thermal expansion (CTE) at different temperatures.

In spite of these studies, temperature dependence of third order elastic constants (TOECs), thermophysical properties such as Debye temperature, heat capacity, thermal conductivity and ultrasonic behaviour of the Ti-Zr-Hf HCP alloy are yet to be studied. This motivates us to investigate these properties.

This study presents the theoretical investigation of temperature dependent SOECs, TOECs, ultrasonic velocities along different direction in the crystal, Debye average velocity, Debye temperature, heat capacity, energy density, thermal conductivity and ultrasonic attenuation for Ti-Zr-Hf HCP alloy.

II. THEORY

The temperature dependence of SOECS and TOECs for Ti-Zr-Hf alloy have been evaluated using Lennard-Jones potential method. The formulation for calculation of six independent SOECs and ten independent TOECs is thoroughly discussed in the literature[18]–[21]. The bulk modulus (B), shear modulus (G), Young’s modulus (Y) and Poisson’s ratio have been computed by Voigt-Reuss-Hill method[22] for hexagonal crystals. The ultrasonic velocities in longitudinal (VL), quasi-shear (VS1) and shear (VS2) modes along unique axis (c-axis) at different angles of propagation, Debye average velocity (VD) and Debye temperature (θD)  can be calculated with the help of higher order elastic constants using the formulation discussed in literature[16], [18], [20], [21].

The RHEAs are used in harsh conditions including high pressure and temperature, so it becomes important to study thermophysical properties such as heat capacity, energy density and thermal conductivity at different temperatures. The heat capacity (CV) and thermal energy density (E0) is evaluated using the Debye model for heat capacity[18], [23], [24].

The lattice thermal conductivity κ has been evaluated using the formulation developed by Morelli and Slack[25]:

IV. ACKNOWLEDGMENT

One of us (SY) is thankful to Council for Scientific and Industrial Research – Human Resource Development Group (CSIR-HRDG) for providing financial support in the form of CSIR-Senior Research Fellowship (09/1014(0012)/2019-EMR-I).

Conclusion

Based on the obtained results and discussion, the following conclusions have been drawn: 1) The results obtained are in good agreements with other studies available in literature. This confirms the significance and successful application of the Lennard-Jones potential approach. 2) The alloy shows strong mechanical stability. 3) The heat capacity follows Dulong-Petit law. 4) The ultrasonic attenuation in longitudinal mode is predominant over shear mode and thermoelastic attenuation.

References

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Copyright

Copyright © 2022 Ramanshu P. Singh, Shakti Yadav, Devraj Singh, Giridhar Mishra. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.