Investigations on the Thermo-elastic,Hardness and Thermodynamic Properties of Rh3Ta Intermetallic Compound under Various Pressures and Temperatures*

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(Key Laboratory of Environmentally Friendly Chemistry and Applications of Ministry of Education,College of Chemistry, Xiangtan University, Xiangtan 411105 China)

Abstract：The thermo-elastic, hardness and thermodynamic properties of Rh3Ta intermetallic compound were investigated under various pressures and temperatures by first-principle calculations based on density functional theory (DFT). The equilibrium lattice parameter of Rh3Ta was derived by the total energies as a function of volume. The optimized lattice constant a0, optimized volume V0 were obtained,moreover, cohesive energy Ec, enthalpy H and density ρ at 0 GPa of Rh3Ta were calculated, which were consistent with experimental and the other theoretical results. The elastic constants Cij and the values of other modulus (B, E, G)，B/G ratio, Poisson’s ratio υ, the anisotropic index A, hardness H and compressibility K for this intermetallic compound were also evaluated. Additionally, the variations of thermodynamic properties such as thermal expansion α, heat capacity CV, CP, entropy S, Gibbs energy G, internal energy U, Debye temperature Θ were all successfully obtained through the quasi-harmonic Debye model in pressure ranges from 0 GPa to 60 GPa and temperature ranges from 0 K to 1 800 K.

Key words：density functional theory (DFT); elastic properties; electronic properties; intermetallics; thermodynamic properties

In recent years, the applications of intermetallic compounds, especially Rh-based and Ir-based, have been increasing academic interests due to their unique mechanical properties, such as high resistance to corrosion, good oxidation resistance, high melting temperature and strength, etc[1-3].Comparing these two Cu3Au type structural intermetallic compounds, Rh-based superalloys have lower density[4]. Hence, they have been regarding as promising high-temperature structural materials, applying to aircraft engines and gas turbines engines.

Up to now,many theoretical and experimental investigations of Rh-based superalloys have been undertaken. For example, the structural, mechanical, elastic, electronic, thermodynamic, phase transition and thermal properties of Ir3Ta and Rh3Ta alloys have been researched by Arikan et al.[5], using different experimental and theoretical methods. Watanabe et al.[6]have tested the crystallization of an Ir-Ta alloy film by increasing Ta content. Miura et al.[7]have concluded that Rh3Ta compound has a variation of strength with a weak positive temperature at around 1 273 K, which is about half of the melting point for Rh3Ta alloy. According to Nuttens et al.[8], Rh3Ta was formed at 800～900 ℃ and it can be used as a diffusion barrier lower than 825 ℃, which is one of high-temperature applications. In addition, the elastic properties of Rh3Ta have been studied by Chen et al.[9]using the tight-binding (TB) and linear augmented plane wave methods. For thermodynamic investigations, Terada et al.[10]have measured the thermal expansion and thermal conductivity of Ir3Ta and Rh3Ta alloys, concluding that these materials have a smaller thermal expansion and a larger thermal conductivity than other intermetallic compounds. Besides, some people also analyzed the electronic structures of Ir3Ta and Rh3Ta intermetallic alloys by means of the self-consistent tight-binding linear muffin tin orbital (TB-LMTO)[11]. As we all know, there is a large number of studies on Rh-based alloys, but most of the them were preformed at 0 K or at the pressure of 0 GPa. It is obvious that there is no works concerning the thermo-elastic, hardness and thermodynamic properties under various pressure and temperatures. Therefore, it is necessary to study systematically how pressure and temperature will affect the elastic, hardness and thermodynamic properties. This work provided some information about them under various pressures and temperatures of Rh3Ta intermetallic compound in cubic structure by the first-principle calculations and the quasi-harmonic Debye model. To understand the influences of pressure and temperature in more detail, the elastic properties, hardness and thermodynamic properties of Rh3Ta under the pressure ranging from 0 to 60 GPa and temperature ranging from 0 K to 1 800 K are analyzed.

1 Computational details and theory

The first-principle calculations for Rh3Ta are used via Cambridge Serial Total Energy Package (CASTEP)[12]in this work and the generalized gradient approximation (GGA) in the form of Perdew-Burke-Ernzerhof (PBE)[13]is chosen for exchange-correlation functional. In order to achieve the relaxation zero strain and best convergence, the cut off energy of 400 eV is used for the calculations and the Brillouin-zone is carried out with the 12×12×12 Monkhorst-Pack grid[14]of k-point. Furthermore, Broyden-Fletcher-Goldfarb-Shanno minimization (BFGS)[15-16]is operated to optimize the structures. Finally, SCF tolerance is limited to 5.0×10-7eV/atom, maximum ionic force within 0.01 eV/Å and maximum stress tensor within 0.02 GPa by using the finite basis-set corrections[17], all forces in the geometrical optimization are converged to less than 0.001 eV/Å. What is noteworthy is that the total energy was minimized by a series of different values of lattice constants. The unit cells were compressed homogeneously with a step of 10 GPa so as to simulate the external pressure.

The stress-strain method[18]was used to calculated the elastic constants at the optimized structure, moreover, the pressure dependence of the elastic constants for cubic Rh3Ta areC11,C12,C44, obtained by a linear fit to the stress-strain relationship. The quasi-harmonic Debye model[19], mainly used for studying the thermodynamic properties of Rh3Ta under various pressure and temperatures, can be introduced by several equations, the non-equilibrium Gibbs functionG*(V;P,T) takes the form:G*=(V;P,T)=E(V)+PV+AVib[Θ(V);T],whereE(V) represents the total energy for per unit cell,PVcorresponds to the constant hydrostatic pressure condition,Θsignifies the Debye temperature andAVibcan be written as[20-23]:

γrefers to the Grüneisen parameter and can be obtained by:γ=-d lnΘ(V)/d lnV. The equilibrium state has been acquired, the internal energyUand entropyScan also be given in the quasi-harmonic Debye model by using the corresponding equilibrium volume[25]:

2 Results and discussion

2.1 Structural and elastic properties

Fig.1 shows that Rh3Ta compound with a cubic crystal structure belongs to Pm3m space group, where Rh (0, 0.5, 0.5) and Ta (0, 0, 0), respectively.

The optimized lattice constanta0, optimized volumeV0, cohesive energyEc, enthalpyHand densityρof Rh3Ta at 0 GPa have been plotted in Tab.1 and compared with the experimental[5]and theoretical results[26]. Unfortunately, there is no date to compare with several experimental outcomes, but other date of Rh3Ta are satisfied with the reference results.

Tab.1 The calculated values of the lattice parameters a0, equilibrium volume V0, cohesive energy Ec, enthalpy H and density ρ at 0 GPa, compared with available experimental and theoretical values from references

In our work, the equilibrium lattice parameter is obtained by the total energy as a function of volume for Rh3Ta compound, which is consistent with the third-order Birch-Murnaghan equation of state (EOS)[27]. From the Fig.2, the structure with the lowest energy when the volume is close to 68.26 Å3, which well fits to the computational value[28]. On the other hand, it indicates the present optimization of Rh3Ta is reliable.

The variations of the normalized volume ratioV/V0(V0is the equilibrium volume at 0 GPa) with pressure from 0 GPa to 60 GPa for Rh3Ta are plotted in Fig.1, whereV/V0decrease along with the pressure increasing, indicating that there are some expected behaviors when the pressure increase. Additionally, the variations of the volume with pressure and temperature for Rh3Ta compound are illustrated in Fig.3(a) and (b), where the volume decreases quickly with the increase of pressure and slightly raises up along with the increasing temperature. This indicates the effects of pressure and temperature for volume are different.

The elastic constants are good indicators to offer some valuable information about stability and can well describe the mechanical and thermodynamic behaviors of materials. For cubic structural intermetallic compound, there are only three independent elastic constants:C11,C12,C44, calculated through the stress-strain method. Generally speaking, the structural stability at 0 GPa should satisfy the Born stability criteria[29]:C11-C12> 0,C11> 0,C44> 0,C11+2C12> 0. Obviously, the results in this work are satisfied with the criteria, referring to the Tab.2.

Otherwise, the variations of elastic constants of Rh3Ta at 0～60 GPa pressure range are displayed in Fig.4. It is noted that they all raise up linearly with increasing the pressure, indicating Rh3Ta is a mechanical stable material andC11is the most susceptible compared toC12andC44.Pettifor[30]suggested that atomic bonding in metals and compounds can be described by the Cauchy pressure(C12-C44). WhenC12-C44is positive, the bond is metallic, otherwise, non-metallic bond. From the Tab.2, we can found that the Cauchy pressure of Rh3Ta is positive completely. On the other hand, the Cauchy pressure is calculated with pressure from 0 to 60 GPa in Fig.5 (a), it is gradually increasing with enhancing the external pressure, which means the metallic feature of Rh3Ta compound becomes stronger as the increasing pressure. The hardness parameterHis given late (1), measuring the resistance of objects against compression. Tab.3 lists the hardness of Rh3Ta at 0 GPa. One can see that it can be considered as a high hardness material above 30 GPa, applying to machinery industry.Fig.5 (b) further describes the calculated hardness with an increasing trend as the pressure rises. Besides, it demonstrates that the pressure would also has a significant impact on the hardness.

Tab.2 The calculated values of Cij for Rh3Ta at different pressures

Tab.3 The calculated bulk modulus B, shear modulus G, Young’s modulus E, B/G ratio, Poisson’s ratio υ,anisotropy index A, hardness H and compressibility K of Rh3Ta at for Rh3Ta at 0 GPa

Tab.3 has listed the calculated values of bulk modulusB, shear modulusG, Young’s modulusE,B/Gratio, Poisson’s ratioυ, the anisotropic indexAand compressibilityKfor Rh3Ta at 0 GPa, estimated on the basis of the Voigte-Reusse-Hill approximation[31], for a cubic system:

B=(BV+BR)/2，G=(GV+GR)/2，E=9BG/(3B+G).

Where subscript R is Reuss date, V is Voigt date. TheB,E,Gincreasing monotonically with various pressures of Rh3Ta compound are shown in Fig.6 and the bulk modulusBis most sensitive under high pressure.

Otherwise, Poisson’s ratioυand the anisotropic indexAcan be expressed by[31]:

υ=(3B-2G)/(6B+2G)，A=(2C44+C12)/C11-1，H=(1-2υ)/6(1+υ)E.

If Poisson’s ratioυis under 0.26, suggesting Rh3Ta with a larger lateral expansion when compressed.Agives a criterion to access the degree of elastic anisotropy. When the value is zero, indicating completely isotropic, otherwise, it exhibits anisotropy behavior. The larger theB/Gratio is, the better the ductility is. Generally speaking, ifB/G>1.75, materials ductile behavior. From Fig.7(a) and (b), it can be seen that the values of Poisson’s ratioυandB/Gratio increase slightly, but the anisotropic indexAin Fig.7(c) decreases as pressure rises up. Combined with Tab.3,υis under 0.26,Ais nonzero andB/G<1.75. Thus, Rh3Ta intermetallic compound is a material with brittleness and anisotropy.

Finally, the variations of the Bulk modulus with different pressure and temperature for Rh3Ta are displayed in Fig.8 (a) and (b). It can be seen clearly that theB-Pcurve increases rapidly with enhancing pressure, whereas it gradually decreases with increasing temperature. It is obvious that the influences of temperature and pressure are also opposite to the bulk modulus.

2.2 Thermodynamic properties

We focus on the variations of the thermodynamic properties of Rh3Ta with different pressure and temperature. All of thermal parameters are investigated with pressure ranging from 0 to 60 GPa and temperature from 0 to 1 800 K throught the quasi-harmonic model. The calculated thermal expansionαof Rh3Ta compound as a function of temperature up to 1 800 K at various pressure (0, 10, 20, 30, 40, 50, 60 GPa) are plotted in Fig.9 (a) and (b). There are some decreasing curves with pressure increasing at a given temperature and they decrease sharply at higher temperature. In addition,whenT<500 K, the thermal expansionαraises up quickly as the increasing temperature,whenT>500 K, it is closes to a slow linear increase at a given pressure. Thus, it is clear that pressure and temperature have are opposite effects on the thermal expansion.

The heat capacityCV,CPat different pressure and temperature for Rh3Ta compound are shown in Fig.10 and Fig.11, respectively. As we can see, from Fig.10 (a) and Fig.11 (a),CV,CPgradually decrease with increasing pressure for a given temperature, however, decreasing gently at higher temperature. From Fig.10 (b), there is a rapid raise with temperature below 800 K, whereas there are some lines approach to the Dulong-Petit limit[32]applied to all solids at higher temperature. Also, as seen from Fig.11 (b), it can be foundCP-Tcurves are well consistent with the trend ofCV, however, there is a difference thatCPis dissatisfied with the Dulong-Petit limit betweenCVandCPabove 1 200 K. Therefore, it is evident that the effects of pressure and temperature for Rh3Ta onCVandCPare homologous and the influence of temperature on the heat capacity is more crucial.

The entropySis an indicator to judge the states of heat vibration. Fig.12 shows the entropySas a function of pressure (a) and temperature (b) for Rh3Ta. It is seen that the calculated entropy decreases with pressure increasing at a given temperature. Nevertheless, it increases sharply following the change of temperature, especially ranging from 200 K to 900 K, which derives from the promotion of temperature to the vibrations and then contribute to the entropy increasing. Therefore, the effects of pressure and temperature on entropy are opposite.

The calculated Helmholtz free energyAof Rh3Ta at various pressure and temperature is shown in Fig.13 (a) and (b), respectively. According to the results, for a given temperature,Aincreases with a slightly slope following the change of temperature, conversely, under the same pressure, their values decrease strongly with temperature increasing. For instance, when the changes of temperature from 0～900 K, 900～1 200 K, 1 200～1 800 K, theAat 0 GPa decreases by 32.07%, 8.45%, 59.48%, respectively. Therefore, it indicates there is a great change for Helmholtz free energy under a higher temperature and the impacts of temperature onAis larger than that of pressure.Gibbs energy, one of the most important thermal parameters, reflects the ability of a thermodynamic reaction. The variations of Gibbs energyGwith different pressure and temperature for Rh3Ta compound have been illustrated in Fig.14. It can be seen that theG-Pcurves almost linearly increase with pressure increasing and there is just a tiny difference among various temperature, especially at the lower temperature. As can be seen from Fig.14 (b), for a given temperature, theGdecreases monotonically with increasing temperature and the tendency is without distinction among various pressure. Meanwhile, the higher the pressure is, the larger the Gibbs energy is, which proves the lowest value obtained when pressure at 0 GPa in all considered pressure range .

The internal energyUof Rh3Ta at different pressure and temperature are shown in Fig.15 (a) and (b) for. It is obvious that there is almost no change forUthough the pressure gradually rise at a given temperature, but it is worthy to notice that temperature is different from that of pressure, there is a strong shift following the change of pressure. For example, when the changes of temperature from 200～900 K, 900～1 200 K, 1 200～1 800 K, theUat 60 GPa increases by 39.07%, 6.47%, 39.20%, respectively. It follows that the influence of temperature on the internal energy is more significant.

Debye temperatureΘ, a fundamental parameter of thermodynamic properties, relates to some physical properties, such as melting temperature and thermal expansion coefficient, etc. Fig.16 has displayed the variations ofΘwith different pressures(a) and temperatures(b). It is clear that theΘincreases monotonically and almost linearly among various temperature with altering the pressure, in other words, it refers to the change of atomic vibration frequency under pressure. As can be seen from Fig.16 (b), we can found thatΘis gently decreasing with the raising temperature and allΘ-Tcurves present a similar tendency under the same pressure. So we can conclude that the impacts of pressure on Debye temperatureΘis opposite and greater.

3 Conclusion

In summary, the first-principle calculations are used to calculate the thermo-elastic and thermodynamic properties of Rh3Ta intermetallic compound under various pressures and temperatures. The equilibrium lattice parameter has been derived by the total energies as a function of volume, and the optimizeda0,V0at 0 GPa were obtained, which is found to agree with the reference date. In addition, the dependence of the elastic constantsCijand the values of other modulus (B,E,G)，B/Gratio, Poisson’s ratioυ, the anisotropic indexA，hardnessHand compressibilityKon various pressure are studied. We conclude that Rh3Ta intermetallic compound is a material with brittleness, anisotropy, brittle behaviors. Finally, the thermodynamic properties of Rh3Ta intermetallic compound, such as thermal expansionα, heat capacityCV,CP, entropyS, Helmholtz free energyA, Gibbs energyG, internal energyU, Debye temperatureΘin wide pressure and temperature ranges are also studied successfully through quasi-harmonic Debye model, which offer a useful reference for Rh3Ta and the analytic results are valuable for the future technological applications.