Case Report, J Nanomater Mol Nanotechnol Vol: 12 Issue: 2

Thermo Elastic Properties of Nanomaterials under High Compression

Anjani K Pandey^*, Prachi Singh, Shivam Srivastava, Shipra Tripathi and Chandra K Dixit

Department of Physics, Shakuntala Misra National Rehabilitation University, Lucknow, India

*Corresponding Author: Anjani K Pandey
Department of Physics, Shakuntala Misra National Rehabilitation University, Lucknow, India; Tel: 8318482960; E-mail: anjani_phys@yahoo.in

Received date: 03 April, 2023, Manuscript No. JNMN-23-98279;
Editor assigned date: 07 April, 2023, PreQC No. JNMN-23-98279 (PQ);
Reviewed date: 21 April, 2023, QC No. JNMN-23-98279;
Revised date: 28 April, 2023, Manuscript No. JNMN-23-98279 (R);
Published date: 26 May, 2023, DOI: 10.4172/2324-8777.1000356

Citation: Pandey AK, Singh P, Srivastava S, Tripathi S, Dixit CK (2023) Thermo Elastic Properties of Nanomaterials under High Compression. J Nanomater Mol Nanotechnol 12:2.

Abstract

Keywords: Equation of States (EOSs), Nanomaterial, High-Pressure, Bulk Modulus, Gruneisen parameter

Introduction

High pressure research on nanomaterials are getting more attention between theoretical as well as experimental researchers, as at sufficiently high pressure a lot of structural effects happen inside nanomaterials viz. pressure ionization, change in electronic properties of nanomaterials etc. This can lead nanomaterials to change their physical, chemical and optical properties. Pressure is also connected with the Volume compression ratio (V/V₀), bulk modulus (K_T) and coefficient of volume expansion. Due to high thermal expansion, high chemical durability, strength and transparency of nanomaterials, it has also attracted attention in different commercial areas. The equation of state is a most useful tool to find the effect of pressure or compression over nanomaterials which leads to knowing the effect on thermoplastic properties such as volume compression, bulk modulus and effect of pressure on nanomaterials. The physical properties of nanomaterials depend on its structure and interatomic distances. if pressure is applied to the nanomaterials, due to pressure nanomaterial shows various effects, such as modification of interactions between the nano-object and pressure transmitting medium, transformations of non-constitutive elements and transformation of the interactions between the nanoobjects [1].

The gruneisen parameter (γ) is also directly related with EOSs at different pressure or compression. This theoretical approach is much useful for nanomaterials at different compressions for which no experimental data is available so for. Theoretical researchers have developed different EOSs for these purposes. In present work, we have used three different isothermal equation of states based on interatomic potential model viz. Birch- Murnaghan equation of state, Brennan-Stacey equation of state and Vinet-Rydberg equation of state to calculate the thermoelastic properties of 3C-SiC (Cubic Silicon- Carbide), Zr_0.1Ti_0.9O₂ (Zr-doped nano anatase), ε-Fe (Epsilon iron) and Rb₃C₆₀ (Rubidium-doped C60) at high pressure [2].

Case Presentation

We have used three different equations of states, derived from lattice potential theory:

Where K₀ and K₀’ are isothermal bulk modulus and its first pressure derivative at zero pressure and

V is the volume at Pressure (P) and V0 is the volume at zero pressure. Equation (1) is based on Finite strain theory called Birch- Murnaghan (III order) EOS. Equation (2) based on thermodynamic Formulation for the Gruneisen parameter called Brennan-Stacey EOS. Equation (3) is the Vinet-Rydberg EOS and it is based on universal relationship between binding energy and interatomic separation for solids [3,4].

Form the equations (1), (2) and (3) we can obtain the expression for isothermal bulk modulus at Pressure (P):

Using equations (4), (5) and (6) we can obtain the expression for K'_T= (∂K_T⁄∂P) the first derivative of isothermal bulk modulus at pressure P given as

Thus equation (1), (4) and (7) represents the Birch-Murnaghan (III) EOS, equation (2), (5) and (8) represents Brennan-Stacey EOS and equation (3), (6) and (9) represents Vinet-Rydberg EOS respectively.

Grüneisen parameter (γ):

The expression for Gruneisen parameter as suggested by borton and Stacey given as,

Where, F=2.35

Results and Discussion

In this work we have describe three different isothermal equation of states viz. Birch-Murnaghan equation of state, Brennan-Stacey equation of state and Vinet-Rydberg equation of state for calculating pressure, isothermal bulk modulus, first pressure derivative of isothermal bulk modulus and Grüneisen parameter at different compressions the value of Pressure (P) are calculated by using equation (1-3). The input value of K₀ and K’₀ are shown in Table 1. Further using the value are P in equation (4-6) We find value of K_T further substituting the values are P and K_T calculated by using equation (1-6) in equation (7-9) we find the values of K’_T further substituting the values of P, K_T and K’_T obtain from equation (1-9) in equation (10) we obtain the value of Gruneisen parameter. The graphs are plotted between the calculated values of P at different compressions by using Birch-Murnaghan EOS, Brennan-Stacey EOS and Vinet-Rydberg EOS are shown in Figures 1-4. Further the graph plotted between V/V₀ and K_T are shown in Figures 5-8 the graph plotted between V/V₀ and K’_T are shown in Figures 9-12 and also the graph plotted between V/V₀ and γ are shown in Figures. Equation (1), (4) and (7) represent the Birch-Murnaghan 3rd order EOS’s. Equation (2), (5) and (8) represent the Brennan-Stacy EOS’s and Equation (3), (6) and (9) represent the Vinet-Rydberg EOS’s and equation (10) represent the Gruneisen parameter respectively [5-8].

Table 1: Values of input data for K₀ and K’₀ is shown.

A graph plotted between P vrs V/V₀ for 3C-SiC, Zr_0.1Ti_0.9O₂, ε-Fe and Rb₃C₆₀ are shown in Figures 1-4.

From Figures 1-4 it is clear that on increasing the pressure, the value of compression increases for all the four nanomaterials viz. 3CSiC, Zr_0.1Ti_0.9O₂, Rb₃C₆₀ and ε-Fe. Further for 3C-SiC all the three EOSs corresponds well upto volume compression range (V/V₀)=0.7 at pressure 144 GPa then after the Brennan-Stacey EOS and Vinet- Rydberg EOS corresponds well with each other but the Birch- Murnaghan EOS starts deviating above the pressure 144 GPa with other two EOSs. Whereas for Zr_0.1Ti_0.9O₂ and Rb₃C₆₀ all three EOSs corresponds well with each other upto compression range (V/V₀)=0.95 at pressure 17.22 GPa and compression range (V/V₀)=0.7 at Pressure 12.05 GPa respectively then after all the three EOSs starts deviating with each other. For ε-Fe the Brennan-Stacey EOS, Vinet-Rydberg EOS and Birch-Murnaghan EOS corresponds well with each other over entire compression range [9-11].

Further the graph for isothermal bulk modulus versus compression for 3C-SiC, Zr_0.1Ti_0.9O₂, ε-Fe and Rb₃C₆₀ are shown in Figures 5-8.

From Figure 5-8 it is clear that on increasing the pressure, the value of compression increases for all the four nanomaterials viz. 3C-SiC, Zr_0.1Ti_0.9O₂, Rb₃C₆₀ and ε-Fe. Further for 3C-SiC all three EOSs corresponds well upto compression range (V/V₀) 0.8 at isothermal bulk modulus 440 GPa but after that the Birch-Murnaghan EOS starts deviate with other two EOSs. Whereas for Zr_0.1Ti_0.9O₂, ε-Fe and Rb₃C₆₀ all three EOSs corresponds well upto compression range (V/ V₀)=0.97 at isothermal bulk modulus 354.82 GPa, compression range (V/V₀)=0.75 at isothermal bulk modulus 438.44 GPa and compression range (V/V₀)=0.85 at isothermal bulk modulus 30.88 GPa respectively but all the Brennan-Stacey EOS, Vinet-Rydberg EOS and Birch- Murnaghan EOS starts deviating above the isothermal bulk modulus 354.82 GPa, 438.44 and 30.88 with each other [12-15].

Further the graph for first pressure derivative of isothermal bulk modulus versus compression for 3C-SiC, Zr_0.1Ti_0.9O₂, ε-Fe and Rb₃C₆₀ are shown in Figures 9-12.

From Figures 9-12 it is clear that on decreasing first pressure derivative of isothermal bulk modulus, the value of compression increases for all the four nanomaterials 3C-SiC, Zr_0.1Ti_0.9O₂, Rb₃C₆₀ and ε-Fe. Further for 3C-SiC the Brennan-Stacey EOS and Vinet- Rydberg EOS corresponds well with each other but the Birch- Murnaghan EOS starts deviating below compression range V/V₀=0.95 at first pressure derivative of isothermal bulk modulus 2.7 with other two EOSs. Whereas for Zr_0.1Ti_0.9O₂, ε-Fe and Rb₃C₆₀ all the three EOSs (Brennan-Stacey EOS and Vinet-Rydberg EOS and Birch- Murnaghan EOS) does not Corresponds with each other entire compression range. Further the graph for Gruneisen parameter (γ) versus compression for 3C-SiC, Zr_0.1Ti_0.9O₂, ε-Fe and Rb₃C₆₀ are shown [16].

Conclusion

It is clear that on decreasing Grüneisen parameter, the value of compression increases for all the four nanomaterials 3C-SiC, Zr_0.1Ti_0.9O₂, Rb₃C₆₀ and ε-Fe. Further for 3C-SiC all three EOSs corresponds well upto compression range=0.9 at Grüneisen parameter 0.45 then after the Brennan-Stacey EOS and Vinet-Rydberg EOS are corresponds well with each other but Birch-Murnaghan EOS start deviating below upto Grüneisen parameter 0.45 with each other two EOSs. Whereas for Zr_0.1Ti_0.9O₂, ε-Fe and Rb₃C₆₀ all the three EOSs does Corresponds with each other entire compression range.

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