Study of Thermophysical Properties at High Temperatures for Some Nanomaterials

Nanomaterials are the central importance in the field of science and technology. The properties of nanomaterials are modified significantly at high temperatures [1,2]. Kumar et al. [3] have studied the temperature dependence of volume expansion of some nanomaterials viz. Zirconia, Ag, Zno, NiO and Al compounds using the Anderson formulation [4] which is based on the assumption that the Anderson- Grüneisen parameter Tδ is independent of temperature, i.e. Tδ does not change with the variation in temperature. It was found by Liu [5,6] that to take account of the temperature dependence of the Anderson- Grüneisen the Anderson formulation is necessary in order to improve the agreement between theoretical results and available experimental data.

Anderson and Zou [7] used the basic calculus to study the thermal properties of materials

Where 𝛼 is the volume thermal expansion coefficient or simply the thermal Expansivity,

And B the isothermal bulk modulus given below

Eq. (1) has the status of an identity. The Anderson-Grneisen parameter δ_T is related to the temperature derivative of bulk modulus as follows [4]

Using another identity between thermodynamic quantities [8,9]

In Eq. (4), we get

Using Eq. (5) in Eq. (1), we get

For most of the materials, the product remains constant in the high temperature region above the Debye temperature. Hence, Eq. (7) gives (dα/dT)_V=0, and then Eq. (1) yields.

Equations (6) and (8) takes together yield

On integrating Eq. (9) with respect to temperature T between the limits T₀ and a value of high temperature T, and taking a constant value of δ_T equal to δ⁰_T , we get

Taking Eq. (2) in Eq. (10), and then further integrating, we find

Where V₀ is the volume V at T−T₀, unified temperature. It is found [3] that the results for volume thermal expansion versus temperature calculated from Eq. (11) for the nanomaterials under study derivative significantly from the experimental data. This discrepancy appears due to the assumption that δ_T giving the following expression [5]

As in Eq. (12) the integral on left hand side, cannot be evaluated exactly with the help of an analytical method, and therefore [5,10] used an approximation procedure by truncating the expansion of the exponential function up to quadratic term in

Where . “s” is the dimensionless thermoelastic parameter whose [4] value is about s=+2, Eq. (12a) is in fact an isobaric equation of state determining V/V₀ as a function of temperature along an isobar at P = 0 for different types of materials.

Value of δ⁰_T , the Anderson- Grneisen at room temperature and atmospheric pressure remain almost between 4 and 6 for different types of materials [7]. For nanomaterials, accurate values of δ⁰_T are not known, we have calculated V/V₀ , taken three representative values of δ⁰_T equal to 4, 5, and 6 for each material. Values of 0 α used as input in calculations are given in Table 1 of Kumar et al. [3],[11].

Table 1:Value of input parameters used in present work [3,11].

In order to demonstrate the validity of Eq. (13) and (11) are calculated as a function of temperature. Because one should check for consistencies using the same set only with Eq. (11) with experimental data [3], here we have used seven nanomaterials in present study, the value of α₀ , δ⁰_T are used as input parameters which are given in Table 1. Eq. (13) and (11) are used to compute temperature dependence of V/Vo at different temperature of nanomaterials. The result are compared with experimental data [3] we found Liu Model (13) is very consistent with experimental data [3] we can say that Liu Model can reproduced the experimental data in figure of merit The present work is simple and straightforward method to study the effect of temperature on nanomaterials due to the simplicity and applicability it may be the current interest to the researchers engaged in this field. To the best of our knowledge such simple methods are not yet available in the literature for nanomaterials (Tables 2-4; Figures 1-3).

Table 2:Temperature dependence of V/Vo of Eqs. (13) and (11) with the value δ⁰_T=4

Table 3:Temperature dependence of V/Vo of Eqs. (13) and (11) with the value δ⁰_T=5.

Table 4:Temperature dependence of V/Vo of Eqs. (13) and (11) with the value δ⁰_T=6.

Figure 1:

Figure 2:

Figure 3:

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