Abstract
Recently developed isothermal Kholiya’s EOS is modified to study the temperature dependent volume expansion and applied for NaCl crystal. The results obtained with the present model are in quite close agreement to the experimental values. The model is therefore extended to study the variation of bulk modulus and the coefficient of volume thermal expansion with temperature. Comparison of the obtained results with the experimental data demonstrates that an isothermal EOS may also be modified to study the temperature dependent elastic properties. The present study also reveals that the quasi harmonic approximation, that is, the product of bulk modulus and the coefficient of volume thermal expansion as constant, is valid in case of NaCl crystal.
1. Introduction
The behavior of solids under the effect of high pressure and high temperature has truly developed into an interdisciplinary area which has important implications for an application in the area of physics, biology, engineering, and technology apart from the discovery of various novel and unexpected phenomena, high pressure high temperature research has provided new insight into the behavior of matter [1]. Strength and elastic properties of a solid depend on the strength of its interatomic forces. Therefore, the application of temperature which changes the interatomic distance of the substances changes its physical properties. The EOS gives us valuable information about the relationship between the changes in thermodynamic variable, namely, pressure, volume, and temperature. Every thermodynamic system has its own EOS, independent of others. An EOS expresses the peculiar behavior of one individual system which distinguishes it from the others. In order to determine the EOS of a system, the thermodynamic variables of the system are accurately measured and a relation is expressed between them. Attempts have been made to derive a compressibility equation from molecular theory, but none of them has resulted in convenient equation expressing the results of experiments with adequate accuracy. To meet this need some empirical equations have been proposed, the sole justification of which is that it works.
In spite of impressive advances on the theoretical front over the past decades, the need for the search of an EOS continues to exist. Although, modern electronic band structure calculation allows the predictions of EOS for solids yet the calculation is time consuming as well as expansive. In the literature, there are number of equations of states, and these arise from an unchecked and improvable assumption concerning an assumed interatomic potential, an assumed strain function, or an assumed boundary condition that is not testable [2]. Recently, Kholiya have developed an isothermal EOS to study the high pressure behavior of nanomaterials [3]. The main advantage of this EOS is that it follows the basic requirements revealed from the fundamental thermodynamics for solids in the limit of extreme compressions, as given by Kholiya et al. [4]. In the present paper our aim is to modify isothermal Kholiya’s EOS to study the temperature dependent elastic properties of the solid. For understanding the thermodynamic behavior of solids at high temperature it is useful to have reliable values of volume thermal expansion coefficient (), isothermal bulk modulus () [5–8]. For this purpose NaCl is taken as an example because NaCl is the most widely used internal pressure standard in high pressure diffraction experiments due to the availability of the large body of experimental data. NaCl has a stable structure (B1) up to a pressure of about 30 GPa and its melting temperature is nearly 1074 K. Thus we have a wide range of pressures and temperatures for studying the equation of state and thermoelastic properties of NaCl. Besides this, numerous attempts have been made to understand the high pressure and high temperature [9–16] behavior of NaCl using equation of state and thus the experimental/theoretical data is available which can be useful to check the validity of our proposed temperature dependent EOS.
2. Method of Analysis
Recently, to study the high pressure elastic properties of nanomaterials Kholiya [3] has expended pressure in powers of density up to the quadratic term and achieved the EOS as Hear and are the bulk modulus and first order pressure derivative of bulk modulus at and , respectively.
Equation (1) can also be written in the inverted form as where .
This EOS may be used to determine the pressure for different compression or vice-versa at the reference temperature (which is generally room temperature) and it is shown by Kholiya [3] that (2) gives the better results for the compression behavior of nanomaterials than some well-known EOS, namely, usual Tait, Murnaghan, and Shanker EOS. To make (2) temperature dependent we have used the basic equation of state which is in terms of thermal pressure as [2] where thermal pressure may be given as In the quasi harmonic approximation considering Constant = , (5) gives Using (4) and (6), (3) may be written as where .
The bulk modulus is defined as Thus, (6) gives The coefficient of volume thermal expansion is defined as Therefore, (6) gives If the pressure the expression for, , bulk modulus , and coefficient of volume thermal expansion comes out to be where .
3. Results and Discussion
To study the volume change, bulk modulus, and coefficient of volume thermal expansion at high temperature, the present model requires three input parameters, namely, , , and . For NaCl, the experimental values of , , and are 24 GPa, 5.5, and 11.8 × 10−5, respectively [2]. Equation (11) is used to calculate the variation of volume () with temperature for NaCl crystal. The obtained results are given in Table 1 along with other theoretical and experimental data. From the table it is clear that our calculated values are quite close to the experimental findings in comparison to the other theoretical results. This close agreement between our theoretical results and the experimental findings motivated us to apply the model for the study of temperature dependence of bulk modulus and coefficient of volume thermal expansion. Tables 2 and 3 represent our calculated, experimental, and other theoretical values of bulk modulus and coefficient of volume thermal expansion at different temperatures. From these tables it may be revealed that the present model gives the results better than other theoretical models. The present study also shows that the quasi harmonic approximation, that is, the product of bulk modulus and the coefficient of volume thermal expansion as constant, is valid in case of NaCl crystal. From the overall discussion it may be concluded that an isothermal EOS may also be modified to study the temperature dependent elastic properties.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.