Equation of State and Thermo Dynamic Behaviour of C60 under High Pressure

In the present study, four equations of state (EOSs) were used, namely Birch-Murnaghan, Dodson, Bardeen and modified Lenard Jones for evaluating relative compression volume VP/Vo, isothermal bulk modulus B and spinodal pressure Psp for C60 under high pressure. Required constant parameters in the EOSs were taken from literature data. Starting from the definition of spinodal pressure, as the pressure at which bulk modulus of a material vanishes (i.e. B=0), A new approach for the evaluation of spinodal pressure have been established, in present work, by extrapolating variation of bulk modulus under high pressure data to the point where B=0, and formulate a new mathematical expression for spinodal pressure. With the aim of finding validity of these equations of state of applying on C60, the calculated results from the entire equations of state are compared with themselves and with other available experimental and theoretical published data. The results for VP/Vo, B, and Psp are found to be in agreement with each other for whole pressure ranges. Some other theoretical and empirical data are brought in this study and give good agreements with our results. The new spinodal pressure approach and high pressure studies might represent a promising entrance to the formalism of universal equation of state for solids.


Introduction
Carbon C 60 is a new crystalline form of solid Carbon; it is coincidently discovered in 1985. Different from diamond and graphite, it formed from a different arrangement of Carbon atoms. Its molecules are assembled to form solid by weakly bound force. C 60 is stable and chemically unreactive. It can be made chemically from graphite (Beiser,[4]). The structure of C 60 at room temperature is face-centered cubic (fcc) and it has been determined by X-ray and neutron diffraction.
High pressure research has got great attention by researchers to investigate thermo elastic and thermodynamic properties of solid phase at high pressure (Decker [8]). For this purpose, numerous attempts have been done for formulating various EOSs of solid matter in order to determine their characteristics under high pressure. Eventually diverse methods have resulted in formulation various forms of EOSs for solids.
Intensive studies have been performed to analysis various thermodynamic properties of C 60 . (Ludwig et al [14]) used X-ray scattering techniques to perform an experimental study on C 60 , with the aim of determining its bulk modulus at ambient pressure and volume compression ratio V P /V o . Pressure induced phase transition from fcc was also observed. In that experiment the lattice parameter of C 60 as a function of pressure have been calculated. (Rekhviashvili [16]) presented a new equation of state of fullerene C 60 using quantum statistical frame work. The intermolecular vibrations of C 60 were calculated with help of Debye heat capacity theory. Structural phase transition of C 60 under high pressure using X-ray diffraction performed in literature, is presented by (Murga and Hodeau [15]). (Duclos et al [10]) performed a very precise experimental measurement on C 60 . In that experimental technique, volume compression ratio V p /V 0 was measured up to pressure of about 20GPa. (Goyal and Gupta [11]), have performed a comparative study of compression volume for C 60 under high pressure using different equations of state such as Tait's EOS, Murnaghan EOS, Vinet EOS, Suzuki EOS and (Sharma and Kumar) EOS to make comparison with the other experimental data obtained by (Duclos et al [10]). In their study, Tait's formula and Vinet EOS are in good agreement 60 Equation of State and Thermo Dynamic Behaviour of C60 under High Pressure with the presented result achieved in the earlier experimental data. While, Suzki and (Sharma &Kumar) approach were given results do not fit the experimental result. Therefore, their data with Murnaghan EOS are a bit close to the empirical values at low pressure ranges.
In the present study, an analysis of high pressure compression in C 60 has been done using B-M, Dodson, Bardeen and modified Lenard-Jones (mL-J) EOSs. The calculated values using different equations of state in this work are found to be in good agreement with (Duclos et al [10]) measurements. This shows that B-M, Dodson, Bardeen and mL-J EOSs are conformed as standard equations of state of C 60 . Regarding to the bulk modulus of C 60 under high pressure, (Sharma et al [18]) has derived a new formula for bulk modulus in terms of Anderson-Grűneisen parameter, by using Murnaghan EOS under a certain condition. The data of bulk modulus achieved by (Sharma et al [18]) is shown to be close to our results over a wide range of pressures.
Further calculation in the present study is a new approach to evaluate spinodal pressure of C 60 using EOSs. Negative pressure results in decreasing of bulk modulus and from a particular value of this pressure, the bulk modulus tend to become zero, and this pressure is called spinodal pressure.

Theoretical Approach
The relationship between pressure (P), volume (V) and temperature (T) known as equation of state, with the help of EOSs, we can understand various properties of matter under varying conditions of pressures and temperatures.
In order to find solutions to a variety of problems in earth science and condensed phase of physics, an EOS that accurately predicts solids behavior at high pressure and temperature is required. As a more precise definition, the pressure-volume relationship at constant temperature is termed as isothermal equation of state.

Types of equations of State
The study of (P-V) EOSs of relevant material is one of the most basics that are needed for pressure calibration (Al-Saqa and Al-Taie [2]). Many different EOSs have been derived based on different physical assumptions, for isothermal description of solids under strong compression: The following are some famous isothermal equations of state that have been used in this work: i) Birch-Murnaghan EOS (Birch[6]) The most famous EOS for solids is the Birch-Murnaghan equation. The Birch-Murnaghan EOS is derived from the internal potential energy in a solid, based on finite strain theory and pressure derivative of internal potential energy, also from the definition of the bulk modulus.

Birch-Murnaghan EOS is given as
P Do : pressure according to Dodson EOS iii) Bardeen EOS [Bardeen [3]): based on the interatomic potentials, such as: P Bard: pressure according to Bardeen EOS iv) Modified Lenard-Jones (mL-J) EOS (Jiuxen [13]): The total potential energy of a solid is derived from generalized Lenard-Jones potential equation. Latterly on differentiating the total potential energy formula with respect to volume and carrying out some mathematical operation, (Jiuxun,[13]) has obtained an EOS as presented:

Isothermal Bulk Modulus (B)
The isothermal bulk modulus, especially its pressure dependence, has been studied by many researchers (Ruoff [17]), (Hofmeister [12]) because of its basic role in the EOS. Therefore, one can define bulk modulus as the pressure increase needed to cause a given relative decrease in volume. The isothermal bulk modulus B can be formally defined by: Experiments have proven that the bulk modulus depends on volume and pressure at a constant temperature (Birch [5]). As the interatomic space decreased as a result of pressure the resistance force against the external agent would increase. Furthermore, to test the validation of the isothermal EOSs, the bulk modulus corresponds to each equation of state is calculated with eq. (5). Therefore, to express bulk modulus under high pressure by using an EOS, B-M EOS, given in eq.(1), has been derived with respect to volume to obtain eq.(6): On substituting (eq.6) into (eq.5) then (eq.7) represents variation of bulk modulus under high pressure according to B-M EOS: ( ) ( ) B Bard. : Bulk modulus according to Bardeen EOS 0 B mL-J : Bulk modulus according to Modified Lenard-Jones EOS.

Spinodal Pressure P sp
Spinodal pressure decomposition agree with dynamics of phase changes in materials which are caused by transferring the material into an initial state (e.g. by rapid cooling and rapid heating) (Al-saqa and Al-sheikh, [1]).
A more interesting case in relating to bulk modulus and pressure is called spinodal pressure. It is defined as the pressure at which B =0, and it has negative value. (Jiuxun [13]) Formulated spinodal pressure as in eq. (11).
In the present work a new approach to evaluate P sp has been established by considering, the slope of the different graphs shown in Fig.(2) and from the definition of P sp as the pressure at which B goes to zero, and on extrapolating variation of B value with high pressure to the point B=0 , it is found that spinodal pressure expressed as: This is different from eq. 11.

Evaluation of Isotherm Properties for C 60
On substituting B 0 and B 0 ' values for C 60 from Table 1 into B-M EOS, given in eq.1, variation of V p /V 0 with high pressure P B-M , for C 60 , has been evaluated and the results are shown in Fig.(1).  5.7 Similarly, on substituting B 0 and B 0 ' values, for C 60 from Table 1, into Dodson EOS, given in eq.2, Bardeen EOS, given in eq.3, and in mL-J EOS given in eq.4 respectively, variations of V p /V 0 with high pressure P Do , P Bard and P mL-J for C 60 , have been evaluated. All the results are shown in Fig.(1) in comparison with the experimental data of (Duclos et al [10]) and theoretical results of (Sharma et al [18]).

Evaluation of Variation of Bulk Modulus for C 60 under High Pressure
Using the parameters from Table 1, and substituting V p /V o data shown in Fig. (1), into equations (7-10), we have obtained the variation of the bulk modulus B for C 60 under high pressure corresponding to each EOS, the results are shown in Fig.(2), in comparison with the data obtained by (Sharma et al [18]). This is an alternative form of (eq.12) Evaluating the slope of the tangent for the different graphs of Fig.(2) and using B 0 value from Table 2, substituting these values in (eq.11) on time and into (eq.12) another time. Values of spinodal pressure for C 60 calculated by using different EOSs according to (Jiuxun [13]) expression and present work expression, eq.(12) or eq.(16), are shown in Fig.(3) and Fig.(4 ) and tabulated in Table 2.   . Spinodal pressure of C 60 calculated with (eq. 12) and (Sharma,et al. [18]) data

Discussion and Conclusions
The present work includes an investigation of some thermo elastic properties of C 60 under high pressure, such as; volume compression ratio V p /V 0 , Bulk modulus and Spinodal pressure.
In terms of V p /V o , Fig.(1) shows that all the EOSs that have been used, give similar results which are in agreement with each other over wide pressure ranges. The obtained results also fit the experimental data taken from (Duclos et al [10]) and theoretical results of (Sharma et al [18]). Therefore, in spite of a little declination of curves of the resulted data at high pressure, it can be claimed that these EOSs that have been used to work on C 60 , are convincingly conformed by the empirical observation. This implies that B-M, Dodson, Bardeen and mL-J EOSs, can be defined as standard equations of state of C 60 .
Further calculation, the isothermal bulk modulus of C 60 under high pressure is evaluated with each EOS and showed from Fig.(2). It reveals that bulk modulus increases with pressure. Under pressures up to around 10 Gpa all curves drawn by the present work are fitted with each other and merely close to the data obtained by (Sharma et al [18]). It can be pointed out that the result achieved by (Sharma et al [18]) deviates largely from our data as pressure increases. This can be interpreted that the data achieved by (Sharma et al. [18]) was calculated from a new expression for bulk modulus = ( ) − , where : is Anderson-Grűneisen parameter. Another interesting analysis is the evaluation of spinodal pressure of C 60 by using (eqs.11and 12) correspond to the entire EOSs. Calculated values of P sp reported in Table 2 and shown in Fig. (3) and Fig.(4).
Evaluated values of P Sp with (eq.11) is around 2.5 Gpa and shown in Fig.(3), while (eq. 12) gave results of just above bout 3Gpa and illustrated in Fig.(4). According to Fig.(3), it is observed that the values of Ps p obtained by (eq.11) is noticeably deviated from the B-P data, whereas our expression of Ps p (eq.12) predicted values which make the B-P graph consistent.