ISSN 1063-7834, Physics of the Solid State, 2017, Vol. 59, No. 4, pp. 835–837. © Pleiades Publishing, Ltd., 2017. Original Russian Text © S.Sh. Rekhviashvili, 2017, published in Fizika Tverdogo Tela, 2017, Vol. 59, No. 4, pp. 816–818.
FULLERENES
Equation of State for Fullerite C60 S. Sh. Rekhviashvili Institute of Applied Mathematics and Automation, Nalchik, 360000 Kabardino-Balkaria, Russia e-mail:
[email protected] Received August 8, 2016
Abstract—A new equation of state for fullerite C60 is derived in the framework of the quantum-statistical method. This equation includes two Grüneisen parameters responsible for vibration–rotational and intramolecular contributions of fullerene molecules, which are represented in the form of isotropic quantum oscillators. The intramolecular vibrations of carbon atoms are described by the Debye heat capacity theory, and the cold contribution to the free energy is calculated using the Lennard–Jones pair potential for fullerene molecules. The theory is in a very good agreement with the experiment. DOI: 10.1134/S1063783417040229
Experimental studies of the thermodynamic properties of fullerenes and their various solutions are frequently concluded by thermal analysis and calorimetric analysis of the temperature dependence of isobaric heat capacity. There are much less reliable and consistent data on wide-range equations of state of these materials. This concerns the most widely used fullerite C60, which is a promising material for nanotechnologies. Studies of physicochemical, electrophysical, and optical properties of fullerites under the action of high compressive pressures are of significant interest for applications and theory [1–4]. In particular, fullerene coatings show promise for development of functional and construction materials with unique mechanical properties. In order to predict the properties of fullerites in a wide range of temperatures, it is necessary to know their equation of state. The first attempts to construct the equation of state of fullerite C60 were made in [5, 6]. The theoretical method used in [5] was based on the calculation of the second virial coefficient and application of the pair potential for fullerene molecules. In [6], the pressure, the volumetric coefficient of thermal expansion, and the compressibility coefficient were calculated in the Debye– Grüneisen approximation. In both cases, the intermolecular potential of the C60–C60 interaction, obtained by the double integration of the Lennard– Jones pair potential for carbon atoms over the spherical surface of the fullerene molecule, was used. These approaches were later developed in [7–12], where the anharmonicity of vibrations and optical phonons were taken into account, the theory of free volume was used, etc. In addition, experimental data can be successfully described by empirical and phenomenological equations of state, e.g., the Birch–Murnaghan equation [13]. Unfortunately, none of these theoretical
models takes into account the possible rotation of fullerene molecules. A qualitatively new model of thermodynamic properties of fullerite C60 was proposed in author’s work [14]. In this model, fullerene molecules are represented by isotropic quantum oscillators performing a vibration–rotational motion. The contribution of thermal vibrations of carbon atoms found on the surface of C60 molecules are described by the standard Debye theory. The adequacy of this model was reliably confirmed by the comparison of calculation results with experimental data on the isochoric heat capacity in a wide temperature range (from 5 to 1000 K). This paper is a continuation of [14] and is concerned with deriving the equation of state for fullerite C60. The free energy of fullerite is the sum of the potential energy U and the vibration–rotational, F1, and intramolecular, F2, contributions. The cold component of the free energy, caused by the attraction and repulsion forces between fullerene molecules, is calculated in the continual approximation using the Lennard–Jones potential; the calculation of the thermal component of the free energy is based on the results of [14]. The separate consideration of the vibration– rotational and intramolecular contributions as an independent description of the modes of the “slow” and “fast” subsystems corresponds, in the given case, to the adiabatic approximation. It is assumed that C60 molecules do not lose their spherical form upon compression and isotropic oscillators can be applied in the entire range of pressure. The energy of a crystalline substance can be calculated by the summation of the pair potential of interaction of atoms and molecules. However, for substances with densely packed particles, it is more convenient to
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use the continual approximation, in which the summation is replaced with the integration over the region occupied by the substance, which makes it possible to obtain simple analytical formulas [15]. In this approximation, the potential energy of one mole of fullerite is ∞
3N A U (z ) = φ(r )r 2dr, 3 2z
∫
(1)
z
where NA is the Avogadro number, z defines the size of the spherical cavity occupied by the fullerene molecule, and φ(r) is the pair potential of interaction for fullerene molecules. It should be noted that z > z0, where z0 = 0.357 nm is the radius of a free fullerene molecule. The potential of interaction of two fullerene molecules should be calculated ab initio by quantumchemical methods with subsequent averaging over molecular orientations. However, for practical purposes, it is expedient to use some approximation of the interaction potential. In a wide range of temperatures and pressures, separate C60 molecules in the composition of a condensed system are identified as inert spherical particles preserving their individuality; therefore the structure of fullerite is similar to the structure of solid noble gases [16]. In consequence, the authors of [17] came to the conclusion that the pair potential of interaction between fullerene molecules can be approximately described by the Lennard– Jones formula, which can be written in the form 6 ⎡ r 12 r ⎤ φ(r ) = D ⎢⎛⎜ 0 ⎞⎟ − 2 ⎛⎜ 0 ⎞⎟ ⎥ , ⎝r⎠ ⎦ ⎣⎝ r ⎠
(2)
where D and r0 are parameters of the potential and r is the distance between the centers of molecules. Substituting (2) into (1) and performing the integration, we find
U (V ) =
3N A D ⎡⎛V 0 ⎞ ⎛V 0 ⎞ ⎢⎜ ⎟ − 2 ⎜ ⎟ 2 ⎣⎝ V ⎠ ⎝V ⎠ 4
2
⎤ ⎥, ⎦
(3)
where V is the volume of the phase and V0 is the volume of the phase corresponding to the minimum potential energy at zero temperature (U0 = –3NAD/2 at V = V0). The bulk modulus is 2 12N A D ⎛ ⎞ . B = ⎜V d U2 ⎟ = V0 ⎝ dV ⎠V0
(4)
The numerical values of the modulus B obtained for fullerite C60 have a substantial scatter. In particular, the experimental values lie in the range of 8.17 to 18.1 GPa and the theoretical values exceed 70 GPa [13].
The formulas derived in [14] for vibration–rotational and intramolecular contributions to the free energy of fullerite read 1 ⎧⎪3θ 2θ 1 F1 = 9R ⎨ + T ln ⎡1 − exp ⎜⎛ − 1 x ⎟⎞⎤ ⎢ ⎣ ⎝ T ⎠⎦⎥ ⎪⎩ 8 0 θ × ⎡1 − exp ⎜⎛ − 1 x ⎟⎞⎤ x 2dx , ⎢⎣ ⎝ T ⎠⎦⎥
∫
(5)
}
1 ⎧⎪θ ⎫⎪ θ 2 F2 = 540R ⎨ + T ln ⎡1 − exp ⎜⎛ − 2 x ⎟⎞⎤ x 2dx ⎬ , (6) ⎣⎢ ⎝ T ⎠⎦⎥ ⎪⎩ 8 ⎪⎭ 0 where R is the universal gas constant and T is the absolute temperature. Formulas (5) and (6) take into account fluctuations in the ground state at T = 0, which exist due to the uncertainty principle. In formulas (5) and (6), at T ≥ 0, the characteristic temperatures θ1 and θ2 are functions of volume V. Differentiating formulas (3), (5), and (6) with respect to the volume, taking into account (4), we find the equation of state
∫
( )
p = − ∂F ∂V
T
⎡ V 5 V 3 ⎤ γ E + γ 2E 2 = B ⎢⎛⎜ 0 ⎞⎟ − ⎛⎜ 0 ⎞⎟ ⎥ + 1 1 , 2 ⎣⎝ V ⎠ ⎝ V ⎠ ⎦ V
⎡ 1 exp ⎛ − θ1 x ⎞⎛ 3exp⎛ − θ1 x ⎞ + 1⎞ x 3dx ⎤ ⎜ ⎟⎜ ⎜ ⎟ ⎟ ⎢ ⎥ ⎝ T ⎠⎝ ⎝ T ⎠ ⎠ E1 = 9Rθ1 ⎢3 − ⎥, θ exp ⎛⎜ − 2 1 x ⎞⎟ − 1 ⎢8 0 ⎥ ⎣ ⎝ T ⎠ ⎦ θ ⎛ (7) 1 exp ⎛ − 2 x ⎞ x 3dx ⎞ ⎜ ⎟ ⎜1 ⎟ ⎝ ⎠ T E 2 = 540Rθ 2 ⎜ − ⎟, θ2 ⎞ ⎛ 8 ⎜ x⎟ −1 ⎟ 0 exp ⎜ − ⎝ ⎝ T ⎠ ⎠ ∂ ln θ i γi = − (i = 1,2), ∂ ln V where γi > 0 are analogs of the Grüneisen parameter responsible for vibration–rotational (i = 1) and intramolecular (i = 2) contributions. Unfortunately, the integrals in Eq. (7) are not expressed in elementary functions; therefore, the calculation of them requires numerical methods. In Eq. (7), it remains to determine the dependence of the characteristic temperatures θi on the volume V. In the simplest case, for γi = const, the characteristic temperatures are obtained in the form of power functions of volume:
∫
∫
γ
V i (8) θ i = θ 0i ⎛⎜ 0 ⎞⎟ , ⎝V ⎠ where θ0i are parameters determined from experimental data on the isochoric heat capacity [14]. In order to test the theoretical model, numerical calculations were performed by formulas (7) and (8) at T = 293 K and the comparison with the experimental data from [13] was made. The results are presented in the figure. For the bulk modulus, the value B =
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EQUATION OF STATE FOR FULLERITE C60
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exchange repulsion forces, extremely large values (>40) were obtained. The problem is that this parameter was determined near the minimum of the potential and describes the “softness” of repulsion not sufficiently correctly. The use of these values of the exponents for the molecular repulsion energies leads to too fast an increase in the potential energy of fullerite with a decrease in the intermolecular distances in the region of decreasing pressures and does not describe the experimental data from [11–13]. ACKNOWLEDGMENTS This work was supported by the Department of Nanotechnologies and Information Technologies of the Russian Academy of Sciences, grant no. 5 “Fundamental Problems of Physics and Technology of Epitaxial Nanostructures and Devices on their Basis.” Isotherm of fullerite C60: (solid curve) calculation by Eq. (7) and (circles) experimental data from [13].
REFERENCES
15.6 GPa was taken [13]. The equilibrium volume at zero temperature was calculated through the molar mass and density of fullerite V0 = M/ρ = 4.4 × 10–4 m3/mol. The characteristic temperatures are θ01 = 47 K and θ02 = 1630 K [14]. The parameters γi were fitted to reach the best agreement between the calculated and experimental results. Finally, within the computational error, they took equal values γ = γ1 = γ2 = 0.22. At the given numerical values of all parameter, the transcendental equation p = 0 has an approximate solution V/V0 ≈ 1.035, which was taken into account when plotting the graph. It is seen from the figure that the calculations as a whole are in a very good agreement with experimental data. The mean relative error is ∼1%. The calculations performed show that, at not high temperatures and large compressing pressures, the intramolecular dynamics of carbon atoms makes the dominant contribution to the equation of state as compared to the vibration-rotational dynamics of fullerene molecules. From the physical viewpoint, this can be related to the fact that, with a reduction in the distance between C60 molecules, their thermal motion in the crystal lattice is hampered but the intramolecular dynamics of carbon atoms changes not so essentially. The Grüneisen parameter corresponding to intramolecular phonon modes of the C60 crystal was determined in [1] by Raman spectroscopy. The maximum value is γ = 0.23, which practically coincides with the result of the present work. As to the vibration–rotational dynamics of fullerene molecules, as far as the author is aware, its influence on the Grüneisen parameter or the equation of state has not been hitherto studied experimentally. In conclusion, it should be noted that, in [7, 18], for the pair potential of interaction of fullerene molecules, the so-called Mie–Lennard–Jones potential was used. For the exponent characterizing the PHYSICS OF THE SOLID STATE
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Translated by E. Chernokozhin
