Mechanical and thermodynamic properties of CO2 and N2O molecular cryocrystals under pressure

JOURNAL OF SCIENCE OF HNUE Mathematical and Physical Sci., 2014, Vol. 59, No. 7, pp. 119-125 This paper is available online at MECHANICAL AND THERMODYNAMIC PROPERTIES OF CO2 AND N2OMOLECULAR CRYOCRYSTALS UNDER PRESSURE Nguyen Quang Hoc1, Bui Duc Tinh1 and Nguyen Duc Hien2 1Faculty of Physics, Hanoi National University of Education 2Mac Dinh Chi Secondary School, Chu Pah District, Gia Lai Province Abstract. The mechanical and thermodynamic properties (such as the nearest neighbor distance, the molar volume, the adiabatic and isothermal compressibilities, the thermal expansion coefficient and the heat capacities at constant volume and at constant pressure) of molecular cryocrystals of many atoms with a face-centered cubic structure such as -CO2, -N2O, at various temperatures and at pressures up to 10 GPa are investigated using the statistical moment method (SMM) in statistical mechanics and compared with the experimental data. Keywords:Molecular cryocrystal, statistical moment method. 1. Introduction Molecular crystals are characterized by strong intramolecular forces and much weaker intermolecular forces. High-pressure spectroscopic studies provide useful data for refining the various model potentials which are used to predict of the physical properties of such systems as well as for the formation of various crystalline phases. CO2 is an important volatile component of the earth as well as other planets in the solar system. Its high-pressure behavior is therefore of fundamental importance in planetary science. CO2 is one of the model systems involving the π bonding and the hybridization properties of the carbon atom, which are strongly affected by high pressure conditions. Pressure-induced transitions from molecular to nonmolecular CO2 crystals are systematically investigated using first-principle lattice dynamics calculation. Geometrically, likely transition pathways are derived from the dynamical instability of the molecular crystals under high pressures. The phase diagram of CO2 consists of 5 phases. CO2-I phase or phase α, known as dry ice) has the face-centered cubic Pa3 structure. CO2-II has the P42/mnm symmetry. Received August 20, 2014. Accepted October 1, 2014. Contact Nguyen Quang Hoc, e-mail address: hocnq@hnue.edu.vn 119 Nguyen Quang Hoc, Bui Duc Tinh and Nguyen Duc Hien CO2-III has the orthorhombic Cmca symmetry. CO2-IV has Pbcn symmetry. CO2-V is the polymeric phase of a tridymite-like structure. In [1], Bonev et al. performed a series of first-principle calculations, including full structural optimizations, phonon spectra and free energies, in order to study the stability and properties of the phases proposed experimentally up to 50 GPa and 1500 K. The DFT calculations were carried out within the Perdew-Burke-Ernzerhof generalized gradient approximation (CGA) [2] using the ABINIT code which implements plane-wave basis sets. Le Sar et al. [3] presented an ab initio method, based on the modified Gordon-Kim (MGK) electron-gas model which worked well in calculating the structure and properties of molecular crystals. A constant pressure Monte Carlo formalism, lattice dynamics and classical perturbation theory are used to calculate the thermal expansion, the pressure-volume relation at room temperature, the temperature dependence of zone center libron frequencies and the pressure dependence of the three vibron modes of vibration in solid CO2 at pressures 0 ≤ p ≤ 16 GPa and temperatures 0 ≤ T ≤ 300 K [4]. Properties of solid N2O at pressures p ≤ 15Gpa and at T = 0 and 300 K have been calculated using energy optimization and Monte Carlo methods in an (N, p, T ) ensemble with periodic, deformable boundary conditions and lattice dynamics. α-N2O is consistent with the known low-pressure low-temperature ordered cubic form, space group Pa3, up to 4.8 GPa where transition to a new solid occurs [5]. Cryocrystals N2O and CO2 are ideal systems on which to have a study of the influence of quantum effects on condensed matter. There has been considerable interest in structural and thermodynamic properties of these crystals under temperature and pressure and in line with this general interest and encouraged by the essential success of our calculations, we tried to consider the mechanical and thermodynamic properties of cryocrystals of many atoms with face-centered cubic structure such as α-N2O, α-CO2 at various temperatures and pressures up to 10 GPa. Heat capacities at constant volume for these crystals are studied by combining the SMM and the self-consistent field method taking into account the lattice vibration and the molecular rotational motion [6]. 2. Content 2.1. Mechanical and thermodynamic properties of cryocrystals α-CO2 and α-N2O at pressure p = 0 It is known that the interaction potential between two atoms in α phase of molecular cryocrystals of N2 type such as solids N2, CO, CO2 and N2O is usually used in the form of the Lennard-Jones pair potential ϕ(r) = 4ε [(σ r )12 − (σ r )6] (2.1) where σ is the distance in which ϕ(r) = 0 and ε is the depth of the potential well. 120 Mechanical and thermodynamic properties of CO2 and N2O molecular cryocrystals... The values of the parameters ε, σ are determined from the following experimental data. ε/kB = 218.82K, σ = 3.829.10−10m for β-CO2 and ε/kB = 235.48K, σ = 3.802.10−10m for α-N2O [8]. Therefore, using the coordinate sphere method and the results in [7], we obtain the values of parameters for α-CO2 and α-N2O as follows: k = 4ε a2 (σ a )6 [ 265.298 (σ a )6 − 64.01 ] , γ = 16ε a4 (σ a )6 [ 4410.797 (σ a )6 − 346.172 ] γ1 = 4ε a4 (σ a )6 [ 803.555 (σ a )6 − 40.547 ] , γ2 = 4ε a4 (σ a )6 [ 3607.242 (σ a )6 − 305.625 ] . (2.2) where a is the nearest neighbor distance at temperature T. Our calculated results for the nearest neighbor distance a, the adiabatic and isothermal compressibilities χT , χS, the thermal expansion coefficient β and the heat capacities at constant volume and constant pressure CV , Cp of α-CO2 and α-N2O at different temperatures and pressure p = 0 are shown in [7]. In general, our calculations are in qualitative agreement with experimental results. 2.2. Mechanical and thermodynamic properties of cryocrystals α-CO2 and α-N2O under pressure In order to determine thermodynamic quantities at various pressures, we must find the nearest neighbor distances. The equation for calculating the nearest neighbor distances at pressure P and at temperature T has the form [7] y2 = 1.1948 + [ 0.1717 + 0.0862 θ ε xcthx ] y4 − 0.0087pσ 3 ε y5 − 0, 0019θ ε xcthxy6 + 0.0021 pσ3 ε y7. (2.3) where y = ( a σ )3 , θ = kBT (kB is the Boltzmann constant), x = ~ω2θ . This is a nonlinear equation and therefore, it has only an approximate solution. From that, the equation for calculating the nearest neighbor distances at pressure p and at temperature 0 K has the form y2 = 1.1948 + 0.1717y4 − 0..0087pσ 3 ε y5 + 0.0021 pσ3 ε y7. (2.4) After finding the solution a (P, 0 K) from (2.4), we can calculate a (P, T) and other thermodynamic quantities. This means is applied to crystal at low pressures. For crystal at high pressures, we must find the solution using (2.4). 121 Nguyen Quang Hoc, Bui Duc Tinh and Nguyen Duc Hien For example in the case of α-CO2 at p = 0.5 kbar, T = 0 K, (2.4) becomes y2 = 1.1948 + 0.17y4 − 0.00807y5 + 0.082y7. (2.5) The solution of this equation is y = 1.281967, i.e. the nearest neighbor distance under the condition p = 0.5 kbar, T = 0K takes a value a = 4.1578.10−10 m. At temperature 0 K and pressure p, the parameters of α-CO2 and α-N2O are summarized in Table 1. Our calculated results for thermodynamic quantities of α-CO2 and α-N2O at different temperatures and pressures up to 10 GPa are shown in Figures 1-11. According to the experimental data, α-CO2 exists in the pressure range of 0 to 12 GPa and in the temperature range of 0 to 120 K and α-N2O exists in the pressure range of 0 to 4.8 GPa and in the temperature range of 0 to 130 K. Our numerical results are carried out in these ranges of temperature and pressure. We have only the experimental data for the phase diagram and the molar volume of α-CO2 and α-N2O under pressure. The dependence of thermodynamic quantities on temperature for α-CO2 and α-N2O crystals at pressure p = 0 and at preesure p ̸= 0 have same behaviour. Our results would be more consistent with experiments if we take molecular rotation and intermolecular motion into account. Our obtained results can be enlarged to cases in higher pressures. Our calculated results for molecular crystals α-CO2 and α-N2O show that at same pressure, when temperature increases heat capacities CV and Cp increase. At same temperature, when pressure increases the heat capacities CV and Cp decrease. In the interval of pressure shown in figures, when temperature T < 20 K, heat capacities CV and Cp approximately are equal to zero. At mentioned pressures, in the range from 50 K to 110 K, heat capacities CV and Cp increase strongly. At same temperature, when pressure increases the value of heat capacity CV comes to the value of heat capacity CP . Table 1. Parameters of −CO2 and −N2O at p = 0.5 kbar, 1 kbar and T = 0 K Crystal p, kbar k, J/m2 !,1013s1 , 1021J/m2 1, 1021J/m2 2, 1021J/m2 a0, 1010m CO2 0.5 4.1687 2.2869 2.3117 0.1108 0.4671 4.1578 1 4.4444 2.3613 2.4559 0.1176 0.4964 4.1430 N2O 0.5 4.5225 2.3649 2.5446 0.1220 0.5141 4.1299 1 4.7967 2.4356 2.6900 0.1288 0.5437 4.1163 122 Mechanical and thermodynamic properties of CO2 and N2O molecular cryocrystals... Figure 7. Graphs of CV (T ), Cp(T ) for α−CO2 at p = 0, p = 0.5 kbar and p = 1 kbar 123 Nguyen Quang Hoc, Bui Duc Tinh and Nguyen Duc Hien Figure 8. Graphs of CV (T ), Cp(T ) for α−N2O at p = 0, p = 0.5 kbar and p = 1 kbar Figure 11. Dependence of relative change of molar volume on pressure at temperature 77 K for α−CO2 from our calculated result (SMM) and experiments (EXPT) [9, 10] 124 Mechanical and thermodynamic properties of CO2 and N2O molecular cryocrystals... 3. Conclusion In this paper, we calculate thermodynamic properties such as the nearest neighbor distance, the isothermal and adiabatic compressibilities, the thermal expansion coefficient, the heat capacities at constant volume and at constant pressure of cryocrystals α-CO2 and α-N2O with fcc structure at low pressures p = 0, 0.5 and 1 kbar and the nearest neighbor distance of cryocrystals CO2 and N2Owith fcc structure at high pressures p = 2, 6 and 10 GPa at different temperatures. Our calculated result for the relative change of molar volume versus pressure at temperature 77 K for α-CO2 is compared with the experimental data. 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