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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.
JOURNAL OF SCIENCE OF HNUE Mathematical and Physical Sci., 2014, Vol 59, No 7, pp 119-125 This paper is available online at http://stdb.hnue.edu.vn MECHANICAL AND THERMODYNAMIC PROPERTIES OF CO2 AND N2 O MOLECULAR CRYOCRYSTALS UNDER PRESSURE Nguyen Quang Hoc1 , Bui Duc Tinh1 and Nguyen Duc Hien2 Faculty of Physics, Hanoi National University of Education Mac 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 α-CO2 , α-N2 O, 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 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 phases CO2 -I phase or phase α, known as dry ice) has the face-centered cubic P a3 structure CO2 -II has the P 42 /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 P bcn 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 ≤ p ≤ 16 GPa and temperatures ≤ T ≤ 300 K [4] Properties of solid N2 O at pressures p ≤ 15Gpa and at T = 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-N2 O is consistent with the known low-pressure low-temperature ordered cubic form, space group P a3, up to 4.8 GPa where transition to a new solid occurs [5] Cryocrystals N2 O 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-N2 O, 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] Content 2.1 Mechanical and thermodynamic properties of cryocrystals α-CO2 and α-N2 O at pressure p = 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 N2 O is usually used in the form of the Lennard-Jones pair potential ] [( ) σ 12 ( σ )6 − ϕ(r) = 4ε r r where σ is the distance in which ϕ(r) = and ε is the depth of the potential well 120 (2.1) Mechanical and thermodynamic properties of CO2 and N2 O molecular cryocrystals The values of the parameters ε, σ are determined from the following experimental data ε/kB = 218.82K, σ = 3.829.10−10 m for β-CO2 and ε/kB = 235.48K, σ = 3.802.10−10 m for α-N2 O [8] Therefore, using the coordinate sphere method and the results in [7], we obtain the values of parameters for α-CO2 and α-N2 O as follows: [ ] ( σ )6 4ε ( σ )6 k= 265.298 − 64.01 , a a a [ ] ( σ )6 16ε ( σ )6 4410.797 − 346.172 γ= a a a [ ] ( σ )6 4ε ( σ )6 γ1 = − 40.547 , 803.555 a a a [ ] ( σ )6 4ε ( σ )6 3607.242 − 305.625 (2.2) γ2 = a a a 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-N2 O at different temperatures and pressure p = 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-N2 O 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] [ ] θ pσ y = 1.1948 + 0.1717 + 0.0862 xcthx y − 0.0087 y ε ε θ pσ − 0, 0019 xcthxy + 0.0021 y (2.3) ε ε ( )3 where y = σa , θ = kB T (kB is the Boltzmann constant), x = 2θω 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 K has the form y = 1.1948 + 0.1717y − 0087 pσ pσ y + 0.0021 y ε ε (2.4) After finding the solution a (P, 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 = K, (2.4) becomes y = 1.1948 + 0.17y − 0.00807y + 0.082y (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 K and pressure p, the parameters of α-CO2 and α-N2 O are summarized in Table Our calculated results for thermodynamic quantities of α-CO2 and α-N2 O 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 to 12 GPa and in the temperature range of to 120 K and α-N2 O exists in the pressure range of to 4.8 GPa and in the temperature range of 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-N2 O under pressure The dependence of thermodynamic quantities on temperature for α-CO2 and α-N2 O crystals at pressure p = and at preesure p ̸= 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-N2 O 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 Crystal CO2 N2 O 122 Table Parameters of α−CO2 and α−N2 O at p = 0.5 kbar, kbar and T = K ω, γ, γ1 , γ2 , a0 , p, kbar k, J/m2 1013 s−1 1021 J/m2 1021 J/m2 1021 J/m2 10−10 m 0.5 4.1687 2.2869 2.3117 0.1108 0.4671 4.1578 4.4444 2.3613 2.4559 0.1176 0.4964 4.1430 0.5 4.5225 2.3649 2.5446 0.1220 0.5141 4.1299 4.7967 2.4356 2.6900 0.1288 0.5437 4.1163 Mechanical and thermodynamic properties of CO2 and N2 O molecular cryocrystals Figure Graphs of CV (T ), Cp (T ) for α−CO2 at p = 0, p = 0.5 kbar and p = kbar 123 Nguyen Quang Hoc, Bui Duc Tinh and Nguyen Duc Hien Figure Graphs of CV (T ), Cp (T ) for α−N2 O at p = 0, p = 0.5 kbar and p = 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 N2 O molecular cryocrystals 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-N2 O with fcc structure at low pressures p = 0, 0.5 and kbar and the nearest neighbor distance of cryocrystals CO2 and N2 O with fcc structure at high pressures p = 2, 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 In comparison with the experimental data, for some values of quantities such as the nearest neighbur distance, the thermal expansion coefficient, our obtained results are relatively good but for quantities such as the adiabatic and isothermal compressibilities, the heat capacities at constant volume and at constant pressure our obtained results only agree in the order of magnitude The dependence of thermodynamic quantities on temperature for α-CO2 and α-N2 O under pressure is in physical agreement with that at zero pressure Our results will be more consistent with experiments by taking account of molecular rotation and intermolecular motion Our obtained results can be enlarged to cases in higher pressures REFERENCES [1] S A Bonev, F Gygi, T Ogitsu and G Galli, 2003 High-pressure molecular phases of solid carbon dioxide Phys Rev Lett 91, No 6, p 065501 [2] J P Perdew, K Burke and M Ernzerhof, 1996 Generalized Gradient Approximation Made Simple Phys Rev Lett.77, No 18, pp 3865-3868 [3] R Le Sar and R G Gordon, 1982 Electron-gas model for molecular crystals Application to the alkali and alkaline-earth hydroxides Phys Rev B 25, No 12, pp 7221-7237 [4] C S Yoo, H Cynn, F Gygi, G Galli, V Iota, M Nicol, S Carlson, D Hausermann and C Maihiot, 1999 Crystal structure of carbon dioxide at high pressure: “superhard” polymeric carbon dioxide Phys Rev Lett 83, No 26, pp 5527-5530 [5] R L Mills, B Olinger, D T Cromer and R Le Sar, 1991 Crystal structures of N2 O to 12 GPa by X-ray diffraction J Chem Phys 95, No 7, pp 5392-5398 [6] Nguyen Quang Hoc and Tran Quoc Dat, 2011 Specific heat at constant volume for cryocrystals of nitrogen type Journal of Research on Military Technology and Science No 11, pp 81-86 [7] Nguyen Quang Hoc, Do Dinh Thanh and Nguyen Tang, 2000 Some thermodynamic properties of the CO2 and N2 O cryocrystals, VNU Journal of Science, Nat Sci, 16, No 4, pp 22-26 [8] B I Verkina, A Ph Prikhotko (ed.), 1983 Cryocrystals, Kiev, pp 1-526 (in Russian) [9] R Stevenson, 1928 Compressions and Solid Phases of CO2 , CS2 , COS, O2 and CO J Chem Phys 27, No 3, pp 673-764 [10] P W Bridgman, The melting curves and compressibilities of nitrogen and argon Proc Amer Acad Arts Sci 70, No 1, pp 1-32 125 ... constant pressure of cryocrystals α -CO2 and α-N2 O with fcc structure at low pressures p = 0, 0.5 and kbar and the nearest neighbor distance of cryocrystals CO2 and N2 O with fcc structure at high pressures... 2.6900 0.1288 0.5437 4.1163 Mechanical and thermodynamic properties of CO2 and N2 O molecular cryocrystals Figure Graphs of CV (T ), Cp (T ) for α? ?CO2 at p = 0, p = 0.5 kbar and p = kbar 123 Nguyen... distance in which ϕ(r) = and ε is the depth of the potential well 120 (2.1) Mechanical and thermodynamic properties of CO2 and N2 O molecular cryocrystals The values of the parameters ε, σ are