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 N2O MOLECULAR CRYOCRYSTALS UNDER PRESSURE
Nguyen Quang Hoc^, Bui Due Tinh^ and Nguyen Due Hien^
^Faculty of Physics, Hanoi Natioruil 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 heal capacities at constant volume and at constant pressure) of molecular cryocrystals of many atoms with a face-centered cubic structure such as Q-CO2, Q-N2O, at various temperatures and at pressiues 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 stiong intiamolecular forces and much weaker intermolecular forces. High-pressure spectioscopic 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 crystalhne 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 die TT bonding and die hybridization properties of die carbon atom, which are stiongly affected by high pressure conditions.
Pressure-induced transitions from molecular to nonmolecular CO2 crystals are systematically investigated using fhst-principle lattice dynamics calculation.
Geometrically, likely tiransition 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 a, known as dry ice) has the face-centered cubic P a 3 stiuctiue. C 0 2 - n has the P42/mnm symmetiy.
Received August 20, 2014. Accepted October 1, 2014.
Contact Nguyen Quang Hoc, e-mail address: [email protected]
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Nguyen Quang Hoc, Bui Due Tinh and Nguyen Due Hien
CO2-III has the orthorhombic Cmca symmeti-y. CO2-IV has Pbcn symmetty. CO2-V is the polymeric phase of a tridymite-like stiucture. In [1], Bonev et al. performed a series of first-principle calculations, including full sttuctural optimizations, phonon spectia 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) electton-gas model which worked well in calculating the stiucture 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 < IbGpa and at T — 0 and 300 K have been calculated using energy optimization and Monte Carlo methods in an (A'^, p, T) ensemble with periodic, deformable boundary conditions and lattice dynamics. a-N20 is consistent with the known low-pressure low-temperature ordered cubic form, space group PaZ, 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 smdy of the influence of quantum effects on condensed matter. There has been considerable interest in stiuctural 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 widi face-centered cubic stiuctine such as Q:-N20, o-COa 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 a-C02 and Q : - N 2 0 at pressure p = 0
It is known that the interaction potential between two atoms in a phase of molecular cryocrystals of N2 type such as solids Ng, CO, CO2 and N2O is usually used in the form of the Lennard-Jones pair potential
m-.[{^r-a
(2.1)where <T is the distance in which ipir) = 0 and s is the depth of the potential well.
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The values of the parameters e, CJ are determined from the following experimental data. E/kg = 21S.82K,a = 3.829.1G-'°m for /S-COa and E/AB = 235.48/f,CT = 3.802.10"'°7rt for Q-N2O [8]. Therefore, using the coordinate sphere method and the results in [7], we obtain the values of parameters for Q-CO2 and Q-N2O as follows:
o'hHd'
a^\Q
^ = ^ ( ^ V | 4 4 1 Q . 7 9 7 n ° - 3 4 6 . 1 7 2
S ( D ° 3607.242(9^-305.625]. (2.2)
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where a is the nearest neighbor distance at temperature T. Our calculated results for the nearest neighbor distance a, the adiabatic and isothermal compressibihties XT^.XS^
die thermal expansion coefficient /? and the heat capacities at constant volume and constant pressure Cv-, Cp of Q-CO2 and a-N20 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 a-C02 and a - N 2 0 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]
1/2 = 1.1948-F 0.1717+ 0.0862-xc(/ixU'^-0.0087—y^
- 0,0019-xcf/ix/ + 0.0021—j/^ (2.3) where y = [^Y,9 = ksT (ks is the Boltzmann constant), x = ^. This is a nonlinear
equation and therefore, it has only an approximate solution. From that, the equation for calculating the nearest neighbor distances at pressine p and at temperature 0 K has the form
1/2 = 1.1948 + 0.1717/ - 0..0087—y^ -h 0.0021^?/^ (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 Uie solution using (2.4).
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Nguyen Quang Hoc, Bui Due Tinh and Nguyen Due Hien
For example in the case of Q-CO2 atp = 0.5 kbar, T = 0 K, (2.4) becomes
y'^ = 1.1948 + Q.17y'^ - 0.00807j/^ -b 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 = OK takes a value t 4.1578.10-1" ^ ^ ( temperature 0 K and pressure p, the parameters of a-COa and a - N 2 0 are summarized in Table 1. Our calculated results for thermodynamic quantities of a-C02 and a-N20 at different temperatiues and pressures up to 10 GPa are shown in Figures 1-11. According to die experimental data, Q-CO2 exists in the pressure range of 0 to 12 GPa and in the temperature range of 0 to 120 K and a;-N20 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 Q-CO2 and a - N 2 0 under pressure. The dependence of thermodynamic quantities on temperature for Q-CO2 and a - N 2 0 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 Q-CO2 and Q ' - N 2 0 show that at same pressure, when temperature increases heat capacities Cy and Cp increase. At same temperature, when pressure increases the heat capacities Cy and Cp decrease. In the interval of pressiu:e shown in figures, when temperatiu-e T < 20 K, heat capacities Cy and Cp approximately are equal to zero. At mentioned pressiues, in the range from 50 K to 110 K, heat capacities Cy and Cp increase sU^ongly. At same temperature, when pressure increases the value of heat capacity Cy comes to the value of heat capacity Cp.
Table L Parameters ofa-CO^ and ot~N20 atp = 0.5 kbar, I kbar andT = OK Crystal
CO2 N2O
p, kbar 0.5
1 0.5 1
k, ilm' 4.1687 4.4444 4.5225 4.7967
1 0 " s - i 2.2869 2.3613 2.3649 2.4356
7 . WHittf
2.3117 2.4559 2.5446 2.6900
7 l , WStrtf
0.1108 0.1176 0.1220 0.1288
72, 1 0 " I / m 2
0.4671 0.4964 0.5141 0.5437
10-'"in 41578 4.1430 4.1299 41163
Figure 1. Graphs ofaiT) for a-CO, at p = 0,p = 0.S kbar and p = 1 kbar
Figure 2. Graphs of a{T) for a-Nfi al p = 0,p = 0.5kbarandp = 1 kbar
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20 40 60 80 100 120 140
Figure 3. Graphs of XT (?") (curves I, 3, 5) Figure 4. Graphs of Xi (T) (curves I, 3, S) and Xs (T) (curves 2, 4, 6) for a-COi and Xs (^) (curves 2, 4, 6) for a-NiO
alp = 0,p = 0.5 kbar andp " / kbar atp = 0,p = 0.5 kbar andp = I kbar
G
t'
(Q. 2
2 0 _fcbar
4 0 60
.--"•"t^^^*-^^:^^^^
•X^
80 1D0 12
Figure 5. Graphs of PiT) for a-COj atp = Q,p = 0.5 kbar andp = I kbar
Figure 6. Graphs of P{T) for ct-IWjO atp = 0,p = 0.5 kbar andp = I kbar
C„ (P = 0) i^p (P = 0 ) C„ (p = 0.5 kbar) Cp (p = 0.5 kbar) C^ (p = 1 kbar) C, (p = 1 kbar)
Figure 7. Graphs ofCviT), C^iT)for a-CO, atp = 0,p = 0.5 kbar andp = I kbar
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Nguyen Quang Hoc, Bui Due Tinh and Nguyen Due Hien
C>, (P = 0) Cp (P = 0) C^ (p = 0.5 kbar) Cp (p = 0.5 kbar) C„ (p = 1 kbar) Cp (p = 1 kbar)
Figure 8. Graphs ofCviT), CfiT)for a-N20 atp =0,p= 0.5 kbar andp = 1 kbar
100 120 140 100 120 140
Figure 9, Graphs of a(T) for OrCOi Figure 10. Graphs of a(T) for a-Nfi alp = 2 GPa, p = 6GPa andp = 10 GPa "tp = 2 GPa, p = 6GPa andp = 10 GPa
Figure 11. Dependence of relative change of molar volume on pressure at temperature 77 Kfor a—C02from our calculated result (SMM) and experiments (EXPT) [9,101
3. Conclusion
In this paper, we calculate thermodynamic properties such as the nearest neighbor distance, the isothermal and adiabatic compressibihties, the thermal expansion coefficient, the heat capacities at constant volume and at constant pressure of cryocrystals Q'-C02 and Q ; - N 2 0 with fee sttucture at low pressures p = 0, 0.5 and 1 kbar and the nearest neighbor distance of cryocrystals CO2 and N2O with fee sttucture 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 a-C02 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 a-C02 and Q-N2O 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.
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