Mechanical and thermodynamic properties of Co2 and N2O molecular cryocrystals under pressure

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.

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MECHANICAL AND THERMODYNAMIC PROPERTIES

Nguyen Quang Hoc1, Bui Duc Tinh1and 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 α-CO2, α-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 CO2consists of 5 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

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 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 P a3, 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.

The values of the parameters ε, σ are determined from the following experimental data ε/k B = 218.82K, σ = 3.829.10 −10 m for β-CO2 and ε/k B = 235.48K, σ = 3.802.10 −10 m 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

]

where a is the nearest neighbor distance at temperature T Our calculated results for the nearest neighbor distance a, the adiabatic and isothermal compressibilities χ T , χ S ,

the thermal expansion coefficient β and the heat capacities at constant volume and constant pressure C V , 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− 0.0087 pσ3

ε y

5

− 0, 0019 θ

ε xcthxy

6+ 0.0021 pσ

3

ε y

where y = (a

σ

)3

, θ = k B T (k B is the Boltzmann constant), x = ~ω 2θ 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 0087 pσ3

ε y

5 + 0.0021 pσ

3

ε y

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)

For example in the case of α-CO2at 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 C V and C p increase At same temperature, when pressure

increases the heat capacities C V and C p decrease In the interval of pressure shown in

figures, when temperature T < 20 K, heat capacities C V and C p approximately are equal

to zero At mentioned pressures, in the range from 50 K to 110 K, heat capacities C V and C pincrease strongly At same temperature, when pressure increases the value of heat

capacity C V comes to the value of heat capacity C P

Table 1 Parameters of α −CO2 and α −N2 O at p = 0.5 kbar, 1 kbar and T = 0 K

10 13 s−1 1021γ,J/m 2

γ1,

10 21 J/m 2

γ2,

10 21 J/m 2

a0,

10−10m

CO 2

Figure 7 Graphs of C V (T ), C p (T ) for α −CO2 at p = 0, p = 0.5 kbar and p = 1 kbar

Figure 8 Graphs of C V (T ), C p (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]

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 CO2and N2O with 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 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-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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