Equation of state and melting temperature for N2, CO, CO2 and N2O molecular cryocrystals under pressure

The equation of state, the absolute stability temperature of crystalline state and the melting temperature for N2, CO, CO2 and N2O molecular cryocrystals under pressure are determined using the statistical moment method and are compared with the experimental data and other calculations.

JOURNAL OF SCIENCE OF HNUE Mathematical and Physical Sci., 2014, Vol 59, No 7, pp 67-75 This paper is available online at http://stdb.hnue.edu.vn EQUATION OF STATE AND MELTING TEMPERATURE FOR N2 , CO, CO2 AND N2 O MOLECULAR CRYOCRYSTALS UNDER PRESSURE Nguyen Quang Hoc1 , Dinh Quang Vinh1 , 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 equation of state, the absolute stability temperature of crystalline state and the melting temperature for N2 , CO, CO2 and N2 O molecular cryocrystals under pressure are determined using the statistical moment method and are compared with the experimental data and other calculations Keywords: Molecular cryocrystal, statistical moment method, limiting temperature, absolute stability Introduction Molecular crystals are characterized by their 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 the physical properties of such systems as well as the formation of various crystalline phases In the most cases, the melting temperature of crystals is described by the empirical Simon equation ln (P + a) = c ln T + b, where a, b and c are constant and P and T, respectively, are the melting pressure and the melting temperature [1] However, this equation cannot be used for crystals at extremely high pressure On the theoretical side, in order to determine the melting temperature we must use the equilibrium condition of the liquid and solid phases However, a clear expression of the melting temperature has not yet been obtained in this way Notice that the limiting temperature of absolute stability for the crystalline state at a determined pressure is not far from the melting temperature Therefore, some researchers had identified the melting curve with the curve of absolute stability for the crystalline state In order to better determine the limiting temperature of absolute stability for the crystalline state, the correlation effects are calculated using the one-particle distribution function method [4, 10] Because the difference between these two temperatures is large Received January 7, 2014 Accepted September 30, 2014 Contact Nguyen Quang Hoc, e-mail address: hocnq@hnue.edu.vn 67 Nguyen Quang Hoc, Dinh Quang Vinh, Bui Duc Tinh and Nguyen Duc Hien at high pressure, this approximation is effective only at low pressure Other researchers have concluded that it is impossible to find the melting temperature using only the limit of absolute stability for the solid phase because the obtained results on the basis of the self-consistent phonon method and the one-particle distribution function method are larger than the corresponding melting temperatures by a factor of to and 1.3 to 1.6, respectively [5] On the basis of the statistical moment method (SMM) in statistical mechanics, some authors have determined the limiting temperature of absolute stability for the crystalline state at various pressures and then they adjust this temperature in order to find the melting temperature [9, 10] The melting temperature is obtained by this way for low as well as high pressures The calculated results for the inert gas crystals agree rather well with the experimental data [2] In the present study, we apply the SMM to investigate the equation of state and the melting temperature of solid N2 , CO, CO2 and N2 O We will calculate the pressure dependence of the lattice constant and the melting temperature of these crystals Content 2.1 Equation of state, limiting temperature of absolute stability and melting temperature for molecular cryocrystals The equation of state of a crystal with a face-centered cubic (fcc) structure can be written in the following form [2]: ( ) a ∂u0 Pv = − + 3γGT θ, ∂a T √ ∑ a ∂k ω v= a , u0 = ϕi0 (|⃗ai |) ,γGT = − X, X ≡ x coth x, θ = kB T, x = , 6k ∂a 2θ i 1∑ k= i ( ∂ ϕi0 ∂u2iβ ) ≡ mω , β = x, y, z (2.1) eq here P is the hydrostatic pressure, v is the volume of the fcc lattice, a is the nearest neighbor distance of the fcc crystal, ⃗ai is the vector determining the equilibrium position of the ith particle, ϕi0 is the interaction potential between the ith particle and the 0th particle, γGT is the Gruneisen constant and kB is the Boltzmann constant From the limiting condition of absolute stability for the crystalline state ( 68 ∂P ∂v ) T ( ∂P = 0, i.e ∂a ) = 0, T (2.2) Equation of state and melting temperature for N2 , CO, CO2 and N2 O molecular cryocrystals we find the corresponding expression of the limiting temperature Ts as follows: ( ) 2P v + a6 ∂∂au20 T ] [ ( T) Ts = ∂γG 3kB a ∂a − γGT (2.3) T If we take the values of the parameters a, k, ω at the same limiting temperature of absolute stability Ts then (2.3) can be transformed into the form [2] [( { ( } ) ) ( )2 ] 4k 2P v ω ∂ 2k ∂k ∂ u0 Ts = + + − (2.4) ( )2 ∂a2 T 4k ∂a2 T 2k ∂a T a kB ∂k ∂a T In the case of P = 0, it gives { ( [( ) ) ( )2 ]} ∂ u0 4k ω ∂k ∂2k Ts = + − ( )2 ∂a2 T 4k ∂a2 T 2k ∂a T kB ∂k (2.5) ∂a T The nearest neighbor distance a is determined by a = a0 + ux0 , where ao denotes the distance a at temperature K and is determined from the experimental data The displacement ux0 of a particle from the equilibrium position is calculated by √ ux0 = ( ) ( )  4 ∑ ∂ ϕi0  ∂ ϕi0 ,β +6 A, γ = 3k 12 i ∂uiβ ∂u2iβ ∂u2iγ 2γθ2 eq ̸= γ, β, γ = x, y, z, A = a1 + eq )i ( ∑ γθ i=2 k2 (2.6) where (i = 1−6) is determined in [2] The equation for calculating the nearest neighbor distances at pressure P and at temperature K has the form [2] y = 1.1948 + 0.1717y − 0087 ( a )3 P σ3 P σ3 y + 0.0021 y ,y = ε ε σ (2.7) We notice that the nearest neighbor distance am corresponding to the melting temperature Tm of the crystal is approximately equal to the nearest neighbor distance as corresponding to limiting temperature Ts In addition, from (2.1) we see that temperature T is a function of nearest neighbor distance a when pressure P is constant, i.e T = T(a) Therefore, we can expand temperature Tm according to the distance difference am - as and keep only the first approximate term [2] ) ( am − as P vs ∂u0 ∂ u0 Tm ≈ Ts + + + as , (2.8) kB γGs as 18 ∂as ∂as 69 Nguyen Quang Hoc, Dinh Quang Vinh, Bui Duc Tinh and Nguyen Duc Hien √ where am = a (Tm P ) , as = a (Ts , P ) , vs = ∂u0 a, s ∂as = ( ∂u ) ∂a , ∂ u0 = a=as ∂a2 s ( ∂ u0 ∂a2 ) , a=as the Gruneisen parameter γGs is regarded in the interval from T to Tm because ( a ∂k as invariable ) s it changes very little and γG = − 6k ∂a x coth x T =Ts ,a=as The equation of state of a crystal with a hexagonal close-packed (hcp) structure can be written in the following form [9]: a ∂u0 c ∂u0 − + 12γGT θ, ∂a ∂a √ ( ) ( ) T ∂kx Xz ∂kz X ∂kx ∂kz v= a − a , a c, γG = − + 2c + 2c 24kx ∂a ∂c 48kz ∂a ∂c Pv = − X ≡ x coth x, Xz ≡ xz cthxz , [( ) ) ] ( ωx ωz 1∑ ∂ ϕi0 ∂ ϕi0 = mωx2 , x= , xz = , kx = + 2θ 2θ i ∂uix eq ∂uix ∂uiy eq ( ) ∑ ∂ ϕi0 kx = = mωz2 (2.9) i ∂u2iz eq here a and c are the lattice constants of the hcp crystal, v is the volume and γGT is the Gruneisen constant From the limiting condition of absolute stability of the crystalline state ( ( ) ) ) ( ∂P ∂P ∂P = 0, i.e = 0or =0 (2.10) ∂v T ∂a T ∂c T we find the corresponding expression of the limiting temperature Ts as follows: 2 o P v + a4 ∂∂au2o − 2c ∂u ∂c ] [ ( T) Ts = ∂γ 12kB a ∂aG − γGT (2.11) x ∂kx If we take the values of the parameters a, c, kx , kz , ∂k , ∂a , at the same limiting ∂a temperature of absolute stability Ts then (2.11) can be transformed into the form [9] ( ) ( ) ω x ∂ kx ωz ∂ kz ωx ∂kx ωz ∂kz o P v + a2 ∂∂au2o − 2c ∂u + a + − 2c + ∂c kx ∂a2 2kz ∂a2 kx ∂c 2kz ∂c [ ] Ts = ( ) ( ) ∂kx ∂kx ∂kz ∂kz ∂kz x kB a k22 ∂k a + 2c + a + 2c ∂a ∂c kz2 ∂a ∂a ∂c x ∂a (2.12) In the case of P = 0, it gives ( ) ( ) ∂uo ω x ∂ kx ωz ∂ kz ωz ∂kz ωx ∂kx ∂ uo a ∂a2 − 2c ∂c + a kx ∂a2 + 2kz ∂a2 − 2c kx ∂c + 2kz ∂c [ Ts = (2.13) ( ∂kx ) ∂kz ( ∂kz )] ∂kx ∂kz x kB a k22 ∂k a + 2c + a + 2c ∂a ∂a ∂c k ∂a ∂a ∂c x 70 z Equation of state and melting temperature for N2 , CO, CO2 and N2 O molecular cryocrystals The nearest neighbor distance or the lattice constant a is determined by a = a0 + ux0 , where a0 denotes the distance a at temperature K and is determined from the experimental data The displacement ux0 of a particle from the equilibrium position in direction x or y is calculated from [( ) ) ] ( ]i [ ∑ γθ ∂ ϕi0 ∂ ϕi0 1∑ , (2.14) ux0 ≈ +6 ,γ ≡ ∂u ∂u ∂u k ix x ix iy eq eq i=1 i where (i = − 6) is determined in [9] Lattice constant c is determined by c = c0 + uz0 , where co denotes the distance c at temperature K and is determined from the experimental data Displacement uz of a particle from the equilibrium position in direction z is calculated from [ )i ]1/2 ( 1∑ θ uz0 ≈ bi , i=1 kz ( ) ( ) ( ) 1∑ ∂ φi0 1∑ ∂ φi0 ∑ ∂ φi0 , τ2 ≡ , τ3 ≡ τ1 ≡ 12 i ∂u4iz eq i ∂u2ix ∂u2iz eq i ∂uix ∂uiy ∂u2iz eq (2.15) here bi (i = − 6) is determined in [9] The equation for calculating nearest neighbor distances at pressure P and at temperature K has the form [9] P σ3 P σ3 P σ3 y − 0.0316y + 0.0007 y − 0.0001 y , ε ε ε ( a )3 y= (2.16) σ The melting temperature Tm of the hcp crystal is approximately equal to [9] [ ( [ ( )] )] am − as 2P vs ∂u0 ∂ u0 ∂ u0 cm − cs 2P vs ∂u0 Tm ≈ Ts + + + as + + as + 6kB γGs as ∂as ∂as 6kB γGs cs ∂cs ∂cs (2.17) √ where cm = c (Tm P ) , cs = c (Ts , P ) , vs = as cs and other quantities are as in (2.8) y = 0.9231 + 0.3188y − 0.0015 2.2 Numerical results and discussion For α-CO2 and α-N2 O with a fcc structure and for β-N2 and β-CO with a hcp structure, the interaction potential between two atoms is usually used in the form of the Lennard-Jones pair potential ] [( ) σ 12 ( σ )6 − , (2.18) ϕ(r) = 4ε r r where σ is the distance in which ϕ(r) = and ε is the depth of the potential well The values of the parameters ε and σ are determined from the experimental data kεB = 218.82 71 Nguyen Quang Hoc, Dinh Quang Vinh, Bui Duc Tinh and Nguyen Duc Hien K, σ = 3.829.10−10 m for α-CO2 , kεB = 235.48 K, σ = 3.802.10−10 m for α-N2 O, kεB = 95.05 K, σ = 3.698.10−10 m for β-N2 and kεB = 100.1 K, σ = 3.769.10−10 m for β-CO [3] Therefore, using two coordinate spheres and the results in [2, 9], we obtain the values of the crystal parameters for α-CO2 and α- N2 O as follows: [ ] [ ] ( σ )6 ( σ )6 4ε ( σ )6 16ε ( σ )6 k= 265, 298 − 64, 01 , γ = 4410, 797 − 346, 172 , a a a a a a (2.19) where a is the nearest neighbor distance of the fcc crystal at temperature T and the crystal parameters for β-N2 and β-CO are as follows: ] ] ( )6 [ ( )6 ( )6 [ ( )6 614.6022 σa − 162.8535 , kz = a4ε2 σa 286.3722 σa − 64.7487 , kx = a4ε2 σa ] ] ( )6 [ ( )6 ( )6 [ ( )6 γ = − a4ε3 σa 161.952 σa − 24.984 , τ1 = a4ε4 σa 6288.912 σa − 473.6748 , [ ] [ ] ( σ )6 ( σ )6 4ε ( σ )6 4ε ( σ )6 τ2 = 11488.3776 − 752.5176 , τ3 = 8133.888 − 737.352 a a a a a a (2.20) Our calculated results for the limiting temperature of absolute stability and the melting temperature of α-CO2 , α-N2 O, β-N2 and β-CO at different pressures (low pressures) are expressed in Figures 1-4 Figure The limiting temperature of absolute stability and the melting temperature at different pressures for α-CO2 The discrepancy in the melting temperature of α-CO2 that exists between our calculated results and the experimental data [7] is 5.3% at P = and increases to 18% at P = 1000 bar 72 Equation of state and melting temperature for N2 , CO, CO2 and N2 O molecular cryocrystals Figure The limiting temperature of absolute stability and the melting temperature at different pressures for α-N2 O The discrepancy in the melting temperature of α-N2 O that exists between our calculated results and the experimental data [7] is 0.04% at P = and increases to 7% at P = 1000 bar Figure The limiting temperature of absolute stability and the melting temperature at different pressures for β-N2 73 Nguyen Quang Hoc, Dinh Quang Vinh, Bui Duc Tinh and Nguyen Duc Hien The discrepancy in the melting temperature of β-N2 that exists between our calculated results and the experimental data [8] is 2.46% and between our calculated results and the experimental data [3] it is 2.69% at P = and increases to 5% compared with the experimental data [8] at P = 100 bar Our calculation is better than that in [10] Figure The limiting temperature of absolute stability and the melting temperature at different pressures for β-CO The discrepancy in the melting temperature of β-CO that exists between our calculated results and the experimental data [8] is 5.75% at P = and increases to 7.68% at P = 100 bar Conclusion From the SMM and the limiting condition of absolute stability for the crystalline state, we find the equation of state, the limiting temperature of absolute stability for the crystalline state and the melting temperature for crystals with fcc and hcp structures at zero pressure and under pressure, the equation for calculating the nearest neighbor distances at pressure P and at temperature K for fcc and hcp crystals These results are analytic and general Theoretical results are applied to determine the melting temperature for molecular cryocrystals of nitrogen type (N2 , CO, N2 O, CO2 ) with fcc and hcp structures in the interval of pressure from to 100 bar for β-N2 , β-CO and from to 1000 bar for α-CO2 , α-N2 O In general, our numerical calculations are in good agreement with the experimental data [3, 6-8] and other calculation [10], especially for α-N2 O, β-N2 and β-CO molecular cryocrystals Our obtained results can be enlarged to cases in higher pressures 74 Equation of state and melting temperature for N2 , CO, CO2 and N2 O molecular cryocrystals REFERENCES [1] F Simon, M Ruhermann, W A M Edwards, 1930 Die schmelzkurven von wasserstaff, neon, stickstoff und argon Z Phys Chem 6, No 5, ss 331-342 [2] Nguyen Tang and Vu Van Hung, 1988 Investigation of the thermodynamic properties of anharmonic crystals by the momentum method Phys Stat Sol.(b)149, pp 511-519 [3] B I Verkina, A Ph Prikhotko (editor), 1983 Cryocrystals Kiev, pp 1-528 (in Russian) [4] I P Bazarov and P N Nikolaev, 1981 Correlation Theory of Crystal MGU (in Russian) [5] N M Plakida, 1969 Stability condition of anharmonic crystal Solid State Physics 11, No 3, pp 700-707 (in Russian) [6] Iu A Freiman, 1990 Molecular cryocrystals under pressures Low Temp Phys 16, No 8, pp 956-1004 (in Russian) [7] O Schnepp, N Jacobi, 1972 The lattice vibrations of molecular solid Adv Chem Phys 22, pp 205-313 [8] S E Babb, 1963 Parameters in the Simon equation relating pressure and melting temperature Rev Mod Phys 35, No 2, pp 400-413 [9] Nguyen Quang Hoc, Do Dinh Thanh and Nguyen Tang, 1996 The limiting of absolute stability, the melting and the α-β phase transition temperatures for solid nitrogen Communications in Physics 6, No 4, pp 1-8 [10] I P Bazarov, 1972 Statistical Theory of Crystalline State MGU (in Russian) 75 ... investigate the equation of state and the melting temperature of solid N2 , CO, CO2 and N2 O We will calculate the pressure dependence of the lattice constant and the melting temperature of these crystals... = and increases to 18% at P = 1000 bar 72 Equation of state and melting temperature for N2 , CO, CO2 and N2 O molecular cryocrystals Figure The limiting temperature of absolute stability and. .. of these crystals Content 2.1 Equation of state, limiting temperature of absolute stability and melting temperature for molecular cryocrystals The equation of state of a crystal with a face-centered