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

Abstract. 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.

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JOURNAL OF SCIENCE OF HNUE Mathematical and Physical Sci., 2014, Vol. 59, No. 7, pp. 67-75 This paper is available online at EQUATION OF STATE AND MELTING TEMPERATURE FOR N2, CO, CO2 AND N2OMOLECULAR CRYOCRYSTALS UNDER PRESSURE Nguyen Quang Hoc1, Dinh Quang Vinh1, 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 equation of state, the absolute stability temperature of crystalline state and the melting temperature for N2, CO, CO2 and N2Omolecular 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. 1. 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 lnT + 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 3 to 4 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 N2O. We will calculate the pressure dependence of the lattice constant and the melting temperature of these crystals. 2. 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]: Pv = −a 6 ( ∂u0 ∂a ) T + 3γTGθ, v = √ 2 2 a3, u0 = ∑ i ϕi0 (|⃗ai|) ,γTG = − a 6k ∂k ∂a X,X ≡ x coth x, θ = kBT, x = ~ω 2θ , k = 1 2 ∑ i ( ∂2ϕi0 ∂u2iβ ) eq ≡ mω2, β = x, y, z. (2.1) here P is the hydrostatic pressure, v is the volume of the fcc lattice, a is the nearest neighbor distance of the fcc crystal, a⃗i is the vector determining the equilibrium position of the ith particle, ϕi0 is the interaction potential between the ith particle and the 0th particle, γTG is the Gruneisen constant and kB is the Boltzmann constant. From the limiting condition of absolute stability for the crystalline state( ∂P ∂v ) T = 0, i.e. ( ∂P ∂a ) T = 0, (2.2) 68 Equation of state and melting temperature for N2, CO, CO2 and N2O molecular cryocrystals... we find the corresponding expression of the limiting temperature Ts as follows: Ts = 2Pv + a 2 6 ( ∂2u0 ∂a2 ) T 3kB [ a ( ∂γTG ∂a ) T − γTG ] . (2.3) 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] Ts = 4k2 kB ( ∂k ∂a )2 T { 1 6 ( ∂2u0 ∂a2 ) T + ~ω 4k [( ∂2k ∂a2 ) T − 1 2k ( ∂k ∂a )2 T ] + 2Pv a2 } . (2.4) In the case of P = 0, it gives Ts = 4k2 kB ( ∂k ∂a )2 T { 1 6 ( ∂2u0 ∂a2 ) T + ~ω 4k [( ∂2k ∂a2 ) T − 1 2k ( ∂k ∂a )2 T ]} . (2.5) The nearest neighbor distance a is determined by a = a0 + ux0, where ao denotes the distance a at temperature 0 K and is determined from the experimental data. The displacement ux0 of a particle from the equilibrium position is calculated by ux0 = √ 2γθ2 3k3 A, γ = 1 12 ∑ i (∂4ϕi0 ∂u4iβ ) eq + 6 ( ∂4ϕi0 ∂u2iβ∂u 2 iγ ) eq  , β ̸= γ, β, γ = x, y, z, A = a1 + 6∑ i=2 ( γθ k2 )i ai. (2.6) where ai(i = 1−6) is determined in [2]. The equation for calculating the nearest neighbor distances at pressure P and at temperature 0 K has the form [2] y2 = 1.1948 + 0.1717y4 − 0..0087Pσ 3 ε y5 + 0.0021 Pσ3 ε y7, y = (a σ )3 . (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] Tm ≈ Ts + am − as kBγsG ( Pvs as + 1 18 ∂u0 ∂as + as ∂2u0 ∂a2s ) , (2.8) 69 Nguyen Quang Hoc, Dinh Quang Vinh, Bui Duc Tinh and Nguyen Duc Hien where am = a (Tm.P ) , as = a (Ts, P ) , vs = √ 2 2 a3s, ∂u0 ∂as = ( ∂u0 ∂a ) a=as , ∂ 2u0 ∂a2s = ( ∂2u0 ∂a2 ) a=as , the Gruneisen parameter γsG is regarded as invariable in the interval from T to Tm because it changes very little and γsG = − ( a 6k ∂k ∂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]: Pv = −a 4 ∂u0 ∂a − c 2 ∂u0 ∂a + 12γTGθ, v = √ 3 2 a2c, γTG = − X 24kx ( a ∂kx ∂a + 2c ∂kx ∂c ) − Xz 48kz ( a ∂kz ∂a + 2c ∂kz ∂c ) , X ≡ x cothx,Xz ≡ xzcthxz, x = ~ωx 2θ , xz = ~ωz 2θ , kx = 1 2 ∑ i [( ∂2ϕi0 ∂u2ix ) eq + ( ∂2ϕi0 ∂uix∂uiy ) eq ] = mω2x, kx = 1 2 ∑ i ( ∂2ϕi0 ∂u2iz ) eq = mω2z . (2.9) here a and c are the lattice constants of the hcp crystal, v is the volume and γTG is the Gruneisen constant. From the limiting condition of absolute stability of the crystalline state( ∂P ∂v ) T = 0, i.e. ( ∂P ∂a ) T = 0or ( ∂P ∂c ) T = 0 (2.10) we find the corresponding expression of the limiting temperature Ts as follows: Ts = Pv + a 2 4 .∂ 2uo ∂a2 − c 2 ∂uo ∂c 12kB [ a ( ∂γTG ∂a ) − γTG ] . (2.11) If we take the values of the parameters a, c, kx, kz, ∂kx∂a , ∂kx ∂a , ... at the same limiting temperature of absolute stability Ts then (2.11) can be transformed into the form [9] Ts = Pv + a2.∂ 2uo ∂a2 − 2c∂uo ∂c + a2 ( ~ωx kx ∂2kx ∂a2 + ~ωz 2kz ∂2kz ∂a2 ) − 2c ( ~ωx kx ∂kx ∂c + ~ωz 2kz ∂kz ∂c ) kBa [ 2 k2x ∂kx ∂a ( a∂kx ∂a + 2c∂kx ∂c ) + 1 k2z ∂kz ∂a ( a∂kz ∂a + 2c∂kz ∂c )] . (2.12) In the case of P = 0, it gives Ts = a2.∂ 2uo ∂a2 − 2c∂uo ∂c + a2 ( ~ωx kx ∂2kx ∂a2 + ~ωz 2kz ∂2kz ∂a2 ) − 2c ( ~ωx kx ∂kx ∂c + ~ωz 2kz ∂kz ∂c ) kBa [ 2 k2x ∂kx ∂a ( a∂kx ∂a + 2c∂kx ∂c ) + 1 k2z ∂kz ∂a ( a∂kz ∂a + 2c∂kz ∂c )] . (2.13) 70 Equation of state and melting temperature for N2, CO, CO2 and N2O 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 0 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 ux0 ≈ 6∑ i=1 [ γθ kx 2 ]i ai,γ ≡ 1 4 ∑ i [( ∂3ϕi0 ∂u3ix ) eq + 6 ( ∂3ϕi0 ∂uix∂u2iy ) eq ] , (2.14) where ai(i = 1 − 6) is determined in [9]. Lattice constant c is determined by c = c0 + uz0, where co denotes the distance c at temperature 0 K and is determined from the experimental data. Displacement uz0 of a particle from the equilibrium position in direction z is calculated from uz0 ≈ [ 1 3 6∑ i=1 ( θ kz )i bi ]1/2 , τ1 ≡ 1 12 ∑ i ( ∂4φi0 ∂u4iz ) eq , τ2 ≡ 1 2 ∑ i ( ∂4φi0 ∂u2ix∂u 2 iz ) eq , τ3 ≡ 1 2 ∑ i ( ∂4φi0 ∂uix∂uiy∂u2iz ) eq . (2.15) here bi(i = 1 − 6) is determined in [9]. The equation for calculating nearest neighbor distances at pressure P and at temperature 0 K has the form [9] y2 = 0.9231 + 0.3188y4 − 0.0015Pσ 3 ε y5 − 0.0316y6 + 0.0007Pσ 3 ε y7 − 0.0001Pσ 3 ε y9, y = (a σ )3 . (2.16) The melting temperature Tm of the hcp crystal is approximately equal to [9] Tm ≈ Ts+am − as 6kBγsG [ 2Pvs as + 1 8 ( ∂u0 ∂as + as ∂2u0 ∂a2s )] + cm − cs 6kBγsG [ 2Pvs cs + 1 4 ( ∂u0 ∂cs + as ∂2u0 ∂c2s )] . (2.17) where cm = c (Tm.P ) , cs = c (Ts, P ) , vs = √ 3 2 a2scs and other quantities are as in (2.8). 2.2. Numerical results and discussion For α-CO2 and α-N2O 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 ϕ(r) = 4ε [(σ r )12 − (σ r )6] , (2.18) 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 and σ are determined from the experimental data. ε kB = 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, εkB = 235.48 K, σ = 3.802.10 −10 m for α-N2O, εkB = 95.05 K, σ = 3.698.10−10 m for β-N2 and εkB = 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- 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 ] , (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: kx = 4ε a2 ( σ a )6 [ 614.6022 ( σ a )6 − 162.8535] , kz = 4εa2 (σa)6 [286.3722(σa)6 − 64.7487] , γ = − 4ε a3 ( σ a )6 [ 161.952 ( σ a )6 − 24.984] , τ1 = 4εa4 (σa)6 [6288.912(σa)6 − 473.6748] , τ2 = 4ε a4 (σ a )6 [ 11488.3776 (σ a )6 − 752.5176 ] , τ3 = 4ε a4 (σ a )6 [ 8133.888 (σ a )6 − 737.352 ] . (2.20) Our calculated results for the limiting temperature of absolute stability and the melting temperature of α-CO2, α-N2O, β-N2 and β-CO at different pressures (low pressures) are expressed in Figures 1-4. Figure 1. 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 = 0 and increases to 18% at P = 1000 bar. 72 Equation of state and melting temperature for N2, CO, CO2 and N2O molecular cryocrystals... Figure 2. The limiting temperature of absolute stability and the melting temperature at different pressures for α-N2O The discrepancy in the melting temperature of α-N2O that exists between our calculated results and the experimental data [7] is 0.04% at P = 0 and increases to 7% at P = 1000 bar. Figure 3. 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 = 0 and increases to 5% compared with the experimental data [8] at P = 100 bar. Our calculation is better than that in [10]. Figure 4. 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 = 0 and increases to 7.68% at P = 100 bar. 3. 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 0 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, N2O, CO2) with fcc and hcp structures in the interval of pressure from 0 to 100 bar for β-N2, β-CO and from 0 to 1000 bar for α-CO2, α-N2O. In general, our numerical calculations are in good agreement with the experimental data [3, 6-8] and other calculation [10], especially for α-N2O, 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 N2O 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. 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