International Conference on Gas Hydrates (ICGH) (6th : 2008)


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  FIRST-PRINCIPLES STUDY ON MECHANICAL PROPERTIES OF CH4 HYDRATE   Caetano R. Miranda∗ Toshifumi Matsuoka Department of Civil and Earth Resources Engineering JAPEX – Energy Resources Engineering Group Kyoto University - Room – C1-2-155 – Kyotodaigaku- Katsura – Nishikyoku Kyoto, 615-8540 JAPAN   ABSTRACT The structural and mechanical properties of s-I methane hydrate have been investigated by first principles calculations. For the first time, the fully elastic constant tensor of s-I methane hydrate is obtained entirely ab-initio. The calculated lattice parameter, bulk modulus, and elastic constants were found to be in good agreement with experimental data at ambient pressure. The Young modulus, Poisson ratio and bulk sound velocities are estimated from the calculated elastic constants and compared with wave speed measurements available.  Keywords: Mechanical properties, Elastic constants, Methane hydrate, First-principles calculations   ∗  Corresponding author: Phone: 81 75 383 3397 – Fax: 81 75 383 3400 E-mail: NOMENCLATURE a0  Equilibrium lattice parameter [Å] BS  Bulk modulus     [GPa] Cij Elastic constant  [GPa] CS  Shear   constant  [GPa] G   Shear   modulus [GPa] E   Young modulus [GPa] VP  Compressional wave speed [km/s] VS  Shear wave speed [km/s] U Internal energy [eV] δ Strain ν Poisson ratio  INTRODUCTION  Methane hydrates has been the subject of extensive experimental and theoretical investigations over decades [1,2]. It consists of methane molecules trapped on clathrate structures of water [1]. Recently, the interest on this system has increase mainly due to its possible use as an energy resource and its environmental implications such as global carbon cycle and greenhouse effects[1,2]. Besides their importance, very little is known about the elastic properties of clathrate hydrate systems[1]. For instance, in order to estimate the amount of methane on hydrate deposits by seismic surveys, one need to know a priori the elastic properties of methane hydrate and its variation with temperature, pressure and hydrate composition [2]. In particular, the role of the guest molecules and the effect of CO2 replacement on CH4 hydrates on the mechanical properties of hydrates are still open problems. From the experimental side, Brillouin spectroscopy[3] compressional and shear-wave speed measurements[4] have been reported for the methane hydrate type-I. However, very few attempts have been done to obtain the fully elastic moduli of methane hydrates and the lack of information is even more critical for CO2 hydrate and other hydrate structures such as types s-II and hexagonal forms. At present, theoretical estimation has been limited to longitudinal acoustic velocities estimation for phase I of methane hydrate [5] and lattice dynamics calculations using simple models Proceedings of the 6th International Conference on Gas Hydrates (ICGH 2008), Vancouver, British Columbia, CANADA, July 6-10, 2008. to describe water and methane molecules [6]. An accurate determination on the elastic properties of these systems is highly desired. In this work, we present for the first time, a first-principle calculation of the elastic moduli of methane hydrates.   COMPUTATIONAL METHODS  A. First principles calculations In this paper, we have focused on the cubic sI structure of methane hydrate. This structure is composed by a combination of six tetrakaidecahedra with 24 H2O molecules and two dodecahedra with 20 H2O molecules cages per unit cell with a total of 46 molecules in the primitive cell [1]. Methane molecules can occupy both large and small cages. Due to the computer effort involved, we have considered only the primitive cell (46 H2O molecules) with fully methane cage occupancy (8 CH4 molecules). The initial structure was prepared by considering the oxygen positions determined experimentally and reported in [7]. The hydrogen positions were displaced taking in account the Bernal-Fowler ice rules and methane molecules were enclosed on the cages. The total energy calculations were performed using density functional theory (DFT) as implemented in the PWSCF code [9]. Exchange and correlation were treated with the Becke-Lee- Yang-Parr (BLYP) functional [9,10] and ionic cores for H, C and O [11] were described by Vanderbilt ultrasoft pseudopotentials [12]. Only Γ point has been applied for the Brillouin-zone sampling and plane waves with kinetic energy cutoff of 680 eV were used. Fully ionic relaxation was performed by using BFGS quasi-Newton algorithm as included in PWSCF code [8]. The supercell volume was fixed during the calculations and all atoms were allowed to relax during the structure optimization process. The systems were considered fully relaxed when the forces on all atoms were smaller than 0.012 eV/Å and total energies converged within less than 0.015 meV.  B. Elastic constants determination The three independent elastic constants C11, C12 and C44 are needed to fully characterize the elastic constants tensor of the cubic methane hydrate structure. C11 and C12 can be obtained from the bulk modulus BS and the shear constant CS, which is related by:  ( ) 32 1211 CCBS +=   (1)   ( ) 21211 CCCS −=   (2)  The bulk modulus BS was determined from the results of total energy change with respect to the hydrostatic variation of volume and fitted to the fourth-order Birch-Murnaghan equation of state. By using the fact that the free energy of the system can be expanded in powers of the strain tensor, the shear constant CS and C44 can be obtained by applying volume conserving tetragonal and orthorhombic strains, respectively. Following [13], we have applied the isochoric strain for a tetragonal distortion in a cubic system:  ( )           −− = 1100 00 00 2δ δ δ ε tetr , where δ is the magnitude of the strain. The strain energy density for the tetragonal deformation is related with the shear constant by:  ( )326 δδ OCU Stetr +=    (3)  For C44, one can apply the orthorhombic deformation:   ( )          − = 22 1/00 00 00 δδ δ δ ε orth ,   with the strain energy density given by  ( )42442 δδ OCU orth +=    (4)  Both, Utetr and Uorth, are obtained directly from the first principles calculations as a function of the strain and the quadratic coefficients estimated from a polynomial fitting. Combining (1), (2), (3) and (4) the elastic constants C11, C12 and C44 were determined  RESULTS AND DISCUSSION  We first present the results for structural properties. In order to obtain the equilibrium volume, we calculated the total energy for the methane hydrate by varying hydrostatically the volume and allowing fully relaxation of the ions. This result is shown in Figure 1 (a).  11.6 12.0 12.4 12.8 0.0 0.5 1.0 (a) Methane Hydrate - sI cubic structure   To ta l E n er gy  [eV ] Lattice parameter [Ang]  This work   -0.02 -0.01 0.00 0.01 0.02 0.00 0.02 0.04 0.06 0.08 0.10 (b) Tetragonal distortion   St ra in  e n e rg y de n si ty  (m e V/ An g3 ) δ  This work   -0.02 -0.01 0.00 0.01 0.02 0.00 0.01 0.02 0.03 0.04 0.05 (c) Orthorombic distortion  This work   St ra in  e n e rg y de n si ty  (m e V/ An g3 ) δ  Figure 1  Total energy (a) and strain energy density in tetragonal (b) and orthorhombic (c) distortion. The energies are shown with respect to the minimum energy value. The equilibrium lattice parameter was found to be 12.11 Å using the GGA - BLYP functional. This value is slightly larger but agrees with the experimental value (11.82 Å) and our calculation using GGA – PBE functional (11.98 Å) [14]. This result is expected since the BLYP functional tends to overestimate the O-H length [ref]. We now turn the attention to the elastic properties of the methane hydrate. Also from Figure 1(a), the bulk modulus was obtained by fitting the fourth-order Birch-Murnaghan equation of state. The result is presented in Table 1 and compared with the experimental data. The calculated value is valid at 0K while the experimental values have been measured at room temperature. The agreement between the calculated and experimental value is very reasonable. Also, it is expected that the bulk modulus value decreases with increasing temperature. Compared with our calculations with guest-free and CO2 hydrates [14], the methane hydrate is slightly more compressible. The strain energy density for the tetragonal and orthorhombic distortions as function of the strain are depict in Figure1 (b) and (c). Using the procedure discussed in the methodological section, we could extract from the strain energy density variation with the strain, the shear constant and the C44 elastic constant. The remained elastic values C12 and C11 were found by solving the equation systems in (1) and (2) using the obtained values of bulk modulus and shear constant. The calculated elastic constant values are summarized in Table 2 and compared with Brillouin spectroscopy measurements at room temperature. It is interesting to note that the elastic constant values obtained by lattice dynamics calculations [6] were also in agreement with the ab-initio ones reported in this paper. Having determined the static elastic constant tensor, we can derive the other elastic properties of the methane hydrate. Considering the cubic system, the shear (G) and the Young`s (E) modulus, and the Poisson ration are given by:  5 3 121144 CCCG −+=   (5) GB GB E S S + = 3 9   (6)        −= B E 3 121ν   (7)  The calculated elastic properties are also shown in Table 1 and compared with experimental data. Note that the calculated values are systematically larger than the experimental ones for the shear and Young`s modulus. It is important to remind that our values are valid for 0K and with the fully occupancy of methane molecules on the cages. Any deviation of these conditions would decrease both the shear and Young`s modulus.  Property This work Exp ρ(g/cm3) 0.89 0.90[3,7] BS (GPa) 8.3 8.0[3] a0(Å) 12.11 11.82[7] E (GPa) 11.07 8.5[3] G(GPa) 4.3 3.3[3] ν 0.2776 0.317[4] VP (km/s) 3.981 3.778[4] VS (km/s) 2.209 1.964[4] VP/ Vs 1.80 1.93[4]  Table 1. First principles structural and elastic properties of s-I Methane hydrate compared with experimental data.   Elastic constant This work (0 K - 0 GPa) Exp (296 K and 0.02 GPa) C11(GPa) 15.1 11.9 C12(GPa) 4.9 6.0] C44(GPa) 3.8 3.4  Table 2. Elastic constants of methane hydrate s-I obtained from first principles and compared with experimental data [3].  In order to compare with compressible and shear wave measurements, we have calculated the longitudinal (VP) and shear (VS) wave velocities by assuming an isotropic media with methane hydrate density ρ:        + = ρ GB V SP 3 4   (8)        = ρ GVS    (9) The calculated results for the wave velocities are summarized in the Table 1. We have observed a good agreement between the first-principles results and the recent experimental data. It remains to be checked the geophysical implications of these results. In particular, it would be important to compare the elastic properties of methane hydrate obtained here with the guest-free and CO2 hydrates, as well to quantify the effect of partial guest occupancy. These would give us not only a clue about the role of the guest in the mechanical stability of the clathrate structure, but more importantly, it may also suggest us a method to estimate the amount of methane in reservoirs.  CONCLUSIONS In summary, we have presented first principles calculations of the structural and elastic properties of s-I methane hydrates. Using DFT with BLYP functional, the lattice constants are in agreement with experimental data, but we have observed a slightly increase with respect to our calculations with GGA-PBE functional. The strain energy densities were calculated to obtain the elastic constant tensor by applying tetragonal and orthorhombic distortions. The calculated elastic constants are in good agreement with available Brillouin spectroscopy and wave speed experimental data. The sound velocities are estimated from the elastic constants results and a good agreement with recent experimental data was found. The set of elastic properties reported here can be used in Rock Physics models to evaluate their geophysical implications and estimate in-situ methane hydrate from seismic surveys. Currently, we are investigating from first-principles calculations the partial methane occupancy and the effect of CO2 substitution on the elasticity properties of clathrate hydrates.  REFERENCES [1] Sloan ED Jr., Clathrate Hydrates of Natural gases, 2nd ed. (Dekker, New York, 1998) [2] Max MD, Johnson AH, Dillon WP, Economic Geology of Natural Hydrate, Coast System and Continental Margins, vol. 9, Springer (2006). [3]Shimizu H, et al, Elasticity of single-crystal methane hydrate at high pressure, Phys. Rev. B 65 (2002) 212102. [4]Helgerud MB et al, Measured temperature and pressure dependence of Vp and Vs in compacted, polycrystalline sI methane and sII methane-ethane hydrate, Can. J. Phys. 81, 47 (2003) [ 5]Whalley E, J. Geophys. Res. 85, 2539 (1980) [ 6]Shapakov VP et al, Elastic moduli calculation and instability in structure I methane clathrate hydrate, Chem. Phys. Lett. 282, 107 (1998) [7] Gutt C et al., The structure of deuterated methane–hydrate, J. Chem. Phys. 113, 4713 (2000) [8] P. Giannozzi et al., http://www.quantum- [9] A.D. Becke, Phys. Rev. A 38 (1988) 3098. [10 ] C. Lee, W. Yang, R.G. Parr, Phys. Rev. B 37 (1988) 785. [11] We used the pseudopotentials: H.blyp- van_ak.UPF, O.blyp-van_ak.UPF and C.blyp- van_ak.UPF from the http://www.quantum- distribution. [12 ] D. Vanderbilt, Phys. Rev. B 41 (1990) 7892. [13] Gülseren O. and Cohen R. E., High-Pressure thermoelasticity of body-centered-cubic tantalum, Phys. Rev. B 65 064103 (2002) [14] Miranda CR and Matsoka T, unpublished. [15] Kuo JL et al, The effect of proton disorder on the structure of ice-Ih: A theoretical study, J. Chem. Phys. 123, 134505 (2005) 


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