Studies on the Structure and Optical Properties of Different Cage Occupancy on s

DENG Rong, WANG Zhao, YANG Liang, LI Jin, YANG Zejin

(1.College of Materials Science and Engineering, Hainan University, Haikou 570228, China;2.School of science, Hainan University, Haikou 570228, China;3.School of science, Zhejiang University of technology, Hangzhou 310023, China)

Abstract: To reveal the effects of different cage occupations on hydrate structure and related properties. The relaxation of the structure, electronic density of state and optical properties of the SI methane hydrates with three different configurations are calculated by first-principle, including (cI) only one large cage lacks its unique methane molecules; (cII), only one of the small cage lacks its unique methane molecule; (cIII) each cage is fully filled with methane molecules. The results show that the hydrate is most stable in the cIII due to its perfect structure, the cII is less stable, whereas the cI is least stable owing to its larger structural distortion after the loss of methane. On the contrary, loss of a methane molecule from cII causes negligible deformation. The relative change in cell volume is 0.56% and 2.1%, corresponding to the cII and the cI, respectively. The electronic density of states and the energy band gap of the cII is almost the same with those of cIII, differing obviously with those of cI. The calculation results display that the contribution of electronic transition is small, and proton disorder is the dominant hydrate permittivity. Methane hydrate is only responsive to light in the ultraviolet region, revealing their similar properties, regardless of their structural discrepancies, or/and the different ratios of water and methane molecules, 46/8=5.75 versus 46/7=6.57. Our calculations demonstrate that the lack of one methane in the cII causes negligible influence to the lattice structure and therefore to the electronic and optical properties in comparison with the cIII, whereas the lack of one methane in the cI can cause detectable changes. These results might provide valuable reference to the industrial exploration.

Key words: methane hydrate; cage occupancy; optical properties; the first-principle

1 Introduction

Since gas hydrate plays an important role in potential energy, global climate change, submarine geological environment, and natural gas pipeline transportation, it has always been an important issue to research[1-6]. Although natural gas hydrates are widely found in seabed sediments and terrestrial frozen soils, there are currently no commercially viable mining methods[7-9]. Over the last several decades, even though significant knowledge has been gained regarding its diverse properties, many aspects of the hydrates remain still puzzling or unknown[10-12].

The methane hydrates are crystalline solids composing of water and methane, in which methane molecules (guests) are trapped by water cavities (host) that are composed by hydrogen-bonded water molecules. Hydrates have three different kinds of crystal structures, cubic structure I (sI), cubic structure II (sII), or hexagonal structure H (sH), respectively. The structure I consists of six large cavities 51262and two small cavities 512per unit cell. Ideally, all cavities are filled with one methane molecule. For ideal hydrates, the hydration number (water molecules per guest molecules) is related to the fraction filling of the large and small cavities. The hydration number would be 5.75 (46/8). In fact, it shouldn’t affect the stability of hydrates when one cavity lacks the guest molecule. Amadeu K. Sum et al have measured cage occupancy and hydration number for CH4hydrate at different equilibrium conditions (temperature and pressure) and found about 90% occupation in the small cavities and almost fully filled in the large cavities[13]. By the powder X-ray diffraction technique, Satoshi Takeya et al obtained that CO2occupies 99% in the large cages and 69% in the small cages per unit cell, respectively[14]. The value for the hydration number ranges from about 5.8 to 6.3 reported in above literatures. The theoretical prediction for cage occupancy and hydration number have been reported based on the statistical van der Waals-Platteeuw model together with ab initio calculations and molecular dynamics method[15-17].

Seismic velocity techniques as well as bottom simulating reflections are the main remote means for hydrate exploration[18-19]. The methods of drilling and coring enable refinement of estimation of reservoir hydrate content. Logging tools such as caliper, gamma ray, density, resistivity, and neutron porosity determine the hydrate depth, and to some extent the concentration[20-23]. With Time Domain Reflectometry, the gas hydrate amount is measured based on the bulk dielectric properties[24]. These probes generally require parameters such as dielectric constant. However, these parameters are relatively lacking and controversial. In some literature, the dielectric constant of hydrates is replaced by the dielectric constant of ice[25].

In this work, we evaluate the effects of different cage occupations on cell stability, electronic density, and optical properties such as dielectric properties by first-principles calculations. The hydrates are non-stoichiometric substances, and the absence of guest molecules in a cage is permitted. Our calculations were performed for two occupancy rates of sI hydrates, 100% (hydration number 46/8 = 5.75) and 87.5% (hydration number 46/7 = 6.57), respectively. The cage occupancy rate of 87.5% means that there is a cage lacking methane molecules, which may occur in small cages or in large cages. Therefore, we calculate and evaluate the following three occupations: i) each cage is filled with methane molecules, ii) a small cage lacks methane molecule, iii) one large cage lacks methane molecules.

2 Computational Setup

The hydrate atomic coordinates are determined according to the available literature[26]. The first-principles calculations were performed using Vienna ab initio simulation package (VASP)[27]and the projector-augmented wave (PAW) function. The energy cutoff for plane-wave expansion was set to 600 eV. The first Brillouin zone (BZ) sampling was chosen as 2×2×2 k-point mesh per unit cell for bulk sI methane hydrate. And for optical calculation, we set the k-points to be 4×4×4, the NBANDS equals 800. The structures of bulk sI methane hydrate were optimized until the forces on all atoms were below 0.02 eV/Å. Electronic relaxations were converged to within 1×10-8eV.

The imaginary part of the dielectric constant, which is given by

(1)

Where u is the vector defining the polarization of the induced electric field, ε2can be considered as detailing the real transitions between occupied and unoccupied electronic states. Since the dielectric constant describes a causal response, the real and imaginary parts are linked by a Kramers-Kronig transform. This transform is used to obtain the dielectric function ε1of the real part. The complex refractive index, N, is determined by the following formula：

N2=ε1+iε2

(2)

N=n+ik

(3)

The detailed calculation process can be found in its manual, for simplicity, we ignore the concrete description.

3 Results and Discussion

We carry out first-principles calculations for three types of I-hydrate structures. The three types I hydrate structures are: i) A large cage is empty, the remaining cages are filled with a methane molecule and the hydration number is still 46/7; ii) A small cage is empty, the remaining cages are filled with a methane molecule with a hydration number of 46/7; iii) each cage is filled with a methane molecule with a hydration number of 46/8. The calculation results and discussion are as follows.

3.1 The optimized hydrate structure and energy

The ideal structure is that each cavity is fully filled with one methane molecule, which has a hydration number of 46/8, as shown in Figure 1. And the two building blocks of the hydrate structure are depicted in Figure 2 as one small cage and one large cage are empty, respectively.

The relaxed cell parameters are shown in Table 1, from which we can see that optimization results (a=11.63 Å, b=11.64 Å, c=11.61 Å) of the ideal structure are in good agreement with the experimental data, which obtained from low-temperature neutron scattering data by Gutt et al. (11.821±0.001 Å, CD4/D2O at 2K)[28]. The absence of one methane molecule from the small cage did not induce evidently changes relative to that of ideal structure, whereas the structure distorted obviously in the case of large cage. The large deformation might relate to the large hollow cavity, which corresponds to small resistance to the neighbor squeezing interaction.

图1 完全占据的甲烷水合物优化后的结构：白色小球代表H原子，蓝色代表C原子，红色代表O原子(下同)。其中绿色代表组成小笼的O原子

Fig.1 The optimized hydrate structure, each cavity is fully filled by one methane molecule. The white balls represent H, the blue represent C, the red represent O (the same below). And green represent the O atoms of small cage

图2 两种不同的水合物笼子的结构(a)小笼;(b)大笼

Fig.2 The two different building blocks of the hydrate structure, consisting of one small cage (a) and one large cage (b), the nearly spherical shape of small cage is clearly seen

表1 计算得到的三种结构的晶胞参数、能量和能带

Table.1 Cell parameters, energy and bandgap for the three cases after calculation

Structure(units)Emptylargecage(cI)Emptysmallcage(cII)Fullcage(cIII)a(Å)10.8511.6111.63b(Å)12.7211.6211.64c(Å)11.2611.5911.61α(°)86.8690.1290.08β(°)96.1189.8989.92γ(°)94.6590.0590.11Cellvolume(Å3)1540.721564.681573.51EnergyEc(eV)-23078.97-23084.47-23304.78ΔEs(eV)-20.69-15.19-14.54Bandgap(eV)3.815.235.27

The ideal lattice energy is -23303.82 eV, lower than those of defective structures. 23084.47 and -23078.97 eV are the energies for the methane lack from small and large cages, respectively. It is also clear that the hydrate is more stable when methane lack from the small cage than from the large, which might be a possible explanation to the experimental observation of the higher occupancy rate of the small cage other than the large cage, which is consistent with the experimental observation that the small cage is formed first and the large cage subsequently.

The fully filled cage (cIII) has a distance of 3.77Å between the C and O atoms in the small cage, slightly smaller than that of experimental data, 3.95Å[29]. These parameters are almost identical with those of the case (cII), the experimental lattice parameters[29]are also slightly larger than the calculated values, for example, a:11.73～11.93Å. Moreover, all of the nearest neighbors of the small cage are large cages, such higher symmetric environment should be one of the reasons for its nearly unchanged lattice structure when the methane is removed. The undistorted structure should also be attributed partially to its nearly spherical shape building blocks in the unit cell. However, the inequivalent nearest neighbors of the large cage, comprising of large and small cages simultaneously, results in its lower symmetry in chemistry, which is quite difficult to keep its original shape when lack of one methane molecule. Owing to the giant deformation of the lattice, it is therefore not meaningful to compare the inter-atomic distance of case (cI) with the ideal lattice. The elongated lattice parameter b could be illustrated by its shorter vertical distance of two hexagonal planes than that of the diameter of its perpendicular plane (nearly circle), that is, totally depending on the building block orientations of the large cage in the lattice, as is shown in Figure 2 (b), vice versa to the case of the shortened a/c.

The stabilization energy (ΔEs) can be defined as[30]

ΔEs=-(Ec-NEH2O-nEguest)

(4)

图3 优化后三种结构的XRD图

Where N and n are the number of water molecules and guests, respectively.Ec,EH2O, andEguestare energies of the (CH4)n@(H2O)Ncomplex, water monomer and methane monomer, respectively. The greater stabilization energy (ΔEs) is, the more stable the system will be. Obviously, the system is more stable without a guest molecule in small cages than in large cages.

The energy change caused by the loss of a methane molecule from a small cage is 220.31 eV, while the loss of a methane molecule in a large cage requires 225.81 eV. This implies that small cages are easier to form than the large cages, which is consistent with the experimental observation by Raman spectroscopy: the small cage is formed first and then the large cage is formed subsequently[31]. Compared with the ideal structure, the relative change in cell volume is 0.56% and 2.08%, corresponding to cII and cI.

Based on the optimization results, we calculated the powder XRD patterns for the three cases which are showing in Figure 3. It can be seen that the structure of lack of guest molecules in the small cage still maintains symmetry. The absence of a guest molecule in a large cage results in a large structural deformation and almost loss of symmetry. These results suggest that the cage occupancy rate around 80% is the critical point, and the occupancy rate of less than 80% will result in the hydrate being unable to keep stable.

3.2 The electronic density of states

Based on the relaxed structure, we calculated its electronic density of state for three cases as show in Figure 4. From the topography of the electronic density curve, the electronic distribution looks almost the same between the case (cII) and case (cIII). However, the electronic distribution in the case with a large cage lacking a methane molecule is quite different from the former two structures. Based on the electronic density, we obtained the bandgap data for the three cases, which are listed in Table 1.

The difference of bandgaps is small between the case (cII) and case (cIII). However, when a large cavity is empty in the unit cell, that is the case (cI), its bandgap dramatically decreases to 3.81 eV. The bandgap varies greatly when the large cage lacks a methane molecule. The first two cases can be considered as insulators, and the latter can be regarded as semiconductors. It can be seen that the larger deformation of the hydrate structure leads to a large change in the distribution of the electronic states.

The average electronic density of states for each atom is plotted in Figure 5. Due to the covalent bond, the hydrogen atom will partially occupy the p electron. It can be seen from the figure that there is almost no difference in the density of state between the lack of methane molecules in a small cage and that filling of the methane molecules in each of the cages. The slight difference is mainly due to the fact that the former has a higher degree of degeneracy, that is, there are more electronic states with the same energy, especially carbon atoms. The lack of methane molecules in the cage results in less perturbation of the atoms, and the degeneracy increases. When the large cage lacks a methane molecule, average electronic density of states has a large change. In particular, the density of state of the oxygen atom in the water molecule has the largest change. Large structural deformation leads to large changes in the density of electronic states.

图4 三种结构的态密度图

图5 三种结构中各个原子的分态密度图：(a) H2O的分态密度；(b)CH4的分态密度

Fig.5 The electronic density of states for each atom for the three cases: (a) partial electronic density of states for H2O; (b) partial electronic density of states for CH4

3.3 Dielectric constant

The dielectric coefficient versus photon energy curve for the three different kinds of structures is shown in Figure 6. The curve topography is similar between the case with full cage and that with the small cage lacking methane, and the curve morphology is quite different when large cage lacking methane. From the curves for the real part of dielectric constant, the peak value of the dielectric coefficient is 2.58, 1.35 and 1.35，correspond to cI, cII and cIII cases,. The static dielectric constants of the hydrates are 1.005 and 1.009, correspond to cII and cIII cases. However, when the large cage lacks methane molecules, the dielectric constant is 2.02. In some literatures, the dielectric of the hydrate is higher, about 58[1,32]. This is mainly due to the consideration of the contribution of proton disorder to the dielectric constant. The dielectric constant appears to be insensitive to the guest species and depends primarily on the structure of the host molecule. For the dielectric constant of ice or hydrate, the proton disorder has its dominant role[32-35]. And calculation formula of the dielectric constant is[32-34].

(5)

Where ε0is the optical dielectric constant and M is the electric dipole moment. The optical dielectric constant ε0is set equal to 1.592[32]. The high dielectric constant is only for pure ice or hydrates that do not contain impurities. When the ice contains impurities, the dielectric constant is about 3[36]. Wright J F, et al. give the apparent dielectric constant of hydrates as a function of sample moisture content θ[24].

Ka=4.0556+0.1132θ+0.007869θ2+0.00002169θ3

(6)

Where θ is the moisture content. Here, the apparent dielectric constantKais the dielectric constant of the mixture and is not the dielectric constant of the pure hydrate. When the water content is zero, the apparent dielectric constant is 4.0556. In reality, the dielectric properties of hydrates are more complex. The dielectric constant obtained according to our calculation method is only for the electrons in the system, which did not take the effects of the proton disorder into ac- count. The results obtained herein can be used as a contribution or reference to the value of ε0in the Equation (5).

图6 三种结构的介电系数与频率曲线

3.4 The reflectivity index and other optical properties

The reflectivity index curves for the three structures are shown in Figures 7. Other optical properties, such as absorption, refractive indices, optical conductivity and energy loss spectrum are shown in Fig.8. The reflected light of the gas hydrate is in the ultraviolet region. In terms of absorption and other optical properties, hydrates respond to light in the ultraviolet region as well. The position of the absorption peak reflects the transition from 2p electrons, mainly the 2p transition of water molecules. The static refractive index is 1.004. As can be seen from these curves, cage occupancy has little effect on the main properties of optics. In fact, since there are few differences among the host structures of the three cases, the reflectance curves are also similar. There is almost no difference in optical properties between the small cage lacking guest molecules and each cage filling guest molecules. The optical properties vary, only when the large cage lacks guest molecules. It can be inferred that the hydrate properties are dominated by the clathrate structure compose of water molecules. It should be emphasized again that we are only calculating the response of electrons to light in the system, without considering the spectrum of the molecule.

图7 三种情况的反射率曲线

4 Conclusions

The lattice structure is more stable when a methane molecule is removed from a small cage than that of large cage due to nearly spherical symmetry and equivalent chemical environment of the small cage. Loss of a methane molecule from a small cage induces subtle change in structural distortion, the shift of electronic density of states, bandgap reduction, and optical related properties response. However, such electronic and optical properties change obviously owing to the larger structural deformation once lose a methane from a large cage. The present investigation also provides valuable insight to other defective cage framework structure with similar variations. The obtained dielectric function shows nearly same profiles with that of refractivity index. The absorption coefficient, loss function, reflectivity index, and the real part of the conductivity index also present similar response. The sensitivity response range is about 5～35 eV for all of the optical properties, the corresponding wavelength is range from 35.45nm to 248.16nm,which corresponds to the ultraviolet region. The current investigations demonstrate the correlation of the structural change of optical and electrical properties with the degree of structural distortion. The negligible structural and electronic change of methane lack from small cage demonstrates the extreme stability of the water framework, independent on the trapped guest molecules, occupancy or not, which is the true origin of the substance storage capacity.

图8 三种情况的其他光学性质：(a)吸收系数, (b) 折射率, (c) 光导, (d) 能量损失谱

Fig.8 The calculated curves of other optical properties for three cases:(a) absorption, (b) refractivity indices, (c) optical conductivity, (d) energy loss spectrum

Acknowledgment: This work is supported by the Key Research and Development Program of Hainan Province, China (Grant No. ZDYF2017098) and the Natural Science Foundation of Zhejiang Province, China (Grant No. LY18E010007).

Conflicts of Interest: The authors declare no conflict of interest.