prediction of structure, composition and phase ...

Proceedings of the 7th International Conference on Gas Hydrates (ICGH 2011), Edinburgh, Scotland, United Kingdom, July 17-21, 2011.

PREDICTION OF STRUCTURE, COMPOSITION AND PHASE BEHAVIOR OF HELIUM HYDRATES Vladimir Belosludov , Oleg Subbotin, Yulia Bozhko Nikolaev Institute of Inorganic Chemistry Siberian Branch of the Russian Academy of Sciences Lavrentiev av. 3, Novosibirsk, 630090 RUSSIA Rodion Belosludov, Hiroshi Mizuseki, Yoshiyuki Kawazoe Institute For Materials Research Tohoku University Katahira 2-1-1, Aoba-ku, Sendai JAPAN Vasily Fomin Institute of Theoretical and Applied Mechanics Siberian Branch of the Russian Academy of Sciences Institutskaya Str., 4/1, Novosibirsk, 630090 RUSSIA ABSTRACT On the basis of statistical thermodynamic model of clathrate hydrates, the existence of helium clathrate hydrates of cubic structures I (sI), II (sII) and helium hydrate based on the framework of ice II has been predicted and their formation pressures have been estimated. We employ here the quasiharmonic lattice dynamics method in order to estimate free energy and Gibbs free energy of helium clathrate hydrates sI, sII and helium-filled ice II. This method allows one to examine hydrates of different structures with multiple occupancies of cages by helium and so to establish the conditions for stable multiple occupancy. The method permits also to estimate the chemical potentials of guest and host molecules. On the molecular level the curves of monovariant equilibrium ‘gas phase-helium clathrate hydrate(helium-filled ice II)-ice (water)’ and the degree of filling of the large and small cavities for hydrates sI, sII and helium-filled ice II in a wide range of pressure and temperature have been determined. It has been shown that the sI hydrate and heliumfilled ice II are metastable relative to the sII hydrate, but at higher pressures structural transformation of the sII hydrate into helium-filled ice II can occur. Keywords: Modeling, helium hydrates, Gibbs free energy, monovariant equilibrium, degree of filling



Corresponding author: Phone: +7-383 3308057 Fax +7-383 3309489, E-mail: [email protected]

INTRODUCTION Clathrate hydrates constitute a group of inclusion

a clathrate phase of hydrogen hydrate was formed

compounds in which cavities of the water host

at these conditions [11]. T his hypothesis was

lattice are occupied by guest molecules. There

confirmed in [13] where formation of hydrogen

exists the relation between van der Waals diameter

hydrate sII at pressures Р =2-3 kbar and low

of guest molecules and free volume of the large

temperature range Т=240-249 K was shown. T he

cavities which defines the structure of the formed

data obtained in this work on the composition of

host lattice. Guest molecules, with van der Waals

the sample had shown fourfold filling of large

diameters in the range 4.2–5.8 Å (examples

cages and twofold filling of small cages by

include СН4 and С2 Н6 ) form the cubic structure I

hydrogen

(sI) hydrates having the smallest volume of the

investigations performed by different methods

large cavities among hydrates. Larger

guest

showed that the possibility of clathrate hydrate

molecules having van der Waals diameters of 5.8–

formation by hydrogen is associated with multiple

7.0 Å, such as С3 Н8 and tetrahydrofuran (THF),

filling of cavities by hydrogen molecules. On the

form the cubic structure II (sII) hydrates [1].

molecular

Special ca se among clathrate hydrates form

equilibrium for H2 hydrates of the structure sII in a

hydrates

wide

of

Ar,

Kr,

N2

and

O2 .

These

molecules.

level

range

of

curves

pressure

monovariant

temperature

smallest sizes of the large cavities among hydrates

monovariant

(with average 4.2 Å), but still form sII hydrate at

experiment . The calculation results are also shown

low pressure [2], [3]. As shown previously in

that small cavities of hydrate sII can be singly or

papers [4-7] the noble gases argon and krypton can

doubly filled

form clathrate hydrates sII with partial double

occupancies from one to four hydrogen molecules.

occupancy of large cages.

For the time being clathrate hydrates formed by

It is known that helium and neon can form solid

noble gases helium and neon, having the smallest

solutions with ices I h , I c and II [8-12]. Light noble

diameters at most 3.5Å, have not been discovered.

gases, such as helium and neon, have been

However, in the work [11] there have been also

generally believed to be unable to form clathrate

observed an anomalous behavior of decomposition

hydrates because their atoms are too small to

curves in the H2 O-Ne system at pressures 80-300

stabilize

hydrates.

MPa and temperatures 253 - 263 К. It was also

However, Yu. A. Dyadin et al. [11-12] had found

shown that in the system H2 O-He the melting

recently, an anomalous behavior of decomposition

temperature exceeds the melting point of ice Ih ,

curves of solid solution H2 O-H2 in the range of

what allows the possibility of clathrate hydrates

hydrogen pressures 50-400 MPa and temperatures

formation by neon and helium.

of

clathrate

263-283 K. It had allowed them to hypothesize that

calculated

equilibrium

agree

curves

were

determined

cavities

The

of

theoretical

atoms/molecules have diameters less than the

the

[14].

the

Indeed,

with

of the

while large cavities can have

To test this hypothesis further, both experimental

Free energy of helium-filled ice II (one type of

and

guests and one type of cages in this case):

theoretical

studies

are

necessary.

Our

F  F1 V , T , y  

calculations presented here permit to answer on the

kTN 1  y ln 1  y   y ln y 

fundamental question about the existence of helium

clathrate

hydrate.

Because

the

intermolecular interaction parameters of helium and hydrogen are close to each other one can think that helium clathrate hydrate can form as well as hydrogen

clathrate

hydrate

and

can

have

,

(1)

where F1 is the part of the free energy independent of transpositions of guest molecules. The second term is the entropic part of free energy of the guest subsystem, y  N

He

N -

degree of cavities

maximally fourfold filling of large cages and

occupation by He, N – the number of cavities,

twofold filling of small cages.

N He - the number of guest molecules in cavities.

In this paper previously suggested generalized model of clathrates [15] is applied to helium hydrates of cubic structures I (sI), II (sII) and helium hydrate based on the framework of ice II. The goal of this work is to investigate the conditions

(pressure

and

temperature)

Microscopic Model of Helium Clathrate Hydrates Free energy of helium clathrate hydrates (there is one type of guests, two types of cages t=1, 2; clusters of i=1, 2... kt He atoms in cages of type t): F  F1 V , T , y  

for

kt kt m     kt y ib  kT  N t  1   y ibt  ln 1   y ibt    y ibt ln t  i!  t 1 i 1 i 1    i 1 

formation of helium clathrate hydrate.

2.1

Statistical

thermodynamic modeling of

helium hydrates The molecular level statistical thermodynamic model used in the calculations is based on the theory of van der Waals [9] but allows one to account influence of guest molecules on the host lattice and the possibility of multiple filling of

.

(2)

In Eq. (2) F1 – is the part of the free energy of clathrate hydrate for the cases when some types of cages and guest molecules exist, and a cavity can hold more than one guest molecule. The second term is the entropic part of free energy of the guest subsystem,

y ti  N ti N t -

the

degree

of

occupation of t - the type cages by clusters of i i

cages by guest molecules. The main assumption of

molecules, N t - number of such clusters, N t –

the model is that the free energy of clathrate

number of t-type cavities.

hydrate is independent from arrangements of guest molecules within cages of given type at fixed hydrate composition. The details of the model can be found in our earlier works [7],[14],[15].

2.2 Computational de tails The calculation procedure of the thermodynamic functions for clathrate hydrates can be represented by the following flowchart:

Microscopic model of helium ice II-based hydrate



Optimization of molecular coordinates in the unit cell



Calculation

of

eigenfrequencie s

of

the

II and in helium clathrate hydrates sI and sII.

structure 

Free energy, equation of state P(V, T) and Gibbs free energy calculation







Parameters for water molecules have been chosen equal to σ =3.1556 Å, ε = 0.65063 kJ/mol. Charges on the hydrogen atoms (q H=+0.4238|e|) and

Calculation of chemical potentials for guest

oxygen atoms (q O= –0.8476|e|) were the same as in

molecules in a gas phase and clathrate hydrate

the

Calculation of equilibrium concentrations of

interactions were calculated by the Ewald method.

guest molecules at the divariant equilibrium

Free energy and its derivatives were calculated at

‘hydrate –– gas phase’

the center of Brillouin zone. Parameters of

Calculation of chemical potentials of host (water) molecules for different water structures and determination of the line of monovariant equilibrium ‘ice I h ( II, water)– gas phase – hydrate’

In

modeling interactions of water molecules in ices I h ,

all structures

SPC/E

model.

Long-range

electrostatic

potential for helium: σ = 2.556 Å, ε = 0.085 kJ/mol [16]. Interaction host-guest wa s modeled by the Lennard-Jones

potential

using

standard

combination rules. Parameters of this potential for water were chosen to reproduce thermal expansion of ice I h as check

hydrogen

atoms

of

water

molecules have been placed in accordance with the

point for the verification. As was shown before, the values of unit-cell volume calculated with the

‘ice rule’: there is always one hydrogen atom

modified SPC/E water-water interaction potential

between any two neighboring oxygen atoms and

agree quantitatively with experimental data at low

this hydrogen atom forms covalent bond with one

temperatures, while the results of calculations

of these oxygen atoms and hydrogen bond with

using SPC/E potential exceed the experimental

another.

values [7]. Deviation between the results of our

Water molecules were oriented so that total dipole

calculations and experimental data does not exceed

moment of the unit cell was vanishing.

1.5%, which also proves the correctness of selected

Parameters of initial unit cells were:

potential.

Chemical

potential

of

water

was

calculated using formula (12) of paper [17].

1) For ice II: a = b = 12.9828 Å, c = 12.9828 Å, α=β=90˚, γ=120˚;

3. RESUTS AND DISCUSSION

2) For sI hydrate: a = b = c = 12 Å, α=β= γ=90˚; 3)

T he properties of helium clathrate hydrates of

For sII hydrate: a = b = c = 17 Å, α=β= γ=90˚; 4)

cubic structures I and II and ice II-based hydrates

For ice I h a = b = 18.0316 Å, c = 14.7252 Å,

have been determined. The optimization was

α=β=90˚, γ=120˚;

performed for a series of unit cell volumes. After

The modified SPC/E potential [7] was used for

the structure optimization we have obtained

a.

b.

Fig. 1 The helium-filled ice II lattice as viewed (a) along and (b) orthogonal to the hexagonal cH axis. T he channels are fully occupied to show the define position occupied by the helium atoms. the set of dynamically stable structures which correspond

to

different

pressures.

At

the

calculations it was assumed that helium clathrate hydrates can contain up to four helium atoms in the large cages and one atom in small cages. 3.1 He hydrates based on ice II a. Structure First, the optimized structure of ice II with complete filling of channels has been found using selected calculation model. The atoms of helium had been positioned in the ice II channels as shown in Figure 1. Distance between nearest helium

Figure.2 Free energy of ice II and helium hydrates at T=220 K. c. Divariant equilibrium ‘gas phase –hydrate ’ The pressure dependence of the filling of helium sites of ice II at T=250 K is presented in Figure 3.

atoms in cavities inside the channel is equal to 2.89 Å and 7.54 Å outside the channel. b. Free energy Figure 2 shows the free energy of ice II with and without helium atoms. The empty ice II lattice at negative pressures is stable at the region to the right from the minimum. Guests expand the lattice which becomes stable already at positive pressures.

Fig.3 Degree of filling of helium sites in ice IIbased helium hydrates

3.2 He clathrate sI and sII hydrates a. Structure. For hydrates of cubic structures I and II optimized structures with different filling of large and small cavities had been found. Figure 4 shows the clusters of four helium atoms in large cavities and two He atoms in small cavities in optimized sII structure. Distance between helium atoms in small 5 12 cavities is equal to 2.49 Å (2.62 Å for hydrogen molecules) and the average distance between He atoms in large 5 12 6 4 cavities is 2.75 Å (2.96 Å for H2 ).

b. Divariant equilibrium 'gas phase – hydrate' Filling of large cages varies consequently from 1 helium atom in a large cage to 4 atoms what leads to increase of configuration term of entropy and decrease of chemical potential. The results of calculation of degree of filling of large and small cages

in

hypothetical

helium

sI

hydrate

independence on pressure are presented in Figure 5. For the helium sII hydrate filling of large cages grows in the same way as for sI hydrate, but less rapidly. At pressure increasing, it leads to slower increasing of entropic part of energy. Therefore, helium sI hydrate at low pressures is metastable relative to sII hydrate.

a. cavity in sII hydrate.

a.

b.

Fig. 4 (a) Cluster of two helium atoms in 5 12 cavity; (b) cluster of four helium atoms in 5 126 4 b. Fig. 5. Degrees of cage filling in helium hydrates a) sI, b) sII at T=250 K.

c. Monovariant equilibrium ‘gas phase –

d. Phase T-P diagram

hydrate -Ice Ih, II

The present theoretical results can support the

Figure 6 shows the chemical potentials of water

hypothesis of existence of helium sII hydrate

molecules in helium sII hydrate depending on

proposed by Dyadin et al.[12] in pressure range

pressure at 250 K presented in comparison with

between 0.6 and 2.5 kbar.

the ices I h and II, and also with the sI hydrate.

Figure

The experiments at this pressure region are very difficult and till now there are doubts about the existence of clathrate phase. At pressures above 630 bar helium hydrate sII becomes more stable than Ih ice. He –filled ice II remains metastable relative to sII hydrate up to 2500 bar. At this pressure there can occur transformation of He clathrate hydrate sII into He-filled ice II.

7

shows

the

calculated

data

of

monovariant equilibrium in comparison with experimental data [12]. It can be seen clearly the formation of the helium sII hydrate forms at the pressure range 0.8-1.5 kbar and 220 K. At the same temperature and pressures greater than 1.5 kbar, the helium-filled ice II is constituted. In order to clarify the range of existence for helium-filled ice II, a new experimental data require. From theoretical prediction the curves (I, III and IV) bound the region of existence for helium clathrate sII hydrate while the curves (IV, V and VI) show the region of existence for helium-filled ice II. However, in this region of pressure stabilization of ice II is possible and hence one needs further comparison in this case of chemical potentials of sII hydrate and the helium clathrate on the base of ice II structure. Corresponding calculations will be a subject of our future investigations.

Fig. 6 Chemical potentials of water molecules  Q for ice I h , liquid water, He clathrate hydrates sII and He ice II-based hydrates at 250 K and 260 K

ACKNOWLEDGMENTS In Russia the work was supported by grant No. 7, Program 18 of Presidium of Russian Academy of Sc iences.

The authors also are grateful for the

continuous support of the HITACHI SR11000K2/51 supercomputing facility by the Computer Sc ience Group at the Institute for Materials Fig. 7 Calculated and experiment [12] data of monovariant equilibrium: I-(-◆-) 'gas phasewater-He clathrate sII hydrate';III-(-▲-) 'gas phase-He clathrate sII hydrate-ice Ih'; IV-(-●-) 'gas phase- He clathrate sII hydrate-helium-filled ice II'; VI-(-★-) 'gas phase-water- helium-filled ice II'; II-(-◯-) admittedly 'gas phase-ice Ihwater' ; V-(- ▢ -) ‘gas phase- water-helium-filled ice II' experiment data [12]; (+) 'gas phasewater-He clathrate sII hydrate' experiment data [12]. 4. SUMMARY 



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