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Molecular dynamics evaluation of cell models for type I gas hydrate crystal dynamics
J. Phys. Chem. Solids Vol. 49, No. 5, pp. 587-588. Printed in Great Britain.
1988
0022-3697/88 $3.00 + 0.00 Pergamon Press plc
TECHNICAL
NOTE
MOLECULAR DYNAMICS EVALUATION OF CELL MODELS FOR TYPE I GAS HYDRATE CRYSTAL DYNAMICS P. Department
of Physics,
University
K.
BASU
of the District
of Columbia,
Washington,
D.C. 20008, U.S.A.
and RAYMOND Thennophysics
Division,
National
(Received
Bureau
D.
MOUNTAIN?
of Standards,
14 July 1987; accepted
Gaithersburg, 2 September
Maryland
20899, U.S.A.
1987)
Abstract-The computer simulation technique of molecular dynamics has been used to compare a cell model description of guest atoms in type I gas hydrate crystals with a fully dynamical description. The density of states for the guest molecules predicted by the cell model is in poor agreement with the density of states generated by the full dynamics of the system. This indicates that the motion of the guest molecules is strongly coupled to the motion of the host molecules. This implies that a rigid cell model is not a satisfactory basis for the study of guest molecule dynamics. Keywords:
Cell model,
gas hydrate
crystal,
molecular
The subject of this note is an examination of the accuracy of a cell model description of the dynamics of guest atoms in Type I gas hydrate crystals [l]. Cell models have proven to be quite useful in developing theories for the thermodynamic properties of these crystals [2] and it would be desirable to know if the cell model picture provides an adequate description of the dynamics of the guest molecules as well. We have used the computer simulation technique of molecular dynamics [3] to examine this topic. Specifically, we have determined the vibrational density of states of a guest atom moving in a cell potential derived from a realistic representation of the structure of a Type I hydrate crystal [4] and from the guest-host interaction potential function [5]. We find that the cell model density of states developed here is in poor agreement with the density of states generated using the full dynamics of the guest-host system [5]. This means that the characteristic features of the motion of the guest atoms cannot be examined by quite simple cell-model computations. Instead, the dynamics of the host water molecules must be considered concurrently as the coupling between the motions of the guest and host molecules is significant. The structure of Type I hydrates has been determined by von Stackelberg and Muller [4], using X-ray diffraction. The water molecules are arranged so that the structure contains two small cages in the form of pentagonal dodecahedra and six large cages in the form of tetrakaidecahedra. The guest molecules are located in these cages. The overall unit cell is cubic with a lattice parameter of I.2 nm and contains 46 water molecules. Tse et al. [5] have used molecular dynamics to examine the vibrational density of states of Xe and CH, hydrates. They used
dynamics,
Lennard-Jones
587
potentials,
e,,,,(r)
= A + Br2 + Cr4,
where r is the distance of the guest atom from the center of the cage. The coefficients A, E, and C are fitting parameters and are listed in Table I for Xe, CH,, and Ar guests, along with the Lennard-Jones parameters of the guest-host potentials [5].
Table model
Xe4 CH,G Ar+
should be addressed, at A105 Bureau of Standards, GaiU.S.A.
simulation.
to represent the guest-oxygen interactions and the SPC interaction [6] to represent the water-water interactions. In this model, the guests do not interact with the hydrogen atoms. We have used these interactions to develop cell model potentials for guest atoms located in the small cages of the hydrate crystal by performing lattice sums over the guest-host interactions. The resulting single particle cell potentials are functions of the position of the guest atom in the cage. Since these potentials are only weakly dependent on the direction of the displacement of the guest from the center of the cage, we represent these lattice sums in the form
1. Guest-host interaction parameters for the cell potential expressed in terms of the Lennard-Jones parameters c and o
Interaction t To whom correspondence Physics Building, National thersburg, Maryland 20889,
computer
A (6) -20 -19 -12
B (c/c? 143 79 -11
c (c/e?
d (nm)
L (J/m00
3340 1960 174
0.349 0.333 0.305
897 762 646
588
Technical
Table 2. Summary of the peak positions, a,,,,,, in the vibrational densities of states for Xe, CH,, and Ar guests calculated using the cell model and for Xe and CH, guests using the fully dynamical model Cell model Guest
T (K)
Dynamical
%I,, (cm-‘)
model f%,x (cm-‘)
Xe
165 208 322
31 32 34
160
41
CH,
117 254 383
66 75 87
145
78
Ar
125 222 333
22 26 29
The cell potentials provide the force fields for the motion of otherwise isolated atoms. Molecular dynamics has been used to generate the vibrational densities of states for the guests, using the formulation of Tse et al. [5] for the guests listed in Table 1. The results of these computations are listed in Table 2, along with the corresponding results obtained from the fully dynamical simulations for Xe and CH, Type I hydrate crystals. We have checked the adequacy of the cell potential by performing simulations using lattice sums to determine the forces in place of the cell potential. The results are essentally identical to those obtained using the cell potential.
Note Note that the peak positions for the cell model densities of states are in poor agreement with the spectra obtained from the fully dynamical simulations for Xe and CH,. The cell model predictions for the peak positions are systematically lower than those obtained from the fully dynamical simulations. From this, we conclude that the motion of the guest molecules is strongly correlated with that of the host molecules and that it is necessary to consider the guest-host system when studying the dynamical properties of gas hydrate crystals. Thus, we must state that rigid cell models are not satisfactory models for the dynamics of the guest molecules in hydrate crystals.
REFERENCES B., Gus Hydrates. Elsevier, 1. Berecz E. and Balla-Achs Amsterdam (1983). This book contains an extensive discussion of hydrate crystals. 2. van der Waals J. H. and Platteeuw J. C., Advances in Chemical Physics (Edited by I. Prigogine), Vol. II. Interscience, New York (1959). Tse J. S.. Klein M. L. and McDonald I. R., J. phys. Chem. 87, 4198 (1983). von Stackelbere M. A. and Muller H. R., Elektrochemie 58, 25 (1959). Tse J. S., Klein M. L. and McDonald I. R., J. them. f’hys. 78, 2096 (1983). Berendsen H. J. C., Postma J. P. M., von Gunsteren W. F. and Hermans J., Inrermolecular Forces (Edited by B. Pullman), p. 331. Reidel, Dordrecht (1981).
1988
0022-3697/88 $3.00 + 0.00 Pergamon Press plc
TECHNICAL
NOTE
MOLECULAR DYNAMICS EVALUATION OF CELL MODELS FOR TYPE I GAS HYDRATE CRYSTAL DYNAMICS P. Department
of Physics,
University
K.
BASU
of the District
of Columbia,
Washington,
D.C. 20008, U.S.A.
and RAYMOND Thennophysics
Division,
National
(Received
Bureau
D.
MOUNTAIN?
of Standards,
14 July 1987; accepted
Gaithersburg, 2 September
Maryland
20899, U.S.A.
1987)
Abstract-The computer simulation technique of molecular dynamics has been used to compare a cell model description of guest atoms in type I gas hydrate crystals with a fully dynamical description. The density of states for the guest molecules predicted by the cell model is in poor agreement with the density of states generated by the full dynamics of the system. This indicates that the motion of the guest molecules is strongly coupled to the motion of the host molecules. This implies that a rigid cell model is not a satisfactory basis for the study of guest molecule dynamics. Keywords:
Cell model,
gas hydrate
crystal,
molecular
The subject of this note is an examination of the accuracy of a cell model description of the dynamics of guest atoms in Type I gas hydrate crystals [l]. Cell models have proven to be quite useful in developing theories for the thermodynamic properties of these crystals [2] and it would be desirable to know if the cell model picture provides an adequate description of the dynamics of the guest molecules as well. We have used the computer simulation technique of molecular dynamics [3] to examine this topic. Specifically, we have determined the vibrational density of states of a guest atom moving in a cell potential derived from a realistic representation of the structure of a Type I hydrate crystal [4] and from the guest-host interaction potential function [5]. We find that the cell model density of states developed here is in poor agreement with the density of states generated using the full dynamics of the guest-host system [5]. This means that the characteristic features of the motion of the guest atoms cannot be examined by quite simple cell-model computations. Instead, the dynamics of the host water molecules must be considered concurrently as the coupling between the motions of the guest and host molecules is significant. The structure of Type I hydrates has been determined by von Stackelberg and Muller [4], using X-ray diffraction. The water molecules are arranged so that the structure contains two small cages in the form of pentagonal dodecahedra and six large cages in the form of tetrakaidecahedra. The guest molecules are located in these cages. The overall unit cell is cubic with a lattice parameter of I.2 nm and contains 46 water molecules. Tse et al. [5] have used molecular dynamics to examine the vibrational density of states of Xe and CH, hydrates. They used
dynamics,
Lennard-Jones
587
potentials,
e,,,,(r)
= A + Br2 + Cr4,
where r is the distance of the guest atom from the center of the cage. The coefficients A, E, and C are fitting parameters and are listed in Table I for Xe, CH,, and Ar guests, along with the Lennard-Jones parameters of the guest-host potentials [5].
Table model
Xe4 CH,G Ar+
should be addressed, at A105 Bureau of Standards, GaiU.S.A.
simulation.
to represent the guest-oxygen interactions and the SPC interaction [6] to represent the water-water interactions. In this model, the guests do not interact with the hydrogen atoms. We have used these interactions to develop cell model potentials for guest atoms located in the small cages of the hydrate crystal by performing lattice sums over the guest-host interactions. The resulting single particle cell potentials are functions of the position of the guest atom in the cage. Since these potentials are only weakly dependent on the direction of the displacement of the guest from the center of the cage, we represent these lattice sums in the form
1. Guest-host interaction parameters for the cell potential expressed in terms of the Lennard-Jones parameters c and o
Interaction t To whom correspondence Physics Building, National thersburg, Maryland 20889,
computer
A (6) -20 -19 -12
B (c/c? 143 79 -11
c (c/e?
d (nm)
L (J/m00
3340 1960 174
0.349 0.333 0.305
897 762 646
588
Technical
Table 2. Summary of the peak positions, a,,,,,, in the vibrational densities of states for Xe, CH,, and Ar guests calculated using the cell model and for Xe and CH, guests using the fully dynamical model Cell model Guest
T (K)
Dynamical
%I,, (cm-‘)
model f%,x (cm-‘)
Xe
165 208 322
31 32 34
160
41
CH,
117 254 383
66 75 87
145
78
Ar
125 222 333
22 26 29
The cell potentials provide the force fields for the motion of otherwise isolated atoms. Molecular dynamics has been used to generate the vibrational densities of states for the guests, using the formulation of Tse et al. [5] for the guests listed in Table 1. The results of these computations are listed in Table 2, along with the corresponding results obtained from the fully dynamical simulations for Xe and CH, Type I hydrate crystals. We have checked the adequacy of the cell potential by performing simulations using lattice sums to determine the forces in place of the cell potential. The results are essentally identical to those obtained using the cell potential.
Note Note that the peak positions for the cell model densities of states are in poor agreement with the spectra obtained from the fully dynamical simulations for Xe and CH,. The cell model predictions for the peak positions are systematically lower than those obtained from the fully dynamical simulations. From this, we conclude that the motion of the guest molecules is strongly correlated with that of the host molecules and that it is necessary to consider the guest-host system when studying the dynamical properties of gas hydrate crystals. Thus, we must state that rigid cell models are not satisfactory models for the dynamics of the guest molecules in hydrate crystals.
REFERENCES B., Gus Hydrates. Elsevier, 1. Berecz E. and Balla-Achs Amsterdam (1983). This book contains an extensive discussion of hydrate crystals. 2. van der Waals J. H. and Platteeuw J. C., Advances in Chemical Physics (Edited by I. Prigogine), Vol. II. Interscience, New York (1959). Tse J. S.. Klein M. L. and McDonald I. R., J. phys. Chem. 87, 4198 (1983). von Stackelbere M. A. and Muller H. R., Elektrochemie 58, 25 (1959). Tse J. S., Klein M. L. and McDonald I. R., J. them. f’hys. 78, 2096 (1983). Berendsen H. J. C., Postma J. P. M., von Gunsteren W. F. and Hermans J., Inrermolecular Forces (Edited by B. Pullman), p. 331. Reidel, Dordrecht (1981).