Theoretical prediction of low-frequency vibrations of extra-framework cations in mordenite zeolites
Received
22nd September 2003
, Accepted 10th November 2003
First published on 26th November 2003
Abstract
Molecular dynamics simulations, using classical potential models to represent the cation-framework interactions, were performed in order to predict the low frequency region of vibrational spectra for mordenite zeolites. The position and the shape of the bands assigned to the cation vibrations have been studied as a function of the nature of the extra-framework charge balancing cations (alkali, alkaline earth) and of the Si/Al ratio characterizing the zeolite framework. The critical role of the forcefield is also demonstrated by computing the low frequency spectra using two different forcefields which include the flexibility of the host framework.
Introduction
Aluminosilicate zeolites characterised by pore systems consisting of channels and cavities, have a considerable number of potential applications in a wide range of chemical science and technology such as catalysis, selective separation, gas storage and ion exchange.1–4 The aluminium content of these porous frameworks controls the electric charge of the lattice which is neutralised by the introduction of extra-framework cations located in surface sites. The location and the dynamical properties of these counter ions are of crucial importance because they strongly influence the catalytic and adsorption behaviours of the zeolites.5,6 Furthermore, they possess the ability to be exchanged easily by various kinds of alkali or alkaline earth ions without altering the framework structure.
In our previous works,7–12 we investigated the static properties of the extra-framework cations by means of energy minimisation techniques and Monte Carlo simulations in typical Na+-mordenite zeolites, as a function of the framework Si/Al ratio,7–10 and of the water content,11,12 which play key roles in acidity properties13 and in ion exchange carried out in aqueous solution14 respectively. These simulations were performed by selecting suitable pair potentials for describing the interactions within the whole system and demonstrated that both the distribution of the extra-framework cations and their interaction with the zeolite host lattice are strongly modified by varying these two parameters. These results were then compared with experimental data obtained by X-ray diffraction15,16 and dielectric relaxation spectroscopy.17,18 The quantitative agreement obtained between theory and experiment reinforced the validity of the potential model used, and leads us to extend this study to the dynamical properties of the extra-framework cations by applying the molecular dynamics technique.19 This powerful tool has been intensively used to study the diffusion of both adsorbed molecules20 and exchangeable cations21–24 in zeolites including either a rigid or flexible framework model. This method allows a direct correlation between experimental spectra, i.e. infra-red, Raman, or inelastic neutron scattering, and atomic motions by evaluating the Fourier transform of the appropriate auto-correlation function for selected atoms.20,25 It is thus particularly useful in providing interpretation of the vibrational spectra when the assignment of the bands is not straightforward in complex systems such as zeolites. A more detailed description of the atomic motions at the microscopic level can thus be obtained.
Using such methods, some attempts have already been made to study the vibrational spectra both of zeolite frameworks, including sodalite, zeolite A and faujasite,26–31 and of adsorbed molecules32 in these host materials. Bougeard et al.33 carried out molecular dynamics calculations in order to simulate the low frequency region of the vibrational spectra (below 200 cm−1) ascribed to the vibrations of the extra-framework cations in zeolites. They focused on the hexagonal EMT zeolites exchanged with various alkali extra-framework cations and reported good agreement with the infrared and Raman features assigned to the cation motions. This work clearly showed the accuracy of the molecular dynamics method, in comparison with other approaches based on local mode analysis or the resolution of the secular equations, previously adopted to determine the normal vibrational modes for the alkali and alkaline earth cations in exchanged zeolites A, faujasite and sodalite.34–37
In this present work, we investigate the vibrational behaviour of the extra-framework cations in Mordenites as a function of many parameters which characterise this zeolite system. In a first step, we study the influence of the force field selected to model the cation–framework interactions in Na-mordenite, by using two different pair potential models and including both flexibility and polarizability of the framework. Furthermore, although a number of molecular dynamics studies have suggested that a flexible framework does not make any significant difference to the diffusion of guest molecules in zeolites,20,38 it is informative to run the dynamics with a fixed framework in order to evaluate this effect on the vibrations of the cations, which are usually located closer to the framework than adsorbed molecules. We then consider the influence of the Si/Al ratio characterising the framework which was previously investigated from a structural point of view. Finally, we also carry out simulations on K+, Mg2+ and Ca2+ exchanged mordenites in order to examine in details the effects of the mass, charge and size of the cations on the frequencies of cation modes. This study requires the choice of suitable potentials for each of the cations. These are firstly validated by reproducing the equilibrium positions of these counter-ions in mordenites observed experimentally, before starting the dynamics investigation. Further to this, the trend deduced from the molecular dynamics runs is compared with local oscillator calculations.34,37,39
Computational methodology
Prior to any molecular dynamics simulation, the various systems investigated, underwent energy minimisation by means of the energy minimisation code GULP
(general utility lattice program).40
Energy minimisation procedure
In a first step, we selected typical (Si,Al) mordenite lattices, characterised by Si/Al ratios of 5 and 11 from representative aluminium configurations previously obtained by combining solid state 29Si NMR spectroscopy data and Monte Carlo simulation.7,8 The location of the extra-framework cations was then determined by means of energy minimisation consisting of a combination of using the Newton–Raphson method to update the Hessian matrix by the BFGS approach and the RFO technique.40 Taking into account that the extra-framework cations are always located close to the aluminium atoms,8 we introduced these cations in such a way that they were in the same channels as the aluminium atoms. The calculations were performed using periodic boundary conditions and the following crystallographic unit cells of mordenite41 corresponding to the various cases investigated in this study.
NaxSi48−xAlxO96 with x
=
4 and 8, for describing the different Si/Al ratios 5 and 11 and Mn+8/nSi40Al8O96, with Mn+
=
Na+, K+, Mg2+, Ca2+ for investigating the various cations. It is noteworthy that, although minimisations were carried out without symmetry constraints, in each case the optimised structure retained the space group of the starting configuration. The potential energy surface of the system was evaluated by appropriate pair potential models. In the first step, we used two different forcefields for describing the interactions within the Na+-Mordenite (Si/Al
=
5) system: the first one was reported by Jackson and Catlow42
(which includes the silica potential of Sanders et al.43) denoted FF1 and the second one corresponds to those developed by Bell44 denoted FF2. These potentials describe the interactions within the zeolite framework and between the framework and the extra-framework cations by means of the Buckingham potential. Furthermore, in these two models, the zeolite framework is considered to be fully flexible by introducing three body terms for O–Al–O and O–Si–O and polarisable via a shell model for the oxygen atoms in the case of FF1. FF2 uses a rigid ion, partial charge model. The pair potential parameters and the charges carried by the atoms for the two force fields are listed in Table 2 and Table 3. Next, we used the pair potential parameters reported by Jackson and Catlow42 to model the interactions between K+, Mg2+, Ca2+ and the zeolite framework (see Table 4), the interactions within the framework remaining the same as those described by FF1 (Table 2). For each of these minimisation procedures, the Ewald summation19 was used for the calculation of the Coulomb interactions and the short range contribution was evaluated by introducing a cut off at 16 Å.
Table 1 Physico-chemical characteristics of the extra-framework cations investigated in this work
Type of cations |
Mass/g mol−1 |
Radius/Å |
Charge/e |
Na+ |
23.0 |
0.97 |
1.0 |
K+ |
39.0 |
1.33 |
1.0 |
Mg2+ |
24.0 |
0.66 |
2.0 |
Ca2+ |
40.0 |
0.99 |
2.0 |
Table 2 Pair potential parameters and charges carried by the atoms in the force field reported by Jackson and Catlow (FF1)
Species |
Charge/e |
Core–shell interactions/eV Å−2 |
Al |
3.00000 |
|
Si |
4.00000 |
|
Na |
1.00000 |
|
O (core) |
0.86902 |
|
O (shell) |
−2.86902 |
74.92 |
Buckingham potential (short range cut off: 16 Å): V(r) = A exp(−ρr) + C/r6 |
Ion pair |
A/eV |
ρ/Å |
C/eV Å6 |
Si–O |
1283.907 |
0.32052 |
10.66158 |
Al–O |
1460.300 |
0.29912 |
0.00000 |
Na–O |
1226.840 |
0.30650 |
0.00000 |
O–O |
22764.000 |
0.14900 |
27.88000 |
Harmonic three-body potential: V(θijk) = k(θijk − θ0)2 |
|
k/eV rad−2 |
θ
0/° |
O–Si–O |
2.09724 |
109.47 |
O–Al–O |
2.09724 |
109.47 |
Table 3 Pair potential parameters and charges carried by the atoms in the force field reported by Bell (FF2). It is noteworthy that the three body term is based on a screened harmonic function including the effects of both interatomic distances and bending interactions
Species |
Charge/e |
Al |
1.40000 |
Si |
2.40000 |
Na |
1.00000 |
O |
−1.20000 |
Buckingham potential (short range cut off: 16 Å): V(r) = A exp(−ρr) + C/r6 |
Ion pair |
A/eV |
ρ/Å |
C/eV Å6 |
Si–O |
30023.00 |
0.16210 |
12.840 |
Al–O |
26998.00 |
0.16220 |
12.840 |
O–O |
894.60 |
0.32440 |
0.000 |
Na–O |
8200.00 |
0.21800 |
11.800 |
Screened harmonic three-body potential: V(θijk) = k(θijk − θ0)2exp[−(rij/ρ1 + rik/ρ2)] |
|
k/eV rad−2 |
θ
0/° |
ρ
1/Å |
ρ
2/Å |
O–Si–O |
12.100 |
109.47 |
1.7 |
1.7 |
O–Al–O |
2.200 |
109.47 |
1.7 |
1.7 |
Table 4 List of parameters compatible with FF1, used to describe the interactions between the various extra-framework cations and the framework
Buckingham potential (short range cut off: 16 Å): V(r) = A exp(−ρr) + C/r6 |
Ion pair |
A/eV |
ρ/Å |
C/eV Å6 |
Na–O |
1226.840 |
0.30650 |
0.0000 |
K–O |
1000.300 |
0.36198 |
10.5690 |
Mg–O |
946.627 |
0.31810 |
0.0000 |
Ca–O |
1227.700 |
0.33720 |
0.0000 |
Molecular dynamics simulations
The various force fields previously described were implemented in the DLPOLY program.45 We selected the optimised structures obtained by the minimisation procedure as starting configurations and the minimised cell dimensions were kept fixed during the molecular dynamics (MD) runs. The MD simulations were run in a NVT ensemble using the Hoover thermostat19 for 100 ps, including a 40 ps equilibration period where the extra-framework cations were maintained fixed in their initial positions. The equations of motion were integrated using the Verlet algorithm implemented in the DLPOLY program.45 It is noteworthy that in a first step the framework was treated as being flexible, before considering the case of a fixed framework. Furthermore, the shell model which is included on oxygen of the framework in the force field reported by Jackson and Catlow (FF1),42 was treated by the adiabatic shell model approach which consists of assigning a small mass to the shell (0.2 u).19 Due to this latter point, and in order to maintain a stable system, we used a very small time step of 0.2 fs. In the case of the potential developed by Bell44
(FF2) where the atoms are characterised by partial charges, we used a time step of 1 fs. We then stored the velocities every 20 fs, i.e. such that the frequency of saving the trajectory gave a time period much smaller than the period of vibrations for the cations. From these saved data, the velocity auto-correlation functions (VACF) were computed for the selected type of extra-framework cations (Na+, K+, Mg2+, Ca2+) by the following equation:19 |  | (1) |
where n corresponds to the number of extra-framework cations j considered in the estimation of the VACF, and we use multiple time origins in order to improve the statistics of the calculation.
Fig. 1 reports a typical velocity auto-correlation function of Na+ cations obtained for Na+-mordenite (Si/Al
=
5) using the force field FF1. We notice that the MD simulations must be run for long enough in such a way that the VACF is dampened to 0.
 |
| Fig. 1 Typical velocity auto-correlation function of Na+ cations in Na+-mordenite (Si/Al = 5) evaluated from the FF1 force field. | |
Finally, the corresponding Fourier transform of the VACF allows us to extract the density of vibrational states of extra-framework cations, also called the VACF power spectrum.
Results and discussions
As previously mentioned, the first step of this work was to define the equilibrium positions of the extra-framework cations by means of an energy minimisation procedure, in order to check the validity of the force fields used by direct comparisons with experimental data, and in this way to generate starting configurations for the molecular dynamics runs. The location of the Na+ extra-framework cations was then investigated using both FF1 and FF2 force fields, which led to the same qualitative distribution of the cations for the selected (Si,Al) mordenite lattice Si/Al
=
5, represented in Fig. 2a. We find that four cations are split equally between sites IV and VI in the main channels, and four are located on sites I in the small channels. It is thus apparent that our chosen (Si,Al) configuration is particularly judicious because it leads to a cation distribution close to those observed experimentally by X-ray diffraction15,16 and dielectric relaxation spectroscopy.17 In the same way, the location of Na+ cations determined for a typical (Si,Al) mordenite lattice Si/Al
=
11 (see Fig. 2b) was in good agreement with experimental data.17 Next, we defined the positions of the K+, Ca2+ and Mg2+ extra-framework cations for selected (Si,Al) mordenite lattices Si/Al
=
5 by using the various force fields previously described.42 We thus found that the potassium ions are distributed among three distinct sites named II, IV and VI (Fig. 3a) whereas the divalent calcium and magnesium ions occupy four sites named I, III, IV and VI (Fig. 3b and 3c). These calculations were in good qualitative agreement with X-ray diffraction data reported in the literature.15
 |
| Fig. 2 Calculated distribution of the Na+ extra-framework cations among the three distinct crystallographic sites I, IV and VI for selected (Si,Al) mordenite unit cells characterised by Si/Al = 5 (a) and Si/Al = 11 (b). | |
 |
| Fig. 3 Simulated partition of the extra-framework cations for selected (Si,Al) Mordenite lattices: a. K+, b. Mg2+, c. Ca2+. | |
From this initial work based on the lattice energy minimisation technique, we have proved from a structural point of view the validity of the forcefields used in this study to model the interactions between the extra-framework cations and the framework. We now will test this forcefield for the prediction of the vibrational spectra from the following dynamical investigations. In these studies, we will deal systematically with the various factors affecting the vibrational behaviours of the extra-framework cations.
Influence of the force field
We first calculated the VACF power spectrum of the sodium cations in Na-mordenite Si/Al
=
5 using the FF1 force field. We can observe in Fig. 4 that this spectrum shows two main peaks at 91 cm−1 and 169 cm−1, one shoulder centred around 221 cm−1 and one artefact due to the Fourier transform algorithm used. This domain of frequency is in qualitative agreement with those previously reported both theoretically and experimentally in other zeolite systems33–37 such as Na-faujasite and Na,K-EMT where bands ranging from 110 cm−1 to 190 cm−1 in the first case and from 100 cm−1 to 210 cm−1 in the later case were attributed to cation vibrations. We then compare this result with the spectrum evaluated from the FF2 force field (see Fig. 4). We can see in the latter case that the signal is significantly up-shifted and broadened with the presence of two peaks at 130 cm−1 and 182 cm−1 and three shoulders around 221 cm−1, 286 cm−1 and 325 cm−1. The difference both in shape and in frequency, observed for the two calculated spectra is likely to be due to the fact that the polarizability of the framework is treated in a different way in these two force fields, with a core-shell model in FF1 and with partial charges in FF2.
 |
| Fig. 4 Comparison between VACF power spectra of Na+ cations in Na+-Mordenite depicted in Fig. 2a, obtained from FF1 (dashed line with square symbols) and FF2 (solid line with circle symbols) force fields. | |
Influence of framework composition
Fig. 5 reports the densities of vibrational states of Na+ obtained for two Si/Al ratios 5 and 11 characterising the mordenite framework calculated from the FF1 force field. We can observe that the initial peak at 169 cm−1 is replaced by a small shoulder when Si/Al ratio increases. A possible explanation of this observation could be a minimisation of the correlation and interaction effects between more well dispersed cations at the higher Si/Al ratio. This diminution of interactions between extra-framework cations is clearly underlined in Fig. 2 where we can observe that each cavity, main and small channel contains only one cation for Si/Al
=
11 (Fig. 2b) whereas, for instance, the small channels host two closely spaced extra-framework cations at the lower Si/Al ratio (Fig. 2a). The decrease of the number of peaks in the VACF power spectra when the Si/Al ratio increases, also reported for other zeolite EMT systems,33 could also be due to less pronounced interactions between extra-framework cations.
 |
| Fig. 5 Comparison between VACF power spectra of Na+ cations in Na+-Mordenites characterised by two different Si/Al ratios 5 (dashed line with circle symbols) and 11 (solid line with square symbols) represented in Fig. 2. | |
We then compared the previous VACF spectrum calculated for a flexible lattice (Fig. 4) with those computed whilst maintaining a fixed zeolite framework. We notice in Fig. 6, that, in this latter case, the frequencies are slightly up-shifted and that the number of peaks becomes more important. This observation indicates that a significant coupling between the framework and the cations exists and that a flexible framework can play a role in the dynamics of the cations. A similar behaviour was recently simulated in NaY46 and was attributed to a thermalisation effect of the cations induced by the flexibility of the framework, which means that a possible energy transfer can occur from the cations to the framework. This result is particularly interesting because it underlines the importance of introducing the flexibility of the framework when studying the dynamics of the cations, its role being much more important than is usually reported for the diffusion of various guest molecules in this kind of host material.
 |
| Fig. 6 VACF power spectra of Na+ cations in Na+-Mordenites Si/Al = 5 obtained both for a fixed (solid line and square symbols) and flexible (dashed line and circle symbols) framework. | |
Influence of the extra-framework cation environment
As it is not straightforward to distinguish the different contributions of each of the cation sites in the global VACF power spectrum of sodium cations, it is more efficient to calculate directly the spectrum for each individual cation site. In this way, Fig. 7 reports the various densities of vibrational states of the three distinct crystallographic sites occupied by sodium ions in Mordenite. As can be seen, we obtain different signals as a function of the nature of the cation sites. It is not surprising that such differences are observed if we consider that these sites are characterised by different geometrical environments and hence by different crystallographic symmetry.15
 |
| Fig. 7 VACF power spectra of the different crystallographic sites I (solid line with square symbols) IV (dashed line with circle symbols) and VI (dotted line with triangle symbols) occupied by the Na+ cations in Na+-mordenite Si/Al = 5 represented in Fig. 2a obtained from FF1 force field. | |
Influence of the nature of extra-framework cation
Fig. 8 reports the calculated VACF power spectra for each of the selected alkali and alkaline earth cations. We notice that these spectra exhibit a progressive increase in vibrational frequency in the series K+, Na+, Ca2+, Mg2+. In order to rationalise this trend, we performed local oscillator calculations34,37,39 which lead to an estimation of the frequencies of vibrations as a function of the nature of the extra-framework cations. Once the secular equations are set up by using the Wilson matrix formalism,47 the following equation is usually reported:34 |  | (2) |
where C is a constant, k the force constant for a simple harmonic oscillator, which can be written as a function of the charges q0, qMn+ on the oxygen of the framework and the cation respectively, and the ionic distances r0 between these two atoms. Finally μ corresponds to the reciprocal mass of the oxygen–cation system.
 |
| Fig. 8 Evolution of the VACF power spectrum as a function of the nature of the alkali and alkaline earth extra-framework cations K+
(solid line with square symbols), Na+
(dashed line with circle symbols), Mg2+
(dotted line with up-triangle symbols) and Ca2+
(dashed-dotted line with down triangle symbols). | |
From this expression and considering the data provided by Table 1 for each of the cations, we estimated the following vibrational frequency ratios:
This calculation concurs with the trend revealed by our simulations. In addition, we can examine separately the effect of both mass and charge by considering the pairings K
+/Na
+
(charge constant,
mK+![[thin space (1/6-em)]](https://www.rsc.org/images/entities/char_2009.gif)
∼
![[thin space (1/6-em)]](https://www.rsc.org/images/entities/char_2009.gif)
2 m
Na+), and Mg
2+/Na
+
(mass
![[thin space (1/6-em)]](https://www.rsc.org/images/entities/char_2009.gif)
∼
![[thin space (1/6-em)]](https://www.rsc.org/images/entities/char_2009.gif)
constant,
qMg2+![[thin space (1/6-em)]](https://www.rsc.org/images/entities/char_2009.gif)
=
![[thin space (1/6-em)]](https://www.rsc.org/images/entities/char_2009.gif)
2
qNa+) respectively. Then, we observe in
Fig. 8, that the vibrational frequency increases with either decreasing mass or increasing charge of the
cations. Furthermore by comparing the differences in frequency between the VACF power
spectra of interest, we notice that the up-shift induced by the charge is clearly more pronounced than those due to the mass. This latter observation is justified by considering the
νK+/
νNa+ and
νMg2+/
νNa+ vibrational ratios obtained from the local mode analysis.
Conclusion
Our simulations, based on selecting reliable potential models and employing them in molecular dynamics simulations, predict significant changes in the position and shape of the bands assigned to the extra-framework cation vibrations in a typical mordenite zeolite, dependent on both the framework Si/Al ratio and on the local environment and site symmetry of the cations. The substantial effects of ignoring framework flexibility are also shown. Furthermore, our calculations reproduced the evolution of the vibrational frequency as a function of mass and charge of the extra-framework cations. The contribution of this work is a preliminary step before studying experimentally in more details the vibrational behaviour of the extra-framework cations in mordenite by inelastic neutron scattering (INS) and far-infrared spectroscopies. The vibrational spectra predicted in this article could then be directly compared with the collected INS data. To model the infra-red spectra, some enhancements to our simulation methods would be necessary, for instance the calculation of the dipole autocorrelation function. Furthermore, we have investigated zeolite systems which don't contain any residual water molecules. The next step of this work will be to include the hydration effect on the calculations of the low frequency vibrations of the cations because as we have previously reported,11,12 we know that the water molecules induce significant perturbations on the cation–framework interactions leading to the detrapping of the cations from their initial sites for the highest hydration rates.
Acknowledgements
This research was supported through a European Communities Marie Curie Individual Fellowship: G.M. with EC Contract Number HPMF-CT-2001-01379. R.B. acknowledges the Leverhulme Trust for funding.
References
-
G. Gottardi and E. Galli, Natural Zeolites, Springer-Verlag, Berlin, 1985, p. 223 Search PubMed.
- C. N. R. Rao, S. Natarajan and S. Neeraj, J. Am. Chem. Soc., 2000, 122, 2810 CrossRef CAS.
- P. J. Davis, H. Van Bekkum and E. N. Coker, J. Chem. Educ., 1999, 76, 469 CAS.
- M. Adabbo, D. Caputo, B. De Gennaro, M. Pansini and C. Collela, Microporous Mesoporous Mater., 1999, 28, 315 CrossRef CAS.
- E. Bosch, S. Huber, J. Weitkamp and H. Knozinger, Phys. Chem. Chem. Phys., 1999, 1, 579 RSC.
- G. Vitale, L. M. Bull, R. E. Morris, A. K. Cheetham, B. H. Toby, C. G. Coe and J. E. Mac Dougall, J. Phys. Chem. B, 1995, 99, 16
087 Search PubMed.
- G. Maurin, P. Senet, S. Devautour, F. Henn, J. C. Giuntini and V. E. Van Doren, Comput. Mater. Sci., 2001, 22, 106 CrossRef CAS.
- G. Maurin, P. Senet, S. Devautour, P. Gaveau, F. Henn, V. Van Doren and J. C. Giuntini, J. Phys. Chem. B, 2001, 105, 9157 CrossRef CAS.
- G. Maurin, P. Senet, S. Devautour, F. Henn and J. C. Giuntini, J. Chem. Phys., 2002, 117, 4, 1405 CrossRef.
- G. Maurin, P. Senet, S. Devautour, F. Henn and J. C. Giuntini, J. Non-Cryst. Solids, 2002, 307, 1050 CrossRef.
- G. Maurin, R. G. Bell, S. Devautour, F. Henn and J. C. Giuntini, J. Phys. Chem. B Search PubMed , accepted.
-
G. Maurin, R. G. Bell, S. Devautour, F. Henn and J. C. Giuntini, Proceedings of the 14th International Zeolite Conference, submitted Search PubMed.
- V. R. Chumbhale, A. J. Chandwadkar and B. S. Rao, Zeolites, 1992, 12, 6663 CrossRef CAS.
- P. Pissis and D. Daoukaki-Diamanti, J. Phys. Chem. Solids, 1993, 54, 701 CrossRef CAS.
-
W. J. Mortier, Compilation of Extra-framework Sites in Zeolites, Butterworth, Guildford, 1982 Search PubMed.
- B. Coughlan, W. M. Carrol and A. McCann, J. Chem. Soc., Faraday Trans., 1977, 73, 1612 Search PubMed.
- M. Pamba, G. Maurin, S. Devautour, J. Vanderschueren, J. C. Giuntini, F. Di Renzo and F. Hamidi, Phys. Chem. Chem. Phys., 2000, 113, 11, 4498 Search PubMed.
- S. Devautour, A. Abdoulaye, J. C. Giuntini and F. Henn, J. Phys. Chem. B, 2001, 105, 9297 CrossRef CAS.
-
M. F. Allen and D. Tildesley, Computer Simulation of Liquids, Oxford, Science Publication, Oxford, 1987 Search PubMed.
- P. Demontis and G. B. Suffritti, Chem. Rev., 1997, 97, 2845 CrossRef CAS.
- D. Faux, W. Smith and T. R. Forester, J. Phys. Chem. B, 1997, 101, 1762 CrossRef CAS.
- D. Faux, J. Phys. Chem. B, 1999, 103, 7803 CrossRef CAS.
- F. Manon Higgins, N. H. de Leeuw and S. C. Parker, J. Mater. Chem., 2002, 12, 124 RSC.
- P. Demontis and G. B. Suffritti, Mol. Phys., 1997, 91, 669 CrossRef CAS.
-
L. J. Lewis and M. L. Klein, Dynamical Properties of Solids, G. K. Horton and A. A. Maradudin, Elsevier, Amsterdam, 1990 Search PubMed.
- P. Demontis, G. B. Suffritti, E. S. Fois and S. Quartieri, J. Phys. Chem. B, 1992, 96, 482 Search PubMed.
- J. B. Nicholas, A. J. Hopfinger, F. R. Trouw and L. E. Iton, J. Am. Chem. Soc., 1991, 113, 4792 CrossRef CAS.
- K. S. Smirnov and D. Bougeard, J. Phys. Chem. B, 1993, 97, 9434 Search PubMed.
- K. S. Smirnov and D. Bougeard, Catalysis Today, 2001, 70, 243 CrossRef CAS.
- P. Bornhauser and D. Bougeard, J. Phys. Chem. B, 2001, 105, 36 CrossRef CAS.
- H. Jobic, K. S. Smirnov and D. Bougeard, Chem. Phys. Lett., 2001, 344, 147 CrossRef CAS.
- F. Jousse, S. M. Auerbach, H. Jobic and D. P. Vercauteren, J. Phys. IV, 2000, 10, Pr7–147 Search PubMed.
- D. Bougeard, C. Bremard, D. Dumont, M. Le Maire, J. M. Manoli and C. Potvin, J. Phys. Chem. B, 1998, 102, 10
805 CrossRef.
- J. Godber, M. D. Baker and G. A. Ozin, J. Phys. Chem. B, 1989, 93, 1409 Search PubMed.
- J. Godber, G. A. Ozin and M. D. Baker, J. Phys. Chem. B, 1988, 92, 2841 Search PubMed.
- A. Boumiz, J. Cartigny and E. Cohen de Lara, J. Phys. Chem. B, 1992, 96, 5419 Search PubMed.
- M. D. Baker, J. Godber and G. A. Ozin, J. Am. Chem. Soc., 1985, 107, 3033 CrossRef CAS.
- P. Demontis, G. B. Suffritti, E. S. Fois and S. Quatieri, J. Phys. Chem. B, 1990, 94, 4329 Search PubMed.
- G. A. Ozin, M. D. Baker, J. Godber and W. Shihua, J. Am. Chem. Soc., 1985, 107, 1995 CrossRef CAS.
- J. D. Gale, J. Chem. Soc. Faraday. Trans., 1997, 93, 629 RSC.
- W. M. Meier, Z. Kristallogr., 1961, 115, 439 CAS.
- R. A. Jackson and C. R. A. Catlow, Mol. Simul., 1988, 1, 207 Search PubMed.
- M. J. Sanders, M. Leslie and C. R. A. Catlow, J. Chem. Soc., Chem. Commun., 1984, 1271 RSC.
-
R. G. Bell and N. A. Ramsahye, submitted.
- W. Smith and T. R. Forester, J. Mol. Graphics, 1996, 14, 3; W. Smith and T. R. Forester, J. Mol. Graphics, 1996, 14, 136 CrossRef CAS.
-
N. A. Ramsahye, PhD Thesis, University of London, UK, 2003.
-
E. B. Wilson, J. C. Decius and P. C. Cross, Molecular Vibrations, McGraw-Hill, New York, 1955 Search PubMed.
Footnote |
† Present address: Laboratoire Madirel, UMR CNRS 6121, Faculté des Sciences St Jérôme, Av. Escadrille Normandie Niemen, 13397 Marseille cedex 20, France. E-mail: maurin@up.univ-mrs.fr. Tel: +33 4 91 63 71 17. Fax: +33 4 91 63 71 11. |
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