4618
J. Phys. Chem. C 2009, 113, 4618–4622
Drastic Phase Transition in Ionic Liquid [Dmim][Cl] Confined Between Graphite Walls: New Phase Formation Maolin Sha,†,‡ Guozhong Wu,*,† Yusheng Liu,†,‡ Zhongfeng Tang,†,‡ and Haiping Fang*,† Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai 201800, P.O. Box 800-204, China, and Graduate School of the Chinese Academy of Sciences, Beijing 10039, People’s Republic of China ReceiVed: December 12, 2008; ReVised Manuscript ReceiVed: January 20, 2009
Confinement can induce unusual behavior in the properties of matter. Using atomistic molecular dynamics simulations, we report here a liquid-to-solid transition of a bilayer of ionic liquid 1,3-dimethylimidazolium chloride ([Dmim][Cl]) confined between graphite walls in order to mimic the phase transition of an ionic liquid confined to hydrophobic nanospace. It was found that the ionic liquid bilayer undergoes a clear and drastic phase transition at a wall distance of about 1.1 nm, forming a new high-melting-point solid phase with different hydrogen bonding networks. In the new phase, each cation is surrounded by the three nearestneighbor anions, and each anion is also encircled by the three nearest-neighbor cations. Strong π-π stacking interactions are found between the cations of the bilayer solid. The anions can be formed into a hexagonal ring around the cations. The new bilayer solid is a high-melting-point crystal possessing a melting point of 825-850 K, which is higher than that of the bulk crystal by more than 400 K. 1. Introduction The crystalline phases of confined hydrogen-bonded systems are of considerable interest in many fields such as biology, chemistry, and materials science.1-8 The confinement of water in solids has been intensively investigated theoretically and experimentally. In particular, the rich phases of nanoconfined ice extend our knowledge of nanoconfinement.1-5 Like water, room-temperature ionic liquids (ILs) also consist of a hydrogen bond network but have much larger molecular weights. ILs are low-melting-point organic salts whose desirable properties are leading to an increasing number of applications.9,10 Moreover, their chemical and physical properties may be tuned by choosing a specific combination of cations and anions among numerous possibilities. When ionic liquids are in contact with a solid, there are strong interactions from wetting, immobilization, or confinement. In some real applications, the interactions between ILs and solids are complex but very useful. For example, ILs can be used to disperse entangled nanotube bundles11 and synthesize zeolite analogues as templates.12 Confined ILs also attract great attention in organometallic catalysis processes, fuels, and solar cells.13-17 For the applications of electrolyte membranes and catalysts, there is a need to immobilize ionic liquids on solid supports or within a solid matrix. Several research groups have dedicated themselves to the study of confining ILs within the solid nanospace and have found the melting-point depression of the confined ILs.18-23 Meanwhile, the transformation of an ionic liquid into a high-melting-point crystal when confined in multiwalled carbon nanotubes has been reported.24 These inconsistent observations of melting-point variation for confined ILs in nanospaces are interesting and deserve attention. However, there is currently no detailed understanding of the microstructure and phase behavior of ILs confined in nanospaces. * To whom correspondence should be addressed. E-mail: wuguozhong@ sinap.ac.cn;
[email protected]. † Shanghai Institute of Applied Physics. ‡ Graduate School of the Chinese Academy of Sciences.
To improve our understanding of confined ILs, we have investigated the microstructure and possible melting-point variation of ionic liquid 1,3-dimethylimidazolium chloride ([Dmim][Cl]) confined between two parallel graphite walls using fully atomistic molecular dynamics simulations. In our previous simulation,25 a monolayer solid of ionic liquid [Dmim][Cl] was found when confined between graphite walls that were 0.7 nm apart. However, what will happen with a less-confining space (i.e., larger wall distance) is not clear. Hence, a bilayer of [Dmim][Cl] was applied to mimic a confined ionic liquid between walls a long distance apart in this work. Because [Dmim][Cl] has a small anion size and short alkyl chains on the imidazolium ring, the motion of the IL molecule is relatively easy to simulate. This can be considered a model system for the study of H-bonding molecules confined in hydrophobic nanospaces. The microstructure of the IL bilayer was studied by varying the graphite wall distance. A liquid-to-solid phase transition of bilayer [Dmim][Cl]] was observed at 425 K in this confined system, whereas the melting point of the bulk [Dmim][Cl] crystal is 399 K. Further calculation indicates a high melting point of the confined IL: melting point ∼825-850 K. The imidazolium ring of the solid bilayer forms a strong π · · · π stacking structure in which each cation is surrounded by the three nearest-neighbor anions. The bilayer solid is a new phase of [Dmim][Cl] differing from the monolayer solid25 under identical confinement conditions or the bulk crystal. 2. Methodology and Simulation Details The [Dmim][Cl] molecule was treated as a systematic allatom force field, as developed by Lopes et al.26 This method was successfully used in our previous simulations.25,27 The model system consists of 112 [Dmim][Cl] molecules placed in a tetragonal box. Molecules are confined between two parallel graphite walls that are perpendicular to the principal axis of the cell (z direction). The graphite wall is constructed from 22 × 13 elementary cells leading to surface dimensions of lx ) 56.56 Å and ly ) 55.38 Å. The [Dmim]+-wall interactions were represented by a 6-12 Lennard-Jones potential with the same
10.1021/jp810980v CCC: $40.75 2009 American Chemical Society Published on Web 02/25/2009
Drastic Phase Transition in Ionic Liquid [Dmim][Cl]
Figure 1. Lateral diffusion coefficients of cation and anion as a function of the distance between the confining walls at T ) 425 K.
parameters used in our previous work.25,27 The anionic chloride atoms were also modeled using an OPLS-AA force field.28 The cross interactions were computed according to the conventional combination rule: σij ) (σi · σj)1/2 and εij ) (εi · εj)1/2 for OPLSAA. The melting point of bulk [Dmim][Cl] at ambient pressure (0.1 MPa) was estimated to be approximately 399 K.29 The first series of simulations was performed at a fixed temperature at 425 K using Gromacs 3.230 to make sure that the ionic liquid is in the liquid state. The simulation was further carried out over a wide temperature range in order to find the melting point of the bilayer IL solid. Periodic boundary conditions were applied in the x and y directions. In all of the simulations, the bond length was constrained with the LINCS algorithm. The cutoff of LennardJones interactions was taken at 12 Å. The long-range Coulomb interactions were handled by PME with a cutoff of 15 Å and a grid spacing of 1.2 Å. The Berendsen thermostat was used to mimic the weak coupling; cations and anions were separated into two heat baths, each with a temperature coupling constant of 0.1 ps. All of the systems were run for 1 ns at 1000 K and then annealed from 1000 to 425 K in three stages: 1 ns at 800 K, 1 ns at 600 K, and 1 ns at 425 K. The initial configuration was prepared by taking a thin out-of-order layer (H ) 0.9 nm) of [Dmim][Cl] from the bulk configuration. At each thermodynamic point (achieved by a sequential increase in H), the system was again equilibrated for 5 ns, and the data were collected for an additional 3 ns. 3. Results and Discussion 3.1. Diffusion Coefficient and Potential Energy. The diffusion of molecules in a confined liquid is known to be different than in the bulk fluid. It has been reported that the diffusion coefficient of water decreases by 3 to 4 orders of magnitude when it transforms into a frozen state.1-3 Hence, the lateral mean-square displacements of the ions were calculated to predict the phase transition. Figure 1 shows the cation and anion lateral diffusion coefficients for wall separations in the range of 1.0 nm e H e 1.4 nm. For a wall separation in the range of 1.15 nm < H < 1.3 nm, [Dmim][Cl] is a liquid bilayer as indicated by the large diffusion coefficient. At H < 1.15 nm, the liquid bilayer transforms into a frozen state as indicated by a value of nearly zero for the lateral diffusion coefficient of the anions and cations. In this state, the dynamics of the ion pairs are best described in terms of small-amplitude fluctuations around fixed positions with occasional cooperative jumps, rather than as free diffusion. At H ) 1.3 nm, the liquid bilayer transforms into a
J. Phys. Chem. C, Vol. 113, No. 11, 2009 4619
Figure 2. Distance dependence of potential energies at 425 K. The potential energy consists of the cation-cation, cation-anion,and anion-anion intermolecular interactions and the cation-wall and anion-wall interactions.
three-layer structure. This liquid-to-solid transition trend is similar to that reported by Gao et al.31 for an atomic fluid between corrugated walls, where evidence of a confinementinduced transition to a solidlike phase was observed. When the wall separation was continually increased beyond 1.3 nm, a multilayer solid could not be obtained. This phenomenon is in good agreement with the result of Pinilla et al.23 The phase transition of ionic liquid [Dmim][Cl] is also confirmed by the wall separation dependence of the potential energy. Figure 2 shows the dependence of the potential energy on wall separation (H) at 425 K. As H is increased, the potential energy suddenly increases by about 30 kJ/mol within a 0.15 nm range (from 1.10 to 1.25 nm). Moreover, the lateral diffusion coefficients also increased steeply with increasing H in this region (Figure 1). For H e 1.10 nm, the phase is a solid bilayer. The distance dependences of the diffusion coefficient and the potential energy both indicate a strong first-order phase transition of the confined [Dmim][Cl]. 3.2. Radial Distribution Function. To obtain a better understanding of the structure of the solid bilayer, the 2D inplane radial distribution function g(r) was computed for the solid structure. The lateral radial distribution function (RDF) in Figure 2 was averaged over the two independent layers. There is clear long-range ordering for the solid bilayer (H ) 1.0 nm) in both the cation-anion and anion-anion distributions. It was observed that the first shell of the cation-anion distribution occurs uniformly at about 0.45 nm. This is due to the dominant interaction between cations and anions in the first shell in both the liquid32-34 and solid states. However, the second and third shells are very different. The RDF of the bulk liquid is more liquidlike compared to the solid bilayer and monolayer. The differences between the RDFs of the solid and the liquid states can also be seen in the anion-anion distributions (Figure 2b). The split peaks of the solid bilayer and monolayer are evidence of solid ordering. It is interesting to find that the first shell of the anion-anion distribution in the solid bilayer, the solid monolayer, and the bulk liquid occurs at 0.81, 0.72, and 0.57 nm, respectively. This result implies that the anion-anion distribution is very sensitive to the structure change. Because the cation-anion interaction determines the basic structure of the system, the anion-anion distribution can be modulated by different cation-anion distributions. The RDFs can also indicate the coordination number for the shell in different systems. By integrating g(r) out to the location of the first minimum, the coordination number of the 2D first
4620 J. Phys. Chem. C, Vol. 113, No. 11, 2009
Figure 3. Two-dimensional mass-centered radial distribution functions of the cation-anion and anion-anion in the bulk phase, monolayer of solid (H ) 0.70 nm), and bilayer of solid (H ) 1.0 nm). (a) Cation-anion distribution. (b) Anion-anion distribution. The data for the monolayer solid are from ref 25.
shell, N, can be calculated via the following equation: N ) ∫0r shellFg(r) 2πr dr, where F is the 2D number density and rshell is the first minimum in g(r). Using rshell ) 0.73 nm, the calculated coordination numbers of the bilayer and monolayer solid for the cation-anion first shell are 2.8 and 3.9, indicating that each ion is surrounded by nearly three or four counterions, respectively. Similarly, the anion-anion coordination numbers of the bilayer and the monolayer solid in the first shell are 6.3 and 4.0, respectively. It is obvious that the bilayer solid has a different number of nearest neighbors than the monolayer solid. The bilayer solid is a new solid phase of [Dmim][Cl]. 3.3. Bilayer Solid Structure. Figure 4a,b shows the inherent structures of the new bilayer solid and liquid phases, respectively. In the solid bilayer, the two molecular layers are nearly flat; the cations and anions in both layers are well situated spatially in a periodic unit cell. These two features do not show up in the liquid phase. These features are also consistent with our previous monolayer solid simulations.25 However, the nearest-neighbor structure of the bilayer solid is different from the monolayer solid confined in between graphite walls and the bulk crystal of [Dmim][Cl].35 In the bilayer solid, each cation is surrounded by three nearest-neighbor anions, and each anion is also encircled by three nearest-neighboring cations. In contrast, there are four and two nearest-neighboring counterions around each ion in the monolayer solid and the bulk crystal, respectively. This is in good agreement with the aforementioned coordination number for the first shell, in which the coordination numbers of the bilayer and monolayer solids for the cation-anion first shell are 2.8 and 3.9, respectively.
Sha et al. To analyze the detailed configuration of the bilayer solid further, corresponding snapshots of the cations and anions in the bilayer solid are shown in Figure 5. For the cations, there is an obvious ordering of the imidazolium ring between the top and bottom layers, which are parallel to each othersa feature of the strong π-π stacking interaction. Such π-π stacking is also observed in bulk crystals of [Dmim][Cl].35 The cations of the bilayer solid are formed via a strong π-π stacking interaction in order to reduce the potential energy and entropy cost. This preference can also be found in the liquid state, but it is more dispersed and out of order.33,34 Similarly, there is also an ordering of the anions in the bilayer solid. When all of the anions are connected by dotted lines, we can find that there are many hexagonal rings in the bilayer solid. Each hexagonal ring consists of three anions in the top layer and three more in the bottom layer. The π-π stacking cations are located in the center of the hexagonal ring. Therefore, in each layer, there are three anions around a cation. Each anion is encircled by the six nearest-neighbor anions in each layer. This result is in accordance with the coordination number of the anion-anion distribution in which the N for the first shell is 6.3. Moreover, compared with the three anions of a hexagonal ring in the bottom layer, the three anions of the hexagonal ring in the top layer are rotated 60 ° along the z axis to minimize the anion-anion repulsive potential. 3.4. Melting-Point Variation. It has been revealed that ILs not only readily fill in the solid material of the pore wall18-22 but also undergo a phase transition when confined in certain spaces.24 If ILs fill in the solid material of the pore wall and remain in the liquid state (i.e., fast dynamics and weak interaction with the walls), then the change in the melting point is caused by capillarity, which reduces the melting point.36 The melting point can be estimated by the Gibbs-Thomson equation. These results are confirmed by Kanakubo et al.18 for ionic liquids confined in controlled-pore glasses (CPGs). However, strong fluid wall interactions always lead to an increase in melting temperature.37-40 We found that the interaction energy between the graphite walls and the ionic liquid is very large.25,27 Hence, if a phase transition of ILs can be induced in confined graphite walls, then the melting point will increase sharply. To determine the melting point of the [Dmim][Cl] bilayer solid between the graphite walls compared to that of the bulk crystal, we performed a simulation of the bilayer liquid at a fixed load (Pzz ) 1.0 atm, normal pressure) between two plane-parallel graphite walls to obtain the melting point. The graphite walls are not fixed in this simulation and can fluctuate along the z axis with a fixed load (i.e., 1.0 atm). The temperature is first dropped in steps from 950 to 625 K and then raised in steps from the freezing point until melting is completed. Figure 6 shows the temperature dependence of the potential energy at a pressure of 1.0 atm. As the system is cooled, the potential energy first decreases gradually and then suddenly drops by 15 kJ/mol-1 at about 725-750 K. In the reverse heating process, the potential energy jumps to a higher value at about 825-850 K, following the trace of the cooling process. These sharp changes in energy and the large hysteresis appearing in the cooling and heating processes confirm the observation of a strong first-order phase transition for confined [Dmim][Cl]. The melting point (ca. 825-850 K) of the confined ionic liquid is much higher than that of the bulk crystal (399 K). Although the melting point computed in the simulation may be an overestimation because the free-energy barrier to the formation of the solid-liquid interface causes superheating,41 there is no doubt that the bilayer IL solid has a higher melting point than
Drastic Phase Transition in Ionic Liquid [Dmim][Cl]
J. Phys. Chem. C, Vol. 113, No. 11, 2009 4621
Figure 4. Structure of the confined [Dmim][Cl] in the solid and liquid states. (a) A solid bilayer, H ) 1.10 nm. (b) A liquid bilayer, H ) 1.25 nm. The top and bottom layesr are depicted in blue and green, respectively. The methyls of the cation are omitted for the display.
Figure 5. Snapshots of the cations and anions of [Dmim][Cl] in the solid bilayer. (a) Cation configurations extracted from the solid bilayer. (b) Anion configurations extracted from the solid bilayer. The top and bottom layers are depicted in blue and green, respectively. The methyls of the cation are omitted, and the anions are connected by a dotted line for the display.
nanotubes, in which the wall of the carbon nanotube is also the graphite. We believe that our finding will be useful for the preparation of IL/nanoparticle hybrid materials and that it is possible, in practice, to make a high-melting-point crystal of ionic liquids by confining them in suitable nanoporous materials or nanotubes. 4. Conclusions
Figure 6. Temperature dependence of the potential energy. The potential energy consists of the cation-cation, the cation-anion, and anion-anion intermolecular interactions and the cation-wall and the anion-wall interactions. Filled and unfilled circles indicate the cooling and heating processes, respectively.
the bulk crystal. It can be explained by simple mean field theory that the increased average number of nearest neighbors in the new solid bilayer leads to a melting-point value higher than that of the bulk.42 The liquid-to-solid phase transition and high melting point of the [Dmim][Cl] solid simulated in this work are in good agreement with the experimentally observed high melting point of the [Bmim][PF6] crystal in multiwalled carbon
A new high-melting-point solid phase of ionic liquid [Dmim][Cl] confined between graphite walls has been found using atomistic molecular dynamics simulations. The new bilayer solid can be formed at a wall distance of about 1.1 nm. In the new phase, each cation is surrounded by the three nearestneighbor anions, while each anion is also encircled by the three nearest-neighbor cations. Strong π-π stacking interactions are found between the cations of the bilayer solid. The anions are formed in a hexagonal ring around the cations. The meltingpoint computation shows that the bilayer solid is a high-meltingpoint crystal. However, our model is still limited, akin to those based on the classical mechanics, and is not currently available for a quantum effect. Hence, these results also need a corresponding experiment to confirm it. If confirmed, it will inspire more studies on ionic liquids confined in solid materials (i.e., molecular sieves and a solid matrix), as well as on the potential fabrication of new high-melting-point ionic liquid crystals in confined systems.
4622 J. Phys. Chem. C, Vol. 113, No. 11, 2009 Acknowledgment. This work was supported by the National Science Foundations of China (20573130 and 20673137) and Shanghai Municipal Committee of Science and Technology and Shanghai Supercomputer Center of China. We thank Dr. G. L. Zhu for his valuable discussions. References and Notes (1) Koga, K.; Tanaka, H.; Zeng, X. C. Nature 2000, 408, 564. (2) Koga, K.; Zeng, X. C.; Tanaka, H. Phys. ReV.Lett. 1997, 79, 5262. (3) (a) Zangi, R.; Mark, A. E. Phys. ReV. Lett. 2003, 91, 025502. (b) Zangi, R.; Mark, A. E. J. Chem. Phys. 2003, 119, 1694. (4) Leng, Y. S.; Cummings, P. T. Phys. ReV. Lett. 2005, 94, 026101. (5) Zhu, Y. X.; Granick, S. Phys. ReV. Lett. 2001, 87, 096104. (6) Wilson, M. Nano Lett. 2004, 4, 299. (7) Meyer, R. R.; Sloan, J.; Dunin-Borkowski, R. E.; Kirkland, A. I.; Novotny, M. C.; Bailey, S. R.; Hutchison, J. L.; Green, M. L. H. Science 2000, 289, 1324. (8) Kalugin, O. N.; Chaban, V. V.; Loskutov, V. V.; Prezhdo, O. V. Nano Lett. 2008, 8, 2126. (9) Welton, T. Chem. ReV. 1999, 99, 2071. (10) Rogers, R. D.; Seddon, K. R. Science 2003, 302, 792. (11) Fukushima, T.; Kosaka, A.; Ishimura, Y.; Yamamoto, T.; Takigawa, T.; Ishii, N.; Aida, T. Science 2003, 300, 2072–2074. (12) Cooper, E. R.; Andrews, C. D.; Wheatley, P. S.; Webb, P. B.; Wormald, P.; Morris, R. E. Nature 2004, 430, 1012–1016. (13) Xu, W.; Angell, C. A. Science 2003, 302, 422–425. (14) Noda, A.; Susan, M. A. B. H.; Kudo, K.; Mitsushima, S.; Hayamizu, K.; Watanabe, M. J. Phys. Chem. B 2003, 107, 4024–4033. (15) Wang, P.; Zakeeruddin, S. M.; Humphry-Barker, R.; Graetzel, M. Chem. Mater. 2004, 16, 2694–2696. (16) Seki, S.; Kobayashi, Y.; Miyashiro, H.; Ohno, Y.; Usami, A.; Mita, Y.; Kihira, N.; Watanabe, M.; Terada, N. J. Phys. Chem. B 2006, 110, 10228–10230. (17) Kuang, D.; Ito, S.; Wenger, B.; Klein, C.; Moser Jacques, E.; Humphry-Baker, R.; Zakeeruddin Shaik, M.; Gratzel, M. J. Am. Chem. Soc. 2006, 128, 4146–4154. (18) Kanakubo, M.; Hiejima, Y.; Minami, K.; Aizawa, T.; Nanjo, H. Chem. Commun. 2006, 1828. (19) No´uze, M.; Bideau, J. L.; Gaveau, P.; Bellayer, S.; Vioux, A. Chem. Mater. 2006, 18, 3931. (20) Shimano, S.; Zhou, H.; Honma, I. Chem. Mater. 2007, 19, 5216– 5221. (21) Lunstroot, K.; Driesen, K.; Nockemann, P.; Go¨rller-Walrand, C.; Binnemans, K.; Bellayer, S.; Bideau, J. L.; Vioux, A. Chem. Mater. 2006, 18, 5711–5715.
Sha et al. (22) Hanabusa, K.; Fukui, H.; Suzuki, M.; Shirai, H. Langmuir 2005, 21, 10383–10390. (23) Pinilla, C.; Po´polo, M. G. D.; Lynden-Bell, R. M.; Kohanoff, J. J. Phys. Chem. B 2005, 109, 17922. (24) Chen, S. M.; Wu, G. Z.; Sha, M. L.; Huang, S. R. J. Am. Chem. Soc. 2007, 129, 2416. (25) Sha, M. L.; Wu, G. Z.; Fang, H. P.; Zhu, G. L.; Liu, Y. S. J. Phys. Chem. C 2008, 112, 18584. (26) Canongia Lopes, J. N.; Deschamps, J.; Po´dua, A. A. H. J. Phys. Chem. B 2004, 108, 2038. (27) Sha, M. L.; Zhang, F. C.; Wu, G. Z.; Fang, H. P.; Wang, C. L.; Chen, S. M.; Zhang, Y.; Hu, J J. Chem. Phys. 2008, 128, 134504. (28) Jorgensen, W. L.; Maxwell, D. S.; Tirado-Rives, J. J. Am. Chem. Soc. 1996, 118, 11225. (29) Fannin, A. A.; Floreani, D. A.; King, L. A.; Landers, J. S.; Piersma, B. J.; Stech, D. J.; Vanghn, R. L.; Wilkes, J. S.; Williams, J. L. J. Phys. Chem. 1984, 88, 2614. (30) Lindahl, E.; Hess, B.; Van der Spoel, D. J. Mol. Mod. 2001, 7, 306. (31) (a) Gao, J.; Luedtke, W. D.; Landman, U. Phys. ReV. Lett. 1997, 79, 705. (b) Gao, J.; Luedtke, W. D.; Landman, U. J. Phys.Chem. B 1997, 101, 4013. (32) Hanke, C. G.; Price, S. L.; Lynden-Bell, R. M. Mol. Phys. 2001, 99, 801. (33) Bhargava, B. L.; Balasubramanian, S. Chem. Phys. Lett. 2006, 417, 486. (34) Hardacre, C.; McMath, S E J.; Nieuwenhuyzen, M.; Bowron, D. T; Soper, A. K. J. Phys.: Condens. Matter 2003, 15, 159. (35) Arduengo, A. J.; Dias, H. V. R.; Harlow, R. L.; Kline, M. J. Am. Chem. Soc. 1992, 114, 5530. (36) Christenson, H. K. J. Phys.: Condens. Matter 2001, 13, R95-R133. (37) Watanabe, A.; Kaneko, K. Chem. Phys. Lett. 1999, 305, 71. (38) Sliwinska-Bartkowiak, M.; Dudziak, G.; Sikorski, R.; Gras, R.; Radhakrishnan, R.; Gubbins, K. E. J. Chem. Phys. 1999, 110, 4867. (39) (a) Raviv, U.; Laurat, P.; Klein, J. Nature 2001, 413, 51. (b) Raviv, U.; Giasson, S.; Frey, J.; Klein, J. J. Phys.: Condens. Matter 2002, 14, 9275. (40) Xia, Y.; Dosseh, G.; Morineau, D.; Alba-Simionesco, C. J. Phys. Chem. B 2006, 110, 19735. (41) (a) Alavi, S.; Thompson, D. L. J. Chem. Phys. 2005, 122, 154704. (b) Velardez, G. F.; Alavi, S.; Thompson, D. L. J. Chem. Phys. 2004, 120, 9151. (42) Gelb, L.; Gubbins, K. E.; Radhakrishnan, R.; Sliwinska-Bartkowiak, M. Rep. Prog. Phys. 1999, 62, 1573.
JP810980V
