Kinetics of Molecular Transport in a Nanoporous Crystal Studied by

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2007, 111, 12339-12344 Published on Web 10/09/2007

Kinetics of Molecular Transport in a Nanoporous Crystal Studied by Confocal Raman Microspectrometry: Single-File Diffusion in a Densely Filled Tunnel Javier Martı´-Rujas,† Arnaud Desmedt,‡ Kenneth D.M. Harris,*,† and Franc¸ ois Guillaume*,‡ School of Chemistry, Cardiff UniVersity, Park Place, Cardiff CF10 3AT, Wales, United Kingdom, and UniVersite´ Bordeaux 1, Institut des Sciences Mole´ culaires, CNRS UMR 5255, 351 cours de la Libe´ ration, 33405 Talence Cedex, France ReceiVed: August 14, 2007; In Final Form: September 13, 2007

Confocal Raman microspectrometry has been used as an in situ probe of the transport of guest molecules along the one-dimensional tunnels in a crystalline urea inclusion compound, under conditions of guest exchange in which “new” guest molecules (pentadecane) are introduced at one end of the tunnel and displace the “original” guest molecules (1,8-dibromooctane). The Raman spectra, recorded as a function of position along the tunnel direction and as a function of time, have been used to establish details of the kinetics of the guest transport process. In particular, the transport of the new pentadecane guest molecules along the tunnel is found to exhibit a linear dependence on time, with the rate of the process in the region of 70-100 nm s-1. Mechanistic aspects relating to the guest transport process are discussed.

Introduction Molecular transport through nanoporous tunnels is the basis of a range of processes of biological, industrial, and technological importance.1-3 In spite of wide-ranging interest in such materials, few experimental studies have probed the molecular transport processes in an in situ, time-resolved manner. In this regard, model systems have an important role to play in establishing a fundamental understanding of such transport processes. Urea inclusion compounds4 (Figure 1) are crystalline materials in which the urea molecules form a hydrogen-bonded host structure5 that contains one-dimensional, nonintersecting tunnels (tunnel diameter ca. 5.5 Å5c). Guest molecules are densely packed within these tunnels, forming a periodic guest substructure that is usually incommensurate5b,6 with the periodicity of the host substructure along the tunnel. A feature of these materials, which has a direct bearing on many of their properties, is the fact that the urea tunnel structure is stable only when the tunnels are densely filled with guest molecules. Removal of the guest molecules leads to collapse of the “empty” urea tunnel structure and formation of the pure crystalline phase of urea. As a consequence, guest exchange in urea inclusion compounds cannot be carried out via the empty host tunnel structure, unlike the situation for other solid host materials that are stable in the empty form, such as zeolites.2 Nevertheless, it is possible, as shown previously,7 to change the identity of the guest molecules in urea inclusion compounds while maintaining the integrity of the host structure, provided that the tunnels remain filled throughout the guest exchange process. This type of guest exchange process can be carried out by inserting “new” guest * Authors for correspondence. E-mail: [email protected]; [email protected]. † Cardiff University. ‡ Universite ´ Bordeaux.

10.1021/jp076532k CCC: $37.00

Figure 1. Structure of a urea inclusion compound at ambient temperature, showing nine complete tunnels (with van der Waals radii) viewed along the tunnel axis. The guest molecules (alkanes) have been inserted into the tunnels illustrating orientational disorder.

molecules at one end of the crystal (e.g., by putting the crystal in contact with the liquid phase of the new guest molecules), with the “original” guest molecules expelled from the other end of the crystal. Net transport of guest molecules thus occurs in one direction along the host tunnel structure. To understand fundamental aspects of this process, it is clearly essential to establish details of the way in which the spatial distribution of the two types of guest molecule within the crystal changes as a function of time, and confocal Raman microspectrometry has been shown8-10 to be a viable in situ probe of such guest exchange processes. In the present work, we exploit this in situ experimental approach to probe kinetic aspects of the guest exchange process in a urea inclusion compound. The results have much wider relevance to other cases of molecular transport in densely filled tunnel systems. © 2007 American Chemical Society

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Letters

Experimental Strategy Previous in situ studies of guest exchange in urea inclusion compounds using confocal Raman microspectrometry8,9 have focused on the system with 1,8-dibromooctane and pentadecane as the original and new guest molecules, respectively. These investigations focused on the variation in the intensity of the Raman band due to the C-Br stretching mode [for the predominant trans end-group conformation; denoted ν(CBrT)] of the 1,8-dibromooctane guest molecules as a function of position in the crystal and time.11 The relative amount of 1,8dibromooctane guest molecules in different regions of the crystal is assessed from the ratio RT ) I(CBrT)/I(CN), where I(CBrT) and I(CN) are the integrated intensities of the ν(CBrT) and νs(CN) bands, respectively, in the Raman spectrum.11 During the guest exchange process, advancement of the new pentadecane guest molecules through the crystal and the corresponding loss of the original 1,8-dibromooctane guest molecules can be monitored from the time dependence and spatial dependence of RT. In practice, the normalized ratio RN,T ) RT/Ro is used, where Ro is the value of R (averaged over the probed area) for the original crystal of the 1,8-dibromooctane/urea inclusion compound before the start of the guest exchange process. By definition, RN,T ) 1 for the original 1,8-dibromooctane/urea crystal and RN,T ) 0 would correspond to the situation after complete guest exchange (i.e., for the inclusion compound containing only pentadecane guest molecules). At a given value of time, RN,T has a sigmoidal distribution, with plateau regions corresponding to the 1,8-dibromooctanerich (high X; RN,T ≈ 1) and pentadecane-rich (low X; RN,T ≈ 0.2) regions of the crystal.13 Between these plateau regions, there is a “boundary region” in which RN,T varies significantly with X. Such plots, studied as a function of time, demonstrate8 that the centroid of the sigmoidal distribution translates along the crystal, and provides a basis for quantifying the progress of the guest exchange process, with the potential to yield kinetic and mechanistic information relating to the transport process. For example, such data have revealed9 that the guest exchange process perturbs the conformational properties of the original (1,8-dibromooctane) guest molecules within the “boundary region” such that the proportion of 1,8-dibromooctane guest molecules with the gauche end-group conformation is increased. In the present paper, we explore kinetic aspects of the guest exchange process, allowing a kinetic model describing the transport of guest molecules in this system to be elucidated. Experimental Section Confocal Raman microspectrometry14 was carried out using a Labram II spectrometer (Jobin-Yvon), an Ar/Kr 2018 SpectraPhysics laser (514.5 nm), and a grating of 1800 lines/mm (spectral resolution ca. 6 cm-1). The laser was focused on the sample through a microscope with a 50× Olympus objective of 0.55 numerical aperture and 700 µm confocal pinhole diameter. The diameter of the probed area was 10 µm (defining the radial resolution of the experiment). The crystals of 1,8-dibromooctane/urea were prepared by standard procedures8,9 and have long hexagonal needle morphology. For the in situ confocal Raman microspectrometry experiments (Figure 2), one end of a single crystal (ca. 1 × 1 × 14 mm3) was attached (using Araldite as sealing system) to a reservoir containing liquid pentadecane and mounted on the XY-motorized table of the Raman microscope. In the laboratory reference frame (X, Y, Z), the Z axis defines the direction of the incident laser beam. The scattered radiation was collected in the same direction as the incident radiation (backscattering

Figure 2. Schematic representation of the experimental assembly for in situ Raman microspectrometry of guest exchange in a urea inclusion compound, comprising a single crystal of the urea inclusion compound (green), initially containing 1,8-dibromooctane guest molecules, attached to a reservoir containing liquid pentadecane.

geometry) and was analyzed through a polarizer along the X direction. The long axis (tunnel direction) of the needle-shaped crystal was parallel to the X axis and polarized spectra were recorded (Z(XX)Z h ) in Porto notation.15 Raman spectra were recorded by scanning along the X axis (tunnel axis; step size 100 µm; scan range 5 mm) at a depth Z ) 175 µm beneath the upper surface of the crystal (axial resolution 50 µm). Four separate scans were carried out at Y coordinates separated by 200 µm (the width along Y for each scan was ca. 10 µm). The time to record each Raman spectrum, each scan, and the complete Raman micrograph of the probed area were ca. 8 s, 10 min, and 40 min, respectively (including the time required for movement of the motorized sample stage). As discussed below, these times are significantly shorter than the overall time scale of the guest exchange process. Intensities of Raman bands were determined by numerical integration of the peak area, with the baseline defined as a straight line between the specified extrema of the spectral window. For a given Raman band, the same extrema of the spectral window were defined for all spectra. All analysis in this paper is based on relatiVe intensities of Raman bands. The results from the four independent scans (each carried out at a fixed value of Y, as discussed above) are presented separately, allowing an assessment of the extent to which the guest exchange process occurs uniformly in different regions of the crystal. The different scans are labeled A-D. Results and Discussion To quantify the progress of the guest exchange process, we measured RN,T(X,t) as described above. Representative plots of RN,T(X,t) versus X at different values of time during the guest exchange process are shown in Figure 4. At a given time t, the experimental data RN,T(X,t) can be fitted by a sigmoidal function:

RN,T (X, t) )

1 - A1 1 + exp[-(X - X0(t))/∆(t)]

+ A1

(1)

with centroid (i.e., inflection point) Xo(t) and width ∆(t). Equation 1 was fitted to the experimental data (Figure 4) to obtain values of Xo(t) and ∆(t). In the fitting process, A1 was fixed at the value of RN,T for the material after completion of the guest exchange process. We note that the upper plateau region of the sigmoidal distribution has RN,T ) 1 (which is, by definition, the value of RN,T for the original 1,8-dibromooctane/ urea inclusion compound). A schematic sigmoidal distribution, including definitions of Xo and A1, is shown in Figure 5. The time dependence of Xo(t) is used here to monitor the progress of the guest exchange process, and in particular to establish kinetic aspects of this process. We note that there is no systematic variation of ∆(t) as a function of time, which remains

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Figure 3. Raman spectra of (a) a crystal of the 1,8-dibromooctane/ urea inclusion compound (before guest exchange) and (b) the urea inclusion compound after partial exchange of the 1,8-dibromooctane guest molecules by pentadecane guest molecules.

Figure 5. Schematic sigmoidal distribution, showing the definitions of Xo and A1.

“single-file diffusion” in which the molecules within a pore are unable to pass or “overtake” each other. As a consequence, the motions of different molecules become correlated, even at long time, because the diffusion of a given molecule requires appropriate displacements of many other molecules within the pore. This correlation is reflected in the time dependence of the mean-squared molecular displacement, which is predicted for a one-dimensional system of infinite length to be

〈r2(t)〉 ) 2 F t1/2

Figure 4. Representative plots showing the sigmoidal distribution RN,T(X,t) vs X at different fixed values of time (t) during the guest exchange process for scan A: (a) t ) 0 min, (b) t ) 840 min, and (c) t ) 1200 min. Clearly the sigmoidal distribution translates along the crystal as a function of time, reflecting the evolution of the guest exchange process.

essentially constant within the experimental error in the estimation of ∆(t). The estimated errors in the values of Xo(t) extracted from the experimental data by fitting the sigmoidal distribution discussed above have been used to establish appropriate error bars in the subsequent analysis of data (Figures 6-8). Initially, we assess whether the Xo(t) data exhibit the characteristic time dependence of models that have been used previously to describe transport in one-dimensional systems: (i) conventional diffusion, and (ii) single-file diffusion. Conventional Diffusion. At sufficiently long time, the meansquared molecular displacement (denoted 〈r2(t)〉) for the conventional Einstein diffusion model in a one-dimensional system has a linear dependence on time

〈r2(t)〉 ) 2 Ds t

(2)

where Ds is the diffusion coefficient. Single-File Diffusion. Diffusion processes in many nanoporous materials differ from conventional diffusion when the dimensions of the diffusing molecules are comparable to the diameter of the pores. Such processes are often described as

(3)

where F is the single-file mobility. Models of single-file diffusion have been applied successfully to describe diffusion in a range of nanoporous systems,16 including zeolites and carbon nanotubes. We note that these systems generally have significantly lower loading of molecules within the pores than the dense loading characteristic of urea inclusion compounds. In the case of guest exchange in urea inclusion compounds, a dense filling of the tunnels must be maintained at all stages of the transport process (indeed, evidence from ex situ singlecrystal X-ray diffraction studies7 confirms that both the original and new guest molecules maintain periodic arrangements along the tunnel direction). Under these circumstances, the time dependence of the displacements of all guest molecules within the tunnel must approximate to the same time dependence as Xo(t). The displacement of each guest molecule from its initial position at t ) 0 is thus given by [Xo(t) - Xo(0)] and the term 〈r 2(t)〉 in eqs 2 and 3 may therefore be equated with [Xo(t) - Xo(0)]2. In the data analysis, t ) 0 is defined as the time of the first measurement (rather than the time at which the liquid pentadecane is put in contact with the original 1,8-dibromooctane/ urea crystal). Measurements of the Raman spectra are not possible near the end of the crystal at which the new pentadecane guest molecules enter the tunnels as this end of the crystal is obscured by the reservoir containing the liquid pentadecane (see Figure 2). Thus, it is not possible to probe the early stages of the guest exchange process with the experimental setup used here. Furthermore, it is clearly only possible to define the sigmoidal distribution (and hence to define Xo) after a sufficient extent of transport of the new guest molecules along the crystal has occurred because at the early stages of the guest transport process the distribution of guest molecules represents an incomplete sigmoidal distribution.

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Figure 6. Plots of [Xo(t) - Xo(0)]2 vs t for each independent scan along X (i.e., parallel to the tunnel axis): (a) scan A, (b) scan B, (c) scan C, and (d) scan D.

To test the applicability of the above models to describe the molecular transport process in urea inclusion compounds, graphs of [Xo(t) - Xo(0)]2 versus t (which should be linear in the case of conventional diffusion) and [Xo(t) - Xo(0)]2 versus t1/2 (which should be linear in the case of single-file diffusion) are shown in Figures 6 and 7, respectively, for each of the four independent scans along X (i.e., along the tunnel axis). Clearly, the plots in Figures 6 and 7 deviate significantly from linearity, from which it is clear that the molecular transport associated with guest exchange in urea inclusion compounds is not described by either conventional diffusion or single-file diffusion. However, as shown in Figure 8, good linear fits are obtained for graphs of [Xo(t) - Xo(0)] versus t for each of the four independent scans along X. From the gradient of the linear fits to the [Xo(t) - Xo(0)] versus t data in Figure 8, the rate of the process is determined to be (70.3 ( 2.2) nm s-1 for scan A, (85.7 ( 5.5) nm s-1 for scan B, (100.2 ( 6.0) nm s-1 for scan C, and (69.7 ( 3.3) nm s-1 for scan D (the quoted errors in the gradient are estimated only from the error in linear regression). Given that the periodic repeat distance for pentadecane guest molecules in the pentadecane/urea inclusion compound is 2.137 nm,17 these rates of the guest exchange process correspond to between ca. 33 (scan D) and 47 (scan C) new guest molecules entering the tunnel per second. We now consider the physical basis for the linear timedependence observed for [Xo(t) - Xo(0)]. In assessing appropriate transport models, it is important to recall that urea inclusion compounds have a dense packing of guest molecules along the tunnel, necessitated (at least in part) by the fact that the empty urea tunnel structure is unstable. Clearly, if individual guest molecules or groups of guest molecules were to move independently of their neighbors, a local gap would open up within the guest substructure, which would be expected to lead to local

collapse of the host tunnel and should hence either hinder (if reversible) or terminate (if irreversible) the guest transport process. Thus, the movement of the different guest molecules along the tunnel must be highly correlated, with the complete set of guest molecules moving along the tunnel together and all guest molecules exhibiting essentially the same displacements as a function of time. Indeed, ex situ single-crystal X-ray diffraction studies7 provide evidence that strong positional correlations (both intra-tunnel and inter-tunnel) are maintained during the guest transport process. We describe this situation as single-file transport in a densely filled tunnel. According to the above description, displacement of the complete set of guest molecules along the tunnel relies on insertion of new guest molecules at one end of the tunnel and removal of the original guest molecules at the other end of the tunnel. Because translation of the complete guest substructure along the tunnel in an incommensurate inclusion compound approximates to a situation of “activation-less” transport,18 the rate-limiting step of the guest exchange process must correspond either to the entry of the new guest molecules at one end of the tunnel or to the expulsion of the original guest molecules at the other end of the tunnel. Under the conditions of the experiments carried out here, a constant supply of new guest molecules is maintained at the surface of the crystal at which new guest molecules enter the tunnel, and this interfacial process should exhibit zeroth-order kinetics, with the rate of this process independent of time. Similarly, expulsion of the original guest molecules at the other end of the crystal should also be a zerothorder process. Thus, whether the entry of the new guest molecules or the expulsion of the original guest molecules is the rate-limiting step, the overall process should occur at constant rate, giving rise to a linear dependence of [Xo(t) Xo(0)] on t, as observed from our experimental data.

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Figure 7. Plots of [Xo(t) - Xo(0)]2 vs t1/2 for each independent scan along X (i.e., parallel to the tunnel axis): (a) scan A, (b) scan B, (c) scan C, and (d) scan D.

Figure 8. Plots of [Xo(t) - Xo(0)] vs t for each independent scan along X (i.e., parallel to the tunnel axis): (a) scan A, (b) scan B, (c) scan C, and (d) scan D.

12344 J. Phys. Chem. B, Vol. 111, No. 43, 2007 Concluding Remarks The results reported here provide a detailed understanding of kinetic and mechanistic aspects of the molecular transport process associated with guest exchange in urea inclusion compounds. It is clear that the guest transport process is influenced significantly by the fact that the urea host structure is stable only when the tunnels are filled by a dense packing of guest molecules, and thus the guest transport process intrinsically involves a high degree of correlation between the motions of different guest molecules along the tunnel. Indeed, as discussed above, the motion of the guest molecules along the tunnel is also strongly connected to the entry of new guest molecules at one end of the tunnel and the expulsion of the original guest molecules at the other end of the tunnel, and recalling that translation of the guest substructure along the tunnel of an incommensurate inclusion compound corresponds to activation-less transport, it is reasonable to propose that the entry of the new guest molecules into the tunnel or the expulsion of the original guest molecules from the tunnel is the rate-limiting aspect of the complete guest exchange process. The linear time-dependence of the guest transport process reported here for urea inclusion compounds may also be expected to be observed for transport processes in other materials comprising a dense filling of tunnels. The work reported here further demonstrates the applicability of confocal Raman microspectrometry for probing details of guest exchange processes in urea inclusion compounds. At the present time, our studies using this technique have been confined to ambient temperature because this is a limitation of the experimental setup employed in the present work. It will clearly be of considerable interest to extend these studies in the future to consider the temperature dependence of the guest exchange process. Several other aspects of guest exchange in urea inclusion compounds merit further investigation, including a more detailed understanding of the processes occurring at the (001) crystal surfaces, corresponding to entry of new guest molecules into the tunnels at the end of the crystal in contact with the liquid phase of the new guest, and expulsion of the original guest molecules from the tunnels at the other end of the crystal. Studies of these issues are currently the focus of our experimental investigations in this field. Acknowledgment. We are grateful to the European Union (Marie Curie Training Fellowship to J.M.-R.), Cardiff University (studentship to J.M.-R.) and Universite´ Bordeaux 1 for financial support, to D. Talaga and J.L. Bruneel (ISM, Bordeaux) for technical assistance, and to Dr. C.E. Hughes for discussions. References and Notes (1) (a) Eisenberg, B. Acc. Chem. Res. 1998, 31, 117. (b) Borgnia, M.; Nielsen, S.; Engel, A.; Agre, P. Annu. ReV. Biochem. 1999, 68, 1015. (c) MacKinnon, R. FEBS Lett. 2003, 555, 62. (d) Agre, P. Biosci. Rep. 2004, 24, 127. (e) MacKinnon, R. Angew. Chem., Int. Ed. 2004, 43, 4265. (f) Agre, P. Angew. Chem., Int. Ed. 2004, 43, 4278. (2) (a) Karger, J.; Ruthven, D. M. Diffusion in Zeolites and Other Microporous Solids; Wiley: New York, 1992. (b) Thomas, J. M. Angew. Chem., Int. Ed. 1999, 38, 3588. (c) Nelson, P. H.; Auerbach, S. M. J. Chem. Phys. 1999, 110, 9235. (d) Papadopoulos, G. K.; Jobic, H.; Theodorou, D. N. J. Phys. Chem. B 2004, 108, 12748.

Letters (3) (a) Lee, S. B.; Martin, C. R. J. Am. Chem. Soc. 2002, 124, 11850. (b) Hod, O.; Rabani, E. Proc. Natl. Acad. Sci. 2003, 100, 14661. (c) Kalra, A.; Garde, S.; Hummer, G. Proc. Natl. Acad. Sci. 2003, 100, 10175. (d) Wei, C. Y.; Srivastava, D. Phys. ReV. Lett. 2003, 91, 235901. (e) Lutz, C.; Kollmann, M.; Bechinger, C. Phys. ReV. Lett. 2003, 93, 026001. (f) Lee, K. H.; Sinnott, S. B. J. Phys. Chem. B 2004, 108, 9861. (g) Valiullin, R.; Kortunov, P.; Karger, J.; Timoshenko, V. J. Chem. Phys. 2004, 124, 11804. (h) Nednoor, P.; Chopra, N.; Gavalas, V.; Bachas, L. G.; Hinds, B. J. Chem. Mater. 2005, 17, 3595. (4) (a) Fetterly, L. C. In Non-Stoichiometric Compounds; Mandelcorn, L., Ed.; Academic Press: New York, 1964; p 491. (b) Takemoto, K.; Sonoda, N. In Inclusion Compounds; Atwood, J. L.; Davies, J. E. D.; MacNicol, D. D., Eds.; Academic Press: New York, 1984; Vol. 2, p 47. (c) Hollingsworth, M. D.; Harris, K. D. M. In ComprehensiVe Supramolecular Chemistry; MacNicol, D. D.; Toda, F.; Bishop, R., Eds.; Pergamon Press: New York, 1996, Vol. 6, p 177. (d) Harris, K. D. M. Chem. Soc. ReV. 1997, 26, 279. (e) Guillaume, F. J. Chim. Phys. (Paris) 1999, 96, 1295. (f) Hollingsworth, M. D. Science 2002, 295, 2410. (g) Harris, K. D. M. Supramol. Chem. 2007, 19, 47. (5) (a) Smith, A. E. Acta Crystallogr. 1952, 5, 224. (b) Harris, K. D. M.; Thomas, J. M. J. Chem. Soc., Faraday Trans. 1990, 86, 2985. (c) George, A. R.; Harris, K. D. M. J. Mol. Graphics 1995, 13, 138. (6) (a) Rennie, A. J. O.; Harris, K. D. M. Proc. R. Soc. London, Ser. A 1990, 430, 615. (b) Schmicker, D.; van Smaalen, S.; de Boer, J. L.; Haas, C.; Harris, K. D. M. Phys. ReV. Lett. 1995, 74, 734. (c) Ollivier, J.; Ecolivet, C.; Beaufils, S.; Guillaume, F.; Breczewski, T. Europhys. Lett. 1998, 43, 546. (d) Lefort, R.; Etrillard, J.; Toudic, B.; Guillaume, F.; Breczewski, T.; Bourges, P. Phys. ReV. Lett. 1996, 77, 4027. (7) Khan, A. A.; Bramwell, S. T.; Harris, K. D. M.; Kariuki, B. M.; Truter, M. R. Chem. Phys. Lett. 1999, 307, 320. (8) Marti-Rujas, J.; Desmedt, A.; Harris, K. D. M.; Guillaume, F. J. Am. Chem. Soc. 2004, 126, 11124. (9) Martı´-Rujas, J.; Harris, K. D. M.; Desmedt, A.; Guillaume, F. J. Phys. Chem. B 2006, 110, 10708. (10) Martı´-Rujas, J.; Harris, K. D. M.; Desmedt, A.; Guillaume, F. Mol. Cryst. Liq. Cryst. 2006, 456, 139. (11) Previous Raman studies of the 1,8-dibromooctane/urea inclusion compound have shown12a that the ν(CBrT) band is at ca. 660 cm-1. The Raman bands due to the urea host structure are identical for 1,8dibromooctane/urea and pentadecane/urea, and the symmetric C-N stretching vibration νs(CN) of urea gives a very intense band at ca. 1024 cm-1.12 For pentadecane guest molecules (in the all-trans conformation) in the pentadecane/urea inclusion compound, the methyl rocking r(CH3) band is at ca. 900 cm-1.12b The methyl rocking band also provides a basis for probing the guest exchange process, as shown previously;9 however, because this band is weak, it represents a less-accurate approach in comparison with study of the ν(CBrT) band. These features of the Raman spectra of urea inclusion compounds containing 1,8-dibromooctane and pentadecane guest molecules are shown in Figure 3 (in this figure, ν(CBrG) denotes the C-Br stretching mode for 1,8-dibromooctane guest molecules with a gauche endgroup conformation). (12) (a) Smart, S. P.; El Baghdadi, A.; Guillaume, F.; Harris, K. D. M. J. Chem. Soc., Faraday Trans. 1994, 90, 1313. (b) El Baghdadi, A.; Guillaume, F. J. Raman. Spectrosc. 1995, 26, 155. (13) As discussed previously,8,9 complete guest exchange is not actually observed and the final value of RN,T in the pentadecane-rich regions does not fall to zero. (14) Bruneel, J. L.; Lasse`gues, J. C.; Sourisseau, C. J. Raman Spectrosc. 2002, 33, 815. (15) Damen, T. C.; Porto, S. P. S.; Tell, B. Phys. ReV. 1966, 142, 570. (16) (a) Ka¨rger, J.; Petzold, M.; Pfeifer, H.; Ernst, H.; Weitkamp, J. J. Catal. 1992, 136, 283. (b) Hahn, K.; Ka¨rger, J.; Kukla, V. Phys. ReV. Lett. 1996, 76, 2762. (c) Berezhovskii, A.; Hummer, G. Phys. ReV. Lett. 2002, 89, 064503. (17) Shannon, I. J.; Harris, K. D. M.; Rennie, A. J. O.; Webster, M. B. J. Chem. Soc., Faraday Trans. 1993, 89, 2023. (18) As discussed previously,6a-c,7 translation of the complete guest substructure along the tunnel in an incommensurate system (such as a urea inclusion compound) is, in principle, associated with no change in the total energy of the system, and is therefore equivalent to “activation-less” transport.