J. Phys. Chem. B 2004, 108, 929-935
929
NMR Studies of Guest Dynamics in Clathrate Hydrates: Spherical Tops SF6, SeF6 and CH4 in Structure II Hydrate John A. Ripmeester,* Chris I. Ratcliffe, and Ian G. Cameron† Steacie Institute for Molecular Sciences, National Research Council of Canada, Ottawa, Ontario, Canada K1A 0R6 ReceiVed: September 8, 2003; In Final Form: NoVember 4, 2003
Spin-lattice relaxation time measurements, obtained between 2 and ∼150 K, are presented for 19F in two structure II clathrate hydrates, SF6‚17D2O and SeF6‚17D2O, as well as for 1H in a double structure II deuteriohydrate of methane and tetrahydrofuran-d8. The 19F results were analyzed in terms of three relaxation mechanisms related to the guest motions, including fluctuations in nuclear dipolar couplings and the anisotropic chemical shift, plus a contribution from spin rotation interactions above ∼50 K. For quantitative agreement, the incorporation of a distribution function into the relaxation equations is required to account for the proton disorder in the hydrate cages, in this case a normal distribution in activation energies. The dynamic parameters derived from relaxation measurements also account for the motional narrowing of the 19F spectra. This is the first instance where a single motional model has accounted for both spectral narrowing and relaxation times for guest dynamics in clathrate hydrates over a wide range of temperatures. The 1H relaxation times, measured for CH4 in the small cages of structure II, are generally consistent with the above model, except that, because of quantum symmetry effects, only intermolecular dipolar couplings appear to be effective in providing relaxation. Trends in activation energies for guest reorientation and the associated distributions, as well as previously published work, are discussed in terms of our understanding of guest dynamics in clathrate hydrates.
Introduction Guest dynamics in clathrate hydrates have been studied since the 1960s, first with dielectric relaxation techniques,1,2 later also with NMR spectroscopic methods.2 In most instances, the results have illustrated the remarkable motional mobility of the guest molecules in the clathrate cages. Although the cages are often described as nearly spherical, this is far from true, as judged by guest dynamics3,4 and more recently also from a direct observation of the anisotropically disordered guests by diffraction techniques.5 In part, the early conclusions on motional freedom may have been influenced by the particular characteristics of dielectric relaxation, as the averaging of molecular dipoles involves only first rank tensors so that even in-plane rotation of a molecular dipole looks like isotropic motion. However, it is very clear from NMR spectroscopy, where the averaging involves spin interactions described by second-rank tensors, that the dynamics are far from isotropic except at the highest temperatures in the truly spherical cages. However, when this inherent difference in techniques is taken into account, the results from the two methods are generally in good agreement.2 Each type of cage, with its characteristic symmetry elements, is a test site capable of giving some insight into guest dynamics.6 For instance, from crystallography, the large cage (51264) in sII hydrate has tetrahedral symmetry, so that even a highly asymmetric guest should show isotropic dynamics. The fact that this is not observed, e.g., from 1H NMR moment analysis2 or from 2H NMR measurements2,3,7 suggests that the individual cages do not have the high symmetry derived from diffraction. * To whom correspondence should be addressed. Telephone: 613 9932011. Fax: 613 998-7833. E-mail:
[email protected]. † Current address: MRI Unit, Ottawa Hospital, General Campus, 501 Smyth Road, Ottawa, ON K1H 8L6, Canada.
Of course, as described a number of times, this is due to the fact that the hydrate lattice is disordered in the same fashion as ice and that diffraction studies show that there are two halfhydrogens in each O-O bond.2 This picture arises because of space averaging of different cage conformations, and we can conclude that each distinct cage conformation gives the encaged guest a somewhat different guest-host potential.2,8 This is consistent with the observation of broad dielectric relaxation peaks and spin-lattice relaxation minima that formally can be described by distributions.1,2 2H NMR and 13C NMR line shapes2,3,7 are consistent with such a description, although models that describe experimental data quantitatively from low temperature up to ∼200 K are not as yet available. The distribution in dynamics is also completely consistent with the observation of methyl tunneling line shapes that show a distributed lattice contribution that adds to the intramolecular gas-phase barrier.9 So far, a comprehensive and consistent model that accounts quantitatively for both line narrowing and spinlattice relaxation has not been presented. In fact, such an analysis is not straightforward to carry out, as complete averaging of the spin interactions for molecules of low symmetry of necessity involves the presence of inequivalent sites and a large number of parameters, even for the most symmetrical cage (51264). For instance, for a molecule like tetrahydrofuran (THF), a popular hydrate guest for study, even reorientation in free space involves three distinct moments of inertia. The simplest case for analysis would be one where the motion is likely to be describable by a single correlation time and where the motion must always take place between equivalent sites.10 For spherical tops such as SF6, SeF6, and CH4 in hydrate cages, this is indeed the case. Here we analyze relaxation time data for these guests in the cages of structure II hydrate (see Scheme 1), methane in the small 512 cage of a double hydrate of THF-d8 and D2O, and the
10.1021/jp036679l CCC: $27.50 Published 2004 by the American Chemical Society Published on Web 12/19/2003
930 J. Phys. Chem. B, Vol. 108, No. 3, 2004
Ripmeester et al.
SCHEME 1: Detail of Structure II (sII) Hydrate (Unit Cell, 16MS‚8ML‚136 H2O), Showing Small (MS ) 512) and Large (ML ) 51264) Cage Sitesa
a For SF6 and SeF6 hydrate, the small cages are empty, giving a composition of Guest‚17H2O. For the double sII hydrate with methane and tetrahydrofuran (THF), the composition is THF‚(CH4)2x‚17H2O, where x, the fraction of small cages filled, depends on the P, T conditions of sample preparation.
other guests in the large 51264 cage of sII. We examine the observed trends in the hope that this will guide the interpretation of the more complex dynamics observed in guests of lower symmetry, and we also discuss the results in light of some recent studies on guest dynamics in clathrate hydrates.11,12 We also note that the dynamics of the water lattice have been well described by dielectric and NMR measurements.1,2 1H, 2H, 13C, and other measurements have shown conclusively that the dynamics of the water lattice and the motional averaging of the guest spin interactions are strongly coupled: only when the water motions are rapid enough do the individual cages take on the high symmetry shown by diffraction, as both space and time averaging occur. In this work, water motion is always in the slow limit, so that the hydrate cages must be fixed in their low-symmetry configurations. Experimental Section The preparation of SF6 and SeF6 deuteriohydrates has been described previously.13 Briefly, enough SF6 and SeF6 together with a slight excess of heavy water were taken through several freeze-pump-thaw cycles and sealed into 10 mm o.d. Pyrex tubes. Care was taken to remove oxygen by degassing both guest and host components, as indicated above, as it was shown that earlier measurements were affected by contamination by molecular oxygen.2,14 Since paramagnetic O2 is more soluble in the clathrate (by residing in the vacant small 512 cage) than in aqueous solution, it can have a considerable effect on relaxation times, especially at low temperatures.2,14 The water-guest mixtures were then frozen and the temperature cycled through 0 °C many times over several weeks. 19F NMR lines were recorded as derivatives with a low-level marginal oscillator at 20.6 MHz using field modulation and lockin detection. Second moments were corrected for modulation broadening (∼2.5% maximum correction). 19F spin-lattice relaxation times were measured using 90-t-90 pulse sequences at 12 and 56.4 MHz for SF6‚17D2O and 10 and 56 MHz for SeF6‚17D2O. The T1 recovery curves were observed to be exponential. Some dipolar relaxation times (T1D) were measured as well for SF6‚17D2O in order to probe slow motions; however, the relaxation plots became highly nonexponential at temperatures below about the T1 minima and were not analyzed. Temperature variation and control were achieved with a liquid helium cryostat; it and details of the data acquisition have been given previously.14 The double deuteriohydrate of methane and THF-d8 was prepared by subjecting a degassed solution of THF-d8‚17D2O,
Figure 1. Temperature dependence of the spin-lattice relaxation time T1 for SF6‚17D2O at Larmor frequencies of 12 MHz (triangles) and 56.4 MHz (circles). Panel b gives a detail of the crossover at the T1 maximum. The solid line gives the full fit to the data as discussed in the text, whereas the other lines show the contributions from T1SR (dot dash), T1DD (long dashes), and T1∆σ (dot dot dash). For T1∆σ the upper curve corresponds to the contribution for 12 MHz.
placed in a 10 mm o.d. Pyrex tube with a ground glass joint to a pressure of ∼20 atm of CH4 for several weeks in a pressure vessel. The vessel was cooled and opened to remove the sample tube, which was then connected to a vacuum line. The sample was flame-sealed, meanwhile keeping the hydrate sample in the bottom of the sample tube immersed in liquid nitrogen at all times and including a low pressure of helium gas for thermal contact. 1H relaxation times were measured at 9.2 and 60 MHz with 90-t-90 pulse sequences. Results and Analysis of Data 19F Relaxation Time Results. The T vs 1/T curves for SF 1 6 and SeF6 in the large cage of structure II are shown in Figures1 and 2. In each case there is a high-temperature frequencyindependent maximum (48 and 63 K for SeF6 and SF6, respectively), and a broad, relatively weak asymmetric minimum (9 and 12 K at 12 and 56.4 MHz for SF6, 18.9 and 15.6 K at 10 and 56 MHz for SeF6). At and below the minima, the relaxation times are strongly frequency dependent, whereas at the maximum there is weak frequency dependence and above the maximum relaxation is frequency independent. Note that in each case the relaxation times for the two frequencies cross over on the low-temperature side of the T1 maximum. The T1 minima are very different from those for the standard relaxation time curves for relaxation resulting from only modulation of dipolar couplings as described by BPP theory,15,16 i.e., a V-shaped dependence on inverse temperature of log T1 with equal slopes on either side of the minimum, frequency-squared dependence on the low-temperature side, and a linear dependence on frequency at the minimum. The relaxation time plots show that in addition to the presence of a number of relaxation time processes, the behavior at low temperatures is typical of the presence of broad distributions.
Studies of Guest Dynamics in Clathrate Hydrates
J. Phys. Chem. B, Vol. 108, No. 3, 2004 931 second moment involved in the motion. The temperature dependence of τc is assumed to follow the Arrhenius equation according to
τc ) τ0eEa/RT
(4)
where τ0 is a constant, R is the universal gas constant, T the absolute temperature, and Ea is the activation energy. Note that the CSA contribution has a field-squared dependence at all rates and hence can be seen to be partly responsible for the departures from the dipolar BPP type relaxation curve noted above. A proper theoretical account of the spin-lattice relaxation due to spin-rotation interactions in solids is not yet available and consequently we use an expression derived by McClung for fluids.17 For this model, referred to as the extended rotational diffusion model, in the limit of small τj, the angular momentum correlation time, the J-diffusion version of McClung’s equation for T1SR of spherical top molecules can be written as
T-11SR ) (I0 /3h2)τR(C2av + 2∆C2) ) CSRτR-1
(5)
Where I0 is the moment of inertia of the molecule, τR is the correlation time, and Cav and ∆C are spin-rotation constants defined as Figure 2. Inverse temperature dependence of the spin-lattice relaxation time T1 for SeF6‚17D2O at 10 MHz (triangles) and 56.4 MHz (circles). Panel b gives a detail of the crossover at the T1 maximum. The solid line gives the full fit to the data as discussed in the text, while the other lines show the contributions from T1SR (dot dash), T1DD (long dashes), and T1∆σ (dot dot dash). For T1∆σ the upper curve corresponds to 10 MHz.
Methods of T1 Analysis. For 19F relaxation in solids, the dominant relaxation mechanism is usually the result of modulation of dipole-dipole and chemical shift anisotropy interactions.15 As well, because of the rapid rate of reorientation of the relatively small guest molecules in their cages, there may be a contribution from spin-rotation interactions where the nuclear spins interact with the molecular magnetic moment.17 As discussed earlier, a distribution of correlation times is expected that is associated with the frozen-in orientational disorder of the water molecules in the lattice. With these considerations in mind, the equation that was used to fit the temperature-dependent experimental relaxation time data is of the general form
T1-1 )
∫F(Ea)[T1-1DD + T1-1CSA] dEa + ∫T1-1SRF(E′a) dE′a
(1)
Where F(Ea) and F(E′a) are the distributions of the activation energies Ea and E′a and the subscripts on T1 stand for dipoledipole, chemical shift anisotropy and spin rotation, respectively. The expressions for T1-1DD and T1-1CSA are given by15,16
Cav ) (C| + 2C⊥)/3
(6)
∆C ) (C| - C⊥)/3
(7)
C| and C⊥ are the components of the spin-rotation interaction tensor parallel and perpendicular to the X-F bond, respectively. It should be noted that the M-diffusion version of McClung’s equation17 for T1SR could have been used instead of eq 5; however, for spherical top molecules in the small τj limit the M-diffusion equation version yields T1SR values larger by a factor of 3, of little consequence here. Note that SR relaxation does not have a field dependence and thus is seen to be the dominant mechanism at the high-temperature end above the T1 maximum in Figures 1 and 2. As discussed in the Introduction, there is good evidence from a variety of measurements that guest dynamics are distributed as a result of the orientational disorder of the water molecules. Although the distribution can be treated in various ways, it is usual to consider some functional form for the distribution in either correlation times or activation energies. A log-normal distribution, F(z), describes the correlation times for SF6‚17D2O in the line narrowing region successfully, and we will use a similar description here. This is actually equivalent to a normal distribution of activation energies F(Ea). The expression for F(z) is
F(z) ) exp[z2(RT/σEa)2/2]/(2π)0.5σEa
(8)
Here z is defined as
T1-1DD ) (2/3)γF2∆M2[f (ω0τ) + 4f(2ω0τ)]
(2)
T1-1CSA ) (2/15)(ω0∆σ)2[f (ω0τ)]
(3)
with
z ) ln(τc /τm)
(9)
with τm the median value of the distribution. Again, τm is assumed to follow an Arrhenius relationship according to
f (ω0τc) ) τ/(1 + ω02τc2)
τm ) τ0eEm/RT
where γF is the gyromagnetic ratio for 19F, τc is the reorientational correlation time, ω0 is the Larmor frequency, ∆σ is the chemical shift anisotropy, and ∆M2 is the change in the dipolar
It should be noted that F(z) has a built-in temperature dependence, giving the equivalent distribution of activation energies a very broad distribution at low temperatures and a
(10)
932 J. Phys. Chem. B, Vol. 108, No. 3, 2004
Ripmeester et al.
TABLE 1: Parameters Derived from the Spin-Lattice Relaxation Time Data and Literature Values guest
10-9CDD, s-2
10C∆σ, s-2
SF6
4.4-4.8 4.46(17)a 2.5-2.75 3.10(11)a
1.28 1.28(0.25)b 2.4-2.8 1.83(20)b
SeF6
1013CSR, s-2
τ0, ps
τ′0, ps
Em, J m-1
E′m, J m-1
σEa
2.2-3.6
0.075-0.090
532-586
1591-1758
268-297
2.25-2.4
0.12-0.145
858-942
2156-2260
473-532
7.1c 9.0c
a Determined from CDD ) (2/3)γF2∆M2, where ∆M2 is the change in second moment associated with the motion; see Table 2. b Determined from C∆σ ) (2/15)(∆σ)2, where ∆σ is the chemical shift anisotropy; see Table 2. c Determined from eq 5.
delta function in z at high temperatures. In all cases, the median value of z will be zero. Upon substituting expressions for F(z) and F(z′) into eq 1, along with eqs 2-5, written in terms of z and z′, eq 1 becomes
T1-1 ) CDD
∫-∞∞[τm ez/(1 + ω02τm2e2z) + 4τmez/(1 + 4ω02τm2e2z)]F′(z) dz +
∫-∞∞[τmez/(1 + ω02τm2e2z)]F′(z) dz + ∞ CSR∫-∞[e-z′/τ′m]F′(z′) dz′
C∆σω02
(11)
where CDD ) (2/3)γF2∆M2, C∆σ ) (2/15)(∆σ)2, CSR is defined in eq 5, and F′(z) ) exp(-z2(RT/x2σEa)2)/(2π)0.5σEa The expression in eq 11 was used for fitting the T1 temperature dependences shown in Figures 1 and 2. Since the integrals in eq 11 are not analytic, the integration was performed numerically using Simpson’s one-third rule. The integration limits were normally set equal to (5σz ) (5σEa/RT; however, for those cases where -5σEa/RT corresponded to a negative activation energy, the lower limit was taken as
z0 ) ln(τ0/τm)
(12)
which is the Ea ) 0 value of z. From eq 5 it should be clear that CSR and τ′0 can be determined only as their quotient. However, according to the extended rotational diffusion model CSR is dependent only on the moment of inertia tensor and the spin-rotation interaction tensor, both of which are insensitive to intermolecular influences. Thus, reliable values of τ′0 can be determined using values of CSR given in Table 1. Because of the large number of parameters it was necessary to simplify procedures by setting σEa ) σ′Ea. Since both distributions arise from the disorder of water molecules, this should be a reasonable approximation, and since F(E′a) only is important at relatively high temperatures, the distribution width is also small. Values for I, ∆σ, Cav, and ∆C were taken from ref 14. To obtain the fits presented in Figures 1 and 2, the values of the parameters τ0, τ′0, Em, E′m, CDD, C∆σ, and σEa were all systematically adjusted until satisfactory agreement between model and experiment was achieved. This procedure is valid in the present case despite the large number of adjustable parameters, since all of these quantities except Em and σEa affect the final result in different ways and can therefore be adjusted independently. To deal with Em and σEa, the value of σEa was held fixed, while Em and the other parameters were varied until a satisfactory fit was achieved. Then, by systematically changing the value of σEa together with the procedure outlined, it was possible to determine a range of values outside of which a fit could not be found to represent the data. The resulting spread in values turns out to be
