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DOI: 10.1039/C9CP04379A
(Paper)
Phys. Chem. Chem. Phys., 2019, Advance Article

Pär Håkansson*^{a},
Muhammad Asadullah Javed^{a},
Sanna Komulainen^{a},
Linjiang Chen^{b},
Daniel Holden^{b},
Tom Hasell^{b},
Andrew Cooper^{b},
Perttu Lantto^{a} and
Ville-Veikko Telkki^{a}
^{a}NMR Research Unit, University of Oulu, P. O. Box 3000, 90014 Oulu, Finland. E-mail: par.hakanson@oulu.fi
^{b}Department of Chemistry and Centre for Materials Discovery, University of Liverpool, Crown Street, Liverpool L69 7ZD, UK

Received
7th August 2019
, Accepted 21st October 2019

First published on 22nd October 2019

The porous solid formed from organic CC3 cage molecules has exceptional performance for rare gas separation. NMR spectroscopy provides a way to reveal the dynamical details by using experimental relaxation and diffusion measurements. Here, we investigated T_{1} and T_{2} relaxation as well as diffusion of ^{129}Xe and SF_{6} gases in the CC3-R molecular crystal at various temperatures and magnetic field strengths. Advanced relaxation modelling made it possible to extract various important dynamical parameters for gases in CC3-R, such as exchange rates, activation energies and mobility rates of xenon, occupancies of the cavities, rotational correlational times, effective relaxation rates, and diffusion coefficients of SF_{6}.

Nuclear magnetic resonance (NMR) relaxation and diffusion experiments provide extraordinarily versatile information about dynamics of fluid molecules in porous media.^{11,12} Combined with a molecular level modelling, a detailed picture about the gas and framework dynamics can be revealed.^{13} The modelling provides deep explanations for the observations and, overall, the analysis may guide in further system design.

This study focuses on the dynamics of sulphur hexafluoride (SF_{6}) and Xe gas in porous organic cages. SF_{6} is a gas that has long atmospheric life span. It absorbs infrared light that makes it a potent greenhouse gas.^{14} SF_{6} acts as an excellent insulator for electrical equipment, thermos-acoustic insulator for windows, contrast agent in medical applications, and etchant plasma in semi-conductor industry.^{15,16} However, due to its high global warming potential—23900 times higher than CO_{2} over a hundred year span^{17}—it should be handled careful and recovered and stored in industrial usage.^{16} Many techniques, including pressure swing adsorption/desorption processes, have been proposed to separate SF_{6} from other gas mixtures and prevent its release to the atmosphere but the process requires high size selectivity and adsorption capacity of adsorbents.^{18} Several extended porous solids^{19,20} have been utilised for the separation and adsorption of SF_{6}. Moreover, cucurbituril in aqueous solution^{21} has also been utilised for the reversible trapping of SF_{6}.

Xenon is used in light appliances due to its glow effects and in medical imaging field. The inertness of Xe and its low abundance in air (0.087 ppm by volume)^{22} makes it commercially expensive. Due to its inert nature, a precise size selectivity is required to capture xenon gas in porous materials.^{23}

Porous organic cages are discrete cage molecules having permanent internal voids accessed by well-defined windows that provide porosity for guest molecules.^{24} These cages provide a way for the analytical separation of rare gases and chiral molecules by selective bindings.^{25} It has been shown that one of the organic cage molecules, CC3,^{26} exhibits extraordinary characteristics for the selective separation of rare gases and SF_{6} over gas mixtures.^{26,27} The tetrahedral CC3 cage is composed of imine bonds that connect the rigid aromatic rings to the flexible cyclohexane linkers (Fig. 1A). These cages pack into a crystalline structure to form an interconnected 3D pore structure accessible via windows of the cages (Fig. 1B). The size of cage cavities is 4.4 Å while the narrowest point between the cage and window cavities is 3.6 Å in diameter. Adsorbed xenon (with diameter of 4.3 Å) was shown to have fast exchange, on the order of 10^{8} s^{−1}, between these cavities,^{28} and even SF_{6} with kinetic diameter of 5.5 Å can enter the cage through the window cavity because of thermal and vibrational motion of the porous organic crystals.^{27}

Fig. 1 (A) Chemical structure of CC3 cage. (B) 3D crystal and cavity structure of CC3 material. The cage and window cavities are illustrated by green and yellow, respectively. |

The fluorine atoms in SF_{6} have an NMR active ^{19}F isotope with spin-1/2, high gyromagnetic ratio and 100% natural abundance. The presence of six magnetically equivalent ^{19}F atoms in each SF_{6} molecule makes the spin density of the gas high. The stable isotopes of S have a spin zero and natural abundance of 99.2%. Chemically SF_{6} is inert, non-toxic, non-explosive, incombustible and thermally stable, which makes it a suitable NMR probe in various applications in chemistry, materials science and medicine.^{29} It has been utilized for the investigation of porous media,^{30} MRI of lungs,^{31} liquid crystals,^{32} and ethylene-propylene rubbers.^{29} In addition, it has been used also as a spy molecule for highlighting the cavities in proteins.^{33}

Xenon has an NMR active ^{129}Xe isotope with spin-1/2 nucleus and relatively high natural abundance (26%). It is an inert, monoatomic noble gas with an easily polarizable electron cloud, which renders its chemical shifts extremely sensitive to the local environments. The sensitivity of the adsorbed ^{129}Xe can be enhanced by several orders of magnitude by applying a spin-exchange optical pumping (SEOP) hyperpolarization technique. Therefore, it is widely used in NMR and magnetic resonance imaging (MRI) applications in chemistry, biochemistry, materials science and medicine.^{34} It has been used in MRI of lungs,^{35} microfluidic flow imaging,^{36} investigation of liquid crystals,^{37} polymers,^{38} cements and shales^{39} and ionic liquids,^{40} for the determination of pore sizes of porous networks^{41,42} and also as an biosensors.^{43,44}

In our previous work,^{28} we performed a thorough ^{129}Xe NMR analysis of adsorption and dynamical phenomena of xenon in CC3. We calculated chemical shifts of ^{129}Xe in cage and window cavities, and, based on a simple two-site exchange model, we were able to interpret observed exchange averaged chemical shifts and T_{2} relaxation times as well as extract populations of cage and window cavities and exchange rates.

In the present work, we extend the previous work by investigating T_{1}, T_{2} and diffusion of both Xe and SF_{6} in homochiral CC3-R material (see Note on cage nomenclature and chirality in ESI†). Quantitative insights on the gas-mobility in CC3-R are deduced by analysing the data with a relaxation and diffusion model. We perform a global analysis of T_{1}, T_{2} and diffusion data using an extensive set of relaxation mechanisms for ^{129}Xe and SF_{6}. The model benefits from a reduced number of adjustable parameters by incorporating quantum chemical information and configurational sampling. Three main aspects are covered here: first, extraction detailed information on the dynamics of the xenon and SF_{6} gases in CC3-R; second, exploration of a model development where a reduced set of adjustable parameters are required by making use of potential energy surface (PES) to precompute relaxation model parameters; finally, we show how global analysis in the parameter estimation is performed with Bayesian Monte Carlo approach.^{45}

2.2.1 NMR spectra. ^{19}F and ^{129}Xe spectra were measured using a 90° pulse angle with single scan. ^{19}F chemical shifts were referenced with trifluorotoluene Bruker standard (−63.72 ppm) ^{129}Xe chemical shifts were referenced with respect to low pressure Xe gas (0 ppm).

2.2.2 T_{1} relaxation experiments. T_{1} relaxation times were measured using the inversion recovery pulse sequence. ^{19}F: the relaxation delay was 4 s and 8 scans were collected. ^{129}Xe: the relaxation delays were 80, 80 and 150 s, and the number of accumulated scans was 1, 2 and 1 for the LL, ML and HL samples, respectively.

2.2.3 T_{2} relaxation experiments. T_{2} relaxation times were measured using CPMG pulse sequence. ^{19}F: the relaxation delay was 4 s and 8 scans were collected. ^{129}Xe: the relaxation delays were 60, 60 and 200 s, and the numbers of accumulated scans were 8, 4 and 2 for the LL, ML and HL samples, respectively.

2.2.4 Diffusion experiments. Diffusion measurements were carried out using a PGSTE experiment with bipolar gradients. ^{19}F: the relaxation delay was 4 s and 32 scans were collected. Diffusion delay Δ was 0.2 s and length of the gradient pulse d varied from 7 to 3 ms. ^{129}Xe: the relaxation delays and numbers of accumulated scans were 60, 55 and 150 s, and 16, 1 and 8 for the LL, ML and HL samples, respectively. Diffusion delay Δ was varied from 0.05 to 25 s and length of the gradient pulsed from 0.3 to 2 ms.

Mechanism | DDinter | Shift-ex | DDHXe-ex | DDHXe-L | CSA-L |
---|---|---|---|---|---|

Amplitude | d_{HXe}, N_{H} |
Δδ, X_{S}, B_{0} |
X_{S}, b^{ex}_{HXe} |
X_{S}, b^{ex}_{HXe} |
X_{S}, Δσ_{S}, B_{0} |

Modulation | τ_{D} |
(k_{ex})^{−1} |
(k_{ex})^{−1} |
τ_{L} |
τ_{L} |

The diffusion of gas molecules is a molecular process causing intermolecular dipole–dipole relaxation (DDinter) due to the interaction with the protons of the organic structure. This is illustrated with motion between unit cells in Fig. 2A, lower panel. The DDinter ^{129}Xe–H relaxation model was originally derived for liquid state relaxation.^{46,47} In the implementation, the protons of the CC3-R structure are assumed to be fixed in space with the proton spin density of N_{H} (computed in ESI†). The model assumes hard sphere diffusion with the spin located at the centre of a sphere, which is reasonable for ^{129}Xe. The amplitude of the DDinter interaction (see Table 1) depends on the length of closest distance (d_{HXe}) between ^{1}H and ^{129}Xe spins (in the liquid state derivation),^{47} which can be explicitly computed by eqn (4). The DDinter mechanism is dependent on diffusion correlation time τ_{D} = d^{2}_{HXe}/D, where D is the experimentally known ^{129}Xe translational diffusion constant (see Fig. 3D).

The chemical shift difference of ^{129}Xe in the cage and window cavities (Δδ = δ_{C} − δ_{W})^{28} provides the shift-ex mechanism^{48} that has a dominant role in T_{2} relaxation. The amplitude (Table 1) is determined by Δδ, the cavity molar fractions of ^{129}Xe (X_{S}, S = C, W) and the strength of external magnetic field (B_{0}). The relaxation is modulated by the exchange rate (k_{ex}).

There are two local dipole–dipole mechanisms. The first one, DDHXe-ex, is due to exchange process and dependent on the b^{ex}_{HXe} interaction (see eqn (7)). The second one (DDHXe-L) is modulated by characteristic time τ_{L} due to local cavity motion (see Fig. 2A) and is dependent on the in-cavity dipole–dipole interaction (b^{S}_{HXe}, S = C, W; the subscripts refer to the ^{129}Xe–^{1}H pairs in window and cage cavities, see eqn (7)).

The CSA-L mechanism is due to the same local cavity motion (τ_{L}) and have amplitude dependent on the chemical shielding anisotropy (Δσ_{S}), populations (X_{S}) and B_{0}.

The relaxation mechanisms are assumed additive (statistically uncorrelated) such that cavity specific relaxation rates (T_{i}^{−1})_{m} m = {DDHXe-L, CSA-L}, (i = {1, 2}) are population weighted:

(T_{i}^{−1})_{m} = (T_{i}^{−1})_{m,C}X_{C} + (T_{i}^{−1})_{m,W}X_{W}.
| (1) |

Mechanism | DD-inter | DDFF-R | DDFH-R | DDFH-ex | CSA-R | SR |
---|---|---|---|---|---|---|

Amplitude | d_{HF}, N_{H} |
ScaleR, b^{C}_{FF}, X_{C} |
ScaleR, b^{C}_{HF}, X_{C} |
b^{ex}_{HF}, X_{C} |
X_{C}, Δσ_{C} |
C_{SR}I, X_{C} |

Modulation | τ_{D} |
τ_{R} |
τ_{R} |
(k_{ex})^{−1} |
τ_{R} |
τ^{−1}_{R} |

Three dipole–dipole mechanisms are included, DDFH-ex, DDFH-R and DDFF-R, where (b^{ex}_{HF}), F–H (b^{C}_{HF}) and F–F (b^{C}_{FF}) couplings are considered, respectively. The first accounts for the two-site exchange modulation. The following two mechanisms are local and assumed to be modulated by the rotational motion with correlation time (τ_{R}) in the cage cavity (C), weighted by the molar fraction X_{C}. Simulation of SF_{6} metadynamics in cage structure suggest there is a significant energy barrier and a rotation in passing the window cavity.^{27} The barrier translates to a low window population and motivates not to include a specific window site relaxation process. The passing through the window cavity is accounted for locally with DDFH-ex and via the proton concentration dependent diffusion process DDinter. The local DD amplitudes have an additional parameter scaleR as compared to ^{129}Xe. Since there is only QC equilibrium and not temperature dependent trajectory information simulated, this parameter serves to rescale F–H and F–F distances in order to resemble the configuration averaged values (see “r” in eqn (7)).

The relatively complex local DDFH interaction is thus modulated by two correlation times due to rotation and site exchange. In the analysis (Section 4.3) the latter is approximated with τ_{D}. An additional fast (sub-picoseconds) liberation process can be anticipated, however, is not explicitly included other than in the final DD amplitude. For this process, an effect on the T_{1} model cannot be ruled out, but a negligible influence on T_{2} is estimated due to the short timescale.

The chemical shielding anisotropy (CSA-R) with respect to zz axis (along a bond) is large for SF_{6} as shown by the quantum chemical (QC) calculations (see Section 3.4.2) and the relaxation is modulated by the correlation time (τ_{R}) in the cage cavity. For the window cavity no large amplitude modulation is expected, and no CSA mechanism is included. Potential DD-CSA cross correlations are not included in the model as discussed in the ESI† (see Section S2.1).

For similar reason, nuclear spin-rotation (SR) mechanism is estimated for SF_{6} rotation in the cage cavity following the Hubbard SR model.^{49} In this case, the modulation is dependent on the inverse rotational correlation time and thus has a reverse temperature dependence. The SR tensors are estimated by using QC calculations (Section 3.4.2).

Strictly speaking, contrary to Xe, SF_{6} molecule is not a perfect sphere, as the model assumes, and this has some effects on its dynamics in CC3. For example, it has been shown by simulations that, when SF_{6} exits the cage, it has to align its outermost triangular face with the triangular face of the window. Thereafter a rotation of about 60° is required so that the second face can align the cage window and pass through.^{27} However, the details of the dynamics in the window cavity is not really important in the relaxation modelling, as the population of SF_{6} in the window cavity is very low. Furthermore, the transport of SF_{6} enters the relaxation model via the diffusion coefficients. Thus, the details of SF_{6} deformations are not explicitly required. Therefore, the spherical approximation should be well justified.

(T^{ex}_{2}(ω_{A}))^{−1} = (ω_{A}Δδ)^{2}X_{C}X_{W}J_{ex}(0); X_{W} = (1 − X_{C}),
| (2) |

The latter depends on correlation time τ_{ex} = 1/k_{ex} and in general resonance frequency ω_{A} for spin A. The spectral densities for local DD, CSA and shift-ex are computed as a Lorentzian from the correlation times τ_{Y}, Y = L, R, ex:

(3) |

3.4.1 ^{129}Xe relaxation amplitudes. In this work we extend the use of the previously developed PES^{28} to reduce the number of adjustable parameters in the relaxation model. The PES allows for sampling of spatial xenon configurations at the chosen temperature in the cage and windows cavities. Furthermore, a property surface of non-relativistic (NR) nuclear shielding tensor was developed,^{28} from which the evaluation of shielding tensor at the spatial location is done at low computational cost. As explained in detail in ref. 28 a static reference nuclear shielding tensor was computed for xenon at the centre of both cage and window cavity (see Fig. 1B) including spin–orbit (SO) and scalar relativistic corrections at SO-ZORA/DFT(BHandHLYP) level.

where d_{S,i} (S = C, W) are the H–Xe distances, and X_{S} are the molar fractions known from experiments.^{28} The inverse power of 6 weights the distances relevant to DD contribution. Equilibrium, as opposed to sampled Xe–H distances, are motivated by the fact that the local dipole–dipole interaction is accounted for with a separate mechanism. Eqn (4) gives d_{HXe}: 5.52, 5.68, 5.72 Å, for LL, ML and HL samples, respectively.

with REL terms denote relativistic effect at SO-ZORA level (SO-ZORA-NR) at the static reference geometries.^{28} Here, it is added on top of the sampled nonrelativistic (NR) chemical shift difference. The relaxation rate can be calculated after mean square averaging of Δδ (see eqn (2)). The averaging leads to Δδ = 181.9 ppm (the latter static relativistic contribution being 32.2 ppm). According to the simulation, the ^{129}Xe chemical shifts both in cage and window cavity increase slightly with temperature.^{28} However, the difference changes only by 1 ppm over the relevant 80 K temperature interval and could affect relaxation amplitude (and rate) less than 1.5%. Thus, Δδ evaluated at 298 K by eqn (5) was used at all the temperatures.

where SD denotes sampled (j) standard deviation over equilibrium Xe–H_{i} distances (i). The result of eqn (6) is r_{ave}: 5.40, 5.36, 5.51 Å for the LL, ML and HL samples, respectively. The interaction (with r = r_{ave}) is given by:^{48}

where μ_{0} is the permeability of vacuum, γ_{A} is the gyromagnetic ratio of nuclei A and ħ is Planck constant in angular units.

computed with SD for each sampled Xe_{i}–H_{j} distance. It is noted that population weight is taken in eqn (1) for individual cavity contributions. The results are r_{C}: 3.05 Å, r_{W}: 2.46 Å and they take the place of r in eqn (7).

resulting in Δσ_{S}: 12.7 and 21.3 ppm for S = W and C, respectively.

The list of ^{129}Xe relaxation mechanisms is provided in Table 1. In this section the amplitude parametrisation is considered. The computed relaxation amplitudes follow from the hypothesis that there is statistically independent cavity dipole–dipole interaction and exchange and diffusion processes. Reasons for this separation may originate from a separation of timescale as well as approximately orthogonal processes. The latter processes are thus sensitive to averaged interaction tensors. Furthermore, our previous work^{28} provides the mole fractions of Xe in the cage and window cavities (X_{C}, X_{W}) that are here used as fixed parameters. Thus, the current relaxation model provides a validation of the two-site model used in our previous work.^{28}

In order to compute the estimates of shielding tensors, the Monte Carlo NVT sampling^{28} was performed at 298 K. The calculations provided spatial Xe configurations and shielding trajectories in the cage and window cavities (not including the loading effect of the neighbouring cavities). The trajectories allow for relaxation modelling with molecular details but only with few adjustable parameters.

For DDinter, the average (denoted with 〈·〉_{i}) of the proton (i) to Xe distance at the equilibrium is computed as:

(4) |

The shift-ex mechanism has a chemical shift difference sampled with the configuration trajectory as:

Δδ = [〈(δ^{NR}_{C} − δ^{NR}_{W})^{2}〉]^{1/2} + |(δ^{REL}_{C} − δ^{REL}_{W})|,
| (5) |

To arrive at local DDHXe-ex amplitude for the cage-window exchange process, the starting point is the standard deviation of [(b^{C}_{HA})^{2} − (b^{W}_{HA})^{2}] and, for comparison, computing the ensemble average as a population weighted effective distance:

(6) |

(7) |

The local cavity interaction (b^{S}_{HXe}, S = C, W) is provided in similar way from the effective distance:

(8) |

The CSA-L is obtained from shielding tensors trajectory (see page 106 of ref. 48) for the cage and window cavity in crystal frame and compute:

Δσ_{S} = std[Δσ^{traj}_{S}].
| (9) |

3.4.2 SF_{6} quantum chemistry. The QC calculations to optimize SF_{6} structure inside the cage/window cavities were carried out using Turbomole^{50} code at non-relativistic density functional theory (DFT) level with hybrid B3LYP^{51–53} functional. The DFT-D3 dispersion correction^{54} was used in order to realistically describe the attraction between F atoms of SF_{6} and H atoms in surrounding cage/window cavity structure. The def2-TZVP^{55} basis set was used for the atoms of SF_{6} molecule, while for the atoms in the surrounding cage/window cavities we used the smaller def2-SVP.^{55} Only the atomic positions of SF_{6} were optimized, while the coordinates of cage/window atoms were fixed to the DFT/PBE-TS optimized crystal structure (see details and structures in ESI†) that was used already in Xe NMR modeling of ref. 28.

The nuclear shielding tensors calculations with the Turbomole code were carried out using the B3LYP functional. For S atom in SF_{6} molecule we used def2-TZVP basis set but for F atoms a completeness-optimized co-r primitive basis set (18s10p6d2f) was used, which provides result for all NMR parameters practically at complete basis set limit for all NMR parameters.^{56} The def2-SVP basis set was used for the atoms in the surrounding cage/window structure. The obtained shielding tensors were symmetrized and diagonalized with python code to yield the isotropic, anisotropic and asymmetric parts in principal axis system (PAS) of the tensor for the relaxation models. The computed chemical shielding anisotropy (CSA) in the cage cavity was Δσ_{C} = 390 ppm and the chemical shift difference between the cage and window cavities was Δδ = 10.7 ppm (see Table S5 in ESI†).

The nuclear spin-rotation (SR) constant (mean of tensor trace) was computed with DFT vacuum optimized geometry (for details see Section S4 in ESI†). The purpose of SR tensor is to explore the order of magnitude contribution of SR-relaxation of SF_{6} in the cage.

3.4.3 SF_{6} relaxation amplitudes. For SF_{6}, an optimized structure in CC3-R was computed, allowing for some information on amplitudes of relaxation contributions (see Table 2).

The relaxation due to F–H dipole–dipole interaction in the cage cavity has a modulation due to overall rotation of SF_{6} with correlation time τ_{R}. The amplitude (b^{C}_{HF})^{2} may be computed following eqn (7) and (8), however, with the SD estimation from the fluoride equilibrium positions as opposed to NVT configurations. The effective distance was found to be r_{C}: 2.72 Å. For the F–F interaction (b^{2}_{FF}) the sum over the five intramolecular 1/d^{6}_{FF} distances were computed leading to an effective distance r_{FF} = 1.8 Å, to be compared with the shortest equilibrium F–F distance of 2.2 Å. Thus, the sum of F–F relaxation channels provides an efficient relaxation mechanism. Due to the approximation of equilibrium distances these distances were rescaled with a fitted parameter scaleR. Amplitude for DDFH-ex is obtained from eqn (6), with the mean computed over F-distances.

Related to nuclear spin-rotation (SR) and CSA the QC values (Section 3.4.2) where used.

3.4.4 Parameter estimation by MCMC. First, we summarize parameter estimation with Markov-chain Monte Carlo^{57} (MCMC) and then discuss the relaxation model results for ^{129}Xe and SF_{6} in Section 4. The deviation from experimental data is defined as the χ^{2}:

where N_{exp} is the number of experimental data, (σ^{2}_{i}) is the uncertainty (variance) of an experimental observation, i, estimated for the experimental data (see Section S4 in ESI†). The model is parametrized with the sets x_{Xe} and x_{SF6} defined in Section 3.3. For xenon the χ^{2} is computed from the {1/T_{1}, 1/T_{2}} data with diffusion constant interpolated from measurements. In the SF_{6} study, diffusion experiments were recorded at higher temperature and χ^{2} was computed from the whole set {1/T_{1}, 1/T_{2}, Diff}, and diffusion was estimated with the Arrhenius model in the set x_{SF6}. Thus, one activation energy is estimated based on data of the whole temperature interval.

(10) |

The parameters in x are estimated together with confidence intervals by sampling the so-called posterior probability of the hypothesis x. The form of this distribution is given in Section S3 in ESI.† It consists of the likelihood function, i.e. the negative exponential of eqn (10) constrained by the prior assumptions of x. In practise, the distribution of x is obtained by generating MCMC-trajectories for the four and seven-dimensional parameter space for ^{129}Xe and SF_{6}, respectively.

First the parameters are assumed to be of uniform distribution prior to the study, with their boundaries (cf. Tables S1 and S2 in ESI†), chosen based on physical constraints, for instance shortest plausible atom distance or alternatively, if the computational cost is too large, the boundary domain is reduced. The MCMC sampling was done following the Metropolis–Hastings algorithm,^{45,57} where initial configurations are generated uniformly and kept within the boundaries (cf. Tables S1 and S2 in ESI†) by implementing reflective boundary condition. After calibrating MCMC step length such that approximately 50% of steps are accepted and simulating an initial “burn in” period the productive estimation follows. For Xe and SF_{6}, 25 and 5 independent trajectories were simulated, with parameters initiated from uniformly distributed random numbers within the boundaries. In order to reduce chance of getting stuck in local minima during the “burn in” sequence an inverse “temperature” (χ^{2} → β·χ^{2}, β = {10^{−7}, 10^{−3}, 1}) was used. Hence, first an equilibration at “high temperature” (small β) is performed, followed by the sampling of the distribution related to eqn (10) (i.e. β = 1).

Consensus is confirmed between the trajectories (with different starting parameters) in that the distribution of χ^{2} is similar for each trajectory as well as the trajectories predicts similar parameter distribution. Thus, from multiple trajectories it is seen that the final χ^{2}-distribution does not depend on the initial condition.^{45} Production run of the trajectories are performed with 10^{6} MCMC steps, as deemed necessary considering the parameter self-correlation in the MCMC-trajectory. The statistically best estimates of parameters are given by the mean over MCMC configurations. Their corresponding 95% confidence interval are estimated by numerically integrating each parameter cumulative probability distribution. Scripts are implemented in MATLAB.

Contrary to the xenon samples, the ^{19}F chemical shift of SF_{6} does not change almost at all with temperature or loading (see also Fig. S1A in ESI†). According to the quantum chemical calculations described in Section 3.4.2, the ^{19}F chemical shift difference between the cage and window cavities (about 11 ppm, see also Table S5 in ESI†) is much smaller than in the case of ^{129}Xe (about 190 ppm). Furthermore, SF_{6} (5.5 Å) is larger than Xe (4.3 Å), and therefore it strongly favors the larger cage cavity over the window cavity, and changes in the temperature or loading does not change much the relative occupancies of SF_{6} in the cavities. Consequently, ^{19}F chemical shift remains almost constant.

The transverse relaxation rate of both ^{19}F and ^{129}Xe decreases with increasing temperature due to the increased mobility (see Fig. 3C). T_{2} of ^{129}Xe is also strongly dependent on the magnetic field strength whereas T_{2} of ^{19}F is not. The predominant mechanism of T_{2} relaxation of ^{129}Xe is the changing chemical shift due to the exchange between the cage and window cavities,^{28} while in the case of ^{19}F it is intermolecular dipolar mechanism. The former depends on the square of the field strength (cf. eqn (2)), while the latter has vanishing field dependence at the slow ^{19}F mobility and high fields of this study (see eqn (S2) in ESI†). The T_{2} of ^{19}F is approximately equal for the LL and ML loading (see Fig. S1C in ESI†) suggesting the same homogeneous broadening in their spectra. However, the spectral line width in Fig. 3A is almost double for the ML sample. Considering the few ppm larger chemical shift for gas phase SF_{6} (QC estimate Table S6 in ESI†), this is consistent with a inhomogeneously broadened ML spectrum due to unresolved gas phase. The longitudinal SF_{6} relaxation rate goes down by a factor of 5 with increased loading (see Fig. S1B in ESI†), most probably due to the same gas pool. Thus, the T_{1} values, chemical shifts and linewidths are implying that not all the SF_{6} molecules are absorbed in CC3-R (see also Section S1.1 in ESI†). As it is an additional complication to consider the effect of non-adsorbed gas, the quantitative modeling was conducted only for the LL SF_{6} sample.

Fig. 3D shows the self-diffusion coefficients of SF_{6} and xenon in CC3-R. The diffusion coefficient of SF_{6} is very low, about 4.8 × 10^{−13} m^{2} s^{−1} at 335 K, and it is almost the same for the LL and ML samples (see also Fig. S2 in ESI†). This reflects very slow diffusional motion of the large SF_{6} molecule in the small cavities of CC3-R. The diffusion coefficient of xenon in the HL sample is more than an order of magnitude higher than the diffusion coefficient of SF_{6} because of the smaller size of xenon. On the other hand, the diffusion of xenon in the LL and ML samples is more than an order of magnitude larger than in the HL loading sample, because diffusion of xenon is significantly restricted in the almost saturated HL sample as xenon atoms cannot pass each other in the channels.

Gas load | τ_{L} (ns) |
k_{ex} (10^{8} s^{−1}) |
τ^{−1}_{ex0} (10^{8} s^{−1}) |
E_{ex} (kJ mol^{−1}) |
τ^{−1}_{L0} (10^{10} s^{−1}) |
E_{L} (kJ mol^{−1}) |
---|---|---|---|---|---|---|

HL | 25 [23, 29] | 0.8 [0.7, 0.9] | 45 [30, 65] | 9.8 [8.7, 11] | 47 [35, 61] | 23 [22, 24] |

ML | 0.36 [0.3, 0.4] | 4.9 [4, 6] | 51 [11, 117] | 5.5 [3, 8] | 55 [24, 105] | 13 [11, 14] |

LL | 0.5 [0.4, 0.7] | 2.9 [2.7, 3.2] | 8.9 [3, 18] | 2.5 [0.7, 4] | 88 [3, 416] | 13 [7, 18] |

The 1/T_{1} increases with temperature for the HL sample and decreases for the ML and LL samples (see Fig. 4B, D and F). The higher temperature interval for HL cannot be the single explanation. It is found that both diffusion and local cavity dynamics are slower in the HL sample. It is noted that T_{1} is particularly sensitive to dynamics close to resonance frequency (see eqn (3), with τ_{L}ω_{0} ∼ 1). Both the net transport (τ^{−1}_{D} factor 3–5) and local cavity dynamics (cavity correlation time τ_{L} factor 70) is decreased by the high loading.

The slow local dynamics at HL and the cause of 1/T_{1} change of temperature dependence may be rationalized with high pressure increasing the influence of slow crystallite undulations on the shape of cavity. Hence, only at the timescale of these modes can the Xe gas fully explore the cavity, leading to the longer correlation time and increased activation energy. Table 3 shows transition from sub-ns to 25 ns, which is still too fast to provide a large change in 1/T_{2}.

In Fig. 5A–C the exchange and diffusion mobility rates k_{ex} and τ^{−1}_{D} of xenon are displayed versus temperature. Both have similar magnitude; however, the temperature dependence is less pronounced for k_{ex}. This is also reflected in the activation energies, which are larger for diffusion than for exchange; the experimentally determined diffusion activation energies (see Fig. S6 in ESI†) are 33 ± 1, 11 ± 0.5 and 19 ± 5 kJ mol^{−1} for the HL, ML and LL samples, respectively, which are larger than the E_{ex} values shown in Table 3.

Fig. 5 Exchange and diffusion mobility rates for xenon in the (A) LL, (B) ML and (C) HL samples. (D) Diffusion mobility rate for SF_{6} in the LL sample, extracted from Arrhenius law. |

In Table 4 the estimated parameters are provided with 95% confidence interval. The closest contact H–F spin distance is estimated to be 4.1 Å. This is relatively close to the static QC configuration estimate, 5.7 Å, following eqn (4). For the local DD mechanisms (DDFH and DDFF) the equilibrium distances were scaled down by 43% in the model; this is expected to be within a plausible range of distance deviations due to thermal motion. The SF_{6} molecule has a rotational correlation time of 6.7 ps and a diffusion constant of 1.08 × 10^{−13} m^{2} s^{−1} at 298 K. These are computed from the estimated Arrhenius parameters (see Table 4). From Fig. 5D it is noted that SF_{6} has more than two orders of magnitude slower diffusion as compared to the xenon samples. The model predicts 0.999 gas molar fraction of the cage cavity. The computed ^{19}F chemical shift difference between the cage and window cavities of 10.7 ppm together with the uneven populations gives a small shift-ex contribution (Fig. 6C). The Xe study suggest that exchange rate is in the regime of inverse diffusion correlation time (Fig. 5). Hence, k_{ex} is approximated as τ^{−1}_{D} in the SF_{6} shift-ex and DDFH-ex to keep adjustable parameters at a minimum.

d_{HF} (Å) |
ScaleR | X_{C} |
τ_{R} (ps) |
D (10^{−13} m^{2} s^{−1}) |
---|---|---|---|---|

4.09 [4.06, 4.13] | 0.5673 [0.5656, 0.5688] | 0.999 [0.99, 1] | 6.7 [6.6, 6.8] | 1.08 [1.06, 1.1] |

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## Footnote |

† Electronic supplementary information (ESI) available: Additional details of experimental methods and results as well as computational modelling. See DOI: 10.1039/c9cp04379a |

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