Interactions and Diffusion of Methane and Hydrogen in Microporous Structures: Nuclear Magnetic Resonance (NMR) Studies

Abstract

Measurements of nuclear spin relaxation times over a wide temperature range have been used to determine the interaction energies and molecular dynamics of light molecular gases trapped in the cages of microporous structures. The experiments are designed so that, in the cases explored, the local excitations and the corresponding heat capacities determine the observed nuclear spin-lattice relaxation times. The results indicate well-defined excitation energies for low densities of methane and hydrogen deuteride in zeolite structures. The values obtained for methane are consistent with Monte Carlo calculations of A.V. Kumar et al. The results also confirm the high mobility and diffusivity of hydrogen deuteride in zeolite structures at low temperatures as observed by neutron scattering.

1. Introduction

There is currently wide interest in the thermodynamic properties of light gases constrained to the interior of microporous structures that include metal organic frameworks [1,2,3] and classical zeolitic structures [4] because of the potential use of these structures for storage and transport [5,6,7,8], and catalytic conversion [9] of light molecular gases (H₂, CH₄, CO₂, CO, etc.). In addition to these practical applications, studies of these systems are of special fundamental interest for exploring the novel properties of quantum fluids when the constraining geometry has dimensions comparable to the thermal de Broglie wavelength or the thermal phonon wavelength [10,11]. While there is a large body of information about the total adsorption of light gases in many microporous structures, much less information is available about the strength of the interactions of the molecules with the confining walls, and detailed dynamics [12] of the gases inside the porous materials. Of particular interest is the thermal activation of diffusion from one micropore to a neighboring pore. We describe a novel application of some well known nuclear spin-lattice relaxation processes that are relevant when the excitation of the molecular systems can be described in terms of coupled thermal reservoirs associated with the internal degrees of freedom of the molecules (molecular rotations, quantum tunneling, etc.) that have well defined but weak inter-connections and weak nuclear spin couplings to the lattice or between different degrees of freedom. This multiple bath model has been used to describe the nuclear spin-lattice and spin-spin relaxation times in solid ³He at low temperatures [13], solid deuterium at intermediate temperatures [14], and the diffusion of hydrogen deuteride impurities in solid hydrogen [15,16]. As a general example we will follow the arguments of Guyer, Richardson, and Zane [13], and consider three energy baths A, B, and C whose internal degrees of freedom can be considered as quasi-independent (e.g., molecular rotations) and that are weakly coupled to one another while only one of the energy baths is strongly coupled to the lattice, as illustrated in Figure 1.

Figure 1. Schematic representation of three energy baths A, B, and C with a weak link (bottleneck) between B and C with energy flow from C to the thermal bath. In this example, there is a bottleneck in the vertical path R_BC.

Figure 1. Schematic representation of three energy baths A, B, and C with a weak link (bottleneck) between B and C with energy flow from C to the thermal bath. In this example, there is a bottleneck in the vertical path R_BC.

If these energy baths (A, B, C) can reach internal thermal equilibrium on a time scale smaller than that of the couplings between the baths, the observed overall relaxation will depend on the heat capacities of the individual baths. In particular, if as in Figure 1, the rate R_BC is the weakest (slowest) thermal link, the observed relaxation rate is given by

$$T_{1,observed}=\frac{C_A+C_B}{C_B}{T}_{BC}$$

where $T_{BC}=R_{BC}^{-1}$. A detailed derivation of this expression and the corresponding amplitudes of the components of the relaxation is given in Appendix A. We first review the application of this bottlenecked relaxation for the cases of solid ³He, and for light molecular gases (hydrogen deuteride and CH₄) trapped in zeolitic structures.

In the case of solid ³He (which is well understood [13]), A is the nuclear spin system, B represents the excitations due to atomic motion (quantum tunneling motions and/or vacancy motion), and C is the phonon system for solid ³He, which is in contact with the containing wall, which in turn is linked to the thermal bath. Following a perturbation of the nuclear spin system (bath A) with, for example, the application of an RF pulse, the thermal bath A will come to a common internal temperature (a nuclear spin temperature) in a time ${(M_2)}^{-\frac{1}{2}}\simeq 0.1$ ms where M₂ designates the second moment of the internal nuclear dipole-dipole interactions [13]. The transfer of energy from bath A to bath B (the tunneling degrees of freedom) is in typical cases much slower (~0.01–0.1 s) because the atom-atom exchange frequencies are much higher than the Zeeman frequency for modest magnetic fields. The weakest link at low temperatures (T ~ 0.1–0.3 K) is the connection between the tunneling motions and the phonons of the ³He lattice because of the enormous difference between the tunneling energies (~mK) and the phonon energies (10–30 K). Typical time constants for this energy exchange are ~1.0–100 s [13]. In this scenario, the observed relaxation is a function of both the intrinsic relaxation rates and the internal heat capacities of the different baths.

For solid ³He,

$$C_A=\frac{1}{2}N{k}_B{(ћ{\omega }_L/k_{B}T)}^2$$

where k_B is Boltzmann’s constant, T is the absolute temperature, and ${\omega }_L$ is the nuclear Larmor frequency.

$$C_B=\frac{3}{8}zN{k}_B{(ћJ/k_{B}T)}^2$$

where J is the exchange frequency due to quantum tunneling and $C_C=234N{k}_B{(T/{\text{Θ}}_D)}^3$ where ${\text{Θ}}_D$ is the Debye temperature for solid ³He. (We use the low temperature limit for the phonon heat capacity.)

For methane, in addition to the nuclear spin degrees of freedom (bath A) we need to consider the rotational motion (bath B) and translational motion (bath C) of the molecules and their thermal contact with the wall (D) of the zeolite (phonons of zeolite), shown in Figure 2. For bath A, as in the case of solid ³He, the nuclear spins reach internal equilibrium on a time scale of ~0.1 ms as determined by the nuclear spin-spin interactions. The coupling between baths A and B is determined by the spin-rotation coupling $H_{SR}=CIJ$ with C = 42 kHz [17]. I is the nuclear spin, and J the rotational angular momentum. The AB coupling time constant is found to be ~0.1 ms, and baths A and B come to a common temperature very rapidly. The same is also true for the coupling between baths B and C, which is determined by electric octupole-octupole interaction between molecules. This octupolar coupling is given by H_OO = I²/R⁷ [17] where I is the electric octupole moment and R the separation of the molecules. For solid methane R = 4.6 Å and H_OO ~ 3 K. For well-separated molecules in zeolite-like structures, R ~ 10 Å and H_OO ~ 10⁻³ K. On expanding H_OO in terms of small displacements we find a coupling time constant τ_BC ~ 10 ms.

Figure 2. Multiple bath model for relaxation of methane molecules in zeolite cages. A represents the nuclear spin energies plus the ground rotational state, B represents the excited rotational states (only the T states have nuclear spin), C consists of the translational molecular motions, and the wall represents the excitations (phonons) of the zeolite lattice. If R_CW is the thermal bottleneck, the observed relaxation time is ${\tau }_{Obs}=(1+\frac{C_A+C_B}{C_C})R_{CW}^{-1}$.

Figure 2. Multiple bath model for relaxation of methane molecules in zeolite cages. A represents the nuclear spin energies plus the ground rotational state, B represents the excited rotational states (only the T states have nuclear spin), C consists of the translational molecular motions, and the wall represents the excitations (phonons) of the zeolite lattice. If R_CW is the thermal bottleneck, the observed relaxation time is ${\tau }_{Obs}=(1+\frac{C_A+C_B}{C_C})R_{CW}^{-1}$.

The large bottleneck at low temperatures is between the translational modes of the molecules and the phonons in the wall of the zeolite. This Kapitza resistance at the wall of the mesoscopic structures results from the large difference between the velocity of translational modes of methane (ν_t ~ 200 m/s) compared to those of the solid zeolite, ν_s ~ 5000 m/s. If ρ_t and ρ_s are the respective densities of the methane and zeolite atoms, only a fraction $f~\frac{{\rho }_t}{{\rho }_s}{\left(\frac{v_t}{v_s}\right)}^3~{10}^{-5}$ of the translational excitations of the CH₄ can transfer energy to the phonons in the wall, leading to thermal time constants of 0.1–1.0 s.

For simplicity in the case of methane we include in bath A both the nuclear Zeeman energy and the ground state for the rotational degrees of freedom. There are three molecular states for a methane molecule, A, T, and E, corresponding to the different spin-symmetry configurations for a tetrahedral group of fermions. The E state has no total nuclear spin and plays no role in the nuclear relaxation processes. The excited states are therefore given by the T species, of which there are at least two, T₁ and T₂, corresponding to the two different symmetry sites in a zeolite cage. The nuclear Zeeman energy and the rotational ground state of the A species are considered as one single bath because they are tightly coupled. The heat capacity for bath B is the sum of the contributions for the T₁ and T₂ excitations, with $C={\displaystyle {\sum }_{i=1,2}{g}_i{N}_i}{k}_B{({\text{Δ}}_i/T)}^2\exp (-{\text{Δ}}_i/T)/{(1+\exp (-{\text{Δ}}_i/T))}^2$ where ∆_i are the excitation energies for the T₁ and T₂ states and g_i are the relative degeneracies.

For hydrogen deuteride molecules trapped in mesoporous structures, A is the combined bath for the Zeeman and the ground state for the translational degrees of freedom, and B is the bath corresponding to the excited states for the hydrogen deuteride molecules (wavenumber non-zero). In this case, C_A is expected to be the low temperature Debye heat capacity (for the ground state) and C_B is the sum of the Schottky heat capacities for each excited energy level.

2. Experimental Section

The samples of zeolite-13X (Na₈₆[(AlO₂)₈₆(SiO₂)₁₀₆]·264H₂O) were prepared from commercially available material [18] that was in the form of hard pellets (1.5 mm outer diameter). The pellets were lightly crushed to facilitate gas penetration and then activated by heating to 200 °C in a high vacuum (<10⁻⁵ Torr.) for seven hours. Following activation, nitrogen adsorption isotherms were measured [19] to characterize the sample and to check that the expected surface area per unit mass was realized. The samples were then placed inside a Teflon^® [20] coil former that was inserted into a groove in a cold finger, Figure 3. The end of the Teflon^® chamber was sealed with a plug of glass wool to prevent motion of the crushed zeolite while providing for gas entry. The sample cell was enclosed by a copper vacuum shroud using an indium metal seal to ensure vacuum integrity at low temperatures. A short brush of very fine copper wires (0.1 mm outer diameter) provided the thermal linkage from the sample to the copper enclosure. The latter had a calibrated carbon glass thermometer affixed to the copper cold block (Figure 4) to measure the temperature. A twisted bifilar heater was wound around the cell to regulate the temperature using a current derived from an error signal between the thermometer readout and a desired temperature determined by a setting on a resistance bridge. Helium exchange gas was admitted to the space separating the metal shroud from a liquid helium bath (Figure 5) to provide a weak thermal link from the helium bath to the sample cell. The cell itself was suspended by an insulating rod fabricated from bakelite. This design allowed us to hold the temperature to within 0.1% of the set temperature for 2 < T < 150 K.

In order to determine the precise coverage for methane on the interior surfaces of the zeolite we measured the adsorption isotherm for methane (on the sample) at T = 77 K. The results for the isotherm studies have been reported elsewhere [21] and are summarized in Appendix B. The isotherm exhibits a distinct jump at the value of adsorbed gas corresponding to the saturation of the α-cages of zeolite-13X for 1.1 × 10⁻² mol/g. The latter was estimated from the mass of the sample and the known surface area of zeolite-13X of 660–800 m²/g [18,22].

A pulse NMR spectrometer was used to measure the nuclear spin relaxation times. A relatively low frequency (5.5 MHz, corresponding to the proton nuclear Larmor frequency in an applied magnetic field of 0.129 T) was chosen to realize the condition for which the Larmor frequency matched the anticipated tunneling rate for one molecule to pass from one cage to another as estimated by recent Monte Carlo calculations [12]. This condition allows one to observe the Bloembergen-Purcell-Pound [23] minimum in the relaxation time from which one can infer the tunneling rate directly in addition to the thermal activation energy. The NMR coil was connected via a low-loss cryogenic cable to a fast-recovery RF duplexer similar to the design of Deschamps et al. [24,25], but altered to work with a high impedance parallel resonance circuit. Two different pulse sequences were employed: (i) 90_x–90_y and (ii) 90_x–180_x sequences in order to observe solid and liquid-like echoes, respectively, with the latter appropriate to the “quasi-melting” behavior expected at the temperatures corresponding to thermal activation of intercage diffusion. Because of the low filling factors employed, typically less than one molecule per cage, the signal to noise ratios are weak and the signals were therefore accumulated and averaged using a computer interfaced digital storage oscilloscope.

Figure 3. (Color on line) Schematic illustration of sample cell showing NMR coil and thermal linkage to sample from cold cap (green) and copper shroud (red).

Figure 3. (Color on line) Schematic illustration of sample cell showing NMR coil and thermal linkage to sample from cold cap (green) and copper shroud (red).

Figure 4. (Color on line) Schematic representation of the support structure and cooling path for the sample cell. The exterior copper can sits in a liquid helium bath that can be pumped to 1.4 K.

Figure 4. (Color on line) Schematic representation of the support structure and cooling path for the sample cell. The exterior copper can sits in a liquid helium bath that can be pumped to 1.4 K.

Figure 5. (Color on line) Variation of the CH₄ proton spin-lattice relaxation with temperature for molecules confined to α-cages of zeolite: diamonds, 1.0 molecule per cage; triangles, 0.5 molecules per cage; and squares, 0.05 molecules per cage. The Shottky dependence is lost for low filling factors.

Figure 5. (Color on line) Variation of the CH₄ proton spin-lattice relaxation with temperature for molecules confined to α-cages of zeolite: diamonds, 1.0 molecule per cage; triangles, 0.5 molecules per cage; and squares, 0.05 molecules per cage. The Shottky dependence is lost for low filling factors.

3. Results and Discussion

3.1. CH₄ Molecules

The values of the relaxation times measured for different quantities of CH₄ adsorbed on zeolite-13X are shown in Figure 5. Three features are observed: two peaks at 27 K and 46 K, and a deep minimum at 78 K. The two peaks are attributed to two tunneling states for the T molecular species associated with two different sites for localization of the CH₄ molecules. In the two-bath model, bath B consists of the thermal excitations to the distinct rotational states T₁ and T₂. The ground state for the molecular rotations referred to in the literature as the symmetric A state with the nuclear spin excitations forms bath A. The heat capacity for bath B therefore consists of two Shottky-like heat capacities giving rise to the two peaks in the observed relaxation times. The values of the energy peaks (27 K and 49 K) are close to those reported for earlier heat capacity measurements that were seen as weak bumps in the total heat capacity [26]. The peaks in the observed relaxation times are much sharper than those for simple Shottky heat capacities and this is expected because the excited states are expected to be pocket states that undergo small amplitude librational motion about their equilibrium orientations.

The broad minimum at high temperatures is interpreted as a classic Bloembergen-Purcell-Pound minimum [23] associated with a thermally activated diffusion for passage from one α-cage to adjacent cages with an intercage tunneling of $\tau ={\tau }_0\text{exp}(-\frac{E_A}{k_{B}T})$. The fit shown in Figure 6 yields an activation energy E_A = 20.8 ± 1.5 kJ/mol. and a tunneling rate τ₀ = 1.2 × 10¹⁵ s⁻¹. This value deduced for the activation energy is comparable to the value of 22 kJ/mol. estimated from Monte Carlo studies [27].

Figure 6. (Color on line) Fit of variation of the observed relaxation time with temperature for CH₄ molecules in zeolite-13X at 78 K to thermal activation of intercage diffusion for 0.5 molecules per cage. The solid green line corresponds to an activation energy of 20.8 ± 1.5 kJ/mol. and a microscopic tunneling rate of 1.2 × 10¹⁵ s⁻¹.

Figure 6. (Color on line) Fit of variation of the observed relaxation time with temperature for CH₄ molecules in zeolite-13X at 78 K to thermal activation of intercage diffusion for 0.5 molecules per cage. The solid green line corresponds to an activation energy of 20.8 ± 1.5 kJ/mol. and a microscopic tunneling rate of 1.2 × 10¹⁵ s⁻¹.

3.2. Hydrogen Deuteride Molecules

In order to carry out a more fundamental study of the molecular relaxation and diffusion in mesoporous structures we carried out experiments on hydrogen deuteride trapped in cages of zeolite. While methane has several distinct molecular species (ortho, meta, and para), corresponding to the different combinations of rotational symmetry and nuclear spin symmetry, hydrogen deuteride molecules do not have these properties and can be regarded as spherical molecules with a weak electric dipole moment. We chose hydrogen deuteride rather than H₂, because like CH₄, H₂ has two molecular species, ortho-H₂ (with total nuclear spin I = 1 and orbital angular momentum J = 1) and para-H₂ (with I = 0 and J = 0). Only ortho-H₂ can be observed by nuclear magnetic resonance and this species converts to para-H₂ slowly via magnetic interactions. In addition, ortho-H₂ has an electric quadrupole moment that results in interesting molecular alignment configurations, but these can be difficult to interpret [28]. The relaxation of hydrogen deuteride molecules, however, is determined by their translational degrees of freedom and the spin-spin interactions between these molecules and the interactions with the walls of the cages. Hydrogen deuteride experiments are therefore ideal for determining the molecular diffusion in mesoporous materials and for exploring the effects of confinement on the molecules. The experiments can test if the translational degrees of freedom are quantized as expected for a perfect spherical cage.

Figure 7 shows the values of the CH₄ proton spin-lattice relaxation times observed for temperatures 1.5 < T < 15 K for two different densities, (i) 1.0 molecule per cage and (ii) 0.5 molecules per cage. The densities were determined from the adsorption isotherms [21]. Distinct peaks are seen for each filling, x, but at different temperatures for the different fillings: at 2.9 K, 4.9 K and 7.3 K for x = 1.0; and at 2.1 K, 4.5 K and 12.3 K for x = 0.5. In Figure 8 we show the fit assuming distinct Shottky heat capacity contributions (illustrated by the broken lines) for the data for 0.5 molecules per cage. The Shottky heat capacity form $C_i=N{k}_B{(\frac{{\text{Δ}}_i}{T})}^2\exp (-\frac{{\text{Δ}}_i}{T})/{[1+\exp (-\frac{{\text{Δ}}_i}{T})]}^2$ (for each excited state ∆_i) can only be considered as an approximation for the weakly trapped molecules and is valid only if the excited states can, to a good approximation be regarded as discrete energy levels. In the bath model of Figure 2, $C_B={\displaystyle \sum _i{C}_i}$ for the i excited states, and C_C is the low temperature heat capacity for the molecular translational motions. In Figure 7 we have assumed an arbitrary amplitude for each contribution to fit the data. The values for the excitation levels shown in Figure 7 are very different from the principal adsorption energies of ~80 K and ~40 K, respectively, for the S1 and S2 bonding sites for hydrogen in zeolite-13X as determined from neutron scattering experiments [29,30]. The S1 sites are at the centers of the six-membered rings adjacent to the eight-sided opening of the α-supercages of the zeolite structure, and the S2 sites are located close to the octagonal openings of the α-supercages. For a detailed summary of the positions of these sites relative to the supercages the reader is referred to Figure 6 of [29], Figure 2 of [31], and Figure 2 of [32]. These discrete energies could be due to: (i) small energy barriers between the potentials at the S1 and S2 sites that allow molecules to jump form one site to another; and (ii) a number of closely spaced energy levels inside the binding potentials.

Figure 7. Observed variation of the hydrogen deuteride proton spin-lattice relaxation with temperature for molecules adsorbed in zeolite 13× with coverages of 1.0 molecule per cage (squares) and 0.5 molecules per cage (triangles). The solid line shows the contribution assuming Shottky level specific heats for three discrete energy levels. Each contribution is shown by the broken lines.

Figure 7. Observed variation of the hydrogen deuteride proton spin-lattice relaxation with temperature for molecules adsorbed in zeolite 13× with coverages of 1.0 molecule per cage (squares) and 0.5 molecules per cage (triangles). The solid line shows the contribution assuming Shottky level specific heats for three discrete energy levels. Each contribution is shown by the broken lines.

3.3. Discussion

If we consider a simple molecule of mass m trapped in a spherical cage of radius r, the energy levels corresponding to the translational motion of the molecule are quantized [33,34] with energy $E_{l,n}={\beta }_{l,n}^2{ћ}^2/2m{r}^2$ where β_l,n is nth root of the l_th order spherical Bessel function. As illustrated in Figure 8, the energy levels for a 15 Å cage for m = 3 are of the order of a few K and should be observable.

Figure 8. (Color on line) Schematic representation of the translational energy levels E_l,n for hydrogen deuteride molecules constrained to a cage 15 Å in diameter.

Figure 8. (Color on line) Schematic representation of the translational energy levels E_l,n for hydrogen deuteride molecules constrained to a cage 15 Å in diameter.

The spherical cage is a very poor approximation to the mesoporous cage of zeolite, which has several open channels but the model does provide an order of magnitude estimate for the translational energies. The values observed experimentally are indeed comparable to those expected in this model. In addition the energies appear to decrease as the filling is reduced and this is attributed to an effective increase in the bounding dimension for the lower x values with reduction of the blocking of the tunnels to the cages and, thus, leading to a decrease in the expected energies. These estimates are very approximate, and computer simulations for the zeolite geometries could provide a much better fit to the observed features.

It is noteworthy that, despite the high values for the absorption energy of the S1 and S2 sites, the molecules are still highly mobile. The relaxation times decrease exponentially below 12 K (Figure 9), according to a thermal activation process. The tunneling rate ${\tau }_T^{-1}={\tau }_0^{-1}\exp (-E_A/k_{B}T)$ where the activation energy $E_A/k_B=73\pm 3$ K and τ₀ = 1.5 × 10⁻¹¹ s⁻¹. These values lead to a diffusion rate of (8.2 ± 2.8) × 10⁻⁶ cm² s⁻¹ at T = 19.5 K, which is to be compared with the value of 4.5 × 10⁻⁶ cm² s⁻¹ obtained at 17.4 K from the recent inelastic neutron scattering measurements of Coulomb et al. [35]. The small difference is accounted for by the small difference in temperatures of the two measurements and the experimental errors. The important feature of these results is that they clearly show that there exists a high mobility for low densities of hydrogen in zeolite down to very low temperatures.

Figure 9. (Color on line) Variation of the nuclear spin-lattice relaxation time with temperature at high temperatures. The solid line (red) is a fit for a thermal activation of 73 ± 3 K and an intrinsic turneling rate of 1.5 × 10¹¹ s⁻¹.

Figure 9. (Color on line) Variation of the nuclear spin-lattice relaxation time with temperature at high temperatures. The solid line (red) is a fit for a thermal activation of 73 ± 3 K and an intrinsic turneling rate of 1.5 × 10¹¹ s⁻¹.

4. Conclusions

Experimental investigations of the temperature variation of the proton spin-lattice relaxation time have shown the existence of distinctive peaks in the relaxation times for both CH₄ and hydrogen deuteride molecules confined to the microporous cages of zeolite for concentrations equal to or less than one molecule per cage. This temperature dependence is interpreted in terms of discrete low energy excitations for the molecules. The origin of the excitations is different for CH₄ and hydrogen deuteride. By using a well-known description of the coupling between different energy baths corresonding to these excitations, the peaks in the relaxation times are related to the peaks in the heat capacities of the relevant baths contributed by these excitations. This analysis is only valid when bottlenecks occur in the energy flow between the baths due to the weak couplings.

For the case of CH₄ the levels are interpreted in terms of excited states for the T-rotational states of CH₄. For hydrogen deuteride, the inferred energy levels are comparable to the values estimated for the quantized translational levels of molecules limited to motion inside spherical cages of 15 Å in diameter. Detailed theoretical calculations for the molecular motion in the connected cages of Zeolite like structures are needed to verify this interpretation. The results also show a strikingly high mobility for hydrogen molecule in these structures down to T ~ 10 K in agreement with recent neutron scattering studies.