Introduction

Water is the most extensively studied molecule on Earth. A precise determination of its intermolecular potential would allow accurate predictions of its properties in gas, liquid and solid phase. However, despite huge theoretical efforts1,2, a full agreement with experiments3,4,5,6 has not yet been achieved. Nuclear magnetic resonance (NMR) of molecules in gas phase7 has some unique features. The coupling between nuclear spins and magnetic moments induced by molecular rotation implies that collisions between molecules lead to a relaxation, i.e. to the return of the longitudinal magnetization Mz to its equilibrium after a perturbation, through a mechanism known as spin-rotation. If spin-rotation is the dominant mechanism, cross-sections for the transfer of angular momentum can be obtained from NMR relaxation rates in gas phase. Such relaxation rates have been measured over a wide range of pressures and temperatures8,9,10. Experimentally determined cross-sections can be used to refine intermolecular potentials11,12,13. In methane, isotopic substitution14,15,16 affects relaxation rates associated with the different isotopomers such as CH4, CH3D, CH2D2, and CHD3. In supercritical water17,18,19, spin-rotation significantly contributes to NMR relaxation despite the high density. In the context of our attempts to prepare para-water20,21,22,23,24,25,26 and related spin states in other molecules27,28,29,30,31,32, we have measured longitudinal relaxation rates R1 = 1/T1 of gaseous H2O and HDO at different temperatures and pressures. To the best of our knowledge, this is the first time that such observations are reported. Our measurements are useful to refine intermolecular potentials for water vapour. These may be compared with water confined in matrices33,34 or in fullerene cages21,35 where a gas-phase like behaviour can be observed.

Theory

Collisions between molecules can induce transitions between rotational quantum states. As a result, spin-dependent interactions vary as a function of time and, if the fluctuations occur at frequencies in the vicinity of the nuclear Larmor frequency ω0, longitudinal NMR relaxation takes place. Comprehensive theoretical treatments of NMR relaxation can be found elsewhere36,37,38,39,40,41,42,43,44. In this article, we shall only mention some aspects of spin-rotation and dipole-dipole relaxation mechanisms that are relevant to longitudinal relaxation in gas phase.

Spin-rotation (SR) relaxation is due to collisions that modulate local fields induced at the sites of the nuclei by the rotation of the surrounding electronic cloud.

Relaxation induced by spin-rotation can be described by ref. 45:

where:

τJ is the spin-rotation correlation time, Ceff (in Hz) the spin-rotation constant, ωJ the rotational frequency (in rad/s)46, is the number density of molecules, v is the average thermal velocity, σJ is the collisional cross-section for the transfer of angular momentum, I0 is the moment of inertia, grot is the g-factor, μN is the nuclear magneton, H is the magnetic field and μ is the reduced mass of the two colliding particles. The correlation time τJ is related to the lifetime of the rotational quantum states. The relaxation process can be described by characterizing the cross-section for the transfer of angular momentum. Intermolecular potentials used to model the interaction mostly consist of an isotropic part, usually a radial function, depending only on the distance between particles (e.g., Lennard-Jones potential) and an anisotropic part, depending also on the orientation of the molecules with respect to each other. The intermolecular potential of a molecule can be written by considering its axial symmetry47 and can be linked to relaxation rates via the Bloom – Oppenheim theory48.

Dipole-dipole (DD) relaxation is due to fluctuations of the interaction between magnetic dipoles, which are induced by physical rotation. The DD interactions are described by a correlation time τC that is related to the mean time needed for the molecule to undergo a rotation through one radiant. DD relaxation can occur between spins in the same molecule (intramolecular DD) or between spins in different molecules (intermolecular DD). Relaxation by the intramolecular DD interaction between the two protons of water is described by ref. 36:

where r is the distance between the protons, γH is the gyromagnetic ratio of protons and μ0 is the magnetic permeability in vacuum.

In liquid phase, the rotational correlation time τC is linked to τJ by the Hubbard relation49 τC/τJ = 1/6. For dilute gases (τJ → ∞), the ratio of correlation times τC/τJ varies from 5/4 (Ivanov model) to 1/4 (extended diffusion model) or 1/24.4 (Langevin model)14,15.

Results

We measured longitudinal relaxation rates R1 by the conventional inversion-recovery method. Experiments were carried out at temperatures T = 353, 363, and 373 K and pressures 9 < p < 101 kPa. The translational diffusion of water molecules does not affect our measurements of longitudinal relaxation rates R1 (see Methods), although it might interfere with measurements of transverse relaxation rates R2. The rates R1 observed in neat water (samples 1–4, H2O-H2O collisions) and in a mixture of HDO and D2O (sample 5, HDO-D2O collisions) are reported in Table 1.

Table 1 Longitudinal relaxation rates R1 for gaseous H2O (samples 1 to 4) and gaseous HDO (sample 5) at 800 MHz and at different temperatures and pressures.
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We shall initially consider spin-rotation to be the dominant relaxation mechanism, neglecting dipole-dipole relaxation. Under our experimental conditions, water vapour is mainly monomeric50,51,52 and the extreme narrowing regime is not fulfilled: R1 shows a maximum at a pressure pmax where τJ = 1/(ωO − ωJ) (see Fig. 1).

Figure 1
figure 1

(Points) Experimental rates R1 of gaseous H2O at 800MHz and at pressures 9 < p < 101 kPa.

(Lines) Estimates of R1 arising from spin-rotation, using Eq. 1 with the parameters in Table 2.

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The number density at pressure p can be estimated via the ideal gas law (see Methods) yielding τJ = RT/(pvσJ). Hence, it is possible to calculate the cross-section for H2O:H2O collisions at pmax as . If we assume to be independent of p (i.e. ) at the low pressures used in our experiments, we can substitute in to find . This last relationship can be used to predict the dependence of R1 on p, at a given T, by using Eq. 1 and the parameters in Table 2. All three curves result from fitting a single parameter (pmax), all the other parameters being fixed to the values given in Table 2. The fitted value pmax = (17 ± 3) kPa provides a fair agreement between experimental relaxation rates (points) and predicted rates (lines) (Fig. 1). The spin-rotation tensor depends on the symmetry of the molecule: in our approximation we take into account only the isotropic constant Ceff35,53,54 which we consider to be independent of both pressure and temperature. The fitted value of pmax is constant while Ceff is fixed to values comprised in its confidence range.

Table 2 Parameters used to calculate cross-sections via Eq. 1.
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In a more refined analysis we included contributions due to the intramolecular dipole-dipole interaction (Eq. 2). We fixed τC to values predicted by the Ivanov model (τC = 5/4τJ), extended diffusion model (τC = 1/4τJ) or Langevin model (τC = 1/24.4τJ)14,15. Our experimental data are compatible with a negligible dipole-dipole contribution or with the Langevin model (τC ≪ τJ), according to which significant contributions of only occur at low pressures p < 10 kPa.

For HDO-D2O mixtures, experimental relaxation rates R1 (sample 5) are reported in Table 1. In this case, we can safely neglect DD contributions. The experimental rates R1 in Table 1 and the parameters in Table 2 are substituted into Eq. 1 to calculate the collision cross-sections.

Cross sections and correlation times for the transfer of the angular momentum in H2O:H2O and HDO:D2O collisions are reported in Table 3.

Table 3 Correlation times τJ and cross-sections σJ for the angular momentum transfer in H2O:H2O and HDO:D2O collisions, calculated with Eq. 1 and parameters in Table 2.
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Discussion

Our analysis provides information about H2O-H2O and HDO-D2O collisions at pressures below 101 kPa and temperatures between 353 and 373 K. The ratio of cross-sections differs from the ratio of the moments of inertia . This discrepancy suggests that there must be appreciable differences between the intermolecular potentials for HDO:D2O and H2O:H2O collisions. This hypothesis is compatible with the fact that H2O and D2O have almost equal electric dipole moments55 while the electric dipole moment of HDO differs in intensity and orientation from those of H2O and D2O55. NMR relaxation studies on the influence of hydrogen/deuterium isotopic substitution on collisional cross-sections have been reported for methane15,16. However, by isotopic substitution on methane only the moment of inertia is markedly altered. A direct comparison with isotopic substitution on the highly polar H2O is therefore not possible.

The collisional cross-sections calculated from our NMR data can be used to refine the anisotropic part of the intermolecular potentials for collisions in gas phase56,57,58 via the Bloom – Oppenheim theory59. However, such calculations are beyond the scope of this work.

Our findings may be relevant for Dissolution Dynamic Nuclear Polarization (D-DNP)60 where a frozen sample is rapidly heated by injecting a burst of superheated D2O (T > 373 K) into the cryostat, and the liquid HDO ‘bolus’, usually containing a hyperpolarized solute, is pushed by pressurized helium gas (typically at 1 MPa) through a polyethylene tube with a 1 mm inner diameter running through a “magnetic tunnel”61, with a length of ca. 4 m between the polarizer and the NMR or MRI system. Attempts to monitor the speed of the bolus moving through the tube by optical means have shown that it tends to break up into small droplets during the transfer. This increases the surface area where water molecules can exchange between the liquid and gaseous phases. If the liquid/gas exchange is fast, the averaged longitudinal relaxation rates are likely to be much shorter than those in liquid water. The shortening of T1 would lead to a rapid loss of hyperpolarization during the transfer between the polarizer and the NMR magnet. Note that the viscosity and surface tension of the transferred liquid are difficult to control, since it consists of an aqueous solution containing analytes, polarizing agents like TEMPOL and glass-forming agents such as glycerol.

To summarize, we reported NMR relaxation rates due to binary H2O:H2O and HDO:D2O collisions in the gas phase and evaluated the cross-sections for the transfer of the angular momentum which can be used to refine the intermolecular potentials.

Methods

Our experimental setup consisted of a pair of coaxial glass tubes (Fig. 2).

Figure 2
figure 2

Schematic view of the coaxial tubes: outer tube (10 mm outer diameter) filled with toluene-d8 and inner tube (5 mm outer diameter, held by Teflon spacers) containing water vapour sealed under vacuum.

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The inner tube with 5 mm outer diameter was held in the center of a 10 mm tube by holders made of PTFE (Teflon). The outer tube contained about 2 mL of deuterated toluene-d8 (boiling point Tbp = 384 K). Its deuterium signal allows one to lock the static field and to shim its homogeneity. The inner tube contained water that was frozen and flame-sealed under vacuum (p = 1 kPa). Four tubes of 3.5 to 4 cm length, labeled as samples 1, 2, 3 and 4, were filled with ca 0.1, 0.2, 0.3 and 4.5 mg H2O, determined with a precision balance (±0.1 mg, max. tara 31 g). A fifth tube (sample 5) was filled with 2 mg of 98% D2O and 2% H2O (v:v), hence containing ca. 2% HDO. The inner tube was completely immersed in the solvent contained in the outer tube (Fig. 2) in order to have a homogeneous temperature and to avoid condensation of water on the walls of the inner tube in regions outside the area where the temperature is accurately controlled. Before and after inserting the samples into the spectrometer, the temperature in the probe was determined with a platinum PT-100 resistance thermometer (“iTRON 08” by JUMO)62 using a similar set of two concentric tubes with toluene-d8 in the outer tube. After each experiment the maximum temperature variations were ±1 K. Two typical 1H NMR spectra are shown in Fig. 3: the peak near 3.2 ppm (w.r.t. TMS) is due to water in the gas phase at T = 363 K.

Figure 3
figure 3

Proton NMR spectra of samples described in Fig. 2 at 800MHz and at 300K (top) or 363K (bottom).

The signals at 2.1 and 7 ppm are attributed to residual protons in incompletely deuterated toluene-d8. The signal at 3.2 stems from water in gas phase. Small peaks between 0.3 and 2 ppm are due to impurities in toluene-d8. We diluted TMS in toluene-d8 to use its resonance at 0 ppm as chemical shift reference.

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NMR instrumentation

All NMR experiments have been performed on a Bruker Avance-II 800 MHz spectrometer equipped with a 10 mm BBO probe.

Evaluation of pressure and density

To determine the pressure p and the number density of the water in samples 1 to 5 we measured the mass of water and estimated the volume of the inner tubes. Samples 4 and 5 contain saturated vapour (p = psat). In that case the pressure psat can be calculated using Antoine’s equation63:

where T is the temperature and A, B and C are sample-specific constants. When expressing the pressure in bar and the temperature in K, we assumed64 A = 5.08354, B = 1663.125 and C = −45.622 for both H2O and D2O, since their vapour pressures are similar within 1% over our range of temperatures65. The number density at a pressure p can be estimated provided that the equation of state of the gas is known a priori. We have compared predicted by the ideal gas law with obtained from a second-order virial expansion. The deviation is always below 2% in the range of pressures and temperatures under investigation, so that the use of the ideal gas law is legitimate.

The quantity of water vapour in samples 1 to 3 has been determined by integration of the relevant signals in the NMR spectra. As a reference for integration we added 1,1,2,2-tetrachloroethane (C2H2Cl4, 0.2% v:v) to the toluene-d8 in the outer sample tube. We calibrated the integral of the C2H2Cl4 reference peak (near ~ 6 ppm) with respect to the number density of sample 4 (saturated vapour). The pressures in samples 1 to 3 are then determined by scaling the peak intensities of the vapour peak with respect to sample 4. The error on the pressures is assumed to be ±10%. The active volume of the 5 mm inner tube has been estimated from documentation by the manufacturer (Wilmad) to be 0.4 cm3.

Translational diffusion and convection

Translational diffusion of water molecules in gas phase is very fast. Translational motion of water molecules between the active volume of the 1H NMR coil and the space outside the coil can affect inversion-recovery measurements of T1 relaxation. Indeed, molecules that carry inverted magnetization −Mz = −Mzeq within the active volume may be contaminated with molecules than come from areas outside the rf coil that carry magnetization in equilibrium Mzeq that has not been inverted. To ascertain the relevance of these effects on the time scale of the T1 measurement (max. 5 · T1 = 140 ms) we performed the following test. The inner tubes were only a few mm longer than the active region of the 1H coil of the 10 mm probe which is about 3 cm long. We measured R1 at the highest temperature T = 373 K (where the effects of diffusion are most pronounced) in two arrangements. First, we centered the inner tube with respect to the active region of the 1H coil. In this configuration, molecules can diffuse to and from the volumes above and below the active region. Secondly, we moved the inner tube up so that its bottom end was aligned with the lower end of the active region of the coil. In this manner, only molecules that cross the limit of the active region of the rf coil from above can influence the NMR signal. Any difference in R1 observed with these two configurations should be due to diffusion or convection. We found the R1 values to be identical within their errors, suggesting that contributions from diffusion can be neglected. Since we immersed the inner tube completely in a liquid with a controlled temperature, we assumed that there was no significant temperature gradient, so that convection due to differences in density should be negligible. Nevertheless, the experimental errors of the relaxation rates were doubled to take into account uncertainties stemming from diffusion and convection.

Additional Information

How to cite this article: Mammoli, D. et al. Collisional cross-section of water molecules in vapour studied by means of 1H relaxation in NMR. Sci. Rep. 6, 38492; doi: 10.1038/srep38492 (2016).

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