Theory and modeling of molecular modes in the NMR relaxation of fluids.

Traditional theories of the nuclear magnetic resonance (NMR) autocorrelation function for intra-molecular dipole pairs assume a single-exponential decay, yet the calculated autocorrelation of realistic systems displays a rich, multi-exponential behavior, resulting in anomalous NMR relaxation dispersion (i.e., frequency dependence). We develop an approach to model and interpret the multi-exponential intra-molecular autocorrelation using simple, physical models within a rigorous statistical mechanical development that encompasses both rotational diffusion and translational diffusion in the same framework. We recast the problem of evaluating the autocorrelation in terms of averaging over a diffusion propagator whose evolution is described by a Fokker-Planck equation. The time-independent part admits an eigenfunction expansion, allowing us to write the propagator as a sum over modes. Each mode has a spatial part that depends on the specified eigenfunction and a temporal part that depends on the corresponding eigenvalue (i.e., correlation time) with a simple, exponential decay. The spatial part is a probability distribution of the dipole pair, analogous to the stationary states of a quantum harmonic oscillator. Drawing inspiration from the idea of inherent structures in liquids, we interpret each of the spatial contributions as a specific molecular mode. These modes can be used to model and predict the NMR dipole-dipole relaxation dispersion of fluids by incorporating phenomena on the molecular level. We validate our statistical mechanical description of the distribution in molecular modes with molecular dynamics simulations interpreted without any relaxation models or adjustable parameters: the most important poles in the Padé-Laplace transform of the simulated autocorrelation agree with the eigenvalues predicted by the theory.

Effect of Nanoconfinement on NMR Relaxation of Heptane in Kerogen from Molecular Simulations and Measurements.

Kerogen-rich shale reservoirs will play a key role during the energy transition, yet the effects of nanoconfinement on the NMR relaxation of hydrocarbons in kerogen are poorly understood. We use atomistic MD simulations to investigate the effects of nanoconfinement on the 1H NMR relaxation times T1 and T2 of heptane in kerogen. In the case of T1, we discover the important role of confinement in reducing T1 by ∼3 orders of magnitude from that of bulk heptane, in agreement with measurements of heptane dissolved in kerogen from the Kimmeridge Shale, without any models or free parameters. In the case of T2, we discover that confinement breaks spatial isotropy and gives rise to residual dipolar coupling which reduces T2 by ∼5 orders of magnitude from the value for bulk heptane. We use the simulated T2 to calibrate the surface relaxivity and thence predict the pore-size distribution of the organic nanopores in kerogen, without additional experimental data.

Thermal and concentration effects on 1H NMR relaxation of Gd3+-aqua using MD simulations and measurements.

Gadolinium-based contrast agents are key in clinical MRI for enhancing the longitudinal NMR relativity (r1) of hydrogen nuclei (1H) in water and improving the contrast among different tissues. The importance of MRI in clinical practice cannot be gainsaid, yet the interpretation of MRI relies on models with severe assumptions, reflecting a poor understanding of the molecular-scale relaxation processes. In a step towards building a clearer understanding of the relaxation processes, here we investigate thermal and concentration effects on r1 of the Gd3+-aqua complex using both semi-classical molecular dynamics (MD) simulations and measurements. We follow the MD simulation approach recently introduced by [Singer et al., Phys. Chem. Chem. Phys., 2021, 23, 20974], in which no NMR relaxation model or free-parameter is assumed to predict r1, thereby bringing new insights into the physics of r1 on a molecular scale. We expand the autocorrelation function G(t) in terms of molecular modes and determine the thermal activation energies of the two largest modes, both of which are consistent with the range of literature values for rotational diffusion. We also determine the activation energies for translational diffusion and low-field electron-spin relaxation, both of which are consistent with the literature. Furthermore, we validate the MD simulations at human body temperature and concentrations of the paramagnetic ion used in clinical MRI, and we quantify the uncertainties in both simulations and measurements.

Freezing of the Lattice in the Kagome Lattice Heisenberg Antiferromagnet Zn-Barlowite ZnCu_{3}(OD)_{6}FBr.

We use ^{79}Br nuclear quadrupole resonance (NQR) to demonstrate that ultraslow lattice dynamics set in below the temperature scale set by the Cu-Cu superexchange interaction J (≃160 K) in the kagome lattice Heisenberg antiferromagnet Zn-barlowite. The lattice completely freezes below 50 K, and ^{79}Br NQR line shapes become twice broader due to increased lattice distortions. Moreover, the frozen lattice exhibits an oscillatory component in the transverse spin echo decay, a typical signature of pairing of nuclear spins by indirect nuclear spin-spin interaction. This indicates that some Br sites form structural dimers via a pair of kagome Cu sites prior to the gradual emergence of spin singlets below ∼30 K. Our findings underscore the significant roles played by subtle structural distortions in determining the nature of the disordered magnetic ground state of the kagome lattice.

Predicting 1H NMR relaxation in Gd3+-aqua using molecular dynamics simulations.

Atomistic molecular dynamics simulations are used to predict 1H NMR T1 relaxation of water from paramagnetic Gd3+ ions in solution at 25 °C. Simulations of the T1 relaxivity dispersion function r1 computed from the Gd3+-1H dipole-dipole autocorrelation function agree within ≃8% of measurements in the range f0 ≃ 5 ↔ 500 MHz, without any adjustable parameters in the interpretation of the simulations, and without any relaxation models. The simulation results are discussed in the context of the Solomon-Bloembergen-Morgan inner-sphere relaxation model, and the Hwang-Freed outer-sphere relaxation model. Below f0 ≲ 5 MHz, the simulation overestimates r1 compared to measurements, which is used to estimate the zero-field electron-spin relaxation time. The simulations show potential for predicting r1 at high frequencies in chelated Gd3+ contrast-agents used for clinical MRI.

NMR 1H-1H Dipole Relaxation in Fluids: Relaxation of Individual 1H-1H Pairs versus Relaxation of Molecular Modes.

The intramolecular 1H NMR dipole-dipole relaxation of molecular fluids has traditionally been interpreted within the Bloembergen-Purcell-Pound (BPP) theory of NMR intramolecular relaxation. The BPP theory draws upon Debye's theory for describing the rotational diffusion of the 1H-1H pair and predicts a monoexponential decay of the 1H-1H dipole-dipole autocorrelation function between distinct spin pairs. Using molecular dynamics (MD) simulations, we show that for both n-heptane and water this is not the case. In particular, the autocorrelation function of individual 1H-1H intramolecular pairs itself evinces a rich stretched-exponential behavior, implying a distribution in rotational correlation times. However, for the high-symmetry molecule neopentane, the individual 1H-1H intramolecular pairs do conform to the BPP description, suggesting an important role of molecular symmetry in aiding agreement with the BPP model. The intermolecular autocorrelation functions for n-heptane, water, and neopentane also do not admit a monoexponential behavior of individual 1H-1H intermolecular pairs at distinct initial separations. We suggest expanding the autocorrelation function in terms of modes, provisionally termed molecular modes, that do have an exponential relaxation behavior. With care, the resulting Fredholm integral equation of the first kind can be inverted to recover the probability distribution of the molecular modes. The advantages and limitations of this approach are noted.

Elucidating the 1H NMR Relaxation Mechanism in Polydisperse Polymers and Bitumen Using Measurements, MD Simulations, and Models.

The mechanism behind the 1H nuclear magnetic resonance (NMR) frequency dependence of T1 and the viscosity dependence of T2 for polydisperse polymers and bitumen remains elusive. We elucidate the matter through NMR relaxation measurements of polydisperse polymers over an extended range of frequencies (f0 = 0.01-400 MHz) and viscosities (η = 385-102 000 cP) using T1 and T2 in static fields, T1 field-cycling relaxometry, and T1ρ in the rotating frame. We account for the anomalous behavior of the log-mean relaxation times T1LM ∝ f0 and T2LM ∝ (η/T)-1/2 with a phenomenological model of 1H-1H dipole-dipole relaxation, which includes a distribution in molecular correlation times and internal motions of the nonrigid polymer branches. We show that the model also accounts for the anomalous T1LM and T2LM in previously reported bitumen measurements. We find that molecular dynamics (MD) simulations of the T1 ∝ f0 dispersion and T2 of similar polymers simulated over a range of viscosities (η = 1-1000 cP) are in good agreement with measurements and the model. The T1 ∝ f0 dispersion at high viscosities agrees with previously reported MD simulations of heptane confined in a polymer matrix, which suggests a common NMR relaxation mechanism between viscous polydisperse fluids and fluids under nanoconfinement, without the need to invoke paramagnetism.

Critical Role of Confinement in the NMR Surface Relaxation and Diffusion of n-Heptane in a Polymer Matrix Revealed by MD Simulations.

The mechanism behind the NMR surface-relaxation times (T1S,2S) and the large T1S/T2S ratio of light hydrocarbons confined in the nanopores of kerogen remains poorly understood and consequently has engendered much debate. Toward bringing a molecular-scale resolution to this problem, we present molecular dynamics (MD) simulations of 1H NMR relaxation and diffusion of n-heptane in a polymer matrix. The high-viscosity polymer is a model for kerogen and bitumen that provides an organic "surface" for heptane. Diffusion of n-heptane shows a power-law dependence on the concentration of n-heptane (ϕC7) in the polymer matrix, consistent with Archie's model of tortuosity. We calculate the autocorrelation function G(t) for 1H-1H dipole-dipole interactions of n-heptane in the polymer matrix and use this to generate the NMR frequency (f0) dependence of T1S,2S as a function of ϕC7. We find that increasing molecular confinement increases the correlation time, which decreases the surface-relaxation times for n-heptane in the polymer matrix. For weak confinement (ϕC7 > 50 vol %), we find that T1S/T2S ≃ 1. Under strong confinement (ϕC7 ≲ 50 vol %), we find that T1S/T2S ≳ 4 increases with decreasing ϕC7 and that the dispersion relation T1S ∝ f0 is consistent with previously reported measurements of polydisperse polymers and bitumen. Such frequency dependence in bitumen has been previously attributed to paramagnetism; instead, our studies suggests that 1H-1H dipole-dipole interactions enhanced by organic nanopore confinement dominate the NMR response in saturated organic-rich shales.

Molecular dynamics simulations of NMR relaxation and diffusion of bulk hydrocarbons and water.

Molecular dynamics (MD) simulations are used to investigate 1H nuclear magnetic resonance (NMR) relaxation and diffusion of bulk n-C5H12 to n-C17H36 hydrocarbons and bulk water. The MD simulations of the 1H NMR relaxation times T1,2 in the fast motion regime where T1=T2 agree with measured (de-oxygenated) T2 data at ambient conditions, without any adjustable parameters in the interpretation of the simulation data. Likewise, the translational diffusion DT coefficients calculated using simulation configurations agree with measured diffusion data at ambient conditions. The agreement between the predicted and experimentally measured NMR relaxation times and diffusion coefficient also validate the forcefields used in the simulation. The molecular simulations naturally separate intramolecular from intermolecular dipole-dipole interactions helping bring new insight into the two NMR relaxation mechanisms as a function of molecular chain-length (i.e. carbon number). Comparison of the MD simulation results of the two relaxation mechanisms with traditional hard-sphere models used in interpreting NMR data reveals important limitations in the latter. With increasing chain length, there is substantial deviation in the molecular size inferred on the basis of the radius of gyration from simulation and the fitted hard-sphere radii required to rationalize the relaxation times. This deviation is characteristic of the local nature of the NMR measurement, one that is well-captured by molecular simulations.

A more accurate estimate of T2 distribution from direct analysis of NMR measurements.

In the past decade, low-field NMR relaxation and diffusion measurements in grossly inhomogeneous fields have been used to characterize pore size distribution of porous media. Estimation of these distributions from the measured magnetization data plays a central role in the inference of insitu petro-physical and fluid properties such as porosity, permeability, and hydrocarbon viscosity. In general, inversion of the relaxation and/or diffusion distribution from NMR data is a non-unique and ill-conditioned problem. It is often solved in the literature by finding the smoothest relaxation distribution that fits the measured data by use of regularization. In this paper, estimation of these distributions is further constrained by linear functionals of the measurement that can be directly estimated from the measured data. These linear functionals include Mellin, Fourier-Mellin, and exponential Haar transforms that provide moments, porosity, and tapered areas of the distribution, respectively. The addition of these linear constraints provides more accurate estimates of the distribution in terms of a reduction in bias and variance in the estimates. The resulting distribution is also more stable in that it is less sensitive to regularization. Benchmarking of this algorithm on simulated data sets shows a reduction of artefacts often seen in the distributions and, in some cases, there is an increase of resolution in the features of the T(2) distribution. This algorithm can be applied to data obtained from a variety of pulse sequences including CPMG, inversion and saturation recovery and diffusion editing, as well as pulse sequences often deployed down-hole.

Low magnetic fields for flow propagators in permeable rocks.

Pulsed field gradient NMR flow propagators for water flow in Bentheimer sandstone are measured at low fields (1H resonance 2 MHz), using both unipolar and bipolar variants of the pulsed gradient method. We compare with propagators measured at high fields (1H resonance 85 MHz). We show that (i) measured flow propagators appear to be equivalent, in this rock, and (ii) the lower signal to noise ratio at low fields is not a serious limitation. By comparing different pulse sequences, we study the effects of the internal gradients on the propagator measurement at 2 MHz, which for certain rocks may persist even at low fields.