Microstructural geometry revealed by NMR lineshape analysis

We introduce a technique for extracting microstructural geometry from NMR lineshape analysis in porous materials at angstrom-scale resolution with the use of weak magnetic field gradients. Diverging from the generally held view of FID signals undergoing simple exponential decay, we show that a detailed analysis of the line shape can unravel structural geometry on much smaller scales than previously thought. While the original q-space PFG NMR relies on strong magnetic field gradients in order to achieve high spatial resolution, our current approach reaches comparable or higher resolution using much weaker gradients. As a model system, we simulated gas diffusion for xenon confined within carbon nanotubes over a range of temperatures and nanotube diameters in order to unveil manifestations of confinement in the diffusion behavior. We report a multiscale scheme that couples the above MD simulations with the generalized Langevin equation to estimate the transport properties of interest for this problem, such as diffusivity coefficients and NMR lineshapes, using the Green-Kubo correlation function to correctly evaluate time-dependent diffusion. Our results highlight how NMR methodologies can be adapted as effective means of structural investigation at very small scales when dealing with complicated geometries. This method is expected to find applications in materials science, catalysis, biomedicine, and other areas.

Methods
Methods for Data Collection and Processing

1. Molecular Dynamics (MD) Simulations:

The primary data for this work were generated using Molecular Dynamics (MD) simulations conducted with the open-source software package LAMMPS. The following steps outline the simulation setup and execution:

System Configuration:

Carbon nanotubes (CNTs) with radii ranging from 8 Å to 40 Å were constructed using a cylindrical lattice arrangement of carbon atoms. Xenon (Xe) gas particles were introduced into the CNTs at densities consistent with experimental conditions. The number of Xe atoms was adjusted to maintain uniform gas density across all CNT geometries.

Interaction Potentials:

The interactions between Xe atoms were modeled using a Lennard-Jones (LJ) potential with parameters \epsilon = 1.77 \, \text{kJ/mol} and \sigma = 4.1 \, \text{Å} . Similarly, Xe-CNT interactions were described using an LJ potential with \epsilon = 0.71 \, \text{kJ/mol} and \sigma = 3.7 \, \text{Å}. The carbon atoms forming the CNT walls were held fixed during the simulation.

Simulation Parameters:

Simulations were performed under periodic boundary conditions along the CNT axis to mimic infinite tube lengths. The time step was set to 1 femtosecond, and each simulation was run for 10 nanoseconds to capture sufficient particle dynamics. Initial conditions, including velocities, were generated using a Maxwell-Boltzmann distribution corresponding to the specified temperatures (240–400 K).

Data Collection:

The Green-Kubo autocorrelation function for the pressure tensor components was computed directly from the MD simulations. These correlation functions provided the foundation for calculating time-dependent viscosity and diffusion coefficients. Position and velocity data for Xe particles were recorded at regular intervals to enable detailed analysis of molecular trajectories.

2. Data Processing and Analysis:

Viscosity Calculations:

The time-dependent viscosity of Xe gas inside CNTs was calculated using the Green-Kubo formalism. The pressure tensor autocorrelation function, sampled during the MD simulations, was integrated over time to determine the shear viscosity \eta(t). Frequency-dependent viscosity \eta(s) was subsequently obtained via a numerical Laplace transform.

Diffusion Coefficient Calculations:

Using the generalized Stokes-Einstein equation in the Laplace domain, the diffusion coefficient D(s) was computed as a function of frequency. An inverse Laplace transform was then applied to recover the time-dependent diffusion coefficient D(t). This approach ensured an accurate representation of memory effects and time-dependent transport properties.

NMR Lineshape Predictions:

The time-domain diffusion coefficients were incorporated into the generalized Langevin equation (GLE) framework to predict NMR lineshapes. By combining MD-derived parameters with the GLE, we successfully modeled signal attenuation and lineshape broadening as functions of molecular transport and confinement effects.

Validation and Sensitivity Analysis:

The results were validated by comparing simulated data to theoretical expectations and trends reported in the literature. Sensitivity analyses were conducted to ensure the robustness of the results with respect to simulation parameters, such as tube radius, temperature, and gas density.

3. Data Accessibility:

The database includes:

Raw MD Simulation Data:

• Positions and velocities of all Xe particles recorded during simulations.

• Pressure tensor components used to calculate Green-Kubo autocorrelation functions.

Processed Data Files:

• Time-dependent viscosity and diffusion coefficients derived from the Green-Kubo approach.

• Frequency- and time-dependent diffusion data used for NMR lineshape modeling.

Analysis Scripts:

• Python scripts for computing autocorrelation functions, viscosity, and diffusion coefficients.

• Numerical Laplace and inverse Laplace transform scripts for processing frequency-domain data.

• GLE-based NMR lineshape modeling code.