Nuclear induction lineshape: Non-Markovian diffusion with boundaries

The dynamics of viscoelastic fluids are governed by a memory function, which is critical yet computationally intensive to determine, particularly when diffusion is restricted by boundaries. We introduce a computational method that effectively captures the memory effects by analyzing the time-correlation function of the pressure tensor, an indicator of viscosity, through the analytic continuation of the Stokes-Einstein equation to the Laplace domain. This equation is integrated with molecular dynamics (MD) simulations to obtain necessary parameters. Our method computes NMR lineshapes by employing a generalized diffusion coefficient that incorporates the influences of temperature and confinement geometry. This approach establishes a direct link between the memory function and thermal transport parameters, enabling precise computation of NMR signals for non-Markovian fluids in confined geometries.

Methods
MD simulations were conducted to calculate the shear viscosity of gaseous xenon (Xe), as defined by Eq. (7). We utilized the Lennard-Jones (LJ) pair potential, expressed as U(r) = 4ϵ[(σ/r)12 − (σ/r)6], where interactions between xenon atoms were characterized by ϵ = 1.77 kJ/mol, the depth of the potential well, and σ = 4.1 Å, the distance at which the potential energy becomes zero. Our simulations of bulk fluid were conducted for isotropic diffusion by placing 2000 xenon atoms within a box defined by periodic boundary conditions. Throughout the simulations, we maintained a consistent particle count, volume, and temperature, adhering to the canonical ensemble (NVT ensemble). Each set of simulations was repeated for ten different random seeds for the initial positions and velocities of the particles to ensure robust statistical sampling and accuracy of the results. In the simulations of restricted diffusion (i.e., diffusion limited by the nanotube geometry), nanotubes of a fixed length and various diameters were employed, and the number of particles was adjusted to maintain a constant particle density.

For the simulations, we used xenon (Xe) particles. The interactions between the Xe particles and the cylindrical boundary were modeled using the Lennard-Jones potential, with parameters ϵ = 0.3 kJ/mol and σ = 4.295 Å, representing a silica tube. To determine the viscosity coefficient, we integrated all three components of the pressure tensor, referred to as Cαβ(τ). Although the Cxy component showed significantly higher values than the Cyz and Cxz components, given the tube’s orientation along the z-axis, we opted to integrate all three components together for a thorough analysis.