Nuclear Induction Lineshape: Non-Markovian Diffusion with Boundaries

Abstract

The dynamics of viscoelastic fluids are governed by a memory function, essential yet challenging to compute, especially when diffusion faces boundary restrictions. We propose a computational method that captures memory effects by analyzing the time-correlation function of the pressure tensor, a viscosity indicator, through the Stokes-Einstein equation's analytic continuation into the Laplace domain. We integrate this equation with molecular dynamics (MD) simulations to derive necessary parameters. Our approach computes NMR lineshapes using a generalized diffusion coefficient, accounting for temperature and confinement geometry. This method directly links the memory function with thermal transport parameters, facilitating accurate NMR signal computation for non-Markovian fluids in confined geometries.

Figures (9)

• Figure 1: This figure illustrates the timescales pertinent to fluid dynamics analysis via MD simulations. The interactions between fluid particles are captured with a time-step in the femtosecond (fs) range. Particle positions and velocities are sampled at 10-fs intervals, with correlation functions tracked until they decay, typically reaching times well into the hundreds of picoseconds (ps). For the decay of the NMR signal due to diffusion to be calculated, particle diffusion must be monitored over the course of hundreds of milliseconds (ms).
• Figure 2: The autocorrelation function of the pressure tensor, as specified in Eq. (\ref{['eq:etaGK']}), was calculated via MD simulation and plotted as a function of time for a series of temperatures. The inset provides a semi-logarithmic view of the initial data points, illustrating that the autocorrelation function follows a clear exponential decay with a consistent rate across different temperatures. The viscosity coefficient, obtained from the integral of the autocorrelation function values, correlates directly with the amplitude of the initial point on the decay curve.
• Figure 3: The blue data points represent the viscosity coefficient obtained by numerically integrating the pressure correlation function at various temperatures. The consistent decay rate in the correlation function suggests that the amplitude of the initial data point provides an accurate and reliable approximation of the viscosity coefficient, as indicated by the orange data points.
• Figure 4: Linewidth of the NMR signal calculated using Eq. (\ref{['eq:lw0freq']}) and derived from the simulated viscosity coefficient. A line-narrowing effect is observed. The inset shows a comparison with experimental results.
• Figure 5: Viscosity coefficient evaluated for cylinders of uniform length but varying diameters. This coefficient was ascertained by integrating all three components of the pressure tensor cross-correlation function. Notably, the temperature trend observed closely parallels that of bulk gas diffusion. Furthermore, the viscosity coefficient demonstrates a clear sensitivity to the diameter of the cylindrical tube.