DOI: 10.5061/dryad.m905qfv7n
title: Nuclear induction lineshape: Non-Markovian diffusion with boundaries

description: 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.
description: 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
 <em>U</em>(<em>r</em>) =
 4<em>ϵ</em>[(<em>σ</em>/<em>r</em>)<sup>12</sup> − (<em>σ</em>/<em>r</em>)<sup>6</sup>], where interactions between xenon atoms were characterized by <em>ϵ</em> = 1.77 kJ/mol, the depth of the potential well, and <em>σ</em> = 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 (<em>NVT</em> 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 <em>ϵ</em> = 0.3 kJ/mol and <em>σ</em> = 4.295 Å, representing a silica tube. To determine the viscosity coefficient, we integrated all three components of the pressure tensor, referred to as <em>C</em><sub><em>αβ</em></sub>(<em>τ</em>). Although the <em>C</em><sub><em>xy</em></sub> component showed significantly higher values than the <em>C</em><sub><em>yz</em></sub> and <em>C</em><sub><em>xz</em></sub> components, given the tube’s orientation along the <em>z-axis</em>, we opted to integrate all three components together for a thorough analysis.
description: # Nuclear Induction Lineshape: Non-Markovian Diffusion with Boundaries
 The simulation was
 executed on the Hoffman2 cluster at UCLA. The "submit_job.sh"
 script was used for running a batch of simulations. A sample of LAMMPS
 code for viscosity evaluation can be found in each folder,
 "in.visc_Bulk" and "in.visc_SiO2" for bulk and
 restricted simulations, respectively. For data analysis in Jupyter Lab, we
 provide a Python code in the "Viscosity Analysis.ipynb"
 notebook. ### Description of the data and file structure The various
 folders are marked to indicate different boundary conditions. The
 "Xegas.zip" folder contains the dataset for simulating Xe gas
 particles in bulk. Folders named A11-A75 hold datasets for cylindrical
 boundary conditions, with the tube radii in Angstroms represented by two
 digits in each folder's name. Each folder contains a file with the
 LAMMPS code named "in.visc_SiO2", a file titled
 "submit_job.sh" for batch submission of simulations to the HPC
 facility, several log files documenting code performance, and 90 data
 files. The data file names have a specific structure: the first part
 indicates the simulation phase and the monitored parameter, the second
 part denotes the temperature, and the third part specifies the random seed
 used for the simulation. Each simulation is conducted with 10 different
 random seeds across 9 temperatures: 240, 260, 280, 300, 320, 340, 360,
 380, and 400 K. For instance, a file named "viscg_280_1.dat"
 represents the results of a viscosity simulation for Xe gas particles at
 280 K using the first set of random distributions. These data are
 generated using a "ave/correlate" fix within the LAMMPS code.
 The resulting output includes rows displaying the timestep in the first
 column, followed by the simulated parameters in subsequent rows.
 Specifically, the components of the pressure tensor correlations, denoted
 as "v_pxy," "v_pxz," and "v_pyz," are
 presented in the last three columns. The Jupyter notebook contains a
 function demonstrating how to import and work with such a file. ###
 Code/Software The Lammps code has the following sections: 1. Variable
 assignments: This section involves defining crucial variables, such as
 temperature (T), simulation dimensions (L), the number of particles
 (npart), and sampling intervals. 2. Unit conversion to SI: Here, unit
 conversions are encoded to ensure consistency with the International
 System of Units (SI). 3. Molecular Dynamics simulation setup: This part
 involves configuring the molecular dynamics simulation by specifying
 boundary conditions, defining the simulation region, setting coupling
 constants, and determining the time steps. 4. NVT canonical ensemble fix:
 A fix is defined to maintain the NVT canonical ensemble during the
 simulation. 5. Equilibration (Prerun): Prior to data collection, a prerun
 of 1e6 steps is executed to allow the system to reach an equilibrium
 state. 6. Correlation term computation: This section of the code focuses
 on calculating the correlation terms for the pressure tensor based on the
 Kubo formulation. The necessary summations are performed. 7. Main
 simulation run: The code proceeds to run for the desired number of steps,
 during which preliminary viscosity coefficient data is collected and
 printed for analysis. The simulation was performed for various
 temperatures in the range of 200-400 K for xenon particles in gas phase
 for both bulk and restricted diffusion.