.. include:: ../additional/links.rst
.. _lennard-jones-label:

Simple fluid
============

Here, NMR relaxation rates are measured from a Lennard-Jones fluid, and compared
to the results from Ref. :cite:`grivetNMRRelaxationParameters2005`.

System
------

.. image:: lennard-jones-fluids/lj-dark.png
    :class: only-dark
    :alt: LJ fluid simulated with LAMMPS - Dipolar NMR relaxation time calculation
    :width: 250
    :align: right

.. image:: lennard-jones-fluids/lj-light.png
    :class: only-light
    :alt: LJ fluid simulated with LAMMPS - Dipolar NMR relaxation time calculation
    :width: 250
    :align: right

The system consists of 16,000 particles interacting via the classical 
Lennard-Jones (LJ) 12-6 potential and simulated using
LAMMPS :cite:`thompsonLAMMPSFlexibleSimulation2022`.
Each particle has a mass 
:math:`m = 1\,\text{g/mol}`, and LJ parameters 
:math:`\sigma = 3\,\text{Å}` and :math:`\epsilon = 0.1\,\text{kcal/mol}`.
All reduced parameters were taken to match the study by Grivet 
:cite:`grivetNMRRelaxationParameters2005`. Specifically,
a cutoff of :math:`4 \sigma` was used for the LJ interactions, the 
simulation box has a volume of :math:`(26.9~\sigma)^3` to match 
the reduced density of :math:`\rho^* = 0.84`.
Production runs were performed in the microcanonical (NVE) ensemble, 
during which 10,000 timesteps were executed, equivalent to 50 times 
the reference time :math:`\sqrt{m \sigma^2/\epsilon}`. Configurations 
were recorded every 10 timesteps. A timestep of 
:math:`0.005\,\sqrt{m \sigma^2/\epsilon}` was used.
The imposed temperatures ranged from :math:`T = 30` to 
:math:`160\,\text{K}`, corresponding to reduced temperatures from 
:math:`T^* = 0.8` to :math:`3.0`.

All LAMMPS input scripts and analysis scripts written in Python are provided
on GitHub; see |dataset-LJ-fluid|.

Reproducing Grivet's results for a simple fluid
-----------------------------------------------

The correlation function :math:`G_{ij}^{(0)}` was first extracted for all
temperatures. For the two extreme values of :math:`T`, namely
:math:`T = 50` and :math:`140\,\text{K}`, the functions
:math:`G_{ij}^{(0)}` are compared with the correlation functions reported by
Grivet :cite:`grivetNMRRelaxationParameters2005`. Our results show good
agreement with those of Grivet, with however some differences observed at the lowest
temperature. The results show that :math:`G_{ij}^{(0)}` shifts to longer times as the
temperature decreases, as expected from the slowing down of molecular motion.

.. image:: lennard-jones-fluids/nmr-correlation-functions-dm.png
    :class: only-dark
    :alt: Correlation functions of a LJ fluid simulated with LAMMPS

.. image:: lennard-jones-fluids/nmr-correlation-functions.png
    :class: only-light
    :alt: Correlation functions of a LJ fluid simulated with LAMMPS

.. container:: figurelegend

    Figure: A) Correlation function :math:`G_{ij}^{(0)}` as extracted from the LJ
    fluid simulation for all temperatures. B) :math:`G_{ij}^{(0)}` for two
    different temperatures compared with the data from Grivet
    :cite:`grivetNMRRelaxationParameters2005` (gray symbols). The dashed
    line shows :math:`t^{-3/2}`.

The NMR relaxation rate spectra :math:`R_1` and :math:`R_2` were extracted for
all temperatures using ``NMRDforMD``. For all temperatures, the spectra show
a decrease with increasing frequency :math:`f`.

.. image:: lennard-jones-fluids/nmr-relaxation-rates-spectra-dm.png
    :class: only-dark
    :alt: NMR relaxation rate of a LJ fluid simulated with LAMMPS

.. image:: lennard-jones-fluids/nmr-relaxation-rates-spectra.png
    :class: only-light
    :alt: NMR relaxation rate of a LJ fluid simulated with LAMMPS

.. container:: figurelegend

    Figure: NMR relaxation rates :math:`R_1` (A) and :math:`R_2` (B)
    as a function of the frequency :math:`f`.

The NMR relaxation rates :math:`R_1` and :math:`R_2` were also extracted for
all temperatures, at a frequency :math:`f_0 = 150\,\text{GHz}` (or 0.07 in
dimensionless units). :math:`R_1(f_0)` shows a maximum at intermediate
temperature and a minimum at the lowest temperature.
:math:`R_1(f_0)` decreases with increasing temperature. Our results show good
agreement with the data from Grivet :cite:`grivetNMRRelaxationParameters2005`.

.. image:: lennard-jones-fluids/nmr-relaxation-rates-at-target-dm.png
    :class: only-dark
    :alt: NMR relaxation rate of a LJ fluid simulated with LAMMPS
    :width: 50%

.. image:: lennard-jones-fluids/nmr-relaxation-rates-at-target.png
    :class: only-light
    :alt: NMR relaxation rate of a LJ fluid simulated with LAMMPS
    :width: 50%

.. container:: figurelegend

    Figure: NMR relaxation rates :math:`R_1` (A) and :math:`R_2` (B) at
    a frequency 0.07 (dimensionless), or :math:`f_0 = 151\,\text{GHz}`.
    The data from Grivet :cite:`grivetNMRRelaxationParameters2005` are shown
    with gray symbols.