# Structure and Dynamics of Argon¶

Difficulty: Medium

## Summary¶

In this the structural and dynamical properties of liquid argon are studied. The radial distribution function (RDF) and structure factor, $$S(k)$$, are used to extract the mean or static structure, and the mean squared displacement (MSD), van Hove self correlation function, $$G_s(r,t)$$, see ref [Hove1954], and dynamic structure factor, $$S(k,ω)$$ are used to examine dynamical properties.

## Background¶

Liquid argon was one of the first liquids to be studied by molecular dynamics, ref [Rahman1964], on account of its alleged simplicity. As this exercise will show however, even simple liquids have complex properties. In this exercise we shall use some fairly typical ways of investigating the properties of liquids, beginning with probably the simplest: the radial distribution function. Next we shall look at diffusion by computing the mean squared displacement and seek to explain the mechanism of this using the van Hove self correlation function. Finally we shall calculate the dynamic structure factor and determine the velocity of sound in the liquid and from that estimate the elastic modulus.

## Instructions¶

Copy the Exercise7.tar.xz file into the $DL_POLY/data directory and unpack: cd$DL_POLY/data
wget https://ccp5.gitlab.io/dlpoly/Exercise7.tar.xz
tar -xvf Exercise7.tar.xz


Now go to the $DL_POLY/execute directory and start up the GUI: cd$DL_POLY/execute


Copy the contents of the subdirectory Exercise7 into the execute subdirectory. To do this using the GUI, select from the main menu Execute > Store/Fetch Data and in the Data Archive window type Exercise7 in the Fetch box and click Fetch. The CONFIG file contains a typical argon liquid configuration simulated at 85 K. Proceed as follows:

1. Take a look at the DL_POLY input data files CONTROL, FIELD and CONFIG (the latter may be visualised using the GUI). Make sure you understand what these files do. Run the simulation using the Run DL_POLY panel from the Execute menu of the GUI, see . The GUI submits a background job, so you will not be automatically told when it is complete. Monitor it as it runs using the STATUS button. Note the job, when complete, will deposit a large HISTORY file in your execute subdirectory.

2. To obtain the RDF, you should use the Structure submenu of the GUI Analysis menu. The simplest (and quickest) choice is to use the RDF-Read option and simply plot the contents of the RDFDAT file. Study the RDF and try to relate it to the force field specification in the FIELD file. You should understand the meaning of the features of the RDF, for instance: is the RDF liquid-like? The structure factor is obtained in a similar way using the $$S(k)$$ option and works by calculating a Fourier transform of the RDF data. This function is of course related to the experimental determination of the RDF by x-ray or neutron scattering. An interesting question is how much can you believe the results at low k vector. Can you think of a way to investigate this?

3. To obtain the MSD, use the MSD option from the Dynamics submenu of the GUI Analysis menu. Invoke the MSD program panel to run the calculation interactively (i.e. within the GUI) and calculate the MSD directly from the HISTORY file (set it to run over 2800 configurations). It make take a minute or two, so be patient. The MSD plot appears automatically when complete. From this estimate the diffusion constant. Note the MSD at short time is not linear, why is this?

4. The next issue is the nature of the diffusion. Two mechanisms suggest themselves: are the atoms are hopping from place to place or they are following a continuous random walk without ‘resting’ in any particular place. We all know it is the latter, but how? Use the self correlation function option inder the van Hove submenu to see how the self correlation varies with time. How does this show that the diffusion is a continuous random walk?

5. Finally use the $$S(k,ω)$$ option appearing under the van Hove submenu of the GUI Analysis menu (this invokes the $$S(k,ω)$$ panel. Run this for 2800 configurations and kmax=2 (no more!). (Once again this is a background job and will need to be monitored.) When complete, you can use the PLOT button to invoke the plotter panel and plot the various functions $$S(k,ω)$$ in which k is fixed and $$ω$$ is the ordinate. Locate the Brillouin peak (if any) appearing in some of these functions. From the (approximate) position of this peak determine the velocity of sound in the liquid. Next attempt to determine the elastic modulus associated with this using the Newtonian rule that the velocity of sound (c) is equal to the square root of the ratio of the elastic modulus ($$γ$$) to the density ($$ρ$$) i.e.

$c=\sqrt{\frac{γ}{ρ}}$

Check the result for more than one k vector. Do the results agree? What does the result mean in terms of the bulk properties of the liquid?

Hove1954
1. Van Hove, Correlations in space and time and born approximation scattering in systems of interacting particles, Phys. Rev., 95, p. 249, Jul 1954, doi: 10.1103/PhysRev.95.249