Your piece of phase space

The calculation to be carried out is assigned by Username below. Each student is asked to carry out simulations at three values of the reduced temperature, T* at the indicated reduced density, r*.

Your individual assignment is listed here

You should carry out simulations at three or more values of the reduced temperature T*. These values should include a value of 0.6, 1.0, and 1.4. You may wish to try one or two more (e.g. 0.8 and 1.2) to see the trends. Keep in mind that these temperatures span the boiling point of argon at the densities studied.

Organization of Analysis

The creation of a randomized configuration

            The initial configuration you will be given is a face-centered cubic (fcc) lattice of argon atoms.  The configuration of the atoms in a liquid is random.  The first part of the assignment is to use either MD or MC to obtain a randomized configuration.  Hint: in both MC and MD configuration space is sampled much more efficiently at a high temperature than at a low temperature.

 

Monte Carlo Simulation

Given your points in phase space answer the following questions.

  1. Perform an initial run to equilibrate the system. Use a cutoff radius of 5 (reduced units).  Plot the energy vs. the number of blocks. How many MC moves does it take to equilibrate the system (approximately)?
  2. Ensemble averages: Once the system is equilibrated, collect 120 blocks of 5,000 MC moves. Calculate the energy (J/mol) and the pressure (kPa) of the system. You should perform the ensemble averages using only the equilibrated configurations.
  3. Fluctuations: The fluctuations of the energy in the canonical ensemble are directly related with the specific heat by the following equation:

                                

         

Calculate the specific heat of the system.

  1. Plot the Ar-Ar radial distribution function.

 

Molecular Dynamics Simulation

  1. What are the required unit cell parameters?  Once you have determined the unit cell parameters, use these values to modify the car file for a molecular dynamics run using DISCOVER.
  2. Set up the input file to allow for equilibration for 20 ps followed by production dynamics of 20 ps.  Note that the protocols in the DISCOVER program automatically perform the equilibration.  However, this phase must be explicitly included in the input file.
  3. Read the output history file into DECIPHER and perform averages over the trajectory to determine the energy and pressure as a function of time.
  4. Determine the fluctuations in the energy as a function of time as use this to calculate the specific heat of the system.
  5. Determine the radial distribution function.
  6. Compare the values that you have obtained by the two different methods in terms of the magnitude of the fluctuations, the magnitude of the energy, the agreement of the radial distribution function with the published value.  Note: you will need to perform appropriate unit conversions for the comparison.
  7. Determine the diffusion coefficient of argon.  Use the velocity autocorrelation function method and the mean square displacement.

 

Comparison of methods

      Write a one page summary of your results.  On separate pages show a comparison of the radial distribution function plots and the plots of the energy and pressure as a function of time.  In your overall summary, discuss the advantages of each method (MC or MD) for determining thermodynamic and dynamical properties of the liquid.  Use a cubic equation of state to calculate the pressure at the three densities. Are the simulation results in good agreement with the values predicted by the equation of state? Comment on the differences.