### Abstract:

This project calculates the melting temperature as a function of pressure for a system defined by the Lennard-Jones 12-6 potential using molecular dynamic simulations. In a single atomistic simulation, the initial conditions are manipulated such that both a liquid and solid phase exist when thermal equilibrium is reached. Phase coexistence indicates that these simulations must be at the melting point, which means the average pressure and temperature provides a data point for the coexistence line between the liquid and solid phase. For this project, over 100 coexistence point simulations were created, and the interpolation of the average pressure and temperature from these simulations agree to within 1% of literature values that were obtained with a number of different simulation techniques.
The molecular dynamic simulations were performed using the software package LAMMPS. Several additional algorithms and codes were needed for this project. Two primary tools that were developed include a liquid/solid phase quantifying algorithm that determined if an atom was in a liquid or solid region, and a re-sizing method that could be performed automatically by the computer to create many coexistence point simulations.
The liquid/solid phase quantifying algorithm allowed the computer to cull any simulations which did not occur at a coexistence point; later in the project, this algorithm also allowed the density of the liquid and solid within a coexistence point simulation to be calculated using a modified Monte Carlo integration algorithm.
The re-sizing method simply changed the volume of a previous coexistence point simulation while simultaneously changing the temperature to prevent the simulation from collapsing to a liquid or solid. It was found that a simulation requires about 200k iterations to reach equilibrium after the re-sizing method was implemented. Despite the long time necessary to reach equilibrium, the re-sizing method successfully created a coexistence point simulation every time it was tested, which is to say, the re-sizing method successfully created over 100 coexistence point simulations.

### Description:

ABSTRACT 7
I. INTRODUCTION 8
The Purpose 8
Molecular Dynamics Using Newtonian Mechanics 8
Tools Needed for Project 9
Coexistence Point Simulations 9
LAMMPS 9
Sub Goal: Automated Creation of Coexistence Point Simulations 10
Lennard-Jones 12-6 potential 10
Previous Results From Mastny et al. 11
II. METHODS 12
A. Overview 12
B. Setup Phase 13
Tedious Method 13
Re-Sizing Method 13
Reasonable Initial Variables 14
Cutoff Radius 14
Reasonable Δt 14
C. Convergence Phase 15
Psuedo-Convergence 15
Equilibrium 15
D. Data Collection Phase 16
Average Pressure and Temperature 16
Solid and Liquid Density 16
Phase Quantifying Algorithm 16
Modified Monte Carlo Integration 18
Verifying Modified Monte Carlo Integration Algorithm 20
Culling Collapsed Simulations 21
III. RESULTS 22
A. Setup 22
B. Equilibrium 23
C. Convergence 25
1. convergence with respect to cutoff radius 25
2. convergence with respect to Δt 26
D. Final Results 27
1. Pressure Vs Temperature 27
2. Density Vs Temperature 29
IV. DISCUSSION 30
A. Setup Phase 30
B. Convergence Phase 30
C. Data Collection and Final Results 31
V. CONCLUSION 32
VI. FUTURE WORK 33
VII. ACKNOWLEDGMENTS 34
VIII. BIBLIOGRAPHY 35
IX. APPENDIXES 36
Appendix A: verification of the modified Monte Carlo integration algorithm 36
Appendix B: a demonstration of the smoothing algorithm 38
Appendix C: Final results in table form 39