Liquid-Solid Phase Transition in Lennard-Jones Particles
Michael A. Lovette
CHE210D Spring 2009 Final Project
Summary
The liquid-solid transition in Lennard-Jones particles was investigated through a series of Monte Carlo simulations in the liquid-solid coexistence region. These simulations were stitched together using WHAM and the phase boundary between the liquid-solid coexistence region and the solid region of the phase diagram was observed.
Background
In this study Lennard-Jones 6-12 particles were simulated at a density (all quantities are dimensionless) of r = 1.0 and over a range of temperatures from T = 0.65 – 1.30. This range has previously been determined by Hansen and Verlet (1969) to enclose the liquid-solid coexistence region (for constant r) of the r,T phase diagram. The goal of these simulations was to observe nucleation. While nucleation, which is a rare event (and likely unobservable given the methods used) was not observed, the transition from the coexistence region to the solid phase was demonstrated.
Simulation methods
The Lennard-Jones 6-12 interaction potential was used to conduct 25 independent Monte Carlo simulations at 14 temperatures incremented from T = 0.65 – 1.30, with N = 200 particles and density of r = 1. The simulations implemented cubic periodic boundary conditions and began from a minimum energy state arrived at using a conjugate gradient search method followed by 500 Monte Carlo Sweeps (MCS). Monte Carlo moves consisted solely of particle displacements and were accepted or rejected based on Metropolis criterion.
After equilibration the simulations proceeded for 19,500 MCS and the potential energies after each attempted move were recorded. The particle positions were also recorded every 50 MCS. At the end of each simulation a histogram of the potential energy was compiled which spanned the energy ranges found during the simulation and was constructed with evenly spaced bins (dU = Ne). The 25 individual histograms for each temperature were then compiled and the weighted histogram analysis method (WHAM) was used to stitch the resulting histograms for each temperature together, yielding S(U) and A(T). The radial distribution functions, g(r), were then constructed (using a histogram method with 50 bins) from the particle coordinates at the end of the simulations for selected runs at T = 0.65, 0.85, 1.05 and 1.25.
Results and interpretation
The simulation results are summarized in Figure 1. The solid/solid-liquid boundary in the phase diagram was determined to occur between T = 0.75 and 0.80 due to the discontinuity in the potential energy between simulations run above and below these temperatures (Figure 1a). Furthermore the negative slope in Figure 1c, indicating a phase transition, occurs at U ~ -5.8, which is in good agreement with the expected potential energy for the system between T = 0.75 and 0.80 (Figure 1a).
WHAM calculations using the method outlined in the course notes converged rapidly to within a low tolerance ~10-14. The resulting free energy increased linearly with temperature (Figure 1b).
Figure 1. Summary of results for Monte Carlo simulations of 200 Lennard-Jones particles over a range of T = 0.65 - 1.30 at r = 1.
The histograms from each simulation were either sharply peaked or exhibited a bimodal distribution and furthermore at all temperatures the combined histograms were. The radial distribution functions shown in Figure 1d, were determined from simulations that contained the most occurrences of the highest probability state for each given temperature and had mono-modal histograms.
The radial distribution functions were sharply peaked at r = 1 and demonstrated increasing order with decreased temperature. While it appears that g(r) for T = 0.65 and T = 0.85 are both consistent with ordered phases the difference in locations of the second and third coordination shells may indicate different crystalline structures (BCC,FCC,HCP).
While the goal of this study was to observe nucleation, it was unlikely to occur given the approach taken. A directed sampling technique, such as umbrella or transition path sampling, would have proven beneficial in that pursuit.
Movie
Monte Carlo simulation of Lennard-Jones 6-12 particles at r = 1.0 and T = 0.85. The movie reverse loops the simulation trajectory, going from disordered-ordered-disordered.
Source code
References
Hansen, J. & Verlet, L. Phase Transitions of the Lennard-Jones System Phys. Rev., American Physical Society, 1969, 184, 151.