Cluster Formation in Charged Particles
David Hassan
CHE210D Spring 2009 Final Project
Summary
In this project I studied how the formation of Lennard-Jones clusters was effected by electrostatic repulsion. In particular, I wanted to test if the so called “magic cluster numbers” were affected by the presence of a Coulomb interaction. The problem was studied using a MD simulation, and the amount of charge present was varied over the simulations.
Background
Cluster formation is important to the study of nano-scale systems. In particular, many nano-systems are metallic, and so understanding the effect of electric charge to these systems can be important.
Simulation methods
I implemented a reduced-units Lennard-Jones potential as the
dominant interaction in the system. To this was added a small biasing potential
to the center of the coordinate system, as well as the Coulomb interaction.
Very small values of charge were used in order that the L-J interactions remain
the dominant interaction type. The pair-wise potential energy was of the form: 
For the “macroscopic energy” the data was fit to a curve of
the form 
.
The energy was minimized using a conjugate-gradient minimization algorithm. 20000 minimizations were performed for each particle number from 2-25, and for values of charge from 0-0.05. These values were chosen because cluster formation stops at approximately q = .1. The pair-wise potential was computed using FORTRAN code and imported into the simulation written in Python. For N = 13, a histogram of energies was computed.
Results and interpretation
As the above graphs show, for the levels of charge simulated, very little changed when the Coulomb interaction was turned on. In particular, the minimum energy increased only very slightly as charge was turned up, and there was no change to the U minus U-macroscopic curve at all. The presence of charge did seem to change the energy landscape significantly, however. For example, for q = .05 it seems that it was more difficult for the system to find the global minimum, as is evidenced by the larger disparity between the minimum and mean energies. Also, for N = 25, which was the largest system size investigated, there is a large difference between minimum and mean energies for the smallest amounts of charge present, indicating a large number of deeper local minima in the energy function.
Very small amounts of charge seem not to effect the formation of L-J clusters in appreciable ways. The dominant effect seems to be a roughening of the energy landscape in certain cases, making the global minimum more difficult to access.
Obviously, much more of the charge parameter space should be simulated. Also, an attempt to quantify at what amount of charge cluster formation stops happening altogether would be useful. Further, many more simulation runs with more detailed histograms could help chart how the energy landscape is affected by turning on charged interactions.
Movie
This is a video of a single energy minimization for N = 19 particles with a charge of q = .03.
Source code