# Simulating the Self-Assembly of Particles Comprising the Haberdasher Dissection

My research centered around the haberdasher dissection, or a method of cutting up an equilateral triangle into four different parts that can then be rearranged into a perfect square, thus capable of perfectly tiling space in two different ways. By using the HOOMD-blue particle simulation toolkit in Python, I attempted to see if the four disparate haberdasher particles would self-assemble into either larger shape by performing Monte Carlo simulations. First, I precisely defined every angle and length of each of the four different particle shapes, as well as determined how to specify each shape in HOOMD. Then, I used the hard particle Monte Carlo simulation (hpmc) package in HOOMD to first verify the accuracy of HOOMD’s scale distribution function, which provides an extrapolation of pressure from the trial move acceptance ratio. By compressing the system at very low packing fractions, plotting βP vs. number density should yield a function equivalent to y=x, as per the ideal gas law, before diverging, which was verified.

After being confident about the values HOOMD provided, I performed simulations of systems composed of these particles in an attempt to find a crystal. By using HOOMD, I was able to compress a thermalized and randomized system to a certain packing fraction, and then allow the system to run for as many as 50 million timesteps in an attempt to find the crystal. Using OVITO/flowws, I was able to visualize the systems. For each trajectory file generated for the various packing fractions, I performed a calculation of the mean-squared displacement as the system equilibrated. For lower packing fractions, the mean-squared displacement was linear, as the system was entirely diffusive. The method of running for long stints of time at higher packing fractions didn’t yield a crystal, though sometimes did yield situations where three particles nearly formed the square formation of the haberdasher particles. We shifted to performing constant pressure simulations, or consistently ramping up the pressure of a system until reaching the target pressure equivalent to a certain packing fraction, using the Walnut computer cluster to perform many simulations at once at larger system sizes.

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