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mmeineke |
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\section{Results} |
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\label{sec:results} |
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\subsection{Octopi} |
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The jamming limit coverage, $\theta_{J}$, of the of the off lattice continuum |
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simulation was found to be 0.531. This value is typical for simulations of |
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circles on a 2D plane.\cite{Evans93} Once the system is constrained by the |
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lattice, $\theta_{J}$ drops to 0.489, showing that the lattice has some small, |
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but mostly inconsequential, effect on the coverage at the jamming limit. |
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Upon examining the radial distribution curves, g(r), in Fig. \ref{fig:octgofr}, |
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it is observed that the lattice has no signifigant contribution to the |
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distribution other than to add noise to the correlation. Also, the features |
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of both curves are quite simple. An initial peak at twice the radius of the |
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octopi corresponding to the first population shell being the closest two |
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circles can approach without overlapping each other. The second peak at |
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four times the radius simply extends this shell out one more circle length. |
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\subsection{Tilted Umbrellas} |
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In the case of the tilted umbrellas, $\theta_{J}$ for the off lattice continuum |
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simulation was 0.924, and 0.918 for the simulation on the lattice. It can |
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be seen from this data that the lattice has even less effect on the |
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tilted umbrellas as on the octopi. Also, the angular overlap allowed by the |
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tilted umbrellas allows for almost total surface coverage based on random |
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parking alone. |
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Because a particles sticking success is closely linked to its orientation, |
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the radial distribution function and the angular distribution function show |
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some very interesting features (Fig. \ref{fig:tugofr}). The initial peak is |
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located at aproximately one radius of the umbrella. This corresponds to the |
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closest distance a second particle that is perfectly aligned with the particle |
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may aproach without overlapping. A quick look at the angular distribution |
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confirms this, showing a maximum angular correlation at this distance. The |
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location of the second peak in the radial distribution corresponds to twice |
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the radius of the umbrella. This peak is accompanied by a dip in the angular |
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ditribution, showing that this peak corresponds to the case of closest approach |
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for two anti-aligned particles. The alignments associated with both peaks are |
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illustrated in Fig. \ref{fig:tuallign}. |