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mmeineke |
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%%%% Computational Methods |
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The simulation size was 4,000 repeated hcp units |
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in both the x and y direction. This gave a rectangular plane, to which periodic |
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boundary conditions were applied. The particle's attachment point was then |
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randomly assigned a location on the plane. This location was then checked |
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against the underlying lattice to see if they were within $\epsilon$ of |
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one of the lattice gaps. If the attachment point was indeed close enough, |
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the particle was said to stick at that location, and the particle's new |
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attachment location was specified to be the lattice gap coordinate. |
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All failures resulted in a new random location for the attachment point. |
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Once the particle was found to stick to the lattice, the particle was tested |
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against the pre-existing particles for overlap. In the case of the octopi |
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model, the test was a simple distance formula test. Here the centers of the |
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particles were specified to be at least $2\sigma$ apart. Where $\sigma$ is |
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the radius of the particle. |
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The test for overlap in the case of the tilted umbrella particle, is slightly |
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more complex. For these particles, several sequential tests are made. The |
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first test is the simplest, and checks to make sure that the new umbrella's |
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attachment point, or ``handle'', does not lie within the elliptical projection |
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of a previously attached umbrella's top onto the xy-plane. |
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If the particle passes this first |
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screening, it is then subjected to a 3-dimensional evaluation of whether the |
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two umbrella tops intersect. This involves using the normals of both |
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umbrellas, and computing the parametric line equation from the intersection |
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of the two planes specified by the umbrella tops. This line is |
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then tested for intersection with the circles defined as the umbrella tops. |
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If there are points of intersection, these points must be tested against |
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both circles, such that the line intersects each circle sequentially. In |
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other words, the line must enter then leave one circle before it can enter |
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the next. |
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To speed up the overlap tests, a modified 2-D neighbor list method was |
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employed. The plane was divided into a 500 x 500 grid of equally sized |
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rectangular bins. The overlap test then cycled over all of the particles within |
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the bins located in a 3 x 3 grid centered on the bin in which the test |
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particle lied. |