1 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
2 |
%%%% Computational Methods |
3 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
4 |
|
5 |
The simulation size was 4,000 repeated hcp units |
6 |
in both the x and y direction. This gave a rectangular plane, to which periodic |
7 |
boundary conditions were applied. The particle's attachment point was then |
8 |
randomly assigned a location on the plane. This location was then checked |
9 |
against the underlying lattice to see if they were within $\epsilon$ of |
10 |
one of the lattice gaps. If the attachment point was indeed close enough, |
11 |
the particle was said to stick at that location, and the particle's new |
12 |
attachment location was specified to be the lattice gap coordinate. |
13 |
All failures resulted in a new random location for the attachment point. |
14 |
|
15 |
Once the particle was found to stick to the lattice, the particle was tested |
16 |
against the pre-existing particles for overlap. In the case of the octopi |
17 |
model, the test was a simple distance formula test. Here the centers of the |
18 |
particles were specified to be at least $2\sigma$ apart. Where $\sigma$ is |
19 |
the radius of the particle. |
20 |
|
21 |
The test for overlap in the case of the tilted umbrella particle, is slightly |
22 |
more complex. For these particles, several sequential tests are made. The |
23 |
first test is the simplest, and checks to make sure that the new umbrella's |
24 |
attachment point, or ``handle'', does not lie within the elliptical projection |
25 |
of a previously attached umbrella's top onto the xy-plane. |
26 |
If the particle passes this first |
27 |
screening, it is then subjected to a 3-dimensional evaluation of whether the |
28 |
two umbrella tops intersect. This involves using the normals of both |
29 |
umbrellas, and computing the parametric line equation from the intersection |
30 |
of the two planes specified by the umbrella tops. This line is |
31 |
then tested for intersection with the circles defined as the umbrella tops. |
32 |
If there are points of intersection, these points must be tested against |
33 |
both circles, such that the line intersects each circle sequentially. In |
34 |
other words, the line must enter then leave one circle before it can enter |
35 |
the next. |
36 |
|
37 |
To speed up the overlap tests, a modified 2-D neighbor list method was |
38 |
employed. The plane was divided into a 500 x 500 grid of equally sized |
39 |
rectangular bins. The overlap test then cycled over all of the particles within |
40 |
the bins located in a 3 x 3 grid centered on the bin in which the test |
41 |
particle lied. |