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Percolation algorithm

In any simulation study of the percolation of randomly-placed, overlapping objects, there are three pieces of information that must be known: 1) the position and shape of each object, 2) which, if any, objects overlap each other, and 3) does a connected path composed of overlapping objects exist through the unit cell of the simulation. The first piece of information, the position and shape of an object, can be stored in several ways. The objects can be stored digitally, in terms of occupied pixels in a digital image [29], which requires a large amount of computer memory, depending of course on the resolution with which each shape is represented. A second way, which is used in this simulation study, is, for an Euclidean object, to store the object geometrically, as a set of Cartesian coordinates for the location and orientation of the particle and sufficient numbers to describe the shape [68]. The lengths and orientations of the three semi-axes and the position of its center completely describe a triaxial ellipsoid, for example. In this paper, we consider ellipsoids of revolution, whose shape can also be expressed geometrically. For very oblate or prolate objects in 3-D, digital image methods do not have sufficient resolution at the present time, due to computer memory limitations, to give accurate percolation thresholds. We examine the largest range in aspect ratio that is compatible with our computational resources. Periodic boundary conditions were also employed to minimize finite size effects. The second piece of information that must be known is whether two given objects overlap or touch each other. This is carried out via a contact function. For Euclidean objects, a function can be computed that unambiguously tells if the two objects overlap each other at all, given the centers, orientations, and sizes of the pair of objects. Such a contact function has been worked out for ellipsoids of revolution, for the purposes of carrying out Monte Carlo simulations of hard- core ellipsoid gas problems [69]. Such an algorithm would be very inefficient if every object had to be compared to every other object, so a binning system is used to subdivide the computational cell. To check for overlaps of a given object with other objects then requires only checking the contents of a limited number of bins. To test the accuracy of our implementation of the contact function described in Ref. [69], we used the algorithm to numerically compute the excluded volume of pairs of identical ellipsoids of various aspect ratios. To do this, we placed a single ellipsoid, oriented along the x-axis, in a box, and repeatedly placed a second, identical ellipsoid in the box, with a random center and orientation. Count was kept of those times when the two particles overlapped, as determined numerically by the overlap function. After typically one million trials, the fraction of the trials that resulted in an overlap times the volume of the box was the excluded volume. Fig. 2 shows a graph of the numerically determined excluded volume plotted vs. the exact excluded volume [see eq. (11)], for oblate and prolate ellipsoids of revolution. The dashed line is the line of equality.