The basic unit of all the models to be described in this paper is the bar, which is a rectangle in 2-D and a rectangular parallelepiped in 3-D. The aspect ratios of the bars were taken to be 7:1, roughly based on average values found in experiments on porous gypsum plasters and on numerical considerations - the size of the unit cell needed to be about 5 to 10 times larger than the longest dimension of the bars, so too high an aspect ratio would have meant too large of a unit cell for reasonable computational turnaround rates of the many results required. These bars were taken to be totally solid, and were placed randomly in several ways in a periodic unit cell. By "periodic," we mean that if a bar extended past the side of the unit cell, it was completed periodically on the other side of the cell. The pore space then was the space in the unit cell not occupied by solid.
In 2-D and 3-D, the first kind of models studied were built by randomly placing the center of the bars, and allowing the bars to be oriented in one of the principal directions (x,y in 2-D and x,y,z in 3-D) (O-HV). Figure 1a shows an example of this arrangement in 2-D. When the system size was 2002, the typical size used in the 2-D studies, the bars were 21 x 3 pixels in size. In 3-D, the bars were 21 x 3 x 3 pixels in dimension when the system size was 1003. In 3-D, a model where the bars were allowed to have random orientations was also briefly studied (O-R).
Another kind of model was generated by first randomly placing bars in the principal directions that were not allowed to overlap (hard cores), and then by expanding each bar around its center through adding a "soft shell" that could freely overlap both phases. The hard cores were placed with random jamming statistics [5,6]. This model version was inspired by the "hard core/soft shell" model usually used with spheres and ellipsoids and is denoted in the text as HCSS-HV [5, 6]. Figure 1b shows an example of this kind of model in 2-D. In 2-D, at a system size of 2002 pixels, the hard core was a 19 x 1 pixel rectangle, and the soft shell was 1 pixel in width, so that the total size of the complete object was the same as for the O-HV case. Similarly, in 3-D, at a system size of 1003 pixels, the hard core was 19 x 1 x 1, with a 1 pixel thick soft shell.
We note here that having a model microstructure split up into individual pixels gives the model great flexibility. For example, it is simple to monitor percolation quantities of both the solid and pore phases using a burning algorithm [4, 7]. Also, to calculate the overlap volume between two ellipsoids is a fairly formidable mathematical problem. To analytically calculate the overlap between three or more ellipsoids is almost impossible. But in a digital model, it is easy to keep track of how many bars have been placed on the same pixel, so the total overlap area (2-D) or volume (3-D) can be readily known. Figure 2 shows the overlap solid fraction vs. the porosity for the two different 2-D models. The 2-D HCSS-HV model has somewhat less overlap area at a given porosity than does the O-HV model. This is because the hard core regions cannot overlap each other, while in the O-HV model, all pixels have the potential of participating in the overlaps. The overlap fraction was investigated as a function of size of the system and digital resolution, and was found to give reasonable results at the typical sizes used in 2-D.
Figure 2: Variation of overlap fraction versus porosity fraction for the 3-D models described in Fig. 1.