Program: PERSHELL.EXE
In this exercise, the user will examine the percolation characteristics of a variety of hard core/ soft shell two-dimensional systems consisting of monosize circles, as illustrated in Fig. 2. Hopefully, the exercise will provide some insight into what happens to the interfacial zones in mortar and concrete as more sand and rocks are added to the mix. As stated in the introduction, we think of the aggregates as the hard cores, and the interfacial zones as the soft shells. If the interfacial zones percolate, that may have important implications for transport and strength properties, as the biggest pores tend to be in the interfacial zones.The major variable the user will be examining is b/a, the ratio of the hard core diameter to the total (hard core + soft shell) diameter. For a given value of b/a, the user will be computing the number of circles, and area fractions of hard cores and soft shells necessary for the soft shells to form a percolating pathway across the 2-D microstructure (both top to bottom and left to right).
The program is called PERSHELL.EXE. Simply switch to the appropriate disk drive and type PERSHELL at the DOS prompt to begin execution. The program is interactive, and is set up for randomly placing circular particles on a square unit cell that is 300 x 300 pixels in dimension. Periodic boundary conditions are used during particle placement and during percolation assessment (in the appropriate directions). Initially, the program will request values for the hard core (2b) and total (2a) diameters of the circles. The user should enter the two values on the same line separated by a single space. The goal is to execute the program for the values of 2b and 2a given in Table 5. After inputting values for b and a, the program requests how many additional particles should be placed this round. If the user requests more particles (hard cores) than can be randomly parked in a non-overlapping arrangement, the placement routine will exit after 10,000 random attempts are made to place a single particle. After placing the particles, the program determines the connectivity of the soft shells from left to right, and from top to bottom and outputs a flag variable (burnt) which is 1 if a percolated pathway is present and 0 otherwise. It also reports the hard core and soft shell areas and the total number of particles placed so far. The user should record these values for when percolation is first achieved in each direction for each set of (2b) and (2a) values.
As an example, microstructures for monosize circles (with 2b=13 and 2a=21) are shown in Figure 3. The top left corner figure shows a system with 160 hcss particles. The hard cores are shown in dark grey and the soft shells are in light grey. As seen in the upper right corner figure, the soft shells are not yet percolated from top to bottom; here, the soft shells which are connected to the top of the system are shown in white. By adding 40 more hcss particles to the system, for a total of 200 particles, percolation from top to bottom is achieved as shown in the two bottom figures. Even at percolation, there still exist some hcss particles which are not accessible from the top of the system as exemplified by the light grey soft shells in the lower right corner figure.
The output of this exercise will be a graph of the various area fractions occupied (hard core, soft shell, and total particle) at the onset of percolation as a function of b/a. Averaging results over several separate executions will increase the accuracy of these graphs so multiple runs should be executed if possible.
Figure 2. Schematic of hard core/soft shell model. b is the inner radius, a is the outer radius.
| 2b | 2a | b/a |
| 0 | 21 | 0 |
| 1 | 21 | 0.0476 |
| 3 | 21 | 0.1429 |
| 5 | 21 | 0.2381 |
| 7 | 21 | 0.3333 |
| 9 | 21 | 0.4286 |
| 11 | 21 | 0.5238 |
| 13 | 21 | 0.6190 |
| 15 | 21 | 0.7143 |
| 17 | 21 | 0.8095 |
Tips (1) The best value for pc for a given run is the average of the critical values for left- right and top-bottom percolation. (2) Since the program requires additional particles to be added, the user must start out below the percolation threshold and approach it slowly by adding fewer new particles as the percolation point is neared.
