We wish to demonstrate, beyond the needle problem discussed above, that this lattice approach to continuum problems gives accurate results in cases where nI and nc are known from a true continuum analysis. We give one test case for nI, and three for nc.
The test case for nI that we consider involves elliptical defects with semi-major axis a and semi-minor axis b, placed in a conducting matrix, as described in Section IV. The exact solution for the effect on the conductivity of a single defect was given in eq. (1), but is rewritten here in a slightly different form. The change in conductivity in an N x N ( all lengths in units of pixels or lattice spacing) sheet with host conductivity σo is given, using eq. (1), by:
| δσH/σo ≡ 1 − σH /σo = πb(a + b)/N2 | |
| δσV/σo ≡ 1 − σV /σo = πa(a + b)/N2 | (11) |
when the defect is insulating, and H and V again stand for horizontal and vertical orientation of the ellipse with respect to the horizontally applied voltage.
For N = 200, a = 21, and b = 9, the calculated values of δ σH / σo and δ σV / σo differ from the exact values by an average 7.2%. The same percentage difference was obtained for an N = 400 pixel lattice, so that this disagreement was not due to the ratio a/N being too small, which would be a finite size effect, but rather must be caused by the finite resolution of the conductor-insulator boundary. To confirm this, another computation was done using N = 400, a = 45, and b = 19, which gives approximately the same aspect ratio hole as in the previous case, but with twice the resolution. This time the relative error for δ σH / σo and δ σV / σo was cut in half, to 3.6%, denoting a boundary effect. We expect that the percentage error in δ σH and δ σV goes to zero linearly with the number of pixels used per unit length of the ellipse. We note that, as in the case of the lattice needle, digitized ellipses always cause a greater reduction in the conductivity than would be expected from eq. (11). The ellipses could be represented reasonably well, electrically, by an effective continuum ellipse, with aeff = a + 0.5, and beff = b + 0.5. Since no electrical connection is made between a boundary defect pixel and a boundary host pixel, electrically the defect appears slightly bigger than its geometric size.
Computing an accurate value of nc would seem to be more problematical than computing nI, since the percolation threshold is determined by tortuous, thin paths of connected material. These paths will certainly be affected by the finite resolution, in the sense that the minimum path thickness is limited to be one pixel, or 1/N in terms of the system size N. Fortunately, there are three (at least) true continuum computations of critical thresholds against which the digital-image techniques can be compared.
The percolation threshold of randomly-centered, overlapping circles has been much studied [6,15,22]. Ref. [6] lists a number of values for pc based on various independent computations, which range from a high of 0.38 to a low of 0.31, with the best estimate probably being about 0.32. Using a 1000 x 1000 lattice, our results for pc are 0.35 ± 0.02 (21 pixel diameter circles), and 0.33 ± 0.02 (41 pixel diameter circles). For either size circle, the critical threshold results are within 10% or less of the best estimate for pc. The value of pc was determined by using a burning algorithm [3] to check for continuity in both directions for 20 configurations, and taking the average of the values of pc computed for left-right and up-down percolation over 20 configurations [17].
Dubson and Garland have experimentally measured the critical threshold for randomly-placed, overlapping parallel squares [23], and found a rather high value of pc = 0.39 ± 0.01. Pike and Seager [15] did a continuum simulation of the same problem, and found pc = 0.33. Gawlinski and Redner also obtained [24] pc = 0.33. Our value, obtained using 20 x 20 squares placed on a 1000 x 1000 lattice, is pc = 0.35 ± 0.02.
The final comparison case is the dependence of pc on the aspect ratio of randomly-placed, overlapping ellipses, the problem discussed in Section III. The continuum analysis carried out in II was for randomly oriented ellipses and for ellipses that were only oriented in the x and y directions. The lattice results, obtained on 500 x 500 lattices with ellipses all having 2a = 41 pixels and various aspect ratios b/a, although not shown here, reproduce the two-orientation ellipse continuum results shown in Fig. 3 within the computational uncertainty.
Having demonstrated the kind of accuracy to be expected from our digital image simulations, we proceed to study the values of nI and nc for a collection of arbitrary shapes.