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The success of the normalized variable [10, 16] x = n Leff2, in unifying the two sets of data shown in Fig. 2 led us to re-analyze the data in II on the percolation thresholds of sheets with random elliptical holes. The value of Leff for an ellipse with semi-major and semi-minor axes a and b, respectively, is defined by combining eqs. (3) and (4):
| Leff = 2(a + b) | (6) |
Table I shows the results for ellipses that were only allowed to lie in the horizontal and vertical directions, and Table II shows the results for randomly oriented ellipses. The numbers in the column marked xI = nI Leff2 are the same, because the exact value of the initial slope for this problem was used to define Leff. Within error bars of ± 0.2, the value of xc seems to be invariant with respect to the ellipse aspect ratio. The third and fourth columns shows nI and nc, made dimensionless by multiplying with the excluded area [9,18], aex. In these columns there is clearly a monotonic decrease in the values as the aspect ratio of the ellipses decrease, while in the xc column there seems to be only random scatter about an average value of about 5.7 in Table I, and about 5.5 in Table II, with the aforementioned small but significant rise in the needle limit in Table I.
| b/a | nILeff2 | ncLeff2 | nIaex | ncaex |
| 1.0 | 2.55 | 5.6 | 2.00 | 4.4 |
| 0.9 | 2.55 | 5.6 | 2.00 | 4.4 |
| 0.8 | 2.55 | 5.7 | 2.00 | 4.4 |
| 0.7 | 2.55 | 5.6 | 1.98 | 4.3 |
| 0.6 | 2.55 | 5.7 | 1.97 | 4.3 |
| 0.5 | 2.55 | 5.6 | 1.93 | 4.3 |
| 0.4 | 2.55 | 5.6 | 1.90 | 4.2 |
| 1/3 | 2.55 | 5.5 | 1.85 | 4.0 |
| 0.25 | 2.55 | 5.6 | 1.78 | 3.9 |
| 0.20 | 2.55 | 5.7 | 1.72 | 3.9 |
| 0.15 | 2.55 | 5.5 | 1.65 | 3.5 |
| 0.10 | 2.55 | 5.9 | 1.55 | 3.6 |
| 1/15 | 2.55 | 5.6 | 1.47 | 3.2 |
| 0.05 | 2.55 | 5.6 | 1.43 | 3.2 |
| 0.04 | 2.55 | 5.7 | 1.40 | 3.1 |
| 1/30 | 2.55 | 5.9 | 1.38 | 3.2 |
| 0.025 | 2.55 | 5.9 | 1.35 | 3.1 |
| 0.0125 | 2.55 | 6.0 | 1.31 | 3.1 |
| 0.005 | 2.55 | 6.2 | 1.29 | 3.1 |
| Table I: This table lists the aspect ratios, xI = nI Leff2, xc= nc Leff2, n I < aex >, and nc < aex > for randomly-centered ellipses, oriented in the horizontal and vertical directions, using data taken from II. | ||||
| b/a | nILeff2 | ncLeff2 | nI aex | nc aex |
| 1.0 | 2.55 | 5.6 | 2.00 | 4.4 |
| 0.9 | 2.55 | 5.6 | 2.00 | 4.4 |
| 0.8 | 2.55 | 5.7 | 2.00 | 4.4 |
| 0.7 | 2.55 | 5.8 | 1.98 | 4.5 |
| 0.6 | 2.55 | 5.8 | 1.97 | 4.5 |
| 0.5 | 2.55 | 5.6 | 1.94 | 4.3 |
| 0.4 | 2.55 | 5.6 | 1.91 | 4.2 |
| 1/3 | 2.55 | 5.6 | 1.88 | 4.1 |
| 0.25 | 2.55 | 5.5 | 1.83 | 4.0 |
| 0.20 | 2.55 | 5.7 | 1.80 | 4.0 |
| 0.15 | 2.55 | 5.4 | 1.76 | 4.0 |
| 0.10 | 2.55 | 5.5 | 1.71 | 3.7 |
| 1/15 | 2.55 | 5.3 | 1.68 | 3.7 |
| 0.05 | 2.55 | 5.4 | 1.67 | 3.5 |
| 0.04 | 2.55 | 5.4 | 1.66 | 3.5 |
| 1/30 | 2.55 | 5.5 | 1.65 | 3.5 |
| 0.025 | 2.55 | 5.5 | 1.64 | 3.5 |
| 0.0125 | 2.55 | 5.5 | 1.63 | 3.5 |
| 0.005 | 2.55 | 5.5 | 1.63 | 3.5 |
| Table II: Same as Table I, except for ellipses that were randomly oriented in all directions. The data is taken from II. | ||||
It therefore appears that the area defined by Leff2 leads to a better invariant than does the excluded area aex. The quantity Leff2 is designed to lead to an invariant initial slope xI. However, the constancy of xc was not expected a priori, but is dramatically demonstrated in Tables I and II.
Fig. 3 shows simulation [12] and experimental [19,20] conductivity data for randomly-centered circles, replotted using the variable x = nc Leff2, where Leff for a circle is given by (6) to be twice the diameter. The solid line is the same universal curve as that plotted in Fig. 2 for the needle data, and agrees with the simulation data very well and with the experimental data fairly well.
Figure 3: Showing the conductivity vs. x = n Leff2 data for circles. The circles from are the blind-ant algorithm [12], the squares are the experimental results of Lobb and Forrester [19], and the triangles are the experimental results of Sofo et.al. [20]. The solid line is the universal conductivity curve (15).
In II it was shown that pc for the data in Tables I and II was a good fit to the formula:
where y = b/a + a/b. However, we find that pc can be equally well fitted to the relation
as shown in Fig. 4. This is important to the present discussion, because using pc = exp(− a b nc) together with (8) implies that
is a universal value for all aspect ratios b/a. Using the initial slope variable xI = nI Leff2, along with eqs. (6) and (9), leads to xc/xI = 2 ln (3) = 2.20 for all ellipses. The only exception is for horizontal and vertical needles where xc rises to 6.2 from 5.6 as was noted earlier.
Figure 4: Showing the percolation threshold pc vs. aspect ratio b/a for the problem of randomly- centered elliptical holes. The data points are from Table I, and the solid line is eq. (8).