From the results reported in the previous sections, it appears that the quantity xc = nc Leff2 is a constant, suggesting that the conductivity σ(x) may be a universal function of x = n Leff2. Given our larger data base, it seems appropriate to improve on the two interpolation formulae (A) and (B) given in I, in order to develop an analytical form for this universal function. Denoting these two formulae by σA and σB, we form an improved interpolation formula σ as
where 0 < α < 1 is an adjustable parameter. The result (13) may be written as
where the critical exponent t = 1.30 for all geometries in two dimensions [13]. The quantity xI = nI Leff2 = 8/ π and so we must choose xc and α. In I, it was suggested on the basis of eq. (7) that xc = 18/π = 5.73. However, eq. (8) suggests that xc = 5.60 is an equally good fit. Examination of Tables III and IV suggest a value of xc = 5.9 would be most appropriate overall, and we adopt this value. The value of α is adjusted to optimize the fits in Figs. 2 and 8, to give α = 0.7. This leads to a universal conductivity curve for holes in two dimensions,
where x = n Leff2. For ellipses, Leff = 2(a+b), and so it is straightforward to plot the data for needles and circles against the universal conductivity curve as shown in Figs. 2 and 3. The agreement is clearly excellent. We have held all the parameters fixed in (14) and varied xc, in order to check the dependence on xc. The quality of the overall fit is insensitive to small changes in xc for values in the range 5.6 < xc < 6.2.
Figure 8 collects together the experimental data on squares [23] with random centers and rectangular slits [1] with an aspect ratio of about 50:1, also with random centers and oriented horizontally and vertically. Ellipses that are oriented horizontally and vertically, or randomly, represent the only defect shape for which Leff is known in closed form. There is therefore an unknown scale factor for the abscissa in Fig. 8, since we do not analytically know the scale factor between nc and xc.
Figure 8: Showing the experimental results for squares [23] and slits [1]. The solid line is the universal conductivity curve (15).
For the slit data, we used careful measurements of the initial slopes SH and SV from I, where H denotes all horizontal and V denotes all vertical slits, respectively, to give Leff via
which is equivalent to fitting the measured initial slope to that of the universal conductivity curve (13) as was done in I. As was shown in I, the slits can be replaced by an ellipse of the same area and same length, 2a = L. This implies that the aspect ratio b/a of the equivalent ellipse is larger than the aspect ratio w/L of the slit by a factor π/4, as found within the error bars in eqs. (6) and (7) of I.
For the square data, we adopted a slightly different procedure and chose the scaling parameter for the abscissa so that the critical points exactly coincided. We fit the critical point to give xc = 5.9 = (28,440)Leff2, where nc = 28,440 [23]. This leads to Leff−1 = 69.4. Using the result [15] that the percolation number concentration is the same for squares and circles of equal area, and the result (6) that Leff = 2D for a circle of diameter D, we find that the size of the square L is given by L−1 = 4 / (Leff π½) = 157. This is rather smaller than the (corrected) value of L−1 = 172 ± 2 given by Dubson and Garland [23,25]. We feel that the value of L−1 = 157 is more accurate than the value of 172, because it leads to a value of pc = exp(−nc L2) = 0.31 that is in better agreement with other authors' values [15,24] of 0.33 [26].
The quality of the overall agreement in Fig. 8 is comparable to Figs. 2 and 3. In practice, it makes little difference if Leff is chosen by adjusting the initial slope, the critical point, or the overall fit.