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In I, the continuum percolation-conduction problem of a conducting sheet with random insulating needle-shaped holes was studied, both experimentally and using computer simulation. The applied voltage is always in the horizontal direction. The needles were equal length, zero width cuts through which no electrical current could flow.
The experimental study was carried out using needles that were aligned horizontally or vertically with equal probability, and placed with random centers throughout the conducting sheet. Because of the experimental method used, the needles were actually very thin rectangles, with a width to length ratio of about 50. Periodic boundary conditions were not employed. A needle whose center would fall outside the sheet boundary was not allowed to be placed. However, there were many needles, with centers near the boundary, which when placed overlapped the boundary. More experimental details are given in I.
Computer simulations for the electrical conductivity were carried out using the "blind ant"
algorithm [11]. A random array of horizontal and vertical needles was created, using
periodic boundary conditions, and then many random walkers, called ants, were "parachuted"
down onto the sheet at random positions, and made to carry out random walks in the open areas,
with the restriction that the ants could not cross a needle. The blind- ant results presented in this
paper represent improvements on the work described in I. The algorithm used incorporated the
first passage method of Tobochnik [12], whenever the walker was farther away than a
predetermined distance from the nearest needle. We also used a variable step length in the
random walk to allow for the possibility of passing through arbitrarily small "necks" between
needles. The correlation length ξ was computed from the mean-squared radius of gyration of
the needle clusters, and was found to be a good fit to the formula
(ξ/L)2 = (1 − n / nc)−8/3 / 12.
For each ant, the number of steps was chosen so that the mean squared distance
covered was (5 ξ)2 from the starting point.
A lattice computation was also carried out for this problem. A square lattice with N-1 rows and N columns of conductors was set up, with the first and last column of conductors set to infinite conductivity to serve as the electrodes. This is the arrangement needed to ensure a unit conductivity for the entire lattice, and is convenient for the implementation of the Y − ∇ conductivity algorithm [13]. To place a needle, the appropriate bonds are simply removed. A horizontal needle is represented by removing a line of vertical bonds and vice versa for a vertical needle. When the desired number of needles were placed, the Y − ∇ algorithm was used to compute the new effective conductivity. The well-documented speed of this algorithm [13] enabled large (N = 1000) lattices to be used, which gave an acceptable degree of resolution in the lattice representation of the continuum needle problem. The needle length used was L = 20 lattice spacings.
Since needles have zero width and thus zero area, the usual remaining area fraction variable p cannot be used, and so we switch to the variable n, which is the number of needles per unit area. The percolation threshold is then denoted nc, and the initial slope of the conductivity vs. n curve, which is defined for a single needle, is conveniently quantified as nI, where nI is the point on the n axis at which the conductivity curve would extrapolate to zero, if the conductivity continued to follow the initial slope.
Again taking the applied voltage to be in the horizontal direction, then the conductivities σH and σV for a few horizontal or vertical elliptical holes in a sheet with conductivity σo are
where a horizontal ellipse means that the semi-major axis a is aligned in the horizontal direction. Averaging the pair of equations shown in eq. (1) gives the result
which requires nI to be defined by
where a and b are the semi-major and semi-minor axes of the elliptical hole. When b goes to zero, we have the needle result that
where L = 2a is the length of the needle. For a few vertical-only needles, eq. (4) changes to
The results in eqs. (1) - (5) are exact. In the lattice approach to the needle problem, however, since the end of a needle is not well-defined, one must examine the computed initial slope to see how the nominal length L of a needle, in terms of how many bonds are removed, compares to the length that is defined by the computed slope. We simulated a single vertical needle in a 1000 x 999 lattice by removing a column of L horizontal bonds, and computing the new conductivity via the Y − ∇ algorithm (approximately two hours of CPU time on a Cyber 205). It turns out that for any L, the conductivity is reduced slightly more by a lattice needle than the continuum result predicts. This difference is due to the finite resolution of the lattice, and can be interpreted in one of two ways. We can either say that the lattice imparts a finite thickness to the needle, and so think of the lattice needle as an effective ellipse, or we can assume the lattice needle is a true zero-width needle, but with an effective length Leff slightly longer than L. The former interpretation is inconsistent with the fact that a single horizontal needle on the lattice does not change the conductivity, and so does act like a true zero-width needle. The latter interpretation, however, will only be useful if the difference between Leff and L is independent of L, for reasonable values of L. Fig. 1 shows δL i≡ Leff − L plotted against L, with 1 < L < 40. Leff is defined using eq. (5) by equating the computed lattice result for nI to 4 / (π Leff2). It is clear that for all values greater than 6, δL is approximately equal to the constant value of 0.68. The data point at L = 1 is just the exact single-bond-defect result [14], δL = (8/π)½ = 0.596. It is therefore valid to think of the lattice needle electrically as a true continuum needle, but with an effective length Leff = L + 0.68 (in units of lattice spacings) for L greater than or equal to 6.
Figure 1: Showing ∇L = Leff − L for insulating needles on a lattice. L is the nominal length, and Leff is the effective length determined electrically. After L = 6, the graph asymptotes to δL = 0.68.
Fig. 2 displays the blind ant and lattice results plotted against the variable x = n Leff2, with Leff = L for the blind ant simulation, and Leff = L + 0.68 for the lattice simulation. This variable makes the initial slopes of the two sets of data the same. The solid line is the universal conductivity curve to be discussed in Section VI.
Figure 2: Showing the conductivity vs. x = n Leff 2 data for needles. The squares are the improved blind-ant data, and the circles are the lattice simulation data points. The appropriate Leff for each method is discussed in the text. The solid line is the universal conductivity curve (15).
It was not possible to extend the blind ant algorithm beyond n/nc = 0.8, as the CPU time required to transverse 5ξ became too long. For this reason, the results for the blind ant algorithm for n/nc > 0.8 are omitted from Fig. 2.
The value of xc = nc Leff2, where the appropriate Leff was used, differs somewhat for different methods. The lattice result, for 40 configurations of L=20 needles on a 1000 x 999 lattice, was 5.7 ± 0.2. Other results from the literature include 5.7 (for randomly oriented needles [15]), 6.3 (horizontal and vertical needles [10]), 5.8 (randomly oriented needles[16]), and 6.56 (horizontal and vertical needles[16]). The fairly small differences between these values can be attributed to statistics (in all cases), finite resolution (for the lattice model), and to the expected small differences in percolation thresholds between completely randomly oriented objects and objects randomly oriented in two orthogonal directions [2,10]. It should be kept in mind that the results in Refs. [10,15,16] were for small systems (unit cell length to needle length was 20 or less), so that finite-size effects must also be taken into account. Also, periodic boundary conditions were not used in Refs. [15,16].
For the horizontal and vertical needle problem, we have made a much higher accuracy determination of nc Leff2 using the same method as Yonezawa et al. [17]. This involves an examination of the percolation probabilities for increasing system sizes and average, intersection, and union probabilities. The percolation concentration is plotted against L−1/ν in the usual finite-size scaling manner, where ν = 4/3. For averages over 1000 samples, of up to 80 000 needles, we found that xc = nc L2 = 6.205 ± 0.006. Note from Table I that n c Leff2 does approach 6.2 from below for horizontal and vertical needles, even in the less accurate data from II. The randomly-oriented ellipse data in Table II do not show a similar rise in the needle limit. This higher value of xc may then be an artifact due to having only two orientations.