Percolation Threshold of Clustered, Oriented and Polydisperse Sticks in a Plane

Monte‐Carlo methods are used to study the influence of stick polydispersity, orientation and clustering on the percolation of sticks in the plane. The results show that for polydisperse stick lengths, the percolation threshold exhibits a linear dependence on the reciprocal of the length‐weighted mean stick length. Reinterpretation of data from the literature confirms this dependence and shows it to persist for oriented networks. Also, the number of intersections per stick at percolation, Bc=3.6$B_c = 3.6$ is independent of orientation for percolation perpendicular to the direction of preferential orientation; in the direction of preferential orientation, it is a linear function of an orientation parameter. For networks of sticks clustered according to a compound Poisson process, a cluster intensity, Ψ is introduced; when this is less than the percolation threshold of a random network of sticks, the percolation threshold increases linearly with cluster intensity and is independent of cluster size. Above this, non‐linear size‐dependent relationships are observed.


DOI: 10.1002/adts.202300131
There is a rich literature providing models for percolation of objects with arbitrary geometries in two and three dimensions and relating these to material properties. [4][5][6] A system of such objects is said to be 'percolated' when there are sufficient of them per unit area to create a cluster that spans the boundaries of a finite region, or yields an 'infinite cluster' in a theoretical infinite system. Probably the earliest Monte-Carlo estimates of the statistics of percolation in 2D for objects with a variety of shapes are those of Pike and Seager. [7] In this contribution, we restrict our discussion to percolation of 1D straight rods, or 'sticks' in the plane, though note that more complex geometries are required to model, for example, nanoplatelet materials such as graphene. [8] For a random network of sticks with length, , Pike and Seager obtained the critical total line length per unit area required for percolation, which is termed the percolation threshold, c as c = k (1) with k = c ≈ 5.71. Subsequent refinements in computational approach and efficiency have resulted in high-precision calculations of k; Li and coworkers [9,10] report k = 5.637 26 (2), and Mertons and Moore [11] give k = 5.637 285 8 (6). While computational approaches to determining the percolation threshold are inevitably the default, Khanarian et al. [12] obtained an experimental value of k = 5.67 for planar networks of silver nanowires. As expected, for curved or wavy sticks, the percolation threshold is higher. [13,14] The classical basis for simulations is a random network of sticks deposited according to a point Poisson process in 2D with a uniform distribution of orientations to any arbitrary axis in the plane, as defined in the seminal work of Kallmes and Corte. [15] We expect real materials to exhibit some structural differences from this family of assumptions: very often manufacturing methods impart shear to particles, resulting in them having a preferential orientation to the direction of manufacture; [16] further, particle interactions result in clustering such that the condition of independence of location associated with a Poisson process is breached and the network exhibits greater non-uniformity. [17] Also, processing of materials typically results in breakage of nanoparticles, resulting in a distribution of lengths. [18] Several workers have addressed these effects through simulation. Balberg and Binenbaum [19] found that the percolation threshold always increases with increasing anisotropy and for a system of sticks with a lognormal distribution of lengths decreased as the variance of stick lengths increased because, 'long sticks in the system determine the onset of percolation'. For sticks with finite width, Kortschot and Woodhams [20] showed that the sensitivity of the percolation threshold to orientation increased with increasing stick aspect ratio. Tarasevich and Eserkepov [21] provide polynomial regressions to give the combined effects of orientation and polydispersity for sticks with lognormally distributed lengths.
While the effects of orientation and polydispersity on the percolation threshold are broadly understood, analysis linking the dependences to intuitively meaningful structural parameters is largely absent. Statistical geometry provides expressions for the number of intersections per stick in oriented stick networks in terms of an orientation parameter; [22] we provide these in our subsequent discussion and show that they allow prediction of the percolation threshold for oriented networks. Given Balberg and Binenbaum's [19] observation that for polydisperse stick lengths, long sticks dominate the onset of percolation, we hypothesize that the percolation threshold depends on the length-weighted average stick length,̄w = 2 ∕̄, and present simulations and analysis to test this. Finally, although we have identified literature that reports experimental observations of clustering and its negative effects on percolation and conductivity, [17] we are not aware of simulations that directly address this important aspect of network structure. Accordingly, we report a family of Monte-Carlo simulations that probe the extent that clustering of sticks influences percolation and introduce a simple clustering parameter that captures the dominant behaviors.

Simulations
We used Mathematica [23] to probe the percolation behaviors of 2D stochastic networks of sticks in a plane. We first describe the approach for the reference case of a random network of widthless sticks (lines) with uniform length and no preferential orientation before elaborating on the approaches to studying polydispersity, preferential orientation, and clustering. Graphical representations of the examples of the networks generated are given in Figure 1.

Reference Case
Widthless sticks of unit length were deposited in a square of side L = 50 with uniform random coordinates of the stick centers. Each stick was assigned an orientation, drawn from a uniform distribution on the interval 0 ≤ < and the coordinates of its ends were computed. Periodic boundary conditions were then applied, such that we consider percolation on a torus. [21] Sticks were added sequentially. For a given stick, those sticks with centers separated by one stick-length from its center were considered candidates for contact. To improve computational efficiency, each of these sticks was considered the diagonal of a rectangle and a simple straddling test was applied only to those pairs of sticks where these rectangles overlapped. After the addition of each stick, the resultant structure was tested for percolation in the x-direction with each additional stick until a single cluster of sticks spanned the vertical boundaries of the square; at this point the simulation was stopped and the percolation threshold, computed. For each condition, we computed the percolation threshold, for multiple evaluations with random seeds and estimated the mean, c by fitting the CDF of the Gaussian distribution to the cumulative data: [24] where is the standard deviation and is frequently reported as the width of the percolation transition, Δ(L) = √ 2 , which scales as a function of system size such that Δ(L) ∝ L 1 , [24] where = 4∕3 for a 2D system of sticks. [9] Rigorous high precision determination of the percolation threshold for an infinite system of sticks requires computation of c at a range of scales L and linear regression to obtain the intercept of a plot of c against L − 1 . Here, our interest is whether departures from the reference case yield significant differences in the percolation threshold. Accordingly, given the large family and range of variables that we sought to consider, and in the interests of computational efficiency, throughout this study we report percolation data obtained from fitting Equation 2 to 10 3 repeats of simulations with L = 50; for the reference case of isotropic random sticks with unit length we obtain a percolation constant of k = c = 5.63768 (cf. [9][10][11] ) confirming the validity of the approach. Should applications require knowledge of the sensitivity of the percolation threshold to very high precision, then a broader family of simulations covering a range of L and extrapolating [24] would be required.

Polydispersity
Stick lengths, were selected at random from a lognormal distribution with a unit mean; this has probability density with standard deviation and length-weighted mean,̄w = 2 ∕̄= (1 + 2 ). [25]

Preferential Orientation
The angles of sticks, , relative to the vertical, were selected at random from the 1-parameter Cosine distribution, with probability density where 0 ≤ ≤ 1 such that Equation 4 recovers the uniform distribution when = 0 and orientation increases with . The distribution has variance, 2 ( ) =

Clustering
Sticks of unit length were dropped in clusters of n sticks with centers distributed randomly within circles of radius r c ; the centers of these circles were distributed randomly within the square. We define a cluster intensity, Ψ as the total stick length in a cluster divided by the area of the circle, i.e.
The number of sticks in a given cluster was selected from a Poisson distribution with mean,n and the radius r c of a given cluster varied such that Ψ was fixed for a given family of simulations and we model a compound Poisson process of stick centers.

Results and Discussion
Recall our earlier hypothesis that for polydisperse stick lengths, the percolation threshold should depend on the length-weighted mean length. We noted that for a lognormal distribution of stick lengths with a unit mean, the length-weighted mean stick length is given bȳw = (1 + 2 ). To probe the dependence of c on the length-weighted stick length, this is plotted against 1∕̄w = 1∕(1 + 2 ) in Figure 2a for 0 ≤ ≤ 0.8. Figure 2b provides some illustrative probability densities for the lognormal distribution with unit mean and a range of . Although Figure 2a does show a linear dependence, as expected, a linear regression on the data does not pass through the origin, as would be expected for uniform sticks with increasing (cf. Equation 1). Figure 2c shows the same behavior for data recovered from the regressions provided by Tarasevich and Eserkepov, [21] who considered systems of lognomally distributed sticks with differing degrees of orientation, including the randomly oriented case considered here. Again, we observe a linear dependence with a non-zero intercept. The gradient, a and intercept b from linear regression on these data is provided in Table 1; the coefficient of determination r 2 > 0.99 in all cases and we note that (a + b) ≈ c (0), as expected given that 0 < 1∕̄w ≤ 1. For the random case, we attribute the small differ- ences in a and b between this study and that of Tarasevich and Eserkepov [21] to that fact that they truncated their input distribution of sticks and discarded any sticks with > L, whereas here these were retained. For uniform stick lengths, we expect c → 0 as → ∞, since in the limit only one stick is required for percolation; for polydisperse sticks with unit mean, although the number of sticks in the infinite cluster diminishes rapidly with increasing , the presence of short sticks gives rise to the positive www.advancedsciencenews.com www.advtheorysimul.com The effect of the orientation parameter, on c is shown in Figure 3 for networks with sticks preferentially oriented in the y-direction. Inevitably, increasing orientation reduces the probability of contact between any given pair of sticks and therefore increases c , and the number of sticks required to percolate in the y-direction (direction of preferential orientation) is fewer than that required in the x-direction. We computed the number of crossings per stick at percolation, B c and this is shown in the inset plot: in the x-direction B c is constant and equal to 3.6, whereas in the y-direction number of crossings decreases such that B c ≈ (3.6 − ∕2). For a 2D network of interpenetrating random rods with high aspect ratio, Mutiso and Winey [26] obtain B c = 3.7; Balberg [4] shows that for 2D systems of interpenetrating rods B c = c V ex , where V ex is the excluded volume of a rod. For 1D sticks on a plane, the excluded area is 2∕ such that B c = (2 c ∕ ) ≈ (2 × 5.637∕ ) = 3.6, as observed here.
The expected number of crossings per stick in a network with orientation given by Equation 4 is [22] B = 2 − 2 3 (6) such that, at the percolation threshold, when = c and B = B c we obtain In the main plot in Figure 3, the black and red lines are given by Equation 7 with B c = 3.6, and B c = (3.6 − ∕2), respectively; both give excellent agreement with the data.
The effect of the clustering intensity, Ψ as given by Equation 5 on the percolation threshold is shown at the top of Figure 4; graphical representations of clustered networks at the percolation threshold are provided in a gallery at the bottom of the figure to guide interpretation. As anticipated, clustering of sticks reduces their efficiency in building a connected network and increases the percolation threshold. We observe that when Ψ is less than a threshold, which seems to be close to c for random stick networks, i.e. Ψ < 5.637, all data can be described by a linear relationship, c = 5.502 + 0.256 Ψ with r 2 = 0.959. Above this threshold, we observe some non-linearity and a dependence on the size of the cluster, as indicated by the broken lines in Figure 4.

Conclusion
We have provided data for the percolation of sticks in the plane and characterized the influence of stick orientation, polydisersity and clustering on some simple and intuitively meaningful parameters. For a lognormal distribution of stick lengths with standard deviation, , our data, and those from the literature [21] show that the percolation threshold decreases linearly with 1∕(1 + 2 ), essentially demonstrating an inverse proportionality to the length-weighted stick length. We have found this dependence to persist for other distributions of length. For oriented networks of sticks, we confirm the established result that the percolation threshold increases with increasing orientation, and is lower in the direction of orientation than in the perpendicular direction. The dependence of these percolation thresholds on the number of crossings (bonds) per stick has been demonstrated to agree with theory for the number of crossings per stick in oriented random networks and to be constant and given by B c = 3.6 in the perpendicular direction. We have introduced a dimensionless 'cluster intensity' for clustered networks and shown a linear dependence of percolation threshold on this parameter when it is lower than the percolation threshold of a random stick network. Below this threshold, the relationship is independent of cluster dimensions, whereas above it, we see some sensitivity to cluster dimensions and a nonlinear dependence.

Supporting Information
Supporting Information is available from the Wiley Online Library or from the author.