(a) Schematic representation of two nanotubes with orientations u and u′, lengths L i and L j , and diameters D α and D β, separated by a distance r between their centerlines and skewed at an angle θ. Charge transport between the rods requires r to be smaller than D + λ = Δ: the dashed cylinders of diameter Δ enclosing the rods must overlap. (b) Solid line: the connectedness potential u + for the idealized “cherry-pit” model between two particles in the same cluster versus their distance r for ideal (ε = 0) and hard particles (ε → ∞). The dashed line shows an alternative connectedness potential βu + = (r − D)/λ for r > D that may provide a more realistic description of an exponentially decaying electron-tunneling probability with a decay length λ. Within the second-virial approximation described in the main text, both connectedness potentials produce identical results.
For a tetradisperse mixture of thick, thin, long, and short rods the percolation threshold ϕ p is shown as a function of the number fractions x L of long rods and x D of thin rods. The tunneling length λ is taken as a constant and drops out of the description. The graphs are for different length ratios (n ≡ L long/L short) and width ratios (D thick/D thin), which are taken equal to n. From top to bottom: n = 2, 4, 8, and 16. Inset: the nonlinear behavior of ϕ p is demonstrated by a cross section for constant ϕ p (x L , x D )/ϕ p (0, 0) = 0.25 (0.15) for the solid (dashed) lines. Pairs of line from top right to bottom left: n = 2, 4, 8, and 16.
For a bidisperse mixture of long and short rods the ratio of the actual percolation threshold ϕ p (x L ) and that of the corresponding monodisperse solution ϕ p (x 0) with the same mean length 〈L k 〉 k is shown as a function of the number fraction x L of long rods. The tunneling length λ is taken as a constant and in that case drops out of this ratio. The graphs are for different length ratios n ≡ L long/L short and we find x 0 = nx/(1 + (n − 1)x). From top to bottom, n = 2, 4, 8, and 16. Inset: the largest reduction of the ratio of PTs is ϕ p (n)/ϕ p (n 0) = 4n/(n + 1)2, with n 0 = (3n − 1)/(n + 1), obtained for x = 1/(n + 1), i.e., the minima in the main graph.
(a) The influence of length polydispersity on the ratio of the PTs of the Gamma distribution and that of the corresponding monodisperse distribution with equal mean length, ϕ p (k)/ϕ p (∞) = k/(k + 1). The tunneling length λ is taken as a constant and in that case drops out of the equation. The ratio of the PTs depends only on the shape parameter k, which is infinitely large for a monodisperse distribution, and the effect can be strong for small k values. (b) For the values k = 1/2, 2, and 8 from (a) the distributions are shown that all have the same mean 〈L k 〉 k = k θ = 1 and the PT values relative to that in the monodisperse limit are ϕ p (k)/ϕ p (∞) = 1/3 (solid), 2/3 (dashed), and 8/9 (dashed-dotted). Inset: the impact of the length polydispersity can also be expressed in terms of the skewness of the distribution, showing that for a large skewness the reduction of the PT is significant. The three points marked on the graph correspond to the three distributions.
(a) For a Weibull distribution the effect of length polydispersity on the ratio of the PTs of the Weibull distribution and that of the corresponding monodisperse distribution with equal mean length, ϕ p (b)/ϕ p (∞) = Γ(1 + 1/b)2/Γ(1 + 2/b) with Γ the Gamma function (Ref. 45). The tunneling length λ is taken as a constant and in that case drops out of the equation. The ratio depends only on the shape parameter b. The scaled variance s goes to zero in the monodisperse limit that b → ∞, in which case the skewness γ → −1.14. A large reduction of the PT is observed for small b. (b) Three Weibull distributions with 〈L i 〉 i = 1 are shown for the values b = 1/2, 1, and 2, for which ϕ p (k)/ϕ p (∞) = 0.17 (solid), 0.5 (dashed), and 0.79 (dashed-dotted). Inset: the impact of polydispersity can also be expressed in terms of the skewness γ of the distribution, showing that for a large skewness the reduction of the PT is significant. The three points marked on the graph correspond to the three distributions.
The percolation threshold ϕ p for a polydisperse distribution relative to its value ϕ p (x 0) for the corresponding monodisperse case with the same mean length and diameter is shown for length and diameter distributions that are linearly correlated, i.e., L i = αD i , with α being a constant. For a constant tunneling length λ, polydispersity raises the PT for the Weibull, Gamma, and exponential distributions, and most predominantly for small values of the shape parameter b and k of the Weibull and Gamma distributions, meaning a large skewness and large spread. For the exponential distribution the PT is raised by a factor of 3, regardless of the shape parameter η that equals the reciprocal of the mean value of the distribution.
The percolation threshold of a binary mixture of large (D large) and small disks (D small) of equal thickness is shown as a function of the mole fraction x L of large plates relative to its value ϕ p (x 0) = 2L/λ(5π + 6) of that of the corresponding monodisperse distribution with equal mean length. The tunneling distance λ is presumed to be a constant and drops out of the equation. From top to bottom: D large/D small = 2, 4, 8, and 16. The ratio of PTs is lowered substantially by adding a small fraction of large plates to a dispersion of small ones.
For a bidisperse mixture of plates with diameter W and thickness T and rods with length L and diameter D, the percolation threshold ϕ p relative to its value of that of the corresponding monodisperse case consisting of only rods is shown as a function of the mole fraction x R of rods. The arrows indicate increasing values of W/L: 0.1, 0.3, 1 and 3. (a) For typical values for graphene and CNTs, , where λ is the hopping distance, a mixture with only rods gives the lowest PT. For sufficiently large rod aspect ratios L/D, the shape of the curves is almost insensitive to changes in L/T and L/λ, which only change the vertical scale. (b) For rods with a smaller L/D, a mixture with plates can have a lower PT than the one for only rods, as shown for L/T = 60, L/λ = L/D = 15.
Five diagrams consisting of three points can be formed such that points 1 and 2 are connected via a continuous path of f + bonds (dashed lines), where the wavy lines represent f* bonds. This path between 1 and 2 can be either direct, as in the top three and the bottom left diagram, or via a third particle, shown by the bottom right diagram. These diagrams give rise to the five terms in the expression (A2) for .
Possible configurations of three disks that are mutually connected and that contribute to the three-body direct connectedness function . The disks have diameter D and the tunneling distance between them is λ ≪ D. If we fix disk 1 and if the difference between the orientations of disk 1 and 2 are almost perpendicular to each other, the overlap criterion is met in a triangle (a) or a branched configuration (b). If this difference is very small, on the other hand, we distinguish between almost complete (c) and limited overlap (d) between particles 1 and 2. In the latter case the angle α that disks 2 and 3 make is of the order λ/D. λ and D denote the ranges of motion in three directions for the disk 2 so that it is connected to disk 1 and for the disk 3 so that it connects to 1 and 2. In all cases we find .
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