^{1,2,a)}and Paul van der Schoot

^{1,3}

### Abstract

We present a generalized connectedness percolation theory reduced to a compact form for a large class of anisotropic particle mixtures with variable degrees of connectivity. Even though allowing for an infinite number of components, we derive a compact yet exact expression for the mean cluster size of connected particles. We apply our theory to rodlike particles taken as a model for carbon nanotubes and find that the percolation threshold is sensitive to polydispersity in length, diameter, and the level of connectivity, which may explain large variations in the experimental values for the electrical percolation threshold in carbon-nanotube composites. The calculated connectedness percolation threshold depends only on a few moments of the full distribution function. If the distribution function factorizes, then the percolation threshold is raised by the presence of thicker rods, whereas it is lowered by any length polydispersity relative to the one with the same average length and diameter. We show that for a given average length, a length distribution that is strongly skewed to shorter lengths produces the lowest threshold relative to the equivalent monodisperse one. However, if the lengths and diameters of the particles are linearly correlated, polydispersity *raises* the percolation threshold and more so for a more skewed distribution toward smaller lengths. The effect of connectivity polydispersity is studied by considering nonadditive mixtures of conductive and insulating particles, and we present tentative predictions for the percolation threshold of graphene sheets modeled as perfectly rigid, disklike particles.

The work of R.O. forms part of the research programme of the Dutch Polymer Institute (DPI, project 648). The authors thank Dr. Mark Miller, Dr. Tanja Schilling, Dr. Claudio Grimaldi, Dr. Cor Koning, Evgeniy Tkalya, Dr. Joachim Loos, and Marcos Ghislandi for stimulating discussions.

I. INTRODUCTION

II. CLUSTER-SIZE CALCULATION

III. APPLICATION TO CARBON NANOTUBES

IV. TETRADISPERSE DISTRIBUTION

V. REALISTIC SIZE DISTRIBUTIONS

VI. MIXTURES OF CONDUCTIVE AND INSULATING PARTICLES

VII. DISCUSSION AND CONCLUSIONS

### Key Topics

- Carbon nanotubes
- 73.0
- Percolation
- 30.0
- Hopping transport
- 13.0
- Graphene
- 10.0
- Networks
- 10.0

## Figures

(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 β*u* ^{+} = (*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.

(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 β*u* ^{+} = (*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 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}) = 4*n*/(*n* + 1)^{2}, with *n* _{0} = (3*n* − 1)/(*n* + 1), obtained for *x* = 1/(*n* + 1), i.e., the minima in the main graph.

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}) = 4*n*/(*n* + 1)^{2}, with *n* _{0} = (3*n* − 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) 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.

(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 } = α*D* _{ 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 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 } = α*D* _{ 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}) = 2*L*/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.

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}) = 2*L*/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.

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 .

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 .

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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