Computational Study of the Curviness Percolation Threshold in Nanotube/Nanowire Networks for Flexible and Transparent Conductors
Authors: Wang Yunong , Ural Ant 
Published: July 2021

Two-dimensional (2D) or quasi-2D networks of one-dimensional (1D) nanoelements, such as carbon nanotubes, metal nanowires, and graphene nanoribbons, have attracted significant research interest recently for next-generation flexible and transparent conductors, as a replacement for indium tin oxide (ITO), which suffers from brittleness, scarcity, high cost, and slow deposition. These nanotube/nanowire networks can be used in many device applications, such as touch screens, flat panel displays, solar cells, light-emitting diodes (LEDs), and wearable flexible electronics. They also have applications in thin film transistors and resistive switching memory.
At the high optical transmittance values required for flexible and transparent conductors, the electrical properties of nanotube/nanowire networks are governed by percolation transport, which deals with the formation of long-range connectivity in random networks. As a result, Monte Carlo simulations need to be employed to theoretically calculate, predict, and optimize the electrical properties of 1D nanoelement networks [1,2]. In most computational work, nanotubes/nanowires in such networks have been modeled as straight sticks. However, experimentally deposited nanotubes/nanowires exhibit some degree of curviness.
There has been some computational work on the density percolation threshold of carbon nanotube/nanofiber networks and nanocomposites at fixed curviness, where the network undergoes an insulator-to-conductor phase transition [3-7]. These studies have shown that curviness increases the density percolation threshold of random networks. However, the curviness percolation threshold at fixed density has not been investigated.
In this work, we employ Monte Carlo simulations to compute the curviness percolation threshold in 2D networks consisting of curvy nanotubes or nanowires. We generate curvy nanotubes/nanowires using third order Bezier curves, which are parametric functions specified by the values of four control points. We introduce the concept of the curviness angle, which puts a geometrical constraint on the locations of the two intermediate control points of the Bezier curve, and hence on the statistical distribution of nanotube/nanowire curviness in the network. For a more universal definition of the degree of curviness, we define the curl ratio as the ratio of the actual length of the nanotube/nanowire to the effective straight length between its two ends. Using simulation data, we establish an empirical relationship between curviness angle and curl ratio.
We first study the percolation probability as a function of curviness angle at fixed density for randomly distributed nanotube/nanowire networks. We extract the curviness percolation threshold at fixed density characterized by the critical curviness angle and the critical curl ratio using finite size scaling analysis. We find that the critical curviness angle and curl ratio increase with increasing density. We also find that there is a minimum (maximum) density below (above) which the percolation probability is 0 (1) regardless of the curviness angle. Second, we find that nanotube/nanowire alignment significantly changes the curviness percolation threshold, resulting in a reversal of the percolation probability versus curviness angle curve.
These results show that nanotube/nanowire curviness plays a significant role in determining the percolation threshold in 2D or quasi-2D networks, films, or nanocomposites consisting of 1D nanoelements. These results further show that computational studies are an essential tool for providing insights into the insulator-to-conductor transition in nanotube/nanowire networks, which are promising candidates for a wide range of applications such as flexible transparent conductors, thin film transistors, and resistive switching memory.
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Figure 1